Modelling of Structure and

Properties

of Soft Carbons with Application

toCarbon Anode Baking

Dr. ing. thesis

�yvind Gundersen

Department of Engineering CyberneticsNorwegian University of Science and Technology

N-7034 Trondheim, Norway

1998

Report 98-10-W

Preface

This work is submitted in partial ful�lment of the requirements for the degree ofdoktor ingeni�r at the Norwegian University of Science and Technology (NTNU).The research was carried out between July 1991 and August 1996 at the De-partment of Engineering Cybernetics at NTNU. Parts of the work were done incooperation with Hydro Aluminium a.s, Technology Centre �Ardal, Norway. Theproject was initiated by Sigmund Gj�rven at the Technology Centre and my aca-demic supervisor Professor Jens G. Balchen in 1991.

Carbon anodes are used in electrolysis for production of aluminium. The anodesare heat treated in large baking furnaces to obtain the desired properties. Anodebaking is a process of high energy demand, and there are potentials for bothenergy and anode quality optimization. This study deals with modelling of carbonproperties as well as modelling and control of ring furnaces for the baking of carbonanodes. Based on a model of the baking process, a model based control strategyhas been suggested. One of Hydro Aluminium's furnaces in �Ardal was used asbasis for development of the model.

During the work with this study, I have had the opportunity to interact with manypeople.

First of all, I want to thank my supervisor Professor Jens G. Balchen. Over along period of time I have learnt to know a charismatic person; highly skilledand also with warm personal qualities. Professor Balchen has always been veryenthusiastic and his encouraging attitude has been an incentive and a source ofinspiration during the research work. In the part of the work dealing with theformulation of the control strategy, Professor Balchen's ideas and advices werevery valuable. Thanks for patience and support over such a long period of time!

I am grateful for the opportunity to work on an industrial application within theframe of a doctoral degree. I wish to thank Sigmund Gj�rven, Trygve Foosn�s andHogne Linga at Hydro Aluminium a.s for o�ering the scholarship. The �nancialsupport from Hydro Aluminium a.s during three years of work with the project isvery much acknowledged. I want to thank Trygve Foosn�s for valuable discussionson important subjects in carbon technology and anode manufacturing. I also wantto thank Robert J�rgensen, Kjell Arne Nerland, Georg Neumann and Arne W.Olsen at the Technology Centre in �Ardal for valuable discussions on ring furnaceoperation and control. Furthermore, I want to thank Magnar Asperheim, also at

ii PREFACE

the Technology Centre, for his e�orts during full scale measurements.

I also wish to thank my former colleagues and the sta� at the Department ofEngineering Cybernetics. Considering the challenges in development of the bakingprocess model, the discussions with Hilde Meisingset and Emil Edwin were verymuch valuable. I'm grateful for the good cooperation with both of them. Specialthanks are given to my former colleague Stein Wasb�, with whom I shared o�ceduring the last two years I spent at the Department of Engineering Cybernetics.In our discussions I have got a lot of valuable inputs. I will also express my thanksto Kjell Eidem and Jan Leistad who keep the computer network running both dayand night.

Most of the time, this work has been interdisciplinary and I have had many inter-esting and helpful discussions with specialists at other faculties at NTNU. I wantto thank Mona Jacobsen at the Department of Thermal Energy and Hydropowerfor interesting discussions in modelling of carbon properties and ring furnace phe-nomena. I also want to thank Hallvard Svendsen at the Department of ChemicalEngineering for many discussions on topics in chemical engineering with relevanceto the development of the model of the baking process.

During my stay in USA in November 1993, I had the opportunity to meet ProfessorHarry Marsh. I will thank him for his interest in this research and for introducingme to many researchers during my short stay at the Southern Illinois University(SIU), Carbondale, Illinois. In our discussions, his broad experience gave ideas forthe �nal formulation of the baking process model.

I will also express my thanks to research manager Odd Magne Akselsen at SINTEFDepartment of Joining Technology for his cooperative attitude during the last twoyears spent to complete this document while working full time at the Departmentof Joining Technology.

Finally, my deepest gratitude goes to my wife Inger and my four children Elin,�ystein, Erlend and Irene; all of them born during the period of work with thisstudy. My wife has been very patient and encouraging and they have all contin-uously reminded me about the very important aspects of life; family and friends.We have all indeed looked forward to the completion of this work.

�yvind GundersenTrondheim, 24. November 1998

Summary

This work deals with topics related to modelling and control of ring furnaces forthe baking of carbon anodes used in aluminium electrolysis.

Anodes made of a granular coke and coal tar pitch are used in aluminium electroly-sis. The anode properties are imperative for successful operation of the aluminiumsmelters. After mixing and forming the anode paste, heat treatment of the carbonblocks takes place in so-called ring furnaces. A ring furnace consists of a series ofheat treatment sections where each section is loaded with a batch of anodes. Theheat treatment of the anodes in a section consumes a lot of energy, and the anodeproperties partly depend on the heat treatment program. Previous work in the�eld of ring furnace modelling, operation and control is shortly reviewed.

Both petroleum coke and coal tar pitch belong to the group of soft carbons. Modelsfor structural parameters and porosity of soft carbons are developed. Furthermore,a new model for pyrolysis of coal tar pitch is proposed. Based on the modelsfor pyrolysis, structure and porosity, new models for properties of single phasecarbons and composite anodes are developed. These models are suitable for usein optimization of the baking process.

A detailed mathematical model of a part of the heat treatment process is formu-lated in three spatial dimensions. The model is based on �rst principle descriptionsof fundamental physical and chemical phenomena and the resulting model appearsas a set of partial di�erential equations. The spatial di�erential operators are dis-cretized by using the �nite volume approach. In this way, a high dimensionalnonlinear state space model is obtained. The model has been simulated using themethod of lines.

A vector of quantities which describes the anode properties is de�ned. This prop-erty vector constitutes a systematic de�nition of anode quality where the qualityparameters are calculated as nonlinear transformations of the state space vector.Models are derived for some anode properties. Anode properties are not directlymeasurable during normal process operation but they can be calculated by usingthe proposed property model. In this way, it has been possible to study anodeproperty evolution as function of the furnace operation strategy.

The ring furnace model has not been veri�ed, but there seems to be a satisfactorycorrespondence between measurements and model calculations.

iv SUMMARY

A model based control strategy based on the proposed model is presented. Openloop optimization of process economy is suggested for calculation of the nominalheat treatment trajectory. Since a certain heat treatment program can be used formany batches of anodes, the solution of the optimization problem can be solvedo� line. In real process operation, deviations from the nominal disturbances mayoccur. A control corrector based on classic control theory is suggested to copewith non-nominal disturbances.

The main contributions in this study are as follows:

� A model of pyrolysis of coal tar pitch derived with basis in �rst principles

� Models of carbon properties derived from fundamental principles

� A detailed �rst principles mathematical model of a part of an anode bakingfurnace

� A two-level model based control strategy for the ring furnace:

{ O�-line optimization of process economy and anode quality

{ A basic level control corrector for on-line tracking along the optimaltrajectory

One of Hydro Aluminium's ring furnaces in �Ardal has been considered during thedevelopment of the mathematical model for the ring furnace. Still, however, manyof the results presented in this study have general signi�cance.

Contents

Preface i

Summary iii

Nomenclature xxxix

1 Introduction 1

1.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Mathematical Modelling of Ring Furnaces . . . . . . . . . . 3

1.1.2 Control of Ring Furnaces . . . . . . . . . . . . . . . . . . . 5

1.1.3 Fields of further Research . . . . . . . . . . . . . . . . . . . 9

1.1.4 Carbon Structure and Properties . . . . . . . . . . . . . . . 9

1.2 Outline of this Work . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Contributions of this Work . . . . . . . . . . . . . . . . . . . . . . 11

I Description of Anode Manufacturing 13

2 Introduction to Anode Manufacturing 15

3 Green Anode Production 19

3.1 Anode Raw Materials . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Petroleum Coke and Coal Tar Pitch . . . . . . . . . . . . . 19

3.1.2 Recycled Butts . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.2 Anode Paste Recipe . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Steps in Green Anode Production . . . . . . . . . . . . . . . . . . . 24

vi CONTENTS

4 Baking Furnace Process Description 27

4.1 Ring Furnace Concepts . . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Open Furnaces . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.2 Closed Furnaces . . . . . . . . . . . . . . . . . . . . . . . . 28

4.1.3 Retro�tted Riedhammer Furnace . . . . . . . . . . . . . . . 28

4.1.4 Furnace Construction and Geometry . . . . . . . . . . . . . 30

4.1.5 Symmetric Properties of a Section . . . . . . . . . . . . . . 30

4.2 Ring Furnace Operation Principle . . . . . . . . . . . . . . . . . . . 33

4.2.1 Description of the Baking Cycle . . . . . . . . . . . . . . . 33

4.2.2 Heat Treatment Design . . . . . . . . . . . . . . . . . . . . 40

4.3 Ring Furnace Instrumentation . . . . . . . . . . . . . . . . . . . . . 42

4.3.1 Temperature Measurements . . . . . . . . . . . . . . . . . . 42

4.3.2 Draught Pressure Measurement . . . . . . . . . . . . . . . . 42

4.3.3 Burner Equipment . . . . . . . . . . . . . . . . . . . . . . . 42

4.3.4 Draught- and Cooling Fans . . . . . . . . . . . . . . . . . . 43

4.4 Process Levels in Anode Baking . . . . . . . . . . . . . . . . . . . . 43

4.5 External Process Variables . . . . . . . . . . . . . . . . . . . . . . . 44

4.5.1 Input Variables . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.5.2 Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.5.3 Disturbances and Interactions . . . . . . . . . . . . . . . . . 46

5 Anode Quality and Carbon Consumption 49

5.1 Optimization of Aluminium Production . . . . . . . . . . . . . . . 49

5.2 Conventional Anode Quality Parameters . . . . . . . . . . . . . . . 51

5.3 Anode Behaviour During Smelting . . . . . . . . . . . . . . . . . . 54

5.3.1 Carbon Consumption . . . . . . . . . . . . . . . . . . . . . 54

5.3.2 Consumption of Electrical Energy . . . . . . . . . . . . . . 59

5.3.3 Anode Porosity vs. Overvoltage . . . . . . . . . . . . . . . . 59

5.3.4 Anode Cracking . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3.5 Anode E�ect . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.3.6 Anode Changing . . . . . . . . . . . . . . . . . . . . . . . . 60

5.4 The Concept of Anode Quality . . . . . . . . . . . . . . . . . . . . 60

CONTENTS vii

6 Discussions and Conclusions 65

II Structure and Properties of Soft Carbons 67

7 Classi�cation of Carbon Forms 69

7.1 Soft Carbons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.2 Hard Carbons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.3 Further Classi�cation of Carbons . . . . . . . . . . . . . . . . . . . 70

7.4 Carbon Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

8 Carbon Structure and Fundamental Carbon Properties 73

8.1 The Fundamental Carbon Properties . . . . . . . . . . . . . . . . . 73

8.2 Structure in Non-Graphitic Carbons . . . . . . . . . . . . . . . . . 74

8.3 Structure of Soft Carbons . . . . . . . . . . . . . . . . . . . . . . . 78

8.3.1 Structural Changes During Fusion . . . . . . . . . . . . . . 78

8.3.2 Structural Changes During Liquid Phase Pyrolysis . . . . . 79

8.3.3 Structural Changes During Calcination and Graphitization 83

8.3.4 Macrostructure vs. Microstructure . . . . . . . . . . . . . . 85

8.4 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

8.4.1 Classi�cation of Porosity . . . . . . . . . . . . . . . . . . . . 86

8.4.2 The Unavoidable Porosity . . . . . . . . . . . . . . . . . . . 86

8.4.3 Frozen-in Stresses . . . . . . . . . . . . . . . . . . . . . . . 88

8.4.4 Pore Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 88

8.4.5 Coke Porosity vs. Coke Structure . . . . . . . . . . . . . . . 88

8.5 Elemental Composition and Impurities . . . . . . . . . . . . . . . . 89

8.6 Limitations in Carbon Production . . . . . . . . . . . . . . . . . . 89

8.7 X-ray Di�raction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.7.1 Conventional Parameters for Average Crystallite Size . . . 90

8.7.2 Structural Information Obtained from Line Pro�les . . . . . 92

8.7.3 Order Parameters vs. Disorder Parameters . . . . . . . . . 92

8.7.4 Limitations of X-ray Di�raction Techniques . . . . . . . . . 93

9 Mathematical Modelling of Structure and Porosity 95

viii CONTENTS

9.1 The Porosity Concept . . . . . . . . . . . . . . . . . . . . . . . . . 95

9.1.1 Total Porosity vs. Open and Closed Porosity . . . . . . . . 95

9.1.2 The Crystalline Density . . . . . . . . . . . . . . . . . . . . 100

9.1.3 Microporosity and Intercrystalline Porosity vs. Closed Poros-ity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

9.1.4 Macro- and Mesopores vs. Open Porosity . . . . . . . . . . 101

9.1.5 The Evolution of Porosity across the Mesophase Transition 101

9.1.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

9.2 A Structural Model of Solid Coke . . . . . . . . . . . . . . . . . . . 102

9.2.1 Turbostratic Carbons: The Bulk Carbon vs. The GranularStructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

9.2.2 Density and Porosity of the Granular Structure . . . . . . . 103

9.2.3 The Relationship Between Volatile Loss, xdm and �dm . . . 108

9.2.4 State Space Vector for Carbon Porosity and Microstructure 111

9.3 Modelling Structural Evolution During Carbonization . . . . . . . 112

9.3.1 The General Population Balance Equation . . . . . . . . . . 112

9.3.2 Modelling The Evolution of Physical Properties Across theMesophase Transition . . . . . . . . . . . . . . . . . . . . . 113

9.3.3 Mechanisms of Crystallite Growth in Solid State Carboniza-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

9.3.4 The Crystallite Growth Model . . . . . . . . . . . . . . . . 121

9.3.5 Weight Fraction xdm vs. Formation Rate and Mass of Crys-tallites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

9.3.6 Derivation of Moment Equations for Calculation of AverageCrystallite Parameters . . . . . . . . . . . . . . . . . . . . . 130

9.3.7 Lumped Models for Crystallite Growth . . . . . . . . . . . 133

9.3.8 Simulation Case I: Population Balance Approach to Growthof La . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

9.3.9 Simulation Case II: Lumped Model Approach to Growth ofLa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

9.4 Conclusions: Porosity and Structure Models . . . . . . . . . . . . . 143

10 Crystallite Growth Modelled as a Thermally Activated Process 151

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

10.2 Di�erent Modelling Approaches . . . . . . . . . . . . . . . . . . . . 153

CONTENTS ix

10.3 The Recommended Approaches . . . . . . . . . . . . . . . . . . . . 153

10.3.1 Models for Realistic Prediction of Crystallite Parameters . 154

10.3.2 Simpli�ed Models for Crystallite Parameters . . . . . . . . 154

10.4 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

10.4.1 Models Based on two Ensembles of Multiple Parallel Processes155

10.4.2 Models Based on two Single Reactions with Conversion De-pendent Activation Energies . . . . . . . . . . . . . . . . . . 156

10.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

11 Modelling of Physical Properties and Carbon Quality 161

11.1 General Structure of the Model . . . . . . . . . . . . . . . . . . . . 161

11.1.1 Model Assumptions . . . . . . . . . . . . . . . . . . . . . . 161

11.1.2 Modelling Principle . . . . . . . . . . . . . . . . . . . . . . 162

11.1.3 Porosity and Pore Geometry . . . . . . . . . . . . . . . . . 162

11.1.4 Crystallite Parameters . . . . . . . . . . . . . . . . . . . . . 163

11.1.5 Texture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

11.1.6 Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

11.1.7 Model Structure . . . . . . . . . . . . . . . . . . . . . . . . 164

11.2 Application to Anode Baking . . . . . . . . . . . . . . . . . . . . . 165

III Mathematical Modelling of Pyrolysis 167

12 Qualitative Description of Pitch Pyrolysis 169

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

12.2 Volatile Compounds in Pitch Pyrolysis . . . . . . . . . . . . . . . . 171

12.3 E�ect of Temperature Rate . . . . . . . . . . . . . . . . . . . . . . 173

12.4 E�ect of Coke Additives . . . . . . . . . . . . . . . . . . . . . . . . 176

12.5 E�ect of Ambient Atmosphere and Pressure . . . . . . . . . . . . . 176

12.6 E�ect of Secondary Coking Reactions . . . . . . . . . . . . . . . . 177

12.7 The Pyrolysis Reactions' impact on Bond Coke Structure . . . . . 178

12.8 Heat of Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

12.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

x CONTENTS

13 Modelling Approaches in Pyrolysis 181

13.1 Approaches in Pyrolysis Modelling . . . . . . . . . . . . . . . . . . 181

13.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 181

13.1.2 Single Reaction Schemes . . . . . . . . . . . . . . . . . . . . 183

13.1.3 Consecutive Reaction Schemes . . . . . . . . . . . . . . . . 190

13.1.4 Multiple Reaction Schemes . . . . . . . . . . . . . . . . . . 194

13.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

14 The Pyrolysis Model 201

14.1 Developing the Model . . . . . . . . . . . . . . . . . . . . . . . . . 201

14.1.1 Requirements and Capabilities of the Pyrolysis Model . . . 201

14.1.2 Model Parameters vs. Experimental Data . . . . . . . . . . 203

14.1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

14.2 The Model for Devolatilization of Condensables . . . . . . . . . . . 205

14.2.1 The Reaction Scheme . . . . . . . . . . . . . . . . . . . . . 205

14.2.2 Estimated Parameters . . . . . . . . . . . . . . . . . . . . . 206

14.2.3 Simulating the Pyrolysis Model . . . . . . . . . . . . . . . . 207

14.2.4 Average Molar Mass of Condensables . . . . . . . . . . . . 208

14.3 The Model for Degassing of Non-Condensables . . . . . . . . . . . 209

14.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 209

14.3.2 Calculation Scheme for Ultimate Yields of Gases . . . . . . 210

14.3.3 Sensitivity of Ultimate Yields for Changes in Coke Yield . . 214

14.3.4 A Model for Degassing of Non-Condensables Based on anExtension of the Model for Low Temperature Pyrolysis . . 216

14.4 A Hydrogen Balance Equation . . . . . . . . . . . . . . . . . . . . 225

14.5 Simulating The Pyrolysis Model . . . . . . . . . . . . . . . . . . . 228

14.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

IV Mathematical Modelling of Carbon Properties 233

15 Density and Porosity of Binder Pitch and Pitch Coke 235

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

15.2 Measurements of Densities and Porosity . . . . . . . . . . . . . . . 236

CONTENTS xi

15.3 Real Density of the Isotropic and Anisotropic Pitch Phases . . . . 238

15.3.1 Real Density of Isotropic Pitch . . . . . . . . . . . . . . . . 238

15.3.2 Density of Mesophase and Semi-Coke . . . . . . . . . . . . 239

15.4 Calculation of Total Porosity in the Anode During Baking . . . . . 242

15.5 A Model for Apparent Density which includes both Low and HighTemperature Pyrolysis . . . . . . . . . . . . . . . . . . . . . . . . . 244

15.5.1 The Swelling Regime . . . . . . . . . . . . . . . . . . . . . . 246

15.5.2 The Collapse Regime . . . . . . . . . . . . . . . . . . . . . . 246

15.5.3 The Shrinkage Regime . . . . . . . . . . . . . . . . . . . . . 247

15.5.4 The Compound Rate Law for Apparent Density . . . . . . 248

16 Pitch Viscosity 251

16.1 Viscosity in Pitch as a Liquid Mixture . . . . . . . . . . . . . . . . 251

16.2 Applications of the Viscosity Model . . . . . . . . . . . . . . . . . 252

16.3 Reversible Viscosity of the Isotropic Pitch Fractions . . . . . . . . 253

16.4 Irreversible Viscosity of the Anisotropic Pitch Fraction . . . . . . . 253

17 Thermal Properties of Mesophase, Semi-Coke and Coke 255

17.1 Speci�c Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . 255

17.1.1 Constant Pressure- vs. Constant Volume Speci�c Heat Ca-pacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

17.1.2 The Constant Volume Speci�c Heat Capacity . . . . . . . . 256

17.1.3 A Semi-Empirical Approach to Modelling of Speci�c HeatCapacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

17.2 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . . . . . 263

17.2.1 The Wiedemann-Franz Law . . . . . . . . . . . . . . . . . . 263

17.2.2 The Model for Thermal Conductivity of Carbons . . . . . . 264

17.2.3 A Comment on the Presented Model . . . . . . . . . . . . . 265

17.3 Thermal Properties of the Bulk Pitch Phase . . . . . . . . . . . . . 266

17.3.1 Speci�c Heat Capacity . . . . . . . . . . . . . . . . . . . . . 266

17.3.2 Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . 266

18 A Model for Soft Carbon Pyrolysis and Property Development 269

18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

xii CONTENTS

18.2 Discussing the Derivation of the Model . . . . . . . . . . . . . . . . 271

18.2.1 The Evolution of Physical Properties Across the MesophaseTransition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271

18.2.2 The Development of Porosity Across the Mesophase Transition272

18.3 Features and Simpli�cations of the Model . . . . . . . . . . . . . . 272

18.4 Model Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

18.5 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

18.6 State Space Formulation of the Model . . . . . . . . . . . . . . . . 284

19 Simulation of Soft Carbon Properties 287

19.1 The Dependence of Physical Properties on the Fundamental CarbonProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287

19.2 Summary of Model Parameters . . . . . . . . . . . . . . . . . . . . 288

19.3 A Comment on the Calculation of Physical Properties . . . . . . . 289

19.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 291

20 Introduction to Modelling of Anode Properties 299

20.1 Two Aspects of Anode Property Modelling . . . . . . . . . . . . . 299

20.1.1 Model-Data vs. Anode Properties . . . . . . . . . . . . . . 299

20.1.2 Anode Quality vs. Anode Properties . . . . . . . . . . . . . 300

20.2 Factors with Impact on Anode Properties . . . . . . . . . . . . . . 300

20.3 The Importance of Optimum Pitching . . . . . . . . . . . . . . . . 301

20.4 Physical Structure of Anodes . . . . . . . . . . . . . . . . . . . . . 301

20.4.1 The Classical View . . . . . . . . . . . . . . . . . . . . . . . 301

20.4.2 A Simpli�ed View of Anode Structure . . . . . . . . . . . . 303

20.5 A Note on Calculation of Bulk Properties of Carbon Anodes . . . 304

20.6 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

20.6.1 Classical Models of Baked Carbon Properties . . . . . . . . 305

20.6.2 Application of the Two-Component Models in this Work . . 311

20.6.3 Future Modi�cations of the Two Component Models . . . . 312

20.6.4 Summary of Property Models . . . . . . . . . . . . . . . . . 312

20.6.5 Contributions from the Aluminium Industry . . . . . . . . . 313

21 Porosity, Density, Permeability and Surface Area of Anodes 319

CONTENTS xiii

21.1 Total Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

21.1.1 Signi�cance of Total Porosity . . . . . . . . . . . . . . . . . 319

21.1.2 Di�erent Types of Porosity in the Anode . . . . . . . . . . 320

21.1.3 Two Views of Total Porosity . . . . . . . . . . . . . . . . . 321

21.1.4 Shrinkage during Cooling vs. The Unavoidable Microporosity326

21.1.5 Thermal Expansion of Coke Phases During Baking . . . . . 327

21.2 The Open Porosity Model . . . . . . . . . . . . . . . . . . . . . . . 328

21.3 The Transport Porosity Model . . . . . . . . . . . . . . . . . . . . 331

21.4 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

21.4.1 Review of Permeability Models . . . . . . . . . . . . . . . . 335

21.4.2 The Ordinary Kozeny Equation for Non-uniform Pore Textures341

21.4.3 The Chosen Approach . . . . . . . . . . . . . . . . . . . . . 341

21.4.4 Models for Average Particle Diameter and Hydraulic Diameter342

21.5 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

21.6 Summary of Model Assumptions . . . . . . . . . . . . . . . . . . . 347

22 Anode Crystallite Parameters 349

23 Simulation of Anode Properties 351

23.1 Overview of Properties . . . . . . . . . . . . . . . . . . . . . . . . . 351

23.2 Discussing the Simulation Results . . . . . . . . . . . . . . . . . . . 352

24 Conclusions 359

V Modelling and Control of Baking Furnaces 361

25 Modelling the Baking Process 363

25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

25.1.1 Modelling from First Principles . . . . . . . . . . . . . . . . 363

25.1.2 Subprocesses in the Baking Process . . . . . . . . . . . . . 364

25.2 System Decomposition and Modelling Strategy . . . . . . . . . . . 366

25.2.1 Main Decomposition of the Ring Furnace System . . . . . . 366

25.2.2 Modelling Strategy . . . . . . . . . . . . . . . . . . . . . . . 373

25.3 Chemical Reactions in the Solid Materials . . . . . . . . . . . . . . 375

xiv CONTENTS

25.3.1 Pyrolysis of Binder Pitch . . . . . . . . . . . . . . . . . . . 375

25.4 Combustion Reactions in the Gas Phase . . . . . . . . . . . . . . . 377

25.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 377

25.4.2 Combustion of General Hydrocarbons . . . . . . . . . . . . 378

25.4.3 Combustion of Volatiles . . . . . . . . . . . . . . . . . . . . 378

25.4.4 Combustion of Fuel Oil . . . . . . . . . . . . . . . . . . . . 381

25.4.5 Combustion of Packing Coke . . . . . . . . . . . . . . . . . 381

25.4.6 Assumptions in Combustion Modelling . . . . . . . . . . . . 381

25.4.7 Stoichiometry of Combustion Reactions . . . . . . . . . . . 382

25.5 Mass Transfer in the Ring Furnace . . . . . . . . . . . . . . . . . . 384

25.5.1 Interphase Mass Transfer in the Anodes and Coke Bed . . . 384

25.5.2 Mass Transfer in the Gaseous Phase . . . . . . . . . . . . . 387

25.5.3 A Simpli�ed Model for Mass Transfer in the Anodes and theCoke Bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

25.6 Heat Transfer in the Ring Furnace . . . . . . . . . . . . . . . . . . 389

25.6.1 Heat Transfer Modes . . . . . . . . . . . . . . . . . . . . . . 389

25.6.2 Heat Transfer Processes in the Ring Furnace . . . . . . . . 390

25.6.3 Conduction Heat Transfer . . . . . . . . . . . . . . . . . . . 392

25.6.4 Heat Transfer Between Combustion Gas and Solid Surfaces 393

25.6.5 Convection Heat Transfer . . . . . . . . . . . . . . . . . . . 393

25.6.6 Radiation Heat Transfer . . . . . . . . . . . . . . . . . . . . 396

25.7 The Conservation Laws in the Ring Furnace . . . . . . . . . . . . . 402

25.7.1 The Mass Balances . . . . . . . . . . . . . . . . . . . . . . . 403

25.7.2 The Momentum Balances . . . . . . . . . . . . . . . . . . . 404

25.7.3 The Energy Balances . . . . . . . . . . . . . . . . . . . . . . 405

25.8 Volatile Transport in the Porous Coke Bed and Anodes . . . . . . 411

25.8.1 Phenomenological Description . . . . . . . . . . . . . . . . . 411

25.8.2 Transport Regime in the Coke Bed and Anodes . . . . . . . 411

25.9 Thermal Phenomena in Solid Materials . . . . . . . . . . . . . . . 412

25.9.1 Thermal Properties of Solid Materials . . . . . . . . . . . . 412

25.9.2 Application of the Ordinary Heat Conduction Equation inthe Ring Furnace . . . . . . . . . . . . . . . . . . . . . . . . 414

25.9.3 The Energy Equation for the Brickwork Materials . . . . . 414

CONTENTS xv

25.9.4 The Energy Equation for the Coke Bed . . . . . . . . . . . 415

25.9.5 The Energy Equation for the Anodes . . . . . . . . . . . . . 417

25.9.6 The Reaction Enthalpies . . . . . . . . . . . . . . . . . . . . 420

25.9.7 The Heat Conduction Equation with Boundary Conditions 422

25.9.8 Heat Losses Through Furnace Lid and Foundation . . . . . 424

25.10The Combustion Gas . . . . . . . . . . . . . . . . . . . . . . . . . . 425

25.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 425

25.10.2Characteristic Features of the Gas Path . . . . . . . . . . . 426

25.10.3Summary of Gas Model Simpli�cations . . . . . . . . . . . 428

25.10.4Mass and Energy Balance Equations for the Gas . . . . . . 430

25.10.5Summary of Gas Path Equations . . . . . . . . . . . . . . . 433

25.10.6Gas Properties . . . . . . . . . . . . . . . . . . . . . . . . . 436

25.11Discussions and Conclusions . . . . . . . . . . . . . . . . . . . . . . 438

26 Simulating the Baking Cycle 439

26.1 Development of the Numerical Model . . . . . . . . . . . . . . . . . 439

26.2 Computational Procedure . . . . . . . . . . . . . . . . . . . . . . . 440

26.3 The Explicit Integration Scheme . . . . . . . . . . . . . . . . . . . 441

26.4 RF3D Program System . . . . . . . . . . . . . . . . . . . . . . . . 441

26.5 Simulation Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

26.6 Discussions and Conclusions . . . . . . . . . . . . . . . . . . . . . . 446

27 Model Based Control of Ring Furnaces 453

27.1 Tuning of the Calcining Level . . . . . . . . . . . . . . . . . . . . . 453

27.1.1 The Conventional Approach . . . . . . . . . . . . . . . . . . 453

27.1.2 A New Concept for Calcining Level Tuning . . . . . . . . . 454

27.2 Conventional Control Strategy . . . . . . . . . . . . . . . . . . . . 455

27.3 Hydro Aluminium Control System . . . . . . . . . . . . . . . . . . 455

27.4 Model Based Control Strategy . . . . . . . . . . . . . . . . . . . . 456

27.4.1 The Structure of a General Process Model . . . . . . . . . . 456

27.4.2 The Selected Control Strategy . . . . . . . . . . . . . . . . 459

27.4.3 The Optimization Problem . . . . . . . . . . . . . . . . . . 462

27.4.4 The Control Problem . . . . . . . . . . . . . . . . . . . . . 464

xvi CONTENTS

27.5 Controllability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 467

27.5.1 Two Step Controllability Analysis . . . . . . . . . . . . . . 468

27.5.2 Analysis of a Finite Time Control Problem . . . . . . . . . 469

27.5.3 Step I: Controllability of Pitch Properties in a Control Volume470

27.5.4 Step II: Controllability of Temperature and Properties in aChain of Sections . . . . . . . . . . . . . . . . . . . . . . . . 471

27.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

References 481

A The Conservation Laws 509

A.1 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . 510

A.2 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . 510

A.3 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . 510

A.4 A Closed Set of Equations . . . . . . . . . . . . . . . . . . . . . . . 511

B Density, Porosity and Surface Area 513

B.1 Dense Composite with n Components . . . . . . . . . . . . . . . . 513

B.2 Porous Material with Open Porosity . . . . . . . . . . . . . . . . . 515

B.3 Porous Material with Open and Closed Porosity . . . . . . . . . . 516

B.3.1 Polycrystalline Material . . . . . . . . . . . . . . . . . . . . 516

B.3.2 Packed Bed of Polycrystalline Material . . . . . . . . . . . . 518

B.4 Porous Composite with n Polycrystalline Components . . . . . . . 521

B.5 Surface Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522

B.5.1 Mercury Porosimetry . . . . . . . . . . . . . . . . . . . . . . 522

B.5.2 Surface Area of Materials . . . . . . . . . . . . . . . . . . . 523

C Basics from Aluminium Electrolysis 525

C.1 The Hall-Heroult Process . . . . . . . . . . . . . . . . . . . . . . . 525

C.1.1 The Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

C.1.2 The Anode . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

C.1.3 The Cathode . . . . . . . . . . . . . . . . . . . . . . . . . . 528

C.1.4 The Modern Electrolytic Cell . . . . . . . . . . . . . . . . . 528

C.2 Current E�ciency and Carbon Consumption . . . . . . . . . . . . 530

CONTENTS xvii

C.3 Energy Consumption and Energy E�ciency . . . . . . . . . . . . . 531

C.4 The Anode E�ect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532

xviii CONTENTS

List of Figures

1.1 Cell for aluminium electrolysis with prebaked anodes. Based onGrjotheim & Kvande (1993, Fig. 2). . . . . . . . . . . . . . . . . . 2

1.2 Cost estimates in Aluminium production. Based on Keller & Oder-bolz (1985). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 The relationship between di�erent parts of the work and ring fur-nace modelling and control. The texts associated with the arrowsrefer to the corresponding parts of this work. . . . . . . . . . . . . 12

2.1 Anode baking costs. Based on Keller & Oderbolz (1985, Fig. 2, 3)and Keller & Fischer (1992). . . . . . . . . . . . . . . . . . . . . . . 17

3.1 Origin of petroleum coke and coal tar pitch. . . . . . . . . . . . . . 20

3.2 Ideal and actual screening curves. From Grjotheim & Welch (1988,Fig. 4.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Optimal pitching level. From Grjotheim & Welch (1988, Fig. 4.4). 23

3.4 Basic steps in green anode production. . . . . . . . . . . . . . . . . 25

4.1 The open furnace concept (schematic diagram). . . . . . . . . . . 28

4.2 The closed furnace concept (schematic diagram). . . . . . . . . . . 29

4.3 The Hydro Aluminium retro�t furnace concept. . . . . . . . . . . . 29

4.4 A ring furnace seen from above. . . . . . . . . . . . . . . . . . . . . 31

4.5 Top view of an uncovered section which shows the arrangement ofthe cassettes. Anodes are positioned side by side surrounded withpacking material to give support and contribute to heat transferfrom the owing gases to the anodes. The drawing does not havethe correct scale and the number of ue channels does not �t withthe actual geometry of the furnace. . . . . . . . . . . . . . . . . . . 32

xx LIST OF FIGURES

4.6 Detailed top view of a cassette in an uncovered section. The positionof the coordinate system in the xy-plane is shown in the drawing.The drawing does not have the correct scale and the number of uechannels does either not �t with the actual furnace geometry. . . . 33

4.7 Pit xy-plane viewed from three di�erent levels along the verticaldirection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.8 Side view (yz-plane, part A) of anodes surrounded with packingmaterial. The gas ow is in vertical upward direction in part A.Anodes are stacked in three levels. The drawing does not have thecorrect scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.9 Side view from xz-plane of anodes surrounded with packing ma-terial. Gas ow directions in part A and part B are shown in thedrawing. It can be seen that anodes are stacked in three levels. Thedrawing does not have the correct scale. . . . . . . . . . . . . . . . 36

4.10 Nominal orientation of an anode in the ring furnace. . . . . . . . . 37

4.11 Typical �recurve used for ring furnace operation. . . . . . . . . . . 38

4.12 Typical draught pressure pro�le along a �re zone. . . . . . . . . . . 39

4.13 Typical O2 pro�le along a �re zone. . . . . . . . . . . . . . . . . . 39

4.14 Periodicity-property of the baking cycle: During ideal �re zone op-eration, the baking curves in subsequent sections are repeated witha lag of one �restep. . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.15 Arrangement of burners in a section of the retro�tted furnace. . . . 43

4.16 Anode-baking process view. . . . . . . . . . . . . . . . . . . . . . . 45

4.17 Fire zone with �ve sections in preheat and direct �re (cooling sec-tions not included). . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.18 Disturbances which act on a section in a ring furnace �re zone. . . 48

5.1 Factors in anode-fabrication and -use which a�ect anode behaviour.Based on Jones (1990). . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2 Optimization of aluminium production. . . . . . . . . . . . . . . . 51

5.3 Factors contributing to anode consumption. Based on Grjotheim &Kvande (1993). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.4 Dissecting the carbon consumption. From Forum Roundtable: Dis-cussing the Issues in Carbon Anode Technology (1990). Note thatthe electrolytic consumption depends on the current e�ciency. . . 56

5.5 Chemical reaction zones on (below) the surface of the anode. . . . 57

LIST OF FIGURES xxi

8.1 Schematic representation of the mutual orientation of the graphiticlamellae in isotropic and anisotropic carbons. In anisotropic car-bons, the aromatic layers are larger, less defective with a smalleramount of heteroatoms than the aromatic layers in isotropic carbons. 76

8.2 A crystallite is an assembly of mutually ordered graphitic layerplanes. Parameters d, La and Lc are used to describe the geometryof the crystallites. The crystallites constitute the microstructureof the carbon material. Crystallites are surrounded by microporesbetween large assemblies (i.e. macrocrystals) of microcrystals. . . . 77

8.3 Simple schematic representation of a macrocrystal. The size of amacrocrystal is typically in the order of 1:0�m or below. The sizeof the crystallites is in the order of 10 �A (i.e. 10�9m). . . . . . . . 77

8.4 Reduction in structural order as function of temperature duringfusion of a soft carbon precursor. Stacking order is lost up to thesoftening point Ts (usually, Ts is below 140�C). After Turner (1995,Fig. 12, 14). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

8.5 Structure of a Brooks & Taylor mesophase sphere. . . . . . . . . . 83

8.6 The properties of the carbon precursor material is the main con-straints with impact on the physical properties which can be achievedby heat treatment of the precursor. During heat treatment, theoptical texture is established within the range of the mesophasetransition which generally is completed at a temperature in the or-der of 500�C. For continued heat treatment of the carbon materialat temperatures in the range between 500 and 1400�C, the opti-cal anisotropy may be considered unchanged. In this temperaturerange, mainly evolution of porosity and crystallinity (structure ofmicrocrystals; i.e. crystallites), takes place. . . . . . . . . . . . . . 90

9.1 Five stages are identi�ed in the pyrolysis process. In the structuralmodel, the basic structural units play a role on every stage in theprocess. In the semicoke stage, the disordered carbon phase con-sists of single graphitic layer planes and peripheral groups linked tothe these layers. The term (QI) symbolizes that primary quino-line insolubles often play a role in nucleation and coalescence of themesophase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

xxii LIST OF FIGURES

9.2 The granular structure of disordered carbons. According to themodel of coal structure given in Hirsch (1954, pp. 137), the crosslinkscorrespond to the disorganized carbon phase. Ideally, the disorga-nized phase consists only of carbon. On the other hand, a certainfraction of inorganic matter belongs to the disorganized phase. Inthis work, we assume that the disorganized phase consists of onlycarbon and a negligible amount of hydrogen. The model of the gran-ular structure is also used by other authors (Brown & Hirsch 1955),(Cartz & Hirsch 1960), (Diamond & Hirsch 1958), (Diamond 1959),(Diamond 1960), (van Krevelen 1981, pp. 337), (Emmerich, DeSousa & Luengo 1987), (Emmerich & Luengo 1993). . . . . . . . . 104

9.3 The relationship between densities and volumes in the bulk car-bon. Here, the bulk carbon consists of open pores and the granularstructure. Within the granular structure, micropores, disorganizedcarbon and crystallites coexist. Thus, the granular structure cor-responds to the real volume as de�ned in this work. Except fromcrosslinks and peripheral groups, the disorganized carbon phase alsoincludes single layers. Crystallite growth partly occurs on the ex-pense of the disorganized carbon phase. . . . . . . . . . . . . . . . 105

9.4 Steady state values of Young's modulus as function of heat treat-ment temperature. It is quite common in the carbon literature toplot properties as function of the heat treatment temperature. Heattreatment at the given temperature must then take place over a timeinterval long enough for the property to settle at a stationary value. 109

9.5 Three sequential regimes in the pyrolysis of pitch. The transitionfrom liquid pitch to solid coke is a dynamic process with activationenergy such that the transition occurs nominally as temperatureapproaches 400�C. . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

9.6 Qualitative evolution of average crystallite parameters during pyrol-ysis. This trend in development of crystallite parameters has beenobserved by several authors (Diamond & Hirsch 1958), (Diamond1959), (Diamond 1960), (Marsh & Stadler 1967), (Marsh 1973),(H�uttinger 1971), (Auguie, Oberlin & Hyvernat 1980), (Whittaker,Miller & Fritz 1970), (Kocaefe, Charette & Castonguay 1993), (Turner1995). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

9.7 Development of structural parameters that characterize the behaviourof a single crystallite during carbonization as function of heat treat-ment temperature. The vertical arrows denote that several initialvalues of the curves are possible. The unchanged sign of the deriva-tives is an assumed feature of the curves. The dotted lines in thepanels for Lc and d002 denote the loss in structure due to fusion ofthe carbon precursor. . . . . . . . . . . . . . . . . . . . . . . . . . . 119

9.8 Two views of carbon structure: Franklin's independent crystalliteconcept and Mizushima's parallel sheet concept. . . . . . . . . . . 121

LIST OF FIGURES xxiii

9.9 Binary coalescence of crystallites with equal values of La. Lc isnot additive (i.e. not conserved) in the coalescence process since anadditional interlayer spacing is added in the coalesced crystallite. . 125

9.10 The Lennard - Jones potential function analogy. . . . . . . . . . . 126

9.11 Heat treatment program used in the simulation of the populationbalance model. The heating rate is a = 10�C=hr. . . . . . . . . . . 141

9.12 The plot shows consumption of the disorganized phase due to growthof La and the corresponding increase in the mass fraction of crys-tallites. The plot of Va=Va;� shows that the bulk volume shrinksduring heat treatment. . . . . . . . . . . . . . . . . . . . . . . . . . 142

9.13 Plot of density of the solids (i.e. real density and apparent density). 143

9.14 Plot of open- (��) and closed (�c) porosity and crystallite diameterLa. The open porosity increases. The closed porosity (intercrys-talline pores) decreases in such a way that the total porosity �T (asa sum of open and closed pores) also decreases. . . . . . . . . . . . 144

9.15 During heat treatment, the mass of crystallites (lower curve) ap-proaches the total mass of the carbon sample (upper curve). . . . . 145

9.16 Heat treatment program used in simulation of the lumped modelfor La. The heating rate is a = 10�C=hr. . . . . . . . . . . . . . . . 147

9.17 Consumption of the disorganized phase due to growth of La andthe corresponding increase in the mass fraction of crystallites. . . . 147

9.18 Plot of density of the solids (i.e. real density and apparent density). 148

9.19 Plot of open- (��) and closed (�c) porosity and crystallite diameterLa. The open porosity increases. The closed porosity (intercrys-talline pores) decreases in a way such that the total porosity (�T )as a sum of open and closed pores also decreases. . . . . . . . . . . 148

9.20 During heat treatment, the mass of crystallites (lower curve) ap-proach the total mass of the carbon sample (upper curve). . . . . . 149

10.1 Lc as function of time. The model for Lc is realized as two parallelprocesses with temperature dependent activation energies. . . . . . 157

10.2 Lc as function of temperature. The model for Lc realized as twoparallel processes with temperature dependent activation energies.Lc decreases between approximately 400 and 800�C. . . . . . . . . 157

10.3 La as function of time. The model for La is realized as two paral-lel processes with temperature dependent activation energies. Thegrowth of La almost halts in a certain part of the heat treatmentprogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

xxiv LIST OF FIGURES

10.4 La as function of temperature. The model for La is realized as twoparallel processes with temperature dependent activation energies.The growth of La is not active between approximately 400 and 800

�C.159

11.1 The principle of modelling physical properties of pitch and bindercoke. The fundamental properties belong to the state vector xsince they are described by di�erential equations of time, temper-ature and space. Furthermore, the physical properties belong tothe property space z: Nonlinear transformations exist between thestate space and the property space. . . . . . . . . . . . . . . . . . . 163

12.1 Transformation in the pitch during pyrolysis. The isotropic pitchmainly consists of bi- and oligoaryls. The mesophase constituteslarge peri-condensed aromatic systems. . . . . . . . . . . . . . . . . 170

12.2 Mechanism of polycondensation reactions. From Ko�st�al, Pr _u�sa &Malik (1994). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

12.3 Rate of release of non-condensables in pitch pyrolysis. A minimumin the gas release rate is observed just below 600�C which is re-lated to a change in the kinetics of the carbonization reactions.The heating rate is 120�C=hr. The minimum in the gas release rateobserved at approximately 600�C is not associated with a certainreaction step but rather to the solidi�cation of the bulk pitch ma-terial. The peak value corresponds to 0.3 ml/(gmin). Based onPolitis & Chang (1985, Fig. 4). . . . . . . . . . . . . . . . . . . . . 173

13.1 Normalized weight loss curves for pyrolysis of laboratory scale an-odes as function of temperature. Heating rates of 5.0, 10.0 and15.0�C=hr are used. In all plots, the weight loss curves are movedto a higher temperature as the heating rate is increased. The plotsare generated from results obtained by simulation of the model inTremblay & Charette (1988). . . . . . . . . . . . . . . . . . . . . . 187

13.2 Normalized weight loss curves for pyrolysis of laboratory scale an-odes as function of time. Additional information is given in Fig-ure 13.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

13.3 Relative total weight loss during pitch pyrolysis generated by theweight loss model in Tremblay & Charette (1988). The heating rateis a = 5�C/hr. Since the ultimate weight losses in the model donot depend on pyrolysis conditions, the same ultimate weight losswill occur at any heating rate. Note the characteristic decrease inweight loss rate at approximately 500�C. This characteristic featureof the weight loss curve was experimentally observed by Fitzer &H�uttinger (1969). . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

LIST OF FIGURES xxv

13.4 Reaction schemes for cracking of heavy petroleum residua. Thecracking reactions occur in parallel with competing polymerizationreactions which in a stepwise manner �nally form coke. Togetherwith the chemical reactions, vapour-liquid equilibrium exists be-tween the reacting uid and the distillate vapour. . . . . . . . . . . 194

14.1 The interaction between processes in pitch pyrolysis. The term crys-tallite growth means that the size of the small crystallites increasesboth in the a- and c-directions. The average size of a layer planeas represented by parameter La increases due to condensation reac-tions as well as consumption of the disordered carbon phase. Theheight Lc of crystallites increases due to coalescence of crystallitesin the c-direction. The microstructure belongs to a resolution levelin the order of 10�6 m and below: Carbon texture is usually de�nedon a level 100 to 1000 times larger than the size of the microcrystal-lites. This corresponds to a resolution in the range of 10�6 m (i.e.the microstructure range). . . . . . . . . . . . . . . . . . . . . . . . 203

14.2 Weight loss data used for estimation of parameters in the rate lawfor loss of condensables during pyrolysis. Data are taken fromWilkening (1983, Fig. 10). . . . . . . . . . . . . . . . . . . . . . . . 204

14.3 Final reaction scheme used to model low temperature pitch pyrolysis.207

14.4 Relationship between rate constants for volatilization and polymer-ization of the 1-fraction. kv dominates over k 1;a at temperaturesfrom approximately 250�C and above. (kv is estimated and k 1;a istaken from (Ko�st�al et al. 1994)). . . . . . . . . . . . . . . . . . . . 207

14.5 Simulation of the model for low temperature pyrolysis with a heat-ing rate of a = 5:5�C=hr. Polymerization kinetics is taken fromKo�st�al et al. (1994). Initial values are given in Table 14.2. . . . . . 208

14.6 Simulation of the model for low temperature pyrolysis with a heat-ing rate of a = 11:0�C=hr. Polymerization kinetics is taken fromKo�st�al et al. (1994). Initial values are given in Table 14.2. . . . . . 209

14.7 Simulation of the model for low temperature pyrolysis with a heat-ing rate of a = 15:0�C=hr. Polymerization kinetics is taken fromKo�st�al et al. (1994). Initial values are given in Table 14.2. . . . . . 210

14.8 Dependence of the yield of condensables on the heating rate. Threeheating rates of a = 5:0; 11:0 and 15:0�C=hr were used. The weightloss increases with the heating rate. Thus, a low heating rate con-tributes to increasing the coke yield. Polymerization kinetics istaken from Ko�st�al et al. (1994). Initial values are given in Table 14.2.211

14.9 The dependence on heating rate for the kinetic parameters in themodel due to Tremblay & Charette (1988) for pyrolysis of pitch inanodes. Reaction orders are 0.7, 0.8 and 1.1 for tar, methane andhydrogen respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 212

xxvi LIST OF FIGURES

14.10Diagram which shows nominal composition of whole pitch. About50 % of the compounds in coal tar pitch are substituted with methylgroups. The distribution of C-atoms in aromatic and aliphatic sys-tems is shown. Also, the nominal distribution of di�erent aliphaticgroups is given. Finally, the assumed relationship between methylgroups and pitch solvent fractions is shown via dotted and dashedlines. It should be noted that the given numbers for the distribu-tion of C atoms between di�erent functional groups (mass fractions)may vary (signi�cantly). The �gure is based on information foundin Grint, Proud, Poplett, Bartle & Wallace (1988). . . . . . . . . . 213

14.11Ultimate yields for non-condensables in dependence of coke yield forwhole pitch. The yields increase as the coke yield increases. Pitchcomposition data are the same as in Examples 1 and 2 above. . . . 215

14.12Subdivision of �-resins into �p- and �s-resins introduces a modi�-cation of the scheme in Gundersen (1995b, Fig. 3.42). . . . . . . . 217

14.13Simulation of the extended reaction scheme of pitch pyrolysis whichincludes degassing of non-condensables. The heating rate is a =5:0�C=hr. Initial pitch data is given in Table 14.4. . . . . . . . . . 220

14.14Simulation of the extended reaction scheme. Heating rate is a =10:0�C=hr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

14.15Simulation of the extended reaction scheme. Heating rate is a =15:0�C=hr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

14.16Individual coke fractions C1 and C2 are introduced instead of thesingle coke fraction C in the scheme presented in Figure 14.12. . . 223

14.17Simulation of the extended reaction scheme with conversion depen-dent activation energy. The plot shows predicted and measuredconversion of hydrogen vs. time in coal tar pitch pyrolysis for aheating rate of a = 5:0�C=hr. The data vectors were generatedby the model in Tremblay & Charette (1988). Initial pitch data isgiven in Table 14.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

14.18Simulation of the extended reaction scheme with conversion depen-dent activation energy for a heating rate of a = 10:0�C=hr. . . . . . 224

14.19Simulation of the extended reaction scheme with conversion depen-dent activation energy for a heating rate of a = 15:0�C=hr. . . . . . 224

14.20Simulation of the extended reaction scheme of pitch pyrolysis whichincludes degassing of non-condensables. A conversion dependentactivation energy is used in the reactions for generation of non-condensables. The heating rate is a = 5:0�C=hr. . . . . . . . . . . . 225

14.21Simulation of the extended reaction scheme of pitch pyrolysis withconversion dependent activation energy for the non-condensables.The heating rate is a = 10:0�C=hr. . . . . . . . . . . . . . . . . . . 226

LIST OF FIGURES xxvii

14.22Simulation of the extended reaction scheme of pitch pyrolysis withconversion dependent activation energy for the non-condensables.The heating rate is a = 15:0�C=hr. . . . . . . . . . . . . . . . . . . 227

14.23Simulation of the total pyrolysis model B. The model for formationof non-condensables in the high temperature pyrolysis is based onan extension of the low-temperature pyrolysis reaction scheme. Ac-tivation energies which depend on the formation of coke fractionsC1 and C2 are used in the reaction steps for generation of methaneand hydrogen. Model parameters can be found in Section 14.2 andSection 14.3. The heating rate is a = 5:5�C=hr. . . . . . . . . . . . 229

14.24Simulation of the total pyrolysis model B. The heating rate is a =11:0�C=hr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

14.25Simulation of the total pyrolysis model B. The heating rate is a =15:0�C=hr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

14.26Simulation of the total pyrolysis model B. The yield of gases asfunction of temperature for three di�erent heating rates is pre-sented for di�erent heating rates. The yield of non-condensablesdoes not vary very much with heating rate. The heating rates area = 5:5; 11:0 and 15:0�C=hr. . . . . . . . . . . . . . . . . . . . . . . 231

14.27Simulation of the total pyrolysis model B. Yield of gases, coke yield,hydrogen content and mass average molar mass as function of tem-perature is presented for three di�erent heating rates. The cokeyield is most sensitive to the heating rate in the low temperatureregime (i.e. up to a temperature of 450�C). Also, hydrogen con-tent and average molar mass seem quite independent of the heattreatment programme used during pyrolysis. The heating rates area = 5:5; 11:0 and 15:0�C=hr. . . . . . . . . . . . . . . . . . . . . . . 231

15.1 Real and apparent volumes of coking binder pitch. Heating ratescomparable to nominal baking rates are used. The volumes arealways less than the initial pitch volume. Thus, swelling is modest incoking of coal tar pitch. Volumes were measured at a temperature of15.5�C. If measured at working temperatures, the relative volumeswould be unchanged if the coe�cient of thermal expansion variesonly negligibly with temperature. The �gure is based on Darney(1958, Fig. 6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

15.2 Evolution of porosity in the binder coke. The �gure is based onDarney (1958, Fig. 5). Note that the evolution of binder cokeporosity is di�erent from the porosity formed in calcined petroleumcoke (Rhedey 1967). In the same temperature range, porosity incalcined petroleum coke monotonously increases with temperature;there is no intermediate maximum in porosity. This may be due tothe signi�cantly higher heating rates used in petroleum coke calci-nation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

xxviii LIST OF FIGURES

15.3 Real density �r;p of binder coke as function of temperature for heat-ing rates of 3:3�C=hr (up to 600�C) and 6:6�C=hr (600 to 1000�C).It should be noted that the at portion of the curve between roomtemperature and 400�C is due to the fact that the measurementsare done at room temperature. At elevated temperatures, the den-sity would have a lower value due to thermal expansion of the pitch.One may conclude that the susceptibility of the room temperaturedensity for pitch heated up to 400�C is relatively una�ected bythe polymerization reactions which take place in the pitch (mainly ! �). The �gure is taken from Wilkening (1983, Fig. 11). . . . . 240

15.4 Real density for binder coke as a function of temperature for threedi�erent heating rates. The initial- and �nal temperatures are 450and 1000�C respectively. Three di�erent heating rates were used:5, 10 and 15�C=hr. The highest density is achieved with the lowestheating rate, i.e. the highest residence time. See also Figure 15.5below. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

15.5 Real density for binder coke as function of time for three di�er-ent heating rates. Initial temperature is 450�C and �nal tempera-ture is 1000�C. Three di�erent heating rates were used: 5, 10 and15�C=hr and the �nal value of density increases as the heating rateis decreased. The density depends on the residence time as well astemperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

15.6 Total porosity in anode as function of temperature for three di�erentheating rates of a = 5; 10 and 15�C=hr. Maximum temperature is1250�C. The lowest total porosity is obtained by using the lowestheating rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

15.7 Heat treatment programs used for simulating the evolution of totalporosity in the anode. . . . . . . . . . . . . . . . . . . . . . . . . . 243

15.8 Total porosity curves which belong to the time-temperature histo-ries shown in Figure 15.7. The lowest total porosity is achieved witha low heating rate up to approximately 500�C and a signi�cantlyhigher heating rate from 500 up to 1250�C. . . . . . . . . . . . . . 244

15.9 Total porosity as function of temperature for a nonlinear (i.e. piece-wise linear) heat treatment program. . . . . . . . . . . . . . . . . . 245

15.10Cooperation of three di�erent regimes for the formation of porosity(apparent density) of pitch coke. . . . . . . . . . . . . . . . . . . . 249

LIST OF FIGURES xxix

16.1 The plot shows typical lapse of viscosity as function of temperaturefor pyrolysis of coal tar pitch for a heating rate of a = 15�C=hr. Themass fraction based pyrolysis model (model B) was used in the sim-ulation. Initial pitch data were [x ; x� ; x�p ]

T = [0:65; 0:27; 0:08]T .The curve qualitatively equals the experimentally measured viscosi-ties shown in Ho�mann & H�uttinger (1993, Fig. 3): As formationof secondary �-resins commences, there is a sudden increase in vis-cosity. Ho�man, however, used a heating rate of 24�C=hr. Datafor the reversible (isotropic) viscosity was found in Ho�mann &H�uttinger (1993, Tab. 4). For the anisotropic (irreversible) viscosity,set �c = 60 Ns/m2. An even better �t with experimental data wouldbe obtained if the irreversible (anisotropic) viscosity was modelledas a thermally activated process. . . . . . . . . . . . . . . . . . . . 254

17.1 Changes in speci�c heat capacity cv introduced via compositionalchanges of the coke material during coke calcination. The initial hy-drogen content is fH = 6% and hydrogen decomposition is assumedto take place linearly dependent of temperature between 500 and1200�C. The Einstein characteristic temperature is �E = 1800K. 261

17.2 Model parameters as function of crystallite size La. Due to theregular shape of curves for parameters ~c and ~d, it is possible toexpress these parameters as function of La. Apparently, a stepchange occurs in parameter b as function of La for La in the orderof 25 �A. This is merely a numerical then a physical phenomenonsince b tends to zero for low values of La. Thus, it is possible alsoto parameterize b as a function of La. Using linear functions for theparameters, a model with six parameters is obtained. . . . . . . . . 265

18.1 The coupling (represented by arrows) between submodels in themodel for simulation of pyrolysis of soft carbons. The tempera-ture is the driving force in the models. As a general principle,the fundamental carbon properties (densities, crystallite parameterLa and viscosity) are calculated as weighted average values of theisotropic and anisotropic pitch fractions. d002 and Lc are modelledas thermally activated processes to represent the values for the bulkpyrolysing material. . . . . . . . . . . . . . . . . . . . . . . . . . . 276

19.1 Heat treatment program used in the simulations. . . . . . . . . . . 294

19.2 Development of crystallite parameters. . . . . . . . . . . . . . . . . 295

19.3 Development of densities. . . . . . . . . . . . . . . . . . . . . . . . 295

19.4 Development of apparent- and real volumes and bulk apparent- andreal densities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

19.5 Development of porosities. Note that both open- and closed poros-ity start at the value zero. . . . . . . . . . . . . . . . . . . . . . . . 296

xxx LIST OF FIGURES

19.6 Development of the mass fractions of isotropic phase, anisotropicphase, hydrogen and disorganized carbon. . . . . . . . . . . . . . . 297

19.7 Development of coke yield, average molar mass, viscosity and elec-trical resistivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

19.8 Development of speci�c heat capacity, thermal conductivity, Young'smodulus and mechanical strength. . . . . . . . . . . . . . . . . . . 298

20.1 Radius of the binder bridge between two �ller particles. Based onMrozowski (1956b, Fig. 1). . . . . . . . . . . . . . . . . . . . . . . 307

20.2 The dependence of electrical resistivity on apparent density for vary-ing pitch contents. According to the �gure, 1=�2el vs. apparent den-sity deviates from a linear behaviour for the lowest binder contents.Thus, apparently the validity of the two-component model breaksdown for low binder contents. As the apparent density increases,the resistivity curves all merge into one curve since the apparentdensity approaches the theoretical maximum limit. This, however,is not explicitly shown in the �gure. Based on Seldin (1959, Fig. 1). 309

20.3 Physical properties as function of binder contents show the samequalitative behaviour as the apparent density: They are not uniquefunctions of the binder content. Corresponding curves exist forphysical properties as function of apparent density showing that theproperty functions are not unique; see Figure 20.4 below. Based onOkada & Takeuchi (1960, Fig. 18). . . . . . . . . . . . . . . . . . . 310

20.4 Physical properties are not uniquely correlated with apparent den-sity as demonstrated in the case of electrical resistivity. This featureof baked carbons was not clearly stated by Mrozowski. The phe-nomenon is due to the fact that the same apparent density can beachieved at two di�erent pitch contents just by varying the moldingpressure. Based on Okada & Takeuchi (1960, Fig. 19). . . . . . . . 311

20.5 Qualitative dependence of c�el on the initial pitch contents. fp;c isthe critical pitch level. . . . . . . . . . . . . . . . . . . . . . . . . . 312

21.1 Simple view of anode structure used in the �rst approach to mod-elling total porosity of the anode. . . . . . . . . . . . . . . . . . . . 323

21.2 Geometric view of anode structure used in the second approach tomodelling total porosity of the anode. . . . . . . . . . . . . . . . . 325

21.3 The anode volume consists of �ller coke, binder coke and gas �lledvoids. Each coke phase is modelled separately even though it isin reality very di�cult to separate the two coke phases. By usingthe real density concept, a model for open porosity of the anode isderived. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

21.4 Distribution of porosity in the anode control volume. . . . . . . . . 334

LIST OF FIGURES xxxi

21.5 Typical elapse of BET surface area as function of the baking tem-perature. From Grjotheim & Welch (1988, Fig. 4.7 p. 88). . . . . . 346

23.1 Plot of the heat treatment program used in the simulation of an-ode property development. From 20 to 500�C, the heating rate is7�C=hr. From 500 to 1200�C, the heating rate is 10�C=hr. Finally,there is a 10 hours hold time at maximum temperature 1200�C .Total heat treatment time is 150 hours. . . . . . . . . . . . . . . . 354

23.2 Development of porosities. . . . . . . . . . . . . . . . . . . . . . . . 355

23.3 Development of densities and crystallite sizes. . . . . . . . . . . . . 355

23.4 Development of anode grain diameter, hydraulic radius, surface areaand permeability. Unfortunately, the current model for the surfacearea is not able to calculate a realistic estimate of the surface areain the anode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

23.5 Development of speci�c heat capacity and thermal conductivity. . . 356

23.6 Development of the reactivity index. . . . . . . . . . . . . . . . . . 357

23.7 Development of mechanical strength and normalized thermal shockresistance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

23.8 Development of normalized electrical resistivity. . . . . . . . . . . . 358

25.1 Schematic description of the ring furnace modelling approach. . . . 367

25.2 Ring furnace system decomposition. In the �gure, q00 denotes heat ux. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

25.3 Pit seen from the xy-plane. Each section contains in the order of5 to 7 adjacent pits. A dividing wall in the under-pit region forcesthe gas ow vertically upwards in part A of the pits. Each sectionis covered by a lid which forces the gas to ow downwards in thevertical ues of part B. Note the de�nition of coordinate axes. . . . 369

25.4 Pit seen from the xz-plane. The coordinate system in the pit regionis shown along with the local coordinate systems used for calculationof heat losses through foundation and furnace lid. It should be notedthat the origin in the pit coordinate system lies in the bottom ofthe headwall in part A at the surface of the ue wall on the righthand side of a cassette when viewed in the positive x-direction. . . 370

25.5 Pit seen from the yz-plane. The vertical ue channels correspond tothe situation in part A of the pit since the gas ow is vertically up-wards. In part B, gas ow in the ues occurs vertically downwards.Note the de�nition of coordinate axes. . . . . . . . . . . . . . . . . 371

xxxii LIST OF FIGURES

25.6 In the subsystems of the ring furnace, several phases are present.Some of the phases are multicomponent mixtures (i.e. the gasphases and the pitch phase). The letter "S" denotes a solid subsys-tem and the letter "G" denotes a gaseous subsystem. . . . . . . . . 373

25.7 Mass transfer paths in the ring furnace system. Devolatilization ofthe pitch component in the anode creates gases which pass throughthe porous anode and packing coke bed to enter the combustion ues. The letter "S" denotes a solid subsystem and the letter "G"denotes a gaseous subsystem. . . . . . . . . . . . . . . . . . . . . . 374

25.8 The matrix of stoichiometric coe�cients. . . . . . . . . . . . . . . . 383

25.9 Interphase mass transfer from the solid to the gaseous phase of theanode. It is assumed that the transfer of carbonization gases in thepitch fraction to the void fraction goes via an entrapped gas phasewhich belongs to the solid phase (i.e. pitch and �ller coke) of theanode. In this way, the volatile components exists in both the solidand the gas phase and the entrapped gas may be considered to bea reservoir of volatiles which are released from the pitch. . . . . . . 386

25.10Heat uxes from gas to solid surfaces along the gas path. . . . . . 393

25.11Radiation diagram for two �nite plates with intermediate combus-tion gas as a refractory surface. From Kreith & Black (1980). . . . 402

25.12Interaction between packing coke and gas in the packing coke bed. 418

25.13Solid control volume seen from yz-plane . . . . . . . . . . . . . . . 422

25.14Boundary conditions for a half pit. . . . . . . . . . . . . . . . . . . 425

25.15Boundary conditions for the ring furnace foundation and lid. . . . 425

25.16Typical under-pit gas computational cell. Note the indexing of theboundary control volume surfaces. Boundary surfaces in control vol-umes both along the gas path and in the solid materials are indexedin this way (see also Figure 25.13). Structurally the same kind ofcontrol volume was used for establishing equations for mass- andenergy conservation in all the zones along the gas path (i.e. head-wall part A, under-pit part A, ue channels part A, ue channelspart A and under-pit part B) except the under-lid zone (part B). . 432

26.1 Program system developed for simulation of a half cassette in a ringfurnace of Hydro Aluminium design. . . . . . . . . . . . . . . . . . 444

26.2 Normalized under-lid fuel mass ow. The burner is switched onafter approximately 80 hours of baking, since the gas temperaturehas to reach a certain level before ignition can occur. . . . . . . . . 445

26.3 Normalized �re curves for ring furnace operation. In traditionalring furnace operation, tracking along these curves is implementedeither by manual or automatic control. . . . . . . . . . . . . . . . . 446

LIST OF FIGURES xxxiii

26.4 Temperature history in the center of anodes (xz- plane) after 180hours. Data are normalized such that maximum value 1 correspondsto the maximum temperature of the anodes during the whole bakingcycle. The axes denote grid points in x- and z-directions. To beable to show the details in the temperature pro�les, the intervalalong the normalized temperature axis varies between the di�erentpanels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447

26.5 Comparison of headwall boundary conditions: Original �reshaftboundary conditions vs. adiabatic headwall boundary conditions.The data are normalized according to the maximum di�erence oc-curring. The interval along the normalized temperature axis changesfrom panel to panel. . . . . . . . . . . . . . . . . . . . . . . . . . . 448

26.6 Normalized temperature di�erence in the xz-plane between anodesurface (7 cm below anode surface) and anode center. The maxi-mum di�erence is found in part B. The temperature is normalizedsuch that maximum value 1 corresponds to the maximum temper-ature di�erence occurring during the baking cycle. . . . . . . . . . 449

26.7 Typical lag between normalized temperatures in the gas (uppercurve) and anodes (surface (intermediate curve) and center (lowercurve)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450

26.8 Typical tar volatile generation rate in the anodes as function of time 450

26.9 Tar volatile generation in the center (xz-plane) of the anodes after94 hours. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

27.1 A hierarchical control system with two levels is suggested for thebaking process: At the upper level, open loop optimization is per-formed to optimize ring furnace operation. On line control is per-formed around the optimized process trajectory . . . . . . . . . . . 457

27.2 The suggested control philosophy for the ring furnace. Open loopoptimization is performed for a whole �rezone. A control correc-tor is designed for tracking along the optimal trajectories. In thesuggested scheme, the control corrector operates on the section level.461

27.3 Block diagram for the control corrector which acts on the sectionlevel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

27.4 The chain of baking sections used as model system for the control-lability analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474

A.1 Control volume used for establishing the balance equations. . . . . 509

B.1 Composite medium which consist of n components. . . . . . . . . . 513

B.2 Volume with an open void fraction. . . . . . . . . . . . . . . . . . . 516

xxxiv LIST OF FIGURES

B.3 Volume with an open and a closed void fraction. . . . . . . . . . . 517

B.4 Porosity relationships in a packed bed with porous particles. . . . . 520

B.5 Mercury porosimetry for pore diameter measurement. . . . . . . . 523

C.1 Steps in aluminium production. . . . . . . . . . . . . . . . . . . . . 526

C.2 Details from the electrolytic bath. From Grjotheim & Kvande (1993).526

C.3 Electrolytic cell with prebaked anode. . . . . . . . . . . . . . . . . 529

List of Tables

3.1 Consequences of butts impurities. From NIF (1986). . . . . . . . . 21

3.2 Example of dry aggregate recipe. From Murgia & Bello (1978). . . 22

5.1 Property-value ranges for prebaked anodes. After Jones (1990), As-perheim, Foosn�s, Naterstad & Werge-Olsen (1993a), Asperheim,Foosn�s, Naterstad & Werge-Olsen (1993b) and Asperheim, Foos-n�s, Naterstad & Werge-Olsen (1993c). For the reactivities, ex-cellent anodes have airburn and CO2-reactivities below 20 and 10mg/(cm

2hr) respectively. Correspondingly, poor anodes have reac-

tivities above 80 and 50 mg/(cm2hr). . . . . . . . . . . . . . . . . . 53

5.2 Overall carbon consumption. From (Grjotheim & Welch 1988, Tab.4.5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.1 Classi�cation of optical texture according to Marsh (1989, pp. 20).d, l and w denote diameter, length and width of the optical domain. 77

9.1 Parameters used in simulation of a simpli�ed model for carbonstructure and porosity. Carbon structure as represented by thelayer plane diameter La changes during heat treatment due to con-sumption of the disorganized carbon phase. . . . . . . . . . . . . . 146

9.2 Parameters used in simulation of the lumped model for developmentof La. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

12.1 Experimentally observed coke yields in pitch pyrolysis as functionof heating rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

12.2 Experimentally observed weight loss of a baked anode. . . . . . . . 175

12.3 Classi�cation of anisotropic texture according to the InternationalCommittee of Coal Petrography. From Krebs, Elalaoui, Mareche& Furdin (1995). An alternative nomenclature is given in Marsh(1989, pp. 20). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

xxxvi LIST OF TABLES

13.1 Di�erent formulations of independent reaction schemes. Single re-action schemes have 4 and 4nv parameters for total and individualvolatiles respectively. In multiple reaction schemes, the correspond-ing number of parameters is 4nr and

Pnvi=1 4nr;i. . . . . . . . . . . 197

14.1 Nominal composition of pitch used for parameter estimation. . . . 204

14.2 Pitch composition used in simulations of the model for low temper-ature pyrolysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

14.3 Nominal coke yield, yield of non-condensables and sensitivities ofthe yield of non-condensables with respect to coke yield. The yieldsare linear functions of pitch coke yield. The sensivities depend onlyon the pitch composition. Composition data are the same as inExamples 1 and 2 above. . . . . . . . . . . . . . . . . . . . . . . . . 215

14.4 Pitch composition used in simulations of the extended reactionscheme for modelling of low-temperature and high temperature py-rolysis of coal tar pitch. . . . . . . . . . . . . . . . . . . . . . . . . 219

14.5 Pitch composition used in simulations of the extended reactionscheme for modelling of low-temperature and high temperature py-rolysis of coal tar pitch. Conversion dependent activation energieswere used for decomposition of the �-resins. . . . . . . . . . . . . . 221

15.1 Parameters in the model for real density of pitch coke. Data weretaken from Wilkening (1983, Fig. 11). A heating rate of 5:0�C=hrwas used in the estimation. By using another set of data, di�erentmodel parameters would be obtained in the estimation. . . . . . . 240

16.1 Parameters in the model for the viscosity of binder coke as a ther-mally activated process. . . . . . . . . . . . . . . . . . . . . . . . . 254

17.1 Thermodynamic data for graphite and amorphous carbon. In thecalculation is used �v = 3�l. Some data can be found in Komatsu& Nagamiya (1951) and Kelly (1981). . . . . . . . . . . . . . . . . 256

17.2 Data suggested for the model of speci�c heat capacity. . . . . . . . 263

17.3 Parameters used in the thermal conductivity model based on anempirical Lorenz-function. . . . . . . . . . . . . . . . . . . . . . . 265

19.1 Parameters for decomposition of the fraction in pitch pyrolysisbased on data for evolution of condensables found in Wilkening(1983, Fig. 10). Data for heating rates of 5.5 and 11.0�C=hr wereused in the estimation. The reaction scheme was discussed by Gun-dersen (1995b) and kinetic data from Ko�st�al et al. (1994) was usedfor the polymerization reactions. . . . . . . . . . . . . . . . . . . . 288

LIST OF TABLES xxxvii

19.2 Parameters for pitch polymerization kinetics taken from Ko�st�al et al.(1994). The kinetics for decomposition of �-resins is also used fortransformation of the primary �-resins (i.e. �p-resins). . . . . . . . 288

19.3 Stoichiometric parameters for the pyrolysis model. . . . . . . . . . 288

19.4 Parameters in �rst order rate laws with conversion dependent ac-tivation energy used for the reaction steps for generation of non-condensables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

19.5 Kinetics for d002, Lc, nL and xdm. . . . . . . . . . . . . . . . . . . 289

19.6 Parameters in the model for real density of the isotropic phase. . . 289

19.7 Molar masses of pitch fractions vs. La. For �p- and �s-resins,La;i = 0:85 f(Mi) was used which gives M�p = M�s = 14:0265 �A.f(�) corresponds to the hexagonal model for aromatic moleculesdescribed in Ollivier & Gerstein (1986). . . . . . . . . . . . . . . . 290

19.8 Initial values for state variables in the simulation model for softcarbon pyrolysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

19.9 Other parameters in the model for high temperature pyrolysis. . . 291

25.1 Heat of combustion for volatiles in pitch pyrolysis. . . . . . . . . . 379

25.2 Subsystems in the ring furnace. . . . . . . . . . . . . . . . . . . . . 403

27.1 Assumed values and ranges for chemical, electrical, mechanical andother physical properties of prebaked anodes which are used in op-timization of anode quality. For the thermal conductivity, the givenrange was taken care of only by speci�cation of hard constraints.Based on Table 5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . 464

27.2 Thermal properties for the solid materials used in the linear control-lability analysis. The values were taken from Charette & Bourgeois(1984). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471

xxxviii LIST OF TABLES

Nomenclature

Abbreviations

BSU Basic Structural UnitCCD Critical Current DensityCE Current E�ciencyEE Energy E�ciencyHHT Heat Treatment TemperatureHHt Heat Treatment timeNPDL Nonlinear Partial Di�erential equation(s)QI Quinoline InsolublesSSPC State Space Predictive Control

Subscripts

Subscript Description

a, i;a anode, anode property indices

A part A

bm binder matrix

B part B

c convection

CH4methane

CHntar volatiles

chn;A ue channel(s) in part A

chn;B ue channel(s) in part B

CO2carbon dioxide gas

cr crystallite

dm disorganized carbon

fc �ller coke

g gas

gr graphite

H2hydrogen gas

xl Nomenclature

hdw headwall

lid section lid (cover)

O2oxygen gas

pc pitch coke

ulid under-lid

up;A under-pit, part A

up;B under-pit, part B

r radiation

s solid

Chemical Compounds and Materials

Compound Description

Ac, ACHn, di�erent phases (compounds) in the anode

ACH4, AH2

Afc, Apc

� �-resins�p primary �-resins�s secondary �-resins�s;CH4

secondary �-resins which release methane�s;H2

secondary �-resins which release hydrogen� � -resins -resins 1 �rst subfraction of -resins (may release volatiles) 2 second subfraction of -resinsAl aluminiumAl2O3 aluminaC carbonCfc �ller cokeCpc binder cokeCO2 carbon dioxideCO carbon monoxideO2 oxygen gas

Symbols from Control Theory

Symbol Description

A, B, C, D matrices in linear (linearized) modelf state space function

d property function

Nomenclature xli

h measurement functionG1, G2 control matricesJ , Ji performance index (objective functional)

used in ring furnace optimizationL objective functionu control vectorx state space vector�q anode quality vector

y measurement vector

~z, z property vectors for anode qualityx�

nominal state vectoru�

nominal control vectorz�

nominal anode quality vector; property vectorv�

nominal disturbance vectorS1, Q1, P matrices used in control objective functionS2, Q2, P

Mathematical Symbols and Variables

Roman Description Unit

Letters

Ar, Ar;� nozzle area m2

A;Ag ; As surface cross sectionA Helmholtz free energy Jb instrumental broadening -B birth rate of countable entities 1/(m3 s)Bc coalescence birth rate 1/(m3 s)Bn nucleation birth rate 1/(m3 s)B1 half peak width -ci concentration of component i mol/m3

cp, cp;i speci�c heat capacity at constant pressure J/(kgK)const. pressure spec. heat cap. of isotropic phase J/(kgK)

cp;c const. pressure spec. heat cap. of anisotropic phase J/(kgK)cv, cv;i speci�c heat capacity at constant volume J/(kgK)cv;el electronic contribution to cv J/(kgK)cv;lat lattice contribution to cv J/(kgK)Cel electric carbon consumption -Cex theoretical electric carbon consumption -Cn net carbon consumption -

dm diameter of mesophase sphere �AdAr actual surface area -dAa super�cial surface area -di diameter of coke aggregate m

dgr interlayer spacing in graphite �A

xlii Nomenclature

dm diameter of mesophase sphere �A

d, d002 interlayer spacing �A

di contribution to interlayer spacing in �Amultiple reaction scheme

d002(1) �nal value of interlayer spacing �A

d002;i(1) �nal value of interlayer spacing �Ain multiple reaction scheme

D death rate of countable entities 1/(m3 s)Dh hydraulic diameter mDi di�usion coe�. for component i m2

Di;e e�ective di�usion coe�. for component i m2

�Di;em e�ective multicomp. di�usion coe�. for component i m2

Di;k mutual di�usion coe�. for components i and j m2

Dk;i Knudsen di�usion coe�. for component i m2

D matrix of di�usion coe�ents m2

Eb� monochromatic emissive power of a blackbody WEb total emissive power of a blackbody WEi apparent activation energy J/molEk kinetic energy J/kgEp potential energy J/kgf Moody friction factor for pipes -f�;i initial fraction of volatile comp. i in green anode -fa mass fraction of catalyzing agents (ash) in anode -fd(�) distribution function for activation energy -fg(�) Gaussian distribution function -f , fi relative yield of volatiles -f�, f�i (relative) ultimate yield of volatiles -f uidity parameter in the collapse-law for �-resins -f 1;b stoichiometric coe�cients in swelling rate law -f 2 , f fw;iso, weight functions in rate law for ��s -fw;c, fw;shrfCH4;� ultimate yield of methane -fH2;� ultimate yield of hydrogen -

f i;CH3

C mass fraction of carbon bonded in methyl -within a pitch fraction or whole pitch (i = ; p)

f iCH3mass fraction of methyl -within a pitch fraction or whole pitch (i = ; p) -

f v ultimate yield of condensables -f; g; h; p; q; s stoichiometric coe�cients -

fd(�) rate law function for d modelled as a �A/sthermally activated process

fLa(�) rate law function for La modelled as a �A/sthermally activated process

fLc(�) rate law function for Lc modelled as a �A/sthermally activated process

fp;� binder contents in green anode -

Nomenclature xliii

fp binder (coke) contents in anode -f1;La functions in expression for vLa 1/hr

f2;La m3�A2

F radiation factor -Fgs radiation factor; rad. betw. gas and solid surfaces -F12 radiation factor; rad. betw. solid surfaces 1 and 2 -g, gi relative conversion in multiple reaction schemes -gA mass ow (gas to vertical ues part A) kg/(m s)

per length in direction of owgB mass ow (gas from vertical ues part B) kg/(m s)

per length in direction of owgf mass ow of fuel per length (in dir. of gas ow) kg/(m s)gv;i volatile mass ow per length of ue channel kg/(m s)G mass ow kg/sGA;i mass ow to under-lid zone from vert. ues in part A kg/sGf under-lid fuel mass ow kg/sh speci�c enthalpy J/kg

heat transfer coe�cient W/(m2K)

hc convective heat transfer coe�cient W/(m2K)

hf height of foam layer in swelling material mhg speci�c enthalpy for gas J/kghg;B spec. enthalpy for gas at outlet of vert. ues in part B J/kghl e�ective height under the lid mJk;i, jk;i intraphase mass di�usion ux vector kg/(m2 s)

~|k;i

intraphase mass di�usion ux vector kg/(m2 s)

J radiosity -ka parameter in the Warren equation -kc parameter in the Scherrer equation -kb rate constant in backward reaction -kf rate constant in forward reaction -ka;iso isotropic shrinkage parameter in model for �a -kr;iso isotropic shrinkage parameter in model for �r -kp viscous permeability m2

kp;a viscous permeability of an anode m2

kp;cl viscous permeability of a close packed medium m2

K total permeability -K� slip ow permeability -KK Carman factor -kt thermal conductivity W/(mK)kt;a thermal conductivity of an anode W/(mK)k�;i pre-exp. factor in Arrhenius-law for volatiles J/moll pore length mL air inleakage ow kg/hr

La layer plane diameter �A

La;i contribution to layer plane diameter in �Amultiple reaction scheme

xliv Nomenclature

La(1) �nal value of layer plane diameter �A

La;i(1) �nal value of layer plane diameter �Ain multiple reaction scheme

Lc stacking height �A

Lc;i contribution to stacking height in �Amultiple reaction scheme

Lc(1) �nal value of stacking height �A

Lc;i(1) �nal value of stacking height �Ain multiple reaction scheme

mb mass of binder pitch kgmc mass of crystallite carbon kgmcr mass of a crystallite kgmfc mass of �ller coke kgmpc mass of pitch coke kgmt total mass of carbon material kgmdm mass of disorganized carbon kgM , Mi molar mass g/moln dynamic section number -na total number of anodes in a section -n�a;�s exp. in collapse-law for �s-resins -nf exponent in uidity factor in collapse-law for �s-resins -ni apparent reaction order in model for volatiles -nL;cr number of layer planes in a crystallite -nL tot. number of layer planes in the carbon material -NA Avogadro`s number -Nx, Ny, Nz spatial resolution along x, y and z - axes -

~p tuning parameter in model for ~X -p pressure Papg pressure in the gas ues Papg;out pressure in out owing gas from a section Papg;in pressure in incoming gas to a section Papf price for fuel krpba price for baked anode krpga price for green anode krPi compressive pressure Pa

q00 total heat ux (convective and radiative) J/(m2s)

q00c convective heat ux J/(m2s)

q00r radiative heat ux J/(m2s)

Q tot. heat transfer per unit length in dir. of gas ow W/mr pore radius m

rc rate of generation of crystallite carbon kg/(m3hr)

rdm rate of consumption of disorganized carbon kg/(m3hr)

ri rate of emission of volatile component i kg/(m3hr)

rv volatilization rate kg/(m3hr)

rr residue from CO2 reactivity test -rCO2;a O2 reactivity of anode mg/(cm2 s)

Nomenclature xlv

rCO2;a CO2 reactivity of anode mg/(cm2 s)

rsw swelling gas release rate kg/(m3hr)

rsw;eff , ~rswR gas constant -

Rav;a rate of release of volatiles as related kg/(m

3hr)

to the apparent volume of the anode

Rrv;a rate of release of volatiles as related kg/(m

3hr)

to the real volume of the anodesi;a dusting index of anode -S geometric shape factor -Sa surface area m2

S, Sg speci�c surface area m2=gSp surface area per unit pore volume 1/mSv surface area per unit bulk (apparent) volume 1/mS stoichiometric matrix -T temperature KTa temperature of ambient air KTc batch time, �re cycle time hrT� reference temperature Kv gas velocity m/su speci�c internal energy J/kgva, vb parameters in speci�c heat capacity model -vl geometric velocity vector m/svi kinetics of the property space -/sv, vi velocity vector m/svk velocity of phase k (phase velocity) m/sV k;i vk;i velocity of component i in phase k m/s

vk;i;d di�usion velocity of phase k m/s

vLa growth rate for La �A/s

vLc growth rate for Lc �A/sV volume m3

Va apparent volume m3

Vr real volume m3

Vb, Vb;� bulk volume m3

Vc, Vc;� crystallite volume m3

�V volume of solid carbon m3

Vc volume of crystallite carbon m3

Vdm volume of disorganized carbon m3

Vulid volume under the lid m3

wa mass ow of air inleakage kg/hrwg;in mass ow of gas into a section kg/hrw(T ) weight function in speci�c heat capacity model -

W work term in energy balance equation W/m3

xa mass fraction of anisotropic pitch fraction -xi mass fraction of isotropic pitch fraction -xf mass fraction of �nes in anode �ller coke -Xi total porosity in �ller coke -

xlvi Nomenclature

Xi;� open porosity in �ller coke -Xi;c closed porosity in �ller coke -XCH4

conversion parameter for CH4 -XCHn

conversion parameter for CHn -XH2

conversion parameter for H2 -Xi fractional conversion of volatiles -~X conversion parameter used in the model for �dm -x; y; z spatial coordinates myi mass fraction of gas component i -Ye interparticular porosity in packing coke bed -

Greek Description Unit

Letters

� absorptivity -� coe�cient of thermal expansion 1/K�a shrinkage correction parameter in the model for �a -�r shrinkage correction parameter in the model for �r -�j correction factor for line broadening (j = a; c) -�i solubility parameter (i = 1; 2) -�1 dimensionless parameters in the slip ow equation -�fc �ller coke particle density within the anode 1/mm3

�a anode grain particle density within the anode 1/mm3

�� gas emissivity correction -�Hi heat of vaporization J/kg�Hf fuel oil combustion entalphy J/kg�Hv;i combustion entalphy of volatile i J/kg�p capillary pressure Pa�x, di�erential increments in space m�y,�z� emissivity -�a volume fraction of anisotropic pitch fraction -�i volume fraction of isotropic pitch fraction -�g emissivity of combustion gas -�s emissivity of solid materials -�0c volume fraction of crystallites within -

the volume of the granular structure�0dm volume of disorganized carbon within -

the volume of the granular structure� current e�ciency -

viscosity kg/(m s)�(�) number density function 1/m3

�i i'th order moment of � �Ai/m3

surface tension N/m� pitch coke yield -

Nomenclature xlvii

�� open porosity, bulk level -�c closed porosity, bulk level -�0c closed porosity, granular structure level -�T total porosity, bulk level -� foaming constant -��;fc open porosity of �ller coke -�dm (possible) porosity within volume Vdm -�c;c induced closed porosity of the �ller coke aggregate -

ie. open �ller coke porosity which correspond tosealed pores

�c;� porosity of the �ller coke aggregate -�a;fc vol. fraction of the anode consisting of �ller coke -��a;fc vol. fraction of the anode not consisting of �ller coke -�f transport (open) porosity in the anode -�T;a total porosity of the anode -�c;a closed porosity of the anode -��;a open porosity of the anode -�T;pc total porosity of binder pitch coke -�c;pc closed porosity of the binder pitch coke -��;pc open porosity of the binder pitch coke -

� density kg/m3

re ectivity -

�� density at reference temperature T� kg/m3

�a apparent density kg/m3

�r real density kg/m3

reduced density kg/m3

��c density of solid carbon without pores kg/m3

�c density of crystalline carbon kg/m3

�abd aggregate bulk density kg/m3

�add aggregate displacement density kg/m3

�b;i;� initial bulk density kg/m3

�a;a apparent density of anode kg/m3

�a;a;b apparent density of baked anode kg/m3

�a;a;g apparent density of green anode kg/m3

�a;pc apparent density of binder pitch (pitch coke) kg/m3

�r;a real density of anode kg/m3

�r;a;b real density of baked anode kg/m3

�r;a;g real density of green anode kg/m3

�r;p real density of pitch kg/m3

�r;p;i;� parameter in density model for isotropic pitch kg/m3

�fc density of �ller coke kg/m3

�gr density of graphite kg/m3

�i mass concentration of component i kg/m3

� Stefan-Boltzmann Constant W/(m2K4)�(�) standard deviation -

xlviii Nomenclature

�b;a bending strength of anode Pa�c;a compressive strength of anode Pa� , �� transmissivity -�D Debye characteristic temperature K�E Einstein characteristic temperature K

Dimensionless Numbers

For ow in a pipe with diameter d, the following dimensional numbers can bede�ned:

Number Formula Description

Nu hcdk Nusselt number

Pe �vdD

Peclet numberPr

cp�

k Prandtl number

Redv�g�

Reynolds number

Chapter 1

Introduction

In a typical electrolysis cell1 as shown in Figure 1.1, several anodes are dippedinto the bath. During electrolysis, the carbon anodes are consumed under the for-mation of gaseous CO2 and liquid aluminium. The prebake-anode technology isdominating in today's aluminium production. Even with several decades of exten-sive progress in carbon science and increased understanding of anode behaviour inthe Hall-Heroult cells, the carbon anode is still one of the most critical componentsof alumina reduction technology. This is due to a new generation of highly auto-mated cells and reduced quality of anode raw materials. The production of highquality anodes depends on both the supply of raw materials as well as successfuloperation in every step along the anode production line. Anode baking whichtakes place in specially designed furnaces is an important and expensive process.In the ring furnace, there is a lack of measurements of process variables in theanodes and also limited knowledge of the complicated transformations which takeplace in the anode during baking. The heat treatment takes place in speciallydesigned sections which are linked together in a chain. A successful control sys-tem for anode baking has to take into consideration the interactions between thesections.

Alumina, electrical energy and anodes are the most cost intensive elements inaluminium production as shown in Figure 1.2. Anode production costs are evenlydivided between raw material and manufacturing costs. Therefore, the productionof high quality anodes at low cost is an important factor in optimization of smelteroperation.

In Hydro Aluminium a.s, research and development in ring furnace technology hasbeen an important activity for several years. The research has mainly been donewith basis in mechanical engineering and thermal engineering. The contributionsin this work are presented also from a control engineering point of view.

1In the Hall-H�eroult process, liquid aluminium is produced by electrolytic reduction of alu-mina. Alumina is dissolved in an electrolyte which mainly contains cryolite. The Hall-H�eroultprocess is reviewed in Appendix C.

2 Introduction

Thermal Insulation

Cathode Block

Current Collector Bars

Busbars

Alumina

BreakerCrust

Molten Aluminium

InsulationThermal

HangerAnode

PrebakedAnode

AluminaCrust

Electrolyte

Casing

Steel

FrozenLedge

Carbon Lining

Figure 1.1: Cell for aluminium electrolysis with prebaked anodes. Based on Gr-jotheim & Kvande (1993, Fig. 2).

Al O2 3

EnergyElectrical

Anodes15 %Labour Costs

Capital Costs

Figure 1.2: Cost estimates in Aluminium production. Based on Keller & Oderbolz(1985).

1.1 Previous Work

In this section, a review of previous work on modelling and control of baking ringfurnaces is provided to summarise state of the art in ring furnace technology. Thissection is based on a more detailed review given in Gundersen (1995a). Summariesof previous work in mathematical modelling of pyrolysis and carbon properties

1.1 Previous Work 3

can be found in Gundersen (1995b), Gundersen (1996f), Gundersen (1996c) andGundersen (1996d) or parts II, II and IV of this study.

Anodes can be baked in three di�erent types of furnaces: Tunnel furnaces, hori-zontal - and vertical ue ring furnaces. Commonly used are the horizontal - andvertical ue ring furnaces. In horizontal furnaces, the gases ow along horizontal ues, giving rise to the name. In vertical furnaces, gases ow in vertical ductsin each section. Process cycle time is long, between 10 and 15 days, making ex-periments both time consuming and expensive. These constraints have promoteddesign and computer implementation of models. For a long time, mathematicalmodels have been used for optimizing both ring furnace construction and opera-tion procedures. Traditionally, manual control has been used for operation of thefurnaces. Many installations have introduced automatic control to improve anodequality and process-economy. Both modelling and control of ring furnaces startedin the late seventies, and the sophistication of both models and control strategieshave grown along with advances in computer technology.

Modelling of ring furnaces is discussed in Subsection 1.1.1. Literature on control ofring furnaces is reviewed in Subsection 1.1.2. Subsection 1.1.3 gives perspectivesfor further research in the �eld of modelling and control of ring furnaces.

1.1.1 Mathematical Modelling of Ring Furnaces

The baking of anodes is mainly a thermal process. Heat is transferred betweengases and solids (brick, coke and anodes). A general description of basic thermalphenomena can be found in Racunas (1980) and Martirena & Marletto (1980) withdiscussion of open and closed furnaces respectively. Ring furnace modelling hasmainly focussed on mathematical description of the thermal phenomena duringbaking.

Dreyer (1988, pp. 303-304) reports extensive use of mathematical models used forboth the design of ring furnaces and ring furnace simulation studies. Models maybe classi�ed in three groups:

� Design models

� Models for furnace simulation studies

� Models for on-line control purposes

In Stevenson (1988), a design model is described in detail. The model was usedfor optimization of ue design in an open ring furnace. Simulation models are use-ful for studying the e�ect of changing furnace operation parameters. Simulationmodels for full ring furnace systems are presented in Keller & Disselhorst (1981),Bui, Dernedde, Charette & Bourgeois (1984), Thibault, Bui, Charette & Dernedde(1985), Bui, Charette & Bourgeois (1987), Lopez, Castillo & Vera (1989) and Bour-geois, Bui, Charette, Sadler & Tomsett (1990). In contrast to Keller & Disselhorst(1981), Bui et al. (1984) and Thibault et al. (1985), Bui et al. (1987) o�er more

4 Introduction

details about the modelling approach. The model presented consists of di�er-ent submodels describing phenomena of heat transfer, combustion, air-inleakageand volatile release. Lopez et al. (1989), Bourgeois et al. (1990) and Gundersen& Balchen (1993b) have developed models for vertical furnaces. Dreyer (1988),Lopez et al. (1989), and Demange (1991) report use of mathematical models foron-line control of ring furnaces.

For modelling purposes, the ring furnace is usually decomposed into subsystems.Stevenson (1988, pp. 307) suggest the following decomposition along the ues ofa horizontal furnace:

1. The combustion ues

2. The ue wall

3. A packed bed (pit) of granular coke and anodes (green, partly carbonized orbaked)

In contrast to the simple gas path subdivision of an open furnace, Bourgeois et al.(1990, pp. 549) identify four calculation areas along the gas path of a closedfurnace:

1. The �reshaft zone

2. The under-lid zone

3. The pit zone

4. The under pit zone

In each calculation area there is heat transfer between solids and gases. A similarprinciple is used in the gas path model presented in Gundersen & Balchen (1993b)for a vertical ring furnace of Hydro Aluminium type.

Keller & Disselhorst (1981) model a horizontal ring furnace as a (semi) continuouscounter ow heat exchanger in which the gases ow in one direction and furnacesolids " ow" in the other direction. This modelling technique is also applied inBui et al. (1984).

A closed furnace cannot be modelled as a counter ow heat exchanger. For aclosed furnace, usually one section is isolated, and a symmetric part of the sectionis modelled.

Due to the large time constants for the temperature �eld of the solid materials, adynamic heat balance is calculated for the solids in ring furnace models. On theother hand, gas phenomena have small time constants and stationary models givesatisfactory results as well as reduced computational time (Stevenson 1988, pp.306).

In ring furnaces, gas or oil may be used as a fuel. In combustion calculations,the simple approach of ideal mixing of fuel and oxygen with complete and ideal

1.1 Previous Work 5

conversion to CO2 and H2O is undertaken. Charette & Bourgeois (1984) give amodel of fuel oil combustion based on these assumptions.

Air inleakage may cause severe problems in ring furnace operation. At high suctionpressures, the air inleakage may be signi�cant and the thermal e�ciency of the ringfurnace decreases, since energy is consumed for heating the excess air. Dernedde& Bourgeois (1987) present a model for air inleakage based on the ori�ce equation.

Anode heat conduction is governed by the Fourier Law. Both one-, two- andthree- dimensional heat conduction models for ring furnace solids can be found inthe literature. Keller & Disselhorst (1981) and Thibault et al. (1985) use a one-dimensional model, but a two-dimensional model is presented in Bui et al. (1984).Three-dimensional models are presented in Furman & Martirena (1980), Hurlen,Lid, Naterstad & Utne (1981), Monica, Marletto & Martirena (1983), Hercules &Sarmiento (1988) and Gundersen & Balchen (1993b) for a symmetric part of a pit.The three-dimensional approach is only applied in closed furnace modelling, butBourgeois et al. (1990) uses a two-dimensional model for a closed furnace pit ofRiedhammer type.

During baking, chemical reactions take place in the pitch fraction of the anodes.The �rst model-based approach to pitch pyrolysis was given by Tarasiewicz &Stumpf (1984). By using general rate equations for the decomposition of chem-ical compounds, the composition of the volatile gases was calculated. Based onthermogravimetric experiments on laboratory scale anodes, Tremblay & Charette(1988) gained new insight into the pyrolysis of anodes during baking by using amodel with Arrhenius type rate laws. The kinetic parameters in the model aregiven as functions of the heating rate. Jacobsen (1997) used the model in Tremblay& Charette (1988) as basis for calculation of coupled mass- and heat transfer inanodes during baking. The pressure distribution of volatiles in a two-dimensionalcross section of the anodes in a vertical ue ring furnace was calculated.

During pitch pyrolysis, gaseous components are released from the pitch, and trans-ported through the porous anode. In the anodes, secondary reactions may causegrowth of carbonaceous material on the walls of the pores. Finally the volatilegases leave the anode to enter the packing material which surrounds the anodes.After passing the packed bed of coke aggregate particles, the volatiles enter the ues to be combusted. The importance of easy evacuation of the volatiles from thepacked bed surrounding the anodes is stressed in Oprescu & Semenescu (1992). Atoo dense packed bed (i.e. small particle sizes) might cause coking of gases in theaggregate surrounding the anodes. In this way, aggregate particles become stickedto the anodes, and one may have problems with cleaning the anodes when bakingis completed

1.1.2 Control of Ring Furnaces

The baking of anodes is a time consuming and fragile process. Due to the chemicaland mechanical transformations in the anodes during baking, careful heating hasto take place in certain parts of the heating cycle. Therefore, certain constraints

6 Introduction

are put on the allowable heating rates during baking. These constraints are partlybased on long experience in furnace operation but often there are model basedtools available to give guidelines in this respect. The models discussed in thissection are powerful tools for design of new �ring strategies. The design of �ringstrategies takes place in open loop. Based on experience, experiments, improve-ments in furnace technology and model calculations, improved �ring strategies canbe implemented.

Anode baking is realized by maintaining a certain ue gas temperature as functionof heat-treatment-time in each section. In real furnace operation, tracking alongthe desired gas temperature curve is implemented either by manual control orclosed loop automatic control using mainly conventional control methods. Assuch, there is usually no monitoring of the state of the anodes during baking, butspecially designed experiments can be used to obtain this important information.

Traditionally, manual control has been used for ring furnaces. In Pinner & Barrier(1977), improvements in manual control was achieved due to optimization of

1. Pit �nishing temperature

2. Exhaust gas temperature

The improved �ring strategy gave both increased anode quality and reduced fuelconsumption. In Auchterlonie & Van der Toorn (1977), temperature in the �rstpreheat section was controlled by the draft (and to a lesser extent by the control ofthe �res). In the direct �red sections, computerized registration of temperatureswas used. These are compared to the desired target temperatures and a messageto the operator is given if adjustments are needed. The furnace control strategymay be summarized as follows:

1. The temperature curve in preheat sections is controlled by furnace draft (i.e.suction pressure)

2. The temperature curve in direct �red sections is controlled by fuel ow

Still there are baking plant facilities which use manual control. The control phi-losophy reported in Auchterlonie & Van der Toorn (1977) is also used today butusually, however, ring furnace operation is automated. In the late seventies, the�rst articles discussing automatic control of ring furnaces appeared.

The use of automatic control in ring furnaces is most often discussed in the con-text of open ring furnaces. In the articles of Adams (1977) and Jurges (1983), adescription of a complete system for both gaseous fuel (Adams) and furnace draftcontrol (Jurges) is presented. Individual target temperature curves are establishedfor each ue. Computer detection of the end of pitch burn is used to determinewhen to ignite burners. Anode temperatures are constantly monitored by at leastone thermocouple located in one of the pits in each section under direct �re. Theburner control algorithm is described in Adams (1977, pp. 371-373). For the draftcontrol system, a description is given in Jurges (1983, pp. 795-796). The most

1.1 Previous Work 7

successful operation was obtained by using a linear target temperature curve forthe �rst preheat section and use furnace draft to maintain ue temperature invicinity of the target temperature. The draft control system gave lowered fuelconsumption, higher bake �nishing temperatures, and better overall temperaturecontrol.

In Berry & Hu (1986) a detailed description of the hardware con�guration of anopen ring furnace control system is given. Two movable burner bridges are usedfor each �re, one for the preheat section and one on the heat section with burnersof impulse type. An infrared temperature sensor is used (Berry & Hu 1986) fortemperature measurements. In contrast to Jurges (1983), draft control is notdiscussed.

Keller & Oderbolz (1985, pp. 1112-1119) describe a control system used at Alu-chemie in the Netherlands. The baking furnace process control system was devel-oped by Alusuisse. The main operation principles are the same as discussed inAdams (1977) and Jurges (1983). However, the ring furnace is divided into fourzones. Zone 1 is controlled by means of the draft and zone 4 by means of fuelquantity. Operation in zones 2 and 3 is controlled by means of the joint in uenceof the draft and fuel quantities.

Similarly, in Dreyer (1988) both draft and fuel injection are used as control param-eters. On the other hand, three important aspects of furnace control is introduced:

1. For each ue line, the static pressure is measured at a point upstream ofthe burners. The pressure at this reference point is regulated around theatmospheric to ensure e�ective fuel injection.

2. An opacimeter mounted at the exhaust fan measures ue gas opacity. Thedraft setpoint for minimum opacity is calculated from measurements. In thisway, optimal combustion is secured.

3. A simulation program continuously computes the rate of release of volatilesand the amount of oxygen needed for their combustion. In this way, thecontrol system is based partly on information obtained from a model runningin real time with the process.

The Bake furnace Process Control System (BPS) presented in Mannweiler, Sulz-berger & Oderbolz (1991) is also based on Alusuisse technology. As reported inKeller & Oderbolz (1985), the ring furnace is divided into four zones for controlpurposes. In contrast to the version of the Aluchemie control system presentedin Keller & Oderbolz (1985), a zero pressure is maintained just behind the lastburner bridge. In principle, this is the same technique for e�ective fuel injectionas discussed in Dreyer (1988).

Descriptions of closed furnace automatic control are more scarce. In Murgia &Bello (1978) and Serra, Murgia & Cherchi (1982), automatic control is appliedfor a ring furnace at Alsar. Baking is controlled automatically by an IBM 1800computer. The quantity of fuel and/or suction pressure may be adjusted accordingto the temperature set point curve. In Jakobsen, Lid & Schreiner (1987, pp. 499),

8 Introduction

a brief description of a control strategy is given. The presented systems are basedon the same principles as �rst discussed in Adams (1977) and Jurges (1983).

Control systems for closed furnaces generally are not described in as much detailas the open furnace systems discussed above. On the other hand, more detailedinformation on control philosophy can be found in Gheorghiu & Oprescu (1987)and Lopez et al. (1989) where a kind of model based approach to furnace controlis taken. These articles are discussed below.

Gheorghiu & Oprescu (1987) state that controlled combustion of volatiles is aprerequisite for successful automation of the baking process. The combustion ofvolatiles depends on the O2-content of the gas in the section, i.e. the excess air inthe actual section. For a controlled combustion of the volatiles, the O2-level mustbe kept below a certain limit. This can be obtained in two ways:

� Increase fuel owrate into the furnace �re-train while keeping the suctionpressure in the furnace constant (i.e. increase O2-consumption).

� Reduce suction pressure while keeping fuel owrate into the furnace constant(i.e. reduce excess air in the furnace sections).

Suction pressure is kept constant, and fuel ow is tuned to obtain controlled com-bustion of the volatiles. A correlation between the amount of fuel and suctionpressure in a section is derived in Gheorghiu & Oprescu (1987, pp. 614-615). Tocope with situations where the temperature in the direct �red (equalizing) section(and the neighbouring downstream section) reaches a temperature above allow-able limits, Gheorghiu & Oprescu (1987) suggest to introduce a so-called cold airby-pass system at the entrance to the direct �red section.

Lopez et al. (1989) operate a model which calculates both anode and gas thermalphenomena in real process time. Block diagrams for the control algorithm ispresented. The control loop checks real temperatures against target temperaturesin sections under direct heat and preheat respectively.

In Demange (1991), a ring furnace control system has been developed based onmodern multivariable control theory. The control system relies on models basedon simple equations giving the correct "cause and e�ect" relationship of the phe-nomenon studied2. Model identi�cation is necessary to �t model parameters tooptimal values. The model is not based on the physical laws governing the process.It is argued that models based on physical knowledge are often ine�cient regard-ing real-time regulation, either because of computer requirements or unavailablemeasurements needed (Demange 1991, pp. 662). A prototype of the multivariablecontrol system has been installed on an old ring furnace and shown successfuloperation characteristics.

Generally, impulse type burners are used for both gas and oil combustion in thedirect �red sections. The duty cycle of the burner is adjustable (Macpherson 1989,pp. 581).

2"Black box models"

1.1 Previous Work 9

At the exhaust manifold, a suction fan sets up the draft in the ues. Open furnacesneed a suction fan for each ue, while only one fan is needed in a closed furnacedue to the furnace construction principle. Forced cooling of anodes is achieved byusing cooling which fans operate at the back end of the �re-train.

For temperature measurements, usually thermocouples are used for gas and solidtemperature measurements. In Macpherson (1989), pyrometers are used for mea-surement of brick surface temperatures. Flue suction pressure is measured with amanometer at the gas exhaust manifold or in the individual sections.

1.1.3 Fields of further Research

A model based approach to description of anode quality, would give new oppor-tunities for evaluation of new baking strategies that are needed due to changes inboth raw materials and furnace design. An automated procedure for design of thenew baking strategy could be implemented.

For a certain baking strategy, a multivariable approach to control of the ringfurnace would take care of interactions between sections as well as be able topredict anode quality during baking. This gives new perspectives in control ofsuch systems. Multivariable control of ring furnaces based on empirical modelshas been reported in the literature, but still progress can be done in the �elds ofboth control theory and modelling based on �rst principles.

1.1.4 Carbon Structure and Properties

Parts II, III and IV of this work deal with structure, pyrolysis, carbonization andthe evolution of properties of soft carbons. This constitutes a very comprehensive�eld of research and is was decided to associate information on previous work inthese �elds in parts II, II and IV together with the descriptions of the models ofthis work. One general comment, however, is o�ered here. In the carbon literature,the evolution of structure and properties is very often presented as a function ofheat treatment temperature. This is frequently done in two situations:

� The evolution of carbon properties as measured in experiments with anisothermal hold temperature.

� The evolution of properties in during heating at a linear temperature rate.

In the �rst case, the steady state value of the property at the set of isothermalhold temperatures is reported. In the second case, there is a linear relationshipbetween time and temperature and it seems as there is a preference in the carbonliterature to present the property as function of temperature rather than as afunction of time. Frequently in this study, the evolution of properties is related totemperature in the same way.

10 Introduction

1.2 Outline of this Work

This work is divided into �ve parts as follows:

� Part I: Description of Anode Manufacturing (Gundersen 1996a)

{ Anode manufacturing is reviewed in Chapter 2.

{ Chapter 3 deals with green anode production.

{ Chapter 4 gives a description of the baking process.

{ The concept of anode quality is reviewed in Chapter 5.

� Part II: Structure and Properties of Soft Carbons (Gundersen 1996f)

{ A summary of di�erent carbon forms is presented in Chapter 7.

{ Chapter 8 gives a review of carbon structure.

{ Models for the development of structure and porosity is derived inChapters 9 and 10.

{ A model for anode quality is presented in Chapter 11.

� Part III: Mathematical Modelling of Pitch Pyrolysis (Gundersen 1995b)

{ Chapter 12 summarizes characteristic features of pitch pyrolysis.

{ Pitch pyrolysis is reviewed in Chapter 13.

{ In Chapter 14, the pyrolysis model is presented.

� Part IV: Mathematical Modelling of Carbon Properties (Gundersen 1996c),(Gundersen 1996d)

{ Models for properties of single phase carbons are presented in chapters15 to 19.

{ Chapters 20 to 23 deal with modelling of anode properties.

� Part V: Modelling and Control of Baking Furnaces (Gundersen 1996b)

{ The ring furnace process model is presented in Chapter 25.

{ Ring furnace simulations are presented in Chapter 26.

{ The control philosophy is presented in Chapter 27.

Three appendices are included as follows:

� Appendix A: Summary of conservation laws

� Appendix B: Density and porosity relationships

� Appendix C: Features of the Hall - H�eroult process

1.3 Contributions of this Work 11

1.3 Contributions of this Work

The contributions of the study are:

� Mathematical models for porosity and crystallinity in carbon materials dur-ing carbonization and graphitization.

� Mathematical models for physical properties of carbon materials based onthe porosity- and crystallinity models.

� A model based formulation of anode quality parameters to be used in controlof ring furnaces.

� A mathematical model of a ring furnace of Hydro Aluminium design (closedtype furnace).

� A model based strategy for o�-line calculation of nominal baking strategies.

� A model based control corrector to be used in on-line tracking control alongthe nominal trajectory.

The following publications have been written:

1. Conference papers:

� Gundersen & Balchen (1993b)

� Gundersen & Balchen (1993a)

� Jacobsen & Gundersen (1997)

2. An extended paper on modelling of the baking process was presented inGundersen & Balchen (1995).

3. Internal reports:

� A review of research in modelling and control of carbon anode bakingis presented in Gundersen (1995a).

� A report on features of the baking process is given in Gundersen (1996a).

� A description of raw materials used in anode production is presentedin Gundersen (1996e).

� Details on modelling of pitch pyrolysis is presented in Gundersen (1995b).

� A state variable approach to the modelling of carbon microstructure ispresented in Gundersen (1996f)3.

� Mathematical modelling for the development of microstructure, poros-ity and physical properties of single phase soft carbons is presented inGundersen (1996c).

3The traditional carbon properties like mechanical strength, thermal conductivity etc. areviewed as functions of the so-called fundamental carbon properties (i.e. state variables) whichare used to de�ne the structure of the carbon material.

12 Introduction

� Gundersen (1996d) deals with mathematical modelling of physical prop-erties in carbon anodes.

� The baking furnace process model is presented in Gundersen (1996b)

The di�erent parts of the work can be put into the context of anode baking asshown in Figure 1.3.

The author hopes that this study contributes to better understanding of the phe-nomena that take place during baking, and that the suggested control strategycan serve as a basis for improvements in ring furnace operation.

Carbonproperties

Raw materials

-petroleum coke

-coal tar pitch

-composites-single phase

Ring Furnace Modelling and Control

Pyrolysis

-carbonization

in carbonsStructure in

-texture-porosity

-microstructure

Models for

Control theory

Models for

Control modelsfor a zone

for a zone

a section

-design

-simulation

-simulation -surveillance

-model basedcontrol

-optimization

common practicein

ring furnace operation

-on line controlsimulation

-training

-evaporation

Part I

Part IV

Part II

Part III

Part V

Part I

Part I

Part V

Figure 1.3: The relationship between di�erent parts of the work and ring fur-nace modelling and control. The texts associated with the arrows refer to thecorresponding parts of this work.

Part I

Description of Anode

Manufacturing

Chapter 2

Introduction to Anode

Manufacturing

There has been a tremendous development during the last two decades in carbonanode production. During the seventies, development in anode technology wasforced by quality deterioration of raw materials and increasing fuel costs. Duringthe last years, focus has been on total cost-e�ectiveness, improved anode qualityand manufacturing environment. These improvements are consequences of thesevere recession in the primary aluminium market in the early eighties (82-83),the introduction of advanced computer systems and new cell designs as well asenvironmental awareness (Hurlen & Naterstad 1991).

The basic steps in anode manufacturing are similar to those of the ceramic andconcrete industry: An aggregate �ller is mixed with a binder and the mixture ismoulded before hardening.

Anode production is subject to the following constraints (Jones 1990, pp. 2):

� Raw materials

� Plant facility (prebake vs. S�derberg)

� Laboratory facilities

� Economy

Within these limitations, the following factors are detrimental for proper anodebehaviour during electrolysis (Jones 1990, pp. 12):

� Anode design

� Connections (strength and conductivity)

� Carbon quality (physical and chemical properties)

16 Introduction to Anode Manufacturing

Parts of this work deal with carbon quality of the anodes. Design and connectionswill not be discussed.

The weigth of a typical unbaked (green) anode is between 0.5 and 1.0 ton. Theanode is made from a petroleum coke aggregate and coal tar pitch which acts asa binder to hold the coke particles together. The binder appears in an amountof maximum 18-20 % by weight. The coke aggregate used for anodes must becalcined (heat treated) up to approximately 1300�C before it can be mixed withpitch. Mixing and forming of anodes take place in separate process stages. Batchor continuous mixing and press and vibration forming of the anodes have beenused. Anode baking is the last and most expensive step in the anode preparationprocess. During baking, chemical and physical bonds between the petroleum cokeand pitch develop. The most important task is to create a uniform structure inthe petroleum coke and pitch binder matrix. Anode manufacturing includes thefollowing process stages:

� Anode paste manufacturing

� Green anode moulding

� Anode baking

In green anode production, raw materials contribute to approximately 90 % of thedirect costs; processing costs are only 10 %. Overviews of traditional and modernanode manufacturing are given in Mannweiler & Keller (1994).

Anode baking is a time consuming process which determines anode productioncapacity (Engelsman & Sommer 1992). According to Jones (1990, pp. 2), aprebaked anode consists of the following main constituents:

� Carbon1

� Inorganic metal impurities2

� Voids3

In anode baking, coking and calcination takes place in one step. During baking,the crystallites grow in size and anode properties is continuously changing duringbaking. Impurities act as oxidation catalysts and also contaminate the �nal metalproduct. They may also cause environmental pollution. Increased void fractionreduces anode density. Pores in the range of 1-10 �m are easily accessible duringelectrolytic carbon consumption

According to Engelsman & Sommer (1992), 70 % of investment costs in an anodeplant is due to installation of the bake furnace. Due to these high costs, alternativesto new installations are retro�tting of existing equipment (Hurlen & Naterstad

1Carbon in the state of disordered graphitic crystallites.2This is typically elements like Na, S, V a etc.3Prebaked anodes contains between 20 % to 30 % of both coarse and �ne; open and closed

voids.

Introduction to Anode Manufacturing 17

1991). Also the production costs in anode baking are tremendous. As shown inFigure 2.1, anode baking costs contribute to 50 and 70 % of total anode productioncosts and is the most expensive stage in the anode manufacturing process. Thecost factors depend on each other and in furnace optimization, all factors mustbe considered (Keller & Oderbolz 1985), (Keller & Fischer 1992). However, the

Anode Baking

20 %

30 %30 %

20 %Capital CostsGas Cleaning System

diverse

10 %

20 %

(50 to ) 70 %

Paste Production

Moulding

Maintenance

PersonnelCost

Filling MaterialincludingEnergy

Figure 2.1: Anode baking costs. Based on Keller & Oderbolz (1985, Fig. 2, 3)and Keller & Fischer (1992).

above cost estimates do not include raw material costs: Raw materials contributeto between 70 and 80 % of the prebaked anode production costs (NIF 1986),(Hurlen & Naterstad 1991).

18 Introduction to Anode Manufacturing

Chapter 3

Green Anode Production

In a carbon plant, separate facilities take care of green anode production andanode baking. A successful green anode production is detrimental for �nal prebakeanode quality; in fact the majority of important physical characteristics of the�nal anode product are determined at the green stage (Vanvoren 1987), (Keller &Fischer 1992). After 1973, raw material quality has deteriorated and prices haveincreased. More isotropic petroleum cokes with increased contents of sulphur andvanadium have been used and the carbon producers no longer can rely on a singleraw material supplier. Monitoring of anode properties due to raw material changesis an important aid in optimisation of anode quality (Keller & Fischer 1992).

3.1 Anode Raw Materials

3.1.1 Petroleum Coke and Coal Tar Pitch

In green anode production, petroleum coke used as a dry material is mixed witha binder. The following materials are used, with coal tar pitch acting as binder:

� Petroleum coke (50 to 60 %)

� Coal tar pitch (13.5 to 18 %)

� Recycled materials:

{ Butts (20 to 30 %)

{ Green scrap anodes (< 5 %)

{ Prebaked scrap anodes

{ Tar emissions and dust from gas scrubber

Certain minimum amounts of petroleum coke and coal tar pitch is always needed;the rest comes from recycled materials. The degree of recycling depends on local

20 Green Anode Production

conditions. Coal tar pitch is the preferred binder used in production of anodes forprimary aluminium production. Coal tar is obtained as the residue in distillationof coal tar. Coal tar is one of the main products in coal carbonization. Optimalbinder performance is achieved with coal tar pitch but the use of petroleum pitch inanode production has also been analysed (Alexander, Bullough & Pendley 1971).

Most frequently, sponge coke from the delayed coking process is used in anodemanufacturing. Sponge coke is regular grade coke containing small pores with nointerconnections. Sponge coke is calcined to achieve to properties needed in anodeproduction.

Descriptions of petroleum coke and coal tar pitch is given in Gundersen (1996e);no further details are supplied in this study.

Dead animalsDead leaves & plants

Coal Petroleum

Distillate

Coke

Residue

Pyrolysis

Residue

Coal tar pitch

Distillation

Petroleum coke Petroleum pitch

Heavy Petroleum Fractions

Coal tar

ResidueDistillation

DelayedCoking

Figure 3.1: Origin of petroleum coke and coal tar pitch.

3.1.2 Recycled Butts

The prebaked anodes take part in the chemical reactions in the cell. The anodeis consumed and it has to be replaced with a new one after about 20 to 22 days.After cleaning (removal of slag and cast iron), the butts are crushed and recycled.Since the butts are pre-impregnated, the butts content will contribute to increaseddensity. However, the butts' structural homogeneity might degrade anode quality.Also, butts contain impurities which may cause increased reactivity (carboxy-and oxyreactivity) and aggressive consumption of anodes. Table 3.1 shows theconsequences of di�erent impurities present in the anode. Furthermore, metalimpurities contaminate the metal. Most impurities are introduced via the buttsbut traces of impurities are also present in coke and pitch. Nominally, the ashcontent should be less than 1 % of the anode paste.

Butts recycling is important due to the high raw material costs in anode baking.Thus, it is important to select coke and pitch qualities that allow for the highest

3.2 Anode Paste Recipe 21

Metal quality Si, T i, V , Fe, Pb

Anode consumption Na, K, Ca, V , Ni, S

Current e�ciency P , V

Environmental H , Pb, S

Table 3.1: Consequences of butts impurities. From NIF (1986).

possible fraction of butts in anodes. Recycling of butts is the cheapest way ofgetting rid of a high-grade material without commercial value.

Tar Emissions and Green Scrap Most of the volatiles escaping from the an-odes during baking su�er combustion along the ue channels of the ring furnace.However, some of the volatiles escape combustion and passes the gas scrubber sys-tem via the main duct. In the gas scrubber, the tar condenses and then separatesfrom the ue gas. Although this tar has a varying quality, it may be recycled.

Raw (green) anodes which do not pass the quality test are crushed and recycled.

3.2 Anode Paste Recipe

Theoretically, the �ller coke aggregate should follow a linear cumulative mass percent versus logarithmic particle size relationship from 10 �m up to between 5 to20 mm; the so-called screening curve. The real aggregate size curve is in uencedby several factors like coke porosity and strength, surface texture, interparticularporosity as well as other factors (Grjotheim & Kvande 1993). A typical screeningcurve is shown is Figure 3.2.

According to Benton (1990), this theoretical size distribution minimizes interspa-tial distance between particles and maximizes packing (apparent/bulk) density fora given distribution of coke particles. Theoretical studies on the determinationof optimum size fractions and quantities which yield maximum packing densityof coke aggregates and molded anodes exist in the literature (Walter, Morris &Joo 1980), (Furnas 1931) and (McGeary 1961). A typical assumption is that ifsmall particles are introduced into a bed of larger particles, the smaller particles�ll up the voids without changing the total volume. The realization of screen-ing curves is implemented through proper control of the screening, grinding andblending steps of green anode production. Wright & Peterson (1989), however,argue that in green anode production a trade o� exists between packing e�ciencyof the coke aggregate and free space available for binder pitch.

Murgia & Bello (1978) give an example of a typical composition of anode paste: 16% pitch is used; the rest (84 %) is the dry mixture which consists of four di�erentgranulometries as speci�ed in Table 3.2.

Dry mixture was made either of only petroleum coke or a mixture of coke andbutts. Butts contributed casually to the formation of the speci�ed size fractions

22 Green Anode Production

20

40

60

80

100

retained on sieveCumulative %

Particle size [mm]10 1 0.1 0.01

Target screening curve

Blendrealized in plant

Figure 3.2: Ideal and actual screening curves. From Grjotheim & Welch (1988,Fig. 4.3).

Fraction name Fraction size Amount [wt.%]

Coarse -10.00 mm + 3.00 mm 25

Medium -3.00 mm + 1.00 mm 20

Fine -1.00 mm + 0.10 mm 15

Dust 50 % > 60 �m 40

Table 3.2: Example of dry aggregate recipe. From Murgia & Bello (1978).

according to the butts size distribution obtained after crushing.

In some recipes, a fraction called "medium coarse" is also used. In general, 3 to6 size fractions is used. A usual constraint put on the dry aggregate is that 25to 35 % must be larger than 2.8 mm. Coarsest materials used is about 20 mmin diameter; the �ne fraction approach 20 �m. Algorithms exist for calculation ofthe amount of each size fraction needed to implement a speci�ed screening curve.

For prebaked anodes made of aggregate of a certain packing density and a giventype of binder pitch, the optimum amount of binder will accurately �ll all poresand void without expanding the aggregate. In this way a maximum product densityis obtained. Most product properties bene�t from high density.

The amount of pitch mixed with the dry aggregate depends on pitch composi-tion, coke porosity and structure. Pitch content should be adjusted until thereis negligible volume change during subsequent baking. However, all dry aggre-

3.2 Anode Paste Recipe 23

gate particles must be wetted and the open pores (diameter greater than 6.0 �m)�lled with pitch to optimize density. The extent of pore �lling depends on theviscosity-temperature relationship and the amount of primary quinoline present inthe pitch. Furthermore, a thin layer of pitch between the coke particles is desired.An increased layer thickness may cause ow-deformations during heating throughthe critical soft region. Pitching level is important for the mechanical properties ofthe anode. The best mechanical properties is achieved at optimum pitching level.Okada & Takeuchi (1960) studied the dependence of density and other properties1

on the binder contents in the green mix. It was found that the apparent densityincreases approximately linearly with the pitch content up to a maximum valuebefore levelling o� again in a linear manner. Both the position of the maximum ofthe density along the binder content axis as well as the slope of the curve dependon the type of �ller, the �ller size distribution and the molding pressure. It wasfound, however, that both mechanical properties and electrical resistivity obtain amaximum at the same pitch level as for the density. Thus, the pitch content of theanode is a very critical parameter to obtain the best possible anode properties.

[kg/m ]31500

1400

1600

10 15 20

Apparent DensityBaked

+1.0

-1.0

Optimum Pitching

Green Anode

Baked Anode

0.0

Binder in anode [wt %]

Linear ExpansionDuring Baking

[%]

Figure 3.3: Optimal pitching level. From Grjotheim & Welch (1988, Fig. 4.4).

Two factors are important in anode paste preparation:

� Particle size distribution: Aggregate density should be maximized

� Pitching level: Volume change during baking should be minimized

Due to changes in raw material properties, modi�cation of the paste recipe isnecessary to maintain stable production of anodes with certain speci�ed properties.

1Young's modulus, bending strength, crushing strength and electrical resistivity.

24 Green Anode Production

3.3 Steps in Green Anode Production

A schematic drawing of green anode production is given in Figure 3.4. Petroleumcoke is usually stored in big silos. Coal tar pitch is stored at a temperature between100 and 200�C to keep the pitch uid. For a description of the production facilitiesin �Ardal; see Farmen, Bakken & Naterstad (1982). Raw material selection andsuccess in the baking process are of crucial importance considering the purity andtexture of the baked carbon. Still, however, the majority of the requisite physicalcharacteristic properties of the �nal baked product are determined in the greenstage (Vanvoren 1987, pp. 525). It is important to have tight quality controlof the product at this stage. Green anodes not �tting the quality criteria, arerecycled. Typical green anode quality parameters are physical dimensions andgreen apparent density.

The steps for making green anodes may be summarized as follows:

1. Upgrading: Raw material cleaning, crushing, sieving, drying

2. Fractionation: Chrushing, screening and milling to achieve 1 to 7 dif-ferent size fractions

3. Dosage/metering according to the actual receipt

4. Aggregate preheating

5. Mixing: Preheated coal tar pitch is mixed into the coke aggregate bythe use of a reciprocating mixer

6. Green anode forming by the use of a vibrocompacting weighing machine

Aggregate preheating takes place either in a batch- or continuous preheater attemperatures between 150 to 190�C.

Mixing of aggregate and binder takes place batch-wise or continuously at temper-atures of 50 to 90�C above binder softening point. During mixing, two parametersare critical:

� Mixing temperature (high enough to allow for pitch ow)

� Mixing time (long enough to obtain paste uniformity)

Typical problems occurring during mixing are particle size segregation and spatialvariations in pitch content. It has been shown that paste rheology is very impor-tant for the behaviour of the anodes during baking and the �nal anode properties.The best paste has a granular appearance like made of small brilliant black spheres.Paste with this kind of rheology gave non-deformable green anodes with an irreg-ular surface which did not crack. It seems as if the compactability of the paste isdetermined by the mixing process (Martirena 1983).

Previously, a hot pressing technique was used for anode forming. In new com-paction techniques, vibro-compaction equipment is applied. Pressures between 3and 15 MPa is used for moulding the anode. Compaction takes place at temper-atures of 110 to 150�C. After compaction, the green anode keeps its structuralform during baking due to a balance between viscosity and surface tension.

3.3 Steps in Green Anode Production 25

Roll

Crushers

Silos PetroleumButtsCoke

Dust

Ball Mill

Dry Aggregate Mixing and Preheating

Coal TarPitch

Coarse Medium Coarse Medium Fine Dust

Dosage & Metering

(Continous or Batch)after recipe

Sieve

From Smelter

PreheatPitch

Vibro-compaction Green Anode

Silos

Dry aggregate and pitch mixing

Sieves

CoarseMedium

Figure 3.4: Basic steps in green anode production.

26 Green Anode Production

Chapter 4

Baking Furnace Process

Description

4.1 Ring Furnace Concepts

In anode baking technology, two ring furnace concepts dominate: Horizontal uering furnaces1 and vertical ue ring furnaces2. Those furnaces are often denotedopen and closed furnaces for short, and this terminology is adopted in this work.Ring furnaces are large industrial units which operate in batch mode. Processcycle time is very long; each batch takes between 10 to 15 days to complete. Inboth types, operation is usually done with two �re zones on each furnace. Eachzone includes from 9 to 14 sections each. Each section is divided into a certainnumber of cassettes (pits) with cassette walls separating them. In vertical ringfurnaces, gas ows vertically in the ues. Di�erences in gas ow patterns hasgiven name to the di�erent furnace construction principles.

In principle, a ring furnace operates like a heat exchanger: Fresh air ows intothe furnace in the cooling end; the cold air cools the baked anodes and the gas isheated. Even more heat is gained in the sections with oil combustion. In this way,the combustion gas obtains enough energy to heat the green anodes put into thefurnace in the draught-fan end of the furnace. The furnace operates as a doubleheat exchanger giving energy to the gas and releasing it again. This operationprinciple is common to both the furnace technologies.

In part V of this work, focus is put on modelling and control of ring furnaces ofthe closed type. A brief description of the open furnace concept is given below. Inthe rest of the study, focus is put on furnaces of the closed type.

1Horizontal (ring) furnace, open (ring) furnace.2Vertical (ring) furnace, closed (ring) furnace.

28 Baking Furnace Process Description

4.1.1 Open Furnaces

The open furnace technology is widely used in France, Australia, Switzerlandand the USA. The ues are closed in both top and bottom and the gas owshorizontally in the ues. The cassettes are not covered; the packing coke are indirect contact with the open air. A schematic diagram of the gas ow pattern inopen furnaces is shown in Figure 4.1.

Gas Flow

Burner

Gas Flow

Figure 4.1: The open furnace concept (schematic diagram).

4.1.2 Closed Furnaces

In closed or vertical type furnaces, the vertical gas ues are open in both top andbottom. Each section is covered by a concrete shield (cover/lid). By setting upa draught in the one end of the �re zone, the gas ows between the sections in aclosed gas path. The gas ow pattern through a section of a conventional closedfurnace is shown in Figure 4.2. Conventional closed type kilns are also denotedRiedhammer furnaces; originally a German furnace patent.

4.1.3 Retro�tted Riedhammer Furnace

A Riedhammer-type furnace retro�tted according to Hydro Aluminium's construc-tion principles (HAL design) was used as basis for development of the ring furnacemodel presented in this work. The furnace is described in Foosn�s, Jarek & Linga(1989).

Each section has 5 pits with a certain number of layers of anodes in each pit. Thecapacity is approximately 120 anodes per section which is an increase of 33 %compared to the conventional Riedhammer design. The �re shafts are removed,and oil combustion takes place under the section cover and in the headwall (crosswall). An important di�erence between Riedhammer and the retro�tted furnace is

4.1 Ring Furnace Concepts 29

Burner

Fire

Shaft

Gas

FlowGas-

Gas Flow

Flow

Headwall

Figure 4.2: The closed furnace concept (schematic diagram).

the ow pattern of the combustion gases. A sealed wall is installed asymmetricallyin the bottom of the section. This forces the combustion gases in a loop in thepit wall as shown in Figure 4.3. In each ue wall a number of ues are stacked inparallel. In the HAL-concept, the position of the sealed wall is in the middle ofthe bottom of the section).

x

z

y

BottomBurner

Top BurnerLid

Under Lid - Zone

-Headwall

Pit

Part APart B

Zone

FoundationUnder pit-

GasFlow

Zone

Figure 4.3: The Hydro Aluminium retro�t furnace concept.

After introduction of the HAL construction principle, anode quality-variance hasdecreased at a reduced speci�c energy consumption (Foosn�s et al. 1989, pp. 572).Less than 3 % of the anodes have an equivalent temperature below 1100 �E. Theprinciples of the Hydro Aluminium patented design is described in Jakobsen et al.

30 Baking Furnace Process Description

(1987).

4.1.4 Furnace Construction and Geometry

For a full description of a section, a model in three spatial dimensions is needed.In a Cartesian coordinate system, the coordinate axes are de�ned in the followingway:

� The positive x-axis is always directed in the �ring direction of a �re-zone.

� The y-axis and z-axis orientation follow from the de�nition of a right handcoordinate system with the positive z-axis pointing upwards.

The position of the origin in the coordinate system depends on the part of thesystem modelled.

A typical cross section in the xz-plane of closed furnaces is shown in Figures 4.2and 4.3. The gas ow pattern is complicated both in the retro�tted furnace andthe Hydro Aluminium furnace due to the dividing wall in the under-pit region. Acloser description of construction and geometry of ring furnaces is given based onthe retro�tted furnace in �Ardal.

A schematic description of a ring furnace seen from above is given in Figure 4.4.The sections are arranged in two parallel lines connected with a cross-over channelat the ends of the section lines.

A top view of a separate uncovered section is shown in Figure 4.5. Details of a cas-sette is given in Figure 4.6. The cross section of the vertical ues is approximatelyquadratic. The �gure shown, describes the retro�tted furnace in which there is anasymmetry in the positioning of the dividing wall in the under-pit region. Thisexplains the larger extension of part B compared to part A. In each section thereare 5 cassettes (pits) with anodes stacked on top of each other.

The stacking of anodes on top of each other with gas owing in the vertical ueson each side of the anode stack can be seen in Figure 4.8 showing a cross sectionthrough the xy-plane (part A). The xz-plane through y = y1 is approximatelyadiabatic due to thermal symmetry in the yz-plane.

Both under the headwall and pit oor, there are support bricks which rest onthe furnace foundation. These brick-arrangements cause pressure loss because offriction. A complicated gas ow pattern is obtained.

4.1.5 Symmetric Properties of a Section

One pit with (half) pit wall segments on each side may be considered as a symmet-ric block in a section. Furthermore, with ideal thermal balance in the section, ahalf pit with wall segment may be considered to be the smallest symmetric unit inthe solid materials. Using this symmetric property, it should be necessary to modelonly a half cassette to obtain a representative temperature �eld in the section.

4.1 Ring Furnace Concepts 31

3029

2827

2625

2423

2221

1918

1716

20

67

89

1011

1215

1314

12

34

5

67

54

32

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atic

Sec

tion

Num

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Cro

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er-

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Val

ves

to s

crub

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Flue

gas

SEC

TIO

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UN

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R P

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ING

SEC

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NS

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OM

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SEC

TIO

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NO

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TIV

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(Zon

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(Zon

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Dir

ecti

on o

f fi

rex

y

Dyn

amic

Sec

tion

Num

bers

Dir

ecti

on o

f fi

rex

yR

ing

furn

ace

for

baki

ng o

f ca

rbon

ano

des

Fire

Tra

in 2

Fire

Tra

in 1

Figure 4.4: A ring furnace seen from above.

32 Baking Furnace Process Description

Part B Part A

Packing Coke

Anodes

Brick/Refractory

Fire Direction

Vertical Flue Channel

Figure 4.5: Top view of an uncovered section which shows the arrangement of thecassettes. Anodes are positioned side by side surrounded with packing material togive support and contribute to heat transfer from the owing gases to the anodes.The drawing does not have the correct scale and the number of ue channels doesnot �t with the actual geometry of the furnace.

If gas mass ow was equally distributed between the vertical ues, one did nothave to calculate the full half cassette either. Only a certain fraction of the pitwould keep to give a satisfactory description of the thermodynamical state of thesolid materials.

4.2 Ring Furnace Operation Principle 33

Packing Coke

Anodes

x (0,0)

Part B Part A

Brick/Refractory

Fire Direction

y

Width y

n=9 vertical x

y

m=12 vertical

Length

1

flue channels in flue channels in

Figure 4.6: Detailed top view of a cassette in an uncovered section. The positionof the coordinate system in the xy-plane is shown in the drawing. The drawingdoes not have the correct scale and the number of ue channels does either not �twith the actual furnace geometry.

Unfortunately these symmetry-features are not present due to nonuniform distri-bution of mass ow within cassette walls as well as nonuniform heat ux distri-bution on the section boundaries. To give a satisfactory description of full sectionbehaviour, it seems necessary to derive a full section model. Here the ideal sym-metry condition is used and focus is put on giving a description of phenomenawhich occur in a half cassette.

Modelling of a half pit is the simplest approach that can be taken and at the sametime obtain a proper qualitative view of the �ring strategies' impact on anodetemperature- and property �elds.

4.2 Ring Furnace Operation Principle

Although the construction details di�er in closed and open furnaces, the furnaceoperation principle and terminology is approximately the same. In the following,a description of a typical ring furnace baking cycle is given.

4.2.1 Description of the Baking Cycle

A ring furnace comprises several sections arranged in two parallel lines connectedin a ring fashion. The section lines are connected at each end via the crossoverchannel. There are normally between 30 and 50 sections in a ring furnace. Theneighbouring sections are separated by a wall of hollow brickwork; the so-calledheadwall. Via this hollow wall, gas is led between the sections and heat can be

34 Baking Furnace Process Description

x

Flue channels Part A

(0,0,0)

y

z

Packing coke covering the anodes

in three layersAnodes

Packing cokefor supportand heat transfer

Figure 4.7: Pit xy-plane viewed from three di�erent levels along the vertical di-rection.

distributed to make the anode temperature pro�le uniform. In some constructions,it also allows for vertical �ring. Each section is divided into several pits where theanodes are stacked in each pit. The pits are separated with pit walls, in whichthe hot gases ow. Approximately 100 anodes are loaded in each section. Theanodes are surrounded by a packing material (petroleum coke or anthracite) whichgives physical support and acts as a heat transfer medium. This packing materialalso prevents O2 oxidation of the anode carbon. In the preheating- and heatingsections, hot gases ow in the pit wall ues for transmission of heat through thebrickwork and packing coke to the anodes. In the cooling sections, ambient air isused for cooling the anodes in a similar way. Thus, the ring furnace acts as a heatexchanger with preheating, heating and cooling sections linked in series. In owingambient air is preheated by cooling the anodes. The heated air supplies oxygenfor the combustion of oil in the heating sections. In the preheating sections, coldanodes are heated by hot gas and combustion of hydrocarbon volatiles coming

4.2 Ring Furnace Operation Principle 35

Brick

Packing Coke

Anodes

Vertical

Flue Channel

Gas Flow

Width

Height z

y

y (0,0) 1

(Part A)

Figure 4.8: Side view (yz-plane, part A) of anodes surrounded with packing ma-terial. The gas ow is in vertical upward direction in part A. Anodes are stackedin three levels. The drawing does not have the correct scale.

from the binder pitch. Thus, the heat mainly comes from combustion of liquidor gaseous fuel as well as hydrocarbon volatiles which are mainly tar, methaneand hydrogen. Some heat also comes from combustion of the packing coke whichsurrounds the anodes. The anodes remain stationary in the pits, but the burners

36 Baking Furnace Process Description

Packing Coke

Anodes

Brick/Refractory

Fire Direction

x (0,0)

zPart B Part A

Directions of gas flow

Figure 4.9: Side view from xz-plane of anodes surrounded with packing material.Gas ow directions in part A and part B are shown in the drawing. It can be seenthat anodes are stacked in three levels. The drawing does not have the correctscale.

are moved at regular intervals in time known as the �re-step3 time (typically 20-40 hours). At each �re-step, one section �lled with anodes is connected to the�re-train in the front of the section line and a section is linked out at the backend of the chain of sections, the cover removed and the anodes unloaded fromthe pits. Typically there are two �re-trains running on each furnace. During onebatch, a section passes a sequence of natural preheat, pitch burn, forced heatingby fuel combustion, heating at peak temperature (hold time) and cooling as shownin Figure 4.11. In closed furnaces, each section is covered with a lid which givesa closed gas path from one section to the other of the �re-train. In Figure 4.4,an overview of a typical ring furnace system is shown. In the �gure, two zones

3The term "�re-step" is often used in two contexts: At �rst, the expression is used to describethe permutation of sections in the �re zone and also the interval in time between permutations.Secondly, it denotes actual time which this permutation takes place and also the action of re-moving one of the section's cover; either the section at the front of the zone or �nal section oncooling.

4.2 Ring Furnace Operation Principle 37

ha

la

wa

xy

z

Figure 4.10: Nominal orientation of an anode in the ring furnace.

operate simultaneously as mentioned above.

Both delayed coke and uid coke has been used as packing medium. Delayed cokeis rather coarse and has low bulk density which allows for signi�cant air inleakagein open furnaces. In open furnaces, the use of uid coke as packing materialhas reduced air inleakage and improved anode support and thermal conductivity.This is due to the small size of the uid coke grains as well as its dense structureand low surface porosity. Stuart (1974) gives a comprehensive discussion of theadvantages of using uid coke as a packing material. Some of the positive e�ectscan be mentioned here: The baking �nishing temperature increased and anodequality variance was decreased due to more uniform heat treatment.

There are two ways of numbering the sections in a ring furnace. The static sectionnumbers are shown in Figure 4.4 (sections 1 to 30). The dynamic section numberrefers to the section's position within an operative �re zone on the furnace. Thesection which has dynamic number 1 (d1) is connected to the draught control fan.With n sections in the zone, the last section (on cooling) has dynamic number dn.Thus section dn is the next section to be linked out of the �re train. Referringto zone 2 in Figure 4.4, static numbered section 27 has dynamic number d1. Atthe next �re-step4, static numbered section 28 will have dynamic number d1 andsection 27 will advance to dynamic number d2.

The ue gas system consists of the ring duct surrounding the furnace and a gasscrubbing system. The pressure on the ring duct is approximately 30-40 mm W(300-400 Pa). Ideally, the ring duct should be isolated against leakage so that gasenters the duct from the zones via the exhaust manifold. Each ring furnace has itsown ring duct connected to the gas scrubber system. In the gas scrubber systemthere is usually an electrostatic �lter used for cleaning the gas. In the �lter, a highvoltage electrostatic �eld is used for separating the polar tar-molecules from the

4At each �re-step, there is a permutation of the sections in a zone.

38 Baking Furnace Process Description

bulk gas ow. The tarry substance collected in the �lter is recycled to the greenanode production facility.

The ring furnace may be described as a nonlinear distributed parameter system.There is a one-way connection of the gas ow from one section to the next on the�re-train. Complex interactions between fuel combustion, gas ow and volatile-emission and -combustion takes place in the furnace. Simultaneously there aretransformations going on within the anodes during the heat treatment. Due tothe distributed nature of the temperature �eld, there exists a spatial distributionof the anode properties. For e�cient automatic control of a ring furnace, one musttake into account the interactions between sections: There is a common gas owthrough the chain of sections.

The usual operation principle is to maintain a prescribed gas temperature as func-tion of time in each section. A typical gas temperature pro�le is schematicallyshown in Figure 4.11. Typical draught pressure- and O2-pro�les along a �re-zoneare shown in Figure 4.12 and Figure 4.13 respectively. In Figure 4.12, the dashedline denotes the desired pressure pro�le. Because of the air inleakage, there is adeterioration of the pressure pro�le. The lowest O2-content in the gas occurs in the�rst direct �red section. In the preheating sections, there is a gradual increase inthe O2-content due to the air inleakage. The gas ow through the zone is usuallymeasured in normal cube meters per hour [Nm3] for easy comparison of di�erent�ring situations.

1300

200 Natural

Preheat

HeatingForcedPeak

Temp.

Cooling

Volatile

Section Fire Cycle

Combust.

Heating

t [hr]

Tgas[oC]

Figure 4.11: Typical �recurve used for ring furnace operation.

The baking cycle can be divided into the four main phases:

1. Preheating

� Heating by forced convection

� Heating by forced convection and volatile combustion

2. Oil combustion

4.2 Ring Furnace Operation Principle 39

Draught Pressure

Dn

Desired

Profile9

SectionDynamic

NumberD1

[mm W]

0

Figure 4.12: Typical draught pressure pro�le along a �re zone.

O Fraction2 [%]

Dn

5

21

NumberSectionDynamic

D1

0

Figure 4.13: Typical O2 pro�le along a �re zone.

3. Oil combustion - Soaking time

4. Cooling

During preheating, volatile combustion takes place due to distillation and crack-ing reactions in the anodes. Both tar-gases, methane and hydrogen drift o� aretransported across the anode and coke bed into the ues. The volatile combustionstops at temperatures between approximately 800 and 1000�C. Approximately50 % of the energy consumption comes from volatile gases. It is important tonot induce too large temperature gradients (heating rates) in the anodes duringbaking. This may cause cracks in the carbon structure which result in low qualitycarbon anodes. Oil combustion takes over and continues until baking is �nished.At the end of the baking cycle there is a hold time at top temperature. This istypically 60 hour at 1280�C.

During cooling, the section cover is put on a ramp 10 to 20 cm above oor level.In this way, the cooling rate is increased at the cost of increased packing cokecombustion.

40 Baking Furnace Process Description

4.2.2 Heat Treatment Design

Until now, success in anode baking has depended on proper tracking along the gastemperature target curves. Thus, �re curve design is very important for obtainingoptimal ring furnace operation. In Figure 4.11, a schematic �re curve (gas tem-perature target curve) for the baking process is shown. The design of such �recurves is performed o� line based on experience and model based tools.

Success in �re curve design depends mainly on the available experience as well asthe ability to express constraints in a manner suitable for the model based tools.Heating rate and spatial temperature gradients in the anodes are constrained.Most frequently, it is a manual task to check whether the constraints are ful�lledor not. Oprescu, Gheorghiu & Georgescu (1988) discuss manual design of bakingcurves based on furnace measurements. State of the art in �re curve design seemsto be a manually performed iterative procedure:

1. De�ne input variable pro�les5.

2. Run model.

3. Repeat steps 1 and 2 until the constraints are ful�lled.

4. Perform baking using by the new �re curve.

5. Adjust �recurve based on laboratory measurements on the produced batchof anodes. If necessary, adjust model parameters and repeat the above steps.

The number of iterations mainly depends on the computer model that is available.In models for open ring furnaces, calculation is usually performed for the whole�re zone; the connection between sections is included in the model. In �re curvedesign using such models, the iterative process only has to consider whether theconstraints on the temperature �eld are ful�lled. But as far as reported in theliterature, there are yet no models available that can explicitly predict anodequality. Most models for closed ring furnaces presented in the literature6 performcalculation for a part of a section only. Thus, it is not possible to calculate theresponse for a certain input parameter pro�le for the whole �re zone. In this case,nominal input data (as function of time) has to be speci�ed for the section andcalculation has to be repeated until the following condition is ful�lled7:

The baking curves (both gas and solid temperature pro�les) in two pointsseparated by a pit length are identical except from a time shift of one �re cycle

i.e. the time interval between permutations.

This property of the baking cycle for two subsequent sections is shown in Fig-ure 4.14 below. In the �gure, time t = 0 is de�ned as the point in time where

5Draught pressure and burner fuel ows.6An exception is the model presented by Bourgeois et al. (1990).7See Keller & Disselhorst (1981, pp. 614) for a discussion of this property of the baking

process.

4.2 Ring Furnace Operation Principle 41

the permutation which introduced section d3 into the �re train took place. Attimes Tf and 2Tf permutations were performed which introduced sections d2 andat last d1 into the �re train. Often, a model of one section is used as an aid indesign of the �recurve. Then, to manually guess the section gas ow input condi-tions to achieve this property, makes the use of such models for �re curve designconsiderably more complicated.

oT [ C]

x

z

y

Part APart B Part APart B Part APart B Part APart B Part APart B

TcTcTc Tc t [hr]

Gas FlowCommon

d1 d2 d3 d4 d5

3 4 5

Figure 4.14: Periodicity-property of the baking cycle: During ideal �re zone op-eration, the baking curves in subsequent sections are repeated with a lag of one�restep.

As mentioned above, no models with the capability of predicting anode qualityhas been reported in the literature. Thus, after designing a new �re curve, onedoes not have explicit knowledge of the actual anode quality produced by us-ing the new �recurve. To achieve this knowledge, a batch of anodes has to bebaked and the usual routine laboratory experiments has to be performed. Often,the concept of equivalent temperature is used: Samples of low temperature heattreated petroleum coke is loaded in the green anode stud holes and follows theanode block during baking. By measuring the crystallite size of the samples afterbaking, an idea of the baking uniformity can be obtained. Iterative changes in the�re curve parameters also might be necessary based on information obtained fromthe laboratory data.

To simplify �re curve design, the models and algorithms should have the followingproperties:

� The model should predict anode quality.

� The model should be able to verify process constraints.

� If possible, veri�cation of constraints should be automated.

42 Baking Furnace Process Description

4.3 Ring Furnace Instrumentation

Conventional ring furnace instrumentation is reviewed in Gundersen (1995a). Basedon J�rgensen (1991), some details on the instrumentation of the retro�tted Ried-hammer furnace are presented below.

4.3.1 Temperature Measurements

Gas temperature is continuously measured in each section by two thermocouplespositioned in the gas ow under the lid (part B; TB) and in the gas ow atthe entrance of the section (headwall part A; TA). The under-lid measurementis performed by a thermocouple which can be easily replaced when the cover isnot in use. In part A, the thermocouple is situated in the outer region of thegas ow. Sometimes, a local hot spot may occur in the region surrounding thethermocouple. Thus, the measured temperature can be misleading. There is noon-line measurement of temperature in the anode charge in a section.

4.3.2 Draught Pressure Measurement

Draught pressure is measured in the section at the front of each �re-zone.

4.3.3 Burner Equipment

In each section, a maximum of 6 burners can operate simultaneously. All burnersare of impulse type. Two burners are situated under the lid (part B) for horizontal�ring at right angles into the gas ow. Fuel oil supply to part A of the section isthrough burners positioned in vertical �reshafts (peepholes) at equidistant posi-tions in the oor above the headwall. Burner con�guration for a typical section isshown schematically in Figure 4.15.

Fuel oil no. 6 (5) is used for energy supply8. The burners are operated viaLeisenberg burner-carriers and Leisenberg pulse-generators; one unit per �re zone.Each Leisenberg-unit has two pulse-generators connected to the two top burners(part B) and four bottom burners (part A) respectively. In this way, it is possibleto control the fuel ow independently in part A and part B. The Leisenberg-unitsare serially connected. A heat exchanger in each unit is used for preheating theoil to a temperature of approximately 150�C to reduce the viscosity of the oil.A compressor gives a pressure build-up in the oil of approximately 10 bar. Thispressure drives the oil-pulse when the magnet-valve is released.

During optimal combustion, the under-lid ame �lls almost the whole volume.Headwall �ring is done from equidistant positions on the headwall oor via burnersfreely resting in the headwall peepholes. Since the burner is not �xed, this might

8Waste oil has also been used.

4.4 Process Levels in Anode Baking 43

Under Lid Burners

HeadwallBurners

Gas FlowPart B Part A

Figure 4.15: Arrangement of burners in a section of the retro�tted furnace.

give a bad direction of the ame and sometimes cause local hot spots on theheadwall brickwork and di�culties for the oil spurt to vaporize.

4.3.4 Draught- and Cooling Fans

The exhaust manifold9 together with the fans in the gas scrubbing system main-tains the gas ow through the �re-zone. The manifold is situated between sectionsd1 and D(�1). A draught pressure of approximately 10 �6mmH2O is maintainedin section d1. At large draught pressures, the problem of air inleakage can besigni�cant. Air inleakage ow is largest in the sections with low dynamic num-bers, i.e. sections of preheating and volatile combustion. In these sections, thetemperature is relatively low, and air inleakage can signi�cantly deteriorate theuniformity of the temperature pro�le in the solid blocks.

4.4 Process Levels in Anode Baking

A characteristic feature of the baking process is that the operative baking units(�re zones) are physically changed as time passes by. Figure 4.4 gives a schematicdescription of this feature of the process.

Nominally, two �re zones operate simultaneously. Each section in a �re zone canbe considered physically identical10. Anyway, the carbon properties in a section

9Also called elephant fan due to the characteristic shape of the manifold.10There will be di�erences both due to constructional details as well as individual instrumen-

tation, wear and tear.

44 Baking Furnace Process Description

di�er in dependence of the section's position in the baking cycle. Whether themanipulable variables in a section are active or not, depends on how far the sectionhas advanced in the baking cycle.

A description of anode baking, depends on the level from which ring furnaceoperation is considered. According to Figure 4.16, the baking operation can beconsidered at least from three points of view. To study the full baking cycle andinter-section phenomena, our approach to the process has to start on level 2. Tostudy intra-section phenomena, it would be satisfactory to assume certain nominalinput conditions for the section (gas temperature, gas ow and draught pressure).Actually, the �rst approach to the baking process has to be an intra-section one,thus studying the phenomena taking place in a single section; i.e. level 1. By usingthis simpli�cation, it is possible to focus on the qualitative relationships within asection.

To understand global phenomena and interactions in a �re zone the whole �rezone has to be considered; ie. level 2. Furthermore, there are dynamic interactionsbetween the �re zones on a ring furnace and also there will be interactions betweenthe ring furnace units connected a gas scrubber system.

In this work, baking is considered from both level 1 and level 2 as follows:

� Section (level 1) - approach:

{ Modelling of ring furnace phenomena

{ Simulation of section behaviour using nominal section input parameters

� Fire zone (level 2) - approach:

{ Overall �re zone simulations

{ Control of ring furnaces (o�-line optimization and on-line control)

In the general case, on-line control should be performed on level 2 to be able totake into account interactions between the sections. It is possible, however, todevelop simpli�ed strategies for on-line control which operates only on level 1.

4.5 External Process Variables

Baking process instrumentation was discussed in Section 4.3 and in the reviewin Gundersen (1995a). With reference to the instrumentation of the furnace in�Ardal, a discussion of main input and output variables is given below. Included isalso a consideration on disturbances and interactions in a �re zone.

4.5.1 Input Variables

Most input parameters act mainly locally in a section with interaction to thedownstream sections. The draught pro�le, however, established by the exhaust

4.5 External Process Variables 45

Level 2

Level 3

Level 1

Ring Furnace

Fire zones

Sections

Figure 4.16: Anode-baking process view.

manifold has global e�ect on the system. The main input variables are as follows(di is dynamic section number):

1. Local (section) input variables:

� udi;j (j = 1; 2): Two top (under-lid) burner fuel ows [kg/hr]

� udi;k (k = 3; 4; 5; 6): Four bottom (headwall) burner fuel ows [kg/hr]

� udi;7: Cooling fan air ow

2. Global (�re zone) input variable:

� udp: Draught pressure at section d1 outlet [Pa]

The draught pressure at the zone outlet is due to an asynchronous motor operatingin the exhaust manifold. In sections other than d1, the draught pressure act as adisturbance.

If the baking process viewed from the section level, input conditions (gas tem-perature and composition) and some output conditions (draught pressure) for thesection may be considered as disturbances. This con�guration of input conditions,gives a two-point boundary value problem.

4.5.2 Measurements

The measurements are as follows:

1. Local (section) measurements:

� ydi;1 - Top (under-lid) gas temperature [�C]

� ydi;2 - Bottom (headwall) gas temperature [�C]

46 Baking Furnace Process Description

Part APart B Part APart B Part APart B Part APart B

u dp

x

z

y

yd2,2

Gas FlowCommon

to u ud2,3 d2,6andu ud2,1 d2,2

yd2,3 yd2,1

Part APart B

ManifoldExhaust

Draught Pressure at Manifold Bottom gas temperature:

Top burners

Top gastemperatureDraught pressure

Headwall burners

d1 d2 d3 d4 d5

Figure 4.17: Fire zone with �ve sections in preheat and direct �re (cooling sectionsnot included).

� ydi;3 - Draught pressure measurement (usually in section d2 only) [Pa]

2. Global (�re zone) measurement:

� None

4.5.3 Disturbances and Interactions

Section Disturbances

Two qualitatively di�erent types of disturbances may act on a section:

� Some disturbances occur regularly but at intervals in time which are con-siderably less frequent than the average batch time for a section. Typicallythese disturbances act via the green anodes as changes in physical properties(i.e. quality) of the raw materials. Such disturbances may be:

{ Petroleum coke properties11:

� Real density

� Porosity

� Crystallinity level

{ Coal tar pitch properties:

� Density and uidity

� Composition (solvent fractions, elemental composition)

� Wetting capabilities

11Disturbances may also be introduced via the butts.

4.5 External Process Variables 47

� Behaviour during pyrolysis (distillation, mesophase-transition, crys-tallite growth behaviour etc.)

Disturbances of this kind may also be introduced via:

{ Anode quality speci�cations

{ Weights factors in the performance index used for optimization

{ Cost of fuel oil

{ Personnel and maintenance costs

When such disturbances occur, there may be a need to perform a reoptimiza-tion of the baking process to achieve a corrected target curve for the gastemperature.

� Another type of disturbances may occur randomly but more frequently thanthe above mentioned disturbances or they may be slowly varying aroundan average (nominal) value. Typically, these disturbances are (see Figure(4.18)):

{ Temperature of the inleaking air

{ Mass ow of air inleakage

{ Heating value of the fuel oil

{ Heat transfer characteristics of the ue gas:

� Gas emissivity

� Heat transfer coe�cient

{ Temperature of gas ow at the section inlet

{ Mass ow of gas at the section inlet

{ Changes in draught pressure at the section outlet

On-line control action is needed to cope with such disturbances.

Furthermore, some disturbances are of an intermediate character. Typically, thesemay be:

� Slow changes in packing coke granularity

� Slow changes in brickwork quality

Such disturbances may be accounted for by changing the corresponding modelparameters when reoptimization of the process is performed.

48 Baking Furnace Process Description

gas flow direction wg;in

Tg;in

�g

h

wa

Ta�Hf

pg;out

Figure 4.18: Disturbances which act on a section in a ring furnace �re zone.

Interaction Between Sections and Fire Trains

Fuel combustion in a certain section interacts with the downstream section in the�re train due to transportation of energy with the gas ow. The draught pressureat the exhaust manifold have impact on all sections in the �re train. The mostsigni�cant e�ects, however, occur in the preheating sections.

Interaction between �re zones may occur via the ring duct due to changes in thedraught pressure.

Chapter 5

Anode Quality and Carbon

Consumption

During the last twenty �ve years, progress in both carbon manufacturing and celldesign has reduced net carbon consumption by approximately 20 %; the largestprogress has been in the last ten years (Hurlen & Naterstad 1991). This is dueto reductions in both primary (electrolytic) carbon consumption as well as dust-ing. Carbon usage is now close to 400 kg carbon per metric ton of Al produced.Improved understanding in coke-pitch interactions and changes in baking pro-cedures have contributed to improvements in anode formulation (Grjotheim &Kvande 1993, cht. 9.6).

5.1 Optimization of Aluminium Production

Raw material selection, green anode production and anode baking have impact onanode quality. Proper aggregate sizing and pitching as well as mixing and formingare important for green anode quality. Green anodes made of high quality rawmaterial may su�er quality deterioration if baking is not properly conducted. Forpoor green anodes due to either second class raw materials or production irregular-ities, even optimized anode baking cannot give prebaked anodes with satisfactoryproperties (Engelsman & Sommer 1992). To fully exploit the properties of goodanodes, cell operation conditions is also of importance. Factors with impact onanode behaviour are summarized in Figure 5.1. As can be seen, success on thebaking stage depends on green anode properties. In the same way, optimal anodeutilization also depends on cell and bath conditions.

The overall goal is to produce high quality aluminium at maximum pro�t. Ac-cording to Keller & Oderbolz (1985), it has been shown that anode quality canhave an e�ect on metal price which is just as signi�cant as the costs of the bakingprocess shown in Figure 2.1.

50 Anode Quality and Carbon Consumption

Cell

Operation

Coal Tar Pitch

Forming

PasteMixing

Anode

Coating

Rodding

Baking

Green Anode

Prebaked Anode

SizePressureTemperature

& Shape

Time Temperature

PorosityUniformity

Adherence

Resistance

Aggregate Pitch

LevelMeteringSizing &

Petroleum Coke

Butts

Baking Strategy

Furnace Design & Anode Packing

Butts Cell Design & Ore Cover

Housekeeping

Anode-Stub

Ohmic Heatup & Anode-Stub Heatup Bath Condition & Exhaust Rate

Figure 5.1: Factors in anode-fabrication and -use which a�ect anode behaviour.Based on Jones (1990).

Anode quality has impact on the metal price but the costs of anode baking isapproximately the same in both optimal and suboptimal operation of the bakingfurnace. Possibly, there is a trade o� between cost of anode raw material and metalprice, but this aspect will not be discussed in this study. For optimizing pro�t inaluminium production, a hierarchical concept as shown in Figure 5.2 would beattractive. This strategy is hardly applied in aluminium production. Aluminiumproduction is more commonly organized as local units with separate budgets andeconomy. Thus, pro�t is locally optimized.

Based on these considerations, it is reasonable to view anode baking as a localoptimization problem. Then the ultimate goal of anode baking can be stated asfollows:

� Conversion of green anodes (of a certain quality) to prebaked anodes of a

5.2 Conventional Anode Quality Parameters 51

Raw

Material

max(J)

Coordinating

CarbonPlant

Raw Production

Anode

Level

Baking

Anode

CarbonPlant

MetalPlantPlant

Metal

GlobalLocal StructureHierarcical Structure

max(J )c max(J )m max(J )c max(J )m

max(J ) bamax(J )ra

Anodes Metal

Market

Figure 5.2: Optimization of aluminium production.

certain speci�ed quality.

� Anode baking should occur at the lowest possible costs.

To achieve this goal, anode quality has to be properly de�ned and the costs ofanode baking have to be stated.

5.2 Conventional Anode Quality Parameters

According to Foosn�s et al. (1989) and Branscomb (1966), the following require-ments for prebaked anode production can be speci�ed:

1. Lowest possible price

2. High chemical purity

3. Adequately high electrical conductivity

4. Uniform electrical current density

5. Satisfactory thermal conductivity

6. Su�cient mechanical strength

52 Anode Quality and Carbon Consumption

7. High thermal shock resistance

8. High degree of homogeneity and consistency

9. Low consumption rates and dusting properties

10. Low CO2 and air reactivities

These qualitative statements have been quanti�ed by Jones (1990, Tab. II, pp.12), Grjotheim & Kvande (1993, Tab. 4.9, pp. 116) and Asperheim et al. (1993a)as shown in Table 5.1. It should be noted that two expressions for thermal shockresistance is given in the table. The alternate expression used by Hydro Aluminiumis based on the fact that thermal conductivity and electrical resistance are nearlyinversely proportional. Thus, according to Nerland (1994), thermal shock resis-tance is best calculated by k�c

��eY. Furthermore, a comprehensive discussion of

anode quality is also given in NIF (1986).

Total porosity should be less than 30 %. A large fraction of the pores are closedwith diameters below 1:0�m which cannot be wetted by the electrolyte.

In Table 5.1, su�cient mechanical strength of the anode is taken care of by con-straints on compressive strength and bending strength. In addition, the anodeshould have no internal cracks which might cause anode failure during electroly-sis. Earlier, the absence of internal cracks was veri�ed by knocking on the anode.

Through research and experience, one has found that anodes with parameters inthe speci�ed range, behave satisfactorily in the cell. Most of these parametersare indirect measures of anode performance; only the starred parameters give adirect measure of electrolytic anode performance. The table also partly containredundant information; super uous parameters can be omitted without loss ofinformation. It is necessary to state that homogeneity of anodes is very important.A few bad anodes used in an operative cell can cause severe problems in electrolysis.Also, optimal use of good anodes is di�cult if the variance in quality is too high.According to Hurlen & Naterstad (1991, pp. 21), anode consistency is de�nedas two times the standard deviation (2�) of any property recorded. Dependingon raw-material supplies and the considered properties, long term consistencies ofbetter than 5 to 20 % is di�cult to achieve. The achievable cell control precisionsigni�cantly depends on consistent anode quality.

In Wilkening (1993), it is stressed that the anode plant capacity is dependent onthe apparent density of the baked anode; an increased density gives an increasedbaked anode throughput and thus increased e�ciency and improved economy inanode production. Thus, baked apparent density should be maximized.

For traditional evaluation of baking uniformity, the concept of equivalent temper-ature is often used. A calibration curve of a certain property of a coke sample isobtained in the laboratory when the coke sample is exposed to a given temper-ature program. Usually, crystallite size represented by Lc as well as real densityare used in the de�nition of equivalent temperature. Foosn�s et al. (1989) usemeasurement of Lc, Dreyer (1989) and Thomas (1989) use measurements of realdensity of a coke sample.

5.2 Conventional Anode Quality Parameters 53

Property Variable Range/ magn. Unit

Baked app. density �a;a;g 1400-1650 [kg/m3]

> 1555

Baked real density �r;a;b 2000-2200 [kg/m3]

Total porosity �T;a < 0.30

BET Surface Area Sa 1-2 [m2=g]

Nitrogen perm. kp;a;N20.49-1.48 [�1011m2]

Permeability kp;a < 2 [nPm]

Thermal cond. kt;a 3.5-5.5 [W/(mK)]

Spec. El. Res. �e 50-75 [�m]

< 60

Coe�. therm. exp. �, CTE 3.5-5.0 [�10�61=�C]

Compr. strength �c;a 350-500 [kg=cm2]

> 30 [MPa]

Bending strength �b;a 60-80 [kg=cm2]

Young's modulus Ya (8000� 1000)� �b [kg=cm2]�c;aY

> 30 [MPa]

Therm. shock resist. Rts;a =k�b�Y

� 150 [W/m]

Rts;a =�c

��eY� 16 [K/(�m)]

Airburn; 550�C rO2;a * 20-80 [mg/(cm2hr)]

< 60

CO2 oxid.; 970�C rCO2;a * 10-50 [mg/(cm

2hr)]

< 40

Dusting index si;a * < 25 [%]

Weight loss 6-8 [%]

Heat capacity cp;a 1300 [J/(kgK)]

Ash < 0.30 (0.40) [%]

Ave. cryst. height Lc;a � 30:0 [�A]

Ave. cryst. size La;a � 30:0 [�A]

Interlayer spacing d002;a � 3:44 [�A]

Electrolytic test (lab, ra = 0) * 110-115 [%]

Table 5.1: Property-value ranges for prebaked anodes. After Jones (1990), Asper-heim et al. (1993a), Asperheim et al. (1993b) and Asperheim et al. (1993c). Forthe reactivities, excellent anodes have airburn and CO2-reactivities below 20 and10 mg/(cm

2hr) respectively. Correspondingly, poor anodes have reactivities above

80 and 50 mg/(cm2hr).

Good anode quality is characterized by low non-electrolytic consumption in thereduction cell. The most important parameters in this context is CO2 reactivity,airburn and thermal shock resistance. It is realized that these properties dependon several of the classical anode properties (Keller, Schmidt-Hatting, Kooijman,Fischer & Perruchoud 1990). To de�ne a concept of anode quality, it is necessary toconsider anode performance during electrolysis even though carbon consumption

54 Anode Quality and Carbon Consumption

depends on bath conditions, cell design as well as anode properties.

5.3 Anode Behaviour During Smelting

An introduction to the Hall-H�eroult process is given in Appendix C. In the fol-lowing, the concept of anode quality is established based on a discussion of anodebehaviour during electrolysis. Carbon anodes made of petroleum coke and coaltar pitch are heterogeneous materials although both �ller and binder coke havegraphitic structure.

To achieve proper operation of the cells, it is important that the anodes haveuniform properties. Anode properties have impact on the following phenomena inelectrolysis:

� Carbon consumption vs. metal production

� Energy consumption

� Anode cracking tendency

� Anode e�ect tendency

� Frequency of anode changing

A discussion of the relationship between anode properties and these phenomenais given below.

5.3.1 Carbon Consumption

Net carbon consumption is one of the parameters most frequently used in charac-terizing anode performance in an aluminium reduction cell (Dreyer 1989). Factorswith impact on carbon consumption are shown in Figure 5.3. Carbon consumptionis due to four main factors:

� Raw materials

� Anode production

� Conditions during smelting (represented by current e�ciency)

� Cell design

Thus, the anode parameters are only partly responsible for the carbon consump-tion. Therefore, direct measurement of anode quality is more attractive thanmonitoring of anode quality through measurement of net anode carbon consump-tion (Keller & Fischer 1992). On the other hand, studies of the behaviour of theanodes during smelting give insight into anode properties important for specifying

5.3 Anode Behaviour During Smelting 55

Consumption

Carbon

Cell

Operation

Anode

Design

Cell

Design

Anode

Parameters

Figure 5.3: Factors contributing to anode consumption. Based on Grjotheim &Kvande (1993).

anode quality. A discussion of carbon consumption during electrolysis is thereforegiven in the following.

Carbon consumption can be dissected as shown in Figure 5.4. Net carbon con-sumption Cn is the sum of two terms:

Cn = Cel + Cex (5.1)

where Cel is electrolytic carbon consumption and Cex is excess carbon consump-tion. In the industry, multiple regression has been used to obtain formulas for thecarbon consumption. By using data from anode samples logged over a decade,Rhedey (1982) presents the following formula for net carbon consumption:

Cn = Cel + Cex = 347 +334

�+ 9:3kp;a � 3:7rr (5.2)

where Cn is expressed as consumed carbon (kg) per ton of produced aluminium.rr is the residue from a CO2 reactivity test. � and kp;a denote current e�ciencyand permeability respectively. The parameters vary slightly with reduction celldesign and operation conditions. The value of such correlations are limited sincethey are purely empirical with no information on the mechanisms contributingto carbon consumption. Typical �gures for reaction mechanisms responsible foroverall carbon consumption is given in Table 5.2.

Electrolytic Carbon Consumption. The electrolytic carbon consumption is

related to the theoretical carbon consumption Ct;el = 0:333kgCkgAl

. Ct;el is the

mass of carbon used when the only anode reaction considered is electrochemicalformation of CO2 and assuming metal is produced at current e�ciency � = 100%.Cel and Ct;el is related via the current e�ciency �:

Cel =Ct;el

�(5.3)

56 Anode Quality and Carbon Consumption

Reaction kg C/kg Al (CE = 0:94)

Electrolytic CO2 0.353-0.350

Electrolytic CO 0.002-0.005

Airburn 0.028-0.075

CO2 oxidation (pores) 0.014-0.028

Dusting 0.002-0.014

Total 0.394-0.468

Table 5.2: Overall carbon consumption. From (Grjotheim & Welch 1988, Tab.4.5).

Al O + 3 C = 4 Al + 3 CO 3 22

Theoretical Consumption

Electrolytic Consumption

Net Consumption

Gross Consumption

Butts

Excess Consumption

0.400 0.5500.333

0.333 kg C/kg Al

0.333/CE kg C/kg Al

Figure 5.4: Dissecting the carbon consumption. From Forum Roundtable: Dis-cussing the Issues in Carbon Anode Technology (1990). Note that the electrolyticconsumption depends on the current e�ciency.

Values of � in the range of 0.85 to 0.95 can occur. Thus electrolytic carbonconsumption is in the range of 0.400 kgC/(kgAl) down to 0.360 kgC/(kgAl)1.

As discussed below, both dusting and thermal shock problems lead to losses incurrent e�ciency which has impact on the electrolytic consumption of carbon aswell as excess carbon losses. At the electrolytic interface, gas formation takes placeaccording to the reactions:

1

2Al2O3(d) +

3

2C(s) ! Al(d) + 3 (5.4)

1

2Al2O3(d) +

3

4C(s) ! Al(d) + 3 (5.5)

The reaction's thermodynamic feasibility depends on the anode voltage. At lowcurrent densities (< 50mA=cm2), CO(g) formation is favoured. At nominal con-ditions using higher current densities, formation of CO2 dominate with a volumepercent in the order of 80 %. For a given current density, less crystalline carbonspromote CO-formation. Current e�ciency is also related to the presence of anodeimpurities (P and V ).

1Gross consumption includes the mass of butts and amounts to between 0.500 and 0.550kgC/(kgAl).

5.3 Anode Behaviour During Smelting 57

Excess Carbon Consumption. Excess carbon consumption Cex is due to sec-ondary reaction with air and anode gases. The anode is porous, and the reactingsurface is inside the pores of the anode. In measuring Cex, three mechanisms mustbe taken into consideration (Dreyer 1989):

� Airburn

� CO2 oxidation

� Dusting (due to selective oxidation of binder coke bridges)

A discussion of each mechanism is given below.

2CO - Oxidation

Electrolyte

Alumina

Airburn

Metal

Cathode

Figure 5.5: Chemical reaction zones on (below) the surface of the anode.

Two reactions are thermodynamically possible in the airburn-zone:

C(s) +O2(g) ! CO2(g) (5.6)

2C(s) +O2(g) ! 2CO(g) (5.7)

Calculation of Gibbs free energies for the reactions shows that at low tempera-tures, formation of carbon dioxide is the favoured reaction; anode upper surfacetemperature lies between 430 to 630�C. There is a change from reaction controlto mass transfer control with increasing temperature. At nominal surface temper-atures, the reaction is reaction controlled. Spray-coating the anodes and isolationby a layer of frozen alumina (crust) are means to avoid or reduce airburn. Also,anode thermal conductivity has impact on the temperature level reached at thetop surface of the anode. A low thermal conductivity of the anode is preferable(Fischer & Keller 1993). Airburn may contribute to between 12 % and 15 % (8-20%) of anode consumption. For good anodes, airburn measured in laboratory has

to be less than 50 mg/(cm2hr). Airburn is nominally responsible for 17 % of the

total prebake carbon consumption.

CO2 is formed at the anode working face. CO2 oxidation mainly takes place in theextended pore system above the working face of the anode or as the gas sweeps thesides of the anode as gas escapes from the bath. Secondary consumption accountsfor 5 % to 10 % of the consumption at the working face. CO2 formation is governedby the Boudouard reaction:

CO2(g) + C(s)! 2CO(g) (5.8)

58 Anode Quality and Carbon Consumption

The signi�cance of the reaction depends on the gas permeability and the hydraulicgas pressure at the anode working face. The Boudouard reaction sometimes givesrise to dusting due to selective oxidation of the binder coke. The Boudouard reac-tion occur at a rate much lower than the airburn rate at comparable temperaturesand are of minor importance at temperatures below 800�C. CO2 oxidation isresponsible for approximately 4 % of the total carbon consumption.

Airburn and CO2 oxidation is in principle determined by the same parameters.Following Grjotheim & Kvande (1993), the rate of reaction of both airburn andCO2 oxidation can be modelled as:

ri = ri(Sa;1

Lc;a; kp;a; fa) (5.9)

where i = O2; CO2 denote airburn and CO2 oxidation respectively. Sa is speci�csurface area, Lc;a is average stacking height of the anode crystallites, kg is the gaspermeability of the carbon sample and fa is the fraction of catalyzing agents (ash)present in the sample. Inherent in this equation are e�ects due to carbon materialcrystallinity and structure as well as di�erent transport mechanisms which can berate-controlling in the secondary reactions. CO2-oxidation is responsible for 4 to 8% of anode consumption and should be less than 20 mg/(cm

2hr). To obtain simple

models for the oxidation rates, one could assume that the impact on one variableon the oxidation rate is independent of the other variables. Then a reasonablemodel can be represented as a product of single valued functions:

ri = krifi;1(Sa)fi;2(1

Lc;a)fi;3(kp;a)fi;4(fc) (5.10)

kri is a constant.

Dusting is mainly due to oxidation and wear at the working face of the anode aswell as selective burning on the surface of the anode above the electrolyte. Sinceairburn has a higher reaction rate than CO2 oxidation at low temperatures, airburnis the major cause of dusting. The dusting level in the electrolyte is nominallybelow 0.1 %. The presence of dust increases bath temperature and lowers theinterpolar distance; both reduce current e�ciency as discussed in Appendix C.Thus, the following qualitative model can be used:

si;a = si;a(rO2;a; fa) (5.11)

to represent the dependence of dusting on airburn and presence of impurities inthe sample. In the literature, the following qualitative relationship has also beenreported (NIF 1986);

si;a = si;a(Lc;a; fa) (5.12)

It has been shown that the presence of Na, K, Ca, V , Ni and S increase carbonconsumption. Usually below 4 % of anode consumption is due to dusting.

Correlations for Carbon Consumption. Through many years, it has beencommon practice in industry to relate carbon consumption to anode physical prop-erties: Correlations between net carbon consumption and anode parameters has

5.3 Anode Behaviour During Smelting 59

been derived as shown in Equation (5.2) and Forum Roundtable: Discussing theIssues in Carbon Anode Technology (1990).

The derived correlations are based on multivariate statistical analysis and thusgive no insight into the mechanisms responsible for an actual net consumption.No further comments on these correlations will be given.

5.3.2 Consumption of Electrical Energy

Electrical energy consumption in the anode is due to contact resistance in theanode-stub contact as well as electrical resistance in the bulk anode. The coe�cientof thermal expansion of the anode has impact on the stub-anode resistance. Sincethe anode expands during heat-up in the electrolytic bath, the anode and stubwill come closer to each other and thus reduce the contact resistance. Bulk anoderesistance is proportional to the electrical resistivity. Reduced electrical resistivityimplies reduced energy costs. But there is a trade-o� between reduced electricalresistivity and increased thermal conductivity which promotes airburn of the anodesurface. Typical voltage drop in the anode is 0.2 V which is approximately 4 %of the total cell voltage drop. Voltage drop due to anode-stub contact resistanceis in the same order of magnitude; 0.10 V. Based on these �gure, it seems as ifonly marginal improvements can be achieved by reduced resistivity and anode-stubcontact resistance. Anyway, every contribution is of importance.

A certain amount of energy is required for keeping the chemical reactions opera-tive; the energy required is due to the polarization voltage. Anode polarization2 gives a signi�cant contribution to the overall energy consumption (Grjotheim& Welch 1988, pp. 76, 176-177). Anode polarization depends on both bath con-ditions and anode parameters. Current e�ciency in uences anode polarization.Thus, anode overvoltage can be decreased by increasing alumina concentrationand bath temperature. Reduced current density and increased anode surface areaalso reduces anodic overvoltage.

5.3.3 Anode Porosity vs. Overvoltage

If the fraction of pores with diameter larger than dcrit = 7:5�m is less than�crit � 12%, the ohmic voltage drop is essentially equal the Faradaic ohmic volt-age drop through the electrolyte (Jones 1990). The ohmic voltage drop tends toincrease with the fraction of pores with diameter larger than dcrit. This is due tothe presence of gas bubbles in the macropores with diameter larger than 7.5 �m.To avoid excessive bubble voltage losses, it is necessary that �crit � 14%. The ex-perimentally obtained anode overvoltage must be corrected for the correspondingohmic voltage drop to obtain the true anodic overvoltage. True overvoltage forbaked anodes is between 0.2 and 0.4 V for �crit � 14%.

2Synonymous with anode overvoltage which is in the order of 0.4 to 0.5 V.

60 Anode Quality and Carbon Consumption

5.3.4 Anode Cracking

Anode cracking is mainly due to thermal shock problems as discussed above. Ther-mal shock problems can decrease current e�ciency in the order of 1.5 % (ForumRoundtable: Discussing the Issues in Carbon Anode Technology 1990, pp. 53).

The thermo-mechanical properties of the anode are explicitly expressed as coldcrushing strength, bending strength, Young's modulus and thermal shock resis-tance.

5.3.5 Anode E�ect

The main cause for an anode e�ect is the depletion of alumina from the electrolyticbath. Probably, the geometric uniformity of the anode working face will havesome impact on the ease with which the gas bubbles during an anode e�ect can beevacuated. On the other hand, it has not been possible to reveal if certain anodeproperties has direct impact on an anode e�ect.

5.3.6 Anode Changing

The anode changing operation is described in Grjotheim & Kvande (1993, pp.201). Anode density has direct impact on anode change frequency: A high densitygives a low net volumetric consumption rate which implies that anodes can bechanged less frequently. The frequency of anode changing also depends on thecarbon consumption rate. It is a well known fact that high density anodes reducescarbon consumption. A high density anode allows for less porosity and reducedpermeability which means a reduced carbon consumption. According to Wilkening(1993, pp. 1167), decrease in anode height (and thus anode consumption) duringelectrolysis is inversely proportional to baked apparent density. Another reasonfor wanting a high anode bulk density is the impact of anode density on the bakingcapacity of the ring furnace as well as reduced net production costs in green anodemanufacturing.

Finally, anode size directly in uences the frequency of anode change. Anode sizehas to do with anode design, and will not be further discussed.

5.4 The Concept of Anode Quality

To summarize, the important relationships between anode parameters and phe-nomena taking place during electrolysis have been indicated:

� Electrolytic carbon consumption is related to the current e�ciency. Currente�ciency is directly in uenced by dusting and thermal shock problems whichcause anode cracking.

5.4 The Concept of Anode Quality 61

� Excess carbon consumption indirectly relates to the following anode param-eters:

{ Speci�c surface area Sa

{ Crystallite structure as represented by Lc;a

{ Gas permeability kp;a

{ Content of anode impurities (ash) fa

{ Thermal conductivity kt;a

{ Coe�cient of thermal expansion �

The excess consumption directly relates to airburn, CO2-oxidation and dust-ing which all depend on the above listed parameters.

� Anode voltage drop depends on the electrical resistivity �e and coe�cient ofthermal expansion.

� Anode overvoltage is reduced by increasing the real surface area of the anode.

� Anode change frequency is related to baked apparent density.

The set of anode quality parameters which describes the outcome of the bakingprocess is divided into groups of chemical, electrical, mechanical and transportparameters and parameters for other physical properties. Then, the following listof anode quality parameters is obtained (see Table 5.1):

1. Chemical properties:

(a) Ash content fa

(b) Airburn rO2;a

(c) CO2-oxidation rCO2;a

(d) Dusting index si;a

2. Mechanical Properties:

(a) Young's modulus Ya

(b) Bending strength �b;a

(c) Compressive (crushing) strength �c;a

(d) Thermal shock resistance Rts;a

3. Transport properties:

(a) Electrical resistivity �el;a

(b) Gas permeability kp;a

(c) Thermal conductivity kt;a

4. Other physical properties:

62 Anode Quality and Carbon Consumption

(a) Baked apparent density �a;a;b

(b) Surface area Sa

(c) Coe�cient of thermal expansion

(d) Crystallite structure (represented by d002;a, La;a and Lc;a)

(e) Micro-structure (i.e. structure of baked binder coke)

5. (Net) Price of baked anode

These parameters have impact on anode performance during baking. Bounds canbe speci�ed for the tolerable value ranges for each parameter and the parameterset can be used for optimization of the baking process to achieve anodes that to acertain degree of uniformity satisfy the speci�ed quality criteria.

Except for the ash content which is mainly determined by the amount of buttsused in manufacturing the green anode3 , the quality �gures on the list is a�ectedby the baking process. Compressive strength (cold crushing strength) is includedseparately on the list to be able to specify the anodes strength during transportand handling.

Some may �nd it strange that apparent baked density occurs on the list. Al-though apparent baked density is easy measurable, it does not directly relateto anode quality (Forum Roundtable: Discussing the Issues in Carbon AnodeTechnology 1990, pp. 54). However, the average operation time of the anodein the cell is directly related to the bulk density and apparent density is importantfor determining the frequency of anode changes as well as baking e�ciency.

The relevant anode properties should be reduced to a low dimensional set of prop-erties which directly relate to anode behaviour during mechanical handling andelectrochemical consumption. Also, some of the anode properties listed in Ta-ble 5.1 correlate. This introduces a certain degree of redundancy in the list ofproperties. Now, it is assumed that the following set of properties can be used tode�ne the anode quality:

1. Chemical properties:

(a) Airburn

(b) CO2-oxidation

(c) Dusting index

2. Mechanical properties:

(a) Compressive (crushing) strength

(b) Thermal shock resistance

3. Transport properties:

(a) Electrical resistivity

3The impurities are mainly due to the butts content in the �ller coke. Ash in petroleum cokeand coal tar pitch only play a negligible role.

5.4 The Concept of Anode Quality 63

(b) Thermal conductivity

4. Other physical properties:

(a) Baked apparent density

(b) Coe�cient of thermal expansion

These properties may be considered to depend on a set of more fundamental statevariables. In this way, one may also expect a certain degree of dependency betweenthese properties that are used to de�ne anode quality.

The costs of anode production also come into consideration as stated earlier inthis discussion. At present, this parameter is omitted from the list. In the abovelist, the chemical parameters are directly related to excess carbon consumption.The mechanical properties represent the anode's ability to withstand mechanicalhandling and introduction into the smelter.

This list of anode properties can be used as basis for selection of a subset of anodeproperties used in model based optimization and control of the baking process.This subject is further discussed in parts II and V of this study.

64 Anode Quality and Carbon Consumption

Chapter 6

Discussions and Conclusions

In this part, a description of the anode production line has been given. Anodebaking is the most cost-intensive part of the anode manufacturing process. There-fore, optimization of the baking process is necessary to obtain satisfactory processeconomy. Adequate quality of raw materials of petroleum coke and coal tar pitchis a prerequisite for high quality anodes.

Two ring furnace concepts dominate in anode production today. In open andclosed ring furnaces, gas ow patterns vary, but the furnaces operate according tothe same principles. Focus is put on ring furnaces of Hydro Aluminium design;a kind of closed ring furnace. Conventional ring furnace operation rely on theapplication of a gas temperature target curve. There seems to be potentials for in-creased instrumentation in conventional ring furnaces of both types. Manipulablevariables are local burner fuel ows and exhaust manifold draught pressure. On-line measurement is restricted to logging of gas temperature and draught pressurein section d1.

A set of anode quality parameters was established with basis in the behaviourof the anodes in the electrolysis. Still a lot of work can be done to increase theunderstanding of the interactions between the carbon anode and the electrolyte.

66 Discussions and Conclusions

Part II

Structure and Properties of

Soft Carbons

Chapter 7

Classi�cation of Carbon

Forms

It the following, a survey of carbon forms derived from organic precursors bypyrolysis is given. In the context of carbon technology, pyrolysis is often termedcarbonization: Carbonization is a heat treatment process for formation of carbonmaterial with increasing carbon content from organic material (ICCTC 1982).Thus, the term carbon covers solid materials ranging from the condensed polymericsolid materials called raw cokes to natural and arti�cial graphites.

In the process of graphitization, non-graphitic carbon is turned into graphite bythermal activation (ICCTC 1982). Solid carbons derived from organic precursorsby carbonization, are divided into two forms:

� Graphitic carbons: Carbons which consist of elemental carbon in the al-lotropic form of graphite (ICCTC 1982). Natural- and synthetic graphitesbelong to this group.

� Non-graphitic carbons: These carbons have two-dimensional long range or-der in an approximately planar hexagonal network. In the c-direction, thereis mainly parallel stacking with no crystallographic order (ICCTC 1982). Ina lot of these carbons, the structure is turbostratic.

The non-graphitic carbons belong to one of two categories:

� Graphitizable carbons (also denoted anisotropic carbon or soft carbons)

� Non-graphitizable carbons (also denoted isotropic carbons or hard carbons)

70 Classi�cation of Carbon Forms

7.1 Soft Carbons

Soft carbons have low porosity and are derived from materials with relativelyhigh contents of hydrogen and low contents of oxygen. During graphitization, softcarbons can be transformed into (synthetic) graphitic carbon. Most soft carbonspass through a uid (plastic) stage during carbonization. While in the uid state,formation of mesophase takes place in the liquid phase. The mesophase consistsof small anisotropic spheres in which planar aromatic molecules are more or lessstacked on top of each other. The formation of mesophase is a prerequisite forfurther growth of the material into an ordered structure. In soft carbons, completegraphitization can be achieved by thermal treatment alone, i.e. the graphitizationprocess is a function of time and temperature.

Pitches, pitch coke, petroleum coke and polynuclear aromatic systems belong tothe soft carbons.

7.2 Hard Carbons

Fusion does not take place during pyrolysis of hard carbons. This is due to themaintenance of a crosslinked molecular structure during the heat treatment pro-cess. Hard carbons maintain imperfect structure even after long heat treatmenttime at high temperature. Complete graphitization of hard carbons can only beachieved during treatment at both high pressure and temperature. Glassy (vitre-ous) carbons, wood (cellulose) and non-fusing coals belong to the hard carbons.Small sized and defective aromatic layers and consequently irregular stacking ar-rangements create spaces of microporosity between the crystallites.

7.3 Further Classi�cation of Carbons

Solid carbons of industrial importance can be classi�ed as follows:

� Coal

� Coke

� Pitch

� Carbon �bres

� Other carbon materials

Coke consists of non-graphitic carbon and is almost pure carbon. Coke is theresidue from carbonization of organic compounds. Coke can be derived from bothpetroleum fractions and coal tar pitch which gives petroleum coke and (coal de-rived) pitch coke respectively. Raw coke is usually termed green coke.

7.4 Carbon Composites 71

Pitch is the result of pyrolysis of organic precursors at temperatures below 500�C.Two types of pitch are of main importance in production of carbon materials. Coaltar pitch is the residue in distillation of coal tar. It consists of thousands of organiccompounds in mixture where the aromatic compounds dominate. Petroleum pitchis obtained in the re�ning of petroleum fractions. Petroleum pitch is more aliphaticthan coal tar pitch.

7.4 Carbon Composites

Composites are carbon artifacts consisting of two or more carbon forms. Thestrength of the composite depends on the interfacial forces between the carbonforms of the composite. Usually the adhesion forces can be of both physical andchemical nature. Often it is di�cult to identify the di�erent carbon phases in acomposite material. Techniques available for investigation of pure carbon formscan also be applied to composites. In composites with a high degree of graphiti-zation, identi�cation of the original carbon forms is impossible.

72 Classi�cation of Carbon Forms

Chapter 8

Carbon Structure and

Fundamental Carbon

Properties

In this chapter, the concept of fundamental carbon properties for description ofcarbon structure is introduced. The text is based on the general carbon litera-ture. Speci�cally, Marsh (1989, cht. 1) and reviews from Walker, Walker, Jr. &Thrower (1965-1991) have given theoretical background for this presentation ofcarbon structure.

The properties of a composite carbon material is determined by the properties ofthe carbon components which constitute the carbon. For carbon anodes, propertiesof �ller - and binder coke must be taken into account.

8.1 The Fundamental Carbon Properties

Fortunately, most physical properties of a carbonaceous material are related to amoderately sized set of fundamental properties:

� Elemental composition (CH-ratio, impurities etc.)

� Group composition (solvent fractions)

� Structure:

{ Macrostructure of the carbon (i.e. macropores etc.)

{ Microstructure of the carbon (i.e. texture, preferential orientation, d002,La, Lc etc.)

� Porosity (or variables related with porosity)

74 Carbon Structure and Fundamental Carbon Properties

� Impurities

In this study, this set of properties constitutes the fundamental carbon properties.These properties span a subset of the state vector in a model for the carbonproperties. Depending on the formulation of the model, however, some of theproperties may appear as functions of the state variables.

In general, it is assumed that models for the physical properties of the carbonprecursor and coke residue can be formulated from the fundamental properties. Infact, the fundamental carbon property-concept is quite similar to the internal statevariable concept presented in Ashby (1992) which has been used for modelling ofmicrostructure evolution in solid state processes.

8.2 Structure in Non-Graphitic Carbons

In carbons, microcrystals (or microcrystallites) are the basic building blocks. Thecrystallites1 are connected by valence bonds in a more or less random manner.The microcrystallites are either bonded directly to each other or indirectly viasingle carbon atoms in the disordered carbon phase. The disordered carbon phaseconsists of carbon atoms not yet systematically introduced into the organizedphase of microcrystallites. In principle, the solid carbon is a three dimensionalarrangement of small crystallites connected by valence bonds between free carbonvalences on the periphery of layer planes. Within the crystallites, however, acertain degree of order exists: Each crystallite is composed of a number of aromaticplanes which essentially consist of a network of benzene rings. In most carbons,highly ordered graphitic structures can be observed on a short range scale. Inlarger domains of the carbon sample, however, the graphitic layer or -lamellae, isin several respects very defective as compared to the Bernal (graphitic) structure:

� Layer planes (layers, lamellaes) can be bent and/or twisted

� Mutual orientation of layers di�ers from graphite

� Interlayer spacing is larger than in graphite

� Crystallite size and stacking height is less than in graphite

� Holes may be present in the layers

� Structural defects occur in the layers

� Crosslinkings occur between layers

� Heteroatoms may be bonded into the graphitic lamellaes

� Intercalation of atoms and molecules may occur between the layers

1Crystallite: A small and often imperfectly formed crystal (Walker 1991).

8.2 Structure in Non-Graphitic Carbons 75

� Disordered material is bonded on the edges of the layer planes and as crosslinksbetween neighbouring layer planes

The crystallites (also called basic structural unit (BSU)), however, resembles thatof graphite.

Within the crystallites, the layer planes are stacked on top of each other withweak Van der Waal's forces of attraction between the layer planes. The valencebonds connecting the crystallites and the individual carbon atoms in the aromaticlayer planes are much stronger than the Van der Waal's type of bonding whichexists between the layer planes. Therefore the sensitivity to both compressionand thermal expansion is greatest in the direction normal to the layer planes.For the bulk material however, the observed anisotropy is an average value of thein-layer plane property (along the a-axis) and the property normal to the planes(along the c-axis) due to the random criss-cross hooking of crystallites in threedimensions. The physical properties of carbon depend in a complicated way onthe properties of the crystallites (microcrystalline properties) and the way they arelinked together. The microstructure is partly determined by the selection of theraw materials but also by the processing conditions used to manufacture the carbon(mixing time and temperature, i.e. heat treatment program). The macrostructureof the carbon material is also an important factor for the physical properties: Themacrostructure however, is mainly determined by the modi�cations introducedduring manufacturing (Okada & Takeuchi 1960).

Large variations in structural characteristics may occur. This results in a largerange of carbon forms which all have the graphitic structure. To classify the varietyin carbon structure, Franklin (1951a) de�ned two extreme forms of carbon:

� Soft (graphitizing/graphitizable) carbons are formed from organic materialswhich fuses during heat treatment and then solidi�es at temperatures in theorder of 500�C due to the thermal setting of the mesophase. The formationof a liquid phase, allows for a certain degree of order to be established in thematerial before solidi�cation. Thus, the carbon consists of nearly parallelcrystallites with weak crosslinks between layer planes. A carbon with low(macro)porosity and anisotropic structure is formed.

� Hard (non-graphitizing/non-graphitizable) carbons do not melt (or alterna-tively they solidify at very low temperatures). The structural order of thesematerials is therefore isotropic. A strong network of crosslinks exists betweenrandomly oriented crystallites. The hard carbons have a high (micro)porosityand show a slower increase in crystallite size upon heat treatment.

The di�erence between isotropic and anisotropic carbons is illustrated in Fig-ure 8.1. Later, Monthioux, Oberlin & Bourrat (1982) showed that all kinds ofintermediate carbon forms exist between the soft and hard carbons. In this work,however, Franklin's classi�cation is used and the focus is on soft carbons (boththe binder coke and the �ller coke belongs to the soft carbons).

The geometry of a crystallite is usually described by three parameters as follows(see Figure 8.2):

76 Carbon Structure and Fundamental Carbon Properties

Isotropiccarbon

Anisotropiccarbon

Figure 8.1: Schematic representation of the mutual orientation of the graphiticlamellae in isotropic and anisotropic carbons. In anisotropic carbons, the aromaticlayers are larger, less defective with a smaller amount of heteroatoms than thearomatic layers in isotropic carbons.

� d : Interlayer spacing

� La: Diameter of stack

� Lc: Height of stack

If there are ncr layers in the stack, Lc = (ncr�1)d. X-ray di�raction is often usedto obtain average values of these parameters. In this study, Lc is selected as acrystalline parameter. Alternatively, ncr could serve as a parameter. Ideally, then,ncr is a parameter which takes only discrete values. In a bulk volume of carbon,however, ncr on the average will be a continuous variable. Thus, both ncr (andLc) may vary continuously.

Sometimes, the intercrystalline porosity is also needed as a parameter for thedetermination of physical properties. The closed porosity (i.e. porosity below thereal density level) will be used as an approximation of the intercrystalline porosity.

The mutual orientation of crystallites may be such that the range of order may beorders of magnitude larger than the size of the crystallites themselves. An assemblyof more or less oriented crystallites is termed a macrocrystal. In general, thedomains which constitute the macrocrystals have sizes in the range from 0.5 up to500 �m; i.e. a factor of 1000 larger than the size of the crystallites themselves. Themacrocrystals comprise the microstructure of the carbon (Marsh & Clarke 1986)and d002, La and Lc can be used to described the structure of the macrocrystals.

For a certain level of resolution, the degree of anisotropy can be found by theestimation of a so-called optical texture index: In optical microscopy, di�erentisochromatic domains are assigned certain optical texture indices according toa speci�ed nomenclature. Several nomenclatures exist for description of opticaltexture. An example of such a nomenclature from Marsh (1989) is shown inTable 8.1. The optical texture of the whole investigated sample can be found as aweighed average value of the optical texture indices of the individual domains.

Two factors are of main importance for the unique properties of carbons:

� The anisotropy of single (micro)crystallites

8.2 Structure in Non-Graphitic Carbons 77

La

Lc

nc

d

Figure 8.2: A crystallite is an assembly of mutually ordered graphitic layer planes.Parameters d, La and Lc are used to describe the geometry of the crystallites.The crystallites constitute the microstructure of the carbon material. Crystallitesare surrounded by micropores between large assemblies (i.e. macrocrystals) ofmicrocrystals.

Figure 8.3: Simple schematic representation of a macrocrystal. The size of amacrocrystal is typically in the order of 1:0�m or below. The size of the crystal-lites is in the order of 10 �A (i.e. 10�9m).

Type of domain Abbr. Size [�m] OTI

Isotropic I No optical activity 0

Fine mosaic F d < 0:8 1

Medium mosaic M 0:8 < d < 2:0 3

Coarse mosaic C 2:0 < d < 10:0 7

Granular ow GF l > 2:0; w > 1:0 7

Coarse ow CF l > 10:0; w > 20:0 20

Lamellar L l > 20:0; w > 10:0 30

Table 8.1: Classi�cation of optical texture according to Marsh (1989, pp. 20).d, l and w denote diameter, length and width of the optical domain.

78 Carbon Structure and Fundamental Carbon Properties

� The polymeric nature of the intercrystalline C � C valence bonds

These properties lead to the following characteristic bulk properties of carbons:

� A very low value (but still a broad range) of the coe�cient of thermal ex-pansion

� The formation of unavoidable microporosity and frozen-in stressed upon cool-ing of the heat treated carbon

8.3 Structure of Soft Carbons

For soft carbons, the formation of structure depends on:

� Characteristics of the feedstock:

{ Aromaticity

{ Presence of alkyl side chains

{ Presence of reactive functional groups (hydroxyl, carboxyl etc.)

{ Presence of heteroatoms

� Characteristics of the pyrolysis process:

{ Time-temperature program

{ Ambient pressure

8.3.1 Structural Changes During Fusion

In the carbon precursor, a certain degree of order exists in the solid state. Duringfusion, however, some structural order is lost due to both thermal vibrations ofthe molecules in the melt as well as the di�usional motion of the molecules. Thisphenomenon was studied for pitches in the fused state by Korai & Mochida (1992)and Turner (1995). It was found that structure as represented by the stackingheight Lc and interlayer spacing d002 deteriorated during fusion: Lc decreased andd002 increased (Turner 1995). Furthermore, it was found that the weight fractionof stacked molecules was reduced during fusion. A corresponding increase in theweight fraction of non-stacked molecules was observed (Korai & Mochida 1992).

Above the softening point, however, this trend was reversed: Lc increases andd002 decreases (Turner 1995, Fig. 12, 14). This may be due to a relatively moree�ective vaporization of the non-organized molecules than the stacked molecules.At temperatures corresponding to the onset of carbonization reactions, the size ofthe molecules increase due to polymerization.

8.3 Structure of Soft Carbons 79

T [�C]Ts Ts T [�C]

d002[�A]Lc[�A]

Figure 8.4: Reduction in structural order as function of temperature during fusionof a soft carbon precursor. Stacking order is lost up to the softening point Ts(usually, Ts is below 140�C). After Turner (1995, Fig. 12, 14).

8.3.2 Structural Changes During Liquid Phase Pyrolysis

As the carbonization temperature increases, progressive volatilization of light com-pounds occurs in parallel with carbonization reactions. At a certain stage, the sizeof the aromatic molecules become large enough to allow formation of stable ag-gregates of molecules.

Formation of Carbon Texture. During pyrolysis, the aromatic layers grow insize both due to volatilization of light compounds and polymerization reactions.At a certain stage, the layers become large enough to associate in stacks due tothe balance between forces of attraction and thermal movement. This usuallyoccurs at temperatures between 400 and 500�C: A reversible formation of a liquidcrystalline state takes place in the uid phase. Upon further heat treatment, theliquid crystals segregate into a separate state called mesophase. The mesophaseconsisting of spheres of aggregated aromatic molecules constitutes an intermediatestage between uid pitch and solid coke since the mesophase is not completelyamorphous and does either not inhabit the state of structural order found inpyrolytic semicoke.

The existence of large stable free radicals is a prerequisite for the maintenance of uidity which allows for the formation of mesophase. The association of moleculesis a physical process which may occur when the chemical polymerization hasreached a certain stage. The liquid phase should be maintained over the broadestpossible temperature range to allow for favourable steric orientation of the planarmolecules. Liquid state pyrolysis therefore leads to well ordered carbon residues.

To achieve a well ordered pyrolytic residue, the molecules must have a planarstructure. Only aromatic compounds have a planar molecular geometry, and thefavourable orientation of the molecules seems to be a parallel arrangement in stacksof a certain height. It has been shown that molecules with a size of 20 to 30 atoms,have the capability of attracting other atoms. This property increases with molec-ular size. For studying the course of the liquid phase pyrolysis, two dimensional(hk)-re ections obtained in X-ray analysis, can be analysed by Ruland's disorderparameters (Ruland 1965). Chemical analysis combined with X-ray analysis givegood insight into the progression of pyrolysis.

80 Carbon Structure and Fundamental Carbon Properties

The size of the optical texture depends mainly on the aromaticity of the feedstock:More reactive feedstocks lead to crosslinking between aromatic planes which canbe observed as a reduced size of the optical texture. On the other hand, Elalaoui,Krebs, Mareche & Furdin (1995) and Krebs et al. (1995) showed that for a certaincarbon precursor, the best possible orientation is obtained if the maximal gasevolution rate occurs just at the mesophase solidi�cation stage. For this to happen,an intimate balance between carbonization temperature and ambient pressure hasto be maintained. Since industrial carbonization mainly takes place at ambientpressure the most important parameter to improve coke texture is the quality ofthe feedstock: The size of the texture can be improved only within narrow limits.

For pitch based cokes, the presence of primary quinoline insolubles inhibits thegrowth and coalescence of the mesophase. This is observed as a more isotropictexture than the texture found in petroleum based cokes.

It is the degree of chemical crosslinkage2 which determines whether a pyrolyticcarbon turns into a soft or a hard carbon. Crystalline preorder depends on theagglomeration of the aromatic components. Ring additions cause formation oflarge planar aromatic structures which is a prerequisite for parallel arrangementof molecules. In some molecules, di�erent local sterical orientations may preventparallel orientation (biphenyl bonds).

The presence of oxygen, nitrogen and sulphur can in uence preorder formationduring pyrolysis both as heteroatoms in the organic precursor material as wellas elementary additives during pyrolysis. Usually, the presence of these elementscontributes to formation of carbons of poor graphitizability.

During the uid stage and mesophase formation, crystallite parameters La, Lcand d improve (Honda, Kimura, Sugawara & Futura 1970, Fig. 4,5). At thesame time, the viscosity of the carbon residue increases due to the polymerizationreactions leading both to increased molecular sizes as well as crosslinking betweenmolecules (Tillmans 1986). The increase in viscosity leads to solidi�cation of theresidue. It is, however, di�cult to decide at which moment solidi�cation occurs:In the process of solidi�cation the residue continuously develops from a highlyviscous carbon material to a solid coke. If the process of solidi�cation is related tothe glass transition temperature, solidi�cation could be de�ned to occur when theglass transition temperature is higher than the actual carbonization temperature.

Formation and Structure of Brooks and Taylor Mesophase. The discov-ery and explanation of the mesophase transition took place during the 1960's.Since the pioneering work by Brooks & Taylor (1965a) and Brooks & Taylor(1965b), signi�cant progress in the understanding of mesophase formation3 andmicrostructure evolution has been gained. Several reviews give general discussionsof structural, chemical and kinetic aspects of mesophase formation (Brooks &Taylor 1968), (Fitzer, Mueller & Schaefer 1971), (Marsh 1976), (Marsh & WalkerJr. 1979), (Marsh & Latham 1986), (Honda 1988). Explicit focus on mesophase

2Autocatalytic reactions, Friedel-Kraft reactions etc.3i.e. the relationship between chemical and physical processes occurring in the carbonizing

substance.

8.3 Structure of Soft Carbons 81

structure is given in Honda et al. (1970), White (1974) and Auguie et al. (1980).

Several changes can be observed during carbonization: At �rst the organic pre-cursor melts into a pitch like material with (plastic) liquid consistency4.

At temperatures of approximately 400�C, the formation of small and stronglyanisotropic spheres can be observed. The spheres have an approximate diameter ofd = 0:1�m but both smaller and larger spheres can be found. Initially, formationof these spheres is due to dispersion forces between lamellar molecules in theliquid pitch. The spheres as well as the isotropic pitch behave as a liquid atthe temperature of mesophase formation. The mesophase5 is a kind of liquidcrystalline system, and the mesophase is denoted nematic (discotic) liquid crystals.Interlamellar spacings of approximately 3.47 �A have been determined from electrondi�raction measurements. The crystallographic order is in the range of 0.5 to 500�m. The raw mesophase has a density of approximately 1400kg/m

3which shows

that the molecular packing between spheres is poor. Also, the layer planes maycontain one or several holes. Thus, the molecular packing in the spheres is poorand each molecular plane has one or several holes. The mesophase segregates fromthe bulk isotropic pitch in stagnant reaction media.

Polymerization processes introduce growth of large planar molecules in the isotropicpitch. The concentration (and size) of these molecules increases with time andtemperature. The mobility of the molecules is reduced as the molecular weightincreases. At a certain molecular size, the strength of the van der Waal's forceswhich act between the lamellar molecules are large enough to cause transition intoa liquid crystal phase. The initial formation of the crystals has been shown tobe a temperature dependent reversible process. Continued polymerization withinthe liquid crystals creates an insoluble phase; the mesophase. Usually one doesnot distinguish between the initial lamellar nematic liquid crystal system and itspartly polymerized derivative; the mesophase. The two most important factorsin the formation of liquid crystals in the pitch system is the size of the moleculesand their planarity. Also the chemical reactivity of the pitch constituents �ts intothis picture. The reactivities promote polymerization of molecules permitting theformation of the liquid crystals. The reactivity is low enough to maintain the uidphase over a temperature range wide enough to allow for the formation of largeanisotropic areas of coke of high graphitizability. A too high rate of reaction wouldcause formation of isotropic coke with only a short range order.

The spheres grow with time and temperature at the expense of the surroundingpitch-like material. This causes an increase in viscosity and �nally the conversioninto anisotropic spheres will be complete. Initially, the mesophase bodies are veryclose to the spherical shape. But as the fraction of mesophase becomes larger,the bodies start to interfere with each other and deviations from the sphericalform might occur. Deviation from regular spherical shapes can be promoted bythe presence of primary quinoline insolubles (QI) (Brooks & Taylor 1968, pp.255). In fact, the presence of QI seems to restrict the growth of the spheres since

4Coal tar pitches and its fractions, petroleum bitumina etc.5Greek mesos - intermediate i.e. intermediate between isotropic liquid pitch and solid semi-

coke.

82 Carbon Structure and Fundamental Carbon Properties

the primary QI particles introduce a physical resistance to mesophase growth.Thus, presence of QI hinders formation of large anisotropic domains. But theinitial nucleation and growth of single spheres is not in uenced. Thus, the overallkinetics of the mesophase conversion is not in uenced but presence of primary QIhas signi�cant impact on the carbon residue texture.

Coalescence of spheres occurs in three dimensions and a complicated pattern ofcurved lamellaes can be seen in any plane across the mesophase: The lamellaemay curve around sharply between fairly uniform areas corresponding to previousindividual spheres.

Precursors capable of forming mesophase during carbonization have a very com-plicated chemical structure. The original molecules in pitch materials are mainlypolynuclear aromatic compounds and the dominating hydrogen present is bondedto aromatic carbon atoms. The more reactive hydrocarbons are converted bydehydrogenation and condensation reactions to larger and more complex specieswhich segregates from the pitch as mesophase. Also, a large portion of the origi-nally present molecules seems to take place in the formation of mesophase spheresand there is not much di�erence between the original pitch and the residue (i.e.hydrogen content etc).

The mechanism of mesophase formation can be summarized in the following steps(Brooks & Taylor 1968):

1. Formation of planar molecules by building reactions in the liquid pitch.

2. Parallel arrangement of the aromatic planar molecules due to dispersionforces (van der Waal's type forces) acting between the molecules.

3. Preferential growth of molecules with La > 25 �A.

4. Mesophase nucleation: Reversible parallel stacking of large aromatic molecules.This seems to take place adjacent to any solid surface (primary QI) presentin the pitch. The nucleation e�ect depends on the available speci�c surfacearea of the solid particles. The QI particles provide a site for nucleation butare excluded from the spheres in the growth process.

5. Growth: This is a kinetic process in which time and temperature are im-portant parameters. QI particles are excluded during mesophase growth.At low rates of carbonization, mesophase appears as few and large spherescompared to the many smaller spheres at high rates of carbonization.

6. Coalescence growth: Coalesence of mesophase spheres (possibly in uencedby the presence of primary QI) leads to the formation of bulk mesophase.The radial arrangement of molecules at the surface of each sphere tends topromote initial fusion, see Figure 8.5.

7. Conversion of interstitial isotropic pitch into mesophase: The size of theordered regions depends on the time the reacting medium is kept in a liquidstate.

8.3 Structure of Soft Carbons 83

There is no obvious change in the material structure during conversion from thehighly viscous liquid mostly containing mesophase and the semicoke obtained byfurther carbonization of the carbon residue.

Several parameters have in uence on the composition and morphology of the meso-phase:

� Pitch precursor (i.e. reactive properties)

� Heat treatment programme

� Content of primary QI

� Gas atmosphere (both composition and pressure)

� Mechanical agitation.

Pole

Pole

Lamellae direction

Figure 8.5: Structure of a Brooks & Taylor mesophase sphere.

8.3.3 Structural Changes During Calcination and Graphiti-

zation

Solidi�cation of the pyrolytic residue gives a raw coke in which further carboniza-tion and subsequent graphitization cause crystallite growth to take place within themosaic units from the mesophase formation. The mosaic structure and lamellarorientation remain in the residue and there is no major reordering of the carbonicmatter.

Carbonization of Raw Coke. At temperatures in the order of 500�C, perfec-tion of the structure takes place in domains surrounding the crystallites. Usually,the growth amounts to a steady increase in La as the temperature increases. Ford002 and Lc, however, deterioration in these parameters accompany the releaseof carbonization gases between 500 and 800�C (Marsh & Stadler 1967, Fig. 1),(Marsh 1973), (Auguie et al. 1980). It seems as if reorganization (solidi�cationand shrinkage) of the mesophase leads to a reduction in the apparent crystallitesize (i.e. structural order). The minimum value observed in Lc (and the maximum

84 Carbon Structure and Fundamental Carbon Properties

value in d002) correspond approximately to the cease of H2 (and other volatiles)degassing from the carbon residue. Above 800�C, however, organization processesagain prevail. For temperatures up to 1500�C, Lc changes more easily than La.

For temperatures between 450 and 1300�C, these structural arrangements aremainly due to uptake of the so-called disorganized carbon phase at the peripheryof the microcrystallites. The consumption of the disorganized phase is accom-panied by the release of light gaseous compounds between 450 and 1000�C. Ata temperature in the order of 1300�C, the disorganized carbon phase is totallyconsumed and further growth occurs at the expense of the smallest crystallites.Carbons heat treated to temperatures in the order of 1300�C have a turbostraticstructure: Within the crystallites, the layer planes are oriented in parallel but themutual orientation of layer planes in the c-direction is still random. On the aver-age, the interlayer spacing is 3.44 �A. Typically, the diameter of the crystallites aswell as the height of the stack of parallel layer is in the order of 30 �A. Probablythe turbostratic crystallites are very imperfect since a lot of unoccupied positionswill form when small layer planes coalesce into one larger plane.

Upon further heat treatment in the pregraphitization regime, the layer planes comecloser to each other since the layer planes are allowed to rotate around the c-axisas well as being shifted relatively to each other.

Graphitization. Usually, the graphitization regime is considered to start as allthe foreign gases (non-condensables) have escaped from the carbon residue. Thedisorganized carbon atoms are absorbed by the growing aromatic layers, and anaccelerated evolution of the crystallite parameters may occur. In soft carbons, thischange in heat treatment regime usually takes place between 1200 and 1400�C.In graphitization, signi�cant three dimensional ordering of the carbon structure isgradually introduced.

The rotation and displacement of the layers in the graphitization process trans-forms the turbostratic carbon into a graphitic structure. In this process, also thediameter and the height of the layer plane stacks increase. Finally, in graphitethe distance between layer planes is 3.3534 �A (Franklin 1951a), (Franklin 1951b).Mrozowski (1956a) suggested that the main driving force for crystallite growth insoft carbons is the relaxation of induced thermal stresses by rearrangement andgrowth of carbon crystallites. This leads to an increased crystallite size as thetemperature increases, and at each temperature there also seems to exist a limit-ing value of the average crystallite size since the crystallite growth ceases as theinduced stresses are relaxed.

Studies of Microstructure in Solid Carbons. A lot of studies exist whichdeal with the development of microstructure in solid carbons, the impact of carbontexture on graphitizability and the mechanisms for improvement of the microtex-ture during carbonization and graphitization6. Franklin showed that the structure

6Franklin (1951a), Franklin (1951b), Mrozowski (1956a), Mrozowski (1958), Akamatu & Ko-ruda (1960), Oberlin & Oberlin (1983), Monthioux et al. (1982), Oberlin (1984) and Oberlin,

8.3 Structure of Soft Carbons 85

obtained during carbonization depends on both the carbon precursor and the heattreatment programme which is used. This gave rise to the division of solid carbonsinto soft and hard depending on their ability to transform into arti�cial graphitein the process of graphitization.

During carbonization and graphitization, the texture in soft carbons is improvedby total disappearance of the basic structural units or crystallites present in thesolid raw coke. According to Oberlin and coworkers, the process of crystallitedisappearance and formation of arti�cial graphite occurs via four stages (Oberlin1984, pp. 521-527, Fig. 8, p. 531), (Oberlin 1984, pp. 88-91, Fig. 4):

� Stage 1: The texture at this stage consists of single crystallites as a resultof the mesophase transition occurring in the liquid phase. Depending on thedegree of anisotropy, the crystallites are more or less mutually oriented.

� Stage 2: Between 600 and 1500�C, the single crystallites arrange into dis-torted columns. The crystallites, however, retain their individual character.As heteroatoms (mainly hydrogen) and defects disappear, the columns getmore closely packed. Due to single misoriented crystallites entrapped be-tween the columns, Lc cannot increase fast.

� Stage 3: Above 1500�C, the entrapped crystallites disappear and the columnscoalesce into stacks of wrinkled layers. Between 1500 and 1900�C, Lc in-creases fast. La, however, cannot increase so fast due to in-plane defectsfrozen in at the boundaries of otherwise perfect graphitic layers. A zigzag tex-ture still present in the planes prevent the rapid increase in La. At 1700

�C,the zigzag texture disappears and rapid dewrinkling and disappearance ofthe turbostratic structure occur between 1900 and 2100�C.

� Stage 4: Above 2100�C, sti� and nearly perfect layers exist in the carbon.Rapid development of three dimensional order may occur.

Other features of graphitization are reviewed in Gundersen (1996e, App. E).

8.3.4 Macrostructure vs. Microstructure

The macrostructure (anisotropy due to processing conditions and large pores) ofa certain carbon material is mainly established during the manufacturing stage(Okada & Takeuchi 1960).

Domains in the order of �m in the carbon material are macrocrystals consistingof microcrystals with a certain degree of mutual ordering (constituting the degreeof anisotropy of the carbon material). These domains are discontinuous due tothe presence of micropores and layer plane defects as well as the surroundingmacropores.

Here, it is assumed that the macrostructure mainly depends on the carbon pre-cursor (and partly the processing conditions applied to achieve the green coke).

Bonnamy, Bourrat, Monthioux & Rouzaud (1986) among others.

86 Carbon Structure and Fundamental Carbon Properties

8.4 Porosity

In arti�cial carbons, pores are a part of the intrinsic microstructure of the rawmaterial but they are also introduced during the carbon manufacturing process.

8.4.1 Classi�cation of Porosity

A distinction is made between open- and closed porosity as follows:

� Open porosity: The pore is connected to the external surface of the carbonartifact

� Closed porosity: The pore has no connection to the external carbon surface

According to IUPAC, porosity is classi�ed according to the pore sizes in the fol-lowing way (Marsh 1989):

� Macropores: Pore diameter greater than 50 nm

� Mesopores: Pore diameter between 2 and 50 nm

� Micropores: Pore diameter less than 2 nm downto sizes comparable to thesize of crystallites

Pore shapes vary from slit shaped - to bubble shaped pores as follows:

� Bubble sized pores with a wide size distribution are introduced by the car-bonization gases which are released during the uid stage of the carboniza-tion process.

� Small slit shaped intercrystalline pores are introduced by anisotropic shrink-age occurring during heat treatment and subsequent cooling of the solid car-bon artifact. The porosity formed during cooling was denoted unavoidableporosity by Mrozowski (1956a). These pores are shrinkage cracks inherentlyintroduced during cooling of the carbon.

In this work, total porosity as the sum of open and closed porosity is considered.Noattempt is given to model the size distribution of the pores.

8.4.2 The Unavoidable Porosity

During cooling, so-called unavoidable microporosity is formed in a manufacturedcarbon due to the di�erence in the thermal contraction of the bulk carbon materialas compared to the single crystallites. An estimate of the order of magnitude of theunavoidable microporosity introduced during cooling of the heat treated carboncan be found by comparing the bulk volume VB of the carbon and the actual

8.4 Porosity 87

volume VC of the crystallites as measured at the heat treatment temperature. LetVB;� denote the bulk volume at room temperature. Correspondingly, VC;� denotesthe volume of the crystallites at room temperature. Use the de�nition of thecoe�cient of thermal expansion to obtain:

VB;� = VB(1� �v;B�T ) (8.1)

VC;� = VC(1� �v;C�T ) (8.2)

where �v;B and �v;C are the bulk expansion coe�cients for the bulk volume anda single crystallite respectively. Also:

VC;� = VB;�(1� �T;�) (8.3)

VC = VB(1� �T ) (8.4)

where �T;� and �T denote the total porosity in the carbon in the cooled and heatedstates respectively. The bulk volume thermal expansion coe�cient is related tothe linear thermal expansion coe�cient �B for the bulk volume by �v;B � 3�B .Also, �v;C � 2�a + �c for the bulk expansion of the microcrystallites. It can beshown that (Gundersen 1996f):

�T;� � �T = (1� �T )(�v;C � �v;B)�T (8.5)

If the carbon at the heat treatment temperature is perfectly dense, we have �T = 0.Then the unavoidable microporosity formed during cooling to the room tempera-ture is in the order of :

�T;� = (�v;C � �v;B)�T (8.6)

For baked carbons, �T is approximately 1300�C. This gives �T;� = 0:036, i.e.in the order of 4 %. Compared to a total porosity in baked carbons which is inthe order of 30 %, the unavoidable porosity amounts to approximately 12 percentof the total porosity. At the highest heat treatment temperature, the carbon isusually not completely dense. Some regular porosity is usually present; this meansthat �T > 0 at the heat treatment temperature. From the model, is can beseen that the relative impact of unavoidable porosity on the di�erence �T;� � �Tis reduced as �T increases. The same analysis can be used for composite bakedcarbons (anodes) if it is assumed that the thermal expansion coe�cient is equal forboth the �ller coke and binder coke crystallites.

According to Mrozowski (1956a), the presence of unavoidable microporosity is ofminor importance for transport properties and physical properties which dependon the number of peripheral bonds between crystallites.

During heat treatment of a carbon artifact, a certain increase in the regular poros-ity takes place at temperatures up to 1000�C due to the loss of gases when thepitch is in the volatile range (excavation e�ect). A certain in ation can also occurdue to the released volatiles. Above 500�C, however, a certain decrease in theregular porosity may also occur due to shrinkage of the coke which takes placeafter solidi�cation of the mesophase (shrinkage process).

88 Carbon Structure and Fundamental Carbon Properties

The formation of unavoidable porosity formed as a consequence of di�erentialshrinkage of the bulk volume and individual crystallites during cooling often com-pensates for the reduction in regular porosity which occurs as the bulk volumeshrinks during the heat treatment process. When porosity is observed in thecooled state, it seems as if a kind of excavation e�ect has been responsible for theformation of porosity in the carbon artifact.

The concepts of unavoidable microporosity and regular porosity are physicallyrelevant, but cannot be easily measured. Therefore, the de�nition of total porosityof a baked carbon will be based on the concepts of real and apparent densities andsome assumptions concerning the behaviour of the bulk volume of the carbonduring baking.

Nevertheless, it is of importance to have in mind the physical interpretation ofunavoidable and regular porosity to be able to understand the basic physics ofcarbon materials.

8.4.3 Frozen-in Stresses

During cooling, large frozen-in stresses will also be formed due to the di�erentialshrinkage of the bulk volume and individual crystallites and since viscous creep ofcarbon does not occur at temperatures below 2500�C. These stresses remain in thecold carbon material until it is reheated. Thus, the frozen-in stresses are reversible.Due to the formation of frozen-in stresses, the mechanical strength of the carbonmaterial decreases during cooling; i.e. the mechanical strength increases withtemperature. Mrozowski (1956a) showed that the stresses are so large that manyof the bonds between microcrystallites in the binder coke formed during bakingwill break during the �rst cooling process. This leads to a further microcrackingof the carbon material. In composite carbons, microcracks are also formed in thebinder coke as a consequence of the shrinkage occurring in the later stages of thebaking process.

8.4.4 Pore Geometry

Both pore shape and pore-wall (microscopic) geometry is of importance for thephysical properties of carbon materials. This is especially the case for mechanical- and reactive properties. Patrick, S�orlie & Walker (1988) and Patrick, Soerlie& Walker (1989) correlated tensile strength of electrode carbons to several pa-rameters for classi�cation of pore size and shape. Carbon reactivity depends in acomplicated way on the microstructural arrangement of so-called active sites onthe carbon surface (Marsh 1989, pp. 108-110).

8.4.5 Coke Porosity vs. Coke Structure

In carbon materials, an extensive range of pore sizes and shapes exists. Theporosity could be considered to be a part of the general structure of the carbon

8.5 Elemental Composition and Impurities 89

material. Then it would be reasonable to consider macropores to be a part of themacrostructure of the carbon. Also, pores with sizes down to intercrystalline poresexists in the carbon material. Such pores can be assumed to constitute a part ofthe microstructure of the carbon material.

In this work, however, a distinction between porosity and structure is made asfollows: Coke structure is related to the structure of the non-porous part of thecoke material.

8.5 Elemental Composition and Impurities

A green carbon material consists mainly of carbon and hydrogen. Heteroatoms ofnitrogen, oxygen and sulphur are also present. During carbonization, hydrogen islost via the light non-condensable gases. At heat treatment temperatures in theorder of 1300�C, most of the hydrogen has been removed from the material.

Traces of metallic impurities (Cr, Mn, V a, Na, Si) are typically present in thecarbon, and may have severe impact on the reactive properties of the carbon.

8.6 Limitations in Carbon Production

The quality of the �nal carbon product depends on all stages of the productionprocess as illustrated in Figure 8.6. The quality of the coke precursor is of mainimportance for the �nal product quality: Failure at a certain stage in the pro-duction line can usually not be accounted for in later steps of the manufacturingprocess.

8.7 X-ray Di�raction

Methods for interpretation of X-ray di�raction pro�les from disordered (i.e. paracrys-talline) carbons signi�cantly evolved between 1910 and 1960. Due to the work ofWarren (1941), Franklin (1951b) and others, the turbostratic model of carbon ma-terials was established. Since then, the turbostratic model has been widely usedas a means of explaining the nature of X-ray di�raction pro�les for disorderedcarbons.

Several theories as reviewed in Gundersen (1996e, App. E) have been presentedto explain the transition of a disordered carbon into arti�cial graphite during theprocess of graphitization. This work, however, mainly deals with disordered car-bons and focus on description of X-ray di�raction as applied to carbons havingnegligible three dimensional ordering of the graphitic layers. Intensity pro�lesobtained by X-ray di�raction of such carbons are di�use with broad peaks corre-sponding to the incompletely developed structural order. The similarity, however,with di�raction pro�les from graphite is obvious. The powder method (Azaro� &

90 Carbon Structure and Fundamental Carbon Properties

Ash (impurities)

Degree of anisotropy (OTI)

Carbon Precursor

Calcined Carbon

Green Carbon

Porosity

Property

transformations

Physical properties

of calcined carbon

500

1250Crystallite size

T [�C]

Figure 8.6: The properties of the carbon precursor material is the main constraintswith impact on the physical properties which can be achieved by heat treatment ofthe precursor. During heat treatment, the optical texture is established within therange of the mesophase transition which generally is completed at a temperaturein the order of 500�C. For continued heat treatment of the carbon material attemperatures in the range between 500 and 1400�C, the optical anisotropy maybe considered unchanged. In this temperature range, mainly evolution of porosityand crystallinity (structure of microcrystals; i.e. crystallites), takes place.

Buerger 1958) is used for obtaining the X-ray line pro�les.

8.7.1 Conventional Parameters for Average Crystallite Size

The crystallites or basic structural units constituting the organized phase of dis-organized carbons are frequently classi�ed by the average interlayer spacing d andaverage crystallite sizes La and Lc. La and Lc correspond to layer plane diameterand stacking height respectively.

According to Short & Walker (1963), these parameters are commonly determinedby direct application of the Bragg-, Warren- and Scherrer- equations on the X-rayline pro�les.

The Interlayer Spacing d002

d002 can be found from Bragg's equation:

2d002 sin � = n� n is an integer (8.7)

where � is the Bragg-angle, i.e. the angle of re ection of the X-ray rays scatteredfrom the crystal-lattice. � is the wave-length of the X-rays.

8.7 X-ray Di�raction 91

The Stacking Height Lc

Lc can be found from a measurement of the half maximum intensity breadth(half-peak width) of the (002) re ection and calculation of the Scherrer equation(Scherrer 1918):

Lc = kc�

Bc(2�) cos(�)(8.8)

kc = 2

rln 2

�� 0:94

Bc is the half-peak width of the (002) di�raction line in radians.

The Layer Plane Diameter La

La is obtained from the half-peak width of a (hk) re ection (usually two-dimensionalre ections (10) and (11)) and the two-dimensional lattice equation due to Warren(1941):

La = ka�

Ba(2�) cos(�)(8.9)

ka � 1:94

A comment should be given to the interpretation of the La-value: La obtainedwith this and other related techniques may underestimate the real crystallite sizesince La is a measure of the defect-free crystallite size. Thus, only a small anomalyin an otherwise perfect layer will contribute to lowering the measured La-value.Thus, La gives an estimate of apparent crystallite sizes.

Instrumental Line Broadening

Usually, the half-peak widths Ba and Bc are corrected for instrumental line broad-ening by using quantities �a and �c instead of Ba and Bc in the above equations.�a and �c are calculated from:

�2i = B2i � b2 (8.10)

where i = a; c. b is the instrumental broadening.

Improved Measurement of Crystallite Parameters

A number of constraints have to be ful�lled to be able to calculate correct param-eters with the above equations.

Short & Walker (1963) showed that the equations give satisfactory values only forLc. Correct values of La and d002 was obtained by a more elaborate analysis ofthe intensity pro�le as developed by Short & Walker (1963). According to thisextended analysis, the following structural information was obtained:

92 Carbon Structure and Fundamental Carbon Properties

1. Weight fraction of crystallographically amorphous carbon

2. Weight fraction of single layers

3. d002

4. La (4 values)

5. Lc (2 values)

Obviously, the crystallite population consists of crystallites which can be charac-terized by distributions of parameters (La; d002; Lc). If knowledge of these sizedistributions of the crystallite sizes are needed, even more sophisticated methodsmust be used.

8.7.2 Structural Information Obtained from Line Pro�les

According to Cartz & Hirsch (1960) and Short & Walker (1963), the followingstructural information can be obtained from X-ray di�raction analysis:

1. Weight fraction of crystallographically amorphous carbon

2. Characteristics of the aromatic/graphitic layer planes: �La

(a) Average layer diameter �La

(b) Distribution of layer diameters

(c) Average bond length within the layers

3. Characteristics of the stacking height: �Lc = (�ncr � 1) �d

(a) Average interlayer spacing �d

(b) Variations in the interlayer spacing

(c) Average number �ncr of layers in a stack

(d) The distribution of the number of layers ncr in a stack

4. Texture, homogeneity and porosity

8.7.3 Order Parameters vs. Disorder Parameters

Fitzer et al. (1971) make a distinction between parameters of order and disor-der. The order parameters are used for describing the deviation of the crystallinestructure of carbon residues (after heat treatment) from the ideal graphite lattice.Thus, the order parameters are used for a description of deviation from crystallineorder in a three-dimensional lattice.

In this work, however, a disordered carbon is de�ned as consisting of small basicstructural units (crystallites) which can be classi�ed by the structural parametersd002, La and Lc. Depending on the method of determination, La is an order or

8.7 X-ray Di�raction 93

disorder parameter. d002 and Lc, however, are order parameters and the applica-tion of these parameters for description of structure in disordered carbons may becontroversial.

8.7.4 Limitations of X-ray Di�raction Techniques

The crystallites constitute the di�racting units for the transmitting X-rays. Oneshould be aware that a certain spatial averaging always occurs during the X-raymeasurements since the minimum di�racting volume has a size in the order of0:00013m3. In the same manner, electron- and optical microscopy gives resolutionin the micron range. Only transmission electron microscopy gives structural infor-mation at a level comparable with the size of single crystallites (Oberlin et al. 1986,pp. 86).

94 Carbon Structure and Fundamental Carbon Properties

Chapter 9

Mathematical Modelling of

Structure and Porosity

An approach to modelling of carbon structure and porosity is presented. Thepresentation is based on concepts well established in the carbon literature andfrequently used in the carbon industry. The model for microcrystalline structure isderived within the framework of general population balance equations. Crystallinestructure and porosity belong to the set of fundamental carbon properties (statevariables). State space models for these properties are derived.

9.1 The Porosity Concept

A set of parameters which describe the distribution of pore sizes and shapes isneeded to give a realistic description of the in uence of porosity on various physicalproperties. A vector p

�of such parameters could be de�ned as follows:

p�=

0BBBB@Volume porosity

Pore sizePore wall sizePore shapePore density

1CCCCA

Pores may be classi�ed as macro-, meso- and micropores depending on their sizeaccording to the nomenclature stated by Marsh (1989).

9.1.1 Total Porosity vs. Open and Closed Porosity

In this work, total porosity as divided into open- and closed porosity is used tomodel the impact of porosity on physical properties. Pore shape is not included

96 Mathematical Modelling of Structure and Porosity

BSU

Single layers

Single layers

Fusion

BSUs

Growth

(QI)

macrocrystals

BSUs

of arrangementAnisotropic

Coalescence

Solid Carbon Precursor

Solid (Semi)Coke

carbon phaseDisordered

transferSelective

Nucleation

Ordered

arrangementof BSUs

Isotropic Liquid Mesophase Spheres

of molecules

(QI)

Distillation lossLoss of carbonization gases

Aggregation of

Spheres=

= Bulk Mesophase

(Texture)

carbonization gasesLoss of

carbonization gasesLoss of

carbonization gasesLoss of

BSU

Figure 9.1: Five stages are identi�ed in the pyrolysis process. In the structuralmodel, the basic structural units play a role on every stage in the process. Inthe semicoke stage, the disordered carbon phase consists of single graphitic layerplanes and peripheral groups linked to the these layers. The term (QI) symbolizesthat primary quinoline insolubles often play a role in nucleation and coalescenceof the mesophase.

in the porosity model. Thus:

p�=

����c

�(9.1)

9.1 The Porosity Concept 97

where �� is the open porosity and �c is the closed porosity. The total porosity isde�ned by:

�T = �c + �� (9.2)

The porosities are often related to di�erent measures of density as follows:

�� = 1� �a

�r(9.3)

�0c = 1� �r

�c(9.4)

�a, �r and �c are the apparent- , real- and crystalline densities respectively. Inthe above equations, the open porosity is related to the apparent volume. Theclosed porosity �0c however, is related to the real volume of the carbon material.As shown in Appendix B, the closed porosity �c related to the apparent volumeis given by:

�c =�a

�r� �a

�c(9.5)

The development of �� is related to the evolution of the real and apparent densities.As shown above, porosity is modelled as function of di�erent measures of density.Thus, the porosities are not modelled as state variables; the densities belong tothe set of fundamental properties. Models for the densities are discussed in thefollowing.

As discussed in Gundersen (1996e), shrinkage of carbons is anisotropic (Pratt1958), and the following shrinkage law is used for the apparent density:

d�a

dt= �aka;iso

d�r

dt(9.6)

ka;iso = (1� ��) (9.7)

In Gundersen (1996e) a more general shrinkage law was used to take into accounttwo other e�ects: The pu�ng- and excavation e�ects of the released volatiles.In this case, the pu�ng e�ect is negligible due to the low heating rates mostfrequently used for heat treatment of anode carbon blocks. Also, the excavatione�ect is neglected. In the general case, �a is a time-varying function which has tobe less than one to give a realistic shrinkage behaviour; i.e. �a is always less than�r and �a lags �r in a way such that �� increases. Note that with �a = 1, thelaw accounts for isotropic shrinkage; i.e. �� is constant during the heat treatmentsince �a = (1� ��)�r. In general, one may set:

�a = �a(T; ��) (9.8)

since it is reasonable to assume that the shrinkage process depends on both tem-perature and the current open porosity. In the shrinkage regime during anodebaking, the open porosity increases. This means that �a is less than one. Forsimplicity, assume that �a is a constant parameter between zero and one.

98 Mathematical Modelling of Structure and Porosity

In this work, it is assumed that the same kind of shrinkage law can be used todescribe the evolution of the real density (solid material density level):

d�r

dt= �rkr;iso

d��cdt

(9.9)

kr;iso = (1� �0c) (9.10)

�0c = 1� �r

��c(9.11)

It should be noted that a slight modi�cation of the traditional de�nition of �0c isintroduced here. ��c denotes the density of the solid carbon material within thebulk (apparent) carbon volume: ��c includes the density of the crystalline carbon(crystallites) and the disorganized matter; see Equation (9.31). The disorganizedcarbon is consumed due to the growth of the layer planes and �nally when thedisorganized phase is completely lost, ��c equals �c. At this stage, our de�nition of�0c is in accord with the de�nition in Equation (9.4) usually met in the literature.The de�nition of �c is changed correspondingly such that:

�c =�a

�r� �a

��c(9.12)

Still, �T = �� + �c.

Qualitatively, �r serves the same purpose as �a. Also for �r, a dependence oftemperature and porosity (i.e. closed porosity) is assumed as follows:

�r = �r(T; �0

c) (9.13)

According to Loch & Austin (1956), however, the closed porosity (i.e. the micro-scopic pore volume) decreases during heat treatment. Since the closed porosity iscalculated from (1� �r

��c), �r has to be larger than one to obtain a decreasing closed

porosity. One should be aware that if �r > 1 and constant, there is a risk that�r may become larger than ��c which is a non-physical situation. Thus, is seemsreasonable that �r has a more complicated structure than only being a constant.Here, we assume that when �r approaches ��c, it becomes more an more di�cult forshrinkage to occur. This may be realized by the following mathematical structure:

�r = f1(T ) f2(�0

c) (9.14)

f1(T ) = �r;�(1 + k(T � T�))

f2(�0

c) = (�0c)n�r

which shows that �r goes to zero as �r approaches ��c; the closed porosity approachzero. It is reasonable to assume that f1(T ) increases with temperature; we haveassumed that f1(T ) depends linearly on temperature. In practice, however, aconstant value for f1(T ) was used:

f1(T ) = �r;� (9.15)

Depending on the selected value for n�r , di�erent types of shrinkage pro�les maybe realized. When f1(T ) is a constant, one should intuitively select a value of n�rto avoid that f2(�

0

c) goes too quickly to zero.

9.1 The Porosity Concept 99

One feature of the above expression for �r should be commented. Since the closedporosity should have a continuous decrease within the shrinkage regime, �r mustalways be larger than one as long as �r < ��c. This constraint is in con ict withthe behaviour of �r for small values of �

0

c; then �r goes to zero. Assume now thatf1(T ) = �r;�, and obtain:

�r > 1) �r;�(�0

c)n�r > 1 (9.16)

Substitution of �0c = 1� �r��c

gives:

�r <

1�

�1

�r;�

� 1n�r

!��c (9.17)

As an example. select �r;� = 30 and n�r = 0:5 and obtain 1�

�1

�r;�

� 1n�r

!� 0:9989

Thus, �r may come very close to ��c before �r falls below 1 and consequently �0�

starts to increase. In this way, it seems realistic that the development of closedporosity can be controlled by the use of this kind of expression for the function�r.

Furthermore, the use of a function �r which is less than one may be exploitedin a model for the development of closed porosity during sulphur pu�ng: Closedporosity increases during pu�ng. Then �r should be related to the rate of releaseof sulphur in a way that predicts �r < 1 as long as desulphurization occurs. With�r < 1, the closed porosity will increase.

So far, the shrinkage law for the real density level accounts for changes in real den-sity due to changes in the density of the solid material of the carbon. Such changesare introduced via the consumption of the disorganized carbon phase (polymer-ization, condensation and dehydrogenation). In the limit, when the disorganizedcarbon is consumed, ��c equals �c. Since changes in the interlayer spacing are onlynegligible, there will only be small changes in �c. These changes probably are notlarge enough to introduce further shrinkage in the real volume which leads to alarge enough increase in �r.

Therefore, it is reasonable to assume that also other mechanisms are responsiblefor introducing changes in �r. In this context, we have not considered the im-pact of coalescence of crystallites on the micropore space, i.e. the real density. Itis reasonable to assume that when crystallites coalesce along either the a- or c-directions (or both directions), a certain impact on the volume of micropores mustbe expected: Probably, the volume of micropores is reduced. This is discussedin more detail in Gundersen (1996c). The correlation between real density andcrystallite parameters is frequently expressed in the literature: Measurements ofcrystallite parameters sometimes replaces the measurement of real density. Themodels presented here, gives a more fundamental description of the relationshipbetween quantities describing the granular structure of the carbon material. Fi-nally, a certain pu�ng e�ect from released volatiles (sulphur pu�ng) may also be

100 Mathematical Modelling of Structure and Porosity

present. The impact of pu�ng on porosity was discussed in Gundersen (1996e)and is not repeated here.

In summary, the following processes seem to contribute to changing the microp-orosity within the granular structure:

1. Consumption of the disorganized phase

2. Pu�ng from released volatiles

3. Coalescence of crystallites along the c-axis

4. Coalescence of crystallites along the a-axis

9.1.2 The Crystalline Density

The crystalline density can be found from X-ray measurement of the interlayerspacing. As shown in Figure 8.2, the volume of a small crystallite is given by:

Vc = �

�La

2

�2

Lc (9.18)

where La and Lc are the sizes of the crystallite in the a- and c-directions respec-tively. La is the diameter of the layer planes (graphitic planes). From the �gure,it is also clear that:

Lc = d(nL;cr � 1) (9.19)

where nL;cr is the number of layer planes in the crystallite stack. d is the distancebetween graphitic planes (interlayer distance). By introducing this quantity in theequation for the volume, one obtains:

Vc = �

�La

2

�2

d(nL;cr � 1) (9.20)

where n is the number of layer planes in the crystallite stack. d002 is the distancebetween graphitic planes (interlayer distance). In graphite, d002 = d002;gr = 3:354�A which gives:

Vc;gr = �

�La

2

�2

dgr(nL;cr � 1) (9.21)

Thus:

Vc;gr

Vc=dgr

d(9.22)

At the same time, the following is also valid:

Vc;gr

Vc=

�c

�c;gr

Hence, the �nal relationship for the crystalline density becomes:

�c =dgr

d�c;gr (9.23)

9.1 The Porosity Concept 101

9.1.3 Microporosity and Intercrystalline Porosity vs. Closed

Porosity

Depending on the uid used for measurement of the real density, a certain porevolume is inaccessible for the uid used in the pycnometry (displacement) method(Franklin 1953). This closed pore volume is associated with the intercrystallinepores due to small separations which occur between the crystallites. These poreshave diameter in the order of 10 �A and belong to the micropore range (Loch &Austin 1956).

In addition to these pores, the disorganized carbon phase also occupies spacebetween the crystallites. This is clearly demonstrated in the model of carbonstructure which is applicable to solid coke as presented in the next section.

9.1.4 Macro- and Mesopores vs. Open Porosity

The open porosity as determined from measurement of apparent and real densitiesincludes pores in both the macro- and mesopore range. Also here, the pore volumedepends on the methods used for measurement of density.

9.1.5 The Evolution of Porosity across the Mesophase Tran-

sition

So far, the de�nitions of open- and closed porosity apply to a solid carbon material.

In the general case, when the solid carbon material is formed in liquid phasepyrolysis, swelling of the carbon residue may occur. Here, assume that in theliquid state, �a � �r and that swelling introduces a certain amount of open poresin the carbon residue. Thus, �� > 0 which is equivalent to �a < �r. Thus, the openporosity �� starts on the value zero and settles as a �nite value greater than zeroat the end of the pyrolysis process. This gives a physically reasonable extension ofthe model for open porosity to cover both the low- and high temperature pyrolysisregimes.

For the closed porosity, a quantity ��c is needed such that ��c � �r in the lowtemperature pyrolysis regime since it is reasonable to assume that both �� and�c are zero in the liquid phase before onset of the carbonization reactions whichleads to swelling of the pyrolysing substance and formation of semicoke. It is thenreasonable to assume that:

�0c = 1� �r

~�c(9.24)

where the density ~�c has the following property:

~�c ���r in the isotropic (liquid) phase��c in the anisotropic (semicoke) phase

(9.25)

102 Mathematical Modelling of Structure and Porosity

In this way, �0c = 0 in the initial part of low temperature pyrolysis. A densityfunction with this property is presented in part IV of this work. Recover that thede�nition of �c is modi�ed correspondingly:

�c =�a

�r� �r

~�c(9.26)

Still �T = �� + �c is assumed valid.

9.1.6 Conclusions

In this work, porosity is calculated from measurements of three di�erent densities.This gives rise to two levels of porosity; i.e. the open and closed porosities. Openporosity relates to the macro- and mespores and closed porosity is assumed torepresent the micropore volume.

Appendix B contains a presentation of relationships between densities and porosi-ties which are applicable to solid carbon and carbon particle aggregates.

9.2 A Structural Model of Solid Coke

9.2.1 Turbostratic Carbons: The Bulk Carbon vs. The

Granular Structure

The main features of the turbostratic carbon structure are as follows (Warren1941), (Biscoe & Warren 1942), (Franklin 1950):

� The carbon contains microcrystallites.

� The crystallites consist of successive graphitic layer planes which are orderedapproximately in parallel and reside in equidistant positions.

� The planes are randomly rotated and displaced (in parallel) with respect toeach other.

The crystallites coexist with pores of varying sizes and a certain amount of disor-ganized matter (mainly carbon in aromo-aliphatic compounds) which partly con-stitutes crosslinks between the crystallites. Thus, three di�erent phases coexist inthe carbon:

� Crystallites

� Disorganized matter (crosslinks)

� Pores (of macro-, meso- and micro-size)

9.2 A Structural Model of Solid Coke 103

In Emmerich & Luengo (1993, pp. 333), the bulk carbon matrix of heat treatedcarbons is composed of the crystallites, cross-links, micropores, mesopores andmacropores. Also a small fraction of hydrogen and mineral matter may be aggre-gated to the carbon. The components of crystallites, crosslinks and microporesappear mutually interdispersed within the so-called granular structure. This de�-nition of the granular structure is also used in Emmerich et al. (1987) where thegranular structure is used as basis for calculation of the volume fraction of theconducting phase of the granular structure. In summary, the bulk carbon materialthen consists of two phases (Emmerich & Luengo 1993, p. 336):

� The granular structure

� The open pore space

The open porosity constitutes pores in the macro- and meso- size ranges. Theporosity within the granular structure (i.e. microporosity) corresponds to theclosed porosity. Thus, this view of carbon structure is in accord with the conceptsof open and closed porosity presented in the previous section.

The structural arrangement of the phases within the granular structure is schemat-ically comparable with the structure of coals as presented in Hirsch (1954). Typicalfeatures of the model is shown in Figure 9.2. The granular structure model has pre-viously been used as basis for derivation of mathematical models of mechanical andelectrical carbon properties (Emmerich et al. 1987), (Emmerich & Luengo 1993),(Emmerich 1993), (Emmerich 1995).

In the following, it is assumed that:

� The disorganized phase consists of mainly carbon and a fraction of hydrogenwhich is lost in the carbonization process.

� The crystallites are made of ideal graphitic planes (i.e. pure carbon).

9.2.2 Density and Porosity of the Granular Structure

The relationship between densities, volumes and apparent (bulk) volume Va ofthe granular structure is shown in Figure 9.3. The average density ��c of the non-porous part of the granular structure (i.e. including only the crystallite volumeand the volume of disordered carbon) can be derived from:

mt = mdm +mc (9.27)

by the use of the relationships:

mt = ��c �Vc (9.28)

mdm = �dmVdm (9.29)

mc = �cVc (9.30)

104 Mathematical Modelling of Structure and Porosity

=Crystallite

Single layer

Crosslinks

PoreIntercrystalline

Group of layers

Figure 9.2: The granular structure of disordered carbons. According to the modelof coal structure given in Hirsch (1954, pp. 137), the crosslinks correspond tothe disorganized carbon phase. Ideally, the disorganized phase consists only ofcarbon. On the other hand, a certain fraction of inorganic matter belongs tothe disorganized phase. In this work, we assume that the disorganized phaseconsists of only carbon and a negligible amount of hydrogen. The model of thegranular structure is also used by other authors (Brown & Hirsch 1955), (Cartz &Hirsch 1960), (Diamond & Hirsch 1958), (Diamond 1959), (Diamond 1960), (vanKrevelen 1981, pp. 337), (Emmerich et al. 1987), (Emmerich & Luengo 1993).

m, � and V denote mass, density and volume respectively. Subscripts t, c anddm denote total mass, disorganized matter and crystalline (organized) carbonrespectively. Note that �c and Vc denote density and volume of the dense partof the carbon. As shown in Appendix B, the density of a solid carbon compositewhich consists of organized and disorganized phases is given by:

��c =1

xcr�c

+ xdm�dm

(9.31)

�dm represents the density of a dense phase of disorganized carbon and in thesimplest case, �dm may be assigned a constant value. These assumptions may becritized as follows:

� As the disorganized phase is consumed (xdm decreases), it is reasonable that�dm also changes. In a more detailed approach, �dm will be modelled as akind of thermally activated process.

9.2 A Structural Model of Solid Coke 105

CarbonBulk

StructureGranular

mc

�c

�o

Va

Vr

�a

�r

mdm

�dmVdm

mc

�cVc

Figure 9.3: The relationship between densities and volumes in the bulk carbon.Here, the bulk carbon consists of open pores and the granular structure. Withinthe granular structure, micropores, disorganized carbon and crystallites coexist.Thus, the granular structure corresponds to the real volume as de�ned in this work.Except from crosslinks and peripheral groups, the disorganized carbon phase alsoincludes single layers. Crystallite growth partly occurs on the expense of thedisorganized carbon phase.

� If a low value of �dm is used, one may argue that this amounts to the existenceof a separate pore space within the disorganized pore space:

�dm = 1� �dm

~�(9.32)

Thus, �dm is the local pore space within the disorganized carbon material.~� is the density of the dense disorganized phase, i.e. the disorganized phasewithout pores. As an approximation, assume that ~� � �c:

�dm � 1� �dm

�c(9.33)

�dm may contribute to extending the micropore space �0c already de�nedwithin the real volume. If mapped over to the real volume range, the poros-ity �dm amounts to a microporosity of �0dm �dm (an expression for �0dm isgiven below Equation (9.42)). The total microporosity �c;T of the granular

106 Mathematical Modelling of Structure and Porosity

structure is then:

�0c;T � �0c + xdm�r

��c(1� �dm

�c) (9.34)

Since �0dm is directly proportional with xdm, the role of the additional micro-porosity becomes very low when xdm approaches zero. Also, since a part ofthe disorganized phase contributes to the microporosity, the volume fraction�0dm has to be reduced with a value corresponding to the porosity :

�0dm ! �0dm � xdm�r

��c(1� �dm

�c) = xdm

�r

��c

�dm

�c(9.35)

In this work, �0dm �dm is not included in the microporosity space since it is assumedthat this void space does not belong to the porosity concept. Rather, this voidspace is ranked in the same category as the space between layers of the crystallitephase. �dm should be adjusted to achieve a qualitatively reasonable behaviour ofthe microporosity �0c.

In the expression for ��c, xcr and xdm denote mass fractions of organized carbon (i.e.crystallites) and disorganized matter respectively. Then, the following relationshipis valid for the mass fractions:

1 = xcr + xdm (9.36)

In the carbon material, three phases of porosity, disorganized matter and crys-talline carbon coexist. The relationship between volume fractions is given from:

�dm + �c + (�� + �c) = 1 (9.37)

where the terms on the left hand side of the equation represent volume fraction ofdisorganized matter, organized carbon, open porosity and closed porosity respec-tively. Earlier, it was found that:

�� = 1� �a

�r

The following relationship can also be used:

�c =�a

�r� �a

��c(9.38)

��c is the density of the solid phase of the bulk carbon material consisting of orderedcarbon and disordered matter. The volume fractions in the real volume domainare related by:

�0c + �0dm + �0c = 1 (9.39)

�0c, �0

dm are volume fractions of the organized and disorganized phases within thereal volume of the carbon. �0c is the closed porosity as related to the real volumeof the carbon:

�0c = 1� �r

��c(9.40)

9.2 A Structural Model of Solid Coke 107

The solid material constitutes a volume fraction of �r��c

of the granular structure.Then:

�0c = xcr�r

��c(9.41)

�0dm = xdm�r

��c(9.42)

By summing up the last three equations, one can see that Equation (9.39) isful�lled.

As the disorganized carbon phase is consumed in the crystallite growth process,Equation (9.31) shows that ��c approaches �c, the crystalline density. Thus, af-ter complete consumption of the disorganised carbon ��c equals �c; the value ofcrystalline density commonly used in the literature.

In order to calculate the closed porosity (�c, �0

c) an estimate of xdm (or xcr) isneeded for evaluation of the expression for ��c. An expression for xcr is found bymodelling the mass balances for the solid phases present in the carbon:

dmdm

dt= �rdmVa

Since rdm = rc + rv , one obtains:

dmdm

dt= �(rv + rc)Va (9.43)

dmc

dt= rcVa (9.44)

dmt

dt= �rvVa (9.45)

This result can be used to obtain:

dxdm

dt= � 1

�a((1� xdm)rv + rc) (9.46)

In this equation, mdm = xdmmt has been used. Alternatively, an equation for themass fraction of crystalites can be found:

dxcr

dt=

1

�a(xcrrv + rc) (9.47)

by usingmcr = xcrmt. rv is the rate of loss of carbonization gases and rc is the rateof mass transfer between the disorganized matter and the organized (crystalline)phase of carbon. Since xcr = 1� xdm, a separate state equation for the organizedcarbon material is not needed.

In the solid coke, texture is already established as a result of processes whichoccur during the mesophase transition. Thus, no state variable for carbon textureis needed. The structure is then described by the crystallite parameters. Thedensities are needed to obtain measures of the porosities. The crystalline densitycan be calculated from the interlayer spacing. Models for the real- and apparent

108 Mathematical Modelling of Structure and Porosity

densities are needed to describe the shrinkage occurring during carbonization inthe solid state. The density of the disorganized carbon phase can be consideredto be a parameter (of constant value) in the model. At this stage, the hydrogenstill left in the carbon residue is not explicitly taken into account: For simplicity,the mass consists of carbon either in the organized or disorganized phase. A ratelaw must take care of the transfer of disorganized carbon to the organized phasedue to crystallite growth.

From the total mass balance, an equation for the apparent volume can be foundsince mt = �aVa:

dVa

dt= �Va

�a(rv +

d�a

dt) (9.48)

9.2.3 The Relationship Between Volatile Loss, xdm and �dm

Valuable insight into the relationship between the quantities describing the gran-ular structure was obtained in simulations of the state space model given in Sub-section 9.2.4. The model was used for prediction of the development of Young'smodulus Y of a soft carbon according to a model presented in Emmerich & Luengo(1993) and Emmerich (1995):

Y � f1(L) f2(X) f3(Lc; d) (9.49)

f1(L) =1

L2(9.50)

L =

�

�La

2

�2

Lc

! 13

(9.51)

f2(X) =X2=3

1�X1=3(9.52)

X = (1� xdm)�r

�c(9.53)

f3(Lc; d) = (Lc

d+ 1) (9.54)

The development of Y as a function of heat treatment temperature depends on therelationship between the functions fi(�) shown in Figure 9.4. The product f1 f3contributes to increasing Y whereas f2 gives a decrease in Y . At temperatures inthe order of 1000�C, there is a change in dominance of these two factors whichcauses a maximum in Young's modulus. In the �rst attempt, an e�ort was doneto realize this behaviour in Young's modulus in the following way:

� As shown in Subsection 9.3.7, the growth of La is related to the consumptionof the disorganized carbon phase as represented by mass fraction xdm. Inthis growth process, it was assumed that the number nL of layer planes isconserved.

� Since �dm was selected as a constant parameter, the kinetics of �r also de-pends on the consumption of disorganized carbon since d�r

dt� d��c

dtwhere

9.2 A Structural Model of Solid Coke 109

d��cdt

is positive when xdm decreases. In this discussion, this contribution isomitted due to the decrease in d which occurs in parallel with the decrease inxdm. In fact, when the decrease in d is also taken into account, d��c

dtbecomes

more positive.

� For Lc, a model with a positive value fordLcdt

was used; i.e. the model did nottake into account the decrease in Lc which occurs between 400 and 800�C.

� In the same way as for Lc, a model for d with a negative value ford(d)dt was

used; i.e. the model did not take into account the increase in d occurringbetween 400 and 800�C.

500 1000 1500

500 1000 1500

500 1000 1500500 1000 1500 T �C

T �C

T �CT �C

f2(X)f1(L)

Y

f3(Lc; d)

Figure 9.4: Steady state values of Young's modulus as function of heat treatmenttemperature. It is quite common in the carbon literature to plot properties asfunction of the heat treatment temperature. Heat treatment at the given temper-ature must then take place over a time interval long enough for the property tosettle at a stationary value.

In this case, the increase in f2 and the decrease in f1 is mainly related to thesame phenomenon (i.e. the consumption of the disorganized phase) and it was notpossible to obtain a realistic evolution of Y as function of time and heat treatmenttemperature. This is due to the strong coupling between xdm and La: Whenxdm starts to decrease, f2 starts to increase. As a consequence of the decreasein xdm (disorganized carbon is consumed), La starts to increase which leads toa corresponding decrease in f1(L). This occurs in a way such that the productf1 f2 is approximately constant and then starts to decrease. At the same time, theincrease in f3 is not able to compensate for the decrease in f1 f2.

An apparently correct progression of Young's modulus could be achieved by tuningthe kinetics of xdm so that xdm starts to decrease at temperatures in the order of800 to 900�C. Then f3 gives an increase in Y , before the product f1 f2 introducesa decrease at temperatures in the order of 1000�C. Most probably, however, xdmstarts to decrease even at lower temperatures than 800 to 900�C. Therefore, thissolution to the problem was not found satisfactory. An alternative solution wasused to improve the situation:

110 Mathematical Modelling of Structure and Porosity

� It was thought that the density �dm of the disorganized matter (i.e. mainlycarbon) depends on the weight loss of non-condensables between 400 and1000�C:

{ �dm increases as the conversion of non-condensables goes by. The fol-lowing model was used for �dm:

�dm = �dm;�(1� ~X) + �dm;f~X; �dm;� < �dm;f (9.55)

~X =p

2XCH4

+ (1� p

2)XH2

(9.56)

p is a tuning parameter with nominal value p = 1.

{ XCH4and XH2

are conversion parameters for methane and hydrogenwhich increases from 0 to 1 as conversion is completed (between 400and 800�C for methane and 400 and 1000�C for hydrogen). Traditionalthermogravimetric models as reviewed in part III were used to modelXCH4

and XH2.

� Also the total weight loss of non-condensables is very much lower than thetotal mass transfer from the disorganized phase to the crystalline phase.Thus, the modelling of the weight loss of non-condensables can be decoupledfrom the main kinetics of xdm: The mass released as non-condensables isneglected in the model for xdm. In this way, the kinetics for xdm can beslightly delayed as compared to the kinetics of XCH4

and XH2. This makes

it possible to obtain an increase in X via �r (see Equation (9.53)) before theconsumption of the disorganized phase sets in (xdm decreases, La increasesand there is a further increase occurring inX as a consequence of the decreasein xdm).

� In the model for Lc, the decrease in Lc between temperatures of 400 and800�C was implemented by a model with two parallel reactions with conver-sion dependent activation energy. This gives a delayed decrease in f1 as wellas a delayed increase in f3. On the other hand, the maximum in Y becomeshigher with a model for Lc which do not take into account the decrease inLc between 400 and 800�C.

� So far, the increase in d between 400 and 800�C was not implemented.

By using this model, a distinct maximum in Y at a temperature in the order of800�C was achieved by careful tuning of the model parameters: The decrease inxdm and corresponding increase in La starts at approximately 700�C. If realisticdata were available, one could probably obtain an even better performance of themodel for Young's modulus. Nominally, the maximum in Young's modulus occurat higher temperatures than 800�C; typically a maximum is observed between 900and 1200�C.

9.2 A Structural Model of Solid Coke 111

9.2.4 State Space Vector for Carbon Porosity and Microstruc-

ture

A general state space model for description of microstructure and porosity devel-opment includes the following state variables:

x =

26666666666664

x1x2x3x4x5x6x7x8x9

37777777777775=

26666666666664

d

LanL;cr�r�axdmXCH4

XH2

cy

37777777777775=

0BBBBBBBBBBBB@

Interlayer spacingLayer diameter

Average number of layers in a crystalliteReal density

Apparent densityWeight fraction of disorganized matter

Conversion of methaneConversion of hydrogen

Coke yield

1CCCCCCCCCCCCA(9.57)

In this case, the coke yield is calculated from a total mass balance on the pyrolysingmaterial by taking into account the loss of methane and hydrogen. Alternatively, itis possible to formulate a di�erential equation for Va (the apparent volume) fromthe total mass balance. In another formulation of the state space vector, nL;crcould be replaced with Lc. Assume that the temperature is a control variable; inother cases it may be reasonable to assume that temperature also belongs to thestate vector.

Often, the following measurements are of interest:

y =

2666666664

y1y2y3y4y5y6y7

3777777775=

2666666664

�a�r

(1� �a�r)

(�a�r� �a

��c)

d(nL;cr � 1)cyVa

3777777775=

2666666664

x5x4

(1� x5x4)

(x5x4� xa

��c)

x1(x3 � 1)x9

cym(0)

x5

3777777775=

0BBBBBBBB@

Apparent densityReal densityOpen porosityClosed porosityCrystallite height

Coke yieldApparent volume

1CCCCCCCCA(9.58)

The following quantities are also needed:

��c =1

(1�x6)�c

+ x6�dm

(9.59)

�c =x1

dgr�gr (9.60)

�dm = �dm;�(1� ~X) + �dm;f~X (9.61)

~X =~p

2x7 + (1� ~p

2)x8 (9.62)

dgr and �gr are interlayer spacing and density of graphite. �dm is related to theconversion of non-condensables; i.e. hydrogen and methane. In the compoundconversion parameter ~X , p is a parameter with value in the order of 1.0.

112 Mathematical Modelling of Structure and Porosity

9.3 Modelling Structural Evolution During Car-

bonization

Carbon structure develops within two regimes during carbonization:

� The formation of carbon texture during the mesophase transition in theliquid phase regime up to temperatures in the order of 500�C.

� The evolution of crystallite parameters within the texture domains in thesolid state regime above temperatures in the order of 500�C.

A model for the growth of crystallites within the framework of the granular struc-ture model of solid carbons is presented. The granular structure model is assumedto be a valid description of carbon microstructure in the temperature interval be-tween approximately 450 and 1500�C. To develop the crystallite growth model,knowledge of the mechanisms for crystallite growth is needed. The model is de-veloped with basis in the principles of a general population balance.

The main purpose of this subsection, is to show that the process of crystallitesize growth can be described by well known modelling concepts. In Gundersen(1996f), models for development of both carbon texture and microcrystallites arepresented. Texture models are not discussed in this work.

9.3.1 The General Population Balance Equation

The general population balance equation for the number density function �(xe; xi; t)for countable entities in a microscopic volume often appears in the following form(Himmelblau & Bischo� 1968, pp. 67), (Randolph 1964), (Hulburt & Katz 1964):

@�

@t+r(ve�) +r(vi�) = B �D (9.63)

xe and xi denote spatial (i.e. external) and property (i.e. internal) coordinatesrespectively, which is used to describe the state of an entity in position (xe; xi) inthe so-called phase space (Hulburt & Katz 1964, pp. 557). Furthermore:

ve =dxedt

(9.64)

vi =dxidt

(9.65)

Thus, ve is the geometric velocity vector and vi is the kinetics of the propertyspace.

Consider a control volume V (of de�nite size) in which � is uniform in space. Thecontrol volume is allowed to vary in size. A certain amount of mass is allowedto leave the control volume (i.e. escape of gaseous volatiles) but no particulatematter is included in this mass loss.

9.3 Modelling Structural Evolution During Carbonization 113

The population balance equation for such a macroscopic control volume can bederived by integration of Equation (9.63) over volume V :Z

V

@�

@tdV +

ZV

r(ve�)dV +

ZV

r(vi�)dV =

ZV

(B �D)dV (9.66)

Himmelblau & Bischo� (1968, pp. 67-68) developed the �rst term in Equa-tion (9.66) by using Leibnitz' rule for di�erentiating de�nite integrals:Z

V

@�

@tdV =

@

@t

ZV

�dV +

��ZV

r(ve�)dV�

(9.67)

By following the derivation due to Himmelblau & Bischo� (1968, pp. 67)1, it canbe shown that the sum of the second term in the above equation and the termRVr(ve�)dV in Equation (9.66) equals zero since there is no mass ow of entities

across the system boundary:

@(�V )

@t+

ZV

r(vi�)dV =

ZV

(B �D)dV (9.68)

Finally, r(vi�), B and D are uniform in space which gives:

@(�V )

@t+ Vr(vi�) = (B �D)V (9.69)

Resolution the time derivative gives the following equation:

@�

@t+

1

V

@V

@t� +r(vi�) = B �D (9.70)

This equation gives a �rst principles description of crystallite growth in solid car-bons.

9.3.2 Modelling The Evolution of Physical Properties Across

the Mesophase Transition

During pyrolysis, both chemical and physical processes contribute to change theaverage physical properties of the pyrolysing material. As an example, the deriva-tion of a model for evolution of the crystallite parameter La is presented.

Changes in La are due to the following processes:

1. Devolatilization of light components from the liquid pitch

2. Polymerization reactions in the liquid pitch

3. Continued polymerization reactions/ consumption of the disorganized car-bon phase within the solidi�ed pitch residue

1Gauss' divergence theorem is applied.

114 Mathematical Modelling of Structure and Porosity

As shown in Figure 9.5, development of La can be considered to go via threeprocess regimes:

� Solid/ uid pitch

� Mesophase transition

� Solidi�ed pitch residue

Solid pitchresidueLiquid pitch

Regime 1 Regime 3Regime 2

Mesophase

400 T [�C]

Figure 9.5: Three sequential regimes in the pyrolysis of pitch. The transition fromliquid pitch to solid coke is a dynamic process with activation energy such thatthe transition occurs nominally as temperature approaches 400�C.

The modelling of La as it develops across these regimes is not a straight forwardtask. Johansen (1994) has designed methods for development of models based onmultiple regimes. In this case, however, the modelling task is even more compli-cated:

� Most possible, the structure of the model changes from one regime to thenext.

� There is a sequential dependence of the states in one regime on the statevariables in the previous regime.

Thus, the use of a regime-based modelling approach is not straight forward in thiscase. The formally correct approach for modelling the evolution of La would be togo via a population balance approach. Still, however, it must be assumed that thestructure of the model depends on the process regime that currently is operative.

9.3 Modelling Structural Evolution During Carbonization 115

In the population balance approach, however, it is easier to implement a structuraldependence of the rate laws by assuming that rate laws (i.e. activation energy)may depend on the value of La as follows:

� The ability for a certain molecule to devolatilize depends on its size (La) andthe concentration of the molecules of this size.

� The polymerization activity of the molecules in the pitch is taken care of byassuming that only binary coalescence of molecules is possible.

� The growth rate of a certain layer plane in the solid residue depends on theamount of disorganized carbon present in the residue and the size of thelayer plane.

Provided that the rate laws which correspond to the above processes can be found,the population balance equation gives a systematic approach to model the devel-opment of La. At this stage, however, no e�ort will be given to specify the math-ematical structure of the required rate laws. The same kind of approach can beused if the property space is extended to include also d and Lc.

If the development of the parameters is modelled by a lumped approach and si-multaneously take into account the presence of two pitch phases (the isotropic andanisotropic phases), a problem is encountered that is di�cult to resolve theoreti-cally:

� The evolution of La;i has impact on the value of La;c.

The coupling between La;i and La;c is di�cult to implement. A simple ad-hocapproach was chosen elsewhere in this work to resolve the problem:

� Integration of La;i and La;c starts simultaneously at time zero.

� The initial value for La;c is selected at time zero, and no correction in La;cdue to changes in La;i is taken into account.

An alternative approach was suggested by Johansen (1996) as described in Gun-dersen (1996f). Details on this approach is not repeated here.

9.3.3 Mechanisms of Crystallite Growth in Solid State Car-

bonization

Up to temperatures in the order of 500�C, the texture of a soft carbon material isformed during the mesophase transition which takes place in the liquid precursor.At higher temperatures, growth processes take place on the crystallite level but theorientation of macrocrystals obtained in the liquid stage is generally maintained.

In Figure 9.6, the qualitative evolution of the crystallite parameters during py-rolysis is shown. The explanation of the observed rise and fall in Lc and the

116 Mathematical Modelling of Structure and Porosity

corresponding fall and rise in d002 for temperatures between 400 and 900�C iscontroversial. Two theories seem to exist:

1. The phenomenon is due to structural disorder introduced during solidi�-cation of the mesophase as well as crosslinking and microcracking of thesemi-coke.

2. The fall in Lc (and rise in d) is due to the appearance of many small crys-tallites which contributes to reducing the observed average value of Lc.

It is agreed upon that the initial rise (and corresponding fall in d002) is attributedto the ordering processes which occur during the mesophase stage.

The �rst theory is supported by experiments described in Diamond & Hirsch(1958), Diamond (1959), Diamond (1960), Marsh & Stadler (1967), Marsh (1973),H�uttinger (1971) and Auguie et al. (1980).

According to Diamond, growth of the layer size (on the expense of disorganizedcarbon) may lead to interaction between the layers (buckling, changes in orien-tation) which causes a reduced value of Lc. Above 700�C, the layers are largeenough to allow for forces of alignment to turn the layers into parallelism. Theinterlayer spacing increases due to crosslinks preventing the layers from being asclose as possible. Marsh, however, attributes the reduced structural order in Lcas a consequence of layer disruption due to gas release; a kind of pu�ng e�ect(Marsh 1973), as well as the fact that crosslinking may tear layers out of align-ment (Marsh & Stadler 1967). H�uttinger (1971) argues that resolidi�cation andaccompanied microcracking leads to reduced values of Lc. Auguie et al. (1980)also observed reduced structural order across the stage of resolidi�cation.

In support of the second alternative, Whittaker et al. (1970) and Kocaefe et al.(1993) argue that a large number of small undetectable crystallites are presentin the green coke. As crystallite growth (i.e. coalescence in the c-direction) setsin, they become large enough to be detected by the X-rays. In this way, theycontribute to reducing the observed mean value of Lc. Turner (1995) in additionargues that a relatively rapid shrinkage of the carbonized sample also contributes toexplaining the reduced value in Lc. This argument indirectly allows for crystallitefracturing as a means to explain the reduced Lc value partly invalidating theprevious postulate that no loss in structural order has occurred.

In summary, the �rst hypothesis assumes that disordering processes is responsi-ble for a certain disintegration of crystallites to increase the number of smallercrystallites. In the other hypothesis, it is suggested that many small undetectablecrystallites suddenly become large enough to be detected.

To draw a conclusion, it seems most likely that the �rst hypothesis is correct: Crys-tallites may fracture in the temperature range between 500 and 800�C. Thus, theforces of attraction between layers are not strong enough to resist the disorderinge�ects due to:

� The release of carbonization gases (pu�ng e�ect)

9.3 Modelling Structural Evolution During Carbonization 117

� The drag introduced between layers via crosslinks

� The increase in layer diameter due to growth of the layers

� The formation of microcracks during resolidi�cation and subsequent shrink-age (observe the behaviour of real density in Wilkening (1983, Fig. 11))

At temperatures in the order of 700 to 800�C, the layers in average become largeenough to overcome the above e�ects. Even more important, the release of car-bonization gases is about with ceasing. Thus, no more pu�ng from released gasesmay occur.

On the other hand, to be in accord with the most recent speculations, it is reason-able to adopt the second hypothesis. By accepting this hypothesis, the di�erentmodes of crystallite behaviour during carbonization is simpli�ed since disintegra-tion of crystallites is not allowed during carbonization. Thus, a single crystalliteis only allowed to grow in size as observed by a steady increase in La and Lc. Atthe same time, there is a steady reduction in d002 due to increased forces of at-traction between layers of growing size as well as mutual ordering processes withinthe crystallites (i.e. layer planes become more and more oriented with respect toeach other). This ideal behaviour of a single crystallite is shown in Figure 9.7.

In conclusion, the curves for the crystallite parameters as shown in Figure 9.6 isthe e�ect of an averaging process in which the appearance of many small crystal-lites reduces the observed values of Lc between 500 and 800�C (correspondinglyfor d002). Furthermore, the appearance of many small crystallites can only be dueto the mutual coalescence of many single layer planes to form crystallites which onthe average are smaller than the ones already existing in the material. Previously,a simpli�cation was introduced by assigning both crystallographic amorphous car-bon and single layers to the same phase (disordered matter) and coalescence ofsingle layers was not taken into account.

Having concluded that crystallites only are able to increase their size, rules arenow needed to describe the mutual interaction between crystallites as well astheir interaction with the disorganized carbon phase2. Signi�cant insight intocarbon structure and its impact on crystallite growth was gained by Franklin(1951b): Solid carbon consists of crystallites (more or less oriented) surroundedby a non-organized phase acting as crosslinks between the crystallites. The degreeof orientation of crystallites and the strength of crosslinks lead to the divisionof carbons into graphitizing (soft) and non-graphitizing (hard): The soft carbonhave nearly parallel-oriented crystallites (in the c-direction) with weak crosslinksbetween them which is a prerequisite for easy development of the crystalline orderduring heat treatment (van Krevelen 1981, pp. 340).

Franklin proposed a mechanism of crystallite growth as follows:

� It was found that at temperatures below 1300�C, the increase in layer sizeoccurs at the expense of the non-organized carbon phase. The non-organized

2Franklin (1951a) Mrozowski (1956a, pp. 36-39), Mrozowski (1958, pp. 8), Mizushima (1960),Akamatu & Koruda (1960, pp. 355), van Krevelen (1981, pp. 340), Oberlin (1984), Oberlin et al.(1986), Emmerich et al. (1987), Emmerich (1993) and Emmerich (1994).

118 Mathematical Modelling of Structure and Porosity

20

30

10

1

2

5

3

4

10

20

30

200 400 600 800 1000

3.4

3.5

3.6

3.7

200 400 600 800 1000

200 400 600 800 1000 200 400 600 800 1000

nc[1]

d002[�A]

La[�A]

T [�C]

Lc[�A]

T [�C]

T [�C]

T [�C]

Figure 9.6: Qualitative evolution of average crystallite parameters during pyroly-sis. This trend in development of crystallite parameters has been observed by sev-eral authors (Diamond & Hirsch 1958), (Diamond 1959), (Diamond 1960), (Marsh& Stadler 1967), (Marsh 1973), (H�uttinger 1971), (Auguie et al. 1980), (Whittakeret al. 1970), (Kocaefe et al. 1993), (Turner 1995).

phase is attached to edge atoms of the layer planes to form crosslinks betweenneighbouring layer planes. In this temperature interval, neither Lc nor Laincrease very much (typically they approach values in the order of 30 to 35�A).

� After consumption of the disorganized carbon phase is completed, furthercrystallite growth takes place by the gradual movement of whole layer planesor groups of layer planes (i.e. crystallites). Typically this takes place attemperatures between 1300 and 1500�C. The parallelism of crystallites al-lows for easy coalescence of the crystallites. At this stage, both La and Lcgrow fast but Lc faster than La. The more rapid growth in Lc as comparedto La is due to the higher probability of a migrating layer plane to �x at thebasal plane of a crystallite than to the edge atoms of a layer plane.

Suggested mechanisms for crystallite growth presented after Franklin's contribu-

9.3 Modelling Structural Evolution During Carbonization 119

1

2

5

3

4

10

20

30

20

30

10

200 400 600 800 1000

3.4

3.5

3.6

3.7

200 400 600 800 1000

200 400 600 800 1000 200 400 600 800 1000

nc[1]

La[�A]

d002[�A]

Lc[�A]

T [�C]

T [�C]

T [�C]

T [�C]

Figure 9.7: Development of structural parameters that characterize the behaviourof a single crystallite during carbonization as function of heat treatment tem-perature. The vertical arrows denote that several initial values of the curves arepossible. The unchanged sign of the derivatives is an assumed feature of the curves.The dotted lines in the panels for Lc and d002 denote the loss in structure due tofusion of the carbon precursor.

tion is usually based on the original ideas due to Franklin.

Mrozowski agrees that crystallite growth up to temperatures in the order of 1300�Coccurs at the expense of the disorganized phase. At temperatures above 1300�C,small and less advantageously crystallites are absorbed by neighbouring crystal-lites. In this temperature interval, crystallite size grows much faster than at tem-peratures below 1300�C. Mrozowski argues that at this stage, thermal activationis not the only driving force: The main process responsible for the growth isthe movement of layers to relieve thermal stresses (due to thermal expansion) inthe carbon material. In this way, a limiting value in crystallite size seems to beachieved at a certain temperature if the holding (soaking) time is large enough.Thus, the main driving force consumes itself in the process.

Also Akamatu & Koruda (1962) adhere to the above stated assumptions of mech-anisms in crystallite growth. In addition, Akamatu states that crystallite growth

120 Mathematical Modelling of Structure and Porosity

is relatively more dependent on the maximum heat treatment temperature thanon the residence time. Akamatu compares carbonization and graphitization tooccur more like phase change processes rather than as rate processes due to thevery rapid change in crystallite size that occurs after a steep increase in the heattreatment temperature. On the other hand, several authors have stressed the im-portance of residence time as a parameter in crystallite growth ((Fischbach 1971),(Fair & Collins 1962), (Mizushima 1963), (Foosn�s, Kulset, Linga, N�umann &Werge-Olsen 1995) etc.). Crystallite growth is indeed a rate process.

Finally, Mizushima (1960) presents a modi�cation of the structural perception of asolid carbon material (see Figure 9.8). Franklin's independent crystallite conceptwas abandoned. Instead it was suggested that molecules (layers) in nearly thesame plane in the carbon connect with each other to form extensive sheets whichmay be at or twisted, bent or in other ways incomplete. Structurally, the carbonconsists of a nearly parallel packing of such sheets with the disorganized phaseand pores distributed within this arrangement.

Also in this structure, growth of the layer planes �rstly occurs at the expenseof the disorganized phase. Accompanying this process, layers in suitable relativeorientations may union.

After consumption of the disorganized phase, coalescence of adjoining layers mayoccur. In this regime, the layers are not too big and there is space surroundingthe layers that is large enough to allow for rotation of the planes into suitableorientations with respect to neighbouring layer planes. This kind of union amonglayer edges seems to take place below 1500�C. The apparent growth in crystalliteheight is a direct consequence of the union of the layer planes since the layers tendto unite within the same sheet which contributes to the removal of irregularitieswithin the sheet pile.

At approximately 1500�C, hydrogen is completely removed from the carbon, andcovalent bonds exist between all layers in a sheet. Then crystallite growth hasto occur by an atomic process. Also, when the layers become large enough, rota-tion cannot easily occur. These facts call for growth occurring in a new regime.Mizushima suggested that at this stage, the increase in layer diameter occursprincipally by atomic displacement of the boundaries between neighbouring lay-ers. Again, the growth in crystallite height is a consequence of the two-dimensionalgrowth which contributes to straightening out the sheets. Furthermore, thermalexpansion leads to stress which also is relieved by straightening out the sheets. Ata certain stage, these two e�ects' impact on the rate of growth in the c-directionbecome less e�ective and the rate of growth in the c-direction is smaller than thegrowth in the a-direction.

In addition, Mizushima presents an energetic explanation of why extensive threedimensional ordering occurs after the layer diameter has reached a size in theorder of 150 �A. At this layer size, the energy release due to 2D-growth equals andbecomes less than the energy release in 3D-ordering.

Emmerich did not allow union (coalescence) of layer planes for heat treatment ofsoft carbons at temperatures below 1400�C since coalescence along the a-direction

9.3 Modelling Structural Evolution During Carbonization 121

has a much higher activation energy than coalescence in the c-direction (Emmerich1994), (Emmerich et al. 1987). In this regime, the number of layer planes isconserved but the number of crystallites decreases due to the coalescence in thec-direction.

Franklin Mizushima

"Crystallite"

Figure 9.8: Two views of carbon structure: Franklin's independent crystalliteconcept and Mizushima's parallel sheet concept.

Franklin's concept of independent crystallites is well suited as basis for mathemati-cal modelling of the crystallite growth process. On the other hand, the mechanismdue to Mizushima must be considered to give a more realistic description of carbonstructure and its modi�cation during heat treatment. Based on the above review,the following simpli�ed mechanism for crystallite growth is adopted:

1. Temperatures below 1400�C:

� La increases mainly due to consumption of the disorganized phase. Thenumber of layer planes is conserved3.

� Lc increases due to coalescence in the c-direction. This includes alsoassociation of single (disorganized) layers to existing crystallites.

2. Temperatures above 1400�C:

� La and Lc increase due to coalescence of crystallites along both the c-and a- directions.

During baking and calcination, crystallite growth occurs mainly according to themechanism in the �rst regime.

9.3.4 The Crystallite Growth Model

Based on the crystallite growth mechanism and the general population balanceequation, a model for crystallite growth in solid coke (initially heat treated to

3On the other hand, simulations of a simpli�ed model for development of microstructure incarbons has shown that it is reasonable to assume that a certain degree of reduction in thenumber of layer planes also takes place below 1400�C.

122 Mathematical Modelling of Structure and Porosity

temperatures in the order of 500�C) is derived:

@�

@t+

1

Va

@Va

@t� +

@

@La(vLa�) +

@

@Lc(vLc�) +

@

@d(vd�) = (B �D) (9.71)

vi denotes the growth rates for the crystallite parameters. Shrinkage of the ap-parent (bulk) volume is taken into account by the term 1

Va

@Va@t� which shows that

shrinkage (rate of change of apparent volume is negative) leads to an increase inthe value of @�

@t .

To solve the equation, an initial condition for � and boundary conditions for theproperties are needed for evaluation of the partial derivatives of the properties.Theoretically, La may vary from zero to in�nity. Practically, however, the lowestvalue of La corresponds to a single benzene ring. In crystalline perfect carbonmaterials both La and Lc may become very large. According to our de�nitionof crystallite size, a crystallite consists of at least two layer planes. Then theminimum value in Lc corresponds to the lowest value possible in the interlayerspacing, i.e. 3.44 �A in turbostratic carbon. Finally, values of d larger than 3.75 �Aare rarely measured.

In summary:

La 2 [7;1) �A (9.72)

Lc 2 [3:44;1) �A (9.73)

d 2 [3:44; 3:75) �A (9.74)

�(La; Lc; d; 0) = �� (9.75)

limLa!0;1

@�

@La

����(La;Lc;d)

= 0 (9.76)

limLc!0;1

@�

@Lc

����(La;Lc;d)

= 0 (9.77)

limd!3:44;3:75

@�

@(d)

����(La;Lc;d)

= 0 (9.78)

Growth in La proceeds by both consumption of the disorganized matter as well ascoalescence of neighbouring layer planes (or crystallites). Here, however, growthonly due to consumption of the disorganized phase is considered. Actually, thereis a gradual transition from this growth regime into the regime with growth dueto coalescence.

Birth- and Death Rates B and D

At �rst, it is assumed that there is no death (and corresponding birth) of crystal-lites due to rupture of larger crystallites. Secondly, pure nucleation of crystallitesdoes either not occur. On the other hand, one may argue that nucleation mayoccur as a consequence of coalescence of single layers to form a simple crystalliteconsisting of only two layer planes. It is reasonable to assume that the mass frac-tion of single layers is low enough to be assigned to a lump denoted disorganized

9.3 Modelling Structural Evolution During Carbonization 123

matter (Gundersen 1996f). Therefore, birth of crystallites due to nucleation is notconsidered.

Birth of crystallites is only due to coalescence along the c-direction and at temper-atures above approximately 1400�C also along the a-direction. The correspondingdeath rates must also be accounted for. Now, coalescence exclusively along thec-axis is considered as described in the paragraph on vLc given below.

Growth Rate vLa

In crystallization processes, it is commonly assumed that the crystallite size growthrate depends on both the concentration of the solvent and the crystallite size. Acorresponding relationship will be assumed here:

vLa = f1;La(T; �dm;a)f2;La(La) (9.79)

f1;La(�dm) = kLa(T )�dm;a (9.80)

kLa(T ) = kLa;�e�

ERT (9.81)

�dm;a = xdm�a (9.82)

Here, �dm;a is the (apparent) density of disorganized carbon within the bulk carbonvolume. Thermal activation is needed to be able to consume the disorganizedphase. A �rst order proportionality to the bulk density of disorganized matteris assumed. So far, the mathematical structure of the size-dependence functionf2(La) has not been speci�ed. Based on experience from industrial crystallization,two types of simple size dependence can be assumed (Hulburt & Katz 1964, pp.562-563), (Sherwin, Shinnar & Katz 1969, pp. 61-63):

f2;La(La) = (a+ �La) (9.83)

f2;La(La) =�

La(9.84)

The �rst relationship is most commonly applied in crystallization whereas thesecond form applies to certain types of di�usional growth processes (Hulburt &Katz 1964, pp. 563). Since the growth process takes place in solid state, it isreasonable to assume that the second form of the function is the most realisticone. This is due to the reduced probability of presence of disorganized carbon atthe periphery of layer planes as crystallization proceeds i.e. the disorganized phasein the closest neighbourhood is consumed.

Furthermore, by considering the derivative of the volume Vcr of a single crystallitewith respect to time, i.e.:

dVcr

dt=

2

3�LaLc

dLa

dt=

2

3�LaLcvLa

it can be observed that at a certain constant consumption rate of disorganizedmatter, the growth in La is inversely proportional to La (Lc considered constant).Therefore, it is reasonable to assume that:

vLa = kLa(T )�dm;a�

La(9.85)

124 Mathematical Modelling of Structure and Porosity

with kLa(T ) as before.

To simplify the model even more, the size- and temperature dependence may belumped together by the application of a size dependent activation energy:

vLa = kLa(T; La)�dm;a (9.86)

kLa(T; La) = kLa;�e�

ELaRT (9.87)

ELa = b+ cLa (9.88)

Thus, by letting the activation energy increase with the size of the layer, one is ableto implement a kind of temperature dependent maximum value for the crystallitesize that can be achieved at a certain temperature.

In the most simple case, assume that:

vLa = kLa(T )�dm;a (9.89)

thus only taking into account the dependence on the amount of disorganized carbonand the temperature. In this case, kLa is again assumed to be an Arrhenius typefunction exclusively dependent on temperature. With such a simple growth law,the derivation of moment equations is feasible.

Coalescence Along the c-Axis

Since increase in Lc occurs by coalescence of crystallites (or nucleation of singlelayer planes), there is no intrinsic growth rate for Lc; i.e. vLc = 0. Instead a modelfor the process of coalescence that leads to increase in Lc is needed.

In general, an agglomeration type approach could be used to model the coalescenceof crystallites in both the a- and c-direction. In the case of carbon crystallites,however, the situation is more complicated since the mutual orientation, size andshape of crystallites may also play a role. For simplicity, assume that an ensembleof crystallites exists in which all crystallites have the same layer plane diameterLa. Also assume that coalescence behaviour is independent of crystallite size,shape and orientation. Following the approach due to Hulburt & Katz (1964, pp.569), binary coalescence is assumed proportional to the number density values forthe two values of Lc. Figure 9.9 demonstrates binary coalescence of crystalliteswith equal layer plane diameters. During formation of a crystallite with (n1+n2)layers from crystallites no. 1 and 2 with n1 and n2 layers respectively, the stackingheight Lc di�ers with one interlayer spacing from the pure addition of the stackingheights of the coalescing crystallites. Thus:

Lc = Lc;1 + Lc;2 + d (9.90)

Crystallites with stacking height Lc is formed by coalescence of crystallites withstacking height Lc � d� Lc;1 and Lc;1 at the following (birth) rate:

B =1

2

Z1

0

a�((Lc � d� Lc;1); t) �(Lc;1; t)dLc;1 (9.91)

9.3 Modelling Structural Evolution During Carbonization 125

Hulburt introduced the factor 12to avoid counting coalescence between crystallites

twice. a is a so-called accommodation factor which describes the e�ciency of thecoalescence process. In this case, a is independent of the crystallite properties buta may be time varying (i.e. temp. dependent).

Crystallites with stacking height Lc is also lost due to coalescence at the following(death) rate:

D = a�(Lc; t)

Z1

0

�(L0c; t)dL0

c (9.92)

The net rate of coalescence is therefore:

B �D = a

�1

2

Z1

0

�((Lc � d� Lc;1); t)�(Lc;1; t)dLc;1�

�(Lc; t)

Z1

0

�(L0c; t)dL0

c

�(9.93)

In the general case, La may also vary. If coalescence is constrained to only occurbetween crystallites having equal values of La, the same type of model as derivedabove can be used. In this case � needs an additional argument:

B �D = a

�1

2

Z1

0

�(La; (Lc � d� Lc;1); t) �(La; Lc;1; t)dLc;1�

�(La; Lc; t)

Z1

0

�(La; L0

c; t)dL0

c

�(9.94)

In this case, variations in d is not considered.

+

n1

n1 + n2

Lc;1 = (n1 � 1)d

Lc = (n1 + n2 � 1)d 6= (n1 + n2 � 2)d

Lc;2 = (n2 � 1)d n2

Figure 9.9: Binary coalescence of crystallites with equal values of La. Lc is not ad-ditive (i.e. not conserved) in the coalescence process since an additional interlayerspacing is added in the coalesced crystallite.

Perfection of the coalescence-growth model is needed to be able to predict curvesfor the crystallite parameters as shown in Figure 9.6. Coalescence of single layersmust be allowed to form many small crystallites. To achieve this, the single layersmust be considered as a separate carbon phase (i.e. not lumped together with the

126 Mathematical Modelling of Structure and Porosity

amorphous material) and a separate mass balance on this phase must be derivedto take account of the consumption of single layers due to mutual coalescence inthe a and c-directions as well as coalescence in the c-direction of single layers withlarger crystallites4.

Coalescence Along the a-Axis

Nominally, coalescence along the a-axis does not take place at temperatures be-low 1400�C. On the other hand, this limit temperature seems rather arti�cial,and a certain reduction in the number of layer planes is also expected even attemperatures below 1400�C.

At this stage, however, no e�ort has been done to model the birth- and death rateswhich correspond to coalescence of layer planes.

Growth Rate vd

In the turbostratic regime, the layer planes are randomly oriented with respect toeach other. It may then be assumed that only the size of the layers have impact onthe interlayer distance i.e. an e�ect of van der Waal's attraction between the layers.One may also assume that thermal motion of the planes contributes to tearingthe layers apart. Analogously with the Lennard-Jones potential function (Reid,Prausnitz & Poling 1987, pp. 393), there is a certain equilibrium interlayer spacingwhich depends on the size of the layer and the temperature; see Figure 9.10.

Pote

ntia

l fun

ctio

n

0

Equilibrium position

d [�A]

Figure 9.10: The Lennard - Jones potential function analogy.

This is a rather ideal situation. In reality, attraction of layers may be hindered bythe presence of heteroatoms and amorphous groups between and within the layerplanes. In a very simple approach, it may be assumed that:

d = xdm ~d+ (1� xdm)dtc (9.95)

where xdm is the mass fraction of disorganized matter present in the carbon (ac-tually not including the single layer planes). ~d is a parameter; typically ~d � 3:75

4Alternatively, one should consider to use nL;cr as parameter instead of Lc.

9.3 Modelling Structural Evolution During Carbonization 127

�A and dtc = 3:44 �A; the interlayer spacing in turbotratic carbon. As the inter-layer spacing approaches the value found in turbostratic carbon the amount ofdisorganized matter disappears.

In more highly ordered carbon materials, several theories for graphitization havecontributed to explain the gradual decrease in d downto the value of 3.354 �A foundin graphite (Gundersen 1996e, App. E). In summary, the following rate law forchanges in d is suggested:

vLd = �ddxdm

dtf1;d(La)f2;d(T ) (9.96)

�d = ( ~d� d) (9.97)

where f1;d decreases with La and f2;d increases with T . Thus, a modulation aroundthe value predicted by Equation (9.95) for a certain fraction of disorganized matteris performed as La and T varies. Here, it is assumed that d is constant. Thus,vd = 0.

In the simplest case, growth of La is considered (i.e. Lc and d constant) as de-scribed by the equation:

@�

@t+

1

Va

@Va

@t� +

@

@La(vLa�) = 0

vLa = f(T; �dm;a)

where � = �(La; t)

If also Lc is allowed to grow via coalescence of crystallites along the c-direction(i.e. only d constant), the population balance model must be changed as follows:

@�

@t+

1

Va

@Va

@t� +

@

@La(vLa�) = (B �D)

(B �D) = a

�1

2

Z1

0

�(La; (Lc � d� Lc;1); t)�(La; Lc;1; t)dLc;1

� �(La; Lc; t)

Z1

0

�(La; L0

c; t)dL0

c

�

where � = �(La; Lc; t). vLa is as before.

9.3.5 Weight Fraction xdm vs. Formation Rate and Mass of

Crystallites

The growth law for La depends on the weight fraction of disorganized matter.Now assume that a fraction � of the total mass ow from the disorganized phase

128 Mathematical Modelling of Structure and Porosity

is released as gas. The rest is transferred to the crystallite phase:

dmdm

dt= �rdmVa (9.98)

dmc

dt= rcVa (9.99)

dmt

dt= �rvVa (9.100)

rc = (1� �)rdm

rv = �rdm

mdm, mc and mt denote mass of disorganized matter, crystallites and bulk ma-terial respectively. The reaction rates are speci�ed on an apparent volume basis.Introduction of rc gives:

dmdm

dt= � 1

1� �rcVa (9.101)

dmt

dt= � �

1� �rcVa (9.102)

which can be used to derive (see also Equation (9.46)):

dxdm

dt=

1

�a

1

(1� �)(xdm�� 1)rc

In general, � may be a time varying function. Furthermore, we have not speci�edthe kind of gases released from the pyrolysing sample. Typically, methane andhydrogen dominate at temperatures above 500�C (Politis & Chang 1985). Insteadof using a stoichiometric parameter �, one could calculate rv from a model of thevolatile loss from the pyrolysing material. Such a model is derived in part III.

An expression for rc can be found by evaluation of an integral of the numberdensity function � (Hulburt & Katz 1964, pp. 561), (Sherwin et al. 1969, pp. 60).Here, the principle is demonstrated by assuming that La is the only crystalliteparameter that varies ; i.e. � = �(La; t). The mass of a crystallite is given by:

mcr = �cVcr (9.103)

Vcr =4

3�

�La

2

�2

Lc (9.104)

If La is the only parameter that may vary, the total mass of crystallites can befound from:

mc =

Z1

0

�cVcr(�(La; t)Va)dLa (9.105)

where again �(La; t)Va denotes the total number of crystallites within the bulkvolume with property between La and La+dLa. Introduction of known quantitiesgives:

mc =1

3��cVaLc

Z1

0

L2a�(La; t)dLa (9.106)

9.3 Modelling Structural Evolution During Carbonization 129

The integral in the above equation corresponds to the second order moment �2 inLa of the number density function �. Substitution of �2 gives:

mc =1

3��cVaLc�2 (9.107)

The formation rate of crystallites can be found by derivation of 1Va

dmc

dtwhere dmc

dt

can be found from the above expression for mc. Alternatively, one can use:

rc =

Z1

0

�(La; t)dmcr

dtdLa

since �(La; t) corresponds to the number of crystallites per unit volume and changein property unit. dmcr

dtis given by:

dmcr

dt=

2

3��cLaLc

dLa

dt

which gives:

dmcr

dt=

2

3��cLaLcvLa

since dLadt

= vLa . Introduction of this expression into rc gives:

rc =2

3��cLc

Z1

0

�(La; t)vLadLa

since variations in d and Lc are not considered. This expression gives the couplingbetween the crystallite population balance and the mass balance equations.

In the simplest case, vLa depends only on the mass fraction of disorganized carbonand temperature:

vLa = f(T; xdm)

This gives:

rc =2

3��cLcf(T; xdm)

Z1

0

La�(La; t)dLa (9.108)

The integral in the above equation corresponds to the �rst order moment �1 in Laof the number density function �. Hence:

rc =2

3��cLcf(T; xdm)�1 (9.109)

The volume fraction of crystallites can be found from:

�c =

Z1

0

Vcr�dLa (9.110)

130 Mathematical Modelling of Structure and Porosity

if is is assumed that Lc and d are constant. Substituting for Vcr gives:

�c =1

3�Lc

Z1

0

La�(La; t)dLa (9.111)

Again introduce the �rst order moment and obtain:

�c =1

3�Lc�1 (9.112)

If in addition, Lc and d are allowed to vary, integration has to be performed overall possible values of these parameters too.

9.3.6 Derivation of Moment Equations for Calculation of

Average Crystallite Parameters

Often, it is not necessary to have knowledge about the time variation of the prop-erty dependence of the number density function. If a knowledge of the variationsof its moment in time is satisfactory, the equation for the number density functioncan be transformed into a set of ordinary di�erential equations for the moments.Hulburt & Katz (1964) and Sherwin et al. (1969, pp. 61) discuss derivation ofmoment equations.

To demonstrate the principle, an equation for the weight average value of La willbe calculated. Again, if only growth of La in existing crystallites is considered5,the following population balance equation is obtained:

@�

@t+

1

Va

@Va

@t� +

@

@La(vLa�) = 0 (9.113)

where � = �(La; t). Also here, assume that vLa = f(T; xdm), and get:

dxdm

dt=

1

�a((1� xdm)

�

1� �+ 1)rc (9.114)

with rc as in Equation (9.108).

The weight average crystallite size6 �La can be found as the ratio of total diameterof crystallites per unit mass 1 to the total number of crystallites per unit mass �. These quantities are de�ned by:

� =

Z1

0

�cVcr(�(La; t)Va)dLa

1 =

Z1

0

La�cVcr(�(La; t)Va)dLa

5La grows due to consumption of the disorganized phase. Lc and d002 do not vary.6According to Mason (1958), one should rather use a harmonic mean value of the crystallite

diameters to obtain the mean value.

9.3 Modelling Structural Evolution During Carbonization 131

Here �(La; t)Va denotes the total number of crystallites within the bulk volumewhich have property between La and La + dLa. Then:

�La = 1

�(9.115)

Now introduce the expression for Vcr and recall that �c is assumed constant in thiscase. Hence:

� =1

3��cVaLc

Z1

0

L2a�(La; t)dLa

1 =1

3��cVaLc

Z1

0

L3a�(La; t)dLa

which shows that �La can be calculated from knowledge of the second and thirdorder moments of the number density function:

�La = 1

�=

R1

0L3a�(La; t)dLaR

1

0L2a�(La; t)dLa

=�3

�2(9.116)

A moment equation is obtained from the population balance equation by multi-plication with Lna and integration from zero to in�nity. This gives:

Z1

0

Lna@�

@tdLa +

Z1

0

Lna1

Va

@Va

@t�dLa +

Z1

0

Lna@

@La(vLa�)dLa = 0 (9.117)

Use vLa = f(T; xdm) to obtain:Z1

0

Lna@

@La(vLa�)dLa = f(T; xdm)

Z1

0

Lna@�

@LadLa

= �f(T; xdm)Z1

0

nLn�1a �dLa (9.118)

where the last expression is obtained by partial integration. The n'th order mo-ment of � is de�ned by:

�n =

Z1

0

Lna�dLa (9.119)

For values of n = 0; 1; 2; 3, the moments are as follows:

� �0 : The total number of crystallites per unit volume

� �1 : The total crystallite diameter per unit volume

� �2 : The total square of crystallite diameter per unit volume

� �3 : The total cube of crystallite diameter per unit volume

132 Mathematical Modelling of Structure and Porosity

Ordinary di�erential equations are obtained for the moments:

d�n

dt+

1

Va

@Va

@t�n � nf(T; xdm)�n�1 = 0 (9.120)

For calculation of �La, �2 and �3 are needed. The moment equations generated forn = 0; 1; 2; 3 are:

d�0

dt+

1

Va

@Va

@t�0 = 0 (9.121)

d�1

dt+

1

Va

@Va

@t�1 � f(T; xdm)�0 = 0 (9.122)

d�2

dt+

1

Va

@Va

@t�2 � 2f(T; xdm)�1 = 0 (9.123)

d�3

dt+

1

Va

@Va

@t�3 � 3f(T; xdm)�2 = 0 (9.124)

In this case, �La was calculated as a weighed average value. Other types of averagevalues may also be calculated by using the same principle (i.e harmonic meansetc.).

The above four equations together with Equation (9.114) constitute the model forcalculation of �La. In the model, rc as expressed in Equation (9.108) is needed.With a knowledge of the moments, a simple expression for rc can be obtained:

rc =2

3��cLcf(T; xdm)

Z1

0

La�(La; t)dLa =2

3��cLcf(T; xdm)�1 (9.125)

as previously derived in Equation (9.109).

Hulburt & Katz (1964, pp. 568-571) has shown that in some cases, a closed setof moment equations can also be found in particle agglomeration processes. Inprinciple, this approach could be used to �nd moment equations corresponding topopulation balance equations for coalescence of crystallites. No further elaborationof this subject will be given here.

Some �nal comments are needed:

� The same principle can be applied if the property space has a higher di-mension (i.e. if calculation of the average values of �d and �Lc is performed).Examples of moment equations for two dimensional property spaces are givenin Himmelblau & Bischo� (1968, pp. 193) and Hulburt & Katz (1964, pp.566-568).

� In general, however, the derivation of moment equations for general growthfunctions is not so straight forward as in this example. Successive approxi-mations for � may be needed (Hulburt & Katz 1964).

� The quantity �La denotes the weight average value of La. In the rest of thestudy, La, Lc and d is used for simplicity to denote the spatial average valueof the crystallite parameters.

9.3 Modelling Structural Evolution During Carbonization 133

9.3.7 Lumped Models for Crystallite Growth

Increase in crystallite size is due to consumption of the disorganized carbon phaseas well as coalescence along the a- and c-axes. Two di�erent models for thedevelopment of La for temperatures up to 1400�C is presented below. In the �rstcase, coalescence of crystallites is allowed to occur only along the c-axis. In a moregeneral case, coalescence must be allowed to occur primarily along the c-axis andsecondarily along the a-axis.

Coalescence of Crystallites Occur Only Along the c-Axis.

A very simple model for the development of the average values of d, La and Lccan be found if the following is assumed:

� Under the assumption that no coalescence occur in the a-direction, the num-ber nL of layer planes is conserved.

� The evolution of average stacking height Lc and interlayer spacing d aremodelled as thermally activated processes:

dLc

dt= fLc(Lc; T; t) (9.126)

d(d)

dt= fd(d; T; t) (9.127)

� Conversion dependent activation energies can be used in the models for dand Lc. Note here that the rate laws used in this case are analogous withrate laws for properties used in traditional population balance modelling.

� A rate law rdm exists for the conversion of the disorganized matter (i.e.mainly carbon) to the crystallograpic phase.

A mass balance over the crystallograpic phase gives:

dmc

dt= rcVa (9.128)

rc = rdm � rv (9.129)

Below, it is shown that mcr depends on La, d, Lc (nL;cr) and ncr where nL;cr andncr denotes the number of layer planes within an average crystallite and the totalnumber of crystallites respectively. This can be used to obtain a rate law for La.

If there are ncr crystallites with volume Vcr in the material, the total mass ofcrystallites is obtained from:

mc = �cncrVcr (9.130)

nL;cr is the number of layer planes in an average sized crystallite. Use Vcr =

��La2

�2Lc where Lc = (nL;cr � 1)d to obtain:

mc = �cncr

�

�La

2

�2

(nL;cr � 1)d

!(9.131)

134 Mathematical Modelling of Structure and Porosity

This expression should be elaborated on to obtain a convenient expression forcalculation of La. The product ncrnL;cr corresponds to the total number nL oflayer planes in the crystallographic phase which in this case was assumed to bea constant during the pyrolysis process since no coalescence occurs along the a-direction7. Then nL = ncrnL;cr = nL;� which gives:

mc = �c�

�La

2

�2

(nL;� � ncr)d (9.132)

An expression for ncr can be obtained from the relationship:

Lc = (nL;cr � 1)d) (nL;cr � 1) =Lc

d(9.133)

and

ncr(nL;cr � 1) = (nL;� � ncr) (9.134)

Thus,

ncrLc

d= (nL;� � ncr) (9.135)

which gives:

ncr = nL;�1

Lcd + 1

(9.136)

Substitution of this expression for ncr in the above expression for mcr gives:

mc = ��c

�La

2

�2

nL;�Lc

Lcd+ 1

(9.137)

In addition, mc may be expressed by the mass fraction xdm of disorganized carbonas follows:

mc = xcrmt = (1� xdm)�aVa (9.138)

Equating the two expressions for mc gives:

nL;� =(1� xdm)�aVa

��c�La2

�2 Lc(Lcd+1)

(9.139)

or:

La = 2

vuut (1� xdm)�aVa

��cnL;�Lc

(Lcd+1)

(9.140)

7In this case, a crystallite concept is used in which all crystallites are assumed to have thesame size. Single layer planes which before has been considered to belong to the disorganizedphase does not exist in this situation.

9.3 Modelling Structural Evolution During Carbonization 135

when solved for nL;� and La respectively. In simulations, nL;� can be obtainedfrom the initial values of the crystallite parameters, xdm(0), �a(0) and Va(0).When the value of nL;� is known, La can be calculated from the above equationand the knowledge of states xdm, �a, Va, d and Lc (�c depends on d). In this case,La belongs to the property space due to the algebraic dependence of the statevariables mentioned above.

nL;� is a large number. This can be demonstrated by assuming that for a certain

carbon material, d = 3:44 �A, La = 46 �A, Lc = 20 �A, xdm = 0:6, �a = 1200 kg/m3

and Va = 1 cm3 = 10�6m3. From the above expression for nL;�, we then obtainnL;� = 4:4657� 1019.

In this modelling approach, the problem of modelling the coalescence process wassolved by assuming that Lc can be represented by a thermally activated process.No fundamental description of the development of nL;cr was given. nL;cr wasinstead related to Lc and d.

Coalescence Both Along the a- and c-Axes

Some experience has been obtained by simulation of the above model for growth ofLa. It turned out that the growth rate was too low to give a realistic progressionof La between 400 and 1200�C for a certain combination of kinetics and initialvalues of xdm, Lc and d. Furthermore, it was not possible to predict a realisticevolution of Young's modulus as function of heat treatment temperature based onthe model presented in Emmerich & Luengo (1993).

It turned out that a realistic development of both La and Young's modulus couldbe achieved if a certain degree of coalescence of crystallite layer planes along the a-axis was allowed to occur; coalescence along the a-axis corresponds to a reductionin the number of layer planes. This seems to be in contrast with the commonassumption that coalescence along the a-axis nominally starts at temperatures inthe order of 1400�C. This kind of coalescence, however, is of a more dramaticcharacter which promotes a very rapid growth of La and a much more signi�cantreduction in the number of layer planes takes place.

In this case, it was decided to model the development of the total number nL oflayer planes in the simplest possible manner by assuming that nL is a function oftemperature only:

nL = nL(T ) = nL;�(1� k(T � T�)) (9.141)

In reality, the evolution of nL should rather be represented by a di�erential equa-tion.

The derivation of the model for La just presented is still valid but now nL is nota constant number; nL;� should be replaced with nL. Then the model for La

136 Mathematical Modelling of Structure and Porosity

becomes:

La = 2

vuut(1� xdm)�aVa

��cnLLc

(Lcd+1)

nL = nL;�(1� k(T � T�)) (9.142)

nL;� can be found from initial values of the state variables and La as before. Itis also possible to formulate a rate law for La. Furthermore, in the case whenthe evolution of nL is governed by a di�erential equation, the derivation of a ratelaw for La would be quite logic: Use La as a state variable instead of nL which isanalogous to the use of Lc as a state variable instead of nL;cr. The derivation of adi�erential equation for La should be based on Equation (9.128) with substitutionof the expression:

mc = ��c

�La

2

�2

nLLc

Lcd + 1

(9.143)

where nL in general is a rate process. The rate equation for mc is then:

dmc

dt=

1

4��cd

�L2a

Lc

Lc + d

dnL

dt+ nL

Lc

Lc + d2La

dLa

dt

+nLLcd

Lc + d

L2a

Lc + d

�1

Lc

dLc

dt� 1

d

dd

dt

��(9.144)

This expression can be used in Equation (9.128) and solved with respect to dLadt

to give:

dLa

dt=

1

2

Lc + d

LaLc

�rcVa

�4�cd nL

� L2a

Lc

Lc + d

1

nL

dnL

dt

� Lcd

Lc + d

L2a

Lc + d

�1

Lc

dLc

dt� 1

d

dd

dt

��(9.145)

As nL decreases, the relative signi�cance of the term rcVa�4�cdnL

increases. Thus, La

grows faster than in the previous model if the �rst term on the right hand sideof the equation dominates over the term L2

aLc

Lc+d1nL

dnLdt . Simulations have shown

that this is the case.

Since nL is an algebraic function of temperature, nL (and La) is no state variable.In the more general case, one may assume that nL obeys the law:

dnL

dt= knL(nL(1)� nL) (9.146)

knL = knL;�e�

EnLRT (9.147)

EnL = EnL;min +

�nL � nL;�

nL(1)� nL;�

�(EnL;max �EnL;max) (9.148)

A corresponding type of model is recommended for the consumption of the disor-ganized carbon phase.

9.3 Modelling Structural Evolution During Carbonization 137

The Consumption Rate of Disorganized Carbon

To complete the above models, an expression for rdm is needed. rdm must be usedto calculate xdm according to Equation (9.46):

dxdm

dt= � 1

�a((1� xdm)rv + rc)

Here, rv is the rate of release of carbonization gases (mainly methane and hydro-gen). For rdm, a �rst order expression was assumed as follows:

rdm = kdm�a;dm (9.149)

kdm is an Arrhenius expression and �a;dm is the density of disorganized carbonwithin the apparent (bulk) volume of the pyrolysing material. This gives:

mdm = �a;dmVa = xdmmt = xdm�aVa

which gives

�a;dm = xdm�a (9.150)

A conversion dependent activation energy was used in the Arrhenius-type expres-sion for kdm:

kdm = kdm;�e�

EdmRT

Edm = Edm;min + (1� xdm

xdm(0))Edm;max

Edm;min is less than Edm;max to achieve that the activation energy increases asxdm decreases and it becomes more and more di�cult to consume the disorganizedphase as xdm decreases.

rv may be calculated as a certain fraction � of rdm or rv may be obtained from aseparate model. An alternative model can be based on the expressions commonlyused in modelling of solid phase composition reactions; see the review in part III.This gives the following model:

mdm = ~xdmmdm;� (9.151)

mdm;� = xdm;�mt;� (9.152)

~xdm = (1�Xdm) (9.153)

dXdm

dt= kdm(1�Xdm)

ndm (9.154)

kdm = kdm;�e�

EdmRT (9.155)

Edm = (1�Xdm)Edm;min +XdmEdm;max

Here, xdm;� is the initial mass fraction of disorganized carbon present in the ma-terial used in the pyrolysis experiment. Xdm is a conversion parameter whichinitially has the value zero and increases up to one when complete conversion of

138 Mathematical Modelling of Structure and Porosity

the disorganized phase is achieved. The Arrhenius expression kdm used in this caseis in general di�erent from kdm in the expression for rdm; see Equation (9.149).Also note the di�erence between Edm used in this case as compared to the abovede�nition of Edm.

Since loss of carbonization gases occurs in parallel with the conversion of disorga-nized carbon, ~xdm is not the value needed for xdm in our model for the granularstructure. In fact, one must use:

xdm =mdm

mt(9.156)

A total mass balance is used to obtain mt:

dmt

dt= �rvVa (9.157)

By introducing mt = cymt;� and Va =mt

�a, the following rate equation is obtained

for the coke yield cy:

dcy

dt= �cy

rv

�a(9.158)

In this case, cy will not change very much since only the loss of non-condensablegases is taken into account. Substituting for mdm and mt = cymt;� in the expres-sion for xdm gives:

xdm =~xdmxdm;�mt;�

cymt;�= xdm;�

(1�Xdm)

cy(9.159)

It should be noted here that cy(0) = 1 to achieve that xdm(0) = xdm;�. The massfraction of crystallites can be found from:

xcr = 1� xdm (9.160)

as before. Even better performance of the model can be obtained if a multiplereaction model with distribution of activation energies is used to model the dy-namics of Xdm. Then it becomes possible to realize a temperature dependent limitvalue for La (via the temperature dependent limit for Xdm).

9.3.8 Simulation Case I: Population Balance Approach to

Growth of La

Based on the previous results, the following state space model for developmentof density, porosity and crystallite diameter during solid coke carbonization is

9.3 Modelling Structural Evolution During Carbonization 139

established:

1

Va

dVa

dt= � 1

�a(rv +

d�a

dt) (9.161)

rv =�

1� �rc (9.162)

d�a

dt= �aka;iso

d�r

dt(9.163)

ka;iso = (1� ��) (9.164)

�� = 1� �a

�r(9.165)

d�r

dt= �rkr;iso

d��cdt

(9.166)

kr;iso = (1� �0c) (9.167)

�0c = 1� �r

��c(9.168)

�c =�a

�r� �a

��c(9.169)

��c =1

xcr�c

+ xdm�dm

(9.170)

�c =dgr

d�c;gr (9.171)

�dm = constant (9.172)

dxdm

dt=

1

�a

1

(1� �)(xdm�� 1)rc (9.173)

rc =2

3��cLc f(T; xdm) �1 (9.174)

d��

dt= � 1

Va

dVa

dt�� (9.175)

d�1

dt= � 1

Va

dVa

dt�1 + f(T; xdm) �� (9.176)

d�2

dt= � 1

Va

dVa

dt�2 + 2f(T; xdm) �1 (9.177)

d�3

dt= � 1

Va

dVa

dt�3 + 3f(T; xdm) �2 (9.178)

f(T; �dm;a) = k�e�

ERT �dm;a (9.179)

�dm;a = xdm�a (9.180)

The following variables are included in the state vector x:

x = [Va

Va(0); �r; �a; xdm; ��; �1; �2; �3]

T

In general, �a may vary with temperature, and �a is therefore not linearly depen-dent of �r. �a is therefore also included in the state vector. In this case no modelsfor Lc and d are included. Furthermore, no detailed model of the devolatilization

140 Mathematical Modelling of Structure and Porosity

is given; it is only assumed that a constant fraction of the current mass transferbetween the disorganized and crystalline phase is lost as volatiles. Finally, �dm isassumed to be a constant parameter and a constant value is used for �r.

The porosities and crystallite diameter are obtained by:

�� = 1� �a

�r

�c =�a

�r� �a

��c

�La =�3

�2

For simulation of the model, initial values for the moments �i (i = 0; 1; 2; 3) of thenumber density function are especially needed. In this case, the initial values wereobtained as follows:

� For simplicity, all crystallites are assumed to be of the same size (La(0); Lc(0)).

� �� corresponds to the total number of crystallites per unit volume. Thecrystallites constitute a fraction (1� xdm) of the total mass of carbon. Themass of one crystallite is mcr. The total number of crystallites per unitvolume can then be found from:

�� =(1� xdm)�a

mcr(9.181)

The use of mcr = ��La2

�2Lc�c gives:

�� =(1� xdm)�a

��La2

�2Lc�c

(9.182)

� The moments of the number density function are calculated from:

�i =

Z1

0

Lia�(La; t)dLa (9.183)

For simplicity, it is assumed that: �(La; 0) = ���(La(0)� La) which gives:

�i(0) =

Z1

0

Lia���(La(0)� La)dLa = La(0)i�� (9.184)

The model was simulated with the initial values and parameters shown in Table 9.1.

As shown in Figure 9.14, the total porosity as a sum of open- and closed (i.e.intercrystalline) porosity decreases during the simulation. Whether this is realisticremains to be discussed. In general, however, the initial value in �c is lower thanin this case.

The kinetic parameters for growth of La are critical for the range of values for La.A higher �nal value of La is obtained if xdm is increased. In this case, growth

9.3 Modelling Structural Evolution During Carbonization 141

0 20 40 60 80 100 120500

600

700

800

900

1000

1100

1200

Time [hr]

T [K

]Heat treatment program

Figure 9.11: Heat treatment program used in the simulation of the populationbalance model. The heating rate is a = 10�C=hr.

of La due to the consumption of disorganized carbon is the only active growthmechanism. Since La is relatively low, it may be concluded that other growthprocesses (i.e. coalescence) may occur in parallel with the consumption of thedisorganized phase. A very simple growth law was used for La and the rate lawfor La is not capable of realizing a limit value for La which depends on the heattreatment temperature.

A constant value (larger than one) was used for �r. This eventually leads to�r > ��c which in the limit gives �r > �c when xdm = 0. This is a non-physicalsituation. Thus, actually �r is a more complicated function which depends onboth �r and ��c in such a manner that changes in �r retard when �r approaches��c.

Coalescence along the c-axis would lead to a reduction in the total number ofcrystallites. In this case, however, the total number of crystallites remains con-stant, thus ��Va is constant. Finally, it should be stressed that the coalescence ofcrystallites along the c-axis is not active; Lc has a constant value

8.

8Coalescence along the a-axis is either not considered. This is a phenomenon that mainly isactive at temperatures in the order of 1400�C in soft carbons.

142 Mathematical Modelling of Structure and Porosity

0 50 100 1500

0.1

0.2

0.3

0.4

Time [hr]

xdm

Mass fraction of disorganized phase.

0 50 100 1500.6

0.7

0.8

0.9

1

Time [hr]

xc

Mass fraction of crystallites

0 50 100 1500.7

0.75

0.8

0.85

0.9

0.95

1

Time [hr]

Va/

Va(

0)

Shrinkage of the apparent volume.

Figure 9.12: The plot shows consumption of the disorganized phase due to growthof La and the corresponding increase in the mass fraction of crystallites. The plotof Va=Va;� shows that the bulk volume shrinks during heat treatment.

9.3.9 Simulation Case II: Lumped Model Approach to Growth

of La

In Subsection 9.3.7, a simple model for growth of La based on the consumption ofthe disorganized carbon phase was derived. In this subsection, this model is usedto simulate the evolution of La. d is assumed constant. For simplicity, it is alsoassumed that a constant value can be used for Lc and that both nL and �dm remainconstant. As in the previous case, a simple model of the volatilization kinetics isgiven. The following model can then be used for prediction of the growth of Ladue to consumption of the disorganized phase:

dxdm

dt= � 1

�a((1� xdm)rv + rc) (9.185)

rc = (1� �)rdm (9.186)

rv = �rdm (9.187)

rdm = kdmxdm�a (9.188)

kdm = kdm;�e�

EdmRT (9.189)

Edm = Edm;min + (1� xdm

xdm(0))(Edm;max �Edm;min) (9.190)

The apparent volume Va is needed for evaluation of the right hand side of thedi�erential equation for La. Therefore, simulation of the development of the total

9.4 Conclusions: Porosity and Structure Models 143

0 50 100 1501900

1950

2000

2050

2100

2150

2200

Time [hr]

rhoc

_b [k

g/m

^3]

Density of the disorganized and crystalline carbon.

0 50 100 1501400

1600

1800

2000

2200

Time [hr]

rho_

r [k

g/m

^3]

real density of the material.

0 50 100 1501200

1300

1400

1500

1600

1700

Time [hr]

rho_

a [k

g/m

^3]

Apparent density of the material.

Figure 9.13: Plot of density of the solids (i.e. real density and apparent density).

mass balance and the corresponding development of densities and porosities isneeded. This was achieved by extending the model to include equations (9.161)to (9.172) in Subsection 9.3.8. The evolution of the following state vector wassimulated:

x = [Va

Va(0); �r; �a; xdm]

T

La is calculated from:

La = 2

vuut (1� xdm)�aVa

��cnL;�Lc

(Lcd+1)

The parameters used in the simulations are given in Table 9.2. Simulation plotswhich correspond to the simulations in Subsection 9.3.8 are presented below.

9.4 Conclusions: Porosity and Structure Models

Based on the granular structure model of a solid carbon, models for porosity andcrystalline structure evolution have been suggested:

� Models for open- and closed porosity were based on the concept of anisotropicshrinkage and the application of a shrinkage parameter. Three levels of

144 Mathematical Modelling of Structure and Porosity

0 50 100 1500.14

0.16

0.18

0.2

0.22

Time [hr]

phi_

oOpen porosity.

0 50 100 1500.05

0.1

0.15

0.2

0.25

Time [hr]

phi_

c

Closed porosity.

0 50 100 1500.25

0.3

0.35

0.4

Time [hr]

phi_

T

Total Porosity

0 50 100 15045

50

55

60

Time [hr]

La [A

A]

Crystallite diameter La

Figure 9.14: Plot of open- (��) and closed (�c) porosity and crystallite diameterLa. The open porosity increases. The closed porosity (intercrystalline pores)decreases in such a way that the total porosity �T (as a sum of open and closedpores) also decreases.

density were needed for realization of the porosity models: The apparent,real and solid material densities.

� Based on the population balance equation, a crystallite growth model wassuggested.

� To demonstrate the modelling principle, only La was allowed to vary. Underthis assumption, a simple model equation for the development of the averagevalue La during heat treatment was derived.

� Carbon texture is mainly established during the liquid state pyrolysis (i.e.mesophase transition) and the evolution of texture was not considered atthis stage.

� A lumped modelling approach was also suggested for modelling the develop-ment of La.

� In a purely empirical approach, the development of crystallite parametersmay be modelled as thermally activated processes.

If models of crystallite parameter evolution are needed in a submodel for calcula-tion of anode carbon properties, the lumped or thermal activation-based models

9.4 Conclusions: Porosity and Structure Models 145

0 20 40 60 80 100 120700

750

800

850

900

950

1000

1050

1100

1150

1200

Time [hr]

[kg/

m^3

]Total mass and crystalline mass

Figure 9.15: During heat treatment, the mass of crystallites (lower curve) ap-proaches the total mass of the carbon sample (upper curve).

are recommended. Modelling of crystallite parameters as thermally activated pro-cessed is discussed in the next chapter.

146 Mathematical Modelling of Structure and Porosity

Variable Description Unit Value

d [�A] 3.440

dgr [�A] 3.354

La [�A] 46.000

Lc [�A] 20.000

xdm(0) 0.400

k� [1/hr] 1:7� 107 1�1(0)

E [kJ/mol] 110

� 0.04

�a 0.82

�r 2.90

�gr [kg/m3] 2226.00

�c(0)dgrd�gr [kg/m

3] 2170.30

�dm(0) [kg/m3] 1600.00

��c(0)1

xdm(0)

�dm+

(1�xdm(0))

�c

[kg/m3] 1899.50

�a(0) [kg/m3] 1200.00

�r(0) [kg/m3] 1400.00

��(0)(1�xdm(0))�a(0)43�(La2 )

2Lc�c

[no. of crystallites

m3 ] 8:98� 1025

�i(0) La(0)i��(0) [�An]

T (0) [K] 773.15

Table 9.1: Parameters used in simulation of a simpli�ed model for carbon structureand porosity. Carbon structure as represented by the layer plane diameter Lachanges during heat treatment due to consumption of the disorganized carbonphase.

9.4 Conclusions: Porosity and Structure Models 147

0 10 20 30 40 50 60 70 80500

600

700

800

900

1000

1100

1200

Time [hr]

T [K

]Heat treatment program

Figure 9.16: Heat treatment program used in simulation of the lumped model forLa. The heating rate is a = 10�C=hr.

0 20 40 60 800

0.1

0.2

0.3

0.4

Time [hr]

xdm

Mass fraction of disorganized phase

0 20 40 60 800.6

0.7

0.8

0.9

1

Time [hr]

xc

Mass fraction of crystallites

0 20 40 60 800.7

0.75

0.8

0.85

0.9

0.95

1

Time [hr]

Va/

Va(

0)

Shrinkage of the apparent volume

Figure 9.17: Consumption of the disorganized phase due to growth of La and thecorresponding increase in the mass fraction of crystallites.

148 Mathematical Modelling of Structure and Porosity

0 20 40 60 801900

1950

2000

2050

2100

2150

2200

Time [hr]

rhoc

b [kg/

m3 ]

Density of the disorganized and crystalline carbon

0 20 40 60 801400

1500

1600

1700

1800

1900

2000

2100

Time [hr]

rho r [k

g/m

3 ]

Real density of the material

0 20 40 60 801200

1300

1400

1500

1600

1700

Time [hr]

rho a [k

g/m

3 ]

Apparent density of the material

Figure 9.18: Plot of density of the solids (i.e. real density and apparent density).

0 20 40 60 800.14

0.15

0.16

0.17

0.18

0.19

0.2

Time [hr]

phi o

Open porosity

0 20 40 60 800

0.05

0.1

0.15

0.2

0.25

Time [hr]

phi c

Closed porosity

0 20 40 60 800.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

Time [hr]

phi T

Total Porosity

0 20 40 60 8046

48

50

52

54

56

58

Time [hr]

La [A

A]

Crystallite diameter La

Figure 9.19: Plot of open- (��) and closed (�c) porosity and crystallite diameterLa. The open porosity increases. The closed porosity (intercrystalline pores)decreases in a way such that the total porosity (�T ) as a sum of open and closedpores also decreases.

9.4 Conclusions: Porosity and Structure Models 149

0 10 20 30 40 50 60 70 800.7

0.8

0.9

1

1.1

1.2

1.3x 10

−3

Time [hr]

[kg/

m3 ]

Total mass and crystalline mass

Figure 9.20: During heat treatment, the mass of crystallites (lower curve) approachthe total mass of the carbon sample (upper curve).

150 Mathematical Modelling of Structure and Porosity

Variable Description Unit Value

d [�A] 3.440

dgr [�A] 3.354

La [�A] 46.000

Lc [�A] 20.000

xdm(0) 0.400

nL (1� xdm(0))�a(0)�c

4

�La(0)2Lc

Lcd

+1

Va(0) 3:81� 1019

k� [1/hr] 3:7� 107

Emin [kJ/mol] 200

Emax [kJ/mol] 420

� 0.04

�a 0.82

�r 3.10

�gr [kg/m3] 2226.00

�c(0)dgrd�gr [kg/m

3] 2170.30

�dm(0) [kg/m3] 1600.00

��c(0)1

xdm(0)

�dm+

(1�xdm(0))

�c

[kg/m3] 1796.90

�a(0) [kg/m3] 1200.00

�r(0) [kg/m3] 1400.00

Va(0) [m3] 10�6

Table 9.2: Parameters used in simulation of the lumped model for development ofLa.

Chapter 10

Crystallite Growth Modelled

as a Thermally Activated

Process

In the following presentation, crystallite growth is modelled as thermally activatedprocesses. In the models, an activation energy which changes during crystallitedevelopment plays an important role. Except for this physical basis, the modelsare of purely empirical nature. A more thorough treatment of the subject is givenin Gundersen (1996c, cht. 2).

10.1 Introduction

The following qualitative behaviour of crystallite growth is used as basis for themodels presented in this chapter:

� Growth processes in the liquid phase:

{ Devolatilization of light components from the liquid pitch has directimpact on the average size of the molecules (i.e. layer plane diameter)in the pitch.

{ Polymerization reactions in the liquid pitch increase the average layerplane diameter.

{ As the molecules grow in size, they become able to associate in stacksof molecules (i.e. these stacks are precursors to the microcrystallitestructures). Thus, Lc and d are changed.

� Growth processes in the solid carbon residue:

{ Growth of the layer planes takes place by consumption of the so-calleddisorganized carbon phase (i.e. crosslinks and single carbon atoms).

152 Crystallite Growth Modelled as a Thermally Activated Process

Polymerization reactions may also occur in the more or less solidi�edcarbon residue.

{ Coalescence of crystallites takes place mainly along the c-axis of thecrystallites i.e. it is assumed that temperature is low enough to notintroduce coalescence in the a-direction.

{ In this case, the apparent deterioration of crystallite order (as observedby a decrease and increase in parameters Lc and d002 respectively) be-tween temperatures of 400 and 800�C is attributed to the release ofcarbonization gases. The temperature interval corresponding to theapparent structural deterioration more or less correspond to the tem-perature interval over which methane CH4 is released from the residue.Therefore, the gas release rate may be used as a means of implement-ing the apparent structural deterioration in the models for Lc and d002(Gundersen 1996f).

In this context, it should be mentioned that there is a tendency for the growthof La to partly level o� (i.e. growth rate is low) between 400 and 600�C. Thisstatement is in accord with observations reported elsewhere (Ruland 1965).

In the carbon literature, two views of the relationship between the magnitudeof crystallite parameters, heat treatment temperature and residence time can befound (Fischbach 1971), (Fair & Collins 1962):

1. In the early carbon literature (literature on graphitization) heat treatmenttemperature was considered to be the only variable which determines crys-tallite structure of the carbon material.

2. Crystallite growth is an ordinary rate process in which time and heat treat-ment temperature both play a role. In some cases, one of the parametersseems to play a more important role than the other; especially heat treat-ment temperature. In general, however, the impact of both parameters issigni�cant.

Today, the second hypothesis is considered to be correct. In practice, however, itis observed as time goes to in�nity, that an apparent maximum (minimum) valueof the crystallite parameter is achieved at a certain heat treatment temperature. Ifthe maximum temperature is slightly increased, a corresponding change is observedin the �nal value of the crystallite parameter.

This assumption of an apparent limiting value gives us the opportunity to use theconcept of thermal activation and �nite limiting values of the crystallite parame-ters1 as basis for the models presented in this chapter.

1The limiting values have to be adjusted to �t with the maximum temperature at which themodels should operate.

10.2 Di�erent Modelling Approaches 153

10.2 Di�erent Modelling Approaches

With basis in the state space concept used in the model of pitch pyrolysis (see partIII), three di�erent approaches can be considered for the modelling the evolutionof crystallite parameters as thermally activated processes:

1. Crystallite parameters for each pitch fraction ( ; �; �p; �s) are modelled asthermally activated processes. Weight average values of these crystalliteparameters are used as a measure of the average crystallite structure of thepitch. To implement the apparent reduction in crystallite order which takesplace in the temperature interval between 400 and 800�C, the models for Lcand d002 are coupled to the rate of release of methane that occurs in theapproximately same temperature interval.

2. In principle, the same approach as described in item 1 above is used exceptthat now only the isotropic and an anisotropic phases of the pitch are consid-ered. Crystallite parameters can be assigned to each phase. The crystalliteparameters for each phase are modelled as thermally activated processes.Also here, weight average values are used to calculate the average crystallitestructure of the pitch and the models are coupled to the release of methaneto implement the apparent reduction in crystallite order.

3. The third approach has to be divided into three alternatives:

(a) La is modelled by using a weighed average value of the La values ofan isotropic and anisotropic pitch phase. Models for Lc and d002 isdecoupled from the development of the isotropic- and anisotropic pitchfractions. Either a single reaction model (which may have a conversiondependent activation energy) or a multiple reaction approach is used tomodel Lc and d002. A certain coupling to the release of methane is alsoincluded to implement the apparent reduction in structural order.

(b) La is modelled as just described. Lc and d002 are modelled as individualthermally activated processes with no mutual coupling and no couplingto the pyrolysis model.

(c) All crystallite parameters are modelled as individual thermally acti-vated processes with no mutual coupling and no coupling to the pyrol-ysis model.

10.3 The Recommended Approaches

Based on a discussion presented in detail in Gundersen (1996c), it was concludedthat the most relevant modelling approaches are of two kinds as described in thefollowing.

154 Crystallite Growth Modelled as a Thermally Activated Process

10.3.1 Models for Realistic Prediction of Crystallite Param-

eters

The following approaches for modelling of realistic development of crystallite pa-rameters during heat treatment are recommended:

1. Approaches for modelling La:

(a) A model for La based on submodels for an isotropic and an anisotropicpitch phase. For the isotropic phase, La;i is calculated by weight aver-aging of assumed constant values for La of the individual pitch fractions , � and �p. For the anisotropic pitch fraction, the model for La;c isbased on submodels for the mass fraction of disorganized carbon andthe total number of layer planes.

(b) Two single reaction models with conversion dependent activation energywhich operate in parallel or two ensembles of multiple reaction modelswith distributed activation energy. The ensembles operate in parallel.

2. Approaches for modelling d002 and Lc:

(a) A single reaction model with conversion dependent activation energyor a multiple reaction model with distributed activation energy. It hasbeen shown that only the multiple reaction model successfully can becoupled to the methane release rate calculated in the pyrolysis model(Gundersen 1996f).

(b) Two single reaction models with conversion dependent activation energyoperating in parallel or two ensembles of multiple reaction models withdistributed activation energy. The ensembles operate in parallel.

10.3.2 Simpli�ed Models for Crystallite Parameters

Even simpler models can be used which still has a realistic prediction of the impactof maximum heat treatment temperature and hold time on the magnitude of thecrystallite parameters d002, La and Lc:

1. Single reaction model with conversion dependent activation energy

2. Multiple reaction model with a distributed activation energy

Some of the recommended mathematical models are described in Section 10.4below.

10.4 The Models

The modelling principle for crystallite parameters is demonstrated by developingmodels for La and Lc.

10.4 The Models 155

These crystallite parameters are modelled as independent thermally activated pro-cesses. Using this approach, one can still succeed in giving a realistic prediction ofthe evolution of La (with almost absence of growth in La for temperatures between400 and 600�C). For Lc (and d002), the deterioration of crystallite order between450 and 800�C can be achieved by using two ensembles of parallel processes andadding the response of the ensembles. Careful tuning of model parameters isneeded.

10.4.1 Models Based on two Ensembles of Multiple Parallel

Processes

The suggested principle will be demonstrated by giving the structure of a model forLc. By using a model based on two ensembles of parallel processes, it is possible tolet one of the ensembles have activation energies which correspond to an operativerange within the temperature interval from 200�C to maximum temperatures inthe order of 1300 to 1400�C. The second ensemble may have rate constants tunedvia the activation energy to be operative in the range from 400 to 800�C.

For Lc, the model appears as follows:

Lc = x1 � x2 (10.1)

where x1 is calculated from a multiple reaction model of the kind:

dx1;i

dt= k1;i(x1;i(1)� x1;i) (10.2)

x1;i(0) = wix1(0)

x1;i(1) = wix1(1)

k1;i = k1;i;�e�

E1;i

RT (10.3)

1 =

nrXi=1

wi (10.4)

x1 =

nrXi=1

x1;i (10.5)

A corresponding model is used for x2. The operation-principle of the model isexplained as follows:

� Initially, there is an increase in Lc coming from the steady increase in x1.

� At temperatures in the order of 400�C, x2 becomes active. If the growth ratefor x2 is larger than the growth rate for x1, the result is a reduction in Lc(i.e. negative growth rate for Lc). This can be achieved by proper selectionof the kinetics in the processes for x1 and x2.

� As the dynamics of x2 fades out at temperatures in the order of 800�C,there is a renewed increase in Lc as the rate of change of x1 again starts todominate over the rate of change of x2.

156 Crystallite Growth Modelled as a Thermally Activated Process

Models of the same kind of structure exist for d002 and La.

In principle, the same behaviour of the model for Lc is achieved if single reactionmodels for x1 and x2 are used instead of the multiple reaction models. Thereforeno simulations of the parallel ensemble of multiple reactions will be given here.The qualitative performance of the modelling principle is demonstrated below bythe use of single reaction models with conversion dependent activation energies.

10.4.2 Models Based on two Single Reactions with Conver-

sion Dependent Activation Energies

For Lc, the model now becomes:

dx1

dt= k1(x1(1)� x1) (10.6)

k1 = k1;�e�E1(x1)

RT (10.7)

E1(x1) = E1;min +(E1;max �E1;min)

x1(1)� x1(0)(x1 � x1(0)) (10.8)

dx2

dt= k2(x2(1)� x2) (10.9)

k2 = k2;�e�E2(x2)

RT (10.10)

E2(x2) = E2;min +(E2;max �E2;min)

x2(1)� x2(0)(x2 � x2(0)) (10.11)

Lc = x1 � x2 (10.12)

The model parameters are xi(0); xi(1); Ei;min; Ei;max, (i = 1; 2). This gives atotal of eight parameters in the model. For simplicity, x2(0) = 0. Also, x1(1) >x2(1) is needed for simulation of nominal baking temperature programs. Theactivation energies have to ful�ll Ei;min < Ei;max and the activation energies andpreexponential factors have to be tuned to achieve the correct shape of the Lccurve: Initially, there is a rise in Lc due to the increased value of x1. x2 shouldbecome active at temperatures in the order of 400�C. The growth in x2 shouldbe aggressive enough to facilitate a decrease in Lc. x2(1) should be reached by atemperature in the order of 800�C and for higher temperatures, x1 gives a furtherincrease in Lc.

The model was simulated for a heating rate of a = 15�C=hr up to a temperatureof 1250�C. The result is shown in Figure 10.1.

Also, the absence of growth of La in the temperature range corresponding to thestage of mesophase solidi�cation can be modelled in this way. Then, correspond-ingly:

La = x1 � x2 (10.13)

As shown in Figure 10.3, the impact on La from state variable x2 is slightlyrelaxed to avoid that the growth rate of La goes negative in the temperature

10.4 The Models 157

0 50 100 150 2000

200

400

600

800

1000

1200

Tem

p [o

C]

a = 15 oC/hr; Tf = 1100 oC

0 50 100 150 20010

15

20

25

30Lc = x1 − x2

Lc [A

A]

0 50 100 150 20010

20

30

40

50x1

Time [hr]

x1 [A

A]

0 50 100 150 2000

5

10

15

20x2

Time [hr]

x2 [A

A]

Figure 10.1: Lc as function of time. The model for Lc is realized as two parallelprocesses with temperature dependent activation energies.

0 200 400 600 800 1000 120010

12

14

16

18

20

22

24

26Lc = x1 − x2

Temperature [oC]

Lc [A

A]

Figure 10.2: Lc as function of temperature. The model for Lc realized as twoparallel processes with temperature dependent activation energies. Lc decreasesbetween approximately 400 and 800�C.

158 Crystallite Growth Modelled as a Thermally Activated Process

interval corresponding to the reduced growth activity. If another relationshipbetween initial- and �nal values of states x1 and x2 was selected, La could also becalculated by:

La = x1 + x2 (10.14)

Then, the following must be ful�lled:

� x1(0) < x1(1)

� x2(0) > x1(1)

The same modelling principle can also be used to model the rise in d002 betweenapproximately 400 and 800�C. For d002, we could select model parameters for x1and x2 which �ts with the expression for d002 as follows:

d002 = x1 + x2 (10.15)

where x1 and x2 are state variables in single reaction models with conversiondependent activation energies. Qualitatively, the parameters must apply to thefollowing speci�cations:

� x1(0) > x1(1)

� x2(0) < x1(1)

� E1;min < E1;max

� E2;min < E2;max

This approach to modelling of d002 is similar to the concept used for Lc andLa. Therefore, no parameters and simulations for this type of model for d002 isdiscussed in this chapter.

10.5 Conclusions

The combination of two processes with conversion dependent activation energiesallows for prediction of the deterioration in structure which occurs between 450and 800�C. This was demonstrated for Lc. Qualitatively, the same type of modelcan be used for d002. This approach was also used for modelling the absence ofgrowth in crystallite size La during the mesophase transition as demonstrated inFigure 10.3 and Figure 10.4.

In conclusion, the model with conversion dependent activation energy seems ex-ible enough to give qualitatively correct predictions of crystallite parameters overthe whole baking temperature range.

10.5 Conclusions 159

0 50 100 150 2000

200

400

600

800

1000

1200

Tem

p [o

C]

a = 10 oC/hr; Tf = 1250 oC

0 50 100 150 20010

15

20

25

30La = x1 − x2

La [A

A]

0 50 100 150 20010

15

20

25

30

35

40x1

Time [hr]

x1 [A

A]

0 50 100 150 2000

2

4

6

8

10x2

Time [hr]

x2 [A

A]

Figure 10.3: La as function of time. The model for La is realized as two paral-lel processes with temperature dependent activation energies. The growth of Laalmost halts in a certain part of the heat treatment program.

0 200 400 600 800 1000 1200 140010

12

14

16

18

20

22

24

26

28

30La = x1 − x2

Temperature [oC]

La [A

A]

Figure 10.4: La as function of temperature. The model for La is realized as twoparallel processes with temperature dependent activation energies. The growth ofLa is not active between approximately 400 and 800�C.

160 Crystallite Growth Modelled as a Thermally Activated Process

Chapter 11

Modelling of Physical

Properties and Carbon

Quality

A property vector z is de�ned and related to the concept of carbon quality. Typ-ically, electrical- and thermal conductivity, mechanical properties and reactiveproperties are important quantities when considering carbon quality: Carbon qual-ity is usually de�ned by specifying bounds on the ranges for physical properties.The physical properties involved in the quality concept will be included in z.

11.1 General Structure of the Model

11.1.1 Model Assumptions

The following assumptions are needed for derivation of the property models:

� The physical properties are studied at a macroscopic (bulk) level much largerthan the dimensions of the crystallites.

� On the macroscopic level, the physical properties are considered isotropic.

� Models partly appear as modi�cations of results found in the literature andpartly as the original results as they are presented in the literature.

� The physical properties are modelled as mathematical transformations of asubset of the state space vector; the so-called fundamental carbon proper-ties. In the set, the following fundamental properties (state variables) areincluded:

162 Modelling of Physical Properties and Carbon Quality

1. Parameters for classi�cation of porosity (i.e. pore size distribution andpore geometry)

2. Crystallite parameters

3. Coke texture

4. Coke impurities

11.1.2 Modelling Principle

The model based quality concept is based on mathematical models of the physicalproperties of the carbon material. The derivation of the quality concept goes viatree stages:

1. Derivation of models for the fundamental carbon properties. The mostchallenging task in this respect is the formulation of a state space which isextensive enough to give an adequate description of the basic transformationswhich occur in the carbon during heat treatment.

2. The derivation of mathematical transformations between the state space andproperty space.

3. The application of the property models in the carbon quality concept.

Physical properties viewed as transformations from a state space to a propertyspace is illustrated in Figure 11.1. This formulation is very attractive in thecontext of process control when focus is put on optimization and maintenanceof product quality and process economy. Mathematical objective functions forcarbon quality to be used in process optimization can easily be formulated withbasis in the property concept.

In general, properties and state variables are related as follows:

_x = f(x; u; v) (11.1)

z = d(x) (11.2)

where x and z are state- and property vectors respectively. u and v are vectorsof control- and disturbance variables respectively. The fundamental carbon prop-erties are included in the state space vector. In principle, this way of propertymodelling is applicable to any kind of material.

In the following, a description of the fundamental carbon properties is given.

11.1.3 Porosity and Pore Geometry

According to a previous discussion, total porosity as divided into open and closedporosity is assumed to give a satisfactory description of carbon porosity. Then:

p�=

����c

�(11.3)

11.1 General Structure of the Model 163

transformations

StructurePorosityPurityComposition

Physical

Chemical-

Mechanical-properties

Electrical-

Thermal-

Basicproperties of carbon: properties of carbon:

etc.

Mathematical

Rn! Rr

xd(x)

z

Figure 11.1: The principle of modelling physical properties of pitch and bindercoke. The fundamental properties belong to the state vector x since they aredescribed by di�erential equations of time, temperature and space. Furthermore,the physical properties belong to the property space z : Nonlinear transformationsexist between the state space and the property space.

This de�nition is well suited for modelling of porosity in single phase carbons.Here, the open- and closed porosities are assigned to the fundamental carbonproperties. Before, however, we have related the porosities to the apparent-, real-and crystalline- densities which are the real state variables.

Models for baked anode properties is also presented. In this case, the real (andapparent) density of the binder pitch and total porosity of the anode may belongto the set of fundamental carbon properties. Then, the apparent density and realdensity of the anode belong to the property space.

11.1.4 Crystallite Parameters

The crystallite parameters d, La and Lc are considered adequate for a descriptionof crystalline structure. The vector p

cof crystallite parameters is then given by:

pc=

24 d

LaLc

35 (11.4)

In general, more realistic descriptions of carbon structure based on other statisticalparameters can be given (Ruland 1965).

164 Modelling of Physical Properties and Carbon Quality

11.1.5 Texture

Coke texture can be parameterized by the average length of the axial componentof the anisotropic unit vectors and anisotropy ratio of physical properties of thecoke. This can be done for directions both parallel and perpendicular to thegraphite-like planes (Elalaoui et al. 1995), (Krebs et al. 1995). On the other hand,since coke texture is mainly given by the grade of the raw carbon material, andthe properties are modelled on a macroscopic level, no detailed discussion of theanisotropic parameters will be given in this work. In general, it is assumed thatthe in uence of anisotropy can be represented by a vector p

a.

In many cases, an index IOTI of optical texture is used for classi�cation of thedegree of anisotropy. IOTI increases as the degree of anisotropy increases. Insome cases, IOTI is used as a parameter to represent the in uence of coke textureon the physical properties of a carbon material.

11.1.6 Impurities

As for coke texture, the presence of coke impurities also depends on the rawmaterial used for manufacturing of the carbon. The impact of impurities likesulphur, ash and traces of metals with catalytic features can be represented by avector p

i.

11.1.7 Model Structure

The following type of transformation is de�ned for the physical properties:

z = d(p�; p

c; p

a; p

i) (11.5)

where:

� z: Vector of physical properties used for the de�nition of coke quality

� p�: Vector of porosity related parameters

� pc: Vector of crystalline parameters

� pa: Vector of anisotropy parameters

� pi: Vector of impurity concentrations

Introducing p�= [��; �c]

T , the model can be linearized as follows:

�z =@d

@x

266664��

�

��c

�pc

�pa

�pi

377775 (11.6)

11.2 Application to Anode Baking 165

Impurities and anisotropy are mainly determined by the raw materials. Anisotropycan also be manipulated during the liquid phase pyrolysis but only to a smallextent. As a good approximation �p

aand �p

iis close to zero. For the crystallinity

parameters, only small variations occur in the interlayer spacing parameter. Also,assume that �d002 � 0. Instead of �p

c, �La and �Lc are introduced to get:

�z = D

2664�����c�La�Lc

3775 (11.7)

where D =@d

@x. Controllability of �z depends on the rank of the matrix D

and features of the underlying state space model. Thus, the intrinsic nature ofthe process has impact on the controllability of the process. Controllability isdiscussed in part V of this work.

Impurity- and anisotropy vectors cannot be in uenced very much during the man-ufacturing process, and it is reasonable to assume that:

z � d(p�; p

c; p

a;�; p

i;�) (11.8)

where pa;�

and pi;�

are nominal values of the anisotropy and impurity vectors.

Thus, the important contributions to carbon property evolution during manufac-turing comes from the porosity and crystallinity parameters:

z � ~d(p�; p

c) (11.9)

Thus, the properties depend only on the porosity and the crystallinity parameters.

11.2 Application to Anode Baking

In part V, a method for model based control of ring furnaces is presented. Themethod includes optimization and control of anode properties. For anodes, thequality concept may include airburn, CO2-reactivity, dusting, compressive strength,thermal shock resistance, thermal conductivity, speci�c electrical resistivity andapparent density. Thus, the following set of anode properties can be used as basisfor de�ning the anode quality concept (dim(z) = 8):

z =

266666666664

�rO2

�rCO2

�si��c��ts�kt��e��b;a

377777777775

(11.10)

166 Modelling of Physical Properties and Carbon Quality

The anode quality as expressed by the property vector plays an important role inboth o�-line (i.e. open loop) optimization and on-line control of the ring furnace.If z is used for control purposes, the actual dimension of z must be related tothe controllability of z. In a ring furnace, control of z must be performed so asto arrive at desired values of z in the last section of the ring furnace. In partV of this work, a controllability analysis for a linearized model of z is discussed.The analysis is performed according to the principle described in Balchen (1984,pp. 377, Ex. 8.7.1). The state space model in part IV of this work was used forcalculation of the fundamental carbon properties needed for calculation of z.

The controllability of the property space depends on the rank of matrix D andproperties of the underlying state space model. Since, according to Equation (11.7),the rank of D can at most be four or below, controllability of the full propertyvector de�ned by Equation (11.10) cannot be achieved. To achieve controllability,a transformed property space must be used for on-line control of the properties.Therefore:

�~z = ~D�z (11.11)

dim( ~D) = ~r � 8 and ~r � 4.

As an example, consider the control a weighed reactivity index (or dusting index)and thermal shock resistance. A linearly transformed property space can be usedin on-line control by specifying matrix ~D as follows:

~D =

"f

�rO2;�

(1�f)�rCO2;�

1 0 0 0 0 0

0 0 0 0 1 0 0 0

#(11.12)

where �rO2;� and �rCO2;� are the optimal reactive properties of the anode carbon. Itis reasonable to set f = 0:5 thus putting equal weight on the reactivity measures.The �nal propery vector �~z is related to the state space x vector by:

�~z = ~DD�x (11.13)

The role of the underlying state space model is discussed in part V of this work. Itwill be shown that to achieve controllability in carbon anode baking, the propertiesz cannot be chosen arbitrarily.

Part III

Mathematical Modelling of

Pyrolysis

Chapter 12

Qualitative Description of

Pitch Pyrolysis

Both chemical and physical processes are active during pitch pyrolysis. In thefollowing, a survey of characteristic features of pitch pyrolysis is given. The focusis on pitch pyrolysis of coal tar pitch in general. Still, the presentation is relevantfor the process of pitch pyrolysis that occurs in carbon anodes during baking.

12.1 Introduction

A commonly used approach in studies of pitch pyrolysis is the use of non-isothermalthermogravimetric weight loss experiments performed at a linear temperaturerate1:

T = T� + at (12.1)

a is the temperature rate nominally in the order between 5�C=hr and 50�C=hrduring anode baking (Wilkening 1983). During experimental work, however, heat-ing rates are higher than these. t is time and T� is the sample's initial temperature.Cumulative curves of total weight loss or weight loss of individual pyrolysis gases(detected by the use of chromatographic techniques) are measured. Focus is puton formation of volatiles rather than describing the state of the pyrolytic residue.Most often the weight loss curves are normalized. Thus, it is di�cult to obtaininformation from the literature on actual weight loss and corresponding coke yieldin dependence of varying experimental conditions.

Experiments are most frequently done with pure pitch. Sometimes, however, acertain fraction of petroleum coke of a certain granularity is mixed with the pitchto obtain a green paste which is pyrolysed. Experiments are also performed tostudy pyrolysis of laboratory scale anodes.

1Often called heating rate in the carbon literature.

170 Qualitative Description of Pitch Pyrolysis

Experimental conditions in the laboratory may di�er very much from real bakingconditions. According to Fitzer & H�uttinger (1969), the following mechanismsdetermine the allowed baking rate:

� Rate of heat transport

� Rates of pyrolysis reactions

� Rates of gaseous transport through the anode

The mechanisms' impact on baking may change during the baking cycle.

As illustrated in Figure 12.1, pyrolysis of coal tar pitch sequentially goes througha series of process regimes in sequence as the temperature increases:

� Distillation of low molecular weight pitch fractions which is observed as evol-ution of tar (up to 400�C). This contributes to increasing the average molec-ular weight of the pitch.

� Decomposition (cracking), polymerization and polycondensation of the bulkpitch with formation of liquid and gaseous products (350 to 550�C). Poly-merization and polycondensation also contribute to increase the averagemolecular weight of the pitch. As the average molecular weight of thepitch increases, van der Waal's forces become stronger and liquid crystal-lites (mesophase) is formed in the isotropic pitch due to these associativeforces between planar aromatic molecules.

� Solidi�cation of the highly viscous pitch residue (between approximately 550to 600�C). The formation of mesophase and continued polymerization withinthe mesophase is responsible for this phase transition.

� Transformation of the pitch semicoke into pitch coke (above 600�C and up toapproximately 1000�C). This transformation occurs under dehydrogenationand demethylization and release of light hydrocarbons (Born 1974a).

Mesophase

Condensables

+

Non-condensables

Non-condensables

Semicoke+

Solidification

Non-condensables

Carbonization

Liquid isotropic pitchSolid pitchMelting

Evaporation

starts

+Solid Coke

200�C

600�C

400�C

Figure 12.1: Transformation in the pitch during pyrolysis. The isotropic pitchmainly consists of bi- and oligoaryls. The mesophase constitutes large peri-condensed aromatic systems.

Pitch coke yield and quality severely depend on the processing conditions in thetemperature interval between 350 to 550�C. Not only the temperature programme

12.2 Volatile Compounds in Pitch Pyrolysis 171

is of importance: For a certain heat treatment programme, it was found that thestructural transformations taking place in the pitch fraction during baking are notonly dependent on the properties of the original pitch but also very much on theambient atmosphere during pyrolysis (Fitzer & Terwiesch 1973), (Svirida, Markina& Fedoroseev 1976), (Sverdlin, Priezzhaya & Yanko 1991), (Sverdlin, Priezzhaya& Yanko 1992).

Figure 12.2: Mechanism of polycondensation reactions. From Ko�st�al et al. (1994).

12.2 Volatile Compounds in Pitch Pyrolysis

The pitch constitutes approximately 15 to 20 % of the total mass of the greenanode. During heat treatment of the green blocks, volatiles start to develop. Itis not possible to give a detailed picture of the kinetics of each component of thevolatile gases. Instead, the volatiles are grouped into two main groups (or lumps)of hydrocarbons:

� Condensable hydrocarbons which are mainly polycyclic aromatic hydrocar-bons). A description of some of the components which belong to this groupof volatiles may be found in Charette, Ferland, Kocaefe, Couderc & Saint-Romain (1990).

� Non-condensable hydrocarbons which are mainly hydrogen and methane.

The complex kinetics of the volatile formation is transformed into that of study-ing the kinetics of three fractions; tar, methane and hydrogen. The condens-able hydrocarbons appear during distillation of the lightest fractions of the pitch.The non-condensables appear in complex chemical reactions: polymerization andcracking at temperatures above 400�C. Thus, the volatiles appear as three majorcomponents at di�erent temperature intervals:

172 Qualitative Description of Pitch Pyrolysis

� Tar (CHn) is released at temperatures between approximately 200 and 500�C.

� Methane (CH4) is released at temperatures between approximately 350 and400 and up to 800�C.

� Hydrogen (H2) is released at temperatures between approximately 350 and400 up to 1000�C.

It should be noted that traditional thermoanalytical methods cannot distinguishpolymerization from vaporization between 300 and 500�C (Boenigk & Nieho�1993). Thus, gas chromatography is needed to separate condensables from non-condensables.

Charette et al. (1990) observed 14 PAH-compounds in the molecular weight rangefrom 166 to 276 g/mol. It was shown that PAH constitute in the order of 70 %of the total mass of hydrocarbons. The presence of medium- and high molecu-lar weight PAH increases with temperature and the low molecular componentsobserved in the pyrolysis gas seems to level o� as temperature increases.

According to Politis & Chang (1985), H2 and CH4 �rst appear at temperaturesbetween 400 and 500�C. Romovacek (1983) also argues that thermal decomposi-tion of pitch does not occur before approximately at 400�C. Politis observed twopeaks in the degassing rate: The �rst occurred at 460�C due to polymerizationreactions taking place in the liquid pitch. After a signi�cant decrease in degassingrate and a minimum value observed at 580�C, the second maximum was observedat 730�C. The decrease in the degassing rate is due to solidi�cation of the pitch.The qualitatively same observation was done by Tremblay & Charette (1988), andit was shown that the second maximum which occurs at 700�C was due to therelease of H2. This is also in accord with observations done by Born (1974a) andBorn (1974b, �g. 7). However, Born's measurements show that the minima indegassing rates of methane and hydrogen occur at 650 and 570�C respectively.Politis & Chang (1985) found that the volumetric ratio between H2 and CH4 wasfairly constant between 500 and 800�C. However, according to Kocaefe, Charette,Ferland, Couderc & Saint-Romain (1990), the amount of hydrogen is small below500�C.

Other non-condensable gases than CH4 and H2 are also released during pyroly-sis. Born (1974a) registered traces of ethane and small traces of higher aliphaticcompounds at temperatures between 450 and 600�C. In the same temperatureinterval, traces of CO and CO2 were also observed. However, these compoundsoccur only in negligible quantities.

Many authors have recognized that the volatilization of pitch seems to occur intwo distinct regimes. First there is release of condensables up to approximately500�C. Above 500�C, mainly non-condensablesH2 and CH4 are released where theamount of H2 dominates over CH4 (Charette, Kocaefe, Ferland & Couderc 1989),(Charette et al. 1990)(Kocaefe et al. 1990), (Charette, Kocaefe, Couderc & Saint-Romain 1991), (Politis & Chang 1985). However, it must be stressed that a smallamount of the non-condensables is released between 400 and 550�C due to the poly-merization reactions in the pitch. At higher temperatures, the non-condensables

12.3 E�ect of Temperature Rate 173

are released in cracking reactions which take place in the solidi�ed pitch. Thus,there is a gradual change in the reaction mechanism as solidi�cation goes by andcarbonization changes from reactions mainly in liquid state to reactions mainlyin solid state (Greinke 1986). In the later stage of carbonization, the processchanges from a chemical process to become a physical process in which the mobil-ity of the carbon lamellae is rate determining. Thus, carbonization is no longer amolecular process. This observation is also con�rmed by the data from Politis &Chang (1985): A minimum in the gas release rate occurred at 580�C as shown inFigure 12.3.

Based on information available in the literature, the main part of the weight lossin pitch pyrolysis is due to the vaporization of condensables at temperatures below500�C. In Tremblay & Charette (1988) loss of non-condensables amounts to lessthan 10 % of the total mass loss. On the other hand, Politis & Chang (1985)report that 80 % of the weight loss occurs below 500�C. The fraction of the totalweight loss of non-condensables depends on the chemical constitution of the pitch.Coal tar pitch is mainly aromatic with a very low hydrogen content, and thereforethe condensables are the most important source for the weight loss.

100 200 300 400 500 600 700 800 900 1000

460 580 730

r[kgm3s ]

T [�C]

Figure 12.3: Rate of release of non-condensables in pitch pyrolysis. A minimumin the gas release rate is observed just below 600�C which is related to a changein the kinetics of the carbonization reactions. The heating rate is 120�C=hr. Theminimum in the gas release rate observed at approximately 600�C is not associatedwith a certain reaction step but rather to the solidi�cation of the bulk pitch ma-terial. The peak value corresponds to 0.3 ml/(gmin). Based on Politis & Chang(1985, Fig. 4).

12.3 E�ect of Temperature Rate

One commonly observed e�ect of a varying temperature rate is that the kineticparameters in traditional single reaction weight loss models vary as the heating ratevaries. Both Collett & Rand (1980) and Tremblay & Charette (1988) observedthat the activation energy and preexponential factors increased with increasingtemperature rate. A general increase was also observed for increased temperatures(Collett & Rand 1980), (Chistyakov, Denisenko & Itskov 1985). H�uttinger (1970)

174 Qualitative Description of Pitch Pyrolysis

used average activation energies for the decomposition of pitch oils and N -resinsand allowed for variation of preexponential factors with the heating rate. Again,the pyrolysis kinetics depends on experimental conditions.

This is in contrast with the observations done by H�uttinger (1971) and Wallouch,Murty & Heintz (1972) who studied non-isothermal pyrolysis kinetics at heatingrates of 66 to 288�C=hr and 25, 50 and 100�C=hr respectively: No parametricdependence of k� and E on the heating rate was found. However, other authorshave also observed a parametric dependence on the heating rate in non-isothermalexperiments. Schucker (1983) concluded that no model with single activationenergy can describe the decomposition reactions taking place in coking of ArabHeavy Vacuum Residuum.

Instead of a single reaction model, a model based on distributed kinetic parametersshould be used. Buttler (1975), in studies of coal tar pitch, also adhere to thisview. Since pitch pyrolysis can be divided into the following main stages (Collett& Rand 1980, pp. 154), (Buttler 1975, pp. 568):

� Pure vaporization (distillation) of low molecular compounds

� Vaporization and chemical reactions in parallel

� Phase transition (formation of mesophase)

� Reaction in solid state

it is not likely that a single reaction model should be able to describe pyrolysis fordi�erent operation conditions.

Not only do the kinetic parameters depend on the heating rate. The ultimateweight loss or coke yield also depends on the heating rate. According to H�uttinger(1970) and Buttler (1975), weight loss was slightly increased as the heating ratedecreased. The following explanation of the phenomenon was put forward: Agreater proportion of the volatile material escape without undergoing thermaldegradation when the heating rate is low. Faster heating gives an earlier formationof solid material through which the volatile matter has to di�use and this willretard weight loss. The heating rates were 80, 160, 360 and 55�C=hr i.e. therates are signi�cantly higher than those used during baking. The same e�ect wasobserved by Collett & Rand (1980) by using heating rates of 24 up to 600�C=hr.Also, Kocaefe et al. (1990) observed a decreased coke yield for lower heating rateswhen studying weight loss characteristics of pitch impregnated electrodes. Heatingrates of 3, 12 and 60�C=hr were used with isothermal intervals of 15 hour attemperatures of 493 and 673 K.

However, these results are in contrast with other observations in the literature.Wilkening (1983, Fig. 10 ) presents weight loss curves for heating rates of 5, 11,25 and 50�C=hr which show that coke yield decreases as heating rate is increasedfor pyrolysis of pure pitch. The qualitatively same behaviour was also observedduring pyrolysis of small green anode samples. Gyoerkoes (1971) also argues thata low heating rate increases coke yield in the anode. These observations were also

12.3 E�ect of Temperature Rate 175

a [�C=hr] CY [wt.%]

10 49.1

15 46.7

20 41.8

Table 12.1: Experimentally observed coke yields in pitch pyrolysis as function ofheating rate.

a [�C=hr] Weight loss [wt.%]

10 6.5

15 6.9

20 7.1

Table 12.2: Experimentally observed weight loss of a baked anode.

con�rmed in laboratory experiments performed at Hydro Aluminium, �Ardal asreported in Table 12.1 and Table 12.2. Finally it must be mentioned that Fitzer& H�uttinger (1969) and Fitzer & Terwiesch (1973) used heating rates of 51 to564�C=hr and 35 to 600�C=hr respectively but did not �nd a correlation betweenheating rate and coke yield.

The fact that the weight loss curve is moved to higher temperatures as the heatingrate increases is simply an e�ect of the reduced residence time needed to reacha certain temperature level: Vaporization which depends on the residence time,does not proceed long enough to reach the same yield as obtained with a lowerheating rate.

In this work, it is assumed that coke yield increases as heating rate decreases. How-ever, as seen from the above discussion, a physical explanation for this behaviouris not obvious. One might assume that there is a change in the mechanisms re-sponsible for the net mass loss as the heating rate is varied. Volatilization seems tobe relatively more important than coking rate as the heating rate increases withinreasonable bounds (i.e. below 50�C=hr). This seems to be due to the dominanceof volatilization over coking reactions at high temperatures. With a high heatingrate, high temperatures are more quickly reached than with a low heating rate andthis favours cumulative weight loss via volatilization. On the other hand, it wouldbe plausible that the increased residence time for low heating rates would favourthe volatilization and thus would decrease coke yield as heating rate decreases.This does not seem to be the case: On the contrary, coking reactions seems to befavoured by a low heating rate.

In summary, the heating rate have impact on both coke yield and kinetic param-eters.

176 Qualitative Description of Pitch Pyrolysis

12.4 E�ect of Coke Additives

In Fitzer & H�uttinger (1969), green anode paste made from certain fractions ofcoke of varying granularity was pyrolysed. The shape of the normalized weightloss curve was not in uenced by varying coke granularity for pyrolysis of greenpaste. However, the absolute coke yield increased with increasing grain size. Alsoit was found that the coking reactions were chemically activated by the presence of�ller coke. This contributes to move the weight loss curve to a lower temperatureas compared to pyrolysis of pure pitch (Fitzer & H�uttinger 1969, Fig. 5). Theincreased coke yield in coarse grained green paste was explained as being the resultof increased importance of secondary coking reactions. The increased capillarityof large sized �ller grains gives the opportunity for secondary coking reactions totake place and thus contribute to the increased coke yield. The same qualitativebehaviour is observed in Born (1974a, �g. 2).

For small grained �ller, capillarity is low and pyrolysis occurs for the most directlyinto ambient gas with less opportunity for secondary coking reactions to takeplace. It should be noted that the presence of �ller coke opposes the displacementof the normalized weight loss curves to higher temperatures as a result of increasedpartial pressure of pyrolysis gases.

It is important to note that the catalyzing e�ect of the coke grains will be more orless lost in baking of large scale anodes. The catalytic e�ect will be dominated bythe e�ect of secondary coking reactions as introduced via the size of the anodes. Itwas observed that coke yield increased with the diameter of laboratory scale anodesimplying that secondary reactions become more important in the larger samples.This is in contrast with the observations done in (Tremblay & Charette 1988) whodid not observe a dependence of weight loss on the size of the anode sample.

12.5 E�ect of Ambient Atmosphere and Pressure

According to Svirida et al. (1976):

� Yield of coke, tar and gases

� Volatile matter in coke residue

� Reaction kinetics

� Air reactivity of coke residue

depend on both coking temperature and ambient atmosphere. Atmospheres ofpitch volatiles, forced air and forced nitrogen were used. For a given coking tem-perature, coke yield decreases for atmospheres of forced air, pitch volatiles andforced argon respectively. The composition of tar showed that the amount of as-phaltenes and phenols was higher in an atmosphere of forced air. This shows thedependence of the reaction kinetics on the ambient gas.

12.6 E�ect of Secondary Coking Reactions 177

In Fitzer & H�uttinger (1969), it was found that during pyrolysis of pure pitch, the(normalized) cumulative weight loss curve is moved to higher temperatures whenpyrolysis takes place in an atmosphere of pitch volatiles. The in uence on cokeyield was not explicitly stated.

In a separate study reported in Fitzer & Terwiesch (1973), it was shown thatthe total gas pressure has severe impact on pyrolysis. However, the in uence ofpressure is mainly con�ned to temperatures below 550�C and the e�ect of totalgas pressures (nitrogen) above 25 bar is negligible. Internal pressure in the anodelies below 2 bar during baking (Jacobsen & Log 1995) and this may contribute tochanges in coke yield both due to increased partial pressure of pitch volatiles aswell as the level of the total pressure. The importance of gas pressure is re ectedin two ways:

� Vaporization of light molecular weight compounds

� Chemical reactions during formation of non-volatile polyaromatics

Later, H�uttinger (1986), H�uttinger (1988) and H�uttinger (1989) explained thee�ect of ambient atmosphere and total pressure on pyrolysis from a chemical en-gineering point of view. In H�uttinger (1986) and H�uttinger (1988), equationsfor vaporization under isochoric and isobaric conditions were derived. Ideal equi-librium (Raoult's law) was assumed and vapour pressure correlations for purehydrocarbons were used for deriving the equations. It was shown how coke yielddepends on both process parameters as well as the properties of the hydrocarbon.Gas phase reactions were neglected since they are slow compared to liquid phasereactions.

In a closed system (isochoric case) the amount of vaporized material increases withtemperature due to an increased vapour pressure. The initial amount of material inthe reactor is also of importance: A decreased reactor �lling ratio gives a increasedratio of vaporized substance. The initial pressure of inert gas is of no importancefor the relative amount of matter which vaporizes. At the same temperature,the amount which vaporizes from a (semi)-open system (isobaric case) will belarger then in a closed system. Vaporization kinetics was studied separately. Itwas shown that the amount of volatiles which vaporize depends indirectly on thetotal pressure via the di�usion coe�cients. H�uttinger (1989) found a correlationbetween coke yield and total pressure for pyrolysis of coal tar pitch. The impactof reaction kinetics was not directly taken into account in the model.

An equation for the vaporization ux in non-equilibrium open systems was alsogiven.

12.6 E�ect of Secondary Coking Reactions

In the above discussion on the catalytic e�ect of coke grains, it was shown that sec-ondary coking reactions may be of importance for the overall coke yields observed

178 Qualitative Description of Pitch Pyrolysis

in anode samples. These secondary reactions contribute to �lling the porous net-work in the anode. This phenomenon was studied by Fitzer & H�uttinger (1969)who observed a radially decreasing permeability in cylindrically shaped anode sam-ples. The decrease in permeability depends on the baking level. For a baking levelof 800�C, permeability increased with approximately 600 % from anode peripheryto anode center.

12.7 The Pyrolysis Reactions' impact on Bond

Coke Structure

The texture of green coke formed during carbonization depends on the viscositychanges, the pitch solidi�cation process and the gas evolution during carbonization.The rate of change of viscosity and the solidi�cation process depend on the rateof volatilization of the light pitch components as well as the rate of condensationreactions. The evolution of light gases is mainly due to cracking reactions. Theformation of coke texture mainly goes via two stages:

� The formation of bulk mesophase which accompany the growth and coales-cence of anisotropic Brooks & Taylor spheres.

� The uniaxial arrangement of the mesophase caused by the escape of car-bonization gases more or less syncronized with the solidi�cation of the meso-phase.

The best possible orientation is obtained if the maximal gas evolution rate occursjust at the mesophase solidi�cation stage (Elalaoui et al. 1995), (Krebs et al. 1995).For this to happen, an intimate balance between carbonization temperature andambient pressure has to be maintained. It has been shown that for a given car-bonization temperature, the optical texture anisotropy increases regularly withincreased carbonization pressure. A nomenclature for classi�cation of optical tex-ture is given in Table 12.3. A lot of authors have shown that correlations existbetween optical anisotropy as represented by the average length of the axial com-ponent of the anisotropic unit vectors and anisotropy ratio of physical propertiesof the green coke (CTE, electrical resistivity etc.). This is valid for directionsboth parallel and perpendicular to the the graphitic planes (Elalaoui et al. 1995),(Krebs et al. 1995).

In baked carbon anodes, however, an anisotropic texture of the binder coke isnot desired since anisotropy tends to reduce the strength of the coke as well asincreases the coke reactivity. Therefore, binder pitches have a content of primaryquinoline insolubles (QI) in the order of 10 % to obtain a binder coke with amore disordered optical texture. The presence of primary QI prevents size growth(i.e. not nucleation) and coalescence of mesophase spheres and contributes to theformation of a disordered binder coke.

In this study, it is assumed that the degree of isotropy of the binder coke is mainlydetermined by the speci�cation of the primary QI contents of the pitch rather than

12.8 Heat of Reactions 179

Optical texture Abbreviation Size [�m]

Isotropic I No optical activity

Fine-grained mosaic Mf < 1

Medium-grained mosaic Mm 1-5

Coarse-grained mosaic Mc 5-20

Weak ow anisotropy Fw < 5 in width, length is 3 � width

Medium ow anisotropy Fm 5-10 in width, length is 3 � width

Strong ow anisotropy Fs > 10 in width, length is 3 � width

Table 12.3: Classi�cation of anisotropic texture according to the InternationalCommittee of Coal Petrography. From Krebs et al. (1995). An alternative nomen-clature is given in Marsh (1989, pp. 20).

the proper conduction of the baking process. Therefore, a model of binder coketexture development is not needed.

Qualitatively, results from the literature support our assumption about a more orless constant optical texture (isotropic texture dominate in the binder coke) of thebaked anode. It has been shown that there is a strong linear correlation betweenCTE and the degree of optical texture anisotropy (Elalaoui et al. 1995). Also,it is commonly accepted that CTE for anodes is more or less constant and nota�ected very much by the baking strategy (Schneider & Coste 1993). This allowsus to conclude that the texture established in the binder coke at the end of themesophase solidi�cation stage, is more or less constant throughout the anode.

12.8 Heat of Reactions

According to Jones & Hildebrandt (1975), the early stage of pitch pyrolysis isbasically endothermic. This is mainly due to the energy needed for volatilizationas well as primary pyrolysis. However, at temperatures between 300 and 400�C,volatilization occurs in parallel with the conversion of -resins to �-resins. Theseare exothermic polymerization reactions, but the overall reaction heat is probablyendothermic. At temperatures above 400�C, the conversion of to �-resins occursin parallel with conversion of � - resins to �-resins. These are exothermic poly-merization reactions and the overall reaction heat is exothermic. At temperaturesabove 400�C exotherms have been observed. Qualitative observations of exother-mal and endothermal e�ects in pitch pyrolysis during anode baking has also beendone (Sverdlin et al. 1991), (Sverdlin et al. 1992). It was found that exothermale�ects were more pronounced in ambient air than in an ambient inert atmosphere(argon) due to oxidizing e�ects. Otherwise, several combinations of exothermaland endothermal e�ects were observed.

Little information is available in the literature on the magnitude of the heat ofreaction in pitch pyrolysis. Some work has been done on coal pyrolysis (Burke& Parry 1927), (Davis & Place 1924) which supports that the heat of reaction is

180 Qualitative Description of Pitch Pyrolysis

generally exothermic in the order of 40 to 160 kJ/kg for temperatures between400 and 600�C. Also, Davis & Place (1924) studied heat of reaction as function oftemperature. Up to 400�C, �H is generally endothermic. The interval from 400and 700�C is exothermic, but probably �H corresponds to endothermic reactionsabove 700�C. Based on these references, Howard (1981) concludes that the netheat of pyrolysis reactions is small enough to be neglected since reaction heatintroduces only second-order e�ects in most heat treatment strategies.

12.9 Conclusions

In pyrolysis of coal tar pitch, The following phenomena are generally observed:

� For an increasing heating rate, a normalized conversion curve is shifted to ahigher temperature but keeps essentially the same shape.

� The yield of total volatiles increases with an increasing heating rate.

� The release of condensables and non-condensables seems to cluster in twodistinct temperature intervals:

{ Condensables appear mainly between 200 to 500�C.

{ Non-condensables appear at temperatures above 500 and up to 1000�C.Methane evolution ceases between 700 and 800�C. The release of hydro-gen ceases just below 1000�C as can be seen from the curves reportedin Tremblay & Charette (1988).

� The presence of �ller coke dust in the pyrolysing pitch seems to catalyse thechemical reactions, i.e. the weight loss curve is shifted to a lower tempera-ture.

� An increasing pressure in the surroundings shifts the weight loss curve tohigher temperatures.

The following parameters have impact on the progression of pitch pyrolysis:

� Heating rate: Most likely, the coke yield decreases with increasing heatingrate for heating rates nominally used in anode baking.

� Ambient pressure: An increased ambient pressure increases coke yield. Thise�ect is active at pressures up to approximately 25 bar.

� Coke additives: Both weight fraction and granularity are of importance.Coking reactions are catalyzed and coke yield increase.

The kinetics of pyrolysis is complex and no single activation energy model is able todescribe pitch pyrolysis. Still, however, state of the art in mathematical modellingof pitch pyrolysis uses thermogravimetric weight loss models with single activationenergies.

Chapter 13

Modelling Approaches in

Pyrolysis

Most often, pitch pyrolysis models are (semi-) empirical based on Arrhenius-likerate laws. Experimental data are most often obtained by thermogravimetric meth-ods. A general introduction to thermal analysis is given in Wendlandt (1986). Inthis chapter, focus is put on techniques commonly used for modelling pyrolysis ofcoal tar pitch.

13.1 Approaches in Pyrolysis Modelling

13.1.1 Introduction

Progress in computer technology has allowed for complex models to be used inmany branches of hydrocarbon processing. Large reaction schemes for pyrolysisof gaseous hydrocarbons based on free radical mechanisms are commonly used.However, since pitch consists of thousands of components, a detailed descriptionof pitch pyrolysis based on the component's kinetic behaviour is not feasible. Adescription of pitch pyrolysis suitable for engineering purposes should be based ona kind of lumped reaction scheme of a semi-empirical nature. On the other hand,such models have limited interest since they are only suited for use in predictionunder the same process conditions as used for obtaining the experimental basis forcurve-�tting the model. These models are not suited for predictions outside theirnominal operation range even though they are often strongly nonlinear. This isdue to the lack of fundamental physics built into the model.

Research on coal pyrolysis may serve as basis for studying di�erent modellingtechniques for describing devolatilization during pyrolysis. Smith, Smoot, Fletcher& Pugmire (1994, cht. 5) systematically discuss di�erent modelling approaches.The following types of pyrolysis models, relevant also for studying pitch pyrolysis,

182 Modelling Approaches in Pyrolysis

may be distinguished:

1. Phenomenological models:

� Schemes of independent reactions:

{ Models based on single reactions

{ Models based on multiple (parallel) reactions

� Schemes of dependent reactions:

{ Consecutive reaction schemes (series decomposition)

{ Competitive reaction schemes

2. Chemical models:

� Functional Group Devolatilization Models / Free Radical Schemes

The single- and multiple reaction models may be used for modelling both total1 -and component-wise weight loss. The application of such models in the context ofcoal pyrolysis is reviewed in Anthony & Howard (1976), Howard (1981), Saxena(1990) and Smoot (1991).

Phenomenological models do not take into account the actual chemical structureand elementary reactions. Still reliable predictions of total or component-wisevolatile loss can be achieved if the model is used in conditions comparable to theexperimental basis used for deriving the model parameters.

In the chemical models, actual structure as represented by the functional groupsare used for specifying a set of elementary reaction mechanisms. A lot a kineticparameters can be calculated from thermochemical methods. In contrast to this,parameters in the phenomenological models can only be derived by curve �ttingof model predictions to experimental data.

In coal pyrolysis, there are probably no intrinsic competing reactions since exper-iments on small coal particles have shown that product yields are independent ofthe time-temperature history during pyrolysis (Suuberg, Peters & Howard 1978).In this case, independent reaction models will give acceptable predictions. Forlarger particles, however, secondary pyrolysis reactions play a role. Intraparticlemass transport competes with the secondary reactions and there is a need for morecomplicated models.

So far, only phenomenological models have been used for modelling of coal tarpitch pyrolysis. Both single reaction models and consecutive reaction schemeshave been used both in modelling of low (i.e. liquid phase)- and high temperaturepyrolysis for predicting the release of volatile gases.

In low temperature pyrolysis, the single reaction schemes are not capable of pre-dicting the dependence of volatile yield on processing conditions (ambient atmo-sphere and pressure, temperature, heating rate, secondary reactions in the de-volatilized tar etc). In high temperature pyrolysis of the solidi�ed semi-coke, in-dependent reaction mechanisms seems more probable. Aromatic layers are linked

1These models are often denoted global or overall reaction models in the literature.

13.1 Approaches in Pyrolysis Modelling 183

together in a network with aliphatic groups or single hydrogen atoms on the pe-riphery. In the reactions, these peripheral units are released. If secondary crackingcan be neglected, it is reasonable that the ultimate yield of gases is independentof the time-temperature history of the pyrolysis.

Consecutive reaction schemes have been used in prediction of the changes in theresidue during pyrolysis. In general, the description of volatile release is not fo-cused on in these models.

In reality, physical processes and chemical reactions occurring during pyrolysisare neither purely independent or competing and a model with features of bothindependent- and consecutive schemes seems to be needed.

13.1.2 Single Reaction Schemes

Model Equations

Commonly, the pyrolysis process is assumed to be an irreversible chemical reactionsymbolically represented by the following equation (Tremblay & Charette 1988,pp. 87):

S(s)! rR(s) + (1� r)V (g)

where

� S is the initial solid phase of mass m� containing both volatiles and non-volatiles.

� R is the solid residue after pyrolysis with mass mr;�.

� V is the volatile matter developed during pyrolysis. The total mass ofvolatiles bonded in the solid material is denoted msv;�.

� r is a stoichiometric coe�cient.

Often, single volatile components are considered so that2:

S(s) =

nvXi=1

Si(s)

R(s) =

nvXi=1

Ri(s)

V (g) =

nvXi=1

Vi(g)

where nv is the number of volatile components escape during pyrolysis. Nowde�ne:

msv;� = f�m� (13.1)

2The sums refer to summation of component masses.

184 Modelling Approaches in Pyrolysis

for the total volatile mass and

msv;i;� = f�i m� (13.2)

for the total mass of component i capable of being released during pyrolysis. f�iis the fraction of volatile component i initially present in mass m�. Since msv;� =Pnv

i=1msv;i;�, the following applies:

nvXi=1

f�i = f� (13.3)

In modelling decomposition of solid materials, the reaction mechanisms usually areconsidered too complex to be modelled from "�rst principles". A semi-empiricalapproach is very often used. A single n'th order reaction is used to model therelease of volatile components. The modelling is performed either on a total massbasis or by considering individual volatile compounds.

The mass msv;i of volatile component i still left in the solid material is given by:

dmsv;i

dt= �~kimni

sv;i (13.4)

~ki = ~ki;� exp(�Ei

RT)

msv;i(0) = msv;i;�

The model gives satisfactory prediction of weight loss behaviour for certain speci-�ed operation conditions. As soon as the operation conditions are changed, anotherparameter set is needed to properly describe the weight loss. This is due to themodel's rather approximate description of the actual processes which take placeduring mass decomposition.

Often a normalized quantity is used to represent the degree of conversion duringpyrolysis. mv;i is the cumulative amount of volatile component i which is releasedduring pyrolysis. Hence:

msv;i;� = mv;i +msv;i (13.5)

Introduce conversion Xi and solve with respect to msv;i:

msv;i = msv;i;� �mv;i = msv;i;�(1�Xi) (13.6)

where Xi =mv;i

msv;i;�is the relative amount of volatile component i released up to

time t. Xi is a normalized quantity where:

Xi(0) = 0

Xi(1) = 1

Substitution of Equation (13.6) into Equation (13.4) gives:

dXi

dt= ~kim

ni�1sv;i;�(1�Xi)

ni

+dXi

dt= ki(1�Xi)

ni (13.7)

13.1 Approaches in Pyrolysis Modelling 185

which is the standard form of the model most often encountered in the literature.ki is a compound parameter. Either �rst order models are used or a certainreaction order is assumed. The model gives no direct information of the absoluteweight loss. The model can also be formulated as a di�erential equation for therelative weight loss. Set msv;i;� = f�i m� which gives:

msv;i = f�i m�(1�Xi)

+msv;i = m�(f

�

i � f�i Xi) (13.8)

Now introduce f�i Xi = fi and obtain:

msv;i = m�(f�

i � fi) (13.9)

which gives:

dfi

dt= ~kim

ni�1�

(f�i � fi)ni

+dfi

dt= ki(f

�

i � fi)ni (13.10)

With this model, conversion with respect to the total initial mass is calculated.ki used in this equation must not be confused with the quantity ki used in Equa-tion (13.7). In the literature on pitch pyrolysis, the formulation in Equation (13.7)is most commonly encountered.

Weight loss experiments are usually performed non-isothermally at a linear heatingrate a:

dT

dt= a

By introducing temperature as the independent variable, the following equationfor the normalized weight loss parameter Xi is obtained:

dXi

dT=ki

a(1�Xi)

ni (13.11)

By de�ning:

msv;i = (1�Xi)f�

i m� = Xif�

i ��V�

an expression for the weight loss rate is obtained as follows:

rsv;i = � 1

V�

dmsv;i

dt= �f�i ��

Xi

dt(13.12)

rv;i = �rsv;i (13.13)

The same type of model is often used for modelling of total volatile weight loss.The model equations for total weight loss is obtained from the above expressionsby removal of subscript i. Using either of the model formulations, the followingparameters must be estimated: ki;�, Ei, ni and f

�

i .

186 Modelling Approaches in Pyrolysis

Model Limitations

A single reaction model for relative total weight loss can be obtained from Equa-tion (13.11) by removing subscript i. Howard (1981, pp. 730) and Saxena (1990,pp. 54) have pointed out some limitations of this kind of model. Let f� denotethe ultimate yield (total weight loss). At any temperature, the model predicts3:

limt!1

f = f�

At any reasonable pyrolysis temperature, calculated yield f monotonically in-creases and the ultimate yield f� will theoretically be approached in a reasonabletime. This is in contrast with experimental observations which show that theactual (apparent) yield is a function of pyrolysis top temperature. Thus:

fa = fa(T ) � f�

where fa denotes apparent (i.e. actual) yield. The variation of actual yield withtemperature is a feature not inherent in this kind of model and therefore a seriousshortcoming of the single reaction model. The monotonic increase in apparent yieldwith temperature causes di�erent values of E and k� to be estimated in di�erentexperiments. Also values of the kinetic parameters are forced to unphysically lowvalues to account for the variation in apparent yield. According to Howard (1981),the single reaction model is in some cases better suited for modelling weight loss ofindividual volatile species since devolatilization of individual volatiles often occurover more narrow temperature intervals and parameters E and k� can be foundthat gives a good �t in each stage.

However, in the previously discussed models on pitch pyrolysis it was observedthat a single parameter was unable to predict weight loss under varying processconditions. Therefore the limitations of the model also exists in modelling compo-nent volatile losses. In spite of these limitations, the use of single reaction modelsin pyrolysis modelling is widespread.

Models Applied in Industry

Dernedde, Charette, Bourgeois & Castonguay (1986) assumedki;�a

to be constantfor heating rates between 5.0 and 15.0�C=hr which are typically encounteredduring anode baking. Weight loss data taken from petroleum coke and S�derberg-paste were assumed to be representative for pyrolysis of green carbon anodes.Losses of components of tar, methane and hydrogen were modelled.

The model in Tremblay & Charette (1988) is of the same type. However, weightloss characteristics for laboratory scale green anodes are used for the parame-ter estimation. The kinetic parameters obtained are the most reliable presentlyavailable in the literature on modelling weight loss during anode baking. Preexpo-nential factors and activation energies for tar, methane and hydrogen were givenas functions of the heating rate.

3Subscript i is omitted when modelling total weight loss.

13.1 Approaches in Pyrolysis Modelling 187

In Tremblay's model, ultimate losses of volatiles are measured on an anode massbasis. Since the test anodes contain fp = 16 % pitch, the mass fractions are fi

fp

on a mass of pitch basis. Mass fractions of 0.3094, 0.0075 and 0.0219 are foundfor tar, methane and hydrogen respectively. Using these ultimate weight losses,the total relative weight loss (based on mass of pitch) is shown in Figure 13.2. Atapproximately 500�C, the weight loss rate seems to approach zero before obtaininga positive value at slightly higher temperatures. This corresponds to the shift inpyrolysis regime: The tar volatiles have been extinguished and non-condensablesCH4 and H2 appear. The same characteristic feature was experimentally observedby Fitzer & H�uttinger (1969, Fig. 5) in pyrolysis experiments on pure pitch.Other authors have measured a slightly less decrease in weight loss rate in thesame temperature interval (Wilkening 1983), (Politis & Chang 1985), (Boenigk &Wildf�orster 1989).

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1Normalized release of tar volatiles

Temperature [oC]

a = 5 oC/hr a = 10 oC/hra = 15 oC/hr

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1Normalized release of methane

Temperature [oC]

0 200 400 600 800 10000

0.2

0.4

0.6

0.8

1Normalized release of hydrogen

Temperature [oC]

Figure 13.1: Normalized weight loss curves for pyrolysis of laboratory scale anodesas function of temperature. Heating rates of 5.0, 10.0 and 15.0�C=hr are used. Inall plots, the weight loss curves are moved to a higher temperature as the heatingrate is increased. The plots are generated from results obtained by simulation ofthe model in Tremblay & Charette (1988).

Using the tabulated expressions, one �nds that when the heating rate is variedfrom 5 to 15�C=hr, the activation energies increases by approximately 10 %. Onthe other hand, the preexponential factors are much more sensitive to changesin heating rate: If release of methane is considered, the preexponential factor in-creases with more than 1000 %. The use of heating rate as a parameter in themodel, makes the model suited for prediction of weight loss in pyrolysis experi-

188 Modelling Approaches in Pyrolysis

0 50 100 150 2000

0.2

0.4

0.6

0.8

1Normalized release of tar volatiles

Time [hr]0 50 100 150 200

0

0.2

0.4

0.6

0.8

1Normalized release of methane

Time [hr]

0 50 100 150 2000

0.2

0.4

0.6

0.8

1Normalized release of hydrogen

Time [hr]

Figure 13.2: Normalized weight loss curves for pyrolysis of laboratory scale anodesas function of time. Additional information is given in Figure 13.1.

ment with varying heating rate. The dependence on the heating rate causes nodi�culties, since the energy balance and mass balance equations for the volatilecomponents have a structure as follows:

dT

dt= f1(T; u)

dXi

dt= f2(Xi; T; a)

where u represents a control variable. Since the heating rate4 a is equal to dTdt, the

model becomes:

dT

dt= f1(T; u)

dXi

dt= f2(Xi; T; f1(T; u)) = ~f2(Xi; T; u)

On the other hand, it is not evident that these parameters will give correct pre-dictions of weight loss in such cases since the model contains no description offundamental chemistry.

Tarasiewicz & Stumpf (1984) combine this approach with a general rate equationfor product formation. According to a set of assumed reactions taking place in

4Tremblay uses a linear heating rate in the experiments.

13.1 Approaches in Pyrolysis Modelling 189

0 100 200 300 400 500 600 700 800 900 10000

0.05

0.1

0.15

0.2

0.25

0.3

0.35Relative weight loss in pyrolysis of coal tar pitch

Temperature [oC]

Figure 13.3: Relative total weight loss during pitch pyrolysis generated by theweight loss model in Tremblay & Charette (1988). The heating rate is a = 5�C/hr.Since the ultimate weight losses in the model do not depend on pyrolysis condi-tions, the same ultimate weight loss will occur at any heating rate. Note thecharacteristic decrease in weight loss rate at approximately 500�C. This charac-teristic feature of the weight loss curve was experimentally observed by Fitzer &H�uttinger (1969).

the pitch and the gaseous volatiles during baking, the concentration of volatiles asfunction of time and space in a ring furnace is modelled.

Other authors have also determined kinetic parameters for pyrolysis of coal tarpitch based on the same kind of model (H�uttinger 1970), (H�uttinger 1971), (Wallouchet al. 1972), (Collett & Rand 1980), (Charette et al. 1989), (Charette et al. 1990),(Charette et al. 1991), (Kocaefe et al. 1990). H�uttinger (1970) studied conversionof pitch fractions using Mallison's fractionation scheme. The other authors concen-trated on decomposition of whole pitch and the modelling of total or component-wise weight loss.

Some de�ciencies of the single reaction model are as follows:

� Actually, coke yield depends on the baking conditions (Wilkening 1983, Fig.10). A model of this kind does not predict the coke yields dependence onpyrolysis conditions.

� The total weight loss depends on the �nal temperature. For this kind ofmodel, the ultimate weight loss will be achieved at any reasonable temper-

190 Modelling Approaches in Pyrolysis

ature if the pyrolysis time is long enough. However, this is a qualitativelyerroneous behaviour since the cumulative weight loss is less than the ultimateyield at low enough temperatures: The weight loss increases (up to a certainlimit value; the ultimate yield) if the temperature is slightly increased.

� In most cases, a model based on a single reaction of order n is not adequatefor describing pyrolysis under varying conditions.

� The weight loss from anodes depends on the pitch content of the anode asshown in Boenigk & Wildf�orster (1989). In a model for anode properties,the initial weight fraction of pitch will be of importance and the lack of thiskind of parameter in the traditional weight loss models is another objection.Nominally, the total weight loss observed in baked anodes is between 6 and8 %. The nominal weight loss in Tremblay's model is 5.42 % which is withinacceptable limits.

The calculated weight loss is only a nominal value which well serve the purpose ofcalculating a nominal heat balance for a ring furnace by taking into account theheating values of the volatile components and their time dependent introductioninto the ues. The application of this kind of model, however, is probably not sowell suited as a basis for calculation of anode properties and their dependence onthe baking conditions.

13.1.3 Consecutive Reaction Schemes

The models discussed so far have dealt with the calculation of total or component-wise weight losses during pyrolysis. In these models, no focus is put on the changesin the chemical composition of the pitch residue during pyrolysis. The group com-position of pitch radically changes during low temperature pyrolysis from 300 upto 550�C. In this temperature interval, light components are distilled o� in com-petition with thermal cracking and condensation of the pitch. Furthermore, thepitch passes the mesophase transition in which the pitch changes from being anisotropic liquid to become a solid semicoke. Traditionally, the kinetics of the pitchgroup composition has been studied via consecutive (serial) reaction schemes. Inmost of the cases, the schemes are exclusively serial, but branching into parallelreaction paths may also occur.

In these models, the main focus has been to give a description of group composi-tion. Modelling the loss of volatiles which occurs in parallel with the changes ingroup composition has in general not been focussed on in this case. Thus, thereis a lack of models in the literature which describe both weight loss features andchanges in group composition at the same time. Consecutive reaction schemesare scarce in Western literature on pitch pyrolysis. On the other hand, authorsin Russia and Eastern Europe seem to have been more attracted to this type ofmodels in studies on pitch upgrading by thermal and thermo-oxidative methods.Schemes based on consecutive (serial) or competing (parallel) reaction or combi-nation thereof have been applied.

13.1 Approaches in Pyrolysis Modelling 191

Also, very common in studies of thermal chemistry of whole pitch is to measurecertain collective properties (formation of mesophase, changes in group compo-sition, molecular weight distribution etc.) as function of time and temperature(Introduction to Coal-Tar and Petroleum Pitch 1993).

By the aid of the traditional pitch solvent fractionation scheme used in aluminiumindustry (Pechiney-scheme), Ko�st�al et al. (1994) studied the kinetics of the se-quential decomposition and formation of pitch fractions during heat treatment:

(l)k1! �(l)

k2! �(l) (13.14)

First order rate laws were shown to give good correspondence between experimen-tal and calculated pitch fractions. Kinetic parameters are available in the article.Although it is stated that there is a loss of volatiles during heat treatment, thisvolatile loss is not accounted for in the reaction scheme. The results on pitch poly-merization kinetics con�rm the qualitative results presented in McHenry, Baron& Saver (1993) and McHenry, Baron & Saver (1994). The result is also supportedby H�uttinger (1970) who stated that decomposition of pitch up to 500�C is dueto thermal decomposition of m-oils, m-oils and N -resins according to Mallison'sscheme. These fractions correspond to +� - fraction above. However, H�uttinger(1970) used reaction orders n = 1:2; 1:5 and 1:2 for the m-oils, m-oils and N -resinsrespectively.

Models from the Eastern and Russian literature are mostly concerned with study-ing pitch upgrading by thermal and thermo-oxidative treatment. Based on De-mann's scheme for pitch fractions, Chistyakov (1979) studied formation of tolueneinsolubles and loss in volatile matter of the coke residue of a coal tar pitch. Firstorder rate laws were used to describe thermal treatment and an e�ective activa-tion energy of 41 kcal/mol for both change in toluene insolubles and loss in volatilematter was observed. Thermo-oxidation showed a lower activation energy. It wasstated that thermal condensation and polymerization as well as oxidation of pitchproceed by a radical chain mechanism.

The fact that pitch pyrolysis is a free radical mechanism was directly used byBelkina, Lur'e & Stepanenko (1981) in a consecutive reaction scheme for the an-thracene fraction of a coal tar pitch. , � and �2 -fractions were consumed inchain generation and propagation steps. Formation of �, �1 and �2 -fractionstake place in chain termination steps. The kinetic parameters were determinedby minimization of a quadratic merit function. The ow rate of O2 used in theoxidation experiments were used as parameter in the rate expressions.

Very often, the mesophase transition is modelled by representing the formationof mesophase with the amount of secondary QI (Honda et al. 1970). However,it is well known that a part of the mesophase is soluble in quinoline. Thereforesecondary QI is only an approximation of the real mesophase content. In severalpublications5, H�uttinger and co-workers modelled the growth of mesophase. InH�uttinger & Wang (1991), the tetrahydrofuran insoluble fraction THF-IS was used

5(Wang 1991), (H�uttinger & Wang 1991) (H�uttinger & Wang 1992a), (H�uttinger & Wang1992b), (H�uttinger, Bernhauer, Christ & Gschwindt 1992), (Bernhauer, Christ, Gschwindt &H�uttinger 1993).

192 Modelling Approaches in Pyrolysis

as an approximation for the mesophase content. The increase in THF-IS followed�rst order kinetics up to a content of 0.9 by mass. In the later publications, thefollowing consecutive reaction scheme was used:

A+ a! Ak1! PA

k2!MA =MP (13.15)

a is the volatile part of the pitch which is assumed to vaporize during preheating.Mesogenic aromatics MA were assumed to form mesophase MP instantaneously.The mesophase transition of pitch A goes via polyaromatics PA to mesogenicaromatics. The proposed reaction scheme is very much similar to the mechanismof mesophase formation suggested by Marsh & Latham (1986) and the formationof mesophase is a result of consecutive reactions. In H�uttinger's scheme, volumepitch fractions were calculated. The volume fraction of mesophase was determinedaccording to a counting procedure presented in Chawastiak, Lewis & Ruggiero(1981). Along with the mesophase content, several pitch properties were alsomeasured (glass transition temperature, coke yield, density and viscosity). For acoal tar pitch, it was found that k1 = 0:5k2

6.

In contrast with H�uttinger's consecutive scheme, Azami & Yamamoto (1994)showed that the formation of mesophase in a petroleum pitch is the result ofan autocatalytic reaction in which the isotropic fraction of the pitch is consumedin formation of the mesophase. However, a plateau in mesophase formation wasobserved below a weight fraction of 0.05 of mesophase which is not characteristicfor an autocatalytic reaction. The activation energy was found to be 237 kJ/molwhich is higher than the value reported by Honda et al. (1970). Mesophase for-mation was thought to occur in three stages:

� Formation of mesogens by polymerization reactions in the isotropic pitch

� The di�usion of mesogens through the isotropic pitch to the mesophase

� The rearrangement (stacking) of the mesogens in the mesophase

The high activation energy could be explained if the mesogenic rearrangementprocess is rate-determining for mesophase formation in petroleum pitches.

Qualitative descriptions of carbonization models based on the view that pyrolysisis a combination of physical and chemical processes are given in Tillmans (1985)and Tillmans (1986). Conversion dependent activation energy and preexponentialfactor are used in the model. The reaction order is assumed to depend on tem-perature. No mathematical equations are given, but a systematic classi�cation ofthe processes which occur during pyrolysis is given.

The modelling of transitions in the group composition of heavy hydrocarbon mix-tures seems to be more developed in the petroleum industry. As seen below,several schemes based on serial and parallel reactions exist for modelling crackingof petroleum residua. References are both from Western and Eastern literature.

6k1 = k1;� exp(�E1RT

) and k1;� = 2:2� 1010 1/hr, E = 150 kJ/mol in argon atmosphere.

13.1 Approaches in Pyrolysis Modelling 193

1. Levinter, Medvedeva, Panchenkov, Aseev, Nedoshivin, Finkels'thein

& Galiakbarov (1966) present a quasi free radical scheme for cracking ofresins, asphaltenes, carbenes (carboids) to form volatiles and coke. Thescheme includes separate branches for the alkyl- and benzyl radicals.

2. Kono (1984) present a reaction scheme for coking of residual oils used byUbe Industries in Japan. The reaction scheme is based on solvent fraction-ation. The BS, BI-QS and QI fractions correspond to the resin, semi-cokeand coke parts of the petroleum residuum. The scheme is presented in Fig-ure 13.4.

3. Valyavin, Fryazinov, Gimaev, Syanyaev, Vyatkin &Mulyukov (1979)

proposed a reaction scheme based on reversible reactions for cracking of themacromolecular part of crude oil. It is assumed that condensation - and de-composition reactions are bimolecular and monomolecular respectively. Thescheme contains �ve lumps describing the group composition of the residuum.The volatiles are collected in a sixth lump.

4. Washimi (1984): In this review, three reaction schemes for thermal crack-ing of heavy petroleum residua are presented. The group composition ofthe petroleum residua was analyzed by solvent fractionation. The reactionscheme presented in Kono (1984) is included in the review. The other twoschemes are developed by researchers at Daikyo Oil and Hokkaido Univer-sity and are both represented in Figure 13.4. The reaction schemes have incommon that the formation of intermediate fraction BI-QS occurs in parallelwith cracking reactions which form lighter components represented by thegas/oil lump. At the same time, there is vapour-liquid equilibrium betweenthe reacting liquid and the distillate vapour. As a consequence, it is di�cultto distinguish between primary vaporized gases and cracking products whichin the next turn are vaporized.

5. Wiehe (1993): A reaction scheme was suggested that takes into account theinduction period observed before coke (as represented by toluene insolubles)is formed during pyrolysis of petroleum residua:

Aromatics! Resins! Asphaltenes! Coke (13.16)

This was achieved by assuming that the concentration of asphaltenes have toreach a critical level (i.e. the maximum that can be held in solution) beforecoke forms at an in�nite rate. This coke formation mechanism is basicallythe same as proposed by Valyavin et al. (1979).

The use of lumped kinetic schemes is well known from other �elds of petroleum up-grading too. Skala, Kopsch, Neumann & Jovanivi�c (1987), Skala, Kopsch, Soci`c,Neumann & Jovanivi�c (1989) and Skala, Kopsch, Soci`c, Neumann & Jovanivi�c(1990) present reaction schemes for pyrolysis of oil shale. More well known arethe lumped schemes for uid catalytic cracking presented in (Weekman 1969),(Weekman & Nace 1970) and (Jacob, Gross, Voltz & Weekman 1976).

194 Modelling Approaches in Pyrolysis

Oils / Volatiles

Oils / Volatiles

Oils / Volatiles

Daikyo Oil

Ube Industry

Hokkaido University

BI-QSk1 k2 k3

k1

k2 k3

k4

k3

k2

k�

PS

BS

BS

BI-QS QI

QI

QIPI-BS

BI-QS

k5

k1

Figure 13.4: Reaction schemes for cracking of heavy petroleum residua. The crack-ing reactions occur in parallel with competing polymerization reactions which in astepwise manner �nally form coke. Together with the chemical reactions, vapour-liquid equilibrium exists between the reacting uid and the distillate vapour.

So far, methods from thermochemical kinetics have not been used for studyingkinetics of coal tar pitch pyrolysis. The application of such methods are more de-veloped in other branches of carbon science, but the applicability of such methodsin studying kinetics of liquid phase pyrolysis of hydrocarbons was demonstratedby Stein (1981) in studying anthracene pyrolysis. A reaction mechanism was pro-posed in which anthracene pyrolysis is a free radical polymerization process whichto a certain degree is product catalyzed (auto-catalytic).

13.1.4 Multiple Reaction Schemes

It is not likely that thermal decomposition of coal tar pitch and coal occurs by asingle reaction because of the complex composition of the pitch. At each instant intime, one might assume that a large number of decomposition reactions occur inparallel and the relative importance of an individual reaction for the contributionto total weight loss vary over the pyrolysis time horizon. The assumption thata certain number of reactions occur in parallel, gives a plausible explanation ofwhy a single reaction model cannot be expected to adequately describe the wholeweight loss history. In the following, it is assumed that nr reactions occur inparallel. In each reaction, a mass msv;j is decomposed. As before, it is assumedthat decomposition of msv;j obeys the law:

dmsv;j

dt= �~kjmnj

sv;j (13.17)

13.1 Approaches in Pyrolysis Modelling 195

Now, introduce the reaction dependent conversionXj =mv;j

msv;j;�, which can be used

to obtain:

msv;j = msv;j;�(1�Xj) (13.18)

Substitution of Equation (13.17) into Equation (13.18) gives:

dXj

dt= kj(1�Xj)

nj (13.19)

An equation for the global (overall) normalized conversion X is obtained by de�n-ing msv;j;� = g�jmsv;�. This gives:

msv;j = g�jmsv;�(1�Xj)

Also de�ne g�jXj = gj and obtain:

msv;j = msv;�(g�

j � gj)

From:

nrXj=1

msv;j;� = msv;� (13.20)

the following holds:

nrXj=1

g�j = 1

Substitution of Equation (13.20) into the decomposition model gives:

dgj

dt= kj(g

�

j � gj)nj (13.21)

kj = kj;� exp(�Ej

RT) (13.22)

1 =

nrXj=1

g�j (13.23)

X =

nrXj=1

gj (13.24)

The model can also be de�ned by the global (overall) relative conversion by usingmsv;� = f�m�:

msv;j = f�m�(g�

j � gj)

Now de�ne f�g�j = f�j and f�gj = fj , which gives:

msv;j = m�(f�

j � fj)

196 Modelling Approaches in Pyrolysis

and obtain the following form of the general multiple reaction model for totalvolatiles:

dfj

dt= kj(f

�

j � fj)nj (13.25)

kj = kj;� exp(�Ej

RT) (13.26)

f� =

nrXj=1

f�j (13.27)

f =

nrXj=1

fj (13.28)

Index j denotes a speci�c reaction. This formulation of the multiple reaction modelis often encountered in literature on coal pyrolysis. Each reaction has an individualultimate weight loss which contributes to the overall weight loss. Furthermore,each reaction has four unknown parameters f�j , kj;�, Ej and nj and which givesa total number of 4nr parameters. No a priori values of the parameters exist; allof them have to be estimated from experimental data. This may be di�cult andtime consuming if the number nr of reactions is large.

The multi-reaction approach can also be used for modelling devolatilization ofindividual volatile species. The previous nomenclature is used as basis. Nowassume that a total number of nv volatile components with the individual speciesmodelled by nr;i number of parallel reaction by the multi-reaction approach. Theequation for the global conversion Xi of volatile component i is given by:

dgi;j

dt= ki;j(g

�

i;j � gi;j)ni;j (13.29)

ki;j = ki;j;� exp(�Ei;j

RT) (13.30)

1 =

nrXj=1

g�i;j (13.31)

Xi =

nrXj=1

gi;j (13.32)

Again, the rate of release of volatiles is given by Equation (13.12). The equationfor gi;j is separable and has the following solution:

gi;j =

8><>:g�i;j � 1�

1

g�i;j

ni;j�1+(ni;j�1)Rt

0ki;jdt

�ni;j�1 if ni;j 6= 1

g�i;j(1� exp(�R t0ki;jdt)) if ni;j = 1

(13.33)

A summary of quantities used in the independent reaction schemes is given inTable 13.1.

A continuous variant of the �rst order multi-reaction scheme was derived by Pitt(1962) with basis in the pioneering work of Vand (1943). In the literature, the

13.1 Approaches in Pyrolysis Modelling 197

Scheme Total Individual volatilevolatiles species

(i = 1; : : : ; nv)

single X Xi

reaction f fischeme

multiple gj gi;jreaction X =

Pnrj=1 gj Xi =

Pnr;ij=1 gi;j

scheme f =Pnr

j=1 fj fi =Pnr;i

j=1 fi;jj = 1 : : : nr

orj = 1 : : : nr;i

Table 13.1: Di�erent formulations of independent reaction schemes. Single reac-tion schemes have 4 and 4nv parameters for total and individual volatiles respec-tively. In multiple reaction schemes, the corresponding number of parameters is4nr and

Pnvi=1 4nr;i.

continuous model is often derived on a relative overall conversion basis. In thecontinuous variant of the multi-reaction approach to modelling the loss of totalvolatiles, the following assumptions are needed:

� All reactions are of �rst order (this is a necessary condition)

� A common preexponential factor is used: kj;� = k�

� The number of parallel reactions is large enough to represent the activationenergy as a continuous distribution function fd(E):Z

1

0

fd(E)dE = 1 (13.34)

where fd(�) is the distribution function.

� In the continuous case, the ultimate weight loss for a single reaction is anin�nitesimal quantity. Hence, from Equation (13.34):

f� =

Z1

0

f�fd(E)dE

Set f� =R f�0df� and obtain:

df� = f�fd(E)dE (13.35)

� By discretizing the axis of activation energies, the ultimate weight loss f�jfor a single reaction is given by:

f�j = f�fd(Ej)�E (13.36)

which shows that the distribution of ultimate yields between the parallelreactions is given from the distribution function.

198 Modelling Approaches in Pyrolysis

Pitt obtained a numerical approximation for the actual distribution function fd(E)from experimental data from isothermal experiments by using the observation thatthe function:

F (t; E) = exp(�kt)

k = k� exp(�E

RT)

changes abruptly from zero to one at the value Es = RT ln(k�t). In isothermalexperiments, the abrupt change is time dependent. This gives:

Z Es

0

f(E)dE � f(t)

f�

from Equation (13.37) below since F (t; E) can be replaced by a step functionwhich is zero for E < Es and unity for E � Es.

Later, Anthony, Howard, Hottel &Meissner (1975) assumed that fd(E) is Gaussianwith mean activation energy E� and standard deviation �E :

fd(E) =1p2��E

exp

�� (E� �E)2

2�2E

�

Other authors have used other distribution functions (Merrick 1983a), (Fu, Zhang,Han & Wang 1989).

A similar expression as given for gi;j in Equation (13.33) also exists for fj :

fj = f�j (1� exp(�Z t

0

kjdt))

fj can be represented with a di�erential quantity df . This gives:

df = df�(1� exp(�Z t

0

kjdt)) = f�(1� exp(�Z t

0

kjdt))f(E)dE

Integrating over all activation energies and obtain:

f =

Z1

0

f�(1� e�

Rt

0kdt

)f(E)dE (13.37)

If in this model, a Gaussian distribution function is used, the total number ofparameters is only four: k�, E�, �E and f�.

Models corresponding to Equation (13.19) and Equations (13.25) to (13.28) existfor the individual normalized conversion and the global relative conversion respec-tively. In these schemes, a total of

Pnvi=1 4nr;i parameters must be estimated.

Chermin & van Krevelen (1956) in studies of secondary coal conversion, proposedto use an activation energy which monotonically increase as conversion goes by.In this way one was able to introduce into the single reaction model that pyrolytic

13.1 Approaches in Pyrolysis Modelling 199

release of volatiles becomes more and more di�cult as conversion passes by. Thefollowing equation for the relative conversion was used:

df

dt= k� exp(�

E

RT)(f� � f)

E = Emax ��E(1� f

f�)

�E = Emax �Emin

By using f = f�X , the dependency of E on X can be shown:

E = Emax ��E(1�X)

This shows that the model can also be expressed in the overall normalized conver-sion. According to Howard (1981), this model is a special case of the isothermalmulti-reaction scheme with a uniform distribution for the activation energy. Nowuse:

fd(E) =1

�E

and obtain:

f =

Z f

0

df� =

Z E

0

f�1

�EdE

+

E = Emax ��E(1� f

f�)

which shows that for a uniform distribution, the activation energy depends on theconversion.

Finally three comments are necessary:

� Some multi-reaction schemes use distribution of both preexponential factorand activation energy. It is also possible to �x the activation energy andallow for distribution of log(k�). Methods used for estimating the distributedparameters will not be discussed, but details can be found in the literature(J�untgen & van Heek 1969), (Hanbaba, J�untgen & Peters 1968).

� In the multi-reaction schemes, the time needed to achieve full conversionincreases strongly with decreasing temperature and thus contributes to syn-thetizizing a kind of temperature dependent or apparent yield. Predictedweight losses in Chermin & van Krevelen (1956) and Pitt (1962) show thatthe multi-reaction scheme is capable of achieving an apparently steady statevalue in reasonable time which correspond to the apparent yields measuredexperimentally.

� The phenomenon of a temperature dependent apparent yield has very muchin common with the temperature dependent development of graphitic struc-ture in carbon materials during heat treatment. In graphitization, the so-called degree of graphitization seems to approach an asymptotic value for

200 Modelling Approaches in Pyrolysis

isothermal heat treatment over a long enough period of time. However, ithas been observed that the degree of graphitization is increasing even afterlong residence times, so that the asymptotic value is most likely an apparentvalue (Fair & Collins 1962). Still the same mathematical framework can beused for modelling of both pyrolysis and growth of crystallites due to thesimilarity in qualitative behaviour and also the close relationship betweenthe phenomena.

13.2 Conclusions

State of the art in modelling of pitch pyrolysis may be summarized as follows:

� Weight loss models based on a single n'th order reaction seems to be preferredboth for total weight loss and loss of volatile components.

� Consecutive reaction schemes are used for the study of pitch group compo-sition.

� Only few models combine the calculation of pitch group composition andvolatile weight loss due to vaporization and chemical reactions.

So far, multi-reaction schemes has not been used in modelling of pitch pyrolysis.The exibility introduced in such schemes allows for prediction of temperaturedependent weight loss at di�erent heating rates. The modelling principle is alsoapplicable for modelling changes in pitch properties both during carbonizationand graphitization. Crystallite growth can preferably be modelled as a thermallyactivated process with varying activation energy.

Consecutive reactions schemes are interesting alternatives to be used for modellingof low temperature pyrolysis where the most important characteristics are thevolatile weight loss mainly via vaporization and the mesophase transition whichsigni�cantly changes the properties of the solidifying pitch residue.

A model of pitch pyrolysis should include characteristic features from both themultiple- and consecutive reaction schemes to allow for inclusion of the most im-portant phenomena.

Chapter 14

The Pyrolysis Model

A new model for coal tar pitch pyrolysis is presented. Even though the modelis speci�cally developed for pyrolysis of coal tar pitch, the model is based ongeneral principles which may be applicable also for description of pyrolysis ofother hydrocarbon materials.

14.1 Developing the Model

It is agreed upon that pitch pyrolysis at temperatures up to 1300�C goes throughdi�erent successive and partly overlapping regimes. Low temperature pyrolysisoccurs in liquid pitch via a free radical mechanism. At higher temperatures, thepitch solidi�es, and the mobility of the carbon-lamellaes become rate determining.Gas release during pyrolysis seems to appear in mainly two temperature intervals;below and above 550�C. The minimum in degassing rate observed at approxi-mately 550�C is due to solidi�cation of the pitch. Below 550�C, the released gasesconsist mainly of condensables from the pitch distillation. Above 550�C, methaneand hydrogen dominate.

14.1.1 Requirements and Capabilities of the Pyrolysis Model

For the pyrolysis model, the following was required:

� The model must be simple.

� The model must include the most important physical and chemical phenom-ena.

� The model must be based on kinetic data from the literature (i.e. experi-mental data are scarse).

202 The Pyrolysis Model

� The model should be based on a nomenclature well known to the aluminiumindustry.

The following capabilities should be included in a model for pitch pyrolysis:

� Prediction of coke yield. The coke yield should decrease as heating rateincreases; see Wilkening (1983, Fig. 10).

� Dynamic prediction of component-wise volatile release.

� Prediction of phase change occurring in pitch during carbonization whichmay serve as a basis for calculation of pitch properties.

Ideally, both the physical and chemical processes which occur during pyrolysisshould be represented in a nonlinear state space model that takes care of thecoupling between the processes:

1. Chemical processes/ Carbonization:

� Cracking (dealkylation, dehydrogenation)

� Condensation

� Polymerization

2. Physical processes:

� Vaporization of light components (distillation)

� Mesophase nucleation, growth and coalescence

� Crystallite growth

� Pitch coke shrinkage and formation of microcracks

The relationship between processes with impact on pitch coke yield, yield of gasesand pitch coke properties is shown in Figure 14.1. From the �gure, one mayconclude that the following qualitative information is needed to give a satisfactorydescription of pyrolysis:

1. Pitch composition during pyrolysis

2. Volatile losses

3. Porosity (mainly macroporosity)

4. Structure of liquid pitch and binder-coke

� Microstructure (micro crystallites and micropores)

� Texture (mesophase spherules and their coalescence)

The pyrolysis model should serve as a basis for the model of anode quality. Themodel will include changes in pitch composition and release of volatile gases. Mod-els for prediction of porosity and structure are presented in parts II and IV of thisstudy.

14.1 Developing the Model 203

Mesophase formationEvaporation Carbonization

chemicalmechanicalthermalother

PYROLYSIS

average molar mass

ordering processes

textureporosity (pore size distr.)

Physical Properties

Purity

porosity

MicrostructureMacrostructure

microporositycrystallite parameters

increase in

shrinkage pores/formation

of microporesgas-releasepores

formationof macropores

crystallite growth

structure

Figure 14.1: The interaction between processes in pitch pyrolysis. The term crys-tallite growth means that the size of the small crystallites increases both in the a-and c-directions. The average size of a layer plane as represented by parameterLa increases due to condensation reactions as well as consumption of the disor-dered carbon phase. The height Lc of crystallites increases due to coalescence ofcrystallites in the c-direction. The microstructure belongs to a resolution level inthe order of 10�6 m and below: Carbon texture is usually de�ned on a level 100to 1000 times larger than the size of the microcrystallites. This corresponds to aresolution in the range of 10�6 m (i.e. the microstructure range).

14.1.2 Model Parameters vs. Experimental Data

The model is not based on fundamental kinetic modelling. Therefore, it is not pos-sible to estimate the kinetic parameters from general thermochemical methods. In

204 The Pyrolysis Model

� [wt.%] �� [wt. %] �� [wt.%]

0.65 0.27 0.08

Table 14.1: Nominal composition of pitch used for parameter estimation.

this situation, it is necessary to partially base the model on parameters availablein the literature and allow for estimation of a few critical model parameters. Es-timation of model parameters was based on data found in Wilkening (1983, Fig.10), Ko�st�al et al. (1994) and Tremblay & Charette (1988). Details are given inGundersen (1995b).

100 200 300 400 5000

10

20

30

40Heating rate is a = 5.5 oC/hr

Temperature [°C]

Wei

ght l

oss

[%]

100 200 300 400 5000

10

20

30

40Heating rate is a = 11.0 oC/hr

Temperature [°C]

Wei

ght l

oss

[%]

100 200 300 400 5000

10

20

30

40Heating rate is a = 25.0 oC/hr

Temperature [°C]

Wei

ght l

oss

[%]

100 200 300 400 5000

10

20

30

40All experiments in same plot

Temperature [°C]

Wei

ght l

oss

[%]

Figure 14.2: Weight loss data used for estimation of parameters in the rate lawfor loss of condensables during pyrolysis. Data are taken from Wilkening (1983,Fig. 10).

14.1.3 Conclusions

For the low temperature pyrolysis, the aromatic polymerization reactions are rep-resented by a consecutive reaction scheme based on pitch solvent fractionation:

1 ! Vc (14.1)

1 ! 2 ! � ! � (14.2)

In the scheme, the -fraction undergoes a polymerization reaction in competitionwith devolatilization (of the low molecular part of the pitch); i.e. the scheme is

14.2 The Model for Devolatilization of Condensables 205

an extension of the traditional Pechiney scheme. Thus, competition between thephysical process of vaporization and the chemical polymerization is implemented inthe scheme and the overall kinetics allows for calculation of heating rate dependentweight loss during low temperature pyrolysis. The scheme also serves as a basisfor the pitch property model since the formation of secondary QI is thought toapproximate mesophase and subsequent semi-coke formation. The kinetics of pitchpolymerization is based on the results presented in Ko�st�al et al. (1994). Theparameters for the distillation sub-model was based on experimental data foundin Wilkening (1983).

Degassing of non-condensables in high temperature pyrolysis was also incorporatedin the scheme. Based on a reaction mechanism from the literature (Greinke 1992)and elemental data for coal tar pitch, two approaches were used to model therelease of methane and hydrogen:

� The use of two independent reaction schemes with constant ultimate yieldsof non-condensables. The kinetics was estimated from measurement datagenerated by the model presented in Tremblay & Charette (1988).

� The extension of the low temperature pyrolysis scheme to allow for forma-tion of coke and non-condensables. Stoichiometric coe�cients were �tted toobtain realistic yields of the non-condensables. Also here, the kinetics wasestimated based on measurement data generated by the model presented inTremblay & Charette (1988).

In this study, the second approach is used. In this way, it was possible to incor-porate both low- and high temperature pyrolysis into the same reaction scheme.

14.2 The Model for Devolatilization of Conden-

sables

14.2.1 The Reaction Scheme

A competing reaction scheme for pitch polymerization and devolatilization ofcondensables was formulated. The scheme is shown in Figure 14.3 and the follow-ing mass balance equations can be derived from the scheme:

dm 1

dt= �rvVa;p � r 1;aVa;p � r 1;bVa;p (14.3)

dm 2

dt= r 1;aVa;p � r 2Va;p (14.4)

dm�

dt= r 1;bVa;p + r 2Va;p � r�Va;p (14.5)

dm�

dt= r�Va;p (14.6)

206 The Pyrolysis Model

Balance equations for total mass and the mass of volatile components are:

dmp

dt= �rvVa;p (14.7)

dmv

dt= rvVa;p (14.8)

Only the release of condensables is considered.

All reaction rates are of �rst order:

ri = ki;�e�

EiRT �b;i (14.9)

Since �b;iVa;p = mi, the model can also be formulated in absolute masses:

dm 1

dt= �kvm 1 � k 1;am 1 � k 1;bm 1

dm 2

dt= k 1;am 1 � k 2m 2

dm�

dt= k 1;bm 1 + k 2m 2 � k�m�

dm�

dt= k�m�

dmv

dt= kvmp

Introduce mass fractions and coke yield via:

mi = ximp

mp = cymp(0)

and get the following model as derived from the mass balance equations for thepitch fraction and total mass of pitch:

dx 1dt

= (x 1 � 1)kvx 1 � k 1;ax 1 � k 1;bx 1 (14.10)

dx 2dt

= x 2kvx 1 + k 1;ax 1 � k 2x 2 (14.11)

dx�

dt= x�kvx 1 + k 1;bx 1 + k 2x 2 � k�x� (14.12)

dx�

dt= x�kvx 1 + k�x� (14.13)

dcy

dt= �kvx cy (14.14)

cy(0) = 1 (14.15)

14.2.2 Estimated Parameters

Kinetic data from (Ko�st�al et al. 1994) was used for the polymerization reactions.A quadratic objective function based on data from Wilkening (1983, Fig. 10) was

14.2 The Model for Devolatilization of Condensables 207

�r 2r 1;a

1r�

Vc

rv

r 1;b

2 �

Figure 14.3: Final reaction scheme used to model low temperature pitch pyrolysis.

used to estimate parameters in the rate laws rv and r 1;a . Constraints were puton the end points of the predicted weight loss curves. The estimated parameterscan be found in Gundersen (1995b).

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

x 10−3

−7

−6

−5

−4

−3

−2

−1

0Arrhenius plot of rate constants kv and kg1a

ln(k

i)

1/T [1/oC]

ln(kv)

ln(kg1a)

Figure 14.4: Relationship between rate constants for volatilization and polymer-ization of the 1-fraction. kv dominates over k 1;a at temperatures from ap-proximately 250�C and above. (kv is estimated and k 1;a is taken from (Ko�st�alet al. 1994)).

14.2.3 Simulating the Pyrolysis Model

The model was simulated for three di�erent heating rates a = 5:5; 11:0 and 15:0�C=hr.Simulation results are shown in Figures 14.5 to 14.8. Initial values for the pitchcomposition is given in Table 14.2.

208 The Pyrolysis Model

x 1(0) x 2(0) x�(0) x�(0)

0.65 0.00 0.27 0.08

Table 14.2: Pitch composition used in simulations of the model for low temperaturepyrolysis.

0 50 1000

0.5

1

Time [hr]

Mas

s fr

actio

ns

Mass fractions vs. time

100 200 300 400 5000

0.5

1

Temperature [oC]M

ass

frac

tions

Mass fractions vs. temp.

0 50 1000

0.5

1

Time [hr]

Tot

al g

amm

a fr

actio

n

100 200 300 400 5000

0.2

0.4

0.6

Temperature [oC]

Tot

al g

amm

a fr

actio

n

100 200 300 400 50070

80

90

100

Temperature [oC]

Cok

e yi

eld

[%]

100 200 300 400 5000

10

20

30

Temperature [oC]

Yie

ld o

f vol

atile

s [%

]

Figure 14.5: Simulation of the model for low temperature pyrolysis with a heatingrate of a = 5:5�C=hr. Polymerization kinetics is taken from Ko�st�al et al. (1994).Initial values are given in Table 14.2.

14.2.4 Average Molar Mass of Condensables

The condensables emanate from the light pitch fraction (i.e. distillate oils andcrystalloids). According to (Barrillon 1971), one can assume the ratio of carbon-to hydrogen atoms in a molecule to be C

H� 1:70 for whole pitch. For the lighter

fractions, one can set CH� 1:50. The ring index is R � 5 for the lighter com-

ponents. The molecular weight of the tar fraction can then be calculated fromMcNeil (1981):

M =(24:02 + 2:016(CH )

(�1))

(1� (CH)(�1))

(R� 1) (14.16)

This gives a molar mass of approximately 304 g/mol for the lighter fraction. Inthis work, the average molar mass for the condensables is equal to 300 g/mol. Themolecular formula is then (C1:5H)n with n = 16 (i.e. C24H16).

14.3 The Model for Degassing of Non-Condensables 209

0 20 40 600

0.5

1

Time [hr]

Mas

s fr

actio

nsMass fractions vs. time

100 200 300 400 5000

0.5

1

Temperature [oC]

Mas

s fr

actio

ns

Mass fractions vs. temp.

0 20 40 600

0.5

1

Time [hr]

Tot

al g

amm

a fr

actio

n

100 200 300 400 5000

0.2

0.4

0.6

Temperature [oC]T

otal

gam

ma

frac

tion

100 200 300 400 50060

80

100

Temperature [oC]

Cok

e yi

eld

[%]

100 200 300 400 5000

20

40

Temperature [oC]

Yie

ld o

f vol

atile

s [%

]

Figure 14.6: Simulation of the model for low temperature pyrolysis with a heatingrate of a = 11:0�C=hr. Polymerization kinetics is taken from Ko�st�al et al. (1994).Initial values are given in Table 14.2.

This estimate for the molar mass of the tar fraction can be tested by assumingthat whole pitch contains 4.5 % hydrogen. Weight loss data for anodes (Tremblay& Charette 1988), gives:

� The anodes contain 16 % pitch. If all hydrogen is lost, this amounts to aweight loss of 0:16 � 0:045 = 0:0072 gH/(g anode).

� 5.42 % of the anode mass is lost during baking. This corresponds to a weightloss of 0.0495 g, 0.0012 g and 0.0035 g of tar, methane and hydrogen per g ofanode respectively. Using a molar mass of 304 g/mol for the condensables,this weight loss amounts to 0.0064 gH per g anode which is reasonably closeto our prediction of 0.0072 gH based on data for whole pitch.

14.3 TheModel for Degassing of Non-Condensables

14.3.1 Introduction

At present, the most sophisticated model of coal tar pitch pyrolysis in the contextof anode baking is due to Tremblay & Charette (1988). Furthermore, the generalliterature on carbonization present no other more complete models of pitch pyrol-ysis. Tremblay uses a single reaction scheme with constant ultimate yields for tar,

210 The Pyrolysis Model

0 20 40 600

0.5

1

Time [hr]

Mas

s fr

actio

nsMass fractions vs. time

100 200 300 400 5000

0.5

1

Temperature [oC]

Mas

s fr

actio

ns

Mass fractions vs. temp.

0 20 40 600

0.5

1

Time [hr]

Tot

al g

amm

a fr

actio

n

100 200 300 400 5000

0.2

0.4

0.6

Temperature [oC]T

otal

gam

ma

frac

tion

100 200 300 400 50060

80

100

Temperature [oC]

Cok

e yi

eld

[%]

100 200 300 400 5000

20

40

Temperature [oC]

Yie

ld o

f vol

atile

s [%

]

Figure 14.7: Simulation of the model for low temperature pyrolysis with a heatingrate of a = 15:0�C=hr. Polymerization kinetics is taken from Ko�st�al et al. (1994).Initial values are given in Table 14.2.

methane and hydrogen. In the model, the kinetic parameters are allowed to varywith the heating rate. The dependence of the parameters on the heating rate isshown in Figure 14.9. The kinetic parameters are determined from experimentsperformed with a linear heating rate; the heating rate is kept constant within theexperimental time. In this chapter, degassing of non-condensables is modelled byan extension of the model for low temperature (liquid phase) pyrolysis. But �rstof all, a calculation scheme for prediction of ultimate yields of gases is presented.

14.3.2 Calculation Scheme for Ultimate Yields of Gases

The method presented for calculation of ultimate yields gases in pitch pyrolysiswas inspired by an approach due to Merrick (1983a) for obtaining ultimate yieldsin coal pyrolysis.

In the presented analysis, the ultimate yields of condensables and non-condensablesare calculated from the coke yield of whole pitch and pitch composition data forwhole pitch and the -fraction of the pitch.

Greinke & Singer (1988) presented a simple free radical scheme for thermal poly-merization of petroleum pitches in which methyl (CH3�) was considered to bethe most important alkyl group. For a coal tar pitch, Grint et al. (1988) showedthat the average chain length of alkyl groups is 1.30 which is close to methyl.

14.3 The Model for Degassing of Non-Condensables 211

100 150 200 250 300 350 400 450 500 5500

5

10

15

20

25

30

35Yield of condesnables for three different heating rates

Temperature [oC]

Yie

ld o

f Con

dens

able

s [%

]

a = 5.50 oC/hr a = 11.00 oC/hra = 15.00 oC/hr

Figure 14.8: Dependence of the yield of condensables on the heating rate. Threeheating rates of a = 5:0; 11:0 and 15:0�C=hr were used. The weight loss increaseswith the heating rate. Thus, a low heating rate contributes to increasing the cokeyield. Polymerization kinetics is taken from Ko�st�al et al. (1994). Initial values aregiven in Table 14.2.

Also, formation of higher order alkanes is negligible in pyrolysis of coal tar pitch.Therefore, the free radical mechanism was considered to give a qualitatively cor-rect description of the formation of non-condensables also during coal tar pitchpyrolysis.

Methyl is formed by breaking bonds in the �-position to an aromatic ring:

Ar � CH3 ! Ar �+CH3� (14.17)

According to Greinke & Singer (1988), cleavage of methyl groups occur moreeasily than cleavage of C �H-bonds. Methyl free radicals stabilize by abstractingaromatic hydrogen from the liquid pitch molecules and mesophase matrix (whichthen gradually turns into solid coke):

CH3 �+Ar �Har ! CH3 �Har(g) +Ar� (14.18)

Har denote aromatic bonded hydrogen. It is assumed that no hydro-aromaticrings can contribute to abstraction of hydrogen. Formation of free hydrogen gastakes place according to the following mechanism:

Ar �Har ! Ar �+Har � (14.19)

Har �+Ar �Har ! arH �Har +Ar� (14.20)

212 The Pyrolysis Model

5 10 1540

60

80

100

120Eo for CHn, CH4 and H2 (Upper=CH4,Lower=CHn)

Heating rate [oC/hr]

Eo

[kJ/

mol

e]

5 10 15100

200

300

400

500

600ko − CHn

Heating rate [oC/hr]

ko_C

Hn

[1/h

r]

5 10 150

5

10

15x 10

4 ko − CH4

Heating rate [oC/hr]

ko_C

H4

[1/h

r]

5 10 150

200

400

600

800

1000ko − H2

Heating rate [oC/hr]

ko_H

2 [1

/hr]

Figure 14.9: The dependence on heating rate for the kinetic parameters in themodel due to Tremblay & Charette (1988) for pyrolysis of pitch in anodes. Reac-tion orders are 0.7, 0.8 and 1.1 for tar, methane and hydrogen respectively.

The polymerization mechanism was supported by experimental results obtainedfrom depolymerization experiments of polymerized pitch as well as pyrolysis gaschromatography. The mechanism is valid until complete extinction of carboniza-tion gases1. This kind of chemical reactions occur in parallel with the volatilizationof condensables but as known from the literature, generation of non-condensablesmainly occur after complete volatilization of condensables.

The ultimate yields of CH4 and H2 depend on the �nal yield of tar since methyl-groups bonded in tar molecules are lost in volatilization. If not lost as vapour, thisfraction of methyl could contribute to the formation of CH3� and �nally generationof CH4.

In an analysis given in Gundersen (1996b), it was shown that the ultimate yieldsof non-condensables may be calculated from the following relationships:

fCH4;� =MC + 4MH

(MC + 3MH)(fp;CH3

C � f v x ;�f ;CH3

C ) (14.21)

fH2;� = fPH � f v x ;�f H

� MH

MC

(MC + 4MH)

(MC + 3MH)(fp;CH3

C � f v x ;�f ;CH3

C ) (14.22)

1Free hydrogen may probably also form by other mechanism, but the reaction step to obtainmethane is needed to be able to calculate a simple elemental balance on hydrogen.

14.3 The Model for Degassing of Non-Condensables 213

AromaticAliphatic

N

Oash

> 93%

C

C

Ar � CH3

� 94:6%

� 1:7%� 0:6%

H

4� 5% C

68%

8%

24%

�

�

CH CH2

� 0:7%

� 5:4% other CH3

� 2:4%

Figure 14.10: Diagram which shows nominal composition of whole pitch. About50 % of the compounds in coal tar pitch are substituted with methyl groups. Thedistribution of C-atoms in aromatic and aliphatic systems is shown. Also, thenominal distribution of di�erent aliphatic groups is given. Finally, the assumedrelationship between methyl groups and pitch solvent fractions is shown via dottedand dashed lines. It should be noted that the given numbers for the distributionof C atoms between di�erent functional groups (mass fractions) may vary (signif-icantly). The �gure is based on information found in Grint et al. (1988).

Furthermore, the yield of condensables can be found from:

f v =(1� cp)� fCH4;� � fH2;�

x ;�(14.23)

A linear set of three equations for f v , fCH4;� and fH2;� is obtained and explicitsolutions for the ultimate yields can be found. In summary, the ultimate yieldsdepend on the following variables (a total of �ve parameters):

� For both whole pitch and -fraction:

1. Mass fraction of carbon bonded to aromatic methyl (fp;CH3

C , f ;CH3

C )

2. Mass fraction of elemental hydrogen (fpH , f H)

214 The Pyrolysis Model

� Pitch coke yield cp

Mass fractions are related to the mass of whole pitch. For a pitch with a certaincomposition, it is only the coke yield that varies during baking. It is interesting tostudy the dependence of ultimate yield for changes in the coke yield. An exampleof use of the scheme as well as a study of sensitivities of ultimate yield with respectto di�erent parameters is given below.

The presented scheme allows for prediction of ultimate yields from data on thechemical composition of the pitch. When the composition of the binder pitchvaries, probably also the kinetic parameters will be perturbed. In an experimentalsetup to obtain data for determination of kinetic parameters, nominal ultimateyields also can be measured. In this case, there will probably be no need for aprediction of yields since they can be measured directly by gas chromatography.Such equipment is costly and time consuming to use (i.e. at low heating rates). Ifproved to perform well, the suggested scheme may save both time and money: Theonly parameters needed are overall coke yield and pitch composition data. Also,the presented scheme is interesting since it gives a theoretical explanation of thedependence of ultimate yields on chemical composition and varying coke yield.

Finally, some comments are necessary:

� During baking, the coke yield depends on the heating rate. Below, it isshown that nominal heating rates used in baking introduces variations inthe pitch coke yield. Thus, variations in yields of both condensables andnon-condensables occur. It is shown that the variations introduced in theyields of non-condensables are negligible (Gundersen 1995b).

� The scheme is based on the assumption that an average molecular massis representative for the whole -fraction. These molecules are capable ofeither vaporizing ( ! V ) or reacting according to ! �. The assumptionof an average molar mass may be a matter of controversy. Still, however,the assumption is needed to be able to calculate ultimate yields accordingto the suggested scheme.

14.3.3 Sensitivity of Ultimate Yields for Changes in Coke

Yield

As shown in Figure 14.11, the ultimate yields for non-condensables are linearfunctions of the pitch coke yield: When the coke yield increases, the yield of non-condensables increases. In consequence, the yield of condensables decreases bothdue to a reduced overall weight loss of the pitch as well as the increased loss ofnon-condensables.

Actually, this is an interesting observation since an increased coke yield have twoconsequences:

� A high coke yield is desired due to the positive impact on anode properties.

14.3 The Model for Degassing of Non-Condensables 215

0.6 0.65 0.7 0.75 0.81

1.5

2

2.5

3

3.5

4Methane

Pitch coke yield

f_C

H4

[%]

0.6 0.65 0.7 0.75 0.81

1.5

2

2.5

3

3.5

4Hydrogen

Pitch coke yield

f_H

2 [%

]

0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 0.76 0.78 0.83

3.5

4

4.5

5Non−Condensables / Methane and hydrogen

Pitch coke yield

f_C

H4

+ f_

H2

[%]

Figure 14.11: Ultimate yields for non-condensables in dependence of coke yield forwhole pitch. The yields increase as the coke yield increases. Pitch compositiondata are the same as in Examples 1 and 2 above.

�cY �fCH4;� [wt.%]�fH2;� [wt.%]

@fCH4;�

@cY[%]

@fH2;�

@cY[%]

0.70 1.15 2.98 1.87 5.46

Table 14.3: Nominal coke yield, yield of non-condensables and sensitivities ofthe yield of non-condensables with respect to coke yield. The yields are linearfunctions of pitch coke yield. The sensivities depend only on the pitch composition.Composition data are the same as in Examples 1 and 2 above.

� Non-condensables have a higher heating value than the condensables. Thus,also from an energy optimization point of view, a high coke yield seems tobe desirable.

The sensitivities of the yield of non-condensables with respect to coke yield isgiven in Table 14.3. If a nominal coke yield of �cp = 0:7 is assumed with maximumdeviation of ��cp � 0:1, the yields will vary with �fCH4;� = 0:2% and �fH2;� =0:5% for methane and hydrogen respectively. The corresponding relative errorsare 16% and 18%. This is slightly above limits commonly accepted for modellingerrors. In industrial baking, coke yields nominally may change only with half ofthe values given here. This gives errors below 10 % which is acceptable.

216 The Pyrolysis Model

14.3.4 A Model for Degassing of Non-Condensables Based

on an Extension of the Model for Low Temperature

Pyrolysis

So far, the models for degassing of non-condensables have been decoupled from thedegassing of condensables in the low-temperature pyrolysis regime. Also, the lowtemperature pyrolysis for conversion of pitch fractions , � and � was modelled byrate laws as intensive quantities. In the multiple reaction approach, the e�ectivereaction rate is an extensive quantity.

The low temperature pyrolysis reaction scheme was extended to also include thereactions for degassing of non-condensables. Rate laws as intensive quantities wereused for the conversion of pitch fractions. So far, the dynamics of the evolutionof the density in the pyrolysing liquid pitch and the binder coke which evolvesduring pyrolysis is not known. In this work, �rst order reaction kinetics is used.This makes it possible to express the model as di�erential equations for the massfractions of the pitch and binder-coke components without the need to know theapparent volume Va;p of the pitch (coke).

Since the model will later be used as basis for the development of a model for realdensity of the pitch during pyrolysis, the �-resins is divided into two subfractions:

� Initial content of QI in the pitch is primary QI denoted �p. This constitutesa separate fraction in the reaction scheme.

� During pyrolysis, primary QI (�p) is converted to secondary QI; i.e. subfrac-tion �s;H2

. The kinetics is only �ctive: the reaction kinetics is approximatedby the polymerization kinetics of the conversion of �-resin to �-resin.

The step �p ! �s;H2is mainly introduced because the primary QI is considered

to belong to the isotropic part of the pitch together with -resins and �-resins inthe model for real density of pyrolysing pitch; the density model is not discussedhere. To let the model �t data from the literature, it is convenient to have aninitial value of zero for the anisotropic phase of the pitch as represented by the�s-resins. � resins are divided into �p and �s-resins as shown in Figure 14.12.

The following mass balance equations can be obtained from the scheme in Fig-ure 14.12:

14.3 The Model for Degassing of Non-Condensables 217

LowTemperature Pyrolysis

HighTemperature

Pyrolysis 1 2

rv

Vc

r 2r 1;a

r 1;b

�

pC + (1� p) � CH4

qC + (1� q) �H2

sr�

�s;CH4

�s;H2

r�s;CH4

r�s;H2

�p

r�p

(1� s)r�

Figure 14.12: Subdivision of �-resins into �p- and �s-resins introduces a modi�-cation of the scheme in Gundersen (1995b, Fig. 3.42).

dm 1

dt= �rvVa;p � r 1;aVa;p � r 1;bVa;p

dm 2

dt= r 1;aVa;p � r 2Va;p

dm�

dt= r 1;bVa;p + r 2Va;p � r�Va;p

dm�p

dt= �r�pVa;p

dm�s;CH4

dt= sr�Va;p � r�s;CH4

Va;p

dm�s;H2

dt= (1� s)r�Va;pr�pVa;p � r�s;H2

Va;p

dmC

dt= pr�s;CH4

Va;p + qr�s;H2Va;p

For the total mass balance, the following equation applies:

dmp

dt= �rvVa;p � (1� p)r�CH4

Va;p � (1� q)r�H2Va;p

For the volatile gases, the following component balances apply:

dmc

dt= rvVa;p

dmCH4

dt= (1� p)r�s;CH4

Va;p

dmH2

dt= (1� q)r�s;H2

Va;p

All the rate laws ri are on the form:

ri = ki�b;i (14.24)

218 The Pyrolysis Model

such that riVa;p = kimi. Iintroduction of fractions xi where mi = ximp gives:model equations:

dx 1dt

= x 1(kvx 1 + (1� p)k�s;CH4x�s;CH4

+ (1� q)k�s;H2x�s;H2

) (14.25)

�kvx 1 � k 1;ax 1 � k 1;bx 1

dx 2dt

= x 2(kvx 1 + (1� p)k�s;CH4x�s;CH4

+ (1� q)k�s;H2x�s;H2

) (14.26)

+k 1;ax 1 � k 2x 2

dx�

dt= x�(kvx 1 + (1� p)k�s;CH4

x�s;CH4+ (1� q)k�s;H2

x�s;H2) (14.27)

+k 1;bx 1 + k 2x 2 � k�x�

dx�p

dt= x�p(kvx 1 + (1� p)k�s;CH4

x�s;CH4+ (1� q)k�s;H2

x�s;H2) (14.28)

�k�px�pdx�s;CH4

dt= x�s;CH4

(kvx 1 + (1� p)k�s;CH4x�s;CH4

+ (1� q)k�s;H2x�s;H2

)

+sk�x� � k�s;CH4x�s;CH4

(14.29)

dx�s;H2

dt= x�s;H2

(kvx 1 + (1� p)k�s;CH4x�s;CH4

+ (1� q)k�s;H2x�s;H2

)

+(1� s)k�x� + k�px�p � k�s;H2x�s;H2

(14.30)

dxC

dt= xC(kvx 1 + (1� p)k�s;CH4

x�s;CH4+ (1� q)k�s;H2

x�s;H2) (14.31)

+pk�s;CH4x�s;CH4

+ qk�s;H2x�s;H2

The coke yield is given from:

dcy

dt= �(kvx 1 + (1� p)k�s;CH4

x�s;CH4+ (1� q)k�s;H2

x�s;H2)cy (14.32)

For the ultimate yields of condensables and non-condensables, the following ap-plies:

dyc

dt= kvx 1cy (14.33)

dyCH4

dt= (1� p)k�s;CH4

x�s;CH4cy (14.34)

dyH2

dt= (1� q)k�s;H2

x�s;H2cy (14.35)

Based on the model for mass fractions of binder pitch components and pitchcoke, the kinetic parameters for formation of non-condensables were estimated.The same data as before (data generated by the model presented in Tremblay &Charette (1988)) were used also in this case. But since Tremblay's data are onlynormalized weight loss curves (or the weight loss of a certain nominal ultimateyield), the weight losses predicted with the model was normalized before used inthe parameter estimation algorithm. Model parameters can be found in Gundersen

14.3 The Model for Degassing of Non-Condensables 219

x 1(0) x 2(0) x�(0) x�p(0) x�s;CH4(0) x�s;H2

(0) xC(0)

0.65 0.00 0.27 0.08 0.00 0.00 0.00

Table 14.4: Pitch composition used in simulations of the extended reaction schemefor modelling of low-temperature and high temperature pyrolysis of coal tar pitch.

(1995b). It should be noted that the accuracy in ultimate yield of non-condensablespredicted with this model depends on the accuracy of the prediction of the yield ofcondensables. There is a chance that the yield of non-condensables can be over-estimated with the present accuracy of prediction of condensables. Initial pitchdata is given in Table 14.4.

To improve the performance of the prediction of non-condensables, two modi�ca-tions of the scheme could be done:

1. The introduction of several reactions in series instead of the single step�s;CH4

! pC + (1 � p)CH4. The same could be said about the step forformation of hydrogen.

2. The use of higher order reactions ri = ki�nib;i for i = �CH4

; �H2.

The �rst approach will make the model more complex since more states are in-troduced in the reaction scheme. In the second alternative, knowledge of thecoke density is needed to be able to calculate the reaction rates for the loss ofnon-condensables.

Still one approach is possible: A kind of conversion dependent activation energycould be introduced for the reaction steps:

�s;CH4

r�s;CH4! pC + (1� p)CH4

�H2

r�s;H2! qC + (1� q)H2

to synthesise a multiple reaction approach in the scheme. As the mass fractionsof �CH4

and �H2decreases, conversion to CH4 and H2 become more and more

di�cult. This approach is discussed in the next paragraph.

The Use of Conversion Dependent Activation Energies for Improvement

of The Reaction Scheme

So far, the extended reaction scheme's predictive performance is comparable withthe use of single �rst order reactions to describe the development of non-condensables.The predictive properties of the model would be signi�cantly improved if a kindof conversion dependent activation energy for the generation of methane and hy-drogen could be introduced: The activation energy in reaction rates r�s;CH4

andr�s;H2

should increase as coke C is formed.

220 The Pyrolysis Model

200 400 600 8000

0.2

0.4

0.6

0.8

1All pitch fractions

Temperature [oC]

xi [1

]

200 400 600 8000

0.2

0.4

0.6

0.8

1Alpha−resin/Pitch Coke

Temperature [oC]

xi [1

]

200 400 600 8000.7

0.75

0.8

0.85

0.9

0.95

1Coke Yield

Temperature [oC]

cy [1

]

200 400 600 8000

1

2

3

4Yields of hydrogen and methane

Temperature [oC]

y_i [

%]

Figure 14.13: Simulation of the extended reaction scheme of pitch pyrolysis whichincludes degassing of non-condensables. The heating rate is a = 5:0�C=hr. Initialpitch data is given in Table 14.4.

200 400 600 8000

0.2

0.4

0.6

0.8

1All pitch fractions

Temperature [oC]

xi [1

]

200 400 600 8000

0.2

0.4

0.6

0.8

1Alpha−resin/Pitch Coke

Temperature [oC]

xi [1

]

200 400 600 800

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Coke Yield

Temperature [oC]

cy [1

]

200 400 600 8000

1

2

3

4Yields of hydrogen and methane

Temperature [oC]

y_i [

%]

Figure 14.14: Simulation of the extended reaction scheme. Heating rate is a =10:0�C=hr.

14.3 The Model for Degassing of Non-Condensables 221

200 400 600 8000

0.2

0.4

0.6

0.8

1All pitch fractions

Temperature [oC]

xi [1

]

200 400 600 8000

0.2

0.4

0.6

0.8

1Alpha−resin/Pitch Coke

Temperature [oC]

xi [1

]

200 400 600 800

0.65

0.7

0.75

0.8

0.85

0.9

0.95

Coke Yield

Temperature [oC]

cy [1

]

200 400 600 8000

1

2

3

4Yields of hydrogen and methane

Temperature [oC]

y_i [

%]

Figure 14.15: Simulation of the extended reaction scheme. Heating rate is a =15:0�C=hr.

x 1(0) x 2(0) x�(0) x�p(0) x�s;CH4(0) x�s;H2

(0) xC1(0) xC2

(0)

0.65 0.00 0.27 0.08 0.00 0.00 0.00 0.00

Table 14.5: Pitch composition used in simulations of the extended reaction schemefor modelling of low-temperature and high temperature pyrolysis of coal tar pitch.Conversion dependent activation energies were used for decomposition of the �-resins.

In the former model with conversion dependent activation energy, the increase inactivation energy was linked to the conversion of the component considered:

E = Emin + (Emax �Emin)X

where X is the conversion of the component which changes from zero to one duringthe reaction. It is of main importance that X increases monotonously during thereaction. This is not the case for the �-resins which give basis for formation ofnon-condensables: First there is an initial build-up of the �-subfractions before themain decomposition commences; see Figure 14.20, Figure 14.21 and Figure 14.22.Then there is a steady decrease in the � subfractions. But since the �-subfractionsboth increases and decreases, we cannot connect the increased activation energydirectly to the mass fractions of �-resins. In the simulations, pitch compositiondata as given in Table 14.5 were used.

On the other hand, the conversion of �s;CH4and �s;H2

is equivalent with the

222 The Pyrolysis Model

formation of coke C. Therefore, the increase in activation energy can be connectedto the build-up of the coke fraction.

Furthermore, individual coke fractions for coke formed from �s;CH4and �s;H2

isalso needed. This is solved by introducing coke-fractions C1 and C2 instead of thesingle fraction C:

dxC1

dt= xC1

(kvx 1 + (1� p)k�s;CH4x�s;CH4

+ (1� q)k�s;H2x�s;H2

) (14.36)

+pk�s;CH4x�s;CH4

dxC2

dt= xC2

(kvx 1 + (1� p)k�s;CH4x�s;CH4

+ (1� q)k�s;H2x�s;H2

) (14.37)

+qk�s;H2x�s;H2

The following two rate laws are used:

r�s;CH4= k�s;CH4

�b;�s;CH4(14.38)

k�s;CH4= k�s;CH4

;�e�

E�s;CH4(xC1

)

RT (14.39)

E�s;CH4= E�s;CH4

;min + (E�s;CH4;max � E�s;CH4

;min)xC1

cCH4

(14.40)

cCH4=s p(1� xc(1))

1� xc(1)(14.41)

r�s;H2= k�s;H2

�b;�s;H2(14.42)

k�s;H2= k�s;H2

;�e�

E�s;CH4(xC1

)

RT (14.43)

E�s;H2= E�s;H2

;min + (E�s;H2;max �E�s;H2

;min)xC2

cH2

(14.44)

cH2=

(1� s) q(1� xc(1))

1� xc(1)(14.45)

xc(1)) is the ultimate yield of condensables. Thus, e�ectively:

E�s;CH4= E�s;CH4

;min + (E�s;CH4;max �E�s;CH4

;min)xC1

s p(14.46)

E�s;H2= E�s;H2

;min + (E�s;H2;max �E�s;H2

;min)xC2

(1� s) q(14.47)

The idea is that there is only negligible loss of non-condensables in parallel withthe evolution of condensables. Therefore (1 � xc(1)) constitutes a reservoir forthe evolution of non-condensables. Furthermore, the stoichiometry gives estimatesof the maximum fractions of C1 and C2 as s p and (1 � s) q respectively. In thelimit, limt!1 ci < 1:0 where i = CH4; H2.

The modi�ed reaction scheme is shown in Figure 14.16.

Summarizing the Characteristics of the Model

� State vector: x = [x 1 ; x 2 ; x� ; x�p ; x�s;CH4; x�s;H2

; xC1; xC2

; cy; yc; yCH4; yH2

]T

14.3 The Model for Degassing of Non-Condensables 223

LowTemperature Pyrolysis

HighTemperature

Pyrolysis 1 2

rv

Vc

r 2r 1;a

r 1;b

�

sr�

�s;H2

pC1 + (1� p) � CH4

qC2 + (1� q) �H2

�s;CH4

r�s;H2

r�s;CH4

�p

(1� s)r�

r�p

Figure 14.16: Individual coke fractions C1 and C2 are introduced instead of thesingle coke fraction C in the scheme presented in Figure 14.12.

0 20 40 60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Predicted (−) and Measured (−−) Conversion of H2

Time [hr]

X_H

2

Figure 14.17: Simulation of the extended reaction scheme with conversion depen-dent activation energy. The plot shows predicted and measured conversion ofhydrogen vs. time in coal tar pitch pyrolysis for a heating rate of a = 5:0�C=hr.The data vectors were generated by the model in Tremblay & Charette (1988).Initial pitch data is given in Table 14.5.

� The fraction was divided into 1 and 2 to be able to predict loss ofcondensables.

� �-resins were divided into �p- and �s -resins; i.e. primary and secondary(mesophase) QI. This was mainly done to have the model capable of pre-

224 The Pyrolysis Model

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Predicted (−) and Measured (−−) Conversion of H2

Time [hr]

X_H

2

Figure 14.18: Simulation of the extended reaction scheme with conversion depen-dent activation energy for a heating rate of a = 10:0�C=hr.

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Predicted (−) and Measured (−−) Conversion of H2

Time [hr]

X_H

2

Figure 14.19: Simulation of the extended reaction scheme with conversion depen-dent activation energy for a heating rate of a = 15:0�C=hr.

14.4 A Hydrogen Balance Equation 225

200 400 600 800 10000

0.5

1Gamma, beta, alpha_p

xi [1

]

200 400 600 800 10000

0.5

1Alpha_s_ch4, alpha_s_h2, C1, C2

xi [1

]

200 400 600 800 10000.7

0.8

0.9

1

cy [1

]

200 400 600 800 10000

2

4

y_i [

%]

200 400 600 800 1000250

300

350

Temperature [oC]

E_c

h4 [k

J/(m

ol K

)]

200 400 600 800 1000140

160

180

200

Temperature [oC]

E_h

2 [k

J/(m

ol K

)]

Figure 14.20: Simulation of the extended reaction scheme of pitch pyrolysis whichincludes degassing of non-condensables. A conversion dependent activation energyis used in the reactions for generation of non-condensables. The heating rate isa = 5:0�C=hr.

dicting the real density of the binder pitch and the pitch coke.

� Secondary �s-resins were subdivided into �s;CH4and �s;H2

to give basis fora submodel for generation of non-condensables.

� Methane forms during decomposition of �s;CH4-resins.

� Hydrogen forms during decomposition of �s;H2-resins.

� A kind of conversion dependent activation energies were used in the reactionsteps for generation of methane and hydrogen. To implement the conversiondependent activation energies, coke subfractions C1 and C2 were needed.

� Total coke formation is represented by C = C1 + C2.

14.4 A Hydrogen Balance Equation

To complete the model of pitch pyrolysis, an equation of the mass balance ofhydrogen in pitch during pyrolysis is derived. The following assumptions are made:

� The pitch is assumed to consist of only hydrogen and carbon. The presenceof heteroatoms as oxygen, nitrogen and sulphur is neglected.

226 The Pyrolysis Model

200 400 600 800 10000

0.5

1Gamma, beta, alpha_p

xi [1

]

200 400 600 800 10000

0.5

1Alpha_s_ch4, alpha_s_h2, C1, C2

xi [1

]

200 400 600 800 1000

0.7

0.8

0.9

1

cy [1

]

200 400 600 800 10000

2

4

y_i [

%]

200 400 600 800 1000250

300

350

Temperature [oC]

E_c

h4 [k

J/(m

ol K

)]

200 400 600 800 1000140

160

180

200

Temperature [oC]

E_h

2 [k

J/(m

ol K

)]

Figure 14.21: Simulation of the extended reaction scheme of pitch pyrolysis withconversion dependent activation energy for the non-condensables. The heatingrate is a = 10:0�C=hr.

� The binder pitch contains a nominal hydrogen content of 5.0 % by mass.

� An average molar mass for the light part of the -fraction which is lost byvaporization is derived from an assumed molecular formula of (C1:5H)n withn = 16.

Hydrogen is lost from the pitch via condensables and non-condensables (methane

and hydrogen). By using the rate laws rv;c [kg/(m3hr)], rv;CH4

[kg/(m3hr)] and

rv;H2[kg/(m

3hr)], the hydrogen balance can be calculated as follows:

� Since:

mi = niMi ) ni =mi

Mi(14.48)

rate laws ri expressed in [kg/m3 hr] can be expressed in [mol/m3 hr] bycalculation with the inverse of the molar mass Mi.

� One mol of the light -fraction corresponds to n mol of hydrogen.

� One mol of methane corresponds to four mol of hydrogen.

14.4 A Hydrogen Balance Equation 227

200 400 600 800 10000

0.5

1Gamma, beta, alpha_p

xi [1

]

200 400 600 800 10000

0.5

1Alpha_s_ch4, alpha_s_h2, C1, C2

xi [1

]

200 400 600 800 1000

0.7

0.8

0.9

1

cy [1

]

200 400 600 800 10000

2

4

y_i [

%]

200 400 600 800 1000250

300

350

Temperature [oC]

E_c

h4 [k

J/(m

ol K

)]

200 400 600 800 1000140

160

180

200

Temperature [oC]

E_h

2 [k

J/(m

ol K

)]

Figure 14.22: Simulation of the extended reaction scheme of pitch pyrolysis withconversion dependent activation energy for the non-condensables. The heatingrate is a = 15:0�C=hr.

For the mass mH of hydrogen in the mass of pitch, the following equation applies:

dmH

dt= �nMH

1�M

rv;cVa;p � 4MH1

MCH4

rv;CH4Va;p (14.49)

�1MH1

MHrv;H2

Va;p

where �M , MCH4and MH are the average molar mass for the light part of the

-fraction, the molar mass of methane CH4 and the molar mass of elementalhydrogen H respectively. In Gundersen (1995b), it is shown that this equationalso can be written on the following form:

dxH

dt= xH (kv;cx 1 + (1� p)k�s;CH4

x�s;H2+ (1� q)k�s;H2

x�s;H2) (14.50)

�f Hkv;cx 1 � 4MH1

MCH4

(1� p)k�s;CH4x�s;H2

�(1� q)k�s;H2x�s;H2

In this context, it is assumed that pitch only consists of hydrogen and carbon.Hence:

xH + xC = 1 (14.51)

228 The Pyrolysis Model

where xC and xH are mass fractions of carbon and hydrogen respectively.

It should be noted that successful integration of the elemental hydrogen balancedepends on careful selection of parameters for the hydrogen content in the volatiliz-able part of the -fraction ( �M ; f

H) and the initial value of hydrogen in whole

pitch. This is due to the fact that there is no dependence on the hydrogen contentxH and the rate of release of gases. The yield of gases is taken care of via thecompetition between vaporization and polymerization to give condensables as wellas the stoichiometric coe�cients to give non-condensables. In this case, an initialvalue of xH(0) = 0:05 was used for whole pitch. This gives simulation resultswhich �t well to the yields of gases2.

f H is a parameter which could be tuned to obtain the best possible match withactual pitch data. In general, the determination of �M should be coordinated withthe algorithm for prediction of ultimate yields and the parameter estimation toobtain the kinetics for the release of gases.

In the simulations below, the dynamics of xH seems con�dent due to the extensiveuse of realistic pitch data to obtain the model parameters. No further elaborationon this subject will be given here.

14.5 Simulating The Pyrolysis Model

The model was simulated for heating rates of a = 5:0; 11:0 and 15:0�C=hr. Thesame data as used in simulation of model A Gundersen (1995b), was used in thesesimulations3. A mass average molar mass for the pitch can be found from:

�M =1

xHMC

+ xCMH

(14.52)

The average molar mass is of importance for the calculation of the speci�c heatcapacity of the binder pitch during pyrolysis.

14.6 Conclusions

In spite of the limitations of the model, the pyrolysis model gives a satisfactorydescription of some of the main phenomena which take place during pyrolysis. Atthe present stage, the model is capable of predicting:

� Yield of condensables

� Yield of non-condensables (methane and hydrogen)

� Pitch coke yield

2On the other hand, xH = 0:044 for whole pitch in the derivation of stoichiometric coe�cientsin the scheme.

3In model A, the yield of non-conensables is constant.

14.6 Conclusions 229

200 400 600 800 10000

0.5

1Gamma, beta, alpha_p

xi [1

]

200 400 600 800 10000

0.5

1Alpha_s_ch4, alpha_s_h2, C1, C2

xi [1

]

200 400 600 800 10000.7

0.8

0.9

1

cy [1

]

200 400 600 800 10000

2

4

y_i [

%]

200 400 600 800 10000

2

4

6

Temperature [oC]

fh [%

]

200 400 600 800 10008

9

10

11

Temperature [oC]

Mav

e [g

/mol

]

Figure 14.23: Simulation of the total pyrolysis model B. The model for formationof non-condensables in the high temperature pyrolysis is based on an extension ofthe low-temperature pyrolysis reaction scheme. Activation energies which dependon the formation of coke fractions C1 and C2 are used in the reaction steps for gen-eration of methane and hydrogen. Model parameters can be found in Section 14.2and Section 14.3. The heating rate is a = 5:5�C=hr.

230 The Pyrolysis Model

200 400 600 800 10000

0.5

1Gamma, beta, alpha_p

xi [1

]

200 400 600 800 10000

0.5

1Alpha_s_ch4, alpha_s_h2, C1, C2

xi [1

]200 400 600 800 1000

0.7

0.8

0.9

1

cy [1

]

200 400 600 800 10000

2

4

y_i [

%]

200 400 600 800 10000

2

4

6

Temperature [oC]

fh [%

]

200 400 600 800 10008

9

10

11

Temperature [oC]

Mav

e [g

/mol

]

Figure 14.24: Simulation of the total pyrolysis model B. The heating rate is a =11:0�C=hr.

200 400 600 800 1000−0.5

0

0.5

1Gamma, beta, alpha_p

xi [1

]

200 400 600 800 10000

0.5

1Alpha_s_ch4, alpha_s_h2, C1, C2

xi [1

]

200 400 600 800 1000

0.7

0.8

0.9

1

cy [1

]

200 400 600 800 10000

2

4

y_i [

%]

200 400 600 800 10000

2

4

6

Temperature [oC]

fh [%

]

200 400 600 800 10008

9

10

11

Temperature [oC]

Mav

e [g

/mol

]

Figure 14.25: Simulation of the total pyrolysis model B. The heating rate is a =15:0�C=hr.

14.6 Conclusions 231

0 500 1000 15000

10

20

30

40Yield of condensables vs. heating rate

y_c

[%]

0 500 1000 15000

0.5

1

Yield of methane vs. heating rate

y_c

h4 [%

]

0 500 1000 15000

1

2

3

4Yield of hydrogen vs. heating rate

y_h

2 [%

]

Figure 14.26: Simulation of the total pyrolysis model B. The yield of gases asfunction of temperature for three di�erent heating rates is presented for di�erentheating rates. The yield of non-condensables does not vary very much with heatingrate. The heating rates are a = 5:5; 11:0 and 15:0�C=hr.

0 500 1000 150060

70

80

90

100Coke yield vs. heating rate

cy

[100

]

0 500 1000 15000

1

2

3

4

5Mass percent of hydrogen vs. heating rate

fh[%

]

0 500 1000 15007

8

9

10

11

12Average molar mass vs. heating rate

Mav

e [g

/mol

]

Figure 14.27: Simulation of the total pyrolysis model B. Yield of gases, coke yield,hydrogen content and mass average molar mass as function of temperature ispresented for three di�erent heating rates. The coke yield is most sensitive tothe heating rate in the low temperature regime (i.e. up to a temperature of450�C). Also, hydrogen content and average molar mass seem quite independentof the heat treatment programme used during pyrolysis. The heating rates area = 5:5; 11:0 and 15:0�C=hr.

232 The Pyrolysis Model

Part IV

Mathematical Modelling of

Carbon Properties

Chapter 15

Density and Porosity of

Binder Pitch and Pitch Coke

Models for apparent- and real densities and volumes of the binder pitch and thecorresponding pitch coke are presented in this chapter. Few measurements ofdensity and porosity for heat treatment up to temperatures in the order of 1400�Ccan be found in the literature. The qualitative basis for the models is supportedby experimental results from Darney (1958) and Wilkening (1983).

15.1 Introduction

The open porosity of a carbon material is obtained from the following relationships:

�� = 1� �a;p

�r;p(15.1)

�� = 1� vr;p

va;p(15.2)

� and v denote density and speci�c volume respectively. The real density can becalculated from the following relationships as shown in Appendix B:

�r;p =

nXj=1

�r;j�r;j (15.3)

�r;p =1Pn

j=1

xj�r;j

(15.4)

�r;j , xj and �r;j are the real volume fractions, mass fractions and real densitiesrespectively of the pitch components. The rate of change of the real density with

236 Density and Porosity of Binder Pitch and Pitch Coke

respect to time can be derived from one of Equations (15.3) or (15.4). This gives:

d�r;p

dt=

nXj=1

(�r;jd�r;j

dt+ �r;j

d�r;j

dt) (15.5)

d�r;p

dt= �

Pnj=1

(�r;jdxj

dt�xj

d�r;j

dt)

�2r;j�Pn

j=1

xj�r;j

�2 (15.6)

In Appendix B, it was shown that:

xj�r;p = �r;j�r;j (15.7)

Corresponding expressions exist for the apparent density:

�a;p =

nXj=1

�a;j�a;j (15.8)

�a;p =1Pn

j=1

xj�a;j

(15.9)

d�a;p

dt=

nXj=1

(�a;jd�a;j

dt+ �a;j

d�a;j

dt) (15.10)

d�a;p

dt= �

Pnj=1

(�a;jdxj

dt�xj

d�a;j

dt)

�2a;j�Pn

j=1

xj�a;j

�2 (15.11)

xj�a;p = �a;j�a;j (15.12)

15.2 Measurements of Densities and Porosity

The evolution of density and porosity is related to the processes of irreversibleexpansion and shrinkage which occur during baking.

Some studies on expansion and shrinkage of petroleum coke during calcinationis reviewed in Gundersen (1996e). For pitch coke and more isotropic carbons,however, most of the studies have been performed in the graphitization regime(Loch & Austin 1956), (Fishbach 1977), (Wagner, Hammer & Wilhelmi 1981).On the other hand, Franklin (1949) studied the development of apparent andreal density during carbonization of coals. More relevant is the study performedby Darney (1958) on the development of apparent and real volume of a binderpitch during coking. Heating rates comparable to nominal baking rates were used.The schematic dependence of volumes on temperature are shown in Figure 15.1.Signi�cant swelling occurs between 320 (350) and 420�C (480�C). The swellingseems to coincide with the onset of thermal condensation (polymerization) of thepitch and the accompanying increase in pitch viscosity. Due to swelling, there is a

15.2 Measurements of Densities and Porosity 237

signi�cant increase in the apparent volume of the binder pitch. Correspondingly,the apparent density decreases. The evolution of densities and porosity proceedas follows:

� As long as the escape of low molecular weight compounds proceeds, theviscosity is low enough to allow for the release of gases without formation ofbubbles in the bulk material.

� At the onset of condensation reactions, pitch viscosity increases due to an in-creased molecular weight of the bulk pitch material. Escape of gases (conden-sables and mainly non-condensables formed in the transition ! � and� ! �s) is obstructed and bubbles (porosity) form in the bulk pitch mate-rial.

� At temperatures in the order of 450�C, a maximum occur in the porosityand apparent volume of the binder pitch. This maximum seems to coincidewith the intermediate maximum in the release rate of non-condensables atapproximately 460�C as observed by Politis & Chang (1985).

� Further increase in apparent volume does not occur. Rather, porosity de-creases slightly up to temperatures in the order of 650�C. This is due eitherto the release of some of the gas bubbles or anisotropic shrinkage of thebinder coke volume (with shrinkage factor � > 1). It may also be that therate of released gases decays above 450�C and that up to approximately600�C (650�C), there is not enough power in the released gas to maintainthe expansion of the binder pitch.

� At temperatures above 650�C, porosity again increases due to anisotropicshrinkage of the solidi�ed binder coke (shrinkage factor � < 1).

Over the whole temperature range, there is a steady decrease in the real volumeof the pitch; i.e. the real density increases. This is in accord with the measuredreal density in Wilkening (1983, Fig. 11), except that Wilkening's real densitycurve has an approximately constant value up to temperatures in the order of400�C. Between 450 and 500�C (600�C), there is relatively little increase in thereal density; see Figure 15.3. According to Darney (1958), �nal porosity in thepure binder coke is in the order of 30 to 35 % at 900�C.

A thorough description of the properties of the pitch fractions is given in Gunder-sen (1996e), and it was especially noted that �-resins have pronounced swellingcapacity during the coking process (Gemmeke, Collin & Zander 1978). Swelling oc-curs in parallel with the condensation reactions which start with the formation of�-resins. The maximum in porosity at approximately 450�C more or less coincidewith the total conversion of and �-resins to �s- resins (mesophase). At 450

�C,the mesophase is still not solid. Now the porosity decreases until solidi�cation ofthe binder coke is complete at about 600 �C (650�C). Above this temperature,the porosity again increases.

238 Density and Porosity of Binder Pitch and Pitch Coke

100

90

80

70

60

50

40

30

20

10

200 300 400 500 600 700 800 900

Loss of volatiles

Swelling

Post plastic shrinkage

VaVa;�

T [�C]

VrVr;�

VV�

[%]

Figure 15.1: Real and apparent volumes of coking binder pitch. Heating ratescomparable to nominal baking rates are used. The volumes are always less than theinitial pitch volume. Thus, swelling is modest in coking of coal tar pitch. Volumeswere measured at a temperature of 15.5�C. If measured at working temperatures,the relative volumes would be unchanged if the coe�cient of thermal expansionvaries only negligibly with temperature. The �gure is based on Darney (1958, Fig.6).

15.3 Real Density of the Isotropic and Anisotropic

Pitch Phases

15.3.1 Real Density of Isotropic Pitch

As reviewed in Gundersen (1996e), correlations for pitch density as function oftemperature between 20 and 300�C is available in the literature. Typically, thedensity is calculated from:

�r;p;i = �r;p;i;�(1� �(T � T�)) (15.13)

15.3 Real Density of the Isotropic and Anisotropic Pitch Phases 239

60

50

40

30

20

10

200 300 400 500 600 700 800 900

Post plastic shrinkage

Swelling

�o[%]

T [�C]

Figure 15.2: Evolution of porosity in the binder coke. The �gure is based onDarney (1958, Fig. 5). Note that the evolution of binder coke porosity is di�erentfrom the porosity formed in calcined petroleum coke (Rhedey 1967). In the sametemperature range, porosity in calcined petroleum coke monotonously increaseswith temperature; there is no intermediate maximum in porosity. This may bedue to the signi�cantly higher heating rates used in petroleum coke calcination.

with � � 0:00044 1/K.

15.3.2 Density of Mesophase and Semi-Coke

Below, the development of real density is modelled as a thermally activated pro-cess. Due to the lack of data, the simplest approach is to assume that the densitycan be calculated from a single n'th order reaction as follows:

d�r;p;c

dt= kr;p(�gr � �r;p;c)

n (15.14)

kr;p = kr;p;�e�

Er;p

RT (15.15)

�r;p;c(0) = 1430:00 kg/m3

�gr is the density of graphite. By using the data in Wilkening (1983, Fig. 11) andthe criterion:

J =

mXi=1

c(�r;p;ci � �r;p;ci)2 (15.16)

where m is the number of data-points and c is a constant, model parameters wereestimated as shown in Table 15.1. Constraints were put on the estimated densityto secure that the density curve is not too steep at high temperatures. Also,

240 Density and Porosity of Binder Pitch and Pitch Coke

200 300 400 500 600 700 800 900 1000

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

T [�C]

�r;p[g

cm3 ]

Figure 15.3: Real density �r;p of binder coke as function of temperature for heatingrates of 3:3�C=hr (up to 600�C) and 6:6�C=hr (600 to 1000�C). It should benoted that the at portion of the curve between room temperature and 400�C isdue to the fact that the measurements are done at room temperature. At elevatedtemperatures, the density would have a lower value due to thermal expansion ofthe pitch. One may conclude that the susceptibility of the room temperaturedensity for pitch heated up to 400�C is relatively una�ected by the polymerizationreactions which take place in the pitch (mainly ! �). The �gure is taken fromWilkening (1983, Fig. 11).

kr;p;� [1/hr] Er;p [J/mol] n

4:4828� 10�5 76310 3.3486

Table 15.1: Parameters in the model for real density of pitch coke. Data weretaken from Wilkening (1983, Fig. 11). A heating rate of 5:0�C=hr was used inthe estimation. By using another set of data, di�erent model parameters wouldbe obtained in the estimation.

it is important that the density has a moderate rate of change with respect totemperature in the temperature interval between 500 and 600�C. The developmentof density for three di�erent heating rates is shown in Figure 15.4 and Figure 15.5.It seems as if the model predicts densities over a too wide range as compared tothe range that would occur in real experiments. This has to do with the lack ofdata used for the parameter estimation as well as the structure of the model.

In summary, the model has the following features:

� The model is capable of giving a qualitatively correct response in real densityvs. residence time/ heating rate: Long residence time/low heating rate gives

15.3 Real Density of the Isotropic and Anisotropic Pitch Phases 241

400 500 600 700 800 900 10001400

1500

1600

1700

1800

1900

2000

2100Real density for binder pitch as function of temperature

Temperature [oC]

rho_

rp [k

g/m

^3]

Figure 15.4: Real density for binder coke as a function of temperature for threedi�erent heating rates. The initial- and �nal temperatures are 450 and 1000�Crespectively. Three di�erent heating rates were used: 5, 10 and 15�C=hr. Thehighest density is achieved with the lowest heating rate, i.e. the highest residencetime. See also Figure 15.5 below.

a higher density compared to the density for a shorter residence time/higherheating rate when densities are compared at the same temperature.

� At low heating rates, the density might be over-estimated.

� At constant low temperature, the estimated density approach the density ofgraphite. This however, can only happen only for very high temperatures.This is an inherent weakness of this kind of model.

To improve the model, several approaches are possible:

� The use of a model with conversion dependent activation energy1.

� The use of a model based on multiple reactions and distributed parameters.

1Also called van Krevelen type of activation energy.

242 Density and Porosity of Binder Pitch and Pitch Coke

0 20 40 60 80 100 1201400

1500

1600

1700

1800

1900

2000

2100Real density for binder pitch as function of time

Time [hr]

rho_

rp [k

g/m

^3]

Figure 15.5: Real density for binder coke as function of time for three di�erentheating rates. Initial temperature is 450�C and �nal temperature is 1000�C. Threedi�erent heating rates were used: 5, 10 and 15�C=hr and the �nal value of densityincreases as the heating rate is decreased. The density depends on the residencetime as well as temperature.

15.4 Calculation of Total Porosity in the Anode

During Baking

The pyrolysis model can be used as a part of a model for calculation of totalporosity during baking. It can be shown that the following equation is valid forthe total porosity:

@�T

@t=

(1� �T � )

�r;p

0@ 3Xj=1

rj +@�r;p

@t

1A (15.17)

rj represents the rates of gas devolatilization; i.e. release of condensables, methaneand hydrogen.

The total porosity was simulated for three di�erent heating rates. The results arepresented in Figure 15.6.

The development of total porosity for four di�erent heat treatment programs areshown in Figure 15.7 and Figure 15.8. Figure 15.7 shows the heat treatmentprograms and Figure 15.8 shows the corresponding total porosity curves.

15.4 Calculation of Total Porosity in the Anode During Baking 243

a = 5 oC/hr

a = 10 oC/hr

a = 15 oC/hr

0 200 400 600 800 1000 1200 14000.16

0.18

0.2

0.22

0.24

0.26

0.28

Temperature [oC]

phi_

T [k

g/m

3]Anode total porosity for different heating rates

Figure 15.6: Total porosity in anode as function of temperature for three di�erentheating rates of a = 5; 10 and 15�C=hr. Maximum temperature is 1250�C. Thelowest total porosity is obtained by using the lowest heating rate.

0 50 100 150 2000

500

1000

1500

Time [hr]

Tem

pera

ture

[oC

]

20−500 oC: a=5; −>1250 oC: a=5

0 50 100 150 2000

500

1000

1500

Time [hr]

Tem

pera

ture

[oC

]

20−500 oC: a=15; −>1250 oC: a = 15

0 50 100 150 2000

500

1000

1500

Time [hr]

Tem

pera

ture

[oC

]

20−500 oC: a=5; −>1250 oC: a=15

0 50 100 150 2000

500

1000

1500

Time [hr]

Tem

pera

ture

[oC

]

20−500 oC: a=15; −>1250 oC: a=5

Figure 15.7: Heat treatment programs used for simulating the evolution of totalporosity in the anode.

244 Density and Porosity of Binder Pitch and Pitch Coke

subplot(221)

subplot(222)

subplot(223)

subplot(224)

0 200 400 600 800 1000 1200 1400

0.2

0.25

0.3

Temperature [oC]

phi_

T [1

]Total porosity for different heat treatment programs

Figure 15.8: Total porosity curves which belong to the time-temperature historiesshown in Figure 15.7. The lowest total porosity is achieved with a low heatingrate up to approximately 500�C and a signi�cantly higher heating rate from 500up to 1250�C.

15.5 AModel for Apparent Density which includes

both Low and High Temperature Pyrolysis

In Gundersen (1996c), it was shown that a model for apparent density basedon the behaviour of pyrolysis at temperatures up to 450�C is only capable ofgenerating a plateau value of porosity (or apparent density). From Figure 15.2,however, it can be seen that there is only an intermediate maximum occurring inporosity: The porosity starts to decay to reach an intermediate minimum valueat approximately 600�C. This minimum in porosity corresponds to the point ofsolidi�cation of the pitch. For increasing temperature, the porosity again starts toincrease. To explain the decrease and subsequent increase in porosity, two di�erentmechanisms in addition to swelling will be introduced.

The maximum in porosity corresponds to a maximum in the release rate of conden-sables. After the maximum is reached, a collapse in the swelled substance seemsto occur probably since the reduced gas release rate is not capable of maintainingthe expansion of the pitch coke apparent volume. The apparent density increases,but there is yet no corresponding increase in real density. In e�ect then, a de-crease in porosity occurs. The decrease continues up to temperatures in the orderof 600�C. According to Politis & Chang (1985) there is a new maximum in the

15.5 A Model for Apparent Density which includes both Low and

High Temperature Pyrolysis 245

0 20 40 60 80 100 120 140 160 180 2000

500

1000

1500

Time [hr]

Tem

pera

ture

[oC

]Temperature as function of time

0 200 400 600 800 1000 1200 1400

0.2

0.25

Temperature [oC]

phi_

T [1

]

Total porosity vs. temperature

Figure 15.9: Total porosity as function of temperature for a nonlinear (i.e. piece-wise linear) heat treatment program.

release rate of gas occurring approximately at this temperature. This maximumis thought to be due to solidi�cation of the binder coke: At higher temperatures,polymerization reactions occur in solid carbon with a di�erent kinetics than thepolymerization reactions which occur in liquid pitch (Greinke 1986). Anisotropicshrinkage of the solid carbon material leads to increased porosity. In summary,the following mechanisms seem to be active:

� The increase in porosity is due to swelling of the �-resins.

� The decay in porosity for temperatures between 400 and 600�C is related tothe gradual solidi�cation of the mesophase (a decay in gas release rate).

� The increase in porosity above temperatures in the order of 600�C is due toanisotropic shrinkage of the solidi�ed binder coke: Above 600�C, the increasein real density of the �-resins becomes more pronounced.

The development of porosity seems to go via three regimes which appears in serieswith partial overlapping: Swelling, collapse and anisotropic shrinkage as shown inFigure 15.10.

The model developed here for �a;�s also corresponds to the apparent density �a;cof the anisotropic pitch phase. If the model of �a;�s is used to represent �a;c, the

246 Density and Porosity of Binder Pitch and Pitch Coke

following replacement of variables must be performed:

�a;�s ! �c (15.18)

x�s ! xc (15.19)

�a;�s ! �a;c (15.20)

�r;�s ! �r;c (15.21)

15.5.1 The Swelling Regime

Theoretically, swelling of �-resins happens as long as evolution of non-condensablesoccur in the uid pitch. The de�nition of uidity used in this context is presentedbelow: Binder pitch remains uid up to temperatures in the order of 600�C. Theswelling behaviour is strongly dependent on the release rate of gas, and after themaximum in swelling rate occurring at approximately 450�C, the collapse e�ect isrelatively more important than the swelling e�ect.

The following model equations describe the swelling regime (Gundersen 1996c):

�a;� = �r;� � ~kswrsw; �-resins (15.22)�d(�b;a;�s va;�s)

dt

�sw

= f�r� va;�s + r�p va;�s ; �s-resins (15.23)

rsw is the swelling gas release rate. f� is a stoichiometric coe�cient for the for-mation of �-resins.

15.5.2 The Collapse Regime

Here, the collapse e�ect is due to the reduction in gas release rate which happensat approximately 450�C (depending on heating rate). The gas ow is no longerable to expand the �s-resins, and the rate of decay in the apparent volume de-pends on the uidity of the pitch as well as the open porosity of the �s-resins.Below, the uidity is de�ned as a transformed measure of viscosity: The pointof zero uidity corresponds to the point of solidi�cation of the pitch. The glasstransition temperature Tg is used as a measure of the temperature at which solid-i�cation occur. At the glass transition point, the viscosity equals 1012 Pa s whichcorresponds to the maximum viscosity. The uidity function equals the di�erencebetween the logarithms of the maximum viscosity and current viscosity. Since theviscosity of �s-resins increases during pyrolysis, the uidity is a decaying functionwhich �nally becomes zero.

In the collapse regime, the rate of change of apparent density of �s-resins is pro-portional to the following parameters:

� The uidity f of the pitch (coke) raised to a certain power nf .

� The amount of pitch coke (�s) present in the pyrolysing system. In this case,the parameter used is the apparent volume fraction �a;�s of �s-resins. To beused in the rate law, it must be raised to a certain power n�a;�s .

15.5 A Model for Apparent Density which includes both Low and

High Temperature Pyrolysis 247

� The void fraction of the �s-resins as represented by ��;�s raised to a certainpower n��;�s .

The collapse tendency is reduced if the uidity and porosity is reduced. Formulatedin mathematical terms:�

d�a;�sdt

�c

= kc�n�a;�sa;�s fnf�

n��;�s�;�s (15.24)

f = log �p(1)� log �p (15.25)

��;�s = 1� �a;�s�r;�s

(15.26)

where kc is a constant. At temperatures when the collapse e�ect is dominating,�p � ��s .

The use of exponents ni (i = �a;�s ; f; ��;�s) makes it possible to give the decay inporosity a correct shape in the collapse regime. Typically:

� n�a;�s > 1

� nf < 1

This contributes to relaxing the collapse e�ect in the �rst period after the max-imum in the swelling release rate since still a certain amount of gas is releasedwhich have a certain swelling potential, i.e. it resists collapse.

One could argue that the collapse e�ect could be modelled by using a large valueof the anisotropic shrinkage factor �shr; anisotropic shrinkage is described andmodelled below. This, however, may be di�cult to achieve since

d�r;�sdt

is relativelylow for temperatures below 600�C. Also, it would be di�cult to implement afunction which changes the shrinkage factor from a value greater than one fortemperatures below approximately 600�C (porosity decreases) to a value below1 for temperatures above approximately 600�C (porosity increases). A simpletemperature e�ect is too simple since the shift in shrinkage behaviour dependson the heating rate used in the pitch pyrolysis: The collapse e�ect is physicallydi�erent from the anisotropic shrinkage occurring mainly at temperatures above600�C.

15.5.3 The Shrinkage Regime

Models for the open porosity of petroleum coke (Gundersen 1996e). In additionto anisotropic shrinkage, pu�ng and excavation was also thought to be importantmechanisms for the development of open porosity in petroleum coke. In the caseof pitch pyrolysis, one nominally uses heating rates so low that no pu�ng e�ect isobserved on the open porosity level. Therefore, the pu�ng e�ect can be neglected.On the other hand, the release of non-condensables leads to deterioration of thecrystalline structure developed in the mesophase formation since d002 increases

248 Density and Porosity of Binder Pitch and Pitch Coke

between 450 and 750�C. Thus, the crystalline density decreases and there is a de-crease in the microporosity. There is also a decrease in Lc in the same temperatureinterval.

Possible excavation is taken into account by adjusting the e�ective shrinkage pa-rameter �shr . Then, the following model is valid in the shrinkage regime:

d�a;�

dt= �shr(1� ��;�)

d�r;�

dt(15.27)

�shr = �shr;��n�shr�;�s (15.28)

��;�s = 1� �a;�s�r;�s

(15.29)

As shrinkage occurs, porosity and microcracks develop. Thus, as �r;�s increases,the ability of �a;�s to track �r;�s may be reduced due to the formation of microc-racks. Thus, set n�shr = �1.

15.5.4 The Compound Rate Law for Apparent Density

When modelling development of open porosity in petroleum coke during calcina-tion, it was thought that anisotropic shrinkage, pu�ng and excavation occur inparallel. Weight factors were used to balance the e�ects in the overall rate law forthe apparent density of the petroleum coke.

In principle, the same technique can be used for modelling the rate law for appar-ent density of the binder coke but here the mechanisms of swelling, collapse andanisotropic shrinkage interact to develop the apparent density of the binder coke.

In this case, however, the swelling behaviour is partially overlapped by the collapsee�ect and subsequent shrinkage. Constant weight factors cannot be used in thiscase since they do not allow for the dominance of a certain mechanism within agiven temperature interval: In general, we need weight functions which dependson the state variables used for modelling the pyrolysis. Now, de�ne three weightfunctions fw;i (i = sw; c; shr) which corresponds to the swelling, collapse andshrinkage regimes respectively. The compound rate law for the apparent densityof �s-resins is now de�ned by:

d�a;�sdt

= fw;sw

�d�a;�sdt

�sw

+ fw;c

�d�a;�sdt

�sc

+ fw;shr

�d�a;�sdt

�shr

(15.30)

Designing the Weight Functions

For designing the weight functions, the following characteristic features of pitchpyrolysis is used:

� fw;sw vs. fw;c:

15.5 A Model for Apparent Density which includes both Low and

High Temperature Pyrolysis 249

100 300 400 500 600 700 800 900200

0.5

1.0

Shrinkage

CollapseSwelling

��

T �C

Figure 15.10: Cooperation of three di�erent regimes for the formation of porosity(apparent density) of pitch coke.

{ Swelling leads to a maximum in porosity (minimum in apparent density)which corresponds to a sudden decay in the apparent volume fractionof �-resins. At the same time, there is a considerable increase in theapparent volume fraction �a;� of �-resins.

� fw;c vs. fw;shr:

{ Theoretically, the collapse e�ect is active until the binder coke hasturned into solid coke. This occurs as approximately 600�C.

{ Up to about 600�C, there is only a negligible shrinkage occurring in theshrinkage of the real volume of �-resins; thus d��

dt is low.

The following weight functions are then selected:

fw;sw =�a; + �a;� + �a;�p

�a; + �a;� + �a;�p + �a;�s= �a; + �a;� + �a;�p (15.31)

fw;c = 1� fw;sw = �a;�s (15.32)

fw;shr = 1 (15.33)

Characteristic Properties of the Weight Functions

The following features of the weight function synthesizes a dominance of a certainporosity developing mechanism within a certain temperature interval:

� As the formation of �s-resins commences, there is a gradual decrease in theapparent volume fraction �a;� of �-resins and then fw;sw changes from oneto zero.

250 Density and Porosity of Binder Pitch and Pitch Coke

� Correspondingly, as the apparent volume fraction �a;�s starts to increase,there is a change in fw;c from zero to one.

� Above 600�C, the collapse e�ect disappears since the uidity of the bindercoke becomes zero.

� Finally, fw;shr = 1 sinced�r;�sdt

is low for temperatures below 600�C.

There exist general techniques for deriving weight functions capable of interpolat-ing between di�erent operating regimes (Johansen 1994).

Chapter 16

Pitch Viscosity

For modelling of pitch viscosity, the pitch is considered to be a binary mixture ofan isotropic and an anisotropic phase. The method due to Grunberg and Nissan(Reid et al. 1987) can be used to calculate the viscosity of liquid mixtures. For abinary mixture, they use:

ln �m = y1 ln �1 + y2 ln �2 + y1y2G12 (16.1)

yj and �j are molar fractions and component viscosities respectively. �m is theviscosity of the liquid mixture. G12 is a function of the pitch components, tem-perature and sometimes the composition of the liquid.

16.1 Viscosity in Pitch as a Liquid Mixture

The following assumptions are considered valid for pitch:

� -resins, �-resins and primary �p-resins are assumed to belong to the isotropicpart of the pitch. The viscosity �i of the isotropic part of the pitch can bemodelled by a formula for reversible change of viscosity commonly found inthe literature:

�i = �i;�eE�iRT (16.2)

� Viscosity �c of secondary �-resins is assumed to develop as a thermally ac-tivated process:

d�c

dt= k�c�c (16.3)

k�c = k�c;�e�

E�cRT (16.4)

The modelling of this viscosity as a thermally activated process is assumedto describe the phenomenon of the sudden increase in viscosity as polymer-ization reactions commence.

252 Pitch Viscosity

� The interaction parameter G12 = Gic is set equal to zero.

Based on Grunberg and Nissan's model, the pitch viscosity can be modelled by:

ln �p = yi ln �i + yc ln �c (16.5)

where

yi = y + y� + y�p

yc = y�s

16.2 Applications of the Viscosity Model

The following applications of the viscosity model is possible:

� Gundersen (1995b)modelled the impact of pitch viscosity on the mass transfercapabilities of the liquid pitch. In this case, however, a very simple viscositymodel was used by assuming that:

�p =k

x (16.6)

where k is a constant. x is the mass fraction of -resins present in the pitch.Viscosity increases to in�nity as the -resins escape from the solution. This isqualitatively correct since polymerization reactions in the pitch contributesto increasing the viscosity.

� Modelling of the impact of viscosity on the swelling behaviour during pitchpyrolysis. The swelling behaviour of �-resins is a�ected by the viscosity sincethe residence time of escaping gas is proportional to the viscosity. Also, onemay argue that the secondary �-resins resist collapse if the viscosity becomeshigh enough. These phenomena occur at temperatures above 300�C. Molarfractions are related to mass fractions by xj �Mp = yj �Mj ; Mp is the massaverage molar mass. Typical molar masses for �-resins and �-resins are 600and 1200 g/mol respectively. Since 1

Mp=Pnc

j=1

xjMj

(nc is the number of

pitch components), the molar- and mass fractions will not di�er very muchand for simplicity one may use:

ln �p � xi ln �i + xc ln �c (16.7)

xi = x + x� + x�p (16.8)

xc = x�s (16.9)

where xj are pitch mass fractions. Swelling is typically active at tempera-tures in the order of 300 up to 450�C. For simplicity, constant values for �iand �c can be used:

�i � 2:50 � 10�6Ns/m2

�c � 1000 �i

16.3 Reversible Viscosity of the Isotropic Pitch Fractions 253

The value for �i was calculated by the equation used by Ho�mann & H�uttinger(1993) for the reversible viscosity:

�i = �i;�eEiRT (16.10)

�i;� = e�46:88Ns/m2 (16.11)

Ei = 1:62� 105 J/mol (16.12)

16.3 Reversible Viscosity of the Isotropic Pitch

Fractions

For the reversible viscosity, an Arrhenius-type expression as shown in Equation (16.2)is commonly used.

16.4 Irreversible Viscosity of the Anisotropic Pitch

Fraction

In our model, the anisotropic phase of the pitch is represented by the secondary QImaterial. The irreversible viscosity is modelled as a thermally activated processas follows:

dy

dt= ky(ym � y)ny (16.13)

ky = ky;�e�

Ey

RT (16.14)

�c = 10y (16.15)

A general n'th order process is used to describe the development of the logarithmof the viscosity. The model parameters are shown in Table 16.1. The modelparameters were manually tuned according to the following constraints:

� At temperatures between 400 and 500�C, there is a sudden increase in thepitch viscosity due to the polymerization reactions in the pitch. Typically,values for the isotropic viscosity just before the increase occurs in the orderof 1 Ns/m2. Now set nc(0) = 10y(0) with y(0) in the order of 2.0 to achievethe sudden increase in viscosity.

� Solidi�cation of the mesophase is completed at temperatures in the order of600�C. This corresponds to the glass transition temperature at which theviscosity equals 1012Ns/m2 by de�nition. This explains the selected valuefor ym.

254 Pitch Viscosity

ym [1] 12

ky;� [1/hr] 2:0806� 1012

Ey [J/mol] 200 000 / 220 000

ny [1] 1

y(0) [1] � 2

Table 16.1: Parameters in the model for the viscosity of binder coke as a thermallyactivated process.

0 100 200 300 400 500 600−20

−15

−10

−5

0

5

10

15

20

Temperature [oC]

ln(e

ta)

[ln(N

s/m

^2)]

natural log. of viscosity as function of temperature

Figure 16.1: The plot shows typical lapse of viscosity as function of temperaturefor pyrolysis of coal tar pitch for a heating rate of a = 15�C=hr. The massfraction based pyrolysis model (model B) was used in the simulation. Initial pitchdata were [x ; x� ; x�p ]

T = [0:65; 0:27; 0:08]T . The curve qualitatively equals theexperimentally measured viscosities shown in Ho�mann & H�uttinger (1993, Fig.3): As formation of secondary �-resins commences, there is a sudden increasein viscosity. Ho�man, however, used a heating rate of 24�C=hr. Data for thereversible (isotropic) viscosity was found in Ho�mann & H�uttinger (1993, Tab. 4).For the anisotropic (irreversible) viscosity, set �c = 60 Ns/m2. An even better �twith experimental data would be obtained if the irreversible (anisotropic) viscositywas modelled as a thermally activated process.

Chapter 17

Thermal Properties of

Mesophase, Semi-Coke and

Coke

The objective of this chapter is to derive models for prediction of speci�c heatcapacity and thermal conductivity of single phase carbon materials in dependenceof the heat treatment temperature. The models can be used for prediction of thethermal properties of the binder pitch as it is transformed into pitch coke.

Correlations for thermal properties of pitches exist in the literature. In this work, itis assumed that these correlations give satisfactory prediction of the pitch proper-ties in the uid (isotropic) part of the pitch during pyrolysis and that extrapolationto temperatures slightly higher than nominal introduces only negligible errors.

Einstein's theory will be used as basis for the model of speci�c heat capacity ofcoke. A semi-empirical model is used for the thermal conductivity.

17.1 Speci�c Heat Capacity

In the following, a general background for modelling of heat capacity based onthermodynamics and statistical mechanics is given.

17.1.1 Constant Pressure- vs. Constant Volume Speci�c

Heat Capacity

Using methods from thermodynamics, an expression for constant pressure speci�cheat capacity cp and constant volume speci�c heat capacity cv can be found. Based

256 Thermal Properties of Mesophase, Semi-Coke and Coke

� �l �

[m2/N] [1/K] [kg/m3]

� 2:9� 10�11 � 6� 10�6 � 2000

Table 17.1: Thermodynamic data for graphite and amorphous carbon. In thecalculation is used �v = 3�l. Some data can be found in Komatsu & Nagamiya(1951) and Kelly (1981).

on Sears & Salinger (1980), it can be shown that:

cp � cv =�2v�Tv (17.1)

where the thermal coe�cient of volume expansion �v and coe�cient of isothermalcompressibility � are de�ned as:

�v =1

v

�@v

@T

�v

(17.2)

� = �1

v

�@v

@p

�T

(17.3)

A rough estimate of �c = cp � cv for amorphous carbon can be found by usingthermodynamic data for graphite and an assumed average density as shown inTable 17.1. It is found that:

cp � cv � 0:0056T (17.4)

Therefore it is reasonable to assume that:

cp � cv (17.5)

for amorphous carbon materials; in this work heat treatment is performed at peaktemperatures less than 1300�C.

17.1.2 The Constant Volume Speci�c Heat Capacity

The constant volume speci�c heat capacity can be derived from models of theinternal energy. Classical kinetic theory and statistical mechanics (thermody-namics) based on Newton mechanics have not succeeded in giving satisfactorymodels for the properties of solid materials. The introduction of quantum meth-ods in statistical mechanics (quantum statistics) has given better models, and thesimplest models have been used for modelling heat capacity in carbon systems(Merrick 1983b).

Usually, two phenomena contribute to the internal energy in a crystal:

� Contribution ulat to internal energy from thermal vibrations in the crystallattice (phonon gas)

17.1 Speci�c Heat Capacity 257

� Contribution uel to internal energy from free electrons (electron gas)

These energy forms are the basis for deriving the lattice speci�c heat cv;lat andthe electronic speci�c heat cv;el. Thus:

u = ulat + uel (17.6)

cv =

�@u

@T

�v

= cv;lat + cv;el (17.7)

Both the lattice and electronic contribution can be modelled by the aid of quantummethods.

In the following, the lattice contribution is discussed. In quantum statistics, accu-mulation and transmission of energy in solid materials partially occur by coupledvibrations of the atoms in the crystal lattice. The energy of lattice vibrationsis quantized, and the energy quantum is called a phonon. Elastic- and thermalvibrations in crystals are due to excited phonons. Thus, the phonon is the particlein the �eld of mechanical energy of the crystal. The thermal vibrations are char-acterized in terms of normal modes of oscillation where each mode at frequency �has an energy:

� = (n+1

2)h� (17.8)

when the mode is excited to a quantum number n if the mode is occupied by nphonons. The total energy is a sum of the energies of all phonon modes. Dependingon the density of states (the number of phonon modes per unit frequency range)di�erent expression for the total mechanical energy can be found (Kittel 1986, pp.102).

In Einstein's approach, the atoms in the solid can be approximated by an assemblyof linear oscillators all vibrating with the same frequency �; each phonon mode(energy mode) has the same frequency. Also, the atoms are free to move in threedimensions. Based on these assumptions, the constant volume speci�c heat wasfound to be:

cv;lat = 3R

��E

T

�2e�ET

(e�ET � 1)2

(17.9)

where �E = h�k

is the Einstein characteristic temperature and � is the vibrationenergy of the atoms.

This model was improved by Debye who assumed that the atoms of the solidconstitute a system of coupled oscillators with a continuous spectrum of natural

258 Thermal Properties of Mesophase, Semi-Coke and Coke

frequencies. This assumption leads to:

cv;lat = 9RT

�T

�D

�3 Z xm

0

x4ex

(ex � 1)2dx (17.10)

x =h�

kT(17.11)

xm =h�m

kT(17.12)

�D =h�m

T(17.13)

�m is the maximum frequency of the spectrum and �D is the Debye temperature.A simple approximation exists only for low temperatures; the Debye T 3-law:

cv;lat =12�4

5

�T

�D

�3

(17.14)

This is a signi�cant improvement for low temperatures when compared to Ein-stein's theory which states that the speci�c heat capacity decreases exponentiallyas temperature approaches absolute zero. It can be shown that both Einstein's andDebye's theories predict a value of cv � 3R for high temperatures; the empiricalDulong-Petit value.

In the classical assumption (Sears & Salinger 1980, pp. 271) the free electrons insolids (metals) contribute to the molal speci�c heat capacity with 3R

2: Energy is

needed to excite the electrons as well as the atoms in the crystal lattice. This value,however, does not �t with experimental results. A satisfactory explanation of theelectronic contribution to the internal energy was achieved by the introductionof quantum methods (Kittel 1986, pp. 135-139), (Sears & Salinger 1980, pp.407-413). For graphite, Komatsu & Nagamiya (1951) showed that the electronicspeci�c heat capacity is given by1:

cv;el =1

Mc2:8 � 10�7T (3:29 + 0:49 � 10�2 T + 0:14 � 10�5 T 2 + � � � ) (17.15)

where Mc is the molar mass of carbon. At T = 1598K, cel � 0:0055 J/(kgK)which certainly is negligible.

In summary, only the lattice speci�c heat capacity signi�cantly contributes to thetotal speci�c heat capacity at constant volume:

cv � cv;lat (17.16)

In the following, a discussion of the lattice contribution to speci�c heat in carbonsis given. According to Komatsu & Nagamiya (1951) and Agroskin (1980), speci�cheat capacity of amorphous carbons di�ers from the heat capacity of ideal graphitedue to structural di�erences. It has been shown that graphite at low temperatures

1Komatsu & Nagamiya (1951) supply an expression for heat capacity in unit cal/(Kmol).

17.1 Speci�c Heat Capacity 259

changes nearly proportional to T 2 than according to Debye's T 3-law. Most analyt-ical studies of heat capacity of carbonaceous materials have been done on graphite(Magnus 1923), (Komatsu & Nagamiya 1951), (Komatsu 1955).

According to Komatsu & Nagamiya (1951), the lattice contribution to the speci�cheat consists of two parts:

� Two dimensional longitudinal and transverse in-plane vibrations giving re-spectively the contributions cv;a;l and cv;a;t to the speci�c heat.

� One dimensional bending vibrations normal to the layer planes with contri-bution cv;c to the speci�c heat.

This assumption was based on the fact that the in-plane atoms lie much more close(distance between atoms is � 1:42 �A) than atoms in di�erent neighbouring layers;the interlayer spacing in graphite is d002 = 3:354 �A. Now assume that the in-planevibrations have the same characteristic temperature2. Thus, cv;a;l = cv;a;t = cv;aand �nally:

cv;lat = 2cv;a + cv;c (17.17)

for the total lattice heat capacity. Thus, two characteristic temperatures are nec-essary to calculate the speci�c heat capacity.

17.1.3 A Semi-Empirical Approach to Modelling of Speci�c

Heat Capacity

In general, the literature reports empirical studies on heat capacity. Constantpressure speci�c heat capacity for petroleum cokes has been thoroughly discussedin Gundersen (1996e). The following qualitative observations were observed:

� For raw cokes, the true heat capacity increases monotonously with temper-ature up to temperatures between 500 and 600�C. For higher temperatures,a decrease in heat capacity was observed.

� The temperature dependent heat capacity decreases with calcination tem-perature but seems to be almost una�ected for calcination temperaturesbetween 1200 and 1400�C.

These observations were con�rmed by Agroskin (1980) in a study of cokes madefrom coal related precursors. On the other hand, it was shown that the e�ect of avarying calcination temperature is more dominant in such coals. It was also shownthat heat capacity depends on coal rank: An increased coal rank is accompaniedwith a decrease in heat capacity. Both calcination temperature and soaking timereduces heat capacity. Also an increased ash content reduces heat capacity. The

2Another approach is to use an average characteristic temperature representing both modessince they actually have individual characteristic temperatures (Komatsu & Nagamiya 1951).

260 Thermal Properties of Mesophase, Semi-Coke and Coke

ash e�ect, however, is very low and can be neglected. Finally, it was stressed thatspeci�c heat capacity of coke as function of temperature cannot be modelled bya single temperature function; this would lead to signi�cant errors. In summary,the following factors are important for coke heat capacity:

1. Feedstock

2. Coke calcining temperature and soaking time

3. Content of volatile matter

Factors 1 and 3 has to do with the elemental composition of the coke as well ascoke structure whereas factor 2 mainly have impact on the coke structure.

Based on the approach in Komatsu & Nagamiya (1951) and Merrick (1983b), amodel which takes into account both compositional and structural dependencies ofheat capacity cab be formulated. Based on Einstein's theory for speci�c heat of asolid at constant volume, a model for speci�c heat capacity of coal was suggested.In the �nal model, two Einstein characteristic temperatures �E;a and �E;c wereused. �E;a was applied as an average value for the in-plane vibrations and �E;c ap-plied for vibrations normal to the layer planes. Since the covalent in-layer bindingforces are stronger than the van der Waal's interlayer binding forces, �E;a > �E;c.Merrick successfully selected �E;a = 1800K and �E;c = 380K. As earlier shown inEquation (17.17), the �nal model was:

cv;lat = 2cv;a + cv;c (17.18)

where

cv;a =R�M

��E;a

T

�2e�E;a

T�e�E;a

T � 1�2 (17.19)

cv;c =R�M

��E;c

T

�2e�E;cT�

e�E;c

T � 1�2 (17.20)

R is the ideal gas constant. �M is the average atomic mass given by:

�M =1Pn

i=1yiMi

(17.21)

where yi and Mi are mass fractions and molar masses of elements carbon, hydro-gen, oxygen, nitrogen and sulphur (i = C;H;N;O; S).

Merrick (1983b) showed that since �M increases during coking, the simultaneous in-teraction of the temperature dependent part and the �M dependent part of the heatcapacity functions leads to a maximum in heat capacity as shown in Figure 17.1.An increase in hydrogen content reduces the mean atomic weight and this leads toan increase in heat capacity: An increase in cv is observed as the volatile matter

17.1 Speci�c Heat Capacity 261

0 200 400 600 800 1000 1200 14007

8

9

10

11

12

13Mean average weight as function of calcination temperature

Temperature [oC]

M [g

/mol

e]

0 200 400 600 800 1000 1200 14000

500

1000

1500

2000

2500Heat capacity as function of calcination temperature

Temperature [oC]

cv [J

/kg

K

Figure 17.1: Changes in speci�c heat capacity cv introduced via compositionalchanges of the coke material during coke calcination. The initial hydrogen con-tent is fH = 6% and hydrogen decomposition is assumed to take place linearlydependent of temperature between 500 and 1200�C. The Einstein characteristictemperature is �E = 1800K.

(i.e. hydrogen) content increases. In this way, the e�ect of compositional changesare accounted for in the heat capacity model.

At high temperatures, cv approaches the Dulong-Petit limit of 3R�Ma. The room

temperature heat capacity must be tuned by selecting a proper �E .

In the following, structural changes and their impact on heat capacity are dis-cussed. It has been shown that crosslinking between carbon atoms of layer planesin amorphous carbons contributes to increasing the strength of the interlayer bind-ing forces. This is characterized by a much higher Debye (Einstein) temperaturethan for the interlayer interactions found in graphite. Values in the order of�D;c = 600K can be found in the literature (Agroskin 1980).

For the in-plane vibration, the opposite seems to be true: Structural defects andsmall layer planes contribute to lower in-plane bond strengths than the one foundwithin the layers of graphite. Thus, the in-plane vibration modes are character-ized by a lower Debye/Einstein characteristic temperature than the characteristictemperature found in graphite. The characteristic temperature can be used forevaluation of structural strength of the coke. During heat treatment, structuralchanges occur in the crystallites of the amorphous carbon:

262 Thermal Properties of Mesophase, Semi-Coke and Coke

1. Crystallite growth leads to a decrease in the amounts of crosslinking bonds.The carbon strength is reduced and the Debye temperature �D;c character-izing the vibrational mode normal to layer planes decreases:

� �C;c>! �C;c;gr

where �C;c;gr is the interlayer characteristic temperature in graphite.

2. The characteristic temperature for the in-plane vibrational modes increasesdue to modi�cation of layer defects as well as increased crystallite size (La)which lead to increased strength of the in-plane bonds:

� �C;a<! �C;a;gr

�C;a;gr is the in-plane characteristic temperature in graphite.

Assume that �C;a and �C;c depend on the structural changes in the carbon material,and set:

�C;a = �C;a(La) (17.22)

�C;c = �C;c(La) (17.23)

since La is a direct measure of the structural modi�cations occurring in the carbonduring heat treatment. One could argue that changes in �C;c rather should becorrelated with the interlayer spacing d002. However, assume that the reductionin crosslinking-bonds can be correlated with the perfection of the graphitic layersas represented by an increase in La. Linear correlations for the characteristictemperatures are assumed:

�C;a = �C;a;� + va(La � La;�) (17.24)

�C;c = �C;c;� + vc(La � La;�) (17.25)

where

va =�C;a;f � �C;a;�

La;f � La;�

vc =�C;c;f � �C;c;�

La;f � La;�

where �C;i 2 [�C;i;�; �C;i;f ] and La 2 [La;�; La;f ]. The structure of this modelmight appear speculative; yet no comparison of predicted and measured data hasbeen performed. Suggested values for the parameters are given in Table 17.2.

The model is based Einstein's approach for modelling heat capacity. Thus, theEinstein characteristic temperature must be used. In summary, the model con-sists of Equations (17.5), (17.16), (17.18), (17.19), (17.20), (17.24) and (17.25).Einstein characteristic temperatures and crystallite sizes are given in Table 17.2.

17.2 Thermal Conductivity 263

�C;a;� �C;a;f �C;c;� �C;c;f La;� La;fK [K] [K] [K] [�A] [�A]

950 1800 600 350 12 35

Table 17.2: Data suggested for the model of speci�c heat capacity.

17.2 Thermal Conductivity

Modelling of thermal conductivity of carbons from �rst principles is described inGundersen (1996c). It was shown that �rst principles modelling of the thermalconductivity leads to a very complicated model and the analysis will not be re-peated here. Instead, the development of a semi-empirical model based on theWiedeman-Franz relationship is described.

17.2.1 The Wiedemann-Franz Law

In the model presented in Gundersen (1996c), it was assumed that the electroniccontribution to the thermal conductivity could be neglected.

For good conductors, however, thermal vibrations contribute to less than one per-cent of the total heat ow (Morse 1969, pp. 357). Metals are typically goodconductors, and the Wiedemann-Franz law states that for a metal at not too lowtemperatures, the ratio of thermal- to electrical conductivity is directly propor-tional to temperature (Kittel 1986):

k

�=�2

3

�kB

e

�T (17.26)

kB and e are Boltzmann's constant and the electronic charge respectively. � isthe electrical conductivity. Equation (17.26) is derived from the assumption thatthe conduction electrons behave like a free electron Fermi gas, and the classicalformula k = 1

3cv� was used to derive the model for the thermal conductivity. k

� isoften denoted the Wiedemann-Franz ratio. Ideally, the so-called Lorenz-number:

L =k

�T(17.27)

is a constant. Experiments show that L is a function of both the metal and thetemperature but the calculated values are in good agreement with theory. Forgraphite, the temperature dependence of L is signi�cant (Bowman, Krumhansl &Meers 1958).

According to Powell (1958), the electronic conductivity is slightly more importantin carbons than in graphites and the relative contribution of the electronic partof thermal conductivity increases with temperature. Still, however, the main con-tribution comes from the lattice component. An empirical model for the Lorenz-function was postulated:

L = aT�b (17.28)

264 Thermal Properties of Mesophase, Semi-Coke and Coke

Inspired by this model for the Lorenz-function, correlations of the following typehave been used for graphite:

k = a(T�b)� + c (17.29)

where � = 1�and � is the electrical resistivity. In both Equation (17.28) and

Equation (17.29), di�erent parameters exist for each individual carbon material.At best, a correlation to crystallite size can be found for each parameter.

If a model exists for the electrical resistivity, Equation (17.28) or Equation (17.29)can be used as basis for deriving a model for the thermal conductivity.

17.2.2 The Model for Thermal Conductivity of Carbons

The work by Mason (1958) on electrical resistivity, is discussed in Gundersen(1996c). Based on the model suggested for �el, Equation (17.28) was used to esti-mate parameters in a model for prediction of thermal conductivity data generatedby Log's model (Log & �ye 1989):

L = aT�b (17.30)

�el = cT +d

LaT(17.31)

Then, the following model is obtained for k:

k =aT 1�b

cT + dLaT

(17.32)

Since focus is not on calculation of resistivity, the model can be simpli�ed bydivision by the parameter a in numerator and denominator. Also set 1 � b = ~bwhich gives:

k =T~b

~cT +~d

LaT

(17.33)

A good parametric �t was obtained using a quadratic criterion for each individualdata-vector. The model parameters turn out to be functions of La as shown inFigure 17.2.

On the other hand, since La explicitly is a parameter in the model, it shouldbe possible to obtain a good �t to the whole set of data-vectors for constantvalues of b, ~c and ~d. In this case, the parameters was found as tabulated inTable 17.3. The error was of one order of magnitude greater than the error obtainedin the individual data-vector �t. However, since eight data-vectors were used, theindividual error is approximately the same. Thus, this three parameter modelis the most simple and it also seems as if it is capable of giving a satisfactoryprediction of thermal conductivity in dependence of structural development of thecarbon material (i.e. changes in La) and varying temperature.

17.2 Thermal Conductivity 265

b 1.42745

~c 2:08212� 10�6

~d 1:87708� 102

Table 17.3: Parameters used in the thermal conductivity model based on an em-pirical Lorenz-function.

10 20 30 4010

−12

10−10

10−8

10−6

10−4

10−2

100

Parameter b

La [Aa]

b

10 20 30 400

0.05

0.1

0.15

0.2

0.25Parameter c

La [Aa]

~c

10 20 30 400

1

2

3

4

5

6x 10

5 Parameter d

La [Aa]

~d

Figure 17.2: Model parameters as function of crystallite size La. Due to the regularshape of curves for parameters ~c and ~d, it is possible to express these parametersas function of La. Apparently, a step change occurs in parameter b as functionof La for La in the order of 25 �A. This is merely a numerical then a physicalphenomenon since b tends to zero for low values of La. Thus, it is possible also toparameterize b as a function of La. Using linear functions for the parameters, amodel with six parameters is obtained.

17.2.3 A Comment on the Presented Model

So far, thermal conduction has been attributed to either thermal vibrations orconduction electrons or combinations of these heat conduction modes. These arethe most important mechanisms in nearly all substances at nearly all temperatures,but other mechanisms do exist (Parrott & Stuckes 1975, pp. 86). Thus, in general:

k = kel + kph (17.34)

where kel and kph are the electronic- and phonon contributions to the thermalconductivity. In this treatment, however, these contributions have been discussedseparately and the conclusion is that kph is dominant in carbonaceous materials.

266 Thermal Properties of Mesophase, Semi-Coke and Coke

17.3 Thermal Properties of the Bulk Pitch Phase

During pitch pyrolysis and carbonization, the pitch is transformed from being abrittle solid to a solid carbon with this transformation going through a liquidnearly isotropic state and the mesophase transition. It is assumed that the bulkproperties of the pitch phase of the anode can be calculated as weighted averagevalues of the individual pitch phases present.

17.3.1 Speci�c Heat Capacity

The following weighed average value is used for calculation of speci�c heat capacityof the pitch:

cp;p = x1cp;i + x2cp;c (17.35)

x1 =x + x� + x�1

xT(17.36)

x2 =x�2xT

(17.37)

xT = x + x� + x�1 + x�2 (17.38)

where cp;i is speci�c heat capacity of the initial pitch material consisting of and �-resins and primary QI (primary �-resins). cp;c is thermal conductivity of secondaryQI (mesophase) and pitch coke formed during pyrolysis and carbonization. xjwith j = ; �; �1; �2 are mass fractions of the di�erent pitch fractions. At �nalconversion of the pitch, the bulk heat capacity equals:

cp;p = cp;c (17.39)

This is typically the case at temperatures above 550�C.

17.3.2 Thermal Conductivity

A geometric mean value is used for calculation of the e�ective thermal conductivityof pitch. Thus:

kp = k�1p;ik�2p;c (17.40)

�1 =� + �� + ��1

�T(17.41)

�2 =��2�T

(17.42)

�T = � + �� + ��1 + ��2 (17.43)

where kp;i is thermal conductivity of the initial pitch material consisting of and�-resins and primary QI (�p-resins). kp;c is thermal conductivity of secondary QI(mesophase) and later pitch coke formed during pyrolysis and carbonization. �j

17.3 Thermal Properties of the Bulk Pitch Phase 267

with j = ; �; �1; �2 are volume fractions of the di�erent pitch fractions. At �nalconversion of the pitch, the bulk thermal conductivity equals:

kp = kp;c (17.44)

This is typically the case at temperatures above 550�C.

268 Thermal Properties of Mesophase, Semi-Coke and Coke

Chapter 18

A Model for Soft Carbon

Pyrolysis and Property

Development

In industrial production of soft carbons, it is often desirable to have knowledgeof the dynamic development of carbonization gases to secure optimal utilizationof the internally released combustibles (condensables, methane and hydrogen).Furthermore, one wants to produce a carbon material of a certain quality. Cokeyield and quantities like texture, porosity and microstructure are all importantcharacteristics with impact on the physical properties (i.e. quality) of the �nalcarbon material. In this chapter, a model for prediction of both gas generationand coke properties is presented.

18.1 Introduction

Ideally, a model for pyrolysis of a soft carbon should be able to predict the follow-ing:

� The release of condensable and non-condensables

� Coke yield

� Formation of mesophase and development of the corresponding coke texture

� Development of porosity

� Development of microstructure (i.e. structure of microcrystallites)

� The model should serve as basis for calculation of critical parameters usedfor speci�cation of carbon quality (i.e. physical properties)

270 A Model for Soft Carbon Pyrolysis and Property Development

It has not been possible to establish such a model entirely based on �rst princi-ples. On the other hand, the combination of �rst principles and semi-empiricalapproaches has lead to models for some of the features described above. Themodel for pyrolysis of soft carbons is based on the following contributions givenin Gundersen (1995b):

� The model of pyrolysis presented in part III is used for prediction of meso-phase formation (i.e. phase transition), coke yield, released gases (conden-sables, methane and hydrogen) and the hydrogen content in the coke mate-rial.

� The models for real- and apparent density is based on the work presented inChapter 15 in this work.

� The porosity model is based on the apparent-, real- and crystalline densityconcepts established in part II.

� Development of crystallite parameters within the granular structure (i.e. d,La and Lc) is modelled as thermally activated processes as recommended inpart II. d and Lc are modelled as individual thermally activated processes.La is assumed to grow on the expense of the disorganized carbon phase anddue to a certain degree of coalescence of crystallites which occurs along thea-axis; see part II.

� The rate of formation of crystallites is denoted rcr. Here, rcr is obtained asthe di�erence between the rate rdm of consumption of the disorganized car-bon phase and the total rate rv of loss of volatile gases (i.e. non-condensables)under the assumption that a certain fraction of hydrogen also is included inthe disorganized phase:

rdm = rv;nc + rcr (18.1)

If prediction of the hydrogen content in the carbon residue is not needed in themodel, the low temperature pyrolysis model may be used (i.e. model A in Gun-dersen (1995b)). Then, the release of methane and hydrogen may be modelledby single reaction models based on nominal yields of gases and the use of conver-sion dependent activation energies. Here, however, the high temperature modelpresented in part II of this work is used.

Probably, it would be possible to incorporate into the pyrolysis model the evolutionof coke texture as represented by the growth of the mesophase spheres. The averagesize of spheres could serve as a measure of coke texture. Then, the kinetics ofmesophase formation as a �rst order rate process dependent on the mass fractionof �-resins must be reformulated. Alternatively, a population balance approachwith certain nucleation-, growth and coalescence laws for the mesophase spheresis needed. In this work, however, no e�ort is done to extend the model to alsoinclude prediction of coke texture.

18.2 Discussing the Derivation of the Model 271

18.2 Discussing the Derivation of the Model

The �nal structure of the model presented in the next section was achieved aftera thorough simulation study of the development of Young's modulus during hightemperature pyrolysis and the behaviour of the physical properties across thetemperature interval corresponding to the mesophase transition and formation ofsemicoke.

18.2.1 The Evolution of Physical Properties Across the Meso-

phase Transition

The physical properties of the isotropic phase has impact on the physical prop-erties of the anisotropic phase which is formed in the mesophase transition. Thefollowing quantities have impact on development of properties across the meso-phase transition:

� The crystallite parameters: d002; Lc; La;c; nL

� Mass fraction of disorganized carbon: xdm

� Densities: �r;c; �a;c; �dm;�

An exact mathematical description of the dependence of the properties of theanisotropic phase on the properties of the isotropic phase has not been derived.However, an approximate solution to the problem was found in the following way:

� For some of the physical properties (d002 and Lc), the problem was avoidedby representing the parameters with thermally activated processes that areassumed to be valid across the mesophase transition.

� For the real- and apparent densities of the anisotropic phase (�r;c and �a;crespectively), initial values of the densities were assumed according to a prioriknowledge available in the literature.

� To obtain a formally correct model, the disorganized carbon is exclusivelyassociated with the anisotropic carbon phase. An approximate initial valuenL(0) of the total number of layer planes can then be achieved by the formula:

nL(0) =(1� xdm(0))�a;c(0)Va(0)

�c(0)��La;c(0)2

2

�2Lc(0)

Lc(0)

d002(0)+1

(18.2)

� Note that xdm is the mass fraction of disorganized carbon within the bulkapparent volume and not within the apparent volume of the anisotropiccarbon phase.

272 A Model for Soft Carbon Pyrolysis and Property Development

18.2.2 The Development of Porosity Across the Mesophase

Transition

The following expressions are used for the total, open- and closed porosities:

�� = 1� �a

�r(18.3)

�c =�a

�r� �a

~�c(18.4)

�0c = 1� �r

~�c(18.5)

�T = �� + �c = 1� �a

~�c(18.6)

~�c = xi�r + xc��c (18.7)

��, �c and �T denote open, closed and total porosity respectively. �c is mea-sured on the granular structure level. Below carbonization temperatures (i.e. noswelling), �� � 0 since �a � �r. Furthermore, �c � 0 and �0c � 0 since xi � 1(and xc � 0) which gives ~�c = �r. Thus, both open and closed porosity have azero starting value.

18.3 Features and Simpli�cations of the Model

A summary of features and simpli�cations of the model is described in the follow-ing:

� The pyrolysis model is based on a traditional pitch fractionation schemewhere the pitch- and coke fractions are grouped into isotropic fractions( 1; 2; �; �p) and anisotropic fractions (�s;CH4

; �s;H2; C1; C2).

� To give a realistic prediction of weight loss as function of heating rate, the -resins were grouped into subfractions 1 and 2-resins.

� It was assumed that only a small amount of non-condensables evolves fromthe reaction steps 1 ! 2, 1 ! � and 2 ! � in the pyrolysis reactionscheme. The formation of hydrogen and methane in these reaction steps wastherefore neglected.

� To predict yields of hydrogen and methane, the �s-resins were divided intosubfractions �s;H2

and �s;CH4to give basis for hydrogen and methane for-

mation respectively.

� A conversion dependent activation energy was used for the hydrogen- andmethane generating reaction steps.

� QuantitiesXCH4andXH2

represent the conversion of hydrogen and methane

respectively. In the calculation ofdXCH4

dt anddXH2

dt , it is assumed that xchas a constant value (xc � 1).

18.3 Features and Simpli�cations of the Model 273

� As basis for calculation of the hydrogen balance equation, we assume that thepyrolysing carbon material consists of mainly carbon with a small fraction ofhydrogen. This view is also used for calculation of the e�ective molar mass�M of the carbon material (xC + xH = 1).

� Furthermore, for calculation of the hydrogen balance, a constant value (care-fully tuned) for the molar mass of the volatilizing -resins is used.

� Real- and apparent densities of the pyrolysing material are calculated as acertain average values (mass fractions of the pitch- and coke components areused as weight factors).

� The model for real density �r;c of the anisotropic phase is related to thegranular structure concept reviewed in part II. The increase in �r;c is re-lated to the decrease in mass fraction xdm of disorganized carbon and thedevolatilization of non-condensables via �dm. The dynamics of d002 also haveimpact on �r;c via the dependence of �r;c on �c. In the temperature inter-val between 400 and 800�C, however, there is a decrease in �c due to anincreased value of d002. This may theoretically lead to a decrease in �r;c via

the rate lawd�r;cdt

= �rkr;isod��cdt. Therefore, the rate law was modi�ed to:

d�r;c

dt= �rkr;iso

d(��c)0

dt

Here, the parameter c in d(��c)0

dt changes from zero to one as temperatureapproach 800�C. Alternatively, the real density could be modelled as athermally activated process realized either as a single reaction model (withconversion dependent activation energy) or a multiple reaction model. InChapter 15, a n'th order single reaction model for the real density was de-rived.

� The function f(�0c) in the model for �r was so selected since it is assumed thatthe ability of the material to shrink decreases as closed porosity decreases.

� For the apparent density, the following applies:

{ The swelling capability (between 300 and 500�C) of �-resins is includedin the model.

{ Three di�erent operating regimes is taken into account in the rate lawfor the apparent density of the anisotropic pitch phase (swelling, col-lapse and shrinkage regimes).

{ The apparent density is calculated as a weighed average value of theapparent densities of the , �p and �-fractions and the anisotropic pitchphase.

{ The equations ford�a;�dt

andd�a;cdt

are implicitly given in the formulationof the model equations presented in this chapter.

274 A Model for Soft Carbon Pyrolysis and Property Development

� Note that the isotropic shrinkage parameters ka;iso and kr;iso for apparent-and real density of the anisotropic pitch phase are based on the quantities

��;c = 1� �a;c

�r;c(18.8)

�0c;c = 1� �r;c

��c(18.9)

i.e. the open and closed porosity within the anisotropic phase and the gran-ular structure part of the anisotropic phase.

� For the crystallite parameters, the following approach is used:

{ The model for La is based on the assignment of constant values forLa in the isotropic pitch phase. Development of La for the anisotropicpitch phase is related to the consumption of disorganized carbon andthe assumption that a certain decrease in the total number of layerplanes (i.e. coalescence along the a-axis) is allowed to occur below1400�C.

{ d002 and Lc are modelled as individual thermally activated processes bytwo parallel single reaction models with conversion dependent activationenergy. This combination of parallel models is able to synthesise thedecrease in Lc and increase in d002 which occur between 400 and 800

�C.

� For calculation of viscosity, it is assumed that mass fractions of the isotropicand anisotropic pitch fractions can be used instead of mole fractions.

� In the model for ��c, it is assumed that the carbon material consists of purecarbon present in two phases: The disorganized- and crystalline carbonphases (xcr + xdm = 1).

� The concept of porosity used in the model is related to the concept ofapparent-, real- and crystalline density.

� Simulations have shown that rdm � rv;nc such that for simplicity, one canset rcr � rdm and neglect the mass loss of non-condensables in the submodelfor xdm.

� Both for the apparent- and real densities and the layer plane diameter La,the same general simpli�cation is used:

{ Property models (�a;j , �r;j and La;j) for the pitch fractions are assumedto be independent thermally activated processes.

The meaning of the last simpli�cation is more clearly explained as follows: Thereal density of the mesophase (here: secondary �-resins) depends on the mesophaseforming compounds (here �-resins). So far, it has not been possible to incorporatethis dependence into the model. The initial values of the density of the anisotropicphase is assumed to take into account the impact of the isotropic phase. The sameargument is valid for the other properties of the anisotropic phase (i.e. �a;c, La;cetc.)

18.4 Model Structure 275

The most severe defect of the above model is the limited ability to predict temper-ature dependent limiting values of the mass fraction xdm of disorganized carbon(and therefore La, d002 and Lc). This was achieved by the use of conversion de-pendent activation energies in the models for xdm, d002 and Lc. To improve theprediction capability of these sub-models, a multiple reaction approach with dis-tributed activation energy can be used. Still, however, the overall structure of themodel remains the same.

18.4 Model Structure

In the following, the coupling between mechanisms in the model is summarized:

� The pyrolysis submodel is used for calculation of pitch mass fractions. Themass fractions give basis for calculation of the mass fractions of isotropic- andanisotropic pitch phases which is used in the calculation of average physicalproperties of the pyrolysing material. The pyrolysis submodel is also used incalculation of ultimate yields of gases, coke yield and the overall hydrogenbalance.

� A separate model is used for calculation of the mass fraction xdm of disorga-nized carbon present in the pyrolysing material. From the equations, it canbe seen that there is a one way (weak) coupling from the pyrolysis model tothe submodel for xdm via the term rv;nc.

� Crystallite parameter submodels: The development of La is linked to theconsumption of disorganized carbon. A certain degree of coalescence alongthe a-axis is allowed to occur. The apparent reduction in crystallite orderrepresented by the reduction and increase in Lc and d002 respectively fortemperatures between 400 and 800�C was realized by the application of twoparallel single reaction models with conversion dependent activation energies.

� The density models are linked to the consumption of disorganized carbon.As the density of the solid carbon material increases due to the growth ofcrystallite size (i.e. La), there is an increase also in real density �r due toanisotropic shrinkage (with shrinkage factor greater than one) of the granularstructure. Wee see that the structure of the model also takes into accountswelling of �-resins occurring at temperatures in the order of 300�C.

The interactions between the submodels are summarized in Figure 18.1.

18.5 The Model

The following submodels are included:

� The model of pitch pyrolysis from part III.

276 A Model for Soft Carbon Pyrolysis and Property Development

Disorganized carbon

TemperatureProgram

(Energy balance)

Crystallite Parameters

Coke Yield(Total mass balance)

of gasesUltimate yields

(Layer plane)(mass balance)

Pyrolysis

(Pitch component)(Mass balances)

Real

Isotropicfraction

densitiesApparentdensities

Anisotropic Anisotropic

Isotropicfraction

fraction

Densities

Anisotropicfraction

Viscosity

fractionAnisotropicIsotropic

fraction

fraction Bulkmaterial

Lc Lad002

�p�

( 1; 2; �; �p;�s;CH4

; �s;H2; C1; C2)

Figure 18.1: The coupling (represented by arrows) between submodels in the modelfor simulation of pyrolysis of soft carbons. The temperature is the driving force inthe models. As a general principle, the fundamental carbon properties (densities,crystallite parameter La and viscosity) are calculated as weighted average values ofthe isotropic and anisotropic pitch fractions. d002 and Lc are modelled as thermallyactivated processes to represent the values for the bulk pyrolysing material.

� The models for real density, apparent density and viscosity as presented inpart IV.

� The models for real density of the granular structure and growth of La asrelated to consumption of disorganized carbon as presented in part II.

18.5 The Model 277

In principle, the model has all the features (except for the ability to predict devel-opment of carbon texture) needed for use in most industrial processes dealing withproduction of carbon materials. A comprehensive overview of the model equationsis given in the following.

Pyrolysis model:

dx 1dt

= x 1(kvx 1 + (1� p)k�s;CH4x�s;CH4

+ (1� q)k�s;H2x�s;H2

) (18.10)

�kvx 1 � k 1;ax 1 � k 1;bx 1

dx 2dt

= x 2(kvx 1 + (1� p)k�s;CH4x�s;CH4

+ (1� q)k�s;H2x�s;H2

) (18.11)

+k 1;ax 1 � k 2x 2

dx�

dt= x�(kvx 1 + (1� p)k�s;CH4

x�s;CH4+ (1� q)k�s;H2

x�s;H2) (18.12)

+k 1;bx 1 + k 2x 2 � k�x�

dx�p

dt= x�p(kvx 1 + (1� p)k�s;CH4

x�s;CH4+ (1� q)k�s;H2

x�s;H2) (18.13)

�k�px�pdx�s;CH4

dt= x�s;CH4

(kvx 1 + (1� p)k�s;CH4x�s;CH4

+ (1� q)k�s;H2x�s;H2

)

+sk�x� � k�s;CH4x�s;CH4

(18.14)

dx�s;H2

dt= x�s;H2

(kvx 1 + (1� p)k�s;CH4x�s;CH4

+ (1� q)k�s;H2x�s;H2

)

+(1� s)k�x� + k�px�p � k�s;H2x�s;H2

(18.15)

dxC1

dt= xC1

(kvx 1 + (1� p)k�s;CH4x�s;CH4

+ (1� q)k�s;H2x�s;H2

) (18.16)

+pk�s;CH4x�s;CH4

dxC2

dt= xC2

(kvx 1 + (1� p)k�s;CH4x�s;CH4

+ (1� q)k�s;H2x�s;H2

) (18.17)

+qk�s;H2x�s;H2

Coke yield and yield of gases:

dcy

dt= �(kvx 1 + (1� p)k�s;CH4

x�s;CH4(18.18)

+(1� q)k�s;H2x�s;H2

) cy

dyc

dt= kvx 1cy (18.19)

dyCH4

dt= (1� p)k�s;CH4

x�s;CH4cy (18.20)

dyH2

dt= (1� q)k�s;H2

x�s;H2cy (18.21)

278 A Model for Soft Carbon Pyrolysis and Property Development

The hydrogen balance:

dxH

dt= xH((1� p)k�s;CH4

x�s;H2+ (1� q)k�s;H2

x�s;H2) (18.22)

+xHkv;cx 1

�nMH1�M

kv;cx 1 � 4MH1

MCH4

(1� p)k�s;CH4x�s;H2

�2MH1

MH2

(1� q)k�s;H2x�s;H2

xC + xH = 1

Average molar mass of the pyrolysing substance:

�M =1

xCMC

+ xHMH

(18.23)

Consumption of disorganized carbon (in mesophase and solid semicoke/coke):

dxdm

dt= � 1

�a((1� xdm)rv;nc + rcr) (18.24)

xdm + xcr = 1 (18.25)

Some of the rate terms used in the model:

rcr = rdm � rv;nc � rdm (18.26)

rdm = kdmxdm�a (18.27)

rv;nc = rv;CH4+ rv;H2

(18.28)

rv;CH4= (1� q)k�s;CH4

x�s;CH4�a (18.29)

rv;H2= (1� p)k�s;H2

x�s;H2�a (18.30)

rv;c = k 1x� 1 �a (18.31)

rv = rv;c + rv;nc (18.32)

18.5 The Model 279

Crystallite parameters:

La = xiLa;i + xcLa;c (18.33)

xiLa;i = x La; + x�La;� + x�pLa;�p (18.34)

dLa;c

dt=

1

2

Lc + d

LaLc

�rcVa

�4�cd nL

� L2a

Lc

Lc + d

1

nL

dnL

dt(18.35)

� Lcd

Lc + d

L2a

Lc + d

�1

Lc

dLc

dt� 1

d002

d(d002)

dt

��dnL

dt= knL(nL(1)� nL)

knL = knL;�e�

EnLRT

EnL = EnL;min +

�nL � nL;�

nL(1)� nL;�

�(EnL;max �EnL;max)

dLc;1

dt= kLc;1(Lc;1(1)� Lc;1) (18.36)

kLc;1 = kLc;1;�e�ELc;1

RT

ELc;1 = ELc;1;min +(ELc;1;max �ELc;1;min)

Lc;1(1)� Lc;1(Lc;1 � Lc;1(0))

dLc;2

dt= kLc;2(Lc;2(1)� Lc;2) (18.37)

kLc;2 = kLc;2;�e�ELc;2

RT

ELc;2 = ELc;2;min +(ELc;2;max �ELc;2;min)

Lc;2(1)� Lc;2(Lc;2 � Lc;2(0))

Lc = Lc;1 � Lc;2 (18.38)

d(d002;1)

dt= kd002;1(d002;1(1)� d002;1) (18.39)

kd002;1 = kd002;1;�e�Ed002;1

RT

Ed002;1 = Ed002;1;min +(Ed002;1;max �Ed002;1;min)

d002;1(1)� d002;1�d002;1

�d002;1 = (d002;1 � d002;1(0))

d(d002;2)

dt= kd002;2(d002;2(1)� d002;2) (18.40)

kd002;2 = kd002;2;�e�Ed002;2

RT

Ed002;2 = Ed002;2;min +(Ed002;2;max �Ed002;2;min)

d002;2(1)� d002;2�d002;2

�d002;2 = (d002;2 � d002;2(0))

d002 = d002;1 + d002;2 (18.41)

280 A Model for Soft Carbon Pyrolysis and Property Development

Mass fractions of isotropic- and anisotropic pitch:

xi + xc = 1

xi = x + x� + x�p

xc = x�s + xC1+ xC2

x = x 1 + x 2

x�s = x�s;CH4+ x�s;H2

Relationship between real- and apparent volume fractions and mass fractions ofpitch components ((j = 1; 2; �; �p; �s;CH4

; �s;H2; C1; C2)):

�r;j = xj�r

�r;j(18.42)

�a;j = xj�a

�a;j(18.43)

Volume fractions of isotropic- and anisotropic pitch:

�q;i + �q;c = 1 (q = a; r)

�q;i = �q; + �q;� + �q;�p

�q;c = �q;�s + �q;C1+ �q;C2

�q; = �q; 1 + �q; 2

�q;�s = �q;�s;CH4+ �q;�s;H2

Viscosity model:

ln � � xi ln �i + xc ln �c (18.44)

�i = �i;�eEiRT (18.45)

dy�

dt= ky�(ym;� � y�)

ny� (18.46)

ky;� = ky;��e�

Ey�

RT

�c = 10y�

18.5 The Model 281

Real density models:

�r =1

xi�r;i

+ xc�r;c

(18.47)

d�r

dt= �

(�r;idxidt� xi

d�r;idt

)

�2r;i�2r �

(�r;cdxcdt� xc

d�r;cdt

)

�2r;c�2r (18.48)

�r;i = �r;i;� (1� �(T � T�))

d�r;i

dt= ��r;i;��

dT

dt(18.49)

�r;i;� � constant

d�r;c

dt= �rkr;iso

d(��c)0

dt(18.50)

�r = �r;�f(�0

c)

f(�0c) = (�0c)n�r

kr;iso = (1� �0c;c)

�0c;c = 1� �r;c

��c

��c =1

xcr�c

+ xdm�dm

d��cdt

= ��1

�c

dxcr

dt+

1

�dm

dxdm

dt� xdm

�2dm

d�dm

dt

+xcr

�c

1

d002

d(d002)

dt

���2c

d(��c)0

dt= �

�1

�c

dxcr

dt+

1

�dm

dxdm

dt� xdm

�2dm

d�dm

dt

+cxcr

�c

1

d002

d(d002)

dt

���2c

c =

�(d002;2 � d002;2(0))

(d002;2(1)� d002;2(0))

�nd�dm = (1� ~X)�dm;� + ~X�dm;f

~X =~p

2XCH4

+ (1� ~p

2)XH2

XCH4� xc

xC1

sp

XH2� xc

xC2

(1� s)q

d�dm

dt= (�dm;� � �dm;f )

d ~X

dt

d ~X

dt=

~p

2

dXCH4

dt+ (1� ~p

2)dXH2

dt

282 A Model for Soft Carbon Pyrolysis and Property Development

dXCH4

dt� xc

sp

dxC1

dt

dXH2

dt� xc

(1� s)q

dxC2

dt

�dm;�; �dm;f � constant

�c =dgr

d002�gr

Apparent density models:

�a =1

x �a;

+x��a;�

+x�p�a;�p

+ xc�a;c

(18.51)

d�a

dt= �

(�a; dx dt� x

d�a; dt

)

�2a; �2a �

(�a;�dx�dt� x�

d�a;�dt

)

�2a;��2a (18.52)

�(�a;�p

dx�pdt

� x�pd�a;�pdt

)

�2a;�p�2a

�(�a;c

dxcdt� xc

d�a;cdt

)

�2a;c�2a

�a; = �r;i

�a; 1 ; �a; 2 = �a;

�a;�p = �r;i

�a;� = �r;� � ~kswrsw

�r;� = �r;i

~ksw � constant

rsw � r + r�

r = r 1;b + r 2

r 1;b = k 1;bx 1;b�a

r 2 = k 2x 2�a

r� = k�x��a

d�a;

dt=d�r;i

dt(18.53)

d�a;�p

dt=d�r;i

dt(18.54)

d�a;�

dt=d�r;�

dt� ~ksw

drsw

dt(18.55)

d�a;c

dt� fw;sw

�d�a;c

dt

�sw

+ fw;c

�d�a;c

dt

�c

(18.56)

+fw;shr

�d�a;c

dt

�shr

18.5 The Model 283

fw;sw = �i

fw;c = �c

fw;shr = 1�d�a;c

dt

�sw

= �a;c(k��a;�

�a;c(�a;�

�a;c� 1)� k�p

�a;�p

�a;�s� 1

Va

dVa

dt) (18.57)�

d�a;c

dt

�c

= kc�n�a;ca;c fnf�

n��;c�;c (18.58)

n�a;c ; nf ; n��;c = constant

��;c = 1� �a;c

�r;c�d�a;c

dt

�shr

= �aka;isod�r;c

dt

�a = �a(��;c)

ka;iso = (1� ��;c)

Real- and apparent volumes:

Vr = Va(1� ��) (18.59)

dVr

dt= �Vr

�r(

rv

(1� ��)+d�a

dt) (18.60)

dVa

dt= �Va

�a(rv +

d�a

dt) (18.61)

Porosities in the pyrolysing material:

�T = �� + �c (18.62)

�� = 1� �a

�r(18.63)

�c =�a

�r� �a

~�c(18.64)

�0c = 1� �r

~�c(18.65)

~�c = xi�r + xc��c (18.66)

Temperature program:

dT

dt= f(t) (18.67)

Total porosity �T;a in anode:

@�T;a

@t=

(1� �T;a � )

�r

0@ 3Xj=1

rj +@�r

@t

1A (18.68)

3Xj=1

rj = rv;c + rv;CH4+ rv;H2

284 A Model for Soft Carbon Pyrolysis and Property Development

In the mass fraction formulation of the mass balance equations, only �rst orderreactions are used to describe the polymerization processes in the pitch. Thisgives a simple structure of the mass balance equations; i.e. they are independentof the apparent volume (Gundersen 1995b, pp. 46, 111). Also, by using the massfraction formulation, the dependency of the rate of change of the apparent (real)density that would be introduced if the equations were formulated in the volumefractions, is avoided. Still, however, the rate of change of real- and apparent

density are needed; see the equations for�d�a;cdt

�sw

and@�T;a@t .

18.6 State Space Formulation of the Model

A state space formulation of the model may be formulated as follows:

_x = f(x; u) (18.69)

y = h(x) (18.70)

z = g(x) (18.71)

where x, y and z are state-, measurement and property vectors respectively. Nowde�ne:

x = [x1; x2; x3]T

x1 = [x 1 ; x 2 ; x� ; x�p ; x�s;CH4; x�s;H2

; xC1; xC2

]T

x2 = [cy; yc; yCH4; yH2

; xH ; xdm; La;c; d002;1; d002;2; Lc;1; Lc;2; y�]T

x3 = [�r; �

a]T

�r= [�r;c]

T

�a= [�a;� ; �a;c]

T

u = T

y = [y1; y

2]T

y1= [x ; x� ; x�p ; x�s ; cy; �; �a; �r; �T ; ��; �c]

T

y2= [Va; Vr; yc; yCH4

; yH2; d002; La; Lc; xdm]

T

z = [ �M;k; cp; Y; �c; �el; rCO2; rO2

]T

Since apparent volume Va can be calculated from coke yield cy, the initial mass

m(0) and apparent density (Va = cym(0)

�a), Va is no state variable in this case. In

this case, �r;i depends linearly on temperature and is therefore not included in thestate vector. In general, however, �r;i is actually a state variable since it dependson the temperature history during pyrolysis due to the dependence of density onthe polymerization processes in the pitch. A total of twenty three state variablesare thus de�ned.

At this stage only temperature T can be used as control variable in this system.In general however, it is possible also to use ambient pressure and mass transferconditions as control variables.

18.6 State Space Formulation of the Model 285

Other candidates may also be included in the vector y of measurements. Ther-mal, (thermo-) mechanical, electrical and chemical properties are included in theproperty vector z.

286 A Model for Soft Carbon Pyrolysis and Property Development

Chapter 19

Simulation of Soft Carbon

Properties

A simulation study of the model described in Chapter 18 for evolution of den-sity, porosity, crystallite parameters, speci�c heat capacity, thermal conductivity,electrical resistivity, mechanical strength and Young's modulus for a soft carbonduring pyrolysis is presented. Models for electrical resistivity, (thermo)mechanicalproperties and reactivity of soft carbons are given in Gundersen (1996d). Modelsfor the other properties were presented previously in this part of the work.

19.1 The Dependence of Physical Properties on

the Fundamental Carbon Properties

In part II, the concept of fundamental carbon properties was introduced. Theseproperty variables constitute a subvector of the vector of state variables describingthe system.

By the fundamental properties is meant a set of properties to which most physicalproperties are related. Typically, crystallite parameters, densities and elemen-tal composition belong to the fundamental carbon properties. As shown in theprevious chapters, Gundersen (1996d) and Gundersen (1996b), the following rela-tionships exist between physical properties and fundamental properties:

cp = cp(xH ; La) (19.1)

k = k(La; �a; ��c) (19.2)

� = �(xdm; �a; �r; ��c; �c; d002; La; Lc) (19.3)

Y = Y (xdm; �a; �r; ��c; �c; d002; La; Lc) (19.4)

�el = �el(xdm; �a; �r; �c; La) (19.5)

The model from Chapter 18 is well suited for calculation of these properties. Incor-

288 Simulation of Soft Carbon Properties

kv;� [1/hr] k 1;a;� [1/hr] Ev [J/mol] E 1;a [J/mol]

0:2149� 103 0:3959� 101 0:3573� 105 0:1847� 105

Table 19.1: Parameters for decomposition of the fraction in pitch pyrolysis basedon data for evolution of condensables found in Wilkening (1983, Fig. 10). Datafor heating rates of 5.5 and 11.0�C=hr were used in the estimation. The reactionscheme was discussed by Gundersen (1995b) and kinetic data from Ko�st�al et al.(1994) was used for the polymerization reactions.

k ;�[1/hr] k�;�[1/hr] E [J/mol] E� [J/mol]

7:4789� 108 1:5793� 109 1:3333� 105 1:3567� 105

Table 19.2: Parameters for pitch polymerization kinetics taken from Ko�st�al et al.(1994). The kinetics for decomposition of �-resins is also used for transformationof the primary �-resins (i.e. �p-resins).

s p q

0.5000 0.9689 0.9194

Table 19.3: Stoichiometric parameters for the pyrolysis model.

porated into these mathematical models for the physical properties, the in uenceof:

� Porosity as represented by �� = 1� �a�r.

� Volume fraction X of crystalline carbon within the granular structure ascalculated from xdm.

� Crystallite parameters La, Lc and d002. In our model, the kinetics of La iscoupled to the kinetics of xdm and nL.

is represented.

19.2 Summary of Model Parameters

A summary of model parameters is given in Tables 19.1 to 19.8. In addition comeparameters in the models for the physical properties (i.e. thermal conductivity,heat capacity, Young's modulus etc.). Recommended values for such parametershas been given in the previous chapters explicitly which deals with the derivationof the property models.

19.3 A Comment on the Calculation of Physical Properties 289

Component i J ki;� Ei;min Ei;max

[1/hr] [J/mol] [J/mol]

Methane CH4 1.5311 3:1623� 1016 277600 328800

Hydrogen H2 0.8467 5:9946� 107 1:4809� 105 1:8102� 105

Table 19.4: Parameters in �rst order rate laws with conversion dependent activa-tion energy used for the reaction steps for generation of non-condensables.

Parameter Preexp. factor Min. act. energy Max. activ. energy

i ki;� Ei;min Ei;max

[1/hr] [J/mol] [J/mol]

Lc;1 1:0� 108 80000 320000

Lc;2 3:5� 1011 165000 225000

d002;1 105000 80000 230000

d002;2 45000 90000 100000

nL 5:0� 1012 190000 500000

xdm 3:7� 1012 300000 900000

Table 19.5: Kinetics for d002, Lc, nL and xdm.

�r;p;i;� [kg/m3] 1310.00

� [1/K] 0.00044

Table 19.6: Parameters in the model for real density of the isotropic phase.

19.3 A Comment on the Calculation of Physical

Properties

For most of the physical properties, a weighed average value of the isotropic andanisotropic phases is used:

� = xi�i + xc�c (19.6)

where �i and �c are the physical properties respectively of the isotropic andanisotropic phases of the pyrolysing material. xi and xc are the mass fractionsof the isotropic and anisotropic pitch phases. In general, the use of the massfractions xi and xc for calculation of the weighed average properties may seemto be a too simple approach; observe the peculiar shape of Young's Modulus fortemperatures in the order of 400�C (50 hr) in Figure 19.8. This observation willnot be elaborated on here.

290 Simulation of Soft Carbon Properties

Pitch fraction Mi [g/mol] La;i = f(Mi) [�A]

1 304.16 7.2547

2 304.16 7.2547

� 600.00 11.0471

�p 1200.00 16.5018

�s 1200.00 16.5018

Table 19.7: Molar masses of pitch fractions vs. La. For �p- and �s-resins, La;i =0:85 f(Mi) was used which givesM�p =M�s = 14:0265 �A. f(�) corresponds to thehexagonal model for aromatic molecules described in Ollivier & Gerstein (1986).

Initial mass fractions of pitch components x 1(0) 0.65x 2(0) 0.00x�(0) 0.27x�p(0) 0.08

x�s;CH4(0) 0.00

x�s;H2(0) 0.00

xC1(0) 0.00

xC2(0) 0.00

Pitch coke yield cy(0) 1.00

Yield of condensables yc(0) 0.00

Yield of methane yCH4(0) 0.00

Yield of hydrogen yH2(0) 0.00

Initial hydrogen content xH(0) 0.05

Initial mass fr. of disorganized carbon xdm(0) 0.40

Initial La;c for anisotropic carbon La;c(0) [�A] 16.00

Initial interlayer spacing d002;1(0) [�A] 3.60

d002;2(0) [�A] 0.00

Final interlayer spacing d002;1(1) [�A] 3.36

d002;2(1) [�A] 0.08

Initial stacking height Lc;1(0) [�A] 10.00

Lc;2(0) [�A] 00.00

Final stacking height Lc;1(1) [�A] 49.00

Lc;2(1) [�A] 13.00

Apparent density of beta resins �a;�(0) [kg/m3] 1310.00

Real density of uid pitch �r;i(0) [kg/m3] 1310.00

Apparent density of pitch coke �a;c(0) [kg/m3] 1430.00

Real density of pitch coke �r;c(0) [kg/m3] 1430.00

Table 19.8: Initial values for state variables in the simulation model for soft carbonpyrolysis.

19.4 Simulation Results 291

Par.; mod. for �dm r 1.000

Interl. spac.; graphite dgr [�A] 3.354

Density of graphite �gr [kg/m3] 2260.000

Init. dens. disorg. C �dm;� [kg/m3] 1750.000

Fin. dens. disorg. C �dm;f [kg/m3] 1860.000

Init. no. of layer pl. nL;�(1�xdm(0))�a;c(0)Va(0)

�c(0)�

�L2a2

�2Lc(0)

Lc(0)

d002(0)+1

2:3 � 1021

Fin. no. of layer pl. nL(1) q = 0:000005 q � nL;�Crystalline density �c(0) [kg/m

3]

dgrd �gr 2105.60

Dens. of solid carbon ��c(0) [kg/m3] 1

xdm(0)

�dm+

(1�xdm(0))

�c

1947.30

Init. app. vol. Va(0) [m3] m(0)

�a(0)3:8 � 10�6

Init. real vol. Vr(0) [m3]m(0)

�r(0)3:8 � 10�6

Init. rel. app. vol. VaVa(0)

1.00

Init. rel. real vol. VrVr(0)

1.00

Init. temp. T� [�C] 20.00

Final temp. Tf [�C] 1200.00

Heating rate a 1/K 10.0

Heat treatm. time tf [hr] 150.00

Isoth. hold time th [hr] 10.00

Mass of pyrolys. C m(0) [kg] 0.005

Par.; mod. for cp La;c(1) [�A] 32.000

swelling factor ~ksw [s] ksw(1� f ) 3.0

Stoichiometry f 1;b f f 2 f f 0.95f� 0.95

b1�f�1�f

1.00

Collapse coe�. kc 40.00n�a;� 5.00nf 0.60

n��;� 1.00

Anis. shrinkage, �a �a 0.700

Anis. shrinkage, �r �r;� 25.000n�r 0.700nd 1.000

Table 19.9: Other parameters in the model for high temperature pyrolysis.

19.4 Simulation Results

As basis for the simulations, model parameters were obtained by di�erent means:

292 Simulation of Soft Carbon Properties

� Reaction kinetics and other parameters from the literature

� Reaction kinetics and other parameters obtained by estimation (i.e. minimi-sation of an objective function)

� Reaction kinetics obtained by hand tuning of parameters and comparisonwith experimental results from the literature

In general, it was not straight forward to establish a proper set of model pa-rameters which gives realistic prediction of both the individual state variables aswell as their compound interaction via the mathematical models for the physicalproperties of the carbon. In particular, this was di�cult in the submodels forconsumption of disorganized carbon (xdm) and the corresponding growth modelfor La;c. The kinetics for xdm (and La) has to "interact" with the kinetics of Lcand d002 in a way that gives a realistic progression of Young's modulus (as wellas mechanical strength and electrical resistivity). As described in Chapter 18,this was achieved by introducing a dependence of �dm on the devolatilization ofnon-condensables. A qualitatively realistic progression of Young's moduluswas achieved by using a kinetic model of xdm which gives a moderate loss in xdmstarting at about 700�C. It is possible, however, that xdm starts to decrease ateven lower temperatures. On the other hand, simultaneous data for both La andxdm only exists for temperatures down to 800-900�C1. Most probably, xdm shouldstart to level o� at even lower temperatures.

To explicitly study the development of the properties of the granular structure(see description in part II where a model for Young's modulus in high temperaturepyrolysis2 was simulated. This simulation study contributed to establishing the�nal structure of the model presented in Chapter 18 as described in the simulationstudy in Chapter 18.

The following observations were obtained in the simulations:

� Proper dynamics of xdm is a key to qualitatively correct prediction of thephysical properties which depends on the volume fraction X of crystalliteswithin the granular structure.

� It may be necessary to use a multiple reaction approach to achieve a reason-able dynamics of xdm; i.e. the levelling o� in xdm should start between 500and 800�C and a temperature dependent limit value should be realized.

� In general, the interplay between the parameters which characterize thegranular structure is of main importance for the evolution of the physicalproperties.

� Evolution of La is related to the consumption of the disorganized carbon andto a certain degree of coalescence in the a-direction. Thus, the total numberof layer planes is not conserved. Also, it is possible that there is a couplingbetween La and Lc even at temperatures below 1000�C; see description in

1xdm as calculated from the values of Emmerich's quantity X.2i.e. Heat treatment of a pitch coke in the temperature range between 400 and 1200�C.

19.4 Simulation Results 293

Mizushima (1963). The increased area of layer planes may introduce anincrease in the average stacking height Lc.

� The simulations shows that the open porosity increases in the high temper-ature regime. At the same time, there is a decrease in the closed porosityin such a manner that the total porosity decreases. By careful tuning of themodel parameters, it is possible to obtain an increasing total porosity in thehigh temperature regime; the parameters in the submodels for �r;c, �dm aswell as the kinetics for xdm then has to interact with the model for �a in acertain manner. No e�ort was made in this case to obtain a parameter setwith this property.

� Young's modulus Y and mechanical strength � achieve an intermediate max-imum value at temperatures in the order of 400 to 500�C. This kind of be-haviour of the modulus and strength is not supported by experiments; it israther a result of the use of a very simple model for obtaining the weightedaverage values for the physical properties as well as the extrapolation of themodel for the physical properties of the solid carbon material to low tem-peratures. Thus, more work is needed to improve the prediction of Young'smodulus across the interval of mesophase transition and formation of semi-coke.

� The curve for the electrical resistivity seems rather arti�cial. This may partlybe due to the use of model parameters derived to �t a hard carbon material;another value of ~Xc most probably is needed since this is a material depen-dent constant. The curve shape may also have to do with the uncertainty inthe progression of xdm up to temperatures in the order of 800�C: xdm hasimpact on the value of X used for calculation of �el. There is also a possibil-ity that the structure of the model for electrical resistivity of the anisotropicphase is valid also for calculation of the electrical resistivity of the isotropicpitch fraction.

� If an arti�cially low �nal value of the real density was achieved in the sim-ulations, a higher value of the real density can be obtained by changing theparameters which control the evolution of the function �r. Care should betaken, however, to avoid a situation where �r integrates higher than ��c. Thisis achieved by using the model speci�ed for �r as described in Chapter 18.

It may be concluded that so far, the models for speci�c heat capacity and thermalconductivity seem most successful. The models for Young's modulus, mechanicalstrength and electrical resistivity (i.e. properties which depend on X) have to befurther improved.

A plot of the temperature program used in the simulations and the correspondingproperties is shown below3. In principle, all the physical properties are calculatedas a weighed average value of properties of an isotropic and anisotropic pitch phase.So far, reliable models for both the isotropic and anistropic pitch phases have beenobtained only for the thermal conductivity and speci�c heat capacity.

3The cooling cycle is not considered in this simulation.

294 Simulation of Soft Carbon Properties

For Young's modulus, mechanical strength and electrical resistivity, constant val-ues for the properties of the isotropic pitch phase were used or the quantities arealgebraic function of temperature. The value selected for the electrical resistivityof the isotropic phase was based on information found in J�ager, Wagner & Wil-helmi (1987). For Young's modulus, a qualitatively correct value was used basedon data for ramming paste used with cathode blocks. For mechanical strength, avalue was selected by pure handwaving4.

0 50 100 1500

200

400

600

800

1000

1200

Time [hr]

T [o

C]

Heat treatment program

Figure 19.1: Heat treatment program used in the simulations.

4Young's modulus for the isotropic phase was selected to Yi = 2:0M Pa. The constant c inthe model for Young's modulus was selected as c = 7:5� 10�10N. For the mechanical strength,�i = 1:0 in arbitrary units was used. From J�ager et al. (1987), �el;i = 2:0� 1010�m was used.

19.4 Simulation Results 295

0 50 100 150

3.5

3.55

3.6

Time [hr]

d00

2 [A

A]

Interl. spac.

0 50 100 1505

10

15

20

25

30

Time [hr]

La [A

A]

Layer plane diam.

0 50 100 15010

15

20

25

30

Time [hr]

Lc [A

A]

Stacking height

0 50 100 1500.5

1

1.5

2

2.5

3x 10

21

Time [hr]

nL [1

]

Tot. no. of layer planes

Figure 19.2: Development of crystallite parameters.

0 50 100 1501700

1800

1900

2000

2100

2200

Time [hr]

rho_

c, r

ho_d

m [k

g/m

^3]

Crystalline − and disorg. carbon densities

0 50 100 1501900

1950

2000

2050

2100

Time [hr]

rho_

cb [k

g/m

^3]

Density of solid carbon phase

0 50 100 1501000

1200

1400

1600

1800

2000

Time [hr]

rho_

r [k

g/m

^3]

Real density

0 50 100 150800

1000

1200

1400

1600

1800

Time [hr]

rho_

a [k

g/m

^3]

Apparent density

Figure 19.3: Development of densities.

296 Simulation of Soft Carbon Properties

0 50 100 1500.4

0.6

0.8

1

1.2

Time [hr]

Va/

Va(

0) [1

]Relative apparent volume

0 50 100 1500.4

0.6

0.8

1

1.2

Time [hr]

Vr/

Vr(

0) [1

]

Relative real volume

0 50 100 150800

1000

1200

1400

1600

1800

Time [hr]

rho_

a [k

g/m

^3]

Apparent density

0 50 100 1501000

1200

1400

1600

1800

2000

Time [hr]

rho_

r [k

g/m

^3]

Real density

Figure 19.4: Development of apparent- and real volumes and bulk apparent- andreal densities.

0 50 100 1500

0.1

0.2

0.3

0.4

Time [hr]

phi_

o [1

]

open porosity

0 50 100 1500

0.2

0.4

0.6

Time [hr]

phi_

T [1

]

Total porosity

0 50 100 1500

0.05

0.1

0.15

0.2

Time [hr]

phi_

c [1

]

Closed porosity− bulk level

0 50 100 1500

0.1

0.2

0.3

Time [hr]

phi_

cg [1

]

Closed porosity − gran. struct. level

Figure 19.5: Development of porosities. Note that both open- and closed porositystart at the value zero.

19.4 Simulation Results 297

0 50 100 1500

0.2

0.4

0.6

0.8

1

Time [hr]

x_i [

1]Mass fraction of isotropic phase

0 50 100 1500

0.5

1

Time [hr]

x_c

[1]

Mass fraction of anisotropic phase

0 50 100 1500

0.01

0.02

0.03

0.04

0.05

Time [hr]

x_h

[1]

Mass fraction of hydrogen

0 50 100 150

0.35

0.4

Time [hr]

x_dm

[1]

Mass fraction of disorganized carbon

Figure 19.6: Development of the mass fractions of isotropic phase, anisotropicphase, hydrogen and disorganized carbon.

0 50 100 1500.6

0.7

0.8

0.9

1

Time [hr]

cy [1

]

Coke yield

0 50 100 1507

8

9

10

11

12

Time [hr]

M_a

ve [g

/mol

]

Average molar mass

0 50 100 15010

−10

10−5

100

105

1010

1015

Time [hr]

eta

[N/(

m^2

s)]

Viscosity

0 50 100 15010

2

104

106

108

1010

1012

Time [hr]

rho_

el [m

u O

hm m

]

Electrical Resistivity

Figure 19.7: Development of coke yield, average molar mass, viscosity and electri-cal resistivity.

298 Simulation of Soft Carbon Properties

0 50 100 1501000

1500

2000

2500

3000

Time [hr]

cp [J

/(kg

K)]

Specific heat capacity.

0 50 100 1500

1

2

3

4

5

Time [hr]

k [W

/(m

K)]

Thermal conductivity

0 50 100 1502000

2500

3000

3500

4000

4500

Time [hr]

Y [M

Pa]

Youngs modulus

0 50 100 1503

3.5

4

4.5

5

5.5

6x 10

4

Time [hr]

sigm

a [a

rb. u

nit]

Mech. Strenght

Figure 19.8: Development of speci�c heat capacity, thermal conductivity, Young'smodulus and mechanical strength.

Chapter 20

Introduction to Modelling of

Anode Properties

The chemical and physical transformations in the anode which occur during bakingare mainly due to pyrolysis and coking of the coal tar pitch. Transformation inthe �ller coke are negligible since the coke usually is heat treated at a temperatureof 1300�C.

Based on the properties of petroleum coke and coal tar pitch and models for pitchpyrolysis and coking, models for di�erent anode properties are presented in thischapter. The anode property models are needed both for the bake process simula-tion model as well as for the model based control strategy for ring furnaces. Thischapter and the following chapters in part IV which deal with anode propertiesare based on Gundersen (1996d).

20.1 Two Aspects of Anode Property Modelling

Anode properties are needed both as input for the simulation models as well asfor prediction of anode quality. In general, only a few models for anode propertiesderived from fundamental principles are systematically used in the industry. Onthe other hand, a lot of work on modelling of baked carbon properties is avail-able in the literature. Also, a lot of information exists which not yet has beenformalized into mathematical model descriptions. In this chapter, a summary ofexisting models as well as suggestions for model improvements based on observedphenomena described mainly in the open literature is given.

20.1.1 Model-Data vs. Anode Properties

In a process model for anode baking, anode porosity, density, speci�c heat capacity,thermal conductivity and permeability are needed for calculation of simultaneous

300 Introduction to Modelling of Anode Properties

heat- and mass transfer in the anode.

20.1.2 Anode Quality vs. Anode Properties

The concept of anode quality is an important basis for the anode baking controlstrategy presented in part V. The quality-concept was presented in parts I and IIbased on a systematic treatment of anode properties into �ve property groups asfollows:

� Reactive (i.e. chemical) properties

� Electrical properties

� (Thermo-) mechanical properties

� Other physical properties

20.2 Factors with Impact on Anode Properties

Both raw materials and process conditions have impact on the �nal properties ofthe baked anode:

� Raw material properties (i.e. �ller coke and binder pitch properties)

� Green anode recipe

� Green anode production

� Baking procedure

For the models presented in this chapter, a certain recipe and nominal raw materialproperties are assumed. Furthermore, it is assumed that green anode productiontakes place under optimal conditions and with optimal results.

Except for the properties of the individual coke phases, pore size distribution,microstructure and macrostructure classify the �nal anode properties. The poresize distribution has impact on both micro- and macrostructure. Micro- andmacrostructure is characterized as follows:

� Microstructure mainly depends on the type of �ller and binder used as wellas the mutual interaction of the coke phases. The following parameters areof importance:

{ Crystallite parameters

{ Intercrystalline and closed porosity

{ Degree of isotropy in the binder coke which depends on the size of themesophase spheres (texture)

20.3 The Importance of Optimum Pitching 301

� Macrostructure mainly depends on techniques and equipment used duringpreparation as well as the preparation conditions.

{ Pore size distribution

{ Total porosity

A certain view of anode structure as discussed in Section 20.4 is used for derivationof models for the anode properties.

20.3 The Importance of Optimum Pitching

For an increasing pitch content, baked anode properties have been shown to im-prove before reaching a maximum or minimum value before deterioration occurif the pitch content is even more increased. At least for baked apparent density,electrical resistivity and mechanical properties, the optimum of these propertiesas function of pitch level seem to occur at the same pitch level for a certain bakedcarbon (Okada & Takeuchi 1960).

Also, it has been shown that theoretical models available in the literature areapplicable only for anodes with a pitch content downto the critical content whichcorresponds to the optimal property value (Seldin & Mrozowski 1959). Here, itis assumed that the pitch content is optimized and thus high enough to obtainanodes with properties that follow the available theoretical correlations describedin the literature. Okada & Takeuchi (1960) showed that a typical critical pitchcontent corresponds to 21 parts of binder for each 100 part of �ller coke. Thiscorresponds to a mass fraction of pitch in the order of 21

(21+100)� 0:175. This is

approximately in same order of magnitude as reported in industrial applications.There is a tendency to use too little rather than to much pitch and in general,typical mass fractions of pitch used in the industry ranges between 0.11 to 0.14and 0.20 (Peterson & Seger 1980).

20.4 Physical Structure of Anodes

20.4.1 The Classical View

A picture of the physical structure of baked carbons was established mainly in the1950's by contributions from pioneers like Mrozowski, Collins, Okada, Seldin andothers. More recent studies also apply to this view of anode structure1. Mrozowski(1956b) derived a model for mechanical properties and electrical resistivity of bakedcarbon based on the assumption that the �ller coke particles were uniformly sizedwith a spherical shape. The binder was uniformly distributed on the particles'

1Mrozowski (1956a), Mrozowski (1956b), Mrozowski (1958), Collins (1959), Mrozowski (1959),Seldin (1959), Seldin & Mrozowski (1959), Okada (1960b), Okada (1960a), Okada & Takeuchi(1960), Wiggs (1960), Fischer (1973), Martirena (1983), Wright & Peterson (1989), Jones & Bart(1990).

302 Introduction to Modelling of Anode Properties

surface. The particles were considered to always remain in direct contact witheach other. Bridges of binder coke extend the contact area between the particles.Finally, both the binder coke and �ller coke phases were assumed to be of maximumdensity except for the presence of the so-called unavoidable porosity (Mrozowski1956a). Penetration of pitch into the �ller coke pores and thereby alteration of the�ller coke properties was not taken into account in the model. Also the in uenceof particle angularity and -size on crystallite growth during baking were neglected.Later, di�erent kinds of modi�cations to this view have been presented in theliterature (Mrozowski 1956b).

The structural picture of an anode as presented in the literature can be summarizedas follows:

� The green anode is a composite medium of two carbon phases (components):Calcined �ller coke and binder pitch. During heat treatment, the binder pitchis transformed to binder (pitch) coke. Thus, the baked anode is a two phasecomposite of �ller- and binder coke.

� During mixing and molding of the carbon blocks, the binder pitch acts as alubricant and contributes to uniform the packing of the molded block.

� The anode mainly consists of a rigid framework of �ller petroleum coke par-ticles in direct contact even when pitch is present. There is also indirectcontact via the binder coke bridges. The rigid framework is usually calledthe �ller coke matrix.

� The �ller coke particle size distribution is an important parameter in greenanode manufacturing that contributes to optimizing the bulk density of thecoke aggregate.

� A fraction xf of the �ller coke consists of coke dust and coke �nes whichmixes with the binder pitch to form the so-called binder matrix. The bindermatrix interacts with the larger particles of the �ller coke.

� The binder matrix penetrates parts of the open porosity and pits of the �llerparticles thereby forming roots into the �ller coke. These roots contributeto making strong connections between the �ller and binder matrices.

� However, at nominal pitch levels, binder pitch and dust are mainly presenton the surface of the �ller coke particles which sometimes contribute tocushioning the particles.

� Anode structure can be compared with a packed bed of coke particles sur-rounded and penetrated by pitch/ binder matrix.

� The interaction between �ller and binder is mainly of physical nature (in-terkeying).

� Chemical bonds, however, also play a role (Jones & Bart 1990).

� In the literature, it is stated that the penetration of pitch into the cokeparticles modi�es the particles' structure and properties.

20.4 Physical Structure of Anodes 303

� Many of the properties of the carbon block depend on both the macroscopicand microscopic structure of the carbon. Some properties however, dependmainly on the microscopic structure (i.e. thermal expansion).

20.4.2 A Simpli�ed View of Anode Structure

A modi�cation of the above view of anode structure is presented in the following:

� The �ller coke is assumed to consist of particles of mainly two sizes: Coarseparticle and coke �nes/dust:

1. A fraction (1� xf ) of coarse �ller particles.

2. A fraction xf of �ne �ller particles and coke dust.

� The pore size distribution is assumed to be equal in both the coarse and �neparticle fractions2:

{ Only a certain fraction (1 � s) of the open �ller coke porosity is pen-etrable by the binder coke matrix. A �lling factor � is introduced todescribe the amount of binder matrix penetrating this open porosity. �may vary between zero and one.

{ The fraction s of the open �ller coke porosity consists of pores whichare too small to be penetrated by the binder pitch/binder coke matrix.These pores transform into closed porosity in the anodes due to thesealing of the pore mouths by the binder pitch.

� The �ne coke fraction mixes with the binder pitch to form the binder matrix.

� The �ller coke matrix consists of the coarser part of the �ller coke particles.

� An average diameter dp;fc is supposed to represent the coarse-sized particlesof the aggregate �ller coke.

� dp;fc is in the order 0.5 to 1.0 mm. The coke �nes and coke dust havediameter in the order of 0.05 mm and below.

� It is assumed that the average particle diameter dp;fc of the coarse �ller cokeis conserved during mixing and moulding of the anode.

� It is assumed that in the binder coke matrix, binder pitch and �ne �llerparticles are uniformly mixed together.

� Partial penetration of pitch occurs both in the �ne and coarse �ller cokefraction.

� The pitch content has uniform properties at each position in space of thegreen anode. Pyrolytic behaviour depends on space only via the di�erencesin absolute temperature and rate of temperature increase during baking.

2This assumption is needed in the derivation of a simple model for the so-called transportporosity.

304 Introduction to Modelling of Anode Properties

The dominating part of the two phase mixture of coke and binder pitch consistsof �ller coke. The pitch already contains small particles called primary quinolineinsolubles in an amount of approximately 15 % (Jones 1990, pp. 612). Thequinoline insoluble fraction is the most important crosslinking agent present in thebinder coke. At a temperature of approximately 400�C, spherical liquid crystalstructures called mesophase start to form in the isotropic liquid of pitch. The liquidmesophase solidi�es at approximately 800�C (Jones & Bart 1990, pp. 615). Thepitch has a more or less typical particulate structure in all temperature intervalsduring baking. As basis for the model, the plastic behaviour of pitch is neglectedand the anode paste is assumed to have a spongy structure already from the initialphase of the pyrolysis process. The anode paste is modelled as a porous structurewith both porosity and tortuosity comparable to the structure of an unconsolidatedpacked coke bed. Therefore, the initial establishment of a porous network in thelique�ed pitch is not described in the mathematical model.

20.5 A Note on Calculation of Bulk Properties of

Carbon Anodes

In the models for the physical properties of anodes, properties of both the �ller-and binder coke must be taken into account. Since the �ller coke is thermally stableduring baking, only the pitch (coke) fraction of the anode undergoes chemical andstructural changes during baking. Models for the physical properties of anodes arebased on a subset of the state vector as explained in part II.

At this stage, it is assumed that properties of �ller coke and binder coke areavailable. Then, this part of the work is not concerned with the modelling ofphysical properties of the individual coke phases. Two approaches may be used toobtain the average property values of the composite anode:

� Approach 1: The structural arrangement of the pitch coke with respect tothe �ller coke3 is of no importance for calculation of the bulk property ofthe anode. Then the average property is usually calculated by the followingkind of equation:

pi;a = pi;a(xpc; xfc; pi;pc; pi;fc) (20.1)

xpc + xfc = 1 (20.2)

xfc denotes the mass fraction of �ller coke and xpc is the mass fractionof pitch coke (binder pitch). pi;fc and pi;pc are the corresponding physicalproperties. This approach is used for calculation of crystallite parametersand speci�c heat capacity.

� Approach 2: Sometimes the structural arrangement of the binder pitch/pitch coke with respect to the �ller coke is important for calculation of the

3A thin layer of pitch coke surround the �ller coke particles. A certain amount of �ller cokedust is mixed with pitch coke fraction.

20.6 Previous Work 305

bulk anode property. Then the concepts of �ller- and binder coke matricescome into use: The binder matrix is a mixture of small �ller coke particles,�ller coke dust and pitch coke. Within the binder matrix, the physical prop-erties of the pitch coke changes during baking. The mass fraction of thebinder- and �ller coke matrices are denoted xbm and xfm respectively. Then4:

xbm = xpc;bm + xfc;bm (20.3)

xpc;bm = xpc (20.4)

xfc;bm = xfxfc (20.5)

xfm = (1� xf )xfc (20.6)

The model for the bulk property of the anode now becomes:

pi;a = pi;a(xbm; xfm; pi;bm; pi;fm) (20.7)

pi;bm = pi;bm(xpc;bm; xfc;bm; pi;pc; pi;fc) (20.8)

pi;fm = pi;fc (20.9)

In some cases, volume fractions of the coke fractions or coke-matrices shouldbe used instead of mass fractions. This approach is used in calculation ofthe thermal conductivity. It can be seen that if xf = 0, the two approachesare similar.

20.6 Previous Work

20.6.1 Classical Models of Baked Carbon Properties

Mrozowski (1956b) derived mathematical models for some physical properties ofbaked carbons. Later, Seldin (1956), Seldin (1959) and Okada & Takeuchi (1960)discussed the validity of the proposed models. In the following, modelling ofphysical properties is reviewed based on these original references.

Mrozowski derived the models based on the following assumptions:

� The carbon is a two component system which consists of �ller coke andbinder coke. d denotes the apparent density of the baked carbon.

� The �ller particles are assumed to be of the same size. They also have aspherical geometry with radius rp. The density of the particles is denoteddp.

� In the green carbon, there areN �ller particles per unit volume of the carbon.The �ller density is denoted d� and equals N 4

3�r3pdp. The density of the

binder pitch is (d� d�).

4Sometimes, the volume fractions �j;pc;bm and �j;fc;bm (j = a; r) are needed instead of thecorresponding mass fractions. Most frequently then, the volume fractions as calculated on a realvolume basis (j = r) are needed.

306 Introduction to Modelling of Anode Properties

� In the green carbon, there is W parts of binder pitch (by weight) per 100parts of �ller coke.

� The �ller particles are uniformly coated with pitch in the mixing process.

� In the carbon, the �ller particles always remain in direct contact via fusedbinder bridges5.

� The density �� of the �ller coke is as high as possible except for the presence ofunavoidable microporosity. This density and some other physical properties6

are used as reference values in the model.

� The density as well as other physical properties of the binder coke equalsthat of the �ller coke after baking is completed.

� The binder bridges are well bonded to the �ller particles.

� During baking, the amount of binder coke is reduced to �W parts of bindercoke per 100 parts of �ller coke. � is the binder coke yield.

Mrozowski derived expressions for the radius a of an average binder bridge and theaverage thickness � of binder layer surrounding the �ller particles. In addition, twoother parameters play an important role in the derivation of the property models:

� The e�ective number n of contacts for a particle (i.e. the number of binderbridges per particle).

� The average distance h between contacts in one particle in the direction ofcompression.

Mrozowski assumed that:

h = 4

3r

�dp

d�

�q(20.10)

n = �

�d�

dp

�p(20.11)

Simple models for Young's modulus, shear modulus, modulus of volume expan-sion, Poisson's ratio, compressive strength, bending strength, electrical resistivity,thermal conductivity and permeability were derived. Some important assumptionswere made to be able to derive some of these models:

� Young's modulus Y : For a carbon under compression, it was assumed thatthe change in length of the material is entirely due to the compression ofbinder bridges. The (elastic) contribution from the �ller particle was notconsidered. This approximation is reasonable as long at a << rp.

5The macroscopic structure of the carbon equals the structure found in a compressed carbonpowder except for the much wider and fused interparticle contacts.

6Young's modulus �Y , crushing strength �S and electrical resistivity ��el.

20.6 Previous Work 307

� Crushing strength Sc: The strength of the carbon material is proportionalto the cross section of the binder bridge.

� Electrical resistivity �el : The main contribution to the resistance to current ow takes place at the constriction as long as the area of the contact (binderbridge) is small in comparison with the diameter of the particles (Holm 1967).

It was found that:

Y = cydz�(d� d�) (20.12)

cy = �Y4

3

L� 2

f �d1+z

z = p� 2q

S = csdy�(d� d�) (20.13)

cs = k1

r2sp

�el =c�el

dx�

pd� d�

(20.14)

c�el =3

2���d1+p+2q

� 2

x =1

2+ p� 2q

Mrozowski's notation is retained in these expressions.

a thin layer of binder pitchFiller particle surrounded by

is displacedThe binder pitch

r

�

2a

Figure 20.1: Radius of the binder bridge between two �ller particles. Based onMrozowski (1956b, Fig. 1).

Mrozowski and Seldin's Criticism of the Two-Component Model

In Seldin (1959), a thorough analysis of the validity of the model for the electricalresistivity for low and high binder contents as well as �ller particle mixtures isreported.

308 Introduction to Modelling of Anode Properties

The model of the electrical resistivity, with a constant �ller coke density becomes:

�el = c�el;�1p

d� d�(20.15)

c�el;� is a constant. In general, both �ller density and apparent density of greencarbons vary when molding pressure and pitch content are varied. But, if insteadbaked carbons are impregnated, only the binder density (and thus the apparentdensity) vary while the �ller density is kept constant7. According to the model,1

�2el

as function of apparent density should be a single linear relationship.

Experimental data, however, show that measured values of electrical resistivitylie on the same curve for a certain binder content. Furthermore, for lower bindercontents, the curve is not linear. On the other hand, the position of the curvesrelative to each other is systematic: Electrical resistivity increases as the bindercontent decreases as shown in Figure 20.2. This may be explained as follows: Lowbinder contents lead to inhomogeneous distribution of �ller particles and severanceof binder bridges upon release of the molding pressure. Also when the pitch contentis low, pitch distribution may not be satisfactory. These phenomena are observedas increased resistivity in carbon samples with a low binder content. Both e�ectscan be present at the same time but Seldin concluded that the increased resistivityis mainly due to an inhomogeneous particle distribution. Seldin also found thatbinder bridge severance occurs most easily in carbons made from small sized �llerparticles. Okada & Takeuchi (1960) disagreed with Seldin's conclusions and alter-natively suggests that binder bridges break due to shrinkage of the binder coke inthe late stages of the baking process.

In conclusion, c�el is not a constant but actually depends on the binder contents insuch a way that the constant increases as binder contents decreases. The structuraldependence of c�el on the binder contents is yet not revealed.

Furthermore, it was shown that the electrical resistivity as function of bindercontent achieves a maximum value. The critical binder contents in this case wasnear to 36 parts binder by weight per each 100 parts of �ller (by weight). This isin accord with the result obtained by Okada & Takeuchi (1960) who also showedthat the physical properties of baked carbons achieve a optimum value for a certaincritical binder content. This phenomenon is thoroughly discussed in Gundersen(1996d). Since the electrical resistivity achieves a minimum for p = 36%, it seemsas if Seldin's carbons with p < 36% have pitch contents below the critical range.

Finally Seldin showed that for carbons made of a mixture of large and small �llerparticles, parameter c�el is more or less insensitive to the particle size. This wasexplained to be a result of the interaction between two mechanism:

� Since large particles have a smaller surface area, the thickness of the bindercoke layer is increased. This introduces a decrease in resistivity.

� Presence of small particles leads to a reduced �ller density (Mrozowski 1956b)which further leads to an increased resistivity.

7For each binder content, the molding pressure was varied to obtain samples of varying ap-parent density.

20.6 Previous Work 309

Apparent Density

Binder contentincreases

0

1�2el

Figure 20.2: The dependence of electrical resistivity on apparent density for vary-ing pitch contents. According to the �gure, 1=�2el vs. apparent density deviatesfrom a linear behaviour for the lowest binder contents. Thus, apparently the va-lidity of the two-component model breaks down for low binder contents. As theapparent density increases, the resistivity curves all merge into one curve since theapparent density approaches the theoretical maximum limit. This, however, is notexplicitly shown in the �gure. Based on Seldin (1959, Fig. 1).

Okada's Criticism of the Two-Component Model

Mrozowski found that the models for the physical properties do not �t the ex-perimental data at very low and very high binder contents. A low binder contentleads to an inhomogeneous particle distribution. Also for low binder contents, sev-erance of binder bridges may occur (Seldin 1956), (Seldin 1959). For high bindercontents, the �ller particles are cushioned by the pitch an the concept of narrowbinder bridges looses its validity.

Later, Okada & Takeuchi (1960) thoroughly explained the dependence on physi-cal properties of green and baked carbon with the pitch contents as reviewed inGundersen (1996d). Okada showed that a physical property as function of thebinder content is not a monotonous function: Holding the molding pressure con-stant, physical properties improve up to a certain critical binder contents at whichexpansion of the �ller matrix takes place. Above the critical pitch contents, phys-ical properties start to deteriorate as the binder contents increases as shown inFigure 20.3. Actually, a physical property as function of binder contents is nounique function (see Figure 20.4). This ambiguity in physical properties is intro-duced via the green manufacturing process as a consequence of the behaviour ofthe �ller coke density as the pitch contents increases. An qualitative explanationfor the phenomenon is given in Gundersen (1996d) based on Okada's model forthe apparent density as function of pitch contents.

In contrast to Seldin (1959), Okada suggested that for low binder contents, cracks

310 Introduction to Modelling of Anode Properties

10 20 30 40 50 10 20 30 40 50

10 20 30 40 50

fp[%] fp[%]

fp[%]

�el[�m]

Y [MPa] S[MPa]

Figure 20.3: Physical properties as function of binder contents show the samequalitative behaviour as the apparent density: They are not unique functions ofthe binder content. Corresponding curves exist for physical properties as func-tion of apparent density showing that the property functions are not unique; seeFigure 20.4 below. Based on Okada & Takeuchi (1960, Fig. 18).

are created in the binder bridges during baking. Thus, severance of the binderbridges during baking rather than after relaxation of the molding pressure was thesource of discrepancy between the two-component model predictions and experi-mental data for low pitch contents (i.e. below the so-called critical pitch contents).This partly con�rms the results presented by Seldin (1959) who also studied car-bons with pitch content below the critical range. Seldin, however, could explainhis results by allowing the parameters ci to depend slightly on the binder contents.

For high pitch contents p > pc, Okada con�rmed the general validity of the two-component models but concluded that x 6= 1

2+ z. On the other hand, the rela-

tionship y = z was found to be valid. According to Okada, the theoretical basisfor the two component model corresponds best with the situation in carbons withpitch content below the critical limit (thin layer of binder surrounding the �llerparticles). Thus, the agreement between models and experimental data for highpitch contents was not obviously expected.

Based on Mrozowski, Seldin and Okada's results, it seems as if the original two-component models need considerable re�nement to obtain general validity.

20.6 Previous Work 311

1300 1400 1500 1600 �[ kgm3 ]

�el[�m]

Figure 20.4: Physical properties are not uniquely correlated with apparent densityas demonstrated in the case of electrical resistivity. This feature of baked carbonswas not clearly stated by Mrozowski. The phenomenon is due to the fact thatthe same apparent density can be achieved at two di�erent pitch contents just byvarying the molding pressure. Based on Okada & Takeuchi (1960, Fig. 19).

20.6.2 Application of the Two-Component Models in this

Work

In this study, however, it is assumed that the two component models are validalso in the range around the critical pitch contents and slightly below. Averagevalues for parameters ci is assumed applicable, but parameters dependent on thebinder pitch contents (binder coke yield) should be considered if necessary. Thisassumption is supported by the following:

� Seldin (1959) argued that for a certain initial pitch contents, the modelsare valid for varying binder densities (i.e. binder coke yields) except at thelowest binder contents.

� The models may be used to predict changes in properties during baking.

� Reasonable correlation obtained between experimental property data andpredictions done with the two-component models as shown in Gundersen(1996d, Cht. 8). Thus, average values for ci also exists in cases with bindercontents below the critical limit.

� As shown in Gundersen (1996d, Cht. 8), the physical properties also cor-relate with microstructural properties (crystallite size etc.). So far, such

312 Introduction to Modelling of Anode Properties

phenomena are not included explicitly in the models presented here.

20.6.3 Future Modi�cations of the Two Component Models

To achieve general validity of the two-component model also for binder contentsbelow the critical limit, empirical functions for the parameters ci as function ofbinder content could be suggested. It seems as if these parameters are constant forpitch contents above the critical limit. For pitch contents below critical, changesoccur. In the case with electrical resistivity, parameter c�el increases to cope withthe observed increase in electrical resistivity which occur when the pitch contentdecrease even when �ller density and apparent density do not change8. Typicallapse of c�el is shown in Figure 20.5.

fp[%]fp;c

c�e[�m]

Figure 20.5: Qualitative dependence of c�el on the initial pitch contents. fp;c isthe critical pitch level.

20.6.4 Summary of Property Models

In the original derivation, it is assumed that the �ller particles are dense exceptfor the existence of so-called unavoidable porosity. The properties of the bindercoke was assumed to equal that of the �ller coke. Other assumptions were madeconcerning the distribution of binder pitch, packing of particles, contact betweenparticles etc. Based on these assumptions, models for Young's modulus, mechan-ical strength and electrical resistivity were derived.

8This is possible by using di�erent values of molding pressure.

20.6 Previous Work 313

Due to the existence of a porous binder coke, the following corrections are intro-duced:

� Deformations in the porous binder coke are larger than in dense binder coke.Thus, Young's modulus decreases slightly as compared with the originalmodel.

� A porous binder coke is mechanically more weak that a dense binder coke.This leads to a decrease in strength as as compared with the original model.

� In the case of the electrical resistivity, the e�ective constriction area dependson the pores and cracks existing in the binder coke. This gives an increasedresistivity as compared with the original model.

The fact that �ller particles are of no de�nite shape and size does not invalidatethe original model structure. The use of elongated particles however, introducesa geometrical anisotropy in the physical properties. Still, the most importantcorrection is introduced via porous �ller particles. This leads to increased valuesof strength and electrical resistivity. The Young's modulus, however, obtain alower value than calculated by the original model.

If the porous binder coke and the porous �ller particles are considered to give themost important modi�cations, the �nal structure of the models becomes:

S = fscs;pdy�(d� d�) (20.16)

Y = fycy;pdy�(d� d�) (20.17)

�el =c�el;p

f�el

1

dx�

pd� d�

(20.18)

In these models, the importance of macroscopic properties on the physical prop-erties is introduced via the �ller - and binder coke densities as represented by d�and d� d� respectively. The in uence of microscopic properties is introduced viathe parameters fi (i = s; y; �el) and ci;p.

For pitch contents below the critical, dependence of parameters ci on the bindercontents should also be allowed.

20.6.5 Contributions from the Aluminium Industry

A lot of work on baked carbon properties has been done by the industry. Of specialinterest is research conducted by the aluminium industry either individually or incooperation with research institutes. Each year, a number of interesting resultsare reported in the Proceeding from the annual Light Metals conference.

Properties of both bench scale and full scale anodes are studied. In general, moststudies are concerned with the impact of the following factors on anode properties:

� Raw materials:

314 Introduction to Modelling of Anode Properties

{ Filler coke (type, density and granularometry)

{ Binder pitch (type, treatment, QI-content)

{ Butts (cleaning and impurities)

� Green anode production:

{ Anode paste formulation (�ller content vs. pitch content, butts con-tent)

{ Preheating

{ Mixing process (time and temperature)

{ Molding process (molding pressure)

{ Vibrocompaction process

� Baking process:

{ Heating rate

{ Top (hold) temperature

{ Holding time

Many studies are exclusively concerned with anode reactivity (CO2 and air) andthermal shock resistance. Most of the work has been done to achieve insight intothe in uence of raw material quality on anode properties. Also e�orts has beendone to obtain insight into the e�ect on anode properties of varying green anoderecipe. Finally, the importance of varying baking parameters has been thoroughlydiscussed.

The production steps in anode manufacturing are reviewed in Keller & Fischer(1992). During the last three decades, anode baking has become relatively moresophisticated than green anode production. Advances in process control, uedesign and furnace size have been signi�cant and the most recent ring furnacesare quite complex systems. Quality of raw materials is one of the most importantfactors for anode quality. On the other hand, consistent anode quality can onlybe achieved by automatic process control at each production step.

Tests on full scale anodes are both time consuming and expensive, and evaluationof production strategies most cheaply takes place by laboratory experiments. Onthe other hand, it is not always possible to achieve realistic process conditions inthe laboratory. Also, experiment planning is very important to achieve the bestadvantage from expensive experiments (Rey Boero 1988).

Even though both the green anode manufacturing (Belitskus 1985), (Coste 1988),(Bin Brek & Vaz 1995) and the baking stage (Hurlen & Naterstad 1991) are wellunderstood, the general conclusion seems to be as follows: At each stage in themanufacturing process, optimal conditions must exist. A mistake at a certainprocess stage can usually not be corrected at a later process stage. Selection ofraw materials occur as a trade-o� between raw material cost and anode productioneconomy. The coupling between the di�erent anode production stages has been

20.6 Previous Work 315

realized for a long time but systematic use of this knowledge in anode productionhas been di�cult.

New anode production technology, however, optimally combines the individualprocess steps (Mannweiler & Keller 1994). Furthermore, routines for process op-timization are designed from a global approach with the goal of optimizing anodeproduction from a multivariable point of view. The recent trend is that sensitiv-ities of properties for variation of a lot of critical process parameters both fromgreen paste production, compaction and the baking step is collected and a globaloptimization is performed. Small sized anodes are used and implementation of anoptimized production strategy is supposed to take a few weeks only. Finally thegreen mill - and baking parameters are selected which gives the lowest price ofaluminium production costs (B�uhler & Perruchoud 1995).

The �nal goal is to improve anode quality. Demands in anode quality have consid-erably increased during the recent years, and some companies have implementedquality control according to ISO standards (Schmidt-Hatting, Baak & Blom 1992).This enables revision of the whole quality control system and contributes to achiev-ing a more constant anode quality and signi�cant reduction in anode scrap.

As stated above, optimal production of anodes demands optimal consistency ateach process stage as well as the use of high quality process equipment. Recently,Fischer & Keller (1993) reviewed the impact of baking parameters on the �nalanode quality. Important parameters during baking are as follows:

� Heat-up rate

� Final baking temperature

� Soaking time9

However, success in anode baking also depends on:

� Constant quality of green anodes

� Flue design and refractory maintenance

� State of the baking furnace

� E�cient baking-process control

Since the baking parameters depend on the raw materials used, optimization ofbaking parameters should be performed after each signi�cant change in raw ma-terials, furnace construction and anode dimensions. Still not answered is the de-termination of optimal baking parameters for a given green anode brand and ringfurnace design. On the other hand, some qualitative observations are well known:

9De�ned as the interval in time between time for reached top temperature in the ue gas andtime for switching o� the burner(s).

316 Introduction to Modelling of Anode Properties

� Heat-up rate: A too high heat-up rate may cause propagation of cracks (eveninvisible for the eye) due to a high pressure of volatiles inside of the anode.This is observed as a decrease in exural strength as well as increased spe-ci�c electrical resistivity. A maximum temperature rate of 12�C=hr duringvolatile release is recommended (slightly varying with green anode recipe).

� Final baking temperature: The �nal baking temperature increases crys-tallinity of the binder coke. On the other hand, there is a risk of post-calcination of the �ller coke which leads to desulphurization. This leads toa certain deterioration of the CO2-reactivity and the porosity introducedduring sulphur loss adversely a�ects airburn. Also the increase in thermalconductivity at high �nal baking temperatures signi�cantly increases air-burn. Therefore optimal �nal baking temperatures lie between 1050 and1200�C depending on the soaking time.

� Soaking time: A soaking time of a certain length is needed to equalize thetemperature �eld in the pits.

One should be aware of that a reduced soaking time cannot be compensated by anincreased �nal baking temperature. This is due to the fact that a pit needs timefor temperature equalization to take place since heat conduction in the anodes is avery slow dynamic process. In conclusion, optimality of heat-up rate, �nal bakingtemperature and soaking time are related in a complicated manner dependent onboth furnace design and raw materials. The �nal goal is to produce anodes ofoptimal properties such as low speci�c electrical resistivity and low reactivity butalso achieve a high productivity at low energy consumption.

Since there is a signi�cant time lag between the production of anodes and their usein the reduction cells, evaluation of anode quality from the anode's behaviour inthe cell is not very well suited for feedback to the anode production line. Rather,anode quality should be evaluated as early as possible.

In Hydro Aluminium, methods for evaluation of bake furnace calcining level in-volves the concept of equivalent temperature which gives a measure of the time- temperature history of a reference coke sample which follows the anodes dur-ing baking (Foosn�s et al. 1995). Measured equivalent temperature and anodeproperties are correlated, and preferable ranges of equivalent temperature can bedetected. It is important to achieve the optimum calcining level since a low cal-cining level gives high reactivities and dusting index. A too high level also leadsto deteriorated anode properties partly re ected by increased energy consumptionin the cell. The equivalent temperature distribution is a characteristic property ofa baking furnace. Optimization of the baking process then amounts to tuning theburners so as to achieve the most narrow distribution curve.

In another approach, so-called quality �gures for anode properties is used to eval-uate the anode production. Except for constraints on:

� Electrical resistivity

� Purity

20.6 Previous Work 317

� Consistency (small deviations in the range of physical properties)

the anode performance is mainly de�ned by:

� Thermal shock resistance

� CO2-reactivity

� Airburn

Quality �gures for the three last properties were de�ned to be used in predictionof behaviour of the anodes in the reduction cells (Keller et al. 1990). Simpleexpressions were obtained based on previous studies on carbon consumption andthermal shock resistance.

318 Introduction to Modelling of Anode Properties

Chapter 21

Porosity, Density,

Permeability and Surface

Area of Anodes

Apparent density, porosity, permeability and surface area are modelled as bulkphenomena by taking into account contributions from the binder- and �ller cokephases. In the derivation of the models, it is assumed that the apparent volumeof the anode is constant since the ideal anode does not swell and has negligibleshrinkage during baking.

21.1 Total Porosity

21.1.1 Signi�cance of Total Porosity

At least two reasons motivate for modelling porosity of anodes during baking:

� Porosity has impact on physical properties of the anode.

� Knowledge of porosity and pore structure of the anode during baking isneeded to evaluate transport parameters for volatile gas ow in the anode.

A lot of work has been done on baked carbon properties and several correlationsexist in the literature in which porosity is an important parameter. Models forelectrical resistivity, (compressive) strength and Young's modulus have been dis-cussed by di�erent authors like Mrozowski, Seldin and Okada.

E�ective di�usion coe�cients for the volatiles as well as anode permeability dependon the transport porosity �f in the anode. It will be shown that anode transportporosity is related to the open porosity of the anode.

320 Porosity, Density, Permeability and Surface Area of Anodes

21.1.2 Di�erent Types of Porosity in the Anode

In a baked anode, both the �ller- and binder pitch coke contribute to the totalporosity.

In the �ller coke, qualitatively three di�erent types of porosity exist:

1. Intraparticular open porosity Xai;�;fc

2. Intraparticular closed porosity (which consists of pores partly exposed byreal density measurements on the �ller)

3. Intercrystalline and closed microporosity (which consists of small pores notdetectable in real density measurements)

To simplify this picture, the last two kinds of porosity may be lumped into onegroup of closed porosity denoted Xa

i;c;fc. To simplify even more, assume that thewhole closed porosity is not detectable in real density measurements. Then Xa

i;c;fc

may be de�ned according to the de�nition of closed porosity in Appendix B:

(Xa)0i;c;fc = 1� �r;fc

��c;fc(21.1)

Xai;c;fc = (Xa)0i;c;fc(1�Xa

i;�;fc) (21.2)

Here, (Xa)0i;c;fc) is related to the real volume (of the �ller coke) whereas Xai;c;fc is

related to the apparent volume of the �ller coke. The intraparticular open porosityXai;�;fc can be calculated according to the de�nition:

Xai;�;fc = 1� �a;fc

�r;fc(21.3)

The total porosity of the �ller coke is then:

Xai;T;fc = Xa

i;c;fc +Xai;�;fc (21.4)

For simplicity, Xai;T;fc, X

ai;c;fc and X

ai;�;fc are denoted X

ai , X

ai;c and X

ai;� in the

following.

A fraction of the intraparticular open porosity of the �ller coke is �lled with bindercoke due to partial penetration of pitch during mixing, molding and baking of theanode. Therefore a certain fraction of the open porosity of the �ller coke appearsas induced closed porosity �c;a;in in the anode.

Within the pitch coke of the binder coke matrix, one must expect to �nd qualita-tively the same kind of porosity as in the �ller coke:

1. Open porosity ��;pc

2. Closed porosity

3. Intercrystalline and closed microporosity

21.1 Total Porosity 321

Again, the last two groups of porosity are lumped together in closed porosity �c;pc.The porosity de�nitions applied for the pitch coke volume gives:

�0c;pc = 1� �r;pc

��c;pc(21.5)

�c;pc = �0c;pc(1� ��;fc) (21.6)

��;pc = 1� �a;pc

�r;pc(21.7)

�T;pc = �c;pc + ��;pc (21.8)

The open porosity in the binder coke constitutes a part of the open porosity in theanode. In addition comes the voids in the open pore space in the �ller coke matrix(if any) not occupied by binder coke. Finally, a certain open (interconnected) porespace exists within the part of the �ller coke matrix which is not impregnated withpitch. In summary, the following types of porosity exist in the anode:

� Intrinsic closed porosity in the �ller and binder pitch coke

� Induced closed porosity in the open pores of the �ller coke

� Open porosity in the �ller- and binder pitch coke

� Interconnected pores within the �ller matrix

21.1.3 Two Views of Total Porosity

Based on the mathematical de�nition, the following equations de�ne total, openand closed porosity of the anode:

�T;a = �c;a + ��;a (21.9)

��;a = 1� �a;a

�r;a(21.10)

�c;a = �0c;a(1� ��;a) (21.11)

�0c;a = 1� �r;a

��c;a(21.12)

��;a and �c;a denote open and closed anode porosity on the apparent volume level.�0c;a is closed porosity in the anode as de�ned on the real volume level. �a;a, �r;aand ��c;a are the apparent, real and solid densities of the anode. The correspondingvolumes are Va;a, Vr;a and �Vc;a. This de�nition makes no consideration of thedi�erent coke phases which exist in the anode. But as discussed in the previoussubsection, the �ller and binder pitch coke phases interact with each other andcreate a complicated network of pores. To obtain a model of this pore space, twoaspects must be resolved:

� The concept of anode porosity should be related to the porosity of the purecoke phases.

322 Porosity, Density, Permeability and Surface Area of Anodes

� A simple geometric view of the interaction between the coke phases is neededto distribute the total porosity into closed- and transport (interconnected)porosity.

Intuitively, both the �ller coke and the binder pitch coke contribute both to theclosed and open porosity of the anode. In addition, some open porosity must alsobelong to the pore space between the �ller particles due to incomplete compaction.Finally, some porosity is generated due to the interaction between the pure cokephases.

Two approaches to modelling of total porosity is discussed below.

Approach 1: Total Porosity vs. Pure Coke Phases

In the simplest approach to modelling of total porosity, the �ller and pitch cokephases do not interact; see Figure 21.1. The models for real- and apparent den-sity of the anode presented later in this chapter is also in accord with this view.Therefore it seems reasonable to give a de�nition of open- and closed porosity ofthe anode from this view. Thus:

�T;a = �c;a + ��;a (21.13)

��;a = 1� �a;a

�r;a(21.14)

�c;a = �0c(1� ��;a) (21.15)

�0c;a = 1� �r;a

��c;a(21.16)

�r;a = �r;fc + (1� ��;a � )�r;p

1� ��;a(21.17)

�a;a = �r;fc + (1� ��;a � )�r;p (21.18)

The open porosity obtained in this case includes all pores outside the real densitylevel of each coke phase. Furthermore, the closed porosity includes only the closedporosity within the pure coke phases. In the following, this kind of closed porosityis denoted intrinsic closed porosity.

Modelling the Intrinsic Closed Porosity

Based on the interpretation of Figure 21.1 for the anode and the fact that the anodeconsists of �ller- and pitch coke, an expression for the intrinsic closed porosity1 ofthe anode (on the apparent volume level) can be found as it relates to the intrinsicclosed porosity of the pure coke phases.

The volume of solid carbon within the anode is given from:

�Vc;a = (1� �0c;a)Vr;a = (1� �0c;a)(1� ��;a)Va;a (21.19)

1By closed porosity, is meant �c;a; i.e. not including closed porosity due to sealing of �llercoke pores by penetrating binder pitch.

21.1 Total Porosity 323

�c;a

Va;pc

��;a

mfc

��c;fc�Vc;fc

mpc

��c;pc�Vc;pc

Va;a

Vr;a

X;ai;c

��;pc

�0c;pc

Vr;fc

Va;fc

Xai;�

Vr;pc

Figure 21.1: Simple view of anode structure used in the �rst approach to modellingtotal porosity of the anode.

since Vr;a = (1 � ��;a)Va;a. The solid carbon consists of tw carbon phases: i.e.�ller- and binder coke:

�Vc;a = �Vc;fc + �Vc;bc (21.20)

where:

�Vc;fc = (1� �0c;fc)Vr;fc (21.21)

�Vc;pc = (1� �0c;pc)Vr;pc (21.22)

The real volumes of �ller- and pitch coke are de�ned by:

Vr;fc = Va;a (21.23)

Vr;pc = (1� ��;a � )Va;a (21.24)

and:

�Vc;fc = (1� �0c;fc) Va;a (21.25)

�Vc;pc = (1� �0c;pc)(1� ��;a � )Va;a (21.26)

and (1� ��;a � ) denote volume fractions of �ller coke and binder pitch/ pitchcoke respectively; see shown in Section 21.2. Finally, obtain:

�Vc;a =�(1� �0c;fc) + (1� �0c;pc)(1� ��;a � )

�Va;a (21.27)

324 Porosity, Density, Permeability and Surface Area of Anodes

From the two expressions for �Vc;a one may obtain:

(1� �0c;a)(1� ��;a) = (1� �0c;fc) + (1� �0c;pc)(1� ��;a � )

This gives the following expression for the closed prosity in the anode:

�c;a = 1� ��;a � (1� �0c;fc)� (1� ��;a � )(1� �0c;pc) (21.28)

Approach 2: Total Porosity vs. Geometric View of The Anode

The next view used for obtaining the model of the total porosity of the anode isshown in Figure 21.2.

A fraction of the pitch coke penetrates the open porosity of the �ller coke aggregate.This leads to a certain volume of induced closed porosity �c;a;in in the anode. Inaddition comes the intrinsic closed porosity which belongs to the two pure cokephases. Thus, two kinds of closed porosity appear in the anode volume:

~�c;a = �c;a + �c;a;in (21.29)

A certain fraction of accessible porosity exists in the pore space outside the impreg-nated �ller particles. This accessible porosity is interconnected and constitutes theso-called transport porosity denoted �f .

Finally, in the pore trunks which contain plugs of binder pitch coke (binder cokematrix), there exist the opportunity that di�erent kinds of porosity coexists mainlydue to the presence of pitch coke (�ller coke dust) and partly due to incomplete�lling of the pore trunk. At least a fraction of this porosity belongs to the intrinsicclosed porosity already mentioned. Furthermore, there may be open porositypresent from both pure coke phases as well as open space due to incomplete �llingof the pore trunk. This porosity is denoted the residual porosity �r;a.

In summary, then:

~�T;a = �f + ~�c;a + �r;a (21.30)

Figure 21.2 is slightly simpli�ed in the way that the �ne �ller grains that are mixedwith the binder pitch to constitute the binder matrix is not explicitly shown inthe �gure.

Comparing the Two Models

Ideally, the open porosity ��;a calculated in the �rst approach should correspondto the following porosities in the second approach:

� The transport porosity �f

� The induced closed porosity �c;a;in since this porosity actually belongs tothe intrinsic open porosity of the �ller coke

21.1 Total Porosity 325

Va;a

Va;fc

Va;pc

�c;a;in

�r;a

�c;a

Figure 21.2: Geometric view of anode structure used in the second approach tomodelling total porosity of the anode.

� The residual porosity �r;a since this includes parts of the open porosity ofthe �ller coke as well as accessible (open) porosity in the coke which plugsthe pores2

The real volume of pitch coke which resides in the interconnected pore space orin the trunks has no impact on the magnitude of the open porosity since the realvolume needs the same amount of space in each of the two positions. Therefore, thereal volume of pitch coke has only impact on the distribution of the open porositybetween the transport porosity and the induced closed porosity. In summary, then:

��;a = �f + �c;a;in + �r;a (21.31)

If the residual porosity is negligible3, the following approximation is valid:

��;a � �f + �c;a;in (21.32)

A model for �c;a;in is derived in Section 21.3. It should be noted here that the openporosity ��;a is open according to the mathematical de�nition of open porosity.Thus, the porosity which is open in the context of being accessible for a owing uid, corresponds to the transport porosity �f .

2Situations with "recursive" �lling of coke pores is neglected; i.e. neglect the fact that theremay exist �ne coke particles within the �ller coke pore trunks that have open porosity �lled withpitch.

3This may happen if the �lling of the pore trunks is very dense.

326 Porosity, Density, Permeability and Surface Area of Anodes

In the ideal case, the two di�erent approaches is able to calculate the same totalporosity since:

�T;a = ��;a + �c;a = ~�T;a (21.33)

according to the relationship recently presented for ��;a. In fact, the geometricalview has made it possible to classify the distribution of open porosity within theanode.

21.1.4 Shrinkage during Cooling vs. The Unavoidable Mi-

croporosity

Mrozowski (1956a) thoroughly discussed the intrinsic presence of a so-called un-avoidable microporosity in carbons at room temperature. The unavoidable micro-porosity is a consequence of the di�erence in bulk contraction of the whole carbonbody and a single crystallite.

Using estimates for volume expansion coe�cients for the bulk carbon material anda single crystallite, an estimate of the unavoidable microporosity was obtained.Assuming that the carbon have low bulk contraction during cooling (�v � 3:0�10�61=�C), Mrozowski showed that the di�erence in volume expansion coe�cientslead to an unavoidable microporosity between 5 and 10 % depending on heattreatment temperature. The unavoidable microporosity is due to intercrystallinepores formed during cooling and will appear even if the carbon material at thehighest heat treatment temperature is perfectly dense.

The unavoidable microporosity puts an upper limit on the theoretical density ��of the carbon material. This is a typical example of limitations in the carbonmanufacturing process. Mrozowski often uses �� as a kind of reference density inmodelling properties of carbon materials. �� cannot be measured directly; only anestimate can be obtained via estimates of the di�erence in bulk expansion coe�-cients. Also, according to Mrozowski (1956b, pp. 197), this density is considerablylower than the corresponding real density of carbon crystallites. This is in contrastwith Seldin (1959, pp. 678) who uses �� = 2100 kg/m3 for a calcined petroleumcoke. This shows that the magnitude of �� is very close to the magnitude of realdensity for a calcined coke. This is not surprising since most of the intercrystallineporosity and closed porosity is included in the real density concept.

On the other hand, Mrozowski (1956a) argues that carbons at the highest heattreatment temperature are never perfectly dense4: A certain degree of so-calledregular porosity exists which induces a contraction during cooling which over-balances the formation of unavoidable microporosity. Thus, the whole carbonbody shrinks at the expense of regular porosity. Thus, the shrinkage which occurduring cooling is anisotropic leading to a reduced total porosity (i.e. the apparentshrinkage parameter � > 1:0).

In summary, the room temperature anode porosity consists of qualitatively twodi�erent kinds of porosity (Mrozowski 1956a):

4In this context, perfectly dense means that the material has got the density of graphite.

21.1 Total Porosity 327

� The unavoidable microporosity due to formation of intercrystalline pores

� The regular porosity which is porosity still present at the highest baking tem-perature and theoretically can be eliminated by some manufacturing process

In this report, the concept of unavoidable microporosity and the correspondingdensity �� is not explicitly used. Instead, the concept of porosity is linked to thereal density/bulk density relationship as measured by methods well establishedin industry. Actually, the unavoidable microporosity is a concept with a physicalinterpretation whereas the porosity-concept derived from the de�nitions of real-andapparent densities is not an absolute quantity since it depends on the laboratoryprotocol used to determine real and apparent densities.

For simplicity, assume that the shrinkage which occur during cooling takes placein an approximate isotropic manner meaning that there is no change in the totalporosity of the anode during cooling.

21.1.5 Thermal Expansion of Coke Phases During Baking

To achieve an understanding of macro- and micro expansion and contraction duringbaking and cooling, the coke phases have to be studied individually.

Filler Coke

It was found that changes in the bulk anode volume and �ller coke expansionduring baking can be neglected. Using relevant values for expansion- and shrinkagebehaviour, it can be shown that that these assumptions are reasonable (Gundersen1996d).

Binder Coke

Thermal expansion of the liquid pitch is taken into account. For temperaturesabove 400�C however, only irreversible shrinkage of the binder coke is taken intoaccount for calculation of the anode total porosity. Actually, the e�ective shrinkageof binder coke at elevated temperatures is somewhat lower than the one predictedby the binder coke real density model since this density is assumed to predict theroom temperature real density not taking into account thermal expansion.

If at elevated temperature there is a certain reversible thermal expansion opposingthe irreversible shrinkage, the e�ective shrinkage rate is reduced. To make thingseven more complicated, the thermal expansion coe�cient most probably also de-pends on the heat-treatment history of the sample. If the thermal expansioncoe�cient of the binder coke equals that of the �ller coke, the thermal expansionat elevated temperature of the binder is negligible compared to the irreversibleshrinkage due to structural modi�cation of the binder during baking. Therefore,thermal expansion of the binder coke is not taken into account.

328 Porosity, Density, Permeability and Surface Area of Anodes

Conclusion

The room temperature real density of �ller coke is used in calculation of thetotal anode porosity. Also, expansion of the solidi�ed binder coke at elevatedtemperature above approximately 500�C is not taken into account. Only thermalexpansion of liquid pitch is included in the model.

21.2 The Open Porosity Model

With a knowledge of the rate laws ri which describe the release of volatiles fromthe pitch fraction of the anode, it is possible to establish a mass balance for theanode. An equation for the open porosity of the anode can be derived from themass balance if models for real density of the anode components are available.

For derivation of the open porosity equation, an apparent control volume Va;a inthe anode is considered. Within the apparent volume Va;a, real volumes Vr;fc andVr;p of �ller coke and binder pitch (binder coke) exist with masses mfc and mp

respectively:

ma = mfc +mp = �r;fcVr;fc + �r;pVr;p (21.34)

ma is the mass of the anode volume. �r;fc and �r;p are real densities of �ller cokeand binder pitch (pitch coke) respectively. The anode's open porosity is repre-sented by a function ��;a(r; t) (the void fraction in the anode) with a dependenceof both space- and time coordinates. Because of degassing of volatiles, open poros-ity changes with time. The density �r;a(r; t) of the solid mass also changes dueto the volatile release. This may be expressed mathematically by writing a massbalance for the solid phase over a volume Va;a taking into account the main volatilecomponents (Aziz & Settari 1979). If ri [kg/(m

3 s)] is the rate of release of volatilecomponent i, the mass balance becomes:

@ma

@t= �

Xi=CHn;CH4;H2

riVa;a (21.35)

Since there is no loss of mass from the �ller aggregate, mfc is constant duringbaking. Thus:

@mp

@t= �

3Xi=1

riVa;a (21.36)

By introducing mp = �r;pVr;p, one obtains:

@(�r;pVr;p)

@t= �

3Xi=1

riVa;a (21.37)

where �r;p is the real density of pitch. Also use Vr;p = Vr;a�Vr;fc, which leads to:

Vr;p = (1� ��;a � )Va;a (21.38)

21.2 The Open Porosity Model 329

Voids

Voids

Voids

Filler Cokewith

withBinder Coke

Isotropic

pitch Mesophase

Binder -(Semi) Coke

Anode

Binder Pitch Coke Filler Coke

Voids

which consists of

Filler Coke

Voids

Binder Pitch Coke

�a;fc

Va

Vr;fc

�r;fc

Vr;pc

mfc

�a;pc�r;pc

��;a

mpc

Figure 21.3: The anode volume consists of �ller coke, binder coke and gas �lledvoids. Each coke phase is modelled separately even though it is in reality verydi�cult to separate the two coke phases. By using the real density concept, amodel for open porosity of the anode is derived.

where =Vr;fcVa;a

. can be measured or calculated from raw-material parameters.

By using this expression for the pitch volume, the following equation for the totalporosity is obtained:

@(�r;p(1� ��;a � )Va;a)

@t= �

3Xi=1

riVa;a (21.39)

330 Porosity, Density, Permeability and Surface Area of Anodes

Resolved the equation for for ��;a gives:

@��;a

@t= �@

@t+

1

�r;p

3Xi=1

ri + (1� ��;a � )

�� 1

�r;p

@�r;p

@t� 1

Va;a

@Va;a

@t

�(21.40)

It can be shown that (Gundersen 1996d):

� The rate of volatilization jPrij is greater than the rate of change of the

anode control volume and the real density of pitch

���� 1Va;a

@@tVa;a

��� << ��� 1�r;p

@�r;p@t

���� @

@t is negligible since the �ller coke is thermally inert.

Therefore the following equation can be used to calculate the development of totalporosity of the anode during baking:

@��;a

@t=

1

�r;p

(3Xi=1

ri + (1� ��;a � )@�r;p

@t

)(21.41)

In thermogravimetric analysis/ pyrolysis of laboratory anodes, Tremblay & Charette(1988) related volatile release represented by rates ri to the total volume of theanode. In modelling pyrolysis of pure pitch, the pyrolysis rates rp;i is related tothe bulk (i.e. apparent) volume of pitch/ pitch coke. In Gundersen (1995b) kineticparameters from the literature was used to simulate pyrolysis of pure pitch. Touse this kinetics in simulations, the rate laws have to be speci�ed on an apparentvolume basis only taking into account the pitch volume. This gives the followingrelationship between the reaction rates:

nX1

riVa;a =

nX1

rp;iVa;p (21.42)

rp;i = ki�nb;a;i (21.43)

In Gundersen (1995b, pp. 42), �rst order reactions were used, and then the fol-lowing is valid:

rp;iVa;p = ~rp;iVr;p (21.44)

where ~rp;i is the reaction rate as related to the real volume of the pyrolysing pitch.Then, an expression for ri can be found:

nX1

riVa;a =

nX1

~rp;iVr;p

+

ri = ~rp;iVr;p

Va;a= ~rp;i(1� ��;a � )

21.3 The Transport Porosity Model 331

This expression for ri must be used in the porosity equation:

@��;a

@t=

(1� ��;a � )

�r;p

3Xi=1

~rp;i +@�r;p

@t

!(21.45)

By using this approach, the open porosity of the anode is estimated as a bulkproperty. It is not possible to distinguish between interconnected and induced closedporosity. The interconnected porosity (transport porosity) �f is needed in themass balance equations for volatile gases in the porous anode; only the transportporosity is available for mass transport. One could assume that the induced closedporosity contributes only to a small fraction of the open porosity. Then the errorintroduced by using open porosity as transport porosity is negligible. In this work,however, a model for the transport porosity is derived in the next subsection.

The quantity can be calculated from green anode parameters. Use Vr;fc =mfc

�r;fc

and mfc = ffcma;� to obtain:

=ffcma;�

�r;fc

1

Va;a= ffc

�a;a;g

�r;fc(21.46)

where �a;a;g is the apparent density of the green anode.

In the di�erential equation for the open anode porosity, two phenomena contributeto the formation of porosity: Loss of volatile material from the pitch (i.e. pitch cokeyield) and shrinkage of the residual pitch coke (i.e. the increase in real density).Then, porosity can be viewed as a function of coke yield and real density of thepitch. In an anode property model, one must decide whether to use anode openporosity or coke yield as a state variable. If coke yield is used as a state variable,the open porosity should be treated as a derived property. This is brie y discussedin Chapter 23.

21.3 The Transport Porosity Model

In the coke bed, the transport porosity corresponds to the interparticular porosity.An analogy can be found for the anode if the pitch partly surrounds the �ller cokeparticles causing the particle diameter to increase. The interparticular porosity(which is open porosity) is then an approximation of the transport porosity in theanode.

As before, assume that the intraparticular porosity Xai of the �ller coke aggregate

is given by:

Xai = Xa

i;� +Xai;c

where Xai;� and Xa

i;c denote open and closed intraparticular porosity of the cokeaggregate. It is assumed that the pore size distribution function is the same foreach coke �ller size fraction. In reality, however, the distribution function varieswith the size of the coke grain.

332 Porosity, Density, Permeability and Surface Area of Anodes

The open porosity ��;o of the anode is divided into transport porosity �f andinduced closed porosity �c;a;in. The occurrence of so-called blind pores (Marsh1989, pp. 155) which do not e�ectively contribute to the e�ective pore space isneglected.

Induced closed porosity is introduced during pitch impregnation (sealing of pores).Here, it is assumed that there is no physical and chemical changes of the �ller coke.Therefore, both intrinsic closed �ller porosity as well as the closed porosity due toimpregnation could be conserved in the baking process.

In summary, the apparent anode volume of coke aggregate impregnated with liquidpitch is the sum of the following volumes as shown in Figure 21.4:

1. Volume Vr;p occupied by pitch divided into:

(a) Volume Vr;p � �(1 � s)��;fcVa;a of pitch in the volume between theinterstices of the coke aggregate. � is a �lling factor that may changeduring baking. If � = 0, no pitch penetrates the open porosity ofthe coke. If � = 1, the open pores are completely �lled with pitch.This factor has to be experimentally determined. According to Lahaye,Ehrburger, Saint-Romain & Couderc (1987, pp. 190) not all coke porescan be penetrated by pitch: Pores with diameter below 5 to 6 �m arenot pitch penetrable. This has been coped with by assuming that onlya fraction (1�s) of open coke porosity is penetrable by pitch. Thereforevolume Vr;p��(1�s)��;fcVa;a of pitch reside in the space between cokeparticles. This pitch volume contains closed porosity (see item 7 below)and is also a part of a certain apparent volume of pitch which containsaccessible open porosity.

(b) Volume �(1 � s)��;fcVa;a of pitch �lling initial open porosity of the�ller coke. As mentioned above, a certain fraction s of the open poros-ity contains pores with diameter so small that the pore space is notaccessible for pitch penetration. Thus, only volume �(1 � s)��;fcVa;ais penetrated by pitch. This pitch volume also contains closed pores(see item 7 below) and belongs to a certain apparent volume of pitchwith inaccessible open porosity since the pitch volume is stuck in thepores of the �ller coke. This inaccessible open porosity is of magnitudeq �(1� s)��;fcVa;a

2. Volume Vr;fc occupied by the coke itself (real volume excluding porosity)5.

3. Volume s��;fcVa;a of induced closed porosity in the coke aggregate due tosealing of the smaller �ller coke pores (i.e. no pitch penetration due to asmall pore diameter). This induced closed porosity entraps a certain volumeof gases.

4. Volume (1� �)(1� s)��;fcVa;a of induced closed porosity in coke aggregatedue to partial �lling of �ller coke pores with pitch. Remember that only afraction (1� s) of open porosity of coke is penetrable by pitch. Within thisclosed porosity, a certain volume of gas is entrapped.

5Total real volume is Vr;a = Vr;p + Vr;fc, i.e. real volume of coke and pitch

21.3 The Transport Porosity Model 333

5. Free volume �fVa;a between coke particles not �lled with pitch. This volumeis due to incomplete compaction as well as free space in interstices betweencoke particles.

6. Volume �c;fcVa;a of intrinsic closed porosity of the �ller coke.

7. Volume �c;pcVa;a of intrinsic closed porosity of the binder coke.

��;fc and �c;fc are open and closed porosity of individual coke particles as relatedto the apparent volume of the anode. For the total porosity �T;a, the followingrelationship is valid:

�T;aVa;a = ��;aVa;a + �c;fcVa;a + �c;pcVa;a (21.47)

Here, �c;fc + �c;pc = �c;a. The open porosity is composed as follows:

��;aVa;a = �fVa;a + s��;fcVa;a + (1� �)(1� s)��;fcVa;a (21.48)

+q�(1� s)��;fcVa;a

The term q�(1 � s)�c;�Va;a represents the open porosity in the apparent volumeof pitch plugging the open (accessible) porosity of the �ller coke; residual porosity�r;a. As before, it is assumed that �r;a is negligible. Thus:

��;aVa;a � �fVa;a + s��;fcVa;a + (1� �)(1� s)��;fcVa;a (21.49)

Parameters ��;fc and �c;fc can be calculated from data for the coke aggregate:

�c;fcVa;a = Xai;cVa;fc

��;fcVa;a = Xai;�Va;fc

where Va;fc is the apparent volume of the �ller particles. For the open porosity ofthe anode, this gives:

��;aVa;a = sXai;�Va;fc + (1� �)(1� s)Xa

i;�Va;fc + �fVa;a

Also use:

Vr;fc = (1�Xai;�)Va;fc

where Vr;fc is the real volume of �ller coke particles. The following expression for��;a is obtained:

��;aVa;a = sXai;�

1�Xai;�

Vr;fc + (1� �)(1� s)Xai;�

1�Xai;�

Vr;fc + �fVa;a

From before, =Vr;fcV

which gives:

��;a = (sXa

i;� + (1� �)(1� s)Xai;�)

1�Xai;�

+ �f (21.50)

334 Porosity, Density, Permeability and Surface Area of Anodes

The transport porosity can be calculated from the above equation if the �llingfactor � is known. In this model, the transport porosity equals the interconnectedporosity of the anode.

If s = 0, the whole open porosity of the �ller aggregate is penetrable by pitch andan elegant expression for ��;a is obtained.

��;a = (1� �)Xa

i;�

1�Xai;�

+ �f (21.51)

Note the assumption that pitch penetration occurs both in the coke �nes and thecoarse particles. Therefore the transport porosity model is derived by consideringthe �ller coke aggregate Va;fc as a whole not taking into account the subdivisioninto �nes (dust) and coarse particles.

Closed Porosity

Intraparticular

Pitch SurroundingFiller Particles

Induced Closed Porosity

Partly FilledCoke Pore

Figure 21.4: Distribution of porosity in the anode control volume.

21.4 Permeability

Anode permeability is an important quality parameter of baked anodes. Anodereactivities depend on the gas permeability of the anode. The observation thatthe green mix behaves as if it had two phases (Martirena 1983) serve as basis forthe permeability model:

1. Large and medium sized �ller coke particles (�ller matrix).

21.4 Permeability 335

2. A mixture of coke �nes and coal tar pitch surrounding the larger particles(binder matrix).

Based on the packed bed analogy for anodes (Seldin 1959), a model for the viscouspermeability has been derived. The model is based on the Kozeny equation wherepermeability depends on the transport porosity �f and the hydraulic radius Rh;a

of the anode. Since the anode is a consolidated medium, the application of theKozeny equation for permeability calculations is not straight forward.

21.4.1 Review of Permeability Models

In studying uid transport in carbons, the Carman equation takes into consider-ation both viscous ow and the e�ect of slip ow and free molecular6 di�usion(Marsh 1989, pp. 188):

K =v1p1L

�p=B�

��p+

4

3K��v (21.52)

K, B� and K� are the total-, viscous and slip ow permeability coe�cients respec-tively7. �v is the mean thermal molecular velocity:

�v =

r8RT

�M(21.53)

The phenomenon of slip ow is due to the fact that molecules do not have zerovelocity at the capillary wall. Slip ow occurs in capillaries with diameter dccomparable to the mean free path � of the molecules. � is given by:

� =�

p

r�RT

2M(21.54)

As dc�

decreases, a transition from slip ow to free molecular ow occurs. Forordinary capillaries, slip ow and Knudsen di�usion occur only at very low pres-sures. However, many porous media have diameters comparable with the meanfree path even at atmospheric pressure. In general, the equation has to be usedwhen studying gas ow through carbon materials. For liquid ow, however, onecan set K� = 0 and thus keeping only the viscous term. Equation (21.52) is ap-plicable to porous media of any kind and can be used to obtain correct values ofB� when slip occurs.

In Peggs, Mill & Stadnyk (1976), the Carman-Arnell Equation (21.52) was com-bined with the Adzumi equation (Carman 1956, pp. 69) to obtain expressions forB� and K� in dependence of the number of capillaries per unit area of cross sectionin composite carbon bodies.

In consolidated media, the solid part forms a continuous and permanent struc-ture. Such media are often made from particles and pore textures are much more

6Also denoted Knudsen di�usion.7Their SI-units are m2/s, m2 and m respectively.

336 Porosity, Density, Permeability and Surface Area of Anodes

complex than those found in granular beds. Even when formed from particles, thepore texture depends on the way that the particles are cemented together as wellas size distribution and shape of the particles. Aggregation of particles occurs andeven in consolidated media of uniform pore texture the concept of tortuosity doesnot have a physical explanation: Experimental values of permeability are oftenlower than expected. This is mainly due to high tortuosities; tortuosity factorsbetween 1 and 10 has been experimentally determined. For consolidated media,B� and K� generally do not give information on pore texture. Also, the Kozenyequation is generally not valid in consolidated media.

A modi�ed Kozeny-equation was suggested to cope with slip ow in unconsolidatedmedia with a uniform pore texture. Still the porous medium is modelled as abundle of non-circular capillaries. By using:

B� =�d2ck��2

(21.55)

K� =�

k1�2�dc (21.56)

the following equation is obtained:

K =�d2ck��2�

�p+4

3

�

k1�2�dc�v (21.57)

The tortuosity factor � is usually set top2. According to Wyllie and Rose,

k� � 2:5 for all porous media (Carman 1956, pp. 51). For pure slip ow, k1 = 1.dc is the capillary diameter which can be replaced with the hydraulic diameterin unconsolidated porous media. The dimensionless factor � approaches constantvalues of �� and �1 corresponding to Knudsen - and slip ow respectively:

�� =2� f�

f�(21.58)

�1 =3�

16

2� f1

f1(21.59)

f� is the fraction of molecules undergoing di�use re ection at the capillary walls.f� � 1. f1 is also constant but depends on the properties of the gas and thesolid surface. f1 decreases from 1 through the transition range from Knudsen ow to true slip ow. The modi�ed Kozeny equation with speci�ed values ofk� and � is only valid for unconsolidated media with uniform pore textures andnormal tortuosity factors. The validity of the Kozeny equation breaks down inporous media of non-uniform pore textures since the concept of an equivalentmean hydraulic radius no longer exists. This also applies to the modi�ed Kozenyequation. Details on the derivation of the modi�ed Kozeny equation can be foundin Carman (1956, pp. 62-64,85) and Hutcheon & Warner (1958).

The modi�ed Kozeny equation also applies for consolidated media of uniform porestructure (Carman 1956, pp. 76). However, the tortuosity no longer have a phys-ical relevance. Introducing the hydraulic radius, the equation becomes:

K =�R2

h

k�q2��p+

4

3

�

k1q2�Rh�v (21.60)

21.4 Permeability 337

using q instead of the traditional symbol � for the tortuosity. Values of q between 1and 20 have been determined experimentally for consolidated bodies with uniformpore texture.

In studies of gas ow through graphite, Hutcheon & Warner (1958) and Hutcheon& Price (1960) argue that even though the range of pores in graphite are too largeto be considered uniform, a similar equation as Equation (21.60) should applyfor porous bodies. The meaning of the hydraulic diameter may change. Thisstatement cannot be physically proved but must rely on pure empirism. In theexperiments in Hutcheon & Price (1960), it was found that the in uence of slip ow could be neglected. Thus:

K � �R2h

k�q2��p (21.61)

Rh and q was determined from the experimental values of B� and k�. It was foundthat q � 4:0. Which average value of the hydraulic radius should be used in thecalculations on porous media of non-uniform pore texture? The correct expressionfor the viscous permeability is due to Wyllie and Spangler:

B� =� < R2

h >

k�q2(21.62)

where:

< R2h >=

1

�

Z �

0

R2hd� (21.63)

under the assumption that the factor k�q2 is the same for capillaries of all sizes

(Carman 1956, pp. 34,53). The integral can be evaluated by using data from mer-cury porosimetry pore size distribution curves. This mean value of the hydraulicradius is not the same as the one used in the ordinary Kozeny equation.

Based on Carman's equation for consolidated bodies with uniform pore size, Wiggs(1958) de�ned permeability as B = K �

�pwhich gives:

B =�R2

h

k�q2+4

3

�

�p

�

k1q2�Rh�v (21.64)

Based on Equation (21.53) and Equation (21.54) for the mean molecular velocityand mean free path respectively, one may �nd:

�

�p�v = �

r2M

�RT

r8RT

�M= �

4

�(21.65)

Substitution into Wiggs' equation gives:

B =�R2

h

k�q2+

16

3�

�

k1q2�Rh�

k1 = 1 and � = �1 for slip ow which gives:

B =�R2

h

k�q2+(2� f1)

f1

�

q2Rh� (21.66)

338 Porosity, Density, Permeability and Surface Area of Anodes

Typical values for f1 are between 0.8 and 1.0. Substituting Rh =dc4for the pore

diameter in Wigg's equation:

B =�d2c

16k�q2+ �

(2� f1)

4f1

�

q2dc (21.67)

Wiggs also realize that uniformity of pore size does not exist in porous carbons.Instead of assuming that the porous network consists of bundles of parallel (or se-rially connected) pores, Wiggs generalized a result by Maxwell for the resistivity ofa uniform matrix containing a fraction of spherical particles of di�erent resistivity.It was assumed that gas permeability is analogous with electrical conductivity. Agraphical technique was developed for solving:Z

1

0

(dc � �d)

(dc + 2�d)

�d�

d(dc)

�d(dc) = 0 (21.68)

where �d denote dv and dm which are (average) pore diameters in viscous andslip ow respectively. A graphical technique was used for obtaining dv and dmfrom cumulative porosimetry curves. Wiggs obtained satisfactory match betweenexperimental- and predicted values of permeability B:

B =�d2v

16k�q2+ �

(2� f1)

4f1

�

q2dm (21.69)

Using the permeability equation presented in Wiggs (1958), Born (1974a) obtaineda criterion for allowable baking rates in an anode ring furnace. For the baked anodesamples with pore sizes in the micron range, slip ow was responsible for about 10% of the permeability K. Thus, the contribution of slip ow could be neglectedin mass transfer calculations:

B =�R2

h

k�q2(21.70)

Rh = 4dv and dv is the average size of the pores responsible for viscous ow asdetermined by Wiggs' graphical technique.

The following summarized the discussion on the Carman-like equation so far:

� The modi�ed Kozeny equation is valid for consolidated media with a uniformpore texture. The interpretation of tortuosity is non-physical; large valuesof the generalized tortuosity q may occur.

� Experimental results support the extension of the theory to also apply toconsolidated media of non-uniform pore sizes. However, the hydraulic diam-eter concept does not necessarily have a physical interpretation in this case.Several techniques exist for calculating the proper mean hydraulic diame-ters to be used in the viscous- and slip ow permeabilities. The assumptionthat the pore space consists of bundles of capillary pores of di�erent sizeseems to dominate (Wyllie and Spangler). However, alternative techniquesfor determining pore size like those suggested by Wiggs also exist.

21.4 Permeability 339

Seldin (1959) studied permeability in baked carbons and beds of carbon powders.He rejected the Kozeny-equation as a model for the permeability of porous media.A model based on the theory of constriction resistances was developed to explainthe variation of permeability with particle size, �ller- and apparent densities.

This permeability model is based on two contributions:

� Permeability in a medium with closed packing

� Permeability in a medium for an expanded porous medium

The permeability was show to depend on the particle size dp;fc of the �ller cokeas follows:

kp / d43

p;fc (21.71)

This was valid also for carbons made from a mixture of two particle sizes if theweighed average value of particle size was used. It was shown that the size depen-dence of permeability was independent of binder content in the green mix whenbinder constitutes between 20 and 40 parts of the green mix. The permeability ofpacked beds turned out to be a power function of apparent density, or equivalentlythe porosity:

kp / �8 (21.72)

However, for a given apparent density, the permeability is lower as the bindercontent is lower. Seldin concluded that the permeability is more sensitive to asmall change in the number of constrictions per unit volume than to a smallchange in the sizes of the constrictions. By extrapolation of experimental data, itwas observed that for a �xed aggregate bulk density8, the permeability decreasesas the apparent density increases by adding binder coke. The permeability seemsto be zero for a certain apparent density ~�a which corresponds to the densityat which all constrictions between particles are �lled with binder coke. Seldinconcluded that baked carbons has a structure of continuous regions of closelypacked particles completely enclosing gaps or more loosely packed regions. Itwas stressed that the value ~�a could not be experimentally determined since theconstrictions can never be completely �lled with binder coke. Some objectionsagainst the Kozeny equation was stated:

� The porosity function in the Kozeny equation does not depend on particleshape.

� There is no reason to require zero permeability for zero porosity. Rathershould the closed packing porosity serve as a reference value for the perme-ability.

� The Kozeny equation does not lead to anisotropy in permeability.

� The particle size dependence (k / d2p;fc) is only approximately correct.

8Aggregate bulk density is denoted �ller density by Seldin (1959).

340 Porosity, Density, Permeability and Surface Area of Anodes

Seldin's model try to overcome some of these de�ciencies. It was stated thatin granular packed beds, the particles have to touch each other to be supportedand thus determine an average pore size which will always equal a fraction of theparticle size. Thus, the closed packing with porosity �cl is responsible for a certainpermeability kp;cl. Seldin assumed that:

kp;cl = ksnclhcl (21.73)

where ks is the permeability of a single pore cell in the closed packing and ncl and1hcl

are the number of pore cells per unit cross section of the packed bed and the

number of particle layers per unit length of the bed respectively9. Since:

ncl /1

d2p;fc

hcl / dp;fc

it was found that ks / d73

p;fc since kp;cl / d43

p;fc. Beds that are not optimally packed(porosity larger than in the closed packing) are equivalent to a closely packedbed where some of the particles have been removed. This suboptimal packing isresponsible for a second kind of pores much larger than in a closely packed bedand of a less de�nite size. Seldin suggested that the increase in permeability dueto the pores of second kind depends on the factor (���cl) where � is the porosityin the suboptimally packed system10. In describing the e�ect of factor (���cl) onpermeability, the statistics of the distribution of particles has to be introduced. Anexponential function was experimentally shown to be appropriate, and the �nalmodel had the structure:

kp / (�dp;fc)43 e

g(���cl)

(1��cl) (21.74)

where g is a factor which represents the number of additional regular size pore cellchannels of ow present when a particle is removed from a layer. Experimentallyit was shown that g � 11. Here � denotes a factor which takes care of the in uenceof binder on the reduction in the size of constrictions. Thus � is a monotonousfunction of the binder coke level in the porous carbon. The factor � was introducedunder the assumption that the statistical distribution of particles does not dependon the binder content. This was also observed experimentally: The permeabilitycurve was shifted to higher densities but remains similar in shape. Seldin gave nosuggestion for the shape of function � in dependence of the binder content.

9hcl is the distance between the layers.10Seldin only discusses open porosity by arguing that closed porosity easily can be corrected

for.

21.4 Permeability 341

21.4.2 The Ordinary Kozeny Equation for Non-uniform Pore

Textures

The Kozeny equation cannot directly be applied to non-uniform pore textures,since the mean hydraulic radius calculated from the equation is:

�Rh;K =1

1�

R �0

d�Rh

(21.75)

whereas the correct mean value to be used should be:

�Rh;mK =q< R2

h > =

s1

�

Z �

0

R2hd� (21.76)

Carman (1956, pp. 35) stressed that for the same type of pore-size distribution,the ratio between these two average values will be the same. It doesn't matter ifthe Carman factor k = k�q

2 has a "wrong" value since it can be used for tuningthe model to match experimental data of permeability. In general, if anothermeasure of mean hydraulic diameter �Rh;K is used, the ratio of mean hydraulicdiameters is constant provided that the pore size distribution remains unchanged.Now, assume that the viscous permeability B� has been experimentally measuredand �Rh;mK calculated from an experimentally determined pore size distribution(mercury porosimetry). If also the porosity � is known, a Carman factor can befound from Wyllie and Spangler's equation:

B� = ��R2h;mK

kmK) kmK = �

�R2h;mK

B�(21.77)

By using another average value �Rh;K one can also predict B�:

B� = ��R2h;K

kK(21.78)

Equating the two expressions for B� gives:

kK = kmK

�R2h;K

�R2h;mK

!(21.79)

where kK is the proper Carman factor to be used. Based on this realization, theordinary Kozeny equation can also be applied for a porous medium with non-uniform pore texture provided the Carman factor has been properly tuned. Itmust be stressed that in this case, �Rh;K does not necessarily have a physicalinterpretation as pore size.

21.4.3 The Chosen Approach

Since both the proportionality constant and the function � in Seldin's model areuncertain as well as the uncertainty in parameter g, a model based on the Kozeny

342 Porosity, Density, Permeability and Surface Area of Anodes

equation with a specially tuned Carman factor will be used. The permeabilitychanges during baking due to changes in the e�ective particle diameter of theanode �ller matrix and porosity. In the model, variations in hydraulic diameterand transport porosity contributes to the varying permeability:

kp;a =�fR

2h;a

kk(21.80)

where kk = k�q2 is a specially tuned Carman factor. An estimate of the size of kk

was obtained by using k� = 2:5 and "tortuosity factor" q � 1 for the anode.

For the model to be valid, the pore size distribution during baking has to beconserved. Most probably this will only be partly correct and an average pore sizedistribution has to be used too.

21.4.4 Models for Average Particle Diameter and Hydraulic

Diameter

The hydraulic diameter Dh;a (hydraulic radius Rh;a)11 depends on the transport

porosity �f and "mean particle diameter" da;p of the mixture of coke and pitch.The assumptions needed to model the hydraulic diameter in the anode were givenin Subsection 20.4.2. Further assume:

1. Pitch is uniformly distributed around the coke particles which gives �ctiveparticles of a certain diameter larger than the average coke aggregate diam-eter.

2. There is a negligible volume expansion of the coke aggregate when mixedtogether with pitch. This means that the particle density is more or lessconserved.

3. The average diameter for coke aggregate is much larger than the diameterof the primary QI particles in the pitch. According to Jones & Bart (1990),primary QI particles have diameters of approximately 1�m. Mesophaseparticles formed at approximately 400�C have diameters of approximately 0to 50 �m.

Filler Coke Particle Density

Based on measurement on the green anode and the �ller coke aggregate, an esti-mate of the number of coke particles per unit volume can be obtained. The volumeVa;fc of �ller coke particles constitutes a fraction �a;fc of the total anode volume:

Va;fc = �a;fcVa;a (21.81)

11The hydraulic diameter Dh may be replaced by the pore diameter if the pores have uniformpore diameter.

21.4 Permeability 343

Use Va;fc =1

(1�Xai;�

)Vr;fc and Vr;fc = Va;a and obtain:

Va;fc

Va;a=

(1�Xai;�)

(21.82)

and therefore:

�a;fc =

(1�Xai )

(21.83)

Let ��a;fc represent the volume fraction of the anode that does not contain �llercoke. Then:

��a;fc = 1� �a;fc (21.84)

��a;fc can be used as basis for de�ning the particle density �fc.

Now assume that there is a number dnfc of coke particles of average radius rp;fcin a di�erential volume dVa;a of the anode. If the particles are spherical is de�nedby the following relationship:

dnfc4

3�r3p;fc = (1� ��a;fc)dVa;a (21.85)

The particle density �fc is de�ned as �fc =dnfcdVa;a

, which gives:

�fc =6

�d3p;fc

(1�Xai;�)

(21.86)

An estimate of the particle density can be found by assuming that the whole �llercoke aggregate is be represented by an average particle diameter. On the otherhand, the �ller coke consists of qualitatively two particle sizes: Coarse and �neparticles as discussed in Subsection 20.4.2. It was assumed that only the coarseparticles constitute the �ller matrix. The mixture of coke �nes and binder pitchconstitutes the binder matrix. Therefore, only a fraction of Va;fc should be usedfor de�ning the particle density. An equation for the reduced volume of �ller coke~Va;fc can be found:

~Va;fc =1

(1�Xai;�)

~Vr;fc (21.87)

where

~Vr;fc =~mfc

�r;fc= (1� xf )

mfc

�r;fc(21.88)

where ~mfc is the mass of �ller coke which belongs to the �ller matrix. As before:

mfc = (1� fp;�)ma = (1� fp;�)�a;b�Va;a (21.89)

Since = (1� fp;�)�a;b��r;fc

, one �nally arrives at the following expression for~Va;fcVa;a

:

~Va;fcVa;a

= (1� xf )1

(1�Xai;�)

=1

(1�Xai;�)

~ (21.90)

344 Porosity, Density, Permeability and Surface Area of Anodes

where ~ = (1�xf ) . To account for the fraction xf of the �ller coke which belongsto the binder matrix, the quantity ~ should replace in the above presentedequations. Thus:

�fc =6

�d3p;fc

~

(1�Xai;�)

(21.91)

Particle Density and Particle Diameter

In the anode, transport porosity corresponds to the intraparticular porosity. In-spired by the particle density for the coke aggregate, an analogous particle densityconcept can be de�ned for the anode:

dna4

3�r3p;a = (1� �f )dVa;a (21.92)

The particle density of the anode is denoted �a = dnadVa;a

. As an approximation,

particle density is assumed to be conserved during mixing and molding the anode.Thus, �a � �fc

12 so that:

r3p;a =3

4�

1

�fc(1� �f ) (21.93)

This equation explicitly takes into account that a fraction of the uid pitch pene-trates the open porosity of the particles; this e�ect is represented by the transportporosity. The rest of the pitch is assumed uniformly spread over the surface areaof the coke particles. In contrast to this model for e�ective particle radius in theanode, Mrozowski (1956b, pp. 197) calculated the particle diameter without tak-ing pitch penetration into account. Substitution of �fc into the expression for rp;agives:

r3p;a = r3p;fc(1� �f )(1�Xa

i;�)

~

or:

dp;a = dp;fc3

s(1� �f )(1�Xa

i;�)

~ (21.94)

Here, ~ is used since only a fraction (1� xf ) of the �ller coke belongs to the �llermatrix.

The Hydraulic Radius

The hydraulic radius of the anode is:

Rh;a =dp;a

6

�f

(1� �f )(21.95)

12Alternatively, dnfc � dna.

21.5 Surface Area 345

The hydraulic diameter is related to the hydraulic radius by Dh;a = 4Rh;a, whichgives:

Dh;a =2dp;a3

�f

(1� �f )(21.96)

for the hydraulic diameter. Substitution of dp;a gives:

Dh;a =2dp;fc3

3

s�3f (1�Xa

i;�)

~ (1� �f )2(21.97)

21.5 Surface Area

The speci�c surface area has impact on the reactivity of the anode carbon. Nom-inally, the surface area of pores accessible for oxygen and carbondioxide decreaseswith the baking temperature. In some cases, post-calcination and subsequent pu�-ing (release of inorganic compounds) of the �ller coke may occur at high bakingtemperatures. This leads to formation of micropores and consequently increasedsurface area. Ideally, post-calcination should not occur. However, there is alwaysa risk of post-calcination which may severely deteriorate the reactive properties ofthe anode carbon.

Based on the model for pore size in an anode, an expression for anode surface areacan be found. For a pore with average diameter dc and length l, the pore surfacearea Ap and volume Vp are given by:

Ap = 2

�dc

2

�l

Vp = �

�dc

2

�2

l

(21.98)

Surface area Sp per pore volume is:

Sp =Ap

Vp=

1

dc(21.99)

If there are n pores per unit volume, the surface area Sv per unit volume V is:

Sv =nVpSp

V(21.100)

Here, nVp = �V and:

Sv =�V Sp

V=

�

dc(21.101)

The speci�c surface area Sg de�ned by:

Sg =Sv

�a=

�

dc�a(21.102)

346 Porosity, Density, Permeability and Surface Area of Anodes

For the anode, the speci�c surface area becomes:

Sg;a =�f

Dh�a;a(21.103)

since Dh � dc. According to Grjotheim & Welch (1988, pp. 92), the crystalliteorder and porosity increase during baking. At the same time, there is a reductionin the surface area of the anode as shown in Figure 21.5. In Gundersen (1996d),this trend in Sa is qualitatively veri�ed by using typical data for green and bakedanodes. But, as discussed below, the order of magnitude of the predicted surfacearea is far too low to be realistic.

800 1000 1200 1400

5

10

15

20

Baking temperature

BE

T S

urfa

ce a

rea

T [�C]

[m2

g ]Sg

Figure 21.5: Typical elapse of BET surface area as function of the baking temper-ature. From Grjotheim & Welch (1988, Fig. 4.7 p. 88).

Typical surface area of anodes obtained by mercury dilatometry is in the orderof 1m2=g corresponding to 1000m2=kg. The application of the Brunauer, Em-mett and Teller (BET) method gives surface areas in the range between 1000 and2000m2=kg .

In comparison to these numbers, our surface area model gives severe underes-timates since true speci�c surface area is approximately 2000 times larger thanactually predicted by the model. An increased value for the surface area can onlybe obtained with a reduced value for the hydraulic diameter. On the other hand,this deteriorates the predicted permeability. Thus, to achieve realistic estimatesfor both surface area and permeability, a uniform pore diameter valid for the wholepore volume of the anode cannot be used: Pores in the range downto 0.1 �m areusually considered to belong to the accessible pore volume and contributes to thereactive surface area. Pores with diameter above 50 �m belong to the permeabilityrange.

Pore diameters in the range of 10 �m belong to large macropores. If pores existwith diameters in the lower macro-, meso- and micro range, a typical pore diametermay lie in the range 10�6m downto 10�8m and even below 10�8m. If the porosityin this range is 0.1 and the e�ective pore doameter is set dp;fc = 5 � 10�8m, a

21.6 Summary of Model Assumptions 347

speci�c surface area of 1290m2=kg in the meso- and micropore range is obtained

with baked anode density 1550 kg/m3. The smallest pores contribute signi�cantly

to the total speci�c surface area if an almost equal distribution of the pore volumeamong the range of pore diameters is assumed. This is a good approximationfor the binder coke (Grjotheim & Kvande 1993, �g 4.10). Thus, the contributionof the largest macropores to the total speci�c surface area is almost negligible.Actually, to obtain a good model of speci�c surface area of the anode, a moresophisticated pore model for the pore distribution in the anode is needed.

21.6 Summary of Model Assumptions

The most important assumptions for the models presented in this chapter are asfollows:

� No segregation of anode coke-aggregate and pitch is assumed during heattreatment.

� Volume expansion of anodes during heat treatment is neglected.

� Swelling of pitch during baking is absorbed mainly by the coke interparticleinterstices (and partly by coke-intraparticle open porosity).

� Porosity introduced by incomplete compaction is neglected when determiningthe hydraulic radius of the anode.

� The anode paste is modelled as (a binary) mixture of particles of two sizes.

� A transport porosity and hydraulic radius is calculated by assuming no di-lation of the coke aggregate during the mixing process.

� Tortuosity is assumed to be equal top2.

� The transport porosity model:

{ The anode control volume is assumed to remain constant. The thermalexpansion of the �ller coke particles is neglected and the impact ofsecondary coking reactions as described in Fitzer & H�uttinger (1969) isnot taken into account.

{ The total porosity is modelled as a mass balance on the pitch content.

{ The �ller coke consists of a coarse and a �ne fraction with a commonpore size distribution. Only a fraction (1� s) of the pores is penetrableby pitch.

� The particle density - /permeability - / surface area models:

{ An average diameter dp;fc for the coarse �ller coke fraction is used. Thecoarse �ller coke constitutes the �ller matrix.

{ The �ne fraction mixes with the binder pitch to constitute the bindermatrix.

348 Porosity, Density, Permeability and Surface Area of Anodes

{ An average pore diameter based on a packed bed analogy for the hy-draulic diameter analogy is used to calculate both permeability andsurface area.

Chapter 22

Anode Crystallite

Parameters

Mason (1958) studied boundary scattering processes and argued that the harmonicmean value of crystallite size should be used. Thus, for the crystallite size, thefollowing model is recommended (Mason 1958, pp. 72):

1

La=

nXi=1

ni

La;i(22.1)

n is the total number of crystallite sizes La;i. ni is the proportion of crystalliteswithin phase i. If m is the number of coke phases, and La;j constitutes the averagecrystallite size of a certain coke phase, an alternate expression is:

1

La=

mXj=1

xj

La;j(22.2)

xj is the mass fraction of each phase.

Mason pointed out that general X-ray methods provide the harmonic mean esti-mates of crystallite sizes. Most probably, the correct estimate for crystallite sizelies somewhere between the arithmetic and harmonic mean values. Mason showedthat the binder phase may dominate the scattering processes in carbons. There-fore the harmonic mean value gives the most realistic estimate of crystallite sizein carbons with signi�cant di�erence between crystallite size of the carbon phases.This is reasonable, since the mean free path in boundary scattering is determinedby the smallest crystallites. On the other hand, since the size range of the crys-tallites in common carbon materials does not di�er very much, there is also littledi�erence between the two mean values. In this work, the harmonic mean valueof crystallite size is used.

It is assumed that the baked anodes consist of binder - and �ller coke exclusively.Also, the �ller coke is assumed to be thermally stable so crystalline structural

350 Anode Crystallite Parameters

changes occur only in the binder coke fraction. The crystalline parameters for thecomposite anode can be calculated as an weighed average value of the pure carbonphases present in the anode. The following expressions were found in Grjotheim& Kvande (1993):

La;a = xpcLa;pc + xfcLa;fc (22.3)

Lc;a = xpcLc;pc + xfcLc;fc (22.4)

for the arithmetic mean value. xpc and xfc denote mass fractions of binder and�ller coke respectively. In the same manner Li;pc and Li;fc denote the crystal-lite parameters where i = a; c. The corresponding harmonic mean value for thecrystallite size is:

1

La;a=

xpc

La;pc+

xfc

La;fc(22.5)

When La;pc and La;fc are close to each other, the two weighting formulas giveapproximately the same value for La;a. For large di�erences, however, the twopredictions may di�er signi�cantly.

In the same manner, the stacking height Lc;a of the bulk anode material can becalculated from:

1

Lc;a=

xpc

Lc;pc+

xfc

Lc;fc(22.6)

Chapter 23

Simulation of Anode

Properties

The physical properties of anodes depend on the corresponding physical propertiesof the �ller and binder coke phases. Previously, a simulation case for developmentof pitch coke (i.e. soft carbon) properties was presented. The same heat treatmentprogram and the corresponding physical properties of the pitch and pitch coke areused for calculation of the physical properties of the anode. Some of the modelsfor anode properties are presented in Gundersen (1996d). The rest of the modelswere discussed in the two previous chapters.

23.1 Overview of Properties

Plots of the following properties are presented:

� Open porosity ��;a

� Transport porosity �f;a

� Induced closed porosity �c;a1

� Anode grain diameter dp;a

� Permeability kp;a

� Speci�c surface area Sa

� Real density �r;a

� Apparent density �a;a

1In the porosity concept, the closed porosities within the pitch- and �ller coke phases of theanode is not included.

352 Simulation of Anode Properties

� Speci�c heat capacity cp;a

� Thermal conductivity ka

� Crystallite parameters La;a and Lc;a

� Electrical resistivity �el;a

� Young's modulus Ya

� Mechanical strength �a

� Chemical reactivity ri;a

� Thermal shock resistance Rts;a

23.2 Discussing the Simulation Results

For calculation of anode properties, parameters given in Gundersen (1996c, cht.10) as well as the following properties characterizing the binder pitch coke wereneeded:

� Real density �r;pc for the pitch coke

� Pitch coke yield cy

� Crystallite height Lc;pc for pitch coke

� Layer plane diameter La;pc for pitch coke

� Young's modulus Ypc for pitch coke

� Mechanical strength �pc for pitch coke

� Speci�c heat capacity cp;pc for pitch coke

� Thermal conductivity kt;pc for pitch coke

� Electrical resistivity �el;pc for pitch coke

Open porosity of the anode was calculated by using the following relationships:

�a;a = (1� (1� cy)xpc;�)�a;a;g (23.1)

��;a = 1� xfc�a;a

�r;fc� xpc

�a;a

�r;pc(23.2)

xfc + xpc = 1

�a;a;g is the apparent density of the green anode. xpc;� is the mass fraction of pitchin the green anode.

23.2 Discussing the Simulation Results 353

Temperature dependent data for �ller coke speci�c heat capacity cp;fc and ther-mal conductivity kt;fc are needed. The following thermally dependent polynomialfunctions for these properties (see Gundersen (1996c, cht. 3.26)) were used:

cp;fc = 175:03+ 20:04T � 6:0 � 10�4T 2 (23.3)

kt;fc = 1:3kcg;e (23.4)

Temperature in Kelvin should be used in the polynomial for cp;fc. kcg;e is givenin Gundersen (1996c, eqn. 3.119, 3.120).

It is assumed that the electrical resistivity of the �ller coke was 80 % of theminimum value of the electrical resistivity of the pitch coke. For a lot of the anodeproperties, only normalized values are presented:

� The reactivity index ra;i

� Young's modulus Ya

� Mechanical strength �a

� Electrical resistivity �el;a

� Thermal shock resistance Rts;a

As discussed in part I, chemical-, mechanical- and transport properties su�cientto describe anode quality. According to the simulations, these properties evolveas follows:

� Chemical properties: ra;i

{ The reactivity index decreases and thus improves the reactive propertiesof the anode. It should be noted that the e�ective reactivity also willdepend on the surface temperature of the anode.

� Mechanical properties: �a and Rts;a

{ Mechanical strength decreases and thermal shock resistance increases.Thus, mechanical strength deteriorates and shock resistance improves.

� Transport properties: �el;a and kt;a

{ Electrical resistivity decreases and thermal conductivity increases. Thedecrease in electrical resistivity is bene�cial. The increase in thermalconductivity may lead to increased surface temperature and subsequentairburn problems.

� Other properties: �a;a

{ The apparent density decreases during baking due to mass loss fromthe coking binder pitch (coke). Pitch coke yield should be as low aspossible.

354 Simulation of Anode Properties

In this case, the simulations show that almost all the mentioned properties stillimprove at the end of the heat treatment interval; the mechanical strength, how-ever, is about with decreasing. Thus, a trade o� exists in the design of the heattreatment program since some properties improve and other properties deteriorateat the end of the heat treatment cycle. Thus, anode quality may be subject tooptimization.

In the model, it is assumed that that no sulphur pu�ng takes place in the �llercoke.

0 50 100 1500

200

400

600

800

1000

1200Heat treatment program

Time [hr]

T [o

C]

Figure 23.1: Plot of the heat treatment program used in the simulation of anodeproperty development. From 20 to 500�C, the heating rate is 7�C=hr. From 500to 1200�C, the heating rate is 10�C=hr. Finally, there is a 10 hours hold time atmaximum temperature 1200�C . Total heat treatment time is 150 hours.

23.2 Discussing the Simulation Results 355

0 50 100 1500.14

0.16

0.18

0.2

0.22

0.24

0.26Total porosity

Time [hr]

phi_

T [1

]

0 50 100 1500

0.05

0.1

0.15Transport porosity

Time [hr]

phi_

f [1]

0 50 100 150−1

−0.5

0

0.5

1

1.5Closed porosity

Time [hr]

phi_

c [1

]

Figure 23.2: Development of porosities.

0 50 100 1501500

1520

1540

1560

1580

1600Apparent density

Time [hr]

rho_

a,a

[kg/

m3]

0 50 100 1501850

1900

1950

2000

2050Real density

Time [hr]

rho_

r,a

[kg/

m3]

0 50 100 15022

24

26

28

30

32Layer plane diameter

Time [hr]

La_a

[AA

]

0 50 100 15024

26

28

30

32

34Crystallite height

Time [hr]

Lc_a

[AA

]

Figure 23.3: Development of densities and crystallite sizes.

356 Simulation of Anode Properties

0 50 100 1501.12

1.14

1.16

1.18Relative anode grain diameter

Time [hr]

dp_a

/dp_

fc [

mm

]

0 50 100 1500.5

1

1.5

2

2.5

3x 10

−5 Hydraulic diameter

Time [hr]

Rh_

a [k

g/m

3]

0 50 100 1500

2

4

6

8x 10

−12 Permeability

Time [hr]

kp_a

[m2]

0 50 100 1508.2

8.3

8.4

8.5

8.6

8.7

8.8x 10

−4 Surface area

Time [hr]

So_

a [m

2/g]

Figure 23.4: Development of anode grain diameter, hydraulic radius, surface areaand permeability. Unfortunately, the current model for the surface area is not ableto calculate a realistic estimate of the surface area in the anode.

0 50 100 150500

1000

1500

2000Specific heat capacity

Time [hr]

cp_a

[J/

(kg

K)]

0 50 100 1502

3

4

5

6

7

8Thermal conductivity

Time [hr]

kt_a

[W/(

m2

K)]

Figure 23.5: Development of speci�c heat capacity and thermal conductivity.

23.2 Discussing the Simulation Results 357

0 50 100 1500.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Normalized reactivity index

Time [hr]

ra_i

./max

(ra_

i) [1

]

Figure 23.6: Development of the reactivity index.

0 50 100 1500.5

0.6

0.7

0.8

0.9

1Normalized mechanical strength

Time [hr]

S_a

/max

(.)

[1]

0 50 100 1500.4

0.5

0.6

0.7

0.8

0.9

1Normalized thermal shock resistance

Time [hr]

(S_a

*kt_

a/(Y

_a*C

TE

_a))

/max

(.)

[1]

Figure 23.7: Development of mechanical strength and normalized thermal shockresistance.

358 Simulation of Anode Properties

0 50 100 15010

−2

10−1

100

Normalized electrical resistivity

Time [hr]

log(

rho_

el_a

)/m

ax(.

) [1

]

Figure 23.8: Development of normalized electrical resistivity.

Chapter 24

Conclusions

Physical properties of carbon anodes are modelled as nonlinear transformations ofthe corresponding physical properties of the binder- and �ller coke phases of theanode. Derivation of models for single phase and composite carbons was discussedin two reports (Gundersen 1996c), (Gundersen 1996d). Models for chemical-,mechanical-, transport properties as well as other physical properties were derived.The results presented in this part of the work based on these reports.

Simulations show that the models give qualitatively correct prediction of boththe binder pitch (binder coke) properties and the corresponding anode properties.The models are suited for use in optimization of heat treatment programs usedin anode baking: A performance index for the baking process may be formulatedwhich directly takes into account anode quality. In the future, more frequentvariations in raw material quality is expected. Then this way of optimizing ringfurnace production may be an alternative to the traditional equivalent temperaturebased method since the new models allow for direct control of anode properties.

360 Conclusions

Part V

Modelling and Control of

Baking Furnaces

Chapter 25

Modelling the Baking

Process

In part I, descriptions of the design- and operation principles for ring furnaces weregiven in Section 4.1 and Section 4.2 respectively. In this chapter, a mathematicalmodel is derived from a simpli�ed process view by focusing on the ring furnacephenomena in a half cassette and the belonging gas path. The HAL-design is usedas basis for implementing the model. Measures and dimension were taken fromthe geometry of a ring furnace in �Ardal. The material presented in this chapter isbased on Gundersen & Balchen (1995).

25.1 Introduction

A model for the ring furnace system is established. A model of the baking processshould include descriptions of the following phenomena:

� Describe the thermodynamic state of the furnace (i.e. energy consumption)

� Give a description of the state of the anode blocks (i.e. product quality)

In this presentation, focus is put on modelling the thermodynamic state of thefurnace to perform heat balance calculations for solid materials (brickwork, cokebed and anodes) and combustion gas owing in the ues. Models for anode carbonproperties were discussed in Part IV

25.1.1 Modelling from First Principles

In process modelling, the evolution of composition (density), temperature andpressure in systems is described by applying the principles of:

364 Modelling the Baking Process

� Conservation of mass

� Conservation of energy

� Conservation of momentum

which may be transformed into a set of coupled equations for mass fractions,temperature and pressure. Equations are established for both the gas phase andsolid materials. General derivations of the equations of change may be found inBird, Steward & Lightfoot (1960), Fogler (1986), Kuo (1975), Froment & Bischo�(1990), and Munson, Young & Okiishi (1990). The equations are summarized inAppendix A.

The most di�cult part in modelling is to determine the complexity of the modelthat is needed to describe the important phenomena in the system and how toestablish proper boundary conditions for the �nal model.

Equations of change are often established by considering an in�nitesimal volumeelement of the system. Spatial integration of the equations of change give balanceequations which have only time as the dependent variable. A �nite set of coupledmacroscopic balance equations constitutes the �nal numerical model. Sometimesit is more convenient to give a lumped description of larger parts of a systemalready from the start of the modelling procedure. Both approaches are used inderivation of the model for the ring furnace.

25.1.2 Subprocesses in the Baking Process

To overcome the challenges in process modelling, the process needs to be dividedinto subprocesses. Modelling of each subprocess may be considered separately. Adescription of subprocesses in the ring furnace is given in the following.

Temperature distribution in the solid materials. During baking, the anodetemperature is the driving force for the chemical and physical transformationswhich take place in the pitch fraction and thus in uences the �nal anode quality.A reliable description of the temperature �eld based on an energy balance for thesolids is a prerequisite for achieving proper control of the baking process. Criticalparameters in this context is thermal properties such as speci�c heat capacityand thermal conductivity. Since both the packing coke bed which surrounds theanodes and the anodes themselves are porous media, an important heat transfermechanism at high temperatures (above 600�C) will be radiation. Thus, a model ofthermal conductivity which includes thermal radiation will be needed. Importantis also the amount of energy consumed in the chemical transformations takingplace in the pitch fraction. Boundary conditions for the outside surfaces of thefurnace (furnace lid etc.) has to include free convection.

Chemical transformations in the pitch. During baking, pyrolysis takes placein the pitch fraction. The petroleum coke is already calcined and thus can be

25.1 Introduction 365

assumed chemically and thermally stable. Thus, anode quality development isdue to transformations in the pitch. The pyrolysis model should give an estimateof the volatiles released as well as serve as a basis for the anode quality model asdiscussed below.

Evolution of anode properties. In the anodes, the pitch fraction is trans-formed to solid coke during baking. In parallel with this transformation, thedevelopment of the anode properties occur. The anode quality parameters andother parameters (heat capacity, density) should be implemented in a separatesubmodel.

Mass transfer in anodes and packing coke. The volatiles which are releasedduring pyrolysis, escape from the anodes via the packing coke to be combustedin the ues. This mass transfer phenomenon probably occur as viscous ow andpartly di�usive ow and should be studied in detail to get an impression of thespatial distribution of volatiles in the combustion ues. Here, volatile combustionis assumed to take place in the vertical ues of a closed furnace.

Gas ow phenomena. The Navier- Stokes equations govern the gas ow in thefurnace. In the ues, the gas ow is considered one-axial or the gas compartmentsare considered ideally mixed with uniform temperature.

Combustion of fuel oil, volatiles and packing coke. The main energysources are fuel oil and volatiles. Both fuel oil and volatile combustion mustbe taken into account to achieve a realistic description of the spatial dependentconversion of energy along the ues. Fuel oil is introduced into the combustioncompartments by the use of impulse type oil-burners. Volatile combustion can beconsidered instantaneous except for the tar-fraction for which a rate law frequentlyis used. Packing coke consumption is most important in the cooling part of thebaking cycle since during cooling, the access of O2 is the highest. In this model,packing coke combustion is ignored.

Gas temperature. The ue gas carries the energy of the system. A gas energybalance which takes into account both heat transfer with the solid materials aswell as heat losses to the surroundings must be developed. An important partof the gas energy balance is the combustion submodel discussed above. At hightemperatures, radiation becomes an important phenomenon in the energy balance.Radiation calculations are based on the grey gas approximation and the use of wellknown electric circuit analogies for description of the radiation heat uxes.

Draught pressure. A model for the draught pressure and its in uence on thegas ow through the sections goes via an impulse balance for the gas as discussedabove.

366 Modelling the Baking Process

Air inleakage Air inleakage gives a decreased thermal e�ciency of the ringfurnace. Thus, air inleakage should be kept on a minimum level. Defects inthe furnace brickwork and inadequate insulation at peepholes and the connectionbetween section-foundations and -lid are the main sources for air inleakage. Forcalculation of the inleakage ow, the gas impulse balance combined with di�erentvariants of the ori�ce equation for nozzle uid ow may be used.

General Comments Modelling is done from �rst principles in combination withsemi-empirical techniques when necessary. First principles modelling goes in detailinto the chemical and physical reality of the process. By rigorously sticking to thebasic physical laws governing the subprocesses, usually a model based on nonlinearpartial di�erential equations (NLPDE) will appear. A nonlinear state space modelcan be derived by discretizing the NLPDEs according to the method of lines. Firstprinciples modelling is very attractive due to the following facts:

� Nonlinearities give a realistic description of the phenomena

� The state space description has a physical meaning

25.2 System Decomposition and Modelling Strat-

egy

A ring furnace is a challenging system to model due to a complex constructiongeometry as well as the complexity of the physical and chemical transformationswhich occur in the binder pitch during baking. A ring furnace process model shouldbe derived with basis in a systematic division of the process into tractable units.Within these units, subprocesses can be identi�ed and quali�ed simpli�cations bemade. A block diagrammatic description of the approach to modelling the ringfurnace is given in Figure 25.1. As shown, modelling of the gas- and solid phasesare considered separately. The modelling approach is described below.

25.2.1 Main Decomposition of the Ring Furnace System

In a ring furnace, two subsystems interact with each other: The ue gases exchangeenergy with the solid materials of brickwork, packing coke and anodes. Based on asuggestion by Stevenson (1988, pp. 307), the following decomposition of a sectionin a vertical ring furnace can be done:

1. Combustion ues with owing gases

2. Ring furnace solid materials:

� Brickwork ue walls

� Packed bed of granular coke and green, partly carbonized or bakedanodes

25.2 System Decomposition and Modelling Strategy 367

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atio

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rack

ing

Pyr

olys

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(pac

king

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e &

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Figure 25.1: Schematic description of the ring furnace modelling approach.

368 Modelling the Baking Process

Coke Anode

Volatilegeneration

Brickwalls

Flue

Channel

hc

v�g �s

_dmvTg

dV

Ts;o

q00r = �Fgs(T4g � T 4

s;o)

Fgs =1

1�s+ 1�g�1

q00c = hc(Tg � Ts)

q00 = q00c + q00r

Figure 25.2: Ring furnace system decomposition. In the �gure, q00 denotes heat ux.

As shown in Figure 25.2, the main interaction mode between these systems is dueto energy transfer. Furthermore, mass transfer occur as transport of the releasedvolatile gases to the ues. This transfer of mass represents an important source ofenergy during baking, since the volatile gases are combusted in the ues.

Characteristics of the Gas Path

In the ues, combustion gases interact in multicomponent transfer of mass, energyand momentum. Volatile gases which escape during pitch pyrolysis are consumedin combustion reactions along the gas path together with fuel oil (supplied in theheadwalls and under-lid).

Due to the geometric structure of the gas path through a furnace section, a con-venient division of the gas path into six zones is shown in Figure 25.4:

� Zone 1: Headwall (in part A)

� Zone 2: Under-pit in part A

25.2 System Decomposition and Modelling Strategy 369

� Zone 3: Vertical ue channels in part A

� Zone 4: Under-lid

� Zone 5: Vertical ue channels in part B

� Zone 6: Under-pit in part B

The same technique was used by Bourgeois et al. (1990) in modelling the gas pathof a conventional closed furnace (Riedhammer type ring furnace). Details of onepit with ue channels seen from the xy- and yz-planes can be seen in Figures 25.3and 25.5 respectively. There is a dividing wall approximately in the middle ofthe section in the under-pit zones. The dividing wall is positioned normal to thegas ow and separates part A and part B. The wall's purpose is to force the gascoming from the headwall in part A to enter the vertical ue channels in part A.Thus, in the ues of part A the gas ow is vertically upwards. After entering theunder-lid region, the gas is forced to ow downwards in the vertical ue channelsin part B and �nally to enter the under-pit region in part B. Thus, the terms partA and part B is used to describe the parts of the pit with vertically upward anddownward gas ow respectively.

Along the gas path, mass is supplied to the bulk gas ow due to inleaking air andreleased volatile gases from the pitch pyrolysing in the anodes. Convective andradiative heat transfer between solids and gases occur at the brick surface in eachzone.

Length

Packing Coke

Anodes

x (0,0)

Part B Part A

Brick/Refractory

Fire Direction

Width y

Vertical Flue Channel

y1

Lx;3Lx;4 Lx;2 Lx;1

Ly;3

Ly;2

Ly;1

Figure 25.3: Pit seen from the xy-plane. Each section contains in the order of5 to 7 adjacent pits. A dividing wall in the under-pit region forces the gas owvertically upwards in part A of the pits. Each section is covered by a lid whichforces the gas to ow downwards in the vertical ues of part B. Note the de�nitionof coordinate axes.

370 Modelling the Baking Process

Part APart B

Under pit-

Under Lid - Zone

Zone

Headwall

z

x

z

x

z

FlowGas

Foundation

y

Lid

Zone-Pit

T

Tg

Tl

Tf

T

Figure 25.4: Pit seen from the xz-plane. The coordinate system in the pit region isshown along with the local coordinate systems used for calculation of heat lossesthrough foundation and furnace lid. It should be noted that the origin in thepit coordinate system lies in the bottom of the headwall in part A at the surfaceof the ue wall on the right hand side of a cassette when viewed in the positivex-direction.

Characteristics of the Solid Subsystem

So far, the ring furnace has been divided into a gaseous- and solid subsystemcorresponding to the ue gas path and the solid materials of brickwork, coke andanodes respectively. A gaseous phase also exists within the anode- and packingcoke parts of the ring furnace.

Within the subsystem of solid materials, the situation varies depending on themedium considered. The brick may be considered impermeable and passage of thevolatile mass uxes across the brickwork occur through cracks and joints in thewalls. Therefore, we only need to take into account traditional heat conduction inthe brickwork. Thermal properties (thermal conductivity, speci�c heat capacity)for the brickwork characterizes the heat conduction process.

Both in the coke bed and anode, the �ller coke material is considered chemicallyinert during heat treatment. On the other hand, energy- and mass transfer occuras a multicomponent and multiphase phenomenon in both the coke bed and theanodes:

� Coke Bed : In the permeable bed of solid coke particles, convection anddi�usion of carbonization gases occur. Thus, a gaseous and solid phase mustbe considered. Furthermore, the gas is a mixture of several gas components.Thus, the coke bed is a two phase system.

25.2 System Decomposition and Modelling Strategy 371

Brick

Packing Coke

Anodes

Gas Flow

Width

Height z

y

y (0,0) 1

(Part A)

Vertical Part of

Flue Channels

Lz;3

Lz;2

Lz;1

Figure 25.5: Pit seen from the yz-plane. The vertical ue channels correspondto the situation in part A of the pit since the gas ow is vertically upwards. Inpart B, gas ow in the ues occurs vertically downwards. Note the de�nition ofcoordinate axes.

� Anodes : The pitch material consists of thousands of components and upto three phases of pitch may be present simultaneously: Fluid pitch, pitchvapour, carbonization gases and mesophase/pitch semicoke. Convection anddi�usion of carbonization gases occur in the anode. Each of the three phasesconstitutes a multicomponent mixture. To simplify the situation, pitch inthe liquid is still associated with the solid phase. Thus, also the anode isconsidered to be a two phase system of solids (�ller- and binder coke) andvolatile gases. The carbonization gases constitute a multicomponent mixture

372 Modelling the Baking Process

in which only components CHn (tar), methane and hydrogen are taken intoaccount.

Thus, for both the coke bed and in the anodes, a "solid" and a gaseous phase existwhich communicate via both energy and mass transfer. The mass transfer pathsbetween the gaseous and solid subsystems are shown in Figure 25.7.

From the pitch fraction of the anodes, volatile gases are released in carbonizationreactions. These gases are mainly heavy polynuclear aromatic hydrocarbons, lighthydrocarbons (methane) and hydrogen. These gases escape from the solid phaseinto the void fraction of the porous anodes. In the voids, secondary coking reac-tions (of heavy hydrocarbons) may lead to deposition of coke on the surface of theporewalls. Thus, interphase mass transfer may occur between both the solid andthe gaseous phase in the anodes.

In part III, pitch pyrolysis was modelled in detail from �rst principles and semiem-pirical approaches. Based on the pitch pyrolysis model, the anode consists of thefollowing components:

� Solid phase:

{ Filler coke: Cfc

{ Binder pitch:

� Gamma-resins: 1 and 2� Beta-resins: �

� Alpha-resins: �p and �s1

� Pitch coke: Cpc which consists of coke fractions C1 and C2

� Gaseous phase:

{ Tar volatiles: CHn

{ Methane: CH4

{ Hydrogen: H2

In this chapter, however, a more simple view is used for modelling the pyrolysisprocess by assuming that the pitch fraction of the anode contains certain reser-voirs of volatile components that are released during heat treatment by parallelconsumption of these reservoirs.

The packing- and �ller coke contents in the coke bed and anodes are consideredthermally inert. Since carbonization gases are transported across the coke bed,there is a chance that secondary coking reactions can occur in the gas phase. Thismay lead to deposition of coke on the surface of the coke particles. In the coke bedthen, interphase mass transfer may occur only from the gas- to the coke phase.

For both the coke bed and anodes, a momentum balance can be established in thegas phase for calculation of the gas pressure. The structure of the subsystems ofbrickwork, coke bed and anodes is summarized in Figure 25.6.

1�s consists of subfractions �s;CH4and �s;H2

capable of generating methane and hydrogenrespectively.

25.2 System Decomposition and Modelling Strategy 373

:

Single phaseheat conduction

Multicomponent gas phase

Solid coke phase

Multicomponent gas phase

Solid

Pitch

Pitch

1

2

PitchLiquid

Multicomponent Combustion

FluesSolid

materials

Coke Bed

gas phase

S

G

G

Vap

oriz

edG

S

S G

phasecoke Solid

S

Ring Furnace

Brickwork Anodes

Gas phase

Figure 25.6: In the subsystems of the ring furnace, several phases are present.Some of the phases are multicomponent mixtures (i.e. the gas phases and thepitch phase). The letter "S" denotes a solid subsystem and the letter "G" denotesa gaseous subsystem.

25.2.2 Modelling Strategy

During baking, the anode charge acts as a heat sink since energy is needed inthe chemical transformations of the binder from pitch into coke. In modelling the ue gas heat balance, three energy sources and sinks are considered (Bourgeoiset al. 1990):

� Heat sources:

1. Fuel-oil

2. Pitch volatiles

3. Packing coke

� Heat sinks:

1. Solid materials of brick, coke and anodes

2. Exhausted combustion gases

3. Heat losses to surroundings via cover, section sidewalls and foundation

374 Modelling the Baking Process

WallsFlue

Anodes

VoidsCoke

Filler coke

Binder coke

Flues

GS

GS

Brickwork

GFuel oil

Air inleakage

Packing coke

Voids

S

S

S

Figure 25.7: Mass transfer paths in the ring furnace system. Devolatilization ofthe pitch component in the anode creates gases which pass through the porousanode and packing coke bed to enter the combustion ues. The letter "S" denotesa solid subsystem and the letter "G" denotes a gaseous subsystem.

Here, only fuel oil and pitch volatiles are used as energy sources since the coolingpart of the anode baking cycle is not modelled, and packing coke combustion ismost signi�cant during cooling. The heat balances are established via �rst prin-ciples modelling. The model for generation of volatiles, however, is an empiricalmodel from the literature. Volatile transport in the anodes and the coke bed is notmodelled, and the released volatiles are instantaneously combusted in the ues.

For the heat balances of the gas and solid materials, several thermally dependentproperties are needed. For the solid materials, data were partly found in theliterature. Average gas property data along the gas path where calculated fromwell known correlations and tabulated values for properties of the gas components.

25.3 Chemical Reactions in the Solid Materials 375

Due to the large time constants in the solid temperature �eld, a dynamic heatbalance is calculated for the solids. On the other hand, gas phenomena have smalltime constants, and a stationary model gives satisfactory results as well as reducedcomputing time (Stevenson 1988, pp. 307).

The modelling of heat conduction in the packing coke and anodes is done for ahalf cassette only under the assumption that the xz-plane through the center ofthe anodes is an adiabatic plane and that all pits experience approximately thesame heat treatment.

25.3 Chemical Reactions in the Solid Materials

During baking, thermal decomposition of the pitch fraction of the anodes takesplace.

25.3.1 Pyrolysis of Binder Pitch

The use of a calcined petroleum coke implies that no or at least negligible chemicaland physical changes occur in the �ller coke during baking since the baking tem-perature should be lower than the soaking temperature used in coke calcination2.The purpose of the baking process is to convert the coal tar binder pitch into highquality binder coke which holds the �ller coke particles together.

Mathematical models for pitch pyrolysis and the accompanying development ofphysical properties of the pitch and corresponding pitch coke material were pre-sented in parts III and IV of this work.

In the model presented in this chapter, focus is put on calculation of the heatbalance of the ring furnace. Therefore, a traditional model of pyrolysis with ki-netics derived from measurements of gas release dynamics in thermogravimetricexperiments is used (Tremblay & Charette 1988).

Pyrolysis Chemistry

The pitch softens at approximately 100�C. Volatile hydrocarbon compounds va-porize and liquid crystal growth takes place between approximately 300 and 600�C.At temperatures above 600�C, further crystal growth takes place in the solidi�edpitch fraction under the release of methane and hydrogen. The carbonizationprocess �nishes between 1100 and 1300�C. The volatiles that escape during car-bonization are usually divided into two groups:

1. Condensable hydrocarbons (mainly polycyclic aromatic hydrocarbons; tar)

2. Non-condensable hydrocarbons (mainly methane and hydrogen)

2One should also be aware of the signi�cant e�ect of the longer soaking time used in anodebaking.

376 Modelling the Baking Process

Simpli�ed Pyrolysis Model

Frequently, thermogravimetric data is used for estimation of the kinetic parametersof pyrolysis. Here, a kinetic model3 derived by Tremblay & Charette (1988) is used.The fractional conversion of volatile compounds are traditionally modelled withArrhenius-like expressions:

dXi

dt= ki(1�Xi)

ni (25.1)

ki = k�;ie�

EiRT (25.2)

Xi is the fractional conversion of volatile component i where i denotes tar, methaneor hydrogen. k�;i and Ei denote apparent preexponential factor and apparentactivation energy respectively. ni is the apparent reaction order. Relevant datafor the anode baking process was taken from Tremblay & Charette (1988). Basedon the equations for dXi

dt, an expression for the volatile emission rate rv;i [kg/(m

3 s)]is as follows:

rv;i = f�;i�a;a;gdXi

dt(25.3)

f�;i is the initial mass fraction of volatile component i in the green anode. �a;a;gis the apparent density of the green anode.

In this case, the rate laws are related to the apparent volume of the anodes. Thissuggests that the anode mass can be viewed as consisting of a thermally inert�ller coke phase and thermally active components capable of releasing hydrogen,methane and tar during pyrolysis. The reaction scheme can be represented by:

A =

8>>>><>>>>:

Ac

ACHn

kCHn! CHn

ACH4

kCH4! CH4

AH2

kH2! H2

(25.4)

Ac = Afc +Apc (25.5)

Ac denotes the coke content of the anode which consists of thermally inert �llercoke and binder coke.

An equation for the density �b;i of the components in the anode can be found byde�ning:

�b;i;� = f�;i�a;a;g (25.6)

�b;i = �b;i;�(1�Xi) (25.7)

�b;i;� is the bulk density of the components in the green anode. Then:

�b;i;�(0)dXi

dt= �d�b;i

dt(25.8)

3The model is based on the use of simultaneous thermogravimetry and gas chromatographyto measure weight loss curves of tar, hydrogen, and methane.

25.4 Combustion Reactions in the Gas Phase 377

since �b;i;� is constant. Introduction of the expression for dXi

dtgives:

� d

dt(�b;i) = �b;i;�ki(1�Xi)

ni (25.9)

Now de�ne the conversion parameter Xi = (1� �b;i�b;i;�

), and obtain:

d�b;i

dt= ��b;i;�ki

��b;i

�b;i;�

�ni(25.10)

The apparent density of the anode is given by:

�a;a = �a;c +

nvXi=1

�b;i (25.11)

�a;c = �a;fc + �a;pc (25.12)

�a;fc = fp�a;a;g (25.13)

�a;pc = (1� fp)(1�X

f�;i)�a;a;g (25.14)

is the volume fraction of �ller coke in the green anode. �r;fc is the real density of�ller coke aggregate. �a;pc is the apparent density of the thermally inert fractionof the binder pitch.

25.4 Combustion Reactions in the Gas Phase

25.4.1 Introduction

Several complex combustion processes generate heat in the ring furnace:

� Turbulent combustion of fuel oil takes place under the cover and in theheadwall.

� Volatile gases generated during pitch pyrolysis are burned in the ue gas dueto gas temperatures above the ignition point of the volatiles.

� Combustion of packing coke takes place under the cover. Combustion de-pends on coke temperature, ash content and granularometry as well as thesurrounding gas temperature.

It is important to know the amount of heat which comes from each combustionsubprocess to be able to calculate the total heat balance for a section. In industrialcombustion processes, one should make sure that combustion takes place in excessair to obtain complete combustion and thereby optimal utilization of the fuel. Ina ring furnace, the content of oxygen in the gas ow has to be shared by all thesections in the �retrain (also remember that there is a certain amount of newoxygen entering the ues via inleaking air). A satisfactory O2 pro�le should bemaintained along the gas path.

378 Modelling the Baking Process

Impulse type oil burners with variable pulse frequency are used in the headwalland under the lid. There is no steady ow of oil into the combustion chambers;the pulse operation causes both the temperature �eld and other properties (likegas composition etc.) to oscillate.

Two burners are used under the lid. The burners �re normal to the bulk gas owacross the under-lid compartment. This introduces turbulence and good mixingof the gases. Up to four impulse type burners are used for fuel oil combustion inthe head wall. Oil burners are symmetrically positioned between cassette walls.

Both waste oils and heavy fuel oils are used for the oil burners in the ring furnace.The heavy fuel oil used, may be assumed to be of type no. 5. The waste oil islighter and can approximately be considered as type no. 3 or 4. The sulphurcontent of the oils is quite low.

In the preheating end of the zone, concentrations of approximately 2000 ppm(0:2%) CO has been measured. Incomplete combustion of volatiles causes forma-tion of CO. Non-ideal combustion of volatiles and fuel oil may occur in the ringfurnace. Here, however, ideal combustion is assumed.

25.4.2 Combustion of General Hydrocarbons

As a �rst approach, complete combustion was assumed. For complete stoichio-metric oxidation of a hydrocarbon CxHy, the following relationship holds:

CxHy + (x +y

4)O2 ! xCO2 +

y

2H2O (25.15)

By using this equation, mass balances along the gas path for the combustion gascomponents were calculated.

25.4.3 Combustion of Volatiles

Dernedde et al. (1986) studied the kinetics of volatile combustion. The maximumevolution rates for methane and hydrogen occur at temperatures above gas ignitiontemperature. Therefore, we assume that CH4 and H2 combustion rates equal theevolution rate from the carbon blocks, since volatile di�usion through the anodeand packing coke was neglected. As a preliminary approach, complete combustionwas also assumed for the tar fraction. Volatile combustion was assumed to takeplace in the vertical ues due to injection of pitch volatiles through cracks andjoints in the brickwork. Ignition temperatures and heats of combustion for thevolatiles are given in Table 25.1. In the following, a general description of volatilecombustion is given.

Combustion of Methane and Hydrogen

The most simple approach is to allow for complete combustion of both methaneand hydrogen.

25.4 Combustion Reactions in the Gas Phase 379

Component i Ignition temperature Heat of combustionTig �H[�C] [J/kg]

Condensables CHn 425� 50 �39� 106

Methane CH4 630 �50� 106

Hydrogen H2 575 �120� 106

Table 25.1: Heat of combustion for volatiles in pitch pyrolysis.

Combustion of hydrogen is represented by the equation:

H2(g) +1

2O2(g)! H2O(g) (25.16)

Hydrogen reacts with oxygen to form water. The heat of combustion for this re-action is �2:42� 105 J/mol.

Combustion of methane is represented by the equation:

CH4(g) + 2O2(g)! CO2(g) + 2H2O(g) (25.17)

Carbon dioxide and water is formed in the reaction. The heat of combustion forthis reaction is �8:03� 105 J/mol.

Rate laws for these reactions can be found in Perron, Bui & Nguyen (1992):

ri = �k�;ie�EiRT �g;i�g;O2

(25.18)

where �g;i and �g;O2are the mass concentrations of gas component i and oxygen

in the ue gas respectively.

Combustion of Tar

According to data supplied in Charette et al. (1990)4, one can assume that themolar mass of the tar gas is 232 g/mol. We also assume that the general formulaof the tar molecules is (C6H6)n and �nd that n � 3:0 for the tar molecules whichgives an average formula of C18H18 for the tar. Thus, x = 18 and y = 18 inEquation (25.15). It may be argued that this molecule has a too high hydrogencontent.

Dernedde et al. (1986) assume that tar combustion occur in two steps: The rapidoxidation of tar CxHy to carbon monoxide and hydrogen followed by the slowoxidation of carbon monoxide to carbon dioxide:

CxHy(g) + (x

2+y

4)O2(g) ! xCO(g) +

y

4H2(g) +

y

4H2O(g) (25.19)

CO(g) +H2O(g) ! CO2(g) +H2(g) (25.20)

4A mean value of molar masses of the aromatic hydrocarbons was determined.

380 Modelling the Baking Process

Hydrogen may further react with oxygen according to the single step reactionsuggested previously:

H2(g) +1

2O2(g)! H2O(g)

Dernedde assumed the following rate-law for combustion of tar:

�rCHn= �kfcCHn

c12

O2c12

H2O+ kbcCO2

cH2(25.21)

or

�rCHn= �kf

�cCHn

c12

O2c12

H2O� kb

kfcCO2

cH2

�(25.22)

where ci are concentrations of the gas components. The rate law is expressed inmol/(cm3 s). kb

kfis the equilibrium constant de�ned as:

Keq =kb

kf=cCHn

c12

O2c12

H2O

cCO2cH2

(25.23)

The �rst term in the rate law represents the forward reaction. The second termrepresents the water-gas shift reaction, which can be neglected if the reactionis far from equilibrium. Dernedde argues that the backward reaction cannot beneglected in the preheating zone of the ring furnace. Still, if the backward reactionis neglected, a reduced value for the reaction rate constant may be used to obtaina realistic global reaction rate:

�rCHn= �k0fCCHn

C12

O2C

12

H2O(25.24)

k0f = k0f;�e�

ERT (25.25)

k0f;� = 1:8� 1013 and E = 31500 cal/mol was used by Dernedde et al. (1986)5.

For simplicity, the overall reaction obtained by addition of the three reaction stepsabove is used:

C18H18 + 22:5O2 ! 18CO2 + 9H2O (25.26)

A single step reaction with a rate law of the type given in Equation (25.18) wasalso used by Perron et al. (1992) for combustion of tar in a model for petroleumcoke calcination. In such a kiln, however, O2 may be more abundant than in asection in a ring furnace when the tar combustion is active.

Conclusion

� Instantaneous and complete combustion seems to be a good approximationfor hydrogen and slightly more weak for methane.

� A rate law should be used for combustion of tar.5The values of the preexponential factor ko and the activation energy E was selected from

the literature and adjusted to available data from a ring furnace which burned 99 % of the tar(Dernedde et al. 1986).

25.4 Combustion Reactions in the Gas Phase 381

25.4.4 Combustion of Fuel Oil

Combustion of fuel oil droplets is controlled by heat transfer for vaporization of theoil as well as di�usion of oxygen to the ame core. The chemical reaction in itselfis instantaneous. A di�usion ame model that can be used to describe industrialcombustion processes was developed by Spang (1972). The general nature ofspray combustion is discussed in Williams (1976, pp. 1-5). A burning spraydi�ers from a premixed, combustible gaseous system in that it is not uniform incomposition. The lack of uniformity in the unburnt mixture results in irregularitiesin the propagation of the ame through the spray and it is di�cult to give ageometric description of the combustion zone. Palmer (1974, pp. 380) discussesreaction times for hydrocarbon fuels at normal pressures. Average reaction timesfor fuel oils is as follows:

� Heavy fuel oils (carbon forming) : 100 ms

� Light fuel oils (large drops) : 10 ms

� Light fuel oils (small drops) : 1 ms

Since fuel oil no. 5 ranges between light and heavy oil, the reaction will be com-pleted within 100 ms. The di�usion processes and reaction may be consideredinstantaneous. Therefore, combustion processes are modelled as stationary.

It was assumed that a fuel oil with approximate stoichiometric formula6 CH1:8 anda lower heating value of 42000 kJ/kg was representative for typical oils used foroperation of the furnace. The following single step reaction model is representativefor the stoichiometry in the combustion reaction:

CH1:8 + 1:45O2 ! CO2 + 0:9H2O (25.27)

25.4.5 Combustion of Packing Coke

In Grjotheim & Welch (1988), calcined coke is described to consist of 98 % to 99% pure carbon by weight. Impurities (like ash) contribute to less than 0:5% andhydrogen to less than 0:1% of the coke mass. The amount of hydrogen in calcinedcoke is negligible and the packing coke can be assumed to consist of pure carbon.A model for combustion of packing coke is derived in Gundersen (1996b). Thismodel is not discussed here, however, since packing coke combustion mainly takesplace during the cooling period of the baking cycle.

25.4.6 Assumptions in Combustion Modelling

Combustion is assumed to occur under the following assumptions:

� O2 is in excess in the combustion zones.

6C:H � 87:13 by mass

382 Modelling the Baking Process

� The impact pulsing behaviour of the burners is neglected. Average mass ow of fuel oil is used for calculation of gas composition and the amount ofcombustion energy supplied.

� Presence of SO2 in the combustion gas is neglected.

� Complete and instantaneous combustion of fuel oil occur in the headwallpart A and under-lid part B. Combustion is assumed to occur in turbulent ames. Turbulence causes good mixing of the gas and the combustion cham-bers can be treated like well stirred reactors with uniform temperature andconcentrations7.

� Combustion of methane, hydrogen and tar is assumed to be complete andinstantaneous.

� Heat from combustion of packing coke is neglected since packing coke com-bustion mainly takes place in the cooling period of the �re cycle.

� For temperatures below 1400 K, dissociation of combustion products is nottaken into account8. For simplicity, dissociation is neglected in this work.

25.4.7 Stoichiometry of Combustion Reactions

To keep track of consumption and formation of components due to chemical reac-tions (i.e. combustion reactions) it is useful to establish a matrix of stoichiometriccoe�cients which can be used during derivation of mass- and energy balance equa-tions in the system. n gas components interact with each other via m chemicalreactions. Then the net rate of formation per unit volume of component i is:

~ri = (ST r)i (25.28)

~ri is the net rate of formation of component i. r is a vector of reaction rates. Inthe ring furnace, �ve reactions and eight gas components are taken into account:

� Reaction no. Rj (j = 1; : : : ; 5):

{ R1: Combustion of hydrogen as represented by Equation (25.16)

{ R2: Combustion of methane as represented by Equation (25.17)

{ R3: Combustion of tar volatiles as represented by Equation (25.26)

{ R4: Fuel oil combustion as represented by Equation (25.27)

{ R5: Combustion of packing coke

� Component no. i (i = 1; : : : ; 8):

1. c1: Nitrogen N2

7For this assumption to be valid, the entering gas ow (and air supply ow) may have su�cientmomentum to justify the assignment of constant values for temperature and concentration ofgases in the combustion chambers (Hottel & Saro�m 1967, pp. 459).

8If dissociation occur, this should be restricted to CO2 and H2O.

25.4 Combustion Reactions in the Gas Phase 383

2. c2: Oxygen O2

3. c3: Carbon dioxide CO2

4. c4: Water vapour H2O

5. c5: Methane CH4

6. c6: Hydrogen H2

7. c7: Tar gas C18H18

8. c8: Fuel oil CHn

The stoichiometric coe�cients in each reaction can be put into the matrix S whereeach line corresponds to a reaction and each column corresponds to a component.

In general, the rate laws are nonlinear expressions. In the case with full conversionof the fuel within a combustion chamber of volume V , the e�ective rate expressionis:

~ri =Gf;i

V(25.29)

where Gf;i is the mass ow of fuel (i.e. key component) in reaction Rj and V isthe volume of the combustion chamber.

When total conversion of the fuel (oil and gases) is assumed, the combustion gasonly contains N2, O2, CO2 and H2O; i.e. no non-consumed fuel is present.

CompoundReaction Reaction rate

r1r2

rj

rm

s1;1s2;1 s2;2

s1;2

c2c1

j

rj cnci

sj;i

sm;n

Figure 25.8: The matrix of stoichiometric coe�cients.

384 Modelling the Baking Process

25.5 Mass Transfer in the Ring Furnace

Mass transfer processes play an important role in many subsystems in the ringfurnace:

� Transfer of mass from the carbonizing pitch to the void fraction of the anodes.

� Convection and di�usion of pitch volatiles and carbonization gases in theporous anode and coke bed.

� Possible transfer of mass from the gas phase in the porous anodes and cokebed to the solid surfaces due to secondary coking reactions.

� Transfer of oxygen and reaction products from the reaction zone in combus-tion reactions.

Convective mass transfer dominates along the combustion ues. Therefore, thediscussion is limited to mechanisms of interphase mass transfer between the solidand gaseous phase and intraphase mass transfer in the gas phase within the porousanodes and packing coke bed.

25.5.1 Interphase Mass Transfer in the Anodes and Coke

Bed

During baking, evaporation of light pitch components (condensables) and releaseof carbonization gases (methane and hydrogen) as reaction products from thecarbonization reactions occur at temperatures between 300 and 1000�C.

A fundamental description of the mass transfer of condensables from the liquidpitch phase to the gaseous phase has been developed previously: From considera-tion of liquid - vapour equilibria and two-�lm theory, an expression for the mass ow rate of condensables was found (Gundersen 1995b, eq. (3.120)). It was alsoshown that a volume based rate law could be used to represent this mass trans-fer. Thus, a di�usion type of process was represented by a volume reaction rate.This will also be done here, but in this case with an even more simple model; i.e.the reaction rate for generation of condensables in the model due to Tremblay &Charette (1988). The rate of release of tar is given by:

~rCHn= ~r00CHn

Sa;a�a;a (25.30)

where Sa;a and �a;a denote speci�c surface area and apparent density of the anoderespectively.

The transport of methane and hydrogen from the bulk pitch volume of the anodesto the void fraction of the anodes is also a kind of di�usion process. Possiblehold-up of carbonization gases within the bulk pitch volume is neglected and itis assumed that carbonization gases are released directly into the void volumein the anodes. Since methane and hydrogen are released in volume reactions in

25.5 Mass Transfer in the Ring Furnace 385

the pitch phase, the mass transfer from pitch to voids of carbonization gases canbe represented by the reaction rate expressions for generation of methane andhydrogen. The rates of release of methane and hydrogen are denoted ~rCH4

and~rH2

.

The rate of release of volatiles can be calculated by:

~ra;i = ( ~Sara)i (25.31)

where ~Sa is a stoichiometric matrix for the reactions in the anode (i.e. pitchfraction) and ra is a vector of reaction rates. i denotes the volatile component.

The rate terms can be used to represent the interphase mass transfer from thesolid phase to the gaseous phase of the anode as follows:

@

@ (�aja;i; ) = ��ara;i (25.32)

@

@ (�aja; ) = �

X�ara;i (25.33)

The partial derivative is taken with respect to the interphase space coordinate.ja;i; are ja; are mass ux terms. The interphase mass ux is represented by asource term. The mass transfer in the opposite direction (i.e. from gas to anode)is the given by:

@

@ (�gjg;i; ) = � @

@ (�aja;i; )

@

@ (�gjg; ) = � @

@ (�aja; )

In this view of the mass transfer processes, the volatile gases reside in the solidvolume after reaction and are transferred to the gas �lled voids by a kind ofdi�usion process. This view is re ected in Figure 25.9.

There is also the possibility that condensation and secondary coking reactions mayoccur in the carbonization gases which reside in the void space within anodes andthe packing coke. This may lead to deposition of coke on the solid surfaces in theanode and coke beds. The mass transfer from the gaseous to the solid phase canbe modelled in the same manner as before:

@

@ (�gjg;i; ) = ��grg;i (25.34)

@

@ (�gjg; ) = �

X�grg;i (25.35)

rg;i = ( ~Sgrg)i (25.36)

rg is a vector of reaction rates in the gas phase. Here, @@ (�aja;i; ) represents the

fraction of gas component i which is lost via interphase mass transfer.

386 Modelling the Baking Process

Pitch

PitchIsotropic

Solid phase

Entrapped Gas

Anisotropic

ANODE

Gas

Fillercoke

�s;H2

�s;CH4

C1

C2

�p

CHn

CH4

H2

Cfc

1

H2

CH4

CHn

2

�

Figure 25.9: Interphase mass transfer from the solid to the gaseous phase of theanode. It is assumed that the transfer of carbonization gases in the pitch fractionto the void fraction goes via an entrapped gas phase which belongs to the solidphase (i.e. pitch and �ller coke) of the anode. In this way, the volatile componentsexists in both the solid and the gas phase and the entrapped gas may be consideredto be a reservoir of volatiles which are released from the pitch.

The total interphase mass transfer is a sum of the contributions from solid phaseand gas phase reactions. Thus:

@

@ (�aja;i; ) = ��ara;i + �grg;i (25.37)

@

@ (�aja; ) = �

X�ara;i +

X�grg;i (25.38)

In this work, secondary coking reactions are neglected.

25.5 Mass Transfer in the Ring Furnace 387

25.5.2 Mass Transfer in the Gaseous Phase

Below, the mechanisms that are active during mass transport in the gas phase inthe void fraction of anodes and coke bed are described.

In a multicomponent one phase medium, mass is transferred by convection anddi�usion. For component i, this is represented as follows:

�ivi = �iv + ji

(25.39)

vi is the velocity of component i. jiis the di�usion ux of component i. v is the

mass average velocity de�ned as:

v =1

�

nXi=1

�ivi (25.40)

� =

nXi=1

�i (25.41)

In a owing multicomponent medium, several mechanical forces which act on themedium tend to introduce separation of the components (Bird et al. 1960, cht. 18).In this work, only the concentration gradients contribution to the mass di�usion ux j

iis considered; i.e. Fickian type of di�usion. Then:

ji= De;ir�i (25.42)

in a constant density system. De;i is the e�ective di�usion coe�cient in the mul-ticomponent di�usion process. De;i takes into account the coupling between thedi�usion uxes in the multicomponent mixture; derivation of expressions for De;i

is not straight forward.

In a system consisting of several (here two) phases, mass transport of componenti in phase k is governed by the equation:

�k�k;iV k;i = �k�k;ivk;i + ~|k;i

(25.43)

where:

�k�k;ivk;i = �k�k;ivk + jk;i

(25.44)

Thus:

�k�k;iV k;i = �k�k;ivk + jk;i

+ ~|k;i

(25.45)

V k;i is the velocity of component i. vk is the phase velocity or interstitial velocity

jk;i

is the intraphase di�usion ux of component i within phase k9. ~|k;i

denotes

the interphase mass transfer vector. Now de�ne:

jk;i

= �k�k;ivk;i;d (25.46)

~|k;i

= �k�k;i~vk;i (25.47)

9Note that the sum over all components of the intraphase di�usion uxes equals zero.

388 Modelling the Baking Process

In this work, the interphase uxes are represented by volume based rate expressionsas follows:

r~|k;i

= ~rk;i (25.48)

As before, the interstitial- or phase velocity vk is de�ned by:

vk =1

�k

nXi=1

�k;ivk;i (25.49)

�k =

nXi=1

�k;i (25.50)

In this context, the super�cial velocity10 vs;k is de�ned as:

vs;k = �kvk (25.51)

The super�cial velocity is the average linear velocity of the phase if no other phaseswere present in the system; i.e. the super�cial velocity is related to the total crosssection of the ow direction.

Di�usion mass transfer in the porous coke bed and anodes is modelled in Gun-dersen (1996a). Based on the Stefan-Maxwell equations, an expression to be usedfor determination of the di�usion coe�cient matrix can be found. For simplic-ity, one often uses the Wilke formula in multicomponent di�usion calculations(Wilke 1965). Details on the Stefan-Maxwell and Wilke approaches will not begiven here.

25.5.3 A Simpli�ed Model for Mass Transfer in the Anodes

and the Coke Bed

In this study, no e�ort is done to model three dimensional volatile transport inthe anodes and packing coke bed. To justify a simple approach to the calculationof the mass ux of volatiles which enter the combustion gas phase, the followingassumptions were used:

1. The dominating gradients (temperature and pressure) occur in the yz-plane.Therefore mass transport is assumed to occur along the y-axis neglectingpossible mass ux components parallel with the x- and z-axes.

2. Transport time for the volatiles can be neglected.

3. Secondary coking reactions along the gas path in the solid anodes and cokebed can be neglected.

4. Partial combustion of the volatiles in the coke bed (or anodes) are neglected.

10Also denoted Darcy velocity, �lter velocity, averaged velocity etc.

25.6 Heat Transfer in the Ring Furnace 389

In this approach, the volatile generation rate in the xz-plane through the anodesis spatially integrated along the y-direction. This integrated generation rate isdirectly transferred to the corresponding part of the vertical ues of the pit forimmediate combustion. The mass ux of volatiles entering the ues at coordinate(x; y; z) = (x; 0; z) is then found from:

�w00i (x; 0; z; t) =

Z Ly;3

Ly;2

~rs;i(x; y; z; t)dy (25.52)

x 2 [Lx;2; Lx;3]

z 2 [Lz;1; Lz;2]

Ly;1 and Ly;2 denote the positions of the surface and center xz-planes respectivelyof the anodes (see Figure 25.3 and Figure 25.5 for speci�cation of the domain overwhich the spatial integration is performed). �w00i (x; z; t) denotes the average mass ux of component i across the xz plane. At present, it is di�cult to estimate themodelling error introduced via this approximation. The net rate of accumulationof the volatile component in the gas phase depends on the combustion reactionrate of the volatile component.

25.6 Heat Transfer in the Ring Furnace

25.6.1 Heat Transfer Modes

In general, heat transfer occurs in three di�erent modes as follows (Incropera &DeWitt 1990, cht. 1):

1. Conduction: Heat conduction is associated by the microscopic activity ofatoms or molecules in matter. Conduction heat transfer is the transfer ofenergy from the more energetic to the less energetic particles due to interac-tions between the particles. Since energy is associated with a temperature,the driving force in conduction is a temperature gradient as expressed inFourier's law. Conduction heat transfer is in general more pronounced insolids than in liquids and gases.

2. Convection: Convection heat transfer occurs in owing uids as a super-position of two mechanisms: Energy is transferred due to random molecularmotion of the molecules (di�usion) as well as macroscopic (bulk) motion ofthe uid11. Fluid motion may be due to external means (forced convection)or the result of di�erences in uid density caused by temperature gradients(free convection). Of special interest is the heat transfer which occurs be-tween a owing uid and a solid surface when the uid and surface havedi�erent temperatures. Boundary layer theory is used for studying this phe-nomenon.

11Advection refers to heat transfer exclusively due to the bulk motion of uid.

390 Modelling the Baking Process

3. Radiation: In radiative heat transfer, energy is transported by electro-magnetic waves (photons). Transfer of energy by conduction or convectionrequires the presence of a material matter but radiative heat transfer is moste�ective in vacuum. The maximum radiative heat ux which can be emittedfrom a solid depends on the temperature as given by the Stefan-Bolzmannlaw.

Transfer of heat which due to energy released in chemical reactions is not discussedhere.

25.6.2 Heat Transfer Processes in the Ring Furnace

Heat transfer in the ring furnace involves all three heat transfer modes. Fourdi�erent situations can be classi�ed:

1. Heat transfer between a owing gas and surfaces of the solid materials(mainly brickwork- and the coke bed surface)

2. Heat transfer between solid surfaces

3. Heat transfer within the solid materials

4. Heat transfer within the gas

A certain heat ux q00 [W/m2] is associated with each of the situations.

Heat transfer of the �rst kind occurs both in the combustion ues as well as in theporous coke bed and anodes. The important heat transfer modes are convectionand radiation.

Heat transfer between solid surfaces occurs as radiation due to temperature dif-ferences between the surfaces. This occurs along the gas path as well as in theporous coke bed and anodes.

Within the solid materials, heat conduction is the most important mode of heattransfer. In a porous solid, radiation will also occur but radiative heat transferis usually incorporated in the conductive mode by the application of an e�ectivethermal conductivity. In principle, heat conduction also occurs in the gas phasebut the conduction mechanism plays a minor role in gases.

To simplify the picture, the following assumptions are made:

� Convection heat transfer in the porous coke bed and anodes due to gas which ows in the connected pore space is neglected due to the fact that gas owoccur at very low velocities12.

12The heat transfer coe�cient depends on the gas velocity via the Nusselt- and Reynoldsnumbers. Also, temperature di�erences between gas and solid surface is assumed to be low.

25.6 Heat Transfer in the Ring Furnace 391

� Radiation heat transfer between solid surfaces and between gas and surfacesin the porous coke bed and anodes is taken into account by the applicationof an e�ective thermal conductivity in Fourier's law for the conduction heat ux.

� Heat conduction is not considered in the gas phase since convection andradiation are the more dominating heat transfer modes in gases.

The remaining heat transfer processes and corresponding heat uxes (see Fig-ure 25.10) is due to radiation between gas and surfaces. This heat transfer occursas simultaneous convective and radiative heat transfer or exclusively as radiativeheat transfer as discussed in the next subsection.

Heat transfer (convection and radiation) between gas and surfaces:

� q00g;hwd: Heat ux between the gas and the surfaces in the headwall in partA and B (i.e. q00g;hwd;A and q00g;hwd;B).

� q00g;up: Heat ux between the gas and the surfaces in the channel under thepit (furnace foundation and under-pit surface) in part A and part B (i.e.q00g;up;A and q00g;up;B):

{ q00g;up;p: Heat ux between the gas and the under-pit surface in theunder-pit channel.

{ q00g;up;f : Heat ux between the gas and the foundation surface in theunder-pit channel

� q00g;chn: Heat ux between the gas and the brickwork in the ue channels inpart A and B (i.e. q00g;chn;A and q00g;chn;B).

� q00g;ulid: Heat ux between the gas and the surfaces in the under-lid compart-ment (coke bed surface and inner surface of lid):

{ q00g;ulid;c: Heat ux between gas and the top surface of the coke bed.

{ q00g;ulid;l: Heat ux between gas and the inner surface of the lid.

� q00g;a: Heat ux between outer surface of the lid and the ambient air.

Each heat ux between gas and brickwork surfaces is due to both convection andradiation:

q00 = q00c + q00r (25.53)

where q00c and q00r represent convective and radiative heat ux respectively. In theunder-pit region, the heat ux is the sum of a heat ux to the foundation and a ux to the bottom surface of the pit:

q00g;up;A = q00g;up;A;p + q00g;up;A;f (25.54)

q00g;up;B = q00g;up;B;p + q00g;up;B;f (25.55)

392 Modelling the Baking Process

Subscripts p and f denote pit and foundation respectively. In the under-lid com-partment, the heat ux is the sum of a heat ux to the top surface of the coke bedand the inner surface of the lid:

q00g;ulid = q00g;c + q00g;l (25.56)

Heat transfer (radiation) between surfaces:

� q00s;hdw: Heat ux between the surfaces in the headwall

� q00f;p: Heat ux between the foundation and the surface under the pit in partA and part B

� q00s;chn: Heat ux between the surfaces in the ue channels in part A and partB

� q00c;l: Heat transfer between the top surface of the coke bed and the innersurface of the lid

These heat uxes are exclusively due to radiation heat transfer. In principle, thecon�guration of the heat uxes between surfaces in the headwall and the uechannels is very complex. A certain assumption was used to avoid the analysis ofthe radiation heat transfer between these surfaces; see the subsection on radiationheat transfer.

25.6.3 Conduction Heat Transfer

The conduction heat ux is proportional to the temperature gradient as expressedby Fourier's law (Incropera & DeWitt 1990, pp. 4, 45):

q00 = �krT (25.57)

The proportionality constant is a transport property (material constant) known asthe thermal conductivity [W/mK]. The minus sign indicates that heat transportgoes in the direction of decreasing temperature.

In the ring furnace, thermal heat conduction is the dominating mode of heattransfer. Furthermore, the resistance to conduction within the solid materials ismuch bigger than the resistance to convection across the boundary layer betweenthe bulk gas ow and the brickwall surface. Therefore a non-uniform temperaturedistribution develops within the solid materials (i.e. the Biot number for theprocess is large (Incropera & DeWitt 1990, pp. 230)). The spatial dependence ofheat conduction is therefore taken into account as follows:

� Brickwork:

{ Flue walls: Three dimensional heat conduction

{ Furnace foundation: Two dimensional heat conduction

{ Furnace lid: One dimensional heat conduction

� Anodes and packing coke: Three dimensional heat conduction

25.6 Heat Transfer in the Ring Furnace 393

Gas

Part B Part A

q00g;up;A

q00g;chn;A

q00g;l

q00g;c

q00g;a

q00g;chn;B

Tl

Tf

Tp = T

Tgq00g;hdw;A

q00g;up;B

q00g;hdw;B

Figure 25.10: Heat uxes from gas to solid surfaces along the gas path.

25.6.4 Heat Transfer Between Combustion Gas and Solid

Surfaces

Since the temperature range of the combustion gases is between 100 and 1300�C,both convective and radiative heat transfer play a role. At low temperatures, theconvective heat transfer mode is the most dominant one. However, at temperaturesabove 500�C, radiation becomes important dominates at high temperatures. Ingeneral, convection and radiation contribute to the heat ux:

q00 = q00c + q00r

where q00, q00c and q00r denote total, convective, and radiative heat uxes respec-tively with unit W=m2. Convective and radiative heat transfer is discussed in thefollowing two subsections.

25.6.5 Convection Heat Transfer

Convection heat transfer is calculated from:

q00c = hc(Tg � Ts) (25.58)

394 Modelling the Baking Process

where q00c is the convective heat ux from gas to solid material. Tg is the gastemperature and Ts is the surface temperature of the brickwork. An equation ofthis type was established for each position along the gas path:

q00c;hdw = hc;hdw(Tg � Thdw) (25.59)

q00c;upit = hc;up(Tg � Ts(x; y; 0)) + hc;up(Tg � Tf ) (25.60)

q00c;chn = hc;chn(Tg � Ts(0; y; z)) (25.61)

q00c;ulid = hc;ulid(Tg � Tl(0)) + hc;ulid(Tg � Ts(x; y; Lz)) (25.62)

q00c;ga = hc;ga(Tg � Ta(1)) (25.63)

hc;i denote the heat transfer coe�cients along the gas path. Thdw is the averagebrickwall surface temperature which is used in the heat transfer calculation in theheadwall. Ts denotes the temperature in the pit. Tf , Tl and Ta is the foundation,lid and ambient temperatures respectively.

Along the gas path, the Reynolds number for the gas ow is in the order of 104

which shows that forced convection takes place in the turbulent regime. Convectiveheat transfer coe�cients were calculated from correlations for the Nusselt number:

Nu =hcDh

kg(25.64)

hc is the convective heat transfer coe�cient, Dh is the hydraulic diameter of thegas-duct and kg is the gas thermal conductivity. The hydraulic diameter wascalculated as follows:

Dh = 4cross section of gas duct

wetted perimeter(25.65)

i.e. four times the hydraulic radius of the gas duct. The hydraulic diameter ande�ective heat transfer area were determined for each region along the gas path.

A correlation attributed to Pethukov et. al. was used for the Nusselt number(Incropera & DeWitt 1990, pp. 457) in the headwall, under pit channels and the ue channels:

Nu =f8RePr

1:07 + 12:7( f8)(

12 )(Pr

23 � 1)

(25.66)

This correlation is valid for:

0:5 < Pr < 2000 (25.67)

104 < Re < 5 � 106

Re is the Reynolds number, Pr denotes the Prandtl number. f denotes the Moodyfriction factor. The correlation is widely used for turbulent ow. Heat transfercoe�cients were calculated from mean values of the gas properties.

For the under-lid compartment, the Nusselt number was calculated from a corre-lation valid for an array of slot nozzles (Incropera & DeWitt 1990, pp. 398-405,

25.6 Heat Transfer in the Ring Furnace 395

eq. (7-76)):

Nu =2

3A0:75r;�

0@ 2Re

ArAr;�

+Ar;�Ar

1A

2=3

Pr0:42 (25.68)

In this case, the array of nozzles corresponds to the uewalls with serially ar-ranged ue channels. Ar;� and Ar are the reference and e�ective nozzle area ratioscalculated from:

Ar;� =

(60 + 4

�H

2W� 2

�2)�1=2

(25.69)

Ar =W

S(25.70)

W is the width of the slot array (i.e. ue channel). S is the distance between slotarrays (i.e. ue channels). H is the e�ective under-lid height calculated from:

H = hl =Vulid

Ac(25.71)

where Vulid is the volume of the under-lid compartment and Ac is the surfacearea of the coke bed (including cross section areas of the ue walls). Incropera &DeWitt (1990) lists several restrictions for the use of this correlation that will notbe reported here. The geometric arrangement of individual ue channels as wellas the curvature of the furnace lid was neglected in the calculations.

For the external lid surface, an appropriate correlation for the Nusselt number forlaminar free convection was used (Incropera & DeWitt 1990, eq. (9.32), pp. 506):

NuL = 0:27Ra1=4L (25.72)

Lc =As

P(25.73)

RaL = GrL Pr (25.74)

RaL =g� (Ts � Ta;1)L

3c

��(25.75)

� =1

2(Ta;1 + Ts) (25.76)

�g =kg

�gcp;g(25.77)

� =�g

�g(25.78)

Lc is the characteristic length of the surface with area As and perimeter P . NuLand RaL are Nusselt's and Rayleigh's numbers respectively. � is the volumetricthermal expansion coe�cient (here given for an ideal gas). � is the thermal dif-fusivity of the gas and � is the kinematic viscosity. Ts is the surface temperatureand Ts;1 is the gas temperature outside the boundary layer. In this case, the gaswhich surrounds the lid is air. The correlation is valid in the range:

105 < RaL < 1010

396 Modelling the Baking Process

Thus, forced convection occurs along the gas path through the furnace and freeconvection takes place above the outer surface of the furnace lid.

25.6.6 Radiation Heat Transfer

Solid Body Radiation Properties

A black body emits the maximum amount of radiation energy. Planck's law givesthe variation of the monochromatic (spectral) emissive power Eb� [W/m3] of ablackbody with the wavelength:

Eb� =C1

C2eC2�T � 1

(25.79)

The total amount of radiative energy per unit area and time (i.e. energy over allwavelengths) which leaves a surface at absolute temperature T is called the totalemissive power. If the surface is a blackbody, the total emissive power denoted Eb[W/m2] is obtained by integration of Eb� over all wavelengths:

Eb = �T 4 (25.80)

� is denoted the Stefan Boltzmann constant.

Radiation properties describe the interaction between radiation energy and thesurface of a material. A surface or a medium may emit, re ect, absorb or transmitradiant energy. In general, radiation properties are functions of:

� wavelength (spectral properties)

� direction (directional properties)

� temperature

� geometric and physical characteristics of the radiant surface or medium

If radiation properties are averaged over all wavelengths and directions, they arecalled total radiation properties. The total incident energy (total irradiation G)on a surface is either absorbed, re ected or transmitted according to:

�s + �s + �s = 1 (25.81)

where

� �s is the fraction of the irradiation which is absorbed

� �s is the fraction of the irradiation which is re ected

� �s is the fraction of the irradiation which is transmitted

25.6 Heat Transfer in the Ring Furnace 397

The emissivity �s is another important radiative property de�ned as the ratio ofthe total emitted energy E of a surface to the total energy that would be emittedby a blackbody at the same temperature:

�s =E

Eb) E = ��T 4 (25.82)

Kirchho�'s law establishes an important relationship between the absorptivity andemissivity of a surface which is at thermal equilibrium with its surroundings:

�s(T ) = �s(T ) (25.83)

This means that a good radiation absorber is also a good emitter of thermalradiation.

In a more fundamental approach, monochromatic (spectral) radiation propertieswhich apply only at a single wavelength could be de�ned. Total properties couldthen be calculated from spectral properties as shown for the absorptivity andemissivity:

�s =

R1

0�s;�G�d�R1

0G�d�

(25.84)

�s =

R1

0�s;�Eb�d�R1

0Eb�d�

(25.85)

�s;� =E�

Eb�(25.86)

For the spectral properties, the following relationships are valid:

�s;� + �s;� + �s;� = 1 (25.87)

�s;� = �s;� (25.88)

The last expression represents Kirchho�'s law on a spectral basis. In this case,the law is not restricted to the conditions of thermal equilibrium. From the aboveexpressions, it can be seen that the total properties depend on surface propertiesand temperature. In addition, �s is a function of all the surrounding surfacesthat contribute to the spectral irradiation G�. The emissivity is not a function ofsurrounding surfaces, only the surface material of the emitting body itself. Thus,Kirchho�'s law is only valid for the total properties in the thermal equilibriumcase.

To simplify the analysis, the concept of a gray body is introduced which is de�nedto have constant monochromatic radiation properties for all wavelengths. Then:

�s = �s;� (25.89)

The other total and spectral radiation properties are related as follows:

�s = �s;� (25.90)

�s = �s;� (25.91)

�s = �s;� (25.92)

398 Modelling the Baking Process

Finally, it can be shown that:

�s = �s (25.93)

which is Kirchho�'s law for a gray body. This relationship is also valid when theemitting body is not in thermal equilibrium with the surroundings.

Directional radiation properties are de�ned in terms of the intensity. To simplifythe radiation calculations, the concept of a di�usively re ecting surfaces is intro-duced. A di�use surface re ects a single incident ray such that the re ected energyhas a equal intensity for all re ected angles.

Gas Radiation Properties

Emitted radiation from a gas di�ers quite a lot from the emitted radiation from asolid surface. In contrast to the relatively continuous emissive power (as function ofwavelength) for a solid substance, emission and absorption of gases occur in narrowwave length bands. Emission and absorption from gases with nonsymmetricalmolecular structures can be signi�cant. It is usual to consider the contributionfrom CO2 and H2O to the radiative properties of gases. Radiation properties forgases depend on:

� Gas temperature

� Gas composition

� Total pressure and partial pressures of gas components

� Shape of gas volume

The emissivity of a gas is found by correlations of the following form:

�g;i = Ci �g;i(Tg; piLc) (25.94)

�g =Xi

�g;i ���g (25.95)

Lc is the mean beam length of the gas compartment and pi is the partial pressureof the gas component considered. Ci is a correction factor which depends on thetotal pressure of the gas.

As for solid bodies, the total absorptivity of a gas depends on the gas propertiesas well as the emitting surfaces which surround the gas. Thus:

�g;i = Ci �g;i(Tg ; piLc; Ts) (25.96)

where Ts is the temperature of the surface which interacts with the gas. Ap-proximate expressions for absorptivity of the gas components can be found basedon the expressions used for calculation of the emissivity of the gas components.Kirchho�'s law for a gray gas gives:

�g = �g (25.97)

25.6 Heat Transfer in the Ring Furnace 399

Shape Factors and Radiation Between Gray Surfaces

To calculate the radiative heat transfer between two surfaces, the percentage of thetotal radiant energy which leaves one surface and arrives directly on the secondsurface must be considered. Several algebraic principles have been derived fordetermination of shape factors:

� Reciprocity relationship: AiFj = AjFi

� Enclosure relationship:P

j Fij = 1

� Crossed string method

These rules can be used to show that:

� Opposite planes with in�nite surface area both have shape factor 1

� Both concave planes which form an enclosure have shape factor 1

When radiative exchange between gray surfaces is studied, one generally makesthe following assumptions:

� Surfaces are isothermal

� Surfaces are di�usively re ecting

� Irradiation on a surface is uniformly distributed

� Surfaces are opaque

The concept of radiosity J (the total energy leaving a gray surface) plays a keyrole in establishment of the thermal circuits used as basis for the calculations:

J = �Eb + �G (25.98)

J is the sum of the emitted e�ect per unit surface and the re ected irradiation.Sometimes it is necessary to take into account radiation through partially trans-mitting and absorbing media. This may be gases which reside in the open spacebetween surfaces. The presence of a gas can be incorporated into the thermalcircuits as a refractory surface under the assumption that there is no externalradiation energy supply to the gas (except from energy from the surrounding sur-faces).

In the literature, form factors are also denoted form-, view- or radiation factors.

Summary of Assumptions used in Radiation Heat Transfer Calculations

� Total radiation properties are used.

400 Modelling the Baking Process

� Solid surfaces are assumed gray, di�usely re ecting and opaque which gives:

�s = �s

�s = 0) �s = 1� �s = 1� �s

� The combustion gas is assumed gray, transmitting and non-re ecting whichgives:

�g = �g

�g = 0) �g = 1� �g = 1� �g

� Shape factors along the gas path are considered to be unity.

� Radiation properties of the gas are calculated by taking into account onlypresence of CO2 and H2O; the most abundant non-symmetric gases.

� The impact of luminuous soot particles in the ames on the gas emissivityis taken into account by adding a certain correction term to the sum of theemissivities of the gas components.

� Radiation heat transfer is taken into account only in the direction normal tothe gas ow.

Radiation Heat Flux Calculation in the Ring Furnace

Radiative heat transfer was studied in four types of regions along the gas pathwith qualitatively di�erent geometry:

� Headwalls (part A and part B)

� Under-pit regions (part A and part B was assumed similar)

� Flue channels (part A and part B are similar)

� Under-lid region (the concavity of the inner surface of the lid was neglected)

Shape factors Fij and mean beam lengths Lc were calculated for the gas compart-ment and ue channels. For the under-lid compartment, the inner surface and topsurface of the pit both were assumed to be in�nite plates with shape factors equalto 1. The basic reason for this assumption is that the surfaces in the under-pitgas channel constitute parts of larger surfaces; the total under-pit surface and thetotal foundation surface. The unity shape factors were used in the analysis leadingto the e�ective shape factors shown below.

The same assumption was used for the under-pit regions. Headwalls and uechannels were treated in the same manner and expressions for e�ective shapefactors are supplied below. In fact, the basic assumption was that the surfacesin both the headwalls and the ue channels locally have the same temperature.

25.6 Heat Transfer in the Ring Furnace 401

Then, no radiative heat transfer between the surfaces occur; only between thesurfaces and the gas.

Mean beam lengths were calculated partly based on the rules given in Kreith &Black (1980, Tab. 6-2) and the expression:

Lc = 3:6V

Ag(25.99)

for geometries not mentioned in Kreith's table. V is the volume of the gas and Ag

is the surface area of the gas. For the ue channels and headwall it can be shownthat the radiative heat ux q00r;gs [W=m2] from gas to surface As is given by:

q00r;gs =qgs

As= �Fgs(T 4

g � T 4s ) (25.100)

if the brickwork surfaces have approximately the same temperature. Fgs can befound from:

Fgs =1

1�g+ 1

�s� 1

(25.101)

Tg and Ts denote average gas- and surface temperatures respectively. Fgs is theradiation factor between gas and surface, �g is the gas emissivity, and �s is thesurface emissivity. The expression for radiative heat transfer under-lid and under-pit are given by more complicated expressions. For the under-pit area, a thermalcircuit as shown in Figure 25.11 for two �nite plates was used, and the combustiongas was modelled as a refractory surface. Analytical expressions13 for the net ra-diation heat uxes were obtained by simpli�cation of the thermal circuit assumingthe plates to have equal surface areas A and emissivities �s:

q00r;gs = �Fgs(T 41 � T 4

g ) + �Fg(T 42 � T 4

g ) (25.102)

q00r;1 = �Fg(T 41 � T 4

g ) + �F12(T41 � T 4

2 ) (25.103)

where:

Fg =1

1�g+ 1

�s� 1

(25.104)

F12 =�2s(1� �g)

[�s + �g(1� �s)][1 + (1� �g)(1� �s)](25.105)

q00r;g is net radiated heat ux from gas to plates. q00r;1 is net radiated heat ux fromplate 1. T1 and T2 denote average surface temperatures.

For the under-lid combustion chamber, the surfaces have di�erent areas and emis-sivities. Therefore, slightly more complicated expressions for the radiation heat uxes were obtained even though the same thermal circuit was used as basis.These expressions are not reported here.

13Calculations were done in MathematicaTM .

402 Modelling the Baking Process

For the outer surface of the lid, air emissivity and absorptivity were neglected andthe radiative heat ux calculated from:

q00r;g;a = ��lid(T4lid � T 4

a ) (25.106)

Ta is the average temperature of roof, walls etc. in the furnace hall. This expres-sion is valid for a small convex object dwelling in a large cavity.

Gas

Plate 2Plate 1

Eb2

q2

q1g

1A2F2g�g

J2

�2�2A2

J1

Ebg

1A1F1g�g

Eb1�1�1A1

1A1F12�g

q12q1

q1g

Figure 25.11: Radiation diagram for two �nite plates with intermediate combustiongas as a refractory surface. From Kreith & Black (1980).

25.7 The Conservation Laws in the Ring Furnace

General balance equations which can be used for the brickwork, packing coke, an-odes and the combustion gas in the ues are summarized in this section. Accordingto Table 25.2, balance equations must be derived for subsystems consisting of upto two phases where each phase may be a multicomponent system.

For a system of np phases, it is assumed that the following relationships apply forthe phase- and total bulk volumes and surfaces areas:

npXk=1

Ak = A (25.107)

npXk=1

Vk = V (25.108)

such that:

npXk=1

�k = 1 (25.109)

25.7 The Conservation Laws in the Ring Furnace 403

Material No. of No. ofphases componentsnp

Brickwork 1 Single

Packing coke 2 Multicomp.

Anodes 2 Multicomp.

Combustion gas 1 Multicomp.

Table 25.2: Subsystems in the ring furnace.

where np is the number of phases and �k =AkA

= VkV

is the volume fraction of phasek. These relationships are also assumed to hold for di�erential control volumes.

In the following, balance equations are derived for a certain phase k which consti-tutes a volume fraction �k of the total volume considered.

25.7.1 The Mass Balances

By using Equation (A.1), the mass balance for component i of phase k may beexpressed as:

d

dt

ZZZVk

�k;idVk = �ZZAk

�k;iVTk;indAk +

ZZZVk

Rk;idVk(25.110)

Rk;i [kg/(m3 hr)] is a net generation rate of component i in the phase as a result

of the chemical reactions in the phase represented by the vector Rk of reactionrates. Now, use dVk = �kdV and dAk = �kdA which gives:

d

dt

ZZZV

�k�k;idV = �ZZA

�k�k;iVTk;indA+

ZZZV

�kRk;idV

(25.111)

The use of Gauss' theorem on the �rst and second terms on the right hand side ofthe equation leads to the following partial di�erential equation for the componentmass balance:

@

@t(�k�k;i) = �r(�k�k;iV k;i) + �kRk;i (25.112)

Introduce the expression for �k�k;iV k;i from Equation (25.45) which gives:

@

@t(�k�k;i) = �r(�k�k;ivk)�r(�kjk;i)�r(�k~|k;i) + �kRk;i (25.113)

In this equation, the nabla operator in the third term on the right hand side ofthe equation is taken with respect to the spatial coordinate for interphase masstransfer.

404 Modelling the Baking Process

Assume that he generation rate of component i is given by:

Rk;i = (STk Rk)i (25.114)

Sk is a stoichiometric matrix for the reactions in phase k. Sk is corrected fordi�erences in the molar masses of the components14. This gives:

@

@t(�k�k;i) = �r(�k�k;ivk)�r(�kjk;i)�r(�k~|k;i) + �k(S

Tk Rk)i (25.115)

The total mass balance for the gaseous phase is obtained by summation of themass balance equations for the components:

nkXi=1

@

@t(�k�k;i) = �

nkXi=1

r(�k;ivk)�nkXi=1

r(�kjk;i) (25.116)

�nkXi=1

r(�k~|k;i) +nkXi=1

�k(STk Rk)i

UsePnk

i=1 jk;i = 0 andPnk

i=1(STk Rk)i = 0 to obtain:

@(�k�k)

@t= �r(�k�kvk)�

nkXi=1

r(�k~|k;i) (25.117)

The second term on the right hand side of the equation represents the net masstransfer from phase k to the other phase in this case represented by:

r(�k~|k;i) = �k( ~STk Rk)i (25.118)

This gives:

@

@t(�k�k;i) = �r(�k�k;ivk)�r(�kjk;i)� �k(S

Tk Rk)i + �k( ~S

Tk Rk)i(25.119)

@(�k�k)

@t= �r(�k�kvk)�

nkXi=1

�k( ~STk Rk)i (25.120)

where ~Sk is the matrix of stoichiometric coe�cients of the chemical reactionsassociated with the interphase mass transfer.

25.7.2 The Momentum Balances

In the ring furnace, two gaseous subsystems and their pressure distributions areof interest:

� The pressure distribution in the anodes is controlled by the heating rateacross the critical temperature interval between 150 and 500�C when thebinder pitch changes from a uid to a solid coke.

14Rk is related to the volume of the phase.

25.7 The Conservation Laws in the Ring Furnace 405

� In the combustion ues, knowledge of the pressure is necessary to be ableto calculate the air inleakage mass ow. In this work however, pressurevariations along the ues are neglected.

Under the assumption that the gas ow in the porous anodes and coke bed islaminar, a more simple relationship between pressure and velocity than found fromthe momentum balance equation may be used. In porous media, the pressure dropdepends on the permeability of the solid materials. In porous materials, Darcy'slaw is often used to model the dependence between pressure and the super�cialvelocity vs of the owing medium (Bear & Bachmat 1991), (Aziz & Settari 1979):

vs = �kp�(rp+ �g) (25.121)

kp and � denote the permeability of the porous material and the viscosity of thegas respectively. Thus, Darcy's law is a simpli�ed stationary momentum balanceequation applicable for porous media with a laminar ow regime.

25.7.3 The Energy Balances

The Heat Term

For the heat term, the followinq expression is used:

Q = �ZZA

q00ndAk (25.122)

Here, Q includes both intraphase and interphase heat transfer. Contributions toq00 is expressed by the equation:

q00 = q00c+ q00

cd+ q00

r(25.123)

where:

� q00cdenotes heat transfer due to convection at the boundary between phases

or at the system boundaries. Heat transfer by convection may play a rolein mass transport of gases in the porous coke bed and anodes. In the ringfurnace, convection heat transfer between combustion gases and the surfaceof the ues is important at temperatures below 600�C.

� q00cd

represents heat transfer due to conduction as governed by Fourier's law:

q00cd

= �krT (25.124)

q00cd

is the heat ux and k is the thermal conductivity. Heat conduction isthe most important mode of energy transport in the solid materials. In thegas phase, however, heat conduction is not very signi�cant.

� q00rdenotes heat transfer due to radiation. At phase- and system boundaries,

q00rmay be treated in the same way as q00

c. In solid materials with gas �lled

voids, radiation is often lumped with the heat conduction term by applicationof an e�ective thermal conductivity (Froment & Bischo� 1990, pp. 301).

406 Modelling the Baking Process

The Work Term

Here, only work performed by a normally acting stress force (pressure) is includedin the work term. Thus, shaft work and viscous work is neglected. This gives:

W =

ZZA

�vTnp1

�dAk (25.125)

The Energy Equation

By application of Equation (A.3), the energy balance equation for phase k afterintroduction of Q and W becomes:

d

dt

ZZZVk

�kEkdVk = �ZZAk

�kvTk nEkdAk �

ZZAk

~|TknEkdAk

�ZZAk

(q00)TndAk �ZZAk

�kvTk n

1

�kpkdAk

The energy transfer due to mass ow is represented by two terms: One term rep-resenting the intraphase energy transfer 15 and a term representing the interphaseenergy transfer 16. In the equation, the work term due to interphase mass transferis not explicitly included.

Again introduce dVk = �kdV and dAk = �kdA and use Gauss' theorem to obtain:

@

@t(�k�kEk) = �r(�k�kvkEk)�r(�k~|kEk) (25.126)

�r(�kq00)�r(�kvkp)

The nabla operator in the second term on the right hand side of the equation istaken with respect to the spatial coordinate for interphase transfer.

Total vs. Internal Speci�c Energy. Several forms of energy are included inthe total energy concept:

� Speci�c internal energy U (i.e. thermal energy and chemically bonded en-ergy)

� Speci�c potential energy Ep = gz

� Speci�c kinetic energy Ek =12v2

15The �rst term on the right hand side of the equation.16The second term on the right hand side of the equation.

25.7 The Conservation Laws in the Ring Furnace 407

Then, the total energy is represented by:

E = U +Ep +Ek (25.127)

U , Ep and Ek have unit [J/kg]. This gives:

E = U + gz +1

2v2 (25.128)

The speci�c internal energy U is a function of thermodynamic properties (statevariables) like pressure p, temperature T , volume V and composition. By ignoringthe dependence of internal energy on composition, one obtains:

dU =

�@U

@T

�v

dT +

�@U

@v

�p

dv (25.129)

v denotes the speci�c volume. From thermodynamics, it is known that (Sears &Salinger 1980, eq. 6-10): �

@U

@T

�v

= cp (25.130)�@U

@v

�p

= T

�@p

@T

�v

� p (25.131)

which gives:

dU = cvdT +

�T

�@p

@T

�v

� p

�dv (25.132)

In the ring furnace system, temperature is the most important state variable. Forsolids and liquids17, it can be shown that the last term in the equation for dUhas negligible contribution (for varying temperature and pressure) to the internalenergy. Furthermore, for an ideal gas, the last term equals zero18. Thus, it isreasonable to use dU � cvdT which gives:

U � Ur +

Z T

Tr

cvT = �cv(T � Tr) = �cvT (25.133)

since �cvTr = Ur. �cv is an average value of cv:

�cv =1

T � Tr

Z T

Tr

cvT (25.134)

17Use a linearized state equation:

v = v�(1 + �(T � T

�)� �(p� p

�))

� and � denote expansivity and compressibility respectively.18Use the ideal gas law.

408 Modelling the Baking Process

�cv is in the order of 1000 J/(kgK) for gases, liquids and solids which shows that Uis the dominating term in all the subsystems of the ring furnace19. Contributionsfrom the kinetic- and potential energy terms can be neglected and the followingenergy equation can be used:

@

@t(�k�kUk) = �r(�k�kvkUk)�r(�k~|kUk) (25.135)

�r(�kq00)�r(�kvkp)

The Energy Equation in a Multicomponent Phase. In a one phase multi-component system, the speci�c internal energy of each component can be explicitlytaken into account. Thus:

�kUk =

nkXi=1

�k;iUk;i (25.136)

which gives:

Uk =

nXi=1

xk;iUk;i (25.137)

where xk;i is the mass fraction of component i in phase k. Now introduce thespeci�c internal energy of the components and obtain the equation:

@

@t(�k

nXi=1

�k;iUk;i) = �r(�knXi=1

�k;ivk;iUk;i)�r(�knXi=1

~|k;iUk;i) (25.138)

�r(�kq00)�r(�kvkp)

For the gaseous subsystems in the ring furnace, the di�usion velocity is usually solow that the energy related to the di�usive mass transport is negligible compared tothe other terms in the energy equation. Furthermore, in cases when the di�usion-energy term �k;ivk;i;dUk;i is comparable with the advection-energy term �k;ivkUk;i,these terms are both negligible compared to other heat transport mechanisms inthe ring furnace. Then, it is assumed that �k;ivk;i can be replaced with �k;ivkwhich gives the following equation:

@

@t(�k

nXi=1

�k;iUk;i) = �r(�knXi=1

�k;ivkUk;i)�r(�knXi=1

~|k;iUk;i) (25.139)

�r(�kq00)�r(�kvkp)

19ep and ek contribute to the total energy in the parts of the ring furnace system containinggas. Estimated bounds on Ep and Ek are as follows: Ep = gz < 50 J/kg and Ek =

12v2 < 50 J/kg

in all the subsystems of the ring furnace.

25.7 The Conservation Laws in the Ring Furnace 409

The component uxes have been introduced also for the interphase mass transfer.Di�erentiation can be performed after summation which gives:

nXi=1

@

@t(�k�k;iUk;i) = (25.140)

�nXi=1

r(�k�k;ivkUk;i)�nXi=1

r(�k~|k;iUk;i)�r(�kq00)�r(�kvkp)

The Enthalpy. The enthalpy can be introduced by considering the sum of termsPni=1r(�k;iUk;ivk) and r(pvk). This gives:

nXi=1

r(�k;iUk;ivk) +r(pvk) = r �k(

nXi=1

xk;iUk;i +pk

�k)vk

!(25.141)

Now use the following relationships for the gas components:

�k;i = xk;i�k

pk;i = xk;ipk

Furthermore, change the order of the summation and nabla operators and usePnk=1 xk;iUk;i = Uk and Hk = Uk +

pk� to obtain:

nXi=1

r(�k;iUk;iv) +r(pkvk) = r(�kHkvk) (25.142)

Now, use Hk =Pn

i=1 xk;iHk;i and obtain:

r(�kHkvk) =

nXi=1

r(�k;iHk;ivk) (25.143)

A similar expression could be derived for the interphase mass transfer if an inter-phase work term was included.

The Final Equation. By introducing the enthalpy into Equation (25.140), thefollowing energy equation is obtained:

nXi=1

@

@t(�k�k;iUk;i) = �

nXi=1

r(�k�k;iHk;ivk)�nXi=1

r(�k~|k;iHk;i) (25.144)

�r(�kq00k)

or alternatively:

@

@t(�k�kUk) = �

nXi=1

r(�k�k;iHk;ivk)�nXi=1

r(�k~|k;iHk;i) (25.145)

�r(�kq00k)

410 Modelling the Baking Process

Use Uk = Hk � pk�k

and get:

@

@t(�k�kHk) = �

nXi=1

r(�k�k;iHk;ivk)�nXi=1

r(�k~|00k;iHk;i) (25.146)

�r(�kq00k) +@

@t(�kpk)

Alternatively, the equation can be expressed as::nXi=1

@

@t(�k;iHk;i) = �

nXi=1

r(�k;iHk;ivk)�nXi=1

r(�k~|k;iHk;i) (25.147)

�r(�kq00k) +@

@t(�kpk)

Introduce the heat ux terms, and get the following energy equation which is usedas basis for modelling thermal processes in the ring furnace:

nXi=1

@

@t(�k�k;iHk;i) = �

nXi=1

r(�k�k;ivkHk;i)�nXi=1

r(�k~|k;iHk;i) (25.148)

+r(�kkrT )�r(�kq00k;c)�r(�kq00

k;r) +

@

@t(�kpk)

The equation may also be written:

@

@t(�k�kHk) = �r(�k�kHkvk)�r(�kjk; Hk) +r(�kkrT ) (25.149)

�r(�kq00k;c)�r(�kq00

k;r) +

@

@t(�kpk)

where:

�kHk =

nXi=1

�k;iHk;i

�kvkHk =

nXi=1

�k;ivkHk;i

jk; Hk =

nXi=1

~|k;iHk;i

In modelling of solid heat conduction, r(q00r) is usually lumped with the conduction

term by application of an e�ective thermal conductivity. In a gaseous system,however, the heat conduction term may be neglected. Convection and radiationis usually taken into account at the boundary of the gas volume.

Based on Equation (25.149), the following energy balance can be used in calcula-tion of the temperature in a solid material:

�s�scp;s@T

@t= r(�sks;erT ) (25.150)

Based on Equation (25.145) and Equation (25.149) a mixture of ideal gases andmultiphase systems were derived in Gundersen (1996b). Details are not repeatedhere.

25.8 Volatile Transport in the Porous Coke Bed and Anodes 411

25.8 Volatile Transport in the Porous Coke Bed

and Anodes

The formation of volatiles depends on the temperature pro�le in the anodes. Thevolatiles mainly escape from the anodes via the coke bed into the gas ues. How-ever, a fraction of the volatiles are cracked in secondary reactions in the gas phase.A knowledge of the volatile transport is needed both for obtaining an estimateof the amount of volatile combustion energy released in the ues as well as theamount of secondary coking in the pores of the anodes.

25.8.1 Phenomenological Description

Volatile transport through the anode- and coke- bed is a complex process. Notonly will there be mechanisms contributing to the overall di�usion and/ or bulktransport of volatiles. There will also be a continuous change in the porosity ofthe anode. In the ring furnace, the anodes are surrounded by packing coke in thecassettes. The anode is kept in a porous bed of packing coke. As the temperature inthe anode rises, volatile gases develop in the anodeblocks. In sequence and partiallyin overlapping temperature intervals, tar-gases, methane and hydrogen escape asreaction products from the pyrolysis reactions. Due to pressure-gradients (Darcy'slaw), concentration gradients (Fick's law) and temperature-gradients (Soret e�ect)the volatiles migrate through the anode blocks and coke bed. At last they reachthe cassette-walls to be burned in the ues. There is no generation of volatiles inthe coke bed, only passive transport. The porosity in the coke bed is constant.An average value for the coke-bed porosity was given in Dernedde & Bourgeois(1987). The anode-bed porosity increases as the volatiles are pyrolysed. Thereis a pressure build-up in the anodes during volatile generation. If the pressurebecomes too large, cracks may occur at the anode surface due to a steep pressuregradient occurring near the anode surface (B�ottger 1990).

25.8.2 Transport Regime in the Coke Bed and Anodes

The maximum volatile velocity across the coke-bed has been measured to 0:8 �10�4m/s (Dernedde & Bourgeois 1987). Is this velocity due to a convective ordi�usive transport mechanism? A simpli�ed one - dimensional analysis presentedin Gundersen (1996a) gave an answer to this question. In the analysis it is assumedthat convective and di�usive mass transport mechanisms are additive.

According to the analysis, it was shown that the convective ow is laminar and thatthe convective velocity is a least an order of magnitude larger than the di�usivevelocity.

In principle, the same kind of analysis can be done for the anodes but in theanodes, pressure gradients are more pronounced. Still however, it is reasonableto assume that the convective ow is laminar. Then Darcy's law can be used formodelling of convective gas transport both in the coke bed and the anodes. In

412 Modelling the Baking Process

Gundersen (1996a), mass balance equations for the gas ow were established. Thesimpli�ed procedure described earlier was used for modelling the evacuation ofvolatiles from the anode- and coke bed. Therefore the mass balance equations arenot repeated here. Jacobsen &Melaaen (1995) calculated one dimensional pressuredistributions in anodes. A more extensive simulation case with coupled heat andmass transfer in anodes are presented in Jacobsen (1997). Such calculations arenot presented in this work.

25.9 Thermal Phenomena in Solid Materials

During baking, there is a continuous transformation of the liquid pitch into bindercoke under the release of volatile gases. This transformation is thermally inducedand has to be controlled to obtain anodes with the desired properties. Also, theheat from the volatiles is returned to the process during combustion in the ues.Therefore, it is of importance to have insight into the shape of the temperaturepro�le in the anodes and its impact on the physical properties of the anodes.

In part IV of this work, models for physical properties of pure- and composite car-bons were modelled as functions of a set of state variables. In the model presentedin this chapter, however, a more simpli�ed approach is used: Apparent values ofthe physical properties of anodes as represented by polynomials of temperature areused. For the anodes, this may be a crude approximation. For the other materials(brickwork and packing coke) the use of such polynomials may be adequate.

25.9.1 Thermal Properties of Solid Materials

For calculation of the solid materials heat balance, data for density, speci�c heatcapacity and thermal conductivity for the brickwork, packing coke, and anodesare needed.

Brickwork

During heating, the brickwork expands and gives a reduction in brickwork density.If � is the coe�cient of thermal expansion, the temperature dependent density isgiven by:

� = ��(1� �(T � T�)) (25.151)

where T� is the room temperature. For the brickwork, � is in the order of10�4 1=�C. For the heat capacity and thermal conductivity, higher order polyno-mial functions of temperature were used to describe the temperature dependency.Low order polynomials (2. or 3. order) were used to �t the brickwork data.

Di�erent types of brickwork is used in the ring furnace construction. This is takeninto account in the model by using di�erent correlations for the thermal propertiesfor each type of brickwork (i.e. insulating and conducting brickwork).

25.9 Thermal Phenomena in Solid Materials 413

Packing Coke

Calcined coke with approximately constant bulk density over the baking tem-perature interval is used as packing coke. The bulk density includes inter- andintraparticular voids of the coke bed which surrounds the anodes. These voids al-low for a signi�cant radiation heat ux in the coke bed at high temperatures. Thiscould be modelled by adding a radiation term in the energy equation. Accord-ing to the procedure described in Froment & Bischo� (1990), however, this termis generally lumped with the heat conduction term by application of an e�ectivethermal conductivity. Also, for the coke bed, polynomial functions of temperaturewere used for the speci�c heat capacity and thermal conductivity.

Anodes

The density of the porous anodes depends on the transformations which take placein the pitch fraction of the anode. In the pitch fraction, a crystalline structuregrows during heat treatment and increases the degree of structural order and there-fore real density. The heat of vaporization and heat of reaction was not explicitlymodelled in the heat equation, but apparent values of speci�c heat capacity andthermal conductivity were used to model the total e�ect of heat conduction andheat of chemical reactions.

In future ring furnace models, models for the thermal properties as derived in partIV should be used instead of the property correlations applied in this case.

Mathematical Structure of Property Correlations

To summarize, the density, speci�c heat capacity, and thermal conductivity dependon the temperature and thus on time. The temperature dependency is modelledby the use of polynomial functions as follows:

�i =

naiXj=0

ai;jTj (25.152)

cp;i =

nbiXj=0

bi;jTj (25.153)

ki =

nciXj=0

ci;jTj (25.154)

where i = brick, coke and anode. Some of the functions were derived from leastsquares polynomial �t to data found in the literature.

414 Modelling the Baking Process

25.9.2 Application of the Ordinary Heat Conduction Equa-

tion in the Ring Furnace

The ordinary heat conduction equation appears as Equation (25.150). In this case,Equation (25.150) is nonlinear due to the temperature dependent properties. Oneshould also note that the property functions are spatially discontinuous due toa regular but complex geometric structure of the pit where brickwork and cokesurround the anodes. The validity of Equation (25.150) for calculation of the ringfurnace temperature is not obvious; an explanation seems necessary.

During baking, apparent density, speci�c heat capacity and thermal conductivityin anode, coke bed and brick material change with temperature. Separate modelsare needed for the thermal properties in each medium.

For the brickwork, density changes are negligible. Equation (25.150) is directlyapplicable for modelling heat conduction in the brickwork.

Also in the coke bed, density changes are negligible. In the coke and anode,gases and solids coexist due to the release and transport of volatiles from anodesto the combustion ues. Thus, the system is heterogeneous system with twophases for which separate mass and energy balances for the gas and solids couldbe derived. However, a uniform temperature is assigned to each control volume.The volatile mass transport takes place at velocities in the order of 10�4m/s(Dernedde et al. 1986). Below, it is shown that the energy contribution due toconvective or di�usive transport of pitch volatiles in the coke bed and anode canbe neglected. Energy transfer in connection with possible chemical reactions inthe di�using gases is assumed negligible. Then Equation (25.150) without sourceterm is a good approximation for the temperature calculations in brickwork andpacking coke.

Also for the anodes, density changes are moderate. It can be shown that a sourceterm can represent the energy consumed during generation of the volatiles. Tra-ditionally one has lumped this source term into an e�ective heat capacity for theanode material. It can be shown that the source term only introduces secondarye�ects in the temperature distribution in the anode. Thus, Equation (25.150) with-out the source term can also be used for modelling the development of temperaturein the anode.

25.9.3 The Energy Equation for the Brickwork Materials

In the brickwork, there is no source term. Thus:

�brcp;br@T

@t= r(kt;brrT ) (25.155)

where �br, cp;br and kt;br denote density and speci�c heat capacity of the brickwork.

25.9 Thermal Phenomena in Solid Materials 415

25.9.4 The Energy Equation for the Coke Bed

In the coke bed, the volatile gases interact with the coke particles as they owfrom the anodes to the combustion ues via the packing coke bed. Coking of theheavy volatile components on the porous surface may occur in both the anode andthe packing coke bed; a certain mass transfer term is associated with such cokingreactions. Finally, condensation of heavy components in the volatile gas may alsooccur. Simulations have shown that the pressure drop across the coke bed is verylow (Jacobsen & Melaaen 1995). Also the average pressure does not vary verymuch.

For the solid coke phase, pressure variations are neglected and the following energybalance applies:

(1� �c)�a;ccp;c@Tc

@t= r((1� �c)kt;crT ) + �chc(Tg � Tc)

+Xi

�ccp;g;i( ~Sgcrg)i(Tg � Tc)

�c is the void fraction of the coke bed. The last term on the right hand siderepresents transfer of energy as a result of interphase mass transfer.

The gas phase is assumed to consist of ideal gases. An energy balance equationfor a mixture of ideal gases was derived in Gundersen (1996b, Eqn. (1.260)). Itwas shown that the following energy balance applies for the gas:

�c�gcv;g@T

@t=X

�c�g;ivgcp;g;irT �Xq

�Hg;qrq �X

�c( ~ST rg)icp;g;i(Tg � Tc)

+r(�ckt;grT )� �chg(Tg � Tc)

For the coke bed, the following assumptions are made:

� Heat transfer between gas and solid is neglected and solid and gas is assumedto have the same temperature. Thus, an average temperature is used for thecoke bed.

� Mass- and heat transfer associated with coking reactions and partial con-densation are neglected.

� Heat of coking reactions is neglected.

This gives:

((1� �c)�a;ccp;c + �c�gcp;g)@T

@t= r(((1� �c)kt;c + �ckt;g)rT ) (25.156)

��gvcp;grT

The e�ective thermal conductivity kt;c;e for the coke bed is introduced which gives:

(1� �c)kt;c + �ckt;g = kt;c;e

416 Modelling the Baking Process

Then, the energy equation for the coke bed becomes:�(1� �c)�a;ccp;c

@T

@t+ �c�gcp;g

@T

@t

�= r(kt;c;erT )� �gvcp;grT (25.157)

The multicomponent composition of the gas which actually ows across the cokebed is represented by the bulk gas density �g and cp;g given by:

�g =

nXi=1

xipMi

RT(25.158)

cp;g =

nXi=1

xicp;g;i (25.159)

In the equation, heat transfer due to radiation is included in an e�ective thermalconductivity for the coke bed. �c denotes the interparticular void space in thecoke bed in which the gas ow may move freely. In addition, gas also occupy partsof the volume fraction (1 � �c) in open and closed porosity in the packing cokeparticles. It is assumed that this porosity is not accessible for the gas ow and thepresence of gas within these pores can be neglected.

The following parameter values and conditions apply for the coke bed:

� �a;c � 1600 kg/m3

� �g � 0:6 kg/m3for air at temperatures (above) 300�C20

� cp;c � 1000 J/(kgK)

� cv;g � cp;g = 1000 J/(kgK)

� �c � 0:3

� @T@y� 10:0K/cm = 1000K/m

� @T@t � 10K/hr � 2:0� 10�3K/s

� v � 0:8� 10�4m/s assuming that this is the super�cial velocity (Dernedde& Bourgeois 1987)

This gives:

� j(1� �c)�a;ccp;cj � 1:0� 106 J/(m3K)

� j�c�gcv;gj � 2:0� 102 J/(m3K)

20In general, the packing coke bed is hotter than the interior of the anode during heat-up sincethe packing coke surrounds the anode. If the gases are assumed ideal, density is proportional withmolar mass. Thus condensables have higher density and non-condensables have lower densitythan air.

25.9 Thermal Phenomena in Solid Materials 417

In this case, (1� �c)�a;ccp;c is the most dominating term. Thus, a good approxi-mation seems to be:

(1� �c)�a;ccp;c@T

@t= r(kt;c;erT )� �gvcp;grT (25.160)

From the data, it can be shown that:

� j(1� �c)�a;ccv;c@T@tj � 2240 J/(m

3s)

� j�gvcp;grT j � 48 J/(m3 s)

which shows that the conduction term r(kt;c;erT ) must dominate the right handside of the equation. Thus, the �nal energy equation for the coke bed is: have:

(1� �c)�a;ccp;c@T

@t= r(kt;c;erT ) (25.161)

for the energy equation in the coke bed. In this case, the energy equation isdecoupled from the continuity equation.

25.9.5 The Energy Equation for the Anodes

The same kind of argument that was used for the energy equation for the cokebed can be used to establish the energy equation for the anodes. In the anodes,however, some characteristic features must be taken into account. As the anodesare heated, the pitch fraction melts and partly vaporize. Mass also escapes in theform of gaseous reaction products from the carbonization reactions. In this case,the pitch both in the solid and liquid state is assumed to belong to the "solid" stateof the anode. It should be noted here that (1� �a;�) changes during baking bothdue to the mass lost from the pitch fraction as well as expansion and shrinkagein the liquid binder pitch and pitch coke respectively. It is assumed that possiblemass transfer associated with secondary coking reactions which take place in thegas phase is negligible compared to mass transfer from the pitch fraction of theanode to the gas phase due to thermal decomposition.

For the solid phase of the porous anode, the following energy balance equationapplies:

(1� �a;�)�r;a@Ta

@t= r((1� �a;�)kt;arTa)�

X(1� �a;�)( ~S

Ta ra)icp;a;i(Ta � Tg)

+�aha(Ta � Tg)�Xq

(1� �a;�)�Hqra;q

For the gas phase, the following balance equation can be used:

�f�gcv;g@Tg

@t= r(�akt;grT )�

X�f�g;ivgcp;g;irTg

�X

(1� �a;�)( ~STa rg)icp;a;i(Ta � Tg)� �chg(Tg � Ta)

418 Modelling the Baking Process

Particles

Voids

ParticlesVoidswithgas

Realistic configuration of phases

Simplified configuration of phases

�y

r00c

y

y

vg;y

�c (1� �c)

Tc

Tg

r00c

vg

(1� �c) �c

�Vc = (1� �c)�V�Vg = �c�V

Figure 25.12: Interaction between packing coke and gas in the packing coke bed.

As for the coke bed, a uniform temperature is used for the gas and the solid phaseof the anode and the energy transfer associated with interphase mass transfer isnegligible. Hence:

((1� �a;�)�r;acp;a + �f�gcv;g)@T

@t= r(((1� �a)kt;a + �akt;g)rT )� �f�gvgcp;grT

�Xq

(1� �a;�)�Hqra;q

Again, an e�ective thermal conductivity kt;a;e is introduced which gives:

((1� �a;�)�r;acp;a + �f�gcv;g)@T

@t= r(kt;a;erT )� �f�gvgcp;grT

�Xq

(1� �a;�)�Hqra;q

25.9 Thermal Phenomena in Solid Materials 419

In the anode, conditions are more extreme than in the coke bed. The permeabilityis lower, and gases are generated within the anode. This leads to a signi�cantlyhigher pressure in the anode and the role of the term �gvgcp;grT is not so easy toquantify. The order of magnitude of the terms in the equation can be estimatedby using the following data for the anodes:

� �r;a � 2000 kg/m3

� �g � 0:6 kg/m3for air at temperatures (above) 300�C21

� cp;a � 1000 J/(kgK)

� cp;g � 1000 J/(kgK)

� �a;� � 0:20

� @T@y � 10:0K/cm = 1000K/m

� @T@t � 10K/hr � 2:0� 10�3K/s

� vg � 0:8� 10�3m/s assuming that the velocity of volatiles is 10 times fasterthan in the coke bed

This gives:

� j(1� �a;�)�r;acp;aj � 1:6� 106 J/(m3K)

� j�f�gcv;gj � 1:2� 102 J/(m3K)

The term j(1� �a;�)�r;acp;aj is the most dominating and the equation simpli�es to:

(1� �a;�)�r;acp;a@T

@t= r(kt;arT )� �gvgcp;grT + (��H)Rv

From the data presented above, there may be potentials for simplifying the equa-tion even more. As for the coke bed, the convective energy term can be comparedto the right hand side of the equation:

� j(1� �a;�)�r;acp;a@T@t j � 3200 J/(m3 s)

� j�gvcp;grT j � 480 J/(m3 s)

This shows that the convective term contributes to 15 % of the magnitude ofthe rate of change of energy. Since (in the y-direction) the gas ows againstan increasing temperature, �gvcp;grT contributes to consumption of energy. Inextreme situations, a not negligible amount of energy may be needed to heatthe gas. On the other hand, one should be aware that anodes are heated in a

21In general, the packing coke bed is hotter than the interior of the anode during heat-up sincethe packing coke surrounds the anode.

420 Modelling the Baking Process

conservative manner; heating rates should not be too high as the anodes pass thecritical stages of devolatilization. Therefore, the convective term in the energybalance is neglected and �nally the following energy equation is obtained for theanode:

(1� �a;�)�r;acp;a@T

@t= r(kt;arT ) + (��H)Rv (25.162)

The energy term (��H)Rv expresses the amount of energy needed for volatiliza-tion and carbonization of pitch. The signi�cance of this term (including both heatof vaporization and heat of reaction of pyrolysis reactions in the pitch) is discussedbelow.

25.9.6 The Reaction Enthalpies

Signi�cance of the Heat of Vaporization and Heat of Reaction

Vaporization of low weight compounds from the pyrolysing pitch in the tempera-ture interval between 100 up to 400�C has an endothermal e�ect on the thermalsystem. At temperatures in the order of 350�C, exothermal carbonization reac-tions occur. This is in accord with Jones & Hildebrandt (1975) who concludedthat early stages in pitch pyrolysis are basically endothermic whereas exothermicreactions occur due to the onset of carbonization reactions (i.e. polymerizationreactions).

According to Fitzer & Fritz (1975), most polymerization reactions occur exother-mally with a heat of reaction �Hr in the order of 125 kJ/mol22. This is in accordwith the ranges of heat of reactions recommended for the exothermal reactions incoal carbonization as studied by Burke & Parry (1927) and Davis & Place (1924)if we assume that the reacting molecules have a molar mass in the order of 1000g/mol.

To quantify the order of magnitude of the heat of vaporization of light compoundsin the binder pitch, a correlation due to Briggs & Popper (1957) can be used:

�Hv � 0:486��(1� 0:0012Tb) [kJ/kg] (25.163)

�� and Tb are the room temperature density (20�C) and average boiling point of

the pitch respectively. If �� = 1320 kg/m3and Tb = 603K, the heat of vaporization

becomes:

�Hv � 177 kJ/kg

Assume that the average molar mass of the light part of the binder pitch is in theorder of 300 g/mol, and get23:

�Hv � 53 kJ/mol

22Fitzer & Fritz (1975) supplies the data as 30 kcal/mol.23Since 1 mol of volatiles corresponds to 0.3 kg or 1 kg of volatiles corresponds to (1/0.3) mol.

25.9 Thermal Phenomena in Solid Materials 421

The heat of vaporization and the heat of reaction contributes to an energy con-sumption and energy release respectively which are in the same order of magni-tude24.

If the average molar mass is in the order of 1000 g/mol, the heat of vaporizationand the heat of reaction have approximately the same order of magnitude:

j�Hrj; j�Hv j � 100 kJ/kg

Simulations of pitch pyrolysis, have shown that the maximum release rate ofvolatiles is in the order of rm = 3:0 kg/(m3 hr) where rm =

Pni=1 ri includes both

condensables and non-condensables. At temperatures above 350�C, rm is signi�-cantly lower than the value supplied here since light compounds like methane andhydrogen starts to dominate above this temperature. This gives:

nXi=1

j(��Hi)j ri �nXi=1

j(��H)j ri = (��H) rm (25.164)

and �nally:

nXi=1

j(��Hi)j ri = 3� 102 kJ/(m3 hr) � 103 kJ/(m3 hr) (25.165)

This estimate ofPn

i=1 j(��Hi)j ri should be compared to the rate of change ofthe energy density as represented by the term �cp

@T@t. If � � 1000 kg/m3, cp �

1000 J/(kgK) and @T@t� 10K/hr, the rate of change of energy becomes:

�cp@T

@t� 104 kJ/(m3 hr)

This is one order of magnitude larger than the chemical contribution. Thus, thethermal e�ects due to vaporization and carbonization reactions introduce onlysecondary e�ects which may be neglected in the calculation of the temperaturepro�le in the anodes. This is in accord with the conclusions in Howard (1981).

Conservative Estimation of Anode Temperature

When the convective energy term and the source term due to vaporization andchemical reactions are neglected, this leads to overestimation of the anode temper-ature. This gives an upper bound on the anode temperature which is preferableif the energy equation is to be used in optimization of ring furnace operation. Atpresent, the model does not take into account the reaction dependent source termand energy transfer due to convection.

24A molar mass of 1000 g/mol gives �Hv � 177 kJ/kg.

422 Modelling the Baking Process

25.9.7 The Heat Conduction Equation with Boundary Con-

ditions

Heat conduction in the pit refractory, the coke bed, and the anodes is modelledin three dimensions by the application of the heat equation without source term.In each section, there are �ve pits. Due to symmetry, the three dimensionaltemperature �eld is calculated for half a pit using the following equation:

�cp@T

@t=

@

@x(k@T

@x) +

@

@y(k@T

@y) +

@

@z(k@T

@z) (25.166)

AnodesCoke

z

y

Brick

x

z

yPit Centerline

(i,j,k)

�xi�zi

�yi

Figure 25.13: Solid control volume seen from yz-plane

The control volume as seen from the yz-plane is shown in �gure 25.13. It has theshape of a parallelepiped with geometry as follows: the height Lz (z-direction)goes from the pit refractory bottom to the coke bed top. The extension in widthLx (x-direction) is from center to center of adjacent headwalls. The thickness Ly(y-direction) is from the surface of the brickwork of the pit wall to the center ofthe pit. Boundary conditions for the six surfaces of the solid pit must be speci�ed:

1. The xz-plane:

� The xz-plane through the pit center is an adiabatic plane (pit symme-try):

@T

@yjy=Ly = 0 (25.167)

� At the xz-plane parallel to the pit wall there is convective and radiativeheat transfer (vertical ues):

�k@T@yjy=0 = hc(Tg � T jy=0) + Fgs�(T 4

g � T 4jy=0) (25.168)

Fgs is the radiation factor between gas and brick surface.

25.9 Thermal Phenomena in Solid Materials 423

2. For the xy-planes through pit top (under-lid) and bottom (under-pit), thereis convective and radiative heat transfer. The heat ux expressions are sim-ilar to the boundary conditions for the xz-plane except for the addition of aterm which represents heat transfer between the surfaces facing each other,as shown in Subsection 25.6.6. Boundary conditions for the heat transferunder-lid and under-pit are then:

� For the under-pit boundary condition:

�k@T@zjz=0 = hc(Tg � T jz=0) +Fgs�(T 4

g � T 4jz=0) (25.169)

+Fz=0;f�(T4f� � T 4jz=0)

Fz=0;f is the radiation factor between the under-pit surface and thefacing surface corresponding to the ring furnace foundation. Tg and Tf�are the gas temperature and the surface temperature of the foundation.

� Under the lid, the boundary condition is correspondingly:

�k@T@zjz=Lz = hc(Tg � T jz=Lz) +Fgs�(T 4

g � T 4jz=Lz)(25.170)

+Fz=Lz;l�(T 4l� � T 4jz=Lz)

Fz=Lz;l is the radiation factor between the pit surface under-lid and thefacing surface corresponding to the lid of the section. Tg and Tl� arethe gas temperature and the surface temperature of the lid facing thetop of the pit.

3. Constituting a part of the headwall, a vertical �re-shaft with a rectangularcross section is used for �ring in part A (see �gure 25.14). In the numericalmodel, outblocking is used for modelling the presence of the �re-shaft. Inthe outblocked region, a linear gas temperature pro�le along the z-directionis calculated. In the top of the �reshaft, the temperature equals the ambienttemperature. In the bottom of the �reshaft, the temperature corresponds tothe gas temperature in the headwall. Convective and radiative heat transferis assumed to occur in the �reshaft. For the rest of the headwall, an adiabaticboundary condition is assumed25:

@T

@xjx=0 = 0 (25.171)

@T

@xjx=Lx = 0 (25.172)

Contact resistances between the di�erent materials in the pit (brick vs. coke bed,anode vs. brick and anode vs. coke bed) were neglected.

25In another case studied, the whole yz-plane (without outblocking) through the center of theheadwall is assumed adiabatic.

424 Modelling the Baking Process

25.9.8 Heat Losses Through Furnace Lid and Foundation

Heat loss through the foundation was modelled as a two dimensional phenomenonin the xz-plane. For the foundation refractory which faces the combustion gas,convective and radiative heat transfer boundary conditions were speci�ed. For thefoundation bottom, a constant temperature was speci�ed. The following equationswere used to describe the foundation heat loss:

�br;f cp;br;f@Tf

@t=

@

@x(kbr;f

@Tf

@x) +

@

@z(kbr;f

@Tf

@z) (25.173)

�kbr;f@Tf

@xjx=0 = 0 (25.174)

�kbr;f@Tf

@xjx=Lx = 0 (25.175)

�kbr;f@Tf

@zjz=0 = hc(Tg � Tf jz=0) +Fgs�(T 4

g � T 4f jz=0) (25.176)

+Ff;p�(T 4p jz=0 � T 4

f jz=0)

Tf;z=1 = Tf;1 (25.177)

Tf denotes the temperature in the foundation. Tpjz=0 denotes the under-pit surfacetemperature. Ff;p is the form factor for radiation between the under-pit surfaceand the top surface of the foundation. The local coordinate system used for thefoundation is given in Figure 25.15.

The curved nature of the furnace lid was neglected and the lid was assumed to bean in�nite plate. Thus, distortion of the temperature distribution at the sides ofthe lid was neglected. A one-dimensional heat conduction model was used due tothe assumption of uniform temperature in the gas under the lid and the in�niteplate assumption. The lid boundary heat uxes were speci�ed as follows: At thelid surface facing the combustion gases, there is forced convection and radiationheat transfer. At the external lid surface, free convection- and radiation heattransfer were taken into account. The following equations apply:

�br;lcp;br;l@Tl

@t=

@

@z(kbr;l

@Tl

@z) (25.178)

�kbr;l@Tl

@zjz=0 = hc(Tg � Tljz=0) +Fgs�(T 4

g � T 4l jz=0) (25.179)

+Fl;p�(T 4l jz=0 � T 4

p jz=Lz)

�kbr;l@Tl

@zjz=Lz;l = hc(Tg � Tljz=Lz;l) + ��l(T

4a;1 � T 4

l jz=Lz;l) (25.180)

Tl is the temperature in the lid. Tp;z=Lz denotes the pit surface temperature. Ta;1is the temperature in the surrounding air. Fl;p is the form factor for radiationbetween the pit top surface and the inner surface of the lid. Fgs is the form factorfor radiation between the combustion gas and the inner lid surface. The localcoordinate system used in the lid is shown in Figure 25.15.

25.10 The Combustion Gas 425

y

x

z

y Part A

Part B

Convection and Radiation

Convection and RadiationFire-shafts @T

@y= 0

@T@x

= 0 @T@x

= 0

Figure 25.14: Boundary conditions for a half pit.

z

x

Gas

flow

Foundation

Convection and Radiation

Lid

z

Forced convection and Radiationfrom the under-lid region

Free convection and Radiation

Constant Temperature

from the under-pit region

@T@x

= 0@T@x

= 0

Figure 25.15: Boundary conditions for the ring furnace foundation and lid.

25.10 The Combustion Gas

25.10.1 Introduction

Due to constraints in ring furnace design, it is impossible to achieve an idealgas ow pattern in the furnace with an equal amount of gas distributed to the

426 Modelling the Baking Process

ues. Rather, the gas ow pattern is quite complicated and di�cult to describeby models. Gas transport is turbulent, and state variables which describe the gas(temperature, pressure, density and composition) varies at every point in spaceand time. The important aspect is to achieve knowledge of the amount of energyavailable and how it is spent; i.e. how much energy is:

� Transferred to solid materials

� Lead away with the gas ow

� Lost to the surroundings

The division of the gas path into six calculation zones as discussed in Subsec-tion 25.2.1 is repeated in each section. Both convectional and radiative heattransfer is modelled.

25.10.2 Characteristic Features of the Gas Path

Spatial Temperature Gradients in the Gas Phase

During furnace operation, large temperature gradients in space exist in each com-bustion chamber (i.e. the headwall in part A and under the cover in part B) of asection. A description is outside the scope of this work. The temperature asym-metry depends mainly upon the �ring strategy used in each combustion chamber.

For �ring under the section cover, combustion is very turbulent and good mixingtakes place. A uniform distribution of temperature to the ue channels in part Bmight be a good approximation.

Headwall �ring is more complicated. It seems possible to achieve an almost uniformdistribution of temperature to the vertical ues in part A of the cassette wall byusing a proper �ring technique. Since the coldest parts of the section is situated atthe sides of the section, more fuel should be lead to the sidewall burners than thetwo center-burners. This may contribute to making the temperature distributionin the ues in part A more unifrom. The main di�culty is to establish this �ringstrategy as a natural part of the furnace-operation procedure.

Combustion phenomena in the ring furnace are complex. Oil is injected intothe combustion chamber by the aid of oil burners of impulse type. Combustiontakes place in turbulent di�usion ames. As a �rst approximation, the followingassumptions are made:

� Ideal mixing is assumed in the combustion chambers in the headwall andunder-lid-zones. Thus, spatial temperature gradients are neglected.

� The gas entering the under-pit region in part A is assumed to have the sametemperature for each cassette.

� The gas entering the ue walls in part B of each pit wall is assumed to havethe same temperature.

25.10 The Combustion Gas 427

The reasons that justify these assumption are as follows:

� Since impulse type burners are used, there will be large temperature uc-tuations in the combustion chambers. The average temperature in time,however, will be quite steady.

� Combustion under the section-cover takes place perpendicular to the direc-tion of the gas ow and combustion occurs in a streched ame. Firing is donefrom both sides of the section, and ames cover almost the whole volumeunder the section when the combustion takes place at optimum conditions(i.e. abundant access of O2). Also, the gas ow has a lot of momentum andis able to induce mixing in the ames. Then there is a good mixing of thegas ow and temperature gradients are smoothed out over time.

� Combustion in the headwall uses a maximum of four burners of impulse typeat equidistant positions. Assuming all burners operative, there will also bea good mixing of the gas ow in the headwall.

� The validity of these assumptions is strongly depend on abundant access ofO2 and symmetric �ring conditions.

The Vertical Flues

The vertical ues constribute to the dominating part of the boundary conditionsfor calculating the temperature �eld in the solid materials of brickwork, packingcoke and anodes. Therefore, it is important to give a correct model of this part ofthe boundary conditions. Each pit wall consists of a certain number of vertical uechannels. The calculation of heat transfer coe�cients (convection and radiation)depends on both channel geometry and channel mass ow. To get a realisticmodel of the heat transfer conditions, it is necessary to base the model on correctgeometrical data as well as realistic local mass ows of gas.

On the other hand, the heat uxes from the ues into the solid materials (brick,coke and anodes) will be �ltered along the path into the anodes. Thus, fromthe anodes point of view, is seems as if the heat ux is more or less uniformlydistributed over the vertical cross section in parallel with the pit wall. In this wayone may argue that there is no need to model the heat uxes coming from eachsingle vertical ue.

Two approaches therefore seem possible:

� Detailed approach: Individual treatment of each ue channel in the ue wallto obtain realistic boundary conditions for the solid material energy balance.

� Lumped approach: In each part of the pit wall, some ues could be lumpedtogether in a rectangular slit. In the simplest case, all ue channels in a partA are lumped together in a single rectangular slit. The same is done for partB.

In the ring furnace model, both approaches are implemented.

428 Modelling the Baking Process

Heat Losses

Heat loss occurs across three main areas:

� Section-cover

� Section foundations

� Section sidewalls (both inner and outer sidewall)

� Headwalls (which give tranport of energy between sections)

The headwalls do not contribute very mush to the heat loss (Bourgeois et al. 1990,pp. 548). From the top view in Figure 25.3 the following may be observed:

� There are two cassette walls which act as sidewalls in the section. Sidewalllosses take place across these walls

� There are four inner cassette walls in the section

� There are �ve cassettes �lled with anodes in the section

Heat Transfer Between Sections

During ring furnace operation, there will be a heat ux between neighbouringsections via the dividing wall which separates the sections. Depending on thesections' position in time along the �recycle as well as the relative position to theother �re zone, there will be varying net heat transport to a section.

Gas Flow Distribution in Flue Channels

To obtain the gas ow distribution along the ring furnace gas path, a solutionthe Navier- Stokes equations in three dimensions is needed. This is an unsuitableapproach, and the following procedure was used as a good alternative:

� The pressure in the ring furnace is assumed constant.

� Mass ow distribution to the ues in part A and B of the ue wall is cal-culated by an empirical model based on experimental results and numericalsimulations of a real furnace of Hydro Aluminium design.

25.10.3 Summary of Gas Model Simpli�cations

Along the gas path, submodels based on the heat balance in Equation (25.145)were derived for each zone described in section 25.2. Except for the under-lidzone, channel ow was assumed for the gas ow. In the headwall in part A, fuelis sprayed into the main gas ow via vertically positioned burners. In the vertical

25.10 The Combustion Gas 429

ues, pitch volatiles enter via cracks and joints in the brickwork. In summary, theother assumptions are as follows:

1. The gas phase is modelled as an ideal gas.

2. A stationary gas model is used.

3. Fuel combustion is assumed to complete in the headwall and under-lid re-gions.

4. Uniform temperature is assumed for the gas in the under-lid region.

5. Volatile combustion is assumed to complete in the vertical ues.

6. There is no model for air inleakage. Air in�ltration is taken care of byadjusting gas ow at section inlet.

7. Gas ow typically occur at high Peclet numbers Pe = �vdD. Pe is the ratio

between strength of convection and di�usion (Patankar 1980). Here Pe �1 and convection dominates compared to the dispersion (di�usion) termsrepresented by dispersion coe�cient D. Dispersion is thus omitted from thethe equations.

8. The mass ow distribution in the vertical ue channels is known from velocity�eld calculations performed in the commercial uid ow package FLUENT(also veri�ed experimentally). Based on these results an empirical mass owdistribution can be calculated.

9. Pressure along the gas path in a section is assumed constant.

10. The in uence of the the mass in the system of pillars which supports theunder-pit surface is neglected both with respect to disturbances in the gasvelocity pro�le as well as in the calculation of the heat balance in the under-pit zones.

11. The presence of a dividing wall in the x-direction between individual uechannels is neglected in calculation of the heat balance for gas owing in thechannels.

12. Average temperatures on the solid surfaces under the lid (top surface ofpacking coke and inner surface of lid) are used as driving temperatures inthe heat transfer calculations between gas and surfaces.

Since the mass ow distribution along the gas path is known a priori, calculationof impulse balances (the Navier- Stokes equations) do not have to be performed.This greatly simpli�es the simulations. Mass ow and temperature along thegas path can be found from coupled mass and energy balances. To obtain thesebalances, idealized model concepts were used for the calculation zones along thegas path: In the under-lid compartment, ideal mixing and uniform gas temperatureis assumed. Plug ow or branched ow of gas was assumed to occur in the othergas compartments. However, the under-pit region has side ows of gas from the

430 Modelling the Baking Process

vertical ues leaving or entering the zone in part A and B respectively. From theunder-pit zone in part A, gas enters the vertical ues in the pit wall belonging topart A. Gas from the under-lid region enters the vertical ues in part B and �nallyexit into the under-pit region of part B.

In the combustion zones, gas component mass balances is updated to obtain cor-rected values for the gas properties discussed in Subsection 25.10.6.

25.10.4 Mass and Energy Balance Equations for the Gas

General mass- and energy balance equations suitable for modelling of mass- andheat ow in the gas phase of the ring furnace were derived earlier in this chapter.Along the gas path, two di�erent types of models are used:

� The headwalls, under-pit channels and ue channels are modelled as a onedimensional gas channel with side ow of either combustion gas, volatiles orfuel oil.

� The under-lid compartment is modelled as a volume with uniform gas tem-perature and gas composition.

Conservation Equations for The Combustion Gas

The combustion gas is a multicomponent mixture of ideal gases. Then, the samekind of model as used for the combustion gas was also used for modelling themixture of volatile gases in the porous anodes and packing coke bed26. However,some characteristics must be speci�cly taken into account:

� Along the gas path, there is only one phase present; the gas.

� Combustion reactions take place along the gas path.

� Mass ows of inleaking air and fuel or combustible volatiles may enter thegas ow.

� Convective and radiative heat transfer occur between the gas and the brick-work surfaces in the gas channel.

Below, model equations for one dimensional gas ow along a channel is presented.

The Mass Balances. The total mass balance equation becomes:

@�g

@t+r(�gv) =

1

Agf (25.181)

26If the gas phase �lls the whole volume (and there is no interface mass transfer to the sur-rounding) the previous gas equations apply with Rsg;i = 0 (i = 1; : : : ; n) and �g = 1.

25.10 The Combustion Gas 431

where gf is the feed mass ow per unit length of the gas channel. A is the crosssection of the gas channel. In this case, the nabla operator denotes one of @

@x or@@z

and v is one of [vx; 0; 0] or [0; 0; vz].

For the gas components, the mass-conservation equations become:

@(�g;i)

@t+r(�g;ivi) = Rg;i +

1

Agf;i (25.182)

Rg;i = (ST rg)i (25.183)

Rg;i is the net rate of generation of component i in the combustion reactions.

It may be convenient to convert the component mass balance equations to a setof equations for the mass fractions. Introduce �g;i = xi�g and obtain:

@�g;i

@t= xi

@�g

@t+ �g

@xi

@t(25.184)

Now, use the expression for the total mass balance and get:

�g@xi

@t= �xi(�r(�gv) +

1

Agf;i)�r(�gvixi) + (ST rg)i +

1

Agf;i (25.185)

This gives:

�g@xi

@t= xir(�gv)�r(�gvixi) + (ST rg)i + (1� xi)

1

Agf;i (25.186)

This formulation is suited for use with the control volume formulation. By re-solving the spatial derivatives in the above equation, the following equation isobtained:

�g@xi

@t= ��gvrxi + (ST rg)i + (1� xi)

1

Agf;i

or:

@xi

@t= �vrxi +

1

�g(ST rg)i + (1� xi)

1

A�ggf;i

Often, it is convenient to retain the mass ux in the equation since the velocity isof no interest. Thus:

@xi

@t=

1

�g

��grxi + (ST rg)i + (1� xi)

1

Agf;i

�(25.187)

can be used as an alternative. Here, g = �v denotes the mass ux vector.

The Momentum Balance. For simplicity, the gas pressure is assumed to beatmospheric and the mass ow distribution (of volatiles) along the boundaries ofthe gas path is calculated from an empirical model; i.e. Darcy's law. Therefore,a separate momentum balance equation for the combustion gas is not needed atthis stage.

432 Modelling the Baking Process

The Energy Balance. Heat transfer along the gas path is due to convection andradiation at the gas channel surface; heat conduction in the gas can be neglected.Also, a certain amount of energy accompany the mass ow which enters the gaschannel. Hence:

@

@t(�gHg) = �r(�gHgv) +

1

Agf Hf +Q+

@p

@t(25.188)

Hf is the enthalpy of the in owing medium. Q is a heat transfer term due toconvection and radiation at the gas path boundaries.

Along the combustion gas path, pressure variations are not dramatic and the term@p@t

may be neglected. This gives:

@

@t(�gHg) = �r(�gHgv) +

1

Agf Hf +Q (25.189)

Example: Conservation Equations For the Under-Pit Channel in Part

B

To illustrate the modelling procedure which was used for the mass- and energybalances for the under-pit zone in part B is presented. The main gas ow is in thex-direction of the furnace with side ows from the vertical ues. As mentionedabove, the ue channel mass ow distribution is known from calculations andexperiments.

x x +�x

Gx = GLGx+�x

hg;x hg;x+�x

gBhg;B

Qup;B

j j + 1j

�x

Figure 25.16: Typical under-pit gas computational cell. Note the indexing of theboundary control volume surfaces. Boundary surfaces in control volumes bothalong the gas path and in the solid materials are indexed in this way (see also Fig-ure 25.13). Structurally the same kind of control volume was used for establishingequations for mass- and energy conservation in all the zones along the gas path(i.e. headwall part A, under-pit part A, ue channels part A, ue channels part Aand under-pit part B) except the under-lid zone (part B).

25.10 The Combustion Gas 433

The total mass balance is given by:

@�g

@t= �@(�gv)

@x+

1

AggB

For the component mass balances, the equation becomes:

@�g;i

@t= � @

@x(�g;iv) + (Srg)i +

1

AggB;i

The term (Srg)i represents formation or loss of component i due to chemicalreactions (i.e. combustion). In this case, combustion is completed before the gasreaches the under-pit region in part B. Therefore rg = 0.

The stationary mass balance equations become:

@G

@x= gB

@

@x(xiG) = Ag(Srg)i + gB;i

G = �gvAg [kg/s] denotes mass ow where �g is gas density, v is gas velocity,and Ag is the cross section of the gas ow. Ag has di�erent values for each zonealong the gas path. gB [kg/(m s)] is mass ow per unit length in x-direction whichcomes from the vertical ue channels. gB is positive for gas ow entering thecontrol volume. xi is the mass fraction of component i.

The energy balance equation is as follows:

@

@t(�gug) = � @

@x(�gvhg) +

1

AggBhg;B +

1

AgQup;B

ug is the speci�c internal energy of the gas, hg [J/kg] is speci�c enthalpy of thegas ow, and hg;B is speci�c enthalpy27 of in owing gas from the vertical uechannels. Qup;B [W/m] is heat transfer (conductive and radiative) from refractorybrickwork to gas per unit length of the furnace in the under-pit region of part B.

The mass ow G = �gvAg can be used to obtain:

�@(Ghg)@x

+@G

@xhg;B +Qup;B = 0

for the stationary heat balance since gB = @G@x .

25.10.5 Summary of Gas Path Equations

An overview of dynamic mass- and energy balances along the gas path is givenbelow. Stationary versions of the equations, as shown in Subsection 25.10.4 wereused in the simulations. In the list of equations, the two �rst equations correspondto the total- and component mass balance equations respectively.

27For an ideal gas, hg = ug + p 1�g

where p is the gas pressure.

434 Modelling the Baking Process

1. Headwall:

@�g

@t= �@(�gv)

@x+

1

Aggf (25.190)

@�g;i

@t= �@(�g;iv)

@x+ (ST rg;A)i (25.191)

rg;A =1

Aggf (25.192)

@

@t(�gug) = �@(�gvhg)

@x+

1

AggfHf +

1

AgQhdw (25.193)

gf [kg/(m s)] is the fuel ow per unit length (in the direction of gas ow)entering the headwall normal to the main gas ow. The term (ST rg)i rep-resents the the conversion of fuel oil into gaseous products where rg is thee�ective fuel oil reaction rate corresponding to immediate and total conver-sion of the oil in the headwall.

2. Under-pit in part A:

@�g

@t= � @

@x(�gv) +

1

AggA (25.194)

@�g;i

@t= � @

@x(�g;iv) +

1

AggA;i (25.195)

@

@t(�gug) = �@(�gvhg)

@x� gAhg +

1

AgQup;A (25.196)

gA [kg/(m s)] is the mass ow per unit length of the furnace (in the directionof gas ow) entering the vertical ues in part A (i.e. leaving the under-pit region of part A). gA is negative for gas exiting the under-pit zone.Combustion is assumed to complete in the headwall. Therefore no termrelated to chemical reactions occurs in the equation.

3. Vertical ue channels in part A:

@�g

@t= � @

@z(�gv) +

1

Ag

nvXi=1

gv;i (25.197)

@�g;i

@t= � @

@z(�g;iv) + (ST rg;chn;A)i (25.198)

rg;chn;A =

�1

Aggv;i

�(25.199)

@

@t(�gug) = � @

@z(�gvhg) +

1

Ag

nvXi=1

gv;iHv;i +1

AgQchn;A (25.200)

gv;i [kg/(m s)] is the volatile mass ow per length of ue channel. Qchn;A

[W/m] is the heat transfer per length of ue channel. rg;chn;A correspondsto the e�ective reaction rates for the volatiles.

25.10 The Combustion Gas 435

4. Under-lid:

d(�gVulid)

dt=

nchn;AXj=1

GA;j �G+Gf (25.201)

d(�g;iVulid)

dt=

nchn;AXj=1

GA;j;i �Gi + (ST rg;ulid)i (25.202)

rg;ulid =Gf

Vulid(25.203)

d(�gVulidug)

dt=

nchn;AXj=1

GA;jhA;j �Ghg +GfHf (25.204)

GA;j = �gvA;jAg is the mass ows from the vertical ue channels in part Awhich enter the under-lid zone. vA;i is the ue channel exit velocity. HereAg denotes the cross section of the ue channels in part A.

5. Vertical ue channels in part B:

@�g

@t= � @

@z(�gv) +

1

Ag

nvXi=1

gv;i (25.205)

@�g;i

@t= � @

@z(�g;iv) + (ST rg;chn;B)i (25.206)

rg;chn;B =

�1

Aggv;i

�(25.207)

@

@t(�gug) = � @

@z(�gvhg) +

1

Ag

nvXi=1

gv;iHv;i +1

AgQchn;B (25.208)

6. Under-pit part B:

@�g

@t= � @

@x(�gv) +

1

AggB (25.209)

@�g;i

@t= � @

@x(�g;iv) +

1

AggB;i (25.210)

@

@t(�gug) = �@(�gvhg)

@x+ gBhg;B +

1

AgQup;B (25.211)

gB [kg/(m s)] is the gas ow per unit length (in direction of gas ow) enteringthe under-pit zone from the vertical ues in part B.

The equations for the under-pit zone in part B are repeated for convenience. De-tails on the boundary conditions for the above equations are not supplied in thistext. For each of the gas compartments, the relationship between enthalpy and

436 Modelling the Baking Process

internal energy can be used as before:

�gUg =

ngXj=1

�g;jUg;j (25.212)

�gHg =

ngXj=1

�g;jHg;j (25.213)

Furthermore, the enthalpy hg can be introduced on the left hand side of the energyequations by the application of:

Ug = Hg �p

�g(25.214)

In the resulting equations, the term @p@t

should be neglected from in the energyequations since the rate of change of pressure along the ues is negligible comparedto the other thermal energy terms.

Di�usion is neglected in this case since advection dominates compared to di�usionin the ow of combustion gas along the gas path.

It should be noted that the combustion enthalpies for fuel oil and volatiles donot explicitly appear in the energy equations. This is formally correct since theenthalpy of combustion will only appear after resolution of the left hand side ofthe energy equations to obtain a temperature explicit model.

25.10.6 Gas Properties

For heat transfer calculations, the following properties were needed:

1. Density: �g

2. Speci�c heat capacity: cp;g

3. Thermal conductivity: kg

4. Viscosity: �g

5. Emissivity: �g

Mean gas properties were calculated from knowledge of the gas composition. Frac-tions of nitrogen N2 , oxygen O2, carbon dioxide CO2 and water vapour H2O(g)were considered since the combustion calculations were based on instantaneousand ideal combustion as discussed in Subsection 25.4. Gas density is calculatedfrom the ideal gas law:

�g =pM

RT(25.215)

25.10 The Combustion Gas 437

where M is the average molar mass of the gas mixture:

M =

nXi=1

1xiMi

(25.216)

xi is component mass fraction and n is the number of gas components. Speci�cheat capacity is calculated as a weighted average value as follows:

cp;g =

nXi=1

xicp;g;i (25.217)

Correlations for gas component heat capacities were taken from Reid et al. (1987).

Gas mixture thermal conductivity and viscosity were calculated from the Wassil-jewa equation (Mason & Saxena modi�cation) and Wilke correlation respectively(Reid et al. 1987, pp. 530, 531, 407):

kg =

nXi=1

xikg;iPnj=1 xj�i;j

(25.218)

�g =

nXi=1

xi�g;iPnj=1 xj�i;j

(25.219)

�i;j =[1 + (�i

�j)12 (

Mj

Mi)14 ]2

[8(1 + Mi

Mi)]

12

(25.220)

Component thermal conductivities and viscosities were obtained from Reid et al.(1987, Tab. 10-3, pp. 515-516 ) and Lydersen (1983, App. 5, pp. 314).

For calculation of mean gas emissivity, only emissivities of CO2 and H2O wereconsidered since they are among the most important nonsymmetrical gases forradiative emission. Diagrams for �H2O and �CO2

as function of temperature, partialpressure and mean beam length were found in Kreith & Black (1980, pp. 353-354).The emissivity of the gas mixture was calculated from:

�g = �H2O + �CO2��� (25.221)

at a total pressure of 1 bar. �� accounts for the overlapping radiative wavelengthbands of H2O and CO2. The correction term �� was set to zero. Calculation ofgas emissivity was done by taking into account variations in partial pressures ofCO2 and O2, mean beam length and temperature.

The gas emissivity calculated by the above expression is not valid in the amezone. For luminous ames, emissivity data should be derived from experimentswith ames similar to that of interest. In this case, experimental results werenot available. But since ame radiative heat transfer is based on a mean ametemperature, emissivity was calculated by adding 0.1 to the non-luminous gasemissivity to include luminous soot emissivity in the ame zone (Perry & Green1984, pp. 10-63).

438 Modelling the Baking Process

25.11 Discussions and Conclusions

In this chapter, a model for the heat balance in a Hydro Aluminium ring furnaceis presented. This kind of ring furnaces have a modi�ed gas path compared to theconventional closed furnaces and thus allows for a better heat distribution alongthe ues.

In the model, the basic ring furnace phenomena are included. Focus is put on cal-culation of the heat balance for the furnace. To arrive at a model with a reasonablecomplexity, a lot of simpli�cations were introduced and the e�ect of these simpli-�cations can only be tested by comparing experimental data with simulations. Atpresent, no systematic comparison of the model and experimental data has yetbeen performed, but the simulation results are comparable to measurements doneon a real furnace of Hydro Aluminium type.

Chapter 26

Simulating the Baking Cycle

The implementation of the ring furnace model is described in this chapter. At�rst, a brief description of the steps in development of the model is given. Then,the RF3D program system is described along with some simulations.

26.1 Development of the Numerical Model

The development of the model for calculation of the heat balance of a half cassetteand the belonging gas path evolved in several steps brie y summarized as follows:

1. One dimensional heat conduction and mass transport in y-direction: Gas,brickwork, coke bed and anode.

2. Two dimensional heat conduction analysis in the xy-plane of a quarter of a ue channel together with brickwork, packing coke and anode.

3. Mass transport in a ue channel and corresponding heat conduction in thesolid materials of brickwork, packing coke and anode; i.e. analysis in thexz-plane.

4. Full three dimensional heat conduction in the solid material of a half cassette.The corresponding gas path was divided into six zones and models derivedfor each zone. In the model, the ue channels are modelled individually orlumped together in groups (with three or four channels in each group).

This stepwise procedure contributed to avoid problems in the programming of themodel as well as for studying the in uence of di�erent geometric simpli�cationsalong the gas path. In this context, three major simpli�cations were performed:

� The in uence of the brickwork support system in the under-pit region onboth the gas ow pro�le and the heat balance was neglected.

440 Simulating the Baking Cycle

� The in uence of the wall in the under-pit region (which separates part Aand part B) on the heat balance was neglected.

� The in uence of the dividing walls which separate the ue channels wasneglected. This leads to a reduction in the total heat transfer area betweengas and brickwork surface. On the other hand, simulations have shown thatthe temperature �eld of the solid materials is overestimated with up to 50�Cas compared to the situation when the dividing wall between ue channels isincluded in the conduction domain. The largest deviation occurs in the earlystage of the heat treatment cycle. By neglecting the presence of the dividingwalls, the geometry of the calculation domain was signi�cantly simpli�ed.

26.2 Computational Procedure

Inlet conditions for a section were obtained by using nominal values for gas tem-perature and composition in the headwall. The most time consuming part of thealgorithm was calculation of the three dimensional temperature �eld in the pit. Anexplicit integration method was used for solving the heat conduction equations.The time step had to be chosen in the order of 100 seconds to assure stability ofthe numerical scheme. Initial values for the numerical solution were obtained asfollows:

1. Initial value for solid materials: Room temperature T = 20�C.

2. Initial value for gas: Solve gas equations for given input conditions in head-wall and use the initial solid temperature �eld as boundary condition.

The computations were performed as two main tasks at each time step:

1. Based on speci�ed refractory surface temperature and headwall gas inputconditions:

(a) Calculate boundary conditions for the gas model equations.

(b) Solve the mass- and energy balance equations along the gas path in thedirection of gas ow.

2. Based on speci�ed gas temperature and composition along the gas path:

(a) Calculate solid boundary conditions.

(b) Integrate the heat conduction equations using an explicit integrationmethod and temperature varying properties.

Alternating between steps 1 and 2, the simulation was performed over a timeinterval which covers the preheat and heating periods of a �re cycle. The coolingperiod of the baking cycle was not simulated. For the under-lid zone, the amountof fuel necessary to maintain a predetermined temperature pro�le in time, wascalculated.

26.3 The Explicit Integration Scheme 441

In the present model implementation, solid and gas properties are updated at acertain number of time steps (which can be selected by the user). The temperature�eld is calculated by using the explicit Euler method with no time step control.

26.3 The Explicit Integration Scheme

Integration of the pit temperature �eld was performed by application of Euler'smethod. Compared to other explicit integration schemes, Euler's method workswell for the ring furnace system due to two reasons:

� Accuracy : The accuracy of Euler's method is good enough since the deriva-tives of the temperature �eld are very smooth. Tests with Runge Kutta- andpredictor-corrector-methods gave no improved accuracy.

� Simulation time: The stability criterion for some high-order integrator schemesare comparable to the stability criterion for Euler's method. Therefore, ap-plication of high-order explicit integration schemes may increase simulationtime several times since the most time consuming part of the integration iscalculation of the derivatives.

Comparison of explicit and implicit schemes on this problem has not been done.However, the application of an implicit integration scheme would necessitate si-multaneous solution of a large number of nonlinear equations. This is often a verytime-consuming operation.

So far, no time step control is used in the simulator. Implementation of time stepcontrol would reduce simulation time.

The solution procedure of alternating solution of gas- and solids-equations has alsobeen used elsewhere (Bourgeois et al. 1990).

26.4 RF3D Program System

An overview of the �le structure of the simulation program for the ring furnaceis given in Figure 26.1. The simulator was written in C. No e�ort was done todevelop a user interface for the program. Simulations are performed batch-wiseand data stored on a MATLAB readable �le format. Postprocessing of data isdone in MATLAB.

In the program, certain parameters can be set which controls the behaviour of thesimulation:

� Parameters which determine the type of simulation:

1. The mass ow, composition and temperature of the gas which entersthe section in headwall part A are assigned nominal values. It is as-sumed that the fuel ow to the headwall and the gas temperature in

442 Simulating the Baking Cycle

the headwall follow values according to the �recurve speci�ed for partA. This alternative is useful for simple comparison of �re strategies.

2. Mass ow, composition and temperature which enter the section at theheadwall in part A vary freely. This alternative must be used when thesection model is used in simulation of a chain of sections. The model isyet not upgraded to treat simulation of a chain of sections.

� Grid layout:

1. The grid-layout follows Patankar's practice A with special treatment ofthe control-volumes at the brickwork boundary in the headwall.

2. The grid along y and z directions may be selected freely except whenthe grid in the y-direction must �t with outblocking of the headwall�reshaft.

3. The grid along x-direction must be coordinated with the geometricalarrangement of ue channels as well as the type of headwall geometryused:

{ Individual or groups of ue channels may be simulated. This isdirectly taken care of by specifying the number of grid points1.

{ The headwall may be adiabatic or non-adiabatic. If the headwallis non-adiabatic, outblocking is used to implement the geometry ofthe headwall �reshaft.

4. The grid is uniform within each type of solid material.

� Simulation with adiabatic headwall (i.e. no outblocking) with no heat trans-fer from the �reshaft to the brickwork in the �reshaft:

1. Nominally, a uniform grid within each material of the pit is used.

2. A non-uniform grid is also allowed along x- and y-axes. This grid iscoordinated with the grid used for simulation of the outblocked headwallin the non-adiabatic headwall. This was allowed to be able to easilycompare simulations with an adiabatic and non-adiabatic headwall.

� Simulation with a nonadiabatic headwall is performed either with or withoutoutblocking:

1. In the case with no outblocking, a certain heat ux is assigned to thepart of the headwall surface in the yz-plane which corresponds to theposition of the �reshaft.

2. In the outblocked case, a realistic geometry of the headwall �reshaft isused. This calls for the need of using a non-uniform grid in the xy-planethrough the headwall.

� Summary of other status variables:

1In part A the grid is coordinated with individual ue channels or groups of channels withthree channels in each group. Part B is treated in the same way but with groups of three orfour channels. This gives a total of three realistic alternatives; the total number of grid points is(9,12), (3,4) and (3,3) in part A and B respectively.

26.5 Simulation Case 443

1. A uniform or linear mass ow pro�le may be used in the ue channels.This is useful to study the impact of the use of one �ctive ue channelboth in part A and part B.

2. Calculation of the pyrolysis reactions and combustion of carbonizationgases may be switched o�. This is useful for studying the impact ofcombustion energy coming from the volatiles.

3. Either average or time-varying solid material properties may be used.

4. The number of samples between each update of solid properties and gasproperties can be individually speci�ed.

5. Di�erent �recurves may be selected.

26.5 Simulation Case

Baking behaviour during preheating and heating parts of the �re-cycle was sim-ulated. Gas ow input conditions (temperature and composition) were speci�ed.Furthermore, a time varying mass ow input pro�le was used to represent the in-crease in mass ow of gas due to air inleakage. A �re-step of 36 hours was chosenand the model was simulated over a time interval of 180 hours corresponding to5 �re-steps. The under-lid fuel consumption needed for maintaining the under-lidtemperature tracking a certain trajectory, was calculated (see Figure 26.2). Thedesired temperature trajectories (normalized values) in headwall and under-lid (parts A and B) are shown in Figure 26.3.

In the numerical model, the grid resolution can be speci�ed by the user. In theresults presented, a �ne mesh was used with approximately 6000 grid points inthe solid materials of pit refractory, coke and anode. Correspondingly, along thegas path there were in the order of 500 grid points. One computing cycle takesapproximately one hour on a Sparc Work Station. Data from the simulations arestored on a MATLAB-readable format. Postprocessing and graphical presentationare done in MATLAB.

In Figure 26.4, the time history of the (normalized) temperature in the xz-planethrough the anode center is shown. Early in the baking cycle, the hottest parts ofthe anodes face the brickwork foundation in the pit. This tendency is maintainedup to baking times of approximately 130 hours. Then the e�ect of the under-lidburner now operative for approximately 50 hours becomes visible: the temperature�eld become more uniform. After 180 hours, the highest temperatures occur onthe top surface of the anodes in part B of the pit. The lowest temperature in thecenter of the anodes (xz-plane) was detected in part A (close to the headwall) ofthe pit. Even though the burner in part A is operated by vertical �ring through a�re-shaft in the headwall, there is loss of heat through the headwall. The coldestposition occur in the headwall region of part A. The temperature pro�le has asaddle-like shape, and it can be concluded that the pro�le is sensitive to theheadwall boundary condition.

444 Simulating the Baking Cycle

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26.5 Simulation Case 445

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

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1Normalized under−lid fuel consumption

Time [hr]

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Figure 26.2: Normalized under-lid fuel mass ow. The burner is switched on afterapproximately 80 hours of baking, since the gas temperature has to reach a certainlevel before ignition can occur.

The sensitivity of the temperature distribution to an alternative headwall bound-ary condition was tested. The simulations were repeated with an adiabatic bound-ary condition speci�ed for the whole headwall. The temperature di�erence (orig-inal vs. new headwall boundary condition) in the xz-plane through the anodecenter is shown in �gure 26.5. The adiabatic headwall boundary condition causesan even lower temperature in the anode in regions facing the headwall than ob-tained with the original boundary condition. The e�ect is most marked in thepart B region of the pit.

The di�erence between anode surface temperature (7 cm below anode surface) inthe xz-plane and anode center temperature (xz-plane) is shown in Figure 26.6.Up to 60 hours of baking, the highest gradients (xz-plane) occur in part A. After90 hours of baking, the largest gradients are found in part B. After 180 hours ofbaking, the minimum temperature di�erence is found at the top of the anodes inpart A. The maximum temperature di�erence is found in the bottom of part B.

The typical lag between gas-, anode surface-, and anode-center (normalized) tem-peratures is shown in Figure 26.7. The lag between gas and solid temperatures issigni�cant. It may be inferred that design of ring furnace baking strategies is nota straightforward task. To arrive at anodes with uniform properties, care must betaken during critical parts of the baking process.

Finally, some plots of volatile generation (source terms) based on Tremblay's modelare presented. In Figure 26.8, a maximum of 4.3 kg=(m3 s) tar volatiles is released

446 Simulating the Baking Cycle

0 20 40 60 80 100 120 140 160 1800.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Normalized firecurves: Headwall part A (lower) and under−lid (upper)

Figure 26.3: Normalized �re curves for ring furnace operation. In traditional ringfurnace operation, tracking along these curves is implemented either by manual orautomatic control.

from a certain position in the xz-plane through the center of the anodes. Atypical snap-shot of the tar volatile generation (after 94 hours of baking) is seenin Figure 26.9. In parts of the anode facing brickwork and top coke layer, volatilegeneration has ceased. Also in part A and B volatile generation is about to cease.Note that the generation rate in the cold parts neighbouring the headwall areashas a certain lag compared to the generation rates observed in the central part ofthe center xz-plane.

26.6 Discussions and Conclusions

A lot of parameters is needed in the model and the modelling of boundary con-ditions for the pit heat conduction problem is not straightforward. Two di�erentheadwall boundary conditions were tested and it was shown that the model issensitive to the boundary conditions used. A lot of work remains in the study ofmodel sensitivity to parameter variations.

The equations for the subsystems of gas and solids were solved by using a sta-tionary model for the gas equations. The control volume approach was used forderiving the numerical model, and integration of the pit temperature �eld equa-tions was performed by the application of Euler's method.

26.6 Discussions and Conclusions 447

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Figure 26.4: Temperature history in the center of anodes (xz- plane) after 180hours. Data are normalized such that maximum value 1 corresponds to the max-imum temperature of the anodes during the whole baking cycle. The axes denotegrid points in x- and z-directions. To be able to show the details in the temper-ature pro�les, the interval along the normalized temperature axis varies betweenthe di�erent panels.

448 Simulating the Baking Cycle

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Figure 26.5: Comparison of headwall boundary conditions: Original �reshaftboundary conditions vs. adiabatic headwall boundary conditions. The data arenormalized according to the maximum di�erence occurring. The interval alongthe normalized temperature axis changes from panel to panel.

26.6 Discussions and Conclusions 449

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200

510

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Temp. diff. (xz−plane) : t = 180 hr

x−axisz−axis

T/m

ax(T

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Figure 26.6: Normalized temperature di�erence in the xz-plane between anodesurface (7 cm below anode surface) and anode center. The maximum di�erenceis found in part B. The temperature is normalized such that maximum value 1corresponds to the maximum temperature di�erence occurring during the bakingcycle.

450 Simulating the Baking Cycle

0 20 40 60 80 100 120 140 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Normalized flue channel temp. v.s anode temp. (surface and center)

Time [hr]

T/m

ax(T

)

Figure 26.7: Typical lag between normalized temperatures in the gas (upper curve)and anodes (surface (intermediate curve) and center (lower curve)).

0 20 40 60 80 100 120 140 160 1800

0.5

1

1.5

2

2.5

3

3.5

4

4.5Tar generation rate

Time [hr]

Tar

gen

erat

ion

rate

[kg/

hr]

Figure 26.8: Typical tar volatile generation rate in the anodes as function of time

26.6 Discussions and Conclusions 451

05

1015

20

0

5

10

0

1

2

3

4

Tar generation rate in xz−plane through anode center

x−axisz−axis

r_ta

r [k

g/m

^3 h

r]

Figure 26.9: Tar volatile generation in the center (xz-plane) of the anodes after94 hours.

452 Simulating the Baking Cycle

Chapter 27

Model Based Control of

Ring Furnaces

A model based approach to control of ring furnaces is presented. A discussionof previous work on control of ring furnaces was given in Subsection 1.1. Thediscussion on control of ring furnaces from the point of view of modern controltheory in Subsections 27.2, 27.3 and 27.4 is based on Gundersen (1995a). Therelated modelling problem was summarized in Subsection 25.1.2.

27.1 Tuning of the Calcining Level

In operation of ring furnaces, tuning of the calcining level is an important aidfor optimization of anode quality. Changes in the calcining level is achieved byadjustment of the heat treatment program.

Two di�erent approaches for tuning the calcination level is discussed in the fol-lowing.

27.1.1 The Conventional Approach

Often, mathematical models are used as an aid for studying the evolution of thetemperature distribution in a section during baking (Gundersen 1995a). Suchmodels allow for e�ective computerized testing of di�erent �ring strategies. Still,however, the procedure of �nding an optimal �ring strategy has not been auto-mated.

The anode quality resulting from a certain heat treatment strategy has to beevaluated either via equivalent temperature related methods or by full scale mea-surement of intrinsic anode properties.

In summary, the implementation of a new �ring strategy and tuning of the cal-

454 Model Based Control of Ring Furnaces

cining level implies both computer assisted and laboratory assisted activity. Arelatively large amount of work is needed in the laboratory in routine analysis ofproduct properties. Thus, there are potentials to reduce this e�ort by introducingmore sophisticated computer based model tools.

27.1.2 A New Concept for Calcining Level Tuning

In this work, a model based approach to calcining level tuning is suggested. Withinthis approach, at least two possibilities exists:

� Model based prediction of the spatially equivalent temperature distributionin a ring furnace bake section

� Model based prediction of the spatial distribution of anode properties in aring furnace bake section

Since equivalent temperature TE is related to Lc of the standard petroleum coke,TE can be calculated from knowledge of Lc if Lc can be predicted from a model.In part II, modelling techniques for development of crystallite size in carbon ma-terials were presented. A model for Lc for the standard coke can be derived basedon these techniques. In this way, a valuable tool for model based prediction ofstructural evolution of the standard coke or equivalently TE is obtained. Such amodel is presented in Gundersen (1996d). The model includes a di�erential equa-tion for the evolution of Lc of standard petroleum coke as function of temperatureand a mapping between Lc and TE . The model has the same structure as themodels presented in this study for evolution of Lc as a thermally activated pro-cess. A model with van-Krevelen type activation energy seems to give satisfactoryprediction capability.

Based on the bake furnace model and additional calculation of the equivalent tem-perature history in each spatial point in the anode load of a section, the in uenceon the baking process of varying heat treatment programs could be studied bypure model simulations. Thus, explicit measurement of TE is not necessary pro-vided that the model has satisfactory predictive capabilities. This approach seemsattractive due to two reasons:

� The method is based on a concept well established in the aluminium industry.

� The concept is very simple since it is based on a dynamic model for devel-opment of Lc in the standard coke and knowledge of the calibration curvegiving equivalent temperature as a function of Lc (TE = TE(Lc)).

On the other hand, there is still room for improvements. Due to the lack of rawmaterial supplies, one has to expect that future anode production has to be donewith raw materials of frequently varying quality. After certain changes in rawmaterial characteristics, one cannot expect that a previous optimal calcining levelfor a section will give optimal anode properties. Actually, correlations between

27.2 Conventional Control Strategy 455

TE and anode properties are needed in each case. A lot of experimental work isneeded to obtain data to be used for establishment of such correlations.

Alternatively, one could perform direct calculation of physical properties of theanodes based on mathematical models for the anode properties. These propertymodels may be used in a model based approach designed for tuning the calcininglevel and achieve automatic control of baking furnaces. Such a model based ap-proach is discussed in Gundersen & Balchen (1993a) where model based controlof a chain of ring furnace-like sections is studied. This approach is attractive sinceit gives an opportunity to obtain direct control of the anode properties. Still,however, there is a need for updating the property model parameters after a rawmaterial change. In general, this is not a straight forward task, and a lot of workmay be needed in the laboratory for tuning the new model parameters.

In this kind of model based control approach, a systematic de�nition of anodequality is needed. Modelling of anode quality was discussed in part II of this study.For proper solution of the control problem, one needs to analyze the controllabilityof the baking process. The controllability analysis is presented at the end of thischapter.

27.2 Conventional Control Strategy

The conventional control philosophy for ring furnaces rely on the principles statedin Auchterlonie & Van der Toorn (1977): The temperature in the �rst preheatingsection(s) is mainly in uenced by the draught and to a lesser extent by the thecontrol of the �res. Based on this observation, the implemented control philosophyis as follows:

1. The gas temperature target curve in the preheat sections is controlled byfurnace draft (i.e. suction pressure).

2. The gas temperature target curve in the direct �red sections is controlled byfuel ow.

In most anode baking control systems, this kind of strategy is chosen. Generally,monovariable control loops are used for tracking along the reference temperaturecurve. Interactions from direct heated sections to the preheating sections via theburners and in the opposite direction via the draught is not explicitly taken careof. A typical gas reference temperature curve was shown in Figure 4.11.

27.3 Hydro Aluminium Control System

Previously, the furnaces at Hydro Aluminium in �Ardal were operated with a man-ual control strategy according to the conventional principle (J�rgensen 1991). Forone of the furnaces, computerized monitoring of the �res was installed to alert

456 Model Based Control of Ring Furnaces

the operators if deviations from the �re target curve occur. Then manual tun-ing of draught (exhaust manifold) and oil burners must take place in an iterativefashion until acceptable tracking is established (Jakobsen et al. 1987). Later, au-tomatic control of the burners in the direct-�red sections has successfully beenimplemented. In this way, ring furnace operation is partly automated.

The ring furnace at HA's plant Sunndals�ra was built according to the principles ofHydro aluminium ring furnace technology. The ring furnace installation in Dubai,represents state-of-the-art in both furnace design and control philosophy of theHydro Aluminium ring furnace technology.

At Sunndals�ra, a decentralized control system is used. A computer networklinks the local control systems together1. There are options for both manual andautomatic control. Today, a new furnace is contructed at Sunndals�ra accordingto the latest pronciples of HAL design.

27.4 Model Based Control Strategy

Application of modern multivariable control theory has been applied for ring fur-nace operation as discussed in Demange (1991). Focus was put on basic on-linecontrol with no discussion on optimization of the baking process and anode quality.

In this study, however, a two level control system is suggested which takes into ac-count both open loop optimization (optimization level) and on-line control (controllevel) of the baking process as illustrated in Figure 27.1.

The cooling part of the baking cycle is not included in the control strategy de-scribed in this work.

27.4.1 The Structure of a General Process Model

Design of a control strategy for the baking process depends on the level fromwhich the process is viewed. To give a description which takes into account inter-actions between sections, the whole �re train has to be considered. This is alreadycommon practice in the design of gas temperature target curves as discussed inSubsection 4.2.2. The input variables were presented in Section 4.5. Each �rezone, have the following input variables:

u = [ud1; : : : ; udn; udp]T (27.1)

1This is in contrast to the system in Dubai which is a centralized system using ABB Master-piece 100, Masterpiece 200 and Masterview 800.

27.4 Model Based Control Strategy 457

Process trajectories

Ring Furnace

Process

Optimization

Basic

Control

Figure 27.1: A hierarchical control system with two levels is suggested for the bak-ing process: At the upper level, open loop optimization is performed to optimizering furnace operation. On line control is performed around the optimized processtrajectory

where the local section control variables are represented with:

udi =

8>>>>>><>>>>>>:

udi;1 = wb1;di 1st under-lid burner fuel mass owudi;2 = wb2;di 2nd under-lid burner fuel mass owudi;3 = wa1;di 1st headwall burner fuel mass owudi;4 = wa2;di 2nd headwall burner fuel mass owudi;5 = wa3;di 3rd headwall burner fuel mass owudi;6 = wa4;di 4th headwall burner fuel mass ow

(27.2)

di denotes dynamic section number in the zone and n is the number of sections inthe �re train. udp denotes the draught pressure at the outlet of the section in frontof the section numbered d1. This suction pressure is established by manipulationof the exhaust manifold. Burner input variables are not available during the wholebaking cycle since burners cannot ignite at temperatures below a certain criticalvalue. Commonly, only two headwall burners are used in each section under �re.

The main purpose of the control strategy is to secure production of anodes ofa certain quality and optimal operation of the baking process. In Chapter 5,anode quality parameters were de�ned based on a discussion of anode behaviourduring electrolysis. In a section, there are several anodes stacked in the pits.Let q denote the spatially dependent anode quality vector. Furthermore, de�ne aproperty vector �q, which is supposed to describe the average anode quality in asection. The mapping between q and �q is a kind of spatial averaging. Thus, therelationship between fundamental carbon properties as a subset of state vector x,

458 Model Based Control of Ring Furnaces

spatial dependent quality q and average quality �q is de�ned as follows:

q = s(x) (27.3)

�q(t) =1

V

ZZZV

q(x; y; z; t) (27.4)

x without an underscore denotes spatial coordinate in the x-direction; x denotesthe state vector.

In each section, a batch of anodes are under heat treatment so the a global propertyvector consists of an augmentation of the local section properties. In general, thefollowing property vector ~z is supposed to describe the anode quality in a �re zone:

~z = [�qd1; : : : ; �q

dn]T (27.5)

�qdiincludes a subset of the following quality parameters:

�qdi=

8>>>>>>>>>>>><>>>>>>>>>>>>:

�qdi;1 = �rO2;di Airburn�qdi;2 = �rCO2;di CO2-reactivity�qdi;3 = �si;di Dusting index�qdi;4 = ��c;di Compressive strength�qdi;5 = �rts;di Thermal shock resistance�qdi;6 = �kt;di Thermal conductivity�qdi;7 = ��e;di Electrical resistivity�qdi;8 = ��b;di Baked apparent density�qdi;9 = ��di Coe�cient of thermal expansion

(27.6)

Again di denotes dynamic section number in the zone and n is the number ofsections. The subset of the property vector that can be controlled, depends on thecontrollability of the process as discussed in the �nal section of this chapter.

In this case, however, it seems reasonable to de�ne the property vector z as thespatially averaged quality parameter �q

dnsince section dn is the last direct �red

section which is considered. Thus:

z = �qdn

(27.7)

To implement the control strategy, a process model is needed. The �rst stepin model based control design is the development of a mathematical model forthe process. As a part of the contributions in this work, the development andsimulation of a state space model for the baking process were described in the twoprevious chapters. The model describes the relationship between input variablesand process states, (internal) quality and outputs. Using vector quantities, thefollowing de�nitions are used for these variables:

� Input vectors:

{ Control/manipulable variables: u

{ Disturbance vector: v

27.4 Model Based Control Strategy 459

� State vector: x

� Internal quality vector: q

� Output vectors:

{ Measurement vector: y

{ Property vector: z

A general model description which uses these variables is as follows:

_x = f(x; u; v) (27.8)

q = s(x) (27.9)

y = h(x) (27.10)

z = g(q) (27.11)

where x 2 Rnx , v 2 Rnv , u 2 Rnr , q 2 Rnq , y 2 Rm and z 2 Rnz . As canbe seen from the above equations, the de�nition of q seems super uous sincez = g(s(x)) = ~g(x) = d(x). However, it is appropriate to de�ne the property z viathe quantity q since this mapping contributes to a deeper understanding of theactual property-concept used in this case.

27.4.2 The Selected Control Strategy

In this context, the control strategy includes algorithms for both open loop opti-mization and feedback control of the baking process.

Traditional operation of the ring furnaces mostly rely on measurements from thegas phase; information of internal anode-block states has not been available inroutine furnace operation.

For control of the ring furnaces, there exist constraints on the control variables.For such systems, Pontryagin's minimum principle can be used for calculation ofthe open loop optimal control variable (Pontryagin 1962). Several applicationsof this principle is reported in the literature (Athans & Falb 1966), (Bryson &Ho 1969).

Characteristic features of the baking process should be taken into considerationbefore selecting an appropriate control strategy. Such features are summarizedbelow:

� Baking is repeated along nominal trajectories (compare the use of a gastemperature target curve).

� Constraints exist for the baking process (both for section states and �nalquality measures).

� A process model exists as a set of nonlinear partial di�erential equation fromwhich a high-dimensional nonlinear state space model can be derived.

460 Model Based Control of Ring Furnaces

� On-line process measurements are scarce.

� The process has very long time constants.

Based on these considerations, the process may be a candidate for State SpacePredictive Control (SSPC). This control strategy is discussed in Strand (1991). InSSPC, control calculations are divided into two parts. At �rst, an open loop op-timal control problem is solved to obtain the optimal process trajectories (Strand1991, pp. 31). Such calculations are very time consuming. Due to model errorsand non-nominal disturbances, a level of basic control is also needed to keep thedependent process variables close to the nominal trajectories found by optimiza-tion. Strand (1991, pp. 40), suggests that a so-called control corrector is designedfor this purpose. Also on the control corrector level, constraints may exist on bothmanipulated and dependent process variables. By neglecting model changes andconstraints on states, a classical LQ controller can be used. In this work, however,a modi�cation of the classical theory will be used (Balchen 1993).

The following characteristics apply to the chosen control strategy:

� O�-line calculation of open loop optimal trajectories for the process. Re-optimization has to be performed if the baking conditions are su�cientlychanged.

� O�-line calculation of control matrices based on linearization along the cal-culated open loop trajectories.

� On-line control is performed locally in each section by a modi�ed LQ(G)controller. At this control level, gas ow input (temperature, mass ow rateand composition) and draught pressure at the section output are consideredto be disturbances.

A control system based on SSPC was presented for a general thermal process inGundersen & Balchen (1993a). A pro�t based performance index was used forthe optimization of process economy. Product quality was secured by specify-ing constraints on the allowable range of the quality parameters. In constrainedparametrization, the constraints can be speci�ed in two ways:

� Anode quality speci�ed as hard constraints

� Anode quality speci�ed via penalty functions

Hard constrains were put on the quality parameters and the control vector waslinearly parameterized as a function of time. Alternatively, certain costs could beassigned to deviations from a certain nominal and desired quality. In this work, theoptimization problem is slightly reformulated thereby taking into account speci�cfeatures of the baking process. The structure of the control system is given inFigure 27.2.

By using a model based control approach of this kind, the following can beachieved:

27.4 Model Based Control Strategy 461

for a fire zone

Open Loop Optimization

Control Corrector Calculationsfor a section

Section Section Section Section

disturbancesNominal process

Constraints

Green anode qualityQuality specifications

Closed Loop(On-line)

modelSection Section Section Section

model model model

Open Loop(Off-line)

Optimizing disturbances

dn

x�

; u�

; G1; G2

v�

(t)

x�

(t); u�

(t)

G1(t); G2(t)

v(t)

x�

; u�

v�

(t)

d1 d2

y1

y2

yd(n�1)

ydn

u1 u2 ud(n�1) udn

_x = f(�)

d(n� 1)

Figure 27.2: The suggested control philosophy for the ring furnace. Open loopoptimization is performed for a whole �rezone. A control corrector is designedfor tracking along the optimal trajectories. In the suggested scheme, the controlcorrector operates on the section level.

� Optimized anode quality

� Minimized fuel consumption (i.e. costs of anode production)

� Maximized production quantity

Below, the optimization- and control problems are brie y stated.

462 Model Based Control of Ring Furnaces

27.4.3 The Optimization Problem

The goal in anode production is to achieve production of prebaked anodes of acertain quality at the lowest possible cost. It is also possible to optimize anodeproduction rate. In this case, the production rate depends on the number ofsections n in a �re-zone and the �re-step time Tf . Here, n and Tf will be �xedat constant values and are not subject for optimization. Thus, anode productionrate is �xed.

The need for optimization is due to changes in optimizing disturbances as follows(see Section 4.5):

� Changes in raw material quality

� Changes in anode quality speci�cations (type of speci�cations and range ofallowed values)

� Changes in fuel cost

In Gundersen & Balchen (1993a), the product quality was secured by specify-ing hard constraints on the quality parameters. In this formulation, however,optimization of anode quality is performed by explicitly including the quality pa-rameters in the performance index used for optimization. Here, the following costsand revenues is included in the performance index:

� Cost of green anodes

� Fuel costs

� Price of prebaked anodes

No cost was assigned to the gas ow rate. Income is related to the quality of thebaked anodes. The following performance index can be formulated (Strand 1991,pp. 48):

J(z; u; x) = �(Tc) +

Z Tc

0

L(t)dt (27.12)

�(Tc) = pbana � pgana (27.13)

L(t) = �pfwf (27.14)

Tc = nTf (27.15)

pba =

nzXi=1

fi(zi; ~zi) (27.16)

na is the number of anodes in a section. pga and pba denote price of green andbaked anodes respectively. Anode quality is introduced in the performance indexby assuming that pba is a sum of functions of the anode quality parameters: Thefunctions fi are designed such that the value of fi for �zi;dn = ~zi is pba =

Pnzi=1 ai;

ai is the maximum value for fi. Suggested values for ~zi is given in Table 27.1. nz

27.4 Model Based Control Strategy 463

is the number of properties considered. pf is the price of fuel. It should be notedhere, that in Table 27.1, the coe�cient of thermal expansion is not included sincethis property is mainly determined by the quality of the raw materials.

The actual spatial dependence of the anode quality parameter is not included inthe index. Rather, spatially averaged values are used and standard deviations arekept track of by specifying constraints on the consistency parameters as shownbelow. Tc is the time needed to complete production of a batch of anodes; anodecooling not included.

Optimization is performed by maximizing J :

maxu J(z; u; x) (27.17)

subject to:

_x = f(x; u; v) (27.18)

y = h(x) (27.19)

z = d(x) (27.20)

x(t�) = x�

(27.21)

u 2 [umin; umax] (27.22)

h(x; u) � 0 (27.23)

The symbols are:

t� initial timeTc �nal timeL cost (pro�t) function� vector of constraints on �nal states

(i.e. quality parameters)x(t) state vectory(t) measurement vector

z(t) property vectorx(t�) initial state vectoru control vectorumin; umax bounds on the control vectorh vector of state-, control trajectory

and property constraint functionsTc �nal batch time

Typically included in the constraint vector are constraints on the consistency pa-rameters:

2�(zi) < bi~zi (27.24)

where bi is a parameter used for de�ning the bounds on the anode property pa-rameters. Typically bi is between 0.05 and 0.20 (Hurlen & Naterstad 1991).

Constraints may be speci�ed for other variables too:

464 Model Based Control of Ring Furnaces

Property Variable Recommended Range Unitlimit value

~ziAirburn at 550�C rO2;a < 60 20-80 [mg/(cm

2hr)]

CO2 oxid. at 970�C rCO2

< 40 10-50 [mg/(cm2hr)]

Dusting index si;a < 25 10-30 [%]

Compr. strength �c;a > 30 30-50 [MPa]

Therm. shock resist.kt;a�b;a�Ya

� 150 100-200 [W/m]

Thermal cond. kt;a 3.5-5.5 [W/(mK)]

Speci�c el. res. �el;a < 60 50-75 [�m]

Baked app. density �a;a;b > 1555 1400-1650 [kg/m3]

Table 27.1: Assumed values and ranges for chemical, electrical, mechanical andother physical properties of prebaked anodes which are used in optimization ofanode quality. For the thermal conductivity, the given range was taken care ofonly by speci�cation of hard constraints. Based on Table 5.1.

� Gas temperatures (maximum value)

� Temperature gradients within anodes (i.e. related to thermal stresses)

Gas temperatures usually are measured (i.e. elements in the measurement vector).

The output from the optimization procedure is a set of optimal trajectories to beused in nominal process operation and for realization of an on-line control system.

27.4.4 The Control Problem

Due to process disturbances and modelling errors, a control corrector is needed tokeep the process trajectories along optimal paths.

For design of the basic control level, several well developed strategies are available(Balchen & Mumm�e 1988), (Seborg 1994). In a model based control system,a state estimator can be used to predict section states. The amount of availableinformation is increased and the manipulable variables can be operated by feedbackfrom an estimate of the process state vector. In general, to control x and z

by feedback from an estimate x of the state vector x, x has to be controllablefrom u and observable from y. In ring furnaces, the measurement vector is lowdimensional, and an state estimator is needed.

Now assume that the whole state vector can be measured. Thus, no estimator isneeded and classic LQ-control can be used. Furthermore, the control corrector isdesigned for control of a single section as shown in Figure 4.18. In a more generalapproach, a multivariable control system should be designed for a whole �re zonefor better being able to cope with interactions between sections.

27.4 Model Based Control Strategy 465

The Control Corrector

A control corrector is speci�ed for each section separately. A nonlinear state spacemodel for each section is given:

_x = f(x; u; v; t) (27.25)

where x, u, v, denote states, disturbances and manipulable variables in the sectionrespectively. The nonlinear model is linearized around the open loop nominaltrajectories to give a linear time varying system on the form:

� _x = A(t)�x +B(t)�u+ C(t)�v (27.26)

for deviations along the nominal trajectories. In this equation, we have used:

A(t) =@f(�)@x

j(xo;uo;vo)

B(t) =@f(�)@u

j(xo;uo;vo)

C(t) =@f(�)@v

j(xo;uo;vo)

A LQ-problem is de�ned for a linear combination of the state vector deviation:

�e = E�x (27.27)

In this case, e is typically derived from the property function z. The followingperformance index is used:

J1 = �eT (T )S1�e(T )+Z T

0

(�eTQ1�e+1

��uTP�u)dt (27.28)

In the performance index, Q1 and P are non-negative and positive de�nite matricesrespectively. Let Q1 and P be diagonal matrices with:

Q1 = fqii;1g; qii;1 =1

�e2

i

(27.29)

P = fpiig; pii =1

�u2

i

(27.30)

�ei and �ui are acceptable variances of ei and ui respectively (Balchen & Mumm�e1988, pp. 65). By using equation (27.27), the performance index can be writtenas:

J2 = �xT (T )S2�x(T )

+

Z T

0

(�xTQ2�x+1

��uTP�u)dt (27.31)

466 Model Based Control of Ring Furnaces

with S2 = ETS1E and Q2 = ETQ1E. By tuning the parameter �, one is able toput weight on the whole control vector deviation relative to the whole state vectordeviation. An increase in � gives an increase in the magnitude of the controlvariables.

A time dependent Riccati-equation:

_R = �RA�ATR+ �RBP�1BTR�Q2

with boundary condition:

R(T ) = S2

must be solved.

to obtain the control corrector �u:

�u = G1�x (27.32)

G1 = �P�1BTR (27.33)

This is a well known result from elementary optimal control theory (Athans &Falb 1966).

Integral Control of Speci�c Properties

Integral control can be achieved for a subset of the property vector. In the fol-lowing, this subset of properties is represented by the same notation z = ~g(x) asbefore.

Although this is a �nite time process, it is possible to have integral state feed-back due to the large batch time Tc compared to the largest time constants inthe process. An outer control loop takes care of the integral state feedback asshown in Figure 27.3 below. In this �gure, a state estimator is included and dis-turbance v is assumed measurable. Calculation of G2 is discussed below. Now,the performance index is slightly modi�ed:

Jz = �zT (T )Sz�z(T ) + J2

which gives:

Jz = �xT (T )S02�x(T )

+

Z Tc

0

(�xTQ2�x+1

��uTP�u)dt (27.34)

where S02 = S2 +GT3 SzG3 and:

G3 =@g(:)

@x(27.35)

where the term GT3 SzG3 appears due to weighting of the �nal value of �z. To

obtain G2, two approaches can be used:

27.5 Controllability Analysis 467

dim(z) = dim(u): In this case, G2 is quadratic and it is possible to calculateG2 by specifying the eigenvalues of the integral loops. Since these loops are slowerthan the proportional loops, the following equation is a good approximation forthe integral loops:

� _p = G2G3(A+BG1)�1B�p = �p�p (27.36)

�p is a diagonal matrix with speci�ed eigenvalues for the integral loops. The choiceof the elements in �q may be done as in SISO-systems (Ziegler - Nichols). Equa-tion (27.36) may be solved with respect to G2.

dim(z) � dim(u): In this case, G2 does not have to be quadratic. A LQ prob-lem with an augmented state vector can be solved to obtain G1 and G2 directly.We de�ne an augmented state vector �~x by:

�~xT = [�x �p] (27.37)

J = �~xT (T ) ~S�~x(T )

+

Z T

0

(�~xT ~Q�~x+1

��uTP�u)dt (27.38)

where:

~S =

�S02 00 0

�(27.39)

~Q =

�Q2 00 Qz

�(27.40)

Values for the elements in Qz can be found by setting 1qii;z

� �ziTm where Tmis the largest time constant in the proportional part of the closed loop system(Valderhaug, Di Ruscio & Balchen 1990).

In both cases, the control corrector is a sum of two terms: a proportional control-and an integral control- term:

�u = G�~x = G1�x+�p (27.41)

27.5 Controllability Analysis

Control of the baking process according to the proposed SSPC-based control strat-egy is only possible if the baking process is controllable. Controllability is anintrinsic property of a dynamic system and thus a fundamental property of thesystem. For real systems, however, one should also check whether controllability isfeasible in practice. The term \degree of controllability" is often used to evaluatewhether controllability can be realized in practice.

468 Model Based Control of Ring Furnaces

PROCESS

Calculations

Off-line

Controller

K

v

z

G2

zo

�u

G1

zo

uo

uo

xo

f(�)(G3)

u

xh(�)

y

xo

G1

y

p

g(x)

Figure 27.3: Block diagram for the control corrector which acts on the sectionlevel.

27.5.1 Two Step Controllability Analysis

In the following, an analysis of the controllability of the baking process is discussed.The analysis is performed in two steps as follows:

1. Check controllability of the properties of coal tar pitch (anode properties)by using a nonlinear model of the properties of the pitch (anode properties).The properties depend on the evolution of the temperature �eld.

2. Check controllability of the temperature �eld and property �eld in a chainof sections. A linear model for the gas temperature and the one-dimensionaltemperature �eld in the solid materials of brick, packing coke and anode isused in the analysis.

The objective for doing the analysis in two steps is that the main in uence onthe evolution of the anode properties during baking is via the temperature �eld inthe baking section. Therefore, controllability of the temperature �eld is explicitlyneeded. It is necessary to perform a controllability analysis of a mathematicalmodel of anode properties in a control volume to determine the size of the propertyvector z that can be controlled. The property vector that is subject to controlshould be designed according to the result of this controllability analysis.

In the following, controllability of a linear �nite time control problem is analyzed.The result from the analysis is then applied to the two-step controllability analysisof the baking process.

27.5 Controllability Analysis 469

27.5.2 Analysis of a Finite Time Control Problem

Analysis of controllability was performed according to the principles presentedin Example 8.7.1 in Balchen (1984, pp. 377). In this example, a �nite timecontrol problem for a discretized linear system is presented. The discrete systemis described by the equation:

x((k + 1)T ) = �x(kT ) + �u(kT ) (27.42)

where k = 0; : : : ; N � 1. T is the sample time for the discretized process. Thediscrete controllability matrix is de�ned as follows:

N = [�N�1�... �N�2�

... � � �... �] (27.43)

It can be shown that the discrete system is controllable if:

rank(N ) = n (27.44)

n is the dimension of the state vector x. Now, assume that the values of the controlvariables are kept constant in the time interval (kT; (k+1)T ). Then, if u is scalar,the number of transitions in u must at least equal the dimension n of the systemto be controllable. Thus:

rank(N ) = n � N (27.45)

In the general case, the dimension of u = r which gives:

rank(N ) = n � Nr (27.46)

To achieve control of a linear combination z = Dx of the state variables, thefollowing is needed:

rank(DN ) = m � Nr (27.47)

Thus, controllability of the system depends on properties of the dynamic system,i.e. matrices (�;�; D)2 and is thus a fundamental property of the system. Here,the calculation of u is not discussed. If the system is controllable, u can becalculated. A calculation scheme for u has already been discussed in the previoussection.

In some cases, control of a subset or a linear transformation of the state vectoris desired. Typically, a property vector z can be related to the state vector asfollows:

z = Dx (27.48)

It can be shown that controllability of z is achieved if:

rank(DN ) = m (27.49)

2Matrices (A;B;D) in the continuous case.

470 Model Based Control of Ring Furnaces

where m is the dimension of the property vector z.

If a process is controllable according to the above description, it means that thereis a theoretical solution to the controllability problem. The question of whetherthe process is controllable in practice has not yet been answered. The \degree ofcontrollability" has also been discussed in the literature (Balchen, Fjeld & Solheim1978, pp. 168). In this study, the condition number of the controllability matrixwill be used as a qualitative measure of the degree of controllability of the process.

27.5.3 Step I: Controllability of Pitch Properties in a Con-

trol Volume

The �ller coke component of the composite anode is thermally stable during thebaking process. Thus, the �ller coke behaves like an inert phase and manipulationof the anode properties can only occur via the properties of the coal tar pitch.Anode quality was related to a set of anode properties presented in Chapter 11.Models for these properties were presented in part III and Gundersen (1996d).Based on these models, it was found that the set of anode properties used forde�ning anode quality can be related to �ve properties of the pitch coke fractionof the anodes3. The model for properties of soft carbons presented in Chapter 18was used as basis for a controllability analysis for the pitch properties since con-trollability of the pitch coke properties is needed to achieve control of the anodeproperties.

The property model is nonlinear and was linearized to be used in the controlla-bility analysis. The model was simulated along the same temperature trajectoryas used for the simulations in Chapter 19. Then, the model was linearized atcertain operating points along the trajectory. It should be noted, that along someparts of the temperature cycle, the rate of change of some state variables may bepractically equal to zero. Thus, the rank of the system matrix becomes less thanthe total number of states in the general nonlinear model. Such state variableswas omitted from the linear model before the controllability of the process waschecked. Controllability was analyzed at operating points in both the low- andhigh temperature regimes of the trajectory4. A sampling time of 0.05 hr was usedover an interval of 1 hr; i.e. assuming that the linear model was valid over thistime interval.

According to the rank-criterion for controllability, the individual pitch propertiesare controllable. However, by using the condition number of the controllabilitymatrix, it was shown that controllability can only be expected for a weightedaverage value of the properties. This is due to the fact that the temperature is theonly driving force for evolution of the properties.

3These properties are real density, interlayer spacing, layer plane diameter, stacking height,coke yield and the volume fraction of disorganized carbon.

4Between temperatures 370 and 530�C in the low temperature regime and at a temperatureof 1100�C in the high temperature regime.

27.5 Controllability Analysis 471

Property Unit Brick Packing Coke Anodes

Thermal cond. [W/(mK)] 1.44 0.40 3.00Spec. heat capacity [J/(kgK)] 1047 1520 1600

Density [kg/m3] 2400 800 1320

Table 27.2: Thermal properties for the solid materials used in the linear control-lability analysis. The values were taken from Charette & Bourgeois (1984).

27.5.4 Step II: Controllability of Temperature and Proper-

ties in a Chain of Sections

Finite time control of the temperature distribution of the baking process wasanalysed with basis in a linear model of the temperature �eld in a section. Below,a linear model for the ring furnace is derived. Based on this model, step II of thecontrollability analysis is performed.

A Linear Ring Furnace Model

It was assumed that the temperature could be represented by a model for one-dimensional heat conduction along the y-axis (i.e. normal to the pit wall). Sevenstates were used to represent the temperature in the solids (2, 2 and 3 statevariables for the brickwork, packing coke and anodes respectively) and one statevariable for the gas temperature. The thermal properties of the solid materialswere assumed to be constant as shown in Table 27.2. The submodel for the thermalsystem was derived by discretizing the one-dimensional heat conduction equation.The following model was obtained:

_� = � + �Tg (27.50)

� is the seven-dimensional vector of solid temperatures (brickwork, coke and an-

ode). � = [�; 0; 0; 0; 0; 0; 0]T and Tg is the gas temperature.

The gas compartment of the section was modelled under the assumption of idealmixing in the gas phase. Thus, a uniform gas temperature was assumed. Possibleair inleakage was neglected, and constant gas properties were used. Then, thefollowing equation can be used to represent the gas temperature:

�gcp;gVdTg

dt= wg;incp;gTg;in + wf (��Hf ) (27.51)

�~h(Tg � Tw)� wg;outcp;gTg

where:

wg;out = wg;in + wf

V is the volume of the gas compartment. wg;in and wg;out are the mass ows ofcombustion gas into and out of the gas compartment. In a chain of sections wg;in

472 Model Based Control of Ring Furnaces

comes from the closest upstream section of the chain. Tg;in is the temperatureof the entering gas ow. wf is the mass ow of fuel. �Hf is the combustion

enthalpy. Tw is the temperature of the brickwall surface. ~h is the e�ective heattransfer coe�cient for the energy exchange between the gas and the brickwall.

The gas temperature equation was linearized by using the fact that the mass owof combustion gas through the section is much larger than the mass ow of fuelthat is added to a section5. Thus, wg;out � wg;in and the following model for thegas temperature is obtained:

dTg

dt= �

wg;incp;g

�gcp;gV+

~h

�gcp;gV

!Tg (27.52)

+~h

�gcp;gVTw +

(��Hf )

�gcp;gVwf +

wg;incp;g

�gcp;gVTg;in

Let Tg = x1 and Tw = x2 and wf = u to obtain:

dx1

dt= �

wg;incp;g

�gcp;gV+

~h

�gcp;gV

!x1 (27.53)

+~h

�gcp;gVx2 +

(��Hf )

�gcp;gVu+

wg;incp;g

�gcp;gVTg;in

Tg;in constitutes the connection of the section to the upstream neighbouring sec-tion. Note that due to the linearization of the model, this is the only (one way)connection between the sections. The equation may be simpli�ed as follows:

dx1

dt= a11x1 + a12x2 + bu+ cTg;in (27.54)

De�ne � = [x2; : : : ; x8]T and x = [x1 �]

T , and get the following augmented statespace model for a section of the ring furnace:

_x =

�a11 �

�

�x+

266666666664

b

0000000

377777777775+ cTg;in (27.55)

where � = [a12; 0; 0; 0; 0; 0; 0]. In the model x1 and x2; : : : ; x8 denote gas tempera-tures and temperatures in the solid materials respectively. Tg;in is the temperatureof the gas ow which enters the section; locally Tg;in may be considered to be a

5It is assumed that the nominal gas ow through the ring furnace is in the order of 18000Nm3/hr which is distributed to 5 cassettes. This corresponds to a mass ow of 24500 kg/hr. Thenominal fuel ow rate is assumed to be in the order of 18 kg/hr which shows that the assumptionis valid.

27.5 Controllability Analysis 473

disturbance. There is a total of eight state variables in a section. Within each sec-tion, controllability of the anode temperatures is desired. The following propertyvector is used:

z = Dx (27.56)

where:

D =

24 0 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1

35 (27.57)

i.e the anode temperatures. Controllability of the average anode temperature, canbe studied by using the following D-matrix:

D =�0 0 0 0 0 1

313

13

�(27.58)

One could of course use a general weighted average value instead of an arithmeticmean value. Other property vectors could also be tried.

A more general case, would be to have:

~z = ~D(Dx) (27.59)

Here, ~z represents a linearized property model where the properties depend onthe anode temperatures. For simplicity, however, the controllability of the anodetemperatures was studied in this work.

Five sections were connected in series which gives forty state variables in the modelfor the chain of sections. Let x1 to x40 represent the state variables in the modelfor the chain of sections, and obtain the following augmented model for the ringfurnace system: 2

66664_x1_x2_x3_x4_x5

377775 = diag(A;A;A;A;A)

266664x1x2x3x4x5

377775+ ~A

266664x1x2x3x4x5

377775 (27.60)

+

266664b1 0 0 0 00 b2 0 0 00 0 b3 0 00 0 0 b4 00 0 0 0 b5

377775

266664u1u2u3u4u5

377775+ cTg;in;� (27.61)

A is the system matrix for the section model as described above. The matrix ~Acontains the coe�cient c in elements ~a9;1, ~a17;9, ~a25;17 and ~a33;25 which representsthe serial connection between the sections, i.e. sections 5 to 4, 4 to 3, 3 to 2, and 2to 1 respectively. 0 denotes an eight dimensional column vector �lled with zeros. cis a column vector with coe�cient c positioned in line 1. Tg;in;� at the inlet of the

474 Model Based Control of Ring Furnaces

chain of sections. The following vector represent the state variables within eachsection:

x1 = [x1; : : : x8]T

x2 = [x9; : : : x16]T

x3 = [x17; : : : x24]T

x4 = [x25; : : : x32]T

x5 = [x33; : : : x40]T

The ordering of the section state variables increases in the direction of gas ow inthe chain of sections6. Within each subsystem, the �rst state variable representsthe gas temperature and the next seven state variables represent the temperaturesin the solid materials. An overview of the total system is given in Figure 27.4.The property vector is calculated as follows:

z = �x (27.62)

where:

� = diag(D;D;D;D;D) (27.63)

Based on this state space model, a discrete model was developed for use in thecontrollability analysis.

Section 1 Section 5Section 4Section 3Section 2

x1; : : : x8x9; : : : x16x17; : : : x24x25; : : : x32x33; : : : x40

x33 x25 x17 x9 x1

Tg;in;�u5 u4 u3 u2 u1

Figure 27.4: The chain of baking sections used as model system for the controlla-bility analysis.

Controllability of the Anode Temperatures

In the discrete model of the ring furnace, the sampling time T was related to the�re step as follows:

TN = Tc (27.64)

Thus, N is the number of time steps needed to complete one �re step. In theanalysis, Tc = 36 hr was used. One special feature of the baking process is thepermutation of the sections that takes place at the end of each �re-step. Thismeans that a cold section is inserted at the front of the �retrain and the sectionat the end of the chain is linked out of the chain. In practice, the permutation

6Note that the dynamical section numbers increase in the direction opposite of the gas ow.

27.5 Controllability Analysis 475

introduces a step change in the values of the state vector x. Therefore, it wasdecided to analyse controllability over a time interval corresponding to the �re stepTc and the changes that takes place due to the permutation was not considered. Byknowing the result of the controllability analysis that is valid for the period of timebetween the permutations, it is possible to give a conclusion on the controllabilityof the whole batch time which in this case is 5Tc. In practice, the temperature inthe section with dynamic number one is so low that one cannot expect that thefuel control variable can be continuously in use. This is so since the oil (gas) willonly ignite if the temperature in the combustion gas is high enough. Therefore, thecontrol variable u5 in the section with dynamic number one should be omitted. Inpractice, however, the gas draught can be used as a control variable in this section.The analysis could be limited to only consider controllability of the properties inthe sections with dynamic number 2, 3, 4 and 5, i.e. corresponding to statevariables x1 to x32. Here, however, it was decided to study controllability of theproperties within each section simultaneously.

The rank and condition number of the controllability matrix N were calculatedfor values T = 0:1; 1; 3; 6; 12; 36 hr in for some con�guration of property vectors z.Now, use NT = Tc to obtain N = 360; 36; 12; 3; 1 for the given values of T . Thereason for using several values for T was to observe how the rank and conditionnumber of the controllability matrix changes as a function of T . T must be selectedsuch that Nr = m. Also, the condition number will settle asymptotically to aminimum value for a given Tc as T decreases. The number of control variables isr = 5.

At �rst, the following situation was studied:

� Control of all anode temperatures in sections 1, 2, 3, 4 and 5: m = 15

For N � 3, (i.e. Nr � m) the rank of the controllability matrix was equal to 15(the number of properties) which is in accord with the theoretical result. But thecondition number of the controllability matrix is so high (minimum value was inthe order of 845) that the process cannot be controlled in practice.

In the next case, a more simple situation was considered:

� Control of the average anode temperature in each section: m = 5

In this case, Nr � m for all values of T and the rank of the controllability matrixequals 5 for every value of T . Furthermore, the condition number of the control-lability matrix is in the order of 5. Then, it may be expected that the process iscontrollable in practice.

Controllability of two individual anode temperatures was also considered. Controlof the anode temperatures closest to the surface of the anode gave a conditionnumber in the order of 45. The same analysis for the two temperatures closest tothe center of the anode gave a condition number in the order of 125. Thus, thereis a signi�cant decrease in the condition number of the controllability matrix whencontrol of the average temperature is considered.

476 Model Based Control of Ring Furnaces

This shows that a weighted average value of the temperatures in a section of thebaking furnace is controllable. Thus, only a certain average value of anode prop-erties can be controlled.

It should be noted that there is a one-way coupling via the gas temperature fromsection to section in this system. Therefore, controllability could also have beenstudied by considering only one section with one control variable. In principle, thisgives the same result as studying the controllability of the chain of �ve sections witha total of �ve control variables. The model of the section chain, however, is neededto simulate the behaviour of the ring furnace.

This analysis is very simple, but it gives important insight into the controllabilityof the baking process. The same kind of analysis can be done with a model ofthe two- or three dimensional temperature �eld in the solid materials. In such amodel, however, there will be at least one more control variable (i.e. oil burner) foreach section. Furthermore, one could experiment with the impact of the numberand positioning of the control variables on the controllability of the process. Thus,in general, controllability of a process can be improved by changing the design ofthe process.

In this study, no further analysis of the case with two control variables in a sectionwill be done. Control of a weighted average value of the anode properties maystill be possible. Furthermore, one may expect to obtain a more exible controlsystem if two control variables are used.

Controllability of Properties within the Anode Control Volumes

Above, the controllability of the temperature within the anodes of the ring furnacewas studied. The analysis showed that in practice, controllability can only beexpected for a weighted average value of the temperature within the anode blockswhen one control variable is used for control of a one dimensional temperaturepro�le in the solid materials of the anode baking section.

In practice, however, control of a set of properties in the anode as discussed inpart I of this analysis is desired. It is only possible to control a weighted averagevalue of the temperature �eld in the anodes, and one may expect that this willalso be the case with a set of anode properties. To verify this, the following modelwas used to represent an anode property:

_x = k�(1 + �T )(xf � x) (27.65)

z = x (27.66)

This model inherits a typical feature of many of the state variables used as basisfor de�ning the anode properties: The derivative of the state variable is a functionof temperature and the state variable7. Thus, the model is nonlinear8. If the

7Recall the models for crystallite height and interlayer spacing.8The term k

�(1 + �T ) can be viewed as a �rst order Taylor approximation around T = T

�

for the term k�exp (� E

RT).

27.5 Controllability Analysis 477

model is linearized, one obtains:

� _x = a�x+ b�T (27.67)

a = �k�(1 + �T�)

b = k��(xf � x�)

The model was linearized around (x�; T�) = (0; 650�C) for values xf = 100, � =0:001 and k� = 1:0. Such a model was associated with each control volume inthe anode; neglecting the fact that the operating points will vary slightly in thedi�erent control volumes. Then an augmented eleven dimensional state spacemodel was used for analysing the controllability of the augmented system where thenew state variables were considered as properties to be controlled. Controllabilitywas analyzed with a sampling time T = 0:01 hr over a time interval of 1 hr.

According to the theory, individual control of the three properties is possible. Inpractice however, control cannot be realized since the condition number in theorder of 1200. Other combinations were also tried all complying with the rank cri-terion for controllability. The condition number, however, was too high to achievecontrollability in practice. The only situation that is practically controllable, isthe control of one of the properties or a weighted average value of the properties.In all other cases, the minimum value of the condition number was larger than100.

Controllability was also checked in an augmented system with two properties ineach anode control volume9. The analysis showed that individual control of theproperties in a control volume could not be achieved.

Thus, in this system, it is only possible to control a weighted average value of oneanode property per section.

Concluding Remark

This analysis is very simple, but it gives important insight into the controllabilityof the baking process. The same kind of analysis can be done with a model ofthe two- or three dimensional temperature �eld in the solid materials. In such amodel, however, there is at least one more control variable (i.e. oil burner) foreach section. Furthermore, one could experiment with the impact of the numberand positioning of the control variables on the controllability of the process. Thus,in general, controllability of a process can be improved by changing the design ofthe process.

In this study, no further analysis of the case with two control variables in a sectionwill be done. Control of a weighted average value of the anode properties maystill be possible. Furthermore, one may expect, to obtain a more exible controlsystem if two control variables can be used.

9Parameters (x�; T

�) = (30; 650�C) and xf = 500, � = 0:002 and k

�= 2:0 were used for the

second property model.

478 Model Based Control of Ring Furnaces

27.6 Conclusions

In this chapter, a model based control strategy for the baking process has beensuggested. Controllability of the baking process was analysed in two steps.

In the �rst step, controllability of a linearized anode property model was analysed.It was shown that controllability can be achieved only for a subset of the anodeproperties. Also, it was shown that the degree of controllability of the anodeproperties depends on the state of the baking process (i.e. the operating point ofthe process).

Secondly, controllability of the temperature in a chain of sections was analyzedwith basis in a simple linear model for the gas temperature and the one-dimensionaltemperature �eld in the chain of sections. It was shown that controllability of aweighted average value of the properties in a section can be achieved.

According to the results of the two step analysis, it was concluded that the bakingprocess is controllable.

References

References

Adams, M. (1977), Computer control of anode bake oven �ring, in K. B. Hig-bie, ed., `Light Metals 1977: Proceedings of sessions. 106th AIME AnnualMeeting', The Metallurgical Society of AIME, Minerals, Metals & MaterialsSociety, Atlanta, Georgia, pp. 367{383.

Agroskin, A. A. (1980), `Thermophysical properties of coke', Koks i Khimiya 2, 14{25.

Akamatu, H. & Koruda, H. (1960), On the sub-structure and the crystallite growthin carbon, in `Proceedings of The Fourth Conference on Carbon', PergamonPress, Bu�alo, New York, pp. 355{368.

Akamatu, H. & Koruda, H. (1962), X-ray small-angle scattering by carbon blacksand heat-treated carbons, in `Proceedings of The Fifth Conference on Carbon,Volume 1', Pergamon Press, Bu�alo, New York, pp. 381{387.

Alexander, C. D., Bullough, V. L. & Pendley, J. W. (1971), Laboratory and plantperformance of petroleum pitch, in T. G. Edgeworth, ed., `Light Metals 1971:Proceedings of sessions. 100th AIME Annual Meeting', The MetallurgicalSociety of AIME, Minerals, Metals & Materials Society, New York, New York,pp. 365{375.

Anthony, B. D., Howard, J. B., Hottel, H. C. & Meissner, H. P. (1975), Rapiddevolatilization of pulverized coal, in `Fifteenth Symposium (International)on Combustion', The Combustion Institute, pp. 1303{1317.

Anthony, D. B. & Howard, J. B. (1976), `Coal devolatilization and hydrogasi�ca-tion', AIChE Journal 22(4), 625{656.

Ashby, M. F. (1992), `Physical modelling of materials problems',Materials Scienceand Technology 8, 102{111.

Asperheim, M., Foosn�s, T., Naterstad, T. & Werge-Olsen, A. (1993a), Dubalperformance evaluation. Part 1 of 3. Kiln 2 (HAL-design), Con�dential, HydroAluminium Technology Centre �Ardal, �Ardal, Norway.

Asperheim, M., Foosn�s, T., Naterstad, T. &Werge-Olsen, A. (1993b), Dubal per-formance evaluation. Part 2 of 3. Kiln 1 (Riedhammer-design), Con�dential,Hydro Aluminium Technology Centre �Ardal, �Ardal, Norway.

482 REFERENCES

Asperheim, M., Foosn�s, T., Naterstad, T. & Werge-Olsen, A. (1993c), Dubalperformance evaluation. Part 3 of 3. Kiln 2 (Hal-design) versus Kiln 1 (Ried-hammer design), Con�dential, Hydro Aluminium Technology Centre �Ardal,�Ardal, Norway.

Athans, M. & Falb, P. L. (1966), Optimal Control, McGraw-Hill, New York.

Auchterlonie, D. & Van der Toorn, C. H. G. (1977), Productivity gain in horizontal ue ring furnace operation, in K. B. Higbie, ed., `Light Metals 1977: Proceed-ings of sessions. 106th AIME Annual Meeting', The Metallurgical Society ofAIME, Minerals, Metals & Materials Society, Atlanta, Georgia, pp. 353{366.

Auguie, D. Oberlin, M., Oberlin, A. & Hyvernat, P. (1980), `Microtexture ofmesophase spheres as studied by high resolution conventional transmissionelectron microscopy (ctem)', Carbon 18, 337{346.

Azami, K. & Yamamoto, S. (1994), `Kinetics of mesophase formation of petroleumpitch', Carbon 32(5), 947{951.

Azaro�, L. V. & Buerger, M. J. (1958), The Powder Method in X-Ray Crystallog-raphy, McGraw-Hill Book Company, New York.

Aziz, K. & Settari, A. (1979), Petroleum Reservoir Simulation, Applied SciencePublishers LTD, New York. ISBN 0-85334-787-5.

Balchen, J. G. (1984), Reguleringsteknikk. Bind 1, Tapir Forlag.

Balchen, J. G. (1993), `A modi�ed LQG algorithm (MLQG) for robust con-trol of nonlinear mulitvariable systems', Modeling, Identi�cation and Control14(3), 175{180.

Balchen, J. G., Fjeld, M. & Solheim, O. A. (1978), Reguleringsteknikk. Bind 3,Tapir Forlag. 2nd Edition.

Balchen, J. G. & Mumm�e, K. (1988), Process Control Structures and Application,Van Nostrand Reinold Company Inc., New York.

Barrillon, E. (1971), In uence of the quality of pitch on the characteristics ofanodes for the electrolytic production of aluminium, in T. G. Edgeworth, ed.,`Light Metals 1971: Proceedings of sessions. 100th AIME Annual Meeting',The Metallurgical Society of AIME, Minerals, Metals & Materials Society,New York, New York, pp. 351{363.

Bear, J. & Bachmat, Y. (1991), Introduction to modeling of transport phenomenain porous media, Vol. 4 of Theory and applications of transport in porousmedia, Kluwer Academic Publishers, Dordrecht. ISBN 0-7923-1106-x.

Belitskus, D. (1985), E�ects of mixing variables and mold temperature on prebakedanode quality, in H. O. Bohner, ed., `Light Metals 1985: Proc. of the technicalsessions sponsored by the TMS Light Metals Committee at the 114th TMSAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, New York, New York, pp. 915{925.

REFERENCES 483

Belkina, T. V., Lur'e, M. V. & Stepanenko, M. A. (1981), `An investigation of thekinetics of the change in the group composition of the anthracene fraction onheat treatment', Solid fuel chemistry : Khimiya Tverdogo Topliva 15(4), 143{149.

Benton, M. (1990), Anode mix pitch demand, in C. M. Bickert, ed., `Light Metals1990: Proc. of the technical sessions presented by the TMS Light MetalsCommittee at the 119th AIME Annual Meeting', The Metallurgical Societyof AIME, Minerals, Metals &Materials Society, Anaheim, California, pp. 657{659.

Bernhauer, M., Christ, K., Gschwindt, A. & H�uttinger, K. J. (1993), The kineticsof mesophase formation during the polymerization of pitches, in `Carbon'92.5. International Carbon Conference', German Carbon Group, American Car-bon Society, State University of New York, pp. 39{40.

Berry, J. S. & Hu, A. C. (1986), Anode bake furnace control system, in R. E.Miller, ed., `Light Metals 1986: Proc. of the technical sessions presented bythe TMS Light Metals Committee at the 115th TMS Annual Meeting', TheMetallurgical Society of AIME, Minerals, Metals & Materials Society, NewOrleans, Louisiana, pp. 643{654.

Bin Brek, A. S. S. & Vaz, M. H. (1995), Process optimization in the green mill, inJ. W. Evans, ed., `Light Metals 1995: Proc. of the technical sessions presentedby the TMS Light Metals Committee presented at the 124th TMS AnnualMeeting', The Metallurgical Society of AIME, Minerals, Metals & MaterialsSociety, Las Vegas, California, pp. 611{616.

Bird, B. R., Steward, W. E. & Lightfoot, E. N. (1960), Transport phenomena,John Wiley & Sons, New York.

Biscoe, J. & Warren, B. E. (1942), `An X-ray study of carbon black', Journal ofApplied Physics 13, 364{371.

Boenigk, W. & Nieho�, A. (1993), Preparative determination of the thermalreactivity of pitch, in `21st Biennal Conference on Carbon, Extended Ab-stracts and Program', American Carbon Society, State University of NewYork, pp. 157{159.

Boenigk, W. & Wildf�orster, R. (1989), Weight loss of green anodes during thebaking process, in P. G. Campbell, ed., `Light Metals 1989: Proc. of thetechnical sessions by the TMS Light Metals Committee at the 118th AIMEAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, Las Vegas, Nevada, pp. 501{503.

Born, M. (1974a), `Pyrolysis and behaviour in baking of industrial carbons', Fuel,London 53, 198{203.

Born, M. (1974b), Untersuchungen zur Optimierung des Brennprozesses Technis-cher Kohlen, in `Techniche Kohle', VED Deutcher Verlag f�ur Grundsto�nd-ustrie, Leipzig, pp. 5{92. Series: Freiberger Forschungshefte A Verfahren-stechnik. A 542.

484 REFERENCES

B�ottger, C. (1990), Ans�atze f�ur die Mathematische Modellierung des Bren-nprozesses von Gro�formatigen Feink�ornigen Graphitwerksto�en, in `En-twicklung und Bewetung von Kohlensto�werksto�en h�oheren Qualit�at',VED Deutcher Verlag f�ur Grundsto�ndustrie, Leipzig, pp. 71{84.Series: Freiberger Forschungshefte A. Grundsto�-Verfahrenstechnik-Brennsto�technik. Vol. A 803.

Bourgeois, T., Bui, R. T., Charette, A., Sadler, B. A. & Tomsett, A. D. (1990),Computer simulation of a vertical ring furnace, in C. M. Bickert, ed., `LightMetals 1990: Proc. of the technical sessions presented by the TMS LightMetals Committee at the 119th AIME Annual Meeting', The MetallurgicalSociety of AIME, Minerals, Metals & Materials Society, Anaheim, California,pp. 547{552.

Bowman, J. C., Krumhansl, J. A. & Meers, J. T. (1958), X-ray and low-temperature thermal conductivity study of defects in graphite, in `Conferenceon Industrial Carbons and Graphite, 1957', The Society of Chemical Indus-try, Staples Printers Limited, Imperial College of Science and Technology,pp. 52{59.

Branscomb, J. A. (1966), Carbon electrode pitches, in A. J. Hoiberg, ed., `Bitu-minous Materials: Asphalts, Tars, and Pitches', Interscience Publishers, NewYork, chapter 12, pp. 359{384. Vol. 3: Coal Tars and Pitches.

Briggs, D. K. H. & Popper, F. (1957), `Data on the vaporization of coal tar andits fractions Part I: Equilibrium ash vaporization', Transactions of the In-stitution of Chemical Engineers 35, 369{373.

Brooks, J. D. & Taylor, G. H. (1965a), `Formation of graphitizing carbons fromliquid phase', Nature 206, 697{699.

Brooks, J. D. & Taylor, G. H. (1965b), `The formation of graphitizing carbonsfrom liquid phase', Carbon 3, 185{193.

Brooks, J. D. & Taylor, G. H. (1968), The formation of some graphitizing carbons,in P. L. Walker Jr., ed., `Chemistry and Physics of Carbon: A Series ofAdvances', Marcel Dekker, New York, pp. 243{286. Vol. 5.

Brown, J. K. & Hirsch, P. B. (1955), `Recent infra-red and X-ray studies of coal',Nature 175(4449), 229{233.

Bryson, A. E. & Ho, Y. (1969), Applied Optimal Control, John Wiley & Sons, NewYork.

B�uhler, U. & Perruchoud, R. C. (1995), Dynamic process optimization, in J. W.Evans, ed., `Light Metals 1995: Proc. of the technical sessions presentedby the TMS Light Metals Committee presented at the 124th TMS AnnualMeeting', The Metallurgical Society of AIME, Minerals, Metals & MaterialsSociety, Las Vegas, California, pp. 707{714.

REFERENCES 485

Bui, R. T., Charette, A. & Bourgeois, T. (1987), `Performance analysis of the ringfurnace used for baking industrial carbon electrodes', The Canadian Journalof Chemical Engineering 65, 96{101.

Bui, R. T., Dernedde, E., Charette, A. & Bourgeois, T. (1984), Mathematicalsimulation of a horizontal ue ring furnace, in J. P. McGeer, ed., `LightMetals 1984: Proc. of the technical sessions sponsored by the TMS LightMetals Committee at the 113th TMS Annual Meeting', The MetallurgicalSociety of AIME, Minerals, Metals & Materials Society, Los Angeles, Califor-nia, pp. 1033{1040.

Burke, S. P. & Parry, V. F. (1927), `The heat of distillation of coal', Industrialand engineering chemistry 19, 15{20.

Buttler, F. G. (1975), Studies on thermal decomposition of electrode pitch, inI. Buzas, ed., `Thermal analysis : proceedings of the fourth internationalconference on thermal analysis, vol. 1-3,1974', Heyden, Budapest, pp. 567{576.

Carman, P. C. (1956), Flow of Gases through Porous Media, Butterworths Scien-ti�c Publications, London.

Cartz, L. & Hirsch, P. B. (1960), `A contribution to the structure of coals fromX-ray di�raction studies', Philosophical transactions of the Royal Society ofLondon. A, Mathematical and physical sciences 252A, 557{604.

Charette, A. & Bourgeois, T. (1984), `Modeling the heat transfer in a ring furnace',IEEE Transactions on industry applications 1A-20(4), 902{907.

Charette, A., Ferland, J., Kocaefe, D., Couderc, P. & Saint-Romain, J. L. (1990),`Experimental and kinetic study of volatile evolution from impregnated elec-trodes', Fuel, London 69, 194{202.

Charette, A., Kocaefe, D., Couderc, P. & Saint-Romain, J. L. (1991), `Comparisonof various pitches for impregnation in carbon electrodes', Carbon 29(7), 1015{1024.

Charette, A., Kocaefe, D., Ferland, J. & Couderc, P. (1989), Kinetic study of thepyrolysis of impregnated electrodes, in P. G. Campbell, ed., `Light Metals1989: Proc. of the technical sessions by the TMS Light Metals Committeeat the 118th AIME Annual Meeting', The Metallurgical Society of AIME,Minerals, Metals & Materials Society, Las Vegas, Nevada, pp. 489{494.

Chawastiak, S., Lewis, R. T. & Ruggiero, J. D. (1981), `Quantitative determinationof mesophase content in pitch', Carbon 19(5), 357{363.

Chermin, H. A. G. & van Krevelen, D. W. (1956), `Chemical structure and prop-erties of coal XVII - A mathematical model of coal pyrolysis', Fuel, London35, 85{104.

Chistyakov, A. N. (1979), `The kinetics of thermal and thermo-oxidative conversionof coal tar pitch', Coke and chemistry USSR : Koks i chimija 28(11), 38{40.

486 REFERENCES

Chistyakov, A. N., Denisenko, V. I. & Itskov, M. L. (1985), `Kinetics of the thermaltransformation of pitches in a nonisothermal process', Solid fuel chemistry :Khimiya Tverdogo Topliva 19(6), 133{138.

Collett, G. W. & Rand, B. (1980), `Thermogravimetric investigation of the pyrol-ysis of pitch materials. A compensation e�ect and variation in kinetic param-eters with heating rate', Thermochimica Acta 41, 153{165.

Collins, F. M. (1959), Thermal expansion of polycrystalline carbons and graphitesII, in `Proceedings of The Third Conference on Carbon', Pergamon Press,Bu�alo, New York, pp. 659{674.

Coste, B. (1988), Improving anode quality by separately optimizing mixing andcompacting temperatures, in L. G. Boxall, ed., `Light Metals 1988: Proc.of the technical sessions by the TMS Light Metals Committee at the 117thTMS Annual Meeting', The Metallurgical Society of AIME, Minerals, Metals& Materials Society, Phoenix, Arizona, pp. 247{252.

Darney, A. (1958), Pitch binder for carbon electrodes, in `Conference on Indus-trial Carbons and Graphite, 1957', The Society of Chemical Industry, StaplesPrinters Limited, Imperial College of Science and Technology, pp. 152{161.

Davis, J. D. & Place, P. B. (1924), `Thermal reactions of coal during carbonization',Fuel in Science and Practice 3, 434{439.

Demange, L. (1991), Anode baking process control: Recent industrial implemen-tation and developments, in E. Rooy, ed., `Light Metals 1991: Proc. of thetechnical sessions presented by the TMS Light Metals Committee presentedat the 120th TMS Annual Meeting', The Metallurgical Society of AIME,Minerals, Metals & Materials Society, New Orleans, Louisiana, pp. 657{665.

Dernedde, E. & Bourgeois, T. (1987), The leakage of air in horizontal ue ringfurnaces, in R. D. Zabreznik, ed., `Light Metals 1987: Proc. of the technicalsessions sponsored by the TMS Light Metals Committee at the 116th TMSAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, Denver, Colorado, pp. 591{595.

Dernedde, E., Charette, A., Bourgeois, T. & Castonguay, L. (1986), Kinetic phe-nomena of the volatiles in ring furnaces, in R. E. Miller, ed., `Light Metals1986: Proc. of the technical sessions presented by the TMS Light MetalsCommittee at the 115th TMS Annual Meeting', The Metallurgical Societyof AIME, Minerals, Metals & Materials Society, New Orleans, Louisiana,pp. 589{592.

Diamond, R. (1959), A study of carbonized coals using new X-ray techniques, in`Proceedings of The Third Conference on Carbon', Pergamon Press, Bu�alo,New York, pp. 367{375.

Diamond, R. (1960), `X-ray studies of some carbonized coals', Philosophical trans-actions of the Royal Society of London. A, Mathematical and physical sciences252A, 193{223.

REFERENCES 487

Diamond, R. & Hirsch, P. B. (1958), X-ray scattering from carbonized coals, in`Conference on Industrial Carbons and Graphite, 1957', The Society of Chem-ical Industry, Staples Printers Limited, Imperial College of Science and Tech-nology, pp. 197{203.

Dreyer, C. (1988), Horizontal- ue ring furnace design and process control, in L. G.Boxall, ed., `Light Metals 1988: Proc. of the technical sessions by the TMSLight Metals Committee at the 117th TMS Annual Meeting', The Metallurgi-cal Society of AIME, Minerals, Metals & Materials Society, Phoenix, Arizona,pp. 303{306.

Dreyer, C. (1989), Anode reactivity in uence of the baking process, in P. G. Camp-bell, ed., `Light Metals 1989: Proc. of the technical sessions by the TMS LightMetals Committee at the 118th AIME Annual Meeting', The MetallurgicalSociety of AIME, Minerals, Metals & Materials Society, Las Vegas, Nevada,pp. 595{602.

Elalaoui, M., Krebs, V., Mareche, J. F. & Furdin, G. (1995), `Carbonization ofcoal-tar pitch under controlled athmosphere-part i: E�ect of temperature andpressure on the electrical property evolution of the formed green coke', Carbon33(5), 653{656.

Emmerich, F. G. (1993), Application of a cross-linking model to the Young's modu-lus of graphitizing and nongraphitizing carbons, in `21st Biennal Conferenceon Carbon, Extended Abstracts and Program', American Carbon Society,State University of New York, p. 651.

Emmerich, F. G. (1994), Evolution of the total volume of the carbon crystallitesby heat treatment, in `Carbon'94', The Spanish carbon Group, University ofGranada, Granada, Spain, pp. 124{125.

Emmerich, F. G. (1995), `Application of a cross-linking model to the Young'smodulus of graphitizable and non-graphitizable carbons', Carbon 33(1), 47{50.

Emmerich, F. G., De Sousa, J. C. & Luengo, A. (1987), `Application of a granu-lar model and percolation theory to the electrical resistivity of heat treatedendocarp of babassu nut', Carbon 25(3), 417{424.

Emmerich, F. G. & Luengo, C. A. (1993), `Young's modulus of heat treated car-bons: A theory for nongraphitizing carbons', Carbon 31(2), 333{339.

Engelsman, R. & Sommer, P. (1992), High-quality anode production with state-of-the-art baking technology, in `Advances in Petroleum and Fabrication inLight Metals and Metal Matrix Composites', Canadian Institute of Mining,Metallurgy and Petroleum, Edmonton, Alberta, Canada, pp. 137{150.

Fair, F. V. & Collins, F. M. (1962), E�ect of residence time on graphitization atseveral temperatures, in `Proceedings of The Fifth Conference on Carbon,Volume 1', Pergamon Press, Bu�alo, New York, pp. 503{508.

488 REFERENCES

Farmen, I., Bakken, A. E. & Naterstad, T. (1982), Integrert prosess-styring M/A-fabrikk, Technical Report 83-56-1, �Ardal og Sunndal Verk, �Ardal. In Norwe-gian.

Fischbach, D. B. (1971), The kinetics and mechanism of graphitization, in P. L.Walker Jr., ed., `Chemistry and Physics of Carbon: A Series of Advances',Marcel Dekker, New York, pp. 1{105. Vol. 7.

Fischer, K. H. (1973), Di�erences in the baking process and its economy of oper-ation, as related to closed and open ring pit furnace system, in `Light Metals1973: Proceedings of sessions. 102th AIME Annual Meeting', The Metallur-gical Society of AIME, Minerals, Metals & Materials Society, Chicago.

Fischer, W. K. & Keller, F. (1993), Baking parameters and the resulting anodequality, in S. K. Das, ed., `Light Metals 1993: Proc. of the technical sessionspresented by the TMS Light Metals Committee presented at the 122nd TMSAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, Denver, Colorado, pp. 683{689.

Fishbach, D. B. (1977), `Glassy carbon graphitization: Density change', HighTemperatures-High Pressure 9, 199{205. Carbon'76.

Fitzer, E. & Fritz, W. (1975), Technische Chemie: Eine Einf�uhrung in die Chemis-che Reaktionstechnik, Springer.

Fitzer, E. & H�uttinger, K. J. (1969), `Zur Kinetik des Brennprozess von GrunenKohlek�orpern', Carbon 7, 447{458.

Fitzer, E., Mueller, K. & Schaefer, W. (1971), The chemistry of the pyrolytic con-version of organic compounds to carbon, in P. L. Walker Jr., ed., `Chemistryand Physics of Carbon: A Series of Advances', Marcel Dekker, New York,pp. 237{383. Vol. 7.

Fitzer, E. & Terwiesch, B. (1973), `The pyrolysis of pitch and the baking of pitchbonded carbon/carbon composites under nitrogen pressure up to 100 bar',Carbon 11, 570{574.

Fogler, H. S. (1986), Elements of chemical reaction engineering, Prentice HallInternational, Inc. ISBN 0-13-263666-2.

Foosn�s, T., Jarek, S. & Linga, H. (1989), Vertical ue baking furnace rebuildingconcepts and anode quality, in P. G. Campbell, ed., `Light Metals 1989: Proc.of the technical sessions by the TMS Light Metals Committee at the 118thAIME Annual Meeting', The Metallurgical Society of AIME, Minerals, Metals& Materials Society, Las Vegas, Nevada, pp. 569{574.

Foosn�s, T., Kulset, N., Linga, H., N�umann, G. R. & Werge-Olsen, A. (1995),Measurement and control of the calcining level in anode baking furnaces, inJ. W. Evans, ed., `Light Metals 1995: Proc. of the technical sessions presentedby the TMS Light Metals Committee presented at the 124th TMS AnnualMeeting', The Metallurgical Society of AIME, Minerals, Metals & MaterialsSociety, Las Vegas, California, pp. 649{652.

REFERENCES 489

Forum Roundtable: Discussing the Issues in Carbon Anode Technology (1990),JOM : the journal of the Minerals, Metals & Materials Society. Vol. 42,Pages 52-55.

Franklin, R. E. (1949), `A study of the �ne structure of carbonaceous solids bymeasurements of true and apparent densities', Transactions of the FaradaySociety 45, 668{682.

Franklin, R. E. (1950), `The interpretation of di�use X-ray diagrams of carbons',Acta Crystallica 3, 107{121.

Franklin, R. E. (1951a), `Crystallite growth in graphitizing and non-graphitizingcarbons', Proceedings of the Royal Society A209, 196{218.

Franklin, R. E. (1951b), `The structure of graphitic carbons', Acta Crystallica4, 253{261.

Franklin, R. E. (1953), `Graphitierende und nichgraphitierende Kohlenstof-fverbindungen Bildung, Struktur und Eigenschaften', Brennsto� Chemie34, 359{361.

Froment, G. F. & Bischo�, K. B. (1990), Chemical reactor analysis and design, 2edn, John Wiley & Sons, Inc., New York. ISBN 0-471-51044-0.

Fu, W., Zhang, Y., Han, H. & Wang, D. (1989), `A general model of pulverizedcoal devolatilization', Fuel, London 68, 505{510.

Furman, A. & Martirena, H. (1980), A mathematical model simulating an anodebaking furnace, in C. J. McMinn, ed., `Light Metals 1980: Proc. of the tech-nical sessions sponsored by the TMS Light Metals Committee at the 109thAIME Annual Meeting', The Metallurgical Society of AIME, Minerals, Metals& Materials Society, Las Vegas, Nevada, pp. 545{552. ISBN 0-89520-359-6.

Furnas, C. C. (1931), `Grading aggregates I - mathematical relations for beds ofbroken solids of maximum density', Industrial and Engineering Chemistry23(9), 1052{1058.

Gemmeke, W., Collin, G. & Zander, M. (1978), The characterization of binderpitches (II), in J. J. Miller, ed., `Light Metals 1978: Proceedings of sessions.107th AIME Annual Meeting', The Metallurgical Society of AIME, Minerals,Metals & Materials Society, Denver, Colorado, pp. 355{366.

Gheorghiu, F. & Oprescu, I. (1987), Pressure di�erence combustible ow ratecorrelation, a basic prerequisite in the complex automation of ring bakingfurnace for carbon electrodes, in `Light Metals 1987: Proc. of the technicalsessions sponsored by the TMS Light Metals Committee at the 116th TMSAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, Denver, Colorado, pp. 613{618.

Greinke, R. A. (1986), `Kinetics of petroleum pitch polymerization by gel perme-ation chromatography', Carbon 24(6), 677{686.

490 REFERENCES

Greinke, R. A. (1992), `Chemical bond formed in thermally polymerized petroleumpitches', Carbon 30(3), 407{414.

Greinke, R. A. & Singer, L. S. (1988), `Constitution of coexisting phases in meso-phase pitch during heat treatment: Mechanism of mesophase development',Carbon 26(5).

Grint, A., Proud, G. P., Poplett, I. J. F., Bartle, K. D. & Wallace, S. Matthews,R. (1988), `Characterization of pitches by solid state nuclear magnetic reso-nance', Fuel, London 68, 1490{1491.

Grjotheim, K. & Kvande, H. (1993), Introduction to Aluminium Electrolysis, 2edn, Aluminium-Verlag, D�usseldorf. ISBN 0-471-51044-0.

Grjotheim, K. & Welch, B. J. (1988), Aluminium Smelter Technology: A Pure andApplied Approach, 2 edn, Aluminium-Verlag, D�usseldorf. ISBN 0-471-51044-0.

Gundersen, �. (1995a), Modeling and control of ring furnaces. A review of re-search, Unrestricted 95-66-W, Department of Engineering Cybernetics. Nor-wegian Institute of Science and Technology, O.S. Bragstadspl. 8, N-7034Trondheim, Norway. Unrestricted.

Gundersen, �. (1995b), Modelling and simulation of pitch pyrolysis, Unrestricted95-116-W, Department of Engineering Cybernetics. Norwegian Institute ofScience and Technology, O.S. Bragstadspl. 8, N-7034 Trondheim, Norway.

Gundersen, �. (1996a), Carbon anode production. A process description, Un-restricted 96-79-W, Department of Engineering Cybernetics. Norwegian In-stitute of Science and Technology, O.S. Bragstadspl. 8, N-7034 Trondheim,Norway. ISBN 82-7842-004-1.

Gundersen, �. (1996b), Mathematical modelling of a ring furnace for baking of car-bon anodes, Unrestricted 96-58-W, Department of Engineering Cybernetics.Norwegian Institute of Science and Technology, O.S. Bragstadspl. 8, N-7034Trondheim, Norway.

Gundersen, �. (1996c), Mathematical modelling of physical properties in soft car-bons, Restricted 96-36-T, Department of Engineering Cybernetics. NorwegianInstitute of Science and Technology, O.S. Bragstadspl. 8, N-7034 Trondheim,Norway.

Gundersen, �. (1996d), Mathematical modelling of physical properties of carbonanodes, Restricted 96-47-T, Department of Engineering Cybernetics. Norwe-gian Institute of Science and Technology, O.S. Bragstadspl. 8, N-7034 Trond-heim, Norway.

Gundersen, �. (1996e), Raw materials used in carbon anode production, Un-restricted 96-35-W, Department of Engineering Cybernetics. Norwegian In-stitute of Science and Technology, O.S. Bragstadspl. 8, N-7034 Trondheim,Norway.

REFERENCES 491

Gundersen, �. (1996f), Structure and properties of soft carbons, Unrestricted 96-92-W, Department of Engineering Cybernetics. Norwegian Institute of Scienceand Technology, O.S. Bragstadspl. 8, N-7034 Trondheim, Norway.

Gundersen, �. & Balchen, J. G. (1993a), A control strategy for large thermalprocesses, in `AIChE 1993 Annual Meeting', AIChE, St. Louis, Missouri. 6pages.

Gundersen, �. & Balchen, J. G. (1993b), Modeling and simulation of a ring furnacefor the baking of carbon anodes, in T. Iversen, ed., `Applied Simulation inIndustry. Proceedings of the 35th SIMS Simulation Conference', SIMS, NTH-Trykk, Kongsberg College of Engineering, Kongsberg, Norway, pp. 255{265.

Gundersen, �. & Balchen, J. G. (1995), `Modeling and simulation of an anodecarbon baking furnace', Modeling, Identi�cation and Control 16(1), 3{33.

Gyoerkoes, T. (1971), Some aspects of prebaked anode manufacture, in T. G.Edgeworth, ed., `Light Metals 1971: Proceedings of sessions. 100th AIMEAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, New York, New York, pp. 377{384.

Hanbaba, P., J�untgen, H. & Peters, W. (1968), `Nicht-Isotherme Reaktionskinetikder Kohlenpyrolyse', Brennsto�-Chemie 49(12), 22.

Hercules, J. R. L. & Sarmiento, V. (1988), Mathematical model for a ring typeanode baking furnace, in L. G. Boxall, ed., `Light Metals 1988: Proc. of thetechnical sessions by the TMS Light Metals Committee at the 117th TMSAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, Phoenix, Arizona, pp. 315{323. ISBN 0-89520-359-6.

Himmelblau, D. M. & Bischo�, K. B. (1968), Process Analysis and Simulation.Deterministic Systems, John Wiley & Sons, New York.

Hirsch, P. B. (1954), `X-ray scattering from coals', Proceedings of the Royal Societyof London. A, Mathematical and physical sciences 226, 143{169.

Ho�mann, W. R. & H�uttinger, K. J. (1993), `Modeling of the apparent viscosity ofpitches and mesophases at linear temperature increase up to 500 oC', Carbon31(2), 263{268.

Holm, R. (1967), Electric Contacts, Almquist & Wiksells akademiska Handb�ocker,Hugo Gebers F�orlag, Stockholm.

Honda, H. (1988), `Carbonaceous mesophase: History and prospects', Carbon26(2), 139{156.

Honda, H., Kimura, H., Sugawara, S. & Futura, T. (1970), `Optical mesophase tex-ture and X-ray di�raction patterns of the early-stage carbonization of pitches',Carbon 8, 181{189.

Hottel, H. C. & Saro�m, A. F. (1967), Radiative heat transfer, McGraw Hill BookCompany.

492 REFERENCES

Howard, J. B. (1981), Fundamentals of coal pyrolysis and hydropyrolysis, in M. A.Elliot, ed., `Chemistry of Coal Utilization. Second Supplementary Volume',John Wiley & Sons, New York, chapter 12, pp. 665{785.

Hulburt, H. M. & Katz, S. (1964), `Some problems in particle technology. A sta-tistical mechanical formulation', Chemical Engineering Science 19, 554{574.

Hurlen, J., Lid, O., Naterstad, T. & Utne, P. (1981), Operation characteristics fora vertical ue ring furnace, in G. M. Bell, ed., `Light Metals 1981: Proc. ofthe technical sessions sponsored by the TMS Light Metals Committee at the110th AIME Annual Meeting', The Metallurgical Society of AIME, Minerals,Metals & Materials Society, Chicago, Illinois, pp. 569{581. ISBN 0-89520-359-6.

Hurlen, J. & Naterstad, T. (1991), `Recent developments in carbon baking technol-ogy', JOM : the journal of the Minerals, Metals & Materials Society 43, 20{21:24{25.

Hutcheon, J. M. & Price, S. T. (1960), The dependence of the properties of graphiteon porosity, in `Proceedings of The Fourth Conference on Carbon', PergamonPress, Bu�alo, New York, pp. 645{656.

Hutcheon, J. M. & Warner, R. K. (1958), The ow of gases through a �ne-poregraphite, in `Conference on Industrial Carbons and Graphite, 1957', TheSociety of Chemical Industry, Staples Printers Limited, Imperial College ofScience and Technology, pp. 259{271.

H�uttinger, K. J. (1970), `Eine Kinetische Analyse der Steinkohlenteerpechpyrolyse.Teil 2.', Bitumen-Teere-Asphalte-Peche und verwandte Sto�e 12, 528{531.

H�uttinger, K. J. (1971), Kinetics of non-isothermal coal tar pitch pyrolysis, inJ. G. Gregory, ed., `Third Conference on Industrial Carbons and Graphite,1970', The Society of Chemical Industry, Staples Printers Limited, ImperialCollege of Science and Technology, pp. 136{141.

H�uttinger, K. J. (1986), `Fl�ussigphasenpyrolyse von Kohlenwassersto�en.Physicalisch-chemische und reactiontechnische Grundlagen-Teil I', Erd�ol undKohle-Erdgas vereinigt mit Brennsto�-Chemie 39(11), 495{500.

H�uttinger, K. J. (1988), `Coke yield in liquid phase pyrolysis of hydrocarbons fromthe standpoint of chemical reaction engineering science', Carbon 26, 477{484.

H�uttinger, K. J. (1989), `A general theoretically derived equation for the coke yieldin liquid phase pyrolysis', Carbon 27, 477{484.

H�uttinger, K. J., Bernhauer, M., Christ, K. & Gschwindt, A. (1992), `Kinetics ofmesophase formation in a stirred tank reactor and properties of the products- III. carbon dioxide atmosphere', Carbon 30(6), 931{938.

H�uttinger, K. J. & Wang, J. P. (1991), `Kinetics of mesophase formation in astirred tank reactor and properties of the products', Carbon 29(3), 439{448.

REFERENCES 493

H�uttinger, K. J. & Wang, J. P. (1992a), `Kinetics of mesophase formation in astirred tank reactor and properties of the products - II. discontinuous reactor',Carbon 30(1), 1{8.

H�uttinger, K. J. & Wang, J. P. (1992b), `Kinetics of mesophase formation in astirred tank reactor and properties of the products - III. continuous reactor',Carbon 30(1), 9{15.

ICCTC (1982), `First publication of 30 tentative de�nitions', Carbon 20, 445.

Incropera, F. P. & DeWitt, D. P. (1990), Introduction to heat transfer, John Wiley& Sons, New York.

Introduction to Coal-Tar and Petroleum Pitch (1993), Abstracts, Southern Illi-nois University at Carbondale. Composition, Applications, Toxicology, andFuture.

Jacob, S. M., Gross, B., Voltz, S. & Weekman, Jr., V. W. (1976), `A lumping andreaction scheme for calatytic cracking', AIChE Journal 22(4), 701{713.

Jacobsen, M. (1997), Heat and Mass Transfer in a Vertical Flue Ring Furnace,Dr. ing. thesis, Norwegian University of Science and Technology, Departmentof Thermal Energy and Hydro Power. ITEV-rapport 1997:04.

Jacobsen, M. & Gundersen, �. (1997), Anode property development during heattreatment, in `Light Metals 1997: Proc. of the technical sessions presentedby the TMS Light Metals Committee presented at the 126th TMS AnnualMeeting', The Metallurgical Society of AIME, Minerals, Metals & MaterialsSociety, Orlando, Florida, pp. 597{604.

Jacobsen, M. & Log, T. (1995), Thermal conductivity of green carbon materialsduring baking, in J. W. Evans, ed., `Light Metals 1995: Proc. of the technicalsessions presented by the TMS Light Metals Committee presented at the124th TMS Annual Meeting', The Metallurgical Society of AIME, Minerals,Metals & Materials Society, Las Vegas, California, pp. 673{680.

Jacobsen, M. & Melaaen, M. C. (1995), Heat and mass transfer in anode materialsduring baking, in J. W. Evans, ed., `Light Metals 1995: Proc. of the technicalsessions presented by the TMS Light Metals Committee presented at the124th TMS Annual Meeting', The Metallurgical Society of AIME, Minerals,Metals & Materials Society, Las Vegas, California, pp. 681{690.

J�ager, H., Wagner, M. H. & Wilhelmi, G. (1987), `Interrelation between viscosityand electrical conductivity of pitches', Fuel, London 66, 1554{1555.

Jakobsen, O., Lid, O. & Schreiner, P. A. (1987), A new ring furnace consept:Design and operation, in R. D. Zabreznik, ed., `Light Metals 1987: Proc. ofthe technical sessions sponsored by the TMS Light Metals Committee at the116th TMS Annual Meeting', The Metallurgical Society of AIME, Minerals,Metals & Materials Society, Denver, Colorado, pp. 497{503.

494 REFERENCES

Johansen, T. A. (1994),Operating regime based process modeling and identi�cation,Dr. ing. thesis, Norwegian Institute of Technology, Department of EngineeringCybernetics. ITK-report 1994: 109-W.

Johansen, T. A. (1996), `The use of a state space estimator to obtain improvedinitial values in serial regime models'. Communication with T. A. Johansen.

Jones, S. S. (1990), Production of high-quality anodes for aluminium electrolysis,Technical report, Allied-Signal, Inc.

Jones, S. S. & Bart, E. F. (1990), Binder for the ideal anode carbon, in C. M.Bickert, ed., `Light Metals 1990: Proc. of the technical sessions presentedby the TMS Light Metals Committee at the 119th AIME Annual Meeting',The Metallurgical Society of AIME, Minerals, Metals & Materials Society,Anaheim, California, pp. 611{627.

Jones, S. S. & Hildebrandt, R. D. (1975), Electrode binder pyrolysis and bond-cokemicrostructure, in `Light Metals 1975: Proceedings of sessions. 104th AIMEAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, pp. 291{322.

J�rgensen, R. (1991), Dokumentasjon brennovn. Brennovn-3, �ardal. Funksjons-beskrivelse prosesstyring., Con�dential, Hydro Aluminium, Teknologisenteret�Ardal. In Norwegian.

J�untgen, H. & van Heek, K. H. (1969), `Fortschritte der Forschung auf dem Gebietder Steinkohlenpyrolyse', Brennsto�-Chemie 50(6), 172{179.

Jurges, C. D. (1983), Anode baking furnace draft control, in E. M. Boxall, ed.,`Light Metals 1983: Proc. of the technical sessions sponsored by the TMSLight Metals Committee at the 112th AIME Annual Meeting', The Met-allurgical Society of AIME, Minerals, Metals & Materials Society, Atlanta,Gheorgia, pp. 791{796. ISBN 0-89520-359-6.

Keller, F. & Disselhorst, J. H. M. (1981), Modern anode furnace developments, inG. M. Bell, ed., `Light Metals 1981: Proc. of the technical sessions sponsoredby the TMS Light Metals Committee at the 110th AIME Annual Meeting',The Metallurgical Society of AIME, Minerals, Metals & Materials Society,pp. 611{621. ISBN 0-89520-359-6.

Keller, F. & Fischer, W. K. (1992), Anode manufacturing in a changing envi-ronment, in E. R. Cutshall, ed., `Light Metals 1992: Proc. of the technicalsessions presented by the TMS Light Metals Commitee presented at the 121stTMS Annual Meeting', The Metallurgical Society of AIME, Minerals, Metals& Materials Society, San Diego, California, pp. 673{686.

Keller, F. & Oderbolz, S. (1985), Process controlled operation of baking furnaces,in H. O. Bohner, ed., `Light Metals 1985: Proc. of the technical sessionssponsored by the TMS Light Metals Committee at the 114th TMS AnnualMeeting', The Metallurgical Society of AIME, Minerals, Metals & MaterialsSociety, pp. 1107{1123.

REFERENCES 495

Keller, F., Schmidt-Hatting, W., Kooijman, A., Fischer, W. & Perruchoud, R.(1990), Characterizing anode properties by quality �gures, in C. M. Bickert,ed., `Light Metals 1990: Proc. of the technical sessions presented by the TMSLight Metals Committee at the 119th AIME Annual Meeting', The Metal-lurgical Society of AIME, Minerals, Metals & Materials Society, Anaheim,California, pp. 479{483.

Kelly, B. T. (1981), Physics of Graphite, Applied Science Publishers LTD.

Kittel, C. (1986), Introduction to Solid State Physics, 6 edn, John Wiley & Sons,New York.

Kocaefe, D., Charette, A. & Castonguay, L. (1993), Investigation of the structuralchanges during green coke calcination, in `Carbon'92. 5. International Car-bon Conference', German Carbon Group, American Carbon Society, StateUniversity of New York, pp. 72{73.

Kocaefe, D., Charette, A., Ferland, J., Couderc, P. & Saint-Romain, J. L. (1990),`A kinetic study of pyrolysis in pitch impregnated electrodes', The CanadianJournal of Chemical Engineering 68, 988{996.

Komatsu, K. (1955), `Theory of the speci�c heat of graphite II', Journal of thePhysical Society of Japan 10(5), 347{356.

Komatsu, K. & Nagamiya, T. (1951), `Theory of the speci�c heat of graphite',Journal of the Physical Society of Japan 6(6), 439{443.

Kono, H. (1984), Chemical engineering approach to cracking and gasi�cation pro-cesses of residual oils, in `Symposium of Advances in Petroleum ProcessingPresented Before the Division of Petroleum Chemistry, Inc. Joint Meeting ofthe American Chemical Society and Chemical Society of Japan', Honolulu.

Korai, Y. & Mochida, I. (1992), `Molecular assembly of mesophase and isotropicpitches at their fused states', Carbon 30(7), 1019{1024.

Ko�st�al, D., Pr _u�sa, Z. & Malik, M. (1994), An investigation into the kinetics duringthe thermal treatment of pitches, in U. Mannweiler, ed., `Light Metals 1994:Proc. of the technical sessions presented by the TMS Light Metals Committeepresented at the 123nd TMS Annual Meeting', The Metallurgical Societyof AIME, Minerals, Metals & Materials Society, San Fransisco, California,pp. 541{545. kinetics, solvent fractionation model.

Krebs, V., Elalaoui, M., Mareche, J. F. & Furdin, G. (1995), `Carbonizationof coal-tar pitch under controlled atmosphere-Part I: E�ect of temperatureand pressure on the structural evolution of the formed green coke', Carbon33(5), 645{651.

Kreith, F. & Black, W. Z. (1980), Basic heat transfer, Harper & Row, Publishers,New York.

Kuo, K. (1975), Principles of combustion, 2 edn, John Wiley & Sons, New York.ISBN 0-471-62605-8.

496 REFERENCES

Lahaye, J., Ehrburger, P., Saint-Romain, J. L. & Couderc, P. (1987), `Physic-ochemical characterization of pitches by di�erential scanning calorimetry',Fuel, London 66, 1467{1471.

Levinter, M. E., Medvedeva, M. I., Panchenkov, G. M., Aseev, Y. G., Nedoshivin,Y. N., Finkels'thein, G. B. & Galiakbarov, M. F. (1966), `Mechanism ofcoke formation in the cracking of component groups in petroleum residues',Chemistry and Technology of Fuels and Oils : Chimija i tekhnologija Toplivi Masel 9, 628{632.

Loch, L. D. & Austin, A. E. (1956), Fine pore structure-crystallite size relation-ships in carbons, in `Proceedings of The First and Second Conferences onCarbon', Pergamon Press, Bu�alo, New York, pp. 65{.

Log, T. & �ye, H. A. (1989), `Thermal conductivity of prebaked anodes', Alu-minium (Duesseldorf) Aluminium : Fachzeitschrift der Aluminiumindustrie65(5), 508{511.

Lopez, H. J., Castillo, F. M. & Vera, A. (1989), Dynamic control of gases in aclosed ring type anode baking furnace, in P. G. Campbell, ed., `Light Metals1989: Proc. of the technical sessions by the TMS Light Metals Committeeat the 118th AIME Annual Meeting', The Metallurgical Society of AIME,Minerals, Metals & Materials Society, Las Vegas, Nevada, pp. 563{567.

Lydersen, A. (1983), Mass transfer in Engineering Practice, John Wiley & Sons,New York.

Macpherson, C. (1989), A new automatic anode baking system using ue bricktemperature control, in P. G. Campbell, ed., `Light Metals 1989: Proc. of thetechnical sessions by the TMS Light Metals Committee at the 118th AIMEAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, Las Vegas, Nevada, pp. 569{574.

Magnus, A. (1923), `Die spezi�sche W�arme des kohlensto�s, Siliziums und Siliz-iumskarbids bei hohen Temperaturen', Ann. Physik 70, 303{330.

Mannweiler, U. & Keller, F. (1994), `The design of a new anode technology for thealuminium industry', JOM : the journal of the Minerals, Metals & MaterialsSociety 46, 15{21.

Mannweiler, U., Sulzberger, P. & Oderbolz, S. (1991), Process control in an an-ode bake furnace �red with heavy oil, in E. Rooy, ed., `Light Metals 1991:Proc. of the technical sessions presented by the TMS Light Metals Commit-tee presented at the 120th TMS Annual Meeting', The Metallurgical Societyof AIME, Minerals, Metals & Materials Society, New Orleans, Louisiana,pp. 667{671.

Marsh, H. (1973), `Carbonization and liquid-crystal (mesophase) development:Part 1. the signi�cance of the mesophase during carbonization of coking coals',Fuel, London 52, 205{211.

REFERENCES 497

Marsh, H. (1976), A review of the growth and coalencence of mesophase (nematicliquid crystals) to form anisotropic carbon, and its relevance to coking andgraphitization, in `Fourth London International Conference on Carbons andGraphite, 1974', The Society of Chemical Industry, Glantave Printers Lim-ited, Imperial College, London, pp. 2{38.

Marsh, H. & Clarke, D. E. (1986), `Mechanisms of formation of structure withinmetallurgical coke and its e�ect on coke properties', Erd�ol und Kohle - Erdgas- Petrochemie vereinigt mit Brennsto� -Chemie 39(3), 113{123.

Marsh, H., ed. (1989), Introduction to Carbon Science, Butterworths, London.

Marsh, H. & Latham, C. S. (1986), The chemistry of mesophase formation, inJ. D. Bacha, J. W. Newman & J. L. White, eds, `Petroleum-derived carbons',American Chemical Society, Washington, DC, chapter 1, pp. 1{28. Series:ACS symposium series. Vol. 303.

Marsh, H. & Stadler, H. P. (1967), `A review of structural studies of the carboniza-tion of coal', Fuel 56, 351{359.

Marsh, H. & Walker Jr., P. L. (1979), The formation of graphitizable carbonsvia mesophase: Chemical and kinetic considerations, in P. L. Walker Jr. &P. A. Thrower, eds, `Chemistry and Physics of Carbon: A Series of Advances',Marcel Dekker, New York, pp. 229{299. Vol. 15.

Martirena, H. (1983), Laboratory studies on mixing, forming and calcining carbonbodies, in E. M. Adkins, ed., `Light Metals 1983: Proc. of the technicalsessions sponsored by the TMS Light Metals Committee at the 112th AIMEAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, Atlanta, Georgia, pp. 749{764. ISBN 0-89520-359-6.

Martirena, H. & Marletto, J. (1980), Thermal phenomena in anode baking fur-naces, in C. J. McMinn, ed., `Light Metals 1980: Proc. of the technical sessionssponsored by the TMS Light Metals Committee at the 109th AIME AnnualMeeting', The Metallurgical Society of AIME, Minerals, Metals & MaterialsSociety, Las Vegas, Nevada, pp. 531{543. ISBN 0-89520-359-6.

Mason, I. B. (1958), The electrical resistance of polycrystalline carbons andgraphites, in `Conference on Industrial Carbons and Graphite, 1957', TheSociety of Chemical Industry, Staples Printers Limited, Imperial College ofScience and Technology, pp. 60{73.

McGeary, R. K. (1961), `Mechanical packing of spherical particles', Journal of theAmerican Chemical Society 44(10), 513{522.

McHenry, E. R., Baron, J. T. & Saver, W. E. (1993), The e�ect of thermal treat-ment on industrial pitch and carbon anode properties-Part 1, in S. K. Das,ed., `Light Metals 1993: Proc. of the technical sessions presented by the TMSLight Metals Committee presented at the 122nd TMS Annual Meeting', TheMetallurgical Society of AIME, Minerals, Metals & Materials Society, Denver,Colorado, pp. 663{668.

498 REFERENCES

McHenry, E. R., Baron, J. T. & Saver, W. E. (1994), The e�ect of thermal treat-ment on industrial pitch and carbon anode properties - Part 2, in U. Mann-weiler, ed., `Light Metals 1994: Proc. of the technical sessions presented by theTMS Light Metals Committee presented at the 123nd TMS Annual Meeting',The Metallurgical Society of AIME, Minerals, Metals & Materials Society,San Fransisco, California, pp. 525{533. see part 1 in LM 93.

McNeil, D. (1981), High-temperature coal tar, in M. A. Elliot, ed., `Chemistry ofCoal Utilization. Second Supplementary Volume', John Wiley & Sons, NewYork, chapter 17, pp. 1003{1083. KjemiL, Ola S. Raaness.

Merrick, D. (1983a), `Mathematical models of the thermal decomposition of coal.1: The evolution of volatile matter', Fuel, London 62, 534{539.

Merrick, D. (1983b), `Mathematical models of the thermal decomposition of coal.2: Speci�c heats and heats of reaction', Fuel, London 62, 540{546.

Mizushima, S. (1960), On the crystallite growth of carbon, in `Proceedings of TheFourth Conference on Carbon', Pergamon Press, Bu�alo, New York, pp. 417{421.

Mizushima, S. (1963), Rate of graphitization of carbon, in `Proceedings of TheFifth Conference on Carbon, Volume 2', Pergamon Press, Pennsylvania StateUniversity, Pennsylvania, pp. 439{447.

Monica, E. d. F., Marletto, J. & Martirena, H. (1983), Combined mathematicalsimulation and experimental studies on a closed baking furnace, in E. M.Adkins, ed., `Light Metals 1983: Proc. of the technical sessions sponsoredby the TMS Light Metals Committee at the 112th AIME Annual Meeting',The Metallurgical Society of AIME, Minerals, Metals & Materials Society,Atlanta, Georgia, pp. 805{817. ISBN 0-89520-359-6.

Monthioux, M., Oberlin, M. & Bourrat, X. (1982), `Heavy petroleum products:Microtexture and ability to graphitize', Carbon 20(3), 167{176.

Morse, P. M. (1969), Thermal Physics, 2 edn, W. A. Benjamin Inc., New York.

Mrozowski, S. (1956a), Mechanical strength, thermal expansion and structures ofcokes and carbons, in `Proceedings of The First and Second Conferences onCarbon', Pergamon Press, Bu�alo, New York, pp. 31{45.

Mrozowski, S. (1956b), Physical properties of carbons and the formulation of thegreen mix, in `Proceedings of The First and Second Conferences on Carbon',Pergamon Press, Bu�alo, New York, pp. 195{215.

Mrozowski, S. (1958), The nature of arti�cial carbons, in `Conference on Indus-trial Carbons and Graphite, 1957', The Society of Chemical Industry, StaplesPrinters Limited, Imperial College of Science and Technology, pp. 7{18.

Mrozowski, S. (1959), Studies of carbon powders under compression I, in `Pro-ceedings of The Third Conference on Carbon', Pergamon Press, Bu�alo, NewYork, pp. 495{508.

REFERENCES 499

Munson, B. R., Young, D. F. & Okiishi, T. H. (1990), Fundamentals of uidmechanics, John Wiley & Sons. ISBN 0-471-855626-X.

Murgia, A. & Bello, V. (1978), Some experience in prebaked anode production,in J. J. Miller, ed., `Light Metals 1978: Proceedings of sessions. 107th AIMEAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, Denver, Colorado, pp. 325{339.

Nerland, A. M. (1994), Thermal Shock Resistance of Anode Carbon for AluminiumElectrolysis, Dr. ing. thesis, Norwegian Institute of Technology, Departmentof Inorganic Chemistry. IUK-report 1994:71.

NIF (1986), Karbonmaterialer i aluminiumsindustrien, NIF, Storefjell h�yfjell-shotell, Gol. In Norwegian.

Oberlin, A. (1984), `Carbonization and graphitization', Carbon 22(6), 521{541.

Oberlin, A., Bonnamy, S., Bourrat, X., Monthioux, M. & Rouzaud, J. N. (1986),Electron microscopic observation on carbonization and graphitization, in J. D.Bacha, J. W. Newman & J. L. White, eds, `Petroleum-derived carbons',American Chemical Society, Washington, DC, chapter 5, pp. 85{98. Series:ACS symposium series. Vol. 303.

Oberlin, A. & Oberlin, M. (1983), `Graphitizability of carbonaceous materials asstudied by TEM and X-ray di�raction', Journal of Microscopy 103(3), 353{363.

Okada, J. (1960a), Dependence of thermal expansion of bonded carbons on theheat treatment temperature, in `Proceedings of The Fourth Conference onCarbon', Pergamon Press, Bu�alo, New York, pp. 553{558.

Okada, J. (1960b), Thermal expansion of pitch bonded carbons, in `Proceedingsof The Fourth Conference on Carbon', Pergamon Press, Bu�alo, New York,pp. 547{552.

Okada, J. & Takeuchi, Y. (1960), Dependence of the density and other propertiesof bonded carbons on the binder proportion in the green mix, in `Proceedingsof The Fourth Conference on Carbon', Pergamon Press, Bu�alo, New York,pp. 657{669.

Ollivier, P. J. & Gerstein, B. C. (1986), `Aromaticity and ring condensation inpyrolyzed pitches', Carbon 24(2), 151{158.

Oprescu, I., Gheorghiu, F. & Georgescu, M. (1988), Temperature �elds in furnacechambers, a basic factor in establishing baking conditions (baking ring furnacefor carbon electrodes), in L. G. Boxall, ed., `Light Metals 1988: Proc. of thetechnical sessions by the TMS Light Metals Committee at the 117th TMSAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, Phoenix, Arizona, pp. 909{915. ISBN 0-89520-359-6.

500 REFERENCES

Oprescu, I. & Semenescu, A. (1992), Study of the volatile matter evacuation pro-cess through the package layers in the baking furnaces of carbonic electrodes,in E. R. Cutshall, ed., `Light Metals 1992: Proc. of the technical sessionspresented by the TMS Light Metals Commitee presented at the 121st TMSAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, San Diego, California, pp. 751{755.

Palmer, H. (1974), Combustion technology: Some modern developments, AcademicPress, New York. ISBN 0-12-544750-7.

Parrott, J. E. & Stuckes, D. (1975), Thermal Conductivity of Solids, AppliedPhysics Series, Pion Limited, 207 Brondesbury Park, London NW2 5JN.

Patankar, S. V. (1980), Numerical Heat Transfer and Fluid Flow, HemispherePublishing Corporation. Mc-Graw Hill Book Company. ISBN 0-07-048740-5.

Patrick, J. W., Soerlie, M. & Walker, A. (1989), `Strenght/structure relationshipsin baked carbon materials', Carbon 27(3), 469{474.

Patrick, J. W., S�orlie, M. & Walker, A. (1988), Strength-structure relationshipsin baked carbon materials, in B. McEnaney & T. J. Mays, eds, `Carbon'88',University of Newcastle upon Tyne, UK, pp. 461{463.

Peggs, I. D., Mill, R. W. & Stadnyk, A. M. (1976), Some observations of graphiteand composite permeabilities, in `Fourth London International Conferenceon Carbons and Graphite, 1974', The Society of Chemical Industry, GlantavePrinters Limited, Imperial College, London, pp. 420{421.

Perron, J., Bui, R. & Nguyen, T. (1992), `Modelisation d'un dour de calcination ducoke de petrole', The Canadian Journal of Chemical Engineering 70, 1108{1119.

Perry, R. H. & Green, D. W. (1984), Perry's Chemical Engineers' Handbook, 6edn, McGraw Hill Book Company. ISBN 0-07-Y66482-X.

Peterson, R. W. & Seger, E. J. (1980), Pitch control of prebaked anodes by slumpmeasurement, in C. J. McMinn, ed., `Light Metals 1980: Proc. of the technicalsessions sponsored by the TMS Light Metals Committee at the 109th AIMEAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, Las Vegas, Nevada, pp. 443{453. ISBN 0-89520-359-6.

Pinner, A. O. & Barrier, W. F. (1977), Operation of ring type carbon anode bakingfurnaces, in K. B. Higbie, ed., `Light Metals 1977: Proceedings of sessions.106th AIME Annual Meeting', The Metallurgical Society of AIME, Minerals,Metals & Materials Society, Atlanta, Georgia, pp. 343{351.

Pitt, G. J. (1962), `The kinetics of the evolution of volatile products from coal',Fuel, London 41, 267{274.

Politis, T. G. & Chang, C. F. (1985), Pitch carbonization, in `17th Biennal Con-ference on Carbon', American Carbon Society, pp. 8{9.

REFERENCES 501

Pontryagin, L. S. (1962), The Mathematical Theory of Optimal Processes, McGrawHill.

Powell, R. W. (1958), The thermal conductivity of carbons and graphites at hightemperatures, in `Conference on Industrial Carbons and Graphite, 1957', TheSociety of Chemical Industry, Staples Printers Limited, Imperial College ofScience and Technology, pp. 47{51.

Pratt, G. C. (1958), The shrinkage of high-temperature cokes, in `Conferenceon Industrial Carbons and Graphite, 1957', The Society of Chemical Indus-try, Staples Printers Limited, Imperial College of Science and Technology,pp. 145{151.

Racunas, B. J. (1980), Anode baking furnace thermal balance, in J. M. Curtis,ed., `Light Metals 1980: Proc. of the technical sessions sponsored by theTMS Light Metals Committee at the 109th AIME Annual Meeting', TheMetallurgical Society of AIME, Minerals, Metals & Materials Society, LasVegas, Nevada, pp. 525{530.

Randolph, A. D. (1964), `A population balance for countable entities', The Cana-dian Journal of Chemical Engineering 42, 280{281.

Reid, R. C., Prausnitz, J. M. & Poling, B. E. (1987), The properties of gases andliquids, 4 edn, Mc-Graw-Hill, New York.

Rey Boero, J. (1988), The planning of experiments in the laboratory study of anodeproperties, in L. G. Boxall, ed., `Light Metals 1988: Proc. of the technicalsessions by the TMS Light Metals Committee at the 117th TMS AnnualMeeting', The Metallurgical Society of AIME, Minerals, Metals & MaterialsSociety, Phoenix, Arizona, pp. 211{216.

Rhedey, P. (1967), `Structural changes in petroleum coke during calcination',Transactions of the Metallurgical Society of AIME 239, 1084{1091.

Rhedey, P. J. (1982), Carbon reactivity and anode reduction cell anodes, in J. E.Andersen, ed., `Light Metals 1982: Proc. of the technical sessions sponsoredby the TMS Light Metals Committee at the 111th AIME Annual Meeting',The Metallurgical Society of AIME, Minerals, Metals & Materials Society,Dallas, Texas, pp. 713{725. ISBN 0-89520-359-6.

Romovacek, G. R. (1983), New analytical procedures for characterization of pitchesI, in E. M. Adkins, ed., `Light Metals 1983: Proc. of the technical sessionssponsored by the TMS Light Metals Committee at the 112th AIME AnnualMeeting', The Metallurgical Society of AIME, Minerals, Metals & MaterialsSociety, Atlanta, Georgia, pp. 683{697. ISBN 0-89520-359-6.

Ruland, W. (1965), `X-ray studies on the carbonization and graphitization ofacenaphtylene and bi uorenyl', Carbon 2, 365{378.

Saxena, S. C. (1990), `Devolatilization and combustion characteristics of coal par-ticles', Progress in Energy and Combustion Science 16(1), 55{94.

502 REFERENCES

Scherrer, P. (1918), `Bestimmung der Gr�osse und der inneren struktur vonkolloidteilchen mittels r�ontgenstrahlen', Nachrichten von der K�oniglichenGesellschaft der Wissenschaften zu G�ottingen 2, 6{8.

Schmidt-Hatting, W., Baak, F. & Blom, E. (1992), Quality assurance in anodeproduction, in E. R. Cutshall, ed., `Light Metals 1992: Proc. of the technicalsessions presented by the TMS Light Metals Commitee presented at the 121stTMS Annual Meeting', The Metallurgical Society of AIME, Minerals, Metals& Materials Society, San Diego, California, pp. 711{714.

Schneider, J. P. & Coste, B. (1993), Thermal schock of anodes: In uence of rawmaterials and manufacturing parameters, in S. K. Das, ed., `Light Metals1993: Proc. of the technical sessions presented by the TMS Light MetalsCommittee presented at the 122nd TMS Annual Meeting', The MetallurgicalSociety of AIME, Minerals, Metals & Materials Society, Denver, Colorado,pp. 611{619.

Schucker, R. C. (1983), `Thermogravimetric determination of the coking kinetics ofarab heavy vacuum residuum', Industrial & Engineering Chemistry. ProcessDesign and Development 22, 615{619.

Sears, F. W. & Salinger, G. L. (1980), Thermodynamics, Kinetic Theory, andStatistical Thermodynamics, 3 edn, Addison Wesley, New York.

Seborg, D. (1994), `A perspective on advance strategies for process control', Mod-eling, Identi�cation and Control 15(3), 179{189.

Seldin, E. J. (1956), Density-resistivity relationships for baked anodes, in `Pro-ceedings of The First and Second Conferences on Carbon', Pergamon Press,Bu�alo, New York, pp. 217{223.

Seldin, E. J. (1959), Electrical resistivities and crushing strengths of baked car-bons, in `Proceedings of The Third Conference on Carbon', Pergamon Press,Bu�alo, New York, pp. 675{686.

Seldin, E. J. & Mrozowski, S. (1959), Permeabilities of baked carbons and beds ofcarbon powders to the ow of a gas, in `Proceedings of The Third Conferenceon Carbon', Pergamon Press, Bu�alo, New York, pp. 687{703.

Serra, A., Murgia, A. & Cherchi, S. (1982), Improvements in the operations of aclosed-type furnace for baking anodes, in J. E. Andersen, ed., `Light Metals1982: Proc. of the technical sessions sponsored by the TMS Light MetalsCommittee at the 111th AIME Annual Meeting', The Metallurgical Societyof AIME, Minerals, Metals & Materials Society, Dallas, Texas, pp. 765{778.ISBN 0-89520-359-6.

Sherwin, M. B., Shinnar, R. & Katz, S. (1969), Dynamic behaviour of the wellmixed isothermal crystallizer, in J. A. Palermo & M. A. Larson, eds, `Crystal-lization from Solutions and Melts', American Institute of Chemical Engineers,pp. 59{74.

REFERENCES 503

Short, M. A. & Walker, P. L. (1963), `Measurement of interlayer spacings andcrystallite sizes in turbostratic carbons', Carbon 1, 3{9.

Skala, D., Kopsch, H., Neumann, H. J. & Jovanivi�c, J. (1987), `Thermogravimet-rically and di�erential scanning calorimetrally derived kinetics of oil shalepyrolysis', Fuel, London 66, 1185{1191.

Skala, D., Kopsch, H., Soci`c, M., Neumann, H. J. & Jovanivi�c, J. (1989), `Mod-elling and simulation of oil shale pyrolysis', Fuel, London 68, 1107{1111.

Skala, D., Kopsch, H., Soci`c, M., Neumann, H. J. & Jovanivi�c, J. (1990), `Kineticsand modelling of oil shale pyrolysis', Fuel, London 60, 490{496.

Smith, K. L., Smoot, L. D., Fletcher, T. H. & Pugmire, R. J. (1994), The Structureand Reaction Processes of Coal, The Plenum Chemical Engineering Series,Plenum Press, New York.

Smoot, L. D. (1991), Coal and char combustion, in W. Bartok & A. F. Saro�m,eds, `Fossil Fuel Combustion: A Source Book', Wiley, New York, chapter 10,pp. 653{781.

Spang, III, H. A. (1972), `A dynamic model of a cement kiln', Automatica 8, 309{323.

Stein, S. E. (1981), `Thermochemical kinetics of anthracene pyrolysis', Carbon19(6), 421{429.

Stevenson, D. T. (1988), Anode baking furnace hydrodynamic ue modeling, inL. G. Boxall, ed., `Light Metals 1988: Proc. of the technical sessions by theTMS Light Metals Committee at the 117th TMS Annual Meeting', The Met-allurgical Society of AIME, Minerals, Metals & Materials Society, Phoenix,Arizona, pp. 307{314. ISBN 0-89520-359-6.

Strand, S. (1991), Dynamic Optimization in State Space Predicitive ControlSchemes, Dr. ing. thesis, Norwegian Institute of Technology, Department ofEngineering Cybernetics. ITK 91-11-W.

Stuart, A. N. (1974), The use of calcined uid coke as packing material in anodebaking pits, in `Light Metals 1974: Proceedings of sessions. 103th AIMEAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, pp. 1065{1073.

Suuberg, E. M., Peters, W. A. & Howard, J. B. (1978), `Product compositionand kinetics of lignite pyrolysis', Industrial & Engineering Chemistry. ProcessDesign and Development 17(1), 37{46.

Sverdlin, V. A., Priezzhaya, I. A. & Yanko, E. A. (1991), Study of kinetics of ther-mal transformations of carbon anodes for aluminium electrolysis, in E. Rooy,ed., `Light Metals 1991: Proc. of the technical sessions presented by the TMSLight Metals Committee presented at the 120th TMS Annual Meeting', TheMetallurgical Society of AIME, Minerals, Metals & Materials Society, NewOrleans, Louisiana, pp. 673{678.

504 REFERENCES

Sverdlin, V. A., Priezzhaya, L. A. & Yanko, E. A. (1992), Study of kinetics ofthermal transformation of carbon anodes for aluminium electrolysis, in E. R.Cutshall, ed., `Light Metals 1992: Proc. of the technical sessions presentedby the TMS Light Metals Commitee presented at the 121st TMS AnnualMeeting', The Metallurgical Society of AIME, Minerals, Metals & MaterialsSociety, San Diego, California, pp. 1363{1368.

Svirida, L. V., Markina, M. S. & Fedoroseev, S. D. (1976), `A study of the car-bonization of coal tar pitch under various conditions', Solid fuel chemistry :Khimiya Tverdogo Topliva 10(3), 131{134.

Tarasiewicz, S. & Stumpf, A. (1984), A computer model for the evolution ofvolatiles during the baking of anodes, in `Proceedings of the Fifteenth AnnualConference on Modelling and Simulation', pp. 747{750.

Thibault, M. A., Bui, R. T., Charette, A. & Dernedde, E. (1985), Simulatingthe dynamics of the anode baking ring furnace, in H. O. Bohner, ed., `LightMetals 1985: Proc. of the technical sessions sponsored by the TMS LightMetals Committee at the 114th TMS Annual Meeting', The MetallurgicalSociety of AIME, Minerals, Metals & Materials Society, New York, New York,pp. 1141{1151.

Thomas, J. C. (1989), Improvements of anode baking in open furnaces, in `Tech-nology in the Aluminium Industry. Second Symposium', Sao Paulo, Brazil,pp. 565{575.

Thonstad, O. (1989), `Over ate- og porefordelingsm�alinger', Institutt for Indus-triell Kjemi. In Norwegian.

Tillmans, H. (1985), `Carbonization of pitch', Fuel, London 64, 1197{1203.

Tillmans, H. (1986), Carbonization and characterization, in J. D. Bacha, J. W.Newman & J. L. White, eds, `Petroleum-derived carbons', American ChemicalSociety, Washington, DC, chapter 16, pp. 215{232. Series: ACS symposiumseries. Vol. 303.

Tremblay, F. & Charette, A. (1988), `Cin�etique de d�egagement des mati�eresvolatiles de la pyrolyse d'electrodes de carbone industrielles', The CanadianJournal of Chemical Engineering 6(2), 86{96.

Turner, N. R. (1995), Tracking the behaviour of pitch at elevated temperature (a)vapour loss at sub-carbonization temperature, (b) carbonization tracked bycarbon-hydrogen content, crystallite height Lc and d-spacing measurements,in J. W. Evans, ed., `Light Metals 1995: Proc. of the technical sessions pre-sented by the TMS Light Metals Committee presented at the 124th TMSAnnual Meeting', The Metallurgical Society of AIME, Minerals, Metals &Materials Society, Las Vegas, California, pp. 573{582.

Valderhaug, A., Di Ruscio, D. & Balchen, J. G. (1990), Synthesis of robust LQcontroller, International Federation of Automatic Control, IFAC, Tallinn,pp. 334{339.

REFERENCES 505

Valyavin, G. G., Fryazinov, V. V., Gimaev, R. N., Syanyaev, Z. I., Vyatkin, Y. L.& Mulyukov, S. F. (1979), `Kinetics and mechanism of the macromolecularpart of crude oil', Chemistry and Technology of Fuels and Oils : Chimija itekhnologija Topliv i Masel 8, 562{565.

van Krevelen, D. W. (1981), Coal: Typology, Chemistry, Physics and Constitu-tion, Coal Science and Technology, Elsevier Scienti�c Publishing Company,Amsterdam.

Vand, V. (1943), `A theory of the irreversible electrical resistance changes of metal-lic �lms evaporated in vacuum', The Proceedings of the Physical Society, Lon-don A55, 222{246.

Vanvoren, C. (1987), Recent improvement in paste plant design, industrial ap-plication and results, in R. D. Zabreznik, ed., `Light Metals 1987: Proc. ofthe technical sessions sponsored by the TMS Light Metals Committee at the116th TMS Annual Meeting', The Metallurgical Society of AIME, Minerals,Metals & Materials Society, Denver, Colorado.

Wagner, M. H., Hammer, W. & Wilhelmi, G. (1981), `Irreversible expansionand shrinkage during graphitization of isotropic carbon materials', HighTemperatures-High Pressure 13(2), 153{158. Carbon'80.

Walker, P. L., Walker, Jr., P. L. & Thrower, P. A., eds (1965-1991), Chemistryand physics of carbon : a series of advances, Vol. 1-23, Marcel Dekker.

Walker, P. M. B., ed. (1991), Chambers Science and Technology, W & R ChambersLtd, Edinburgh.

Wallouch, R. W., Murty, H. N. & Heintz, E. A. (1972), `Pyrolysis of coal tar pitchbinders', Carbon 10, 729{753.

Walter, J. J., Morris, E. G. & Joo, L. A. (1980), Packing characteristics ofpetroleum coke particles in relation to density of molded bodies, in `ExtendedAbstracts. Program for Biennal Conference on Carbon', pp. 601{602.

Wang, J. (1991), Kinetik der thermischen Polymerisation von �ussigen Kohlen-wassersto�gemischen zu Polyaromaten-Mesophase und Eigenschaften derProdukte: Vergleich von diskontinuerlich und kontinuerlich betriebenemR�uhrkesselreaktor, PhD thesis, Technischen Universit�at Karlsruhe.

Warren, B. E. (1941), `X-ray di�raction in random layer lattices', The PhysicalReview 59(9), 693{698.

Washimi, K. (1984), Review of studies on the properties and reaction kinetics ofheavy oil thermal cracking residue, inW. O. Meyer, ed., `Future Heavy CrudeTar Sands, International Conference 2nd, 1982', McGraw Hill.

Weekman, Jr., V. W. (1969), `Kinetics and dynamics of catalytic cracking se-lectivity in �xed-bed reactors', Industrial & Engineering Chemistry. ProcessDesign and Development 8(3), 385{391.

506 REFERENCES

Weekman, Jr., V. W. & Nace, D. M. (1970), `Kinetics of catalytic cracking selec-tivity in �xed, moving and uid bed reactors', AIChE Journal 16(3), 397{404.

Wendlandt, W. W. (1986), Thermal Analysis, Vol. 19 of Chemical Analysis, JohnWiley, New York. ISBN 0-471-88477-4.

White, J. L. (1974), `The formation of microstructure in graphitizable materials',progress in Solid State Chemistry 9, 59{103.

Whittaker, M. P., Miller, F. C. & Fritz, H. C. (1970), `Structural changes ac-companying coke calcination', Industrial and Engineering Chemistry. ProductResearch and Development 9(2), 187{190.

Wiehe, I. A. (1993), A phase separation kinetic model for coke formation, in`Symposium on Resid Upgrading. Presented befor the Division of PetroleumChemistry, Inc. 205th natioanl Meeting', American Chemical Society, Denver,Colorado, pp. 428{433.

Wiggs, P. K. C. (1958), The relation between gas permeability and pore sizedistribution in consolidated bodies, in `Conference on Industrial Carbons andGraphite, 1957', The Society of Chemical Industry, Staples Printers Limited,Imperial College of Science and Technology, pp. 252{257.

Wiggs, P. K. C. (1960), Prediction of optimum binder content of a carbon mixingby use of the mercury porosimeter, in `Proceedings of The Fourth Conferenceon Carbon', Pergamon Press, Bu�alo, New York, pp. 639{643.

Wilke, C. R. (1965), Chemical Engineering Progress 46, 95.

Wilkening, S. (1983), Properties and behaviour of green anodes, in E. M. Ad-kins, ed., `Light Metals 1983: Proc. of the technical sessions sponsored by theTMS Light Metals Committee at the 112th AIME Annual Meeting', The Met-allurgical Society of AIME, Minerals, Metals & Materials Society, Atlanta,Georgia, pp. 727{740. ISBN 0-89520-359-6.

Wilkening, S. (1993), Apparent density of prebaked anodes, in S. K. Das, ed.,`Light Metals 1993: Proc. of the technical sessions presented by the TMSLight Metals Committee presented at the 122nd TMS Annual Meeting', TheMetallurgical Society of AIME, Minerals, Metals & Materials Society, Denver,Colorado, pp. 1157{1171.

Williams, A. (1976), Combustion of sprays of liquid fuels, Topics in Fuel Science,Elek Science, London. ISBN 0-236-31044-5.

Wright, K. B. & Peterson, R. W. (1989), Pitch content optimization model foranode mixes, in P. G. Campbell, ed., `Light Metals 1989: Proc. of the techni-cal sessions by the TMS Light Metals Committee at the 118th AIME AnnualMeeting', The Metallurgical Society of AIME, Minerals, Metals & MaterialsSociety, Las Vegas, Nevada, pp. 479{488.

Appendices

Appendix A

The Conservation Laws

In this appendix, the conservation laws are presented on an integral formulation.The �nal goal is to establish conservation equations as represented by partialdi�erential equations in time and space for the brickwork, packing coke, anodeand combustion gas subsystems of the ring furnace.

Both single phase and multiphase subsystems exist within the ring furnace. Inmost cases, each phase is a multicomponent mixture; i.e. the binder pitch contentin the anodes, the mixture of volatile gases in the void fraction of the packing cokeand anodes and the combustion gas in the ues.

Rigorous derivation of the mathematical representation of the conservation lawscan be found in general textbooks as Bird et al. (1960) and Munson et al. (1990).

n

dV dA

V

A

Figure A.1: Control volume used for establishing the balance equations.

510 The Conservation Laws

A.1 Conservation of Mass

The principle of conservation of mass states that the rate of change (net rateof accumulation) of mass in a control volume equals the net rate at which massleaves the control volume plus the rate at which mass is generated inside thecontrol volume.

The principle can be stated in mathematical form as follows:

d

dt

ZZZV

�dV = �ZZA

�vTndA+

ZZZV

rdV (A.1)

� is the bulk density of the substance inside the control volume. v is the velocityat the boundary A of the control volume and n is the unitary normal vector tothe boundary surface pointing outwards.

Note that the principle of conservation of mass can be applied for the total massas well as the mass of a certain component within the control volume.

A.2 Conservation of Momentum

The principle of conservation of momentum states that the rate of change of mo-mentum in a control volume equals the net rate at which momentum leaves thecontrol volume plus the sum of all forces acting on the control volume.

The principle can be stated in mathematical form as follows:

d

dt

ZZZV

�vidV = �ZZA

�vTnvidA+Xk

Fk (A.2)

vi is the component of the velocity vector v parallel with coordinate axis i (i.e. oneof the axes x, y or z). Thus, the equation for i = 1 gives the rate of accumulation ofx-momentum �v1 in the control volume.

Pk Fk is the sum of all body- and surface

forces acting on the control volume. Gravity is the only body force included.Pressure and friction forces belong to the surface forces.

It should be noted here that the three momentum equations are vector equations;one for each coordinate direction.

A.3 Conservation of Energy

The principle of conservation of energy states that the rate of change of energyin a control volume equals the net rate at which energy leaves the control volumeplus the net rate of heat added to the system from the surroundings minus the netrate of work energy performed by the system on the surroundings.

A.4 A Closed Set of Equations 511

The principle can be stated in mathematical form as follows:

d

dt

ZZZV

�EdV = �ZZA

�vTnEdA+Q�W (A.3)

E is the total energy per unit mass inside the control volume; also denoted totalspeci�c energy. Usually, the total energy includes internal-, kinetic - and gravita-tional energy1. Q denotes heat energy into the system2. W denotes work done bythe system on the surroundings3. Generally, heat is transferred by the mechanismsof conduction, convection and radiation. The work term generally includes bothpressure work, gravitational work and friction (viscous) work.

A.4 A Closed Set of Equations

Mass balance- (total mass and component balances), energy balance and momen-tum balance equations for the conservation principles. These equations togetherwith an equation of state and initial- and boundary conditions can be simultane-ously solved to obtain temperature, density, pressure and the velocity componentsin a system.

1Electric, magnetic and nuclear energy is not included.2Heat is positive into the system.3Work is positive when done by the system on the surroundings.

512 The Conservation Laws

Appendix B

Density, Porosity and

Surface Area

B.1 Dense Composite with n Components

Consider a composite of n components where each component may have a denseor porous structure. The mass and volume of the composite is denoted m and Vrespectively. The density of the composite is denoted �c. Each component con-tributes with mass mi and volume Vi to the composite's mass and volume respec-tively. Each pure component has density �i. In the case with porous components,�i includes the porosity. The situation is schematically described in Figure B.1.

mi

Vi�i

mc

Vc

�c

Figure B.1: Composite medium which consist of n components.

514 Density, Porosity and Surface Area

By using masses mi and volumes Vi, one obtains:

mc =

nXi=1

mi (B.1)

Vc =

nXi=1

Vi (B.2)

Let xi denote mass fraction of component i within the composite. In the samemanner, let �i denote volume fraction of component i. xi and �i are de�ned asfollows:

xi =mi

mc(B.3)

�i =Vi

Vc(B.4)

The following constrains are valid for mass fractions xi and volume fractions �i:

nXi=1

xi = 1 (B.5)

nXi=1

�i = 1 (B.6)

The mass of the composite can be calculated from known density �c and volumeVc:

mc = �cVc (B.7)

The composite density �c can be expressed as a function of mass- and volumefractions xi and �i and pure densities �i. By using the mass- and volume fractionsxi and �i, the mass of each component can be calculated in two ways:

mi = ximc = xi�cVc (B.8)

mi = �iVi = �i�iVc = �i�iVc (B.9)

The following relationships between mass- and volume fractions for component iare valid:

�i�iVc = xi�cVc

+�i = xi

�c

�ior xi = �i

�i

�c

Now usePn

i=1 xi = 1 and get:

�c =

nXi=1

�i�i (B.10)

B.2 Porous Material with Open Porosity 515

The same can be done by usingPn

i=1 �i = 1 to obtain an expression for �c basedon the mass fractions xi:

nXi=1

�i =

nXi=1

xi�c

�i= 1

+

�c =1Pn

i=1xi�i

(B.11)

Hence, the composite density can be calculated either by volume fractions or massfraction combined with density of the pure components.

B.2 Porous Material with Open Porosity

Porous materials consist of both voids and solid matter. As shown in Figure B.2the total volume V is given by:

V = Vv;o + Vr (B.12)

Vv;o is the volume of the voids and Vr is the volume of solid matter; the so-called real volume. Let � denote the void fraction within total volume V . ThenVv;o = �V which gives:

Vr = V (1� �) (B.13)

� is also denoted (total) porosity. Two measures of density are commonly de�nedfor porous materials, assuming that the void fraction contain no closed pores1:

� Apparent (bulk) density: Density of the material including the voids

� Real density: Density of the material without voids (determined by pycno-metric methods on �nely grounded material; grain diameters usually below10 �m)

The apparent density �a is calculated from known mass m and outer volume V :

�a =m

V(B.14)

In the same manner, the real density can be expressed by:

�r =m

Vr(B.15)

1Explanations of apparent and real densities can be found in the glossary of anode bakingfound in Gundersen (1996e).

516 Density, Porosity and Surface Area

Vr

V

� 1� �

Figure B.2: Volume with an open void fraction.

The real and apparent densities can be related to the void fraction � by usingVr = V (1� �) as follows:

�aV = �rVr

+�r =

�a

1� �or � = 1� �a

�r(B.16)

B.3 Porous Material with Open and Closed Poros-

ity

In some materials both open and closed porosity occur. The total porosity � canbe separated into an open porosity �o and a closed porosity �c. This gives:

� = �c + �o (B.17)

Pores may be classi�ed according to their size and shape. The following de�nitionis recommended by IUPAC (Marsh 1989):

� Macropores: Pore diameter greater than 50 nm

� Mesopores: Pore diameter between 2 and 50 nm

� Micropores: Pore diameter less than 2 nm

B.3.1 Polycrystalline Material

Examples of polycrystalline materials are carbons and graphites. For such mate-rials, real density is de�ned as including the closed voids. The material is usually

B.3 Porous Material with Open and Closed Porosity 517

L1

L2

L3

V

�o

Vc

Vr

�0c

Figure B.3: Volume with an open and a closed void fraction.

grinded before pycnometric measurement of real density. It is assumed that mostof the closed pores are preserved during grinding. In this case, the open porosityis given by the following equation:

�o = 1� �a

�r(B.18)

Total porosity consists of closed and open porosity. To calculate the total porosityof a polycrystalline material, one needs knowledge of the crystal density �c. Thecrystal density is the density of a void free material measured on the microscopiclevel. For total porosity �, the following relationship is valid:

� = 1� �a

�c(B.19)

By using Equation (B.17), the closed porosity can be calculated as follows:

�c = �� �o =�a

�r� �a

�c= �a(

1

�r� 1

�c) (B.20)

In this case, the closed void fraction is inherent in the solid material. Thus:

Vr = Vr;v + Vr;c (B.21)

Vr;v is the closed porosity in the solid material and Vr;c is the volume of thecrystalline dense material. Vr;c and the fraction of closed voids �0c in the solid arerelated as follows:

Vr;c = Vr(1� �0c) (B.22)

518 Density, Porosity and Surface Area

The mass of the material can be calculated from three di�erent densities andvolumes:

m = �aV

m = �rVr

m = �cVr;c

By equating:

Vr;c = V (1� �) (B.23)

and:

Vr;c = Vr(1� �0c) = V (1� �o)(1� �0c) (B.24)

one can derive:

1� � = (1� �o)(1� �0c) (B.25)

+� = �o + �0c(1� �o) (B.26)

+�c = �0c(1� �o) (B.27)

since � = �o + �c. Equation (B.27) gives the relationship between bulk closedporosity �c and closed porosity �0c inherent in the solid material. Now, use Equa-tion (B.19) to obtain:

1� � =�a

�c=�a

�r

�r

�c= (1� �o)

�r

�c(B.28)

Comparison with Equation (B.25) gives �r�c

= (1� �0c) which gives and �nally:

�0c = 1� �r

�c(B.29)

When �0c � 0, �r � �c. Thus, in cases with negligible closed porosity, real density�r is a good approximation of crystal density �c.

In summary, porosities � and �o are found from measurements of �a, �r and �c andEquations (B.19) and (B.18). �c is found from Equation (B.20). Equation (B.27)relates closed porosity in the solid volume to closed porosity as related to the bulkvolume.

B.3.2 Packed Bed of Polycrystalline Material

Packed beds of granular material also inhabit a certain void fraction. The void frac-tion is due to interparticular and intraparticular porosity. Interparticular porosityis open; intraparticular porosity might be both open and closed. Thus, the open

B.3 Porous Material with Open and Closed Porosity 519

void fraction consists of both interparticular voids as well as open intraparticularvoids.

Let �b;pb denote bulk (apparent) density of a sample of the granular material.Then the total porosity in a packed bed can be calculated from Equation (B.19):

� = 1� �b;pb

�c(B.30)

Equation (B.18) and Equation (B.20) for open and closed porosity respectively,are still valid.

Let Ye denote interparticular porosity and Vp the volume of particles includingboth open and closed intraparticular porosity. Then the volume of particles isgiven by:

Vp = V (1� Ye) (B.31)

To describe intraparticular porosity, measurements on the particles are needed. Inthe following, �a;p denotes apparent density of single coke particles. Let Xi denotethe intraparticular porosity. Xi exist as both open and closed porosity:

Xi = Xi;o +Xi;c (B.32)

Xi;o denotes open porosity and Xi;c denotes closed porosity. Let Vp;r denote theparticle volume including closed voids. Vp;c represents the volume of the non-porous particulate material. As before:

Vp;c = Vp(1�Xi)

and:

Vp;r = Vp(1�Xi;o)

Vp;c = Vp;r(1�X 0

i;c)

which gives:

Xi = Xi;o +X 0

i;c(1�Xi;o) (B.33)

This relationship is analogous to Equation (B.26). Thus, Xi;c = X 0

i;c(1 � Xi;o).Since X 0

i;c =�r�c, one obtains X 0

i;c = �0c. Also:

Xi = 1� �a;p

�c

Xi;o = 1� �a;p

�r

Ye and Xi are related to the total porosity. The volume of the crystalline phaseis:

Vp;c = Vp(1�Xi)

520 Density, Porosity and Surface Area

Now use Equation (B.31) for Vp to get:

Vp;c = V (1� Ye)(1�Xi)

Furthermore, use Vp;c = V (1� �) to obtain:

� = Ye +Xi(1� Ye) (B.34)

Substituting for Xi and Xi;c, the total porosity can be expressed by inter- andintraparticular porosities as follows:

� = Ye +Xi;o(1� Ye) +X 0

i;c(1�Xi;o)(1� Ye) (B.35)

Since � = �o + �c, open- and closed porosity can be found from:

� Open porosity: �o = Ye +Xi;o(1� Ye)

� Closed porosity: �c = X 0

i;c(1�Xi;o)(1� Ye)

X 0

i;c is analogous with quantity �0c in Subsection B.3.1 above. Figure B.4 showsthe relationship between the di�erent measures of porosity.

L4

L1

L2

L3

Open porosityIntra-particular porosity

Closed porosityInter-particular porosity

L1

L2

L3

L4

Ye

Xi;o

V

�0c = X 0

i;c

�o

X 0

i;c

Vp;c

Vp;r

Vp

V

Vp;r

Vp;c

Figure B.4: Porosity relationships in a packed bed with porous particles.

The interparticular porosity Ye can be found from Equation (B.34):

Ye =��Xi

1 +Xi(B.36)

where Xi = Xi;o +Xi;c.

In summary, porosity in packed beds is classi�ed as follows:

B.4 Porous Composite with n Polycrystalline Components 521

� Total porosity � consist of open- �o and closed- �c porosity.

� Interparticular void fraction Ye is due to open porosity.

� Intraparticular void fraction Xi is due to partly open and closed porosity.

These quantities can be calculated from measurements of:

� Packed bed (apparent) bulk density �b;pb

� Granular apparent (bulk) density �a;p

� Granular real density �r

� Crystalline density �c

This procedure for obtaining porosity does not give a measure of pore size distribu-tion. To obtain such information, standard techniques for pore size measurementsmust be used (mercury porosimetry etc.). However, parameters �, �o and �c aswell as Ye and Xi;c will be used for classifying the pore structure in a packed bed.

B.4 Porous Composite with n Polycrystalline Com-

ponents

In a porous composite, both porosity within the di�erent components of the com-posite as well as voids between the components may contribute to the total voidfraction. One may use the above derivations to obtain:

�c;a = �c;c(1� �) (B.37)

where � is total porosity. �c;a and �c;c are composite apparent- and compositecrystal densities respectively. If the intracomponent closed porosity is negligible,�c;c � �c;r where �c;r is the composite real density:

�c;a � �c;r(1� �) (B.38)

�c;c and �c;r can be calculated in the same manner as shown in Section B.1:

~�c; =

nXi=1

�i� ;i

~�c; =1Pn

i=1xi� ;i

where = c; r. This approach was used for modelling total porosity in the anodeduring baking. However, during anode production, the situation is even morecomplicated since the uid pitch partly �lls the petroleum coke open porosity aswell as surrounds the coke grains with a very thin layer of pitch.

522 Density, Porosity and Surface Area

B.5 Surface Area

Anodes belong to the macroporous carbons. Pore classi�cation was discussed inSection B.2; macropores have diameter above 50 nm.

Macropores in the binder phase are formed due to gas evolution during binderpyrolysis. These spherically shaped pores are often linked together by a networkof shrinkage cracks formed in the later stages of the baking process. Some of thesecracks are also due to anisotropic contraction during cooling of the anodes; so-called Mrozowski cracks (unavoidable porosity). The macroporosity constitutes anopen interconnected network of pores. However, the shrinkage cracks in the �llerparticles are closed since they are usually sealed during the mixing and mouldingstages of green anode production. There is also a certain degree of penetration ofpitch during early stage baking.

B.5.1 Mercury Porosimetry

Macropores is most commonly characterized by mercury porosimetry as shown inFigure B.5. Mercury does not wet the pore walls (contact angle below 90�) andcan only enter the pores by application of a certain pressure. The pressure pmapplied to the liquid mercury is related to the radius r of an assumed cylindricalpore according to the Washburn equation (Marsh 1989):

r = �2 cos �

�p(B.39)

where �p = pm�pg. pg is the pore gas pressure, is the surface tension of mercuryand � is the contact (wetting) angle. The sample should be evacuated prior tomeasurement and therefore pg << pm. Thus, pg can be neglected. The pores areassumed to be non-intersectioning and cylindrical. Usually � 480mN/m and� � 140�. These are all assumptions with a considerable degree of uncertainty.The use of high pressures might cause breakdown of the porous network. Mercuryporosimetry is not suitable for characterizing micropores. In modern porosimetry,pressures from sub-atmospheric to 5000 atmospheres is used which correspond topore radii in the range from 7.5 �m down to 1.5 nm. Use the above mentionedvalues and obtain:

r � 75000

p(B.40)

r is measured in �Angstr�om [�A]. Pressure p is expressed in kp/cm2.

Micropores have pore diameter below 2 nm. To give an idea of this pore size, itmight be compared to the interlayer spacing in turbostratic carbons which equals3.44 �A and corresponds to 0.344 nm.

The pore size distribution can be found from mercury porosimetry. Total speci�csurface area of a sample is assumed to be Vg [cm

3/g]. Then:

Vg = Vi + V (r) (B.41)

B.5 Surface Area 523

p

�

r

Figure B.5: Mercury porosimetry for pore diameter measurement.

where Vi = Vi(p) is the volume of mercury which penetrates the pores at pressurep. V (r) is the pore volume containing pores with radius up to r:

V (r) =

Z r

0

dV (r)

drdr (B.42)

where:

dV (r)

dr=dV (r)

dp

dp

dr(B.43)

V (r) and Vi are related as follows:

dV (r)

dp= �dVi

dp

dp

dr= �2 cos �

r2= �p

r

which gives:

dV (r)

dr=dVi

dp

p

r=

1

r

dVi

d ln p(B.44)

Thus, V (r) can be determined by integration of Equation (B.44) using the exper-imental values of dVi

d(ln p) . The above derivation is based on Thonstad (1989).

B.5.2 Surface Area of Materials

Speci�c surface area Sg [m2/g] can be determined from porosimetry-measurements.

The work performed due to penetration of a certain volume dVi of mercury is givenby:

dW = p�r2dL = pdVi (B.45)

The surface of this volume is dS = 2�dL which gives dL = dS2�. Substitution for

dL gives:

pdVi =1

2prdS

+

dS =2dVir

524 Density, Porosity and Surface Area

Substituting r = � 2 cos �p

, gives:

dS =p

cos �dVi (B.46)

which is used to determine:

Sg =

Z Vi

o

dS (B.47)

The above derivation is based on Thonstad (1989).

Based on the Kozeny-equation, Carman proposed a direct method for measuringsurface area in uniformly packed unconsolidated beds. Thus, the method is notapplicable for measuring speci�c internal surface of consolidated porous media.Therefore the method cannot be used for anode surface area measurements. How-ever, the method should be considered for determination of the average diameterof coke particles which can be calculated from the speci�c surface of the particlesif they are assumed spherical.

Appendix C

Basics from Aluminium

Electrolysis

Aluminium is the most abundant metal in the earth's crust but it does not occurnaturally in elemental form. Due to its strong a�nity to oxygen, the metal isbonded in oxides and silicates. At about 1850, the �rst commercial process foraluminium manufacturing was operative. Sodium aluminium chloride reacted withmetallic sodium in a reduction process.

The Hall H�eroult process has been dominating in aluminium production since itsindependent invention by Charles Hall (USA) and Paul Heroult (France) in 1886.Materials needed in the Hall-H�eroult process is shown in Figure C.1.

Based on the readable text by Grjotheim & Kvande (1993), this chapter gives anintroduction to the Hall Heroult process.

C.1 The Hall-Heroult Process

In the Hall Heroult process, liquid aluminium is produced by electrolytic reductionof alumina dissolved in an electrolyte (bath) which mainly contains cryolite. Thebath temperature is approximately 970�C1 and alumina concentrations is nomi-nally between 2 to 4 %. Several carbon anodes are dipped into the bath. Oxygenfrom the alumina is electrolytically discharged onto the anode. Immediate reactionwith the solid carbon leads to formation of carbon dioxide. Under the electrolyte,there is a pad of liquid aluminium resting on a preformed carbon lining. Thebath/metal interface acts as the cathode with formation of aluminium at the in-terface. The arrangement of the anode and cathode with the bath and metal padin between is shown in Figure C.2.

1The melting point of cryolite; 1012�C is lowered by the addition of alumina.

526 Basics from Aluminium Electrolysis

PetroluemCoal

CarbonBayer

Bauxite

Cryolite

Hall Heroult

Aluminium

Refining

Deleyed Coking

Calcining

Coal Tar Pitch

Petroleum Coke

Alumina

Manufacturing

Oven

Coke

Process

Process

Figure C.1: Steps in aluminium production.

Aluminium

Electrolyte

Cathode

Anode

Solid Electrolyte(Side Ledge)

Figure C.2: Details from the electrolytic bath. From Grjotheim & Kvande (1993).

C.1.1 The Bath

During electrolysis, dissolved alumina is electrolytically decomposed to liquid alu-minium and carbon dioxide. The following primary reaction equation describesthe reduction of dissolved alumina to aluminium and carbon dioxide under theconsumption of the carbon anode:

1

2Al2O3(d) +

3

4C(s)! Al(l) +

3

4CO2(g) (C.1)

In the bath, cryolite (Na3AlF6) is the main component. In addition, the bath inmodern cells contains the following components on a mass basis:

C.1 The Hall-Heroult Process 527

� 6 to 13 % aluminium uoride (AlF3)

� 4 to 6 % calcium uoride (CaF2)

� 2 to 4 % alumina (A2O3)

During electrolysis, there is no consumption of the bath. Bath losses are mainlydue to vaporization. The bath temperature in modern cells is in the range of 940to 970�C. The bath height is rather constant in the order of 20 cm. The interpolardistance2 is typically between 4 and 5 cm.

Alumina is the raw material in the process. Theoretically, 1.89 kg Al2O3 gives1.00 kg of aluminium according to the stoichiometry in Equation (C.1).

In modern cells, alumina is fed continuously to the bath by the use of so-called"point-feeders". Two to �ve feeders successively add 1 to 2 kg of alumina every 1to 2 minutes. The alumina is quickly dissolved and mixed into the bath to avoidsludge-formation. A too high alumina concentration promotes sludge (muck) for-mation. A too low alumina concentration may lead to an anode e�ect. During ananode e�ect, there is a distortion of normal cell operation due to rapid increase ofthe cell voltage. Voltages in the order of 30 to 50 V may occur. Fluoride compo-nents are electrolytically decomposed and leads to the formation of an insulatinggas layer at the anode working face. To terminate an anode e�ect, the bath has tobe agitated to remove the sub-anode gas layer. Furthermore, alumina is rapidlyadded to restore satisfactory alumina concentration.

On the top of the bath, a layer of thermally insulating "crust" (frozen bath) isformed. The "crust" prevents anode airburn and reduces heat losses from thebath.

C.1.2 The Anode

The anode has two main functions in the Hall-H�eroult process:

� Lead electric current

� Act as reductant in the cleavage of alumina dissolved in cryolite

Carbon is consumed during electrolysis according to Equation (C.1). Thus, eachanode in the cell has to be replaced after a certain time of operation. Anodereplacement occurs in a regular sequence. The frequency of anode changes isdetermined by the anode size and usually the spent anode is less than one quarterof the original size at replacement. Usually, at least one anode has to be replacedeach day. Anode consumption rate is reasonably uniform for a given anode quality;there is only minor variations in the size of spent anodes (butts).

Two basic anode designs exist:

2Distance between the anode working face and the surface of the metal pad.

528 Basics from Aluminium Electrolysis

� S�derberg anodes

� Prebaked anodes

Prebaked anodes give the best metal quality and are also associated with the low-est carbon consumption. Theoretical carbon consumption is 333 kg C/ton Al. Netcarbon consumption in prebaked cells is approximately 400 kg C/ton Al. Grossconsumption which also includes the butts (spent anodes recycled from the elec-trolysis) raises carbon consumption to between 500 and 550 kg C/ton Al.

S�derberg anodes are continuous and self baking. This advantage,however , cannotcompete with the superior prebaked anode technology. Anyway, still 40 % of theworld's total aluminium production takes place in S�derberg cells.

Prebaked anodes contain between 13 to 18 % coal tar pitch; the rest is butts andpetroleum coke aggregate3. The carbon paste is moulded and baked in speciallydesigned baking furnaces. Maximum heat treatment temperature is approximately1200�C.

C.1.3 The Cathode

The bottom carbon lining on which the liquid metal rests, serves as the cath-ode during electrolysis. The cathode consists of carbon blocks baked at semi-graphitizing temperatures. Anthracite is the main carbon component in the blocks.At certain intervals in time, cell linings have to be changed due to penetration ofsodium and molten bath into the lining and subsequent swelling of the lining. Cellrelining takes place at considerable costs; the economic consequences of short celllife is rather dramatic: Average cathode life time is in the order of 1000 to 2000days but single cells have been operative for more than 4000 days. The best celllines can have average life time in the order of 2500 days.

C.1.4 The Modern Electrolytic Cell

Prebaked cells in modern lines now carry currents of 300 kA. Daily metal produc-tion is in the order of 2300 kg/day for these large cells. There is approximately 20anodes in each cell and an anode has to be replaced usually after 22 to 26 days.

In aluminium plants, the cells are arranged in long rows called cell (pot) lines.Modern cells are commonly placed side-by-side to reduce the problem of interactionbetween induced magnetic �elds and current ow through the cell. Between 150to 288 cells are serially arranged with the cathode of one cell electrically connectedto the anode part of the downstream neighbouring cell. Across each cell, there isa voltage drop of 4 to 5 V contributing to a total voltage drop between 1000 Vand 1500 V for a typical line. The magnetic �eld causes stirring of the metal pooland thus height variations and instabilities in the pad. Progress in cell design and

3In S�derberg anodes, more than 25 % coal tar pitch is used.

C.1 The Hall-Heroult Process 529

mutual cell arrangement has reduced this problem. A schematic description of amodern electrolytic cell is given below.

Thermal Insulation

Cathode Block

Current Collector Bars

Busbars

Alumina

BreakerCrust

Molten Aluminium

InsulationThermal

HangerAnode

PrebakedAnode

AluminaCrust

Electrolyte

Casing

Steel

FrozenLedge

Carbon Lining

Figure C.3: Electrolytic cell with prebaked anode.

Since in modern cells, alumina is automatically feed into the electrolyte, the mostimportant manual routine cell operations are anode changing and metal tapping.

Two main parameters describe the performance of the cell:

� Current E�ciency (CE)

� Energy E�ciency (EE)

Current e�ciency is a measure of the percentage of supplied electric current thatis used for metal production. It is de�ned as the ratio of the mass of actual metalproduced and the theoretically possible mass which could be formed accordingto Faraday's �rst law. Theoretically, current e�ciency should be 100 %. Themain reason for loss in current e�ciency is the metal solubility in the electrolyte.Dissolved metal reacts with carbon dioxide that is formed at the anode and thusreoxidized by the following back-reaction:

2Al(d) + 3CO2 ! Al2O3(d) + 3CO(g) (C.2)

The back-reaction leads to lowered metal production, increased carbon consump-tion and is the main source of formation of carbon monoxide during normal elec-trolysis.

Energy consumption includes both current e�ciency and cell voltage. Thereforeenergy consumption is a better cell-performance criterion. Typically, modern cells

530 Basics from Aluminium Electrolysis

consume in the order of 14.5 kWh/kgAl. Theoretical consumption is only 6.34kWh/kgAl at 977�C. Energy e�ciency de�ned as the ratio of actual and theoret-ical energy consumption is usually below 50 %. A lot of energy is lost as heat tothe surroundings. Heat conservation is an important aspect of modern cell design.

C.2 Current E�ciency and Carbon Consumption

During aluminium electrolysis, aluminium dissolved in the electrolyte will be partlyresponsible for a not insigni�cant back reaction to form dissolved alumina andcarbon monoxide. As mentioned above, current e�ciency CE = � 100% is de�nedas the ratio of the mass of actual metal produced and the theoretically possiblemass which could be formed. In the back reaction, CO is formed according to theequation:

(1� �)Al(l) +3

2(1� �)CO2(l)! (C.3)

1

2(1� �)Al2O3(d) +

3

2(1� �)CO(g)

(1� �) is the fraction of dissolved metal converted in the back reaction.

Based on Equation (C.1) and Equation (C.3), the overall reaction, taking intoaccount the back reaction of a fraction (1� �) of the aluminium, will be:

1

2Al2O3(d) +

3

4�C(s)! (C.4)

Al(l) +3

4(2� 1

�)CO2(g) +

3

2(1

�� 1)CO(g)

neglecting the possibility that CO(g) may react to C(s). (1 � �) is the fractionof aluminium which is reoxidized, under the assumption that the possibility of afurther reaction of CO to C is ignored. Metal solubility is in the order of 0.1%. The reaction reduces � since it causes aluminium consumption and increasedanode consumption. The reaction is also the reason for most of the CO presentduring normal cell operation. Usually � > 0:92, but values of � in the range of0.85 to 0.95 can occur.

An important factor a�ecting � is the bath temperature. � will be improvedby lowering bath temperature. Probably, a lowered bath temperature directlyin uences the recombination of aluminium and carbon dioxide in the electrolyte.Inorganic metal impurities in the anode due to recycling of butts also have impacton � (occurrences of phosphorus P and vanadium V a are mostly unwanted). Theash content of the anode gives a measure of anode impurities. Finally carbon lossdue to selective oxidation of the binder coke (dusting) also have impact on �4. Thisselective oxidation causes fragments of the anode working face to drop into thebath without being consumed electrolytically, thus directly leading to increasedcarbon consumption. Finally, anode thermal shock problems can lead to loss in �

4Dusting re ects underbaking of the anodes and/or de�ciency of pitch.

C.3 Energy Consumption and Energy E�ciency 531

of approximately 1.5 %. This is also due to fragmentary consumption of the anodewhich leads to carbon loss. To summarize, � is in uenced by:

� Bath temperature

� Anode ash content

� Dusting

� Thermal shock problems

It can be seen that current e�ciency also expresses the electrolytic consumptionof carbon. Let Ct;el be the theoretical electrolytic consumption of carbon; 333kg C/ton Al as mentioned above. Also let Cel be the real electrolytic carbonconsumption. Then Cel and Ct;el is related via the current e�ciency �:

Cel =Ct;el

�(C.5)

During the recent years, current e�ciency has increased due to progress in elec-trolyte chemistry and process control. � continuously approach the theoreticalvalue of 1.00. A practical limit seems to lie between 0.96 to 0.98 due to the solu-bility of the dissolved metal pieces in the electrolyte. Low bath temperature andstable metal pad has resulted in current e�ciency in the order of 0.95 over shortperiods of time.

C.3 Energy Consumption and Energy E�ciency

Both electrical energy (electrochemical and "Ohmic" work) and thermal energyin the form of carbon is consumed in the Hall-H�eroult process. The energy con-sumption is distributed between three terms:

� Electrochemical energy : 6.5 kWh/(kgAl)

� Heat losses from the cell : 8.1 kWh/(kgAl)

� External (busbar) heat losses : 0.6 kWh/(kgAl)

This gives a total energy consumption of 15.2 kWh/(kgAl). The energy e�ciencyEE is the ratio of the theoretical and the actual energy consumption:

EE = (1 +WOhm

zFE)� (C.6)

EE is in the order of 50 % for modern cells. WOhm is the ohmic resistance workin the cell.

About 20 % of the heat produced in electrolysis is used for metal production.The rest of the energy is consumed in keeping the bath and metal at working

532 Basics from Aluminium Electrolysis

temperature, heating reactants and keeping the alumina dissolved and lost as heat.Current density and cell voltage are heat generating parameters. For operativecells, the heat lost is determined by the cell voltage. The cell voltage is determinedby the interpolar distance; the width of the reaction zone.

C.4 The Anode E�ect

This phenomenon is normally manifested by a sudden increase in the cell voltage.The anode e�ect depends on the alumina concentration in the bath and the bathtemperature. For a certain alumina concentration and bath temperature, theanode e�ect occurs at a given current density; the critical current density (CCD).Both lowered bath alumina concentration and bath temperature lowers the criticalcurrent density. CCD also depends on the anode orientation but bath propertiesare the most important factors for the anode e�ect.

The unstable condition that occur when the cell is "on light" (sparkling and tur-bulence) is not predictable. A number of explanations exist. Which (or whether)speci�c anode properties play a role in this phenomenon is not clear. Most proba-bly deteriorated wetting characteristics cause the build-up of an insulating gas �lmon the anode working side. The size of the gas bubbles depends on the aluminaconcentration.

Two positive features are due to the anode e�ect: It gives control of the aluminaconcentration and cleans the working side of the anode as well as brings carbondust to the surface of the bath.