106487559 Process Plant Simulation Babu

Process Plant

Simulation B.V. Babu

Birla Institute of Technology and Science, Pilani, Rajasthan

OXFORD UNIVERSITY PRESS

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It was the discovery of crude oil in 1859 by Drake that led to the thought that there was the need to have a separate discipline of chemical engineering. Until then, chemists and mechanical engineers also served as chemical engineers. In the early stages of the evolution of chemical engineering as a separate discipline, it was thought to be a combination of unit operations and unit processes. Since then, chemical engineering has been dynamic and has undergone many changes. The beauty of this discipline lies in its versatility and ability to adapt itself to various new and interdisciplinary fields of engineering and science. Environmental engineering, biochemical engineering, energy engineering, optimization, and process control and instrumentation can all come under the single umbrella of chemical engineering. Taking these dynamic changes and the versatility of this discipline into account, it has been redefined as comprising the concepts of process synthesis, analysis, and optimization. Process synthesis deals with the selection of the best process alternative out of the millions of alternate process flowsheets available for manufacturing any product. A systematic methodology has been developed based on the conceptual design principles for this process synthesis activity. It includes making process design decisions at various levels of hierarchy, such as batch versus continuous, input-output structure, recycle structure, general structure, and energy integration. The complexity of the decisions increases as the level goes up from batch versus continuous to energy integration. Process analysis deals with the detailed simulation of process plants for the best process alternative selected from the process synthesis activity. Optimization, as the name suggests, deals with finding the optimum design parameters for maximizing the profit or minimizing the total investment on a process plant (trade-off between capital and operating costs).

Process plant simulation, as a subject, deals with concepts on process analysis and optimization. It includes the concepts of modelling, optimization, decomposition of networks, modular and equation-solving approaches, data regression, convergence promotion, specific-purpose simulation and dynamic simulation, etc.

I have been teaching the modelling part of this course for the last 17 years of my teaching career. Earlier, modelling was part of a course I was teaching at Gujarat University during 1985-1996 as chemical systems modelling, and now at BITS, Pilani, it is part of process plant simulation, which I have been teaching for the last five years.

About the Book There is no consolidated literature available which covers all these aspects of process plant simulation. The idea of writing a book on process plant simulation originated in my mind two years ago. I thought it appropriate and felt the necessity to bring all the concepts and aspects of this subject together in the form of a single book. The purpose and objectives of this book are fourfold:

to provide a textbook for the course on process plant simulation, bringing all related concepts together, offered basically in undergraduate and postgraduate programmes at various engineering colleges and universities across the country and abroad

vi Preface

I to introduce a generalized approach to modelling various engineering systems, so that these principles can be used to model any new situation or system in various engineering, science and other applied disciplines

I to introduce non-traditional optimization techniques, which can serve as a research monograph and guide to the methodology for various evolutionary optimization techniques (population-based search algorithms)

I to familiarize students with professional software packages such as HYSYS and FLUENT

Content and Structure The book is divided into five parts. A brief introduction to process synthesis, process analysis, and optimization is followed by the presentation of process plant simulation as a subject in Chapter 1. Subsequently, Part I gives an overview of modelling aspects in Chapter 2, and the classification of mathematical modelling in Chapter 3.

Part II discusses chemical systems modelling in four chapters: modelling mass-transfer systems in Chapter 4, heat-transfer systems in Chapter 5 , and fluid mechanics and reaction engineering systems in Chapters 6 and 7, respectively. Apart from focusing on modelling various chemical engineering systems, various mathematical techniques to be used for solutions of the models proposed (such as algebraic equations, ordinary differential equations, partial differential equations, finite difference equations, Laplace transformations, solution by series, etc.) have also been dealt with.

Part 111 deals with the treatment of experimental results, wherein error propagation and data-regression techniques (Chapter 8) are discussed in detail.

Part JY focuses on traditional optimization techniques (analytical methods, Lagrangian multiplier method for constraint optimization, gradient methods such as steepest descent and sequential simplex methods, etc.) in Chapter 9, and non-traditional optimization techniques (such as simulated annealing, genetic algorithms, differential evolution, evolutionary strategies, etc.) in Chapter 10.

Finally, Part V, consisting of five chapters, gives a detailed overview .of all aspects related to simulation. Chapter 1 1 covers the modular and equation-solving approaches. Chapter 12 deals with the decomposition of networks and the various associated tearing algorithms available in literature. Chapter 13 covers convergence promotion and the physical and thermodynamic properties. Chapter 14 discusses case studies on specific- purpose simulation and dynamic simulation. Chapter 15 gives an overview of two most widely used professional software packages, namely, HYSIS of HyproTech and FLUENT, and demonstrates the step-by-step procedure of solving problems using these software packages.

Key Features I Offers exhaustive coverage of all topics related to plant simulation I Covers traditional as well as non-traditional optimization techniques I Discusses case studies on specific-purpose and dynamic simulation

Preface vii

Includes an overview of professional software packages used in plant simulation,

rn Includes a CD-ROM containing program codes and related useful information Comes with a Solutions Manual which provides the solutions to selected

such as HYSIS and FLUENT

problems

Acknowledgements Many people have contributed directly or indirectly in bringing this book to its present form since its conception in the year 2000. To begin with, I would like to accord my deep sense of gratitude and thanks to Prof. S. Venkateswaran, Vice Chancellor, BITS, Pilani, for encouraging this course development work. I must acknowledge the active guidance and encouragement I received from Prof. L.K. Maheshwari, Director, BITS Campus, Pilani; Prof. K.E. Raman, Deputy Director (Administration), BITS, Pilani, who constantly inspired me in time management and encouraged me to complete this work; and Prof. V.S. Rao, Deputy Director (Off-campus Programmes), BITS, Pilani, for being a fatherly figure for me. I must thank Prof. R.K. Patnaik, Prof. G.P. Srivastava, Prof. A.K. Sarkar, Dean (Instruction Division), BITS, Pilani, and Prof. R.N. Saha, Dean (Educational Development Division), BITS, Pilani, who made the necessary arrangements for the manuscript to be ready in time. I am deeply grateful to my colleagues Mr Rakesh Angira, Mr Ashish Chaurasia, Mr V. Ramakrishna, Mr Nitin Maheshwari, Mr Sheth Pratik, and Mr Suresh Gupta, who are doing their PhD under my guidance, for helping me in reviewing portions of the manuscript and providing valuable comments apart from making the figures and proofreading. I am indebted to all my students at BITS, Pilani (especially Mubeen, Pallavi, Ravindra, Gautam, Shobhana, Vishnu, Simli, Mandar, Angarasu, Amol, and Subha Jyothi), for their tireless effort and help in bringing out the preliminary draft of the book. I cannot find words to describe the debt I owe to Dr Rainer Storn of ICSI, Berkeley, Prof. David E. Goldberg of the University of Illinois at Urbana- Champaign, and Prof. Kalyanmoy Deb of IIT, Kanpur, whose inputs have helped me to understand non-traditional optimization techniques better. Last but not the least, this work would not have been completed without the understanding and support I got from my wife Shailaja and my children Shruti and Abhinav.

I heartily welcome any constructive suggestions, criticism, and appraisal by the readers, which would help me in improving the quality of this book in subsequent editions.

B.V. BABU

Contents Preface

1. Introduction 1.1 Process Synthesis 3 1.2 Process Analysis 5 1.3 Optimization 8 1.4 Process Plant Simulation 10 1.5 Organization of the Text 10

Exercises 12

V

1

PART I MODELLING 2. Modelling Aspects

2.1 Deterministic Versus Stochastic Processes 15 2.1.1 Deterministic Process 15 2.1.2 Stochastic Process 16

2.2 Physical Modelling 16 2.3 Mathematical Modelling 17 2.4 Chemical Systems Modelling 20

2.4.1 Model Formulation Principles 21 2.4.2 Fundamental Laws used in Modelling 22

2.5 Cybernetics 31 2.6 Controlled System 3 1 2.7 Pri,iciples of Similarity 32

2.7.1 Geometric Similarity 32 2.7.2 Kinematic Similarity 32 2.7.3 Dynamic Similarity 33

Exercises 38

15

3. Classification of Mathematical Modelling 39 3.1 Independent and Dependent Variables, and Parameters 39 3.2 Classification based on Variation of Independent Variables 42

3.2.1 Distributed Parameter Models 42 3.2.2 Lumped Parameter Models 42

3.3.1 Static Model 42 3.3.2 Dynamic Model 43 3.3.3 The Complete Mathematical Model 44

3.4 Classification Based on the Type of the Process 45 3.4.1 Rigid or Deterministic Models 45 3.4.2 Stochastic or Probabilistic Models 45 3.4.3 Comparison Between Rigid and Stochastic Models 45

3.3 Classification Based on the State of the Process 42

3.5 Boundary Conditions 46 3.6 The Black Box Principle 46

Contents ix

3.7 Artificial Neural Networks 47 3.7.1 Network Training 48 3.7.2 Modes of Training 49 3.7.3 Network Architecture 49 3.7.4 Back-propagation Algorithm 50 3.7.5 ANN Applications 51 3.7.6 Example Problems 51

Exercises 57

PART II CHEMICAL SYSTEM MODELLING 4. Models in Mass-transfer Operations

4.1 Steady-state Single-stage Solvent Extraction 6 1 4.2 Steady-state Two-stage Solvent Extraction 65 4.3 Steady-state TWO-stage Cross-current Solvent Extraction 69 4.4 Unsteady-state Single-stage Solvent Extraction 73 4.5 Unsteady-state Mass Balance in a Stirred Tank 79 4.6 Unsteady-state Mass Balance in a Mixing Tank 82 4.7 Unsteady-state Mass Transfer (Ficks Second Law of Diffusion) 85 4.8 Steady-state N-stage Counter-current Solvent Extraction 90 4.9 Multistage Gas Absorption (Kremser-Brown Equation) 94

4.10 Multistage Distillation 98 Exercises 103

61

5. Models in Heat-transfer Operations 107 5.1 Steady-state Heat Conduction Through a Hollow Cylindrical

Pipe (Static Distributed Parameter Rigid Analytical Model) 108 5.2 Unsteady-state Steam Heating of aLiquid

(Dynamic Lumped Parameter Rigid Analytical Model) 115 5.3 Unsteady-state Heat Loss Through a Maturing Tank

(Dynamic Lumped Parameter Rigid Analytical Model) 117 5.4 Counter-current Cooling of Tanks 124 5.5 Heat Transfer Through Extended Surfaces (Spine Fin) 129 5.6 Temperature Distribution in a Transverse

Cooling Fin of Triangular Cross Section 135 5.7 Unsteady-state Heat Transfer in a Tubular Gas Preheater 138 5.8 Heat Loss Through Pipe Flanges 139 5.9 Heat Transfer in aThermometer System 141

5.10 Unsteady-state Heat Transfer by Conduction 145 Exercises 148

6. Models in Fluid-flow Operations 6.1 The Continuity Equation 152 6.2 Flow Through a Packed Bed Column 155 6.3 Laminar Flow in a Narrow Slit 156

152

x Contents

6.4 Flow of a Film on the Outside of a Circular Tube 160 6.5 Choice of Coordinate Systems for the Falling Film Problem 162 6.6 Annular Flow with Inner Cylinder Moving Axially 164 6.7 Flow Between Coaxial Cylinders and Concentric Spheres 166 6.8 Creeping Flow Between Two Concentric Spheres 169 6.9 Parallel-disc Viscometer 172

6.10 Momentum Fluxes for Creeping Flow into a Slot 174 Exercises 178

7. Models in Reaction Engineering 7.1 Chemical Reaction with Diffusion in a Tubular Reactor 181 7.2 Chemical Reaction with Heat Transfer in a Packed Bed Reactor 185 7.3 Gas Absorption Accompanied by Chemical Reaction 191 7.4 Reactors in Series-I 192 7.5 Reactors in Series-I1 196

Exercises 199

PART 111 TREATMENT OF EXPERIMENTAL RESULTS 8. Error Propagation and Data Regression

8.1 Propagation of Errors 203 8.1.1 Propagation Through Addition 203 8.1.2 Propagation Through Subtraction 204 8.1.3 Propagation Through Multiplication and Division 204

8.2.1 Errors of Measurement 205 8.2.2 Precision Errors 205 8.2.3 Errors of Method 205 8.2.4 Significant Figures 205

8.3 Data Regression (Curve Fitting) 205 8.3.1 Theoretical Properties 205 8.3.2 Data-regression Methods 206 8.3.3 Problems on Data Regression 21 1

8.2 Sources of Errors 204

Exercises 216

PART IV OPTIMIZATION 9. Tradition a I 0 p t i m i za t i o n Tech n i q u es

9.1 Limitations of Optimization 221 9.2 Applications of Optimization 222 9.3 Types of Optimization 224

9.3.1 Static Optimization 224 9.3.2 Dynamic Optimization 224

9.4.1 Analytical Methods of Optimization 226 9.4.2 Optimization with Constraints (Lagrangian Multipliers) 227

9.4 Methods of Optimization 225

181

203

22 1

Contents xi

9.4.3 Gradient Methods of Optimization 231 9.4.4 Rosenbrock Method 252 9.4.5 Other Methods 252

Exercises 255

10. Non-traditional Optimization Techniques 10.1 Simulated Annealing 258

10.1.1 Introduction 258 10.1.2 Procedure 258 10.1.3 Algorithm 259

10.2 Genetic Algorithms 260 10.2.1 Introduction 260 10.2.2 Definition 260 10.2.3 The Problem 261 10.2.4 Techniques used in GA 10.2.5 Operators in GA 262 10.2.6 Comparison of GA with Traditional Optimization Techniques 10.2.7 Applications of GA 265 10.2.8 Stepwise Procedure for GA Implementation 267

10.3.1 Introduction 272 10.3.2 XOR Versus Add 273 10.3.3 DE at a Glance 273 10.3.4 Steps Performed in DE 273 10.3.5 Choice of DE Key Parameters (NP, F, and CR) 10.3.6 Strategies in DE 275 10.3.7 Innovations on DE 276 10.3.8 Applications of DE 276 10.3.9 Stepwise Procedure for DE Implementation 283

10.4.1 Evolution Strategies 286 10.4.2 Evolutionary Programming 287 10.4.3 Genetic Programming 287 10.4.4 Other Population-based Search Algorithms

261

265

10.3 Differential Evolution 272

275

10.4 Other Evolutionary Computation Techniques 286

288 Exercises 290

PART V SIMULATION 11. Modular Approaches and Equation-solving Approach

1 1.1 Modular Approaches to Process Simulation 295 1 1.1.1 Analysis Versus Design Mode 295

11.2 The Equation-solving Approach 300 1 1.2.1 Precedence-Ordering of Equation Sets 301 11.2.2 Disjointing 305 11.2.3 Tearing a System of Equations 305 11.2.4 The SWS Algorithm 305 11.2.5 Maintaining Sparsity 307

Exercises 3 1 1

257

295

xii Contents

12. Decomposition of Networks 12.1 Tearing Algorithms 3 13 12.2 Algorithms Based on the Signal Flow Graph 314

12.2.1 The Barkley and Motard Algorithm 315 12.2.2 The Basic Tearing Algorithm 319

12.3 Algorithms Based on Reduced Digraph (List Processing Algorithms) 321

12.3.1 Kehat and Shacham Algorithm 321 12.3.2 M&H Algorithms 323 12.3.3 Comparison of Various Tearing Algorithms 331

Exercises 332

13. Convergence Promotion and Physical and Thermodynamic Properties

13.1 Convergence Promotion 337 13.1.1 Newtons Method 338 13.1.2 Direct Substitution 339 13.1.3 Wegsteins Method 341 13.1.4 Dominant Eigenvalue Method 346 13.1.5 General Dominant Eigenvalue Method 346 13.1.6 Quasi-Newton Methods 347 13.1.7 Criterion for Acceleration 348

13.2.1 Sources 349 13.2.2 Databanks 350 13.2.3 Modularity and Routing 352

13.2 Physical and Thermodynamic Properties 349

Exercises 353

14. Specific-purpose Simulation and Dynamic Simulation 14.1 Auto-thermal Ammonia Synthesis Reactor 355

14.1.1 Ammonia Synthesis Reactor 355 14.1.2 Problem Formulation 357 14.1.3 Simulated Results and Discussion 361 14.1.4 Optimization 365 14.1.5 Conclusions 370

14.2 Thermal Cracking Operation 371 14.2.1 Thermal Cracking 371 14.2.2 Problem Description 372 14.2.3 Problem Reformulation 374 14.2.4 Simulated Results and Discussion 375 14.2.5 Conclusions 376

14.3 Design of Shell-and-Tube Heat Exchanger 377 14.3.1 The Optimal HED Problem 377 14.3.2 Problem Formulation 378 14.3.3 Results and Discussion 380 14.3.4 Conclusions 395

31 3

337

355

Contents xiii

14.4 Pyrolysis of Biomass 396 14.4.1 Stepwise Procedure 396 14.4.2 Problem Formulation 413 14.4.3 Method of Solution and Simulation 415 14.4.4 Results and Discussion 416 14.4.5 Conclusions 429

Exercises 430

15. Professional Simulation Packages 15.1 HYSIS-A Professional Software Package 432

15 , l . 1 Integrated Simulation Environment 433 15.1.2 HYSIS Products 433 15.1.3 Intuitive and Interactive Process Modelling 434 15.1.4 Open and Extensible HYSYS Architecture 435 15.1.5 Stepwise Methodology of HYSYS Usage for Problems 435

15.2 FLUENT-A Software Package for Computational Fluid Dynamics 479

15.2.1 Structure of the Program 480 15.2.2 Capabilities of FLUENT 481 15.2.3 Using FLUENT-An Overview 482 15.2.4 Physical Models in FLUENT 15.2.5 Stepwise Methodology of FLUENT Usage for a Problem 484

483

Exercises 5 13

432

References Index

520 532

CHAPTER 1

INTRODUCTION

The transformation of raw materials into desired products usually cannot be achieved in a single step in any chemical process. The overall transformation is broken down into a number of steps that provide intermediate transformations. These are carried out through reaction, separation, mixing, heating, cooling, pressure change, particle size reduction and enlargement, etc. So, the synthesis of a chemical process involves two broad activities. The first is the selection of individual transformation steps, and the second is finding the interconnections between these individual transformations to form a complete structure that achieves the required overall transformation, leading to a flowsheet, which is a diagrammatic representation of the process steps with their interconnections. The simulation of the process can be carried out after the flowsheet structure is defined.

Simulation is a mathematical model of a process, which attempts to predict how the process would behave if it was constructed. After creating a model of the process, the flow rates, compositions, temperatures, and pressures of the feeds are assumed. The simulation model can then be used to predict the flow rates, compositions, temperatures, and pressures of the products. Simulation also allows the sizing of the individual items of equipment in the process and prediction of the amount of raw material, energy, etc. required. The performance of the design can then be evaluated. Once the basic performance of the design has been evaluated, changes can be made to improve the performance, i.e., optimization. These changes might involve the synthesis of alternate structures, i.e., structural optimization

As energy consumption expands and resources of material and energy become more expensive and less certain, substantial changes will be required in many of

(Smith 2000).

2 Process Plant Simulation

the conventional chemical processing plants (Husain 1986). These have to incorporate a high degree of energy integration and achieve greater efficiencies through process modifications, thus conserving material and energy resources. At the same time, safety standards are tightening, therefore lower temperatures and pressures are desired. It is obvious that higher reaction rates, fewer stages, and smaller plants will reduce the burden of the material in a process, leading to safer plants.

In order to meet these challenges, greater effort on the part of the process engineers towards evaluation of the process flowsheets is necessary. Today, the emphasis is on tackling a process design problem from a broader perspective. This is facilitated by the use of computers and computer-aided design (CAD) software. The computer-aided design and simulation of a variety of systems is catching up very fast and being applied extensively in all fields of science, arts, engineering, and medicine.

In addition, chemical process engineers need to carry out process plant simulation for visualizing plant processes, performing heat and material balances of process flowsheets, designing new plants, suggesting modifications and expansions of existing plants (retrofitting), helping engineers to develop a better understanding of how their plants really operate, reducing costs and increasing profits, predicting operating efficiencies and anticipating problems, troubleshooting process and control problems, assisting operators in planning for production changes and disruptions, training operators, etc.

The design, development, and control of chemical plants and equipment are an essential part of chemical engineering. From an industrial standpoint, the economic feasibility and profitability of any process should be favourable for its implementation. It is therefore important for a chemical engineer to combine plant design and economics while formulating a prospective design plan. Process plant simulation constitutes the process analysis and optimization aspects of process design.

Chemical engineering discipline has been evolving continuously over the years without any limits, both in depth as well as in breadth. The beauty of this discipline lies in its ability to be versatile and its adaptability to various interdisciplinary areas (environmental engineering, biochemical engineering, energy engineering, optimization, to name a few). In order to appreciate the aspects involved in process plant simulation and to understand the importance of this subject, we have to go back to the history of evolution of chemical engineering as a discipline. Until Drake discovered crude oil in 1859, the chemists and mechanical engineers used to do the work of chemical engineers. It was the discovery of crude oil which led to the feeling among the scientists and academicians that there was a need to have a separate discipline in order to cater to the needs of processing and purifying various useful products from crude oil. The sequence of events that led to the birth

Introduction 3

of chemical engineering as a discipline is as follows: The first course in chemical engineering was offered by Prof. Lewis M. Norton in 1888 at MIT, USA for chemical major students. A full-fledged course was offered for the first time by the University of Pennsylvania in 1908; but a methodology on which chemical engineering could develop as a distinct discipline was still lacking. Two other developments, which had a bearing on the growth of chemical engineering as a discipline, took place in the late nineteenth century and the early twentieth century. In 1915, Arthur D. Little introduced the unit operations (mechanical operations, fluid-flow operations, heat-transfer operations, mass-transfer operations, etc.) concept. Subsequently, in 1922, the unit processes (oxidation, hydrogenation, nitration, sulphonation, etc.) concept was introduced by Groggins. Therefore, chemical engineering is defined as a combination of unit operations and unit processes. The roots of chemical engineering are physics, chemistry, and mathematics.

A broad-based definition for chemical engineering given by AIChE is Application of the principles of physical sciences, together with the principles of economics and human relations, to fields that pertain directly to processes and process equipment in which matter is treated to effect a change in state, energy, content, or composition.

A working definition of a chemical engineer could be one who can develop, design, construct, control, and manage any process involving physical and/or chemical changes (chemical changes include biochemical changes).

Keeping the history and versatility of chemical engineering in the background, considering the AIChE and working definitions of improved and matured chemical engineering, and bearing the challenges ahead of a chemical engineer in mind, the complete process design can be viewed as being carried out in the following three stages:

1. Process synthesis 2. Process analysis 3. Optimization

1.1 Process Synthesis Process synthesis is the first stage of the process design activity, in which a flowsheet of the process is constructed and this includes all equipment and their interconnections. For this task, information from several sources has to be gathered. One of the most commonly used sources is experience gathered over the years in solving both simple and complex design problems in a particular area of a process industry. In other words, process synthesis involves the use of design heuristics. But this may not provide the best efficiency for a given system. There may be other flowsheets that give higher levels of efficiency. Therefore, in recent times, the


approach has been to develop a generalized and systematic approach to process synthesis.

The major feature that distinguishes design problems from other types of engineering problems is that they are under-defined (i.e., only a very small fraction of the information needed to define a design problem is available from the problem statement) and open-ended (to supply the missing information, assumptions are to be made and there are numerous ways of achieving the same goal).

Process synthesis deals with the conceptual design of chemical processes, following a systematic procedure (Douglas 1988). The objectives of conceptual design are to find the best process flowsheet (Lee, to select process units and their interconnections), and optimum design conditions. This is very difficult, as experience shows that there are to the tune of 104-105 possible alternatives to be considered in process synthesis activity. In addition, the success rate of new ideas and subsequent designs ever becoming commercialized is very less (in fact it is less than 1%). That means there are many possibilities to consider, with a small chance of success.

In many cases the processing costs associated with the various process alternatives differ by an order of magnitude or more, so that short-cut calculations have to be used to screen the alternatives. At the same time, one should be certain that the solution is in the neighbourhood of the optimum design conditions for each alternative, in order to prevent discarding an alternative because of poor choice of design variables. So, cost studies must be used as an initial screening to eliminate ideas for designs that are unprofitable.

If a process appears to be profitable, other factors such as safety, environmental constraints, controllability, etc. should be considered. Because of the under-defined and open-ended nature of design problems, and because of the low success rate, it is useful to develop a strategy for solving design problems. This process synthesis activity has to be carried out by establishing a hierarchy of design decisions.

A systematic procedure has been laid down for screening the alternatives by following five levels of design decision hierarchy. With this approach, a very large and complex problem can be decomposed into a number of smaller problems that are much simpler to handle. By focusing on the decisions that must be made at each level in the hierarchy, one can identify the existing technologies that could be used to solve the problem without precluding the possibility that some new technology might provide a better solution. Moreover, by listing the alternate solutions that are proposed for each decision, a list of process alternatives can be systematically generated. In some cases, it is possible to use design guidelines (rules of thumb or heuristics) to make decisions about the structure of the flowsheet and/or to set the values of some of the design variables. Order-of-magnitude arguments are used to derive many of these heuristics, and a simple analysis of this type is used to identify the limitations of the heuristics. In cases where no heuristics

Introduction 5

are available, short-cut design methods are used as a basis for making decisions. The five levels of decision hierarchy in conceptual design are

1. Batch versus continuous production 2. Input-output structure of the flowsheet 3. Recycle structure of the flowsheet 4. General structure of the separation system

(a) Vapour recovery system (b) Liquid recovery system

5. Heat-exchanger network synthesis (energy integration) A beginner can substitute the evaluation of a number of extra calculations for

experience during the development of a conceptual design by following this hierarchical decision procedure. However, the penalty paid in the form of time required to screen more alternatives is not very high, as short-cut calculations are used. As a designer gains experience, it is possible to recognize alternatives that should not be considered for a particular type of process, and thereby improve efficiency.

This entire process synthesis activity is carried out prior to the process analysis stage to find the best process alternative by systematically screening millions of possible process alternatives following the above-mentioned hierarchy of design decisions. But this process synthesis is beyond the scope of the present course on process plant simulation. Process synthesis itself is vast and is offered as a separate compulsory course named Process Design Decisions.

1.2 Process Analysis Process analysis is the next stage in process design after process synthesis. Once the process flowsheet is synthesized, an analysis is required for the following purposes:

rn Solving material and energy balances for a steady-state process rn Sizing and costing the equipment rn Evaluating the worth of the flowsheet

Chemical process simulation, also known as flowsheeting, is represented by a mathematical model in order to obtain information about the response of a plant to various inputs. The salient features and requirements of general purpose simulation are (Husain 1986)

rn Modular approaches (sequential and simultaneous) to process simulation rn Equation-solving approach (unconstrained and constrained material

rn Decomposition of networks (partitioning, tearing algorithms, etc.) balances)


w Convergence promotion w Physical and thermodynamic properties w Specific-purpose simulation and w Dynamic simulation

Simulation can be carried out either in the design mode or in the analysis mode. In the design mode the outputs are specified and the corresponding inputs are simulated using the system model equations for various units. On the contrary, in the analysis or performance mode, the inputs are specified and the corresponding outputs are simulated. Ideally, with an aim of using process simulation as a design tool, the system inputs andor design parameters should be calculated from the specified outputs (i,e,, the design mode should be used). But such a simulation in a design mode is numerically less stable than that in the analysis or performance mode. The latter is characterized by the fact that all system inputs and design parameters for the units are specified and the outputs are calculated using the given information. Thus, the information flow in the analysis mode is in the same direction as the energy and material flow in a chemical plant. Being numerically more stable, the analysis mode is more often used to perform design calculations iteratively for making case studies. Hence, a majority of application packages are written in this mode of simulation.

Calculating heat and mass balances is the most tedious and repetitive problem of process design. As mentioned in the previous section on process synthesis, during the initial stages of a flowsheet study, of a new process or an existing one, simple material balances alone may suffice. At this level, specifications may be set from the plant data if an existing plant is being considered, or from engineering experience or pilot plant data if a new process flowsheet is under development. In the later stages of design, however, heat and mass balances must be calculated along with the equilibrium and rate equations, P-V-T relationships, and relations governing counter-current operations. All these equations are generally strongly non-linear. Moreover, in the simplest case, if a process is of sequential configuration, it is easy to proceed from the feed streams until the products are obtained, calculating sequentially for one process unit after the other. Unfortunately, most chemical plants are of complex configurations involving recycling of streams, mass and/or energy; they represent interlinked networks of units. To make the calculation procedure sequential in such situations, it is essential to decompose the network. This, in turn, requires convergence promotion.

The necessity of incorporating constraints or specifications other than the natural parameters while modelling a flowsheet has, of course, been recognized for quite some time in computer-aided process design. The importance of simulating in the design mode compared to the analysis mode has been outlined above. This differentiates rating calculations from design calculations. In the former case, each

Introduction 7

unit is specified and its performance is calculated. In the latter case, performance is specified and each unit is designed to give the required performance. In the sequential modular approach, constraints can be accommodated using additional iteration loops around the module, provided these constraints involve stream quantities associated with that module. However, if the constraints involve streams not incident to the underspecified module, then iterations in the outer loops involving the entire flowsheet are needed; this procedure is quite cumbersome and tedious. For this reason, the modular approach has not gained popularity for simulation in the design mode. In the equation-solving approach, the mathematical model of a steady-state process is organized and handled as one large global set of equations representing the entire process. This is in contrast to subsets of equations, called modules, used in the modular approach according to the process units that appear in a given process. Hence, in the global approach of equation-solving, any number of constraints can be added in the form of equations to the set defining the problem. Because the global approach analyses all the equations representing an entire chemical process, it takes full advantage of the specific features of these equations, which are ignored in the modular approach. It also generates a tailored computer program for each new problem, while the same modular-oriented simulator is used for all the problems. The tailored computer program executes faster and uses less memory compared to the modular-oriented simulator. However, the major disadvantage of the global approach is the necessity of manually preparing a new input description for all the equations of each new problem. This preparation can become tedious and highly error-prone, since chemical processes can be modelled using hundreds and thousands of non-linear equations.

Several design organizations started building libraries of computer programs for various unit operations in the early 1960s. It then became evident that many of these could be put into a system enabling one to direct calculations for an entire flowsheet, thus saving considerable engineering time. This resulted in the development of specific-purpose programs to simulate a particular plant or part of a plant; these were naturally more detailed in nature but at the same time rigid in structure. Upon examining these programs, it became clear that a major portion could be common to all types of plants, i.e., calculations involved in various unit operations, methods used to compute different physical and thermodynamic properties, decomposition and convergence algorithms, cost information, as well as a library of numerical routines. Furthermore, a common approach to diagnostics might also be beneficial. Ultimately, this gave rise to several general-purpose simulation packages.

Over the last few years, steady-state simulation has become a significant input to process analysis. It represents the system under consideration by a mathematical model and obtains information about the system response by applying different sets of inputs to the model. The model is often applied in the form of a computer


program for a steady-state deterministic system simulation. Dynamic simulation (unsteady-state simulation) is necessary during the startup and shutdown of a process plant.

1.3 Optimization This is the final step in the development of a process flowsheet. Many constrained and unconstrained optimization techniques are employed for this purpose. Of late, nontraditional optimization techniques such as genetic algorithms and differential evolution are being used. All these interrelated activities should obviously lead to an optimal design and safe operation of a chemical plant.

The chemical industry has undergone significant changes during the past 15 years due to the increased cost of energy and increasingly stringent environmental regulations. Modifications of both plant design procedures and plant operating conditions have been made in order to reduce costs and meet the constraints. Most industry observers believe that the emphasis in the near future will be on improving the efficiency and profitability of existing plants rather than on plant expansion (Edgar & Himmelblau 1989). One of the most important engineering tools that can be employed in such activities is optimization. As a result of computers having become more powerful, the size and complexity of problems, which can be simulated and solved by optimization techniques, have correspondingly expanded.

The goal of optimization is to find the values of the variables in a process which yield the best value of the performance criterion. This usually involves a trade-off between capital and operating costs. Typical problems in chemical engineering design or plant operation have many, and possibly infinite number of, solutions. Optimization is concerned with selecting the best among the entire set by efficient quantitative methods. Unfortunately, no single method or algorithm of optimization can be applied efficiently to all problems. The method chosen for any particular case will depend primarily on (i) the character of the objective function and whether it is known explicitly, (ii) the nature of the constraints, and (iii) the number of independent and dependent variables. The general objective in optimization is to choose, keeping in view the various constraints, a set of values of the variables that will produce the desired optimum response for the chosen objective function.

There are two distinct types of optimization algorithms in use today. First, there are algorithms that are deterministic, with specific rules for moving from one solution to the other. These algorithms (also known as traditional methods) have been successfully applied to some of the engineering design problems. Second, there are algorithms that are stochastic in nature, with probabilistic transition rules. These are comparatively new and are gaining popularity due to certain properties which deterministic algorithms do not have. As stated earlier, most of the traditional optimization algorithms based on gradient methods are susceptible to getting trapped

Introduction 9

at local optima depending upon the degree of non-linearity and the initial guess. Unfortunately, none of the traditional algorithms are guaranteed to find the global optimal solution, but genetic and simulated annealing algorithms are found to have a better global perspective than traditional methods (Deb 1996). Moreover, when an optimal design problem contains multiple global solutions, designers are interested in finding not just one global optimum solution, but as many as possible for various reasons. First, a design suitable in one situation may not be so in another situation. Second, designers may not be interested in finding the absolute global solution. They might instead be interested in a solution that corresponds to a marginally inferior objective function value but is more amenable to fabrication. Thus, it is always prudent to know about other equally good solutions for later use. However, if the traditional methods are used to find multiple optimal solutions, they need to be applied a number of times, each time starting from a different initial guess and hoping to achieve a different optimal solution.

During the past two decades there has been a growing interest in algorithms, that are based on the principle of evolution (survival of the fittest). A common term, coined recently, refers to such algorithms as evolutionary algorithms (EAs) or evolutionary computation (EC) methods. The best-known algorithms in this class include genetic algorithms, evolutionary programming, evolution strategies, and genetic programming. There are many hybrid systems which incorporate various features of the above paradigms and consequently are hard to classify, and can be referred to as just EC methods (Dasgupta & Michalewicz 1997).

Simulated annealing (SA) is a probabilistic nontraditional optimization technique, which mimics the cooling phenomenon of molten metals to constitute a search procedure. Rutenbar (1989) gave a detailed discussion of the working principle of SA and its applications. Since its introduction, SA has diffused widely into many diverse applications.

Genetic algorithms (GAS) are computerized search and optimization algorithms based on the mechanics of natural genetics and natural selection. They mimic the survival of the fittest principle of nature to make a search process. The key control parameters in GA are N , the population size; pc , the crossover probability; and pm, the mutation probability (Goldberg 1989).

Price and Storn (1997) have given the working principle of differential evolution (DE), which is an improved version of GA, along with its application to polynomial fitting problems. They have also suggested some simple rules for choosing the key parameters such as the population size NP, crossover constant CR, and the weight F applied to the random differential (scaling factor) of DE for any given application.

The three interrelated steps mentioned above (process synthesis, process analysis, and optimization) thus help to generate an optimal process plant design for any chemical process starting from the initial stage of conception. The flowsheet shown in Fig. 1.1 describes the strategy for process engineering.

10 P m e s s Plant Simulation

Define process objective

I Total R&D effort Collect and store Control analysis

Equipment Safety analysis

information

I _ _ _ _ _ _ _ ------- I Create alternate I I processconcept I L---,--,,,------ I I _ _ _ _ _ _ _ ------- I Synthesize alternate I I flowsheets > L--- , - - - - - - - - - - -

I

operating costs

Optimization - Is the project

Detailed plant design + Construction and installation +

1 startup 1 + Operation

1 Product

Fig. 1.1 Strategy for process engineering

1.4 Process Plant Simulation It constitutes process analysis and optimization stages of process design. This book deals with the various aspects of process analysis and optimization, in other words, process plant simulation.

1.5 Organization of the Text As has been discussed in the process analysis activity, modelling plays an important role. Any process plant comprises a number of interconnected units. These units may be reactors, heat exchangers, pumps, compressors, distillation columns, absorbers, adsorbers, evaporators, extractors, leaching units, driers, mixers, etc. The output of one unit becomes the input of another unit. So, in order to carry out process plant simulation, the outputs of the various units are to be predicted from

Introduction 11

the known inputs or from simulated outputs of the previous unit. This can be done using models of the individual units.

Similarly, optimization is also a very important aspect in process design. With reference to some of the other process analysis aspects-such as modular approaches to process simulation, the equation solving approach, decomposition of networks, convergence promotion, physical and thermodynamic properties-specific-purpose simulation and dynamic simulation are discussed in brief in this text for completeness of the subject under consideration. There has been no consolidated literature available on modelling and optimization aspects which covers the entire gamut of these primarily important process design aspects. Hence, the present text focuses on these two aspects of process plant simulation, and also covers the associated aspects on treatment of experimental results.

Keeping these objectives in mind, the text is organized as follows. Part I deals with various aspects of modelling: Chapter 1 gives a brief introduction of the subject. Chapter 2 gives a brief introduction to modelling, deterministic versus stochastic processes, physical and mathematical modelling, model formulation principles, cybernetics, controlled systems, and principles of similarity. Chapter 3 discusses the classification of mathematical modelling, the black box principle, artificial neural networks, dependent and independent variables, parameters, and boundary conditions.

Part I1 deals with chemical systems modelling, focusing on different areas of chemical engineering. The required mathematical techniques for solutions are emphasized as and when required. Chapters 4, 5 , and 6 focus on models in the areas of mass-transfer, heat-transfer, and fluid-flow (momentum-transfer) operations. Chapter 7 deals with models on reaction engineering.

Part I11 deals with the treatment of experimental results. Chapter 8 includes various types of errors and their propagation, and data regression techniques such as the method of averages and the method of linear least squares.

Part IV focuses on various optimization techniques. Chapter 9 covers the traditional optimization techniques including analytical methods, constrained optimization (Lagrangian multiplier), gradient methods such as the method of steepest descent (ascent) and the sequential simplex method, random search methods such as the box complex and Rosenbrock methods, etc. Chapter 10 discusses the nontraditional optimization techniques such as simulated annealing, genetic algorithms, differential evolution, evolutionary strategies, etc.

PartV deals with the remaining aspects of process analysis. Chapter 11 focuses on modular approaches to process simulation and the equation solving approach. Chapter 12 deals with the decomposition of networks and the corresponding tearing algorithms such as the Barkley and Motard algorithm, basic tearing algorithm, and list processing algorithms such as the Kehat and Shacham algorithm and various forms of the Murthy and Husain algorithm. Chapter 13 discusses the convergence


promotion, and physical and thermodynamic properties. Chapter 14 covers the specific-purpose simulation and dynamic simulation, discussing industrial problems as case studies. Chapter 15 discusses two professional simulation software packages (HYSYS and FLUENT), and the stepwise methodology for using these packages is demonstrated by solving four engineering problems.

EXERCISES 1.1 Discuss various stages of the development of chemical engineering as a discipline. 1.2 What are the various unit operations and unit processes? What are the analogies among these? 1.3 Describe various levels of decision hierarchy in process synthesis? 1.4 What are the aspects covered in process analysis? 1.5 What is the role of optimization in process design? 1.6 Discuss the strategy of process engineering with the help of a neat flow chart. 1.7 What is process plant simulation?

PART I

MODELLING

IN THE NEXT two chapters we will understand the importance of modelling in general and the model formulation principles for applying to chemical engineering systems in particular. We will see the types of modelling (physical and mathematical), understand the relationship among cybernetics, computers, and mathematical modelling, and classify various processes and mathematical modelling. Then we will discuss the black box principle and artificial neural networks. We will also focus on the related aspects of modelling such as the controlled system, principles of similarity, dependent and independent variables,. parameters, and boundary conditions.

CHAPTERS 2. Modelling Aspects 3. Classification of Mathematical Modelling

CHAPTER 2

MODELLING ASPECTS

Most of the engineering processes that demand accurate product quality, and proceed at high rates, high temperatures, and high pressures, are distinct for their utmost complexity. A simple change in one of the variables may bring about complex and non-linear changes in other variables.

The external potential of information about any engineering process is extremely high. This complex situation can be handled diligently with very narrow channels of perception by gaining an insight into a particular process using models. A model is a simplified representation of those aspects of an actual process that are being investigated (Kafarov & Kuznetsov 1976).

The flow of information is broken down into two stages. In the first stage, the model is compared with the real process and considered adequate if the discrepancy is negligible. In the second stage, the expectations are compared with the indications of the model. This procedure is called modelling. Modelling is subdivided into two groups:

rn Physical modelling Mathematical modelling

We will also focus on specific applications of mathematical modelling in chemical engineering, which is generally referred to as chemical systems modelling.

Before actually going into the details of and differences among physical, mathematical, and chemical systems modelling, we need to have a good grasp of different types of processes, such as deterministic and stochastic processes, and their differences.

2.1 Deterministic Versus Stochastic Processes 2.1.1 Deterministic Process In this process the observables take on a continuous set of values in a well-defined (or definable) manner, while the output variable most representative of the process


is uniquely determined by the input variable. These processes can be adequately described by classical analysis and numerical methods. An example is the process that takes place in a simple continuous stirred tank reactor (CSTR).

2.1.2 Stochastic Process This is a process in which observables change in a random manner and often discontinuou.sly. The output variable is not directly related to the input variable. These processes are described in terms of statistics and probabilistic theory. Examples are the contact-catalytic process (packed beds) in which the yield of the product diminishes with decrease in the activity in the catalyst as it ages with time and the pulse properties (such as pulse frequency, pulse velocity, pulse height, base hold-up, and pulse hold-up) in trickle bed reactors (Babu 1993).

A deterministic process is one whose behaviour (with respect to time, since it is of interest) can be predicted exactly. However, for stochastic processes, we can predict its response only approximately (generally invoking probabilistic notions). Simple CSTR and hydrocracking in trickle bed reactors are quoted as examples for deterministic and stochastic processes, respectively. However, it may be noted that any process, whether it is a process in CSTR or hydrocracking or fluid catalytic cracking (FCC), can be modelled as a deterministic or stochastic process, depending on how much trust we repose in the model we have constructed and how well we can describe all the inputs that affect the process behaviour. While a CSTR is a simpler process to describe than an FCC, this does not in any way imply that we can model CSTR behaviour exactly or that we know all the inputs that affect the response of a CSTR. It is more a matter of our faith than of the process itself.

2.2 Physical Modelling In physical modelling, the experiment is carried out directly on the real process. The process of interest is reproduced on different scales, and the effect of physical features and linear dimensions is analysed. The experimental data are reduced to relationships involving dimensionless groups made up of various combinations of physical quantities and linear dimensions. The relationships determined with this dimensionless presentation can be generalized to classes of events having these dimensionless groups, or similarity criteria. The resulting models are also known as empirical models.

Physical modelling consists in seeking the same or nearly the same similarity criteria for the model and the real process. The real process is modelled on a progressively increasing scale, with the principal linear dimensions scaled up in proportion (the similarity principle). Thus, a physical model is restrained directly within the system where the real process of interest takes place. This approach requires that the process be modelled up to the commercial scale, along with the

Modelling Aspects 17

complex systems that one has to deal with in chemical engineering. For relatively simple systems (such as single-phase fluid-flow or heat-transfer systems) the similarity principle and physical modelling are justified because the number of criteria involved is limited. However, with complex systems and processes described by a complex system of equations, one has to deal with a large set of similarity criteria that are not simultaneously compatible and, as a consequence, cannot be realized.

Let us consider the example of designing an industrial heat exchanger. For computing the heat-transfer coefficients that are required for designing a heat exchanger, the empirical correlations (of the form Nu = c Re"Pu", where Nu is the Nusselt number, Re is the Reynolds number, and Pr is the Prandtl number; c, m, and n are constant and the exponents are determined experimentally) developed at laboratory scale could be scaled up to industrial scale using geometric and dynamic similarities. Section 2.7 contains a detailed discussion on similarity principles.

The similarity principle has proved its worth in the analysis of deterministic processes that obey the laws of classical mechanics and proceed in bounded single- phase systems (within solid walls as a rule). It is, however, difficult to apply physi.cal similarity to an analysis of probabilistic processes involving multivalued stochastic relationships between the events, an analysis of two-phase unbounded systems, and to processes complicated by chemical reactions.

2.3 Mathematical Modelling A mathematical model of a real chemical process is a mathematical description which combines experimental facts and establishes relationships among the process variables. Mathematical modelling is an activity in which qualitative and quantitative representations or abstractions of the real process are carried out using mathematical symbols. In building a mathematical model, a real process is reduced to its bare essentials, and the resultant scheme is described by a mathematical formalism (formulation) selected according to the complexity of the process. The resulting models could be either analytical or numerical in nature depending upon the method used for obtaining the solution.

The objective of a mathematical model is to predict the behaviour of a process and to work out ways to control its course. The choice of a model and whether or not it represents the typical features of the process in question may well decide the success or failure of an investigation.

A good model should reflect the important factors affecting a process, but must not be crowded with minor, secondary factors that will complicate the mathematical analysis and might render the investigation difficult to evaluate. Depending on the process under investigation, a mathematical model may be a system of algebraic or differential equations or a mixture of both. It is important that the model should also represent with sufficient accuracy both quantitative and qualitative properties


of the prototype process and should adequately fit the real process. For a check on this requirement, the observations made on the process should be compared with the predictions derived from the model under identical conditions. Thus, a mathematical model of a real chemical process is a mathematical description combining experimental facts and establishing relationships between the process variables. For this purpose, it uses both theory and experimentation (Babu & Ramakrishna 2002a). When one is lacking information about a process or a system, one begins with the simplest model, taking care not to distort the basic (qualitative) aspects of the prototype process.

Mathematical modelling involves three steps:

gation (mathematical formulation) w formalization-the mathematical description of the process under investi-

w development of an algorithm for the process w testing of the model and the solution derived from it

In this method of analysis, the model itself is restrained through simulation on a computer, rather than with the real process or plant, as is the case with physical modelling. For this purpose, the variables that affect the course of a process are changed from the computer control console to a predetermined program (algorithm) and the computer represents the resultant outputs. With a conservative capital outlay, mathematical modelling coupled with present-day computers makes it possible to investigate various plant configurations in order to trace process behaviour under different conditions, and to find ways and means for improvement. Furthermore, this approach always guarantees an optimum solution within the framework of the model being used. However, it should be stressed that mathematical modelling is in no way set to oppose physical modelling. Rather, the former supplements the latter with its wealth of mathematical formulation and numerical analysis. The importance of lateral mixing from experimental evidence in thermal resistance models (Babu 1993; Shah et al. 1995) is a very good example of this combination.

Mathematical modelling involves the simulation of a process on a computer by changing in the interlinked variables. Using this technique, all promising alternatives can be simulated in order to arrive at an optimum model and, as a consequence, to optimize the process itself within a relatively short time. Mathematical modelling is economic and less time consuming than physical modelling. Mathematical modelling also uses the principles of analogies, or correspondence between different physical phenomena, dekribed by analogous mathematical equations. An example is the analogy among energy, heat, mass and electricity transport as is demonstrated below.


Energy or momentum transport (force of friction) Newtons law of viscosity

z = -.(%) can be rearranged to

where v p is the momentum per unit volume. In addition, in terms of driving force hp itbecomes

g, D M 2fpv2 = - L

Heat transport (heat flux) Fouriers law of heat conduction

- Q = - k ( z ) A

can be rearranged to

(2 .3)

(2.4)

where pCpT is the heat per unit volume. Similarly, heat flux QLA in terms of driving force AT is

- = - h A T Q A

Mass transport (mass flux) Ficks first law of diffusion

q m = J = - D ( z )

can be rearranged to

where c is the mass per unit volume. Similarly, mass flux NA in terms of driving force Ac is

N , = - k , A c (2.9)


Electricity transport (current) Ohms law

(2.10)

2.4 Chemical Systems Modelling Performing experiments and interpreting the results is routine in all applied sciences research. This may be done quantitatively by taking accurate measurements of the system variables, which are subsequently analysed and correlated, or qualitatively by investigating the general behaviour of the system in terms of one variable influencing another. The first method is always desirable, and if a quantitative investigation is to be attempted, it is better to introduce the mathematical principles at the earliest possible stage, since they may influence the course of investigation. This is done by looking for an idealized mathematical model of the system. The second step is the collection of all relevant physical information in the form of conservation laws and rate equations. The conservation laws of chemical engineering are material balances, heat balances, and other energy balances; whilst the rate equations express the relationships between flow rate and driving force in the fields of fluid flow, heat transfer, and diffusion of matter. These are then applied to the model, and the result should be a mathematical equation which describes the system. The type of equation (algebraic, differential, finite difference, etc.) will depend upon both the system under investigation and its model. For a particular system, if the model is simple, the equation may be elementary; whereas if the model is more refined, the equation will be more complex. Appropriate mathematical techniques are then applied to this equation and a result is obtained. This mathematical result must then be interpreted using the original model in order to give it physical significance.

Most chemical engineering processes that proceed at high rates, high temperatures, and high pressures in multiphase systems are distinct in their utmost complexity. A change in one system variable may bring about non-linear changes in other variables. This complexity becomes still more formidable in the case of multiple feedback loops. In addition, random disturbances are superimposed on the process. The external potential of information about chemical engineering process is very high. An analysis of the system and system variables may therefore be carried out with very narrow channels of perception by gaining insight into a particular process through modelling. The development of new processes in the chemical industry is becoming more complex and increasingly expensive. If the research and development of a process can be carried out with confidence, the ultimate design will be more exact and the plant will operate more economically.


Mathematics, which is the language of quantitative analysis, plays a vital role in all facets of such a project. Therefore, training in mathematical methods is of utmost importance to chemical engineers.

The most important result of developing a mathematical model of a chemical engineering system is the understanding that is gained of what really makes the process work. This insight enables one to strip away from the problem many extraneous confusing factors and get to the core of the system. It is basically trying to find cause-and-effect relationships between the variables. Mathematical models can be useful in all phases of chemical engineering, from research and development to plant operations, and even in business and economic studies (Luyben 1990). In research and development: determining chemical kinetic mechanisms and parameters from laboratory or pilot-plant reaction data; exploring the effects of different operating conditions for optimization and control studies; aiding in scale-up calculations. In design: exploring the sizing and arrangement of processing equipment for dynamic performance; studying the interactions of various parts of the process, particularly when material recycling or heat integration is used; evaluating alternative process and control structures and strategies; simulating startup, shutdown, and emergency situations and procedures. In plant operations: troubleshooting control and processing problems; aiding in startup and operator training; studying the effects of and requirements for expansion (bottleneck-removal) projects; optimizing plant operation. It is usually much cheaper, safer, and faster to conduct the kinds of studies listed above on mathematical model simulations than experimentally on an operating unit. This is not to say that plant tests are not needed. As we will discuss later, they are vital for confirming the validity of a model and for verifying important ideas and recommendations that evolve from model studies.

2.4.1 Model Formulation Principles Basis The bases for mathematical models are the fundamental physical and chemical laws, such as the laws of conservation of mass, energy, and momentum. To study dynamics we will use them in their general form with time derivatives included. Assumptions Probably the most vital role an engineer plays in modelling is in exercising his engineering judgement as to what assumptions can be validly made. Obviously an extremely rigorous model that includes every phenomenon down to microscopic detail would be so complex that it would take a long time to develop and might be impractical to solve, even on the latest supercomputers. An engineering compromise between a rigorous description and getting an answer that is good enough is always required. This has been called optimum sloppiness. It involves making as many simplifying assumptions as are reasonable. In practice, this


optimum usually corresponds to a model, which is as complex as the available computing facilities will permit. The development of a model that incorporates the basic phenomena occurring in a process requires a lot of skill, ingenuity, and practice. It is an area where the creativity and innovativeness of the engineer is a key element for the success of the process. The assumptions that are made should be carefully considered and listed. They impose limitations on the model that should always be kept in mind when evaluating its predicted results. Mathematical consistency of model Once all the equations of the mathematical model have been written, it is usually a good idea, particularly with big, complex systems of equations, to make sure that the number of variables equals the number of equations: The so-called degrees of freedom of the system must be zero in order to obtain a solution. If this is not true, the system is underspecified or overspecified and something is wrong with the formulation of the problem. This kind of consistency check may seem trivial, but experience shows that it can save many hours of frustration, confusion, and wasted computer time. This is required for a simulation exercise. For an optimization exercise, there should be some degrees of freedom available for optimizing, that is, an optimization problem is an underspecified problem. Checking to see that the units of all terms in all equations are consistent is perhaps another trivial and obvious step, but one that is often forgotten. It is essential to be particularly careful of the time units of parameters in dynamic models. Any unit can be used (seconds, minutes, hours, etc.), but these cannot be mixed. Dynamic simulation results are frequently in error because the engineer has forgotten a factor of 60 somewhere in the equations. Solution of the model equations The available solution techniques and tools must be kept in mind as a mathematical model is developed. An equation without any way to solve i t is not worth much. Verification An important but often neglected part of developing a mathematical model is proving that the model describes the real-world situation. At the design stage, this sometimes cannot be done because the plant has not yet been built. However, even in this situation there are usually either similar existing plants or a pilot plant from which some experimental dynamic data can be obtained. The design of experiments to test the validity of a dynamic model can sometimes be a real challenge and should be carefully thought out.

2.4.2 Fundamental Laws used in Modelling Some fundamental laws of physics and chemistry are required for modelling a chemical engineering system. These laws are reviewed in their general time- dependent form in this section.


2.4.2.1 Continuity equations Total continuity equation (mass balance) The principle of conservation of mass applied to a dynamic system is

[Mass flow into system] - [mass flow out of system] = [time rate of change of mass inside system] (2.1 1)

The units of Eq. (2.11) are mass per time. Only one total continuity equation can be written for one system. According to the normal steady-state design equation we use, we say that what goes in, comes out. The dynamic version of this says the same thing with the addition of the word eventually. For any property in a system, if we assume that the property does not vary with respect to spatial location, we obtain an ordinary differential equation (ODE). Otherwise we obtain a partial differential equation (PDE). If we assume that the property does not change with time (steady-state assumption), we get an algebraic equation. The right-hand side of the above equation will be either a partial derivative or an ordinary derivative of the mass inside the system with respect to the independent variable t. Component continuity equations (component balances) Unlike mass, chemical components are not conserved. Again, to be precise, the total mass for a reacting/ non-reacting system is conserved. However, in general, for a reacting system, the total number of moles is not a conserved quantity. For a reacting system, while the masses of individual elements are conserved, masses/moles of individual components are not conserved due to molecular rearrangement during a reaction. If a reaction occurs inside a system, the number of moles of an individual component will increase if it is a product of the reaction or decrease if it is a reactant. Therefore the component continuity equation of the jth chemical species of the system is

[Flow of moles of jth component into system] - [flow of moles of jth component out of system] + [rate of formation of moles ofjth component from chemical reactions] = [time rate of change of moles ofjth component

The units of Eq. (2.12) are moles of componentj per unit time. The flows in and out can be both convective (due to bulk flow) and molecular (due to diffusion). We can write one component continuity equation for each component in the system. If there are NC components in a system, there are NC component equations. However, the total mass balance and these NC component balances are not all independent, since the sum of all the moles times their respective molecular weights equals the total mass. Therefore a given system has only NC independent continuity equations. We usually use the total mass balance and NC - 1 component balances. For example, in a binary (two-component) system, there would be one total mass balance and one component balance.

inside system] (2.12)


There are some exceptions to the above rule. Consider a non-reacting system first. If we consider a flow balance involving S streams connected to a process unit (either in terms of moles or mass), NC component balances and S normalization relations (one for each stream), which describe how the total mass of a stream is related to the composition of the stream, then together these constitute NC + S + 1 equations, out of which one equation is redundant. Generally, it is not possible to work with NC - 1 components (and implicitly use the normalization equation for computing it) because the specified compositions may be distributed arbitrarily.

The same holds for a reacting system, but we should note that the equations involve r parameters corresponding to the extents of reaction (with known stoichiometry) assumed to occur in the reactor. A further complication is that element balances are not equivalent to component balances unless some conditions are satisfied (Reklaitis 1983; Felder & Rousseau 1999).

2.4.2.2 Energy equation The first law of thermodynamics puts forward the principle of conservation of energy. Written for a general open system (where flow of material in and out of the system can occur) it is

[Flow of internal, kinetic, and potential energy into system by convection or diffusion] - [flow of internal, kinetic, and potential energy out of system by convection or diffusion] + [heat added to system by conduction, radiation, and reaction] - [work done by system on surroundings (shaft work and PVwork)] = [time rate of change of internal, kinetic, and potential energy inside system] (2.13)

2.4.2.3 Equation of motion According to Newtons second law of motion, force F is equal to mass M times acceleration a for a system with constant mass M , i.e.,

Ma F = - g c

(2.14)

where g, is the conversion constant needed when FPS or MKS units are used = 32.2 (lb,ft)/lb,s2 in FPS (English engineering) units = 9.81 kg,m/kg,s2 in MKS units = 1 kg,m/N s2 in SI units = 1 g,cm/dyns2 in CGS units

This is the basic relationship used in writing the equations of motion for a system. In a slightly more general form, where mass can vary with time,

(2.15)


where vi is the velocity in the i direction and Fii is thejth force acting in the i direction. From Eq. (2.15), we can say that the time rate of change of momentum in the i direction (mass times velocity in the i direction) is equal to the net sum of the forces pushing in the i direction. It can be thought of as a dynamic force balance, or more eloquently it is called the conservation of momentum.

In the real world there are three directions: x, y, z. Thus, three force balances can be written for any system. Therefore, each system has three equations of motion (plus one total mass balance, one energy equation, and NC - 1 component balances). Instead of writing three equations of motion, it is often more convenient (and always more elegant) to write the three equations as one vector equation. The field of fluid mechanics makes extensive use of the conservation of momentum.

2.4.2.4 Transport equations The equations discussed so far are the laws governing the transfer of momentum, energy, and mass. These transport laws all have the form of a flux (rate of transfer per unit area) being proportional to a driving force (a gradient in velocity, temperature, or concentration). The proportionality constant is a physical property of the system (such as viscosity, thermal conductivity, or diffusivity). However, for transport on a molecular level, the laws bear the familiar names of Newton, Fourier, and Fick [Eqs (2.1), (2.4), and (2.7), respectively].

Transfer relationships of a more macroscopic overall form are also used; for example, film coefficients and overall coefficients in heat transfer. Here the difference in the bulk properties between two locations is the driving force [Eqs (2.3), (2.6), and (2.9), respectively]. The proportionality constant is an overall transfer coefficient.

2.4.2.5 Equations of state In mathematical modelling we need equations for physical properties, primarily density and enthalpy, as a function of temperature, pressure, and composition. Liquid density,

(2.16)

(2.17)

(2.18)

(2.19)

Occasionally, these relationships have to be fairly complex to describe the system accurately. But in many cases simplification can be made without sacrificing much


overall accuracy. Some of the simple enthalpy equations that can be used in energy balances are

h = C,T

H = C,T + 2, The next level of complexity

(2.20)

(2.21) would be to make the Cps functions of temperature:

(2.22)

A polynomial in T is often used for Cp:

Then Eq. (2.22) becomes C, (T) = A1 + 4 T

= A,(T - To) + ! L ( T ~ - T~ 0 ) 2

(2.23)

(2.24) L

Of course, with mixtures of components, total enthalpy is needed. If heat-of-mixing effects are negligible, the pure-component enthalpies can be averaged:

NC c xjhjMj (2.25)

where xj is the mole fraction of the jth component, Mj is the molecular weight of the jth component, and hj is the pure-component enthalpy of the jth component (energy per unit mass). The denominator of Eq. (2.25) is the average, molecular weight of the mixture.

Liquid densities can be assumed constant in many systems unless large changes in composition and temperature occur. Vapour densities usually cannot be considered invariant and some sort of PVTrelationship is almost always required. The simplest and most often used is the perfect-gas law:

PV = nRT (2.26) where P is the absolute pressure (Wa), V is the volume (m3), n is the number of moles (kmol), R is a constant = 8.314 kPam3/kmolK, and T is the absolute temperature (K). Rearranging to get an equation for the density p, (kg/m3) of a perfect gas with molecular weight M, we get

nM MP p v = y = i @ - (2.27)


2.4.2.6 Equilibrium relations The basis for the equations that give the conditions of a system when equilibrium conditions prevail is the second law of thermodynamics. Chemical equilibrium Equilibrium occurs in a reacting system when

NC c v j p j = 0 (2.28) j = 1

where vj is the stoichiometric coefficient of thejth component with reactants having a negative sign and products a positive sign, and p is the chemical potential of the jth component. The usual way to work with this equation is in terms of the equilibrium constant for a reaction. For example, consider a reversible gas-phase reaction of A to form B at a specific rate k , and B reaching back to A at a specific reaction rate k2. The stoichiometry of the reaction is such that va moles ofA react to form vb moles of B.

(2.29) V,A.&-N V,B (rate constant for reverse reaction is k2)

v b p b - v a p a = (2.30) From Eq. (2.28), equilibrium will occur when

The chemical potential for a perfect-gas mixture can be written as

p j = py + RT lnPj (2.31) where is the standard chemical potential (or Gibbs free energy per mole) of the jth component, which is a function of temperature only, Pj is the partial pressure of the j th component, R is the perfect-gas law constant, and T is the absolute temperature. Substituting into Eq. (2.30) and simplifying, we get

0 0 in( = - ' b p B

RT (2.32)

The right-hand side of this equation is a function of temperature only. The term in parentheses on the left-hand side is defined as the equilibrium constant Kp, which gives us the equilibrium ratios of products and reactants.

(2.33)

Phase equilibrium potential of each component is the same in the two phases:

Equilibrium between two phases occurs when the chemical

(2.34)


where pj is the chemical potential of thejth component in phase I and py is the chemical potential of the jth component in phase 11.

Since a vast majority of chemical engineering systems involve liquid and vapour phases, many vapour-liquid equilibrium relationships are used. They range from very simple to the very complex. Some of the most commonly used relationships are listed below. More detailed treatments are presented in many thermodynamics texts. Basically, we need a relationship that permits us to calculate the vapor composition if we know the liquid composition, or vice versa.

The most common problem is a bubble point calculation: calculate the temperature T and vapour composition yp given the pressure P and the liquid composition xi. This usually involves a trial-and-error, iterative solution because the equations can be solved explicitly only in the simplest cases. Sometimes we have bubble point calculations that start from known values of xj and T and we want to find P and yi. This is frequently easier than when pressure is known, because the bubble point calculation is usually non-iterative. Dew point calculations must be made when we know the composition of the vapour y j and P (or 7J and want to find the liquid composition xj and T (or P). Flush calculations must be made when we know neither xj nor y j and must combine phase equilibrium relationships, component balance equations, and an energy balance to solve for all the unknowns.

Let us assume ideal vapour-phase behaviour in our examples, i.e., the partial pressure of the jth component in the vapour is equal to the total pressure P times the mole fraction of thejth component in the vapour y j (Daltons law):

q=Py j (2.35) Corrections may be required at high pressures. As far as the liquid phase is concerned, several approaches are being used, which are shown below. Ruoults law Mixtures that obey Raoults law are called ideal mixtures.

NC

P = Cx,P; j = 1

x jp ; y . =- P

(2.36)

(2.37)

where P, is the vapour pressure of pure component j . Vapour pressures are functions of temperature only. This dependence is often described by

A . T

lnPj S =>+ Bj (2.38)


Relative vohtiZity The relative volatility av of component i to component j is defined as

(2.39)

Relative volatilities are fairly constant in a number of systems. They are convenient and hence are frequently used. In a binary system the relative volatility a of the more volatile component compared with the less volatile component is

Rearranging,

m = 1 + (a - 1)x

(2.40)

(2.41)

K vahes in the petroleum industry.

Equilibrium vapourization ratios of K values are widely used, particularly

(2.42)

The Ks are functions of temperature and composition and, to a lesser extent, pressure.

Activity coencients For non-ideal liquids, Raoults law must be modified to account for the non-ideality in the liquid phase. The fudge factors used are called activity coeficients.

NC P = c x j p ; r , (2.43)

j=l

where r j is the activity coefficient for the jth component. The activity coefficient is equal to 1 if the component is ideal. The ys are functions of composition and temperature.

2.4.2.7 Chemical kinetics In modelling of chemical reactors, we must be familiar with the basic relationships and terminology used in describing the kinetics (rate of reaction) of chemical reactions. Arrhenius temperature dependence The effect of temperature on the specific reaction rate k is usually found to be exponential:

k = Cre-E/RT (2.44)


where k is the specific reaction rate, a is the preexponential factor, E is the activation energy [shows the temperature dependence of the reaction rate, i.e., the bigger E is, the faster the increase in k with increasing temperature (caVgmol) is], T is the absolute temperature, and R is the perfect-gas constant = 1.99 caVgmol K.

This experimental temperature dependence represents one of the most severe non-linearities in a chemical engineering system. It may be noted that the apparent temperature dependence of a reaction may not be exponential if the reaction is mass-transfer limited and not chemical-rate limited. If both zones are encountered in the operation of the reactor, the mathematical model must obviously include both reaction-rate and mass-transfer effects. Law of mass action Let us define an overall reaction rate 93, with the conventional notation, as the rate of change of moles of any component per volume due to the chemical reaction divided by that component's stoichiometric coefficient:

(2.45)

The stoichiometric coefficients vj are positive for products of the reaction and negative for reactants. It may be noted that CH is an intensive property and can be applied to systems of any size. For example, assume we are dealing with an irreversible reaction in which components A and B react to form components C and D .

(2.46)

(2.47)

From the law of mass action, the overall reaction rate CH will vary with temperature (since k is temperature-dependent) and with the concentrations of reactants raised to some powers:

CH = k ( T ) (cA>a ( C B > b (2.48) where C, is the concentration of component A and cB is the concentration of component B. The constants a and b are not, in general, equal to the stoichiometric coefficients v, and vba The reaction is said to be first-order in A if a = 1. It is second-order in A if a = 2. The constants a and b can be fractional numbers.


As indicated earlier, the units of the specific reaction rate k depend on the order of the reaction. This is because the overall reaction rate 3 always has the same units (moles per unit time per unit volume). For a first-order reaction of A reacting to form B, the overall reaction rate 3, written for component A , would have units of moles of Nmin ft3.

% = k C A (2.49) If cA has units of moles of A/ft3, k must have units of min-'. If the overall reaction rate for the above system is second-order in A ,

3 = kc i (2.50) 3

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