Development and validation oftwo-dimensional mathematical model ofboron carbide manufacturing process

A Thesis submitted

for the degree of

Master of Science (Engg.)

in the Faculty of Engineering

by

Rakesh Kumar

Department of Materials Engineering

Indian Institute of ScienceBangalore 560 012 (India)

2006

Dedicated to my parents

i

Synopsis

Boron carbide is produced in a heat resistance furnace using boric oxide and petroleum

coke as the raw materials. In this process, a large current is passed through the graphite

rod located at the center of the cylindrical furnace, which is surrounded by the coke and

boron oxide mixture. Heat generated due to resistance heating is responsible for the

reaction of boron oxide with coke which results in the formation of boron carbide. The

whole process is highly energy intensive and inefficient in terms of the production of

boron carbide. Only 15% charge gets converted into boron carbide. The aim of the

present work is to develop a mathematical model for this batch process and validate the

model with experiments and to optimize the operating parameters to increase the

productivity.

To mathematically model the process and understand the influence of various operating

parameters on the productivity, existing simple one-dimensional (1-D) mathematical

model in radial direction is modified first. Two-dimensional (2-D) model can represent

the process better; therefore in second stage of the project a 2-D mathematical model is

also developed. For both, 1-D and 2-D models, coupled heat and mass balance equations

are solved using finite volume technique. Both the models have been tested for time step

and grid size independency. The kinetics of the reaction is represented using nucleation

growth mechanism. Conduction, convection and radiation terms are considered in the

formulation of heat transfer equation. Fraction of boron carbide formed and temperature

profiles in the radial direction are obtained computationally.

Experiments were conducted on a previously developed experimental setup consisting of

heat resistance furnace, a power supply unit and electrode cooling device. The heating

furnace is made of stainless steel body with high temperature ceramic wool insulation. In

order to validate the mathematical model, experiments are performed in various

conditions. Temperatures are measured at various locations in the furnace and samples

are collected from the various locations (both in radial and angular directions) in the

ii

furnace for chemical analysis. Also, many experimental data are used from the previous

work to validate the computed results. For temperatures measurement, pyrometer, C, B

and K type thermocouple were used.

It is observed that results obtained from both the models (1-D and 2-D) are in reasonable

agreement with the experimental results. Once the models are validated with the

experiments, sensitivity analysis of various parameters such as power supply, initial

percentage of B4C in the charge, composition of the charge, and various modes of power

supply, on the process is performed. It is found that linear power supply mode, presence

of B4C in the initial mixture and increase in power supply give better productivity

(fraction reacted). In order to have more confidence in the developed models, the

parameters of one the computed results in the sensitivity analysis parameters are chosen

(in present case, linear power supply is chosen) to perform the experiment. Results

obtained from the experiment performed under the same simulated conditions as

computed results are found in excellent match with each other.

iii

Acknowledgment

I don’t find adequate words to express my feelings and gratitude for the institute. To

me IISc is a place where I’ve realised my dreams and have seen a great future ahead.

It’s a real encouragement to watch passionate professors and students contributing

to the research field to the utmost of their dedication.

I am deeply indebted to my advisor, Prof. Govind S. Gupta for his unending guid-

ance and support throughout my graduate study. It is my tremendous honor to com-

plete this research work under his supervision. Advice from him has extended far

beyond the technical realm. His emphases on creativity, perseverance, written and

oral communications, and experimental skills are the most valuable treasures that I

have learnt from him, and I will implement them in my future work. ”Multi-prong

approach” is the most common word that I have always heard from Prof. Gupta.

Under his guidance I have overcome my all time fear of taking many task at hand

and doing equal justice to them all.

I am thankful to the Chairman in the Department of Metallurgy for allowing me to

use the lab facilities whenever required. I am also thankful to Prof. Subranmanian ,

Prof. Vikram Jayram, Prof. Choksi and Prof. Subodh Kumar in the Department of

Metallurgy for letting me use their lab facilities time to time. I am thankful to Mr.

Babu for helping me out with experimental setup problems.

I would like to thank Prof. R.V. Ravikrishana, and Prof. J. Srinivasan in the De-

partment of Mechanical Engineering for their valuable comments drawn from their

vast research experiences to enhance my dissertation. Moreover, I appreciate Prof.

N. Balakrishanan in the Department of Aerospace Engineering for his valuable tips

on writing efficient simulation code.

iv

Certainly my stay in IISc would not have been so delightful and fruitful without the

friends around. I would like to express my gratitude to all my friends and colleagues

who have supported my effort in the graduate study and academic research, includ-

ing Vikrant, Sabita, Rao, Manjunath, Manish, Suman, Abhishek, Arvind, Sachin,

Santosh, Rami, Ankit, Rathore, Neelam and Sunita. Without all these people contri-

bution, in one way or other, I would have never completed my work. Special thanks

are also due to Azeem and Ashwini for helping me out in carrying experiments time

to time.

In addition, I highly appreciate presence of Vishal and Foram around while I needed

some help on critical issues related to mathematical modeling and error handling.

More then that they worked as a stress-buster in the hour of peak tension.

Finally, my family deserves my warmest appreciation. I am thankful to god for be-

stowing me a loving and caring parents. I am thankful to my brother and sister for

being a source of constant love and inspiration. It is their patience, understanding,

encouragement, and help that gave my faith and strength to complete my graduate

studies at Indian Institute of Science, Banglore.

Contents

1 Introduction 1

1.1 Research background . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Physical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Chemical properties . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5 Typical usages and applications of boron carbide . . . . . . . . . . 5

1.6 Boron carbide manufacturing routes . . . . . . . . . . . . . . . . . 6

1.6.1 Direct from elements . . . . . . . . . . . . . . . . . . . . . 6

1.6.2 Magnesiothermic route . . . . . . . . . . . . . . . . . . . . 7

1.6.3 Gaseous route . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.6.4 Carbothermal reduction route . . . . . . . . . . . . . . . . 8

1.7 Current understanding of the process . . . . . . . . . . . . . . . . . 12

1.8 Modeling of boron carbide manufacturing process . . . . . . . . . . 14

1.9 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.10 Outline of present work . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Thermodynamics and reaction kinetics 18

v

Contents vi

2.1 Thermo-chemistry of B-O-C system . . . . . . . . . . . . . . . . . 18

2.2 Phase diagram for boron-carbon system . . . . . . . . . . . . . . . 20

2.3 Reaction kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.1 Rate of reaction . . . . . . . . . . . . . . . . . . . . . . . . 23

3 Mathematical modeling 25

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Process description . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Mathematical formulation . . . . . . . . . . . . . . . . . . . . . . 30

3.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.1 Overall 2-D heat balance equation . . . . . . . . . . . . . . 32

3.4.2 Boundary conditions for overall 2-D heat balance equation . 33

3.4.3 Overall 2-D mass balance equation for CO . . . . . . . . . 35

3.4.4 Boundary conditions for overall 2-D mass balance . . . . . 35

3.5 Determination of properties . . . . . . . . . . . . . . . . . . . . . 36

3.5.1 Determination of CO generation (o

W ) . . . . . . . . . . . . 36

3.5.2 Determination of Dco-air . . . . . . . . . . . . . . . . . . . 37

3.5.3 Determination of effective properties . . . . . . . . . . . . 37

3.5.4 Determination of the rate of heat consumption (∆Hr) . . . 39

3.6 Non-dimensionalization . . . . . . . . . . . . . . . . . . . . . . . 39

3.6.1 Overall 2-D heat balance equation . . . . . . . . . . . . . . 40

3.6.2 Overall 2-D mass balance equation . . . . . . . . . . . . . 41

Contents vii

3.7 1-D model equations . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.8 Computational technique . . . . . . . . . . . . . . . . . . . . . . . 44

3.8.1 Discretization and solution methodology . . . . . . . . . . 44

3.8.2 Solution methodology for 2-D model . . . . . . . . . . . . 45

4 Physical modeling and process description 48

4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1.1 Resistance heating furnace . . . . . . . . . . . . . . . . . . 50

4.1.2 Power supply unit with control panel . . . . . . . . . . . . 51

4.1.3 Thermocouples and pyrometer . . . . . . . . . . . . . . . . 52

4.1.4 Data recording device . . . . . . . . . . . . . . . . . . . . 53

4.1.5 Safety accessories . . . . . . . . . . . . . . . . . . . . . . 54

4.2 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . 55

4.2.1 Chemical analysis . . . . . . . . . . . . . . . . . . . . . . 57

5 Results and discussion 61

5.1 Results obtained from 1-D model . . . . . . . . . . . . . . . . . . . 61

5.1.1 Comparison between fully explicit and implicit scheme . . . 63

5.1.2 Grid independency . . . . . . . . . . . . . . . . . . . . . . 64

5.1.3 Time independency . . . . . . . . . . . . . . . . . . . . . . 65

5.1.4 Validation of 1-D model . . . . . . . . . . . . . . . . . . . 66

5.2 Results obtained from 2-D model and its validation . . . . . . . . . 75

5.2.1 Comparative study of 1-D and 2-D models with experimen-

tal results . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Contents viii

5.3 Yield analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.4 Sensitivity analysis / optimization of the process . . . . . . . . . . . 83

5.4.1 Effect of input power supply . . . . . . . . . . . . . . . . . 84

5.4.2 Effect of mode of heating cycle . . . . . . . . . . . . . . . 84

5.4.3 Effect of varying charge composition . . . . . . . . . . . . 88

5.5 Comparison of experimental results with sensitivity analysis . . . . 92

6 Conclusions and scope of future work 95

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2 Scope for future work . . . . . . . . . . . . . . . . . . . . . . . . . 96

A Modeling details 97

A.1 Finite volume discretization technique for PDE’s . . . . . . . . . . 97

A.2 Correlation for property data . . . . . . . . . . . . . . . . . . . . . 98

A.2.1 Enthalpy correlations . . . . . . . . . . . . . . . . . . . . 99

A.2.2 Thermal conductivity correlations . . . . . . . . . . . . . . 99

A.2.3 Specific heat correlations . . . . . . . . . . . . . . . . . . . 101

A.2.4 Porosity correlation . . . . . . . . . . . . . . . . . . . . . . 101

List of Tables

1.1 Physical properties of boron carbide . . . . . . . . . . . . . . . . . 3

1.2 Conversion timing findings by various researchers . . . . . . . . . . 12

3.1 Non-dimensional parameters . . . . . . . . . . . . . . . . . . . . . 39

5.1 Simulation parameters used for fully explicit and implicit schemes . 62

5.2 Simulation parameter used for 2-D model . . . . . . . . . . . . . . 75

5.3 Standard data used for sensitivity analysis . . . . . . . . . . . . . . 83

ix

List of Figures

1.1 Rhombohedral crystalline structure of B4C . . . . . . . . . . . . . 2

1.2 Schematic of apparatus using pulsed-laser . . . . . . . . . . . . . . 8

1.3 Schematic of resistance heating furnace . . . . . . . . . . . . . . . 11

2.1 B-C phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1 Computational domain for 1-D and 2-D mathematical model . . . . 31

3.2 Flow diagram for 1-D program . . . . . . . . . . . . . . . . . . . . 46

3.3 Flow diagram for 2-D program . . . . . . . . . . . . . . . . . . . . 47

4.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Internal construction of the furnace . . . . . . . . . . . . . . . . . . 51

5.1 Effect of fully explicit and implicit scheme on core temperature us-

ing 1-D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Effect of grid size on core temperature using 1-D model and fully

implicit scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3 Effect of time step on core temperature using 1-D model and fully

implicit scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

x

List of Figures xi

5.4 Variation in power and primary current supply to the transformer -

(Exp. 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.5 Temperature variations at different locations with time - (Exp. 1) . . 68

5.6 Product formation with distance – (Exp. 1) . . . . . . . . . . . . . . 69

5.7 Fraction of material reacted with time . . . . . . . . . . . . . . . . 71

5.8 Enlarged view of figure 5.7 . . . . . . . . . . . . . . . . . . . . . . 71

5.9 Power supply and primary current variation with time – (Exp. 2) . . 73

5.10 Temperature variation at different locations with time – (Exp. 2) . . 74

5.11 Product formation with distance – (Exp. 2) . . . . . . . . . . . . . . 74

5.12 Typical 2-D plot for temperature variation with time at different

locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.13 Angular variation in product formation at various locations, as ob-

tained from 2-D model . . . . . . . . . . . . . . . . . . . . . . . . 76

5.14 Temperature variation at different locations with time – (Exp. 3) . . 79

5.15 Comparison of 1-D and 2-D computed core temperature with ex-

perimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.16 Comparison of 1-D and 2-D computed temperatures away from core

with experimental data . . . . . . . . . . . . . . . . . . . . . . . . 81

5.17 Temperature variation with time for different power input . . . . . . 85

5.18 Effect of power supply on conversion radius . . . . . . . . . . . . . 85

5.19 Different modes of power supply . . . . . . . . . . . . . . . . . . . 86

5.20 Effect of mode of power supply on core temperature . . . . . . . . . 87

5.21 Effect of mode of power supply on percentage conversion . . . . . . 87

5.22 Effect of excess B2O3 on core temperature . . . . . . . . . . . . . . 89

List of Figures xii

5.23 Conversion radius with excess B2O3 . . . . . . . . . . . . . . . . . 89

5.24 Effect of initial B4C on computed temperature . . . . . . . . . . . 90

5.25 Effect of initial B4C content on final product formation . . . . . . . 90

5.26 Comparison between 1-D and 2-D model with experiment . . . . . 91

5.27 Comparison between 1-D and 2-D model for conversion radius . . . 93

A.1 Control volume in polar coordinates . . . . . . . . . . . . . . . . . 98

Chapter 1

Introduction

Boron carbide was discovered about one and half century ago which has made a

tremendous impact on science and technology. The exceptional features of boron

carbide, e.g., specific gravity, extreme hardness, wear resistance, high mechanical

strength at both low and high temperature, thermal and chemical resistance, nuclear

properties, and chemical reactivity, makes it an outstanding material for material

processing and for nuclear and military applications.

1.1 Research background

Boron carbide is one of the hardest materials known, ranking third behind the dia-

mond and cubic boron nitride. It is the hardest material produced in tonnage quan-

tities. B4C was originally discovered in mid 19th century around 1858 [1] as a

by-product in the production of metal borides. Joly in 1883 and Mossian in 1894

synthesized B4C in a purer form [2] and identified boron-carbon compounds of

different composition as B3C and B6C respectively. Boron carbide was studied

in detail after 1930 and the first phase diagram was published in 1934 [3]. Stoi-

chiometric formula (B4C) for boron carbide was assigned in 1934 [4]. After that

many other diverse formulae were proposed by Russian authors; which have not

1

Chapter 1 2

been confirmed yet [1, 5] but today a homogeneity range from B4.0C to B10.4C has

been established [2, 6]. A high B/C molar ratio, as high as B51C, is also reported in

literature [7].

1.2 Crystal structure

Boron carbide can be considered a compound of α - rhombohedral boron which

include B12C3, B12S, B12O2, B12P2 etc. [6]. The lattice belongs to D3d5 − R 3

space group [2]. The rhombohedral unit cell contains 15 atoms corresponding

to B12C3 as shown below in figure 1.1 (a). The boron carbides are composed of

(a) (b)

Figure 1.1: Rhombohedral crystalline structure of B4C (a) Structure of D3d5 − R 3 space

group [2], (b) B4C - shaded icosahedra are in the background [8].

twelve-atoms (icosahedral clusters) which are linked by direct covalent bonds and

through three-atom inter icosahedral chains [2]. As per recent study the sequence

Chapter 1 3

C-B-C can be assumed for the chain. In addition to these two carbon atoms per unit

cell, as shown in figure 1.1 (b), carbon likely replaces boron at the boron sites in

the icosahedra. It is pointed out that four sites are available for a total of 15 boron

and carbon atoms, so the most widely accepted structural model for B4C has B11C

icosahedra with C-B-C inter icosahedral chains. Further details about the lattice

parameters can be found in literature [2, 8 -10].

Table 1.1: Physical properties of boron carbide

Properties Unit Value

Melting point K 2720

Boiling point K 3770

Fracture toughness MPa.m1/2 2.9 - 3.7

Bulk density kg/m3 2520

Hardness (Knoop, 100g) kg/mm2 2900 - 3580

Young’s Modulus GPa 450 - 470

Shear Modulus GPa 180

Poisson’s ratio – 0.21

Electrical conductivity Ωm−1 140

Thermal conductivity W/(m − K) (298 K) 29 - 67

Expansion coefficient K−1 (298 K - 1273 K) 4.5 x 10−6

Neutron capture cross-section Barns 600

1.3 Physical properties

Important physical properties of B4C are listed in table 1.1 [11, 12]. Because of its

high hardness and strength, B4C is inferior in abrasive resistance only to diamond.

B4C is a high temperature p-type semiconductor [2]. The electrical conductivity

depends on the B:C ratio and impurity content [6]. The electrical conductivity of

Chapter 1 4

boron carbides increases with temperature [2, 13]. Boron carbide has a negative

temperature coefficient of resistivity, similar to other ceramic material. Thermal

conductivity of boron carbide decreases with temperature; however, it has low re-

sistance to thermal shocks [6]. After 1950, more attention was paid towards boron

carbide applications based on its structural properties. In 1954 it was concluded

that B4C has a rhombohedral structure with crystal lattice periods a = 5.598 A, c =

12.12 A [10]. The hardness of B4C is known to depend heavily upon stoichiom-

etry with maximum hardness at molar B/C = 4.0. However, while stoichiometry

plays a major role in determining the hardness of boron carbide, other factors such

as microstructure and additives (T iB2), or impurities (Fe), have been found to be

important. The larger grain size and boron-enriched stoichiometry contribute to the

lower hardness [14].

1.4 Chemical properties

Boron carbide is supposed to be one of the most stable compounds to acids. It is not

dissolved by mineral acids or aqueous alkali; however, it is decomposed slowly in a

mixture of HF − H2SO4 and HF − HNO3 acids [2]. Molten alkali decomposes

boron carbide to give borates. At high temperature boron carbide reacts with many

metal oxides to give carbon monoxide and metal borides [6]. It also reacts with

many metals that form carbides or borides at 1000 oC, i.e., iron, nickel, titanium

and zirconium. Above 1800 oC it reacts with nitrogen to give boron nitride. B4C

can be attacked by chlorine at about 600 oC and bromine attacks it at above 800 oC,

giving boron trihalides [2, 6]. Thus, it is a way to prepare boron halides. Boron

carbide can be melted without decomposition in a CO atmosphere, but it reacts in

the temperature range 600-750 oC with CO2 to form B2O3 and CO. Boron carbide

has good oxidation properties in the air up to about 600 oC.

Chapter 1 5

1.5 Typical usages and applications of boron carbide

Boron carbide is one of the hardest materials that have been widely used in appli-

cations requiring a great hardness such as armor plating, bearings, dies and cutting

tools [15]. Boron carbide and boron suboxide have high potential opportunities to

be successfully used in nuclear power engineering and chemical industries, in ther-

moelectric energy converters and composites [16]. Major usage of B4C are based

on its specific gravity, extreme hardness, wear resistance, high mechanical strength

at both low and high temperature, thermal and chemical resistance, nuclear proper-

ties and chemical reactivity. The majority of commercial B4C goes into abrasive

slurries, blast nozzles, and neutron absorbing materials [11].

Sintered-B4C wheel-dressing sticks are used to produce new cutting edges on sur-

face grains of grinding wheels with minimum wear. The combination of extreme

hardness and low density of boron carbide has made it suitable material for uses

such as lightweight boron carbide armor in helicopter and fighter planes. Besides,

the lightweight coupled with a large heat of combustion (51900 J/g) of boron car-

bide makes it a useful solid propellant for rocket.

Boron has two principal isotopes, B10 and B11. The effectiveness of boron as neu-

tron absorber is due to the high absorption cross section of B10 isotope [17]. Boron

has another advantage over other potential neutron absorber materials.

5B10 + on

1 → 2He4 + 3Li7 + 2.4MeV

The reaction products of neutron absorption namely helium and lithium are formed

as stable, non-radioactive isotopes. Also, no high-energy, secondary radioactive

products are produced. B4C is both cheaper and easier to fabricate than the ele-

mental boron itself. As a result, it has found almost exclusive usage as a control rod

material, neutron poison, shutdown balls, and as neutron shielding material.

Chapter 1 6

The thermoelectric properties of B13C2 are such that it could be an interesting ma-

terial for high-temperature thermoelectric conversion. Thermo-elements made of

the couple B4C − C can be used for temperature measurement up to 2300 oC.

1.6 Boron carbide manufacturing routes

Boron carbide can be prepared by a variety of high temperature methods. They can

be grouped in the following major categories:

• Synthesis of boron carbide using virgin elements

• Synthesis of boron carbide by magnesiothermic route

• Synthesis of boron carbide by gaseous route

• Synthesis of boron carbide by carbothermal reduction route

1.6.1 Synthesis of boron carbide using virgin elements

Elemental route of B4C production gives the best quality product. Boron, in its

elemental form, can be synthesized by the following routes [17- 20].

i) By reduction of boron halides with H2

ii) By reduction of boron halides with Zn

iii) By reduction of boron oxide with Mg

The reaction between amorphous boron and carbon is kinetically fast compared

to crystalline form. Initially the raw material is thoroughly mixed in ball mill to

get homogeneous product. The reaction between boron and carbon is completely

diffusion controlled. High temperature, of the order of 1800 K, is required for the

Chapter 1 7

preparation of boron carbide. Due to the susceptibility of boron for oxidation, the

reaction is carried out under vacuum of the order of 10−3 mbar.

1.6.2 Synthesis of boron carbide by magnesiothermic route

Gray et al., [21] discovered a method of producing fine particle size boron carbide

by heating a mixture of boron oxide, carbon and magnesium. Overall reaction can

be written as:

2B2O3 + C + 6Mg −→ B4C + 6MgO (1.1)

Oxidation of Mg is strongly exothermic and the heat liberated during the oxidation

is used for the reaction to form boron carbide from boric acid. The boron carbide,

produced by the this method, is unsatisfactory for high purity applications because

the boron carbide is contaminated with the magnesium, the starting material, and

even after repeated digestions with hot mineral acids the magnesium is difficult to

remove. More details are given in reference [22].

1.6.3 Synthesis of boron carbide by gaseous route

Very fine powders of boron carbide have been produced by vapor phase reactions

of boron compounds with carbon or hydrocarbons, using laser or plasma energy

sources. These reactions tend to form highly reactive amorphous powders. Due to

their extreme reactivity, handling in inert atmospheres may be required to avoid

the contamination by oxygen and nitrogen. These very fine powders have ex-

tremely low bulk densities, which make loading of hot press dies and processing

greenware very difficult. More details about the procedure can be found elsewhere

[23- 25].

Pulse laser technique is used for the synthesis of boron carbide crystallite encap-

sulated in graphite particles via chemical vapor deposition of C6H6 + BCl3 gas

Chapter 1 8

mixture. Gas mixture consisting of C6H6 + BCl3 or C6H6 + CCl4 is introduced

into the Pyrex glass reactor chamber which is further connected to a vacuum sys-

tem [26]. Before the introduction of gas mixture into the reactor chamber, it is first

baked out. Raw material is then irradiated with Nd:YAG laser which is focused

with a lens (fl = 200 mm). The reaction gets completed because of intense laser

pulse. Schematic of the apparatus is shown in the figure 1.2. The interaction of IR

Figure 1.2: Schematic of apparatus using pulsed-laser

laser radiation with gases and gaseous mixture for the synthesis of boron carbide is

described by Bastl et al., [27]. Francis et al., [28] has described a gaseous phase

reaction of acetylene (C2H2) and diborane (B2H6) in a closed chamber at a tem-

perature of less than 80 oC, to produce amorphous porous boron carbide having a

mean particle size of few µm in diameter. Details are given elsewhere [12].

1.6.4 Synthesis of boron carbide by carbothermal reduction route

Carbothermal reduction of boric acid has scientific and economic advantages over

the other methods of boron carbide production. Powders prepared by carbothermic

reduction have excellent morphology and surface characteristics [29].

In carbothermal reduction process, boric acid or boron oxide as a source of boron

Chapter 1 9

and carbon active or petroleum coke as reducing agent is used as the main raw ma-

terial. Depending upon the method or process adopted, there are many ways of

producing boron carbide. Few of such methods for the synthesis of boron carbide

are as follow:

i) Using boron oxide and carbon black

ii) Using arc furnace process

iii) Resistance-heating furnace process

Using boron oxide and carbon black

Scott et al., [30] and Smudski et al., [31] have produced boron carbide by the car-

bothermic reduction of boron oxide. For carrying out the carbothermic reduction

reaction, a reactive mixture of a boric oxide source such as boric acid and a carbon

source such as carbon black, is prepared by mechanically mixing them together.

This reactive mixture is then heated at a reaction temperature for a sufficient length

of time to form B4C. The temperature of firing the reactive mixture is in the range

of 1700 – 2100 oC.

The particle size of boron carbide can range anywhere between 0.5 and 150 µm

with no control of particle size distribution. Another shortcoming of this process

is the non homogeneity of the end product. The product samples, taken from the

various parts of the furnace, vary markedly in their composition such as, high free

carbon, or unreacted boric acid etc. Substantially complete reaction of the carbon

is desired to eliminate any ”free carbon” in the boron carbide product.

Arc furnace process

The reactant used for the process is a mixture of old mix and fresh charge. Old

mixture is from the previous run and differs from the fresh charge in that some

Chapter 1 10

part of it is partially converted B4C material [30]. In other words, the old mix

has some boron compounds having less oxygen than B2O3 and some boron carbide

having more carbon than B4C. The arc furnace in comparison to resistance furnace

requires only 59% of the power. Production rate in this method is much greater than

that of the resistance furnace. The major drawback of this process is that the control

of the temperature above 2300 K is not possible. This leads to the vaporization of

boron from the system affecting the B/C ratio. Therefore, more than 65% of excess

B2O3 was used to compensate for the loss of boron during the process [12].

Resistance-heating furnace process

Industrially, boron carbide on large scale is produced by carbothermal reduction

process using boric acid and petroleum coke in graphite resistance-heating furnace

[29]. Operation and design wise this furnace is similar to the Acheson furnace

which is used for SiC synthesis. Resistance heating furnace is cylindrical in shape,

with a graphite electrode as the heating element. Since, graphite has a very high

melting point so it is an ideal choice for heating electrode in the resistance heating

furnaces. The interior is lined with high temperature ceramic bricks and glass wool

and the outer shell of the furnace is made of stainless steel [12]. Schematic of

the resistance heating furnace is shown in figure 1.3. Heat is generated due to the

application of voltage across the heating electrode and it is based on the Ohmic law

of resistance heating. According to the Ohmic law of resistance heating, the power

converted into heat is given by:

P = I2R

Where, ’I’ is the amount of current flowing through the electrode and ’R’ is the

resistance of the heating electrode.

Boron carbide reaction is highly endothermic reaction, therefore, the heat gener-

ated due to resistance heating is responsible for heating the charge which surrounds

Chapter 1 11

Figure 1.3: Schematic of resistance heating furnace

the electrode. Once the reaction temperature is reached, the raw material reacts to

form B4C. The overall carbothermal reduction reaction is described by

2B2O3 (l, g) + 7C (s) → B4C (s) + 6CO (g) (1.2)

Because of the slow rate of heat conduction [32] that controls the heat transfer

process, the cool-down period of the furnace is long. Formation of B4C is a

very complex process. It involves both physical and chemical phenomena such

as condensation, vaporization, decomposition and recrystallization of many chem-

ical species [12]. Upon heating there is a continuous phase change of the reacting

material. Softening of raw material (H3BO3) starts at about 600 K, whereas it melts

down at 725 K and further down the line it makes various sub-oxides at 1550 K in

reducing atmosphere and it boils at 2133 K [32]. At higher reaction temperature

vaporization loss of boron occurs in the form its oxide/sub-oxides. Therefore, ex-

cess B2O3 is used in the starting mixture than required by stoichiometrically. The

Chapter 1 12

loss of boron can be minimized if the reaction is carried out at lower temperature

[33]. CO gas, which evolves during the reaction, diffuses out through the charge

and burns at the top of the furnace.

The kinetics of B4C formation is not understood properly. Several researchers have

investigated the reaction kinetics of the overall carbothermal reduction reaction us-

ing various carbon sources . The comparative results are summarized in table 1.2.

Intimate mixing of B2O3 and C may improve the kinetics of the process [33]. More

Table 1.2: Conversion timing findings by various researchers

Researcher Carbon source Conversion Temperature Proposed

time range mechanism

Weimer [14], Carbon Less than Greater than Nuclei-growth

Rafaniello (From calcined 1 sec 2200 K control

and Moore [34] corn starch)

Pikalov [35] Technical 90 min 1870 K Phase boundary

carbon 15 min 2070 K reaction control

Carbohydrates, 30 min 2373 K –

Smudski [31] resins and

polyhydric 180 min 1973 K –

compounds

about reaction kinetics of carbothermal reduction reaction is discussed in chapter 2.

1.7 Current understanding of the process

As mentioned previously that the resistance heating furnace is used for the mass

production of B4C. No other process compete it. Commercially, B4C is produced

with boric acid and carbon by a carbothermal process at temperatures near the melt-

Chapter 1 13

ing point of B4C in a batch resistance heating furnace [15, 33, 36]. Unfortunately,

not many efforts have been made to improve this process either experimentally or

theoretically. Similarly, the reaction kinetics of its formation is not well understood

yet.

Thevenot et al. [18] discusses about the significant contribution made by Prof.

Jean Cueilleron in the field of boron and refractory borides. Prof. Cueilleron de-

voted all his energies to resolve the difficult analytical problems associated with

boron and refractory borides. He established correlations between boron purity and

mechanical (Knoop microhardness) and electrical (Seebeck coefficient, resistivity)

properties. J. Cueilleron was one of the first in the world to perfect the fabrication

of boron fibers by continuous deposition of boron, obtained through reduction of

BCl3 by hydrogen, on a heated tungsten filament. He prepared boron carbide by

using BCl3 and methane in a plasma reactor. Choong-Hwan et al. [15] discusses

about the carbothermal reduction route adopted for the production of carbon free

B4C. Several experiments were conducted to determine the minimum amount of

excess B2O3 or the deficiency of carbon needed for the complete conversion to B4C

at low temperature. Tsuneo et al. [13] discusses the simultaneous measurement of

the heat capacities and the electrical conductivities of BxC (X = 3, 4 and 5) in the

temperature range of 300 to 1500 K.

As it is evident from the above discussion that there is almost no research or ex-

perimental data available on the production of boron carbide using heat resistance

furnace which is used for mass production of B4C. It is bit unfortunate that this

century old process did not get much attention of the scientific community either in

understanding or in optimizing the process. It is true, that it is not easy to study this

process due to many reasons such as: black box nature of it, very high temperature

involvement in the process, hazardous nature of the process and emission of poi-

sonous products such as CO. Also, if product gases do not come out properly then

Chapter 1 14

high pressure will build up in the furnace which may lead to explosion. Recently

only our group has taken a challenge to study this process in a systematic way using

both experiments and mathematical model. No one else has attempted the process

until now in this way. In the previous study [12] the focus was on the development

of experimental facility for the production of B4C using the resistance heating fur-

nace. Many experiments were conducted with the aims to measure the emissivity

of graphite in order to get accurate core temperature using pyrometer, porosity vari-

ation of raw material mixture (B2O3 + C) with temperature and distance from the

core, calibration of temperature measuring devices and thermocouple sheaths. Cur-

rently this is the only known experimental study which has been done in detail to

understand the process. Though it is lacking a systematic study, it has been suc-

cessful in measuring high temperature and revealing some interesting features of

the process. A simplified mathematical model in 1-D was also developed in this

study.

1.8 Modeling of boron carbide manufacturing process

The Acheson process is similar to B4C manufacturing process. This carbothermal

reduction process is used for the synthesis of SiC. The earliest mathematical model

of any carbothermal process is reported by Gupta et al. [37] in 2001 for SiC manu-

facturing. Gupta et al. [37] have developed a simplified 1-D mathematical model,

as shown by equation 1.3, to describe the Acheson process.

ρeCPe

(∂T

∂t

)︸ ︷︷ ︸Heat accumulation

=1

r

[∂

∂r

(rke

∂T

∂r

)]︸ ︷︷ ︸

Condution in r-direction

+ QR︸︷︷︸Source term

(1.3)

Where, QR is a source term which, in this case, includes the heat of reaction and

rate of reaction for the Acheson process. However, this model has ignored the effect

of radiation and convection on the process. Nevertheless, it was the first model for

this process to understand and led the foundation to develop it further.

Chapter 1 15

Because of the impossibility of seeing what is going on in the furnace [12], math-

ematical model becomes a valuable tool to explore the process. It is mentioned

previously that B4C synthesis process is similar to the Acheson process for which

a simplified 1-D model was developed in the previous study [12, 38]. The model

was in the same lines as the Acheson process model developed by Gupta et al. [37].

In this model, in addition to conduction, convection and radiation terms were also

considered in the mathematical formulation. This model was having some conver-

gence and stability problems. Also, this model was lacking in validation especially

for heat transfer and mass transfer was not studied well. Therefore, a sound math-

ematical model both in 1-D and 2-D is lacking.

1.9 Objectives

This dissertation is a continuation of the previous work initiated by our group to look

into the physical and mathematical modeling aspects of the boron carbide manufac-

turing process. Previous research [12] was mainly focused on the development

of experimental facilities and carrying out the experiments in various conditions.

Apart from this a simplified 1-D mathematical model was also developed as dis-

cussed above. It is also mentioned that experimentally the process is hazardous in

nature and it is very difficult to perform the experiments and many precautions have

to be exercised during the experiment. So, it is thought to develop a good mathe-

matical model to study the process.

Therefore, the objectives of the present work are:

• To develop a more robust 1-D mathematical model of heat and mass transfer

for the B4C manufacturing process and validate it with experimental results.

• To develop a 2-D mathematical model for heat and mass transfer which rep-

Chapter 1 16

resents the physical process more closely and validate the computed results

with experiments.

• To conduct more experiments to validate the model’s predictions on heat and

mass transfer.

• To optimize the process using mathematical model and conduct more experi-

ments to compare the optimised results.

1.10 Outline of present work

In chapter 2, an overview of the thermo-chemistry and kinetics of carbothermal

reduction reaction is discussed. This chapter, in particular, is focused toward the

thermodynamics of B-O-C system and the reaction kinetics of carbothermal reduc-

tion process.

In chapter 3 the mathematical formulation of the carbothermal reduction process

is presented. Various assumptions are discussed in details to justify them. Also,

the non-dimensional form of the governing heat and mass transfer equations with

their relevant boundary conditions are presented. Various computational techniques

which have been used to solve the governing equations are also discussed.

A brief discussion about the experimental setup and methodology adopted for con-

ducting the experiments is presented in chapter 4. This chapter in particular talks

about the physical modeling of carbothermal reduction process and about the intri-

cate details of the phenomenon taking place during the boron carbide manufactur-

ing operation. Finally, the complexities involved with the operation are discussed.

This chapter also discusses the various techniques of temperature measurement and

chemical analysis of the product.

Chapter 1 17

Chapter 5 is dedicated toward the results and discussion of 1-D and 2-D mathe-

matical modeling. Comparison between the experimental and computed results is

shown in this chapter. Optimisation of the process, using mathematical model, is

also discussed in this chapter.

Finally, Chapter 6 summarizes the research findings and suggests future research

directions.

Chapter 2

Thermodynamics and reaction

kinetics

This chapter presents an overview of the thermo-chemistry and kinetics of carboth-

ermal reduction reaction to produce boron carbide. Thermodynamics can tell the

feasibility of a reaction to occur, however, activation energies, diffusional resis-

tances, and other reaction kinetic considerations may prevent a reaction which oth-

erwise should occur [7].

2.1 Thermo-chemistry of B-O-C system

Thermodynamically, the overall reactions are not favorable unless the standard free

energy change become negative (i.e. ∆ G < 0). Therefore, at atmospheric pres-

sure, the minimum temperature required for the various overall reactions in B-O-C

18

Chapter 2 19

system to occur at equilibrium ∗ is as follow:

B2O3(l) + C(s) → B2O2(g) + CO(g) T∆G=0 = 2069K (2.1)

2B2O2(g) + 5C(s) → B4C(s) + 4CO(g) T∆G=0 = 1339K (2.2)

B4C(s) + 5B2O3(l) → 7B2O2(g) + CO(g) T∆G=0 = 2245K (2.3)

7C(s) + 2B2O3(l, g) → B4C(s) + 6CO(g) T∆G=0 = 1834K (2.4)

2B2O3(l) + 2B(s) → 3B2O2(g) T∆G=0 = 2246K (2.5)

2B4C(s) + B2O2(g) → 10B(s) + 2CO(g) T∆G=0 = 2231K (2.6)

B2O3(l) + 3B4C(s) → 14B(s) + 3CO(g) T∆G=0 = 2242K (2.7)

In practice, temperature above the minimum is required to promote the reactions

at a reasonable rate. Since the reactions are reversible, it is desirable to remove

the by-product CO produced in the process [7]. Unless CO produced is removed

from the process; a higher temperature is needed to promote reaction at reasonable

rate [14]. Reactions ( 2.1), ( 2.2) and ( 2.3) add up to give the reaction ( 2.4) and

reaction ( 2.7) can be expressed as the sum of reactions ( 2.3), ( 2.5) and ( 2.6). From

matrix theory, it can be shown that the rank of the matrix formed considering the

coefficients of the components involved in the reactions (from reaction 2.1 - 2.7) is

3. Therefore, only three reactions are independent. Thus, main reactions describing

the manufacturing process of boron carbide system are:

B2O3 (s) + C (s) → B2O2 (g) + CO (g) (2.8)

2B2O2 (g) + 5C (s) → B4C (s) + 4CO (g) (2.9)

B2O3 (l) + 3B4C (s) → 14B (s) + 3CO (g) (2.10)

Here reaction ( 2.8) and ( 2.9) are the main product formation reactions, whereas

the reaction ( 2.10) is the product dissociation reaction which initiates at very high

∗Equilibrium is said to exist in a system when it reaches a state in which no further change is

perceptible, no matter how long one waits [39]. This could happen if the system sinks into a very

deep free energy minimum.

Chapter 2 20

temperature to give elemental boron. The overall reaction, combining reaction ( 2.8)

and ( 2.9) can be written as follow:

2B2O3 (l, g) + 7C (s) → B4C (s) + 6CO (g) (2.11)

Figure 2.1: B-C phase diagram [2]

2.2 Phase diagram for boron-carbon system

Although numerous studies are available, not all parts of the B-C system have yet

been fully elucidated. Samsonov, Shuralov, et al. [6] reported the compounds B13C

and B12C3, both with a large homogeneity range, in addition to the carbon-rich BC2

and the boron-rich phases. Elliott et al., [40] reported the solid solubility of boron

carbide from ≈ 8 to 20 mol % C over the temperature range from room temperature

to the melting point of 2450 oC. The B4C-C eutectic temperature was reported

to be 2375 oC, at 29 mol % carbon (see figure 2.1). Recent measurements have

supported this broad range of solid solubility. In additions to the compounds given

in these publications, B25C [20], B8C [21], and B13C3 [22] have been reported

Chapter 2 21

recently. These are likely low-temperature phases, which are often observed in

chemical vapor deposition [6].

2.3 Reaction kinetics

The formation of boron carbide is highly dependent upon the phase change of re-

actant boron oxide from solid to liquid to gaseous boron sub-oxides and the effect

of reaction environment (i.e., heating rate and ultimate temperature) on the rate at

which the phase change occur [32]. Although final reaction equilibrium products

are determined solely from the temperature, pressure and chemical species and the

reaction mechanism. The reaction rate depends on a number of additional variables

like particle size, the degree of mixing of reactants, diffusion rates, porosity and the

presence and level of impurities or catalyst [7].

There are many types of reaction rate expressions reported by various researchers

based upon their investigation for carbothermal reduction process. These are gener-

ally based upon the kind of rate controlling mechanism considered for the reaction

rate. It could be internal or external diffusion of reactants, nuclei growth or chem-

ical reaction that controls the overall process [7]. All these mechanisms can be

represented by the equation 2.12

F (X) = Kt (2.12)

Where ,

K = Ko exp (−Ea/RgT )

Values of K for the reaction ( 2.11) are given as follows

K = 3.86 × exp [− (301000 ± 55000) /RgT ] for 1803 ≤ T ≤ 1976K

K = 2.00 × 1020 × exp [− (820000 ± 89000) /RgT ] for 1976 ≤ T ≤ 2123K

Chapter 2 22

The reaction rate constant, K, accounts for the effect of temperature on the reaction,

while the form of expression, F (X), accounts for virtually all other effects includ-

ing composition, diffusion and particle size. As such the proper reaction kinetics

of reaction ( 2.11) is not understood properly till now. However, the carbothermal

reduction reaction ( 2.11) generally takes place via fluid-solid and fluid-fluid mech-

anism rather than by solid - solid mechanism [7] and generally agreed mechanism

of boron carbide reaction is given by nucleation growth kinetics. The nucleation

kinetics mechanism is based on the activation of reaction sites, followed by growth

of the ’product’ nuclei through chemical reaction.

The nucleation and growth effects are combined into a single mechanism called nu-

cleation kinetics. An extensive explanation of this mechanism is given by Avrami

et al. [41 -- 43]. Tompkins et al. [44] indicates that the Erofeyev [45] approxim-

ation of Avrami’s expression is adequate for describing most kinetic data of the

nucleation type. Thus the form of Erofeyev [45] equation is

ln (1 − X) = − (Ktm)m (2.13)

Here,

– X is the fraction of carbon reacted

– K is the rate constant of the reaction, s−1

– tm is the reaction time, s

– m is the index which is 4 when the nucleus activation is rate limiting and 3

when isotropic 3-D nucleus growth is rate limiting. For 1-D rod-like growth

m → 1, while for 2-D planer growth m → 2 and m → 3 has been considered

for the reaction ( 2.11).

Above nucleation kinetic reaction model is well accepted [7, 32, 46]. In this present

study a similar approach is adopted. The free energy of the reaction ( 2.11) is posi-

Chapter 2 23

tive till 1834 K. Unless the CO produced is removed from the system, a higher tem-

perature is needed to promote reaction at a reasonable rate [7]. Reaction mechanism

is highly dependent on heating rates. Little nucleation occurs at lower temperature.

Then large crystallites are formed when growth takes place. But at large heating

rates or at higher temperatures the increased nucleation is the reason for many small

crystallites. At high temperatures, vaporization of boron oxide/suboxide may com-

pete with direct reaction of liquid oxide with carbon [32].

Boron carbide is both time dependent and temperature dependent process. Also

the reactant molar feed B/C ratio is crucial to the manufacturing of stoichiomet-

ric B4C at temperature above 2300 K. It is studied that formation of B4C is heat

transfer controlled and heating rate has a substantial influence on the mechanism

of overall reaction. It is also observed that, the carbon conversion increase with

increasing the temperature and in excess of boron oxide. For slow heating rates,

reactants react via classical nucleation - growth mechanism due to the reaction pro-

ceeding through a liquid boron path [32]. For higher temperature range the reaction

proceed via gaseous route. There is a change in mechanism at about 1976 K, which

is believed to be the result of competition between B2O3 (l) and B2O2 (g) reacting

with carbon. The liquid phase reaction dominates at lower temperature while the

gas phase reaction dominates at higher temperature [7].

2.3.1 Rate of reaction

The rate of reaction is rate of change in number of moles of reacting components

due to chemical reaction in its various forms, be it on unit volume or unit area basis

[47]. Equation 2.13 may be written in the following form:

X(t) = 1 − exp(−Kmtmm) (2.14)

Chapter 2 24

By differentiating equation 2.14 with respect to the time, we get

dX

dtm= Kmm tm

m−1 exp(−Kmtmm) (2.15)

Since

rA=CodX

dtm

= CoKmm tm

m−1 exp(−Kmtmm)

= ComK(Km−1tm

m−1)exp (−Kmtm

m)

Using ln (1 − X) = − (Ktm)m, above reaction can be written [46]

rA= Co K m(1 − X)

[ln

(1

1 − X

)] (m−1)m

Putting the value of m = 3 taken from literature [32], above equation can be written

as

rA= 3 × CoK(1 − X)

[ln

(1

1 − X

)]2/3(2.16)

Where,

X is the fraction of carbon/graphite reacted and rate of reaction is expressed in terms

of initial concentration of carbon/graphite.

Equation 2.16 is the desired equation for the rate of reaction for reaction ( 2.11)

and this would be used further in our mathematical modeling chapter 3.

Chapter 3

Mathematical modeling

3.1 Introduction

In today’s age where processor speed and memory usage is no longer a constraint,

numerical solutions are taking a big leap over physical experiments. In situations

where actual experiments are expensive and difficult to do, then mathematical mod-

eling is a best tool available to understand the complexities of the system. But it’s

not always true that numerical solutions of the complex process are cheap and easy

to do. In situations like capturing effects of turbulence, sometime even the best

available model will take months to make any meaningful prediction. Any model

available is not a good model if it cannot be validated with experimental results.

This chapter describes the formulation of mathematical model for B4C process.

3.2 Process description

The formation of boron carbide is least understood since last one century. Hardly

any systematic experimental or theoretical study is available in the open literature.

Therefore, any contribution toward understanding this process would be very ben-

25

Chapter 3 26

eficial. Many researchers [7] have given various reaction kinetics models as dis-

cussed in chapter 2 and still it’s a matter of debate in scientific community. More-

over there are so many changes taking place simultaneously into the reacting system

like condensation, vaporization and re-crystallization etc., which makes it more dif-

ficult to understand. This study is a step forward in the direction of understanding

the various controlling parameters in B4C manufacturing process with the use of

mathematical modelling.

Though furnaces can either be in rectangular shape or cylindrical shape. A cylindri-

cal shaped furnace is considered in the present study as shown in the figure 1.3. Heat

generated is transferred to the charged material surrounding the electrode by con-

duction, convection and radiation. Heat may get consumed during decomposition

and vaporization, while it may be recovered during recrystallization and condensa-

tion. Therefore, as a first approximation it can be assumed that these phenomena do

not have significant effect on heat transfer process. The reactive mixture has very

low thermal conductivity. Also, the specific heat of boron carbide is very high. So

one can expect steep temperature gradient around the reacting core. As such, far

from the electrode, above discussed phenomena would be absent.

Until the reaction temperature is reached, heat is chiefly transferred to the reac-

tive mixture by conduction. Thus the standard Fourier heat conduction equation

can be considered for heat transfer. During the chemical reaction some gaseous

products/by-products are produced. Some of them diffuses out (such as CO) through

the reactive mixture and burn at the top of the charge. Diffusion of these hot gaseous

products through the unreacted charge further add to the heat transfer. Mostly CO

gas, as by-product of the reaction comes out. So convective heat transfer can be

modeled considering diffusive heat flux of the by-product CO gas.

As discussed in the section 1.6.4, the reactive mixture in the furnace near the core

Chapter 3 27

is at very high temperature, and one may expect a good contribution of radiations in

heat transfer. However, the charge is very fine and is packed nicely around the core

and the void fraction is also low, therefore one may not expect a significant con-

tribution of radiation in the heat transfer process except from the outer surface of

the charge which is open to the atmosphere. Nevertheless, radiation effect has been

considered in the present case using Rosseland approximation. A brief discussion

is given below.

Opposite to heat conduction and convection radiation is a nonlocal phenomenon,

which can be described by an integro-differential equation – the so-called radia-

tive transfer equation. An additional complication in numerical solution is raised

that the relatively large grid size which is reasonable for integrating the radiation

terms without extensive computation are often not adequate to give good accuracy

for the local conduction and/or convection terms [48]. Apart from the mathemat-

ical complexities, there are difficulties in determining accurate physical properties

that are to be inserted into the integro-differential equations. Moreover, in case of

participating media the problem becomes more difficult because the participating

media are capable of absorbing, emitting and scattering thermal radiations. Thus,

this again limits the accuracy even if the exact mathematical solutions of integro-

differential equations are available. Hence, Rosseland approximation comes handy

to overcome such a difficulty.

Rosseland approximation neglects all the geometric information about the medium.

Therefore, the Rosseland approximation is valid only for very highly absorbing me-

dia. According to this approximation, also known as diffusion approximation, the

Chapter 3 28

net radiative heat flux for the case of optically thick medium ∗ in near thermody-

namic equilibrium can be approximated by a simple correlation given as follow

[49]:

− 16η2σT 3

3κ(grad T )

Here,

– η is the refractive index

– κ is Rosseland mean extinction coefficient, 1/m

– σ is the Stephen Boltzmann constant , W/m2 − K4

– T is the temperature, K

The extinction coefficient for a particular substance is a measure of how well it

absorbs electromagnetic radiation (EM waves). If the EM wave can pass through

very easily, the material has a low extinction coefficient. Conversely, if the radiation

hardly penetrates the material, but rather quickly becomes ”extinct” within it, the

extinction coefficient is high. The value of extinction coefficient data (κ) for B4C

is not available in the literature. Hence, the extinction coefficient value available for

the SiC [50] and carbon particles [51], which resembles our system is considered.

Thus, reported value of κ for SiC and carbon particles which is of the order of 103

has been considered here for B4C. η is taken as 1.

Using the above result, 1-D steady state energy equation with simultaneous con-

duction and radiation terms without any source term can be written [52] as

d

dy

(λT

d

dy

)= 0

∗A medium is said to be optically thick if the radiation mean free path i.e. reciprocal of the

extinction coefficient is very small compared to the characteristic dimension of the medium.

Chapter 3 29

Here, λT is the total thermal conductivity of the medium which further can be writ-

ten as

λT = λC + λr = λC +16η2σT 3

3κ

Where

– λC is effective thermal conductivity, W/m − K

– λr is radiative conductivity, W/m − K

This approximation gives good results with optically thick medium [50, 52]. There-

fore, radiation transport can be characterized as a diffusion process in the optically

thick limits. In our case, Rosseland approximation is used to account for radiative

heat transfer development of the mathematical model.

While considering the mass balance, mass transfer is occurring due to CO diffu-

sion through the reacting material and due to chemical reactions. Fick’s law of

mass diffusion is solved in its transient form coupled with energy equation. While

considering the advection of the by-product gas, which is a combined phenomenon

of fluid flow due to pressure gradient and diffusion due to difference in chemical

potential of the species, here, we consider only diffusional flow. The amount of CO

produced is not much to cause significant convection. CO produced during early

stages comes out of the system freely. At high temperature when reaction starts

and the reacting mass becomes viscous, CO gets entrapped into the bed and thus

diffusion process takes over the convective mass transfer.

Based on the above physico-chemical description of the process, the following as-

sumption have been made into the development of 1-D and 2-D mathematical mod-

els.

Chapter 3 30

3.3 Mathematical formulation

Before proceeding to 1-D and 2-D mathematical model formulation of the B4C

process, it is essential to make a few assumption which are listed below. The justi-

fication of their assumption have been given wherever it is necessary.

3.3.1 Assumptions

• Axisymmetry along vertical plane passing through the central axis of the fur-

nace is assumed.

Looking at figures 1.3 and 3.1 it is reasonable assumption which reduces the

domain of consideration for solution and thus adds in achieving the solution

faster without affecting the accuracy of the results.

• Continuum model based approach is adopted.

• Diffusion model is considered for the by-product gas i.e. CO (carbon mono-

oxide).

• The effect of high temperature phenomena, like condensation, vaporisation

and recrystallisation in the overall process is negligible.

It is thought whatever heat is consumed in phenomena like vaporisation is re-

covered during other phenomena such as condensation. Therefore, the overall

effect of these processes would be negligible. Moreover, it would be very dif-

ficult to model these phenomenon in absence of the availability of the proper

physics.

• Temperature dependence of density variation of the reacting mixture is not

considered.

The obvious reason behind this assumption is lack of data. Since during reac-

tion there is lot of physical and chemical changes occurring into the reacting

furnace, so practically it becomes very difficult to get the density variation

Chapter 3 31

with temperature and so constant density approach is fair enough for the

mathematical model development. Same can be later incorporated into the

model as per the availability of the requisite density data.

• Numerical domain of consideration is shown in the figure 3.1 for both 1-D

and 2-D model respectively.

• Rosseland approximation is applied to include the radiation effects into the

model.

(a) (b)

Figure 3.1: Computational domain for (a) 1-D and, (b) 2-D mathematical model

Here,

– ro is the inner periphery radius of the furnace, m

– r3 is the radius of furnace with refractory lining, m

– r4 is the radius of furnace with refractory lining and steel shell, m

Chapter 3 32

3.4 Governing equations

3.4.1 Overall 2-D heat balance equation

Using the first principle of the heat balance and applying it across a radial elemental

ring of size ”dr” present at distance r from the core of furnace (as shown in figure 3.1),

the heat balance equation can be written as follow:

⎡⎢⎢⎢⎣

Rate of accumulation

of enthalpy in the

control volume (CV)

⎤⎥⎥⎥⎦ =

⎡⎢⎢⎢⎣

Enthalpy entering the

CV due to conduction,

convection and radiation

⎤⎥⎥⎥⎦−

⎡⎢⎢⎢⎢⎢⎢⎣

Enthalpy leaving

the CV due to

cond., convection

and radiation

⎤⎥⎥⎥⎥⎥⎥⎦

+

⎡⎢⎢⎢⎣

Rate of heat generation

in the CV by chemical

reaction

⎤⎥⎥⎥⎦

Inside the furnace, both, solid and gas are assumed to be at the same temperature

i.e., Tg = Ts = T . Therefore the final shell balance equation for heat transfer in 2-D

co-ordinate can be written as:

ρeCPe

(∂T

∂t

)︸ ︷︷ ︸Heat accumulation

=1

r

[∂

∂r

(rke

∂T

∂r

)]︸ ︷︷ ︸

Condution in r-direction

+1

r

[∂

∂r

(rDCO−air,eCPgT

∂CCO

∂r

)]︸ ︷︷ ︸

Diffusion in r-direction

+1

r

[∂

∂r

(r16ση2T3

3κ

∂T

∂r

)]︸ ︷︷ ︸

Radiation in r-direction

+1

r

[∂

∂θ

(ke

r

∂T

∂θ

)]︸ ︷︷ ︸Conduction in θ-direction

+1

r

[∂

∂θ

(DCO−air,eCPgT

r

∂CCO

∂θ

)]︸ ︷︷ ︸

Diffusion in θ-direction

+1

r

[∂

∂θ

(16ση2T3

3rκ

∂T

∂θ

)]︸ ︷︷ ︸

Radiation in θ-direction

− ∆Hr (1−ε)︸ ︷︷ ︸Heat generation

(3.1)

Chapter 3 33

Each term in the above equation has units as Jm3−s

.

Where,

– CCO is the concentration of carbon mono-oxide (CO), kgmol/m3

– CPe is effective specific heat of gas-solid mixture, J/kgmol − K

– CPg is effective specific heat of CO gas, J/kgmol − K

– DCO−air,e is the effective mass diffusivity of CO in air, m2/s

– ∆Hr is the rate of heat consumption during reaction, W/m3

– ke is the effective thermal conductivity of raw material, W/m − K

– L is the length of the furnace, m

– ε is the porosity of raw material

– ρe is the effective density of solid mixture, kgmol/m3

Estimation of effective properties will be explained later in the section 3.5. The

correlations used for finding the property data are given in appendix A.2.

3.4.2 Boundary conditions for overall 2-D heat balance equation

• At time t = 0, for all r and θ, T = Ti

• At time t > 0, for r = ri at all θ (at electrode surface)

−ke∂T

∂r=

[Heat input

2πriL

]• At time t > 0, for θ = 0 and θ = π for all r (axisymmetry boundary

condition)

∂T

∂θ= 0

Chapter 3 34

• At time t > 0, for r = ro (at the inner periphery of furnace)

i) For 0 < θ ≤ 2π

3; −ke

∂T

∂r=

[Heat loss23× 2πroL

]

ii) For2π

3< θ < π ; − ke

∂T

∂r=

⎡⎢⎢⎢⎢⎢⎢⎣

Heat loss due to

conv. and radiation13× 2πroL

⎤⎥⎥⎥⎥⎥⎥⎦−

⎡⎢⎢⎢⎢⎢⎢⎣

Heat recovered

due to CO burning13× 2πroL

⎤⎥⎥⎥⎥⎥⎥⎦

Where,

Heat loss =2πL (Tref − Tamb)[(

nr3ro

λref

)+

(n

r4r3

λSS

)+(

1hr4

)] ; h [53]= 1.32

[(TSS−Tamb)

2r4

]0.25

⎡⎣ Heat loss due to

conv. and radiation

⎤⎦ = σA(Tg

4 − Tamb4) + hA(Tg − Tamb)

⎡⎣ Heat recovered

due to CO burning

⎤⎦ =

⎡⎣ Heat of combustion

of CO per mole

⎤⎦×

⎡⎣ No. of moles of CO

reaching the top surface

⎤⎦

Here,

– TSS is the temperature of outer steel shell, K

– Tref is the reference temperature, K

– Tamb is the ambient temperature, K

– Tg is the gas temperature burning at the top surface of furnace, K

– h is the convective heat transfer coefficient for heat loss between the surface

of steel shell and atmospheric air, W/m2 − K

– σ is Stefan-Boltzmann constant, W/m2 − K4

– r3 and r4 are defined in figure 3.1, m

Chapter 3 35

3.4.3 Overall 2-D mass balance equation for CO

Considering only the diffusive flux of CO through the reacting mixture and adopting

the similar approach for mass balance as for heat balance, we can formulate the mass

balance equation for CO as given below:⎡⎣ Rate of CO accumulation

per unit control volume (CV)

⎤⎦ =

⎡⎣ Rate of CO entering

the CV by diffusion

⎤⎦−

⎡⎣ Rate of CO leaving

the CV by diffusion

⎤⎦

+

⎡⎣ Rate of CO generation

per unit control volume

⎤⎦

ε

(∂CCO

∂tm

)︸ ︷︷ ︸

Mass accumulation

=1

r

[∂

∂r

(rDCO−air,e

∂CCO

∂r

)]︸ ︷︷ ︸

Diffusion in r-direction

+1

r

[∂

∂θ

(DCO−air,e

r

∂CCO

∂θ

)]︸ ︷︷ ︸

Diffusion in θ-direction

+o

W (1−ε)︸ ︷︷ ︸Mass generation

(3.2)

Here,o

W is the rate of CO generation.

Each term in the above equation has units askg moles of CO

m3 − s

3.4.4 Boundary conditions for overall 2-D mass balance

• At time t = 0, for all r and θ (inside the furnace)

CCO

= 0

• At time t > 0, for r = ri at all θ (at the surface of electrode)

∂CCO

∂r= 0

• At time t > 0, for θ = 0 and θ = π at all r (axisymmetry boundary

condition)∂C

CO

∂θ= 0

Chapter 3 36

• At time t > 0, for r = ro (at the inner periphery of furnace)

i) For 0 < θ ≤ 2π

3;

∂CCO

∂r= 0

ii) For2π

3< θ < π ; C

CO= C

CO−air

3.5 Determination of properties

3.5.1 Determination of CO generation (o

W )

From the overall reaction ( 2.11) of boron carbide formation we know that

2B2O3 (l, g) + 7C (s) → B4C (s) + 6CO (g) (3.3)

For the above reaction we can write the reaction rate with respect to different com-

ponents involved i.e. in terms of depletion of C/B2O3 or in terms of formation of

B4C/CO. Therefore from stoichiometry we can write [47]

−rc

7= − rB2O3

2=

rB4C

1=

rCO

6(3.4)

Here,

• (−rc) is the rate of consumption / depletion of carbon

• (−rB2O3) is the rate of consumption / depletion of boron oxide

• rB4C is the rate of formation of boron carbide

• rCO is the rate of formation of carbon mono - oxide

Hence, using equation 3.4 we can write,

rCO = − 6

7× rc

i.e. Rate of formation of CO (o

W ) =6

7× Rate of depletion of carbon/graphite.

Here, the rate of depletion of carbon is found using equation 2.13. Detailed expla-

nation is given in section 2.3.1.

Chapter 3 37

3.5.2 Determination of Dco-air

The diffusion coefficient of CO in air is found using Gilliland equation [54, 55],

which is a function of temperature (oC) and pressure (atmosphere).

DCO−air =

⎡⎢⎢⎢⎣ 0.0606T 1.78

P

(VCO)

1/3 + (Vair)1/32

⎤⎥⎥⎥⎦[1 +

√MCO + Mair

60√

MCO × Mair

]E − 4

Here,

– DCO−air represents CO diffusivity in air, m2/s

– MCO and Mair represents molar mass of CO and air respectively, kg/mol

– VCO and Vair represents molar volume of CO and air respectively, l/mol

3.5.3 Determination of effective properties

Effective properties used in the equations 3.1, 3.2, 3.7 and 3.8 are calculated on the

weighted average basis. Thus the effective properties of gas – solid mixture can be

expressed as follows:

Pe = εPg + (1 − ε) Ps

Where, Ps =n∑

i=1

xipi

Here,

– Pg represents physical properties of the gas.

– Ps represents physical properties of the solid.

– xi represents mole fraction of ith solid component.

– pi represents the physical property of ith solid component.

– n represents the total number of solid component.

Weighted mole fraction average is used to determine the molar specific heat whereas

weighted volume average is used to determine the thermal conductivity of gas-solid

mixture [37]. For example, the molar specific heat based on weighted mole fraction

Chapter 3 38

average is given by

CP,M = ε × CP,g+ (1−ε)×CP,s

CP,s = CP,C×XC+CP,BO×XBO+CP,BC×XBC

Here,

– CP,M is the total specific heat of solid-gas mixture, J/kgmol − K

– CP,g is the specific heat of the by-product gas, J/kgmol − K

– CP,s is the specific heat of solid mixture, J/kgmol − K

Where,

XC = MC/(MC+MBO+MBC)

XBO = MBO/(MC+MBO+MBC)

XBC = MBC/(MC+MBO+MBC)

Here, XC , XBO and XBC are the mole fraction of carbon/graphite, boric acid and

boron carbide into the reacting mixture. MC , MBO and MBC are used for molar

masses of carbon, boric acid and boron carbide respectively. Similarly, thermal

conductivity of the gas-solid mixture based on weighted volume average can be

given by

KT,M = ε×KT,g+ (1−ε)×KT,s

KT,s = KT,C×XVC+KT,BO×XVBO+KT,BC×XVBC

Here,

– KT,M is the total thermal conductivity of solid-gas mixture, W/m − k

– KT,g is the thermal conductivity of the by-product gas, W/m − k

– KT,s is the thermal conductivity of solid mixture, W/m − k

– XVC , XVBO and XVBC are volume fraction of the reacting raw material.

Where,

XVC=

(MC

ρC

MC

ρC+MBO

ρBO+MBC

ρBO

)

XVBO and XVBC can also be calculated on the same line as discussed above.

Chapter 3 39

3.5.4 Determination of the rate of heat consumption (∆Hr)

The rate heat consumption for the reaction 3.3 at any temperature T can be deter-

mined as

∆Hr = ∆H × rate of reaction (rA) (3.5)

Here, ∆H is the heat of formation for the reaction 2.11 which can be found from

enthalpy difference of ’product’ minus ’reactants’. In mathematical terms we write

∆H = (1 × HB4C + 6 × HCO) − (2 × HB2O3 + 7 × HC) (3.6)

In other words heat of the reaction at temperature T is the heat transferred from

source to the reacting system where, say, ’x’ moles of reactant disappear to form

’y’ moles of product at the same temperature and pressure before and after reac-

tion [47]. Enthalpy data are given in the appendix A.2.1.

Table 3.1: Non-dimensional parameters

Quantity Dimensional quantity Non-dimensional quantity

Temperature T T ∗ = T/Ti

Radius r r∗ = r/ro

Concentration of CO CCO C∗CO = CCO/CCO−air

Time (in energy balance) t t∗= (tαe,i)/r2o

Time (in mass balance) tm tm∗= (tmDCO−air,e,i)/r2

o

Diffusion coefficient DCO−air,eD∗

CO−air = DCO−air,e/DCO−air,e,i

Thermal diffusivity αe = ke/(ρeCp,e) α∗ = αe/αe,i

3.6 Non-dimensionalization

The non-dimensional parameters which have been used to represent the governing

equations are given in table 3.1. Here, subscript ’e’ denotes the effective properties

Chapter 3 40

and subscript ’i’ denotes the initial property i.e. property at time t = 0. Any property

with superscript ’*’ is used to denote non-dimensional quantity.

3.6.1 Non-dimensionalization of overall 2-D heat balance equation

Using the non-dimensional quantities, as given in table 3.1, the overall heat balance

equation 3.1 can be written in non-dimensional form as

(∂T ∗

∂t∗

)=

1

r∗

[∂

∂r∗

(r∗α∗∂T ∗

∂r∗

)]+

1

r∗

[∂

∂r∗

(r∗D∗

CO−airN∗T ∗∂CCO

∗

∂r∗

)]

+1

r∗

[∂

∂r∗

(r∗k∗T ∗3 ∂T ∗

∂r∗

)]+

1

r∗

[∂

∂θ

(α∗

r∗∂T ∗

∂θ

)]

+1

r∗

[∂

∂θ

(D∗

CO−air

r∗N∗T∗∂C∗

CO

∂θ

)]+

1

r∗

[∂

∂θ

(k∗T ∗3

r∗∂T ∗

∂θ

)]− ∆H∗

r

Here,

N∗ =

[CCO−airDCO,e,iCp,g

ρeCp,eαe,i

]

∆H∗r =

[∆Hr (1 − ε) r2

o

TiρeCp,eαe,i

]

k∗ =

[16η2σTi

3

3κρeCp,eαe,i

]

Non-dimensional boundary conditions for 2-D heat balance equation

• At time t = 0, for all r∗ = r/ro and θ, T ∗ = 1

• At time t > 0, for r∗ = ri/ro at all θ (at electrode surface)

−∂T ∗

∂r∗=

[Heat input2πr∗LkeTi

]

Chapter 3 41

• At time t > 0, for θ = 0 and θ = π for all r∗ (axisymmetry boundary

condition)

∂T ∗

∂θ= 0

• At time t > 0, for r∗ = 1 (at the inner periphery of furnace)

i) For 0 < θ ≤ 2π

3; −∂T ∗

∂r∗=

[Heat loss

23× 2πr∗LkeTi

]

ii) For2π

3< θ < π ; − ∂T ∗

∂r∗=

⎡⎢⎢⎢⎢⎢⎢⎣

Heat loss due to

conv. and radiation13× 2πr∗keLTi

⎤⎥⎥⎥⎥⎥⎥⎦−

⎡⎢⎢⎢⎢⎢⎢⎣

Heat recovered

due to CO burning13× 2πr∗keLTi

⎤⎥⎥⎥⎥⎥⎥⎦

3.6.2 Non-dimensionalization of overall 2-D mass balance equation

Using the non-dimensional parameters as discussed in table 3.1, the 2-D mass bal-

ance equation in its non-dimensional form can be written as follows:

ε

(∂C∗

CO

∂tm∗

)=

1

r∗

[∂

∂r∗

(r∗D∗

CO−air

∂C∗CO

∂r∗

)]+

1

r∗

[∂

∂θ

(D∗

CO−air

r∗∂C∗

CO

∂θ

)]+W∗

Here,

W∗ =

o

W r2o (1 − ε)

CCO−airDCO−air,e,i

Non-dimensional boundary conditions for 2-D mass balance equation

• At time t = 0, for all r∗ and θ (inside the furnace)

C∗CO

= 0

• At time t > 0, for r∗ = ri/ro, (at the surface of electrode)

∂C∗CO

∂r∗= 0

Chapter 3 42

• At time t > 0, for θ = 0 and θ = π at all r∗ (axisymmetry boundary

condition)∂C∗

CO

∂θ= 0

• At time t > 0, for r∗ = 1 (at the inner periphery of furnace)

i) For 0 < θ ≤ 2π

3;

∂C∗CO

∂r∗= 0

ii) For2π

3< θ < π ; C∗

CO= 1

3.7 1-D model equations

As discussed in section 1.8, that a simplified 1-D mathematical model was devel-

oped in a previous study [12] where validation of the model was lacking. In particu-

lar the mass transfer model was not validated at all. Also instability and convergence

problems were reported in the earlier model. Therefore, first 1-D model was stud-

ied in detail particularly from numerical stability view point. Adopting the same

approach as used for the development of 2-D heat and mass balance equations in

this chapter, 1-D heat and mass balance equations can be developed. For the sake of

brevity, only the non-dimensional form of 1-D heat and mass balance equations are

shown here. Therefore, 1-D heat balance model equation with relevant boundary

conditions can be expressed as follows:

(∂T ∗

∂t∗

)︸ ︷︷ ︸

Heat accumulation

=1

r∗

[∂

∂r∗

(r∗α∗∂T ∗

∂r∗

)]︸ ︷︷ ︸

Conduction in r-direction

+1

r∗

[∂

∂r∗

(r∗D∗

CO−airN∗T∗∂C∗

CO

∂r∗

)]︸ ︷︷ ︸

Diffusion in r-direction

+1

r∗

[∂

∂r∗

(r∗k∗T ∗3∂T ∗

∂r∗

)]︸ ︷︷ ︸

Radiation in r-direction

− ∆H∗r︸︷︷︸

Heat generation

(3.7)

• At time t = 0, for all r∗ = r/ro, T ∗ = 1

Chapter 3 43

• At time t > 0, for r∗ = r/ro (at the surface of electrode)

−∂T ∗

∂r∗=

[Heat input2πr∗LkeTi

]• At time t > 0, for r∗ = 1 (at the inner periphery of the furnace)

−∂T ∗

∂r∗=

[Heat loss

2πr∗LkeTi

]

Similarly, non-dimensional form of 1-D mass balance equation for CO can be ex-

pressed as:

ε

(∂C∗

CO

∂tm∗

)=

1

r∗

[∂

∂r∗

(r∗D∗

CO−air

∂C∗CO

∂r∗

)]+ W∗ (3.8)

• At time t = 0, for all r∗ (inside the furnace)

C∗CO

= 0

• At time t > 0, for r∗ = ri/ro, (at the surface of electrode)

∂C∗CO

∂r∗= 0.0

• At time t > 0, for r∗ = 1 (at the inner periphery of furnace)

C∗CO

= 1

In 1-D model, no resistance is considered for the CO diffusion at the furnace bound-

ary whereas for heat transfer a series of resistances are considered based upon the

thickness of glass-wool, fire bricks and steel shell etc. In other words, wall is as-

sumed porous for CO diffusion in case of 1-D mathematical model.

Chapter 3 44

3.8 Computational technique

As a part of the solution methodology, initially, 1-D model is first discretized in its

non-dimensional form using Finite Volume Method † (FVM). The resulting algebric

equations were solved using tri-diagonal matrix algorithm (TDMA) method, using

a computer code written in FORTRAN 95. The computed results obtained from

1-D model are then validated with some experimental results. Similarly, the results

obtained from 2-D model are validated with experimental results.

3.8.1 Discretization and solution methodology

Discretization is a way of replacing the continuous information in discrete points.

These discrete points are called grid points. It is this systematic discretization of

space and of the dependent variables that makes it possible to replace the governing

differential equations with simple algebraic equations, which can be solved with

relative ease [56]. There are techniques like Finite Element Method (FEM), Finite

Difference (FD) and Finite Volume Method (FVM) etc. to do this discretization

job. Each of them, in certain way has advantage over other technique and they

have their own drawbacks also. Like, finite difference (FD) discretization of the

partial differential equation (PDE) is inappropriate near discontinuities because the

PDE does not hold there, whereas with finite volume method (FVM) discretization,

which implies integral conservation, is still valid even for discontinuous solution.

But unfortunately the integral form is more difficult to work with than the differen-

tial equation, especially when it comes to discretization. Since the PDE continues

to hold except at discontinuities, another approach is to supplement the differential

equation by additional ”jump condition” that must be satisfied across discontinu-

ities. These can be derived by again appealing to the integral form [58].

†A brief explanation of FVM technique is given in appendix. Detailed explanation is available in

literature [56, 57].

Chapter 3 45

Thus applying the integral form of discretization scheme for coupled heat and

mass balance equations, we get a set of discretized coupled algebraic equations in

the cylindrical calculation domain, which further can be solved directly by apply-

ing TDMA. Both, fully explicit and fully implicit formulation schemes have been

solved for 1-D model. A computer code, in FORTRAN 95 [59], has been devel-

oped to solve the system of equations. Computational procedure adopted here in

1-D case is shown in a flow chart (see figure 3.2).

3.8.2 Solution methodology for 2-D model

Fully explicit and fully implicit formulation schemes have been used to solve the

1 - D model whereas for 2 - D model only fully implicit scheme has been used

along with the line-by-line TDMA. ‡ A computer code, in FORTRAN 95 , has been

developed to obtain the solution of 2-D model. Figure ( 3.3) shows the flowchart

adopted for the solution of discretized equations in their non-dimensional form in

2-D case. The results obtained are later converted into their dimensional form.

‡In line-by-line TDMA, for a particular time step we first find the converged solution in one

direction; say in r-direction at a particular θ, assuming the quantities to be constant in the neighbor-

hood of ’r’ in θ-direction. Thus we keep applying TDMA in r-direction and sweep in θ-direction.

By doing so we cover the full domain of consideration. Same procedure is adopted while applying

TDMA in θ-direction and sweeping in r-direction.

Chapter 3 46

Figure 3.2: Flow diagram for 1-D program

Chapter 3 47

Figure 3.3: Flow diagram for 2-D program

Chapter 4

Physical modeling and process

description

As discussed in chapter 1 and 3 that experimental setup for the B4C manufacturing

process was developed in the previous study [12]. Therefore, only a brief discus-

sion on the experimental setup is given in this chapter followed by the experimental

procedure which has been adopted in the current study to perform the desired ex-

periments. A detailed discussion on the experimental setup with the accuracy of

various instruments and data are given in reference [12].

4.1 Experimental setup

Carbothermal reduction process and various other routes of manufacturing B4C are

discussed briefly in chapter 1. The reaction of formation of B4C is strongly en-

dothermic in nature with a favorable free energy change at high temperature which

at its best is carried out in a specially designed graphite resistance furnace at tem-

perature above 1834 oC, using boric acid and graphite/petroleum coke as starting

materials. A schematic diagram of the experimental setup is shown in the figure 4.1.

48

Chapter 4 49

Chapter 4 50

The experimental setup mainly consists of the following:

1. Resistance heating furnace

2. Power supply unit with control panel

3. Thermocouples and pyrometer

4. Data recording device

5. Safety accessories

A brief discription of each equipment is given below.

4.1.1 Resistance heating furnace

Resistance heating furnace is cylindrical in shape with opening at the top so as to

provide a way out for the by-product gases generated during the operation. Also

an exhaust blower is provided with hood assembly on the top of the furnace for

fast removal of the by-product gases. The outer body of the furnace is made-up

of stainless steel sheet of 3 mm thickness . Inside part of the furnace is lined with

high temperature fire bricks and glass wool. A special electrode holding arrange-

ment is developed in-house with cooling facility. The electrode holding assembly

is connected to a water pump for continuous water supply for cooling purpose. Ar-

rangement of two water tanks in tandem with water pump works as a source of

continuous water supply system. One of the electrode holder assemblies is on a

small movable trolley that gives it an advantage to move it and fix the electrode

into the holder assembly properly. Otherwise loose electrode connection is a ma-

jor reason for failure of the experiments. When high potential is applied across

the graphite electrode, it generates lots of heat based on the principle of resistance

heating. Through-holes, as shown in figure 4.2, are provided along circumferential

line at various θ locations in the furnace to note down the temperatures during the

Chapter 4 51

Figure 4.2: Internal construction of the furnace

experiments via the use of various types of thermocouples and two-colour radiation

pyrometer.

4.1.2 Power supply unit with control panel

The whole unit consists of the following:

• II-phase oil cooled transformer (75 kVA, manufactured by Universal Trans-

formers, Banglore)

• Variac assembly fixed on the top of transformer for an on load power supply

variation

• LED displays for voltmeter and ampere meter

• Thermocouple temperature display unit

Chapter 4 52

To meet the energy requirements a customized transformer was designed by M/s

Universal Transformers, Banglore with 75 kVA rating. It is a step down transformer,

which converts the 440 V coming from control panel to maximum 35 V output. A

variac wheel is used to control the power supplied to the heating furnace. By rotat-

ing this wheel in clockwise or anti-clockwise direction one can control the amount

of power fed to the heating furnace. Also, this variac assembly gives us advantage

of doing the experiments with different modes of power supply like constant power

supply or with stepwise change in power supply. To record the secondary voltage

and current, suitable voltmeter and ammeter assembly is given at the top of trans-

former. Control panel displays primary current, voltage displays and all the ther-

mocouple readings to calculate the temperature during the experiment. The supply

coming from mains goes to the transformer via control panel. For safety purposes,

molded case circuit breaker (MCCB) is installed along with the power supply unit

which trips when the system withdraws more power then the rated capacity.

4.1.3 Thermocouples and pyrometer

Thermocouples and pyrometer setup is the backbone of the experiment and data

obtained using these assemblies play an important role in validation of the mathe-

matical models. Since there exists a wide range of temperature (100oC to 2500oC)

in the heating furnace, so a verity of thermocouples along with two-colour radi-

ation pyrometer are used to capture the temperature in different ranges. C-type

thermocouple (Tungsten-5% Rhenium v/s Tungsten-26% Rhenium) has been used

to measure temperature in the range of 1473-2473 K. To avoid the oxidation of

thermocouple wires, it is placed inside a graphite tube (5.5 mm thick) which is con-

nected with a continuous UHP (ultra high pure) N2 supply. B-Type thermocouple

(Pt-30 % Rh v/s Pt) is used with 12 mm thick re-crystallized alumina sheath, for

the temperature range of 1173 K to 1973 K. B-type thermocouples are placed at

different locations into the furnace. For measuring temperature rise at the surface

Chapter 4 53

of heating electrode, a non-contacting temperature measuring device, such as py-

rometer, is used. It works in the temperature range of 1173-3273 K. Pyrometer is

focussed on the heating electrode via a graphite tube (sighting tube) running through

the furnace and touching the heating electrode in the furnace. The view field of py-

rometer should be free from smoke, dust or any other kind of scattering particles.

Thus, UHP N2 purging is provided into the sighting tube so as to remove the CO or

any other gaseous product produced during the reaction. M/s Mikron infrared Inc.,

U.S.A, supplied the pyrometer. It works on the principle of two-color radiation py-

rometery. Extensive literature is available on this subject [60 - 62].

As there is no published data available on relative emissivity of the graphite at

very high temperature so to capture the accurate core temperature using pyrometer,

experiments were conducted with a calibrated C-type thermocouple and emissiv-

ity value was adjusted online for pyrometer to match the temperature as obtained

using C-type thermocouple. Once the emissivity variation in the operating tempera-

ture range in known, suitable emissivity is set for pyrometer during the experiment.

Experiments were conducted with ±10 K accuracy in temperature value between

pyrometer and C-type thermocouple reading. More details are provided in refer-

ence [12]. Both graphite tube and recrystallized alumina sheaths, which are used

as thermo well for C-type and B-type thermocouples respectively were also cali-

brated. About other experimental details on calibration and other findings one can

go through the reference [12].

4.1.4 Data recording device

The data recording system consists two major parts which are:

• Thermocouple amplifier and,

• Data logger

Chapter 4 54

The voltage produced by the thermocouples is in the range of µV to mV. In order

to get a reading that is easy to record, thermocouple voltage amplifier is used. The

input to the thermocouple amplifier comes straight from the thermocouple and out-

put of the amplifier goes into the input junction of data logger ∗. The data logger

can be interfaced with a computer for the analysis of recorded data. Later, using the

temperature-voltage correlation for particular thermocouple, the data can be con-

verted back into temperature.

4.1.5 Safety accessories

For any high temperature experiment the safety of the working personnel is must.

For the safety purposes, the main accessories used are as follows:

• Heat resistant gloves

• Apron

• Goggles and face shield

• Breathing masks

• CO detector

A good amount of CO is produced as a by-product of carbothermal reduction re-

action. If inhaled in large quantity, it can cause giddiness, lose of sight, vomiting

etc. Thus a CO detector (product supplied by M/s Cole Parmer Instrument Com-

pany, U.S.A) is used to give a audio-visual warning signal. Using this instrument,

the working personnel can maintain a good distance from the furnace to avoid the

inhalation of CO.∗Data logger is a sort of mini computer with a programmable chip and a hard drive to store the

online data.

Chapter 4 55

4.2 Experimental procedure

Before starting the experiment, an intimate mixture of reactants (boric acid and

graphite/petroleum coke) is prepared. The composition of the product is highly

dependent on temperature and the initial molar ratio of C and B2O3 [7, 63]. Ther-

modynamic study shows that B2O3(s) is highly hygroscopic in nature. Chemically

bounded water (as H3BO3) may react with C, which results in the reduction of the

amount of feed C available for the reaction with B2O3. So, it is first dried in an

oven for about 3-4 hours at a temperature of about 150-200 oC so as to remove

the bounded water with it. Once the mixture is ready to charge in the furnace,

electrode and its assembly along with sighting tube assembly is fixed in the fur-

nace. Before charging the mixture into the furnace, exhaust fan is switched-on and

thermocouples are inserted at the desired locations and distance from the core into

the pockets provided through furnace. After this, material is charged into the fur-

nace carefully without disturbing the heating electrode and sighting tube assembly.

Thermocouples are connected to data logger via thermocouple amplifier to record

the data. Water pump is switched-on for the cooling of electrode holder and then de-

sired power is supplied to the resistance-heating furnace using variac wheel. Variac

wheel is rotated as per the power supply requirement. Heat generated at the surface

of the electrode is transferred to the surrounding reacting material. With time the

reacting mass becomes hard, sticky and viscous and the level of top surface starts

receding toward the heating electrode. Thus we keep on adding fresh charge from

the opening provided at the top of the furnace. During the experiment, depending

upon the power supply, the core temperature of the furnace varies between 2200-

3000 K. CO produced during the early stage of heating comes out at the top of the

furnace where it gets burnt. At later stage in the heating CO is gets entrapped in the

region around the core because of the formation of viscous mass. As the viscous

mass has the least porosity, so poking is done intermittently to provide a passage

for CO to go out. Otherwise, the pressure because of CO accumulation keeps on

Chapter 4 56

building inside the furnace which may lead to hazardous situations. First, this may

results in the lateral movement of the electrode holding assembly that ultimately

results into the loose connection between electrode holder and graphite rod (heating

electrode). This loose connection further leads to sparking at the joints that may

cause the electrode breakdown during the experiment. Second, due to the presence

of gas around the core, it pushes the reacting mass away from the core and there is

swelling of the reacting mass. It has been observed during experiments that in such

situation the electrode gets consumed/melts down, which again leads to breakdown

of the electrode. Third, from safety point of view, if the entrapped gases come out

at their own then there are chances of spillage of reacting mass because of sudden

bursting. So one has to be careful while conducting experiments and should take

care of all these points as explained above. The CO detector meter when exposed

to these fumes confirms the presence of CO by its audio-visual alarm.

Core temperature is measured using pyrometer which is focused on the electrode

through the sighting tube. Depending upon the power supply, after about an hour

or so, green flames are observed at the top of the reacting mass. These green flames

are the indication of the start of the main reaction and occurs due to the oxidation

of boron oxide gas. After sometime the temperature at core becomes constant at

around 2300-2400 K. Still the firing is done for 4-5 hours. Once the green and blue

flames are diminished at the top of the furnace, which is an indication of completion

of reaction, the power supply through control panel is switched-off thereafter and

the furnace is left for a day for cooling. Cooling pump and exhaust fan are kept

in on position till the temperature at the core reaches around 500-600 K. After the

cooling, the side gate of furnace is opened to collect the samples of B4C formed

from different locations in the furnace. After cleaning, the furnace is prepared again

for the next run.

Boron carbide produced this way (mixing the charge in stoichiometric quantity)

Chapter 4 57

contains about 10-12% of free carbon [29]. This is due to the loss of boron in the

form of B2O2(g) at high temperature. Thus excess of boric acid (10-20% more

then the stoichiometric amount) is added while preparing the raw material mixture

to compensate the losses due to volatilization during the reaction. Boron carbide

with 10-12% free carbon is not suitable for certain applications, e.g., in the nuclear

industry. So, control of the reactant feed B/C ratio and temperature is crucial to

manufacture stoichiometric B4C at temperature above about 2300 K [14].

4.2.1 Chemical analysis

For the analytical investigations of B-C system, there are many techniques avail-

able in literature [2, 5]. These techniques are broadly classified into two major

categories i.e. destructive methods and non-destructive methods for quantitative

analysis. Chemical analysis is a destructive method technique used for total boron,

total carbon and free-boron and free-carbon analysis. The same approach is adopted

in this study also.

As such the sample may have a mixture of B4C, free carbon and boron oxides.

In order to determine the percentage of each constituent in the mixture, the follow-

ing procedure has been adopted. The samples of B4C collected from the various

locations in the furnace after the experiment was done. Boron being a light element,

chemical analysis is the best tool available for the determination of total boron con-

tent of the product sample.

Dissolution step forms the most important step in the chemical analysis. Conven-

tional fusion technique using sodium carbonate is adopted. Boron carbide is totally

oxidized by fusion with alkaline carbonate. The resulting fused mass is dissolved by

HCl. The total boron content is then determined as orthoboric acid H3BO3, com-

plexed by manitol and titrated by soda using potentiometry [2]. In simple words,

the boron in the melt is converted to boric acid with the aid of excess mineral acid,

Chapter 4 58

which is neutralized by NaOH solution to a pH of 7 by using indicator. Mannitol is

added to convert the weakly acidic boric acid to a relatively stronger acid complex

of manitol-boric acid, which is estimated by visual alkalimetry to a phenolphthalein

end-point using standard NaOH solution. Discussed chemical analysis is a well de-

veloped and standardized technique [2, 5]. Steps followed for chemical analysis

are provided by Dr. A.K. Suri [64] in a documented form. The steps followed in

the analysis are

1. Take a measured weight of B4C sample (about 0.13-0.14 gm) in a the plat-

inum crucible.

2. Add Sodium Carbonate (Na2CO3) in sufficient quantity in the above sample

and mix it properly.

3. Keep the lid covered platinum crucible in a muffle furnace at 900oCfor about

an hour. This is to ensure that Na2CO3 can fuse with B4C completely.

4. After 1 hour of heating, take the crucible out of the furnace and let it cool to

reach the room temperature.

5. Take 100-150 ml of deionized water into beaker and heat it till it starts boiling.

6. Put the platinum crucible into the heated beaker to dissolve the diffused mass.

7. Add 1-2 drops of methyl red indicator into the final solution. Appearance

of yellow color corresponds to basic nature of the solution which is due to

the formation of sodium borate. If the color becomes red, it means that the

final solution obtained is acidic in nature and boron is present as boric oxide.

Generally one gets yellow color after the addition of methyl red indicator.

8. If the obtained solution is yellow in color, add HCl solution(1:1 by vol.) to

make the solution red. After it becomes red, boil the solution for 2-3 minutes

to expel CO2.

Chapter 4 59

9. After cooling, titrate the solution against NaOH solution (40 gm solid NaOH

+ 1 liter of deionized water). Continue titration till a neutral color is reached

i.e., neither red nor yellow.

10. Add manitol into the solution which will change the color to pink-red. After

dissolution of manitol, it again becomes neutral.

11. Add phenolphthalein indicator.

12. Titrate the final solution against previously prepared NaOH (in step 9) solu-

tion till the color of solution turns back to slightly pink. Note down the total

volume of NaOH solution consumed in step 9.

Total boron ( % ) = Vol. of NaOH solution consumed ×(

0.011×0.1×100Wt. of sample taken

)

Thus, from the above procedure one can find out the percentage of total boron

present in the sample taken.

For analysis of water soluble boron there is not much difference in the procedure

as discussed above. Following is the procedure to obtain the percentage of water

soluble boron in the sample.

1. Take a measured weight of B4C sample (about 0.13-0.14 gm) in a platinum

crucible.

2. Dissolve the sample into hot deionized water.

3. Filter the above solution into a beaker.

4. With filtrate solution follow the same procedure from steps 7 to 12 as fol-

lowed for ’total boron analysis’. Repeat the same calculation methodology as

discussed above to find out the water soluble boron present into the sample.

Chapter 4 60

Water soluble boron ( % ) = Vol. of NaOH solution consumed ×(

0.011×0.1×100Wt. of sample taken

)Free carbon can modify the physiochemical properties of boron carbide and in nu-

clear applications it is suspected that it carburizes the metallic cladding materials

[65]. It is thus important to determine the amount of free carbon present. Determi-

nation of free carbon in boron carbide using chemical methods are unreliable and

irreproducible / give poor results [65]. For successful estimation of free carbon

in the boron carbide samples, there are number of conditions that need to to be

fulfilled. These conditions are discussed in detail in literature [65, 66]. Seeing the

complexity and time constraints, no chemical analysis for free carbon determination

is carried out.

Chapter 5

Results and discussion

This chapter describes the typical results obtained from model’s simulation. The

temperature profiles and the fraction reacted profiles are explained in detail. Results

have been presented for both 1-D and 2-D models. Predictions have been verified

against the experimental data. Experimental data have been obtained by performing

various experiments which are reported here. The sensitivity analysis of the model

is presented to know the effect of various initial parameters on the process and

to optimize the process. Though the model developed has been solved in non-

dimensional form, the results are converted back into their respective dimensions

while presenting the results. It is advised that reader should go through figure 3.1

in order to understand the various locations in the process for which the results have

been presented.

5.1 Results obtained from 1-D model

Before proceeding to the model’s result for the process, it becomes mandatory to

make the model independent of time-step and grid-size because the dependency

on time-step and grid-size is a major source of numerical errors. In case of fully

explicit method, the ease of solution and less memory requirement comes with an

61

Chapter 5 62

additional drawback of system instability. As we go on refining the grid-size, the

time-step which is a function of grid-size also reduces and at any point if the Courant

stability condition is violated, physically unrealistic results could emerge [56], thus

fully explicit scheme is conditionally stable scheme. So in case of explicit schemes,

time step is dictated by numerical stability but not by the physics of the problem

whereas this is possible in case of fully implicit scheme, which is an uncondition-

ally stable scheme.

If a scheme is unconditionally stable then it does not mean that the solution ob-

tained for any time-step or grid size is a desired result. Unconditional stability

ensures that the errors because of time-step or grid-size are bounded; they will not

grow with time as they do in case of fully explicit scheme. Therefore, check of grid

and time independency is necessary for fully implicit scheme also so as to get more

accurate results. In case of 1-D model, both fully implicit and fully explicit schemes

are being made time-step independent and grid-size independent.

Table 5.1: Simulation parameters used for fully explicit and implicit schemes

Index Simulation Exp-1 Exp-2

Radius of electrode (mm) 17.5 17.5 17.5

Furnace in radius (mm) 180 180 180

Inner wool thickness (mm) 20 20 20

Thickness of furnace steel shell (mm) 3 3 3

Refractory bricks thickness (mm) 300 300 300

Length of furnace (mm) 360 360 360

Furnace heating time (min) 360 340 345

Total material added (kg) 33 32 31

Power input ( kWh) 55.45 106.20 97.82

Chapter 5 63

5.1.1 Comparison between fully explicit and implicit scheme

Simulated results obtained from 1-D mathematical model for fully implicit and

explicit schemes are shown in figure 5.1. From this figure, it is seen (in explicit

scheme) that after a certain time step, in this case more than 0.10162 s, results starts

diverging and system become unstable. Parameters used in the simulation are given

in table 5.1. Grid size is taken as 0.3202 mm. Comparing the results, it can be

-50 0 50 100 150 200 250 300 350 400

0

500

1000

1500

2000

2500

All the profiles are for simulated core temperatureTem

pera

ture

(K

)

Time (min)

Explicit with time-step 0.10162 s Explicit with time-step 0.11799 s Implicit with time-step 10.8 s

Figure 5.1: Effect of fully explicit and implicit scheme on core temperature using 1-D model

concluded that even though the implicit code is run for much coarser time step still

the results obtained are in very much agreement with the results obtained from ex-

plicit scheme. Thus, using the fully implicit scheme we can get the results much

faster on finer grid without losing much accuracy. Therefore, this scheme has been

adopted in all simulation here. Any change in grid size for explicit scheme asks

Chapter 5 64

for the repetition of above process to get the converged solution on new grid. For

convergence criterion, the relative error is considered to be less then 0.1%.

5.1.2 Grid independency

Simulations were carried out using implicit scheme for grid size independency

with the same parameters as given in table 5.1. Figure 5.2 shows the effect of grid

size on core temperature. It can be seen that there is no significant change in the

temperature profile at core for grid size finer than 0.2717 mm so we can say that the

system has converged for this grid size. Hence, for further simulations this size i.e.

0.2717 mm is used as standard grid size.

-50 0 50 100 150 200 250 300 350 400

0

500

1000

1500

2000

2500

Tem

pera

ture

(K

)

Time (min)

3.3854 mm 0.2717 mm 0.1628 mm

Figure 5.2: Effect of grid size on core temperature using 1-D model and fully implicit

scheme

Chapter 5 65

5.1.3 Time independency

Again the simulation study using implicit scheme is carried out to establish the time

step independency using same simulation parameters as described in table 5.1. An

interesting phenomenon can be seen from figure 5.3 that when the value of time step

goes beyond a critical value, the temperature profile bifurcates and starts diverging.

A smaller time step is thus chosen for further simulations. However, there is not

much noticeable difference in the temperature profiles when time step is less than

42 s. Here, in all simulations, 25.42 s time step has been chosen.

-50 0 50 100 150 200 250 300 350 400

0

500

1000

1500

2000

2500

Temperature - time plot for different time steps for implicit scheme

Tem

pera

ture

(K

)

Time (min)

Time step : 60 sec Time step : 41.14 sec Time step : 25.42 sec

Figure 5.3: Effect of time step on core temperature using 1-D model and fully implicit

scheme

Chapter 5 66

5.1.4 Validation of 1-D model

Experiments were conducted following the methodology as explained in section

4.2. The detailed analysis of the results obtained is presented here. Tempera-

ture profiles obtained from experiments are validated with the simulation results.

The product obtained after the experiment was analyzed for the percentage of B4C

present in the samples taken from different locations away from the core of the

furnace. Distance up to which the presence of product is seen through chemical

analysis is also validated with simulation results.

Experiment - 1

All experimental parameters for this experiment are given in the table 5.1 under

’Exp-1’ column and the same parameters have been used for simulation. The power

supply and primary current to the transformer during the experiment is shown in the

figure 5.4. Primary voltage is almost constant around 238 V. Temperature variations

with time during the experiment at various locations are shown in figure 5.5. ”Exp”

and ”Sim” legends indicate experimental and simulated data respectively. Surface

of the heating electrode is labeled as ”core” of the furnace, which is the source of

the heat required for the reaction. Power fed to the system is increased in steps till

100 minutes of the run and then it is kept constant as shown in figure 5.4. Small

fluctuations shown in power supply curve after 100 minutes of run are because of

voltage fluctuations in the main supply. Total power supplied during the experi-

ment is ≈106 kWh for 360 minutes run. Approximately 18-20 kg of raw material

is charged into the furnace at the beginning in one go and rest ≈13 kg is charged

during the experiment with intermittent poking. As explained in chapter 4, pyrome-

ter focused through the sighting tube is used to capture the temperature variations at

the core. Since pyrometer temperature measurement range is in between 1173-3273

K, so the core temperature data at the beginning of the experiment are not available.

It can be seen from the figure 5.5 that temperature of the core is 1600 K after 75

Chapter 5 67

0 50 100 150 200 250 300 350

0

5,000

10,000

15,000

20,000

25,000

Power Primary current

Time (min)

Pow

er s

uppl

ied

(W

)

0

20

40

60

80

100

Prim

ary current (A)

Figure 5.4: Variation in power and primary current supply to the transformer - (Exp. 1)

min, which keeps on rising with time.

One can clearly see from this figure 5.5 that after 120 minutes of the start of ex-

periment, the temperature of the core is nearly 2200 K and temperature at 100 mm

and 115 mm away from the core is 510 K and 360 K respectively. These large

temperature gradients are because of poor thermal conductivity of the precursor

material. It is also obvious from the figure 5.5 that temperature at 100 mm and 115

mm away from the core starts rising only after 100 minutes of the start of experi-

ment. The main reason for this is that when the core temperature is more than 1834

K, the main reaction (reaction 3.3) responsible for B4C formation takes place and

Chapter 5 68

0 50 100 150 200 250 300 350

0

250

500

750

1,000

1,250

1,500

1,750

2,000

2,250

2,500

2,750

3,000

Tem

pera

ture

(K

)

Time (min)

Exp - Core Sim - Core Exp - 100 mm Sim - 100 mm Exp - 115 mm Sim - 115 mm

Figure 5.5: Temperature variations at different locations with time - (Exp. 1)

the CO produced as by-product starts coming out of the system. This hot CO while

coming out from the system gives the heat to the surrounding material which is at

lower temperature and thus gives rise to the temperature at locations away from the

core. After reaching to the top this CO get burnt which again adds to the recovery

of heat into the system.

After the experiment is over, samples are collected from the different locations

and are analysed for the boron content. The results from the chemical analysis

are shown in figure 5.6. Left, right, top and bottom directions are shown in the fig-

ure 5.6 which indicates that the samples are taken from these locations. In case of

Chapter 5 69

samples, taken from the top and bottom side of the core, there is an initial rise in the

total boron content of the sample and then it decreases as one moves away from the

core. Similarly, samples taken from, both, left and right side show the same trend

of decreasing boron content as one moves away from the core. Simulation study

1 2 3 4 5

10

20

30

40

50

60

70

80

90

100

Left

Rig

ht

Bottom

Top

% B

oron

in B

4C a

s pe

r ch

emic

al a

naly

sis

Distance from core (cm)

Average of % B in B4C - Right

Average of % B in B4C - Bottom

Average of % B in B4C - Left

Average of % B in B4C - Top

Conversion radius - simulation

Figure 5.6: Product formation with distance – (Exp. 1)

for the above experiment is carried out using the same simulation parameters as in

experiment and are shown in table 5.1 under column ’Exp-1’. Computed results

under the same experimental conditions are shown in figure 5.5. One can see that as

the power supply is increased, the core temperature also increases with the time and

as soon as the reaction temperature (1834 K) is reached, a drop in the temperature is

observed after 45 minute. As explained earlier in section 1.6.4 (resistance-heating

furnace process), B4C formation is highly endothermic process. So, after the re-

action temperature is reached, there is a sudden temperature drop at the core. This

Chapter 5 70

drop in the temperature indicates that the reaction (reaction 3.3) between B2O3 and

C has begun and B4C is being formed.

In fact, from the fraction reacted profile in figure 5.7, it can be seen that at the

surface of the core all the charge has been converted into B4C in less than 50 min-

utes from the start of the experiment. It should be noted, from figure 5.8, that within

few minutes (about 4 minutes), when temperature becomes favorable for reaction

to occur, the fraction of B4C becomes 1.

Because there is a continuous power supply from the electrode to the material, so

temperature again starts rising. After reaction at the core is completed, no other

sudden drop in temperature is observed in the figure 5.5. Once the power sup-

ply becomes constant, no significant rise in the simulated core temperature is seen

from figure 5.5. Temperature profile becomes flatter. It can also be seen from the

figure 5.7 that the drop in temperature is observed at the same time when the con-

version reaches from 0 to 1. From the enlarged view as shown in figure 5.8, it is

clear that the main reaction takes about 4 min for completion.

Experimentally observed temperature profile at the core is in very good match with

the simulated temperature profile except at the beginning. The simulated profiles of

temperature at 100 mm and 115 mm away from the core do not show the prominent

rise with time as observed during the experiment. Main reason of this discrepancy

in the result is due to neglecting θ-direction effect on the results as it is 1-D model

only. There is a strong effect of θ-direction on temperature which will become evi-

dent in the later section where we present 2-D model’s results. Also, the prominent

drop in core temperature is not that prominent in 2-D results. It should be noted that

pyrometer can measure the temperature above 1200K and not below this. As such

the reliable reading starts above 1400 K. Therefore, experimental core temperatures

are reported after 1400 K in figures 5.5 and 5.10. Due to this, pyrometer is not able

Chapter 5 71

0 50 100 150 200 250 300 350

0

0.2

0.4

0.6

0.8

1

Time (min)

Con

vers

ion

Figure 5.7: Fraction of material reacted with time

25 30 35 40 45 50 55 60 65 70

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (min)

Con

vers

ion

Figure 5.8: Enlarged view of figure 5.7

Chapter 5 72

to detect this sudden drop which is a limitation of the instrument and experimental

conditions under which it is being used. In fact in a carefully performed experiment

in a previous study [12], it has been shown that pyrometer is able to detect this

fluctuation in temperature though not prominent.

From simulated results it is clear that one can get boron carbide formation till 3.8

cm distance away from the core surface for experiment one i.e ’Exp-1’. Simulation

results will always give formation of boron carbide as 100% after which it becomes

zero due to the reaction kinetics which has been adopted in the modeling. However,

in experiments this is not the case. But it seems model and experimental results

both show that after some distances product formation is negligible. Experiment

results show that formation of boron carbide becomes negligible between distance

3 and 4 cm away from the core. We say negligible formation of boron carbide when

percentage of boron carbide experimentally below 20%. Simulation results at 3.8

cm distance from the core show that B4C formation is zero which is again due to

the reaction kinetics. It is obvious that this reaction kinetics is not perfect and one

has to put the proper reaction kinetic expression in order to get the better agreement

between the simulated and experimental results. However, it should be noted that

qualitatively still the present reaction kinetics gives reasonable results. Therefore,

in absence of any other reaction kinetics for this process, the present expression for

the reaction kinetics seems good enough.

Experiment - 2

Figure 5.9 shows the power and current supply curves for experiment two. Raw

materials used are in stoichiometric ratio. Simulation parameters are shown in ta-

ble 5.1 under ’Exp-2’. Trend of experimental temperature profiles and B4C fraction

can be explained in a similar way as it is done for experiment-1 above. Therefore,

they are not repeated again. Figure 5.10 shows a comparison between computed

and experimental temperature profiles at various locations. One can clearly see

Chapter 5 73

0 50 100 150 200 250 300 350

0

5000

10000

15000

20000

25000

Time (min)

Pow

er s

uppl

ied

(W)

current Power

Prim

ary current (A)

Figure 5.9: Power supply and primary current variation with time – (Exp. 2)

from figure 5.10 that the core temperature trend predicted by the model is in good

agreement with the temperature profiles obtained from the experiment. The trends

obtained from the model at other positions are also in reasonable agreement with

the experimental values. As such there is a variation in experimental core tempera-

ture which is due to having a difficulty in maintaining constant power supply during

the experiment. The small deviation in computed results, besides the θ-direction

effect, may be because of the errors in the material properties like thermal con-

ductivity and specific heat taken for the simulation, porosity variation, which is

dynamic with both temperature and position in the furnace, poking of the charge

and addition of the fresh charge into the furnace during the experiment etc., which

are not accounted in the model. As such results obtained from 1D model are quite

encouraging lending a good support to the model. Computed fraction reacted or

Chapter 5 74

0 50 100 150 200 250 300 350 400

500

1000

1500

2000

2500

Exp - Core Sim - Core Exp - 80 mm Sim - 80 mm Exp - 100 mm Sim - 100 mm Exp - 115 mm Sim - 115 mm

Tem

pera

ture

(K

)

Time (min)

Figure 5.10: Temperature variation at different locations with time – (Exp. 2)

1 2 3 4 5

10

20

30

40

50

60

70

80

90

100

Average of % B in B4C - Top

Average of % B in B4C - Left

Average of % B in B4C - Right

Average of % B in B4C - Bottom

Conversion radius - simulation

Rig

ht

Left

Bottom

Top

% B

oron

in B

4C a

s pe

r ch

emic

al a

naly

sis

Distance from core (cm)

Figure 5.11: Product formation with distance – (Exp. 2)

Chapter 5 75

Table 5.2: Simulation parameter used for 2-D model

Index 2-D Simulation Exp-3 Exp-4

Radius of electrode (mm) 17.5 13.5 17.5

Furnace in radius (mm) 180 205 180

Inner wool thickness (mm) 20 30 20

Thickness of furnace steel shell (mm) 3 3 3

Refractory bricks thickness (mm) 300 300 300

Length of furnace (mm) 360 360 360

Grid size in r-direction (mm) 0.8125 0.8125 0.8125

Grid size in θ-direction (180o = π radians) 0.36o 0.36o 0.36o

Time step for simulation (s) 20 20 20

Furnace heating time (min) 250 205 285

Total material added (kg) 32 41 33

Power input (kWh) 88.75 27.96 86.35

formation of boron carbide along with experimental data for experiment two are

shown in figure 5.11.

5.2 Results obtained from 2-D model and its validation

Like 1-D model, 2-D model is also tested for grid size and time step independency

before obtaining the desired results. Most of the findings which are true in 1-D case,

apply to 2-D case also. Therefore, these results are not reported here again. Fully

implicit scheme has been used for 2-D model. Standard simulation parameters are

used in 2-D model and they are mentioned in table 5.2. 2-D model is computa-

tionally intensive and takes a long time to get the results. Therefore, only salient

Chapter 5 76

0 50 100 150 200 250

0

500

1000

1500

2000

2500 Theta = 180o

Theta = 0o

Tem

pera

ture

(K

)

Time (min)

At core (theta = 0o)

At core (theta = 180o)

Inner periphery (theta = 0o)

Inner periphery (theta = 180o)

9.45 cm (theta = 0o)

9.45 cm (theta = 180o)

6.45 cm (theta = 0o)

6.45 cm (theta = 180o)

Figure 5.12: Typical 2-D plot for temperature variation with time at different locations, as

obtained from 2-D model

-20 0 20 40 60 80 100 120 140 160 180 200

0.0

0.2

0.4

0.6

0.8

1.0

2.87 cm away from core 2.93 cm away from core 3.00 cm away from core

Con

vers

ion

Angle

Figure 5.13: Angular variation in product formation at various locations, as obtained from

2-D model

Chapter 5 77

results have been presented in this section. 2-D results have also been compared

with experiments and 1-D results.

Figures 5.12 and 5.13 show the typical temperature and conversion profiles ob-

tained from 2-D mathematical model. The simulation parameters used are given

in table 5.2 under the heading ’2-D simulation’. Stoichiometric composition of the

raw material is used for computational purpose. Figure 5.12 shows that with time

there is an increase in temperature at all locations in the furnace. Maximum temper-

ature is achieved at the core of the furnace. This is because of the continuous power

supply to the heating electrode. Once the temperature at the core becomes favorable

for reaction i.e, 1834 K, a sudden temperature drop is seen at the core temperature

profile. This drop in temperature indicates that the formation of boron carbide is

taking place. It is already discussed that reaction 2.4 is highly endothermic and

as this reaction occurs, it will absorb an enormous amount of heat. This leads to

sudden temperature drop at points where reaction occurs. Because of the continu-

ous heat supply to the core, temperature keeps on increasing and the reaction at the

neighboring node starts. Temperature reduces very fast as we move away from the

core. This is because of the poor thermal conductivity of the reacting charge. From

figure 5.12, one can see that temperature at inner periphery near the open boundary,

i.e., at θ = 180o, is more than what is computed at θ = 0o. This is because of the

heat recovered due to CO burning at the open boundary. Trends of all the tempera-

ture profile can be explained in a similar way as it is done for 1-D results in section

5.1.4.

However, it should be noted that sudden drop in core temperature, during the re-

action, is not that much as it is in 1-D model which is clearly showing the effect

of second dimension (i.e. θ-direction) on heat transfer mechanism. Same finding is

true for other temperature profiles at other locations.

Chapter 5 78

Also, it should be noted that 1st drop in the core temperature is observed at longer

time than the corresponding 1-D model, though there is not a big difference in power

supply especially in the starting of experiment. This point will be discussed further

during the validation of 2-D model.

Figure 5.13 shows the angular variation of conversion at different locations in the

furnace at the end of the computational run. It is clear that the charged material is

completely reacted till 2.87 cm away from the core. Angular variation in conver-

sion can be seen in the fraction profile which is at 2.93 cm away from the core. As

we move from θ = 0o to 180o, there is a decrease in conversion but near the open

boundary again the conversion rise is seen. Heat carried away by CO diffusion is

responsible for this partial increment in conversion. Also, heat is recovered at the

top surface by burning CO.

Experiment - 3

Figure 5.14 shows the variation in experimental and computed temperature with

time at different locations in the furnace. Raw material used is in stoichiometric

ratio. Other simulation parameters used in the calculations are given in table 5.2

under ’Exp-3’ column. One can see very clearly from this figure that there is an

excellent match between the simulated and experimental temperature profiles at the

core. The deviation of computed results from experimental data is almost negligible

considering the magnitude of the temperature and complex nature. Though the core

temperature profile is simulated at two different θ values but there is no visible dif-

ference in the core profile with respect to θ-direction at the core which is expected as

core is a source of temperature. From the computed results as shown in figure 5.14,

one can see that there are little humps in the temperature profile at the core which

may be due to the uncertainties in the material properties like thermal conductivity

and specific heat taken for simulation. Also, the power supply considered for com-

puted results is taken from experimental data which itself has some fluctuations.

Chapter 5 79

The trends obtained from the model at other positions also agree reasonably with

the experimental values. This excellent match of temperature between the exper-

0 50 100 150 200

0

500

1000

1500

2000

2500 113o

20o

67o

Exp - Core (113 o)

Sim - Core (180 o)

Sim - Core (113 o)

Exp - 94 mm (67 o)

Sim - 94 mm (67 o)

Exp - 184 mm (20 o)

Sim - 184 mm (20 o)

Tem

pera

ture

(K

)

Time (min)

Figure 5.14: Temperature variation at different locations with time – (Exp. 3)

iment and theory at various locations in the furnace clearly demonstrates that this

complex process has been modeled quite successfully in terms of heat transfer and

all the assumptions which have been made in developing the model are reasonable.

Certainly, these results are lending a good support to the developed model.

It is thought that we should compare the results of both the models 1-D and 2-D

against experimental data so that one can know how much deviation these model

have with experiment. The idea of this comparison is that if there is no significant

difference between 1-D and 2-D results with experiments than sensitivity analysis /

optimization of the process could be done using 1-D model as 2-D model is highly

computational intensive and takes very long time.

Chapter 5 80

5.2.1 Comparative study of 1-D and 2-D models with experimental results

Figure 5.15 and 5.16 shows the computed temperature plots using 1-D and 2-D

model and the computed results are compared with experimental data. The simu-

lation study is done with same parameters as discussed in table 5.2 under column

’Exp-3’. Experimental data is taken from figure 5.14 as discussed in section 5.2

under the heading ’Experiment - 3’.

Comparison between the computed results obtained from 2-D model and experi-

ment 3 has already been explained. One can see from the figure 5.15 that in early

stage there is a reasonable match of 1-D model result with experimental data and

with computed 2-D results. The only discrepancy appearing in 1-D and 2-D result

is the shift of temperature dip observed during endothermic reaction. The probable

caused for this has already been explained in section 5.2. However, the other pos-

sible reason for higher temperature at the early stage in case of 1-D model is that

we have considered conduction resistance all through the inner periphery of heating

furnace. So heat is lost due to conduction only, whereas in case of 2-D model be-

sides the enclosed furnace periphery, there is an open boundary from which the heat

is lost. Also, heat flow from other dimension (θ-direction) is taking place which is

giving more realistic profiles. In the later stages, there is almost no difference in

temperature profiles obtained using 1-D and 2-D model, which are in an excellent

match with experimental results.

Also, from figure 5.16 it is clear that the computed results obtained using 1-D

and 2-D models, away from the reacting core, have a similar trend of temperature

variation. There is a minor temperature difference in the computed profiles obtained

at 94 mm away from the core. The reason for this has been explained above. How-

ever, at 184 mm away from core there is almost no difference between the computed

results obtained from both the models. As 2-D model is computationally more in-

Chapter 5 81

0 50 100 150 2000

500

1000

1500

2000

2500

113o

67o

20o

Exp - Core Sim - Core (1-D)

Sim - Core (2-D, 113 o)

Sim - Core (2-D, 180 o)

Tem

pera

ture

(K

)

Time (min)

Figure 5.15: Comparison of 1-D and 2-D computed core temperature with experimental

data

0 50 100 150 200

300

350

400

450

500

550

Exp - 94 mm (67 o) Sim - 94 mm (1-D)

Sim - 94 mm (2-D, 67 o)

Exp - 184 mm (20 o) Sim - 184 mm (1-D)

Sim - 184 mm (2-D, 20 o)

Tem

pera

ture

(K

)

Time (min)

20o

67o

113o

Figure 5.16: Comparison of 1-D and 2-D computed temperatures away from core with

experimental data

Chapter 5 82

tensive compared to 1-D model, so this comparative study gives us a confidence that

1-D model can be used for faster results for sensitivity analysis without any serious

drawback. Finally, the experiment is conducted based on the predictions made with

sensitivity analysis and the results are discussed later in section 5.5. Here again, 1-

D and 2-D models are validated with the experimental data to have more confidence

in the developed models.

5.3 Yield analysis

Yield of B4C formation is an important issue for the optimization of manufactur-

ing process. Theoretical power requirement for B4C production is approximately

40,000 kJ/kg of B4C [67]. Considering the percentage of B4C as per our chem-

ical analysis in the samples taken after experiment and taking account of porosity

effects, the actual power requirement for B4C production comes out to be approxi-

mately 1,22,000 kJ/kg of B4C. This is almost 3 times the theoretical power require-

ment.

Not enough experimental data, on industrial process, are available in the litera-

ture in order to calculate the power requirement for the formation of boron carbide.

However, the data made available through a private communication [64] shows

that one can get about 12 kg B4C from 75 kg charge. Considering these data, the

actual power requirement at large scale production of B4C is about 2,57,000 kJ/kg

of boron carbide which is nearly 8 times the theoretical power requirement. This

indicates that industrial process can be improved substantially if those operations

are adopted which are used at laboratory scale. Probably, it may not be possible to

adopt the all operations which have been used at laboratory scale due to constraints

in the working environment of the industry. However, it is felt that some operations

certainly can be implemented at industrial scale to improve the process which may

enhance its efficiency. These observations are made based on limited number of

Chapter 5 83

Table 5.3: Standard data used for sensitivity analysis

Index Values

Radius of electrode (mm) 17.5

Furnace in radius (mm) 180

Inner wool thickness (mm) 20

Thickness of furnace steel shell (mm) 3

Refractory bricks thickness (mm) 300

Length of furnace (mm) 360

Furnace heating time (min) 380

Total material added (kg) 33

experiments.

As such sensitivity analysis (using mathematical model) shows that the produc-

tivity / yield is affected by charge composition and mode of power supply. It further

shows that that the yield can be improved further even at laboratory scale exper-

iments. Therefore, mathematical model becomes a very useful tool to study this

dangerous and unexplored process.

5.4 Sensitivity analysis / optimization of the process

Once both the computer codes (1-D and 2-D) are validated against experiments,

sensitivity analysis is done in order to optimize the process. Simulations are con-

ducted for various parameters like input power supply, mode of power supply and

initial charge composition on the process. The results are presented in this section.

The standard parameters used for sensitivity analysis are given in table 5.3.

Chapter 5 84

5.4.1 Effect of input power supply

Figure 5.17 shows the core temperature variations with time for different power in-

puts. The corresponding product formation profile is given in figure 5.18. All the

results obtained are for stoichiometric charge composition i.e., B2O3 : C :: 1.69 : 1

with no amount of B4C into the initial charge. It is clear from figure 5.18 that at

low power supply i.e., at 63.54 kWh, the conversion radius ∗ is less in comparison

with other two cases. It is because, as the temperature reaches to reaction tempera-

ture, it drops and goes through a dip in the profile which takes longer time to reach

to the reaction temperature again as it is seen in figure 5.17.

As discussed earlier in chapter 1, the conversion rate is temperature controlled

phenomenon. So, lower power supply demands the furnace to be run for longer

time. One can see from figure 5.17 that as the power supply is increased from 63.54

to 129.56 kWh, the temperature at the core and other places in the furnace is also

increased. Thus the effect of higher power supply is directly reflected in the con-

version radius. If the power supply is not sufficient then it ends up just heating of

the charge and no product is formed, because the charge does not to reach to the

reaction temperature.

5.4.2 Effect of mode of heating cycle

Figure 5.19 shows three different modes of power supply keeping the total

power supply and run time constant. In case of regular power supply, which is the

usual case during experiments, the power is gradually increased from 0 to 16350

watts in the duration of 150 minute and then kept constant for 3 hours. Afterwords

the power is further increased 20700 watts in 20 minutes and kept constant. In case

of linear power supply the power is gradually kept increasing with time whereas∗Conversion radius is the distance from the reacting core surface up to which the product is

formed.

Chapter 5 85

0 100 200 300 400

0

500

1000

1500

2000

2500

3000

63.54 KWH 121.93 KWH 129.56 KWHT

empe

ratu

re (

K)

Time (min)

Figure 5.17: Temperature variation with time for different power input

60 70 80 90 100 110 120 130

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

Con

vers

ion

radi

us fr

om th

e co

re s

urfa

ce (

cm)

Power Supply (kWh)

Figure 5.18: Effect of power supply on conversion radius

Chapter 5 86

0 40 80 120 160 200 240 280 320 360 400 4400

5000

10000

15000

20000

25000

30000

Regular Step Linear

Time (min)

Pow

er (

Wat

ts)

Figure 5.19: Different modes of power supply keeping the total power supply and time

constant

in case of constant power supply it is suddenly increased to 15000 watts and kept

constant thereafter till the end of the run as shown in figure 5.19.

It can be seen from figure 5.20, in case of linear power supply the core tempera-

ture profile keeps on rising with time whereas in case of constant and regular power

supply it becomes almost flat after about 150 minute. For all the three cases the

total power supply is 63.58 kWh. From figure 5.21, one can see that in case of lin-

ear power supply the percentage conversion is higher. This indicates that for higher

product formation, adoption of linear power supply mode is beneficial.

Chapter 5 87

0 100 200 300 400

0

500

1000

1500

2000

2500

Linear supply Regular supply Step supply

Tem

pera

ture

(K

)

Time (min)

Figure 5.20: Effect of mode of power supply on core temperature

Regular Step Linear

7.4

7.6

7.8

8.0

8.2

8.4

% c

onve

rsio

n of

rea

ctan

t

Mode of power supply

Figure 5.21: Effect of mode of power supply on percentage conversion

Chapter 5 88

5.4.3 Effect of varying charge composition

Effect of excess of B2O3 in the initial charge

Simulations are performed for three different initial boric oxide percentages besides

stoichiometry content in the charge. The results are shown in figures 5.22 and 5.23.

One can see from figure 5.22 that in all the cases there is a sudden drop in the tem-

perature during the reaction almost at the same time. Figure 5.23 shows that as the

boric acid content are increased in the charge from 0 to 20% there is only a slight in-

crease in conversion radius. When the boric acid content is increased to 30%, there

is almost no change in the conversion radius. This is because with the increased

B2O3 content the amount of graphite/charcoal per unit control volume is decreased

and hence the endothermic requirement for the reaction per unit control volume is

decreased. Also, increased B2O3 results in the change in physical properties of the

reacting mixture and thus affects the temperature and reaction kinetics.

Effect of B4C

Figure 5.24 shows the temperature profiles at the center of the furnace with time for

five different initial percentages (weight) of the boron carbide in the charge. Except

the initial boron carbide content, all other simulations parameters are kept constant

for all the runs. It is clear from the fact that as the initial percentage of boron carbide

is increased from 0% to 50% the time for the first drop increases from 29 minutes to

36 minutes. This may be explained by the fact that boron carbide being a refractory

material has higher specific heat content compared to the other two materials (B2O3

and C). The thermal conductivity of boron carbide is higher than the other two ma-

terials. Because of these properties, as one increase the boron carbide content in the

initial precursor material, the time needed for heating up the material to the reaction

temperature is much higher. As the initial boron carbide content is increased, the

reacting material per control volume decreases and hence, the endothermic require-

Chapter 5 89

0 100 200 300 400

0

500

1000

1500

2000

2500

Temperature - time plot for various B2O

3feed composition

Tem

pera

ture

(K

)

Time (min)

No excess 10% B

2O

3

20% B2O

3

30% B2O

3

Figure 5.22: Effect of excess B2O3 on core temperature

0 10 20 30 40

2.6

2.8

3.0

3.2

3.4

Con

vers

ion

radi

us (

cm)

% excess of B2O

3 into feed

Figure 5.23: Conversion radius with excess B2O3

Chapter 5 90

0 50 100 150 200 250 300 350 400 450

0

500

1000

1500

2000

2500

As per stoichiometry 10 % excess of B

4C

20 % excess of B4C

30 % excess of B4C

40 % excess of B4C

50 % excess of B4C

Tem

pera

ture

(K

)

Time (min)

Figure 5.24: Effect of initial B4C on computed temperature

0 10 20 30 40 50 60 70 80

2.0

2.5

3.0

3.5

4.0

4.5

Con

vers

tion

radi

us fr

om c

ore

surf

ace

(cm

)

% excess of B4C into feed

Figure 5.25: Effect of initial B4C content on final product formation

Chapter 5 91

ment per control volume also decreases correspondingly.

Figure 5.25 shows the effect of initial percentage of boron carbide present in the

mixture on the total formation of boron carbide. One can clearly see from the graph

that as the initial percentage of boron carbide increases, the conversion increases.

The conversion of the reactant into the product is about 2.7 cm from the core surface

at 0% initial boron carbide while, the same is 4.1 cm when the initial boron carbide

content is 50% in raw materials. This may be, due to the relatively increased in ther-

mal conductivity of the boron carbide as compared to the boron oxide and carbon.

As the reaction of boron carbide formation is dependence on heat transfer also, if

the thermal conductivity of the mixture increases the conversion also increases as

one can gets favourable reaction temperature at longer distance from the core.

0 50 100 150 200 250 300

0

300

600

900

1200

1500

1800

2100

2400

2700

3000

3300

Experimental temperature profile 1-D simulated temperature profile

2-D simulated temperature profile (113 o)

Tem

pera

ture

(K

)

Time (min)

Figure 5.26: Comparison between 1-D and 2-D model with experimental data

Chapter 5 92

5.5 Comparison of experimental results with sensitivity analysis

Figure 5.26 shows the comparison between the computed results obtained from

both, 1-D and 2-D, mathematical models and experimental results for linear case

power supply case as discussed above. Although both the models have been vali-

dated against experiments and sensitivity analysis has also been done, it is thought

that in order to have more confidence in using the developed model in real practice

to pick up any one optimization condition from this sensitivity analysis and verify

the theoretical results by doing an experiment under the similar conditions in which

the theoretical predictions have been made. Therefore, we have picked up the mode

of power supply condition (section 5.4.2) in which it is mentioned that linear power

supply gives better yield / product. For this purpose we have taken standard param-

eters mentioned in table 5.2 under ’2-D simulation’ column. Total power in this

case is 88 kWh, which is taken in linear mode. Experiment is done under similar

conditions and we tried to keep the total power in linear mode, same as in theory.

Experimental parameters are given in table 5.2 under ’Exp-4’ column. One can

see from this table that there is almost no difference between theoretical and exper-

imental parameters. Results are reported below. The data under the column ’Exp-4’

given in table 5.2 are used for simulating the results for 1-D model. In both the

simulations (1-D and 2-D), linear power supply is assumed. Figure 5.26 shows the

comparative plots only for the core temperature with time. One can see that there

is an excellent match between the results obtained from 2-D model and experiment.

However, comparison between 1-D computed results and experiment is reasonable.

After the completion of the reaction 1-D model also gives good results.

Figure 5.27 shows a comparison between the optimized computed fraction reacted,

obtained using 1-D and 2-D models with experiment which has been performed un-

der same conditions as the computed results. One can see that the conversion radius

obtained from 1-D model is little better than the 2-D model. The probable reason of

Chapter 5 93

1 2 3 4 5

0

10

20

30

40

50

60

70

80

90

100

Left

Bottom

Top

% B

oron

in B

4C a

s pe

r ch

emic

al a

naly

sis

Distance from core (cm)

B in B4C - Bottom

B in B4C - Right

B in B4C - Top

B in B4C - Left

1-D model result 2-D model result (Bottom) 2-D model result (Top) 2-D model result (Left/right)

Figure 5.27: Comparison between 1-D and 2-D model results for conversion radius

these discrepancies have already been discussed in section 5.2 and therefore, they

are not reported here. It is clear from figure 5.26 that the core temperature as per

1-D model is higher than that of the 2-D model core temperature so in 1-D model

the temperature penetrates deeper into the bed. Also one can see that in case of

2-D, better conversion radius is achieved from the bottom side. This is because the

top side of the core is loosing much heat toward the open boundary and hence is

the reason for better conversion from bottom side. There is not much change in the

conversion radius from the left/right side. Since axisymmetry is considered so ’left

side’ and ’right side’ conversion radius comes out to be same. Nevertheless, it is

good to see such a pleasing match when a set of computed optimized conditions are

Chapter 5 94

verified against experiment. Indeed, it is giving a much desired boost in applying

these models in actual practice. We consider it as big milestone in developing a

realistic model which can be used in practice.

Chapter 6

Conclusions and scope of future work

6.1 Conclusions

From this work it is conclude that

• One and two-dimensional mathematical models have been developed suc-

cessfully considering conduction, convection and radiation heat transfer mech-

anisms.

• Both the models give reasonable predictions with respect to temperature and

boron carbide formation. However, two-dimensional model gives more real-

istic results.

• Many high temperature experiments have been performed in order to validate

both the models with respect to temperature and boron carbide formation.

Both the data match very well lending a good support to the developed mod-

els.

• From sensitivity analysis one can say that having some amounts of boron car-

bide in the starting material increases the heat transfer inside the furnace and

hence the conversion is increased. Increase in the power supply increases the

95

Chapter 6 96

conversion of boron carbide. Simulation results indicate that linear heating of

the charge gives better conversion.

• In order to get more confidence in the developed model so that they can be

used in real situations, one of the optimized conditions from the sensitivity

analysis was reproduced experimentally and it is found that both the results

are matching very well.

• Preliminary analysis of limited laboratory scale experiments shows that yield

can be improved in substantial way in the industry if proper procedure is

adopted in the production of boron carbide as they were adopted during the

laboratory scale experiments.

6.2 Scope for future work

• Models can be improved as per the availability of more accurate property data

and reaction kinetics to achieve the better results.

• The future work should be directed toward a systematic study of improving

the process yield further.

Appendix A

Modeling details

A.1 Finite volume discretization technique for PDE’s

To illustrate the control volume discretization scheme, an example of an unsteady

heat conduction with source term in polar coordinates is considered.

ρCP∂T

∂t=

1

r

∂

∂r

(rk

∂T

∂r

)+

1

r

∂

∂θ

(k

r

∂T

∂θ

)+

o

S (A.1)

The grid and control volume in r − θ coordinates are shown in figure A.1. In z-

direction the thickness of the control volume is assumed to be unity. The discretized

equation is obtained by multiplying equation A.1 by r and by integrating it with

respect to r and θ and over the time interval from t to ∆t in the control volume. This

way we get a volume integral, where rdrdθ represents the volume of the element

with unit thickness. The discretization equation can be written as

aP TP = aETE + aW TW + aNTN + aSTS + b (A.2)

Where

aE =ke∆r

re (δθ)e

aW =kw∆r

rw (δθ)w

97

Appendix A 98

aN =knrn∆θ

(δr)n

aS =ksrs∆θ

(δr)s

aoP =

ρCP ∆V

∆t

b = Sc∆V + aoP T o

P

aP = aE + aW + aN + aS + aoP − Sp∆V

Here ∆V is the volume of the control volume. ∆V not necessarily be equal to

rP ∆θ∆r, unless P lies midway between ’n’ and ’s’.

Figure A.1: Control volume in polar coordinates

A.2 Correlation for property data

Temperature dependent variation is considered for the property data for the species

namely B4C, B2O3, C and CO. For each species the correlation was developed

from the literature available [12].

Appendix A 99

A.2.1 Enthalpy correlations

Here HB2O3 , HC , HBC and HCO represents the enthalpy of boric acid, car-

bon/graphite, boron carbide and carbon mono-oxide respectively.

For 298 ≤ T ≤ 723

HB2O3=

(−4170 + 8.73T + 12.70 × 10−3T2+

1.31 × 105

T

)−305400

×4.184 × 1000

For 3000 > T > 723

HB2O3= (−7590 + 30.50T)−305400×4.184 × 1000.0

For 3000 > T ≥ 298

HC=

−1972 + 4.10T + 0.51 × 10−3×T2+

2.10 × 105

T

×4.184 × 1000

For 3000 > T ≥ 298

HBC=

(−10690 + 22.99T + 2.70 × 10−3T2+

10.72 × 105

T

)−13800

×4.184 × 1000

For 3000 > T ≥ 298

HCO=

(−2100.0 + 6.79T + 0.49x10−3T2+

0.11x105

T

)−26416.0

×4.184 × 1000

A.2.2 Thermal conductivity correlations

Here KB2O3 , KC and KBC represents the thermal conductivity of boric acid,

carbon/graphite and boron carbide respectively.

Appendix A 100

For 298 < T < 700 KB2O3= 1.07

For 700 ≤ T < 1250

KB2O3= 14.10751 − 0.04231T + 4.33252 × 10−5T2−1.34861 × 10−8T3

For T ≥ 1250 KB2O3= 2.65

For 298 ≤ T < 1000

KC=(4.12045 × 10−4+7.96255 × 10−5T − 8.26172 × 10−8T2+3.31235 × 10−11T3

)×100

For 1000 ≤ T < 1300 KC= 0.027 × 100

For 1300 ≤ T < 1480

KC=(−4.51076 + 0.00975T− 6.99371 × 10−6T2+1.67613 × 10−9T3

)×100

For 1480 ≤ T < 1700

KC=(−89.81462 + 0.17062T − 1.07898 × 10−4T2+2.27232 × 10−8T3

)×100

For 1700 ≤ T < 2400

KC=(40.6564 − 0.08689T + 6.96269x10−5T2−2.47679x10−8T3+3.30059x10−12T4

)×100

For 298 < T < 3250

KBC =(0.09061 − 7.78777 × 10−5T − 1.01299 × 10−8T2+6.94771 × 10−11T3

)×4.184 × 102

+(−4.575927 × 10−14T4+1.24536 × 10−17T5−1.25502 × 10−21T6

)×4.184 × 102

For thermal conductivity of gas the correlation used is

KTG=

(−0.00815 + 0.00101T− 5.319 × 10−7T2+1.73778 × 10−10T3

10

)

Appendix A 101

A.2.3 Specific heat correlations

Here CPC, CPB2O3

and CPBCrepresents the specific heat of carbon/graphite,

boric acid and boron carbide respectively. For 298 < T < 3000

CPC=(0.04245 + 6.60261 × 10−4T − 3.57548 × 10−7T2+7.07561 × 10−11T3

)×4184 × 12

For 298 ≤ T ≤ 723

CPB2O3=(−0.1518 + 0.00181T− 2.42373 × 10−6T2+1.27257 × 10−9T3

)×4184 × 68

For 723 < T < 1000

CPB2O3=(0.18398 + 2.53222x10−4T

)×4184×68

For 1000 ≤ T < 3000

CPB2O3= 0.4372 × 4184.0 × 68

For 298 ≤ T ≤ 2300

CPBC=(0.12376 + 3.67825x10−4T − 5.41582x10−8T2−4.82847x10−12T3

)×4184 × 52

For 2300 < T < 3000 CPBC= 136000.0

A.2.4 Porosity correlation

Porosity of the reactive mixture is represented by ε. Experiments were con-

ducted to correlate the porosity variations with temperature.

For 298 < T ≤ 725

ε=(0.51088− 3.04455 × 10−4T + 7.25663 × 10−8T2

)For 725 < T ≤ 1834

ε= 0.23895 + 1.20336 × 10−4T

Appendix A 102

For 1834 < T ≤ 1960

ε= −1.58536 + 0.00112T

For T > 1960 ε = 0.59

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