1. Development and validation of
two-dimensional mathematical model of
boron 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 Science
Bangalore 560 012 (India)
2006
3. 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
4. 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.
5. 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.
6. 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.
11. 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
12. 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
13. 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
14. 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
15. 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
16. 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
17. 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
18. 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 (TiB2), 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 o
C, i.e., iron, nickel, titanium
and zirconium. Above 1800 o
C it reacts with nitrogen to give boron nitride. B4C
can be attacked by chlorine at about 600 o
C and bromine attacks it at above 800 o
C,
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 o
C with CO2 to form B2O3 and CO. Boron carbide
has good oxidation properties in the air up to about 600 o
C.
19. 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
+ on1
→ 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.
20. 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 o
C.
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
21. 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
22. 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 o
C, 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
23. 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 o
C.
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
24. 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 = I2
R
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
25. 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
26. 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-
27. 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
28. 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.
29. 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-
30. 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.
31. 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.
32. 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
33. 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.
34. 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 o
C. The B4C-C eutectic temperature was reported
to be 2375 o
C, 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
35. 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
36. 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-
37. 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(−Km
tm
m
) (2.14)
38. Chapter 2 24
By differentiating equation 2.14 with respect to the time, we get
dX
dtm
= Km
m tm
m−1
exp(−Km
tm
m
) (2.15)
Since
rA=Co
dX
dtm
= CoKm
m tm
m−1
exp(−Km
tm
m
)
= ComK Km−1
tm
m−1
exp (−Km
tm
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.
39. 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
40. 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
41. 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
42. 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]:
−
16η2
σT3
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.
43. Chapter 3 29
Here, λT is the total thermal conductivity of the medium which further can be writ-
ten as
λT = λC + λr = λC +
16η2
σT3
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.
44. 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
45. 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
46. 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,eCPg T
∂CCO
∂r
Diffusion in r-direction
+
1
r
∂
∂r
r
16ση2
T3
3κ
∂T
∂r
Radiation in r-direction
+
1
r
∂
∂θ
ke
r
∂T
∂θ
Conduction in θ-direction
+
1
r
∂
∂θ
DCO−air,eCPg T
r
∂CCO
∂θ
Diffusion in θ-direction
+
1
r
∂
∂θ
16ση2
T3
3rκ
∂T
∂θ
Radiation in θ-direction
− ∆Hr (1−ε)
Heat generation
(3.1)
47. Chapter 3 33
Each term in the above equation has units as J
m3−s
.
Where,
– CCO is the concentration of carbon mono-oxide (CO), kgmol/m3
– CP e is effective specific heat of gas-solid mixture, J/kgmol − K
– CP g 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
48. 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 loss
2
3
× 2πroL
ii) For
2π
3
< θ < π ; − ke
∂T
∂r
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Heat loss due to
conv. and radiation
1
3
× 2πroL
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
−
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Heat recovered
due to CO burning
1
3
× 2πroL
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Where,
Heat loss =
2πL (Tref − Tamb)
n
r3
ro
λref
+
n
r4
r3
λSS
+ 1
hr4
; h [53]= 1.32
(TSS−Tamb)
2r4
0.25
⎡
⎣
Heat loss due to
conv. and radiation
⎤
⎦ = σA(Tg
4
− Tamb
4
) + 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
49. 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 as
kg 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)
∂CCO
∂θ
= 0
50. 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) For
2π
3
< θ < π ; CCO
= CCO−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.
51. 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 (o
C) and pressure (atmosphere).
DCO−air =
⎡
⎢
⎢
⎢
⎣
0.0606T1.78
P (VCO)
1/3 + (Vair)
1/3
2
⎤
⎥
⎥
⎥
⎦
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
52. 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.
53. 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)/r2
o
Time (in mass balance) tm tm
∗
= (tmDCO−air,e,i)/r2
o
Diffusion coefficient DCO−air,e
D∗
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
54. 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−air
N∗
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∗
=
16η2
σTi
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 input
2πr∗LkeTi
55. 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
2
3
× 2πr∗LkeTi
ii) For
2π
3
< θ < π ; −
∂T∗
∂r∗
=
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Heat loss due to
conv. and radiation
1
3
× 2πr∗keLTi
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
−
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
Heat recovered
due to CO burning
1
3
× 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 r2
o (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
56. 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) For
2π
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−air
N∗
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
57. Chapter 3 43
• At time t > 0, for r∗
= r/ro (at the surface of electrode)
−
∂T∗
∂r∗
=
Heat input
2πr∗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.
58. 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].
59. 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.
62. 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 o
C, 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
64. 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
65. 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
66. 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 (100o
C to 2500o
C)
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
67. 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
68. 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.
69. 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 o
C 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
70. 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)
71. 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,
72. 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 900o
Cfor 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.
73. 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 × 0.011×0.1×100
Wt. 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.
74. Chapter 4 60
Water soluble boron ( % ) = Vol. of NaOH solution consumed × 0.011×0.1×100
Wt. 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.
75. 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
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