REMOVAL OF IMPURITIES FROM TITANIUM-BEARING SOURCES
AND THEORETICAL AND EXPERIMENTAL
KINETIC STUDIES
by
Jaehun Cho
A thesis submitted to the faculty of
The University of Utah
in partial fulfillment of the requirements for the degree of
Master of Science
Department of Metallurgical Engineering
The University of Utah
August 2016
Copyright © Jaehun Cho 2016
All Rights Reserved
T h e U n i v e r s i t y o f U t a h G r a d u a t e S c h o o l
STATEMENT OF THESIS APPROVAL
The thesis of Jaehun Cho
has been approved by the following supervisory committee members:
Michael L. Free , Chair 05/06/2016
Date Approved
Zhigang Zak Fang , Member 05/06/2016
Date Approved
Amarchand Sathyapalan , Member 05/06/2016
Date Approved
and by Manoranjan Misra , Chair/Dean of
the Department/College/School of Metallurgical Engineering
and by David B. Kieda, Dean of The Graduate School.
ABSTRACT
This thesis work mainly evaluates methods of obtaining pure titanium hydride
from different sources and methods and its theoretical and experimental kinetics using
model formulation.
The first route was to use Direct Reduction of Titanium Slag (DRTS) in which
metallic titanium can be obtained by dehydrogenation of titanium hydride after impurities
are removed. The leaching characteristics of iron removal from the reduced upgraded
titanium slag was studied with mild hydrochloric and boric acids under ambient pressure
and elevated pressure. Under the constraint that 1% (w/w) of titanium hydride loss is the
maximum amount tolerable, 0.1 M hydrochloric acid at 140 °C was found to be the most
effective condition for iron removal (87.63%). A factorial design of experiment for
equation modeling with three main factors (temperature, concentration of hydrochloric,
and boric acids) was performed and associated modeling results were in good agreement
with experimental data.
Additional study was carried out to justify the assumption, which utilized the
evaluation of the effects of three empirical size distributions, the Gate-Gaudin-Schuhmann
distribution, the Rosin-Rammler-Bennett distribution, and the Gamma distribution, on the
fluid-solid reaction kinetics. The expressions for overall conversion rate of entire particle
assemblages were derived mathematically, and calculated by a technical computing
language, “MATLAB”. According to the calculation, the assumption that particle size is
iv
uniform can be valid in the determination of fluid-solid kinetics in the case where the
coefficient of variation (CV) is less than 0.5. Based on the theoretical kinetic study for the
effect of particle size distribution, it was assumed that the effect of particle size distribution
of reduced upgraded titanium slag (UGS) does not have to be considered in the calculation
of kinetics. Based on this calculation, a rate-controlling process can be found and it seems
to follow interfacial reaction controlled kinetics. The activation energy of the reaction was
determined to be 73.9 kJ/mole. Also, the other mechanism of the reaction-controlling
process (solid-state diffusion) is suggested.
An additional way of obtaining pure titanium involves the extraction of titanium
from ilmenite using tannic acid. The experimental and modeling results showed the
feasibility of the new process to obtain titanium.
TABLE OF CONTENTS
ABSTRACT………………………………………………...........………………….…..iii
LIST OF TABLES………………………………………………………………………..vii
LIST OF FIGURES…………………………………………………………......……...viii
ACKNOWLEDGEMENTS……………………………………………………………….x
Chapters
1 INTRODUCTION………………..…………………………………………………...…1
1.1 Direct reduction of titanium slag……………………...………...…......…….1
1.2 Theoretical kinetic study………………………………………….……...…...4
1.3 Selective extraction of titanium from ilmenite using tannic acid………...……6
2 REVIEW OF THE LITERATURE…………………………………………………...….7
2.1 Processes for titanium production……………………………………...…...7
2.1.1 Kroll process…………………………………....…………...….….7
2.1.2 Alternative processes………………………………………...…….8
2.1.3 Extraction of titanium from ores………………………………......11
2.2 Effect of particle size distribution on kinetics…………………..……………12
2.3 Tannic acid……………………………………...……………...……………16
3 LEACHING CHARACTERISTICS OF IRON REMOVAL FROM REDUCED
UGS……………………………………………...………………………………….….18
3.1 Experimental methods………………………………………………….….18
3.1.1 Preparation of reduced UGS by reduction and acetic acid leaching.18
3.1.2 Experimental procedures........................................................……18
3.2 Experimental results and discussion……………………………….………19
3.2.1 Feed analysis……………………………………………………19
3.2.2 Experimental results…………………………………………….21
3.2.2.1 Effect of hydrochloric acid…………………….……….21
3.2.2.2 Effect of boric acid…………………………….……….25
3.2.3 Effect of temperature…………………..……….…………………30
vi
3.2.4 Factorial design of experiments for equation modeling…………32
3.2.5 Removal of iron without acetic acid leaching……..…….......…….37
4 EFFECT OF PARTICLE SIZE DISTRIBUTION ON KINETICS…………………..…40
4.1 Model formulation…………..………………………………………………40
4.1.1 Shrinking unreacted core model………………………………...40
4.1.2 Mathematical formulation for size distribution…...………………44
4.2 Results and discussion……………………………….…………………...….50
4.2.1 Gate-Gaudin-Schuhmann distribution………………………….52
4.2.2 Rosin-Rammler-Bennett distribution…………………………...55
4.2.3 Gamma distribution………...…………………………………….57
4.2.4 Comparison of the effect of three size distributions on the kinetics.59
4.2.5 R2 values for the effect of particle size distributions………………61
4.2.6 Comments on particles with nonbasic shapes…………….…...…63
5 KINETIC STUDY OF REDUCED UGS WITH EFFECT OF PARTICLE SIZE
DISTRIBUTION………………………………………………………………….……64
5.1 Particle size distribution of reduced UGS…………...…………………….…64
5.2 Determination of rate-controlling process of the reduced UGS…...…………66
5.2.1 Interfacial reaction……………………………………………...66
5.2.2 Solid state diffusion……………………………………………….69
5.2.2.1 Mathematical formulation……………………….…….69
5.2.3 Effect of external mass transport………………………………….73
6 SELECTIVE RECOVERY OF TITANIUM FROM LEACHED ILMENITE USING
TANNIC ACID………………………..…………………………………………………78
6.1 Materials and methods…………………......………………………………...78
6.1.1 Preparation of titanium solution from ilmenite……………………78
6.1.2 Synthesis of titanium-tannic acid complex………………….…….78
6.1.3 Methodology of DFT simulation………………………………….79
6.2 Results and discussion……………………………………..………………...79
6.2.1 Experiment……………...………………………………………...79
6.2.2 Simulation…………………………...……………………………83
7 CONCLUSIONS……………………………………………………………………...87
REFERENCES…………………….……………………………………………………92
LIST OF TABLES
Table Page
2.1 Mean, variance, and CV of three distributions……………………………………….14
3.1 Chemical composition of reduced UGS before and after acetic acid washing……….20
3.2 Fe removal % and Ti loss % at different leaching parameters with a solid to liquid ratio
of 1 to 100 and 4 h of leaching time……………...………………………....……......…....24
3.3 Leaching parameters………………………………………………………………...27
3.4 Chemical composition of reduced UGS leached by 0.1 M HCl at 140oC……….….33
3.5 Composition of reduced UGS leached by 0.4 M hydrochloric acid…………………...39
4.1 Mean, variance and CV of three distributions……………………………………....47
4.2 X(t) of three distributions for interface-reaction-controlled system…………………49
4.3 X(t) of three distributions for pore-diffusion-controlled system…………………….51
4.4 Maximum differences of conversion rate between uniform and nonuniform particles
with different distributions, particle shapes, CV, and rate controlling steps…………….54
4.5 R2 values of each case……………………………………….………………………62
5.1 Titanium loss with leaching time. Leaching parameters: .1 M HCl, 1000 rpm, 90oC, no
boric acid………………………………………….………………………....…………...74
LIST OF FIGURES
Figure Page
1.1 Flow sheet of new route for production of metallic titanium …….………...……….3
1.2 The shrinking unreacted-core system …………………………………….…….….5
2.1 The scheme of TiO2 reduction in the FCC Cambridge process ……………...........10
2.2 The normal and log-normal distributions (CV=0.3)…………………………...15
2.3 The structure of (a) tannic acid and (b) dissociation of tannic acid in aqueous solution
as digallic acid……………………………………………...……………………...….….17
3.1 XRD pattern of the reduced UGS after acetic acid washing ……………….……...22
3.2 SEM image of reduced UGS after acetic acid washing …………………...……....23
3.3 Titanium loss versus HCl concentration …………………………………….……26
3.4 Comparison of iron removal and time for 0.05 M and 0.1 M HCl at 90oC……...….28
3.5 Comparison of iron removal versus time with/without boric acid at 90oC………...29
3.6 Fe removal and Ti loss versus temperature with 0.1 M HCl and with/without boric
acid………………………………………………...……………………………….......31
3.7 XRD pattern of the reduced UGS after 0.1 M HCl leaching at 140oC......................34
3.8 SEM image of reduced UGS after hydrochloric acid leaching……………………35
3.9 Residual plot for iron removal……………………………………………………36
3.10 Predicted and actual iron removal versus HCl concentration (a) Fe removal versus
HCl concentration at 90oC (b) Fe removal versus Temperature in 0.1 M HCl……………38
4.1 The three size distributions investigated in this work and the normal distribution
(CV=0.3)…………...…..........……………………….............….…………………...46
4.2 Fraction reacted versus normalized time for the GGS distribution (a) Interface reaction
ix
control and (b) Pore-diffusion control………….………………………………………...53
4.3 Fraction reacted versus normalized time for the RRB distribution (a) Interface reaction
control and (b) Pore-diffusion control………...………………………………………….56
4.4 Fraction reacted versus normalized time for the Gamma distribution (a) Interface
reaction control and (b) Pore-diffusion control…………...………………………………58
4.5 Comparison for the three size distributions; Fp=3 and CV=0.3 and 1.5 (a) Interface
reaction control and (b) Pore-diffusion control………………………………………….60
5.1 Particle size of reduced UGS after acetic acid washing………………………….65
5.2 Rela t ionship be tween the values o f (1 - (1-X)) 1 / 3 and t ime a t t h ree
temperatures……………………………………..……………………………………….67
5.3 Arrhenius Plot………………………...………………………………........................68
5.4 Possible schematic diagram of the reduced UGS (Brown: TiH2, Black: FexOy, or Fe,
blue: other impurities, gray: hydrogen ion). In this scenario, the reaction is controlled by
solid state diffusion of iron……………………………...……………………………...70
5.5 Comparison of predicted values produced from the equation for solid state diffusion
and experimental value; 0.1 M HCl, 1,000 rpm, 90oC, no boric acid………….………….72
5.6 Comparison of predicted values produced from the equation for solid state diffusion
and experimental value until 4 h of leaching time…………………………..…………….75
5.7 Iron removal versus time at three rotation speed………………………………....…...76
6.1 (a) SEM image of titanium-tannic acid complex, (b) the composition of titanium-tannic
acid complex analyzed by EDS………………………………………………………...81
6.2 FT-IR spectra of (a) tannic acid (b) titanium-tannic acid complex……….…………...82
6.3 (a) The schematic diagram of titanium and tannic acid and (b) optimized structure by
DFT simulation at B3LYP with 6-31G: gray (carbon), red (oxygen), white and big atom
(titanium), white and small atom (hydrogen)………………………………...………...…84
6.4 The molecular orbitals’ energy levels of titanium-tannic acid complex at LUMO+1,
LUMO, HOMO, and HOMO-1: green (positive isosurface of molecular orbitals at the
energy level of each molecular orbital), red (negative isosurface of molecular orbitals at
the energy level of each molecular orbital)……….………………………………………85
ACKNOWLEDGEMENTS
The research in this thesis was carried out with financial support of the Direct
Reduction of Titanium Slag (DRTS) by Advanced Research Projects Agency-Energy
(ARPA-E) of the US Department of Energy (DOE) (Grant # 55800707).
Firstly, I would like to thank my supervisor, Dr. Michael L. Free for his meticulous
guidance and support on the research. It was very fortunate to meet him and become one
of his students. What I learned from him will be one of the most lasting experiences and
lessons in my life. I would like to express sincere appreciation to Dr. H.Y. Sohn for giving
me an opportunity to work on one of the most fascinating fields, fluid-solid reaction
engineering, in my course work. Also, I would like to thank Dr. Z. Zak Fang and Dr.
Amarchand Sathyapalan for their academic guidance for two years and for helping me
complete this thesis.
Secondly, I would like to thank my lab mate, Syamantak Roy who works on the
same project as me. We have gone through several difficulties in the research and course
work together. Also, I would like to thank other lab mates, Dr. Prashant Sarswat, Yakun
Zhu, Gaosong Yi, Weizhi Zeng, Joshua Werner, Sayan Sarkar, and Erik Sundberg for their
encouragement, training, and priceless suggestions.
CHAPTER 1
INTRODUCTION
1.1 Direct reduction of titanium slag
Pure metallic titanium is of great industrial importance because of its use in various
applications such as aviation, aerospace, biomedical, marine, and nuclear waste storage due
to its corrosion resistance, high specific strength, light weight, relatively high melting point,
and chemical/heat stability.1-5 Moreover, it has great mechanical, and electronic properties,
as well as biocompatibility.6 Although it is predominantly found as rutile (TiO2) and
ilmenite (FeTiO3), which are abundant in the earth’s crust, the production cost is relatively
high compared to competing metals such as steel and aluminum.7 In general, metallic
titanium is produced from the Kroll process in which greenhouse gas is generated8 and it
requires an upgraded starting material, which increases the cost of production. Thus, it can
be said that an alternative route, which does not involve the environmental and cost
problems for production of metallic titanium, is desired. Recently, Fang et al. suggested
titanium production from titanium hydride (TiH2), which can be obtained by reduction of
upgraded titanium slag (UGS) or synthetic rutile under a controlled environment in the
presence of metallic magnesium and salts.9 After making titanium hydride, metallic
titanium is produced by dehydrogenation of titanium hydride at about 500oC9 with auxiliary
processes such as size, shape control, and deoxygenation.
In this new route, production cost and environmental problems can be significantly
2
reduced and productivity can be improved compared to the Kroll process. The feed material,
upgraded titanium slag (UGS), is obtained by reduction of ilmenite ore in a blast furnace
followed by roasting and acid leaching. However, one of the concerns in this process is the
level of residual impurities such as iron oxide.
In this work, investigation of the leaching characteristics was carried out using
hydrochloric and boric acids associated with the removal of iron from UGS which has been
reduced using hydrogen gas, magnesium powder, and salts at 750oC for 5 h.9 Reduced UGS
is predominantly composed of titanium hydride and oxide impurities involving iron, silicon,
aluminum, and/or magnesium. According to the American Society for Testing and
Materials (ASTM), the maximum tolerable amount of impurities in titanium sponge for
general purpose is 0.08, 0.15, 0.05, and 0.04% by weight for magnesium, iron, aluminum,
and silica, respectively.10 Thus, one of the goals of this thesis research is iron removal from
reduced UGS to meet specifications. The flow chart shown in Figure 1.1 illustrates the
overall production of pure titanium from UGS. In this investigation, modest concentrations
of hydrochloric and boric acids were used to reduce product loss. Boric acid was used to
accelerate iron removal kinetics, because boric acid and borate tend to form complexes
with hydrous iron oxides as shown in Equation 1.1.11
𝐹𝑒3+ + 𝐻3𝐵𝑂3 + 𝐻2𝑂 = 𝐹𝑒𝐵(𝑂𝐻)42++ 𝐻+ (1.1)
The stability constant of iron borate complexes is 1.0 ± 0.2 × 10−2 25oC for an
ionic strength of 0.68.12
3
Figure 1.1 Flow sheet of new route for production of metallic titanium
4
1.2 Theoretical kinetic study
Fluid-solid reaction kinetics is important in chemical and metallurgical processes
such as the extraction of metals, the combustion of solid fuels, and coal gasification.
Various reaction models have been investigated for various fluid-solid reaction systems.
The shrinking unreacted-core model is one of the simplest models commonly used for an
initially nonporous solid producing a porous product layer.13-15 In this model the reaction
occurs at a sharp interface after reactant fluid diffuses through the porous product layer. As
the reaction proceeds, the size of the unreacted core diminishes. However, the overall size
of the solid remains constant. Figure 1.2 schematically shows the reaction of a nonporous
solid according to this model.
It is well known that the particle size of solid materials is rarely uniform. In most
cases, the particle size of solid materials follows certain distributions. Among, the most
commonly used forms of empirical size distributions in industry are the Gate-Gaudin-
Schuhmann (GGS) distribution, the Rosin-Rammler-Bennett (RRB) distribution, and the
Gamma distribution. All of the size distributions have two parameters that determine mean,
variance, and the coefficient of variation (CV), which is the ratio of the standard deviation
to the mean of the size distribution, which is also called relative standard deviation. The
best fit values of the parameters for each distribution can be determined directly from plots
or from computed values of the mean and variance.16-18
This study investigated the overall fluid-solid reaction kinetics when the particle
size distribution follows the above-mentioned three empirical size distributions with either
interface-reaction or pore-diffusion-controlled kinetics for the shrinking unreacted-core
model.
5
Figure 1.2 The shrinking unreacted-core system
6
The CV of the particle size was varied over a wide range from 0 to 1.5. Also, three
basic particle shapes were considered. The results of this study would be applied to the
leaching characteristics of reduced UGS.
1.3 Selective extraction of titanium from ilmenite using tannic acid
The merits of a metal production process have to be judged not only by the
metallurgical quality of the product obtained but also by its impact on environment and
health. It has been reported that abnormalities in pulmonary function and pleural disease
among people associated with titanium manufacturing can be correlated to exposure to
titanium tetrachloride, titanium dioxide, and duration of work in titanium production.19
In this regard, additional work was carried out to evaluate the ability of tannic acid
to selectively chelate with titanium ions from a multielement solution containing iron,
vanadium, silicon, aluminum, and other related ions, prepared by dissolving a complex ore
such as ilmenite. This route can decrease the significant number of processes for the
preparation of titanium pigment, which is a starting material for the route of direct reduction
of titanium slag (DRTS). That is, metallic titanium can be produced by dehydrogenation of
titanium hydride (TiH2), obtained by the reduction of titanium-tannic acid complex in the
presence of hydrogen. Using the special ability of tannic acid, titanium can be extracted
more environmentally and economically. Also, to acquire a deeper scientific insight into
the complexation and selectivity for titanium chelation, density functional theory (DFT)
simulations have been performed on the complex in addition to the experimental work.
CHAPTER 2
REVIEW OF THE LITERATURE
2.1 Processes for titanium production
2.1.1 Kroll Process
The current commercial method for production of metallic titanium is the Kroll
process in which metallic titanium is obtained by magnesiothermic reduction of titanium
tetrachloride (TiCl4).20 Titanium dioxide obtained by treating rutile and ilmenite is reduced
in the presence of cokes and chlorine gas in a fluidized bed reactor at 1,000oC. The resulting
titanium tetrachloride and impurities are separated using continuous fractional distillation.
Then, titanium tetrachloride is reduced to titanium sponge by magnesium at 850oC for
several days, depending on the size of the reactor, and liquidized magnesium chloride is
obtained as the main by-product. The separation of magnesium chloride from titanium
sponge can be performed by vacuum distillation.21 The chemical reactions can be expressed
as Equations 2.1 and 2.2.
𝑇𝑖𝑂2(𝑠) + 𝐶(𝑠) + 2𝐶𝑙2(𝑔) → 𝑇𝑖𝐶𝑙4(𝑔) + 𝐶𝑂2(𝑔) (2.1)
𝑇𝑖𝐶𝑙4(𝑔) + 2𝑀𝑔(𝑙) → 𝑇𝑖(𝑠) + 𝑀𝑔𝐶𝑙2(𝑙) (2.2)
Although the Kroll process allows industry to produce high-purity metallic
titanium with low oxygen levels, there are several challenges. Firstly, developing a
continuous reduction process is difficult due to the high vacuum, long-term processing, and
8
the generation of titanium deposits on the inner wall of the reactor. Secondly, batch-type
processing for long periods of time results in high labor and power, and low productivity.21
Furthermore, environmental issues such as generation of greenhouse gas and corrosive
intermediate products like titanium tetrachloride are involved.6, 8 In summary, the Kroll
process has several major drawbacks such as extensive time and energy consumption, low
productivity, and environmental problems.
2.1.2 Alternative processes
To mitigate the challenges of the Kroll process, several new alternative techniques
with low cost and high purity, and productivity have been suggested.
The FFC Cambridge process is the one of the alternative processes developed to
replace the Kroll process, in which electrochemistry is utilized unlike other processes
which use pyrometallurgy.23 Titanium dioxide obtained from either sulfate or chloride
processes is used to form a sintered rectangular cathode using traditional ceramic
processing through which a conductive wire is inserted. The titanium dioxide cathode is
immersed in calcium chloride electrolyte with an anodic carbon electrode. As voltage is
applied, the oxygen in the titanium dioxide cathode is ionized in the molten calcium
chloride, transported to the carbon electrode and gasified as forms of oxygen, carbon
monoxide, and carbon dioxide.23 After that, pure metallic titanium remains in the cathode.
This process does not involve the complicated chemical reactions and corrosive titanium
tetrachloride to reduce titanium dioxide that are common in other pyrometallurgical
processes. However, the production cost of the sintered titanium dioxide cathode is fairly
high and the electrochemical processing time is long in order to achieve low levels of
oxygen in the titanium cathode. Moreover, this process includes greenhouse gases such as
9
carbon monoxide and carbon dioxide. Figure 2.1. shows a schematic diagram of the FCC
Cambridge process.
Another alternative process to the Kroll processes is the Armstrong process. This
process adopted sodium as a reducing agent which is also utilized in the Hunter process.
However, the Armstrong process is a continuous process, rather than a batch process used
for the Hunter process.24 Titanium dioxide is reduced to titanium tetrachloride in the
presence of chlorine gas and cokes. The titanium tetrachloride is then reduced by liquid
sodium in a titanium reactor where continuous processing takes place. Sodium can be
removed using distillation and sodium chloride can be separated from titanium sponge by
washing. All sodium and sodium chloride are recycled in this process. The reaction that
takes place can be expressed as Equation 2.3.
𝑇𝑖𝐶𝑙4(𝑔) + 4𝑁𝑎(𝑙) → 𝑇𝑖(𝑠) + 4𝑁𝑎𝐶𝑙(𝑙) (2.3)
This process can become a viable commercial process if it can overcome
equipment durability, optimization of each process, and particle size and morphology
control.23 Titanium subchlorides such as titanium dichloride (TiCl2) and titanium
trichloride (TiCl3) have been investigated as potential replacements for titanium
tetrachloride (TiCl4) feed. Processing using subchlorides keeps the advantages of the Kroll
process such as an oxygen-free environment and easy impurity-control and may increase
the process speed22 However, concerns such as handling volatile titanium tetrachloride
require further technical and practical studies before industrial application.
Another alternative technique is direct reduction from titanium dioxide by
electrodeposition of titanium from ionic solutions, but this approach has several problems
like difficulties in handling the redox cycle of multivalent titanium ions and reactive
10
Figure 2.1 The scheme of TiO2 reduction in the FCC Cambridge process
11
dendritic products23. This process which directly deposits titanium on the cathode from the
titanium ion bearing solution is different from the FFC Cambridge process where oxygen
is removed from titanium dioxide cathode.
Although new technologies for production of metallic titanium without drawbacks
of Kroll process have been suggested and investigated extensively, none of these
technologies have been industrialized on a larger scale due to their intrinsic disadvantages
and limits.
2.1.3 Extraction of titanium from ores
In general, the extractions of titanium dioxide (TiO2) and titanium pigment are
performed using mainly two routes, sulfate and chloride processes. In the sulfate process,
low grade ilmenite (FeTiO3) can be used as a feed stock which is treated by concentrated
sulfuric acid (H2SO4). In the next step, impurities are precipitated by cooling down the
sulfuric acid solution and titanium dioxide can be obtained by performing hydrolysis on
titanyl sulfate (TiOSO4) at 95-110oC, and calcination at 1,000oC.25 On the other hand, the
chloride process requires a high grade feed stock such as rutile (TiO2). The oxygen in the
feed material rutile is removed by a reducing agent, usually carbon, and titanium
tetrachloride (TiCl4) is synthesized using chlorine gas. After that, pure titanium dioxide can
be obtained by oxidizing titanium tetrachloride at 1200-1700oC.26 However, there are
critical drawbacks in these processes such as environmental problems and high cost of
production. Alternate routes are still required to produce low cost, environmentally friendly
titanium from its naturally occurring ores such as ilmenite, by reducing prior treatments.
Alternatively, extraction of titanium using hydrochloric acid has been investigated
extensively due to its low cost, reduced environmental issues, and potential for recycling
12
compared to competing lixiviants such as sulfuric acid. However, most of the work has
focused on extraction of titanium from ores, not reduced UGS, which is the starting
material for extraction of titanium in this thesis.27-29
Titanium can be extracted from ilmenite dissolution using hydrochloric acid. The
reaction can be described as Equation 2.4.
𝐹𝑒𝑇𝑖𝑂3 + 𝐻𝐶𝑙 → 𝐹𝑒2+ + 𝑇𝑖𝑂2+ + 2𝐶𝑙− + 2𝑂𝐻− (2.4)
According to El-Hazek et al., complete dissolution of titanium and iron could be
achieved using 12 M hydrochloric acid at a solid/liquid ratio of 1/20 at 80oC for 2.5 h.27
However, when the resistance of titanium hydride to acidic media is taken into account,
hydrochloric acid concentration should be much lower than 12 M in our work.
2.2 Effect of particle size distribution on kinetics
Generally, the most basic probability distributions are known to be normal and log-
normal distribution. The probability density function of the normal distribution can be
given as
ℎ(𝑅) =1
𝜎√2𝜋𝑒𝑥𝑝(−
(𝑅−𝜇)2
2𝜎2) (2.5)
where 𝜇 is the mean and 𝜎 is the standard deviation. Although the normal distribution is
important in statistics and social science, it is rarely encountered in a practical system of
engineering. The probability density function of log-normal distribution, the logarithm of
which is normally distributed can be expressed as Equation 2.6.
ℎ(𝑅) =1
𝜎√2𝜋𝑅𝑒𝑥 𝑝 (−
(𝑙𝑛𝑅−𝜇)2
2𝜎2) , 𝑅 > 0 (2.6)
If variable R is a particle size of a certain system that we are interested in, then the
13
probability distribution can be understood as the particle size distribution. The adjustable
parameters for the normal and log-normal distribution can be used to determine the mean
and the standard deviation. To compare these distributions, the CV is used to normalize the
mean and variance of each distribution. Table 2.1 shows mean, variance, and mathematical
relationship among adjustable parameters and CV. Also, when CV is fixed at 0.3, two
probability distributions can be plotted as shown in Figure 2.2.
It is known that the effect of particle size distribution of solid materials treated in
mineral processing occupies an important position in determining the kinetics of the
reaction. Nonetheless, in most models, it is assumed that the particle size and shape of
reacting solid materials are uniform. Only few researchers have studied the effect of the
size distribution on the fluid-solid reaction kinetics. Hulburt, Katz, Randolph, and Larson
observed that a mathematical expression for a reaction taking place in a system cannot
represent the actual system which has a distribution of particle properties.30-32 They
suggested a concept of population balance where changes in the distribution of chosen
particle properties can be described. Mcllvried and Massoth narrowed down the focus of
the particle properties to the particle size, and particularly considered normal and log-
normal particle size distributions for nonporous spherical particles and either a chemical-
reaction-controlled or pore-diffusion-controlled process.33
The conclusion was that the overall reaction rate of a particle assemblage with a
normal or log-normal particle size distribution was not significantly different from those
with a uniform mass-average size as long as the CV of the distribution was less than 0.3.
The particle size distribution found in nature or formed by mineral processing rarely
follows a normal particle size distribution. In fact, it is convenient to use empirical
14
Table 2.1 Mean, variance, and CV of three distributions
Name Normal distribution Log-normal distribution
Adjustable
parameters 𝜇, 𝜎 𝜇, 𝜎
Mean 𝜇 𝑒𝜇+𝜎2/2
Variance 𝜎 (𝑒𝜎2− 1)𝑒2𝜇+𝜎
2
CV 𝜎
𝜇 √𝑒𝜎
2− 1
15
Figure 2.2 The normal and log-normal distributions (𝑪𝑽 = 𝟎. 𝟑)
16
relationships which offer concise mathematical expressions of size distributions.34
2.3 Tannic acid
It has been reported that hydroxyaromatic compounds can selectively form a
complex with titanium compounds at pH 4-535-37. Tannic acid is such a compound and it
possesses carboxylic acid groups expected to show a strong affinity towards metal ions
having high oxidation states.35, 38 Also, tannic acid in water is hydrolyzed to digallic acid
and glucose.39 Figure 2.3 shows the molecular structure of tannic and digallic acids. By
adding tannic acid in the titanium bearing solution and adjusting the pH between 4 and 5
during precipitation, a titanium-tannic acid complex can be precipitated selectively from a
multielement solution, containing dissolved impurities with iron, magnesium, vanadium,
and manganese as ions. Moreover, tannic acid is commonly found in beverages, food, and
plant materials.40 Using such an ecofriendly material to separate titanium can minimize the
environmental and health concerns associated with current industrial titanium production
processing.
17
Figure 2.3 The structure of (a) tannic acid and (b) dissociation of tannic acid in
aqueous solution as digallic acid
CHAPTER 3
LEACHING CHARACTERISTICS OF IRON REMOVAL
FROM REDUCED UGS
3.1 Experimental methods
3.1.1 Preparation of reduced UGS by reduction and acetic acid leaching
UGS is composed of about 96% titanium dioxide and 4% impurities. A mixture of
UGS (less than 40 μm ), magnesium powder, and salts, such as magnesium chloride
(MgCl2) and potassium chloride (KCl), with a ratio of 1:1:1 by weight were placed in a
furnace under hydrogen atmosphere and reduced for 5 h at 750oC. Excess magnesium and
salts are used to help increase kinetics.9 The product resulting from the reduction is washed
using 4.3 M acetic acid (CH3COOH) with a solid to liquid ratio of 1 g to 40 ml for 2 h at
70oC to remove magnesium compounds and salts from reduced UGS. After acetic acid
washing and filtration, the residue was washed with deionized water and 0.05 M
hydrochloric acid. The sample so obtained is used as the feed in this study.
3.1.2 Experimental procedures
A solution of 200 ml which contained 0.05 or 0.1 M hydrochloric acids with or
without 1 M boric acid was prepared in a flask made of semiclear polymethylpentene. A
chemical and temperature resistant plug with a hole that allows for a thermometer insertion
was applied on the top of the flask to minimize evaporation losses. The solution was mixed
19
by a magnetic stirring bar at a rotational speed of 1,000 rpm and heated to the
predetermined temperature, which is lower than the boiling point of water (70, 80, and
90oC). Once the temperature reached the desired level, 2 g of reduced UGS obtained after
acetic acid washing was added into the flask. The leaching experiment was run for 4 h and
samples were collected every hour for the study of reaction kinetics. In the cases where the
experiments were conducted at higher temperature than the boiling point of water (140 and
190oC), a different experimental setup, which involved a pressure reactor, was used. A
solution of 200 ml which contained a low concentration of hydrochloric and boric acids
with 2 g of reduced UGS after acetic acid washing was prepared in a
polytetrafluoroethylene beaker and experiments were carried out in the pressure reactor.
The desired temperature was achieved after 1 h, and the experiments were carried out for
3 h at the desired temperature. The pressure reactor was slowly cooled for 2 h. After that,
the solution was filtered and residue was washed with deionized water and 0.05 M
hydrochloric acid. The solution samples were analyzed by inductively coupled plasma-
optical emission spectroscopy (ICP-OES). The solid samples were characterized using X-
ray diffraction (XRD) and scanning electron microscopy (SEM).
3.2 Experimental results and discussion
3.2.1 Feed analysis
Table 3.1 shows the composition of reduced UGS immediately after reduction.
Magnesium compounds and salts in Table 3.1 come from magnesium oxide, excess
magnesium powder, magnesium chloride, and potassium chloride. The composition of the
acetic acid washed reduced UGS is also listed in Table 3.1, showing the impurities which
predominantly consist of iron, aluminum, magnesium, and silicon oxides.
20
Table 3.1 Chemical composition of reduced UGS before and after acetic acid washing
Component (%) Mg compounds
and salts Ti Fe Al Si
Before washing 79.79 19.52 0.22 0.07 0.41
After washing 0.36 96.20 1.07 0.35 2.02
21
Figure 3.1 presents the mineralogical phases of the acetic acid washed; reduced
UGS analyzed by XRD shows that the crystalline phases of the products predominantly
consist of titanium hydride and ferric oxide.
Figure 3.2 shows the surface morphology of reduced UGS after acetic acid
washing as analyzed by SEM. This result can be used for comparison of surface
morphology with that of the product after leaching in hydrochloric and boric acids.
Although the sizes of individual particles seem very small, Figure 3.2 reveals larger
agglomerates.
3.2.2 Experimental results
Table 3.2 shows the effect of each parameter on iron removal and titanium loss
after 4 h of leaching. Again, the leaching temperature, hydrochloric acid concentration, and
boric acid concentration were chosen as main factors to verify the optimized leaching
parameters. Three values for leaching temperature, two values for hydrochloric acid, and
boric acid concentrations give 12 different experiments in total. By analyzing those
experimental results, the most important leaching parameter affecting iron removal can be
investigated.
3.2.2.1 Effect of hydrochloric acid
One of the difficulties in the leaching of reduced UGS is that titanium hydride is
easily dissolved in strong acid. In particular, titanium hydride tends to be converted to
titanium trichloride in high concentrations of hydrochloric acid. The dissolution of titanium
hydride can be expressed as41
2𝑇𝑖𝐻2 + 6𝐻+ + 6𝐶𝑙− = 2𝑇𝑖𝐶𝑙3 + 5𝐻2 (3.1)
22
Figure 3.1 XRD pattern of the reduced UGS after acetic acid washing
23
Figure 3.2 SEM image of reduced UGS after acetic acid washing
24
Table 3.2 Fe removal % and Ti loss % at different leaching parameters with a solid to
liquid ratio of 1 to 100 and 4 h of leaching time
Temp (oC) HCl (M) Boric acid (M) Fe removal (%) Ti loss (%)
90 0.05 0 27.30 0.07
90 0.05 1 43.80 0.08
90 0.1 0 55.01 0.50
90 0.1 1 61.42 0.60
140 0.05 0 79.59 0.06
140 0.05 1 76.01 0.15
140 0.1 0 87.63 0.21
140 0.1 1 87.82 0.25
190 0.05 0 79.19 0.22
190 0.05 1 81.17 0.22
190 0.1 0 88.26 0.12
190 0.1 1 88.51 0.34
25
Thus, before conducting the regular experiments with hydrochloric acid, the acceptable
quantity of titanium hydride loss during the leaching process needs to be determined. Our
research group determined arbitrarily that 1% (w/w) of titanium loss is the maximum
amount tolerable. Test results for titanium loss as a function of hydrochloric acid
concentration are presented in Figure 3.3 under the leaching parameters in Table 3.3.
Based on the results in Figure 3.3, the maximum concentration of hydrochloric
acid that meets the requirement of less than 1% titanium loss is approximately 0.1 M. Thus,
0.05 M and 0.1 M hydrochloric acids were chosen to study the influence of hydrochloric
acid concentration on the removal of iron as shown in Figure 3.4 under the leaching
parameters in Table 3.3. The leaching with 0.1 M hydrochloric acid shows 55.0% of iron
removal after 4 h along with 0.5% of titanium loss, whereas 0.05 M hydrochloric acid
removed only 27.3% of the iron. Much of the iron removal taking place during leaching
can be expressed generally as,
𝐹𝑒2𝑂3 + 6𝐻+ + 6𝐶𝑙− = 2𝐹𝑒𝐶𝑙3 + 3𝐻2𝑂 (3.2)
3.2.2.2 Effect of boric acid
Figure 3.5a shows the effect of boric acid in 0.05 M hydrochloric acid. The iron
removal efficiency significantly increased from 27.3% to 43.8% with the increase of boric
acid. However, in 0.1 M hydrochloric acid, the iron removal efficiency increased by only
6.4% when boric acid was added according to Figure 3.5b.
According to Peak et al., trigonal boric acid reacted with ferric iron would be
converted to tetrahedral surface complexes as a result of the Lewis acidity of the boron
metal center.11 Then hydrogen ions would be displaced from the tetrahedral boron, and
ferric borate complexes are obtained as shown in Equation 1.1.11, 12 However, this reaction
26
Figure 3.3 Titanium loss versus HCl concentration
27
Table 3.3 Leaching parameters
Temperature
(oC)
Time
(h)
Stirring speed
(rpm)
S/L
(g : ml)
Particle size
(𝛍𝐦)
90 4 1,000 1:100 0.05 – 0.2
28
Figure 3.4 Comparison of iron removal and time for 0.05 M and 0.1 M HCl at 90oC
29
Figure 3.5 Comparison of iron removal versus time with/without boric acid at 90oC
30
is only favorable in weak acid, which might explain why boric acid does not affect iron
removal much at the higher concentration of hydrochloric acid (Figure 3.5).
3.2.3 Effect of temperature
The previous results show that the concentration of hydrochloric acid is restricted
to 0.1 M to prevent significant titanium loss, and boric acid does not help iron removal
much in the presence of 0.1 M hydrochloric acid in the solution. Thus, the effect of leaching
temperature on iron removal was studied in the broad temperature range of 70 – 190oC,
with 0.1 M hydrochloric acid and with/without 1 M boric acid. Figure 3.6 shows that the
iron removal increases with the increase of temperature. It is noted that once the
temperature reaches about 140oC, no additional iron was removed. Thus, it appears that the
limit of iron removal is near 90% if the titanium loss is kept below 1%. It is also noted that
iron removal without boric acid shows almost the same iron removal result as that with
boric acid as shown in Figure 3.6. This result suggests high temperature and high acidity
are not favorable for boric acid to form tetrahedral surface complexes with iron.
Therefore, the parameters that show the best results for iron removal based on the
series of tests presented are 140oC, 0.1 M hydrochloric acid and no boric acid. Weight
percentage of iron removal under those optimized parameters is 87.63%, which means that
only 0.13% of iron remains in the leached, reduced UGS. The percentage of titanium was
significantly changed from 96.2% to 99.45% after substantial impurity removal. According
to the ASTM, the maximum tolerable amount of iron impurities in titanium sponge for
general purpose is 0.15% by weight.10 Thus, the reduced UGS that has been purified under
the optimized parameters meets ASTM B299-13 iron specifications for general purposed
titanium sponge. The whole composition of the reduced UGS leached by 0.1 M
31
Figure 3.6 Fe removal and Ti loss versus temperature with 0.1 M HCl and with/without
boric acid
32
hydrochloric acid with/without boric acid at 140oC is tabulated in Table 3.4. The sample
was analyzed by ICP-OES.
Figure 3.7 shows the XRD pattern of the reduced UGS obtained after 0.1 M
hydrochloric acid leaching at 140oC. According to Figure 3.7, little titanium hydride was
oxidized to titanium dioxide or metallic titanium, and most of impurities were below the
detection limit for XRD.
Figure 3.8 shows the surface morphology of the reduced UGS after 0.1 M
hydrochloric acid leaching at 140oC. Comparing it to the SEM image of reduced UGS
before hydrochloric acid leaching (Figure 3.2), it is apparent that the surface morphology
is nearly the same. The particle shape remains granular, and agglomerates are formed.
3.2.4 Factorial design of experiments for equation modeling
A model that can predict iron removal with the three parameters was established.
Three levels of temperature (90, 140, 190oC) and two levels of concentration of
hydrochloric (0.05, 0.1 M), and boric acids (0, 1 M) were used as shown in Table 3.2. The
factorial design of experiments was analyzed by the statistical software “Minitab”. The
equation modeling for prediction of iron removal with three parameters is described in
Equation 3.3.
Iron removal (wt%) = -164.116 + 2.56685 T + 677.07 [HCl] + 18.0998 [H3BO3] -
0.00687205 T2 - 2.89343 T [HCl] - 0.103388 T [H3BO3] (3.3)
where T, [HCl], and [H3BO3] represent temperature in degrees Celsius, molarity of
hydrochloric and boric acids, respectively, and the R2(adj.) value of the equation is 0.9541.
Also, in Figure 3.9, the residual versus order plot shows randomness and unpredictability
where the sense of accuracy of the model is observed.
33
Table 3.4 Chemical composition of reduced UGS leached by 0.1 M HCl at 140oC
Component (%) Mg Ti Fe Al Si
0.1 M HCl 0.05 99.45 0.13 0.12 0.25
0.1 M HCl + 1 M H3BO3 0.07 99.13 0.13 0.10 0.57
34
Figure 3.7 XRD pattern of the reduced UGS after 0.1 M HCl leaching at 140oC
35
Figure 3.8 SEM image of reduced UGS after hydrochloric acid leaching
36
Figure 3.9 Residual plot for iron removal
37
Figure 3.10a shows a comparison of iron removal at different concentrations of
hydrochloric acid for predicted and actual results at 90oC and Figure 3.10b presents a
comparison of iron removal as a function of temperature with 0.1 M hydrochloric acid.
Overall, Figures 3.10a and b show good agreement between predicted iron removal and
actual iron removal. Note that although some of the data in Figure 3.10a show high iron
removal, concentrations above 0.1 M hydrochloric acid result in significant titanium loss.
3.2.5 Removal of iron without acetic acid leaching
The starting material in this research was reduced UGS which was washed by
acetic acid to remove magnesium oxide and salts introduced from the reduction. Another
way of removing magnesium oxide and salts is to use hydrochloric acid instead of acetic
acid which allows us to remove iron and magnesium oxides together. This concept can help
reduce the number of leaching steps required for removal of impurities and results in
reduction of the cost for production of metallic titanium. Using this concept experimentally,
2 g of the powder taken out from the reduction furnace was prepared and leached in 0.4 M
hydrochloric acid at 70oC for 2 h to remove magnesium oxide and salts. After that, the
mixture of the leach liquor and the powder was put in the pressure reactor and leached at
140oC for 3 h to remove iron oxide. Table 3.5 shows the composition of reduced UGS
leached by 0.4 M hydrochloric acid at 70oC and 140oC for 2 h and 3 h, respectively. The
amount of iron and magnesium in the reduced UGS leached by 0.4 M hydrochloric acid
meets the ASTM B299-13 iron and magnesium specifications for general purposed
titanium sponge.10 Thus, the result indicates that this route has a strong potential to reduce
the production cost as well as produce the qualified titanium sponge which meets the
ASTM specifications.
38
Figure 3.10 Predicted and actual iron removal versus HCl concentration (a) Fe removal
versus HCl concentration at 90oC (b) Fe removal versus Temperature in 0.1 M HCl
39
Table 3.5 Composition of reduced UGS leached by 0.4 M hydrochloric acid
Composition Ti (%) Fe (%) Si (%) Al (%) Mg (%)
w/o acetic acid
leaching 99.25 0.06 0.49 0.20 <0.03
ASTM
Specific cation - 0.15 0.04 0.05 0.08
CHAPTER 4
EFFECT OF PARTICLE SIZE DISTRIBUTION ON KINETICS
4.1 Model formulation
4.1.1 Shrinking unreacted core model
Consider a reaction where the product solid forms a porous layer around an
unreacted nonporous core:
𝐴(𝑓) + 𝑏𝐵(𝑠) → 𝑐𝐶(𝑓) + 𝑑𝐷(𝑠) (4.1)
If the overall rate of the reaction is controlled by the interface chemical reaction,
the progress of the reaction is represented for the three basic geometries by13
𝑑𝜉
𝑑𝑡∗= −1 (4.2)
where
𝜉 ≡ (𝐴𝑝
𝐹𝑝𝑉𝑝) 𝑟𝑐 (4.3a)
𝑡∗ = (𝑏𝑘
𝛼𝐵𝜌𝐵) (
𝐴𝑝
𝐹𝑝𝑉𝑝) [𝐶𝐴𝑂
𝑛 −𝐶𝐶𝑂𝑚
𝐾] 𝑡 (4.3b)
where 𝐴𝑝 and 𝑉𝑝 are the original surface area and volume; 𝑟𝑐 the position of the
reaction interface in the distance coordinate perpendicular to the solid surface; and 𝐹𝑝 is
the shape factor, which has the value 1, 2, or 3 for infinite slabs, long cylinders, or spheres,
41
respectively. 𝑘 is the heterogeneous rate constant, 𝛼𝐵 the fraction of pellet volume
occupied by solid reactant B, 𝜌𝑠 the molar density of the solid B, K the equilibrium
constant, 𝐶𝐴𝑂 and 𝐶𝐶𝑂 are the concentrations of A and C in the bulk fluid stream,
respectively. The fraction reacted of an individual particle, β, is given by
β = 1 − ξ𝐹𝑝 (4.4)
Integration of Equation 4.2 and substitution in Equation 4.4 gives
β(𝑡, 𝑅) =
{
1 − (1 −
𝑘𝐶𝑅𝑡)𝐹𝑝
at 𝑡 <𝑅
𝑘𝐶 (4.5a)
1 𝑎𝑡 𝑡 ≥𝑅
𝑘𝐶 (4.5b)
where
𝑘𝐶 = (𝑏𝑘
𝛼𝐵𝜌𝑠) [𝐶𝐴𝑂
𝑛 −𝐶𝐶𝑂𝑚
𝐾] (4.6)
𝑅 =𝐹𝑝𝑉𝑝
𝐴𝑝 (4.7)
𝑘𝐶 is a new constant incorporating system parameters, 𝑅 is the initial half thickness of
an infinite slab and initial radius of a long cylinder or a sphere, and β(𝑡, 𝑅) stands for the
fraction reacted of a particle with initial radius 𝑅 at a certain time 𝑡 . Equation 4.5b
represents the fact that once a particle is completely reacted, there is no longer any reaction
taking place in it.
If the overall rate of reaction is controlled by diffusion through the porous product
layer of a spherical particle, the mass balance for the reactant fluid species in the porous
product layer yields:
42
𝑑
𝑑𝑟(𝑟𝐹𝑝−1𝑁𝐴) = 0 (4.8)
where
𝑁𝐴 = 𝑥𝐴(𝑁𝐴 + 𝑁𝐶) − 𝐷𝑒𝐶𝑇∇𝑥𝐴 (4.9)
𝑁𝐴 is the molar flux of species A, 𝑁𝐶 the molar flux of species C, 𝑥𝐴 mole fraction of
fluid A, 𝐷𝑒 effective diffusivity, and 𝐶𝑇 total molar concentration of fluid. The boundary
conditions are given by
𝐶𝐴 = 𝐶𝐴𝑂 𝑎𝑛𝑑 𝐶𝐶 = 𝐶𝐶𝑂 at 𝑟 = 𝑅 (4.10)
𝐶𝐴 = 𝐶𝐶/𝐾 at 𝑟 = 𝑟𝑐 (4.11)
The first boundary condition assumes rapid external mass transfer, which has been
shown to provide only a secondary effect to pore diffusion for the reaction of an individual
solid,12 and thus should not significantly affect the conclusions on the effect of particle size
distribution. For further mathematical simplicity, we will consider the case in which c in
Equation 4.1 is unity. Then, from the mole balance of species A and C, the following
equation is obtained:
𝐶𝐴 + 𝐶𝐶 = 𝐶𝐴𝑂 + 𝐶𝐶𝑂 (4.12)
Integrating Equation 4.8 with the boundary conditions and combining Equation
4.8 and Equation 4.12 gives42
𝑘𝐷
𝑅2𝑡 = 1 −
𝐹𝑝(1−𝛽)2/𝐹𝑝−2(1−𝛽)
𝐹𝑝−2 (4.13)
where
43
𝑘𝐷 = (𝐾
1+𝐾)(𝐶𝐴𝑂 −
𝐶𝐶𝑂
𝐾)(2𝐹𝑝𝑏𝐷𝑒
𝜌𝑠) (4.14)
𝑘𝐷 is a new constant incorporating system parameters. The conversion versus time
relationship for spheres, long cylinders, and infinite slabs under pore-diffusion control can
be obtained in more familiar forms from Equation 4.13:
for 𝐹𝑝 = 3 (spheres),
𝑘𝐷
𝑅2𝑡 = 1 − 3(1 − β)
2
3 + 2(1 − β) (4.15)
for 𝐹𝑝 = 2 (long cylinders), by applying L’Hospital’s rule,
𝑘𝐷
𝑅2𝑡 = β + (1 − β)ln(1 − β) (4.16)
and for 𝐹𝑝 = 1 (slabs),
𝑘𝐷
𝑅2𝑡 = β2 (4.17)
It is necessary to convert Equations 4.15, 16, and 17 into β as an explicit function
of time and 𝑅 to allow integration with respect to particle size.
Equation 4.15 is thus converted to43
β(𝑡, 𝑅) =
{
1 − {sin [
1
3arcsin (1 −
2𝑘𝐷𝑅2
𝑡)] +1
2}3
𝑎𝑡 𝑡 <𝑅2
𝑘𝐷 (4.18a)
1 𝑎𝑡 𝑡 ≥𝑅2
𝑘𝐷 (4.18b)
The new forms of Equations 4.16 and 4.17 are
44
β(𝑡, 𝑅) =
{
1 − exp[𝑊−1(
𝑘𝐷𝑡𝑅2
− 1
𝑒) + 1] 𝑎𝑡 𝑡 <
𝑅2
𝑘𝐷 (4.19a)
1 𝑎𝑡 𝑡 ≥𝑅2
𝑘𝐷 (4.19b)
and
β(𝑡, 𝑅) =
{
√𝑘𝐷𝑅2𝑡 𝑎𝑡 𝑡 <
𝑅2
𝑘𝐷 (4.20a)
1 𝑎𝑡 𝑡 ≥𝑅2
𝑘𝐷 (4.20b)
where 𝑊−1 is the 𝑊 ≤ −1 portion of the Lambert W function defined by44 𝑧 =
𝑊(𝑧)𝑒𝑊(𝑧).
Again, Equations 4.18b, 19b, and 20b represent the fact that once a particle is
completely reacted, there is no longer any reaction taking place in it.
4.1.2 Mathematical formulation for size distribution
As mentioned previously, the most commonly used empirical size distributions,
that is, the GGS distribution, the RRB distribution, and the Gamma distribution, were
chosen to investigate the effect of particle size distribution on the fluid-solid reaction. Let
ℎ(𝑅) represent the probability density function in terms of mass of radius of a long
cylinder or a sphere, and the half thickness of an infinite slab.
For the GGS distribution,45
ℎ(𝑅) = 𝑚𝑅𝑚−1
𝑅𝑚𝑎𝑥𝑚 (4.21)
for the RRB distribution,46
45
ℎ(𝑅) =𝑚
𝑙𝑚𝑑𝑚−1𝑒𝑥𝑝[− (
𝑅
𝑙)𝑚
] (4.22)
and for the Gamma distribution,47
ℎ(𝑅) =𝑏𝑝𝑅𝑝−1𝑒−𝑏𝑅
𝛤(𝑝) (4.23)
where Γ(𝑝) is the gamma function of 𝑝 and {𝑚, 𝑅𝑚𝑎𝑥}, {𝑚, 𝑙}, 𝑎𝑛𝑑 {𝑏, 𝑝} represent
adjustable parameters for the three distributions, respectively. The mutual comparison
among the three size distributions can be achieved by adjusting the parameters that regulate
the mean (μ) and variance (𝜎2). For this study, the CV is used to standardize the effect of
the mean and variance of the different distributions. Figure 4.1 shows a plot of the three
size distributions and the normal distribution for comparison when the CV of each size
distributions is identical at 0.3 and Table 4.1 shows mean, variance, and CV expressed with
the adjustable parameters of the three distributions.
The mass fraction of particles within the range of radius 𝑅 and R+𝑑𝑅 is given
by ℎ(𝑅) ∙ 𝑑𝑅. As stated above, β(𝑡, 𝑅) stands for the fraction reacted of a particle with
initial radius R at a certain time t. Thus, the overall fraction reacted of the entire particle
assemblage at a certain time t, 𝑋(𝑡), is given by
𝑋(𝑡) = ∫ β(𝑡, 𝑅)∞
0ℎ(𝑅)𝑑𝑅 (4.24)
In order to determine the overall fraction reacted, β(𝑡, 𝑅) (depending on controlling
mechanism), ℎ(𝑅) (depending on the size distribution), and the particle shape factor must
be known. For interface-reaction control, the GGS distribution, and the particle shape
factor 𝐹𝑝, substitution of Equations 4.5 and 4.21 into Equation 4.24 gives the following
equation,
46
Figure 4.1 The three size distributions investigated in this work and the
normal distribution (𝐶𝑉 = 0.3)
47
Table 4.1 Mean, variance, and CV of three distributions
Name GGS RRB Gamma
Adjustable
parameters 𝑚,𝑅𝑚𝑎𝑥 𝑚, 𝑙 𝑏, 𝑝
Mean (𝛍) 𝑚
𝑚 + 1∙ 𝑅𝑚𝑎𝑥 𝑙 ∙ Γ(
𝑚 + 1
𝑚)
𝑝
𝑏
Variance
(𝝈𝟐) [
𝑚
(𝑚 + 2)−
𝑚2
(𝑚 + 1)2] ∙ 𝑅𝑚𝑎𝑥
2 𝑙2 ∙ [Γ (𝑚 + 2
𝑚) − Γ(
𝑚 + 1
𝑚)]
𝑝
𝑏2
CV (=𝝈
𝝁)
√𝑚
(𝑚 + 2)−
𝑚2
(𝑚 + 1)2
𝑚𝑚 + 1
√Γ(𝑚 + 2𝑚
) − Γ(𝑚 + 1𝑚
)
Γ(𝑚 + 1𝑚 )
1
√𝑝
48
𝑋(𝑡) = ∫ β(𝑡, 𝑅)𝑅𝑚𝑎𝑥
0[𝑚 ∙
𝑅𝑚−1
𝑅𝑚𝑎𝑥𝑚 ] 𝑑𝑅 (4.25)
where β(𝑡, 𝑅) is given by Equations 4.5a and b. The final solution is given by
𝑋(𝜏) = 𝑚(𝑚
𝑚+1)𝑚 [∫ [1 − (1 −
𝜏𝐶
𝑎)𝐹𝑝] 𝑎𝑚−1
𝑚+1
𝑚𝑎𝛽=1
𝑑𝑎 + ∫ 𝑎𝑚−1𝑎𝛽=10
𝑑𝑎] (4.26)
where
𝑎 =𝑅
𝜇 (4.27)
𝜏𝐶 =𝑘𝐶𝑡
𝜇 (4.28)
𝑋(𝜏) is a function of three parameters, 𝑚,𝐹𝑝, and 𝜏. The adjustable parameter, 𝑚, can
be determined by fixing the CV, for mutual comparison with the effects of the other two
size distributions. For example, the value of 𝑚 is 2.48 or 1.24 when CV is 0.3 or 0.5,
respectively. Following a similar mathematical procedure, the final forms of 𝑋(𝜏) for the
RRB distribution and the Gamma distribution under the interface-reaction control were
derived and are tabulated in Table 4.2 together with Equation 4.26 for the GGS distribution.
Unlike the interface-reaction-controlled case, the pore-diffusion-controlled reaction has
different forms of 𝛽(𝑡, 𝑅) according to the particle shape factors, as shown in Equations
4.18, 4.19, and 4.20. For the case of pore-diffusion control, the GGS distribution, and
spherical particles, substitution of Equations 4.18 and 4.21 into Equation 4.24 and
expressing 𝑋(𝜏) with the normalized parameters in Equations 4.27 and 4.32 give the final
form of 𝑋(𝜏) as follows:
𝑋(𝜏) = 𝑚(𝑚
𝑚+1)𝑚 [∫ [1 − {𝑠𝑖𝑛 [
1
3arcsin(1 −
2𝜏𝐷
𝑎2)] +
1
2}3
] 𝑎𝑚−1𝑚+1
𝑚𝑎𝛽=1
𝑑𝑎 + ∫ 𝑎𝑚−1𝑎𝛽=10
𝑑𝑎] (4.31)
49
Table 4.2 X(t) of three distributions for interface-reaction-controlled system
Distr. 𝑿(𝒕)
GGS 𝑚(𝑚
𝑚 + 1)𝑚 [∫ [1 − (1 −
𝜏𝐶𝑎)𝐹𝑝] 𝑎𝑚−1
𝑚+1𝑚
𝑎𝛽=1
𝑑𝑎 + ∫ 𝑎𝑚−1𝑎𝛽=1
0
𝑑𝑎]
RRB 𝑚{Γ(𝑚 + 1
𝑚)}𝑚
[ ∫ [1 − (1 −
𝜏𝐶𝑎)𝐹𝑝]
∞
𝑎𝛽=1
𝑎𝑚−1𝑒−{𝑎Γ(
𝑚+1𝑚
)}𝑚
𝑑𝑎
+∫ 𝑎𝑚−1𝑒−{𝑎Γ(
𝑚+1𝑚
)}𝑚𝑎𝛽=1
0
𝑑𝑎]
Gamma 𝑝𝑝
Γ(𝑝)[∫ [1 − (1 −
𝜏𝐶
𝑎)𝐹𝑝]
∞
𝑎𝛽=1
𝑎𝑝−1𝑒−𝑝𝑎𝑑𝑎 + ∫ 𝑎𝑝−1𝑒−𝑝𝑎𝑎𝛽=1
0
𝑑𝑎]
50
where
𝜏𝐷 =𝑘𝐷𝑡
𝜇2 (4.32)
For the same rate-control and the size distribution but different particle shape factors, it can
be presented as
𝑋(𝜏) = 𝑚(𝑚
𝑚+1)𝑚 [∫ [1 − 𝑒
𝑤−1(𝜏𝐷 𝑎2⁄ −1
𝑒)+1] 𝑎𝑚−1
𝑚+1
𝑚𝑎𝛽=1
𝑑𝑎 + ∫ 𝑎𝑚−1𝑎𝛽=1
0𝑑𝑎] (4.33)
𝑋(𝜏) = 𝑚(𝑚
𝑚+1)𝑚 [∫ √
𝜏𝐷
𝑎2𝑎𝑚−1
𝑚+1
𝑚𝑎𝛽=1
𝑑𝑎 + ∫ 𝑎𝑚−1𝑎𝛽=1
0𝑑𝑎] (4.34)
where Equations 4.33 and 4.34 are for the particle shape of a long cylinder and an infinite
slab which takes the value of 2 and 1, respectively. 𝑋(𝜏) for the other two size
distributions with the three shape factors can be obtained in the same way and are tabulated
in Table 4.3.
4.2 Results and Discussion
The effects of size distributions and shape factors on the kinetics for different CV
values are visually illustrated in the following figures. The first five curves from the top
represent the fraction reacted versus normalized time for spherical particles (𝐹𝑝 = 3) with
CV values of 1, 0.75, 0.5, 0.3, and 0. The next five curves are for flat particles (𝐹𝑝 = 1). It
is noted that the behavior of cylindrical particles (not shown) is in between those of
spherical particles and slabs. It is also noted that the case of flat particles with a distribution
of thickness may be less often encountered in a practical problem than spherical particles.
On the other hand, modeling of spherical particles can be applied to real particles that have
similar sizes in the three dimensions even if they are not exactly spherical.
51
Tab
le 4
.3 X
(t)
of
thre
e dis
trib
uti
ons
for
pore
-dif
fusi
on-c
ontr
oll
ed s
yst
em
X(t
)
𝑚(𝑚
𝑚+1)𝑚[∫
[1−{𝑠𝑖𝑛[1 3𝑎𝑟𝑐𝑠𝑖𝑛(1−2𝜏 𝐷 𝑎2)]+1 2}3
]𝑎𝑚−1
𝑚+1
𝑚
𝑎𝛽=1
𝑑𝑎+∫
𝑎𝑚−1
𝑎𝛽=1
0
𝑑𝑎]
𝑚(𝑚
𝑚+1)𝑚[∫
[1−𝑒𝑤−1(𝜏𝐷𝑎2
⁄−1
𝑒)+1]𝑎𝑚−1
𝑚+1
𝑚
𝑎𝛽=1
𝑑𝑎+∫
𝑎𝑚−1
𝑎𝛽=1
0
𝑑𝑎]
𝑚(𝑚
𝑚+1)𝑚[∫
√𝜏 𝐷 𝑎2𝑎𝑚−1
𝑚+1
𝑚
𝑎𝛽=1
𝑑𝑎+∫
𝑎𝑚−1
𝑎𝛽=1
0
𝑑𝑎]
𝑚{𝛤(𝑚
+1
𝑚)}𝑚
[∫[1−{𝑠𝑖𝑛[1 3𝑎𝑟𝑐𝑠𝑖𝑛(1−2𝜏 𝐷 𝑎2)]+1 2}3
]𝑎𝑚−1𝑒−{𝑎𝛤(𝑚
+1
𝑚)}𝑚
∞
𝑎𝛽=1
𝑑𝑎+∫
𝑎𝑚−1𝑒−{𝑎𝛤(𝑚
+1
𝑚)}𝑚
𝑎𝛽=1
0
𝑑𝑎]
𝑚{𝛤(𝑚
+1
𝑚)}𝑚
[∫[1−𝑒𝑤−1(𝜏𝐷𝑎2
⁄−1
𝑒)+1]𝑎𝑚−1𝑒−{𝑎𝛤(𝑚
+1
𝑚)}𝑚
∞
𝑎𝛽=1
𝑑𝑎+∫
𝑎𝑚−1𝑒−{𝑎𝛤(𝑚+1
𝑚)}𝑚
𝑎𝛽=1
0
𝑑𝑎]
𝑚{𝛤(𝑚
+1
𝑚)}𝑚
[∫√𝜏 𝐷 𝑎2𝑎𝑚−1𝑒−{𝑎𝛤(𝑚
+1
𝑚)}𝑚
∞
𝑎𝛽=1
𝑑𝑎+∫
𝑎𝑚−1𝑒−{𝑎𝛤(𝑚
+1
𝑚)}𝑚
𝑎𝛽=1
0
𝑑𝑎]
𝑝𝑝
𝛤(𝑝)[∫
[1−{𝑠𝑖𝑛[1 3𝑎𝑟𝑐𝑠𝑖𝑛(1−2𝜏 𝐷 𝑎2)]+1 2}3
]𝑎𝑝−1𝑒−𝑝𝑎
∞
𝑎𝛽=1
𝑑𝑎+∫
𝑎𝑝−1𝑒−𝑝𝑎
𝑎𝛽=1
0
𝑑𝑎]
𝑝𝑝
𝛤(𝑝)[∫
[1−𝑒𝑤−1(𝜏𝐷𝑎2
⁄−1
𝑒)+1]𝑎𝑝−1𝑒−𝑝𝑎
∞
𝑎𝛽=1
𝑑𝑎+∫
𝑎𝑝−1𝑒−𝑝𝑎
𝑎𝛽=1
0
𝑑𝑎]
𝑝𝑝
𝛤(𝑝)[∫
√𝜏 𝐷 𝑎2𝑎𝑝−1𝑒−𝑝𝑎
∞
𝑎𝛽=1
𝑑𝑎+∫
𝑎𝑝−1𝑒−𝑝𝑎
𝑎𝛽=1
0
𝑑𝑎]
Fp
3
2
1
3
2
1
3
2
1
Dis
tr.
GG
S
RR
B
Gam
ma
52
4.2.1 Gate-Gaudin-Schuhmann distribution
The effect of the GGS distribution for interface-reaction-controlled and pore-
diffusion-controlled cases is shown in Figures 4.2. The fraction reacted for nonuniform
particles is higher than for uniform particles at the beginning of the reaction. After some
time, the fraction reacted for nonuniform particles is lower than for uniform particles. This
is due to the fact that reaction of small particles is completed rapidly at the beginning of
the reaction and large particles take longer to complete the reaction. The maximum
differences of conversion between uniform and nonuniform particles for different
distributions, particle shapes, CVs, and rate-controlling steps are tabulated in Table 4.4.
The difference of fractional conversion greater than 0.15 is considered as a significant
difference in this work. When the CV values are less than 0.5 or are greater than 0.75, the
differences of fractional conversion are much less than or are much greater than 0.15.
Therefore, only the differences at CV of 0.5 and 0.75 are shown in Table 4.4. For the
particle shape factor of 3, which is most likely to be encountered in practical systems, and
the overall rate control by interface reaction, the effect of particle size distribution is small
until CV increases to 0.5 where the maximum difference of conversion rates is 0.086 at a
normalized time of 0.1. Once CV increases to 0.75, the maximum difference is 0.171 which
is a significant difference. In case of cylindrical particles, the difference is somewhat higher
than those for spherical particles. The maximum difference is 0.091 at a CV value of 0.5.
When CV is 0.75, differences tend to exceed 0.15. Also, flat particles show the maximum
difference of 0.148 at the CV value of 0.5 and a normalized time of 1. Like spherical and
cylindrical cases, the difference is higher than 0.15 at the CV value of 0.75.
For pore-diffusion control, significant differences appear as CV increases to 0.75,
53
Figure 4.2 Fraction reacted versus normalized time for the GGS distribution (a) Interface
reaction control and (b) Pore-diffusion control
54
Table 4.4 Maximum differences of conversion rate between uniform and nonuniform
particles with different distributions, particle shapes, CV, and rate controlling steps
𝑭𝒑 CV
GGS RRB Gamma
Interface
reaction
Pore
diffusion
Interface
reaction
Pore
diffusion
Interface
reaction
Pore
diffusion
3 0.5 0.086 0.093 0.071 0.080 0.065 0.075
0.75 0.171 0.180 0.135 0.146 0.128 0.141
2 0.5 0.091 0.098 0.078 0.085 0.072 0.081
0.75 0.178 0.185 0.143 0.152 0.137 0.147
1 0.5 0.148 0.141 0.130 0.130 0.123 0.116
0.75 0.198 0.198 0.169 0.169 0.165 0.165
55
which is the same trend as in the case of interface-reaction control. It is observed that the
difference of fractional conversion between uniform and nonuniform particles increases
with increasing CV value and decreasing shape factor. In brief, the differences caused by
the dispersion of particle size for the case of the GGS distribution are not significant until
CV increases to 0.5 irrespective of the type of rate-controlling step. When CV is higher
than 0.5, it is recommended to consider the effect of particle size distribution on the overall
kinetics.
4.2.2 Rosin-Rammler-Bennett distribution
Figure 4.3 represent the RRB distribution effects for interface-reaction-controlled
and pore-diffusion-controlled cases. The fraction reacted versus a normalized time for the
RRB distribution shows the same trend as in the case of the GGS distribution. However,
the differences of fractional conversion between uniform and nonuniform particles for the
RRB distribution is smaller than those for the GGS distribution with the same values of
CV and particle shape factor. For the case of interface-rate control in Figure 4.3a, the
maximum differences of 0.135, 0.143, and 0.169are observed for the CV value of 0.75
when the shape of particles are spheres, cylinders, and slabs, respectively. For pore-
diffusion control in Figure 4.3b, the maximum differences are 0.146, 0.152, and 0.169 for
the CV value of 0.75 for the three particle shapes. Thus, it can be said that the effect of the
RRB distribution on the kinetics for spherical and cylindrical particles under interface-
reaction control can be considered as insignificant until the CV is increased to 0.75. For
flat particles under interface-reaction control, the distribution effects are tolerable up to the
CV value of 0.5. In case of pore-diffusion control and spherical particles, the effect of
dispersion of particle size can be considered as tolerable up to the CV value of 0.75.
56
(a) Figure 4.3 Fraction reacted versus normalized time for the RRB distribution (a)
Interface reaction control and (b) Pore-diffusion control
57
However, for cylindrical and flat particles under pore-diffusion control, the effect of size
distribution is somewhat greater.
4.2.3 Gamma distribution
The effect of the Gamma distribution for interface-reaction-controlled and pore-
diffusion-controlled systems is illustrated in Figure 4.4. For interface-rate control in Figure
4.4a, 0.128, 0.137, and 0.165 are the maximum differences observed for the CV value of
0.75 when the shape of particles are spheres, cylinders, and slabs, respectively. For pore-
diffusion control and different particle shapes, the maximum differences are 0.141, 0.147,
and 0.165 at the CV value of 0.75. It shows very similar results to those of the RRB
distribution. The gamma distribution effects on the kinetics for spherical and cylindrical
particles under interface-reaction control are minor up to the CV value of 0.75. On the other
hand, in case of flat particles under interface-reaction control, the distribution effects are
acceptable only up to the CV value of 0.5. For spherical particles and cylindrical particles
under pore-diffusion control, the effect of particle size distribution is tolerable up to the CV
value of 0.75. However, for flat particles under pore-diffusion control, the effects are
insignificant only up to the CV value of 0.5. In summary for the Gamma distribution effects,
the effects are small until the CV increases to 0.75 for spherical and cylindrical particles
irrespective of the type of rate-controlling mechanism. When the CV value is higher than
0.75, it is recommended to consider the effect of particle size distribution on the overall
kinetics. For flat particles under both rate-controlling steps, the size variation effects on the
kinetics for a CV value higher than 0.5 should be considered significant.
Overall, the difference in conversion values between uniform and nonuniform
particles under pore-diffusion control is slightly higher than those under interface-reaction
58
Figure 4.4 Fraction reacted versus normalized time for the Gamma distribution (a)
Interface reaction control and (b) Pore-diffusion control
59
control. Thus, it can be surmised that when a system is rate-controlled simultaneously by
both chemical kinetics and pore diffusion along with the variation of particle size, the size
distribution effects in this system will be in between those for interface-reaction-controlled
case and pore-diffusion-controlled cases.
Another case frequently encountered in industry is that of porous solids. If the
overall reaction of a porous solid is slow and thus the fluid concentration is relatively
uniform throughout the porous solid, the fraction reacted will be the same regardless of the
pellet size. When pore diffusion controls the overall rate of a porous solid, the effects of
size will be the same as those for nonporous solids.12, 48-50
4.2.4 Comparison of the effect of three size distributions on the kinetics
The effects on reaction kinetics for the three types of size distributions are similar
as can be seen in Figures 4.2a, 4.3a, and 4.4a for interface-reaction control and Figures
4.2b, 4.3b, and 4.4b for pore-diffusion control although their probability density function
for size distribution presented in Figure 4.1 is quite different, especially for the GGS
distribution. If the CV value and the particle shape factor are the same, the mathematical
evaluation shows almost the same effect irrespective of the type of particle size distribution.
It is somewhat surprising especially for the GGS distribution, considering that its
probability density function is monotonous with particle size unlike for the other two size
distributions. Figure 4.5 describe the similarity of the reaction kinetics for the three size
distributions for spherical particles under interface-reaction control and pore-diffusion
control when the value of the CV is 0.3 and 1.5. For a CV value of 0.3 with both interface
reaction and pore-diffusion control, the difference among the three different curves is very
small. For a CV value of 1.5, the kinetics for the RRB and the Gamma distribution seems
60
Figure 4.5 Comparison for the three size distributions; 𝐹𝑝 = 3 and 𝐶𝑉 = 0.3 and 1.5
(a) Interface reaction control and (b) Pore-diffusion control
61
identical. The kinetics for the GGS distribution deviates rather slightly from the two other
curves. It is noted that the CV value of 1.5 is rarely encountered in practical systems. It has
thus been determined that the effect of particle size distribution on the reaction kinetics is
similar regardless of the type of distribution function as long as the CV values of the
distribution functions are the same. Thus, even when the effect of particle size distribution
is significant, it can be represented just by the CV value.
4.2.5 R2 values for the effect of particle size distributions
Another way to quantify the effects of particle size distribution on the kinetics is
to calculate R2 values which stand for the degree of the deviation of the overall conversion
rate from the case where the particle size is uniform. If R2 value is one, the conversion rate
of entire particle assemblages is the same as the case where the particle size is uniform.
When R2 value approaches to zero, the conversion rate shows the different trend from the
conversion rate of uniform particle size.
The R2 values in each case are tabulated in Table 4.5. As can be seen in the table,
R2 values are greater than 0.9 in every case where CV value is less than or equal to 0.5. If
we arbitrarily determine that R2 of 0.9 is the lowest value tolerable to be considered for the
particle size as uniform, the assumption that particle size is uniform can be valid in the
determination of fluid-solid kinetics in the case where CV is less than 0.5. On the other
hand, in the case where CV value is larger than 0.5, R2 values can be less than 0.9. Thus, it
can be said that the effects of particle size distribution should be considered in the fluid-
solid kinetics when CV is larger than 0.5.
62
Tab
le 4
.5 R
2 v
alues
of
each
cas
e
CV
1.5
0.6
36
3
0.5
32
7
0.1
30
4
0.5
32
5
0.4
71
2
0.3
10
8
0.8
08
7
0.7
40
6
0.4
75
3
0.7
01
8
0.6
52
2
0.5
22
9
0.7
75
0
0.6
99
7
0.4
07
5
0.6
66
3
0.6
14
8
0.4
80
5
1
0.8
52
5
0.8
02
6
0.6
19
9
0.7
54
6
0.7
27
2
0.6
50
8
0.9
11
1
0.8
75
8
0.7
40
4
0.8
36
9
0.8
11
0
0.7
39
2
0.9
11
1
0.8
75
9
0.7
40
4
0.8
36
9
0.8
11
0
0.7
39
2
0.7
5
0.9
32
3
0.9
04
5
0.8
07
5
0.8
60
5
0.8
42
8
0.7
96
4
0.9
54
6
0.9
34
4
0.8
57
9
0.9
02
6
0.8
86
3
0.8
41
2
0.9
57
9
0.9
38
8
0.8
65
0
0.9
08
5
0.8
92
1
0.8
46
5
0.5
0.9
80
8
0.9
70
2
0.9
32
0
0.9
46
4
0.9
35
8
0.9
10
6
0.9
85
4
0.9
77
3
0.9
45
8
0.9
58
8
0.9
50
2
0.9
27
2
0.9
87
2
0.9
79
9
0.9
50
2
0.9
63
0
0.9
54
7
0.9
31
5
0.4
0.9
91
0
0.9
85
3
0.9
62
7
0.9
70
5
0.9
62
9
0.9
45
1
0.9
92
8
0.9
88
2
0.9
69
4
0.9
76
1
0.9
70
1
0.9
54
5
0.9
93
7
0.9
89
7
0.9
72
1
0.9
78
7
0.9
73
1
0.9
57
6
0.3
0.9
96
8
0.9
94
4
0.9
83
1
0.9
87
1
0.9
82
7
0.9
71
8
0.9
97
3
0.9
95
2
0.9
85
7
0.9
88
9
0.9
85
4
0.9
76
2
0.9
97
6
0.9
95
8
0.9
87
1
0.9
90
2
0.9
87
0
0.9
78
1
0.2
0.9
99
3
0.9
98
7
0.9
94
6
0.9
96
3
0.9
94
5
0.9
89
7
0.9
99
4
0.9
98
8
0.9
95
3
0.9
96
7
0.9
95
2
0.9
91
1
0.9
99
5
0.9
98
9
0.9
95
8
0.9
97
0
0.9
95
7
0.9
91
9
0.1
1.0
00
0
0.9
99
9
0.9
99
3
0.9
99
6
0.9
99
3
0.9
98
4
1.0
00
0
0.9
99
9
0.9
99
3
0.9
99
7
0.9
99
4
0.9
98
5
1.0
00
0
0.9
99
9
0.9
99
4
0.9
99
7
0.9
99
5
0.9
98
7
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
1.0
00
0
Fp
3
2
1
3
2
1
3
2
1
3
2
1
3
2
1
3
2
1
Rea
ctio
n
Ch
emic
al
Dif
fusi
on
Ch
emic
al
Dif
fusi
on
Ch
emic
al
Dif
fusi
on
Dis
tr.
GG
S
RR
B
Gam
ma
63
4.2.6 Comments on particles with nonbasic shapes
Although many real particles have shapes that may be approximated by the three
basic shapes used in the above analysis, other particles have shapes that are intermediate
of the three basic shapes or more irregular. Examples are finite cylinders, parallelepipeds,
ellipsoids, completely irregular particles, or a mixture thereof. Assemblages made up of
particles with some of these shapes can be treated based on first principles, but even in such
cases the computation will be lengthy. Furthermore, with variations in particle shape, size
distribution, and reactivity even for the same solid as well as the errors typically present in
rate measurements, it is not worthwhile to spend such complicated computational efforts.
Here, the following suggestions for systems consisting of particles of nonbasic shapes can
be offered. Express the particle size in terms of 𝐴𝑝 and 𝑉𝑝 as defined by Equation 4.7.
This requires an estimation of the value of 𝐹𝑝, which depends on the ratios of the three
major dimensions of the particle and can be a noninteger between 1 and 2 or between 2 and
3. The exact appropriate value was found to be different by 0.1-0.3 depending on the rate-
controlling step.51 Further, differences of this extent in the value of 𝐹𝑝 have only modest
effects on the conversion versus time relationships regardless of the rate-controlling
mechanism. The use of such estimated 𝐹𝑝 values is made greatly more convenient by the
expression of conversion versus time relationships with 𝐹𝑝 as the parameter for different
particle shapes, especially for diffusion-controlled reactions. Thus, Equation 4.13 can be
used for any intermediate shapes, although this will involve evaluating the values of
β(𝑡, 𝑅) by an implicit method.
CHAPTER 5
KINETIC STUDY OF REDUCED UGS WITH EFFECT OF
PARTICLE SIZE DISTRIBUTION
5.1 Particle size distribution of reduced UGS
To study the effect of particle size of reduced UGS washed by acetic acid on the
reaction kinetics, the particle size distribution was analyzed as shown in Figure 5.1 by a
laser diffraction particle size analyzer. As can be seen in Figure 5.1, the particle size is not
uniform, but follows a particle size distribution. Again, the types of particle size distribution
do not affect the kinetics significantly according to Figure 4.5. Especially when CV value
is 0.3, the effect of types of particle size distribution is negligible. In this regard, the mean
and standard deviation of the distribution were found to be 0.096 and 0.032 μm ,
respectively, which gives the CV value (defined as standard deviation divided by mean) of
0.33. According to Table 4.5, R2 values for the CV value of 0.3 are larger than 0.97 in every
case. Especially, when we consider that the particle shape of reduced UGS washed by acetic
acid seems to be spherical based on the SEM image in Figure 3.2, the smallest R2 value for
shape factor of 3 is 0.9871. Thus, it will be assumed that the effect of particle size
distribution of reduced UGS washed by acetic acid is negligible in the calculation of
kinetics. In other words, it will be assumed that the particle size is uniform in the calculation
of the reaction kinetics of reduced UGS washed by acetic acid.
65
Figure 5.1 Particle size of reduced UGS after acetic acid washing
66
5.2 Determination of rate-controlling process of the reduced UGS
5.2.1 Interfacial reaction
The rate-controlling process and kinetic parameters were investigated on the basis
of the discussion in Chapter 5.1. If the overall rate of reaction is controlled by the interfacial
reaction, Equation 4.5 can be expressed for a normalized time parameter, t*, with respect
to conversion rate, X as follows13, 52 which is only valid under the assumption that the
particle size is uniform:
𝑡∗ = 1 − (1 − 𝑋)1/3 (5.1)
Experimental data obtained at three different temperatures, 70, 80, and 90oC were
applied to Equation 5.1 to evaluate interfacial reaction controlled kinetics as illustrated by
plotting 1 − (1 − 𝑋)1/3 versus time as shown in Figure 5.2. The value of 1 − (1 − 𝑋)1/3
is linearly related to leaching time, which suggests that the overall rate of reaction can be
controlled by the interfacial reaction in the leaching process.
The activation energy can be used to evaluate the rate-controlling step. The
Arrhenius equation, when rearranged for determining the activation energy is described in
Equation 5.2.52
𝑙𝑛𝑘 = 𝑙𝑛𝐴 −𝐸𝑎
𝑅𝑇 (5.2)
where k is the rate constant of a chemical reaction, A the preexponential factor, Ea the
activation energy, R the universal gas constant, and T the temperature in degrees Kelvin.
Figure 5.3 shows the relationship between lnk and 103/T in which the activation energy can
be determined from the slope of the line. The activation energy was determined to be 73.9
kJ/mole, suggesting interfacial reaction controlled kinetics.
67
Figure 5.2 Relationship between the values of 𝟏 − (𝟏 − 𝐗)𝟏/𝟑 and time at three
temperatures
68
Figure 5.3 Arrhenius Plot
69
5.2.2. Solid state diffusion
It is worthwhile to note that the iron removal in the reduced UGS is not increased
although the leaching temperature was increased from 140oC to 190oC. If the overall
reaction was controlled by interface reaction, iron removal would be increased with
increase of temperature.
In this regard, the other possibility of rate-controlling process of the reduced UGS
is solid state diffusion of iron in the titanium hydride, which also has high value of the
activation energy. The average particle size of the reduced UGS is 0.1 μm, which is fairly
small for the reduced UGS to retain enough pore size. Thus, overall rate of reaction might
be controlled by the diffusion of iron in the titanium hydride. If leaching temperature is
high enough for iron to be stably dissolved in titanium hydride, increasing the leaching
temperature might oppress iron removal. In fact, it was observed that iron removal in 0.05
M hydrochloric acid rather decreased from 79.59% to 79.19% as temperature increased
from 140oC to 190oC.
Overall, several experimental evidences indicate that the rate-controlling process
of iron removal from the reduced UGS might be controlled by solid state diffusion. Figure
5.4 shows the probable micro-scale structure of the reduced UGS. This figure indicates that
the effectiveness of iron removal may be limited by locking up of iron as interstitial
substitutions.
5.2.2.1 Mathematical formulation
As can be seen in Figure 3.8, the overall particle shape is assumed to be sphere.
Solid state diffusion equation in a spherical particle according to Fick’s second law can be
given as,
70
Figure 5.4 Possible schematic diagram of the reduced UGS (Brown: TiH2, Black: FexOy, or
Fe, blue: other impurities, gray: hydrogen ion). In this scenario, the reaction is controlled
by solid state diffusion of iron.
71
𝜕𝐶
𝜕𝑡= 𝐷(
𝜕2𝐶
𝜕𝑟2+2
𝑟
𝜕𝐶
𝜕𝑟) (5.3)
where C is concentration and D is diffusivity. By setting u = Cr in Equation 5.3, it can be
rewritten as,
𝜕𝐶
𝜕𝑡= 𝐷
𝜕2𝑢
𝜕𝑟2 (5.4)
Under the condition that concentration changes with variation of time, the solution of
Equation 5.4 can be expressed as,53
𝐶−𝐶1
𝐶0−𝐶1= 1 +
2𝑎
𝜋𝑟∑
(−1)𝑛
𝑛𝑠𝑖𝑛
𝑛𝜋𝑟
𝑎exp(−
𝐷𝑛2𝜋2𝑡
𝑎2)∞
𝑛=1 (5.5)
where the boundary conditions and initial condition are
𝑎𝑡 𝑟 = 0, 𝑢 = 0, 𝑡 > 0 (5.6)
𝑎𝑡 𝑟 = 𝑎, 𝑢 = 𝑎𝐶𝑜 , 𝑡 > 0 (5.7)
𝑎𝑡 𝑡 = 0, 𝑢 = 𝑟𝐶1 (5.8)
where a is radius of a particle, Co is surface concentration, and C1 is initial uniform
concentration. When surface concentration of iron in the reduced UGS, Co, is 0 and the
initial uniform concentration if iron in the reduced UGS, C1, is 1, the total amount of
diffusing iron which leaves the titanium hydride can be given as,54
𝑴𝒕
𝑴∞= 𝟏 −
𝟔
𝝅∑
𝟏
𝒏𝟐𝐞𝐱𝐩(−
𝑫𝒏𝟐𝝅𝟐𝒕
𝒂𝟐)∞
𝒏=𝟏 (5.9)
The diffusion coefficient of iron in titanium hydride is not found in reality, so it is
difficult to judge whether iron removal from the reduced UGS follows solid state diffusion.
However, predicted and experimental iron removal curves were plotted in Figure 5.5 under
72
Figure 5.5 Comparison of predicted values produced from the equation for solid state
diffusion and experimental value; 0.1 M HCl, 1,000 rpm, 90oC, no boric acid
73
the assumption that the value of diffusion coefficient of iron in titanium hydride is 3E-16
cm2/s at 90oC, which gives the best fit of values using the Equation 5.9 to fit the
experimental data obtained under the condition in Table 3.3.
It should be noted that titanium loss was drastically increased at 8 h as shown in
Table 5.1. Decreasing the trend of titanium loss after 8 h can be due to the precipitation of
titanium ion as titanium dioxide. Dissolution of titanium hydride might allow hydrochloric
acid to contact with iron easily due to the disappearance of the titanium hydride shell, which
was supposed to prevent hydrochloric acid from reacting with iron. Therefore, predicted
and experimental iron removal curves until 4 h was plotted again in Figure 5.6 to mitigate
the effect of titanium dissolution by hydrochloric acid when the diffusion coefficient of
iron in titanium hydride is 1E-16 cm2/s, which gives the best fit. Figure 5.6 shows fairly
good agreement of experimental data with theoretical data. Thus, it can be said that solid
state diffusion of iron in titanium hydride can be the rate-controlling process which governs
the overall rate of reaction.
However, due to the fact that the additional reaction such as the dissolution of
titanium hydride in hydrochloric acid can affect the iron removal rate after 4 h of leaching
time, it is difficult to judge the rate-controlling process of iron removal from the reduced
UGS.
5.2.3 Effect of external mass transport
The influence of external mass transport on the reaction rate of leaching with 0.1
M hydrochloric acid at 90oC was investigated by varying the stirring speed in the range of
100 – 1,000 rpm. The data in Figure 5.7 show that the reaction rate did not change
significantly with increasing stirring speed, which implies that the overall rate of reaction
74
Table 5.1 Titanium loss with leaching time. Leaching parameters: .1 M HCl, 1000 rpm,
90oC, no boric acid
Leaching time (hr) Titanium loss (%)
1 0.44
2 0.29
4 0.35
8 2.04
12 3.46
16 4.30
20 2.10
24 1.63
75
Figure 5.6 Comparison of predicted values produced from the equation for solid state
diffusion and experimental value until 4 h of leaching time
76
Figure 5.7 Iron removal versus time at three rotation speed
77
is not controlled by mass transport through an outer boundary layer.
To sum up, the overall rate of reaction seems to follow interfacial reaction
controlled kinetics. The rate-controlling process of reaction was determined under the
assumption that the diameter of every single particle is identical. This assumption believed
to be valid due to the fact that the CV value of the particle size distribution is 0.33, which
gives R2 value of more than 0.97. When we consider that the reaction takes place based on
hydrometallugical processing, which is known as a slower process than pyrometallurgical
processing, the fact that the reaction is controlled by interfacial reaction seems to be
reasonable.
CHAPTER 6
SELECTIVE RECOVERY OF TITANIUM FROM LEACHED
ILMENITE USING TANNIC ACID
6.1 Materials and methods
6.1.1 Preparation of titanium solution from ilmenite
Sixteen molar sulfuric acid was chosen as a lixiviant to prepare a titanium
containing multielement solution from ilmenite. Ten grams of ilmenite was added into 50
ml of sulfuric acid in a 500 ml round-bottom boiling flask at the room temperature. Twenty
grams of ammonium sulfate ((NH4)2SO4) was also added to elevate the boiling point of the
sulfuric acid solution during leaching. A 200 ml Graham condenser which has an inner coil
for additional surface area was applied on the neck of the flask to maintain the volume. The
solution was refluxed and the leaching was performed for 8 h. A magnetic stirring bar at
1,000 rpm was used for homogenizing the solution. After the digestion process, the solution
is allowed to settle. A dark brownish multielement sulfate solution was obtained as
supernatant liquid was collected for chelation.
6.1.2 Synthesis of titanium-tannic acid complex
Fifty grams of tannic acid in 200 ml water was placed in a 500 ml three-neck
flask and constantly stirred by a magnetic stirring bar at a rotational speed of 500 rpm. A
200 ml separatory funnel fitted with a stopcock, containing titanium bearing solution, was
installed on one of necks of the flask. A pH probe was also inserted through another neck
79
and the third neck was sealed. Ammonium hydroxide was added from the separatory funnel
so that the pH was maintained at 4.5 during the precipitation of the dark orange-colored
precipitate. The experiment was carried out at room temperature. The precipitate was
collected by vacuum filtration, washed with 2% hydrochloric acid solution, followed by
cold deionized water and isopropyl alcohol, dried at room temperature, and ground into
powder. The filtrate was investigated by ICP-OES. The solid samples were characterized
using SEM, energy-dispersive X-ray spectroscopy (EDS), and Fourier transform infrared
spectroscopy (FTIR).
6.1.3 Methodology of DFT simulation
DFT simulation using general ab initio quantum chemistry package, Gaussian 09,
was carried out to determine optimized geometry of titanium-tannic acid complex and the
stability based on highest occupied molecular orbital (HOMO) and lowest unoccupied
molecular orbital (LUMO). B3LYP55, 56 for the exchange-correlation energy functional
calculation was employed with a basis of 6-31G, which has been known as an effective
method for the simulation of organic molecules.57 Also, the optimization was conducted
without symmetry constraints.
6.2 Results and discussion
6.2.1 Experiment
The multielement solution, which was prepared by dissolving ilmenite in
concentrated sulfuric acid, was composed of 22.64% (w/w) of titanium, 71.28% (w/w) of
iron, and 6.08% (w/w) of other impurities determined by ICP-OES. After precipitating out
titanium-tannic acid complex with tannic acid using ammonium hydroxide to adjust the pH
80
to 4.5, the yield of titanium was found to be 95.58% (w/w). Figure 6.1 shows the surface
morphology and the chemical composition of titanium-tannic acid. The particle shape looks
irregular and it forms an agglomerate. The sizes of individual particle are only few micro
meters. The powder has 98.6% (w/w) of titanium, 0.2% (w/w) of iron, and 1.2% (w/w) of
other impurities. When it is considered that iron is difficult to remove in titanium
production, Figure 6.1b shows an important capability of tannic acid to extract titanium
selectively from the solution containing impurities such as iron, magnesium, and silica.
Figure 6.2 shows FTIR spectra of (a) titanium and (b) titanium-tannic acid
complex. The IR spectra present peaks at a broad range of 3500~3200 cm-1 due to the O-
H stretching vibration in both tannic acid and titanium-tannic acid complex.58 However,
the positive shift and decreased intensity of the peak for O-H bond in the titanium tannic
acid complex were observed due primarily to coordination of phenolic –OH group with
titanium ion by means of deprotonation.59 This can suggest that tannic acid tends to form
complexes with titanium using hydroxyl groups attached to benzene rings. Several peaks
below 1800 cm-1 are due to tannic acid including benzene rings and carboxylic acids. C=O
stretching vibration at 1685 and 1706 cm-1 for tannic acid and titanium-tannic acid complex
was observed.55 The peaks 1500~1400 cm-1 account for C-C stretching vibration in
aromatic rings. Significant changes in the spectrum of titanium-tannic acid complex at a
range of 1700~1450 cm-1 may be due to the chelation process for the formation of metal
complex. The peaks due to C-O stretching vibration and O-H bending in benzene rings and
carboxylic acids appear at 1300~1000 cm-1 in both Figures 6.2a and b.55 The peaks below
900 cm-1 can be explained by C-H bonding in benzene rings. It is worthwhile to be noted
that peaks below 1452 cm-1 in tannic acid match with peaks in titanium-tannic acid complex,
81
Figure 6.1 (a) SEM image of titanium-tannic acid complex, (b) the composition
of titanium-tannic acid complex analyzed by EDS
82
Figure 6.2 FT-IR spectra of (a) tannic acid (b) titanium-tannic acid complex
83
which may mean that there is no significant participation of carboxylic groups in
coordination with titanium ion.
6.2.2 Simulation
Based on the fact that titanium ion forms an ionic bonding with two –OH groups
in gallic acid,37 Figure 6.3 shows a probable schematic diagram of titanium-tannic acid
complex and optimized geometry determined by DFT simulation. The geometry was fully
stabilized.
A LUMO-HOMO gap is known as the important parameter that can describe the
degree of stability of the complexes.60 A molecule having a larger LUMO-HOMO gap is
more stable than a molecule having a smaller LUMO-HOMO gap. Also, according to a
concept of hard and soft Lewis acids and bases (HSAB), the global hardness of a molecule
is the representative stability index of its complex.61 A high value of the global harness
implies a highly stable complex. In the DFT simulation, the LUMO-HOMO gap was
calculated to be 0.08934 a.u (2.43 eV) as shown in Figure 6.4 and the global hardness (h)
was found to be 0.04467 a.u based on Equation 6.1.
h = (𝐸𝐿𝑈𝑀𝑂 − 𝐸𝐻𝑂𝑀𝑂)/2 (6.1)
where h is the global hardness, 𝐸𝐿𝑈𝑀𝑂 the energy of the lowest unoccupied molecular
orbitals, and 𝐸𝐻𝑂𝑀𝑂 the energy of the highest occupied molecular orbitals. The LUMO-
HOMO gap and the global hardness of titanium-tannic acid complex are large enough to
indicate the energetic feasibility and stability for the formation of titanium-tannic acid
complex.
Also, it is interesting to note that electron density mainly consists of carbon and
oxygen from a π-conjucated gallic acid unit located at the end of the molecule in both
84
Figure 6.3 (a) The schematic diagram of titanium and tannic acid and (b) optimized
structure by DFT simulation at B3LYP with 6-31G: gray (carbon), red (oxygen), white and
big atom (titanium), white and small atom (hydrogen).
85
Figure 6.4 The molecular orbitals’ energy levels of titanium-tannic acid complex at
LUMO+1, LUMO, HOMO, and HOMO-1: green (positive isosurface of molecular
orbitals at the energy level of each molecular orbital), red (negative isosurface of
molecular orbitals at the energy level of each molecular orbital).
86
HOMO and HOMO-1 levels in which there is no contribution from the titanium atom.
On the other hand, in the LUMO level, molecular orbitals are uniformly distributed
around the titanium atom and the adjacent gallic acid group, which has empty π-orbitals
of benzene rings. It is also observed in the LUMO+1 level that the distribution of molecular
orbitals is somewhat similar to that in the LUMO level, but there is a more significant
localization in the titanium atom and less contribution from digallic acid group. This can
be explained by the relative energetic alignment between molecular orbitals of titanium
and gallic acid groups. Titanium atoms in LUMO are lower in energy than digallic acid
groups. Thus, titanium in titanium-tannic acid complex has a possibility of forming
titanium hydride in a hydrogen atmosphere as the titanium atom can easily accept excited
electrons.
CHAPTER 7
CONCLUSIONS
In summary, this thesis work mainly evaluates methods of obtaining pure titanium
hydride from different sources and methods. The first route was to use DRTS in which
metallic titanium can be obtained by dehydrogenation of titanium hydride after impurities
are removed. The leaching characteristics of iron removal from the reduced upgraded
titanium slag was studied with mild hydrochloric and boric acids under ambient pressure
and elevated pressure. Under the constraint that 1% (w/w) of titanium hydride loss is the
maximum amount tolerable, 0.1 M turned out to be the most effective concentration of
hydrochloric acid for iron removal from the reduced UGS. Thus, 0.05 M and 0.1 M
hydrochloric acid were chosen as leaching solutions with or without 1 M boric acid. 90oC,
140oC, and 190oC were studied for optimization of leaching temperature.
Effect of boric acid on iron removal from reduced UGS was strongly dependent
on acidity and temperature of the leaching environment. Solutions of 1 M boric acid with
0.05 M hydrochloric acid at 90oC showed significant improvement of iron removal, but
with 0.1 M hydrochloric acid, a reduced effect was observed. At higher temperature (140
and 190oC), boric acid did not affect iron removal, regardless of the concentration of
hydrochloric acid.
The extent of iron removal increased significantly with the increase of temperature.
Once temperature reaches about 140oC, 87.63% of iron removal was achieved with 0.1 M
88
hydrochloric acid. Additional iron removal was not observed at higher temperature (190oC).
To sum up, the best leaching conditions from those studied were found to be 0.1 M
hydrochloric acid, no boric acid, and 140oC.
A factorial design of experiment (FDE) for equation modeling was carried out to
study the influence of three factors (temperature and concentrations of hydrochloric, and
boric acids) on iron removal. The predicted iron removal obtained by the FDE model
showed good agreement with actual iron removal within the range of condition evaluated.
To verify the overall reaction kinetics, shrinking unreacted core model was
introduced with an assumption that every single particle has same diameter. Thus,
additional study was carried out to justify the assumption, which utilized the evaluation of
the effects of three empirical size distributions, the Gate-Gaudin-Schuhmann distribution,
the Rosin-Rammler-Bennett distribution, and the Gamma distribution, on the fluid-solid
reaction kinetics. The effects were analyzed for the two extreme rate-controlling steps,
either interface reaction or pore diffusion, for the shrinking unreacted-core model and three
basic particle shapes. The expressions for overall conversion rate of entire particle
assemblages were derived mathematically, and calculated by a technical computing
language, “MATLAB”.
According to the calculation, as the degree of spread of particle size distribution
increases, the reaction rates deviate progressively from the case of uniform particles. Also,
the effect of particle size distribution is somewhat greater for pore-diffusion-controlled
reactions than for interface-reaction-controlled cases, although the difference is rather
small. The case of mixed control is expected to be in between the two controlling steps.
In the case of spherical particles, the effect of the GGS distribution on the kinetic
89
is small for both rate controlling steps until the CV increases to 0.5. For the RRB and the
Gamma distributions, the effect remains small up to a CV value of 0.75. The effects on
cylindrical and flat particles are similar, if slightly greater in that order. We can also
evaluate the effect of particle size distribution on the kinetics using R2 values. If we
arbitrarily determine that R2 of 0.9 is the lowest value tolerable to be considered for the
particle size as uniform, the assumption that particle size is uniform can be valid in the
determination of fluid-solid kinetics in the case where CV is less than 0.5. On the other
hand, in the case where CV value is larger than 0.5, the effects of particle size distribution
should be considered in the fluid-solid kinetics.
The effects of the types of size distribution on the overall rate is similar. When the
effect of particle size distribution is significant and must be taken into consideration, the
spread can be represented by the value of the CV regardless of the form of the distribution
function. Also, the effect of size distribution on the reaction rate of porous particles is
expected to be smaller than on nonporous particles, because there will be no effect of size
distribution if pore diffusion is fast unlike for nonporous solids, and the effect will be
identical to the case of nonporous particles if pore diffusion controls the overall reaction
rate of porous particles.
Based on the theoretical kinetic study for the effect of particle size distribution,
justification of the assumption suggested for the assessment of reaction kinetics of reduced
UGS can be evaluated. The coefficient of variation of reduced UGS defined as standard
deviation divided by mean was found to be 0.33. R2 values for the CV values of 0.3 are
larger than 0.98 in every spherical particle case. Therefore, it was assumed that the effect
of particle size distribution of reduced UGS does not have to be considered in the
90
calculation of kinetics. Based on this calculation, a rate-controlling process can be found
and it seems to follow interfacial reaction controlled kinetics. The activation energy of the
reaction was determined to be 73.9 kJ/mole.
However, the reaction-controlling process may be controlled by solid state
diffusion of iron in titanium hydride based in part, on the fact that the iron removal in the
reduced UGS is not increased with increase of leaching temperature from 140oC to 190oC.
If leaching temperature is high enough for iron to be stably dissolved in titanium hydride,
increasing the leaching temperature might oppress iron removal. Therefore, the
effectiveness of iron removal may be limited by locking up of iron as interstitial
substitutions. However, the rate-controlling process of this mechanism is difficult to
analyze to completely due to the limited information about the diffusion coefficient of iron
in titanium hydride and involvement of other reactions such as dissolution of titanium
hydride.
An additional way of obtaining pure titanium involves the extraction of titanium
from ilmenite. The strong affinity of tannic acid towards metal ions having high oxidation
states was explored to recover titanium selectively from a multielement solution prepared
by dissolving ilmenite in sulfuric acid and ammonium sulfate. Titanium-tannic acid
complex was selectively precipitated from the multielement solution at pH 4.5. The
precipitate was separated by vacuum filtration and the filter cake was washed with 2%
hydrochloric acid solution, deionized water, and cold isopropyl alcohol after filtration. The
washed filter cake was dried at room temperature and ground into powder on drying. The
titanium-tannic acid complex so obtained was composed of 98.6% titanium, 0.2% iron, and
1.2% other impurities. The density functional theory simulation at B3LYP method with a
91
basis of 6-31G was also conducted to identify optimized complex structure and its stability.
The result showed an energy gap of 2.43 eV between two frontier orbitals (highest occupied
molecular orbitals and lowest unoccupied molecular orbitals), which is large enough to
indicate energetic feasibility of the complex formation.
REFERENCES
1. Zhou, Y.G.; Zeng, W.D.; Yu, H.Q. Mater. Sci. Eng. A 2005, 393, 204-212.
2. Boyer, R.R. Mater. Sci. Eng. A 1996, 213, 103-114.
3. Niinomi, M. Sci. Tech. Adv. Mater. 2003, 4, 445–454.
4. Gorynin, I.V. Mater. Sci. Eng. A 1996, 263, 112-116.
5. Hua, F.; Mon, K.; Pasupathi, P.; Gordon, G.; Shoesmith, D. Corrosion 2005, 61,
987-1003.
6. Amarchand, S.; Rama Mohan, T.R.; Ramakrishnan, P. Adv. Powder Technol. 2000,
11, 415–422.
7. Froes, F.H.; Eylon, D. Int. Mater. Rev. 1990, 35, 162–182.
8. Middlemas, S.; Fang, Z.Z.; Fan, P. Hydrometallurgy 2013, 131, 107–113.
9. Fang, Z.Z.; Middlemas, S.; Fan, P.; Guo, J. J. Am. Chem. Soc. 2013, 135, 18248–
18251.
10. ASTM B299-13; Standard Specification for Titanium Sponge In ASTM
International; West Conshohocken, Pennsylvania, 2013.
11. Peak, D.; Luther III, G.W.; Sparks, D. L. Geochim. Cosmochim. Acta 2013, 67,
2551–2560.
12. Elrod, J.A.; Kester, D.R. J. Solut. Chem. 1980, 9, 885–894.
13. Szekely, J.; Evans, J.W.; Sohn, H.Y.; Gas-Solid Reactions; Academic Press, New
York, 1976.
14. Sohn, H.Y.; Baek, H.D. Metall. Trans. B 1998, 20, 107-110.
15. Homma, S.; Ogata, S.; Koga, J.; Matsumoto, S. Chem. Eng. Sci. 2005, 60, 4971-
4980.
93
16. Herbst, J.A.; Rate Processes In Multiparticle Metallurgical Systems In Rate
Processes of Extractive Metallurgy; Sohn, H.Y., Wadsworth, M.E., Eds.; Plenum,
New York, 1979; pp 53-112,
17. Ahmed, M.M.; Ahmed, S.S. J. Eng. Sci. 2008, 36, 147-166.
18. Macias-Garcia, A.; Cuerda-Correa, E.M.; Miaz-Diez, M.A. Mater. Cha. 2004, 52,
159-164.
19. Garabrant, D.H.; Fine, L.J.; Oliver, C.; Bernstein, L.; Peters, J.M. Scandi. J. Wo.
Envir. Heal. 1987, 13, 47-51.
20. Park, I.; Abiko, T.; Okabe, T.H. J. Phys. Chem. Solids 2005, 66, 410–413.
21. Kroll, W.J. Trans. Electrochem. Soc. 1940, 78, 35–47.
22. Takeda, O.; Okabe, T.H. Mater. Trans. 2006, 47, 1145–1154.
23. Chen, G.Z.; Fray, D.J.; Farthing, T.W. Nature Letter 2000, 407, 361-364.
24. Chen, W.; Yamamoto, Y.; Peter, W.H.; Gorti, S.B.; Sabau, A.S.; Clark, M.B.;
Nunn, S.D.; Kiggans, J.O.; Blue, C.A.; Williams, J.C.; Fuller, B.; Akhtar, K. Pow.
Tech. 2011, 214, 194-199.
25. Büchel, K.H.; Moretto, H.H.; Werner, D.; Woditsch, P.; Industrial Inorganic
Chemistry; Wiley-VCH, Mörlenbach, 2000; pp 548-558.
26. Froes, F.H.; Titanium: Physical Metallurgy, Processing, and Applications; ASM
International, Ohio, 2015.
27. El-Hazek, N.; Lasheen, T.A.; El-Sheikh, R.; Zaki, S.A. Hydrometallurgy 2007, 87,
45–50.
28. Ogasawara, T.; Veloso de Araujo, R.V. Hydrometallurgy 2000, 56, 203–216.
29. Van Dyk, J.P.; Vegter, N.M.; Pistorius, P.C. Hydrometallurgy 2000, 65, 31–36.
30. Hulburt, H.M.; Katz, S. Chem. Eng. Sci. 1964, 19, 555.
31. Randolph, A.D. Can. J. Chem. Eng. 1964, 42, 280.
32. Randolph, A.D.; Larson, M.; Theory of Particulate Processes; Academic Press,
New York, 1971; pp 50-76.
33. Mcllvried, H.G.; Massoth, F.E. Ind. Eng. Chem. Fundam. 1973, 12, 225-229.
34. Svarovsky, L.; Solid-Liquid Separation; Butterworths, London, 1977; pp 15-23.
94
35. Iffat, A.T.; Maqsood, Z.T.; Fatima, N. J. Chem. Soc. Pak. 2005, 27, 174-178.
36. Surleva, A.; Atanasova, P.; Kolusheva, T.; Costadinnova, L. J. Chem. Tech. Met.
2014, 49, 594-600.
37. Araujo, P.Z.; Morando, P.J.; Blesa, M.A. Langmuir 2005, 21, 3470-3474.
38. Borgais, B.A.; Cooper, S.R.; Koh, Y.B.; Raymond, K.N. Inorg. Chem. 1984, 23,
1009-1016.
39. Hem, J.D.; Complexes of Ferrous Iron with Tannic Acid In Chemistry of Iron in
Natural Water; United States Government Printing Office, Washington, 1962; pp
75-94.
40. Finar, I.L.; Organic Chemistry; English Language Book Society & Longman
Group Limited, London, 1975.
41. Astrelin, I.M.; Prokofyeva, G.N.; Suprunchuk, V.I.; Knyazev, Y.V.; Panashenko,
V.M.; Morozov, I.A.; Morozova, R.A.; Interaction of TiHx (x < 2) with solutions
of some acids and alkalies In Hydrogen Materials Science and Chemistry of Metal
Hydrides; Veziroglu, T.N., Zaginaichenko, S.Y., Schur, D.V., Trefilov, V.I., Eds.;
Kluwer Academic Publishers, Netherlands, 2002; pp 133-140,
42. Sohn, H.Y.; Sohn, H.J. Ind. Eng. Chem. Process Des. Dev. 1980, 19, 237-242.
43. Perry, J.H.; Chemical Engineers’ Handbook; McGraw-Hill, New York, 1950; p
67.
44. Corless, R.M.; Gonnet, G.H.; Hare, D.E.G.; Jeffrey, D.J.; Knuth, D.E. Adv. Com.
Math. 1996, 5, 329–359.
45. Schuhmann, R. A.I.M.E. Tech. Publ. 1940, 1189, 1-11.
46. Rosin, P.; Rammler, E. J. Inst. Fuel 1933, 7, 29-36.
47. Belz, M.H.; Statistical Methods for the Process Industries; Wiley, New York, 1973;
p 98.
48. Sohn, H.Y.; Szekely, J. Chem. Eng. Sci. 1972, 27, 763-778.
49. Sohn, H.Y.; Chaubal, P.C. AIChE J. 1986, 32, 1574-1577.
50. Sohn, H.Y. Metall. Trans. B 1978, 9B, 90-96.
51. Sohn, H.Y.; Unpublished Work In Graduate course on Advanced Fluid-Solid
Reaction Engineering; University of Utah, Salt Lake City, Utah, 1981.
95
52. Free, M.L.; Hydrometallurgy: Fundamentals and Applications; John Wiley &
Sons, lnc., New Jersey, 2013.
53. Porter, D.A.; Easterling, K.E.; Phase Transformations in Metals and Alloys; VNR
International, Berkshire, 1988.
54. Crank, J.; The Mathematics of Diffusion; Clarendon Press, Bristol, 1975.
55. Becke, A.D. J. Chem. Phys. 1993, 98, 5648-5652.
56. Lee, C.; Yang, W.; Parr, R.G. Phys. Rev. B 1988, 37, 785-789.
57. Tirado-Rives, J.; Jorgensen, W.L. J. Chem. Theo. Comp. 2008, 4, 297-306.
58. Larkin, P.J.; IR and Raman Spectroscopy: Principles and Spectral Interpretation;
Elsevier, Amsterdam, 2011.
59. Peres, R.S.; Cassel, E.; Azambuja, D.S. ISRN Corrosion 2012, 9-19.
60. Zhou, Z.; Parr, R.G. J. Am. Chem. Soc. 1990, 112, 5720–5724.
61. Pearson, R.G. J. Am. Chem. Soc. 1963, 85, 3533–3539.
