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The thermodynamics and kinetics are the fundamental knowledge in physical chemistry. In the next following passages, the fundamentals of thermodynamic modelling and kinetics study is reviewed, particularly for the application in high temperature processing such as the silicothermic processes.
3.1 Thermodynamic Modelling
Thermodynamic modelling has been widely used in high temperature materials processing. The purpose of thermodynamic modelling in materials processing is to predict multi-phase equilibria, which will specify the limits of the process and also hopefully lead to a better understanding of complicated processes. For example, predicting the partitioning of phosphorus between slag and metal in a steelmaking reactor . The chemical composition of phases can be predicted from thermodynamic models, in conditions far from the condition studied experimentally. This is clearly beneficial for process optimisation and determining the favourable working parameters in an industrial process . This also means that thermodynamics can be used as a tool for process development . Thermodynamic modelling can also be used for constructing phase diagram for alloy development, equilibria prediction in metal production such as iron, steel, rare earth metal and non-ferrous metal . The use of thermodynamic modelling has increased significantly in the last three decades, as the availability of computing power and data storage has fuelled the general growth of computational techniques. There has been significant development of thermochemical databases and commercial software for ready use by researchers and practitioners alike. However, like many computational techniques, various assumptions and simplifications are required to deal with complex phenomena. In the case of thermodynamic modelling, there are significant challenges in modelling the solution behaviour, particularly for oxide and metallic phases.
In this chapter, the fundamental of thermodynamic modelling will be described, particularly in reference to modelling high temperature systems, where the assumption of equilibrium is often valid because of the fast chemical kinetics and high rates of mass and heat transfer associated with these conditions. An example of thermodynamic modelling of magnesium production will also be presented in this paper. In this context, thermodynamic modelling is a valuable tool for assessing the impact of different processing routes and feed materials on product purity.
3.1.1 Gibbs Energy Minimisation
The Gibbs energy  minimisation technique [6, 7] is a powerful approach for determining phase equilibria. When mixture of components in multi-phase system are not in equilibrium, the Gibbs energy of such a system is high and reactions that will lower the total free energy of the system to a minimum are thermodynamically favoured. In the Gibbs minimisation method, the total Gibbs energy of all phases in the system is at a minimum and follows Eq. (1).
Where ni is the mol of species i and Gi is the Gibbs energy of i in ï¦ phase. The ni-values must be non-negative and mass balance constraints, an equation summing up the total moles of that element, must be satisfied as in Eq. (2).
aji is the number of g atoms of element j and bj is the total atom of element j. In the method described by Eriksson , Lagrange's method of undetermined multipliers is used for determining the constrained minimum, and the logarithmic equation thus obtained are expanded in a Taylor series about initially estimated ni-values, neglecting second and higher orders. The equilibrium amounts are therefore obtained after a series of iterations. This procedure is repeated until the constant ni-values is achieved, and the set of phases and composition can be calculated.
The Gibbs energy itself has temperature dependence which can be expressed as a power series, as shown in Eq. (3.3).
In real multi-component and multi-phase systems, the Gibbs energy of each phase can be divided into three contributions:
Go is a contribution of the Gibbs energy of pure element in ï¦ phase. Gideal corresponds to the Gibbs energy of mixing of ideal solution, while Gxs is the excess term, which relates to real behaviour of solution. The excess Gibbs energy of mixing of the real solution is associated with activity coefficient of species by:
where ï§ is the activity coefficient of component i in the i - j solution. Thus the Gibbs energy excess will represent the activity behaviour. The contribution of Gibbs energy excess relates to solution activities and depends on the type of solution, which will be explained in the solution models section.
3.1.2 Database Development
Reliable thermochemical databases for species are vital inputs for thermodynamic calculations. Thermochemical databases was developed from critical analysis of experimental data primarily obtained using calorimetric techniques but also from other techniques such as Differential Thermal Analysis (DTA), Electromotive Force (EMF) technique, vapour pressure measurement , metallography, X-Ray Diffraction , as well as high temperature phase equilibration followed by quenching and electron microprobe x-ray analysis . The Gibbs energy data is expressed mathematically as a function of temperature, as shown in Eq. (3). Entropy, enthalpy, and heat capacity data is expressed as per Eq. (6) to (8).
These property databases are collected and assessed in thermochemical software as data sources for thermodynamic calculations.
In the past thirty years, there has been a significant development of thermochemical database through systematic collation and assessment of data. The most widely used is the SGTE (Scientific Group Thermodata Europe) database. The development of this database started in 1979 (Dinsdale, 1991). This common databank was formed through collaboration by NPL (National Physical Laboratory), AEA Harwell in the UK, the University of Grenoble in France, and RWTH in Aachen, Germany. The SGTE Unary Database contains data of around 78 elements and their compounds (Dinsdale, 1991). The NPL database was developed in the 1980s in England, but was later merged with the SGTE .
Table 3. 1 Example of various thermodynamic models
Example of Application
No enthalpy mixing and volumetric change
Vapour at atmospheric pressure
Dilute solution with Henrian Activity
Henry's law for dilute solution
Carbon in liquid iron
Interaction parameter between binary solution is independent of composition (0L)
MgO-FeO oxide solid solution 
Interaction parameter between binary solution is 0L and 1L (function of temperature and linear to composition)
Liquid Fe-Ni solution
Redlich-Kister Polynomial 
Disordered phase and substitutional solid solution. Interaction parameter is function of compositional and temperature (0L, 1L, 2L, etc)
Disordered liquid, hcp, fcc and bcc solid solution 
two different sublattice
Ni-Al solid solution
Compound Energy Formalism  
Development of on sublattice model
Solid solution with interstitial and vacancy. Spinel, double oxide, double salt 
Modified Quasichemical model 
Molten oxides and silicates 
Formation of associate compound
Ti-O System 
Canadian scientists at the Ecole Polytechnique in Montreal also developed their own databank system, F*A*C*T or Facility for the Analysis of Chemical Thermodynamics . The FACT53 compound database contains data for over 4500 compounds and contains selected data for thousands of compounds taken from standard compilations which have been optimised .
3.1.3 Solution Models
In high temperature systems it is common for species to dissolve into each other to form multi-component phases such as slags, mattes, and alloys. In some cases, this solution behaviour is quite complex, with complex interactions between different species in the phase strongly influencing the distribution of elements between different phases. Unfortunately, there is no single approach to modelling multi-component solution behaviour that will satisfy all systems. There is also no comprehensive scientific treatment of solution behaviour, instead a combination of theory and empiricism is used to deal with the thermodynamics of these systems . Some of important models are summarised in Table 1. This includes ideal solution model and real solution models, and example of its application to high temperature systems. The models are described in more detail in the following section.
220.127.116.11 Ideal Solution Model
In an ideal solution it is assumed that there is no interaction between molecules, which means there is no enthalpy of mixing and volumetric changes resulting from mixing. Raoult's law , states that the vapour pressure of an ideal solution is dependent on the vapour pressure of each chemical component and the mole fraction of the component present in the solution. The Gibbs energy for A-B ideal binary solution phase is written as Eq. (9) and (10):
Ideal solution can be used for mixture of vapour in atmospheric pressure because the compression factor can be negligible. However, in condensed phase such as slag and metal phases, interaction between molecules is likely to occur and necessitate more sophisticated models. It is also common for many researchers to use the ideal solution assumption as a starting point for their calculations.
18.104.22.168 Dilute Solution Model
In applications such as liquid iron processing, some species such as C, Mn, Si, and P are dissolved in liquid iron in dilute concentration. Thermodynamic properties of this system can be described by dilute solution model. In dilute solutions, behaviour of real solutions can be explained using Henry's law . In Henry's law, the solute activity is assumed to have a linear relationship with concentration (Eq. (11)).
ï§oi is Henrian activity coefficient or Henrian Constant.
Assuming that Henry's law holds at 1 mass %, the Gibbs energy of solution of pure component i in iron at 1 mass % is
where Mi is the molecule weight of solute. The use of interaction coefficients  allows this model to be applied to higher order systems. This approach is commonly used in liquid metal applications, for example, in calculating the equilibrium between liquid steel and inclusion chemistry .
22.214.171.124 Substitutional Solution Model / Random Mixing Solution Model
The behaviour of real solution deviates from ideal solution, i.e. there is interaction between species that called excess terms. The random mixing solution model is used for substitutional solutions and assumes that different species occupy random positions within a defined lattice . The excess Gibbs energy is represented using the Redlich-Kister equation :
with (3. 4)
where Li,j is a binary interaction parameter, an and bn are the model parameter. When n = 0, the excess Gibbs energy become regular and the model is called a regular solution model, while n = 1 means the excess Gibbs energy become subregular. The Redlich-Kister polynomial equation is widely used in metallic systems for substitutional phase, such as liquid, b.c.c, f.c.c, etc. However, when there is short-range ordering in the liquid (such as in liquid metals with a tendency to form intermetallics and most slags) this equation is not sufficient, and other model have to be considered .
126.96.36.199 Sublattice Model
In substitutional solution, all lattices site are assumed to be equivalent. However, some crystalline species are formed in two or more different lattice structures. Therefore, it may be advantageous to model the multi-component solution using a sublattice model. In a sublattice model, a fractional site, yi, is defined as the total number of component i (nis) in sublattice S divided by the total component (NS) in the same sublattice  as in Eq. (17). It is related to mole fraction xi by Eq. (18)
(3. 4) ;
where yVa indicates site fraction of vacancies. The interaction parameter can be modelled using a Redlich-Kister polynomial. The Gibbs energy of solution is defined by Eq. (3.17).
188.8.131.52 Compound energy formalism
Compound Energy Formalism (CEF) is based on a sublattice model for solid solutions that has two sublattices . The variation for different phases has been constructed by a number of researchers with wide-range application . CEF can describe thermodynamic properties of phase with vacancy and interstitial defect in crystal structure. It can also be applied to ionic solutions with application for solid oxides, such as spinel and pyroxenes . The Gibbs energy expression in the CEF per formula unit of solution is written as follow:
Where M is the sites and SC is configurational entropy. For application of metal with interstitial, for example Fe and Cr in fcc crystal form with C as intersititial, the phase can be modelled as (Fe,Cr)1(C,Va)2 . This means Fe and Cr are distributed in the first sublattice, while C and vacancy in the second sublattice.
The Gibbs energy of formation, ideal mixing and excess term is written as:
(3. 4) (3. 4)
The sublattice model or Compound Energy Model is similar to the random mixing solution model but uses two different sublattice. These models can be applied for a wide-range application of metals and alloys .
184.108.40.206 Modified quasichemical system
In some phases, the species in solution are not randomly distributed. For example, in molten CaO-SiO2 slags, there is a tendency for short range ordering to occur at around specific conditions and compositions. Modified quasichemical models for short-range ordering liquid solution has been developed and derived from quasichemical theory . For a liquid binary solution, atoms or molecules A and B are distributed over the sites of a quasilattice. If we consider a pair exchange reaction:
The Gibbs energy for the liquid using this model can be written as:
Where ni and nj are the number of moles of the component i and j, nij is the number of (i-j) pairs, and ï„SC is the configurational entropy of mixing given for randomly distributing the (i-i), (i-j), and (i-j) pairs. The molar and entropy change ï„gAB is noted as (ï·-ï¨T). This idea can be extended to multi-component systems .
There are still some other proposed models, including the ionic liquid sublattice model, which is a modification of the Compound Energy model which has been used for ionic liquids such as molten salts . Models must be applied for phases that have specific criteria. For example, ideal solution model may be applied for mixture of gas. Regular solution model may be sufficient for simple binary solution that has similar crystal structure and similar cation/anion size. More complex system requires more sophisticated models, such as solid solutions that have interstitial and vacancy properties generally requires Compound Energy Formalism model .
3.1.4 Thermochemical Packages
The development of thermochemical packages has had a significant impact on material processing and the use is now common among engineers and researchers alike. The key features of four thermochemical packages are summarised in Table 2 and described in the following section.
220.127.116.11 Chemix (CSIRO-SGTE Thermodata System)
Chemix is a module part of CSIRO-SGTE Thermodata System, using Solgasmix , a Gibbs energy minimisation subroutine, to calculate equilibrium of multi-component and multi-phase system. This thermochemical package was developed in the 1980s by CSIRO Minerals. Chemix has been used for modelling of laboratory and industrial chemical processes by means of equilibrium calculation . A number of applications in extractive metallurgy has been reported, such as direct smelting of zinc concentrate , bauxite purification system , solid solution formation between arsenic and antimony oxides , and carbothermic of magnesium production .
The activity coefficient models for solution calculation in Chemix are comprehensive, such as fixed activity coefficient, polynomial, Redlich-Kister, Margules, Virial, Redlich-Kwong, and Pitzer dilute solution. In this way, the user has a wide range choice of models depending on their own system. The activity coefficient for each phase must be entered by user, which can be obtained from a private database or available literature; this flexibility is a significant advantage. However, this software is based on DOS and not user friendly in terms of generating files and printing results. Whilst, widely used in Australia during the 1990s, CSIRO has stopped providing technical support for the software. Nevertheless, Chemix thermochemical software is still useful for calculating equilibrium in complex systems and its use continues.
Table 3. 2 Comparison of four thermochemical packages
SGTE 1977, JANAF, NPL, CSIRO
Barin , JANAF
SGTE 2009, FACT 2009, binary solution
SGTE Unary and Solution database, NPL solution database (Alloy, Oxide, salt, solder, aqueous, Al, Dilute solution), ThermoCalc Fe and Ni Solution
Solution models/ Activity Coefficients
Fixed, Polynomial (three version), Debye-Huckel, Interpolation, Virial, Bethelot, Subregular, Redlich, Margules, Redlich-Kister, Lupis-Elliot, Virial Full, Pitzer, Redlich-Kister, Regular, Redlich-Kwong
Wagner-Interaction Formalism, Quasichemical, Sublattice, Pitzer, Polynomial (Muggianu), Polynomial (Kohler/Toop), Compound Energy Formalism
Ideal gas, associated solutions, CEF, Redlich-Kister polynomials, two-Sublattice ionic models, quasichemical model. Can also use an extended Kapoo-Frohberg model for liquid slags.
Equilibrium, Phase Diagram, optimisation, solidification
Equilibrium, Phase Diagram, optimisation, solidification
Yes (MATLAB, COMSOL)
Own Pure Species and Solution Data
HSC Chemistry was developed by Outokumpu Technology in 1974. The database is taken from Barin  and JANAF . The Solgasmix routine  based on Gibbs energy minimisation is used in the equilibrium module. In HSC, definition of system is crucial step and carried out by the user. Users must specify the substances that may be present at equilibrium, thought the software can readily identify the possible combinations. The activity coefficients for individual species can be entered as a constant number, or as a polynomial function of composition and temperature. The limitation of HSC lies on the configuration of activity coefficients. For systems with complex solution behaviour, such as would be expected for slag, matte and metallic phases over a range of temperatures, more complex treatments of solution behaviour are required to obtain accurate results. HSC has wide applications and is widely used in industry because of its user-friendliness and calculating power. For example, it is excellent for calculating heat and mass balances for process flowsheets.
FactSage is an integrated database computing system for chemical thermodynamic. This package has optimized database for solutions, such as alloys, liquid and solid oxides, and slags. For pure components, the data are from JANAF Thermochemical Tables  and thermodynamic properties data . The solution model for liquid slag phase is using a modified quasi-chemical model .
The details of this thermochemical package, such as the databases and various calculation modules, can be found elsewhere. Solution models and databases for common systems have been optimised by the developer. However, the user still can use his/her own private database in FactSage software using the "Compound" module for the species properties (G, H, S, Cp) and the "Solution" module for solution interaction parameters. Thermochemical solution models for various systems are available, as listed in Table 2, for example random solution model, CEF, and modified quasichemical models. This provision of solution models is a significant advantage over Chemix and HSC for rapid calculation of phase equilibria (i.e. you do not need to find your own solution data), though this may result in a non-critical attitude to using solution models. Though similar comments could be made for all thermodynamics software (i.e. naÃ¯ve users can easily make mistakes).
In the "Equilib" module, the Gibbs energy minimization technique is used to calculate the concentration of chemical species when specified elements or compounds are react to reach the state of equilibrium. Phases from the compound and solution databases are retrieved and listed as possible products in the equilibrium result. FactSage offers user-friendly software and a complete database for elements and solutions, but also give flexibility for user to use their own data. Other module are available, such "React" module which calculates the enthalpy and Gibbs energy of reaction, and a "Phase Diagram" module for generating phase diagram.
MTDATA stands for Metallurgical and Thermochemical Databank, developed by National Physical Laboratory, England. The principle of MTDATA is similar with other thermochemical packages, which is a software/data package for the calculation of phase equilibria in multi-component and multiphase systems using critically assessed thermodynamic data . It has numerous applications in the fields of metallurgy, chemistry, materials science, and geochemistry depending on the available data. Computational interface for thermodynamic calculations with MATLAB also has been reported . It uses Gibbs energy minimisation routine to predict equilibrium and has a number of different modules allowing presentation and analysis of its predictions in different formats such as Pourbaix, Kellogg or phase diagrams. It primary calculation module for complex equilibria in metallurgical systems is called Multiphase. The specifics of the minimisation routine used are dependent on the level of accuracy required in the calculation. The highest accuracy minimisation routine is essentially consistent with Solgasmix .
Each different thermochemical packages has its different features and limitations. These thermochemical packages may be used for different purposes and systems. For example, HSC can be used for simple processes that do not require multi-component solutions. FactSage and MTDATA can be utilised to generate phase diagram as predictive tools for alloy development. One important part that cannot be neglected in all thermodynamic calculation is the definition of phases and possible species to be considered by the thermodynamic models, Theses choices will have a significant influence on the results generated and the repercussions of these choices need to be considered when using these valuable tools.
3.2 Reaction Kinetics
Knowledge of reaction rates or reaction kinetics is important for better operation and control of processes as well as for process design and analysis. The reaction kinetics is more difficult than thermodynamics because it is time dependent. The rate of reaction depends on the path it takes to move from initial state to final state, the one that not considered by thermodynamics. Metallurgical reactions are largely heterogeneous reactions. For example, the Pidgeon process involves three different solids and one gas and another solid.
3.2.1 The rate of reactions
Reactions can happen in one step or several elementary steps. The mechanisms of reaction determine the path of several elementary steps the reaction travel until it goes to final state. Amongst the several steps, the slowest part is usually control the rate of reaction, which is called the rate controlling steps.
For the reactants A and B which produce C and D, the rate of reaction can be expressed as , , , or , where C is the concentration of species. The rate of reaction can be function of concentration of reactants or products, even both of them, which can be expressed by:
where the kc is the rate constant. kc is a function of temperature which follow the Arrhenius equations:
where Ea is energy activation and A is Arrhenius constant. The value of energy activation has been used empirically to estimate the reaction mechanism, such as solid state diffusion reactions have energy activation between 80 - 400 kJ/mol; and mineral solution alteration process has energy activation between 40 - 80 kJ/mol .
Figure 3. 1 Energy Activation 
3.2.2 Kinetics of Heterogeneous Reactions
It has been mentioned that heterogeneous reactions usually occurs in metallurgy and the reactions can only happens at the interface of different phase, for example gas solid interface or solid A - solid B interface. For solid A- solid B reaction, the reactions may take several steps :
Self diffusion of reactant A species
Diffusion of reactant A species through product layer
Its diffusion and reactions between A and B in B sites
For gas solid reactions, the reaction may take these steps :
Gas phase mass transfer of the gaseous reactant from the bulk of the gas stream to the external surface of the solid particle
Several steps that may take simultaneously in a diffuse spatia; domain:
Diffusion of the gaseous reactant through the pores of the solid matrix, which could consist of a mixture of solid reactants and products
Adsorption of the gaseous reactant on the surface of the solid matrix
Chemical reaction at the surface of the solid matrix
Desorption of the gaseous product from the surface of the solid matrix
Diffusion of gaseous reaction product through pores of the solid matrix
Gas phase mass transfer of the gaseous product from the external surface of the solid to the bulk of the gas stream.
From the phenomena mentioned above, it is clear that in heterogeneous reaction, there are several phenomena that must be considered besides intrinsic chemical reaction, such as diffusions, mass transfer, and heat transfer. Self diffusion is particularly important for solid state reaction. For the Pidgeon process reactions particulary, the reactions involve three different reactants produce one gaseous phase and other solids, a combination of solid-solid reaction and solid-gas reaction steps. That means it will involve the self diffusion of solid reactants, the reactions at interface, and the diffusion of gaseous phase produced through the pores of solid, and finally the mass transfer of gaseous product from the surface of solids to the bulk gas phase.
3.2.3 Kinetics Theory of Solid-Solid Reactions
The mechanism of solid state reaction as in the Pidgeon process is evidently very complicated. It involves three different solid reactants and produce one gas phase and another solid product. In the literature there are plenty knowledge about gas-solid reaction and solid-solid reaction [45-48]. There several publish works attempted to describe the phenomena by modelling the gas-solid reaction such as reduction of hematite in hydrogen and carbon monoxide , and solid-solid reaction, such as reaction between nickel oxide and alumina .
Gas solid reaction and solid-solid reaction has been treated as different category in modelling the systems. Fundamentals of gas solid reactions includes the basic component of gas-solid reactions as well as models to describe gas-solid reaction has been thouroughly explained by Szekely et al . Furthermore, there has been a review on gas-solid reactions  which analyse various models to describe gas-solid non catalytic reactions, which is generally classified as sharp interface model, volume reaction model, and particle-pellet model. In the field of solid-solid reaction, Doraiswamy and Sharma  and later on Tanhamkar and Doraiswamy  reviewed some important aspects of the analysis of solid-solid reaction, with the emphasis on the diffusional aspect of reactions. In essence, gas-solid reaction and solid-solid reaction differ from the controlling factor of the process, since in the solid-solid reaction, solid state diffusion has a dominant role, while in gas-solid reactions, diffusion of gas in bulk phase and inside the solid may take part and add the controlling factor. The form of solid, i.e. non-porous, porous, or a mixed powder will also affect the kinetic of system.
In Pidgeon process system, the solids are in the mixed powder forms. Powder reactions are substantially more complex and not as yet amenable to treatment based on " first principle" . Tamhankar and Doraiswamy summarised that in essence, in any powder reaction, the solid particles of reactants should contact one another, and at least one of them must diffuse though an increasing product shell after initial surface reaction. There are several reaction models for mixed powder reaction has been proposed and it is based on three different rate-limiting controls:
Product layer diffusion control
In these models, the rate limiting steps is the diffusion of reactant species in product layer. The Jander model is the simplest one and the Valensi-Carter is the most advanced because it takes account the volume changes. All of these models assume a shrinking homogeneous core of reactant surrounded by homogeneous shell of product.
Jander . Reactant particle are sphere, with a continuous product layer, spherical geometry is made by approximating the shell as plane sheet (constant reaction cross section)
Serin- Ellickson . Modify Jander equation by eliminating assumption of constant reaction cross section.
Ginstling-Brounstein . Diffusion in a spherical shell
Valensi-Carter . Change in volume is accounted by introducing Z to eliminate constant particle size
Z is volume of product formed per unit volume reactant consumed.
Nuclei growth control
The theory of nucleation and growth of product phases has been formulated by Avrami, which is mainly employed for decomposition reaction. These based on two steps: formation of nuclei at active sites, and the growth of these nuclei. The parameter m is depends on reaction mechanism, number of nuclei present, composition of initial species, and geometry of nuclei. This model is rare to be applied in general solid-solid reaction.
Phase boundary reaction control
These models assume that the diffusion of species through product layer is faster compared to reaction. The models have been developed for different geometries.
Sphere. Sphere reacting from the surface inwards
Cylinder. Circular disk reacting from the edge inwards or cylinder
Contracting cube. Contracting cube Equation
There are still several aspects that affecting the kinetic behaviour of solid-solid reaction, for example particle size distribution, compaction pressure of briquette, geometry of briquette, geometry of particle, and flux addition. However, little consideration has been given to the effect of particle size distribution in solid-solid mixed powder reaction studies. Some researchers reported the effect of different variable qualitatively.
Apart from solid-solid gas reactions, many attempts have been made to describe gas-solid reactions. There are three different basic models that have different assumptions, which are: sharp interface model, volume reaction model, and grain model. Sharp interface model is applicable for nonporous solid reactant and assumes that reaction occurs at a sharp interface that divides between reactant core and ash/product at the outer part of the shell. Several assumption used in this model is isothermal, first order reaction, and equimolar counter diffusion. Volume reaction model is based on a porous solid and the reaction occurs in all over the place because gas can penetrate into the solids. In grain model, the solid pellet is assumed as consisting of a number of grains or small particle. At surrounding of the grains, gas can diffuse and react with solids at the surface. The grain model is applicable to pellet that have a very fine size of grains.
All three models do not take account of structural changes due to reactions. In fact, chemical reaction and sintering will change the structural of solids. These will have an effect to effective diffusivity and change in porosity of the solids. There are several models that have been proposed to describe gas-solid reaction which take accounts the structural changes, namely: modified volume reaction model, modified particle pellet model, single pore model, and distributed pore model. Based on the name, the modified volume reaction model is based on volume reaction model but the porosity is assumed to change with local solid conversion according to proposed empirical equation. Intra-particle diffusivity is also changed based proposed empirical equation.
However, since the reaction in the Pidgeon process between calcined dolomite and ferrosilicon occurs in solid state and do not involve gas reactant at initial step, the solid-solid reaction model will be used in the kinetic modelling in order to describe the intrinsic kinetics of reaction.
Diffusion is the movement of a species from a region of high concentration to a region of low concentration, or in general, the rate of diffusion is proportional to the concentration gradient. The diffusion happens because of the difference of chemical potential in the system. Fick's first law of diffusion states that species A diffuses in the direction of decreasing of concentration of A. The rate equation is expressed as:
where JAx is the molar flux of A in the x-direction (mol.s-1.cm-2), and DA is diffusion coefficient or diffusivity of A, cm2s-1.
Fick's second law described the accumulation or depletion of concentration,C, when steady-state are not not achieved, and is obtained from spatial derivation of flux:
The Fick's second law can be solved using boundary conditions that determined by experiment.
Self diffusion rate is the rate at which atom meanders through the lattice of a pure metal. It can be measured using radioactive atoms as tracers. The self diffusion coefficient in a simple cubic lattice can be expressed by:
where d is inter-atomic spacing and v is jump frequency.
For a system that have more than one species, interdiffusion is occurs. Thus, interdiffusion coefficient may be used rather than intrinsic diffusion coefficient. Interdisivity of A-B species for monoatomic gas can be determined using Chapman-Enskog theory:
where ï³AB: collision diameter,
â„¦AB: collision integral for A-B mixture at dimensionless Temperature. TAB for the Lennard-Jones potential (a function of KT/ï¥AB), .
The diffusion in non-metals similar with the diffusion in metals, but since there is varying degrees of polarisation between cations and anions, there is additional effect. Figure 3.2 shows the inter-diffusion coefficient in common ceramics. The diffusion coefficient in solids is also strongly dependent to the temperature based Arrhenius law.
Figure 3. 2 Diffusion of Common Ceramics. The Activation Energy Q can be Estimated from the Slope and Insert. 
Diffusion of gaseous species in porous solid is more complicated and much less well understood. The actual diffusion path which will not follow the straight line, the pores that may be small enough to the species, and pressure gradient affect the actual diffusion . The effective diffusivity is introduced which is function of porosity, , and tortuosity :
Mass transfer closely related to diffusion. The mass transfer between single particles and a moving gas stream can be espressed as:
Kc, is mass transfer coefficient, which in the majority practical case is obtained empirically. Kc relates to the flow of the gas and the physical properties of the particle. It has empirical correlation in dimensionless form with Reynold number, Schmidt number, and Grashoft number. One expression that widely used for the system of particle and gas are the Ranz and Marshall relations :
For the mass transfer in a tube where gas passed through, the Warner relation  can be applied:
3.2.3 Kinetics of Condensation
When gaseous solution, pure gases, pure liquids, or liquids reach some degree of supersaturation or supercooling, i.e the gases stream reach below condensation temperature of gas, condensation to form solid crystalline phases will occur. The condensation process from vapour streams comprise of several steps as schematically described in Figure 3.3:
generation of reactants, can be from reaction, such as from silicothermic reaction, or vaporation of pure condensed phase,
transport of vapours to the growth surface,
boundary layer transport,
formation of crystal nuclei, and
the growth of crystal
Figure 3. 3 Basic Steps of Crystal Growth from Condensation of Vapours. A. Generation. B. Bulk Transport. C. Boundary Layer Transport. D. Adsorption/Desorption. E. Migration. F. Nucleation 
18.104.22.168 Homogeneous Nucleations
The condition of supersaturation alone is not enough for a system to condense or begin to crystalise, nuclei or must exist before the growth of crystal started. From the classical theory of nucleation from the work of Gibbs, Volmer, and Becker and Doring, nucleation is based on the condensation of vapor to liquid phase or solid phase. When a group of molecules becomes aggregated to more condense state, in which the molecular movement is restricted, a quantity of energy is released. On the other hand, the formation of solid particle demands a quantity of energy to form solid surface. Therefore, the quantity of work required to form a stable nucleus is the sum of the work to form the bulk of particle and the work required to form the bulk of particle.
A measure of supersaturation, S, is defined as these equation:
If the r value is substituted, so the W becomes:
It means that when the system is just saturated (S= 1), the amount energy for nucleation is infinitive. That is also suggests that any supersaturated system or solution can create a homogeneous nucleation.
The free energy of homogenous nucleation is a summation of surface excess free energy, Gs, i.e the excess energy between the surface and the bulk of particle, and volume excess free energy, Gv, i.e the excess energy between a very large particle and the solute in the solution. The Gs is positive, while Gv is negative. The maximum value corresponds to the critical free energy to form homogeneous nucleation, which is written as:
rc is the critical nucleus, which represent the minimum size of a stable nucleus. Particles smaller than rc will dissolve, or evaporate if the particle is in a supersaturated vapour. Similarly, particle larger than rc will continue to grow.
The rate of nucleation, J, e.g. the number of nuclei formed per unit time per unit volume, can be expressed as Arrhenius equation:
k is Bolzmann Constant and R is gas constant
The rate of nucleation can be arranged from Gcrit to be:
This equation indicates that the rate of nucleation is affected by three variables: temperature, degree of saturation, and interfacial tension. The supercooling or undercooling has an effect to the rate of nucleation. The Gv can be expressed by:
T* is the equilibrium temperature (in K) and delta T is the supercooling, while L is latent heat of fusion. Thus, the rate of nucleation can be expressed by:
Tr is reduced temperature which is defined as T/T*
22.214.171.124 Heterogeneous Nucleations
Heterogeneous occurs when nucleation takes place in the special sites in the materials that can be capable to lowering the Gv. In laboratory scale, the sign of condensation often appears in one regions of the vessel, which usually have the high degree of supersaturation or supercooling.
To form heterogeneous nucleation, the overall free energy change associated with the formation of critical nucleus under heterogeneous condition, ï„Gcrit, must be less than corresponding homogeneous free enery change, ï„Gcri:
where ï¦ is less than unity. The factor that controlling heterogeneous nucleation is interfacial energy, ï³, that closely related to contact angle.
The factor ï¦ can be expressed as:
where ï± is contact angle
As soon as stable nuclei have been form in supercooled system, they will grow to a larger crystal. There have been several theories attempted to describe the growth of single crystal. Surface energy theories comes from Gibbs who suggested that the growth of a crystal could be considered based on the principle that the total of free energy of a crystal in equilibrium with its surrounding at constant temperature and pressure would be a minimum for a given volume. Wulff  suggested that crystal faces would grow proportionally to the surface energy, and in addition Laue  pointed that all possible combination of faces must be considered.
Adsorption layer theories from Volmer's theory  states that when units of crystallising substance arrive at the crystal face they are not merely integrated to lattice and then migrate over the crystal face through surface diffusion. Therefore, there will be a loosely adsorbed layer at the interface, which play important role in growth phenomena.
The physical properties and environment condition such as pressure and temperature of the species will affect the growth, direction, crystal morphology and structure.