# Determination of a Complete Conversion Model for Gasification of Lignite Char
## 2 Theory  
### Kinetic Models  
Generally, external mass transport from the gas phase to the outer particle surface, the intra-particle diffusion and/or the chemical reaction at the char surface determine the rate of char-gas reactions, depending on temperature and particle properties. For temperatures below 1000 \({}^{\circ}\)C and particles in the order of magnitude of 0.1 mm, the reaction rate is controlled by the chemical reaction [9]. Equation (1) is a general expression for the chemical reaction rate, given by [PERSON] et al. [10]. \[\frac{dX}{dt}=k\Big{(}T,\overline{p}_{g}\Big{)}\,f(X) \tag{1}\]  
Here, \(k\) is the apparent reaction rate depending on temperature \(T\) and the partial pressures, of the gasifying agents and gas phase products, described by the vector \(\overline{p}_{g}\), according to a reaction model. \(f(X)\) describes the change in physical or chemical properties of the char with ongoing char conversion, \(X\), according to a conversion model. A simple representation of the apparent reaction rate during gasification is the Arrhenius reaction model, which only considers the partial pressure \(p_{g}\) of the gasifying agent and the temperature. \[k_{Arr}\Big{(}T,p_{g}\Big{)}=p_{g}\,k_{0}\,e^{-\frac{E_{g}}{K\,T}} \tag{2}\]  
The kinetic parameters for this model are the pre-exponential factor \(k_{0}\) and the activation energy \(E_{a}\). The inhibitive influence of the product, which has been observed in several studies [11; 12], is considered when applying the Langmuir-Hinshelwood reaction model (L-H model) to the gasification mechanism [13]. Here, the rate-determining step is the formation of occupied sites on the carbon surface. Equations (3) and (4) apply to the CO\({}_{2}\) and H\({}_{2}\)O gasification respectively. \[k_{LH,CO_{2}}\Big{(}T,\overline{p}_{g}\Big{)}=\frac{p_{CO_{2}}\,k_{1}\,e^{- \frac{E_{n,1}}{kT}}}{1+p_{CO}\,k_{2}\,e^{-\frac{E_{n,2}}{kT}}+p_{CO_{2}}\,k_{3} \,e^{-\frac{E_{n,3}}{kT}}} \tag{3}\]  
\[k_{LH,H_{2}O}\Big{(}T,\overline{p}_{g}\Big{)}=\frac{p_{H_{2}O}\,k_{1}\,e^{- \frac{E_{n,1}}{kT}}}{1+p_{H_{2}}\,k_{2}\,e^{-\frac{E_{n,2}}{kT}}+p_{H_{2}O}\,k_ {3}\,e^{-\frac{E_{n,3}}{kT}}} \tag{4}\]  
Here, the kinetic parameters are the three pre-exponential factors \(k_{1}\), \(k_{2}\) and \(k_{3}\) and the three activation energies \(E_{n,1}\), \(E_{n,2}\) and \(E_{n,3}\). They have to be determined separately for steam and CO\({}_{2}\) gasification. In this work, four conversion models for the change in char properties with progressing char conversion are investigated with respect to their applicability for the fuel sample: the volumetric model (VM), the grain model (GM), the random pore model (RPM), which are the most common models used in gasification kinetics [14], and the Johnson model (JM). According to Equation (5), the VM assumes a decreasing reaction surface proportional to the remaining volume or mass of the particle. \[\frac{dX}{dt}=k\Big{(}T,\overline{p}_{g}\Big{)}\left(1-X\right) \tag{5}\]  
In Equation (6), the GM or shrinking core model considers the particles as an assembly of nonporous spheres with constant density and decreasing diameter [15]. The reaction only takes place at the surface. \[\frac{dX}{dt}=k\Big{(}T,\overline{p}_{g}\Big{)}\left(1-X\right)^{\frac{2}{3}} \tag{6}\]  
The RPM was proposed as an semi-empirical model by [PERSON] and [PERSON] [16]. It considers arbitrary pore size distributions in the reacting solid and is able to predict a first increasing and then decreasing reaction rate due to the growth and later the coalescence of pores. In the according equation, Equation (7), \(\psi\) is a parameter related to the pore structure of the unreacted sample. \[\frac{dX}{dt}=k\Big{(}T,\overline{p}_{g}\Big{)}\left(1-X\right)\,\left(1-\psi \ln(1-X)\right)^{0.5} \tag{7}\]  
The JM is another semi-empirical approach by [PERSON] [17].