## 5 Modeling of Solid-Gas Reactions in Packed Beds
### Shrinking Core Model  
The shrinking core model (SCM), also referred to in the literature as unreacted core model, unlike the deactivation model, is based on a mechanistic view for the sorbent conversion inside the pellets. Initially the fresh sorbent on the pellet surface reacts with H\({}_{2}\)S and forms a layer of solid product (consumed sorbent, colloquially called the ash layer). In the continuation of the process, H\({}_{2}\)S diffuses through the consumed sorbent layer to reach the fresh sorbent. Since the reaction rate is considered to be much faster than the diffusion rate, the reaction takes place on a sharp front separating the consumed sorbent from the fresh sorbent, as seen in Figure 4 as the reaction interface. The two main transport mechanisms in the SCM are: diffusion through the consumed sorbent layer and surface reaction. The fluxes attributed to each of these phenomena are equal on the reaction \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sorbent** & \(\mathbf{n^{\prime}}\) & \(\mathbf{m^{\prime}}\) & \(\mathbf{k_{s}}\) **(m/s)** & \(\mathbf{k_{a}}\) **(s-1)** & **Ref.** \\ \hline Cu, M and V oxides & 0 & 1 & \(\sim\)10\({}^{-2}\) & \(\sim\)10\({}^{-2}\) & [99] \\ Zn-Mn oxides & 0 & 1 & \(\sim\)10\({}^{-2}\) & \(\sim\)10\({}^{-2}\) & [116] \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Deactivation model in the literature. Figure 4: Schematic of the shrinking core model. The light gray is the fresh sorbent and the dark gray is the consumed sorbent. A sharp interface separates the fresh and consumed sorbent. As the conversion proceeds the consumed sorbent shell grows inward. interface since there is no accumulation of H\({}_{2}\)S on the interface. Therefore, we can derive an expression for the flux to the pellets that reads as [118]:  
\[ \begin{array}{ll}N_{p}=&\left(\frac{1}{k_{Rx}}+\frac{1}{k_{Df}}\right)^{-1}C_{ p,s}\\ k_{Rx}=&k_{s}(1-X)^{2/3}\\ k_{Df}=&\frac{D_{a}}{R_{p}}\frac{(1-X)^{1/3}}{1-(1-X)^{1/3}}\end{array}  \tag{22}\]  
where \(k_{Rx}\) and \(k_{Df}\) are the mass transfer coefficients of reaction and diffusion, respectively, \(k_{s}\) is the reaction rate function of the surface reaction, \(R_{p}\) is the characteristic length of diffusion which in case of spherical pellets corresponds to the pellet radius, \(D_{a}\) is the apparent diffusivity of H\({}_{2}\)S in the consumed sorbent and \(X\) is the sorbent conversion. Notice that the \(k_{Rx}\) is derived by assuming that the surface reaction is of first order with respect to the gas. The sorbent conversion is obtained from a mass balance over the sorbent:  
\[ \begin{array}{ll}\frac{\partial X}{\partial t}=\frac{\gamma^{\alpha_{p}}}{C _{\text{MO}}}N_{p}\\ \text{I.~{}C.:~{}}X(x,0)=0\end{array}  \tag{23}\]  
where \(C_{\text{MO}}\) is the molar concentration of the sorbent (calculated from the density and the molecular weight of the metal oxide) and \(\gamma\) is the stoichiometric coefficient of the sorbent. Equation (17) together with Equations (22) and (23) are solved simultaneously to obtain the solid conversion \(X(x,t)\) and the H\({}_{2}\)S concentration \(C(x,t)\) along the bed at different times. Solving the coupled equations can be numerically expensive. To mitigate this problem, a method referred to as constant pattern method was frequently adopted in the literature. [PERSON] et al. [119] presents a simple description of the constant pattern method applied to the SCM. Table 7 gives the apparent diffusivity for different metal oxides obtained from applying the SCM to fit experimentally measured breakthrough curves. As pointed out in earlier works [120], the estimated diffusivities are relatively large when compared with values obtained from correlations for molecular or Knudsen diffusion (the value of \(D_{a}\) for iron oxyhydroxide is much lower compared to those of the other sorbents in Table 7 which is due to the low operating temperature in which the measurements for this sorbent were done).