## 5 Modeling of Solid-Gas Reactions in Packed Beds
### Grain Model  
The grain model considers the pellets to consist of smaller grains, as shown in Figure 5. The void spaces between the grains represent the pores in the pellet and the grains mimic the non-porous part of the pellet. H\({}_{2}\)S diffuses through the pellet pores to reach the grain surfaces. There, the grains follow a shrinking core type of behavior, i.e., H\({}_{2}\)S diffuses through the non-porous consumed sorbent that covers the grain surface (i.e., the ash layer) to reach the fresh sorbent where it reacts, forming a sharp reaction interface that separates the fresh sorbent from the consumed sorbent [118]. The grain model thus adds another scale to the modeling, namely the grain scale on which the ash diffusion and the reaction take \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sorbent** & **T (\({}^{\circ}\)C)** & \(D_{a}\) **(m/s\({}^{2}\))** & **Ref.** \\ \hline Zinc titanate & 550–700 & \(\sim\)10\({}^{-7}\) & [119] \\ Zinc ferrite & 500–600 & \(\sim\)10\({}^{-5}\) & [119; 121] \\ Perovskite-type sorbents & 300–500 & \(\sim\)10\({}^{-7}\) & [122] \\ Zinc oxide & 250 & \(\sim\)10\({}^{-8}\) & [123] \\ Calcium oxide & 560–1100 & \(\sim\)10\({}^{-6}\) & [124] \\ Manganese based & 850 & \(\sim\)10\({}^{-6}\) & [125] \\ Iron oxyhydroxide & 0–100 & 10\({}^{-9}\) & [126] \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Apparent diffusivity in the shrinking core model for different sorbents at given temperature range. place. The main transport phenomenon on the pellet scale is the intra-pellet diffusion, while the convective flow through the packed bed is the main phenomenon on the bed scale. Hence, Equation (17) is still valid to describe the bed scale and only the modeling of the pellet flux \(N_{p}\) is refined. The latter is derived from the concentration distribution inside the pellets that is governed by a mass balance over the pellet. For a spherical pellet containing non-porous grains the mass balance of H\({}_{2}\)S reads as:  
\[ \begin{split}&\varepsilon_{p}\frac{\partial C_{p}}{\partial t}= \frac{1}{r^{2}}\frac{\partial}{\partial r}\Big{(}D_{e}r^{2}\frac{\partial C_{ p}}{\partial r}\Big{)}-(1-\varepsilon_{p})a_{g}N_{g}\\ &\text{B. C.:}\left\{\begin{array}{ll}\frac{\partial C_{p}}{ \partial r}=0&\text{at}&r=0\\ C_{p}=C_{p,s}&\text{at}&r=R_{p}\end{array}\right.\\ &\text{I. C.:}\;C_{p}=0\end{split}  \tag{24}\]  
where \(C_{p}\) is the H\({}_{2}\)S concentration inside the pellet, \(\varepsilon_{p}\) is the pellet porosity, \(a_{g}\) is the specific surface area per unit volume of the grains (\(a_{g}=6/d_{g}\) for spherical grains of diameter \(d_{g}\)), \(D_{e}\) is the effective intra-pellet diffusivity and \(N_{g}\) is the flux of H\({}_{2}\)S to the grains. It is suggested that effective intra-pellet diffusivity follows \(D_{e}\sim\varepsilon_{p}^{2}D_{p}\) to count for the effect of the porous media and tortuosity of the pores [39; 127]. Pore diffusivity \(D_{p}\) is typically expressed as a combination of the molecular diffusivity \(D_{M}\) and the Knudsen diffusivity \(D_{K}\):  
\[D_{p}=\!\!\left(\frac{1}{D_{M}}+\frac{1}{D_{K}}\right)^{-1} \tag{25}\]  
The values of \(D_{M}\) and \(D_{K}\) are calculated from the expressions found in the literature [45]. To find the flux to the grains \(N_{g}\) we establish a shrinking core model on the grain scale. Following a similar argument that lead to Equation (22) results in:  
\[ \begin{split}& N_{g}=\ \ \ \Big{(}\frac{1}{k_{Rx}}+\frac{1}{k_{Df}} \Big{)}^{-1}C_{p}\\ & k_{Rx}=\ \ \ \ k_{s}(1-X)^{2/3}\\ & k_{Df}=\ \ \ \ \ \frac{D_{\text{sh}}}{k_{g}}\frac{(1-X)^{1/3}}{1-(1-X)^{ 1/3}}\end{split}  \tag{26}\]  
Figure 5: Schematic of the grain model. On the grain scale, the light gray is the fresh sorbent and the dark gray is the consumed sorbent. Grains undergo a shrinking core-like of behavior. Grains closer to the pellet surface are consumed sooner as the conversion proceeds in a pellet.