## 5 Modeling of Solid-Gas Reactions in Packed Beds  
In this section, modeling methods to simulate the removal of H\({}_{2}\)S from a diluted gas stream by using a packed bed of metal oxides is discussed. The bed is filled with pellets that contain metal oxides. As the gas stream passes through the packed bed, H\({}_{2}\)S transfers from the bulk of the fluid to the surface of the pellets. H\({}_{2}\)S then diffuses through the pellets to reach the metal oxides (MO) where it finally reacts with them according to reaction (Equation (1)). At operating temperatures higher than 100 \({}^{\circ}\)C, the products of the sulfidation reaction (Equation (1)) are water vapor and metal sulfide (MS). The water vapor transfers back to the fluid bulk, while the metal sulfide substitutes the metal oxide in the solid phase. The transfer of water vapor to the fluid bulk has negligible effects on the overall rate of H\({}_{2}\)S removal and therefore is usually not included in the modeling process. Moreover, \begin{table}
\begin{tabular}{c l l l} \hline \hline
**Sorbent** & \multicolumn{2}{c}{**Regeneration Agent**} & \multicolumn{2}{c}{**Operating Condition**} & **Ref.** \\ \hline Ca-based & Oxidation with O\({}_{2}\) followed by reduction with CO & 1050 \({}^{\circ}\)C, tested for 10 cycles & [80] \\ Fe-based & Mixture of 15\% steam, 5\% O\({}_{2}\) and 80\% N\({}_{2}\) & 400–700 \({}^{\circ}\)C, tested for 5 cycles & [111] \\ Zn-based & Mixture of 50\% steam, 3–6\% O\({}_{2}\) and balance N\({}_{2}\) & 650–750 \({}^{\circ}\)C, tested for 20 cycles & [112] \\ Cu-based & Oxidation in air followed by regeneration in a mixture & 375 \({}^{\circ}\)C, tested for 50 cycles & [113] \\  & of steam and H\({}_{2}\) & \(>\) 600 \({}^{\circ}\)C, tested for 110 cycles & [107] \\ Mn-based & SO\({}_{2}\), O\({}_{2}\), steam or a combination of them. & \(>\) 900 \({}^{\circ}\)C, tested for 5 cycles & [114] \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Regeneration of spent metal oxides. despite the exothermic nature of the sulfidation reaction, temperature changes are negligible for the low H\({}_{2}\)S concentrations typically experienced in sorption processes employing metal oxides. Also, the equilibrium of reaction (Equation (1)) at the operating temperature is to the right for most of the metal oxides such that reaction (Equation (1)) can be considered irreversible. Consider a packed bed of fresh sorbent of length \(L\) to which a stream containing H\({}_{2}\)S at a concentration \(C_{0}\) is fed. Assuming a constant gas flow, a mass balance for the bulk fluid concentration reads as:  
\[ \begin{array}{l}\frac{\partial C}{\partial t}+u_{int}\frac{\partial C}{ \partial x}=E\frac{\partial^{2}C}{\partial x^{2}}-va_{p}N_{p}\\ \text{B. C.:}\left\{\begin{array}{ll}C_{0}=C-\frac{E}{u_{int}}\frac{\partial C }{\partial x}&\text{at}&x=0\\ \frac{\partial C}{\partial x}=0&\text{at}&x=L\end{array}\right.\\ \text{I. C.:}\ C=0\end{array}  \tag{17}\]  
where \(C\) is the concentration of H\({}_{2}\)S in the bulk fluid, \(u_{int}=Q/(\varepsilon_{b}A)\) is the interstitial velocity of the gas stream, with \(Q\) denoting the volumetric gas flow rate, \(\varepsilon_{b}\) is the bed porosity and \(A\) is the cross-sectional area of the packed bed, \(E\) is the axial dispersion in the bed, \(\
u=(1-\varepsilon_{b})/\varepsilon_{b}\) is the solidity of the packed bed, \(a_{p}\) is the specific surface area per unit volume of the pellets (for spherical pellets with diameter \(d_{p}\) we have \(a_{p}=6/d_{p}\)) and \(N_{p}\) is the flux of H\({}_{2}\)S from the bulk fluid to the pellets. Boundary conditions (B.C.) are formulated as Danckwerts boundary conditions [44], while initial conditions (I.C.) assume the bed is initially free of H\({}_{2}\)S.  
The flux \(N_{p}\) depends on the H\({}_{2}\)S concentration at the pellet surface \(C_{p,s}\). In order to find \(C_{p,s}\), we revisit the transport phenomena occurring on the pellet surface and its vicinity. H\({}_{2}\)S transfers from the bulk fluid to the pellet surface, from where it diffuses into the pellet.