## 2 Process Design
### Reactor Scale  
There are three common reactor configurations for conducting reaction (Equation (1)): entrained flow reactor [40], fluidized reactor [41] and packed bed reactor [7]. In an entrained flow reactor, the sorbent particles are immersed into the gas stream which carries them through the reactor. Thus, the residence time of the gas and the sorbent in the reactor are of a similar order of magnitude. In a packed bed reactor, the gas stream flows through the reactor while the sorbent particles remain in the reactor. The residence time of the sorbent is considerably larger than that of gas. Hence, from the viewpoint of the solid sorbent the packed bed reactor resembles a batch reactor. A fluidized bed reactor lies in between these two extremes and, depending on the operating mode, from the viewpoint of the solid sorbent the reactor resembles a well mixed continuous reactor or a batch reactor. The parameter that can be used to characterize the different reactor configurations is the particle volume fraction \(\phi\), that relates to the bed porosity \(\varepsilon_{b}\) as follows:  
\[\phi=\frac{\text{Total volume of solid particles}}{\text{Total volume of reactor}}=1- \varepsilon_{b}. \tag{2}\]  
The entrained flow reactor has the lowest solid volume fraction while the packed bed reactor has the highest solid volume fraction among the three reactor configurations (\(\phi>40\%\)) [42].  
There are different empirical correlations to predict the bed porosity for a packed bed reactor. A simple correlation with acceptable accuracy is given by [PERSON] and [PERSON] [43]:  
\[\varepsilon_{b}=\frac{A}{\left(d_{0}/d_{p}+C\right)^{2}}+B \tag{3}\]  
where \(d_{p}\) is the equivalent sphere diameter of the particles, \(d_{0}\) is the reactor diameter and \(A\), \(B\) and \(C\) are empirical parameters. Table 1 gives these parameters for common particles shapes. Notice that the first term on the right hand side (rhs) of Equation (3) represents a correction for the case where the particle size is comparable to the reactor diameter; for \(d_{p}\ll d_{0}\) the first term vanishes and the bed porosity becomes a constant.  
Figure 1: Schematic of a packed bed reactor for H\({}_{2}\)S removal with metal oxides: (**a**) reactor scale, (**b**) pellet scale.  
Important parameters in choosing the reactor configuration are pressure drop, attrition of particles and contact surface. In this specific case, the available contact-surface between the solid particles and the fluid is of great importance since the kinetics of the removal reaction is not very fast and more importantly a high level of sulfur removal is often required. Among the three reactor configurations, the packed bed reactor provides the largest contact surface area for a fixed reactor volume. This is one of the reasons why the packed bed is the reactor of choice in most applications.  
To estimate the pressure drop in a packed bed reactor Ergun equation is used, which reads as [44]:  
\[\frac{\Delta p}{L}=150\frac{(1-\varepsilon_{b})^{2}}{\varepsilon_{b}^{3}} \frac{\mu_{g}u_{g}}{\left(\Phi d_{p}\right)^{2}}+1.75\frac{(1-\varepsilon_{b} )}{\varepsilon_{b}^{3}}\frac{\rho_{g}u_{g}^{2}}{\Phi d_{p}} \tag{4}\]  
where \(\Delta p/L\) is the pressure drop per unit length, \(u_{g}\) is the superficial gas velocity, \(\Phi\) is the particle sphericity, \(\rho_{g}\) is the density of the gas mixture and \(\mu_{g}\) is the viscosity of the gas mixture. Expressions for estimating \(\rho_{g}\) and \(\mu_{g}\) for a gas mixture are given in Appendix A. The Ergun equation accounts for the different flow regimes, i.e., the first term on the rhs of Equation (4) is dominant when the flow is laminar while the second term is dominant when the flow is turbulent.