## 3 Challenges and Limitations
### Sorbent Utilization  
Low sorbent utilization is one of the main issues when operating a packed bed of metal oxide, in particular when using regenerated sorbents [59]. The sorbent utilization can be estimated from the breakthrough curve. A breakthrough curve of a packed bed shows the outlet concentration of the reactor as a function of time. A schematic of a breakthrough curve is shown in Figure 3a. Although simple in concept, breakthrough curves carry a lot of information about the various phenomena taking place in the packed bed. The breakthrough time \(\tau_{b}\) is defined as the time at which the concentration of H\({}_{2}\)S in the outlet stream of the reactor reaches a predefined threshold value. In experiments, the latter is typically taken close to the detection limit. The theoretical maximum breakthrough time, denoted by \(\tau_{2}\) in Figure 3, is calculated by assuming full utilization of the sorbent. Considering a sorbent that consists of several different metal oxides, the time for full sorbent utilization reads as:  
\[\tau_{2}=\frac{m_{\text{s}}}{QC_{0}}\sum_{i}\frac{x_{i}}{\gamma_{i}M_{i}}=\frac {\tau_{0}(1-\varepsilon_{b})\rho_{p}}{C_{0}}\sum_{i}\frac{x_{i}}{\gamma_{i}M_{ i}} \tag{7}\]  
where \(m_{\text{s}}\) is the total mass of sorbent in the packed bed, \(x_{i}\) is the mass fraction of metal oxide \(i\) in the sorbent, \(M_{i}\) is the molecular weight of metal oxide \(i\), \(\rho_{p}\) is the pellet density (see Equation (A1) in Appendix A), \(\varepsilon_{b}\) is the bed porosity, \(Q\) is the volumetric flow rate of the gas stream and \(C_{0}\) is  
Figure 3: Analysis of breakthrough curves. (a) Schematic of a breakthrough curve indicating the breakthrough time \(\tau_{b}\), the surface utilization time \(\tau_{1}\) and the time for complete sorbent utilization \(\tau_{2}\). (b) Schematic illustration of the sorbent utilization at \(t=0\), \(\tau_{1}\) and \(\tau_{2}\). the concentration of H\({}_{2}\)S in the inlet stream. Moreover, \(\tau_{0}=V/Q\) is the superstitial residence time, where \(V\) is the volume of the empty reactor (i.e., the reactor in the absence of sorbent). The full sorbent utilization time \(\tau_{2}\) given by Equation (7) together with the breakthrough data allows for calculating the removal capacity, defined as the ratio of the actual amount of H\({}_{2}\)S that is removed divided by the maximum amount that can be removed:  
\[\zeta=\frac{1}{\tau_{2}}\int_{0}^{\infty}\!\!\left(1-\frac{C(t)}{C_{0}}\right)dt \tag{8}\]  
where \(C(t)\) is the reactor outlet concentration. The integral in Equation (8) is proportional to the amount of H\({}_{2}\)S that is removed by the packed bed. In graphical terms, it corresponds to the area above the breakthrough curve as indicated by the hatched area in Figure 3a. Sometimes, \(C(t)\) for large \(t\) has to be extrapolated or modelled as experiments are often terminated soon after H\({}_{2}\)S is detected at the reactor outlet. Reasons for a low removal capacity can be blockage of sorbent due to deposition of other material, blockage of pores or inactivation of sorbent due to reduction of the metal oxide. The time for full sorbent utilization presents an upper bound to the breakthrough time \(\tau_{b}\). This motivates to define a removal efficiency of a packed bed as:  
\[\eta=\tau_{b}/\tau_{2}. \tag{9}\]  
Compared to the removal capacity, the removal efficiency defined in Equation (9) can readily be deduced from breakthrough curves that are terminated soon after breakthrough. Notice that in general, the removal efficiency is different than the removal capacity. For a practical assessment of the sorbent utilization, the removal efficiency is of higher relevance. Another characteristic time that helps for the interpretation of breakthrough data can be derived by considering the onset of the sulfidation reaction. Inspired by the work of [PERSON] et al.