## 3 Results and Discussion
### Effect of the Inlet Gas Temperature  
From this point on, our attention will be focused on the curve at \(\mathrm{V_{in}}\) = 1 m/s of Figure 10 and, in particular, on its lower branch. In order to identify operating conditions that allow for the further decrease of the maximum temperature and/or the time for filter regeneration, simulations were run at lower values of inlet gas temperature (\(\mathrm{T_{in}}\) \(<\) 813 K). In Figure 11, the results in terms of \(\mathrm{T_{max}}\) and \(\mathrm{t_{reg}}\) are plotted versus \(\mathrm{T_{in}}\), as obtained at \(k\) = 35, 50, 100, and 200. There are several interesting features shown in this figure. First of all, at each value of \(k\), a transition from fast regeneration to slow regeneration is predicted with decreasing \(\mathrm{T_{in}}\). As the value of \(k\) is increased, this transition shifts towards lower values of \(\mathrm{T_{in}}\). \(\mathrm{T_{max}}\)/\(\mathrm{t_{reg}}\) monotonically increases/decreases with increasing \(\mathrm{T_{in}}\) until a nearly constant plateau is reached. In addition, at high inlet temperatures, \(\mathrm{T_{max}}\) decreases with increasing \(k\), whereas \(\mathrm{t_{reg}}\) slightly increases. Conversely, at low inlet temperatures, \(\mathrm{T_{max}}\)/\(\mathrm{t_{reg}}\) increases/decreases with increasing \(k\). These trends suggest that, at high inlet temperatures, regeneration takes place according to the oxygen transport-limited mode, whereas at low inlet temperatures, it takes place according to the kinetics-limited mode. Once a maximum allowable catalyst temperature is assumed, \(\mathrm{T_{max\_allowable}}\), of 1023 K (i.e., 750 \({}^{\circ}\)C) [24], the optimal inlet gas temperature, \(\mathrm{T_{in\_optimal}}\), can be identified as the temperature that minimizes regeneration time while still ensuring temperature control (i.e., \(\mathrm{T_{max}}\) \(\leq\) \(\mathrm{T_{max\_allowable}}\)). As an example, Figure 12 shows this criterion applied to the case of \(k\) = 100. Figure 10: Operating map of the filter in the plane \(\mathrm{T_{max}}\) versus \(\mathrm{t_{reg}}\): simulations run by varying the catalyst activity, \(k\), at different inlet gas velocities. It can be seen that \(\mathrm{T_{in\_optimal}}\) lies on the boundary between the oxygen transport-limited mode and the kinetics-limited mode. This also occurs in the cases of \(k=35\), \(50\), and \(200\) (not shown). Under such optimal conditions, the regeneration process is not controlled either by mass transport or by kinetics. The results of Figure 11 were also rearranged in terms of the objective function, \(\phi=\mathrm{t_{reg}}\)\(\mathrm{T_{max}}\). Interestingly, it has been found that, at each value of \(k\), \(\phi\) exhibits a minimum at \(\mathrm{T_{in}}=\mathrm{T_{in\_optimal}}\). Figure 13 shows the plot of \(\mathrm{T_{in\_optimal}}\) versus \(k\). In Figure 14, \(\mathrm{T_{max}}\) and \(\mathrm{t_{reg}}\) at \(\mathrm{T_{in}}=\mathrm{T_{in\_optimal}}\) are shown as a function of \(k\). Figure 11: Maximum temperature, \(\mathrm{T_{max}}\), and time for filter regeneration, \(\mathrm{t_{reg}}\), versus inlet gas temperature, \(\mathrm{T_{in}}\), at different values of \(k\). In Figure 15, the results of Figure 14 are shown in the plane T\({}_{\rm max}\) versus t\({}_{\rm reg}\) (circles). In this map, the lower branch of the curve at V\({}_{\rm in}\) = 1 m/s of Figure 10 is also shown. In these computations, T\({}_{\rm in}\) was set equal to 813 K.  
At \(k\) = 35 and 50, the main effect of decreasing the inlet gas temperature from 813 K to the optimal value is lowering T\({}_{\rm max}\). Conversely, at \(k\) = 100 and 200, the main effect is lowering t\({}_{\rm reg}\). It is worth remembering that our simulations were run assuming the cake to be absent, and that all the soot trapped inside the wall of the filter is in contact with the catalyst.