## 2 Mathematical Model  
The CFD-based model developed in Reference [14] was used to run simulations of the regeneration process for a single-channel catalytic DPF. Figure 1 shows the two-dimensional computational domain. The soot was assumed to be trapped only inside the porous wall of the filter, and not even in the form of a cake layer deposited on top of the wall. The model is described in detail in Reference [14] and here is only briefly presented. The governing fluid flow equations are the mass, momentum, species, and energy conservation equations. In the porous wall, these equations were coupled to the soot conservation equation, and local thermal equilibrium was assumed between the fluid and solid phases (in other words, it was assumed that fluid and solid are at the same temperature). The model equations were solved in \"time-dependent\" form. The consumption rate of soot was expressed according to the kinetics obtained by [PERSON] et al. [19] from experiments of non-catalytic (i.e., thermal) and catalytic oxidation of real diesel soot with oxygen:  
\[r_{total}=r_{slow\_oxidation}+r_{fast\_oxidation} \tag{1}\]  
with \(r_{slow\_oxidation}\) contribution of slow (i.e., thermal) oxidation:  
\[\frac{r_{slow\_oxidation}}{m_{total,t=0}}=A\;\;e^{(-E_{a}/RT)}\;\;\;\left(x_{ Oxygen}\right)^{a}\;\;\;\left(\frac{m_{total,t}}{m_{total,t=0}}\right)^{b} \tag{2}\]  
and \(r_{fast\_oxidation}\) contribution of fast (i.e., catalytic) oxidation--this contribution involves only the soot in contact with the catalyst:  
\[\frac{r_{fast\_oxidation}}{m_{cataly\_total,t=0}}=A^{\prime}\;\;e^{(-E_{a}^{ \prime}/RT)}\;\;\;\left(x_{Oxygen}\right)^{a^{\prime}}\;\;\;\left(\frac{m_{ catalyst\_t,t}}{m_{cataly\_total,t=0}}\right)^{b^{\prime}} \tag{3}\]  
\(x_{Oxygen}\) (Equations (2) and (3)) is the oxygen mole fraction in the gas phase, \(m_{total}\) (Equation (2)) is the local concentration of soot (all the soot present in the system), and \(m_{cataly\_2}\) (Equation (3)) is the local concentration of soot in contact with the catalyst. The values of the kinetic parameters for Equations (2) and (3) are given in Table 1.  
Figure 1: Two-dimensional computational domain (not to scale): single-channel filter (length of the inlet and outlet channels = 45 mm; thickness of the wall = 0.38 mm). In the model, all the soot was assumed to be in contact with the catalyst (\(m_{catalyzed}=m_{total}\)) and, thus, both contributions (Equations (2) and (3)) were considered for the regeneration kinetics. The (intrinsic) catalyst activity was varied by changing the pre-exponential factor, \(A^{\prime}\), in Equation (3). In particular, the ratio, \(k\), between \(A^{\prime}\) and the pre-exponential factor of the kinetics by [PERSON] et al. [19], \(A^{\prime}\)_[PERSON]_, was increased from 1 to 200. The model was developed in the frame of the porous medium approach of the CFD code ANSYS Fluent 15.0 [20]. A user-defined scalar was used to specify the local soot concentration, whereas the regeneration kinetics was implemented via user-defined subroutines. The model equations were discretized through the finite-volume method on a grid having about \(2.2\times 10^{5}\) square cells with size equal to 0.025 mm. The spatial discretization and the time integration were performed by using second-order schemes (also for convective terms) and the second-order implicit Crank-Nicholson scheme, respectively. Fixed flat profiles were assumed as inlet boundary conditions for velocity, species concentration, and temperature. At the outlet boundary, a condition of fixed static pressure was assigned. All the remaining boundaries were assumed to be adiabatic walls. If not otherwise specified, the simulation (inlet and initial) conditions are those listed in Table 2.  
Table 3 gives the values of the properties of the support material (Silicon Carbide, SiC) [21], along with the values of porosity and permeability of the (clean) wall of the filter. A similar value of porosity is reported in Reference [21] for SiC DPFs.