## 2 Methods
### Multi-Region Framework  
The adopted modeling strategy relies on the identification of the different domains that are considered for the case under investigation. In particular, the different types of simulation domains can be classified as follows: (a) fluid region, which describes the exhaust gas flow through the exhaust line; (b) solid regions, representing the solid walls of the generic metal pipes and cannings; (c) porous regions, which account for the presence of the equipped after-treatment devices, eventually catalyzed, in the exhaust line. Such components are modeled through the porous media approach, relying on overlapping meshes, associated with the fluid and the specific device computational domains. A block representation of such an approach for a simplified exhaust line is reported in Figure 1, highlighting the different equations solved for each computational domains. In addition, it is possible to appreciate the coupling achieved between the fluid and the porous domain (namely Substrate Domain ) through the inter-phase heat and mass transfer source terms, \(Q^{w\to g}\) and \(J_{i}^{w\to g}\) respectively. With regard to the other depicted source terms, \(S_{Y_{i},g}\) describes the rate of the gas phase homogeneous reactions; \(S_{E,g}\) models the gas phase reaction heat; \(S_{E,s}\) is considered to describe the eventual external heating and the catalytic surface reaction heat; \(S_{Y_{i},w}\) accounts for the catalytic surface reactions rate. The effects induced by the presence of the after-treatment component, in terms of inter-phase heat transfer, inter-phase mass transfer, and the flow resistance due to the permeability of the porous domain, are accounted for by means of the addition of specific source terms. The computation of such terms relies on the proper coupling models available in the literature for common honeycomb monoliths. In this context, the flow resistance component \(R_{g}\) is typically modeled by the Hagen-Poiseuille theory for laminar flows [14] or exploiting the [PERSON] [15] or Darcy-Forchheimer correlations. The inter-phase heat and mass transfer terms (Equations (1) and (2)) are typically computed resorting to Hawthorn correlations [16; 17] for the estimation of Nusselt and Sherwood numbers as a function of Reynolds, Prandtl, and Schmidt numbers:  
\[Q^{w\to g} =Nu(Re,Pr)\frac{\lambda}{d_{c}}S_{V}V_{C}(T_{g}-T_{w}) \tag{1}\] \[J_{i}^{w\to g} =Sh(Re,Sc)\frac{D}{d_{c}}S_{V}V_{C}(C_{g}-C_{w}) \tag{2}\]where subscripts \(w\) and \(g\) denote the washcoat and gas phases, \(\lambda\) denotes the fluid thermal conductivity, \(d_{c}\) denotes the fluid characteristic dimension inside the after-treatment device channels, \(S_{V}\) denotes the characteristic surface-to-volume ratio of the component, \(V_{\text{C}}\) denotes the considered computational element volume, \(T\) denotes the temperature, \(D\) denotes the fluid mass diffusivity, and \(C\) denotes the species \(i\) concentration. Typically, in order to reduce the computational burden, the mass diffusivity is computed, assuming the Schmidt number is equal to 1. In addition, an alternative approach relies on the adoption of constant user-defined Nusselt and Sherwood numbers. In the context of POCS structures, the described approach is still valid, upon a reformulation of the coupling source terms. In particular, the POCS-associated region is treated as a standard solid domain. In case a coated structure is considered, the multi-region methodology relies on the identification of the washcoat layer as a simi-porous domain. The first layer of the fluid computational cells next to the POCS solid region is selected for the creation of the washcoat layer. The fluid flow-induced resistance is neglected, while the inter-phase heat and mass transfer terms are computed, resorting to the following formulations:  
\[Q^{w\to g}=\frac{\lambda A}{L}(T_{g}-T_{w}) \tag{3}\]  
\[J_{i}^{w\to g}=\frac{DA}{L}(C_{g}-C_{w}) \tag{4}\]  
where \(L\) and \(A\) represent the considered grid element length and boundary face area, respectively.