# On the Impact of Particulate Matter Distribution on Pressure Drop of Wall-Flow Particulate Filters
## 2 Wall-Flow DPF Model
Both the porous wall and particulate layer permeabilities are obtained as a function of the porosity of the medium (\(\varepsilon\)), the collector unit diameter (\(d_{c}\)) and the slip-flow correction given by the Stokes-Cunningham Factor (SCF) as [23]:  
\[k=f\left(\varepsilon\right)d_{c}^{2}SCF \tag{7}\]  
In particular, the permeability of the porous wall is calculated considering that the soot penetration is only partial [24; 25], so that Equation (7) is applied to a soot-loaded porous wall thickness and to the complementary one, which is considered to be kept fully clean:  
\[k_{w,\varepsilon}=\frac{k_{w}k_{w_{0}}}{f_{w}k_{w_{0}}+\left(1-f_{w}\right)k_ {w}} \tag{8}\]  
In Equation (8), \(f_{w}\) represents the fraction of porous wall thickness where soot is collected; \(k_{w_{0}}\) is the permeability of the clean porous wall; and \(k_{w}\) is the permeability of the loaded porous wall. The properties of the soot loaded porous wall are obtained from clean conditions applying the packed spherical particles theory [26]. The diameter of the collector unit as soot is collected is obtained applying Equation (9)  
\[\frac{\partial\left(\rho_{j}u_{j}F_{j}\right)}{\partial t}+\frac{\partial \left(\rho_{j}u_{j}^{2}F_{j}+p_{j}F_{j}\right)}{\partial x}-p_{j}\frac{dF_{j} }{dx}=-F_{w}u_{j}u_{j}\]0  
where the apparent density of the collected soot is defined as the product of the density of soot aggregates of mean fractal dimension (\(\rho_{s,w}\)) [27] and a shape factor (\(\chi\)) representing the irregular deposition of the soot around the collector unit [23]. The variation of the collector unit diameter as the porous wall is loaded involves the change of the porosity. Knowing the cell unit diameter (\(d_{cell,w}\)) from clean conditions,  
\[\frac{\partial\left(\rho_{j}u_{j}F_{j}\right)}{\partial t}+\frac{\partial \left(\rho_{j}u_{j}^{2}F_{j}+p_{j}F_{j}\right)}{\partial x}-p_{j}\frac{dF_{j} }{dx}=-F_{w}u_{j}u_{j}\]1  
the porosity under soot loading conditions is obtained as:  
\[\frac{\partial\left(\rho_{j}u_{j}F_{j}\right)}{\partial t}+\frac{\partial \left(\rho_{j}u_{j}^{2}F_{j}+p_{j}F_{j}\right)}{\partial x}-p_{j}\frac{dF_{j} }{dx}=-F_{w}u_{j}u_{j}\]2  
Besides permeability, the change in porous wall properties also determines the fluid-dynamic field in the inlet channels as the soot loading takes places and governs the variation in filtration efficiency [28]. Brownian diffusion, interception and inertial deposition mechanisms are considered to compute the filtration efficiency of an isolated collector unit. Integrating within the packed bed control volume using the pore velocity as characteristic velocity for the particles due to the proximity among collectors [29], the filtration efficiency of the porous wall yields:  
\[\frac{\partial\left(\rho_{j}u_{j}F_{j}\right)}{\partial t}+\frac{\partial \left(\rho_{j}u_{j}^{2}F_{j}+p_{j}F_{j}\right)}{\partial x}-p_{j}\frac{dF_{j} }{dx}=-F_{w}u_{j}u_{j}\]3where \(\eta_{DRI}\) is the filtration efficiency of an isolated collector unit due to the combination of the related collection mechanisms. Once the transition from deep bed to cake filtration regimes is completed, the porous wall properties remain constant, and the particulate layer acts as a barrier filter. Thus, all collected soot is assumed to be deposited on the particulate layer, varying its thickness. The amount of soot loading is determined every time-step accounting for the balance between filtrated and regenerated soot mass. Incomplete soot oxidation due to oxygen (O\({}_{2}\)) and nitrogen dioxide (NO\({}_{2}\)) is considered [30].