# On the Impact of Particulate Matter Distribution on Pressure Drop of Wall-Flow Particulate Filters
## 2 Wall-Flow DPF Model  
The computational study performed in this work is based on the use of a wall-flow DPF model developed in previous works, which is next briefly described. The model is integrated into OpenWAM(tm)(Version 2.2, CMT-Motores Termicos, Valencia, Spain), which is an open-source gas dynamics software developed at CMT-Motores Termicos [17; 18]. The model solves the conservation equations in a single pair of inlet and outlet channels assuming non-homentropic one-dimensional unsteady compressible flow:  
* Mass conservation: \[\frac{\partial\left(\rho_{j}F_{j}\right)}{\partial t}+\frac{\partial\left(\rho_ {j}u_{j}F_{j}\right)}{\partial x}=(-1)^{j}4\left(\alpha-2w_{pl}j\right)\rho_{ j}u_{w_{j}}\] (1)
* Momentum conservation: \[\frac{\partial\left(\rho_{j}u_{j}F_{j}\right)}{\partial t}+\frac{\partial \left(\rho_{j}u_{j}^{2}F_{j}+p_{j}F_{j}\right)}{\partial x}-p_{j}\frac{dF_{j} }{dx}=-F_{w}u_{j}u_{j}\] (2)
* Energy conservation: \[\frac{\partial\left(e_{0j}\rho_{j}F_{j}\right)}{\partial t}+\frac{\partial \left(h_{0j}\rho_{j}u_{j}F_{j}\right)}{\partial x}=q_{j}\rho_{j}F_{j}+(-1)^{j }4\left(\alpha-2w_{pl}j\right)h_{0w}\rho_{j}u_{w_{j}}\] (3)
* Chemical species conservation: \[\frac{\partial\left(\rho_{j}Y_{j}F_{j}\right)}{\partial t}+\frac{\partial \left(\rho_{j}Y_{j}u_{j}F_{j}\right)}{\partial x}=(-1)^{j}4\left(\alpha-2w_{ pl}\right)\rho_{j}u_{wj}Y_{j}\] (4)  
In Equations (1)-(4), \(j\) identifies the type of monolith channel (0 = outlet, 1 = inlet) and takes into account the existence of the particulate layer. Figure 1 shows schematically the way in which the DPF channels are discretized in the axial direction, as well as the cross-section of an inlet monolith channel identifying the main geometrical characteristics of the cell. The flow field of inlet and outlet channels is conditioned by the source terms related to flow across the porous media, which are governed by [PERSON]'s equation applied along the particulate layer and the porous wall [16]. Thus, the filtration velocity at every axial node of the inlet channel can be calculated as a function of the pressure difference between the inlet and outlet channel, the cellular geometry and the permeability of every porous medium:  
\[u_{w_{in}}=\frac{p_{in}-p_{out}}{\frac{\mu_{in}w_{w}}{k_{w}}\frac{\rho_{in} \left(\alpha-2w_{pl}\right)}{\rho_{out}\alpha}+\frac{\mu_{in}\left(\alpha-2w_ {pl}\right)}{2k_{pl}}\ln\left(\frac{\alpha}{\alpha-2w_{pl}}\right)} \tag{5}\]  
Accordingly, the filtration velocity corresponding to the outlet channel is then obtained considering the continuity equation between the inlet and outlet interface of the porous media:  
\[u_{w_{out}}=\frac{u_{w_{in}}\rho_{in}\left(\alpha-2w_{pl}\right)}{\rho_{out} \alpha} \tag{6}\]  
Figure 1: Scheme of the axial and cross-sections of Diesel Particulate Filter (DPF) channels. The solution of the governing equations is obtained applying finite difference methods. In particular, the two-step Lax and Wendroff method (2 LW) [19] adapted to porous medium channels is coupled with the Flux-Corrected Transport (FCT) technique [20]. The monolith channels are coupled to inlet and outlet volumes, which are included to account for the inertial pressure drop contribution because of flow contraction and expansion. The volumes are solved by a filling and emptying method and its connection to the monolith applying the Method of Characteristics (MoC) [21] adapted to solve the boundary conditions of inlet and outlet porous channels [22], as indicated in Figure 1.  
According to Equation (5), the filtration velocity is dependent on the gas and porous media properties and the monolith meso-geometry.