**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™(Version 2.2, CMT-Motores Térmicos, Valencia, Spain), which is an open-source gas dynamics software developed at CMT-Motores Térmicos [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:

$$\frac{\partial \left(\rho\_{\dot{f}} F\_{\dot{f}}\right)}{\partial t} + \frac{\partial \left(\rho\_{\dot{f}} u\_{\dot{f}} F\_{\dot{f}}\right)}{\partial x} = (-1)^{\dot{f}} 4 \left(\alpha - 2w\_{p\dot{l}} \dot{j}\right) \rho\_{\dot{j}} u\_{w\_{\dot{j}}} \tag{1}$$

• Momentum conservation:

$$\frac{\partial \left(\rho\_{\dot{j}}u\_{\dot{j}}F\_{\dot{j}}\right)}{\partial t} + \frac{\partial \left(\rho\_{\dot{j}}u\_{\dot{j}}^2F\_{\dot{j}} + p\_{\dot{j}}F\_{\dot{j}}\right)}{\partial x} - p\_{\dot{j}}\frac{dF\_{\dot{j}}}{dx} = -F\_w\mu\_{\dot{j}}u\_{\dot{j}}\tag{2}$$

• Energy conservation:

$$\frac{\partial \left(\varrho\_{0j}\rho\_{j}F\_{\bar{j}}\right)}{\partial t} + \frac{\partial \left(h\_{0j}\rho\_{j}u\_{\bar{j}}F\_{\bar{j}}\right)}{\partial x} = q\_{j}\rho\_{j}F\_{\bar{j}} + (-1)^{j}4\left(a - 2w\_{pl}j\right)h\_{0w}\rho\_{j}u\_{w\_{\bar{j}}}\tag{3}$$

• Chemical species conservation:

$$\frac{\partial \left(\rho\_{\rangle} \mathbf{Y}\_{\rangle} \mathbf{F}\_{\rangle}\right)}{\partial t} + \frac{\partial \left(\rho\_{\rangle} \mathbf{Y}\_{\rangle} u\_{\rangle} \mathbf{F}\_{\rangle}\right)}{\partial \mathbf{x}} = (-1)^{j} \mathbf{4} \left(\mathbf{a} - 2w\_{pl}\right) \rho\_{\rangle} u\_{w\rangle} \mathbf{Y}\_{\rangle} \tag{4}$$

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.

**Figure 1.** Scheme of the axial and cross-sections of Diesel Particulate Filter (DPF) channels.

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 Darcy'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:

$$\mu\_{\text{IVw}} = \frac{p\_{\text{in}} - p\_{\text{out}}}{\frac{\mu\_{\text{in}} w\_{\text{pr}}}{k\_w} \frac{\rho\_{\text{in}} \left(a - 2w\_{\text{pl}}\right)}{\rho\_{\text{out}} a} + \frac{\mu\_{\text{in}} \left(a - 2w\_{\text{pl}}\right)}{2k\_{pl}} \ln\left(\frac{a}{a - 2w\_{\text{pl}}}\right)}\tag{5}$$

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:

$$u\_{w\_{\rm out}} = \frac{u\_{w\_{\rm in}} \rho\_{\rm in} \left(a - 2w\_{pl}\right)}{\rho\_{\rm out} \alpha} \tag{6}$$

The solution of the governing equations is obtained applying finite difference methods. In particular, the two-step Lax and Wendroff method (2LW) [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. Both the porous wall and particulate layer permeabilities are obtained as a function of the porosity of the medium (*ε*), the collector unit diameter (*dc*) and the slip-flow correction given by the Stokes–Cunningham Factor (SCF) as [23]:

$$k = f\left(\varepsilon\right)d\_c^2 \mathcal{S} \mathcal{C} F \tag{7}$$

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:

$$k\_{w\mu} = \frac{k\_{w}k\_{w\_0}}{f\_w k\_{w\_0} + (1 - f\_w)k\_w} \tag{8}$$

In Equation (8), *fw* represents the fraction of porous wall thickness where soot is collected; *kw*0 is the permeability of the clean porous wall; and *kw* 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)

$$d\_{c,w} = 2\left(\frac{d\_{c,w\_0}^3}{8} + \frac{3m\_{s\_{cell}}}{4\pi\chi\rho\_{s,w}}\right)^{\frac{1}{9}},\tag{9}$$

where the apparent density of the collected soot is defined as the product of the density of soot aggregates of mean fractal dimension (*ρ<sup>s</sup>*,*<sup>w</sup>*) [27] and a shape factor (*χ*) 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 (*dcell*,*<sup>w</sup>*) from clean conditions,

$$d\_{cell,w} = \frac{d\_{c,w\_0}}{(1 - \varepsilon\_{w\_0})^{\frac{1}{\beta}}} \tag{10}$$

the porosity under soot loading conditions is obtained as:

$$
\varepsilon\_w = 1 - \frac{d\_{c,w}^3}{d\_{cell,w}^3}.\tag{11}
$$

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:

$$E\_{f,w} = 1 - \varepsilon^{-\frac{3\eta\_{DR,l}(1-\varepsilon\_w)w\_fw\_Rs\_\xi}{2x\_wd\_\xi,w}}\tag{12}$$

where *ηDRI* 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 (O2) and nitrogen dioxide (NO2) is considered [30]. The variation of these reagents is solved separately across the particulate layer and the porous wall thickness for every reagen<sup>t</sup> [31] as:

$$\frac{\partial X\_{\hbar}}{\partial z} = -\frac{S\_{p}k\_{\hbar}a\_{\hbar}}{u\_{w}},\tag{13}$$

where subscript *n* identifies O2 or NO2, *X* is the molar fraction, *Sp* is the soot specific surface, *αn* the stoichiometric coefficient and *kn* the kinetic constant, which is temperature dependent according to an Arrhenius-type equation. Knowing the depletion rate of every gaseous reagen<sup>t</sup> across the particulate layer and the porous wall, the amount of regenerated soot per time-step and control volume can be obtained as:

$$
\Delta \mathfrak{u}\_{\mathfrak{n}} = \Delta X\_{\mathfrak{n}} \mathfrak{u}\_{\mathfrak{n}^\sigma} A\_f \mathbb{C}\_{\mathfrak{J}^{\mathfrak{u}\mathfrak{s}}} \Delta t \tag{14}
$$

$$m\_{s, \text{reg}} = M\_{\text{C}} \left( -\frac{\Delta n\_{\text{NO}\_2}}{a\_{\text{NO}\_2}} - \frac{\Delta n\_{\text{O}\_2}}{a\_{\text{O}\_2}} \right) \tag{15}$$

where *n* represents the reagen<sup>t</sup> moles, *Af* is the filtration area, *Cgas* is the gas concentration and *MC* is the soot molecular weight, which is assumed to correspond to carbon.

The rate of heat generated by the soot oxidation is included in the thermal balance solved in the porous substrate of every control volume in which the channels are axially discretized. The heat transfer model [32] is based on a bi-dimensional discretisation of the porous media between a pair of inlet and outlet channels. Besides the regeneration heat source, the model accounts for thermal inertia, convection gas to solid heat transfer, heat conduction across the substrate both in the axial and the tangential directions, as well as radial conduction towards the external canister, whose wall temperature is also computed taking into account heat transfer to the ambient environment.
