**1. Introduction**

Emission levels for automotive engines are submitted to increasingly stringent limits. From September 2015, all new European diesel cars must be compliant with the Euro 6 emissions standards, which set a particle number emission limit of 6 <sup>×</sup> 1011 particles km−<sup>1</sup> and a limit of 4.5 mg km−<sup>1</sup> for the mass of particulate [1]. In the United States, the phase-in period of Tier 3 (2017–2025) applies at a federal level, and requires automakers in the US to certify an increasing percentage of their fleet are complying with the new emissions standard (3 mg/mi of particulate matter) [2]. US standards are led by the California low emission vehicle (LEV) legislation, which

sets stricter emission limits to cope with its exceptional smog problems [3]. In the near future, new standards will likely be developed for even stricter emissions. In 2017, a new worldwide harmonized light-vehicle test procedure (WLTP) came into force, and it is compulsory for all new car registrations from September 2018 [4]. Also, for non-road mobile machinery, the Stage V of the EU Regulation 2016/1628 will be effective from 1 January 2019, reducing the particles mass limit for all the engines above 19 kW, and introducing a new limit for particle number emissions [5].

Different strategies to control emissions are developed by different manufacturers. New designs on combustion chamber, injection and supercharging systems, and control are introduced in new engines to reduce particle emissions, but they are not enough to ensure the compliance with current regulation thresholds [6]. Switching to cleaner alternative fuels can be also an effective way to reduce pollutant emissions in internal combustion engines (ICEs). The introduction of biofuels in the automotive sector provide an all-inclusive solution to the dependence on fossil fuels, and the associated environmental impact [7]. Although there are multiple biofuel formulations and many different studies on the emissions derived from their application to ICEs [8–10], there is a general agreement in their contribution to reduce gas emissions (CH, CO) and particulate matter (PM) [11–13] and overall life cycle carbon dioxide (CO2) [14]. Some research suggests, nonetheless, that increasing the blending rate of biofuel in an engine may increase particulate emissions depending on the engine design [15], on the biofuel properties [16], on the engine operating conditions [17], and even on the measurement method [18]. Despite the progress in the engine technology and the development of new biofuels, complying with the current emissions standards requires highly effective aftertreatment systems in the abatement of soot particles [6,19]. Nowadays, the most popular aftertreatment system for the abatement of particulate emissions in ICEs is the wall-flow diesel particulate filter (DPF). The requirements for the proper performance of a DPF are, mainly, high filtration efficiency, low pressure drop [20], and high capacity to resist the regeneration processes [21]. With an appropriate design, and a suitable substrate, the wall-flow DPF is able to comply with the current PM legislation [20]. The challenge for a DPF is the correct balance between filtration efficiency and pressure drop for an adequate engine performance. These parameters are generally closely related, an increase in filtration efficiency brings an increase in pressure drop, and vice versa. Also, the quick saturation of the filter, and the thermal stress produced by the regeneration cycles leads, in many cases, to the cracking and collapse of the DPF structure [22], increasing engine backpressure and penalizing the engine operation.

The introduction of biodiesel or other alternative fuels in the engine operation brings a new variable to the post-combustion systems. Biofuels alter the particle size distribution in the exhaust gases [23,24], and this affects in turn the performance of the DPF [25]. The combustion of biodiesel reduces the primary particle diameter [26] and shifts the distribution curve towards smaller particles [24,27]. The behavior of a wall-flow DPF is given by its geometry (diameter, length, wall thickness, cell density) at the macroscopic scale [28], and by the properties of the material used as substrate (permeability, porosity, pore size, tortuosity) at the microscopic scale [29]. For a given geometry, the microstructural properties of the substrate define the filtration efficiency and the pressure drop of the filter. The soot size has a significant influence on the initial deep-bed loading process [30]. The search for new materials that improve DPFs permeability, thermal properties, and filtration performance is a recurrent research topic today. The alterations that new biofuel blends introduce in vehicular PM emissions should be also taken into account when designing or developing new substrates for DPFs.

Recent studies have presented biomorphic silicon carbide (bioSiC) as a viable candidate for use as a substrate in hot gas filtration applications [31,32] and, specifically, as a substrate in DPFs for automotive diesel engines [33]. BioSiC is a porous bioceramic material characterized by preserving the hierarchical biological microstructure of the wood precursor from which it was made [34] so it is considered a bio waste or biomass. In this sense, the microstructure of this material can be tailored, to some extent, by the choice of the precursor to fit any application [35]. As opposed to traditional ceramic granular media, bioSiC can be good candidate to optimize the pressure drop/efficiency balance of DPFs while coping with the variability that new biofuel blends introduce in the engine operation. BioSiC

can be manufactured from a wide variety of plant species or precursors, including vegetal waste or biomass. Each precursor leads to a different microstructure. If the microstructural characterization of a precursor is known, numerical simulation techniques can be applied to predict the performance of the global filtration system and to identify the potential for its use at real scale with different fuel inputs.

Mathematical and physical models used for internal combustion engine particulates filter performance prediction, evolve from the pioneering work of Bissett and Shadman [36,37], published in 1985. The original approach proposed by Bissett was one-dimensional model with two channels, based on the basic principles of fluid-mechanics, and the application of an energy balance in the solid wall and gas phase. A transient filtration model was implemented later by Konstandopoulos et al. [38], who used exhaust conditions, including particle size distribution and wall microstructure properties, to calculate filtration efficiency in discretized wall slabs. These models have been further refined by Tandon et al. [29] who gave emphasis on the efficiency evolution during transition to cake filtration, and Bollerhof et al. [39], who studied filtration in inhomogeneous wall structures. The latter model and several variations are available in the commercial software package, named Axisuite [40], which is employed in this work. A comprehensive review of DPF modeling is given in references [41,42]. In this work, a numerical model of wall-flow DPFs has been adapted to the experimental microstructural features of the bioSiC material, in order to establish a starting point in the generalized analysis of different precursors, used for bioSiC generation. The objective is to predict the filtration performance of a bioSiC wall-flow DPF with a systematic procedure that allows to eventually improve the system performance and to fit different fuel/biofuel inputs through the identification of optimum precursors. In this study, the experimental microstructural features of bioSiC made from medium density fiberboard (MDF) were used. The validation of the model was made in a small prototype of bioSiC wall-flow DPF. It was designed, manufactured, and tested under controlled conditions, with the aid of a soot generator [33]. The resulting experimental measurements of filtration efficiency and pressure drop were then used to calibrate the real scale model, with the adjusted microstructural parameters to simulate the performance under NEDC driving cycle conditions. The same procedure might be used in the future to fit the microstructure of any other non-granular substrate.

This paper is structured in four sections. In Section 1, the motivations and objectives of the work are presented, and the general background of the model is introduced. Section 2 summarizes the main aspects and equations of the numerical model. It will describe the calibration-validation process and how the results of a small prototype were extrapolated to a full-size system. In Section 3, the main results are reported and a prediction about the performance of the filter, compared to that of other commercial systems, is presented. Finally, Section 4 summarizes the main results and conclusions.

#### **2. Materials and Methods**

The numerical model used for this work is applicable to any wall-flow DPF with constant cross-section and straight channels. A large part of the governing equations in the model act at the macroscopic scale and do not depend on the material the filter is made of, so these could be applicable to any DPF with the same geometry regardless of the composition of its substrate. There is a group of equations that model the walls of the filter and the passage of the gas through them, and here is where the microstructural characteristics of the substrate may have an effect. This group of equations depend on features of the porous material such as porosity, pore size, and permeability. Thus, to apply this latter group of equations, a previous knowledge of the biomorphic substrate is needed.

#### *2.1. Model Description*

The numerical model used in this study is built in two levels. The first level is the 'single channel problem'. In the 'single channel problem', the spatial discretization is comprised of one inlet channel and four quarters of outlet channels. Figure 1 presents the frontal area of the control volume, which is marked with a dashed red line. The background image is a SEM micrograph of the MDF bioSiC filter used for the calibration and validation of the model, as explained in Section 2.2. At this level, all the equations for a single channel are solved. The second level is the 'multi-channel problem', which extrapolates the results of the single channel to the whole system, including the additional phenomena associated with the physics of the complete monolith.

**Figure 1.** Front area of the control volume in the 'single channel problem' represented over a SEM image of a real MDF-bioSiC filter.

The evolution of pressure and velocity in each single channel is described mainly through two interrelated groups of equations. On the one hand, along the free path of the inlet or the outlet channels, the behavior of the flux is governed by the mass and momentum conservation equations of fluid dynamics. On the other hand, across the porous mediums, such as the walls of the filter or the soot layer, the behavior of the flux is governed by Darcy's law and the inertial effects (Forchheimer). Table 1 summarizes these equations. The full model is described in detail in [43].


**Table 1.** Summary of the governing equations of the flux in the single channel control volume.

In these equations, the subscript *i* is the identifier of the channel: 1 for inlet channels, and 2 for outlet channels. *N* is the number of walls of the channel, 4 in this case, and *d* is their width. The pressure drop through the substrate wall is set considering that the flow velocity *vw* is constant. The result is the direct application of Darcy's law with the Forchheimer's extension. On the contrary, the pressure drop through the soot layer is calculated, considering that the gas velocity varies along the soot layer, due to changes in gas density and flow area. Taking into account the geometrical definitions shown in Figure 2, the pressure drop through the soot layer is calculated by expressing *v* and ρ as a function of

the coordinate *w*, and integrating along the thickness of the soot layer [43]. The result depends on the permeability of the particulate deposit *kp*, and on the soot layer thickness *wp*, the value of which is recalculated in every successive step with the accumulated amount.

**Figure 2.** Scheme of the channel with geometrical definitions used in the model [43].

Equations for the porous medium summarized in Table 1 depend on permeability values that must be calibrated based on experimental results. The pressure drop across the soot layer depends on the permeability of the particulate deposit *kp*, while the pressure drop across the substrate walls depends on the permeability of the substrate wall *kw*. The permeability of the particulate deposit may be expressed as a function of the local temperature and pressure, using the correlation of Pulkrabek [44]

$$k\_p = k\_{p,0} \left( 1 + C\_4 \frac{p\_0}{\overline{p}} \mu \sqrt{\frac{T}{\mathcal{M}\_\mathcal{S}}} \right) \tag{1}$$

The local pressure *p* depends initially on the position *w*, but introducing this variable in the analytical approach would extremely complicate the solution. Instead, a mean value of the pressure *p* is considered, calculated as the average between the inlet and the outlet pressure. *C*<sup>4</sup> and *kp*,0 are characteristic parameters of the porous media and must be estimated based on experimental data.

Similarly, the permeability of the wall may be expressed as

$$k\_{\rm av} = \frac{1}{\frac{1}{k\_{\rm av0}} + C\_1 \rho\_p + C\_2 \rho\_p^2} \cdot \left(1 + C\_4 \frac{p\_0}{\overline{p}} \mu \sqrt{\frac{T}{M\_\odot}}\right) \tag{2}$$

The permeability of the clean wall *kw*,0 is the permeability of the clean bioSiC for each particular precursor under consideration. For calibration and validation purposes, this study has used a real MDF-bioSiC DPF as a model, the main features of which are summarized in Section 2.2 including its permeability in the clean stage. *C*<sup>1</sup> and *C*<sup>2</sup> are also characteristic parameters of the porous substrate but they govern the behaviour of the medium as it becomes loaded with particles. The permeability of the loaded wall is a function of the amount of soot trapped in the wall, ρ*<sup>p</sup>* is the instantaneous concentration of soot in the wall. Therefore, the values of *C*<sup>1</sup> and *C*<sup>2</sup> must also be estimated, based on experimental data. In the next section, the calibration procedure is explained for both the soot layer permeability and the wall permeability. With regard to the filtration model, the filtration efficiency of a porous medium is the result of the behavior of its unit collectors at the micro-scale. In this work, the model was specifically prepared to be applied to a bioSiC filter made from MDF. Thus, to model

the filtration efficiency, the fiber microstructure of bioSiC made from MDF was taken into account. The filtration efficiency of a clean fiber unit collector by diffusion and interception exposed to aggregate particles is summarized in Table 2 [43,45].


**Table 2.** Single collector filtration efficiency by diffusion and interception for fiber unit collectors.
