*4.1. Pressure Drop*

Figure 2 shows the DPF pressure drop during the soot loading tests up to 30 g in soot mass. According to the experimental analysis presented in [12], the restructuring of the soot deposits in the particulate layer was hypothesized as the main cause to explain the decrease in pressure drop after every injection event and the capability to limit maximum pressure drop regardless of the amount of collected soot. This hypothesis has been analyzed in this work diagnosing the main particulate layer properties. A variety of soot mass distributions in the particulate layer with different effective porosity have been computed applying the wall-flow DPF model. The porous wall has been kept saturated with a soot penetration thickness of 2%. These characteristics are based on the results obtained from the modeling of the soot loading process up to the first water injection event, which are described in [28]. Therefore, the objective has been to identify the main trends in particulate layer properties' variation that provide the pressure drop after a pre-DPF injection event.

**Figure 2.** Soot loading test in DPF #A defining the conditions of the parametric studies.

In particular, the pressure drop at the end of the injections marked with a red circle in Figure 2 has been analyzed. The selected pressure drop value is the one after the end of the thermal transient that follows a pre-DPF water injection. This value determines the benefit in pressure drop reduction [12]. For the sake of simplicity, the properties of all inlet channels have been assumed to be the same, which implies the assumption of a homogeneous water distribution within the monolith cross-section. Therefore, the results of the parametric study must be understood as lumped representative properties of the collected soot.

According to the results presented in [28] and as the boundary condition for the modeling work, the particulate layer porosity before the first water injection event is known to be 0.65. This data were obtained assuming that the representative collector unit diameter in the particulate layer is that of the mode of the particle size distribution (69 nm). Nevertheless a range of particulate layer porosity from 0.4–0.97 has been considered in this work. It provides a wide range to analyze the possible effect of particulate layer compaction at the same time that the maximum values of porosity reported in the literature [35] have been covered.

Concerning the soot mass distribution, an increase of the particulate layer thickness has been assumed along the inlet channels till the plug end. In every axial location, the thickness of the particulate layer is assumed to be homogenous on all of the channel walls. Linear and parabolic laws to define the rate of thickness increase have been explored. In addition, the particulate layer thickness is assumed to be very thin and constant from the inlet cross-section up the a given distance from which the increasing thickness profile is imposed. This distance, which will be referred to as the onset of the soot mass distribution (*δp<sup>l</sup>*) from now on, has been also varied in order to determine its impact on the

pressure drop. This study is based on the conclusions obtained from Scanning Electron Microscope (SEM) analysis [34]. Figure 3 shows SEM pictures corresponding to two different samples of DPF #B, which was loaded with 44.6 g. The pictures show the cross-section of an inlet channel at two different distances from the monolith entrance. One of the samples was subjected to a pre-DPF water injection. Both cases show that the penetration is very small, as concluded from experimental [25] and modeling [23] studies, just affecting the porous wall rugosity. On the one hand, the baseline sample shows a similar thickness of the soot cake both at the inlet and rear end. On the other hand, the case of the sample subjected to water injection shows an irregular thin particulate layer at the inlet region of the channel. The soot tends to be accumulated in the end region of the channel close to the plug. This is confirmed by Figure 4, which shows three pictures of the rear end (21.5 cm from monolith inlet) in different inlet channels. A clear random accumulation of soot layer fragments can be observed. Its effect can be assumed equivalent to a fast increase of the particulate layer thickness in this region, thus decreasing the inlet channel effective cross-section area.

**Figure 3.** Scanning Electron Microscope (SEM) pictures of the cross-section of DPF #B inlet channels with the same soot loading at different locations in baseline and after water injection conditions: (**a**) baseline at 2.5 cm; (**b**) baseline at 20.5 cm; (**c**) pre-DPF water injection at 2.5 cm; (**d**) pre-DPF water injection at 20.5 cm.

**Figure 4.** Pictures of soot agglomerates accumulation in the rear end of three inlet channels (21.5 cm from monolith inlet) after water injection in DPF #B.

The swept-in particulate layer porosity and onset of the soot mass distribution provide an extensive family of particulate layer structures. Figure 5a shows how the particulate layer porosity changes the soot thickness along the inlet channels. The grey dashed series represents the particulate layer thickness just before the first injection event, the particulate layer porosity being 0.65. In all remaining cases, the soot mass distribution is exactly the same, i.e., at a given point, the amount of soot is the same at any particulate layer porosity. In these examples, the onset of the soot mass distribution is at *δpl* = 0.1 m from the inlet monolith cross-section imposing a parabolic profile. Therefore, the change in porosity is the responsible of the different cake thickness according to Equation (16):

$$w\_{pl,i} = \frac{a\_{in} - \sqrt{a\_{in}^2 - \frac{m\_{spl,i}}{\Delta x \rho\_{pl}}}}{2} \tag{16}$$

where subscript *i* identifies the node of computation along the channel. The density of the particulate layer (*ρp<sup>l</sup>*) is a function of the carbon density and the porosity of the particulate layer:

$$
\rho\_{pl} = \rho\_{\mathbb{C}} \left( 1 - \varepsilon\_{pl} \right) \tag{17}
$$

**Figure 5.** Effect of particulate layer porosity and soot mass distribution onset on the particulate layer thickness profile in DPF #A.

Below a porosity of 0.6, the particulate layer thickness is very thin along the whole channel. However, as the porosity increases, the thickness undergoes a faster growth, as clearly observed for 0.95 in porosity. The differences in thickness become evident from the very beginning of the soot mass distribution onset and increase towards the inlet channel rear end. In contrast to porosity, the onset of the soot mass distribution in the particulate layer, whose effect is represented in Figure 5b, gives rise to quite homogeneous cake thickness, most of the differences being concentrated in the rear end region. This is especially evident as the soot mass distribution is moved back, which produces a sharp rate of thickness increase from 0.15 m in the onset length.

The DPF pressure drop resulting from the parametric study imposing experimental inlet flow conditions and soot loading after the water injection is shown in Figure 6 for Injections 1, 5, 9 and 13. A parabolic soot mass distribution is considered in these computations. In all plots, the white line represents the solution domain corresponding to the experimental pressure drop value for every injection.

**Figure 6.** DPF pressure drop as a function of the particulate layer porosity and the soot mass distribution onset after different injection events in DPF #A.

As observed in all cases, moving back the soot mass distribution onset provides an almost linear decrease of the pressure drop regardless of the particulate layer porosity. This is shown in Figure 7, which presents the pressure drop dependence on the onset of the soot mass distribution for the parabolic profile and different water injections. The maximum pressure drop is always found when the onset of the particulate layer is placed close to the monolith inlet. This means that the worst loading conditions of the DPF are determined by an homogeneous particulate layer along the whole channel, which is the natural profile towards soot loading processes' convergence [28]. These results evidence the interest for soot and ash accumulation in the rear end of the inlet channels. On the other hand, given any onset for the growth of the particulate layer thickness, the pressure drop decreases as the porosity increases. These kinds of conditions would be caused by an engine operating at high mass flow, thus at medium-low temperature, thus resulting in a high Peclet number [35]. In addition, the impact is greater in homogenous soot mass distribution and as soot loading increases. Higher soot loading correlates with the injection number according to Figure 2, i.e., the 13th water injection takes place at higher soot loading conditions.

Keeping as a reference a particulate layer porosity of 0.65, the analysis of the plots in Figure 6 reveals that the experimental pressure drop obtained in DPF #A after the first pre-DPF water injection can be only attained provided that the particulate layer begins its growth several centimeters after the channel inlet (∼4.5 cm). This trend is more clear as the number of injections increases, even taking into account that the pressure drop after the thermal transient related to the injection event grows up from 8500 Pa–9200 Pa. In the 13th water injection, the growth of the particulate layer should begin at 12.5 cm from the monolith inlet, assuming that the particulate layer porosity is kept in 0.65. Therefore, the onset of the soot mass distribution is progressively moved towards the channel end as the amount of soot increases. This fashion in soot mass distribution in the particulate layer explains the need to increase the target pressure drop after the 13th water injection that was necessary to impose during the soot loading test shown in Figure 2. In order to ensure the effectiveness of the water injection technique [12], it is necessary to allow the particulate layer thickness to grow again along the inlet channels before to perform the next water injection. In addition, it is worth noting how the rear soot accumulation should be more intensive if the porosity of the particulate layer decreases as a result of a water compaction process. According to Darcy's law, this effect indicates that the permeability decrease caused by the porosity reduction has much more negative impact than the benefits brought by a thinner particulate layer. Figure 8 represents the particulate layer permeability as a function of the porosity, according to Equation (7). In contrast, the thickness is a function of the porosity, the soot distribution profile and, consequently, the axial location along the inlet channel, in agreemen<sup>t</sup> with Figure 5.

**Figure 7.** Impact of the onset of the soot mass distribution on the DPF pressure drop as a function of the particulate layer porosity and the soot mass loading (number of injections) in DPF #A.

**Figure 8.** Particulate layer permeability as a function of the porosity.

Based on these magnitudes, the resulting pressure drop is finally defined by the filtration velocity. Figure 9a shows the filtration velocity profile along the inlet channel for a particular onset of the soot mass distribution (*δp<sup>l</sup>* = 0.11 m) as a function of the particulate layer porosity after the first water injection. As the porosity decreases, the filtration velocity gets reduced in the initial inlet channel region. However, the filtration velocity is higher for low porosities in the rear end region. The flow tends to accumulate in the rear end of the inlet channel increasing the gas pressure because of the lower permeability related to low porosity. As a consequence, more mass flow passes across the thicker particulate layer region at higher velocity when the porosity decreases, thus leading to higher pressure drop. Complementarily, the filtration velocity profile for a particular porosity (*ε pl* = 0.75) is represented in Figure 9b as a function of the soot mass distribution onset. In this case, the filtration velocity gets reduced in the thin particulate layer region as the onset is moved back. Despite an intermediate region with the highest filtration velocity, it gets the minimum value also in the rear end, where the particulate layer has the maximum thickness. This kind of profile leads progressively to the almost linear pressure drop decrease shown in Figure 7 as the soot mass distribution onset is moved back.

**Figure 9.** Filtration velocity profile as a function of the soot mass distribution onset and the particulate layer porosity after the first water injection event in DPF #A.

Several combinations of values of particulate layer porosity and onset of the soot mass distribution provide the experimental pressure drop, i.e., the white line represented in the plots of Figure 6. This is due to the influence of these variables on filtration velocity and particulate layer permeability and thickness. Figure 10 represents a set of filtration velocity profiles determined by combinations of particulate layer porosity and the onset of soot mass distribution that reproduce the experimental pressure drop after every water injection event. In all cases, the compaction of the particulate layer must be accompanied by moving back the soot mass distribution since it leads to lower filtration velocity both in the thin layer region and in the rear end region, reducing the length of the transition from fast to slow velocities. This trend is more apparent as the amount of soot and the number of injections increase, since the soot is progressively dragged towards the rear end of the inlet channels.

**Figure 10.** Filtration velocity profiles for different soot mass distribution onsets and particulate layer porosities providing the experimental pressure drop after every modeled injection event in DPF #A.

The analysis of the pressure drop after the water injection events provides general trends on the change of the particulate layer properties. In order to describe with higher detail its characteristics after every injection, several pairs of particulate layer porosity and soot mass distribution onset were selected to model the soot loading process following the injection event. Figure 11 represents the cases for the first and the 13th water injections. As shown in (a), the slope of the pressure drop after the first water injection is very sensitive to the properties of the particulate layer. In fact, the best fitting for the soot loading process is obtained for the case of no effects on the particulate layer porosity, whose reference value before injection is 0.65, just a minimum drag of the particulate layer with an onset of the soot mass distribution in 0.045 m being required. Shorter drag would have minor effects on the pressure drop increasing rate, but requiring higher porosity of the particulate layer than baseline. By contrast, noticeable drag of the particulate layer after the first injection would require grea<sup>t</sup> compaction of the particulate layer, leading to small porosity values and to an excessive slope of the pressure drop increase as a function of the soot loading. As the number of injections increases, the slope of the pressure drop loss sensitivity to the particulate layer properties since the onset of the soot mass distribution increases for all possible porosities. As shown in the 13th water injection, in which the effectiveness of the injection process is limited, the onset of the soot mass distribution is within a small range. Nevertheless, it is interesting to note that there is still a valid solution for the baseline porosity, which confirms that the compaction effect of the water can be considered negligible.

**Figure 11.** Influence of the soot mass distribution onset and the particulate layer porosity on the rate of increase of the pressure drop after water injection events in DPF #A.

To finish the analysis of the pressure drop reduction causes, Figure 12 represents the obtained results for the first and 13th water injections imposing a linear soot mass distribution instead of a parabolic profile. As observed, all of the general trends previously described can be considered independent of the kind of soot mass distribution law defining the particulate layer. The main difference is related to the onset of the soot mass distribution, which should be delayed in the case of a linear distribution. This is why this kind of soot distribution imposes, for a particular onset, less soot moved towards the rear end in comparison to a parabolic distribution. Nevertheless, the order of magnitude of the solutions is very similar and proves that a lumped representation of the inlet channels cross-section provides good accuracy to understand the mechanism leading to pressure drop reduction after pre-DPF water injection.

**Figure 12.** DPF pressure drop as a function of the particulate layer porosity and the soot mass distribution onset in DPF #A imposing a linear soot mass distribution.
