## 4 Active Regeneration Analysis  
Knowing the efficiency of the late fuel post-injection and the flow properties at the DPF inlet, the active regeneration process is next discussed. The gas composition, mass flow, pressure and temperature at the DOC outlet as a function of time, i.e., including the effect of the thermal transient due to the post-injected fuel heat release, were imposed as boundary conditions for the simulation of the DPF response. It was assumed that the initial DPF soot load of the filter was 11 g/L, which is a usual upper limit for the start of the regeneration [29]. The dynamics of the active regeneration is described based on the definition of the pre-heating, maximum reactivity and late oxidation stages [38]. Every stage is defined as a function of the derivative of the pressure drop across the DPF and the soot depletion rate. Figure 11a shows the evolution of the soot loading corresponding to the active regeneration used for the model calibration (Figure 5) in Section 2. The three regeneration stages are identified in Figure 11b. Figure 10: Post-injection efficiency as a function of the post-injection timing and the post-injected fuel mass rate. The active regeneration event starts with an increase of the pressure drop across the DPF, i.e., positive pressure drop derivative, corresponding to the pre-heating stage. This increase is related to the increase in exhaust gas temperature. The deviation between the experimental and modelled pressure drop shown in Figure 5a at the beginning of the process causes the difference in pressure drop derivative observed in Figure 11b. Nevertheless, as the substrate increases its temperature the soot oxidation initiates, which causes a progressive decrease of the pressure drop rate as observed in the experimental and modelled results. Consequently, a sudden increase of the soot depletion rate derivative takes place. Once the pressure derivative becomes equal to zero, the maximum reactivity stage is started. This second stage lasts until the minimum in pressure drop derivative. Its duration coincides with the maximum to zero derivative in soot depletion rate. Finally, the regeneration is slowly completed during the late oxidation stage, which is characterised by a gradual smooth increase towards zero of the pressure drop and soot depletion rate derivatives. Considering the parametric study for the late fuel post-injection parameters, Figure 12 depicts the duration of each active regeneration stage to achieve a consumption of 95% of the initial soot mass load. Both operating points confirm that a high delay in the post-injection is more convenient from the point of view of the velocity of the regeneration, independently of the post-injected fuel mass rate. Particularly, the best result is achieved when the post-injection is performed at 60 CAD aTDC for both engine operating points. These results are coincident with the trend shown in post-injection efficiency with respect to post-injection timing. This conclusion is also consistent with the work presented by [PERSON] et al. [24], who established that the HC oxidation in the catalyst induces more efficient regeneration than the combustion inside the cylinders. It is also interesting to note that some post-injection strategies are not successful to complete the regeneration. Several phenomena are leading to this result. On the one hand, the lower efficiency in  
Figure 11: Dynamics of (**a**) the soot mass load and (**b**) the derivatives of the DPF pressure drop and the soot depletion rate along an active regeneration event. Figure 12: Effect of the variation of the post-injection timing and fuel mass on the duration of each regeneration stage. terms of DPF inlet temperature benefit of the earliest post-injections is combined with a significant increase of the soot emission, as shown in Figure 8. It leads the DPF to the equilibrium condition between soot filtration and oxidation rates.