For the diesel particulate filter, the loading–regeneration level of the model cor- responds to the kinetics and washcoat level of the monolithic catalytic converter model. The target of the modeling at this level is to quantitatively describe the fundamental phenomena involved in the operation of the DPF. These are:
• Loading
• Pressure drop
• Regeneration
• VOF adsorption–desorption
Filter loading modeling implies prediction of the filtration efficiency of the filter and the characteristics of soot deposit. In principle, the filtration efficiency of the filter should be predicted, so as to determine (a) the particle concentration finally emitted from the vehicle, and (b) the mass of the soot that is accumulated in the filter. Ideally, not only the amount of the accumulated soot has to be determined but also its morphology, because it significantly affects filtration efficiency itself, pressure drop and regeneration.
The pressure losses of the exhaust gas flow depends on the DPF’s geometry, the distribution of the deposit layer, the exhaust gas flow rate and temperature. It is noteworthy that the distribution of flow resistances within the DPF affects the flow at its inlet, since flow adjusts itself and is channeled through the low-resistance regions of the filter. Thus, the accurate computation of the induced pressure drop is important for the prediction of the exhaust gas flow and the further evolution of the accumulated soot within the filter. Furthermore, the pressure drop of the DPF is needed for the design of the exhaust line, because it affects the operation of the engine.
For an active regeneration system such as an additive assisted regeneration sys- tem, the pressure drop induced by the DPF is continuously monitored. Modeling can be used to connect it with the soot mass loading of the filter. Work towards the feasibility of this approach has been presented by Stratakis and Stamatelos [37].
For the chemical reactions that take place during the regeneration phase (i.e. car- bon and hydrocarbons thermal and catalytic oxidation), the rate expressions deter- mine the soot consumption and the accompanying heat release. Arrhenius-type rate expressions are typically employed in all models that have been presented in the literature, e.g. [38, 5, 39].
Finally, the importance of the VOF content of the soot lies in two facts:
• Pressure drop depends on the amount of VOF in the soot deposit [40], and
• VOF is oxidized at temperatures considerably lower than the respective tem- peratures of dry soot. Local, partial regenerations may therefore occur due to heat release from VOF oxidation [41].
Soot particles of the diesel exhaust contain VOF from the moment they are formed in the engine’s combustion chamber, but the VOF content of the soot deposit can
Sec. 3.2 Overview of the DPF modeling problem 57 change depending on the operation point of the filter. Specifically, at low tempera- tures, heavy hydrocarbons in the gaseous phase can be adsorbed on the deposited soot, increasing its VOF portion; at increased temperatures, VOF desorbs or is ox- idized (burned) from exhaust gas oxygen. Adsorption, desorption and oxidation of soot VOF are still obscure phenomena. Hence, their modeling is considerably dif- ficult and the VOF content effect has been neglected in almost all DPF modeling efforts.
It must be noted that, in order to model a diesel particulate filter at this or any other level, the geometry of the filter must be known. This is straightforward only in the case of the wall-flow DPF. For fiber or foam filters, where the structure of the DPF is not well defined, the complexity of the geometry is accounted for via averaging quantities and statistical distributions. For example, the filter void fraction and the distribution of fiber diameters are employed for the characterization of the geometry of the fiber filter [10, 11].
3.2.2 Channel level
At the channel level, the objectives of modeling are to determine:
1. the distribution of the exhaust gas flow along the channel of a wall-flow filter or along the inner channel of any radial-flow deep-bed filter, and
2. the convective heat transfer between the exhaust gas and the solid phase (DPF ceramic and deposit of soot)
The flow and pressure distribution along the channel of the DPF depends on flow resistances along both the axial direction and the direction normal to the soot layer and ceramic wall, the latter being much higher than the former. Therefore, the accuracy of the flow field predictions at the channel level depends on the pressure drop submodel accuracy. In its turn, convective heat transfer between the exhaust gas flow and the solid phase depends on the flow distribution computations.
Thus, modeling at the channel level depends strongly on the lower level of pressure-drop and regeneration modeling, just as the catalytic converter’s channel model depends on the washcoat and kinetics modeling. Again, the DPF’s channel model is developed using a quasi-steady-state approach: the transient terms from the mass and energy balances for the gas flow are omitted. For more details, see the corresponding discussion for the catalytic converters model (Section 2.2.4).
3.2.3 Reactor level
Similarly to the monolithic catalytic converter, modeling at the reactor level deals only with the problem of heat transfer at the solid phase of the DPF. The input data of the reactor level model are the heat sources that were computed at the loading– regeneration level (due to VOF adsorption–desorption and soot combustion) and at the channel level (due to convective heat transfer between the exhaust gas and the DPF). Output is the temperature field of the DPF.
In fact, every aspect of DPF modeling at the reactor level is analogous to the monolithic catalytic converter modeling at the same level because, in both reactors, the only interaction between channels is through heat conduction. In principle, the same computational model implementation could be used for both reactors.
Thus, in a manner analogous to the monolithic catalytic converter models, the reactor model can be one-, two- or three-dimensional. One-dimensional DPF models
assume uniform flow and temperature distributions across the monolith’s inlet and consider their variations only in the axial direction. Multi-dimensional models work on clusters of channels and accounted for flow/temperature variations between the channel clusters.
Contrary to the monolithic catalytic converter, though, DPF modeling suggests the use of a three-dimensional reactor model, because the DPF operation is strongly three-dimensional. This is particularly true for regeneration modeling, which re- quires three-dimensional computation of the temperature field and the soot deposit distribution. To back this argument, one should consider that regeneration is not a local phenomenon but forms a combustion front that propagates in the filter. Thus, one- or two-dimensional models are expected to be a considerable compromise in accuracy of DPF behaviour prediction.
Nevertheless, only one-dimensional reactor models for the DPF have been pre- sented in the literature until now [38, 5, 4]. In this work, we formulate both a 1D and a 3D heat transfer model for the DPF, each with different scope and applica- tion range. The 3D model is implemented by interfacing our 1D channel model with commercial FEM software. The ABAQUS and ANSYS FEM software packages have been linked with the 1D DPF model, creating two instances of a 3D model [42, 43]. The 3D model has been applied for the prediction of the behaviour of a modular SiC DPF that inherently exhibits prominent three-dimensional heat tranfer behaviour.