In this section, we present in detail the models for the prediction of loading, pressure drop and regeneration for the wall-flow particulate filters. The theory lying under the loading and pressure drop model has been already presented in the literature for the filtration and pressure drop of aerosols flowing through porous media. Here, we apply is for the case of the wall-flow filter.
3.3.1 Loading
Calculation of loading in this category of filters presents no difficulties. Typical filtration efficiency of a wall-flow DPF is about 90–95% and it remains approximately constant, regardless of the exhaust gas flow rate. Thus, filtration efficiency of a wall- flow DPF is assumed to be constant and no particularly complicated model is needed. Instead, a fixed percentage of the particulate mass of the exhaust gas is assumed to be filtered and deposited in the channel walls.
The soot deposit forms a layer on the walls of the inlet channels of the wall-flow DPF. The model does not consider the details of the structure of the deposit layer. Of interest is only the axial distribution of the layer’s porosity and thickness for each channel because both are important for the correct estimation of the induced pressure drop.
For some fixed amount of filtered mass, the thickness of the deposit layer that it is formed depends on its porosity. The porosity of the particulate layer is found to be dependent on the loading conditions, (exhaust gas flow rate, particulate composition etc.) [40]. In practice, though, we have no theoretical background to calculate the porosity as a function of the loading conditions. Therefore, it is assumed constant, its value is considered a tunable parameter of the model, and all phenomena that influence it are lumped into this value.
Sec. 3.3 Loading and pressure drop model 59 We may recall from Section 3.1.2 that the actual filtration process of the wall- flow DPF is much more complicated and the approach may seem oversimplified. In fact, we could formulate a more sophisticated model for filtration in the wall and the deposit of this filter type. In concept, such a model should be built on the grounds of the filtration mechanisms described in Section 3.1.2—but there are at least three reasons that such an effort is presently out of place.
First, since the filtration efficiency of the wall-flow DPF is already very high, building such a model would provide additional insight only regarding the porosity of the soot deposit. The problem here is that it is doubtful if one could ever have the additional data required for reliable calculations, namely characteristics of the particles and how these correlate to the filtration efficiency of the soot deposit. Such difficulties are evident in the case of the deep-bed particulate filter (foam or fibrous DPF) [10, 11], where the formulation of a filtration model cannot be bypassed.
Second, it is probably the case that a simplified pressure drop model—for which porosity data are significant—can give acceptable accuracy for our engineering model. For an engineering work, it is vital to keep things as simple as possible.
Third, there are presently much more urgent problems in DPF modeling, which are connected to the regeneration process in the filter. Of these problems, this work attempts to address the issues of VOF content and 3D reactor modeling but much work remains to be done. A detailed filtration model for the DPF is therefore of the lowest priority.
3.3.2 Pressure drop
In the wall-flow DPF, pressure drop is induced as the exhaust gas flows through the deposit and the wall of the filter. This is essentially a problem of fluid flow through a porous medium, where pressure drop should be correlated with the thickness of the medium and geometrical characteristics of the porous medium and the temperature and mass flow rate of the gas. In our problem, we have two resistances to the flow (the deposit layer and the filter wall), connected in series. The total pressure drop is the sum of these resistances, i.e.:
∆p = ∆p1+ ∆p2 (3.1)
As a first approximation, the classical Darcy’s law can be used in this regard. It was postulated in 1856 as a result of measurements of pressure drop of water flowing through packed beds of sands. In differential form, the Darch’s flow equation can be written as:
−dp dx =
µu
K (3.2)
where dp is the differential change in pressure over a length dx, µ is the dynamic viscosity of the fluid, u is the fluid velocity and, finally, K is the permeability of the porous medium. The permeability is conceived to be a property of the solid porous medium and independent of the fluid; it depends on such properties of the porous medium as porosity, pore size distribution and surface area.
The Forchheimer relationship is an alternative to the Darcy’w law; it extends the latter by adding a term for nearly-quadratic dependence upon fluid velocity:
−dp dx = µu K + ρun B (3.3)
In the above equation, n is a number close to 2, ρ is the fluid density, and B is a second permeability-like property, that is again dependent only on the porous
material. The Forchheimer relationship is supposed to give better results when applied to the case of high mass flow rates (high fluid velocities).
The first (linear) term in the Forchheimer equation has been designated as the viscous term, while the second (approximately quadratic) term has been designated as the inertia term. It is suggested by theory [44] that the viscous term is connected to pressure losses due to viscous forces exerted in the fluid, while the inertia term is connected to repeated expansions, contractions and direction changes experienced by the fluid due to its complex motion through the pores and channels of the porous medium.
Further variations of the Darcy and Forchheimer relationshiops have been devel- oped to account for slip flow effects. When the pore diameter of the medium dporeis much greater that the mean free path of the fluid molecules λ, it is assumed that the flow velocity at the wall is zero. The Darcy’s and Forchheimer relationships have been developed for exactly such flows. Nevertheless, the gas flow at the wall may be greater than zero in case the mean free path is comparable or greater that the pore diameter of the porous medium; this is the case of slip flow.
The Knudsen number Kn = λ/dpore is the dimensionless quantity that should be checked for the presence of slip flow. The mean free path of the gas molecules can be calculated by the formula:
λ = µ p
r πRT
2M
Obviously, slip flow can be neglected for Kn À 1 while it becomes prominent for Kn ≈ 1.
Slip flow conditions are usually met in rarified or slow velocity gas flows through fine-grained porous media. The observed effect of slip flow is that the pressure drop is reduced to values lower that those predicted by the Forchheimer equation. This gives permeability values that are lower than those obtained for non-slip-flow conditions, which is unacceptable because both K should be dependent on porous medium properties only.
In the original work of Bissett and Shadmann [3], the Forchheimer relationship has been used for the prediction of pressure drop through the DPF soot layer and wall, neglecting slip flow effects. Nevertheless, slip flow is relevant in the case of the DPF, since the Knudsen number for the flow through the soot layer may be calculated to be approximately equal to 1, assuming mean pore diameter of the soot layer equal to 0.1 µm [32].
In a recent publication, Versaevel et al. [45] have performed an experimental and computational study on the premeability of the soot layer of a DPF, where the effect of slip flow has been included. They provide experimental evidence that the porosity of the soot deposit is not constant but depends on flow conditions. Their work includes a small review of a number of relationships that have been proposed to correct the Forchheimer equation for the case of slip flow. Such relationships define the permeability as a function of either the mean free path of the gas molecules λ or the Knudsen number Kn. It appears that there is no consensus regarding the correction to account for slip flow effects.
For the case of the DPF soot layer, Versaevel et al. used the Darcy’s law. Com- bining (3.1) and 3.2, the total pressure drop becomes:
∆p = µum A(ρ1K1) +
µuws
Sec. 3.4 Regeneration model 61