CFD Simulation of Soot Filtration in Diesel

CFD Simulation of Soot Filtration in Diesel Particulate Filters S. Bensaid, D. L. Marchisio, D. Fino, G. Saracco, V. Specchia Dipartimento di Scienza dei Materiali e Ingegneria Chimica - Politecnico di Torino, ITALY

1. Introduction Soot emissions from automotive applications can be reduced through on-board diesel particulate filters (DPFs): the filter gets progressively loaded by filtering the soot laden flue gases, thus causing an increasing pressure drop, until an oxidative regeneration is induced by temperature rise. Due to the complexity of the involved phenomena, a fully predictive and detailed mathematical model represents a very useful tool that could be profitably used during trap design and in the engineering of the regeneration phase. In particular, a detailed analysis of the filtration efficiency with respect to particle size and filter through-wall velocity was performed via the estimation of the different particle collection mechanisms. The EulerianEulerian multi-phase approach was employed in this work due to its flexibility and low computational costs, especially when compared to Eulerian-Lagrangian models. 2. Governing equations The investigation of soot filtration via DPFs can be conceptually decomposed into three subproblems: the first one concerns the computation of the multi-phase flow field inside the DPF, the second one is related to the description of the evolution of the filter characteristics as soot is deposited, whereas the last one focuses on the formation of a soot cake in the DPF channels. An Eulerian-Eulerian approach, known as multi-fluid model, was employed in this work. The evolution of the multi-phase system is described by well-known volume-averaged balance equations of mass and momentum for the continuous and the dispersed phase. The interaction between the two phases is given by the drag force, calculated according to Schiller and Naumann [1]. It is worth noticing that the influence of Brownian forces on the drag force via the Stokes-Cunningham coefficient, and on particles trajectories in the channels, is not considered in this work, since for this particular configuration they can be neglected [2]. In fact, the gas streamlines crossing the wall filter are mainly responsible for the deposition location of the particles, while Brownian forces act just as a random shift. As it will become clearer below, however, Brownian motions are indeed considered in the deposition model, since this is the dominant phenomenon and defines the collection efficiency η. The Darcy equation was adopted to describe the pressure drop across the filter walls (modeled as an isotropic porous media), and the filter permeability k is obtained from the Ergun equation for laminar flows. The deposition rate of soot into the filter walls can be calculated as reported in the source term of the soot balance equation (Eq. 1) and was implemented into the code via user defined functions: U ∂ (α p ρ p ) + ∇ ⋅ (α p ρ p Up ) = −α p ρ pη 1 − ε 3 p , ε 2 Dc ∂t

(1)

where αp, ρp and Up are the soot volume fraction transported by the fluid, the particle density and the particle velocity vector, respectively. The collector diameter Dc represents the equivalent diameter of a bed of spheres constituting the porous walls and that has the same particle collection efficiency of the real porous medium. It is calculated as a function of the

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porosity and of the mean pore diameter as follows [3]: Dc =

3 (1 − ε ) d pore . 2 ε

(2)

A parameter of major importance in particle filtration is the global collection efficiency η, namely the probability of successful collections of soot particles onto fictitious spherical collectors constituting the channel walls, largely due to deposition from Brownian motions and interception. In the evaluation of this efficiency, calculated as indicated by [3], the size of the soot particles has a considerable impact, as shown later. The assessment of the efficiency, and more generally of the soot deposition rate, in the first stages of filtration is very important because it is representative of the capacity of the bare filter to collect particles of various diameters when filtration is not enhanced by the soot cake, yet. The Brownian efficiency ηD (see Eq. 4) and the interception efficiency ηR (see Eq. 5) can be considered as independent of each other and the resulting expression of the global collection efficiency η can be hypothesized as follows:

η = η D + η R − η Dη R

(3)

The contribution of Brownian motions to the collection efficiency is calculated through the evaluation of the Brownian diffusivity, taking into account the deviation due to the StokesCunningham coefficient. Hence, the Brownian efficiency ηD is calculated as a function of the Peclet number (Pe) and of the Kuwabara function g(ε) [3]:

η D = 3.5 g( ε ) Pe −2 / 3

(4)

The other contribution to the collection efficiency is given by the interception mechanism, depending on the parameter N, namely the ratio between the particle and the collector diameters, and of the coefficient s:

η R = 1.5 N 2

g (ε ) 3 (1 + N ) s

where N =

dp Dc

and s =

3 − 2ε

ε

(5)

In order to quantify the ability of a filter with a wall depth of L to collect soot particles, it is very convenient to calculate an “integral” filtration efficiency E, defined as the integral of the right hand side of Eq. 1 at steady state for a simple filter characterized only by a single spatial coordinate, resulting in: ⎛ 1− ε 3 L ⎞ ⎟ E = 1 − exp⎜⎜ − η ε 2 Dc ⎟⎠ ⎝

(6)

This filtration efficiency represents the volume (or mass) fraction of the incoming soot particles that is collected by the filter. The accumulation of soot particles into the filter pores modifies its characteristics as follows: 9 1 ⎡ ⎤ 1 2 2 − ( 1 − ε )1 3 − ε − ( 1 − ε )2 ⎥ 3 ⎛ Dc ⎞ ⎢⎣ − + ε α 1 6 ⎛ ⎞ 5 5 0 d ⎦ 1− ε0 ε = ε 0 − α d , Dc = ⎜ ⎟ ⎟ , k = k0 ⎜⎜ ⎟ ⎡ π⎠ 9 1 D ⎝ 1− ε ⎝ c 0 ⎠ 2 − ( 1 − ε 0 )1 3 − ε 0 − ( 1 − ε 0 )2 ⎤ 1 − ε ⎢⎣ ⎥⎦ 5 5

(7)

where ε0 , Dc0 and k0 are the properties of the clean filter, while αd is the deposited soot volume fraction. During filter loading, the porosity of the filter decreases until a minimum value is attained, corresponding to the maximum volume fraction of deposited soot in the filter. The regions of the filter where this limit is reached become impermeable to the

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dispersed phase, and as a consequence a soot layer onto the porous filter surface is built up. This operation is modeled by setting equal to zero the velocity of the dispersed phase in the computational cells where the maximum soot packing limit is attained. During cake formation, the soot layer (which is characterized by a constant permeability and a constant maximum packing density, if no compaction effects due to pressure are considered), grows with time and generates an additional pressure drop, increasing with the soot layer thickness. 3. Numerical details and operating conditions

The three-dimensional geometry adopted to simulate the behavior of the system is depicted in Fig. 1. In particular, the four modeled channels are visible in the xy plane: two of them allow the gas to enter the filter, while the other two are plugged. At the end of the filter, plugs are reversed, and the gas is thus forced to flow through the channel walls of the DPF. Before the inlet of the filter, there is an upstream region, that allows to set a flat velocity profile Uc0. Similarly, after the filter, there is a downstream region that allows the flow to fully develop. The mesh is scaled with respect to the channel width: the lengths of the upstream region, of the filter, and of the downstream region are 36, 181 and 33 times the channel width, respectively. Each square channel is 1.4 mm in side and the filter wall is 0.38 mm in depth. The final computational domain is discretized in about 3×105 computational cells, guaranteeing a grid independent solution, which was proven by successive refinements of an initial grid. As far as the boundary conditions of the domain are concerned, the lateral faces are set as periodic. This allows to model the behavior of real DPFs, characterized by a periodic structure of hundreds of cells per square inch.

Fig. 1

Sketch of the grid used to simulate the flow field in the four channels constituting the computational domain.

Simulations are carried out under laminar flow conditions and with an initial filter porosity ε0 equal to 0.45, a resistance coefficient (1/k0) equal to 2×1012 m-2, a cake resistance coefficient (1/kcake) equal to 4×1013 m-2 and an initial collector diameter Dc0 equal to 15.8 μm. The soot laden flue gas is fed at 3 m/s with a typical solid volume fraction of 5×10-8 and the evolution of the filter characteristics are included in the model through user defined sub-routines. Mono-dispersed populations of soot particles with size equal to 100 nm, 200nm, 500nm and 2μm are here considered. It is worth noticing that the sub-micronic particles are more representative of the typical particulate matter emitted by diesel engines. The computational test is however very interesting, because it allows to verify the performance of the code for different particle inertia-to-drag ratios. Most of the simulations are carried out using the multifluid approach already described, however for the 100 nm diameter particles, when inertial effects are negligible, a pseudo-single phase approach is adopted as well, by resorting to user

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defined scalars for the continuous phase. 4. Results and discussion

Simulations showing the trend of the pressure drop between the inflow (at the beginning of the upstream region) and the outflow (at the end of the downstream region) of the system at different inlet velocities and for different filter resistances were already reported in [4]. Here, the behavior of different mono-dispersed populations of soot particles is displayed, allowing to verify the behavior of the DPF for different particle inertia-to-drag ratios. Fig. 2 displays the axial velocities (a) and radial velocities (b) of the continuous and the dispersed phase, at the initial stages of filtration, keeping constant the filter permeability throughout the simulation. The cases with particle size equal to 2 μm and 100 nm are depicted. By observing the behavior of axial velocities in Fig. 1-a, one can see that the 100 nm soot particles completely follow the gas streamlines, while for the 2 μm ones inertial effects occur at in the channel inlet. These discrepancies between the velocities of the two phases produce an uneven soot volume fraction distribution along the channel, and this can be verified by comparing the profiles of the soot through-wall velocity with the wall soot mass flow rate in Fig. 2-b. Instead, for particles with a small inertia such as the 100nm soot particles, the soot mass flow rate has the same profile as the through-wall velocity and it is just scaled by the soot mass concentration.

Fig. 2

(a) Axial velocity for gas (solid line) and soot particles (dots); (b) Soot through-wall velocity (solid line) and wall soot mass flow rate (dashed line).

The results reported so far refer to soot particle distributions along the channel length but not on the real amounts of soot filtered. In order to estimate these quantities it is crucial to VII-4, 4

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properly calculate the global collection efficiency η, that is a function of the gas and soot properties, notably gas temperature, soot particle size and velocity. Fig. 3 highlights the effect of the filtration efficiency E by comparing the wall soot mass flow rate of particles ranging from 100 nm to 2 μm, with the actual soot mass deposition rate. As it is seen, large particles (dp = 2 μm) are completely filtered by the filter walls, and due to particle inertia soot is driven to the end of the channel resulting in an uneven soot distribution. For smaller particles the inertia effects are negligible and the soot deposition profiles along the channel length are quite uniform. This is due to the fact that, although at the inlet and at the outlet the soot flow rate is quite high (see Fig. 2-b), the filtration efficiency E in these regions is low (see Fig. 3-b, right axis) and higher efficiencies are obtained only when a soot layer is formed on the filter walls in the successive filtration steps.

Fig. 3

(a) Wall soot mass flow rate; (b) Soot mass deposition rate (solid line - left axis) and filtration efficiency E (dashed line - right axis).

The evolution of the filter characteristic parameters is affected by the soot deposition rate, which is determined by the initial flow field of the continuous phase, and which progressively modifies the flow field itself by the effect of a pressure drop related to the interstitial soot loading. Moreover, after the soot volume fraction has reached the maximum packing limit the cake starts building up, and an additional contribution to the total pressure drop is generated, affecting again the flow field of both the continuous and dispersed phases. Fig. 4 depicts how the through-wall velocity (and the mass deposition rate as a consequence) is influenced by the soot loading first, and then by the cake formation. For sake of brevity, only the case with characteristic particle size of 100 nm is here presented. At the beginning of the filtration operation (Fig. 4-a), the through-wall velocity of the gas corresponds to the one of Fig. 2-b for 100nm. Afterwards, it changes according to soot loading: it appears that the most involved regions are located at the end of the channel, close to the plug (Fig. 4-b). As filtration proceeds, soot reaches the maximum packing limit inside the porosities of the filter and the cake starts building (Fig. 4-c), so creating a pressure drop highlighted by the decrease of the through-wall velocity. As the cake grows with time, the through-wall velocity profile becomes flatter and flatter (Fig. 4-d, 4-e and 4-f), corresponding to a slightly constant cake thickness along the axial coordinate.

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Fig. 4

Through-wall gas velocity for different amounts of collected soot (normalized). dp=100 nm.

The cake resistance coefficient is generally higher than that of the filter; therefore, after the cake has started to build onto the surface of the whole channel length, it immediately becomes the only considerable resistance. As a result, the soot cake deposition profile does not correspond to the wall “soot mass deposition rate” as presented in Fig. 3-b at the beginning of the filtration operation, but evolves assuming an “iso-permeability” profile characterized by a constant thickness of the soot layer. 5. Conclusions

A model for the description of soot filtration through DPFs was presented and discussed. Firstly, initial deposition is considered, treating as constants the parameters of the filter and evaluating its filtration efficiency: this condition represents the ability of the bare filter to trap soot particles, as a function of particle diameters and axial position into the channel. Then the modification of the porous structure due to soot loading and the formation of a soot cake are modeled, thus computing the local soot deposition rate onto the DPF walls. In conclusion, poly-disperse soot particle distributions, as well as an exhaustive comparison with experimental data, will be the next investigated issues. 6. References

1. 2. 3. 4.

Schiller, L., Naumann., Z.: Z. Ver. Deutsch. Ing., 77:318 (1935). Sbrizzai, F., Faraldi, P., Soldati, A.: Chemical Engineering Science, 60:6551 (2005). Konstandopoulos, A.G., Skaperdas, E., Papaioannou, E., Zarvalis, D., Kladopoulou, E.: SAE 2000-01-1016 p.593 (2000). Bensaid S., Marchisio D.L., Fino D., Saracco G., Specchia V.: Proceedings of The Italian Section of the Combustion Institute, I-7 (2007).

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