Predicting Pressure Drop, Temperature, and Particulate Matter Distribution of a Catalyzed Diesel Particulate Filter Using a Multi-Zone Model Including Cake Permeability

Abstract

A multi-zone particulate filter (MPF) model was developed to predict pressure drop and PM oxidation of a catalyzed diesel particulate filter (CPF). The MPF model builds upon our previous work (Mahadevan et al., J Emiss Control Sci Technol 1:183–202, 2015; Mahadevan et al., J Emiss Control Sci Technol 1:255–283, 2015) by adding a new multi-zone version of a classical 1-D filtration model (Konstandopoulos and Johnson, 1989) to account for PM filtration within the substrate wall and PM cake of a CPF. In addition, pressure drop (∆P) simulation capability was also developed for the MPF model in order to simulate the pressure drop across the substrate wall and PM cake of the CPF. A cake permeability model was developed based on fundamental research findings in the literature. The PM cake and wall pressure drop simulation accounts for the wall and cake permeability variation during loading, PM oxidation, and an additional post-loading after oxidation. This extended MPF model was calibrated using 18 runs of experimental data from a Cummins ISL engine that consisted of passive and active regeneration data sets for ULSD, B10, and B20 fuels. The validation results show that the new MPF model can predict PM loading with a maximum root mean square (RMS) error of 7.4% and predict (∆P) across the filter with an RMS error of within 7.2%. It is found that the permeability of the PM cake layer increases rapidly during PM oxidation. The increase in permeability was attributed to the damage in the PM cake and was simulated using the newly developed cake permeability model. The increased permeability of the damaged PM cake layer and oxidation of cake PM leads to near zero cake PM pressure drop during PM oxidation for the passive and active regeneration experiments.

Appendices

Appendix 1. Experimental Data

Table 8 Passive oxidation experiments used for the calibration of the MPF model [5, 12]

Table 9 Active regeneration experiments used for the calibration of the MPF model [6, 12]

Appendix 2. Pressure Drop Sub-model

In the pressure drop sub-model, the pressure drop at each radial section is calculated by starting out with exit pressure P ₂| _x = L  = P _Baro and then traversing through all possible streamlines (dashed lines) shown in Fig. 30.

Fig. 30

Schematic of the streamlines (shown as dashed lines) used for calculating the pressure drop across CPF for the 3 × 1 zone model (4 axial and 1 radial discretization)

The absolute pressure of radial section (i) is calculated by following the streamlines ( s1 , s2 , and s3) shown below

$$ {\left.{P}_1\right|}_{i, s1}={O}_4\to {{\varDelta P}_{\mathrm{wall}}}_{i, j}\to {{\varDelta P}_{\mathrm{cake}}}_{i, j}\to {I}_3\to {I}_2\to {I}_1 $$

(29)

$$ {\left.{P}_1\right|}_{i, s2}={O}_4\to {O}_3\to {{\varDelta P}_{\mathrm{wall}}}_{i, j}\to {{\varDelta P}_{\mathrm{cake}}}_{i, j}\to {I}_2\to {I}_1 $$

(30)

$$ {\left.{P}_1\right|}_{i, s3}={O}_4\to {O}_3{\to {O}_2\to \varDelta {P}_{\mathrm{wall}}}_{i, j}\to {{\varDelta P}_{\mathrm{cake}}}_{i, j}\to {I}_1 $$

(31)

The pressure drop in the outlet channel stream lines (O ₄, O ₃, O ₂, and O ₁) are calculated using the following equation

$$ {P}_2\left|{}_{i, j}={P}_2\right|{}_{i, j+1}+{\left.\rho {v}_2^2\right|}_{i, j+1}-{\left.\rho {v}_2^2\right|}_{i, j}+ F\varDelta x\frac{\mu {v}_2}{a^2}\Big|{}_{i, j} $$

(32)

The pressure drop in the inlet channel stream lines (I ₄, I ₃, I ₂, and I ₁) are calculated using the following equation

$$ {P}_1\left|{}_{i, j}={P}_1\right|{}_{i, j+1}+{\left.\rho {v}_1^2\right|}_{i, j+1}-{\left.\rho {v}_1^2\right|}_{i, j}+ F\varDelta x\frac{\mu {v}_1}{a^2}{\left(\frac{a^{\ast }}{a}\right)}^2\Big|{}_{i, j} $$

(33)

The wall pressure drop at each zone is calculated using the following equation

$$ {\varDelta P}_{{w all}_{i, j}}={\mu}_{i, j}{v}_{w_{i, j}}\frac{w_s}{{k_{\mathrm{wall}}}_{i, j}} $$

(34)

The cake pressure drop at each zone is calculated using the following equation

$$ {\varDelta P}_{{\mathrm{cake}}_{i, j}}={\mu}_{i, j}{v}_{s_{i, j}}\frac{w_{p_{i, j}}}{k_{{\mathrm{cake}}_{i, j}}} $$

(35)

The pressure drop across each radial section is calculated as

$$ {\varDelta P}_{\mathrm{CPF}, i, s1}={\left[{P}_1|{{}_{x=0}-{P}_2}_{x= L}\right]}_{i, s1} $$

(36)

$$ \begin{array}{l}\varDelta {P}_{\mathrm{CPF}, i, s2}={\left[{P}_1|{{}_{x=0}-{P}_2}_{x= L}\right]}_{i, s2}\hfill \\ {}\varDelta {\mathrm{P}}_{\mathrm{CPF},\mathrm{i},\mathrm{s}3}={\left[{\mathrm{P}}_1|{{}_{\mathrm{x}=0}-{\mathrm{P}}_2}_{\mathrm{x}=\mathrm{L}}\right]}_{\mathrm{i},\mathrm{s}3}\hfill \end{array} $$

(37)

The overall pressure drop of the CPF is calculated using the following equation

$$ \varDelta {P}_{\mathrm{CPF}}=\frac{\sum_{s1}^{s \max}\left[{\sum}_{i=1}^{i= i \max }{\mathrm{VF}}_i\varDelta {P}_{\mathrm{CPF}, i}\right]}{3} $$

(38)

Appendix 3. Filtration Sub-model

In the filtration sub-model, the substrate wall is divided in to ${n}_{\max }$ (${n}_{\max }$ = 4) number of slabs. Each slab consists of several spherical wall collectors [4, 22]. The diameter of unit collector increases as the PM accumulates into the collector. The initial diameter of the unit collector is given as

$$ {d}_{c0, s}=\frac{3}{2}\left(\frac{1-{\varepsilon}_{0, s}}{\varepsilon_{0, s}}\right){d}_{\mathrm{pore},\mathrm{wall}} $$

(39)

The number of pores in each zone of the substrate wall is given as [24]

$$ {Np}_{i, j}=\frac{Veo_{i, j}}{\frac{4\pi}{3}{\left(\frac{d_{\mathrm{pore},\mathrm{wall}}}{2}\right)}^3} $$

(40)

The empty volume of the substrate wall is given as

$$ { V eo}_{i, j}={\varepsilon}_{0, s}{V_f}_{i, j} $$

(41)

The number of pores in each slab at each zone is calculated as

$$ {\left[{Np}_{i, j\ }\right]}_n=\frac{Np_{i, j}}{n_{\max }} $$

(42)

where n is 1, 2, 3, and 4.

Wall collector efficiency at each slab is calculated as

$$ {\eta}_{{\mathrm{wall}}_{i, j\ slab\ n}={\left[{\eta}_{D_{i, j}+}{\eta}_{R_{i, j}-}{\eta}_{D_{i, j}}{\eta}_{R_{i, j}}\right]}_{\mathrm{wall}\ \mathrm{slab}\ n}} $$

(43)

The filtration efficiency of a unit collector in the PM cake layer is calculated as

$$ {\eta}_{{\mathrm{cake}}_{i, j} = {\left[{\eta}_{D_{i, j}+}{\eta}_{R_{i, j}-}{\eta}_{D_{i, j}}{\eta}_{R_{i, j}}\right]}_{\mathrm{cake}}} $$

(44)

The partition coefficient is used to determine transition from deep bed filtration regime to cake filtration regime, and it is calculated as

$$ \Phi =\frac{{d_{c\mathrm{wall},\mathrm{slab}1}^2}_{i, j}-{d}_{c0,\mathrm{wall}\ \mathrm{slab}\ 1}^2}{{\left(\varPsi b\right)}^2-{d}_{c0,\mathrm{wall}\ 1}^2} $$

(45)

where d _{c wall, slab 1} is the unit collector diameter in the first slab of the substrate wall at a given axial and radial direction, Ψ is the percolation factor, and b is the unit cell diameter, and it is calculated as

$$ \frac{d_{c0, s}^3}{b^3}=1-{\varepsilon}_{0, s} $$

(46)

The detailed formulation of Eqs. ((33)) to ((35)) is explained in [12, 13].

Appendix 4. Post-Loading Permeability

During post-loading of PM in the CPF, the cracks and holes that formed in the PM cake during PM oxidation is filled by the incoming PM. This damage recovery process of the PM cake reduces the permeability. Figure 31 shows the change in permeability during post-loading for the passive oxidation experiments. For this analysis, all the passive oxidation experiment runs listed in Table 8 were used except PO-B10-14 because of the very low PM oxidation rates causing gain in PM mass retained during PM oxidation. The post-loading permeability ratios for the PO-B10-14 experiment were in the range of 1 to 1.10.

Fig. 31

Change in permeability ratio during the post-loading for the passive oxidation experiments

From the data presented in Fig. 31, the PM cake permeability during post-loading is calculated as

$$ {k}_{d_{i, j}}={k}_{p_{i, j}}\left({C}_{10}{\sum}_{i=1, j=1}^{i= i \max, j= j \max }{mc}_{i, j}+{C}_{11}\right) $$

(47)

where ${k}_{di, j}$ is the PM cake layer permeability accounting for the damage in the PM cake during PM oxidation (passive oxidation and active regeneration), ${k}_{pi, j}$ is the PM cake layer permeability accounting for the changes in mean free path length of the gas at each zone, C ₁₀ is the slope of the post-loading cake permeability equation, C ₁₁ is the constant for the post-loading cake permeability equation, and mc _i , j is the mass of cake PM in each zone.

Figure 32 shows the relative change in permeability during the post-loading for the active regeneration experiments. In Fig. 32, it can be seen that the permeability ratio changes are non-monotonic and non-linear indicating that the PM cake appears to exhibit a kind of “deep bed” filtration during the damage recovery with PM being primarily in the cracks at lower PM cake masses at the beginning of the post-loading. This concept needs further research and modeling. For the MPF model presented in this research, it was decided to use the same equation as used for the passive oxidation experiments (Eq. (47)) to calculate PM cake layer permeability during post-loading for the active regeneration experiments. The pressure drop simulation error with this assumption is within −0.2 to −0.5 kPa at the end of post-loading for the active regeneration experiments as shown in Appendix 6.

Fig. 32

Relative change in permeability ratio during the post-loading for the active regeneration experiments

Appendix 5. MPF Model Validation Results: PM Mass Retained Summary

Table 10 Comparison of experimental and model PM mass retained using calibration parameters (Table 5)

Table 11 Fractional PM mass oxidized during passive oxidation and active regeneration phases of all experiments

Appendix 6. MPF Model Validation Results: Pressure Drop Summary

Table 12 Comparison of experimental and model pressure drop using single set of calibration parameters (Tables 3, 4, 6 and 7)