Abstract
A multizone 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 multizone version of a classical 1D 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 postloading 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.
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Abbreviations
 AR:

Active regeneration
 B10:

Diesel blend (ULSD) with 10% biodiesel
 B20:

Diesel blend (ULSD) with 20% biodiesel
 CPF:

Catalyzed particulate filter
 CO_{2} :

Carbon dioxide
 DOC:

Diesel oxidation catalyst
 DPF:

Diesel particulate filter
 ECU:

Electronic control unit
 HC:

Hydrocarbons
 MPF:

Multizone particulate filter
 MTU:

Michigan Technological University
 NO_{2} :

Nitrogen dioxide
 NO:

Nitrogen monoxide
 O_{2} :

Oxygen
 OBD:

Onboard diagnostics
 PO:

Passive oxidation
 PM:

Particulate matter
 RMS:

Root mean square
 RU:

Ramp up
 SCR:

Selective catalytic reduction
 ULSD:

Ultralowsulfur diesel
 ∆P:

Pressure drop
 1D:

Onedimensional
 3D:

Threedimensional
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Acknowledgments
The authors would like to thank Kenneth Shiel and James Pidgeon of Michigan Technological University for collecting the temperature distribution data presented in this work and Dr. Kiran Premchand for assistance in understanding his 1D model simulation of the data presented in this work. We would like to extend our appreciation to the reviewers, whose comments helped us to significantly improve the content.
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Appendices
Appendix 1. Experimental Data
Appendix 2. Pressure Drop Submodel
In the pressure drop submodel, 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.
The absolute pressure of radial section (i) is calculated by following the streamlines ( s1 , s2 , and s3) shown below
The pressure drop in the outlet channel stream lines (O _{4}, O _{3}, O _{2}, and O _{1}) are calculated using the following equation
The pressure drop in the inlet channel stream lines (I _{4}, I _{3}, I _{2}, and I _{1}) are calculated using the following equation
The wall pressure drop at each zone is calculated using the following equation
The cake pressure drop at each zone is calculated using the following equation
The pressure drop across each radial section is calculated as
The overall pressure drop of the CPF is calculated using the following equation
Appendix 3. Filtration Submodel
In the filtration submodel, 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
The number of pores in each zone of the substrate wall is given as [24]
The empty volume of the substrate wall is given as
The number of pores in each slab at each zone is calculated as
where n is 1, 2, 3, and 4.
Wall collector efficiency at each slab is calculated as
The filtration efficiency of a unit collector in the PM cake layer is calculated as
The partition coefficient is used to determine transition from deep bed filtration regime to cake filtration regime, and it is calculated as
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
The detailed formulation of Eqs. ((33)) to ((35)) is explained in [12, 13].
Appendix 4. PostLoading Permeability
During postloading 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 postloading for the passive oxidation experiments. For this analysis, all the passive oxidation experiment runs listed in Table 8 were used except POB1014 because of the very low PM oxidation rates causing gain in PM mass retained during PM oxidation. The postloading permeability ratios for the POB1014 experiment were in the range of 1 to 1.10.
From the data presented in Fig. 31, the PM cake permeability during postloading is calculated as
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 postloading cake permeability equation, C _{11} is the constant for the postloading 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 postloading for the active regeneration experiments. In Fig. 32, it can be seen that the permeability ratio changes are nonmonotonic and nonlinear 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 postloading. 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 postloading 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 postloading for the active regeneration experiments as shown in Appendix 6.
Appendix 5. MPF Model Validation Results: PM Mass Retained Summary
Appendix 6. MPF Model Validation Results: Pressure Drop Summary
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Mahadevan, B.S., Johnson, J.H. & Shahbakhti, M. Predicting Pressure Drop, Temperature, and Particulate Matter Distribution of a Catalyzed Diesel Particulate Filter Using a MultiZone Model Including Cake Permeability. Emiss. Control Sci. Technol. 3, 171–201 (2017). https://doi.org/10.1007/s4082501700626
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Keywords
 Diesel particulate filter model
 Particulate matter oxidation
 Pressure drop
 Cake permeability modeling
 Substrate wall permeability modeling