Development of a Catalyzed Diesel Particulate Filter Multizone Model for Simulation of Axial and Radial Substrate Temperature and Particulate Matter Distribution
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Abstract
The catalyzed particulate filter (CPF) is an important exhaust aftertreatment subsystem that is managed by the electronic control unit (ECU) of an engine. CPFs need periodic regeneration to avoid temperature exotherms and excess engine back pressure. To this end, a multizone particulate filter (MPF) model was developed in this research to serve as a simulation tool to provide onboard diagnostics (OBD) data for managing CPF active regeneration (AR). The MPF model runs in real time within the ECU to provide feedback on temperature and particulate matter (PM) loading distribution within each axial and radial zone of the filter substrate. The MPF model accounts for the internal and external heat transfer mechanisms, inlet temperature distribution using the fully developed boundary layer concept, and PM oxidation by thermal (O_{2}) and NO_{2}assisted oxidation mechanisms. A calibration procedure was developed to calibrate the PM kinetics and heat transfer coefficients of the MPF model. The model shows the good capability to predict temperature and PM loading distribution within the filter.
Keywords
Diesel particulate filter model ECU based particulate filter model Temperature distribution Particulate loading distribution Diesel exhaust emission controlAbbreviations
 AR
Active regeneration
 B10
Diesel blend (ULSD) with 10% Biodiesel
 B20
Diesel blend (ULSD) with 20% Biodiesel
 CFD
Computational fluid dynamics
 CPF
Catalyzed particulate filter
 CO_{2}
Carbon dioxide
 DOC
Diesel oxidation catalyst
 DPF
Diesel particulate filter
 ECU
Electronic control unit
 MPF
Multizone particulate filter
 MTU
Michigan Technological University
 NO_{2}
Nitrogen dioxide
 NO
Nitrogen monoxide
 OBD
Onboard diagnostics
 O_{2}
Oxygen
 PO
Passive oxidation
 PM
Particulate matter
 SCR
Selective catalytic reduction
 ULSD
Ultralowsulfur diesel
 1D
One dimensional
 2D
Two dimensional
 3D
Three dimensional
Nomenclature
 A_{amb}
Surface area of outer surface [m^{2}]
 Ā
Average crosssectional area [m^{2}]
 A_{02}
Preexponential for thermal (O_{2}) PM oxidation [m K^{−1} s^{−1}]
 A_{N02}
Preexponential for NO_{2}assisted PM oxidation [m K^{−1} s^{−1}]
 Af_{i,j}
Crosssectional area perpendicular to direction of heat transfer [m^{2}]
 Ar_{i,j}
Area normal to direction of heat transfer in the radial direction [m^{2}]
 As_{i,j}
Combined surface area of both inlet and outlet channels [m^{2}]
 C
Constant [−]
 c_{f}
Specific heat of filter material [J kg^{−1} K^{−1}]
 C_{NO2}
CPF inlet NO_{2} concentration [ppm]
 C_{O2}
CPF inlet O_{2} concentration [ppm]
 c_{p}
Constant pressure specific heat [J kg^{−1} K^{−1}]
 C_{PM}
CPF Inlet PM concentration [mg m^{3}]
 c_{s}
Specific heat of PM cake [J kg^{−1} K^{−1}]
 d
Side length of square channels [m]
 D
Overall diameter of the CPF [m]
 \( {E}_{O_2} \)
Activation energy for thermal (O_{2}) PM oxidation [J gmol^{−1}]
 \( {E}_{{\mathrm{NO}}_2} \)
Activation energy for NO_{2}assisted PM oxidation [J gmol^{−1}]
 F1
Temperature factor [−]
 F2
Mean to surface temperature ratio [−]
 F_{3–1}
Radiation view factor between inlet of the channel to filter wall [−]
 F_{3–2}
Radiation view factor between outlet of the channel to filter wall [−]
 h_{amb}
Ambient convective heat transfer coefficient [W m^{−2} K^{−1}]
 h_{g}
Convective heat transfer coefficient [W m^{−2} K^{−1}]
 ΔH_{reac}
Heat of reaction for carbon oxidation via O_{2} [J kg^{−1}]
 J1
Radiosity of channel inlet surface [W m^{−2}]
 J2
Radiosity of filter wall surface [W m^{−2}]
 J3
Radiosity of channel outlet surface [W m^{−2}]
 \( {k}_{{\mathrm{o}}_2} \)
Rate constant for thermal (O _{2}) PM oxidation [m s^{−1}]
 \( {k}_{{\mathrm{NO}}_2} \)
Rate constant for NO_{2} assisted
 k_{g}
Thermal conductivity of channel gas [W m^{−1} K^{−1}]
 L
Axial length [m]
 L_{t}
Total length of CPF [m]
 ∆L
Effective zone length [m]
 ∆t
Solver Time step [s]
 \( \overset{\cdot }{m} \)
Instantaneuos exhaust mass flow rate [kg s^{−1}]
 \( {\overset{\cdot }{m}}_i{,}_j \)
Mass flow rate entering each zone [kg s^{−1}]
 \( m{s}_{i,j} \)
Mass of PM in each zone [kg]
 \( m{s}_t \)
Total mass of PM retained [kg]
 \( {\overset{\cdot }{m}}_{{}_{\mathrm{total}}} \)
Total mass flow rate into CPF [kg s^{−1}]
 Nc_{i}
Number of cells in each radial zone [−]
 Nc_{t}
Total number of cells [−]
 Nu_{avg}
Average Nusselt number of the inlet and outlet channel
 Nu_{inlet}
Nusselt number of the inlet channel
 Nu_{outlet}
Nusselt number of the outlet channel
 Pe_{w}
Peclet number of wall [−]
 P_{in}
CPF inlet gas pressure [kPa]
 \( {\overset{\cdot }{Q}}_{\mathrm{cond},\mathrm{axial}} \)
Axial conduction [W]
 \( {\overset{\cdot }{Q}}_{\mathrm{cond},\mathrm{radial}} \)
Radial conduction [W]
 \( {\overset{\cdot }{Q}}_{\mathrm{conv}} \)
Convection between channel gases and filter wall [W]
 \( {\overset{\cdot }{Q}}_{\mathrm{rad}} \)
Radiation between channel surfaces [W]
 \( {\overset{\cdot }{Q}}_{\mathrm{reac}} \)
Total energy released during PM cake exothermic reactions [W]
 \( {\overset{\cdot }{Q}}_{{\mathrm{reac},\mathrm{NO}}_2} \)
Energy released during NO_{2}assisted PM cake exothermic reactions [W]
 \( {\overset{\cdot }{Q}}_{{\mathrm{reac},\mathrm{O}}_2} \)
Energy released during thermal (O_{2}) PM cake exothermic reactions [W]
 rc_{i}
Radial distance of a zone from centerline [m]
 Δr
Effective zone radius [m]
 R_{u}
Universal gas constant [J gmol^{−1} K^{−1}]
 \( {\overset{\cdot }{S}}_{c_{(th)}} \)
Thermal (O_{2})assisted PM cake oxidation rate [kg C_{(s)} m^{−3} s^{−1}]
 \( {\overset{\cdot }{S}}_{c_{(NO2)}} \)
NO_{2}assisted PM cake oxidation rate [kg C_{(s)} m^{−3} s^{−1}]
 S_{p}
Specific surface area of PM [m^{−1}]
 t
Time [s]
 T
Average gas temperature of channels [K]
 T_{amb}
Ambient temperature [K]
 T_{exit}
Filter exit gas temperature [K]
 Tf
Temperature of combined filter and PM cake [K]
 T_{in}
CPF inlet temperature [ºC]
 T_{m}
Mean exhaust gas temperature [K]
 T_{s}
Wall inner surface temperature [K]
 T_{r}
Temperature at a given radial location [K]
 ts_{i,j}
Average PM cake thickness in each zone [m]
 \( \overline{ts} \)
Average PM cake thickness across entire CPF [m]
 \( \overline{t{s}_{\mathrm{i}}} \)
Average PM cake thickness in each radial zone [m]
 u
Representative velocity in each zone [m s^{−1}]
 u_{I}
Average inlet channel velocity [m s^{−1}]
 u_{II}
Average outlet channel velocity [m s^{−1}]
 u_{s}
Average velocity through PM layer [m s^{−1}]
 u_{si}
Average velocity through PM layer in each radial zone [m s^{−1}]
 u_{w}
Average velocity through wall layer [m s^{−1}]
 u_{wi}
Average velocity through wall layer in each radial zone [m s^{−1}]
 V
Total volume of a zone [m^{3}]
 V_{e}
Empty volume in each zone [m^{3}]
 V_{es}
Empty volume accounting for PM cake [m^{3}]
 V_{f}
Volume of filter in each zone [m^{3}]
 V_{s}
PM cake volume in each zone [m^{3}]
 V_{t}
Total volume of the filter [m^{3}]
 W
Exhaust gas molecular weight [kg kmol^{−1}]
 \( {W}_{c_{(s)}} \)
Molecular weight of carbon [kg kmol^{−1}]
 \( {W}_{{\mathrm{O}}_2} \)
Molecular weight of oxygen [kg kmol^{−1}]
 \( {W}_{N_{{\mathrm{O}}_2}} \)
Molecular weight of nitrogen dioxide [kg kmol^{−1}]
 x
Diameter ratio of CPF or DOC [−]
 Y
Mass fractions [−]
 y
Axial location [−]
Greek Variables
 a
Thermal diffusivity [m^{2} s^{−1}]
 \( {a}_{O_2} \)
O_{2} oxidation partial factor [−]
 \( {a}_{N{O}_2} \)
NO_{2} oxidation partial factor [−]
 ε_{r}
External radiation coefficient [−]
 λ
Effective thermal conductivity of PM cake and filter [W m^{−1} K^{−1}]
 λ_{f}
Thermal conductivity of filter [W m^{−1} K^{−1}]
 λ_{s}
Thermal conductivity of PM cake [W m^{−1} K^{−1}]
 ρ
Exhaust gas density [kg m^{−3}]
 ρ_{i}
Exhaust gas density in each radial zone [kg m^{−3}]
 ρ_{f}
Filter density [kg m^{−3}]
 ρ_{s}
PM cake density [kg m^{−3}]
 σ
StefanBoltzmann constant [W m^{−2} K^{−4}]
Subscripts and Superscripts
 i
Radial direction
 j
Axial direction
1 Introduction
Diesel particulate filters play a key role in meeting current and future particulate emission standards for diesel engines. One of the disadvantages of the diesel particulate filter is the need for periodic active regeneration. Regeneration is necessary to avoid an increase in engine exhaust back pressure as the particulate matter (PM) accumulates in the filter substrate. The active regeneration oxidizes PM which reduces engine back pressure and fuel consumption caused by increased back pressure/pumping work. However, active regeneration also consumes fuel. An experimental study by Rose et al. showed that there are overall fuel consumption penalties due to the increase in back pressure and the extra fuel required for active regeneration. The results showed a 3.3 % fuel penalty for a Euro 5 compliant 1.4L turbocharged diesel engine with B10 fuel during the New European Driving Cycle [1].
The knowledge of PM mass retained as a function of time is the vital input for an effective and efficient active regeneration strategy. This is mainly because the PM loading beyond an acceptable level can lead to a damaged catalyzed particulate filter (CPF) and underloading leads to frequent regeneration events and hence excess fuel consumption and CO_{2} emissions. The current engine electronic control unit (ECU) controls the PM loading using an internal PM estimator model. The PM estimator relies on the calibrated engine PM maps; pressure drop across the CPF and temperature measurements to determine PM mass retained and regeneration frequencies [2].
Advanced regeneration strategies involve simplified CPF models that run real time within the ECU to provide more accurate feedback on current PM loading of the filter substrate. By applying simplified models that are similar to the ones used during design, development, and application of CPFs, the regeneration frequency and duration can be optimized based on vehicle operating conditions which determine engine operating conditions. This would lead to fuel consumption savings and resulting CO_{2} emission reductions and also a potential increase in the durability of the CPF.
The optimum regeneration frequency and duration rely on the accurate prediction of temperature distribution within the filter. Due to the heat transfer and cake PM oxidation, CPF filter temperature varies spatially (both axial and radial directions) and affects the regeneration efficiency of the filter as the PM oxidation reactions are highly temperature dependent. Hence, the accurate prediction of temperature distribution within the filter substrate will aid in the optimum regeneration frequency and duration. The prediction of temperature distribution within the filter also provides robust diagnostics capability by monitoring the CPF temperature (as a virtual sensor) at several locations within the filter. The other use of the temperature distribution prediction could be as an alternative to a conventional CFD model to calculate axial and radial temperature distribution of the substrate. This could reduce significant simulation time and resources during the design and development phase of diesel particulate filters.
Hence, the modelbased approach presented in this paper would enable the use of simplified MPF models that run fast enough in the ECU to predict temperature distribution within the filter and the PM cake oxidation. This paper describes a new MPF model for modeling temperature distribution in a CPF and the local cake PM oxidation.
The overall objective of this study is to develop a computationally efficient MPF model for predicting CPF temperature distribution and PM oxidation. The model should require limited calibration effort and needs to be validated with experimental data.
 1.
Review the literature related to modeling of heat transfer in the CPF substrate and the PM density distribution including experimental data.
 2.
Develop a MPF model that is capable of simulating filter substrate temperature and PM oxidation during passive oxidation and active regeneration within the axial and radial zones of the CPF.
 3.
Develop a calibration procedure for the heat transfer within the substrate and to the ambient including PM kinetics (NO_{2} and O_{2}assisted PM oxidation) locally.
 4.
Calibrate the MPF model using the experimental data [3, 4, 5, 6] to arrive at a common set of MPF model calibration parameters for all experiments.
2 Background and Literature Review
The literature review is organized in two parts. First, the use of simplified models (0D and 1D) for ECUbased controls from previous studies is explored including the heat transfer models and their assumptions. Second, the simplified multidimensional modeling efforts (2D and 3D) in CPF modeling were studied along with their benefits and limitations related to modeling accuracy and computational efficiency.
2.1 CPF Models for ECUBased Controls
CPF models can be incorporated in the ECU to monitor and optimize CPF performance along with the engine performance. Such a CPF model for CPF regeneration was described by Kladopoulou et al. [7] using a lumped parameter model. This was a 0D model and the simulation relied on time dependence of input parameters. The spatial dependence (axial and radial directions) was assumed to be negligible. This lumped model included an external heat transfer mechanism by considering external ambient heat transfer through convection. Subsequently, further advanced modelbased control techniques were explored by many researchers to simplify conventional 1D models for realtime ECU application with reasonable accuracy and computational speed compared to conventional mapbased control approaches as presented by Rose et al. [2].
The realtime implementation of a 0D CPF model along with 1D DOC model was presented by Nagar et al. [8] and showed that the 0D CPF model was able to predict the average filter substrate temperature within 25 °C. They also highlighted the difficulty of initiating regeneration based on ∆P (difference between inlet and outlet pressure of CPF) measurement and concluded that PM loading provides a more reliable criterion to trigger CPF filter regeneration than using ∆P values. The CPF model by Nagar et al. assumed internal convective heat transfer from filter substrate to exhaust gas as a mechanism to dissipate the energy release during PM oxidation. However, the model ignores any conductive heat transfer within substrate.
Mulone et al. presented the 1D CPF model for ECU application for steadystate [9] and transient operating conditions [10]. The model is based on the single channel representation of the CPF. The model was able to predict axial variation in the PM loading. However, the model ignored radial temperature gradients and the radial PM loading distribution in the CPF.
The resistance node methodology presented by Depcik et al. [11] provides a simplified and computationally efficient modeling approach to predict axial and radial temperatures of the filter. However, this model assumes uniform inlet temperature and ignores the inlet temperature variation along the radial direction of the CPF.
2.2 Simplified Multidimensional CPF Modeling
Konstandopoulos et al. [12] developed a multichannel model using a multiphase continuum approach to simulate spatial nonuniformities in the filter (axial and radial directions). The multiphase continuum model was derived from the discrete multichannel description of the CPF. The model worked in a CFD code framework to include the partial differential equations in the CPF model. The conduction, convection, and radiation heat transfer within the filter were considered in the model. With inlet radial nonuniformities in velocity and temperature, the model showed partial regeneration of the filter. The partial regeneration mainly occurred at the periphery of the filter as the PM at the periphery of the CPF was not oxidized due to lower temperatures in this region. This continuum model is computationally expensive as it involved a system of several ordinary and partial differential equations and the equations have to be solved in a 3D domain.
Yi [13] developed a 3D macroscopic model for predicting PM loading within the filter. The model was based on grouping the channels with reasonably uniform inlet conditions and solving each group using 1D model equations. This established a link between 1D and 3D models and reduced the complexity of detailed 3D simulations. The model did not consider PM oxidation (passive oxidation and active regeneration) within the filter and also neglected heat losses from the substrate can of the filter. The model showed PM distribution within the filter over a period of time and also indicated that PM distribution evolves to uniform when simulated without inlet and outlet connections (uniform inlet velocity). The model showed significant nonuniform radial and axial distribution of PM when simulated with inlet and outlet connections attached to the substrate and highlighting the need for multidimensional analysis to determine the actual PM distribution within the filter.
2.3 Proposed MPF Model
The nonuniform flow and temperature at the inlet of the filter affect both axial and radial PM distribution within the filter. This would cause localized heating and excess thermal stress due to heterogeneous oxidation of PM under high temperature and a high PM loading condition. This affects the durability of CPF systems. Previous 0D and 1D studies ignore these effects; thus, this study focuses on developing a multizone modeling approach (axial and radial zones) of the CPF for ECU applications.
The new MPF model proposed in this work is capable of predicting both temperature and PM loading distribution within the CPF in the axial and radial directions. The model considers varying CPF inlet temperature conditions to account for the thermal boundary layer development at the inlet of the CPF, heat transfer within the filter, and also heat dissipation to the ambient including conduction, convection, and radiation heat transfer. PM cake oxidation by O_{2} and NO_{2} is also considered.
As will be shown later in the paper, the proposed model can predict nonuniform distribution of the filter loading and temperature within the filter substrate and it requires significantly less simulation time compared to the multidimensional CPF models in the literature.
3 Experimental Data
Specifications of the CPF used in the experiments
CPF  Units  

Substrate material  Cordierite  – 
Cell geometry  Square  – 
Diameter  267  mm 
Length  305  mm 
Cell density  31 (200)  cells cm^{−2} (cells in.^{−2}) 
Cell width  1.49  mm 
Wall density  0.45  g cm^{−3} 
Specific heat  891  J kg^{−1} K^{−1} 
Thermal conductivity  0.84  W m^{−1} K^{−1} 
Experimental data collected by Shiel et al. [3, 5] and Pidgeon et al. [4, 6] at 18 different operating conditions are used to calibrate and validate the model developed in this study. The test summary of 18 experiments (6 passive oxidation and 12 active regeneration experiments) used in this study is shown in reference [14]. The experiments were performed with three fuels—ULSD, B10, and B20 blends. The properties of test fuels used for the experiments are documented in references [3, 4]. The detailed test setup, test matrix, and instrumentation are explained in reference [3] for the passive oxidation experiments and reference [4] for the active regeneration experiments.
The passive oxidation and active regeneration experiments start with 600 °C cleanup phase followed by stage 1 loading of filter with a DOC inlet temperature of 265 ± 10 °C for 30 min. Then the stage 2 loading starts and continues to achieve a target filter loading of 2.2 ± 0.2 g/L. Upon completion of stage 2 loading, the test was continued further with the ramp up phase (RU) for 15 min. Following the RU phase, for the passive oxidation experiments, the engine was operated at the passive oxidation test conditions (for POB1016 experiment, the CPF inlet conditions are temperature 408 °C, 61 ppm NO_{2}, 209 ppm NOx, and 7.1 % O_{2}) for a specified duration (43 min for POB1016 experiment). For active regeneration experiments, following the RU phase, the engine was operated at an active regeneration ramp phase for 10 min (or until DOC inlet temperature has stabilized at 325 ± 10 °C) and then the active regeneration phase for a predetermined duration (26 min for ARB101 experiment) at the specified CPF inlet conditions (temperature 530 °C, 4 ppm NO_{2}, 119 ppm NOx, and 7.8 % O_{2} for ARB101 experiment). Upon completion of passive oxidation phase for passive oxidation experiments and active regeneration phase for active regeneration experiments, the filter was loaded at stage 3 loading condition for 30 min and then the test was continued further with stage 4 loading for 60 min. The filter was weighted at the end of each stage. The engine operating conditions for all four stages of the loading are the same.
The experimental data from these tests (temperature distribution and PM loading) are used in this study for the calibration of the internal and external heat transfer of the MPF model and also the PM kinetic parameters.
3.1 CPF Temperature Distribution
Having temperature distribution data is critical for this study to validate the MPF model since predicting temperature distribution within the CPF is one major contribution from this study. The experimental temperature distribution data were collected from references [5, 6]. For each of the 18 experiments in this study, temperature distribution within the CPF was measured using 16 ungrounded Ktype thermocouples to determine temperature distribution within the filter during each test. These data were used for the heat transfer model calibration and validation of the MPF model simulation data. The CPF thermocouple layout and specifications of the thermocouples are shown in references [5, 6].
From Figs. 1 and 2, the following trends are observed. The temperatures are varying in both the axial and radial directions. The radial variation of temperature is comparatively higher (up to 40 °C) than the axial variation in temperature (up to 12 °C). The radially decreasing temperature is attributed to external ambient heat transfer of the filter and inlet flow/temperature maldistribution, and the axially increasing temperature is attributed to the oxidation of PM within the filter during active regeneration along with the heat transfer. Without the PM oxidation, the temperature would drop axially due to heat transfer.
4 Model Development
4.1 Overview
 1.
The inlet PM deposits uniformly over the entire volume of the filter substrate and all the PM deposits as a cake.
 2.
The PM inlet rate into the each zone is assumed to be the ratio of volume of each zone to the total volume of the filter. In other words, no maldistribution of inlet PM is considered.
 3.
Species concentrations (O_{2} and NO_{2}) are assumed to be uniform in each zone of the filter and are equal to inlet concentrations.
 4.
Back diffusion of NO_{2} due to the catalyst wash coat is not considered.
 5.
Gaseous hydrocarbon oxidation within CPF is not considered.
 6.
PM cake layer and substrate wall are at the same temperature. In other words, no temperature gradient across the PM cake layer and substrate wall is considered.
 7.
A fully developed boundary layer exists at the inlet of the CPF.
 8.
The exhaust gas mixture is assumed to be an ideal gas.
 9.
The exhaust gas has the same properties as air at 1 atm pressure. Properties are considered as a function of temperature. CPF inlet species concentrations (CO_{2}, O_{2}, N_{2}, and H_{2}O) are used for the calculation of molecular weight of the exhaust gas.
4.2 Discretization
4.3 Filter Temperature Equations
The detailed formulation for the terms used in Eq. (1) is explained in Appendix A.
 Nu _{avg}

Average Nusselt number of the inlet and outlet channel
 Nu _{inlet}

Nusselt number of the inlet channel
 Nu _{outlet}

Nusselt number of the outlet channel
 k _{ g }

Thermal conductivity of channel gas
 Pe _{ w }

Peclet number of wall
 Re _{ w }

Reynolds number of wall
 As _{ i,j }

Combined surface area of inlet and outlet channels.
4.4 PM Oxidation
In Eq. (5), the pressure term for computing the exhaust gas density (ρ _{ i,j }) is assumed to be constant.
The details of formulation for the terms used in Eq. (5) is explained in Appendix A.
4.5 Velocity Equations
The detailed formulation of Eqs. (6) to (9) are explained in Appendix A.
4.6 Temperature Distribution at Filter Inlet
The radial temperature distribution at the inlet of the filter is due to the thermal boundary layer development as explained in references [12, 19, 20, 21]. In order to account for the thermal boundary layer development, the empirical temperature factor profile is determined by analyzing experimental data from the 18 runs.
The temperature profile at the inlet of the CPF is not constant across the radial direction for the data in Figs. 1 and 2. This is mainly because of the thermal boundary layer development in the upstream of exhaust pipes and the DOC. The thermal boundary layer develops when the exterior surface of the pipe is exposed to a different temperature than the fluid flowing through the pipe. If the air temperature of outer surface is lower than the exhaust gas temperature, then the temperature of the exhaust gas in contact with the inner surface decreases and causes a subsequent drop in temperature of the exhaust gas in other regions of the pipe. This leads to the development of thermal boundary layer (similar to velocity boundary layer).
 T _{ m }

Mean exhaust gas temperature
 T _{s}

Wall inner surface temperature
 T _{ r }

Temperature at a given radial location
 y

Axial location.
The diameter ratio is the ratio of CPF diameter at a given measurement location to the maximum CPF diameter. From Fig. 6, the temperature factor is almost constant up to CPF diameter ratio of 0.4 (indicating uniform temperature) and drops to 0 value (minimum temperature) at the CPF diameter ratio of 1.0 (outer radius of the filter). The maximum gradient in the temperature factor is observed at the CPF diameter ratio of 0.8 to 1.0, showing that 50 % of the radial temperature reduction is in the 20 % of the filter section closest to the outer radius of the filter. From Appendix B, the minimum substrate temperature at the outer surface (R = 133 mm) is 4.3 % lower than mean substrate temperature at the inlet \( \left(\frac{T_m}{T_s}=1.043\right) \).
The detailed procedure for developing the thermal boundary layer temperature factor and other coefficients used in the MPF model are explained in Appendix B. Using Eqs. (11, 12) and knowing the temperature at one radial location/zone of the CPF inlet, the temperatures at the other radial locations can be determined. The MPF model uses one CPF inlet temperature sensor data (T_{in}) to develop thermal boundary layer profile for the remaining zones at the CPF inlet.
4.7 Numerical Solver
where
∆t is the time step for the solver.
 k _{ g }

Thermal conductivity of exhaust gas
 \( \rho \)

Density of exhaust gas
 c _{ p }

Specific heat capacity of exhaust gas
 h=h _{ g }

Convective heat transfer between filter substrate and channel gas.
Using the Eqs. (14) and (15), the initial values of time step can be determined. The MPF model includes conduction, convection, and radiation terms together. By iteration from the above initial guess, the stable time step of 0.01 s was determined to work for up to a 20 × 20 multizone formulation in this work.
5 Model Calibration
5.1 Inputs and Outputs
 1)
Instantaneous exhaust mass flow rate \( \left(\overset{\cdot }{m}\right) \)
 2)
CPF inlet concentrations (C_{PM}, C_{NO2}, and C_{O2})
 3)
CPF inlet temperature (T _{in})
 4)
CPF inlet gas pressure (P _{in}).
MPF model constants
Symbol  Description  Units  Values 

F  Radiation view factor  [−]  0.011 
C1  DOC radial temperature distribution factor 1  [−]  −2.493 
C2  DOC radial temperature distribution factor 2  [−]  1.0585 
C3  DOC radial temperature distribution factor 3  [−]  −0.3285 
C4  DOC radial temperature distribution factor 4  [−]  1.7631 
C5  DOC mean to surface temperature ratio  [−]  1.0425 
C6  DOC temperature sensor offset  [−]  0.0273*Tin−2.4996 
ρ _{pm}  PM density  kg m^{3}  104 
Sp  Specific surface area of PM  m^{−1}  5.5 × 10^{7} 
Cp_{air}  Specific heat of air  J kg^{−1} K^{−1}  Using Eq. (66) 
μ  Dynamic viscosity of air  N s m^{−2}  Using Eq. (67) 
λ _{ f }  Thermal conductivity of substrate wall  W m^{−1} K^{−1}  1 
λ _{ p }  Thermal conductivity of PM cake layer  W m^{−1} K^{−1}  2.1 
α _{O2}  O_{2} combustion partial factor  [−]  0.8 
α _{NO2}  NO_{2} combustion partial factor  [−]  1.75 
5.2 Calibration Process
For the model calibration and results presented in this work, the 10 × 10 MPF model was used which runs about 12 times faster than real time on a laptop with 12GB RAM, 64bit and inter core i7 processor. However, the optimum number of zones for an ECU application can be determined by running the model at different levels of discretization (axial and radial zones). This discretization study along with the detailed model validation is ongoing work and will be covered in the next research paper.
 1)
Calibration of PM oxidation: PM kinetic parameters (A _{NO2}, E _{NO2}, A _{O2}, and E _{O2}) are determined in this step. The calibration of PM kinetic parameters from the engine experiments are preferred over synthetic gasbased lab reactor experiments because the engine experiments provide representative PM composition, residence time, and operating temperatures of the filter for calibration. Hence, the PM kinetic parameters determined from the engine experiments can be directly used in the MPF model. The objective of the first step is to minimize the error between the simulation and the total experimental PM mass retained.
 2)
Calibration of heat transfer coefficients: convective and radiation heat transfer coefficients (h _{amb} and ε _{ r }) are determined from this step. In addition, the filter substrate density (ρ _{ f }) is found. The substrate density along with other thermophysical properties (thermal conductivity, substrate density, and PM density) of the filter changes during the filter loading. Hence, in order to simulate the change in thermophysical properties of the filter, the filter substrate density is also considered as the one of the calibration variables in the MPF model. Depcik et al. [11] showed in his model calibration efforts that the model accuracy could be improved during temperature rise portion of the experiment by optimizing the filter substrate density. The objective of this step is to minimize the RMS temperature error between the simulation and the experimental data measured by the 16 CPF thermocouples during the stage 1 and 2 loading phases of the experiments.
The calibration process starts with the initial assumption of calibration parameters. The initial values of the calibration parameters were determined from the references [11, 14, 23, 24]. All 18 runs are simulated and the results are compared with the experimental data. To improve the model accuracy, the calibration process is repeated if the PM mass loading error exceeds 3 g and the RMS temperature distribution error exceeds 15 °C. Details about the calibration process for the two steps are explained next.
5.2.1 Step 1: PM Kinetics Calibration Procedure
 1.
Determine the NO_{2}assisted PM kinetics (A _{NO2} and E _{NO2}) from the passive oxidation experiments keeping other parameters constant. Optimization is done in Matlab® using NelderMead Simplex method [25]. Matlab function fminsearch is used to minimize the error between MPF model simulation and the experimental PM mass retained. The error value of 1 g is used as the target for the Simulink design optimization at the end of each of the stages of loading.
 2.
Use the NO_{2}assisted PM kinetics (from task 1) to determine the thermal (O_{2})assisted PM kinetics from the active regeneration experiments keeping other parameters constant in the model.
 3.
From the PM kinetics determined from tasks 1 and 2, use the Arrhenius plots to determine the optimum PM kinetic parameters for each type of fuel.
 4.
From tasks 1, 2, and 3, determine one set of PM kinetic calibration parameters (A _{NO2}, E _{NO2}, A _{O2}, and E _{O2}) for each type of fuel (ULSD, B10, and B20)
5.2.2 Step 2: Heat Transfer Coefficients Calibration
Similar to step 1, NelderMead Simplex optimization method [24] is used to calibrate the heat transfer coefficients in the MPF model. The heat transfer coefficients and filter density values are varied keeping all other parameters constant in the model. The objective of the optimization routine is to minimize the RMS temperature error between the simulation and the experimental temperature data measured by the 16 thermocouples during stages 1 and 2 of the loading phase of the experiment. The RMS error of 2 °C is used as the target for optimization.
Calibrated parameters of the MPF model for PM kinetics for three different fuels
PM oxidation  Symbol  Description  Units  ULSD  B10  B20 

NO_{2}assisted  A _{NO2}  Preexponential for NO_{2}assisted PM oxidation  m K^{−1} s^{−1}  0.0088  0.0083  0.0085 
E _{NO2}  Activation energy for NO_{2}assisted PM oxidation  kJ gmol^{−1}  60.8  64.1  63.1  
Thermal (O_{2})  A _{O2}  Preexponential for thermal (O_{2}) PM oxidation  m K^{−1} s^{−1}  0.982  1.004  1.029 
E _{O2}  Activation energy for thermal (O_{2}) PM oxidation  kJ gmol^{−1}  158.1  151.8  148.9 
Calibrated heat transfer coefficients and filter density for the MPF model
Symbol  Description  Units  Values 

h_{amb}  Convection heat transfer coefficient  W m^{−2} K^{−1}  8.1 
ε _{r}  External radiation coefficient  [−]  0.16 
ρ _{filter}  Filter density  kg m^{−3}  449 
PM kinetics parameters standard deviation summary between runs
PM oxidation  Symbol  Description  Units  ULSD (%)  B10 (%)  B20 

NO2 assisted  A _{NO2}  Preexponential for NO_{2}assisted PM oxidation  m K^{−1} s^{−1}  3.8  0.6  NA 
E _{NO2}  Activation energy for NO_{2}assisted PM oxidation  kJ mol^{−1}  1.1  1.2  NA  
Thermal (O2)  A _{O2}  Preexponential for thermal (O_{2}) PM oxidation  m K^{−1} s^{−1}  1.4  2.5  4.5 % 
E _{O2}  Activation energy for thermal (O_{2}) PM oxidation  kJ mol^{−1}  0.8  0.2  0.9 % 
6 Results and Discussion
Using the single set of calibration parameters (Tables 3 and 4) determined from the calibration process, the MPF model simulations of one passive oxidation (POB1016) and one active regeneration (ARB101) experiment were studied. The results are presented in the following section.
6.1 Temperature Distribution
From Fig. 13, the experimental filter substrate temperature shows an increase in temperature (10–12 °C at filter radiuses below 40 mm) axially due to the local PM oxidation, and the substrate temperature close to the wall shows a lower increase in temperature axially (5 °C approximately) due to the dominant convection and radiative heat loss to the ambient compared to the local PM oxidation. From Fig. 13, the MPF simulation model also shows an increase in temperature (4.4 °C) in the axial direction due to PM oxidation; however, the magnitude of the increase is low compared to the experimental data.
6.2 PM Mass Retained
The measured PM masses at the end of each stage of the experiment (stage 1 to 4) are marked with plus markers in Fig. 14. NO_{2}assisted PM oxidation is the dominant mode of PM oxidation shown by a dashed line because of the NO_{2} concentration (61 ppm). The thermal (O_{2})assisted PM oxidation is low (<1.3 g) compared to NO_{2}assisted PM oxidation due to the lower exhaust gas temperature during the PM oxidation (408 °C at 4.49 to 5.22 h).
Comparison of experimental PM retained measured to MPF model output PM retained at the end of each stage for POB1016 experiment
Stage  Time  Expt.  Model  Diff. 

[hrs]  [g]  [g]  [g]  
1  0.47  4.3  4.3  0 
2  4.23  36.5  37.8  1.3 
3  5.70  26.0  26.6  0.6 
4  6.70  35.1  35.9  0.8 
Comparison of experimental PM retained measured to MPF model output PM retained at the end of each stage for ARB101 experiment
Stage  Time  Expt.  Model  Diff. 

[hrs]  [g]  [g]  [g]  
1  0.49  4.0  4.8  0.8 
2  4.97  43.2  46.0  2.8 
3  6.34  18.3  17.0  −1.3 
4  7.34  27.9  27.6  −0.3 
From the above simulation and experimental results analysis, the MPF model shows good capability in predicting temperature distribution, PM mass retained, and filter loading distribution within the filter substrate. For an ECU application, computational time is the key requirement for successful implementation of modelbased control approach such as one presented in this paper. In general, the model with fine discretization is good for accuracy but takes long computational time. Hence, it is necessary to carry out a parametric study to determine the optimum required number of zones in the MPF model for aftertreatment control applications. This discretization study along with the detailed model validation is an ongoing work and will be covered in the next research paper.
7 Summary and Conclusions
A new MPF model was developed for simulating filter substrate temperature and PM loading distribution within the filter. The model was parameterized through a twostep calibration procedure using 18 sets of data from the passive oxidation and active regeneration experimental data from references [3, 4, 5, 6]. The model was calibrated with sample passive oxidation and active regeneration experimental data, and the model shows good capability in predicting the temperature distribution and filter loading within the filter substrate.
 1.
The MPF model to predict axial and radial temperature and filter loading distribution within the filter substrate was developed. The model accounts for heat transfer within and external to the filter and PM cake oxidation by O_{2} and NO_{2}.
 2.
The radial temperature distribution initiates well before the inlet of the CPF and affects the distribution for the entire length of the CPF. This radial temperature distribution can be characterized using thermal boundary layer equations. For the CPF presented in this study, 50 % of overall radial temperature reduction is in the 20 % of filter section close to the outer radius of the filter.
 3.
The detailed twostep MPF model calibration procedure was developed. A single set of PM kinetic parameters for each fuel type (Table 3) was found, and a single set of heat transfer coefficients (Table 4) was found for the all 18 runs.
 4.
The new MPF model simulation results show good capability in predicting temperature distribution (within −24/+6 °C), PM mass retained (within −1.3/+2.8 g), and filter loading distribution within the CPF.
Notes
Acknowledgments
The authors would like to thank Kenneth Shiel and James Pidgeon of Michigan Technological University for recording the temperature distribution data presented in this work and Dr. Kiran Premchand for the assistance in 1D model simulation data presented in this work.
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