# Understanding Factors Affecting the Balance Point (and Rate of Balance Point Approach) of a Diesel Particulate Filter: an Analytical Expression for the Balance Point Soot Loading

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## Abstract

Diesel particulate filters (DPF) are commonly used to remove harmful particulate matter (PM) from the exhaust of diesel engines. If the DPF is subjected to a constant inlet condition, under favourable conditions, the soot loading will eventually stabilise at a constant value, when the rate of soot accumulation is matched by the rate of soot oxidation by NO_{2}; this is known as the balance point. The balance point soot loading (BPSL) is commonly used as a measure of the effectiveness of passive soot oxidation. Generally, the DPF will take a long time to reach the balance point, making determining BPSLs, experimentally or using a 1-dimensional model, extremely time-consuming. This paper offers an alternative. By making some assumptions (constant temperature and through-wall gas velocity along the DPF), an equation allowing instantaneous BPSL prediction is derived, as is an equation predicting the variation in soot loading with time. Both give comparable predictions to a 1-dimensional model. The equation predicts that the BPSL is independent of the substrate, is proportional to the space velocity (but independent of DPF size) and is dependent on NO_{2}/PM ratio (but independent of NO_{2} concentration). Finally, this approximate approach is applied to the prediction of BPSL and evolution of soot loading for a DPF subjected to a repeated transient drive cycle. In this case, it is no longer possible to obtain a simple equation, but still the prediction is obtained much more quickly than with a 1-dimensional model. The prediction is in excellent agreement with the 1-dimensional model.

## Keywords

0-dimensional model Analytical expression Balance point Diesel particulate filter (DPF) Passive regeneration Simulation## Abbreviations

- 0-D
0-dimensional

- 1-D
1-dimensional

- BPSL
Balance point soot loading

- DPF
Diesel particulate filter

- HDD
Heavy-duty diesel

- PM
Particulate matter

- ODE
Ordinary differential equation

- uDPF
Uncoated diesel particulate filter

- WHTC
World harmonised transient cycle

## Nomenclature

*A*_{PF}Cross-sectional area of the DPF (m

^{2})*B*M

_{S}S_{V}C_{S}(kg m^{−3}s^{−1})*C*1-γ, -

*C*_{NO2}Concentration of NO

_{2}; from Eq. (7) onwards, the concentration of NO_{2}entering the DPF (mol m^{−3})*C*_{NO2,In}Concentration of NO

_{2}entering the DPF (mol m^{−3})*C*_{NO2,Out}Concentration of NO

_{2}leaving the soot cake (mol m^{−3})*C*_{S}Concentration of PM in the gas entering the DPF as moles of carbon per unit volume (mol m

^{−3})*d*Width of soot-free channel (Fig. 1b) (m)

- Da′
Pseudo Damköhler number, \( {k}_{NO2}^{\prime }/{S}_V \) (m

^{3}kg^{−1})*f*(*y*)Width of the soot cake at a distance y into the soot cake (Fig. 1b) (m)

*k*_{NO2}Rate constant for soot oxidation by NO

_{2}(s^{−1})*k*_{NO2,Lit}Equivalent of k

_{NO2}from Kandylas et al. [1] (s^{−1})- \( {k}_{\mathrm{NO}2}^{\prime } \)
*k*_{NO2}/*ρ*_{S,C}(m^{3}kg^{−1}s^{−1})- \( k{{\prime\prime}}_{\mathrm{NO}2,\mathrm{Lit}} \)
Kandylas et al.’s [1] rate constant, as published (m s

^{−1})*L*DPF length (m)

*M*_{S}Molar mass of soot (0.012 kg mol

^{−1})*R*Molar (or ideal) gas constant (J mol

^{−1}K^{−1})*S*Soot loading, i.e. mass of soot per unit volume of DPF (kg m

^{−3}) (≡ g L^{−1})- \( \overline{S} \)
Average soot loading over a drive cycle (kg m

^{−3})*S*_{0}Initial soot loading (kg m

^{−3})*S*_{∞}Balance point soot loading (kg m

^{−3})*S*_{CO}Selectivity to CO of PM oxidation by NO

_{2}, 0.20*S*_{Max}Maximum soot loading that could possibly fit in the DPF (kg m

^{−3})*s*_{p}Surface area per unit volume of soot (m

^{−1})*S*_{t}Soot loading at time

*t*(kg m^{−3})*S*_{V}Space velocity at DPF temperature, \( \dot{V}/{V}_{PF} \) (s

^{−1})*T*Temperature (K)

*t*Time (s)

- \( \dot{V} \)
Volumetric gas flow into the DPF (m

^{3}s^{−1})*V*_{PF}Volume of the DPF,

*A*_{PF}*L*(m^{3})*V*_{W}Superficial gas velocity through the soot cake (m s

^{−1})*w*_{S}Thickness of the soot cake (Fig. 1b) (m)

*x*\( {e}^{-D{a}^{\prime}\kern0.1em S} \), -

*y*Distance through the soot cake (Fig. 1b) (m)

*α*_{2}CO/CO

_{2}selectivity of soot oxidation from Kandylas et al. [1], -- Δ
*t*_{j} Time interval of the

*j*th data point (s)- δ
*y* Thickness of an element of soot cake (Fig. 1b) (m)

*γ*Stoichiometric factor,

*C*_{NO2}/*C*_{S}(2 −*S*_{CO}), -*ρ*_{Cell}Cell density, i.e. number of channels per unit cross-sectional area for the

*whole*DPF (m^{−2})*ρ*_{S,C}Packing density of the soot in the soot cake (kg m

^{−3})- \( {\rho}_{\mathrm{S},\mathrm{C}}^{\prime } \)
Soot packing density from Kandylas et al. [1] (kg m

^{−3})

## Subscript

*j*Data point index or

*j*th time point in the model input data

## Notes

### Acknowledgements

The author wishes to thank Johnson Matthey Plc for permission to publish this paper. The reviewers are thanked for their helpful comments, as well as the many people I discussed this work with at MODEGAT VI.

### Compliance with ethical standards

The authors declare that they have no competing interests.

## Supplementary material

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