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

  • Timothy C. WatlingEmail author
SPECIAL ISSUE: 2019 MODEGAT September 8-10, Bad Herrenalb, Germany


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 NO2; 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 NO2/PM ratio (but independent of NO2 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.


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







Balance point soot loading


Diesel particulate filter


Heavy-duty diesel


Particulate matter


Ordinary differential equation


Uncoated diesel particulate filter


World harmonised transient cycle



Cross-sectional area of the DPF (m2)


MS SV CS (kg m−3 s−1)


1-γ, -


Concentration of NO2; from Eq. (7) onwards, the concentration of NO2 entering the DPF (mol m−3)


Concentration of NO2 entering the DPF (mol m−3)


Concentration of NO2 leaving the soot cake (mol m−3)


Concentration of PM in the gas entering the DPF as moles of carbon per unit volume (mol m−3)


Width of soot-free channel (Fig. 1b) (m)


Pseudo Damköhler number, \( {k}_{NO2}^{\prime }/{S}_V \) (m3 kg−1)


Width of the soot cake at a distance y into the soot cake (Fig. 1b) (m)


Rate constant for soot oxidation by NO2 (s−1)


Equivalent of kNO2 from Kandylas et al. [1] (s−1)

\( {k}_{\mathrm{NO}2}^{\prime } \)

kNO2/ρS,C (m3 kg−1 s−1)

\( k{{\prime\prime}}_{\mathrm{NO}2,\mathrm{Lit}} \)

Kandylas et al.’s [1] rate constant, as published (m s−1)


DPF length (m)


Molar mass of soot (0.012 kg mol−1)


Molar (or ideal) gas constant (J mol−1 K−1)


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)


Initial soot loading (kg m−3)


Balance point soot loading (kg m−3)


Selectivity to CO of PM oxidation by NO2, 0.20


Maximum soot loading that could possibly fit in the DPF (kg m−3)


Surface area per unit volume of soot (m−1)


Soot loading at time t (kg m−3)


Space velocity at DPF temperature, \( \dot{V}/{V}_{PF} \) (s−1)


Temperature (K)


Time (s)

\( \dot{V} \)

Volumetric gas flow into the DPF (m3 s−1)


Volume of the DPF, APFL (m3)


Superficial gas velocity through the soot cake (m s−1)


Thickness of the soot cake (Fig. 1b) (m)


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


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


CO/CO2 selectivity of soot oxidation from Kandylas et al. [1], -


Time interval of the jth data point (s)


Thickness of an element of soot cake (Fig. 1b) (m)


Stoichiometric factor, CNO2/CS(2 − SCO), -


Cell density, i.e. number of channels per unit cross-sectional area for the whole DPF (m−2)


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)



Data point index or jth time point in the model input data



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

40825_2019_146_MOESM1_ESM.pdf (625 kb)
ESM 1(PDF 624 kb)


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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Johnson MattheyReadingUK

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