Modelling the Size Distribution of Soot Particles

When modeling the formation of soot in flames, one is interested in the spatial and temporal evolution of the particle size distribution function (PSDF) of soot. Hence the problem of solving the population balance of soot particles has to be studied. For this purpose we have to model particle formation, growth and oxidation by solving the corresponding population balance equation. The knowledge of the shape of the size distribution is desired in particular in light of recent findings that ultra-fine particles might be responsible for effects of diesel exhaust on human health.

Our present work focuses on the application of an alternative approach to solve the population balance of soot particles, that removes some limitations in methods developed previously. The method is based on a stochastic description of the particle ensemble. Among others the main advantage of this method is that the modeling of higher dimensional distributions is straightforward and the history of single soot particles is known explicitly.

Comparison of measured (columns) and computed size distributions (lines)
Figure 1: Comparison of measured (columns) and computed size distributions (lines) of soot particles at different heights above the burner in a premixed laminar flame. Measurements are taken from Bockhorn, H., Fetting, F., and Heddrich, F., Proc. Comb. Inst., 21:1001 (1986).

In our method, all processes of soot formation and oxidation are treated probabilistically using Monte-Carlo techniques. The method makes use of the new concept of fictitious jumps employing a majorant kernel and hence is much more efficient than previously used Monte-Carlo methods. The stochastic particle method is coupled to a detailed kinetic soot model to simulate soot formation and oxidation in laminar premixed flames. The detailed kinetic soot model we use was developed by Frenklach, Wang, and coworkers and by Mauss et al. in fuel rich laminar premixed and counter-flow flames.

Figure 1 shows an example of PSDFs obtained with the stochastic method in a C2H2/O2/Ar laminar premixed flame (p = 0.12 atm, C/O = 1.1). Calculated PSDFs are compared to measurements by Bockhorn et al.

Calculated soot particle size distribution
Figure 2: Calculated soot particle size distribution as function of residence time in an ethylene/air premixed laminar flame.

The PSDF develops a log-normal shape at large height above the burners and agrees well to the measurements. The simulations however predict a peak in the small size region early in the flame, which was not found in the measurements. This peak is due to nucleation and was also predicted in a simulation of a fuel rich, atmospheric C2H2/O2/Ar flame as shown in Fig. 2. Measurements of the PSDF by Zhao et al. (Zhao et al., Combust. Flame 133 (2003) 173-188) revealed the same feature. It is expected that this feature might have significant impact on the morphology of soot aggregates.