The effect of turbulence on chemical reactions is known to be important in many practical combustion systems, in fact, it strongly influences concentration profiles of fuel, oxidiser, secondary and intermediate products, as well as of some important pollutants. The chemical pathways leading to the formation of the major air-pollutants are known with acceptable accuracy, but the formation of soot is the most complex and least understood phenomenon. Soot emissions are considered to have significant negative effects on human health and global warming. Soot formation is caused by incomplete oxidation of the species that constitute the fuel, with the production of benzene molecules, which grow to give polycyclic aromatic hydrocarbons (PAH), considered key compounds in the reactions involved in the first stages of the process. The objective of the present study is to develop a fully predictive numerical model for the prediction of soot formation and evolution based on Population Balance Equations (PBE). In this first part of the study the different approaches available (i.e., in terms of turbulence models, micro-mixing approaches, soot particle nucleation, aggregation, surface growth and oxidation models) are tested and results are compared with experimental data [1] for a first screening. The investigation carried out in this work is limited to the Reynolds-averaged Navier-Stokes approach (RANS). The closure problem is then solved with the k- turbulence model, which has been shown to give results in good agreement with experimental data for non-swirling flames [2]. Moreover, two different micro-mixing models have been tested (i.e., the beta-pdf and the multiple laminar flamelet approach) coupled with the same kinetic scheme involving twenty different chemical species [3]. The core of the mathematical model is the PBE that tracks the evolution of soot particles. Particle nucleation, aggregation, surface molecular growth, oxidation and restructuring were taken into account into the model, the rate of each process being evaluated by simplified kinetic expressions available in the literature [4,5,6,7]. Aggregation was instead described in terms of the Fuchs theory [8], that interpolates between the continuum and the free molecule regimes. All the rate expressions were adapted to take into account the fractal nature of the particle; the simplest choice is to fix an unique value for all the particles but it is known that aged soot aggregates might undergo some restructuring processes that compact the aggregates increasing their final fractal dimension [9]. The radiative transfer due to soot particles was taken into account by introducing an additional source term for the energy equation and an effective absorption coefficient for soot was determined. Since the presence of soot particles, through radiation, has a strong effect on the flame structure, the PBE must be solved together with the continuity, Navier-Stokes, enthalpy balance, and species transport equations. A convenient approach for the solution of the PBE is based on the use of methods based on quadrature approximations. In fact, it is possible to show that, given 2Nd moments of the Number Density Function (NDF) one can calculate the Nd values of the weights and the Nd values of the abscissas of the quadrature approximation, directly solving transport equations of weights and abscissas themselves [10]. In this work a commercial Computational Fluid Dynamics (CFD) code (Fluent 6.2) is used and the PBE is directly implemented in the CFD code by resorting to the so-called the Direct Quadrature Method of Moments (DQMOM).

In the figure model predictions for soot volume fraction (fv) and total soot particle number density (m0) along the axial coordinate of the flame are reported and compared with experimental data. Experimental data taken from the literature [1] are indicated in the Figure with filled symbols whereas the different continuous lines represents model predictions obtained by using different choices of micro-mixing models and kinetics expressions for soot particle nucleation, surface growth, and oxidation. As it is seen the agreement with experimental data is acceptable showing that this quite simple modelling approach is suitable for predicting soot properties with sufficient accuracy. Moreover in this work the issue of the number of the internal coordinates is also addresses. In fact, by adopting the very same kinetics expressions DQMOM in a pseudo-monovariate and in a truly bi-variate formuations are compared and the advantage of using two internal coordinates instead than one (at least for this particular case) is eventually assessed. References 1. Hu B.,Yang B., Koylu U. O.: Combustion and Flame 134: 93 (2003). 2. Kent J.H., Honnery D.: Combustion Science and Technology, 54:383 (1987). 3. Peeters. T.: PhD thesis, Delft Technical University, Delft, The Netherlands, (1995). 4. Fairweather M., Jones W. P., Lindstedt R. P.:Combustion and Flame, 89: 45 (1992). 5. Moss J.B., Stewart C.D., Young K. J.: Combustion and Flame, 101: 491(1995). 6. Park S. H., Rogak S. N.: Aerosol Science and Technology, 37:947 (2003). 7. Lee K., Thring M. W., Beer, J.: Combustion and Flame, 6:.137 (1962). 8. Fuchs N. A.: Mechanics of Aerosols, Pergamon, New York, U.S.A., (1964). 9. Di Stasio S., Kostandopoulos A.G., Kostoglou M. Journal of Colloid and Interface Science, 247, 33-46.(2002). 10. Marchisio D.L., Fox R. O., Journal of Aerosol Science, 36, 43 (2005).