In this thesis, a discretised population balance eqaution (PBE) with a comprehensive model of soot formation processes has been coupled with the computational fluid dynamics (CFD) to predict the soot evolution in laminar diffusion flames. Contributions have been made in terms of methodology, modelling and applications.

First of all, a conservative finite volume method is proposed to discretise the PBE with regard to the coagulation process. This method rigorously calculates the double integrals arising from the coagulation terms via a geometric representation, and exactly balances the coagulation source and sink terms to conserve moments. It proves that the proposed method is able to accurately predict the distribution with a small number of sections and conserve the first moment (or any other single moment) in the coagulation process, in an extensive test of various coagulation kernels, initial distributions and 'self-preserving' distributions, by comparison with analytical solutions and direct numerical solutions of the discrete PBE. Moreoever, the method is also flexible to an arbitrary non-uniform grid.

Later on, the proposed method is also coupled with the CFD program to simulate Santoro flame, a laminar ethylene diffusion flame, for the validation on its accuracy, economy and robustness. Furthermore, the simulation results have been compared with simultaneous multiple diagnostics measurements drawn from a single data source, providing guidance on soot kinetic models. Three well-established PAH-based chemical reaction mechanisms, ABF, BBP and KM2, are employed to model the inception of soot precursors and oxidants. The physical model involves the nucleation by PAH dimerisation, surface growth by HACA mechanism and PAH condensation, size-dependent coagulation. Experimental signals are directly modelled, including the line-of-sight attenuation (LOSA) for the integrated soot volume fraction, planar OH laser-induced fluorescence (OH-PLIF) and elastic light scattering (ELS) for the soot distribution. The comparisons between model predictions and experimental measurements reflect the predictive capability for soot formation in laminar diffusion flame in terms of the flame structure, soot appearance and amount of soot production. The background gas phase chemistry clearly affects the soot modelling and a sensitivity analysis suggests that coordinating the rates of nucleation and surface growth help adjust the soot production on the centreline and sooty wings.

Finally, the same soot model has been extended to two studies of diffusion flames with blends oxygen-containing surrogates: (1) methyl decanoate (MD) with the addition of dibutyl ether (DBE); (2) four practical methyl ester-based real biodiesels and their blends with petroleum diesel. In the first case study, aiming to reproduce an experiment which was to investigate the effects of dibutyl ether (DBE) addition to the biodiesel surrogate (methyl decanoate, MD), a combined and reduced MD-DBE-PAH mechanism from three sub-mechanism sources has been employed in the simulation. Due to the heavy molecular weight of the biodiesel fuel, the terms of the effect of molecular weight, thermophoresis and Dufour effect in the energy equation exhibit a similar magnitude with the original diffusion term, especially in the region of high temperature and a large gradient of the average molecular weight. Predicted temperature profiles are in good comparison with the experiment in terms of position and absolute value. The swallow-tail shape of the soot region and the absolute soot production are correctly predicted by the simulation. In terms of soot suppression, the model predicts 33\% reduction of soot as the DBE addition ranges from 0\% to 40\%, in contrast to around 55\% reduction measured in the experiment. In the second study, a semi-detailed kinetic mechanism for the pyrolysis and combustion of a large variety of biodiesels fuels are considered. The model successfully captures the reduction of soot formation by addition of biodiesels, but not necessarily the rate of decrease with blending. The current investigation offers pioneer and encouraging results on modelling soot formation in biodiesel flames, which has been fewly explored.