On the stability of the process of formation of combustion-generated particles by coagulation and simultaneous shrinkage due to particle oxidation

A mathematical stability analysis is conducted to define the domain of operating conditions under which a combustion-generated particle will be formed by coagulation of incipient particles (or clusters). Two opposing concurrent mechanisms are considered, namely: (i) growth by coagulation, and (ii) simultaneous shrinkage due to sublimation or oxidation of the particle. These two mechanisms are incorporated in the aerosol discrete coagulation equations. Then, summations are carried out over all discrete sizes, once for the total number of particles and once for the total mass. A particle is formed under conditions of growth in mass and a decrease in the total number of incipient particles. The stability analysis is based on Liapunov's theorem. Various mathematical forms of collisional kernels are analyzed. Our results indicate that only enhanced collision rates (e.g. kernels of the form of β ij ∼ i· j) will lead to an unstable rapid growth of paticles. This observation supports the use of enhancement factors in the coagulation kernels by Harris et al. [1,2], who found such enhancement factors to be necessary in order to fit their theoretical soot size distribution calculations to the data on soot particles obtained in a premixed ethylene flame. A nondimensional Damkohler-like number which represents the ratio of the characteristic shrinkage (or sublimation) rate and the characteristic coagulation rate is defined here. The domain of operating conditions under which a combustion-generated particle will be formed via an unstable rapid growth process is then calculated in terms of the aforementioned Damkohler-like number. It is shown that for any given value of the Damkohler-like number, the initial average mass of the incipient particles must be larger than a certain value in order for a rapid growth of combustion-generated particles to occur.