Analytical approximation schemes for mean nucleation rate in turbulent flows

Nucleation rate is a very sensitive function of the temperature and vapor mole fraction. Analytical approximation schemes for the mean nucleation rate in turbulent flows are derived using Laplaces approximation method. The schemes only require the derivative of the nucleation rate function and the probability density function (pdf) of the vapor mole fraction and/or temperature at the point of maximum nucleation rate. Based on the relation between the mole fraction and temperature, i.e., linearly correlated or not, different approximation schemes are developed. Numerical examples are constructed to investigate the accuracy of these approximation. In the examples, the pdfs of mole fraction and/or temperature in various turbulent flows are assumed to come from the beta distribution with five distinct forms. The mean nucleation rate of dibutyl phthalate (DBP) aerosol in these turbulent flows are calculated from the approximation schemes, and compared with exact numerical integration. The relative errors are less than 1% for cases when nucleation rate diminishes at the bounds of temperature fluctuations, and no more than 50% for all studied examples. Furthermore, the approximation schemes are not sensitive to the precise form of the pdfs. Hence, these developed approximation schemes can be used to estimate the mean nucleation rate in a broad range of turbulent flows conveniently.