Beyond OPOSPM: A Corrected Maximum Entropy Weibull Distribution for Solving Population Balances

01.Sep.2020

​The Population Balance Equation is a transport equation which accommodates the evolution of particle size distribution due to convection, nucleation, growth, breakage and coagulation in space-time with no general analytical solution. As a reduced model, OPOSPM is a two-moment model for solving this equation and finds its way in modelling real chemical engineering equipment ranging from pilot extraction columns to annular centrifugal extractors with no fundamental principle to predict the full-size distribution. To overcome this problem, we decoded the underlying distribution that is consistent with the two moments of OPOSPM by maximizing the Shannon entropy. The analytical form of this distribution is found to be a Weibull distribution. This distribution is evolved by the two moments of OPOSPM which may move faster or slower than the exact solution. To correct the prior Weibull distribution, we minimized the relative entropy as represented by the Kullback-Leibler divergence (KLD) with Weibull distribution as the most uncommitted prior probability distribution. The posterior distribution, viewed as a correction to the prior Weibull distribution, is found by expanding the minimum KLD solution using a set of orthogonal Legendre polynomials. This sequence of continuous approximations is found to converge exponentially to the exact solution in the sense of RMSE, KLD and mean properties.

https://www.researchgate.net/publication/345343811_Beyond_OPOSPM_A_Corrected_Maximum_Entropy_Weibull_Distribution_for_Solving_Population_Balances​