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Solution of the Population Balance Equation using the Differential Maximum Entropy Method (DMaxEntM): An application to liquid extraction columns


​The population balance equation (PBE) is an integro-partial differential equation with nonlinear source term. The PBE is known to admit analytical solutions only for a few cases with restricted forms of interaction kernels. We propose for the first time a novel converging sequence of continuous approximations to the number concentration function as a solution to the population balance equation (PBE). These approximations are internally consistent with respect to any finite number of desired moments. The uniqueness and convergence of such a sequence are assured by being an optimal solution to the constrained NLP, which maximizes the constrained Shannon entropy function. The solution is an optimal functional containing the maximum missed information about the distribution. This entropy maximization problem is a convex program and is solved by converting the constrained NLP into a set of transport equations in terms of the optimal Lagrange multipliers. Since differential form of the Lagrange multipliers is used, the method is given the name the Differential Maximum Entropy (DMaxEnt) method. The DMaxEnt method is tested using many standard and even complex liquid–liquid extraction processes, where the population balance modeling is needed.