SOLUTION OF THE POPULATION BALANCE EQUATION: FROM GLOBAL TO LOCAL CONSTRAINED MAXIMUM ENTROPY METHOD

01.May.2018

​The population balance equation (PBE) which governs the space-time evolution of discrete phase systems, at micro and/or macro state levels, finds numerous applications including inanimate particles and animate cells. The numerical modelling of such systems is essential to understand and predicted the intrinsic kinetics and behaviour of such systems. In this work, we propose a numerical solution to the PBE based on maximization of the Shannon entropy function subject to mean integral properties of the particle size distribution. The solution of this maximization problem is derived and found to take an exponential form with particle size independent parameters that are determined from a suitable set of orthogonal functions. The solution is then used to close the integrals appearing in the source term of the population balance equation with careful sampling of the particle size distribution at a set of prescribed points using the roots of N th-degree Legendre polynomial. This sampling is considered optimal in the sense that it minimizes the truncation error of functional approximation through which the PBE is transformed into a set of conservation laws with distinct velocities in the external space. The derived local maximum entropy functional works as an interpolative vehicle to accommodate newly born particles by aggregation and breakage events. This is achieved by exchanging continuous functional information among the sampling points in an efficient way to minimize the computational efforts. As being a maximum entropy functional, it is unique and converges rapidly to the exact solution as the number of sampling points increases. This convergence ensures the accurate calculations of its integral properties including the low order moments of the particle size distribution. The present solution method is not restricted to any particle kinetics including coupled particle growth and aggregation. The accuracy and efficiency of the method are demonstrated by trying different case studies of particle aggregation, coupled aggregation and growth, breakage and coupled hydrodynamics and mass transfer in a pilot plant RDC liquid extraction column with fifty compartments.

https://www.researchgate.net/publication/325107807_SOLUTION_OF_THE_POPULATION_BALANCE_EQUATION_FROM_GLOBAL_TO_LOCAL_CONSTRAINED_MAXIMUM_ENTROPY_METHOD​