A Hybrid Scheme for the Solution of the Bivariate Spatially Distributed Population Balance Equation

The advantages of the generalized fixed pivot technique as extended to mass transfer and the quadrature method of moments are hybridized to reduce the bivariate spatially distributed population balance equation describing the coupled hydrodynamics and mass transfer in liquid-liquid extraction columns. The key idea in the hybridization technique is to use the available moments furnished by the generalized fixed pivot technique to find the abscissa and weights for the Gaussian-quadrature based approach, in an attempt to evaluate the integrals over unknown droplet densities. To implement the quadrature method of moments efficiently, an explicit form for the abscissas and weights is derived based on the product-difference algorithm as described by McGraw [1]. The proposed technique is found to reduce the discrete system of partial differential equations from 2Mx + 1t oMx + 2, where Mx is the number of pivots or classes. The spatial variable is discretized in a conservative form using a couple of recently published central difference schemes. The numerical predictions of the detailed and reduced models are found to be almost identical, accompanied by a substantial reduction of the CPU time as a characteristic of the hybrid model.