Numerical solution of moment-transformed population balance equation with fixed quadrature points

Abstract

A solution to the moment-transformed population balance equation is presented. The method is based on calculating integrals with fixed quadrature points and corresponding moment-conserving weights. The quadrature points are fixed in the sense that their relative locations along the internal coordinate are determined based on a pre-determined condition. One such condition is choosing them as zeros of orthogonal polynomials. The collocation points are then distributed along the physical internal coordinate by using knowledge about the ratio of maximum possible internal coordinate value to its mean. The most promising feature of the fixed quadrature point method is that the selected family of orthogonal polynomials used for calculating the quadrature points can be further optimized based on the problem at hand. The method applies also to an arbitrary set of collocation points, which can be chosen based on the known features of the distribution evolution. Such example related to agglomeration of solid particles of constant size is shown. The present method is compared to the QMOM method of McGraw [1997. Description of aerosol dynamics by the quadrature method of moments. Aerosol Science and Technology 27, 255–265] where quadrature points are calculated using the PD algorithm of Gordon [1968. Error bounds in equilibrium statistical mechanics. Journal of Mathematical Physics 9, 655–663]. Predictions are compared to analytical solutions in breakage, agglomeration and growth. Excellent agreements are found for both tested methods. The fixed quadrature method, however, is more accurate in most cases, but with a minor increase in computational cost, related to a slightly larger number of closure function calls. In the present method, an odd as well as an even number of moments can be used; in contrary to the PD algorithm. The quadrature weights are calculated from a linear transformation that is always non-singular. Then the time derivatives for the population balance equation can be uniquely solved for each time step, which increases the robustness of the present method. This is a very desirable feature when large systems are modeled, such as in process flowsheet and fluid dynamic simulations. The method is also extended to problems with multiple internal coordinates.