A numerical technique for the solution of integrodifferential equations arising from balances over populations of drops in turbulent flows

This paper describes a numerical technique designed to solve certain forms of partial differential equations. The method is applied to the partial int egrodifferential population balance equations presented by Jairazbhoy [Jair azbhoy, V., Tavlarides, L. L., & Lewalle, J., (1995) A cascade model for ne utrally buoyant two-phase homogeneous turbulence - part I. Model formulatio n. International Journal of Multiphase Flow, 21(3), 467] that describe the behavior of dense liquid dispersions of interacting drops in isotropic turb ulence. In the successively contained semi-discretization scheme developed, the drop number density functions are discretized into non-uniform interva ls corresponding to Gaussian quadrature points. The governing equations are assumed to hold identically at all the discretization points, generating a set of ordinary integrodifferential equations that are solved by an integr ator package. The integrals in each function evaluation are calculated by G aussian quadrature. The results show that, in some cases, as many as fiftee n quadrature points are required to achieve grid independence. Each additio nal discretization point results in an additional ordinary integrodifferent ial equation. To achieve comparable accuracy with a uniform discretization scheme, many more discretization points would be required, resulting in an inordinately large number of ordinary integrodifferential equations. The co mputations also show that, in every run, there appears to be an optimum num ber of discretization intervals around which incremental increases in the r esolution do not increase the CPU time or perceivable accuracy of the solut ion. (C) 2000 Elsevier Science Ltd. All rights reserved.