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

Abstract

This paper describes a numerical technique designed to solve certain forms of partial differential equations. The method is applied to the partial integrodifferential population balance equations presented by Jairazbhoy [Jairazbhoy, V., Tavlarides, L. L., & Lewalle, J., (1995) A cascade model for neutrally buoyant two-phase homogeneous turbulence – part I. Model formulation. International Journal of Multiphase Flow, 21(3), 467] that describe the behavior of dense liquid dispersions of interacting drops in isotropic turbulence. In the successively contained semi-discretization scheme developed, the drop number density functions are discretized into non-uniform intervals 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 integrator package. The integrals in each function evaluation are calculated by Gaussian quadrature. The results show that, in some cases, as many as fifteen quadrature points are required to achieve grid independence. Each additional discretization point results in an additional ordinary integrodifferential 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 computations also show that, in every run, there appears to be an optimum number of discretization intervals around which incremental increases in the resolution do not increase the CPU time or perceivable accuracy of the solution.