Resolving aggregation processes using a novel radial basis function method with two internal variables

Aggregation, a key in flocculation, aerosol, and chemical reactor multiphase flows, is mathematically described by the Smoluchowski coagulation equation (SCE)—a special case of the population balance equation (PBE). Coupling PBE with Navier–Stokes equations constitutes population balance modeling in multiphase flows. In this work, a novel numerical scheme utilizing the radial basis function method is first proposed to solve SCEs with two internal variables. These governing equations form the foundation for modeling two-component aggregation systems. The number density function is approximated by a weighted combination of two-dimensional Gaussian radial basis functions. The integral terms are evaluated using Gaussian quadrature, and the SCEs are transformed into a system of ordinary differential equations through the collocation method. Solving this system yields weight matrices at discrete time steps, enabling construction of the particle size distribution over time. The accuracy of this method is validated by comparisons with analytical solutions and the Taylor-series expansion method of moments. This method has been shown to outperform the widely used method of moments by directly resolving the evolution of the particle size distribution rather than only its statistical moments, achieving a maximum relative error below 5% while maintaining an acceptable computational cost.