An opportunity for streamlined computational fluid dynamics integration via a semi-analytical method for weighted finite volume fragmentation equations

Over the past decade, finite volume schemes have significantly advanced, becoming well-regarded for solving linear and nonlinear population balance equations (PBEs). These schemes are highly accurate and efficient, making them ideal for applications like liquid-liquid dispersion, bubble and droplet fragmentation, in the chemical and pharmaceutical industries. Solving PBEs in continuous form remains challenging, particularly with complex fragmentation kernels and selection functions. Typically, these problems are tackled by forming discretized ordinary differential equations, with accuracy depending on the mesh type and cell count. To address these challenges, a new semi-analytical approach for solving the weighted finite volume scheme breakage equations has been developed [Kumar et al. (2015), “Development and convergence analysis of a finite volume scheme for solving breakage equation,” SIAM J. Numer. Anal. 53(4), 1672-1689]. This approach can replace traditional numerical schemes using the fourth order Runge-Kutta method. The mesh-independence with respect to time of finite volume schemes allows efficient coupling with computational fluid dynamics (CFD) tools. The accuracy and efficiency of the proposed method have been validated with analytically tractable and physically relevant fragmentation kernels and selection functions, demonstrating high accuracy in estimating number density functions and their integral moments. This new approach reduces computational time by approximately 60%, making it an excellent option for integration with CFD software due to its efficiency.