Executive Summary : | The population balance equation (PBE) is used in many areas of science and engineering for modelling the space-time evolution of the number density function of disperse particle phases. The PBE is also coupled with the Navier-Stokes equation in multiphase flow models in the field of computational fluid dynamics, giving it a lot of popularity in academia and industry. The major complexity of the PBE lies in its nonlinear and integro-differential nature which appear in the source term of the equation which includes terms such as particle growth, nucleation, dispersion, breakage and aggregation. Solving this equation analytically becomes almost impossible in a general sense and some special solutions have been proposed in the literature. Such solutions are extremely useful when verifying numerical methods implemented in computer codes. The internal dispersion term is an important constituent of the source term in modelling many physical processes in the PBE, such as the particles growing at different rates. This dispersion term has always been modelled using a Fourier's (or Fick's) first-order diffusion law. Although Brownian effects can be successfully modelled using such first-order diffusion law, there is no empirical or theoretical reason for the internal dispersion to follow first-order diffusion. It should ideally be modelled using a more general space fractional-order derivative resulting in a source term containing (Caputo form of the) Riemann--Liouville fractional differential operator of fractional order. This results in a fractional-order population balance equation, as referred to in this proposal. The aim of the present research is to parametrically study and develop solutions to the PBE containing a fractional-order internal dispersion model through analytical and numerical approaches. |