Direct discretizations of bi-variate population balance systems with finite difference schemes of different order

The accurate and efficient simulation of bi-variate population balance systems is nowadays a great challenge since the domain spanned by the external and internal coordinates is five-dimensional. This report considers direct discretizations of this equation in tensor-product domains. In this situation, finite difference methods can be applied. The studied model includes the transport of dissolved potassium dihydrogen phosphate (KDP) and of energy (temperature) in a laminar flow field as well as the nucleation and growth of KDP particles. Two discretizations of the coupled model will be considered which differ only in the discretization of the population balance equation: a first order monotone upwind scheme and a third order essentially non-oscillatory (ENO) scheme. The Dirac term on the right-hand side of this equation is discretized with a finite volume method. The numerical results show that much different results are obtained even in the class of direct discretizations