Fast High-Resolution Method for Solving Multidimensional Population Balances in Crystallization

Abstract

High-resolution (HR) finite-volume methods can provide an accurate numerical solution of population balance equations describing crystallization processes. To satisfy the stability requirements, the time step (Δt) for the available HR methods needs to be selected conservatively, rendering them computationally expensive for processes with size-dependent growth rates. In this paper, we propose a coordinate transformation scheme to improve the solution time of HR methods applied to batch crystallization processes with nucleation and size-dependent growth. The coordinate transform takes advantage of the functional form that is used to describe the size dependency of the growth rate. Using benchmark examples, we demonstrate that the proposed method provides a more significant reduction in the solution time than existing HR methods, while retaining the same level of accuracy.

Measurement, modelling, and closed-loop control of crystal shape distribution: Literature review and future perspectives

“…Therefore, various numerical algorithms have been developed for solving PB equations such as method of moment [114][115][116][117], method of characteristics [108,[118][119][120][121], Monte Carlo techniques [122,123], and discretization methods including finite element technique [119,124,125], cell average methods [107], hierarchical solution strategy based on multilevel discretization [126], method of classes [82,95], fixed and moving pivot method [127,128], and finite difference/volume methods [90,119,[129][130][131]. Table 1 summarises these numerical solution methods with the further reviews below.…”

“…Monte Carlo simulations are most suitable for stochastic PB equations, especially for complex systems, but typically very computationally expensive. In the method of finite difference/volume discretisation, the PB equation is approximated by a finite difference scheme [90,127,[129][130][131]. Numerous discretisation methods for the PB equations with different orders of accuracy have been investigated and applied to various particulate systems (see for example [82,90,108,[129][130][131]).…”

“…In the method of finite difference/volume discretisation, the PB equation is approximated by a finite difference scheme [90,127,[129][130][131]. Numerous discretisation methods for the PB equations with different orders of accuracy have been investigated and applied to various particulate systems (see for example [82,90,108,[129][130][131]). An alternative approach, high resolution finite volume method, was proposed [90,129,132] to provide high accuracy while avoiding the numerical diffusion (that is, smearing) and numerical dispersion (that is, nonphysical oscillations) associated with other finite difference and finite volume methods.…”

“…An alternative approach, high resolution finite volume method, was proposed [90,129,132] to provide high accuracy while avoiding the numerical diffusion (that is, smearing) and numerical dispersion (that is, nonphysical oscillations) associated with other finite difference and finite volume methods. This method has been shown to be a promising method and applied to various crystallisation systems [90,105,111,129,130] and further developed for effectively reducing the computation time using either an adaptive mesh technique [130] through redistributing the mesh by moving the spatial grid points iteratively and obtaining the corresponding numerical solution at the new grid points by solving a linear advection equation or a coordinate transformation technique [131] which utilizes a coordinate transformation technique to convert a size-dependent growth rate process into a size-independent growth rate problem with a larger time step to be allowed.…”

Graphical processing unit (GPU) acceleration for numerical solution of population balance models using high resolution finite volume algorithm

“…Gunawan et al (2008) proposed parallelized solution using a master/slave structured CPU cluster. Majumder et al (2010) developed the Fast HR-FVM method which uses a coordinate transformation to speed up the simulation by maintaining its accuracy. Prakash et al (2013) …”