High-Resolution Simulation of Multidimensional Crystal Growth

Abstract

Many of the crystals in the pharmaceuticals, photographic, and other industries are multidimensional; that is, their growth is associated with the change of multiple internal coordinates. The main governing equation for such systems is a highly nonlinear multidimensional population balance equation that must be solved for a wide range of length scales. For population balance equations, it is well-known that the standard first-order schemes give diffusive solutions while the commonly used second-order schemes give spurious oscillations. This paper presents a highresolution simulation algorithm that provides short computation times and high accuracy. The high-resolution algorithm is compared to the upwind difference and Lax-Wendroff methods through simulations of potassium dihydrogen phosphate (KDP, KH 2 PO 4 ) crystal nucleation and growth. No spurious oscillations or numerical diffusion occurred, in contrast to the upwind method and Lax-Wendroff methods. The numerical stability of the algorithm is assessed using the Courant-Friedrichs-Lewy condition.

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

“…The kinetic parameters were chosen based on literature data (Ma, Tafti, and Braatz 2002), originally valid for the crystallization of potassium nitrate (KNO3). Nevertheless, the objective of this work is not the analysis of a particular chemical system but the PBE solution method, thus the kinetic parameters in some batches were modified and a different size dependent growth rate equation is defined, which fits better the original purpose of the given work.…”