High resolution algorithms for multidimensional population balance equations

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

“…Other crystals such as potassium dihydrogen phosphate, L-glutamic acid and hen egg-white lysozyme can also be represented by two-size parameters, and thus, 2D PB models can be developed (e.g. [20,21,54,74,[89][90][91][92][93]). By defining the size of a face as its normal distance from the geometrical center of the crystal, Ma et al [1,[94][95][96] proposed a morphological PB model to simulate the dynamic size evolution of crystal population in faceted directions.…”

“…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.…”