Morphological population balance for modeling crystal growth in face directions

Abstract

in Wiley InterScience (www.interscience.wiley.com).A morphological (or polyhedral) population balance (PB) model is presented for modeling the dynamic size evolution of crystals grown from solution in all crystal growth directions. The morphological PB approach uses the crystal shape information for a single crystal obtained from morphology prediction, or experiment as the initial face locations, as well as face growth rates to predict the shape evolution of the crystal population. For each time instant during the crystallization process, the prediction uses its shape at the previous time moment, and the growth rate for the specific crystal habit plane. The methodology is demonstrated through a study of potash alum (KAl (SO 4 ) 2 Á12H 2 O), for which literature data is available for comparison and validation. The discretization method, method of classes, was used to convert the equations to ordinary differential form with the computational domain being discretized into small meshes. The ordinary differential equations ([1.5 million) were then solved simultaneously using the Runge-Kutta-Fehlbergh 4th/5th-order solver with automatic time-step control. The results obtained clearly demonstrate that the morphological PB model developed can predict crystal growth and surface area of individual habit faces in detail, together with crystal shape and size evolution.

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

“…[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. Shape evolution of 3D faceted crystals involving morphological changes was also studied in the work of Zhang et al [80], where the entire family of possible discrete shape evolution events, such as vertices bifurcating into edges or faces were exhaustively numerated using a set of simple testable conditions.…”

“…The generic mathematical formulation for multidimensional PB modeling, no matter employing a single-size dimension plus other external variables, or multiple-size dimensions, can be given by the following equation [74,95] (1) where x is the internal variable vector with n components, which can be parameters related to crystal size, shape, and other properties; y is the external variable vector such as spatial coordinates (y 1 , y 2 , y 3 );  is the number population density function of crystals;  is the gradient operator for the y coordinates. For the left-hand side of Eq.…”

“…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.…”

“…The kinetic parameters for the crystal growth rates of each face can be estimated by experimental studies including single crystal investigations and batch crystallisation using advanced techniques, such as specialized imaging and image analysis, and also by model identification techniques. The method [95] involves several steps including the identification of independent faces using the crystal morphology information, the formulation of a morphological multi-dimensional PB model, and the solution of the corresponding PB equation to predict the location variations for each independent crystal face, and the temporal reconstruction of crystal shapes during the crystallisation process with post-processing enabling extraction of shaperelated crystal properties and their relationships with the operating conditions. It is worth noting that if the homogeneous conditions, and/or symmetry of crystal shape cannot be applied to one face due to variations of hydrodynamics within a reactor, and/or addition of additives and impurities into the reactor for crystal shape manipulation, that face can be treated as an independent face, which therefore can be modelled as an additional independent dimension within the morphological PB model.…”

“…Therefore, a three-dimensional MPBM, the corresponding three parameters, x, y, z, as shown in Figure 4, can be formed to simulate the morphological evolution of potash alum crystals. This complex inorganic hydrate compound, potash alum, was specially chosen in several PB studies [94][95][96]111], which displays no extensive attrition and without observed significant particle/particle agglomeration. By solving the morphological PB equation for a time interval of t, the growth of individual faces, {111}, {100} and{110}, can be obtained, and, hence, used to calculate the new locations of the corresponding crystal faces, with respect to their previous locations.…”