Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_5637_Библиотеки_им_академика_М_И_Перельмана

Several types of solution techniques have been developed for solving the pop­ulation balance models, including the method of moments [10–12], discretization methods [9, 13–15], and Monte Carlo methods [16, 17]. The standard method of moments (SMM) and the quadrature method of moments (QMOM) convert the partial integrodifferential equation into a set of ordinary differential equations (ODEs). Very often these moment equations do not form a closed finite number of ODEs. The QMOM is a popular method for deriving a closed set of ODEs in the method of moments, by applying a quadrature approximation of the distribution function [11]. The moment equations alone only allow average properties of the particles to be computed (such as mean length), and techniques for reconstruction of the size distribution from moments have been developed [18, 19]. Discretization methods such as finite difference and finite volume methods solve the PBMs directly and can simulate the size distribution dynamically, but are usually computationally expensive and can exhibit numerical diffusion or dispersion [9]. More advanced numerical schemes such as the parallel high-resolution finite volume method have been developed to reduce the computational time and increase numerical accuracy [13, 14, 20]. Another well-known discretization method is the method of classes [21, 22], which is a variation of a first-order-accurate finite volume method and the numerical method of lines. While some discretization methods are much faster and/or more accurate than others, such methods still exhibit some distortion in the shape of the distribution.

Advances in particle sensor technology have enabled the online acquisition of data that has inspired an increase in efforts to utilize such information in real-time optimization and control [7]. Real-time simulation and optimization of the PSD would be facilitated by having even faster PBM solvers, which has resulted in a resurge in interest in a classical method known as the method of characteristics (MOCH) [18, 23–27]. An early work that employed MOCH discretized the size into bins and integrated the population density function within each bin to generate a set of ODEs [23]. This combined MOCH-discretization approach had improved accuracy compared with some finite discretization methods that had been previously proposed in the literature [24, 28]. Anotherapproach combines the MOCH andfinite volume methods for modeling processes with nucleation, growth, and aggregation [29]. Those studies assumed constant supersaturation (and, as a result, growth rates). In growth-only crystallization processes with size-independent growth, the MOCH was used to solve a PBM that included the effect of impurities [25]. To take into account the mass conservation constraint, MOCH has also been combined with SMM to simulate crystallizations with size-independent growth and nucleation [26]. An extension combined the QMOM and the MOCH for simulating crystallizations with size-dependent growth and nucleation [18, 27]. Such combined moments-MOCH approaches require two steps: pre-solve the moment equations to obtain the supersaturation and the lower order moments, and subsequently use the supersaturation information in the MOCH simulation.

This paper revisits the implementation of the method-of-characteristics approach for efficiently modeling the size distribution evolution in typical crystallization processes without combination with other methods such as finite volume methods, SMM, and QMOM. The PBM is converted into a differential-algebraic equation (DAE) system that allows the simultaneous simulation of both the entire size distribution and the solute concentration. The solution technique is demonstrated with examples of representative mechanisms that arise in particulate systems, including multidimensional growth, size-dependent growth and nucleation, and growth with agglomeration.