safyan mukhtar | COMSATS Institute of Information Technology (original) (raw)
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This paper presents an efficient and accurate numerical method for solving batch crystallization ... more This paper presents an efficient and accurate numerical method for solving batch crystallization models with fines dissolution. Fines removal is useful for controlling fines production to improves the product quality and to minimize the production cost. The growth rate can be size-dependent and a time-delay in the dissolution unit is also included in the model. The proposed method has two parts. In the first part, a coupled system of ordinary differential equations (ODEs) for moments and solute mass is numerically solved in the time domain of interest. The resulting discrete values are used to get growth and nucleation rates. In the second part, the discrete growth and nucleation rates along with the initial crystal size distribution (CSD) are used to construct the final CSD. The analytical expression of the CSD is obtained by applying the method of characteristics and Duhamel’s principle on the given population balance model (PBM). A Gaussian quadrature method, based on orthogonal ...
An alternative quadrature method of moments (QMOM) is proposed for solving one and two-dimensiona... more An alternative quadrature method of moments (QMOM) is proposed for solving one and two-dimensional length-based population balance models incorporating various kinetic processes such as nucleation, size-dependent growth, aggregation and breakage. The method is based on orthogonal polynomials for calculating pairs of quadrature abscissas (points) and weights. Several numerical test cases with different combination of processes are considered in this manuscript. It was found that the proposed method is efficient, accurate and easy to program. To validate the performance, the numerical results of QMOM are compared with the analytical solutions and the finite-volume schemes results. Excellent agreements were observed in all test cases.
An efficient numerical method is introduced for solving multi-dimensional batch crystallization m... more An efficient numerical method is introduced for solving multi-dimensional batch crystallization models with size-independent growth rates. The proposed numerical technique involves two steps. In the first step, a coupled system of ordinary differential equations (ODEs) for moment and solute concentration is solved. In the second step, the numerical solution of ODEs along with the initial crystal size distribution (CSD) are exploited to construct the final CSD by using the method of characteristics and Duhamel's principle on the given population balance model (PBM). The proposed method is efficient, accurate and easy to implement in the computer. For validation, the numerical results of the proposed scheme are compared with those obtained from the high resolution finite volume scheme.
Journal of Crystal Growth, 2010
Chemical Engineering …, Jan 1, 2009
This paper presents an efficient and accurate numerical method for solving batch crystallization ... more This paper presents an efficient and accurate numerical method for solving batch crystallization models with fines dissolution. Fines removal is useful for controlling fines production to improves the product quality and to minimize the production cost. The growth rate can be size-dependent and a time-delay in the dissolution unit is also included in the model. The proposed method has two parts. In the first part, a coupled system of ordinary differential equations (ODEs) for moments and solute mass is numerically solved in the time domain of interest. The resulting discrete values are used to get growth and nucleation rates. In the second part, the discrete growth and nucleation rates along with the initial crystal size distribution (CSD) are used to construct the final CSD. The analytical expression of the CSD is obtained by applying the method of characteristics and Duhamel’s principle on the given population balance model (PBM). A Gaussian quadrature method, based on orthogonal ...
An alternative quadrature method of moments (QMOM) is proposed for solving one and two-dimensiona... more An alternative quadrature method of moments (QMOM) is proposed for solving one and two-dimensional length-based population balance models incorporating various kinetic processes such as nucleation, size-dependent growth, aggregation and breakage. The method is based on orthogonal polynomials for calculating pairs of quadrature abscissas (points) and weights. Several numerical test cases with different combination of processes are considered in this manuscript. It was found that the proposed method is efficient, accurate and easy to program. To validate the performance, the numerical results of QMOM are compared with the analytical solutions and the finite-volume schemes results. Excellent agreements were observed in all test cases.
An efficient numerical method is introduced for solving multi-dimensional batch crystallization m... more An efficient numerical method is introduced for solving multi-dimensional batch crystallization models with size-independent growth rates. The proposed numerical technique involves two steps. In the first step, a coupled system of ordinary differential equations (ODEs) for moment and solute concentration is solved. In the second step, the numerical solution of ODEs along with the initial crystal size distribution (CSD) are exploited to construct the final CSD by using the method of characteristics and Duhamel's principle on the given population balance model (PBM). The proposed method is efficient, accurate and easy to implement in the computer. For validation, the numerical results of the proposed scheme are compared with those obtained from the high resolution finite volume scheme.
Journal of Crystal Growth, 2010
Chemical Engineering …, Jan 1, 2009