Development and Convergence Analysis of a Finite Volume Scheme for Solving Breakage Equation

Kumar, Jitendra; Saha, Jitraj; Tsotsas, Evangelos

doi:10.1137/140980247

Cited by 33 publications

(20 citation statements)

References 16 publications

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“…Various numerical methods have been applied to solve Equation (1). Examples for the most commonly used methods are: the method of moments [25][26][27][28], the fixed pivot technique [29,30], the Monte Carlo simulation method [31][32][33][34], the finite element method [35][36][37], approaches based on a separable approximation of the kernel function with the fast Fourier transformation [38][39][40], the cell average technique [41,42], and the finite volume method [43,44]. In this work, the finite volume method [43,44] is applied for the spatial discretization, and the second-order Runge-Kutta method is used for the time variable to get the full discretized form of the PBE (1), suitable for formulating transformation matrices.…”

Section: Use Of Population Balance Equations For Particulate Processesmentioning

confidence: 99%

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Application of Transformation Matrices to the Solution of Population Balance Equations

et al. 2019

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The development of algorithms and methods for modelling flowsheets in the field of granular materials has a number of challenges. The difficulties are mainly related to the inhomogeneity of solid materials, requiring a description of granular materials using distributed parameters. To overcome some of these problems, an approach with transformation matrices can be used. This allows one to quantitatively describe the material transitions between different classes in a multidimensional distributed set of parameters, making it possible to properly handle dependent distributions. This contribution proposes a new method for formulating transformation matrices using population balance equations (PBE) for agglomeration and milling processes. The finite volume method for spatial discretization and the second-order Runge–Kutta method were used to obtain the complete discretized form of the PBE and to calculate the transformation matrices. The proposed method was implemented in the flowsheet modelling framework Dyssol to demonstrate and prove its applicability. Hence, it was revealed that this new approach allows the modelling of complex processes involving materials described by several interconnected distributed parameters, correctly taking into consideration their interdependency.

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Section: Use Of Population Balance Equations For Particulate Processesmentioning

confidence: 99%

“…Proceeding in the same way as Kumar et al [43], the discretized form with two weight functions can be obtained as…”

Section: Breakagementioning

confidence: 99%

Application of Transformation Matrices to the Solution of Population Balance Equations

et al. 2019

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“…In literature the numerical methods like, finite volume methods [4,5,10,13], stochastic methods [19,23], moment methods [7,8], sectional methods [9,16] etc. have been used widely to approximate different forms of the population balance equations (PBEs).…”

Section: Existing Literaturesmentioning

confidence: 99%

Development of a mass conserving discretization technique for breakage problems and its convergence analysis

Saha

Kumar

2015

Int J Adv Eng Sci Appl Math

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In this work an efficient and accurate numerical scheme based on the finite volume method has been introduced to approximate the pure breakage population balance equations. The scheme is designed to conserve the total mass of the particles in the system. The simplicity of both the discrete formulation and its coding are the key features of the new method. It is seen that besides conserving the total mass, the new scheme also gives a better prediction of the total number of the particles in the system as compared to the finite volume scheme of Kumar et al. (Appl Math Comput 219(10):5140-5151, 2013). Unlike [13], the scheme in this paper is computationally very efficient and robust to apply on both the uniform and nonuniform meshes. The development of the new scheme is completed by providing a detailed consistency and convergence analysis of the numerical solution. It is observed that the new scheme is of second order convergent independently of the type of meshes. Moreover, numerical results are compared against several test problems, which include the problems whose solutions are analytically tractable and those which are practically oriented. The mathematical results on convergence analysis of the new scheme has also been verified numerically.

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“…The discretization methods are a collection of several sub methods including the sectional method (Gelbard, Tambour and Seinfeld 1980;Kumar et al 2006;Park and Rogak 2004), the finite volume method (Filbet and Laurencot 2004;Kumar, Saha, and Tsotsas 2015;Qamar and Warnecke 2007), and the finite element method (Ganesan 2012;Rigopoulos and Jones 2003). These methods discretize the continuous number density function into a finite number of sections, pivots, finite elements, or finite volumes.…”

Section: Introductionmentioning

confidence: 99%

A new method for solving population balance equations using a radial basis function network

Wang

Peng

2020

Aerosol Science and Technology

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This study presents a new method for solving the population balance equation (PBE) for particle coagulation. The method introduces a radial basis function network to approximate the number density function. The approximate solution should satisfy the PBE at the collocation points in a continuous manner. The coagulation terms in the PBE are directly handled using the Gaussian quadrature techniques. The final solution is a bivariant analytical function of particle volume and time, which can be easily used in any subsequent calculation. Then the method is validated by comparing with analytical solutions and the sectional method for four numerical test problems. These problems are selected to vary in coagulation kernels and initial conditions. The comparative results show that the present method can accurately predict the time evolution of the number density function. Moreover, the computational efficiency of the present method is quite acceptable considering the high accuracy.

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Development and Convergence Analysis of a Finite Volume Scheme for Solving Breakage Equation

Cited by 33 publications

References 16 publications

Application of Transformation Matrices to the Solution of Population Balance Equations

Application of Transformation Matrices to the Solution of Population Balance Equations

Development of a mass conserving discretization technique for breakage problems and its convergence analysis

A new method for solving population balance equations using a radial basis function network

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