Adomian decomposition method for solving fragmentation and aggregation population balance equations

Singh, Randhir; Saha, Jitraj; Kumar, Jitendra

doi:10.1007/s12190-014-0802-5

Cited by 29 publications

(33 citation statements)

References 36 publications

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“…(26) and proceeding in the similar manner as done in the proof of Theorem 2.1 and using (27), we will easily get that d dt…”

Section: Theorem 22mentioning

confidence: 90%

“…The weight x i is independent of the number density function and the breakage function has been assumed to be constant in time, therefore it can be calculated prior to the computation of the system of differential equations (26). But in general, the breakage function is time dependent.…”

Section: Remark 21mentioning

confidence: 99%

“…In the first case, a test problem having an exact solution is taken. The solution can be found in [26,31]. Since, the exact solution is available we plot the numerical results of particle number density function, the first and the zeroth moment against the exact results.…”

Section: Numerical Comparisonmentioning

confidence: 99%

“…(1) in different measure spaces can be found in [3,17,18,21,22,25,28,29], with suitable bounds over the selection and the breakage functions. But it is seen in the articles [2,26,30,31] that analytical solutions are available only for a limited number of problems with simple forms of selection and breakage functions. In general, the BPBEs with complicated kernels are used widely for various practical experiments.…”

Section: Introductionmentioning

confidence: 99%

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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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“…(26) and proceeding in the similar manner as done in the proof of Theorem 2.1 and using (27), we will easily get that d dt…”

Section: Theorem 22mentioning

confidence: 90%

Section: Remark 21mentioning

confidence: 99%

Section: Numerical Comparisonmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…Analytical solutions of the PBE are limited to very few and simple forms of kernels. [19][20][21] A detailed discussion about numerical approaches is given by Bayraktar. 22 Stochastic methods [23][24][25] like Monte-Carlo simulations are comparatively slow but very exact.…”

Section: Introductionmentioning

confidence: 99%

A discrete population balance equation for binary breakage

Liao

Oertel

Kriebitzsch

et al. 2018

Numerical Methods in Fluids

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Summary The numerical solution of the population balance equation is frequently achieved by means of discretization, ie, by the method of classes. An important concern of discrete formulations is the preservation of 2 chosen moments of the distribution, eg, numbers and mass, while remaining flexible on the grid and kernels applied. Existing formulations for breakage are either able to perserve only one moment or restricted by the choice of grid and kernels. Two types of kernel functions for the description of breakup rate exist, ie, total breakup model and partial breakup model. The first type states the total breakup rate of a mother particle and requires a daughter size distribution function. The other type gives the breakup rate between a mother and a daughter particle directly. Existing formulations are known to work well for the former type but inefficiently for the latter one due to the need of additional numerical integrations. The particular focus of the present work lies in developing an efficient formulation for this type of kernels. A discrete formulation of the breakup terms due to binary breakage is proposed, which allows a direct implementation of both types of breakup kernels and an efficient solution of the population balance equation, making it favorable for the coupling to computational fluid dynamics codes. The results for a pure breakage process obtained by using the new formulation and various kernels are compared with those by a reference formulation as well as analytical solutions. Good agreement is achieved in all test cases, and numerical efficiency of the new formulation for partial breakup models is evidenced.

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A volume‐consistent discrete formulation of aggregation population balance equations

Singh

Kumar

Bück

et al. 2015

Math Methods in App Sciences

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Adomian decomposition method for solving fragmentation and aggregation population balance equations

Cited by 29 publications

References 36 publications

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

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

A discrete population balance equation for binary breakage

A volume‐consistent discrete formulation of aggregation population balance equations

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