Analytical approach for solving population balances: a homotopy perturbation method

Kaur, Gurmeet; Singh, Randhir; Singh, Mehakpreet; Kumar, Jitendra; Matsoukas, Themis

doi:10.1088/1751-8121/ab2cf5

Cited by 44 publications

(40 citation statements)

References 57 publications

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“…For example, Ji-Huan He et al applied the HPM to the generalized N/MEMS oscillators, Duffing oscillator, Fangzhu oscillator, nonlinear oscillators with coordinate-dependent mass, which were all elucidated by ordinary differential equations (ODEs) [2,3,11,12]. And there was much research on its application to partial differential equations(PDEs) [8,[13][14][15][16][17][18][19]. Gurmeet Kaur et al applied the HPM to the fragmentation as well as aggregation population balance equation [16].…”

Section: The Models Solved By the Hpmmentioning

confidence: 99%

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The Application of Homotopy Perturbation Method: An Overview

Yang

2022

ARJOM

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As an effective method for solving linear and nonlinear equations, the homotopy perturbation method is usually applied to solving relevant problems. We analyze 74 studies on the application of the homotopy perturbation method and present a comprehensive review of them with the conclusion obtained: (1) Homotopy perturbation method is generally applied to solving the problems of ordinary differential equations; (2) Homotopy perturbation method is usually combined with the technology of transform when it is used to solve more complicated equations; (3) By comparing homotopy perturbation method with other similar methods, many researchers sought that homotopy perturbation method is rapidly convergent, highly accurate, computational simple; (4) Some studies point out that when homotopy perturbation method is applied, some parameters including the number of terms, time span, time step must be prescribed carefully. Finally, two suggestions on the further study of the application of the HPM are provided.

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Section: The Models Solved By the Hpmmentioning

confidence: 99%

“…And there was much research on its application to partial differential equations(PDEs) [8,[13][14][15][16][17][18][19]. Gurmeet Kaur et al applied the HPM to the fragmentation as well as aggregation population balance equation [16]. Sumit Gupta et al applied it for solving convection-diffusion equations [17].…”

Section: The Models Solved By the Hpmmentioning

confidence: 99%

The Application of Homotopy Perturbation Method: An Overview

Yang

2022

ARJOM

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“…In particular two different cases will be tested (a) analytically tractable kernels for which the analytical results for both moments and number density function are available in the literature, and (b) practically oriented kernel for which analytical results are not available. The analytical solutions of moment and number density functions corresponding to the different initial conditions are available in literature [16,45,46]. For our comparison, monodisperse 1 initial conditions are considered for analytically tractable cases, whereas for the practically oriented problem, the following initial condition is considered:…”

Section: Numerical Validationmentioning

confidence: 99%

“…The other studies related to the existence [7,25] and uniqueness [2,24,32] of the fragmentation equation are discussed in detail in these references. Despite of complex behavior of the fragmentation equation, some analytical solutions of fragmentation equation are derived by [16,45,46]. Other investigations concern scattering, self similarity and shattering are examined and discussed by [4,5,11].…”

Section: Introductionmentioning

confidence: 99%

Finite volume approach for fragmentation equation and its mathematical analysis

Singh

Walker

2021

Numer Algor

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This work is focused on developing a finite volume scheme for approximating a fragmentation equation. The mathematical analysis is discussed in detail by examining thoroughly the consistency and convergence of the numerical scheme. The idea of the proposed scheme is based on conserving the total mass and preserving the total number of particles in the system. The proposed scheme is free from the trait that the particles are concentrated at the representative of the cells. The verification of the scheme is done against the analytical solutions for several combinations of standard fragmentation kernel and selection functions. The numerical testing shows that the proposed scheme is highly accurate in predicting the number distribution function and various moments. The scheme has the tendency to capture the higher order moments even though no measure has been taken for their accuracy. It is also shown that the scheme is second-order convergent on both uniform and nonuniform grids. Experimental order of convergence is used to validate the theoretical observations of convergence.

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“…Finding analytical (exact) solutions of the population balance equation (PBE) ( 1) is difficult due to the presence of a nonlinear integral in the equation. However, still, for some simple structured kernels, a few analytical solutions are listed in [20][21][22][23]. Therefore, in this exercise, we choose numerical approximations to solve bivariate pure aggregation PBE (1).…”

Section: Introductionmentioning

confidence: 99%

Characterization of Simultaneous Evolution of Size and Composition Distributions Using Generalized Aggregation Population Balance Equation

et al. 2020

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The application of multi-dimensional population balance equations (PBEs) for the simulation of granulation processes is recommended due to the multi-component system. Irrespective of the application area, numerical scheme selection for solving multi-dimensional PBEs is driven by the accuracy in (size) number density prediction alone. However, mixing the components, i.e., the particles (excipients and API) and the binding liquid, plays a crucial role in predicting the granule compositional distribution during the pharmaceutical granulation. A numerical scheme should, therefore, be able to predict this accurately. Here, we compare the cell average technique (CAT) and finite volume scheme (FVS) in terms of their accuracy and applicability in predicting the mixing state. To quantify the degree of mixing in the system, the sum-square χ2 parameter is studied to observe the deviation in the amount binder from its average. It has been illustrated that the accurate prediction of integral moments computed by the FVS leads to an inaccurate prediction of the χ2 parameter for a bicomponent population balance equation. Moreover, the cell average technique (CAT) predicts the moments with moderate accuracy; however, it computes the mixing of components χ2 parameter with higher precision than the finite volume scheme. The numerical testing is performed for some benchmarking kernels corresponding to which the analytical solutions are available in the literature. It will be also shown that both numerical methods equally well predict the average size of the particles formed in the system; however, the finite volume scheme takes less time to compute these results.

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Analytical approach for solving population balances: a homotopy perturbation method

Cited by 44 publications

References 57 publications

The Application of Homotopy Perturbation Method: An Overview

The Application of Homotopy Perturbation Method: An Overview

Finite volume approach for fragmentation equation and its mathematical analysis

Characterization of Simultaneous Evolution of Size and Composition Distributions Using Generalized Aggregation Population Balance Equation

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