Analytical solution for a three-dimensional non-homogeneous bivariate population balance equation—a special case

Bhutani, Gaurav; Brito-Parada, Pablo R.

doi:10.1016/j.ijmultiphaseflow.2016.11.005

Cited by 6 publications

(4 citation statements)

References 16 publications

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“…To investigate exact solutions of any complex non-linear partial differential equations and examine the behavior of the solutions is very interesting. Many effective methods for constructing the exact solutions are proposed, including Bäcklund transformation method [4] (G′/G) expansion method [5,6], Hirota bilinear method [7], Homogeneous balance method [8,9], Lie symmetry method [10][11][12], Inverse scattering method [13], F-expansion method [14], Exp-function method [15,16], Darboux transformation method [17], Riemann-Hilbert method [18,19] and so on.…”

Section: Introductionmentioning

confidence: 99%

Lie symmetry analysis and exact solutions of the (3+1)-dimensional generalized Shallow Water-like equation

Yang

Song²,

Wang³

2023

Front. Phys.

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In this article, (3+1)-dimensional generalized Shallow Water-like (gSWl) equation is discussed. The infinitesimal generators of the equation are derived by using the Lie symmetry analysis method. The optimal system is obtained based on the adjoint table of the generators of the equation. Exact solutions of the equation are constructed by applying symmetry reduction, Exp−ϕ(ξ) expansion method, Exp-function expansion method, Riccati equation method, and G′/G expansion method. For analyzing the dynamical behavior of the solutions, we derive the physical structures of dark soliton, kink wave, and periodic solutions via numerical simulations.

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Section: Introductionmentioning

confidence: 99%

Lie symmetry analysis and exact solutions of the (3+1)-dimensional generalized Shallow Water-like equation

Yang

Song²,

Wang³

2023

Front. Phys.

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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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“…As for energy transfer between two phases, Bhutada et al [5] investigated the relationship between throat structure and the change of pressure and velocity. Dopkin et al [6] considered that longer multiphase-mixed tube could make consumption of momentum increased due to friction force.…”

Section: Introductionmentioning

confidence: 99%

Investigation of Droplets Distribution and Reaction Kinetics on SO₂ Removal in Flue Gas via Venturi Scrubber

et al. 2020

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This paper investigates the problem of gas-liquid flow SO2 removal in nonquiescent flue gas. Using venturi scrubber as the prototype, the population balance model (PBM) combined with the CFD is implemented to characterize the droplets behaviors. Discrete methods as class model (CM) and various quadrature-based moment models (QBMMs) are applied to numerically solve the population balance equations (PBEs). Taking NaOH solution as the reaction kinetics, the sulfur removal efficiency simulation with CM and different QBMMs methods is validated through the operation measurements. The comparison results show that the CM can achieve better accuracy with more bins, which showed the minimal error 3.6%, consisting 30 bins. However, the computational time of the CM is approximately 19.3 times as long as QBMMS. Among the QBMMs, the ECQMOM approach enjoys the best balance between the simulation efficiency and accuracy, while EQMOM shows the least computational load and CQMOM wins the minimal calculation precision. This result will provide sufficient reference for engineers working in the field of the droplets distribution in the venturi scrubber design.

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Analytical solution for a three-dimensional non-homogeneous bivariate population balance equation—a special case

Cited by 6 publications

References 16 publications

Lie symmetry analysis and exact solutions of the (3+1)-dimensional generalized Shallow Water-like equation

Lie symmetry analysis and exact solutions of the (3+1)-dimensional generalized Shallow Water-like equation

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

Investigation of Droplets Distribution and Reaction Kinetics on SO₂ Removal in Flue Gas via Venturi Scrubber

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Analytical solution for a three-dimensional non-homogeneous bivariate population balance equation—a special case

Cited by 6 publications

References 16 publications

Lie symmetry analysis and exact solutions of the (3+1)-dimensional generalized Shallow Water-like equation

Lie symmetry analysis and exact solutions of the (3+1)-dimensional generalized Shallow Water-like equation

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

Investigation of Droplets Distribution and Reaction Kinetics on SO2 Removal in Flue Gas via Venturi Scrubber

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Product

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About

Investigation of Droplets Distribution and Reaction Kinetics on SO₂ Removal in Flue Gas via Venturi Scrubber