Efficient solution of population balance equations with discontinuities by finite elements

Mahoney, Alan W.; Ramkrishna, Doraiswami

doi:10.1016/s0009-2509(01)00427-4

Cited by 141 publications

(55 citation statements)

References 22 publications

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“…The resulting equations are often partial integrodifferential equations with integral boundary conditions that rarely admit analytical solutions, therefore the use of numerical techniques is necessary for obtaining a solution [26][27][28]. Consequently, the method of discretization of the continuous PBE has emerged as an attractive alternative to the various other numerical methods of solutions [29][30][31] and has been successfully employed, starting with the work of [32], to render accurate numerical solutions of the PBE [16,[33][34][35].…”

Section: Modeling Of Breakage and Coalescence In Screen-type Static Mmentioning

confidence: 99%

Turbulently flowing liquid–liquid dispersions. Part I: Drop breakage and coalescence

Azizi¹,

Taweel²

2011

Chemical Engineering Journal

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Section: Modeling Of Breakage and Coalescence In Screen-type Static Mmentioning

confidence: 99%

Turbulently flowing liquid–liquid dispersions. Part I: Drop breakage and coalescence

Azizi¹,

Taweel²

2011

Chemical Engineering Journal

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“…Gelbard and Seinfeld (1978) proposed a finiteelement method combined with orthogonal collocation to solve the population balance equation. The main drawbacks of the collocation method are a relatively high computational cost and the nonpreservation of the integral properties of the DSD (Mahoney and Ramkrishna 2002). Alternatively, the approach proposed by Kovetz and Olund (1969) preserves both number and mass of the DSD but suffers from the same numerical diffusivity as that of other approaches (e.g., Bleck's method, Gelbard and Seinfeld's method).…”

Section: Introductionmentioning

confidence: 99%

A Robust Numerical Solution of the Stochastic Collection–Breakup Equation for Warm Rain

Prat

Barros

2007

Journal of Applied Meteorology and Climatology

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The focus of this paper is on the numerical solution of the stochastic collection equation-stochastic breakup equation (SCE-SBE) describing the evolution of raindrop spectra in warm rain. The drop size distribution (DSD) is discretized using the fixed-pivot scheme proposed by Kumar and Ramkrishna, and new discrete equations for solving collision breakup are presented. The model is evaluated using established coalescence and breakup parameterizations (kernels) available in the literature, and in that regard this paper provides a substantial review of the relevant science. The challenges posed by the need to achieve stable and accurate numerical solutions of the SCE-SBE are examined in detail. In particular, this paper focuses on the impact of varying the shape of the initial DSD on the equilibrium solution of the SCE-SBE for a wide range of rain rates and breakup kernels. The results show that, although there is no dependence of the equilibrium DSD on initial conditions for the same rain rate and breakup kernel, there is large variation in the time that it takes to reach steady state. This result suggests that, in coupled simulations of in-cloud motions and microphysics and for short time scales (Ͻ30 min) for which transient conditions prevail, the equilibrium DSD may not be attainable except for very heavy rainfall. Furthermore, simulations for the same initial conditions show a strong dependence of the dynamic evolution of the DSD on the breakup parameterization. The implication of this result is that, before the debate on the uniqueness of the shape of the equilibrium DSD can be settled, there is critical need for fundamental research including laboratory experiments to improve understanding of collisional mechanisms in DSD evolution.

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“…If we assume binary aggregation to be the dominant aggregation mechanism (i.e. no more than two particles aggregate simultaneously), it is possible to develop fairly simple expressions for the two coagulation terms [7], [7,10]. Unfortunately, there is a fair amount of confusion in the literature regarding this issue.…”

Section: Principles Of Pbesmentioning

confidence: 99%

Modeling particle size distribution in emulsion polymerization reactors

Vale

McKenna

2005

Progress in Polymer Science

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A review of the use and limitations of Population Balance Equations (PBE) in the modeling of emulsion polymerisation (EP), and in particular of the particle size distribution of the dispersed system is presented. After looking at the construction of the general form of PBEs for EP, a discussion of the different approaches used to model polymerization kinetics is presented. Following this, specific applications are presented in terms of developing a two-dimensional PBE for the modeling of more complex situations (for example the particle size distribution, PSD, and the composition of polymerizing particles). This review demonstrates that while the PBE approach to modeling EP is potentially very useful, certain problems remain to be solved, notably: the need to make simplifying assumptions about the distribution of free radicals in the particles in order to limit the computation complexity of the models; and the reliance of full models on approximate coagulation models. The review finishes by considering the different numerical techniques used to solve PBEs.

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Efficient solution of population balance equations with discontinuities by finite elements

Cited by 141 publications

References 22 publications

Turbulently flowing liquid–liquid dispersions. Part I: Drop breakage and coalescence

Turbulently flowing liquid–liquid dispersions. Part I: Drop breakage and coalescence

A Robust Numerical Solution of the Stochastic Collection–Breakup Equation for Warm Rain

Modeling particle size distribution in emulsion polymerization reactors

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