Least Squares Higher Order Method for the Solution of a Combined Multifluid-Population Balance Model: Modeling and Implementation Issues

Borka, Zsolt; Jakobsen, Hugo A.

doi:10.1016/j.proeng.2012.07.504

Cited by 15 publications

(11 citation statements)

References 12 publications

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“…A comprehensive discussion of the least-squares method can be found in Jiang [24] and Gunzburger and Bochev. [45] Previous applications of the leastsquares method to PB problems may be found in, for example Dorao, [46] Zhu, [47] Nayak et al, [30] Sporleder, [48] Patruno, [49] Borka and Jakobsen, [32][33][34] and Solsvik and Jakobsen. [35] Consider the following generalised problem formulation:…”

Section: The Spectral Least-squares Methodsmentioning

confidence: 99%

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Population balance model: Breakage kernel parameter estimation to emulsification data

et al. 2013

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Many processes used across, for example the cosmetics, pharmaceutical, food and chemical industries involve two-phase liquid-liquid interactions. The quality of liquid-liquid emulsification systems may be related to the droplet size distribution. The population balance equation (PBE) can be used as a modelling tool when accurate description of the dispersed phase is required. Still, the key challenge with the formulation of predictive population balance (PB) models is experimental determination of unknown breakage and coalescence functions. The complexity in the processes and phenomena governing the changes of dispersed systems makes the derivation of the corresponding models a significant challenge.The present study considers a PBE optimisation problem to allow parameter identification to experimental data. The experimental data are measured for a breakage dominated liquid-liquid emulsification system in a stirred tank. Parameter identifications to the breakage frequency models proposed by Coulaloglou and Tavlarides, [26] Alopaeus et al. [28] and Baldyga and Podgorska [27] are performed. The PBE is numerically solved using the high-order least-squares method. Moreover, the nonlinear parameter identification algorithm is based on the minimisation of the residual between the experimental data and the numerical solution in a least-squares sense. The problem has been implemented in the programming language MATLAB where the fmincon function has been used.Parameter estimation can sometimes be straight forward, for instance when the process and formulated model are relatively simple and sufficient data are available. These conditions are not always met, which may result in difficulties in determining accurate parameter values. A thorough statistical analysis is required in order to explore the actual accuracy of the estimated parameter values. The present study presents a relative comprehensive statistical study of the fit compared to what has been provided in previous PBE parameter estimation studies. Moreover, the optimisation algorithm and challenges associated with parameter estimation are discussed. The present study revealed, by systematically assessing the problem formulation and the fit, that a better understanding of the model and more successful parameter estimation can be achieved, or a limitation of the model is unveiled.

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Section: The Spectral Least-squares Methodsmentioning

confidence: 99%

“…A transient diameter-based PBE holding convective, breakage and coalescence terms can be written as follows [30][31][32][33][34][35] :…”

Section: The Population Balance Equationmentioning

confidence: 99%

Population balance model: Breakage kernel parameter estimation to emulsification data

et al. 2013

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“…Based on the work of Nayak et al, [5] Sporleder et al, [6] Borka and Jakobsen, [7][8][9] and Solsvik and Jakobsen [10] the PBE can be presented as follows:…”

Section: The Population Balance Equationmentioning

confidence: 99%

“…Based on the work of Nayak et al., Sporleder et al, Borka and Jakobsen, and Solsvik and Jakobsen the PBE can be presented as follows:

\begin{matrix} right * - 6 italicpt & center & left \frac{\partial f_{d, m} (ξ, boldr, t)}{\partial t} + \nabla_{r} \cdot [v_{r} (ξ, boldr, t) f_{d, m} (ξ, boldr, t)] \\ right * - 6 italicpt & center & left - \frac{f_{d, m} (ξ, boldr, t)}{ρ_{d} (boldr, t)} ([], \frac{\partial ρ_{d} false(r, t false)}{\partial t} + {boldv}_{boldr} false(ξ, r, t false) \cdot \nabla_{boldr} ρ_{d} false(r, t false)) \\ right * - 6 italicpt & center & left + \frac{\partial}{\partial ξ} [v_{ξ} (ξ, boldr, t) f_{d, m} (ξ, boldr, t)] - \frac{3}{ξ} v_{ξ} (ξ, boldr, t) f_{d, m} (ξ, boldr, t) \\ right * - 6 italicpt & center & left = - B_{D} (ξ, boldr, t) + B_{B} (ξ, boldr, t <...> \end{matrix}

…”

Section: The Population Balance Equationmentioning

confidence: 99%

Evaluation of breakage kernels for liquid–liquid systems: Solution of the population balance equation by the least‐squares method

et al. 2013

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The breakage frequency and daughter size distribution functions by Coulaloglou and Tavlarides[1] are frequently adopted closures in population balance (PB) modelling. A survey of the extensions and modifications of the Coulaloglou and Tavlarides[1] breakage frequency function is provided. Furthermore, the daughter size distribution functions within the statistical category, herein the model proposed by Coulaloglou and Tavlarides[1], are outlined. Most of the breakage models available in literature commonly assume binary breakage only. Thus, the daughter size distribution function suggested by Diemer and Olson[2] is of interest as higher order breakage can be modelled. The breakage closures are evaluated solving the population balance equation (PBE) for a liquid–liquid emulsification system in a stirred tank. The results obtained from a least‐squares solver are compared with the experimental data when available.

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“…Based on the work of Nayak et al [7] , Borka and Jakobsen [32][33][34] , and Solsvik and Jakobsen [14] the PBE is presented in terms of a mass based distribution function f d,m with diameter as the internal coordinate: in which the source terms B D , B B , C D and C B are the birth and death of particles due to breakage and coalescence events: Breakage death:…”

Section: The Population Balance Equationmentioning

confidence: 99%

On the solution of the dynamic population balance model describing emulsification: Evaluation of weighted residual methods

et al. 2013

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Numerical techniques in the family of weighted residual methods; the orthogonal collocation, Galerkin, tau and least‐squares, are evaluated for the solution of transient population balance (PB) models describing liquid–liquid emulsification systems in stirred batch vessels. The numerical solution techniques are compared based on (i) a breakage dominated system with experimental data available, and (ii) a breakage–coalescence test case. Two numerical approaches are studied for the transient term: (i) time‐differencing by a low order finite difference approximation, and (ii) the spectral‐element technique. Both approaches use spectral approximations in the phase space dimension. Based on a residual measure, computational costs, and implementation complexity the combined finite difference–spectral approach is recommended above the spectral‐in‐time‐spectral‐in‐space approach. Within this recommended solution framework, it is not necessary to use a more mathematical complex spectral method than the orthogonal collocation technique.

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Least Squares Higher Order Method for the Solution of a Combined Multifluid-Population Balance Model: Modeling and Implementation Issues

Cited by 15 publications

References 12 publications

Population balance model: Breakage kernel parameter estimation to emulsification data

Population balance model: Breakage kernel parameter estimation to emulsification data

Evaluation of breakage kernels for liquid–liquid systems: Solution of the population balance equation by the least‐squares method

On the solution of the dynamic population balance model describing emulsification: Evaluation of weighted residual methods

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