Population balance model: Breakage kernel parameter estimation to emulsification data

Abstract

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.

Solution of the dynamic population balance equation describing breakage–coalescence systems in agitated vessels: The least‐squares method

“…Otherwise, without any attention to optimisation of the algorithm, the computational time may easily be extended to hours. In the study by Solsvik et al, the computational time of the least‐squares solver is analysed and also compared to the orthogonal collocation and Galerkin techniques.…”

“…The choice of the parameters used in the simulations were selected to showcase the capabilities of the least‐squares solver. However, for breakage dominated emulsification systems, the least‐squares solutions to the PBE are compared to the experimental data by Solsvik et al, The breakage birth and death source terms require closure laws for the breakage frequency and daughter size redistribution functions. The function profile of these functions are presented in Figures –.…”

How do you write and present research well? Q4 – Do not metastasize with metadiscourse

“…Is it predicting the composition of the intermediates, the extent of the reaction, the speed of the reaction, what initiates it, or the mechanism? Expressions containing hard are vague.[p. 1063]The complexity in the processes and phenomena governing the changes of dispersed systems makes the derivation of the corresponding models a significant challenge This is a vague statement with two self‐conscious words: complexity and challenge .[p. 1082]Although lignite is difficult to be gasified in supercritical water, the results indicated that gasification efficiency about 47 % was achieved at 30 min, 550 °C with 10 % KOH …”

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

“…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:…”

“…During the last decade, the least-squares method has gained increasing interest in the chemical engineering field and is considered as a good candidate for solving reactor modelling problems. The least-squares method has been adopted for PB problems, [1][2][3][4][5][6][7][8][9][10][11][12][13][14] pseudo-homogeneous fixed bed reactor models, [15] heterogeneous fixed bed reactor models, [16] fluidised bed by Kunii-Levenspiel type of model [17] and pellet problems. [18][19][20] Because the leastsquares method is associated with the most complicated linear algebra theory and thus involved implementation issues, it is of interest to investigate the performance of the least-squares technique to chemical reactor problems relative to the more frequently used Galerkin, tau and orthogonal collocation methods.…”