Population balance model: Breakage kernel parameter estimation to emulsification data

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.