Keywords: simulation of liquid-liquid systems, mixer-settler, parameter tuning, Hooke-Jeeves algorithm.

This paper describes recent work on a model for the shallow-layer settler unit, in a mixer-settler system, which is able to describe the hydrodynamic phenomena of the transient state of a liquid-liquid system. The mathematical model used includes parameters of the drop transport process as well as of the drop-drop and drop-interface coalescence with the active interface. In order to find the most appropriate values for the model parameters, we have linked the mixer-settler simulation algorithm to an optimization algorithm. A direct numerical resolution technique had already been proposed by the authors [1,2] for the simulation of liquid-liquid systems.

In order to tune these parameter values of the mixer-settler model we have used the Hooke-Jeeves algorithm [3]. This algorithm uses a deterministic pattern search method and, due to its simplicity and robustness, it is widely used for non-smooth objective functions. The objective function (to be minimized) is defined as the sum of squares of the differences between the computed and given target values for the thickness of the dispersion band. Such target values are being determined in experiments conducted in our liquid-liquid systems laboratory.

The program, written in C, needs a starting guess for the parameter values. With this set of parameters, the simulation program runs the transient state until a steady state is reached. At this point, the optimization program evaluates the objective function. Next, it tries to minimize the objective function by finding a new set of parameters and running the simulation again. The optimization algorithm stops when the value of the objective function is below a given threshold or a maximum number of iteration is reached.

Preliminary results show that the optimization approach is able to significantly reduce the value of the objective function, even if we start from a relatively good initial guess. In the previous studies of our simulation program we've seen that the computational results were found to be meaningful and indicate that the model and the simulation provide adequate qualitative predictions of the settler dynamic behaviour and we knew the possible range for these four parameters.

Our main current difficulty is, the low reliability of the experimental data obtained up until now. Experimental work has been made difficult by the existence, in the settler unit, of crud at the interface between the two liquids. In the future, given more and better sets of target values, we will be able to work on other aspects of the simulation algorithms. In particular, the behaviour of the thickness and shape of the dispersion band needs further analysis and validation. We have now an optimization program to fit these model parameters to the experimental work.

1

E.F. Gomes, M.M.L. Guimarães, L.M. Ribeiro, "Numerical modelling of a gravity settler in dynamic conditions", Journal of Advances in Engineering Software, 38, 810-817, 2007. doi:10.1016/j.advengsoft.2006.08.033

2

E.F. Gomes, M.M.L. Guimarães, G.A. Pinto, L.M. Ribeiro, "Dynamical model for a settler unit using kinetic formulation", in Proceedings of The Fifth International Conference on Engineering Computational Technology, B.H.V. Topping, G. Montero and R. Montenegro, (Editors), Civil-Comp Press, Stirlingshire, United Kingdom, paper 176, 2006. doi:10.4203/ccp.84.176

3

R. Hooke, T.A. Jeeves, "Direct search solution of numerical and statistical problems", Journal of the ACM, vol. 8, no. 2, pp. 212-229, 1961. doi:10.1145/321062.321069

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