Keywords: simulation of liquid-liquid systems, mixer-settler system, shallow layer gravity settler.

In this paper we describe a dynamical model for the shallow-layer settler unit of a liquid-liquid system which describes its transient state, including hydrodynamic phenomena of drop-drop and drop-interface coalescence for the shallow layer settler. This dynamical model uses a kinetic formulation in which the movement of the dispersion is modelled as caused by gravity. The model is also suitable to consider the mass transfer phenomena.

A direct numerical technique adapted from the one already used by the authors for the simulation of other liquid-liquid systems is used for the implementation of the mathematical model [1,6] and is also described in this work.

In our approach, we model the phenomena that occur in each volume element (vertical slice) of the dispersion band without resorting to the empirical expressions previously used by Ruiz [7] and previously tested by the present authors [1,2,4]. In the case of the work presented in this paper, the modelling of the longitudinal transport of the drops is set-up by taking into account the gravitational instability caused both in the dispersion band and in the drops being compacted by the non-uniform thickness of the band. The viscous character of the draining of the continuous phase is also taken into account in this model.

The calculation of the drop volume variation through time is made for each volume element , corresponding to each longitudinal position x in the dispersion band in the settler, where H is the thickness of the dispersion band and w is the width of the settler. For each time step, the events that occur in each volume element of the dispersion band are studied. These events are the input of dispersion from the mixer into the settler, the drop-drop coalescence and the drop-interface coalescence, the draining of the continuous phase to the passive interface and the transport of the dispersion between consecutive volume elements in the dispersion band (from the thicker to the thinner). A mathematical model to describe these events has been proposed by the authors [2]. In this work we present the corresponding numerical model and algorithm.

A direct numerical approach has been used by the authors to simulate the dynamics of liquid-liquid vessels [6] and columns [3,5]. The same method was adopted for the settler. A phase space-time discretization is used. The phase space coordinates are length position in the dispersion band and the size (volume) of the drops. For each time step we study the dispersion changes in each volume element (slice) of the dispersion band, taking into account the variations of the volume of the continuous phase and of the properties of the dispersed phase due to the discrete phenomena present. We will present and discuss the numerical method used, in particular the choice of time and phase space discretization that are crucial for the accuracy achieved.

Preliminary computational results where found to be meaningful and indicate that the model and the simulation provide adequate qualitative predictions of the settler dynamic behaviour. In particular, the length of the band changes in a plausible fashion when input parameters (incoming hold up and volume flow, agitation speed at the mixer) are given appropriate step changes and the system converges to the predictable steady state if the process variables are kept unchanged. However, the behaviour of the thickness and shape of the dispersion band needs further analysis and validation. At the same time, experimental work is being carried out in our laboratory using a pilot-scale model of the mixer-settler. We hope to obtain the presently lacking experimental data necessary for a more accurate validation of the mathematical model and for parameter tuning.

1

Gomes E.F, Simulação de Regimes Permanentes e Transitórios de Misturadores-Decantadores, Ph.D. Thesis, Universidade do Porto, Portugal, 2006.

2

Gomes E.F., Madureira C.M.N., Guimarães M.M.L., Ribeiro L.M.; "Numerical modelling of a gravity settler in dynamic conditions", in Topping B.H.V., Mota Soares C.A., (Editors), Proceedings of the Fourth International Conference on Engineering Computational Technology, Civil-Comp Press, Stirling, UK, 2004. doi:10.4203/ccp.80.59

3

Gomes, E.F., Ribeiro, L.M, Regueiras, P.F.R, Cruz-Pinto, J.J.C, "A Parallel Algorithm for the Simulation of the Dynamic Behaviour of Liquid-Liquid Agitated Columns"; Proceedings of Vector and parallel processing VECPAR 2000, LNCS, Springer, 2000. doi:10.1007/3-540-44942-6_44

4

Pinto, G.A; Guimarães, M.; Madureira, C.M.N.; Regueiras, P.F.; Cruz-Pinto, J.J., "A New Model and Algorithm for the Mixer-Settler Unit". Chempor'2001, Aveiro, Portugal, 2001.

5

Regueiras P.F.R, Gomes M.L, Ribeiro L.M, Guimarães M.M.L, Cruz-Pinto JJC. "Efficient Computer Simulation of the Dynamics of an Agitated Liquid-Liquid Extraction Column", Proceedings of the 7th International Chemical Engineering Conference, Lisboa, Portugal, 1998.

6

Ribeiro L.M. "Simulação Dinâmica de Sistemas Líquido-Líquido, Um novo Algoritmo com Potencialidades de Aplicação em Controlo", Ph.D. Thesis, Universidade do Minho, Portugal, 1995.

7

Ruiz M.C, "Mathematical Modelling of a Gravity Settler", Ph.D. Thesis, University of Utah, 1985.

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