**A MOVING BOUNDARY PROBLEM AND ORTHOGONAL COLLOCATION IN SOLVING A DYNAMIC LIQUID SURFACTANT MEMBRANE MODEL INCLUDING OSMOSIS AND BREAKAGE**

E.C.Biscaia Junior^{1*}, M.B.Mansur, A.Salum and R.M.Z.Castro

Departamento de Engenharia Química, Universidade Federal de Minas Gerais

Rua Espírito Santo, 35, 6^{o} andar, CEP 30160-030, Phone +55 (31) 3238-1780,

Fax +55 (31) 3238-1789, Belo Horizonte - MG, Brazil

E-mail: mansur@deq.ufmg.br

^{1}Programa de Engenharia Química, COPPE, Universidade Federal do Rio de Janeiro

Cx. P. 68502, CEP 21945-970, Phone +55 (21) 590-2241,

Fax +55 (31) 590-7135, Rio de Janeiro - RJ, Brazil ]]>
E-mail: evaristo@peq.coppe.ufrj.br

(Received: October 10, 2000 ; Accepted: May 14, 2001)

**Abstract - **A dynamic kinetic-diffusive model for the extraction of metallic ions from aqueous liquors using liquid surfactant membranes is proposed. The model incorporates undesirable intrinsic phenomena such as swelling and breakage of the emulsion globules that have to be controlled during process operation. These phenomena change the spatial location of the chemical reaction during the course of extraction, resulting in a transient moving boundary problem. The orthogonal collocation method was used to transform the partial differential equations into an ordinary differential equation set that was solved by an implicit numerical routine. The model was found to be numerically stable and reliable in predicting the behaviour of zinc extraction with acidic extractant for long residence times.

*Keywords*: liquid surfactant membranes, moving boundaries, orthogonal collocation, metal extraction, dynamic modelling.

**]]> Liquid surfactant membrane separation is complementary to solvent extraction in mineral processing and is suitable for treatment of dilute streams from leaching or industrial wastewater. The technology combines three phases – external aqueous feed, organic and internal aqueous stripping – in the form of multiple emulsions to allow simultaneous extraction and stripping of the metal in the same equipment. The overall process, schematically shown in Figure 1, consists of four different steps. A water-in-oil emulsion, referred to as (W/O) emulsion, is prepared in the first step (emulsification) by the creation of a dispersion of the stripping phase droplets within the organic phase that is stabilised by the addition of a surfactant appropriate for the organic phase. The separation of the metal starts in the permeation step, when this (W/O) emulsion is dispersed in the feed phase containing the metal to be separated, giving (W/O/W) multiple emulsion globules. The metal is extracted by chemical reaction with a suitable extractant also added to the organic phase, producing a metal complex soluble in the organic phase that permeates towards of the stripping phase. After extraction of the metal, the complex-loaded (W/O) emulsion is separated from the feed phase (settling step) and sent to the splitting step to recover the metal-concentrated stripping phase and return the organic phase to the emulsification step.**

INTRODUCTION

Unlike solvent extraction, separation with liquid membranes does not depend on the thermodynamic equilibrium between the aqueous feed and the organic phases, but it is governed by the chemical reaction and the mass transfer rates at the liquid-liquid interfaces. In other words, the organic phase works as a membrane, and not as a temporary receptor; hence less extractant is required for an equivalent separation level. For example, the same level of cobalt extraction from nickel-cobalt solutions has been obtained by liquid membranes using 10% of the extractant (Cyanex 272) required by solvent extraction (Salum et al., 1999). Furthermore, it is easier to obtain concentrated solutions with membranes because they extract solutes despite the difference in concentration between the inner and the outer aqueous phases. However, some undesirable phenomena related to the stability of the membrane – the swelling and the break-up of the droplets – have to be controlled during operation. These phenomena may seriously affect the efficiency of separation mainly for long residence times.

The swelling of the globules is caused by the existence of different ionic strengths in the aqueous phases separated by the membrane. Therefore, in order to dilute the concentrated internal phase, water from the feed phase is transferred through the membrane in the form of inverse micelles with the molecules of the surfactant and/or by solvation of the transferred metal complex. This phenomenon results in the dilution of the stripping phase and consequent enlargement of the globules, leading to the break-up of the membrane and leaking out of the concentrated internal aqueous solution. The break-up of the membrane is also influenced by the mixing conditions of the system.

In order to get a reliable mathematical formulation of the liquid surfactant membrane process valid for long residence times, the breakage rate and the swelling of the globules have to be incorporated into suitable models that are more difficult to solve. Both phenomena affect the volumes of the aqueous phases, and hence discontinuity occurs in the spatial position of the liquid-liquid interface where chemical reactions take place. These discontinuities are not easily handled as the position where the model equations have to be switched changes with time. This kind of problem – the moving boundary problem – is not uncommon in chemical engineering systems modelling. For instance, it has been observed in regenerative air-to-air heat exchangers used in air conditioning systems and in problems regarding gas-solid reactions in a pellet and chemical reactions with a zero-order reaction rate for one of the reactants (Bischoff, 1963; Frauhammer et al., 1998). Furthermore, dynamic models for liquid surfactant membranes require suitable numerical procedures to solve the partial differential equation set that describes the mass transfer rates inside the emulsion globule (Mansur et al., 1995). The orthogonal collocation method has been applied for diffusion-reaction problems in several chemical engineering applications (Villadsen and Michelsen, 1978; Finlayson, 1980). In this method, a trial function is taken as a series of orthogonal polynomials whose roots are used as collocation points (thus avoiding an arbitrary choice by the user) and the dependent variables become the solution values at these collocation points. The accuracy of the method increases rapidly with the order of the trial function but a first-order approximation usually gives good results. In addition, the method has an additional advantage of reducing by 50% the number of unknown variables if solution of the model is symmetric.

In this work, a dynamic model considering the break-up of internal droplets and the swelling of the emulsion globules is proposed. The numerical procedure applied to solve the moving boundary problem created by the incorporation of the swelling of the globules is presented in detail. Theoretical concentration profiles inside the globule are symmetric in relation to the centre of the globule, so the method of orthogonal collocation was chosen to solve the model.

**]]> A DYNAMIC LIQUID SURFACTANT MEMBRANE MODEL FOR METAL EXTRACTION**

The metal transfer through the liquid membrane is usually known as carrier-mediated transfer. Basically, the metal ion is extracted from the feed phase by chemical reaction with the extractant at the external surface of globules and stripped out by reverse reaction at the surface of the internal droplets, while the regenerated extractant comes back to the external surface to start the process again. Actually the driving force for the transfer of the metal complex through the organic phase is the difference of proton concentration between the aqueous phases separated by the membrane. In this context, the following assumptions are made:

(1) The kinetics of extraction is typical of acidic extractants like organic acids derived from phosphoric acid, such as MTPA (mono tio-di-2-ethylhexyl phosphoric acid) or D2EHPA (di-2-ethylhexyl phosphoric acid). Counter-transportation occurs throughout the membrane, with the metal transferring from the external to the internal interface and protons transferring in the opposite direction. In the case of zinc extraction, the following mechanism and kinetic rate expressions for extraction and stripping reactions were proposed by Lorbach and Marr (1987), respectively:

(2) The moving front model proposed by Ho et al. (1982) is adopted to describe the mass transfer inside the globule. An effective diffusion coefficient accounts for the metal-complex transfer in the polydisperse medium of the membrane.

(3) The droplets inside the globule are immobilised by the action of the surfactant and all droplets and globules are uniform in size and distribution. No coalescence effect is considered.

(4) The process is isothermal at perfect mixing and all physical properties are constant during operation. No loss of complex and extractant to the aqueous phases is considered.

(5) The pH value in the aqueous feed phase is kept constant throughout the process by the use of appropriate buffers.

(6) Although swelling and break-up of the droplets are considered, the number of (W/O/W) emulsion globules remains constant. Actually, as schematically shown in Figure 2, membrane breakage is assumed to occur only on the outermost droplets of the globules. Thus, as the volume of the membrane phase does not change during operation (the organic phase is insoluble in the aqueous phase), the film of membrane that covers these outermost droplets will not resist and will break but the number of globules remains constant over time.

]]>

The volume of the aqueous phases and the interfacial area (internal and external) change with the swelling and the break-up rate of the droplets, so the transient superficial area of N spherical (W/O/W) emulsion globules as a function of the radius of the globule is

and the transient volume of the feed phase is

Here the breakage constant (k_{b}) is included to account for the interfacial area of the internal droplets and the number of droplets:

The following relationship between the volume of the internal phase and the number of internal droplets that changes with time is assumed:

]]>so the average radius and the interfacial area of the internal droplets as a function of R(t) are

The following mass balance equations may be formulated with the previous assumptions:

(a) Metal ion in aqueous feed:

(b) Reaction at external surface of the globules:

(c) Complex and extractant inside the globules:

]]>(d) Metal ion and protons in the internal aqueous phase:

(e) Swelling of the globules (Kinugasa et al., 1992). The average concentration of the species in the internal phase with R(t) is calculated by the Radau quadrature.

Initial conditions:

Boundary conditions:

in the centre of the globule:

]]>on the external surface of the globule:

NUMERICAL SOLUTION OF THE MODEL

The moving boundary problem occurs as a consequence of the transfer of water into the droplets during process operation. This water transfer leads to the swelling of the globules, which is accounted for by the variation in globule radius (eq. 15), and change of the spatial position of the interfacial reactions and boundary condition (eq. 18) over time. The following non-dimensional variables and parameters have been introduced into the model:

The moving boundary problem was introduced into the model by differentiation of the independent variables as a function of the radius of the globule, so the derivatives with respect to radius and time were substituted by

]]> so the mass balance equations for the complex (eq. 11) and the metal in the internal aqueous phase (eq. 13) are rewritten asrespectively. The spatial discretization of eqs. 22 and 23 by the orthogonal collocation method at the collocation points inside the emulsion globule results in a system of ordinary differential equations over time. A trial function incorporating the first boundary condition (eq. 17) was chosen (physically the concentration profiles are symmetric to the centre of the globule). Therefore, only even powers of r are necessary and an orthogonal polynomial written as a function of u = h^{2} was adopted. The other boundary condition (eq. 18) was also incorporated into the trial function by the inclusion of the superior extreme (u_{n+1} = 1) as a collocation point. Further details about the application of the orthogonal collocation method to the model are given in Appendix A. In summary, the interpolation expression of the concentration of the complex inside the globules is given by

where n is the number of internal collocation points or the degree of the orthogonal polynomial, P_{n}^{(1,1/2)}(h^{2}), whose roots are the collocation points. The *extremum* h = 1 was also included in eq. 24 as a collocation point, and the term a(q) is a function of q given in the Appendix A. Therefore, the concentration of the complex at the internal collocation points (i = 1,2,...,n) is given by

and the complex concentration at the *extremum* u_{n+1} = 1 including eq. 18 by

The concentration of the metal ion in the internal aqueous phase is given by the following equation (i = 1,2,...,n+1):

]]> and the variation in the radius of the globule byThe metal concentration in the feed phase is

and at the external surface of the globule, y_{5}(q), is given by

The concentration of extractant inside the globule is

and the concentration of protons in the internal aqueous phase is

]]> The ordinary differential equations set (eq. 25 to 29) was integrated numerically by the fourth-order Rosenbrock method for stiff systems after introducing the algebraic equations (eq. 30 to 32). The concentration of species was calculated at the collocation points inside the globule and average concentrations were given by the Radau quadrature to allow comparison between calculated concentrations and experimental data. The n internal collocation points, obtained as zeros of the Jacobi polynomial P_{n}

^{(1,1/2)}(h

^{2}) with n > 12, proved to be sufficient to obtain accuracy higher than 10

^{-6}. As shown in Figure 3, the convergence to the solution increases with the number of points, but greater computational effort is required. Twelve collocation points were used to simulate the experimental data for zinc extraction.

SIMULATION OF ZINC EXTRACTION

The model was evaluated using experimental data obtained by Haselgrübler (1990) for the transient extraction of zinc with MTPA (mono tio-di-2-ethylhexyl phosphoric acid) in an agitated tank reactor. Swelling and break-up of droplets were measured by the relative change in the volume of the globules and droplets over time. The simulation was conducted using parameters estimated from experimental data (shown in Table 1) and calculated using semi-empirical correlations given by Lorbach and Marr (1987), Bart et al. (1992) and Mansur (1994). The parameters P_{o} and k_{b} were estimated according to Kinugasa et al. (1992). The numerical procedure adopted to solve the model was stable for the operational conditions studied experimentally.

]]>

If operational conditions are favourable to swelling, i.e., a high concentration of surfactant in the membrane phase, for example, the transfer of water from the feed phase to the internal droplets dilutes the stripping phase and a considerable decrease in the zinc concentration is observed in this phase. As shown in Figure 4 (points are experimental data), no satisfactory prediction is obtained after 20 minutes of permeation if the swelling and the breakage of the droplets are not considered in the model. Swelling enhances the breakage rate of the droplets, increasing the leaking of the stripping phase to the feed, so incorporation of such effects into transient models does improve the capability of predicted experimental data. The model is able to predict the experimental behaviour for long residence times of permeation, and it is possible to use it to fit experimental data to improve estimation of the parameters in order to achieve a better prediction. Figure 5 shows the influence of extractant concentration on zinc concentration in the stripping phase for one hour of permeation. In this situation, no reliable prediction is obtained if the swelling and the break-up rates are not included.

**]]> CONCLUSIONS**

A dynamic liquid surfactant membrane model including the swelling and the breakage of the droplets to simulate the extraction of metallic ions from the aqueous phase at long residence times was presented and solved numerically. Incorporation of these undesirable inherent phenomena results in moving discontinuities under the boundary conditions of the model equations during the simulation time. The method of orthogonal collocation was used to solve the model and a stable numerical procedure was found for practical ranges of application in zinc extraction. Simulations showed a considerable improvement in the prediction of the concentration profiles of the species at long residence times when the swelling and the breakage of the membrane are incorporated into the transient model, so this model may contribute towards improving the estimation of the parameters of the system.

ACKNOWLEDGEMENTS

The authors are grateful to the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES – Brazilian Government) for its financial support in the form of a Master of Science grant.

NOMENCLATURE

C | concentration, kmol/m^{3} |

D_{eff} | effective diffusivity of complex, m^{2}/s |

g | osmotic coefficient |

k_{A} | mass transfer coefficient of metal in feed phase, m/s |

k_{b} | leakage coefficient, 1/s |

K_{eq} | equilibrium constant for extraction reaction (-) |

K_{req} | equilibrium constant for stripping reaction (-) |

k_{F} | forward reaction rate constant, m/s |

k_{R} | reverse reaction rate constant, m^{4}/(kmol.s) |

n | number of droplets (-) |

molar flux, kmol/(m^{2}s) | |

N | number of globules (-) |

P_{o} | water transfer coefficient, m/s |

r | radial coordinate in the globule, m |

R | radius of emulsion globule, m |

R_{B} | reaction rate of stripping, kmol/(s.m^{2}) |

R_{F} | reaction rate of extraction, kmol/(s.m^{2}) |

S | outer interfacial area of globules, m^{2} |

S’ | internal interfacial area, m^{2} |

t | time, s |

V | volume, m^{3} |

V_{H2O} | partial molar volume of water, m^{3}/kmol |

Subscript |

A | metal |

B | extractant |

C | complex |

H | hydrogen |

1 | internal aqueous phase |

2 | membrane phase |

3 | aqueous feed phase |

Superscript |

* | at interface |

^{o} | at initial time |

^{¾} | average value |

REFERENCES

Bart, H., Ramaseder, C., Haselgrübler, T. and Marr, R., The Investigation of Osmosis in Liquid Membrane Technique. Influence of Key Parameters, Hydrometallurgy, 28, 253-267 (1992). [ Links ]

Biscaia Junior, E. C., O Método dos Resíduos Ponderados com Aplicações em Simulação de Processos, Proc. XV Congresso Nacional de Matemática Aplicada e Computacional, São Carlos, Brazil (1992). [ Links ]

Bischoff, K. B., Accuracy of the Pseudo-Steady-State Approximation for Moving Boundary Diffusion Problems, Chem. Eng. Sci., 18, 711-713 (1963). [ Links ]

Finlayson, B. A., Nonlinear Analysis in Chemical Engineering, McGraw-Hill (1980). [ Links ]

Frauhammer, F., Klein, H., Eigenberger, G. and Nowak, U., Solving Moving Boundary Problems with Adaptative Moving Grid Method: Rotary Heat Exchangers with Condensation and Evaporation, Chem. Eng. Sci., 53 (19), 3393-3411 (1998). [ Links ]

Haselgrübler, T., Osmose und Zinktransportuntersuchungen im Rühr kessel in der flüssigmembranpermeation, Master’s thesis, Technische Universität Graz, Austria (1990). [ Links ]

Ho, W. S., Hatton, T. A., Lightfoot, E. N. and Li, N. N., Batch Extraction with Liquid Surfactant Membranes: A Diffusion Controlled Model, AIChE J., 28 (4), 662-670 (1982). [ Links ]

Kinugasa, T., Watanabe, K. and Takeuchi, H., Stability of (W/O) Emulsion Drops and Water Permeation through a Liquid Membrane, J. Chem. Eng. of Japan, 25 (2), 128-133 (1992). [ Links ]

Lorbach, D. and Marr, R., Emulsion Liquid Membranes. Part II: Modelling Mass Transfer of Zinc with bis(2-ethylhexyl)dithiophosphoric Acid, Chem. Eng. Process., 21, 83-93 (1987). [ Links ]

Mansur, M. B., Modelagem e Simulação da Técnica de Membranas Líquidas Surfatantes Aplicada a Extração de Zinco, Master’s thesis, Universidade Federal de Minas Gerais, Brazil (1994). [ Links ]

Mansur, M. B., Salum, A. and Castro, R. M. Z., Modeling Mass Transfer of Divalent Cations through Liquid Surfactant Membranes, Brazilian J. of Chem. Eng., 12 (2), 88-97 (1995). [ Links ]

Salum, A., Mansur, M. B., Miranda, T. L., Silva, G. L. and Ciminelli, V. S. T., Effects of Some Parameters on Cobalt/Nickel Separation and on Cobalt Permeation by a Surfactant Liquid Membrane Process, Proc. The International Solvent Extraction Conference (ISEC’99), Barcelona, Spain (1999). [ Links ]

Villadsen, J. and Michelsen, M. L., Solution of Differential Equations Models by Polynomial Approximation, Prentice-Hall (1978). [ Links ]

APPENDIX A

Equation 22 is a partial differential equation that describes the transient profile of concentration of the metal complex in the organic phase as a function of the radius of the globule. Basically, application of the method of orthogonal collocation to discretise the radial coordinate of eq. 22 results in a set of ordinary differential equations. In this approach, the concentration of the complex is no longer dependent on the radius of the globule, but is calculated at specific points inside the globule, namely collocation points. The development of the orthogonal collocation method presented here considers only the partial component of eq. 22.

The boundary condition in the centre of the globule (eq. 17) shows that the concentration profiles inside the globule are symmetrical, so it is possible to use a trial function to interpolate eq. A1 in terms of u = h^{2}. Writing eq. A1 using this new independent variable,

The polynomial approximation that uses the modified Galerkin method (Biscaia Junior, 1992) imposes the condition of no residue on the differential equation at the n collocation points and an average residue equal to zero. Including the *extremum* u_{n+1} = 1 results in

where

to j = 1,2,...,n+1

to j = 1,2,...,n

and p_{NT}(u) is the nodal polynomial, using the Galerkin method with spherical geometry. The following property is valid:

Combining eq. A4 with eqs. A2 and A3:

]]>The residue in eq. A5 is zero at the points of collocation. The derivative equations are written as

Matrices **A** and **B** are calculated using the Lagrange polynomial, so

A general expression is obtained substituting eq. A5 into eqs. A2 and A1:

The term a(q)P_{n}^{(1,1/2)}(q) is obtained by differentiation of eq. A3:

and substitution of the boundary condition (eq. 18):

]]>

* To whom correspondence should be addressed

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