Faraday Discuss. Chem. Soc., 1984, 77, 75-84 A General Model to Account for the Liquid/Liquid Kinetics of Extraction of Metals by Organic Acids BY MICHAEL A. HUGHES" Schools of Chemical Engineering, University of Bradford, Bradford, West Yorkshire AND VLADIMIR ROD Czechoslovak Academy of Sciences, Prague, Czechoslovakia Received 9 th November, 1983 A model based on two-film theory is developed for the case of the rate of extraction of a divalent metal from an aqueous acid phase with an organic acid HR held in a second immiscible solvent phase. The model involves a reaction zone of variable thickness so that the case of reaction at an interface of molecular dimensions can be accommodated as well as the case of reaction extending into the diffusion film on the aqueous side of the interface.Four parameters are used, one involving a partition coefficient for the acid, one involving diffusivities in the films together with the two-film mass-transfer coefficients. Rate data from three techniques used in laboratory liquid/liquid contacting are fitted by the model. The important commercial liquid/liquid extraction systems for the metals copper, cobalt and nickel involve contacting an aqueous acid phase with some organic acid, HR, held in a diluent.' Typically, HR is the hydroxyoxime LIX64, SME529, P5000 for copper or di-2-ethylhexylphosphoric acid for cobalt and nickel. Several papers describe the kinetics of extraction in these systems, but a variety of concentration conditions, in both the aqueous and organic phases, have been employed together with different contacting techniques.The latter range from the single-drop experi- ment2 through the constant-interface stirred cell3 to the rotating diffusion cell.4 Danesi' has shown how a combination of experimental technique with concentration conditions can result in true chemical-kinetic control or true mass-transfer control or mixed control. Rod et aL6*' have proposed a model for mass transfer with a fast reversible reaction and product extraction and have applied it to the extraction of copper(I1) by hydroxyoximes. It is now necessary to show how this model is generally applied and also how examples proposed by other workers are specific cases of the general model. THE MODEL The concentration profiles for this model are shown in fig. 1. The concentrations shown at the interface are arbitrary; in some cases a high interfacial concentration may exist if the extractant is surface active.It is taken that reaction must be at the interface and/or extending out into the aqueous diffusion film. Note that as P H R -+ 00 and P M R 2 - 00 then the profiles for cHR and c M R ~ , in the aqueous phase, are coincident at zero. Also, under these conditions, the cH and cM profiles in the aqueous phase become linear, as shown by the dotted lines in fig. 1. In fig. 1 A d is the diffusion-film thickness and Ar is the reaction-zone thickness. 7576 EXTRACTION OF METALS BY ORGANIC ACIDS organic aqueous I I I I For a divalent metal M2+: M2'+2HR MR,+2H' K E X = Z M R 2 ~ k + / ~ M 2 + C ; R . (4) Reaction (2) or (3) may be rate controlling. In the derivations, superscript bar will indicate the organic phase, subscript i will denote an interface condition and N will denote a flux.The film coefficients will be shown as k, for organic and ka, for water. D is a diffusivity within the films. It can be shown that if reaction (2) is rate determining then: (i) by Astarita's method: (ii) for the organic phase: CHR,i = CHR- N H R / k H R , ~ CMR2,i = CMR2 + N H R / 2 k M R 2 , 0 (iii) for the aqueous phase:M. A. HUGHES AND V. ROD 77 Table 1. Special cases of the general model values of key parameters particular case reaction in the film finite finite finite equilibrium reaction in the film a3 finite co reaction at the interface co a3 finite instantaneous reaction at the inter- 00 a3 face (Chapman's model) Some special cases may be noted which are summarised in table 1.The model of Chapman et a1.' is of particular interest but is only one special case, for under their assumed conditions then kR+ a, e = kRDHJPLRKHR+ 00, PHR+ 00, K E X is finite, thus eqn (5) transforms to: - 1 =o. ck,iCMRz,i K E X c i R , i C M , i Eqn (6) and (7) remain as before. Eqn (8) transforms to cHR = 0. (13) Eqn (9) transforms to c M R ~ = 0. Eqn ( 10) transforms to CM,i = cM - 0.5 NHR/ kM,aq. (14) (15) Eqn (11) transforms to cH,i = CH -k NHR/ kH,aq. (16) It may be noted here that if reaction (3) is the rate-determining step then eqn ( 5 ) becomes: but the other equations remain as before. IMPORTANT PARAMETERS OF THE MODEL The most sensitive parameters in the model are: together with PHR.Their values decide the location of the reaction and characteristics of the transfer process; table 2 illustrates this in a general way.78 EXTRACTION OF METALS BY ORGANIC ACIDS Table 2. Location of the reaction and characteristics of the transfer process P H R 0, or 6; particular case description of process finite finite reaction in the film diffusion coupled with kinetics of reaction in the film finite +co equilibrium reaction in the film diffusion coupled with equili- brium in the film -bm finite reaction at the interface diffusion and reaction kinetics at the surface +m --*a instantaneous reaction at the diffusion and equilibrium at the interface surface THE POSSIBILITY OF BULK PHASE REACTION In order to consider the possibility of reaction in a bulk aqueous phase, the thickness of the reaction zone j must be considered.Now: (18) only if i> 1 can reaction occur in the bulk phase. Suppose that the partition coefficient of the extractant was relatively low at ca. 100 and the mass-transfer coefficients relatively high at kHR+, = kHR,, == m s-l then with CHR = kmol m-2 s-l which i s too low for practical purposes. Extractants of this nature, which react in the bulk, would be of no practical use. kmol mP3 eqn (18) gives N H R , i < 3 x MATHEMATICAL TREATMENT OF THE SIMULTANEOUS EQN (5)-(11) The implicit function. for NHR is: F[ NHR,Cj,(KEX, K H R , PHR, PMRZ,kR,Dj)(kaq, ko)] = (19) in which cj is the bulk concentration.and Dj the molecular diffusivity of species j . The function is therefore made up of: (1) flux, (2) concentration and (3) physical and hydrodynamic parameters.The physical parameters are regrouped to give: in which el = k R D H R / P h R & R and e2= DHRPMR~/DMR~PHR. The ratio D~/DHR refers to a transferring speciesj with diffusivity Dj, and this ratio is relatively easy to obtain from generalised correlations: in any case the flux N H R is not very sensitive to this ratio. Thus in any problem KEX, D H R P M R 2 / D M R 2 P H R and D , / D H R are known or may be estimated for a given system and the data from the technique (be it single drop, stirred cell etc.) may be fitted by the model optimising the best value of O,, k,, and k,. It is k,, and k,, the hydrodynamic parameters, which change from one technique to another.A computer program is written using the Runge-Kutta-Merson method for numerical integrations and the Marquardt method for the optimisation technique to give parameter estimation. In the case of HR= hydroxyoxime, the programM. A. HUGHES AND V. ROD 79 Table 3. Most important parameters of F [eqn (19)] dictated by positiorl of rate and bulk concentrations far from equilibrium, e.g. initial k,, 0, ko near to equilibrium, e.g. in real KEX, k,, KEX, 8 , K E X rates contactors includes the chemical model of Whewell and Hughes'' to calculate the thermo- dynamic concentrations, cHR and cH+, cM2+. A reasonably large number of points taken from kinetic experiments on a particular system can be fitted to the model. Alternatively, the values of constants making up certain parameters can be estimated or measured separately and the parameters can be inserted into the model to calculate fluxes which can be compared with experiment.THE SENSITIVITY OF THE MODEL TO THE PARAMETERS The sensitivity of the model to the parameters depends on whether the rate is measured near to equilibrium or far away from equilibrium together with the bulk concentration conditions ; the most important parameters for the varying conditions are highlighted in each case in table 3. The behaviour of the model can be demonstrated using assumed parameter values and selected concentrations for the aqueous metal ion, the aqueous proton and the 'free' organic ligand, HR. In fig. 2 the flux (or extraction rate) is seen to depend upon the extractant concentration at variable values of the rate parameter 8,.As the values of 61 increase, the rate approaches the maximum theoretical diff usion-controlled transfer curve A. This theoretical curve A would take on new positions in the plot when the cM2+, k,, and k,, values are altered. Note that with increasing extractant concentration the extraction rate is approaching the region where the mass transfer of metal ions is the rate-controlling step. Only at very low values of is chemical control possible, and for < The dependence of the extraction rate on the concentration of the metal ion is shown in fig. 3. As cM2+ increases the rate approaches the limit of k,cHR, and for a given cM2+ value the flux is higher as the el increases. Again chemical control becomes significant when 61 becomes very small, e.g.ca. The importance of the influence of the reverse reaction is partly measured in the KEX value, and for set values of all the parameters this influence on the rate is illustrated in fig. 4, where cHR and KEX are varied. As expected, for a given cHR value the rate increases as KEX increases, chemical control is forced upon the system when KEX becomes very small and, in any case, as cHR increases the diffusion- controlled limit at 2 kaqcM2+ is approached. The sensitivity of the flux to the parameters can be best summarised in fig. 5 , in which a fractional change in flux produced by a fractional change in a parameter is plotted. The left-hand side of the graph represents excess metal in the aqueous phase and the right-hand side represents excess extractant in the organic phase.The sensitivity of the flux with respect to the mass-transfer coefficient k,, increases (or near) the model is not very sensitive to this parameter. or less.80 EXTRACTION OF METALS BY ORGANIC ACIDS 0 20 40 60 cHR/g mol m-3 Fig. 2. Dependence of the flux of HR on the parameter 8, and the concentration of HR in the organic bulk phase. 8, = ( a ) lov6, ( b ) (c) lo-'' and ( d ) 2k,,, c, *+ kCH, 0 1 2 3 Fig. 3. Dependence of the flux of HR on the parameter 8, and the concentration of metal in the aqueous bulk phase. 8, = ( a ) lop6, ( b ) loe8, (c) lo-'' and ( d ) 10-l2. with the extractant concentration and with the ratio koC~~/2kaqC~2+. On the other hand, the sensitivity with respect to k, decreases with increasing cHR and the ratio above.A point on the graph at cHR = 20 kg mol m-3, corresponding to a value of 1 .O for p = k&HR/2kaqCM2+, is a stoichiometric point for the cM2+ value chosen for this calculation. It is now seen that the sensitivity of the flux to kaq is high if p >> 1 and its sensitivity to k, is high if p << 1. The sensitivity of the flux to is a maximum if p = 1. These observations demonstrate that in order to obtain good estimates ofM. A. HUGHES AND V. ROD 81 er,Jg moI m-3 Fig. 4. Dependence of the flux of HR on the value of log K,, and the concentration of HR in the organic bulk phase. log KEX = ( a ) 1 .O, ( h ) 2.0 and ( c ) 3.0. 20 40 60 c I q R l g mol m-3 Fig. 5. Dependence of the rate of change in flux of HR on the fractional change in the parameters ( a ) k,, ( b ) kaq and ( c ) 8,.the three major parameters of the model then experimental data should be obtained from three different regions. Experiments in the region cM2+ >> cHR will provide good estimates of k, and experiments in the region cHR >> cM2+ will provide good estimates of kaq. The reaction rate parameter O1 is best estimated if the experimental conditions are such that82 EXTRACTION OF METALS BY ORGANIC ACIDS 0 f l 5 Fig. 6. Typical fit of the model (solid line) to experimental points from the rising-drop experiment. The oxime concentration is constant at 20 vol '/o HR and the aqueous copper concentration is 8 g dm-3. [H2S04]/g dm-3 = (a) 2, ( b ) 3.5, ( c ) 5, (d) 8 and (e) 12. 0.08 m I Q E - E" 3 0.OL 5 U 0 L 00 800 1200 r/min Fig.7. Typical fit of the model (solid line) to experimental points from the gauze-cell experiment. The oxime concentration is constant at 20 vol % HR and the aqueous copper concentration is 8 g dm-3. [H2S04]/g dm-3 = ( a ) 1, ( b ) 2, ( c ) 4, (d) 6 and (e) 8. APPLICATION OF THE MODEL TO DATA FOR THE LIX64N +CUSO~ + H2SO4 SYSTEM We have tested this model on the LIX64N +CuS04 + H2S04 system using data from two entirely different techniques of contacting the two phases but using bulk concentrations in both phases which are of commercial interest. In particular, the pH range is commercially more realistic than that adopted by Albery and F i ~ k . ~ The model was fitted to the initial rate data from the single-drop experiment^,^ see fig. 6.Rate data may be obtained over more extensive times using a 'gauze cell': where a fixed volume of organic phase is continually stirred at an interface, of known area, with an aqueous phase of constant composition continuously flowing throughM. A. HUGHES AND V. ROD 83 Table 4. Parameter estimates for copper extraction systems data Cu + LIX64N gauze cell 7.0 x 6.7 3.3 11.0 Cu + LIX64N rising drops 7.0 x 150 100 11.0 Cu + P5000 6.8 X 10.4 21.8 1 .o membrane cell (Albery and Fisk) 0 50 100 150 200 c,,/mmol dm-3 Fig. 8. Fit of the model to the data from the rotating diffusion cell. Data reported by Albery and F i ~ k . ~ Curve B, the concentration of aqueous copper is constant at 10 mmol dmP3 but HR is varied. Curve A, the concentration of HR is constant at 68 mmol dmP3 but aqueous copper is varied; the pH is 4.1.the cell and in contact with the organic phase at that interface. Data from these experiments, which involve concentrations in the same range as those for the single drop, have been reported elsewhere.’ The model fitted these data, see fig. 7. The parameters are reported in table 4. In the gauze cell the rate of mass transfer is controlled by diffusion with chemical reaction. The parameters O,, k, and k,, are determined with confidence because the data are measured in concentration-time ranges which allow the model to be sensitive to all three parameters. The model is not sensitive to e2 in this range. It is probable that the relatively low k, and k,, values are caused by inefficient stirring near to the interface.In the case of the rising-drop experiments the rate is mainly controlled by chemical reaction. Because these are initial rate data the range over which they are measured means that the rate is not so sensitive to k, and kaq, so these are not as well determined as in the case of the gauze-cell work. Note that the value of if it is determined independently for the drop data alone. The model appears successful especially since it accounts for rates measured both at initial times and near to equilibrium. We now turn to the experiments of Albery and F i ~ k , ~ who used a rotating diffusion cell to study the extraction of copper with P5000 from slightly acid media. The present model also accounts for their results, see fig. 8. In table 4 the value of 8, is is 6.9 x84 EXTRACTION OF METALS BY ORGANIC ACIDS greater than that found for the Cu +LIX64N system; this is to be expected since P5000 has a lower partition coefficient than LIX64N.l o A theoretical value of k, = 1 1.4 can be calculated for this system using the classical equation for a rotating disc, developed by Levich. So the value of 10.4 found by optimisation of the parameters is satisfactory. The relatively low value of k, shows that there is a diffusion resistance but it is not possible to say if this is due to the membrane itself or some film on the organic side of the membrane. In all the above cases the equation involving the addition of the first ligand [eqn (2) and thus (S)] gave the best fit. ' P. J. Bailes, C. Hanson and M. A. Hughes, Chem. Eng., 1976, 86. ' R. J. Whewell, M. A. Hughes and C. Hanson, J. Inorg. Nucl. Chem., 1975, 87, 2323. C. A. Fleming, Narl Insr. Merafl., Repub. S. Afr., Rep. no. 1793, 1976. W. J . Albery and P. R. Fisk, in HydrornetaNurgy '81 (SOC. Chem. Ind., London, 1981), F5/1-F5/ 15. P. R. Danesi and R. Chiarizia, in Critical Reviews in Analytical Chemistry, ed. B. Campbell (CRC Press, Boca Raton, Florida, 1980), chap. 10, p. I. V. Rod, Chem. Eng. J., 1980, 20, 131. V. Rod, L. Stmadova, V. HanEil and Z. Sir, Chem. Eng. J., 1981, 21, 187. M. Bhaduri, C. Hanson, M. A. Hughes and R. J. Whewell, in ISEC'83 (American Institute of Chemical Engineers, 1983), p. 293. l o R. J. Whewell, M. A. Hughes and C. Hanson, in ISEC'77 (Canadian Institute of Mining and Metallurgy, 1979), vol. 21, p. 185. ' W. Chapman, R. Caban and M. Tunison, Am. Inst. Chem. Eng. Symp. Ser., 1975, 71, 152.