Ind. Eng. Chem. Res. 1998, 37, 1943-1949
1943
Role of the Extraction Equilibrium Constant in the Countercurrent Multistage Solvent Extraction-Stripping Process for Metal Ions Mikiya Tanaka,*,† Kazuya Koyama,† and Junji Shibata‡ Materials Processing Department, National Institute for Resources and Environment, MITI, 16-3 Onogawa, Tsukuba, Ibaraki 305-8569, Japan, and Department of Chemical Engineering, Kansai University, 3-3-35 Yamate-cho, Suita, Osaka 564-8680, Japan
Computer simulation of a steady-state coutercurrent multistage metal solvent extractionstripping process (ESP) using cation-exchange reagents shows that there is a value of extraction equilibrium constant, K, yielding the maximum metal recovery when the other operational parameters are constant. Qualitatively, this is because the larger K makes stripping more difficult. Steady-state local linearization reveals the symbolic relation behind the numerical results. The coefficient of d(ln K) in the differential equilibrium relation extended to the countercurrent multistage process is the weighted average of the partial derivative, in each stage, of the equilibrium organic-metal molarity with respect to ln K under constant equilibrium aqueous-metal molarity, where the weight is related to the partial derivative of the equilibrium organic-metal molarity with respect to the equilibrium aqueous-metal molarity under constant K. The balance of these coefficient values for the extraction and stripping sections of ESP determines the trend of the recovery with varying K: at the maximum recovery, these values are equal to each other. The stagewise plot of the partial derivative of the equilibrium organicmetal molarity with respect to ln K under constant equilibrium aqueous-metal molarity versus organic-metal loading ratio clarifies the relation between the trend of the recovery with varying K and the organic-metal loading. Introduction Metal solvent extraction is now widely used in extractive metallurgy. In plant operation of metal solvent extraction, the extraction process is usually combined with the stripping process; namely, the loaded organic phase leaving the extraction section is sent to the stripping section, and the barren organic phase leaving the stripping section is recycled to the extraction section. In this process, the extraction and stripping sections are often operated, respectively, by a countercurrent multistage sequence. In the present paper, we call such a process a countercurrent multistage extraction-stripping process (ESP). Many operational parameters for ESP can be considered; however, the compositions and the flow rates of the aqueous inlets of the extraction and stripping sections are usually determined by the close relation to the adjacent processes. A leaching process is usually employed before the extraction section and an electrowinning process after the stripping section. Thus, operational parameters actually adjustable in ESP are rather limited. The extraction equilibrium constant, K, is one of such operational parameters and, under the assumption that the activity coefficients of the species in the organic phase are unity, represents the extraction capability of the extractant. When K is too small, the metal transfer from the aqueous to organic phases in * To whom correspondence should be addressed. Telephone: +81-298-58-8486. Fax: +81-298-58-8458. E-mail: m
[email protected]. † National Institute for Resources and Environment. ‡ Kansai University.
the extraction section will be insufficient. When K is too large, the metal transfer from the organic to aqueous phases in the stripping section will be insufficient. Therefore, there would be an optimum K in a given ESP, and computer simulation of a model ESP would produce such a situation. Previously, computer simulations of ESP at steady state were done by some researchers. Rod (1984) simulated multistage extraction-stripping processes and compared the process efficiencies between countercurrent and cross-flow stage sequences. Extraction of copper with hydroxyoximes was dealt with as a model system. Similar studies were also presented by Hughes and Parker (1985) and Bogacki and Szymanowski (1990). Bogacki and Szymanowski (1992) investigated copper extraction-stripping processes containing different hydroxyoximes in various extraction-stripping loops. Bogacki et al. (1992, 1993) dealt with the nickel extraction-stripping processes using di-n-butyl phosphorodithioate as the extractant and examined the efficiencies of various stage sequences. In our previous paper, we carried out steady-state local linearization of the countercurrent multistage extraction process (EP) with cation-exchange reagents and presented a normal form of a total differential equation expressing the relation of the metal-ion molarities in the aqueous outlet to the operational parameters capable of independent continuous variation (Tanaka et al., 1997). In the present paper, the role of K in ESP at steady state is investigated by the computer simulation of a model system, and the symbolic relations behind the results are derived on the basis of the local linearization extended from our previous results (Tanaka et al., 1997).
S0888-5885(97)00723-9 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/04/1998
1944 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998
Figure 2. Importance of K in metal recovery.
Figure 1. Schematic diagram of the countercurrent multistage extraction-stripping process. Table 1. Conditions for Computer Simulation l m p CM,in/kmol m-3 C′M,in/kmol m-3 CH,in/kmol m-3 C′H,in/kmol m-3 CoHA/kmol m-3 F:S:F′
2 2 1 0.05 0.5 0.01 2 0.2 1:1:0.2
Process Considered, Assumptions, and the Definition of the Extraction Equilibrium Constant Figure 1 presents a schematic diagram of the ESP. The solid and broken lines denote the aqueous and organic phases, respectively. The assumptions for the computer simulation, the stoichiometric relation of the extraction with a cationexchange reagent, and the definition of K are the same as those in our previous paper (Tanaka et al., 1997). We dealt with the hypothetical ESP with some operational parameters fixed at the values listed in Table 1. The log K value was varied from -3 to 4. Various total numbers of the stages (N and N′) were examined. This condition was set up in reference to the leaching-solvent extraction-electrowinning process for copper oxide ores (Szymanowski, 1993); namely, we supposed that the target metal ion was Cu(II) and the extractant was β-hydroxyoxime. Method of Computer Simulation The relaxation method proposed by Rose et al. (1958) for the continuous distillation calculations was applied to the simulation of ESP. Rod (1984) and Hughes and Parker (1985) described the application of the relaxation method to the simulation of ESP; thus, we followed their procedure. In the actual calculation, an equation solver EQUATRAN (Peterson et al., 1978) was used. The RungeKutta-Verner method (Verner, 1978) was applied for the numerical solution of differential equations.
Figure 3. Relation among L1, L′1, and log K for (N,N′) of (2,2).
Results of Computer Simulation Figure 2 shows the results of the simulation of ESP at various combinations of N and N′. In this figure, the Y-axis represents the overall metal recovery (denoted by recovery, hereafter) defined as the percentage of the amount of the metal ion transferred from the extraction inlet to the stripping outlet to the amount of the metal ion in the extraction inlet. The plotted points in this figure represent the calculated values, and the two adjacent points are connected by the line segments. As shown in Figure 2, the higher recovery is achieved by the larger N and N′ at the same log K value. At the fixed N and N′, the recovery increases with increasing K when K is small, reaches the maximum at the log K in the range from 0 to 1, and decreases thereafter. The K value giving the maximum recovery, Kmax, depends on N and N′. The values of log Kmax are 0.6, 0.6, 0.8, 0.8, and 0.6 for (N, N′) of (1, 1), (2, 1), (2, 2), (3, 2), and (4, 3), respectively. The increase in K enhances the extraction capability of the extractant but makes stripping more difficult. As a result, CMh ,1 and C′Mh ,1 increase with increasing K. Figure 3 shows the relation among L1, L′1, and log K for (N,N′) of (2,2), where the metal loading ratios, Ln and L′n, are defined as the ratios of the metal-ion molarities in the organic phases from the nth stages, respectively, in the extraction and the stripping sections to the loading capacity of the organic phase. Thus, Ln is written as
Ln ) CMh ,n/(CoHA/l)
(1)
The difference of L1 from L′1 is proportional to the recovery. As shown in Figure 3, when log K exceeds 0, L′1 rapidly increases. This reduces the net metal transfer from the aqueous to organic phases in the extraction section.
Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1945
dCM,N ) [1/(r + θ)] dC′Mh ,1 - ψlnK/(r + θ) d(ln K) (12)
Theoretical Analysis Extension of Differential Equilibrium Relation to ESP. In our previous paper, we derived, on the basis of the differential equilibrium relation for each stage of EP (relation I) (eq 6 in Tanaka et al. (1997)), the total differential equation expressing the relation of the metal-ion molarity in the aqueous outlet of the EP to the operational parameters capable of independent continuous variation (relation II) (eq 18 in Tanaka et al. (1997)). When the operational parameters other than K and CMh ,in are constant, the relation I and II are written, respectively, as
dCMh ,n ) Rn dCM,n + βlnK,n d(ln K)
(2)
dCM,N ) BMh ,in dCMh ,in + BlnK d(ln K)
(3)
where
Rn ) (1/CM,n + m2/CH,n)/[1/CMh ,n + l2/(p2C(HA)p,n)] (4) βlnK,n ) 1/[1/CMh ,n + l2/(p2C(HA)p,n)] i-1
N
∑ ∏ i)1 j)1
BMh ,in ) [
(
∑ ∏ i)1 j)1
i-1
N
∑ ∏ i)1 j)1
BlnK ) -[
(
i
N
Rj)/ri]/{1 +
[(
Rj)/ri]}
(6)
i
N
Rj)βlnK,i/ri]/{1 +
(5)
Rj)/ri]} ∑ ∏ i)1 j)1 [(
(7)
i-1 Here, ∏j)1 Rj ) 1 when i ) 1. Combining eq 3 with the differential material balance of the metal ion around EP yields
dCMh ,1 ) θ dCM,N + ψlnK d(ln K)
(8)
where N
∏ i)1
θ ) (1 - rBMh ,in)/BMh ,in ) [(
i-1
N
Rj)/ri] ∑ ∏ i)1 j)1
Ri)/rN]/[
(
(9)
[
i-1
N
(10)
Equation 8 can be called an extended differential equilibrium relation because eq 8 is an extension of eq 2 to EP. Equation 10 states that ψlnK is the weighted average of βlnK of each stage in EP, where the relative weight of the nth stage, Fn, is expressed as n-1
∏ i)1
i-1
N
Rj)/ri-1] ∑ ∏ i)1 j)1
Ri)/rn-1/[
(
(11)
Equations 8, 10, and 11 mean that the contribution of the infinitesimal variation in ln K of the nth stage in EP to dCMh ,1 at constant CM,N is equal to FnβlnK,n d(ln K). Steady-State Local Linearization of ESP. On the basis of eqs 3, 9, and 10, the following equation holds true in the extraction section of ESP, when the operational parameters other than K are constant.
(13)
The dC′M,N′ in eq 13 is replaced by -(F/F′) dCM,N on the basis of the differential material balance around ESP. By substituting the resulting equation into eq 12, we obtain
dCM,N ) -
ψ′lnK (F/F′)θ′ dCM,N + d(ln K) r+θ r+θ ψlnK d(ln K) (14) r+θ
Here, the sum of the first and second terms on the right side of eq 14 is equal to the first term on the right side of eq 12 and arises as a result of the infinitesimal variation in C′Mh ,1. By rearranging eq 14, we obtain
dCM,N )
ψ′lnK - ψlnK r + θ + (F/F′)θ′
d(ln K)
(15)
The denominator on the right side of eq 15 is always positive; therefore, the variation in the recovery shown in Figure 2 is connected to the following relations between ψlnK and ψ′lnK, because the recovery (%) is equal to 100(1 - CM,N/CM,in): (i) When the recovery is increasing, ψlnK > ψ′lnK. (ii) At the maximum recovery, ψlnK ) ψ′lnK. (iii) When the recovery is decreasing, ψlnK < ψ′lnK. These relations are confirmed by calculating the ψlnK and ψ′lnK values from the results of the computer simulation. Figure 4a,b presents the results of such calculations for (N, N′) of (1, 1) and (4, 3), respectively. The ψlnK and ψ′lnK values intersect when K is equal to Kmax. The signs of ψ′lnK - ψlnK are in agreement with the variation in the recovery in Figure 2. Interpretation in Terms of Metal Loading in the Organic Phase. Equation 5 is rewritten by using Ln as
CoHA
i-1
(∏Rj)βlnK,i/ri-1]/[∑(∏Rj)/ri-1] ∑ i)1 j)1 i)1 j)1
Fn ) (
dC′Mh ,1 ) θ′ dC′M,N′ + ψ′lnK d(ln K)
βlnK,n
ψlnK ) -BlnK/BMh ,in ) N
Here, dC′Mh ,1 is expressed by the extended differential equilibrium relation in the stripping section of ESP as
)
pLn(1 - Ln) l[(l - p)Ln + p]
(16)
It is readily proved that βlnK,n/CoHA has a maximum in the range of Ln between 0 and 1. In our hypothetical system, this function is illustrated by the solid curve in Figure 5. When Ln varies from zero to x2 - 1 (denoted by L* hereafter), the value of βlnK,n/CoHA monotonically increases from zero. When Ln reaches L*, βlnK,n/CoHA has a maximum value. When Ln varies from L* to unity, the value of βlnK,n/CoHA monotonically decreases to zero. In Figure 5, the relation between βlnK,1 and L1 for the extraction and stripping sections when (N, N′) ) (1, 1) is also illustrated. In this case, ψlnK ) βlnK,1 and ψ′lnK ) β′lnK,1 from eq 10. The log K values of -1, 0.6, and 2 are selected corresponding to the typical situations of K < Kmax, K = Kmax, and K > Kmax, respectively. Figure 5 shows the following relations: (i) When log K ) -1, L′1 < L1 < L*. (ii) When log K ) 0.6, L′1 < L* < L1. (iii) When log K ) 2, L* < L′1 < L1. As far as the metal ion is transferred from the extraction to the stripping
1946 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998
Figure 4. Variations in ψlnK with log K.
Figure 5. βlnK,n/CoHA vs Ln plot at the extraction and stripping sections of ESP when (N, N′) ) (1, 1).
sections, L1 > L′1; thus, the two sets of the coordinates of (L1, βlnK,1/CoHA) and (L′1, β′lnK,1/CoHA) are never placed at the same position. Therefore, for ESP under the conditions where (N, N′) ) (1, 1) and K ) Kmax, the inequality of L′1 < L* < L1 holds true as shown in the relation (ii) mentioned above, because βlnK,1 ) β′lnK,1 (that is, ψlnK ) ψ′lnK) from eq 15. Figure 5 also indicates that the L1 or L′1 value, respectively, when the ψlnK or ψ′lnK is at its maximum in Figure 4a is equal to L*. Figure 6 shows the βlnK,n/CoHA versus Ln plots when (N, N′) ) (4, 3). As shown in Figure 4, the behavior of ψlnK with the variation in K when (N, N′) ) (4, 3) is more complicated than that when (N, N′) ) (1, 1). There are two maxima of ψlnK in Figure 4b; thus, the following six points were investigated: log K of -2 (before the first maximum of ψlnK, Figure 6a), -0.8 (near the first maximum of ψlnK, Figure 6b), 0 (near the minimum between the first and second maxima of ψlnK, Figure 6c), 0.6 (between the minimum and the second maximum of ψlnK, Figure 6d), 1.2 (near the second maximum of ψlnK, Figure 6e), and 2.2 (after the second maximum of ψlnK, Figure 6f). In Figure 6, the numbers denote the stage number in the stripping and extraction sections. The Fn value at each stage is shown in the figure. The open square in Figure 6 denotes the ψlnK/CoHA value (weighted average of βlnK,n/CoHA’s) against L1 (metal
loading as an extraction section), while the filled square denotes the ψ′lnK/CoHA value (weighted average of β′lnK,n/ CoHA’s) against L′1 (metal loading as a stripping section). As seen in eq 11, the Fn value is closely related to the values of Rn’s. The Rn is equal to the slope of the tangent to the equilibrium point of the nth stage on the extraction isotherm (CMh vs CM diagram) (Tanaka et al., 1997). The values of F′2 and F′3 (in particular, F′3) in Figure 6 are much lower than that of F′1. This is because, under the conditions of stripping, (i) the increase in CMh with increasing CM is very slow (the R′n values are very low) and (ii) the R′n values decrease in the sequence: R′3 > R′2 > R′1. From Figure 6, we can see the variation in the relation among Ln, βlnK,n, ψlnK, and Fn with the variation in K as follows: (i) Figure 6a: The K is so small that Ln and L′n of each stage is low. (ii) Figure 6b: The L1 is about to reach L*. This increases the ψlnK value and results in the first maximum of ψlnK in Figure 4b. (iii) Figure 6c: After L1 passes through L*, the βlnK,1 decreases. In addition, the value of F4 has become larger than before due to the increase in R2 and R3. These facts lead to the decrease in ψlnK. The L′n values are still low. (iv) Figure 6d: With the increase in Ln of each stage, βlnK,2, βlnK,3, and βlnK,4 show larger values than before, although βlnK,1 decreases. This results in the reincrease in ψlnK. The L′n of each stage has considerably increased. As a result, the ψ′lnK value has become equal to the ψlnK value. (v) Figure 6e: The L4 is about to reach L*. The F4 value is by far the largest among the Fn values; thus, ψlnK is near the second maximum in Figure 4b. The L′n of each stage is very near to L*; thus, the ψ′lnK value becomes very large. (vi) Figure 6f: After L4 exceeds L*, the decrease in βlnK,n is very rapid; thus, the ψlnK value has become very low. The L′3, L′2, and L′1 have also exceeded L*; thus, ψ′lnK is also decreasing. Discussion Generality of Equations 8 and 15. As discussed in our previous paper (Tanaka et al., 1997), in EP, the differential equilibrium relation for each stage and the total differential equation hold true even when the activity coefficients of the species relevant to the extraction are not constant (assumption (vii) in our previous paper), although the expressions of Rn and βi,n (the subscript i denotes one of the various operational parameters capable of independent continuous variation) need some corrections. This is understood by considering that eq 2 expresses the infinitesimal movement of the equilibrium point on the CMh versus CM diagram caused by the infinitesimal variation in ln K. Accordingly, the extended differential equilibrium relation (eq 8) and the differential equation for ESP (eq 15) hold true even when the assumption (vii) in our previous paper does not hold true, although eqs 4 and 5 need some corrections. Here, to make the discussion rigorous, we add the following comment. In real organic solution, the extraction capability depends not only on K but also on the activity coefficients of the extractant and the metalextractant complex in the organic phase. Therefore, for
Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1947
Figure 6. (a-f) βlnK,n/CoHA vs Ln and ψlnK/CoHA vs L1 plots at the extraction and stripping sections of ESP when (N,N′) ) (4,3). Numerals on the curve denote n or n′: O, βlnK,n/CoHA; b, β′lnK,n/CoHA; 0, ψlnK/CoHA; 9, ψ′lnK/CoHA.
a given two organic phases with the same stoichiometric relation of the extraction, even if the K value and the molarities of the extractant and the metal-extractant complex are, respectively, equal to each other, their extraction capabilities are generally different. As a result, when we consider the variation in K, we have to specify the direction of the variation (e.g., where in the chemical structure of the extractant is modified or how the composition of the diluent is varied) to define the variation in the extraction capability. Extraction Capabilities of Ketoxime and Aldoxime Reagents in Relation to the Copper SXEW Process. A typical example showing the importance of the extraction capability of the extractant in ESP is the copper SX-EW process, where the copper ion contained in the sulfuric acid solution caused by the leaching of copper oxide ores is extracted with a hydroxyoxime reagent and stripped with spent electrolyte (Szymanowski, 1993). In the early stage of the copper SX-EW process, ketoximes were used as a hydroxyoxime reagent. Later, aldoximes, another type of the hydroxyoxime reagents, having stronger extraction capability, were developed. However, the extraction capabilities of the aldoximes are so strong that it is
difficult to economically strip loaded copper with spent electrolyte, when the aldoximes are used without adjusting their extraction capabilities by adding a modifier. When the appropriate amount of modifier is added to the organic phase containing an aldoxime, the stripping efficiency is markedly improved with little change in performance under extraction conditions. As a result, the recovery by the ESP is improved, or the number of stages in the ESP is reduced to attain a certain specified recovery (Dalton and Seward, 1984; Kordosky, 1992). Due to our simplistic assumptions, it is difficult to quantitatively compare the results of the computer simulation in the present paper with the real systems. In the experimental studies of the metal-extraction equilibrium with hydroxyoximes, the values of extraction equilibrium constant are usually obtained by considering the aggregation of hydroxyoximes in the organic phase. Since the aggregating tendencies of hydroxyoximes generally depend on the kinds of hydroxyoximes, the real extraction capabilities cannot be represented only by the values of the extraction equilibrium constant, which is in accord with the situation mentioned at the last paragraph in the preceding section.
1948 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998
We should, therefore, regard the X-axis in Figure 2 as the extraction capability. In this sense, the curves in Figure 2 qualitatively represent the relation between the extraction capabilities of ketoximes and aldoximes, indicating the importance of appropriately selecting the extractant: The recoveries with ketoximes and aldoximes are regarded to be drawn, respectively, at the left and right sides of the maxima in Figure 2, when the industrial diluents are used. A semiquantitative comparison between the extraction capabilities of ketoximes and aldoximes in terms of extraction equilibrium constants can be done only when the aggregating tendencies are similar to each other on the condition that the aqueous-phase model for the equilibrium analysis is consistent with each other. Although such literature data are scarce, the extraction equilibrium constant of copper with ketoxime (anti-2-hydroxy-5-nonylacetophenone oxime) is 100.39 m3 kmol-1 with the stepwise aggregation constant (Whewell et al., 1977) of 0.9 m3 kmol-1 (Tanaka, 1992), while that with aldoxime (2-hydroxy-5-nonylbenzaldehyde oxime) is 101.81 m3 kmol-1 with the stepwise aggregation constant of 0.8 m3 kmol-1 (Tanaka, 1991). These constants, obtained by using xylene as the diluent, consider the activities for the aqueous species and the molarities for the organic species. Thus, the extraction capabilities of these organic phases is comparable with each other by the extraction equilibrium constants. There is a difference of a 1.4 unit on the common logarithmic scale between the two extraction equilibrium constants. Nevertheless, at present, we refrain from stating that the extraction capability of 2-hydroxy-5nonylbenzaldehyde oxime in xylene is greater than that giving a maximum recovery in the ESP because, in comparison with xylene, industrial diluents are usually more aliphatic, resulting in stronger extraction capability. A more realistic simulation is required to clarify the exact position of the extraction capability for a given organic phase. Conclusions Computer simulation of a hypothetical steady-state ESP using a cation-exchange reagent has been carried out, on the basis of the simplified assumptions, to investigate the relation between process efficiency and K. There is a value of K, which gives the maximum recovery in ESP, when other operational parameters are kept constant. This is because stripping becomes difficult with increasing K. The symbolic relation lying behind the numerical results has been derived on the basis of the steady-state local linearization; namely, ψlnK > ψ′lnK, ψlnK ) ψ′lnK, and ψlnK < ψ′lnK, when the recovery is, respectively, increasing, at the maximum, and decreasing with increasing K. Here, the ψlnK, the coefficient of d(ln K) in the extended differential equilibrium relation for EP (eq 8), is the average of the βlnK,n values n-1 Ri/rn-1. The βlnK,n versus Ln with the weights of ∏i)1 plot together with the ψlnK versus Ln plot is useful to interpret the results of the computer simulation. Acknowledgment We express our appreciation for the financial support by the Metal Mining Agency of Japan. Nomenclature BMh ,in ) coefficient in eq 3 and is defined by eq 6 BlnK ) coefficient in eq 3 and is defined by eq 7, kmol m-3
C ) molarity, kmol m-3 CoHA ) total molarity of HA, kmol m-3 F ) flow rate of the aqueous phase, m3 s-1 K ) extraction equilibrium constant HA ) monomeric species of the extractant L ) metal loading ratio defined by eq 1 L* ) metal loading ratio at the maximum βlnK l ) number of extractant anions in the metal-extractant complex M ) metal to be extracted m ) valency of the metal cation N ) total number of stages n ) stage number p ) aggregation order of the extractant r ) ratio of flow rates defined by F/S S ) flow rate of the organic phase, m3 s-1 Rn ) coefficient in eq 2 and is defined by eq 4 βlnK,n ) coefficient in eq 2 and is defined by eq 5, kmol m-3 θ ) coefficient in eq 8 and is defined by eq 9 F ) relative weight defined by eq 11 ψlnK ) coefficient in eq 8 and is defined by eq 10, kmol m-3 Subscripts in ) inlet max ) value at the maximum recovery ′ ) stripping section Superscript ) organic-phase species
Literature Cited Bogacki, M. B.; Szymanowski, J. Modeling of extraction equilibrium and computer simulation of extraction-stripping systems for copper extraction by 2-hydroxy-5-nonylbenzaldehyde oxime. Ind. Eng. Chem. Res. 1990, 29, 601. Bogacki, M. B.; Szymanowski, J. Computer simulation of copper extraction-stripping systems containing different hydroxy oxime extractants in various extraction-stripping loops. Ind. Eng. Chem. Res. 1992, 31, 328. Bogacki, M. B.; Cote, G.; Szymanowski, J.; Bauer, D. Modeling of extraction equilibrium and computer simulation of extractionstripping systems for nickel extraction by di-n-butyl phosphorodithioate. Ind. Eng. Chem. Res. 1992, 31, 1768. Bogacki, M. B.; Cote, G.; Szymanowski, J. Modeling of nickel extraction between di-n-butyl phosphorodithioate and acid. Ind. Eng. Chem. Res. 1993, 32, 2775. Dalton, R. F.; Seward, G. W. Modified aldoxime reagents for the solvent extraction of copper. In Reagents in the Mineral Industry; Jones, J. M., Oblat, R., Eds.; IMM: London, 1984; p 107. Hughes, M. A.; Parker, N. A computer study of liquid-liquid stagewise calculations in typical and new countercurrent contacting. J. Chem. Technol. Biotechnol. 1985, 35A, 255. Kordosky, G. A. Copper Solvent Extraction: the State of the Art. J. Metals 1992 (May), 40. Peterson, J. N.; Chen, C.-C.; Evans, L. B. Computer programs for chemical engineers: 1978-part 4. Chem. Eng. 1978 (Aug 28), 107. Rod, V. Unconventional extraction-stripping flowsheets for the separation of metals by liquid-liquid extraction. Chem. Eng. J. 1984, 29, 77. Rose, A.; Sweeny, R. F.; Schrodt, V. N. Continuous distillation calculations by relaxation method. Ind. Eng. Chem. 1958, 50, 737. Szymanowski, J. Hydroxyoximes and Copper Hydrometallurgy; CRC: Boca Raton, 1993. Tanaka, M. Prediction of Cu(II) solvent extraction behavior with hydroxyoxime in the presence of alkali-metal chloride. J. Min. Mater. Process. Inst. Jpn. 1991, 107, 639. Tanaka, M. Application of Pitzer’s equations to the analysis of extraction equilibria of Cu(II) from hydrochloric acid solutions with hydroxyoximes under practical conditions. In Solvent Extraction 1990. Proceedings of the International Solvent Ex-
Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1949 traction Conference (ISEC 90); Sekine, T, Ed.; Elsevier: Amsterdam, 1992; p 1157. Tanaka, M.; Koyama, K.; Shibata, J. Steady-state local linearization of countercurrent multistage extraction process for metal ions. Ind. Eng. Chem. Res. 1997, 36, 4353. Verner, J. H. Explicit Runge-Kutta methods with estimates of the local truncation error. SIAM J. Numer. Anal. 1978, 15, 772. Whewell, R. J.; Hughes, M. A.; Hanson, C. Aspects of the kinetics and mechanism of the extraction of copper with hydroxyoximes. In Proceedings of the International Solvent Extraction Confer-
ence 1977; Lucas, B. H., Ritcey, G. M., Smith, H. W., Eds.; Canadian Institute of Mining and Metallurgy: Montreal, 1979; p 185.
Received for review October 20, 1997 Revised manuscript received February 6, 1998 Accepted February 7, 1998 IE9707230
