Calculation of Multistage Multicomponent Liquid-Liquid Extraction by Relaxation Method J. Jelinek' and V. HlavaEek' Department of Chemical Engineering, Institute of Chemical Technology, Prague 6, Czechoslovakia
An iterative method for the evaluation of stage concentrations based on relaxation techniques is proposed to find a rigorous solution of a multicomponent multistage liquid-liquid extraction problem. Four practical problems are presented to demonstrate the feasibility of the algorithm.
Introduction Computation of extraction processes is often an important point in the design of a chemical plant. Since the equations describing material balances are strongly nonlinear, the calculation of compositions is customarily carried out by some suitably chosen iterative procedure. While the calculation of fractionating columns has been the subject of extensive investigation in chemical engineering in the past decade, the area of rigorous multicomponent multistage calculations of extraction processes remains unexplored so far. There have been only a few attempts to develop rigorous calculation techniques which are capable of evaluating the stage concentrations. For a three-component system the extraction process can be calculated by means of graphical techniques; for more than three components it is necessary to make use of approximation (short-cut) methods or numerical procedures. Apparently the oldest calculation technique is that proposed by Smith and Brinkley (1960, 1963) which is based on the utilization of stripping factors. The calculation is very simple; however, it is sometimes difficult to guess representative values of the distribution coefficient. The stage-by-stage procedure was used by Eubanks and Lowe (1968) for calculation of extraction of inorganic salts (curium extraction) and by Hanson et al. (1962) for organic systems. The former authors did not propose any procedure for improving a new guess from the old profiles while Hanson proposed a simple corrective algorithm. The B method for extraction calculations was devised by Hutton and Holland (1973). Hanson's method is used as a subroutine in a general simulation program FLOWTRAN (Seader et al., 1974). Though only a few examples were reported in the literature, this group of methods may be conveniently used for a simple extraction column; for complex columns with more than two feeds, for extraction columns with associated solvent recovery etc., difficulties may be expected. Cohen et al. (1972) modified the Wang-Henke procedure (1966) to a multistage extraction process; however, they did not report any practical experience with the procedure. Tierney and Bruno (1967) used the Newton-Raphson method to correct the interstage flow rates while Roche (1969) proposed to make use of the Newton-Raphson method to solve simultaneously material balances and equilibrium relationships. Of course, the Newton-Raphson method exhibits a rapid convergence; however, it is necessary to calculate the values of the first derivatives. Since the relations describing the concentration dependence of the activity coefficient are rather complex, the analytical development of first derivatives leads to formidable expressions while a numerical enumeration is time consuming. Olander (1961) developed a computer
' Currently at Department of Applied Mathematics, Research Institute of Chemical Equipment, Brno, Kiiiikova 70, Czechoslovakia.
routine which makes it possible to solve the design problem, i.e., to calculate the number of theoretical stages required for the desired separation. Mills (1965) reported several computer programs for calculation of extraction problems of inorganic salts. The purpose of this paper is to show that the method of false transients (erroneously also called the relaxation technique) can be used for rigorous calculation of extraction towers. The approach developed in this work makes use of simultaneous solution of each component balance. The false transient method can be used for complex extraction towers as well as for an interconnected set of extraction and distillation columns. This paper demonstrates a successful application of the false transient concept to the extraction problems. Method of Computation by t h e False Transient Technique Consider an extraction column shown in Figure 1.A fresh solvent, S, is sent to the top tray of the column; the raffinate product, D, is withdrawn from this stage. A part of the raffinate product can be mixed with the fresh solvent and fed back to the column. The fresh feed, F, is sent to the stage f. The liquid leaving the column at the bottom stage is fractionated in a column to give the regenerated solvent, SE,and the extract product, B. A part of the extract product can be mixed with the second fresh feed. This is the most complicated process which will be considered. The false transient mass balances are of the same form as for the rectification process ( J e h e k and HlavBEek, 1973);here V, and L, are the flow rates of the raffinate and extract, respectively. For the stage j
udX,J - L,-lX,-1,1 + V,+lY,+l,, ' dt For a stage provided with feed or side stream an additional term on the right-hand side of eq 2 occurs. For the top and bottom stages the balances must be slightly modified. The solvent Lo entering the top stage is given by
D = - V1 (3) R+1 The false transient equation for the upper stage results from eq 1-3
R
+ v1 (R+1- 1) y1.i
(i = 1,2,. . . , c ) (4)
Ind. Eng. Chem., Process Des. Dev., Vol. 1 5 , No. 4, 1976
481
D I
IS
Table I. Example 1. Number of Iterations and CPU Computer Time
(maxlxJ,it+l- rJ,Ltl< J,, ~~~~
Lf
ComputNo. of er iterations time. s
Method
w
Hanson's Method I1 Newton-Raphson Relaxation Relaxation Relaxation Relaxation
-
19 7
5
21
10 100 1000
~
ComDuter
17 20
73 43 74.04 11.55 65.59
IBM704 IBM7094 IBM360/40 IBM370/145 IBM360/40
24
-
-
Table 11. Example 2. Number of Iterations and CPU Computer Time
I
F'
(maxlxJ,,t+l- ~
/ B
Figure 1. Schematic diagram of an extraction column.
Method
For the bottom stage the concept of separation factors is useful; the separation factor gi is given as the ratio of component i in the regenerated solvent to the bottom product, i.e.
Based on this equation the component balance around the lower stage is
On combination of eq 1and 6 we have
~< 10-4) , ~
~
l
1.1
Newton-Raphson Relaxation Relaxation Relaxation Relaxation a
No. of Computer iterations time, s
w -
8
35
a
-
1 0.01 0.008
45 44
0.005
100
Computer IBM 7094
25.41 23.65
IBM 370/145 IBM 370/145
-
-
Overflow after sixth iteration.
For the feed stage the additional term (ht/u,)FxF,i must be written on the right-hand side of (10). In an analogous way we have for (4) and (7)
The equilibrium ratio K , is given by
(i = 1 , 2 , . . . , c ) (11) The functional form of the activity coefficients is nonlinear and is dependent on the composition of the extract or raffinate phases. The auxiliary transient equations can be integrated by virtue of the Euler implicit method
where 0 < 6 I 1. Making use of the integration formula (9), eq 1yields an iteration formula
- (?AtLj-]'+l xj- I,, t+ 1 UJ
The equilibrium coefficient Kj,, is a strongly nonlinear function of composition in both phases; cf eq 8. To simplify eq 9 through 12 the following assumptions are made
K,,,t+l + K J , l.t LJt+l
* LJt
VJt+l = u 1
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=
vJ
up = . . . = U N = u
(134 (1%)
(13~) (13d)
Table 111. Example 2. Mole Fractions in the Extract and Raffinate Phase
Extract phase
Raffinate phase
Stage
Furfural
n-Heptane
Cyclohexane
Furfural
n -Heptane
Cyclohexane
1 2
0.93852 0.93760 0.93593 0.93192 0.92547 0.91710 0.90830 0.90081 0.88819 0.87514
0.05913 0.05851 0.05789 0.05544 0.05171 0.04672 0.04124 0.03638 0.02783 0.01842
0.00235 0.00394 0.00610 0.01261 0.02276 0.03614 0.05044 0.06281 0.08400 0.10641
0.05706 0.05732 0.05715 0.05801 0.05889 0.05974 0.06049 0.06103 0.06168 0.06246
0.92482 0.91195 0.89587 0.84455 0.76777 0.67110 0.57'2 33 0.49048 0.35743 0.22490
0.01844 0.03084 0.04753 0.09733 0.17287 0.26873 0.36693 0.44847 0.58140 0.71315
3 5 7 9 11
13 15 16
Table V. Example 4. Number of Iterations and CPU Computer Time
Table IV. Example 3. Number of Iterations and CPU Computer Time W
No. of iterations
CPU
W
No. of iterations
44 13 10
13.77 5.03 4.24
1
40
10
12
1 10
100
CPU 19.33 6.09
Illustrative Examples
Using these approximations a set of linear equations with a tridiagonal structural matrix for a component i results. The computational algorithm is as follows. 1. Problem specification: number of trays; location, flow rate, and composition of feeds, extract and raffinate reflux R' and R , respectively, and recovery fractions gi. 2. Guess composition of both phases and flow rates on each stage. 3. Calculate the activity coefficients in both phases by means, e.g., of Margules or NRTL equations. 4. Solve component balances by the false transient method. The composition of the extract phase x j , i t + ' results. 5. Calculate the composition of the raffinate phase y,,if+' by means of (14) (14)
6. Calculate new values of all activity coefficients. 7 . Correct phase flows by summation equations
L j f + l= Lit
,e
xj,it+'
1=1
Vjt+l =
Vjt
C
yj,it+l
1=1
8. Repeat steps 4-7 as long as the convergence criterion is not satisfied. The raffinate flow rate is calculated from (3) and the extract flow rate is given by
A simple mass balance yields for the flow rate of the recovered solvent
+
SE = L N - (1 R')B (17) The composition of the extract phase being withdrawn as a product is
The composition of the recovered solvent is given by eq 5.
T o illustrate the algorithm suggested four examples are presented. Example 1. The first example is the 15-stage, four-component extraction problem described by Hanson et al. (1962). An equimolar mixture of acetone and ethanol is to be separated by extraction in a column using two solvents, chloroform and water. Chloroform is fed to stage 1,water to stage 15. The feed is fed to stage 5 . The flow rates are: feed, 0.2 kmol/h; chloroform, 0.8 kmol/h; water, 1.0 kmol/h. Here R = R' = SE = 0. The functional form of the equilibrium data used in this example is the Margules equation. The numerical values of the constants in the Margules equation are given in the Hanson book. Using the same initial assumptions as Hanson, we obtained the results shown in Table I. These compare favorably with results reported by Hanson. We can notice that the number of iterations is only weakly dependent on the value of the relaxation factor. Example 2. The second example is the 16-stage, threecomponent extraction problem described by Roche (1964). We consider an extraction tower including extract reflux for the separation of n-heptane and cyclohexane by means of a single solvent-furfural. The amounts of each component in the feed are: n-heptane 50 kmol/h and cyclohexane 50 kmolh; the feed is sent to stage 13. Pure furfural, 1210 kmol/h, is fed to stage 16. The recovery fractions g, are [component, g, (%)I: furfural, 99.874; n-heptane, 52.631; cyclohexane, 0.369. The equilibrium data are described by the Margules equation and are presented in the Roche paper. Using the same initial assumptions as Roche, we obtained the results given in Table 11. Here also a comparison with the NewtonRaphson method is presented. Unlike the preceding example, the number of iterations depends strongly on the value of the relaxation factor. The mole fractions in the extract and raffinate phases are given in Table 111. Example 3. The system of two components-diethylene glycol and benzene-is to be separated by extraction. The column is a 10-stage extraction tower. The feed 1 (1kmol/h) is an equimolar mixture of diethylene glycol and benzene and is sent to stage 6. The amounts of the feed 2 which are fed to stage 10 are: 0.4 kmol/h of ethylene glycol and 0.1 kmol/h of Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
483
Table VI. Example 4. Mole Fractions in the Extract and Raffinate Phase
Extract phase
Raffinate phase
Stage
Phenol
n-Butyl acetate
Water
Phenol
n-Butyl acetate
Water
1
0.00616 0.00418 0.00418
0.07164 0.07872 0.07872 0.10076 0.10076
0.92219 0.91707 0.91700 0.89920 0.89919
0.22188 0.17103 0.17103
0.77055 0.82404 0.82404 0.99905 0.99906
0.00755 0.00492 0.00492 0.00093 0.00093
4 7 11 15
0.00000 0.00000
benzene. The extraction occurs a t 50 "C. In this example the Renon-Prausnitz NRTL equation is used. The coefficients used for calculation are presented by Guffey (1971). The number of iterations and the CPU computer time on IBM 370/145 is given in Table IV. Example 4. In an extraction tower a mixture of phenol and water is separated. The column is provided with two feeds. Feed 1(2 kmol/h, 8%phenol and 92% water) is fed to the upper stage; feed 2 ( 5 kmol/h, 10%phenol and 90% water) is sent to stage 8. The solvent is a pure n-butyl acetate and is fed to the lower stage. The flow rate of n-butyl acetate is 3 kmol/h. The tower is provided with 15 theoretical stages. The temperature in the column is 44.4 "C. The reflux of the raffinate product is R = 2. The separation factor gi is [component, g, (%)I: phenol, 50.0; n-butyl acetate, 98.0; water, 10.0. The NRTL equation was used to calculate the activity coefficients. The values of constants in the NRTL equation are presented by Guffey. The number of iterations and the CPU computer time on IBM 370/145 are found in Table V. The mole fractions in the extract and raffinate phase are reported in Table VI. Conclusions
The false transient method has been used in the calculation of extraction towers. The method is suitable to handle various types of extraction columns involving multiple feeds, multiple side-streams, two solvents, etc. Solvent recovery is usually accomplished by distillation or azeotropic distillation. Recently, we have proposed using the false transient method for calculation of interconnected rectification columns (Jelinek and HlavAEek, 197313);an extension to interlinked extraction and rectification columns is obvious. The false transient method is capable of handling azeotropic distillation while very often the other methods fail to calculate this problem (Jelinek and HlavAEek, 1974). Evidently, the false transient method may easily handle a liquid extraction process with associated solvent recovery. The three substantial advantages of the false transient methods for extraction problems are (i) convergence from arbitrary initial guesses, (ii) the speed of convergence is high, and (iii) the first derivatives are not necessary. As a result, any arbitrary new relation for the activity coefficient may be readily incorporated. The false transient method can also be used toward solution of inorganic extraction problems as, e.g., extraction of rare earth, extraction of nuclear materials, etc. Since the false transient method is capable of calculating the proklem of equilibrium stage distillation with a reaction (Jelinek and HlavAEek, 1976) the extraction process with a simultaneous chemical reaction may be calculated. The flexibility along with the rigor of the method have proved useful in solving complicated industrial problems. The false transient method may be used as a fail-safe subroutine for
484
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 4, 1976
0.00001 0.00000
calculation of extraction processes in a general simulation program. Notation
B c D F
= extract flow rate = number of components = raffinate flow rate = feed flow rate
f = feed stage g, = separation factor, see eq 5 K, = equilibrium ratio for component i L, = extract flow rate on the stage j N = number of stages R = reflux ratio in raffinate part of column R' = reflux ratio in extract part of column S = feed of fresh solvent S E = side stream of regenerated solvent t = time U, = relaxation parameter VI = raffinate flow rate x,,~ = molar fraction of component i on stage j = weightingfactcr y, = activity coefficient o = relaxation factor = At/U
Subscripts and Superscripts F = feed i = component j = stage L = extract phase S = fresh solvent SE = regenerated solvent V = raffinatephase Literature Cited Cohen, G., et al., Chem. Eng. Sci., 26, 2051 (1971). Eubanks, J. D., Lowe, J. T., Ind. Eng. Chem., Process Des. Dev., 7, 172 (1968). Guffey, C. G., Ph.D. Thesis, The Louisiana State University, 1971. Hanson, D. N., Duffin, J. H.. Somerville, G. F., "Computation of Multistage Separation Processes," Reinhold, New York, N.Y.. 1962. Huvon. A. E., Holland, C. D., Chem. Eng. Sci., 27, 919 (1972). Jeljnek, J., Hlavacek, V., Chem. Eng. Sci., 28, 1825 (1973a). Jeljnek, J., Hlavacek, V., Chem. Eng. Sci., 28, 1833 (1973b). Jeljnek, J., Hiavacek. V., Chem. Prum., 24, 440 (1974). Jelinek, J., Hlavacek. V.. Chem. Eng. Commun., 2, 79 (1976). Mills, A. L., "Review of Computer Programs for Solvent ExtractionCalculations," TRG Rept. 902(D) 1965, Reactor Group, United Kingdom, Atomic Energy Authority, 1965. Olander. D. R., Ind. Eng. Chem., 53, l(1961). Roche, E. C.. Brit. Chem. Eng., 14, 1393 (1969). Seader, J. D., Seider, W. D., Pauls, A. C., "FLOWTRAN Simulation-an Introduction," Ulrich's Bookstore, Ann Arbor, Mich., 1974. Smith, B. D.. Brinkley, W. K., AIChEJ., 6, 451 (1960). Smith, B. D., "Design of Equilibrium Stage Processes," McGraw-Hill, New York, N.Y., 1963. Tierney, J. W., Bruno, J. A., AIChEJ., 13, 556 (1967). Wang, J. C., Henke. G. E., Hydrocarbon Process., 45 (a), 155 (1966).
Received for review December 26, 1974 Accepted March 6,1976
