Ind. Eng. Chem. Process Des. Dev. 1986, 25, 631-634
Method for Equilibrium-Stage Calculations in Liquid-Liquid Extraction. Application to Countercurrent Cascade Design for Quaternary System
631
a
Francisco Ruiz,* Antonio Marcilla, and Vicente Gomls Departamento de Quimica Tecnica, Untversidad de Alicante, Apdo 99, Alicante, Spain
An iteration-free method for equilibrium-stage calculations in multicomponent systems has been developed. The method is based on the interpolation method suggested in this paper which involves the fitting of the solubility surface of two polynomials (one for each phase) and their use together with the experimental tie-line data in order to calculate new equilibrium data. The method has been applied to a quaternary extraction example for the system water-acetone-acetic acid-chloroform at 25 °C. The suggested method has the advantages of any computer method as compared with the graphical methods. Compared to the other computer methods, it is much faster, since no iterations are needed to solve any equilibrium stage. The method is easily extended to higher order multicomponent systems.
In recent years, the applications of multicomponent systems in liquid-liquid extraction processes have received increased interest in chemical technology. Stage calculations become complex when the feed is a mixture of three components or when a mixture of two solvents (either miscible (mixed solvent) or immiscible (double solvent) is used. In those cases, the corresponding quaternary liquid-liquid equilibria have to be considered. For isothermal conditions, it is possible to carry out the tie-line calculations by solving the material balances and equilibrium conditions simultaneously at each stage. Calculations can be done stage by stage by using equilibrium graphs. Thus, various graphical methods have ben proposed for quaternary systems, such as those described by Hunter (1942), Smith (1944), Cruickshank et al. (1950), Powers (1954), and Treybal (1963). However, these procedures all require extensive graphical construction and, in some cases, the use of a simplifying hypothesis to solve each stage and thus prove rather tedious in practice. Alternatively, calculations may be done by computer, provided that there is a suitable method for correlating
nonideal multicomponent vapor-liquid mixtures. One of the essential assumptions was to treat multicomponent mixtures as pseudo-binary solutions over a region of composition and temperature. This enabled the number of parameters required to model the composition dependence of the various activity coefficients to be drastically reduced. The model parameters were obtained by a linear regression fit of the approximate functions to the rigorous A'-value data generated by the thermophysical property subroutines. This concept was extended to include liquid-liquid systems. But they had to develop an empirical equation for representing K for ternary systems, which is not easily applicable to multicomponent systems. Moreover, this method requires iterations to solve each equilibrium stage since K's are a function of the unknown compositions. The objective of the present work is to develop a new iteration-free method for equilibrium-stage calculations in multicomponent systems, suitable for the liquid-liquid extraction design. It is based on a method for interpolating multicomponent liquid-liquid equilibrium data. The method has been applied to a quaternary extraction example for the system water-acetone-acetic acid-chloroform at 25 °C.
the equilibria data. Hence, thermodynamic correlation methods such as the NRTL, Wilson, and UNIQUAC have proven to be useful. For example, Renon et al. (1971) reported a comprehensive computer calculation method for a staged extraction process using the NRTL model to correlate the equilibrium data. However, Sorensen et al. (1979) have concluded that such models, including NRTL and UNIQUAC, are not fully successful for liquid-liquid equilibria. For example, in the case of type I ternary systems, it is often not possible to represent both the binodal curve and the solute equilibrium ratios over the entire two-phase region with sufficient accuracy for extraction design purposes. When objective functions which emphasize the solute concentrations are used, the situation is somewhat improved. Obviously, this inaccuracy becomes more evident when multicomponent systems are involved. On the other hand, these methods require iterations at any equilibrium stage since they have to make equal the activity of both phases in equilibrium. Recently, Chimowitz et al. (1983) developed nonlinear local models for equilibrium ratios (K values) for highly *
Proposed Method In an earlier work, Ruiz et al. (1984) suggested a new graphical method for stage calculations in liquid-liquid extraction for quaternary systems. Further work has led us to develop a method independent of graphical interpolations and free of iterations unlike the ones previously listed. Moreover, it requires less computing time as compared with other computing methods by eliminating the calculation through activity coefficients or equilibrium
ratios. The suggested interpolation method is based on very simple geometrical principles. The first step is to correlate the equilibrium data. Instead of the usual means of establishing a thermodynamic equation involving activity coefficients and equating them in both phases, etc., the separate functions of the two equilibrium surfaces corresponding to the raffinate and extract phases are derived. Such functions are obtained by the least-squares fittings of the compositions of each set of ends of the tie lines to two series of the type
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1986 American Chemical Society
632
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986
AjXj
+ A2X2 +
+
...
An_iXn„i + BjXj2 +
jB2X22 +
B^X^2
+
...
=
(1)
1
where X, denotes the weight fraction of component i, A, and Bt fitted parameters, and N is the number of components of the system. We have now at our disposal the solubility surface fitted to two polynomials and the experimental tie-line data, which, in addition to the interpolation method that we suggest, will allow us to solve any problem related to stage
calculations in liquid-liquid extraction. Any stage calculation involves the solution of the following problem: given the composition of a phase, find the composition of the conjugated phase. In order to solve this kind of problems, we suggest the following procedure: let z1 = (zi, z\,..., z^-i) be the composition vector of one point on the solubility surface I in a N-component system from which we want to obtain the other end of the corresponding tie line. First, we calculate the euclidean distance (d,) from the point z1 to the extremes of the experimental tie lines p) = (p}v p}2..... + (z^-i PiN-1)2]1/2 di = [(zl p\i)2 + (z‘ p{)2 + -
-
“
....
(2)
Let p\, p\,..., ph-i be the N- 1 experimental extremes of the tie lines closest to the z1 coordinates and pf.....$-1 their corresponding conjugated ones. From the euclidean distances (du d2, djy-i) from the point z1 to the N 1 points p\, p\,..., pl/-i, we obtain the point zm (assumed to belong to the tie line sought) on the hyperplane defined -
..,
by the set p?, p2,
...,
pf-i,
satisfying the equation
N-1
z»
=
£ (pp/di) -for
i
d; ^ 0
=
1,2,...,1V
-
1
£ (l/d;) zm
=
pf
if
dj
=
0
(3)
This equation gives greater weight to the nearest experimental points since they are weighted according to the reciprocals of their distances. Finally, the solution is given by the intersection of the straight line passing through the points z1 and zm with the solubility surface opposite to the z1 point (i.e., surface II). In order to illustrate this interpolation method, Figure la shows the solubility surface of the quaternary system A-B-C-D in a tetrahedrical representation where the points z1 in the D-rich phase (phase I) are also shown. By taking apart the shaded zone surrounding the point zl, we can draw the extremes of the three experimental tie lines (p{, p\, and pi) closest (those with lowest euclidean distance) to the z* point (Figure lb). The end points of these tie lines (pf, pf, and pf) define the corresponding plane where the point zm is located as the one that satisfies eq 3. The sought point z11 is obtained as the intersection of the straight line zlzm with the solubility surface II (A-rich phase).
The proposed interpolation method is the main contribution of the present work. Once the interpolaion problem has been overcome, any problem involving the extraction-stage calculation in multicomponent systems can be solved by alternating the material balances with equilibrium conditions. As can be seen, the proposed
procedure is simple and does not require any iteration as used by the previously mentioned methods. The suggested method can be easily implemented in a computer and requires less calculating time than the graphical and thermodynamic methods. Obviously, the method yields progressively better results as the number of available experimental data increases.
Table I. Tie-Line Data (Weight Percent) for the Quaternary System Water(W)-Acetone(A)-Acetic
Acid(AC)-Chloroform(C)
at 25 °C
aqueous phase
w
A
96.8 93.8 90.0 86.2 81.8 77.5 73.0 68.0 95.6 91.3 86.9 82.5 78.0 72.4 66.2 61.2 94.0 88.9 83.6 78.4 73.1 67.0 61.2 54.5 93.0 87.1 81.0 75.0 69.5 63.0 57.2 49.6 91.7 84.6 78.1 71.7 65.0 58.5 51.8 44.4 90.1 82.4 75.3 68.4 61.6 55.1 48.2 41.1
2.55 5.60 9.44 13.2 17.5 21.7 25.9 30.8 2.00 4.39 7.30 10.3 13.4 17.5 22.2 25.5 1.50 3.35 5.55 7.90 10.4 13.6 16.5 19.9 0.84 1.89 3.40 4.96 6.41 8.40 10.3 12.5 0.43 1.03 1.70 2.25 3.32 4.31 5.50 6.84
AC
C
W
0.70 0.62 0.60 0.64 0.71 0.84
0.24 0.45 0.65 1.00 1.35 2.00 2.70 3.70 0.22 0.48 0.77 1.15 1.50 2.31 3.38 4.77 0.17 0.35 0.59 0.99
1.02 1.23 1.81 3.49 5.01 6.38 7.81 8.90 9.80 10.8 3.71 6.89 9.81 12.4 15.0 17.3 19.3 21.2 5.50 10.1 14.5 18.5 22.1
25.7 28.6 31.2 7.08 13.3 18.9 24.2 29.1
33.4 37.2 40.3 9.0 16.5 23.2 29.4 35.2 40.3 44.7 48.2
0.65 0.77 0.79 0.84 0.92 1.20 1.83 2.40 0.79 0.81 1.03 1.27 1.50 2.15 3.00 4.40 0.70
0.90 1.10 1.51 2.01 2.90 3.89
6.70 0.83 1.03 1.30 1.91 2.57 3.79 5.50 8.50 0.87 1.09 1.53 2.22 3.20 4.63 7.06 10.7
1.57 2.41 3.75 5.45 0.17 0.30
0.50 0.91 1.51
2.09 3.39 5.10 0.15 0.24 0.42 0.62 1.00 1.51 2.31
3.70 0.12 0.18 0.22 0.36 0.48 0.73 1.03 1.51
organic phase A AC 7.34 14.0 20.0 25.9 31.2 36.5 41.7 46.4 5.95 11.6 16.5 21.3 25.0 29.6 33.0 37.1 4.60 8.83 12.7 16.2 19.5 22.5 25.5 28.0 3.20 6.25 8.85 11.4 14.1 16.2 18.4 20.4 1.61 3.10 4.55 6.18 7.20 8.40 9.35 10.1
0.21 0.55 1.10 1.80 2.61 3.50 4.62 5.81 0.33 1.12 2.09 3.19 4.81 6.50 8.50 10.8 0.40 1.62 3.00 4.71 6.92 9.10 11.8 14.9 0.72 2.02 3.80 5.82 8.18 10.7 13.6 17.0 0.60 2.28 4.50 6.81 9.50 12.3 15.5 19.0
C
92.4 85.6 79.4 73.2 67.5 61.5 55.6 49.9 93.6 87.4 81.6 75.8 70.9 64.6 59.0 52.3 94.9 89.7 84.7 79.6 74.1 68.6 62.2 55.8 96.3 91.8 87.7 83.0 77.5 72.6 66.4 59.6 97.5 94.7 91.2 87.4 83.6 79.4 74.8 69.2 99.3 97.5 95.3 92.8 90.0 87.0 83.5 79.5
However, the method does not really depend on the source of the equilibrium data, so long as it is reliable. When experimental data for a given system are not available, the physically more sound methods of estimating activities could generate a set of equilibrium data of sufficient accuracy to justify using this method for the sake of speeding up the computations in subsequent applications.
Application to Countercurrent Liquid-Liquid
Extraction Calculation of the Number of Theoretical Stages for Quaternary Systems A computer program has been developed in order to solve, as an example, the problem of calculating the number of theoretical stages in a countercurrent extraction operation for a quaternary system. A numerical example for the quaternary system wateracetone-acetic acid-chloroform at 25 °C (Ruiz and Prats, 1983) has been solved by means of the above-mentioned program. The equilibrium data for this system are reproduced in Table I. The conditions (weight percent) of the example are the following: feed composition, 25%
Ind. Eng. Chem. Process Des. Dev., Vol. 25, No. 3, 1986
Table II. Compositions of the Various Currents in the Countercurrent Acetone (A), and Acetic Acid (AC) with Chloroform (C) F fli r2 R*
Xw xA XAC
Xc
25.0 20.0 55.0 0.0
58.7 2.72 34.7 3.88
83.1
0.10 15.8 0.97
Extraction
90.0 0.009 9.21 0.79
Example of
a
633
Mixture of Water (W),
S
El
Ei
E3
0.0 0.0 0.0 100.0
1.29 4.66 12.4 81.7
0.19 0.25 2.72 96.8
0.17 0.001 0.50 99.3
B
Figure
3.
Countercurrent multistage extraction. A
1. Illustration of the interpolation method in a schematic representation of a quaternary system, (a) Representation of the three experimental tie lines (p, pj1, p[, p2, p\ p") closest to z1. (b) Amplification of the shaded zone showing the calculation of zn, the conjugated point of z1.
Figure
-
-
-
Figure 4. Quaternary system countercurrent multistage extraction. Graphical construction up to the first equilibrium stage.
raffinate with less than 9.4% acetic acid and 0.1% acetone is desired (R„). The calculation procedure is completely parallel to the procedure described by Hunter and Nash (1932) which has been extended to quaternary systems. The flow chart of the computer program (shown in Figure 2) can be easily understood if combined with Figure 3, a flow diagram of the countercurrent extraction process, and Figure 4 where a schematic graphical construction of the calculation on the equilibrium diagram is shown. For the sake of clarity, Figure 4 only illustrates the construction up to the first equilibrium stage. The computer program involves the following subroutines which solve the corresponding calculation problems: INTERPOLATION which calculates a tie line following the suggested procedure, and intersection which calculates the intersection of a straight line with a solubility surface by geometrical methods. The results obtained for the example are shown in Table II. It can be seen that three theoretical stages are needed to obtain the desired raffinate.
Nomenclature A
= acetone AC = acetic acid A, = parameters of the solubility-surface-fitted polynomials fi; = parameters of the solubility-surface-fitted polynomials C = chloroform d, = euclidean distance from z’ to pj E, = extract flow leaving stage i
F K M N
feed
=
equilibrium ratio sum of two flows = number of components of the system pj composition vector of one of the N 1 points solubity surface j closest to the points z; pj* = component k of the composition vector pj raffinate flow leaving stage i Rt =
=
=
2. Flow
chart of the computer program for countercurrent Figure multistage extraction.
-
=
water, 20% acetone, 55% acetic acid (F); solvent composition, 100% chloroform (S); solvent/feed ratio, 3.5; a
S
W
solvent
= =
water
on
the
634
Xi
Ind. Eng. Chem. Process Des. Dev. 1986, 25, 634-642
weight percent of component i composition vector pf one point on the solubility surface 3
Hunter, T. G.; Nash, A. W. J. Soc. Chem. Ind. 1932, 51, 285 T. Hunter, T. G. Ind. Eng. Chem. 1942, 34, 963. Powers, J. E. Chem. Eng. Prog. 1954, 50, 291. Renon, H.; Asselineau, L.; Cohen, G.; Raimbault, C. “Calcul sur Ordinateur des Equillbres Llquide-Vapeur et Liquide-Liqulde’’; Technip: Paris, 1971. Ruiz, F.; Prats, D. Fluid Phase Equilib. 1983, 10, 77. Ruiz, F.; Prats, D.; Marcilla, A. F. Fluid Phase Equilib. 1984, 15, 257. Smith, J. C. Ind. Eng. Chem. 1944, 36, 68. Sorensen, J. M.; Magnussen, T.; Rasmussen, P.; Fredenslund, A. Fluid Phase Equilib. 1979, 3, 47. Treybal, R. E. “Liquid Extraction", 2nd ed.; McGraw-Hill: New York, 1963.
Chlmowitz, E. H.; Anderson, T. F.; Macchieto, S.; Stutzman, L. F. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 217. Cruickshank, A. J. B.; Haertsh, N.; Hunter, T. G. Ind. Eng. Chem. 1950, 42, 2154.
Received for review December 19, 1984 Revised manuscript received September 19, 1985 Accepted October 24, 1985
z>
=
=
i
component k of the composition vector zJ composition vector of the point which satisfies eq Greek Letters A = net flow
z{
zm
=
=
Literature Cited
Aerosol Reactor Design: Effect of Reactor Type and Process Parameters on Product Aerosol Characteristics Sotlrls E. Pratslnls^ Tolvo T. Kodas, Mllorad P. Dudukovlc,* and Sheldon K. Friedlander* Department of Chemical Engineering, University of California, Los Angeles, Los Angeles, California 90024
A systematic study is made of the effect of the reactor type on aerosol product characteristics. The behavior of Knudsen aerosols in the absence of coagulation is modeled in three basic types of aerosol reactors: batch, CSTR, and tubular with laminar and plug flow. Five indexes of reactor performance are introduced and expressed in terms of the moments of the aerosol size distribution. Four dimensionless groups determining the characteristics of the product aerosol are identified. The reactors are compared with respect to their potential to produce high concentrations of monodisperse aerosols at high yields.
1.
Introduction
Three standard configurations (batch, CSTR, and tubular flow) have been used for aerosol reactors. A batch reactor was used to study the dynamics of the Los Angeles photochemical smog (McMurry and Friedlander, 1978). Well-mixed conditions (CSTR) have been reported in flow reactor studies by Badger and Dryden (1939). Tubular flow reactors have been used far more than any other reactor configuration in industrial and environmental studies (Kasahara and Takahashi, 1976; Dahlin et al., 1981; Flagan and Alam, 1982). Most studies of aerosol reactors have involved the investigation of a specific physical or chemical process. Notable exceptions are the analysis of a stirred tank aerosol reactor (Friedlander, 1977, Chapter 10; Crump and Seinfeld, 1980), the modeling of aerosol formation and growth in laminar flow (Pesthy et al., 1983), and the description of constant-rate aerosol reactors (Friedlander, 1983). However, no systematic attempt has been made to examine the effect of the reactor type (or even reactor parameters) on the aerosol product properties. The purpose of this work is to examine and compare the performance of the standard aerosol reactors (batch, CSTR, and tubular). The
Aerosol reactors have long been the basic research tool in studies of gas-to-particle conversion. Most of these studies have been oriented toward the investigation of smog aerosol formation. However, aerosol reactors also present an attractive route in the manufacture of fine particles and are used for this purpose in industry. Typical processes include the production of various pigments (Ulrich and Subramanian, 1977; George et al., 1973), carbon black, and optical fibers (Miller et al., 1984). Potential applications include the production of silicon (Flagan and Alam, 1982) and various ceramic powders (Sanders, 1984). On the other hand, in chemical vapor deposition processes, particle formation is undesirable and must be minimized (Murthy et al., 1976; Sarma and Rice, 1981).
Earlier studies identified the major physicochemical phenomena involved in aerosol formation and growth. Generally, aerosol precursors (usually molecules of gaseous reactants) react to produce a monomer (molecule of a condensable species). The monomer molecules form unstable clusters and stable particles. The newly formed stable particles grow by monpmer condensation. When the concentration of these particles becomes high enough, coagulation takes place, providing another mechanism for particle growth which, however, reduces the number of particles (Friedlander, 1982).
dimensionless groups governing reactor performance are identified. The goal is to establish a procedure and a set of measures which can be used to select the reactor type and operating conditions suitable for producing an aerosol of specified properties. In this study, we examine systems in which a single condensable species (monomer) is formed from gaseous precursors by thermal, photochemical, or chemical reaction at a constant (zeroth order) rate. It is assumed that the precursors are well-mixed on the molecular level prior to the start of the reaction so that micromixing of reactants does not affect the rate. Wall losses of particles and mo-
* To whom correspondence should be addressed. +Present address: Department of Chemical and Nuclear Engineering, University of Cincinnati, Cincinnati, OH 45221. 1 On sabbatical leave from Washington University, St. Louis,
MO 63130.
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1986 American Chemical Society
