A small exponent will assure success but requires more time. Further study will be needed to define a general and perhaps more efficient method of handling this exponent for complicated problems. Study of additional problems of this type will also ascertain the usefulness of this method. While stochastic search techniques may never be as efficient in solving simpler optimization problems, these procedures may be found to be the best means (perhaps in conjunction with conventional techniques) of optimizing complex chemical processes. Summary I t has been shown that complex process optimization studies can be carried out using flow sheet simulation programs with realistic economic criteria. These studies were completed in 7-12 min of IBM 370/165 computer time with minor modifications to an existing simulation system. As expected, process optimization with an economic objective function produces different results from optimization of the same process using an objective function from material or energy balance data. The adaptive random search method, while inefficient, found the optimum in this problem. The complex and pattern search methods failed because of their inability to follow constraints.
Barneson, R. A., Brannock, N. F., Moore, J. G., Morris, C.. Chem. Eng., (July
27, 1970). Box, M. J., ComputerJ., 8, 1 (1965). Bracken, J., G. P. McCormick. "Selected Applications of Nonlinear Programming," p 37,Wiley. New York. N.Y., 1968. Dibella, C. W.. W. F. Stevens, lnd. Eng. Chem., Process Des. Dev., 4, 9
(1965). Evans, L. B., Steward, D.G.. Sprague. C. R.. Chem. Eng. frog., 64, 4 (1968). Flower, J. R., Whitehead, B. D., Chem. Eng., 273 (1973). Friedman, P., Ph.D. Dissertation, University of British Columbia, 1971. Friedman, P.. Pinder, K. L., lnd. Eng. Chem.. Process Des. Dev.. 11, 512
(1972). Gaddy, J. L., Chem. Eng. Educ., Vlll, 3 (Summer, 1974). Gall, D. A., "A Practical Multifactor Optimization Criterion," in "Recent Advances in Optimization Techniques." A. Lavi and T. P. Vogl. Ed., Wiley, New York, N.Y., 1966. Gottfried, B. S . , Bruggink, P. R.. Harwood, E. R.. Ind. Eng. Chem., Process Des. Dev., 9, 4 (1970). Guthrie, K. M.. Chem. Eng., 65, (June 15, 1970). Hooke, R., Jeeves, T. A,, J. Assoc. Comput. Mach., 8, (1961). Hughes, R. R., Chem. Eng. Educ., 1, (1973). Komatsu, S.,Ind. Eng. Chem., 60,2 (1968). Mason, J. T., Ph.D. Dissertation, University of Missouri, Rolla, Mo., 1971. Motard. R. L., Lee, H. M., Barkley. R. W.. "CHESS User's Guide." University of Houston, 1969. Motard, R. L.. Lee, H. M., Barkley. R. W., "CHESS User's Guide," University of Houston, 1969. Seader, J. D. D a h , D. E., Chem. Eng. Computing, 1, (AIChE Workshop Series), 87 (1972). Shannon, P. T., Johnson, A. I., Crawe, C. M., Hoffman, T. W., Hamielec, A. E., Woods, D. R., Chem. Eng. frog., 62, 6 (1966). Umeda, T., Ichikawa, A., lnd. Eng. Chem.. ProcessDes. Dev., 10, 2 (1971). Williams, T. J., Otto,R. E., A./.EE. Trans., 79, l(1960). Zellnik, H. E., Sondak, N. E., Davis, R . S., Chem. Eng. frog., 58, 8 (1962).
L i t e r a t u r e Cited Adelman. A,, Stevens, W. F., AlChE J., 18, 1 (1972). Ahlgren, T. D.. Stevens, W. F., hd. Eng. Chem., Process Des. Dev., 5, 290
Received for review August 8,1974 Accepted September 3, 1975
( 1966).
Calculation of High-pressure Phase Equilibrium from Total Pressure-Liquid Composition Data lsamu Nagata' and Taisuhiko Ohia Department of Chemical Engineering, Kanaza wa University, Kanazawa, 920,Japan
A numerical method is presented to calculate vapor compositions from isothermal total-pressure data up to the critical region. The method uses the isothermal general coexistence equation based on the Gibbs-Duhem equation and the Redlich-Kwong equation of state (Prausnitz modification) to the calculation of fugacities of components in high-pressure mixtures. Calculated results for 22 systems are in good agreement with experimental data.
Introduction Van Ness e t al. (1973) described the numerical methods by which one may accomplish the reduction of binary vapor-liquid equilibrium data a t constant temperature and low pressure to yield a correlation of the liquid-phase activity coefficients and suggested that the recommended method for data reduction is the one based on just P-x data. Recently some authors extended this method to the calculation of high-pressure phase equilibrium from P-x data. Won and Prausnitz (1973) used Barker's method for binary systems containing one supercritical component. They expressed the binary activity coefficient equat.ion with three parameters in the unsymmetric convention as a function of composition. Christiansen and Fredenslund (1975) presented an extension of the method of Van Ness e t al. (1973) t o
high-pressure systems using the orthogonal collocation method. Several investigators have discussed numerical integration of the coexistence equation to calculate vapor compositions at low pressures (for example, Van Ness, 1964). Lu et al. (1968, 1974) applied this method to ternary systems u p t o 8 atm using the virial equation of state. Manley and Swift (1971) utilized the isothermal general coexistence equation and the Redlich-Kwong equation of state to obtain relative volatilities of propane-propene mixtures up to 22 atm. The purpose of the present work is t o extend the isothermal general coexistence equation to high-pressure binary vapor-liquid equilibria up to the critical region of interest because it is unnecessary to assume a particular liquidphase model. Extension t o high-pressure systems requires Ind. Eng. Chem.. Process Des. Dev., Vol. 15, No. 1 . 1976 211
Table I. Dimensionless Constants in Redlich-Kwong Equation of State for Saturated Vapors and Acentric Factors
%
Component
0.4278 0.0867 Methane Ethane 0.4340 0.0880 0.4380 0.0889 Propane n-Butane 0.4450 0.0906 0.4510 0.0919 n-Pentane Ethylene 0.4323 0.0876 Nitrogen 0.4290 0.0870 0.4340 0.0882 Hydrogen sulfide 0.4470 0.0911 Carbon dioxide Difluoromethane 0.4550 0.0927 0.4532 0.0923 Trifluoromethane 0.4400 0.0894 Trifluorochloromethane a Calculated from wi = - log (PiS/Pci)- 1.0 at T/Tci= 0.7. accurate information about the vapor-phase fugacity coefficients of components along the saturation line and the liquid molar volume. So we use a modified Redlich-Kwong equation of state as discussed by Prausnitz and Chueh (1968).
High-pressure Isothermal General Coexistence Equation Van Ness (1964) derived an exact and general form of the Gibbs-Duhem equation. The isothermal form for a single-phase binary system is V
E d P = Z I d In f l
+ z 2 d In j 2
Since the criterion of equilibrium requires that the fugacities in the vapor and liquid phases be identical, writing eq 1 for both phases and subtracting the two resulting equations gives
where $=
(vv- VL)/RT
(3)
By definition fi
(4)
= GiYiP
Substituting eq 4 into eq 2 gives
where
r = In (+d&)
(6)
w
Data source
0.013 0.105 0.152 0.200 0.252 0.085 0.040 0.100 0.225 0.2770 0.2654 0.1703
Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Adams and Stein (1971) Stein and Proust (1971) Stein and Proust (1971)
Table 11. Characteristic Constant k ,* for Binary Systems System, 1-2 Me thane-ethane
1
Me thane-propane
2
Ethane-propane
0
Ethane-n-pentane
2
Nitrogen-methane
3
Hydrogen sulfiden-pentane Carbon dioxide-ethane
12
Carbon dioxide-propane
11
Carbon dioxide-n-butane
16
Ethylene-carbon dioxide
6
Carbon dioxidedifluoromethane Trifluoromethanetrifluorochloromethane (at 31.9"F) (at -0.1"F) (at -55.1"F) (at -100.05"F)
0
(7) Combining eq 5 and eq 7, and rearranging gives the isothermal general coexistence equation
IL-(") ]
-dY1 =
dP
aP
Yl-Xl
I
(
3
P
+
8
9.1
9.1 8.1 6.4
212
Proust Proust Proust Proust
V
\-,
(1971) (1971) (1971) (1971)
(10)
The molar volume of a saturated vapor mixture is obtained from eq 10. The fugacity coefficient of component k in a vapor mixture is expressed by
+--
2 5 yiaik
In @k = In - b k V- b V- b
y ] (1 - y l )
Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 1, 1976
Stein and Stein and Stein and Stein and
AI
Yl
y ] (Il- y1)i
Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Prausnitz and Chueh (1968) Adams and Stein (1971)
T".5V(V+b)
R T3I2b where
+
Data source
a
p=--RT V-b
which may be arranged to be
I ( 3 P
lo2
Equation of State for Vapor Mixtures The Redlich-Kwong equation is used to calculate molar volumes and fugacity coefficients in the vapor phase. That equation is
as a function of P , T , and y leads to the
Defining In (GI/&) following
k,, x
The constants are related to
i= 1
V+b+ V
In -
RT3I2b V+b V V+b
Table 111. Average Deviations for Vapor-Liquid Equilibria of Binary Systems Average absolute deviation
No. of Pressure No.
System, 1-2
Temp
data points
aP,
range, atm
atm
Ay, ~1000
Data source
Wichterle and Kobayashi (1972b) Methane-ethane 189.65 K 13 1.3-44.1 0.04 4 Wichterle and Kobayashi (1972b) 186.11 K 11 Methane-ethane 1.1-39.4 0.03 2 Wichterle and Kobayashi (1972a) 187.54 K Methane-propane 0.1-41.3 9 0.06 3 Djordjevich and Budenholzer (1970) 2.6-14.8 0" F 9 14 0.02 4 Ethane-propane Reamer et al. (1960) 0.3-26.2 40°F 7 6 0.03 5 Ethane-n-pentanea Stryjek et al. (1974) 2.2-27.4 9 -240°F 11 0.02 6 Nitrogen-methane Reamer et al. (1951) 8.5-58.6 70°F 15 8 0.03 7 Carbon dioxide-propanea Reamer et al. (1951) 5 5.4-38.6 40°F 0.02 10 8 Carbon dioxide-propanea Hakuta et al. (1969) 0.08 1 4 23.8-39.3 9 0" Cb 9 Carbon dioxide-ethane Hakuta et al. (1969) 1 2 34.4-43.4 7 0.03 0" Cb 1 0 Ethylene-carbon dioxide 0.08 Adams and Stein (1971) 50°F 12 11.0-44.3 6 11 Carbon dioxidedifluoromethane 4 Adams and Stein (1971) 13 6.5-28.7 0.04 20°F 12 Carbon dioxidedifluoromethane 12 2.8-14.6 -20°F 3 Adams and Stein (1971) 0.01 1 3 Carbon dioxidedifluoromethane 4 Adams and Stein (1971) 1.0-6.4 0.01 11 -60" F 14 Carbon dioxidedifluoromethane 1 4 19.4-27.3 Stein and Proust (1971) 0.01 31.9"Fb 3 1 5 Trifluorome thanetrifluorochloromethane 4 0.01 -0.1 0"Fb 1 6 12.0-16.8 Stein and Proust (1971) 1 6 Trifluorome thanetrifluorochloromethane Stein and Proust (197 1) -5 5.1 0"Fb 1 6 0.004 3 4.4-6.1 17 Trifluoromethanetrifluorochlorome thane Stein and Proust (1971) 19 1.5-2.1 -100.05"Fb 0.003 7 18 Trifluoromethanetrifluorochloromethane Reamer et al. (1960) 7 1.1-51.4 100°F 0.08 13 19 Ethane-n-pentanen 14 6.5-88.6 0.03 220°F 20 Hydrogen sulfide-n-pentanea 13' Reamer et al. (1953) Reamer et al. (1951) 20 100"F 18 12.8-68.2 0.04 21 Carbon dioxide-propanea 8c Old et al. (1949) 20 100°F 3.5-74.5 0.02 22 Carbon dioxide-n-butanea a Using liquid-phase volumetric data. b Forming azeotrope. c Calculated with linear rule for uC,*. 1 2 3
where f m is defined as fl
= 1; f 2 = X I ;
f3
= xi2;f4 = x i 3 ; . . . ; f M = xiM-' (14)
T h e coefficients D m p are calculated from the P-x data according to the article of Hutchison and Fletcher (1970). T h e values of dPldx1 can be obtained by using the fitted P-x relation from eq 13. Prausnitz and Chueh (1968) proposed mixing rules to calculate ail. Table I lists the dimensionless constants, Q , and Qb, and the acentric factor, w . Table 11represents the binary interaction constant k 12, which indicates the deviation from the geometric mean of the critical temperatures for each component. The critical constants of pure components are available elsewhere (Canjar and Manning, 1967). (ar/ayl)pand (JrlJP),, in eq 9 are analytically obtained from eq 11. Molar Volume of Liquid M i x t u r e s Experimental liquid molar volume data along the saturation line are fitted by using the orthogonal polynomials of Hutchison and Fletcher (1970) described below. When experimental volumetric data are not available, they are assumed to be a linear function of mole fraction between the values of molar volume for the pure components. Pure liquid molar volumes are obtained from Francis (1959). Correlation of T o t a l - P r e s s u r e D a t a We (Nagata and Ohta, 1974) represented the experimental total-pressure data by a set of orthogonal polynomials described by Hutchison and Fletcher (1970). T h a t is
N u m e r i c a l Integration Equation 13 was fitted t o P vs. x1 data. The resulting equation was then differentiated analytically t o yield d P l dxl values. The fourth-order Runge-Kutta-Gill method was used to integrate eq 9. Equation 9 becomes indeterminate when XI = y1 = 0 and x1 = y1 = 1. However, limiting values are determined by application of L'Hospital's rule
The numerical integration of eq 9 proceeds in the direction of increasing pressure (Van Ness, 1970; White and Lawson, 1970). For a system exhibiting a maximum pressure azeotrope, computations were made from the pure component end points toward the azeotrope. The incremental size, Ax,, was taken to be 0.001 mole fraction. Calculated Results T o demonstrate the capability of this method, we have made calculations for 22 binary systems. Table I11 presents the mean deviations in vapor mole fraction and total pressure for 18 systems (no. 1 to 18) in the subcritical region Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
213
50 o
EXPERIMENTAL
-
FITTED
30
40
. _ _ .PREDICTED __
l-----
30
/
5 + a
a'
I
I I-
n.
20
EXPERIMENTAL
0
- FITTED .___ PREDICTED -__ IO
0 ' 0.2
0
0.4
0.6
x,
0.8
0
10
06
0.4
02
XI I
Yl
Figure 1. P-x-y diagram for methane(l)-ethane(Z) a t 189.65OK (data of Wichterle and'Kobayashi, 1972b)
1
08
I O
Y,
Figure 3. P-r-y diagram for CHF3(1)-CCiF3(2) a t 31.9OF (data of Stein and Proust, 1971)
60
100 e
0
-
EXPERIMENTAL
-
FITTED
50
FITTED
.__.._ PREDICTED
80
( Vc,*
: ARITHMETIC MEAN
PREDICTED ( Vc,. : LORENTZ MEAN
40
,ICRlT ICAL POINT
EXPERIMENTAL
0
)
j/
I'
I
60 5
z a .30
a 40
20
20
IO
0
0
02
06
04 XI.
0 0
02
06
04
08
I O
x,. Yo
08
I O
Y,
Figure 4. P-r-y diagram for hydrogen sulfide(1)-n-pentane(2) a t 220'F (data of Reamer et al., 1953)
Figure 2. P-r-y diagram for carbon dioxide(l)-propane(Z) a t 70°F (data of Reamer et al., 1951)
and for 4 systems (no. 19 to 22) containing one supercritical component. The table demonstrates clearly that the present method gives a good agreement with the experimental data in the subcritical region and that the agreement with the experimental data u p to the critical point is fairly good. Figures 1 t o 3, respectively, illustrate P-x-y diagrams for methane-ethane at 189.65OK, carbon dioxidepropane a t 70°F, and trifluoromethane-trifluorochloromethane a t 31.9'F. When no experimental liquid-phase volumetric data along the saturation line are available for systems whose components are condensable, they are estimated by a linear combination of pure component saturated liquid volumes. This assumption seems not to affect the accuracy of the present method on the basis of the systems studied. How214
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
ever, when one component is noncondensable, experimental phase equilibrium and liquid-phase volumetric data are obtained from the same source and volume data a t saturation are interpolated using the orthogonal polynomials of Hutchison and Fletcher. The present calculated results present a test of the applicability of the modified Redlich-Kwong equation of state t o high-pressure vapor-phase behavior. Won and Prausnitz pointed out that a more fundamental difficulty is the choice of a vapor-phase equation of state from which vapor-phase fugacity coefficients are obtained. The great importance of this choice is shown in some cases where the components differ significantly in molecular size. The fugacity coefficients strongly depend on the mixing rule in obtaining the mixture parameter uCl2,and the calculated vapor compositions are closely related to methods of calculation of fugacity coefficients. Figure 4 gives such results
for the system hydrogen sulfide-n-pentane a t 220'F. The figure shows that an arithmetic mean, rather than a Lorentz mean, for uCl2leads to better results as shown by Won and Prausnitz. In case of disagreement between calculated and experimental data for other high-pressure systems different from those described here, the data cannot be declared inaccurate; the equation of state may be inadequate for representation of the systems under study. It is recommended that the reliability of liquid and vapor phase thermodynamic properties be checked carefully.
Conclusions The isothermal general coexistence equation based on the Gibbs-Duhem equation is extended to calculate highpressure binary vapor-liquid equilibria u p t o the critical point. Vapor compositions are calculated by the integration of the coexistence equation using the P-x data. Since this method does not need an analytical expression for the liquid-phase activity coefficient to obtain the best fit to the P-x data, it is simple and direct for evaluating vapor compositions. Calculated results for 22 binary systems are in good agreement with the experimental data. Especially for the subcritical systems, the present method gives results with a high degree of accuracy. The proposed method can be extended to higher pressures with a sufficient accuracy, provided a correct representation of vapor-phase properties is available. Acknowledgment The authors are grateful to the Data Processing Centers, Kyoto and Kanazawa Universities, for the use of their facilities. Nomenclature a, b = constants in the Redlich-Kwong equation of state Dmp = coefficient of eq 13 j?i = fugacity of component i in solution, atm f m = simple polynomial term k l 2 = binary interaction constant defined by k l 2 = 1
M P R T
T,,2/(TclTc2)1'2 = maximum number of polynomial terms = total pressure, atm = gasconstant = temperature, K
V = molar volume, l./mol x = liquid-phase mole fraction
y = vapor-phase mole fraction z = mole fraction in liquid or vapor phase
Greek Letters r = In $ = coefficient as defined in eq 3 b = vapor-phase fugacity coefficient in solution w = acentric factor %$b = dimensionless constants in the Redlich-Kwong equation of state Subscripts c = critical state i , i i = componenti j, k, 1 or 2 = component j, k, 1 or 2 12 = binary pair of components 1 and 2 Superscripts L = liquid phase V = vapor phase
Literature Cited Adams, R. A,, Stein, F. P., J. Chem. Eng. Data, 16, 146 (1971). Canjar, L. N., Manning, F. S., "Thermodynamic Properties and Reduced Correlations of Gases," p 202,Gulf Publishing Co.,Houston, Texas, 1967. Chang, S.-D., Lu, 0 . C.-Y., Can. J. Chem. Eng.. 46,273 (1968). Christiansen, L. J., Fredenslund, A,, AlChE J., 21,49 (1975). Djordjevich, L., Budenholzer, R. A,, J. Chem. Eng. Data, 15, 10 (1970). Francis, A. W., Chem. Eng. Sci., 10, 37 (1959). Hakuta, T.. Nagahama, K., Suda, S., Kagaku KOgaku, 33, 904 (1969). Hutchison, H. P., Fletcher. J. P., Chem. Eng. (London), NO. 235, CE29 (1970). Manley, D. B.. Swift, G. W., J. Chem. Eng. Data, 16, 301 (1971). Nagata, I., Ohta, T., Ind. Eng. Chem., Process Des. Dev., 13, 304 (1974). Old, R. H.. Reamer, H. H.. Sage, 0 . H., Lacey, W. N., hd. Eng. Chem., 41,
475 (1949). Polak, J., Lu, 0 . C.-Y., Can. J. Chem. Eng., 52,88 (1974). Prausnitz, J. M., Chueh, P. L., "Computer Calculations for High-pressure Vapor-Liquid Equilibria," p 16,Prentice-Hall, Englewood Cliffs, N.J., 1968. Reamer, H. H., Sage, 0 . H., Lacey, W. N., Ind. Eng. Chem., 43,2515 (1951). Reamer, H. H.. Sage, 0 . H., Lacey, W. N., Ind. Eng. Chem., 45,1805 (1953). Reamer, H. H.,Sage, 0. H., Lacey, W. N., J. Chem. Eng. Data, 5, 44 (1960). Stein, F. P., Proust, P. C., J. Chem. Eng. Data, 16, 389 (1971). Stryjek. R.. Chappelear, P. S..Kobayashi, R., J. Chem. Eng. Data, 19, 334
(1974).
-
Van Ness, H. C.. "Classical Thermodynamics for Non-Electrolyte Solutions," p 136,Pergamon. Elmsford, N.Y., 1964. Van Ness, H. C., AlChEJ., 16, 18 (1970). Van Ness, H. C.. Byer, S. M., Gibbs, R . E., AIChEJ., 19,238 (1973). White, N., Lawson, F.,Chem. Eng. Sci., 25, 225 (1970). Wichterle, I., Kobayashi, R., J. Chem. Eng. Data, 17, 4 (1972a). Wichterle, I., Kobayashi, R.,J. Chem. Eng. Data, 17, 9 (1972b). Won, K. W. Prausnitz, J. M., Ind. Eng. Chem., Fundam., 12, 459 (1973).
Received for review J u n e 25, 1975 Accepted August 18, 1975
Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 1, 1976
215
