Ind. Eng. Chem. Process Des. Dev. 1983, 22, 385-391
n - 1 = tray n - 1 i = component i
385
King, C. J. “Separation Process”, 2nd ed.; McGraw-Hill, Inc; New York, 1981; p 472 Lewis, W. K. Ind. Eng. Chem. 1936, 28. 399. Mohan, T. M.Tech. Thesis, I.I.T., Kanpur, India, 1981.
D = distillate Literature Cited
Received for review May 27, 1981 Revised manuscript received October 21,1982 Accepted December 16,1982
“BubbleTray Design Manual”; American Institute of Chemical Engineers, New Ycfk, 1958. Colbwn, A. P. Ind. Eng. Chem. 1936, 28, 526.
A Versatile Phase Equilibrium Equation of State Paul 1111. Mathlas’ Energy Laboretfny and Chemical Engineering Department, Massachusetts Instftute of T e c h n o w , C a m b r m , Massachusetts 02139
The Soave modification of the Redlich-Kwong equation has been very effective for conelating the phase equilibrium of systems containing nonpolar and slightly polar substances. In this work further modiflcations are introduced which retain the simplicity and robustness of the Soave equation but extend its application to systems containing highly polar substances such as water. Appllcations shown in the work include systems of interest in coal processing and some common systems containing water.
Thus it would be very useful to have an equation of state applicable to systems containing polar substances. One attempt to achieve this goal was that by Gmehling et al. (1979), who assumed a chemical equilibrium hypothesis. In their work, Gmehling et al. assumed that polar species form dimers, and by fitting data to determine a standard-state enthalpy and entropy of dimerization they correlated vapor-liquid equilibria for many polar mixtures. However, this approach has several disadvantages. Whiting and Prausnitz (1981) point out three of them. First, experimental evidence supports the existence of dimers only in rare cases. Second, application of chemical theory greatly increases the computational load since it requires the solution of chemical-as well as phaseequilibrium. Third, the number of pure-component and binary parameters increases because standard-state enthalpies and entropies of dimerization are required. An important disadvantage not cited by Whiting and Prausnitz (1981) concerns applications at severe conditions, say in the retrograde region or even approaching the critical point. Under these conditions even the use of relatively simple models like Soave (1972) have presented severe computational challenges requiring sophisticated solutions (Michelson, 1980; Asselineau et al., 1979; Mathias et al., 1981; and Chan and Boston, 1981). Models employing chemical theory may well be pathological to computational robustness. A remarkable success in the design of an equation of state for the correlation of fluid-phase equilibrium has been the simple idea of Soave (1972). He recognized that a prerequisite for the correlation of phase equilibria of mixtures is the correlation of the vapor pressures of pure substances. The Soave modification of the Redlich-Kwong (1949) equation has been very successful in correlating the phase behavior of multicomponent systems containing nonpolar and slightly polar substances. In this work, the Soave approach is used to extend the Redlich-KwongSoave equation to systems containing highly polar substances such as water and the alcohols. The work also utilizes the modification of the Soave equation suggested by Boston and Mathias (1980) to im-
Introduction The increased use of computers for chemical process design has stirred great interest in the analytic representation of the phase behavior of multicomponent systems. This analytic description must, of course, be quantitatively accurate, but it should also possess the qualities of few and easily attainable correlation parameters, never predicting physically absurd results and being computationally robust and efficient. Very broadly, practical methods to represent phase behavior can be divided into activity coefficient and equation of state approaches. In the activity coefficient method an activity coefficient model, say the UNIQUAC equation, is used to represent the nonideality of the liquid phase($ while a different model, usually an equation of state, is used to describe the departure from the ideal gas reference state of the vapor phase. The latter approach uses the same equation of state to represent all coexisting phases. The equation of state approach has been effective in describing systems containing only nonpolar and slightly polar components, but the common consensus appears to be that the additional flexibility of activity coefficient models is necessary to correlate highly nonideal systems which contain polar substances. The activity coefficient approach has been found to correlate a wide variety of complicated phase behavior, but this has been at the expense of a large number of parameters which extrapolate poorly with temperature. Another-and more serious-disadvantage is that anomalous results are obtained in the critical region since different models are employed for vapor and liquid phases. On the other hand, the equation of state approach predicta critical regions quite naturally, and even relatively simple equations like those of Soave (1972) and Peng and Robinson (1976) are capable of satisfactory predictions of complicated critical region phase behavior (e.g., see Heidemann and Khalil, 1979). *Air Products and Chemicals, Inc., Box 538, Allentown, PA
18105. 0198-430518311122-0385$01.50/0
@
1983 American Chemical Society
388
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983
Table I. Vapor Pressures of Highly Polar Substances; Comparison between Original and Modified RKS % dev, % dev, p = 0 p fitted temp range, substance T,, K P c , atm w P K max ava max ava water acetone methanol
647.3 508.1 512.64
218.3 46.38 79.91
0.3439 0.307 0.565
0.1277 0.0715 0.2359
273-647 259-508 288-513
-33.6 -9.1 -19.4
10.2 2.5 4.6
0.9 0.9 0.9
0.3 0.4 0.4
ethanol
513.92
60.68
0.646
0.1006
293-514
-9.4
1.5
1.7
0.7
l-pentanol
588.15
38.58
0.585
-0.2615
348-512
17.9
5.3
1.7
0.7
l-octanol
652.50
28.23
0.598
-0.2109
386-554
17.9
4.0
4.5
2.2
reference
Meyer et al. (1967) Ambroseetal. (1974) Ambrose and Sprake (1970) Ambrose et al. (1975) Ambrose and Sprake (1970) Ambrose et ai. (1975) Ambrose and Sprake (1970) Ambrose et al. (1975) Ambrose and Sprake (1970) Ambrose et al. (1975)
Average absolute percentage deviation = Z i Ipiemfl- piCad I/(no. of data points).
prove VLE calculations of systems containing highly supercritical substances. We show that this modification successfully correlates the VLE of systems containing substances such as hydrogen, methane, and carbon dioxide at temperatures much above their respective critical temperatures. Incidentally, these systems are of interest in coal conversion processes. A comment about the importance of fundamental considerations is in order. Many authors (e.g., see Henderson, 1979) have pointed out that the theoretical basis of the Redlich-Kwong equation is weak. There is a valid separation of the attractive and repulsive contributions, but the expression for repulsive contributions is correct only for a one-dimensional system of hard rods. Further, the equation gives a universal value of the critical compressibility factor of 1/3 which is too large for all substances, leading to poor liquid molar volumes. However, perhaps due to a fortunate cancellation of errors and certainly due to the clever parameterization of Soave (1972), it has been successfully used for the prediction of multicomponent phase equilibria in a wide variety of process design applications. It should also be mentioned that the approach of this work will probably yield similar results with the Peng-Robinson (1976) or Martin (Martin, 1979; Joffe, 1981) equations. Description of the Equation Pure Fluids. The Soave equation is U p = - -RT u - b U(U + b ) In order to reproduce the critical temperature and pressure, the constants a and b are given by
ai(TCi)=
R2Pd
= 0.42747-
RTci
bi = 0.08664-
Pci
Pci (3)
Following Soave, we keep b as a constant and make a a function of temperature such that the vapor pressure of the pure substance is well correlated. Thus
ai(T) aciai(T) (4) where ai(7') is an adimensional fador which takes the value of unity at T = Tci. A good correlation for ai is given by aP5 = 1 + mi(l - pi(l - TR)(0.7 - TRi) (5) where T a = T/T,is the reduced temperature of component i. The second term on the right side of eq 5 was introduced by Soave to reproduce the vapor pressures of nonpolar
substances. He correlated the characteristic constant m as a function of the acentric factor. The correlation used in this work was obtained by Graboski and Daubert (1978). It is very similar to that of Soave but is slightly better since it was regressed from a larger data set. The correlation for m is given by mi = 0.48508 + 1 . 5 5 1 9 1 ~ - ~0.15613wt
(6)
The Soave equation for a (eq 6 and only the first two terms on the right side of eq 5) adequately correlates the vapor pressures of nonpolar and slightly polar substances (Graboski and Daubert, 1978). However, it was found that the agreement with vapor pressure data of polar substances is seriously in error. By definition, the Soave equation reproduces the vapor pressures of all substances at the critical temperature and a reduced temperature of 0.7. Comparison with experimental data for polar substances shows that the Soave equation over-predicts the vapor pressure at reduced temperatures between 0.7 and 1.0 and under-predicts it at reduced temperature less than 0.7, or vice versa. One of the simplest empirical terms which corrects this is the third term on the right side of eq 5, and fortunately it is found to work very well. Table I shows the significant improvement in vapor pressure correlation obtained by wing the polar parameter p for some common polar substances. We note that the critical temperature and pressure used are the best experimental value and the acentric factor is that which reproduced the experimental vapor pressure at a reduced temperature of 0.7 using the Soave equation. Finally, the polar parameter p was fitted to the experimental vapor pressure data. The polar parameter p introduced in eq 5 is highly empirical. It is unlikely that p can be correlated in terms of other quantities such as the dipole moment since ita value is probably a lumped result of many different effectaincluding the inadequacy of the Redlich-Kwong equation. (Note that ita value even changes sign as one goes to the higher normal alcohols.) However, there is only one additional parameter whose value could probably be obtained from few experimental vapor pressure data. Further, the simple form of the Redlich-Kwong equation is retained, and this is important for the computational robustness and efficiency essential to process design applications. CY for Supercritical Substances. Since the CY expression was obtained by matching pure component data, use of the same expression a t supercritical temperatures (T> TJ represents extrapolation into an unknown region. Boston and Mathias (1980) have suggested the following extrapolation at supercritical temperatures up5 = exp[ci(l - TRd')] (7) where
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 387 I
3Or
Equations 7,8, and 9 were obtained by splining eq 7 with eq 5 at TRi= 1; the value and first derivative have been matched and the second derivative of with respect to TRihas been set to zero. The Boston and Mathias extrapolation has been adopted in this work. As explained in the Introduction, this work aims to design an equation of state to produce good agreement with experimental phase equilibrium behavior. Given a simple equation such as the Redlich-Kwong equation, this necessarily means a compromise, albeit small, in other quantities such as molar densities. Thus it is not appropriate to choose the supercritical a by enhancing agreement with PVT data. The Boston and Mathias extrapolation provides better agreement with VLE data than the Soave equation for common nonpolar binaries (Boaton and Mathias, 1980). Further, eq 7 produces a qualitatively correct high-temperature limit while the Soave equation does not. This is illustrated in Figure 1, which shows second and third virial coefficients for nitrogen. The Soave a produces better agreement with experimental second virial coefficient data but does go through a qualitatively incorrect maximum a t a temperature of about lo00 K. (We should note that this temperature is high enough to be of little practical interest and this is generally true for all substances.) For the third virial coefficient it is hard to tell whether the Soave a is better than eq 7 given the sparse available data. Again, the Soave a produces an unphysical minimum at a temperature of about 1000 K. Extension to Mixtures. The one-fluid theory has been chosen to extend the pure fluid equation to mixture. Thus the form of eq 1 is retained in the mixture case but the parameters are obtained as simple quadratic mole fraction averages of the pure component values
a = CCXiXjUi, i l
(10)
and
The binary parameter k,O is the one commonly used in the Soave equation. Binary parameters have been assumed for both a and b and each is linear in temperature giving a total of four binary parameters. Use of four binary parameters for each binary might be considered excessive-and indeed this is so-but they have been defmed since different kinds of applications require different subsets. For example, hydrogen VLE is well correlated using the parameters kbo and kbl. Further, all the applications in this work have assumed a zero value for kal. It is expected that nollzero values of k,‘ will be required only for immiscible systems; these systems will be addressed in future work. Recently, more theoretically justifiable mixing rules have been suggested based on the local composition concept (Huron and Vidal, 1979; Whiting and Prausnitz, 1981). These have not been employed since good quantitative success has been obtained with the simple mixing rules
I
Original RKS Points Experimentoi D a t a
(Dymond and S m i t h , 19691
I
I---- GLL 2000
:: 8
1500 -I
>-
IO00
I
1
50
I00
I
I
I
I
200 500 1000 3000 TEMPERATURE, K
I
5
f
t
5000
Figure 1. Virial coefficienta of nitrogen calculated by original and modified Soave equations.
used. Perhaps the results of this work could be used as a benchmark against which the more sophisticated mixing rules may be evaluated. It should also be noted that the simple mixing rules defined by eq 10-13 have been found to work very well. For example, zero values for all binary parameters produce good results for hydrocarbon mixtures, even those with large size disparities (Graboski and Daubert, 1978).
Applications A wide variety of applications are shown in this work which demonstrate that the simple equation of state defined by eq 1-13 is flexible enough to correlate “difficult” phase behavior. No applications are shown for systems exhibiting liquid immiscibility, but indications are that this too can be satisfactorily correlated. Experimental phase equilibrium data have been regreased by finding optimum values of the binary parameter subset chosen through the use of the generalized leastsquares method of Britt and Luecke (1973). Values of standard deviation chosen are 0.1 K for temperature and 2% for the other variables. In cases where vapor composition data were not available, estimated values were used and the standard deviation made extremely large. For mole fraction data there is a possibility that the model values will move outside the range 0-1. Hence these variables have been transformed by
The variable zi is used in the regression instead of xi since it varies from -- to m as x i varies from 0 to 1. While the generalized least-squares method provides a flexible, powerful, and statistically correct way to regress data, it creates ambiguity in reporting quantitatively the goodness of fit. For example, if the standard deviations in temperature and pressure are made large relative to those for vapor and liquid composition, good apparent agreement in K values could be obtained even with a poor model. Hence in reporting average errors in this work, errors in K values are given with the model value calculated at the experimental conditions. Since this equation of state defaults to the Soave equation, all the good agreement with experimental data for systems containing nonpolar, subcritical substances will be obtained here. For systems containing some nonpolar, supercritical substances superior agreement may be expected (Boston and Mathias, 1980). Further applications are shown below.
388
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983
Table 11. Regression of Binary Vapor-Liquid Equilibrium: Systems with Hydrogen Err. Err. temprange, rnaxP, no. solvent kb kb' KH' K.q K atm data methane 0.0335 0.0 2.49 0.89 ethane 0.0230 0.0 9.54 6.25 5.11 12.87 propane 0.0071 0.0 m-butane 0.2770 0.8701 3.13 2.18 n-hexane 0.1998 0.5849 5.78 12.92 n-octane 0.7840 1.7775 3.18 1.20 benzene 0.4451 1.0722 2.38 1.62 tetralin 0.2547 0.6100 3.36 4.96 diphenylmethane 0.1378 0.3338 3.35 5.06 m-cresol 0.2142 0.4534 3.53 2.48 1-methylnaphthalene 0.1495 0.3574 3.49 4.06 m-xylene 0.4095 0.9475 3.79 3.78 quinoline 0.1241 0.2493 3.70 4.52 n-decane 0.3683 0.8204 2.67 6.39 bicy clohexyl 3.19 0.2526 0.5511 6.09 toluene 0.4295 1.0140 3.78 4.58 water 0.5063 0.8450 6.71 2.24 ammonia 0.0640 0.1779 5.62 1.44 methanol 0.2561 0.4677 2.45 0.60 Average absolute error in K value of hydrogen (see Table I).
91-117 103-283 88-297 328-394 278-478 463-543 433-633 463-662 463-702 462-662 462-702 462-582 463-702 462-683 462-702 462-575 273-373 273-373 294-413
125 545 546 167 681 149 176 250 260 250 250 250 250 252 250 250 1000 1000 300
20 38 48 44 47 50 49 24 27 41 27 27 27 26 28 25 20 32 15
reference
Kirk and Ziegler (1965) Williams and Katz (1954) Williams and Katz (1954) Klink et al. (1976) Nichols et al. (1967) Connolly (1962) Connolly (1962) Chao et al. (1980) Chao et al. (1980) Chao et al. (1980) Chao et al. (1980) Chao et al. (1980) Chao et al. (1980) Chao et al. (1980) Chao et al. (1980) Chao et al. (1980) Wiebe and Gaddy (1934) Wiebe and Tremearne (1934) Krichevskii et al. (1937)
Table 111. Prediction of Hydrogen-Coal Oil VLE. Data of Chao (1980) ~~
Tc, K Pc, atm 716.5 33.1 770.2 28.2 772.2 35.2 782.0 45.0 705.9 45.0 617.0 35.0 562.1 48.3 Average absolute error in K value of hydrogen (see Table I). solvent tetralin diphenylmethane 1-methylnaphalene quinoline m-cresol m-xylene toluene
Hydrogen-Containing Systems. There is ambiguity about the values of critical constants for components such as hydrogen and helium since quantum effects are quite significant at their critical points. Chueh and Prausnitz (1967) have suggested using effective classical critical constants. In this work the true critical constants as listed by Reid et al. (1977)have been used: T,= 33.2 K,P, = 12.8 atm, and w = -0.22. Figure 2 shows that the agreement with experimental second and third virial coefficient data is reasonably good, but more importantly, use of the true critical constants leads to valuea of the binary parametere that are very close to zero for binaries with the lower normal alkanes. Actually all this is nothing less than fortunate coincidence since eq 6 was regressed using only substances with positive acentric factors. Incidentally, we note that if the hydrogen critical pressure is changed to 13.55 atm very good agreement is obtained with PVT data. The average absolute error in molar volume for 32 experimental data points covering a temperature range of 223-773 K and pressures up to 400 atm is 0.06% (dataof Deming and Shupe, 1932). However, this value of P, was not used since it was considered important to maintain consistency with reported values and also since the change did not make much difference to the more important VLE applications. The binary parameters kbo and kb' were used to regress the hydrogen binary data. Table I1 shows that good agreement with data is obtained for a wide variety of solvents-even highly polar solvents such as water and methanol. For many process design applications the solvents are pseudocomponents generally representing a set of components which boil within a narrow temperature range. In this case binary parameters must be estimated by using
W
solubil param, (cal/cm3)'' 2
0.316 0.434 0.302 0.333 0.466 0.331 0.212
9.50 9.60 10.02 10.90 10.78 8.82 8.92
Err. K s
Err. KH'
4.86 5.89 4.37 4.62 2.29 3.76 6.42
3.65 8.21 5.39 4.31 3.93 3.80 4.85
O \
700
8 u
600 Paints
z
M odi f i ed R K S Experimental Data
-
1 20
1
50
I 100
I
1
I
I
k
200 504 1000 2000 TEMPERATURE, K
Figure 2. Virial coefficienta of hydrogen calculated by modified Soave equation.
only available quantities. To this end the data of Chao et al. (1980)have been used to correlate the binary parameters with the solvent solubility parameter 6. kbo
= 1.061 - 0.8636
kb'
= -2.79
+ 0.236
(15)
(16)
Table I11 shows that the good agreement with experimental data is retained even when this generalized correlation is used. Further, the results are comparable with those of the activity coefficient-approach correlation developed specifically for hydrogen systems by Chao et al. (1980). Systems ContainingMethane and Carbon Dioxide. Recently some VLE data for methane and carbon dioxide
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 389 Table IV. Correlation of Methane-Coal Liquid VLE. Data of Chao (1980) maX kbo fitted kbo = -0.02 temp range, preso ., no. solvent data atm K kbo EKIa EK, EK, EK, EK, EK, n-decane 311-583 272 64 -0.002 3.38 5.61 2.96 5.62 7.70 5.63 n-hexadecane 462-704 -0.003 5.22 249 24 4.40 5.02 4.43 7.15 9.25 tetralin 462-665 -0.003 3.53 250 24 8.70 4.96 4.64 4.89 4.85 diphenylmethane 462-703 250 -0.016 2.58 25 5.07 5.43 4.88 3.18 5.12 1-methylnaphthalene 464-704 -0.022 2.81 248 27 8.7 7 7.04 8.88 2.70 8.78 quinoline 46 3-70 3 t0.002 3.71 250 28 7.33 3.72 7.36 9.48 7.68 m-cresol 462-666 2 50 25 3.78 -0.017 2.72 3.22 3.83 5.92 3.55 m-xylene 461-582 199 22 4.47 -0.020 4.00 3.98 4.56 4.04 4.47 benzene 421-501 2 39 18 -0.019 3.47 3.73 6.84 6.97 3.49 6.97 toluene 422-543 -0.026 4.60 2 49 26 4.47 5.40 6.17 5.42 5.40 Average absolute error in K value of methane (see Table I). Low-temperature data of Reamer et al. (1942) also included. Table V. Correlation of Carbon Dioxide-Coal Liquid VLE.a Data of Chao (1980)
solvent
temp range, K
n-decane n-hexadecane tetralin diphenylmethane 1-methylnaphthalene quinoline m-cresol m-xylene toluene
463-584 463-664 462 -66 5 463-704 463-704 4 62-7 0 3 463-665 462 -58 3 393-543
maX press., atm
hao fitted
no. data
51 50 51 50 50 50 51 52 51 Average absolute error in K value of CO, (see Table I).
EK, 4.52 7.25 6.41 3.66 2.00 2.62 4.06 4.04 3.22
kao 0.138 0.133 0.178 0.123 0.144 0.043 0.096 0.135 0.166
16 16 15 16 15 15 14 16 21
hao = 0.13 EK, 4.17 7.11 8.53 4.19 2.88 16.55 7.61 3.79 5.06
4.76 8.56 1.49 3.06 6.93 2.74 3.36 1.79 5.10
EK, 5.02 8.60 1.91 3.21 7.39 4.63 3.99 1.77 5.11
Table VI. Correlation of VLE of Binary Systems Containing Water other component
kao
kbo
hydrogen nitrogen methane ethane oxygen methanol ethanol
0 0 0 0 0 -0.0218 0.0079
0.5063 0.4225 0.3849 0.3643 0.4369 0.1494 0.1354
kbl -0.8450 -0.6950 -0.8391 -0.9056 -1.0460 -0.02474 -0.0703
temp range, max K press., atm 273-373 323-513 298-444 311-444 273-313 298-373 313-623
in solvents representative of coal liquids a t elevated temperatures and pressure have become available (Chao et al., 1980). In this section the present equation is used to correlate their data. Table I V shows the results of the correlation for binary systems containing methane. The mast effective binary parameter for the correlation if k$. In general it was found that a value of zero (all binary parameters equal to zero) works quite well for the normal alkane solvents and a value of -0.02 gives a good correlation for the heavy aromatic solvents. An exception to this is quinoline, perhaps because it contains a nitrogen atom. Table V shows similar results for the binary systems containing carbon dioxide. In this case the most effective binary parameter is .:k A universal value of 0.13 appears to work quite well for all heavy solvents. Here two solvents (quinoline and m-cresol) do not follow the common value, perhaps due to relatively high polarity. A tentative conclusion may be made that the binary parameter can be generalized for groups of substances. Systems Containing Water. Systems containing water provide a good test of any model since they generally exhibit significant nonideality. Hence an attempt has been made to correlate data for these systems. Table VI summarizes the results. Figure 3 shows the results obtained for the solubility of nitrogen in liquid water. It is encouraging to note that a
1000.0 300.0 444.0 343.0 1.o 3.0 183.0
no. data
reference
20 18 42 37 3 44 104
Wiebe and Gaddy (1934) Saddington and Krase (1934) Culberson and McKetta (1950) Culberson and McKetta (1951) Tokunaga (1975) Dechema(1977) Barr-David (1959) Dechema (1977)
io-zcPoints
Dolo of Saddington and Calculated
KroSe ( 1 9 3 4 ) by m odi f i ed R K S
300 olm
200 otm
100oIm
300
400 500 T E M P E R A T U R E , 'K
600
Figure 3. Solubility of nitrogen in liquid water.
binary parameter that is linear in temperature (kboand kb') can correlate the minimum in solubility shown by experimental data. Comparisons with experimental VLE data for the ethanol-water system are shown in Figure 4. The model demonstrates good agreement with data over a wide range of conditions. Figures 5 and 6 show that the model can correlate different kinds of data with the same parameters. The model
300
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 62315K Male Fraction average Colculoted b y modified R K S
20000
20000
4-
0000
”
473 15K 1
5000 3000 2000 Point5 Experimental Data Calculated by modlfled R K S
3 Ln
w
I
1000 5000
IO+
3000
.
2 000 1000
5000
10-1
2ooo~ 02
04 06 08 MOLE FRACTION E T H A N O L
0
IO
Figure 4. Vapor-liquid equilibrium of the ethanol-water system (data of Barr-David, 1959; and from Dechema, 1977). 5000r
I
2
Poinls
Experimental Data ( O e c h e m a i bymodified R K S
- Calculated
oc
lo‘
i 0’2 0’4 d6 MOLE FRACTION METHANOL
l o
Figure 5. Vapor-liquid equilibrium of the methanol-water system. can correlate VLE data for the methanol-water syatem and the Henry constant for oxygen in the methanol-water mixed solvent. In the latter case, the model is inadequate at low methanol concentrations but is in good agreement with data for the rest of the concentration range. Conclusions The results shown in the work demonstrate that a simple Redlich-Kwong variant which is computationally robust and economical can be successfully applied to a wide variety of phase equilibria applications. It is expected to be useful for process design applications. However, more testing must be done before the model can be accepted for engineering use. It is possible that the simple mixing rules defined by eq 10-13 will not be able to describe highly nonideal systems (e.g., water-hydrocarbon immiscibility) at both ends of the composition
,
,
,
I
I:f
too0 02 04 06 08 MOLE FRACTION METHANOL
10
Figure 6. Henry constant of oxygen in the methanol-water mixed solvent (binary parameters for oxygen-methanol: k 2 = 0.0527, kbl = -0,1732).
range. Further, good description of binary phase equilibrium does not guarantee a correspondingly improved description of multicomponent systems. These issues will be addressed in future work. Acknowledgment The author wishes to express his thanks to Dr, Joseph F. Boston for helpful discussions. This work was part of Advanced System for Process Engineering Project (ASPEN) supported by the U.S.Department of Energy under Contract Number E(49-18)-2295. Nomenclature a = attraction parameter, eq 1 b = hard core volume, eq 1 c = constant, eq 7 d = constant, eq 7 k = binary parameter, eq 12 and 13 m = characteristic constant, eq 5 p = polar parameter, eq 5 P = pressure R = gas constant T = absolute temperature, K u = molar volume r = mole fraction z = transformed mole fraction, eq 14 Greek Letters a = adimensional factor for a, eq 4 6 = solubility parameter, ( c a l / ~ m ~ ) l / ~ w = acentric factor Subscripts
= critical property i, j = component identifications Literature Cited c
Ambrose, D.; Sprake, C. H. S. J . Chem. Thefmodyn. 1970, 2, 631. Ambrose, D.; Sprake. C. H. S.: Townsend, R. J . Chem. Themrodyn. 1974, 6 , 693. Ambrose, D.; Sprake, C. H. S.; Townsend, R. J . Chem. lhermodyn. 1975, 7, 105. Asseiineau, L.; Bogdanlc, 0.; Videl. J. Flukl Phase €qui&. 1979, 3, 273. Barr-David, F.; Dodge. B. F. J . Chem. Eng. Data 1959, 4 , 107. Boston, J. F.; Mathias. P. M. “phase Equilibria in e TMrd-Generation Process Simulator": Resented at 2nd International Gonference on Phase Equiiibria
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Received for review June 29, 1981 Accepted October 13, 1982
Close Approximations of Global Optima of Process Design Problems C. Y. Lut and J. Welsman" Department of Chemiai & Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio 4522 1
Procedures are devised for transforming process optimization problems Into signomlal programming problems. An investigation of the various algorithms proposed for finding the global optimum of such a signomiai programming problem showed the Falk algorithm to be most suitable. When the Falk procedure is coupled to an efficient optimization program, global optima to small size problems are readily obtained. In a moderate size problem, a close approximation of the global optimum can be obtained with reasonable computing effort.
Introduction The use of formal optimization or mathematical programming procedures has become quite common in process engineering design. A number of large size, nonlinear, constrained optimization computer programs are generally available. Such programs are capable of efficiently solving problems of the form max ( m i d f(x) (1) subject to g i ( m = bi (i = 1, 2, ..., 1) (24 (i = 1 + 1, 1 2, ..., m) (2b) gi(X){S,ZJbi Nonlinear problems with more than 75 variables and a similar number of constraints can now be readily handled. A wide variety of optimal design problems have been successfully solved by this approach. Essentially all of the available optimization programs are hill climbing (descending) techniques. They locate the constrained mRnimum (or minimum) closest to the starting point, but they will not necessarily determine the global maximum (or minimum) if there are multiple extrema.
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Case Western Reserve University, Cleveland, OH 44106. 0196-4305/83/1122-0391$01.50/0
One can be certain that the global optimum has been obtained only if a maximization problem is a concave programming problem or a minimization problem is convex. We have a convex programming problem if (a) the function being minimized, f ( X ) is convex and (b) the contrainta are only inequalities of the form gi(X)Ibi with each g i ( X ) also being convex. A concave programming problem is the maximization of concave objective function subject to g i ( X )I bi where the g i ( X )are concave. Almost all practical design problems are neither convex nor concave and one cannot be certain that the global optimum has been found. Most designers have been willing to accept the fact that they may not have located the global optimum providing the minimum or maximum they have located is considerably better than they would have achieved without the use of the formal optimization procedure. However, in many cases the system being designed is very costly and an expediture of a considerable design effort can be justified to locate a better optimum. A common procedure in this case is to repeat the optimization process for a series of different starting points in the hope that one of the chosen starting points will be in the vicinity of the global optimum. This technique is often successful. However, Wilde (1979) points out that in some real optimization @ 1983 American Chemlcal Society
