Ind. Eng. Chem. Fundam.
m = number of components N = number of data points P = total pressure R = gas constant T = absolute temperature V = total volume u = molar volume x i = liquid-phase mole fraction of component i yi = vapor-phase mole fraction of component i
83
1983,22, 83-86
V = vapor phase Literature Cited
Subscripts i , j = component
Abrams, D. S.; Prausnitz, J. M. AIChe J. 1075, 27, 116. Fredenslund, A; Gmehling, J.; Rasmussen, P. "Vapor-Liquid Equilibria Using UNIFAC"; Elsevier: Amsterdam, 1977; Chapter 1. Hachenberg, H.; Schmidt, A. P. "Gas Chromatographic Headspace Analysis"; Heyden: New York, 1977; p 10. Legret, D.; Richon, D.; Renon, H. Ind. Eng. Chem. Fundam. 1080, 19, 122. Martln, W. R.; Paulaitis, M. E. Ind. Eng. Chem. Fundam. 1979, 78, 423. Renon, H.; Prausnltz, J. M. AIChE J. 1968, 74, 135. Turek, E. A,; Arnold, D. W.; Greenhorn, R. A.; Chas, K. C. Ind. Eng. Chem. Fundam. 1970, 78, 426. Van Ness, H. C.; Soczek, C. A.; Kochar, N. K. J. Chem. Eng. Data 1967, 72, 346. Wagner, M.; Alexander, A,; Abraham, T. Paper presented at the 5th International Congress in Scandlnavia on Chemical Engineering, Copenhagen, Denmark, April 1980. Wlchterle, I.; Hala, E. Ind. Eng. Chem. Fundam. 1963, 2 , 1955. Wilson, G. M. J. Am. Chem. SOC. 1964, 86, 127.
Superscripts L = liquid phase
Receiued for reuiew December 21, 1981 Accepted July 20, 1982
Greek Letters yi = activity coefficient of component i
AXij = Wilson binary parameter
Correlations for Predicting Azeotropic Heat of Vaporization of Multicomponent Mixtures Abraham Tamlr Department of Chemical Engineering, Ben Gurion Universiv of the Negev, Beer Sheva, Israel
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+
+
The new correlations, L(J/kg-mol) = 8.345 X 103T,(K)[(11.944 - 11.476Tr 11.459T:) 4-1.9778 15.456T, - 21.057T;)I for 0.5 I T, I 0.85,with a mean relative deviation from observed values of 3.5%, and L(J/kg-mol) = 8.345 x 10-V,(~) [(:0.52277(Tr - I) - 5.600(~,7- I)) w{9.1047(Tr - I) - IO.IOI(T,~ - I))] for 0.5 5 T , I1, with a mean relative deviation of 4.7%, are recommended for predicting the azeotropic latent heat of vaporization, L , of multicomponent mixtures. The correlations were derived solely on the basis of 81 binary azeotropic mixtures, but they predicted extremely well the azeotropic latent heat for 14 ternary mixtures.
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Introduction Multicomponent azeotropic mixtures, where X i = Yi,i = 1,2, ..., c , may be considered in some respects as a single component. For example: (a) They possess one kind of latent heat, namely, the differential heat is identical with the integral heat, which is the heat required for a complete vaporization of a mixture. This is in contrast to nonazeotropic mixtures which exhibit both kinds of latent heats. (b) The Clausius-Clapeyron equation (C.C.) for multiazeotropes becomes identical with the C.C. equation for a single substance. This can be shown as follows. According to Malesinski (1965), the Clausius-Clapeyronequation for a multiazeotropic mixture reads
i=l
where c is the number of components in the mixture. Denoting the azeotropic latent heat of vaporization by
assuming a perfect gas mixture, and neglecting V / ,gives
Equation 3 is identical with the C.C. equation for a pure substance. The major conclusion drawn from the above behaviors is that it is possible to apply the same kind of equations originally developed for a pure substance so as to obtain the behavior of a multicomponent azeotropic mixture. In the present case, we make use of the Riedel vapor-pressure correlation (eq 4) and apply eq 3 to obtain the correlation for the azeotropic latent heat. The parameters of the correlation are determined from available data of the azeotropic latent heat of binary mixtures provided by Tamir (1980/81).
Correlations for the Latent Heat and Estimations of Mixture Properties Many correlations have been compiled by Reid et al. (1977) for the vapor pressure of pure substances. In the present study, the Riedel correlation has been arbitrarily chosen. Acceptance of the Pitzer idea that, in addition to the critical properties, the acentric factor can be used as an additional correlating parameter leads to the following correlation for the vapor pressure on the basis of the Riedel equation In PI = ( a l + pl/TI
+ y1 In TI + SlT,6) +
w(aZ
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+ & / T , + yZ In T, + hTr6)
The application of eq 3 gives the following result 0 1983 American Chemical Society
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84
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983
Pi, yi,and 6i where i = 1, 2, are adjustable parameters, determined on the basis of m observed data points of L vs. T, by minimizing the objective function m
i=l
(Lobad
-
5 or
di2
An alternative equation may be obtained for L taking advantage of the following conditions
L = 0; P, = 1
(7)
(T, = 1)
X I= A2 + BzT
(13)
where the parameters A2 and B2 were reported by Tamir (1980/81). Equation 13 predicts values of X1 with respect to observed values with a relative deviation of the order of 1.5%. When the composition is known, it is possible to calculate the physical properties by means of eq 9-11. The minimization of eq 6 yielded finally the following correlations
It is obtained that
The properties of the mixture are introduced into eq 5 and 8, by means of mixture critical properties, and the acentric factor as follows. According to Spencer et al. (1973), the Li method (1971) is recommended for calculating the true critical temperature, for the simple reason that it is accurate and simple. For any mixture, Li's equation has the form c
(9) where
CXiVU' i=l
The mixture acentric factor w is given by the approximation (Reid et al., 1977, p 74) C
w
=
cxiwi
i=l
where wiis the acentric factor of the pure substance.
Results and Discussion Equations 5 and 8-11 indicate that the latent heat of vaporization is a function of the following data: Tci,V,,, and wi of the pure components, which are available from Reid et al. (1977). The values of the latent heats, needed to obtain the parameters in eq 5 and 8 according to eq 6, were obtained as follows: Tamir (1980/81) reported parameters for correlating azeotropic data of the total pressure and composition vs. temperature of 108 binary systems. The correlating equation for the total pressure was In P = A , + B , T 1 + C,T. The raw material for obtaining the parameters of AI, B1,and C1were data for the azeotropic pressure, temperature, and composition collected from Hirata et al. (1975) and Gmehling et al. (1977-78) which are the most reliable and comprehensive sources of vapor-liquid equilibria data available today. It was found that the relative deviation between observed and predicted values of P according to the above P-T correlation i s of the order of 0.5%. By applying the ClausiusClapeyron equation, the above P-T correlation yielded the following equation from which values of the latent heat of vaporization, designated as Lobs& could be obtained Lobad
It should be noted that the above procedure is the most acceptable one for obtaining azeotropic heat of vaporization data. The parameters B1 and C1for 108 azeotropic mixtures were reported by Tamir (1980/81). In order to obtain the parameters in eq 5 and 8, according to eq 6, only 81 systems for which the critical properties and wi were available were considered. The 744 observed values of the latent heat corresponding to the 81 binary systems were generated as follows. For each system, Low was calculated for several temperatures within the range of temperatures for which eq 12 is valid. For each temperature the composition of the binary mixture was calculated from
= -R(B1 - ClpZ)
(12)
L(J/kg-mol) = 8.345 X 10-3T,(K)[(11.944- 11.4761: + 11.459Tr7)+ w(-1.9778 + 15.4567', - 21.057T:)I (14) for 0.5 5 T, 5 0.85, with a mean relative deviation from observed values of 3.5% and a sum of squares of 122.4 (J/kg-molY, according to eq 6. L(J/kg-mol) = 8.345 X 10-3Tc(K)[{-0.52277(TI - 1)5.6O0(TI7- 1))+ 0{9.1047(7', - 1) - 10.101(T,7- l))] (15) for 0.5 5 TI 5 1, with a mean relative deviation from observed values of 4.7% and a sum of squares of 217.7 (J/kg-molI2,according to eq 6. It should be noted that an additional term containing w2 was added to eq 5, with three additional adjustable parameters. However, its effect on improving the goodness of the fit of the data by such an equation was negligible. A similar behavior was observed by attempting to improve the prediction ability of eq 8. The prediction of eq 14 and 15 is demonstrated in Table I, which includes 14 ternary azeotropic systems and one binary mixture. It should be emphasized that the systems in Table I were not included in the information, used for obtaining the parameters in eq 5 and 8. The table contains information required for the calculations according to eq 14 and 15, the observed latent heats, and the deviations of the predicted values. The major conclusion drawn from the table is that the new correlations are very good predictors (within experimental accuracy) of the latent heat of multicomponent mixtures. The mixtures comprise organic liquids, such as alcohols, esters, ketones, hydrocarbons, amines, ethers, and acids and inorganic liquids such as water, hydrazine, and sulfur dioxide. A possible explanation for the excellent prediction ability of ternary azeotropes by the correlations, derived from binary information, lies in the fact that azeotropes behave in some respects like a pure substance. Moreover, the 81 binary mixtures are probably a good spectrum of species having properties which might also appear in ternary and other multicomponent mixtures. It is, therefore, expected that eq 14 and 15 will also predict latent heats of quaternary and systems of higher order. However, this hypothesis was not proved for want of data for the latent heat for such systems. In order to evaluate the present correlations, it is worthwhile comparing with the correlation for predicting the latent heat of binary mixtures suggested recently by Santrach and Lielmezs (1978). A t first, it should be noted
Ind. Eng. Chem. Fundam., Vol. 22, No. 1, 1983 85
Table I. Prediction ofthe heotropic Heat of Vaporization by Eq 1 4 and 15 of Binary and Ternary Systems Not Included in the Process of Computing the Parameters of Eq 14 and 15
acetone-methyl acetate" cyclohexane-benzene-ethanol cyclohexane-benzene-propanol
0.607 0.626 0.629 cyclohexane-benzene-2-methyl-1-propanalb 0.629 cyclohexane-benzene-butanol 0.629 benzene-ethanol-waterc 0.604 ethyl acetate-ethanol-waterc 0.642 trichloroethylene-ethanol-water 0.602 propanol-benzene-waterc 0.601 pyridine-acetic acid-heptane 0.673 pyridine-acetic acid-octaned 0.672 pyridine-acetic acid-nonaned 0.666 pyridine-acetic acid-decaned 0.665 pyridine-acetic acid-undecane 0.666 acetone-chloroform-methanole 0.571 " Tamir (1980/81). Swietoslawski (1961). Lic& and Denzler (1948). et al. (1981). D ( % ) = 100[(Lobsd - L&cd)/L&sd]; D = 11 1 4 ~ 1 4 i = 1 (only
32.1 1.7 -0.1 33.9 -1.8 -4.1 32.8 -0.9 -4.1 31.8 0.0 -3.8 31.4 0.0 -4.1 35.0 -1.7 -3.1 35.7 2.8 0.0 36.2 -1.4 -2.5 35.6 3.1 1.4 32.3 -4.3 -8.0 36.8 3.3 -2.4 40.4 5.9 2.0 42.0 6.7 4.0 39.8 1.8 -1.5 35.0 4.3 4.9 Pawlak and Zielenkiewicz (1965). e Tamir
507.6 547.1 553.3 556.3 557.6 558.8 533.2 563.2 565.4 547.9 578.7 601.1 612.4 616.3 522.1
f
that it is not clear from their writing what kind of heat Santrach's correlation predicts, namely, differential or integral heats. Certainly, for azeotropic systems it is the same. Secondly, the correlation is based on 15 binary systems with 135 data points, where the correlations suggested here are based on 81 binary systems and 744 data points. The present correlation needs Tci,Vci,and wi as input data, whereas the correlation suggested by Santrach and Lielmezs (1978) needs also the normal boiling point as well as the normal latent heat of vaporization, not usually available. The present correlation is dm applicable for multicomponent mixtures, whereas the correlation of Santrach is restricted to binary mixtures. The deviation of the predicted latent heats from observed values is more or less the same. Finally, it has also been revealed that instead of calculating T , in eq 14 and 15 on the basis of T , (eq 9), it is possible to calculate this parameter from the pseudocritical temperature, namely
0.315 0.341 0.287 0.243 0.228 0.339 0.388 0.362 0.288 0.342 0.366 0.379 0.376 0.371 0.321
for ternary systems).
definition of the integral heat, results in the following equation for the integral latent heat L,, at a constant pressure C
C
i=l
i=l
L, = CXiliq +
Xi[C:i(F - Ti) - Cbi(P - Ti)]+
m,
- AH1p (19)
where
is the latent heat of the pure substance at Ti. For an azeotropic mixture, with an identical composition as the nonazeotropic mixture, T' = T" = T*, where T* is the boiling temperature of the azeotrope. Equation 19 reduces to
L*, = C
C X i l i ~+,
i=l
C
Cxi(c:i- CLi)(T*- Ti) + M'p
- AH'p
i=l
(21)
The advantage is that the critical volume is not needed in this case. The justification for the above approach is the fact that the azeotropic latent heat of the ternary mixtures in Table I was predicted with an overall relative deviation of about 5%. Recall that correlations 14 and 15 were derived on the basis of binary data and T, calculated from eq 9.
Application of the Correlations to Nonazeotropic Systems Correlations 14 and 15, for the integral azeotropic latent heat of vaporization, may also be used for predicting the integral latent heat of nonazeotropic systems. This is proved as follows. The partial enthalpies of component i, in the liquid and vapor phases, corresponding to an identical composition of both phases, are given by
+ CEi(T" - T;)+ Blip = H'~T,+ Cki(T1- Ti) + i i ? l I ~ l
Pi,
= H';,
Inspection of eq 19 and 21 indicates that the difference between L, and L*,, is not appreciable. This is because the dominant term in the right-hand side of eq 21 is the first one. As a result, correlations 14 and 15, originally developed on the basis of azeotropic data, can also be used to predict-within experimental accuracy-the integral latent heat of nonazeotropic systems. Under isothermal conditions C
L*T
LT = CXiliT i=l
+ AHT'
-
M+
(22)
where T is the isothermal temperature.
Acknowledgment
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The author is grateful to Mr. Moshe Golden for his assistance in the computational work and to Mr. Elisha Elijah for his cooperation in editing this paper.
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Nomenclature
where the effect of the pressure on enthalpy is neglected. CPi'sare mean values of C, with respect to temperature, Ti is the boiling temperature of the pure components of a pressure P where and T" are respectively the bubble point and the dew point of the mixture. The substitution of eq 17 and 18 for the Hi's in eq 2, which is a general
number of components C,i = mean specific heat with respect to temperature of pure component i D(%)= relative deviation in percent from observed value defined by loo(Lot,,d - LcdCd)/Lobsd D(%)= mean relative deviation defined by D = (l/s)~s;=Jlil where s is the number of systems c =
Ind. Eng. Chem. Fundam. 1983, 22, 86-90
86
fZi = molar enthalpy of a pure component i Hi= partial molar enthalpy of component i l i , = latent heat of pure component i at temperature Ti L = azeotropic (integral) latent heat of vaporization, or integral latent heat of vaporization of a non-azeotropic mixture in eq 19 and 22 P = total pressure P, = reduced pressure defined by PIPc P, = critical pressure of the mixture R = universal gas constant T = absolute temperature Ti = boiling temperature of a pure component T, = reduced temperature defined by Tf T , T,, Tci,Tcp= true critical temperature of a mixture; critical temperature of component i ; pseudocritical temperature given by eq 16 y, = dew point or bubble point of a mixture, respectively V , = partial volume of component i Vci = critical volume of component i Xi= mole fraction of component i in the liquid AHi = partial heat of mixing of component i AH = heat of mixing Greek Letters a , p , y , 6 = adjustable parameters
= acentric factor of the mixture defined by eq 11;acentric factor of component i
w , wi
v = of the vapor * = of an azeotropic mixture
Subscripts i = of pure component i
iTi = of component i at a temperature Ti obsd = observed P, T = at a constant pressure or temperature, respectively P, T1 = at temperature TVor T',respectively
Literature Cited Gmehling, J.; Onken, V.; A&, W. "Vapor-Liquid Equilibrium Data Collection", Dechema, Chemistry Data Series, 1977-1978; Voi. 1, Parts 1, 2a, and 2b. Hirata, M.; Ohe, S.; Nagahama, K. "Computer-AMed Data Book of Vapor-Liquid Equiiibrlum"; Eisevier: Amsterdam, 1975. Li, C. C. Can. J. Chem. Eng. 1971, 49, 709. Licht, W.; Denzler, C. G. Chem. Eng. Prog. 1948, 4 4 , 627. Malesinski, W. "Azeotropy and Other Theoretlcai Problems of Vapor-Liquid Equillbrium"; PWN-Polish Scientific Publishers: Warszawa, 1965. Pawlak, J.; Zielenkiewicz, A. Rocz. Chem. Ann. SOC. Chim. Polonorum 1985, 39,314. Reid, R. C.; Prausnitz. J. M.; Sherwood, T. K. "The Properties of Gases and LiquMs", 3rd ed.;McGraw Hill; New York, 1977. Santrach, D.; Lielmezs, J. Ind. Eng. Chem. Fundam. 1978, 17, 93. Spencer, C. F.; Daubert, J. E.; Danner, R. P. AIChE J. 1973, 19, 522. Swietoslawski, W. Rocz. Chem. 1961, 35, 317. Tamir, A. FluidPhase Equilib. 198Ol81, 5 , 199. Tamir, A.; Apelblat, A.; Wagner, M. Fhid Phase Equilib. 1981, 6 , 113.
Superscripts 1 = of the liquid
Received for review November 17, 1981 Accepted October 12, 1982
Estimation of Setchenow Constants for Nonpolar Gases in Common Salts at Moderate Temperatures Ellen M. Pawlikowskl and John M. Prausnltz' Deparfment of Chemical Engineering, University of California, Serkeley, California 94720
A simple technique is suggested for estimating salting-out constants for nonpolar gases in aqueous salt solutions in the range 0 to 60 O C . As suggested by the perturbation theory of Tiepel and Gubbins (1973), the salting-out constant is given by a linear function of the Lennardlones energy parameter of the gas. Data for nine nonpolar gases were examined. Correlation parameters are given for nine nonpolar gases and fifteen salts at 25 O C . Correlation parameters are also reported for individual ions. For a few salts, correlation parameters are given as a function of temperature.
Introduction It has long been recognized that the presence of a salt alters the solubility of a gas in water, and many theories have been suggested for calculating salt effects from independent measurements of the properties of salts in water and gases in water. Early theories were based on electrostatic considerations; gas-water interactions were considered to be affected by a change in the dielectric constant of the medium due to the presence of salt ions. Later, theories utilizing the concept of internal pressure were introduced. A review is provided by Long and McDevit (1952). All of these theories are difficult to use because they require parameters which are not readily available. Tiepel and Gubbins (1973) proposed a statistical-mechanical perturbation theory based on that of Barker and Henderson. However, perturbation theory gives only qualitative agreement between theory and experiment. Calculations using this theory are tedious and require
several parameters that cannot easily be determined from independent experiments. In this paper, we propose a simple correlation, suggested by perturbation theory, for estimating salting-out constants for nonpolar gases in salt solutions over a moderate temperature range, 0 to 60 OC.
Correlation To examine the effect of low concentrations of salt in water on the solubilities of sparingly soluble gases, we express the Gibbs energy G of the liquid phase in the form suggested by Guggenheim (Newman, 1973)
nowoo G / R T = -+ Cnj[ln (mjX?) - 11 + RT jzo C &,mini Debye-Huckel contribution (1) i#O jzo
+
where no is the moles of water, nj is the moles of species
0196-4313/83/1022-0086$01.50/0 $2 1983 American Chemical Society
