I
ROGER GILMONTI1 DAVID ZUDKEVITCH,2 and DONALD F. OTHMER Polytechnic Institute of Brooklyn, Brooklyn, N. Y.
Correlation and Prediction
of. . .
Binary Vapor-Liquid Equilibria This method, which utilizes available or calculable physical properties, fills a long-standing need of the chemical engineer and should stimulate further research
THE
ivork reported here was undertaken to develop a method of predicting vapor-liquid equilibria of binary systems from a minimum of readilv available data on the individual components. At the same time the method would utilize generalized factors obtained from a statistical digestion of all the suitable experimental data on binary systems given in the literature. To accomplish this obiective it was required to develop simultaneously a method of correlating the existing binary data which would objectively determine a minimum number of parameters that could accurately describe the data, within experimental error. In this connection, it was possible to d o this for all binary systems with two independent parameters and still describe the qystem adequately for engineering purposes. In a large number of cases, the s x o n d parameter was not significant, and only one independent parameter was needed. The final step in the process was to convert the parameters obtained from the new method of correlating all available binary systems into a set of generalized factors that could be used in conjunction with individual component data to predict binary vapor-liquid equilibria for new systems.
Correlation Equation The new method of correlation uses the statistical theory of least squares with weighted factors to account for the difference in experimental precision due to change of composition. It is based on a tlvo-parameter equation which automatically obeys the Gibbs-Duhem equation and is thxefore inherently thermodynamically consistent. This equation ( 4 ) expresses the vapor-liquid equilibria in terms of a relative volatility deviation as a function of liquid composition, namely:
z
=
& a t constant temperature), and the systems are arranged alphabetically with respect to the more volatile component. The constant pressure systems include a n average temperature, for comparison purposes, determined as the weighted average of the reciprocal boiling points Z W / ( Z W / T ) . I n addition to the parameters, their standard deviations are included to indicate the extent of confidence that can be given to the data. Quite a large number of these systems are accurately described by a single parameter. An analysis of these 500 systems substantiates the general engineering practice of employing no more than two independent parameters in the correlation of vapor-liquid equilibria. The use of the weighting factor also substantiates the contention that the data near the pure components are not as reliable as those near the midpoint composition. Figure 1 shows that when the mole fraction of one component is less than 0.1 and the relative volatility close to unity the reliability of the data is scarcely more than 0.1 as good as that for data near the midpoint. This condition appeared to be intuitively understood by most investigators, since the experimental measurements of the data in the literature avoided the highly dilute regions below 0.1 mole fraction. The statistical method of correlation not only eliminates the subjective factor but also gives an independent quantitative evaluation of the experimental and thermodynamic reliability of the data as applied to the correlation, besides also providing for the easy application of high-speed computers. The plot in Figure 4 clearly shows how the reliability of the data is determined by the correlation which automatically gives a thermodynamically consistent curve. The correlation of these many systems verified two other general assumptions made in engineering practice, namely:
(PI
0 Corrections for nonideality in the vapor phase are negligible in comparison with experimental error when pressures involved are less than 0.1 of the critical. T h e change in activity coefficient with temperature can be neglected in practically all constant pressure systems. The table of parameters together with their standard errors served as the source for evaluating a set of generalized factors for predicting binary vapor-liquid equilibria from the pure components.
Generalized Field Factors Based on the concept of “internal pressure” developed by Hildebrand (5) and Scatchard (76) and recently elaborated by Erdos ( Z ) , a method was de-
226
In addition to the figures and tables which appear in this condensed version, the unabridged manuscript (see coupon) contains the following: TABLES Experimental Parameters and Factors
Field
FIGURES Analysis of Field Factors for Normal Alcohols in Water Comparison of Experimental and Predicted Vapor Composition Data for 1 -Propanol-Water at Atmospheric Pressure Comparison of Experimental and Predicted Vapor Composition Data for 1-Butanol-Water at Atmospheric Pressure Comparison of Experimental and Predicted Vapor Composition Data for Carbon Tetrachloride-1 -Propanol at Atmospheric Pressure
vised for converting the parameters obtained by the statistical correlation into a generalized set of factors, called “field factors.” I n this new method, the excess entropy of mixing was not neglected but accounted for by a n extension of Kirejev’s principle (7) of assuming proportionality between excess free energy of mixing and excess enthalpy of mixing. This extension (7) assumes a proportionality between each of the parameters and their enthalpy counterparts. The generalized factors may be expressed by the following equations :
The factors k,, and k,, which represent the behavior of the molecules of one species in the field of molecules of another species are appropriately called “field factors.” The proportionality constants z ’ and z” between the free energy parameters and their enthalpy counterparts are defined as follows: z’
e (B, + B,) ~
g’
where
BIB,
+= 1
2.303 R T
(dmj- dbZzi)Z
(12)
B , = [ ~ , 3p (13) [ P , ] = parachor of component z and C, = AH: - R T = internal energy of vaporization of component z AH: = latent heat of vaporization per mole
The energy function L-%may be evaluated by the method of Othmer (73) as follows:
where m = slope of Othmer vapor pressure . . plot
molal latent heat of water at the temperature of the system ( ZG - Z L ) , (2;- 2;)= differences in compressibilities of gas and liquid phases of the component and water, respectively ( 74) L’
=
I
The parachors may be taken from listings given in several compilations (6, 75) or calculated by the method of Mumford and Phillips (72) and tabulated by Wilke and Chang (20).
Sample Calculation of Field Factors. T o illustrate the method of calculating Table 11.
Cross Reference for Field Factors
kij
Group for Com-
Cornponent
ponent
Type
i
1
2
Acids Phenols Water Alcohols Ketones Nitriles Esters Ethers Aromatics Paraffins Sulfides Halogens without active H Z Halogens
1
1.0
1.0
-1.0
2
1.0
1.0
2.2
3
-1.0
1.3
1.0
Group for Component j4 5 6 7
1.0
8
9
10
0.4
-0.06
-0.1
1.0
1.0
2.5
0.4
0.5
0.5
0.5
0.5
0.025 0.1
0.1
-0.035
0.015 0.08 0.06 - 0 . 0 3 5
1.0
1.0
4
1 .o
0.2
0.2
0.2
-1.0
6
1.0 0 . 5
0.02
0.01
0.2
1.0
0.5
0.5
-0.03
7
1.0 0.6
0.08
0.06
0.2
0.5
1.0 1.0
0.5
8
0.4
0.05
0.1
0.2
0.5
1.0 1.0
1.0
0.4
-0.025
5
9
-0.1
0.4
with active H Z Misc. 10
INDUSTRIAL AND ENGINEERING CHEMISTRY
3
0.05
1.3
0.5
-0.025
-1.0 - 0 . 0 2 5 0 . 5
1.0
1.0
B I N A R Y V A P O R - L l O U l D EOUlLlBRlA the field factors from the parameters the same binary system in the previous sample is used, namely chloroformethanol a t 55" C. (chloroform = Component 1). From Equation 14: ( U1)'
= 81.6 and ( C2)12 = 96.7
I PI]'^^ = 5.681 and [ P 2 ] 13 = 5.021
From Equation 12:
6 = 13.013 From Table I : g ' = 0.854 andg" = -0.952
From Equation 11 : = 1.077 and z "
= -0.356
Substituting in Equation 10: k12 =
0.688 and
X?i
=
For predicting the composition of a vapor in equilibrium with a liquid mixture of two components, the information required includes field factors (Table 11) and properties of the pure components:
(73) intercept of theOthmer plot (73) [PI = parachor (6. 75,20) T, = critical temperature, O K. (8, 70, 7 7 ) P, = critical pressure, atmospheres (8.10) Steps in the calculation a t isothermal conditions include the following: m C
From the parachor listings :
t'
Prediction Procedure
1.918
The above procedure \vas applied to the 500 systems, and the field factors Lvere determined for each system using the high speed computer. The next problem was to develop a generalized grouping of these factors based on the tabulated data. Grouping the Field Factors. After due consideration for experimental error: it was generally found that for members of the same homologous series, the field factors could be taken as unity-i.e., k,i = kji = 1. This is in accordance lvith the fact that molecules of a homologous series are of similar size, shape, and field and form regular solutions \vith each other (zero excess entropy of mixing). In setting up a suitable grouping of organic compounds, the classification suggested by Ewe11 and others (3) was used as a basis. The five suggested classes were broken down into 10 groups representing all of the compounds under study (see box a t right). The average rounded field factors for the 10 groups are conveniently arranged in Table 11. Based on the data, the assumption was made that field factors for members of the same group were both equal to unity. Because of insufficient data in the literature, only 75% of the table could be completed. However, if the miscellaneous group is omitted (Group l o ) , over 90%, of the table is complete. As a last resort when no indications are apparent, the field factors may- be taken as unity. Since the final vapor-liquid prediction is usually insensitive to changes in the field factors, in the worst case a fair prediction will result from this assumption. Minor exceptions to Table I1 are listed in the complete manuscript.
= slope of the Othmer plot =
(15)
0 For the desired values of X = '1 2 calculate the relative volatility deviation Z of Equation 1. Calculate the ideal relative volatility:
x,,
logat,
=
1% (P9/P9 logpw(mi - mi) 4- Ct
log
(16)
ai, = 2
+ log a;
Calculate the vapor composition:
0 Calculate the total pressure:
Calculate the function of Equation
ance with the following equations:
C,
Calculate the relative volatility:
Calculate the vapor pressure and reduced vapor pressure of the pure components a t the temperature. 0 Evaluate the energies by Equation 14.
12. 0 Calculate the parameters in accord-
-
wherepw = vapor pressure of water in atmospheres at the temperature.
+ PPxjri
P
PPxi~i
where log 7 ."2 and log
*/j
=
1 ~
(g'
- 9 " ) x:
1 (g' f 9")xf 2 ~
+ 23 g"xy
+ 2jg''xt
(18)
Grouping of Field Factors Class I Liquids capable of forming strong hydrogen bonding. Liquids in this class are also capable of association and formation of networks. Group 7 . Group 2.
Organic acids, phenols, aromatic alcohols, in some cases aniline. Water and alcohols.
Class II Liquids that contain a donor atom (oxygen, nitrogen) but no active hydrogen atoms. Group 3. Ketones and nitriles without a-hydrogen. Group 4 . Esters. Group 5. Ethers and dioxane.
Class Ill Liquids composed of molecules that are not capable of hydrogen bonding. Group 6. Aromatic compounds, including the halogen aromatic compounds that have no active hydrogen-e.g, chlorobenzene, dichlorobenzene, dimethylaniline. Group 7. Nonaromatic hydrocarbons. This group also includes the cycloparaffins with the exception of cyclohexane. Also included are the silicoparaffins, CS2, mercaptans, and other sulfides. Group 8. Halogenated hydrocarbons that d o not contain an active hydrogen atom attached to the same carbon atom--e.g., monochlorides or CCla.
Class IV Liquids composed of molecules containing an active hydrogen atom but no donor atom. Group
9. Halogenated hydrocarbons that have a single hydrogen atom attached
to the same carbon atom to which the halogen atoms are attached-e.g.,
molecules
of the type CC13, CH2C13, CH2CICHCl2.
Class V Miscellaneous grouping which includes those compounds which could not b e classified because o f the lack of sufficient data. Group 7 0.
Picolines,
VOL. 53, NO. 3
MARCH 1961
227
L
OO
0.5
X, (n-octone)
0
--c
Figure 5. Comparison of experimental and predicted vapor composition data a t atmospheric pressure. Left. Ethanol-water system i s a typical alcohol and water binary. Center. trans-Dichloroethylene-methyl ethyl ketone system shows negative deviations from Raoult’s law. Right. n-Octane-p-xylene system i s a nonideal mixture of hydrocarbons
For a constant pressure system, the same equations apply except that the temperature cannot be obtained directly, and a method of trial and error or successive approximation or interpolation must be used. An excellent successive approximation may be made by solving for the vapor pressure of one of the components : pQ
’
YZXl
+
P a:, YtX,
(19)
The only unkno\\n on the right hand side of Equation 19 is which changes very slightly with temperature. A first approximation is made for aiz, and ,b: is calculated. This value of p: gives the corresponding first approximation of temperature. The procedure of continued approximations rapidly converges to give the correct temperature, usually in only two approximations and rarely more than three. The parameters may be calculated a t the average reciprocal boiling point of the two components a t the given pressure. Test of the Method. The prediction procedure was applied to the 500 systems studied to obtain a measure of its validity and limitations. T h e high-speed computer was used to perform the tremendous number of calculations required for such a n analysis. A quantitative comparison showed that the mean average difference for all the systems was 1.7 mole % between the predicted and experimental vapor composition. while the average of the maximum differences for each system was 4.5 mole %. Deviations smaller than the mean devidtion were found in systems that contained hydrocarbons, while larger deviations were found in mixtures containing members of Group 3, such as ketones. The evaluation of the prediction technique is readily visualized when the predicted and experimental data points are plotted. A large deviation in vapor composition when the relative volatility is high may be less significant than a
dp
228
smaller deviation rvhen the relative volatility is close to unity. The comparisons given in Figure i present a few interesting cases. Because many types of deviations from Raoult’s law are linked with hydrogen bonds being formed or destroyed, systems that contain an alcohol. ketone. and water are presented as examples
Discussion Since the method of prediction yields a set of proportionality constants between the corresponding free energy and enthalpy parameters, it should also be possible to use the method for predicting heat of solution of binary mixtures. However, preliminary tests appear to indicate that a higher order of accuracy of prediction is needed for the vaporliquid equilibria before satisfactory prediction of heat of solution can be made. This follows from the Gibbs-Helmholtz relationship Lvhich gives the enthalpy as a derivative function of free energ!with respect to reciprocal temperature. Thus, the order of magnitude of the accuracy of the free energy data must be higher than that of the enthalpy data derived from the Gibbs-Helmholtz relationship. One significant \vay of improving the degree of accuracy of the prediction method is to include the electromagnetic effect between the two species of molecules. Preliminary Ivork indicates that the field factors could probably be correlated in terms of the polarizabilities (9) of the binaries; they would thus be obtained with a much higher degree of accuracJ-.
Acknowledgment The authors are grateful to the Esso Education Foundation for sponsoring this work and to the Esso Research and
INDUSTRIAL AND ENGINEERING CHEMISTRY
Engineering Co. for providing facilities in the computing center of their petroleum development division.
literature Cited (1) Bromberg. I.: M.S. thesis, Polytechnic Institute of Brooklyn, Brooklyn, N. Y., 1950. (2) Erdos, E., Collection Czechosloo. Chem. Communs. 21, 1528 (1956). (3) Ewell: R. H., Harrison. J. M., Berg, L., IND.EXG.CHEM.36, 871 (1944). (4) , Gilmont, R., “Thermodynamic Principles for Chemical Engineers,” p. 246, Prentice-Hall, Englewood Cliffs, N. J., 1959. (5) Hildebrand, J., “Solubility of h-onElectrolytes.” Reinhold, New York, 1950. (6) Jacobson, B.: Acta Chem. Scand. 9, 997 (1955). (7)‘ Kirdjev, V., Acta Phys. C‘.R.S.S. 13, 552 (1940). ( 8 ) Kobe, R.-4., Lynn, R. E.. Chem. Revs. 52, 117 (1953). (9) Lu, B. C., Graydon, W. F., IND.ENG. CHEM.49. 1058 (1957). (10) Lydersen, A. ‘ L., Greenkorn. R. A., Hougen, 0. A., Wisconsin Lniv. Eng. Expt. Sta., Rept. No. 4, 1955. (11) Meissner, H. P., Redding. E. M., IXD.ENG.CHEM.34, 521 (1942). (12) Mumford, S. A,. Phillips, J . W. C., J . Am. Chem. SOL.52, 5295 (1930). (13) Othmer, D. F., Maurer, P. W., Molinary. C. J., Kowalski. R. C., IND. ENG.CHEM.49, 125 (1957). (14) Othmer, D. F., Zudkevitch, D., Ibid.. 51. 791 (1959). (15) Quaile. O.‘R., Chern. Rew. 52, (1953). (16) Scatchard, G., Zbid., 8, 321 (1931). (17) Scatchard, G., Raymond, C. L., J . Am. Chem. Soc. 60, 1278 (1938). (18) Steinhauser, H. W., White, R. P., IND.ENG.CHEM.41, 2912 (1949). (19) Volk, W., “Applied Statistics for Engineers,” McGraw Hill, hTew York, 1958. (20) Wilke, C. R.. Chang, Pin, A.I.Ch.E. Journal 1, 264 (1955). RECEIVED for review December 14, 1959 ACCEPTEDNovember 29, 1960 Division of Industrial and Engineering Chemistry, 136th Meeting, ACS, Atlantic City, N. J., September 1959. Taken from D.Ch.E. thesis of David Zudkevitch. Polytechnic Institute of Brooklyn, Brooklyn, N. Y.. 1959.
