T*
= p*/('Pk,pl*/T,
+ ?Fl,p,*/T,*)
(A-4)
where pI2* = ( P ~ * P , * ) ~ ' ~-( ~A)
(A-5)
The reduced volume of the mixture, U , is obtained from eq 2 once the pressure, temperature, composition, and is (or X z l ) of the mixture are known. A more detailed discussion of mixture parameters is given by Bonner and Prausnitz (1973). The pure-component parameters for ethylene and polyethylene used in this study are functions of temperature and are given by the following equations
+
pz*
pl* = 5764.1 7.848T - 6.9 x 10-'T2 atm = 5124.1 - 5.257T 7.418 X 10-3T2 a t m
+
+ 3.3835T + 1.635 X K T: = 6151.3 + 9.7962T - 1.1742 X 10-*T2 K = 1.2419 + 9.74 X 10-5T + 3.76 X 10-7T2 cm3/g vzsp* = 0.9970 + 1.406 X 10-'T + 2.956 X 10-'T2 cm3/g T,*
= 2454.0
plSp*
where T is in "C. The value of 1 a t 260" is -0.05045. The value of X z l a t 260" is -96.5 atm. Acknowledgment
Society, and to Gulf Oil Chemicals Company for financial support and to the computer centers a t the University of California, Berkeley, and a t Texas Tech University for the use of their facilities. One of us (D. C. B.) also gratefully acknowledges the support of the National Science Foundation through Grant No. GK-37059. Literature Cited Benzler, H., Koch, A. V., Chem.-ing.-Tech., 27, 71 (1955). Bonner, D. C., BazQa, E. R., Prausnitz, J. M., ind. €ng. Chem., Fundam., 12,254 (1973). Bonner, D. C.. Prausnitz, J. M.. Amer. lnst. Chem. Eng. J.. 19, 943 (1973). Chung, C. I., J. Appl. Polym. Sci., 15, 1277 (1971). Ehrlich, P., J. Polym. Sci., PartA-3, 131 (1965). Flory, P. J., J. Chem. Phys., 9 , 6 6 0 (1941). Flory, P. J., J. Chem. Phys., 12,425 (1944). Flory, P. J., J . Amer. Chem. SOC.,87, 1833 (1965) Flory. P. J . , Discuss. faraday Soc., 49, 7 (1970). Foster, G. N. I l l . Waldman, N., Griskey. R. G., J. Appi. Poiym. Sci.. 10, 201 (1966). Hellwege, von K.-H., Knappe, H.. Lehmann, P., Kolloid-Z. 2. Poiym., 183,110 (1962). Huggins, M. L.,J. Chem. Phys.. 9,440 (1941). Koningsveld, R., Staverman, A . J.. J. Polym. Sci., Part A-2, 6 , 305 (1968). Matusoka, S., J. Polym. Sci.. 57, 569 (1962). Orwoll, R . A., Flory, P. J., J. Amer. Chem. SOC.,89, 6814 (1967) Parks, W . , Richards, R. B., Trans. faraday SOC.,45,203 (1949). Patterson, D.,Macromoiecules, 2, 672 (1969). Prigogine, I . , Trappeniers, N., Mathot, V . , Discuss. faraday SOC..15, 93 (1953). Prigogine, I., "Molecular Theory of Solutions," North-Holland Press, Amsterdam, 1957, Chapter X V I . Siow, K. S., Delmas, G.. Patterson, D . , Macromoiecules, 5, 79 (1972). Steiner, R.. Horle, K., Chem.-/ng.-Tech.,44, 1010 (1972).
The authors are grateful to the donors of the Petroleum Research Fund, administered by the American Chemical
Receiced for reuieu, J u l y 26, 1973 Accepted October 23, 1973
COMMUNlCATlONS
A Correlation Equation for Vapor Pressure-Bubblepoint Temperature Data for the Methane-Ethane System
An equation is developed to correlate the vapor ethane system between approximately 100 to 700 byshev polynomial. The average absolute error of including pure components is 0.55%. Slopes of pare well with literature values.
Vapor pressure data of many pure substances have been compiled (Jordan, 1954; Nesmeyanov, 1963) and correlation equations for narrow pressure ranges may be represented by simple equations such as the Antoine equation. Correlation equations for pure substances over a wide pressure range up to the vicinity of critical conditions are often very complex in form ( e . g , Martin, 1959; Strobridge, 1962). It is, therefore, understandable that correlation equations for mixtures t o cover a wide vapor pressure range are almost nonexistent. This work presents the results of a study to obtain such an equation for the methane-ethane system. Such an equation has the potential applications in design and operation of distillation equipment and storage tanks as well as in thermodynamic calculations (Houser and Weber, 1961).
pressure-bubblepoint temperatures for the methanepsia for the entire composition range by using a Chepredicted pressures of 200 points for 1 6 compositions ( a P / d T ) ( X I ) calculated from this equation also com-
For pure substances, Gibson (1967) used Chebyshev polynomials to correlate the vapor pressure of water and Ambrose, Counsell, and Davenport (1970) extended its usage to oxygen, nitrogen, and six organic compounds. They demonstrated the advantages of such orthogonal polynomials over other vapor-pressure equations. Equation l is their best choice with X as the normalized tem-
T log P = a ,
+
a,E,(X)
+
a,E,(X)
+
. . . + a , E , ( X ) (1)
perature between -1 and 1. Equation 1 was applied to vapor pressure-bubblepoint temperature data for mixtures of given compositions of the methane-ethane system (Bloomer, Gami. and Parent, 1953). However, the coeffiInd. Eng. Chem., P r o c e s s D e s . D e v e l o p . , Vol. 13, No. 1, 1974
95
cients were found difficult to correlate as functions of liquid composition. Correlation Equation Among several modifications, eq 2 was found to represent the data satisfactorily
(A
+
T ) log P = a,
+
a,X
where
A = 262.397 - 9 6 7 . 2 2 1 ~+~ 768.444xl' a , = 1800.49 - 2 7 0 5 . 8 8 ~-~ 19.2836x1' + 5236.76Xl3 - 6677.75~: 4 3 3 5 . 7 3 ~-~ ~ 1046.07~~~
-+
a , = 521.474 - 1 3 0 3 . 2 5 ~+~ 4892.93Xl2 1 1 4 5 4 . 5 ~+~ 1~ 4 8 9 2 . 1 ~-~ ~9 9 9 8 . 7 1 ~+~ ~ 2683.15~~~
X
=
(2T
-
(Tmax
+
Tmin))/( Tmax
-
Tmin)
- 3 1 2 . 9 8 ~+~ 1 0 7 . 5 2 ~ ~ ~ 401.75 - 3 7 2 . 4 5 ~+~ 2 3 4 . 3 9 ~ ~ '
T,,,
= 550.19
T,,,
=
Equations 7 and 8 were arbitrarily chosen to include the desirable pressure ranges of the present system. From the data (Bloomer, Gami, and Parent, 1953), coefficient A for a given constant composition liquid is calculated by minimizing the sum squares of ( P c a l c d P d a t a ) of eq 2 with an additional third term of a2(&Y2 1).Therefore, for each liquid composition, there is a set of A, ao, al, and a2. Consequently, these coefficients are plotted cs. the liquid composition and are correlated by using the least-squares method to obtain eq 3, 4, and 5. The optimum values calculated scatter randomly around the curves corresponding to eq 3, 4, and 5. The spread of a2 in a very narrow band of *0.6 implies little composition dependence ofthis coefficient. Discussion Computations were made on 200 points covering 16 compositions including pure components of 9 points each and mixture with 13 points for each given composition. Absolute error of vapor pressures calculated from eq 2 is 0.55% and the average error of a given composition is less than 0.9%. Another sensitive test for eq 2 is to compare its derivatives with those of Houser and Weber (1961) who carefully evaluated ( a P / d T ) ( x l ) from data without using a general equation for the entire composition range. The derivative values vary from 2 to 10 psi/"R. The differences between
96
Ind. Eng. Chem., Process Des. Develop., Vol. 13, No. 1 , 1974
about 110 sets show a maximum, a minimum, and the average as 7.3, 0.0, and 1.6%, respectively. These values indicate the validity of eq 2 for the methane-ethane system. This study, therefore, suggests a method and the form of an equation to correlate vapor pressures of a binary mixture over a wide pressure range. It implies also as a means for data reduction useful for equipment design and thermodynamics calculations. Since calculations including the evaluation of coefficients are compiled within the range of T,,, and T,,, as defined, extrapolation outside this range is uncertain and needs further study. Conclusions A useful equation to represent the vapor pressure-bubblepoint temperature relationship for mixtures is proposed and a systematic method is demonstrated to correlate these data over a wide range of pressures for methaneethane system. Acknowledgment The authors wish to acknowledge financial aid from the Phillips Petroleum Company to Bruce Corn and Mark Young as well as the support of Engineering Research Center. Nomenclature ao, al, a2 = Chebyshev polynomial coefficient A = coefficient in eq 2 El, E2, E,, = Chebyshev polynomial P = vapor pressure in psia T = absolute temperature in "R x, = mol fraction of methane in methane-ethane liquid mixture X = nomalized temperature defined by eq 6
Literature Cited Ambrose, D.,Counsell, J. F.. Davenport, A J.. J. Chem. Thermodyn , 2. 283 (1970) Bloomer, 0. T , Gami, D. C., Parent, J. D , Inst. Gas Techno/.. Res. B u / / . . 22 (1953). Gibson, M R . , Bruges, E. A., Mech Eng. Sci.. 9. 24 (1967). Houser. C. G.. Weber. J. H . J . Chem. Data. 6 . 510 (19611 Jordan, T C . "Vapor Pressure of Organic Compounds,' Interscience. NewYork, N . Y . . 1954 Martin, J. J., "Thermodynamic and Transport Properties of Gases, Liquids and Solids," American Society of Mechanical Engineers, New York, N . Y . , 1959, p 110. Nesmeyanov, A. N., "Vapor Pressure of the Elements, ' Academic Press, New Y o r k , N. Y , 1963, p 401-403 Strobridge, T. R., Ma:. Bur. Stand. i U . S . I . Tech. Note. 129 (1962).
Department of Chemical Engineering L'ni u er 9 1t > of e'\, b rad?a Lincoln, Nebraska 68508
Bruce R . Corn Mark D. Young James H. Weber Luh C . Tao"
R e c e i i e d f o r ret i ~ June u 20, 1973 A t c c p t e d August 20, 1 9 3
