Ind. Eng. Chem. Res. 1987,26, 759-762 exists), generally S is preferable to I as a parameter, while both S and I are superior parameters to the CH ratio. Tb is superior to M , while M is superior to v. Acknowledgment Financial aid from the Refining Department of the American Petroleum Institute is greatly appreciated. Nomenclature a-f = constants in eq 2 and 3 CH = carbon-to-hydrogenweight ratio d = liquid density at 20 "C, g/cm3 I = Huang characterization factor, (n2- l ) / ( n 2 + 2) M = molecular weight n = sodium D-line refractive index at 20 "C and 1 atm P, = critical pressure, psia S = specific gravity at 60160 "F Tb= normal boiling point, "R T , = critical temperature, "R V , = critical volume, ft3/lb Greek Symbols p = correction factor 0 = physical property 01,e, = input parameters in eq 2 and 3 v 1 = kinematic viscosity at 100 "F, constant v2 = kinematic viscosity at 210 "F, constant AHVNB= heat of vaporization at normal boiling point, Btu/(lb-mol)
759
Registry No. Eicosane, 112-95-8;tetradecylbenzene,1459-10-5; nonadecane, 629-92-5.
Literature Cited American Petroleum Institute (API) Technical Data BookPetroleum Refining, 4th ed.; Daubert, T. E., Danner, R. P., Eds.; American Petroleum Institute: Washington, DC, extant 1986. Brule, M. R.; Starling, K. E. Ind. Eng. Chem. Process Des. Dev. 1984, 23, 833. Cavett, R. H. Presented at the 27th Midyear Meeting, API Division of Refining, San Francisco, CA, May 15, 1964. Gray, J. A. Report DOE/ET/10104-7,April 1981; Department of Energy, Washington, DC. Kesler, M. G.; Lee, B. I. Hydrocarbon Process. 1976, 55(3), 153. Kesler, M. G.; Lee, B. I.; Sandler, S. I. Ind. Eng. Chem. Fundam. 1979, 18, 49. Lin, H. M.; Chao, K. C. AIChE J . 1984, 30, 981. Riazi, M. R. Ph.D. Thesis, Department of Chemical Engineering, The Pennsylvania State University, University Park, 1979. Riazi, M. R.; Daubert, T. E. Hydrocarbon Process. 1980,59(3) 115. Riazi, M. R.; Faghvi, A. Ind Eng. Chem. Process Des. Deu. 1985,24, 398. Twu, C. H. Fluid Phase Equilib. 1984, 16, 137. Winn, F. W. Pet. Refiner 1957, 36(2), 157.
Received for review April 29, 1986 Accepted January 13, 1987
The Martin Equation Applied to High-pressure Systems with Polar Components Joseph Joffe* Department of Chemical Engineering and Chemistry, New Jersey Institute Newark, New Jersey 07102
of
Technology,
The Martin equation of state may be applied to a polar substance provided that the equation parameter a is made to fit the vapor pressure data at the temperature of interest. Instead of Martin's procedure for adjusting parameters b and c by means of the value of the experimental critical compressibility factor, it is best to use with a polar compound a liquid density datum point as close to the temperature of interest as possible and to obtain values of b and c by volume translation. This procedure is illustrated with the ethylene-chloroform system. Other high-pressure systems studied are the ethane-acetone system, the hydrogen sulfide-water system, and the carbon dioxide-methanol system. Some of the results are compared with those obtained by Guo and co-workers with the cubic chain-of-rotators equation. The Martin form of the Clausius equation of state (Martin, 1979) is the simplest of the three-constant cubic equations of state. In common with other three-constant equations, such as the Schmidt-Wenzel (1980) and the Patel-Teja (19821, it is capable of representing liquid densities of pure fluids and mixtures over a greater range of fluid types than is possible for two-constant equations, such as the Redlich-Kwong-Soave (Soave, 1972) or the Peng-Robinson (1976). The Martin equation may be written
P = R T / ( V - b ) - a / ( V + c)'
(1)
The equation may also be written in reduced form PR = TR/(z,VR - B ) - A/(z,VR + C)' (2) where B = bP,/RT,, C = cP,/RT,, A = aP,/R2TT,2,and 2, is the compressibility factor at the critical point. The van der Waals conditions at the critical point yield the relations *Present address: 77 Parker Avenue, Maplewood, NJ 07040.
A = 27/64
B = 2,
-
1/4
B
+ C = 1/8
(3)
Martin proposed that the critical point of the Clausius equation be shifted so as to cause coincidence with the experimental critical isotherm at about twice the critical density. This is accomplised by setting B = 0.8572, - 0.1674 (4) where z, is the experimental critical compressibility factor. It follows from eq 3 that C = 0.2924 - 0.85'72, (5) As proposed by Martin, B and C are to be taken as constants for a given substance, while A is made into a temperature function. When the Martin equation is applied to vapor-liquid equilibria, A is best represented by a Soave-type temperature function (Joffe, 1981): A = (27/64)[1 + m(l - TR1'2)]2 (6) In previous work, the Martin equation with a Soave-type temperature function was applied by me and co-workers to the study of vapor-liquid equilibria and fugacity coef-
0888-5885/87/2626-0159$01.50/0 0 1987 American Chemical Society
760 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987
Table I. Values of Parameters in Martin Equation 10-3a1a temp, K components (1)-(2) ethane-acetone hydrogen sulfide-water hydrogen sulfide-water carbon dioxide-methanol carbon dioxide-methanol ethylene-chloroform ethylene-chloroform ethylene-chloroform a in dm6.bar/kmol.
298 311 444 298 313 323 348 373
* b in cm3/kmol.
ec
5.668 15 5.083 24 4.015 31 3.727 14 3.558 38 4.15477 3.903 60 3.664 24
10-3b1* 39.801 37 26.29403 26.29403 23.174 84 23.17484 32.51248 32.512 48 32.51248
10-3c1c 25.18800 17.106 34 17.10634 19.684 77 19.684 77 25.76874 25.768 74 25.76874
10-3az 23.98667 9.346 74 7.651 16 16.19252 15.649 11 21.531 88 20.635 16 19.75880
10-~b~ 56.465 59 13.46422 13.464 22 30.923 05 30.923 05 65.577 28 65.577 28 65.577 28
10-3c, 55.852 98 17.047 52 17.047 52 34.880 93 34.880 93 36.30892 36.30892 36.30892
in cmY/kmol.
ficients of hydrocarbon systems and a few systems containing polar components, such as n-butane-water, ammonia-propane, and ammonia-hydrogen (Joffe, 1981;Joffe et al., 1981, 1984).
Current Work The current investigation centers on high-pressure systems with polar components. In previous work with nonpolar substances, the parameter m in eq 6 has been correlated with acentric factor and temperature (Joffe et al., 1984) as m = 0.49950 1.5618~- 0 . 1 3 1 7 3 ~+~ 0.13904(TR - 0.7) + 0.64344(TR - 0.7)’ (7) with TRI1.0. This correlation is not recommended for polar substances. Instead, the parameter A is best adjusted to fit the vapor pressure datum of the polar compound at the temperature of interest, so that the vapor and liquid fugacities of the pure substance as calculated with the Martin equation become equal. Again, Martin’s procedure for calculating the constants B and C with eq 4 and 5 gives good results for liquid volumes of nonpolar substances but is of questionable validity for polar substances. Sometimes good results are obtained by calculating an “effective”critical compressibility factor from the acentric factor of the polar substance, as with the following correlation due to Hougen and co-workers (1959): z, = 1/(1.280 + 3.41) (8) The effective z, is substituted into eq 4 and 5. When this procedure is used for water and ammonia a t a reduced temperature of 0.700, the liquid volumes calculated with the Martin equation deviate from experimental values by respectively 0.82% and 1.73%. However, this same procedure, when used to calculate the liquid volume of chloroform at 293 K, yields a value of 0.067 642 m3/kmol which is too small by 0.012 539 m3/kmol or by 15.64%. By use of the principle of volume translation as described by Peneloux et al. (1982),the difference of 0.012539 is added to the value of b in the Martin equation (see eq 1)and is substracted from the constant c. The resultant values of b and c are respectively 0.065 577 and 0.036 309 m3/kmol. With these new parameters, the Martin equation yields the liquid volume of chloroform correct to five significant figures at 293 K. It is recommended that with a polar compound, a liquid volume datum point be selected as close as possible to the temperature of interest and values of the parameters b and c can be obtained by volume translation as illustrated above with chloroform. Parenthetically, it should be stated that while volume translation, as described above, can be used to greatly improve predictions of liquid volumes, it has only a minor effect on calculated fugacity coefficients and a hardly noticeable effect on calculated K constants.
+
Systems Studied The high-pressure ethylene-chloroform system was selected in the current study because the investigators (Shim
and Kohn, 1964) determined the volumes along with the compositions of the coexisting phases at several temperatures. In the case of ethylene, the parameter a was obtained from eq 6 and 7. The constants b and c were obtained with eq 4 and 5. In the case of chloroform, a polar substance, experimental vapor pressures were used at each temperature of interest to establish the value of parameter a by fugacity equalization. The parameters b and c were obtained from a liquid density datum at 293 K as described above. Throughout this study critical constants and acentric factors were obtained from Reid et al. (1977). The values of pure component parameters a, b, and c have been entered in Table I for all systems treated in the current study. The application of eq 1 to mixtures was described in a previous paper (Joffe, 1981). For the ethylene-chloroform system, experimental data at each of three different temperatures and each of six different pressures were used to start the calculations. Experimental compositions were substituted into the Martin equation, the latter was solved for the liquid and vapor volumes, the component fugacity coefficients were found for each phase, and the K constants were calculated from the ratios of fugacity coefficients of each component. The K constants were combined with material balance equations, to recalculate the liquid and vapor compositions, which differed in general from the experimental values. The new compositions were used to recalculate the volumes, fugacity coefficients, and K constants, which in turn were used to calculate a new set of compositions. This procedure was iterated until there was no further change in the calculated compositions to five decimal places. The same iteration procedure was used with all systems treated in the current study. A single binary interaction constant, k I 2= -0.050, was used to correlate the data in the ethylene-chloroform system at all three temperatures studied. The results of the calculations are presented in Table I1 as average absolute deviations of calculated from experimental compositions at each of three temperatures. The average absolute percent deviation of calculated volumes from experimental values in the ethylene-chloroform system at each of three temperatures is also shown in Table 11. It is seen that the deviation in volumes is least at the temperature closest to the liquid volume datum point used to establish the constants b and c of chloroform. Other systems investigated in the current study were ethane-acetone a t 298 K (Ohgaki et al., 1976), hydrogen sulfide-water a t 311 and 444 K (Selleck et al., 1952), and carbon dioxide-methanol at 298 and 313 K (Ohgaki and Katayama, 1976). In the ethane-acetone system, the constants of ethane were established in the same manner as for ethylene. Acetone being a polar compound, the vapor pressure and density of acetone a t 298 K were used to establish the parameters a, b, and c, as explained above for chloroform.
Ind. Eng. Chem. Res., Vol. 26, No. 4,1987 761 Table 11. Vapor-Liquid Equilibrium Calculations: Errors in Calculated Compositions and Volumes av, Guo et al. -
av, this work temp, K 298 311 444 298 313 323 348 373
no. of points 8 5 11
components (1)-(2) ethane-acetone hydrogen sulfide-water hydrogen sulfide-water carbon dioxide-methanol carbon dioxide-methanol ethylene-chloroform ethylene-chloroform ethylene-chloroform
a
8 6 6 6
pressure, bar 4.8-39.4 6.9-20.7 13.8-172 7.9-59.5 5.8-77.0 10.1-60.8 10.1-60.8 10.1-60.8
kI2 0.143 -0.016 0.070 0.040 0.060 -0.050 -0.050 -0.050
60
-
50
-
Ix, - xel
Iyc - yel
Iyc - yel
0.06622 0.00020 0.00269 0.07684 0.03'751 0.00706 0.00369 0.00394
0.00207 0.00031 0.01250 0.00099 0.00161 0.00454 0.00638 0.00868
0.01347 0.00093 0.02540 0.00348 0.00712
av abs. % dev. liq vol
0.99 1.60 3.79
1
406
P
f
30-
h 0
1
0
" .I
" .2
"
.3
' .4
" .S
~ .6
.7
.8
.9
Comporltion, x, ,y,
Figure 1. Pressure-composition diagram for ethane(l)-acetone(2) system a t 298 K: (0) data by Ohgaki et al. (1976); (-) equation.
20
-
10
-
1.0
Martin
t
The rather large average absolute deviation of the calculated from the experimental liquid compositions in this system, shown in Table 11, is affected by three datum points which fall in the critical region. The other five calculated points follow the experimental values closely, as may be seen by inspection of Figure 1. In the system hydrogen sulfide-water at 311 K, the parameters a of the two components were calculated from their vapor pressures at 311 K by the method of fugacity equalization previously described. For hydrogen sulfide, a value of m was calculated from the value of a at 311 K with eq 6, and this value of m was then used to calculate a for hydrogen sulfide with eq 6 at 444 K, a temperature at which hydrogen sulfide is supercritical. The parameter a of water was calculated from its vapor pressure at 444 K as before. The constants b and c of water were calcualted from its acentric factor with eq 8,4,and 5,those of hydrogen sulfide from the experimental z, and eq 4 and 5 as in a previous study (Joffe et al., 1981). In this system, the binary interaction constant varies with temperature, resulting in values K I 2 = -0.016 at 311 K and k12 = 0.070 a t 444 K. The results as summarized in Table I1 show remarkably low deviations at 311 K and reasonably low deviations at 444 K. The fourth binary system investigated in the current study is the carbon dioxide-methanol system at 298 and 313 K. The parameters a, b, and c of carbon dioxide were established as described above for ethylene. For the polar substance, methanol, the parameters b and c were established from a density datum by volume translation, while the parameter a was obtained from vapor pressure data a t 298 and 313 K by fugacity equalization. The binary interaction constant for this system was found to vary with temperature, with assumed values of 0.040 at 298 K and
-0
.I
,2
.3
.4
.5
.6
.?
.8
.9
1.0
Conpodtion, x, .Y,
Figure 2. Pressure-composition diagram for carbon dioxide(1)methanol(2) system at 298 K: (0)data by Ohgaki and Katayama (1976); (-) Martin equation.
0.060 at 313 K. The results of vapor-liquid equilibrium calculations for this system are shown in Table 11. The rather large average absolute deviation in liquid compositions at 298 K can be attributed to the fact that four of the experimental points fall in the critical region of this system. The other four points show close agreement between calculated and experimental values. This may be seen by inspection of Figure 2. At 313 K only two of the eight experimental points fall into the critical region, resulting in a considerably smaller average absolute deviation in liquid compositions than at 298 K.
Comparison with Other Work The last three systems covered in this study have also been investigated by Guo et al. (1985b) using the cubic chain-of-rotators equation. Whereas in the current investigation experimentalvalues of system temperature and pressure are assumed as known and the Martin equation is solved for liquid and vapor compositions (Flash calculation), Guo and co-workers assume a known system temperature and liquid composition and solve for system pressure and vapor composition (bubble point calculation). While the results of the two investigations are not strictly comparable, it is nevertheless instructive to compare the results obtained for the vapor compositions in the two studies as shown in Table 11. In all cases the average absolute error in the vapor composition is considerably
762 Ind. Eng. Chem. Res., Vol. 26, No. 4, 1987
larger by the method of Guo and co-workers than in the current work. It should also be pointed out that in cases under consideration, Guo and co-workers used two binary interaction constants in the chain-of-rotators equation as compared with a singly binary interaction constant employed with the Martin equation. In the two systems studied at two different temperatures, hydrogen sulfidewater and carbon dioxide-methanol, the binary interaction constants were found to change with temperature both in the work of Guo and co-workers and in the current study. What may account for the better results obtained in the present study as compared with those of Guo and coworkers? Perhaps it is the fact that the parameter a of a polar component was established in the Martin equation from the experimental vapor pressure at each temperature of interest, whereas in the chain-of-rotators equation the two constants A , and A2 are established from the best fit of the component vapor pressure over a range of temperatures (Guo et al., 1985a). Nomenclature a , b , c = parameters in the Martin equation A , B, C = parameters in the reduced Martin equation
k = binary interaction constant K = vapor-liquid equilibrium ratio m = characteristic parameter in Soave-type temperature function, eq 6 P = absolute pressure R = ideal gas constant T = absolute temperature V = volume x , = calculated mole fraction in liquid phase x , = experimental mole fraction in liquid phase yc = calculated mole fraction in vapor phase ye = experimental mole fraction in vapor phase t = compressibility factor
Greek Symbol = Pitzer acentric factor
w
Subscripts c = critical state
R = reduced 1, 2 = component 1, 2
Literature Cited Guo, T. M.; Kim, H.; Lin, H. M.; Chao, K. C. Ind. Eng. Chem. Process Des. Dev. 1985a,24,764. Guo, T. M.; Lin, H. M.; Chao, K. C. Ind. Eng. Chem. Process Des. Deu. 1985b,24,768. Hougen, 0.A.; Watson, K. M.; Ragatz, R. A. Chemical Process Principles, 2nd ed.; Wiley: New York, 1959;Part 11, p 572. Joffe, J. Ind. Eng. Chem. Process Des. Dev. 1981,20, 168. Joffe, J.; Joseph, H.; Tassios, D. Second World Congress on Chemical Engineering, Montreal, Canada, Oct 4-9, 1981. Joffe, J.; Joseph, H.; Tassios, D. AIChE Meeting, Atlanta, March 11-14,1984. Martin, J. J. Ind. Eng. Chem. Fundam. 1979, 18, 81. Ohgaki, K.; Katayama, T. J. Chem. Eng. Data 1976,21,53. Ohgaki, K.;Sano, F.; Katayama, T. J. Chem. Eng. Data 1976,21,55. Patel, N. C.; Teja, A. S. Chem. Eng. Sci. 1982,37,463. Peneloux, A.;Rauzy, E.; Freze, R. Fluid Phase Equilib. 1982,8,7. Peng, D.Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976,15,59. Reid, R. C.; Prausnitz, J. M.; Sherwood, T. K. The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. Schmidt, G.; Wenzel, H. Chem. Eng. Sci. 1980,35,1503. Selleck, F.T.; Carmichael, L. T.; Sage, B. H. Ind. Eng. Chem. 1952, 44,2219. Shim, J.; Kohn, J. P. J . Chem. Eng. Data 1964,9,1. Soave, G. Chem. Eng. Sci. 1972,27, 1197. Received f o r review January 31, 1986 Accepted September 29, 1986
