Ind. Eng. Chem. Process Des. Dev. 1985, 24, 764-767
764
Glacalone, A. Gazz. Chlm. Ital. 1851, 87, 180. Katlnas. T. G.; Danner, R. P. Hy&ocarbon Rocesslng 1877, 56 (3), 157. Lee. B . 4 ; Erbar, J. H.;Edmister, W. C. AIChE J . 1873, 79, 349. Lee, B.4.; Kesier, M. G. AIChEJ. 1875, 27, 510. Lydwsen, A. L.; Greenkorn, R. A,; Hougen, 0. A. "Generalized Thermodynamic Properties of Pure Fiulds", University of Wisconsin, College of Eng. Exp. Sta: Rep. 4; Madison, WI. Oct 1955. Pltzer, K. S.;Uppman, D. 2.; Curl, R. F.; Hugglns, C. M.; Petersen, D. E. J . Am. Chem. SOC. 1855, 77, 3433. Riedei, L. Chem. Ing. Tech. 1854a, 26, 83. Riedei, L. Chem. Ing. Tech. w1954b, 26, 679.
S. R. S.;" a n a Rao, M. V.; Reddy, K. A.; Doraiswamy, L. K. Br. Chem. €ng. 1888, 74 (7), 959. Spencer, C. F.; Danner. R. P. J . Chem. Eng. Data 1872, 77, 236. Vetere, A. "Modification of the Kistkkowsky Equation for the Calculation of Enthalpies of Vaporization of Pure Compounds", Lavoritori Ricerche Chlmica Industriale, SNAMPROGETTI: San Donato, Milanese, 1973. Watson, K. M. Ind. Eng. Chem. 1831, 23, 360. Sastri.
Received for review June 15, 1984 Accepted October 19, 1984
Cubic Chain-of-Rotators Equation of State. 2. Polar Substances Van-Mln Guo, Hwayong Klm, Ho-Mu Lln, and Kwang-Chu Chao' School of Chemical Engineering, Purdue Universlv, West Lafayette, Indiana 47907
The cubic chainsf-rotators equation of state is a new equation whose application to nonpolar molecular fluids and their mixtures has been reported. In thii work the equation is applied to polar substances. pvTand vapor pressure of water and ammonia are quantltatiiely represented by the equatlon for gas and liquid states when four equation constants are adjusted from their generalized values and are assigned specificvalues. Vapor pressures of 45 polar fluids are likewise represented by adjusting two equation constants. The calculated vapor pressures agree with data generally to within 0.5% in the entire temperature range of the data.
Introduction Equations of state are widely used for the calculation of thermodynamic properties of nonpolar substances and their mixtures, but hardly for polar substances because of deterioration of the equation when applied to the latter class of substances. Nevertheless an equation of state, if indeed applicable to polar substances, has just as much to recommend itself for thermodynamic and fluid phase equilibrium calculations as for nonpolar substances. Thus, starting in recent years much effort has been directed at extending equation of state to polar Substances. Won and Walker (1979) added a polar pressure term to the Soave equation. Bazua (1983) extended the Redlich-Kwong equation to a three-parameter form. Soave (1979) introduced a new temperature function for the parameter a. Yokoyama et al. (1983) added a polar pressure term to their nonpolar equation. In this work we extend the cubic chain-of-rotators equation to polar substances by adjusting several equation parameters from their generalized values.
polyatomic molecules; the rotational freedom has been correlated with Pitzer's acentric factor w);
+
cR = 24.863~- 33.3680~ 5 7 . 2 6 6 ~for ~ w50 cR = 0 for w < 0
(3) expressing the attractive pressure with the last two terms of eq 1. The derivation of the CCOR equation and application to nonpolar substances and mixtures have been reported by Kim et al. (1985). Equation 1is a cubic equation in u which can be solved algebraically at given T and p. The parameters a, b, c, and d of the equation are given in eq 3-6, where they are first scaled with the critical properties.
The Cubic Chain-of-Rotators Equation The cubic chain-of-rotators (CCOR) equation of state is P=
R T ( 1 + 0.77b/u) u - 0.42b
R T b/u + CR 0.055 u - 0.42b
a U(U
bd
U(U
+ C)(U
-
+ C)
0.42b) (1)
The equation was obtained by (1)replacing the van der Waals ultrasimplified expression R T / ( u - b) of the repulsive pressure with the first term on the right-hand side of eq 1which simulates the Carnahan-Starling equation for hard sphere fluids; (2) adding the second term on the right-hand side of eq 1to express the rotational pressure of polyatomic molecules (this term vanishes for monatomic spherical molecules for which the rotational freedom cR is equal to zero; it is greater than zero for nonspherical 0196-4305/85/1124-0764$01.50/0
(2)
a = aR,R2T,2/p,
(3)
d = odR2T:/p,
(6)
The hard-core volume of a molecule, b/4, has been correlated with z, through an expression for Rb as follows: ab
= 0.4756 - 3.3962,
+ 8.2362:
(7)
The constants R,, R,, and f i d are determined by the critical-state conditions u = u,, (ap/au)T = 0, and (d2p/du2)T = 0. Thus, with CY = y = 1 at T,,we have
+ 1.0-3.02, R, = 0.42QbRc+ R, + (0.77 + 0.055cR)Rb+3.022 a d = [(0.77 + 0.055cR)f&Qc + 0.4251,Rb - 2,3]/fib R, = 0.4251b
0 1985 American Chemical Society
(8) (9)
(70)
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 765
Table I. A and A 2 Values and Deviations of Calculated Vapor Pressure of Some Polar Substances no. 1
2 3 4 5
6 7 8 9 10 11 12 13 14 15
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45
substance acetone diethyl ketone methyl ethyl ketone methyl propyl ketone methanol ethanol I-propanol l-butanol 2-methyl-1-propanol 1-pentanol I-hexanol 1-octanol phenol dimethyl ether diethyl ether methyl acetate ethyl acetate n-propyl acetate n-butyl acetate ethyl butyrate ethyl formate acetonitrile acetic acid propionic acid butyric acid ethylene oxide dimethylamine diethylamine trimethylamine triethylamine nitromethane aniline chloroform chlorobenzene ethyl chloride dichloromethane 1,2-dichloroethane 1,1,2-trichloroethane carbon tetrachloride acetaldehyde carbon disulfide hydrogen sulfide ammonia carbon dioxide water
no. of data pointa 47 18 17 14 44 48 19 26 7
35 13 19 6 12 27 15
16 14 15
9 13 9 20 7 8 14 13 13 19 16 15 11 15
23 12 14 26 5 11 10 17 13 16 20 40
temp range, K 259-5OW 330-385 316-362 268-542 288-5121 293-514 333-378 362-563' 493-54w 348-514 325-430 386-480 380-426 202-248 212-46W 275-329 289-349 268-542 332-400 277-394 277-327 246-355 303-400 330-413 364-436 224-285 201-280 240-483 193-277 323-368 329-410 331-528 263-334 335-405 217-286 233-313 248-373 323-387 253-452 273-308 219-529 283-373' 223-323 280-304 273-6471
vap press range, bar 0.04-46.7 0.20-1.33 0.27-1.33 0.01-30.7 0.09-81.0 0.05-61.3 0.20-1.33 0.20-44.2 17.3-42.6 0.07-12.7 0.01-1.01 0.05-1.33 0.08-1.01 0.05-1.01 0.01-36.0 0.09-0.98 0.08498 0.006-30.7 0.09-1.01 0.006-1.01 0.12-1.01 0.006-1.01 0.03-1.33 0.03-1.01 0.06-1.01 0.04-1.07 0.005-1.01 0.01-30.7 0.01-1.04 0.26-1.20 0.20-2.67 0.006-5.07 0.04-0.98 0.09-1.01 0.03-1.02 0.02-1.01 0.004-1.60 0.10-1.01 0.01-10.1 0.44-1.73 0.006-61.3 1.87-12.0 0.41-20.0 41.3-73.8 0.006-220.5
AI
3.921795 4.192108 4.059943 4.389497 4.529835 5.050 658 5.513423 5.684150 4.600311 5.703638 5.556483 5.456386 4.527924 3.638051 3.964629 4.140553 4.302627 4.389497 4.454741 4.000587 4.143337 3.582903 4.015495 5.057775 4.781743 3.642924 4.344359 4.018633 2.708875 4.243223 3.804343 4.359124 3.722032 3.845921 3.631123 3.596007 3.728395 4.181254 3.720429 3.588346 3.218004 2.910896 3.753419 5.916994 3.874242
A2 0.238791 0.267024 0.254417 0.277443 0.229231 0.296685 0.384749 0.426825 0.275333 0.428476 0.378965 0.369277 0.284342 0.248300 0.260113 0.271540 0.280482 0.277443 0.279730 0.172463 0.296815 0.166632 0.185316 0.347024 0.230588 0.247771 0.328445 0.257621 0.262372 0.292443 0.197368 0.282426 0.253524 0.258317 0.252491 0.235908 0.212726 0.335085 0.265234 0.177248 0.223572 0.176604 0.238839 0.537704 0.210442
AAD, % 0.21 0.01 0.01 0.90 1.18 1.29 0.17 0.40 0.19 0.60 0.54 0.69 0.20 0.06 0.38
data source
f a a
b d,e d,e
d a,c C
d,e a d
a a
a,c
0.05 0.05 0.90 0.12 0.30 0.06 0.59 0.14 1.54 0.28 0.07 0.12 0.53
a a
0.10
a a a
b a
b a
b a a a a a
b
0.54 0.05 2.83 0.13 0.06 0.10 0.66 1.33 0.03 1.25 0.59 0.84 0.26 0.08 0.20 0.69
b a a a a a a
b a
b h &?
h
i
"Boublik, Fried, and Hala (1973). *Perry and Chilton (1973). CKayand Donham (1955). dAmbrose and Sprake (1970). eAmbrose, Sprake, and Townsend (1975). fAmbrose, Sprake and Townsend (1974). gvan Wylen and Sonntag (1978). "Sage and Lacey (1955). 'Keenan et al. (1969). jCritical temperature. 1 o5
The critical compressibility factor z, is correlated with the acentric factor W: Z , = 0.291 - 0.080 (11) a and y of eq 3 and 5 y
C r i t i c a l Point
are functions of temperature:
= exp[Cl(l - T,c')]
(12)
At T, I 1 a = [1.0
At T,
+ A1(l/T>25- 1) - A2 (1 - T,2)I2
(13)
b
>1 cy
a
= exp[A,(T, - 1) - A 4 ( f i r - l ) ]
(14)
10 1.o
Triple Point-
where A3 = 0.38349 + 4.88359~+ 14.21130~
A4 = 1.17543 + 14.3109~+ 30.5032~'
(15)
Id' ._
0.001
0.002
0.003
0.004
1 I T , K-'
(16)
Equations 2 through 16 express the CCOR equation parameters in terms of T,.,pe, and W. The parameter calculation is, however, incomplete inasmuch as Al and A2 of eq 13 and Cland C2of eq 12 are still missing. These
data from Keenan et al. Figure 1. Vapor pressure of water: (0) (1969);(-) CCOR equation with AI = 2.732190,A2 = 0.155565,C1 = 4.925244,and C2 = 0.247926.
constants will be determined to fit experimental puT and vapor pressure data as described below.
766
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 6
/
- CCOR o
D o l a from Keenon(1969)
sr
5
U
w E
e
+
2
4
3
I
5
4
A 1 of Table I 10
0 01
01
10
10
1000
100
V , m 3 / kmol
Figure 2. pu diagram of water.
Figure 5. Correlation of A,: (a) acetone; (b) methanol; (c) ethanol; (d) phenol; (e) methyl acetate, (0ethyl acetate; (9) n-propyl acetatq (h) n-butyl acetate; (i) dimethyl ether; 6)diethyl ether; (k) diethyl ketone; (1) methyl ethyl ketone; (m) methyl propyl ketone; (n) diethylamine; ( 0 ) trimethylamine; (p) triethylamine; (9)nitromethane; (r) aniline; (E) chlorobenzene; (t) dichloromethane, (u) carbon tetrachloride; (v) carbon disulfide; (w) ammonia; (x) water; (y) ethyl chloride.
'"4'I
t 102
' 10
10 I 0 002
I 0.003
0 004 I / T , K1
0 005
Figure 3. Vapor pressure of ammonia: (0) data from Reynolds (1978); (-1 CCOR equation with AI = 2.958009, A2 = 0.180720, C1 = 5.310287, and C2= 0.258373. A
A 2 of Table
I
Figure 6. Correlation of A2 (the legend of the symbols is the same as given in Figure 5).
V . m3/kmol
Figure 4. p u diagram of ammonia.
p v T and Vapor Pressure of Water and Ammonia
Water and ammonia are highly polar substances. To describe their pvT and vapor pressure we adjust Al, A2 of
eq 13 and C1,C2of eq 12. Figure 1 shows the calculated vapor pressure of water in comparison with tabulated data (Keenan et al., 1969). With the valuea of the four constants shown in the figure caption, the calculated vapor pressure agrees with the table value to 0.68% on the absolute average in the vapor pressure range from 10 to 22 OOO kPa, with a maximum deviation of 2.2%. Figure 2 shows the pu isotherms and saturated envelope of water. The calculated values generally agree well with the tabulated values with larger than usual deviations occurring at supercritical pressures and temperatures. Similar results for ammonia are shown in Figures 3 and 4. The vapor pressure calculation spans more than 3 orders of magnitude starting from the critical temperature, giving an absolute average deviation of 0.43% and a maximum of 0.85%. Vapor Pressure of 45 Polar Substances For polar substances PUTdata are generally not available, but vapor pressure data'are available, and these are
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 3, 1985 767
ranges are the same as in Table I. The correlated Al and A2 are useful if vapor pressure data are not found. Calculated mixture fluid phase equilibria based on the correlated AI and A2 have been found to be in reasonable agreement with data, as will be discussed in the next part of this series.
Table 11. Vapor Pressure Deviation Calculated from Correlated A and A no. substance AAD, % 1 acetone 0.12 2 diethyl ketone 0.36 3 methyl ethyl ketone 0.63 4 methyl propyl ketone 1.80 5 methanol 5.34 6 ethanol 4.36 7 1-propanol 2.49 8 1-butanol 2.28 9 phenol 2.42 10 dimethyl ether 3.90 11 diethyl ether 0.75 12 methyl acetate 0.75 13 ethyl acetate 0.48 14 n-butyl acetate 0.64 15 ethylene oxide 5.36 16 trimethylamine 1.23 17 carbon tetrachloride 2.23 18 chloroform 1.05 19 ammonia 1.31 20 water 0.7P
Acknowledgment Financial support for this work was provided by Electric Power Research Institute through Research Project RP367 and by National Science Foundation through Grant CPE-8209624.
Nomenclature
Vapor pressure range 10-22 000 kPa.
the most important to have represented by an equation of state in order for the mixture phase equilibrium to be represented by the equation. We have determined values of Al and A’ of eq 13 for the best fitting of the vapor pressure of 45 polar substances while using our previously reported general formulas for C1and C2 (Kim et al., 1985).
C1 = 7.04333 - 5.004220 + 1.885970~
(17)
+ 0.16896~+ 0.031790~
(18)
Cz = 0.23177
Table I presents values of Al and Az that we have determined and the average absolute percent deviation of the calculated vapor pressure in the indicated temperature range and pressure range. The calculated vapor pressures agree with data generally to within 0.5% in the entire temperature range of the data.
Correlation of A and A The values of AI and A2 obtained from vapor pressure fitting have been correlated with the equations A1 = 2.66709 + 5.51328~- 2 . 6 5 3 3 3 ~-~0.01148(p*)2 (19)
A , = 0.18471
+ 0.38357~- 0.32706~’- 0.002078(p*)2 (20)
where
p*
is the reduced dipole moment given by ( P * ) ~= p2/[9.8694 X
10-6(RTc)2/p,]
(21)
The units of the dimensioned quantities are given under Nomenclature. For a nonpolar substance p* = 0, and eq 19 and 20 reduce to their previously reported forms for nonpolar substances (Kim et al., 1985). Figure 5 shows comparison of Al of Table I and eq 19. Similarly Figure 6 shows comparison of Az of Table I and eq 20. Vapor pressures calculated with the correlating equations are in reasonable agreement with data, though deviations are, in general, several times as large as in Table I. Some results are presented in Table I1 where the data
A = constant in eq 13 and 14 a = parameter in eq 1 b = four times the hard core volume, m3/kmol C = constant in eq 12 c = parameter in eq 1 cR = number of degrees of rotational freedom of a molecule d = parameter in eq 1 p = pressure, kPa R = universal gas constant, 8.31434 kPa m3/kmol K T = temperature, K u = volume, m3/kmol z = compressibility factor, pu/(Rn
Greek Letters a = factor in eq 3 y = factor in eq 5 M = dipole moment, debye w* = reduced dipole moment 52 = factor in eq 3, 4, 5, 6 w = acentric factor Subscripts a, b, c, d = for a, b, c, d, respectively c = critcal-state property r = reduced property
Literature Cited Ambrow, D.; Sprake, C. H. S. J . Chem. Thermo@n. 1970, 2 , 631. Ambrose, D.; Sprake, C. H. S.; Townsend, R. J . Chem. Thermodyn. 1974, 6 , 893. Ambrose, 0.; Sprake, C. H. S.; Townsend, R. J . Chem. Thermodyn. 1975, 7 , 185. Bazua. E. R. In ”Chemlcal Englneerlng Thermodynamks”; Newman, S. A., Ed.; Ann Arbor Sclence: Mlchlgan, 1983; Chapter 17. Boubllk, T.; Frled, V.; Hala, E. “The Vapor Pressure of Pure Substances”; E l s e v k Amsterdam, 1973. Kay, W. B.; Donham, W. E. Chem. Eng. Sci. 1955, 4 , 1. Keenan. J. H.; Keyes, F. 0.;HIII, P. 0.;Moore, J. G. “Steam Tables”; Wlley: New York, 1969. Kim. H. Y.; Lln. H. M.: Chao. K. C., submitted for Dubllcatlon in Ind. €ne.
Chem . Fundam.
Perry, R. H.; Chilton, C. H. “Chemlcal Engineers’ Handbook”, 5th ed.; McGraw-HIII: New York, 1973. Reynolds, W. C. “Thermodynamlc Properties In SI”;Department of Mechanlcal Engineering: Stanford University, 1978. Sage, 6. H.; Lacey, W. N. ”Monograph on API Research Project 37 - Some Propertles of the Llghter Hydrocarbons. Hydrogen Sulflde, and Carbon DIoxlde”; American Petroleum Institute: New York. 1955. Soave, 0. S. Int. Chem. Eng. Symp. Ser. ,!979, No. 56, 1.211. Van Wylen, G . S.; Sonntag, R. E. Fundamentals of Classical Thermodynamics”, 2nd ed.; Wlley: New York, 1978. Won, K. W.; Walker, C. K. Adv. Chem. Ser. 1978: 182. 235. Yokoyama, C.; Aral, K.; Salto. S. I n Chemlcal Engineering Thermodynamics”, Newman, S. A., Ed.; Ann Arbor Science: Mlchlgan. 1983; Chapter 23.
Receiued for review December 8, 1983 Accepted August 2, 1984
