Fugacity Coefficients and Isothermal Enthalpy Differences for Pure Hydrocarbon Liquids Byung-lk l e e and Wayne C. Edmister Oklahoma State rniversity, Stillwater, Oklahoma 74074
A new empirical equation has been developed for predicting the fugacity coefficients for pure hydrocarbons in the liquid phase. The parameters of this correlation are reduced pressure, reduced temperature, and acentric factor. There are 13 empirical constants and the equation is convenient for computer solution. Graphical comparisons show that this new equation agrees with values of f/P derived from P-V-T data. In addition to predicting reference state fugacities for activity coefficients, this equation can be used in the calculation of enthalpies of pure components at saturated liquid and compressed liquid states. Tabular comparisons show that the proposed new equation agrees with experimental values (Ho - H L ) and also values calculated from P-V-T data by others.
T h e fugacit'y of a pure component in the liquid phase is a frequently used reference in the activity coefficient expressing departure from ideal solutions. Pure component' fugacit'y coefficients, Le., Y = f / P , have been calculated from pressurevolume-temperature data and generalized by Lydersen, et al. (1955), and by Curl and Pitzer (1958). Lydersen, et al., used the critical compressibilit'y factor, Z,, as a third parameter and covered reduced temperatures down t o 0.5. Curl and Pitzer used the acentric factor, w , as a third parameter and lower limit was T , = 0.8. Chao and Seader (1961) gave equations for Y a t values of T , > 0.5. Xore recently, Chao, et al. (1969), computed Y values for the temperature range 0.35 7 T , 7 0.75 and presented two generalized correlations, w being the third parameter of one and 2, the third parameter of the other.
METHANE
76 7
w=oo13
A\
-:
-;
Objectives
The objective of this work is to present a generalized empirical equation, using w as a t'hird parameter, for the fugacity coefficient of pure hydrocarbons as saturated or compressed liquids. I n the development of this equation the performance requirements mere: (1) agreement with Y values calculated from P-V-T data and (2) accurate values of the isothermal enthalpy differences obtained from temperature derivatives of In Y. This fugacity coefficient equation is limited t o "real" liquid and is not intended for "hypothetical" liquids, thus fixing T , = 1.0 as the upper temperature limit'. The lower temperature limit was set at T , = 0.4 below which sufficiently accurate input d a t a could not be found. Also T , = 0.4 seems to represent a satisfactory lower limit for a general purpose correlation, excluding cryogenic conditions. An analytical equation form for this new generalized Y correlation is a very important objective of t'his work as the correlation is to be used in a computer algorithm for the prediction of vapor-liquid K values. The Chao-Seader (1961) correlation was in equation form but this expression did not represent the actual values of the fugacity coefficient but was derived by fitting a n empirical equation to values of f / P obtained by back-calculations from experimental y/x values, using available expressions for the fugacity coefficients of components in the vapor phase mixtures and the activity coefficients of the liquid phase components. The fugacity
2
LEGEND T H I S WORK, EPN.1 I C 8 S EQN A CHAC et a1 DERIVED VALUES C 8 P TABULATIONS
---
REDUCED
PRESSURE
Figure 1 . Fugacity coefficients from four sources at T, = 0.6 to 1 .O for pure liquid methane Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
229
Table
I. Comparison of liquid Fugacity Coefficients for Propane at l o w Temperature via Calculations from Data and Correlations Calculated Values of Based on d a t a
P,
TT
0.5 1.0 3.0 5.0
0.4
Y
via correlations Chao, e l a / . (1969) This work
la
IP
0.109 x 10-3 0.610 x 10-4 0.320 X 0.303 x 10-4
0.109 x 10-3 0.610 X 0.319 x 0.300 x lo-'
0.105 x 10-3 0.540 x 0.299 X 0.254 x
0 . io8 x 10-3 0.607 X 0.318 x 10-4 0.300 x
0.5 0.380 x 0.381 x 0.381 x lo-? 0.378 x 1.0 0.209 X 0.210 x 10-2 0.210 x 10-2 0.208 X 10-2 3.0 0.102 x 10-2 0.102 x 10-2 0.106 X 0.102 x 10-2 0.895 x 5.0 0.901 x 10-3 0.946 X 0.896 X 10.0 0.118 x 0.113 x IO-? 0.125 x lo-? 0.115 x 10-2 a Method I: Equation 1 with (f/P)8 from Martin (1963) second virial coefficient equation; using liquid volumes from API 44 with an incompressibility assumption in the Poynting effect; and using experimental vapor pressure data. RIethod 11:Equations 1 and 4 with (f/P).from Pitzer and Curl (1957) second virial coefficient, and using experimental vapor pressure data. 0.5
j / P for saturated liquid with the vapor pressure and the Poynting effect as nP
1
Since fugacities are identical for coexisting equilibrium vapor and liquid states of pure components, the values of v S for REDUCED
saturated liquid can be found via f / P computations for the saturated vapor. Values of Y' and p s for the light hydrocarbons (methane through n-pentane) were found from the tabulations of Canjar and RIanning (1967) for the T , range 0.6 to 1.0. The Canjar and Manning tabulations do not include any values of p s , Vs, and Y' for subatmospheric conditions. The vapor REDUCED
PRESSURE
PRESSURE
01
4
6 7 8 9 "
n -DECANE
n-PENTANE
I
5
w :0.467
5 4
:1 ooo;,
LEGEND THIS WORK, EQN. I I C 8, S , EON , , ,,,,
,
,
, ,
CHAO e t a1 DERIVED VALUES C 8 P TABULATIONS
A
2
3
4
5 6 7 8 9
REDUCED
2
3
4
5
,;_I 6
7n9,00001
Figure 2. Fugacity coefficients from four sources at T, = 0.6 to 1.O for pure liquid n-pentane
230
01
PRESSURE
Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
REDUCED
Figure 3.
IO PRESSURE
IO
Fugacity coefficients from four sources at T, =
0.6 to 1.O for pure liquid n-decane
REDUCED REDUCED P R E S S U R E
o.l 0 0Ohl
2
3
4
5
2
6 ? 8 9
3
4
5
0.1
,,
PRESSURE IO
IO
10 , 0001
0.I
6 7 8 9 ~
8
Tr = 0.5 3 4
a
e P 0.001
5
v)
W
a >
3
I 2
Z
E
--A
o'ooooook,l
1
N D , , , , , ,, , , , THIS WORK, EQN. I I C B S EON CHAO e t a1 D E R I V E D V A L U E S T H I S WORK; DERIVED VALUES
2
3
L
5
6 7 8 9
REDUCED
1.0
2
3
4
,
1:
, ,
2
3 6 7 8 9
0.000000I
where Z,(1
- 0 . 8 9 ~ ~exp(6.9547 '~) - 76.2853Tr + 191.306Tr2- 203.5472TTa+ 82.7631TT4) (3)
Integrating Equation 2 with respect t o pressure between the vapor pressure and system pressure gives
, , , , ,,
,
,
,
,
,
, ,
--2
3
4
5 6 7 8 9
01
2
3
4
REDUCED
,;I
3 6 7 a 9
IO
PRESSURE
pressure and liquid volume data in the T, = 0.4 t o 0.6 range were obtained from other sources (hmericaii Petroleum Institute, 1963; Barkelew, et al., 1947; Messerly and Kennedy, 1940; Stearns and George, 1943; Tickner and Lossing, 1951). The u s values in this low-temperature range were from an equation of state (Lee and Edmister, 1971). The latter is justified by the fact that (a) the variation of u s is between 0.97 and 1.0 and (b) the equation of state has a satisfactory performance at the low-temperature vapor pressure. I n evaluating the Poynting effect, L e . , the last term in Equation 1, we used the analytical expression of Chueh and Prausnitz (1969) for the liquid molal volume of paraffinic hydrocarbons as a function of the values of the saturated liquid molal volume and vapor pressure and the values of T,, P,, and w . The frequently made assumption that VL is independent of pressure is not satisfactory a t temperatures above T, = 0.7. The expression is
=
:,
'.
LEGEND THIS WORK, EON. I I C 8. S EON CHAO et a1 DERIVED VALUES THIS W O R K , DERIVED VALUES
0.00001
IO
Figure 4. Fugacity coefficients from four sources at T, = 0.4 for propane and n-decane as pure liquids
P
,
J
00000~
IO
PRESSURE
Figure 5. Fugacity coefficients from four sources at T , = 0.5 for propane and n-decane as pure liquids
& l:
VLdP
P,VS
=
~
8RTp
([l
+ 9p(P, -
p r S ) ] 8 ' 9-
1)
(4)
A combination of Equations 1 and 4 gives the value of the liquid fugacity coefficient for hydrocarbons in the compressed liquid state as well as in the saturated liquid state. This combination of Equations 1 and 4 served a dual purpose in this work, namely: (1) provided values of u for empirical curve fitting, and (2) indicated the functional form of the mathematical model for the empirical equation. Mathematical Model
Referring to Equations 1, 2, and 4, it is noted that v ' , p s , and V s are functions of temperature and independent of pressure. It follows therefore that pressure effects on the liquid fugacity appear in terms of the Poynting effect expression. The form of the desired empirical equation is found by expanding Equation 4 into an infinite series, truncating it after the second term, and then combining with Equation 1. Ln P , is then subtracted from each side to make both sides dimensionless and the expression is rearranged to give
f = F1 In PC
+ F2Pr+ F3PT2
(5)
where F~
=
In
us
+ In prs - RT
Ind. Eng. Chem. Fundam., Vol. 10, NO. 2, 1971
(6) 231
Table II. Comparison of Four Methods for Calculating Isothermal Enthalpy Differences for Methane, Ethane, and Propane in Saturated liquid Phase with Canjar and Manning (1967) Values. Obsda Temp, OF
- 240
- 200 - 180 - 150 - 130
S&Tb
Y&Ac
E-P-Ed
Methane -1.931 -3.424 -3.178 -1.990 -1.382 2.146
7.558 7.329 7.577 7.785 5.409 4.668
9.203 7.556 7.070 7.271 6.419 7.330
-1.588 -1.239 -1.744 -3.242 -8.193 -7.431
2.342
6.721
7.475
3.906
Ethane 0.158 -0.844 -0.980 -0.927
1.642 2.922 3.310 3.301
- 12.855
1.791 0.455 0.339 1.201
Btu/lb
32.4 64.5 115.7 191.5 364.0 527.0
-220
Deviation of calcd values, Btu/lb
( H o - HI,
Pressure, psia
215.6 208.9 200.7 191.1 174.8 156.7
This work
Abs av dev (6 pts) Btu/lb
- 100 -60
31.6 77.4 113.4 220.8
-40
0
205.7 195.2 189.5 176.8
388.1 499.6 632.7 Abs av dev (12 pts) Btu/lb 40 60 80
-20
161 . O 149.4 133.1
25.4 38.4 78.6 143.8
0 40 80
181.1 176.6 167.3 157.0
-0.573 1.524 6.638 1.460
1.527 0.601 -1.817 2.083
Propane -1.332 -1.254 -0.871 -0.036
120 242.2 145.9 133.0 160 383.5 119.6 190 525.1 Abs av dev (12 pts) Btu/lb a Canjar and Manning (1967). * Stevens and Thodos (1963).
0.328 0.259 1.729
1.311 2.412 4.102 5.298 4.904 1.722 -4.496
-11.737 - 10.931 -8.940 -6.472 -3.974 -2.120 7.967
4.286 8.956 17.806 4.759
-8.986 -6.176 -1.603 -1.940
-1.967 -1.968 -1.316 0.466
3.569 2.403 -2.131
2.730 5.986 11.312
0.751 3.381 3.287 3.475 Yen and Alexander (1965); Yen (1968). d Erbar, et al. (1964). ~~
Table 111. Comparison of Four Methods for Calculating Isothermal Enthalpy Differences for n-Butane and lsobutane in Saturated liquid Phase with Canjar and Manning (1967) Values. Obsda TyP,
F
Pressure, psia
(H'
- H)
Btu/lb
This work
'Deviotion of calcd values, Btu/lb S&Tb Y&A'
E-P-Ed
n-Butane 40 60 100 140
17.6 26.0 51.4 92.9
167.0 164.2 158.0 150.9
0.269 -0.678 -2.020 -2.765
-3.825 -3.648 -3.209 -2.767
180 154.7 240 361.9 280 437.3 Abs av dev (12 pts) Btu/lb
143.1 127.9 109.3
-3.477 -3.372 2.688 2.170
-2.917 -3.997 -3.182 3.279
20 60 100
17.8 38.0 72.0
156.8 148.6 141.0
Isobutane 0.732 1.574 1.632
140 180 220
125.2 203.0 312.6
133.1 123.5 111.8
1.440 1.910 2.817
-16.292 -14.590 -11.468 -8.834
-2.019 -3.034 -4.048 -3.858
-7.063 -5.321 -2.399 8.431
-2.985 1.029 11.087 3.588
-11.238 -7.910 -5.923
-22.490
-1.161 -0.267 0.440
-4.695 -3.425 -3.099
- 14.353
-19.240 - 16.742 -11.015 -7.228
1.530 4.048 8.010
260 461.4 95.5 -4.110 15.503 5.867 -6.299 Abs av dev (14 pts) Btu/lb 3.678 2.046 14.328 6.275 Canjar and Manning (1967). Stevens and Thodos (1963). c Yen and Alexander (1965); Yen (1968). Erbar, et al. (1964).
232
Ind. Eng. Chem. Fundam., Vol. 10,
No. 2, 1971
Table IV. Comparison of Four Methods for Calculating Isothermal Enthalpy Differences for Methane in liquid Phase with Values of Jones, et a/.. Pressure, psia
- 200 - 180 - 160 - 140
- 120 Abs av
115 190 298 443 631 dev (10 pts) Btu/lb
- 260
1000 1500 2000 -220 1000 1500 2000 - 180 1000 1500 2000 - 140 1000 1500 2000 - 120 1000 1500 2000 Abs av dev (29 pts) Btu/lb a Jones, et al. (1963). * Stevens and
Obsd"
(Ho- H) Btu/lb
This work
197.8 188.5 177.6 163.4 137.0
217.9 213.4 210.8 204.8 200.6 198.1 189.4 186.0 184,4 169.8 168.8 168.6 155.2 158.2 159.6 Thodos
(1963).
Deviation of calcd values, Btu/lb Y &AC S&Tb
Saturated Liquid -0.277 10.477 0.611 10.385 9.687 1.669 3.296 8.429 12.663 10.171 9.486 3.512 Compressed Liquid 0.075 2.180 2.344 -2.538 -0.178 0.396 -1.049 1.674 2.438 -1.467 0.717 1.829 -3.544 -4.143 -3.487 1.641 c Yen and Alexander (1965); Yen
E-P-Ed
10.071 10.037 9.941 10.024 9.678 9.975
1.157 -0.639 -3.286 -5.147 2.595 2.637 -1.089 2.789 4.766 1.183 4.486 6.089 -3.517 -1.339 -0.960 - 13.325 - 13.920 - 15.316 - 16.935 -21.736 - 24.938 6.156
(1968).
Erbar, et al. (1964).
Table V. Comparison of Four Methods for Calculating Isothermal Enthalpy Differences for Propane in liquid Phase with Yesavage (1969) Values. Obsd" Temp, OF
Pressure, psia
14.1 50 82.7 150 122.5 250 151.4 350 174.7 450 194.4 550 Abs av dev (13 pts) Btu/lb
- 120
(Ho- HI, Btu/lb
174.0 157.2 146.4 134.9 125.5 116.3
1000 199.1 1500 197.4 2000 196.5 -40 1000 183.0 181.3 1500 180.5 2000 40 1000 166.5 1500 165.4 2000 164.5 120 1000 148.5 148.2 1500 148.2 2000 180 1000 131.8 1500 133.7 2000 134.9 Abs av dev (52 pts) Btu/lb Yesavage (1969). Stevens and Thodos (1963).
-
This work
Deviation of calcd values, But/lb Y&AC S&Tb
Saturated Liquid -1.770 2.467 -0.912 4.414 3.570 -0.905 1.396 3.888 2.199 1.000 3.083 -5.385 1.885 3.475 Compressed Liquid 0.670 0.640 -0.209 -1.683 -1.526 -2.312 -1.509 -1.309 -1.393 -1.563 -0.986 -0.861 -4.363 -4.710 -4.581 1.656 c Yen and Alexander (196Fj); Yen (1968).
E-P-Ed
-2.457 -0.316 1.656 6.254 9.637 13.348 5.99
-5.013 1.129 2.324 4.006 2.340 -3.127 3.426
5.991 7.681 8.616 0.483 2.122 2.933 -1.070 -0.123 0.728 -0.583 -0.578 -0.732 1.225 -1.086 -2.556 2.053 Erbar, et al.
(1964).
Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
233
Table VI. Comparison of Four Methods for Calculating Isothermal Enthalpy Differences for n-Pentane in liquid Phase with Values of Brydon, et a l a Temp,
Obsd" ( H o - HI,
Pressure, psia
O F
Deviation of calcd values, Btu/lb
Btu/lb
This work
S&Tb
Y&A"
~E-P-Ed
Saturated Liquid
100 140 200
15.6 31.1 73.9
240 120.3 280 185.6 320 274,4 360 391.9 Abs av dev (15 pts) Btu/lb
156.5 150.9 141.2
0.036 -1.076 -1.834
-7.732 -6.794 -5.257
- 22.337 - 18.734 - 13.848
-3.676 -4.586 -3.908
134,O 125.7 114.5 100.1
-2.195 -2.272 -0.640 2,547
-4.621 -4.307 -3.496 -4.689
-11.270 -9.112 -6.401 -5.378
-2.062 -0.425 4.126 11.033 5.439
2.121
5.699
Compressed Liquid -0.435 -0.623 -0.767
13.093
100
1000 1500 2000
154.6 153.5 152.3
140
1000 1500 2000
149.0 148.0 146.8
-1.006 -1.060 -0.988
-3.118 -2.173 -0.914
220
1000 1500 2000
136.8 136.0 135.1
-1.460 -1.013 -0.589
-3.051 -2.381 -1.463
-2.640 -2.899 -2.393 1.468
0.984 -0.916 -1.613 2,274
1000 113.3 114.9 1500 2000 115.5 Abs av dev (28 pts) Btu/lb a Rrydon, et al. (1953). * Stevens a.nd Thodos (1963). Yen and Alexander (1965); Yen (1968). Erbar, et al. (1964). 340
-2.123 -1.051 0.220
Empirical Equation
(7) P,V8
F 3 = - - P
2RT
F1, F z , and F3 are functions of temperature and independent of pressure. More convenient temperature functions than Equations 6, 7, and 8 are desired, however. An empirical form for these functions was deduced from a fugacity-enthalpy relationship. The isobaric temperature derivative of In f is related to the isothermal enthalpy difference by (9) Integrating a t constant pressure, combining with H = C, dT and C , = bl bzT b3TZ and changing the constants gives the following
+
f
In - = BI PC
+
+ Bz/T + B3 In T + B4T + B6T2
In
(10)
This form is more convenient than Equations 6, 7, and 8 for the temperature effects. It also has semitheoretical justification, as shown above. Thus an empirical equation of this form, after being fitted to derived fugacities, should also fit enthalpy difference values. 234 Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
An empirical generalized equation was derived for the fugacity of the pure liquid hydrocarbons by fitting Equation 10 to the values of Fl and FPobtained by the solution of Equations 6 and 7 for several hydrocarbons and approximating F 3 as a single-constant function. I n this way the proposed mathematical model was fitted to values of the reduced fugacity, Le., f / P e , obtained from an expanded and truncated form of the rigorous expression, Le., a combination of Equations 1 and 4. The constants obtained in this manner were then adjusted by fitting the resulting equation to the values of f / P , obtained from Equations l and 4. The empirical constants were assumed to be linear in the acentric factor and the resulting expression was rearranged. A Tr6 term was added to improve enthalpy prediction near the critical, which prediction is made via temperature derivatives of the In Y expression. The final equation is v =
AI
+ Az/Tr + A3 In T r + A,TT2+ AgTr6+
+ A7 In Tr + AsTr2)Pr + AgTr3Pr2- In Pr + ~ [ (-l Tr)(Alo+ A I I / T ~+) Al*Pr/Tr + A13Tr3Pr21 (11) (AdTr
where A I = 6.32873; A z = -8.45167; A B = -6.90287; = 1.87895; Ag = -0.33448; A6 = -0.018706; A7 = -0.286517; As = 0.18940; Ag = -0.002584; Am = 8.7015; All = -11.201; Alz = -0.05044; and A18 = 0.002255. Aq
Table VII. Comparison of Four Methods for Calculating Isothermal Enthalpy Differences for n-Pentane in liquid Phase with Canjar and Manning (1967) Values. Obsda Temp, OF
Pressure, psia
( H o - HI, Btu/lb
-
This work
Deviation of calcd values, Btu/lb S&Tb Y&AC
E-P-Ed
Saturated Liquid 100 140 200
15.67 31.1 73.9
240 120.3 280 185.6 320 274.4 360 391.9 Abs av dev (13 pts) Btu/lb
156.2 152.6 138.6
0.336 -2.776 0.766
-7.432 -8.495 -2.657
- 22.037 - 20.433 -11.249
-3.376 -6.286 -1.308
134.0 123.9 110.7 96.9
-2.195 -0.473 3.159 5.747 2.102
-4.621 -2.507 0,304 -1.489 4.130
-11.269 -7.312 -2.602 -2.178 11.649
-2.602 1.375 7.926 14.233 5,386
Compressed Liquid 100
1000 1500 2000 3000
155.0 153.9 152.7 150.0
-0.835 -1.023 -1.167 -1.325
-2.523 -1.451 -0.180 2.961
140
1000 1500 2000 3000
152.0 151.0 149.8 147.3
-4.006 -4.060 -3.988 -3.967
-6.118 -5.173 -3.914 -0.953
220
1000 1500 2000 3000
138.3 137.5 133.6 134.3
-2.960 -2.513 0.911 -1.109
-4.551 -3.881 0.037 -0.188
1000 109.1 2000 111.9 3000 112.5 Abs a v dev (27 pts) Btu/lb Canjar and Manning (1967). b Stevens and Thodos (1963). yen and Alexander (1965); Yen (1968). Erbar, et al. (1964).
-4.329 -3.677 -2.163 2.315
1.036 -1.924 -2.130 2,978
360
The above generalized empirical equation for the fugacity coefficient of pure liquids was developed for use in a vaporliquid phase equilibria calculation method. It also serves as a basis for an enthalpy prediction method. Evaluations for both fugacity and enthalpy predictions have been made. Fugacity Predictions
Equation 11 and the 13 universal constants listed above give the fugacity coefficient for pure hydrocarbon as real liquids as a function of reduced temperature, reduced pressure, and acentric factor. This expression may be applied a t temperatures between the critical and -200'F or T , = 0.4, whichever is higher, and at pressures between the vapor pressure and P , = 10. Values of Y from Equation 11 are compared with other values in Figures 1 through 5, Figures 1, 2, and 3 covering the T , = 0.6 to 1.0 range for methane, n-pentane, and ndecane, while Figures 4 and 5 are for propane and n-decane a t T, = 0.4 and 0.5. On these plots the solid curves are from Equation 11 and the dashed curves are from the ChaoSeader (1961) equation. Other v values shown on these plots are the values of Curl and Pitzer (1958), Chao, et al. (1969), and also some values of v that were calculated for propane a t
T , = 0.4 and 0.5 by applying Equations 1 and 4 with p' and V s data (American Petroleum Institute, 1963; Tickner and Lossing, 1951). As can be seen in Figures 1, 2, and 3, the proposed new equation agrees very well with the tabular values of Curl and Pitzer (1958). Agreement between the proposed equation and the Chao (1969) correlation values is only fair a t the lower temperatures despite the fact that the same experimental data were used in both studies, there being an average diff erence of about 5% between the two. Fugacities for liquid propane calculated a t several reduced pressures for T , = 0.4 and 0.5 by four methods are given in Table I. The four calcuation methods are indicated. Method I was used by Chao, et al. (1969), and differs from method I1 in the second virial coefficient equation used and in the way the Poynting effect was figured. Methods I and I1 are in good agreement, as can be seen. Two sets of correlation values are given and it can be seen that the values from this work, L e . , Equation 11, agree with the values calculated from experimental vapor pressure data, while the Chao (1969) correlation values do not agree. A similar comparison was made for n-pentane, for which the Chao (1969) correlation agreed with the data-based Y values as well as did our correlation. Ind. Eng. Chem. Fundam., Vol. 10, No. 2, 1971
235
Enthalpy Differences
h generalized equation for the isothermal enthalpy difference (Le., the enthalpy at temperature and pressure of the system minus the ideal gas-state enthalpy a t zero pressure and system temperature) for pure liquids was obtained by differentiating Equation 11 with respect to temperature, pressure constant, and combining with Equation 9. This derivation gave the following expression, which is in reduced form with the dimensionless isothermal enthalpy difference ratio as a function of T,, P,, and W .
+
H - H” - A2 - A3Tr - 2A4TT3- 6A5TT7 ~-
RT,
F1, F P ,F 3
=
sH
=
T T,
= = = = =
ZC
=
PS
=
temperature functions as defined by Equations 5, 6, 7 and 8 fugacity enthalpy a t T and P of system ideal gas state enthalpy at T of system isothermal enthalpy ~.difference pressure reduced pressure gas constant temperature reduced temperature critical compressibility factor saturation or vapor pressure
= = =
compressibility as defined by Equation 3 fugacity coefficient acentric factor
H” (H - H”) P
P, R
Y =
j/P
W
Conclusions
Empirical generalized equations are presented for predicting the fugacity coefficients and the isothermal enthalpy differences for pure liquid hydrocarbons. These new expressions are applicable between T , = 0.4 and 1.0 and P , values between the saturation pressure and 10, i.e., for real liquids in saturated or compressed states. Graphical comparisons show that the proposed new equation for fugacity coefficient agrees with f / P values that have been derived from experimental data. Tabular comparisons show that the proposed ne\v equation for the isothermal enthalpy difference is more accurate and versatile t’han ot’her equations t’hat are available. Nomenclature dl-A13
=
236 Ind. Eng.
empirical constants for Equation 11
Chem. Fundam., Vol. 10, No. 2, 1971
=
GREEK
P
Equation 12 is applicable to pure substances in saturated and compressed liquid phases. It was developed from hydrocarbon information but is generalized and should be applicable to other nonpolar substances. Isothermal enthalpy differences have been calculated via Equation 12 and three other methods for six hydrocarbons in both saturated and compressed states. The results of these calculations are given in Tables 11-VII, which tabulations show part of the points, the condensation being made to coilserve space. The “observed” data, which are given as (H” - H ) values, in Btu/lb and positive, are from reliable computations or experiments by others. The comparisons in Tables I1 and I11 are for saturated liquid enthalpy differences only. Those in Tables IV-VI1 are for both saturated and compressed liquids. Only two methods were applicable for all conditions. As can be seen from these tabulations, the proposed new equation is better than the previous equation that one of us coauthored (Erbar, et al., 1964). The average absolute deviation for 243 points that were included in the comparisons are 1.8 Btu/lb for Equation 12 and 9.8 for the previous method (Erbar, et al., 1964).
= =
SUPERSCRIPTS S = saturation condition L = liquid SUBSCRIPTS C
=
r
=
critical condition reduced condition
literature Cited
American Petroleum Institute Research Project 44, “Selected Values of Properties of Hydrocarbons and Related Compounds,,’ Texas A&M University, College Station, Texas, 1963. Barkelew, C. H., Valentine, J. L., Hurd, C. O., Trans. A.I.Ch.E. 43, No. 1, 25 (1947). Brydon, J. W., Walen, N., Canjar, L. N., Chem. Eng. Progr. Symp. Ser. 49, 151 (1953). Canjar, L. N., Manning, F. S., “Thermochemical Properties and Reduced Correlation for Gases,” Gulf Publishing Co., Houston, Tex., 1967. Chao, K. C., Greenkorn, R. A,, Olahisi, O., presented at A.1.Ch.E. Meeting at Portland, Aug 1969. Chao, K. C., Seader, J. D., A.I.Ch.E. J . 7,598 (1961). Chueh, P. L., Prausnitz, J. M., A.I.Ch.E. J . 15, 471 (1969). Curl, R. F., Jr., Pitzer, K. S.,Ind. Eng. Chem. 50, 265 (1958). Erbar, J . H., Persyn, C. L., Edmister, W. C., Proceedings of the 43rd Annual Convention Natural Gas Processors Association, March 1964. Lee, B. I., Edmister, W. C., IND.ENG.CHEM.,FUNDAM. 10, 32 (1971).
Jones, M. L., Jr., blage, D. T., Faulkner, Jr., R. C., Katz, D. L., Chem. Eng. Progr. Symp. Ser. 59, 52 (1963). Lydersen, A. L., Greenkorn, R. A., Hougen, 0. A., Engineering Experimental Station Report No. 4, University of Wisconsin, Madison, Wis., 1955. Martin, J. J., Chem. Eng. Progr. Sym . Ser. 59, 120 (1963). Messerly, G. H., Kennedy, R. M., J!Amer. Chem. SOC. 62, 2988 (1940).
Pitzer, K. S., Curl, R. F., Jr., J . Amer. Chem. SOC.79,2369 (1957). Steams, W. V., George, E. J., Ind. Eng. Chem.35,602 (1943). Stevens, W. F., Thodos, G., A.I.Ch.E. J . 9, 293 (1963). Tickner, A . W., Lossing, F. P., J . Phys. Coll. Chem. 55, 733 i19.51). Y&L. C., Alexander, R. E., A.Z.Ch.E. J . 11, 334 (1965). Yen, L. C., rivate communication, Feb 5, 1968. Yesavage, $: R., Ph.D. Thesis, University of Michigan, 1969.
RECEIVED for review July 22, 1970 ACCEPTEDFebruary 12, 1971
