Prediction of Liquid-Phase Enthalpies with the Redlich-Kwong Equation of State Joseph Joffel and David Zudkevitch Esso Research and Engineering Co., Florham Park, .V. J . 07932
The Redlich-Kwong equation with temperature-dependent coefficients, previously used for predicting vaporliquid equilibria, has been applied to the calculation of liquid-phase enthalpies of pure compounds and mixtures. The two parameters of the Redlich-Kwong equation, treated as temperature functions, are obtained for each component from its saturated pure liquid properties. Binary interaction constants are required for describing mixing behavior in binary and multicomponent systems. These constants have been established for a number of systems as part of earlier work in utilizing this equation for predicting vapor-liquid equilibria. The proposed method for predicting enthalpies i s compared with the pseudocritical method of Barner and Quinlan for several systems for which there are experimental enthalpy data.
T h e Redlich-Kwong equation of state combines simplicity with a degree of accuracy satisfactory for calculat'ing therniodynamic properties for engineering work. It has been used for calculating vapor-phase enthalpies (Edniister and Yarborough, 1963) and, in a modified form, for calculating liquidphase enthalpies as well (Wilson, 1966). I n the current study, aimed a t predicting liquid enthalpies, the two constants, a and b, of the origiiial Redlich-Kwong equation are treated as temperature-dependent variables following the procedure recently applied with grat,ifying results to the prediction of vapor-liquid equilibria a t high pressures (Zudkevitch and Joffe, 1970). I n the Redlich-Kwong equation,
p
=
RT
~-
V-b
-
a
T'12V(V
=
+ b)
s2,R2T,2.5/P~
(2)
b = QbRTJPc (3) Redlich and Kwong (1949) recommended two universal constant's for Q, and a,,, while Chueh and Prausnitz recommended two pairs of constants, one for the vapor (1967b) and one for the liquid (19674. Wilson (1964) proposed a generalized formula for the variation of the R-K coefficients, a, with t,emperature. The authors (Zudkevitch and Joffe, 1970) use only one set of temperature-dependent D coefficient's for every compound. The values of Q, and for a pure compound are established iii the authors' method, a t each of a series of temperatures, by simultaneous solution of Equation 1 and bhe follo~viiigequat'ion for the fugacity coefficient a t saturation, In
$Ls =
In R T / P ( V L- b )
In $cs
=
ln R T / P ( V ,
92
+ (PVJRT) - 1 (a/RT3'*b)In (V, + b ) / V ,
(5)
+ b)/V,
= +LS
(6)
Equations 1 to 6 an? solved simultaneously for R, and f i b . hIeasured or correlat'ed values of t'he saturated liquid properties, P and V i ,must be supplied as in the first method, but correlated values of $L' are not needed. This alternative procedure for establishing values of Q , and a b has not been utilized in the current work, since the original method of Zudkevitch and Joffe (1970) appears to be more appropriate when liquid-phase enthalpies are the primary object of the study. When the modified Redlich-Kwong equation is applied to mixtures, coefficients a and b are calculated by combining rules similar to those used by previous workers (Chueh and Prausnitz, 196ia, b; Wilson, 1964) :
and
+ (PVJRT)- 1 (a/RT312b)In ( V ,
- b)
I n view of the condition for phase equilibrium for pure components,
coefficients a and b are ciilculat'ed from the critical properties of the conipound and two proportionality coefficients. Chueh and Prausnitz (1967~1,b) recommended the symbols a, and nb for these proport~ioiialitycoefficients:
a
the pure component a t saturation. The values of +L' are obtained from the generalized correlation of Lyckinaii, Eckert, and Prausnitz (1965). The saturated liquid properties, P and V i ,are taken from tabulated data or calculated by suitable correlations. An alternative procedure has also been used to establish the temperature-dependent values of 9, and Qb (Joffe et al., 1970). I t involves utilizing Equation 4 twice, once for the saturated liquid, as above, and again for the saturated vapor, as follons:
b
X,b,
= i
(4)
Furthermore,
where P is t'he vapor pressure a t temperature T , Til is the sat'urated liquid volume, and +L' is the fugacity coefficient of and Present nddresb, Kewark College of Engineering, Newark, X. J. 07102. T o whom correspondence should be sent. Ind. Eng. Chem. Fundam., Vol.
9, No. 4, 1970 545
? ' l i u ~ , binary interaction constants, Ct3, are needed, one for earlL l h a r y pair in a multicomponent mixture. These constants have been established for a number of systems from experimental binary vapor-liquid equilibrium data (Zudkevitch et al., 1969; Zudkevitch and Joffe, 1970). These or similar C,, values have been used in the current work. The isothermal effect of pressure on the enthalpy of a fluid is given by the thermodynamic relation: ( H O
- H)T
=
RT - PV
- W),= RT - PV
[,I:;G
+ $'
Saturated liquid
Saturated vapor
Pressure,
OF
PSlA
da
In (V
+ b)/V
cxix,,,daii
dQai
dQbi
Din,
Calcd.
1961
227 215 202 185 157 102
226 217 203 186 161 104
- 270.69 - 243.69
- 207.69 -171.69 - 135.69 -ll5,79
8.3 28.0 93.2 231.3 477.8 673.1
-270.69 - 171,69 -135.69
8.3 231.3 477,8
1.2 23.1 46.2
1.2 26.1 52.7
734.8 1469.6
45.3 108.6
44.1 96.1
- 63.69 - 63.69
(12)
and da -=
Supercritical
- H, Btu/Lb
-I-
Equation 12 iq applicable to a pure substance and likewise to mixture.. In the latter case we have, in vien- of Equations 7 and 8,
(14)
Ifi = j ,
If i # j ,
and -, are ohtaitied by numerical or dT dT graphical differentiation of the Q a t m t l ( I b i values for each
The derivatives,
State
Temp.,
e]($) (c)]
+ [_ ,"T2b _ -
($) - 7
dT
Ho
+
Substituting Equation 1 into Equation 11, and in view of the fact that in the modified Redlich-Kwong equation a and b are temperature functions, there is obtained: ( H O
Table 1. Comparison of Isothermal Pressure Effects on Enthalpy of Methane
p i m coinpouiid. For a supercritical cmiponent "" and ~
dT
"" are assumed to be zero, since in t,he method of Zudkevitch
dT :ind Joffe (1970) a and b are taken as constants a t temperature? above the critical. Equation 12 has been applied in bhe current work to the calculation of the enthalpy departure, H" - H , of methane to illust,rate the validity of the method for a pure substance. previously obtained by Zudkevitch The values of a, and aiid Joffe (1970) in V-L-E studies by simultaneous solution of Equations 1 and 4 were utilized. The derivatives of Q, and Q, were obtained by graphical differentiation. Calculated enthalpy departures, H o - H , are compared in Table I with data from Din's tables (Din, 1961). Reasonably good agreement is obtained with Din's values for the enthalpy of the saturated liquid, while the agreement for enthalpies of the saturated vapor and enthalpies of the gas above the critical temperature is not as good as in the liquid phase. From this 546 Ind. Eng. Chem. Fundam., Vol. 9,No. 4, 1970
and other test's, it has been concluded that the modification of the Redlich-Kwong equation following Zudkevitch and Joffe (1970) is more suited to calculating liquid than vapor enthalpies. This appears to stem from the fact that parameters Qa and ab are based on experimental or correlated saturated liquid volumes rather than vapor volumes. I n applying Equation 12 for predicting isothermal pressure effects on enthalpy of liquid mixtures, pure component, properties, vapor pressures, and densities of the saturated liquids are required. These are used to establish the values of Q, and ab and of their derivatives for each pure component. The Redlich-Kwong mixture paramet.ers and their derivatives are calculated from Equations 7 to 10 and 13 to 16. Equation 1 is then solved by trial for the volume of the liquid mixture. These R-K parameters, their derivatives, and the calculated liquid volume of the mixture are substituted into Equation 12 to obtain the enthalpy departure ( H O - H ) of the fluid. Three systems have been studied: methane-propane, pentane-octane, and pent'ane-hexadecane. The vapor pressures of all pure components when not available were calculated from the equations of Sondak and Thodos (1956). Densities of methane and propane were obtained from Din (1961) and the API RP-44 tables (1952). Densities of pentane were obtained from Timmermans (1950). Densities of octane and of hexadecane were taken from the API tables (1952) or established from Riedel's correlation (1954). I n all cases, the derivatives of the Q , and f i b parameters were approximated b y finding the increment in the parameter for a half a degree (Fahrenheit) increment in temperature and equating the derivative to the increment per degree. The results of these calculations are compared in Tables 11, 111, IV, and V with experimental enthalpy values and with results obtained by the method of Barner and Quinlan (1969), which utilizes the extended Curl-Pit'zer correlation (American Petroleum Institute, 1966) and a special procedure for calculat,ing the pseudocritical properties of mixtures. Table I1 lists results for saturated liquid mixtures of methane and propane a t three coniposit'ions. Previous work on V-L-E data for this system indicated that the data may fall into t,wo groups, perhaps due to differences in experimental apparatus (Zudkevitch et al., 1969), one set correlating well with an interaction coefficient C i j = 0, the other set correlating with an average C t , = 0.073. As shown in Table 11, enthalpy departures correlate better with Cij = 0, except for the methane-propane mixt>urecontaining 5.2 mole % propane (Table 11). Results a t - 120°F are adversely affected by the
Table II. Enthalpy Departures from Ideal Gas, Methane-Propane Saturated liquid Mixtures Ho
Male Fraction Propane
Temp., OF
Pressure, PSlA
- 144 - 100 - 62
0.766
c,,
Exptl.
Btu/Lb Calculated Barner and C,, = 0.073 Quinlan, 1969
= 0
12 47 76 102
100 200 300 400 500 600 700 800
204. 5 193.5 184.3 175.6 166.7 158.0 149.5 140.4
203 2 194 0 184 8 174 4 165 2 154 8 146 5 138 2
199 190 181 171 161 151 142 134
0.052
- 260 - 200 - 120
10 102 580
221.5 198.6 157.8
226.4 203.5 170.5
0,120
- 209 - 150 - 124 - 101 -77 -49
- 24
100 210 300 188 500 170 700 159 900 145 1100 127 d v . absolute
6 1 9
211 186 171 4 161 4 144 1 123 deviation 2
3 8 8 6 0 3 7
Table 111. Enthalpy Departures from Ideal Gas, Pentane-Octane Saturated liquid Mixtures
59.7 mole
Temp., F. m-
40
200 300 450 500
Lenoir, et al.,
Pressure, PSlA
1968
161 140 125 89 81 4
0 9 1 1 5 6
152 139 123 95 84 3
0 5 0 2 4 8 5 4
Yesavage et al., 1969
222 5 199.7 163.0
222.0 202 6 163 7
Bhirud and Powers, 1969
204 180 173 153 134 110 5
206 186 176 165 144 114 3
Yesavage, 1968
7 7 6 3 6 6 3 0
197 189 183 175 166 157 150 142
7 3 0 3 6 5 7
5 5 5 5 0 7 4
% n-hesadecane
58.7 mole % n-pentane, 41.3 mole
- H,
Btu/Lb Calculated Barner and C,, = -0.05 Quinlan, 1969
200 157 5 400 142 2 500 126 6 600 98 2 8004 89 0 Av. absolute deviation
Ref.
Table IV. Enthalpy Departures from Ideal Gas, Pentane-Hexadecane
yon-pentane, 40.3 mole yon-octane Ho
a
- H,
Temp., F.
2 1 2 6 8 7
200 280 360 440 500
Compressed liquid
Lenoir, et a l ,
Pressure, PSlA
1969
H o - H, Btu/Lb Calculated C,, = Barner and -0.075 Quinlan, 1969
70 140 3 200 128 4 400 117 7 630 108 2 1400 101 2 A\v. absolute deviation
139 138 121 102 93 3
8 2 2 0 9 5
130 124 114 104 98 4
0 2 8 7 9 6
Table V. Enthalpy Departures from Ideal Gas, Methane-Propane Compressed liquid Mixtures Ho
Mole Fraction Propane
0 766
Temp., OF.
- 40 80 160
0 120
-130 - 100 - 30
Pressure, PSlA
Exptl.
1000 2000 1500
177 8 148 5 123 2
1000 185 1500 165 2000 143 Av. abqolute
5 0 6 deviation
cij
= 0
- H,
Btu/Lb Calculated Barner and C;j = 0.073 Quinlan, 1969
Ref.
177 4 145 1 123 1
174 4 142 5 119 6
178 5 148 2 123 0
Ye*a\ age et a1 1969
185 166 141 1
179 I59 134 5
186 165 139 1
Yeqavage, 1968
6 4 5 25
~
5 7 5 6
9 2 8 1
~
~~~~
Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
547
fact that this temperature is close to the methane critical point, where the derivatives of Q, and iib change rapidly with temperature and are difficult to establish. Table I11 lists results of calculations for mixtures of pentane and octane (Lenoir et al., 1968), while Table I V gives results for mixt.ures of pentane and hexadecane (Lenoir et al., 1969). It is significant that in both systems deviations from experimental values are large above 385”F, the critical temperature of t’he more volatile component, pentane. The method for calculating H” - H from Equation 12 is very sensitive to the assumed values of the derivatives of R, and no.T o assume t’hat these derivatives are zero above the crit’ical temperature of the compound is probably incorrect. Such an assumption introduces little error into vaporliquid equilibrium calculations (Zudkevitch and Joffe, 1970) which do not involve the derivatives of a and b. Enthalpy calculations, however, may be adversely affected. Work is currently ainied a t studying the variation of Qa and a b with temperature in the supercritical region. The proposed method, in its present st,age of development, compares Fell with that of Barner and Quinlan, which is not as good as the proposed niethod a t the lower teniperatures listed in Tables I11 and IV, but better than t’he proposed method a t the higher temperatures, above t,he critical point’of the light compound. Table V list’s experimental and calculated enthalpy departures for compressed liquid mixtures of methane and propane. =is iii t,he case of saturated liquid mixtures breated i n Table 11, the isothermal pressure effects on the enthalpies of methane-pi,opane mixtures correlate better with the interaction parameter of C i j = 0 than with C i j = 0.073. The niethod of Barner and Quinlan compares favorably with the proposed method for this system. Conclusions
The proposed method provides a tool for calculating enthalpy departures of saturated and compressed liquids, both pure compounds and liquid mixtures. However, additional work is required for establishing the temperature dependence of the K-K parameters of compounds above their critical temperatures for accurate results. Pending such development, a pseudocritical method such as that of Barner and Quinlaii may be used for calculating enthalpies of liquid mixtures. Our comparisons indicate that below a pseudoreduced teniperature of 0.8, the extended Curl-Pitzer correlation of enthalpy departures (American Petroleum Institute, 1966) employed by Barner and Quinlan is probably less reliable than the propoqed method. Acknowledgmenl
The authors thank Esso Research and Engineering Co. for releasing this work for publication and gratefully acknowledge the as-istance of George 11.Schroeder in programming and calculations and Gloria A. O’Brien’s assistance in the prepaiation of the manuscript.
548 Ind. Eng. Chem. Fundam., Vol. 9, No. 4, 1970
Nomenclature = = =
= = = =
= = =
coefficients in Redlich-Kwong equation, Equation 1 interaction constant for pair of components i and j enthalpy pressure, vapor pressure gas constant absolute temperature molar volume mole fraction in liquid phase fugacity coefficient dimensionless parameters defined by Equations 2 and 3
SGBSCRIPTS C
i, j G, g L, 1
= = =
=
critical state component i or j vapor or gas phase liquid phase
SCPERSCRIPTS = =
0 S
ideal gas state saturated state
Literature Cited
American Petroleum Instit,ute, New York, Research Project 44 (1932). American Petroleum Institiit.e, S e w Tork, “Technical Data Book, Petroleum Refining,” 1966. Barner, H. E., Quinlan, C. IT.!Znd. Eng. Chem. Proc. Des. Dclop. 8, 407-12 (1969). Bhiriid, 1’. L., Powers, J. E., “Thermodynamic Properties of a 3 Mole Per Cent Propane in 11et.hane Llixture,” Report t o Natural Gas Processors Association, Tulsa, Okla., 4ugust 1969. Chueh, P. L., Pransnitz, J. 11.).I.Z.Ch.E. J . 13, 1099 (1967a). Chueh, P. L., Prausiiitz, J. 11.)I s n . ENG.CHEM.F u m i u . 6, 492 il967h). -, \ - - -
Din, F., “Thermodynamic Functions of G a w , ” pp. 47, 57, Butterworths, London, 1961. Edmister, W.C., Tarborough, L., A.Z.Ch.E. J . 9, 240-6 (1963). Joffe, J.. Schroeder, G. hl., Zudkevitch,. D.,. A.I.Ch.E. J . 16, 496 (197oj. Lenoir, J. X, Robinson, D. R., Hipkin, H. G., 33rd Midyear RIeeting, American Petroleum Institute, Philadelphia, hIay 15, 1968. Lenoir, J. M.,Kuravila, G. K., Hipkin, H. G., 34th Midyear hleeting, A4merican Petroleum Institiite, Chicago, May 12, 1969. Lyckman, E. W.> Eckert, C. h., Prausnitz, J. LI.,Chena. Eng. Sci. 20, 68.5-91 (196.5). Redlich, O., Kwoiig, J. N. S., Chem. X r v . 44, 283 (1949). Riedel, L., C h o n . Zng. Tcch. 26, 83, 257, 679 (19j4). Soiidak, S . E., Thodos, G., i l . I . C h . E . J . 2, 347-53 (1956). Timmermaiis, J., “Physico-Chemical Constants of Pure Organic Compounds,” p. 31, Elsevier, S e w Tork, 1950. ., i l d m n . CrUog. Eng. 9, 168-76 (1964). ., A ~ I I uCrUog. ~ . Eng. 11, 392-400 (1966). F., Ph.11. thesis, University of Jlichigan, 1968. F., Katz, D. L., Poxers, J. E., J . Chem. Eng. Data 14,137-49 (1969). Zudkevitch, D., Joffe, J., .t.Z.Ch.E. J . 16, 112 (1970). Zudkevitch, I),, Joffe, J., Schroeder, G. ll.,“Prediction of VaporLiquid Equilibria within the Critical Region,” Interiiational Symposium on Distillation, Bright on, England, Sept’. 8-10, 1969. 1bc~Ivm for review February 24, 1970 ACCEPTED July 23, 1970 Division of Industrial arid Eiigiiieering Chemistry, 159th Meeting
ACS, Houston, Tex., February 1970.
