Gill, W. N., Sankarasubramanian. R.. Proc. Roy. SOC.London, Ser. A, 322, 101
Sankarasubramanian, R., Gill, W. N., Proc. Roy. SOC.London, Ser. A, 333, 115
Gill, W. N., Sankarasubramanian, R., P r x . Roy. SOC.London, Ser. A, 327, 191
Taylor, G. I., Proc. Roy. SOC.London, Ser. A, 219, 186 (1953). Wasan, D. T., Dayan, J.. Can. J. Chem. Eng., 48, 129 (1970).
(1971). (1972).
(1973).
Harlacher, E. A., Engel, A. J., Chem. Eng. Sci,, 25, 717 (1970). Krantz, W. B.. Wasan, D. T.. lnd. Eng. Chem., Fundam., 13, 56 (1974). Sankarasubramanian, R., Gill, W. N., Roc. Roy. SOC.London. Ser. A, 329, 479
Received for review January 27,1975 Resubmitted M a r c h 18,1976 Accepted June 28,1976
(1972).
A Modified Redlich-Kwong-Soave Equation of State Capable of Representing the Liquid State Gerald G. Fuller’ Department of Chemical Engineering, University of Calgary, Calgary, Alberta, Canada
The modified form of the Redli’ch-Kwong equation of state proposed by Soave is further generalized in order to reproduce saturated liquid volumes and compressed liquid volumes of pure substances with greater accuracy. The additional modifications contain two features; first, the equation of state now leads to a variable critical compressibility factor, and second, a new universal temperature function is incorporated in the equation making both the a and b parameters functions of temperature. The added modifications preserve the ability of the equation to predict vapor pressures with only the critical volume and parachor being required as additional data input. The results of calculations indicate that the proposed equation is capable of describing even polar molecules such as water and ammonia with reasonable accuracy.
Introduction The modification of the Redlich-Kwong equation of state introduced by Soave (1972) has made it possible to predict vapor pressures accurately with a cubic equation in volume. This modification (hereafter referred to as the RKS equation) involved allowing the a parameter of the original RK equation to be a function of temperature and used the acentric factor as a correlating parameter. The success of the modification is, however, restricted to the estimation of vapor pressure. The calculated saturated liquid volumes are not improved and are invariably higher than the measured data. The failure to reproduce the liquid state data can be attributed to two restrictions inherent in the RKS equation of state. First, the critical compressibility factor 2,is fixed at Y3 regardless of the substance being modeled, which forces the VLE volumetric data to attain values which are too large in the critical region. Second, the excluded volume parameter b is invariant to temperature changes, whereas arguments based on the kinetic theory of gases (Loeb, 1961) strongly suggest the parameter should be a decreasing function of temperature. In the sections to follow it will first be shown how an equation of state of the “van der Waals” form can be generalized to incorporate a variable compressibility factor. Following this, a second empirical temperature function, to be used in conjunction with that of Soave’s, is proposed. This effectively makes the b parameter a function of temperature as well as giving the a parameter additional temperature dependence. These changes make it possible to represent the PVT properties of pure compounds accurately a t densities much higher than allowable with the present Rks equation.
Formulation of the Modified Equation of State In order for the liquid state to be adequately represented, it is essential that the equation of state accurately reproduce the 2,value of the substance described. The RKS equation in its present form fails in this respect but this difficulty can be overcome with the addition of a third parameter. For a more general treatment of this problem the reader is referred to the work of Abbott (1973) which considers the four-parameter cubic equation of state. It is, however, only necessary to consider the three-parameter case in order to produce a variable critical compressibility factor. The equation of state then has the form
where the additional parameter added is denoted as e . The specification of the parameters a, b, and c should constrain the equation to produce the following conditions at the critical point
($),.=o Using the above constraints it is convenient to parametrize the three variables a, b, and c in terms of a single unknown (3 which is defined as to take on the following value at the critical point Pc
=
b
-
( T = T,)
U,
carrying out the required manipulations it is found that Address correspondence t o the a u t h o r a t t h e D e p a r t m e n t of Chemical Engineering, California I n s t i t u t e of Technology, Pasadena, Calif. 91125.
254
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
c(P)
=’(&---) 1 3 P
P
4
3 2
(4)
I
IC
10 065
I
I
I
I
I
1
070
075
080
085
090
095
100
Figure 1. The ratio /3//3, vs. the reduced temperature for n-butane.
Equation 10 (the universal temperature function proposed by Soave) is modified through eq 11in order to remove any adverse coupling of the proposed modification onto the Soave modification. This leaves the estimated vapor pressures essentially the same as those calculated with the original RKS equation and in fact causes the RMS error in the vapor pressures to be slightly reduced in most cases. If P is kept invariant with respect to temperature one may reduce eq 1to the RKS equation, the van der Waals equation, and the ideal gas equation by setting p, to take on specific values. Setting PCto 0.259921, we have c = 1, R, = 0.4274802, Q h = 0.0866404, z, = 0.333, and
p=--RT v -b
a(T) 6)
u(u
+
which is the RKS equation of state. Setting P, to a value of 113, the general form reduces to c = 0, R, = 0.421875, Rb = 0.125, Z, = 0.375, and
p = - -RT u-b
a(T) u2
RT
-
a(T)
In
u+cb
(14)
Figure 1 depicts the ratio plotted against reduced temperature for the case of n-butane. All other compounds produced curves of similar shape. Figure 1displays two important characteristics which should be sought after in choosing a correlating functional form. First, there exists an asymptotic limit Po a t low temperatures and second, the representative function should fall off rapidly as the critical point is approached. The function chosen should be well behaved for high temperatures and not produce values equal to or less than zero in view of the constraints imposed by eq 5 . The function chosen to be the best suited was the following
e
P
= Pc
+ (Po - P c )
2 (1 e , / ( T r - l , - 1) +
(15)
025
026 zc
I
I
I
I
027
OZE
029
03
-
Figure 2. The ratio /30//3~ vs. the critical compressibility factor. Refer to Table I in order to associate the numbered points with each respective compound.
The value of 3, can be determined immediately from knowledge of the critical compressibility factor and generalized methods for determination of Po and R have been obtained. Values of PO for each compound were obtained from graphs similar to Figure 1. A good correlation between POand Z, exists as shown by Figure 2. Therefore PO can be estimated from the following polynomial obtained by a least-squares fit of the data presented in Figure 2.
y)C ~ R T (7)
(
024
023
which is the van der Waals equation. The ideal gas equation is recovered by taking @, to infinity: c = 0, Q, = 0 , f i b = 0 , 2, = 0, and P = RT/u. I t is necessary, however, to make /3 a function of temperatures and not simply hold it fixed at its critical point value. In order to determine the temperature dependence of p, saturated liquid volume data of 27 substances were used in reduced temperature ranges from about 0.5 up to the critical point. Based on the liquid volumes a t a given temperature, an iterative procedure was used to vary /3 until the liquid and vapor fugacities were matched. Here it is important to note that the parameters c, Rh, R,, and q vary with @accordingto eq 5,7,9, and 11. The fugacity coeificient is required for such equilibrium calculations and is given by
Pu In Cb = - - 1 - In
1
I20
’ P,
= ‘7.7880 - 36.83162,
+ 50.7O61Zc2
(16)
To calculate optimum values for 0, the Fibonacci search was employed to determine the point of minimum root mean square percent deviation in the saturated liquid volume calculations. These optimum values fell in the range of 11to 27 for the compounds investigated. As shown in Figure 3, R was correlated against the parchor allowing this parameter t o be estimated frcm R = 10.9356
+ 0.0285p
(17)
The parachor has been shown to characterize the molecular volume (Harlacher, 1970) and may be considered to be a “true” measure of acentricity of the molecule. Values for the parachor may be quickly calculated by the group contribution Ind. Eng. Chern., Fundarn., Vol. 15, No. 4, 1976
255
Table I. Comparison of Calculated Vapor Pressures and Calculated Saturated Liquid Volumes
Compound 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
Methane Ethane Propane n-Butane Isobutane n-Pentane Isopentane n-Hexane Cyclohexane 2,3-Dimethylbutane n-Heptane n -Octane Diisobutyl n - Nonane n-Decane Ethylene Propylene 1-Butene Benzene Acetylene Ammonia Carbon dioxide Carbon monoxide Nitrogen Oxygen Sulfur dioxide Water
Saturated liquid volume rms Vapor pressure rms dev, % dev, % Original Original RKS Chaudron, This RKS Chaudron, This equation etal. work equation etal. work 1.61 0.96 1.41 1.18 1.17 1.02 0.73 1.90 0.78 0.82 2.79 3.31 1.07 1.11 2.76 1.79 1.30 1.17 1.33 2.82 1.02 0.54 1.67 1.55 1.02 1.61 13.23
11.2 49.8 5.15 7.18 22.76 11.47 22.3 27.03 65.59 194.9 82.04 51.04 31.48 39.37 31.29 5.67 99.29 15.12 37.98
1.42 0.81 1.09 0.96 1.03 0.74 0.45 1.44 0.51 0.60 1.20 1.33 0.95 1.62 2.0 1.59 1.03 0.99 1.10 2.52 1.03 0.29 1.56 1.36 0.85 1.24 10.99f
9.69 15.73 15.15 14.83 12.98 16.61 14.10 16.90 13.29 15.95 17.76 18.07 19.02 18.65 19.94 9.48 12.17 13.28 14.81 14.34 30.66 15.8 6.52 9.78 6.97 18.15 41.28
5.62 2.23 1.9 2.89 2.19 3.2 9.71 2.95 5.92 2.92 5.28 3.21 10.22 6.71 4.01 1.99 73.9 1.77 4.62
Temp range,K
1.48 2.95 0.83 1.07 2.55 1.06 2.21 1.71 1.18 1.62 1.23 1.21 1.70 2.86 2.58 3.16 1.67 1.22 2.21 1.60 4.73 1.76 0.93 1.74 2.25 1.99 3.63
Pressure range,atm
111-190.7 183-305.4 230-369.9 272-425.2 255-408.1 310-469.5 298-460.4 283-507.3 353-553.2 332-499.9 273-540.3 298-568.8 383-544 298-594.6 298-617.6 169-283.1 225-365.1 273-419.6 310-562.1 192-309.5 243-405.6 216-304.2 67-133 77-126.2 90-154.8 255-430.7 273-647.0
2,
1.O-45.8 1.0-48.2 1.O-42.0 1.0-37.5 0.78-36 1.0-33.3 1.0-32.9 1.0-29.9 1.1-40 1.05-30.9 0.015-27.0 0.0 18-24.6 1.1-24.5 0.006-22.5 0.001-20.8 1.0-50.5 1.0-45.4 1.26-39.7 0.22-48.6 1.27-61.6 1.24-1 12.5 5.11-72.9 0.15-34.5 1.0-33.5 1.0-50.1 0.698-77.7 0.006-218.3
Datap source
0.29 0.285 0.276 0.274 0.283 0.269 0.268 0.264 0.271 0.27 0.259 0.256 0.263 0.25 0.247 0.278 0.276 0.276 0.274 0.274 0.242 0.276 0.294 0.291 0.29 0.268 0.23
a Canjar and Manning (1967). Das and Kuloor (1967). API (1953). Washburn (1928). e Acentric factors and critical properties were obtained from Reid and Sherwood (1966) except in the case of ethylene where values were obtained from the “International ThermodynamicTables of the Fluid State-Ethylene, 1972,” IUPAC, Butterworths, London, 1974. f The large error in the vapor pressure estimation for water is processed mainly by the vapor pressure data below 0.322 atm. Above this point the rms deviation is only 2.74% up to the critical point.
method outlined by Quayle (1953) which only requires the knowledge of the compound’s structural formula. I t should also be added that originally the critical volume was used as the correlating parameter but was replaced by the parachor as the latter is much more reliable and accessible. Another important comment is that the root-mean-square deviation in the saturated liquid volume is to a large degree insensitive to small changes in 0. Results The results of the proposed modification in simulating the VLE data of the 27 compounds are given in Table I. The proposed modification is compared to the original RKS equation as well as the modified RK equation of Chaudron et al. (1973) which claimed considerable accuracy in high density fluid calculations. The Chaudron modification also assumed the parameters a and b to be temperature dependent but does not satisfy eq 2 and 3 a t the critical point. The optimal set of coefficients was used in all computations using the Chaudron modification rather than the generalized one. In all cases the modified RKS equation of state produced a root-mean-square error in saturated liquid volumes of less than 5% and in the majority of cases also improved the vapor pressure deviations of the original RKS equation. The vapor-liquid equilibrium dome for water shown in Figure 4 brings out several features. The asymptotic limit on p a t low temperatures allows the liquid volumes to be very accurately matched in this region and of course the fitting of the critical point data leads to superior representation in the vicinity of the critical region. The matching of the critical 256
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
26 Theta versus the Parachor
15 14
I
I
1
1
point also brings the saturated vapor boundary into better alignment. The modified equation may also prove to be capable of describing the compressed liquid region as well as highly dense supercritical fluids. The original RKS equation usually sets b a t too large a value to allow these regions to be approached. The compressed liquid water system was studied and the use of the proposed modification gave a significant improvement as shown by the three isotherms of Figure 5. When simulating
way as to retain the equation’s ability to simulate vapor pressures. Furthermore, the changes only require the additional specification of the critical volume and the parachor. Unlike some previous attempts to modify the RK equation of state by assuming temperature-dependent a and b parameters (Chaudron et al., 1973; Kennedy and Bhagia, 1969), the equation satisfies the necessary conditions a t the critical point. The short investigation of compressed liquid water suggests that the modified equation can also be used to simulate PVT data in this regime as well.
-eiperimental
Critical Point (experimental
RKS RKS
---original ---modified
‘00 I
10
IO
100
Acknowledgment I t is a pleasure for the author to express his gratitude to R. A. Heidemann and R. G. Moore of the University of Calgary as well as M. Duncan of the California Institute of Technology for many helpful suggestions. The author is especially thankful for the computer time granted by R. A. Heidemann and L. G. Leal of C.I.T.
SPECIFIC VOLUME ( f t 3 / I b-mole )-
Figure 4. V a p o r - l i q u i d
equilibrium dome
Nomenclature
for water.
a = molecular interaction parameter b = excluded volume parameter c = coefficient on “b” in the proposed modification m = constant introduced in the Soave modification P = pressure g = parameter included to modify the Soave “m” constant for variable values R = universal gas constant T = absolute temperature u = specific volume = compressibility factor = Pu/RT P = parachor
5000s
4000-
z
Greek Symbols = Soave temperature function on the “a” parameter /3 = parameter introduced in the proposed modification R = coefficients on the “a” and “b” parameters w = Pitzer acentric factor
-
D
CY
Y
3
Lo X
2000
-
IOOO-
,,
0-
I 02
I 03
,
Subscripts a = refers to the “a” parameter b = refers to the “6” parameter c = refers to the critical point 0 = refers to the “low-temperature limit” on
1-
axprimentol -c-c-original RKS eq
-.---modified
I
04
l
l
05
I OB
R K S el
07
Figure 5. Pressure vs. v o l u m e f o r t h e compressed liquid water system.
this system for temperatures ranging from 32 to 700 OF and pressures in the range of 500 to 5000 psi, the original equation produced an average absolute error of 37.5% while the modified version gave 3.87% for 67 data points. Conclusions The RKS equation of state has been modified in order to allow for a variable critical compressibility factor. In addition, a univ,ersal temperature function has been incorporated into the equation allowing saturated liquid volume data to be accurately computed. These alterations are applied in such a
L i t e r a t u r e Cited Abbott. M. M., AIChf J., 19, 596 (1973). Canjar, L. N., Manning, F. S., “Thermodynamic Properties and Reduced Correlations for Gases”, Gulf Publishing Company, Houston, Texas, 1967. Chaudron, J., Asselineau, L.. Renon, H., Chem. Eng. Sci., 28, 839 (1973). Das, T. R., Kulor, N. R., /ndianJ. Techno/., 5, 69, 75 (1967). Harlacher, E. A,. Braun, W. G., Ind. fng. Chem., Process Des. Dev., 9, 479 (1970). Kennedy, H. T., Bhagia, N. S., S.P.E.J., 9, 279 (1969). Loeb, L. B., “The Kinetic Theory of Gases”, 3rd ed, pp 148, 176. Dover Publications, New York, N.Y., 1961. Quayle, 0. R., Chem. Rev., 53, 439 (1953). Redlich, 0..Kwong, J. N. S.,Chem. Rev., 44, 233 (1949). Reid, R. C., Sherwood, T. K., “The Properties of Gases and Liquids”, 2nd ed, McGraw-Hill, New York. N.Y., 1966. Soave, G., Chem. Eng. Sci., 27, 1197 (1972). Washburn, E. W., Ed., “International Critical Tables of Numerical Data Physics, Chemistry and Technology”, 1st ed, Vol. Ill, McGraw-Hill, New York, N.Y., 1928. “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons”, API research Project 44, 1953. Sontagg, R. E., Van Wylen, G. J., “Introduction to Thermodynamics Classical and Statistical”, Wiley, New York, N.Y., 1971.
Receiued for recieu September 8, 1975 Accepted J u n e 21,1976
Ind. Eng. Chem., Fundam., Vol. 15, No. 4, 1976
257
