3006
Ind. Eng. Chem. Res. 1998, 37, 3006-3011
A Simple Equation of State Valid in the Critical Region for Pure Compounds Gustavo A. Iglesias-Silva Departamento de Ingenierı´a Quı´mica, Instituto Tecnolo´ gico de Celaya Celaya, Guanajuato, CP 38010, Me´ xico
Kenneth R. Hall* Chemical Engineering Department, Texas A&M University, College Station, Texas 77843-3122
A simple, analytic expansion about the critical point appears capable of adequate representation for the compression factor in the single phase region near the critical point. The equation requires TC, PC, VC, and ω to predict the compression factor, and we have used it to represent the behavior of argon, carbon dioxide, ethene, methane, and nitrogen. The scaling hypothesis would indicate that this expansion is invalid; however, the best available data seem to be susceptible of such treatment. The equation can also represent phase behavior in the critical region by utilizing it with the Maxwell criterion. Introduction
and
The behavior of pure fluids in the region of their critical points has received considerable attention recently ranging from the classical description discussed by Landau and Lifshitz (1959) to the development of the scaling hypothesis as evidenced by the works of Griffiths (1965), Fisher (1967), and Levelt Sengers (1970). Confirmation of these latter methods requires high quality experimental data. In this work, we present a functional representation of the compression factor valid near the critical point in the single phase region. The function resembles the classical description of Landau and Lifshitz (1959), but it is explicit in the compression factor. We have tested the equation using the exceptional data collected at Ruhr Universita¨t Bochum in the Institut fu¨r Fluid und Thermodynamik. The Bochum group also modeled their data using accurate equations of state as reported by Setzmann and Wagner (1991), Span and Wagner (1996), Tegeler et al. (1997), and Span et al. (1998). Our equation cannot represent the data with the same accuracy as these accurate equations; however, our equation is much simpler, represents all compounds with the same form, and confirms the observations of Wagner and his co-workers that analytical equations are adequate in the critical region of pure fluids. Development The critical point imposes conditions that are useful for development of equations of state. These conditions in mathematical form are:
∂P (∂V )|
)
T CP
(∂P∂F ) |
T CP
)0
(1)
( )| ( )| ∂2P ∂V2
T CP
∂2P ∂F2
)0
(2)
T CP
where P is pressure, T is temperature, V is molar volume, and F is density. Higher order derivatives may be zero also as discussed by Baher (1963). Equations of state usually have the form:
P ) f(T, V or F) or Z ) f(T, V or F)
(3)
where Z ()P/FRT) is the compression factor and R is the gas constant, because such representations require fewer terms than forms explicit in P and T. The first and second derivatives of P with respect to F expressed in terms of Z and its derivatives are:
(∂P∂F )
) FRT
T
(∂Z∂F )
T
+ ZRT
(4)
and
( ) ∂2P ∂F2
) FRT
T
( ) ∂2Z ∂F2
() ∂Z ∂F
+ 2RT
T
(5)
T
In general, the nth derivative is:
( ) ∂nP ∂Fn
( ) ∂nZ ∂Fn
) FRT
T
+ nRT T
( ) ∂n-1Z ∂Fn-1
n g 1 (6)
T
Often it is convenient to express eq 6 as the derivative of Z with respect to F:
( ) ∂nZ ∂Fn
* Author to whom correspondence should be addressed. Telephone: (409) 845-3357. Fax: (409) 845-6446. E-mail:
[email protected].
)
T
)
Z
n
∑
P i)0
(-1)i+1
n! 1
() ∂iP
i! Fn-i ∂Fi
(7)
T
where n ) 1, 2, ..., ∞. Because density and temperature are inverse variables with respect to Z, the derivative
S0888-5885(97)00594-0 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/28/1998
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3007
( )| ( )|
of Z with respect to Τ at constant F is:
( ) n
∂ Z
∂Tn
F
)
Z
n
n! 1
∑(-1)i+1 i! P i)0
()
d2P dT2
i
∂P
Tn-i ∂Ti
(8) F
[ ( )] ∂nZ
m
∂T Z
m
∂F n
n!
m!
∑ ∑(-1)k+n-i(m - k)! i! Pk)0i)0
1 k n-i
TF
[ ( )] ∂
m-k
∂Fk
m-k
∂T
∂P ∂Fi
(9)
( )
CP
)
CP
k ZCk! {(-1)k + (-1)k-iβi0} i)3 FkC
∑
k ZCk! {(-1)k(R - 1) + (-1)k-i β0i} i)3 TkC
∑
|
)
( ) ∂Z ∂F2
∂Tj∂Fn-j CP
)2
T
F2C
(10)
T-G Plane. On the T-F plane, isobars have shapes similar to isotherms on the P-F plane such that:
(∂T∂F ) |
)0
(12)
∂2T ∂F2
)0
(13)
P CP
( )|
P CP
Following the same procedure as used for the P-F plane, we obtain:
(∂Z∂F ) |
P CP
( )| ∂2Z ∂F2
P CP
)0
)2
ZC F2C
j-2 n-j
(-1) ∑ l)3
βl,0 +
(-1)n-l-1 βl,1 + (R - 1)(-1)n-1} ∑ l)1
(20)
where the β denote cross partial derivatives of P with respect to T and F in reduced form. An Equation of State for the Critical Region If we assume that the equation of state is analytic in the single phase region (which is possible even if the critical point is nonanalytic in the two-phase region) it is possible to make a Taylor series expansion of Z ) Z(T, F) about the critical point: ∞
Z ) ZC +
n
1
∑ ∑ j!(n - j)!
(14)
The derivatives (at least through the second) at the critical point are the same on both the T-F and P-F planes. P-T Plane. On the P-T plane, the slope of the critical isochore equals the slope of the vapor pressure curve at the critical point of a pure component. The slope of the vapor pressure at the critical point is proportional to the Riedel constant:
( )|
n-j
n-l
n)0j)1
(15)
( )|
TC dP σ TC ∂P ) ≈ 5.808 + 4.917ω (16) Rk ≡ PC dT CP PC ∂T F CP where superscript σ denotes the saturation curve (vapor pressure) and ω is the acentric factor. Although this approximation cannot represent the experimental data within its expected uncertainty, it can provide R values for compounds that lack excellent data with enough accuracy for the purposes of this paper. In addition, the critical point is an inflection point for the vapor pressure/critical isochore line:
(19)
∑ ∑(-1)k+n-j-l βl,j-k k)0t)1 +
(11)
(18)
∑
n-j
ZC
(17)
ZC j!(n - j)! j-2 { (-1)k+n-j β0,j-k + n-j j k)0 F C TC
and 2
)0
F CP
and
∂ nZ
For n ) 1, 2 at the critical point, eq 7 becomes:
ZC ∂Z )∂F T FC
|
)
∂Tk CP
i
|
∂kZ
∂kZ
)
n
∂2P ∂T2
)
Equations 7-9 at the critical point are:
The cross derivative of Z with respect to temperature and density is:
∂m
σ
[ ] ∂nZ
× ∂Tj∂Fn-j CP (T - TC)j(F - FC)n-j (21)
Our analysis can start with a single equation such as
( )| ( )|
Z ) ZC +
∂Z ∂T
∂Z ∂F
CP
CP
( )| ( )|
(F - FC) +
(T - TC) +
( )| ∂2Z ∂T∂F
CP
∂2Z ∂F2
∂2Z ∂T2
(F - FC)2 + 2! CP
(T - TC)2 + 2! CP
(T - TC)(F - FC) + ... (22)
Substituting eqs 12, 16, and 17 into eq 22 results in:
( ) ( ) [( ) ( ( )(
F - FC F - FC 2 Z - ZC )+ + ZC FC FC T - TC T - TC 2 + (R - 1) TC TC βFT T - TC F - FC -1 R-1 TC FC
) )( )]
(23)
Equation 23 is the simplest form of a Taylor’s expansion that could fit data in the critical region. Obviously, higher derivative terms, if included, could extend the
3008 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 Table 2. Values of the Parameters Used in the Equation of State for Argona param- esti- standard eter mate error
Figure 1. Region within which eq 24 can predict (using eqs 16 and 25) Z within 0.1%.
β11 β12 β13 β14 β15 β21 β22 β23 β24 β31 β32 β33 β41 β42 β51 β03 β04 β05 β06 β30 β40 β50 β60 a
5.888
Note:
Table 1. Critical Parameters substance
TC, K
argon carbon dioxide ethene methane nitrogen
150.687 304.128 282.350 190.551 126.192
PC, MPa 4.8630 7.3773 5.0418 4.5992 3.3958
FC, kg m-3
ω
535.60 467.60 214.24 162.66 313.30
0.0000 0.2210 0.0865 0.0120 0.0380
applicability of the equation over a wider range. It is possible to estimate βFT from:
( )
∂2Pr ≈ 6.3 + 2.715ω ∂Tr∂Fr
(24)
Thus eq 23 can predict data for the critical region given the critical properties, acentric factor, and eqs 16 and 24. Figure 1 illustrates the region in which eq 24 can predict Z within 0.1% using eqs 16 and 24. Results To test the applicability of this method, we have applied eq 23 as well as the corresponding equations including through 4th and 6th derivatives to predict the compression factor of argon, carbon dioxide, ethene, methane, and nitrogen. A comparison between eq 23 and experimental data validates eq 23. Data sources are Gilgen et al. (1994), Duschek et al. (1990), Nowak et al. (1996), Kleinrahm et al. (1987), and Wagner (1997) for argon, carbon dioxide, ethene, methane, and nitrogen, respectively. Table 1 contains the properties of the substances used in this work. Tables 2-6 contain the values (and standard deviations) of the cross derivatives obtained from fits of the data using Taylor’s expansions through the 2nd, 4th, and 6th derivatives. In each case, the range of data has been restricted such that the deviations are random and within the experimental error estimates. For these data sets, we have estimated the standard deviation to be ∼0.02% for the compression factors. We arrived at this figure by accepting the estimates of the experimenters for the densities of 0.01-
0.08
βij )
estimate
standard error
estimate
standard error
5.811 12.066 -56.29
0.08 1.8 8.4
-9.615 14.33
0.43 4.2
13.08
1.8
3.241 -64.62
0.82 5.6
-0.858 -0.519
0.08 0.51
5.803 15.456 -105.86 -14 471 -9.537 25.38 -259 427 16.709 -69.1 4 13.36 -77.05 10.84 3.165 -96.56 165 94 -0.809 0.012 3.408 3.507
0.04 0.88 6.9 24 36 0.13 2.1 16 36 0.44 4.4 12 1.2 5.4 1.6 0.43 5.0 19 18 0.02 0.07 0.26 0.97
( )
FiC TjC ∂i+jP i!j!PC ∂Fi∂Tj
.
CP
Table 3. Values of the Parameters Used in the Equation of State for Carbon Dioxidea param- esti- standard eter mate error β11 β12 β13 β14 β15 β21 β22 β23 β24 β31 β32 β33 β41 β42 β51 β03 β04 β05 β06 β30 β40 β50 β60 a
6.341
Note:
0.30
estimate
standard error
6.392 32.34 -179
0.09 2.6 16
-11.166 3.72
0.33 4.1
18.13
1.2
29.42 -272
1.4 13
-0.823 -0.143
βij )
( )
FiC TjC ∂i+jP i!j!PC ∂Fi∂Tj
0.03 0.16
estimate
standard error
6.310 0.05 41.78 2.2 -362 39 -274 290 8778 1413 -12.950 0.10 49.32 3.8 -907 99 4097 671 23.657 0.38 -175 13 450 83 17.419 0.72 -146.03 9.0 -22.381 0.71 63.11 3.3 -1105 83 4966 661 -3146 1709 -0.821 0.00 0.212 0.01 2.737 0.03 0.906 0.03
.
CP
0.015%. We then assumed the reported temperatures are accurate within 0.001 K and that the pressures are accurate to ∼0.01%. Then, using propagation of errors, the 1σ uncertainty in Z would be ∼0.02%. Therefore, we have restricted the ranges of data such that the fits replicate about 60-70% of the data within 0.02% and all of the data within 0.06%. Interestingly, it does not appear that the third derivative of pressure with respect to density is zero although, in some cases, the fourth derivative is zero and in all cases it is near zero. Although the βij are actually fit parameters, it is
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3009 Table 4. Values of the Parameters Used in the Equation of State for Ethenea
Table 6. Values of the Parameters Used in the Equation of State for Nitrogena
param- esti- standard eter mate error
param- esti- standard eter mate error
β11 β12 β13 β14 β15 β21 β22 β23 β24 β31 β32 β33 β41 β42 β51 β03 β04 β05 β06 β30 β40 β50 β60 a
5.655
0.70
estimate
standard error
estimate
standard error
5.751 26.93 -1234
0.06 1.7 11
-9.851 12.05
0.26 3.5
16.08
1.3
31.51 -276.55
1.0 9.9
5.708 31.38 -232 10 3221 -10.921 34.81 -440 1329 23.74 -137 107 25.42 -181 -7.72 50.10 -800 3550 -3515 -0.932 0.057 5.097 8.29
0.05 1.5 17 82 139 0.24 5.5 43 112 1.1 14 41 2.9 18 5.4 1.3 26 151 204 0.01 0.03 0.30 1.1
-0.824351 0.115
Note:
βij )
( )
FiC TjC ∂i+jP i!j!PC ∂Fi∂Tj
0.01 0.04
β11 β12 β13 β14 β15 β21 β22 β23 β24 β31 β32 β33 β41 β42 β51 β03 β04 β05 β06 β30 β40 β50 β60 a
.
CP
5.530
Note:
0.18
βij )
estimate
standard error
estimate
standard error
0.06 1.4 7.9
5.683 25.33 -182 123 1244 -11.256 52.53 -415 779 20.975 -60.48 -147 34.41 -135.57 13.61 20.052 -349.75 1339 -1032 -0.935 -0.150778 5.893 10.886
0.03 0.87 8.5 33 48 0.16 3.2 21 48 0.83 6.4 19 2.5 9.3 3.6 0.57 9.7 50 55 0.01 0.02 0.18 0.58
5.684 23.60 -109.99
-9.658694 17.80
0.24 2.8
12.77
1.7
11.799 -127.46
0.51 4.3
-0.805 0.089
0.01 0.03
( )
FiC TjC ∂i+jP i!j!PC ∂Fi∂Tj
.
CP
Table 5. Values of the parameters used in the Equation of State for Methanea param- esti- standard eter mate error β11 β12 β13 β14 β15 β21 β22 β23 β24 β31 β32 β33 β41 β42 β51 β03 β04 β05 β06 β30 β40 β50 β60 a
5.000
Note:
0.22
estimate
standard error
5.082 39.21 -241
0.07 2.3 16
-9.488 30.76
0.28 4.4
19.41
1.4
32.48 -305
1.7 18
-0.803 0.042
βij )
0.01 0.09
( )
FiC TjC ∂i+jP i!j!PC ∂Fi∂Tj
estimate
standard error
5.107 0.04 42.85 1.9 -340 46 -1699 522 26345 3565 -10.549 0.09 64.45 4.0 -1044 86 4792 590 25.259 0.42 -201.26 9.2 396 64 23.688 0.77 -270 12 -30.81 1.5 110.56 8.7 -2786 326 22770 3570 -49705 9731 -0.828 0.00 0.109 0.01 3.283 0.02 2.104 0.07
.
CP
reasonable to assume that, if they have essentially the same values for different fits (e.g., 2nd, 4th, and 6th derivative expressions), they represent acceptable approximations for the actual derivatives as noted by Hall and Canfield (1967). Figures 2-6 illustrate the 2nd, 4th, and 6th derivative fits for each of the substances. The only two-phase data of acceptable quality in this region come from a private communication by Wagner (1997); these data are preliminary and have not yet appeared in the reviewed literature, but they are more than good enough for our
Figure 2. Correlation of Ar data: (A) 2nd derivative equation; (B) 4th derivative equation; (C) 6th derivative equation.
purposes. Furthermore, our equation, coupled with the Maxwell criterion can represent them as well as the single phase data. Obviously, we cannot present plots of these data before they appear under the names of the collectors. All uncertainties in Z reported here are:
∆Z ) (Zobs - Zcal)/Zobs
(25)
where the subscripts obs and cal denote observed and calculated, respectively.
3010 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998
Figure 3. Correlation of CO2 data: (A) 2nd derivative equation; (B) 4th derivative equation; (C) 6th derivative equation.
Figure 5. Correlation of CH4 data: (A) 2nd derivative equation; (B) 4th derivative equation; (C) 6th derivative equation.
Figure 6. Correlation of N2 data: (A) 2nd derivative equation; (B) 4th derivative equation; (C) 6th derivative equation. Figure 4. Correlation of C2H4 data: (A) 2nd derivative equation; (B) 4th derivative equation; (C) 6th derivative equation.
Conclusions A simple Taylor's expansion in temperature and density appears capable of correlating compression factor data in the near critical region for pure substances. Depending on the number of derivatives included in the expansion, the reduced density range varies from ∼0.9-1.02 up to 0.55-1.3. The first nonzero derivative of pressure with respect to density appears to be the third. The expansion can describe both single-
and two-phase data. The equation has the same form for all substances and, given the critical properties and the acentric factor, can even do a reasonable job of predicting the compression factor in the near critical region. Acknowledgment The authors acknowledge financial support from Texas Engineering Experiment Station and Instituto Tecnolo´gico de Celaya for this project.
Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3011
Literature Cited Baher, H. D. Das Verhalten der spezifischen Wa¨rmekapazita¨t cv und der Entropie am kritischen Punkt des Wassers. Brennstoff. Wa¨ rme Kraft 1963, 15, 514. Duschek, W.; Kleinrahm, R.; Wagner, W. Measurement and correlation of the (pressure, density, temperature) relation of carbon dioxide. I. The homogeneous gas and liquid regions in the temperature range from 217 K to 340 K at pressures up to 9 MPa. J. Chem. Thermodyn. 1990, 22, 827. Fisher, M. E. Rep. Prog. Phys. 1967, 30, 615. Gilgen, R.; Kleinrahm, R.; Wagner, W. Measurement and correlation of the (pressure, density, temperature) relation of argon. I. The homogeneous gas and liquid regions in the temperature range from 90 K to 340 K at pressures up to 12 MPa. J. Chem. Thermodyn. 1994, 26, 383. Griffiths, R. B. Ferromagnets and simple fluids near the critical point: some thermodynamics inequalities. J. Chem. Phys. 1965, 43, 1958. Hall, K. R.; Canfield, F. B. Optimal recovery of virial coefficients from experimental compressibility data. Physica Utrecht 1967, 33, 481. Kleinrahm, R.; Duschek, W.; Wagner, W. (Pressure, density, temperature) measurements in the critical region of methane. J. Chem. Thermodyn. 1986, 18, 1103. Kohler, F. Liquid State, Monographs in Modern Chemistry; Verlag Chemie Gmbh: Weinheim, 1972. Landau, L. D.; Lifshitz, E. M. Statistical Physics; Pergamon: London, 1956. Levelt Sengers, J. M. Scaling predictions for thermodynamics anomalies near gas-liquid critical point. Ind. Eng. Chem. Fundam. 1970, 9, 470.
Nowak, P.; Kleinrahm, R.; Wagner, W. Measurement and correlation of the (p, F, Τ) relation of ethylene. J. Chem. Thermodyn. 1996, 28, 1423. Riedel, M. L. Eine neue universelle Dampfdruckformel. Chem.Ing.-Tech. 1954, 26, 83. Setzmann, U.; Wagner, W. A new equation of state and tables of thermodynamic properties for methane covering the range from the melting line to 1100 K at pressures up to 800 MPa. J. Phys. Chem. Ref. Data 1991, 20, 1061. Span, R.; Wagner, W. A new equation of state and tables of thermodynamic properties for carbon dioxide covering the fluid region from the triple point temperature to 625 K at pressures up to 1000 MPa. J. Phys. Chem. Ref. Data 1996, 25, 1509. Span, R.; Lemmeon, F. W.; Jacobsen, R. T.; Wagner, W. A reference quality equation of state for nitrogen. Int. J. Thermophys. 1998, submitted for publication. Tegeler, C.; Span, R.; Wagner, W. Eine neue Fundamentalgleichung fu¨r das fluide Zustandsgebeit von Argon fu¨r Temperaturen von Schmeltzlinie bis 700 K und Dru¨cke bis 1000 Mpa. Fortschr. Ber. VDI; VDI-Verlag: Du¨sseldorf, 1997; Reihe 3, Nr. 480. Wagner, W. Ruhr Universita¨t Bochum, personal communication, 1997 (paper to appear in J. Chem. Thermodyn.).
Received for review August 28, 1997 Revised manuscript received December 10, 1997 Accepted February 3, 1998 IE970594N
