Ind. Eng. Chem. Res. 2009, 48, 6901–6903
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CORRESPONDENCE Comments on “Joule-Thomson Inversion Curves and Third Virial Coefficients for Pure Fluids from Molecular-Based Models” and “Predicted Inversion Curve and Third Virial Coefficients of Carbon Dioxide at High Temperatures” Daniela Espinoza,† Hugo Segura,† Jaime Wisniak,‡ and Ilya Polishuk*,§ Departamento de Ingenierı´a Quı´mica, UniVersidad de Concepcio´n, POB 160-C, Concepcio´n, Chile, Department of Chemical Engineering, Ben-Gurion UniVersity of the NegeV, Beer-SheVa, 84105, Israel, and Department of Chemical Engineering & Biotechnology, UniVersity Center of Ariel, 40700, Ariel, Israel Sir: In the two papers under consideration,1,2 the authors claim the existence of a direct inter-relation between the accuracy of equation of state (EOS) models in predicting the third virial coefficient and the precision in describing the high-temperature range Joule-Thomson inversion curve (JTIC). Inter-relationships between different thermodynamic properties are extremely valuable, both for testing the prediction capability of models and/or for assessing the consistency of experimentally determined data. Sometimes, such inter-relations are obvious consequences of Maxwell’s relationships, and in other cases, they may be suggested by a sort of repetitive cause and effect empirical evidence. In the present case, however, the claimed inter-relation deserves an additional consideration as we discuss in what follows. Among other possibilities, the thermodynamic condition of the JTIC is given by3 ψ ) V˜ - T
( ∂T∂V˜ )
P
)0
(1.a)
Equation 1.a may be conveniently transformed for the purpose of developing alternative relations for describing the JTIC locus. For example, considering some typical thermodynamic differential identities, one can obtain
( ∂T∂V˜ )
)P
∂V˜ ( ∂P ∂T ) ( ∂P ) V˜
( )
P
∂V˜ )0⇒ V˜ ∂P T ∂P ∂P ψ* ) V˜ +T ∂V˜ T ∂T
∂P ) V˜ + T ∂T
( )( ) ( )
˜ ) -S˜ dT + V˜ dP dG
( )
V˜
)0
(1.c)
Equation 1.c is a useful and well-known expression for calculating inversion points from pressure-explicit models4 as in the case of van der Waals-type or molecular based models. With the purpose of further systematizing the calculation of JTICs from traditional thermodynamic potentials, let us consider again eqs 1 as the starting relation. Since the volume of a pure fluid depends on pressure and temperature, an obvious thermodynamic potential to be considered for describing the inversion * To whom correspondence should be addressed. E-mail: polishuk@ ariel.ac.il,
[email protected]. Phone: +972-3-9066346. Fax: +9723-9066323. † Universidad de Concepcio´n. ‡ Ben-Gurion University of the Negev. § University Center of Ariel.
(2)
From eq 2 we can deduce that V˜ )
( ) ˜ ∂G ∂P
T
˜P )G
(3)
Consequently, and considering the formalism of the Gibbs energy function, eq 1.a becomes ˜ P - TG ˜ PT ) 0 ψ)G
(4)
and, from this, we can deduce the following differential expansion: ˜ P2T dT + (G ˜ 2P - TG ˜ 2PT) dP dψ ) ψT dT + ψP dP ) -TG (5) According to eq 5, it follows that the temperature slope of the pressure along the JTIC (i.e., at nil and constant ψ value) is given by
(1.b)
T
from which the JTIC basic relationship may be equivalently written as the following constraint: ∂V˜ ψ ) V˜ - T ∂T
point is the Gibbs energy of the fluid, whose fundamental equation for a closed system is given by
dP dT
( )
)ψ
˜ P2T ψT TG ) ˜ 2P - TG ˜ 2PT ψP G
(6)
In addition, since according to eq 5, ψ depends on temperature and pressure, its derivatives depend on the same set of variables, thus yielding the following relation for the curvature along the JTIC:
( ) d2P dT2
)ψ
ψ2Tψ2P - 2ψTψPψPT + ψT2 ψ2P ψ3P
2 ˜ P3T(G ˜ 2P - TG ˜ 2PT)2 - T2G ˜ P2T ˜ 3P - TG ˜ 3PT) + TG (G
)
˜ 2P - TG ˜ 2PT)(G ˜ 2P - T(G ˜ 2PT - 2TG ˜ 2P2T)) ˜ P2T(G G ˜ 2P - TG ˜ 2PT)3 (G
(7)
Equations 4, 6, and 7 rigorously describe the geometric locus, the slope, and the curvature of the JTIC in terms of a model for the Gibbs energy potential of a fluid system. The quoted equations may be directly applied to numerically calculate the JTIC from the general EOS models by considering the following definition of the Gibbs energy function,
10.1021/ie900275w CCC: $40.75 2009 American Chemical Society Published on Web 06/18/2009
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Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009
˜ ) RT G
∫
P
0
C
C
Z-1 dP + RT xi ln xi + P i)1
∑
∑ x G˜ i
ig i (T)
i)1
(8) ˜ ig where Z corresponds to the compressibility factor and G i is the Gibbs energy of the pure ideal gas. Equation 8 is useful for volume (orsequivalentlysfor compressibility factor) explicit equations of state models, as the case of pressure virial expansions. Clearly, the application of eq 4 to eq 8 yields the following:
ψ)-
RT2 ∂Z P ∂T
( )
P
)0
(9)
Equation 9 is an additional useful relation for determining the JTIC, from which we can deduce that inversion points are directly reflected by stationary points of compressibility isobars along the Z-T projection, as it can be seen in Figure 1. Let us now consider the high-temperature range of the JTIC, which is generally related to a supercritical gas phase with volumetric properties that can be predicted from the following virial expansion in pressure
coefficients of the pressure series are related to the virial coefficients Bm of density series as follows:
∞
∑ B′
Z)1+
Figure 1. Graphical interpretation for the Joule-Thomson inversion, as deduced from eq 9 and the van der Waals EOS: (s) compressibility isobar, (- - -) JTIC.
k k+1P
(10)
k)1
B′2 ) In eq 10, Bm′ corresponds to the mth virial coefficient of the pressure series which, at constant concentration, depends on temperature only. Consequently, the functionality Z ) Z(P,T) is obvious for the EOS in eq 10, and applying eq 8, one can obtain the following model for the Gibbs energy function:
˜ ) RT G
∫
P
0
∞
C
Z-1 dP + RT xi ln xi ) RT Bk+1 ′ Pk + P i)1 k)1
∑
∑
B′3 ) B′4 ) B′5 )
B2 RT B3 - B22 (RT)2 B4 - 3B2B3 + 2B23
(14)
3
(RT) B5 - 4B2B4 - 2B32 + 10B22B3 - 5B24
l
(RT)4
C
RT
∑ x ln x i
i
(11)
i)1
Applying eq 4 to eq 11 allows deducing the following condition for the inversion point:
(∑
Consequently, for the case of the high-temperature inversion limit we obtain lim ψ ) 0 ⇒ -RT2
Pf0
dB2 dB2′ ) B2 - T )0 dT dT
(15)
(13)
Equation 15 corresponds to a well-known result for the Joule-Thomson inversion, from which it is interesting to note that the predicted high-temperature limit rigorously depends on B2 only. Consequently, since volume becomes infinite as the high temperature inversion limit is approached, the best numerical alternative for solving such JTIC point is (a) to analytically determine the second virial coefficient predicted by the EOS model under consideration and, then, (b) to calculate the inversion temperature root by means of eq 15. Using the geometric relationships that we developed before, it is also possible to obtain rigorous relations for the geometry of the JTIC at the high temperature limit.
Equation 13 is usually called the high temperature inVersion limit. At this point it is convenient to note that the virial
Let us now consider the geometry of the JTIC near the high temperature limit. Direct application of eqs 6 and 7 to eqs 11 and 14 leads, after straightforward algebra, to the following results:
∞
˜ P - TG ˜ PT ) RT ψ)G ∞
RT
∑ k)1
∑ B′
∞
k+1P
k)1
)
k-1
-T R
Bk+1 ′ P
k-1
+
k)1
∞ dBk+1 dBk+1 ′ ′ Pk-1 ) -RT2 Pk-1 dT dT k)1
∑
(12)
from which, over the low pressure range (i.e., P f 0), we obtain a known result:
ψ ) -RT2
dB′2 )0 dT
Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009
recently published work of Naresh and Singh. As a final remark, it should be noticed that the experimental data on the high temperature part of JTIC are scarce and usually inexact. At the same time, the conclusions reached relying on the empirical interpolations of these data are not very reliable.
d B2 lim
Pf0
( dP dT )
ψ)0
( )
d2P lim Pf0 dT2
[{
dT
) RT2
2
dB3 dT d2B2
2B3 - T RT
)-
ψ)0
dT2 dB3 2B3 - T dT
[
{〈
]
3
×
〉
} ]
d2B3 dB3 dB2 d2B2 + -4 12B3 + 4TB3 T T 2 2 dT dT dT dT dB3 2 d2B2 dB4 5 T +2 - 3B4 dT dT dT2 + T2 dB3 dB2 d2B2 d2B3 - 2T T + 2 dT dT dT dT2 2
[ ]
〈
〈
6903
5
2
〉
〉
Acknowledgment This work has been financed by ACS Petroleum Research Fund grant PRF No. 47338-B6 and by the FONDECYT, Chile (Project 1080596).
}
-
Literature Cited
3
RT2
d B2
dT3 dB3 2B3 - T dT
[
]
(16)
According to eqs 15 and 16, a physically meaningful prediction of the geometry of the JTIC oVer the high temperature range requires knowing up to the fourth Virial coefficient, together with their temperature deriVatiVes. From eqs 16, it is clearly seen that B2 and B3 are useful for determining the slope of the JTIC; however, in contrast to authors claim,1,2 these virial coefficients only are clearly not yet enough for determining the ending curVature. A similar conclusion has been reached in the
(1) Castro-Marcano, F.; Olivera-Fuentes, C. G.; Colina, C. M. JouleThomson Inversion Curves and Third Virial Coefficients for Pure Fluids from Molecular-Based Models. Ind. Eng. Chem. Res. 2008, 47, 8894. (2) Colina, C. M.; Olivera-Fuentes, C. G. Predicted Inversion Curve and Third Virial Coefficients of Carbon Dioxide at High Temperatures. Ind. Eng. Chem. Res. 2002, 41, 1064. (3) Deiters, U. K.; De Reuck, K. M. Guidelines for Publication of Equations of State. I-Pure Fluids. Pure Appl. Chem. 1997, 69, 1237–1249. (4) Segura, H.; Kraska, T.; Mejia, A. S.; Wisniak, J.; Polishuk, I. Unnoticed pitfalls of soave-type alpha functions in cubic equations of state. Ind. Eng. Chem. Res. 2003, 42, 5662–5673. (5) Naresh, D. J.; Singh, J. K. Virial coefficients and inversion curve of simple and associating fluids. Fluid Phase Equilib. 2009, 279, 47.
IE900275W
