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Ind. Eng. Chem. Res. 2007, 46, 9248-9256
Novel Four-Parameter EOS with Temperature-Independent Parameters Ilya Polishuk* The Department of Chemical Engineering & Biotechnology, The Ariel UniVersity Center of Samaria, 40700 Ariel, Israel
The present study aims at developing a simple, robust, and reliable equation of state (EOS) for pure fluids and their mixtures. It is demonstrated that the temperature dependencies attached to the parameters of the existing engineering EOS models, and which may result in nonphysical predictions, can be replaced by the slight improvement of the theoretical base. The novel approach not only keeps the EOS model simple but also removes the need for the tedious fitting of the temperature-dependent parameters. In addition, it is free of numerical pitfalls and demonstrates a surprising reliability in predicting different thermodynamic properties in a wide range of temperatures and pressures. The new EOS could be a good starting point for further development. Introduction Development of accurate models for predicting thermodynamic properties and phase equilibria of fluids with complex molecular geometry and their mixtures over wide ranges of temperature and pressure is an extremely important problem for process design in the chemical industry that has not been satisfactorily solved yet. The present proposal aims at developing a simple, robust, and reliable equation of state (EOS) for small, large, and associating molecules and their mixtures, suitable for industrial and tutorial purposes. Modern thermodynamics is dealing with theoretically advanced approaches, such as the thermodynamic perturbation theory for chain molecules,1 for answering the question. This theory uses the thermodynamic properties of unconnected monomers of chained molecules as a starting point (reference fluid) and treats the chain connectivity as perturbation. Its engineering application is known as the statistical associating fluid theory (SAFT),2,3 which has attracted great attention because of its practical relevance as a modeling tool. However, we have found4 that SAFT2 yields some undesired predictions. In particular, in addition to the common vapor-liquid coexistence region, it generates a stable pure-compound liquid-liquid split. These nonphysical phase splits may interrupt the ordinary phase equilibria in mixtures, which affects the robustness and the reliability of the modeling. Lately, Yelash et al.5 have investigated with great detail a perturbed-chained modification of SAFT2 (PC-SAFT)6 and discovered that, in addition to the liquid-liquid splits, the model also predicts a gas-gas demixing in pure compounds. The thermodynamic perturbation theory for chain molecules prescribes some temperature dependencies, which are strongly justified by the molecular reality. Nevertheless, these temperature dependencies can be responsible for numerical pitfalls, such as the nonphysical crossing of isotherms and additional stable critical points. If the EOS has many real roots for the molar volume, the chances for such nonphysical behavior substantially increase. Remarkably, similar undesired results can also be predicted by other models derived from molecular theories, such as the Johnson-Zollweg-Gubbins (JZG)7 model for the Lennard-Jones fluids4 and the Lennard-Jones-Devonshire cell model8 (see also ref 9). * E-mail:
[email protected]. Phone: +972-3-9066346. Fax: +972-3-9066323.
Figure 1. HC virial coefficients and their fitting by eqs 7-10 as a function of m: (O), (B), second and third pearl-necklace HC virial coefficients;19 and (3), (1), second and third LTHS model’s HC virial coefficients.19
Table 1. Values of Hard-Sphere Virial Coefficients β2
β3
β4
β5
β6
β7
theoretical16
4
10
18.3648
eq 7 eq 8
4 4
10 10
25 21.3333
28.2245 ( 0.0003 62.5 43.3333
39.38 ( 0.38 156.25 86.9333
56.1 ( 2.3 390.625 173.956
After several years dedicated to the investigation of the pitfalls generated by the theoretically based EOSs, we have concluded that the numerical mechanisms responsible for the phenomena are much more complex than the ones present in well-known cubic EOSs.4 Unfortunately, so far we been unable find a way to guarantee the reliability of the theoretically based EOSs in the entire thermodynamic phase space. Therefore, here we propose a different strategy for developing a reliable EOS for complex molecules. In spite of deriving engineering applications directly from the advanced molecular approaches and being unaware of possible numerical pitfalls, we can move just in the opposite direction, namely, gradual improving the theoretical base of a very simple model while carefully controlling its robustness.
10.1021/ie070799o CCC: $37.00 © 2007 American Chemical Society Published on Web 11/28/2007
Ind. Eng. Chem. Res., Vol. 46, No. 26, 2007 9249
hard spheres (other equations may have different expressions for the covolume). The packing fraction of chained molecules built by spherical segments is extended with the chain parameter m:20
y)
mb 4V
(4)
Describing the repulsive pressure by eq 2 is not practical for developing EOS models. This is because the virial series does not converge to infinite pressure as the volume reaches the value of covolume. The desired behavior can be generated by a fraction function of the following kind:
Figure 2. Experimental vapor pressure data of methane42 (m ) 1), ethane43 (m ) 2), propane44 (m ) 3), and n-butane45 (m ) 4), and their prediction by eq 13 with B2 ) 0.34 Vc,expt and Vc,EOS ) 1.07Vc,expt (solid lines), and by RK-Twu41 EOS (dotted lines).
Theory We propose to start with a simple van der Waals (vdW)-like EOS, which provides only a crude description of molecular interactions, but it avoids multiple volume roots. Thus, if its parameters are not attached with temperature functionalities, this model may guarantee physically reasonable predictions in the entire temperature and pressure range. Although an accurate vdW-like EOS with temperature-independent parameters could be a perfect starting point for further development, so far this option has not received appropriate attention in the literature. Only numerous references have recently dealt with such EOSs (see, for example, refs 10-13). However, an accurate prediction of the vapor pressure data of real fluids has not been considered. It has also been demonstrated14 that the vapor pressures can be fitted combining the vdW and Dieterici potentials, without attaching the parameters by empirical temperature dependencies. In what follows, we will check if such an option is at all possible for the sole vdW-like potential. The vdW-like equations (cubic and noncubic) describe the pressure as the contribution of the repulsive and the attractive intermolecular forces:
P ) Prepulsive - Pattractive
(1)
For the repulsion term, the simple hard-sphere (HS) or hardchain (HC) models can be employed. They are expressed by the virial equation as follows, m)∞
Zrepulsive ) 1 +
βm + 1ym ∑ m)1
(2)
where β2, β3, etc. are the virial coefficients, which are related to the molecular forces that exist between molecules.15 Thus, β2 represents interactions between two molecules, β3 represents interactions between three molecules, and so forth. The values of the virial coefficients in eq 2 for the HS and HC models are available elsewhere16-19 (see also Table 1). In eq 2, y is the packing fraction, which for the spherical molecules is given as
y)
b 4V
(3)
Here, V is a molar volume and b, according to the original vdW EOS, is the covolume, the volume occupied by 1 mol of
Zrepulsive )
F(y) 1 - ζy
(5)
F(y) should be developed in such a way that the virial expansion of eq 5 will yield the theoretical values of the HS or HC virial coefficients (βm). ζ is a packed limit, and the value of covolume is ζmb/4. The packed limit of spheres of the same size is given by21
ζ ) 6/πx2 ≈ 1.35047
(6)
However, this criterion is not satisfied by the vast majority of the existing EOS models, whose theoretical background is based on the HS approach. For example, the typically implemented cubic EOSs22,23 for process design use ζ ) 4, as originally set by van der Waals. The very popular EOS of Carnahan and Starling,24 implemented by some modifications of SAFT,25 uses ζ ) 1. In fact, nonspherical bodies and mixtures of spheres different in size will also not satisfy eq 6. At the present level of knowledge, it does not seem practical to seek for the appropriate values of ζ for all these cases. Thus, in our work, we will concentrate on the other basic criterion assigned to eq 5, namely, avoiding multiple roots for the molar volume. Complex equations that may generate multiple real roots for V in the one-phase region of pure compounds are totally unacceptable. Moreover, we propose to start with the EOS that will yield no more than four real roots for the two-phase region. This is because, even in the temperature-independent cases, the multiple real roots in the two-phase region may result in some not-entirely-robust phenomena, such as multiple spinodal curves. Further attachment of such EOSs by temperature dependencies is a certain recipe for a numerical-complication mess, whose consequences we have described above. This severe limitation will restrict us to the very simple expressions for eq 5, which seem unable to yield the right values for more than the first two HS virial coefficients. Let us consider the two possible expressions of the following kind:
Zrepulsive )
1 + K 1y 1 - K2y
(7)
Zrepulsive )
exp(K1y) 1 - K2y
(8)
The only appropriate values of K1 and K2 that allow matching both β2 and β3 for eq 7 are 1.5 and 2.5, respectively, and for eq 8, K1 ) K2 ) 2. Different approaches for selecting the values of K1 and K2 have been discussed in other studies.26-28 Table 1 compares the outcome of expanding eqs 7 and 8 into series, with the pertinent theoretical16 values of the HS virial coefficients. The table shows that eq 8 is superior in fitting the HS
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Figure 3. Prediction of the PVT data of light alkanes by eq 13: (O), saturated data; (0), single-phase data; solid lines, predicted data.
Figure 4. Saturation data of argon46 (m ) 1, B2 ) 0.34 Vc,expt, and Vc,EOS ) 1.07Vc,expt).
Figure 5. Saturation data of xenon47 (m ) 1, B2 ) 0.34 Vc,expt, and Vc,EOS ) 1.07Vc,expt).
theory; however, eq 7 has some very important advantages. Its incorporation into a vdW-like EOS allows analytical solutions for V. Such a feature is advantageous if a Legendre transformation of the properties is required. The need for the precise fit of the high HS virial coefficients requires additional discussion. The higher HS virial coefficients are important for the description of the condensed phases. Nevertheless, it must be remembered that simple EOSs provide only a very crude approach of the real molecular picture. Hence, it does not seem reasonable to focus on the detailed theoretical
description of only repulsive interactions, while taking into account the attractive ones by a very simple empirical expression. Thus, we should not be surprised by the fact that the adjustment of eq 5 to all the theoretical values given in Table 1 does not yield any visible improvement in describing the experimental data. The only consequence of such a practice is a potentially dangerous increase of the complexity of the equations. Several theories that yield the values for the hard-chain virial coefficients are available in the literature (see, for example, refs
Ind. Eng. Chem. Res., Vol. 46, No. 26, 2007 9251
Figure 6. Vapor pressure data of nitrogen48 (m ) 2) and krypton49 (m ) 1) and the available saturated molar volume data of krypton49 (B2 ) 0.34 Vc,expt and Vc,EOS ) 1.07Vc,expt for both).
Figure 7. PVT data of nitrogen:48 (O), saturated data; (0), single-phase data; solid lines, predicted data.
Figure 8. Saturation data of Twu41 EOS.
toluene:50
solid lines, eq 13; dotted lines, RK-
19 and 29-31). However, as the number of the HC coefficient grows, the deviations between different theories increase. Making eq 7 dependent on the value of the chain parameter m, one could fit the second and the third HC virial coefficients evaluated from these theories without affecting the simplicity of the EOS. We have selected to fit the pearl-necklace theory’s HC virial coefficients reported by Vega et al.19 This theory considers a fully flexible chain of m tangent hard spheres, which
Figure 9. Joule-Thomson inversion curve of toluene:50 solid lines, eq 13; dotted lines, RK-Twu41 EOS; dotted-dashed lines, C4EOS;34 dashed lines, Peng and Robinson EOS.23
Figure 10. Joule-Thomson inversion curve of methane:52 solid line, eq 13; dotted line, RK-Twu41 EOS; dotted-dashed line, C4EOS;34 dashed lines, Peng and Robinson EOS.23
Figure 11. PVT data of water: (O), saturated data;53 (∆)53 and (0),52 singlephase data; solid lines, data predicted by eq 13; m )1, B2 ) 0.36 Vc,expt, and Vc,EOS ) 1.125Vc,expt; dotted lines, RKS-Twu EOS.
represents an idealized reality of polymer solutions. The results of the fitting are presented by eqs 9 and 10 and Figure 1.
K1 ) 0.477341 + 1.022659m0.758763
(9)
K2 ) 1.48380 + 1.01620m0.669909
(10)
Figure 1 shows that the system of eqs 7-10 is capable of a very accurate presentation of the second and the third pearl-
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Figure 12. Saturated liquid molar volumes of water:53 solid lines, eq 13; dotted lines, RK-Twu41 EOS.
necklace HC virial coefficients.19 The data for the fourth virial coefficient are very diverse, depending on the simulation method. Therefore, we do not consider it here. A separate mixing of the chain parameters m through the one-fluid vdW approach creates undesired consequences for both the accuracy and the numerical simplicity. These problems can be avoided by defining the following auxiliary parameters:
B1 )
K1mb 4
(11a)
B2 )
K2mb 4
(11b)
It can be seen that B2 is the covolume of an EOS. Its value should represent the experimental data at very high pressures. In addition, it can be seen that
B1 )
K1B2 K2
(12)
We propose to start with the following four-parameter vdW-like EOS:
P)
RT 1 + (B1/V) a V 1 - (B2/V) (V + c)(V + d)
(13)
Four-parameter EOSs are more accurate and flexible than the widely used two-parameter ones.32-34 Unfortunately, they are less convenient for the engineering practice because, so far, they have not been attached by the analytical expressions for their parameters. However, after some algebra, such expressions can be derived by setting the first and second derivatives of P with respect to V equal to zero at the critical point and implementing the experimental values of Vc as the third condition:
a)
{
(B1RTc + Vc(ψ - PVc))3 RTcχ
(14)
xRTcVc(Vc (ψ - 3PcVc) + B1(3Vc(3B2Pc + RTc) -2 3B2ψ -3/28PcVc )) -
c)
2
2
ξ(B1RTc + PcVc(B2 - Vc) + RTcVc) - B1 B2(RTc)
}
2xRTcχ
{
3B1(RTcVc)2 + ξxRTc(B1RTc + Vc(ψ - PcVc)) + (RTc)2Vc3 - B2(B1RTc)2 3B1B2RTcVc(B2Pc + RTc - 3PcVc) - PcRTcVc3(8B1 + 3Vc - B2) d) 2RTcχ
(15)
}
(16)
where
ξ)
x
(B1B2)2RTc + Vc4(4B2Pc + RTc - 4PcVc) + 2B1Vc(2B22(B2Pc + RTc) B2Vc(8B2Pc + 3RTc) + 2Vc2(6B2Pc + RTc) - 6PcVc3) χ ) B1B22Pc + B1RTc(B1 + B2) - 3B1PcVc(B2 - Vc) + PcVc3
(17) (18)
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and
ψ ) B2Pc + RTc
(19)
It should be noticed that the proposed EOS may have up to four real solutions for the molar volume. Although it is not cubic, it still does not generate the nonphysical phenomena described above. The fugacity coefficients for pure compounds and for the compounds in mixtures according to eq 13 are given by
ln φ ) 1 + Z + ln φ ˆi )
[ ]
V+c - (B1 + B2)* V+d V (c - d)RT ln V - B2
aB2 ln
[
]
B2RT(c - d)
(20)
(B1 + B2)B2i′
+ (V - B2)B2 V ln (B (B +B2 + B1i′) - B1B2i′) V - B2 2 1
[
]
B22 a((V + c)di′ - Vci′ - dci′) ln
- ln[Z]
[ ]
RT(V + c)(V - d)(V + d)
-
V+c ((c - d)ai′ + a(c - d - ci′ + di′)) V+d RT(c - d)2 Qi′ )
+
∂nQ (Q ) a, B1, B2, c, d) ∂ni
- ln[Z] (21) (22)
ni is the number of moles of the compound i in a mixture. As the first approach, we will apply eq 13 to mixtures using the one-fluid vdW model: N
Q)
N
∑ ∑ xixjQij
(23)
i ) 1j ) 1
aij ) xaiiajj
(24)
Ordinary arithmetic average combining rules for the volumetric parameters are less preferable from the theoretical viewpoint35 than the following expression, 3
3
(xQ + xQ ) Q ) ii
ij
3
jj
8
(25)
where Q ) B1, B2, c, and d. These combining rules will be used here for eq 13. Since 1949,36 significant developments of engineering vdWlike EOSs have been achieved by attaching their parameters with empirical temperature dependencies, in order to improve the fitting of the saturation pressures of pure compounds.37 Unfortunately, the impact of these dependencies on predicting other thermodynamic properties has not always received an appropriate attention. Moreover, it has been found4,33 that these dependencies can be responsible for some nonphysical predictions (see also discussion in refs 38 and 39). In what follows, we will demonstrate that the temperature dependencies can be replaced just by a minor improvement of the EOS’s theoretical base. Results As a first very crude approach, we have neglected the rotation of chained molecules and the association of polar fluids by
taking m to be equal to the number of segments for nonbranched molecules, an approximation that works surprisingly well for m < 20. In addition, in order to attach an entirely predictive character to eq 13, we have used for all nonpolar molecules with
