# New attraction term for the Soave-Redlich-Kwong equation of state.

INTRODUCTIONThe knowledge of the vapour-liquid-equilibrium (VLE) is important for the modelling, the design and the operation of refrigeration processes. Cubic equations of state (CEoS) are currently considered as the most appropriate for VLE calculations (Vidal, 1997; Twu et al., 2002).

The general form of a cubic equation of state can be written as (Abbott, 1973):

P = RT/([upsilon] - b) - a/([upsilon]2 + [delta][upsilon] + [epsilon]) (1)

where b, [delta] and [epsilon] are fluid specific constants, and a the usually temperature dependant attraction term.

Equation (1) may be viewed as a generalization of the van der Waals equation, to which it reduces as the simplest case (a = const; [delta] = [epsilon] = 0). The Soave-Redlich-Kwong EoS,

P = RT/([upsilon] - b) - a(T)/[upsilon]([upsilon] + b) (2)

is deduced from (Equation (1)) when [delta] = b and [epsilon] = 0.

It has been found (Twu et al., 2002) that the predictive capability of a CEoS depends strongly--but not uniquely--on the attraction term, a(T), and hence great efforts have been concentrated in the past in the search for the appropriate form of [alpha]([T.sub.r]) in the expression (Geana and Feroiu, 1992; Twu et al., 1995; Twu et al., 2002; Valderrama, 2003; Nasrifar and Bolland, 2004):

a(T) = [a.sub.c][alpha]([T.sub.r]) (3)

where [a.sub.c] is the value of a at the critical point.

In the original [alpha]([T.sub.r]) function proposed by Soave (1972),

[alpha]([T.sub.r]) = [1 + m(1 - [[square root of [T.sub.r])].sup.2] (4)

the adjustable parameter m is obtained by forcing the equation to reproduce the vapour pressure for non-polar compounds at [T.sub.r] = 0:7. It has been correlated to the Pitzer acentric factor [omega]:

m = [M.sub.0] + [M.sub.1][omega] + [M.sub.2][[omega].sup.2] (5)

with

[M.sub.0] = 0.48; [M.sub.1] = 1.574 and [M.sub.2] = -0.176

The Soave-Redlich-Kwong (SRK) equation gives satisfactory results in the calculation of the VLE with non-polar and slightly polar components. But, it represents poorly the supercritical state: the properties of the fluids in this region are inaccurately predicted, in particular, derived properties such as the second virial coefficient [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and the Joule-Thomson inversion curve. Furthermore it predicts very roughly the liquid densities (Abbott, 1973; Twu et al., 2002).

In summary, the SRK equation has the same common failing of other cubic EoS, namely the inability, when subject to the classical constraints conditions on the critical isotherm, to provide simultaneously acceptable values for the second virial coefficient, the critical compressibility factor [Z.sub.c], the saturated liquid densities and the supercritical volumetric behaviour (Martin, 1967; Abbott, 1973).

Such failing has its root parts in the inadequate expression of the attraction term and parts in the mathematical limitations of the cubic equations. In this paper we focus attention solely on the first aspect. We will also discuss the impact of the choice of the data base on the predictive capabilities of the EoS.

To illustrate the problematic, let us take a closer look to the temperature behaviour of the original Soave attraction term (Equation (4)). It decreases with increasing temperature, becomes minimum at [T.sub.r] = [(1 +1/m).sup.2], and then increases again. As has been pointed out in Segura et al. (2003), and because of the relation between the second virial coefficient B and [alpha]([T.sub.r]):

B = b - [a.sub.c] [alpha]([T.sub.r])/RT (6)

such a behaviour leads to multiple Boyle temperatures. This means that beyond the second Boyle temperature, B is attractive again and consequently attractive forces dominate at higher temperatures, which is an incoherent physical behaviour.

Our purpose in the present paper is to propose an equation for [alpha]([T.sub.r]) with a sound physical meaning and that will make the SRK EoS better describe the P - V - T surface in the supercritical region for [T.sub.r] up to 5 or 6 and [P.sub.r] up to 12.

ATTRACTION TERM: MODELS AND DATA BASE

Like the Soave proposal (Equation (4)), most of the known attraction terms (Soave, 1972; Mathias and Copeman, 1983; Soave, 1993; Vidal, 1997; Haghighi et al., 2003) become either negative or increase when extrapolated to extreme conditions (reduced temperature [T.sub.r] up to 5 and more). To avoid such a temperature evolution, the [alpha] function must be finite, positive for all temperatures, equals unity at the critical point, decreases monotonically and approaches a finite value as the temperature approaches infinity (Twu and Bluck, 1991).

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We will see that these conditions are necessary but not sufficient, i.e., it is necessary that the attractive term not only has the adequate functional form, but also that the adjustable parameters in the function are deduced from informations from the entire domain of the fluid state. Usually, these parameters are determined by best fitting the EoS solely to the vapour pressure of the considered substance. The use of the EoS in the supercritical region becomes then a problematic extrapolation.

Let us start, at first, with the following attraction term:

[alpha]([T.sub.r]) = exp[m(1 - [T.sup.n.sub.r)] (7)

proposed by Heyen (1981), where m and n are adjustable fluid specific parameters. This type of function can be extrapolated to the supercritical region without exhibiting any abnormal behaviour: the exponential function ensures the positive sign of the a([T.sub.r]) term, and by affecting the same sign to m and n, the attraction term will be a monotonic decreasing function of temperature.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

The resulting EoS, denoted SRK1-a, is:

P = RT/[upsilon] - b - [a.sub.c]exp(m(1 - [T.sup.n.sub.r]))/[upsilon]([upsilon] + b) (8)

The determination of m and n is performed with the help of a non-linear regression procedure (Daroux, 1991) by minimizing the sum of the squared residuals when, in a first step, as usual only vapour pressure data are used.

To test the different attraction terms, at first a set of eight low acentric factor fuids (Ne, Ar, Kr, C-[H.sub.4], [O.sub.2], [N.sub.2], CO and [C.sub.2]-[H.sub.6]) are considered. Their main characteristics are given in Table 1. [N.sub.[upsilon]p] in the 7th column is the number of vapour-pressure data.

[FIGURE 9 OMITTED]

The obtained values of m and n for the test fluids are listed in Table 2 with the percent average absolute deviation, AAD, between predicted and reference vapour pressure, saturated liquid and vapour volume data.

Figure 1 shows that the percent deviation, denoted dev, lays between -0.9% and 0.5% while in the case of the original SRK (Figure 2), the range is [-2 to 3%] for the same set of fluids.

Before further discussing these results, let us consider for the purpose of comparison a second attraction term, with a correct temperature behaviour also, the function proposed by Twu and Bluck (1991):

[alpha]([T.sub.r]) = [T.sup.N(M-1).sub.r] exp(L(1 - [T.sup.NM.sub.r])) (9)

This function improves remarkably the description of the vapour pressure for the set of the test fluids in comparison with the original SRK equation, from the triple point to the critical point. As Figure 3 illustrated it, this 3-parameter ??function is expectably somewhat better than the 2-parameter function (Equation (7)) to which is reduces for M = 1.

[FIGURE 10 OMITTED]

To compare both attraction terms (Equations (7) and (9)) we focus attention on the prediction of the fluid properties in the supercritical region. As expressed by Equation (6), the second virial coefficient B for the SRK EoS is explicitly related to the [alpha]([T.sub.r])-function and can be used to test the reliability of the attraction term in this region.

Although the SRK equation with the original attraction term gives an accurate B description for [T.sub.r] < 10 (Figure 4), this accuracy is lost and an unacceptable physical behaviour is observed at higher relative temperatures. Beyond this interval [alpha]([T.sub.r]) increases again and a second Boyle temperature appears. Such problems are avoided when the SRK equation with the attraction term proposed by Twu and Bluck (1991) is used, since Equation (9) is continuous and decreases monotonically for all temperatures. However, in the vicinity of the Boyle temperature, the predicted B values diverge from the experimental data (Figure 5). Indeed, when only vapour pressure data are used in the regression procedure to determine M, N and L, Equation (9) gives a positive [alpha]([T.sub.r]) function that vanishes in the neighbourhood of the Boyle temperature ([T.sub.r] [approximately equal to] [T.sub.r,B]). As a consequence, from this temperature up, B becomes a positive constant, equal to the co-volume b.

It should be noticed that even if [alpha]([T.sub.r]) vanishes at a higher temperature, i.e., for [T.sub.r] > [T.sub.r,B,] B is always less than b for all positive, monotonic and decreasing expressions for [alpha]([T.sub.r]) used with the SRK EoS.

As shown by Figure 6, the SRK1-a EoS (Equation (8)) gives best results in predicting B values. Although the required accuracy in the vicinity of the Boyle temperature is not reached, the improvement in the supercritical region is evident.

A further comparison criteria for both attraction terms is the prediction of the Joule-Thomson inversion curve (JTIC). This derivative property is known to be a severe test for the quality of an EoS and can be used to judge its physical soundness, especially in the supercritical region. This curve is the locus of the points on the thermodynamic surface at which the Joule-Thomson coefficient [[micro].sub.JT] vanishes:

[[mu].sub.JT] := [([partial derivative]T/[partial derivative]P).sub.H] = 0 (10)

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It is found by solving the equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Figure 7 shows the inversion curves predicted by the SRK EoS with different attraction terms (Equations (4), (7) and (9)). We notice that the corresponding EoSs describe satisfactorily only the lower branch of the JTIC. The upper branch and the maximum inversion point are far apart from the observed behaviour. However, even when the equations of state based on relations (4) and (7) don't describe the entire inversion curve accurately they exhibit a remarkable, surely not yet fulfilling, improvement of the JTIC as predicted by the SRK with Twu's attraction term.

In fact, most of the expressions for [alpha]([T.sub.r]) were developed considering only experimental data in the region [T.sub.r] < 1 (Gunn et al., 1966). As a result, there has been some uncertainty about whether the deduced expressions can be extrapolated to higher temperatures. Let us consider Figure 8 which shows the temperature behaviour of the SRK equation with the original, the Twu and Heyen attraction terms for nitrogen. It can be observed that for reduced temperatures less than 1.5, all functions show a correct physical behaviour and are in good agreement with each other. At higher temperatures however, they not only diverge from one another but can exhibit an incorrect behaviour. We notice also that [alpha]([T.sub.r]) functions without an exponential form either show a minimum for [T.sub.r] in the range (6, 8) or become negative at higher temperatures (Soave, 1993).

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To ensure an adequate description of the supercritical region and an accurate VLE calculation with Equation (8), we fitted it to fluid data from the sub and the supercritical region. The adjustable parameters m and n are obtained by minimizing as objective function the sum of the quadratic average deviation from the literature values for the vapour pressure, from NIST Chemistry WebBook, and the second virial coefficient (Tillner-Roth, 1998):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

B values in the immediate vicinity of the Boyle temperature are excluded. [M.sub.B] is the number of the second virial coefficient data used in the fitting procedure (Equation (12)) and is given in the 8th column of Table 1.

[FIGURE 16 OMITTED]

As a result, we have a new parameterization of the Equation (8) which is referred to as SRK1-b. The obtained values of m and n for the seven fluids are listed in Table 3 with the percent average absolute deviation, AAD, from the underlying vapour pressure, saturated liquid and vapour volume data and the average absolute deviation, errB, from the second virial coefficient data (Tillner-Roth, 1998).

When SRK1-b is compared to SRK1-a, we find a significant improvement in the supercritical region: a correct global behaviour, an accurate prediction of the second virial coefficient (Figure 9) on one side and a better description of the inversion curve (Figure 10) on the other side. However the vapour pressure is now less accurately predicted (Table 3). In fact, the percent AAD becomes somewhat higher (about 2%) as Figure 11 shows whereas it was within 0.35% of the experimental data when calculated with the SRK1-a. One may conclude that on one hand some supercritical property data should be included, we choose the second virial coefficient B, in the fitting procedure for a correct description of the thermodynamic P(T,V) surface in the supercritical region and on the other hand that the temperature function (Equation (7)) must be made more flexible for accurate VLE calculations.

NEW MODEL

This flexibility may be ensured by adding a third degree of freedom in the [alpha]([T.sub.r]) function. To this purpose, we propose the following general 3-parameter exponential expression for the attraction term:

[alpha]([T.sub.r]) = exp(m(1 - [T.sup.k.sub.r]) + n (1 - [T.sup.k+1/2.sub.r]) + p(1 - [T.sup.k+1.sub.r])) (13)

with k [member of] {1/2, 1, 3/2, 2}.

The finally retained attraction term that gives best results when Equation (12) is minimized corresponds to k = 3/2:

[alpha]([T.sub.r]) = exp(m(1 - [T.sup.3/2.sub.r]) + n(1 - [T.sup.2.sub.r]) + p(1 - [T.sup.5/2.sub.r])) (14)

Because of the exponential function, [alpha]([T.sub.r]) is always positive. To ensure that it decreases with increasing temperature, two constraints are imposed on the values of m, n and p:

0 < [n.sup.2] < 15/4 mp,

p and m : and positive

[FIGURE 17 OMITTED]

The resulting EoS, denoted SRK2, writes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

Like for the previous equation (SRK1-b), the determination of the three coefficients is performed by means of the same fitting method and by using the same data base as for SRK1-b.

The values of m, n and p for the set of eight low acentric factor fluids are listed in Table 4 together with the percent AAD from the underlying vapour pressure, saturated liquid and vapour volume data and errB, the average error on B.

Considering Table 4 and Figure 12, we notice that for the eight low acentric factor fluids (Ne, Ar, Kr, C-[H.sub.4], [O.sub.2], [N.sub.2], CO and [C.sub.2]-[H.sub.6]) the AAD in respect to the vapour pressure is within 0.87% of the experimental data. The predictive capability of the Equation (15) is as good as that of the original SRK EoS while keeping simultaneously a correct behaviour of the second virial coefficient (Figure 13). As a result, the calculated high temperature branches of the inversion curves (Figure 14) are now more realistic.

To test the predictive capability of the Equation (15) for more important, practical and real fluids, a set of seven hydrocarbons ([C.sub.3]-[H.sub.8], i-[C.sub.4]-[H.sub.10], n-[C.sub.4]-[H.sub.10], n-[C.sub.5]-[H.sub.12], n-[C.sub.6]-[H.sub.14], n-[C.sub.7]-[H.sub.16] and n-[C.sub.8]-[H.sub.18]) is considered (main characteristics are included in Table 1). The adjustable parameters m, n and p are determined by using only vapour-pressure data. The obtained values of these parameters are listed in Table 4. As this table together with Figure 12 show, the AAD for the vapour-pressure lays within 0.11% of the experimental data which reflects an accurate P - V - T surface's description for these hydrocarbons.

APPLICATION TO BINARY MIXTURES

To extend the applicability of the equation of state (Equation (15)) to mixtures, the van der Waals one-fluid mixing rules are used:

[a.sub.m] = [summation over (1)][summation over (j)][[zeta].sub.i][[zeta].sub.j][a.sub.ij]

and

[b.sub.m] = [summation over (i)] [[zeta].sub.i][b.sub.i] (16)

For the cross-interaction terms [a.sub.12] and [a.sub.21], the classical vdW-1 mixing rules are adopted (Poling et al., 2001):

[a.sub.12] = (1 - [k.sub.12])[square root of [a.sub.1][a.sub.2]]

[a.sub.21] = (1 - [k.sub.21])[square root of [a.sub.1][a.sub.2]] (17)

We set [k.sub.12] = [k.sub.21] and hence [a.sub.12] = [a.sub.21]. The values of [k.sub.12] are obtained by regressing vapour-liquid equilibrium data. The objective function is then the sum of the absolute average deviation from the literature values for the mixture pressure (Knapp et al., 1982) at a given temperature and liquid molar composition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

To test the obtained EoS, four binary hydrocarbons systems are considered: [C.sub.2]-[H.sub.6] / [C.sub.3]-[H.sub.8], [C.sub.3]-[H.sub.8] / i-[C.sub.4]-[H.sub.10], n-[C.sub.4]- [H.sub.10] / n-[C.sub.7]-[H.sub.16] and [C.sub.3]-[H.sub.8] / n-[C.sub.7]-[H.sub.16].

The optimum values of [k.sub.12] are given in Table (5) with the pressure and the temperature ranges where the [k.sub.12] values can be used. Their corresponding values are provided in the same table for each system.

Figures (15 to 17) show for several isotherms the calculated equilibrium total pressure and the mole fraction of the liquid phase in comparison with bibliographic data. Table (5) shows also a comparison between the original SRK and SRK2 EoSs. The same accuracy in the predicted mixture pressure and the molar liquid fraction of the more volatile component is reached by the two equations. We thus can conclude that the SRK2 equation can be safely and advantageously used to replace the original one.

CONCLUSION

The proposed attraction term (14) ensures a correct physical behaviour particularly in the supercritical region, satisfactory results of the prediction of the Joule-Thomson curve and of the second virial coefficient while keeping a good accuracy in the calculated vapour pressure. The equation of state (Equation (15)) was originally developed for a set of low acentric factor fluids and extended to an other group of real fluids, it was shown that it gives accurate description not only for the VLE surface of the pure fluids but also of their mixtures.

This work shows the importance of incorporating second virial coefficient data for the determination of the equation's adjustable parameters if the equation is to describe correctly the thermodynamic surface of a fluid on a wider range of temperature and pressure.

The next step could be the elaboration of a corresponding state principle of simple correlation for the new attraction.

NOMENCLATURE a attraction term b fluid specific constant B second virial coefficient m3/ mol k binary interaction parameter m, n and p adjustable parameters of the new attraction term L, M and N Twu and Bluck (1991) attraction term's parameters P pressure MPa R universal gas constant J/ mol.K T thermodynamic temperature K [upsilon] molar volume [m.sup.3]/ mol Z compressibility factor Greek Symbols [delta] fluid specific constant [epsilon] fluid specific constant [tau] reduced temperature function [omega] acentric factor [zeta] molar fraction Subscripts B Boyle c critical f liquid phase g vapour phase m mixture inv inversion nbp normal boiling point r reduced sat saturation

Manuscript received October 16, 2006; revised manuscript received February 27, 2007; accepted for publication March 18, 2007.

REFERENCES

Abbott, M. M., "Cubic Equations of State," AIChE J. 19, 596-601 (1973).

Daroux, M., "Analyse Numerique Appliquee," LSGC, ENSIC, Nancy, France (1991).

Geana, D. and V. Feroiu, "Calculation of Joule-Thomson Inversion Curves from a General Cubic Equation of State," Fluid Phase Equilibria. 77, 121-132 (1992).

Gunn, R. D., P. L. Chueh and J. M. Prausnitz, "Inversion Temperatures and Pressures for Cryogenic Gases and Their Mixtures," Cryogenics. 6, 324-329 (1966).

Haghighi, B., M. R. Laee and N. S. Matin, "A Comparison among Five Equations of State in Predicting the Inversion Curve of Some Fluids," Cryogenics. 43, 393-398 (2003).

Heyen, G., "A Cubic Equation of State with Extended Range of Application," Second World Congress of Chemical Engineering, Montreal, QC, Canada (1981).

Knapp, H., R. Doring, L. Plocker and J. M. Prausnitz, "Vapor-Liquid Equilibria for Mixtures of Low Boiling Substances," Chemistry Data Series, Vol. VI, DECHEMA, Frankfurt (1982).

Martin, J. J., "Equations of State," Applied Thermodynamics, American Chemical Society, 65-82 (1967).

Mathias, P. M. and T. W. Copeman, "Extension of the Peng-Robinson Equation of State to Complex Mixture: Evaluation of the Various Forms of the Local Composition Concept," Fluid Phase Equilibria. 13, 91-108 (1983).

Nasrifar, Kh. and O. Bolland, "Square-Well Potential and a New [alpha]Function for the Soave-Redlich-Kwong Equation of State," Ind. Eng. Chem. Res. 43, 6901-6909 (2004).

NIST-server, http://webbook.nist.gov/chemistry.

Poling, B. E., J. M. Prausnitz and J. P. O'Connell, "The Properties of Gases and Liquids," 5th edition, McGraw-Hill (2001).

Segura, H., T. Kraska, A. Mejia, J. Wisniak and I. Polishuk, "Unnoticed Pitfalls of Soave-Type Alpha Functions in Cubic Equations of State," Ind. Eng. Chem. Res. 42, 5662-5673 (2003).

Soave, G., "Equilibrium Constants from a Modified Redlich-Kwong Equation of State," Chem. Eng. Sci. 27, 1197-1203 (1972).

Soave, G., "Improving the Treatment of Heavy Hydrocarbons by the SRK EOS," Fluid Phase Equilibria. 84,. 339-342 (1993). Tillner-Roth, R., "Fundamental Equations of State," Shaker Verlag, Aachen (Germany) (1998).

Twu, C. H. and D. Bluck, "A Cubic Equation of State with a New Alpha Function and a New Mixing Rule," Fluid Phase Equilibria. 69, 33-50 (1991).

Twu, C. H., J. E. Coon and J. R. Cunningham, "A New Generalized Alpha Function for a Cubic Equation of State Part 2. Redlich-Kwong Equation," Fluid Phase Equilibria. 105, 61-69 (1995).

Twu, C. H., W. D. Sim and V. Tassone, "Getting a Handle on Advanced Cubic Equations of State," CEP, 58-65 (2002).

Valderrama, J. O., "The State of the Cubic Equations of State," Ind. Eng. Chem. Res. 42, 1603-1618 (2003).

Vidal, J., "Thermodynamique-Application au Genie Chimique et a l'Industrie Petroliere," Technip, Paris (1997).

O. Chouaieb and A. Bellagi *

U.R.: Thermique et Thermodynamique des Procedes Industriels, Ecole Nationale d'Ingenieurs de Monastir, E. N. I. M Av. Ibn Jazzar, 5060 Monastir, Tunisie

* Author to whom correspondence may be addressed. E-mail address: a.bellagi@enim.rnu.tn

Table 1. Characteristics of the test fluids from NIST Chemistry WebBook: critical data [T.sub.c], acentric factor co and used reduced temperature ranges {[T.sub.r,nbp'] [T.sub.r,max]}. [N.sub.vp] and [M.sub.B] are the number of used vapour-pressure and second virial coefficient B data, respectively. Fluid [T.sub.c] [P.sub.c] [Z.sub.c] [omega] K M Pa Ne 44.49 2.679 0.303 -0.039 Ar 150.69 4.863 0.289 -0.002 Kr 209.48 5.510 0.292 -0.002 C-[H.sub.4] 190.56 4.599 0.286 0.011 [O.sub.2] 154.58 5.043 0.288 0.022 [N.sub.2] 126.19 3.396 0.289 0.037 CO 132.80 3.493 0.291 0.051 [C.sub.2]-[H.sub.6] 305.33 4.872 0.279 0.099 [C.sub.3]-[H.sub.8] 369.82 4.247 0.279 0.152 i-[C.sub.4]-[H.sub.10] 407.81 3.629 0.276 0.184 n-[C.sub.4]-[H.sub.10] 425.13 3.796 0.274 0.201 n-[C.sub.5]-[H.sub.12] 469.70 3.366 0.268 0.251 n-[C.sub.6]-[H.sub.14] 507.82 3.034 0.266 0.299 n-[C.sub.7]-[H.sub.16] 540.13 2.736 0.263 0.349 n-[C.sub.8]-[H.sub.18] 569.32 2.497 0.256 0.393 Fluid [T.sub.r,nbp']-- [N.sub.vp] [M.sub.B] [T.sub.r,max] Ne 0.62 - 7.0 35 32 Ar 0.58 - 4.6 64 38 Kr 0.57 - 3.34 45 27 C-[H.sub.4] 0.59 - 3.25 41 47 [O.sub.2] 0.58 - 2.55 32 32 [N.sub.2] 0.61 - 6.5 50 47 CO 0.61 - 1 & 1.6 - 3.6 52 41 [C.sub.2]-[H.sub.6] 0.60 - 2. 61 20 [C.sub.3]-[H.sub.8] 0.62 - 1. 35 -- i-[C.sub.4]-[H.sub.10] 0.64 - 1. 30 -- n-[C.sub.4]-[H.sub.10] 0.64 - 1. 16 -- n-[C.sub.5]-[H.sub.12] 0.65 - 1. 32 -- n-[C.sub.6]-[H.sub.14] 0.67 - 1. 34 -- n-[C.sub.7]-[H.sub.16] 0.69 - 1. 34 -- n-[C.sub.8]-[H.sub.18] 0.7 - 1. 35 -- Table 2. SRK1-a: Equation (8) parameters and percent average absolute deviation for vapour-pressure, saturated liquid and vapour volume Ne Ar Kr m 0.359 144 4 0.495 429 7 0.510 114 6 n 1.295 394 1 1.017 745 8 0.982 962 4 [AAD_P.sub.sat]% 0.246 0.343 0.325 [AAD_v.sub.f]% 4.271 5.399 5.517 [AAD_v.sub.g]% 0.802 0.601 0.795 [O.sub.2] [N.sub.2] CO m 0.531 861 7 0.551 963 4 0.586 836 7 n 1.020 528 6 1.026 594 3 0.999 909 2 [AAD_P.sub.sat]% 0.354 0.317 0.462 [AAD_v.sub.f]% 5.575 5.900 5.592 [AAD_v.sub.g]% 0.531 0.620 0.861 C-[H.sub.4] m 0.490 147 4 n 1.083 989 1 [AAD_P.sub.sat]% 0.347 [AAD_v.sub.f]% 6.547 [AAD_v.sub.g]% 0.769 [C.sub.2]-[H.sub.6] m 0.638 346 8 n 1.040 662 0 [AAD_P.sub.sat]% 0.368 [AAD_v.sub.f]% 9.663 [AAD_v.sub.g]% 1.199 Table 3. SRK1-b: Equation (8) parameters and percent average absolute deviation for vapour-pressure, saturated liquid and vapour volume and the average absolute deviation for second virial coefficient Ne Ar Kr m 0.511 540 4 0.846 607 7 0.973 779 0 n 0.793 668 5 0.536 363 9 0.459 344 4 [AAD_P.sub.sat]% 2.390 1.409 1.616 [AAD_v.sub.f]% 4.365 5.780 5.915 [AAD_v.sub.g]% 2.373 1.402 1.439 errB [cm.sup.3]=mol 0.621 0.646 0.750 [O.sub.2] [N.sub.2] CO m 0.921 226 6 0.882 169 8 1.897 285 5 n 0.536 603 2 0.573 897 6 0.281 754 9 [AAD_P.sub.sat]% 1.466 2.210 2.474 [AAD_v.sub.f]% 5.874 6.431 5.748 [AAD_v.sub.g]% 1.416 2.270 3.325 errB [cm.sup.3]=mol 1.176 1.250 0.663 C-[H.sub.4] m 0.879 177 7 n 0.539 660 5 [AAD_P.sub.sat]% 1.600 [AAD_v.sub.f]% 6.938 [AAD_v.sub.g]% 1.601 errB [cm.sup.3]=mol 0.767 [C.sub.2]-[H.sub.6] m 1.151 204 5 n 0.523 402 1 [AAD_P.sub.sat]% 1.650 [AAD_v.sub.f]% 10.056 [AAD_v.sub.g]% 1.318 errB [cm.sup.3]=mol 0.711 Table 4. SRK2: Equation (15) parameters, the percent average absolute deviation for vapour-pressure, saturated liquid and vapour volume and the average absolute deviation for second virial coefficient Ne Ar m 0.974 215 9 1.206 710 3 n -0.725 975 5 -0.929 154 8 p 0.166 862 2 0.208 865 7 [AAD_P.sub.sat]% 0.865 0.733 [AAD_v.sub.f]% 4.398 5.599 [AAD_v.sub.g]% 0.947 0.751 errB [cm.sup.3]=mol 0.186 0.413 [O.sub.2] [N.sub.2] m 1.376 043 6 1.398 643 2 n -1.101 080 3 -1.103 789 9 p 0.259 163 5 0.256 273 6 [AAD_P.sub.sat]% 0.908 0.797 [AAD_v.sub.f]% 5.755 6.139 [AAD_v.sub.g]% 0.851 0.845 errB [cm.sup.3]=mol 1.026 0.470 [C.sub.3]-[H.sub.8] i-[C.sub.4]-[H.sub.10] m 5.755 864 7 5.950 636 8 n -8.269 496 2 -8.482 941 3 p 3.480 188 2 3.551 247 2 [AAD_P.sub.sat]% 0.107 0.085 [AAD_v.sub.f]% 11.148 12.229 [AAD_v.sub.g]% 1.420 1.433 n-[C.sub.6]-[H.sub.14] n-[C.sub.7]-[H.sub.16] m 6.100 666 8 6.118 193 2 n -8.537 183 9 -8.501 368 2 p 3.576 330 8 3.568 026 7 [AAD_P.sub.sat]% 0.122 0.166 [AAD_v.sub.f]% 16.585 19.347 [AAD_v.sub.g]% 2.680 3.768 Kr C-[H.sub.4] m 1.347 693 3 1.342 237 4 n -1.106 283 3 -1.081 362 4 p 0.263 137 1 0.256 193 4 [AAD_P.sub.sat]% 0.905 1.041 [AAD_v.sub.f]% 5.728 6.785 [AAD_v.sub.g]% 0.826 1.089 errB [cm.sup.3]=mol 0.511 0.542 CO [C.sub.2]-[H.sub.6] m 1.436 439 3 1.584 092 5 n -1.093 238 4 -1.195 449 2 p 0.232 502 6 0.254 771 2 [AAD_P.sub.sat]% 0.867 0.984 [AAD_v.sub.f]% 5.827 9.907 [AAD_v.sub.g]% 1.448 0.768 errB [cm.sup.3]=mol 0.132 0.505 n-[C.sub.4]-[H.sub.10] n-[C.sub.5]-[H.sub.12] m 5.919 289 8 6.028 122 9 n -8.472 348 6 -8.521 983 9 p 3.576 981 3 3.580 057 3 [AAD_P.sub.sat]% 0.116 0.111 [AAD_v.sub.f]% 13.289 15.467 [AAD_v.sub.g]% 2.552 2.154 n-[C.sub.8]-[H.sub.18] m 10.408 195 2 n -15.373 268 9 p 6.527 212 1 [AAD_P.sub.sat]% 0.092 [AAD_v.sub.f]% 20.968 [AAD_v.sub.g]% 3.964 Table 5. Comparison between SRK (Knapp et al., 1982) and SRK2 EoSs for several binary systems: the percent average absolute deviation for mixture pressure and the percent absolute deviation of the liquid molar fraction of the more volatile component. The [k.sub.12] and the number of used data, [N.sub.m], are also shown. [C.sub.2]-[H.sub.6]/ [C.sub.3]-[H.sub.8] EoSs SRK2 SRK [N.sub.m] 83 84 [k.sub.12] 0.00174 -0.0022 [T.sub.mi] K 255 255 [T.sub.max] K 366 366 [P.sub.min] MPa 0.261 0.261 [P.sub.max] MPa 5.18 5.18 AAD_P % 1.02 0.71 dev_[y.sub.1] mol% 0.63 0.91 [C.sub.3]-[H.sub.6]/ i-[C.sub.4]-[H.sub.10] EoSs SRK2 SRK [N.sub.m] 67 86 [k.sub.12] -0.004 -0.01 [T.sub.mi] K 266 266 [T.sub.max] K 394 394 [P.sub.min] MPa 0.123 0.123 [P.sub.max] MPa 4.17 4.17 AAD_P % 1.02 1.13 dev_[y.sub.1] mol% 0.32 0.65 n-[C.sub.]-[H.sub.10]/ n-[C.sub.7]-[H.sub.16] EoSs SRK2 SRK [N.sub.m] 55 41 [k.sub.12] 0.00016 -0.0004 [T.sub.mi] K 355 355 [T.sub.max] K 538 538 [P.sub.min] MPa 0.689 0.689 [P.sub.max] MPa 3.96 3.96 AAD_P % 0.54 1.06 dev_[y.sub.1] mol% 0.32 0.61 [C.sub.3]-[H.sub.8]/ [C.sub.7]-[H.sub.16] EoSs SRK2 SRK [N.sub.m] 45 44 [k.sub.12] -0.01037 0.0044 [T.sub.mi] K 333 333 [T.sub.max] K 513 513 [P.sub.min] MPa 0.0281 0.0281 [P.sub.max] MPa 4.82 4.82 AAD_P % 1.35 1.56 dev_[y.sub.1] mol% 1.67 2.16

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Author: | Chouaieb, O.; Bellagi, A. |
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Publication: | Canadian Journal of Chemical Engineering |

Date: | Dec 1, 2007 |

Words: | 5600 |

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