Ind. Eng. Chem. Res. 2009, 48, 5901–5906
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Rigorously Universal Methodology of Volume Translation for Cubic Equations of State Chorng H. Twu*,† and Hui-Shan Chan‡ Tainan UniVersity of Technology, 529 Chung Cheng Road, Yung Kang, Tainan 71002, Taiwan, and National Tainan Institute of Nursing, 78, Sec. 2, Minzu Rd., Tainan City, Tainan 70043, Taiwan
To model systems typically found in hydrocarbon production problems and the chemical industry, there are three basic types of calculations are required: phase equilibria, volumetric behavior, and thermophysical properties. Since 1972, cubic equations of state (CEoS) have shown surprising capabilities in the prediction of the phase equilibria of complex nonpolar systems. However, the prediction of the liquid volumes from cubic equations of state, despite the recently important advances in CEoS/AE mixing rules, still remains the weak point of this type of equations. A rigorous, universal, but simple methodology without introducing any regressed parameters is proposed for the volume translation for all types of cubic equation of state. The proposed methodology applies to not only defined components but also petroleum fractions. The translated cubic equations of state reproduce almost exactly the liquid density over the entire temperature range from the triple point to the critical point. Introduction Although van der Waals proposed his pioneer cubic equation of state in 1873, it has been limited to the prediction of thermodynamic properties for vapor phase for almost a full century. The Soave modification of the Redlich-Kwong equation1 has been a major success in extending the application of CEoS from the prediction of vapor thermodynamic properties to vapor-liquid equilibria for nonpolar and slightly polar components. The modern development of combining cubic equations of state with excess Gibbs energy models has advanced CEoS toward becoming a very effective method in correlating and predicting phase equilibrium behavior for highly nonideal systems. However, in spite of these important advances in CEoS/ AE mixing rules, the prediction of the liquid volumes from CEoS for the oil, gas, and chemical systems still remains the problem. We will propose a rigorous, universal, but simple methodology to predict correctly the liquid density from cubic equations of state without introducing any regressed parameters. The proposed methodology without affecting the K-value calculations is applicable to all types of cubic equations of state. The application of the methodology extends naturally from defined components to petroleum fractions. Cubic Equations of State A two-parameter cubic equation of state is considered here: P)
a RT V-b (V + ub)(V + wb)
(1)
where P is the pressure, T is the absolute temperature, and V is the molar volume. The constants u and w are equation-of-statedependent (for the Twu-Sim-Tassone (TST) equation, u ) 3 and w ) -0.5, for the Soave-Redlich-Kwong (SRK) equation, u ) 0 and w ) 1 and for the Peng-Robinson (PR) equation, u ) -0.4142 and w ) 2.4141). * To whom correspondence should be addressed. E-mail: t80060@ mail.tut.edu.tw. Telephone: 886-6-2422603. Fax: 886-6-2433812. † Tainan University of Technology. ‡ National Tainan Institute of Nursing.
There are two parameters a and b in the cubic equation of state to be determined. Generally there are two ways to evaluate them. One is to fit the parameters to experimental data, usually the vapor pressure and liquid density. The other is to derive the parameters from the critical constraints. Since the cubic equations of state do not represent the PVT behavior well, particularly near the critical region, the prediction of liquid densities from such equations is badly in error. This is one of the inherent limitations of any cubic equation of state. Consequently, the major drawback of forcing the equation-of-state parameters to fit the liquid density not only fails to satisfy the critical constraints but also sacrifices the ability of the prediction of more desired K-value property from equations of state. This procedure also leads to an overestimation of critical temperature and critical pressure. The use of critical constraints in determining the cubic equation of state parameters is a very unique feature for CEoS. Since most noncubic equations of state are complex, it is infeasible or almost impossible to derive analytical expressions for their parameters from critical constants. On the other hand, the form of the cubic equation of state is simple enough to allow its parameters to be determined analytically from critical constraints. The critical constraints on the parameters also enhance the accuracy of the cubic equation of state in the prediction of K-values, especially at or near the critical point. The critical constraints, however, result in a constant value of critical compressibility factor Zc for all components. The value of Zc of real fluids is known from experiment to be generally smaller than that predicted from the cubic equation of state. As a result, the predicted liquid densities differ considerably from their experimental values. Fortunately, the underprediction of liquid density from cubic equations of state can be corrected by the method of volume translation without affecting any K-value calculation. Volume-Translated Cubic Equations of State Since the early 1970s, cubic equations of state quickly became popular as modeling and predictive tools for natural gas systems because cubic equations have shown remarkable abilities in the prediction of the phase behavior of hydrocarbon fluids. To
10.1021/ie900222j CCC: $40.75 2009 American Chemical Society Published on Web 05/11/2009
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Ind. Eng. Chem. Res., Vol. 48, No. 12, 2009
extend the use of the CEoS to the retrograde phenomenon of hydrocarbon systems, as well as to the prediction of the liquid fraction condensed for actual gas condensate systems, however, it has to cope with the difficulty existing in CEoS. That is, the common cubic equations of state do not lead to accurate results in all PVT conditions. The introduction of a third parameter, from the translation concept,2 has been shown to improve significantly the accuracy of density prediction from the cubic equations of state. Peneloux et al.3 elaborated the concept of Martin2 and applied it to improve the liquid volumetric predictions from the SRK equation of state. In that procedure, a volume translation is introduced, which does not affect the vapor pressure or calculated phase equilibrium, but does minimize the errors in the saturated liquid volumes as predicted by the SRK equation of state at a reduced temperature of 0.7. The Peneloux procedure separates the problem of K-value calculations from the problem of density calculations. Similar to the way that Peneloux et al.3 elaborated the idea of Martin,2 our idea is also inspired by the work of Peneloux et al. In order to maintain the K-value calculation unaffected by the volume translation, the volume correction has to be either a constant or a temperature-dependent function. The desired volume correction, c, is the difference between saturated liquid volume calculated from any cubic equation of state, Vs,CEoS, and the experimental saturated liquid volume, Vs,exp: c ) Vs,CEOS - Vs,exp
(2)
Numerous empirical correlations for c have been tried to improve the prediction of liquid density. These empirical correlations typically contain several parameters to be regressed from the liquid density data. Since they are empirical with regressed parameters, they all exhibit certain limitations and even some serious problems in the application. In this work, a rigorous, universal, but simple method without introducing any regressed parameters is proposed here for the volume translation for all types of cubic equations of state. Most investigators treated c itself as an purely empirical correlation and ignored the saturated liquid volume terms Vs,CEoS and Vs,exp on the right-hand side of eq 2. Then they optimized several parameters in c using density data from a limited number of components. To be rigorous, we have to deal with Vs,CEoS and Vs,exp separately. Spencer and Danner4 found that the most accurate and simplest means of predicting the effect of temperature on the saturated liquid densities is by the Rackett equation. Hence, the experimental saturated liquid volume, Vs,exp, in eq 2 is proposed to be replaced by the Rackett equation, Vs,RA: c ) Vs,CEoS - Vs,RA
(3)
The Rackett equation of Spencer and Danner4 is written in the following form: Vs,RA )
( )
RTc [1+(1 Z Pc RA
- Tr)2/7]
(4)
Here, ZRA is the Rackett parameter and is a particular constant for the Rackett equation. The value of ZRA is unique to each component and is confined within a small range of certain values, typically in the range of 0.2-0.3. Its value is determined from the saturated liquid density data of the component and can be found in many literature works. The term Vs,CEoS in eq 3 is the saturated liquid volume calculated from any cubic equation of state. At low pressure,
the saturated liquid volumes are adequately assumed to be independent of the pressure. The saturated liquid volume Vs,CEoS is then proposed to be solved from the cubic equation of state by neglecting the pressure and selecting the smaller root: Vs,CEoS )
( )
{(
RTc b*c a* -u-w Pc 2 b* a* 2 a* u+w- 4 uw + b* b*
)
[(
)
(
)] } 1/2
(5)
The parameters a*, b*, and b*c in eq 5 are defined as a* ) Pa/R2T2
(6)
b* ) Pb/RT
(7)
b*c ) Pcbc /RTc
(8)
Equation 5 represents the analytical solution of the cubic equation of state for saturated liquid density for temperature up to normal boiling point and will be used for the volume correction in this temperature range. At temperatures above the normal boiling point and below the critical temperature, the saturated liquid density Vs,CEoS is proposed to be solved from the equation of state, eq 1, with saturated vapor pressure P calculated from the Antoine equation: ln P ) c1 +
c2 T
(9)
The constants c1 and c2 are determined from the normal boiling point Tb and the critical temperature Tc, both in kelvin:
(
ln c1 ) ln(Pc) -
(
ln c2 )
(
(
Pc 1.01325 Tc 1Tb
)
) )
Pc 1.01325 1 1 Tc Tb
)
(10)
(11)
Pc in eqs 10 and 11 is the critical pressure in bar. Equation 1 with saturated vapor pressure P calculated from the Antoine equation can be solved analytically for Vs,CEoS. The continuity of the volume calculations for Vs,CEoS switched from eq 5 to 1 is well-maintained because the same equation of state is used. Equation 9 predicts saturated vapor pressure accurately for the temperature range between Tb and Tc. The use of eq 9 eliminates the need of solving saturated liquid density from the cubic equation of state by use of the equality of fugacities of the component at equilibrium. The methodology of calculating the volume correction at temperatures below the critical point is represented by eqs 1-11. A word should be said about the supercritical region. An extrapolation from the subcritical range is always unreliable, and with the volume translation method, it has no physical significance. Soave5 observed that the volume correction obtained from the critical point is too high to reproduce the critical isotherm. Soave5 found that the best value for the critical isotherm is the value obtained from the liquid densities at low temperature such as the atmospheric boiling temperature. On the basis of the Soave’s finding, the volume correction at or above the critical temperature is to be determined from the
Ind. Eng. Chem. Res., Vol. 48, No. 12, 2009
difference between the saturated liquid volume predicted from the cubic equation of state and the saturated liquid volume predicted from the Rackett equation at the normal boiling point Tb: c ) νs,CEoS
|
- νs,RA T)Tb
|
(12) T)Tb
The saturated liquid volumes Vs,CEoS and Vs,RA are calculated at Tb from eqs 5 and 4, respectively. Equation 12 becomes c)
( ) |
RTc [1+(1 (ν*s,CEoS - ZRA Pc T)T
- Tb /Tc)2/7]
)
(13)
b
|
T)
) Tb
{( ) [(u + w - b*a* ) - 4(uw + b*a* )] }|
b*c a* -u-w 2 b* 2
1/2
(14)
T)Tb
The volume correction, c, in eq 3 is considered to be the third parameter in the two-parameter cubic equations of state. Since the linear mixing rule is used for the cubic equation of state covolume term b, a linear mixing rule should also be used for the volume correction c for the calculation of mixture volume to maintain the internal consistency with b. The predicted mixture volume from cubic equation of state is then corrected by the mixture volume correction term to give the desired volume, Vcorrected: n
Vcorrected ) VCEoS -
the alpha function and critical constants. Twu et al. discovered that the alpha function is a linear function of the acentric factor, not a fourth- or sixth-order function as suggested by Soave and other researchers. The advantage of a linear function in the acentric factor is obvious in the extrapolation of the alpha function to heavy hydrocarbons, petroleum fractions, and gas condensates. To maintain the internal consistency, the acentric factor of the petroleum fraction can be computed from the cubic equation of state of Twu et al.7,8 using the normal boiling point at unit atmosphere. The method described here provides a consistent and accurate approach for the prediction of the volumes of petroleum fractions using a cubic equation of state. Comparison with Peneloux Volume Correction
where ν*s,CEoS
5903
7,8
∑xc
i i
(15)
i
The term ci in eq 15 is the volume correction for the pure component i calculated from eqs 3 and 4 together with 5 for temperature up to Tb or with eqs 1 and 9-11 for temperature between Tb and Tc or with eq 12 for either the temperature at or above the critical temperature or the system at extremely high pressures. Volume Corrections for Petroleum Fractions The methodology proposed here, represented by eqs 1-14, intends to apply to not only defined components but also petroleum fractions. The conventional characterization parameters used for many years in the petroleum refining industry are the normal boiling point (Tb) and the specific gravity (SG). Because critical properties are generally not available for petroleum fractions, problems arise if an undefined petroleum fraction is included in the system. Since Tc, Pc, acentric factor (ω), and ZRA can all enter into the volume translation calculations, special care must be given to properly characterize the petroleum fraction to maintain the internal consistency. To apply our method to petroleum fractions, the critical constants (Tc and Pc) which are not known for this type of compounds, but required by our method, can be predicted based on Tb and SG from the Twu method,6 which is currently widely used in refining industries. The required Rackett parameter ZRA of the petroleum fraction can then be solved from the Rackett equation, eq 4, by giving the SG of the petroleum fractions at the temperature (60 °F) of SG. Our volume translation method also involves the calculation of the equation of state parameter a, which is computed from
The Peneloux volume correction has been widely used for the correction of the volume predicted from SRK equation of state. Peneloux et al.3 found that the best correlating parameter for the pure component volume correction, c, is not the acentric factor, but the Rackett compressibility factor ZRA appearing in Spencer and Danner’s correction of the Rackett equation for saturated liquid volumes. The volume correction c proposed by Peneloux for the SRK equation attempts to reproduce as closely as possible the saturated liquid volumes at a reduced temperature Tr ) 0.7:
( )
c ) 0.40768 )
( )
RTc (0.29441 - ZRA) Pc
RTc (0.120025 - 0.40768ZRA) Pc
(16)
The liquid volumes calculated from eq 16 usually are within 1-2% of the experimental values between the reduced temperature of 0.55 and 0.82 for light hydrocarbons. The Peneloux method has several disadvantages: the volume correction is a constant for individual component, the volume representation is not satisfactory for the heavy components or petroleum fractions, and the volume correction is inappropriate for polar components. It will be interesting to compare the Peneloux volume correction, eq 16, with our rigorous equation for the volume correction, eqs 3-5. If the same reference temperature as that chosen by Peneloux is selected, our volume correction, eq 3 becomes c ) νs,CEoS
|
Tr)0.7
- νs,RA
|
(17)
Tr)0.7
Similar to eq 13, eq 17 is rewritten as c)
( ) |
RTc (ν*s,CEoS - ZRA1.70893) Pc T )0.7
(18)
r
where ν*s,CEoS
|
Tr)0.7
)
{(
b*c a* -u-w 2 b*
[(u + w - b*a* )
)
2
(
- 4 uw +
a* b*
)] }| 1/2
(19)
Tr)0.7
where the equation of state parameter a is evaluated at Tr ) 0.7.
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Table 1. Average Absolute Deviation Percent of Predicted Saturated Liquid Densities for the Temperature Ranging from the Triple Point (Tt) to the Reduced Temperature Tr ) 0.7 from SRK and SRK with the Volume Correction Methods from Peneloux, eq 24 at Tr ) 0.5, eq 23 at Tr ) 0.6, eq 18 at Tr ) 0.7, eq 13 at Tb, and eqs 3-11 from the Method of Twu component
ZRA
SRK
Peneloux
Tr ) 0.50
0.60
0.70
Tb/Tc
Twu
methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-undecane n-dodecane n-tridecane n-tetradecane n-pentadecane n-hexadecane n-heptadecane n-octadecane n-nonadecane eicosane
0.2881 0.2809 0.2764 0.2731 0.2686 0.2642 0.2608 0.2571 0.2543 0.2518 0.2486 0.2467 0.2435 0.2380 0.2390 0.2379 0.2366 0.2323 0.2336 0.2329
1.00 4.20 6.31 7.44 9.50 11.28 12.96 14.30 15.66 16.72 18.31 19.02 20.99 22.94 23.10 23.74 23.92 25.21 25.11 25.37
1.33 1.15 1.00 1.14 0.98 0.92 0.68 0.81 0.70 0.56 0.52 0.49 0.64 0.33 0.66 0.80 0.63 0.53 0.75 0.94
0.51 0.54 0.61 0.38 0.35 0.26 0.32 0.34 0.39 0.35 0.44 0.40 0.65 1.07 0.64 0.72 0.66 1.28 0.79 0.94
0.60 0.29 0.44 0.33 0.34 0.26 0.34 0.25 0.32 0.35 0.38 0.40 0.82 0.63 0.84 1.15 0.78 0.47 0.82 0.98
2.02 1.16 1.18 1.05 0.99 1.06 0.78 0.96 0.80 0.50 0.59 0.43 0.65 1.38 0.63 0.77 0.64 0.74 0.78 1.04
0.56 0.28 0.40 0.32 0.40 0.61 0.58 0.98 1.07 0.91 1.25 1.19 1.19 3.22 1.66 1.53 1.84 3.63 2.14 1.89
0.30 0.17 0.14 0.03 0.10 0.25 0.16 0.40 0.40 0.12 0.24 0.10 0.34 1.09 0.30 0.61 0.13 1.05 0.09 0.08
For the SRK equation of state, the values of u and w are 0 and 1, respectively. The volume correction, eq 18, for the SRK equation becomes c)
( ){ [( RTc Pc
) (1 - b*a* )2 - 4 b*a* ]
b*c a* -1 2 b*
Tr)0.7
ZRA1.70893
}
(20)
For the SRK equation, b*c ) 0.086641. The subscripts “Twu” and “Peneloux” will be used in the following equations, eqs 21 and 22, to refer to the volume correction predicted from our method and that of Peneloux et al.,3 respectively. Comparing eq 20 with eq 16, it reveals that Peneloux et al. have approximated our model with the following:
{(
c)
)
)
]} 1/2
Tr)0.7(Twu)
) 0.120025(Peneloux) (21)
and ZRA1.70893(Twu) ) 0.40768ZRA(Peneloux)
(22)
Equation 21 can be solved for a*/b* at Tr ) 0.7. Before solving eq 21, it is interesting to point out that the value of a*/b* at Tr ) 0.7 is not a constant but increases with the boiling point of the component. For example, the value of a*/b* at Tr ) 0.7 is 8.24 for methane, 8.59 for ethane, 8.79 for propane, 8.98 for n-butane, 9.18 for n-pentane, and 10.12 for n-decane, etc. The solution of eq 21 gives a*/b* ) 8.58, which is essentially the same as that of ethane. Using the ethane’s value of ZRA) 0.2809 in eq 22, we obtain ZRA1.70893 ) 0.28091.70893 ) 0.114186 from Twu and 0.40768ZRA ) (0.40768)(0.2809) ) 0.114517 from Peneloux. The result confirms very well the equality of eq 22. However, this equality is valid only for ethane. For example, substituting the value of ZRA) 0.2518 of n-decane into eq 22, the value of (0.40768)(0.2518) ) 0.102654 from Peneloux is not close to our rigorous value 0.25181.70893 ) 0.094721. Our analysis indicates that Peneloux et al. have applied the volume correction for ethane to all components including heavy oils and polar components. This explains the
( ) |
RTc (ν*s,CEoS - ZRA1.76967) Pc T )0.6
(23)
r
If Tr ) 0.5 is picked, we have c)
0.086641 a* -1 2 b* a* 2 a* 1-4 b* b*
[(
reason why the Peneloux volume translation is not appropriate for heavy as well as polar components. Our rigorous methodology offers a convenient way to select any reference temperature for the volume correction. If the normal boiling point is selected, eq 13 is applied. If Tr ) 0.7 is chosen, eq 18 is used. However, other reference temperatures than Tb or Tr ) 0.7 could be preferred for the volume correction. For example, if Tr ) 0.6 is selected, we have
( ) |
RTc (ν*s,CEoS - ZRA1.82034) Pc T )0.5
(24)
r
We will examine the impact of the chosen reference temperature on the liquid volume prediction. Six volume correction methods are compared in the prediction of liquid densities. They are eq 16 of Peneloux, eq 24 at Tr ) 0.5, eq 23 at Tr ) 0.6, eq 18 at Tr ) 0.7, eq 13 at Tb, and the rigorous equation given by eqs 3-11. In Tables 1 and 2, “SRK” means the liquid densities are predicted from SRK CEoS without any volume correction. And, Peneloux, Tr ) 0.50, Tr ) 0.60, Tr ) 0.70, Tr ) Tb/Tc, and Twu refer to the densities predicted from SRK with the volume correction methods from eqs 16, 24, 23, 18, 13, and 3-11, respectively. Note that the values of Rackett parameters for n-tetradecane and n-octadecane as shown in the tables indicate that the liquid densities of these two components in the DIPPR databank are inconsistent with others. Therefore, the results for these two components are for reference only. Examining the accuracy of reproducing liquid density as shown in Table 1, it was found that the Peneloux procedure is quite effective in correcting volume for temperature up to Tr ) 0.7. However, it is evident from the result given in Table 1 showing that Peneloux correlation produces the largest deviation for methane. As mentioned previously, since Peneloux used ethane as the reference in volume translation, it was really surprised to find that the Peneloux exhibits larger deviation for light components than heavy ones. The same trend of deviation is also found by eq 18 at Tr ) 0.7. On the other hand, using
Ind. Eng. Chem. Res., Vol. 48, No. 12, 2009
another reference temperature at Tb such as eq 13, the results reverse the trend of deviation to give the larger deviation for heavy components. This seems to indicate that the accuracy depends primarily on the selected reference temperature, not the reference component. It is interesting to find out that both of our methods eq 24 at Tr ) 0.5 and eq 23 at Tr ) 0.6 are more appropriate than the Peneloux in correcting the liquid volume prediction from the cubic equation of state. Finally, the Twu method consistently gives the most accurate results for all cases over the entire range of temperatures. Since the Peneloux correlation is based on a fixed value of a*/b* ) 8.58 of ethane, the discrepancy between measured and calculated liquid densities apart from this fixed value cannot be removed and become worse especially when temperature progresses to higher values than Tb. The results shown in Table 2 demonstrate this weakness that the deviation increases significantly when the density data of temperature higher than Tb are included. It is worth mentioning that eq 16 proposed by Peneloux for the volume correction is proper only for SRK equation of state, inappropriate for use in any other cubic equations of state. Since our method is general for any cubic equation of state, the Peneloux-type volume correction for TST cubic equation of state can be derived directly from our eq 18 without the need of applying the procedures described by Peneloux. The TST cubic equation of state was developed by Twu et al.9 to allow better prediction of liquid densities for heavy hydrocarbons and polar components as well as the accurate prediction of vapor pressure of all components in databanks. The TST cubic equation of state is represented by the following equation: a RT P) V-b (V + 3b)(V - 0.5b)
ac ) 0.470507R2Tc2 /Pc
(26)
bc ) 0.0740740RTc /Pc
(27)
Zc ) 0.296296
(28)
where subscript c denotes the critical point. It is worth noting that the values of Zc from the SRK and PR equations are both larger than 0.3, whereas that from the TST equation is slightly below it and is closest to the true value. Substituting the values of b*c ) 0.074074073, u ) 3, and w ) -0.5 for TST equation of state and the value of ZRA ) 0.2809 for ethane with a*/b* ) 10.55 at Tr ) 0.7 from TST equation into eq 18, we have the Peneloux-type volume correction for TST equation as c)
( )
RTc (0.100069 - ZRA1.70893) Pc
(29)
For ethane (ZRA) 0.2809), the last term in eq 29 is replaced by ZRA1.70893 ) 0.406501ZRA
(30)
where the numerical value of 0.406501 is calculated from ZRA1.70893/ZRA ) 0.28091.70893/0.2809 ) 0.406501. Substituting eq 30 into 29 results in c ) 0.406501
( )
RTc (0.246172 - ZRA) Pc
(31)
Similarly for the PR equation of state, substituting the values of b*c ) 0.0777961, u ) -0.4142, and w ) 2.4141 and the value of a*/b* ) 9.90 at Tr ) 0.7 from the PR equation into eq 18, we have the Peneloux-type volume correction for PR equation as c ) 0.406501
(25)
The values of a and b at the critical temperature are found by setting the first and second derivatives of pressure with respect to volume equal to zero at the critical point, resulting in
5905
( )
RTc (0.260484 - ZRA) Pc
(32)
Equations 16, 32, and 31 are the Peneloux-type volume corrections for SRK, PR, and TST equation, respectively. They are different from each other significantly. Conclusions We have developed a rigorous, universal, but simple methodology for the volume translation to remove the inherent
Table 2. Average Absolute Deviation Percent of Predicted Saturated Liquid Densities for the Temperature Ranging from the Triple Point (Tt) to the Critical Temperature (Tc) from SRK and SRK with the Volume Correction Methods from Peneloux, eq 24 at Tr ) 0.5, eq 23 at Tr ) 0.6, eq 18 at Tr ) 0.7, eq 13 at Tb, and eqs 3-11 from the Method of Twu component
ZRA
SRK
Peneloux
Tr ) 0.50
0.60
0.70
Tb/Tc
Twu
methane ethane propane n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-undecane n-dodecane n-tridecane n-tetradecane n-pentadecane n-hexadecane n-heptadecane n-octadecane n-nonadecane eicosane
0.2881 0.2809 0.2764 0.2731 0.2686 0.2642 0.2608 0.2571 0.2543 0.2518 0.2486 0.2467 0.2435 0.2380 0.2390 0.2379 0.2366 0.2323 0.2336 0.2329
6.92 6.69 8.12 9.55 11.18 12.95 14.32 15.80 17.90 17.73 19.08 19.68 21.16 23.90 22.96 23.20 23.70 25.15 25.60 25.60
6.15 3.49 3.03 3.59 3.36 3.65 3.50 3.96 5.09 3.80 3.81 3.80 3.91 5.03 4.00 3.93 3.92 3.08 4.86 4.74
6.43 3.60 3.23 3.60 3.37 3.56 3.53 3.89 5.06 3.82 3.82 3.81 3.89 5.11 3.87 3.80 3.89 3.27 4.84 4.74
6.26 3.36 3.06 3.54 3.35 3.57 3.59 3.92 5.11 3.95 3.91 3.97 4.15 5.04 4.17 4.23 4.19 3.07 5.09 4.93
6.17 3.49 3.10 3.56 3.37 3.69 3.52 3.99 5.10 3.80 3.80 3.80 3.88 5.17 3.90 3.89 4.03 3.13 5.04 4.99
6.29 3.34 2.98 3.43 3.24 3.58 3.48 3.99 5.18 3.88 3.98 3.96 3.92 5.71 4.10 3.89 4.23 4.18 5.35 5.15
0.27 0.28 0.19 0.18 0.15 0.24 0.17 0.42 0.42 0.14 0.36 0.14 0.57 1.30 0.51 0.70 0.24 1.15 0.26 0.31
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limitations of the under prediction of liquid density from any cubic equations of state. The translated cubic equations of state reproduce accurately the liquid density over the entire temperature range from the triple point to the critical point. The proposed methodology can be applied to not only the defined components but also the petroleum fractions. Our recent efforts toward the development of unified CEoS/AE mixing rule10,11 and the Twu alpha function12 for highly nonideal systems in predicting vapor-liquid equilibria (VLE) have been highly successful. The development of the volume translation in this work extends the cubic equation of state further to be able to predict accurately the liquid density for nonpolar, polar, and polymer systems, in addition to phase equilibrium behavior. Acknowledgment We are thankful for the financial support of the National Science Council of Taiwan for the Project No. NSC96-2221E-165-001. Nomenclature a,b ) cubic equation of state parameters a*, b* ) reduced parameters of a and b c ) volume correction P ) pressure R ) gas constant T ) temperature u, w ) cubic equation of state constants V ) molar volume V*s ) reduced liquid volume at saturated pressure xi ) mole fraction of component i ZRA ) Rackett compressibility factor AbbreViations AAD% ) average absolute deviation percent CEoS ) cubic equation of state PR ) refers to the Peng-Robinson equation of state SRK ) refers to the Soave-Redlich-Kwong equation of state TST ) refers to the Twu-Sim-Tassone equation Greek Letters ω ) acentric factor
Subscripts b ) boiling point c ) critical property r ) reduced property s ) saturated property Superscripts * ) reduced property
Literature Cited (1) Soave, G. Equilibrium Constants from a Modified Redlich-Kwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197–1203. (2) Martin, J. J. Cubic Equations of State: Which. Ind. Eng. Chem. Fundam. 1979, 18, 81–97. (3) Peneloux, A.; Rauzy, E.; Freze, R. A Consistent Correction for Redlich-Kwong-Soave Volumes. Fluid Phase Equilib. 1982, 8, 7–23. (4) Spencer, C. F.; Danner, R. P. Improved Equation for Prediction of Saturated Liquid Density. J. Chem. Eng. Data 1972, 17, 236–241. (5) Soave, G. Improvements of the van der Waals Equation of State. Chem. Eng. Sci. 1984, 39, 357–369. (6) Twu, C. H. An Internally Consistent Correlation for Predicting the Critical Properties and molecular Weights of Petroleum and Coal-tar Liquids. Fluid Phase Equilib. 1984, 16, 137–150. (7) Twu, C. H.; Coon, J. E.; Cunningham, J. R. A New Generalized Alpha Function for a Cubic Equation of State: Part 1. Peng-Robinson Equation. Fluid Phase Equilib. 1995, 105, 49–59. (8) Twu, C. H.; Coon, J. E.; Cunningham, J. R. A New Generalized Alpha Function for a Cubic Equation of State: Part 2. Redlich-Kwong Equation. Fluid Phase Equilib. 1995, 105, 61–69. (9) Twu, C. H.; Sim, W. D.; Tassone, V. An Extension of CEOS/AE Zero-Pressure Mixing Rule for An Optimum Two-parameter Cubic Equation of State. Ind. Eng. Chem. Res. 2002, 41, 931–937. (10) Twu, C. H.; Sim, W. D.; Tassone, V. Liquid Activity Coefficient Model for CEOS/AE Mixing Rules. Fluid Phase Equilib. 2001, 183-184, 65–74. (11) Twu, C. H.; Sim, W. D.; Tassone, V. A Versatile Liquid Activity Model for SRK, PR and A New Cubic Equation of State TST. Fluid Phase Equilib. 2002, 194-197, 385–399. (12) Twu, C. H.; Bluck, D.; Cunningham, J. R.; Coon, J. E. A Cubic Equation of State with New Alpha Function and New Mixing Rule. Fluid Phase Equilib. 1991, 69, 33–50.
ReceiVed for reView February 9, 2009 ReVised manuscript receiVed April 24, 2009 Accepted April 28, 2009 IE900222J
