1696
Ind. Eng. Chem. Res. 1998, 37, 1696-1706
Prediction of the Condensation Behavior of Natural Gas: A Comparative Study of the Peng-Robinson and the Simplified-Perturbed-Hard-Chain Theory Equations of State Marianna E. Voulgaris, Cor J. Peters,* and Jakob de Swaan Arons Laboratory of Applied Thermodynamics and Phase Equilibria, Faculty of Chemical Technology and Materials Science, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
For the prediction of the condensation behavior of natural gas, one has to select an equation of state (EoS) which will be accurate in the temperature and pressure range of interest (10 < P/bar < 70 bar and 250 < T/K < 310). Another requirement of the selected EoS is that it easily can be adapted to a characterization procedure for the heavy-end fraction. For that purpose, two equations were tested: the Peng-Robinson (PR), which is one of the most applied cubic EoS, and the simplified-perturbed-hard-chain theory (SPHCT) equation, which is one of the simplest EoS based on sound statistical mechanical principles. In the underlying study, their predictive capabilities for the prediction of saturated vapor pressures of pure compounds and vapor/liquid equilibrium pressures for binary mixtures are compared. Only components present in natural gas are considered. In addition, new pure-component parameters for the SPHCT EoS for n-alkanes are evaluated. Also a method to find the characteristic energy for non-nalkane molecules is proposed in this study. This study revealed that the PR EoS predicts more accurately the liquid phase composition, whereas the SPHCT EoS is superior for the gas phase prediction, especially for asymmetric binary mixtures. It was concluded that, with respect to the purpose of this study, both EoS, when used with optimum binary interaction parameters, have an equivalent descriptive accuracy. Therefore, the simpler PR EoS was preferred to describe natural gas mixtures. Introduction During storage, transportation, and processing of natural gas, small amounts of a hydrocarbon liquid phase may precipitate upon reduction in pressure. This phenomenon is called retrograde condensation (Lammers et al., 1973). To prevent operational problems in all kinds of facilities, caused by accumulation of liquid, the natural gas industry is interested in having available a thermodynamic model that accurately predicts the condensation behavior of natural gas. In such a model an equation of state plays the key role; i.e., an equation of state has to be selected which allows the accurate prediction of the liquid-vapor phase behavior in the temperature and pressure range of interest for multicomponent mixtures resembling natural gas. The equation of state must be accurate in a pressure range 10 < P/bar < 70 and a temperature range 250 < T/K < 310. Under these conditions, which are far beyond the critical point of the mixture, retrograde condensation may occur. As natural gas is composed of components with, on the one hand, very volatile substances such as methane, ethane, nitrogen, carbon dioxide, etc., and, on the other hand, many low volatile heavier hydrocarbons such as paraffins, naphthenes, and aromatics, a specific prerequisite for the equation of state to be selected for the model is its capability to describe accurately the vaporliquid phase behavior of such asymmetric mixtures. For a typical Dutch natural gas, Table 1 gives its major constituents. From this table it becomes apparent that there is a large concentration range in which natural gas components may occur; for instance, the C7+ fraction * To whom correspondence should be addressed.
Table 1. Average Concentration of the Main Constituents of a Typical Dutch Natural Gas component
concentration (% mol)
helium carbon dioxide nitrogen methane ethane propane isobutane
0.05 1.00 14.30 81.00 3.00 0.40 0.07
component
concentration (% mol)
n-butane isopentane n-pentane isohexane n-hexane C7+
0.07 0.03 0.02 0.02 0.01 0.03
may contain detectable compounds as heavy as C16, which, on a molar basis, are typically in the ppb region. From a sensitivity analysis (Voulgaris et al., 1995), it could be concluded that the most important components for the accurate prediction of the retrograde condensation behavior of natural gas are always in the hydrocarbon number range of C7-C12. Since the concentration of the hydrocarbons from C7 and higher cannot be detected separately, another prerequisite for the equation of state to be applied is that the equation can be combined with a suitable characterization procedure for the heavy-end fractions of natural gas. This aspect has been discussed elsewhere (Voulgaris, 1995). In this study, two potential candidate equations of state will be compared on their applicability to describe the condensation behavior of natural gas. From the wide variety of cubic equations of state, the PengRobinson (Peng and Robinson, 1976) was selected as a representative semiempirical equation. The second equation studied in this work was the simplifiedperturbed- hard-chain theory (SPHCT) equation of state. This equation was originally introduced by Kim et al. (1986a) and is one of the most simple equations
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Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1697
of state with a statistical mechanical background. An important feature of this equation is that it is specially designed for the description of highly asymmetric molecules. So far, the latter equation of state has not been extensively applied for practical purposes. Peters et al. (1988), Rijkers (1991), Gasem and Robinson (1990), Ponce Ramirez et al. (1991), and Garcia-Sanchez et al. (1992) have reported studies on the SPHCT equation of state and showed that, for the modeling of vaporliquid equilibria of asymmetric hydrocarbon mixtures, this equation may be a promising alternative. Equations of State The Peng and Robinson equation of state (EoS) (Peng and Robinson, 1976) is one of the most popular and widely used cubic EoS. The original one was based on hydrocarbons with a carbon number (Nc) less than 10; i.e., for species with higher carbon numbers the equation may be less accurate. To overcome this shortcoming, a modified version was proposed by Magoulas and Tassios (1990). This version turned out to be much more accurate in predicting the low saturated vapor pressures of pure heavy n-alkanes, especially at low temperatures. The mathematical structure of this equation reads as
P)
RT a V - b V(V + b) - b(V - b)
terms of the compressibility factor Z, the PHCT equation is defined by SW Z ) 1 + c(ZCS rep + Zatt )
(9)
where ZCS rep is the compressibility factor due to repulsions and is calculated from the Carnahan-Starling EoS for hard spheres (Carnahan and Starling, 1969) and SW is the compressibility factor due to attractions and Zatt is calculated from molecular dynamic results of squarewell molecules, obtained by Alder et al. (1972). The parameter c represents one-third of the total number of density-dependent external degrees of freedom. Kim et al. (1986a) simplified the PHCT EoS by replacing its complicated Zatt term by a simple expression based on the local composition model of Lee et al. (1985). The resulting EoS, which is named simplified-perturbedhard-chain theory (SPHCT) EoS, has the same basic mathematical structure as eq 9. Its repulsive and attractive terms for a pure compound are given by
τ2 τ 4 -2 v˜ v˜ ) τ3 1v˜
(10)
Y ZSW att ) - ZM v˜ + Y
(11)
q (2ckT ) - 1 ) exp(2T1 ) - 1
(12)
v˜ ) V/v*
(13)
ZCS rep
(1)
() ( )
with
where
a ) acR
(2)
ac ) 0.45724RTc/Pc
(3)
R ) [1 + m(1 - Tr0.5)]2
(4)
m ) 0.384401 + 1.522760ω - 0.213808ω2 + 0.034616ω3 - 0.001976ω4 (5) b ) 0.07780RTc/Pc
(6)
For mixtures, the conventional van der Waals one-fluid mixing rules were applied N N
am )
∑ ∑zizjxaiiajj(1 - kij) i)1 j)1
bii + bjj bm ) zizj (1 - lij) 2 i)1 j)1
(7)
N N
∑∑
(8)
where N is the number of compounds in the mixture, zi is the mole fraction of compound i in the mixture (liquid or vapor phase), and kij and lij are adjustable binary interaction parameters. Characteristic pure-component parameters of this EoS are the critical temperature (Tc) and pressure (Pc) and the acentric factor (T). The simplified-perturbed-hard-chain theory EoS (SPHCT) belongs to the family of the perturbed-hardchain theory (PHCT) EoS, first introduced by Beret and Prausnitz (1975) and Donohue and Prausnitz (1977). In
Y ) exp and
T ˜ /T*
ZM is the maximum coordination number and in this study fixed at 36. The constant J is equal to 0.74048. The pure-component parameters of this EoS are the characteristic temperature T*, the closed-packed molar volume v*, and the parameter c, being one-third of the number of external degrees of freedom per molecule. The characteristic volume v* can also be related to characteristic molecular parameters
v* )
NAsσ3
x2
(14)
with NA Avogadro’s number, s the number of segments per molecule, and σ the hard-core diameter of a segment. The product T*c represents the intermolecular interaction energy and is equal to
T*c )
q ) k k
(15)
where * is the well-depth (per segment) of the squarewell potential and the characteristic molecular energy per unit of external surface area. The variable q is the external surface area of the molecule, and k is Boltzmann’s constant. Kim et al. (1986a) have used the definition of T*c in terms of the quantities and q instead
1698 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998
of * and s. The mixing rules for the SPHCT equation read N
v* m )
ziv* ∑ i i)1
(16)
Table 2. Accuracy (% AAD, Eq 24) of the PR and the SPHCT EoS of the Psat of Isoparaffins, Naphthenes, and Aromatics (SPHCT’s Parameters c, v*, and T* Obtained from Eqs 25-27; PR Parameters Tc, Pc, and ω Experimentally Defined) % AAD in Psat temp range (K)
PR EoS
SPHCT EoS
Isoparaffins 250-310 253-313 253-293 253-313 253-313 253-313 253-313 253-313 253-315 253-315 253-315 253-316 253-315 265-315 253-315 253-313 251-313 253-313 253-313 253-313 253-313 253-313 265-312 253-313 253-313
1.9 2.2 2.1 1.0 1.7 1.5 1.3 1.2 1.1 1.0 1.1 1.2 1.2 1.0 0.9 2.8 2.7 3.1 2.8 2.3 3.1 1.3 0.8 4.1 1.6
0.5 0.4 0.7 0.8 1.1 1.7 1.7 0.9 1.2 1.8 2.0 1.4 3.0 0.9 3.5 0.6 0.6 0.7 1.5 2.1 1.0 3.2 2.6 3.7 4.2
Normal Alkanes 253-305 253-313 268-313 293-313 278-313 279-313 280-313
0.4 2.0 4.5 2.7 9.8 40.1 65.3
4.2 2.5 13.4 10.3 13.5 25.4 23.6
Naphthenes cyclopentane 254-317 cyclohexane 280-318 methylcyclopentane 268-314 cycloheptane 284-315 t-1,2-methylcyclopentane 273-298 methylcyclohexane 283-313 1,1-dimethylcyclopentane 273-313 c-1,3-dimethylcyclopentane 283-313 ethylcyclopentane 293-315 ethylcyclohexane 293-313 c,c,t-1,2,4-trimethylcyclopentane 283-313
1.0 3.3 1.2 20.3 5.5 1.2 0.4 1.9 10.3 15.6 1.3
0.4 0.6 0.8 1.1 1.3 1.3 0.8 1.4 1.6 1.0 1.7
Aromatics 280-315 253-313 273-313 273-313 253-313 253-313
3.9 2.8 2.2 2.8 5.8 9.7
0.3 1.2 2.4 3.4 1.7 1.2
N
cm )
zici ∑ i)1
(17)
[ ( ) ] ijqi
N N
(cv*Y)m )
component
∑ ∑zizjciv*ji exp 2c kt i)1 j)1
-1
(18)
i
where m denotes mixture properties. The cross terms are given by
v* ji ) σij )
NAsjσij3
(19)
x2 σii + σjj
(20)
x2
ij ) xiijj(1 - kij)
(21)
For mixtures, the SPHCT EoS reads
Z ) 1 + cm
( ) ( )
τv* τv* m m -2 v v 3 τv* m 1v˜
4
2
- ZM
(cv*Y)m
(cv*mY)m v+ cm
(22)
From eqs 19-22, it can be concluded that, for the calculation of vapor-liquid equilibria of mixtures, knowledge of the values of the three characteristic purecomponent parameters T*, v*, and c is not sufficient. Also qi or ii and si or σii of each compound need to be available. The parameters of the SPHCT EoS are obtained from a simultaneous fit of both saturated vapor pressures and saturated molar liquid densities of each species. Kim et al. (1986a) have found the values of T*, v*, and c of pure n-alkanes and established that v*, c, and T*c are almost linear functions of the carbon number Nc, indicating that each -CH2 segment of a n-alkane contributes almost an equal amount to the hard-core volume, energy, and number of degrees of freedom of each molecule. Based on this observation, Kim et al. (1986a) obtained from the slopes of v* and T*c versus the carbon number Nc the following values for the n-alkanes:
) 62.5 K k NAσ3
x2
(23)
) 0.008 667 L/mol
From the slope of T*c as a function of Nc, the value */k is defined, but not the corresponding /k value. /k was evaluated by supposing that the external surface area of one -CH2 segment is 4/5. This information allows one to evaluate from eq 15 the number of segments s and the external surface area q of each n-alkane. As a
iso-C4 iso-C5 neo-C5 iso-C6 3-methyl-C5 2,2-dimethyl-C4 2.3-dimethyl-C4 2-methyl-C6 3-methyl-C6 2,2-dimethyl-C5 2,3-dimethyl-C5 2,4-dimethyl-C5 3,3-dimethyl-C5 3-ethyl-C5 2,2,3-trimethyl-C4 2-methyl-C7 3-methyl-C7 4-methyl-C7 2,4-dimethyl-C6 3,4-dimethyl-C6 3-ethyl-C6 2,3,4-trimethyl-C5 2,2,4-trimethyl-C5 2,2,3-trimethyl-C6 2,2,4-trimethyl-C6 n-C2 n-C3-n-C9 n-C10 n-C11 n-C12 n-C14 n-C16
benzene toluene o-xylene m-xylene isopropylbenzene butylbenzene
result, phase equilibrium calculations of n-alkane mixtures can be performed. Computational Results for Pure Compounds For both EoS the performance with respect to the prediction of the saturated vapor pressure (Psat) of pure compounds in the temperature region of interest (253 e T/K e 313) has been investigated. For that purpose, hydrocarbons have been selected which are natural gas constituents (see Table 2). The main data sources used are Boublı´k et al. (1984), Daubert and Danner (1986),
Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1699 Table 3. Values of the Well-Depth and Hard-Core Diameter for the CH3 and - CH2 Segments, Based on the T*c and v* of Ethane and Propane (Kim et al., 1986a) segment
* (erg/segment)
σ (Å/segment)
CH3 CH2
1.047 22 × 10-4 6.471 68 × 10-15
3.187 956 2.686 063
Table 4. Values of the Characteristic Parameters for n-Alkanes Based on the Proposed Method of This Study
Figure 1. Comparison between the predictive accuracy (% AAD, eq 24) of the PR and the SPHCT EoS for the Psat of n-alkanes. Characteristic parameters for SPHCT EoS from Kim et al. (1986a).
Reid et al. (1987), Wilhoit and Zwolinski (1971), Dreisbach (1955), and Vargaftik (1975). To compare both equations, the AAD has been calculated: n
∑ i)1 % AAD )
|
|
sat Psat pred - Pexp
Psat exp n
× 100
(24)
In Table 2 for all components tested, the AAD’s are summarized, while Figure 1 compares the results for the n-alkanes graphically. The pure-component parameters of the PR EoS (Tc, Pc, and T) were taken from various sources, for instance, Ambrose (1988) for the n-alkanes, Reid et al. (1987), Daubert and Danner (1986), Ambrose and Townsend (1979), and the data bank of the API (1983) for the other components. Figure 1 shows for the n-alkanes and the PR EoS a significant increase of the deviation with the experimental data. A more or less similar conclusion holds for the isoparaffins, naphthenes, and aromatics (Table 2). However, for the heavier naphthenes and aromatics the large deviations may also be caused by experimental uncertainties of the critical properties of these components with increasing Nc (Simmrock et al., 1985). For these components, an average error of less than 2% in Psat can be achieved by changing the critical properties within their region of experimental uncertainty. As pointed out above, for an appropriate characterization of the heavy-end fraction of natural gas, it is of great interest to find a property, for instance, the boiling point Tb, that accurately relates Tc, Pc, and ω of all hydrocarbons present in natural gas (paraffins, isoparaffins, naphthenes, and aromatics) as a smooth function of such a property. Unfortunately, accurate correlations with only one parameter turned out not to exist. For details of this part of our study, one is referred elsewhere (Voulgaris, 1995). For the simplified-perturbed-hard-chain theory equation of state, Kim et al. (1986a) calculated the purecomponent parameters T*, v*, and c for n-alkanes up to C20 and for some other multipolar fluids such as benzene and carbon dioxide. The parameter estimation was based on experimental saturated vapor pressure and saturated molar liquid volume data. Although less pronounced than the case for the PR EoS, for the
Nc
T*c (K)
T* (K)
c
v* × 105 (m3/mol)
2 3 4 5 6 7 8 9 10 11 12 14 16
151.71 198.59 245.47 292.34 339.22 386.10 432.98 479.86 526.74 573.61 620.49 714.25 808.00
120.54 142.45 159.93 171.66 180.04 186.50 191.66 195.81 199.31 202.40 205.04 208.53 211.75
1.2586 1.3941 1.5348 1.7030 1.8842 2.0702 2.2591 2.4506 2.6428 2.8341 3.0262 3.4251 3.8158
2.7595 3.5848 4.4101 5.2354 6.0607 6.8860 7.7113 8.5366 9.3619 10.1872 11.0125 12.6631 14.3137
n-alkanes (Table 2) again an increasing AAD occurs with increasing Nc. A major reason for the deviation in this case is that the experimental data used sometimes were far beyond our temperature region of interest. The same holds for the values of the characteristic parameters evaluated by Gasem and Robinson (1990) and Ponce-Ramirez et al. (1991). Because the parameters of the SPHCT EoS cannot be determined experimentally, their real values are not known. Moreover, if one is interested in parameters valid in a limited temperature range, a unique set of T*, v*, and c, which satisfies a minimum AAD does not exist; i.e., different sets of parameters may yield approximately the same AAD. For that reason and in order to avoid parameter fitting in the temperature range of interest, we have tried to obtain them from the structure of the n-alkanes by applying the definition of each parameter. Realizing that n-alkanes are composed of two different types of segments CH3 and -CH2, and knowing the hard-core diameter σ and the interaction energy per segment * for both segments, eqs 14 and 15 allow one to obtain v* and the product T*c for all n-alkanes. Using the values of the characteristic parameters v* and T*c of ethane for the CH3 segment and of propane for the -CH2 segment, as reported by Kim et al. (1986a), σ and * of both segments have been obtained (see Table 3). The splitting of the product T*c into T* and c is performed by minimizing the AAD of the experimental vapor pressure data of each n-alkane. It was verified that the pure-component parameters T*c, v*, and c of the n-alkanes also satisfied a linear relationship versus the carbon number Nc. The results for the n-alkanes up to n-C16 are summarized in Table 4. Equations 25-27 give their mathematical representation:
T*c ) 57.9559 + 46.8779Nc
(25)
c ) 0.805634 + 0.185283Nc
(26)
v* × 105 ) 1.10890 + 0.82530Nc
(27)
The parameters presented in Table 4 yield an AAD less
1700 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998
Figure 2. Characteristic temperature T* for the isoparaffins, naphthenes, and aromatics with Nc < 11 as a function of their boiling point: 9, adjusted T* of hydrocarbons in order to minimize the % AAD at Psat; s, correlation based on the proposed parameters for n-alkanes.
than 1% in Psat for all the compounds except for C16 (AAD ) 8%). To find the SPHCT pure-component parameters for the isoparaffins, naphthenes, and aromatics, the “effective” carbon number Nc of each component belonging to this category was calculated by substituting their boiling point Tb in the correlation of Tsonopoulos (Tsonopoulos, 1987):
ln(1071.28) ) 6.97596 - 0.116307Nc2/3
(28)
Applying this approach, it turned out that the average AAD’s of the saturated vapor pressures of these compounds were 5.7%, 8.2%, and 8% for the isoparaffins, naphthenes, and aromatics (P, N, and A), respectively, with larger deviations for the heavier ones. In Figure 2, the optimized values of T* for the P, N, and A hydrocarbons of our data bank are compared with those predicted from the n-alkane correlation. Computational Results for Mixtures The primary objective of this study is to test the two equations of state to represent accurately the vaporliquid phase behavior of binary mixtures present in natural gas. For that purpose, flash calculations have been performed for binary systems consisting of one of the basic constituents of natural gas, e.g., methane (C1), nitrogen (N2), or carbon dioxide (CO2), and of one of the heavier hydrocarbons (paraffinic, aromatic, or naphthenic). The performance of the EoS is based on two objective functions: n
∑ i)1 S1 )
|
|
xpred - xexp i i xexp i n
n
∑ i)1 and S2 )
|
|
ypred - yexp i i yexp i n
havior of its binary mixtures with low volatile hydrocarbons is of crucial importance. The binary mixtures studied are given in Table 5. As pointed out already, to describe mixtures, not only the three characteristic parameters of the pure compounds T*, v*, and c are required but also the number of segments or the diameter of a segment and the interaction energy per unit area or the external surface area q need to be available. If the concept of the welldepth per segment * is used instead of , knowledge of one of the three parameters σ, s, and * is sufficient to define the remaining two parameters. If it is assumed that n-alkanes are comprised of only -CH2 segments, for these molecules characteristic information can easily be obtained from the slope of T*c and v* as a function of Nc. This means that all non -CH2 segments (i.e., CH3 and CH4) are represented by a number of -CH2 segments. For example, according to the values for NAσ3/x2 and v*, the two CH3 groups in ethane are equivalent to 3.18 -CH2 segments. In this work, we have changed the values of the characteristic parameters of the n-alkanes, i.e., NAσ3/ x2 and /k, which are equal to
) 58.6 K k NAσ3
x2
) 0.008 253 0 L/mol
From these values the number of segments and the external surface area which corresponds to each nalkane can be calculated, and the following relationship has been found:
q ) -0.0858761 + 0.8000021s
q ) 0.0471967 + 0.776270s
(32)
According to Donohue and Prausnitz (1977), we assigned the same diameter to the segments of all compounds, hydrocarbons and non-hydrocarbons, because they are simulated by a number of -CH2 segments. Then the external surface area, which corresponds to the number of segments of the compound of interest, can be calculated from eq 31 or eq 32. In addition, the interaction energy can easily be found from eq 15. It should be noted that if /k and NAσ3/x2 are the same for all compounds, their values are not needed because eq 18 simplifies to N N
where yi and xi are the mole fractions of the less volatile compound and n is the number of experimental points. S1 and S2 are not combined together in one sum but are considered separately because their orders of magnitude, most of the time, may differ significantly since the solubilities of the heavy hydrocarbons in the gas phase are extremely low. Because methane is a major constituent, the accurate description of the phase be-
(31)
The equivalent relationship as obtained by Kim et al. (1986a) reads
(cv*Y)m ) (29)
(30)
[ ( ) ] 1
∑ ∑zizjciv*j exp 2T˜ i)1 j)1
-1
(33)
i
In Figures 3 and 4, S1 and S2 (eq 29), of the two EoS for binary mixtures of methane and n-alkanes are compared with kij ) 0. For the SPHCT equation the parameters as calculated in this study have been used. Moreover, in Figures 5-8 we show graphically the performance of both EoS for some selected binary methane mixtures (propane, decane, and hexadecane). From Figure 3 and, more specifically, from Figures 7a and 8a, it can be seen that the liquid phase is better represented by the PR EoS. The erroneous behavior of
Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1701 Table 5. Data Bank of the Binary Systems under Study system
temp range (K)
pressure range (bar)
ref
methane-propane
255-311
7-102
methane-butane
255-311
6-103
methane-pentane
248-310
6-100
methane-hexane
255-310
7-103
methane-heptane
248-310
7-103
methane-octane methane-decane methane-dodecane methane-hexadecane methane-benzene methane-toluene methane-cyclohexane nitrogen-decane carbon dioxide-propane carbon dioxide-butane carbon dioxide-hexane
248-323 253-314 258-314 288-315 279-315 255-278 294-315 263-311 278-313 311 273-313
10-70 5-108 5-100 5-100 5-100 7-105 14-100 5-100 7-69 7-81 8-116
carbon dioxide-benzene
273-313
9-100
Reamer et al. (1950) Price and Kobayashi (1959) Akers et al. (1954) Sage et al. (1940) Kahre (1974) Elliot et al. (1974) Kahre (1975) Chu et al. (1976) Sage et al. (1942) Shim and Kohn (1962) Gunn et al. (1974) Lin et al. (1978) Reamer et al. (1956) Chang et al. (1966) Kohn and Bradish (1964) Rijkers (1991) Rijkers (1991) Rijkers (1991) Rijkers (1991) Lin et al. (1978) Reamer et al. (1958) Pearce et al. (1993) Reamer et al. (1951) Shibata and Sandler (1989) Li et al. (1981) Kaminishi et al. (1987) Gupta et al. (1982) Kim et al. (1986b) Kaminishi et al. (1987)
Figure 3. Comparison between the predictive accuracy of the PR and the SPHCT EoS in the liquid phase composition for binary mixtures of methane with n-alkanes (kij ) 0).
Figure 5. Equilibrium ratio K ) y/x of propane for the system methane-propane (kij ) 0). Exptl data: b, T ) 255 K; 9, T ) 283 K; 2, T ) 311 K; s, SPHCT EoS; ‚‚‚, PR EoS.
Figure 4. Comparison between the predictive accuracy of the PR and the SPHCT EoS in the gas phase composition for the binary mixtures of methane with n-alkanes (kij ) 0).
the SPHCT EoS possibly can be attributed to the behavior of its attractive term at high reduced densities. However, the opposite holds for the gas phase prediction of binary methane mixtures with heavy n-alkanes
(Figures 4, 7b, and 8b), especially with n-alkanes with Nc > 12. From Figure 8 it can be seen that the PR equation overestimates the composition of the heavy compound (C16) in the gas phase, which is approximately as high as some ppm on a molar basis. This is a direct consequence of the inaccurate representation of Psat of the pure compound (Figure 1). Furthermore, from the same figures we can conclude that the higher the temperature, the more accurate are the predictions of the composition of the gaseous phase by the SPHCT equation. The SPHCT pure-component parameters obtained in this study show an improved description of the liquid phase (Figures 7 and 8) when compared to those obtained by Kim et al. (1986a), whereas for the gaseous
1702 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998
Figure 6. Vapor-liquid equilibrium for the system methanepropane (kij ) 0). Exptl data: b, T ) 255 K; 9, T ) 283 K; 2, T ) 311 K; s, SPHCT EoS; ‚‚‚, PR EoS.
Figure 8. Vapor-liquid equilibrium for the system methanehexadecane (kij ) 0). Exptl data: b, T ) 293 K; 9, T ) 303 K; 2, T ) 313 K; s, SPHCT EoS (parameters taken from Kim et al. (1986a); - -, SPHCT EoS (parameters as proposed in this study); ‚‚‚, PR EoS. (a) Liquid phase. (b) Gas phase.
Figure 7. Vapor-liquid equilibrium for the system methanedecane (kij ) 0). Exptl data: b, T ) 253 K; 9, T ) 283 K; 2, T ) 303 K; s, SPHCT EoS (Kim et al. parameters); - -, SPHCT EoS (parameters as proposed in this study); ‚‚‚, PR EoS. (a) Liquid phase. (b) Gas phase.
phase both sets of parameters are almost equally accurate. As an exception, only for the C1-C16 binary
mixture the improvement of the description of the gas phase is also significant (Figure 8b). The parameters presented by Gasem and Robinson (1990) lead to somewhat less accurate results for both phases of all the methane binary mixtures. In Figures 5 and 6 the VLE of the simple mixture C1C3 is shown. It can be seen that, although the K ) x/y values obtained from the SPHCT equation are better than those obtained from the PR equation, especially at higher temperature (Figure 5), the PR equation performs better (Figure 6). It was quite a general finding that the SPHCT equation is not adequate for the simpler mixtures and, as a consequence, large kij values are needed to improve the quality of the description. In addition, also binary mixtures of methane with aromatics, such as benzene (see Figure 9) and toluene, and naphthenes (cyclohexane) have been modeled. The SPHCT parameters provided by Kim et al. (1986b) have been used for benzene and those of Ponce-Ramirez et
Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1703
Figure 10. Vapor-liquid equilibrium for the system carbon dioxide-n-hexane at T ) 313 K (kij ) 0). 9, exptl data (Li et al., 1981); s, SPHCT EoS [(/k)CO2 ) 58.6]; - -, SPHCT EoS [(/k)CO2 ) 149.9]; ‚‚‚, PR EoS.
Figure 11. Comparison between the descriptive accuracy of the PR and the SPHCT EoS for the liquid phase composition for binary mixtures of methane with n-alkanes (optimum kij for each binary). Figure 9. Vapor-liquid equilibrium for the system methanebenzene (kij ) 0). Exptl data: b, T ) 263 K; 9, T ) 283 K; 2, T ) 303 K; s, SPHCT EoS [(/k)benzene ) 58.6]; - -, SPHCT EoS [(/k)benzene ) 79.6]; ‚‚‚, PR EoS. (a) Liquid phase. (b) Gas phase.
al. (1991) for cyclohexane and toluene. The general conclusion is that using a value of the characteristic energy /k for the non-n-alkanes, differing from that of the n-alkanes, the predictive capability of the SPHCT equation for the liquid phase increases appreciably; e.g., for C1-benzene the error reduces with almost 50% (see Figure 9a). However, for the gas phase the differences are negligible (Figure 9b). We have also used the characteristic parameters derived from the correlations for n-alkanes as given in eqs 25-27. It turned out that the error in the liquid phase becomes even larger, whereas in the gas phase the error remains approximately the same. It seems that the characteristic parameters cannot be expressed as a function of only one parameter (for instance, Tb) for all the P, N, and A hydrocarbons. This means that according to this approach all components in the mixture behave as nalkanes with an effective carbon number Nc. Furthermore, binary mixtures have been studied of nitrogen with decane and of carbon dioxide with nalkanes (see Figure 10 for the binary mixture CO2-C6)
and with benzene. Similar conclusions could be drawn as pointed out above. Again the liquid phase predictions obtained from the SPHCT equation improve appreciably if a different /k value for the non-n-alkanes is applied. For some CO2 systems this procedure makes the SPHCT equation even more accurate than the PR equation (see Figure 10). Even for some CO2 systems the gas phase predictions become more accurate (see also Figure 10). This observation encouraged us to try a different /k value for methane as well. The reasoning behind this is that methane does not belong to the n-alkane correlation versus Nc, and its only segment, CH4, is simulated with -CH2 in the same way as was done for N2 and CO2. Calculations with /k ) 45.4 K for methane showed that the liquid phase predictions are improved by 10-15%. On the other hand, in almost all cases approximately the same gas phase accuracy was obtained. Introduction of optimum kij values for each methane binary mixture did result in a more accurate description of the liquid phase applying the PR equation, whereas the predictions for the gas phase hardly change. It is obvious that a second binary parameter is needed for the asymmetrical mixtures (Figures 11 and 12).
1704 Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998
Figure 12. Comparison between the descriptive accuracy of the PR and the SPHCT EoS for the gas phase composition for binary mixtures of methane with n-alkanes (optimum kij for each binary).
and 13a for the system C1-C10). On the other hand, an increase of the kij values results in a shift of the dewpoint curve to leaner concentrations of the heavier compound. Consequently, the gas phase description for the asymmetrical mixtures worsens (Figures 12 and 13b). This is not surprising since, for kij ) 0, the SPHCT equation predicts leaner gas phase compositions compared with experimental data (Figure 7). With kij different from zero, both the SPHCT and the PR EoS show equivalent capabilities for both the liquid and gas phases, with as an exception the gas phase predictions of asymmetrical mixtures (e.g., C1 + C16). In that case, the SPHCT equation performs much better (Figures 12 and 13). For the CO2 systems the optimum kij values for the SPHCT equation (using the same /k for all compounds) and those for the PR equation are of the same magnitude. However, their value decreases 50% when an improved /k is introduced for the non-n-alkanes. Conclusions
Figure 13. Vapor-liquid equilibrium for the system methanedecane. Exptl data: b, T ) 253 K; 9, T ) 283 K; 2, T ) 303 K; ‚‚‚, SPHCT EoS (kij ) 0); s, SPHCT EoS (kij ) 0.063). (a) Liquid phase. (b) Gas phase.
Although the optimum kij values for the SPHCT equation are larger in magnitude, they improve significantly the liquid phase description (e.g., see Figures 11
From this study it became apparent that compared to the PR EoS, the SPHCT EoS is superior in representing the low Psat data of pure compounds heavier than dodecane. Of course, this is a direct consequence of the fact that the characteristic parameters of the SPHCT equation were optimized from saturated vapor pressure data. For binary mixtures of methane with compounds up to dodecane, using optimum kij values for each binary mixture and equation of state applied, both EoS have an equivalent accuracy, although the PR equation performs somewhat better in the liquid phase and the SPHCT equation shows a somewhat better performance in the gas phase. A major observation originating from this study is that, if we are dealing with asymmetrical mixtures, the SPHCT EoS is by far the most superior one. With a minor loss of accuracy of the gas phase description, inaccurate liquid phase predictions simply can be corrected by applying a kij value. Moreover, applying a characteristic energy /k for the non-nalkanes and methane as well that differs from the one used for n-alkanes improves considerably the predictive capability of the SPHCT equation and simultaneously decreases the absolute value of the optimum kij parameter. Neither the PR nor the SPHCT equation have purecomponent parameters that can be expressed as a function of only one property, for instance, the Tb. Since the primary objective of this study was to use an equation of state that accurately predicts the condensation behavior of lean natural gas, i.e., we are dealing with mixtures where the amount of C13+ is negligible, and not its dewpoint, which is very sensitive to even traces of the heavier compounds, this research did not find strong arguments that justify the application of the SPHCT equation instead of the simpler PR EoS. On the other hand, to improve the predictive capabilities of the PR EoS for asymmetrical mixtures, this effectively can be done by introducing a second binary interaction parameter for the covolume bm (eq 9). Also application of alternative mixing rules may lead to an improved description of asymmetric mixtures; for instance, the recently developed mixing rules by Wong and Sandler (1992) could be an option.
Ind. Eng. Chem. Res., Vol. 37, No. 5, 1998 1705
List of Symbols a ) parameter in Peng-Robinson equation of state, eq 1 (bar cm6 mol-1) AAD ) average absolute deviation percent at Psat, eq 29 b ) parameter in Peng-Robinson equation of state, eq 1 (cm3 mol-1) 3c ) number of external degrees of freedom per molecule K ) equilibrium ratio, K ) y/x k ) Boltzmann’s constant kij ) interaction parameter at the attractive term lij ) interaction parameter at the covolume bm m ) parameter in Peng-Robinson equation of state, eq 4 n ) number of experimental points N ) number of compounds in the mixture Nc ) number of carbon atoms for an n-alkane NA ) Avogadro’s number P ) pressure (bar) q ) normalized surface area per molecule, q ) 4/5 for a -CH2 segment R ) universal gas constant (bar m3/(K mol)) T ) absolute temperature (K) T* ) characteristic temperature for intermolecular interactions (K) s ) number of segments per molecule S1 ) average error in liquid phase composition, eq 29 S2 ) average error in gas phase composition, eq 29 v* ) characteristic volume per mole (m3/mol) V ) molar volume (m3/mol) x ) mole fraction of a compound in the liquid phase y ) mole fraction of a compound in the gas phase Y ) parameter of the SPHCT’s attractive term, eq 12 z ) mole fraction of a compound in the mixture Z ) compressibility factor Zm ) maximum coordination number, here equal to 36 Greek Symbols R ) parameter of the Peng-Robinson equation of state, eq 4 ) characteristic energy per unit external surface area of a molecule (erg/m3) * ) characteristic segmental interaction energy (erg/ segment) σ ) hard-core diameter of a segment (Å/segment) τ ) constant equal to 0.74048 (x2π/6) ω ) acentric factor Subscripts att ) attractive b ) atmospheric pressure c ) critical exp ) experimental i, j ) compound i, j m ) mixture property n ) normal conditions (P ) 1 bar and T ) 273.15 K) pred ) predicted r ) reduced in respect to PR characteristic parameters, Tc or Pc rep ) repulsive Superscripts CS ) Carnahan-Starling sat ) saturated SW ) square-well ∼ ) reduced with respect to the SPHCT characteristic parameters, T* or v*
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Received for review September 10, 1997 Revised manuscript received March 9, 1998 Accepted March 9, 1998 IE970641B
