Ind. Eng. Chem. Res. 1996, 35, 905-910
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An Approach for the Application of a Cubic Equation of State to Hydrogen-Hydrocarbon Systems Chorng H. Twu,* John E. Coon, Allan H. Harvey,† and John R. Cunningham Simulation Sciences Inc., 601 South Valencia Avenue, Brea, California 92621
This paper presents an approach for predicting the solubility of hydrogen in hydrocarbons using a cubic equation of state. Because the R function from the Soave-Redlich-Kwong and PengRobinson equations of state is inadequate at temperatures far above the critical temperature, the binary interaction parameters required for hydrogen-hydrocarbon systems using these equations are typically large and strongly dependent on the temperature. An approach is proposed for developing a temperature-dependent R function for hydrogen which minimizes the need for interaction parameters. The minimization of the values of the binary interaction parameters between hydrogen and heavy hydrocarbons is desirable because this produces better predictions when the binary interaction parameters are not known and are set to zero. A universal procedure is also presented for interconverting the binary interaction parameters between the new hydrogen R function developed in this work and the traditional Soave R function or any other type of R function. Introduction Hydrogen is one of the most important gases in refineries. Hydrogen demand has increased substantially due to increased hydrocracking. The hydrocracking has increased the need for basic data on vaporliquid equilibria (VLE) in hydrogen-hydrocarbon systems. The hydrocarbons may constitute light hydrocarbons such as methane, ethane, etc., as well as heavy crudes with petroleum fraction cuts having boiling points as high as 1600 °F. VLE for the hydrogen-hydrocarbon systems is usually calculated in the refinery industry using cubic equations of state (CEOS). However, because these calculations are far above the critical temperature of hydrogen, application of the Soave type of CEOS to these systems results in binary interaction parameters that are unusually large and strongly dependent on temperature. Different equations for correlating hydrogen-hydrocarbon binary interaction parameters for CEOS have been presented (Gray et al., 1985; Moysan et al., 1983, 1985, 1986). These authors regressed VLE data into binary interaction parameters and presented them in a correlation form. The correlation by Gray et al. is a function of the critical temperature of the solvent hydrocarbon. The correlation by Moysan et al. is a function of reduced temperature. Instead of correlating binary interaction parameters themselves, some authors (Mathias, 1983; Boston and Mathias, 1980; El-Twaty and Prausnitz, 1980) recommended specific R functions for hydrogen be used in VLE calculations for hydrogencontaining systems. Other authors (Wang and Zhong, 1989; Graboski and Daubert, 1979) regressed VLE data into the hydrogen R function. Based on systems and binary VLE data selected, different hydrogen R functions with different functional forms were proposed by these authors. Extrapolation of the R value outside the range of data definitely depends on choosing the proper functional form. * Corresponding author. e-mail:
[email protected]. Fax: 714/579-0236. † Present address: Thermophysics Division, National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80303.
0888-5885/96/2635-0905$12.00/0
Another approach is proposed in this work for handling hydrogen-hydrocarbon systems. In addition, a universal correlation is presented for interconverting the binary interaction parameters between the new hydrogen R function developed in this work and Soave’s traditional R function or any other R function type. A Modified Redlich-Kwong Cubic Equation of State The Redlich-Kwong (RK) cubic equation of state can be modified, as suggested by Wilson (1964), by replacing the term a with a more general temperature-dependent term, a(T):
P)
a(T) RT v - b v(v + b)
(1)
where P is the pressure, T is the absolute temperature, and v is the molar volume. 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 to zero at the critical point, resulting in
a(Tc) ) 0.427481R2Tc2/Pc
(2)
b ) 0.086641RTc/Pc
(3)
where subscript c denotes the critical point. b in eq 1 is assumed to be independent of temperature and a is a function of temperature written as:
a(T) ) R(T) a(Tc)
(4)
where the R function, R(T), is a temperature-dependent function. In the original Redlich-Kwong equation (1949):
R(T) ) 1/Tr0.5
(5)
Wilson (1964) was the first to introduce a general form of the temperature dependence of the a parameter (eq 4) in the Redlich-Kwong CEOS. However, the R(T) function that gained widespread popularity and use was © 1996 American Chemical Society
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Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996
Table 1. Component-Dependent Parameters L, M, and N of the r Function (eq 8) To Be Used with the Modified Redlich-Kwong Cubic Equation of State (eq 1) component
Tc (K)
Pc (bar)
methane n-butane n-hexane n-heptane n-octane n-decane hydrogen
190.58 425.17 507.85 540.26 568.83 618.45 33.23
46.04 37.97 30.31 27.36 24.86 21.23 13.16
L
M
0.106 750 0.920 161 0.308 201 0.859 571 0.128 223 0.893 666 0.206 521 0.852 664 0.363 345 0.825 314 0.316 547 0.828 468 1.252 82 13.269 0
unknown region. Therefore, a proper approach is required to derive the R function for hydrogen when correlating hydrogen binaries.
N 3.096 74 2.008 90 4.882 41 3.320 97 2.312 82 2.903 95 0.040 00
Development of an r Function for Hydrogen From the definition of Henry’s constant of solute 1 in a solvent 2, H12, one has
H12 ) lim
x1f0
proposed by Soave (1972) as an equation of the form:
R ) [1 + m(1 - Tr0.5)]2
(6)
The m parameter was obtained by forcing the equation to reproduce the vapor pressure for nonpolar compounds at Tr ) 0.7 and was correlated as a function of the acentric factor ω for the RK CEOS:
m ) 0.480 + 1.574ω - 0.175ω2
(7)
Due to its reasonable accuracy and simplicity, Soave’s type of R function has subsequently been used by many investigators, such as Peng and Robinson (1976). Soave’s development of eq 6 represented a significant milestone in the application of a CEOS. The resulting R(T) from Soave (1972) as a function of the reduced temperature and the acentric factor adequately correlates the vapor pressures of hydrocarbons at high reduced temperatures, but the error increases rapidly at low reduced temperatures. In addition, Soave’s R function becomes zero at some finite temperature above the critical and then rises again with increasing temperature. This behavior is the reason why the binary interaction parameters using Soave’s R function for hydrogenhydrocarbon binaries, where hydrogen is at a very large reduced temperature, are quite large and strongly dependent on the temperature. Although using Soave’s generalized R function in the RK CEOS is quite convenient for estimating vapor pressures of pure hydrocarbons, the deviation of predicted vapor pressures at low temperatures can be quite large. On the other hand, using a component-dependent R function can duplicate the vapor pressures of pure components from the triple point to the critical point very accurately. Therefore, the component-dependent R function proposed by Twu et al. (1991) is used in this work for correlating the vapor pressure data of pure components. NM)
R ) TrN(M-1) eL(1-Tr
(8)
Equation 8 has three parameters, L, M, and N. These parameters are unique to each component and have been determined from the regression of pure-component vapor pressure. All critical constants and vapor pressure data from the triple point to the critical point for hydrocarbons are from DIPPR (Daubert and Danner, 1990). The true critical constants, acentric factor (Tc ) 33.23 K, Pc ) 13.16 bar, ω ) -0.22), and vapor pressure data from Vargaftik (1975) are used for hydrogen. Table 1 lists L, M, and N values of the components for use in the solubility calculations of hydrogen in hydrocarbons in this work. For highly supercritical fluids like hydrogen at refinery temperatures, the R function with parameters derived from the vapor pressure data represents extrapolation into an
f1L x1
) lim Pφ1L
(9)
x1f0
The liquid phase fugacity coefficient, φ1L, can be derived from eq 1. It is then substituted into eq 9 to give Henry’s constant as follows:
B1 ln H12 ) ln P2 + (Z2 - 1) - ln(Z2 - B2) + B2 A2 B1 A1 1/2 B2 -2 (1 - k12) ln 1 + (10) B2 B2 A2 Z2
[
()
] (
)
where P2 and Z2 are vapor pressure and the liquid compressibility factor of pure solvent at saturation. Z2 is the solution of eq 1:
Z23 - Z22 + (A2 - B2 - B22)Z2 - A2B2 ) 0 (11) Ai and Bi in eqs 10 and 11 are defined as:
Ai ) Pai/R2T2
(12)
Bi ) Pbi/RT
(13)
where ai and bi are calculated from eqs 2 and 3. Equation 10 is the relationship between the binary interaction parameter k12 and Henry’s constant H12 at a given temperature. The procedures for deriving the hydrogen R are described here. Let the hydrogen be the solute designated as component 1 and the hydrocarbon be the solvent designated as component 2. The binary interaction parameter k12 in eq 10 is intentionally assigned a zero value. When Henry’s constant of hydrogen in pure hydrocarbon solvent at a given temperature is known, eq 10 can be solved for the A1 value of hydrogen. The hydrogen R value at the temperature for this binary is obtained from A1 by using eqs 12, 4, and 2. This calculation is repeated at other temperatures. The resulting calculated hydrogen R values are a function of hydrogen reduced temperature for this particular binary. This procedure is performed again for other hydrogen-hydrocarbon binaries. The hydrocarbon solvents used range from methane to n-decane. These binaries cover a sufficiently wide range of hydrogen reduced temperature to exhibit the shape of the hydrogen R function. Figure 1 shows the resulting hydrogen R values derived from Henry’s constants of hydrogencontaining binaries as a function of the hydrogen reduced temperature. One of the requirements of the R function is that its value equals unity at the critical point. It can be seen from Figure 1 that it would be very difficult to find a hydrogen R function that passes through the hydrogen R values derived from Henry’s constants of hydrogen in methane as well as other hydrocarbons and at the same time satisfies the critical point requirement. In our experience, when the properties of normal alkanes
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 907
Figure 1. The solid line is the proposed hydrogen R function. Dashed and dotted lines are the Soave R with acentric factors equal to -0.22 and -0.125, respectively. Points are data from Henry’s constants of hydrogen in methane (+), n-butane (solid circle), n-hexane (solid square), n-heptane (solid star), n-octane (blank square), and n-decane (blank triangle).
are plotted against normal boiling point, the first few members of the series (especially mathane) often do not line up with the other members. Therefore, it is not surprising to find that the hydrogen R values derived from methane are not on the same hydrogen R curve, as proposed in this work in Figure 1, as those derived from other hydrocarbons. Because of this, the hydrogen R values derived from Henry’s constants of hydrogen in methane will not be included in the process of developing the hydrogen R function. Ignoring the R values from the hydrogen-methane binary in the process of developing the new hydrogen R will result in a hydrogen R with the proper shape over the entire range of temperature. On the other hand, if the hydrogen R values derived from Henry’s constants of hydrogen in methane are considered and the critical point requirement is ignored, the hydrogen R function obtained from the regression of the R data not only approaches zero very quickly at high reduced temperatures but also cannot be used in the vapor pressure prediction for hydrogen. The R value of almost zero at high reduced temperatures will require an abnormally large binary interaction parameter to correlate the data for hydrogen and heavy hydrocarbon systems, such as petroleum fractions. Therefore, the hydrogen R in this work will be derived from both the vapor pressure data of hydrogen and Henry’s constants of hydrogen in hydrocarbons excluding methane. Combining the hydrogen vapor pressure data and Henry’s constants of hydrogen in hydrocarbons, the shape of the hydrogen R function can then be defined. A multiproperty regression is required to obtain the following hydrogen R function in terms of hydrogen reduced temperature, Tr: 0.530760]
R ) Tr0.490760e1.25282[1-Tr
(14)
The constants in eq 14 are computed from the values of the L, M, and N parameters, which are listed in Table 1, according to eq 8.
Figure 2. Vapor pressures of hydrogen: comparison of predicted vapor pressures from this work (solid line) and Soave (dashed and dotted lines are from acentric factors equal to -0.22 and -0.125, respectively) with data from Vargaftik (1975, solid circles).
Results and Discussion Because the Soave modification of the RedlichKwong (SRK) equation of state has been used widely in correlating hydrogen-hydrocarbon systems (for example, Gray et al., 1985), the new hydrogen R function developed in this work is compared with the Soave R function. The solid line in Figure 1 is from this work (i.e., eq 14), and the dashed line is from the Soave R function with ω ) -0.22 (eq 7). Figure 1 illustrates that the Soave R function substantially overestmates the hydrogen R values. This is the reason why the SRK equation of state needs large binary interaction parameters to correlate hydrogen and hydrocarbon systems. Different values of the acentric factor (other than -0.22) have been used by previous investigators for hydrogen in Soave’s R function. However, we have found that the optimum value (for fitting Henry’s constants) of the acentric factor for hydrogen in Soave’s R function is -0.125, not zero. The dotted line in Figure 1 is Soave’s R function with ω ) -0.125, which is almost as good as the solid line except that it approaches zero quickly at reduced temperatures higher than 16 and its vapor pressure prediction is quite poor (see Figure 2). Figure 2 compares the prediction of hydrogen’s vapor pressure from the triple point to the critical point from this work with that from Soave’s R function. In this case, Soave’s R function underestimates the hydrogen vapor pressures. The solubility of hydrogen in liquid hydrocarbons is determined from the Henry’s constants of hydrogen in hydrocarbons. The calculated values shown in Figures 3-7 are from the RK CEOS using the hydrogen R function developed in this work and Soave’s R without introducing any binary interaction parameters. The solid line is from this work, and the dashed line is from Soave’s R function. With the binary interaction parameter equal to zero, Soave’s R function significantly underpredicts Henry’s constants and the new hydrogen R function from this work matches Henry’s data very well for all hydrocarbons. The results shown in these figures indicate that this new R function provides a way of being able to predict with confidence the hydrogen
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Figure 3. Comparison of the predicted Henry’s constant of hydrogen in n-butane with experimental data (solid circles) from Kline (1975). The solid line is from this work and the dashed line from Soave.
Figure 4. Comparison of the predicted Henry’s constant of hydrogen in n-hexane with experimental data (solid circles) from Young (1981). The solid line is from this work and the dashed line from Soave.
solubility in hydrocarbons without resorting to binary interaction parameters (for most hydrocarbons). Interconversion of Binary Interaction Parameters As mentioned in the previous section, the developed hydrogen R function does not go through the hydrogenmethane data. However, a large amount of hydrogenmethane data have been compiled and correlated using the SRK CEOS; the Gray correlation (1985) is a good example for this system. Here we show that such existing interaction parameters can be used directly to make equation of state predictions. That is, there is no need to regress once again existing data for the use of the new hydrogen R function with an equation of
Figure 5. Comparison of the predicted Henry’s constant of hydrogen in n-heptane with experimental data (solid circles) from Cook (1957). The solid line is from this work and the dashed line from Soave.
Figure 6. Comparison of the predicted Henry’s constant of hydrogen in n-octane with experimental data (solid circles) from Cook (1957) and Brunner (1985). The solid line is from this work and the dashed line from Soave.
state; the existing binary interaction parameters can be used directly in the equation of state with our hydrogen R model. The binary interaction parameter k12, which corresponds to the new hydrogen R function in this work, and the existing binary interaction parameter value k12S, which corresponds to Soave’s R function, have the following simple relationship:
k12 ) 1 - (R1S/R1)1/2(1 - k12S)
(15)
where subscripts 1 and 2 designate hydrogen and hydrocarbon, respectively. R1 is eq 14 and R1S is eqs 6 and 7 with ω ) -0.22. The superscript S means that Soave’s R function is used in the calculation.
Ind. Eng. Chem. Res., Vol. 35, No. 3, 1996 909
Figure 7. Comparison of the predicted Henry’s constant of hydrogen in n-decane with experimental data (solid circles) from Brunner (1985). The solid line is from this work and the dashed line from Soave.
Figure 8. Comparison of the predicted Henry’s constant of hydrogen in methane from this work (solid line) with experimental data (symbols) from Hong and Kobayashi (1981).
To illustrate the calculation for the hydrogenmethane binary, Gray’s correlation is used as an example. Since Gray’s correlation was developed for the SRK equation of state, the binary interaction parameter given by Gray is k12S, which is:
gen-hydrocarbon systems with minimum need for interaction parameters. For hydrogen-methane, the binary interaction parameter values already available in the literature and other sources can be directly converted for use in our R model. No refitting of VLE data is needed. For petroleum fractions, the new hydrogen R function provides a way to predict the hydrogen solubility with confidence without any binary interaction parameters.
k12S ) 0.0067 + X)
0.63375X3 1 + X3
Tc2 - 50 1000 - Tc2
(16)
(17)
where Tc2 is the critical temperature of hydrocarbon in Kelvin. Figure 8 shows the excellent results for the hydrogenmethane binary when eqs 15-17 are applied to the equation of state. Gray’s correlation may be applied to hydrogen and light hydrocarbons such as methane, ethane, and propane. Since the new hydrogen R function minimizes the need for binary interaction parameters for hydrogenpetroleum fraction systems and provides a feasible way of being able to predict with confidence the hydrogen solubility in petroleum fractions, the value of k12 for hydrogen and petroleum fractions can be reasonably assumed to be zero. In that case, eq 15 becomes:
k12S ) 1 - (R1/R1S)1/2
(18)
Equation 18 is another generalized correlation for the binary interaction parameter for hydrogen-hydrocarbon systems for the SRK equation of state. Equation 18 indicates that the binary interaction parameter between hydrogen and hydrocarbon is indeed a very complicated function of temperature when the SRK equation is applied to hydrogen-containing systems. However, if the hydrogen R function developed in this work is used in the SRK equation of state, then the need for any interaction parameters is minimized.
Nomenclature a, b ) Redlich-Kwong equation of state parameters A, B ) reduced parameters of a and b, respectively f ) fugacity H12 ) Henry’s constant k12 ) binary interaction parameter L, M, N ) parameters in the R function P ) pressure R ) gas constant T ) temperature v ) molar volume x ) liquid mole fraction Z ) compressibility factor Greek Letters φ ) fugacity coefficient R ) R function defined in eq 4 ω ) acentric factor defined at Tr ) 0.7 Subscripts 1 ) component 1 2 ) component 2 c ) critical property r ) reduced property i ) component i
Conclusion
Superscripts
Our new approach to the hydrogen R function allows simple cubic equations of state to be applied to hydro-
L ) liquid phase property S ) Soave’s R function used
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Received for review June 20, 1995 Accepted November 14, 1995X IE9503813
X Abstract published in Advance ACS Abstracts, February 1, 1996.
