Ind. Eng. Chem. Res. 2007, 46, 7253-7259
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Methodology for Predicting Water Content in Supercritical Gas Vapor and Gas Solubility in Aqueous Phase for Natural Gas Process Chorng H. Twu* Tainan UniVersity of Technology, 529 Chung Cheng Road, Yung Kang, Tainan 71002, Taiwan
Suphat Watanasiri Aspen Technology, Inc., Ten Canal Park, Cambridge, Massachusetts 02141
Vince Tassone Suite 900, 125 9th AVenue SE, Calgary, Alberta T2G 0P6, Canada
A methodology is proposed to accurately predict the water content in the supercritical gas vapor from the cubic equation of state (CEoS) without the need to calculate the liquid fugacity for the supercritical gas. The van der Waals (vdW) one-fluid mixing rule and our latest development of the advanced CEoS/AE mixing rule are incorporated in Twu-Sim-Tassone (TST) CEoS to investigate their ability in representing simultaneously both the water content in the gas vapor and the gas solubility in the aqueous phase over the application range of temperature and pressure encountered in the natural gas industry. Introduction The streams in the natural gas process contain light hydrocarbons, mainly methane and ethane, associated with nonhydrocarbon supercritical gases (nitrogen, hydrogen, argon, etc.). These streams are often contacted with water under processing conditions, which normally result in gas vapor and water liquid phases. For the natural gas process industries, it is very important to know how much water in the supercritical gas vapor phase, because the water may cause hydrate formation under lowtemperature conditions that may plug the valves and fittings in gas pipelines. In addition, water vapor may cause corrosion difficulties when it reacts with hydrogen sulfide (H2S) or carbon dioxide (CO2), which are commonly present in gas streams. The droplets of acidic water can erode instrumentation and control equipment. The formation of hydrates can reduce the available cross section for natural gas flow and, in some extreme cases, can completely block the gas flow in a pipeline. A reduction in gas flow has a serious financial impact and endangers the supply of gas to customers. To avoid the problems related to trace amounts of water and the resulting financial losses, natural gas is dried before feeding it into a pipeline. To make an optimal design of these drying units, a precise knowledge of the dependency of water content on temperature, pressure, and composition of the gas is needed. Because the water concentration is usually very low, the water content in dry gas vapor is expressed in terms of weight of water per volume of dry gas in the natural gas industries. For example, as shown in Table 1,1 the water content in pure (dry) nitrogen gas at 5 bar and 248.15 K (-25 °C) is 101.6 mg/Nm3. It means that 1 m3 of dry (100% pure) nitrogen under normal conditions of 1 atm and 273.15 K contains 101.6 mg of water when the system is at equilibrium (conditions of 5 bar and 248.15 K). For the purpose of illustration, the system of nitrogen and water is used here as a example for solving the water content in the supercritical nitrogen vapor. At phase equilibrium, the * Author to whom correspondence should be addressed. Tel.: 8866-2422603. Fax: 886-6-2433812. E-mail address: t80060@ mail.tut.edu.tw.
fugacities of nitrogen (N2), denoted as fn, and water (H2O), denoted as fw, in the liquid and vapor phases are the same:
f Ln ) f Vn f Lw ) f Vw
(1) (2)
The fugacity of component i in the solution can be calculated from any equation of state. If the system temperature (T) and the liquid-phase compositions are known, eqs 1 and 2 can be used to determine the equilibrium pressure P and its vaporphase compositions for this binary. Equations 1 and 2 can be rewritten to give the expression of vapor mole fractions of the components, in terms of fugacity coefficients (Φ):
φLn yn ) xn V φn
(3)
φLw y w ) xw V φw
(4)
The general procedure to find the binary interaction parameters in the cubic equation of state (CEoS) is to minimize the difference between the calculated equilibrium pressure (P) and the given experimental pressure to satisfy the summation of the vapor mole fractions of the components calculated from eqs 3 and 4 to be within the tolerance of one. Although this method is applicable to most of systems, it may not be quite appropriate for systems containing supercritical gases, especially often only PTy data are available for this type of systems. For system that contains supercritical gases, the gas solubility in water can be related to the Henry’s law constant. The binary interaction parameter can then be derived from the Henry’s law constant. However, there is no suitable relationship presented in the literature for water content in gas vapor to derive the binary interaction parameter. In this work, we will propose a methodology to calculate the water content in supercritical gas by avoiding the calculation of fLn under supercritical conditions.
10.1021/ie0706212 CCC: $37.00 © 2007 American Chemical Society Published on Web 09/22/2007
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Table 1. Water Content in Dry (100% Pure) Nitrogen at 1 atm and 273.15 K pressure, P (bar)
temperature, T (K)
H2O content in N2 (mg/Nm3)
5.0 5.0 5.0 5.0 5.0 5.0 15.0 15.0 15.0 15.0 15.0 15.0 15.0 15.0 40.0 40.0 40.0 40.0 60.0 60.0 60.0 60.0 60.0 60.0 60.0 60.0 60.0 80.0 80.0 80.0 80.0 80.0 80.0 80.0 80.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0
248.15 253.15 263.15 273.15 278.15 283.15 248.15 253.15 263.15 268.15 273.15 278.15 283.15 293.15 263.15 273.15 283.15 293.15 248.15 253.15 258.15 263.15 268.15 273.15 278.15 283.15 293.15 253.15 258.15 263.15 268.15 273.15 278.15 283.15 293.15 253.15 258.15 263.15 268.15 273.15 278.15 283.15 293.15
101.6 166.8 427.1 1002.0 1433.2 2017.5 38.4 60.6 151.7 233.6 343.0 483.1 674.1 1313.6 64.2 137.3 284.8 525.6 11.5 17.9 27.9 45.0 68.4 101.9 142.7 199.5 375.7 16.7 24.8 38.3 59.7 85.4 121.5 164.4 301.7 14.5 19.8 33.4 47.5 72.9 98.1 141.1 262.5
oL V V f oL w (Td) ) Pφw (Td) ) f w(Td) ) ywPφw(Td)
(6)
or
yw )
φoL w (Td)
(7)
φVw(Td)
where yw is the vapor mole fraction of water in equilibrium with the water solution at the equilibrium P and T values. The water content in pure dry nitrogen at the dew point temperature Td is related to the equilibrium yw, which is related to the number of moles in vapor. The number of moles of a gas is conveniently expressed by the equation of state:
P0 n ) V Z0RT0
(8)
where T0 and P0 are temperature and pressure under normal conditions of 273.15 K and 1 atm, respectively; Z0 is the gas compressibility factor under normal conditions, and n is the number of moles in vapor volume V. The number of moles of water (nw) in the vapor then becomes
( )
P0 nw ) yw V Z0RT0
The derived equations then are used to determine the binary interaction parameters in the mixing rules for the accurate prediction of water content in the natural gas. Finally, the latest development of the advanced CEoS/AE mixing rule is incorporated in Soave-Redlich-Kwong (SRK), Peng-Robinson (PR), and Twu-Sim-Tassone (TST) CEoSs, to show its ability in representing both the water content in the gas vapor and the gas solubility in the aqueous phase simultaneously over the application ranges of temperature and pressure encountered in the natural gas industry. Methodology for Predicting Water Content in Supercritical Gas
(9)
In terms of weight of water in the vapor, designed as Ww, eq 9 is multiplied by the molecular weight of water Mw (18.0152 g/mol) to become
( )
M wP 0 Ww ) yw V Z0RT0
(10)
To get the water content in dry (100% pure) nitrogen, eq 10 must be divided by the quantity (1 - yw):
( )( )
yw M wP 0 Ww ) V 1 - yw Z0RT0
(11)
Rewriting eq 11 yields
In industry, the specification for the equilibrium vapor is normally expressed in terms of its dew point, which is the temperature at which liquid may condense out of the gas phase. The vapor mixture at the equilibrium temperature is at its dew point. For the vapor, which contains water and supercritical gas, the liquid that condenses out of the vapor phase will be almost all water, with only a tiny amount of the supercritical gas dissolved in it. Therefore, a methodology is proposed here: the first drop of liquid condensed from the supercritical gas vapor at the dew point is assumed to be pure water. In terms of an equation, at the dew point, we have V f oL w (Td) ) f w(Td)
where foL w (Td) is the liquid fugacity of pure water at the dew point temperature Td and equilibrium pressure P and fVw(Td) is the vapor fugacity of water at the dew point temperature Td, equilibrium pressure P, and equilibrium vapor-phase compositions yi, which is in equilibrium with the water solution at equilibrium temperature T. Because of the assumption made in eq 5, the calculated dew point temperature Td should be quite similar to, but will not be the same as, the equilibrium temperature T. Rewrite eq 5 as
(5)
( )
f Vw M wP 0 MwP0 ywP Ww ≈ ) V (1 - yw)P Z0RT0 (1 - yw)P Z0RT0
(12)
The vapor fugacity coefficient of water is assumed to be 1.0 in eq 12. Substituting eq 5 into eq 12, we obtain
( )
f oL Ww M wP 0 w ) V (1 - yw)P Z0RT0
(13)
Giving T0 ) 273.15 K, P0 ) 1 atm, Mw ) 18.0152 g/mol and assuming Z0 ) 1.0, we obtain the water content in terms of the liquid fugacity coefficient of pure water at the dew point temperature Td:
Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007 7255
( )
oL Ww 803792 f w 803792 oL ) φ (T ) (14) (mg/m3) ) V (1 - yw) P (1 - yw) w d
If the classical vdW mixing rules are used for the parameters a and b, they are given as
a)
Equation 14 shows that the proposed methodology requires only the liquid fugacity coefficient of pure water to calculate the water content in gas. The methodology avoids the need to calculate the liquid fugacity of gas under supercritical conditions. Prediction of Dew Point and Water Content Equation 7 requires the fugacity coefficients of pure liquid water and the water in the vapor mixture for the calculation of dew point temperature Td. The fugacity coefficient can be derived from any equation of state. A two-parameter cubic equation of state generally can be expressed by the following equation:
P)
RT a V - b (V + ub)(V + wb)
(15)
where P is the pressure, T the absolute temperature, and V the molar volume. The u and w are equation-of-state dependent constants. For the SRK equation, u ) 0, w ) 1; for the PR equation, u ) -0.4142, w ) 2.4141; and for the TST equation, u ) -0.5 and w ) 3.0. SRK predicts better liquid density for methane, PR is best for n-pentane up to n-heptane, and TST is superior for n-octane and higher carbon numbers, as well as for polar components. The parameter a is a function of temperature. The value of a at any temperature a(T) can be calculated from
a(T) ) R(T)ac
(16)
where ac is the value of a at the critical temperature and R(T) represented the alpha function, which is a function only of reduced temperature (Tr ) T/Tc). The original Soave R(T) was developed for light hydrocarbons only, not for components such as water, hydrogen, etc. To predict the accurate vapor pressure for all components over the entire temperature range, the Twu alpha function2 is used:
R(T) ) Tr
N(M-1)
NM
exp[L(1 - Tr )]
(17)
Equation 17 has three parameters: L, M, and N. These parameters are unique to each component and are determined from the regression of pure component vapor pressure data. The L, M, and N parameters have been derived for all components in the Design Institute for Physical Property Data (DIPPR) databank for the SRK, PR, and TST CEoSs. The fugacity coefficient derived from eq 15 gives the following expression:
( ) [ ( )] ( ) {[ ( )] [ ( )]} (
fi 1 ∂nb ) (Z - 1) ln xiP b ∂ni 1 a* w - u b*
1 ∂nb b ∂ni
- ln(Z - b*) +
-
1 ∂n2a na ∂ni
ln
)
Z + wb* (18) Z + ub*
where a* and b* in eq 18 are defined as
a* )
Pa R2T2
Pb b* ) RT
(19) (20)
∑i ∑j xixj xaiaj (1 - kij)
b)
[
1
∑i ∑j xixj 2(bi + bj)
(21)
]
(22)
where kij is the binary interaction parameter. Taking the derivative of eqs 21 and 22, with respect to the number of moles of component i, and then substituting that term into eq 18, the fugacity coefficient becomes
() fi
ln
bi ) (Z - 1) - ln(Z - b*) + xiP b a* bi 2 Z + wb* 1 xjaij ln (23) w - u b* b a j Z + ub*
( ){
∑
}(
)
where aij is
aij ) xaiaj (1 - kij)
(24)
The fugacity coefficient of eq 23 is ready to be used to perform the calculation for the dew point and water content, and the determination of the binary interaction parameter kij in eq 21. Equations 7 and 14 are the working equations for predicting the water content in dry gas vapor. By defining the water content as Ww/V, the value of the vapor mole fraction of water (yw) is calculated from eq 11. Equation 7 is then used to solve for the water dew point temperature (Td) at the given yw value and equilibrium pressure P and T values. After knowing Td, the water content in the dry gas can be calculated from eq 14. The binary interaction parameter kij is then adjusted to minimize the difference between the calculated water content and the experimental data. Prediction of Gas Solubility in the Aqueous Phase The gas solubility in solution is generally expressed in terms of the Henry’s law constant. Therefore, instead of using eqs 1 and 2 to regress the gas solubility in the aqueous phase into a binary interaction parameter, using the Henry’s law constant is more effective and practical to derive the binary interaction parameter for the system. The Henry’s law constant of solute 1 in a solvent 2 is related to the fugacity coefficient of the solute in the liquid mixture (ΦL1 ) by
H12 ) lim PφL1
(25)
x1f0
Substituting the fugacity coefficient of eq 23 into eq 25 for a binary system and taking the limit as x1 approaches zero, one obtains the Henry’s law constant (H12), in terms of the binary interaction parameter (k12):
b*1 ln H12 ) ln P2 + (Z2 - 1) - ln(Z2 - b*2) + b* 2 a*2 b*1 a*1 1/2 Z2 + wb*2 1 -2 (1 - k12) ln (26) w - u b*2 b*2 a*2 Z2 + ub*2
( )[ ( )
](
)
The system pressure and liquid density in eq 26 become the saturated vapor pressure and saturated liquid density of pure solvent 2. Equation 26 illustrates that k12 is explicitly related to
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H12 and can be calculated from H12 without the need of any numerical iteration. Using the Henry’s law constant is the simplest way to derive the binary interaction parameter for the accurate prediction of gas solubility in solution. In case the PTxy data are reported to represent the gas solubility, the PTxy data can be converted to the Henry’s law constants and eq 26 is then used to find k12. Advanced CEoS/AE Mixing Rule The classical vdW mixing rule is used in the previous sections in the equation of state to predict either the water content or the gas solubility. We also want to investigate the ability of our latest development of the advanced CEoS/AE mixing rule to test if the advanced mixing rule can represent both phase properties simultaneously. Twu, Sim, and Tassone (TST) recently developed CEoS/AE mixing rules3,4 to permit the smooth transition of the mixing rules to the conventional vdW one-fluid mixing rules. The incorporation of their proposed GE model and mixing rules into a CEoS allows the equation of state to describe both a van der Waals fluid and highly nonideal mixtures over a broad range of temperatures and pressures in a consistent and unified framework. The TST zero-pressure mixing rules for the CEoS parameters mixture a and b are
a* ) b*
[
)]
(
E E a* vdw 1 A0 A0vdw + b*vdw Cr RT RT
(27)
b ) bvdw
RT
)
E A∞vdw
RT
[
) C1
a* vdw
b*vdw
GE
-
]
a* i
∑i xib* i
1 1+w ln w-u 1+u
(
) (
(29)
)
∑i xi
∑j xjτjiGji
)
(w -1 u) ln(rr ++ wu )
(31)
AE∞ and AE0 in the aforementioned equations are the excess Helmholtz free energies at infinite pressure and zero pressure, respectively. The subscript “vdw” in the terms A∞E vdw and AE0 vdw denotes that the properties are evaluated from the cubic equation of state using the van der Waals mixing rule for its a and b parameters, avdw and bvdw, which are given by eqs 21 and 22. The binary interaction parameter kij is introduced to eq
(32)
n
∑k xkGki
Equation 32 seems to be similar to the NRTL equation, but there is a fundamental difference between them. The NRTL model assumes Aij, Aji, and Rij are the parameters of the model, but our excess Gibbs energy model assumes τij and Gij are the binary interaction parameters. More importantly, any appropriate temperature-dependent function can be applied to τij and Gij. For example, to obtain the NRTL model, τij and Gij are calculated as usual from the NRTL parameters Aij, Aji, and Rij:
τji )
Aji T
(33)
Gji ) exp(-Rjiτji)
(34)
In this way, there is no difference between the NRTL model and our model in the prediction of phase equilibrium calculations. We also note that eq 32 can recover the conventional vdW mixing rules when the following expressions are used for τij and Gij:
1 τji ) (δijbi) 2 Gji )
(30)
where u and w are constants that are dependent on the equation of state and are used to represent a general two-parameter cubic equation of state. For the TST equation of state, for example, u ) 3 and w ) -0.5. Cr in eq 27 is a function of a parameter r, which is the reduced liquid volume at zero pressure. The value of r ) 1.18 is recommended by Twu et al.5 and used in this work.
Cr ) -
n
RT
With this assumption, the zero-pressure mixing rule transitions smoothly to the conventional van der Waals one-fluid mixing rule. The parameter C1 in eq 29 is a constant and is defined as
C1 ) -
( ) n
(28)
The expression of bvdw is given in eq 22. The TST zero-pressure mixing rules assume that AE0vdw, which is the excess Helmholtz energy of van der Waals fluid at zero pressure, can be E approximated by A∞vdw , which is the excess Helmholtz energy of van der Waals fluid at infinite pressure:
AE0vdw
21 to correct the approximation made in eq 29. However, the parameter kij is not needed and is set equal to zero here because, in the regression of data, the binary interaction parameters of the GE model (eq 32) are sufficient to represent the data accurately. Because AE0 in these equations is at zero pressure, its value is identical to the excess Gibbs free energy GE at zero pressure. Therefore, any activity model such as the NRTL equation can be used directly for the excess Helmholtz free energy expression AE0 in the equation. Twu et al.3,4 have proposed a multicomponent equation for a liquid activity model for use in the TST excess energy mixing rules:
(35)
bj bi
(36)
where
δij ) -
C1 RT
[(
) ( )( )]
xai - xaj bi
bj
2
+ 2kij
xai xaj bi
bj
(37)
Equations 35 and 36 are expressed in terms of the CEoS parameters (ai and bi) and the binary interaction parameter (kij). The aforementioned discussion demonstrates that eq 32 is more general in its form than the NRTL model. Both the NRTL equation and vdW one-fluid mixing rule are special cases of this excess Gibbs energy function (eq 32). To cover the entire application range of temperature, eq 33 is modified to include a temperature-dependent binary interaction parameter Bji as follows:
τji )
Aji + BjiT T
(38)
Ind. Eng. Chem. Res., Vol. 46, No. 22, 2007 7257 Table 2. Binary Interaction Parameter (kij) and Average Absolute Deviation Percentage (AAD%) for the Predicted Water Content in Different Gases from the Twu-Sim-Tassone (TST) CEoS, Using kij in the vdW Mixing Rule binary
k12
AAD%
N2-H2O Ar-H2O CO-H2O H2-H2O C1-H2O C2-H2O C3-H2O
0.773676 0.761597 0.765812 0.692402 0.706467 0.531855 0.301727
0.21 0.22 1.53 1.39 0.40 0.65 3.85
The results from the CEoS/AE mixing rule will be compared with those from the vdW mixing rule in the prediction of the water content or the gas solubility. Results The binary interaction parameters kij derived from the proposed methodology for the prediction of water content and the accuracy of the predicted water content in the different gas vapor, in terms of the average absolute deviation percentage (AAD%) from TST CEoS, using kij in the vdW mixing rule, are given in Table 2 for seven systems: N2-H2O, Ar-H2O, C1-H2O, C2-H2O, C3-H2O, H2-H2O, and CO-H2O.
It was observed that kij is a very weak function of temperature in the prediction of water content in the vapor gas; hence, the use of a constant positive value of kij is sufficient for a CEoS to predict accurately the water content in supercritical vapor phase within experimental uncertainty over a wide temperature range. The accuracy of the prediction of water content in the gas vapor in terms of AAD% from the equation of state is generally
