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Ind. Eng. Chem. Res. 2006, 45, 6770-6777
Measurement and Modeling of the Solubility of Water in Supercritical Methane and Ethane from 310 to 477 K and Pressures from 3.4 to 110 MPa Matt Yarrison, Kenneth R. Cox, and Walter G. Chapman* Department of Chemical and Biomolecular Engineering, Rice UniVersity, 6100 S. Main St. MS 362, Houston, Texas 77005
We describe a new flow cell apparatus for measuring the water content of gases up to 120 MPa at 490 K that uses a combination of gravimetric and electrical resistance techniques to determine the solubility of water in the gaseous phase. The new experimental data for the solubility of water in supercritical methane and ethane were obtained with our apparatus spanning pressures from 3.4 to 110 MPa covering a temperature range from 310 to 477 K. We model the experimental results by combining two equations of state; vapor-phase fugacities and fugacity coefficients are calculated with a modified Peng-Robinson equation of state, and aqueous-phase fugacities are calculated using an equation by Wagner and Pruss (J. Phys. Chem. Ref. Data 2002, 31, 387-535) or by a modification of a correlation developed by Saul and Wagner (J. Phys. Chem. Ref. Data 1987, 16, 893-901). We compare the model results with new and existing experimental data and with commercially available simulators. Our model reproduces the experimental data within 2-6% using one adjustable parameter, indicating that the predictions of the model are equal to or superior to the commercially available simulators. 1. Introduction Accurate knowledge of the equilibrium water content of hydrocarbon gas mixtures is of vital importance to the energy industry. This information is necessary to prevent corrosion problems caused by moisture condensation in pipelines or process equipment and to determine the dosage of chemical inhibitor to avoid hydrate crystal formation in subsea and terrestrial pipelines. Hydrates can block fluid transmission lines, can cause serious damage to plant equipment, and are a serious safety concern. Despite the studies of Olds et al.,1 Reamer et al.,2 Deaton and Frost,3 Kobayashi and Katz,4 McKetta and Katz,5 Culberson and McKetta,6-8 and Culberson et al.,9 data for methane and ethane are available in the open literature only to pressures of 70.0 MPa. Only Dhima 10 has measured the solubility of gas in the aqueous phase (the other side of the phase diagram from what we are measuring) between 70 and 100 MPa. Mohammadi et al.,11-13 Chapoy et al.,14 and Gillespie and Wilson15 have performed lower-pressure (∼0.1-34 MPa) and lower-temperature (273-340 K) measurements for similar systems. As oil and gas exploration moves to greater depths both on- and offshore, reservoir pressures are frequently on the order of 100 MPa, beyond the range of reasonable extrapolation from existing data. To gain a better understanding of the phase behavior of these industrially important, highly nonideal systems at all industrially relevant conditions, we systematically study the solubility of water in methane and ethane over a pressure and temperature range that covers reservoir conditions to platform conditions. 2. Experimental Equipment The solubility of water in these supercritical hydrocarbon gases is measured using a flow scheme similar to that of Prausnitz and Benson16 or Rigby and Prausnitz17 or of Song et al.18 Figure 1 shows the important components of the experi* Corresponding author. Phone: (713) 348-4900. Fax: (713) 3485478. E-mail:
[email protected].
mental apparatus, which consists of a gas cylinder (S), a 1000 cm3 boost pump (P1), a 500 cm3 main pump (P2), a temperature equilibration coil (EC), a saturation cell (SC), throttling valves (TV1 and TV2), an analysis train, and a flow meter (F). Both EC and SC are contained in a thermostatically controlled air bath, which is controlled to within 0.3 K using a 1.8 kW heater. Gas enters the system from the cylinder and is compressed to the desired pressure using positive displacement pumps P1 and P2. From pump P2, the gas enters the equilibration coil, where it is heated to bath temperature. All pressure and temperature data from the pumps, heat bath, and saturation cell are recorded using a data logger. A detailed drawing of the saturation cell is given in Figure 2. The cell has a nominal internal volume of 1300 cm3, with an internal diameter of ∼7.5 cm. Gas enters the cell and flows through a glass dispersion frit (PF) which disperses the gas into the liquid water as bubbles of a nominal diameter of 0.01 mm. A stainless steel retaining ring (RR) holds the frit in place at the bottom of the cell. The bubbles travel through the liquid water phase and into the headspace of the saturation cell. A stainless steel baffle and an antientrainment section (shown in gray) prevent any water spray from leaving the saturation cell. Gas exits the saturation cell through a throttle valve (TV1), where the pressure is reduced to ∼1 bar before passing to the analytical train. Three different pressure transducers (PT) monitor cell pressure; pressures below 70 bar use a Heise transducer, pressures from 70 to 700 bar use a Data Instruments transducer, and pressures from 700 to 1400 bar use a Sensometrics transducer. All transducers are calibrated yearly using a Ruska dead-weight tester. A J-type thermocouple (TC) located in a thermowell (TW, shown by crosshatching) monitors cell temperature. The J-type thermocouple was calibrated by OMEGA engineering against the ice and boiling points of water and the melting points of tin and zinc. An Autoclave Engineers rupture disk (RD) mounted in the antientrainment section provides overpressure protection. The dashed line indicates the approximate water level in the cell. Analytical train A consists of the General Electric (GE)Panametrics moisture analyzer (PMA), three desiccant-charged
10.1021/ie0513752 CCC: $33.50 © 2006 American Chemical Society Published on Web 09/02/2006
Ind. Eng. Chem. Res., Vol. 45, No. 20, 2006 6771
Figure 1. Schematic of experimental apparatus. Data from all thermocouples, pressure transducers, and the PMA, if in use, are logged real-time using LabVIEW. U-tubes are weighed using a Sartorius scale accurate to 0.01 mg.
and the analytical train is maintained 10-20 K above air bath temperature using electrical heating tape to prevent condensation. When using the desiccant-charged u-tubes, the density of the sample gas inside the u-tubes will generally be slightly different from the density of the air inside the tubes when charging with desiccant. This buoyancy difference can lead to minor measurement errors if not taken into account. After several experimental tests, we found that purging the u-tubes with dried sample gas (dried using a combination of Dehydrite, magnesium perchlorate, and molecular sieve 4a) before and after the experimental run is sufficient to eliminate buoyancy effects. A Sartorius CP-5000 series scale, with an accuracy of 0.01 mg, is used to weigh the u-tubes. The scale is checked periodically against a set of standard-grade Ohaus weights. 3. Materials The methane and ethane used are ultrahigh-purity grade (99.995% pure) from the Matheson gas company. Standard laboratory-grade, deionized, UV-sterilized water with a maximum conductance of 0.25 micro siemens (µS) is used without further distillation. The anhydrous ACS-grade magnesium perchlorate used is from VWR. 4. Experimental Procedure
Figure 2. Schematic showing the saturation cell. The cell is made from A-286 alloy stainless steel and has a maximum pressure rating of 2000 bar at 204 °C. A standard NIST type high-pressure closure is used to seal the cell with a poly(tetrafluoroethylene) (PTFE) or Aflas O-ring. The equilibrium coil, closure nut, and baffles are omitted for clarity.
stainless steel u-tubes (U), and a gas flow meter (F). The PMA measures the water content by measuring the resistance across an aluminum oxide sensor; as the water present in the gas stream changes, so does the resistance across the sensor, which is converted by the PMA into a mole fraction and a mass of water per volume reading. The unit is calibrated by GE using National Institute of Standards and Technology (NIST) traceable standards and is recalibrated by GE yearly. Both PMA and the desiccant-charged tubes are used at temperatures below 333 K, but temperature limitations of the PMA prevent its use above this temperature. All tubing between the throttle valve (TV1)
Deionized water (2 L) is vacuum boiled in a 4 L sample cylinder to remove dissolved oxygen and nitrogen. Methane or ethane (hereafter referred to as simply “gas”) is bubbled slowly into the cylinder until the pressure in the sample cylinder reaches ∼0.2 MPa. After a 1 h equilibration period, the gas is vented and the process is repeated twice more to ensure that no unwanted oxygen or nitrogen remain dissolved in the water. After degassing, the water is used to completely fill the 1.3 L saturation cell. Gas is used to displace 300-400 cm3 of water in order to create a gas cap in the saturation cell, and sufficient gas is added to pressurize the cell and equilibration coil to ∼2 MPa. The heat bath is set to the desired temperature T and allowed to equilibrate for ∼12 h until the cell internal temperature has stabilized. Methane or ethane from gas source S fills pumps P1 and P2, which fill the saturation cell with gas at the desired pressure. Since compression tends to raise the cell temperature ∼0.5 K, the cell is allowed to come to thermal equilibrium with the bath before an experiment starts. Once the cell is at the same temperature as the bath, the throttle valves are opened so that gas flows out of the cell and into the analysis train at a rate of
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Figure 3. Water content of supercritical methane. Circles are data from this work, diamonds are those of Olds et al.,1 and squares are those of Rigby and Prausnitz.17 Lines are the water contents predicted using the modeling procedure outlined in this paper, using the mixing rules in eq 18. Please note that y-axis is -log(yH2O) and not yH2O.
0.014-0.028 L‚s-1 of expanded gas at ambient temperature and pressure. Cell pressure is maintained through a continuous flow of gas from pump P2. Approximately 17 L of gas is used to equilibrate the analysis train so it is at steady state. Upon equilibration, flow switches from the vent to the chemical absorption traps or PMA. An experiment will use between 17 and 56 L of expanded gas, and a typical experiment will deposit in excess of 100 mg of water in the u-tube traps. Cell and pump pressures are constantly monitored, and the pump P2 flow rate is manually adjusted to maintain cell pressure throughout the experiment. Once the desired volume of gas passes through the analysis train, the throttle and u-tube valves are closed and the traps are removed, sealed, and weighed. Gas volume is recorded using the flow meter, and this volume is corrected from ambient temperature and pressure to 0.1013 MPa and 288.65 K. Each (pressure, temperature) point is repeated a minimum of three times to ensure repeatability and accuracy. The u-tubes are made from 3/8 in. o.d. stainless steel tubing and use Dow-Corning glass wool to hold the solid desiccant in place. At the beginning of each experiment, the tubes are packed with fresh desiccant and flushed with dry sample gas, sealed using Suba-Seal stoppers, and weighed. Experiments have shown that the sealed tubes do not pick up more than 0.01 mg of mass over a 24 h period and that traps do not pick up more than 0.01 mg of mass when the traps are connected and disconnected to/from the expansion train. These results indicate that the measurement technique is adequate. Once enough gas has flowed through the expansion train so that it is at steady state, the tubes are connected to the expansion train and the wet-test flow meter using Tygon tubing and clamps. After the experiment, the tubes are disconnected, sealed, and weighed. The tubes are then flushed with dry sample gas and reweighed. 5. Experimental Results The data covering the temperature range spanning 310-477 K and the pressure range from 3.4 to 110 MPa are presented in Figure 3 for methane and in Figure 4 for ethane. The data of Olds et al.1, Rigby and Prausnitz,17 and Reamer et al.2 are plotted for reference. The experimental data are shown in Table 1 for methane and in Table 2 for ethane, while the raw data from the
Figure 4. Water content of supercritical ethane. Diamonds are data from this investigation, squares are those of Reamer et al.,2 while lines are calculated using the model outlined in this paper. Note that y-axis is -log(yH2O).
electrical resistance and the gravimetric measurements are listed in Tables 3 and 4. Uncertainties are listed along with the data in Tables 1 and 2, but generally range from 2 to 6%. Uncertainties reported are from a propagation-of-error analysis or from the observed deviation in measurements, whichever is greater. The reported results are the average of the experimental runs. 6. Modeling To make the data more useful for engineering practice, we have developed an accurate engineering model for the methane + water and ethane + water systems. The vapor phase is modeled using the Peng-Robinson (PR) equation of state (EoS), while the aqueous phase is modeled using an equation of state based on the work of Saul and Wagner.19 Using the Saul and Wagner formulation allows quick and accurate water fugacity calculations with a minimum of parameters. 6.1. Conditions for Equilibrium. For binary water + hydrocarbon systems, the phase-equilibrium relationship for water is
yH2Oφˆ H2OP ) (1 - xHC)f H(T,P),pure 2O
(1)
where P is the pressure, f H(T,P),pure is the fugacity of pure liquid 2O water evaluated at temperature T and pressure P, yH2O is the mole fraction of water in the vapor phase, φˆ H2O is the fugacity coefficient for water in the vapor phase calculated using the PR equation of state, and xHC is the mole fraction of the hydrocarbon dissolved in the liquid water. Here, we have assumed that the liquid water phase is so rich in water that the activity coefficient of water is ∼1. 6.2. Calculating Water Fugacity. We have used two approaches to calculate the fugacity of pure liquid water: the equation by Wagner and Pruss20 and a modification of the Saul and Wagner19 formulation. Each water model gives similar results over the range of conditions studied here. Here, we only present our modification of the Saul and Wagner formulation. The fugacity of water is written as
ln(fH2O(P,T)pure) ) ln(φHsat2O(T)PHsat2O(T))
+
∫P
P sat
(
VHL 2O(P,T) RT
)
dP (2)
where φHsat2O(T) is the fugacity coefficient of water at saturation,
Ind. Eng. Chem. Res., Vol. 45, No. 20, 2006 6773 Table 1. Experimentally Determined Values for Water Solubility in Supercritical Methanea temperature, K 310.9 104
pressure, MPa 3.45 ( 0.0007 6.89 ( 0.0007 20.68 ( 0.086 41.37 ( 0.086 62.05 ( 0.68 75.84 ( 0.69 89.63 ( 0.69 96.53 ( 0.69 110.32 ( 0.69
22.09 ( 1.1 12.48 ( 0.6 6.997 ( 0.3 5.636 ( 0.3
313.9 104
366.5 103
422.04 102
466.5 10
477.5 10
4.90 ( 0.25 4.72 ( 0.25 4.67 ( 0.25
23.54 ( 0.8 13.25 ( 0.4 6.56 ( 0.2 3.96 ( 0.2 3.44 ( 0.15 3.13 ( 0.15 2.88 ( 0.15
12.46 ( 0.42 7.46 ( 0.37 3.16 ( 0.16 2.04 ( 0.1 1.65 ( 0.08 1.43 ( 0.08 1.37 ( 0.08
4.15 ( 0.2 2.24 ( 0.11 0.84 ( 0.04 0.60 ( 0.03 0.46 ( 0.02 0.40 ( 0.02
5.46 ( 0.17 2.95 ( 0.1 1.13 ( 0.06 0.70 ( 0.04 0.57 ( 0.03 0.51 ( 0.03
0.35 ( 0.02 0.33 ( 0.02
0.494 ( 0.03 0.455 ( 0.02
a Mole fractions and errors are reported multiplied by the power of 10 listed in that column. Uncertainties listed are calculated using a propagation of error analysis or the observed variance, whichever was largest.
Table 2. Experimentally Determined Values for Solubility of Water in Supercritical Ethanea
using the following correlations:19
temperature, K pressure, MPa
314.8 104
366.5 103
3.45 ( 0.0007 22.46 ( 0.67 3.59 ( 0.0007 21.88 ( 1.09 6.89 ( 0.086 12.77 ( 1.09 12.38 ( 0.37 20.68 ( 0.086 8.37 ( 0.41 6.43 ( 0.2 41.37 ( 0.68 7.33 ( 0.36 4.06 ( 0.1 62.05 ( 0.68 5.82 ( 0.25 3.29 ( 0.1 75.84 ( 0.69 4.93 ( 0.25 3.02 ( 0.1 89.63 ( 0.69 4.49 ( 0.25 2.87 ( 0.1 96.53 ( 0.69 110.32 ( 0.69
422.0 102
466.5 10
12.114( 0.36 4.154 ( 0.16 6.957 ( 0.21 2.794 ( 0.08 1.819 ( 0.05 1.583 ( 0.05 1.498 ( 0.04 1.382 ( 0.04
2.163 ( 0.11 0.920 ( 0.06 0.658 ( 0.04 0.510 ( 0.03 0.455 ( 0.03 0.401 ( 0.03 0.372 ( 0.02
ln
[ ] PHsat2O Pc
)
Tc (-7.858 23τ + 1.839 91τ1.5 - 11.781 1τ3 + T 22.670 75τ3.5 - 15.939 3τ4 + 1.775 16τ7.5) (4)
Fsat(T) ) 1 + 1.992 06τ1/3 + 1.101 23τ2/3 Fc 0.512 506τ5/3 - 1.752 63τ16/3 - 45.448 5τ43/3 675 615τ110/3 (5) τ)1-
a
Mole fractions and errors are reported multiplied by the power of 10 listed in that column. Table 3. Experimentally Determined Mole Fractions of Water in Methane from Gravimetric and Electric Resistance Measurements for Methanea pressure, MPa
mol fraction, gravimetric
mol fraction, resistance
reported
3.45 6.89 20.68 41.37
Methane, 310.8 K 2.228E-03 2.150E-03 1.197E-03 1.400E-03 7.076E-04 6.760E-04 5.821E-04 5.080E-04
2.209E-03 1.248E-03 6.997E-04 5.636E-04
62.05 75.84 89.63
Methane, 313.9 K 4.914E-04 4.870E-04 4.733E-04 4.660E-04 4.727E-04 4.510E-04
4.903E-04 4.715E-04 4.673E-04
a
The reported average was calculated using a weight average with a weight of 0.75 assigned to the gravimetric measurement and 0.25 assigned for the resistance measurement.
VHL 2O(P,T) is the liquid water volume, P and PHsat2O(T) are the system pressure and the water saturation pressure, respectively, R is the universal gas constant, and T is the system temperature. The fugacity from eq 2 is plotted against steam table data in Figure 5. φHsat2O is calculated using the following correlation
274 < T < 647 K 274 + T 1.956 70 × 10-3 (3)
( (
φHsat2O ) 1 - 0.001 340 927 477 538 65 exp 9.7 1 -
))
which we fitted to the data of the 1995 steam tables;21 it is shown in Figure 6. The vapor pressure is calculated
T Tc
(6)
F(P,T) ) Fsat(T) + (P - PHsat2O)(8.48 × 10-6T2 - 5.91 × 10-3T + 1.39) + b(T) (7) 274 < T < 510 K
( ( )) Tc - T Tc
b(T) ) 143 exp -14
510 < T < 640 K
( ( ))
b(T) ) 143 exp -14
Tc - T Tc
(8)
erf((P - PHsat2O)a(T))
a(T) ) 7.129 × 10-4T + 0.3692 VL )
1 F(P,T)
(9)
Tc is the critical temperature of water, Fc is the critical density of water, and Pc (MPa) is the critical pressure of water. Equations 3-9 are intended for use with water in the liquid state in the range from 273.15 to 640 K and have been validated against experimental data within that temperature range for pressures up to 200 MPa. Equation 7 is a modification of Saul and Wagner’s19 saturation pressure correlation that makes the density and, consequently, the liquid volume more accurate. Equation 7 is simply a best fit function using least-squares regression. Including the effect of pressure on the liquid density is significant. For comparison, the approximation of using the saturated liquid density at all pressures is shown as a dashed line in Figure 7. A few comparisons are shown with the 1995 NIST/ASME steam tables in Figure 7. Both the NIST implementation21 of Wagner and Pruss’20 equation of state for water
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Table 4. Experimentally Determined Mole Fractions of Water in Ethane from Gravimetric and Electric Resistance Measurements for Ethanea pressure, MPa 3.59 6.89 20.68 41.37 62.05 75.84 89.63
mol fraction, gravimetric
mol fraction, resistance
Ethane, 314.8 K 2.181E-03 2.209E-03 1.239E-03 1.390E-03 8.417E-04 8.220E-04 7.310E-04 7.370E-04 5.798E-04 5.900E-04 4.952E-04 4.850E-04 4.508E-04 4.430E-04
reported 2.188E-03 1.277E-03 8.368E-04 7.325E-04 5.824E-04 4.926E-04 4.488E-04
a The reported average was calculated using a weight average with a weight of 0.75 assigned to the gravimetric measurement and 0.25 assigned for the resistance measurement.
Figure 7. Comparisons of improved correlation from eq 7 vs NIST21 steam table data. The dashed lines are values of the saturated liquid density calculated from eq 5 to show the effect of pressure on liquid density.
Figure 8. Comparison of XHC as calculated by eq 10 with experimental data of O’Sullivan and Smith.25 Error bars are 6%.
Figure 5. Fugacity of water as calculated by eq 2 vs fugacities from the 1995 steam tables.
hydrocarbon in the aqueous phase, y and x are the mole fractions of the hydrocarbon in the vapor and liquid phases, respectively, and Hi,H2O(T,P) is the Henry’s constant for species i in water at temperature T and pressure P. The amount of hydrocarbon gas in the liquid phase can be calculated by explicitly solving for xHC. As the hydrocarbons used in this investigation have low solubilities in the water-rich phase even at pressures up to 110 MPa, they are adequately represented10 using the KrichevskyKasarnovsky22 equation, ∞
VHC,H2O f LHC (P - PHsat2O) ln(HHC,H2O(T,P)) ) ln ) lnHi,0 + xHC RT (11)
Figure 6. Fugacity coefficient at saturation φsat is shown here from 274 to 640 K.
and the Saul and Wagner treatment as modified above give similar results. A relationship similar to eq 1 can be written for the hydrocarbons in the aqueous phase,
yHCφˆ HCP ) f VHC ) f LHC ) xHCHHC,H2O(T,P)
(10)
where H is the Henry’s constant of hydrocarbon in the liquid water phase at temperature T and system pressure P, f VHC is the fugacity in the vapor phase, f LHC is the fugacity of the
where f LHC is the fugacity of the hydrocarbon component i in water, xHC is the mole fraction of component i dissolved in the water at temperature T and system pressure P, HHC,0 is the Henry’s law constant for the hydrocarbon species in water at T ∞ is the partial molar and water saturation pressure, and VHC,H 2O volume of the hydrocarbon in water at infinite dilution. HHC,O is calculated using a method developed by Harvey,23 while ∞ VHC,H is calculated using the corresponding states method of 2O Lyckman et al.24 For reference, the solubility of methane in water is shown in Figure 8 in comparison with the data of O’Sullivan and Smith.25 Model results based on eq 11 have been found to reproduce the experimental data within 0.3-12% of the experimental value. However, this is not to say that using eq 11 will always produce such results. Any error in the value ∞ will produce large deviations in the value of xHC at of VHC,H 2O high pressures, as P - PHsat2O becomes large at high pressures. For example, an error of 10% in the partial molar volume of methane at infinite dilution will create a 0.1% error in the Henry’s constant at 1 MPa, a 6% error at 50 MPa, and a 20% error at 150 MPa at 324 K. We can, therefore, expect the maximum possible uncertainty in xHC to be on the order of 20-
Ind. Eng. Chem. Res., Vol. 45, No. 20, 2006 6775 ∞ 30%, as the uncertainties in VHC,H are on the order of 5 cm3/ 2O mol or ∼16%. 6.3. Equation of State. The gas-phase fugacities of water and hydrocarbon components are calculated using the PengRobinson (PR) equation of state.26 The PR26 equation of state can be written in dimensionless form as
Z)
(
)
BMix/Z AMix 1 (1 - BMix/Z) BMix 1 + 2BMix/Z - (BMix/Z)2
(12)
Table 5. Parameters Used in Peng-Robinson EoSa component
Pc (MPa)
Tc (K)
ω
Fc (kg/m3)
water methane ethane
22.12 4.604 4.88
647.3 190.6 305.4
0.344 0.011 0.099
322 163 207
a
Values taken from Poling et al.31
Table 6. Binary Interaction Parameters (kij) Used for Mixing Rule in Equations 18 and 19 temperature, K
where
310.9
biF ≡ Bi/ZZ ≡
aiiP P Aii ≡ 2 2 FRT RT
(13)
methane ethane
0.48
R2Tc,i2 Tc,i ai ≡ ζiRi; ζi ≡ 0.457 235 53 bi ≡ 0.077 960 7R Pc,i Pc,i
methane
0.045
(14)
and finally, taking the appropriate derivatives, the fugacity coefficient for the binary system is
(
B
(ZV - 1) - ln(ZV - BMix) -
Mix
)(
AMix ln B x8 Mix
ZV + (1 + x2)BMix 2(y1Ai1 + y2Ai2)
ZV + (1 - x2)BMix 2
AMix )
AMix
2
∑i ∑j
-
Bi BMix
)
(16)
2
yiyjAijBMix )
∑i yiBi
(17)
The form of the mixing rules used in eq 17 is sometimes called van der Waals mixing rules; however, the form used here is empirical. Numerous investigators26,27 have suggested the cross-interaction parameter, A12, should be of the form
A12 ) (1 - kij)xA11A22
(18)
where kij is a fitted parameter, component 1 is water, and component 2 is the hydrocarbon. This approximation is fairly accurate if the attractions between molecules are only due to van der Waals interactions. For the water/hydrocarbon system, eq 18 results in large values of kij. This occurs since, in the Peng-Robinson equation of state, the fitted A11 (pure water) includes not only van der Waals attractions but also contributions from hydrogen bonding and strong dipole-dipole interactions. A large kij is needed since the attraction between water and hydrocarbon molecules is only due to van der Waals interactions. We expect that the van der Waals interaction between water and methane should be closer to the methane-methane van der Waals attraction than to A12 from eq 18. Therefore, for the methane/water system, we consider a simple, empirical mixing rule of the form
A12 ) [A11kij + A22(1 - kij)]
(19)
where component 1 is water, component 2 is the hydrocarbon, and kij is a fitted binary interaction parameter, which is a function
366.4
422
Lorentz-Berthelot Mixing Rule 0.48 0.47 0.45 0.49 0.51 0.46 Linear Mixing Rule 0.045 0.039
A × 106
κi ≡ 0.374 64 + 1.542 26ωi - 0.269 93ωi2 T Tr ) (15) Tc,i
Bi
314.8
0.052
466.4
477.6
0.41 0.43
0.37 0.42
0.075
Table 7. Coefficients Used to Interpolate the Binary Interaction Parameter (kij) for the Mixing Rules Used in This Work, Valid from 310 to 477 K
Rii ≡ [1 + κi(1 - xTr,i)]2
ln(φˆ Vi ) )
313.7
methane ethane methane
B × 103
Lorentz-Berthelot Mixing Rule -3.5867 2.3165 -5.5729 3.9974 Linear Mixing Rule 3.0087 -2.1472
C 0.1007 -0.2206 0.4221
of temperature. Equation 19 gives good agreement with the data using kij ) 0; however, better agreement is obtained on fitting kij. Since the best fitted value of A12 is the same regardless of whether eq 18 or eq 19 is used, a small kij in eq 19 for the methane/water system corresponds to a large kij in eq 18. The purpose of introducing eq 19 is to demonstrate that the large kij's produced from the geometric mixing rule are reasonable in this case. Both mixing rules reproduce the solubility data for the methane + water and ethane + water systems to within 2-7% average absolute deviation (AAD) over the experimental pressure and temperature range where
AAD )
1
(
∑n
n
)
|yexp - ycalc| y exp
× 100
(20)
In this case, n is the number of experimental data points and yexp is the observed mole fraction of water in the gas phase. Table 5 lists the parameters used in the model, and Table 6 lists the binary interaction parameters (kij). It should be noted that, while the mixing rule used in eq 19 gives values for the binary interaction parameter for the methane/water system that are closer to zero, we would not expect the model to work well for the ethane/water system. In this case, we might expect that replacing A22 in eq 19 with A22/2, where the 2 represents the carbon number of ethane, would give better results. Equation 19 was presented for illustrative purposes only. We propose to use the geometric mixing rule for multicomponent mixtures. 6.4. Model Results. Calculations made using the mixing rule in eq 18 are shown in Figures 3 and 4. AAD% results of 2-7% are obtained by making kij a weak function of temperature. Binary interaction parameters (kij) are shown in Table 6. For both sets of mixing rules, the binary interaction parameters can be interpolated using kij ) AT2 + BT + C, where T is in Kelvin and A, B, and C are given in Table 7. Comparisons with Multiflash cubic plus association (CpA),28 Colorado School of Mines Gibbs Energy Minimization29 (CSM GeM), and CALSEP’s PVT SIM30 are presented in Figure 9. These calculations are presented using the default, recommended
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Table 8. Experimental Mole Fraction Data and Model Predictions with AAD% for Methane + Water Binary System at 477.5 K data
CALSEP
dev. %
Multiflash
dev. %
0.547 0.295 0.114 0.070 0.057 0.051 0.049 0.046
0.536 0.297 0.134 0.094 0.079 0.074
1.93 0.41 17.83 33.43 38.42 42.76
0.542 0.289 0.107 0.060 0.044 0.037
0.064
40.43
0.028
0.80 2.25 5.39 14.44 24.01 27.50 0.033 38.04
CALSEP AAD%
25.03
Multiflash AAD%
16.06
parameters for these systems. At low temperatures and low pressures, all methods do an adequate job of predicting water solubility in the hydrocarbon-rich phase, but as temperatures and pressures increase, they tend to diverge from the experimental measurements. Multiflash CpA and CSM GeM underestimate the amount of water in the vapor phase at high pressures, while CALSEP tends to overestimate the amount of water present in the vapor phase. CSM GeM gives the best predictions next to the model presented here. It should be noted, however, that the CSM GeM formalism was developed for use at or near hydrate-formation conditions, so the results presented here are an extreme test. It is strongly suspected that the CpA model would provide much better results if fitted to the higher pressure data, as it incorporates the statistical associating fluid theory (SAFT) association term, which would greatly improve calculations for the aqueous liquid phase. We would expect that, for all models, if the model parameters were fitted to highpressure data, the calculated values would show improvement. Table 8 lists the AAD% for methane + water at 477 K for the various models shown in Figure 9.
Figure 9. Solubility of water in methane, including comparisons with CALSEP, CSM GeM, Multiflash CpA, and our model at 366 and 477 K.
7. Conclusions The water content in the hydrocarbon-rich phase of the binary systems methane + water and ethane + water were measured from 310 to 477 K over a pressure range of 3.4-110 MPa. At ∼310 K, both a gravimetric and an electrical resistance method were used to measure the water content of the expanded gas stream. Above 333 K, only gravimetric methods were employed. A calculation method using the Peng-Robinson equation of state and an accurate liquid water model was used to model the experimental data acquired in this investigation and that of other investigators, showing good agreement with binary data. Similar calculations with commercially available process (see Figure 9) simulators show that commercial models can substantially
CSM GeM 0.523 0.277 0.107 0.063 0.046 0.040 32.31 0.030 CSM GeM AAD%
dev. % 4.40 6.08 5.53 10.69 18.99 22.29 0.043 33.57 16.73
this model 0.515 0.277 0.115 0.072 0.056 0.050 13.54 0.039 this model AAD%
dev. % 5.81 6.14 1.07 3.04 1.69 3.31 14.03 6.08
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ReceiVed for reView December 9, 2005 ReVised manuscript receiVed June 15, 2006 Accepted July 5, 2006 IE0513752
