Ind. Eng. Chem. Res. 2008, 47, 459-469
459
Phase Equilibrium in Two-Phase, Water-Rich-Liquid, Hydrate Systems: Experiment and Theory Yi Zhang 1249 Benedum Engineering Hall, Department of Chemical and Petroleum Engineering, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15261
Gerald D. Holder 240 Benedum Engineering Hall, Department of Chemical and Petroleum Engineering, UniVersity of Pittsburgh, Pittsburgh, PennsylVania 15261
Robert P. Warzinski* U.S. Department of Energy, National Energy Technology Laboratory, P.O. Box 10940, Pittsburgh, PennsylVania 15236
Two-phase equilibrium between CO2 hydrate (H) and a water-rich liquid (L) are experimentally measured and theoretically described between 273 and 281 K, at pressures below 30 MPa, and at aqueous CO2 concentrations between 0.0163 and 0.0242 mole fraction. These data represent the conditions where hydrates form from a single-phase aqueous solution of fixed composition. Both theoretical and experimental results indicate that the equilibrium pressure is very sensitive to concentration at all temperatures. The concentrations reported represent the solubility of CO2 in a water phase in equilibrium with hydrate at the given temperature and pressure. When a constant aqueous composition LH curve is extrapolated to the three-phase VLH curve, the composition characterizing the LH curve also represents the solubility of CO2 in water at the VLH conditions. Since the solubility of CO2 in water at hydrate-forming conditions is difficult to obtain, this method provides an excellent way of indirectly measuring this three-phase solubility. The effect of salinity on hydrate formation from water-rich-liquid systems was also studied. A modified model was introduced to describe the experimental results and produced good agreement between calculated and experimental pressures. A simplified version of the model can provide quick and reasonable estimations of the equilibrium conditions of hydrates at low concentrations and medium to low pressures. Interestingly, the increase of salt increases the maximum temperature at which hydrates are stable for a constant pressure and constant composition system. This is because the salt increases the chemical potential of the dissolved gases, which more than offsets the reduction in the chemical potential of the liquid water. The model can also be used for prediction of LH equilibrium for other gas hydrates. An example is given for methane hydrate at three different concentrations of methane in water. 1. Introduction The potential impact of rising greenhouse gas levels in the atmosphere is a current global concern. Large potential sinks include geologic formations, soils and vegetation, the deep ocean,1 and deep-oceanic sediments.2 In both geologic and oceanic systems the CO2 is often in contact with water, seawater, or brines.3,4 Understanding the behavior and fate of CO2 in such aqueous systems is important for developing and assessing many of the potential options and for responding to the impacts of seepage or leakage of CO2 into aqueous environments including the unintentional release of CO2 from suboceanic storage into the deep ocean. Formation of CO2 hydrate from an aqueous solution using only the hydrate former dissolved in the aqueous phase is the focus of this work. The impact of salinity on liquid-hydrate (LH) equilibrium was also investigated. Most theoretical5-13 and experimental studies summarized in Sloan’s book14 for different gas hydrates were conducted under conditions in which hydrates were formed from two-phase systems consisting of liquid water and a hydrate former in a separate gas or liquid phase, i.e., * To whom correspondence should be addressed. Tel.: 412-3865863. Fax: 412-386-4806. E-mail:
[email protected].
vapor-liquid-hydrate equilibrium (VLH) and liquid1-liquid2hydrate equilibrium (L1L2H), respectively. Information in the literature addressing the formation of hydrate from single-phase solutions of hydrate former dissolved in water is limited.4,15-18 Prior work done at the National Energy Technology Laboratory (NETL) has demonstrated that if CO2 hydrate forms from a twophase system of either gaseous or liquid CO2 and water, the bulk hydrate formed was initially less dense than the aqueous solution.17 This is likely due to occluded bubbles or drops of CO2 in the hydrate clusters. With time, this hydrate became more dense than the aqueous solution as any occluded CO2 either escaped or became part of the hydrate matrix. However, if CO2 hydrate forms from a single-phase system, the hydrate formed was initially more dense than the aqueous phase.17 In an oceanic setting, this type of hydrate could transport the CO2 farther down the oceanic water column to greater depths. In this work, experimental two-phase, water-rich-liquid, hydrate (LH) equilibrium for CO2-water systems was performed that extended previous work17 and validated the thermodynamic model9,10 that describes the phase equilibrium of hydrate formation by including the distortion of the hydrate lattice. Further modification and improvement to this model were
10.1021/ie070846c CCC: $40.75 © 2008 American Chemical Society Published on Web 12/18/2007
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Ind. Eng. Chem. Res., Vol. 47, No. 2, 2008 Table 1. Recipe for Artificial Seawater with a Salinity of 35.0019 and the Actual Amount of Salts Used in Our Artificial Seawater standard artificial seawater with a salinity of 35 (1 kg) 19 salt g/kg NaCl Na2SO4 KCl NaHCO3 KBr B(OH)3 NaF
MgCl2 CaCl2 SrCl2
Figure 1. Schematic of the experimental setup.
also performed. Two-phase equilibrium between CO2 hydrate (H) and a water-rich liquid (L) were experimentally measured and theoretically described between 273 and 281 K, at pressures up to 30 MPa and at aqueous CO2 concentrations between 0.0163 and 0.0242 mole fraction. The experiments were conducted both in water and 35 salinity artificial seawater to study the effect of salinity on the hydrate formation. Theoretical calculations for the LH equilibrium of methane hydrate for three different concentrations of methane in water are also presented. 2. Experimental Method The experiments were conducted in a 100-mL Autoclave Engineers EZE-SEAL laboratory-scale stirred autoclave. The original air motor for the impeller was replaced by an electric motor to provide consistent mixing, especially at slower speeds. A stirring speed of 200 rpm was used. This was sufficient to mix the system, as evidenced by a test in which soap flakes were visually observed in a glass beaker of similar diameter as the autoclave, but slow enough to not cause heat to be added to the system from the magnetically coupled stirrer. Figure 1 shows a schematic of the experimental setup. The pressure transducer used was a Heise DXD digital pressure gauge (accuracy: (0.02% full scale, range: 0-52 MPa). The pressure transducer was installed in a connection at the top of the autoclave stirring assembly. Installation at other points on the autoclave required a short section of tubing that would occasionally plug with hydrate. The top connection only required an adapter, which did not experience plugging. An Omega RTD (Model: PR-13) with an accuracy of (0.3 K within our measured range was used as the temperature sensor. The entire system was enclosed in a Tenney T10 temperature programmable environmental chamber that could maintain the temperature of interest to within (0.1 K. Water purified by reverse osmosis and deionization (18 megaohm-cm) and CO2 (SFC grade, 99.99+% purity) were injected into the autoclave through Teledyne ISCO precision high-pressure D series syringe pumps. A 260 mL syringe pump (ISCO 260D) was used for injecting water (flow accuracy of 0.5% of set point, displacement resolution of 16.6 nL, and pressure range: 0-52 MPa). A 100 mL syringe pump (ISCO 100DM) was used for CO2 (flow accuracy of 0.5% of set point, displacement resolution of 4.8 nL, and pressure range of 0-69 MPa). The amounts of liquid CO2 and water injected through the syringe pumps were
23.9849 4.0111 0.6986 0.1722 0.1000 0.0254 0.0029
actual composition used in this work, g/kg 23.9850 4.0100 0.6985 0.1732 0.1001 0.0257 0.0031
volumetric salts
volumetric salts
5.0290 1.1409 0.0143
5.0278 1.1431 0.0147
determined from the volume delivered. The density for CO2 was obtained from the IUPAC International Thermodynamic Tables of the Fluid State for carbon dioxide at the operating pressure and the temperature of the pump. Table 1 contains the recipe followed for preparation of 1 kg of 35.00 salinity artificial seawater.19 In Table 1, the actual amount of salts used in our artificial seawater are also listed. The MgCl2 and CaCl2 were purchased as volumetrically diluted salts; whereas the small amount of SrCl2 was added directly as a powder. A procedure was developed for completely filling the autoclave with water containing dissolved CO2 to achieve the desired concentration of CO2 and at the same time avoid any CO2 trapped inside the CO2 inlet, which could cause erratic pressure spikes by local hydrate formation. After estimating the amount of liquid CO2 and water needed in order to achieve a certain concentration at a high pressure (typically 25.5 MPa) in the autoclave, the autoclave was purged with CO2 and then evacuated using a mechanical vacuum pump to around 1.3 KPa (10 Torr). The autoclave was charged by first adding most of the water through the water syringe pump. The water entered the autoclave through valve #5, valve #4, and valve #3. Then the liquid CO2 was pumped into the autoclave through valve #1 and valve #2 by the CO2 syringe pump. Valve #2 and valve #3 were then shut off so that the autoclave was isolated from the outside. The fittings at point A and point B were disconnected from valve #4 and valve #2, respectively. Then the fittings at these two points were connected with each other as shown by the dashed line in Figure 1. The air in the newly connected system was purged, while valve #2, valve #3, and valve #4 were kept closed. A small amount of water was then added through valve #5, point A, point B and valve #2 to the autoclave to flush any remaining CO2 into the autoclave, which otherwise could be trapped in the inlet tubing. The exact concentration of the CO2 solution was calculated based on the actual input of water and CO2. The temperature of the system was kept at 289 K, and the pressure drop of the system was closely monitored to determine when the dissolution of the CO2 in the water was complete, which usually took 6 days. After total dissolution of the CO2, the system was quickly cooled to 271 K and then heated up to 290 K at the rate of 0.3 K/h. This is called one cycle. The pressure versus temperature trace for a typical experiment is presented in Figure 2 with light arrows indicating the history of the experiment. Two repeated cycles are shown in Figure 2. The marks of A, B, and C in Figure 2 correspond to those in Figure 3, which are described in detail later. Because of metastability in hydrate formation, the hydrate dissociation trace obtained during heating was used
Ind. Eng. Chem. Res., Vol. 47, No. 2, 2008 461
Figure 2. Pressure versus temperature history for two cycles of an experiment in which hydrates were formed and decomposed in a single-phase solution with a mole fraction of CO2 equal to 0.0163. Light arrows indicating the history of the experiment. Data for L1L2H is from the literature.14 Points A, B, and C were obtained in that order during the heating cycle. Note that points B and C are (nearly) indistinguishable on this figure; therefore, Figure 3 is needed to clearly identify the equilibrium points.
Figure 3. dP/dT versus temperature for the heating section of the dissociation of CO2 hydrate into a single-phase solution of CO2 dissolved in water with a mole fraction of CO2 equal to 0.0163. Point C is taken as the equilibrium point.
to evaluate the equilibrium point. The equilibrium point shown in this trace in Figure 2 is located at the inflection point in the pressure-temperature data. Note that the traces in Figure 2 are not consistent with the formation of ice. Ice formation would cause an increase in pressure. Ice formation was not observed in the experiments reported here. After completing the cycles at the highest pressure, the pressure in the reactor was lowered by first disconnecting the fitting at point A in Figure 1, then letting a small amount of water with dissolved CO2 from the reactor into the small section of tubing between valve #3 and valve #4, and then venting this solution through valve #4. This was repeated if necessary. The concentration of CO2 in liquid trapped between valve #3 and valve #4 was assumed to be identical to the concentration in the reactor. In order to determine the equilibrium point from the pressure vs temperature trace more accurately, the slope of dissociation curve (dP/dT) versus temperature was plotted as shown in Figure 3. The composition of the aqueous phase is constantly changing
between the points labeled A and C. The peak of curve (point B) represents the point of maximum dissociation rate but does not represent the equilibrium for the overall CO2 concentration, because the water-phase composition is changing. Point C represents the temperature at which dissociation stops and the composition returns to its original values (0.0163 mole fraction of CO2 in this example). The temperature and pressure at point C represent the equilibrium conditions. This produces an estimated uncertainty of (0.2 K. As can be seen in Figure 3, the traces of two experimental cycles are highly overlapped. In other words, the dissociation point in our experiment is highly repeatable. The location of the equilibrium point is indicated in Figures 2 and 3. The concentration of CO2 in the experiment shown in Figures 2 and 3 was 0.0163 (mole fraction). 3. Results and Discussion We have conducted experiments at seven different concentrations in water solution and three different concentrations in
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Table 2. Experimental Results for the Formation of CO2 Hydrates from Various CO2 Concentrations in Watera CO2 concn (mole fraction)
temp, K
pressure, MPa
calcd temp, K
absolute deviation, %
0.0163 0.0163 0.0163 0.0163 0.0163 0.0163 0.0169 0.0169 0.0169 0.0169 0.0179 0.0179 0.0179 0.0179 0.0179 0.0187 0.0187 0.0187 0.0187 0.0200 0.0200 0.0200 0.0200 0.0200 0.0218 0.0218 0.0218 0.0218 0.0242 0.0242
274.42 274.25 274.20 274.13 274.06 274.10 275.41 274.95 274.80 274.76 276.38 276.05 275.94 275.92 275.86 277.22 277.18 276.77 276.6 278.78 278.47 278.27 278.01 277.98 279.51 279.29 279.05 278.96 281.08 280.86
23.449 16.217 9.655 5.449 3.595 1.874 20.443 10.533 8.751 6.704 23.139 16.354 9.344 5.031 3.099 21.997 14.127 7.722 5.659 23.242 16.085 9.293 3.282 2.502 21.973 14.658 8.032 5.957 23.598 15.088
274.70 274.56 274.44 274.36 274.32 274.29 275.42 275.10 275.04 274.97 276.33 276.10 275.86 275.72 275.65 276.94 276.67 276.44 276.37 277.99 277.73 277.48 277.26 277.23 279.32 279.03 278.77 278.69 281.20 280.82
0.10 0.11 0.09 0.09 0.10 0.07 0.01 0.05 0.09 0.08 0.02 0.02 0.03 0.07 0.08 0.10 0.19 0.12 0.08 0.28 0.27 0.28 0.27 0.27 0.07 0.09 0.10 0.09 0.04 0.02
a The water solution is the only phase present prior to hydrate formation. Calculated values are based upon the theory presented in this paper.
Table 3. Experimental Results for the Formation of CO2 Hydrates from Various CO2 Concentrations in Artificial Seawater Solutiona CO2 concn (mole fraction)
temp, K
pressure, MPa
calcd temp, K
absolute deviation, %
0.0180 0.0180 0.0180 0.0180 0.0180 0.0188 0.0188 0.0188 0.0188 0.0197 0.0197 0.0197 0.0197
276.84 276.57 276.51 276.36 276.34 277.72 277.49 277.30 277.14 278.49 278.20 278.08 277.99
19.199 12.946 8.116 5.783 3.530 21.635 14.993 8.274 4.417 21.370 13.621 8.839 4.417
276.95 276.73 276.55 276.47 276.39 277.71 277.48 277.24 277.11 277.03 276.75 276.58 276.42
0.04 0.06 0.02 0.04 0.02 0.00 0.01 0.02 0.01 0.52 0.52 0.54 0.56
a Detailed artificial seawater composition is listed in Table 1. Calculated values are based upon the theory presented in this paper.
artificial seawater solution. For each concentration, at least four data points were collected, except for a concentration of 0.0242 in which only two data points were able to be obtained. For each data point, at least two cycles were completed as indicated in Figures 2 and 3, and the average values of the temperatures and pressures at the dissociation points were used as the results. The experimental results in water and artificial seawater are listed in Tables 2 and 3, respectively. On average, the standard deviation of the temperatures in the experimental results is 0.02% of the reported values, and the standard deviation of the pressures in the experimental results is 0.16% of the reported values. The induction time of hydrate formation was also recorded in our experiments. The system was set at 289 K and then was programmed to cool down quickly to 271 K. The initial state of the system at 289 K was time zero in our measurements. Table 4 lists experimental hydrate equilibrium conditions and the induction time of hydrate formation from water solutions with three different CO2 concentrations (0.0169, 0.0179, and
Table 4. Experimental Hydrate Equilibrium Conditions and the Induction Time for Hydrate Formation from Water Solutions with Different CO2 Concentrations experimental cycle number
temp, K
pressure, MPa
induction time, h
1 2
xCO2 ) 0.0169 275.4 20.46 275.4 20.43
1.83 1.50
1 2 3 4
275.0 275.0 274.9 274.9
2.83 3.30 2.42 1.83
1 2
274.8 274.8
8.751 8.751
2.83 1.67
1 2
274.8 274.8
6.707 6.701
2.83 1.92
1 2
xCO2 ) 0.0179 276.5 23.14 276.3 23.14
1.50 2.58
1 2 3 4
276.0 276.0 276.1 276.1
2.05 5.50 5.67 2.17
1 2
276.0 275.9
9.344 9.344
3.17 2.17
1 2
276.0 275.9
5.024 5.038
2.83 14.75
1
275.9
3.099
24.00
1 2
xCO2 ) 0.0200 278.8 23.24 278.8 23.24
2.10 1.88
1
278.5
1.87
1 2
278.3 278.2
9.296 9.289
3.43 1.67
1 2
278.0 278.0
3.285 3.278
2.85 2.33
1 2
278.0 278.0
2.491 2.512
2.25 1.87
10.54 10.55 10.52 10.52
16.35 16.35 16.35 16.35
16.09
0.0200). At each temperature and pressure, the experimental cycle was repeated twice, in some cases, four times. On two occasions as shown in Table 4, only one cycle was performed. As can be seen in Table 4, the concentration of CO2 does not have an obvious impact on hydrate formation times. It was suggested in earlier studies that hydrate formation in pure water is characterized by a strong “memory effect”.14,20,21 It was hypothesized that structured water still exists after hydrates dissociate. This residual structured water then promotes more rapid hydrate formation in future cycles because the water remembered the hydrate structuresthe so-called “memory effect”.14 In the experiments reported here, a strong “memory effect” does not always exist, even though our system was heated up to 289 K, not 301 K which was reported to be the temperature that no structured water is left in the solution.14 The salts in seawater solutions did not have an obvious impact on prolonging or reducing the induction time of hydrate formation as shown in Table 5. Our results obtained in water are compared with CO2 solubility in L-H equilibrium obtained by Yang et al.,16 as the L-H phase diagram we obtained also provided the CO2 solubility information under the L-H equilibrium condition. The results are compared at two different pressures, 6.10 and 10.44 MPa in Figures 4 and 5, respectively. It can be seen that the two results are close, but our results are a bit lower than those of Yang et al. A reason for this is that there is a high possibility that in the experiments of Yang et al., some very small CO2 hydrate particles were in the sample they took after hydrate formation. The amount of dissolved CO2 was measured by expanding the sample in an expansion chamber. Therefore, if some hydrate particles were hidden inside the sample, the measured amount of CO2 in water in equilibrium with hydrate
Ind. Eng. Chem. Res., Vol. 47, No. 2, 2008 463
where Cji is the Langmuir constant for species i in cavity j; fi is the fugacity for the hydrate forming species; and θji is the fraction of j-type cavities, which are occupied by i-type gas molecules. The value of ∆µw (the chemical potential difference of water in the water-rich phase) is calculated from the following equation:6,7
Table 5. Experimental Hydrate Equilibrium Conditions and the Induction Time for Hydrate Formation from Artificial Seawater Solutions with Different CO2 Concentrations experimental cycle number
temp, K
pressure, MPa
induction time, h
1 2
x ) 0.0197 278.51 21.401 278.47 21.339
1.83 2.03
1 2
278.18 278.21
13.638 13.603
1.83 1.87
1 2
278.13 278.02
8.853 8.825
4.25 1.98
1 2 3
277.98 278.03 277.96
4.440 4.413 4.399
3.93 5.27 8.97
1 2
x ) 0.0188 277.64 21.621 277.79 21.648
3.87 3.63
1 2 3
277.45 277.57 277.45
14.995 15.009 14.975
1
277.3
8.274
1 2
277.13 277.15
4.068 4.054
1 2 1
x ) 0.0180 276.84 19.240 276.83 19.157 276.59 12.953
2.50 1.50 2.50
2
276.68
12.966
2.50
1 2
276.51 276.62
8.116 8.137
1.73 3.67
1 2
276.4 276.51
5.783 5.811
3.15 3.85
1 2
276.35 276.39
3.540 3.547
7.50 3.50
∆µw ∆µ0w ) RT RTo
6.67 3.97 20.8 9.97 13.4 10.0
-
∑
νj ln(1 -
j,cavities
νj ln(1 -
-
(1)
∑i Cji fi
∆µw
θji) )
RT
(T, P ) 0) +
νj ln(1 - ∑ θVLH ∑ ji ) ) j,cavities i
∆µw RT
(4)
∫0P
( )
(T, P ) 0) +
VLH
∆V
(2)
dP - ln(xVLH w ) (5)
RT
()
Subtracting eq 5 from eq 4, the following equation is obtained:
-
∑
νj ln
1-
j,cavities
∑i θji
∑i
)
θVLH ji
∆V
∫PP
VLH
RT
dP - ln
xw
xVLH w
(6)
Since for single hydrate species
1 - θji
) VLH
[
1 - θji
1 + Cji fVLH ji 1 + Cji fi
]
(7)
Combining eqs 6 and 7, the following equation is obtained
∑j νj ln
1
Cji fVLH i
+
fVLH i
(
)
V h i(P - Psat)
f sat i exp
1 Cji fVLH i
RT
+1
)
∆V(P - PVLH) RT
Cji fi 1+
(3)
At VLH equilibrium, the following relation is obtained:
where νj is the ratio of j-type cavities present to the number of water molecules present in the hydrate phase and
θji )
∑i
1-
∑i θji)
dP - ln(γwxw) ∫0P ∆V RT
∆V
The first model that describes the hydrate-phase equilibrium was developed by van der Waals and Platteeuw (vdW-P model)23 in 1959. The vdW-P model was based on classical statistical thermodynamics with the analogy to classical adsorption theory. This model was later generalized by Parrish and Prausnitz in 19725 to predict single and mixed component hydrate equilibrium conditions. This method was further simplified by Holder et al.6 A lattice distortion theory based on vdW-P was developed by Holder’s group.8-10,24 The model described below for calculation of hydrate formation from a single-phase water-rich liquid is based on this theory. For the water species in the hydrate phase, the value of ∆µH (the chemical potential difference of water in the hydrate phase) is obtained by using the following equation5
∑
∆h dT + RT2
∫0P RT dP - ln(xw)
4. Models for Hydrate Equilibrium
j,cavities
F
o
The terms ∆h and ∆V are the molar enthalpy and volume differences, respectively, between the empty hydrate and liquid water phases. xw is the mole fraction of water in the water-rich phase. γw is the activity coefficient for water, which was usually taken to be 1.0 in our calculation when only water and gas systems are studied due to the low solubility of gas in water. At equilibrium, ∆µH ) ∆µw, hydrates can form. The first two terms of eq 3 on the right represent ∆µw(T, P ) 0), the chemical potential difference at a fixed temperature and zero pressure. At a fixed temperature, hydrate forms from singlephase solution. The following relationship is obtained:25
would be higher. The evaluation conducted by Diamond and Akinfiev concluded that “the precision of the measurement was relatively low” in the experiments of Yang et al.22
∆µH ) -RT
∫TT
( )
- ln
xw
xVLH w
(8)
where Psat and f sat are the pressure and corresponding fugacity of the CO2, which are required to dissolve the experimental
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Figure 4. CO2 solubility at L-H equilibrium at 6.10 MPa from our experiments and ref 16. Our results were also fitted to a second-order polynomial equation.
Figure 5. CO2 solubility at L-H equilibrium at 10.44 MPa from our experiments and ref 16. Our results were also fitted to a second-order polynomial equation.
levels of CO2 in the water phase at the given temperature. The solubilities of CO2 in water at different temperatures and pressures were calculated from Diamond’s model22 to obtain these values. The exponential term is the Poynting correction26 to f sat, giving the fugacity at pressure P. V h i is the partial molar volume of CO2 in liquid water. xVLH is the mole fraction of w water in the water-rich phase at VLH equilibrium. It was calculated as the following VLH xVLH ) 1 - xCO w 2 VLH where xCO is the solubility of CO2 at the temperature of 2 interest and at VLH equilibrium. Equation 8 can be solved for the pressure. In this approach, reference state properties are not directly relevant, but ∆µ0w was used in the calculation of Langmuir constants. We used
the empirical correlation between the shell radii of all cavities, R, and ∆µ0w developed by Zele et al.9,10 as shown in the following
R ) A + B × ∆µ0w
(9)
where A and B are constants for three water shells of each type of cavity. The values of A and B are listed by Lee.10 The Langmuir constants were calculated as the following10,27
C)
4π kT
(
∫0R exp -
)
W1(r) + W2(r) + W3(r) 2 r dr (10) kT
where W1(r), W2(r), and W3(r) are smooth cell potentials of the first, second, and third shells based upon the Kihara potential function.
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Figure 6. The comparison of experimental results and calculated results from our model for CO2 hydrate formed from single-phase water solutions with various CO2 concentrations. (The literature data14 on VLH and L1L2H equilibrium are also shown.)
The values of the pressures from eq 8 can be easily compared to those obtained in our experiments. Further simplification was also applied as the following:
In many cases, Cji fi . 1,
fisat
=
VLH
fi
xw Psat and VLH = 1 VLH P xw
Then, the following simplified equation is obtained:
∑j νj ln
[ ( Psat
PVLH
)]
V h i (P - Psat)
exp
RT
)
∆V(P - PVLH) RT
(11)
In this equation, P is the dissociation point of hydrates formed from single-phase solutions, which is the unknown variable. The values of all the other variables can be obtained from either experiments or literature data. Note that we used 32 cm3/mol as the partial molar volume, V h i, of the CO2 gas in our calculations for all the concentrations except for the concentration of 0.0163 where the value of 30 cm3/mol was used to better represent the trend of the experimental results.7 Figure 6 presents the comparison of experimental and predicted data that are calculated by eq 8. It can be seen in Figure 6 that the calculated results from our model fit the experimental results well. It is very clear that for a given CO2 concentration, the equilibrium temperature for hydrate stability increases with pressure. At constant temperature, the equilibrium pressure for hydrate stability decreases with increasing CO2 concentration. In Figure 7, the results of using simplified model eq 11 and rigorous model eq 8 are shown. It is clear that the simplified model can provide a very good estimation of the equilibrium pressures, especially when the concentration of CO2 is low. At low concentrations, such as 0.0150 and 0.0160, the results from the simplified and rigorous models almost overlap each other. The maximum discrepancy is no more than 6% at the highest pressure. The discrepancy between the simplified and rigorous model increases when the concentration of CO2 increases, and
it also increases with the pressure. As expected, at higher pressures, the error attributable to using pressure to replace fugacity gets greater. When this simplified model is used on systems with the CO2 concentration no higher than 0.0190 and the pressure no higher than 55 MPa, the average error is less than 12%, and the maximum error is less than 20%. This simplified model can therefore provide a quick estimation of the system behavior. The calculation of hydrate formation from seawater requires the activity coefficient of water, γw. The following equation (eq 12) was used.
∑j νj ln
[
1
Cji fVLH i
+
(
)
V h i(P - Psat)
fsat i exp
fVLH i 1
Cji fVLH i
RT
+1
∆V(P - PVLH) RT
](
)
- ln
γwxw
VLH γVLH w xw
)
(12)
Margules expressions for the activity coefficient of water in systems containing inhibitors were used.7 Figure 8 shows that the calculated results fit our experimental results very well in the artificial seawater system. The VLH equilibrium data of CO2 hydrate formation in artificial seawater were obtained from Dholabhai.28 The solubility of CO2 in seawater was obtained from Duan’s program.29,30 The absolute deviations between calculated and experimental temperatures at given pressures for fresh water and seawater systems using our models are listed in Tables 2 and 3. The overall average absolute deviations were less than 0.3 and 0.5 K for water and artificial seawater, respectively. The effect of salinity on the formation of CO2 hydrate from solutions with dissolved CO2 was also studied. The calculated results for a mole fraction of 0.0180 of CO2 in water and
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Figure 7. Comparison of the results from using the simplified model eq 11 and rigorous model eq 8 for CO2 hydrate formation in a water system. (Literature data14 on VLH and L1L2H equilibrium are also shown.)
Figure 8. Comparison of experimental and calculated results from the model for CO2 hydrate formed from single-phase artificial seawater solutions with various CO2 concentrations. (Literature data of VLH equilibrium of CO2 hydrate formation in seawater are also shown.28)
seawater solutions are compared in Figure 9. As can be seen, at the same concentration of CO2 and at the same pressure, hydrate forms at a higher temperature in the seawater than in water with the same level of dissolved CO2. In other words, in single-phase solutions, salts in seawater serve as promoters for hydrate formation rather than inhibitors. This is not the result that would have been intuitively expected, because it is wellknown that in two-phase systems containing excess gas or liquid CO2, salts are inhibitors to hydrate formation, not promoters. To explain this observation, consider a two-phase system in which excess gas exists (a VL system). In this system, the chemical potential of CO2 in the liquid is unchanged by the presence of salt, because the chemical potential of CO2 in the liquid must equal the chemical potential of CO2 in the vapor which is essentially that of pure CO2. The concentration of CO2 in the liquid phase adjusts so that the chemical potential remains
in equilibrium with the vapor. For VL systems, the concentration of CO2 would decrease at constant chemical potential as salinity increases. At constant temperature and pressure, the chemical potential of CO2 is therefore unaffected by salt in the liquid because the aqueous concentration of CO2 adjusts to keep the chemical potential constant. However, the chemical potential of the aqueous water is reduced making the liquid phase more stable relative to the hydrate thus inhibiting the formation of hydrates from VL systems. For three-phase equilibrium (hydrates formed from L and V phases) specifying the temperature and pressure fixed the system. For two-phase (LH) systems, there is no source of CO2 to adjust (buffer) the concentration of CO2 in the liquid. Thus a third independent variable, concentration in this study, is necessary to fix the system. The concentration is set by the experiment and does not adjust to keep the liquid in equilibrium
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Figure 9. Effect of salinity on the formation of CO2 hydrate from the solutions with dissolved CO2. The insert illustrates the relationship of VLH and LH of CO2 hydrate in seawater and water.
Figure 10. Our results of solubility of CO2 in water at VLH equilibrium obtained from the extrapolation of our experimental data are compared with calculated results from the literature.7,22,31
with the (nonexistent) gas. Thus, when salt is added, the concentration of CO2 remains constant, unlike the VL system described above, since there is no other phase to act as a source or a sink. At a constant concentration of CO2, the chemical potential of CO2 increases with salinity. A similar effect was also noted by Handa for methane hydrate.18 Obtaining the data presented in this paper is, therefore, only possible if the increase in CO2 chemical potential more than compensates for the reduction of the aqueous water chemical potential and hydrate formation is promoted. As Zatsepina and Buffett pointed out,15 when hydrate forms from a single-phase system of dissolved gas, salts lower the solubility of hydrate-forming gas in the water. This is physically equivalent to raising the chemical
potential of dissolved CO2 at constant concentration. For the systems studied, at a given temperature and pressure, CO2 hydrate forms at a lower CO2 concentration in seawater than in water as should be expected based upon previous studies.15 The equilibrium data shown in Figures 6 and 8 give the solubility of CO2 in a water phase. If a constant aqueous composition LH curve is extrapolated to the three-phase VLH curve, the composition characterizing the LH curve also represents the solubility of CO2 in water at the VLH conditions. Table 6 lists the solubility of CO2 in water at VLH equilibrium obtained by extrapolating our experimental results to the VLH curve. Since the solubility of CO2 in water at hydrate-forming conditions is difficult to obtain, this method provides an
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Figure 11. Prediction of LH equilibrium of methane hydrate at three different concentrations of methane in water using eq 8. (Literature data of VLH equilibrium of methane hydrate formation in water are also shown.14) Table 6. Solubility of CO2 in Water at Three-Phase VLH Equilibrium Obtained by Extrapolating Our Experimental Results to the VLH Curve T (K)
solubility of CO2 in water, mole fraction
274.1 274.4 275.7 276.5 278.0 278.8 280.2
0.0163 0.0169 0.0179 0.0187 0.0200 0.0218 0.0242
excellent way of indirectly measuring this three-phase solubility. Figure 10 compares the three-phase solubility obtained from our experimental results with the calculated results from models in the literature.7,22,31 It can be seen that the models of Anderson31 and of Diamond and Akinfiev22 predict our experimental solubility data well. The model proposed by Holder et al.7 does not predict the data as well. Our model can also be used for prediction of LH equilibrium for other gas hydrates. For example, in Figure 11, the methane hydrate LH equilibrium was calculated at three different concentrations. The solubility of methane hydrate was obtained from Duan’s program.32 The literature data on VLH of methane hydrate were obtained from Sloan’s book.14 Based upon the accuracy of the CO2 calculations, the LH methane hydrate predictions are expected to also represent any experimental data quite well. 5. Conclusions Two-phase (LH) hydrate formation data on CO2 hydrate were obtained and are well predicted by a theoretical model. A simplified model was also developed that provided good prediction, especially at lower CO2 concentrations. The data and theory show that the addition of salts at fixed CO2 concentration lowers the pressure required to form hydrate. While this result was unexpected, it is easily predicted using well established theoretical models. The model would also be applicable to other hydrate-forming gases. The model also provides a means for estimating the three-phase (VLH) solubility of CO2, or other hydrate-forming gas, in water.
Acknowledgment The authors thank Ronald Lynn and Charles Levander for help in the experimental portion of this research. The participation of Dr. Yi Zhang was supported by NETL through the University/NETL Student Partnership Program. The participation of Dr. Gerald Holder was supported by NETL through the ORISE Visiting Faculty Program. Disclaimer: Reference in this report to any specific product, process or service is to facilitate understanding and does not imply its endorsement or favoring by the United States Department of Energy. Literature Cited (1) Carbon Sequestration Research and DeVelopment; 810722; U.S. Department of Energy, Office of Science, Office of Fossil Energy: December, 1999. (2) House, K. Z.; Schrag, D. P.; Harvey, C. F.; Lackner, K. S. Permanent carbon dioxide storage in deep-see sediments. Proc. Natl. Acad. Sci. U.S.A. 2006, 103 (33), 12291. (3) Holder, G. D.; Cugini, A. V.; Warzinski, R. P. Modeling Clathrate Formation During Carbon Dioxide Injection into the Ocean. EnViron. Sci. Technol. 1995, 29, 276. (4) Zatsepina, O. Y.; Buffett, B. A. Experimental study of the stability of CO2-hydrate in a porous medium. Fluid Phase Equilib. 2001, 192, 85. (5) Parrish, W. R.; Prausnitz, J. M. Dissociation pressure of Gas Hydrates Formed By Gas Mixtures. Ind. Eng. Chem. Process Des. DeV. 1972, 11, 26. (6) Holder, G. D.; Corbin, G.; Papadopoupoulos, K. D. Thermodynamic and molecular properties of gas hydrates from mixtures containing methane, argon and krypton. Ind. Eng. Chem. Fundam. 1980, 19, 282. (7) Holder, G. D.; Zetts, S. P.; Pradhan, N. Phase behavior in systems containing clathrate hydrates. ReV. Chem. Eng. 1988, 5 (1-4), 1. (8) Hwang, M.; Holder, G. D.; Zele, S. R. Lattice Distortion by Guest Molecules in Gas-Hydrates. Fluid Phase Equilib. 1993, 83, 437. (9) Zele, S. R.; Lee, S. Y.; Holder, G. D. A Theory of Lattice Distortion in Gas Hydrates. J. Phys. Chem. B 1999, 103 (46), 10250. (10) Lee, S. Y.; Holder, G. D. Model for Gas Hydrate Equilibria Using a Variable Reference Chemical Potential: Part I. AIChE J. 2002, 48 (1), 161. (11) Klauda, J.; Sandler, S. A Fugacity Model for Gas Hydrate Phase Equilibria. Ind. Eng. Chem. Res. 2000, 39, 3377. (12) Klauda, J.; Sandler, S. Phase behavior of clathrate hydrates: a model for single and multiple gas component hydrates. Chem. Eng. Sci. 2003, 58, 27.
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ReceiVed for reView June 21, 2007 ReVised manuscript receiVed September 18, 2007 Accepted October 1, 2007 IE070846C
