Ind. Eng. Chem. Res. 2008, 47, 451-458
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GENERAL RESEARCH Semiempirical Method for Determining Water Content of Methane-Rich Hydrocarbon Gas in Equilibrium with Gas Hydrates Amir H. Mohammadi and Dominique Richon* Centre Energe´ tique et Proce´ de´ s, Ecole Nationale Supe´ rieure des Mines de Paris, CEP/TEP. CNRS FRE 2861, 35 Rue Saint Honore´ , 77305 Fontainebleau, France
In this report, we present a semiempirical method for determining water content of methane-rich hydrocarbon gas in equilibrium with gas hydrates. A model based on equality of fugacity concept for estimating the water content of methane in equilibrium with gas hydrates is first introduced. The model estimates the water content of methane using the vapor pressure of the empty hydrate lattice and the partial molar volume of water in the empty hydrate as well as pressure and temperature of the system. In order to extend the capabilities of this tool for determining water content of methane-rich hydrocarbon gases in equilibrium with gas hydrates, a correction factor, which is a function of gas gravity and the system pressure, is used. The predictions of the developed technique are found in acceptable agreement with independent experimental data (not used in developing this method) reported in the literature and the result of a previously reported predictive tool, demonstrating its reliability for estimating the water content of methane-rich hydrocarbon gas in equilibrium with gas hydrates. Introduction Determining water content of gases in equilibrium with gas hydrates is necessary to avoid gas hydrate formation risks and to optimize operating conditions of natural gas production, transportation, and processing facilities. Estimation of water content of gas being at equilibrium with gas hydrates is difficult, because of the very low concentration of water in the gaseous phase. The experimental work done to describe the water content of gas in equilibrium with gas hydrates is limited in accuracy due to various factors: the fact that metastable liquid water may extend well into the gas hydrate formation region and sampling and analysis of the gas phase for the small concentrations of water is indeed difficult. It is also necessary to maintain a constant composition in the gas mixtures by enriching or adding gas, which is depleted preferentially during the course of hydrate formation. Furthermore, the hydrate phase should be decomposed and recrystallized to ensure that the hydrate crystal is indeed in equilibrium with the gas phase.1-3 A literature survey therefore reveals the availability of few sets of experimental data for water content of gases in equilibrium with gas hydrates, and all other data represent metastable liquid water-gas equilibrium. Among the data reported for gaseous systems in the literature, those reported by Sloan et al.4 (quoted by Aoyagi et al.5), Aoyagi et al.5 and Song et al.6 for water content of methane, Song and Kobayashi3 for water content of a mixture of methane (94.69 mol %) and propane (5.31 mol %), and Song and Kobayashi7 for water content of carbon dioxide and also a mixture of carbon dioxide and methane seem to represent gas-hydrate equilibria. Few predictive methods for the water content of gases in * To whom correspondence should be addressed. E-mail: richon@ ensmp.fr. Tel.: +(33) 1 64 69 49 65. Fax: +(33) 1 64 69 49 68.
equilibrium with gas hydrates have been reported in the literature as these methods are generally based on experimental data.1,2 The only recommended method (see Kobayashi et al.8 and Sloan1) was presented by Kobayashi et al.8 Carroll9 also presented a chart for estimating the water content of methane in equilibrium with gas hydrates and then extended it to methane-rich hydrocarbon gases with higher gravities, using simple assumptions. The main aim of this work is to develop a semiempirical method based on equality of fugacity concept for estimating the water content of methane-rich hydrocarbon gases in equilibrium with gas hydrates. To develop this method, experimental data on water content of methane and a gas mixture containing methane (94.69 mol %) and propane (5.31 mol %) are used. The results of this technique are compared with independent experimental data and the prediction of a previously recommended predictive tool.8 The results prove that it is convenient way for simply estimating the water content of gas in equilibrium with gas hydrates. Predictive Methods for Estimating Water Content of Methane-Rich Hydrocarbon Gas in Equilibrium with Gas Hydrates2 Method of Kobayashi et al.8 Kobayashi et al.8 presented the following complicated algorithm to calculate the water content of methane-rich hydrocarbon gases in equilibrium with gas hydrates. In order to calculate the water content of gas in equilibrium with hydrates at given temperatures and pressures, the six steps below should be followed:1,2,8 (1) Calculate the metastable water content at the temperature and pressure of interest. This may be done via an appropriate figure, such as the McKetta-Wehe chart1 or by the following
10.1021/ie070372h CCC: $40.75 © 2008 American Chemical Society Published on Web 12/14/2007
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equation for pressures between 1.378 and 13.789 MPa and for temperatures between 233.15 and 322.04 K:
[
yMS ) exp C′1 +
C′2 + C′3(ln P) + T C′4 C′5(ln P) + C′6(ln P)2 (1) + T T2
]
where yMS is metastable water content, T is temperature, P is pressure, and C′i are constants whose values are given in a table reported by the authors.8 (2) Calculate the three-phase liquid water-hydrate-gas (Lw-H-G) temperature at the given pressure and water-free composition, using an appropriate method.1 Obtain a temperature difference ∆T by subtracting the temperature of interest from the calculated three-phase temperature. (3) Calculate the displacement from the metastable water content (∆yw) at the above ∆T value and pressure of interest using the following equation for methane and a 94.69 mol % methane + 5.31 mol % propane mixture:
∆yw ) exp[C′′1 + C′′2(ln P) + C′′3(ln ∆T) + C′′4(ln P)2 + C′′5(ln P)(ln ∆T) + C′′6(ln ∆T)2 + C′′7(ln P)3 + C′′8(ln P)2(ln ∆T) + C′′9(ln P)(ln ∆T)2 + C′′10(ln ∆T)3 + C′′11(ln P)4 + C′′12(ln P)3(ln ∆T) + C′′13(ln P)2(ln ∆T)2 + C′′14(ln P)(ln ∆T)3 + C′′15(ln ∆T)4] (2) Constants for the above equation are reported in a table, reported by the authors8 for methane, with a regression of the methane data in the pressure range between 3.447 and 10.342 MPa and the temperature range of 239.81-269.81 K. The constants for the mixture were generated in the pressure range of 3.447 and 10.342 MPa, and in the temperature range of 234.26 -277.59 K. (4) Calculate the ∆yw value for the gas composition of interest by a linear interpolation between the ∆yw value for methane (gravity 0.554) and the ∆yw value for the mixture containing 5.31 mol % propane (gravity 0.606), using gravity as an interpolation parameter. (5) Calculate the equilibrium water content by subtracting the ∆yw value obtained in step 4 from the metastable water value obtained in step 1. (6) Consider the range of the data used to determine the regression constants of above equations to determine whether the answer obtained in step 5 is within the bounds of the correlation. As can be seen, the above method requires many steps and is not really to be used. Method of Carroll.9 Carroll9 showed that if one examines the correction chart on the McKetta-Wehe chart,1 it is easy to conclude that the gravity correction is unity in the range of temperature encountered with gas hydrate formation (that is, there is no gravity correction). However, it has been observed that gas gravity does affect the water content. The data of Song and Kobayashi3 on the water content of a mixture of methane (94.69 mol %) and propane (5.31 mol %) (which has a gas gravity of 0.606) in equilibrium with gas hydrates encouraged Carroll9 to make some interpretation of the effect of gas gravity on the water content of a gas in equilibrium with gas hydrates. Carroll9 plotted a figure of experimental data of Song and Kobayashi3 along with curves based on the water content of
methane in equilibrium with hydrates reported by Aoyagi et al.,5 where the mixture data were at consistently lower water content than those for pure methane. A close examination of this plot revealed that the difference in the logarithm of the water content is approximately independent of the temperature. However, there appears to be pressure-dependent, albeit a weak dependence. Carroll9 used a regression procedure and assumed that the water content is a linear function of the gas gravity. Carroll’s9 method requires a semilogarithmic chart for estimating the water content of methane, which is a little difficult to read and can introduce significant errors. Furthermore, this method gives higher values of water content for gases with higher gas gravities, which is not in agreement with those experimentally reported where the water content decreases, when increasing the gas gravity.1 In addition, because the existing correlations/charts are based on limited information, they should be used with some caution and the thermodynamic-based methods are more suitable for this region.1 Development of a New Method A preliminary study shows that exact prediction of water content of methane has an important effect on determination of water content of natural gas.1,2 As mentioned earlier, using semilogarithmic charts for estimating the water content of methane, which is a little difficult to read, can introduce significant errors. Our previous study (see Appendix) showed that methods based on thermodynamic concepts can be used for this purpose; i.e., these methods can be based on the equality of water fugacities concept: The gas-hydrate equilibrium of a system is calculated, by equating the fugacities of water in the gas phase, f gw, and in the hydrate phase, f Hw:1,2
f gw ) f Hw
(3)
The fugacity of water in the hydrate phase, f Hw, is related to the chemical potential difference of water in the filled and empty hydrate by the following expression:1,2
f Hw ) f MT w exp
(
)
µHw - µMT w RT
(4)
where f MT w is the fugacity of water in the hypothetical empty represents the chemical potential hydrate phase, µHw - µMT w difference of water in the filled and empty hydrate, and R stands for the universal gas constant. The solid solution theory of van der Waals-Platteeuw10 can 1,2 For methane be employed for calculating (µHw - µMT w )/(RT). hydrates, the following expression can be obtained:
µHw - µMT w RT
)-
∑i Vji ln(1 + Ci fCH ) ) ∑i ln(1 + CifCH )-Vj
i
4
4
(5)
where Vji is the number of cavities of type i per water molecule in a unit hydrate cell (the values of Vji have been reported by Sloan1), Ci stands for the Langmuir constant for methane’s interaction with each type cavity, and fCH4 is the fugacity of methane in the gas phase. For simple gas hydrates, including methane hydrates, fCH4 can be set to pressure with a good
Ind. Eng. Chem. Res., Vol. 47, No. 2, 2008 453
approximation up to relatively intermediate pressures, where nonideality of the system is not very important.1,2,9 Therefore,
(
)
µHw - µMT w
exp
RT
)
∏i (1 + CiP) -Vj
i
(6)
The Langmuir constants accounting for the interaction between the gas and water molecules in the cavities were reported by Parrish and Prausnitz11 for a range of temperatures. The integration procedure was followed in obtaining the Langmuir constants for lower temperatures using the Kihara potential function with a spherical core according to the study by McKoy and Sinanogˇlu.12 The Langmuir constants are consistent with initial gas hydrate data.1 However, since the experimental conditions go far below the initial hydrate formation conditions, the assumptions that are evidently valid at the initial hydrate formation conditions may be invalid elsewhere.1 Nevertheless, until experimental occupation numbers as a function of temperature, pressure, composition, etc., are obtained, there can be no independent check of this assumption as discussed by Davidson.1,13 In this work, the Langmuir constants for methane’s interaction with each cavity type have been determined as a function of temperature, which are expressed from statistical mechanics as well as from data at the three-phase line by Parrish and Prausnitz.11 For pentagonal dodecahedra (small cavity),11
(
)
2.7088 × 103 3.7237 × 10 -3 exp Csmall ) T T
(7)
for tetrakaidecahedra (large cavity),11
Clarge )
(
)
(8)
where T is in Kelvin and C has units of reciprocal atmosphere. The fugacity of water in the empty lattice can be expressed as1
VMT w dP PMT w RT
∫
P
(9)
MT MT where PMT w , φw , Vw , and P are the vapor pressure of the empty hydrate lattice, the correction for the deviation of the saturated vapor of the pure (hypothetical) lattice from ideal behavior, the partial molar volume of water in the empty hydrate given by von Stackelberg and Mu¨ller14 (quoted by Sloan1), and pressure, respectively, and the exponential term is a Poynting type correction. The concept in the above equation of universal empty hydrate vapor pressure for each structure, prompted Dharmawardhana et al.15 to calculate the PMT w from a number of simple hydrate three-phase ice-gas-hydrate equilibrium, which represent the upper concentration limit of the method.1 By equating the fugacity of water in the hydrate phase to that of pure ice at the three-phase line, Dharmawardhana et al.15 obtained the following equation for the vapor pressure of the empty hydrate structure I:1
(
PMT w ) exp 17.440 -
[
MT f MT w ) Pw exp
]
MT VMT w (P - Pw ) RT
(11)
The fugacity of water in the gas phase is expressed by1
f gw ) ywφgwP
(12)
where yw is the water content of gas phase and φgw is the fugacity coefficient of water in the gas phase. Using the above equations, the following expression is obtained for estimating the water content of methane in equilibrium with gas hydrates:
[
]
MT VMT w (P - Pw ) × yw ) g exp RT φ wP
PMT w
[(1 + CsmallP)-Vsmall(1 + ClargeP)-Vlarge] (13) In the above equation,
1.8373 × 10-2 2.7379 × 103 exp T T
MT MT f MT w ) Pw φw exp
(typically, 10-3-10-5 MPa). However, they have been corrected for pressure by Poynting effect. Again, the vapor pressures of the empty hydrate lattice are consistent with initial hydrate data. However, since the experimental conditions go far below the initial hydrate formation conditions, the assumptions that are evidently valid at the initial hydrate formation conditions may be invalid elsewhere.1 Equation 9 may be simplified by two assumptions:1 (1) that the hydrate partial molar volume equals the molar volume and is independent of pressure and (2) that PMT w is relatively small 1 (on the order of 10-3 MPa), so that φMT w ) 1. Therefore,
6003.9 T
)
(10)
where PMT w is given in atmospheres and T is given in Kelvin. The nonideality of water vapor pressure of the empty hydrate at saturation seems to be negligible due to the small pressures
3 VMT w ) 0.022655 m /kg mol
(from von Stackelberg and Mu¨ller14) 1 Vjsmall ) (from Sloan1) 23 3 (from Sloan1) Vjlarge ) 23 The fugacity coefficient of water (φgw) in the gas phase may be calculated as below:2
φgw) exp(B′′P + C′′P2 + D′′P3)
(14)
where B′′-D′′ are temperature dependent. The following relations for B′′-D′′ seem to be satisfactory:2
b T d C′′ ) c + T hf D′′ ) e + T B′′ ) a +
(15) (16) (17)
where a-fh are constants and can be calculated for the watermethane system by regression of the available water content data. The water contents in the gaseous mixtures of hydrocarbons show lower values than that in the pure methane gas. The reasons for this behavior are as follows.2 The shift in the initial hydrate formation condition (the higher the initial hydrate formation conditions, the greater will be the distance between
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Table 1. Water Content of Methane in Equilibrium with Gas Hydratesa
Table 3. Water Content of a Gaseous Mixture of Methane (94.69 mol %) and Propane (5.31 mol %) (Gravity ) 0.606) in Equilibrium with Gas Hydratesa
Water Content/ppm(mol) P/MPa
T/K
exptl5,6
calcd/pred
3.45 3.45 3.45b 6.9 6.9 6.9b 6.9 10.34 10.34 10.34 10.34
240 250 260 270 240 250 260 270 240 250 260 270
Experimental Data from ref 5 12.3 11.3 32.2 30.8 78.2 78.3 178.0 185.9 5.6 5.7 15.5 15.5 39.6 39.4 94.4 93.5 2.72 2.72 8.46 8.47 24.2 24.2 64.2 64.3
3.45c 3.45 3.45c 3.45 3.45 3.45 6.9c 6.9 6.9 6.9 6.9 6.9 6.9
270 254 240 224 213 196 260 254 246 230 213 204 202
Experimental Data from ref 6 173.0 186.0 42.4 45.1 11.7 11.3 2.17 1.88 0.548 0.473 0.0632 0.0417 41.6 39.4 20.7 22.7 9.48 10.5 2.08 1.91 0.232 0.238 0.0695 0.0691 0.0526 0.0517
3.45b
8.1 4.3 0.1 4.4 1.8 0.0 0.5 1.0 0.0 0.1 0.0 0.2 7.5 6.4 3.4 13.4 13.7 34.0 5.3 9.7 10.8 8.2 2.6 0.6 1.7
a
Boldface type indicates independent data (not used in developing the model). b There is some deviation between these data and the data reported in ref 6. c There is some deviation between these data and the data reported in ref 5. Table 2. Constants a-fh in eqs 15-17 value
a b c d e hf
0.54575 -26.58029 -0.01583 -15.21177 -0.00507 2.31292
[
T/K
exptl3
calcd/pred
AD/%
2.07 2.07 2.07 2.07 2.07 2.07 3.45 3.45 3.45 3.45 3.45 6.89 6.89 6.89 6.89 6.89 6.89 10.34 10.34 10.34 10.34 10.34 10.34
234.2 246.2 251.7 260.1 266.5 277.2 234.2 246.2 252.1 263.2 274.7 234.2 246.2 252.1 260 263.2 276.2 234.2 246.2 252.1 260.1 266.5 277.6
6.86 24.3 41.5 85.2 162.0 427.0 3.47 13.9 27.5 78.8 188.0 1.92 7.03 12.3 25.4 35.8 104.0 1.15 3.75 7.33 14.7 26.8 81.2
6.86 24.6 42.4 93.4 165.0 405.0 3.62 12.7 22.6 62.2 164.0 1.96 6.88 12.2 25.3 33.6 99.7 0.93 3.83 7.33 16.9 31.8 88.9
0.0 1.2 2.2 9.6 1.9 5.2 4.3 8.6 17.8 21.1 12.8 2.1 2.1 0.8 0.4 6.1 4.1 19.1 2.1 0.0 15.0 18.7 9.5
a Boldface type indicates independent data (not used in developing the model).
Table 4. Constants m and n in eq 19 constant
value
m n
-11.223 0.040
Predicted Water Content/ppm (mol)
the stable and metastable equilibrium values at a given pressure and temperature, as shown by Aoyagi et al.5), differences in the hydrate crystal filling characteristics of the molecules, and the possibility of the coexistence of structures I and II under the high pressures and low temperatures. Song and Kobayashi3 observed some large deviations in the predictions at low temperatures near the two-phase envelope and recommended the use of experimental measurements at these temperatures rather than the predicted values. In order to estimate water content of methane-rich hydrocarbon gases in equilibrium with gas hydrates, the following equation can be used:2
Fγ ) exp m(γ - 0.554) + n(γ - 0.554)
P/MPa
Table 5. Water Content of a Gas Whose Gravity Is 0.575 in Equilibrium with Gas Hydrates at 6.895 MPa and 260.04 K
constant
yw,γ ) yw,CH4Fγ
Water Content/ppm (mol)
AD/%
(18)
( )] P P0
(19)
where P is the system pressure and P0 represents atmospheric pressure. The gas gravity, γ, is dimensionless and Fγ is a correction factor. yw,CH4 and yw,γ stand for water content of methane and methane-rich hydrocarbon gas in equilibrium with gas hydrates at the same temperature and pressure. Constants m and n can be calculated for the water-gas mixture system
P/MPa
T/K
using the method of Kobayashi et al. 8
using this method
AD/%
6.895
260.04
32.1
33.1
3.0
by regression of the available water content data. Therefore, eq 13 can be used along with eqs 18 and 19 for estimating the water content of methane-rich hydrocarbon gas in equilibrium with gas hydrates. Results and Discussion The experimental data5,6 reported in Table 1 on water content of methane in equilibrium with gas hydrates were used to adjust constants a-fh in eqs 15-17. As can be seen, the temperature range is from 196 to 270 K, and the pressure is up to 10.34 MPa. These constants are reported in Table 2. Table 1 also compares the predictions of this model with independent experimental data (not used in adjusting constants a-fh; see Table 2). As can be seen, the agreement between experimental and predicted/calculated data is generally acceptable and the average absolute deviation (AAD) is equal to 5.5%. To adjust constants m and n in eq 19, the experimental data3 on water content of a mixture of methane (94.69 mol %) and propane (5.31 mol %) in equilibrium with gas hydrates reported in Table 3 were used. As can be seen, the temperatures range is from 234.2 to 277.6 K and the maximum pressure is 10.34 MPa. The constants m and n are reported in Table 4. Table 3 also shows a comparison between the predictions of this model and independent experimental data for water content of this gas
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mixture. As can be seen, the agreement between experimental and predicted data is generally acceptable and the AAD is equal to 7.2%. In order to further evaluate the performance of the model, a comparison has been made in Table 5, between predictions of Kobayashi et al.’s8 method and the model developed in the present work for water content of a gas whose gravity is 0.575 in equilibrium with gas hydrates at 6.895 MPa and 260.04 K. As can be observed, close agreement was achieved and the absolute deviation (AD) relative to the prediction of Kobayashi et al.’s8 method is 3%. Unfortunately, more experimental data are not available to compare the predicted results with tem. In summary, the method developed in this work can estimate water content of methane-rich hydrocarbon gas, whose gravity is in the range of 0.554-0.606 in equilibrium with gas hydrates. It is recommended to use this method for estimating the water content at 234.2-277.6 K and pressures up to 10.34 MPa. Furthermore, the model should not be used at high pressures, as the nonideality of the system is not ignorable at these conditions and the assumption made in eq 6 (fCH4 can be set to P) is conservative. Using more reliable/new water content data in the future for readjusting the model parameters can increase the accuracy of this method. Conclusions A review was made on the predictive methods reported in the literature for determining the water content of methanerich hydrocarbon gas in equilibrium with gas hydrates. The review showed a need for developing a simple and more robust method. A model based on equality of fugacity concept was then proposed for estimating the water content of methane in equilibrium with gas hydrates. A correction factor, which is a function of gas gravity and the system pressure, was then developed and used to estimate water content of methane-rich hydrocarbon gas in equilibrium with gas hydrates. The results of this method were in close agreement with independent experimental data and the result of a previously reported predictive tool, demonstrating the capability of the new technique for estimating the water content of methane-rich hydrocarbon gas in equilibrium with gas hydrates. Appendix Determining Water Content of Natural Gases in Equilibrium with Liquid Water or Ice.2 To develop predictive methods and to improve the accuracy of the estimated water content of gases, experimental data are required (which could also be used for validation of predictive tools). Mohammadi et al.16 have done a comprehensive review on the existing experimental methods for measuring water content/water dew point of gases, gathered the water content data for the main components of natural gases from the literature, and concluded that most of the water content data for hydrocarbons and for non-hydrocarbon gases (e.g., N2, CO2, and H2S) at low temperatures are often inconsistent. These types of uncertainties can lead to large deviations for correlations and models, when using these scattered data for regressing.2 Chapoy et al.17-22 and Mohammadi et al.16,23,24 have measured the necessary data for vapor-liquid water equilibria of natural gas main components, especially at low temperatures, and tuned parameters of a thermodynamic model16 to predict the water content/water dew point of gases. They also concluded that using
only gas solubility in aqueous-phase data for adjusting parameters of their thermodynamic model16 can lead to accurate prediction of water content of gas phase.2,25 Mohammadi et al.26 have also done a review on the correlations/charts existing in the open literature and then developed a semiempirical approach based on equality of water fugacity in equilibrium phases for estimating the water content of natural gases in equilibrium with liquid water or ice:2 The gas-liquid equilibrium of a system is calculated, using the following equation:
f gi ) f Li
(i ) 1, ..., N)
(A.1)
where f is the fugacity, the superscripts “g” and “L” represent the gas and liquid phases, and N is the number of components. The equality of fugacities can be calculated using the following relationship:
yiφiP ) xiγiPsat i exp
∫PP
sat i
VLi dP RT
(A.2)
where y, φ, P, x, γ j , V, R, and T represent mole fraction in the gas phase, fugacity coefficient, the pressure, the mole fraction in the liquid phase, the activity coefficient, the molar volume, the universal gas constant, and the temperature, respectively. The superscript “sat” stands for saturation condition. In the intermediate pressure range, liquid water is an incompressible fluid and gas solubility is very small comparing to unity for hydrocarbons and some gases like nitrogen (solubility of hydrocarbons in water is, in general, considerably less than water in hydrocarbons) and to an approximation activity coefficient of water can be taken unity. However, the nonideality of the liquid phase and gas solubility become important at high pressures. Therefore, the mole fraction of water in the gas phase can be estimated, using the following equation:
yw )
[
]
Psat VLw(P - Psat w w) exp φ wP RT
(A.3)
where the subscript “w” stands for water. As can be seen, water content is determined primarily by the fugacity coefficient of water (φw) in the gas phase, temperature, and pressure. In other words, the nonideality of the gas phase is the critical factor determining water content in the intermediate pressure range. The fugacity coefficient of water in the gas phase up to intermediate pressures may be calculated as below:
hP + C h P2) φw ) exp(B
(A.4)
where B h and C h are functions of temperature. The following relations for B h and C h seem to be satisfactory:
B h ) aj +
bh T
(A.5)
C h ) cj +
dh T
(A.6)
where aj-dh are constants and can be calculated for every watergas system by regressing the water content data for that system. To estimate vapor pressure and molar volume of water in
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Table A-1. Constants aj-dh in eqs A.5 and A.6 for Methane-Water System2,26 a constant
value
aj bh cj dh
0.0693558 -30.904809 -0.0007654 0.3178958
Table A-2. Constants of eq A.142,31
It is recommended that one use constants aj-dh in eqs A.5 and A.6 at temperatures and pressures up to 377.59 K and 13.81 MPa, respectively, because experimental data at these ranges were used to determine these constants.26
eq A.3, the relations reported by Daubert and Danner27 and McCain28 can be used, respectively:
(
7258.2 - 7.3037 ln(T) + T
)
4.1653 × 10-6T2 (A.7) VLw ) Fw )
18.015 Fw
(A.8)
62.368 D hw
(A.9)
D h w ) (1 + ∆Vwt)
(A.10)
5.50654 × 10-7t2 (A.11) L where, T, t, p, Psat w , and Vw are in K, F, psia, MPa, and ft3/lb mol, respectively and Fw, D h w, and ∆Vwt are water density in lbm/ft3, formation volume factor, and volume change due to temperature, respectively. Equations A.10 and A.11 are valid at t < 260 °F, even over a wide range of salt concentration.29 To find constants aj- dh in eqs A.5 and A.6, water content data are used as input for a multidimension regression procedure, in order to reduce the average absolute deviation between experimental and calculated data. Table A-1 shows optimized values for constants aj-dh.2,26 The above approach also can be used for estimating water content of gases in equilibrium with ice. For this purpose, the following relations for molar volume of ice and ice vapor pressure can be used:30
log(Psub w ))-
19.655 + 0.0022364(T - 273.15) 103
(A.12)
1032.5576407 + 51.0557191 log(T) T
-0.03185 -0.01538 0.02772
acid gas
temperature, T (K)
pressure, P (MPa)
maximum concentration of acid gas (mol%)
H2S CO2 CO2 + H2S
310-420 310-420 310-420
0.5-40 0.5-40 0.5-35
30 50 15% H2S, 35% CO2
Table A-4. Constants b1-b3 in eq A.172,31 constant
value
b1 b2 b3
0.17006 -0.15241 -0.04515
correction factor, Fsour, was suggested for taking into account the effect of acid gases on the water content:2,31
[ ( ) ( )( ) ( )] T T P P + c2 + c3 T0 T0 P0 P0
(A.13)
where the superscripts “I” and “sub” refer to ice and sublimation, respectively. In the above equations, T, VIw, and Psub w are given in K, m3/kg mol, and mmHg, respectively. Mohammadi et al.31 developed new correction factors for taking into account the effect of acid gases and also heavy hydrocarbons on water content of sweet gases. Considering the fact that the water content of an acid/sour gas is a function of temperature, pressure, and acid gas mole fraction, the following
(A.14)
In the above equation, P0 and T0 are reference pressure (atmospheric pressure) and reference temperature (273.15 K), respectively, and c1-c3 are constants, which are reported in Tables A-2 and A-3. Fsour is a correction factor due to the preis equivalent mole fraction of acid sence of acid gases and zHequi 2S gases, which is calculated using the following equation:2,31
) zH2S + 0.75zCO2 zHequi 2S
(A.15)
where z is the mole fraction in the natural gas, the subscripts “CO2” and “H2S” refer to carbon dioxide and hydrogen sulfide, respectively, and the superscript “equi” refers to the equivalent H2S concentration. The water content of sour gases is then calculated by multiplying the new correction factor and the water content of the corresponding sweet gas:2,31
yw,sour ) Fsouryw,sweet
(A.16)
where the subscripts “sour” and “sweet” refer to sour and sweet gases, respectively. Fsour is dimensionless, i.e., the two water content terms have the same units. McKetta-Wehe1 suggested a graphical correction factor as a function of gas gravity and temperature for taking into account the effectof heavy hydrocarbons. This correction factor is well represented by the following equation:2,31
()
T + T0 T b3(γ - 0.554)2 T0
Fhh ) 1 + b1(γ - 0.554) + b2(γ - 0.554)
0.0977079751T + 7.035711316 × 10-5T2 98.5115496
c1 c2 c3
Fsour ) 1 + zHequi c1 2S
∆Vwt ) -1.0001 × 10-2 + 1.33391 × 10-4t +
VIw)
value
Table A-3. Application Range of eq A.1431
a
Pwsat ) 10-6 exp 73.649 -
constant
()
2
(A.17)
where Fhh is correction factor to take into account the effect of heavy hydrocarbons, γ stands for gas gravity, and b1-b3 are constants. These constants are tabulated in Table A-4.2,31 Dissolved solids (salts) in the aqueous phase can change water properties such as reducing the water vapor pressure and therefore reducing the water content of natural gases.28,29 To take into account the presence of salts, the extrapolation of the salinity correction factor in the McKetta-Wehe1 chart to high salt concentrations is believed to underpredict the water content of a gas in equilibrium with brine.29 Instead, the graphical correla-
Ind. Eng. Chem. Res., Vol. 47, No. 2, 2008 457
tion of Katz32 for the salinity correction factor is recommended.29 The graphical correlation, developed from water vapor pressure depression due to the presence of salt, can be expressed as2,28,29
FSalt ) 1 - 4.920 × 10-3wSalt - 1.7672 × 10-4wSalt2 (A.18) where FSalt is a correction factor due to the presence of salt, w is the weight percent of salt in brine, and the subscript “Salt” refers to salt. Nomenclature N AAD ) average absolute deviation; AAD ) 1/N Σi)1 (experimental value - calculated/predicted value)/experimental value AD ) absolute deviation; AD ) (experimental value calculated/predicted value)/experimental value B h ) constant B′′ ) constant C ) Langmuir constant C h ) constant C′ ) constant C′′ ) constant D h ) formation volume factor D′′ ) constant F ) correction factor G ) gas H ) hydrate L ) liquid N ) number of components/data P ) pressure R ) universal gas constant T ) temperature V h ) volume exptl ) experimental value calcd ) calculated value pred ) predicted value a ) constant aj ) constant b ) constant bh ) constant c ) constant cj ) constant d ) constant dh ) constant e ) constant f ) fugacity hf ) constant m ) constant n ) constant p ) pressure t ) temperature V ) molar volume Vj ) number of cavities per water molecule in a unit hydrate cell w ) weight percent of salt in brine x ) mole fraction in the liquid phase y ) mole fraction in the gas phase z ) component mole fraction in natural gas Subscripts large ) large cavity small ) small cavity sour ) sour gas sweet ) sweet gas hh ) heavy hydrocarbon i ) component i t ) temperature
w ) water 0 ) reference condition γ ) gas gravity Superscripts MT ) hypothetical empty hydrate H ) hydrate I ) ice L ) liquid equi ) equivalent H2S concentration sat ) saturation condition sub ) sublimation g ) gas Greek Symbols ∆ ) difference γ ) gas gravity γ j ) activity coefficient µ ) chemical potential F ) density φ ) fugacity coefficient Acknowledgment Some parts of this paper were presented at the 85th Gas Processors Association (GPA) Annual Meeting. The authors thank the Centre for Gas Hydrate Research of Heriot-Watt University for their assistance. Literature Cited (1) Sloan, E. D. Clathrate Hydrates of Natural Gases, 2nd Edition; Marcel Dekker: New York, 1998. (2) Mohammadi, A. H.; Chapoy, A.; Tohidi, B.; Richon, D. Advances in Estimating Water Content of Natural Gases. Presented at the 85th GPA Annual Convention, Grapevine, TX, March 5-8, 2006. (3) Song, K. Y.; Kobayashi, R. Measurement and Interpretation of the Water Content of a Methane-Propane Mixture in the Gaseous State in Equilibrium with Hydrate. Ind. Eng. Chem. Fundam. 1982, 21, 391-395. Also: Song, K. Y.; Kobayashi, R. Measurement & Interpretation of the Water Content of a Methane-5.31 mol % Propane Mixture in the Gaseous State in Equilibrium with Hydrate. GPA Research Report 50, Tulsa, OK, January 1982. (4) Sloan, E. D.; Khoury, F. M.; Kobayashi, R. Water Content of Methane Gas in Equilibrium with Hydrates. Ind. Eng. Chem. Fundam. 1976, 15, 318-323 (quoted in ref 5). (5) Aoyagi, K.; Song, K. Y.; Kobayashi, R.; Sloan, E. D.; Dharmawardhana, P. B. (I). The Water Content and Correlation of the Water Content of Methane in Equilibrium with Hydrates, and (II). The Water Content of a High Carbon Dioxide Simulated Prudhoe Bay Gas in Equilibrium with Hydrates. GPA Research Report 45, Tulsa, OK, December 1980. (Also: Aoyagi, K.; Song, K. Y.; Sloan, E. D.; Dharmawardhana, P. B.; Kobayashi, R. Improved Measurements and Correlation of the Water Content of Methane Gas in Equilibrium with Hydrate. Presented at the 58th Annual GPA Convention, Denver, CO, 1979.) (6) Song, K. Y.; Yarrison, M.; Chapman, W. Experimental low temperature water content in gaseous methane, liquid ethane, and liquid propane in equilibrium with hydrate at cryogenic conditions. Fluid Phase Equilib. 2004, 224, 271-277. (7) Song, K. Y.; Kobayashi, R. The Water Content of CO2-rich Fluids in Equilibrium with Liquid Water and/or Hydrates. GPA Research Report 99, Tulsa, OK, June 1986. (8) Kobayashi, R.; Song, K. Y.; Sloan, E. D. Phase Behavior of Water/ Hydrocarbon Systems. In Petroleum Engineering Handbook; Bradley, H. B., et al., Eds.; Society of Petroleum Engineers: Richardson, TX, 1987; Chapter 25. (9) Carroll, J. J. Natural Gas Hydrates: A Guide for Engineers; Gulf Professional Publishing: Boston, 2003. (10) van der Waals, J. H.; Platteeuw, J. C. Clathrate solutions. AdV. Chem. Phys. 1959, 2, 1. (11) Parrish, W. R.; Prausnitz, J. M. Dissociation pressures of gas hydrate formed by gas mixture. Ind. Eng. Chem. Process. Des. DeV. 1972, 11, 2634. (12) McKoy, V.; Sinanogˆlu, O. Theory of dissociation pressures of some gas hydrates. J. Chem. Phys. 1963, 38, 2946-2956.
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(13) Davidson, D. W. In Water: A ComprehensiVe Treatise; Frank, F., Ed.; Plenum Press: New York, 1973. (14) von Stackelberg, M.; Mu¨ller, H. R. Feste Gashydrate. Z. Elektrochem. 1954, 58, 25 (quoted in ref 10). (15) Dharmawardhana, P. B.; Parrish, W. R.; Sloan, E. D. Experimental thermodynamic parameters for the prediction of natural gas hydrate dissociation conditions. Ind. Eng. Chem. Fundam. 1980, 19, 410-414 (quoted in ref 1). (16) Mohammadi, A. H.; Chapoy, A.; Richon, D.; Tohidi, B. Experimental measurement and thermodynamic modeling of water content in methane and ethane systems. Ind. Eng. Chem. Res. 2004, 43, 7148-7162. (17) Chapoy, A.; Mohammadi, A. H.; Richon, D.; Tohidi, B. Gas solubility measurement and modeling for methane-water and methaneethane- n-butane-water systems at low temperature conditions. Fluid Phase Equilib. 2004, 220, 113-121. (18) Chapoy, A.; Mokraoui, S.; Valtz, A.; Richon, D.; Mohammadi, A. H.; Tohidi, B. Solubility measurement and modeling for the system propane-water from 277.62 to 368.16 K. Fluid Phase Equilib. 2004, 226, 255-263. (19) Chapoy, A.; Mohammadi, A. H.; Tohidi, B.; Richon, D. Gas Solubility Measurement and Modeling for the Nitrogen + Water System from 274.18 K to 363.02 K. J. Chem. Eng. Data 2004, 49, 11101115. (20) Chapoy, A.; Mohammadi, A. H.; Chareton, A.; Tohidi, B.; Richon, D. Measurement and Modeling of Gas Solubility and Literature Review of the Properties for the Carbon Dioxide-Water System. 2004, Ind. Eng. Chem. Res. 43, 1794-1802. (21) Chapoy, A.; Mohammadi, A. H.; Tohidi, B.; Valtz, A.; Richon, D. Experimental Measurement and Phase Behavior Modeling of Hydrogen Sulfide-Water Binary System. Ind. Eng. Chem. Res. 2005, 44, 75677574. (22) Chapoy, A.; Mohammadi, A. H.; Tohidi, B.; Richon, D., Water Content Measurement and Modeling for Methane-Water and MethaneEthane-n-Butane-Water Systems Using a New Sampling Device. J. Chem. Eng. Data 2005, 50, 1157-1161.
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ReceiVed for reView March 12, 2007 ReVised manuscript receiVed June 8, 2007 Accepted September 11, 2007 IE070372H
