Energy & Fuels 2000, 14, 19-24
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Representation of Hydrate Phase Equilibria in Aqueous Solutions of Methanol and Electrolytes Using an Equation of State Julian Y. Zuo,* Dan Zhang, and Heng-Joo Ng DB Robinson Research Ltd., 9419-20 Avenue, Edmonton, Alberta, Canada T6N 1E5 Received June 7, 1999
In this work, the method for the calculation of hydrate phase equilibria proposed by Zuo and Stenby (Zuo, Y.-X.; Stenby, E. H. SEP 9816. J. Soc. Petr. Eng., submitted) has been extended to represent incipient equilibrium hydrate formation conditions in aqueous solutions containing methanol and electrolytes. To predict hydrate formation conditions, the parameters of the extended Fu¨rst-Renon electrolyte equation of state (Fu¨rst, W.; Renon, H. AIChE J. 1993, 39, 335-343) have been reevaluated. Interaction parameters between water and methanol have been determined to match experimental hydrate formation data of methane in the water-methanol solutions. The interaction parameters between methanol and ions are associated with cationic Stokes and anionic Pauling diameters. Furthermore, cationic Stokes diameters in methanol are assumed to be a function of temperature. Coefficients of the temperature dependence have been determined by two methods. In method I, the cationic Stokes diameters in methanol are equal to those at 298.15 K. In method II, the coefficients are adjusted so that the predictions match the experimental hydrate formation data of the 80% methane + 20% carbon dioxide mixture in the presence of methanol and electrolytes. The proposed methods have been applied to calculate hydrate formation conditions for a number of systems containing methanol and/or electrolytes. Good agreement has been reached between the calculated hydrate formation temperatures (or pressures) and the experimental data. The overall temperature deviations obtained by methods I and II are within 0.56 and 0.70 K, respectively.
Introduction The formation of gas hydrates can cause serious problems in the production and transportation of wet natural gas. A number of thermodynamic models have been proposed and commercial software packages have been available for predicting hydrate formation conditions to chemical and petroleum industries. However, most models are not suitable for situations where another solvent, like methanol, is present in addition to electrolytes. Recently Bishnoi et al.3 measured hydrate formation conditions for a number of natural gases in electrolyte solutions containing mixed solvents (water and methanol). Mei et al.4 also reported hydrate formation data for a synthetic natural gas mixture in aqueous solutions containing methanol and/or electrolytes. These data are very useful in developing and testing hydrate formation models. To date, few thermodynamic models have been developed for predicting hydrate formation conditions in aqueous solutions containing methanol and electrolytes simultaneously in the open literature. Recently, Nas* Corresponding author. E-mail:
[email protected]. (1) Zuo, Y.-X.; Stenby, E. H. SEP 9816. J. Soc. Petr. Eng., submitted. (2) Fu¨rst, W.; Renon, H. AIChE J. 1993, 39, 335-343. (3) Bishnoi, P. R.; Dholabhai, P. D.; Mahadev, K. N. GPA Research Report RR-156, 1996. (4) Mei, D.-H.; Liao, J.; Yang J.-T.; Guo, T.-M. J. Chem. Eng. Data 1998, 43, 178-182.
rifar et al.5 have developed a method for this purpose. In their model the water activity in the aqueous phase was associated with the enthalpy of hydrate formation by an empirical expression, and the method was tested against a few systems. Zuo and Fu¨rst6 extended the aqueous electrolyte equation of state (EOS) proposed by Fu¨rst and Renon2 to predict the vapor-liquid equilibria for water-methanol electrolyte systems. Zuo and Stenby1 combined the extended EOS with the modified BarkanSheinin hydrate model7 in pure water to develop a predictive method for calculating hydrate formation conditions. The model was tested over a number of natural gas hydrates in aqueous electrolyte solutions and pure carbon dioxide hydrates in water-methanol electrolyte solutions and proved accurate. However, the model cannot be used for hydrates of other pure gas and gas mixtures in aqueous solutions containing methanol and electrolytes. Therefore, the objective of this work is to extend the model proposed by Zuo and Stenby1 to calculate hydrate formation conditions in aqueous solutions containing methanol and/or electrolytes based on a few experimental data. (5) Nasrifar, K.; Moshfeghian, M.; Maddox, R. N. Fluid Phase Equilib. 1998, 146, 1-13. (6) Zuo, Y.-X.; Fu¨rst, W. Fluid Phase Equilib. 1998, 150, 267-275. (7) Barkan, E. S.; Sheinin, D. A. Fluid Phase Equilib. 1993, 86, 111136.
10.1021/ef990110v CCC: $19.00 © 2000 American Chemical Society Published on Web 12/03/1999
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Thermodynamic Hydrate Model For a given system, the gas hydrate phase equilibrium condition is given by:
∆µH ) ∆µW
(1)
∆µH
represents the difference between the chemiwhere cal potential of water in the empty hydrate lattice and in the hydrate phase, and ∆µW denotes the difference between the chemical potential of water in the empty hydrate lattice and in the aqueous solution. ∆µW is evaluated by the following equation:
∆µW ∆µ0 ) + RT RT
∫0P
∆νW dp RT
∆h
∫TT RTW2 dT - ln aW 0
(2)
where T0 ) 273.15 K; ∆hW and ∆vW are the differences of water molar enthalpies and volumes between the empty hydrate lattice and pure water (or ice), respectively; aW denotes the water activity in the aqueous phase (calculated by the extended EOS described in the next section). The profound effect of inhibitors is a nonlinear term ln aw which is added to the equation describing the deviation of hydrate formation conditions from pure water. ∆µH is determined by
∆µH RT
2
)
νj ln(1 + ∑ Cijfi) ∑ j)1 i
(3)
where νj is a crystal chemical constant characterizing a ratio of the number of cavities of the jth type to the number of water molecules in the elementary hydrate cell; fi denotes the fugacity of hydrate former i (also calculated by the extended EOS); Cij represents the Langmuir constant of the hydrate former i in the jth type cavity, which is calculated by the modified Barkan and Sheinin7 expression. Extended Electrolyte Equation of State As mentioned above, when electrolytes or/and methanol are present in the aqueous phase, a thermodynamic model is required to calculate the activity of water in the last term of eq 2. In this work, the extended Fu¨rst and Renon (FR) electrolyte EOS2 has been used. The molar Helmholtz energy of an electrolyte mixture consists of four contributions:
∆a ∆a ∆a ∆a ∆a ) + + + (4) RT RT RF RT SR1 RT SR2 RT LR
( ) ( )
( )
( )
( )
The first term (RF) is related to repulsive forces, the second one (SR1) represents attractive short-range interactions involving no ions. These two terms are expressed by the SRK equation of state. The last two terms are specific to ionic contributions, the first one (SR2) represents solvation interactions and the second one (LR) is related to long-range interactions. The longrange contribution is expressed by a simplified mean sphere approximation (MSA) model. The SR2 term is a specific one and involves symmetrical cation-solvent (Wcs) and cation-anion (Wca.) interaction parameters. The other interactions involving ions (cation-cation, anion-anion, or anion-solvent) are ignored due to
charge repulsive effects and low solvation of anions (compared to cations). The others ionic adjustable parameters are the anionic (ba) and cationic (bc) covolumes. All parameters are related to Stokes diameters σSc (for cations) and Pauling diameters σPa (for anions):
bc ) λ1(σSc )3 + λ2 and ba ) λ1(σPa )3 + λ2
(5)
For interaction parameters
Wcs ) λ3σSc + λ4 and Wca ) λ5(σSc + σPa )4 + λ6 (6) Correlation coefficients λ1-λ6 have been deduced from the data treatment of numerous experimental osmotic coefficients for aqueous halide and nonhalide systems2 and vapor pressures for binary methanol-halide systems.6 To extend the model to mixed-solvent (watermethanol) systems, the Wong-Sandler mixing rule8 with the UNIQUAC model has been introduced. For binary water (1) and methanol (2) mixtures, adjustable parameters are k12 ) k21 (set to zero), τ12 and τ21, where τ12 and τ21 (τij ) ∆Uij/RT) are the energy parameters in the UNIQUAC model. Evaluation of Parameters Zuo and Stenby1 only used the binary vapor-liquid equilibrium data (instead of hydrate formation data) to determine model parameters. However, the model parameters cannot be suitable for other gas (except pure carbon dioxide) hydrates in aqueous solutions containing methanol and electrolytes. That is, the model systematically underpredicts the effect of methanol and electrolytes on hydrate formation pressures for other gas hydrates. In this work, therefore, parameters τ12 and τ21 for the binary water (1) and methanol (2) pairs have been redetermined using the experimental hydrate formation data9 of methane in the presence of methanol up to 50 wt %, instead of vapor-liquid equilibrium data. The optimized values are: τ12 ) -3.8558 and τ21 ) 1.5707. Following the work of Zuo and Stenby1, it is assumed that the cationic Stokes diameters in methanol are temperature dependent. The temperature dependence of the cationic Stokes diameters in methanol is expressed as
1 (T1 - 298.15 )
s + A(T - 298.15) + B σsc ) σc,298.15
(7)
where σc,298.15 is the cationic Stokes diameters at 298.15 K. A and B are two adjustable parameters. A and B were tuned to match experimental cationic Stokes diameters by Zuo and Stenby.1 In this work, however, two methods have been used to determine A and B. In the first one, A and B are equal to zero. That means the cationic Stokes diameters in methanol are independent of temperature. The cationic Stokes diameters at 298.15 K are used. This method is referred to as method I. (8) Wong, D. S.; Sandler, S. I. AIChE J. 1992, 38, 671-680. (9) Robinson, D. B.; Ng, H.-J.; Chen, C.-J. Proceedings of the 66th GPA Annual Convention; Denver, Colorado, March, 1987; pp 16-18.
Representation of Hydrate Phase Equilibria
Figure 1. Hydrate formation conditions of H2S in aqueous electrolyte solutions (experimental data: Bishnoi et al.3).
Energy & Fuels, Vol. 14, No. 1, 2000 21
Figure 2. Hydrate formation conditions of synthetic natural gas in aqueous electrolyte solutions (experimental data: Mei et al.4).
In the other method, A and B are adjusted to match experimental hydrate formation data of the 80% methane + 20% carbon dioxide mixture in the presence of methanol and electrolytes.2 The correlated results are listed in Table 1 (the first several rows) for the 80% methane + 20% carbon dioxide mixture. The average deviations of the correlated hydrate formation temperatures and pressures are 0.40 K and 4.4%, respectively. This method is referred to as method II. The two methods have been used to predict hydrate formation conditions for a number of systems. Results and Discussion First of all, the proposed model has been used to predict hydrate formation conditions in aqueous electrolyte solutions. For the prediction of gas hydrates in aqueous electrolyte solutions, the Zuo and Stenby approach1 was followed. The method was used to predict hydrate formation conditions in a lot of aqueous single and mixed electrolyte solutions.1 In most cases, the predictions match experimental data very well. The overall average absolute deviation (AAD) of the predicted hydrate formation pressures for all the systems is 4.3% (521 data points). In this work, some new experimental data available have been used to test the proposed model for hydrates of a 78% methane + 20% CO2 + 2% propane mixture, a natural gas, a synthetic natural gas, and pure H2S. The results are also shown in Table 1. Figures 1 and 2 show hydrate formation conditions of H2S and the synthetic natural gas in aqueous electrolyte solutions, respectively. It can be seen that the predictions are in good agreement with experimental data. Second, methods I and II have been applied to predict hydrate formation conditions for a number of pure gas and gas mixtures in the presence of methanol and electrolytes. The predicted results are also given in Table 1 (in columns 4 and 5, respectively). The average deviations of the calculated hydrate formation temperatures using methods I and II are 0.70 and 0.56 K, respectively. The average relative deviations of the calculated hydrate formation pressures using methods I and II are 8.5% and 6.9%, respectively. Obviously,
Figure 3. Hydrate formation conditions of the 50% methane + 50% CO2 mixture in aqueous solutions of methanol and electrolytes (experimental data: Bishinoi et al.3).
method II is better than method I because the model parameters have been adjusted to match experimental hydrate data of the 80% methane + 20% CO2 mixture in method II. Figures 3 and 4 show the comparison of the hydrate formation pressure and temperature for the 50% methane + 50% carbon dioxide mixture and the 90% methane + 5% propane + 5% H2S mixture in aqueous solutions of methanol and electrolytes (20 wt % in total inhibitors), respectively. The hydrate formation conditions of the mixtures in pure water were also calculated for both systems shown as references in the figures. The predictions by method I (solid lines) match experimental data well. Figure 5 compares the predicted hydrate formation conditions of the 78% methane + 2% propane + 20% carbon dioxide mixture in aqueous solutions containing methanol and electrolytes with the measured data. The model (method I) slightly under-predicts hydrate formation pressures in a high-pressure region. Considering this is also the trend of hydrate formation predictions
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Table 1. Average Deviations of the Predicted Hydrate Formation Temperatures inhibitor (wt %) 80% methane + 20% CO2 pure water methanol (10) methanol (20) methanol (5) + NaCl (5) methanol (5) + NaCl (15) methanol (5) + KCl (10) methanol (5) + CaCl2 (10) methanol (5) + CaCl2 (15) methanol (10) + NaCl (10) methanol (10) + CaCl2 (10) methanol (5) + NaCl (5) + KCl (5) + CaCl2 (15) overall 50% methane + 50% CO2 methanol (15) + NaCl (5) methanol (10) + NaCl (10) methanol (20) methanol (5) + NaCl (5) + KCl (5) + CaCl2 (5) overall 78% methane + 20% CO2 + 2% propane pure water NaCl (5) NaCl (10) NaCl (20) methanol (10) methanol (20) methanol (5) + NaCl (5) methanol (10) + NaCl (10) methanol (15) + NaCl (5) methanol (5) + CaCl2 (5) methanol (10) + NaCl (5) + CaCl2 (5) overall natural gasa pure water NaCl (20) methanol (20) methanol (5) + NaCl (15) methanol (15) + NaCl (5) overall 90% methane + 5% propane + 5% H2S methanol (10) + NaCl (10) methanol (5) + NaCl (15) methanol (15) + NaCl (5) methanol (10) + CaCl2 (10) overall 85% methane + 15% H2S methanol (10) + NaCl (10) methanol (5) + NaCl (15) methanol (15) + NaCl (5) methanol (10) + CaCl2 (10) overall 95% methane + 5% H2S methanol (10) + NaCl (10) methanol (5) + NaCl (15) methanol (15) + NaCl (5) methanol (10) + CaCl2 (10) overall 100% H2S pure water NaCl (20) NaCl (10) + CaCl2 (10) NaCl (10) + KCl (10) NaCl (10) + KCl (5) + CaCl2 (5) methanol (10) methanol (20) methanol (10) + NaCl (10) methanol (5) + NaCl (15) methanol (15) + NaCl (5) methanol (10) + CaCl2 (10) overall
T-range K
Np
method I AAD, K
method II AAD, K
source of data
274-281 268-281 264-277 268-282 262-274 267-280 266-280 262-274 262-275 262-276 263-276
4 14 10 6 6 6 6 6 6 6 6 76
0.09 0.32 0.63 0.46 0.19 0.29 0.56 1.17 0.36 1.18 0.15 0.51
0.09 0.32 0.63 0.31 0.66 0.58 0.07 0.15 0.54 0.17 0.73 0.40
3 3 3 3 3 3 3 3 3 3 3
266-274 264-274 264-276 264-274
6 6 5 6 23
1.30 0.76 0.24 0.64 0.76
0.74 0.12 0.24 0.14 0.31
3 3 3 3
277-289 280-288 277-282 274-279 277-286 274-280 277-286 274-280 273-280 277-286 274-280
8 6 4 6 6 6 6 12 6 6 6 72
0.65 1.11 0.55 1.24 0.62 0.70 0.47 0.80 0.66 0.76 0.18 0.71
0.65 1.11 0.55 1.24 0.62 0.70 0.40 0.37 0.25 0.57 0.83 0.64
3 3 3 3 3 3 3 3 3 3 3
281-291 276-282 277-284 277-283 277-284
5 6 6 6 6 29
0.61 0.55 0.82 0.96 1.50 0.90
0.61 0.55 0.82 0.55 1.18 0.75
3 3 3 3 3
280-285 279-284 279-283 280-284
6 6 6 6 24
0.31 0.29 0.51 0.74 0.46
0.41 0.44 0.42 0.36 0.40
3 3 3 3
273-283 273-283 273-284 273-283
6 6 6 6 24
1.21 0.93 1.20 2.13 1.37
0.81 0.65 1.04 1.22 0.93
3 3 3 3
272-279 273-279 272-279 272-279
6 6 12 6 30
0.62 0.58 1.55 1.58 1.18
0.24 0.14 1.12 0.63 0.65
3 3 3 3
284-294 272-280 275-284 275-286 276-286 281-291 277-286 278-286 278-284 278-286 278-286
6 6 6 6 6 6 6 6 6 6 6 66
0.41 1.37 0.72 0.26 0.56 0.36 1.12 0.40 0.32 0.21 0.42 0.56
0.41 1.37 0.72 0.26 0.56 0.36 1.12 0.53 0.42 0.38 0.45 0.60
3 3 3 3 3 3 3 3 3 3 3
Representation of Hydrate Phase Equilibria
Energy & Fuels, Vol. 14, No. 1, 2000 23
Table 1 (Continued) inhibitor (wt %) synthetic natural gasb pure water NaCl (10) KCl (10) CaCl2 (10) NaCl (2) + KCl (0.5) + CaCl2 (0.5) methanol (10) methanol (20) methanol (30) methanol (10) + NaCl (10) methanol (10) + KCl (10) methanol (10) + NaCl (2) + CaCl2 (0.5) methanol (20) + NaCl (2) + CaCl2 (0.5) methanol (20) + KCl (10) methanol (20) + NaCl (10) methanol (10) + CaCl2 (10) overall
T-range K 273-281 267-279 269-279 266-279 273-279 266-279 264-277 262-270 264-277 266-279 268-281 266-279 264-271 267-279 267-279
Np
method I AAD, K
method II AAD, K
source of data
8 8 6 7 4 7 7 6 7 6 6 7 4 7 7 96
0.42 0.25 0.37 0.29 0.71 1.07 0.47 0.62 0.36 0.42 0.78 0.89 0.64 1.82 0.34 0.62
0.42 0.25 0.37 0.29 0.71 1.07 0.47 0.62 0.48 0.47 0.90 0.51 0.24 0.15 0.98 0.53
4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
a Natural gas: 82% C , 0.5% CO , 11.3% C , 4.2% C , 0.9% iC , 0.6% nC , 0.1% iC , 0.2% nC , and 0.2% nC . b Synthetic natural gas: 1 2 2 3 4 4 5 5 6 97.25% C1, 1.42% C2, 1.08% C3, and 0.25% iC4.
Figure 4. Hydrate formation conditions of the 90% methane + 5% propane + 5% H2S mixture in aqueous solutions of methanol and electrolytes (experimental data: Bishinoi et al.3).
Figure 5. Hydrate formation conditions of the 78% methane + 2% propane + 20% CO2 mixture in aqueous solutions of methanol and electrolytes (experimental data: Bishinoi et al.3).
of the mixture in pure water, the effect of methanol and electrolytes on hydrate formation is well predicted. The comparison of the predictions by method I for the synthetic natural gas in aqueous solutions of methanol and electrolytes with the measured data is also presented in Figure 6. The good agreement between predicted and measured data is reached. It should be noted that experimental data on pure CO2 hydrates in 10 and 20 wt % methanol measured by Dholabhai et al.10 and by Ng and Robinson11 are inconsistent, which has been discussed by Dholabhai et al.10 The experimental pressures given by Dholabhai et al.10 are systematically lower than those reported by Ng and Robinson. The temperature difference (for a given pressure) between the data of Dholabhai et al. and of Ng and Robinson is 0.5 to 2.0 K. Dholabhai et al. have checked repeatability and reliability of their data. The excellent repeatability of their experiments let them
conclude their data are reliable. As mentioned in the introduction of this paper, the model developed by Zuo and Stenby1 can match pure CO2 hydrate data measured by Dholabhai et al.10 in the presence of methanol and electrolytes very well. However, the model cannot be used for hydrates of other pure gas and gas mixtures in aqueous solutions containing methanol and electrolytes. When the modified model, with either method I or method II, is applied to predict hydrate formation pressures of pure carbon dioxide in aqueous solutions containing methanol and electrolytes, predictions are systematically higher than experimental data of Dholabhai et al.10 The overall pressure deviations are 12% by method I and 20% by method II, respectively. While the modified model can match experimental data of Ng and Robinson very well. For pure CO2 hydrates in pure water and in aqueous electrolyte solutions, on the other hand, the modified model is actually the same as that of Zuo and Stenby which has good performance. The difference between the modified and the Zuo and Stenby models is the only way to determine interaction parameters between water and methanol, especially for method
(10) Dholabhai, P. D.; Parent, J. S.; Bishnoi, P. R. Ind. Eng. Chem. Res. 1996, 36, 819-823. (11) Ng, H.-J.; Robinson, D. B. Fluid Phase Equilib. 1985, 21, 145155.
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of other pure gas and gas mixtures at a given concentration of methanol and electrolytes. Thus, it might appear that there exist special interactions for pure CO2 hydrate in the presence of methanol and electrolytes. To this end, further investigation is required. Conclusion
Figure 6. Hydrate formation conditions of synthetic natural gas in aqueous solutions of methanol and electrolytes (experimental data: Mei et al.4).
I. In the original Zuo and Stenby model, the interaction parameters for the water-methanol pair were determined by matching experimental vapor-liquid equilibrium data of the binary water and methanol system. In this case, the calculated pure CO2 hydrate formation conditions in the presence of methanol are in good agreement with experimental data of Dholabhai et al. On the contrary, in the modified model, the interaction parameters for the water-methanol pair were determined by matching experimental methane hydrate data in the presence of methanol instead of vapor-liquid equilibrium data. The results calculated by the modified model are in good agreement with experimental data of Ng and Robinson11 and the modified model can perform well for hydrates of other pure gas and gas mixtures. Therefore, interaction parameters between water and methanol for pure CO2 hydrates are quite different from those for hydrates of other pure gas and gas mixtures. That is, water activity in the aqueous solutions containing methanol and electrolytes for pure CO2 hydrates is quite different from that for hydrates
The hydrate model proposed by Zuo and Stenby1 has been extended to predict incipient equilibria of gas hydrate formation conditions in aqueous solutions containing methanol and/or electrolytes. After the parameters determined from the hydrate formation data of methane in methanol up to 50 wt % (and those of the 80% methane + 20% carbon dioxide mixture), the method was then used to predict hydrate incipient equilibrium conditions in aqueous solutions containing electrolytes, and containing methanol and electrolytes. The results indicate that the new method is capable of accurately predicting hydrate formation conditions in aqueous solutions containing electrolytes, and containing methanol and electrolytes. The proposed methodology provides a valuable tool for the investigation of the gas hydrate formation conditions. Nomenclature aw AAD C fi hw N Np P R T W ∆ ∆a λ µ σ
activity of water average absolute deviation defined by: exp (1/Np)| Σj|Tcal j - Tj | Langmuir constant fugacity of component i molar enthalpy of water number of components number of data points pressure, Pa gas constant temperature, K interaction parameter property difference molar Helmholtz energy parameters defined in eqs 5 and 6 chemical potential diameter EF990110V
