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Modelling the hydrate formation condition for sour gas and mixtures
Chemical Engineering Science 60 (2005) 4879 – 4885 www.elsevier.com/locate/ces
Modelling the hydrate formation condition for sour gas and mixtures Chang-Yu Sun, Guang-Jin Chen∗ State Key Laboratory of Heavy Oil Processing, University of Petroleum, Beijing 102249, PR China Received 10 January 2005; received in revised form 28 March 2005; accepted 5 April 2005 Available online 25 May 2005
Abstract In order to improve the predicting ability of hydrate formation condition for sour gas contained systems, the dissolution of gas in water and hydrolytic reaction equilibrium existing in aqueous phase was taken into account and two concepts, i.e., the true compositions and apparent compositions of aqueous phase, were introduced. And then the approach to determine the true compositions including ionic components was given and the modified method to calculate the component fugacity of aqueous phase was developed by introducing Debye–Huckel electrostatic contribution term. The new method coupled with Chen–Guo model was successfully used to predict the thermodynamics property of hydrates for carbon dioxide and hydrogen sulfide pure gases, binary, and ternary sour gas mixtures in pure water systems. The calculation precision is superior to that of original Chen–Guo model and CSMHYD hydrate software. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Equilibrium; Hydrate; Hydrolytic reaction; Model; Sour gas; Thermodynamics
1. Introduction Gas hydrates are non-stoichiometric crystalline compounds in which individual guest molecules are caged inside a network of water molecules. It has been known that sour gases such as carbon dioxide (CO2 ) or hydrogen sulfide (H2 S) is a suitable guest molecule that can physically combine with water under the proper temperature and pressure to form gas hydrates. High H2 S and CO2 content sour natural gases are quite common in the world. One such natural gas field is located in Southwest China, where H2 S content up to 30 mol% has been found. As H2 S is capable of forming hydrates under rather low pressure and rather high temperature in the presence of water compared with other gases, such as methane, hydrate formation in these environments may easily occur and cause complications in extracting natural gas. But it is more difficult to measure the hydrate formation condition containing H2 S because of its toxicity and corrosive to the device. And such experimental data are scarce in ∗ Corresponding author. Tel./fax: +86 10 89733252.
E-mail address: [email protected] (G.-J. Chen). 0009-2509/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.04.013
the open literature. So it is important to develop the thermodynamics model to predict reliably the hydrate formation pressure or temperature in the sour gas containing system. Most of the existing thermodynamic models (Parrish and Prausnitz, 1972; Ng and Robinson, 1976a,b; John and Holder, 1985; John et al., 1985) for predicting hydrate formation are various modifications of the vdW-P model proposed by van der Waals and Platteeuw (1959). Chen and Guo (1996, 1998) developed a hydrate model based on the proposed two-step hydrate formation mechanism. These hydrate models assumed that the activity coefficient of water is unity (Sloan, 1998). However, the solubility of CO2 or H2 S is relatively high and CO2 or H2 S will also react with water. There exists hydrolytic reaction equilibrium in the solution, and the water activity coefficient cannot be assumed to be equal to unity. Therefore, the influence of hydrolytic reaction cannot be neglected in the aqueous phase under the hydrate formation conditions. In order to improve the predicting ability of hydrate formation condition for sour gas containing systems, the dissolution of sour gas in water and hydrolytic reaction equilibrium existing in aqueous phase were taken into account in this paper. Two concepts similar to those of
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Rivas and Prausnitz (1979), i.e., the true compositions and apparent compositions of aqueous phase, were introduced. And then the approach to determine the true compositions including ionic components was given and the modified method to calculate the component fugacity of aqueous phase was developed by introducing Debye–Huckel electrostatic contribution term. This method coupled with Chen–Guo model was used to predict the hydrate formation condition for sour gas and gas mixtures in pure water systems.
2. Thermodynamic framework In most hydrate models, the influence of dissolution of gas in water on hydrate formation condition was neglected. It is feasible for the unpolarized gases, such as methane, ethane, and nitrogen, as the solubility is very small. But it will result in considerable deviation for the sour gases, because of the solubility of sour gas in water is higher. In this paper, the hydrolytic reaction due to the presence of the sour gases dissolved in the water is considered. 2.1. Hydrolytic reaction equilibrium It is well known that gas molecules are involved in hydrolytic reaction equilibria during the dissolution of sour gas in water. For example, in the aqueous phase for the (CO2 + H2 S + H2 O) system, the following hydrolytic reaction equilibria should be taken into account: K1
+ CO2 + 2H2 O ←→ HCO− 3 + H3 O , K2
H2 S + H2 O ←→ HS− + H3 O+ .
(1) (2)
Second acid dissociation will take place: K3
2− + HCO− 3 + H2 O ←→ CO3 + H3 O , K4
HS− + H2 O ←→ S2− + H3 O+ ,
K1 K2
C1K
C2K
C3K
235.482 218.599
−12092.1 −12995.4
−36.7816 −33.5471
species and ionic species exist in the aqueous phase. The true compositions in the liquid phase can be obtained based on the hydrolytic reaction equilibrium and material balance. Take (H2 S + H2 O) system for example, if not considering the hydrolytic reaction equilibrium, there are two molecular species in the aqueous phase, H2 S and H2 O, and the apparent compositions are x1 , x2 , respectively. However, there exists two molecular species, H2 S and H2 O, and two ionic species, HS− and H3 O+ , in the equilibrium liquid phase after the hydrolytic reaction. From the material balance, the true compositions of the above four species can then be described as H2 S: z1 = x1 − x1 ,
(6)
H2 O: z2 = x2 − x1 ,
(7)
HS− : z3 = x1 ,
(8)
H3 O+ : z4 = x1 ,
(9)
where is the percent of reaction for sour gas that transforms into ionic species, which can be obtained from the Eq. (2). The true compositions and apparent compositions for CO2 and sour gas mixtures in water systems can also be obtained based on the above method. It should be pointed out that ions are assumed to be not volatile, and are not present in the gas phase. In addition, if no sour gas component exists in the gas phase, there are no hydrolytic reactions between solute and water. In that case, zi is the same as xi .
(3) 2.2. The calculation of the fugacity in the aqueous phase (4)
2− concentrations can be neglected because the CO2− 3 and S second dissociation constant is generally three to four orders of magnitude smaller than the first one (Li and Fürst, 2000). The equilibrium constants are assumed to have the following temperature dependence:
ln K = C1K + C2K /T + C3K ln T .
Table 1 The hydrolytic reaction equilibrium constant coefficients for CO2 and H2 S in water
(5)
Coefficients C1K through C3K for the equilibrium constants in Eqs. (1) and (2) are taken from the work of Edwards et al. (1978), which is listed in Table 1. In this paper, when not considering the hydrolytic reaction, the aqueous phase compositions are named as apparent compositions, xi . Correspondingly, the aqueous phase compositions are named as true compositions, zi , when taking the hydrolytic reaction into account. In this case molecular
There exists ionic species in the aqueous phase because of the hydrolytic reaction, then the fugacity coefficient of component i in the liquid phase consists of two terms: an equation of state term and a Debye–Huckel electrostatic term: ln i = ln EOS + ln DH . i i
(10)
The Patel–Teja equation of state (PT EOS) (Patel and Teja, 1982) is used to calculate the first term on the right-hand side of Eq. (10) and the fugacity coefficient of component i in the gas phase: P=
a[T ] RT − , v − b v(v + b) + c(v − b)
(11)
where a[T ] = a (RT c )2 /Pc (Tr ),
(12)
C.-Y. Sun, G.-J. Chen / Chemical Engineering Science 60 (2005) 4879 – 4885
b = b (RT c /Pc ),
(13)
c = c (RT c /Pc ).
(14)
For the (sour gas + water) system, the PT EOS parameters a, b, c for ionic species are determined similar to the method proposed by Zuo and Guo (1991) for salt-containing electrolyte solutions. For ionic species the parameters b and c are empirically expressed as b = 23 Na 3 ,
(15)
c = b,
(16)
where Na and represent Avogadro’s number and ionic diameter, respectively. The parameter a for ionic species is estimated from the following equation proposed by Hu et al. (1985): a = 2.57012Na2 3 fa ,
(17)
where fa is an empirical constant, which was set to a value of 6 (Zuo and Guo, 1991). The ionic energy parameter () is estimated from dispersion theory (Mavroyannis and Stephen, 1962):
/k = 2.2789 × 10−8 1/2 3/2 −6 ,
(18)
where and stand for the number of electrons in an ion and the polarizability of an ion respectively, and k is Boltzmann constant. Classical van der Waals (VDWs) mixing rules are used for parameter a, b, and c: aVDW = zi zj (ai aj )0.5 (1 − kij ), (19) i
b=
j
zi b i ,
(20)
z i ci .
(21)
i
c=
i
The interaction parameters kij in Eqs. (19) are to be determined from experimental binary phase equilibrium data. The second term on the right-hand side of Eq. (10) is a Debye–Huckel expression (Li and Pitzer, 1986): 1/2 1 + BDH IDH 2Ei2 DH ln i = − ADH ln √ BDH 1 + BDH / 2 1/2 2 3/2 IDH Ei − 2IDH + , (22) 1/2 1 + BDH IDH where IDH = 0.5
i
zi Ei2 ,
(23)
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3/2 1 2Na do 1/2 e2 , ADH = 3 Ms DkT
(24)
BDH = 2150(do /DT )1/2 ,
(25)
where D, e, E, MS , and d0 denote dielectric constant, electronic charge, number of charges, molecular weight of solvent, and solvent density, respectively. The Debye–Huckel term is neglected for the gas phase and it should be noticed that the true compositions of the species are used in the above equations instead of apparent compositions. 2.3. Phase equilibria of hydrates Based on the two-step hydrate formation mechanism proposed by Chen and Guo (1998), there are two kinds of equilibrium existing during the hydrate formation: the quasi-chemical reaction equilibrium for the formation of a stoichiometric basic hydrate, and the physical adsorption equilibrium for the filling of gas molecules in the linked cavities. For pure gas hydrates, using the constraint of chemical equilibrium and the Langmuir adsorption theory, the following thermodynamic relation can be obtained:
0B + 1 RT ln(1 − ) = W + 2 [ 0G (T ) + RT ln f ], (26) where W , 0G (T ), and 0B stand for the chemical potential of water, ideal gas state, and unfilled basic hydrate, respectively. 1 denotes the number of linked cavities (small cavities) per water molecule in the basic hydrate, and 2 denotes the number of gas molecules per water molecule in basic hydrate. represents the fraction of the linked cavities occupied by the gas molecules, which is calculated as follows:
=
Cf , 1 + Cf
(27)
where f denotes the fugacity of the gas species, and C is the Langmuir constant. In dealing with the hydrate formation of gas mixtures, the major equations involved are given as follows: H fi = xiB fi0 1 − j , (28) j
i
j =
j
j f j Cj , 1 + j fj C j
xiB = 1.0,
(29) (30)
where H = 1/3 for structure I and H = 2 for structure II hydrates; fi denotes the fugacity of hydrate former i in the gas phase or hydrocarbon-rich liquid phase; xiB stands for the mole fraction of basic hydrate component i formed by hydrate former i; j denotes the fraction of linked cavities occupied by gas component j; fi0 is the fugacity of hydrate
C.-Y. Sun, G.-J. Chen / Chemical Engineering Science 60 (2005) 4879 – 4885
former in equilibrium with the unfilled pure basic hydrate i, fi0 is formulated as P 0 0
2 fi = fT i exp −1/ , (31) w T − j Aij j Bi 0 , (32) fT i = exp Ai exp T T − Ci where w is the activity of water in the aqueous phase; = 0.4242 K bar −1 , 2 = 3/23 for structure I hydrates, and = 1.0224 K bar −1 , 2 = 1/17 for structure II hydrates. The evaluation of Antoine constants A i , Bi , Ci and binary interaction coefficients Aij are referred to Chen and Guo (1998). The Langmuir constant Cj in Eq. (29) is formulated as Yj Cj = Xj exp , (33) T − Zj where Xj , Yj and Zj are constants of component j, which had been reported by Chen and Guo (1998) for typical hydrate formers.
4.5 4.2 3.9 P (MPa)
4882
3.6 3.3 3.0 2.7 280.0 280.5 281.0 281.5 282.0 282.5 283.0 T (K)
Fig. 1. The hydrate formation condition for (CO2 + H2 O) system: () experimental data (Vlahakis et al., 1972); (—) this paper; (- - -) Chen and Guo (1998); (· · · · · · · · ·) CSMHYD software (Sloan, 1998).
2.1
3. Results and discussion
2.0
3.1. Pure gas hydrates We have used the proposed model to the prediction of the hydrate formation condition for (CO2 + H2 O) system. The experimental data are reported by references (Deaton and Frost, 1946; Larson, 1955; Robinson and Mehta, 1971; Vlahakis et al., 1972; Adisasmito et al., 1991), and the average absolute deviation for 112 data points using the proposed model, the original Chen–Guo model (1998), and CSMHYD software (Sloan, 1998) are 1.65%, 1.90%, and 2.18%, respectively. The comparison of the predicted hydrate formation conditions for CO2 (Vlahakis et al., 1972) based on the proposed model, the original Chen–Guo model, and CSMHYD software is depicted in Fig. 1. It can be seen that the predicting ability is improved after taking the hydrolytic reaction of CO2 into account.
1.9 P (MPa)
We have used the proposed method to calculate the solubility for pure sour gas and gas mixtures in water. The average absolute deviation for (CO2 + H2 O), (H2 S + H2 O), (CO2 +CH4 +H2 O) and (CO2 +N2 +H2 O) systems is about 8%, much lower than that of Zuo and Guo model (1991). In order to examine the influence of hydrolytic reaction on the hydrate formation calculation, the van der Waals mixing rule (Eqs. (19)–(21)) is used. And the interaction parameters kij for gas–water are set to zero, which is accord with the original hydrate model proposed by Chen and Guo (1998). Therefore, no additional parameter is introduced to the proposed method, which is still a predictive model of calculating the hydrate formation condition.
1.8 1.7 1.6 1.5 1.4 298.5
299.0
299.5
300.0
300.5
301.0
T (K) Fig. 2. The hydrate formation condition for (H2 S + H2 O) system: () experimental data (Carroll and Mather, 1991); (—) this paper (van der Waals mixing rule); (– .– .– .) this paper (Kurihara mixing rule); (- - -) Chen and Guo (1998); (· · · · · · · · ·) CSMHYD software (Sloan, 1998).
Calculated with the proposed model, the original Chen–Guo model, and CSMHYD software, the average absolute deviation for the hydrate formation condition data of (H2 S + H2 O) system (Selleck et al., 1952; Carroll and Mather, 1991) are 4.74%, 6.27%, and 6.81%, respectively. The prediction results for experimental data of Carroll and Mather (1991) are shown in Fig. 2. In the proposed model, van der Waals mixing rule is used and the kij for gas–water is set to zero. To describe the phase equilibria of hydrocarbon-water highly non-ideal systems, the mixing rule proposed by Kurihara et al. (1987) can be introduced to EOS parameter a, from which the calculation of the sour
C.-Y. Sun, G.-J. Chen / Chemical Engineering Science 60 (2005) 4879 – 4885
4883
4.0
4.0 3.5
3.0
3.0
2.5
2.5
P (MPa)
P (MPa)
3.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5 274
276
278
280
282
284
T (K)
0.5
0.4
0.5
0.6
0.7
0.8
0.9
1.0
y1
Fig. 3. The hydrate formation condition for (CO2 + C2 H6 + H2 O) system: () y1 = 0.19–0.26; (◦) y1 = 0.60–0.64; () y1 = 0.81–0.84; () y1 = 0.92–0.93; () y1 = 0.96–0.97 (Adisasmito and Sloan, 1992); (—) this paper; (- - -) Chen and Guo (1998).
Fig. 4. Variation of hydrate formation pressure with CO2 concentration for (CO2 + C3 H8 + H2 O) system: () 282.0 K; () 278.2 K (Adisasmito and Sloan, 1992); (—) this paper; (- - -) Chen and Guo (1998); (· · · · · · · · ·) CSMHYD software (Sloan, 1998).
gas solubility can be significantly improved. Similarly, the Kurihara mixing rules can be used to improve the describing of hydrate formation conditions. Fig. 2 also shows the calculated results after the Kurihara mixing rules are adopted, where the interaction parameter for the Kurihara mixing rules is regressed from the solubility data of (H2 S + H2 O) system. It can be seen that the precision can be improved for H2 S hydrate formation condition. But since a new interaction parameter is introduced, the predicting ability of the hydrate model is decreased. Therefore, in this paper, the van der Waals mixing rule is used and the interaction parameters kij for gas–water is set to zero to keep the predicting ability of the original Chen–Guo hydrate model.
It can be seen that when the concentration of CO2 is lower than 60%, the prediction results using the proposed work are similar to those of the original Chen–Guo model. With the increase of CO2 concentration, the solubility of CO2 in the water and the influence of hydrolytic reaction increase. Therefore the prediction precision using the proposed method is improved when the concentration of CO2 is higher than 60% compared with the original Chen–Guo model. The variation of the hydrate formation pressure with the concentration of CO2 for (CO2 + C3 H8 + H2 O) system (Adisasmito and Sloan, 1992) at 278.2 and 282.0 K are shown in Fig. 4. It can also be seen that the proposed method is superior to the original Chen–Guo model and CSMHYD model when the concentration of CO2 is high. When the CO2 is mixed with a small amount of propane (e.g. < 5%), the hydrate structure is I type. But the hydrate structure will change from I to II type with the continuous increase of propane concentration. So there is an inflexion in Fig. 4 when the CO2 composition is about 0.95 and the hydrate formation pressure is no longer linear at this region. Fig. 5 shows the predicted results for the hydrate formation P–T plots of 5 ternary (CH4 +CO2 +H2 S) gas mixtures in the presence of pure water (Sun et al., 2003). The composition of H2 S for the five systems is from 0.0678 to 0.2662. For comparison, the results based on the original Chen–Guo model and the CSMHYD software are also shown in Fig. 5. It can be found that the proposed method gives better predictions although the calculation deviation increases with increasing H2 S concentration in the gas mixture. For systems that the composition of H2 S is 0.1771 and 0.2662, it can be seen that with the decrease of temperature, all three models give poor prediction precision. The experimental measurement of hydrate formation condition may be influenced
3.2. Gas mixture hydrates The predicted results for 10 binary and ternary sour gas containing hydrate systems (Unruh and Katz, 1949; Noaker and Katz, 1954; Adisasmito and Sloan, 1992; Sun et al., 2003) using the proposed model are listed in Table 2. For comparison, the parallel calculation results based on the original Chen–Guo model, and CSMHYD software are also given in Table 2. The comparison shows that the model proposed in this work gives smaller AADPs in most cases. The average absolute deviation of calculated formation pressures (AADP) for 201 data points using the proposed model, the original Chen–Guo model, and CSMHYD software are 5.3%, 6.6%, and 9.8%, respectively. In the proposed model, although no additional parameters are introduced, the predicting ability is improved by considering the effect of hydrolytic reaction equilibria. The hydrate formation condition for (CO2 +C2 H6 +H2 O) system (Adisasmito and Sloan, 1992) is depicted in Fig. 3.
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Table 2 The comparison of hydrate formation conditions by three models for sour gas mixtures in water y1
Gas mixture
H2 S + CH4 CO2 + CH4 CO2 + C2 H6 CO2 + C3 H8 CO2 + nC4 H10 H2 S + CO2 + CH4 H2 S + CO2 + CH4 H2 S + CO2 + CH4 H2 S + CO2 + CH4 H2 S + CO2 + CH4
Temperature (K)
0.01–0.22 0.055–0.71 0.189–0.967 0.099–0.987 0.93–0.994 0.0678 0.0993 0.1498 0.1771 0.2662
276.5–295.4 275.5–285.7 273.5–287.8 273.7–282.0 273.7–278.2 276.2–291.2 278.2–293.2 277.2–295.7 282.2–297.2 281.2–299.7
No.
20 17 40 55 19 12 9 9 11 9
AADP%
Data source
M1
M2
M3
7.99 2.38 2.57 5.84 4.27 4.94 2.48 5.55 7.62 15.25
10.8 3.47 3.78 6.83 4.64 5.81 4.48 6.81 9.96 17.60
6.45 2.61 6.93 13.78 — 9.36 6.98 11.04 12.50 19.02
a b c c c d d d d d
M1: this paper; M2: Chen and Guo (1998); M3: CSMHYD software (1998). a: Noaker and Katz (1954); b: Unruh and Katz (1949); c: Adisasmito and Sloan (1992); d: Sun et al. (2003).
and Guo. Using the proposed method, the predicted results of hydrate formation condition for CO2 and H2 S pure gases is 1.65% and 4.74%, respectively, which is superior to that of the original Chen–Guo model and the CSMHYD hydrate software. When the proposed method is used to the binary, and ternary sour gas mixtures, the average absolute deviation of calculated formation pressures for 201 data points is 5.3%, giving lower error compared with the original Chen–Guo model and the CSMHYD hydrate software.
9 8 7
P (MPa)
6 5 4 3 2
Acknowledgments
1 276
279
282
285
288 T (K)
291
294
297
300
Fig. 5. The hydrate formation condition for (H2 S + CO2 + CH4 ) in water at different H2 S concentration: () y1 = 0.2662; (◦) y1 = 0.1771; () y1 = 0.1498; () y1 = 0.0993; () y1 = 0.0678 (Sun et al., 2003); (—) this paper; (- - -) Chen and Guo (1998); (· · · · · · · · ·) CSMHYD software (Sloan, 1998).
by the higher solubility of H2 S in water at the higher H2 S concentration.
4. Conclusions The dissolution and hydrolytic reaction of sour gas component in water cannot be ignored in the predicting of hydrate formation condition. The true compositions and apparent compositions of aqueous phase are introduced to describe the influence of hydrolytic reaction. Debye–Huckel electrostatic contribution term coupled with PT equation of state is used to calculate the component fugacity of aqueous phase. The phase equilibria of hydrates is built based on the two-step hydrate formation mechanism proposed by Chen
Financial support received from CNPC Innovation Foundation (04E7028), the Key Project of Chinese Ministry of Education (No: 105107), and the National Natural Science Foundation of China (Grant nos: 20490207) is gratefully acknowledged.
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Modelling the hydrate formation condition for sour gas and mixtures Chang-Yu Sun, Guang-Jin Chen∗ State Key Laboratory of Heavy Oil Processing, University of Petroleum, Beijing 102249, PR China Received 10 January 2005; received in revised form 28 March 2005; accepted 5 April 2005 Available online 25 May 2005
Abstract In order to improve the predicting ability of hydrate formation condition for sour gas contained systems, the dissolution of gas in water and hydrolytic reaction equilibrium existing in aqueous phase was taken into account and two concepts, i.e., the true compositions and apparent compositions of aqueous phase, were introduced. And then the approach to determine the true compositions including ionic components was given and the modified method to calculate the component fugacity of aqueous phase was developed by introducing Debye–Huckel electrostatic contribution term. The new method coupled with Chen–Guo model was successfully used to predict the thermodynamics property of hydrates for carbon dioxide and hydrogen sulfide pure gases, binary, and ternary sour gas mixtures in pure water systems. The calculation precision is superior to that of original Chen–Guo model and CSMHYD hydrate software. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Equilibrium; Hydrate; Hydrolytic reaction; Model; Sour gas; Thermodynamics
1. Introduction Gas hydrates are non-stoichiometric crystalline compounds in which individual guest molecules are caged inside a network of water molecules. It has been known that sour gases such as carbon dioxide (CO2 ) or hydrogen sulfide (H2 S) is a suitable guest molecule that can physically combine with water under the proper temperature and pressure to form gas hydrates. High H2 S and CO2 content sour natural gases are quite common in the world. One such natural gas field is located in Southwest China, where H2 S content up to 30 mol% has been found. As H2 S is capable of forming hydrates under rather low pressure and rather high temperature in the presence of water compared with other gases, such as methane, hydrate formation in these environments may easily occur and cause complications in extracting natural gas. But it is more difficult to measure the hydrate formation condition containing H2 S because of its toxicity and corrosive to the device. And such experimental data are scarce in ∗ Corresponding author. Tel./fax: +86 10 89733252.
E-mail address: [email protected] (G.-J. Chen). 0009-2509/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2005.04.013
the open literature. So it is important to develop the thermodynamics model to predict reliably the hydrate formation pressure or temperature in the sour gas containing system. Most of the existing thermodynamic models (Parrish and Prausnitz, 1972; Ng and Robinson, 1976a,b; John and Holder, 1985; John et al., 1985) for predicting hydrate formation are various modifications of the vdW-P model proposed by van der Waals and Platteeuw (1959). Chen and Guo (1996, 1998) developed a hydrate model based on the proposed two-step hydrate formation mechanism. These hydrate models assumed that the activity coefficient of water is unity (Sloan, 1998). However, the solubility of CO2 or H2 S is relatively high and CO2 or H2 S will also react with water. There exists hydrolytic reaction equilibrium in the solution, and the water activity coefficient cannot be assumed to be equal to unity. Therefore, the influence of hydrolytic reaction cannot be neglected in the aqueous phase under the hydrate formation conditions. In order to improve the predicting ability of hydrate formation condition for sour gas containing systems, the dissolution of sour gas in water and hydrolytic reaction equilibrium existing in aqueous phase were taken into account in this paper. Two concepts similar to those of
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Rivas and Prausnitz (1979), i.e., the true compositions and apparent compositions of aqueous phase, were introduced. And then the approach to determine the true compositions including ionic components was given and the modified method to calculate the component fugacity of aqueous phase was developed by introducing Debye–Huckel electrostatic contribution term. This method coupled with Chen–Guo model was used to predict the hydrate formation condition for sour gas and gas mixtures in pure water systems.
2. Thermodynamic framework In most hydrate models, the influence of dissolution of gas in water on hydrate formation condition was neglected. It is feasible for the unpolarized gases, such as methane, ethane, and nitrogen, as the solubility is very small. But it will result in considerable deviation for the sour gases, because of the solubility of sour gas in water is higher. In this paper, the hydrolytic reaction due to the presence of the sour gases dissolved in the water is considered. 2.1. Hydrolytic reaction equilibrium It is well known that gas molecules are involved in hydrolytic reaction equilibria during the dissolution of sour gas in water. For example, in the aqueous phase for the (CO2 + H2 S + H2 O) system, the following hydrolytic reaction equilibria should be taken into account: K1
+ CO2 + 2H2 O ←→ HCO− 3 + H3 O , K2
H2 S + H2 O ←→ HS− + H3 O+ .
(1) (2)
Second acid dissociation will take place: K3
2− + HCO− 3 + H2 O ←→ CO3 + H3 O , K4
HS− + H2 O ←→ S2− + H3 O+ ,
K1 K2
C1K
C2K
C3K
235.482 218.599
−12092.1 −12995.4
−36.7816 −33.5471
species and ionic species exist in the aqueous phase. The true compositions in the liquid phase can be obtained based on the hydrolytic reaction equilibrium and material balance. Take (H2 S + H2 O) system for example, if not considering the hydrolytic reaction equilibrium, there are two molecular species in the aqueous phase, H2 S and H2 O, and the apparent compositions are x1 , x2 , respectively. However, there exists two molecular species, H2 S and H2 O, and two ionic species, HS− and H3 O+ , in the equilibrium liquid phase after the hydrolytic reaction. From the material balance, the true compositions of the above four species can then be described as H2 S: z1 = x1 − x1 ,
(6)
H2 O: z2 = x2 − x1 ,
(7)
HS− : z3 = x1 ,
(8)
H3 O+ : z4 = x1 ,
(9)
where is the percent of reaction for sour gas that transforms into ionic species, which can be obtained from the Eq. (2). The true compositions and apparent compositions for CO2 and sour gas mixtures in water systems can also be obtained based on the above method. It should be pointed out that ions are assumed to be not volatile, and are not present in the gas phase. In addition, if no sour gas component exists in the gas phase, there are no hydrolytic reactions between solute and water. In that case, zi is the same as xi .
(3) 2.2. The calculation of the fugacity in the aqueous phase (4)
2− concentrations can be neglected because the CO2− 3 and S second dissociation constant is generally three to four orders of magnitude smaller than the first one (Li and Fürst, 2000). The equilibrium constants are assumed to have the following temperature dependence:
ln K = C1K + C2K /T + C3K ln T .
Table 1 The hydrolytic reaction equilibrium constant coefficients for CO2 and H2 S in water
(5)
Coefficients C1K through C3K for the equilibrium constants in Eqs. (1) and (2) are taken from the work of Edwards et al. (1978), which is listed in Table 1. In this paper, when not considering the hydrolytic reaction, the aqueous phase compositions are named as apparent compositions, xi . Correspondingly, the aqueous phase compositions are named as true compositions, zi , when taking the hydrolytic reaction into account. In this case molecular
There exists ionic species in the aqueous phase because of the hydrolytic reaction, then the fugacity coefficient of component i in the liquid phase consists of two terms: an equation of state term and a Debye–Huckel electrostatic term: ln i = ln EOS + ln DH . i i
(10)
The Patel–Teja equation of state (PT EOS) (Patel and Teja, 1982) is used to calculate the first term on the right-hand side of Eq. (10) and the fugacity coefficient of component i in the gas phase: P=
a[T ] RT − , v − b v(v + b) + c(v − b)
(11)
where a[T ] = a (RT c )2 /Pc (Tr ),
(12)
C.-Y. Sun, G.-J. Chen / Chemical Engineering Science 60 (2005) 4879 – 4885
b = b (RT c /Pc ),
(13)
c = c (RT c /Pc ).
(14)
For the (sour gas + water) system, the PT EOS parameters a, b, c for ionic species are determined similar to the method proposed by Zuo and Guo (1991) for salt-containing electrolyte solutions. For ionic species the parameters b and c are empirically expressed as b = 23 Na 3 ,
(15)
c = b,
(16)
where Na and represent Avogadro’s number and ionic diameter, respectively. The parameter a for ionic species is estimated from the following equation proposed by Hu et al. (1985): a = 2.57012Na2 3 fa ,
(17)
where fa is an empirical constant, which was set to a value of 6 (Zuo and Guo, 1991). The ionic energy parameter () is estimated from dispersion theory (Mavroyannis and Stephen, 1962):
/k = 2.2789 × 10−8 1/2 3/2 −6 ,
(18)
where and stand for the number of electrons in an ion and the polarizability of an ion respectively, and k is Boltzmann constant. Classical van der Waals (VDWs) mixing rules are used for parameter a, b, and c: aVDW = zi zj (ai aj )0.5 (1 − kij ), (19) i
b=
j
zi b i ,
(20)
z i ci .
(21)
i
c=
i
The interaction parameters kij in Eqs. (19) are to be determined from experimental binary phase equilibrium data. The second term on the right-hand side of Eq. (10) is a Debye–Huckel expression (Li and Pitzer, 1986): 1/2 1 + BDH IDH 2Ei2 DH ln i = − ADH ln √ BDH 1 + BDH / 2 1/2 2 3/2 IDH Ei − 2IDH + , (22) 1/2 1 + BDH IDH where IDH = 0.5
i
zi Ei2 ,
(23)
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3/2 1 2Na do 1/2 e2 , ADH = 3 Ms DkT
(24)
BDH = 2150(do /DT )1/2 ,
(25)
where D, e, E, MS , and d0 denote dielectric constant, electronic charge, number of charges, molecular weight of solvent, and solvent density, respectively. The Debye–Huckel term is neglected for the gas phase and it should be noticed that the true compositions of the species are used in the above equations instead of apparent compositions. 2.3. Phase equilibria of hydrates Based on the two-step hydrate formation mechanism proposed by Chen and Guo (1998), there are two kinds of equilibrium existing during the hydrate formation: the quasi-chemical reaction equilibrium for the formation of a stoichiometric basic hydrate, and the physical adsorption equilibrium for the filling of gas molecules in the linked cavities. For pure gas hydrates, using the constraint of chemical equilibrium and the Langmuir adsorption theory, the following thermodynamic relation can be obtained:
0B + 1 RT ln(1 − ) = W + 2 [ 0G (T ) + RT ln f ], (26) where W , 0G (T ), and 0B stand for the chemical potential of water, ideal gas state, and unfilled basic hydrate, respectively. 1 denotes the number of linked cavities (small cavities) per water molecule in the basic hydrate, and 2 denotes the number of gas molecules per water molecule in basic hydrate. represents the fraction of the linked cavities occupied by the gas molecules, which is calculated as follows:
=
Cf , 1 + Cf
(27)
where f denotes the fugacity of the gas species, and C is the Langmuir constant. In dealing with the hydrate formation of gas mixtures, the major equations involved are given as follows: H fi = xiB fi0 1 − j , (28) j
i
j =
j
j f j Cj , 1 + j fj C j
xiB = 1.0,
(29) (30)
where H = 1/3 for structure I and H = 2 for structure II hydrates; fi denotes the fugacity of hydrate former i in the gas phase or hydrocarbon-rich liquid phase; xiB stands for the mole fraction of basic hydrate component i formed by hydrate former i; j denotes the fraction of linked cavities occupied by gas component j; fi0 is the fugacity of hydrate
C.-Y. Sun, G.-J. Chen / Chemical Engineering Science 60 (2005) 4879 – 4885
former in equilibrium with the unfilled pure basic hydrate i, fi0 is formulated as P 0 0
2 fi = fT i exp −1/ , (31) w T − j Aij j Bi 0 , (32) fT i = exp Ai exp T T − Ci where w is the activity of water in the aqueous phase; = 0.4242 K bar −1 , 2 = 3/23 for structure I hydrates, and = 1.0224 K bar −1 , 2 = 1/17 for structure II hydrates. The evaluation of Antoine constants A i , Bi , Ci and binary interaction coefficients Aij are referred to Chen and Guo (1998). The Langmuir constant Cj in Eq. (29) is formulated as Yj Cj = Xj exp , (33) T − Zj where Xj , Yj and Zj are constants of component j, which had been reported by Chen and Guo (1998) for typical hydrate formers.
4.5 4.2 3.9 P (MPa)
4882
3.6 3.3 3.0 2.7 280.0 280.5 281.0 281.5 282.0 282.5 283.0 T (K)
Fig. 1. The hydrate formation condition for (CO2 + H2 O) system: () experimental data (Vlahakis et al., 1972); (—) this paper; (- - -) Chen and Guo (1998); (· · · · · · · · ·) CSMHYD software (Sloan, 1998).
2.1
3. Results and discussion
2.0
3.1. Pure gas hydrates We have used the proposed model to the prediction of the hydrate formation condition for (CO2 + H2 O) system. The experimental data are reported by references (Deaton and Frost, 1946; Larson, 1955; Robinson and Mehta, 1971; Vlahakis et al., 1972; Adisasmito et al., 1991), and the average absolute deviation for 112 data points using the proposed model, the original Chen–Guo model (1998), and CSMHYD software (Sloan, 1998) are 1.65%, 1.90%, and 2.18%, respectively. The comparison of the predicted hydrate formation conditions for CO2 (Vlahakis et al., 1972) based on the proposed model, the original Chen–Guo model, and CSMHYD software is depicted in Fig. 1. It can be seen that the predicting ability is improved after taking the hydrolytic reaction of CO2 into account.
1.9 P (MPa)
We have used the proposed method to calculate the solubility for pure sour gas and gas mixtures in water. The average absolute deviation for (CO2 + H2 O), (H2 S + H2 O), (CO2 +CH4 +H2 O) and (CO2 +N2 +H2 O) systems is about 8%, much lower than that of Zuo and Guo model (1991). In order to examine the influence of hydrolytic reaction on the hydrate formation calculation, the van der Waals mixing rule (Eqs. (19)–(21)) is used. And the interaction parameters kij for gas–water are set to zero, which is accord with the original hydrate model proposed by Chen and Guo (1998). Therefore, no additional parameter is introduced to the proposed method, which is still a predictive model of calculating the hydrate formation condition.
1.8 1.7 1.6 1.5 1.4 298.5
299.0
299.5
300.0
300.5
301.0
T (K) Fig. 2. The hydrate formation condition for (H2 S + H2 O) system: () experimental data (Carroll and Mather, 1991); (—) this paper (van der Waals mixing rule); (– .– .– .) this paper (Kurihara mixing rule); (- - -) Chen and Guo (1998); (· · · · · · · · ·) CSMHYD software (Sloan, 1998).
Calculated with the proposed model, the original Chen–Guo model, and CSMHYD software, the average absolute deviation for the hydrate formation condition data of (H2 S + H2 O) system (Selleck et al., 1952; Carroll and Mather, 1991) are 4.74%, 6.27%, and 6.81%, respectively. The prediction results for experimental data of Carroll and Mather (1991) are shown in Fig. 2. In the proposed model, van der Waals mixing rule is used and the kij for gas–water is set to zero. To describe the phase equilibria of hydrocarbon-water highly non-ideal systems, the mixing rule proposed by Kurihara et al. (1987) can be introduced to EOS parameter a, from which the calculation of the sour
C.-Y. Sun, G.-J. Chen / Chemical Engineering Science 60 (2005) 4879 – 4885
4883
4.0
4.0 3.5
3.0
3.0
2.5
2.5
P (MPa)
P (MPa)
3.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5 274
276
278
280
282
284
T (K)
0.5
0.4
0.5
0.6
0.7
0.8
0.9
1.0
y1
Fig. 3. The hydrate formation condition for (CO2 + C2 H6 + H2 O) system: () y1 = 0.19–0.26; (◦) y1 = 0.60–0.64; () y1 = 0.81–0.84; () y1 = 0.92–0.93; () y1 = 0.96–0.97 (Adisasmito and Sloan, 1992); (—) this paper; (- - -) Chen and Guo (1998).
Fig. 4. Variation of hydrate formation pressure with CO2 concentration for (CO2 + C3 H8 + H2 O) system: () 282.0 K; () 278.2 K (Adisasmito and Sloan, 1992); (—) this paper; (- - -) Chen and Guo (1998); (· · · · · · · · ·) CSMHYD software (Sloan, 1998).
gas solubility can be significantly improved. Similarly, the Kurihara mixing rules can be used to improve the describing of hydrate formation conditions. Fig. 2 also shows the calculated results after the Kurihara mixing rules are adopted, where the interaction parameter for the Kurihara mixing rules is regressed from the solubility data of (H2 S + H2 O) system. It can be seen that the precision can be improved for H2 S hydrate formation condition. But since a new interaction parameter is introduced, the predicting ability of the hydrate model is decreased. Therefore, in this paper, the van der Waals mixing rule is used and the interaction parameters kij for gas–water is set to zero to keep the predicting ability of the original Chen–Guo hydrate model.
It can be seen that when the concentration of CO2 is lower than 60%, the prediction results using the proposed work are similar to those of the original Chen–Guo model. With the increase of CO2 concentration, the solubility of CO2 in the water and the influence of hydrolytic reaction increase. Therefore the prediction precision using the proposed method is improved when the concentration of CO2 is higher than 60% compared with the original Chen–Guo model. The variation of the hydrate formation pressure with the concentration of CO2 for (CO2 + C3 H8 + H2 O) system (Adisasmito and Sloan, 1992) at 278.2 and 282.0 K are shown in Fig. 4. It can also be seen that the proposed method is superior to the original Chen–Guo model and CSMHYD model when the concentration of CO2 is high. When the CO2 is mixed with a small amount of propane (e.g. < 5%), the hydrate structure is I type. But the hydrate structure will change from I to II type with the continuous increase of propane concentration. So there is an inflexion in Fig. 4 when the CO2 composition is about 0.95 and the hydrate formation pressure is no longer linear at this region. Fig. 5 shows the predicted results for the hydrate formation P–T plots of 5 ternary (CH4 +CO2 +H2 S) gas mixtures in the presence of pure water (Sun et al., 2003). The composition of H2 S for the five systems is from 0.0678 to 0.2662. For comparison, the results based on the original Chen–Guo model and the CSMHYD software are also shown in Fig. 5. It can be found that the proposed method gives better predictions although the calculation deviation increases with increasing H2 S concentration in the gas mixture. For systems that the composition of H2 S is 0.1771 and 0.2662, it can be seen that with the decrease of temperature, all three models give poor prediction precision. The experimental measurement of hydrate formation condition may be influenced
3.2. Gas mixture hydrates The predicted results for 10 binary and ternary sour gas containing hydrate systems (Unruh and Katz, 1949; Noaker and Katz, 1954; Adisasmito and Sloan, 1992; Sun et al., 2003) using the proposed model are listed in Table 2. For comparison, the parallel calculation results based on the original Chen–Guo model, and CSMHYD software are also given in Table 2. The comparison shows that the model proposed in this work gives smaller AADPs in most cases. The average absolute deviation of calculated formation pressures (AADP) for 201 data points using the proposed model, the original Chen–Guo model, and CSMHYD software are 5.3%, 6.6%, and 9.8%, respectively. In the proposed model, although no additional parameters are introduced, the predicting ability is improved by considering the effect of hydrolytic reaction equilibria. The hydrate formation condition for (CO2 +C2 H6 +H2 O) system (Adisasmito and Sloan, 1992) is depicted in Fig. 3.
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Table 2 The comparison of hydrate formation conditions by three models for sour gas mixtures in water y1
Gas mixture
H2 S + CH4 CO2 + CH4 CO2 + C2 H6 CO2 + C3 H8 CO2 + nC4 H10 H2 S + CO2 + CH4 H2 S + CO2 + CH4 H2 S + CO2 + CH4 H2 S + CO2 + CH4 H2 S + CO2 + CH4
Temperature (K)
0.01–0.22 0.055–0.71 0.189–0.967 0.099–0.987 0.93–0.994 0.0678 0.0993 0.1498 0.1771 0.2662
276.5–295.4 275.5–285.7 273.5–287.8 273.7–282.0 273.7–278.2 276.2–291.2 278.2–293.2 277.2–295.7 282.2–297.2 281.2–299.7
No.
20 17 40 55 19 12 9 9 11 9
AADP%
Data source
M1
M2
M3
7.99 2.38 2.57 5.84 4.27 4.94 2.48 5.55 7.62 15.25
10.8 3.47 3.78 6.83 4.64 5.81 4.48 6.81 9.96 17.60
6.45 2.61 6.93 13.78 — 9.36 6.98 11.04 12.50 19.02
a b c c c d d d d d
M1: this paper; M2: Chen and Guo (1998); M3: CSMHYD software (1998). a: Noaker and Katz (1954); b: Unruh and Katz (1949); c: Adisasmito and Sloan (1992); d: Sun et al. (2003).
and Guo. Using the proposed method, the predicted results of hydrate formation condition for CO2 and H2 S pure gases is 1.65% and 4.74%, respectively, which is superior to that of the original Chen–Guo model and the CSMHYD hydrate software. When the proposed method is used to the binary, and ternary sour gas mixtures, the average absolute deviation of calculated formation pressures for 201 data points is 5.3%, giving lower error compared with the original Chen–Guo model and the CSMHYD hydrate software.
9 8 7
P (MPa)
6 5 4 3 2
Acknowledgments
1 276
279
282
285
288 T (K)
291
294
297
300
Fig. 5. The hydrate formation condition for (H2 S + CO2 + CH4 ) in water at different H2 S concentration: () y1 = 0.2662; (◦) y1 = 0.1771; () y1 = 0.1498; () y1 = 0.0993; () y1 = 0.0678 (Sun et al., 2003); (—) this paper; (- - -) Chen and Guo (1998); (· · · · · · · · ·) CSMHYD software (Sloan, 1998).
by the higher solubility of H2 S in water at the higher H2 S concentration.
4. Conclusions The dissolution and hydrolytic reaction of sour gas component in water cannot be ignored in the predicting of hydrate formation condition. The true compositions and apparent compositions of aqueous phase are introduced to describe the influence of hydrolytic reaction. Debye–Huckel electrostatic contribution term coupled with PT equation of state is used to calculate the component fugacity of aqueous phase. The phase equilibria of hydrates is built based on the two-step hydrate formation mechanism proposed by Chen
Financial support received from CNPC Innovation Foundation (04E7028), the Key Project of Chinese Ministry of Education (No: 105107), and the National Natural Science Foundation of China (Grant nos: 20490207) is gratefully acknowledged.
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