Ind. Eng. Chem. Res. 2006, 45, 2131-2137
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Prediction of Gas Hydrate Formation Conditions in the Presence of Methanol, Glycerol, Ethylene Glycol, and Triethylene Glycol with the Statistical Associating Fluid Theory Equation of State Xiao-Sen Li,† Hui-Jie Wu,‡ and Peter Englezos*,‡ Guangzhou Institute of Energy ConVersion, The Chinese Academy of Sciences, Nengyuan Road, Wushan, Tianhe District, Guangzhou, People’s Republic of China 510640, and Department of Chemical and Biological Engineering, The UniVersity of British Columbia, 2360 East Mall, VancouVer, British Columbia, Canada V6T 1Z3
The statistical associating fluid theory (SAFT) equation of state is employed for the prediction of the thermodynamic inhibiting effect of methanol, glycerol, ethylene glycol, and triethylene glycol on gas hydrate formation. The results were found to be in satisfactory to excellent agreement with experimental data. The SAFT equation takes into account hard-sphere repulsion, hard chain formation, dispersion, and association. This enables this model to be able to correlate and predict successfully systems containing water, alcohols, and hydrocarbons for which traditional cubic equations of state fail in general. 1. Introduction Knowledge of the equilibrium hydrate-forming conditions is necessary for the rational and economic design of processes in the chemical, oil, gas, and other industries where hydrate formation is encountered. The practice of prevention of the hydrate formation with the use of methanol and various glycols is called thermodynamic inhibition. Hence, it is important to have available reliable methods for calculating the impact of the addition of these chemicals into the aqueous phase on the equilibrium hydrate formation conditions (inhibiting effect). Hammerschmidt developed the first method used in the industry for predicting the inhibiting effect of methanol.1 The method is empirical, and the reliability of the calculations is variable.2 Anderson and Prausnitz presented a thermodynamicsbased method for calculating the inhibiting effects of methanol.3 They used the van der Waals-Platteuw model for the solid hydrate phase, Redich-Kwong equation of state for the vapor phase, and the UNIQUAC model for the liquid phase. Henry’s constants were used for calculating the fugacities of components in their supercritical state in the liquid phase. Furthermore, empirical correlations were used for calculating the molar volumes, partial molar volumes at infinite dilution, and fugacity of hypothetical liquid water below the ice-point temperature. Robinson and Ng presented a computer program for the calculation of the depression of hydrate formation temperatures due to methanol.4 A computational method based on the Trebble-Bishnoi equation of state was presented by Englezos et al. for calculating the depression effects of methanol and the amounts of methanol required.5,6 All equilibrium hydrate prediction methods use either an equation of state for all fluid phases or an equation of state for the vapor phase and activity coefficient models for the liquid one.7 Cubic equations do not take into account association interactions, and, hence, they are expected to perform poorly in estimating the inhibiting effects of glycols. On the other hand, * To whom correspondence should be addressed. Tel.: (604) 8226184. Fax: (604) 822-6003. E-mail:
[email protected]. † The Chinese Academy of Sciences. ‡ The University of British Columbia.
the statistical associating fluid theory (SAFT) is based on Wertheim’s first-order thermodynamic perturbation theory for associating fluids and has been developed very rapidly in recent years.8,9 Molecular-based equations of state are generally more reliable than empirical models for extrapolation and prediction. Consequently, SAFT has been used to model successfully a wide variety of the thermodynamic properties and phase equilibria for industrially important fluids containing n-alkane mixtures and alcohol aqueous solutions.9-11 It should be noted that recent progress in hydrate equilibrium prediction methods focused on improving the hydrate model.12,13 In this work we focus on taking advantage of progress in modeling the fluid phase behavior. In particular the SAFT equation of state was successfully used to predict the phase equilibria of several water/alcohol/alcohol, water/alcohol/ hydrocarbon, water/alcohol/CO2 ternary systems and their constituent binaries.9,14,15 The predictive success of the SAFT model with the ternary systems motivated the present study where we propose employing SAFT in conjunction with the van der Waals-Platteeuw model for the prediction of the thermodynamic inhibiting effect of methanol, glycerol, ethylene glycol (EG), and triethylene glycol (TEG) on gas hydrate formation. In particular the vapor and liquid phases are modeled using SAFT, and the solid hydrate phase is modeled with the van der Waals-Platteeuw model. 2. Thermodynamic Framework In a system of N components containing a solid hydrate (H), vapor (V), and liquid (L), the thermodynamic equilibrium is represented by
f Li ) f Vi (i ) 1, ..., N)
(1)
f Hj ) f Vj (j ) 1, ..., nc)
(2)
where f is the fugacity of component i or j, N is all the components, and nc is the hydrate-forming components including water. In the above equations, the fugacities in the vapor, liquid, and solid phases may be calculated using a suitable thermodynamic model. In this work, the SAFT equation of state is employed.
10.1021/ie051204x CCC: $33.50 © 2006 American Chemical Society Published on Web 02/17/2006
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2.1. SAFT Equation of State for Vapor and Liquid Phases. The residual Helmholtz free energy for an n component mixture of associating chain molecules can be expressed as the sum of hard-sphere repulsion, hard chain formation, dispersion, and association terms as follows:
Ares ) A - Aid ) Ahs + Achain + Adisp + Aassoc
ghs ii (dii) )
3
Adis
Aid
n
Ahs
ximi ∑ i)1
6 )
πFs
NkT
[
n
)
NkT
1
dis ximi (Adis ∑ 1 + A2 /TR) T i)1
1 - ζ3 ζ23 ζ32
+
]
ζ23/ζ32
where 2 3 Adis 1 ) FR(-8.5959 - 4.5424FR - 2.1268FR + 10.285FR ) (12) 2 3 Adis 2 ) FR(-1.9075 + 9.9724FR - 22.216FR + 15.904FR ) (13)
TR ) kT/x
xσx3 )
n
ximi ln(1 - ζ3) ∑ i)1
σx3 )
where
yi yjijσij3 ∑ ∑ i)1 j)1
∑
n
∑ i)1
ximi
(6)
In the above equations, Fn is the total number density of molecules in the solution, and dii is the hard-sphere diameter of segment i. Its relationship with the soft-sphere diameter (σii) is based on the Barker-Henderson perturbation theory and is expressed by Cotterman et al. as follows:18
1 + 0.2977kT/ii dii ) σii 1 + 0.33163kT/ + 0.001047(kT/ )2 ii ii
xi mi
Achain NkT
xi(1 - mi) ln(ghs ∑ ii (dii)) i)1
(19)
ij ) (1 - kij)xiijj
(20)
NkT
n
)
[
1
]
(ln XA - XA /2) + Mi ∑i xi ∑ 2 A i
i
i
(21)
where Mi is number of associating sites on molecule i. The term XAi is defined as the mole fraction of molecules i not bonded at site A, in mixtures with other components, and is given by
(8)
n
XAi ) [1 + NA
xjFXB ∆A B ]-1 ∑j ∑ B j
i j
(22)
j
where hs gseg ij (dij) ≈ gij (dij) )
σij ) (σii + σjj)/2
where kij is binary interaction parameter. 2.1.4. Association Term. The Helmholtz energy due to association is calculated by the expression of Chapman et al.20
Aassoc
n
)
(18)
n
In the above equations, σij and ij are the cross parameters between different segments and are calculated by the following combining rules
(7)
where ii is the energy parameter of the L-J potential. 2.1.2. Hard Chain Formation Term. The chain term Achain was derived by Chapman et al.19
(17)
xj mj ∑ j)1
(5)
n
Fs ) Fn
(16)
∑ ∑yi yjσij3 i)1 j)1
yi ) π n ζl ) Fn ximid lii (l ) 1, 2, 3) 6 i)1
(15)
n
n
(4)
(14)
6 ζ3 x2π
n
+
(1 - ζ3)2
ln(1 - ζ3) -
(11)
R
FR )
3ζ1ζ2 - ζ23/ζ32
(10)
3
2.1.3. Dispersion Term. This term is calculated by using an expression based on the (L-J) potential18
(3)
where is the free energy of an ideal gas with the same density and temperature as the system, Ahs is the free energy of a hardsphere fluid relative to the ideal gas, Achain is the free energy when chains are formed from hard spheres, and Adisp and Aassoc are the contributions to the free energy of dispersion and association interactions, respectively. The molecules are considered to be chainlike and composed of spherical segments of equal-size and equal-interaction parameters with Lennard-Jones (L-J) potential. 2.1.1. Hard-Sphere Repulsion Term. The hard-sphere term Ahs is calculated with the Boublik-Mansoori-CarnahanStarling-Leland equation as follows:16,17
3diiζ2 dii2ζ22 1 + + 1 - ζ3 2(1 - ζ )2 2(1 - ζ )3
[
]
3diidjj ζ2 diidjj 2 ζ22 1 + + 2 (9) 1 - ζ3 dii + djj (1 - ζ )2 dii + djj (1 - ζ )3 3 3 Equation 9 for like segments becomes
where ∑Bj means summation over all sites on molecule j, Aj, Bj, Cj, ..., F is the total molar density of molecules in the solution, and ∆AiBj is the associating strength given by
∆AiBj ) dij3gij(dij)segκAiBj[exp(AiBj/kT) - 1]
(23)
In eq 23, κAB is the bonding volume and AB/k is the associating
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energy. For cross-associating mixtures, we have the mixing rules20
κAjBi ) κAiBj ) (κAiBi + κAjBj)/2
(24)
AjBi ) AiBj ) xAiBiAjBj
(25)
2.2. Model for Hydrate Phase. The introduction of inhibitor (alcohol) in a liquid water-hydrocarbon mixture alters the prevailing structure in the aqueous phase. Alcohol-water and alcohol-hydrocarbon molecular interactions result in a “less structured” organization of water molecules, thus reducing the possibility of forming a stable hydrate. It is noted, however, that alcohol is not incorporated in the hydrate lattice.21 Hence, the model of van der Waals and Platteeuw based on statistical mechanics is valid and used in the work for the fugacity of water in the hydrate phase.22 It is expressed as follows:
f HW ) f MT W exp
(
)
-∆µMT-H W RT
(26)
where
∆µMT-H W
)
Cmj fj)) ∑ (νm ln(1 + ∑ m)1 j)1
σ (10-10 m)
/k (K)
AB/k (K)
κAB
water methanol EG glycerol TEG methane ethane propane CO2
0.982 1.124 1.043 2.180 3.204 1.186 1.437 2.367 1.833
2.985 3.642 4.232 4.194 3.805 2.990 3.193 3.078 2.654
433.91 309.90 354.65 405.08 252.03 160.84 199.73 174.07 165.80
1195.20 2320.77 2375.26 2195.15 2470.02
0.038 0.019 0.020 0.004 0.061
Table 2. Binary Interaction Parameters for the SAFT Equation14,15 system
kij
system
kij
methane/water ethane/water CO2/water methanol/water
0.0291 0.0068 -0.0452 -0.1043
ethane/methanol CO2/methanol methane/methanol
0.0641 0.2025 0.2204
between the empty hydrate lattice water and the liquid water, 0 is expressed from the following equarespectively. ∆hMT-L W 28 tion:
(27)
∫0R
4π kT
m
exp
(
)
-Wmj(r) 2 r dr kT
(28)
Here, Wmj(r) is the cell potential, obtained from McKoy and Sinanoglu.24 The integral is calculated numerically by using the fifteen-point Gauss-Laguerre quadrature formula.25 All the hydrate structural parameters, required in the calculation of Wmj(r), are taken from Parrish and Prausnitz and Anderson and Prausnitz.26,27 For temperatures between 260 and 300 K, the Langmuir constants are obtained from the expression of Parrish and Prausnitz.26 The fugacity of water in the empty hydrate lattice, f MT W , is obtained from the difference in the chemical potential of water 0 in the empty lattice and that of pure liquid water, ∆µMT-L ) W 0 L 28 µMT W - µW , using the following equation: 0
L f MT W ) f W exp
(
)
0 ∆µMT-L W
RT
(29)
where
∆µ0W ∆µMT-L W ) RT RT0 0
m
0
H Here, ∆µMT-H ) µMT W W - µW, and it represents the difference between the chemical potential of water in the empty lattice (MT) and that in the hydrate lattice (H). fj are the fugacities of the hydrate-forming gases in the vapor phase. Cmj are the Langmuir constants and represent the gas-water interactions. It is given by John and Holder for a single spherical water shell23
0
fluid
∆hMT-L ) ∆h0W + W
nc
2
RT
Cmj )
Table 1. Segment Parameters for Pure Fluids for the SAFT Equation14,15
MT-L0 T∆hW
∫T
0
RT2
0
∆VMT-L P W (30) dT + RT
Here, f LW is the fugacity of pure water, and it can be calculated from the SAFT equation state.14 ∆µ0W is an experimentally determined quantity.29 It is the chemical potential difference between the empty hydrate and pure liquid water at the reference conditions of T0 ) 273.15 K and zero absolute pressure. 0 0 and ∆VMT-L are the enthalpy and volume differences ∆hMT-L W W
∫TT [∆CPW + β(T - T 0)] dT 0
0
(31)
Here, ∆h0W and ∆C0PW are the enthalpy and volume differences between the empty hydrate lattice water and the liquid water at the reference conditions, respectively. They and β were regressed from hydrate formation experimental data and are given elsewhere.29 2.3. SAFT Model Parameters. The SAFT equation requires three pure-component parameters for nonassociating fluids and five parameters for associating fluids. These parameters are the L-J potential well depth (/k), the soft-sphere diameter of segments (σ), the number of segments of the molecule (m), the bonding volume (κAB), and the association energy between sites A and B (AB). In the case of mixtures, the SAFT equation uses van der Waals one-fluid mixing rules with the binary interaction parameter, kij, for the dispersion interactions. In this work, these parameters required in the SAFT are taken from Li and Englezos and are shown in Tables 1 and 2.14,15 As described in that work, the four-site model is used for the hydrogen bonds of the water molecule and the two-site model is used for the hydrogen bonds of each hydroxyl group on the alcohol. It is noted that the binary interaction parameter, kij, for the systems with EG, TEG, or glycerol was taken equal to 0 because it was adequate to give satisfactory results. 3. Prediction Results The following nine systems were examined: methane/water/ methanol, ethane/water/methanol, CO2/water/methanol, methane/ water/glycerol, CO2/water/glycerol, methane/water/EG, methane/ water/TEG, ethane/water/TEG, and propane/water/TEG. The incipient equilibrium hydrate formation pressure was calculated at a given temperature and at a given overall concentration of the inhibitor (methanol, EG, TEG, and glycerol). The inhibitor concentration is usually reported as the water phase concentration possibly because when an experiment is conducted water is mixed with an amount of the inhibitor and the resulting concentration is reported. That concentration is considered the overall concentration in our calculations. It is noted that the overall concentrations must be specified in the isothermal isobaric flash calculation procedure. Table 3 summarizes the results. The absolute average deviation of the predicted pressure
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Table 3. Predictions of the Hydrate Formation Pressures concentration of inhibitor in the aqueous phase (wt %)
T range (K)
P range (MPa)
AAD(P) (%)
10% methanol 20% methanol 35% methanol 50% methanol
266.2-286.4 263.3-280.2 250.9-267.8 233.1-255.3
2.14-18.8 2.83-18.75 2.38-13.68 1.47-16.98
1.87 4.36 11.46 18.53
10% EG 30% EG 50% EG
270.2-287.1 267.6-280.1 263.4-266.5
2.42-15.6 3.77-16.14 9.89-15.24
0.93 0.59 0.50
30
10% TEG 20% TEG 40% TEG
274.6-293 275-293 274.5-283
3.17-25.57 4.37-39.87 7.27-35.17
0.78 0.71 0.82
31
25% glycerol 50% glycerol
273.8-286.2 264.2-276.2
4.39-20.53 4.53-20.53
0.36 15.19
32
10% methanol 20% methanol 35% methanol 50% methanol
268.3-281.4 263.5-274.1 252.6-262.2 237.5-249.8
0.417-2.8 0.55-2.06 0.502-1.48 0.423-1.007
0.63 0.28 0.29 0.48
2, 33, 34
10% TEG 20% TEG 40% TEG
277-282 273.7-283 275-275.8
1.0-1.8 0.79-2.63 1.97-2.3
1.14 1.68 4.64
31
propane
10% TEG 20% TEG 30% TEG
272.3-276.8 271.7-275.2 270.2-272.4
0.18-0.51 0.25-0.50 0.29-0.425
1.69 1.88 0.31
35
carbon dioxide
10% methanol 20% methanol 35% methanol 50% methanol
269.6-274.9 264.0-268.9 242.0-255.1 232.6-241.3
1.58-3.48 1.83-2.94 0.379-1.77 0.496-1.31
0.80 2.11 1.87 0.69
2, 30
10% glycerol 20% glycerol 25% glycerol 30% glycerol
272.3-279.3 270.4-277.1 269.6-276.8 270.1-273.2
1.391-3.345 1.502-3.556 1.48-3.96 2.03-2.981
0.35 2.32 0.43 2.74
32, 36
gas methane
ethane
(AAD(P), %) is defined as follows:
AAD(P) )
1
[
∑| NP
NP i)1
|]
Pcal - Pexp Pexp
× 100 i
where NP is the number of data points. 3.1. Inhibiting Effect of Methanol. Table 3 provides information about the AAD(P) for methane, ethane, and carbon dioxide hydrate. The experimental data along with predictions are also given in Figures 1 and 2 for methane and ethane hydrate. As seen, reasonably good predictions are obtained even at high pressures and at high methanol concentrations. It
Figure 1. Methane hydrate formation in the presence of methanol: data (available in refs 2 and 30) and predictions based on SAFT.
data source (refs) 2, 30
should be noted that the parameters required by the van der Waals-Platteau model also play a role in the quality of the predictions. 3.2. Inhibiting Effect of EG. Table 3 and Figure 3 present the prediction of the inhibiting effect of EG on methane hydrate formation with 10-50% of EG using the SAFT equation. As seen, the predictions compare quite well with the experimental data. The total AAD is 0.67%. The maximum deviation is only 0.93%. This demonstrates the excellent prediction function of the SAFT. The EG molecule has more functional groups for association than methanol. Thus, its associating behavior is expected to be stronger than that of methanol. It can be seen
Figure 2. Ethane hydrate formation in the presence of methanol: data (available in refs 2, 33, and 34) and predictions based on SAFT.
Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 2135
Figure 3. Methane hydrate formation in the presence of EG: data (available in ref 30) and predictions based on SAFT.
Figure 5. Methane hydrate formation in the presence of TEG: data (available in ref 31) and predictions based on SAFT.
propane/TEG/water, respectively. The agreement between predictions and data is quite good. 4. Discussion
Figure 4. Methane hydrate formation in the presence of glycerol: data (available in ref 32) and predictions based on SAFT.
from the above calculations that the stronger the associating behavior of the fluid, the stronger the prediction ability of the molecular-based SAFT. 3.3. Inhibiting Effect of Glycerol. The SAFT equation was also employed to predict the inhibiting effect of glycerol on the hydrate formation of methane and CO2. The results are presented in Table 3 and Figure 4. As seen, the prediction on methane hydrate formation is excellent and the deviation is only 0.36% at 25% of glycerol. However, at 50% of glycerol the model overpredicts the data at the two lower temperatures. The reason is not known. The overprediction might be related to the fact that the binary interaction parameter for the methane/ glycerol binary was set equal to zero. There are not any relevant phase equilibrium data from which the parameter could be estimated. Finally, the calculated pressures for the CO2/glycerol/ water system with SAFT are in quite good agreement with the data at low and high concentrations of glycerol. The total AAD is 1.46%. 3.4. Inhibiting Effect of TEG. The inhibiting effect of TEG on the hydrate formation from methane, ethane, and propane was also computed using SAFT. The results are shown in Table 3. Figure 5 also shows the prediction with the experimental data for methane hydrate. The total predicted ADDs are 0.77, 2.49, and 1.29% for the methane/TEG/water, ethane/TEG/water, and
As seen from the above calculations SAFT results in satisfactory to excellent predictions of the inhibiting effect of methanol, EG, glycerol, and TEG on gas hydrate formation. The SAFT model takes into account chain formation and molecular associating interactions in addition to repulsion and dispersion. In systems containing inhibitors (methanol, EG, glycerol, and TEG) and water, association interactions are expected to be strong. Consequently, a model like SAFT is the proper one to choose even though a classic cubic equation of state might have a very good correlational ability. Finally, it should be noted that in certain industrial applications glycols are added in hydrate-forming systems together with electrolytes and, hence, the need to predict the hydrate formation in this mixed solvent system arises. Clark and Bishnoi presented the first equation-of-state-based method to deal with such situations.37 In principle, that methodology can incorporate SAFT. 5. Conclusions The SAFT equation of state has successfully been employed to predict the gas hydrate formation conditions in the presence of thermodynamic inhibitors including methanol, EG, glycerol, and TEG. Only pure-component parameters are used by SAFT to determine the effect of EG, glycerol, and TEG. The ability of the SAFT model to account for association interactions is the critical factor enabling successful predictions. Acknowledgment The financial support by the National Science and Engineering Research Council of Canada (NSERC) and by the Research Program of the One-Hundred Talent Project of the Chinese Academy of Sciences is greatly appreciated. Nomenclature A ) Helmholtz free energy, J C ) Langmuir constant, MPa-1 Cp ) heat capacity, J mol-1 K-1 d ) hard-sphere diameter, 1 × 10-10 m f ) fugacity, MPa
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Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006
g ) radius distribution function k ) Boltzmann constant, J K-1 mi ) effective number of segments in component i (i ) 1, 2, ..., n) M ) number of associate sites n ) number of components in mixture nc ) number of hydrate-forming substances N ) number of molecules NA ) Avogadro constant, 6.022 17 × 1023 mol-1 r ) radial distance from the center of the hydrate cavity, m R ) gas constant, 8.3143 J mol-1 K-1 Rm ) type m spherical cavity radius, m P ) pressure, MPa T ) absolute temperature, K V ) molar volume, m3 mol-1 W(r) ) cell potential function, J xi ) mole fraction of component i in the liquid phase (i ) 1, 2, ..., n) XAi ) mole fraction of molecule i not bonded at site A yi ) mole fraction of component i in the vapor phase (i ) 1, 2, ..., n) Greek Letters /k ) energy parameter of dispersion, K AB/k ) energy parameter of association between sites A and B, K κAB ) bonding volume ∆AB ) association strength between sites A and B µ ) chemical potential F ) molar density, mol m-3 Fn ) number density, m-3 σ ) soft-sphere diameter, 1 × 10-10 m Vm ) number of cavities of type m Subscripts i, j, k ) components m ) type of cavity W ) water Superscripts assoc ) association interaction A, B ) association site chain ) hard-sphere chain disp ) dispersion interaction hs ) hard-sphere res ) residual term H ) hydrate L ) liquid L0 ) pure liquid water MT ) empty lattice 0 ) reference conditions of 273.15 K and zero absolute pressure V ) vapor Literature Cited (1) Hammerschmidt, E. G. Formation of Gas Hydrates in Natural Gas Transmission Lines. Ind. Eng. Chem. 1934, 26, 851-855. (2) Ng, H.-J.; Robinson, D. B. Hydrate Formation in Systems Containing Methane, Ethane, Propane, Carbon Dioxide or Hydrogen Sulfide in the Presence of Methanol. Fluid Phase Equilib. 1985, 21, 145-155. (3) Anderson, F. E.; Prausnitz, J. M. Inhibition of Gas Hydrates by Methanol. AIChE J. 1986, 32, 1321-1333. (4) Robinson, D. B.; Ng, H.-J. Hydrate Formation and Inhibition in Gas or Gas Condensate Streams. J. Can. Pet. Technol. 1986, 25, 26-30.
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ReceiVed for reView October 28, 2005 ReVised manuscript receiVed January 13, 2006 Accepted January 26, 2006 IE051204X
