Modelling the gas hydrate formation of inhibitor containing systems

Fluid Phase Equilibria 205 (2003) 291–302 Modelling the gas hydrate formation of inhibitor containing systems Qing-Lan Ma, Guang-Jin Chen, Tian-Min Guo∗ High Pressure Fluid Phase Behavior and Property Research Laboratory, University of Petroleum, P.O. Box 902, Beijing 102249, PR China Received 30 April 2002; accepted 15 October 2002 Abstract The accurate evaluation of the activity of water in aqueous phase is of importance in predicting the hydrate formation conditions for polar inhibitor containing systems. In this work, the Patel–Teja equation of state (PT EOS) coupled with the Kurihara mixing rule was used to correlate the gas solubility data and determining the liquid phase composition. Binary interaction parameters between typical natural gas components and hydrate inhibitors have been determined. The PT EOS was then coupled with the hydrate model proposed by Chen and Guo [Chem. Eng. J. 71 (1998) 145] and applied to predict the hydrate formation conditions of various systems containing polar inhibitors. Comparisons with experimental data and the prediction results of typical van der Waals–Platteeuw type hydrate models showed that significant improvement in predicting the hydrate formation of inhibitor containing systems has been achieved. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Hydrate formation condition; Gas solubility; Equation of state; Mixing rule; Chen–Guo hydrate model; Polar inhibitor 1. Introduction In spite of enormous efforts have been made in developing new kinetic hydrate inhibitors in the past two decades, traditional polar inhibitors (e.g. methanol, ethylene glycol, etc.) are still widely used in industrial practices. In order to determine the amount of inhibitor needed for preventing hydrate formation in natural gas pipelines and processing equipments, it is important to predict reliably the hydrate formation conditions in the inhibitor containing systems. The critical factor in modelling the hydrate formation conditions for inhibitor containing systems is the evaluation of the activity of water in the aqueous phase (a highly nonideal liquid solution), which is closely related to the composition of the aqueous phase. In this work, firstly the high-pressure vapour–liquid equilibria (VLE) of (natural gas components + water + polar inhibitor) systems were modelled by the Patel–Teja equation of state (PT EOS) [1] using ∗ Corresponding author. Tel.: +86-10-6234-0132; fax: +86-10-6234-0132. E-mail address: [email protected] (T.-M. Guo). 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S0378-3812(02)00295-9 292 Q.-L. Ma et al. / Fluid Phase Equilibria 205 (2003) 291–302 the Kurihara mixing rule [2]. Binary interaction parameters between typical natural gas components and hydrate inhibitors have been determined. The established VLE model was then coupled with the hydrate model previously proposed by Chen and Guo [3] and applied to predict the hydrate formation conditions of various polar inhibitor containing systems. Extensive comparisons with experimental data and the prediction results of a number of van der Waals–Platteeuw type hydrate models have been performed. The test results indicate that the proposed model is, in general, capable of providing more reliable hydrate formation predictions for inhibitor containing systems. 2. The modelling of high-pressure VLE of polar inhibitor containing systems The conventional approach to handle the high-pressure VLE of a strongly nonideal polar system is coupling an EOS with a sophisticated mixing rule. Based on the previous experience of this group, the combination of a three-parameter cubic EOS, PT EOS [1], with the Kurihara mixing rule [2] has been chosen. The familiar PT EOS is formulated as P = RT v − b − a v(v + b)+ c(v − b) (1) where a, b and c are the three parameters of PT EOS. In the mixing rules proposed by Kurihara, following linear mixing rules are adapted for parameters b and c: b = ∑ i xibi (2) c = ∑ i xici (3) Parameter a is evaluated by the following expression: a = ∑ i ∑ j xixj √ aiaj (1− kij)+ ξbg E ln((3+ ψ − ξ)/(3+ ψ + ξ)) (4) where ψ = ∑ i xiψi = ∑ i xi ci bi (5) ξ = √ ψ2 + 6ψ + 1 (6) The fugacity coefficient expression for component i derived from the PT EOS and Kurihara mixing rule is as follows: RT ln φi =−RT ln(Z − B)+ RT ( bi v − b ) − a ′ 2d ln ( Q+ d Q− d ) + a(bi + ci) 2(Q2 − d2) + a 8d3 [ci(3b + c)+ bi(3c + b)] [ ln ( Q+ d Q− d ) − 2Qd Q2 − d2 ] (7) Q.-L. Ma et al. / Fluid Phase Equilibria 205 (2003) 291–302 293 where a′ is formulated as a′ = 1 n ∂(n2a) ∂ni = 2 ∑ j xj √ ajai(1− kij)+ 1ln((3+ ψ − ξ)/(3+ ψ + ξ)) × { big Eξ + RT ln γibξ − bg E ln((3+ ψ − ξ)/(3+ ψ + ξ)) × [ ψ2 − ψψi + 3ψ − 3ψi ξ + 2(ψ − ψi) ]} (8) In this work, Wilson equation [4] was used to evaluate the excess Gibb’s free energy (gE) and the corresponding activity coefficient (γ i): gE RT = − n∑ i=1 xi ln   n∑ j=1 Λijxj   (9) Λij = vLj vLi exp [−(λij − λii) RT ] (10) There are three binary interaction parameters: kij , (λij − λii) and (λji − λjj) to be determined from binary VLE data. The optimised interaction parameters of some binary systems of interest in this work Table 1 Optimised binary interaction parameters for Kurihara and vdW one-fluid mixing rules Component k12a (λ12−λ11) (J/mol) (λ21−λ22) (J/mol) k12b Data source 1 2 CO2 H2O −0.131 −347.54 97595 −0.044 [5] CO2 CH3OH 0.089 762.46 −1791.7 0.025 [5] C2H6 CH3OH 0.039 −3178.3 84273 0.010 [5] CO2 C2H5OH 0.012 488.33 1690.5 0.078 [5] C2H6 C2H5OH 0.038 1042.4 −1029.8 0.031 [5] C3H8 C2H5OH 0.022 −2529.5 5288.8 0.018 [5] CO2 EG 0.157 −3048.9 5258.6 0.141 [6] CH4 EG −0.110 2217.6 4135.1 0.137 [6] CO2 TEG 0.111 −3395.4 399.8 −0.012 [7] H2S TEG 0.034 −2594.6 −501.72 −0.071 [7] CH4 TEG 0.065 1623.6 −1890.4 −0.013 [7] C2H6 TEG 0.040 1450.0 −1085.0 0.042 [7] C3H8 TEG 0.127 −405.25 −1722.4 0.052 [7] H2O CH3OH −0.150 2085.2 479.59 −0.143 [8] H2O C2H5OH −0.128 4000.3 1600.6 −0.130 [9] H2O EG −0.105 5300.5 −5299.4 −0.138 [9] H2O TEG −0.180 −2141.2 879.02 −0.180 [9] a k12 in Kurihara mixing rule. b k12 in vdW one-fluid mixing rule. 294 Q.-L. Ma et al. / Fluid Phase Equilibria 205 (2003) 291–302 are tabulated in Table 1 along with the kij values of van der Waals (vdW) one-fluid mixing rule. In Table 1, EG and TEG refer to ethylene glycol and triethylene glycol, respectively. 3. The modelling of hydrate formation for inhibitor containing systems The thermodynamic hydrate model proposed by Chen and Guo in 1998 [3] was used in this work. The major equations involved are given as follows: fi = xif 0i  1−∑ j θj   α (11) Fig. 1. Comparison of calculated and experimental solubility of CO2 in ethylene glycol (T = 323.15 K). Fig. 2. Comparison of calculated and experimental solubility of CO2 in methanol (T = 298.15 K). Q.-L. Ma et al. / Fluid Phase Equilibria 205 (2003) 291–302 295 Fig. 3. Comparison of calculated and experimental solubility of H2S in triethylene glycol (T = 298.15 K). ∑ i xi = 1.0 (12) ∑ j θj = ∑ jfjCj 1+∑jfjCj (13) where α = 1/3 structure I and α = 2 for structure II hydrates; fi denotes the fugacity of hydrate former i in the gas phase or hydrocarbon-rich liquid phase (calculated by PT EOS); xi stands for the mole fraction of basic hydrate component i formed by hydrate former i; θj denotes the fraction of linked cavities occupied by gas component j; f 0i is the fugacity of hydrate former in equilibrium with the unfilled pure basic hydrate i (∑ j θj = 0 ) , f 0i is formulated as f 0i = f 0Ti exp [ βP T ] a−1/λ2w (14) f 0Ti = exp [−∑jAij θj T ][ A′i exp ( B ′i T − C ′i )] (15) where aw is the activity of water in the aqueous phase; β = 0.4242 K bar−1, λ2 = 3/23 for structure I hydrates and β = 1.0224 K bar−1, λ2 = 1/17 for structure II hydrates. The evaluation of Antoine constants Ai ′ Bi ′ Ci ′ and binary interaction coefficients Aij are referred to Chen and Guo [3]. The Langmuir constant Cj in Eq. (13) is formulated as Cj = Xj exp [ Yj T − Zj ] (16) where Xj , Yj and Zj are constants of component j, values for typical hydrate formers had been reported by Chen and Guo [3]. 296 Q.-L. Ma et al. / Fluid Phase Equilibria 205 (2003) 291–302 From Eq. (14) we can see that f 0i is sensitive to the activity of water (aw), and its value could be significantly affected by the presence of a polar inhibitor (methanol, EG, TEG, etc.). By definition: aw = fw f 0w (17) where fw denotes the fugacity of water in the aqueous phase (calculated by PT EOS) and f 0w the fugacity of pure water under the system temperature and pressure. In the subsequent section, Table 2 Prediction results of the hydrate formation temperatures Gas phase Concentration of inhibitor in aqueous phase (wt.%) P-range (MPa) T-range (K) AADT (1)a (%) AADT (2)b (%) Np Data source CH4 10% CH3OH 2.14–18.82 266.23–286.40 0.140 0.698 6 [11] 20% CH3OH 2.83–18.75 263.34–280.17 0.146 1.326 6 [11] 35% CH3OH 2.38–20.51 250.90–270.10 0.468 0.508 7 [11] 15% C2H5OH 3.83–13.67 273.30–284.70 0.113 0.256 5 [11] CH4 + C3H8 10% CH3OH 0.53–13.83 265.51–291.23 0.442 0.150 6 [11] 20% CH3OH 0.94–14.10 265.17–286.47 0.423 0.521 5 [11] 35% CH3OH 0.62–20.11 253.10–276.60 0.437 0.539 6 [11] 50% CH3OH 0.69–20.42 241.20–262.60 0.484 – 7 [11] C2H6 10% CH3OH 0.42–2.82 268.28–281.89 0.085 0.474 7 [11] 20% CH3OH 0.55–2.06 263.53–274.07 0.094 1.322 6 [11] 35% CH3OH 0.50–1.47 252.6–262.2 0.588 0.386 4 [11] 50% CH3OH 0.42–1.01 237.5–249.8 1.281 – 4 [11] C3H8 5% CH3OH 0.23–0.47 272.12–274.79 0.149 0.138 5 [11] 10.39% CH3OH 0.19–0.43 268.30–271.82 0.053 0.443 6 [11] 35% CH3OH 0.14–0.21 248.00–250.20 0.132 0.807 2 [11] CO2 10% CH3OH 1.59–3.48 269.49–274.92 0.414 1.264 6 [11] 20.02% CH3OH 1.59–2.94 263.96–268.86 0.465 2.039 7 [11] 10.04% EG 1.15–3.20 270.90–278.30 0.146 0.278 4 [12] 10% CH3OH 1.74–2.35 271.60–273.80 0.060 0.272 2 [12] H2S 10% CH3OH 0.07–1.08 265.69–291.77 0.255 0.604 6 [11] 16.5% CH3OH 0.28–1.50 273.20–290.10 0.625 1.810 3 [11] 20% CH3OH 0.22–0.59 271.79–281.15 0.115 1.218 2 [11] 35% CH3OH 0.22–0.58 263.20–274.20 0.410 0.670 3 [11] 16.5% C2H5OH 0.39–1.48 280.70–291.80 0.890 0.745 3 [11] CO2 + CH4 10% EG 1.14–3.22 268.70–278.00 0.384 0.646 4 [12] CO2 + C2H6 10.6% EG 0.85–2.31 269.10–276.40 0.173 0.199 5 [12] CO2 + N2 13.01% EG 0.93–3.39 267.20–276.90 0.143 0.637 8 [12] NG1 10% CH3OH 1.04–19.03 268.27–288.34 0.653 1.550 9 [11] 20% CH3OH 1.41–19.15 264.41–280.97 1.062 2.617 9 [11] NG2 42.9% TEG 2.31–8.59 275.15–285.25 0.201 0.209 6 –c a AADT (1): model proposed in this work. b AADT (2): vdW-P type model proposed by Zuo et al. c Experimental data measured in this laboratory. Q.-L. Ma et al. / Fluid Phase Equilibria 205 (2003) 291–302 297 we will show the effect of gas solubility in the aqueous phase on the hydrate formation conditions. 4. Test results on VLE and hydrate formation predictions 4.1. VLE—solubility of gas species in pure alcohols/glycols/aqueous methanol solution Comparisons have been made of the calculated and experimental solubility of gas species in pure alcohols, glycols and an aqueous methanol solution using the Kurihara mixing rule and vdW one-fluid Table 3 Compositions of the natural gas samples tested Component Mole fraction NG1 NG2 NG3a CO2 0.1419 0.0200 – N2 0.0596 0.0095 0.0545 H2S – – 0.2500 CH4 0.7160 0.8521 0.6210 C2H6 0.0473 0.0705 0.0430 C3H8 0.0194 0.0316 0.0171 iC4H10 – 0.0073 – nC4H10 0.0079 0.0052 0.0072 iC5H12 – 0.0015 – nC5H12 0.0079 0.0007 0.0072 C6H14 – 0.0005 – C7H16 – 0.0003 – a NG3 is a synthetic natural gas mixture. Fig. 4. Comparison of calculated and experimental hydrate formation conditions of CH4 in an aqueous methanol solution. 298 Q.-L. Ma et al. / Fluid Phase Equilibria 205 (2003) 291–302 Fig. 5. Comparison of calculated and experimental hydrate formation conditions of C2H6 in an aqueous methanol solution. mixing rule. The Kurihara mixing rule provided better agreement with experiment than the vdW one-fluid mixing rule for the solubility of: methane in TEG, carbon dioxide in methanol, carbon dioxide in EG, propane in ethanol, hydrogen sulfide in TEG. Some representative plots of these comparisons are shown in Figs. 1–3; where x1 is the solubility of gas species 1 in mole fraction. 4.2. Hydrate formation conditions The prediction results of the hydrate formation temperatures of 30 inhibitor containing systems obtained by the proposed model and the vdW-P type model proposed by Zuo et al. [10] are listed in Table 2, where NG1 and NG2 are natural gas samples (compositions refer to Table 3). The absolute average deviation Fig. 6. Comparison of calculated and experimental hydrate formation conditions of CO2 in an aqueous methanol solution. Q.-L. Ma et al. / Fluid Phase Equilibria 205 (2003) 291–302 299 Fig. 7. Comparison of calculated and experimental hydrate formation conditions of H2S in an aqueous methanol solution. of predicted temperature (AADT, %) is defined as follows: AADT (%) = ( 1 Np ) Np∑ j=1 [ Tcal − Texp Texp ] j × 100 (18) where Np is the number of data points. Representative comparisons of the calculated hydrate formation conditions based on the proposed model and two typical vdW-P type models developed by Zuo et al. [10] and Sloan (CSMHYD model) [11] are depicted in Figs. 4–8. Fig. 8. Comparison of calculated and experimental hydrate formation conditions of a gas mixture (90.99 mol% CO2 + 9.01 mol% N2) in an aqueous ethylene glycol solution. 300 Q.-L. Ma et al. / Fluid Phase Equilibria 205 (2003) 291–302 Fig. 9. Comparison of calculated and experimental hydrate formation conditions of a gas mixture (90.99 mol% CO2 + 9.01 mol% N2) in an aqueous ethylene glycol solution. Typical effects of neglecting the solubility of gas species in the aqueous phase as well as the choice of mixing rule on the hydrate formation predictions are shown in Figs. 9–11. In addition, a brief comparison with the Du–Guo hydrate model [13] in predicting the hydrate forma- tion conditions of methanol-containing systems is presented in Table 4. Du–Guo model was developed based on the simplified multi-shell hydrate model (vdW-P type) coupled with the Peng–Ronbinson EOS [14] using random–nonrandom mixing rule proposed by Mollerup [15]. The comparison shows that the proposed hydrate model gives better predictions in the higher methanol concentration region. Fig. 10. The effect of gas solubility on the predicted hydrate formation conditions of a natural gas mixture (NG3) in an aqueous methanol solution. Q.-L. Ma et al. / Fluid Phase Equilibria 205 (2003) 291–302 301 Fig. 11. The effect of mixing rule on the predicted hydrate formation conditions of a gas mixture (91.12 mol% CH4 + 8.88 mol% C3H8) in an aqueous methanol solution. Table 4 Comparison with Du–Guo hydrate model in predicting the hydrate formation temperatures for methanol-containing systems Gas phase Concentration of inhibitor in aqueous phase (wt.%) P-range (MPa) T-range (K) AADT (%) Np Data source This work Du–Guo model CH4 10% CH3OH 2.14–18.82 266.23–286.40 0.14 0.12 6 [16] 20% CH3OH 2.83–18.75 263.34–280.17 0.15 0.50 6 [16] C2H6 10% CH3OH 0.73–20.2 272.10–284.45 0.17 0.16 15 [16] 20% CH3OH 1.52– 20.4 271.75–278.61 0.17 0.21 7 [16] C3H8 5% CH3OH 0.23–6.34 272.12–274.97 0.24 0.42 8 [16] 10.39% CH3OH 0.35–6.51 271.07–274.22 0.44 0.56 6 [16] 5. Conclusions (1) The PT EOS using Kurihara mixing rule is adequate for describing the solubility of various natural gas components in polar inhibitor containing aqueous solutions, which alternately contributes to the more reliable prediction of the activity of water in the aqueous phase. (2) Extensive comparison with experimental data and a number of van der Waals–Platteeuw type hydrate models indicates that the model proposed in this work (the combination of Chen–Guo hydrate model with the PT EOS using Kurihara mixing rule) is capable of, in general, giving more reliable hydrate formation predictions for systems containing polar inhibitors. References [1] N.C. Patel, A.S. Teja, Chem. Eng. Sci. 37 (1982) 463–473. 302 Q.-L. Ma et al. / Fluid Phase Equilibria 205 (2003) 291–302 [2] K. Kurihara, K. Kojima, J. Chem. Eng. Jpn. 20 (1987) 227–231. [3] G.-J. Chen, T.-M. Guo, Chem. Eng. J. 71 (1998) 145–151. [4] G.M. Wilson, J. Am. Chem. Soc. 86 (1964) 127–130. [5] H. Knapp, R. Doring, L. Oellrich, Vapor–Liquid Equilibria for Mixtures of Low Boiling Substances, Frankfurt, DECHEMA, 1982. [6] D.-Q. Zheng, W.-D. Ma, R. Wei, T.-M. Guo, Fluid Phase Equilib. 155 (1999) 277–286. [7] H. Knapp, S. Zeck, R. Langhorst, Vapor–Liquid Equilibria for Mixtures of Low Boiling Substances, Frankfurt, DECHEMA, 1988. [8] T.-M. Guo, Multi-Component Vapor–Liquid Equilibria and Distillation (in Chinese), Chemical Industry Press, Beijing, 1983. [9] J. Gmehling, U. Onken, Vapor–Liquid Equilibrium Data Collection: Aqueous–Organic Systems, Frankfurt, DECHEMA, 1991. [10] Y.-X. Zuo, S. Gommesen, T.-M. Guo, Chin. J. Chem. Eng. 4 (3) (1996) 189–202. [11] E.D. Sloan Jr., Clathrate Hydrates of Natural Gases, Marcel Dekker, New York, 1990. [12] S.-S. Fan, G.-J. Chen, Q.-L. Ma, T.-M. Guo, Chem. Eng. J. 78 (2000) 173–178. [13] Y.-H. Du, T.-M. Guo, Chem. Eng. Sci. 45 (1990) 893–900. [14] D.-Y. Peng, D.B. Robinson, Ind. Eng. Chem. Fundam. 15 (1976) 59–64. [15] J. Mollerup, Fluid Phase Equilib. 15 (1983) 189–207. [16] H.-J. Ng, D.B. Robinson, Fluid Phase Equilib. 21 (1985) 145–155. Modelling the gas hydrate formation of inhibitor containing systems Introduction The modelling of high-pressure VLE of polar inhibitor containing systems The modelling of hydrate formation for inhibitor containing systems Test results on VLE and hydrate formation predictions VLE-solubility of gas species in pure alcohols/glycols/aqueous methanol solution Hydrate formation conditions Conclusions References