Fluid Phase Equilibria 220 (2004) 113–121 Gas solubility measurement and modeling for methane–water and methane–ethane–n-butane–water systems at low temperature conditions Antonin Chapoy a , Amir H. Mohammadi b , Dominique Richon a , Bahman Tohidi b,∗ a Centre d’Energétique, Ecole Nationale Supérieure des Mines de Paris CENERG/TEP, 35 Rue Saint Honoré, 77305 Fontainebleau, France b Centre for Gas Hydrate Research, Institute of Petroleum Engineering, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, UK Received 1 October 2003; received in revised form 13 February 2004; accepted 25 February 2004 Available online 12 May 2004 Abstract In this communication, experimental measurements and thermodynamic modeling of solubilities of methane and a hydrocarbon gas mixture (94% methane + 4% ethane + 2% n-butane) in water at low temperature conditions are reported. Methane solubility measurements have been conducted at a temperature range of 275.11–313.11 K and pressures up to 18 MPa. The solubility of the individual components in the gas mixture was measured from 278.14 to 313.12 K and pressures up to 14.407 MPa. A static–analytic apparatus, taking advantage of a pneumatic capillary sampler is used for fluid sampling. The Valderrama modification of the Patel–Teja equation of state combining with non-density dependent mixing rules were used for modeling gas solubilities in water. The data generated in this work are compared with the literature data and the predictions of the thermodynamic model, demonstrating the reliability of the techniques used in this work. © 2004 Elsevier B.V. All rights reserved. Keywords: Gas solubility; Methane; Ethane; n-Butane; Water; Equation of state 1. Introduction Natural gases are normally in contact with water in reservoirs. During production and transportation, dissolved water in the gas phase may form condensate, ice and/or gas hydrate. Forming a condensed water phase may lead to corrosion and/or two-phase flow problems. Also, a condensed water phase in a compressor may lead to damaging of impeller blades. Ice and/or gas hydrate formation may cause blockage during production and transportation. To give a qualified estimate of the amount of water in the gas phase, thermodynamic models are required. Accurate gas solubility data, especially at low temperature conditions, are necessary to develop and validate thermodynamic models. The gas solubility in water is also an important issue from an Abbreviations: AAD, average absolute deviation; BIP, binary interaction parameter; FID, flame ionization detector; FOB, objective function; NDD, non-density dependent mixing rules; PID, proportional integrator derivative controller; TCD, thermal conductivity detector; VPT-EoS, Valderrama modification of Patel–Teja equation of state ∗ Corresponding author. Tel.: +44-131-451-3672; fax: +44-131-451-3127. E-mail address: Bahman.Tohidi@pet.hw.ac.uk (B. Tohidi). 0378-3812/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2004.02.010 environmental aspect, due to new legislations and restrictions on the hydrocarbon content in water disposal. Unfortunately, gas solubility data for most light hydrocarbons at low temperature conditions are scarce and often rather dispersed. The main objective of this work is to provide the much needed solubility data at the above mentioned conditions. In this work, experimental data for the solubility of methane, ethane and n-butane have been assembled from the literature and examined for consistency. Then new solubility measurements of methane and of a gas mixture (94% methane + 4% ethane + 2% n-butane) in water have been generated at low and ambient temperatures. The different isotherms presented herein were obtained using an apparatus based on a static–analytic method taking advantage of a capillary sampler. The Valderrama modification of the Patel–Teja equation of state (VPT-EoS) [1] with non-density dependent mixing rules (NDD) [2] were used to model the gas solubilities in water. The binary interaction parameters (BIPs) between methane and water were tuned using the new experimental results on methane solubility data. Using the previously reported binary interaction parameters [3] for ethane and n-butane, the solubilities of each gas of the mixtures were predicted. Predictions were found to be in good agreement 114 A. Chapoy et al. / Fluid Phase Equilibria 220 (2004) 113–121 with the experimental data, demonstrating the reliability of the experimental technique and model used in this work. Liquide is pure grade with traces of water (3 ppm) and of hydrocarbons (0.5 ppm). The gas mixture, 94% methane, 4% ethane (±2%) and 2% n-butane (±2%) was purchased by Messer Griesheim. 2. Literature review 2.1. Methane Bunsen [4] conducted the first study of the solubility of methane in water in 1855. The solubilities of several gases including methane were measured at atmospheric pressures. The first study reporting intermediate and high-pressure solubility data for methane in water was the study of Frolich et al. [5]. Since then, many studies have been performed, e.g. Michels et al. [6], Culberson et al. [7], Culberson and Mc Ketta [8], Davis and Mc Ketta [9], Duffy et al. [10], O’Sullivan and Smith [11], Sultanov et al. [12], Amirijafari and Campbell [13], Sanchez and De Meer [14], Price [15], Stoessel and Byrne [16] and Abdulgatov et al. [17]. All these authors have measured solubility of methane at intermediate and high-pressures, but only at high temperatures. The number of researcher reporting data of methane solubility in water at low temperature conditions (T ≤ 298.15 K) is rather limited but the more recent ones are, e.g. Cramer [18], Yarym-Agaev et al. [19], Yokoyama et al. [20], Toplak [21], Wang et al. [22], Lekvam and Bishnoi [23], Song et al. [24], Yang et al. [25], Servio and Englezos [26],Wang et al. [27] and Kim et al. [28]. 2.2. Ethane This system has not been so widely examined; only a few authors have conducted the solubility experiments on this system at intermediate/high-pressure conditions. The majority of the authors have reported solubility measurements at temperatures higher than 298.15 K, e.g. Culberson and Mc Ketta [29], Culberson et al. [30], Anthony and Mc Ketta [31,32] and Danneil et al. [33], Sparks and Sloan [34]. Again, the number of authors who have reported data at low temperature conditions is rather limited, e.g. Wang et al. [27] and Kim et al. [28]. 2.3. n-Butane The solubility data for n-butane–water system at intermediate/high-pressure conditions is limited to only a few authors, e.g. Brooks et al. [35], Reamer et al. [36], Le Breton and McKetta [37], Yiling et al. [38] and Carroll et al. [39]. 3. Experimental 3.1. Materials Methane was purchased from Messer Griesheim with a certified purity greater than 99.995 vol.%. Helium from Air 3.2. Apparatus and experimental procedures The apparatus used in this work (Fig. 1) is based on a static–analytic method with fluid phase sampling. This apparatus is similar to that described by Laugier and Richon [40]. The phase equilibrium is achieved in a cylindrical cell made of Hastelloy C276, the cell volume is about 34 cm3 (internal diameter = 25 mm, height = 69.76 mm) and it can be operated up to 40 MPa between 223.15 and 473.15 K. The cell is immersed in an ULTRA-KRYOMAT LAUDA constant-temperature liquid bath that controls and maintains the desired temperature within ±0.01 K. In order to perform accurate temperature measurements in the equilibrium cell and to check for thermal gradients, temperature is measured at two locations corresponding to the vapor and liquid phases through two 100 ohms platinum resistance thermometer devices (Pt100) connected to an HP data acquisition unit (HP34970A). These two Pt100 are carefully and periodically calibrated against a 25 ohms reference platinum resistance thermometer (TINSLEY precision instruments). The resulting uncertainty is less than ±0.02 K. The 25 ohms reference platinum resistance thermometer was calibrated by the Laboratoire National d’Essais (Paris) based on the 1990 International Temperature Scale (ITS 90). Pressures are measured by means of two Druck pressure transducers (type PTX 610 range 0–30 MPa and type PTX611, range 0–1.6 MPa) connected to the HP data acquisition unit (HP34970A). The pressure transducers are maintained at a constant temperature (temperature higher than the highest temperature of the study) by means of a specially-made air-thermostat, which is controlled using a PID regulator (WEST, model 6100). Both pressure transducers are calibrated against a dead weights pressure balance (Desgranges & Huot 5202S, CP 0.3–40 MPa, Aubervilliers, France). Pressure measurement uncertainties are estimated to be within ±0.5 kPa in the 0–2.5 MPa range using the 0–1.6 MPa pressure transducer and within ±5 kPa in the 2.5–38 MPa range using the 0–30 MPa pressure transducer. The HP on-line data acquisition unit is connected to a personal computer through a RS-232 interface. This system allows real time readings and storage of temperatures and pressures throughout the different isothermal runs. The analytical work was carried out using a gas chromatograph (VARIAN model CP-3800) equipped with two detectors in series, a thermal conductivity detector (TCD) and a flame ionization detector (FID), connected to a data acquisition system (BORWIN ver 1.5, from JMBS, Le Fontanil, France). The analytical column is a Hayesep C 80/100 Mesh column (silcosteel tube, length: 2 m, diameter: 1/8 in.). The TCD was used to detect the water; it was repeatedly calibrated by injecting known amount of water A. Chapoy et al. / Fluid Phase Equilibria 220 (2004) 113–121 115 Fig. 1. Flow diagram of the equipment. C: carrier gas; d.a.s.: data acquisition system; DH2 O: degassed water; EC: equilibrium cell; FV: feeding valve; GCy: gas cylinder; HPC: high-pressure compressor; HT: hastelloy tube; LB: liquid bath; LS: liquid sampler; PD: pressure display; PP: platinum resistance thermometer probe; PTh and PTl: high- and low-pressure transducers; SM: sampler monitoring; SV: special valve; Th: thermocouple; TR: temperature regulator; Vi: valve number i; VS: vapor sampler; VSS: variable speed stirrer; VP: vacuum pump. through liquid syringes. The FID was used to detect the hydrocarbons. It was first calibrated by introducing known amounts of methane, ethane, n-butane through a gas syringe in the injector of the gas chromatograph, and then different gas mixtures of known amount of each gas were prepared and loaded in the cell before sampling in order to calibrate the FID in different small proportion for each gas. present in the equilibrium cell (around 10 cm3 ), it is possible to withdraw many samples without disturbing the phase equilibrium. The temperature of the sampling device is set at 473.15 K to avoid condensation of any of the components. 3.3. Experimental procedures 4.1. Pure compound properties The equilibrium cell and its loading lines were evacuated down to 0.1 Pa and the necessary quantity of the preliminary degassed water (approximately 10 cm3 ) was introduced using an auxiliary cell. Then, the desired amount of gas was introduced into the cell directly from the commercial cylinder or via a gas compressor to reach the desired pressure. For each equilibrium condition, at least 10 samples are withdrawn using the pneumatic samplers ROLSITM [41] and analyzed in order to check for measurement repeatability. As the quantity of the withdrawn samples (less than 0.5–2 ␮g) is very small compared to the volume of the liquid phase The critical temperature (Tc ), critical pressure (Pc ), critical volume (vc ) and acentric factor (ω), for each of the four pure compounds are provided in Table 1. 4. Thermodynamic model Table 1 Critical properties and acentric factors [42] Compound Pc (MPa) Tc (K) vc (m3 kg mole−1 ) ω Water Methane Ethane n-Butane 22.048 4.604 4.880 3.797 647.30 190.58 305.42 425.18 0.056 0.0992 0.1479 0.255 0.3442 0.0108 0.09896 0.1931 116 A. Chapoy et al. / Fluid Phase Equilibria 220 (2004) 113–121 4.2. Model F = 0.46283 + 3.58230(ωZc ) + 8.19417(ωZc )2 A general phase equilibrium model based on uniformity of the fugacity of each component throughout all the phases [42,43] was used to model the gas solubility. The VPT-EoS [1] with the NDD mixing rules [2] was employed in calculating fugacities in fluid phases. This combination has proved to be a strong tool in modeling systems with polar as well as non-polar components [2]. The VPT-EoS [1] is given by where Zc is the critical compressibility factor, and ω is the acentric factor. Tohidi-Kalorazi [3] relaxed the α function for water, αw , using experimental water vapor pressure data in the range of 258.15–374.15 K, in order to improve the predicted water fugacity: P= RT a − v − b v(v + b) + c(v − b) ā = b= c= 2 2 a R Tc Pc b RTc (2) a = aC + aA (3) where aC is given by the classical quadratic mixing rules as follows:

aC = xi xj aij (13) i c∗ RTc (5) Pc α(Tr ) = [1 + F(1 − TrΨ )]2 (11) The above relation is used in the present work. In this work, the NDD mixing rules developed by Avlonitis et al. [2] are applied to describe mixing in the a-parameter: (4) Pc αw (Tr ) = 2.4968 − 3.0661Tr + 2.7048Tr 2 − 1.2219Tr 3 (1) with a = āα(Tr ) (10) (6) where P is the pressure, T the temperature, v the molar volume, R the universal gas constant and Ψ = 0.5. The subscripts c and r denote critical and reduced properties, respectively. The coefficients Ωa , Ωb and Ωc∗ , and F are given by Ωa = 0.66121 − 0.76105Zc (7) Ωb = 0.02207 + 0.20868Zc (8) Ωc∗ = 0.57765 − 1.87080Zc (9) (12) j and b, c and aij parameters are expressed by
b= xi bi (14) i c=
x i ci (15) √ aij = (1 − kij ) ai aj (16) i where kij is the standard binary interaction parameter. The term aA corrects for asymmetric interaction, which cannot be efficiently accounted for by classical mixing rules:

aA = xp2 xi api lpi (17) p i Table 2 Experimental and calculated methane mole fractions in the liquid phase of the methane–water system T (K) Pexp (MPa) xexp (×103 ) xcal (×103 ) x (%) 275.11 275.11 275.11 275.11 0.973 1.565 2.323 2.820 0.399 0.631 0.901 1.061 0.361 0.567 0.815 0.969 9.52 10.14 9.54 8.67 283.13 283.12 283.13 283.13 1.039 1.810 2.756 5.977 0.329 0.558 0.772 1.496 0.327 0.553 0.812 1.561 0.61 0.90 −5.18 −4.34 298.16 298.16 298.15 298.13 0.977 2.542 5.922 15.907 0.238 0.613 1.238 2.459 0.238 0.589 1.233 2.498 0.00 3.92 0.40 −1.59 313.11 313.11 313.11 313.11 1.025 2.534 7.798 17.998 0.204 0.443 1.305 2.325 0.205 0.486 1.295 2.346 −0.49 −9.71 0.77 −0.90 A. Chapoy et al. / Fluid Phase Equilibria 220 (2004) 113–121 117 Table 3 Experimental and predicted methane, ethane and n-butane mole fractions in the liquid phase of the gas mixture–water system T (K) Pexp (MPa) x(1) exp (×103 ) x(1) prd (×103 ) x (%) x(2) exp (×104 ) x(2) prd (×104 ) x (%) x(3)exp × 105 x(3)prd × (105 ) x (%) 278.14 278.15 1.032 2.004 0.339 0.646 0.337 0.629 0.59 2.63 0.199 0.387 0.207 0.364 −4.02 5.94 0.703 1.121 0.707 1.115 −0.57 0.54 283.14 283.16 283.16 283.15 283.15 283.14 283.15 0.987 1.038 1.988 2.077 3.079 3.413 3.415 0.295 0.300 0.566 0.593 0.826 0.891 0.896 0.292 0.306 0.566 0.59 0.841 0.921 0.922 1.02 −2.00 0.00 0.51 −1.82 −3.37 −2.90 0.172 0.165 0.285 0.302 0.385 0.428 0.436 0.160 0.168 0.294 0.305 0.411 0.441 0.441 6.98 −1.82 −3.16 −0.99 −6.75 −3.04 −1.15 0.512 0.482 0.754 0.941 1.063 1.121 1.048 0.487 0.507 0.809 0.831 1.013 1.052 1.053 4.88 −5.19 −7.29 11.69 4.70 6.16 −0.48 288.16 288.17 1.038 3.068 0.282 0.755 0.279 0.768 1.06 −1.72 0.172 0.399 0.168 0.43 2.33 −7.77 0.577 0.734 0.507 0.773 12.13 −5.31 298.14 298.14 298.14 298.14 298.14 0.994 2.964 7.257 11.749 14.407 0.218 0.637 1.359 2.014 2.191 0.228 0.637 1.364 1.939 2.215 −4.59 0.00 −0.37 3.72 −1.10 0.147 0.33 0.562 0.674 0.672 0.134 0.338 0.574 0.648 0.657 8.84 −2.42 −2.14 3.86 2.23 0.387 0.811 0.991 0.366 0.773 0.893 5.43 4.69 9.89 313.12 313.12 7.460 12.624 1.157 1.817 1.176 1.748 −1.64 3.80 0.406 0.586 0.498 0.594 −22.66 −1.37 0.694 0.725 −4.47 (18) 1 ln φi = RT api = √ ap ai 0 1 lpi = lpi − lpi (T − T0 )  ∞ V  ∂P ∂ni (19) where p is the index of polar components. Using the VPT-EoS [1] and the NDD mixing rules [2], the fugacity of each component in all fluid phases is calculated from:  T,V,nj =i RT − V  dV − ln Z for i = 1, 2, . . . , M fi = xi φi P (20) (21) where φi , V, M, ni , Z and fi are the fugacity coefficient of component i in the fluid phases, volume, number of 0.003 Methane Mole Fraction 0.0025 0.002 0.0015 0.001 0.0005 0 0 2 4 6 8 10 12 14 16 18 20 Pressure / MPa Fig. 2. Methane mole fraction in water rich phase in the methane–water binary system as a function of pressure at various isotherms. (䊐) 275.11 K; (䉫) 283.12 and 283.13 K; (△) 298.13, 298.15 and 298.16 K; (䊊)313.11 K; solid lines, calculated with the VPT-EoS [1] and NDD mixing rules [2] with parameters from Table 4. 118 A. Chapoy et al. / Fluid Phase Equilibria 220 (2004) 113–121 0.0025 Methane Mole Fraction 0.002 0.0015 0.001 0.0005 0 0 2 4 6 8 10 12 14 16 Pressure /MPa Fig. 3. Methane solubilities in the gas mixture–water system as a function of pressure at various isotherms. (×) 278.14 and 278.15 K; (䊊) 283.14, 283.15 and 283.16 K; (䊐) 288.16 and 288.17 K; (䉬) 298.14 K; (△) 313.12 K; solid lines, predicted with the VPT-EoS [1] and NDD mixing rules [2] with parameters from Table 4. components, number of moles of component i, the compressibility factor of the system and the fugacity of component i in the fluid phases, respectively. and for the gas mixture–water systems, respectively, and plotted in Figs. 2 and 3. As mentioned before, the BIPs between methane–water are adjusted directly to the measured methane solubility data through a Simplex algorithm using the objective function, FOB, displayed in Eq. (22):  N  1
 xi,exp − xi,cal  FOB = (22)   N x 5. Results and discussions The experimental and the calculated/predicted gas solubility data are reported in Tables 2 and 3 for the methane–water i,exp 1 0.0025 (a) 0.0013 Methane Mole Fraction Methane Mole Fraction 0.0015 0.0011 0.0009 0.0007 0.0005 0.0003 (b) 0.002 0.0015 0.001 0.0005 0 0.0001 0 0.5 1 1.5 2 2.5 3 3.5 0 4 2 Pressure /MPa 6 8 10 Pressure /MPa 0.003 Methane Mole Fraction 0.003 Methane Mole Fraction 4 (c) 0.0025 0.002 0.0015 0.001 0.0005 (d) 0.0025 0.002 0.0015 0.001 0.0005 0 0 0 2 4 6 8 10 12 Pressure / MPa 14 16 18 20 0 5 10 15 20 25 Pressure / MPa Fig. 4. Comparison of experimental methane solubilities in water in the methane–water binary system. (a) (䊐) 275.11 K; (䊊) 274.15 K from [24]; (䉬) 274.29 K from [23]. (b) (䉫) 283.13 K; (䊉) 283.15 K from [22]; (△) 283.2 K from [23]; ( ) 283.2 K from [27]; (䉱) 285.65 K from [23]. (c) (△) 298.15 K; (䉱) 298.15 K from [7]; (×) 298.15 K from [8]; (△) 298.15 from [17]; (+) 298.1 from [25]; (䊉) 298.15 K from [6]; (䉬) 298.2 K from [19]; (䊏) 298.15 from [20]; (䊊) 298.15 K from [21]; (䉬) 298.15 K from [20]; (䉫) 298.15 K from [28]. (d) (䊊) 313.11 K; (×) 310.93 from [12]; (䊏) 310.93 from [17]; (△) 313.2 K from [18]; dashed lines, predicted with the VPT-EoS [1] and NDD mixing rules [2] with parameters from Table 4. A. Chapoy et al. / Fluid Phase Equilibria 220 (2004) 113–121 Table 4 BIPs between hydrocarbons, hc and water, w for the VPT-EoS [1] and NDD mixing rules [2] System kw–hc 0 lw−hc l1w−hc (×104 ) Methane–watera Ethane–waterb n-butane–waterb 0.5044 0.4974 0.5800 1.8302 1.4870 1.6885 51.72 45.4 33.57 a For improved accuracy it is recommended to use the following BIPs between water and methane at 273.15 < T ≤ 277.13 K, while continuity conditions for gas solubility at 277.13 K is assured:kw–hc = 0.4969, l0w−hc = 1.8332 and l1w−hc (×104 ) = 58.13. b From [3]. where N is the number of data points, xi,exp the measured solubility and xi,cal the calculated solubility. The binary interaction parameters between ethane–water and n-butane–water are set to those reported by Tohidi-Kalorazi [3]. All the BIPs for hydrocarbon–hydrocarbon are set to zero. Table 4 reports all non-zero BIPs values used in this work. Our isothermal P, x data sets for the methane–water are well represented with the VPT-EoS [1] and NDD mixing rules [2]. Methane solubility data reported in the literature at a temperature range 274.15 and 313.2 K are plotted in Fig. 4. As can be shown, the isotherms in the figures are not exactly identical, but sufficiently close for comparison purposes. At the low temperature conditions (i.e. below 276 K and also around 283.15 K) and at high temperature conditions (i.e. around 313.15 K), there is good agreement between the different authors, with the exception of the data reported by Wang et al. [22] at 283.15 K. Several authors have reported solubility measurements at 298.15 K isotherm. However, some of the reported data show deviations with other authors and with the model predictions, in particular at pressures higher than 6 MPa (e.g. Michels et al. [6] and Kim et al. [28]). The data of Yang et al. [25] are dispersed at this temperature. Our isothermal P, x data sets for the gas mixture–water are also well represented with the VPT-EoS [1] and NDD mixing rules [2] for the majority of the experimental results (Average absolute deviation (AAD) of 1.82% for methane solubilities, AAD of 4.86% for ethane solubilities, and AAD of 5.56% for n-butane solubilities). An increase in AAD with an increase in carbon number is expected, due to low solubility of ethane and n-butane and hence the analytical work is more difficult (higher uncertainty in the calibration of the detector). 6. Conclusions Accurate gas solubility data especially at low temperature conditions are necessary for developing and validating thermodynamic models for water–hydrocarbon systems. New experimental data on the solubility of methane in water were generated at low temperature conditions (i.e. 275.11–313.11 K) and pressures up to 18 MPa 119 using a static–analytic apparatus, taking advantage of a high-pressure capillary sampler. The new experimental data were employed in tuning the BIPs between methane and water, with the aim of improving the reliability of a thermodynamic model. Another set of gas solubility data was generated on a ternary gas mixture containing methane (94%), ethane (4%) and n-butane (2%) at low temperature conditions (i.e. 278.14–313.12 K) and pressures up to 14.407 MPa. Using the tuned BIPs for the methane–water system and the previously reported BIPs for the other compounds, the solubility of each gas in water were predicted. The predictions were in good agreement with the experimental data, demonstrating the reliability of the techniques and model used in this work. List of a b c f F ITS k l M n N P R T u v V w x Z symbols attractive parameter of the equation of state parameter of the equation of state parameter of the equation of state fugacity parameter of the equation of state international temperature scale binary interaction parameter for the classical mixing rules dimensionless constant for binary interaction parameter for the asymmetric term number of components number of moles number of experimental points pressure universal gas constant temperature parameter of the equation of state in general form molar volume volume parameter of the equation of state in general form mole fraction compressibility factor Greek letters α temperature dependent function ∆ deviation φ fugacity coefficient Ψ power parameter in the VPT-EoS Ω parameter in the VPT-EoS ω acentric factor Superscripts A asymmetric properties C classical properties 0 non-temperature dependent term in NDD mixing rules 1 temperature dependent term in NDD mixing rules 120 A. Chapoy et al. / Fluid Phase Equilibria 220 (2004) 113–121 Subscripts a index for properties b index for properties c critical property c∗ index for properties cal calculated property exp experimental property i, j molecular species hc hydrocarbon compound p polar compound prd predicted property r reduced property w water 0 reference property (1) methane (2) ethane (3) n-butane W= Pw RT (A.6) Z= Pv RT (A.7) and A′i = 1 ∂(n2 a) na ∂ni Bi′ = 1 ∂(nb) b ∂ni Ui′ = 1 ∂(nu) u ∂ni Wi′ = 1 ∂(nw) w ∂ni (A.8) T,nj =i (A.9) T,nj =i (A.10) T,nj =i (A.11) T,nj =i The compressibility factor Z is given by the following dimensionless equation [44]: Acknowledgements The financial support by the European Infrastructure for Energy Reserve Optimization (EIERO) provided the opportunity for this joint work which is gratefully acknowledged. Z3 −(1 + B − U)Z2 + (A − BU − U − W 2 )Z Appendix. Calculation of fugacity coefficient using an EoS References The following general form can be used for expressing any cubic equation of state [44]: RT a P= (A.1) − 2 v − b v + uv − w2 The fugacity coefficient for component i in a mixture can be expressed as [44]: Bi′ B A ln φi = −ln(Z − B) + +√ 2 Z−B U + 4W 2   U ′ U 2 + 4Wi′ W 2 × A′i − i 2 U + 4W 2   √ 2Z + U − U 2 + 4W 2 ×ln √ 2Z + U + U 2 + 4W 2    2 (2Z + U) Wi′ W 2 + UZ − 2W 2 Ui′ U   2 −A Z2 + UZ − W 2 U 2 + 4W 2 (A.2) where A= Pa (RT)2 Pb RT Pu U= RT B= (A.3) (A.4) (A.5) −(AB − BW2 − W 2 ) = 0 (A.12) [1] J.O. Valderrama, J. Chem. Eng. Jpn. 231 (1990) 87–91. [2] D. Avlonitis, A. Danesh, A.C. Todd, Fluid Phase Equilib. 94 (1994) 181–216. [3] B. Tohidi-Kalorazi, Ph.D. Thesis, Heriot-Watt University, 1995. [4] R. Bunsen, Ann. Chem. Pharm. 93 (1855) 1–50. [5] P.K. Frolich, E.J. Tauch, J.J. Hogan, A.A. Peer, Ind. Eng. Chem. 23 (5) (1931) 548–550. [6] A. Michels, J. Gerver, A. Bijl, Physica III (1936) 797–808. [7] O.L. 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