Prediction of VaporâLiquid Equilibrium and Thermodynamic

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Thermodynamics, Transport, and Fluid Mechanics

Prediction of vapor-liquid equilibrium and thermodynamic properties of natural gas and gas condensates Nefeli Novak, Vasiliki Louli, and Epaminondas Voutsas Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.9b00756 • Publication Date (Web): 14 Mar 2019 Downloaded from http://pubs.acs.org on March 17, 2019

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Industrial & Engineering Chemistry Research

Prediction of vapor-liquid equilibrium and thermodynamic properties of natural gas and gas condensates

Nefeli Novak*,1, Vasiliki Louli1, Epaminondas Voutsas1

1Laboratory

of Thermodynamics and Transport Phenomena

School of Chemical Engineering, National Technical University of Athens 9, Heroon Polytechniou Str., Zografou Campus 15780 Athens, Greece

*

Corresponding author. Tel: +30 210 772 3230 E-mail: [email protected]

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Abstract In this work the performance of the UMR-PRU model, which combines the Peng-Robinson (PR) equation of state (EoS) with the original UNIFAC through the Universal Mixing Rules (UMR), is further improved and updated through the use of a Mathias-Copeman (MC) function for the attractive term parameter of the PR EoS. The new model, called UMR-MCPRU, utilizes a newly parametrized MC function for the 𝑎 term, which satisfies consistency constrains that ensure its safe extrapolation to the supercritical region. Moreover, a correlation of the MC parameters of hydrocarbons with acentric factor is proposed. The MCPR equation of state (EoS) is then combined with the UMR mixing rules, resulting to the UMR-MCPRU model. All binary interaction parameters relevant to natural gas mixtures are determined by fitting binary vapor-liquid equilibrium data. UMR-MCPRU is used to predict vapor-liquid equilibrium, critical points, liquid dropouts, densities (𝜌) and derivative thermodynamic properties, namely isobaric (𝑐𝑃) and isochoric (𝑐𝑉) heat capacity, Joule-Thomson coefficients (𝜇𝐽𝑇) and speed of sound (𝑤), of natural gas and gas condensate mixtures. The Peneloux volume translation is also examined, wherever relevant. The results reveal that UMR-MCPRU coupled with the Peneloux translation is able to accurately describe both phase equilibria and other important thermodynamic properties of natural gas and gas condensate mixtures.

Keywords: natural gas, gas condensates, phase equilibrium, derivative properties, UMRMCPRU, Mathias-Copeman

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1. Introduction From an engineering point of view, the oil and gas industry covers extraction, refining and transportation of petroleum products and is divided to three main areas: upstream, midstream and downstream. Midstream and downstream covers all processes that follow the fluid extraction and entirely rely on commercial simulators (UniSim, PRO/II, ASPEN, etc.) for process design, operation and optimization. Process simulators solve mathematical problems of variable complexity but the accuracy of their results mainly depends on the quality of the thermodynamic model used in the simulation. These models need to successfully describe the phase equilibrium and other thermophysical properties in a wide range of conditions and at the same time they must have a predictive character in order to overcome the lack of experimental data. Even though in recent years Statistical-Associating-Fluid-Theory (SAFT) 1 type EoS, like PC-SAFT 2, 3 gather the interest of the academic community, the most popular thermodynamic models in the industry are still based on cubic Equations of State (cubic EoS), especially SoaveRedlich-Kwong (SRK)

4

and Peng-Robinson (PR) 5. This stems for their simplicity, that

translates to reduced computational time, their fairly good results in vapor-liquid equilibrium calculations, and last but not least their predictive character, due to the group contribution approach. The most popular groups contribution cubic EoS is PPR78 6, 7, while several more have been introduced in the form of EoS/GE models, such as PSRK 8, LCVM 9-11, VΤPR 12-14 and UMR-PRU 15-20. A predictive model with applications to the petroleum industry should also be able to handle complex multicomponent mixtures, like natural gas mixtures that usually contain components up to normal decane. Nasrifar et. al

21, 22

have published a comparison of many EoS in the

prediction of phase equilibria of natural gas mixtures. Recently they turned their attention to predictive cubic EoS 23 for phase equilibria and critical points of natural gas mixtures, as well as phase envelopes of synthetic and real natural gas mixtures 24. Predictive PR (PPR78) that is able to accurately describe VLE of asymmetric binary mixtures with accuracy comparable to EoS/GE models, when it comes to multicomponent mixtures yields inferior results 6, 25. Another predictive model, that is very successful in predicting dew points of natural gases is UMRPRU, and has been found superior to the most established EoS/GE models (LCVM, PSRK 17 and VTPR 25) as well as cubic EoS and non-cubic EoS, like PR, SRK, PPR78 and PC-SAFT 3 ACS Paragon Plus Environment


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EoS 18, 19, 25. This model combines UNIFAC with PR EoS through the Universal Mixing Rules (UMR) and has been successfully applied to various multicomponent mixtures like natural gases 17-20, polar and associating mixtures 26 and quite recently to mixtures that contain mercury 27.

Apart from natural gas mixtures, that can be classified as asymmetric, the real challenge for any thermodynamic model is gas condensate mixtures, which may contain up to compounds of 200 carbon number

28,

exhibiting therefore very high asymmetry. Recent attempts at

modeling phase envelopes of asymmetric natural gas mixtures with cubic EoS and non-cubic EoS include new fitting of interaction parameters on VLE data

29

and expansion of the VLE

database for fitting binary interaction parameters through Monte Carlo simulation 30, and the use of the three parameter cubic EoS RKPR EoS

31.

RKPR yields improved predictions

compared to PR 31, while the UMR-PRU model is found superior to PR, SRK and PC-SAFT EoS 20. Moreover, the location of the critical point on the phase envelope has been the focus of many studies over the years. The critical locus of binary mixtures of methane with hydrocarbons up to decane has been examined by Polishuk et al. 32 and Jaubert et al. 6, while asymmetric mixtures of methane with heavier alkanes have been studied by Duarte et al. 33 and Nikolaidis et al.

34.

Critical points of synthetic mixtures containing natural gas components

have also been studied

23, 35-39,

although the available experimental data are limited to non-

asymmetric or slightly asymmetric mixtures. Cubic EoS like PR 35, 39 and PT 39 and non-cubic EoS like PC-SAFT and SAFT 35, as well as predictive cubic EoS like PPR78 and PSRK 23 have been used for this purpose. Apart from VLE predictions, accurate prediction of the thermodynamic properties of natural gases is also required for flow metering purposes under custody transfer and pipeline transmission conditions

40.

Such properties include primary thermodymanic properties, like

density (𝜌), as well as derivative properties, like isobaric (𝑐𝑃) and isochoric (𝑐𝑉) heat capacity, speed of sound (𝑤) and Joule-Thomson coefficients (𝜇𝐽𝑇). The usual way to improve the volumetric behavior of cubic EoS is volume translation, initially introduced by Peneloux et al. 41. Many different expressions have been used since to calculate the volume translation 12, 42-45, although only the temperature independent ones retain thermodynamic consistency

46, 47.

For natural gases, SRK and PT seems to yield better 4

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saturation density predictions than PR, as shown by Salehpour et al.

48

and Nasrifar et al. 22.

However, for gas condensate mixtures, PR and three parameter EoS (PT, PTV) have an advantage over SRK, with three parameter EoS being the most accurate

21, 48, 49.

Recently,

Yan et al. 29 showed that PC-SAFT yields on average similar results with the untranslated PR, highly superior to those of SRK. Apart from the saturation densities, for liquid dropout predictions the equilibrium ratio is also required. Such calculations are less accurate near the dew point pressure, especially close to the critical point affected by volume translation

20.

50,

and seem to be insignificantly

Liquid dropout predictions with UMR-PRU, PR and PC-

SAFT indicate that these models have similar accuracy and are superior to SRK 20, 50. Although the saturation pressure and volumetric properties are well studied with cubic EoS, the study of derivative properties is relatively recent. The evaluation of derivative properties predictions are not entirely reliable, since the actual experimental data are very few and most evaluations are based on “pseudo-experimental” data, calculated by reference EoS

51

or

molecular simulations 52-54. Derivative properties of pure components present a variety of extrema with temperature and density, mostly connected to the critical point but not only at near critical conditions 51, 55; these extrema disappear as temperature increases, until the value of the ideal gas state is approached at high temperatures and moderate pressures. However, at very high pressures a large discrepancy with the ideal gas behavior is observed 55. Between cubic EoS and non-cubic EoS (PR and SAFT-BACK EoS) cubic EoS are less accurate in the prediction of second derivative properties

56,

while PC-SAFT was found

superior to SAFT EoS 57. For binary CCS mixtures 58 PC-SAFT is, on average, more accurate than cubic EoS and SAFT, and the same stands for ethylene mixtures59. However, when a binary interaction parameter fitted to the experimental data is used, model correlations with cubic EoS and non-cubic EoS are of comparable accuracy. For natural gas mixtures, Nasrifar et al. 22 report that for 𝑤, PR yields slightly better predictions than SRK; the reverse is observed for 𝑐𝑃, while for 𝜇𝐽𝑇, SRK is superior to PR. Αn appropriate temperature dependency of the cohesive parameter 55 and/or a temperature and pressure dependent co-volume parameter 51 may lead to an improvement of the cubic EoS predictions, while volume translation only affects some of the derivative properties of fluids (



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𝑤 and 𝜇𝐽𝑇), and has no effect on others (𝑐𝑃 and 𝑐𝑉) 60. Such modifications for the improvement of 𝜇𝐽𝑇 61-64, enthalpy of vaporization and heat capacity 45 have been reported in the literature. 2. Temperature dependency of the attractive term of cubic EoS (𝒂(𝑻)) All van der Waals (vdW) type cubic EoS can be written in the following general form 65: 𝑃=

𝑎𝑐 ∙ 𝑎(𝑇) 𝑅𝑇 ― 𝑣𝑡𝑟 ― c ― 𝑏 (𝑣𝑡𝑟 ― c + 𝑏𝛿1)(𝑣𝑡𝑟 ― c + 𝑏𝛿2)

Eq. 1

where 𝑃 is pressure, 𝑇 is temperature, 𝑣𝑡𝑟 is the translated molar volume, 𝑐 is the volume translation, 𝑅 is the universal gas constant, 𝑎(𝑇) is the energy parameter of the EoS accounting for dispersion forces, 𝑎𝑐 is the corresponding value at the critical point and 𝑏 the repulsive parameter, representing the repulsion between molecules when their effective volumes adjoin. In the case of two parameter EoS, 𝛿1 and 𝛿2 are EoS specific constants (for PR EoS: 𝛿2 = 1 + 2 , 𝛿1 = 1 ― 2). The generalized expression for the Peneloux translation 𝑐 for PR EoS 60 is shown in Eq. 2: 𝑅𝑇𝑐

(0.1154 ― 0.4406 𝑧𝑅𝐴) 𝑃𝑐 𝑧𝑅𝐴 = 0.29056 ― 0.08775𝜔

𝑐 =

Eq. 2

Since the temperature dependency of the energy parameter of cubic EoS, 𝑎(𝑇), is highly relevant to vapor pressure predictions, many alternatives have been presented in the literature, the most common of which are polynomial or exponential functions of temperature. The most popular examples of polynomial types of 𝑎 functions is that of Soave 4, Mathias 66, Mathias and Copeman

67

and Stryjek and Vera

68.

Examples of exponential dependency have been

proposed by Boston and Mathias 69, Trebble and Bishnoi 70, Malhem 71 and Twu 72, 73. Recently, Nasrifar and Moshfeghian

74

came up with an 𝑎 function by combining the second virial

coefficient and the square well potential function. The same combination was also used in combination with the Soave function for the subcritical region 75. The 𝑎 functions are usually generalized with the use of the acentric factor. The Soave 𝑎 function was generalized by Soave 4, Peng and Robinson 5, 76, Graboski and Daubert 77, Tassios and Magoulas

44

and recently Pina-Martinez et al.78. The Mathias-Copeman expression has 6 ACS Paragon Plus Environment

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been generalized by Coquelet et al. 79, while the Twu 𝑎 function has been generalized by Twu 80, 81.

A usual practice with 𝑎 functions is to split the subcritical and supercritical domains and utilize different properties for the parametrization of the 𝑎 function, or even different temperature dependencies for each region. This stems from the need to accurately describe both the subcritical and supercritical domains, and not rely on extrapolation to an unknown region for either one of the two domains. Safe extrapolation can be achieved either by fitting the properties for both regions or by introducing a theoretical criterion to safely extrapolate to the unknown temperature domain. Unknown extrapolation into the supercritical region is usually observed. The Soave expression is an extrapolation into the supercritical region in all of the aforementioned publications. Since some 𝑎 functions adopt the Soave expression at the supercritical region, like the Mathias and Copeman 67 and Boston and Mathias 69, the same also applies to them. Twu was the first to use different parametrization schemes for the sub- and supercritical domains in the generalization of the 𝑎 functions 80, 81. For the subcritical domain, he fitted vapor pressure data, as is the common practice, while for the supercritical one, Henry’s constants for hydrogen-hydrocarbon and methane-hydrocarbon mixtures were used instead. Different parametrization for supercritical components is also suggested by Floter et al.

82

who used

methane fugacities to fit the parameters of various 𝑎 functions at supercritical temperatures, while a similar procedure was followed by Morch et al. 83. Unfortunately, the use of different function or different parameters at sub-and supercritical regions may lead to inconsistencies near the critical point, which are important in the modelling of pure fluids, or mixtures with one major component, like natural gas. To address this issue theoretical criteria have been introduced to establish the consistency of 𝑎 functions and safely extrapolate to the unknown temperature domain (Table 1). Twu 73 introduced some consistency requirements: (a) it must be finite and positive for all temperatures, (b) the deviation function 𝑎(𝑇) must be unity at the critical point and (c) it must reach a finite value at the infinite temperature limit. Additional criteria (d in Table 1) for a theoretically sound and at least qualitatively accurate 𝑎 function were introduced by Neau et al.

84, 85,

who observed that the first and second derivative of the 𝑎 function are related to

residual enthalpy and residual isobaric heat capacity; both derivatives should be continuous, 7 ACS Paragon Plus Environment


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while the first derivative should always be negative. Following the same path, Le Guennec et al.

86

recently added more criteria related to the isobaric and isochoric heat capacity: (e) the

second derivative should be positive and (f) the third derivative should always be negative. These criteria, are formulated and briefly explained in Table 1.

Table 1. Consistency constrains for 𝑎 functions 86. The 𝑎 function represents the strength of dispersion forces with a 𝑎 = continuous temperature, therefore should be continuous. lim 𝑎(𝛵) = constant The 𝑎 function should reach a finite value, at the infinite b 𝑇→∞ temperature limit. Dispersion forces are responsible for bringing molecules together; they decrease the pressure of a system hence the attractive term of c 𝑎(𝛵) ≥ 0 𝑎𝑐 ∙ 𝑎(𝑇)

d e f

𝑑𝑎(𝛵) ≤0 𝑑𝑇 𝑑2𝑎(𝛵) ≥0 𝑑𝑇2 𝑑3𝑎(𝛵) ≤0 𝑑𝑇3

the EoS ― (𝑣 + 𝛿1𝑏)(𝑣 + 𝛿2𝑏) should always be negative Since the 𝑎(𝛵) reaches a finite value at the infinite temperature limit, its derivative should become zero at infinite temperature. Inflection points in the second derivative

(

𝑑2𝑎(𝛵) 𝑑𝑇2

)

= 0 cause

crossing isobars in 𝑐𝑣 vs T Wave shape in the 𝑐𝑝 vs T plot that does not agree with reference EoS or known experimental data.

To comply with said criteria, Le Guennec et al.

45

parametrized the Twu 𝑎 function

according to all these constrains and Mahmoodi et al. 87, 88 modified five 𝑎 functions that change forms at the critical point to fix any abnormal behaviors. The aim of this work is to further improve the predictive, cubic EoS/GE model UMR-PRU, by substituting the Soave 𝑎 function with a consistent one based on the Mathias-Copeman expression. The resulting model, called UMR-MCPRU, has therefore consistent supercritical behavior for pure components, a good mixing rule to account for asymmetric mixtures; the volume translation (Eq. 2) proposed by Peneloux is utilized 60, wherever relevant. This model is tested in the prediction of phase equilibria, volumetric and derivative properties for natural gases and gas condensates.


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3. Results and Discussion 3.1. Consistent Mathias-Copeman (MC) 𝒂 function The polynomial form of the Mathias-Copeman function implies a minimum in the 𝑎 function with respect to temperature and therefore cannot comply with the consistency criteria (a-f) described above for all temperatures. However, ensuring that the function is consistent within a safe temperature range for practical applications is a good compromise. In this work, the upper temperature limit for consistency has been set to 1000K. For PR EoS, new MC parameters are calculated and then generalized with acentric factor. To avoid abnormal behavior at the critical point and comply with the aforementioned criteria, unlike the original version of Mathias-Copeman, the MC polynomial expression is used both at subcritical and supercritical temperatures: 2

2 3 𝑎(𝑇) = [1 + 𝐶1(1 ― 𝑇𝑟) + 𝐶2(1 ― 𝑇𝑟) + 𝐶3(1 ― 𝑇𝑟) ] for all Τ

Eq. 3

The parameters are obtained using a least squares unconstrained minimization routine, where the consistency constrains of Table 1 were enforced by means of the internal penalty method. The objective function was the absolute relative deviation of vapor pressures (30 data points) taken from DIPPR 89 spanning the triple point temperature (Ttr) up to the critical point temperature (Tc): 𝑁𝑃

𝐹=

∑ 1

𝑎𝑏𝑠(𝑃𝑒𝑥𝑝 ― 𝑃𝑝𝑟𝑒𝑑) 𝑃𝑒𝑥𝑝

= min from Ttr to Tc

Eq. 4

Critical properties and acentric factors are also taken from the DIPPR database 89. The obtained MC parameters are tabulated in Table 2 for light components present in natural gas, while for the rest hydrocarbons (HC) are given in Supporting Information Table S1.



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Table 2. Component specific Mathias-Copeman parameters of light components for PR EoS. 𝐴𝐴𝑅𝐷𝑃 𝑠𝑎𝑡%a 𝐶1 𝐶2 𝐶3 Ttr [K] Tc [K] Comp CH4 C2H6 N2 CO2

90.69 90.35 63.15 216.58

190.56 305.32 126.10 304.19

𝐴𝐴𝑅𝐷𝑃 𝑠𝑎𝑡% =

-0.07105 -0.00610 -0.01390 0.04926

0.00338 0.05378 0.01435 0.08945

0.5 1.1 0.1 0.4

1 𝑁𝑃|𝑃𝑒𝑥𝑝 ― 𝑃𝑝𝑟𝑒𝑑| ∑ % 𝑁𝑃 1 𝑃𝑠𝑎𝑡 𝑒𝑥𝑝 𝑠𝑎𝑡

a

0.41169 0.53176 0.44213 0.69775

𝑠𝑎𝑡

NP: 30 experimental points from DIPPR 89 spanning a temperature range from Ttr to Tc For generalization purposes, the specific parameters tabulated in the Supporting Information Table S1 have been used for the correlation of the MC parameters of HC heavier than ethane with acentric factor. The resulting equations are the following (Eq. 5a-c): 𝐶1 = 0.396 + 1.3644𝜔 𝐶2 = ―0.0964 + 0.6593𝜔 ― 1.0793𝜔2

Eq. 5a-c

𝐶3 = 0.1656 ― 0.0609𝜔 + 1.1139𝜔2 To the best of our knowledge, the only generalized form of Mathias-Copeman in the literature is the one proposed by Coquelet et al. 79. The comparison between the generalized function of this work with those of Soave and Coquelet per HC type is shown in Table 3 and Supporting Information Table S2. The results with the new correlation are superior than those of Coquelet et al., especially for the high molecular weight normal alkanes. It should be noted that the MC parameters for n-C21 up to n-C30 in this work were not used in the correlation and the presented results are pure predictions. In the rest of this work, the MC parameters used are the component specific values for the light gases present in natural gas, i.e. N2, CO2, CH4 and C2H6 (Table 2), and the ones predicted by the proposed correlation for heavier hydrocarbons.


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Table 3. Deviation (𝐴𝐴𝑅𝐷𝑃 𝑠𝑎𝑡%a ) of predicted vapor pressures from triple to critical point with PR EoS with the generalized Mathias-Copeman parameters of this work, Coquelet and Soave. NCb HC Type Soave Coquelet This work 18 n-alkanes C3 up to n-C20 11.7 5.4 1.6 8 n-alkanes n-C21 up to n-C30 42.7 27.4 5.2 23 branched alkanes 14.5 2.5 2.7 5 cyclo-alkanes 15.4 2.7 4.7 5 Aromatic HC 5.4 1.9 1.8 59 Overall 15.8 6.4 2.6 1 𝑁𝑃 ∗ 𝑁𝐶|𝑃𝑒𝑥𝑝 ― 𝑃𝑝𝑟𝑒𝑑| ∑ % 𝑁𝑃 ∗ 𝑁𝐶 1 𝑃𝑠𝑎𝑡 𝑒𝑥𝑝 𝑠𝑎𝑡

a

𝐴𝐴𝑅𝐷𝑃

𝑠𝑎𝑡

%=

𝑠𝑎𝑡

NP: 30 experimental points from DIPPR 89 spanning a temperature range from Ttr to Tc b NC: number of compounds per HC type

3.2. Prediction of pure compound properties with MCPR Vapor pressure (𝑃𝑠𝑎𝑡), volumetric and derivative property predictions, namely density (𝜌), isobaric (𝑐𝑃) and isochoric (𝑐𝑉) heat capacity, speed of sound (𝑤) and Joule-Thomson coefficients (𝜇𝐽𝑇) of pure components have been performed: (a) along the saturation line from Ttr to Tc (30 points per isotherm), and (b) for 8 isotherms between 150 K - 500 K and 10 bar - 500 bar (50 points per isotherm). Experimental data are derived from NIST

90.

At the

saturation line the deviations with the Soave expression and the new MC 𝑎 function have been calculated separately for the saturated liquid and vapor phase (Table 4). Corresponding values with translated models for 𝜌, 𝑤 and 𝜇𝐽𝑇 are given in Table 5. Results at the one phase region with the untranslated models are shown at Table 6, while their corresponding values with volume translation in Table 7. The ideal gas properties used for the calculation of 𝑐𝑃 and 𝑐𝑉 were taken from DIPPR 89. At the saturation, amongst the studied properties those with the higher uncertainty were 𝜇𝐽𝑇 for both phases (>100% for the liquid phase and apr. 40% for the vapor phase), followed by 𝑤 of the liquid phase (15%). 𝑐𝑃 and 𝑐𝑉 are calculated with similar accuracy for both phases (7%8% for 𝑐𝑃 and 5%-10% for 𝑐𝑉), and so is liquid phase 𝜌 (6%-7%). The most accurate property with the UMR models is 𝑤 of the vapor phase (1%). The most important differences between the models along the saturation curve are observed for the saturation pressure that is naturally 11 ACS Paragon Plus Environment


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better with the MCPR, for the vapor phase 𝜌 which is improved with MCPR and the liquid phase 𝑐𝑉 for which the MCPR yields worse predictions. The translated models yield greatly improved liquid phase 𝜌, while an improvement is also observed in the prediction of 𝜇𝐽𝑇. Moreover, it was observed that for both models 𝑤 and 𝜇𝐽𝑇predictions of light gases are favored by the volume shift, while the opposite behavior is observed for heavier hydrocarbons, as shown in Figure 1. This is peculiar considering that for higher molecular weight alkanes, where the translation yields the most significant improvement of 𝜌 predictions, worst accuracy is observed for property predictions and proves that the EoS predictions for 𝑤 and 𝜇𝐽𝑇 are subjected to a great extent to cancelation of errors.


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50

20

P (MPa)

40

15

30 10 20 5

10 0

0 0

5

10

15

20

25

30

0

1

2

3

4

5

𝜌 (mol/lt) 50

20

40

P (MPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


15

30 10 20 5

10 0

0 200

2000

40

400

4000

𝑤 (K/MPa) Figure 1. Saturated 𝜌 (top) and 𝑤 (bottom) of methane (left) and dodecane (right) with MCPR. Solid black line: MCPR, dashed black line: MCPR-Peneloux, markers: experimental data from NIST 90 at the saturation (+).


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Table 4. Deviations (𝐴𝐴𝑅𝐷%a) in various properties with untranslated MCPR and PR EoS along the saturation line. 𝑃𝑠𝑎𝑡 𝑐𝑃 𝜌 PR MCPR PR MCPR HC T (K) P (bar) PR MCPR Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor CH4 90.8 190.4 0.7 0.5 9.1 1.5 9.2 1.3 8.2 11.1 9.1 11.3 C2H6 90.5 305.1 3.9 0.4 7.0 4.2 7.0 0.8 7.2 12.9 7.6 12.9 C3 85.5 366.8 13.2 5.2 5.3 13.1 5.3 5.4 7.1 10.1 13.8 10.2 n-C4 134.9 424.9 5.0 1.6 4.6 5.3 4.7 2.1 6.5 5.9 4.8 5.8 n-C5 143.5 469.2 6.7 2.1 3.4 6.9 3.4 2.5 6.7 5.3 4.4 5.3 n-C8 216.4 568.5 5.5 1.2 5.5 5.5 5.5 1.4 14.7 9.8 12.2 9.7 n-C10 243.6 617.4 5.2 2.3 7.3 5.5 7.2 2.7 4.6 3.0 2.6 3.0 n-C12 263.6 656.8 8.9 0.8 10 8.7 9.9 1.5 7.0 2.4 4.6 2.4 Overall 6.1 1.8 6.5 6.3 6.5 2.2 7.8 7.6 7.4 7.6 𝑐𝑉 𝜇𝐽𝑇 𝑤 PR Liquid Vapor CH4 4.8 11.3 C2H6 7.9 13.9 C3 8.4 10.1 n-C4 5.8 4.3 n-C5 5.3 3.5 n-C8 3.1 1.4 n-C10 4.3 1.2 n-C12 3.2 1.5 Overall 5.3 5.9 HC

a

𝐴𝐴𝑅𝐷% =

MCPR Liquid Vapor 5.7 11.6 13.8 13.9 24.9 10.2 11.3 4.2 10.8 3.5 6.7 1.4 7.6 1.2 6.0 1.5 10.8 6.0

PR MCPR PR MCPR Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor 14.9 1.5 13.7 1.6 89.7 15.2 86.5 15.3 15.0 1.5 15.3 1.5 121.1 91.1 123.4 91.5 15.9 1.0 16.4 1.0 201.3 33.5 203.9 32.8 15.4 0.8 15.6 0.8 92.1 34.5 89.2 34.1 15.7 0.5 16.0 0.5 248.9 38.1 246.2 37.7 13.7 0.9 13.8 0.9 202.3 40.0 200.5 39.6 13.5 1.3 15.3 1.3 77.4 44.5 73.9 44.0 13.3 1.5 13.3 1.6 68.3 40.7 64.5 40.2 14.7 1.1 14.9 1.1 137.6 42.2 136.0 41.9

1 𝑁𝑃|𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑒𝑥𝑝 ― 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑝𝑟𝑒𝑑| ∑ % |𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑒𝑥𝑝| 𝑁𝑃 1

NP: 30 experimental points from NIST 90 spanning a temperature range from Ttr to Tc


Page 15 of 56 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46


Table 5. Deviations (𝐴𝐴𝑅𝐷%a) in various properties with translated MCPR and PR EoS along the saturation line. 𝜇𝐽𝑇 𝑤 𝜌 PR Peneloux MCPR Peneloux PR Peneloux MCPR Peneloux PR Peneloux MCPR Peneloux HC Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor Liquid Vapor CH4 2.9 1.1 2.8 1.0 9.0 1.7 8.3 2.0 62.8 15.5 62.5 15.5 C2H6 3.2 4.1 3.2 0.7 12.8 1.6 13.1 1.6 99.0 91.0 95.9 91.4 C3 3.2 13 3.1 5.3 14.6 0.9 15.2 0.9 171.2 33.6 165.3 33.0 n-C4 3.0 5.2 3.0 2.0 13.5 0.7 13.8 0.7 84.2 34.6 80.2 34.2 n-C5 3.3 6.8 3.3 2.4 14.6 0.5 15.0 0.5 231.1 38.1 228.0 37.8 n-C8 3.3 5.5 3.3 1.4 15.2 0.8 15.2 0.8 217.8 40.0 216.4 39.6 n-C10 2.8 5.3 2.8 2.6 18.0 1.1 19.7 1.1 85.3 44.4 82.7 43.9 n-C12 3.8 8.4 3.9 1.2 20.3 1.2 19.7 1.2 81.4 40.6 78.5 40.0 Overall 3.2 6.2 3.2 2.1 14.8 1.1 15.0 1.1 129.1 42.2 126.2 41.9 a

𝐴𝐴𝑅𝐷% =

1 𝑁𝑃|𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑒𝑥𝑝 ― 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑝𝑟𝑒𝑑| ∑ % |𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑒𝑥𝑝| 𝑁𝑃 1

NP: 30 experimental points from NIST 90 spanning a temperature range from Ttr to Tc



Page 16 of 56

In the one phase region amongst the studied properties those with the higher uncertainty were again 𝜇𝐽𝑇 (100%), followed by 𝑤 (10%). Deviations for 𝑐𝑃 are lower than 4% and for 𝑐𝑉 lower than 7%, while 𝜌 predictions deviate around 5% from the experimental data. Outside the saturation line 𝑐𝑃 is greatly improved with MC yielding almost half the deviations of the Soave expression, unlike 𝑐𝑉 which is more accurately predicted by the Soave expression of original PR. For 𝜌, 𝑤, and 𝜇𝐽𝑇 the accuracy of the two models is similar. In this region, the introduction of volume translation in the EoS, improves all related properties (𝜌, 𝑤, and 𝜇𝐽𝑇). Table 6. Deviations (𝐴𝐴𝑅𝐷%a) in various isothermal properties with untranslated MCPR and PR EoS. 𝑐𝑃 𝜌 HC T (K) P (bar) NP b PR MCPR PR MCPR CH4 150-500 10-500 399 4.7 3.9 1.1 1.4 C2H6 150-500 10-500 397 5.5 5.5 2.3 1.6 C3 150-500 10-500 397 5.7 5.7 3.3 1.9 n-C4 150-500 10-500 397 5.0 5.0 3.6 1.5 n-C5 150-500 10-500 363 4.9 4.9 6.6 3.7 n-C8 250-500 10-500 296 2.3 2.2 3.9 0.8 n-C10 250-500 10-500 259 4.6 4.5 3.6 0.5 n-C12 300-500 10-500 243 7.7 7.6 4.5 1.2 Overall 5.0 4.9 3.6 1.6 𝑐𝑉 𝜇𝐽𝑇 𝑤 HC PR MCPR PR MCPR PR MCPR CH4 1.6 1.6 6.6 5.6 350.8 389.9 C2H6 3.8 4.7 9.3 9.2 66 68.5 C3 4.7 7.9 11.1 11.1 98.3 99.4 n-C4 4.9 9.2 12 12.2 79.8 76.1 n-C5 3.4 5.1 14 14.1 69.1 65.0 n-C8 3.5 7.2 8.8 9.1 42.6 38.3 n-C10 5.0 8.6 7.8 8.1 34.4 29.9 n-C12 3.9 7.5 9.1 9.6 38.1 33.1 Overall 3.9 6.5 9.8 9.9 97.4 100.0 1 𝑁𝑃|𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑒𝑥𝑝 ― 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑝𝑟𝑒𝑑| ∑ % |𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑒𝑥𝑝| 𝑁𝑃 1

a

𝐴𝐴𝑅𝐷% =

b

NP: experimental points from NIST 90


Page 17 of 56

Table 7. Deviations (𝐴𝐴𝑅𝐷%a) in various isothermal properties with translated MCPR and PR EoS. 𝜇𝐽𝑇 𝜌 𝑤 HC PR MCPR PR MCPR PR MCPR Peneloux Peneloux Peneloux Peneloux Peneloux Peneloux CH4 0.9 1.5 2.5 2.5 185.5 222.4 C2H6 1.3 1.4 6.2 6.2 41.6 43.4 C3 1.7 1.7 8.9 8.9 70.1 70.4 n-C4 2.1 2.1 10.8 11.0 63.6 59.6 n-C5 2.7 2.7 12.6 12.8 64.3 60.0 n-C8 2.0 1.9 9.3 9.5 41.7 37.4 n-C10 2.9 2.9 9.2 9.1 25.1 20.8 n-C12 3.7 3.7 8.8 8.6 18.6 13.7 Overall 2.2 2.2 8.5 8.6 63.8 66.0 1 𝑁𝑃|𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑒𝑥𝑝 ― 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑝𝑟𝑒𝑑| ∑ % |𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑒𝑥𝑝| 𝑁𝑃 1

a

𝐴𝐴𝑅𝐷% =

b

NP: experimental points from NIST 90

Properties of methane are of great significance since it is the major component of natural gas mixtures and the prediction of its properties are also graphically shown in Figure 2 to Figure 5. In Figure 2 the volume translated models are more than capable to accurately describe the 𝜌 of methane, especially at high temperatures in the one phase region.

P (MPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5

50

4

40

3

30

2

20

1

10

0

0 0

10

20

30

0

10

20

30

𝜌 (mol/lt) Figure 2. Saturated (left) and isothermal (right) 𝜌 prediction for pure methane. Solid black line: MCPR, solid red line: PR, dashed black line: MCPR-Peneloux, dashed red line: PR-Peneloux, markers: experimental data from NIST 90 at the saturation (+), 150K (x), 200K (), 300 K (), 400 K (◊), 500 K (*). 17 ACS Paragon Plus Environment


In Figure 3 the characteristic extrema of 𝑐𝑃 and 𝑐𝑉 with density and temperature described by Gregorowicz et al.

51

and Polishuk et al.

55

are observed, which in

accordance with both authors disappear with increasing temperature and cannot be accurately captured with cubic EoS. The observation of Polishuk et al.

55,

that an

appropriate temperature dependency of the cohesive parameter (such as the MC α function used in this work) may result in relatively accurate description at low and moderate pressures, although description of the critical point behavior and the very high pressure region is not achieved via the same modifications.

10000

100

𝑐𝑃 (J/mol*K)

90 80

1000

70 60 100

50 40 30

10 0

𝑐𝑉 (J/mol*K)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 56

10

20

0

30

55

41

50

39

10

20

30

20

30

37

45

35

40

33

35

31

30

29

25

27

20

25 0

10

20

30

0

10

𝜌 (mol/lt) Figure 3. Saturated (left) and isothermal (right) 𝑐𝑃 and 𝑐𝑉 prediction for pure methane. Solid black line: MCPR, solid red line: PR, dashed black line: MCPR-Peneloux, dashed red line: PR-Peneloux, markers: experimental data from NIST 90 at the saturation (+), 150K (x), 200K (), 250 K (), 300 K (), 400 K (◊), 500 K (*).


Page 19 of 56

Methane 𝜇𝐽𝑇 and 𝑤 are reported in Figure 4. In this figure it is shown that the volume translation greatly improves the 𝑤 predictions of methane, especially at low temperatures, as shown in Figure 4 at 150K and 200K. Although not very obvious in the figure, the

P (MPa)

same is also observed for 𝜇𝐽𝑇.

5

50

4

40

3

30

2

20

1

10

0

0 0

1000

2000

300

600

900

1200

1500

𝑤 (m/s)

P (MPa)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5

50

4

40

3

30

2

20

1

10

0

0 -1

9

19

29

39

49

-1

1

3

5

7

9

11

𝜇𝐽𝑇 (K/MPa) Figure 4. Saturated (left) and isothermal (right) 𝑤 and 𝜇𝐽𝑇 prediction for pure methane. Solid black line: MCPR, solid red line: PR, dashed black line: MCPR-Peneloux, dashed red line: PR-Peneloux, markers: experimental data from NIST 90 at the saturation (+), 150K (x), 200K (), 250 K (), 300 K (), 400 K (◊), 500 K (*).



Moreover, an improvement is observed in pure methane fugacity predictions 91 with the new MC parameters (AARD%=1.9%) over the Soave expression (AARD%=2.6%) at the temperature range from 200 K to 620 K and pressure range from 0.25 bar up to 500 bar (Figure 5). Introduction of volume translation also reduces the deviations of the two models to 0.6% for PR-Peneloux and 0.8% for MCPR-Peneloux, which indicates that the volume translation has a greater effect in the fugacity predictions than the 𝑎 function. The effect of these modifications is more pronounced at higher temperatures and pressures. The improvement in methane fugacity is very improvement, as it may lead to improved 82

VLE predictions of asymmetric hydrocarbon mixtures

and natural gas dew point

predictions 83.

600 500 400

f (bar)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 56

300 200 100 0 0

100

200

300

400

500

P (bar) Figure 5. Supercritical fugacity prediction for methane. Solid black line: MCPR, solid red line: PR, dashed black line: MCPR-Peneloux, dashed red line: PR-Peneloux, markers: experimental data from Wagner 91 at 200 K (◊), 250 K (), 300 K (), 400 K (+), 500 K (x). 3.3. New UNIFAC interaction parameters for gases with the UMR-MCPRU model The extension of the UMR-MCPRU to mixtures is accomplished by utilizing the Universal Mixing Rules (UMR), which combine the PR EoS with Original UNIFAC 92, as in the UMR-PRU

16, 17

model. The mixing rules are presented in Eq. 6 a-g. For the

excess Gibbs energy of the activity coefficient model, only the Staverman-Guggenheim 20 ACS Paragon Plus Environment

Page 21 of 56 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


𝑆𝐺 𝐸, 𝑟𝑒𝑠 (𝐺𝐸, 𝐴𝐶 ) and the residual (𝐺𝐴𝐶 ) excess Gibbs energy terms of the original UNIFAC are

used. 𝑆𝐺 𝐸, 𝑟𝑒𝑠 1 𝐺𝐸, 𝐴𝐶 + 𝐺𝐴𝐶 = + 𝑏𝑅𝑇 𝛢 𝑅𝑇

𝑎

𝑎𝑖

∑𝑧 𝑏 𝑅𝑇 𝑖

𝑖

𝑏 = ∑𝑖∑𝑗𝑥𝑖𝑥𝑗𝑏𝑖𝑗 with 𝑏𝑖𝑗1/2 = 𝑆𝐺 𝐺𝐸, 𝐴𝐶

𝑅𝑇 𝑟𝑒𝑠 𝐺𝐸, 𝐴𝐶

𝑅𝑇

∑

=5

𝑖 𝑖 𝑖 𝑘

𝑖

[

(∑ 𝜃

𝑙𝑛𝛤𝑘 = 𝑄𝑘 1 ― 𝑙𝑛

+ 𝑏1/2 𝑗 2

𝜃𝑖 𝑥𝑖𝑞𝑖𝑙𝑛 𝜑𝑖

∑𝑥 𝑣 (𝑙𝑛𝛤

=

𝑖

𝑏1/2 𝑖

𝑚𝛹𝑚𝑘

𝑚

𝑘

)

― 𝑙𝑛𝛤𝑖𝑘)

―

(∑∑ 𝜃 𝛹

Eq. 6 a-g

𝜃𝑚𝛹𝑚𝑘

𝑚

𝑛 𝑛

𝑛𝑚

)]

For component i: 𝜑𝑖 =

𝑥𝑖𝑟𝑖 ∑ 𝑥𝑗𝑟𝑗

, 𝜃𝑖 =

𝑗

𝑥𝑖𝑞𝑖 ∑ 𝑥𝑗𝑞𝑗 𝑗

For group m: 𝜃𝑚 =

𝑄𝑚𝑋𝑚

∑ 𝑥𝑗𝑣(𝑗) 𝑚 , 𝑋𝑚 =

∑ 𝑄𝑛𝑋𝑛 𝑛

𝑗

∑ ∑ 𝑣(𝑗) 𝑥 𝑗 𝑛 𝑛 𝑗

For the determination of the interaction parameters (IPs) of the new model the same procedure as with UMR-PRU is used, therefore gases like CH4, C2H6, N2, CO2 are treated as separate groups. UNIFAC r and q parameters are the same with previous publications 16, 17.

The IPs of the UMR-MCPRU model (Table 8) were fitted to isothermal VLE

experimental data for binary mixtures containing gases (CH4, C2H6, N2, CO2) and hydrocarbon groups (CH2, ACH, ACCH), covering linear and branched chain alkanes, cycloalkanes and aromatics, as well as gas-gas binary mixtures. The database is presented in Supporting Information Table S3 to Supporting Information Table S7. The IPs of original UNIFAC Ψ function of Eq. 7, which has 6 adjustable parameters for the interactions between groups m and n: 𝐴𝑚𝑛, 𝐴𝑛𝑚, 𝐵𝑚𝑛, 𝐵𝑛𝑚, 𝐶𝑚𝑛, 𝐶𝑛𝑚 , are calculated by the objective function of Eq. 8. Depending on the temperature range covered from the VLE experimental data, constant, linear or quadratic IPs are determined.

[

𝛹𝑚𝑛 = exp ―

𝐴𝑚𝑛 + 𝐵𝑚𝑛(𝑇 ― 298.15) + 𝐶𝑚𝑛(𝑇 ― 298.15)2 𝑇

]

Eq. 7



𝑁𝑃

𝐹=

∑|

𝑖=1

|

𝑃𝑒𝑥𝑝 ― 𝑃𝑝𝑟𝑒𝑑 𝑃𝑒𝑥𝑝

Page 22 of 56

Eq. 8

Due to the uncertainty of the critical properties of higher MW alkanes only alkanes up to n-C20 were used in the correlation. For the same reason, components containing fused aromatic rings were also not employed in correlation of the ACH and ACCH groups. Table 8. IPs of the UMR-MCPRU model. Group n Group m Anm (K) Bnm CO2 N2 349.28 1.16856 CO2 CH4 144.97 -0.16801 CO2 C2H6 74.70 -0.25787 CO2 CH2 75.65 -0.16063 CO2 ACH 43.21 -0.71323 CO2 ACCH -210.24 -1.02582 N2 CH4 -80.21 -0.51612 N2 C2H6 -156.37 -0.90370 N2 CH2 266.49 6.08555 N2 ACH 140.62 -0.68638 N2 ACCH 246.52 -0.26071 C H CH4 65.38 0.37581 2 6 CH4 CH2 425.11 2.76663 CH4 ACH 21.16 0.28044 CH4 ACCH 13.90 -1.45341 C2H6 CH2 145.20 0.18139 C2H6 ACH 172.57 1.39270 C2H6 ACCH 2.11 0.57891

Cnm (K-1) 0 0 0 0.009400 0.002012 -0.000679 0 0 0.027083 -0.001617 -0.008579 0 0.001441 0.017254 0.008336 -0.003316 0.009278 0.009681

Amn (Κ) -125.06 64.36 128.85 75.78 60.86 601.83 129.53 302.45 -93.34 141.78 191.30 -67.30 -252.68 37.38 -59.15 -133.10 -114.95 61.07

Bmn -1.52736 -0.43602 -0.97745 -1.18162 -0.45651 15.33254 0.44367 0.93365 -3.00993 -0.72501 0.46784 -0.49062 -1.40663 -1.90988 0.95152 -0.46544 -1.01380 -2.71912

Cmn (K-1) 0 0 0 -0.003070 0.001283 0.140106 0 0 0.005237 0.004422 0.005875 0 -0.000282 -0.002139 -0.007210 0.002289 -0.004638 0.005606

The total AARDP% per gas group is presented in Table 9 and some indicative VLE results are presented in Figure 6 through Figure 8. Both symmetric systems like methaneethane (Figure 6 a), methane-propane (Figure 6 b), ethane-propane (Figure 8 a) and asymmetric mixtures like methane-decane and hexadecane (Figure 6 c and d), ethanedecane and eicosane (Figure 8 b and c) are accurately described at various temperatures. Good agreement with the experimental data is also observed for mixtures of gases with branched alkanes, cyclic and aromatic hydrocarbons (Figure 7 a-d and Figure 8 d).


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Table 9. Correlation results for bubble point pressure of binary mixtures obtained with UMR-MCPRU and UMR-PRU model, expressed as AARDP% and Δy%. Mixture NPa Nyb T range (K) P range (bar) x range AARDP%c Δy%d AARDP%c Δy%d UMR-PRU UMR-MCPRU 216 216 130.37 - 283.15 1.23 - 66.57 0.00600 - 0.99560 1.3 0.4 1.3 0.5 CH4-C2H6 343 343 207.00 298.15 3.29 66.30 0.01500 0.98910 0.8 0.9 0.9 0.8 CO2-C2H6 700 592 173.15 - 301.00 10.78 - 85.21 0.00240 - 0.99950 2.0 1.2 1.6 1.1 CH4-CO2 363 341 88.71 - 183.15 0.96 - 50.62 0.00170 - 1.00000 1.0 0.5 0.9 0.5 N2-CH4 207 197 120.00 - 290.00 3.45 - 134.65 0.00360 - 0.99500 2.5 0.9 2.5 0.9 N2-C2H6 160 160 218.15 - 298.20 12.77 - 167.26 0.00400 - 0.35300 1.7 0.9 1.7 1.0 N2-CO2 2204 1832 130.37 703.55 0.00 703.46 0.00000 1.00000 6.9 1.8 5.7 1.6 CH4-alkanes 387 258 295.00 - 582.35 3.45 - 527.00 0.00967 - 0.82590 8.5 1.9 6.6 1.9 CH4-aromatics 1274 775 144.26 - 510.93 0.00 - 158.90 0.00000 - 1.00000 3.9 1.1 4.0 1.0 C2H6-alkanes 146 122 293.15 - 473.10 4.50 - 132.80 0.10170 - 0.95540 3.0 4.7 3.8 4.9 C2H6-aromatics 2423 1610 177.20 510.93 0.21 344.10 0.00000 1.00000 5.3 0.9 5.2 1.0 CO2-alkanes 566 508 273.15 - 573.15 3.10 - 172.90 0.01270 - 0.98120 3.5 0.5 4.0 0.5 CO2-aromatics 817 721 223.15 - 543.50 2.50 - 997.00 0.00100 - 0.69910 7.1 1.9 7.2 1.7 N2-alkanes 156 156 303.20 - 544.00 20.21 - 1,001.00 0.01800 - 0.39000 5.6 2.2 5.4 2.0 N2-aromatics a: NP is the number of experimental bubble point pressures b: Ny is the number of experimental vapor phase compositions 1 𝑁𝑃|𝑃𝑒𝑥𝑝 ― 𝑃𝑝𝑟𝑒𝑑| c: 𝐴𝐴𝑅𝐷𝑃% = ∑ % 𝑃 𝑁𝑃 1 d:

Δy% =

1 𝑁𝑦 ∑ | 𝑁𝑦 1 𝑦𝑒𝑥𝑝

𝑒𝑥𝑝

― 𝑦𝑝𝑟𝑒𝑑|%



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x1, y1 Figure 6. Prediction of isothermal dew and bubble curves for binary systems containing methane (1) with the UMR-MCPRU model. Solid line: predicted curves with the UMR-MCPRU model, Points: experimental data. a) Top Left: CH4-C2H6: (◊) Τ1=130.4 Κ 93, () Τ2=190.9 Κ 93, () Τ3=227.6 Κ 94, (+) Τ4=270.0 K 95 b) Top right: CH4-C3: (◊) Τ1=144.3 Κ 96, () Τ2=192.3 Κ 96, Τ3=213.7 Κ 96, (+) Τ4=277.6 Κ 97, (x) Τ5=344.3 Κ 97 c) Bottom left: CH4-n-C10: (◊) Τ1= 244.3 Κ 98, () Τ2= 310.9 Κ 99, () Τ3= 410.9 Κ 99, (+) Τ4= 510.9 Κ 99, (x) Τ5= 583.1 Κ 100 d) Bottom right: CH4-n-C16: (◊) Τ1= 290.0 Κ 101, () Τ2= 350.0 Κ 101, () Τ3= 462.5 Κ 102, (+) Τ4= 542.7Κ 102, (x) Τ5= 623.2 Κ 102.


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x1, y1 Figure 7. Prediction of isothermal dew and bubble curves for binary systems containing methane (1) with the UMR-MCPRU model. Solid line: predicted curves with the UMR-MCPRU model, Points: experimental data. a) Top Left: CH4-i-C4: (◊) Τ1= 310.9 Κ 103, () Τ2= 344.3 Κ 103, () Τ3= 377.6 K 103 b) Top right: CH4-cy-C6: (◊) Τ1=294.3 Κ 104, () Τ2=344.3 Κ 104, () Τ3=444.3 K 104 c) Bottom left: CH4-benzene: (◊) Τ1= 323.2 Κ 105, 106, () Τ2= 421.1 Κ 100, () Τ3= 461.9 K 100, (+) Τ4= 501.5 K 100 d) Bottom right: CH4-toluene: (◊) Τ1= 233.2 K 107, () Τ2= 313.2 Κ 108, 109, () Τ3= 442.3 K 100



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x1, y1 Figure 8. Prediction of isothermal dew and bubble curves for binary systems containing ethane (1) with the UMR-MCPRU model. Solid line: predicted curves with the UMR-MCPRU model, Points: experimental data. a) Top Left: C2H6-C3: () Τ1= 199.8 Κ 110, () Τ2= 255.4 Κ 110, (+) Τ3=310.9 K 111, (x) Τ4=355.4 K 111 b) Top right: C2H6-n-C10: (◊) Τ1= 277.6 Κ 112, () Τ2= 310.9 Κ 113, () Τ3=377.6 K 112, (+) Τ4=477.6 K 112 c) Bottom left: C2H6-n-C20: (◊) Τ1= 290.0 Κ 114, () Τ2= 330.0 Κ 114, () Τ3=430.0 K 114, (+) Τ4=572.9 K 115 d) Bottom right: C2H6-M-cy-C6: (◊) Τ1= 313.1 K 116, () Τ2= 393.1 Κ 116, () Τ3= 473.0 K 116


Page 27 of 56 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The prediction of phase equilibria of binary mixtures in the critical region is a very demanding task for any model. However, accurate prediction of the critical lines for binary mixtures gives necessary information about the phase behavior of the mixture. Thorough studies regarding critical lines for mixtures of methane with hydrocarbons obtained with EoS can be found in the literature 32-34. VLE critical lines for representative binary mixtures of methane with alkanes are presented in Figure 9, where the model behaves very well, especially taking into account that no critical points were used in the correlation of the IPs. In this work critical points are calculated from interpolation polynomials based on points on each side of the critical. This procedure usually calculates the critical point with an accuracy of 0.01K in critical temperature and 0.01 atm in critical pressure, as suggested by Michelsen 117. For methane with propane, Type I behavior is observed with both models, according to the classification scheme of van Konynenburg and Scott

118,

which is in accordance

with the experimental data, while methane-pentane is predicted as Type V with UMRMCPRU and Type I with UMR-PRU. According to Polishuk et al. 32, it is not yet clear whether this system is actually Type I or V, therefore we accept both predictions as a correct result. For the asymmetric methane-decane and methane-hexadecane, type III behavior is correctly predicted 33 with both models. In the figure, only VLE critical lines and, in the case of decane-hexadecane mixture only the branch starting from the critical point of the heavy component is shown.



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T (K) Figure 9. VLE critical lines for binary systems containing methane with the UMRMCPRU and UMR-PRU model. Solid black line: predicted curves with the UMR-MCPRU model, dashed red line: predicted curves with the UMR-PRU model, Points: experimental data 32, 33. CH4-C3 (◊), CH4-n-C5 (), CH4-n-C10 (), CH4-n-C16 (+). 3.4. Vapor-liquid equilibrium predictions for multicomponent mixtures 3.4.1. Natural gas mixtures Knowledge of VLE is of great importance to engineers working in the Oil and Gas industry for various reasons. Phase envelopes set the boundary conditions for a lot of processes that require single phase fluids. More important, cricondentherm and cricondenbar points on the phase envelope, i.e. the maximum temperature and pressure that phase split may occur for a certain fluid composition, are used as specifications of the gas transportation. Moreover, when phase split occurs, accurate knowledge of the amount of liquid that will precipitate, i.e. liquid dropout, is also very important for gas engineers. The UMR-MCPRU model is tested in the prediction of experimental dew point data for 30 synthetic natural gas mixtures (SNGs) collected from the literature. The compositions for these mixtures are given in the Supporting Information Table S8. The results with UMR-MCPRU are presented in terms of average absolute deviation (AAD) in the dew point temperature (Table 10). For pressures where the model predicts a single 28 ACS Paragon Plus Environment

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phase, the corresponding experimental point was omitted from the calculation of the AAD and the number of such points is abbreviated as NRP. The same table also contains the absolute deviations from the experimental cricondentherm temperature and cricondenbar pressure. The results are presented in the same way as Louli et al. 18 so as to enable direct comparison between the PR (kij=0), PPR-78 and UMR-PRU models for SNG1-SNG23. Figures 7 and 8 present graphically typical results. The results indicate that the UMR-MCPRU model has similar accuracy with the UMR-PRU model when it comes to the dew points of synthetic natural gases, and is superior to PR and PPR78 presented by Louli et al. 18. UMR-PRU yields slightly better cricondenbar and cricondentherm predictions than UMR-MCPRU. However, lowpressure dew points are more accurately captured by UMR-MCPRU model, which is attributed to the improvement of the pure component vapor pressures predictions (Figure 10, SNG 12). Also, prediction of a CO2 rich natural gas (SNG 10) is also improved, compared to the UMR-PRU model, as shown in Figure 10. The predictions of UMR-MCPRU for the real natural gases (RGs) reported by Skylogianni et al.

19

are shown in Table 11. The new model predicts the cricondenbar

with 0.8 bar average deviation, as compared to 1.6 bar of the UMR-PRU. However the cricondentherm is less accurate with 2.5 K deviation as compared to 1.7 K of the UMRPRU model. Moreover, UMR-MCPRU also yields more accurate dew point predictions since it has a lower AAD than UMR-PRU and less rejected points due to single phase prediction. Therefore, UMR-MCPRU yields more accurate predictions for real natural gases than those of PR, SRK and PC-SAFT EoS presented by Skylogianni et al.

19.

Composition is given in Supporting Information Table S9.



Page 30 of 56

Table 10. Deviation in dew point temperature (𝐴𝐴𝐷b), CricoT (ΔTd) and CricoP (ΔPe) for SNG mixtures using the UMR-MCPRU and UMR-PRU model. ΔTd ΔPe ΔTd ΔPe System NPa AADb NRPc AADb NRPc (K) (bar) (K) (bar) UMR-MCPRU UMR-PRU SNG 1 100 1.6 0 1.1 1.6 0 1.0 SNG 2 18 1.4 2 0.6 1.2 1 0.2 SNG 3 18 1.5 1 1.5 1.4 1 1.4 SNG 4 16 0.7 0 0.6 0.6 0 -0.4 SNG 5 23 1.8 0 0.5 1.7 0 0.4 SNG 6 20 0.8 0 0.7 0.5 0 0.2 SNG 7 15 1.3 3 2.5 3.4 1.1 2 2.0 2.7 SNG 8 19 1.1 2 1.1 1.4 1.4 1 0.8 0.3 SNG 9 22 1.5 4 1.2 5.4 1.3 4 0.7 4.3 SNG 10 63 0.7 0 -0.1 0.9 0 1.0 SNG 11 47 1.6 0 0.5 -2.0 2.3 0 0.8 -2.7 SNG 12 14 0.9 3 0.2 2.6 1.4 2 0.7 2.0 SNG 13 20 2.7 0 -3.1 2.8 0 -2.9 SNG 14 13 2.1 0 -2.2 1.5 0 -1.0 SNG 15 18 3.6 0 -3.6 -2.8 3.8 0 -3.3 -4.0 SNG 16 25 2.5 1 -3.7 1.0 2.1 1 -2.9 1.6 SNG 17 28 3.1 1 -4.9 0.0 2.6 1 -3.9 0.6 SNG 18 26 2.7 0 -4.0 -1.0 1.8 1 -2.9 0.0 SNG 19 17 1.7 4 1.1 7.1 2.0 3 1.8 6.4 SNG 20 18 3.0 0 -1.7 -3.4 1.9 0 -0.4 -1.8 SNG 21 17 2.8 0 -1.4 -4.2 1.8 0 -0.1 -2.8 SNG 22 36 3.3 6 -0.5 6.1 2.9 4 -0.2 3.8 SNG 23 27 0.6 0 0.1 0.0 1.5 0 1.1 -0.5 SNG 24 4 0.4 0 0.7 0.8 0 1.4 SNG 25 4 0.7 0 -0.2 1.0 0 1.1 SNG 26 4 0.5 0 -0.5 1.3 0 1.1 SNG 27 4 2.7 0 -3.1 1.2 0 -1.4 SNG 28 4 2.1 0 -2.4 0.5 0 -0.6 SNG 29 9 2.2 0 -1.1 -1.3 2.2 0 -1.4 -2.9 SNG 30 6 5.3 0 1.4 -5.5 6.2 0 1.1 -7.4 Overall 649 1.8 27 1.5 3.0 1.8 21 1.3 2.7 a NP: number of experimental data points 1 𝑁𝑃 ― 𝑁𝑅𝑃 b 𝐴𝐴𝐷 = ∑ |𝑇𝑒𝑥𝑝 ― 𝑇𝑝𝑟𝑒𝑑| 𝑁𝑃 ― 𝑁𝑅𝑃 1 c NRP: number of rejected data points since a single phase is predicted d ΔT = 𝐶𝑟𝑖𝑐𝑜𝑇 𝑒𝑥𝑝 ― 𝐶𝑟𝑖𝑐𝑜𝑇𝑝𝑟𝑒𝑑 e ΔP = 𝐶𝑟𝑖𝑐𝑜𝑃 𝑒𝑥𝑝 ― 𝐶𝑟𝑖𝑐𝑜𝑃𝑝𝑟𝑒𝑑


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Table 11. Deviation in dew point temperature (𝐴𝐴𝐷b), CricoT (ΔTd) and CricoP (ΔPe) for RG mixtures using the UMR-MCPRU and UMR-PRU model. ΔTd ΔPe ΔTd ΔPe System NPa AADb NRPc AADb NRPc (K) (bar) (K) (bar) UMR-MCPRU UMR-PRU RG 1 13 2.9 1 3.1 0.3 3.5 1 2.2 1.1 RG 2 15 2.2 2 3.4 0.8 2.3 2 0.5 1.6 RG 3 18 1.7 1 1.2 0.7 2.1 3 0.1 1.7 RG 4 12 2.2 2 1.5 1.2 2.7 2 0.6 2.0 RG 5 25 3.3 5 3.2 0.9 3.5 8 2.2 1.7 RG 6 26 2.8 4 2.9 0.7 2.9 6 2.7 1.3 Overall 109 2.6 15 2.5 0.8 2.9 22 1.7 1.6 a NP is the number of experimental data points. 1 𝑁𝑃 ― 𝑁𝑅𝑃 b 𝐴𝐴𝐷 = ∑ |𝑇𝑒𝑥𝑝 ― 𝑇𝑝𝑟𝑒𝑑| 𝑁𝑃 ― 𝑁𝑅𝑃 1 c NRP is the number of rejected data points due to the prediction of a single phase. d ΔT = |𝐶𝑟𝑖𝑐𝑜𝑇 𝑒𝑥𝑝 ― 𝐶𝑟𝑖𝑐𝑜𝑇𝑝𝑟𝑒𝑑| e ΔP = |𝐶𝑟𝑖𝑐𝑜𝑃 𝑒𝑥𝑝 ― 𝐶𝑟𝑖𝑐𝑜𝑃𝑝𝑟𝑒𝑑| 3.4.2. Gas condensate mixtures Gas condensates contain significant amount of heavy hydrocarbons, which shifts the saturation curve to very high temperatures and pressures

119,

and makes the

thermodynamic modelling of these mixtures a difficult task. Previous modelling of VLE 31 ACS Paragon Plus Environment


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and liquid dropouts of such mixtures has shown that the UMR-PRU model is capable of describing these mixtures with sufficient accuracy 20. The compositions of the synthetic gas condensate (SGC) mixtures studied in this work are given in Supporting Information Table S10. The corresponding results with UMR-MCPRU are tabulated in Table 12, where the AARD% results reveal an improvement over UMR-PRU, which is also seen in Figure 11. Table 12. Deviation in saturation point pressure (AARD%b) for SGC mixtures using the UMR-MCPRU and UMR-PRU model. System NPa UMR-MCPRU UMR-PRU SGC 1 8 4.4 4.2 SGC 2 12 12.7 13.1 SGC 3 7 3.4 3.7 SGC 4 12 0.5 2.4 SGC 5 7 3.8 2.4 SGC 6 2 11.7 11.4 SGC 7 4 7.1 6.7 SGC 8 3 7.4 8.9 SGC 9 7 2.0 10.4 SGC 10 10 3.9 13.1 SGC 11 14 17.8 19.1 SGC 12 15 7.6 6.4 SGC 13 13 12.0 12.9 SGC 14 11 9.1 10.6 SGC 15 10 2.8 3.7 SGC 16 10 4.4 8.2 SGC 17 14 1.9 1.2 SGC 18 14 0.4 1.2 SGC 19 14 0.6 0.6 SGC 20 12 0.7 1.6 Overall 199 5.6 6.9 a NP is the number of experimental data points. 100 𝑁𝑃 |𝑃𝑒𝑥𝑝 ― 𝑃𝑝𝑟𝑒𝑑| b 𝐴𝐴𝑅𝐷% = ∑ 𝑃 𝑁𝑃 𝑖 = 1 𝑒𝑥𝑝

Dew points of gas condensate mixtures SGC13-16, have recently been measured by Regueira et al.

50.

The results with the UMR-PRU and UMR-MCPRU model are

presented in Figure 11. Both models seem to overpredict the low temperature saturation points of the ternary mixtures CH4-n-C4-n-C10 and CH4-n-C4-n-C12 but accurately capture the multicomponent mixtures of low and high gas to oil ratio (GOR). Overall UMRMCPRU gives better results than UMR-PRU model, especially for SGC 16 (high GOR).


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The most challenging mixtures are deemed the highly asymmetric condensate mixtures measured by Ungerer et al. that contain n-C36 119. The critical properties for nC36 were taken from Lemmon and Goodwin 120 and the acentric factor was calculated by the method of Constantinou-Gani 121.The predictions for these mixtures with both models are shown in Figure 11. The results of UMR-MCPRU are by far superior to the ones with UMR-PRU and very close to the experimentals. 3.4.3. Critical points The UMR-MCPRU model has been already applied in the prediction of critical lines for binary mixtures and at this point is evaluated in the prediction of critical temperature and pressure of multicomponent mixtures (Table 13). Critical points of multicomponent mixtures have been calculated in the same manner as the binary critical points. The average relative deviation in critical temperature is 2.3 % and in critical pressure 3.5 %, which are very similar to those obtained with the UMR-PRU.


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T (K) Figure 11. Phase envelope predictions for the gas condensate SGC 4 (left), SGC 13-16 (middle) and SGC 9-10 (right). Solid black line: UMR-MCPRU, solid red line: UMR-PRU, markers: experimental data. Left: SGC 4 (◊) Middle: SGC 13 (), SGC 14 (◊), SGC 15 (), SGC 16 () Right: SGC 9 (), SGC 10 (◊)


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Table 13. Deviation in critical point temperature (𝐴𝑅𝐷𝑇𝑐% a) and pressure (𝐴𝑅𝐷𝑃𝑐% b) for multicomponent mixtures using the UMR-MCPRU and UMR-PRU model. EXP UMR-MCPRU UMR-PRU Mixture N2 CH4 C2H6 C3 n-C4 n-C5 n-C6 Source Tc (K) Pc (bar) ARDTc% ARDPc% ARDTc% ARDPc% 94 1 0.8330 0.1300 0.0350 227.6 68.9 0.6 2.4 0.3 3.6 94 2 0.8000 0.0390 0.1610 255.4 89.6 2.0 6.2 1.6 7.5 122 3 0.1930 0.4700 0.3370 354.3 76.4 2.4 3.8 2.5 5.1 122 4 0.0070 0.8790 0.1140 324.5 54.8 2.0 4.1 1.9 4.1 122 5 0.0400 0.8210 0.1390 323.7 57.9 3.3 4.0 3.3 3.9 123 6 0.4610 0.4430 0.0950 310.9 103.4 0.3 2.5 0.3 0.8 123 7 0.1960 0.7580 0.0450 310.9 68.9 0.3 1.1 0.3 0.8 124 8 0.0490 0.4345 0.0835 0.4330 313.7 89.6 1.5 1.5 1.6 0.1 125 9 0.2019 0.2029 0.2033 0.2038 0.1881 387.0 72.2 2.6 2.5 2.6 3.4 122 10 0.3910 0.3540 0.2550 331.5 97.2 2.5 3.9 2.7 6.6 126 11 0.043 0.415 0.542 322.0 86.7 2.3 0.1 2.3 0.1 126 12 0.095 0.36 0.545 322.0 92.0 2.5 0.3 2.5 0.2 126 13 0.0465 0.453 0.5005 313.7 92.3 3.0 0.6 3.0 0.5 126 14 0.0855 0.4115 0.503 313.7 98.0 3.2 1.8 3.2 1.8 126 15 0.1015 0.3573 0.2629 0.1794 0.0657 0.0322 376.4 65.4 2.0 1.4 2.1 1.9 126 16 0.022 0.316 0.388 0.223 0.043 0.008 313.7 78.5 2.4 2.0 2.5 0.7 126 17 0.6626 0.1093 0.1057 0.0616 0.0608 310.5 137.5 3.8 11.7 4.2 8.5 126 18 0.7075 0.0669 0.0413 0.05058 0.1353 308.4 137.0 4.2 12.5 4.6 10.7 2.3 3.4 Overall 2.3 3.5 a b

𝐴𝑅𝐷𝑇𝑐% = 𝐴𝑅𝐷𝑃𝑐% =

|𝑇𝑐𝑒𝑥𝑝 ― 𝑇𝑐𝑝𝑟𝑒𝑑| 𝑇𝑐𝑒𝑥𝑝

%

𝑃𝑐𝑒𝑥𝑝

%

|𝑃𝑐𝑒𝑥𝑝 ― 𝑃𝑐𝑝𝑟𝑒𝑑|



3.5. Liquid dropout Liquid dropout predictions with the UMR-MCPRU model are not significantly different from those of UMR-PRU

20

(Figure 12). The maximum liquid dropout is sometimes

higher and other times lower with the UMR-MCPRU model, and not consistently closer to the experimental data. Compositions for the presented mixtures can be found in Supporting Information Table S11. What is more, the use of volume translation does not significantly affects the calculated dropout, since the various terms of the VpL definition mostly cancel out its effect (Figure 12).

30

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VPL 3, 366.48 K

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P (bar) Figure 12. Liquid dropout predictions for synthetic gas condensate systems. Solid black line: predicted curves with the UMR-MCPRU model, Dashed black line: predicted curves with the translated UMR-MCPRU model, Solid red line: predicted curves with the UMR-PRU model, Dashed red line: predicted curves with the translated UMR-PRU model, Points: experimental data.

3.6. Derivative properties predictions for binary and multicomponent mixtures The property predictions for binary systems are shown in Table 14 and Figure 13 to Figure 15. The prediction of 𝑐𝑃 for mixture CH4-C2H6 is similar between the two models (Figure 14 a), unlike CH4-C3, where great differences are found. For this mixture the liquid phase predictions with the UMR-PRU model are very poor, while the UMRMCPRU model has a much better behavior, even though both models predict a minimum 36 ACS Paragon Plus Environment

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which is not observed in the experimental data (Figure 14 b). This is attributed to the extrapolation of the UMR interaction parameters at much lower temperatures than those of the database used for their fitting. On the other hand, predictions of 𝑐𝑉 with the UMRMCPRU model are slightly worse than those of UMR-PRU, which is in accordance with the trend of pure component calculations. Moreover, both 𝑐𝑃 and 𝑐𝑉 are left unaffected by the volume translation therefore no values are reported. The deviations in 𝜇𝐽𝑇 are drastically improved for the mixture CH4-C3 (Figure 14 d) with UMR-MCPRU, which is again attributed to the extrapolation of IPs at low temperature, while similar results are obtained for the mixture CH4-C2H6. Volume translation greatly improves the predictions for both models, which is in accordance with the observations of pure components, since the mixtures contain only light components. For 𝑤, overall similar results are obtained with UMR-PRU and UMR-MCPRU. However, Figure 13 indicates that UMR-MCPRU performs better at higher temperatures, while at low temperatures, it has an advantage at the lower pressures. Moreover, the introduction of the volume translation has a great effect at higher pressures, where it usually shifts the speed of sound to higher values, and further away from the experimental data, increasing thus the overall deviations. For lower pressures, the volume translation actually improves the predictions of both models. Mixtures containing n-C16 with gases are actually negatively affected by the volume translation except for the n-C6- n-C16 mixture that is highly improved.



470

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P (MPa) Figure 13. Speed of sound predictions for binary systems containing CH4 (1). Solid black line: predicted curves with the UMR-MCPRU model, Solid red line: predicted curves with the UMR-PRU model, Dashed black line: predicted curves with the translated UMR-MCPRU model, Dashed red line: predicted curves with the translated UMR-PRU model, Points: experimental data. a) Top Left: CH4-CO2, z1= 0.95 127, T1=250 K (◊), T2=275 K (), T3=300 K (), T4=325 K (+), T5=350 K (x) b) Top Right: CH4-N2, z1= 0.95 127, T1=250 K (◊), T2=275 K (), T3=300 K (), T4=325 K (+), T5=350 K (x) c) Bottom Left: CH4-C3, z1= 0.90 127, T1=250 K (◊), T2=275 K (), T3=300 K (), T4=325 K (+), T5=350 K (x) d) Bottom Right: CH4-n-C16, z1= 0.68 128, T1=292 K (◊), T2=313 K (), T3=340 K (), T4=373 K (+), T5=413 K (x)


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P(MPa) T (K) Figure 14. Isobaric heat capacity for binary systems containing CH4 (1). Solid black line: predicted curves with the UMR-MCPRU model, Solid red line: predicted curves with the UMR-PRU model, Points: experimental data. a) Left: CH4-C2H6, z1= 0.85 129, T1=250 K (◊), T2=275 K (), T3=300 K (), T4=350 K (+) b) Right: CH4-C3, z1= 0.49 130, P1=1.7 MPa (◊), P2=3.5 MPa (), P3=6.9 MPa (), P4=10.3 MPa (+), P5=13.8 MPa (x)

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P (MPa) Figure 15. Joule-Thomson coefficients for binary systems containing CH4 (1). Solid black line: predicted curves with the UMR-MCPRU model, Solid red line: predicted curves with the UMR-PRU model, Dashed black line: predicted curves with the translated UMR-MCPRU model, Dashed red line: predicted curves with the translated UMR-PRU model, Points: experimental data. Left: CH4-C2Η6 129, T1=250 K (◊), T2=275 K (), T3=300 K (), T4=350 K (+) Right: CH4-C3, z1=0.49, T1=172 K 130 (◊), z2=0.24, T2=202 K 131 ()



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Table 14. Deviation (AARD%) in property predictions for binary mixtures using the UMR-PRU and UMR-MCPRU model with and without Peneloux volume translation. UMRUMRb a b b PRU MCPRUb System (Property) Source NP T (K) P (MPa) UMR-PRU UMR-MCPRU Peneloux Peneloux 129 c CH4-C2H6 (𝑐𝑃) 56 250 - 350 0.6 - 30.0 1.7 1.6 -c 130, 131 CH4-C3(𝑐𝑃) 429 100 - 422 1.7 - 13.8 43.7 7.2 -c -c 132 CH4-C2H6 (𝑐𝑉) 31 101 – 323 4.7 - 18.4 3.8 4.6 -c -c 133 C2H6-n-C5 (𝑐𝑉) 57 309 – 309 3.1 - 7.1 10.7 11.3 -c -c 127, 134 1.8 2.0 CH4-C2H6 (𝑤) 425 250 – 350 0.0 - 20.1 1.5 1.6 127 0.9 1.2 CH4-C3 (𝑤) 75 250 – 350 0.5 - 10.5 0.9 0.7 128 14.6 13.5 CH4-n-C16 (𝑤) 396 292- 413 6.3 - 66.2 6.5 6.8 127 0.9 1.3 CH4-CO2 (𝑤) 239 250- 350 0.5 - 10.8 0.4 0.3 127 0.3 0.7 CH4-N2 (𝑤) 253 250 – 350 0.1 - 10.7 0.8 0.3 127 0.6 0.6 CO2-N2 (𝑤) 65 250 – 350 0.5 - 10.3 1.0 1.0 128 21.2 20.0 CO2-n-C16 (𝑤) 404 293 - 333 2.2 - 55.6 16.6 16.8 128 8.1 7.9 n-C6-n-C16 (𝑤) 340 298 – 373 0.1 - 70.0 14.1 15.1 129 CH4-C2H6 (𝜇𝐽𝑇) 48 250 – 350 0.5 - 30.0 12.5 13.7 7.5 8.3 130, 131 CH4-C3 (𝜇𝐽𝑇) 19 172 – 202 2.1 - 13.8 51.7 22.6 43.1 14.9 a NP is the number of experimental data points. 1 𝑁𝑃|𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑒𝑥𝑝 ― 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑝𝑟𝑒𝑑| b 𝐴𝐴𝑅𝐷% = ∑ % |𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 | 𝑁𝑃 1 𝑒𝑥𝑝

c property

unaffected by volume translation


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For multicomponent mixtures, whose compositions are shown in Supporting Information Table S12, all the results (Table 15 and Figure 16) are clearly in favor of the UMR-MCPRU model, as they are either similar or better than those of the UMR-PRU, apart from 𝜇𝐽𝑇 (the mixture of Ernst

129).

For this mixture, incorporation of the volume

translation improves the predictions of both models. Moreover, volume translation slightly affects the 𝑤 predictions of both models, however no consistent improvement is observed. The 𝜌 predictions of the UMR-MCPRU model are superior to those of UMRPRU, and volume translation as expected improves both models.



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Table 15. Deviation (AARD%) in property predictions for synthetic multicomponent mixtures using the UMR-PRU and UMR-MCPRU model with and without Peneloux volume translation System UMR-PRUb UMR-MCPRUb Source NPa T (K) P (MPa) UMR-PRUb UMR-MCPRUb (Property) Peneloux Peneloux 129 Ernst (𝜇𝐽𝑇) 7.1 8.8 39 250 - 300 0.5 – 30.0 10.8 13.5 127 Amarillo (𝑤) 0.6 1.0 82 250 - 350 0.5 - 23.4 0.9 0.5 134 Costa (𝑤) 0.9 1.5 39 250 - 350 0.0 - 20.2 1.9 1.2 127 Gulfcoast (𝑤) 0.8 1.3 83 35 - 350 0.4 - 10.7 0.7 0.5 127 Statoildry (𝑤) 1.0 1.3 91 250 - 350 0.4 - 10.4 0.9 0.8 127 Statvordgass (𝑤) 0.7 0.8 44 300 - 350 0.4 - 10.4 1.3 0.8 c 129 Ernst (𝑐𝑃) -c 54 250 - 350 0.6 - 30.0 3.6 3.6 c 135 GasA (𝑐𝑃) -c 30 308 - 406 14.9 - 40.0 1.0 1.0 135 GasH (𝑐𝑃) -c -c 30 308 - 406 14.9 - 40.0 1.5 1.1 135 GasA (𝜌) 1.1 1.0 30 308 - 406 14.9 - 40.0 5.1 4.1 135 GasH (𝜌) 3.3 2.3 30 308 - 406 14.9 - 40.0 7.5 6.4 a NP is the number of experimental data points. 1 𝑁𝑃|𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑒𝑥𝑝 ― 𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦𝑝𝑟𝑒𝑑| b 𝐴𝐴𝑅𝐷% = ∑ % |𝑃𝑟𝑜𝑝𝑒𝑟𝑡𝑦 | 𝑁𝑃 1 𝑒𝑥𝑝

c property

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P (MPa) Figure 16. Property predictions for synthetic natural gas mixtures. Solid black line: predicted curves with the UMR-MCPRU model, Solid red line: predicted curves with the UMR-PRU model, Dashed black line: predicted curves with the translated UMR-MCPRU model, Dashed red line: predicted curves with the translated UMR-PRU model, Points: experimental data. Left: Amarillo mixture 127, T1=250 K (◊), T2=275 K (), T3=300 K (), T4=350 K (+) Middle: Ernst mixture 129, T1=250 K (◊), T2=275 K (), T3=300 K (), T4=350 K (+) Right: Ernst mixture 129, T1=250 K (◊), T2=275 K (), T3=300 K (), T4=350 K (+)



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4. Conclusions In this paper, an improved version of the original UMR-PRU model is developed, by substituting the Soave 𝑎(𝑇) function of the EoS part of said model by a consistent version of the Mathias-Copeman 𝑎(𝑇) function. The Mathias-Copeman 𝑎(𝑇) function that is developed in this work satisfies the consistency requirements for the temperature range of most industrial applications. The Mathias-Copeman parameters are calculated by fitting vapor pressures, and generalized expressions for hydrocarbons heavier than ethane are proposed as a function of their acentric factor. This generalized form is compared with others from the literature and is found to be superior, especially for heavy alkanes. The MCPR EoS is combined with UNIFAC via the UMR mixing rules leading to the so-called UMR-MCPRU model, and most of the interaction parameters relevant to the natural gas industry are calculated. Very satisfactory VLE results are obtained for binary systems, as well as critical lines for methane-hydrocarbons mixtures. More importantly, the UMR-MCPRU model has been found to yield very good dew point predictions of natural gases and gas condensates with a varying degree of asymmetry. The results of the UMR-MCPRU model are at least as accurate or superior to the UMR-PRU model, and by extension better than classical EoS, such as PR, SRK and PC-SAFT. Finally, the UMR-MCPRU is used to predict various thermophysical properties such as density, isobaric and isochoric heat capacity, speed of sound and Joule-Thomson coefficients as well as critical points of multicomponent hydrocarbon mixtures, yielding satisfactory results. Wherever relevant the Peneloux volume translation is also applied. In conclusion, the UMR-MCPRU model is an accurate predictive thermodynamic model, that can be used as an alternative to the classical cubic equations of state used in the Oil & Gas industry.


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List of Symbols 𝑐𝑃 𝑐𝑉 cricoP cricoT EoS IPs m, n MW P RG RGC SGC SNG T UMR VpL% 𝑤 𝜌

Isobaric Heat Capacity Isochoric Heat Capacity Cricondenbar Pressure Crincondentherm Temperature Equation of State Interaction Parameters UNIFAC groups Molecular Weight Pressure Real Gas Real Gas Condensate Synthetic Gas Condensate Synthetic Natural Gas Temperature Universal Mixing Rule 𝑉𝐿

Volume percent Liquid, 𝑉𝑝𝐿% = 𝑉𝐿 + 𝑉𝑉% Speed of Sound Density

Greek letters ω 𝜇𝐽𝑇

Acentric factor Joule-Thomson Coefficient

Subscripts/Superscripts tr c r sat exp calc pred

Triple Critical Reduced property Saturation property Experimental value Calculated value Predicted value



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Supporting Information Component specific Mathias-Copeman parameters; deviation in predicted vapor pressures (AARDP sat%) by the new MC generalized parameters with those of Coquelet and Soave with PR EoS; binary VLE database containing gases; binary VLE database containing hydrocarbons and CO2; binary VLE database containing hydrocarbons and CH4; binary VLE database containing hydrocarbons and C2H6; binary VLE database containing hydrocarbons and N2; molar compositions of synthetic natural gas mixtures; Molar compositions of real natural gas mixtures; molar compositions of synthetic gas condensate mixtures; molar compositions of the gas condensate mixtures with measured liquid dropout (VPL); molar compositions of multicomponent mixtures with measured properties (PDF)

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Table of Contents Graphic 8 6 μJT (K/MPa)

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30

P (MPa)


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