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Effective Critical Constants for Helium for Use in Equations of State for Natural Gas Mixtures Darren Rowland,* Thomas J. Hughes, and Eric F. May Fluid Science and Resources Division, School of Mechanical & Chemical Engineering, University of Western Australia, Crawley, WA 6009, Australia ABSTRACT: Most of the world’s helium supply is obtained by the cryogenic distillation of natural gas. Modeling the distillation conditions requires equations of state capable of predicting vapor−liquid equilibria over wide ranges of conditions. Equations of state, including cubic equations and multiparameter Helmholtz models, depend on the critical properties of pure substances for predicting various properties of multicomponent fluid mixtures. Predictions for helium are problematic as its critical point (5.195 K, 0.2275 MPa) is influenced strongly by quantum effects; large, empirical interaction parameters tend to be used in equations of state to compensate for these effects. Prausnitz and co-workers proposed an alternative approach using effective critical constants for quantum gases but demonstrated it for only three helium-containing binary mixtures. Here, we demonstrate that the use of an effective critical point at (11.73 K, 0.568 MPa) for helium substantially improves the prediction of VLE by the Peng−Robinson equation of state for 15 binary and 2 ternary mixtures, including the major components of natural gas. This effective critical point for helium was selected from a critical analysis of pTxy data for the (methane + helium) binary. The effective critical constants for helium are compatible with an acentric factor near zero, as expected for a small spherical molecule and similar to the acentric factors of heavy noble gases. The possibility of applying this approach to address recognized limitations of the GERG-2008 equation of state is discussed.

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INTRODUCTION The market for helium is one of growing demand and uncertain supply.1 At present, natural gas is the most economical and largest source of helium.2 If the helium concentration of the feed gas is 0.3 vol % or higher,3 then a convenient and costeffective method of helium recovery is purification of the endflash gas produced in liquefied natural gas (LNG) plants.2,4 The end-flash gas is often obtained after a nitrogen rejection unit (NRU) and therefore can be N2-rich or CH4-rich (if no NRU is present) and may contain other compounds, including argon and on the order of (1 to 3) vol % He.4 This gas stream may either be vented to the atmosphere3 or undergo further cryogenic distillation to produce a crude helium stream comprising up to 70 vol % He.4 Optimizing the efficiency of the distillation process requires a detailed understanding of the phase behavior of helium in mixed-gas streams; equations of state (EOS) are generally used by engineers to model phase behavior over wide ranges of temperature, pressure, and composition. Improving the accuracy of vapor−liquid equilibrium (VLE) predictions made with equations of state for mixtures containing helium may be valuable for enhancing the commercial recovery of helium. Equations of state for natural gas mixtures are numerous and differ greatly in complexity. Particularly for VLE calculations in multicomponent mixtures of nonpolar compounds, complex equations of state (i.e., multiparameter equations and/or with terms of higher order than cubic) sometimes offer little to no overall advantage compared to their simpler counterparts, as discussed, for example, in references 5−7. Consequently, simple © 2017 American Chemical Society

cubic equations of state are still favored in many industrial applications.8,9 As one of the most popular cubic equations of state, the Peng−Robinson equation of state10 can predict accurately (e.g., within experimental uncertainty) vapor−liquid equilibria of multicomponent gas mixtures successfully with only three parameters for each pure fluid component (critical temperature Tc, critical pressure Pc, and acentric factor ω) and small-valued binary interaction parameters, over a range of conditions that is useful to many engineering applications.11 Helium is an outlier in this regard. Compared to the classical gases, helium and other quantum gases (H2, its isotopic variations, and neon) do not conform well with the law of corresponding states.12−15 In practice, to compensate for this limitation, binary interaction parameters between quantum gases and other compounds tend to be set much larger than normal.16,17 As discussed by Prausnitz and co-workers13,14 and recognized much earlier,12 the corresponding states approach works well for classical gases but fails for gases whose critical properties are influenced by quantum phenomena. This problem is most severe for helium because it has one of the lowest critical temperatures of any gas (5.1953 K).18 Attempts to rectify cubic equations of state to accommodate quantum gases Special Issue: Memorial Issue in Honor of Ken Marsh Received: February 1, 2017 Accepted: March 20, 2017 Published: April 10, 2017 2799

DOI: 10.1021/acs.jced.7b00122 J. Chem. Eng. Data 2017, 62, 2799−2811

Journal of Chemical & Engineering Data

Article

Figure 1. Dependence of the function α = (1 + κ(ω)(1 − Tr ))2 for different values of acentric factor ω (eqs 6 and 7) with Tr ≡ T/Tci ).

Figure 2. Enhancement factor for methane f = yCH4P/Psat CH4 in the binary system (methane + helium) near 124 K.

Table 1. Survey of pTxy Vapor−Liquid Equilibrium Data for Binary Mixtures with Heliuma

and customizability. The degree to which an EOS may be customized within a commercial simulation software package is often limited. For example, although a process simulator may allow for temperature-dependent binary interaction parameters to be defined, only simple functions of temperature (e.g., linear term in T) are typically allowed. Considering these limitations, the method for rectifying cubic equations of state for helium that is seemingly most compatible with commercial software for process simulation is one where the true critical constants are replaced by effective values. The early use of effective critical constants for helium by Prausnitz and co-workers’ considered only binary mixtures (nitrogen + helium), (argon + helium), and (tetrafluoromethane + helium)13,14 and did not relate to methane or any other hydrocarbons occurring in natural gas. Later, Prausnitz and co-workers25 applied their model to helium solubility in cryogenic solvents including methane, nitrogen, ethane, propane, argon, oxygen, carbon dioxide, and ternary mixture (methane + nitrogen + helium). However, their approach required a distinctive mixing rule, making it unsuitable for use with standard modern process simulation software. Their method also relied on tuned binary interaction parameters to achieve good agreement with data. Consequently, for mixtures containing components for which little or no VLE data exist for their respective helium binary system, the lack of reliable binary interaction parameters limits the predictive accuracy achievable with this method. Although natural gas is composed predominantly of methane, heavy compounds possessing high boiling-point temperatures may also be present in gas mixtures, either due to natural occurrence or as chemical additives. Binary mixtures of helium with heavy compounds including paraffins,5 aromatic hydrocarbons,16,26 and fatty alcohols27 were investigated by Lee and co-workers to test the Soave−Redlich−Kwong and Peng− Robinson cubic equations of state at high reduced temperatures. They reported that they could obtain reasonable agreement only between the equations of state and the available pTxy data by employing “extraordinarily large (close to or greater than unity)” binary interaction parameters,27 in some cases reaching 1.9.16 It should be noted that there is sentiment in the literature that values of binary interaction parameters greater than unity are physically unrealistic: Ashour et al. claim that values greater than unity violate the postulates of

mixture (+ helium)

N

methane nitrogen argon oxygen ethane propane isobutane butane carbon dioxide toluene meta-xylene hexadecane ethylene propylene decanol

491 388 146 40 73 62 37 38 59 20 20 19 36 30 14

T/K 93 65 91 103 133 173 193 193 220 423 465 464 130 200 353

to to to to to to to to to to to to to to to

192 126 149 116 273 348 273 273 293 545 584 665 216 255 453

P/MPa 0.1 1 1 3 0.5 1 0.5 0.5 3 5 5 5 2 2 2

to to to to to to to to to to to to to to to

26 83 97 22 12 21 4 4 20 15 15 15 12 12 10

kij

refs

0.7649 0.0685 0 0 1.1232 1.0642 0 0 0.7967 1.337 1.3598 1.139 0 0 1.499

30−34 34−44 45−48 46 49 50 51 51 52, 53 26 16 5 54 54 27

a N is the total number of data points, and kij are binary interaction parameters for mixtures containing helium (from the literature reference shown, otherwise from the HYSYS V9 databank (italics)).

(chiefly hydrogen) fit loosely into four categories: (1) those where the reduced temperature for the quantum gas is calculated with an “effective” critical temperature instead of with the true critical temperature;12−14 (2) those with novel temperature-dependent functions in the attractive term of the equation of state, which remove spurious minima at high reduced temperatures;17,19 (3) those with novel mixing terms developed especially to describe interactions between small molecules (e.g., hydrogen, helium) and large molecules;20 and (4) those with temperature-dependent binary interaction parameters.9,21,22 Specialized noncubic equations of state have also been proposed.23,24 Each of these methods has shown some level of success in the literature. However, to improve the predictions of equations of state for commercial helium distillation, methods that preserve the simplicity inherent in cubic EOS have significant advantages for use within process simulation software. Commercial process simulators (e.g., Aspen HYSYS, VMGSim, ProMax) vary in terms of efficiency, functionality, 2800



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Figure 3. (a) p−xy diagram for (methane + helium). Peng−Robinson equation of state predictions with true critical constants Tc = 5.1953 K, Pc = 0.22746 MPa, ω = −0.365, and kij = 0 (dashed lines) and effective critical constants Tc = 11.73 K, Pc = 0.568 MPa, ω = 0.0, and kij = 0 (solid lines). Data points from Heck and Hiza.31 (b) Deviations (log Kexpt − log Kcalc) where Kexpt values are critically selected data for (methane + helium) (Method section) and Kcalc values are calculated with the Peng−Robinson equation of state using true critical constants Tc = 5.1953 K, Pc = 0.22746 MPa, ω = −0.365 and kij = 0. (c) Deviations (log Kexpt − log Kcalc) where Kexpt are critically selected data (Method section) and Kcalc are calculated with the Peng− Robinson equation of state using effective critical constants Tc = 11.73 K, Pc = 0.568 MPa, ω = 0.0, and kij = 0 for (methane + helium). Symbols: blue ×, methane; orange +, helium.

the kinetic theory of gases (ref 28, p 47). On the other hand, Jaubert and co-workers, whose “predictive” Peng−Robinson equation involves temperature-dependent binary interaction parameters exceeding 1.5,9,22 are convinced that this is not a problem as long as the overall attractive contribution from intermolecular forces remains positive (ref 22, p 67). Although the question of the physical meaning of large binary interaction parameters remains unsettled, the need for large binary interaction parameters when modeling heavy compounds mixed with helium is problematic for gas processing. In general, experimental data for mixtures of helium with heavy compounds are lacking, so binary interaction parameters cannot be regressed. Furthermore, constraints on chemical-analysis techniques mean that the heavy compounds in natural gases and oils are typically grouped into so-called petroleum fractions containing compounds having similar boiling points. It is these petroleum fractions that are used for modeling the properties of oil and gas mixtures. Thus, the binary interaction parameters of most

importance may not be those between helium and individual heavy compounds but those between helium and various petroleum fractions. Unfortunately, meaningful binary interaction parameters between helium and petroleum fractions, which would be expected to be large in conventional frameworks, cannot be estimated easily; the ability to predict a binary interaction parameter for petroleum fractions is important in any framework. The model developed in this work avoids the problem of large parameter values: phase equilibrium data are predicted satisfactorily over wide ranges of conditions without binary interaction parameters. The emphasis of this work is on natural gas mixtures and their process applications. As such, the aims of this work are twofold: (a) to regress effective critical constants for helium from reliable VLE data for the (methane + helium) binary and (b) to demonstrate that the regressed parameters can be used to predict quite accurately phase equilibrium data for binary and ternary mixtures with helium and various other substances from a range of classes. 2801



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We are aware that process simulators contain multiple versions of the Peng−Robinson cubic equation of state and may provide options such as “Modify Tc, Pc for H2, He”. Unfortunately, in our experience such options are generally undocumented, and it is not made clear (i) what the values of “Tc” and “Pc” are modified to, (ii) whether the value of the acentric factor is also modified, and (iii) whether the binary interaction parameters between helium and other components of the mixture are also modified. Using binary interaction parameters with different values of the pure component critical constants compared to those that were applied in their original regression will obviously lead to thermodynamic inconsistencies. Because of all of these concerns and because there is no information regarding whether the model has been tested against appropriate vapor−liquid equilibrium data, we advise caution before using such options for natural gas mixtures containing helium.

Table 2. Deviations between the Peng−Robinson Equation of State and Literature pTxy Data AAD AAD (Tc = 5.1953 K, (Tc = 5.1953 K, AAD c c P = 0.22746 MPa, P = 0.22746 MPa, (Tc = 11.73 K, c ω = −0.365, ω = −0.365, P = 0.568 MPa, mixture kij = 0) (+ helium) kij from Table 1) ω = 0, kij = 0) figure methane nitrogen argon oxygen ethane propane isobutane butane carbon dioxide toluene meta-xylene hexadecane ethylene propylene decanol

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METHOD The Peng−Robinson equation of state10 was used as the basis of this work and is given in standard form by P=

RT a − v−b v(v + b) + b(v − b)

0.78 0.81 0.70

0.79

1.00

3 4 4 4 5 5 5 5 5

0.20 0.19 0.16 0.14 0.13 0.40

6 6 6 7 7 7

(1)

j

aij = b=

0.10 0.08 0.15 1.29 1.11 0.13

1.08 1.14e

0.09 0.35 0.68 0.54 0.22 0.23 0.11 0.05 0.06

30 points (of 128 critically selected) predicted as single-phase (143 of 491 overall predicted as single-phase). b175 of 388 points predicted as single-phase. c186 of 388 points predicted as single-phase. d81 of 146 points predicted as single-phase. e3 of 62 points predicted as single-phase.

∑ ∑ xixjaij i

1.30a 1.45c

a

where a=

0.23 1.37b 1.75d 0.72 1.15 0.18 0.77 0.70 0.25

ai aj (1 − kij)

positive slope with respect to temperature at all temperatures (Figure 1). The cause of this is the large negative acentric factor for helium (ωHe = −0.365). An acentric factor of close to zero seems more reasonable for a small, spherical molecule such as helium and would be consistent with the acentric factor values of other noble gases. With ω ≡ 0, the α function obtains a minimum at around T/Tci = 13, corresponding to T ≈ 65 K for helium. Above this temperature, the α function is increasing. In our tests, the various implementations available for the α function in Aspen HYSYS do not allow this effect to be handled adequately, but the developers of the software package Multiflash (version 4.4) [personal communication with R. Szczepanski] have confirmed that beyond this minimum Multiflash in effect sets the ai function equal to zero. The developers also confirmed that any aij values that are negative are set equal to zero. Accordingly, the Peng−Robinson Advanced equation of state as implemented in Multiflash (version 4.4) was used throughout this work. A survey of literature pTxy vapor−liquid equilibrium data for binary mixtures containing helium is summarized in Table 1. Where available, binary interaction parameters for mixtures with helium are shown. The literature survey was restricted to pTxy data as these have the most information content for validating the phase equilibrium predictions from the equation of state. Overall, a wide range of temperature (65 to 665 K) is covered by the data. The survey includes a wide variety of compounds: short- and long-chain hydrocarbons, aromatics, noble gases, other simple gases, and alcohols. The paper by Lee et al.27 also contains data for binary mixtures (hexanol + helium) and (octanol + helium). We included only the data for (decanol + helium) here as being representative of the fatty alcohols. Decanol is not a regular component in Multiflash, so it was included by creating a pseudofluid with properties of Mw = 158.28 g·mol−1, Tc = 687.0 K, Pc = 2.2 MPa, and ω = 0.61.

(2)

∑ xibi

(3)

i 2

ai = 0.45724

R2Tic αi Pic

(4)

bi = 0.07780

RTic Pic

(5)

αi = (1 + κi(1 −

T /Tic ))2

(6)

κi = 0.37464 + 1.54226ωi − 0.26992ωi 2

(7)

Here, v is the molar volume, and ωi are the critical temperature, critical pressure, and acentric factor of component i, respectively, and kij is a binary interaction parameter applicable to the binary system comprising components i and j. Note that kij is typically set to zero when the interactions between components i and j have not been characterized and kji ≡ kij. The Peng−Robinson equation of state can be rewritten as a cubic equation in the compressibility factor (Z = Pv/RT) as Tci ,

Pci ,

Z3 − (1 − B)Z2 + (A − 3B2 − 2B)Z − (AB − B2 − B3) (8)

=0 2

where A = aP/(RT) and B = bP/RT. The Peng−Robinson equation of state can typically be implemented straightforwardly for pure fluids and mixtures. However, problems arise when helium is present. In contrast to all other fluids and in violation of the requirements stated by Coquelet et al.,29 the α function for helium has a 2802



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Figure 4. Deviations log Kexpt − log Kcalc of binary mixtures of cryogenic gases with helium, as predicted with the Peng−Robinson equation of state using true critical constants Tc = 5.1953 K, Pc = 0.22746 MPa, ω = −0.365, and kij = 0 (left panels) and using effective critical constants Tc = 11.73 K, Pc = 0.568 MPa, ω = 0.0, and kij = 0 (right panels). Sources of data are given in Table 1. Points predicted to be in the single-phase region are not shown. Symbols: blue ×, nonhelium components; orange +, helium.

system for applications in natural gas processing involving helium. Because both methane and helium molecules can be well-approximated as hard spheres, the cubic equation of state should yield good agreement with phase equilibrium data with a negligible binary interaction parameter. Accordingly, during the regression of helium’s effective critical constants and in all other predictive calculations, the acentric factor for helium was set to zero in accord with the approximate values used for other noble gases. One of the methods by which the pTxy data for (methane + helium) were assessed for consistency was by analyzing the vapor enhancement factors of methane. The enhancement factor equals the quotient of the (experimental) partial pressure of methane and the saturation vapor pressure of pure methane 30 at the same temperature, that is, f = yCH4P/Psat For isoCH4. thermal data, the enhancement factor approaches unity in the limit of zero pressure. An example for isothermal data near 124 K is shown in Figure 2. It can be seen that the data sets of Heck and Hiza,31 DeVaney et al.,32 and Rhodes et al.33 are mostly consistent, but the data of Sinor et al.30 are systematically too high and the data from Fontaine34 exhibit large

To regress effective critical constants that will yield accurate predictions in mixtures with multiple components, it is important to select carefully the data against which the parameters will be tuned. The following factors were among those taken into consideration: • experimental measurement technique (equipment and procedures) • reported uncertainty in data (including the purity of the chemicals used) • coherence of data among different authors (where available) • ranges of temperature, pressure, and composition It is important to avoid over-reliance on unverified results during parameter optimization. For example, Rowland and May55 provide examples where estimates of experimental uncertainty assigned by authors to their own data are an order of magnitude too optimistic and systematic errors are hidden within internally consistent data. Often the true reliability of experimental measurements is revealed only by comparing with multiple sources of independent data. The (methane + helium) mixture is the most well studied and most relevant binary 2803



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Figure 5. Deviations log Kexpt − log Kcalc of binary mixtures of light natural gas components with helium, as predicted with the Peng−Robinson equation of state using true critical constants Tc = 5.1953 K, Pc = 0.22746 MPa, ω = −0.365, and kij = 0 (left panels) and using effective critical constants Tc = 11.73 K, Pc = 0.568 MPa, ω = 0.0, and kij = 0 (right panels). Sources of data are given in Table 1. Points predicted to be in the singlephase region are not shown. Symbols: blue ×, nonhelium components; orange +, helium. 2804



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Figure 6. Deviations log Kexpt − log Kcalc of binary mixtures of heavy natural gas components with helium, as predicted with the Peng−Robinson equation of state using true critical constants Tc = 5.1953 K, Pc = 0.22746 MPa, ω = −0.365, and kij = 0 (left panels) and using effective critical constants Tc = 11.73 K, Pc = 0.568 MPa, ω = 0.0, and kij = 0 (right panels). Sources of data are given in Table 1. Symbols: blue ×, nonhelium components; orange +, helium.

the two-phase system observed. Neglecting these data from the AAD could lead to an overly optimistic assessment of the performance of poor models. Instead, any such affected data points had their contribution to the AAD fixed at M, which represents an order-of-magnitude error in K for each component in the mixture.

internal scatter. As cubic equations of state are known to behave poorly near mixture critical points,11 the pTxy data for (methane + helium) above 190 K were also eliminated from consideration. This left 128 data points suitable for parameter regression from the data sets of Heck and Hiza,31 DeVaney et al.,32 and Rhodes et al.33 The objective function used throughout this work was the absolute average deviation (AAD) between the experimental and calculated distribution K factors (Ki ≡ yi/xi) calculated according to AAD =

1 N

N

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RESULTS AND DISCUSSION Values of the effective critical constants for helium that minimized the AAD between the Peng−Robinson equation of state and the critically selected data for (methane + helium) were Tc = 11.73 K and Pc = 0.568 MPa (with ω ≡ 0). Compared to the true critical constants, the effective critical constants yield a huge improvement in the agreement between the Peng−Robinson equation of state predictions and the critically selected data (Figure 3). The effective critical constants developed in this work are reasonably concordant with the early suggestion of Newton12 (13.2 K) and the classical critical constants for helium reported by Gull et al.13 (10.47 K, 0.667 MPa). It is instructive to compare the Peng−Robinson equation of state predictions for all of the binary systems in Table 1 made

M

∑ ∑ |log K i ,expt,n − log K i ,calc,n| n=1 i=1

(9)

where M is the number of components in the mixture. The AAD is preferred compared to other statistical measures (e.g., the sum of squares or root-mean-square deviation) as it is more tolerant of outliers in the data. To minimize the AAD, it is important that the equation of state gives good agreement with both the vapor- and liquid-phase compositions. The equation of state sometimes predicts that the system exists as a single phase at a given pressure, temperature, and composition instead of as 2805



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Figure 7. Deviations log Kexpt − log Kcalc of binary mixtures of chemical process additives with helium, as predicted with the Peng−Robinson equation of state using true critical constants Tc = 5.1953 K, Pc = 0.22746 MPa, ω = −0.365, and kij = 0 (left panels) and using effective critical constants Tc = 11.73 K, Pc = 0.568 MPa, ω = 0.0, and kij = 0 (right panels). Sources of data are given in Table 1. Symbols: blue ×, nonhelium components; orange +, helium.

mixtures containing helium in Table 1, is actually needed for (nitrogen + helium) if the true critical constants for helium were to be retained in such calculations. This is particularly pertinent in helium capture applications where nitrogen is the dominant component in the crude helium stream produced by the nitrogen rejection unit; instead we recommend use of the effective critical parameters and ω = 0 for helium with kij = 0. It is instructive to compare in detail the quality of the predictions with kij = 0 for the reasons discussed above. The results obtained using the effective critical constants for helium are vastly superior to those obtained with the true critical constants for helium. (See Table 2 and compare left-hand panels with right-hand panels in Figures 3 to 8.) In the latter case, the Peng−Robinson EOS underpredicts the K factors for helium in all mixtures considered here, and the K factors of the other components tend to be overpredicted. For (nitrogen + helium), some of the predicted K factors differ by more than an order of magnitude from the experimental results. It should also be noted that around one-third of the data for binary mixtures of helium with methane, nitrogen, and argon were improperly predicted to be single-phase instead of two-phase when the true

with the effective critical constants for helium with those obtained using the true critical constants with and without a binary interaction parameter. The AAD values for each scenario are given in Table 2. No critical selection of data was performed for the calculation of the statistics in Table 2, except for (methane + helium) as described above to determine the effective critical constants for helium. The results obtained with the effective critical constants developed in this work are clearly favorable: the average deviations with the effective critical constants are mostly smaller than the average deviations obtained with true critical constants. In the 5 (out of 15) binary systems where this is not the case, very large binary interaction parameters (from 1.06 to 1.50) are required to achieve good agreement with the data. For the system (nitrogen + helium), the true critical constants and the binary interaction parameter from HYSYS (kij = 0.0685) produce an average deviation that is large compared to those obtained for the other compounds, with 175 out of 388 data points being predicted erroneously as single-phase; this is almost as poor as the results obtained with no binary interaction parameter. This suggests that a much larger binary interaction parameter, in line with those for other binary 2806



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Figure 8. Deviations log Kexpt − log Kcalc versus T/K of (nitrogen + helium) and (argon + helium), as predicted with the Peng−Robinson equation of state using true critical constants Tc = 5.1953 K, Pc = 0.22746 MPa, ω = −0.365, and kij = 0 (left panels) and using effective critical constants Tc = 11.73 K, Pc = 0.568 MPa, ω = 0.0, and kij = 0 (right panels). Sources of data are given in Table 1. Points predicted to be in the single-phase region are not shown. Symbols: blue ×, nonhelium components; orange +, helium.

Table 3. Survey of pTxy Vapor−Liquid Equilibrium Data for Ternary Mixtures with Helium mixture (+ helium) a

methane + nitrogen methane + nitrogenb methane + nitrogenc methane + nitrogen propane + nitrogen

N 104 23 52 126 18

T/K 76 to 76 to 80 to 130 to 273

130 110 164 180

P/MPa 1 1 0.7 4 3

to to to to to

14 8 8 10 21

ref 57 58 58 59 50

a Nominally (0.50He + 0.45N2 + 0.05CH4). bMix A = (0.6465He + 0.3420N2 + 0.0115CH4). cMix B = (0.2543He + 0.2653N2 + 0.4803CH4).

Table 4. Deviations between the Peng−Robinson Equation of State and Literature pTxy Data AAD AAD (Tc = 5.1953 K, (Tc = 5.1953 K, AAD c P = 0.22746 MPa, P = 0.22746 MPa, (Tc = 11.73 K, c ω = −0.365, ω = −0.365, P = 0.568 MPa, mixture kij = 0) (+ helium) kij from Table 1) ω = 0, kij = 0) c

Figure 9. Comparison of literature pTx data for (ethane + helium). Data from Heck56 and Nikitina et al.49

methane + nitrogen methane + nitrogena methane + nitrogenb methane + nitrogen propane + nitrogen

critical constants were used in the Peng−Robinson equation of state. A well-known deficiency of cubic equations of state is their behavior around mixture critical points.11 In this study, this effect is most noticeable for the (nitrogen + helium) and (argon + helium) binary systems where the absolute deviations diverge as log K tends to zero (Figure 4), coinciding with temperatures approaching the critical points of nitrogen and argon, respectively (Figure 8). Nevertheless, the deviations in this region obtained with the effective critical constants for helium are comparable to or smaller than the deviations shown in Figure 4 for helium’s true critical constants, even far from the mixture’s critical region. The effective critical constants gave highly satisfactory results for most of the systems considered here. For the systems

ref

0.76

1.27c

0.28

57

0.71

1.06

0.14

58

0.17

0.80

0.11

58

0.13

1.18d

0.09

59

0.11

0.73

0.15

50

a

Mix A. bMix B. c5 of 104 points predicted as single-phase. d26 of 126 points predicted as single-phase.

(propane + helium), (butane + helium), (hexadecane + helium), and (decanol + helium), a few data points with small K factors had relatively large deviations. This could be due to experimental 2807



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Figure 10. Deviations log Kexpt − log Kcalc of ternary mixtures containing helium for literature K factors, as predicted with the Peng−Robinson equation of state with true critical constants Tc = 5.1953 K, Pc = 0.22746 MPa, ω = −0.365, and kij = 0 (left panels) and those predicted with the Peng−Robinson equation of state with effective critical constants Tc = 11.73 K, Pc = 0.568 MPa, ω = 0.0, and kij = 0 (right panels). Sources of data are given in Table 3. Symbols: green Δ, methane or propane; blue x, nitrogen; orange + , helium. 2808



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relatively rare in the chemical literature. This directly impacts the reliability of thermodynamic models used in process engineering because the empirical binary interaction parameters ordinarily required by these models cannot be regressed against experimental data. Therefore, the binary interaction parameters either must assume standard values, kij = 0 in the case of the Peng−Robinson equation of state, or must be estimated by other means. As many of the binary interaction parameters for helium mixtures have large magnitudes (even greater than unity), the potential for poor-quality predictions when unregressed values are used is substantial. This is well illustrated in Table 2 by comparing the average deviations obtained with and without binary interaction parameters. A key advantage of the model developed in this work is that binary interaction parameters are not necessary to achieve good phase equilibrium predictions. Issues related to large interaction parameters apply not only to the cubic equations of state tested in this work but also to the more-sophisticated multiparameter Helmholtz equations of state recommended for natural gas transmission and sales applications (e.g., GERG-200861 equation of state). In regard to the GERG-2008 equation of state, regressed parameter values for most fluid combinations typically differ from the standard ideal values by −0.1 to 0.3, which is (10 to 30) % of those ideal values.61−63 However, for helium-containing binary systems, regressed parameters vary from the standard ideal values by as much as 2.2! Even so, the correlation of phase equilibrium data for helium-containing binary systems is poor for the GERG-2008 equation of state (Figure 11),64 and predictions of

uncertainties associated with sampling trace amounts of the heavy component in the gas phase. For (ethane + helium), the deviations of the helium K factors near log Kexpt = 1.0 were anomalously large when the effective critical constants were used in the Peng−Robinson equation (Figure 5). Although there are no pTxy data available with which to compare the values of Nikitina et al.,49 the solubility of helium in liquid ethane was measured by Heck.56 A comparison of some of the pTx data of Heck56 and Nikitina et al.49 at nearby isotherms shows large discrepancies between the two sources (Figure 9). As both sets of data appear to be internally consistent, it is plausible that the data of Nikitina et al.49 may be subject to some systematic error, and further independent experimental evaluation of this system is warranted. For the system (oxygen + helium), the pTxy data deviated systematically from the model and were about twice as large as the deviations in the comparable system (nitrogen + helium). These data originate from the same laboratory as the uncertain data for (ethane + helium), so an independent experimental investigation of the system (oxygen + helium) may also be warranted. The Peng−Robinson equation of state with the effective critical constants for helium developed in this work gives especially good results for binary mixtures comprising helium with methane, nitrogen, and carbon dioxide. These components dominate most natural gas mixtures, so it is important to characterize accurately their mixtures with helium. As these binary systems are described well, it gives confidence that the phase behavior of multicomponent mixtures will be predicted accurately. Vapor−liquid equilibrium data are available for a small number of well-characterized ternary mixtures containing helium (Table 3). The system (carbon dioxide + ethylene + helium) has also been studied, but no experimental pTxy data were published.60 Most of the available data relate to the ternary system (methane + nitrogen + helium) that is of greatest relevance for helium purification from natural gas. As for the binary mixtures containing helium, AAD values were calculated for each ternary mixture. These values are given in Table 4. The predictions of the phase equilibria in ternary mixtures further indicate the superiority of the method based on effective critical constants for helium developed in this work (Figure 10). The new model gives especially good results for the (methane + nitrogen + helium) system when the methane content is lower than 5% (data of Boone et al.57 and Mix A of Rhodes et al.58). Note that in the case of Boone et al.57 the methane K factors are subject to very large uncertainty at low pressures as the fraction of methane in the vapor phase was close to the detection limit and could not be determined accurately. By comparison, the approach with nonzero binary interaction parameters and true critical constants performs very poorly for these mixtures. As discussed above, these poor predictions are reflective of the abnormal binary interaction parameter between nitrogen and helium. It is possible that a larger binary interaction parameter value might lead to improved predictions for helium distillation from nitrogen-rich fluid streams when the true critical constants are used. However, such an approach would have limited predictive power and applicability to mixtures encountered in natural gas processing. The issue of availability of binary interaction parameters is especially important for mixtures containing helium. As reflected by the literature survey in Table 1, high-quality pTxy data for binary mixtures of natural gas components with helium are

Figure 11. p−xy diagram for (methane + helium) at T = 170 K. Peng−Robinson equation of state predictions with effective critical constants Tc = 11.73 K, Pc = 0.568 MPa, ω = 0.0, and kij = 0 (green solid lines). GERG-200861 equation of state predictions (blue dashed− dotted lines). Data points from Heck and Hiza.31

density for (methane + helium) deviate from recent experimental data by up to 7%, which is much more than the combined stated uncertainty of the experimental data (relative uncertainty in density is less than 0.1%65,66) and the stated uncertainty of the density calculated with the GERG-2008 equation of state for binary systems without a departure function (i.e., (0.5 to 1) %).61 Improving property correlations and predictions by multiparameter Helmholtz equations of state for helium-containing fluids is an avenue for future work.

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CONCLUSIONS This work demonstrates that problems relating to the prediction of vapor−liquid equilibria for mixtures involving helium 2809



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(10) Peng, D.-Y.; Robinson, D. B. A new two-constant equation of state. Ind. Eng. Chem. Fundam. 1976, 15, 59−64. (11) May, E. F.; Guo, J. Y.; Oakley, J. H.; Hughes, T. J.; Graham, B. F.; Marsh, K. N.; Huang, S. H. Reference quality vapor−liquid equilibrium data for the binary systems methane + ethane, + propane, + butane, and + 2-methylpropane, at temperatures from (203 to 273) K and pressures to 9 MPa. J. Chem. Eng. Data 2015, 60, 3606−3620. (12) Newton, R. H. Activity coefficients of gases. Ind. Eng. Chem. 1935, 27, 302−306. (13) Gunn, R. D.; Chueh, P. L.; Prausnitz, J. M. Prediction of thermodynamic properties of dense gas mixtures containing one or more of the quantum gases. AIChE J. 1966, 12, 937−941. (14) Chueh, P. L.; Prausnitz, J. M. Vapor-liquid equilibria at high pressures. Vapor-phase fugacity coefficients in nonpolar and quantumgas mixtures. Ind. Eng. Chem. Fundam. 1967, 6, 492−498. (15) Leland, T. W., Jr; Chappelear, P. S. The corresponding states principlea review of current theory and practice. Ind. Eng. Chem. 1968, 60, 15−43. (16) Lin, H.-M.; Lee, M.-J.; Lee, R.-J. High-temperature phase equilibria for asymmetric mixtures of helium + m-xylene and nitrogen + m-xylene. Ind. Eng. Chem. Res. 1995, 34, 4524−4530. (17) Twu, C. H.; Coon, J. E.; Harvey, A. H.; Cunningham, J. R. An Approach for the Application of a Cubic Equation of State to Hydrogen− Hydrocarbon Systems. Ind. Eng. Chem. Res. 1996, 35, 905−910. (18) McCarty, R. D.; Arp, V. D. A New Wide Range Equation of State for Helium. Advances in Cryogenic Engineering; Springer: New York, 1990; Vol. 35, pp 1465−1475, doi: 10.1007/978-1-4613-06399_174. (19) Graboski, M. S.; Daubert, T. E. A modified Soave equation of state for phase equilibrium calculations. 3. Systems containing hydrogen. Ind. Eng. Chem. Process Des. Dev. 1979, 18, 300−306. (20) Radosz, M.; Lin, H.-M.; Chao, K.-C. High-pressure vapor-liquid equilibriums in asymmetric mixtures using new mixing rules. Ind. Eng. Chem. Process Des. Dev. 1982, 21, 653−659. (21) Schulze, W. A simple generalization of the binary temperaturedependent interaction parameters in the Soave-Redlich-Kwong and Peng-Robinson equations of state for helium-mixtures. Fluid Phase Equilib. 1993, 87, 199−211. (22) Qian, J.-W.; Jaubert, J.-N.; Privat, R. Phase equilibria in hydrogen-containing binary systems modeled with the Peng− Robinson equation of state and temperature-dependent binary interaction parameters calculated through a group-contribution method. J. Supercrit. Fluids 2013, 75, 58−71. (23) Kato, M.; Tanaka, H. Vapor-liquid equilibrium prediction with pseudo-cubic equation of state for binary mixtures containing hydrogen, helium, or neon. Sekiyu Gakkaishi 1990, 33, 99−103. (24) Wei, Y. S.; Sadus, R. J. Vapour-liquid and liquid-liquid phase equilibria of binary mixtures containing helium: comparison of experiment with predictions using equations of state. Fluid Phase Equilib. 1996, 122, 1−5. (25) Solen, K. A.; Chueh, P. L.; Prausnitz, J. M. Thermodynamics of helium solubility in cryogenic solvents at high pressures. Ind. Eng. Chem. Process Des. Dev. 1970, 9, 310−317. (26) Lin, H.-M.; Lee, M.-J.; Lee, R.-J. Phase-Equilibria of Toluene in Mixtures with Helium or Nitrogen at High-Temperatures. J. Chem. Eng. Data 1995, 40, 653−656. (27) Lee, M.-J.; Cheng, J.-W.; Lin, H.-M. Measurement and Correlation of Vapor− Liquid Equilibria for Asymmetric Mixtures of Helium and Polar Compounds. Ind. Eng. Chem. Res. 1998, 37, 3508− 3514. (28) Ashour, I.; Almehaideb, R.; Fateen, S.-E.; Aly, G. Representation of solid-supercritical fluid phase equilibria using cubic equations of state. Fluid Phase Equilib. 2000, 167, 41−61. (29) Coquelet, C.; Chapoy, A.; Richon, D. Development of a new alpha function for the Peng−Robinson equation of state: comparative study of alpha function models for pure gases (natural gas components) and water-gas systems. Int. J. Thermophys. 2004, 25, 133−158.

can be addressed satisfactorily by replacing helium’s true critical constants in a cubic equation of state with the effective critical constants (Tc = 11.73 K, Pc = 0.568 MPa). This is done on the basis that the true critical constants are affected by quantum phenomena that are not relevant at high temperatures. One of the advantages of this approach is that it can be incorporated into process simulation software relatively simply by modifying the properties of helium or introducing a pseudofluid with the appropriate effective critical constants and an acentric factor equal to zero. The data analysis in this work is the most comprehensive for binary and ternary mixtures containing helium, with 15 distinct binary systems considered in total spanning (65 to 665) K. Predictions are satisfactory for all systems up to pressures of around 20 MPa except near the mixture critical temperatures, which is a known limitation of the cubic equation of state employed in this work. This investigation points the way to an improved modeling of helium recovery from natural gas mixtures, including those dominated by methane, nitrogen, and carbon dioxide.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Darren Rowland: 0000-0002-5772-2608 Notes

The authors declare no competing financial interest. Funding

This research was funded by the Australian Research Council through IC15001019.

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ACKNOWLEDGMENTS The authors thank Richard Szczepanski from Infochem for useful discussions. REFERENCES

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