Behavior and Virial Coefficients for the (Argon + ... - ACS Publications

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Vapor-Phase (p, ρ, T, x) Behavior and Virial Coefficients for the (Argon + Carbon Dioxide) System Mohamed A. Ben Souissi, Markus Richter,* Xiaoxian Yang,† Reiner Kleinrahm, and Roland Span Lehrstuhl für Thermodynamik, Ruhr-Universität Bochum, D−44780 Bochum, Germany ABSTRACT: A two-sinker magnetic suspension densimeter was used to carry out accurate density measurements on binary mixtures (argon + carbon dioxide) with carbon dioxide mole fractions of 0.50000, 0.49975, and 0.75093. The measurements cover a temperature range from T = (273.15 to 323.15) K with pressures up to 9.1 MPa or the dew-point pressure, whichever was lower. The mixtures were prepared gravimetrically. With all measurement uncertainties in temperature, pressure, density, and composition taken into account, the relative combined expanded uncertainty (k = 2) in density was estimated to be less or equal to 0.033%. Relative deviations of the experimental densities from the GERG-2008 equation of state were within 0.31% for the mixtures with 0.50000 and 0.49975 mole fraction carbon dioxide and up to 1.90% for the mixture with 0.75093 mole fraction carbon dioxide. In contrast, the relative deviations from a recently developed multiparameter equation of state (EOS) optimized for combustion gases (EOSCG) were less than 0.30% for the three mixtures under study. Third-order virial equations were fitted to the measured density values. The correlated molar masses at different temperatures agreed within less than 0.01% with the molar masses determined from the gravimetric mixture preparation. Values and uncertainties of the second and third virial coefficients as well as of the second interaction virial coefficient were calculated.

1. INTRODUCTION The GERG-2008 equation of state (EOS),1,2 for example, as implemented in the NIST REFPROP database,3 is the current reference EOS for natural gas and similar mixtures. This mixture model is explicit in the reduced Helmholtz free energy as a function of density ρ, temperature T, and composition x of the respective mixture; it is based on a multifluid approximation and uses accurate fundamental EOS for each considered mixture component in combination with functions for the binary mixtures of the components to take into account the residual mixture behavior. A recently developed Helmholtz energy model for combustion gases (EOS-CG),4,5 for example, as implemented in the TREND software package6 of RuhrUniversity Bochum, adapted the functional form of the GERG2008 EOS. The two mentioned EOS are fully empirical, thus, their accuracy in predicting thermodynamic properties relies on the quality of the underlying experimental data. For binary systems with a comprehensive basis of accurate experimental data, usually binary specific departure functions are developed, and the uncertainty in the prediction of gas-phase densities can be as good as 0.1%. Considering the GERG equation, the focus during development was on natural gas and not on mixtures rich in carbon dioxide. Moreover, the experimental data basis for mixtures with carbon dioxide is usually worse than for binary hydrocarbon mixtures. Hence, within the GERG-2008 EOS the binary system (argon + carbon dioxide) was only fitted in terms of an adjusted reducing function, and the © 2016 American Chemical Society

uncertainty of the calculated gas-phase densities was estimated to be 1.0%. In contrast, for the EOS-CG, a binary specific departure function for the (argon + carbon dioxide) system was developed, and an estimated uncertainty of less than 0.3% was reported. Within the scope of the present project, we recently measured densities of the (0.05049 argon + 0.94951 carbon dioxide) mixture utilizing our two-sinker magnetic suspension densimeter.7 Relative deviations in density of the experimental data from values calculated with the GERG-2008 EOS were as high as 1.0%. A comparison to the EOS-CG revealed relative deviations of only up to 0.2%, as a result of using a binary specific departure function. Nevertheless, the binary (argon + carbon dioxide) system is attracting a great deal of interest in the field of carbon capture and storage (CCS),8,9 which aims at mitigating global carbon emissions.10 For this particular binary mixture system, an obvious lack of accurate experimental data was observed. In this context, the available experimental (p, ρ, T, x) data before the year 2016 were summarized by Yang et al.11 To fill the data gap for the binary (argon + carbon dioxide) system, and for the sake of improving the performance of mixture models, the (p, ρ, T, x) behavior of binary (argon + carbon dioxide) mixtures with 0.50000, 0.49975, and 0.75093 mole fraction carbon Received: July 30, 2016 Accepted: October 28, 2016 Published: November 11, 2016 362

DOI: 10.1021/acs.jced.6b00687 J. Chem. Eng. Data 2017, 62, 362−369

Journal of Chemical & Engineering Data

Article

mixture preparation are described in Table 1; they were used as received without further gas analysis or purification. We

dioxide was investigated. We used the same two-sinker magnetic suspension densimeter as in our previous work7 within the same project. Measurements were carried out over the temperature range from 273.15 to 323.15 K with pressures up to 9.1 MPa or the dew-point pressure, whichever was lower.

Table 1. Sample Information

2. EXPERIMENTAL SECTION 2.1. Apparatus Description. The fundamental technique of the two-sinker magnetic suspension densimeter was developed by Kleinrahm and Wagner12 in the 1980s, and the apparatus we used for the present measurements was built by Pieperbeck et al.;13 it was specifically designed to measure the densities of pure gases and natural gas mixtures over the temperature range from 273.15 to 323.15 K with pressures up to 12 MPa. The apparatus was described in detail in previous papers13,14 and was adapted for recent studies in our group.7,15,16 Details of this general type of instrument are reported elsewhere.17−20 Thus, we just briefly explain the measuring principle here. Two specially matched sinkers (made of stainless steel) with approximately the same surface area were used for density measurement: a hollow sphere (VS ≈ 107 cm3; mS ≈ 123 g; ρS ≈ 1.16 g·cm−3) and a solid ring (VR ≈ 15.6 cm3; mR ≈ 123 g; ρR ≈ 7.90 g·cm−3). Their surfaces were electrolytically polished and afterward gold plated to realize a very smooth surface finish. In order to measure the density, both sinkers, being immersed in the gas under investigation, were weighed alternately using an analytical balance (readability: 0.00001 g) in conjunction with a magnetic suspension coupling. This coupling transmitted the gravity and buoyancy forces on the sinkers to the balance, thus isolating the gas sample (inside the pressure-tight measuring cell) from the balance, which was placed under ambient conditions. The force transmission error of the magnetic suspension coupling19 was relatively small due to the use of nonmagnetic materials of the coupling housing and because of a nearly identical position of the permanent magnet for the magnetic suspension coupling for both sinker weighings. However, a small fluid-specific effect still existed. This remaining error was covered by the uncertainty in density measurement. The density of the gas under investigation was determined by ρgas (T , p) =

source

purity/mol fraction

purification method

argon carbon dioxide

Air Liquid Air Liquid

0.999990a 0.999995b

none none

Impurities (stated by supplier): x(H2O) ⩽ 2.0 × 10−6, x(O2) ⩽ 2.0 × 10−6, x(CmHn) ⩽ 0.5 × 10−6, x(CO2) ⩽ 0.2 × 10−6, x(N2) ⩽ 5.0 × 10−6. bImpurities (stated by supplier): x(H2O) ⩽ 2.0 × 10−6, x(O2) ⩽ 1.0 × 10−6, x(CmHn) ⩽ 0.1 × 10−6, x(N2) ⩽ 2.0 × 10−6, x(CO) ⩽ 0.5 × 10−6, x(NOx) ⩽ 0.1 × 10−6. a

employed 20 L aluminum gas cylinder and filled them with a comparatively high pressure up to approximately 14 MPa. The cylinders were supplied by Scott Specialty Gases with a particular treatment of the inside surface, which was intended to improve the long-term stability of the mixture composition (for further information, see Aculife inerting cylinder treatments of Scott Specialty Gases). Gravimetrically determined mole fractions, molar masses, and the measured isotherms of the studied mixtures are listed in Table 2. The expanded uncertainty (k = 2) in composition due to the mixture preparation was estimated to be 0.0004 mole fraction for all three mixtures, which corresponds to an expanded uncertainty (k = 2) in molar mass of 0.0016 g·mol−1. 2.3. Experimental Procedures. The (T, p) state points of the present density measurements are depicted in Figure 1 in pressure versus temperature phase diagrams. The phase envelopes in Figure 1 were calculated with the GERG-2008 EOS as implemented in the NIST REFPROP database.3 Density measurements of the (argon + carbon dioxide) mixtures were carried out along isotherms at T = (273.15, 283.15, 293.15, and 323.15) K with a carbon dioxide mole fraction of 0.75093, at T = (273.15 and 298.15) K with a carbon dioxide mole fraction of 0.50000, and at T = 323.15 K with a carbon dioxide mole fraction of 0.49975. For temperatures below the maxcondentherms, the pressures extended to the dew-point pressures, while for those above the maxcondentherms, the pressures extended up to about 9.1 MPa. At each (T, p) state point, at least two replicate measurements were carried out. Compared to argon, carbon dioxide is more inclined to adsorb on the solid surfaces inside the measuring cell and in particular at higher pressures. Subsequently, it would desorb back to the bulk gas phase when going to lower pressures. Moreover, carbon dioxide is known for its affinity to diffuse into elastomer seals. The consequence of both is a change in composition of the measured gas and therefore a change of the gas density inside the measuring cell. It is known that the underlying surface phenomena distort the composition of the mixture to be measured especially in the vicinity of the dew line.16 Richter and Kleinrahm16 suggested to investigate sorption effects for thermophysical property measurements on mixtures and demonstrated that waiting for “sorption equilibrium“ and then flushing the measuring cell with the sample gas at the temperature and pressure of interest was an expediend way to yield accurate measurement results. Therefore, in contrast to our previously reported measurements on the (argon + carbon dioxide) mixture with a carbon dioxide mole fraction of 0.94951 in which a delicate experimental procedure to avoid the occurrence of a phase transition had to

* * ) (mS − mR ) − (mS,fluid − mR,fluid VS(T , p) − VR (T , p)

chemical name

(1)

where m and V are the mass and the volume of the sinker, m* is the weighing value of the sinker, and the subscripts refer to the two sinkers. For density measurement, at first the spherical sinker was weighed, and the balance was tared to m*S,fluid = 0.00000 g. Then, the two sinkers were typically exchanged 30 times to improve the accuracy of the difference in weighing values (m*S,fluid − m*R,fluid) and an average value was calculated. The two sinkers had nearly the same mass (mS − mR ≈ 0.00080 g), the same surface area, and the same surface material but had a substantial difference in volume (VS − VR ≈ 91.4 cm3). Accordingly, the buoyancy effect was very large and therefore even the low-density range of gases could be measured accurately. 2.2. Experimental Material. The three binary (argon + carbon dioxide) mixtures were prepared gravimetrically; a detailed description of our sample preparation procedure and our mixture preparation system can be found in Section 6 of the dissertation of Schäfer.21 The pure components we used for 363



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Table 2. Gravimetrically Determined Mole Fraction (xgrav), Molar Mass (Mgrav), and the Measured Isotherms (Tmeas) of the Studied Mixtures mixtures xgrav (argon) xgrav (carbon dioxide) Mgrav/g·mol−1 Tmeas/K

(75/25)

(50/50)

(50/50)

0.24907 0.75093 42.9981 273.15, 283.15, 293.15, 323.15

0.50000 0.50000 41.9789 273.15, 298.15

0.50025 0.49975 41.9779 323.15

Table 3. Uncertainty Budget for the Relative Combined Standard Uncertainty in Density uC(ρ)/ρa uncertainty contribution

standard uncertainty

temperature T 2.5 mK pressure p 0.0035% p density ρ 0.0075% ρ molar mass Mb 0.0008 g·mol−1 reproducibilityc 0.0115% ρ relative combined expanded (k = 2) uncertainty

contribution to uC(ρ)/ρ 0.0041% 0.0065% 0.0075% 0.0016% 0.0115% 0.032%

a

As an example, the contributions to uC(ρ)/ρ were calculated for the binary (argon + carbon dioxide) mixture with a carbon dioxide mole fraction of 0.75093 at a temperature of 283.15 K and a pressure of 7.26379 MPa. bFrom gravimetric sample preparation. cIncluding sorption effects. Figure 1. Pressure−temperature phase diagrams of the investigated (argon + carbon dioxide) mixtures. +, values measured in the present work; *, critical point; ―, liquid−vapor phase boundaries calculated with the GERG-2008 equation of state of Kunz and Wagner.1,2 (a) (0.50000 argon + 0.50000 carbon dioxide) at T = (273.15 and 298.15) K, and (0.50025 argon + 0.49975 carbon dioxide) at T = 323.15 K; (b) (0.24907 argon + 0.75093 carbon dioxide).

was implemented in the TREND software package6 of RuhrUniversity Bochum. Experimental data of Altunin and Koposhilov,24 Abraham and Bennett,25 and Sarashina et al.26 for the same binary system with similar compositions from T = (273.15 to 323.15) K with pressures up to 9.0 MPa are also illustrated in the deviation plots. As shown in Figures 2 and 3, the relative deviations of the present experimental data generally decrease with decreasing pressure along each isotherm and, as expected from theory, converge to zero when approaching the ideal gas limit (at low pressures). In comparison with our new experimental densities with the GERG-2008 EOS, the relative deviations are within 0.31% for the mixtures with carbon dioxide mole fractions of 0.50000 and 0.49975. The relative deviations increase to a maximum of about 1.90% for the mixture with a carbon dioxide mole fraction of 0.75093. In contrast, relative deviations of our experimental data from values calculated with the EOS-CG are less than 0.30% for all three mixtures studied, which is significantly smaller compared to the GERG-2008 EOS. For density calculations in the gas phase of the binary (argon + carbon dioxide) system, the authors of the GERG-2008 EOS reported a relative uncertainty of 0.5% to 1.0%, and the authors of the EOS-CG reported a relative uncertainty of less than 1.0%. However, a few experimental data exceed the given uncertainty boundaries of the GERG-2008 EOS in the vicinity of the dew line; all of our new experimental data agree with the EOS-CG within its stated uncertainty. The reason for the better agreement between the new experimental data and the EOSCG is that the EOS-CG employs a binary specific departure function for the (argon + carbon dioxide) system, whereas the GERG-2008 EOS does not. The experimental data of Altunin and Koposhilov24 and of Abraham and Bennett25 show a similar agreement with the equations of state as our new data. Experimental data reported by Sarashina et al.26 show larger and in part unsystematic relative deviations.

be applied,7 flushing the measuring cell at selected state points was conducted. As a consequence, the uncertainty of the composition due to the sorption effects is significantly smaller compared to our previous measurements.7 Accordingly, the combined uncertainties in density and in the determined virial coefficients are much smaller (see Sections 3.2 and 3.3).

3. RESULTS AND DISCUSSION 3.1. Uncertainty in Measurement. For the determination of the measurement uncertainty we followed the “Guide to the Expression of Uncertainty in Measurement”,22 commonly known as GUM. We applied the methodology reported by Richter and McLinden23 to calculate the combined uncertainty in density. The uncertainty budget for the combined standard uncertainty in density is summarized in Table 3. All individual uncertainty contribution in Table 3 were assumed to be not correlated. The relative combined expanded uncertainties (k = 2) in density are listed in Table 4 for each measuring point; they were all less than or equal to 0.033%. 3.2. Comparison to Equations of State. The experimental (p, ρ, T, x) data are listed in Table 4. Relative deviations of the isothermal experimental data from values calculated with the GERG-2008 EOS1,2 (top diagram) and the EOS-CG4,5 (bottom diagram) are shown in Figure 2 for the binary (argon + carbon dioxide) mixtures with a carbon dioxide mole fractions of 0.50000 and 0.49975, and in Figure 3 for the mixture with a carbon dioxide mole fraction of 0.75093. The EOS-CG is a relatively new Helmholtz energy mixture model specifically developed for humid gases and CCS mixtures. It 364



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Table 4. Experimental (p, ρ, T, x) Data for the (Argon + Carbon Dioxide) Mixtures and Relative Deviations of the Experimental Densities ρexp from Densities Calculated with the GERG-2008 Equation of State1 ρGERG and the EOS-CG3,4 ρEOS‑CG where p is the Pressure, T is the Temperature (ITS-90), and Uc(ρexp)/ρexp is the Relative Combined Expanded Uncertainty (k = 2) in Densitya p/MPa

ρexp/(kg·m−3)

0.51142 1.01315 3.00837 5.01697 6.03111 6.41004

9.60258 19.3275 61.4185 110.665 138.861 150.086

0.50528 1.01459 2.01919 3.02069 4.02440 5.01698 6.02536 7.04516 8.01404

8.65443 17.5824 35.8258 54.9064 75.0005 95.8895 118.229 142.007 165.743

0.51698 1.02195 3.03402 5.04018 7.04004 9.01663

8.14761 16.2440 49.9179 85.8585 124.127 164.336

0.51011 1.00570 2.00728 2.99666 4.02247 4.48860 5.00059

9.89648 20.0055 42.2195 67.1561 97.4841 113.378 133.062

0.50818 1.00200 2.00250 2.98905 3.99224 4.98806 6.07330 6.58797 7.26379

9.48215 19.1090 40.0616 63.0470 89.5274 120.070 160.561 183.670 219.753

0.50206 1.00182 2.02436 3.00106 4.00901 4.99679 6.02400 7.04903 8.01681 9.05106

9.02478 18.3618 38.7149 60.0081 84.3530 111.184 143.207 180.860 223.610 279.404

0.51424 1.02097 2.01219 3.03614

8.34326 16.7960 34.0421 52.9514

100 (ρexp − ρGERG)/ρGERG

100 Uc(ρexp)/ρexp 0.50000 CO2 + 0.50000 0.029 0.030 0.030 0.030 0.030 0.030 0.50000 CO2 + 0.50000 0.029 0.029 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.49975 CO2 + 0.50025 0.029 0.029 0.030 0.030 0.030 0.030 0.75093 CO2 + 0.24907 0.029 0.030 0.030 0.030 0.030 0.030 0.031 0.75093 CO2 + 0.24907 0.029 0.030 0.030 0.030 0.030 0.030 0.031 0.031 0.032 0.75093 CO2 + 0.24907 0.029 0.030 0.030 0.030 0.030 0.030 0.031 0.031 0.032 0.033 0.75093 CO2 + 0.24907 0.029 0.029 0.030 0.030

−1

Ar, Mgrav = 41.9789 g·mol , T = 0.0240 0.0503 0.1498 0.2507 0.2869 0.3077 Ar, Mgrav = 41.9789 g·mol−1, T = 0.0070 0.0160 0.0277 0.0332 0.0384 0.0279 0.0143 −0.0226 −0.0747 Ar, Mgrav = 41.9779 g·mol−1, T = −0.0001 −0.0024 −0.0156 −0.0526 −0.1245 −0.2164 Ar, Mgrav = 42.9981 g·mol−1, T = 0.0681 0.1315 0.3154 0.5494 0.9065 1.1288 1.5292 Ar, Mgrav = 42.9981 g·mol−1, T = 0.0436 0.0927 0.2172 0.3746 0.5801 0.8577 1.3364 1.6848 1.8707 Ar, Mgrav = 42.9981 g·mol−1, T = 0.0367 0.0741 0.1663 0.2701 0.4057 0.5689 0.7905 1.0788 1.3815 1.4551 Ar, Mgrav = 42.9981 g·mol−1, T = 0.0198 0.0430 0.0756 0.1140 365

100 (ρexp − ρEOS‑CG)/ρEOS‑CG

273.15 K −0.0060 −0.0129 −0.0710 −0.1449 −0.1903 −0.1968 298.15 K −0.0054 −0.0104 −0.0290 −0.0538 −0.0751 −0.1049 −0.1276 −0.1601 −0.1928 323.15 K −0.0040 −0.0109 −0.0411 −0.0776 −0.1164 −0.1365 273.15 K 0.0028 −0.0081 −0.0124 −0.0283 −0.0306 −0.0226 0.0863 283.15 K −0.0077 −0.0158 −0.0319 −0.0506 −0.0728 −0.0887 −0.0443 0.0403 0.2736 293.15 K −0.0032 −0.0107 −0.0272 −0.0493 −0.0689 −0.0869 −0.0896 −0.0643 −0.0168 −0.0533 323.15 K −0.0002 0.0017 −0.0117 −0.0247 DOI: 10.1021/acs.jced.6b00687 J. Chem. Eng. Data 2017, 62, 362−369


Article

Table 4. continued p/MPa 4.03780 5.06179 6.05547 7.05484 8.05372 9.04501

ρexp/(kg·m−3) 72.6588 94.2140 116.646 140.911 167.037 195.026

100 (ρexp − ρGERG)/ρGERG

100 Uc(ρexp)/ρexp

0.75093 CO2 + 0.24907 Ar, Mgrav = 42.9981 g·mol−1, T = 323.15 K 0.030 0.1504 0.030 0.1981 0.030 0.2302 0.030 0.2623 0.030 0.2761 0.030 0.2950

100 (ρexp − ρEOS‑CG)/ρEOS‑CG −0.0402 −0.0449 −0.0603 −0.0696 −0.0864 −0.0805

a

The expanded uncertainties (k = 2) of the measurements are 5 mK for temperature, 0.007% for pressure, and 0.015% for density. The mixture samples were prepared gravimetrically with the uncertainty in composition 0.0004 mole fraction.

Figure 2. Relative deviations of the experimental densities ρexp for (argon + carbon dioxide) mixtures from densities ρGERG calculated with the GERG-2008 EOS1,2 (zero line in the top diagram) and from densities ρEOS‑CG calculated with the EOS-CG4,5 (zero line in the bottom diagram). Densities measured in the present work: ○, T = 273.15 K (xCO2 = 0.50000); △, T = 298.15 K (xCO2 = 0.50000); ▽, T = 323.15 K (xCO2 = 0.49975). Densities of other authors are plotted for comparison. Altunin and Koposhilov:24 +, T = 323.15 K (xCO2 = 0.4858); *, T = 303.15 K (xCO2 = 0.4810). Abraham and Bennett:25 ×, T = 323.15 K (xCO2 = 0.464). The error bars of our new experimental data correspond approximately to the vertical size of the plot symbol.

Figure 3. Relative deviations of the experimental densities ρexp for (argon + carbon dioxide) mixtures from densities ρGERG calculated with the GERG-2008 EOS1,2 (zero line in the top diagram) and from densities ρEOS‑CG calculated with the EOS-CG4,5 (zero line in the bottom diagram). Densities measured in the present work (xCO2 = 0.75093): ○, T = 273.15 K; □, T = 283.15 K; ◊, T = 293.15 K; ▽, T = 323.15 K. Densities of other authors plotted for comparison. Altunin and Koposhilov:24 +, T = 323.15 K (xCO2 = 0.8009). Abraham and Bennett:25 ×, T = 323.15 K (xCO2 = 0.721). Sarashina et al.:26 *, T = 288.15 K (xCO2 = 0.752). In the bottom figure, data of Sarashina et al.26 show increasing systematic deviations from 0.5% to 1.2% in the pressure range from p = (2.5 to 9.5) MPa. The error bars of our new experimental data are smaller than the vertical size of the plot symbol.

3.3. Virial Coefficients. In line with our previous publication,7 the present isothermal data were used to fit the third-order virial equation p=

ρR mT ⎛ ⎛ ρ ⎞2 ⎞ ρ ⎜1 + B(T ) · + C(T ) ·⎜ ⎟ ⎟ ⎝M⎠ ⎠ M ⎝ M

excluded from fitting; however, only two data points were affected. Attempts to include the fourth virial coefficient did not substantially improve the goodness of the fit for data with pressures lower than plim. At first, B(T) and C(T) in eq 2 were fitted together with the molar mass M of the mixture. Excellent correlations were obtained. The root-mean-squares (RMSVE) of the relative deviations in pressure of the experimental data from values calculated with the fitted virial equation were smaller than 0.0035% for all isotherms (see Table 5). Relative deviations of all the correlated molar masses MVE from the gravimetrically determined one Mgrav were within 0.0074%. However, this was larger than the combined uncertainty (k = 2) of Mgrav, which was 0.004%. The reasons were that (1) the difference in molar

(2)

where p is the pressure, ρ is the density, T is the temperature, M is the molar mass of the gas mixture, Rm = 8.314472 J·K−1· mol−1 is the molar gas constant, and B(T) and C(T) are the second and third virial coefficients, respectively. All data were weighted equally for the fitting procedure, and the results are given in Table 5. For each isotherm, a pressure limit plim (see Table 5) was determined within the scope of the fitting procedure. Exceeding this limit made the quality of the fit worse. Therefore, data with pressures higher than plim were 366



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Table 5. Summary of the Virial Analysis for the (Argon + Carbon Dioxide) Systema M taken as a fitted parameter T/K

Pim/MPa

RMSVE/%

x(CO2)/ (mol frac)

273.150 298.150

6.5 8.1

0.0020 0.0013

0.50020 0.50029

323.150

9.2

0.0028

0.50017

273.150 283.150 293.150 323.150

4.5 7.3 8.2 9.2

0.0032 0.0028 0.0034 0.0035

0.75045 0.75019 0.75019 0.75173

MVE/ (g/mol)

M equals to Mgrav ΔM/ (g/mol)

RMSgrav/%

B/ (cm3/mol)

U(B)/ (cm3/mol)

(0.50000 Ar + 0.50000 CO2), Mgrav = 41.9789 41.9797 0.0008 0.0022 −68.69 41.9801 0.0012 0.0017 −55.45 (0.50025 Ar + 0.49975 CO2), Mgrav = 41.9779 41.9796 0.0017 0.0034 −44.92 (0.24907 Ar + 0.75093 CO2), Mgrav = 42.9981 42.9962 −0.0020 0.0036 −105.69 42.9951 −0.0030 0.0042 −97.05 42.9951 −0.0030 0.0046 −89.46 43.0014 0.0032 0.0047 −70.65

g·mol−1 0.20 0.18 g·mol−1 0.19 g·mol−1 0.26 0.11 0.11 0.14

C/ (cm3/mol)2

U(C)/ (cm3/mol)2

B12/ (cm3/mol)

U(B12)/ (cm3/mol)

2741 2419

49 38

−51.66 −41.64

0.40 0.35

2159

40

−33.43

0.37

4058 3817 3611 3048

86 20 20 28

−52.52 −47.67 −43.65 −33.30

0.70 0.31 0.34 0.39

M is the molar mass; plim is the pressure limit above which experimental data were not used for fitting the virial equation; RMS is the root-meansquare of the relative deviations in pressure of the experimental data from values calculated with the fitted virial (eq 2); MVE is the molar mass determined by the virial analysis (eq 2); ΔM is the absolute difference of the fitted molar mass MVE from the gravimetrically determined one Mgrav; x(CO2) is the mole fraction of carbon dioxide determined by the fitted mole mass MVE; B and U(B) are the second virial coefficient and its expanded uncertainty (k = 2), respectively; C and U(C) are the third virial coefficient and its expanded uncertainty (k = 2), respectively; B12 and U(B12) are the second interaction virial coefficient calculated by eq 3 and its expanded uncertainty (k = 2), respectively. a

masses of carbon dioxide and argon (MCO2 = 44.0098 g·mol−1 and MAr = 39.9480 g·mol−1) was small, (2) not enough data could be fitted, and (3) adsorption and desorption effects probably distorted the composition. Then, B(T) and C(T) in eq 2 were fitted with the molar mass Mgrav being constant. Thereby, the root-mean-squares (RMSgrav) of the relative deviations in pressure became slightly larger compared to those with M as a fitted parameter, nevertheless, the values were still smaller than 0.0047% for all isotherms (see Table 3). The correlated results for B(T) and C(T) are listed in Table 5 together with the second interaction viral coefficient B12(T), which is defined by B12 (T ) =

B(T ) − x12B11(T )2 − x 2 2B22 (T )2 2x1x 2

(3)

Figure 4. Second interaction virial coefficients B12(T) for the (argon + carbon dioxide) system as a function of temperature. △, this work, carbon dioxide mole fractions of 0.50000 and 0.49975; □, this work, carbon dioxide mole fraction of 0.75093; ◊, our previous work7 in the same project with a carbon dioxide mole fraction of 0.94951; , quadratic equation fitted to all the data of this work and some data from our previous work7 (B12(T)/(cm3·mol−1) = −0.0012294·(T/K)2 + 1.1035·(T/K) − 261.83). Error bars for the combined expanded uncertainty (k = 2) are illustrated for the measurements on the mixture with a carbon dioxide mole fraction of 0.94951; for the other data, the error bars are within the vertical size of the plot symbols.

where xi is the mole fraction of component i, Bii(T) is the second virial coefficient of the pure gas i, and the subscripts 1 and 2 represent carbon dioxide and argon, respectively. Bii(T) was computed with the respective reference equations of state (argon27 and carbon dioxide28). The values for B12(T) from this work and our previous work7 are plotted as a function of temperature in Figure 4. A clear smooth trend of B12(T) changing with temperature can be observed, and the correlated function is (B12(T)/(cm3·mol−1) = −0.0012294·(T/K)2 + 1.1035·(T/K) − 261.83). The uncertainties of the virial coefficients were also calculated, as listed in Table 5. The main contributions to the uncertainty of B(T) and C(T) were the statistical uncertainties in fitting the virial equation, the uncertainty of measurements, and the uncertainty of Mgrav. The uncertainty of B12(T) was determined by applying the error propagation principle to eq 3. As can be seen in Table 5, both B(T) and B12(T) increase with increasing temperature while C(T) decreases with increasing temperature, which is a plausible result. Furthermore, values for B12(T) were calculated with the EOS-CG as well as with the GERG-2008 EOS and are illustrated as a function of composition in Figure 5. Values for B12(T) obtained from the experiment are also shown in Figure 5. Compared to the GERG-2008 EOS, the EOS-CG yields a much better agreement with the B12(T) values from the experiment. As expected from theory,29 values for B12(T) obtained from the experimental data are independent of

composition; whereas, those calculated with the equations of state reveal a composition dependence, which implies a fundamental weakness in the EOS. A similar weakness of this general type of equation of state was shown by Richter and McLinden.23 Further investigations on the reason for the dependence of B12 obtained from both equations of state on composition would be of great interest. The relative deviations of the experimental densities from densities calculated with the virial equation using the gravimetrically determined mole mass Mgrav and the fitted virial coefficients B(T) and C(T) are shown in Figure 6. According to this figure, the virial equation represents the experimental densities up to pressures p = plim (see Table 5) within 0.012%, which is considerably less than the standard uncertainty of the measured densities, being about 0.017%. 367



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1.87% at T = 283.15 K, p = 7.26379 MPa for the mixture with a carbon dioxide mole fraction of 0.75093. The third-order virial equation was fitted to the present isothermal data for the determination of the second and third virial coefficients. The second interaction virial coefficients were calculated as well, and a clear smooth trend with temperature (independent of composition) was observed. In summary, the present experimental data and virial coefficients contribute to an accurate basis to effectively improve the performance of the multiparameter equations of state for mixtures.

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Figure 5. Second interaction virial coefficients B12(T) for the (argon + carbon dioxide) system. ○, T = 273.15 K; □, T = 283.15 K; ◊, T = 293.15 K; △, T = 298.15 K; +, T = 308.15 K; and ▽, T = 323.15 K. The solid curves and dashed curves indicate the B12(T) predicted by the EOS-CG3,4 and the GERG-2008 EOS,1 respectively, for the corresponding temperatures. Data for the mixture with with a carbon dioxide mole fraction of 0.94951 were taken from our previous work;7 error bars were only plotted for selected points of this mixture. For the data from this work, error bars correspond approximately to the vertical size of the plot symbols.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +49-234−32-26395. Present Address †

(X.Y.) State Key Laboratory of Power Systems, Department of Thermal Engineering, Tsinghua University, Beijing, 100084, P.R. China. Funding

We thank Tsinghua University for supporting the stay of X.Y. at Ruhr-University Bochum under Grant 2014037. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7-ENERGY-20121-1-2STAGE) under Grant Agreement 308809 (The IMPACTS project). The authors acknowledge the project partners and the following funding partners for their contributions: Statoil Petroleum AS, Lundin Norway AS, Gas Natural Fenosa, MAN Diesel & Turbo SE, and Vattenfall AB.

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Figure 6. Relative deviations of experimental densities ρexp for the (argon + carbon dioxide) mixtures from densities ρvirial calculated with the virial equation (with the molar mass M = Mgrav) as given in eq 2 (zero line). ○, T = 273.15 K (xCO2 = 0.50000); △, T = 298.15 K (xCO2 = 0.50000); ▽, T = 323.15 K (xCO2 = 0.49975); □, T = 273.15 K (xCO2 = 0. 75093); ◊, T = 283.15 K (xCO2 = 0. 75093); ×, T = 293.15 K (xCO2 = 0. 75093); +, T = 323.15 K (xCO2 = 0. 75093). The two values, which are above plim (see Table 5) were not plotted because these data were not used for fitting the virial equation, and they exceed the figure’s boundaries. For the experimental data, the relative combined expanded uncertainty (k = 2) in density was estimated to be less or equal to 0.033% (see Table 4).

REFERENCES

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4. CONCLUSIONS Accurate experimental (p, ρ, T, x) data for the binary (argon + carbon dioxide) system with carbon dioxide mole fractions of 0.50000, 0.49975, and 0.75093 at temperatures between T = (273.15 and 323.15) K and with pressures up to 9.1 MPa or the dew-point pressure, whichever was lower, were presented. The mixtures were prepared gravimetrically, and the measurements were carried out with a two-sinker magnetic suspension densimeter. Flushing the measuring cell (at selected state points) before starting measurements effectively reduced the compositional uncertainty due to sorption effects. The relative combined expanded uncertainty (k = 2) in density was estimated to be less or equal to 0.033%. Relative deviations of the new experimental data from values calculated with the EOS-CG were less than 0.30%; deviations from the GERG2008 EOS were significantly larger, reaching a maximum of 368



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