Vapor-Phase (p, Ï, T, x) Behavior and Virial Coefficients for the

Article pubs.acs.org/jced

Vapor-Phase (p, ρ, T, x) Behavior and Virial Coefficients for the (Methane + Propane) System Markus Richter‡ and Mark O. McLinden* Applied Chemicals and Materials Division, National Institute of Standards and Technology, 325 Broadway, Mailstop 647.07, Boulder, Colorado 80305, United States S Supporting Information *

ABSTRACT: The (p, ρ, T, x) behavior of three (methane + propane) mixtures was measured with a two-sinker magnetic suspension densimeter over the temperature range of (248.15 to 373.15) K with pressures up to the dew-point pressure or 6 MPa, whichever was lower. The compositions of the gravimetrically prepared mixtures were (0.74977, 0.50688, and 0.26579) mole fraction methane. A detailed uncertainty analysis is presented. The relative combined expanded uncertainty (k = 2) in density considering all effects, including the uncertainty in composition, was 0.05 % or lower for most points, except it was larger at densities less than 5 kg·m−3. Comparisons to the GERG-2008 equation of state for natural-gas mixtures showed significant deviations in density (as large as −1.3 %) that increased with decreasing temperature, with increasing pressure, and with increasing propane fraction in the mixture. The experimental values were also used to calculate interaction virial coefficients B12(T) for this system. The B12(T) agreed well with literature values. They were constant (within experimental uncertainty) with composition, as expected from theory. In contrast, the B12(T) calculated with the GERG-2008 equation of state varied with composition. Wagner and Kleinrahm,2 and our instrument is described in detail by McLinden and Lösch-Will.3 Briefly, two sinkers of nearly the same mass and same surface area, but very different volumes, were each weighed with a high-precision balance while they were immersed in a fluid of unknown density. The fluid density ρ is given by

1. INTRODUCTION The thermodynamic properties of multicomponent mixtures, such as natural gases, are most conveniently calculated with a wide-ranging equation of state, such as the GERG-2008 equation of state of Kunz and Wagner.1 This equation is based on high-accuracy equations of state for the mixture components together with departure functions for binary pairs of the components. These departure functions are fitted to experimental data. The binary pair (methane + propane) is among the most important binary pairs for natural gas systems, and especially so for certain “shale-gas” deposits that are exploited explicitly for their high fraction of heavier hydrocarbons. Yet, the vapor-phase (p, ρ, T, x) data for the (methane + propane) system available in the literature are surprisingly lacking (in terms of quality, if not quantity) at high propane fractions. Against this background, the present project was undertaken to fill this data gap and to examine the performance of the GERG-2008 equation at high propane fractions. The (methane + propane) system is also well-suited for exploring aspects of the virial model (as presented here) and for studying sorption effects as well as condensation phenomena near dewpoint conditions, which will be the topic of future work.

ρ=

(1)

where m and V are the sinker mass and volume, W is the balance reading, and the subscripts refer to the two sinkers. Each sinker had a mass of 60 g; one was made of tantalum (V ≈ 3.6 cm3) and the other of titanium (V ≈ 13.3 cm3). A magnetic suspension coupling transmitted the gravity and buoyancy forces on the sinkers to the balance, thus isolating the fluid sample from the balance. In comparison with other buoyancy techniques, the main advantage of the two-sinker method, as developed by Kleinrahm and Wagner,4,5 is that systematic errors in the weighing and from other sources approximately cancel. Moreover, the two-sinker technique leads to lower uncertainties for measurements at low densities compared to a single-sinker instrument. In addition to the sinkers, two calibration masses (designated mcal and mtare) were also weighed. This provided a calibration of the balance and also the information needed to correct for

2. EXPERIMENTAL SECTION 2.1. Apparatus Description. The present measurements utilized a two-sinker densimeter with a magnetic suspension coupling. This type of instrument applies the Archimedes (buoyancy) principle to provide an absolute determination of the density. This general type of instrument is described by This article not subject to U.S. Copyright. Published 2014 by the American Chemical Society

(m1 − m2) − (W1 − W2) (V1 − V2)

Received: August 25, 2014 Accepted: October 29, 2014 Published: November 12, 2014 4151

dx.doi.org/10.1021/je500792x | J. Chem. Eng. Data 2014, 59, 4151−4164

Journal of Chemical & Engineering Data

Article

Table 1. Sample Information

a

chemical name

source

initial purity/mol fraction

purification method

final purity/mol fraction

analysis method

methane propane

Matheson Scott

0.99999 0.99999

none none

0.99999 0.99999

GC/MSa GC/MS

Gas chromatography/mass spectrometry.

magnetic effects as described by McLinden et al.6 The weighings yield a set of four equations that are solved to yield a balance calibration factor α and a parameter β related to the balance tare (i.e., the magnets and other elements of the system that are always weighed). This analysis yields the fluid density in terms of directly measured quantities:

Aluminum gas cylinders of approximately 10 L internal volume were cleaned with acetone and ethanol. New brass cylinder valves with PTFE gaskets and seats were installed in the cylinders. The cylinders were filled with research-grade nitrogen to a pressure of approximately 2 MPa and then evacuated to a final pressure < 10−4 Pa; this purge−evacuate cycle was repeated three times. The sample cylinder was then connected to the propane or methane supply cylinder via a manifold, and the sample lines were purged and evacuated three times before introducing sample into the receiving cylinder. The sample cylinder sat on a platform balance (with a resolution of 0.1 g), and the sample was slowly introduced until the target mass was reached. The cylinder was then taken to a high-precision mass comparator for the determination of the actual sample mass loaded. The sample mass was determined by a double substitution weighing,7 with a nearly identical “tare” or reference cylinder serving as the main substitution mass. The weighing design consisted of four separate weighings: (1) the tare cylinder and standard masses (as needed to bring the mass comparator into its weighing range), (2) the sample cylinder and standard masses, (3) the sample cylinder and masses used in weighing 2 plus a 20 g sensitivity mass (which served to check the linearity of the mass comparator), and (4) the tare cylinder and masses used in weighing 1 plus the sensitivity mass. The key advantage in this approach was that the air buoyancy effect was reduced to the (small) difference in volumes of the tare and sample cylinders and the relatively small effect of the standard masses. The mass comparator used in these measurements had a total capacity of 10 060 g, an electronic weighing range of 60 g, a resolution of 0.1 mg, and a linearity and repeatability of 0.3 mg according to the manufacturer’s specification. The three sample cylinders were loaded to pressures corresponding to the dew point pressure at T = 298.15 K, which were p = 6 MPa for the (0.74977 methane + 0.25023 propane) mixture, p = 1.9 MPa for the (0.50688 methane + 0.49312 propane) mixture, and p = 1.2 MPa for the (0.26579 methane + 0.73421 propane) mixture. The sample cylinders were continuously heated to T > 313 K for the duration of the testing to prevent condensation. Because of sorption effects, the composition of the sample in the measuring cell could be different from that calculated from the sample masses loaded into the sample cylinder; this is discussed in sections 3 and 4.2. 2.3. Experimental Procedures. Measurements were carried out along isotherms at T = (248.15, 273.15, 293.15, 298.15, 323.15, and 373.15) K; except that measurements at T = 293.15 K and T = 298.15 K were not undertaken for all three mixtures. For isotherms below the maxcondentherm (see Figure 1) for a given mixture, the pressures extended to the dew-point pressure. For isotherms above the maxcondentherm, the pressures extended to the filling pressure of the sample cylinders. We were particularly mindful that sorption effects could change the composition of the sample inside the measuring cell, thus distorting the measured densities, as discussed by Richter and Kleinrahm.8 Tests were carried out to explore this source of

⎡ (W1 − W2)m1 ⎤ ρfluid = ⎢(m1 − m2) − ⎥ W1 − αβ ⎦ ⎣ ⎡ (W1 − W2)V1 ⎤ ⎥ − ρ0 ⎢(V1 − V2) − W1 − αβ ⎦ ⎣

(2)

where ρ0 is the indicated density when the sinkers are weighed in vacuum. In other words, ρ0 is an “apparatus zero.” The average value of ρ0 for the present measurements was 0.0014 kg·m−3 with a standard deviation of 0.0008 kg·m−3. The density given by eq 2 compensates for the magnetic effects of both the apparatus and the fluid being measured. The efficiency of the magnetic suspension coupling is characterized by a “coupling factor” ϕ; values of ϕ different than 1 indicate the magnitude of the force transmission error.6 For the present measurements ϕ varied from 1.000 002 in vacuum to 0.999 994 for the gas at the highest densities. The temperature was measured with a 25-ohm standard platinum resistance thermometer (SPRT) and a resistance bridge referenced to a thermostated standard resistor. Pressures were measured with one of two vibrating-quartz-crystal type pressure transducers having full-scale pressure ranges of (2.8 or 6.9) MPa. The transducers and pressure manifold were thermostated at T = 313.15 K to minimize the effects of variations in laboratory temperature. Our two-sinker densimeter was not optimized for mixture measurements. It had only a single sample inlet, for example, and this precluded flushing the cell. We modified the filling lines by adding additional heat tracing in an attempt to minimize condensation and sorption. We continuously heated the sample cylinder so that the sample was well in the homogeneous gas region. The heating also induced convection currents (vortices) inside the cylinder for mixing. 2.2. Preparation of the Sample Mixtures. The gas mixtures were prepared gravimetrically. The components are described in Table 1. We used the materials as received, but we did confirm their purity with our own analysis. The compositions of the three prepared mixtures are given in Table 2. Table 2. Gravimetric Compositions (Mole Fraction) and Average Molar Masses of the Studied Mixtures mixture component

(75/25)

(50/50)

(27/73)

methane propane M/g·mol−1

0.74977 0.25023 23.0623

0.50688 0.49312 29.8761

0.26579 0.73421 36.6393 4152



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of the sinker masses (m1, m2), and calibration masses (mcal, mtare), and the apparatus zero (ρ0). The uncertainties in the sinker volumes V1 and V2 dominate the overall uncertainty in density. The volumes of the sinkers were determined at T = 293.15 K and atmospheric pressure by the hydrostatic comparator method described by McLinden and Splett.9 The volumes at other temperatures and pressures were calculated using the linear thermal expansion coefficient of tantalum and titanium as well as literature values for the bulk modulus. These volumes were further adjusted by the method of Moldover and McLinden.10 The combined standard uncertainty (k = 1) in the sinker volume difference (V1 − V2) was estimated to be 16·10−6·(V1 − V2) at T = 293.15 K and atmospheric pressure increasing to 23·10−6·(V1 − V2) at T = 373 K and p = 6 MPa, as detailed in ref 9. At low densities, errors in the weighings have a significant effect on the uncertainty in density. The standard deviations observed in replicate weighings ranged from 0.2 μg to 5.2 μg for the different objects, and these contribute an uncertainty in density of 0.0005 kg·m−3. The uncertainty in the masses of the sinkers and calibration masses ranged from 21 μg to 50 μg. Because of the differential nature of the two-sinker method, these contribute only 2·10−6 to the relative uncertainty in density. The magnetic suspension coupling (MSC) transmited the buoyancy force on the sinkers to the balance; therefore, any magnetic materials near the MSC will affect the density measurement. This effect is known as the “force transmission error”.6 The analysis of McLinden et al.6 demonstrated that such errors can be accounted for, reducing the standard uncertainty from this effect to 2·10−6·ρ. The standard uncertainty in the density was

Figure 1. p, T-diagram with phase boundaries of the three (methane + propane) mixtures investigated in the present work; the phase boundaries were calculated with the GERG-2008 equation of state of Kunz and Wagner:1 , (0.74977 methane + 0.25023 propane); - - -, (0.50688 methane + 0.49312 propane); ---, (0.26579 methane + 0.73421 propane).

error. Therefore, we replicated several isotherms with conditions that should show sorption effects: We measured isotherms with both increasing and decreasing pressure; we started isotherms at different pressures; we started isotherms following a brief evacuation or after many hours of evacuation. One typical example for the different measurement conditions we tested is discussed in section 4.1.1. To achieve a quantitative indication of the change in mixture composition due to sorption effects, we calculated the molecular weight of the mixture from a virial analysis of the measured densities (as discussed in section 4.2) and compared this value to the one derived from the gravimetric composition. The standard procedure between the different fillings was to evacuate the measuring cell for (10 to 30) min. This was followed by a purging cycle consisting of filling the measuring cell up to 50% of the dew-point pressure (or at least p = 1 MPa at temperatures above the maxcondentherm); this sample remained inside the cell for approximately 2 min; the cell was then evacuated for 2 min; this routine was repeated three times before filling in the sample for the next isotherm. This procedure was a compromise to minimally disturb the sorption equilibrium, since flushing the measuring cell at constant conditions (isobaric and isothermal) was not possible because the sample inlet and outlet were the same and, moreover, the amount of sample was limited. Evacuation of the measuring cell was important because many of the isotherms ended in the two-phase region. Hence, it was necessary to clear any sample that had condensed inside the measuring cell during the course of the previous isotherm. However, an extended evacuation time was not considered helpful as the material would have probably desorbed completely such that the composition of the new sample would be altered in reestablishing sorption equilibrium. When changing from one mixture to another, the measuring cell was evacuated for 18 h or more. This opportunity was used to check the “apparatus zero” in eq 2.

u(ρ) = [{16}2 + {0.20|(T /K − 293)|}2 kg·m−3 + {0.63p/MPa}2 ]0.5

10−6ρ + 0.0007 kg·m−3

(3)

where the term in brackets is from the uncertainties in the volumes of the two sinkers, and the final, constant term includes all other uncertainties. For the present work, the uncertainty in the density measurement itself was a maximum of 0.0020 kg·m−3 at the highest density measured; the largest relative uncertainty was 0.042 % at the lowest measured density. The SPRT used to measure the temperature of the mixture was calibrated in our laboratory on ITS-90 from 83 K to 505 K by use of fixed-point cells (argon triple point, mercury triple point, water triple point, indium freezing point, and tin freezing point). This was done as a system calibration using the same lead wires, standard resistor, and resistance bridge that were used during the measurements. Such calibrations have been carried out every three to four years. The SPRT was regularly checked in a water triple-point cell between full calibrations; it has proven to be very stable: it has drifted only 1.4 mK over eight years of use. The standard uncertainty of the temperature, including the uncertainty in the fixed point cells, drift in the SPRT and in the standard resistor, and any temperature gradients, is 3 mK. This temperature uncertainty impacts gas densities at the level of approximately 0.0010 %. The uncertainty of the pressure arises from three sources: the calibration of the transducers, the repeatability and temporal drift of the transducers, and the uncertainty in the hydrostatic

3. UNCERTAINTY ANALYSIS McLinden and Lösch-Will3 provide an analysis of uncertainties for gas-phase measurements, and McLinden and Splett9 provide a further detailed uncertainty analysis for this instrument. Those results are summarized here. Uncertainty in the density calculated with eq 2 arises from uncertainties in the sinker volumes (V1, V2), weighings of the sinkers and calibration masses (W1, W2, Wcal, Wtare), knowledge 4153



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ratio of the component masses loaded into the sample cylinder. This term cannot be estimated a priori, but a virial analysis of the data provides an independent estimate of M (see section 4.2). We take the root-mean-square (RMS) average of the difference between the M from the gravimetric preparation Mgrav and that determined from the virial analysis Mvirial to be the standard uncertainty in M arising from sorption effects:

head correction. The pressure transducers were calibrated with a gas-operated piston gauge. This calibration was done in situ by connecting the piston gauge to the sample port of the filling and pressure manifold. The pressure uncertainty arising from the hydrostatic head correction is estimated to be less than 2·10−6·p. We estimated the standard uncertainty in pressure to be (20·10−6·p + 0.025 kPa) for the low-range transducer and (20·10−6·p + 0.06 kPa) for the 6.9 MPa transducer. The complete data tables in the Supporting Information of this paper indicate which transducer was used for a given measurement. The maximum impact of the pressure uncertainty on the gas densities is 0.0023 % at the lowest pressures. To the above uncertainty estimates we added the standard deviations actually observed in the multiple temperature, pressure, and balance readings made over the 12 min necessary to complete a single density determination. For the majority of the tests, these contributions were significantly less than the uncertainties stated above (as detailed in the Supporting Information). For two isotherms, a leaking valve caused a steady downward drift in pressure and density and resulted in much larger standard deviations in these quantities. While this propagated into a larger combined uncertainty, the weighing design, in which each of the sinkers and calibration masses was weighed twice in a time-symmetric fashion for each density determination, compensated for such a linear drift; this point is discussed further by McLinden.11 For a gas mixture a considerable fraction of the overall density uncertainty is due to uncertainty in the composition. To first order, the molar density of a vapor mixture is constant for a given T, p, and x, or equivalently, the mass density is proportional to the average molar mass, so that the uncertainty in density due to uncertainties in the composition is proportional to the uncertainty in M, the molar mass of the mixture. The uncertainty in M due to uncertainties in the gravimetric preparation is given by

u(M )sorption

(5)

where the summation is over the N measured isotherms; this uncertainty value is 0.0052 g·mol−1 (or 0.0114 % to 0.0226 % of M), which is 2.8 to 32 times larger than the effect from the gravimetric preparation of the mixture. The density uncertainty arising from the composition uncertainty (both mixture preparation and sorption effects) is thus u(ρ(x)) = 1 2 2 0.5 ρ · [u(Mgrav )2 + u(M )atomic weights + u(M )sorption ] M (6)

For purposes of comparing (p, ρ, T, x) measurements to a model, it is customary to assume that the temperature, pressure, and composition are known exactly, and to lump all uncertainties into a single value for the density, a so-called state-point uncertainty. This combined expanded uncertainty (k = 2), including all effects, is given by ⎧ ⎪ ⎪ ⎪ UC(ρ) = 2·⎨ ⎪ ⎪+ ⎪ ⎩

0.5 ⎧ ⎡ ⎤2 ⎫ m n n i ⎪ ⎥⎪ ⎪ n ⎢ ∑i = 1 Mi − (∑i = 1 mi)/Mj ⎪ u(Mgrav ) = ⎨∑ ⎢ u ( m ) j ⎥ ⎬ 2 m n ⎥⎪ ⎪ j=1 ⎢ ∑i = 1 Mi ⎪ ⎢⎣ ⎥⎦ ⎪ i ⎩ ⎭

(

0.5 2⎤ ⎡ ∑ N (M grav, i − M virial, i) ⎥ i=1 ⎢ = ⎢⎣ ⎥⎦ N

)

⎫0.5 ⎪ ⎪ ⎪ ⎬ ⎡⎛ ⎞ ⎤2 ⎡ 2⎪ ⎢⎜ ∂ρ ⎟ u(T )⎥ + ⎢ ρ u(M )⎤⎥ ⎪ ⎦ ⎪ ⎣M ⎢⎣⎝ ∂T ⎠ p ⎦⎥ ⎭ ⎡⎛ ⎞ ⎤2 ρ ∂ 2 [u(ρ)] + ⎢⎜ ⎟ u(p)⎥ ⎢⎣⎝ ∂p ⎠ ⎥⎦ T

(7)

where the u are the individual standard uncertainties, and the derivatives are calculated from the GERG equation. This value for each measured point is tabulated in the Supporting Information. The relative combined expanded uncertainty in density ranged from 0.032 % at the highest densities measured to 0.17 % at ρ = 1.66 kg·m−3, except for a limited number of points for which the pressure was drifting due to a leaking valve.

(4)

where the Mi are the molar masses of the components, and the mi are the masses of each of the components loaded into the sample cylinder. With an uncertainty of 0.010 g for each of the component masses, u(Mgrav) ranged from (0.00016 to 0.00185) g·mol−1 for the mixtures studied here. A derivation of eq 4 is presented in the Supporting Information. Given the very high purity of our starting materials, the uncertainty in Mi due to impurities in the components was negligible. The total range of the atomic weights of carbon and hydrogen given by Wieser et al.12 (12.0096 to 12.0116 for carbon and 1.00784 to 1.00811 for hydrogen) yields a total possible range in M for our mixtures of (0.0044 to 0.0069) g·mol−1. However, Wieser et al.12 also state that the variation in M for fossil fuels (i.e., “crude oil” and “methane from tank gases”) is much smaller, resulting in a standard uncertainty from this source of approximately 0.0003 g·mol−1. A further source of uncertainty in the composition arose from the sorption of sample onto the inner walls of the sample cylinder, filling lines, measuring cell, etc. In other words, the composition in the measuring cell was not necessarily a simple

4. RESULTS 4.1. Measured (p, ρ, T, x) Data Along Isotherms. The three gas mixtures were measured over the temperature range of (248.15 to 373.15) K, with maximum pressures of near the dew-point pressure at the measured temperature or the maximum pressure in the sample cylinder, whichever was lower. Figure 1 depicts the phase diagram for each of the mixtures, and Figure 2 shows the location of the measured points. Tables 3 to 5 present the measured values along isotherms. Densities on a total of 27 isotherms, including replicates, were measured. Three or more replicate points were measured at each (T, p) state point, and average values are given in the tables. 4.1.1. Comparison to the GERG Equation of State. Tables 3 to 5 also list the relative deviations of our measurements to 4154



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0.25023 propane) mixture in terms of deviations from the GERG equation. The isotherms proceeded with different sequences of increasing and decreasing pressures, but the maximum difference between replicate points was 0.03 %. These results indicate that sorption phenomena were likely present, but at a low level. 4.1.2. Comparison to Literature Data. Kunz et al.13 provide a comprehensive review of the literature data for the (methane + propane) system existing as of 2004. They list a total of 2920 (p, ρ, T, x) points from 11 sources. The bulk of these data are at high methane fractions or are in the liquid phase, or both. There were 426 density data overlapping the present measurements; that is, p < 6 MPa and xmethane ≤ 0.8. (Additional virial data are discussed in section 4.2.2.) Reamer et al.14 measured two to four vapor-phase points along eight isotherms (T = 277 K to 511 K) for each of eight compositions for a total of 227 points. Sage et al.15 measured three to five vapor-phase points along five isotherms (T = 293 K to 363 K) for each of seven compositions for a total of 185 points. May et al.16 measured 10 points over the temperature range T = (285 to 293) K for xmethane = 0.7931. Huang et al.17 report a single vapor-phase point. These authors also reported measurements at higher methane fractions. A database search did not reveal any additional sources. Deviations of these data from the GERG equation are shown in Figure 7. One immediately notices the large scatter in the dataup to ± 5 % for xmethane ≤ 0.3. The GERG equation fitted the data to the extent that was possible, but it is understandable that systematic deviations of the magnitude revealed in this work would not be detected given the uncertainties in the literature data available at the time. 4.2. Virial Coefficients and Determination of Molar Mass. The measured density values along each of the isotherms were fitted to a virial expansion:

Figure 2. p,T-diagrams of the three investigated (methane + propane) mixtures: +, (p, ρ, T, x) state points measured in the present work; , phase boundaries calculated with the GERG-2008 equation of state of Kunz and Wagner.1 (a) (0.74977 methane + 0.25023 propane); (b) (0.50688 methane + 0.49312 propane); (c) (0.26579 methane + 0.73421 propane).

p=

densities calculated with the GERG equation for natural gas systems: Δ = 100

(9)

where p is the pressure, T is the temperature, and ρ is the mass density. The B(T, x) and C(T, x) are the second and third virial coefficients and are fitted parameters. For isotherms with a maximum density of less than 0.25 mol·L−1, the data were not sufficient to obtain reliable values of C(T, x), and eq 9 was truncated after the second virial coefficient. Attempts to include the fourth virial coefficient D(T, x) did not substantially improve the goodness of the fit and often resulted in values of D(T, x) that were not statistically valid. All of the data were given equal weight, and the fitted values are given in Table 6. The uncertainties listed in Table 6 comprise contributions from (1) the variance in the data, which was the standard deviation in the fitted value returned by the regression software,18 and (2) the effects of possible systematic errors in the experimental quantities, which were determined by varying all the input data by their corresponding uncertainties, rerunning the regression, and adding (in quadrature) the resulting difference in the fitted quantity. The molar mass M appears in eq 9 because densities were measured on a mass basis and, by convention, virial coefficients are expressed on a molar basis. The molar mass is typically computed from the composition, but by letting M be a fitted parameter its value can be determined independently. In this way, density measurements along an isotherm serve as a consistency check on the gravimetric mixture composition.

(ρexp − ρGERG ) ρGERG

ρRT ⎡ ρ ρ2 ⎤ ⎢1 + B(T , x) + C(T , x) 2 ⎥ M ⎣ M M ⎦

(8)

Figures 3 to 5 show the deviations from the GERG equation for the isothermal data. There are clear and systematic deviations, up to −1.3 %. These increase as the fraction of propane is increased going from the (0.74977 methane + 0.25023 propane) mixture (Figure 3) to the (0.50688 methane + 0.49312 propane) mixture (Figure 4) to the (0.26579 methane + 0.73421 propane) mixture (Figure 5). These deviations are greater than the uncertainties in the experimental data for most of the points. The uncertainty reported for the GERG equation is 0.1 % for the conditions considered here, and for the (0.74977 methane + 0.25023 propane) mixture the majority of the points are within this uncertainty. For the other two compositions with higher propane fractions, the differences exceed 0.1 % for most of the points. For each of the mixtures, the deviations follow the same trend, with deviations at the lower temperatures increasing most rapidly with pressure. (The lower temperature isotherms terminate at lower pressures and, thus, often have smaller maximum deviations than the higher temperature isotherms.) Figure 6 presents the results of four replicate measurements of the T = 298.15 K isotherm for the (0.74977 methane + 4155



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Table 3. Experimental (p, ρ, T, x) Results for the (0.74977 Methane + 0.25023 Propane) Mixture and Relative Deviations of the Experimental Densities ρ from Densities Calculated with the GERG-2008 Equation of State ρGERGa T/K

p/MPa

273.150 273.150 273.150 273.150 273.148 273.150 273.150

1.99890 1.50089 1.00041 0.49246 1.01497 1.51094 2.00195

373.154 373.158 373.157 373.156 373.159 373.156 373.157 373.157

5.85963 4.88106 3.94503 2.98742 1.98580 1.46824 0.99105 0.48718

298.150 298.147 298.148 298.146 298.148

0.50020 0.99901 1.99952 3.99054 5.99963

298.148 298.151 298.150 298.151

4.11949 2.00138 1.00061 5.99747

298.146 298.148 298.148 298.148 298.147 298.147 298.146 298.147 298.148 298.146 298.147

4.99386 4.00313 3.00632 4.42424 3.04389 2.07540 1.68660 1.37243 1.09398 0.78506 0.49382

ρ/kg·m−3 Test: 1401no 22.8321 16.5996 10.7338 5.1329 10.9019 16.7231 22.8750 Test: 1401r 48.2829 39.5669 31.4611 23.4236 15.2929 11.1998 7.4967 3.6521 Test: 1401v 4.7420 9.6790 20.2389 44.5527 75.0173 Test: 1401wx 46.3151 20.2558 9.6924 74.9816 Test: 1401zzb 58.9113 44.7224 31.9148 50.5647 32.3714 21.0754 16.8301 13.5070 10.6384 7.5358 4.6835

100((ρ − ρGERG)/ρGERG)

T/K

−0.1390 −0.0892 −0.0498 −0.0355 −0.0280 −0.0792 −0.1241 −0.0088 0.0060 −0.0024 −0.0044 −0.0090 −0.0358 −0.0184 −0.0198 −0.1155 −0.0174 −0.0741 −0.1600 −0.0228 −0.1473 −0.0905 −0.0452 −0.0198 −0.1318 −0.1607 −0.1330 −0.1516 −0.1421 −0.0999 −0.0750 −0.0623 −0.0531 −0.0451 −0.0468

323.158 323.160 323.160 323.160 323.163 323.163 323.162 323.162 323.161

5.92879 4.47538 3.01765 1.98881 1.72099 1.49228 1.22594 0.98192 0.74799

373.156 373.155 373.156 373.154 373.151 373.151 373.150 373.152 373.155 373.152 373.151 373.154

6.04127 4.50302 2.99664 1.99093 3.99843 1.99827 1.49113 1.22244 0.97387 0.73221 0.49425 0.29398

273.154 273.150 273.153 273.151 273.153 273.154

0.50089 0.89981 1.29921 1.70039 1.99971 2.30364

248.169 248.164 248.161 248.159 248.159 248.159 248.160

0.27607 0.39004 0.50033 0.58070 0.68507 0.78624 0.83560

ρ/kg·m−3 Test: 1401zc 62.0416 44.4692 28.5100 18.1586 15.5780 13.4094 10.9233 8.6824 6.5657 Test: 1401zdf 49.9291 36.2589 23.4974 15.3338 31.9150 15.3921 11.3819 9.2859 7.3646 5.5128 3.7047 2.1956 Test: 1401zgh 5.2229 9.5966 14.1894 19.0432 22.8429 26.8673 Test: 1402a 3.1456 4.4826 5.7986 6.7717 8.0551 9.3205 9.9461

100((ρ − ρGERG)/ρGERG) −0.0784 −0.1039 −0.0790 −0.0541 −0.0441 −0.0376 −0.0344 −0.0298 −0.0320 −0.0074 −0.0126 −0.0144 −0.0111 −0.0105 −0.0135 −0.0123 −0.0142 −0.0171 −0.0271 −0.0451 −0.0525 −0.0388 −0.0579 −0.0826 −0.1198 −0.1355 −0.1710 −0.0690 −0.0662 −0.0776 −0.0931 −0.1019 −0.1138 −0.1156

a

Points are presented in the sequence in which they were measured; the different sample fillings are indicated by their test numbers.

adsorbed onto the walls of the sample cylinder, the charging lines, the measuring cell, etc. The two isotherms for which the fitted M was higher than the corresponding gravimetric values were outliers, in the sense that the fitted values of C(T, x) and B12(T) (discussed in section 4.2.1) were significantly lower than the other replicates. 4.2.1. Interaction (Cross) Virial Coefficients. The second virial coefficient for a gas mixture is given by

Moreover, the composition of a binary mixture can be determined from the fitted value of M: x 2 = (M − M1)/(M 2 − M1)

p/MPa

(10)

where the composition x2 is on a molar basis, M1 and M2 are the molar masses of the two components, and x1 = (1 − x2). The molar mass determined by eq 9 for each of the isotherms and their uncertainties are given in Table 6. The deviations of the experimental data from the virial fits of eq 9 are given in Table 6. The relative RMS deviation in pressure ranged from 0.0013 % to 0.0164 % for the fits of the individual isotherms. No obvious systematic deviations were detected, indicating the adequacy of the model. Most of the molar masses determined from eq 9 were slightly below the gravimetric values (with a maximum difference of −0.0271 g·mol−1 or 0.074 %), although in all cases the differences were less than the uncertainty in the fitted M. This would be consistent with a small amount of the propane being

n

B (T , x ) =

n

∑ ∑ xixjBij(T ) i=1 j=1

(11)

where the xi and xj are the compositions (mole fraction) of components i and j, the summations are over the n components in the mixture, and the Bij are the pure-component virials when i = j. B12 represents the interaction of one molecule of species (1) and one molecule of species (2) and is, thus, independent of the composition of the overall mixture (see, for example, the 4156



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Table 4. Experimental (p, ρ, T, x) Results for the (0.50688 Methane +0.49312 Propane) Mixture and Relative Deviations of the Experimental Densities ρ from Densities Calculated with the GERG-2008 Equation of State ρGERG T/K

p/MPa

248.151 248.149 248.150 248.152 248.152 248.151

0.12009 0.18340 0.24440 0.30331 0.36050 0.42487

273.147 273.149 273.150 273.150 273.149 273.149 273.148 273.148 273.148 273.150

0.25840 0.32134 0.44130 0.55633 0.61213 0.71171 0.80319 0.93025 0.97191 1.00954

273.150 273.148 273.147 273.148 273.149 273.150 273.150 273.149 273.150

0.21804 0.34035 0.45352 0.50693 0.60775 0.70131 0.82952 0.90776 1.01545

293.146 293.148 293.148 293.146 293.146 293.148 293.149 293.145

0.39388 0.61382 0.80374 1.00619 1.20944 1.39960 1.60580 1.80719

323.165 323.163 323.164 323.166 323.164 323.163

0.24014 0.43483 0.60943 0.81412 1.02837 1.20620

ρ/kg·m−3 Test: 1401c 1.7641 2.7164 3.6498 4.5661 5.4702 6.5055 Test: 1401d 3.4829 4.3576 6.0553 7.7247 8.5493 10.0475 11.4547 13.4618 14.1337 14.7469 Test: 1401e 2.9261 4.6226 6.2301 7.0019 8.4837 9.8889 11.8643 13.1006 14.8424 Test: 1401f 4.9714 7.8849 10.4890 13.3634 16.3610 19.2777 22.5759 25.9503 Test: 1401g 2.7038 4.9489 7.0050 9.4697 12.1152 14.3650


T/K

−0.0965 −0.1212 −0.1379 −0.1662 −0.2059 −0.2675 −0.0503 −0.0790 −0.1428 −0.1986 −0.2333 −0.2926 −0.3522 −0.4446 −0.4779 −0.5097 −0.0831 −0.1145 −0.1593 −0.1877 −0.2380 −0.2909 −0.3788 −0.4360 −0.5228 −0.1113 −0.1726 −0.2397 −0.3285 −0.4321 −0.5408 −0.6721 −0.8104 −0.0630 −0.0864 −0.1109 −0.1435 −0.1880 −0.2304

classic text of Mason and Spurling19). For a binary mixture, the second interaction virial coefficients (also called cross virial coefficients) are given by B12 (T ) =

B(T , x) − x12B11(T ) − x 22B22 (T ) 2x1x 2

ρ/kg·m−3

p/MPa

323.161 323.164 323.164 323.163

1.40713 1.60344 1.80474 1.91384

373.148 373.149 373.147 373.145 373.148 373.148 373.146 373.148

1.99935 1.74533 1.58145 1.38719 1.17736 0.96914 0.77615 0.58798

373.151 373.147 373.145 373.148 373.147 373.145 373.148 373.149 373.145

0.28624 0.55953 0.73634 0.94267 1.11869 1.29808 1.49697 1.70729 1.90665

273.153 273.150 273.147 273.149 273.147 273.146 273.147 273.149 273.146 273.149 273.151 273.147 273.150 273.151 273.150 273.150 273.149 273.149 273.149

0.26229 0.26926 0.27728 0.30900 0.32308 0.40273 0.40905 0.41533 0.49106 0.49716 0.50323 0.61926 0.62549 0.63166 0.70587 0.78192 0.88898 0.98620 1.04167

Test: 1401g 16.9693 19.5802 22.3318 23.8550 Test: 1401h 20.6289 17.8434 16.0735 14.0021 11.7969 9.6405 7.6695 5.7723 Test: 1401i 2.7813 5.4874 7.2655 9.3682 11.1859 13.0615 15.1688 17.4293 19.6043 Test: 1401j 3.5352 3.6315 3.7427 4.1836 4.3804 5.5040 5.5939 5.6832 6.7718 6.8601 6.9481 8.6551 8.7476 8.8402 9.9584 11.1241 12.8024 14.3651 15.2749

100((ρ − ρGERG)/ρGERG) −0.2846 −0.3465 −0.4109 −0.4518 −0.1685 −0.1415 −0.1263 −0.1105 −0.0914 −0.0755 −0.0641 −0.0621 −0.0613 −0.0674 −0.0723 −0.0777 −0.0890 −0.1031 −0.1214 −0.1459 −0.1702 −0.0880 −0.0976 −0.0970 −0.1097 −0.1149 −0.1403 −0.1427 −0.1469 −0.1757 −0.1809 −0.1832 −0.2424 −0.2486 −0.2460 −0.2919 −0.3425 −0.4166 −0.4953 −0.5407

function of temperature, and a clear, smooth trend with temperature is observed. The B12(T) was computed from both the gravimetric composition and that computed from eqs 9 and 10; the RMS difference was 0.28 cm3·mol−1, indicating that the composition uncertainty did not substantially impact the B12(T). The third virial coefficient for a mixture is given by

(12)

The virial coefficients for the pure components were computed from the equations of state of Setzmann and Wagner20 for methane and Lemmon et al.21 for propane. The B12(T) calculated from the experimental (p, ρ, T, x) data are given in Table 6 and shown in Figure 8. The values are constant (within experimental uncertainty) with composition, as expected from theory.19 Figure 9 shows the B12(T) as a

n

C(T , x) =

n

n

∑ ∑ ∑ xixixkCijk(T ) i=1 j=1 k=1

(13)

For a binary mixture, this becomes 4157



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Table 5. Experimental (p, ρ, T, x) Results for the (0.26579 methane + 0.73421 Propane) Mixture and Relative Deviations of the Experimental Densities ρ from Densities Calculated with the GERG-2008 Equation of State ρGERG T/K

p/MPa

273.150 273.150 273.150 273.149 273.148 273.148 273.149 273.150

0.16749 0.24218 0.34434 0.40645 0.49173 0.56857 0.61559 0.65932

273.149 273.147 273.148 273.148 273.150 273.150 273.151

0.40170 0.43080 0.48610 0.53770 0.56221 0.60884 0.65239

273.151 273.149 273.149 273.149 273.150 273.150 273.149 273.149 273.150 273.150 273.149 273.151

0.63508 0.59852 0.54215 0.49124 0.44543 0.39665 0.34743 0.30007 0.24799 0.19896 0.14935 0.12394

298.145 298.149 298.150 298.149 298.148 298.152 298.148 298.150 298.150

0.16058 0.31148 0.47127 0.60541 0.75838 0.90920 1.05513 1.20117 1.35571

293.144 293.146 293.147 293.145 293.145 293.146 293.147 293.147 293.145 293.145

0.14990 0.25804 0.35508 0.46930 0.56875 0.65565 0.85740 0.95525 1.05630 1.10565

ρ/kg·m−3


Test: 1312q 2.7678 4.0482 5.8500 6.9764 8.5641 10.0392 10.9644 11.8408 Test: 1312r 6.8905 7.4250 8.4581 9.4417 9.9157 10.8307 11.7011 Test: 1312u 11.3537 10.6270 9.5274 8.5547 7.6956 6.7965 5.9050 5.0612 4.1483 3.3031 2.4607 2.0343 Test: 1312v 2.4140 4.7646 7.3487 9.6013 12.2732 15.0282 17.8236 20.7664 24.0659 Test: 1312x 2.2913 3.9961 5.5657 7.4652 9.1680 10.6963 14.4071 16.2977 18.3225 19.3404

T/K

−0.0990 −0.1449 −0.2362 −0.3001 −0.4098 −0.5211 −0.5979 −0.6767 −0.2813 −0.3260 −0.3969 −0.4705 −0.5090 −0.5826 −0.6613 −0.6255 −0.5635 −0.4789 −0.4107 −0.3520 −0.2986 −0.2475 −0.2062 −0.1668 −0.1367 −0.1172 −0.1033 −0.0991 −0.1478 −0.2373 −0.3361 −0.4777 −0.6446 −0.8391 −1.0641 −1.3410 −0.0960 −0.1337 −0.1845 −0.2593 −0.3411 −0.4243 −0.6600 −0.8025 −0.9630 −1.0451

293.146 293.147 293.147

1.15392 1.17497 1.18746

323.165 323.166 323.167 323.166 323.166 323.165 323.167 323.166 323.166 323.167

0.16142 0.27780 0.38081 0.47408 0.55687 0.65358 0.86114 0.95630 1.05764 1.15538

373.164 373.165 373.165 373.165 373.164 373.164 373.164 373.164 373.163 373.163

0.15036 0.28196 0.36195 0.46953 0.56491 0.67604 0.86650 0.96318 1.05718 1.15636

373.162 373.161 373.163 373.162 373.160 373.162 373.164 373.160

1.29310 1.17470 1.03428 0.88343 0.73318 0.59263 0.44247 0.24950

248.155 248.151 248.151 248.152 248.152 248.152 248.151

0.09660 0.14170 0.18456 0.22080 0.25113 0.27199 0.27599

248.151 248.151 248.152 248.151 248.151 248.151

0.25149 0.21778 0.18816 0.15947 0.12919 0.1292

ρ/kg·m−3 Test: 1312x 20.3554 20.8043 21.0718 Test: 1312y 2.2297 3.8766 5.3632 6.7339 7.9706 9.4407 12.6940 14.2336 15.9088 17.5614 Test: 1312z 1.7881 3.3776 4.3551 5.6830 6.8735 8.2772 10.7249 11.9881 13.2303 14.5570 Test: 1312zb 16.4139 14.8058 12.9279 10.9464 9.0070 7.2228 5.3479 2.9840 Test: 1312zc 1.7469 2.5862 3.3988 4.0982 4.6926 5.1060 5.1859 Test: 1401a 4.7012 4.0405 3.4688 2.9223 2.3527 2.353

100((ρ − ρGERG)/ρGERG) −1.1312 −1.1699 −1.1974 −0.0973 −0.1140 −0.1372 −0.1653 −0.1968 −0.2403 −0.3519 −0.4114 −0.4817 −0.5551 −0.0966 −0.0894 −0.0846 −0.0952 −0.1083 −0.1221 −0.1531 −0.1733 −0.1956 −0.2134 −0.2369 −0.2073 −0.1789 −0.1451 −0.1195 −0.1027 −0.0862 −0.0702 −0.1153 −0.1568 −0.1970 −0.2371 −0.2765 −0.3087 −0.3141 −0.2443 −0.2110 −0.1743 −0.1413 −0.1106 −0.1106

Figure 10. (The C112 and C122 could not be calculated at T = 248.15 K because the C(T, x) was fitted only for the (0.74977 methane + 0.25023 propane) mixture. The two isotherms at T = 273.15 K noted above as “outliers” were excluded.) Eubank and Hall22 discuss the optimum mixture compositions for the determination of Cijk(T) based on the uncertainties in the individual C(T) terms. The isotherms with a higher fraction of

C(T , x) = x13C111(T ) + 3x12x 2C112(T ) + 3x1x 22C122(T ) + x 23C222(T )

p/MPa

(14)

The C112 and C122 were obtained by writing out eq 14 at two different compositions at a fixed temperature and solving simultaneously. This was done for a total of 31 pairs of isotherms, and the results are given in Table 7 and shown in 4158



Article

Figure 3. Relative deviations of experimental densities ρexp for the (0.74977 methane + 0.25023 propane) mixture from densities ρGERG calculated with the GERG-2008 equation of state of Kunz and Wagner:1 ○, T = 248.16 K; +, T = 273.15 K; ◇, T = 298.15 K; ×, T = 323.16 K; △, T = 373.15 K. The error bars show the relative combined expanded uncertainty (k = 2) in the measured densities; for clarity, these are shown only for the isotherm at T = 298.15 K. The uncertainty of the GERG equation (zeroline) is ± 0.1 %.

Figure 6. Relative deviations of experimental densities ρexp along the isotherm at T = 298.15 K for the (0.74977 methane + 0.25023 propane) mixture from densities ρGERG calculated with the GERG2008 equation of state of Kunz and Wagner.1 Different measurement sequences are shown to illustrate the reproducibility of the measurement for different apparatus fillings: ×, series 1, increasing pressure; ○, series 2, pressure decreasing, then increasing to p = 6 MPa; ◇, series 3, decreasing pressure; +, series 4, decreasing pressure; →, sequence of measurements.

Figure 4. Relative deviations of experimental densities ρexp for the (0.50688 methane + 0.49312 propane) mixture from densities ρGERG calculated with the GERG-2008 equation of state of Kunz and Wagner:1 ○, T = 248.15 K; +, T = 273.15 K; ◆, T = 293.15 K; ×, T = 323.16 K; △, T = 373.15 K. The error bars show the relative combined expanded uncertainty (k = 2) in the measured densities; for clarity, these are shown only for the isotherm at T = 293.15 K. The uncertainty of the GERG equation (zeroline) is ± 0.1 %.

Figure 5. Relative deviations of experimental densities ρexp for the (0.26579 methane + 0.73421 propane) mixture from densities ρGERG calculated with the GERG-2008 equation of state of Kunz and Wagner:1 ○, T = 248.15 K; +, T = 273.15 K; ◆, T = 293.15 K; ◇, T = 298.15 K; ×, T = 323.16 K; △, T = 373.16 K. The error bars show the relative combined expanded uncertainty (k = 2) in the measured densities; for clarity, these are shown only for the isotherm at T = 298.15 K. The uncertainty of the GERG equation (zeroline) is ± 0.1 %.

Figure 7. Relative deviations of experimental densities from the literature ρexp for the (methane + propane) system from densities ρGERG calculated with the GERG-2008 equation of state of Kunz and Wagner.1 ○, Huang et al.;17 ◇, May et al.;16 ×, Sage et al.;15 +, Reamer et al.;14 − − −, uncertainty bounds of the GERG equation (± 0.1 %).

4159


4160

0.26579 0.26579 0.26579 0.50687 0.50687 0.50687 0.74977 0.74977

0.26579 0.50687

0.26579 0.74977 0.74977 0.74977

0.26579 0.50687 0.74977

273.149 273.149 273.150 273.148 273.149 273.149 273.150 273.152

293.146 293.147

298.149 298.148 298.151 298.147

323.166 323.164 323.161

36.6124 36.6181 29.8641 29.8557 23.0469 23.0550

36.6160 29.8604 23.0545

36.6257 23.0531 23.0427 23.0459

36.6282 29.8631

36.6321 36.6591 36.6182 29.8824 29.8657 29.8640 23.0526 23.0492

36.6122 36.6203 29.8673 23.0505

(g·mol )

−1

Mvir

0.1680 0.1181 0.1899 0.1652 0.1679 0.2201 0.1564 0.1423 0.1050 0.0949 0.0956 0.0499 0.0533 0.0648 0.1271 0.1241 0.0614

−0.0073 0.0197 −0.0211 0.0063 −0.0105 −0.0121 −0.0097 −0.0131 −0.0112 −0.0131 −0.0136 −0.0092 −0.0196 −0.0164 −0.0233 −0.0157 −0.0078 0.1533 0.1205 0.1051 0.1284 0.0943 0.0856

0.2116 0.2115 0.2108 0.2737

−0.0271 −0.0190 −0.0088 −0.0118

−0.0269 −0.0213 −0.0120 −0.0204 −0.0154 −0.0073

(g·mol )

−1

U(M)

(g·mol )

−1

ΔM B −1

−163.79 −163.38 −102.85 −104.11 −56.30 −55.65

−223.89 −143.93 −80.52

−264.56 −97.71 −98.14 −97.87

−273.79 −177.71

−318.36 −313.52 −320.39 −204.32 −206.82 −207.15 −119.03 −119.00

−392.98 −393.55 −250.90 −144.67

(cm ·mol ) 3

−1

3.19 2.08 0.89 1.25 0.36 0.21

2.07 1.04 0.11

1.12 0.11 0.10 0.16

1.40 0.64

3.69 3.33 4.56 2.05 2.08 2.58 0.86 0.71

4.27 4.57 2.58 2.97

(cm ·mol ) 3

U(B)

C −2

15647. 14856. 8273. 9822. 4418. 4180.

16816. 10203. 5051.

14099. 5820. 5921. 5834.

13031. 10091.

9766. 853. 12176. 6526. 9864. 10173. 6519. 6482.

5233.

(cm ·mol ) 6

−2

4655. 2557. 687. 1074. 119. 59.

2496. 769. 26.

1000. 24. 22. 42.

1441. 423.

6230. 6048. 8130. 2342. 2375. 2862. 531. 375.

3695.

(cm ·mol ) 6

U(C)

0.0075 0.0021 0.0021 0.0029 0.0164 0.0100

0.0047 0.0037 0.0022

0.0057 0.0035 0.0057 0.0079

0.0053 0.0013

0.0020 0.0025 0.0039 0.0040 0.0027 0.0023 0.0059 0.0042

0.0035 0.0058 0.0120 0.0040

%

RMSp −1

−86.08 −85.02 −78.80 −81.32 −78.79 −77.05

−118.12 −112.10 −109.10

−135.87 −131.71 −132.87 −132.15

−139.26 −137.33

−159.67 −147.28 −164.87 −153.62 −158.63 −159.29 −159.10 −159.01

−194.37 −195.84 −185.70 −188.89

(cm ·mol ) 3

B12,grav −1

−85.77 −84.78 −78.72 −81.18 −78.69 −77.00

−117.75 −111.94 −109.03

−135.62 −131.61 −132.65 −131.96

−139.04 −137.17

−159.51 −147.71 −164.40 −153.70 −158.48 −159.12 −158.97 −158.83

−193.62 −195.31 −185.55 −188.70

(cm ·mol ) 3

B12,vir

3.80 2.56 1.06 1.54 0.56 0.31

2.80 1.34 0.26

1.62 0.32 0.55 0.53

1.83 0.96

4.02 4.19 5.50 2.23 2.37 2.92 1.13 1.07

5.77 5.61 2.88 3.36

(cm3·mol−1)

U(B12) test no.

1312z 1312zb 1401h 1401i 1401r 1401zdf

1312y 1401g 1401zc

1312v 1401v 1401wx 1401zzb

1312x 1401f

1312q 1312r 1312u 1401d 1401e 1401j 1401no 1401zgh

1312zc 1401a 1401c 1402a

xmethane is the gravimetric composition; Mvir is the molar mass determined by the virial analysis (eq 9); ΔM is the difference between the gravimetric M and that determined by the virial analysis; U(M) is the expanded uncertainty (k = 2) in Mvir; B and U(B) are the second virial coefficient fitted by eq 9 and its expanded uncertainty; C and U(C) are the third virial coefficient fitted by eq 9 and its expanded uncertainty; RMSp is the RMS deviation (in pressure) of the experimental data from the virial fit (eq 9) expressed in parts per hundred; B12,grav is the second interaction virial coefficient calculated using the gravimetric composition; B12,vir is the second interaction virial coefficient calculated using the virial composition; U(B12) is the expanded uncertainty in B12 (applies to both B12,grav and B12,vir); and the “test no.” cross-references with Tables 3, 4, 5, and 7.

a

0.26579 0.26579 0.50687 0.74977

248.152 248.151 248.151 248.161

0.26579 0.26579 0.50687 0.50687 0.74977 0.74977

(mol frac)

K

373.164 373.162 373.148 373.147 373.157 373.153

xmethane

T

Table 6. Summary of the Virial Analysis for the (Methane + Propane) System Sorted by Temperature Ta

Journal of Chemical & Engineering Data Article



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shown in Figure 11. Most of the points are represented within 0.2 % by the virial model, but points for the (0.74977 methane + 0.25023 propane) mixture at T = 298 K and T = 323 K at pressures above 3 MPa have deviations as large as −0.31 %. These points are near the “nose” of the phase envelope (see Figures 1 and 2), and a simple virial expansion truncated at the third virial coefficient would not be expected to accurately represent this region. The isotherms at T = 373.15 K on the (0.74977 methane + 0.25023 propane) mixture show deviations as large as 0.21 % at a pressure of 6 MPa, and the fourth virial coefficient may have been required to fit the data at the higher pressures. As noted above in section 4.2, this was attempted, but the values of the fourth virial coefficient were not statistically valid. Two other single isotherms show somewhat larger deviations than the others: test number 1312r at T = 273.15 K on the (0.26579 methane + 0.73421 propane) mixture was noted as an outlier in section 4.2; here the composition could have been affected by sorption. Test number 1401f at T = 293.15 K on the (0.50688 methane + 0.49312 propane) mixture shows deviations as large as 0.19 %, but there is no obvious reason. Excluding the points with p > 3 MPa (but including tests 1312r and 1401f), the RMS deviation of the virial model is 0.046 %. The deviations shown in Figure 11 are for a virial model where the B12(T), C112(T), and C122(T) were functions only of temperature, yet the experimental data cover a wide range of composition. The RMS deviation of the fit is comparable to the relative combined expanded uncertainty of the experimental densities. This is further confirmation that the interaction virial coefficients are independent of composition. 4.2.2. Comparison to Literature Data and the GERG Equation. Trusler et al.23 determined second and third interaction virial coefficients from speed of sound measurements on a (0.851 methane + 0.149 propane) mixture over the temperature range (225 to 375) K. They determined the interaction virials in conjunction with an intermolecular pairpotential model. Their values for B12(T), shown in Figure 9, agree very well with ours, with a maximum difference of 5.7 cm3·mol−1 at T = 375 K; this provides an indirect verification of our density data against the sound speed data. Dantzler et al.24 experimentally determined values for B12(T) over the temperature range (298.15 to 373.15) K; they also list values from Gunn25 over a similar temperature range. The data of Gunn were fitted to experimental values of Sage and Lacey.26 Both of these agree with the present results within their uncertainties, as shown in Figure 9. Our values for C112 agree with those from Trusler et al.,23 within their uncertainties, although the values for C122 generally do not agree (see Figure 10). The C122 represents the interaction of one methane molecule with two propane molecules, and since Trusler et al. measured a mixture with a high fraction of methane, it is perhaps not surprising that larger differences are seen for C122. Figure 8 shows the B12(T) calculated from the GERG-2008 equation (as implemented in the NIST REFPROP database27). These values show a composition dependence. The B12(T) at high fractions of methane are in good agreement with our experimental values, but they decrease with increasing propane fraction. Extensive, high-quality data at high fractions of methane were available to fit the GERG equation, explaining the agreement in this region. But the vapor-phase density data at high fractions of propane exhibited very large scatter (see Figure 7), and the mixture model was not constrained to give

Figure 8. Second interaction virial coefficients B12(T) for the (methane + propane) system: ◇, T = 248.15 K; □, T = 273.15 K; ×, T = 293.15 K; +, T = 298.15 K; △, T = 323.15 K; and ○, T = 373.15 K; the dashed lines indicate the B12(T) predicted by the GERG-2008 model1 for the corresponding temperatures.

Figure 9. Second interaction virial coefficients B12(T) for the (methane + propane) system as a function of temperature; ○, this work; , quadratic equation fitted to experimental data of this work (B12(T)/cm3·mol−1 = −0.0035903·(T/K)2 + 3.0962·(T/K) − 736.73); ◇, Trusler et al.;23 +, Dantzler et al.;24 ×, Gunn.25 The error bars show the combined expanded uncertainty (k = 2) in the present results and those of Dantzler et al. (Trusler et al. and Gunn did not report uncertainties.).

propane terminate at lower pressures and, thus, have larger uncertainties in C(T) (see Table 6). The Eubank and Hall analysis would suggest measurements at compositions weighted heavily toward methane, but we did not do this. We deliberately chose mixtures to explore the region with high propane fractions. The uncertainty in Cijk(T) is dominated by uncertainties in the C(T, x) that enter into eq 14; these were numerically propagated through the solution for Cijk(T) to obtain the uncertainties in Cijk(T) presented in Table 7. There is large scatter in the values of the third interaction virial coefficients, comparable to the magnitude of the values themselves, but a trend with temperature is evident. The experimental (p, ρ, T, x) data were compared with the values obtained from a simple virial model (eqs 9, 11, and 13). The B12(T) were fitted to a simple quadratic equation (indicated by the curve in Figure 9), and the C112(T) and C122(T) were represented by linear functions of temperature, indicated by the lines in Figure 10. On the basis of the analysis by Eubank and Hall,22 we gave a greater weight to the C112(T) and C122(T) determined from the (0.74977 methane + 0.25023 propane) and (0.50688 methane + 0.49312 propane) data. The virial coefficients for the pure components were computed from the equations of state of Setzmann and Wagner20 for methane and Lemmon et al.21 for propane. The resulting deviations are 4161



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Table 7. Summary of the Analysis for the Third Interaction Virial Coefficients C122 and C112; the Temperature T Is the Average of the Isotherms, and Test-1 and Test-2 Refer to the Two Isotherms Used to Determine the Cijk (See Table 6) C122

U(C122)

C112

U(C112)

K

T test-1

test-2

cm6·mol−2

cm6·mol−2

cm6·mol−2

cm6·mol−2

273.149 273.149 273.149 273.149 273.149 273.149 273.149 273.149 273.149 273.149 273.149 273.149 293.146 298.149 298.149 298.149 323.164 323.164 323.164 373.155 373.155 373.155 373.155 373.155 373.155 373.155 373.155 373.155 373.155 373.155 373.155

1312q 1312q 1312q 1312q 1312u 1312u 1312u 1312u 1401e 1401e 1401j 1401j 1312x 1312v 1312v 1312v 1312y 1312y 1401g 1312z 1312z 1312z 1312z 1312zb 1312zb 1312zb 1312zb 1401h 1401h 1401i 1401i

1401e 1401j 1401no 1401zgh 1401e 1401j 1401no 1401zgh 1401no 1401zgh 1401no 1401zgh 1401f 1401v 1401wx 1401zzb 1401g 1401zc 1401zc 1401h 1401i 1401r 1401zdf 1401h 1401i 1401r 1401zdf 1401r 1401zdf 1401r 1401zdf

4181. 3726. 6622. 6657. 12846. 12389. 13007. 13042. 13456. 13587. 14732. 14863. 9182. 12617. 12510. 12595. 19205. 18460. 16379. 18778. 16476. 16656. 16895. 15915. 13620. 14552. 14790. 10749. 11647. 17158. 18063.

22637. 22755. 16488. 16483. 29422. 29512. 21509. 21504. 9979. 9878. 11956. 11870. 5211. 2645. 2645. 2646. 9038. 6602. 3172. 16761. 16794. 12310. 12312. 9242. 9316. 6763. 6764. 2863. 2837. 4450. 4435.

20910. 22162. 10446. 10348. 12493. 13749. 8319. 8223. 8169. 8041. 7744. 7616. 16728. 7058. 7354. 7120. 7389. 3538. 4232. 2839. 9150. 2972. 2319. 5619. 11921. 3672. 3020. 4937. 4067. 2805. 1929.

31718. 32146. 7869. 7811. 40898. 41225. 10211. 10163. 4867. 4735. 5771. 5658. 7241. 1248. 1248. 1251. 12570. 3112. 1496. 23091. 23210. 5800. 5804. 12822. 13088. 3195. 3191. 1377. 1342. 2113. 2094.

Figure 10. Third interaction virial coefficients C112(T) and C122(T) for the (methane + propane) system as a function of temperature; ○, C112(T), present work; −−−, linear fit to present work (C112(T)/cm6· mol−2 = −46.044·(T/K) + 20514.); □, C122(T), present work; - - -, linear fit to present work (C122(T)/cm6·mol−2 = 7.3896·(T/K) + 11688.); (filled symbols indicate points given heavier weight in the fit); ∗, C112(T), Trusler et al.;23 △, C122(T), Trusler et al.23 The error bars show the combined expanded uncertainty (k = 2) in the present results. (Trusler et al. did not report uncertainties.)

Figure 11. Relative deviations of the experimental pressures pexp from the pressures calculated with the virial model pvirial (eqs 9, 11, and 13) for three compositions of the (methane + propane) system; ▽, T = 248.15 K; ○, T = 273.15 K; +, T = 293.15 K; ×, T = 298.15 K; △, T = 323.15 K; ∗, T = 373.15 K.

preferable to fit only high-quality data, even if over a limited composition range, and constrain the B12(T) and Cijk(T) to be independent of composition.

values for B12(T) that were invariant with composition. The virial model (eqs 9, 11, and 13 with B12(T), C112(T), and C122(T) independent of composition), on the other hand, fit our vapor-phase measurements very well (RMS deviation of 0.046 % for p < 3 MPa). This suggests that it would be

5. CONCLUSIONS Accurate vapor-phase (p, ρ, T, x) data were measured along six isotherms for the (methane + propane) system at three compositions (0.74977, 0.50688, and 0.26579 mole fraction 4162



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Saturated Vapor Densities of Pure Fluid Substances along the entire Phase Boundary). Fortschr.-Ber. VDI, 1984, Reihe 3, No. 92. (5) Kleinrahm, R.; Wagner, W. Measurement and correlation of the equilibrium liquid and vapour densities and the vapour pressure along the coexistence curve of methane. J. Chem. Thermodyn. 1986, 18, 739− 760. (6) McLinden, M. O.; Kleinrahm, R.; Wagner, W. Force transmission errors in magnetic suspension densimeters. Int. J. Thermophys. 2007, 28, 429−448. (7) Harris, G. L.; Torres, J. A. Selected Laboratory and Measurement Practices and Procedures, To Support Basic Mass Calibrations; NISTIR 6969; National Institute of Standards and Technology: Gaithersburg, MD, 2003. (8) Richter, M.; Kleinrahm, R. Influence of adsorption and desorption on accurate density measurements of gas mixtures. J. Chem. Thermodyn. 2014, 74, 58−66. (9) McLinden, M. O.; Splett, J. D. A liquid density standard over wide ranges of temperature and pressure based on toluene. J. Res. Nat. Inst. Stand. Technol. 2008, 113, 29−67. (10) Moldover, M. R.; McLinden, M. O. Using Ab Initio “Data” to Accurately Determine the Fourth Density Virial Coefficient of Helium. J. Chem. Thermodyn. 2010, 42, 1193−1203. (11) McLinden, M. O. P−ρ−T behavior of four lean synthetic natural-gas-like mixtures from 250 to 450 K with pressures to 37 MPa. J. Chem. Eng. Data 2011, 56, 606−613. (12) Wieser, M. E.; Holden, N.; Coplen, T. B.; Böhlke, J. K.; Berglund, M.; Brand, W. A.; De Bievre, P.; Gröning, M.; Loss, R. D.; Meija, J.; Hirata, T.; Prohaska, T.; Schoenberg, R.; O’Connor, G.; Walczyk, T.; Yoneda, S.; Zhu, X. Atomic weights of the elements 2011 (IUPAC Technical Report). Pure Appl. Chem. 2013, 85, 1047−1078. (13) Kunz, O.; Klimeck, R.; Wagner, W.; Jaeschke, M. The GERG2004 Wide-Range Equation of State for Natural Gases and Other Mixtures, GERG TM15. Fortschr.-Ber. VDI 2007, 6. (14) Reamer, H. H.; Sage, B. H.; Lacey, W. N. Phase equilibria in hydrocarbon systems, volumetric and phase behavior of the methanepropane system. Ind. Eng. Chem. 1950, 42, 534−539. (15) Sage, B. H.; Lacey, W. N.; Schaafsma, J. G. Phase equilibria in hydrocarbon systems, II. Methane−propane system. Ind. Eng. Chem. 1934, 26, 214−217. (16) May, E. F.; Miller, R. C.; Shan, Z. Densities and dew points of vapor mixtures of methane + propane and methane + propane + hexane using a dual-sinker densimeter. J. Chem. Eng. Data 2001, 46, 1160−1166. (17) Huang, E. T. S.; Swift, G. W.; Kurata, F. Viscosities and densities of methane-propane mixtures at low temperatures and high pressures. AIChE J. 1967, 13, 846−850. (18) Boggs, P. T.; Byrd, R. H.; Rogers, J. E.; Schnabel, R. B. User’s Reference Guide for ODRPACK Version 2.01 Software for Weighted Orthogonal Distance Regression; NISTIR 4834; National Institute of Standards and Technology: Gaithersburg, MD, 1992; code is available at http://www.netlib.org/odrpack/. (19) Mason, E. A.; Spurling, T. H. The Virial Equation of State; Pergamon Press: Oxford, 1969. (20) Setzmann, U.; Wagner, W. A new equation of state and tables of thermodynamic properties for methane covering the range from the melting line to 625 K at pressures to 1000 MPa. J. Phys. Chem. Ref. Data 1991, 20, 1061−1151. (21) Lemmon, E. W.; Wagner, W.; McLinden, M. O. Thermodynamic properties of propane. III. Equation of state. J. Chem. Eng. Data 2009, 54, 3141−3180. (22) Eubank, P. T.; Hall, K. R. Optimum mixture compositions for measurement of cross virial coefficients. AIChE J. 1990, 36, 1661− 1668. (23) Trusler, J. P. M.; Wakeham, W. A.; Zarari, M. P. Second and third interaction virial coefficients of the (methane + propane) system determined from the speed of sound. Int. J. Thermophys. 1996, 17, 35− 42.

methane) over the temperature range of (248.15 to 373.15) K; the pressures ranged from 0.10 MPa to a maximum pressure of 6 MPa. Comparisons to the GERG-2008 equation of state for natural gas mixtures showed significant deviations in density (as large as −1.3 %) that increased with decreasing temperature, with increasing pressure, and with increasing propane fraction in the mixture. The measured values were also used to calculate second interaction virial coefficients B12(T) for this system. These were constant (within experimental uncertainty) with composition, in contrast to those calculated with the GERG2008 equation of state. The GERG-2008 equation was fitted to high-quality data at high methane fractions (0.90 mole fraction methane and higher), but the vapor-phase literature data available at high propane fractions were of lower quality. Our results suggest that, in developing a mixture equation of state, the B12(T) should be fitted only to high-quality data and constrained, if necessary, to be independent of composition.

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ASSOCIATED CONTENT

S Supporting Information *

All measured values from which the reported single average results were calculated; details on uncertainties including the temperature, pressure, density, molar mass, and combined uncertainties for each measured point; derivation of eq 4 (uncertainty in molar mass arising from uncertainties in the gravimetric mixture preparation). This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +1-303-497-3580. Fax: +1-303-497-6682. Present Address ‡

M.R.: Lehrstuhl fü r Thermodynamik, Ruhr-Universität Bochum, D-44780 Bochum, Germany; E-mail: m.richter@ thermo.rub.de. Tel.: +49−234−32−26395.

Funding

We thank the Deutsche Forschungsgemeinschaft (DFG) for supporting the stay of M. Richter at NIST under Grant No. RI 2482/1-1. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank our NIST colleague Dr. Eric W. Lemmon for providing his database of literature data for our comparisons. Dr. Thomas Bruno of NIST analyzed the methane and propane used to prepare our mixtures.

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REFERENCES

(1) Kunz, O.; Wagner, W. The GERG-2008 wide-range equation of state for natural gases and other mixtures: An expansion of GERG2004. J. Chem. Eng. Data 2012, 57, 3032−3091. (2) Wagner, W.; Kleinrahm, R. Densimeters for very accurate density measurements of fluids over large ranges of temperature, pressure, and density. Metrologia 2004, 41, S24−S39. (3) McLinden, M. O.; Lösch-Will, C. Apparatus for wide-ranging, high-accuracy fluid (p−ρ−T) measurements based on a compact twosinker densimeter. J. Chem. Thermodyn. 2007, 39, 507−530. (4) Kleinrahm, R.; Wagner, W. Entwicklund und Aufbau einer Dichtemeßalnage zur Messung der Siede- und Taudichten reiner fluider Stoffe auf der gesamten Phasengrenzkurve (Development and Setup of a Densimeter for the Measurement of Saturated Liquid and 4163



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(24) Dantzler, E. M.; Knobler, C. M.; Windsor, M. L. Interaction virial coefficients in hydrocarbon mixtures. J. Phys. Chem. 1968, 72, 676−684. (25) Gunn, R. D. Volumetric properties of nonpolar gaseous mixtures. M.S. Thesis, University of California-Berkeley, 1958. (26) Sage, B. H.; Lacey, W. N. Some Properties of the Lighter Paraffin Hydrocarbons; Am. Petrol. Inst.: New York, 1955. (27) Lemmon, E. W.; Huber, M. L.; McLinden, M. O. NIST Standard Reference Database 23, NIST Reference Fluid Thermodynamic and Transport PropertiesREFPROP, version 9.1; Standard Reference Data Program, National Institute of Standards and Technology: Gaithersburg, MD, 2013.

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