Density Measurements for Ethane, Carbon Dioxide, and Methane +

Article pubs.acs.org/jced

Density Measurements for Ethane, Carbon Dioxide, and Methane + Nitrogen Mixtures from 300 to 470 K up to 137 MPa Using a Vibrating Tube Densimeter Martin A. Gomez-Osorio, Robert A. Browne, Mauricio Carvajal Diaz, Kenneth R. Hall,† and James C. Holste* Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843, United States S Supporting Information *

ABSTRACT: This paper presents experimental p−ρ−T measurements for nitrogen + methane mixtures containing approximately 0.25, 0.50, and 0.75 mole fraction methane at T = 300, 350, 400, and 470 K with p = 10−137 MPa. Methane, nitrogen, and argon are used as calibration fluids, and measurements on carbon dioxide and ethane verify the calibration methodology. An error contribution analysis indicates that the expanded uncertainty (95% confidence limit) is 0.1 kg·m−3 for densities lower than 200 kg·m−3 and 0.0005·ρ for higher densities. The GERG-2008 equation of state describes the measurements for all three mixtures to within ±0.0015·ρ with the largest deviations occurring at the highest pressures.

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INTRODUCTION Equation of state (EoS) development requires property measurements covering wide ranges of temperature and pressure. This work uses a vibrating tube densimeter (VTD) to perform measurements of single-phase fluid densities at T = 300−473 K at pressures ranging up to 137 MPa. Quantitative description of the measurement process appears in Apparatus Description and Calibration Methodology. The paper also describes a calibration procedure for the apparatus that uses nitrogen, argon, and methane as calibration fluids. Measurements for ethane and carbon dioxide validate the calibration procedures. New measurements for three nitrogen + methane mixtures along four isotherms extend the range for accurate density data up to p = 137 MPa. Previous data do not exceed 100 MPa, and the data above 35 MPa are less accurate than those at lower pressure. The current data provide a new anchor for extrapolating EoS. The higher pressure conditions are important to producing and processing natural gas from reservoirs encountered in ultra deep-water environments.

The physical principle in this densimeter is the relationship between the vibrating mass and the resonant period of a onedimensional oscillator. The total mass includes the mass of the tube and the mass of the fluid inside the tube, which is related to the fluid density, as described by eq 1. τ2 ∝

(1)

in which τ is the period of oscillation at resonance, ρ is density of the fluid and mo is the mass of the tube. The internal volume (Vi) and stiffness constant (K) are weak functions of temperature and pressure. Equation 1 implies that an appropriate working equation for density as a function of resonant period is ρ = A ( T , p ) τ 2 − B ( T , p)

(2)

in which the parameters A and B are functions of temperature and pressure as noted by Bouchot and Richon.10 Apparatus Description. Figure 1 illustrates the vibrating tube apparatus including the measuring cell, temperature control, compression system, pressure measurement devices, and an evaluation unit (mPDS). The operational limits for these experiments are T = 300 to 473 K at pressures up to 137 MPa. The measurement cell (DMA HPM) is a commercial unit provided by Anton Paar that contains the vibrating tube.

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VIBRATING TUBE DENSIMETER A vibrating tube provides density measurements based upon the resonant frequency of a tube containing a specific fluid. These apparatus consist of a U-shaped tube that vibrates using an oscillatory driving force, described (for example) by McGregor,1 Wagner et al.,2 Holcomb and Outcalt.3 Blencoe et al.4 and Chang and Moldover5 are examples of authors who report having built vibrating tube densimeters, however, Ihmels et al.,6 Lagourette et al.,7 May et al.8 and Outcalt and McLinden9 describe commercial apparatus that provide accurate measurements. © 2016 American Chemical Society

mo + ρVi(T , p) K (T , p)

Special Issue: In Honor of Kenneth R. Hall Received: February 15, 2016 Accepted: June 30, 2016 Published: July 12, 2016 2791

DOI: 10.1021/acs.jced.6b00138 J. Chem. Eng. Data 2016, 61, 2791−2798

Journal of Chemical & Engineering Data

Article

Figure 1. Schematic of the vibrating tube apparatus.

Table 1. Parameters for Vibrating Tube Densimeter Calibration Equation (Equation 3) parameter −3

A0/kg·m αv/K−1 βv/MPa−1 βτ/MPa−1

value

σ

May et al.8

16091.4 4.348 × 10−5 3.148 × 10−5 −3.191 × 10−6

6.24 0.1039 × 10−5 0.1736 × 10−5 0.0060 × 10−6

3.6 × 10−5 2.3 × 10−5 −0.95−1.2 × 10−6

Table 2. Sample Information Tablea

a

chemical names

source

mole fraction purity

purification method

argon nitrogen methane ethane carbon dioxide

Airgas Airgas Airgas Matheson Matheson

0.99995 0.999995 0.99995 0.9995 0.99999

none none none none none

All gases were used as received from the supplier.

measurements for 10 MPa increments require less than 20 min. This means it is possible to complete an isotherm in approximately 8 h. For pressure measurement, the apparatus uses two Paroscientific pressure transducers with full scale ranges of 2000 psi (13.7 MPa) and 20 000 psi (137 MPa). All mixture compositions in this paper are single phase at all thermodynamic conditions in the experiment. Thus, compositions remain intact throughout the experiment. Calibration Methodology. The parameters A and B in eq 2 require calibration with fluids for which the density is known accurately as a function of temperature and pressure. Holcomb and Outcalt3 and Bouchot and Richon10 contend that the calibration methodology defines the accuracy of the apparatus. Various authors have proposed calibration models that include empirical, semiempirical, and physically based approaches. Bouchot and Richon10 suggest a forced-path mechanical calibration (FPMC) model that attempts to set realistic paths for mechanical properties with variations of temperature and pressure. This physically based model limits the influence of calibration measurements in the estimation of densities. However, they developed this model for a low-pressure apparatus with tubes having thin walls. Following May et al.8 this work uses the working equation

Figure 2. Absolute deviations for densities calculated from eq 3 and equations of state for the following: *, nitrogen;12 ○, argon;13 □, methane;14 and (--) expanded uncertainties (eq 7).

The U-shaped tube material of construction is Hastelloy C-276. The output from the Anton Paar device is the period of resonant oscillation. For temperature control, the cell resides within a cylindrical, aluminum shield, which is heated using resistance tape. A resistance thermometer (RTD) located at the surface of the cylinder and a PID control routine integrated into Labview provide feedback control. The system also includes a standard platinum resistance thermometer (SPRT) provided by Fluke Calibration (reference 5686-B glass capsule SPRT, 25 Ω) located at the measurement cell. A Fluke 1594A Super-Thermometer provides temperature readings. The criterion for thermal equilibrium is when the standard deviation of the SPRT temperature readings over 10 min periods is less than ±1.5 mK. The procedure when changing to a new isotherm is to wait ∼5 h after achieving thermal stability at about ±1.5 mK and then to measure τ0 (the period of oscillation at resonance in vacuum). A thermocouple gauge ensures that the pressure is less than 1 mbar, that is, the density of the residual gas is less than 0.01 kg m−3. Then, the sample is introduced to the desired pressure. After stabilizing at ±1.5 mK for each pressure,

⎡ ⎤ ⎛ τ ⎞2 A 0(T0 , p = 0) ⎢ (1 + βτ p)⎜ ρF = ⎟ − 1⎥ ⎥⎦ 1 + αv(T − T0) + βv p ⎢⎣ ⎝ τ0(T ) ⎠ (3) 2792



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Table 3. Methane + Nitrogen Sample Masses, Mixture Compositions, and Molar Fraction Expanded Uncertainties (95% Confidence Limits) sample

mCH4/g

mN2/g

xCH4

xN2

U(x)

25% CH4−75% N2 50% CH4−50% N2 75% CH4−25% N2

540.1 1106.1 1734.6

2831.0 1934.9 1015.6

0.24989 0.49956 0.74889

0.75011 0.50044 0.25111

0.00004 0.00003 0.00002

Table 4. Uncertainty Contributions to Equation 6 coefficients term apparatus, U(ρ)app temperature, U(T) pressure, U(P)

expanded uncertainty (95% confidence limit)) 0.1 kg·m−3 for ρ ≤ 200 kg·m−3 0.0005·ρ for ρ ≥ 200 kg·m−3 10 mK 0.0138 MPa (HPT) or 0.00138 MPa (LPT)

a

derivative

source

equivalent in density/kg·m−3

∂ρ ∂T P , mi

EoS

0.1−0.5 0.001−0.008

( )

EoS

0.015−0.148

( ) ∂ρ ∂p

T , mi

resonant period, U(τ)

0.005 μs

( ∂∂ρτ )P ,T

eq 3

0.060

composition, U(xi)

0.00004

( )

EoS

0.0007−0.004

∂ρ ∂xi P ,T

a The Paroscientific transducer accuracies are specified by the manufacturer as ±0.0001·pmax, where pmax is 13.8 and 138 MPa for the low (LPT) and high (HPT) pressure transducers, respectively.

Table 5. Experimental Values of Temperature T, Pressure p, and Density ρ and Values of Density ρEoS Calculated Using the Span and Wagner EoS17 for Single Phase Carbon Dioxidea T/K

p/MPa

304.303 304.304 304.300 304.297 304.293 304.290 304.287 304.285 304.294 304.294

119.826 136.467 104.549 86.449 69.065 56.603 42.233 31.385 21.349 10.390

398.610 398.617 398.619 398.614 398.610 398.613 398.609 398.616 398.616 398.602

118.142 137.833 119.829 103.727 86.718 70.281 52.521 36.685 36.789 20.324

469.763 469.759 469.738 469.735 469.748 469.740 469.729 469.777 469.775

116.457 137.604 103.717 103.847 86.961 69.479 54.964 36.325 20.740

ρexp/kg·m−3

ρEoS/kg·m−3

T = 304 K 1148.82 1149.13 1169.87 1170.20 1127.06 1127.47 1097.67 1097.98 1063.89 1064.08 1034.73 1034.73 992.73 992.44 950.66 950.15 895.09 894.61 768.95 768.89 T = 399 K 975.99 975.55 1012.46 1012.04 979.32 978.92 944.72 944.24 900.46 900.06 846.15 845.85 764.26 764.02 645.59 645.24 646.63 646.28 391.90 391.67 T = 470 K 861.27 860.52 908.75 907.90 827.38 826.75 827.75 827.13 773.96 773.46 702.01 701.57 621.41 620.95 466.70 466.53 272.48 272.60

100(ρexp − ρEoS)/ρexp −0.027 −0.028 −0.036 −0.028 −0.017 0.000 0.030 0.054 0.054 0.008 0.044 0.041 0.041 0.050 0.044 0.036 0.032 0.054 0.053 0.059

Figure 3. Relative deviations of measured carbon dioxide densities from the Span and Wagner EoS17 for the following: ●, this work (304, 400, and 470 K) and □, Mantilla et al.18 (310, 400, and 450 K).

in which A0 is an apparatus factor that physically is related to the mass, internal volume, and external dimensions of the tube. The parameters αv and βv incorporate the temperature and pressure variations of the internal volume, and βτ incorporates the variations with pressure of the elastic properties of the tube. Finally, τ0 is the resonant period of the empty tube. In practice, all parameters result from calibrations with known reference fluids. Equation 3 differs from that of May et al.8 in that the resonant period at p = 0 is reestablished for each new isotherm via a vacuum measurement. Iglesias et al.11 show that this approach can limit uncertainties to 0.1 kg·m−3 for liquids. However, because the equilibrium temperature may change after each pressure change along the isotherm, τo(T), must be adjusted to compensate for temperature variations. The vacuum resonant period exhibits quadratic temperature dependence

0.087 0.094 0.076 0.075 0.065 0.062 0.074 0.036 −0.042

a Expanded uncertainties are U(T) = 0.01 K, U(p) = 1.4 kPa for p < 14 MPa, U(p) = 14 kPa for p > 14 MPa, U(x) = 0.00001, and the combined expanded uncertainty Uc is Uc(ρ) = 0.1 kg·m−3 for ρ ≤ 200 kg·m−3, Uc(ρ) = 0.001·ρ for ρ ≥ 200 kg·m−3. All expanded uncertainties are at 0.95% level of confidence.

τ(T ) = c0 + c1T + c 2T 2

(4)

Because the temperature variations are small, a first-order Taylor series expansion provides sufficient accuracy. 2793



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Table 6. Experimental Values of Temperature T, Pressure p, and Density ρ and Values of Density ρEoS Calculated Using the Bücker and Wagner EoS19 for Single Phase Ethanea T/K

p/MPa

304.306 304.303 304.300 304.299 304.297 304.295 304.294 304.300 304.299 304.299 304.299

120.907 137.643 113.927 103.841 86.990 69.715 56.004 41.974 30.964 20.724 10.173

399.011 399.014 399.025 399.005 399.002 399.002 399.001 399.004 399.006 399.000 399.004

123.093 136.122 120.965 103.351 86.735 69.356 55.861 42.071 30.973 20.736 10.202

469.996 469.992 469.966 469.956 469.967 469.968 469.969 469.982 469.961 469.948 469.938

120.161 125.610 137.417 105.448 86.507 70.405 55.719 41.149 31.179 21.121 10.882

ρexp/kg·m−3

ρEoS/kg·m−3

T = 304 K 532.80 532.90 542.63 542.77 528.42 528.47 521.63 521.65 509.02 508.94 494.00 493.61 479.29 478.99 460.60 460.41 441.74 441.49 417.38 417.13 372.80 372.51 T = 399 K 479.22 478.86 489.14 488.78 477.50 477.13 462.05 461.62 444.78 444.25 422.40 421.74 399.66 399.28 368.25 368.08 330.57 330.51 270.05 269.92 130.49 130.47 T = 470 K 439.65 439.16 444.69 444.23 454.98 454.45 424.56 424.14 401.16 400.87 376.04 375.81 346.04 345.82 303.71 303.54 260.81 260.56 194.69 194.50 98.22 98.02

100(ρexp − ρEoS)/ρexp −0.019 −0.026 −0.010 −0.002 0.014 0.079 0.063 0.041 0.055 0.059 0.078

Figure 4. Relative deviations of measured ethane densities from the Bücker and Wagner EoS19 for the following: ●, this work (304, 400, and 470 K) and □, Cristancho et al.20 (298, 400, and 450 K).

0.077 0.073 0.078 0.094 0.119 0.156 0.096 0.046 0.018 0.048 0.019

Table 7. Experimental Values of Temperature T, Pressure p, and Density ρ and Values of Density ρEoS Calculated Using GERG-200823 for a Single Phase Gas Mixture of Molar Composition (0.24989 CH4 + 0.75011 N2)a

0.111 0.104 0.116 0.098 0.071 0.060 0.066 0.057 0.098 0.099 0.200

a

Expanded uncertainties are U(T) = 0.01 K, U(p) = 1.4 kPa for p < 14 MPa, U(p) = 14 kPa for p > 14 MPa, U(x) = 0.0005, and the combined expanded uncertainty Uc is Uc(ρ) = 0.1 kg·m−3 for ρ ≤ 200 kg·m−3, Uc(ρ) = 0.001·ρ for ρ ≥ 200 kg·m−3. All expanded uncertainties are at 0.95% level of confidence.

τo(T ) = τo(To) +

∂τo ∂T

(T − To) To

= τo(To) + (c1 + 2c 2T0)(T − To)

(5)

where To is the temperature of the vacuum measurement. The parameters c0, c1, and c2 come from fitting eq 4 to a set of vacuum measurements made at various temperatures. The values used in the application of eq 4 are c1 = 0.2626 μs·K−1 and c2 = 0.000126 μs·K−2. The resonant period calibration data appear in Table S1 of the Supporting Information. Outcalt and McLinden9 mention that uncertainties associated with the reference fluid EoS are among the highest contributors to errors. This paper uses highly accurate EoS for pure nitrogen, argon, and methane to reduce EoS error contributions. The Span et al.12 EoS for nitrogen has an expanded uncertainty of ±0.0002·ρ at temperatures between 240 and 523 K at 2794

T/K

p/MPa

304.170 304.170 304.162 304.160 304.155 304.157 304.156 304.154 304.154 304.160 304.156 304.154

119.970 137.644 119.475 100.150 80.085 69.944 60.032 49.960 40.008 29.009 20.038 10.026

350.052 349.911 350.005 350.022 350.021 350.020 350.022 350.023 350.028 350.026 350.034 350.034

119.603 137.330 120.082 99.993 80.012 69.941 60.017 49.997 39.994 29.993 20.016 10.009

399.916 399.912 399.898 399.891 399.885 399.883 399.882 399.885 399.873 399.857

120.219 137.620 120.086 100.077 80.004 69.977 60.031 49.831 39.948 30.038

ρexp/kg·m−3

ρEoS/kg·m−3

T = 304 K 543.01 542.75 568.44 568.13 542.25 541.99 509.22 509.00 466.50 466.35 440.11 439.99 409.84 409.73 372.83 372.73 327.46 327.39 262.61 262.54 194.30 194.20 101.09 100.96 T = 350 K 503.31 503.04 530.55 530.19 504.14 503.86 467.88 467.67 423.20 423.06 395.97 395.87 364.90 364.83 327.90 327.85 283.49 283.46 229.32 229.30 163.38 163.33 85.31 85.22 T = 400 K 467.40 467.24 494.98 494.73 467.18 467.03 429.90 429.83 384.10 384.09 356.85 356.87 325.98 326.02 289.34 289.39 247.77 247.83 198.74 198.77

100(ρexp − ρEoS)/ρexp 0.047 0.055 0.047 0.043 0.031 0.028 0.028 0.027 0.022 0.027 0.052 0.135 0.054 0.068 0.056 0.045 0.033 0.025 0.019 0.014 0.009 0.012 0.031 0.099 0.034 0.050 0.032 0.017 0.000 −0.004 −0.013 −0.016 −0.022 −0.013



Article

Table 7. continued T/K

p/MPa

399.861 399.867

19.996 10.013

469.864 469.865 469.903 469.886 469.879 469.892 469.885 469.883 469.890 469.900 469.916 469.909

120.802 137.596 119.671 99.985 80.110 69.961 59.957 49.999 40.040 30.012 20.020 10.004

ρexp/kg·m−3

ρEoS/kg·m−3

T = 400 K 140.32 140.31 73.52 73.49 T = 470 K 424.87 424.82 452.08 451.96 422.86 422.84 385.63 385.65 340.44 340.55 313.52 313.63 283.71 283.84 250.17 250.30 212.00 212.12 168.09 168.16 118.12 118.21 61.80 61.84

0.002·ρ for higher pressures. The Setzmann and Wagner14 EoS for methane claims expanded uncertainties of 0.0003·ρ for pressures below 12 MPa and 0.0007·ρ for pressures up to 50 MPa. During development of the reference equations, accurate p−ρ−T data between pressures of 40 to 200 MPa were not available. However, Mantilla et al.15 used a magnetic suspension densimeter (MSD) to show that the EoS for nitrogen reproduces their data within ±0.00025·ρ up to p = 200 MPa. Using the same technique, Cristancho et al.16 showed that the methane EoS reproduces density values within ±0.0004·ρ for pressures up to 200 MPa at temperatures from 300 to 450 K. Using the same MSD, this work uses as yet unpublished argon density measurements up to p = 200 MPa to show that errors in the reference EoS for argon are less than ±0.0005·ρ. Calibration measurements cover isotherms for nitrogen, argon, and methane at temperatures between 303 and 474 K with 14 pressure values between 10 and 137 MPa for each isotherm. The measured values are available in the Supporting Information in Tables S2, S3, and S4 for nitrogen, argon, and methane, respectively. Equation 3 describes the nitrogen, argon, and methane calibration data within ±0.1 kg·m−3 for densities lower than 200 kg·m−3 and ±0.0005·ρ for densities up to 1100 kg·m−3. Table 1 contains the calibration parameters for eq 3. The temperature and pressure coefficients are consistent with those reported by May et al.8 Figure 2 presents the residual errors for calibration measurements of methane, nitrogen and argon including uncertainties caused by calibration.

100(ρexp − ρEoS)/ρexp 0.009 0.051 0.013 0.026 0.004 −0.006 −0.032 −0.036 −0.045 −0.054 −0.054 −0.045 −0.078 −0.067


pressures up to 30 MPa. For higher pressures, it claims errors less than ±0.006·ρ. The Tegeler et al.13 EoS for argon behaves similarly to that for nitrogen with expanded uncertainties in density of 0.0003·ρ for pressures up to 30 MPa and less than

Table 8. Experimental Values of Temperature T, Pressure p, and Density ρ and Values of Density ρEoS Calculated Using GERG200823 for a Single Phase Gas Mixture of Molar Composition (0.49956 CH4 + 0.50044 N2)a T/K

p/MPa

304.320 304.317 304.308 304.301 304.297 304.295 304.294 304.293 304.291 304.296 304.293 304.289

119.878 137.784 120.092 99.969 80.027 69.979 59.992 49.950 40.009 30.001 20.007 9.997

349.847 349.849 349.847 349.844 349.845 349.841 349.840 349.839 349.841 349.837 349.833 349.831

120.005 137.830 120.023 99.951 79.968 70.033 59.961 50.006 40.010 29.954 20.028 9.961

ρexp/kg·m−3

ρEoS/kg·m−3

T = 304 K 480.29 479.73 501.45 500.78 480.60 480.01 452.36 451.89 417.33 416.96 395.67 395.34 370.21 369.93 339.04 338.81 300.09 299.92 248.55 248.43 179.37 179.23 91.69 91.53 T = 350 K 446.62 446.00 469.21 468.46 446.64 446.03 416.45 415.94 378.82 378.52 356.12 355.86 329.19 328.97 297.37 297.21 258.28 258.21 209.28 209.23 149.00 148.95 76.43 76.37

100(ρexp − ρEoS)/ρexp 0.117 0.132 0.123 0.102 0.088 0.083 0.076 0.068 0.057 0.050 0.075 0.166 0.138 0.159 0.138 0.121 0.080 0.075 0.065 0.052 0.029 0.025 0.036 0.072

T/K

p/MPa

399.904 399.909 399.905 399.900 399.904 399.913 399.914 399.920 399.917 399.912 399.915 399.913 399.924

120.559 137.807 120.026 100.082 79.818 69.931 59.961 49.865 39.952 29.983 19.922 9.988 9.987

470.004 470.019 470.021 470.063 470.010 469.998 469.994 469.987 469.982 469.977 469.973 469.973

119.844 137.925 119.963 100.025 80.022 69.917 60.016 49.973 40.008 29.966 19.923 9.992

ρexp/kg·m−3

ρEoS/kg·m−3

T = 400 K 414.70 414.22 437.67 437.03 413.98 413.46 382.51 382.19 343.04 342.84 319.87 319.72 292.96 292.88 261.11 261.07 224.03 224.02 179.50 179.51 126.11 126.10 65.56 65.51 65.54 65.51 T = 470 K 374.90 374.59 399.92 399.47 375.09 374.76 342.70 342.57 303.46 303.42 280.09 280.08 254.20 254.21 224.25 224.31 190.16 190.21 150.62 150.63 105.29 105.27 54.93 54.91

100(ρexp − ρEoS)/ρexp 0.115 0.148 0.125 0.084 0.058 0.046 0.028 0.017 0.003 −0.006 0.012 0.067 0.053 0.084 0.112 0.089 0.038 0.014 0.004 −0.006 −0.024 −0.029 −0.007 0.018 0.031

a

Expanded uncertainties are U(T) = 0.01 K, U(p) = 1.4 kPa for p < 14 MPa, U(p) = 14 kPa for p > 14 MPa, U(x) = 0.00004, and the combined expanded uncertainty Uc is Uc(ρ) = 0.1 kg·m−3 for ρ ≤ 200 kg·m−3, Uc(ρ) = 0.001·ρ for ρ ≥ 200 kg·m−3. All expanded uncertainties are at 0.95% level of confidence. 2795



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Table 9. Experimental Values of Temperature T, Pressure p, and Density ρ and Values of Density ρEoS Calculated Using GERG-200823 for a Single Phase Gas Mixture of Molar Composition (0.74890 CH4 + 0.25110 N2)a

SAMPLE PURITY AND MIXTURE PREPARATION Table 2 shows the characteristics of the gases used for the pure fluid measurements and for the synthesis of the mixtures. All gases were used as received with no additional purification. This work used an in-house gravimetric synthesis apparatus to create nitrogen + methane mixtures. The apparatus contains a Sartorius Combics-IS Series High Precision Scale with a readability of 0.1 g, a high-vacuum system capable of 10−9 bar, and an aluminum cylinder to collect the sample. Before loading a sample, the entire system (including the aluminum cylinder) is evacuated to p < 10−6 bar as measured by an ion gauge. After recording the mass of the evacuated cylinder, methane is loaded into the cylinder and mass again is recorded. Finally, nitrogen is added to the sample and the mass recorded. All lines are disconnected from the collection vessel, and the cylinder is allowed to cool to room temperature before each mass reading. Atmospheric pressure and room temperature are recorded to calculate air buoyancy effects on the cylinder. Table 3 contains the amounts of nitrogen and methane used for each mixture, the resulting mole fractions, and the compositional uncertainties.

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UNCERTAINTY ANALYSIS Outcalt and McLinden9 note that the uncertainty in density measurements using vibrating tube densimeters depends strongly upon the reproducibility of vacuum readings and uncertainties in the reference EoS for the calibration fluids. The various contributions to uncertainty come from the apparatus repeatability, the calibration fluid densities, the temperature and pressure sensors, the variations in the resonance period, and composition effects so that the overall expanded uncertainty (95% confidence limit) is ⎡⎛ ⎞ ⎤2 ∂ ρ 2 app 2 [Uc(ρ)] = [U (ρ) ] + ⎢⎜ ⎟ U (p)⎥ ⎢⎝ ∂p ⎠ ⎥ ⎣ ⎦ T ,m i

⎡⎛ ⎞ ⎤ ⎡⎛ ∂ρ ⎞ ⎤ ∂ρ + ⎢⎜ ⎟ U ( T ) ⎥ + ⎢⎜ ⎟ U ( τ ) ⎥ ⎢⎣⎝ ∂T ⎠ P , mi ⎥⎦ ⎢⎣⎝ ∂τ ⎠ P , T ⎥⎦ 2

⎡ ⎤2 ⎛ ⎞ ⎢ ∂ρ ⎥ + ∑ ⎢⎜ ⎟ U (xi)⎥ ∂ x ⎝ ⎠ i P ,T ,m i=1 ⎣ ⎦ j≠i N

(6)

in which app

U(ρ)

= 0.1 kg·m

−3

−3

for ρ ≤ 200 kg·m ,

0.0005ρ for ρ ≥ 200 kg ·m−3

(7)

The calibration technique uses argon, methane, and nitrogen, whose EoS have density errors of less than ±0.0005·ρ, as reference fluids. Figure 2 demonstrates the goodness of fit for the combined calibration results and confirms that eq 6 adequately describes U(ρ)app. Table 4 presents estimates of the various uncertainty contributions. The SPRT and associated electronics provide temperatures with a resolution in temperature of ±1 mK with standard deviations within ±1.5 mK for each measurement. However, the estimated accuracy of ITS 90 with respect to the true thermodynamic temperature is ±0.01 K in this range. The accuracies of the two Paroscientific pressure transducers with maximum working pressures of 13.7 and 137 MPa are specified

T/K

p/MPa

304.114 304.115 304.103 304.100 304.097 304.096 304.096 304.096 304.094 304.092 304.089 304.097

120.278 137.680 119.811 100.110 80.054 69.976 59.950 49.895 39.907 29.972 19.971 9.981

349.917 349.918 349.915 349.913 349.897 349.899 349.901 349.899 349.898 349.896 349.891 349.888

119.772 137.872 119.963 99.975 79.983 69.727 59.941 50.031 39.935 30.048 20.006 10.027

399.862 399.878 399.878 399.877 399.872 399.882 399.881 399.881 399.886 399.868 399.859 399.862

118.835 137.873 119.993 100.029 79.995 70.064 59.988 50.075 40.061 30.067 20.006 9.992

469.886 469.883 469.876 469.877 469.862 469.863 469.854 469.843 469.835 469.874 469.838 469.851

119.899 137.448 120.092 99.995 80.021 69.975 59.945 50.010 39.994 29.908 20.016 10.026

ρexp/kg·m−3

ρEoS/kg·m−3

T = 304 K 418.82 418.39 435.25 434.73 418.39 417.92 396.33 395.91 368.16 367.83 350.71 350.42 330.09 329.83 304.63 304.41 272.08 271.92 228.08 227.94 165.17 165.04 82.59 82.45 T = 350 K 388.99 388.61 407.63 407.12 389.22 388.82 364.75 364.46 334.02 333.86 314.65 314.53 292.83 292.75 266.14 266.11 232.26 232.24 189.70 189.71 134.29 134.27 68.31 68.27 T = 400 K 359.57 359.31 380.53 380.13 361.01 360.67 335.02 334.82 302.54 302.47 283.02 282.98 260.00 259.98 233.16 233.17 200.52 200.55 160.90 160.92 112.82 112.78 57.76 57.72 T = 470 K 326.99 326.83 347.30 346.99 327.26 327.07 299.96 299.91 266.76 266.78 246.91 246.94 224.29 224.35 198.54 198.59 168.40 168.45 133.12 133.11 93.29 93.24 48.26 48.22

100(ρexp − ρEoS)/ρexp 0.102 0.118 0.112 0.106 0.091 0.081 0.079 0.071 0.062 0.063 0.082 0.178 0.099 0.126 0.103 0.082 0.049 0.039 0.025 0.009 0.009 −0.006 0.009 0.053 0.071 0.103 0.094 0.059 0.023 0.014 0.008 −0.003 −0.013 −0.010 0.037 0.067 0.048 0.089 0.057 0.019 −0.008 −0.011 −0.025 −0.026 −0.028 0.004 0.053 0.096


2796



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by the manufacturer to be ±0.01% of full-scale. The apparatus resonant period varies within a standard deviation of 0.0025 μs. Vacuum readings usually were repeated at the end of each isotherm, verifying that hysteresis effects and drifts were negligible. The values in Table 4 show that the first term in eq 6 dominates the uncertainty, therefore eq 7 provides an excellent working description of the overall expanded uncertainties of the measured densities.

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Tables 7, 8, and 9 present (p−ρ−T) measurements and densities calculated using the GERG-2008 EoS developed by Kunz and Wagner23 for the three mixture compositions. Figure 5 shows that the relative deviations of the GERG-2008 EoS23 from the experimental results for all three mixtures vary slightly with composition. For pressures below 40 MPa, the EoS describes experimental densities within their experimental uncertainties. The deviations increase to slightly larger than the experimental uncertainties at higher pressures but the EoS density estimates are within ±0.0015·ρ for these mixtures. Seitz et al.22 report the only previously existing p−ρ−T measurements for nitrogen + methane mixtures at pressures above 40 MPa and at temperatures up to 470 K. Their highest pressure is 100 MPa with density uncertainties of approximately ±1 kg·m−3. Figure 6 illustrates the relative deviations between their data and

RESULTS

This work uses measurements of carbon dioxide and ethane to validate the calibration procedure. It also presents experimental densities for mixtures of methane and nitrogen at pressures up to 137 MPa. The measurements for carbon dioxide are at three different temperatures with pressures from 20 to 140 MPa. Table 5 presents the experimental results and values of density predicted using the Span and Wagner17 EoS for carbon dioxide. Table 5 also contains the relative differences between the experimental data and this EoS. Figure 3 presents relative deviations of the measurements from the EoS as a function of pressure for T = 304, 400, and 470 K including error bars showing experimental uncertainties. Figure 3 also includes carbon dioxide measurements reported by Mantilla et al.18 at T = 310, 400, and 450 K using a magnetic suspension densimeter, which does not depend upon calibration fluids. The VTD and MSD results agree well, verifying the accuracy of the calibration model. The new data extend the temperature range up to T = 470 K for high-pressure data. The ethane density measurements performed in the VTD at T = 304, 400, and 470 K from p = 10−137 MPa appear in Table 6. The Bücker and Wagner19 EoS estimates for ethane and the relative deviations also appear in Table 6. Figure 4 presents the relative deviations as functions of pressure, including experimental error bars. Figure 4 also presents ethane densities measured using an MSD reported by Cristancho et al.20 The VTD results agree with the MSD densities with some discrepancies at low pressure. Ortiz et al.21 note that at small densities for both apparatus the initial term in eq 6 and an equivalent term for the MSD become increasingly dominant as the pressure decreases.

Figure 6. Relative deviations of Seitz et al.22 densities from GERG-2008 EoS23 for methane + nitrogen samples (○, T = 323.15 K; Δ, T = 373.15 K; □, T = 473.15 K; *, T = 573.15 K).

GERG 2008. GERG-2008 utilized the Seitz et al.22 density data because no other data were available at pressures above 40 MPa for temperatures up to 470 K. GERG-2008 predicts data from the current paper within 0.0015 ρ.

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CONCLUSIONS The measurements for pure carbon dioxide and pure ethane, based upon methane, argon, and nitrogen as calibration fluids for the VTD, confirm that the expanded uncertainties (95% confidence limit) in the measured densities are 0.1 kg·m−3 for ρ ≤ 200 kg·m−3 and 0.0005·ρ for ρ ≥ 200 kg·m−3. The apparatus can perform measurements for temperatures between (300 and 470) K at

Figure 5. Relative deviations of measured densities from GERG-2008 EoS23 for methane (*, x ≈ 0.25; •, x ≈ 0.50; □, x ≈ 0.75) + nitrogen samples for the 304 K, 350 K, 400 and 470 K isotherms. 2797



Article

Sci. Instrum. 2014, 85, 095111; Erratum. Rev. Sci. Instrum. 2015, 86, 049902. (9) Outcalt, S. L.; McLinden, M. O. Automated Densimeter for the Rapid Characterization of Industrial Fluids†. Ind. Eng. Chem. Res. 2007, 46, 8264−8269. (10) Bouchot, C.; Richon, D. An enhanced method to calibrate vibrating tube densimeters. Fluid Phase Equilib. 2001, 191, 189−208. (11) Iglesias-Silva, G. A.; Bravo-Sánchez, M.; Estrada-Baltazar, A.; Bouchot, C.; Hall, K. R. P−ρ−T Data for 1-Butanol and Isobutyl Alcohol from (283.15 to 363.15) K at Pressures up to 66 MPa. J. Chem. Eng. Data 2015, 60, 1076−1090. (12) Span, R.; Lemmon, E. W.; Jacobsen, R. T.; Wagner, W.; Yokozeki, A. A Reference Equation of State for the Thermodynamic Properties of Nitrogen for Temperatures from 63.151 to 1000 K and Pressures to 2200 MPa. J. Phys. Chem. Ref. Data 2000, 29, 1361−1433. (13) Tegeler, C.; Span, R.; Wagner, W. A New Equation of State for Argon Covering the Fluid Region for Temperatures From the Melting Line to 700 K at Pressures up to 1000 MPa. J. Phys. Chem. Ref. Data 1999, 28, 779−850. (14) 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 up to 100 MPa. J. Phys. Chem. Ref. Data 1991, 20, 1061−1155. (15) Mantilla, I. D.; Cristancho, D. E.; Ejaz, S.; Hall, K. R.; Atilhan, M.; Iglesias-Silva, G. A. New P−ρ−T Data for Nitrogen at Temperatures from (265 to 400) K at Pressures up to 150 MPa. J. Chem. Eng. Data 2010, 55, 4227−4230. (16) Cristancho, D. E.; Mantilla, I. D.; Ejaz, S.; Hall, K. R.; Atilhan, M.; Iglesia-Silva, G. A. Accurate PρT Data for Methane from (300 to 450) K up to 180 MPa. J. Chem. Eng. Data 2010, 55, 826−829. (17) Span, R.; Wagner, W. A New Equation of State for Carbon Dioxide Covering the Fluid Region from the Triple-Point Temperature to 1100 K at Pressures up to 800 MPa. J. Phys. Chem. Ref. Data 1996, 25, 1509−1596. (18) Mantilla, I. D.; Cristancho, D. E.; Ejaz, S.; Hall, K. R.; Atilhan, M.; Iglesias-Silva, G. A. P-ρ-T Data for Carbon Dioxide from (310 to 450) K up to 160 MPa. J. Chem. Eng. Data 2010, 55, 4611−4613. (19) Bücker, D.; Wagner, W. A Reference Equation of State for the Thermodynamic Properties of Ethane for Temperatures from the Melting Line to 675 K and Pressures up to 900 MPa. J. Phys. Chem. Ref. Data 2006, 35, 205−266. (20) Cristancho, D. E.; Mantilla, I. D.; Ejaz, S.; Hall, K. R.; Atilhan, M.; Iglesias-Silva, G. A. Accurate PρT Data for Ethane from (298 to 450) K up to 200 MPa. J. Chem. Eng. Data 2010, 55, 2746−2749. (21) Ortiz-Vega, D. O.; Mantilla, I. D.; Acosta, H. Y.; Gomez-Osorio, M. A.; Holste, J. C.; Hall, K. R.; Iglesias-Silva, G. A. Uncertainty estimates for experimental density measurements: Effects of temperature, pressure and sample preparation. J. Chem. Thermodyn. 2013, 58, 14−19. (22) Seitz, J. C.; Blencoe, J. G.; Bodnar, R. J. Volumetric properties for {(1 − x)CO2+xCH4}, {(1 − x)CO2+xN2}, and {(1 − x)CH4+xN2} at the pressures (9.94, 19.94, 29.94, 39.94, 59.93, 79.93, and 99.93) MPa and temperatures (323.15, 373.15, 473.15, and 573.15) K. J. Chem. Thermodyn. 1996, 28, 521−538. (23) 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.

pressures up to 137 MPa, providing a method for accurate and rapid high-pressure measurements. The carbon dioxide and ethane density mesurements also show that a four-parameter calibration equation for the VTD is valid. The new measurements for nitrogen + methane mixtures show that GERG-2008 describes the (p,T,ρ,x) behavior for this system within ±0.0015·ρ for temperatures between 300 and 470 K at pressures up to 140 MPa. These uncertainties are much smaller than those implied by previous measurements at high pressures.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jced.6b00138. Table of measured resonant periods at p = 0 for different temperatures used to determine parameters in eq 4; tables of measured resonant periods at various temperatures and pressures for nitrogen, argon, and methane used to determine the parameter values shown in Table 1.(PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +1.979.845.3384. Fax: +1.979.845.6446. Present Address †

(K.R.H.) Emeritus professor, Texas A&M University.

Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. All authors contributed equally. Funding

The authors grateful acknowledge financial support from the Jack E. and Frances Brown Chair Endowment and the Texas A&M Engineering Experiment Station. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) McGregor, D. R. A High Pressure, High Temperature Vibrating Tube Densimeter: Densities of Toluene, Ethylbenzene, and 2,2,4-trimethylpentane; Texas A&M University: College Station, TX, 1989. (2) Wagner, W.; Kleinrahm, R.; Lösch, H. W.; Watson, J. T. R.; Majer, V.; Pádua, A. A. H.; Woolf, L. A.; Holste, J. C.; De Figueiredo Palavra, A. M.; Fujii, K.; Stansfeld, J. W. Density. In Experimental Thermodynamics; Goodwin, A. R. H., Marsh, K. N., Wakeham, W. A., Eds.; Elsevier: Amsterdam, The Netherlands, 2003; Vol. 6, Chapter 5, pp 125−235. (3) Holcomb, C. D.; Outcalt, S. L. A theoretically-based calibration and evaluation procedure for vibrating-tube densimeters. Fluid Phase Equilib. 1998, 150−151, 815−827. (4) Blencoe, J. G.; Drummond, S. E.; Seitz, J. C.; Nesbitt, B. E. A vibrating-tube densimeter for fluids at high pressures and temperatures. Int. J. Thermophys. 1996, 17, 179−190. (5) Chang, R. F.; Moldover, M. R. High-temperature high-pressure oscillating tube densimeter. Rev. Sci. Instrum. 1996, 67, 251−256. (6) Ihmels, E. C.; Aufderhaar, C.; Rarey, J.; Gmehling, J. ComputerControlled Vibrating Tube Densimeter for Liquid Density Measurement in a Wide Temperature and Pressure Range. Chem. Eng. Technol. 2000, 23, 409−412. (7) Lagourette, B.; Boned, C.; Saint-Guirons, H.; Xans, P.; Zhou, H. Densimeter calibration method versus temperature and pressure. Meas. Sci. Technol. 1992, 3, 699−703. (8) May, E. F.; Tay, W. J.; Nania, M.; Aleji, A.; Al-Ghafri, S.; Martin Trusler, J. P. Physical apparatus parameters and model for vibrating tube densimeters at pressures to 140 MPa and temperatures to 473 K. Rev. 2798


Density Measurements for Ethane, Carbon Dioxide, and Methane +

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