Accurate pâÏâT Data for a Synthetic Residual Natural Gas Mixture

Article pubs.acs.org/jced

Accurate p−ρ−T Data for a Synthetic Residual Natural Gas Mixture (0.95 CH4 + 0.04 C2H6 + 0.01 C3H8) at Temperatures between (135 and 500) K at Pressures to 200 MPa Andrea Tibaduiza, Diego E. Cristancho†, Hugo Acosta Ramirez, Diego Ortiz, James C. Holste,* and Kenneth R. Hall Artie McFerrin Department of Chemical Engineering, Texas A&M University, College Station, Texas 77846-3122, United States ABSTRACT: This paper reports p−ρ−T (pressure, mass density, temperature) data measured for a mixture with molar composition (0.95039 methane + 0.03961 ethane + 0.01000 propane) using a highpressure, single-sinker, magnetic-suspension densimeter (MSD) and high- and low-pressure automated isochoric apparatus at temperatures from (135 to 500) K and pressures to 200 MPa. The composition is representative of a residual natural gas in pipelines, but the range of conditions covers conditions possibly encountered in production and processing. The k = 2 relative uncertainty for the density measurements using the MSD is 5 × 10−4·ρ based upon an uncertainty analysis for the instrument. The isothermal densities measured in the MSD in combination with the low- and high-pressure isochoric data determine additional density data with essentially the same estimated relative uncertainty as the MSD in the high temperature range (above 300 K) and a relative uncertainty of 3 × 10−3·ρ at lower temperatures. The measured densities range from (433.170 to 27.493) kg·m−3. The GERG-2008 equation of state compares well to the density data. Although the data behave in an expected manner, they cover ranges of temperature and pressure beyond those previously examined. Additionally, the paper describes a new, high-pressure isochoric apparatus as well as a methodology for compensating volume changes and the mass interchanges in the high-pressure cell. The latter technique allows experimental determination of the cell distortion coefficients.

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INTRODUCTION Accurate volumetric and isochoric experimental data for natural gas mixtures are fundamental for the development and validation of new thermodynamic models used in industry. A particularly pertinent model is the GERG-2008 equation of state (EOS). This model is probably the most widely used equation for natural gas mixtures, so it appears frequently throughout this paper. However, few high-pressure data were available to the developers. The model allows calculation of additional thermodynamic properties, such as internal and Helmholtz energies, enthalpies and entropies, to use for process design and to create a complete model for mixture properties. A relevant natural gas mixture of interest is so-called residual natural gas. Residual (or pipeline) natural gas is a principal product of a natural gas processing plant. Its importance as an energy source for industrial processes, residential and commercial uses, transportation and generation of electric power is unquestionable.1 Although its composition is variable, a ternary mixture of methane, ethane, and propane is a suitable surrogate. In a previous publication, Cristancho et al.2 reported volumetric, phase boundary, and isochoric information for the same mixture along isotherms at (300, 350, and 400) K for pressures to 170 MPa using a magnetic suspension densimeter and along isochores from (200 to 300) K for pressures to 20 © XXXX American Chemical Society

MPa using a low-pressure isochoric apparatus. This paper reports additional experimental volumetric and isochoric data that span temperatures from (135 to 500) K at pressures to 200 MPa. These data cover ranges useful in natural gas production and processing as well as for equation of state development. The Thermodynamics Laboratory at Texas A&M University has developed a new high-pressure isochoric apparatus to determine high-pressure isochoric data. The characteristics of this new apparatus appear in this paper as well as a methodology to compensate for the not strictly isochoric, nor isomolar, behavior of the system during the experimental measurements.

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EXPERIMENTAL SECTION Table 1 contains information about the residual natural gas mixture. DCG Partnership, Ltd. prepared the ternary mixture gravimetrically with a certified molar composition of 0.95039 methane, 0.03961 ethane, and 0.01000 propane with an estimated uncertainty, u(xi) = ±0.0004, that is NIST traceable Special Issue: In Honor of Kenneth R. Hall Received: February 15, 2016 Accepted: June 28, 2016

A

DOI: 10.1021/acs.jced.6b00137 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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Table 1. Residual Natural Gas Sample Informationa chemical Gas Mixture Methane Ethane Propane

source

mole fraction

DCG Partnership, Ltd.

purification

analysis method

none

GPb

0.95039 0.03961 0.01000

a

Compositions certified by the manufacturer; u(x) = 0.0004. bGP = gravimetric preparation. The mass of each component in the mixture is determined by weighing before and after the introduction of each component.

by weight. The magnetic suspension densimeter (MSD) contains a titanium sinker whose mass and volume are 30.39159 g and 6.741043 cm3 determined using the apparatus and procedure described by McLinden and Splett.3 Patil et al.4,5 describe the single-sinker MSD, and additional modifications to expand the range of measured temperature have appeared.6,7 A typical data point results from averaging five individual measurements. The apparatus requires approximately 2 h to equilibrate after changing conditions. The calibration of the standard platinum resistance thermometer (SPRT) (Minco Products model S1059PAX6) is at the fixed temperature points defined by ITS-90 using a calibrated SPRT traceable to NIST. The temperature stability is ±5 mK and the uncertainty of the SPRT is ±2 mK with respect to the triple point of water.6,7 Two Digiquartz transducers (40 and 200 MPa) from Paroscientific Inc. measure pressure. The uncertainty for these transducers is 10−4·pfs, where pfs is the full scale pressure of the respective transducer. The characteristics of the low pressure isochoric apparatus (LPI) appear in Zhou et al.8,9 who measured a nine component mixture with 91 mol % methane and n- and iso-hydrocarbons through pentane plus carbon dioxide and nitrogen at temperatures from (270 to 340) K for pressures to 20 MPa. The pressure and temperature transducers in this apparatus have the same uncertainties as those for the MSD; however, it operates only up to 20.7 MPa. A new, high-pressure isochoric apparatus also has become available. The data reported in this paper go well beyond conditions reported in the literature. At conditions that do appear in the literature, our data agree with the results reported by the best laboratories. High-Pressure Isochoric Apparatus. Cristancho10 provides details of the high-pressure isochoric apparatus (HPI), which includes several principal and ancillary instruments. Lau11 describes the design details of the isochoric cell, which is constructed of beryllium copper (Cu−Be 175).10 Pressure tests of the cell range up to 340 MPa at room temperature, and the cell volume is 10.5 cm3. Figure 1 is a cut view of the HPI that shows only the internal shield, whereas Figure 2 shows schematically the salient features. The ancillary instruments are a cylinder storage hot box, feed charging and discharging manifolds, temperature control heat exchangers around the high pressure cell, two external isothermal shields, pressure and temperature measurement systems, a compressor, a vacuum system, a heating/cooling liquid constant-temperature circulation bath, and a computer for data acquisition and control. Ejaz6 and Atilhan7 have described the cylinder storage hot box and the feed charging manifold. A system of two three-

Figure 1. Cut view of new isochoric apparatus. The inner isothermal shield is shown.

way/two stem connection valves allows use of the vacuum and the feed charging manifold as desired. A high-pressure manifold allows feeding and discharging the high-pressure isochoric cell. It consists of four high-pressure valves (30 000 psia), a high-pressure POLYPAK B-1372 (30 000 psia) hand pump with a capacity of 11 cm3 for finetuning the pressure and the tubing lines (60 000 psia rating) all from High Pressure Equipment Company (HIP). Finally, it has a high-pressure gauge for monitoring the inlet pressure to the isochoric cell. Valve V3 in Figure 2 is closed during the measurement, so the sample occupies the sample cell, the pressure transducer, the high pressure tee and the connecting lines. The tubing line between the isochoric cell and the pressure transducer has a volume less than 0.1% of the cell volume as recommended by Matabe.12 The pressure transducer is an oil-free, absolute pressure, resonating crystal transducer (Paroscientific, Inc., model 430 K-101). Location in an aluminum block allows temperature control to provide better stability. The aluminum block thermostat system includes a three-lead platinum resistance thermometer (PRT), an autotune PID temperature controller, a solid-state relay (SSR) switch, a cartridge heater (all supplied by Omega Engineering) and a variable AC power supply. The PRT used in the thermostat is a three lead, ceramic encapsulated, 100 Ω PRT (Omega model: RTD-2-1PT100 KN2528-108-T). The temperature stability achieved by this system is ±0.1 °C. Additional temperature control of the feeding line uses a simple PID control scheme with Clayborn precision heat tape. Maintaining the line temperature at 60 °C minimizes the amount of sample contained in the tubing during operation. The high-vacuum system in the isochoric apparatus consists of a mechanical vacuum pump (Welch Duo-Seal model 1402), an oil diffusion pump, and a vacuum gauge model 801 from Varian Inc. A cold trap located between the diffusion pump and the vacuum line going to the isochoric apparatus reduces the backflow of oil molecules. Temperature Control. Christancho10 provides details of the physical and digital control scheme that provides optimum temperature control for the high-pressure isochoric apparatus. The apparatus consists of an external aluminum chamber insulated with a fiberglass layer and an additional layer of spiralB



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Figure 2. Schematic view of high pressure isochoric apparatus.

Figure 3. Temperature control methodology. C



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Figure 4. Visual representation of the experimental p−T conditions: ◇, high pressure isochores; □, low pressure isochores; ●, magnetic suspension densimeter isotherms. The Peng−Robinson EOS provides the phase envelope prediction.

Robinson estimate of the boundary for informative purposes. Because the “isochoric” data is neither isochoric nor isomolar because of apparatus effects, this section contains a description of the procedure to calculate the sample density in the cell for each temperature. The densities measured with the MSD and comparisons to GERG-2008 EOS predictions (as implemented in Refprop 8.09)13 appear in Table 2. The table reports the temperatures at even values. The experimental temperatures are not exactly even, but minor adjustments (within experimental errors) to even temperatures make the table more useful. Figure 5 illustrates the deviations. This figure indicates that GERG200813 has an excellent predictive capability across the range of pressure up to 200 MPa as indicated by Cristancho et al.2 GERG-200813 predicts density data with a relative deviation of approximately 0.02% up to 200 MPa. This result is consistent with those found previously for pure component density data at high pressure.14−17 The earlier measurements covered methane, ethane, carbon dioxide, and nitrogen as pure components. Therefore, it appears that the approach developed by different authors recently13,18,19 to construct multiparameter EOS can predict pure component, high-pressure density data with excellent accuracy, at least up to 200 MPa. Table 3 contains experimental measurements made in the LPI, and Table 4 shows those made in the HPI. Because the isochoric experiment was neither truly isochoric nor isomolar, this work follows the methodology described in detail by Tibaduiza et al.20,21 to compensate for those effects and to determine the experimental isochoric densities shown in Tables 2 and 3. The procedure is described briefly in the next section. Isochoric Densities and Noxious Volume Corrections. Slight changes in the gas density along the experimental path are caused by the variation of the cell volume with temperature and pressure, and the presence of part of the sample in a volume that is external to the cell and at a different temperature. This paper terms the external volume as “noxious volume”. The noxious volume is the transmission line, high pressure tee, and pressure transducer shown in Figure 2. The procedure to determine the isochoric density begins with the definition of a reference state in pressure, temperature, and

on thermal insulated tape made of a high quality cork and synthetic rubber. This insulation provides excellent isolation for the isochoric system from the surroundings. Two heat shields, internal and external, reside between the external chamber and the isochoric cell. These shields are sources or sinks of heat for temperature control. High vacuum applied to the interior of the external chamber makes radiation the predominant mechanism for heat transfer between the shields and the isochoric cell. A four-wire standard platinum resistance thermometer (SPRT) from Minco measures the temperature at the bottom of the high-pressure isochoric cell. The measurement methodology is similar to that described by Zhou,9 Atilhan7 and Ejaz.6 The isochoric system contains four heaters (labeled as H1, H2, H3, and H4 in Figure 2). H1 and H4 control the temperature gradient across the cell. They have a separate PID control loop to keep the temperature gradient below 3 mK. At very low temperatures (around 120 K), this task becomes more difficult because of heat conduction from the feed line and the aluminum platform to the isochoric cell. Therefore, the low temperature range has a gradient of 10 mK. However, a detailed analysis proves that a 10 mK gradient does not have a significant effect upon the measured pressure. Heaters H2 and H3 are responsible for the cell temperature control. Figure 3 represents the implemented methodology for temperature control. The control scheme resides totally in LabView 8.0. The data acquisition and control occurs through a connector block SCB-68 and a data acquisition (DAQ) card PCI-6704 both from National Instrument (NI) Company. The computer uses and sends TTL (transistor-transistor-logic) signals through a PCI-DAQ card to control the on/off action of solid-state relays (SSR) that control the heater.

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RESULTS AND ANALYSIS The experimental data consist of 10 isochores and 5 isotherms covering a range of temperature from (140 to 500) K at pressures up to 200 MPa. This range of conditions far exceeds those of previous, high-accuracy measurements. Figure 4 presents the (p−T) conditions for the MSD, HPI, and LPI experiments. These data do not include measurements of the saturation boundary; however, Figure 4 contains a Peng− D



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Table 2. Experimental Values of Density ρexp at Temperature T, Pressure p, from Magnetic Suspension Densimeter and Values of Density ρGERG Calculated Using GERG-2008 EOS13 for a Single Phase Gas Mixture of Molar Composition (0.95039 CH4 + 0.03961 C2H6 + 0.01000 C3H8)a p/MPa

ρexp/kg·m−3

4.999 7.503 10.002 12.505 14.999 17.508 19.995 29.984 39.987 49.935 59.928 79.952 99.947 119.913 139.983 149.800 159.850 179.730 199.494

37.306 58.646 81.394 104.869 127.822 149.468 168.813 225.810 261.352 286.230 305.314 333.809 354.975 371.914 386.188 392.430 398.411 409.211 418.862

5.000 10.000 12.500 15.006 17.507 19.996 30.059 40.015 49.963 59.992 79.963 99.953 119.937 139.769 149.914 159.898 179.838

33.485 70.876 90.289 109.585 128.112 145.414 201.282 238.616 265.488 286.267 317.017 339.762 357.863 372.872 379.699 385.959 397.373

1.006 2.023 4.998 7.495 10.021 12.526 15.025 17.544 20.021 30.062

5.891 11.983 30.470 46.652 63.431 80.209 96.805 113.084 128.379 180.758

ρGERG/kg·m−3 T = 300.00 K 37.287 58.620 81.357 104.819 127.793 149.437 168.788 225.738 261.289 286.088 305.160 333.646 354.803 371.736 386.007 392.245 398.221 409.025 418.676 T = 325.00 K 33.474 70.845 90.261 109.539 128.058 145.355 201.210 238.539 265.346 286.116 316.863 339.608 357.713 372.709 379.537 385.808 397.223 T = 350.00 K 5.899 11.986 30.465 46.639 63.418 80.209 96.806 113.073 128.364 180.775

100·(ρexp − ρGERG)/ρexp

p/MPa

0.051 0.045 0.046 0.048 0.022 0.020 0.015 0.032 0.024 0.050 0.051 0.049 0.049 0.048 0.047 0.047 0.048 0.045 0.044 0.033 0.043 0.031 0.042 0.042 0.040 0.036 0.032 0.054 0.053 0.049 0.045 0.042 0.044 0.043 0.039 0.038 −0.132 −0.024 0.016 0.027 0.021 0.000 −0.001 0.009 0.012 −0.010

40.025 49.958 59.953 80.013 99.973 119.974 140.057 149.839 159.954 179.786 199.260

218.503 246.513 268.419 301.223 325.311 344.476 360.457 367.377 374.043 385.932 396.370

1.026 5.017 10.019 12.508 14.970 17.509 20.011 30.035 40.061 59.976 80.023 99.910 119.931 140.002

5.591 28.131 57.578 72.366 86.847 101.434 115.297 163.982 201.330 252.342 286.582 311.732 331.832 348.536

2.054 5.009 10.024 12.499 15.018 17.495 20.025 30.033 40.031 49.997 59.979 80.040 100.082 120.026 139.961 150.019

10.539 26.058 52.951 66.231 79.591 92.444 105.175 150.277 186.306 214.747 237.723 272.978 299.242 320.036 337.257 344.956

ρGERG/kg·m−3 T = 350.00 K 218.531 246.486 268.390 301.200 325.284 344.439 360.419 367.333 374.001 385.891 396.333 T = 375.00 K 5.600 28.141 57.575 72.364 86.843 101.431 115.300 163.972 201.321 252.334 286.572 311.730 331.823 348.526 T = 400.00 K 10.543 26.058 52.932 66.209 79.563 92.415 105.148 150.267 186.326 214.774 237.756 273.030 299.289 320.045 337.263 344.960

100·(ρexp − ρGERG)/ρexp −0.013 0.011 0.011 0.008 0.008 0.011 0.011 0.012 0.011 0.011 0.009 −0.160 −0.034 0.005 0.003 0.005 0.003 −0.002 0.006 0.004 0.003 0.004 0.001 0.003 0.003 −0.038 −0.001 0.036 0.033 0.036 0.031 0.025 0.007 −0.011 −0.012 −0.014 −0.019 −0.016 −0.003 −0.002 −0.001

a Standard uncertainties are u(T) = 0.01 K, u(p) = 4 kPa for p < 40 MPa, u(p) = 20 kPa for p > 40 MPa, u(x) = 0.0004, and the combined expanded uncertainty Uc is Uc(ρ) = 0.1 kg·m−3 + 0.001·ρ (0.95 level of confidence).

density followed by the isochoric density determination for each isochore. Definition of Reference State. The total number of moles in the isochoric experiment must be constant in the absence of leaks. The total number of moles in the closed system shown in Figure 2 is n = nc + nl + nt

ρexp/kg·m−3

or in terms of density and volume ρV = ρc Vc + ρl Vl + ρt Vt

(2)

The subscripts c, l, and t in eqs 1 and 2 refer to the cell, connection line, and pressure transducer, respectively. The density on the left-hand side of eq 2 requires a reference state. Using the data from the MSD, it is possible to determine a (p−ρ−T) value that corresponds to a (p−T) point on an

(1) E



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Writing eq 8 in terms of density and volume ρc Vc (ρ*Vt* − ρt Vt) (ρ *Vl * − ρl Vl ) =1+ l + t ρ*Vc* ρ*Vc* ρ*Vc* c

ρc =

Vc* V V ρ* + l (ρl* − ρl ) + t (ρt* − ρt ) Vc c Vc Vc

V = V *exp(α(T − T *) + β(P − P*))

(10)

(11)

The parameters α and β in eq 11 are the volumetric thermal expansion α=

1 ⎛⎜ ∂V ⎞⎟ V ⎝ ∂T ⎠ P

(12)

and isothermal compressibility

GERG 200813 serves as the interpolating EOS for eq 3. Figure 5 shows the relative deviations in density for the five isotherms measured in the MSD. This information along with eq 4 enables calculation of the reference density (ρ*) ρEOS ρ* = 1 − δρrel (4)

β=−

1 ⎛⎜ ∂V ⎞⎟ V ⎝ ∂P ⎠T

(13)

Table 6 contains the values of α and β for stainless steel and Cu−Be reported in the literature.10,11,22 When using eqs 10 and 11 to determine the isochoric densities, the noxious volume and the pressure transducer contributions must be treated separately because of the different operating temperatures. Isochoric Densities for Low-Pressure Isochores. The noxious volume in the LPI is mostly the volume of the pressure transducer. The assumption is that the contribution of the transmission line is negligible because the line is at the same constant temperature as the pressure transducer, and its volume is small compared to those of the cell and pressure transducer. Table 7 presents information about the materials of construction and the physical characteristics of the cell which shows that the noxious volume for the LPI is approximately 0.34% of the volume of the cell. In effect, Vl ≈ V*l and Vt ≈ V*t . Therefore, calculation of the densities for isochores 6 to 10 is

Because the reference state is quite close to measured values, errors introduced by the use of the GERG EOS to establish the reference state are much smaller than the measurement errors.21 Table 5 summarizes the reference condition values for the ten isochores measured in the high- and low-pressure isochoric apparatus for the residual gas sample. Isochoric Density Determination. This section shows the algebraic derivations, assumptions and equations needed to determine the isochoric density, which is the density of gas inside the isochoric cell when at equilibrium conditions in (p− T). The isomolar characteristic of the isochoric experiment and the reference state can correct for the noxious volume effect when calculating the isochoric density. From eq 1, the number of moles in the cell is nc = n − nl − nt (5)

ρc =

Replacing the total number of moles at reference conditions

Vc* V ρ* + t (ρt* − ρt ) Vc c Vc

(14)

Validation of the parameters α and β for stainless steel reported in Table 6 requires two assumptions: the total number of moles for a single isochore is constant, and values from the MSD and GERG 200813 can provide the density in the cell and in the pressure transducer, respectively. This process introduces insignificant errors because the pressure transducer is much smaller than the cell. A system of three equations with three unknowns (n, α, and β) results from choosing three different points (p−T) on an isochore, and determining their densities from the MSD data. Table 8 reports the values of n, α, and β

(6)

Organizing the terms in eq 6 for the transmission line and pressure transducer (7)

Dividing eq 7 by the moles in the cell at reference conditions nc (n * − n t ) (n * − nl) =1+ l + t * * nc nc nc*

(9)

The volume for each subsystem in eq 10 has the general form

isochore. The reference state, denoted by (p*−ρ*−T*) for each isochore is the starting point for determining the isochoric density for each (p−T) pair on its isochoric line. The reference density results from using the relative deviation in density values from the MSD ρ − ρEOS δρrel = MSD ρMSD (3)

nc = nc* + (nl* − nl) + (nt* − nt)

c

The assumption in the determination of isochoric density is that the main contributions to the mass transport in the system are the expansion and contraction of the volume of the cell caused by varying temperature and pressure and the variation of pressure in the connecting lines. Furthermore, the temperatures of the connecting lines and pressure transducer are constant during the isochoric experiment. Therefore, the changes in volume of the noxious volume and the pressure transducer are negligible compared to the changes in the volume of the cell, and eq 9 solved for the isochoric density of the cell (ρc) becomes

Figure 5. Relative deviations of experimental densities measured using the magnetic suspension densimeter (Table 2) from GERG-200813 EOS: ●, T = 300 K; △, T = 325 K; ■, T = 350 K; ○, T = 375 K; ◆, T = 400 K.

nc = (nc* + nl* + nt*) − nl − nt

c

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Table 3. Experimental Values of Temperature T and Pressure p, and Derived Values of Density ρ from the Low-Pressure Isochoric Apparatus, and Values of Density ρEOS Calculated Using GERG-2008 EOS13 for a Single Phase Gas Mixture of Molar Composition (0.95039 CH4 + 0.03961 C2H6 + 0.01000 C3H8)a T/K

p/MPa

300.00 290.00 280.00 270.00 260.00 250.00 240.00 230.00 220.00 210.00 205.00

20.567 19.023 17.471 15.914 14.361 12.795 11.229 9.666 8.109 6.534 5.806

420.00 400.00 390.00 380.00 370.00 360.00 350.00 340.00 330.00 320.00 310.00 300.00 290.00 280.00 270.00 260.00 250.00 240.00 230.00 220.00

20.120 18.790 18.128 17.461 16.791 16.116 15.446 14.766 14.074 13.376 12.677 11.962 11.268 10.559 9.863 9.136 8.413 7.691 6.953 6.193

450.00 440.00 430.00 420.00 410.00 400.00 390.00 380.00 370.00 360.00 350.00 340.00 330.00 320.00 310.00 300.00 290.00 280.00 270.00 260.00 250.00 240.00 230.00

10.683 10.406 10.128 9.850 9.527 9.291 9.010 8.727 8.444 8.126 7.871 7.584 7.297 7.008 6.717 6.421 6.130 5.841 5.549 5.243 4.944 4.644 4.336

ρexp/kg·m−3

ρEOS/kg·m−3

Isochore 6 172.942 172.917 173.065 173.066 173.190 173.200 173.315 173.335 173.441 173.568 173.569 173.698 173.696 173.851 173.824 174.067 173.951 174.411 174.079 173.006 174.141 176.458 Isochore 7 99.115 98.833 99.241 98.989 99.305 99.103 99.369 99.208 99.434 99.315 99.498 99.413 99.563 99.565 99.629 99.678 99.694 99.731 99.761 99.759 99.827 99.802 99.894 99.723 99.960 99.872 100.027 99.915 100.093 100.176 100.160 100.112 100.228 100.170 100.295 100.376 100.362 100.442 100.430 100.106 Isochore 8 48.825 48.820 48.855 48.852 48.885 48.884 48.915 48.920 48.947 48.726 48.976 48.988 49.007 49.021 49.037 49.048 49.068 49.080 49.099 48.896 49.129 49.116 49.160 49.139 49.190 49.170 49.221 49.194 49.252 49.211 49.283 49.196 49.314 49.233 49.344 49.307 49.375 49.375 49.406 49.313 49.437 49.350 49.469 49.412 49.500 49.408

100·(ρexp − ρEOS)/ρexp

T/K

0.015 −0.001 −0.006 −0.011 −0.073 −0.075 −0.089 −0.140 −0.264 0.616 −1.330 0.284 0.255 0.203 0.162 0.119 0.086 −0.001 −0.050 −0.037 0.001 0.025 0.171 0.088 0.112 −0.082 0.048 0.057 −0.081 −0.079 0.323 0.010 0.005 0.002 −0.009 0.451 −0.024 −0.030 −0.022 −0.024 0.413 0.027 0.042 0.042 0.055 0.083 0.177 0.163 0.076 0.001 0.189 0.177 0.114 0.185

p/MPa

220.00 210.00

4.029 3.737

500.00 490.00 480.00 470.00 460.00 450.00 440.00 430.00 420.00 410.00 400.00 390.00 380.00 370.00 360.00 350.00 340.00 330.00 320.00 310.00 300.00 290.00 280.00 270.00 260.00 250.00 240.00 230.00 220.00 215.00 210.00 205.00

9.625 9.414 9.202 8.990 8.777 8.562 8.348 8.133 7.916 7.699 7.481 7.263 7.044 6.825 6.605 6.383 6.161 5.938 5.715 5.490 5.262 5.038 4.812 4.585 4.353 4.122 3.891 3.663 3.455 3.340 3.193 3.088

370.00 360.00 350.00 340.00 330.01 320.00 310.00 305.15 300.00 290.00 280.00 270.00 260.00 240.00 220.00 210.01

4.822 4.682 4.534 4.384 4.214 4.084 3.939 3.868 3.788 3.639 3.489 3.336 3.182 2.874 2.561 2.402

ρexp/kg·m−3

ρEOS/kg·m−3

Isochore 8 49.531 49.479 49.562 49.956 Isochore 9 39.023 38.987 39.046 39.018 39.070 39.047 39.094 39.079 39.118 39.109 39.142 39.133 39.166 39.163 39.190 39.191 39.214 39.213 39.238 39.237 39.262 39.259 39.286 39.284 39.310 39.306 39.334 39.332 39.359 39.357 39.383 39.373 39.407 39.393 39.431 39.411 39.456 39.435 39.480 39.450 39.505 39.447 39.529 39.485 39.553 39.516 39.578 39.551 39.602 39.550 39.627 39.574 39.651 39.622 39.676 39.745 39.700 40.231 39.712 40.357 39.725 39.967 39.737 40.333 Isochore 10 27.493 27.447 27.509 27.506 27.526 27.521 27.543 27.526 27.560 27.394 27.576 27.541 27.593 27.591 27.601 27.612 27.609 27.600 27.626 27.630 27.643 27.659 27.660 27.667 27.677 27.673 27.710 27.708 27.744 27.740 27.761 27.750

100·(ρexp − ρEOS)/ρexp 0.104 −0.794 0.092 0.073 0.058 0.038 0.022 0.024 0.007 −0.004 0.003 0.003 0.009 0.007 0.011 0.005 0.005 0.026 0.036 0.052 0.053 0.077 0.146 0.112 0.094 0.067 0.133 0.132 0.074 −0.173 −1.337 −1.624 −0.610 −1.500 0.168 0.011 0.017 0.061 0.600 0.126 0.008 −0.040 0.033 −0.015 −0.057 −0.027 0.014 0.007 0.016 0.042

a

Standard uncertainties are u(T) = 0.01 K, u(p) = 2 kPa and the combined expanded uncertainty Uc is Uc(ρ) = 0.1 kg·m−3 + 0.001·ρ for T > 300 K and Uc(ρ) = 0.1 kg·m−3 + 0.003·ρ for T < 300 K (0.95 level of confidence).

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Table 4. Experimental Values of Temperature T and Pressure p, and Derived Values of Density ρ from the High-Pressure Isochoric Apparatus, and Values of Density ρEOS Calculated Using GERG-2008 EOS13 for a Single Phase Gas Mixture of Molar Composition (0.95039 CH4 + 0.03961 C2H6 + 0.01000 C3H8)a T/K

p/MPa

300.00 295.00 290.00 285.00 280.00 275.00 270.00 263.06 259.99 250.00 240.00 230.00 220.00 210.00 200.00 190.00 180.00 170.00 160.00 150.00

199.871 195.624 191.147 186.633 182.090 177.422 172.833 166.346 163.152 153.629 143.910 134.009 123.882 113.535 103.186 92.171 81.107 69.828 58.336 46.673

390.00 380.00 375.00 370.00 360.00 350.00 340.00 330.00 325.00 320.00 310.00 300.00 290.00 280.00 270.00 260.00 250.00 240.00 230.00 220.00 210.00 200.00 190.00 180.00 170.00 160.00 150.00

200.509 194.055 190.822 187.502 180.858 174.109 167.260 160.301 156.694 153.255 146.102 138.909 131.404 123.913 116.283 108.572 100.712 92.716 84.574 76.296 67.890 59.332 50.644 41.844 32.998 24.230 15.612

450.00 440.00 430.00 420.00 410.00 400.00 390.00 380.00

193.800 188.558 183.199 177.828 172.351 166.951 161.350 155.672

ρexp/kg·m−3

ρEOS/kg·m−3

Isochore 1 419.037 418.851 419.424 419.219 419.817 419.500 420.213 419.785 420.612 420.080 421.015 420.336 421.418 420.656 421.984 421.084 422.245 421.124 423.076 421.771 423.923 422.439 424.789 423.143 425.676 423.863 426.588 424.619 427.523 425.555 428.512 426.281 429.544 427.194 430.641 428.217 431.833 429.385 433.170 430.769 Isochore 2 379.994 380.661 380.655 381.150 380.986 381.411 381.321 381.638 381.993 382.134 382.671 382.630 383.356 383.133 384.049 383.639 384.400 383.840 384.749 384.164 385.458 384.702 386.174 385.298 386.906 385.778 387.647 386.363 388.401 386.956 389.170 387.606 389.956 388.272 390.765 388.972 391.599 389.708 392.465 390.502 393.371 391.377 394.332 392.333 395.368 393.418 396.516 394.686 397.835 396.265 399.427 398.383 401.402 401.259 Isochore 3 352.941 354.461 353.533 354.909 354.131 355.327 354.731 355.780 355.337 356.212 355.945 356.744 356.560 357.195 357.181 357.647

100·(ρexp − ρEOS)/ρexp

T/K

0.044 0.049 0.076 0.102 0.126 0.161 0.181 0.213 0.265 0.308 0.350 0.387 0.426 0.462 0.460 0.521 0.547 0.563 0.567 0.554 −0.176 −0.130 −0.111 −0.083 −0.037 0.011 0.058 0.107 0.146 0.152 0.196 0.227 0.292 0.331 0.372 0.402 0.432 0.459 0.483 0.500 0.507 0.507 0.493 0.461 0.395 0.261 0.036 −0.431 −0.389 −0.338 −0.296 −0.246 −0.224 −0.178 −0.131

p/MPa

370.00 360.00 350.00 340.00 330.00 325.00 320.00 310.00 300.00 290.00 280.00 275.00 270.00 260.00 250.00 240.00 230.00 220.00 210.00 200.00 190.00 180.00 170.00 160.00

149.924 144.085 138.193 132.236 126.204 123.137 120.066 113.859 107.927 101.518 95.010 91.387 88.411 81.090 74.622 67.751 60.808 53.767 46.602 39.369 32.061 24.878 17.926 10.359

450.00 440.00 430.00 420.00 410.00 400.00 390.00 380.00 370.00 360.00 350.00 340.00 330.00 320.00 310.00

146.879 142.519 138.102 133.630 129.127 124.565 119.962 115.312 110.611 105.850 101.041 96.175 91.252 86.267 81.228

300.00 290.00 280.00 270.00 260.00 250.00 240.00 230.00 220.00 210.00 200.00

35.793 32.993 30.162 27.306 24.432 21.523 18.583 15.612 12.599 9.558 6.517

ρexp/kg·m−3

ρEOS/kg·m−3

Isochore 3 357.808 358.108 358.441 358.563 359.081 359.043 359.727 359.543 360.382 360.058 360.713 360.304 361.046 360.567 361.719 361.104 362.392 361.970 363.089 362.533 363.801 363.112 364.176 363.095 364.530 363.721 365.303 363.732 366.063 364.730 366.865 365.471 367.702 366.309 368.587 367.235 369.540 368.238 370.585 369.421 371.770 370.822 373.144 372.819 374.758 375.738 376.726 378.293 Isochore 4 320.809 322.241 321.348 322.638 321.893 323.029 322.442 323.418 322.995 323.826 323.554 324.230 324.118 324.648 324.688 325.076 325.264 325.515 325.848 325.955 326.439 326.413 327.039 326.881 327.649 327.362 328.270 327.856 328.903 328.373 Isochore 5 248.183 248.112 248.781 248.827 249.412 249.581 250.082 250.407 250.798 251.358 251.568 252.375 252.396 253.493 253.281 254.727 254.209 255.982 255.154 257.272 256.083 258.634

100·(ρexp − ρEOS)/ρexp −0.084 −0.034 0.011 0.051 0.090 0.113 0.133 0.170 0.116 0.153 0.189 0.297 0.222 0.430 0.364 0.380 0.379 0.367 0.352 0.314 0.255 0.087 −0.262 −0.416 −0.447 −0.401 −0.353 −0.303 −0.257 −0.209 −0.163 −0.120 −0.077 −0.033 0.008 0.049 0.088 0.126 0.161 0.029 −0.018 −0.068 −0.130 −0.224 −0.321 −0.434 −0.571 −0.697 −0.830 −0.996

a Standard uncertainties are u(T) = 0.01 K, u(p) = 20 kPa and the combined expanded uncertainty Uc is Uc(ρ) = 0.1 kg·m−3 + 0.001·ρ for T > 300 K and Uc(ρ) = 0.1 kg·m−3 + 0.003·ρ for T < 300 K (0.95 level of confidence).

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Article

Table 5. p−ρ−T Reference Values for Each Isochorea isochore

T*/K

p*/MPa

δρrel

ρEOS/kg·m−3

ρ*/kg·m−3

1 2 3 4 5 6 7 8 9 10

300.00 350.00 350.00 350.00 300.00 300.00 350.00 350.00 350.00 350.00

199.871 174.109 138.193 101.041 35.793 20.567 15.446 7.871 6.383 4.534

0.044 0.011 0.011 0.008 0.028 0.015 −0.001 0.027 0.022 0.016

418.852 382.630 359.043 326.413 248.113 172.917 99.564 49.116 39.374 27.522

419.037 382.671 359.081 326.439 248.183 172.942 99.563 49.129 39.383 27.526

a

EOS values from GERG-2008.13 Figure 6. Relative deviations of experimental densities measured using the low pressure isochoric apparatus (Table 3) from GERG-200813 EOS: ●, isochore 6; △, isochore 7; ■, isochore 8; ○, isochore 9; ◆, isochore 10.

Table 6. Thermal Expansion and Isothermal Compressibility Coefficients from Previous Work stainless steel parameter

Stouffer

−1

10 α/K 105 β/MPa−1 5

4.86 2.53

22

beryllium copper Lau

11

0.185 3.37

Cristancho10

Table 9 reports the lengths of the each segment of the transmission line and the corresponding volumes. Table 10

160 2.53

Table 9. Dimensions of Connecting Lines in the High Pressure Isochoric Apparatus

Table 7. Physical Characteristics and Dimensions of the Low Pressure Isochoric Apparatus isochoric cell

transmission line

pressure transducer (43K-101)

material: volume: temperature: material: volume: temperature: material: volume: temperature:

stainless steel 6.01 × 10−5 m3 200−500 K stainless steel 7.32 × 10−9 m3 350.45 K stainless steel 2.05 × 10−7 m3 350.45 K

segment

L/cm

V/cm3

transducer to HIP cross HIP cross to HIP valve HIP cross to aluminum plate aluminum plate to isochoric cell total

7.62 2.54 19.69 12.7 42.55

0.0154 0.0051 0.0399 0.0257 0.0862

Table 10. Technical Specifications for High Pressure Cross and Valve Manufactured by HIP Co.

Table 8. Parameters for Low Pressure Isochoric Cell isochore

n/mol

105 α/K−1

105 β/MPa−1

7 8 9

0.355258 0.175462 0.140664

4.88 4.86 4.86

2.55 2.53 2.53

high pressure cross model: working pressure: connection: length (e): volume:

60-24HF2 60 000 psi tube 1/8 in. OD 1.5 in. = 3.81 cm 0.0077 cm3 High Pressure Valve

obtained by solving the system of equations for isochores 7 to 9, which correspond to the LPI. The values of α and β found for stainless steel agree with those reported by Stouffer.22 Because α and β are properties of the materials of construction of the cell, they do not depend upon sample composition. Therefore, in this paper, eqs 11 and 14 along with the parameters for stainless steel reported in Table 6 are used to calculate the isochoric densities for the LPI. Figure 6 shows the relative deviation of the isochoric densities in the LPI compared to GERG 2008.13 The measurements at T = (360 and 410) K for isochore 8 and at T = 320 K for isochore 10 appear to be outliers. We have chosen to report these measurements, but they are suspect. Isochoric Densities for High-Pressure Isochores. Quantification of the noxious volume and new values of α and β for the high-pressure isochoric apparatus are new contributions from this work. To determine the noxious volume, it is necessary to estimate the volume of the transmission line that connects the isochoric cell to the pressure transducer. This line is stainless steel and has a working pressure of 60 000 psi, 1/8 in. OD and 0.020 in. ID.

model: working pressure: connection: length (e): volume:

30-11HF2 30 000 psi tube 1/8 in. OD 1.5 in. = 3.81 cm 0.0077 cm3

provides similar information about the high-pressure cross and valve manufactured by HIP Co. The pressure transducer (Paroscientific Inc. Model 430 K-101) has a maximum working pressure of 207 MPa and an internal volume of 0.142 cm3. Table 11 summarizes information about the materials of construction and the physical characteristics of the apparatus. The starting values in the fitting routine for determining the parameters α and β for the Cu−Be cell are those calculated by Lau11 and reported in Table 6. The noxious volume for the HPI corresponds to 2.3% of the volume of the cell, and the volume of the transmission line is approximately equal to the volume of the pressure transducer. In addition, the temperature of operation is different for the two systems. Consequently, both contributions are important I



■

CONCLUSIONS This paper reports accurate experimental p−ρ−T data for a residual natural gas mixture using a high-pressure, single-sinker MSD having experimental uncertainty less than 5 × 10−4·ρ and high- and low-pressure isochoric apparatus with essentially the same estimated relative uncertainty as the MSD in the high temperature range (above 300 K) and a relative uncertainty of 3 × 10−3·ρ in the low temperature range. By combining the isothermal data from the MSD and the isochoric data from the low- and high-pressure isochoric apparatus and the corrections for the volume change and mass interchange in the gas cell, it is possible to determined isochoric densities experimentally with a reasonable experimental error. Finally, the paper contains a description of and presents data from a new high-pressure isochoric apparatus, capable of measuring experimental data in the range of temperatures from (135 to 500) K at pressures up to 200 MPa. The results presented here provide experimental measurements for a model mixture representing natural gases with high methane content. The experimental values are useful for validation and development of the EOS used for design of practical processes that involve high pressures, such as ultra deep water drilling and low temperature gas liquefaction. These results extend to much higher pressures than any existing data of comparable accuracy.

Table 11. Physical Characteristics and Dimensions of the High Pressure Isochoric Apparatus cell

transmission line

pressure transducer (430K-101)

material: beryllium copper 175 volume: 1.05 × 10−5 m3 temperature of operation: 200−500 K material: stainless steel volume: 1.02 × 10−7 m3 temperature of operation: 333.15 K material: stainless steel volume: 1.42 × 10−7 m3 temperature of operation: 309.15 K

when calculating the noxious volume and isochoric densities for the HPI. Because of the broad ranges in temperature and pressure in the HPI and the desire to determine the isochoric densities as accurately as possible, it is imperative to consider the entire noxious volume and to determine new distortion parameters for this particular experiment. The procedure for calculating the parameters α and β is the same as that explained for the LPI isochores. Table 12 has the values found for n, α, and β. The Table 12. Parameters for High Pressure Isochoric Cell isochore 2 3 4 average

n/mol

104 α/K−1

105 β/ MPa−1

0.246437 0.231301 0.209711

1.240 1.520 1.220 1.327

3.370 3.380 3.370 3.373

Article

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +1.979.845.33357. Fax: + 1.979.845.6446.

average in the values of α and β for isochores 2, 3, and 4 determine the isochoric density along with eqs 10 and 11 and the parameters for stainless steel. Figure 7 shows the percentage

Present Addresses

(D.E.C.) DOW Chemical, Freeport, Texas 77541, United States. (K.R.H.) Professor emeritus, Texas A&M University, College Station, Texas 77843, United States. Author Contributions

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

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

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors gratefully acknowledge support for this work from the Jack E. & Frances Brown Chair endowment and from the Texas Engineering Experiment Station.

Figure 7. Relative deviations of experimental densities measured using the high pressure isochoric apparatus (Table 4) from GERG-200813 EOS: ●, isochore 1; △, isochore 2; □, isochore 3; *, isochore 4; ◆, isochore 5.

■

ABBREVIATIONS MSD, magnetic suspension densimeter; LPI, low pressure isochoric apparatus; HPI, high pressure isochoric apparatus; EOS, equation of state

deviation of the isochoric densities from the HPI with respect to GERG 2008.13 The parameters for Be−Cu and the effect of the noxious volume allow calculation of the gas densities from isochoric (p−T) data with an uncertainty of ±3 × 10−3·ρ at temperatures from 150 to 450 K at pressures up to 200 MPa. The numerical values for the isochoric densities from the HPI using the new parameters α and β appear in Table 4.

■

REFERENCES

(1) Annual Energy Review 2009, Markets and End Use; Office of Energy Information Administration, U.S. Department of Energy Publication: Washington, DC, 2009. (2) Cristancho, D. E.; Mantilla, I. D.; Coy, L. A.; Tibaduiza, A.; OrtizVega, D. O.; Hall, K. R.; Iglesias-Silva, G. A. Accurate P-ρ-T Data and

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