Speed of Sound, Density, and Derivative Properties of Tris(2

Article pubs.acs.org/jced

Speed of Sound, Density, and Derivative Properties of Tris(2-ethylhexyl) Trimellitate under High Pressure Jean-Patrick Bazile, Djamel Nasri, and Jean-Luc Daridon* Laboratoire des Fluides Complexes et leurs Réservoirs-IPRA, UMR5150, CNRS/TOTAL/Univ Pau & Pays Adour, 64000, Pau, France

ABSTRACT: Speed of sound measurements were carried out in liquid tris(2-ethylhexyl) trimellitate (TOTM) at pressures up to 200 MPa along isotherms ranging from (293.15 to 373.15) K in order to report data of density and its derivatives. The measurements were performed with a pulse echo technique working at 3 MHz. Additional density and heat capacity measurements were carried out at atmospheric pressure in order to initiate the estimation procedure of volumetric properties from speed of sound integration. Results were compared with density measurements performed with a U-tube densimeter at pressures up to 140 MPa on the same sample. Finally, an equation that correlates density and its derivatives within their estimated uncertainties is given.

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INTRODUCTION Because of the considerable progress achieved in drilling techniques more and more wells are drilled at greater depths than conventional reservoirs which result in higher temperatures as well as higher pressures in downhole environments. Consequently, alongside conventional sources of oil and gas resources, specific reservoirs such as hyperbaric (70−100 MPa), extreme high pressure (100−150 MPa) and ultra-high-pressure reservoirs (above 150 MPa) are now increasingly investigated. The development of fields with high pressure poses scientific and technical challenges in most of the area covered by field operations. In particular, measurement of fluid properties under high pressure requires reference fluids that could be used for calibration of industrial devices that works in relative way. With the aim to propose an industrial viscosity standard for high pressure−high viscosity working fluids, viscosity measurements were recently carried out in tris(2-ethylhexyl) ester (TOTM) by several authors.1−4 For such high pressure− high viscosity reference fluids, volumetric properties must also be known in an extended pressure range because density is needed in most working equations of viscometers. Furthermore, the knowledge of both density and its derivatives with respect to pressure provides experimental information which is required to develop models able to predict thermophysical properties of fluids under pressure. Thus, the aim of this work is to provide a full characterization of volumetric properties of TOTM in the pressure range 0.1 to 200 MPa that covers © XXXX American Chemical Society

most petroleum reservoir applications. On the basis of high pressure speed of sound measurements and additional density and heat capacity data determined at atmospheric pressure, the density of liquid TOTM was determined up to 200 MPa using a numerical integration method. By a combination of density and speed of sound data, the isentropic compressibility was also determined in the same pressure range. Moreover the isothermal compressibility was evaluated from these measurements.

■

EXPERIMENTAL SECTION Materials. The liquid TOTM (CAS-No. 3319-31-1) used in the present work was purchased from Sigma-Aldrich with a nominal minimum purity of 99%. It was used for speed of sound measurement without any further treatment. Table 1 shows the details of the sample description. Table 1. Sample Description chemical name tris(2-ethylhexyl) ester

shorthand designation TOTM

source

purity

SigmaAldrich

0.99

lot number

purification method

MKBT 5164 V

none

Received: February 13, 2017 Accepted: April 7, 2017

A

DOI: 10.1021/acs.jced.7b00162 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

Article

Speed of Sound Measurement. The speed of sound was measured using a pulse-echo technique working in the reflection mode. The apparatus is essentially made up of a double path sensor (Figure 1) composed of a piezoelectric disk and two reflectors.

Measurements were carried out by a double path technique initially proposed by Muringer et al.5 because this method is more appropriate to viscous and dissipative liquids than the successive echoes technique as the second echo travels along a shorter path length. In this technique, the piezoelectric disk emits two sound waves that propagate in opposite directions. After reflection, both echoes come back to the transducer with a time delay Δt caused by the difference in path length. Measurement of this time delay allows determining the speed of sound according to the following relation:

w=

2ΔL Δt

(1)

where ΔL is the difference between the path lengths of the two echoes. This equation ignores the diffraction effect created by the finite size of the piezoelectric disk as this correction accounts for less than 0.005% of the measured time delay for the specific geometry of the acoustic device used here. The time delay Δt is determined by a digital overlap method in which both echoes are recorded and digitalized by a digital oscilloscope with a sampling rate of 1.GS/s and a record length of 2500 points for all time bases. After recording, the first echo is translated along the temporal axis so as it overlaps the second echo and the difference between both signals becomes minimal. During this time translation, the amplitude of the first echo is numerically decreased by multiplying it by an adjustable coefficient in order to compensate the disparity in dissipation caused by the difference in path length and therefore to allow a perfect overlapping. The accurate time delay between echoes is estimated by determining the time translation that minimizes the difference between both echoes. The difference in path length ΔL is calibrated with water as for this reference component reliable data were reported.6−9 During calibration, the effects of dilation and compression on the acoustic wave sensor were taken into account by considering a linear behavior in both temperature and pressure:

Figure 1. Schematic diagram of the acoustic wave sensor: (1) stainless steel reflector; (2) short length cylinder; (3) electric insulator ring; (4) piezoelectric ceramic disc with its electrical connection; (5) long length cylinder; (6) stainless steel reflector.

The piezoelectric element, with a resonant frequency of 3 MHz, is mounted between two hollow cylindrical supports of different lengths. The reflectors are arranged facing the piezoelectric disk at both ends of the cylindrical supports. The different parts are held together by stainless steel screws fixed in axial holes drilled through the walls of the supports. The acoustic sensor is enclosed in an autoclave cell filled with the studied liquid and connected to a volumetric pump whereby pressure is produced (Figure 2). Two pressure transducers (HBM) are mounted between the high pressure vessel and the pump. One is calibrated in the full pressure scale 0.1 to 200 MPa with an uncertainty of 0.1 MPa, whereas the other is only calibrated between 0.1 and 100 MPa in order to achieve a lower uncertainty (0.01 MPa) in this range. The high pressure vessel is itself housed within a thermostatic bath of stability 0.02 K. A temperature probe made up of a Pt 100 of 1.2 mm diameter insert in a metal finger is placed inside the cell. With this probe, temperature is measured with an uncertainty less than 0.1 K. The pressure vessel is closed at the top by a cap in which three hermetic feedthrough assemblies were accommodated to wire the acoustic sensor to an ultrasonic emission/ reception device.

ΔL(p , T ) = ΔL0[1 + a(T − T0)]·[1 + b(p − p0 )]

(2)

The path length difference ΔL0 at the reference temperature (T0 = 293.15 K) and pressure (P0 = 0.1013 MPa) as well as a and b parameters are estimated by calibration with water. The uncertainty of the path length calibrated in this way is evaluated to 0.02% by combining the uncertainty in speed of sound data with uncertainty measurements involved in the calibration (Δtwater, T, p).

Figure 2. Schematic diagram of speed of sound experimental setup: (1) high pressure volumetric pump; (2) pressure sensor (100 MPa); (3) pressure sensor (200 MPa); (4) cell inlet valve; (5) high pressure vessel; (6) acoustic wave sensor ; (7) thermostatic bath ; (8) temperature sensor inserted in a metal finger; (9) electrical feedthrough. B



Article

Table 2. Experimental Values of Speed of Sound w at Temperatures T and Pressures p for the Liquid TOTMa p

T

w

T

w

T

w

T

w

T

w

MPa

K

m·s−1

K

m·s−1

K

m·s−1

K

m·s−1

K

m·s−1

0.1013 10 20 30 40 50 60 70 80 90 100 120 140.1 160.1 180.1 200.1 0.1013 10 20 30 40 50 60 70 80 90 100 120 140.1 160.1 180.1 200.1

293.05 293.05 293.05 293.05 293.05

1421.3 1466.0 1507.2 1545.0 1580.3

303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15

1385.4 1430.5 1473.7 1513.3 1548.7 1583.6 1617.5 1649.8

313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35

1352.8 1396.8 1441.2 1480.4 1518.6 1553.8 1587.7 1621.3 1651.5 1682.7 1712.8 1769.5

323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45

1319.3 1365.3 1410.1 1450.4 1488.3 1525.4 1560.4 1593.6 1626.3 1656.1 1685.2 1740.7 1796.0 1846.7 1896.3

333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55

1286.6 1334.6 1380.2 1421.5 1460.8 1497.8 1533.6 1567.4 1599.7 1631.7 1661.2 1718.9 1773.1 1827.1 1876.8 1918.9

343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75

1255.2 1304.6 1351.5 1394.3 1435.8 1473.9 1509.7 1543.2 1576.1 1606.9 1637.9 1693.4 1746.9 1800.3 1849.1 1897.6

353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85

1223.4 1275.3 1323.4 1367.7 1409.0 1447.4 1483.9 1519.4 1553.0 1585.0 1615.2 1674.1 1727.9 1779.5 1831.0 1881.0

364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15

1193.4 1246.7 1295.6 1341.9 1383.1 1422.8 1460.6 1495.8 1529.5 1562.4 1593.8 1651.8 1706.1 1758.6 1810.3 1856.9

374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25

1164.3 1218.4 1268.4 1315.7 1357.8 1399.1 1437.8 1473.9 1508.5 1541.0 1572.3 1632.1 1686.4 1737.9 1788.7 1837.5

a Standard uncertainties u are u(T) = 0.1 K, u(p) = 0.01 MPa up to 100 MPa, u(p) = 0.1 MPa between (100 and 200) MPa and the combined expanded uncertainties Uc (level of confidence = 0.95, k = 2) are Uc(w) = 0.0006 w up to 100 MPa, Uc(w) = 0.002 w between (100 and 200) MPa.

This uncertainty combined with uncertainty in measurement of time, temperature, and pressure in liquid TOTM leads to a combined uncertainty of 0.03% for speed of sound measurements up to 100 MPa according to the error propagation law. By considering the conventional coverage factor kP = 2 (P = 95%), the expanded uncertainty for speed of sound measurements is therefore 0.06% between (0.1 and 100) MPa. Density Measurement. Density was experimentally determined by two methods. The first one is based on the measurement of the period of oscillation of a U-shape tube (DMA HPM Anton Paar) which related to density by a linear law in terms of the square of the period. The parameters of this linear function are calibrated by the method proposed by Comuñas et al.10 using vacuum and a liquid of known densities. Water11 and decane12 are used as reference fluid depending on the (p,T) domain investigated. As TOTM is highly viscous in the pressure temperature range investigated, a correction, corresponding to 0.04% in average, must be applied to the value provided by the linear law in order to take into account viscous friction on the period of vibration of the U-tube oscillator.13 Taking into account the uncertainty in temperature, pressure, and density of the reference fluid as well as the error in the measurements of the periods of oscillation, the combined uncertainty is estimated to be 0.06% at atmospheric pressure and 0.1% at higher pressures. The second method, based on speed of sound measurements,

Figure 3. Speed of sound w of liquid TOTM as a function of pressure along various isotherms: blue □, 293.05 K; ◆, 313.35 K; red ○, 333.55 K; green ▲, 353.85 K; ●, 374.25K.

consists of integrating the set of two differential equations with respect to pressure at fixed temperature: C



Article

Table 3. Values of Densities ρ at Temperatures T and Pressures p Measured in Liquid TOTM by Using U-Tube Densimetera p

T

ρ

T

ρ

T

ρ

T

ρ

T

ρ

MPa

K

kg·m−3

K

kg·m−3

K

kg·m−3

K

kg·m−3

K

kg·m−3

0.1013 10 20 30 40 50 60 70 80 90 100 110 120 130 140 0.1013 10 20 30 40 50 60 70 80 90 100 110 120 130 140

293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 293.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15

988.0 993.5 998.9 1003.9 1008.7 1013.4 1017.7 1022.0 1026.1 1030.1 1033.9 1037.6 1041.2 1044.8 1048.1 952.6 959.3 965.8 971.8 977.8 983.1 988.3 993.1 997.8 1002.3 1006.6 1010.9 1014.9 1018.8 1022.6

303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15

980.8 986.6 992.3 997.6 1002.6 1007.4 1012.0 1016.3 1020.5 1024.5 1028.4 1032.2 1035.9 1039.4 1042.9 944.2 951.9 958.7 965.1 970.9 976.5 981.9 986.9 991.7 996.2 1000.5 1004.7 1008.8 1012.9 1016.6

313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15

973.5 979.7 985.5 990.9 996.1 1000.9 1005.6 1010.1 1014.5 1018.6 1022.7 1026.7 1030.4 1034.0 1037.5 937.1 945.0 952.1 958.6 964.8 970.5 976.0 981.2 986.2 990.7 995.1 999.5 1003.7 1007.7 1011.7

323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15

966.9 973.1 979.1 984.7 989.8 994.9 999.7 1004.3 1008.7 1012.9 1017.1 1021.0 1025.0 1028.5 1032.3 929.2 937.9 945.3 952.1 958.5 964.5 970.0 975.2 980.2 985.2 989.8 994.5 998.8 1002.8 1006.6

333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15

959.9 966.5 972.8 978.7 984.0 989.2 994.2 999.0 1003.5 1008.0 1012.2 1016.2 1020.1 1023.9 1027.5

a

Standard uncertainties u are u(T) = 0.01 K, u(p) = 0.01 MPa, and the combined expanded uncertainties Uc (level of confidence =0.95, k = 2) is Uc(ρ) = 0.001 ρ.

arbitrary fixed at the value of 1 K in order to represent each isobar by approximately 90 points. This method requires both the density and isobaric heat capacity at initial conditions to initiate the iterative procedure. Atmospheric density data were taken from U-tube measurements, whereas heat capacities were measured as a function of temperature at atmospheric pressure by using a μDSC (Setaram). This calorimeter is based on Calvert’s principle in which the output signal is given by the difference of the heat flux received by the flux meters which completely surround both the reference and measuring cells. The combined expanded uncertainty with a level of confidence 0.95 (k = 2) of this heat capacity measurement method was estimated as 0.5%.16 From this method the expanded uncertainty (kP = 2%) in density is estimated to 0.1% up to 100 MPa and 0.2% above.

Table 4. Experimental Values of Heat Capacity at Temperatures T and Atmospheric Pressure (patm = 0.1018 MPa)a T/K = 297.91

T/K = 307.79

T/K = 317.66

T/K = 327.54

T/K = 337.42

cp (J·K−1·kg−1)

1813 T/K = 347.29

1838 T/K = 357.17

1865 T/K = 367.05

1892 T/K = 376.92

1918 T/K = 386.80

cp (J·K−1·kg−1)

1944

1969

1996

2025

2054

a

Standard uncertainties u are u(T) = 0.1 K, u(patm) = 0.0004 MPa and the combined expanded uncertainties Uc (level of confidence = 0.95, k = 2) are Uc(cp) = 0.005 cp.

⎛ ∂ρ ⎞ Tαp2 1 ⎜ ⎟ = 2 + cp w ⎝ ∂p ⎠T

(3)

⎛ ⎞ ⎛ ∂cp ⎞ T ⎜ 2 ⎛ ∂αp ⎞ ⎟ ⎜ ⎟ = − αp + ⎜ ⎟ ρ ⎜⎝ ⎝ ∂p ⎠T ⎝ ∂T ⎠ p⎟⎠

(4)

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RESULTS AND DISCUSSION The speed of sound of liquid TOTM was measured for a set of nine isotherms in the temperature range (293 to 374) K and from 0.1 to 200 MPa. However, for lower temperatures measurements were limited in pressure due to a high sound adsorption in this domain. The results are listed in Table 2 and represented in Figure 3 as a function pressure for few isotherms. Density was measured with the U-tube densimeter at pressures from (1 to 140) MPa along isotherms space apart 10 K between (293 and 373) K. The results are given in Table 3. Finally, heat capacity

where αp stands for isobaric expansion and cp for isobaric heat capacity. The integration is carried out by a numerical predictor− corrector technique described in detail in previous papers.14,15 A 0.1 MPa pressure step was chosen for the predictor−corrector procedure, whereas the numerical temperature interval was D



Article

Table 5. Values of Densities ρ at Temperatures T and Pressures p Determined from Integration of Speed of Sound Measurements in Liquid TOTMa p

T

ρ

T

ρ

T

ρ

T

ρ

T

ρ

MPa

K

kg·m−3

K

kg·m−3

K

kg·m−3

K

kg·m−3

K

kg·m−3

0.1013 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 0.1013 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200

293.15 293.15 293.15 293.15 293.15

987.9 993.4 998.7 1003.7 1008.4

303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15

980.9 986.7 992.2 997.3 1002.2 1006.9 1011.4 1015.7

313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15 313.15

973.8 979.9 985.7 991.0 996.1 1001.0 1005.6 1010.0 1014.3 1018.4 1022.3 1026.1 1029.8

323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15 323.15

966.8 973.1 979.1 984.7 990.1 995.1 999.9 1004.4 1008.8 1013.0 1017.1 1021.0 1024.7 1028.4 1031.9 1035.4 1038.7 1041.9 1045.1

333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15 333.15

959.6 966.3 972.5 978.4 983.9 989.1 994.1 998.8 1003.4 1007.7 1011.9 1015.9 1019.8 1023.5 1027.1 1030.7 1034.1 1037.4 1040.6 1043.7 1046.8

343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15 343.15

952.3 959.3 965.9 972.0 977.7 983.2 988.3 993.2 997.9 1002.4 1006.7 1010.8 1014.8 1018.6 1022.4 1026.0 1029.5 1032.9 1036.2 1039.4 1042.5

353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15 353.15

944.8 952.2 959.1 965.5 971.5 977.1 982.4 987.5 992.3 997.0 1001.4 1005.7 1009.8 1013.7 1017.5 1021.3 1024.8 1028.3 1031.7 1035.0 1038.2

363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15 363.15

937.1 944.9 952.1 958.8 965.1 970.9 976.5 981.7 986.7 991.5 996.1 1000.5 1004.7 1008.8 1012.7 1016.5 1020.2 1023.8 1027.2 1030.6 1033.9

373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15 373.15

929.1 937.4 945.0 952.0 958.5 964.6 970.4 975.8 981.0 986.0 990.7 995.2 999.6 1003.8 1007.8 1011.7 1015.5 1019.2 1022.7 1026.2 1029.5

a Combined expanded uncertainties Uc (level of confidence = 0.95, k = 2) are Uc(ρ) = 0.001 ρ up to 100 MPa and Uc(ρ) = 0.002 ρ between (100 and 200) MPa.

between both sets of density data. Finally, the observation in Figure 4 of deviations as a function of pressure does not show a recurrent increase of deviations with pressure. This agreement between acoustic and U-tube measurements confirms the capacity of the acoustic method to determine density at high pressures. The knowledge of speed of sound and density in the same (p,T) conditions allows determining derivative properties such as isentropic and isothermal compressibilities according to the following relations:

measurements were carried out at atmospheric pressure from 298 to 387 K. The experimental data concerning this property are reported in Table 4. Using speed of sound measurements and data of both density and heat capacity at atmospheric pressure, densities were estimated at pressure up to 200 MPa. The density data obtained by this second method are reported in Table 5. These data match very well with U-tube measurements. Comparison between both data sets in the common range of pressure covered reveals a satisfactory agreement with an average deviation of a 0.04% and maximum deviation that does not exceed 0.11%. Moreover, comparison of the average deviation −0.014% and the absolute average deviation 0.04% does not exhibit any systematic error

κs = E

1 ρw 2

(5) DOI: 10.1021/acs.jced.7b00162 J. Chem. Eng. Data XXXX, XXX, XXX−XXX


Article

pressure instead of density. With this aim in mind, the following rational function was chosen: ⎛ ∂v ⎞ a + dp ⎜ ⎟ =− b+p ⎝ ∂p ⎠T

(7)

where d does not depend on temperature, whereas a and b parameters were expressed as a function of temperature by considering simple polynomial forms.

Figure 4. Relative deviations between U-tube measurements and speed of sound method calculation densities Δρ/ρ % = 100{ρ(w) − ρ(Utube)}/ρ(U-tube) for TOTM at blue □, 293.15 K; ⧫, 313.15 K; red ○, 333.15 K; green ▲, 353.15 K; yellow ●, 373.15 K.

κT =

Tαp2 1 + ρcp ρw 2

a = a0 + a1T + a 2T 2 + a3T 3

(8)

b = b0 + b1T + b2T 2

(9)

These coefficients were determined (Table 8) by minimizing the following objective function that intercorrelates density and speed of sound measurements: 2 ⎛⎛ exp ⎞2 ⎛ cal ⎞ cal ⎞2 ⎞ ⎛ v ∂ ∂ v T v χ 2 = ∑ ⎜⎜ i exp ⎟ + ⎜ ⎟ + cal ⎜ ⎟ ⎟ ⎜⎝ wi ⎠ ∂ ∂ p T ⎝ ⎠ ⎝ ⎠ p⎟⎠ c p i ⎝ T Nexp

(6)

The results for these properties are listed in Tables 6 and 7. Furthermore, the simultaneous knowledge of speed of sound and density allows correlating the change in volume with respect to

(10)

where superscript exp stands for experimental data and cal represents calculations from eqs 7 to 9. Integration of eq 7 with

Table 6. Values of Isentropic Compressibility κS in Liquid TOTM at Temperatures T and Pressures pa p

T

κS

T

κS

T

κS

T

κS

T

κS

MPa

K

GPa−1

K

GPa−1

K

GPa−1

K

GPa−1

K

GPa−1

0.1013 10 20 30 40 50 60 70 80 90 100 120 140.1 160.1 180.1 200.1 0.1013 10 20 30 40 50 60 70 80 90 100 120 140.1 160.1 180.1 200.1

293.05 293.05 293.05 293.05 293.05

0.501 0.468 0.441 0.417 0.397

303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15

0.531 0.495 0.464 0.438 0.416 0.396 0.378 0.362

313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35

0.561 0.523 0.489 0.460 0.435 0.414 0.395 0.377 0.361 0.347 0.333 0.310

323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45

0.594 0.551 0.514 0.483 0.456 0.432 0.411 0.392 0.375 0.360 0.346 0.322 0.300 0.282 0.266

333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55

0.630 0.581 0.540 0.506 0.476 0.451 0.428 0.408 0.390 0.373 0.358 0.332 0.310 0.290 0.273 0.259

343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75

0.667 0.613 0.567 0.529 0.496 0.468 0.444 0.423 0.404 0.386 0.370 0.344 0.321 0.300 0.282 0.667

353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85

0.266 0.708 0.646 0.596 0.554 0.519 0.489 0.462 0.439 0.418 0.399 0.383 0.353 0.329 0.308 0.289

364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15

0.750 0.681 0.626 0.580 0.542 0.509 0.480 0.456 0.433 0.413 0.395 0.365 0.339 0.317 0.297 0.281

374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25

0.795 0.719 0.658 0.607 0.566 0.530 0.499 0.472 0.448 0.427 0.409 0.376 0.349 0.326 0.306 0.288

a

Standard uncertainties u are u(T) = 0.1 K, u(p) = 0.01 MPa up to 100 MPa, u(p) = 0.1 MPa between (100 and 200) MPa and the combined expanded uncertainties Uc (level of confidence = 0.95, k = 2) are Uc(κS) = 0.002 κS up to 100 MPa and Uc(κS) = 0.005 κS between (100 and 200) MPa. F



Article

Table 7. Values of Isothermal Compressibility κT in Liquid TOTM at Temperatures T and Pressures pa p

T

κT

T

κT

T

κT

T

κT

T

κT

MPa

K

GPa−1

K

GPa−1

K

GPa−1

K

GPa−1

K

GPa−1

0.1013 10 20 30 40 50 60 70 80 90 100 120 140.1 160.1 180.1 200.1 0.1013 10 20 30 40 50 60 70 80 90 100 120 140.1 160.1 180.1 200.1

293.05 293.05 293.05 293.05 293.05

0.584 0.544 0.511 0.482 0.457

303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15

0.617 0.574 0.537 0.505 0.477 0.453 0.432 0.413

313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35 313.35

0.654 0.605 0.564 0.529 0.499 0.473 0.450 0.429 0.410 0.393 0.378 0.351

323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45 323.45

0.693 0.638 0.593 0.555 0.522 0.493 0.468 0.446 0.426 0.408 0.391 0.362 0.337 0.316 0.298

333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55 333.55

0.735 0.674 0.624 0.582 0.546 0.515 0.488 0.464 0.442 0.423 0.405 0.374 0.348 0.325 0.306 0.289

343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75 343.75

0.781 0.713 0.658 0.612 0.572 0.538 0.509 0.483 0.459 0.438 0.420 0.387 0.359 0.335 0.315 0.297

353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85 353.85

0.831 0.756 0.694 0.643 0.600 0.563 0.531 0.503 0.478 0.455 0.435 0.400 0.370 0.345 0.324 0.305

364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15 364.15

0.887 0.802 0.734 0.677 0.630 0.590 0.555 0.524 0.497 0.473 0.452 0.414 0.383 0.356 0.333 0.313

374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25 374.25

0.947 0.853 0.776 0.714 0.662 0.618 0.580 0.547 0.518 0.492 0.469 0.429 0.396 0.367 0.343 0.322

a Standard uncertainties u are u(T) = 0.1 K, u(p) = 0.01 MPa up to 100 MPa, u(p) = 0.1 MPa between (100 and 200) MPa and the combined expanded uncertainties Uc (level of confidence =0.95, k = 2) are Uc(κT) = 0.008 κS up to 100 MPa and Uc(κT) = 0.02 κT between (100 and 200) MPa.

Table 8. Parameters of eqs 7 to 13 for Correlating Volumetric Properties of TOTM from 293 to 373 K and up to 200 MPa a0 a1 a2 a3 b0 b1 b2 d Deviations AAD%a (ρ) MD%b (ρ) AAD%a (c) MD%b (c) a

−4.85690 × 10−5 1.06615 × 10−6 −3.31230 × 10−9 3.73104 × 10−12 3.96673 × 102 −1.23466 1.04120 × 10−3 3.16951 × 10−8

v0 v1 v2 v3 cp,0 cp,1 cp,2

6.20425 × 10−4 2.66947 × 10−6 −6.98960 × 10−9 8.33323 × 10−12 1.18020 × 103 1.68732 1.47113 × 10−3

1.8 × 10−3 4.6 × 10−3 9.9 × 10−2 5.6 × 10−1

AAD%a (κs) MD%b (κs) AAD%a (κT) MD%b (κT)

1.8 × 10−1 4.9 × 10−1 1.5 × 10−1 6.5 × 10−1

AAD = absolute avrage deviation. bMD = maximum deviation.

vatm = v0 + v1T + v2T 2 + v3T 3

respect to pressure allows calculating the volume v by reference to measurements performed at atmospheric pressure: v = vatm

⎛ p+b ⎞ ⎟⎟ − d(p − patm ) + (bd − a) ln⎜⎜ ⎝ patm + b ⎠

(12)

Parameters v0, v1, v2, and v3 are listed in Table 8. By adding eq 12 to eqs 7−9, the following correlation for heat capacity data at atmospheric pressure:

(11)

These atmospheric measurements were correlated as a function of temperature by a polynomial form:

cp ,atm = cp ,0 + cp ,1T + cp ,2T 2 G

(13) DOI: 10.1021/acs.jced.7b00162 J. Chem. Eng. Data XXXX, XXX, XXX−XXX


Article

fluid for high viscosity. Part I: Viscosity measurements at temperatures from (303 to 373)K and pressures up to 65 MPa, using a novel vibratingwire instrument. Fluid Phase Equilib. 2014, 384, 50−59. (2) Diogo, J. C. F; Avelino, H. M. N. T.; Caetano, F. J. P.; Fareleira, J. M. N.A.; Wakeham, W. A. Tris(2-ethylhexyl) trimellitate (TOTM) as a potential industrial reference fluid for viscosity at high temperatures and high pressures: New viscosity, density and surface tension measurements. Fluid Phase Equilib. 2016, 418, 192−197. (3) Bair, S. The temperature and pressure dependence of the viscosity and volume for two reference liquids. Lubr. Sci. 2016, 28, 81−95. (4) Baled, H. O.; Tapriyal, D.; Gamwo, I. K.; Bamgbade, B. A.; McHugh, M. A.; Enick, R. M. Viscosity Measurements of Two Potential Deepwater Viscosity Standard Reference Fluids at High Temperature and High Pressure. J. Chem. Eng. Data 2016, 61, 2712−2719. (5) Muringer, M. J. P.; Trappeniers, N. J.; Biswas, S. N. The Effect of Pressure on the Sound Velocity and Density of Toluene and n-heptane up to 2600 bar. Phys. Chem. Liq. 1985, 14, 273−296. (6) Del Grosso, V. A.; Mader, C. W. Speed of sound in pure water. J. Acoust. Soc. Am. 1972, 52, 1442−1446. (7) Marczak, W. Water as a standard in the measurements of speed of sound in liquids. J. Acoust. Soc. Am. 1997, 102, 2776−2779. (8) Wilson, W. D. Speed of sound in distilled water as a function of temperature and pressure. J. Acoust. Soc. Am. 1959, 31, 1067−1072. (9) Vance, S.; Brown, J. M. Sound velocities and thermodynamic properties of water to 700 MPa and −10 to 100 °C. J. Acoust. Soc. Am. 2010, 127, 174−180. (10) Comuñas, M. J.P.; Bazile, J. P.; Baylaucq, A.; Boned, C. Density of Diethyl Adipate using a Vibrating Densimeter from 293.15 to 403.15 K and up to 140 MPa. Densimeter calibration and measurements. J. Chem. Eng. Data 2008, 53, 986−994. (11) Wagner, W.; Pruß, A. The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. J. Phys. Chem. Ref. Data 2002, 31, 387−535. (12) TRC. Thermodynamic Tables; Texas A&M University: College Station, 1996. (13) Segovia, J. J.; Fandiño, O.; López, E. R.; Lugo, L.; Martín, M. C.; Fernández, J. Automated densimetric system: Measurements and uncertainties for compressed fluids. J. Chem. Thermodyn. 2009, 41, 632−638. (14) Daridon, J. L.; Lagrabette, A.; Lagourette, B. Speed of Sound, Density, and Compressibilities of Heavy Synthetic Cuts from Ultrasonic Measurements Under Pressure. J. Chem. Thermodyn. 1998, 30, 607− 623. (15) Dutour, S.; Daridon, J. L.; Lagourette, B. Pressure and Temperature Dependence of the Speed of Sound and Related Properties in Normal Octadecane and Nonadecane. Int. J. Thermophys. 2000, 21, 173−184. (16) Pauly, J.; Kouakou, C.; Habrioux, M.; Le Mapihan, K. Heat capacity measurements of pure fatty acid methyl esters and biodiesels from 250 to 390 K. Fuel 2014, 137, 21−127. (17) Ndiaye, E. H. I.; Habrioux, M.; Coutinho, J. A. P.; Paredes, M. L. L.; Daridon, J. L. Speed of Sound, Density, and Derivative Properties of Ethyl Myristate, Methyl Myristate, and Methyl Palmitate under High Pressure. J. Chem. Eng. Data 2013, 58, 1371−1377. (18) Diogo, J. C. F; Avelino, H. M. N. T.; Caetano, F. J. P.; Fareleira, J. M. N.A. Tris(2-Ethylhexyl) trimellitate (TOTM) a potential reference fluid for high viscosity. Part II: Density measurements at temperatures from (293 to 373)K and pressures up to 68 MPa. Fluid Phase Equilib. 2014, 384, 36−42.

one can calculate all thermodynamic properties within the experimental range.17 Deviations of these calculations with data given in Table 5 for density shows a good match between calculated and experimental density data with an observed AAD % of 1.8 × 10−3 and MD% of 4.6 × 10−3. Derivative properties are also well represented by the equation with an observed maximum deviation less than experimental uncertainties. Density measurements were carried out previously in TOTM by several authors1,2,18 at pressures up to 95 MPa. Comparison in Table 9 between the proposed correlation (eqs 11 and 12) and Table 9. Deviations with Previous Measurements of Density ref

T range/K

p range/MPa

AAD%

MD%

Diogo et al.18 Diogo et al.1 Diogo et al.2 Bair3 Bair3

293−373 303−371 303−373 313−353 313−353

0.1−68 1−65 4.71−99 0.1−150 250−350

0.06 0.06 0.06 0.05 0.14

0.11 0.10 0.11 0.14 0.19

the data reported by these authors shows a very good agreement with a maximum deviation that never exceeds the estimated uncertainties of reported data. In addition Bair3 reported data of relative volume measured in liquid TOTM at pressures up to 350 MPa by a metal bellows piezometer. Comparison of these data with our measurements in the common experimental range (0.1−150) MPa and (313−353) K exhibits a maximum deviation of 0.14%. Moreover, by extending the comparison to 350 MPa with the proposed correlation (eqs 11, 12) used in extrapolation, one can observe a maximum deviation of 0.19%. This deviation is still below the reported experimental uncertainty. This last result shows the capacity of eqs 7−9 to represent the compressibility of TOTM as a function of pressure.

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CONCLUSION The speed of sound of tris(2-ethylhexyl) trimellitate (TOTM), was measured at temperatures from (293 to 373) K and pressures up to 200 MPa using a pulse echo technique working at 3 MHz. Density and compressibilities were obtained from this measurements. The volumetric data reported in this work were correlated with a function that expressed the derivative of density with respect to pressure by a simple rational function with three parameters, only two of which are a function of temperature. This function allows representing data of both density and compressibilities within the reported experimental uncertainty. The present measurements are in agreement with previous reported density data with deviations that never exceed experimental uncertainties. Comparison of correlation predictions beyond experimental pressure range (350 MPa) has shown the predictive capacities of this simple equation.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Jean-Luc Daridon: 0000-0002-0522-0075 Notes

The authors declare no competing financial interest.

■

REFERENCES

(1) Diogo, J. C. F; Avelino, H. M. N. T.; Caetano, F. J. P.; Fareleira, J. M. N.A. Tris(2-ethylhexyl) trimellitate (TOTM) a potential reference H


Speed of Sound, Density, and Derivative Properties of Tris(2

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