Thermodynamic Properties of Propane. IV. Speed of Sound in the

IV. Speed of Sound in the Liquid and Supercritical Regions. K. Meier* and S. Kabelac ... Publication Date (Web): November 12, 2012. Copyright © 2012 ...
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Thermodynamic Properties of Propane. IV. Speed of Sound in the Liquid and Supercritical Regions K. Meier*,† and S. Kabelac‡ Institut für Thermodynamik, Helmut-Schmidt-Universität/Universität der Bundeswehr Hamburg, Holstenhofweg 85, D-22043 Hamburg, Germany ABSTRACT: Comprehensive and accurate measurements of the speed of sound in pure propane have been carried out in the liquid and supercritical regions by a double-path-length pulse-echo technique. The measured data cover the temperature range from (240 to 420) K with pressures up to 100 MPa. The measurement uncertainties amount to 3 mK for temperature, 0.01 % for pressures below 10 MPa, and 0.005 % for pressures between 10 MPa and 100 MPa, and 0.02 % for speed of sound. The high accuracy of the measurements is demonstrated by comparisons with literature data and equation of state models.

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Standard and Technology in Boulder.6,7 In parallel with this work, a new fundamental equation of state for propane was developed by Lemmon et al.,8 which is presented in the third paper of the series. In the optimization process of this equation of state, our speed of sound data were already used as part of the experimental data set, to which the equation of state was fitted.

INTRODUCTION Propane is a fluid with a wide range of technical applications. For example, it is one of the most important basic substances in many production processes in the chemical industry, it is used as a working fluid in refrigeration cycle processes, and it occurs as a secondary component in natural gas mixtures. Therefore, accurate thermodynamic properties of propane are required for process and plant design. Thermodynamic properties are best represented by fundamental equations of state, from which all thermodynamic properties can be calculated. The development of a fundamental equation of state requires, among other properties, speed of sound data as part of the experimental data set for the optimization process. By including accurate speed of sound data in the optimization process of an equation of state, it can be expected that the liquid region, where many properties depend strongly on density, is described more accurately and that the extrapolation behavior and the representation of caloric properties is improved. Although the speed of sound in propane under high pressures has been measured by several authors,1−4 there are considerable differences between the different data sets, and it was not known which one was correct. To overcome this situation, it was the aim of this work to provide accurate and comprehensive speed of sound data in the liquid and supercritical regions of propane. New accurate speed of sound data can help to differentiate between the different data sets and extend the data basis for a new fundamental equation of state for propane. The speed of sound measurements in propane form part of a larger program in our laboratory to measure the speed of sound in several pure fluids.5 This paper is the fourth paper in a series of papers on the thermodynamic properties of propane. The first two papers describe measurements of the p−ρ−T behavior and isochoric heat capacity of propane carried out at the National Institute of © 2012 American Chemical Society

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APPARATUS Our speed of sound instrument, the calibration procedure, and the analysis of measurements have been described in detail in ref 9. Therefore, only a short description of the experimental setup and measurement analysis is given here. The measurement principle of our acoustic sensor is based on a double-path-length

Figure 1. Measurement principle of the speed of sound sensor.

pulse-echo technique (see Figure 1), which was introduced by Muringer et al.10 A piezoelectric quartz crystal with a resonance Received: April 22, 2012 Accepted: October 9, 2012 Published: November 12, 2012 3391

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frequency of 8 MHz is mounted asymmetrically at distances of L1 = 20 mm and L2 = 30 mm between two stainless steel reflectors. The crystal is excited with a sinusoidal burst signal of 30 to 80 cycles and emits sound signals from its front and back surface into the sample fluid. The signals propagate in both directions in the sample fluid, are reflected at the reflectors, and travel back to the crystal, which now acts as a receiver. Due to the different distances, the echoes from the two reflectors arrive successively at the crystal, separated by a time difference Δt. The speed of sound c is obtained as c = 2(L2 − L1)/Δt. For the precise measurement of the time difference, the crystal is excited by a second burst signal at about the time Δt after the first signal. The second signal has the same shape as the first signal, but of opposite sign, and the amplitude is reduced to account for sound attenuation. The time interval and the amplitude of the second signal are adjusted so that the first echo of the first signal traveling the longer path and the first echo of the second signal traveling the shorter path exactly cancel. This cancellation is monitored on an oscilloscope. When complete cancellation occurs, the time between the emission of the two signals equals the time difference Δt between the arrival of the first and second echoes of the first signal at the crystal. With this technique, a resolution of the time difference measurement better than 5 ppm is achieved. The acoustic path length ΔL = 2(L2 − L1) and the thermal expansion coefficient of the sensor material were determined by calibration measurements in liquid water at ambient pressure. In the analysis of the measurements, corrections for changes of the distances in the speed of sound sensor with temperature, for compression of the sensor with pressure, and for diffraction effects are applied. The uncertainty of the speed of sound measurement is 0.007 % at a 95 % confidence level, excluding contributions from sample impurities and from temperature and pressure measurement uncertainties. The exceptional high accuracy of the speed of sound measurement accomplished with our apparatus has motivated others to utilize the essential features of our acoustic sensor in their speed of sound instruments. This enabled, for example, Gedanitz et al. to achieve a similar high accuracy for speed of sound measurements in liquids.11,12 The speed of sound sensor is housed in a pressure vessel, which is thermostatted in a circulating liquid-bath thermostat. The temperature inside the pressure vessel is kept constant within 0.5 mK. The temperature was measured by a Pt25 sensor calibrated on the ITS-90 located in the wall of the pressure vessel with an estimated uncertainty of 3 mK. The resistance of the thermometer was measured with an ASL F18 bridge system with calibrated reference resistors. The pressure inside the pressure vessel was measured with two nitrogen-operated gas pressure balances with measurement ranges of 5 MPa and 100 MPa. The pressure balances were coupled to the sample liquid via a differential pressure null indicator (Ruska membrane type cell). The uncertainty of the pressure measurement is estimated to be 0.01 % below 10 MPa and 0.005 % between 10 MPa and 100 MPa. Both temperature and pressure measurement uncertainties are also for a 95 % confidence level.

Table 1. Chemical Sample Description chemical name

source

initial volume purification final volume analysis fraction purity method fraction purity method

propane Scott Specialty Gases a

0.99993

none

0.99988

GCa

Gas chromatography.

The first peak was identified as water, while the nature of the second peak could not be identified. This impurity is probably another hydrocarbon, which remains from the production process. No nitrogen or oxygen were detected. The volume purity of the sample is estimated to be 99.988 %. For these impurities, an additional allowance of 0.005 % is added to the uncertainty of the speed of sound data. The reproducibility of the speed of sound, when repeating measurements at the same state point after pressure and temperature cycles, was within 0.002 %. Thermal relaxation phenomena do not significantly influence the propagation of sound waves in propane in the frequency range of our measurements13 so that no dispersion correction is required. Including the reproducibility and the additional contribution due to sample impurities, the total uncertainty of the speed of sound measurement is given by u(c) = (1.4·10−4 + 2.5·10−7·p/MPa)·c, where c is the speed of sound and p denotes pressure. The second term accounts for the uncertainty of the sensor compression with pressure. The uncertainty contributions to the speed of sound due to the uncertainties of temperature and pressure measurements were estimated by the equation of state of Lemmon et al.8 They amount to 0.003 % for temperature measurement and 0.002 % for pressure measurement. For the lowest measured pressures on the supercritical isotherms, the influence of the pressure is larger because the isotherms are rather flat in this state region, and its contribution amounts to 0.005 %. Taking these additional contributions into account, the combined uncertainty of the speed of sound measurement becomes Uc(c) = (1.9·10−4 + 2.5·10−7·p/MPa)·c. At the lowest pressures on supercritical isotherms, the value 1.9·10−4 has to be replaced by 2.2·10−4. The uncertainty estimates are for a 95 % confidence level.

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MATERIALS The propane sample was from the same batch as used by McLinden6 to measure the density of propane and was kindly provided to us for our speed of sound measurements (see Table 1). It was originally purchased from Scott Specialty Gases with a manufacturer specified volume purity better than 99.993 %. A gas chromatograph analysis revealed two peaks with 0.009 % and 0.003 % in area fraction besides the main propane peak.

Figure 2. Distribution of our measurements and literature data for the speed of sound in propane in the p,T plane. The gray area denotes the region of our measurements. ×, this work; □, ref 2; ○, ref 3; +, ref 4; ⧖, ref 14; △, ref 15; ▽, ref 16; ⋈, ref 17 and , vapor pressure. 3392

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Table 2. Results for the Speed of Sound in Liquid and Supercritical Propanea T/K

p/MPa

c/m·s−1

239.9981 239.9977 239.9974 239.9977 239.9948 239.9977 239.9978 239.9971 239.9967 239.9967 239.9967 239.9960 239.9952

1.30427 2.10477 3.10535 4.10604 4.10602 5.10662 6.10773 7.10853 8.10964 9.11030 10.1110 12.6128 15.1144

1113.77 1121.56 1131.11 1140.47 1140.49 1149.65 1158.66 1167.51 1176.20 1184.73 1193.13 1213.53 1233.17

259.9956 259.9958 259.9957 259.9961 259.9960 259.9963 259.9962 259.9966 259.9964 259.9967 259.9966 259.9965 259.9968 259.9971 259.9970

1.30257 2.10222 3.10375 4.10435 5.10507 6.10593 7.10663 8.10702 9.10776 10.1085 12.6102 15.1120 17.6141 20.1158 25.1193

983.541 992.980 1004.49 1015.69 1026.61 1037.26 1047.67 1057.84 1067.78 1077.52 1101.02 1123.43 1144.89 1165.48 1204.42

280.0026 280.0013 280.0015 280.0014 280.0013 280.0013 280.0013 280.0012 280.0016 280.0018 280.0018 280.0016 280.0016 280.0014 280.0016 280.0015 280.0019 280.0014 280.0013

1.30337 1.30177 2.10178 3.10257 4.10327 5.10402 6.10483 7.10557 8.10769 9.10852 10.1092 12.6111 15.1129 17.6148 17.6148 20.1170 25.1205 30.1239 35.1274

851.162 851.142 862.953 877.198 890.908 904.149 916.957 929.366 941.421 953.125 964.520 991.759 1017.45 1041.79 1041.79 1064.97 1108.35 1148.41 1185.77

300.0003 300.0003 300.0002 300.0001 300.0001 300.0005 300.0003 300.0004 300.0004 300.0011

1.30344 2.10401 3.10451 4.10512 5.10562 6.10613 7.10668 8.10732 9.10793 10.1088

713.168 728.746 747.167 764.589 781.134 796.919 812.030 826.541 840.510 853.989

T/K

p/MPa

c/m·s−1

239.9951 239.9948 239.9947 239.9939 239.9941 239.9939 239.9937 239.9937 239.9935 239.9931 239.9927 239.9923 239.9923

17.6162 20.1178 25.1214 30.1250 35.1284 40.1319 45.1353 50.1387 60.1456 70.1526 80.1598 90.1668 100.174

1252.10 1270.40 1305.29 1338.19 1369.39 1399.07 1427.44 1454.62 1505.90 1553.68 1598.50 1640.82 1680.92

259.9974 259.9975 259.9976 259.9973 259.9973 259.9972 259.9973 259.9974 259.9964 260.0024 260.0022 260.0020 260.0017 260.0012 260.0019

35.1262 40.1297 45.1332 50.1368 60.1444 70.1514 80.1584 90.1655 100.173 1.30362 4.10533 10.1098 20.1167 30.1235 40.1304

1275.00 1307.34 1338.08 1367.38 1422.35 1473.22 1520.71 1565.33 1607.49 983.511 1015.67 1077.49 1165.45 1240.76 1307.32

280.0014 280.0011 280.0009 280.0011 280.0009 280.0008 280.0009 280.0013 280.0029 280.0028 280.0033 280.0036 280.0032 280.0034 280.0009 280.0019 280.0020 280.0019 280.0014

40.1309 45.1345 50.1381 60.1451 70.1521 80.1590 90.1660 100.173 1.30288 2.10340 3.10392 4.10459 5.10515 6.10591 1.30358 4.10552 10.1097 20.1165 30.1235

1220.84 1253.96 1285.39 1343.97 1397.84 1447.87 1494.68 1538.77 851.157 862.965 877.196 890.912 904.151 916.955 851.186 890.946 964.529 1064.97 1148.40

300.0013 300.0015 300.0016 300.0016 300.0014 300.0013 300.0015 300.0013 300.0010 300.0038

35.1265 40.1301 45.1335 50.1370 60.1440 70.1509 80.1578 90.1649 100.172 1.30369

1101.93 1139.76 1175.26 1208.79 1270.89 1327.62 1380.04 1428.91 1474.78 713.142

T = 240 K

T = 260 K

T = 280 K

T = 300 K

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Table 2. continued T/K

p/MPa

c/m·s−1

300.0017 300.0020 300.0020 300.0016 300.0016 300.0016

12.6107 15.1124 17.6142 20.1159 25.1194 30.1229

885.817 915.374 943.046 969.124 1017.35 1061.33

320.0044 320.0046 320.0044 320.0041 320.0045 320.0048 320.0048 320.0050 320.0051 320.0050 320.0045 320.0048 320.0051 320.0050 320.0050 320.0049 320.0049 320.0051

2.10294 3.10370 4.10490 5.10503 6.10569 7.10634 8.10700 9.10775 10.1084 12.6102 15.1124 17.6143 20.1161 25.1196 30.1232 35.1267 40.1302 45.1338

583.667 609.720 633.378 655.159 675.415 694.414 712.344 729.354 745.558 783.141 817.342 848.853 878.187 931.653 979.736 1023.66 1064.25 1102.10

339.9981 339.9983 339.9980 339.9982 339.9984 339.9983 339.9983 339.9984 339.9989 339.9990 339.9987 339.9991 339.9993 339.9991

3.60491 4.10532 5.10610 6.10682 7.10849 8.10921 9.10981 10.1104 12.6115 15.1132 17.6148 20.1165 25.1199 30.1235

472.320 490.155 521.884 549.791 574.939 597.929 619.208 639.085 683.978 723.708 759.610 792.530 851.584 903.854

359.9973 360.0044 359.9986 359.9987 360.0004 359.9994 359.9973 359.9971 359.9974 359.9972 359.9967 359.9981 359.9985 359.9986 359.9987 359.9987

3.60291 3.60440 3.60427 3.85408 3.85464 3.95413 3.95462 4.10337 4.60361 5.10390 6.10488 7.10523 7.10518 8.10596 9.10667 10.1073

257.681 257.743 257.833 285.624 285.671 294.945 295.002 307.534 342.105 369.878 414.549 450.816 450.811 481.951 509.541 534.494

380.0005 379.9985

6.10522 6.10531

260.562 260.557

T/K

p/MPa

c/m·s−1

299.9961 300.0039 299.9963 300.0035 300.0033

4.10571 4.10565 5.10643 10.1098 20.1168

764.609 764.571 781.152 853.978 969.117

320.0049 320.0038 320.0039 320.0038 320.0040 320.0041 320.0481 319.9973 319.9975 319.9976 319.9977 319.9974 319.9974 319.9988 319.9993 319.9993 319.9994 319.9996

50.1374 60.1432 70.1502 80.1572 90.1642 100.171 2.10395 2.10355 3.10422 4.10478 5.10545 6.10606 7.10672 8.10788 9.10859 10.1093 12.6111 15.1129

1137.65 1203.10 1262.54 1317.20 1367.95 1415.46 583.334 583.717 609.760 633.412 655.185 675.446 694.448 712.380 729.385 745.591 783.169 817.359

339.9992 339.9994 340.0000 340.0013 340.0014 340.0012 340.0014 340.0014 340.0015 340.0027 340.0022 340.0022 340.0027

35.1268 40.1299 45.1332 50.1367 60.1437 70.1506 80.1575 90.1644 100.171 2.60267 3.10303 3.60340 4.10370

951.083 994.375 1034.49 1071.97 1140.59 1202.52 1259.23 1311.72 1360.70 430.600 452.619 472.248 490.081

359.9967 359.9967 359.9968 359.9967 359.9977 359.9974 359.9976 359.9977 359.9978 359.9974 359.9977 359.9986 359.9988 359.9987 359.9985

12.6105 15.1123 17.6141 20.1158 25.1200 30.1235 35.1269 40.1304 45.1338 50.1371 60.1441 70.1513 80.1584 90.1654 100.172

588.796 635.050 675.843 712.612 777.416 833.859 884.295 930.158 972.400 1011.69 1083.21 1147.41 1205.95 1259.99 1310.30

379.9969 379.9976

15.1127 17.6149

552.604 598.510

T = 300 K

T = 320 K

T = 340 K

T = 360 K

T = 380 K

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Table 2. continued T/K

p/MPa

c/m·s−1

379.9994 379.9978 379.9991 379.9976 379.9994 379.9974 379.9973 379.9976 379.9972 379.9974 379.9971 379.9970 379.9969

6.60555 6.60568 7.10592 7.10610 7.60631 7.60654 8.10695 9.10812 10.1089 11.1097 12.1104 13.1112 14.1120

293.175 293.175 320.396 320.386 344.051 344.035 365.128 401.907 433.606 461.751 487.227 510.611 532.306

400.0014 400.0012 400.0009 400.0006 400.0009 400.0003 399.9994 399.9996 400.0001 399.9997 399.9995 399.9997 399.9996

9.10726 9.60768 10.1081 10.6085 11.1090 12.1099 13.1111 14.1118 15.1125 16.1132 17.1139 18.1146 19.1153

305.170 324.656 342.772 359.724 375.671 404.996 431.581 455.974 478.587 499.713 519.574 538.353 556.186

420.0024 420.0029 420.0024 420.0016 420.0015 420.0031 420.0032 420.0036 420.0039 420.0042 420.0049 420.0050

11.1087 11.6090 12.1100 12.6104 13.1107 14.1108 15.1115 16.1122 17.1130 18.1137 19.1143 20.1150

309.813 324.887 339.471 353.488 366.974 392.473 416.258 438.519 459.463 479.253 498.028 515.910

T/K

p/MPa

c/m·s−1

379.9977 379.9980 379.9983 379.9980 379.9982 379.9980 379.9985 379.9987 379.9989 379.9993 379.9996 380.0000 379.9996

20.1166 22.6183 25.1200 30.1234 35.1268 40.1302 45.1337 50.1371 60.1441 70.1510 80.1579 90.1649 100.172

639.153 675.888 709.562 769.940 823.352 871.565 915.735 956.634 1030.76 1096.96 1157.14 1212.53 1263.99

399.9999 400.0005 400.0004 400.0010 400.0011 400.0013 400.0012 400.0012 400.0018 400.0020 400.0020 400.0021 400.0017

20.1160 22.6179 25.1197 27.6214 30.1232 35.1267 40.1302 45.1337 50.1373 60.1442 70.1511 80.1579 80.1585

573.190 612.644 648.537 681.605 712.356 768.342 818.591 864.418 906.711 983.054 1050.99 1112.57 1112.57

420.0051 420.0051 420.0050 420.0050 420.0055 420.0053 420.0051 420.0051 420.0047 420.0048 420.0048

21.1157 22.6167 25.1183 27.6199 30.1214 35.1260 40.1295 45.1331 50.1367 60.1438 70.1508

532.993 557.302 594.831 629.294 661.258 719.262 771.122 818.281 861.699 939.853 1009.20

T = 380 K

T = 400 K

T = 420 K

a Uncertainty of temperature: u(T) = 3 mK; relative uncertainty of pressure: ur(p) = 1·10−4 for p < 10 MPa, ur(p) = 5·10−5 for p > 10 MPa; and combined uncertainty of speed of sound: Uc(c) = (1.9·10−4 + 2.5·10−7·p/MPa)·c (all uncertainties refer to a level of confidence = 0.95).

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RESULTS The distribution of our measurements and literature data in the p,T plane is shown in Figure 2. Our data cover the subcritical liquid region from 240 K upward and extend up to 420 K into the supercritical region with pressures to 100 MPa. On subcritical isotherms, the lowest pressures, at which measurements were carried out, were chosen close to the vapor pressure. On supercritical isotherms, measurements were started at the lowest pressure where a clear signal cancellation could be observed. The measurement results are reported in Table 2. Figure 3 shows the speed of sound data for the 10 measured isotherms as a function of pressure. In the measured state region, the speed of sound ranges from about 250 m s−1 to 1700 m s−1.

DISCUSSION Experimental data for the speed of sound in propane were published by seven groups. Details of these literature data sets are summarized in Table 3. Terres et al.,14 Trusler and Zarari,15 He et al.,16 and Hurly et al.17 measured the speed of sound in the gas phase, while the data of Noury18 cover the vicinity of the critical point in the supercritical region. Data in the liquid region were published by Lacam,2 Younglove,3 and Niepmann.4 Fundamental equations of state in terms of the Helmholtz free energy as a function of density and temperature were developed by Miyamoto and Watanabe,19 Bücker and Wagner,20 and Lemmon et al.8 Span and Wagner21 developed a set of short equations of state with the same functional form and fixed exponents for a broad range of nonpolar fluids including an 3395

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Figure 3. Speed of sound in propane as a function of pressure for all measured isotherms. ■, 240 K; ◇, 260 K; ◆, 280 K; ▽, 300 K; ●, 320 K; △, 340 K; ▼, 360 K; □, 380 K; ▲, 400 K; ○, 420 K; , speed of sound at the measured isotherms calculated from the fundamental equation of state of Lemmon et al.; and ···, saturated liquid speed of sound calculated from the fundamental equation of state of Lemmon et al.

equation for propane. These equations were primarily designed for application to fluids for which limited experimental data are available and do not compete with the highly accurate equations of state dedicated to a particular fluid. Younglove and Ely22 published a thermal equation of state of the MBWR-type, which together with an equation for the isochoric ideal gas heat capacity can be represented as a fundamental equation of state. The equation of Bücker and Wagner was developed as a temporary equation until the Lemmon et al. equation was finished. However, even as an interim equation, it is still more accurate than the older three equations. In the remainder of this section, our data are compared with the literature data and equation of state models. The equation of state of Lemmon et al.8 is chosen as a reference for these comparisons. In the optimization process of this equation of state, our speed of sound data were already used. Figures 4, 5, and 6 show percentage deviations of our data, literature data at similar temperatures, and the three other equation of state models from the equation of state of Lemmon et al. Our data agree with the equation of state within 0.06 % over the entire range of temperatures and pressures. This excellent agreement is due to the fact that they were used by Lemmon et al. in the optimization process of the equation of state. From

Figure 4. Fractional deviations Δc = c(expt.) − c(calc.) of experimental speeds of sound c(expt.) in propane, literature data, and three equation of state models from values c(calc.) obtained from the fundamental equation of state of Lemmon et al. as a function of pressure at 240 K, 260 K, and 280 K. Experimental data: ○, this work; △, ref 2; □, ref 3; and +, ref 4. Equations of state: , ref 19; −·−, ref 20; ···, ref 21 and −−−, ref 22.

the three literature data sets, the data of Younglove at 240 K, 260 K, 280 K, and 300 K agree best with our data. They are slightly lower than our data, but the agreement is within 0.04 % except for a few data, where the differences are larger. Younglove reported an uncertainty of 0.05 % for his data. Thus, both data sets agree among each other within their quoted uncertainties. The data of Niepmann show some scatter and lie about 0.2 % to 0.8 % below our data. The differences between the data of Niepmann and our data are similar on all isotherms except for the highest isotherm, which was measured by Niepmann,

Table 3. Literature Data for the Speed of Sound in Propane author

year

method

data

T/K

p/MPa

sample purity

uncertainty

He et al.16 Hurly et al.17 Lacam2 Niepmann4 Noury18 Terres et al.14

2002 2003 1956 1984 1954 1957

SRa GRb DLc PEe DL IFf

24 11 200 241 118 99

293−323 298 298−498 200−323 348−398 293−448

0.7 0.85 100 60 15 10

> 99.95 % not reported SPd > 99.95 % SP > 96 %

Trusler and Zarari15 Younglove3

1996 1981

SR PE

68 180

225−375 90−300

0.85 35

> 99.95 % > 99.95 %

< < < < < < < <