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Speed of sound, density, and compressibilities of heavy synthetic cuts from ultrasonic measurements under pressure
J. Chem. Thermodynamics 1998, 30, 607]623 Article No. ct970330
Speed of sound, density, and compressibilities of heavy synthetic cuts from ultrasonic measurements under pressure J. L. Daridon, a A. Lagrabette, and B. Lagourette Laboratoire Haute Pression, Centre Uni¨ ersitaire de Recherche Scientifique, Uni¨ ersite´ de Pau, A¨ enue de l’Uni¨ ersite, ´ 64000 Pau, France Thermophysical properties such as speed of sound, density, and isentropic and isothermal compressibilities have been obtained from ultrasonic speed measurements on two synthetic systems which are representative of distillation cuts with high bubble points ŽT s 520 K and T s 573 K, respectively.. The first system Žcomposition expressed in mass fraction., Ž0.40C 13 H 28 q 0.35C 13 H 26 q 0.25C 13 H 20 ., was investigated at pressures from Ž0.1 to 150. MPa in the temperature range T s Ž293.15 to 373.15. K, whereas the second one, Ž0.40C 17 H 36 q 0.35C 16 H 32 q 0.25C 16 H 26 ., was investigated at pressures up to 100 MPa in the temperature range T s Ž293.15 to 373.15. K. Q 1998 Academic Press Limited KEYWORDS: speed of sound; density; compressibility; pressure; hydrocarbon
1. Introduction Because of the great complexity of most reservoir fluids it has become usual to adopt an analytical representation of reservoir fluids which consists of considering the fluids of interest as a juxtaposition of groups of components and of a heavy fraction represented as C nq. The major weak point in modelling a complete petroleum fluid resides in the inadequate description of the undifferentiated heavy fraction, often assimilated to C 11q. In order to develop effective methods, experimental data are required on systems composed of a number of components which are representative of petroleum distillation fractions with high bubble points, and hence made up of heavy components. In this context, and in order to acquire a substantial body of experimental data on different thermophysical properties over a wide range of pressure Ž0.1 to 150. MPa, we conducted measurements of ultrasonic speed under pressure, on two representative synthetic systems of distillation cuts; in other words, systems composed of substances with identical bubble points and with compositions respecting the average proportions determined by compositional analysis of a large number of real fluids. The first system ŽS 250., made up of Ž0.40C 13 H 28 q 0.35C 13 H 26 q 0.25C 13 H 20 ., has a bubble point of approximately T s 520 K, a
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Q 1998 Academic Press Limited
608
J. L. Daridon, A. Lagrabette, and B. Lagourette
whereas the second ŽS 300., which is composed of Ž0.40C 17 H 36 q 0.35C 16 H 32 q 0.25C 16 H 26 ., has a bubble point of approximately T s 573 K. The ultrasonic speed data acquired in this research, linked with complementary density measurements made at atmospheric pressure, were used to calculate densities, and isentropic and isothermal compressibility coefficients under pressure.
2. Experimental The technique deployed to carry out measurements of ultrasonic speed waves u in systems under high pressure is based on a pulse technique in which the piezoelectric elements are isolated from the liquid studied, and hence from the effects of pressure, by the presence of intermediate bars. The experimental apparatus, which has already been described in detail elsewhere,Ž1. is composed essentially of an autoclave cell which has been further improved so as to extend the domain of measurement to p s 200 MPa. This cylindrical-shaped cell Žfigure 1. is closed at each end by two identical buffers which serve as a connecting medium between the piezo-electric transducers and the test fluid. The ultrasonic speed is deduced from measurement Ž2. of the transit time through the sample t S by means of the following relationship: u s Lrt S .
Ž 1.
In order to achieve satisfactory ultrasonic speed data using this method, the length of passage L must be accurately known. As L is affected by the pressure and temperature constraints it is essential to introduce terms to correct for these effects. This was done by ascribing to the length L the following temperature and pressure dependency: L Ž p, T . s L0 1 q a Ž T y T0 . 4 . 1 q b Ž p y p 0 . 4 ,
Ž 2.
in which the coefficients a and b, that are linked to the properties of the materials used in the manufacture of the autoclave cell, were determined by calibration with water,Ž3 ] 5. and were subsequently checked with pentane.Ž6,7. The error in u due to the determination of the length LŽT, p . and the transit time t S is about "0.06 per cent.Ž2. However, two further sources of error in u have to be taken into account. The first one stems from the uncertainty in temperature and leads to an error in u of about "0.03 per cent.Ž2. The second one is connected to the quality of the pressure gauge ŽHBM P3M. and leads to an error in u of "0.1 per cent at most Žat higher pressures..
3. Ultrasonic speed measurement The two systems studied were restricted to three substances belonging to the families of chemicals found essentially in natural fluids Žparaffins, naphthenes, and aromatics. with a proportion Žby mass fraction. of 0.40 for the component of the alkane family, 0.35 for the naphthene, and 0.25 for the aromatic compound in such
Speed of sound of synthetic cuts
609
FIGURE 1. High pressure measurement cell. 1, spring; 2, backing element; 3, piezo-electric element; 4, sample; 5, temperature probe; 6, autoclave cylinder; 7, inlet valve; 8, buffer.
a way as to match the average compositions observed in reservoir fluids. Moreover, the three compounds Žtable 1. were chosen within each chemical family in such a way as to obtain similar bubble point temperatures. The first system, which is composed of C 13 substances, has a bubble point temperature close to T s 520 K and a mean molar mass of M s 181.6 g . moly1 , whereas the second, which is made up of C16 and C17 substances and is therefore heavier Ž M s 229.0 g . moly1 ., was designed so as to have a bubble point temperature of approximately T s 573 K. The paraffins and aromatic compounds used in the preparation of these systems
610
J. L. Daridon, A. Lagrabette, and B. Lagourette
TABLE 1. Components of the investigated mixtures, their formulae, boiling point temperatures T b and compositions System
Components
Chemical formula
Tb
Mass fraction
Mole fraction
S 250
Tridecane Heptylcyclohexane Heptylbenzene
C 13 H 28 C 13 H 26 C 13 H 20
507.8 K 518.1 K 519.2 K
0.40 0.35 0.25
0.3940 0.3485 0.2575
S 300
Heptadecane Decylcyclohexane Decylbenzene
C 17 H 36 C 16 H 32 C 16 H 26
576.0 K 570.8 K 571.1 K
0.40 0.35 0.25
0.3808 0.3571 0.2621
were supplied by Fluka and Aldrich companies, and had a mass fraction purity higher than 0.99, while the naphthenes, which has a mass fraction purity higher than 0.98, were supplied by Kasei. The ultrasonic measurements were performed at different fixed conditions of pressure and temperature, at regular intervals, to provide a representation of the isothermal Žfigure 2. and isobaric Žfigure 3. behaviour of the ultrasonic speed of the fluid. In all nine isothermal curves 10 K intervals between T s 293.15 K and T s 373.15 K were investigated. As all the systems are single-phase liquids at atmospheric pressure, the pressure measurements were carried out from p s 0.1 MPa up to the maximum pressure covered by the study in 5 MPa steps. This maximum pressure was fixed at p s 150 MPa for the S 250 synthetic cut, whereas
FIGURE 2. Speed of propagation of ultrasound waves u in the S 250 system against pressure p: T s 293.15 K; ', T s 333.15 K; `, T s 373.15 K.
I,
Speed of sound of synthetic cuts
611
FIGURE 3. Speed of propagation of ultrasound waves u in the S 300 system against temperature T : p s 0.1 MPa; ', p s 20 MPa; ^, p s 40 MPa; v, p s 60 MPa; `, p s 80 MPa; l, p s 100 MPa.
I,
for the S 300 cut the appearance of solid deposits under high pressures compelled us to limit the maximum pressure to 100 MPa. The speeds obtained Žassociated with a frequency of 3 MHz. are displayed in tables 2 and 3. The experimental ultrasound wave propagation speed data can be correlated, to a high degree of accuracy,Ž3,8. by means of two-dimensional polynomial forms with 15 or 20 adjustable parameters. However, these polynomial expressions do not yield simple analytical forms for the integral of 1ru 2 which is involved in the indirect determination of density under pressure. For this type of procedure it is therefore preferable to correlate directly the inverse of the square of the ultrasonic speed with temperature and pressure. Unfortunately, although the dependency of this parameter with pressure at a fixed temperature can easily be reproduced by a polynomial form,Ž9. its representation by a double power series in p and T requires a substantial number of adjustable parameters to obtain a satisfactory accuracy. We therefore opted for a rational fraction with a denominator limited to the first degree to describe the behaviour of the term 1ru 2 . This function was used to represent each of the systems studied with an absolute average deviation of the order of 0.02 per cent Žtable 4. despite the presence of only eight adjustable parameters: 1ru 2 s Ž A q Bp q Cp 2 q Dp 3 . r Ž E q Fp . ,
Ž 3.
A s A 0 q A 1T q A 2 T 2 ,
Ž 4.
E s 1 q E1T .
Ž 5.
with
612
J. L. Daridon, A. Lagrabette, and B. Lagourette
TABLE 2. Ultrasonic velocity u in Ž0.40C 13 H 28 q 0.35C 13 H 26 q 0.25C 13 H 20 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150
333.15
343.15
353.15
363.15
373.15
1165.1 1195.5 1225.7 1254.9 1282.6 1309.0 1334.1 1358.8 1382.5 1405.1 1427.3 1448.8 1469.3 1489.6 1509.2 1527.9 1547.4 1565.3 1583.0 1600.7 1617.2 1635.2 1651.0 1667.1 1682.3 1697.9 1713.2 1728.4 1743.3 1757.7 1772.0
1129.5 1162.4 1194.5 1224.2 1253.2 1279.7 1306.3 1330.7 1354.8 1378.7 1400.2 1422.6 1443.5 1464.4 1484.1 1503.0 1522.1 1541.4 1559.6 1577.4 1594.6 1611.1 1629.0 1644.9 1660.4 1676.0 1691.0 1706.7 1721.5 1736.4 1750.3
1095.3 1129.7 1162.7 1193.3 1222.6 1250.5 1277.3 1303.2 1327.9 1352.6 1374.6 1397.7 1418.1 1439.3 1459.6 1479.2 1498.9 1517.5 1535.9 1553.9 1571.6 1588.7 1604.9 1622.6 1638.8 1654.4 1669.9 1684.9 1700.1 1715.5 1729.6
1062.1 1097.1 1131.3 1161.3 1193.1 1222.0 1249.4 1275.9 1301.3 1325.8 1348.1 1371.6 1393.4 1415.0 1435.7 1455.8 1475.2 1494.3 1513.5 1530.9 1549.3 1566.8 1584.2 1601.6 1616.6 1633.0 1648.7 1665.1 1678.7 1692.4 1708.0
urŽm . sy1 .
prMPa 1346.5 1373.2 1398.9 1423.4 1447.1 1469.8 1491.9 1512.8 1534.0 1554.2 1573.8 1592.9 1611.2 1628.9 1647.3 1664.6 1681.6 1698.7 1714.7 1730.8 1746.4 1761.7 1777.0 1792.4 1806.3 1821.1 1835.5 1849.1 1863.0 1876.6 1889.9
1308.7 1336.0 1362.3 1388.0 1412.6 1435.8 1458.3 1480.2 1501.9 1522.3 1542.4 1562.1 1581.3 1599.7 1618.0 1636.2 1653.1 1670.0 1687.0 1703.1 1718.8 1734.4 1749.8 1765.3 1780.6 1794.3 1809.5 1823.6 1837.3 1851.3 1864.3
1271.9 1300.2 1327.8 1353.9 1379.1 1403.5 1426.7 1449.0 1470.9 1492.1 1512.8 1532.5 1552.4 1571.4 1589.8 1607.9 1625.6 1642.8 1659.8 1676.7 1692.5 1708.4 1724.5 1739.4 1755.1 1769.7 1784.2 1798.9 1813.1 1827.1 1840.5
1235.0 1264.9 1293.0 1320.5 1346.4 1370.8 1395.4 1418.0 1440.5 1462.1 1484.3 1504.0 1523.8 1543.1 1562.3 1580.8 1598.7 1616.0 1633.9 1650.4 1666.9 1683.9 1699.9 1715.0 1730.0 1745.1 1760.1 1774.9 1788.8 1803.1 1817.0
1199.1 1229.7 1259.0 1287.2 1313.9 1339.7 1364.2 1388.1 1411.1 1433.1 1454.7 1475.6 1496.4 1516.1 1535.5 1553.8 1572.4 1590.6 1608.5 1627.1 1642.1 1658.5 1674.3 1690.1 1706.0 1721.6 1735.9 1751.1 1765.6 1779.8 1794.1
Moreover, by comparing the average and absolute average deviations it is found that the equation does not introduce any systematic error regarding the calculation of ultrasonic speed. The simple form of the denominator of equation Ž3. allows the integral of 1ru 2 to be expanded in the following analytical form: y2
Hu
dp s p Ž BrF y CErF 2 q DE 2rF 3 . q p 2 Ž CrF y DErF 2 . r2 q p 3 Ž DrF . r3 q
Ž ArF y BErF 2 q CE 2rF 3 y DE 3rF 4 . ln Ž E q Fp . ,
Ž 6.
which represents most of the variation of density linked to pressure change.
4. Density measurement The data on the speed of propagation of ultrasonic waves obtained as outlined
613
Speed of sound of synthetic cuts
TABLE 3. Ultrasonic velocity u in Ž0.40C 17 H 36 q 0.35C 16 H 32 q 0.25C 16 H 26 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
333.15
343.15
353.15
363.15
373.15
1211.7 1241.6 1270.6 1298.0 1324.4 1348.9 1374.3 1397.5 1420.6 1442.0 1463.5 1484.5 1504.5 1524.2 1543.4 1562.1 1580.3 1598.0 1615.4 1632.5 1649.4
1177.8 1209.0 1239.1 1267.5 1294.7 1321.0 1345.8 1369.6 1393.1 1415.4 1437.7 1458.6 1478.7 1498.5 1518.2 1537.1 1555.5 1573.7 1591.5 1608.9 1625.8
1144.9 1177.1 1208.2 1237.1 1265.2 1291.8 1317.6 1343.4 1366.2 1389.3 1411.2 1432.5 1453.6 1474.0 1494.0 1513.3 1532.2 1550.3 1568.4 1585.7 1603.0
1112.5 1146.2 1178.2 1208.2 1237.0 1264.4 1290.5 1315.6 1339.6 1363.8 1386.4 1408.2 1429.3 1450.2 1470.2 1489.7 1509.0 1527.5 1545.8 1563.6 1581.1
urŽm . sy1 .
prMPa 1391.0 1415.7 1440.1 1463.2 1486.1 1508.1 1529.4 1550.1 1570.4 1589.9 1609.2 1627.8 1646.0 1663.7 ] ] ] ] ] ] ]
1353.4 1379.2 1404.4 1428.6 1452.7 1475.1 1497.1 1518.3 1538.7 1558.8 1578.2 1597.2 1616.1 1634.0 1651.8 1669.2 1686.0 1702.7 1718.6 1734.5 1750.4
1316.5 1344.4 1369.8 1394.8 1419.1 1442.3 1465.3 1486.9 1507.9 1528.6 1548.1 1567.6 1586.6 1604.9 1623.0 1640.8 1658.1 1675.1 1691.5 1707.7 1723.7
1280.9 1309.0 1336.3 1362.0 1386.4 1410.4 1433.7 1456.5 1477.6 1498.7 1519.1 1539.0 1558.4 1577.5 1595.9 1614.0 1631.5 1648.7 1665.6 1681.8 1697.9
1246.0 1274.8 1302.5 1329.6 1355.1 1379.6 1403.5 1426.4 1448.7 1470.1 1491.0 1511.2 1531.1 1550.3 1569.3 1587.7 1605.2 1622.6 1639.8 1656.5 1673.0
above can be used to determine density and various thermoelastic coefficients under pressure. However, as the method is based on integration of Ž rr p .T , it is essential to know the behaviour of the density against temperature at a reference pressure. For this reason further density measurements were carried out at atmospheric pressure using an Anton-Paar densitometer ŽDMA 60 model.. The
TABLE 4. Parameters of equation Ž3.
A 0rŽmy2 . s 2 . A1rŽmy2 . s 2 . Ky1 . A 2rŽmy2 . s 2 . Ky2 . BrŽmy2 . s 2 . MPay1 . CrŽmy2 . s 2 . MPay2 . DrŽmy2 . s 2 . MPay3 . E1 rKy1 FrMPay1 10 2 . ADa 10 2 . AADb 10 2 . MDc a
S 250
S 300
5.98781 . 10y09 1.20398 . 10y09 y1.00980 . 10y12 1.25712 . 10y09 y3.43420 . 10y12 6.42239 . 10y15 y1.73011 . 10y03 6.35585 . 10y03 7.8 . 10y04 2.2 . 10y02 1.3 . 10y01
y3.94960 . 10y09 1.21976 . 10y09 y9.91940 . 10y13 1.31159 . 10y09 y4.08580 . 10y12 9.75709 . 10y15 y1.64146 . 10y03 6.45235 . 10y03 3.0 . 10y04 1.4 . 10y02 8.1 . 10y02
AD s average deviation. AAD s absolute average deviation. c MD s maximum deviation. b
614
J. L. Daridon, A. Lagrabette, and B. Lagourette TABLE 5. Parameters of equation Ž7. System
S 250
S 300
r 0rŽkg . my3 . r 1rŽkg . my3 . Ky1 . r 2rŽkg . my3 . Ky2 . r 3rŽkg . my3 . Ky3 .
1.1802 . 10q03 y2.2636 4.6479 . 10y03 4.6633 . 10y06
1.0696 . 10q03 y1.2097 1.5570 . 10y03 y1.5488 . 10y06
principle of operation of this experimental apparatus consists in recording the periods of oscillation of a U-shaped tube and deducing the density which is linked to the square of the period by a linear equation whose two constants are determined by the method defined by Lagourette et al.Ž10. The accuracy obtained by this experimental apparatus can be estimated to be "1 . 10y2 kg . my3 from the calibration, noting that the DMA 60 densimeter can determine periods of oscillation to 6 significant figures and the temperature within the apparatus is measured by means of a platinum probe linked to an AOIP brand calibrating thermometer. For each system, the measurements, which were carried out in 10 K steps between T s 293.15 K and T s 373.15 K, were used to represent Žwith an absolute average deviation less than 0.005 per cent. density with respect to temperature by means of a third degree polynomial:
r s r 0 q r 1T q r 2 T 2 q r 3 T 3 ,
Ž 7.
for which the coefficients are displayed in table 5.
5. Determination of densities and compressibility coefficients at high pressure Ultrasonic wave propagation speeds are only of interest in thermophysics if they coincide with the speed of sound within the low frequency boundary c 0 . Within this frequency domain, the speed of sound represents a purely thermodynamic parameter linked to a reversible adiabatic process, and can be defined by the thermodynamic relationship: 1r2 c 0 s 1r Ž r . k S . ,
Ž 8.
which links it with the isentropic compressibility coefficient k S and the density r . For this to be the case, in other words for the measured speed u to correspond to the theoretical speed c 0 defined by this relationship, all that is needed is that the fluid should not present any dispersive phenomena with regard to speed between the very low frequency domain and the measurement frequency Ž3 MHz.. For the systems studied in this investigation, this condition is satisfied because the hydrocarbons present in the mixture do not generally present dispersive effects in the frequency domain concerned by the experiment.Ž11,12. In these conditions, the ultrasonic speed, assimilated with the speed of sound, can be correlated with
615
Speed of sound of synthetic cuts
various thermophysical properties by means of the isentropic compressibility coefficient k S :
k S s 1rr u 2 ,
Ž 9.
and the thermodynamic relationship which links this coefficient to the isothermal compressibility coefficient k T is:
k T s k S q Ta 2rr C p ,
Ž 10 .
in which a designates the isobaric coefficient of thermal expansion, and C p the isobaric heat capacity. Thus by replacing the product rk S by 1ru 2 , this relationship can be used to express variation in density with pressure during a process at constant temperature in terms of ultrasonic speed by:
Ž rr p . T s 1ru 2 q Ta 2rC p ,
Ž 11 .
which, by integration with respect to pressure, yields a relationship which explicitly links density to the speed of sound:
r Ž p, T . s r Ž p 0 , T . q
p
Hp
uy2 dp q T
0
p
Hp Ž a
2
rC p . dp,
Ž 12 .
0
in which p 0 represents the atmospheric pressure. In this expression, the density with respect to pressure at different measurement temperatures is defined by the sum of three terms. The first one, which provides the main contribution, can be obtained by direct measurement of the density at atmospheric pressure. The first integral, which corresponds to the predominant additive contribution, can be evaluated directly by means of the ultrasonic speeds along the isotherms considered. The second integral, which numerically only represents a few per cent of the first, can be calculated iteratively by Davis and Gordon’s methodŽ13. in which the behaviour with pressure of the parameters a and C p are described by means of the supplementary thermodynamic relationships:
Ž ar p . T s yŽ k Tr T . p ,
Ž C pr p . T s yT a 2 q Ž ar T . p 4 rr .
Ž 13 . Ž 14 .
It is therefore possible, using this method, to extend the determination of densities to pressure domains accessible through ultrasonic measurements. However, when the temperature interval separating the measurement isotherms is too large, calculation of the derivatives with respect to T Žof r , a , and most especially of k T ., which is necessary in this numerical procedure, can introduce substantial uncertainty in the densities calculated at the highest pressures. To avoid this, two modifications were introduced into the procedure. First, the ultrasonic speed measurements were correlated simultaneously against temperature and pressure by the rational function equation Ž3. to Ž5.4 so that the integration
616
J. L. Daridon, A. Lagrabette, and B. Lagourette
can be extended to any temperature between the two extremes of temperature covered by the study and is therefore no longer limited to the experimental isotherms. In other words, the temperature interval used in the numerical procedure is no longer imposed by the experimental protocol but, on the contrary, is fixed in such a way as to reduce the uncertainties linked to the numerical temperature step. Tests carried out with different values of this interval showed that below T s 3 K the results obtained were no longer affected by this parameter. The interval was therefore fixed at the value of 1 K, which meant that the isobars were represented by approximately one hundred points. Finally, in order to avoid having to determine the derivative of k T , the effect of pressure on the thermal expansion coefficient is determined by a ‘‘predictor corrector’’ type approach rather than by using equation Ž13. which reflects the equality of the cross second order derivatives. In this approach variation of the density over a small pressure interval D p is initially
TABLE 6. Densities r of Ž0.40C 13 H 28 q 0.35C 13 H 26 q 0.25C 13 H 20 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150
333.15
343.15
353.15
363.15
373.15
762.29 766.47 770.50 774.34 777.99 781.49 784.84 788.06 791.17 794.17 797.07 799.88 802.61 805.26 807.83 810.34 812.78 815.17 817.50 819.77 821.99 824.17 826.30 828.39 830.44 832.44 834.42 836.35 838.25 840.12 841.96
755.06 759.47 763.72 767.75 771.58 775.23 778.72 782.07 785.30 788.41 791.41 794.32 797.13 799.86 802.51 805.09 807.60 810.05 812.44 814.77 817.05 819.28 821.46 823.59 825.68 827.73 829.75 831.72 833.66 835.56 837.44
747.77 752.45 756.93 761.16 765.17 768.99 772.64 776.13 779.48 782.70 785.81 788.82 791.72 794.54 797.27 799.93 802.51 805.03 807.48 809.87 812.20 814.48 816.72 818.90 821.04 823.13 825.19 827.20 829.18 831.12 833.03
740.39 745.36 750.09 754.55 758.76 762.76 766.56 770.20 773.69 777.03 780.26 783.37 786.37 789.28 792.10 794.83 797.49 800.07 802.59 805.05 807.44 809.78 812.06 814.30 816.48 818.62 820.72 822.78 824.80 826.78 828.72
rrŽkg . my3 .
prMPa 798.55 801.79 804.98 808.04 811.00 813.86 816.64 819.32 821.94 824.48 826.95 829.37 831.72 834.02 836.27 838.47 840.63 842.74 844.81 846.84 848.84 850.79 852.72 854.61 856.47 858.30 860.10 861.88 863.62 865.35 867.05
791.19 794.59 797.92 801.12 804.20 807.17 810.04 812.82 815.52 818.15 820.70 823.18 825.60 827.96 830.27 832.53 834.73 836.89 839.01 841.08 843.12 845.11 847.07 849.00 850.89 852.75 854.58 856.39 858.16 859.91 861.63
783.91 787.49 790.97 794.31 797.52 800.61 803.59 806.47 809.26 811.97 814.60 817.16 819.65 822.08 824.45 826.76 829.02 831.23 833.40 835.52 837.59 839.63 841.63 843.59 845.52 847.42 849.28 851.11 852.91 854.69 856.44
776.69 780.45 784.10 787.59 790.94 794.16 797.25 800.24 803.13 805.93 808.65 811.28 813.85 816.35 818.78 821.15 823.47 825.74 827.95 830.12 832.25 834.33 836.37 838.37 840.34 842.27 844.16 846.03 847.86 849.67 851.44
769.49 773.45 777.29 780.94 784.44 787.79 791.01 794.11 797.11 800.00 802.81 805.53 808.18 810.75 813.25 815.69 818.07 820.39 822.66 824.88 827.05 829.18 831.26 833.31 835.31 837.28 839.22 841.11 842.98 844.82 846.63
617
Speed of sound of synthetic cuts
predicted by assuming the term Ta 2rC p to be constant and determining the density as a function of temperature Ž14. at pressure Ž p q D p . by:
r T , Ž p q D p. 4 s r Ž T , p. q
Hu
y2
dp q Ž Ta 2rC p . D p,
Ž 15 .
and by derivation to predict the coefficient a ŽT . at Ž p q D p .. In a second phase the values of r ŽT . and a ŽT . at Ž p q D p. obtained by this procedure are corrected. To achieve this the density at Ž p q D p . is recalculated by equation Ž12. in which the behaviour with respect to pressure of the coefficient a is interpolated linearly in the interval D p on the basis of known values at p and the predicted values at Ž p q D p ., while C p is considered constant during the integration. This corrective procedure is reiterated until the values of r calculated at Ž p q D p . no longer change. When this objective has been achieved, the heat capacity C p is also determined at Ž p q D p . by equation Ž14. in its integrated form. On the basis of r measurements carried out on the two systems at atmospheric pressure, and the knowledge of C p at atmospheric pressure,Ž15. the determination of densities are extended from atmospheric pressure up to 150 MPa by means of ultrasonic speeds using this numerical procedure. All the values obtained by this method are presented in tables 6 and 7. Simultaneous knowledge of u and r at the same temperatures and pressures TABLE 7. Densities r of Ž0.40C 17 H 36 q 0.35C 16 H 32 q 0.25C 16 H 26 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
prMPa 0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
333.15
343.15
353.15
363.15
373.15
775.22 779.06 782.78 786.33 789.73 792.99 796.13 799.16 802.08 804.91 807.65 810.31 812.89 815.40 817.85 820.24 822.56 824.84 827.06 829.23 831.36
768.34 772.37 776.28 779.99 783.54 786.94 790.21 793.35 796.38 799.31 802.14 804.89 807.55 810.14 812.66 815.12 817.51 819.85 822.13 824.36 826.54
761.43 765.68 769.78 773.67 777.38 780.92 784.32 787.58 790.72 793.76 796.69 799.52 802.27 804.94 807.54 810.06 812.52 814.92 817.26 819.55 821.78
754.50 758.98 763.29 767.37 771.24 774.93 778.47 781.86 785.11 788.25 791.28 794.21 797.05 799.80 802.47 805.07 807.59 810.06 812.46 814.80 817.09
rrŽkg . my3 . 809.74 812.73 815.67 818.51 821.25 823.91 826.49 829.00 831.44 833.81 836.12 838.38 840.58 842.74 ] ] ] ] ] ] ]
802.79 805.93 809.02 811.98 814.85 817.62 820.30 822.90 825.43 827.89 830.29 832.62 834.90 837.12 839.29 841.41 843.49 845.53 847.52 849.48 851.40
795.88 799.18 802.41 805.51 808.50 811.39 814.18 816.88 819.50 822.05 824.52 826.94 829.29 831.58 833.82 836.01 838.14 840.24 842.29 844.30 846.27
788.99 792.45 795.84 799.08 802.20 805.21 808.11 810.92 813.63 816.27 818.83 821.33 823.75 826.12 828.42 830.68 832.88 835.03 837.14 839.20 841.22
782.10 785.75 789.30 792.69 795.95 799.08 802.10 805.01 807.83 810.56 813.21 815.78 818.29 820.73 823.10 825.42 827.68 829.90 832.06 834.18 836.25
618
J. L. Daridon, A. Lagrabette, and B. Lagourette
FIGURE 4. Speed of propagation of ultrasound waves u in the S 250 system against density r : , along isobars; - - - -, along isotherms. v, 0.1 MPa; ^, 80 MPa; l, 150 MPa.
FIGURE 5. Isentropic compressibility coefficient k S of the S 250 system against temperature T : p s 0.1 MPa; ', p s 25 MPa; ^, p s 50 MPa; v, p s 75 MPa; `, p s 100 MPa; l, p s 125 MPa; p s 150 MPa.
I, e,
619
Speed of sound of synthetic cuts
TABLE 8. Coefficients of isentropic compressibility k S in Ž0.40C 13 H 28 q 0.35C 13 H 26 q 0.25C 13 H 20 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150
333.15
343.15
353.15
363.16
373.15
0.9664 0.9129 0.8638 0.8201 0.7814 0.7468 0.7158 0.6873 0.6613 0.6378 0.6159 0.5956 0.5772 0.5597 0.5435 0.5286 0.5138 0.5007 0.4881 0.4761 0.4652 0.4538 0.4440 0.4343 0.4255 0.4167 0.4083 0.4003 0.3925 0.3853 0.3783
1.0382 0.9746 0.9177 0.8691 0.8253 0.7876 0.7526 0.7221 0.6938 0.6673 0.6445 0.6221 0.6020 0.5830 0.5657 0.5499 0.5344 0.5196 0.5060 0.4933 0.4813 0.4703 0.4587 0.4488 0.4393 0.4301 0.4215 0.4128 0.4048 0.3969 0.3898
1.1148 1.0413 0.9773 0.9226 0.8744 0.8315 0.7932 0.7586 0.7275 0.6983 0.6735 0.6489 0.6280 0.6076 0.5888 0.5713 0.5546 0.5394 0.5250 0.5114 0.4985 0.4865 0.4754 0.4638 0.4535 0.4439 0.4346 0.4258 0.4173 0.4089 0.4013
1.1974 1.1146 1.0417 0.9827 0.9259 0.8779 0.8357 0.7976 0.7632 0.7321 0.7052 0.6785 0.6550 0.6328 0.6125 0.5937 0.5762 0.5597 0.5439 0.5300 0.5160 0.5031 0.4907 0.4788 0.4686 0.4581 0.4482 0.4383 0.4303 0.4223 0.4136
k S rGPay1
prMPa 0.6907 0.6614 0.6348 0.6108 0.5888 0.5687 0.5502 0.5333 0.5170 0.5021 0.4882 0.4752 0.4631 0.4519 0.4407 0.4304 0.4207 0.4112 0.4026 0.3942 0.3862 0.3787 0.3714 0.3642 0.3578 0.3513 0.3451 0.3393 0.3336 0.3281 0.3229
0.7379 0.7050 0.6753 0.6479 0.6232 0.6010 0.5805 0.5615 0.5436 0.5274 0.5122 0.4978 0.4844 0.4720 0.4600 0.4487 0.4384 0.4285 0.4188 0.4099 0.4015 0.3934 0.3856 0.3779 0.3707 0.3642 0.3574 0.3511 0.3452 0.3393 0.3339
0.7885 0.7512 0.7171 0.6868 0.6593 0.6341 0.6114 0.5906 0.5711 0.5531 0.5364 0.5210 0.5062 0.4926 0.4799 0.4679 0.4565 0.4458 0.4355 0.4257 0.4168 0.4081 0.3995 0.3918 0.3839 0.3768 0.3699 0.3631 0.3567 0.3505 0.3447
0.8441 0.8008 0.7628 0.7282 0.6975 0.6701 0.6442 0.6215 0.6000 0.5804 0.5613 0.5449 0.5292 0.5144 0.5004 0.4873 0.4751 0.4638 0.4524 0.4423 0.4325 0.4227 0.4138 0.4055 0.3976 0.3899 0.3824 0.3752 0.3686 0.3620 0.3558
0.9039 0.8551 0.8116 0.7729 0.7385 0.7072 0.6793 0.6535 0.6301 0.6086 0.5887 0.5701 0.5526 0.5366 0.5215 0.5078 0.4944 0.4818 0.4698 0.4579 0.4484 0.4384 0.4291 0.4201 0.4113 0.4030 0.3954 0.3877 0.3805 0.3737 0.3669
enabled us to plot bundles of curves uŽ r . along isotherms and isobars Žfigure 4., whose shapes have been found to be fully regular and almost linear. Moreover, this knowledge could be used to deduce adiabatic compressibility coefficients at the different pressures and temperatures of the investigation. These are given in tables 8 and 9, and represented along a few isobars in figure 5. It is observed from this figure that the dependency of the coefficient k S with respect to temperature is almost linear at high pressures and is equal to zero at the highest pressures. Finally, as the numerical procedure is based on an equation which links the isothermal and isentropic compressibility coefficients, it is also possible to calculate the coefficient k T at all the temperatures and pressures studied Žtables 10 and 11.. It can be seen from figure 6, which presents the isobaric behaviour of this coefficient, that the general shape of the bundle of curves is identical to that observed for the isentropic coefficient. The curves are simply translated by the
620
J. L. Daridon, A. Lagrabette, and B. Lagourette
TABLE 9. Coefficients of isentropic compressibility k S in Ž0.40C 17 H 36 q 0.35C 16 H 32 q 0.25C 16 H 26 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
333.15
343.15
353.15
363.16
373.15
0.8786 0.8326 0.7912 0.7548 0.7219 0.6930 0.6650 0.6407 0.6178 0.5975 0.5781 0.5600 0.5435 0.5279 0.5133 0.4996 0.4868 0.4747 0.4633 0.4525 0.4421
0.9383 0.8858 0.8390 0.7980 0.7614 0.7282 0.6987 0.6720 0.6470 0.6245 0.6031 0.5840 0.5664 0.5497 0.5338 0.5192 0.5056 0.4925 0.4802 0.4686 0.4577
1.0020 0.9426 0.8899 0.8445 0.8036 0.7674 0.7344 0.7036 0.6775 0.6527 0.6303 0.6095 0.5900 0.5718 0.5548 0.5391 0.5242 0.5106 0.4974 0.4853 0.4736
1.0708 1.0029 0.9438 0.8927 0.8474 0.8071 0.7714 0.7389 0.7098 0.6821 0.6575 0.6349 0.6142 0.5945 0.5765 0.5597 0.5438 0.5291 0.5151 0.5020 0.4895
k SrGPay1
prMPa 0.6383 0.6139 0.5912 0.5707 0.5514 0.5337 0.5173 0.5020 0.4877 0.4745 0.4619 0.4502 0.4391 0.4287 ] ] ] ] ] ] ]
0.6801 0.6523 0.6267 0.6034 0.5816 0.5621 0.5439 0.5272 0.5117 0.4971 0.4836 0.4708 0.4586 0.4474 0.4367 0.4265 0.4171 0.4079 0.3995 0.3913 0.3834
0.7249 0.6923 0.6641 0.6382 0.6141 0.5925 0.5720 0.5537 0.5367 0.5206 0.5061 0.4921 0.4790 0.4669 0.4553 0.4443 0.4340 0.4242 0.4149 0.4062 0.3977
0.7725 0.7365 0.7037 0.6746 0.6485 0.6243 0.6020 0.5813 0.5629 0.5454 0.5292 0.5140 0.4998 0.4864 0.4740 0.4621 0.4511 0.4406 0.4306 0.4213 0.4123
0.8235 0.7831 0.7468 0.7137 0.6842 0.6575 0.6330 0.6105 0.5898 0.5708 0.5532 0.5368 0.5213 0.5070 0.4933 0.4806 0.4689 0.4577 0.4470 0.4369 0.4272
term Ta 2rr C p which is practically temperature independent and whose contribution falls as the pressure rises. The data obtained from experimental measurements on the two ternary mixtures of hydrocarbon compounds are particularly useful as they are very difficult to predict from corresponding properties of the pure compounds. Actually, even for ideal solutions, the properties u, k S , and k T are not calculated by a simple linear relation but according to the following expression: Ž16.
k Sid s
Ý fi Ž k S q TVi a i2rCpi . y T Ý x iVi Ý fi a i
ž
i
i
u id s Ž r idk Sid .
y1 r2
i
,
/ž
i
2
/ ž Ý x Cp / , i
i
Ž 16 .
i
Ž 17 .
which requires the accurate knowledge of volume fractions f i and the thermodynamic properties a , k S , C p , and V of each pure component under the same conditions of pressure and temperature.
7. Conclusion The work reported in this paper is part of a programme of systematic investigation of synthetic cuts, including heavy hydrocarbon compounds with very close boiling points, in other words cuts simulating fractions of petroleum distillation. These cuts
621
Speed of sound of synthetic cuts
TABLE 10. Coefficients of isothermal compressibility k T in Ž0.40C 13 H 28 q 0.35C 13 H 26 q 0.25C 13 H 20 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150
333.15
343.15
353.15
363.16
373.15
1.1493 1.0820 1.0209 0.9667 0.9189 0.8763 0.8383 0.8034 0.7718 0.7431 0.7166 0.6920 0.6697 0.6487 0.6292 0.6113 0.5938 0.5780 0.5630 0.5487 0.5357 0.5223 0.5106 0.4992 0.4886 0.4783 0.4684 0.4589 0.4499 0.4414 0.4331
1.2311 1.1518 1.0814 1.0212 0.9673 0.9208 0.8781 0.8407 0.8062 0.7742 0.7464 0.7195 0.6953 0.6725 0.6518 0.6327 0.6143 0.5967 0.5805 0.5654 0.5513 0.5381 0.5246 0.5128 0.5016 0.4908 0.4806 0.4704 0.4610 0.4519 0.4435
1.3198 1.2284 1.1490 1.0813 1.0218 0.9693 0.9225 0.8804 0.8426 0.8074 0.7773 0.7478 0.7225 0.6980 0.6755 0.6547 0.6348 0.6167 0.5996 0.5835 0.5683 0.5540 0.5409 0.5274 0.5153 0.5040 0.4931 0.4828 0.4728 0.4630 0.4542
1.4169 1.3134 1.2231 1.1493 1.0800 1.0212 0.9697 0.9233 0.8817 0.8441 0.8114 0.7795 0.7512 0.7247 0.7004 0.6780 0.6572 0.6376 0.6190 0.6025 0.5859 0.5707 0.5562 0.5423 0.5303 0.5179 0.5064 0.4949 0.4854 0.4760 0.4660
k T rGPay1
prMPa 0.8471 0.8096 0.7758 0.7452 0.7173 0.6920 0.6686 0.6473 0.6270 0.6084 0.5911 0.5749 0.5599 0.5459 0.5322 0.5195 0.5076 0.4960 0.4854 0.4751 0.4655 0.4563 0.4474 0.4388 0.4310 0.4231 0.4156 0.4087 0.4018 0.3952 0.3889
0.8970 0.8553 0.8175 0.7830 0.7519 0.7240 0.6984 0.6747 0.6525 0.6324 0.6135 0.5958 0.5793 0.5640 0.5494 0.5356 0.5229 0.5109 0.4991 0.4883 0.4781 0.4683 0.4589 0.4497 0.4410 0.4332 0.4250 0.4176 0.4104 0.4034 0.3969
0.9515 0.9044 0.8615 0.8235 0.7890 0.7577 0.7293 0.7035 0.6795 0.6573 0.6368 0.6179 0.5998 0.5832 0.5677 0.5530 0.5392 0.5262 0.5139 0.5021 0.4913 0.4807 0.4706 0.4612 0.4519 0.4433 0.4351 0.4270 0.4193 0.4120 0.4051
1.0123 0.9580 0.9103 0.8672 0.8289 0.7949 0.7629 0.7348 0.7085 0.6844 0.6612 0.6411 0.6219 0.6040 0.5870 0.5712 0.5565 0.5427 0.5292 0.5170 0.5052 0.4936 0.4829 0.4730 0.4636 0.4544 0.4455 0.4370 0.4291 0.4213 0.4140
1.0786 1.0175 0.9633 0.9152 0.8725 0.8340 0.7996 0.7679 0.7392 0.7130 0.6887 0.6662 0.6450 0.6257 0.6075 0.5909 0.5748 0.5597 0.5454 0.5314 0.5198 0.5079 0.4968 0.4861 0.4757 0.4658 0.4568 0.4478 0.4393 0.4312 0.4233
share two major features: they involve aliphatic, naphthenic, and aromatic chemical families, which predominate in the composition of natural fluids, and from the point of view of their composition they respect the average proportions observed as a result of analysis of a large number of crude oils. The objective pursued in this programme of investigations is to build up a body of data on densities and derived properties for ternary mixtures, or mixtures involving a larger number of components which are normally grouped together indistinctly in the heavy fraction of reservoir fluids. These data, through the diversity of the properties studied and the extend of the pressure and temperature domain covered, will be of use in testing the reliability of models designed to represent heavy fractions. The authors wish to thank F. Montel for his highly appreciated assistance. They are also indebted to the Societe ´ ´ ELF AQUITAINE for financial support.
622
J. L. Daridon, A. Lagrabette, and B. Lagourette
TABLE 11. Coefficients of isothermal compressibility k T in Ž0.40C 17 H 36 q 0.35C 16 H 32 q 0.25C 16 H 26 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
333.15
343.15
353.15
363.16
373.15
1.0356 0.9789 0.9280 0.8833 0.8431 0.8077 0.7739 0.7442 0.7166 0.6919 0.6686 0.6469 0.6270 0.6083 0.5908 0.5745 0.5592 0.5448 0.5312 0.5183 0.5060
1.1023 1.0380 0.9808 0.9306 0.8861 0.8459 0.8101 0.7777 0.7477 0.7206 0.6950 0.6720 0.6509 0.6310 0.6121 0.5947 0.5785 0.5630 0.5485 0.5347 0.5218
1.1738 1.1012 1.0370 0.9817 0.9320 0.8882 0.8485 0.8117 0.7802 0.7504 0.7236 0.6988 0.6755 0.6540 0.6338 0.6152 0.5977 0.5815 0.5660 0.5517 0.5379
1.2513 1.1686 1.0967 1.0346 0.9799 0.9314 0.8883 0.8494 0.8145 0.7816 0.7523 0.7255 0.7008 0.6776 0.6563 0.6365 0.6178 0.6005 0.5841 0.5687 0.5541
k T rGPay1
prMPa 0.7685 0.7375 0.7086 0.6826 0.6583 0.6361 0.6155 0.5964 0.5786 0.5621 0.5465 0.5320 0.5183 0.5054 ] ] ] ] ] } ]
0.8147 0.7797 0.7474 0.7182 0.6909 0.6666 0.6440 0.6232 0.6040 0.5859 0.5692 0.5535 0.5385 0.5247 0.5116 0.4992 0.4876 0.4765 0.4662 0.4562 0.4466
0.8644 0.8238 0.7884 0.7560 0.7261 0.6992 0.6740 0.6514 0.6304 0.6107 0.5928 0.5757 0.5597 0.5449 0.5308 0.5175 0.5049 0.4931 0.4819 0.4712 0.4610
0.9173 0.8725 0.8318 0.7957 0.7633 0.7335 0.7061 0.6809 0.6583 0.6369 0.6171 0.5987 0.5814 0.5652 0.5501 0.5358 0.5224 0.5098 0.4978 0.4866 0.4758
0.9741 0.9240 0.8790 0.8383 0.8021 0.7693 0.7394 0.7120 0.6868 0.6637 0.6423 0.6225 0.6038 0.5865 0.5701 0.5549 0.5407 0.5273 0.5145 0.5024 0.4909
FIGURE 6. Isothermal compressibility coefficient k T of the S 250 system against temperature T : p s 0.1 MPa; ', p s 25 MPa; ^, p s 50 MPa; v, p s 75 MPa; `, p s 100 MPa; l, p s 125 MPa; p s 150 MPa.
I, e,
Speed of sound of synthetic cuts
623
REFERENCES 1. Ye, S.; Alliez, J.; Lagourette, B.; Saint-Guirons, H.; Arman, J.; Xans, P. Re¨ ue Phys. Appl. 1990, 25, 555]565. 2. Daridon, J. L. Acustica 1994, 80, 416]419. 3. Wilson, W. D. J. Acoust. Soc. Am. 1959, 31, 1067]1072. 4. Del Grosso, V. A.; Mader, C. W. J. Acoust. Soc. Am. 1972, 52, 1442. 5. Petitet, J. P.; Tufeu, R.; Le Neindre, B. Int. J. Thermophys. 1983, 4, 35]47. 6. Chavez, M.; Palacios, J. M.; Tsumura, R. J. Chem. Eng. Data 1982, 27, 350]351. 7. Lainez, A.; Zollweg, J. A.; Street, W. B. J. Chem. Thermodynamics 1990, 22, 937]948. 8. Bobik, M.; Niepmann, R.; Marius, W. J. Chem. Thermodynamics 1979, 11, 351]357. 9. Muringer, M. J. P.; Trappeniers, N. J.; Biswas, S. N. Phys. Chem. Liq. 1985, 14, 273]296. 10. Lagourette, B.; Boned, C.; Saint-Guirons, H.; Xans, P.; Zhou, H. Meas. Sci. Technol. 1992, 3, 699]703. 11. Lamb, J. Physical Acoustics, Principles and Methods, Vol. 2. Mason, W. P.: editor. Academic Press: New York. 1965, p. 203. 12. Hakim, S. E. A.; Comley, W. J. Nature 1965, 208, 1082]1083. 13. Davis, L. A.; Gordon, R. B. J. Chem. Phys. 1967, 46, 2650]2660. 14. Denielou, L.; Petitet, J. P.; Tequi, C.; Syfosse, G. Bull. Mineral. ´ 1983, 106, 139]146. 15. Bessieres, D. LHP Universite ´ de Pau. Personal communication. 1997. 16. Handa, Y. P.; Halpin, C. J.; Benson, G. C. J. Chem. Thermodynamics 1981, 13, 875]886.
(Recei¨ ed 16 June 1997; in final form 27 No¨ ember 1997)
WA97r046
Speed of sound, density, and compressibilities of heavy synthetic cuts from ultrasonic measurements under pressure J. L. Daridon, a A. Lagrabette, and B. Lagourette Laboratoire Haute Pression, Centre Uni¨ ersitaire de Recherche Scientifique, Uni¨ ersite´ de Pau, A¨ enue de l’Uni¨ ersite, ´ 64000 Pau, France Thermophysical properties such as speed of sound, density, and isentropic and isothermal compressibilities have been obtained from ultrasonic speed measurements on two synthetic systems which are representative of distillation cuts with high bubble points ŽT s 520 K and T s 573 K, respectively.. The first system Žcomposition expressed in mass fraction., Ž0.40C 13 H 28 q 0.35C 13 H 26 q 0.25C 13 H 20 ., was investigated at pressures from Ž0.1 to 150. MPa in the temperature range T s Ž293.15 to 373.15. K, whereas the second one, Ž0.40C 17 H 36 q 0.35C 16 H 32 q 0.25C 16 H 26 ., was investigated at pressures up to 100 MPa in the temperature range T s Ž293.15 to 373.15. K. Q 1998 Academic Press Limited KEYWORDS: speed of sound; density; compressibility; pressure; hydrocarbon
1. Introduction Because of the great complexity of most reservoir fluids it has become usual to adopt an analytical representation of reservoir fluids which consists of considering the fluids of interest as a juxtaposition of groups of components and of a heavy fraction represented as C nq. The major weak point in modelling a complete petroleum fluid resides in the inadequate description of the undifferentiated heavy fraction, often assimilated to C 11q. In order to develop effective methods, experimental data are required on systems composed of a number of components which are representative of petroleum distillation fractions with high bubble points, and hence made up of heavy components. In this context, and in order to acquire a substantial body of experimental data on different thermophysical properties over a wide range of pressure Ž0.1 to 150. MPa, we conducted measurements of ultrasonic speed under pressure, on two representative synthetic systems of distillation cuts; in other words, systems composed of substances with identical bubble points and with compositions respecting the average proportions determined by compositional analysis of a large number of real fluids. The first system ŽS 250., made up of Ž0.40C 13 H 28 q 0.35C 13 H 26 q 0.25C 13 H 20 ., has a bubble point of approximately T s 520 K, a
To whom correspondence should be addressed.
0021]9614r98r050607 q 17 $25.00r0
Q 1998 Academic Press Limited
608
J. L. Daridon, A. Lagrabette, and B. Lagourette
whereas the second ŽS 300., which is composed of Ž0.40C 17 H 36 q 0.35C 16 H 32 q 0.25C 16 H 26 ., has a bubble point of approximately T s 573 K. The ultrasonic speed data acquired in this research, linked with complementary density measurements made at atmospheric pressure, were used to calculate densities, and isentropic and isothermal compressibility coefficients under pressure.
2. Experimental The technique deployed to carry out measurements of ultrasonic speed waves u in systems under high pressure is based on a pulse technique in which the piezoelectric elements are isolated from the liquid studied, and hence from the effects of pressure, by the presence of intermediate bars. The experimental apparatus, which has already been described in detail elsewhere,Ž1. is composed essentially of an autoclave cell which has been further improved so as to extend the domain of measurement to p s 200 MPa. This cylindrical-shaped cell Žfigure 1. is closed at each end by two identical buffers which serve as a connecting medium between the piezo-electric transducers and the test fluid. The ultrasonic speed is deduced from measurement Ž2. of the transit time through the sample t S by means of the following relationship: u s Lrt S .
Ž 1.
In order to achieve satisfactory ultrasonic speed data using this method, the length of passage L must be accurately known. As L is affected by the pressure and temperature constraints it is essential to introduce terms to correct for these effects. This was done by ascribing to the length L the following temperature and pressure dependency: L Ž p, T . s L0 1 q a Ž T y T0 . 4 . 1 q b Ž p y p 0 . 4 ,
Ž 2.
in which the coefficients a and b, that are linked to the properties of the materials used in the manufacture of the autoclave cell, were determined by calibration with water,Ž3 ] 5. and were subsequently checked with pentane.Ž6,7. The error in u due to the determination of the length LŽT, p . and the transit time t S is about "0.06 per cent.Ž2. However, two further sources of error in u have to be taken into account. The first one stems from the uncertainty in temperature and leads to an error in u of about "0.03 per cent.Ž2. The second one is connected to the quality of the pressure gauge ŽHBM P3M. and leads to an error in u of "0.1 per cent at most Žat higher pressures..
3. Ultrasonic speed measurement The two systems studied were restricted to three substances belonging to the families of chemicals found essentially in natural fluids Žparaffins, naphthenes, and aromatics. with a proportion Žby mass fraction. of 0.40 for the component of the alkane family, 0.35 for the naphthene, and 0.25 for the aromatic compound in such
Speed of sound of synthetic cuts
609
FIGURE 1. High pressure measurement cell. 1, spring; 2, backing element; 3, piezo-electric element; 4, sample; 5, temperature probe; 6, autoclave cylinder; 7, inlet valve; 8, buffer.
a way as to match the average compositions observed in reservoir fluids. Moreover, the three compounds Žtable 1. were chosen within each chemical family in such a way as to obtain similar bubble point temperatures. The first system, which is composed of C 13 substances, has a bubble point temperature close to T s 520 K and a mean molar mass of M s 181.6 g . moly1 , whereas the second, which is made up of C16 and C17 substances and is therefore heavier Ž M s 229.0 g . moly1 ., was designed so as to have a bubble point temperature of approximately T s 573 K. The paraffins and aromatic compounds used in the preparation of these systems
610
J. L. Daridon, A. Lagrabette, and B. Lagourette
TABLE 1. Components of the investigated mixtures, their formulae, boiling point temperatures T b and compositions System
Components
Chemical formula
Tb
Mass fraction
Mole fraction
S 250
Tridecane Heptylcyclohexane Heptylbenzene
C 13 H 28 C 13 H 26 C 13 H 20
507.8 K 518.1 K 519.2 K
0.40 0.35 0.25
0.3940 0.3485 0.2575
S 300
Heptadecane Decylcyclohexane Decylbenzene
C 17 H 36 C 16 H 32 C 16 H 26
576.0 K 570.8 K 571.1 K
0.40 0.35 0.25
0.3808 0.3571 0.2621
were supplied by Fluka and Aldrich companies, and had a mass fraction purity higher than 0.99, while the naphthenes, which has a mass fraction purity higher than 0.98, were supplied by Kasei. The ultrasonic measurements were performed at different fixed conditions of pressure and temperature, at regular intervals, to provide a representation of the isothermal Žfigure 2. and isobaric Žfigure 3. behaviour of the ultrasonic speed of the fluid. In all nine isothermal curves 10 K intervals between T s 293.15 K and T s 373.15 K were investigated. As all the systems are single-phase liquids at atmospheric pressure, the pressure measurements were carried out from p s 0.1 MPa up to the maximum pressure covered by the study in 5 MPa steps. This maximum pressure was fixed at p s 150 MPa for the S 250 synthetic cut, whereas
FIGURE 2. Speed of propagation of ultrasound waves u in the S 250 system against pressure p: T s 293.15 K; ', T s 333.15 K; `, T s 373.15 K.
I,
Speed of sound of synthetic cuts
611
FIGURE 3. Speed of propagation of ultrasound waves u in the S 300 system against temperature T : p s 0.1 MPa; ', p s 20 MPa; ^, p s 40 MPa; v, p s 60 MPa; `, p s 80 MPa; l, p s 100 MPa.
I,
for the S 300 cut the appearance of solid deposits under high pressures compelled us to limit the maximum pressure to 100 MPa. The speeds obtained Žassociated with a frequency of 3 MHz. are displayed in tables 2 and 3. The experimental ultrasound wave propagation speed data can be correlated, to a high degree of accuracy,Ž3,8. by means of two-dimensional polynomial forms with 15 or 20 adjustable parameters. However, these polynomial expressions do not yield simple analytical forms for the integral of 1ru 2 which is involved in the indirect determination of density under pressure. For this type of procedure it is therefore preferable to correlate directly the inverse of the square of the ultrasonic speed with temperature and pressure. Unfortunately, although the dependency of this parameter with pressure at a fixed temperature can easily be reproduced by a polynomial form,Ž9. its representation by a double power series in p and T requires a substantial number of adjustable parameters to obtain a satisfactory accuracy. We therefore opted for a rational fraction with a denominator limited to the first degree to describe the behaviour of the term 1ru 2 . This function was used to represent each of the systems studied with an absolute average deviation of the order of 0.02 per cent Žtable 4. despite the presence of only eight adjustable parameters: 1ru 2 s Ž A q Bp q Cp 2 q Dp 3 . r Ž E q Fp . ,
Ž 3.
A s A 0 q A 1T q A 2 T 2 ,
Ž 4.
E s 1 q E1T .
Ž 5.
with
612
J. L. Daridon, A. Lagrabette, and B. Lagourette
TABLE 2. Ultrasonic velocity u in Ž0.40C 13 H 28 q 0.35C 13 H 26 q 0.25C 13 H 20 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150
333.15
343.15
353.15
363.15
373.15
1165.1 1195.5 1225.7 1254.9 1282.6 1309.0 1334.1 1358.8 1382.5 1405.1 1427.3 1448.8 1469.3 1489.6 1509.2 1527.9 1547.4 1565.3 1583.0 1600.7 1617.2 1635.2 1651.0 1667.1 1682.3 1697.9 1713.2 1728.4 1743.3 1757.7 1772.0
1129.5 1162.4 1194.5 1224.2 1253.2 1279.7 1306.3 1330.7 1354.8 1378.7 1400.2 1422.6 1443.5 1464.4 1484.1 1503.0 1522.1 1541.4 1559.6 1577.4 1594.6 1611.1 1629.0 1644.9 1660.4 1676.0 1691.0 1706.7 1721.5 1736.4 1750.3
1095.3 1129.7 1162.7 1193.3 1222.6 1250.5 1277.3 1303.2 1327.9 1352.6 1374.6 1397.7 1418.1 1439.3 1459.6 1479.2 1498.9 1517.5 1535.9 1553.9 1571.6 1588.7 1604.9 1622.6 1638.8 1654.4 1669.9 1684.9 1700.1 1715.5 1729.6
1062.1 1097.1 1131.3 1161.3 1193.1 1222.0 1249.4 1275.9 1301.3 1325.8 1348.1 1371.6 1393.4 1415.0 1435.7 1455.8 1475.2 1494.3 1513.5 1530.9 1549.3 1566.8 1584.2 1601.6 1616.6 1633.0 1648.7 1665.1 1678.7 1692.4 1708.0
urŽm . sy1 .
prMPa 1346.5 1373.2 1398.9 1423.4 1447.1 1469.8 1491.9 1512.8 1534.0 1554.2 1573.8 1592.9 1611.2 1628.9 1647.3 1664.6 1681.6 1698.7 1714.7 1730.8 1746.4 1761.7 1777.0 1792.4 1806.3 1821.1 1835.5 1849.1 1863.0 1876.6 1889.9
1308.7 1336.0 1362.3 1388.0 1412.6 1435.8 1458.3 1480.2 1501.9 1522.3 1542.4 1562.1 1581.3 1599.7 1618.0 1636.2 1653.1 1670.0 1687.0 1703.1 1718.8 1734.4 1749.8 1765.3 1780.6 1794.3 1809.5 1823.6 1837.3 1851.3 1864.3
1271.9 1300.2 1327.8 1353.9 1379.1 1403.5 1426.7 1449.0 1470.9 1492.1 1512.8 1532.5 1552.4 1571.4 1589.8 1607.9 1625.6 1642.8 1659.8 1676.7 1692.5 1708.4 1724.5 1739.4 1755.1 1769.7 1784.2 1798.9 1813.1 1827.1 1840.5
1235.0 1264.9 1293.0 1320.5 1346.4 1370.8 1395.4 1418.0 1440.5 1462.1 1484.3 1504.0 1523.8 1543.1 1562.3 1580.8 1598.7 1616.0 1633.9 1650.4 1666.9 1683.9 1699.9 1715.0 1730.0 1745.1 1760.1 1774.9 1788.8 1803.1 1817.0
1199.1 1229.7 1259.0 1287.2 1313.9 1339.7 1364.2 1388.1 1411.1 1433.1 1454.7 1475.6 1496.4 1516.1 1535.5 1553.8 1572.4 1590.6 1608.5 1627.1 1642.1 1658.5 1674.3 1690.1 1706.0 1721.6 1735.9 1751.1 1765.6 1779.8 1794.1
Moreover, by comparing the average and absolute average deviations it is found that the equation does not introduce any systematic error regarding the calculation of ultrasonic speed. The simple form of the denominator of equation Ž3. allows the integral of 1ru 2 to be expanded in the following analytical form: y2
Hu
dp s p Ž BrF y CErF 2 q DE 2rF 3 . q p 2 Ž CrF y DErF 2 . r2 q p 3 Ž DrF . r3 q
Ž ArF y BErF 2 q CE 2rF 3 y DE 3rF 4 . ln Ž E q Fp . ,
Ž 6.
which represents most of the variation of density linked to pressure change.
4. Density measurement The data on the speed of propagation of ultrasonic waves obtained as outlined
613
Speed of sound of synthetic cuts
TABLE 3. Ultrasonic velocity u in Ž0.40C 17 H 36 q 0.35C 16 H 32 q 0.25C 16 H 26 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
333.15
343.15
353.15
363.15
373.15
1211.7 1241.6 1270.6 1298.0 1324.4 1348.9 1374.3 1397.5 1420.6 1442.0 1463.5 1484.5 1504.5 1524.2 1543.4 1562.1 1580.3 1598.0 1615.4 1632.5 1649.4
1177.8 1209.0 1239.1 1267.5 1294.7 1321.0 1345.8 1369.6 1393.1 1415.4 1437.7 1458.6 1478.7 1498.5 1518.2 1537.1 1555.5 1573.7 1591.5 1608.9 1625.8
1144.9 1177.1 1208.2 1237.1 1265.2 1291.8 1317.6 1343.4 1366.2 1389.3 1411.2 1432.5 1453.6 1474.0 1494.0 1513.3 1532.2 1550.3 1568.4 1585.7 1603.0
1112.5 1146.2 1178.2 1208.2 1237.0 1264.4 1290.5 1315.6 1339.6 1363.8 1386.4 1408.2 1429.3 1450.2 1470.2 1489.7 1509.0 1527.5 1545.8 1563.6 1581.1
urŽm . sy1 .
prMPa 1391.0 1415.7 1440.1 1463.2 1486.1 1508.1 1529.4 1550.1 1570.4 1589.9 1609.2 1627.8 1646.0 1663.7 ] ] ] ] ] ] ]
1353.4 1379.2 1404.4 1428.6 1452.7 1475.1 1497.1 1518.3 1538.7 1558.8 1578.2 1597.2 1616.1 1634.0 1651.8 1669.2 1686.0 1702.7 1718.6 1734.5 1750.4
1316.5 1344.4 1369.8 1394.8 1419.1 1442.3 1465.3 1486.9 1507.9 1528.6 1548.1 1567.6 1586.6 1604.9 1623.0 1640.8 1658.1 1675.1 1691.5 1707.7 1723.7
1280.9 1309.0 1336.3 1362.0 1386.4 1410.4 1433.7 1456.5 1477.6 1498.7 1519.1 1539.0 1558.4 1577.5 1595.9 1614.0 1631.5 1648.7 1665.6 1681.8 1697.9
1246.0 1274.8 1302.5 1329.6 1355.1 1379.6 1403.5 1426.4 1448.7 1470.1 1491.0 1511.2 1531.1 1550.3 1569.3 1587.7 1605.2 1622.6 1639.8 1656.5 1673.0
above can be used to determine density and various thermoelastic coefficients under pressure. However, as the method is based on integration of Ž rr p .T , it is essential to know the behaviour of the density against temperature at a reference pressure. For this reason further density measurements were carried out at atmospheric pressure using an Anton-Paar densitometer ŽDMA 60 model.. The
TABLE 4. Parameters of equation Ž3.
A 0rŽmy2 . s 2 . A1rŽmy2 . s 2 . Ky1 . A 2rŽmy2 . s 2 . Ky2 . BrŽmy2 . s 2 . MPay1 . CrŽmy2 . s 2 . MPay2 . DrŽmy2 . s 2 . MPay3 . E1 rKy1 FrMPay1 10 2 . ADa 10 2 . AADb 10 2 . MDc a
S 250
S 300
5.98781 . 10y09 1.20398 . 10y09 y1.00980 . 10y12 1.25712 . 10y09 y3.43420 . 10y12 6.42239 . 10y15 y1.73011 . 10y03 6.35585 . 10y03 7.8 . 10y04 2.2 . 10y02 1.3 . 10y01
y3.94960 . 10y09 1.21976 . 10y09 y9.91940 . 10y13 1.31159 . 10y09 y4.08580 . 10y12 9.75709 . 10y15 y1.64146 . 10y03 6.45235 . 10y03 3.0 . 10y04 1.4 . 10y02 8.1 . 10y02
AD s average deviation. AAD s absolute average deviation. c MD s maximum deviation. b
614
J. L. Daridon, A. Lagrabette, and B. Lagourette TABLE 5. Parameters of equation Ž7. System
S 250
S 300
r 0rŽkg . my3 . r 1rŽkg . my3 . Ky1 . r 2rŽkg . my3 . Ky2 . r 3rŽkg . my3 . Ky3 .
1.1802 . 10q03 y2.2636 4.6479 . 10y03 4.6633 . 10y06
1.0696 . 10q03 y1.2097 1.5570 . 10y03 y1.5488 . 10y06
principle of operation of this experimental apparatus consists in recording the periods of oscillation of a U-shaped tube and deducing the density which is linked to the square of the period by a linear equation whose two constants are determined by the method defined by Lagourette et al.Ž10. The accuracy obtained by this experimental apparatus can be estimated to be "1 . 10y2 kg . my3 from the calibration, noting that the DMA 60 densimeter can determine periods of oscillation to 6 significant figures and the temperature within the apparatus is measured by means of a platinum probe linked to an AOIP brand calibrating thermometer. For each system, the measurements, which were carried out in 10 K steps between T s 293.15 K and T s 373.15 K, were used to represent Žwith an absolute average deviation less than 0.005 per cent. density with respect to temperature by means of a third degree polynomial:
r s r 0 q r 1T q r 2 T 2 q r 3 T 3 ,
Ž 7.
for which the coefficients are displayed in table 5.
5. Determination of densities and compressibility coefficients at high pressure Ultrasonic wave propagation speeds are only of interest in thermophysics if they coincide with the speed of sound within the low frequency boundary c 0 . Within this frequency domain, the speed of sound represents a purely thermodynamic parameter linked to a reversible adiabatic process, and can be defined by the thermodynamic relationship: 1r2 c 0 s 1r Ž r . k S . ,
Ž 8.
which links it with the isentropic compressibility coefficient k S and the density r . For this to be the case, in other words for the measured speed u to correspond to the theoretical speed c 0 defined by this relationship, all that is needed is that the fluid should not present any dispersive phenomena with regard to speed between the very low frequency domain and the measurement frequency Ž3 MHz.. For the systems studied in this investigation, this condition is satisfied because the hydrocarbons present in the mixture do not generally present dispersive effects in the frequency domain concerned by the experiment.Ž11,12. In these conditions, the ultrasonic speed, assimilated with the speed of sound, can be correlated with
615
Speed of sound of synthetic cuts
various thermophysical properties by means of the isentropic compressibility coefficient k S :
k S s 1rr u 2 ,
Ž 9.
and the thermodynamic relationship which links this coefficient to the isothermal compressibility coefficient k T is:
k T s k S q Ta 2rr C p ,
Ž 10 .
in which a designates the isobaric coefficient of thermal expansion, and C p the isobaric heat capacity. Thus by replacing the product rk S by 1ru 2 , this relationship can be used to express variation in density with pressure during a process at constant temperature in terms of ultrasonic speed by:
Ž rr p . T s 1ru 2 q Ta 2rC p ,
Ž 11 .
which, by integration with respect to pressure, yields a relationship which explicitly links density to the speed of sound:
r Ž p, T . s r Ž p 0 , T . q
p
Hp
uy2 dp q T
0
p
Hp Ž a
2
rC p . dp,
Ž 12 .
0
in which p 0 represents the atmospheric pressure. In this expression, the density with respect to pressure at different measurement temperatures is defined by the sum of three terms. The first one, which provides the main contribution, can be obtained by direct measurement of the density at atmospheric pressure. The first integral, which corresponds to the predominant additive contribution, can be evaluated directly by means of the ultrasonic speeds along the isotherms considered. The second integral, which numerically only represents a few per cent of the first, can be calculated iteratively by Davis and Gordon’s methodŽ13. in which the behaviour with pressure of the parameters a and C p are described by means of the supplementary thermodynamic relationships:
Ž ar p . T s yŽ k Tr T . p ,
Ž C pr p . T s yT a 2 q Ž ar T . p 4 rr .
Ž 13 . Ž 14 .
It is therefore possible, using this method, to extend the determination of densities to pressure domains accessible through ultrasonic measurements. However, when the temperature interval separating the measurement isotherms is too large, calculation of the derivatives with respect to T Žof r , a , and most especially of k T ., which is necessary in this numerical procedure, can introduce substantial uncertainty in the densities calculated at the highest pressures. To avoid this, two modifications were introduced into the procedure. First, the ultrasonic speed measurements were correlated simultaneously against temperature and pressure by the rational function equation Ž3. to Ž5.4 so that the integration
616
J. L. Daridon, A. Lagrabette, and B. Lagourette
can be extended to any temperature between the two extremes of temperature covered by the study and is therefore no longer limited to the experimental isotherms. In other words, the temperature interval used in the numerical procedure is no longer imposed by the experimental protocol but, on the contrary, is fixed in such a way as to reduce the uncertainties linked to the numerical temperature step. Tests carried out with different values of this interval showed that below T s 3 K the results obtained were no longer affected by this parameter. The interval was therefore fixed at the value of 1 K, which meant that the isobars were represented by approximately one hundred points. Finally, in order to avoid having to determine the derivative of k T , the effect of pressure on the thermal expansion coefficient is determined by a ‘‘predictor corrector’’ type approach rather than by using equation Ž13. which reflects the equality of the cross second order derivatives. In this approach variation of the density over a small pressure interval D p is initially
TABLE 6. Densities r of Ž0.40C 13 H 28 q 0.35C 13 H 26 q 0.25C 13 H 20 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150
333.15
343.15
353.15
363.15
373.15
762.29 766.47 770.50 774.34 777.99 781.49 784.84 788.06 791.17 794.17 797.07 799.88 802.61 805.26 807.83 810.34 812.78 815.17 817.50 819.77 821.99 824.17 826.30 828.39 830.44 832.44 834.42 836.35 838.25 840.12 841.96
755.06 759.47 763.72 767.75 771.58 775.23 778.72 782.07 785.30 788.41 791.41 794.32 797.13 799.86 802.51 805.09 807.60 810.05 812.44 814.77 817.05 819.28 821.46 823.59 825.68 827.73 829.75 831.72 833.66 835.56 837.44
747.77 752.45 756.93 761.16 765.17 768.99 772.64 776.13 779.48 782.70 785.81 788.82 791.72 794.54 797.27 799.93 802.51 805.03 807.48 809.87 812.20 814.48 816.72 818.90 821.04 823.13 825.19 827.20 829.18 831.12 833.03
740.39 745.36 750.09 754.55 758.76 762.76 766.56 770.20 773.69 777.03 780.26 783.37 786.37 789.28 792.10 794.83 797.49 800.07 802.59 805.05 807.44 809.78 812.06 814.30 816.48 818.62 820.72 822.78 824.80 826.78 828.72
rrŽkg . my3 .
prMPa 798.55 801.79 804.98 808.04 811.00 813.86 816.64 819.32 821.94 824.48 826.95 829.37 831.72 834.02 836.27 838.47 840.63 842.74 844.81 846.84 848.84 850.79 852.72 854.61 856.47 858.30 860.10 861.88 863.62 865.35 867.05
791.19 794.59 797.92 801.12 804.20 807.17 810.04 812.82 815.52 818.15 820.70 823.18 825.60 827.96 830.27 832.53 834.73 836.89 839.01 841.08 843.12 845.11 847.07 849.00 850.89 852.75 854.58 856.39 858.16 859.91 861.63
783.91 787.49 790.97 794.31 797.52 800.61 803.59 806.47 809.26 811.97 814.60 817.16 819.65 822.08 824.45 826.76 829.02 831.23 833.40 835.52 837.59 839.63 841.63 843.59 845.52 847.42 849.28 851.11 852.91 854.69 856.44
776.69 780.45 784.10 787.59 790.94 794.16 797.25 800.24 803.13 805.93 808.65 811.28 813.85 816.35 818.78 821.15 823.47 825.74 827.95 830.12 832.25 834.33 836.37 838.37 840.34 842.27 844.16 846.03 847.86 849.67 851.44
769.49 773.45 777.29 780.94 784.44 787.79 791.01 794.11 797.11 800.00 802.81 805.53 808.18 810.75 813.25 815.69 818.07 820.39 822.66 824.88 827.05 829.18 831.26 833.31 835.31 837.28 839.22 841.11 842.98 844.82 846.63
617
Speed of sound of synthetic cuts
predicted by assuming the term Ta 2rC p to be constant and determining the density as a function of temperature Ž14. at pressure Ž p q D p . by:
r T , Ž p q D p. 4 s r Ž T , p. q
Hu
y2
dp q Ž Ta 2rC p . D p,
Ž 15 .
and by derivation to predict the coefficient a ŽT . at Ž p q D p .. In a second phase the values of r ŽT . and a ŽT . at Ž p q D p. obtained by this procedure are corrected. To achieve this the density at Ž p q D p . is recalculated by equation Ž12. in which the behaviour with respect to pressure of the coefficient a is interpolated linearly in the interval D p on the basis of known values at p and the predicted values at Ž p q D p ., while C p is considered constant during the integration. This corrective procedure is reiterated until the values of r calculated at Ž p q D p . no longer change. When this objective has been achieved, the heat capacity C p is also determined at Ž p q D p . by equation Ž14. in its integrated form. On the basis of r measurements carried out on the two systems at atmospheric pressure, and the knowledge of C p at atmospheric pressure,Ž15. the determination of densities are extended from atmospheric pressure up to 150 MPa by means of ultrasonic speeds using this numerical procedure. All the values obtained by this method are presented in tables 6 and 7. Simultaneous knowledge of u and r at the same temperatures and pressures TABLE 7. Densities r of Ž0.40C 17 H 36 q 0.35C 16 H 32 q 0.25C 16 H 26 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
prMPa 0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
333.15
343.15
353.15
363.15
373.15
775.22 779.06 782.78 786.33 789.73 792.99 796.13 799.16 802.08 804.91 807.65 810.31 812.89 815.40 817.85 820.24 822.56 824.84 827.06 829.23 831.36
768.34 772.37 776.28 779.99 783.54 786.94 790.21 793.35 796.38 799.31 802.14 804.89 807.55 810.14 812.66 815.12 817.51 819.85 822.13 824.36 826.54
761.43 765.68 769.78 773.67 777.38 780.92 784.32 787.58 790.72 793.76 796.69 799.52 802.27 804.94 807.54 810.06 812.52 814.92 817.26 819.55 821.78
754.50 758.98 763.29 767.37 771.24 774.93 778.47 781.86 785.11 788.25 791.28 794.21 797.05 799.80 802.47 805.07 807.59 810.06 812.46 814.80 817.09
rrŽkg . my3 . 809.74 812.73 815.67 818.51 821.25 823.91 826.49 829.00 831.44 833.81 836.12 838.38 840.58 842.74 ] ] ] ] ] ] ]
802.79 805.93 809.02 811.98 814.85 817.62 820.30 822.90 825.43 827.89 830.29 832.62 834.90 837.12 839.29 841.41 843.49 845.53 847.52 849.48 851.40
795.88 799.18 802.41 805.51 808.50 811.39 814.18 816.88 819.50 822.05 824.52 826.94 829.29 831.58 833.82 836.01 838.14 840.24 842.29 844.30 846.27
788.99 792.45 795.84 799.08 802.20 805.21 808.11 810.92 813.63 816.27 818.83 821.33 823.75 826.12 828.42 830.68 832.88 835.03 837.14 839.20 841.22
782.10 785.75 789.30 792.69 795.95 799.08 802.10 805.01 807.83 810.56 813.21 815.78 818.29 820.73 823.10 825.42 827.68 829.90 832.06 834.18 836.25
618
J. L. Daridon, A. Lagrabette, and B. Lagourette
FIGURE 4. Speed of propagation of ultrasound waves u in the S 250 system against density r : , along isobars; - - - -, along isotherms. v, 0.1 MPa; ^, 80 MPa; l, 150 MPa.
FIGURE 5. Isentropic compressibility coefficient k S of the S 250 system against temperature T : p s 0.1 MPa; ', p s 25 MPa; ^, p s 50 MPa; v, p s 75 MPa; `, p s 100 MPa; l, p s 125 MPa; p s 150 MPa.
I, e,
619
Speed of sound of synthetic cuts
TABLE 8. Coefficients of isentropic compressibility k S in Ž0.40C 13 H 28 q 0.35C 13 H 26 q 0.25C 13 H 20 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150
333.15
343.15
353.15
363.16
373.15
0.9664 0.9129 0.8638 0.8201 0.7814 0.7468 0.7158 0.6873 0.6613 0.6378 0.6159 0.5956 0.5772 0.5597 0.5435 0.5286 0.5138 0.5007 0.4881 0.4761 0.4652 0.4538 0.4440 0.4343 0.4255 0.4167 0.4083 0.4003 0.3925 0.3853 0.3783
1.0382 0.9746 0.9177 0.8691 0.8253 0.7876 0.7526 0.7221 0.6938 0.6673 0.6445 0.6221 0.6020 0.5830 0.5657 0.5499 0.5344 0.5196 0.5060 0.4933 0.4813 0.4703 0.4587 0.4488 0.4393 0.4301 0.4215 0.4128 0.4048 0.3969 0.3898
1.1148 1.0413 0.9773 0.9226 0.8744 0.8315 0.7932 0.7586 0.7275 0.6983 0.6735 0.6489 0.6280 0.6076 0.5888 0.5713 0.5546 0.5394 0.5250 0.5114 0.4985 0.4865 0.4754 0.4638 0.4535 0.4439 0.4346 0.4258 0.4173 0.4089 0.4013
1.1974 1.1146 1.0417 0.9827 0.9259 0.8779 0.8357 0.7976 0.7632 0.7321 0.7052 0.6785 0.6550 0.6328 0.6125 0.5937 0.5762 0.5597 0.5439 0.5300 0.5160 0.5031 0.4907 0.4788 0.4686 0.4581 0.4482 0.4383 0.4303 0.4223 0.4136
k S rGPay1
prMPa 0.6907 0.6614 0.6348 0.6108 0.5888 0.5687 0.5502 0.5333 0.5170 0.5021 0.4882 0.4752 0.4631 0.4519 0.4407 0.4304 0.4207 0.4112 0.4026 0.3942 0.3862 0.3787 0.3714 0.3642 0.3578 0.3513 0.3451 0.3393 0.3336 0.3281 0.3229
0.7379 0.7050 0.6753 0.6479 0.6232 0.6010 0.5805 0.5615 0.5436 0.5274 0.5122 0.4978 0.4844 0.4720 0.4600 0.4487 0.4384 0.4285 0.4188 0.4099 0.4015 0.3934 0.3856 0.3779 0.3707 0.3642 0.3574 0.3511 0.3452 0.3393 0.3339
0.7885 0.7512 0.7171 0.6868 0.6593 0.6341 0.6114 0.5906 0.5711 0.5531 0.5364 0.5210 0.5062 0.4926 0.4799 0.4679 0.4565 0.4458 0.4355 0.4257 0.4168 0.4081 0.3995 0.3918 0.3839 0.3768 0.3699 0.3631 0.3567 0.3505 0.3447
0.8441 0.8008 0.7628 0.7282 0.6975 0.6701 0.6442 0.6215 0.6000 0.5804 0.5613 0.5449 0.5292 0.5144 0.5004 0.4873 0.4751 0.4638 0.4524 0.4423 0.4325 0.4227 0.4138 0.4055 0.3976 0.3899 0.3824 0.3752 0.3686 0.3620 0.3558
0.9039 0.8551 0.8116 0.7729 0.7385 0.7072 0.6793 0.6535 0.6301 0.6086 0.5887 0.5701 0.5526 0.5366 0.5215 0.5078 0.4944 0.4818 0.4698 0.4579 0.4484 0.4384 0.4291 0.4201 0.4113 0.4030 0.3954 0.3877 0.3805 0.3737 0.3669
enabled us to plot bundles of curves uŽ r . along isotherms and isobars Žfigure 4., whose shapes have been found to be fully regular and almost linear. Moreover, this knowledge could be used to deduce adiabatic compressibility coefficients at the different pressures and temperatures of the investigation. These are given in tables 8 and 9, and represented along a few isobars in figure 5. It is observed from this figure that the dependency of the coefficient k S with respect to temperature is almost linear at high pressures and is equal to zero at the highest pressures. Finally, as the numerical procedure is based on an equation which links the isothermal and isentropic compressibility coefficients, it is also possible to calculate the coefficient k T at all the temperatures and pressures studied Žtables 10 and 11.. It can be seen from figure 6, which presents the isobaric behaviour of this coefficient, that the general shape of the bundle of curves is identical to that observed for the isentropic coefficient. The curves are simply translated by the
620
J. L. Daridon, A. Lagrabette, and B. Lagourette
TABLE 9. Coefficients of isentropic compressibility k S in Ž0.40C 17 H 36 q 0.35C 16 H 32 q 0.25C 16 H 26 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
333.15
343.15
353.15
363.16
373.15
0.8786 0.8326 0.7912 0.7548 0.7219 0.6930 0.6650 0.6407 0.6178 0.5975 0.5781 0.5600 0.5435 0.5279 0.5133 0.4996 0.4868 0.4747 0.4633 0.4525 0.4421
0.9383 0.8858 0.8390 0.7980 0.7614 0.7282 0.6987 0.6720 0.6470 0.6245 0.6031 0.5840 0.5664 0.5497 0.5338 0.5192 0.5056 0.4925 0.4802 0.4686 0.4577
1.0020 0.9426 0.8899 0.8445 0.8036 0.7674 0.7344 0.7036 0.6775 0.6527 0.6303 0.6095 0.5900 0.5718 0.5548 0.5391 0.5242 0.5106 0.4974 0.4853 0.4736
1.0708 1.0029 0.9438 0.8927 0.8474 0.8071 0.7714 0.7389 0.7098 0.6821 0.6575 0.6349 0.6142 0.5945 0.5765 0.5597 0.5438 0.5291 0.5151 0.5020 0.4895
k SrGPay1
prMPa 0.6383 0.6139 0.5912 0.5707 0.5514 0.5337 0.5173 0.5020 0.4877 0.4745 0.4619 0.4502 0.4391 0.4287 ] ] ] ] ] ] ]
0.6801 0.6523 0.6267 0.6034 0.5816 0.5621 0.5439 0.5272 0.5117 0.4971 0.4836 0.4708 0.4586 0.4474 0.4367 0.4265 0.4171 0.4079 0.3995 0.3913 0.3834
0.7249 0.6923 0.6641 0.6382 0.6141 0.5925 0.5720 0.5537 0.5367 0.5206 0.5061 0.4921 0.4790 0.4669 0.4553 0.4443 0.4340 0.4242 0.4149 0.4062 0.3977
0.7725 0.7365 0.7037 0.6746 0.6485 0.6243 0.6020 0.5813 0.5629 0.5454 0.5292 0.5140 0.4998 0.4864 0.4740 0.4621 0.4511 0.4406 0.4306 0.4213 0.4123
0.8235 0.7831 0.7468 0.7137 0.6842 0.6575 0.6330 0.6105 0.5898 0.5708 0.5532 0.5368 0.5213 0.5070 0.4933 0.4806 0.4689 0.4577 0.4470 0.4369 0.4272
term Ta 2rr C p which is practically temperature independent and whose contribution falls as the pressure rises. The data obtained from experimental measurements on the two ternary mixtures of hydrocarbon compounds are particularly useful as they are very difficult to predict from corresponding properties of the pure compounds. Actually, even for ideal solutions, the properties u, k S , and k T are not calculated by a simple linear relation but according to the following expression: Ž16.
k Sid s
Ý fi Ž k S q TVi a i2rCpi . y T Ý x iVi Ý fi a i
ž
i
i
u id s Ž r idk Sid .
y1 r2
i
,
/ž
i
2
/ ž Ý x Cp / , i
i
Ž 16 .
i
Ž 17 .
which requires the accurate knowledge of volume fractions f i and the thermodynamic properties a , k S , C p , and V of each pure component under the same conditions of pressure and temperature.
7. Conclusion The work reported in this paper is part of a programme of systematic investigation of synthetic cuts, including heavy hydrocarbon compounds with very close boiling points, in other words cuts simulating fractions of petroleum distillation. These cuts
621
Speed of sound of synthetic cuts
TABLE 10. Coefficients of isothermal compressibility k T in Ž0.40C 13 H 28 q 0.35C 13 H 26 q 0.25C 13 H 20 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150
333.15
343.15
353.15
363.16
373.15
1.1493 1.0820 1.0209 0.9667 0.9189 0.8763 0.8383 0.8034 0.7718 0.7431 0.7166 0.6920 0.6697 0.6487 0.6292 0.6113 0.5938 0.5780 0.5630 0.5487 0.5357 0.5223 0.5106 0.4992 0.4886 0.4783 0.4684 0.4589 0.4499 0.4414 0.4331
1.2311 1.1518 1.0814 1.0212 0.9673 0.9208 0.8781 0.8407 0.8062 0.7742 0.7464 0.7195 0.6953 0.6725 0.6518 0.6327 0.6143 0.5967 0.5805 0.5654 0.5513 0.5381 0.5246 0.5128 0.5016 0.4908 0.4806 0.4704 0.4610 0.4519 0.4435
1.3198 1.2284 1.1490 1.0813 1.0218 0.9693 0.9225 0.8804 0.8426 0.8074 0.7773 0.7478 0.7225 0.6980 0.6755 0.6547 0.6348 0.6167 0.5996 0.5835 0.5683 0.5540 0.5409 0.5274 0.5153 0.5040 0.4931 0.4828 0.4728 0.4630 0.4542
1.4169 1.3134 1.2231 1.1493 1.0800 1.0212 0.9697 0.9233 0.8817 0.8441 0.8114 0.7795 0.7512 0.7247 0.7004 0.6780 0.6572 0.6376 0.6190 0.6025 0.5859 0.5707 0.5562 0.5423 0.5303 0.5179 0.5064 0.4949 0.4854 0.4760 0.4660
k T rGPay1
prMPa 0.8471 0.8096 0.7758 0.7452 0.7173 0.6920 0.6686 0.6473 0.6270 0.6084 0.5911 0.5749 0.5599 0.5459 0.5322 0.5195 0.5076 0.4960 0.4854 0.4751 0.4655 0.4563 0.4474 0.4388 0.4310 0.4231 0.4156 0.4087 0.4018 0.3952 0.3889
0.8970 0.8553 0.8175 0.7830 0.7519 0.7240 0.6984 0.6747 0.6525 0.6324 0.6135 0.5958 0.5793 0.5640 0.5494 0.5356 0.5229 0.5109 0.4991 0.4883 0.4781 0.4683 0.4589 0.4497 0.4410 0.4332 0.4250 0.4176 0.4104 0.4034 0.3969
0.9515 0.9044 0.8615 0.8235 0.7890 0.7577 0.7293 0.7035 0.6795 0.6573 0.6368 0.6179 0.5998 0.5832 0.5677 0.5530 0.5392 0.5262 0.5139 0.5021 0.4913 0.4807 0.4706 0.4612 0.4519 0.4433 0.4351 0.4270 0.4193 0.4120 0.4051
1.0123 0.9580 0.9103 0.8672 0.8289 0.7949 0.7629 0.7348 0.7085 0.6844 0.6612 0.6411 0.6219 0.6040 0.5870 0.5712 0.5565 0.5427 0.5292 0.5170 0.5052 0.4936 0.4829 0.4730 0.4636 0.4544 0.4455 0.4370 0.4291 0.4213 0.4140
1.0786 1.0175 0.9633 0.9152 0.8725 0.8340 0.7996 0.7679 0.7392 0.7130 0.6887 0.6662 0.6450 0.6257 0.6075 0.5909 0.5748 0.5597 0.5454 0.5314 0.5198 0.5079 0.4968 0.4861 0.4757 0.4658 0.4568 0.4478 0.4393 0.4312 0.4233
share two major features: they involve aliphatic, naphthenic, and aromatic chemical families, which predominate in the composition of natural fluids, and from the point of view of their composition they respect the average proportions observed as a result of analysis of a large number of crude oils. The objective pursued in this programme of investigations is to build up a body of data on densities and derived properties for ternary mixtures, or mixtures involving a larger number of components which are normally grouped together indistinctly in the heavy fraction of reservoir fluids. These data, through the diversity of the properties studied and the extend of the pressure and temperature domain covered, will be of use in testing the reliability of models designed to represent heavy fractions. The authors wish to thank F. Montel for his highly appreciated assistance. They are also indebted to the Societe ´ ´ ELF AQUITAINE for financial support.
622
J. L. Daridon, A. Lagrabette, and B. Lagourette
TABLE 11. Coefficients of isothermal compressibility k T in Ž0.40C 17 H 36 q 0.35C 16 H 32 q 0.25C 16 H 26 . at temperatures T and pressures p TrK:
293.15
303.15
313.15
323.15
0.1 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
333.15
343.15
353.15
363.16
373.15
1.0356 0.9789 0.9280 0.8833 0.8431 0.8077 0.7739 0.7442 0.7166 0.6919 0.6686 0.6469 0.6270 0.6083 0.5908 0.5745 0.5592 0.5448 0.5312 0.5183 0.5060
1.1023 1.0380 0.9808 0.9306 0.8861 0.8459 0.8101 0.7777 0.7477 0.7206 0.6950 0.6720 0.6509 0.6310 0.6121 0.5947 0.5785 0.5630 0.5485 0.5347 0.5218
1.1738 1.1012 1.0370 0.9817 0.9320 0.8882 0.8485 0.8117 0.7802 0.7504 0.7236 0.6988 0.6755 0.6540 0.6338 0.6152 0.5977 0.5815 0.5660 0.5517 0.5379
1.2513 1.1686 1.0967 1.0346 0.9799 0.9314 0.8883 0.8494 0.8145 0.7816 0.7523 0.7255 0.7008 0.6776 0.6563 0.6365 0.6178 0.6005 0.5841 0.5687 0.5541
k T rGPay1
prMPa 0.7685 0.7375 0.7086 0.6826 0.6583 0.6361 0.6155 0.5964 0.5786 0.5621 0.5465 0.5320 0.5183 0.5054 ] ] ] ] ] } ]
0.8147 0.7797 0.7474 0.7182 0.6909 0.6666 0.6440 0.6232 0.6040 0.5859 0.5692 0.5535 0.5385 0.5247 0.5116 0.4992 0.4876 0.4765 0.4662 0.4562 0.4466
0.8644 0.8238 0.7884 0.7560 0.7261 0.6992 0.6740 0.6514 0.6304 0.6107 0.5928 0.5757 0.5597 0.5449 0.5308 0.5175 0.5049 0.4931 0.4819 0.4712 0.4610
0.9173 0.8725 0.8318 0.7957 0.7633 0.7335 0.7061 0.6809 0.6583 0.6369 0.6171 0.5987 0.5814 0.5652 0.5501 0.5358 0.5224 0.5098 0.4978 0.4866 0.4758
0.9741 0.9240 0.8790 0.8383 0.8021 0.7693 0.7394 0.7120 0.6868 0.6637 0.6423 0.6225 0.6038 0.5865 0.5701 0.5549 0.5407 0.5273 0.5145 0.5024 0.4909
FIGURE 6. Isothermal compressibility coefficient k T of the S 250 system against temperature T : p s 0.1 MPa; ', p s 25 MPa; ^, p s 50 MPa; v, p s 75 MPa; `, p s 100 MPa; l, p s 125 MPa; p s 150 MPa.
I, e,
Speed of sound of synthetic cuts
623
REFERENCES 1. Ye, S.; Alliez, J.; Lagourette, B.; Saint-Guirons, H.; Arman, J.; Xans, P. Re¨ ue Phys. Appl. 1990, 25, 555]565. 2. Daridon, J. L. Acustica 1994, 80, 416]419. 3. Wilson, W. D. J. Acoust. Soc. Am. 1959, 31, 1067]1072. 4. Del Grosso, V. A.; Mader, C. W. J. Acoust. Soc. Am. 1972, 52, 1442. 5. Petitet, J. P.; Tufeu, R.; Le Neindre, B. Int. J. Thermophys. 1983, 4, 35]47. 6. Chavez, M.; Palacios, J. M.; Tsumura, R. J. Chem. Eng. Data 1982, 27, 350]351. 7. Lainez, A.; Zollweg, J. A.; Street, W. B. J. Chem. Thermodynamics 1990, 22, 937]948. 8. Bobik, M.; Niepmann, R.; Marius, W. J. Chem. Thermodynamics 1979, 11, 351]357. 9. Muringer, M. J. P.; Trappeniers, N. J.; Biswas, S. N. Phys. Chem. Liq. 1985, 14, 273]296. 10. Lagourette, B.; Boned, C.; Saint-Guirons, H.; Xans, P.; Zhou, H. Meas. Sci. Technol. 1992, 3, 699]703. 11. Lamb, J. Physical Acoustics, Principles and Methods, Vol. 2. Mason, W. P.: editor. Academic Press: New York. 1965, p. 203. 12. Hakim, S. E. A.; Comley, W. J. Nature 1965, 208, 1082]1083. 13. Davis, L. A.; Gordon, R. B. J. Chem. Phys. 1967, 46, 2650]2660. 14. Denielou, L.; Petitet, J. P.; Tequi, C.; Syfosse, G. Bull. Mineral. ´ 1983, 106, 139]146. 15. Bessieres, D. LHP Universite ´ de Pau. Personal communication. 1997. 16. Handa, Y. P.; Halpin, C. J.; Benson, G. C. J. Chem. Thermodynamics 1981, 13, 875]886.
(Recei¨ ed 16 June 1997; in final form 27 No¨ ember 1997)
WA97r046