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A magnetic-suspension apparatus to measure densities of liquids as a function of temperature at pressures up to 100 MPa Application to n-heptane
M-2430 J. (‘hem. Thermodynamics 1989, 21, 1263-1277
A magnetic-suspension apparatus to measure densities of liquids as a function of temperature at pressures up to 100 MPa Application to n-heptane SIR0 TOSCANI,
PIERRE FIGUIERE.
and HENRI
Laboratoire de Chimie Physique des MatPriaus URA 01104 du CNRS, Britiment 490, Universith de Paris-&d, 91405 Orsay. France
SZWARC
Amorphes.
(Received 21 September 1989) A densimeter designed for high-pressure work has been constructed. Densities of liquids have been measured with an accuracy of 0.2 per cent through the determination of the buoyancy of a magnetic float immersed in the samples. Thanks to a microbalance, a dynamic method is used to measure the total downward force of the magnetic-suspension system and of the Boating buoy. The density of n-heptane has been determined from 298.15 to 373.15 K at pressures up to 100 MPa. Our results are compared with those previously published.
1. Introduction Because of the lack of results in high-pressure fluid thermodynamics, our purpose has been to develop an experimental method of determining (p, V, T) behaviour. It can be applied to pressurized liquids and liquid mixtures and covers the pressure range up to 100 MPa, in the temperature range between 295 and 400 K. The apparatus we built is a magnetic-suspension densimeter and allows us directly to determine densities of liquids with an overall accuracy better than 0.2 per cent. It has been checked against some published densities of n-heptane.” ” In a magnetic-suspension densimeter, the force F necessary to float a magnetic buoy of mass mn and volume V, in a fluid of unknown density p is measured. This force is the difference between the free-fall force mg and the buoyant force pV,g, so that F = (me -P l/,kb
(1)
where g is the local acceleration of free fall. In indirect methods, the force F is related, through a calibration procedure, to the current I necessary to suspend the buoy in a stable position in a magnetic field generated by a solenoid. 0021-9614/89/0121263+
15 rSO2.00/0
C 1989 Academic
Press Limited
1264
S. TOSCANI,
P. FIGUIERE.
AND
H. SZWARC
Such densimeters have been developed by Beams and Clarke,‘h’ Clarke,“’ Senter,“’ and Almeida and Crouch,‘“’ and successively improved by Beams.““’ Hodgins and Beams, Cli) Haynes and Stewart,” ” and Haynes et al.” 3’ In Haynes’s, (14’ Haynes and Frederick’s,” 5, and Okada rt a/.‘~(‘~~ works, calibration procedure is restricted to the determination of the mass and the volume of a magnetic buoy, but a very accurate positioning of the buoy is necessary, which seems very difficult to achieve. In direct methods, often called magnetic weighing, the apparent mass (P/g) of the buoy is directly measured by a precision balance to which the magnetic equipment is attached. Masui et .1.‘17’ have developed a densimeter of this kind able to attain pressures up to 15 MPa in the temperature range 290 to 470 K with an uncertainty of 0.1 per cent. In the densimeter recently built by Kleinrahm and Wagner,“*’ an electronically controlled magnetic coupling is achieved between an upper electromagnet, fixed at the balance, and a lower permanent magnet connected by a bar with a cage in the measuring cell. The method we propose is a direct one, following those of Beams and Clarke’@ and of Masui et u1.(17)Our main purposes were to attain pressures of 100 MPa and more, and to be rid of the necessity of obtaining a stable positioning of the buoy after levitation. A magnetic suspension directly attached to a precision balance has been built and a dynamic magnetic weighing based upon an electrical-scanning procedure has been proposed. It consists in determining the value of the apparent mass (F/g) of the buoy at the moment when its levitation starts.
2. Experimental The schematic drawing in figure 1 represents the measuring system, consisting of a microbalance with an attached magnetic suspension, a thermostat equipped with a regulator, a high-pressure cell with a complete pressurizing system, an electronic potential-difference source connected to the solenoid, and a microcomputer driving the experimental procedure. We used a precision microbalance (Mettler AE 163) equipped with a stability detector. The instrument was used in its 0 to 160 g range; the legibility is 0.0001 g, the loss of linearity is +0.0004 g. It was maintained at a controlled room temperature and lay on an anti-vibration support system fixed above the thermostat. The linking between the upper part of the magnetic suspension and its lower portion, situated inside the thermostat, was assured by a length adjuster and an aluminium rod (figure 2). Therefore, the height of this part of the magnetic suspension could easily be adjusted for optimizing the magnetic coupling between the solenoid and the buoy. This lower part was chiefly composed of a main solenoid (i.d., 15 mm; o.d., 31.5 mm; height, 19 mm; mass, 77.28 g; total resistance at 298.15 K, 140 R; 3300 windings of 0.2 mm diameter enamelled copper wire) fixed between two thin dural crowns (i.d., 13.4 mm; o.d., 43 mm). Three homologous holes were drilled at an angle of 2rr/3 from one another in both plates near their edges. In these holes, three 2 mm thick dural
DENSIMETRY
UNDER PRESSURE BY DYNAMIC
MAGNETIC
WEIGHING
1265
FIGURE 1. Schematic drawing of the measuring system: a, microbalance; b, magnetic suspension; c, thermostat; d, high-pressure cell; e, pressurizing system; f, dead-weight gauge; g, electric potentialdifference source and microcomputer.
rods were fixed. They allowed the solenoid to be firmly maintained between the two plates by means of a six-nuts system. The upper extremities of these rods were fixed to an aluminium plate placed about 150 mm above the upper crown of the solenoid. The plate had a hook in its centre so as to allow the vertical linking with rod d in figure 2. Thus the lower portion of the magnetic suspension could easily be unhooked during its dismantling. The high-pressure system included a pressure generator, a high-pressure cell lying inside the thermostat (figure 3), and an outer high-pressure vessel. The latter vessel, placed outside the thermostat, contained a flexible stainless-steel bellows which separated the sample from the pressure-transmitting fluid. The cell was mainly composed of a non-magnetic titanium alloy high-pressure tubing (length, 133 mm; i.d., 5.3 mm; o.d., 9.52 mm). The inner wall of the tubing had been carefully polished to reduce contact frictions with the magnetic buoy during the measurements. The magnetic buoy was placed inside the tubing and rested on a dural 15 mm long pedestal machined with side grooves to make place for the pressurizing fluid. The cell was placed in the centre of the solenoid; its axis was vertical and coincided with that of the solenoid. An aluminium rod (length, 50 mm; diameter, 5 mm) fllled the upper part of the tubing to reduce the volume of the sample. The lower part of the measurement cell included an elbow linking the cell to an S-shaped tubing. The elbow was firmly fastened to a vibration-free support plate which had no contact with the base of the thermostat. A little coil was placed around the lower part of the cell. It was connected to a potential-difference meter for the detection of the buoy displacements by magnetic induction.
1266
S. TOSCANI,
P. FIGU&RE.
AND H. SZWARC
/
d
/Je
h\ ..::. ..:: ...:. . . ,................ .::_ :.: I/ ..:. :;:j ::. ii; 4 rr,::: .:: _ ; , ;q3 ::: /: :.:;IIIf I.‘: fb f
1
i . .._...... :..:,.,.:; ::_., :_: .j:j fl jjij I... I :::: ;:i: .:.: 1:;: a 5 .:. 1 ,; .: e 7. 7 I, :j:j I ;:: :::: /I,I ‘I fl ::::
:I :::: :<: zl;i $:: ,..A:_:_.. .--.A :::: ,._.:.: .. . . . __II
FIGURE 2. Balance and magnetic suspension: a, balance pan; b, balance base; c, anti-vibration support; d, length-adjuster and aluminium rods; e, “thermal silence?; f, thermostat; g, solenoid; h, suspension plate; i, cell; j, detection coil; k, support plate: I, thermal sheath; m. heatproof cloth.
It had been checked that the cell assembly was fully non-magnetic: in the absence of the magnetic buoy the assembled cell did not give any weighing contribution. (Samarium + cobalt) alloy (Recoma 20, Ugimag S.A.) was selected to build the buoy because of its high magnetization M which depends weakly on temperature (d In M/dT = -0.0004 K-l). A (samarium + cobalt) cylindrical bar (diameter, 4 mm; length, 15 mm) was enclosed inside an aluminium cylinder (diameter, 5 mm; length, 50 mm) which was thereafter sealed by electronic welding. Aluminium was chosen because of its low density, its low adsorption of liquid hydrocarbons, and because its isobaric thermal expansivity and isothermal bulk modulus are known.‘19) A calculation based on Lame’s formulae GO) showed that, as far as pressuretemperature corrections to the buoy volume were concerned, it could be considered as entirely made of aluminium. The mass and the volume of the buoy were determined by weighing in the air and in n-heptane. In both cases, two different methods were used: the method of dynamic weighing in the densimeter and the static method proposed by the balance manufacturer. (21) The results from both methods
DENSIMETRY
UNDER PRESSURE BY DYNAMIC
MAGNETIC
WEIGHING
1267
FIGURE 3. High-pressure cell: a, closure plug; b, sample-volume reducing rod; c, suspension rod, d, solenoid; c, buoy magnetic core; f, detection coil; g, coil support; h, buoy; i, buoy pedestal; j, adapter coupling; k, elbow; I, support plate; m, S-shaped tubing inlet; n, thermometer.
agreed within 0.04 per cent for the volume and better than 0.01 per cent for the mass at 298.15 K. For the mass mB and the volume V, of the buoy, there were found:
mB= (3.5507 +_O.OOOl) g (static weighing), V, = (0.9258 ~0.0006) cm3 (dynamic method). Our density determinations were made with respect to the density of n-heptane at 298.15 K {&,H16, 298.15 K) = 683.76 kg.mm3).(22) The results that we shall present have been corrected for the variation of the air buoyancy on the magnetic-suspension device as the temperature changed. This correction can be expressed by the formula: (old,,,,.
= (F/d-
{dak
293.15 K) - Aak
T)) L,
where (F/g) is the measured apparent mass of the buoy, p(air, 293.15 K) and &air, T) are the densities of air at 293.15 K and at temperature T, respectively, and V,, is the volume of the magnetic-suspension apparatus inside the thermostat {V~,=(21.48+0.01)cm3}.
1268
S. TOSCANI.
P. FIGUIkRE.
AND H. SZWARC
FIGURE 4. Thermostat: a, b, and c, outward openings; d, copper crown; e. heatproof cloth; f, support beam; g, copper shield; h, base plate. The thermostat (figure 4) had three openings; the upper one allowed the passage of the magnetic suspension and of the feeding wires of the solenoid; the left-side one allowed the passage of the S-shaped tubing; and the right-side one allowed that of the detection-solenoid feeding wires and of a metal beam supporting the anti-vibration base plate. The thermostat and the plate supporting system were fastened to a great dural plate fixed to a heavy granite table. Three jointed half-cylindrical copper shields (two of them rested on the base plate h in figure 4) and a cylindrical copper crown fixed at the ceiling of the thermostat reduced the thermal gradient between the upper and the lower parts of the cell and protected the latter from air convective currents. The transmission of downward vibrations was prevented by interposing a heatproof cloth. A “thermal silencer” (figure 5) lying between the thermostat top and the balance-support plate maintained a regular thermal gradient so as to reduce the dragging effect of the upward hot-air currents on the suspension plate. In the temperature-stabilization procedure, the power dissipation of the solenoid was taken into account. Care had to be taken that the average power dissipated during a scanning cycle (see procedure) was roughly equal to the power dissipated during temperature stabilization (x0.25 W at room temperature). This led to a temperature scatter equal to 0.04 K. Temperature was measured by means of a metal-sheathed platinum resistance thermometer, which had been calibrated at the L.N.E. (Laboratoire National d’Essais, Sevres, France) between 290 and 470 K (error x0.03 K). The pressure generator was a 400 MPa hand pump (Nova Swiss). Pressure was
DENSIMETRY
UNDER
PRESSURE
FIGURE
BY DYNAMIC
5. “Thermal
MAGNETIC
WEIGHING
1269
silencer.”
measured by a transducer (Autoclave Engineers) with an uncertainty of 0.25 per cent up to 140 MPa. The calibration of the gauge was controlled with respect to a dead-weight balance whose error was smaller than 0.05 per cent up to 110 MPa. The sample of n-heptane was provided by Aldrich (purity, 99.9 mass per cent); it was used without further purification. It was carefully degassed by the freeze-andthaw technique. The sample was introduced into the high-pressure cell under a vacuum, keeping the bottom of the bellows at liquid-nitrogen temperature. A test measurement of the apparent buoy mass was generally performed to check whether the liquid had filled the cell completely. Operating procedure, acquisition, processing, and storage of results were computer driven. Acquisition was allowed by three listener-talker devices, namely, the balance interface, a two-circuit multimeter (Keithley 195A) for the detection of electromotive forces induced by buoy displacements, and a programmable potentialdifference source (Keithley 228) coupled with a stable source (Fontaine MC 2030 C) on the basis of a “floating” method. We said before that our density determination implied the measurement of the apparent mass of the buoy (F/& when levitation started. To perform such a determination of (F/g)r at fixed temperature and pressure, the solenoid-circuit potential difference V was first increased step by step and the controller merely recorded a correlated increase of the (F/g) ratio at the balance.
1270
S. TOSCANI,
P. FIGUIfiRE.
AND H. SZWARC
Subsequently, at each V step of the most sensitive phase of the scan, the controller defined an integer variable n to number the acquisition blocks coming from talker devices until weighing stability was attained. At the last step, V, at which levitation occurred, the curve of (F/y) against n exhibited a sharp break associated with the upward motion of the buoy. At levitation we have F = bag-pkd+W where Sf is the magnetic-force increment that triggers the upward motion of the buoy. We consider that the apparent mass of the buoy was given by the last (F/g) value recorded before levitation. At each temperature, such determinations were performed starting at the highest pressure (z 100 MPa) and decreasing pressure by 5 or 10 MPa decrements. Each (F/g) determination resulted from the statistical average of 9 or 10 measurements. The operating procedure involved a preliminary scanning to determine roughly the apparent mass (F/g)L of the buoy, the corresponding potential difference V, at levitation, and that Vu at the end of levitation. Using these V,, (F/g),, and Vn values, the controller determined five parameters which correspond to the different phases of a scan: v, = v,-0.4
v,
(F/g)3 = (F/gk - O-0165 it,
v* = v,-0.075
v,
(F/d4 = (F/s)L - 0.0030 g>
v, = v,+o.o7
v.
FIGURE 6. Significant phase of a weighing (F/g) scan; a, the break in the curve of (F/g) against n; V,, the solenoid circuit potential difference V at levitation, each step indicating a 1 mV increment.
DENSIMETRY
UNDER
PRESSURE
BY DYNAMIC
MAGNETIC
WEIGHING
1271
Potential difference was increased by the programmable source from zero to V, by 500 mV steps and from V, to V, by 50 mV steps, each step lasting 1.5 s. At V,, a 5 mV step-scan started and the controller began recording the weighing results. When (F/g)3 was attained, a 1 mV step-scan started. From (F/g)4, a further 1 mV step was carried out only if the difference between the last two (F/g) values was smaller than the resolution of the balance (lop4 g). This procedure was repeated until levitation occurred. The final phase of the scan illustrated by figure 6 was thus very similar to a static weighing measurement. Figure 7, which exhibits the break in the (F/g) against n curve at I’,, also shows the values of the induced potential difference ViDaat the detection coil as a function of n. It can be seen that its break occurred at the very same value as that of the former curve. After levitation, the buoy was returned to its rest position. The same positioning was obtained by always following the same procedure: the potential difference was first dropped to I’, in one step and then varied by 10 mV decrements until the buoy fell. After waiting some time at V = 0 to keep thermal stability, the measurement cycle was resumed.
3. Sensitivity Let d(F/g)/dV be the scanning sensitivity. The magnetic force F is equal to F = KI = ICI//R,
(2)
where R, I, and V are the resistance, the current, and the potential difference in the solenoid circuit, respectively, and K is a constant which describes the coupling between the solenoid and the buoy, so that F/g = K VIRg.
Then, the scanning sensitivity is d(F/g)/dV = K/Rg = K’.
(3)
From relation (l), we get P = (me - F/d/b So we obtain WWld
= - l/v,.
(4)
We can also write WdV and by combining
= (d~ld(Fls)}(d(F/sUdV),
(5)
(3), (4), and (5): dp/dV = -(l/VB)K’
= -(l/ViJd(F/g)/dV.
Consequently, the sensitivity depends on the buoy volume and on the scanning sensitivity. In our experimental conditions, in which we tried to get a good compromise between a high resolution and a low power dissipation, sensitivity was roughly equal to 4 x low4 g*cm-3.mV-‘.
1272
S. TOSCANI, ‘5, , , , , ,t;;“:;L*I:*&’
P. FIGUII?RE, AND H. SZWARC I I I ,
I I I I
I I I I I I I
-0
j\ I
.-
15 20 n FIGURE 7. Simultaneous breaks: 0, in the weighing (F/g) against n curves; and A, in the curves of induced potential difference vnd against n. 0
5
10
4. Results Density measurements, reported in table 1, were performed for n-heptane along nine isotherms up to 100 MPa. These isotherms are represented on figure 8 and exhibit the usual features of the curves of density against pressure for liquid hydrocarbons. TOTAL
UNCERTAINTY
The fluid density is P = Cm, - (F/g), + { p(air, 293.15 K) - p(air, 7)}K,,l/ V,, where V, is the buoy volume at T and p. Thence dp/p = (dma + I( - l)d(F/g),l + d{p(air, 293.15 K) - p(air, T)} V,, + d&/,,{p(air, 293.15 K)- p(air, T)} + I( - lXdV,IV,)ICmB-(~/dL + {dair, 293.15 W-&k ~))Y,,‘,,IYP~. In this equation, dVB/VB is the uncertainty on the buoy volume resulting from the measurement of the apparent mass (F/g): of the buoy in the calibration fluid of
DENSIMETRY TABLE T/K:
UNDER 1. Density
298.15
303.15
PRESSURE
p of n-heptane 313.15
BY DYNAMIC as a function
323.15
piMPa
333.15
MAGNETIC
of temperature
1273
WEIGHING T and pressure
343.15
353.15
363.15
640.0
630.6
620.9
p 373.15
p/(kg.m-?
0.1 1 5 10 20 30 40 50 6=0 70 80 90 100
675.8
679.6
684.1 692.9 699.2 705.8 711.9 717.9 723.3 728.3 733.1 737.7
688.2 696.1 703.1 710.2 716.0 721.9
666.3
675.7 684.2 692.1 699.5 705.3 711.5 717.3 722.5 727.4 731.8
658.0
668.1 677.5 685.2 692.7 699.6 705.5 711.9 717.6 722.5 727.6
649.3
659.7 669.9 679.2 686.9 693.6 699.7 705.9 712.4 717.5 722.8
650.8 661.6 671.3 679.7 687.1 694.8 700.9 707.5 712.1 717.4
643.3 655.0 664.2 673.0 681.3 689.7 695.5 701.2 706.4 711.9
615.6 622.3 628.2 640.7 650.4 659.8 668.8 675.8 683.0 689.9 696.8 702.9
634.2 646.4 657.5 666.4 675.2 682.3 689.6 695.9 702.9 707.7
density p*, and dm, is the standard deviation of the mass of the buoy measured in air at 293.15 K and is equal to 10m4 g. Since v, = {MB - (FldEllP*. we have dG’,lJG = Pm,+ I(- l)d(Flg)tO/(m,-(Flg)t) + I(- l)dp*lp*l. By substituting
this expression into the previous one, we can write
kr,+d(F/g)r+
&d{p(air, 293.15 K)-p(air, T)}+{p(air, 293.15 K)-p(air, mB -(F/g)L + { p(air, 293.15 K) - p(air, T)} V,,
+ _______ dw, + W/d,* m,-(F/s)L*
T)}dV&
I dp* p* .
The total uncertainty is thus the sum of two main contributions: (1), the uncertainty corresponding to a routine measurement in the studied fluid ( z 0.12 per cent); and (2), the uncertainty resulting from the calibration of the buoy volume (~0.065 per cent). The other contributions are smaller than 0.005 per cent. Then one can write the simplified expression:
dp -= P
dm, + W/g), mB - (F/g)L + { p(air, 293.15 K) - p(air, T)j
dmB+ W/d,* ’
“‘B -(F/S):
’
Consequently the overall estimated uncertainty is 0.2 per cent. Random errors can be due to buoy friction against the inner wall of the cell and to vibrations in the system. Their occurrence is random, but they result in values of (F/g)L which are quite different from the mean of the measurements performed under
S. TOSCANI,
1274
I
I
II 0
P. FIGUIl?RE, AND H. SZWARC I
I
II
II
I
II
I
I
1
II
II
I
I
I
I
,
I
50
pIMPa FIGURE 8. Density of n-heptane plotted against pressure along nine isotherms: a, 298.15 K; b, 303.15 K; c, 313.15 K; d, 323.15 K; e, 333.15 K; f, 343.15 K; g, 353.15 K; h, 363.15 K; and i, 373.15 K.
the same (p, T) conditions. We have systematically rejected values deviating from the mean by 4 mg or more (5~ or more; (r is the standard deviation). In the case of a loss of alignment of the assembly, due for instance to pressure straightening, errors could be systematic. They could be detected by the existence of
DENSIMETRY
UNDER PRESSURE BY DYNAMIC
MAGNETIC
1275
WEIGHING
TABLE 2. Density of n-heptane at 298.15 K as a function of pressure p. Comparison of our measurements (p*) with densities p(l), p(2), and p(4) from references 1, 2, and 4, respectively
P(l) i&i?
p MPa 0.1 10 20 30 40 50 60
619.6 688.2 696.1
679.5 688.7 696.8 704.2
703.1 710.2 716.0 721.9
711.0 717.3 723.1
Pw--PL o.o1p* - 0.02 0.07 0.10 0.16 0.11 0.18 0.17
P(2) kgm-3 679.6 688.6 696.6 703.9 710.6 716.8 722.6
P(2)-Pp*
P(4)
kgm-’
o.o1p*
679.4
0.00
0.06 0.07 0.11 0.06 0.1 I 0.10
717.6
Pu-P* ~-
o.o1p*
-0.03
0.22 -
a loss of consistency of the results with respect to those obtained at other pressures at the same temperature. 5. Discussion We have compared our results with those previously published. Doolittle’s measurements were performed at the same (p, T) values as ours, so the comparison is direct. As to the results given by Muringer et al., Dymond et al., Eduljee et al. and Boelhouwer, we have applied to each isotherm the interpolation equation of Hudleston(23’ which may be written in the form: ln{p,I$2’3/(1-~1/3)j
= A(T)+B(T)(1-V,“3)
in which pr and V, denote relative pressure and relative volume, and which is widely used to represent the compressions of hydrocarbons for which it is superior to Tait’s equation.
TABLE
3. Density of n-heptane at 303.15 K as a function of pressure p. Comparison of our density measurements p* with densities p(3) and p(5) from references 3 and 5, respectively P
MPa 0.1 5 10 20 30 40 50 60 70 80 90 100
P*
kgm-3 675.8 684.1 692.9 699.2 705.8 711.9 717.9 723.3 728.3 733.1 737.7
P(3) &ii? 675.3 684.7 692.9 700.4 707.2 713.4 719.2 724.7 729.8 734.7 739.3
P(31-p*
o.o1p* -0.07 0.09 0 0.17 0.20 0.21 0.18 0.19 0.21 0.22 0.22
P(5)
igG= 675.3 680.3 684.9 693.3 701.0
@5)-P*
o.o1p* -0.07 0.12 0.06 0.26
714.2 -
0.32
740.6
0.39
1276
S. TOSCANI,
TABLE
4. Densities
0.1 1 5 10 20 30 40 50 60 70 80 90 100
313.15
MPa 0.1 1 5 10 20 30 40 50 60 70 80 90 loo
H. SZWARC
323.15
7‘ and pressure
p,
333.15
P(4) kgm-3
P@-P* o.o1p*
P(5) kg?
Pu-P* o.o1p*
.__P(3) kg.m-3
666.6
0.05
658.3
0.045
649.3
0
Pu-P* o.o1p*
669.5 679.0 687.4
0.21 0.22 0.32
0.33
702.0
0.34
0.45
730.2
0.36
660.9 670.8 679.5 687.3 694.5 701.0 707.1 712.8 718.2 723.2
0.18 0.13 0.04 0.06 0.12 0.19 0.17 0.06 0.10 0.06
-
707.6 -
735.1
T/K:
P
AND
p(3), p(4), and p(5) of n-heptane as a function of temperature Comparison with our density measurements p*
T/K: P MPa
P. FIGUIGRE,
333.15 P(4) kgm3 649.2
694.3
363.15 P(4bP*
o.o1p* -0.02 -
0.1 -
723.5
0.1
P(3)
kgm-3
373.15 PwP* o.o1p*
620.7 -
- 0.032
635.3 647.2 657.4 666.3 674.4 681.4 688.5 694.7 700.6 706.1
0.17 0.12 - 0.02 - 0.02 -0.12 -0.09 -0.16 -0.17 -0.33 -0.23
P(5) kgm-’
621.5 629.7 642.6 653.6 671.7
705.0
P(5)-P* o.o1p* -0.13 0.24 0.30 0.50 0.43 -
0.3
It can be seen that, except for Doolittle’s results and some of Eduljee et al.%, the average standard deviations are less than 0.2 per cent which is within our uncertainty range. As for Doolittle’s and Eduljee et al.% measurements, they had been intended to determine high-pressure values (up to 500 MPa), and so the difference is probably to be assigned to a lower accuracy of their results in the relatively low pressure range (100 MPa) that we have scanned (see tables 2, 3, and 4). Our measurements at 298.15 K are in good agreement with those of Dymond et al., and of Muringer et al. (table 2). The agreement with Muringer et al.‘s results is particularly interesting since their experimental temperature range (175 to 310 K) is complementary to ours. The pressure range (up to 120 MPa) explored by Boelhouwer is comparable with
DENSIMETRY
UNDER
PRESSURE
BY DYNAMIC
MAGNETIC
WEIGHING
1277
ours. Comparison is possible at 303.15, 333.15, and 363.15 K (tables 3 and 4). The average standard deviation between his measurement sets and ours is less than 0.2 per cent (0.13 per cent). To sum up the comparison with Boelhouwer’s results, his densities tend to be higher than ours at low temperatures and lower at 363.15 K. The overall agreement of our results with those obtained by others indicates that the method of dynamic weighing we described is quite suitable to determine the densities of liquids. In the near future, other pure hydrocarbons and binary mixtures will be studied. We thank the Compagnie Francaise des P&roles-C.F.P. Total for financial support of this work and a special research grant for one of us (ST.), and R. Coutouly and J. Jaffre for technical assistance. REFERENCES 1 2. ? 4: 5. 6. 71 8. 9. 10. II. I?.
13. 14. 1.
16. 17.
Muringer. M. P. J.; Trappeniers, N. J.; Biswas. S. N. Phtx Chem. Liq. 1985, 14. 273. Dymond, J. H.: Malhotra. R.; Isdale, J. D.; Glen. N. F. J. Chem. Therm&namics 1988, 20, 603. Boelhouwer. J. W. M. Physicu 1960, 26, 1021, Eduljee. H. E.; Newitt, D. M.; Weale, K. E. J. Chem. Sot. 1951, 4. 3086. Doolittle. A. K. J. Chem. Eng. Data 1964, 9. 27.5. Beams, J. W.: Clarke, A. M. Rev. Sci. Instrum. 1962, 33. 750. Clarke. A. L. J. Sci. Instrum. 1%7, 44, 382. Senter, J. P. Rev. Sci. Instrum. 1%9, 40, 334. Almeida, S. P.; Crouch, T. H. Rev. Sci. Znstrum. 1971, 42. 1344. Beams, J. W. Rev. Sri. Instrum. 1%9, 40, 167. Hodgins, M. G.: Beams, J. W. Rev. Sci. Instrum. 1971, 42, 1455. Haynes, W. M.; Stewart, J. W. Rev. Sci. Instrum. 1971, 42. 1142. Haynes. W. M.; Hita, M. J.; Frederick, N. V. Rev. Sci. Instrum. 1976, 47. 1237. Haynes. W. M. Rev. Sci. Instrum. 1977, 48, 39. Haynes. W. M.; Frederick, N. V. J. Res. Nur. Bur. Stand. U.S. 1983, 88, 241. Okada, M.; Uematsu, M.; Watanabe, K. J. Chem. Thermodynamics 1986, 18, 527. Masui. R.. Haynes, W. M.; Chang. R. F.; Davis, H. A.; Levelt Sengers. J. M. H. Rev. Sci. Instr. 1984, 55. 1 132.
1X. Kleinrahm. R.; Wagner, W. J. Chem. Thermodynamics 1986, 18. 739. 1’). Lincoln. R. C.; Ruoff. A. L. Rev. Sci. Instrum. 1973, 44. 1239. 20. Lame, G.: Clapeyron, B. P. E. Mkm. P&sent& par Divers Savunts. 2nd ser. 1833, 4, 463. 21. Mettler Instrumente AG’S ME-40290 Bulletin. 22. Handbook qf Chemistry and Physics. Weast, R. C.: editor, 54th edition. CRC Press: Baton Rouge. 1973-1974. 23.
Hudleston. L. J. Trans. Faraday Sot. 1937, 33. 97.
A magnetic-suspension apparatus to measure densities of liquids as a function of temperature at pressures up to 100 MPa Application to n-heptane SIR0 TOSCANI,
PIERRE FIGUIERE.
and HENRI
Laboratoire de Chimie Physique des MatPriaus URA 01104 du CNRS, Britiment 490, Universith de Paris-&d, 91405 Orsay. France
SZWARC
Amorphes.
(Received 21 September 1989) A densimeter designed for high-pressure work has been constructed. Densities of liquids have been measured with an accuracy of 0.2 per cent through the determination of the buoyancy of a magnetic float immersed in the samples. Thanks to a microbalance, a dynamic method is used to measure the total downward force of the magnetic-suspension system and of the Boating buoy. The density of n-heptane has been determined from 298.15 to 373.15 K at pressures up to 100 MPa. Our results are compared with those previously published.
1. Introduction Because of the lack of results in high-pressure fluid thermodynamics, our purpose has been to develop an experimental method of determining (p, V, T) behaviour. It can be applied to pressurized liquids and liquid mixtures and covers the pressure range up to 100 MPa, in the temperature range between 295 and 400 K. The apparatus we built is a magnetic-suspension densimeter and allows us directly to determine densities of liquids with an overall accuracy better than 0.2 per cent. It has been checked against some published densities of n-heptane.” ” In a magnetic-suspension densimeter, the force F necessary to float a magnetic buoy of mass mn and volume V, in a fluid of unknown density p is measured. This force is the difference between the free-fall force mg and the buoyant force pV,g, so that F = (me -P l/,kb
(1)
where g is the local acceleration of free fall. In indirect methods, the force F is related, through a calibration procedure, to the current I necessary to suspend the buoy in a stable position in a magnetic field generated by a solenoid. 0021-9614/89/0121263+
15 rSO2.00/0
C 1989 Academic
Press Limited
1264
S. TOSCANI,
P. FIGUIERE.
AND
H. SZWARC
Such densimeters have been developed by Beams and Clarke,‘h’ Clarke,“’ Senter,“’ and Almeida and Crouch,‘“’ and successively improved by Beams.““’ Hodgins and Beams, Cli) Haynes and Stewart,” ” and Haynes et al.” 3’ In Haynes’s, (14’ Haynes and Frederick’s,” 5, and Okada rt a/.‘~(‘~~ works, calibration procedure is restricted to the determination of the mass and the volume of a magnetic buoy, but a very accurate positioning of the buoy is necessary, which seems very difficult to achieve. In direct methods, often called magnetic weighing, the apparent mass (P/g) of the buoy is directly measured by a precision balance to which the magnetic equipment is attached. Masui et .1.‘17’ have developed a densimeter of this kind able to attain pressures up to 15 MPa in the temperature range 290 to 470 K with an uncertainty of 0.1 per cent. In the densimeter recently built by Kleinrahm and Wagner,“*’ an electronically controlled magnetic coupling is achieved between an upper electromagnet, fixed at the balance, and a lower permanent magnet connected by a bar with a cage in the measuring cell. The method we propose is a direct one, following those of Beams and Clarke’@ and of Masui et u1.(17)Our main purposes were to attain pressures of 100 MPa and more, and to be rid of the necessity of obtaining a stable positioning of the buoy after levitation. A magnetic suspension directly attached to a precision balance has been built and a dynamic magnetic weighing based upon an electrical-scanning procedure has been proposed. It consists in determining the value of the apparent mass (F/g) of the buoy at the moment when its levitation starts.
2. Experimental The schematic drawing in figure 1 represents the measuring system, consisting of a microbalance with an attached magnetic suspension, a thermostat equipped with a regulator, a high-pressure cell with a complete pressurizing system, an electronic potential-difference source connected to the solenoid, and a microcomputer driving the experimental procedure. We used a precision microbalance (Mettler AE 163) equipped with a stability detector. The instrument was used in its 0 to 160 g range; the legibility is 0.0001 g, the loss of linearity is +0.0004 g. It was maintained at a controlled room temperature and lay on an anti-vibration support system fixed above the thermostat. The linking between the upper part of the magnetic suspension and its lower portion, situated inside the thermostat, was assured by a length adjuster and an aluminium rod (figure 2). Therefore, the height of this part of the magnetic suspension could easily be adjusted for optimizing the magnetic coupling between the solenoid and the buoy. This lower part was chiefly composed of a main solenoid (i.d., 15 mm; o.d., 31.5 mm; height, 19 mm; mass, 77.28 g; total resistance at 298.15 K, 140 R; 3300 windings of 0.2 mm diameter enamelled copper wire) fixed between two thin dural crowns (i.d., 13.4 mm; o.d., 43 mm). Three homologous holes were drilled at an angle of 2rr/3 from one another in both plates near their edges. In these holes, three 2 mm thick dural
DENSIMETRY
UNDER PRESSURE BY DYNAMIC
MAGNETIC
WEIGHING
1265
FIGURE 1. Schematic drawing of the measuring system: a, microbalance; b, magnetic suspension; c, thermostat; d, high-pressure cell; e, pressurizing system; f, dead-weight gauge; g, electric potentialdifference source and microcomputer.
rods were fixed. They allowed the solenoid to be firmly maintained between the two plates by means of a six-nuts system. The upper extremities of these rods were fixed to an aluminium plate placed about 150 mm above the upper crown of the solenoid. The plate had a hook in its centre so as to allow the vertical linking with rod d in figure 2. Thus the lower portion of the magnetic suspension could easily be unhooked during its dismantling. The high-pressure system included a pressure generator, a high-pressure cell lying inside the thermostat (figure 3), and an outer high-pressure vessel. The latter vessel, placed outside the thermostat, contained a flexible stainless-steel bellows which separated the sample from the pressure-transmitting fluid. The cell was mainly composed of a non-magnetic titanium alloy high-pressure tubing (length, 133 mm; i.d., 5.3 mm; o.d., 9.52 mm). The inner wall of the tubing had been carefully polished to reduce contact frictions with the magnetic buoy during the measurements. The magnetic buoy was placed inside the tubing and rested on a dural 15 mm long pedestal machined with side grooves to make place for the pressurizing fluid. The cell was placed in the centre of the solenoid; its axis was vertical and coincided with that of the solenoid. An aluminium rod (length, 50 mm; diameter, 5 mm) fllled the upper part of the tubing to reduce the volume of the sample. The lower part of the measurement cell included an elbow linking the cell to an S-shaped tubing. The elbow was firmly fastened to a vibration-free support plate which had no contact with the base of the thermostat. A little coil was placed around the lower part of the cell. It was connected to a potential-difference meter for the detection of the buoy displacements by magnetic induction.
1266
S. TOSCANI,
P. FIGU&RE.
AND H. SZWARC
/
d
/Je
h\ ..::. ..:: ...:. . . ,................ .::_ :.: I/ ..:. :;:j ::. ii; 4 rr,::: .:: _ ; , ;q3 ::: /: :.:;IIIf I.‘: fb f
1
i . .._...... :..:,.,.:; ::_., :_: .j:j fl jjij I... I :::: ;:i: .:.: 1:;: a 5 .:. 1 ,; .: e 7. 7 I, :j:j I ;:: :::: /I,I ‘I fl ::::
:I :::: :<: zl;i $:: ,..A:_:_.. .--.A :::: ,._.:.: .. . . . __II
FIGURE 2. Balance and magnetic suspension: a, balance pan; b, balance base; c, anti-vibration support; d, length-adjuster and aluminium rods; e, “thermal silence?; f, thermostat; g, solenoid; h, suspension plate; i, cell; j, detection coil; k, support plate: I, thermal sheath; m. heatproof cloth.
It had been checked that the cell assembly was fully non-magnetic: in the absence of the magnetic buoy the assembled cell did not give any weighing contribution. (Samarium + cobalt) alloy (Recoma 20, Ugimag S.A.) was selected to build the buoy because of its high magnetization M which depends weakly on temperature (d In M/dT = -0.0004 K-l). A (samarium + cobalt) cylindrical bar (diameter, 4 mm; length, 15 mm) was enclosed inside an aluminium cylinder (diameter, 5 mm; length, 50 mm) which was thereafter sealed by electronic welding. Aluminium was chosen because of its low density, its low adsorption of liquid hydrocarbons, and because its isobaric thermal expansivity and isothermal bulk modulus are known.‘19) A calculation based on Lame’s formulae GO) showed that, as far as pressuretemperature corrections to the buoy volume were concerned, it could be considered as entirely made of aluminium. The mass and the volume of the buoy were determined by weighing in the air and in n-heptane. In both cases, two different methods were used: the method of dynamic weighing in the densimeter and the static method proposed by the balance manufacturer. (21) The results from both methods
DENSIMETRY
UNDER PRESSURE BY DYNAMIC
MAGNETIC
WEIGHING
1267
FIGURE 3. High-pressure cell: a, closure plug; b, sample-volume reducing rod; c, suspension rod, d, solenoid; c, buoy magnetic core; f, detection coil; g, coil support; h, buoy; i, buoy pedestal; j, adapter coupling; k, elbow; I, support plate; m, S-shaped tubing inlet; n, thermometer.
agreed within 0.04 per cent for the volume and better than 0.01 per cent for the mass at 298.15 K. For the mass mB and the volume V, of the buoy, there were found:
mB= (3.5507 +_O.OOOl) g (static weighing), V, = (0.9258 ~0.0006) cm3 (dynamic method). Our density determinations were made with respect to the density of n-heptane at 298.15 K {&,H16, 298.15 K) = 683.76 kg.mm3).(22) The results that we shall present have been corrected for the variation of the air buoyancy on the magnetic-suspension device as the temperature changed. This correction can be expressed by the formula: (old,,,,.
= (F/d-
{dak
293.15 K) - Aak
T)) L,
where (F/g) is the measured apparent mass of the buoy, p(air, 293.15 K) and &air, T) are the densities of air at 293.15 K and at temperature T, respectively, and V,, is the volume of the magnetic-suspension apparatus inside the thermostat {V~,=(21.48+0.01)cm3}.
1268
S. TOSCANI.
P. FIGUIkRE.
AND H. SZWARC
FIGURE 4. Thermostat: a, b, and c, outward openings; d, copper crown; e. heatproof cloth; f, support beam; g, copper shield; h, base plate. The thermostat (figure 4) had three openings; the upper one allowed the passage of the magnetic suspension and of the feeding wires of the solenoid; the left-side one allowed the passage of the S-shaped tubing; and the right-side one allowed that of the detection-solenoid feeding wires and of a metal beam supporting the anti-vibration base plate. The thermostat and the plate supporting system were fastened to a great dural plate fixed to a heavy granite table. Three jointed half-cylindrical copper shields (two of them rested on the base plate h in figure 4) and a cylindrical copper crown fixed at the ceiling of the thermostat reduced the thermal gradient between the upper and the lower parts of the cell and protected the latter from air convective currents. The transmission of downward vibrations was prevented by interposing a heatproof cloth. A “thermal silencer” (figure 5) lying between the thermostat top and the balance-support plate maintained a regular thermal gradient so as to reduce the dragging effect of the upward hot-air currents on the suspension plate. In the temperature-stabilization procedure, the power dissipation of the solenoid was taken into account. Care had to be taken that the average power dissipated during a scanning cycle (see procedure) was roughly equal to the power dissipated during temperature stabilization (x0.25 W at room temperature). This led to a temperature scatter equal to 0.04 K. Temperature was measured by means of a metal-sheathed platinum resistance thermometer, which had been calibrated at the L.N.E. (Laboratoire National d’Essais, Sevres, France) between 290 and 470 K (error x0.03 K). The pressure generator was a 400 MPa hand pump (Nova Swiss). Pressure was
DENSIMETRY
UNDER
PRESSURE
FIGURE
BY DYNAMIC
5. “Thermal
MAGNETIC
WEIGHING
1269
silencer.”
measured by a transducer (Autoclave Engineers) with an uncertainty of 0.25 per cent up to 140 MPa. The calibration of the gauge was controlled with respect to a dead-weight balance whose error was smaller than 0.05 per cent up to 110 MPa. The sample of n-heptane was provided by Aldrich (purity, 99.9 mass per cent); it was used without further purification. It was carefully degassed by the freeze-andthaw technique. The sample was introduced into the high-pressure cell under a vacuum, keeping the bottom of the bellows at liquid-nitrogen temperature. A test measurement of the apparent buoy mass was generally performed to check whether the liquid had filled the cell completely. Operating procedure, acquisition, processing, and storage of results were computer driven. Acquisition was allowed by three listener-talker devices, namely, the balance interface, a two-circuit multimeter (Keithley 195A) for the detection of electromotive forces induced by buoy displacements, and a programmable potentialdifference source (Keithley 228) coupled with a stable source (Fontaine MC 2030 C) on the basis of a “floating” method. We said before that our density determination implied the measurement of the apparent mass of the buoy (F/& when levitation started. To perform such a determination of (F/g)r at fixed temperature and pressure, the solenoid-circuit potential difference V was first increased step by step and the controller merely recorded a correlated increase of the (F/g) ratio at the balance.
1270
S. TOSCANI,
P. FIGUIfiRE.
AND H. SZWARC
Subsequently, at each V step of the most sensitive phase of the scan, the controller defined an integer variable n to number the acquisition blocks coming from talker devices until weighing stability was attained. At the last step, V, at which levitation occurred, the curve of (F/y) against n exhibited a sharp break associated with the upward motion of the buoy. At levitation we have F = bag-pkd+W where Sf is the magnetic-force increment that triggers the upward motion of the buoy. We consider that the apparent mass of the buoy was given by the last (F/g) value recorded before levitation. At each temperature, such determinations were performed starting at the highest pressure (z 100 MPa) and decreasing pressure by 5 or 10 MPa decrements. Each (F/g) determination resulted from the statistical average of 9 or 10 measurements. The operating procedure involved a preliminary scanning to determine roughly the apparent mass (F/g)L of the buoy, the corresponding potential difference V, at levitation, and that Vu at the end of levitation. Using these V,, (F/g),, and Vn values, the controller determined five parameters which correspond to the different phases of a scan: v, = v,-0.4
v,
(F/g)3 = (F/gk - O-0165 it,
v* = v,-0.075
v,
(F/d4 = (F/s)L - 0.0030 g>
v, = v,+o.o7
v.
FIGURE 6. Significant phase of a weighing (F/g) scan; a, the break in the curve of (F/g) against n; V,, the solenoid circuit potential difference V at levitation, each step indicating a 1 mV increment.
DENSIMETRY
UNDER
PRESSURE
BY DYNAMIC
MAGNETIC
WEIGHING
1271
Potential difference was increased by the programmable source from zero to V, by 500 mV steps and from V, to V, by 50 mV steps, each step lasting 1.5 s. At V,, a 5 mV step-scan started and the controller began recording the weighing results. When (F/g)3 was attained, a 1 mV step-scan started. From (F/g)4, a further 1 mV step was carried out only if the difference between the last two (F/g) values was smaller than the resolution of the balance (lop4 g). This procedure was repeated until levitation occurred. The final phase of the scan illustrated by figure 6 was thus very similar to a static weighing measurement. Figure 7, which exhibits the break in the (F/g) against n curve at I’,, also shows the values of the induced potential difference ViDaat the detection coil as a function of n. It can be seen that its break occurred at the very same value as that of the former curve. After levitation, the buoy was returned to its rest position. The same positioning was obtained by always following the same procedure: the potential difference was first dropped to I’, in one step and then varied by 10 mV decrements until the buoy fell. After waiting some time at V = 0 to keep thermal stability, the measurement cycle was resumed.
3. Sensitivity Let d(F/g)/dV be the scanning sensitivity. The magnetic force F is equal to F = KI = ICI//R,
(2)
where R, I, and V are the resistance, the current, and the potential difference in the solenoid circuit, respectively, and K is a constant which describes the coupling between the solenoid and the buoy, so that F/g = K VIRg.
Then, the scanning sensitivity is d(F/g)/dV = K/Rg = K’.
(3)
From relation (l), we get P = (me - F/d/b So we obtain WWld
= - l/v,.
(4)
We can also write WdV and by combining
= (d~ld(Fls)}(d(F/sUdV),
(5)
(3), (4), and (5): dp/dV = -(l/VB)K’
= -(l/ViJd(F/g)/dV.
Consequently, the sensitivity depends on the buoy volume and on the scanning sensitivity. In our experimental conditions, in which we tried to get a good compromise between a high resolution and a low power dissipation, sensitivity was roughly equal to 4 x low4 g*cm-3.mV-‘.
1272
S. TOSCANI, ‘5, , , , , ,t;;“:;L*I:*&’
P. FIGUII?RE, AND H. SZWARC I I I ,
I I I I
I I I I I I I
-0
j\ I
.-
15 20 n FIGURE 7. Simultaneous breaks: 0, in the weighing (F/g) against n curves; and A, in the curves of induced potential difference vnd against n. 0
5
10
4. Results Density measurements, reported in table 1, were performed for n-heptane along nine isotherms up to 100 MPa. These isotherms are represented on figure 8 and exhibit the usual features of the curves of density against pressure for liquid hydrocarbons. TOTAL
UNCERTAINTY
The fluid density is P = Cm, - (F/g), + { p(air, 293.15 K) - p(air, 7)}K,,l/ V,, where V, is the buoy volume at T and p. Thence dp/p = (dma + I( - l)d(F/g),l + d{p(air, 293.15 K) - p(air, T)} V,, + d&/,,{p(air, 293.15 K)- p(air, T)} + I( - lXdV,IV,)ICmB-(~/dL + {dair, 293.15 W-&k ~))Y,,‘,,IYP~. In this equation, dVB/VB is the uncertainty on the buoy volume resulting from the measurement of the apparent mass (F/g): of the buoy in the calibration fluid of
DENSIMETRY TABLE T/K:
UNDER 1. Density
298.15
303.15
PRESSURE
p of n-heptane 313.15
BY DYNAMIC as a function
323.15
piMPa
333.15
MAGNETIC
of temperature
1273
WEIGHING T and pressure
343.15
353.15
363.15
640.0
630.6
620.9
p 373.15
p/(kg.m-?
0.1 1 5 10 20 30 40 50 6=0 70 80 90 100
675.8
679.6
684.1 692.9 699.2 705.8 711.9 717.9 723.3 728.3 733.1 737.7
688.2 696.1 703.1 710.2 716.0 721.9
666.3
675.7 684.2 692.1 699.5 705.3 711.5 717.3 722.5 727.4 731.8
658.0
668.1 677.5 685.2 692.7 699.6 705.5 711.9 717.6 722.5 727.6
649.3
659.7 669.9 679.2 686.9 693.6 699.7 705.9 712.4 717.5 722.8
650.8 661.6 671.3 679.7 687.1 694.8 700.9 707.5 712.1 717.4
643.3 655.0 664.2 673.0 681.3 689.7 695.5 701.2 706.4 711.9
615.6 622.3 628.2 640.7 650.4 659.8 668.8 675.8 683.0 689.9 696.8 702.9
634.2 646.4 657.5 666.4 675.2 682.3 689.6 695.9 702.9 707.7
density p*, and dm, is the standard deviation of the mass of the buoy measured in air at 293.15 K and is equal to 10m4 g. Since v, = {MB - (FldEllP*. we have dG’,lJG = Pm,+ I(- l)d(Flg)tO/(m,-(Flg)t) + I(- l)dp*lp*l. By substituting
this expression into the previous one, we can write
kr,+d(F/g)r+
&d{p(air, 293.15 K)-p(air, T)}+{p(air, 293.15 K)-p(air, mB -(F/g)L + { p(air, 293.15 K) - p(air, T)} V,,
+ _______ dw, + W/d,* m,-(F/s)L*
T)}dV&
I dp* p* .
The total uncertainty is thus the sum of two main contributions: (1), the uncertainty corresponding to a routine measurement in the studied fluid ( z 0.12 per cent); and (2), the uncertainty resulting from the calibration of the buoy volume (~0.065 per cent). The other contributions are smaller than 0.005 per cent. Then one can write the simplified expression:
dp -= P
dm, + W/g), mB - (F/g)L + { p(air, 293.15 K) - p(air, T)j
dmB+ W/d,* ’
“‘B -(F/S):
’
Consequently the overall estimated uncertainty is 0.2 per cent. Random errors can be due to buoy friction against the inner wall of the cell and to vibrations in the system. Their occurrence is random, but they result in values of (F/g)L which are quite different from the mean of the measurements performed under
S. TOSCANI,
1274
I
I
II 0
P. FIGUIl?RE, AND H. SZWARC I
I
II
II
I
II
I
I
1
II
II
I
I
I
I
,
I
50
pIMPa FIGURE 8. Density of n-heptane plotted against pressure along nine isotherms: a, 298.15 K; b, 303.15 K; c, 313.15 K; d, 323.15 K; e, 333.15 K; f, 343.15 K; g, 353.15 K; h, 363.15 K; and i, 373.15 K.
the same (p, T) conditions. We have systematically rejected values deviating from the mean by 4 mg or more (5~ or more; (r is the standard deviation). In the case of a loss of alignment of the assembly, due for instance to pressure straightening, errors could be systematic. They could be detected by the existence of
DENSIMETRY
UNDER PRESSURE BY DYNAMIC
MAGNETIC
1275
WEIGHING
TABLE 2. Density of n-heptane at 298.15 K as a function of pressure p. Comparison of our measurements (p*) with densities p(l), p(2), and p(4) from references 1, 2, and 4, respectively
P(l) i&i?
p MPa 0.1 10 20 30 40 50 60
619.6 688.2 696.1
679.5 688.7 696.8 704.2
703.1 710.2 716.0 721.9
711.0 717.3 723.1
Pw--PL o.o1p* - 0.02 0.07 0.10 0.16 0.11 0.18 0.17
P(2) kgm-3 679.6 688.6 696.6 703.9 710.6 716.8 722.6
P(2)-Pp*
P(4)
kgm-’
o.o1p*
679.4
0.00
0.06 0.07 0.11 0.06 0.1 I 0.10
717.6
Pu-P* ~-
o.o1p*
-0.03
0.22 -
a loss of consistency of the results with respect to those obtained at other pressures at the same temperature. 5. Discussion We have compared our results with those previously published. Doolittle’s measurements were performed at the same (p, T) values as ours, so the comparison is direct. As to the results given by Muringer et al., Dymond et al., Eduljee et al. and Boelhouwer, we have applied to each isotherm the interpolation equation of Hudleston(23’ which may be written in the form: ln{p,I$2’3/(1-~1/3)j
= A(T)+B(T)(1-V,“3)
in which pr and V, denote relative pressure and relative volume, and which is widely used to represent the compressions of hydrocarbons for which it is superior to Tait’s equation.
TABLE
3. Density of n-heptane at 303.15 K as a function of pressure p. Comparison of our density measurements p* with densities p(3) and p(5) from references 3 and 5, respectively P
MPa 0.1 5 10 20 30 40 50 60 70 80 90 100
P*
kgm-3 675.8 684.1 692.9 699.2 705.8 711.9 717.9 723.3 728.3 733.1 737.7
P(3) &ii? 675.3 684.7 692.9 700.4 707.2 713.4 719.2 724.7 729.8 734.7 739.3
P(31-p*
o.o1p* -0.07 0.09 0 0.17 0.20 0.21 0.18 0.19 0.21 0.22 0.22
P(5)
igG= 675.3 680.3 684.9 693.3 701.0
@5)-P*
o.o1p* -0.07 0.12 0.06 0.26
714.2 -
0.32
740.6
0.39
1276
S. TOSCANI,
TABLE
4. Densities
0.1 1 5 10 20 30 40 50 60 70 80 90 100
313.15
MPa 0.1 1 5 10 20 30 40 50 60 70 80 90 loo
H. SZWARC
323.15
7‘ and pressure
p,
333.15
P(4) kgm-3
P@-P* o.o1p*
P(5) kg?
Pu-P* o.o1p*
.__P(3) kg.m-3
666.6
0.05
658.3
0.045
649.3
0
Pu-P* o.o1p*
669.5 679.0 687.4
0.21 0.22 0.32
0.33
702.0
0.34
0.45
730.2
0.36
660.9 670.8 679.5 687.3 694.5 701.0 707.1 712.8 718.2 723.2
0.18 0.13 0.04 0.06 0.12 0.19 0.17 0.06 0.10 0.06
-
707.6 -
735.1
T/K:
P
AND
p(3), p(4), and p(5) of n-heptane as a function of temperature Comparison with our density measurements p*
T/K: P MPa
P. FIGUIGRE,
333.15 P(4) kgm3 649.2
694.3
363.15 P(4bP*
o.o1p* -0.02 -
0.1 -
723.5
0.1
P(3)
kgm-3
373.15 PwP* o.o1p*
620.7 -
- 0.032
635.3 647.2 657.4 666.3 674.4 681.4 688.5 694.7 700.6 706.1
0.17 0.12 - 0.02 - 0.02 -0.12 -0.09 -0.16 -0.17 -0.33 -0.23
P(5) kgm-’
621.5 629.7 642.6 653.6 671.7
705.0
P(5)-P* o.o1p* -0.13 0.24 0.30 0.50 0.43 -
0.3
It can be seen that, except for Doolittle’s results and some of Eduljee et al.%, the average standard deviations are less than 0.2 per cent which is within our uncertainty range. As for Doolittle’s and Eduljee et al.% measurements, they had been intended to determine high-pressure values (up to 500 MPa), and so the difference is probably to be assigned to a lower accuracy of their results in the relatively low pressure range (100 MPa) that we have scanned (see tables 2, 3, and 4). Our measurements at 298.15 K are in good agreement with those of Dymond et al., and of Muringer et al. (table 2). The agreement with Muringer et al.‘s results is particularly interesting since their experimental temperature range (175 to 310 K) is complementary to ours. The pressure range (up to 120 MPa) explored by Boelhouwer is comparable with
DENSIMETRY
UNDER
PRESSURE
BY DYNAMIC
MAGNETIC
WEIGHING
1277
ours. Comparison is possible at 303.15, 333.15, and 363.15 K (tables 3 and 4). The average standard deviation between his measurement sets and ours is less than 0.2 per cent (0.13 per cent). To sum up the comparison with Boelhouwer’s results, his densities tend to be higher than ours at low temperatures and lower at 363.15 K. The overall agreement of our results with those obtained by others indicates that the method of dynamic weighing we described is quite suitable to determine the densities of liquids. In the near future, other pure hydrocarbons and binary mixtures will be studied. We thank the Compagnie Francaise des P&roles-C.F.P. Total for financial support of this work and a special research grant for one of us (ST.), and R. Coutouly and J. Jaffre for technical assistance. REFERENCES 1 2. ? 4: 5. 6. 71 8. 9. 10. II. I?.
13. 14. 1.
16. 17.
Muringer. M. P. J.; Trappeniers, N. J.; Biswas. S. N. Phtx Chem. Liq. 1985, 14. 273. Dymond, J. H.: Malhotra. R.; Isdale, J. D.; Glen. N. F. J. Chem. Therm&namics 1988, 20, 603. Boelhouwer. J. W. M. Physicu 1960, 26, 1021, Eduljee. H. E.; Newitt, D. M.; Weale, K. E. J. Chem. Sot. 1951, 4. 3086. Doolittle. A. K. J. Chem. Eng. Data 1964, 9. 27.5. Beams, J. W.: Clarke, A. M. Rev. Sci. Instrum. 1962, 33. 750. Clarke. A. L. J. Sci. Instrum. 1%7, 44, 382. Senter, J. P. Rev. Sci. Instrum. 1%9, 40, 334. Almeida, S. P.; Crouch, T. H. Rev. Sci. Znstrum. 1971, 42. 1344. Beams, J. W. Rev. Sri. Instrum. 1%9, 40, 167. Hodgins, M. G.: Beams, J. W. Rev. Sci. Instrum. 1971, 42, 1455. Haynes, W. M.; Stewart, J. W. Rev. Sci. Instrum. 1971, 42. 1142. Haynes. W. M.; Hita, M. J.; Frederick, N. V. Rev. Sci. Instrum. 1976, 47. 1237. Haynes. W. M. Rev. Sci. Instrum. 1977, 48, 39. Haynes. W. M.; Frederick, N. V. J. Res. Nur. Bur. Stand. U.S. 1983, 88, 241. Okada, M.; Uematsu, M.; Watanabe, K. J. Chem. Thermodynamics 1986, 18, 527. Masui. R.. Haynes, W. M.; Chang. R. F.; Davis, H. A.; Levelt Sengers. J. M. H. Rev. Sci. Instr. 1984, 55. 1 132.
1X. Kleinrahm. R.; Wagner, W. J. Chem. Thermodynamics 1986, 18. 739. 1’). Lincoln. R. C.; Ruoff. A. L. Rev. Sci. Instrum. 1973, 44. 1239. 20. Lame, G.: Clapeyron, B. P. E. Mkm. P&sent& par Divers Savunts. 2nd ser. 1833, 4, 463. 21. Mettler Instrumente AG’S ME-40290 Bulletin. 22. Handbook qf Chemistry and Physics. Weast, R. C.: editor, 54th edition. CRC Press: Baton Rouge. 1973-1974. 23.
Hudleston. L. J. Trans. Faraday Sot. 1937, 33. 97.