T

Bs(p) = P(p)L2(p)

T2 (p)

is the volume expansivity at pres

sure p;

is the pressure-dependent Grüneisen parameter;

is the absolute temperature; and is the adiabatic bulk modulus with p(p) density, L(p) sample length. and (p) ultrasonic transit time.

An iterative procedure is used to compute increasingly better approximations of V(p) and the length L(p) of the sample. The successful solution of this problem requires knowledge of any as a function of pressure. It also requires knowledge of the initial dimensions of the fluid container, the amount of fluid contained, and its density and speed of sound at atmospheric pressure. If a,yT is small we may choose to neglect it. For the fluids investigated here ayT is large at atmospheric pressure and is estimated to change rapidly with pressure. Its value as a function. of pressure and temperature is not sufficiently well known to be used in the present investigation. Therefore it seems more appropriate to determine the density as a function of pressure from dilatometric

op, np are corrections for compression of the piston stack and distortion of other parts;

y[1-e-p] is a correction for initial slack in the setup at low pressures; and

Bey Lop is a correction for the effect of cylinder expansion which adjusts the actual length to the length which the sample would have in a rigid cylinder, permitting the use of L(p)/Lo for VVo. The coefficients o, n, y, e, and Bey are determined in separate experiments described in reference [5]. The zero pressure dial indicator reading, ho, is not known since a small unknown pressure is always generated when the press is closed up. The following procedure is used to determine ho. A value is assumed for h, and the bulk modulus of the sample fluid is computed from the pressure derivative of eq (2) using L(p) from eq (3). We further assume that as a first approximation the bulk modulus of a sample is a linear function of pressure, at least at low pressures. We then fit the experimental bulk modulus over part of the pressure range to a linear function of pressure.

where

V(p) +VPE(P) Dπ/4

IPE (P) = Vo.PE

is the volume of the polyethylene sleeve. We combine eq (3) with eqs (5), (6), and (7) and rearrange to solve for ho

+y[1 − e− ] + BeyiLop

(3)

where ho, h(p) are the dial indicator readings at zero ram pressure and at pressure p:

Bof

Now ho can be extrapolated from any datum point p. This procedure is unsatisfactory, at least in the present case, because the bulk modulus of the very compressible liquids is not a linear function of pressure and furthermore because in many instances the lowest pressure point is at 2000 bar and the extrapolation is rather long.

A low-pressure dilatometer was therefore constructed and measurements of Bo,ƒ were made to aid in the determination of ho.

3. Low-Pressure Dilatometer

A simple low-pressure dilatometer was constructed to measure the isothermal compressibility of liquids at atmospheric pressure. The pressure vessel is shown in figure 2. The lower half is a 5-in o.d. stainless steel tube brazed to a brass bottom plate and closed at its

TRANSPARENT PLASTIC TUBE

GRADUATED

CAPILLARY

TUBE

PLATIMUM
RESISTANCE

THERMOMETER

M

tap. In addition the midplate holds a transparent plastic tube into which the capillary of the dilatometer protrudes. This plastic tube is sealed by O-rings and held to the midplate by bolts from a top closure.

The dilatometer is made from a 500-ml roundbottom flask sealed to the male portion of a groundglass taper joint. A 0.2-ml graduated pipet was sealed to the female portion of the ground-glass taper joint. The upper end of the pipet has an open cup sealed around it to aid in the final filling. When assembled the readable portion of the scale on the pipet covers approximately 0.17 ml, graduated in 0.001 ml. The flask is nested in a perforated brass can with springs attached to the rim which hook over a clamp around the ground-glass joint of the pipet to hold the joint together. The can also serves to locate the dilatometer in the pressure vessel.

The base plate of the pressure vessel was maintained slightly above room temperature to permit thermostatic control of the temperature. The same fluid used in the dilatometer was also used as a bath fluid to fill the pressure vessel around the dilatometer to about one inch below the midplate. The same fluid was chosen to decrease evaporation from the open capillary and to cause similar changes in temperature due to change in pressure. Nitrogen was used above the liquid as the pressure fluid.

The volume of the dilatometer was determined to within 0.2 percent by the weight of water contained in it. The capillary scale was checked with a weighed slug of mercury and was found to be within 0.2 percent of the capillary volume and uniform to within 0.2 division. The pressure was measured with a calibrated bourdon tube gage. The temperature of the liquid surrounding the dilatometer within the pressure vessel was measured with a platinum resistance thermometer. After filling the apparatus with a sample of fluid. the position of the meniscus in the capillary was read at various temperatures and at pressures between 1 and 5 bars.

The data were fitted to eq (9) by OMNITAB FIT for

The estimated systematic uncertainties in the measurements of K, are 0.2 percent due to Vpo; 1 percent for απ and Kg; 0.3 percent for p; 0.2 percent for Ve; and 3 percent for t. The largest uncertainty, the 3 percent for t, results from an estimated uncertainty of 0.02 °C in the measurement of the equilibrium temperature. While this method of measuring the isothermal compressibility at atmospheric pressure gives results good only to about 3 percent, this is sufficient for our present use. The results for 2-methylbutane, aviation instrument oil, and 50/50, 75/25, and 25/75 mixtures of the two are shown in figure 3 by (A) as the isothermal bulk modulus at atmospheric pressure (B1⁄4 ̧ƒ).

are plotted (x) in figure 3 and both coefficients are listed in table 1. A considerable improvement can be made, if Bo, is set equal to B,, the zero-pressure bulk modulus obtained from the low-pressure dilatometer measurements. A straight line is fitted to the same Bpm) as used above and forced through B The B are also plotted (A) in figure 3 and the coefficients B and B are listed in table 1. The values of B, and B, for each mixture were used to obtain an

TABLE 1. Density at atmospheric pressure and coefficients of bulk modulus of mixtures of aviation instrument oil and 2-methylbutane

(10)

Bƒ(p)=Bo,ƒ+B1,ƒp

for the first six non-zero pressures (approximately 1 to 10 kbar) for each fluid sample. The resulting Bo,f

ho from eq (8) by averaging the values of ho from the first six non-zero pressure points for each mixture.