Harwood
Engineering Company, Inc.
Harwood
Engineering Company, Inc.
Our experience has shown that certain aspects of this evaluation could
be improved. Specifically, it seemed to be important to clarify the effect
of the speed of rotation of the piston. Also, in the past we and some other
investigators assumed implicitly2,6 that the stall would not
depend in any way on the properties of the fluids as long as they remain
fluid. Later, evidence was found suggestive that this concept might be
wrong. Some results showing the effect of piston rotation speed and fluids
of various viscosities on stall curves are presented in this article. A
conventional cylindrical piston is denoted as simple, as opposed to two
other kinds of pistons, namely, grooved and spherical ones.
The flow through an annular orifice can be approximately described by the following relation:
(1)From Equation (1) an expression for Vp1/3 which is very useful in the analysis of DWT performance can easily be derived:
(2)The variable Vp1/3 is used in plotting experimental results on piston fall rate versus jacket pressure Pj. The resulting curves (known as fall-rate or float curves) for a given measured pressure are linear, as shown in Figure 1(a).
An important parameter in the theory of controlled-clearance DWT is the slope of m of a fall-rate curve which can be represented in the form
(3)From Eq. (3) it is seen that the slope of a fall-rate curve depends on the viscosity of a particular fluid at the measured pressure under consideration and the pressure, provided that L is constant.
In the determination of an effective area the radial clearance e is involved. Since it is proportional to dPj, the clearance at a fixed fall rate is also proportional to the cube root of viscosity over measured pressure.
The average area of the piston can be presented in the following form7:
(4)At stall, jacket pressure deformation of the bore due to DPjs is supposedly equal to deformation of the bore and piston due to the measured pressure, which means that at this point the bore and piston have essentially the same diameter assuming there is no "residual" film of high-pressure fluid between them. Hence, the strain at the bore due to DPjs equals the strain at the bore due to Pm plus the strain at the piston due to Pm. Thus, from elasticity one may arrive at
(5)
(6)
(7)A signal from the transducer was fed into a recorder. Usually, we measured the fall rate over a fall distance of 1.27 mm. Also, we used a dual-beam oscilloscope (type 502) to monitor quality of the film between piston and cylinder by measuring continuously the electrical resistance of the film. Fine settings of the piston position and jacket pressure were made by means of manually operated venier displacement regulators manufactured by Harwood. Measured and jacket pressures were read out on Heise 300-mm dial gauges. Isolating valves were provided to lock in pressure in the two circuits. Relatively large tubing and fittings were used to minimize pressure drops between the panel and measuring head so that time to pressure equilibrium would be sufficiently small. The piston was made of tungsten carbide within 0.13 mm in roundness and cylindricity. Its surface roughness was better than 0.025 mm, and surface roughness on the bore surface was initially less than 0.15 mm and was improved after a run-in period. Further description of the equipment seems to be redundant since a reader can find details elsewhere.8
In all fall-rate studies the piston and the transducer were set so that
in a given series of experiments, we could observe the fall rate for the
same portion of the piston. Before every run we would determine an approximate
"freezing" pressure, that is the jacket pressure at which the piston is
just seized by the cylinder when the measured pressure is zero. Required
fall rates were achieved conveniently by varying jacket pressure at a given
measured pressure. These fall rates were usually within the range of 1-15
mm/s. In some of these studies even slower fall
rates were used. Fall rates were determined as slopes of experimental fall
distance vs. time curves recorded on strip charts. Since a fixed fall distance
was used in our experiments we could present the fall-rate as the inverted
time to fall through a 1.27 mm distance and plot the fall-rate curves in
the coordinates jacket pressure versus cube root of the inverted time.
The fall-rate curves were found by linear regression technique available
on the Texas Instruments Model SR-51-II. Construction of a stall curve
based on the fall rate data is described in Ref. 8, and for convenience
shown on Figure 1. Here, fall-rate curves for various
values of the measured pressure Pm are extrapolated to zero velocity. Intercepts
of the curves with the jacket pressures can then be plotted versus measured
pressure. A stall curve equation can easily be calculated by the linear
regression technique. For practical purposes operating curves corresponding
to any required fall rate are used. Analysis of the instrumental corrections
and respective uncertainties are given in Ref. 9.
DWT-1000 was used extensively in this study to a maximum pressure of
138 MPa in a range of 0-160 rpm with white gasoline, oil DTE-24, and Spin-Esso.11
Some experimental results on the stall curve and operating curve slopes
for both deadweight testers are summarized in Table I.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
The slopes of the fall-rate curves for DTE-24 oil (curve 1), Spin-Esso (curve 2), Univis P12 (curve 3), and white gasoline (curve 4) are shown graphically as a function of measured pressure in Figure 4. For a given L these curves are unique and can be used to evaluate the effect of pressure on the viscosity of fluids.12
It appears that speed of rotation has a pronounced effect, both on the
slopes of the fall-rate curves and the intercepts of the curves with jacket
pressure axis (Figure 5).
In some of our experiments we were monitoring the fluid film which surrounds the piston by electrical resistance measurements. Our observations show that the piston axis motion might be described as a whirl motion; in other words, the axis does not remain stationary, even when the film is very thin. It means that the actual hydrodynamic behavior and the pressure distribution are more complicated than the usual assumptions imply.
Comparison of stall curves obtained with various fluids makes it quite clear that at a constant measured pressure different stall jacket pressures are required for different fluids. This is a very important result. In fact, until recently a single stall curve was believed to exist for all the fluids.
The slope of a stall curve for white gasoline is approximately 0.7, which compares favorably with calculations [see Eq. (7)], while for more viscous fluids such as DTE-24 oil, Exxon, and Univis P12, the slope is about 0.55.
There are perhaps two possible explanations for these effects. Either the common assumption that essentially metal to metal contact exists when the piston is at stall needs to be reexamined because of the existence of a "residual" fluid film between piston and cylinder which actually increases the effective area, or else the pressure distribution along the axis changes, as our results14 obtained for a DWT with a spherical piston9,15 suggest.
The authors believe that it might only be verified experimentally by measuring either clearance between the piston and cylinder or the pressure distribution along the axis. These measurements are beyond the scope of this paper.
If a residual fluid film does exist, one should be very careful when using more viscous fluids without taking it into account since a considerable error, of the order of 0.05% (Figure 6), might be introduced if calibration results obtained for a different fluid were used. Note the difference between curves 3 and 4 for Univis P12, where hypothetical curve 4 is based on the assumption that its stall curve is the same as for white gasoline.
In any case more viscous fluids, as shown in Figure 6, have far greater influence on the effective area than do fluids like white gasoline or pentane.
Also, the fact that the slope of the stall curve for white gasoline is in agreement with the calculated slope could be accidental, since in the actual cylinder, pressure drop could occur below the level of the upper packing. To clarify this point, a specially designed geometry of the cylinder is required, where the position of the pressure gradient band would be accurately known.
As Figure 5 shows, for more viscous fluids at any given measured pressure there is considerable variation of the float curve slope due to the speed of rotation. The variation for DTE-24 oil is approximately three times as large as for white gasoline.
Thus, in the pressure range investigated in this work, white gasoline appears to be the fluid of choice considering its smaller correction in the calculation of effective area due to the clearance between piston and cylinder and the resulting smaller uncertainty. Also, the time to establish equilibrium conditions is smaller when less viscous fluids are used.
