•
PRESSURE MEASUREMENT IN TRANSIT ION AND FREE MOLECULAR FLOWS USING ORIFICE PROBES
T:C
'f·:tSnT
1l0G~SOlOOl DElFT VU:GïUIGt,QUW;·:UNDE byBIG lI01lIEEK
J. C. Lafrance1 NOV.
1963
..
PRESSURE MEASUREMENT IN TRANSITION AND FREE MOLECULAR FLOWS USING ORIFICE PROBES
by
J. C. Lafrance
JUNE. 1963 UTIA TECHNICAL NOTE NO. 67
"
ACKNOWLEDGEMENT
The author wishes to express his gratitude to Dr. G. N.
Patterson for his supervision and encouragern ent during the progress of this work. The assistance of Dr. deLeeuw during the latter part of the work has been rnosi..helpful.
The additional advice of Mr. S. A. Gordon is gratefully ack-nowledged.
The financial assistance of the Aeronautical Research Labora-tories, Office of Aerospace Research, United States Air Force, under con-\; tract number AF 33(616)-6990, Task 7064-01, Project 7064, has made this
investigation possible.
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Dil
Ion
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SUMMARY
When bodies travel in a gas so rarefied that the mean-free-path of the molecules is much larger than any of the dimensions of the body the aerodynamic properties of the body can be calculated using kinetic theory ignoring the collision between the molecule that have struck the body and those directed towards it. At slightly highe r densities such collisions start to become of importance and deviations from the free-molecular theoretical results will occur.
This report describes a series of experiments performed in a rotating arm apparatus to study the deviations from free-molecular flow that occur at low subsonic speeds. The specific criterion used in these tests was the pressure measured by an orifice probe in the walls of cylinders and flat plates of rectangular and circular shape which were perpendicular to the flow direction.
It was found that the deviations from free-molecular flow depend on both dimensions perpendicular to the flow in relation to the
'
.
1. II. ij ,. 111. IV.V.
VI. VII. TABLE OF CONTENTS NOTATION INTRODUCTION EXPERIMENTAL CONSIDERATIONS 2. 1 Description of the Experiments2.2 Orifice Probe Theory in Free Molecule Flow 2.3 Long Tube Pressure Probe
2.4 Response Time
2.5 Thermal Transpiration
2.6 Effect of Centrifugal Force on Pressure Readings APPARATUS
3.1 Vacuum Vessel 3. 2 Press ure Gauges 3.3 Pressure Probes EXPERIMENT AL PROCEDURE ACCURACY OF DATA DISCUSSION OF RESULTS CONCL USIONS REFERENCES FIGURES v 1 1 1 2 3 3 4 5 5 5 6 6 7 8 9 11 12
.
.
Kn Ue
I
NOTATLON.most probable molecular speed Knudsen number
=
=
Characteristic dimension Mean free pathstatie pressure of free stream
impact pressure corrected for centrifugal force field impact pressure existing in the probe volume near the orifice
pressure as read by impact pressure gauge pressure as read by statie pressure gauge speed ratio U lCm
velocity of body density
•
1. INTRODUCTION
The behaviour of bodies under nearly collision-free conditions has been described theoretically by so-called one-collision theories (Refs .
1 - 2). Such theories are very complicated and results so far are only avail-able for cases where the speed ratio of the flow is very large. In addition the validity of these theoretical results is as yet uncertain.
In physical terms the first deviation from the free-molecular behaviour can be described as a shielding effect of the molecules emitted from the body surface. Baker (Ref. 1) assumes in hls calculations that, when the mean free path of the emitted molecules (in the atmosphere of inc.ident mole-cules) is larger than a characteristic body d.imension, the majority of emitted molecules can be considered to collide only once with incident molecules. The deflected incident molecules will then either miss the body entirely or impinge up on it at an angle other than that of their original direction. One effect of this interaction is a reduction in the expected drag force on the body (Ref. 2).
In the present investigation, a rotating arm appara.tus giving low subsonic flow speeds was used in an attempt to measure analogous effects at low speeds. Special attention was directed towards the effect of both body dimensions perpendicular to the flow on the flow characteristics. This was done by measuring the pressure read by probes of different aspect ratio in flows at different Knudsen numbers. (The Knudsen numbers were based on the' smallest dimension of the probes. ) The results are of interest for the use of orifice probes as measuring instruments, since in many investigations the probe body has to be much longer than it is wide.
Il. EXPERIMENTAL CONSIDERATIONS 2. 1 Description of the Experiments
. The present investigation is concerned with the influence of the Knudsen number, i. e. the ratio of the mean free path to a significant length of the body, on the pressure read by an orifice pressure probe facing the on-coming stream. When the mean-free-,paths are sufficiently larger than the body dimensions, this pressure will be equal to the theoretical value based on an assumed collision-free flow field. However, when the mean-free-path becomes comparable to the body dimensions, deviations from this 'free-molecular' value for the pressure will occur, and the regime of transition flow is entered.
Particular attention was paid to the influence of the shape of the body, as indicated by the aspect ratio, on the value of the Knudsen number that is required to ensure that the deviation from free -molecular behaviour was a specified fraction of the difference between the free-molecular and continuum pressure readings.
Two types of probe bodies were used in the investigation. One set consisted of cylinders with aspect ratios of 30, 22, 17 and 5. In the se-cond set, flat plates of aspect ratios 10, 3.5 and 1 were used. In all cases the long dimension of the probe was perpendicular to the flow and the orifice was located at the geometrie centre of the probe body.
2.2 Orifice Probe Theory in Free Molecule Flow
At the largest mean free paths used in the experiments, the probe behaviour was essentially free molecular. The correspondingtheore-tical values were taken from Rei. 10. The relation for the pressure is briefly summarized here.
s
2.1"t
t
Consider an orifice probe with no inclination to the free
stream and assum.e. that the mean free paths are adequately large so that the oncoming molecules can reach the probe with essentially the distribution function they have at infinity. When this distribution function is a drHting Maxwellian with respect to the probe. then the pressure in the probe is given by the relation
where
s
Cf=
Ji,---=;=;-' _
P,
JT
-is the speed ratio of the flow, defined by the ratio of the tmdisturbed flow velocity to the most probable random speed of the gas molecules. In other words
S
=
?t/Cm,.
=
~2RT
In the case of the rotating arm, the flow speed equals the velocity of the body at the end of the arm. giving a speed ratio
s=
where = angular velocity of the arm
r
=
radius of orifice location.For a rnean temp. of 700F and an arm radius of 14.42'1 we
•
't
find that
-5 S
=
9. 305 x (R. P. M. ) x 10The mean free path,
À ,
on which the Knudsen nurnbers are based can, according to Ref. 5 be written as\
/ 6 ) ( N
I
À
-=
sp:r
7)(
~RT
~
=
the viscosity~
=
the density whereR
=
specific gas constantT
= absolute temperatureFor air, using Sutherland's relation for the viscosity (Ref. 6) we get
-3
T2.
À
=.B·
·
328x
I()
XP
7+2./0-
6
where
~
is in inches, P in microns Hg, and T in degrees Rankine. 2.3 Long Tube Pressure ProbeIn some of the checks on the proper operation of the equipment, long tube pressure probes were used in addition to the orifice probes. The theoretical free molecular relations for such probes were taken from Ref. 4, in which the results for the gauge pressure P2 are indicated by the relation
--Values of the function W(S, D) and W(O, D) are tabulated in Ref. 4, for the case of a tube with an internal diameter, d , rnuch less than its length, L, and a. speed ratio m uch less than unity.
2.4 Response Time
A difficulty in making pressure measurements at low pressures is associated with the fairly long, response times-. Harris, in Ref. 5, has made an investigation of the response time of low pressure probes. He found that the gauge pressure, p, at any time t for an orifice probe is given by the expression:
p - P2 PI - P2
where
P1 = initial press ure
P2 = final pressure t L = time constant and tL
=
cv(~
+
8)
3r3 0 with C-
-
3 1~
27f RT i 4 V = gauge volumeL = length of pressure lead
rt = internal radius of pressure lead
ro = effective radius of orifice
In the present experiments the orifices had a diameter of O. 015",
the pressure lead had a length of 30", an effective radius of 0.05", and the gauge volume was approximately O. 1 cu. in. This leads to a time constant of ab out 75 sec. In the experimental work, enough time was allowed for the error due to the lag of the gauge pressure to become insignificant.
2. 5 Thermal Transpiration
The pressure of the orifice prbe was measured by a pressure gauge that was actually located outside the apparatus. Some consideration has to be given to the relation between the probe pressure P2 and the gauge pressure PI' In all cases the pressures in the line connecting the gauge and probe we re so low that the m ean free path in the tube was considerably larger
than the tube diameter. It is wel! known (see for instance Ref. 7) that under
such conditions the pressures at the two ends of the line obey the simple relation
The pressure gauge was kept at a controlled, wel! known temperature. Hence the actual gauge pressure in conjunction with the gauge temperature, directly gave the quantity P2/f'i'WhiCh was used in the experiments.
A check on the correctness of the use of the collision-free thermal transpiration relation was provided by the test runs with orifice and long tube probes mentioned in Sec. VI.
2.6 Effect of Centrifugal Force on Pressure Readings
Part of the line connecting probe and gauge is necessarily ro-tated with the arm and the gas in this line therefore experiences a centrifugal force field that leads to pressure differences. For a gas at rest, as is the case here, and a uniform temperature of the line, Kennard (Ref. 7) gives the ratio between the pressure at the end of the arm and that at the centre as
.2
:1-/
W t"/2. RT
=
eI<
-
-
Pend of armPcentre of arm
where r = distance from centre to orifice location.
Values of the correction factor K(W) are plotted in Fig. 11, for a temperature of 700F and a radius of 14.42".
It should be noted that the correction is a relatively small one, so that small variations in the temperature along the arm will have little effect on the final result. It was therefore assumed that this temperature could be taken to be constant.
The pressure at the probe can then be found by the relation P 2
=
PDKwhere P D is the pressure as actually measured by the impact pressure gauge.
IIl. APPARA TUS 3. 1 Vacuum Yessel
The vacuum vessel which contains the arm (Fig. 1 - 2) (Ref. 9)
is a cylinder 34 inches in diameter and 11 inches high. An
"a"
ring is used to seal the top of the vessel, and a bronze to steel compressor seal (Fig. 5) seals the drive shaft opening. Other openings in the vessel (Fig. 3) accommo-date the three pressure gauge leads, the leak valve and the pumping inlet.Sealing for the lead from the probe at the tip of the rotating arm to a stationary Pirani pressure gauge is achieved by having the hollow drive shaft of the arm rota te in a container filled with low vapour pressure oil, through which the stationary pressure lead protrudes (Fig. 4). This arrange-ment was found to be satisfactory up to the maximum speed of rotation of 3000 R. P. M. used in these experiments.
A dummy wall is present to confine the model to motion in a rectangular cross section of 4 in. wide and 10 in. high. The function of this wall is to separate the disturbance produced in the gas by the thick arm as much as possible from the region in which the model moves. The vicinity of a wall also aids in damping the wake of the model, although of course the
models have to be kept relatively small compared to the cross section in order to avoid noticeable wall effects.
3. 2 Pressure Gauges
A Philips PHG-09 cold cathode ionization gauge was used to give a general indication of pressure in the vacuum vesseL For absolute pressure measurement, a 700 cc McLeod gauge was especially built. This gauge had a 0-40 mieron Hg range with a high sensitivity in the 10- 4 to 10- 3 mm range.
The pressure P 2 of the probe was measured using a Pirani type gauge with a compensator bulb in a constant voltage bridge circuit.
(Fig. 8) (Ref. 8) A second such circuit with identical components was used to measure the static pressure in the vacuum vessel. Each circuit was temperature compensated by using identical elements in the symmetric
positions of the bridge. The voltage resulting from the out-of-balance current of the bridge was measured across precision wire-wound resistors, stabie to temperature changes to better than 1 part per milllon per oR. The out-of-balance current of the bridge was measured by a Tinsley type 3184-D
potentiom eter.
3.3 Pressure Probes
The probes (Fig. 6 - 7) were manufactured from 0.10" dia-meter stainless steel tubing of needie temper., '. . The long tube probe and statie probe were used to insure that the apparatus was working properly,
i.
e
.
no thermal transpiration or out-gassing effects were present to a noticeable degree, by comparing experimental pressures measured with these probes with theoretical values. The ex?ellent agreement shown in Fig.12 indicated that the equipment was working satisfactorily.
The orifice probes were made by cementing 0.00031" thiek aluminum foil with a O. 015" diameter hole drilled in the centre, on the probe body (Fig. 6).
In all cases the probes were mounted symmetrieally so that the centrifugal forces did not introduce bending moments in the support.
The orifice measuring the pressure experienced by the probe was mounted on two types of probe body. In one case cylindrically shaped bodies of aspect ratio 30, 22,,17 and 5 were used. In the other instance, flat rectangular plates of aspect ratio 10, 3. 5 and 1 were studied. In both cases the longest dimension of these bodies were perpendicular to the flow with the orifice at the geometrical center of the exposed face.
'
.
IV. EXPERIM.ENTAL PROCEDURE
The basic quantity of interest in the investigation is the ratio of the impact pressure P2 to the statie pressure PI (see Sec. 2.2). As stated in Ref. 8, a constant voltage bridge of the type used should give a linear relation between out-of-balance voltage and pressure over a 0-500 micron range of pressure for a bridge resistance ratio R2/Rs up to 10. Actually this linear relationship was found to be applicable only if R2
I
RS is approximately unity as shown by the calibration curves in Fig. 9. The system we used had a ratio of R2/Rs of about unity and Fig. 9 shows that, although the calibration curve is not exactly a straight line, a linear approximation is quite good over a limited range of pressures. Such a linear approximation was used for convenience in reducing the data.Small differences existed in the slope of the calibration curves for the gauge circuit indicating the impact pressure voltage VD' and the cir-cuit indicating static pressure by the voltage VS. These differ.ences were taken into account by a calibration of the impact pressure gauge voltage in terms of the static pressure gauge voltage by means of the curve in Fig. 10. The relationship between VD and V S was found to be linear over the entire pressure range used.
The Pirani gauges were zeroed at pressures in the tank of less than 10- 4 mrn Hg. as indicated by the cold cathode gauge, 4ny residual
pressure due to outgassing was therefore suppressed and the measured pressures actually represent pressures with respect to a possible small out-gassing background pressure. However, these should in any case have been very small, since the vacuum vessel was kept under vacuum almost continu
-ously, except for probe changes.
The relative calibration of the impact pressure gauge and static pressure gauge was checked for each probe, by making measurements at three pressure levels, which always were chosen to include the expected test pressure within their range.
As stated previously, the response time constant for a typieal probe was about 75 seconds. In order to ensure that the lag effects were small, measurements of VD and Vs were taken af ter a period of 3 minutes.
The probes were run at different speeds to get a variation of speed ratio and at different ambient pressures to provide a range of Knudsen nurnbers. In addition some tests were made of probes of different size but sirnilar geornetry to get a comparison of measurernents under different con-ditions but at the same Knudsen number.
Difficulty was sornetimes experienced in getting, consistent re-sults from the larger size probes when these were newly manufactured. It
was felt th at a good reading had to satisfy two general requirements for con-sistency. The !irst was that the measured points should not show a scatter
of more than 3% from the mean curve. And secondly a measurement was con-sidered suspect if it did not follow the general tendency that the pressure de-viates further from the free molecular limit with decreasing Knudsen number. Whenever these conditions were not met the particular measurements were repeated to make certain that they were not in error because of spurious effects.
Especially the latter condition was not always met at first.
Some-times a particular probe under test (especially a larger one) would show a
smaller deviation from the free molecular pressure at smaller Knudsen
num-bers than at larger ones. It was found that this peculiarity was related to the
sequence of operations in the experiment, and an acceptable solution was found by adjusting the procedure.
Before going into this a word must be said about the particular problem of outgassing these larger probes. As mentioned in Section 3.3, the probes were built by cementing thin aluminium foils to the probe body. These were sealed to the body by glyptal paint. This paint was also used to seal alt welds on the probes against the possibility of leaks. On the larger probes,
this meant that an appreciable amount of this paint was present. It was also
noticed that upon drying the surface of the paint became pitted so that an appreciable amount of occluded gas could be present in this paint layer.
The usual procedure for the calibration of the impact and static
pressure probes one against the other in the manner of Fig. 10, consisted of taking three readings of pressure vs. voltage. The first one was taken at low pressure, the second at a high pressure and the third one at the pressure de-sired for the test run, which pressure was in between the other two. Under these conditions, the actual experiments were always carried out on a probe that had been exposed to a higher ambient pressure in the immediate past, with the possibility of more outgassing, especially for the larger probes. Even on the smaller probes it was noticed that under these conditions, the first few pressure readings were higher than anticipated.
To rernedy th is situation, the procedure to bring the apparatus up to the desired pressure for the test was reversed. The highest pressure point of Fig. 10 was obtained first, then the lowest point and finally the posi-tion corresponding to the test condiposi-tions. This procedure eliminated the ab-normally high pressure readings on sorne of the larger probes.
V. ACCURACY OF DATA
Three quantities were measured; the static pressure PI, the impact pressure P2 and the speed ratio "S".
The speed of rotation of the motor was found to be constant to
within O. 60/0 over the range or speed available. The length of the arm could be
measured to within 0.04%. The temperature inside the vessel could not have changed by more than 3% since for any one run, the temperature change of the atmosphere was never more than 3%.
\!
Using these values and the relation S =
CU
r /J
2 RT iwe find that the experimental error in S is approximately 2%.
In Ref. 3 it is stated that for pressures below 4 micron Hg and Knudsen numbers larger than 0.60, there is no aerodynamic heating of the
probes. In this work we have not gone beyond these limits and we have assumed that there was no significant aerodynamic heating and the arm was at constant temperature.
Under the assumption of no thermal transpiration and steady bath temperatures for the gauge heads, experimental errors in PI and P2 are negligible. Figure 12 also shows that the spread in experimental values is of
the order of 2%, in agreement with the above analysis.
VI. DISCUSSION OF RESULTS
A first set of runs were made with a long tube probe of O. 1 in.
external diameter to check the proper operation of the equipment, experi-mental procedure and reduction of the data. The static pressure in these ex-periments was 1 micron Hg. The ratio of mean free path to the external dia-meter of the probe was therefore about 20, ensuring 'free-molecular'
condi-tions. The experimental results for the ratio of impact pressure to static
pressure, when account is taken of the difference in temperature of the gas in
the vessel and the gauge temperature, are plotted in Fig. 12 and compared
with the theoretical values described in Sec. 2.3. The excellent agreement indicates that the experimental technique was sound.
A second set of experiments were made on flat plate probes of circular cross section. Four different probes were used with values for the
disk diameter of 0.7 and 0. 335 in. Figure 13 shows the results of the
mea-surements with these probes at a Knudsen number based on diameter of about 10 as a function of speed ratio. The agreement with the theoretical curves for 'free molecular' conditions is excellent and it should be noted that although different probe sizes and static pressures were used the results c1early show that for the same Knudsen number the probe pressure is the same. This is an
indication that for these probe sizes there are no significant effects of the
vicinity of the walls. Also if wake effects were present, a systematic
varia-tion with the size of the probe should be apparent.
Figure 14 shows similar results for two of the disk probes at a Knudsen number of 4.4. Again the points for the two different size prohes correlate weU, but a clear deviation from the free-molecular value has now occurred.
In Fig. 15 results are shown for the two sizes of disk probes at
value of the speed ratio, the ratio of impact pressure to static pressure is
1. 309, if the temperature of the gas and that in the gauge are equal. On the
other hand, the same ratio for continuum flow and low subsonic conditions is given by
2.
=
1+5
and hence this ratio takes the smaller value 1. 0256. Consequently we see that at a Knudsen number of 2, the value of the pressure ratio is lower than
the free molecular value by about 18
%
of the difference between free-molecularand continuum values. In Fig. 16 the percentage deviation from the free mole-cular values, in this sense, for the disk probes is given as a function of speed ratio.
Results of measurement on a cylindrical probe of aspect ratio 30 are contained in Fig. 17. Data are presented in this figure for four differ-ent Knudsen numbers, based on the diameter of the probe, as a factor of speed
ratio. This increasing deviation from the free molecular values with
decreas-ing Knudsen numbers is quite pronounced and it should be noted that these deviations become appreciable at significantly higher Knudsen numbers then thóse for the disk probes .
Similar results. for a flat rectangular plate bf aspect ratio 3. 5
are presented in Fig. 18, and a comparison of the behaviour of
~
for threedifferent aspect ratios at a fixed speed ratio of S = 0.16 but as a function of
Knudsen number based on width, are given in Fig. 19. A similar comparison
for two aspect ratios at S
=
0.25 are presented in Fig. 20.To get some explicit indication of the effect of aspect ratio on
the deviations from free-molecular flow, the Knudsen numbers required to
have a 10% deviation from free molecular behaviour in terms of the difference between free-molecular and continuum values were obtained from the curves in Figs. 19 and 20. These Knudsen numbers are plotted as a function of
aspect ratios in Figs. 21 and 22 for speed ratios of S
=
0.16 and S=
0.25respectively. In these graphs the results obtained from cylindrical and flat
plate probes are plotted together and it is seen that the results for the two
different shapes lie essentially on the same curve, indicating that the aspect
ratio is the controlling parameter.
The curve in Fig. 21 indicates an effect of aspect ratio up to values of about 25, af ter which the Knudsen number based on width no longer
changes, indicating that at this value the probe body has effectively the same
effect as an infinitely long one. At the higher speed ratio, this levelling off
has apparently not yet occurred at an aspect ratio of 30, the highest value used in the experiments.
\
VII. CONCL USIONS
The experimental results of this investigation show that at low subsonic speeds the shape of the body influences the first deviations from
free-molecular behaviour to a marked degree. In the case of pressure probes
the consequence is that the Knudsen number based on the diameter of the probe body that is required to ensure operation under free-rnolecular condi-tions is a function of aspect ratio and for long probes such Knudsen numbers
have to attain values of about 30.
1. Baker, R.M. L., Jr. 2. Charwat, A. F. 3. Editor: Devienne 4. Harris, E. L. Patterson, G. N. 5. Harris, E. L.
6. Kay and Laby
7. Kennard, E. H. \ 8. Leck, J. H. 9. Muntz, E. P. 10. Patterson, G. N. 11. Patterson, G. N. REFERENCES
Drag Interactions of Meteorites with the Earth's Atmosphere, University of
Cali-fornia (Ph. D. Thesis), 1958.
Lift and Pitching Moment in
Near-Free-Molecule Flow, Project Rand - Aerodynamics of the Upper Atmosphere, R-339.
Rarefied Gas Dynamics, lst Nice Symposium.
Properties of Impact Probes in Free Mole-cular Flow, UTIA Report No. 52.
Investigation of Time Response and Outgass
-ing Effects of Pressure Probes in Free Mole
-cular Flow, UTIA TN No. 6.
Tables of Physical and Chemical Constants,
Longmans, Green and Co., 1958.
Kinetic Theory of Gases, McGraw-Hill Book
Co., 1938.
Pressure Measurements in Vacuum Systems,
The Institute of Physics - London - 1937.
Pressure Measurem ent in Free Molecular Flow with a Rotating Arm Apparatus, UTIA TN NO. 22.
Theory of Free Molecular , Orifice Type Probes, in Isentropic and Non-Isentropic
Flows, UTIA Report No. 41.
Free Molecular Flow, John Wiley and Sons, New York, 1956.
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7 8FIG. 15 PRESSURE RATIO VS KNUDSEN NUMBER
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PLOT OF KNUDSEN NUMBER AT WHICH WE ARE 10% OF "A" AWAY FROM F.M.F.
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