.
-OCTOBER, 1961
by
1. J. Billington, E. Eggmann B. C. Stonehill and J. C. Lafrance
LOW DENSITY AERODYNAMIC STUDIES
by
1. J. Billington, E. Eggmann B. C. Stonehill and J. C. Lafrance
This research was supported partly by the Defence Research Board of Canada and partly by the Aeronautical Research Laboratory, Office of Aerospace Research, United States Air Force, under Contract No. AF 33(616) - 6990 and has been produced in cooperation with
The feasibility and usefulness of a rotating arm facility for aerodyna.mic studies at low density has been studied. Structure of the arm itself appears to be the most critical aspect of the design of such a facility. A theory for optimum arm design is outlined and a number of design charts based on this theory are presented.
This study suggests that it is mechanically feasible to construct a rotating arm facility with a capability of Mach num bers up to about 8 over a range of pressure levels encompassing .most of the low density flow regimes of interest. Although somewhat higher Mach numbers -are theoretically
possible, justification of the attendant increase in facility size and co st appears doubtful.
Based on the theory and design data of this report, facilities with maximum capabilities of 4000 and 8000 ft/sec respectively are
discussed. It is concluded that a facility of the 4000 ft/ sec class could be designed by direct application of the present analysis. For the higher speed facility, however, analytical investigation and probably some related
NOTATION ii
1. INTRODUCTION 1
2. CHARACTERISTICS OF LOW DENSITY FLOW 3
3. MECHANICAL FACTORS INFL UENCING DESIGN
CONCEPT 4
4. INFLUENCE OF TEST PROCEDURES ON
DESIGN CONCEPT 6
5. ROT A TING ARM CAP ABILITIES AND COMPARISON
WITH CONVENTIONAL F ACILITIES 8
6. INSTRUMENTATION PROBLEMS 10
Pressure Measurements 10
Heat Transfer 11
Measurement of Model Aerodynamic Forces 11
Data Transmission 12
7. POSSIBLE REFINEMENTS TO ROTATING ARM
EQUIPMENT 13
Airloek 13
Gases other than Air 13
Refrige ra tion 14
Rotating Nozzle 14
Magnetohydrodynam ie s 14
Counter-Rotation of Shell 15
Re-entry Simulation 15
Rotating Arm Cooling 15
Prevention of Flow Degeneration 15
8. TYPICAL FACILITY CONFIGURATION AND COST 16 Selection of 4000 ft/ sec Facility 16 Description of 4000
ft/
sec Facility 18 Cost Estimate for 4000 ft/sec Facility 198000 ft/ sec Facility 20
9. CONCL USIONS 22
10. SUMMARY AND RECOMMENDATIONS 25
Page
APPENDIX A: Low Density Flow Parameters 27
APPENDIX B: Theory of Rotating Arm Structure 29
Shape of the Arm 29
Centrifugal Acceleration 32
Payload Considerations 32
Weight of the Arm 33
APPENDIX C: Rotating Arm Calculations 35
Material Properties 35
Operational Curves 35
Aerodynamic Drag Estimate and Drive Power
Requirernent 36
APPENDIX D: The UTIA Rotating Arm Apparatus 38
The Vacuum Vessel 38
Rotating Arm 38
Drive Shaft 39
Motor 39
Pressure Gauges 39
NOTATION
A Frontal Area of Arm (ft 2 )
a Cross Sectional Area of Arm (in2 )
C Molecular Mean Random Velocity (ft
I
sec) CD Drag CoefficientD Drag (Ibs)
Fe Centrifugal Force (Ibs)
fc Centrifugal Stress (psi)
g Acceleration of Gravity (32.2 ft/sec2 )
Kn Knudsen Number L Characteristic Length (ft) M Mach Number m Mass (slugs) P Power (H. P~1 p Pressure (psi)
q Dynamic Pressure (lb Ift 2)
R Gas Constant (ft Ib
I
slug OR)Re Reynolds Number
r Arm Radius (ft)
T Temperature (oR)
u Arm Tip Velocity (ft
I
sec)W PayIoad Weight (lb)
WA Arm Weight (lb)
y
oe
(3
r
Radius of Arm Cross Section
Q:uantity Defined by Equation (B. 7) Quantity Defined by Equation (B. 15) max Maximum Taper Angle
Specific Heat Ratio
Molecular Mean Free Path Viscosity Coefficient Air Density (in) (degrees) (ft) (slug/ft sec) (slug/ft3) ProlJability Integral (See Equation (B. 24»
Angular Velocity (rad/ sec)
Subscripts: o Root
t Tip
1. INTRODUCTION
Aerodynamic test facilities fall into two main groups. those where flow is produced by means of an air stream moving past a stationary object and those where flow is produced by motion of an object through stationary air. From the point of view of instrumentation. measurement and recording the former class has considerable advantage since the model under test is rigidly attached to some earth mounted reference. Consequently this type of facility. which includes wind tunnels and shock tubes. has been used very widely. Resort has only been made to the second class of facility which includes such devices as aeroballistic ranges and free flight testing ranges. when the desired test environment could not easily b~ provided around a fixed test model.
The rotating arm test facility lies in an intermediate region between the two classes discussed above. Although the model is not rigidly mounted, it is constrained to move in a predetermined path and is at all times mechanically connected to a.rigid axis. lnstrumentation measure-ment and recording of test data is not as difficult in many respects as in the case of the free flight models, but the presence of the rotating arm does complicate this aspect of the operation considerably compared to wind
tunnel testing. The use of a rotating arm can therefore generally be justified only in cases where the same environment cannot be obtained in a more conventional device.
There are other obvious advantages and disadvantages to the rotating arm facility. On the one hand a precisely determinedspeed can be more easily ach,ieved, maintained and measured with the rotating arm device than with either a wind tunnel or a free flight range. On the other hand the model and its supporting arm are continuously traversing the same region of space and consequently, if the surrounding air density is more than a fraction of standard sea level atmosphere. the significance of the test results may well be obscured because the model is continually running into its own wake. At sufficiently high pressures the air mass surrounding the rotating arm may receive an inducedrotational velocity which is a substantial fraction of the arm velocity, thus cancelling out much of the speed advantage of the arm device.
Because of the above it may be concluded immediate~y that the rotating arm is not a very useful apparatus for the study of most flow phenomena in the continuum regime. However, for the study of flow phenomena at lower densities the picture is rather different. The low density areas of interest are indicated on the chart of Fig. 1. In this region one of the parameters of significance is the Knudsen number. the ratio of mean free path of a gas molecule to a typical linear dimension of a test model. At low values of the Knudsen number. as shown in Fig. 1, the gas flow is in the continuum regime. As the Knudsen number increases various intermediate regimes are traversed and finally, at high Knudsen numbers. free molecule flow is achieved. There is currently considerable
interest in experimental investigations of flow in the area bordering and just within the free molecule flow regime. It caD. be seen from Fig. 1 that this area can be entered at a given density level by an increase of Mach number. However, for conventional facilities, such as low density wind tunnels, it has been found that with decreasing density the available Mach number is also decreased (see Ref. 1). The ability of the rotating arm to penetrate this region is limited only by structural considerations which limit maximum speed and by the pressure obtainable in the chamber surrounding the whirling arm.
Once having attained free molecule flow there appears to be little advantage to be derived from an experimental point of view by pene-trating to even lower densities, since it has been demonstrated that most phenomena of interest are independent of Reynolds number in this regime. There does however appear to be some m erit in study of very high speed motions in the free molecule flow regime which simulate high altitude satellite motions. If sufficient energy can be obtained ionization or dissociation of the gas molecules may result but to reach this point, at least with air, requires extremely high speeds.
The purpose of the present report is to study the feasibility of rotating arm test rigs from several points of view. The areas in which such a device can most usefully b~ employed are established. The
mechanical and operational factors which influence design concept are outlined together with the range of speeds and sizes which appear to be mechanically practical for such a facility. Recommendations concerning the size and cost of a typical rotating arm facility are made and some possible variations and modifications to increase the scope and usefulness of this apparatus are discussed.
2. CHARACTERISTICS OF LOW DENSITY FLOW
The characteristics of a flow under low density conditions can be specified if the values of three flow parameters are known. These para-meters are the Mach number, the Reynolds number, and the Knudsen number. The first two parameters are those commonly used in continuum flow and the last, the Knudsen number, is peculiar to low density flows. The Knudsen number is defined as the ratio of the molecular mean free path of the gas to the characteristic dimension of the model under study. Using the Knudsen number the following regimes of fluid mechanics may be identified, with the
limiting values of Knudsen number as shown (see Ref. 1);
Continuum flow
Slip flow
o.
01<'Kn~ 0.1Transition region O. 1 c::: Kn ~ 10
Free molecule flow
The mathematical relationships between Mach number, Reynolds number and Knudsen number are listed in Appendix A. These relationships have been illustrated in Figure 1 as a plot of Mach number versus Knudsen
num ber per inch with lines of constant Reynolds num ber. It is convenient to
use Knudsen number per inch as a basis for comparing the operation of
different facilities or different test modeis, since for a flow at given pressure the actual flow regime experienced is dependent on the size of the model being tested. The flow regimes which would be experienced by a test model with a 1 inch characteristic dimension are illustrated in Figure 1. For models of other sizes the pressure scale would be shifted relative to the rest of the figure.
In the continuum flow regime the familiar laws of classical fluid mechanics generally apply. At the other extreme, in free molecule flow, the gas can no longer be considered as acontinuous medium and collisions between gas molecules and the surface of a model under study become much more
frequent than inter-molecular collisions. An understanding of this flow regim e is best obtained by application of kinetic theory. The intermediate regimes, exhibit a gradual change in flow phenomena between the two extremes. This variation is illustrated for one flow parameter, the temperature assumed by
.an object in the flow, in Figure 2 (based on Ref. 1). Flow phenomena in the
slip flow and transition regions are at the present time not thoroughly under-stood and there is considerable scope for valuable contributions to fluid
3. MECHANICAL FACTORS INFLUENCING DESIGN CONCEPT
The most important aspect of the mechanical design is the rotating arm itself. The theory of rotating armstructure is investigated in some detail in Appendix B, based on the assumption that a constant stress at all sections fr om root to tip of the arm is desirabie . The analysis leads to an arm shape of the type shown in Figure 3. It may be concluded from this
analysis that if a rotating arm facility is to be design to carry a specific pay-load weight at a specific maximum peripheral speed th en the minimum arm radius is determined by the maximum allowable taper angle on the arm. This angle (
r
max, illustrated in Fig. 3) must be small in order that the trans-verse stresses in any element of the arm are small compared with the radial stresses. The sample calculations leading to the results included in this report were based on values of maximum taper angle which were known to be acceptable for the materials considerE(d (see Appendix C).Based on the maximum allowable taper angle for aluminum, the millimum rotating arm radius has been plotted for this material as a function of payload weight in Fig. 4. A similar plot for fibreglass is presented in Fig. 5. It may be seen by comparison of these two figures that a fibreglass arm offers a substantial decrease in facility radius for a given payload weight.
lf model speed is specified but payload weight is not, then a decrease in tip radius accompanied by a decrease in tip area can be made without exceeding the maximum allowable taper angle. This variation, which is equivalent to following a constant speed line in Figs. 4 or 5, is accompanied by a diminishing allowable payload weight. Ultimately, however, the required tip area becomes too small for effective attachment of any load. For the
calculations of this report a minimum useful tip area of 0.25 square inches has been assumed and on th is basis the limiting "tip area" curves of Figs. 4 and 5 have been plotted.
At any given speed the absolute minimum arm radius ,which can be employed in any rotating arm device -is that arm which has both the
minimum practical tip area and the maximum allowable taper angle. This radius is found at the intersection of the appropriate constant speed line with the tip area limit in Figs. 4 or 5 for aluminum and fibreglass respectively, and has been cross-plotted as a function of speed in Fig. 6. Also included in Fig. 6 for comparison are similar curves for steel and titanium based on the material constants listed in Appendix C. It can be seen from Fig. 6 that the choice of material for arm fabrication will have considerable influence on the overall size arid design of the facility ..
A further comparison of the four materials considered above is given in Fig. 7. In th is figure a constant peripheral speed of 4000ft/ sec has been assumed; this value appears to be a very practical upper speed limit and corresponds to approximately a Mach number of 3.5 referred to the speed of sound at normal atmospheric temperatures. Minimum tip
radius (corresponding to maximum taper angle) has been shown in Fig. 7 as a function of PSlY weight. lt will be noted that three of the curves shown are terminated at the lower end by the minimum tip area limitation mentioned above. For fibreglass this limit lies off the plot to the left.
It is apparent that considerable improvement in the operational envelope of a rotating arm facility, or conversely a considerable reduction in facility size for a desired operating envelope, could be obtained if larger values of the maximum taper angle were tolerabIe. For example, if a 40$ increase in the angle were possible it would allow a doubling of the payload at given radius and tip speed or a reduction in minimum radius of 20% for a given tip speed and payload. The literature does not provide a clear answer to the question of what maximum taper angle is permissible in a rotating arm. Therefore the results of this report are based on values known to be conserv-ative. In the development of a design for a rotating arm to operate at high speed, it appears worthwhile to conduct a further study with a view to increas-ing the taper angle. This study would include a theoretical analysis of arm stress distributions and possibly some simple experim.e:b.tal stress measure-ments of arms of various tapers in a small rotating rig.
Bending of the arm under gravitational force has not been con
-sidered in the present analysis. Although this may be significant for long slender arms at rest or revolving slowly it should not be important under design whirling speeds.
Practically there is an upper limit to the velocity attainable with a rotating arm device because the minimum radius required {and the arm weight) rapidly increases with increasing peripheral speed as shown in the following tabIe.
TABLE I
MINIMUM ARM RADIUS AT VARIOUS MODEL SPEEDS
Aluminum Fibreglass Steel Titanium
u .= 6000
ft/
sec 65 feet 40 15.8 7.000 400 117 75 8000 16.6 1300 400 9000 58.4 10000 140 11000 515It would appear difficult to justify speeds in excess of about 8000 ft/ sec because of the large size and cost penalty associated with further relatively minor increases and the presently known need for study in this range.
Other mechanical factors influencing the design of the facility are the size and shape of the containing vessel, the capacity, ultimate pressure limit, and pumping time of the vacuum system and the provision of adequate seals.
Pumps
Pumps for this facility appear to be quite easily obtainable. Since the pumps will have to handle only the leakage flow into the test facility it appears that reasonable care in test chamber manufacture, and with seals, wiU make the }lse of quite modest pumping capacities feasible. It is therefore most likely that the pumping capacity will be sized almost entirely to provide reasonably rapid pump-down times in order to permit frequent testing and
to minimize the time in which the facility mu·st stand idle. This time, however, cannot be reduced beyond a minimum value because no matter how large a
pump is installed a certain waiting period is unavoidable to provide adequate out-gassing of the test facility, model, and instrumentation.
Commercially available off-the-shelf diffusion pumps wiU easily produce vacuums down to approximately 10- 2 microns which would permit a Knudsen number per inch of approximately 200. For models having characteristic lengths of 1 inch or less this is therefore weU into the free molecule flow region. lf testing is required only into the transition region Knudsen numbers of about 1 are required and this can be achieved with a
1 inch diameter model and pressure· of the order of 1 micron. Pressure in this region can be attained by rotary pumps. It is therefore anticipated that, barring serious leaks or seal problems, the pumps for the facility wiU be a comparatively simple accessory.
4. THE INFLUENCE OF TEST PROCEDURES ON DESIGN CONCEPT The consideration of mechanicallimitations on the design of a rotating arm facility does not yield an unique solution. A range of mechani-caUy practical configurations results, and from this range a fin al arrange-ment must be chosen. In this respect, the operational procedures proposed for the apparatus may be of considerable influence in choosing a final con-figuration. For exa.rnple, if a large number of short runs are likely to be made with a return to atmospheric pressure between each run, then the
pumping system must have a greater capacity than that required for a facility in which slow pump down and long outgassing periods are the rule.
Pump down time can be significantly reduced by stopping the model and arm in a test section airlock so that only a portion of the low density cavity is exposed to atmospheric pressure during model exchange or instrument adjustment.
The choice and design of a test model and its associated
instrumentation are an important part of the test procedures. The allowable stress of the model and the instrumentation are therefore factors to be
con-sidered when choosing the configuration for the test facility. It is evident
that as the arm radius is increased the centrifugal "g" loading at constant
peripheral speed rapidly decreases; this can be seen in Figs. 4 to 7 where
centrifugal "g" curves have been superimposed as dotted lines. lf. for
example. pressure gauges of the type ha ving heated filaments are to be em-ployed it is necessary that the "g" loading be kept within moderate limits.
In fact. the allowable "g" loading appears to be the deciding factor in the
choice of a design point for the facility if freedom in instrumentation selection
is to be provided. If the "g" loading is not considered important from an experimental point of view then there is little justification for a facility other than one having the minimum radius for the design point payload and speed. On the other hand. if an allowable "g" load is postulated then this rather than the payload weight will be the important consideration and willlargely determine the size. material and cost of the facility as will be described in a subsequent section of the report.
In the choice of a facility design point it is necessary to establish first the maximum speed required. The necessity for this choice is
illus-trated by the curve of Fig. 8 which shows the ratio of allowable payload to
design point payload as a function of the ratio of operating tip speed to design point tip speed. As the speed is reduced below the design point the
centri-fugal stresses for a given payload weight decrease as the square of the vel
-ocity. and therefore an increased payload may be carried without overstress-ing the rotatoverstress-ing arm. In the other direction however very little increase in operating speed above the design point can be permitted. even at very reduced payloads. since speed increases would overstress sections of the arm near the root due to the increased centrifugal loading on outboard elements of the
arm. This speed limitation. due to arm stress. is shown as a rapidly
fall-ing off dotted line at speeds above the design point in Fig. 8.
Theoretically. the payload at low speeds could be increased
many times above the design point payload as the operating speed is decreased (see Fig. 8). In practice however there are several reasons for a limitation
to the maximum payload at low speeds. One of these is the bending of the
arm under gravitational loading at low speed or when at rest. The second reason is the tangential stresses which would be induced in the arm during acceleration and deceleration of the model. In this latter connection a sudden }ailure of the vacuum system with the model at full speed could result in a sudden tangential loading on the arm due to aerodynamic drag
and might result in a catastrophic failure of the facility. The ~orm of the
low speed arm Qending limit is shown by a dotted curve in Fig: 'S. but further
The linear dimensions of the test models influence the choice of the operating flow regime, since the Knudsen number depends upon these dimensions. Payload weight at maximum speed also influences the design configuration by placing a minimum value on the allowable tip radius (as may be seen in Fig. 7).
It is shown in Appendix B that the ratio of arm weight to pay-load weight for a given structural material depends only up on peripheral speed, provided the minimum arm radius is chosen. On this basis the
curves of Fig. 9 have been plotted for the four different structural materials considered in this report. This figure serves as an approximate indication of the variation of cost with design point speed. It should be noted that the weights considered in this report are the weight of the arm alone and do not include àny allowance for a counterweight which would undoubtedly b~
necess-ary for dynamic balancing of the apparatus.
5. ROTA TING ARM CAP ABILITIES AND COMPARISON WITH
I CONVENTIONAL F ACILIT lES
·t
The field of operation of a practical rotating arm facility and of other facilities aimed at tests in the same regime, is shown in Fig. 10. In this figure the limits beyond which tests cannot be satisfactorily conducted are shown on a plot of Mach number versus Knudsen number per inch. The Knudsen number per inch has been chosen as one variabIe in order to give a basis for the comparison of various facilities. Many tests in the free mole-cule flow regime, at Knudsen num bers greater than 10, have been achieved by the use of extremely small modeIs, such as 0.001 inch diameter wires.
Referring to Fig. 10, the approximate limitations of the present UTIA low density wind tunnel are shown and it wil! be seen that operation ,at Mach numbers in excess of 1 requires the acceptance of small Knudsen
numbers per inch, bordering on the continuum flow regime. In order to carry out tests in the transition or free molecule flow regimes, the Mach numbers must be progressively reduced in order to obtain saiisfactory operation of the facility. This limitation is typical of a test facility in which flow velocity is obtained by the use of a supersonic nozzle and though the range may be improved somewhat by the use of perforated nozzles etc., the limiting line for all such facilities will be essentially as shown for the UTIA tunnel.
Experimental m easurem ents of heat transfer to a wire in a similar low density tunnel at Ames Laboratory (see Reference 2), were made in a region which is also shown in Fig. 10. It wil! be seen that the field which could be reached by the Ames tunnel lies astride the limit of the UTIA tunnel. In order to achieve this field the Ames Laboratory tunnel employed a perforated nozzle with boundary layer suction, and the experiments were made on wires between 0.001 and 0.080 inches in diameter. Low density tunnels employirig supersonic nozzles can therefore reach the transition or
free molecule flow regions, but at the expense of limiting the tests to comparatively modest Mach numbers and
tb
verys.mall modeis, which must consequently be of a very simple nature.The field of operation available with the rotating arm 'facility is also shown in the same figure, and it will be seen that its limitations are of a quite different nature. First, the maximum Knudsen number per inch available from such a facility is limited only by the capacity of the pumps to maintain a vacuum. The pumps required will not be large (sinee the facility is completely enclosed) and .will have to deal only with leakage; however, they must be able to achieve a reasonably high vacuum. In the event that a vacuum of the order of 10-2 microns can be attained, and this is quite possible with standard commercially available pumps, then Knudsen numbers of 200 may easily be obtained with a 1 inch model. This pressure has arbitrarily been chosen as an operational limit in Fig. 10.
The Mach number obtainable with such a facility is basically limited by the stressing of the rotating arm as discussed elsewhere in th is report. A Mach number of about 7. 25 as shown in Fig. 10 korresponding to -u
=
8, 000 ft/sec) appears to be a reasonable limit and could be achieved with a 17 ft. fibreglass arm. The maximum velocity as permitted by arm stress, may be obtained over a considerable range of Knudsen num bers so long as the arm itself remains in a low density flow. This requires a high Knudsen number based on arm cross-sectional dimensions. As the pressure in the facility is increased, and the KIludsen number decreased, the arm moves from the free molecule flow region through the transition and slip flow regions towards continuum flow and the third limit, that of aerodynamic heating of the arm, is reached.For Mach numbers below about 2, and with the facility walls near room temperature, the stagnation temperature on the arm is not
sufficient to cause a heating problem. ~. For Mach numbers above 2 the stagnation temperatures become higher than that allowable for the very highly stressed material of the arm; These high Mach num hers can thus be maintained only if the density in the facility is kept low enough to avoid these severe heating problems.
The temperature of a wire in a low density flow is known to varyas shown in Fig. 2 (based on data of Ref. 1) as the pressure is changed. The upper and lower limits of these curves can be predicted
with reasonable confidence from continuum and kinetic theories respectively. The allowablearm tem'perature limits have been calculated by the two
theor~es as a function of velocity and pressure, and form the two boundaries of the temperature limit area shown in Fig. 10. ' In practice the arm heating rate is determined by the amount of energy available in the flow as weU as
by the stagnation temperature. The emissivities of the model, arm and waUs are also of importance. In any case there is considerable uncertainty
about the exact temperature boundary of a rotating arm facility, but this limit almost certainly faUs in the shaded region of Fig. 10. Appropriate choice of model and arm dim ensions and chamber pressure could be used to ensure that the arm was operating in a safe temperature region. The entire problem of arm hea ting warrants further consideration in a detailed des ign study.
From Fig. 10 it will be seen that for Knudsen numbers smaller than about 10- 2 per inch the envelope of operation of a rotating arm falls below that of a low density tunnel, due to the temperature limi-tation on the rotating arm device. For tests at Knudsen numbers smaller than this value, therefore, the low density tunnel offers a more versatile test facility and will permit tests to a higher Mach number than would the rotating arm. For Knudsen numbers above about 10- 2 per inch the rotating arm device becomes progressively more attractive and in the transition and free molecule. flow regions it offers a much greater choice of Mach number. The ability of the rotating arm device to penetrate deeply into the free molecule flow region with a model of a fairly large size has little foreseeable advantage at the present moment, since it is considered that there wiU be little change in most flow parameters with increasing Knudsen number once the free molecule flow regime has been reached. In the event that comparatively complex or large models were required, however, the very low density ability of the rotating arm device would be convenient because even a very large model could still be tested at pressures low
enough to ensure data in the transition oi'free molecule regions.
A smal! rotating arm apparatus has been used at UTIA (see Reference 5) for low density pressure m easurements. This apparatus can penetrate to pressures of O. 01 micron, but is limited in speed range to Mach numbers less than 0. 25 (2500 rpm with a tip radius of 11. 9 inches). 6. INSTRUMENTATION PROBLEMS
The details of instrumentation of a test model are properly part of the test program and model design but the general instrumentation concept must be considered during facility design. Some of the more important instrumentation problems have been reviewed in the present study and are briefly discussed below.
Pressure Measurements
The practicability of using pressure probes inlow density flows is weU established (see Refs. 5, 7,8). However, certain difficulties such as outgassing of the interior surfaces of pressure probes and long time constants of probes with large volumes and long tubes are commonly encountered. These problems can be minimized by keeping both interior surface area and tube length as sm all as possible.
If it is desired to connect model pressure probes by means of long leads to sensing devices in the rotating arm hub then centrifugal forces acting on the gas inside the pressure leads will distort the readings obtained. Such effects. although corrections càn be applied. are undesir
-able.
In view of the above it would appear worthwhile to consider the use of a short pressure probe connected to a pressure transducer with
-in the model. An electrical signal could then be used to actuate a pressure recorder or indicator. This in turn require s the use of a reasonably large model. The inclusion of a pressure transducer in the payload introduces a new problem; if sensitive instruments such as Pirani or ionization gauges are to be used, a limitation is placed upon the allowable "g" field. A
preliminary calculation suggests that heated filaments could not be operated at greater than 60, 000 "g". However,' there appears to be reasonable hope for some further development of transducers suitable for this application. Heat Transfer
Heat transfer measurements do not appear to present serious instrumentation difficulties since these would probably be made using
reasonably rugged electrical type gauges which would be able to withstand the "g" loading. It does appear however that very long runs may be re-quired. especially at the lowest densities. in order to achieve a significant energy interchange between a large model and the flow, a problem common to most tests in the low density field.
Measurement of Model Aerodynamic Forces
The measurement of aerodynamic forces on a model is most desirable. but is difficult in the case of a rotating arm device because the rotation subjects the model to forces of the order of thousands of times the model weight. A satisfactory model support must thus withstand these radial forces while, at the same time, re.maining sensitive to the compara
-tively minute forces of lift and drag. Three things that affect the problem are:
(1) Arm radius a,nd tip speed. The lowest speed and the maximum arm radius will minimize the accel-eration loads.
(2) Model density. A low model density will permit a large mode.l of low weight and will provide the best ratio of aerodynamic to acceleration forces. (3) Model Support. A steel wire is an example of a
support with great strength in one direction and only a fraction of that strength in another direction.
Fig.ure 11 shows a simple wedge model mounted on a wire support. The wire support is attached to the model at the model centre of gravity and carries the centrifugal forces in torsion. Aerodynamic forces are carried by the wire in torsion. Since the wire is torsionally weak, a deflection occurs due to the aerodynamic forces and the deflections can be recorded photographically as the model passes an observation window. For this method no deflection wiU occur if the resultant aerodynamic force
passes through the model centre of gravity. If necessary, therefore, the model C. G. must be moved by drilling lightening holes (as shown in Fig.
Ub) or by adding weights.
To illustrate the feasibility of this method the diameters of steel "wires" required to support a 1 lb. model have at various "g" loadings been calculated and are listed in Table Il. Also listed are the torques
required to twist a 6 inch length of each "wire" through O. 10 . TABLE Il
CHARACTERISTICS OF TORSIONAL MODEL SUPPORT "WIRES" Centrifugal "g" Load 50,000 25, 000 10, 000 "Wire" Diameter O. 541 in. 0.381 0.241 Torgue for 0.10 Deflection of 6" Wire 29.5 lb-in 7.2 1.2
The advantage of low "g" operation is immediately evident from Table Il.
Data Transmission
There are two stages of data transmission involved, the first from arm tip to hub and the second from hub to external operator.
The first of the above stages introduces some problem s
because of the high stressing of the arm itself. A metallic conductor loose-ly connected between the hub and the tip either through an internal passage or along the exterior of the arm would be unable to support itself under the centrifugal loading. Such connectors would therefore need to be bonded to the arm in some manner. For a fibreglass arm a number of metallic fila-ments could be moulded into the arm during fabrication. For a metallic arm some electrical insulation would be required. This problem requires further consideration in a detailed arm design. Pressure leads could be provided between tip and hub by manufacturing an arm with internal passages but this would require some refinement to the arm stressing calculations.
The second transmission stage - from arm root to external operator - can be obtainedin several alternative ways including the following;
(1) Transmission of electrical signals through slip rings. This presents a noise problem but should be suitable for some types of data.
(2) Transmission of pressure signals through rotating pressure seals. This method is described in Ref. 5. (3) Telemetry using transmitter mounted on the
rotating arm near the hub. Alternatively. if payload allowance permits. a telemetry set could be inc1uded in the payload.
(4) Recording internally on tape or other recorder mounted on the rotating arm or in the payload.
In addition to the methods described above various other schemes. such as photographic recording are possible.
7. POSSIELE REFINEMENTS TO ROTATING ARM EQUIPMENT Many possible refinements and extensions of the rotating arm method suggest themse1ves. and some of these are outlined below. Airlock
As mentioned previously it is possible that lengthy waiting periods for adequate outgassing may be avoided by the provision of an airlock so that the rotating arm. in some given position. may be enc10sed in an airlock (or at least an airlock provided which will enclose the tip of the arm and the model) and the model and instrumentation thus be available for change or adjustment without losing a high vacuum through the rest of the facility. Such a device could very considerably reduce the waiting time for outgassing and provide a great increase in the available testing time from a given facility.
Gases other than Air
The use of gases other than air will. of course. be simple using a closed facility. The facility may first be evacuated to remove the air and th en a small quantity of any other gas injected into the facility will provide the desired atmosphere. The costs of using even a compara-tively rare gas will thus not be excessive.
The possibilities of using gases other than air appear to be quite' wide. For example the use of freon, xenon or sulphur hexafluoride would, at the sam e model speed, provide Mach numbers approximately double those obtainable with air. Ceriain gases could be em ployed to produce or improve flow visualization techniques. It also appears likely, although this has not been thoroughly in~stigated in the present study, that real gas effects such as dissociation could be studied using gases other than air. This last possibility depends upon the selection of a gas having a
sufficiently low dissociation energy. Dissociation of air itself does not appear to be feasible within the practical speed range...df a rotating arm unless the entire device could be operated at elevated temperatures. Refrigeration
Refrigeration of the entire rotating arm facility appears to be one possibility for obtaining increased Mach numbers at a given peripheral speed. The feasibility of this method has not been considered in detail during the present study, but might be worthy of some investigation when undertaking a specific facility design.
Rotating Nozzle
Some consideration has been given in a previous study to the possibility of' attaching a convergent-divergent nozzle to the end of a
rotating arm. A test model could then be placed in this "revolving super-sonic nozzle". It was determined that this device would operate approxi-mately as a Mach number doubler, although access to the model and measure-ment of flow parameters would be quite difficult. In general this arrange-ment is not weIl suited to low density flows, but it might have some appli-cation in the highest pressure ranges of the facility considered here, since at those pressures the Mach number attainable is limited by arm heating. Magnetohydrodynamics
Study of flight in an ionized gas or plasma appears to be possible in the type of facility considered in this report. Further investigation of .
this aspect of rotating arm operation appears to belong in a specific facility design, however some general comments may be made. It is probable that only a small volume of air need b~ ionized and that the necessary tests could be carried out while the model passed through this particular section of its' circular path. Suitable ionization electrodes or plasma generators need be attached only in the appropriate region.
Counter-Rotation of Shell
It has also· been suggested that h~ghe,r Mach numbers may be obtained by driving the arm in <;>n~ d..irection.,and r~tating the boçiy of the test facility in the other. The gas ,thenrotates with the,fa~ility in the opposite direction to the arm, and the relative velocities are considerably increased. Since the velocities attaiI)able wUh the arm are attainable only l;>y reason of an extremely sophisticated stress analysis and arm design, the speedwith which the facility could be rotated in the other direction is likely to be only a fraction of the speed with which the arm may be rötated, and.at first sight it appears that the gain achiev~ble in this manner would p~obably not be
worth the considerable .increas.e in complexity and cost. This aspect how,ever requires further investigation.in conjunction with the inyestigation of the ~f?e of gases other than air, since it is possible that with some particular gas ' some very real advantage may be gained by a compar~tively modest increase in Mach number ...
: I j Re-entry Simulation
It appears likely that at least some portion of an atmospheric re-entry could ,be simulated by .running a mO,del ini~ially at very low densities and;,t.hen slowly leaking in air to raise the density. The ,Possibilities here are litnited by considerations of tangential forces, arm bending, and aerp-dynamic heating. , "
Rotating Arm Cooling
The operational: limit due to high temperature on the arm may possibly be overcome by the use ,of a thermal shield along the leading edge of the arm. Since the arm is stressed to carry the payload only, the shield would nave to be self-supporting and would in effect be a second slender arm along the leading edge of the first. A further possibility is that of cool-ing jhe leading edge of the arm, but this appears difficult because the cool-ing gas would have to be passed through the arm from root to tip and back and could not be released into the facility. These possibilities both require considerable further investigation to establish their feasibility.
Prevention of Flow Degeneration
In the transition or free molecule flow regimes, the flow is unlikely to be disturbed by the passage of the arm, because between success-ive revolutJons the molecule!;! wiU have enough collisions w:ith the walls of the facility that they w:ill.be completely recovered from the effects of any cont9-ct with the arm ,before it r,eturns agàin. As the tests 'range frpm the
transition region, ,throu,gh slip flow and t~wards the continuum regime,
however, an increasing tenctency for the, gas to rotate with the arm will become apparent. ,-·If testing is, to ,)Je cqntinued into these higher pressure regions, it' is. probabl~, ~h~t the interioT of the facility will.have to be provided with baffle:s j q prev,ent or miJ;limize this flow degeneration.
Development of suitable interior baffles is likely to present::.a number of problems since they must prevent flow rotation without interfering with the flow field around the model. This problem is probably weU suited to some experimental study and development,
8. TYPICAL FACILITY CONFIGURATION AND COST Selection of 4000 ft/sec Facility
It appears useful to apply the analysis of this report to the preliminary design of a facility to meet a !lypothetical performance require-ment. The performance requirement postulated is.as Iollows:
Maxfmum Speed: 4000 ft/ sec
Maximum Centrifugal Load: 100, 000 "g" Minimum Pressure: 10- 2
Payload is not specified. This specification appears to represent a useful operating range. with Mach numbers up toabout 3.5 in a moderate size. The maximum "g" loading may preclude the use of the more elaborate
instrum entation at the highest speeds but not at reduced speed (see Table lIl) TABLE lIL
"g" LOADING AT REDUCED SPEED FOR 4000 ft/sec 100,000 "g" FACILITY
u (ft/ sec) 4000 3000 2000 1000
"g"
100.000 56.000 25.000 6. 200Referring to Fig. 7 it is seen that a minimum tip radius of 4.96 feet is required to meet the specified "g" limit. However, the choice of materials is seen to provide· a wide range of possible maximum payloads at this radius. Note that it is possible to de.sign an arm for any payload to the 1E~ft of the intersection of the 100, 000 "g" line with the appropriate material curve. In general. however, there appears to be no substantial economy in overall facility size or cost by designing for less than the maximum payloàd for a givenmaterial. while considerable operational
flexibility would be sacrificed. On this basis, therefore, the material comparisons listed in Table IV can be obtained from Figs. 7 and 9 and Appendix B.
TABLEIV
COMPARISON OF MATERlALS FOR 4000 ft/sec ARM
Fibreglass Titanium Steel Aluminum
Maximum Payload 15 13.5 3.3 0.62
(Ibs) (Fig. 7)
Arm /Payload 7 86 155 285
Weight Ratio (Fig. 9)
Arm Weight (Ibs) 105 1161 512 176
TiR Cross Section 11. 74 7.51 1. 50 0.89
(in2) (Eq (B. 18) )
On the basis of the above comparison it appears reasonable to reject the aluminum ar.m because of its low payload capacity and the titanium arm because of the unfavourable comparison with fibreglass.
The investigation of steel and fibreglass has been carried further, however, by calculating the profiles of these two arms which are th en compared in Fig. 12. It can be seen from this figure that the root areas of the two alternative arms are approximately the same. Thus either arm could be used in enclosing chamber of the same size but the steel arm is heavier and has a smaller payload capacity.
Although a closer investigation might lead to other conclusions, the following choice ~ppears, on the basis of t4e present study, to be
appropriate for this example. Assuming that a 3.3 lb payload is adequate for test purposes a steel arm of the dimensions shown in Fig. 12 should present a very straightforward fabrication job. Fibreglass, on the other hand, might lead to some unforeseen development problems. Consequently a steel arm is -selected for this 4000
ft/
sec facility. This choice does not preclude the possibility of extending the capacity of the facility, should this appear worthwh,ile at ,a later date, by the manufacture of an alternative fibreglass arm for use in the same chamber. Based on this choice of arm, the facility is described below.Description of 4000
ft/
sec FacilityThe g.eneral configuration and cost of the 100. 000. "g" .
4000 ft/sec facility discussed in the previous section can now be estimated and is described below. Reference is made to Fig. 13 which shows a
schematic illustration of the probable arrangement of the facility.
For safety of operating personnel. in case of accident. the rotating arm and its vacuum chamber are located in a pit. The entire structure is firmly mounted on a concrete pedestal in the centre of the pit.
The general structure and shape of the steel arm for this facility have been discussed in some detail above. The arm iUustrated in Fig. 13 is shown as tapering to a minimum cross-section and then enlarging slightly to the model attachment. This technique suggests itself as a method for increasing the available attachment area. The enlarging section would have a stress less than
ft
and would be considered as part of the payload (thus reducing the maximum aUowable model weight) and might .contain a number of threaded holes for model attachment. The minimum arm cross-section represents the "tip" in the analysis of Appendix B. It should be noted that the model speed wi11 be somewhat higher than the "tip" speed because the model radius is greater than rt. For the same reason the model wiU experience a slightly lower "g" load than the "tip" section.The tip of the rotating arm projects through a narrow slot between two circular baffles designed io inhibit radial air flow due to centrifugal forces. thus enclosing the model in a toroidal test section. The cross-sectional area of this cross-section is dependent upon the shape and size of the largest models to be used in the facility.
It is anticipated that the arm wiU be supported by a shaft with one end passing out through the vacuum vessel. A rotating seal is required for such a shaft; however an adequate seal is considered feasible and this arrangement aUows instrumentation leads from the model to be brought out to exterior slip rings. The drive is also entirely outside the vacuum vessel.
Two bearings are required. The lower bearing must take the fuU weight of the arm as a thrust load; a medium series single row angular contact ball bearing of 40 mm inner diameter would be adequate. The upper bearing must take radialloads due to dynamic unbalance only. In this connection it is important to minimize dynamic unbalance in the system; an unbalance of less than one gram at tip radius appears desirable. A
balancing m echanism probably in the form of a screw adjusted balance weight on the counterweight side. is necessary. After any change in pay
Access to the arm and model will be through the observation windows, one or more of which will function as an access door. The observ-ation windows are positioned as shown in Fig .. 13.
The power required to drive the arm is given byequation (C-5). This equation may be sim plified by substituting a conical arm, that is an arm having a straight taper from root to tip, in place of the real arm shape. Such a simplification has been made in the present case and calculation shows that 5 HP will suffice to drive the arm at a Mach number of 2.0 at pressures up to 103 microns. Higher speeds, up to the design 4000 ft/sec are attain-able at lower pressures where heating is not a problem, but the HP require-ments are less. The drive would be effected through a magnetic coupling, or similar device, to allow accurate control of speed at any fraction of maximum design speed.
It is anticipated that pumping requirements for such a facility will be quite modest. Since an appreciable length of time wi11 be required for outgassing of the facility and instrumentation, the pump-down time wi11 not be critical. Choice of actual pumps will be based upon the desired pump-down time and the probablfi! leakage through the rotating seal. Cost Estimate for 4000 ft/ sec Facility
For the 4000
ft/
sec rotating arm facility described above engineering and design costs are estimated as follows:Basic Design and Layout Final Design, Analysis and Detailing
Engineering Supervision During Construction and Installation
$ 7,000
18,000
5,000
TABLE.V
COST ESTIMATE FOR 4000 FT /SEC 100,000 lig" ROTATING ARM FACILITY
Arm, counterweight and shaft assembly
Motor. starter and control '
Speed Control OYIagnetic
Coupling) Gearbox
Vacuum Vessel
Positive displacement pump Concrete foundations and
walls . Diffusion pumps Building 1500 Ibs at $5/lb Installed cost
"
11 11 11 19, 000 Ibs at $l/lb Installed cost 90 cu. yds at $50/ cu. yd installed Installed costFacility Manufacturing Cost
8000 ft/sec Facility $ 7,500 1,000 1;800 1, 700 19,000 2, 000 4,500 6,000 5,000 $48,500
As discussed above 8000 ft/sec appears to represent a
practical upper limit for tip speed. It is therefore of interest to consider
briefly the likely configuration of a facility designed for operation at this speed.
It is evident from Fig. 6 that fibreglass is the only structural
material of those considered in this report, which would result in a facility of moderate size. For the other three materiais, arm radii of 100 ft. or greater would be required.
Fig. 6 suggests that a minimum radius of about 17 ft. would be possible using fibreglass. At this radius a payload of about 4oz. could
be carried. From Fig. 5 it may be seen that an increase to 26 ft. radius
would be required if a 1 lb. payload was desired. The model on the 17 foot
arm would be subjected to about 125,000 "g" centrifugalloading and that on the 26 foot arm to about 75, 000 "g".
Such a facility would be of the same general configuration as the 4000 ft/sec facility. A drag calculation similar to that described above for the 4000 ft/ sec arm indicates a power requirement approximately double that for the smaller facility. This is a small drive for such a large appar-atus and does not thus present any difficult problems.
This high speed facility would be as much as 5 times the diameter and twice as deep as the 4000 ft/sec facility. The volume and hence pumping requirements, will therefore be increased by about 50, if the pump-down time is to be kept constant. Due to the large increase in pumping,requirements, it is possible that the provision of an airlock, which permits access to the tip of the arm and the model while maintaining a
vaçuum in the remainder of the vessel, would be an economical solution for a large facility. Such an airlock might also have a beneficial effect on the outgassing time for the facility.
The 8000 ft/sec arm would operate at an rpm substantially lower than that of the 4000 ft/sec arm considered in this report as can be seen in Table VI below. This table cornpares the size and performance of the three arms discussed here.
TABLE VI
COMP ARISON OF THREE POSSIBLE F ACILITIES
Maximum tip speed, ft/sec Tip radius, ft.
Payload, Ibs. Arm material Arm weight, Ibs. Arm root dia. inches Arm R. P. M.
Estimated volume of vacuum vessel, cu. ft. Facility diam eter, ft.
4000 5. 3.3 steel 500 11~ 7640 170 12 8000 17. 0.25 fibreglass 900 15~ 4490 4500 36 8000 26 1.0 fibreglass 3500. 24 2940 8500 54
For the 8000 ft/sec facility considerable optimization of the design appears possible and the further study involved should yield substant-ial savings in the total cost and improvements in operation. Consequently a cost estimate for this larger facility has nohbeen included in the present report.
9. CONCLUSIONS
1. It appears technically and economically feasible to design and construct rotating arm facilities for useful high speed. low density aero-dynamic research. Such facilities would appear to be most useflJ,l for research in the lower free molecule flow regime, the transitionregime
. between free molecule flow and slip flow. the slip flow regime, and.in the low density continuum flow regime.
2. The rotating arm technique cOIJlplérnents the balUstic range and low density continuous .flow wind tunnel - overcoming some of the inherent limitations of these types of facility in respect to both aero-dynamic capability (wind tunnels) and instrumentation (ballistic ranges). 3. The rotating arm technique also appears to offer a reasonable
range of versatility in respect to employment of gases of varying physical properties. including the possibility of ionized gases. and the use of
torsional balanees which would permit direct measurement of lift. drag and moment derivatives for models of varying size and shape. It also has the capacity for improvement in operating capability by use of alternate arms.
4. From the stand point of economie feasibility and operational convenience the size of a rotating arm facility is of prime importance and size is basically a function of the radius of the rotating arm.
Minimum arm radius is dictated by three factors. viz: (1) The permissible "g" loading on the
model and its attachment to the tip of the rotating anp.
I
(2) The minimum feasible cross-section of the arm at the tip:
(a) to withstand the model and attachment loads
(b) to accommodate instrumentation leads
(3) The maximum permissible taper angle of the arm.
5. The basic factor which limits speed is the tensile strength to density ratio of available materials and the best of currently used materials (glass fibre) establishes the current theoreticallimit of capability.
It appears to be quite straightforward to design and construct a rotating arm facility capable of achieving model speeds up to about 5, 000 ft. per second with currently available materials such as steel and conventional manufactIiring techniques 'and without entailing a requirement .:for any significant amount of engineering development work.
Within the range of currently available materials but requiring further design refinement and probably a program of development work relating to arm construction. it would appear feasible to achieve model speeds up to about 8, 000 ft. per second . . One avenue of study which offers the possibility of significant improvement in minimum arm radius is an arm design em ploying increased taper angle to that used in this study.
It is theoretically possible to achieve still higher model speeds within the range of available materials. However, increases in speed beyond ab out 8. 000 ft. per se.cond appear to entail excessive costs in either development work or in facility size in relation to the limited additional aerodynamic research benefits which accrue.
6. The primary research requirements which establish the basis for design size and cost of such a facility are:
(a) Model weight
(b) Permissible radial acceleration force on the model and / or its instrumentation (c) Model speed
. Secondary requirements of importance are:
(a) Range of gas pressures and densities in the te st cham ber
(b) Range of physical properties of gases employed as aerodynamic test media (c) Ionization requirements
7. The cost of a typical facility of arbitrarily specified perform-ance requirements (as defined below), is estimated at roughly
are:
The performance requirements established for this facility
Tip Speed Payload We!ght Perrnissible radial acceleration force on tip Aerodynamic Mediu.m Chamber Operating Pressure Range Ionization Requirernent 4, 000 ft/ sec 3.3 Ibs 100, 000
"g"
Air 103 to 10- 2 microns Nil ",10 .. SUMMARY AND RECOMMENDATIONS
This study has been aimed primarily at determining the technical and economic feasibility of low density rotating arm facilities and establishing design criteria and operating speed and size limitations for such facilities .
It has been necessary to give some consideration to the potential field of aerodynamic research work made possible through facilities of this type. The objective of work in this area has, however, been to establish reasonable boundaries for the study and to make possible the selection of basic performance requirements for a typical facility upon which to base an estimate of co st for design and construction. No attempt has therefore been made to study in depth the areas of potential usefulness of these devices - such studies being left for the aerodynamic researcher to undertake.
viz:
Consistant with the above, action is recommended as follows,
1. That the Institute of Aerophysics review and assese~thevarious aerodynamic research objectives which might be achieved
through the use of low density high speed rotating arm facilities of the type studied in this investigation.
2. That based on the foregoing review and employing the material contained in th is report as a guide on technical feasibility and cost, the Institute of Aerophysics establish basic performance criteria upon which to design, estimate cost and construct a facility which would best cover the range of research objectives with which the Institute is concerned.
The basic performance criteria to be established are: (1) Maximum Required Speed
(2) Maximum Model Weight at Maximum Speed (3) Maximum Permissible Radial Acceleration
Force at the Tip
(4) Range of Chamber Test Pressure and Temperature (5) Range of Test Gases to be Employed
(6) Ionization Requirements
1. K. R. Enkenhus 2. J .. R. Stalder G. Goodwin M. O. Creager 3. G. N. Patterson
4.
E. Janke F. Emde 5. E. P. Muntz 6. S. F. Hoerner 7. G. N. Patterson 8. E. L. Harris REFERENCES. "The Design. Instrumentation and Operation of the UTIA Low Density Wind Tunnel", UTIA Report No. 44, Institute of Aerophysics,
University of Toronto, June 1957
"Heat Transfer to Bodies in a High-Speed
Rarefied-Gas Stream", NACA Report 1093., 1952
"Molecule Flow of Gases", J. Wiley & Sons, 1956 "Tables of Functions " Dover, 1945
"Pressure Measurements in Free Molecule Flow with A Rotating Arm Apparatus" UTIA Technical Note No. 22, Institute of Aerophysics, University of Toronto, 1958
"Fluid Dynamic Drag", 1958
"Theory of Free-Molecule, Orifice-Type,
Pressure Probes in Isentropic and Nonisen
-tropie Flows", UTIA Report No. 41, Institute of Aerophysics, University of Toronto, 1956 "Investigation of the Time Response and
Outgassing Effects of Pressure Probes in
Free Molecule Flow", UTIA Technical Note No. 6, University of Toronto, 1955
LOW DENSITY FLOW PARAME.TERS
One of the most significant parameters describing flow at low densities is the Knudsen number, defined by:
(A. 1)
where
~
is the rnolecular mean free path in the gas and L is acharacteristic dim ension of a body in the flow. From Ref. 3 the rn ean free path rnay be expressed as:
= (A. 2)
0.499 pC
where ~ is the viscosity,
f
the density and C_ the mean random velocity of the molecules defined in terms of the gas constant Rand temperature T:(A. 3)
Now the Mach number is defined as
M
=
u (A. 4)where
1
is the specific heat ratio of the gas. Then, substituting (A. 4) in (A. 3) givesc
=
2~
V
7Tl'
u
M
Sirnilarly fr om the definition of Reynolds number,
(A. 5)
Af ter substituting frorn (A.5). (A. 6) and (A. 2) equation (A. 1) can be rewritten. in the case of air
(3'
=
1. 4). in the form=
1:486 M (A. 7)Re
This is the basic relationship between the quantities Kn' Mand Re. are both referred to the same characteristic dimension. Fig. 1 illustrates this relationship.
The Knudsen number mayalso be expressed in terms of gas temperature and pressure. First. substituting (A. 2) in (A. 1):
=
~
fL
JK1"
(A. 8)
Now from the equation of state. with pressure expressed in psi,
f
=
144 PRT
which can be substituted in equation (A. 8) to yield
= 8. 68 x 10- 3. j.( ol
&
' p L(A. 9)
(A. 10)
The pressure conversion factor 1. 93 x 10- 5 psia. = 1 micron Hg. has been used in the above equation to develop the relationship:
Kn per inch
=
1. 93/p (A. 11)for p in microns. Equation (A. 11) has been applied to produce the pressure scale on Fig. 1.
THEORY OF ROTATING ARM STRUCTURE Shape of the Arm
It is desired to calculate the shape of an arm such that all sections from the root to the tip have the same radial stress.
Consider a small element of a rotating arm lying between radii rand r
+
dr as shown in FiH: 3. Let the cross-sectional area of the arm at radius r (ft) be a (in). Then if w is the density of the material (lb/in3 ) the mass of the element is;dm = 12 wadr slugs (B.1)
g
and its centri(ugal force when rotating at angular velocity
CU
is:= 12 wadr W 2r g
Ibs (B.2)
For constant centrifugal stress, fc. it is required that {or every radius
Then da = ·1 fc = (B.3) (B.4)
Integration of equation (B. 4) between the limits rl and r2 leads to the
general relationship .
(B.5)
It is convenient for the prese,nj discussion to consid~r the arm root to be
located at the axis of r.otation (rl
=
0). If r2 is now taken as the tip radius,rt, equation (B. 5) reduces to:
Loge
(:~
)
- 6ww2 rt 2 gf c (B. 6) Now letd;
=
6wttJ2~.t2=
6wu2-gfc 'gfc (B.7)
Where u is the tip peripheral speed. the root to tip area ratio is now
given by:
oe
=
e (B.8)Now assume the arm everywhere to have a circular cross
-section of radius y, and let Yt be the radius of the tip cross-section
(See Fig. 3). Then equation (B. 5) can be written in terms of r, y and
as follows:
2
