Aerodynamic Heating of Miniature Bead Thermistor Thermometers in a Rarified Airstream

Donald C. Thompson New Zealand Meteorological Service, Wellington

Search for other papers by Donald C. Thompson in
Current site
Google Scholar
PubMed
Close
Full access

Abstract

The recovery factors of miniature bead thermistors as used in meteorological rocketsondes were measured at subsonic Mach numbers using a whirling arm device inside a vacuum chamber. Measured values were near 0.75 at sea level pressure and remained relatively constant with pressure up to a pressure altitude of about 30 km. Above a pressure altitude of 40 km the recovery factors steadily increased with altitude and at 65 km values between 1.10 and 1.45 were recorded.

Abstract

The recovery factors of miniature bead thermistors as used in meteorological rocketsondes were measured at subsonic Mach numbers using a whirling arm device inside a vacuum chamber. Measured values were near 0.75 at sea level pressure and remained relatively constant with pressure up to a pressure altitude of about 30 km. Above a pressure altitude of 40 km the recovery factors steadily increased with altitude and at 65 km values between 1.10 and 1.45 were recorded.

504 JOURNAL OF APPLIED METEOROLOGYAerodynamic Heating of Miniature Bead Thermistor Thermometers in a Rarified Airstreamt DONALD C. THO~I'SONNew Zealand Meteorological Service, Wdlington(Manuscript received 26 December 1967)ABSTRACT The recovery factors of miniature bead thermistors as used in meteorological rocketsondes were measuredat subsonic Mach numbers using a whirling arm device inside a vacuum chamber. Measured values were near0.75 at sea level pressure and remained relativdy constant with pressure up to a pressure altitude of about30 kin. Above a pressure altitude of 40 km the recovery factors steadily increased with altitude and at 65 kmvalues between 1.10 and 1.45 were recorded.1. Introduction When an immersion thermometer is ex])osed in a highspeed alrstream it will, in the absence of other sources oferror, record a temperature higher than a thermometermoving along with the flow. This temperature rise is dueto a combination of adiabatic heating and viscous dissipation. It is usually ex~pressed in terms of the stagnanation temperature rise, T,--To, where T, is the temperature which the moving gas would acquire if broughtadiabatically to rest, and To is the free-stream temperature as recorded by the moving thermometer. It is readily shown that r. = r0[l+- (7-1)M~], (1)where M is the Mach number and 7 the ratio of specificheats for air. If T, is the temperature recorded by the thermometer,the thermal recovery factor r is defined by the relation ,-- (r,- r0)/(r.- To). (~) In the majority of applications T, is measured, but ToJs the quantity desired. Use is then made of (1) and (2),together with a knowledge of r and M, to deduce T0.Correction of aircraft temperatures for aerodynamicheating is a routine example of this procedure. The design and exposure of the thermometer is often arrangedso that r is known from physical principles to be veryclose to unity, so that T,=T,. In other cases, for example where conventional thermometers are exposed directly to the airstream, r must usually be determinedby experiment. t The research reported in this paper was performed in theDepartment of Meteorology, Massachusetts Institute of Technology, and was supported by the Air Force Cambridge ResearchLaboratories under contract number A1e 19(628)~4165. The authoralso received financ'ml ass/stance from the New Zealand Meteorological Service. Recently, miniature bead thermistors of diameters between 0.005 inch (5 mils )and 0.015 inch (15 mils) havebeen used as atmospheric temperature sensors. An important application of these thermistors is in meteorological rocket soundings of the stratosphere and mesosphere. Temperature is measured as the rocketsondedescends by parachute. Because of the very low dragoffered the parachute by the air, high fall velocities areinvolved in the upper levels, so that considerable aerodynamic heating occurs. In a typical sounding the correction for aerodynamic heating at 65 km may easily exceed 15C, and it increases rapidly above this level. In the lower atmosphere the recovery factor r of anythermometer is always less than or equal to one, and remains substantially constant with pressure. However,if the air density is reduced to the point where the meanfree path x is no longer very small compared with thecharacteristic dimension D of the thermometer, the heattransfer regime changes and r is no longer constant. Oppenheim (1953) has shown that in the extreme caseof free-molecule flow, where the Knudsen number)~/D>>I, the thermal recovery factor may be calculatedtheoretically for various geometrical shapes. Oppenhelm's results show that the recovery factor of a spherical thermometer in a free-molecule flow is a function ofMach number but approaches a constant value of about1.5 for values of M< 1.0. Under the same conditions therecovery factor of a cylindrical thermometer exposednormal to the flow approaches $ value of about 1.7.Since a rocketsonde thermistor consists of a roughlyspherical bead attached to cylindrical lead wires, onewould expect the recovery factor in free-molecule flowto be intermediate between these values. In the case of meteorological rocket soundings the requirements for free-molecule flow are fully met only ataltitudes somewhat higher than those attained by current operational rockets. However, averyslgniticant portion of the sounding does extend above the level whereJUNE 1968 DONALD C. THOMPSON 505l*io. 1. Rotating arm and reference thermistor mounted on vacuum chamber pump plate.continuum flow may be assumed. In this transition region there is no satisfactory theory for computing therecovery factor, and the amount of relevant experimental data available is extremely limited. To the author'sknowledge there are no published experimental measurements of the recovery factors of miniature beadthermistors, either for high altitudes or at sea level pressures. As part of a comprehensive study of the characteristics of miniature bead thermistor thermometers(Thompson, 1966; Thompson and Kelly, 1967) it wasconsidered desirable to carry out laboratory measurements of recovery factors under conditions approachingthose encountered in meteorological rocket soundings.2. Experimental methods The experimental procedure was to expose the thermistors on the end of a counterbalanced arm which rotated at high speed inside a vacuum chamber. A secondthermistor situated within 3/16 inch of the path of themoving one provided a reference temperature with respect to which the temperature rises were measured. Fig. 1 shows the arm, reference thermistor, etc.,mounted on the vacuum chamber pump plate. The thermistors were mounted by their leads, symmetrically between two 0.05-inch diameter stainless steel supports 2.0cm apart, which were electrically insulated from the arm.The radius of the circle described by the thermistorbead was 6 inches. With this apparatus, thermistorspeeds up to 100 m sec-~ could be simulated at pressuresranging from atmospheric down to a few microns ofmercury. An obvious question arises as to the amount of motioninduced in the air inside the vacuum chamber by therevolving arm. Devienne (1957) has made a study ofthis effect in connection with the use of a revolving armfor low density heat transfer studies. He found that theinduced motion was quite negligible at pressures of 2 mmmercury and lower. At higher pressures the induced motion was noticeable but small. In the present case the arm was designed so as tominimize the induced motion. The latter was measuredat sea level pressure by setting up the probe of a smallthermistor anemometer as close as possible to the pathof the moving thermistor. A linear relation was found tohold quite closely between the thermistor speed and theinduced motion, such that the latter was 4.0% of thethermistor speed. Hence, in calculating the speed of thethermistor relative to the air from the rotation speed ofthe arm, an allowance of --4% was made at sea levelpressure but, in view of Devienne's results, this correction factor was decreased proportionally to the pressurefor lower pressures. Another effect which had to be allowed for was theincrease in temperature of the air in the chamber whertthe arm was rotating at speed. This effect, which is presumably due to frictional heating of the air in the vicinity of the arm, was studied in detail by Devienne forhis apparatus. He found that the temperature rise wasleast when the mean free path was not small comparedwith the arm thickness, and that while the temperaturerise was greater at higher pre ;sures the temperature gradients in the vicinity of the arm became small. Devienneconcluded that heat-transfer measurements were vMid506 JOURNAL OF APPLIED METEOROLOGY VOLUME7 o 0.3 mm mercury / + 760turn mercury /d 0.1 5 10 50 100 Thermistor Airspeed, m/sec,Fzo. 2. Net temperature rise of moving thermistor as a function of airspeed at two different pressures.provided temperature differences with respect to suitably placed reference thermometers were used in thecalculations. Similar effects were noted by the author although thedetails of the temperature distribution in the vacuumchamber were not observed, only one point very near thepath of the thermistor being available. Devienne's results suggest, however, that this one point should havebeen reasonably representative of the free-stream temperature. The temperature rise of the reference thermis tor depended only on the speed of the revolving arm and was found to be proportional to the square of the rota tion speed. At a pressure of 4X 10-2 mm mercury the reference temperature rise was about 3% of that of the moving thermistor. As the pressure was increased it was found that near 0.25 mm mercury the heating increased quite sharply to 15-20% of the moving thermistor's, and at higher pressures this percentage gradually increased, being 30-35% at sea level pressure. Changes in heating were in phase with changes in the rotation speed and there was no tendency for the heating to increase with the length of time that the rotation had been going on. The aerodynamic heating measurements were carried out at Mach numbers between 0.18 and 0.29. Mach- numbers of double these values may be reached in the upper levels of rocket-soundings, but these would have exceeded the design limits of the rotating arm. A major limitation to the speed attainable was the centrifugal acceleration to which the thermistors and their mount ing system were subjected during the tests. For example, it was found that while 15-mil thermistors with leads each 1 cm long were able to withstand centrifugal accel erations of 4000-5000 g for prolonged periods, failures of the leads were liable to occur when the acceleration ex ceeded about 6000 g. On the other hand, 5- and 10-mil thermistors were run without failures at speeds of 100 m sec-~ for long periods, and even faster for brief ones, with corresponding accelerations of 7000-10,000 g. All of the measurements were carried out at room temperature near 22C, whereas very much colder tem peratures occur in the upper atmosphere. It can be shown theoretically, however (Thompson 1966), that the errors involved in applying the results of tests of this type made at room temperature to atmospheric temperatures occurring in a meteorological rocket sound ing are small.2'0 ...... I '1'5 o t= om ~ X ~__ix+ + ~ A t 1(0'5 ,I I I ,i 70 ~O 0 ,~ ,,,,,I , 0'1 1'0[O O x+~ Nora. Diam. Co~tin~ Leod length.o 5 mils None 1.0 cmx 10,mils None 1.0 cm13 15 mils AI. 1.85 cmA 15 mils AI. 1.0'cm+ 5 mils None 1.0 cm (4 le~ds) I i i r,o ,;o ' ~o U. S, Std. Atmosphere, kin:, , , ,,,i , , , , , ,,,I i 1-0 10 PFesst~Pe~ n~m JVJer'cuPy.~ ~0 :lOOFro. 3. Measured recovery factors of thermistors.1968DONALD C. THOMPSON5O781'00 +13~ 0 0 b ' b ..... b ' ' ' ' ' '7 6 50 40 3 20 10 U.S. Std. Atmosphere~ km. o I 1'0 10 100 760 Pressure, mm Mercury.Fro. 4. P. ecovery factors of thermistors, corrected for conduction and radiation losses in the experiments. Symbols have the same meanings as in Fig. 3. The current used in determining the thermistor beadtemperatures was always sufficiently small that therewas no significant electrical heating at any pressure.3. Results Fig. 2 shows the net temperature rise (i.e., the difference between that of the moving thermistor and thatof the stationary thermistor) as a function of relativespeed, plotted on logarithmic scales for a 10-mil thermistor. The slope of the lines is 2.0, indicating the validityof (1) and (2) for a constant r. Fig. 3 shows the measured recovery factors of fiveminiature bead thermistors as a function of pressure.The nominal diameter of the beads is indicated in mils.The 10- and 15-rail thermistors had 0.001-inch diameterplatinum-iridium leads and the 5-mil thermistors had0.0007-inch diameter leads also of platinum-iridium. At pressures greater than 10 mm mercury the measured values are fairly constant with pressure and lienear 0.75. These results are somewhat lower than thevalue of 0.9 assumed for these pressures by other authors(e.g., Barr, 1961; Wagner, 1963) in evaluatinglhe aerodynamic heating correction for rocketsonde thermistors.However, reference to the original papers showed thatthese authors had based their estimates on published experimental data taken at Mach numbers between 2 and4. Moffat (1962) has summarized results of measurements of the recovery factors of butt-welded thermocouples of various wire diameters at sea level pressureand subsonic Mach numbers. For wires normal to theairflow, measured values of r were near 0.68 and forwires parallel to the flow r was near 0.86. Hottel andKalitinsky (1945) made tests on a thermocouple consisting of 0.010-inch diameter wires whose junction wasencased in a 0.07dnch diameter solder ball. Althoughmuch larger, this configuration is similar to a bead thermistor. With airstream parallel to the wire, r was about0.78 and for flow normal to the wire r was 0.73. At pressures below 7 mm mercury the measured recovery factors shown in Fig. 3 increase with decreasingpressures, and in most cases exceed unity below 1 mmmercury. A maximum is reached at about 0.2 mm mercury, at which pressure the Knudsen number is ~ 1 forthe thermistor beads and ~10 for the lead wires. Thereason for this maximum, and the fact that the measured recovery/actors do not achieve the value expectedfor free-molecule flow at very low pressure, is that inthese experiments both radiation to the cooler walls ofthe chamber and thermal conduction of heat to the armthrough the lead wires prevent the thermistor fromachieving the full recovery temperature at low pressures.Thermal inertia and conduction of heat to the mainbody of the rotating arm prevent the relatively heavythermistor support posts from undergoing significantaerodynamic heating during the relatively brief duration of the tests. Thus, during the time required to makea measurement, we can assume that the thermistor supports and the aluminum walls of the vacuum chamberremain at a constant temperature, less than the recoverytemperature of the thermistor. It is possible to estimate a correction for these effects.Consider any element dA of the thermistor bead or leadwire surface. Then for small temperature differences thenet heat lost in unit time by radiation can be written asdQ~ = 4e taTSO~dA,where O~ is the temperature difference between the element and the chamber walls, e~ the emissivity and T theambient temperature. We can therefore define heat508 JOURNAL OF APPLIED METEOROLOGY VOLUME7 transfer coefficients hv'=4~vaTa and hw'=4e~,oaT3 for the heat exchange by radiation of the thermistor bead and lead wires which are analogous to those (hr,hw) describing the heat exchange with the air. If we intro duce these and assume that the recovery temperatures of the thermistor bead and lead-wires individually are not greatly different from that of the thermistor as a whole, then reasoning similar to that used by Thompson and Keily (1967) in computing the response of a ther mistor to solar radiation results in the following expres sion for the difference between the temperature rise ex pected in the absence of conduction and radiation and that actually measured: (rr-- To)--AT= (r~-- ro)K-l2kawp[-a coth pd +(l--a) csch paOq-h~'Av, (3)where K= 2ka,~p coth pd+ (hv+hr')Av.In this equation, a~ is the cross-sectional area of a thermistor lead-wire of thermal conductivity k, radius r,o and'length d, p=2(h~-kh~t)/kr~oi, a=h~'/(h~-Ch,='),and A v is the surface area of the thermistor bead. Thequantity K is the "dissipation rate" of the thermistor asmounted in the chamber, and can be readily measuredindependently (Thompson, 1966). At subsonic Machnumbers both K and h,~ are functions of pressure but aresubstantially independent of airspeed for pressures lessthan 1 mm mercury. The recovery factors of the five thermistors, corrected for conduction and radiation with the aid of (3), are shown in Fig. 4. Measured values of K and Av for the actual thermistors were used, while values of h,o for the thermistor lead wires were taken from Thompson (1966). For uncoated thermistors ~tv was taken as 1.0 and for the two aluminized thermistors ~tv was taken as 0.13 on the basis of experimental results given in the same port. The emissivity of the thermistor lead wires was assumed to be 0.10. No corrections were attempted for pressures lower than 0.05 mm mercury. It is seen that the corrected recovery factors continue to increase with decreasing pressures, and approach val ues more consistent with the free-molecule theory. Be cause it is difficult to assess the accuracy of the individual measurements, it is perhaps not wise to draw conclusions about individual thermistor types. However,the results show a tendency for the smaller themfistorsto have higher recovery factors at low pressures aswould be expected, since at a given pressure the Knudsennumber would be higher. The "four-lead" thermistorwas an experimental type having four lead wires insteadof the usual two and since the bead shape was consequently more deformed it cannot be compared directlywith the others. It should be noted that before applying these resultsto actual rocket soundings it is necessary to ascertainthat the thermistors are exposed sufficiently clear of therocketsonde body so that the airflow over the thermistoris not disturbed. It should also be noted that these results cannot necessarily be applied directly to rocketsoundings in which the recently-developed "thin-film"method of mounting the thermistor is used. As pointedout by Morrissey and Carten (1967), there is considerable coupling between the film and the thermistor sothat the aerodynamic heating error of the sensor as awhole must depend on the degree of aerodynamic heating of the film as well as on that of the thermistor. Thiscoupling has yet to be fully studied. REFERENCESBarr, W. C., 1961: Theoretical considerations in the design of atmospheric temperature-sensing elements. Tech. Rept. 2195, U. S. Signal Research and Development Lab., Fort Monmouth, N. J., 15 pp.Devienne, F. M., 1957: Experimental study of the stagnation temperature in a free molecular flow. J. Aeron. Sci., 24, 403-407.Hottel, H. C., and A. Kalitinsky, 1945: Temperature measure ments in high velocity air streams. Trans. Amer. Soc. Mech. Engrs., 67, A-25.M offat, R. J., 1962: Gas temperature measurement. Temperature: Its Measurement and Control in Science and Industry. Vol. 3., New York, Reinhold Publ. Co., 553-571.Morrissey, J. F., and A. S. Carten, 1967: Importance of thermistor mount configuration to meteorological rocket temperature measurements. Bull. Amer. Meteor. Sot., 48, 684-688.Oppenheim, A. K., 1953: Generalized theory of convective heat transfer in a free-molecule flow. J. Aeron. Sci., 20, 49-58.Thompson, D. C., 1966: The accuracy of miniature bead thermis tors in the measurement of upper air temperatures. Sci. Rept. AFCRL-66-773, Air Force Cambridge Research Laboratories, Bedford, Mass., 264 pp.. , and D. P. Keily, 1967: The accuracy of thermistors in the measurement of upper air temperatures. J. Appl. Meteor., 6, 380-385.Wagner, N. K., 1964: Theoretical accuracy of a rocketsonde thermistor. J. A ppl. Meteor., 3, 461-469.

Save