NASA TECHNICAL NASA TM X-71967MEMORANDUM COPY NO.

--i

MODEL WALL AND RECOVERY TEMPERATUREEFFECTS ON EXPERIMENTAL HEAT

<4 TRANSFER DATA ANALYSIS

D. A. Throckmorton and D. R. Stone

June 1974

(NASA-Tf.-X-719 6 7) MODEL WALL ANDRECOVERY RgERATUE FFECTS ON N7-25815EXPERIiE2TAL HEAT TRNPE CS ONB7Y281EASAI) 45 P HC $5.25RASER DATA ANALYSIS

CSCL 20D UnclasG3/12 40862

This informal documentation medium is used to provide accelerated orspecial release of technical information to selected users. The contentsmay not meet NASA formal editing and publication standards, may be re-vised, or may be incorporated in another publication.

NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONLANGLEY RESEARCH CENTER, HAMPTON, VIRGINIA 23665

https://ntrs.nasa.gov/search.jsp?R=19740017702 2020-05-22T18:34:54+00:00Z

1. Report No. 2. Government Accession No. 3. Recipient's Catalog No.NASA TM X-71967

4. Title and Subtitle 5. Report DateModel Wall and Recovery Temperature Effects on June 1974Experimental Heat Transfer Data Analysis 6. Performing OrganizatiooCode

7. Author(s) 8. Performing Organization Report No.David A. Throckmorton and David R. Stone

10. Work Unit No.9. Performing Organization Name and Address

NASA Langley Research Center 11. Contract or Grant No.Hampton, VA 23665

13. Type of Report and Period Covered12. Sponsoring Agency Name and Address

Technical MemorandumNASA 14. Sponsoring Agency Code

15. Supplementary Notes

Back-up document for AIAA Journal Synoptic.

16. Abstract

Basic analytical procedures are used to illustrate, both qualitativelyand quantitatively, the relative impact upon heat transfer data analysis ofcertain factors which may affect the accuracy of experimental heat transferdata. Inaccurate knowledge of adiabatic wall conditions results in acorresponding inaccuracy in the measured heat transfer coefficient. Themagnitude of the resulting error is extreme for data obtained at walltemperatures approaching the adiabatic condition. High model walltemperatures and wall temperature gradients affect the level and distributionof heat transfer to an experimental model. The significance of each of thesefactors is examined and its impact upon heat transfer data analysis isassessed.

17. Key Words (Suggested by Author(s)) (STAR category underlined) 18. Distribution Statement

Fluid MechanicsUnclassified - Unlimited

Boundary layersHeat transfer

19. Security Qassif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price'

Unclassified Unclassified 17 $3.00

iThe National Technical Information Service, Springfield, Virginia 22151

STIF/NASA Scientific and Technical Information Facility, P.O. Box 33, College Park, MD 20740

/

MODEL WALL AND RECOVERY TEMPERATURE EFFECTSON EXPERIMENTAL HEAT TRANSFER

DATA ANALYSIS

D. A. Throckmorton and D. R. Stone

Summary

Basic analytical procedures are used to illustrate, both qualitativelyand quantitatively, the relative impact upon heat transfer data analysis ofcertain factors which may affect the accuracy of experimental heat transferdata. Inaccurate knowledge of adiabatic wall conditions results in acorresponding inaccuracy in the measured heat transfer coefficient. Themagnitude of the resulting error is extreme for data obtained at walltemperatures approaching the adiabatic condition. High model walltemperatures and wall temperature gradients affect the level and distributionof heat transfer to an experimental model. The significance of each of thesefactors is examined and its impact upon heat transfer data analysis isassessed.

*t

il

INTRODUCTION

Development of the phase-change coating technique (reference 1) hasprovided a valuable tool for obtaining quantitative measurements of theheat transfer to bodies in hypersonic wind tunnels. The technique offerseconomies of both time and money when compared to thermocouple techniques,and also yields measurement of highly detailed heating distributions notpossible with previous methods. These advantages have led to widespreaduse of the phase-change technique in both basic fluid mechanics researchand configurational heating studies. However, the procedures required inutilizing this technique result in data obtained over long test time'intervals and, therefore, at model wall temperature levels and gradientsnot normally encountered in thin-skin testing. In thin-skin testing, dataare obtained simultaneously at all points on the model, at times near testinitiation such that model wall temperatures are nearly uniform - i.e.temperature gradients are generally negligible. During a phase-changecoating test, however, data at various model locations are obtained atdifferent times, with time intervals sufficiently large to allow significanttemperature gradients to exist on the model when the data are obtained.

Increased model wall temperature results in an increased sensitivityof phase-change heat transfer coefficients to adiabatic wall temperature,as compared to thin-skin data. Analysis of data obtained on models withwall temperatures approaching adiabatic conditions, and on models withsignificant surface temperature gradients demands an understanding of theeffects of wall temperature on the heat transfer process, to assureaccurate interpretation of that data. The significance of each of thesefactors (Tw, Taw, Tw gradients) is examined and their impact upon heattransfer data analysis is assessed. The investigation utilized bothexperimental phase-change coating data and "exact" numerical solutions tothe laminar boundary layer equations to observe the nature of the walltemperature effects on the heat transfer to a flat plate, a hemisphere-cylinder, and to the windward centerline of a representative space shuttleorbiter configuration.

NOMENCLATURE

cp model material specific heat

h heat transfer coefficient

k model material thermal conductivity

M Mach number

q heating rate

Pr Prandtl number

r laminar recovery factor

2

S/R non-dimenslonalized surface coordinate

X/L non-dimensionalized longitudinal coordinate

t time

T temperature

a angle of attack

B pressure gradient parameter

y ratio of specific heats of test gas

6 flow deflection angle

e parameter defined by equation (3)

X model material thickness

p model material density

Subscripts:

aw adiabatic wall condition

e boundary layer edge condition

i initial condition

o stagnation point

pc phase change

t total condition

w wall condition

cc free stream condition

ANALYSIS AND RESULTS

Sensitivity of Heat Transfer Coefficient to Adiabatic Wall Temperature

Experimental heating data, obtained using either the thin-skincalorimeter or phase-change coating technique, are usually expressed inthe form of the aerodynamic heat transfer coefficient (h). This parameteris defined by Newton's Law of Cooling as the proportionality constantrelating the local heat transfer rate (4) and the forcing function of the

3

heat transfer process; i.e. the difference between the local adiabatic walltemperature (Taw) and the local wall temperature (Tw).

q = h (Taw - Tw) (1)

For the analysis of experimental data, expressions for the aerodynamic heattransfer coefficient are derived from the equation governing the one-dimensional, transient conduction of heat into a solid, with application ofappropriate boundary and initial conditions. Both the phase-change andthin-skin calorimeter techniques assume a step heat input, usually obtained'by rapid injection of an isothermal model into the airstream. For the thin-skin technique, heat transfer coefficient is based upon heat conduction intoa finite solid of known thermal properties:

h = PCX T w 1 (2)PCp at Taw-Tw

where p, cp, and X are model material density, specific heat and thick-ness, respectively. The measured quantities are the wall temperature (Tw)and its time-rate-of-change (DT/9t). The corresponding equation used toreduce phase-change data is based upon heat conduction into a semi-infinitesolid of known thermal properties:

6 2 T -Th = cepV where 1 - e erfc () = T T(3)=PTaw Ti

The measured quantity is the time (tpc) required for the model surfacetemperature to increase from some initial value (Ti) to a known coatingphase-change temperature (Tpc). (Reference 1.)

The adiabatic wall temperature is rarely measured in thin-skin tests,and the phase-change technique is incapable of indicating this temperature.Computed heat transfer coefficients are extremely sensitive to excursions ofan assumed value of the adiabatic wall temperature. This is illustrated inFigure 1 where the ratio of heat transfer coefficient to that value computedassuming Taw - Tt is presented as a function of Taw/Tt for constantvalues of phase-change temperature ratio and model initial temperatureratios. The initial model-to-stream temperature ratios indicated arerepresentative of conditions for a room temperature model and currenthypersonic wind tunnels. (The phase-change coating technique is presentlyroutinely used in many hypersonic air facilities which operate at initialtemperature ratios of -.35, and has been considered for use in facilitieswhich operate near Ti/Tt = .65). (Reference 2.) The sensitivity of thin-skin calorimeter data to adiabatic wall temperature is indicated by the

4

lower curve on each figure (correspondinq to the phase-chanqe temperatureequal to the initial temperature). The remaining curves indicate thesensitivity of phase-change data, with increasing values of the phase-changetemperature. Analysis of the curves clearly indicates the increasedsensitivity of phase-change derived data, as opposed to thin-skin data, tothe accuracy of the adiabatic wall temperature estimate. In addition,comparison of the plots for the range of Ti/Tt illustrates the magnifica-tion of this sensitivity with increasing initial temperature.

The phase-change temperature is constrained by the test time requiredto effect the coating phase-change; this time must be long enough foraccurate measurement, yet short as compared to the thermal diffusion timeof the model. The shaded areas of the figure approximate those regions ofinterest for practical test operations.

Local adiabatic wall temperatures can be adequately estimated forsimple shapes by use of "exact" numerical computation techniques; however,for complex geometries, such "exact" numerical solutions are presentlybeyond the "state-of-the-art." Consequently, it has become commonpractice to base experimental data on a nominal adiabatic wall temperatureratio (Taw/Tt) assumed constant over an entire configuration. The use ofa nominal value of Taw/Tt = 1.0 in the data reduction results in data whichare in error as indicated in Figure 1. Attempts to reduce this error byassuming a compromise ratio of 0.95 or 0.90 diminish the maximum potentialerror; however, the functional relationship of heat transfer coefficientto deviations of the assumed adiabatic wall temperature from the actualvalue is unchanged. It is therefore necessary to assess the impact ofinaccuracies in adiabatic wall temperature estimation procedures on heattransfer coefficients derived from experimental data.

The value of the adiabatic wall-to-total temperature ratio may beexpressed for an ideal gas, as a function of recovery factor (r) andboundary layer edge Mach number (Me) in the form:

Taw = I + r M 2 (4)Tt I + y1 Me2

Solutions of the compressible laminar boundary layer equations indicate thatrecovery factor is a function of Prandtl number, pressure gradient, and alsoboundary layer edge Mach number. For zero pressure gradient flows with lowedge Mach number (Me < 2), recovery factor (r) is closely approximatedby square root of the Prandtl number (Pr) (Reference 3). The work ofWortman and Mills (Reference 4), ho ever , indicates that for acceleratinglaminar boundary layers, recovery factor is highly dependent upon thepressure gradient parameter, 6, decreasing monotonically to an asymtoter . Pr as 0 -+ ; a weaker dependence upon edge Mach number is indicated.Stone, et.al (Reference 2) illustrated the combined effect which edge Mach

5

number and pressure gradient variations exhibit on recovery factor in highMach number (-20) flows in Helium. The impact of this effect on lower Machnumber hypersonic air flows is less pronounced, as edge Mach number andpressure gradient parameter are normally much lower; yet their significanceshould be ascertained for any specific flow in question.

In the absence of flow field surveys, estimation of the local boundarylayer edge Mach number presents a further obstacle to accurate determinationof adiabatic wall temperature. Two methods for estimating boundary layeredge conditions are the tangent wedge/cone, and the normal-shock expansion.approximations. The tangent wedge/cone technique models the inviscid flowas that occurring on a wedge or cone surface with half angle equal to thelocal flow deflection angle (6); which approximation method (wedge or cone)is more accurate depends upon the geometry of the flow to be modeled. Thevariation of adiabatic wall temperature with flow deflection angle, asdefined by the tangent cone and wedge approximations, is illustrated inFigure 2 for a range of recovery factors typical of hypersonic air flowsover real configurations. Adiabatic wall temperature level is shown to beequally sensitive to both recovery factor and flow deflection angle.

Another approach to approximation of local boundary layer edgeconditions involves use of a measured or analytically determined surfacepressure distribution coupled with a local entropy assumption to define thedesired quantities. Possibly, the most common model of this type assumes aNewtonian pressure distribution with edge conditions resulting from anisentropic expansion of the flow from a stagnation point behind a normalshock. This constant entropy assumption, although commonly used inboundary layer computation, may not be valid for particular flows of interest.

Figure 3 presents adiabatic wall temperature distributions on thewindward center line of a space shuttle delta wing orbiter configuration at300 angle of attack as computed by the tangent cone and the normal shockexpansion techniques. A constant value of recovery factor (r = 0.84) wasused for both calculation methods in order to emphasize the sensitivity toedge Mach number estimation techniques alone; edge Mach number and pressuregradient were low so that the previously discussed effects of these parametersupon recovery factor were small. It is important to note that although themaximum deviation between the adiabatic wall temperatures computed by thesemethods is only about 5 percent, the heat transfer results which aredependent upon these values may differ by 5 to 25 percent as a function ofmodel initial temperature (Ti/Tt) and phase-change temperature (Tpc/Tt) asillustrated in Figure 1.

Wall Temperature Effects

A basic assumption used in derivation of the expressions for aerodynamicheat transfer coefficient, for both the thin-skin and phase-changetechniques, is that the model experiences a step input in heat transfercoefficient to a value which is constant with time. For thin-skin testing,this is a normally valid assumption as test time is short and the model

6

remains essentially isothermal over the duration of the transient test.Phase-change testing, however, may violate this assumption as the data areobtained over relatively long test intervals and, therefore, at modelwall temperature levels and distributions not normally encountered in thin-skin testing.

Chapman and Rubesin (Reference 5) indicated that for laminar boundarylayer flows with variable surface temperature, local boundary layerproperties (and, therefore, heat transfer) depend not only on the localtemperature potential, but on the entire surface temperature distributionupstream of the point in question. The effects of temperature level and

'distribution on the value of heat transfer coefficient are due to what theytermed "the inappropriateness of the conventional heat transfer coefficientwhen applied to flows with variable surface temperature." This"inappropriateness" is most apparent as wall temperature approaches thelocal adiabatic wall condition. For variable wall temperatures (Tw) andwall temperature level of the order of local adiabatic wall temperature(T T aw), heat transfer coefficient may reverse sign and even becomeinfinite. This anomalous behavior is illustrated in Figure 4for a flat plate at zero angle of attack with surface temperature distribu-tion as indicated. These curves result from "exact" solutions to thelaminar boundary layer equations (Reference 6); adiabatic wall temperaturewas calculated from Equation (4) with r = 0.84.

The effect of wall temperature on heat transfer coefficient is againillustrated in Figure 5, by solutions to the laminar boundary layer on ahemisphere-cylinder. In the case of uniform wall temperature, a change inthe temperature level results in an alteration to the heat transfercoefficient distribution due to the changed relationship between the walland adiabatic wall temperatures. As the wall temperature is increased, thelocal heat transfer coefficient decreases. This decrease is negligiblefor low values of the wall-to-total temperature ratio, but becomessignificant as wall temperature approaches the adiabatic condition. Incontrast, a negative gradient of surface temperature along the wall,results in an increase in the local heat transfer coefficient. The predictedincrease shown in Figure 5 is a result of the temperature gradient effectplus the opposing effect of an increased wall temperature level. (Comparingthe heat transfer coefficient distributions for the gradient case and theconstant wall temperature Tw/Tt = 0.1 case, heat transfer coefficientswould be expected to decrease for the gradient case as wall temperature haseverywhere increased above the Tw/Tt = 0.1 level). The temperaturegradient effect predominates and heat transfer coefficient increases. Thewall temperature distribution indicated for this calculation, is typical ofthat on a hemisphere phase-change model at a specific instant in time duringtest in a hypersonic wind tunnel.

For the more practical case of a space shuttle orbiter configuration,wall temperature gradient effects are illustrated in Figure 6. Again, theindicated temperature distribution would exist on a phase-change model at a

7

specific instant during a hypersonic heat transfer test. The predictionsresult from solution of the laminar boundary layer by application of theaxisymmetric analog to the flow on the orbiter lower surface plane ofsymmetry. The inviscid pressure distribution and streamline divergenceinformation were computed by the method of DeJarnette and Hamilton(Reference 7). The impact of variable wall temperature is less significantfor this flow than for that about the hemisphere. This is attributed to thesmaller temperature gradient present on the orbiter lower surface aft ofthe nose region.

Heating Data Impact

Phase-change test derived heating distributions on spheres have beenfrequently used in an inverse method for determining phase-change modelmaterial thermophysical properties (Reference 8). This procedure requirescomparison of the experimental heating distribution with a theoreticaldistribution to obtain that value of the phase-change thermal propertiesparameter, pcpk, which results in the best correlation. The quality ofthe measurement is a direct function of the sophistication of thetheoretical method utilized. Figure 7 presents experimental phase-changeheating data for a hemisphere-cylinder at Mach 20.3 in helium (fromReference 4). The measured heat transfer data were reduced using anadiabatic wall temperature distribution obtained by the non-similar solutiontechnique. Also shown is the theoretical heat transfer distribution for aconstant wall temperature equal to the model initial temperature. As can beseen from Figure 7, the use of a constant wall temperature theory withphase-change data may result in a derived thermal property value which issubstantially in error. Thermal properties so derived will be accurate onlyif the theory utilized adequately models the non-isothermal nature of thephase-change test itself.

Conversely, if experimental heating data are to be used to verifytheoretical calculation procedures, it is important that the theoryaccurately model the experiment which produced that data. Data which mayexhibit significant wall temperature effects should not be used to verifya theory which lacks the sophistication to account for them.

CONCLUDING REMARKS

Basic analytical procedures have been used to illustrate, bothqualitatively and quantitatively, the relative impact upon heat transferdata analyses of certain factors which may affect the accuracy of experi-mental heat transfer data. It is recognized that the physical principlesinvolved, i.e. wall temperature effects on heat transfer, recovery factorand adiabatic wall temperature computation procedures, have all beenpreviously discussed in detail by other investigators. However, recentwidespread adoption of the phase-change coating technique for use in avariety of heat transfer investigations requires a renewed awareness, bythe experimentalist, of the possible error sources (and the significance

8

of each) which exist in regimes common to phase-change testing. Thesubject material leads to the following comments:

1. Experimental heat transfer coefficient data accuracy (for eitherthin-skin or phase-change) is directly dependent upon accurate knowledge ofthe local adiabatic wall temperature. Phase-change coating data exhibitsa significantly greater sensitivity to this quantity than does thin-skincalorimeter data. Errors in heat transfer coefficient resulting frominaccurate knowledge of the adiabatic condition may be diminished by testingat decreasing values of the model initial-to-stream total temperature ratio.In the limit, however, accuracy in computed heat transfer coefficients is.directly proportional to the accuracy of the adiabatic wall temperature.

2. Wall temperature gradients, which may result from model geometrycharacteristics and/or long run times, can significantly affect measuredheat transfer coefficient distributions. Wall temperature gradients andtheir resulting effects are minimized by thin-skin testing or, in the caseof phase-change testing, utilizing phase-change temperatures which approachthe initial condition.

3. If experimental data are to be used to verify theoretical calcula-tion procedures, it is important that the theory accurately model theexperiment which produced that data. Data which may exhibit significantwall temperature effects should not be used to verify a theory which lacksthe sophistication to account for them.

REFERENCES

I. Jones, Robert A.; and Hunt, James L.: Use of Fusible TemperatureIndicators for Obtaining Quantitative Aerodynamic Heat TransferData. NASA TR R-230, February 1966.

2. Stone, D. R.; Harris, J. E.; Throckmorton, D. A.; and Helms, V. T.:Factors Affecting Phase Change Paint Heat Transfer Data ReductionWith Emphasis on Wall Temperatures Approaching Adiabatic Conditions.AIAA Paper No. 72-1030, September 1972.

3. Pfyl, Frank A.; and Presley, Leroy L.: Experimental DeterminationOf the Recovery Factor and Analytical Solution of the Conical FlowField for a 200 Included Angle Cone at Mach Numbers of 4.6 and 6.0and Stagnation Temperatures to 26000 R. NASA TN D-353, June 1961.

4. Wortman, A.; and Mills, A. F.: Recovery Factor for Highly AcceleratedLaminar Boundary Layers. AIAA Journal, Vol. 9, No. 7, July 1971,pp. 1415-1417.

5. Chapman, D. R.; and Rubesin, M. W.: Temperature and Velocity Profilesin the Compressible Laminar Boundary Layer With Arbitrary Distributionof Surface Temperature. Journal of Aero. Sci., 16, 1949, pp. 547-565.

9

6. Hamilton, H. Harris, II: A Theoretical Investigation of the Effectof Heat Transfer on Laminar Separation. M.S. Thesis, VirginiaPolytechnic Institute, May 1969.

7. DeJarnette, Fred R.; and Hamilton, H. Harris, II: Inviscid SurfaceStreamlines and Heat Transfer on Shuttle Type Configurations.AIAA Paper No. 72-703, June 1972.

8. Conners, L. E.; Sparks, V. W.; Bhadsalve, A. G.: Heat Transfer Testsof the NASA-MSFC Space Shuttle Booster at the Langley ResearchCenter Hypersonic Continuous Flow Facility. LMSC/HREC D162722,Lockheed Missiles and Space Company, Huntsville, Alabama,December 1970.

f01.8

Tthactual

h(TawTt) .8

.781.4 "

Tpc=T

1.0 I.85 .90 .95 1.0

Taw/Tt

(a) Ti/Tt = 0.65

1.8 1.8

hactual Tyc_ hactualChactualh(Taw=Tt) .9 Tt h(Ta=Tt)

.8

1.4 \.6 1.4

pc Tpc

T =T.--'

1.0 1.0 I I.85 .90 .95 1.0 .85 .90 .95 1.0Taw/Tt Taw/Tt

(b) Ti/Tt = 0.40 (c) Ti/Tt = 0.10

Figure 1.- Effect of adiabatic wall temperature assumptionon computed heat transfer coefficient.

1.0WEDGE FLOW

.9r

Taw .84

Tt .82.80

.8 .78

0 20 40 606, deg.

1.0CONE FLOW

.9

Taw

Tt .84.82

0 20 40 606, deg.

Figure 2.- Adiabatic wall temperature variation with flow deflectionangle. (M. = 10.0)

1.00- I 10.0

r = .84

.95 _NEWTONIAN PRESSURE

Taw CONSTANT ENTROPY

Tt

.90TANGENT CONE

0 1.0X/L

Figure 3.- Delta-wing shuttle orbiter lower surface centerlieadiabatic wall temperature distribution. a = 30

.9T -- ,- - Taw-T TawTt .8 -

.Tw.7-

5.0

q 0-__

-5.0

-10.0 -

3.0-

2.0

h 1.0

0

-1.0-

-2.0

-3.0-I I I I I I0 .2 .4 .6 .8 1.0

X/L

Figure 4.- Effect of wall temperature on heat transfercoefficient for a flat plate at Mach 10.0in air. a = 00.

1.0 1.0 -f(S/R)

.8- Tw GRADIENT TwTt

hI f I I

.4 -

INCREASED Tw "< T =f(S/R)

8w G.8 .UNIFORM = 9

0 .4 .8 1.2 1.6

S/R

Figure 5.- Theoretical heat transfer distributions on a hemiSphere-cylinderat Mach 10.0 in air.

0.20 1.0 -f (X/L)

Tw -

I I I I 1.00 1.00.10 X/L

ho 0.08 T f (X/L)

0.06

T

0.04

0 .2 .4 .6 .8 1.0X/L

Figure 6.- Theoretical heat transfer distributions on the windward centerlineon a space shuttle orbiter configuration at Mach 10.0 in air.a = 300 .

EXPERIMENTAL DATA Tpc/Tt

(Ref. 2) ---- - .66

.8 .72

.7476

Ti/Tt = 0.625.6 -

h

0

.4-

.2- Theory

T,/Tt = 0.625 _

0 I I I

0 .4 .8 1.2 1.6

S/R

Figure 7.- Effect of wall temperature variation on measured heat transfer on ahemisphere-cylinder at Mach 20.3 in helium.

MODEL WALL AND RECOVERY TEMPERATURE EFFECTSON EXPERIMENTAL HEAT TRANSFER

DATA ANALYSIS

By

D. A. Throckmorton* and D. R. Stone**NASA Langley Research Center

Hampton, Virginia

ABSTRACT

Basic analytical procedures are used to illustrate, both qualita-

tively and quantitatively, the relative impact upon heat transfer data

analysis of certain factors which may affect the accuracy of experi-

mental heat transfer data. Inaccurate knowledge of adiabatic wall

conditions results in a similar inaccuracy in the measured heat

transfer coefficient. The magnitude of the resulting error is extreme

for data obtained at wall temperatures approaching the adiabatic

condition. High model wall temperatures and wall temperature

gradients affect the level and distribution of heat transfer to an

experimental model. The significance of each of these factors is

examined and their impact upon heat transfer data analysis is assessed.

*Aero-Space Technologist, Thermal Analysis Section, Space Systems Division

**Aero-Space Technologist, Hypersonic Analysis Section, Space SystemsDivision

2

NOMENCLATURE

Cp model material specific heat

h heat transfer coefficient

k model material thermal conductivity

M Mach number

4 heating rate

Pr Prandtl number

r laminar recovery factor

S/R non-dimensionalized surface coordinate

X/L non-dimensionalized longitudinal coordinate

t time

T temperature

aangle of attack

8 pressure gradient parameter

y ratio of specific heats of test gas

6 flow deflection angle

e parameter defined by equation (3)

A model material thickness

p model material density

Subscripts:

aw adiabatic wall condition

e boundary layer edge condition

i initial condition

o stagnation-point condition

pc phase change

t total condition

w wall condition

Sfree stream condition

INTRODUCTION

Development of the phase-change coating technique (reference 1)

has provided a valuable tool for obtaining quantitative measurements

of the heat transfer to bodies in hypersonic wind tunnels. The

technique offers economies of both time and money when compared to

thermocouple techniques, and also yields measurement of highly detailed

heating distributions not possible with previous methods. These

advantages have led to widespread use of the phase-change technique

in both basic fluid mechanics research and configurational heating

studies. However, the procedures required in utilizing this technique

result in data obtained over long test time intervals and, therefore,

at model wall temperature levels and gradients not normally encountered

in thin-skin testing. In thin-skin testing, data are obtained

simultaneously at all points on the model, at times near test inifation

such that model wall temperatures are nearly uniform - i.e. temperature

gradients are generally negligible. During a phase-change coating

test, however, data at various model locations are obtained at

different times, with time intervals sufficiently large to allow

4

significant temperature gradients to exist on the model when the data

are obtained.

Increased model wall temperature results in an increased

sensitivity of phase-change heat transfer coefficient data to

adiabatic wall temperature, as compared to thin-skin data. Analysis

of data obtained on models with wall temperatures approaching

adiabatic conditions, and on models with significant surface tempera-

ture gradients demands an understanding of the effects of wall

temperature on the heat transfer process, to assure accurate inter-

pretation of that data. The significance of each of these factors

(Tw,Taw, Tw gradients) is examined and their impact upon heat transfer

data analysis is assessed. The-investigation utilized both experimental

phase-change coating data and "exact" numerical solutions to the

laminar boundary layer equations to observe the nature of the wall

temperature effects on the heat transfer to a flat plate, a hemisphere-

cylinder, and to the windward centerline of a representative space

shuttle orbiter configuration.

ANALYSIS AND RESULTS

Sensitivity of Heat Transfer Coefficient to Adiabatic Wall Temperature

Experimental heating data, obtained using either the thin-skin

calorimeter or phase-change coating technique, are usually expressed in

the form of the aerodynamic heat transfer coefficient (h). This

parameter is defined by Newton's Law of Cooling as the proportionality

constant relating the local heat transfer rate (j) and the forcing

function of the heat transfer process; i.e. the difference between the

local adiabatic wall temperature (Taw) and the local wall temperature

(Tw).

S= h (Taw - T~) (1)

For the analysis of experimental data, expressions for the aerodynamic

heat transfer coefficient are derived from the equation governing the

one-dimensional, transient conduction of heat into a solid, with

application of appropriate boundary and initial conditions. Both the

phase-change and thin-skin calorimeter techniques assume a step heat

input, usually obtained by rapid injection of an isothermal model into

the airstream. For the thin-skin technique, heat transfer coefficient

is based upon heat conduction into a finite solid of known thermal

properties:Tw 1

h = pcp at Taw-Tw (2)

where p, cp, and A are model material density, specific heat and

thickness, respectively. The measured quantities are the wall temper-

ature (Tw) and its time-rate-of-change (3T/at). The corresponding

equation used to reduce phase-change data is based upon heat conduction

into a semi-infinite solid of known thermal properties:

0 2 T -T.h = p k E where 1 - e erfc (8) = Ti (3)

The measured quantity is the time (tpc) required for the model surface

temperature to increase from some initial value (Ti) to a known coating

phase-change temperature (Tpc). (Reference 1.)

6

The adiabatic wall temperature is rarely measured in thin-skin

tests, and the phase-change technique is incapable of indicating this

temperature. Computed heat transfer coefficients are extremely

sensitive to excursions of an assumed value of the adiabatic wall

temperature. This is illustrated in Figure 1 where the ratio of heat

transfer coefficient to that value computed assuming Taw = Tt is

presented as a function of Taw/Tt for constant values of phase-change

temperature ratio and model initial temperature ratios. The initial

model-to-stream temperature ratios indicated are representative of

conditions for a room temperature model and current hypersonic wind

tunnels. (The phase-change coating technique is presently routinely

used in many hypersonic air facilities which operate at initial tempera-

ture ratios of -.35, and has been considered for use in facilities

which operate near Ti/Tt = .65.) (Reference 2.) The sensitivity of

thin-skin calorimeter data to adiabatic wall temperature is indicated

by the lower curve on each figure (corresponding to the phase-change

temperature equal the initial temperature). The remaining curves

indicate the sensitivity of phase-change data, with increasing values

of the phase-change temperature. Analysis of the curves clearly

indicates the increased sensitivity of phase-change derived data, as

opposed to thin-skin data, to the accuracy of the adiabatic wall temper-

ature estimate. In addition, comparison of the plots for the range of

Ti/Tt illustrates the magnification of this sensitivity with increasing

initial temperature.

7

The phase-change temperature is constrained by the test time

required to effect the coating phase-change; this time must be long

enough for accurate measurement, yet short as compared to the thermal

diffusion time of the model. The shaded areas of the figure approxi-

mate those regions of interest for practical test operations.

Local adiabatic wall temperatures can be adequately estimated for

simple shapes by use of "exact" numerical computation techniques;

however, for complex geometries, such "exact" numerical solutions are

presently beyond the "state-of-the-art." Consequently, it has become

common practice to base experimental data on a nominal adiabatic wall

temperature ratio (Taw/Tt) assumed constant over an entire configura-

tion. The use of a nominal value of Taw/Tt = 1.0 in the data reduction

results in data which are in error-as indicated in Figure 1. Attempts

to reduce this error by assuming a compromise ratio of 0.95 or 0.90

diminish the maximum potential error; however, the functional relation-

ship of heat transfer coefficient to deviations of the assumed

adiabatic wall temperature from the actual value is unchanged. It is

therefore necessary to assess the impact of inaccuracies in adiabatic

wall temperature estimation procedures on heat transfer coefficients

derived from experimental data.

The value of the adiabatic wall-to-total temperature ratio may be

expressed for an ideal gas, as a function of recovery factor (r) and

boundary layer edge Mach number (Me) in the form:

1 + r-Me21 + r y-l M 2

Taw 2 eT 1 + - =Me2

8

Solutions of the compressible laminar boundary layer equations indicate

that recovery factor is a function of Prandtl number, pressure gradient,

and also boundary layer edge Mach number. For zero pressure gradient

flows with low edge Mach number (Me < 2), recovery factor (r) is closely

approximated by square root of the Prandtl number (Pr) (Reference 3).

The work of Wortman and Mills (Reference 4) however, indicates that for

accelerating laminar boundary layers, recovery factor is highly

dependent upon the pressure gradient parameter, 8, decreasing monotoni-

cally to an asymtote r + Pr as -+ c; a weaker dependence upon edge

Mach number is indicated. Stone, et.al (Reference 2) illustrated the

combined effect which edge Mach number and pressure gradient variations

exhibit on recovery factor in high Mach number (-20) flows in Helium.

The impact of this effect on lower Mach number hypersonic air flows

is less pronounced, as edge Mach number and pressure gradient parameter

are normally much lower; yet their significance should be ascertained

for any specific flow in question.

In the absence of flow field surveys, estimation of the local

boundary layer edge Mach number presents a further obstacle to accurate

determination of adiabatic wall temperature. Two methods for estimating

boundary layer edge conditions are the tangent wedge/cone, and the

normal-shock expansion approximations. The tangent wedge/cone technique

models the inviscid flow as that occurring on a wedge or cone surface

with half angle equal to the local flow deflection angle (6); which

approximation method (wedge or cone) is more accurate depends upon the

9

geometry of the flow to be modeled. The variation of adiabatic wall

temperature with flow deflection angle, as defined by the tangent cone

and wedge approximations, is illustrated in Figure 2 for a range of

recovery factors typical of hypersonic air flows over real configurations.

Adiabatic wall temperature level is shown to be equally sensitive to

both recovery factor and flow deflection angle.

Another approach to approximation of local boundary layer edge

conditions involves use of a measured or analytically determined

surface pressure distribution coupled with a local entropy assumption

to define the desired quantities. Possibly the most common model of

this type assumes a Newtonian pressure distribution with edge conditions

resulting from an isentropic expansion of the flow from a stagnation

point behind a normal shock. This constant entropy assumption, although

commonly used in boundary layer computation, may not be valid for

particular flows of interest.

Figure 3 presents adiabatic wall temperature distributions on the

windward center line of a space shuttle delta wing orbiter configuration

at 300 angle of attack as computed by the tangent cone and the normal

shock expansion techniques. A constant value of recovery factor

(r = 0.84) was used for both calculation methods in order to emphasize

the sensitivity to edge Mach number estimation techniques alone; edge

Mach number and pressure gradient were low so that the previously

discussed effects of these parameters upon recovery factor were small.

It is important to note that although the maximum deviation between the

10

adiabatic wall temperatures computed by these methods is only about

5 percent, the heat transfer results which are dependent upon these

values may differ by 5 to 25 percent as a function of model initial

temperature (Ti/Tt) and phase-change temperature (Tpc/Tt) as illustrated

in figure 1.

Wall Temperature Effects

A basic assumption used in derivation of the expressions for

aerodynamic heat transfer coefficient, for both the thin-skin and phase-

change techniques, is that the model experiences a step input in heat

transfer coefficient to a value which is constant with time. For thin-

skin testing, this is a normally valid assumption as test time is short

and the model remains essentially isothermal over the duration of the

transient test. Phase-change testing, however, may violate this

assumption as the data are obtained over relatively long test intervals

and, therefore, at model wall temperature levels and distributions

not normally encountered in thin-skin testing.

Chapman and Rubesin (reference 5) indicated that for laminar

boundary layer flows with variable surface temperature, local boundary

layer properties (and, therefore, heat transfer) depend not only on the

local temperature potential, but on the entire surface temperature

distribution upstream of the point in question. The effects of tempera-

ture level and distribution on the value of heat transfer coefficient

are due to what they termed "the inappropriateness of the conventional

heat transfer coefficient when applied to flows with variable surface

temperature." This "inappropriateness" is most apparent as wall temper-

ature approaches the local adiabatic wall condition. For variable wall

temperatures (Tw) and wall temperature level of the order of local

adiabatic wall temperature (Tw x Taw), heat transfer coefficient may

reverse sign and even become infinite in magnitude. This anomalous

behavior is illustrated in Figure 4 for a flat plate at zero angle of

attack with surface temperature distribution as indicated. These

curves result from "exact" solutions to the laminar boundary layer

equations (Reference 6); adiabatic wall temperature was calculated

from Equation (4) with r = 0.84.

The effect of wall temperature on heat transfer coefficient is

again illustrated in Figure 5, by solutions to the laminar boundary

layer on a hemisphere-cylinder. In the case of uniform wall

temperature, a change in the temperature level results in an alteration

to the heat transfer coefficient distribution due to the changed relation-

ship between the wall and adiabatic wall temperatures. As the wall

temperature is increased, the local heat transfer coefficient

decreases. This decrease is negligible for low values of the wall-to-

total temperature ratio, but becomes significant as wall temperature

approaches the adiabatic condition. In contrast, a negative gradient

of surface temperature along the wall, results in an increase in the

local heat transfer coefficient. The predicted increase shown in

Figure 5 is a result of the temperature gradient effect plus the oppos-

ing effect of an increased wall temperature level. (Comparing the heat

12

transfer coefficient distributions for the gradient case and the

constant wall temperature Tw/Tt = 0.1 case, heat transfer coefficients

would be expected to decrease for the gradient case as wall temperature

has everywhere increased above the Tw/Tt = 0.1 level.) The temperature

gradient effect predominates and heat transfer coefficient increases.

The wall temperature distribution indicated for this calculation, is

typical of that on a hemisphere phase-change model at a specific

instant in time during test in a hypersonic wind tunnel.

For the more practical case of a space shuttle orbiter configura-

tion, wall temperature gradient effects are illustrated in Figure 6.

Again, the indicated temperature distribution would exist on a phase-

change model at a specific instant during a hypersonic heat transfer

test. The predictions result from solution of the laminar boundary

layer by application of the axisymmetric analog to the flow on the

orbiter lower surface plane of symmetry. The inviscid pressure distri-

bution and streamline divergence information were computed by the

method of DeJarnette and Hamilton (Reference 7). The impact of variable

wall temperature is less significant for this flow as compared to the

hemisphere. This is attributed to the smaller temperature gradient

present on the orbiter lower surface aft of the nose region.

Heating Data Impact

Phase-change test derived heating distributions on spheres have

been frequently used in an inverse method for determining phase-change

model material thermophysical properties (Reference 8). This procedure

13

requires comparison of the experimental heating distribution with a

theoretical dist:ibution to obtain that value of the phase-change

thermal propertie.; parameter, pcpk, which results in the best correla-

tion. The quality of the measurement is a direct function of the

sophistication of the theoretical method utilized. Figure 7 presents

experimental phase-change heating data for a hemisphere-cylinder at

Mach 20.3 in helium (from Reference 4). The measured heat transfer data

were reduced using an adiabatic wall temperature distribution obtained

by the non-similar solution technique. Also shown is the theoretical

heat transfer distribution for a constant wall temperature equal to the

model initial,temperature. As can be seen from Figure 7, the use of a

constant wall temperature theory with phase-change data may result in

a derived thermal property value which is substantially in error.

Thermal properties so derived will be accurate only if the theory

utilized adequately models the non-isothermal nature of the phase-change

test itself.

Conversely, if experimental heating data are to be used to verify

theoretical calculation procedures, it is important that the theory

accurately model the experiment which produced that data. Data which

may exhibit significant wall temperature effects should not be used to

verify a theory which lacks the sophistication to account for them.

CONCLUDING REMARKS

Basic analytical procedures have been used to illustrate, both

qualitatively and quantitatively, the relative impact upon heat

14

transfer data analyses of certain factors which may affect the accuracy

of experimental heat transfer data. It is recognized that the physical

principles involved, i.e. wall temperature effects on heat transfer,

recovery factor and adiabatic wall temperature computation procedures,

have all been previously discussed in detail byother .investigators.

However, recent widespread adoption of the phase-change coating

technique for use in a variety of heat transfer investigations

requires a renewed awareness, by the experimentalist, of the possible

error sources (and the significance of each) which exist when testing

in the regimes required by the phase-change technique. The subject

material leads to the following comments:

1. Experimental heat transfer coefficient data accuracy (for

either thin-skin or phase-change) is directly dependent upon accurate

knowledge of the local adiabatic wall temperature. Phase-change coat-

ing data exhibits a significantly greater sensitivity to this quantity

than does thin-skin calorimeter data. Errors in heat transfer coeffic-

ient resulting from inaccurate knowledge of the adiabatic condition may

be diminished by testing at decreasing values of the model initial-to-

stream total temperature ratio. In the limit, however, accuracy in

computed heat transfer coefficients is directly proportional to the

accuracy of the adiabatic wall temperature.

2. Wall temperature gradients, which may result from model

geometry characteristics and/or long run times, can significantly

affect measured heat transfer coefficient distributions. Wall tempera-

ture gradients and their resulting effects are minimized by thin-skin

testing or, in the case of phase-change testing, utilizing phase-

change temperatures which approach the initial condition.

3. If experimental data are to be used to verify theoretical

calculation procedures, it is important that the theory accurately

model the experiment which produced that data. Data which may exhibit

significant wall temperature effects should not be used to verify

a theory which lacks the sophistication to account for them.

REFERENCES

1. Jones, Robert A; and Hunt, James L.: Use of Fusible TemperatureIndicators for Obtaining Quantitative Aerodynamic Heat TransferData. NASA TR R-230, February 1966.

2. Stone, D. R.; Harris, J. E.; Throckmorton, D. A.; and Helms, V. T.:Factors Affecting Phase Change Paint Heat Transfer Data ReductionWith Emphasis on Wall Temperatures Approaching AdiabaticConditions. AIAA Paper No. 72-1030, September 1972.

3. Pfyl, Frank A.; and Presley, Leroy L.: Experimental Determinationof the Recovery Factor and Analytical Solution of the ConicalFlow Field for a 200 Included Angle Cone at Mach Numbers of 4.6and 6.0 and Stagnation Temperatures to 26000 R. NASA TN D-353,June 1961.

4. Wortman, A.; and Mills, A. F.: Recovery Factor for HighlyAccelerated Laminar Boundary Layers. AIAA Journal, Vol. 9, No. 7,July 1971, pp. 1415-1417.

5. Chaptan, D. R.; and Rubesin, M. W.: Temperature and VelocityProfiles in the Compressible Laminar Boundary Layer WithArbitrary Distribution of Surface Temperature. Journal of Aero.Sci., 16, 1949, pp. 547-565.

6. Hamilton, H. Harris, II: A Theoretical Investigation of the Effectof Heat Transfer on Laminar Separation. M.S. Thesis, VirginiaPolytechnic Institute, May 1969.

16

7. DeJarnette, Fred R.; and Hamilton, H. Harris, II: Inviscid SurfaceStreamlines and Heat Transfer on Shuttle Type Configurations.AIAA Paper No. 72-703, June 1972.

8. Conners, L. E.; Sparks, V. W.; Bhadsalve, A. G.: Heat TransferTests of the NASA-MSFC Space Shuttle Booster at the LangleyResearch Center Hypersonic Continuous Flow Facility. LMSC/HRECD162722, Lockheed Missles and Space Company, Huntsville, Alabama,December 1970.

LIST OF FIGURES

Figure I.- Effect of adiabatic wall temperature assumption on computedheat transfer coefficient.

Figure 2.- Adiabatic wall temperature variation with flow deflectionangle. (MO = 10.0)

Figure 3.- Delta-Wing shuttle orbiter lower surface centerline adiabaticwall temperature distribution. a = 300.

Figure 4.- Effect of wall temperature on heat transfer coefficient fora flat plate at Mach 10.0 in air. a 00.

Figure 5.- Theoretical heat transfer distributions on a hemisphere-cylinder at Mach 10.0 in air.

Figure 6.- Theoretical heat transfer distributions on the windwardcenterline on a space shuttle orbiter configuration atMach 10.0 in air. a = 30

Figure 7.- Effect of wall temperature variation on measured heat transfer onon a hemisphere-cylinder at Mach 20.3 in helium.

1.8 T r

hactual Tth(Taw=Tt) .8

.7.1.4

1.0.85 .90 .95 1.0

Taw/Tt

(a) Ti/Tt = 0.65

1.8 1.8

hactual T hactualh(T•Tt) .9 Tt h(Taw=Tt )

.8

1.4 / .6 1.4

TpC

1.0 - - - 1.0.85 .90 .95 1.0 .85 .90 .95 1.0Taw/Tt Taw/Tt

(b) Ti/Tt = 0.40 (c) Ti/Tt = 0.10

Figure 1.- Effect of adiabatic wall temperature assumptionon computed heat transfer coefficient.

1.0WEDGE FLOW

.9r

Taw .84

Tt .82.80

.8 .78

I I I0 20 40 60

6, deg.

1.0CONE FLOW

.9

aw r

Tt .8 .8

8.78

0 20 .40 606, deg.

Figure 2.- Adiabatic wall temperature variation with flow deflectionangle. (M = 10.0)

.95 NEWTONIAN PRESSURE

Taw CONSTANT ENTROPY

Tt

.90 -TANGENT CONE

0 1.0X/L

Figure 3.- Delta-Wing shuttle orbiter lower surface centerligeadiabatic wall temperature distribution a = 30

1.0

T ------- TawTt .8 -

Tw.7 -

.8 - x 10-3

B .4

ft2sec

-. 4

-.8

-1.2

1.2 -x 10-4

.8h

[ BTU 4

t2sec] .oR0-

-.4

-.8

-1.2I I I I I I0 .2 .4 .6 .8 1.0

X/L

Figure 4.- Effect of wall temperature on heat transfer co8fficientfor a flat plate at Mach 10.0 in air. a = 0".

Lo - .1.0

.8 -- Tw GRADIENTTw/Tt -

h J

ho 0 L6.4 - S/R

INCREASED Tw ,

.2CONSTANT T,/Tt .9

0 .4 .8 1.2 1.6

S/R

Figure 5. -Theoretical heat transfer distributions on a hemisphere-cylinder at Mach 10.0in air.

0.20 1.0-

STw/Tt. O

0.10 0 X/L 1.

0.08

0. 06 CONSTANT Tw/Tt = 0. 3

0.040 .2 .4 .6 .8 L.0

X/L

Figure 6. -Theoretical heat transfer distributions on the windward centerlineon a space shuttle orbiter configuration at Mach 10.0 in air.a - 300.

1. 0 EXPER IMENTAL DATA Tpc/Tt(Ref. 4) ---------- .66

8 - .67S.72

.74

6 .76

h Ti/Tt = 0. 625

ho.4

Theory.2 _ Tw/Tt = 0. 625

00 .4 .8 1.2 1.6

S/R

Figure 7. - Effect of wall temperature variation on measured heat transfer on a hemispherecylinder at Mach 20.3 in helium.