EXPERIMENTAL SPHERE

AND

RAREFIED

V.

STUDY A FLAT

GAS

Kh.

OF PLATE

HEAT

TRANSFER

OF

A

IN S U P E R S O N I C

FLOW

Avleeva

An experimental study of the mean convective heat t r a n s f e r of a sphere and flat plate at zero angle of attack in a supersonic low-density s t r e a m is d e s c r i b e d . The sphere m e a s u r e m e n t s c o v e r e d the p a r a m e t e r range 3.6 < M < 6.1 and 5 < R < 400, and those on the flat plate c o v e r e d the range 3~ < M < 6.1 and 5 < R < 65. 1. Simultaneous studies of heat t r a n s f e r and a e r o d y n a m i c s were made in the low-density wind tunnel d e s c r i b e d in [1, 2]. The n o z z l e s used in [1, 4] were used to c r e a t e the supersonic flow. The working gas was dried air. The gas was not preheated in the r e c e i v e r . 2. The experimental models were made f r o m copper. The use of copper was based on its high thermal conductivity, which e n s u r e s rapid and uniform heating of the model in the flow. The s c h e m e for m e a s u r i n g the model t e m p e r a t u r e by a t h e r m i s t o r , the technique for embedding the t h e r m i s t o r in t h e model, and the type of t h e r m i s t o r were suggested by N~ A. Kolokolova, who also made the repeated calibrations of the t e m p e r a t u r e s e n s o r . Five models were f a b r i c a t e d for study of sphere heat t r a n s f e r . The four models with d i a m e t e r s D = 2, 4, 8, 12 m m consisted of two h e m i s p h e r i c a l p a r t s : a copper half facing the flow and an ebonite ha!f completing the sphere ( r e a r half of the model). One model with d i a m e t e r D= 4 m m was fabricated entirely f r o m copper. An opening was drilled in the center of the copper h e m i s p h e r e and the t h e r m i s t o r head was inserted. This a r e a was potted with aquadag for good t h e r m a l contact. The leads f r o m the t h e r m i s t o r were brought out through a p a s s a g e in the ebonite half, drilled along a radius perpendicular to the copper part of the model, and then run through the hollow glass base sting to the instrument used to r e c o r d the t e m p e r a t u r e s e n s o r indication. The copper part and the sting were bonded to the ebonite part using carbinol cement. The flat plates were fabricated f r o m copper foil 43 p thick. The study was made using two square plates with sides 2 and 4 mm. The t h e r m i s t o r bead together with the initial segment of the very fine (8-~) c u r r e n t leads, f i r s t t h e r m a l l y insulated with aquadag, were bonded between the two plates. The plate mortaring and s c h e m a t i c of the setup in the s t r e a m a r e shown in Fig. 1, where 1 is the plate model, 2 are kapron filaments, 3 is a m i c a plate, 4 is the f r a m e , 5 is the t r a v e r s i n g mechanism, and 6 is the nozzle.

#

3. The convective h e a t - t r a n s f e r c h a r a c t e r i s t i c s - the mean equilibrium t e m p e r a t u r e and the mean conw~ctive h e a t t r a n s f e r c o e f f i c i e n t - were d e t e r m i n e d by the t r a n s i e n t method d e s c r i b e d in [5]. According to this method the change of the heat content of a small body with uniform t e m p e r a t u r e d i s tribution within it is equal to its t h e r m a l l o s s e s , which in general are made up of l o s s e s resulting f r o m convection and radiation, and the l o s s e s along the sting resulting f r o m t h e r mal conduction. Neglecting the latter, the method can be r e p r e s e n t e d by dT

Fig. 1

c w V - - ~ h~F(r -- re)+ h~F(T- T~ dt

(3.1)

Moscow. T r a n s l a t e d f r o m Izv. AN SSSR. Mekhanika Zhidkosti i Gaza, Vol. 5, No. 2, pp. 191--195, M a r c h - A p r i l , 1970. Original a r t i c l e submitted July 28, 1969. 9 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever witl~out permission of the publisher. A copy of this article is available from the publisher for $15.00.

339

with the boundary conditions T=r0,

t=0;

T----T'e, t = ~

(3.2)

Here c, w, V, and F a r e r e s p e c t i v e l y the heat capacity, specific weight, volume, and surface area of the body being studied (the copper part of the body in our case); h c i s t h e c o n v e c t i v e h e a t - t r a n s f e r coefficient; hr is the radiative h e a t - t r a n s f e r coefficient; T e is the equilibrium t e m p e r a t u r e owing to convection alone; T e' is the equilibrium t e m p e r a t u r e owing to the combined action of convection and radiation; and T ~ is the wall t e m p e r a t u r e of the c h a m b e r surrounding the body. Equation (3.1) can be integrated ff h c and h r are independent of the body t e m p e r a t u r e . In general the coefficient h r is a function of T: T ~ _ TO4 h~ T--T

~

When the difference (T - T ~ is small (for example, less than T~ the coefficient h r can with good a c c u r a c y be c o n s i d e r e d constant. In the e x p e r i m e n t s the difference (T - T ~ was about 0.1 T ~ Integrating (3.1) and considering condition (3.2), we obtain the solution F(ho + h,)

h T

A

- -

-

Te r

In

(3.3)

- t cwV

hcTe -{- h r T ~ T'

To --: T / '

(3.4)

hc + h~

where T J is a quantity derived f r o m the general solution with the second condition in (3.2). We find with the aid of (3.4) the e q u i l i b r i u m t e m p e r a t u r e owing to convection alone: hr -t Te = Te" + - - ( T e - - T ~) hc

(3.5)

In the absence of flow and with such a low p r e s s u r e in the c h a m b e r that t h e r e is no f r e e convection, the solution of (3.1) is Fhrt A

=

- - -

cwV'

T h =

l

n

T~

- -

-

-

To - - T ~

(3.6)

Equations (3.3), (3.5),and (3.6) were used to reduce the experimental r e s u l t s in the experiments with and without flow. 4o The f i r s t p r o c e d u r a l experiments on the study of heat t r a n s f e r in r a r e f i e d gas flow showed that the t e m p e r a t u r e difference (T O - Te') is frequently small, and in this c a s e the slope of the straight line (3.3), which gives the connection between the quantity A and the time t, and t h e r e f o r e the coefficient (hc + hr), is determined with a l a r g e e r r o r . To i n c r e a s e the value of this difference, in the experiments we utilized a small d e g r e e of model preheating. The p r e h e a t e r consisted of a lamp mounted inside a teflon shell in the f o r m of an ellipsoid at one end of the m a j o r axis. At the opposite end t h e r e was a 1 - c m - d i a m hole at which the model was placed for p r e heating. The shell was s u r r o u n d e d by a shield made f r o m a double l a y e r of metallic foil. 5. The e x p e r i m e n t s were conducted with and without preheating of the model. The flow was generated and controlled as d e s c r i b e d in [1]. The model was introduced into the working s t r e a m (the isentropic core at the nozzle exit section) by a t r a v e r s i n g m e c h a n i s m . When the model t e m p e r a t u r e r e a c h e d the equilibrium value Te' it was r e m o v e d f r o m the s t r e a m , placed at the p r e h e a t e r opening, and the p r e h e a t e r was activated. After preheating the model to less than 50 ~ C, the p r e h e a t e r was turned off and the model was again inserted into the same s t r e a m . The moments of model insertion into and r e m o v a l f r o m the s t r e a m and the m o m e n t s of p r e h e a t e r activation and deactivation were noted by signals on the tape of the ]~PP-09 automatic electronic r e c o r d e r , which provided simultaneous r e c o r d i n g s of the model, r e c e i v e r , and c h a m b e r t e m p e r a t u r e s and also the t e m p e r a t u r e s m e a s u r e d by the t h e r m i s t o r s . 6. Curves of A as a function of time were plotted f r o m the r e s u l t s of the model t e m p e r a t u r e m e a s u r e ments in the s t r e a m . In a c c o r d a n c e with (3.3) the slope of the resulting c u r v e s makes it possible to d e t e r mine the coefficient (hc + hr) f o r a body with known g e o m e t r i c dimensions and material. The quantity F / c w V is the same f o r the spherical models made entirely of copper and those consisting of half copper and half ebonite.

340

The coefficient hr was determined s i m i l a r l y f r o m the r e s u l t s of m e a s u r e m e n t s in the ~bsence of flow. The mean equilibrium t e m p e r a t u r e was calculated using (3.5). The r e s u l t s were presented in the f o r m of the dependence of the t e m p e r a t u r e r e c o v e r y factor T and the dimensionless h e a t - t r a n s f e r coefficient N, the Nusselt number, on the s t r e a m p a r a m e t e r s : T,,

-

-

T

"g === ~

hoD ,

To

- -

N

~

~

T

In the expression for N t h e r e is the c h a r a c t e r i s t i c dimension D (the d i a m e t e r of the sphere or the side of the plate) and t h e h e a t - t r a n s f e r coefficient k, which was d e t e r m i n e d for the f r e e s t r e a m t e m p e r a t u r e T, stagnation t e m p e r a t u r e T . , and equilibrium convective t e m p e r a t u r e T e. Using the r e s u l t s of studies of total head probe n o z z l e s [1], the Mach n u m b e r M was determined f r o m the m e a s u r e d p r e s s u r e in the r e c e i v e r . The Reynolds n u m b e r R was calculated using R = 6.t5

MP.(T. +

t22 + 24.4Ma)

(6.1)

T , 2 (i-t- 0.2M ~) i/~-t

where P . is in m m Hg and D is in m e t e r s . This f o r m u l a was derived under the assumption that the gas is perfect, the flow is isentropic, and the viscosity is defined by the Sutherland f o r m u l a / T \t'5273+6

where ~0 = 1.775 9!0 -~ k g - s e e / m 2 and C = 122 for air.

(6.2)

The e s t i m a t e s made showed that the e r r o r s in the e x p e r i m e n t a l d e t e r m i n a t i o n of T and N are, r e s p e c t i v e l y , f o r ~-,~2.5~ on the s p h e r e and ~=3.5% on the fiat plate and for N,* 2 0 - 3 0 ~ . The m a x i m u m e r r o r in the determination of R was ~:10%. 7. Experimental r e s u l t s for the sphere were obtained for 3.6 < M < 6.1 and 5 < R < 400. The t e m p e r a t u r e r e c o v e r y f a c t o r ~- as a function of the p a r a m e t e r @ M is shown in Fig. 2a. Points 1 . . . . . 9 c o r r e spond to the Mach number variation intervals 1(3.6-3.8), 2(3.8-4.2), 3(4.8-5.2), and 4(5.6-6.1); this figure also shows the r e s u l t s of e a r l i e r studies of Drake and Backer [5]: points 5(2.24-2.75), 6(2.78-3.56); Eberly [61: point 7(4-6); and c o n c u r r e n t or l a t e r e x p e r i m e n t s made by I4oshmarov [7]: points 8(5.5-6.25) and 9 (7.5-8). Also shown are the free m o l e c u l a r values (dashed curves). The f r e e molecular values of the heatt r a n s f e r coefficients for a body of given f o r m can be obtained if we use the r e s u l t s of [8] for a s u r f a c e element and integrate over the s u r f a c e of the body. These caleuIations for the sphere were made in [9]. The r e s u l t s of s i m i l a r calculations for all bodies of simple f o r m are s u m m a r i z e d in [10]. The r e s u l t s obtained here extend the study of the r e c o v e r y coefficient to lower p r e s s u r e s . R e g a r d l e s s of the Maeh number, all the experimental points plotted in the coordinates 7 and ~R-R/M lie__on a single curve with s c a t t e r within the experimental limits. We see f r o m Fig. 2a that for values of { R / M > 5, T r e m a i n s constant. With i n c r e a s e of the r a r e f a c t i o n , the r e c o v e r y f a c t o r i n c r e a s e s r values g r e a t e r than 1; i.e., the equilibrium t e m p e r a t u r e exceeds the stagnation t e m p e r a t u r e . This phenomenon is c h a r a c t e r i s t i c for heat t r a n s f e r in a r a r e f i e d gas s t r e a m , in c o n t r a s t with continuum flow, in which the t h e r m a l l y insulated body can acquire at most the gas s t r e a m stagnation t e m p e r a t u r e if t h e r e a r e no heat l o s s e s in the boundary l a y e r . Figure 3a shows the relationship between the Nusselt and Reynolds n u m b e r s obtained in the e x p e r i ments d e s c r i b e d . We note stratification of the r e s u l t s as a function of M. The dashed lines show t]he e x t r a p olation of the r e s u l t s in the f r e e m o l e c u l a r region, with indications of the magnitude of the accommodation

7A

"o:

ZT--

o

0

:<3 9 #

0"0.6........

.

06

.8

"7

V9

l.'~

aa

-

0.8

1.8

1.2

i - ~t

"1

"

L.

2.0

+ +3-J

I ,,sj

CN/M

r

2.2

3.0

3.8

a.G

s

6.2

Fig. 2

341

:o~ Al a .; ~ \ I oooJ- YL &i~l '1-11

9

8 lO

a8 ~ o

Frl--

~

Ill

20

gO

Go

:

~F r f f [ ~

I Fgll

.'o

zo8 o . 8 ~ j ~ o~o ,, ~ ~

,z

O.B 8.8

o

~-

o

4+ % ~

A~ ~ 9

oZ

.

+Z'--

~z c ~ 1 7 6 o : : 5 " . . i

O.ll l

~"

~

++3

""

o

~ ,, q.

Z

L~

8

+o

R. I0

e2_

z/

G 8 lO

20

z/O

Fig. 3 coefficients for each experimental Mach n u m b e r (in the experiments described 0.5 < ~ < 0.9) for which this extrapolation is made. The accommodation coefficient depends on the Mach n u m b e r ; the l a r g e r the value of M the s m a l l e r the value of c~. This conclusion is in qualitative agreement with the results of [11]. It was indicated in [12, 13] that presentation of the r e s u l t s in the f o r m of the dependence of N. on R . (the a s t e r i s k indicates that the t h e r m a l conductivity and viscosity were determined at the stagnation t e m perature) makes it possible to exclude the effect of M as an additional p a r a m e t e r . Figure 3b, in which the experimental r e s u l t s are presented in this form, c o n f i r m s this hypothesis; the experimental points denoted by p r i m e s were obtained with preheating. T h e r e is p r a c t i c a l l y no effect of preheating~ The experimental points obtained on the D = 4 m m spherical model consisting entirely of copper and only half copper lie in the range 7 < R. < 14. The r e s u l t s obtained on the two models under identical flow conditions agree to within experimental e r r o r . Figure 4 shows the r e s u l t s of different authors: points 1-4 were obtained in the present study, points 5 (M = 2.7-4.1) a r e the r e s u l t s of Drake and Backer, points 6 (4-6) are E b e r l y ' s r e s u l t s f r o m [5, 6], and points 7 (6.2-6.35) are tCoshmarov's r e s u l t s [7] in the f o r m of the dependence of Ne (thermal conductivity determined at the t e m p e r a t u r e T e) on R* behind the shock wave. No stratification as a function of M is observed. The data of the different authors agree to within the experimental s c a t t e r throughout the range of flow p a r a m e t e r s . In Fig. 4 the solid curve shows the relation for a continuum as M ~ 0, taken f r o m [14]. With i n c r e a s e of flow r a r e f a c t i o n t h e r e is m o r e marked r e d u c tion of the dimensionless heat t r a n s f e r coefficient than predicted by the dependence of Ne on R* for the continuum flow r e g i m e . Heat t r a n s f e r of a flat plate aligned with the s t r e a m was studied for 3.6 < M < 6.1 and 5 < R < 65. The r e s u l t s are shown in Figs. 2b and 3Co F i g u r e 2b shows the experimental dependence of the r e c o v e r y factor on the p a r a m e t e r vrR--/M; the points are I(M = 3.6-3.8), 2(3.8-4.2), 3(4.8-5.2), 4(5.6-6.1). Also shown are the r e s u l t s of [15], points 5 (M = 2.6-3.2); the open points were obtained in experiments without preheating, the filled points were obtained with preheating, and the a s t e r i s k on the ordinate axis denotes the f r e e molecular value. lO

/.0 O.8 O.a

l

2

Zl 6 8 I0

20

~0 8080100 200 000 GO0 /00D

Fig. 4

342

LITERATURE i. 2. 3. 4. 5. 6. 7~ 8. 9. i0o 11. 12. 13.

14. 15. 16.

CITED

V . A . Sukhnev, "Corrections to total head probe indications in supersonic rarefied gas flow,'~Izv. A N SSSR, O T N , Mekhanika i mashinostroenie, no. 5 (1964). G . A . Evseev, "Experimental study of rarefied gas flow," Izv. A N SSSR, Mekhanika, no. 3 (1965). L . A . Zhukova, N.A.Kolokolova,and V. A~ Sukhnev, "Measurements of small differential pressures in rarefied gases, ~ Izv. A N SSSR,OTN, Mekhanika i mashinostroenie, no. 6 (1961). V.A. Sukhnev, "Experimental determination of the drag coefficient of a sphere in supersonic rarefied gas flow," Izv. A N 8SSR, Mekhanika, no. 3 (1965). R . M . Drake and G. H. Backer, "Heat transfer from sphere to rarefied gas in sup<~:sonic flow," Vopr. raketn, tekhn., no. 2, 1953; Trans. A S M E , vol. 74, no. 7, p. 1211 (October, 1952). S.A. Schaaf and P. L. Chambre, "Flow of rarefied gases," in: Fundamentals of Gas Dynamics [Russian translation], Izd-vo inostr, lit., M o s c o w (1963). Yuo A. K o s h m a r o v and N~ M. Gorskaya, "Heat transfer and equilibrium temperature of a sphere in supersonic rarefied gas flow,, Izv. A N SSSR, M Z h G , vol. i, no. 4 (1966). J. R~ Stalder and D. @ukoff, "Heat transfer to bodies travelling at high speed in upper atmosphere~ ~ Vopr. raketn. [ekhn., no. 5 (1952); N A C A Repro 944 (1950). F . M . Sauer, "Convective heat transfer from spheres in a free molecule flow," J. Aeronaut. Sci., vol. 18, no. 5 (1951). A~ Oppenheim, nOn the general theory of convective heat trm~sfer in free molecular flow" [Russian translation], in: Mekhanika, no. 5 (1953). F . M . Devienne, " E x p e r i m e n t a l study of the stagnation t e m p e r a t u r e in a f r e e m o l e c u l a r flow," J. Aeronaut. Sci., vol. 24, no. 6 (1957). J . R . Stalder, G. Goodwin, and M. O. C r e a g e r , "Heat t r a n s f e r to bodies m h i g h - s p e e d r a r e f i e d - g a s s t r e a m , Vopr. raketn, tekhn., no. 1 (1954); NACA Rept. 1093 (1952). L . V . Baldwin, V. A. Sandborn, and J. C. L a w r e n c e , "Heat t r a n s f e r f r o m t r a n s v e r s e andyawed c y l i n d e r s in continuum, slip, and f r e e m o l e c u l e air flows," T r a n s . ASME, s e r . C, J. Heat T r a n s f e r , vol. 82, no. 2 (1960). L . L . Kavanau, "Heat t r a n s f e r f r o m s p h e r e s to a r a r e f i e d gas in subsonic flow," T r a n s . ASME, vol. 77, no. 5 (1955). S . I . Kosterin, Yu. A. K o s h m a r o v , and N. M. G o r s k a y a , " E x p e r i m e n t a l study of heat t r a n s f e r of a flat plate in s u p e r s o n i c r a r e f i e d gas flow, ~ Inzh. zh., vol. 2, no. 2 (1962). Yu. A. K o s h m a r o v and N. M. G o r s k a y a , "Heat t r a n s f e r of a fiat plate in s u p e r s o n i c r a r e f i e d gas flow," Inzh. zh., vol. 5, no. 2 (1965).

343