Advances in Heat Transfer, Volume 7

Contributors to Volume 7 W. 0. BARSCH RICHARD J. GOLDSTEIN W. B. HALL T. MIZUSHINA GEORGE S. SPRINGER E. R. F. WINTER

Advances in

HEAT TRANSFER Edited by Thomas F. Irvine, Jr.

James P. Hartnett

State University of New York at Stony Brook Stony Brook, Long Island New York

Department of Energy Engineering University of Illinois at Chicago Chicago, Illinois

Volume 7

@ 1971 ACADEMIC PRESS

New York

London

COPYRIGHT 0 1971, BY ACADEMIC PRESS, INC. ALL RJGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

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PRINTED IN THE UNITED STATES OF AMERICA

LIST OF CONTRIBUTORS W. 0. BARSCH, School of Mechanical Engineering, Purdue University, Lafayette, Indiana RICHARD J. GOLDSTEIN, Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota W. B. HALL, Nuclear Engineering Department, University of Manchester, Manchester, England

T. MIZUSHINA, Department of Chemical Engineering, Kyoto University, Kyoto, Japan GEORGE S. SPRINGER, Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan E. R. F. WINTER, School of Mechanical Engineering, Purdue University, Lafayette, Indiana

V

PREFACE T h e serial publication, “Advances in Heat Transfer,” is designed to fill the information gap between the regularly scheduled journals and university level textbooks. T h e general purpose of this series is to present review articles or monographs on special topics of current interest. Each article starts from widely understood principles and in a logical fashion brings the reader up to the forefront of the topic. T h e favorable response to the first six volumes by the scientific and engineering community is an indication that our authors have competently fulfilled this purpose. T h e editors are pleased to announce the publication of Volume 7 and wish to express their appreciation to the current authors who have so effectively maintained the spirit of the series.

ix

Heat Transfer near the Critical Point . .

W B HALL Nuclear Engineering Department. University of Manchester. Manchester. England I . Introduction

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I1. Physical Properties near the Critical Point . . A. Thermodynamic Properties . . . . . . . B. Molecular Structure near the CriticalPoint

....... . . . . . . . .................

C . Transport Properties D . The Implications of Physical Property Variation on Heat Transfer . . . . . . . . . . . . . . . . . . . . . . I11. The Equations of Motion and Energy . . . . . . . . . . . A . Boundary Layer Flow . . . . . . . . . . . . . . . . B. ChannelFlow . . . . . . . . . . . . . . . . . . . . C . The Turbulent Shear Stress and Heat Flux . . . . . . . IV. Forced Convection . . . . . . . . . . . . . . . . . . . A . Methods of Presentation of Data . . . . . . . . . . . . B. Experimental Data . . . . . . . . . . . . . . . . . C . Correlation of Experimental Data . . . . . . . . . . . D . Semiempirical Theories . . . . . . . . . . . . . . . V . Free Convection . . . . . . . . . . . . . . . . . . . . A . Experimental Results . . . . . . . . . . . . . . . . . B . Theoretical Methods and Correlations . . . . . . . . . . VI. Combined Forced and Free Convection . . . . . . . . . . A . Experimental Results . . . . . . . . . . . . . . . . . B. A Proposed Mechanism for the Heat Transfer Deteriorations VII . Boiling . . . . . . . . . . . . . . . . . . . . . . . . A . Nucleate Boiling . . . . . . . . . . . . . . . . . . . B. Film Boiling . . . . . . . . . . . . . . . . . . . . C. PseudoBoiling . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . 1

10 15 17 19 22 25 26 31 43 51 55 55 63

66 67

68 74 76 79

81 82 83

2

W. B. HALL I. Introduction

The rapid growth of research activity in supercritical heat transfer over the past ten or fifteen years is a consequence of several trends in engineering. There has been a steady development of steam plant towards supercritical conditions, and supercritical water has been considered as a coolant for several types of nuclear reactors. Helium is used at nearcritical conditions as a coolant for the conductors of electrical machines, and rocket motors are frequently cooled by pumping fuel through cooling pipes at supercritical pressure. From a fundamental standpoint, the problem has been regarded as one in which the variation of physical properties with temperature becomes extremely important. Effects which, with most fluids, may be treated as small perturbations of the “constant property” idealization, sometimes become dominant, rendering existing theoretical models and empirical correlations useless. In some cases phenomena appear which have no counterpart with constant property fluids. At the same time experimental difficulties have hampered the investigation of these effects. These are not merely the difficulties of operating equipment at high pressures, but also the problems of compressibility (which becomes very high near the critical point and makes the density sensitive to relatively small pressure variations) and of specific heat (which also becomes large and hinders the accomplishment of thermal equilibrium). It might be thought that heat transfer experiments of such complexity would have little to contribute to the understanding of basic mechanisms. It is true that in constructing models of the process one is forced to introduce additional assumptions which are difficult to test; nevertheless, there are some cases where extreme property variations afford a much more stringent test of some aspects of current theories than could be obtained in other ways. An example of this is the interaction between forced and free turbule‘nt convection; with a supercritical fluid the trend of the results is in the opposite sense to that which one would expect. This may well lead to a reexamination of the same problem for fluids with small property variations. The near-critical region may be thought of as that region in which boiling and convection merge. When the pressure is sufficiently subcritical or supercritical, the problem tends towards either a boiling problem or a constant property convection problem; under such conditions existing theoretical and empirical methods are generally adequate. We shall concentrate on the region rather close to the critical point where the property variations are severe and where there are very significant heat transfer effects. Such effects are usually found in a range of pressures

HEATTRANSFER NEAR

THE

CRITICALPOINT

3

from the critical up to about 1.2 times the critical; they are generally largest when the temperatures of the hotter surface and the fluid span the critical temperature. We begin with a brief description of the behavior of thermodynamic and transport properties near the critical point. T h e equations of continuity, momentum, and energy are then examined with a view to revealing the effect of variable properties and deciding whether the same simplifications can be made as are common with a constant property fluid. A discussion of the various modes of heat transfer then follows, particular attention being given to the interaction between forced and free convection. 11. Physical Properties near the Critical Point

A. THERMODYNAMIC PROPERTIES T h e properties of a fluid near its critical point have interested thermodynamicists for the past hundred years. This is hardly surprising in view of the singular behavior in this region: the classical description indicates] for example, that the compressibility and the specific heat at constant pressure both become infinite at the critical point. These factors make experimentation difficult; it is evident that as (avjap),becomes large, the hydrostatic pressure variation in the fluid will lead to significant density variations even for small changes of height and also that the approach to thermal equilibrium will be slow as cp becomes large. T h e present state of knowledge of thermodynamic behavior is not entirely satisfactory, either from a theoretical or from an experimental standpoint; nevertheless, it is probably true to say that an understanding of heat transfer in the critical region is limited more by lack of knowledge of the heat transfer processes (e.g., turbulent diffusion, effect of buoyancy forces) than by uncertainties in the thermodynamic properties. In these circumstances, the classical description of the critical point may still be adequate.

1 . The van der Waals Model I n 1873, van der Waals proposed an explanation of thermodynamic behavior near the critical point. His model, in which an allowance is made for the attractive and repulsive forces between molecules, leads to an equation of state of the following form:

W. B. HALL

4

The physical arguments underlying the equation are well known and need not be repeated here; it is sufficiently to note that the constant b accounts for the strong, short range repulsive forces (imposing a limit to the reduction of volume as pressure is increased), and the term a i r 2 represents the long range attractive forces between molecules. Figure 1 illustrates the shape of isotherms on a p , V diagram, according to van der Waals equation. Consider a particular isotherm, marked abcdef in Fig. 1. The fluid

I

!?' 3

Lo

a !?'

Volume, V

-

FIG. 1. The van der Waals isotherms.

can exist in a homogeneous state along the section of the isotherm marked abc and def; the section cd represents conditions in which the thermodynamic inequality ( W W T

lo0 seems to belong to the transition region. By rotating the cylinder, the local mass transfer coefficients are measured by an isolated cathode prepared on the cylinder surface. T h e results of the local mass transfer measurements are shown in Fig. 10. For Gr . Sc > 3 x los, all the correlating curves have minimum points at the separation points in the range between 130 and 180". Figure 11 indicates that the limiting current to the isolated cathode, i.e., the local transfer coefficient, is fluctuating in the turbulent region but not in the laminar region.

3. Free Convection Mass Transfer at a Sphere Free convection mass transfer at a sphere was also measured by Schutz (15). T h e experimental technique was the same as that for the cylinder. T h e experimental results of the space-time-averaged and local mass transfer coefficients are shown in Figs. 12 and 13, respectively. The correlating equation of the space-time-averaged mass transfer coefficients is Sh

=2

+ 0.59 (Gr - S C ) O ~ ~

(18)

C. FORCED CONVECTION

1. Forced Convection Mass Transfer in Tube Flow a. Fully Developed Mass Transfer in Turbulent Flow. Several equations have been proposed for predicting the mass transfer coefficients between a pipe wall and turbulent flow in the region of a fully developed concentration profile. T h e equations differ from each other in the effect of the Schmidt number on the mass transfer coefficients. I n the empirical correlations of the heat transfer coefficients by Chilton and Colburn (16) and Friend and Metzner (27) the exponent for the Prandtl number is #. The semitheoretical equation of Lin et al. (28) predicts that the mass transfer coefficients are proportional to Sc2I3 for high Schmidt numbers, and a similar treatment of Deissler (19) leads to an exponent of $. Since the concentration gradient in the direction of flow (x-coordinate) is much smaller than that in the direction perpendicular to the wall (y-coordinate), the mass flux in the fully developed mass transfer of turbulent flow along a wall is usually expressed by Eq. (3).

I 200

k 100 2.60 .

Id

3.99 ' 10'

-

0

0

90

45

135

180

e (deg) FIG. 10. Local coefficients of free convection mass transfer on a cylinder.

180'

0.20/&

-

0.15 -

oo

1 20°

-

c

3' 0

aE 0 . 1 0 - w 1 5 5 °

v

.-

0.05

-1

50'

-

u 20 40

O O

t

FIG. 11. a cylinder.

(sec 1

Fluctuation of the local coefficients of free convection mass transfer on

THEELECTROCHEMICAL METHOD

FIG. 12. Free convection mass transfer coefficients on a sphere.

400

I 7.68.10'

2.94.109

5.18. lo8

2.d.108

- 0

0

45

90

135

180

0 (deg)

FIG. 13. Local coefficients of free convection mass transfer on a sphere.

105

T. MIZUSHINA

106

Assuming that diffusion in the direction of flow is ignored, and in addition that the concentration boundary layer is so thin that the wall curvature is negligible, one obtains the following equation for mass balance on the diffusing species: u+-ac+ = _ a [(sc-l+ ax+ ay+

g]

+)

The boundary conditions are c+ = 0

at

1 c+ = 1

at

c+ =

at

x+

> 0, y+ = 0 y+ = a3

x+

(20)

lo00 agree with the solutions of the equation of turbulent flow using the total-viscosity formula of van Driest (43a). Wragg et al. (44) carried out a similar experiment with a flat vertical surface. Their experimental values of the mass transfer coefficients are a little smaller than the predictions of LCv&que'stheoretical equation. Vibrator and velocity transducer

1

voltage power

0-100 kn

FIG.43. Apparatus of Goren and Mani for the mass transfer from an artificially waved liquid layer.

b. Mass Transferfrom an Artificially Waved Liquid Layer. Goren and Mani (45) studied the effect of artificial standing waves of controlled amplitude and frequency on the steady state rate of mass transfer in thin horizontal liquid layers. They measured oxygen transferring through aqueous potassium hydroxide solution to a horizontal silver cathode at the bottom of liquid layer in the diffusion-controlled condition. A schematic view of the apparatus is shown in Fig. 43. The silver cathode in a Plexiglas trough has a porosity of 60-70 yo by volume. The trough was connected to another Plexiglas box which houses a nickel anode. The two electrodes should be separated because the oxygen liberated at the anode by the reverse reaction must be kept away from the cathode. They found that vibrations increased the transfer rate up to

T. MIZUSHINA

134

more than one order of magnitude. Their data at low frequencies are correlated with the following equation:

(i - i0)/(4Fcb/h)= &!UC$~'~T'''&~''

(48)

where ( i - i,) is the increase of electrical current to the cathode by making a wave, A is the surface area of the cathode, a is the amplitude, q5 is the frequency of the wave motion, r is the thickness of the liquid layer on the cathode, and B is the distance between blades of the wave generator.

c. Mass Transfer in Packed Beds. Jolls and Hanratty (46)studied the transition from laminar to turbulent flow in liquid flow through a packed bed of spheres of I-in. diameter. T h e transition was detected by oscillograms of the instantaneous local mass transfer rate to one of the packings. T h e test sphere was plated with nickel and fourteen isolated nickel cathodes were embedded in its surface so as to make a meridian circle around the sphere. T h e solution contained 0.01 mole of K,Fe(CN), and K,Fe(CN), , respectively, and 1 mole of NaOH per liter. Its Schmidt number was 1700. Keeping the main cathodes on the test sphere inactive, they measured the fluctuating mass transfer rates to the isolated cathodes and found that the transition occurred at Re = 110-150 where the Reynolds numbers were based on the diameter of the sphere and the fluid velocity in an empty column. They (46a) also correlated the space-time-averaged mass transfer coefficients to a single sphere in a packed bed by the following equations: Sh = 1.59 Re0eS6Sc1I3

for Re > 140

Sh = 1.44

for

Sc1I3

35

< Re < 140

(49)

(50)

These correlations are in good agreement with the experimental results of the heat transfer coefficients between air and a single sphere in a packed bed but they predict coefficients significantly larger than the experimental values of the mass transfer coefficients in a packed bed in which all of the packing are active in the mass transfer. This difference is easily explained by the fact that the thickness of the concentration boundary layer of a single active sphere is much less than that of a bed of many active spheres. Their measurements of the local mass transfer coefficients varied more around a meridian than on an equator owing to the difference of the variations of the boundary layer thickness. Ito et al. in Osaka University are going to study the mass transfer between packing solids and flowing liquid film in a packed column in

THEELECTROCHEMICAL METHOD

135

which nitrogen gas flows up. This experiment is of practical importance as a model of gas-liquid reaction in a catalyst-packed column.

d. Mass Transfer from a Jet Flow. T h e local mass transfer rates in a wall jet have been studied mainly by means of solid dissolution. But the electrolytic technique has a much larger advantage than the solid dissolution method in measuring the fluctuations of the local mass transfer rates. Kataoka et al. of Kobe University are studying the local mass transfer in a two-dimensional wall jet with a deposition reaction of Cuz+ ions. T h e experimental values of the Sherwood numbers are approximately constant in the impingement region and then decrease in proportion to the dimensionless distance from the stagnation point. Mizushina et al. of Kyoto University studied mass transfer from two-dimensional multiple impinging and sucking jets as shown in Fig. 44. This study was done for a simulation of mass transfer from a Taylor vortex flow which was described in Section 111, C, 4,a. T h e space-time-averaged mass transfer coefficients are correlated as follows: Sh = 0.24 Sc1l3 for lo2 < Re < lo4 (51) where Re

=

2 ~ , ( 7 D / S )B~/ vl ~

u, is the fluid velocity in nozzle,

D is the gap of nozzle, S is the distance between nozzle exit and wall, and B is the distance between impinging and sucking nozzles. T h e distribution of the local time-smoothed mass transfer coefficients

0'

0

5

I

1

I

10

15

20

Z(cm) Embedded ~ o i n tcathodes

FIG.44.

Multiple impinging and sucking jets.

T. MIZUSHINA

136

in laminar vortex flow is shown in Fig. 44.T h e flow pattern was detected by oscillograms of local fluctuating limiting currents to be laminar for Re < 200, laminar vortex for 200 < Re < 500, transition for 500 < Re < 1050 and turbulent for Re > 1050. Weder (47) measured the mass transfer rates from liquid to a horizontal plate around a single circular gas nozzle in the center of the plate. This study was carried out as a model of boiling heat transfer around a bubble evolved on a heated plate. Eleven concentric nickel ring-shaped cathodes are embedded around the gas nozzle to measure the local time-smoothed mass transfer coefficients as a function of the distance from the nozzle.

IV. Application to Shear Stress Measurements

A. PRINCIPLE AND METHOD OF MEASUREMENTS

It is very difficult to measure directly the velocity gradient close to a surface because the boundary layer thickness is so small that any measuring instrument disturbs the flow. However, a diffusion-controlled electrochemical reaction at small cathodes embedded in the wall can be applied to obtain the velocity gradient at the wall. If the test electrode is made quite small in length, the concentration boundary layer is very thin owing to a large value of the Schmidt number. Therefore, the curvature of the surface may be neglected, and it can be assumed that the velocity gradient in the concentration boundary layer is linear. Thus, the electrode is analogous to a constant-temperature hot-wire anemometer with the characteristics that the surface concentration is constant and the electrical current in the circuit depends on the surface shear stress. Furthermore, several limitations on both measurements are also similar. High frequency velocity fluctuations cannot be measured by either method owing to the thermal inertia of the wire of the anemometer and the capacitance effect of the concentration boundary layer over the electrode. Nonlinear response is caused in both systems by large turbulent intensities. I n addition, if there is nonuniform flow over the wire length or the electrode width, it may result in some error. Consider a rectangular cathode embedded in the surface with its short side parallel to the direction of flow. T h e length L of the cathode is much smaller than the width so as to make the concentration boundary layer two-dimensional. Assuming use of a redox system of ferro-ferricyanide, the mass balance for the ferricyanide ion gives aclat

+

ac/ax +

atlay

= 9 a2c/ap

(52)

THEELECTROCHEMICAL METHOD

137

where x and y are the coordinates in the direction of the flow and perpendicular to the surface, respectively. Boundary condition: c=O

at y = O ,

c =cb

at

y =

c=cb

at

x> 1, 2H,/Um2 = 0.05 [R. F. Probstein, Heat transfer in rarefied gas flow, in “Research in Heat Transfer” (J. A. Clark, ed.), 33-60. Macmillan (Pergamon), New York, 19631. y

=

GEORGE S. SPRINGER

200

and 10 for M, = 10. Thus, the transition regime appears to span about two decades in density.

2. Cylinders Although complete solutions do not exist for a cylinder in high speed flow, many excellent experimental data are available on the total heat transfer and recovery factor to cylinders normal to the flow. The data of Stalder, Goodwin, and Craeger (66), Sherman (116), Laufer and McClellan (117), Christiansen (118), Dewey (119). and Vrebalovich (120) are given in Figs. 16 and 17. Additional experimental data are available in (79,121-123). The Nusselt number, based on the cylinder diameter D (see Fig. 16) varies with Reom,n varying from 1 at low Reynolds numbers to 8 at high Reynolds numbers. This behavior is the same as for the stagnation point heat transfer to a blunt body. The data in Fig. 16 imply that the wire thermal accommodation coefficient was close to unity in all the experiments.

CONTINUUM 0

Y \ 0 t

II

0 3

z

w ( L m

I,z 5 w

v)

v)

3

FREE MOLECULE THEORY

z

0.4

(0

!02

REYNOLDS NUMBER, Reo=

PWUWD -

PO

FIG.16. Nusselt number for right circular cylinders in subsonic, supersonic and hypersonic flow of air. Key: (o)Vrebalovich, M, = 0.6-1.6; ( 0 ) Dewey, M m = 5.8; ( x)Laufer and McClellan, M, = 1.334.54;( .)Christiansen(cold wire), Mm = 0.8-1.2. [C. F. Dewey, Hot wire measurements in low Reynolds number hypersonic flows. ARS /. 31, No. 12, 1709-1718 (1963). T. Vrebalovich, Heat loss and recovery temperature of fine wires in transonic transition flow, in “Rarefied Gas Dynamics” (C. L. Brundin, ed.), Vol. 11, 1205-1220. Academic Press, New York, 1966.1

HEATTRANSFER IN RAREFIEDGASES’

201

4.4 1.2 1.0

0.0

> a

0.6

8

0.4

W

>

w

(L

0.2 0

-0.2

0.I

I

(0

INVERSE FREE STREAM KNUDSEN NUMBER, I / Kn=,

(00 D / ,X

FIG. 17. Normalized recovery factor for right circular cylinders in subsonic, superDewey, sonic, and hypersonic flow of air. Key: (0)Vrebalovich, Mm = 0.6-1.6; ).( Mm = 5.8; (+)Sherman, Mm= 2.0,4.0; ( X ) Stalder, Goodwin, and Craeger, Mm = 0.6-2.5; Laufer and McClellan (without end corr), Mm = 1.33-4.59; (+) Laufer and McClellan (with end corr), M m = 3.05. [C. F. Dewey, Hot wire measurements in low Reynolds number hypersonic flows. A R S J. 31, No. 12,1709-1718 (1963). T. Vrebalovich, Heat loss and recovery temperature of fine wires in transonic transition flow, in “Rarefied Gas Dynamics” (C. L. Brundin, ed.), Bol. 11, 1205-1220. Academic Press, New York, 1966.1

(m)

In Fig. 17 the variation of the normalized recovery ratio with the free stream Knudsen number based on the wire diameter is presented. Following Dewey (119) the normalized recovery ratio is defined as 66 r)* = (9 - ~ ~ ) / ( r )vC) ~ ~where is the measured” recovery ratio TJT,,, and vFM and qc are the free molecule and continuum recovery ratio’s, respectively. In reducing the data, yc was taken equal to 0.95 except for the experiments of Stalder et al. (66),where a value of 0.96 was used. From Fig. 17 it can be seen that the free molecule and continuum values are effectively reached at Knudsen numbers of 10 and 0.1. These Knudsen numbers are very close to those suggested in the previous section as the free molecule and continuum limits for the blunt body stagnation point heat transfer.

202

GEORGE S. SPRINGER

For a perfect gas the heat transfer to the stagnation point of a cylinder can be approximated by (224)

For the distribution of the heat transfer around the cylinder, Tewfik and Giedt (125) suggest the following empirical formula q(q)/qo = 0.37

+ 0.48 cos p + 0.15 cos 2q

(44)

where is the angle measured from the forward stagnation point where the heat transfer is qo . Equation (44) agrees with the data within about -6% for M = 1.3-5.7.

3 . Spheres Relatively little experimental data have been reported on the heat transfer to spheres. The most likely reason for this is that it is more difficult to make accurate measurements with spheres than with cylinders. In Figs. 18 a summary is given of the Nusselt numbers Nu, based on the experiments of Drake and Becker (126) and Eberly (127); Nu,. has been computed from the measured heat transfer coefficient, the sphere diameter, and the thermal conductivity evaluated at the recovery temperature. In Fig. 18 the results of continuum theory are also indicated (128).The data tend towards the correct continuum limit. Since the lowest free stream Reynolds number for which data are available is about 50, one cannot evaluate how the heat transfer behaves at lower Reynolds numbers and particularly at near free molecule flow conditions. The recovery factors reported by Drake and Becker and Eberly are shown in Fig. 19. The continuum value of the recovery factor is reached at about Re, = 100. Using Eq. (1) we find the corresponding free stream Knudsen number to be about 0.075 for M, = 5. This value is of the same order of magnitude as the one obtained for cylinders.

4. Sharp Leading Edge Flat Plates For flows over semi-infinite flat plates the appropriate characteristic length is the distance measured from the leading edge. At large distances downstream there is a Prandtl type boundary layer near the plate surface. At hypersonic speeds there is also an inviscid shock over the plate caused by the deflection of the inviscid flow due to the boundary layer. T h e strength of this shock increases towards the leading edge resulting at

HEATTRANSFER IN RAREFIED GASES

'yI"

10

r II

3-

z W n

4: -

-

20

203

--

CONTINUUM THEORY, M = O (DRAKE,SAUER AND SCHAAF)

y 2 w o

m

@

AND

-

FREE MOLECULE THEORY

( a = !I

I

I

0.i

I

I

I

1

1

1

1

io5

iOL

40

i

REYNOLDS NUMBER BEHIND SHOCK Re, =p,U,D/p,

20

I

1

1

1

I

1

1

-

1

"

I

I

be.

10 6 8-

I

I

e e

>&H

,og%

4-

-

NF

4,a

24

-

">-//6'% I

I

, I

I

--

I

I l l

I

I

I

1

FIG. 18. Supersonic and hypersonic flow of air past spheres. The variation of Nusselt number with free stream Reynolds number and with Reynolds number based on conditions behind a normal shock. Data from Eberly: ( 0 ) M, = 3.39-4.05; ( 0 ) M m = 5.44-6.01. (---) Drake and Becker(best fit to data): @ Ma = 2.70-3.22; @ M m = 3.28-4.09. [D. K. Elberly, Forced Convection Heat Transfer from Spheres to a Rarefied Gas. Engineering Research Rept. HE-1 50-140, Univ. of California, Berkeley, California, 1956.1

first in weak and further on in strong interactions between the viscous and inviscid flows (3). Further upstream from the strong interaction regime, the shock wave thickens and merges with the boundary layer. I n this regime, termed the merged layer regime, it is still possible to describe the flow by continuum type analyses. Moving further towards the leading edge one encounters the transition, near free molecule, and free molecule regimes, as shown in Fig. 13. Most previous investigations of the heat transfer to a flat plate have used the continuum Navier-Stokes equations as a model. I n the boundary

GEORGE S. SPRINGER

204 1.2

I

I

I 1 1

I

I

I

l

l

I

I

I

I

-

-

>

B >

8

-

-

-

0.9

-

r

E

0.8

40

I

I

I l l

I

I

I

l

402 FREE STREAM REYNOLDS NUMBER R%=-

l

I

403 P U D

I

I

1

104

Po3

FIG.19. Recovery factor for spheres in supersonic and hypersonic flow of air. Key: MCCs 3.4-6.0; (-) Drake and Becker, M g 2.7-4.1 (best fit to data).

(0) Eberly,

[D. K. Eberly, Forced Convection Heat Transfer from Spheres to a Rarefied Gas. Engineering Research Rept. HE- 150- 140, Univ. of California, Berkeley, California, 1956.3

layer regime, the heat transfer to a highly cooled plate in hypersonic flow may be approximated by the expression (64)

where qFM = 0.5p,U,2 (a = 1). Here, ReOzis defined in the same way as in Eq. (41) except that here the characteristic length is x, the distance from the leading edge. Equation (45) is based on the assumption that the viscosity varies as the square root of the temperature. In the strong interaction viscous layer regime, Oguchi presented a method for calculating heat transfer both with (129) and without (130) the effects of temperature jump and slip at the surface. Oguchi’s model, which assumes a fully viscous shock layer bounded by a thin shock wave satisfying the Hugoniot conditions, was applied to the flat plate problem also by Street (131) and by Jain and Li (132). Pan and Probstein (133) analyzed the merged layer by assuming that the density ratio across the shock is small and that there is only moderate thickening of the shock. T h e experiments of McCroskey et al. (134), Harbour and Lewis (135), and Becker and Boylan (136)indicate that the results of these calculations are somewhat in error. Oguchi (137) analyzed the problem again without assuming a small shock-density ratio and a small shock thickness but taking the shock to be locally straight, i.e., he treated the shock as onedimensional. Recently, Shorenstein and Probstein (238) extended

HEATTRANSFER IN RAREFIEDGASES

205

Oguchi's calculations by considering the shock to be locally circular. Shorenstein and Probstein obtained numerical results for the heat transfer for y = 1.4, M = 10-25, Tw/To= 0.05-0.2, and found that the following expression approximates the heat transfer within at least 5 yo St/St,, = &[l - tanh(0.91 log,, Kn*

+ l.lO)]

for

Kn* < 0.1 (46)

T h e subscript SI denotes the strong interaction solution given by St,,

+ 0.0684)[M,(C,/Re,,)1/2]3/2 (47) - H J ] , and K n * = ( Tw/T0)1~2M,2C,/Re,, .

= (0.368T,/T0

where St = -q[p,U,(H, Here His the enthalpy, C, the proportionality constant in the relationship between temperature and density, and Re,, is based on p,, U , ,p: and X. T h e foregoing formula and also the exact numerical solutions of Shorenstein and Probstein's analysis are compared to the data of Vidal and Bartz (139) in Fig. 20. T h e good agreement between the data and the analytical results suggests that this analysis describes the heat O( 1). transfer satisfactorily up to M,(C,/Re,,)l12

-

FIG. 20. Stanton number for hypersonic flow over a sharp leading edge flat piate. Zero angle of attack. Key: Analytical results of Shorenstein and Probstein, (---) T,/Tn = 0.20, (--.--.-) TWITn= 0.05, numerical solution, merged layer regime; (-) Correlation formula. Data: (1 11 I) Vidal and Bartz, Mm = 19.2-22.4, Re,,/inch = 32G9000, TJT, = 0.059-0.074, y = 1.4. Sts, = (0.368 TWITo 0.0684) [Mm(Cm/RemJ'la]l"~a. [M. L. Shorenstein and R. F. Probstein, The hypersonic leading edge problem. AIAA J. 6, NO.10, 1898-1906 (1968.1

+

206

GEORGE S. SPRINGER

It is worth noting here that the calculations of Pan and Probstein (133) show that the hypersonic heat transfer to a flat plate can be greater than the free molecule value. T h e calculations of Charwat (140) also indicate this effect. There are no experimental data available on flat plates that would substantiate this result, but this phenomenon has been observed in measurements on pointed cones (141) (see the next Section). It can be also shown (64) that for the flow past a flat plate the Nusselt number varies as Re& ,n going from 1 to i,as in the case of heat transfer to the stagnation point of blunt bodies (Fig. 15) and to cylinders (Fig. 16). 5. Cones Similarly as in the case of flat plates the flow along a pointed cone may vary from free molecular at the cone vertex to boundary layer type at large distances downstream from the vertex of the cone. As the Reynolds number decreases, the ratio of boundary layer thickness to body radius (S/R,) increases and the interaction of the boundary layer with the flow becomes significant. Spreading of the boundary layer (transverse curvature effect) also influences the flow field around the cone. T h e flow past a slender, pointed cone was studied in the weak to moderate interaction regime by Probstein (142), Probstein and Elliott (143), Yasuhara (144), Nikolayev (145), and Mirels and Ellinwood (146) and in the strong interaction regime by Stewartson (147), Solomon (144, and Ellinwood and Mirels (149). These analyses are for the condition S/Rw < 1. Analytical results for the fat (i.e., nonslender) cone in the incipient merged layer regime were reported by Cheng (91), and Waldron (150). T h e latter also considered some of the effects neglected by Cheng, namely the effects of transverse curvature, shock curvature, viscous layer displacement, shock angle different from body angle, and surface slip. T h e results of the aforementioned analyses are compared in Fig. 21 to the data obtained by Wilkinson and Harrington (151), and by Waldron (150) in air, and in helium by Horstman and Kussoy (152). All these data are for zero angle of attack. It can be seen that at small values of the rarefaction parameter (M,C,/Re,,)l12/sin2 0, both Cheng’s and Probstein and Elliott’s analytical results agree well with the data of Waldron, and Wilkinson and Harrington. For the ranges of variables in these experiments (15 < M, < 25, 15 < Re,,/in. < 25000, 8, < 20” and TJT,E 0.1) Yasuhara’s, Stewartson’s and Probstein and Elliott’s results are quite close (150) and, therefore, the former are not shown separately in Fig. 21. At higher degrees of rarefaction Cheng’s results, although qualitatively correct, overestimate the heat transfer by about

HEATTRANSFER IN RAREFIED GASES

207

I 1_1

FIG. 21. Stanton number for hypersonic flow past cones at zero angle of attack. (St, is base on T, - Tw).Data: ( 0 ) Horstman and Kussoy (He), M, = 41, Rem,/in. = 5600, T,,,/T,,= 0.35; (0) Waldron (Air), Mm= 19-24 Rem,/in. = 150-4000, T w / T o g O . I ; ( 0 ) Wilkinson and Harrington (Air), Mm = 15-20, Rem,/in. = 4000-19,000, T,/To = 0.1 (0 = 6.3 and 9"). Analyses: @) Probstein and Elliot (viscous layer, transverse curvature); @ Cheng (viscous layer); Waldron (viscous layer, including slip); @ Ellinwood and Mirels (strong interaction); @ Mirels and Ellinwood (weak interaction, similarity solution).

25 yo. Waldron's analysis, which includes some of the effects neglected by Cheng agrees well with the data. T h e agreement is less good between Horstman and KUSSOY'S data and the analytical results of Mirels and Ellinwood (146). A comparison between various measured heat transfer values is given in Fig. 22. This figure serves to illustrate two main points. First, M 0.6) note that up to a rather high degree of rarefaction (M,/(Re,,)1/2

208

GEORGE S. SPRINGER

L

4

-

-

u

-

,

'

G i

I

I

10.~

I

, I

c

I

o-~

I

10-2

Re;,

10.1

11'~~1111,,1,

too

an48
-

-

I

c

-.i L

\ c

40x10-3

9 0.10

0.05

- 10

-

0.

- 25x i0-3

I

I

I

I

I

1

0

-5xlO-3

-0

HEAT LOAD ( B t u / h r )

FIG.46. Calculated heat pipe parameters versus heat transport rate. [J. Schwartz, Performance map of the water heat pipe and the phenomenon of noncondensible gas generation.Presented at the ASME-AIChE Heat Transfer Conf., Minneapolis,Minnesota, Aug. 3-6, 1969, Paper No. 69-NT-15.1

Werner (118) has employed Cotter’s pressure balance, Eq. (29), for Re, 1, and obtained in a computer calculation the axial heat flux and Mach number as a function of length for a 1500°K lithium heat pipe with rP = 0.5 cm and r, = 0.4cm. The results of these computations are shown in Fig. 47 illustrating the high velocities and fluxes which may be obtained in a high temperature heat pipe. Figure 47, of course, is based on the equations for a capillary limited heat pipe and thus the flux rates may well exceed other heat pipe limits such as the entrainment or boiling limit. Werner and Carlson (105) also modified

>

THEHEATPIPE

I

0

I

40 80 120 160 LENGTH OF HEAT PIPE (cm)

295

1

200

FIG.47. Calculated axial heat flux and Mach number versus length for lithium heat pipe at 1500°K. [R. W. Werner, The Generation and Recovery of Tritium in Thermonuclear Reactor Blankets Using Heat Pipes. Lawrence Radiation Lab., Univ. of California, Livermore, UCID-15390, Oct. 3, 1968.1

Eq. (29) for the special case of a heat pipe with a capillary structure consisting of axial grooves covered by a single layer of screen material. If the intervening wall between grooves is negligibly thin at the inner radius, then the pressure drop in the liquid may be written as APi = pi gl cos B

+

(3p1$?1/4(Yw

- y v ) ~hegpirvr2)

(71)

where (Y, - Y,) is the groove depth and rc is the capillary radius (groove half-width). Following a procedure identical to Cotter’s and assuming no gravity effect, Werner inserted Eq. (71) into Eq. (29) and arrived at the following expressions for the optimum capillary radius and the maximum heat transfer rate: (72)

and

(73)

where A = (1

- 4/n2)/8pvr>hfg

,

B

= 3p11/4n(rW- rv) hfgplrv ,

C = 2a cos B

In addition, he found that the maximum axial heat flux is obtained when Y , / Y , = 516 a value approximately 2.3% higher than the one Notice that Werner assumed the same size found by Cotter (+)l/” for the capillary radius and the groove half-width. Consequently the screen mesh size and the groove width cannot be selected independently of one another. Hampel and Koopman (32) solved this by decoupling

E. R. F. WINTERAND W. 0.BARSCH

296

the channel dimension from the pressure supporting mesh dimension. The liquid flow in the channels was treated simply as flow through an annulus and corrected adequately to account for the actual area available to the flow. In addition, a scaling factor a was introduced to account for the additional pressure drop induced by the channel configuration. For the case of Re, 1 and zero gravity effect, Hampel and Koopman derived an equation similar to Eq. (29):

>

(1 - 4/+

Q2

sPvqh;g

+ ePphfgrv(rw-

< 2a cos e l l

2+iQ1

yv>[yw2

+

- (y,*

- r;/1n

(ywbvN1

(74)

where d is the screen mesh opening half-width and e is the ratio of the active channel area to the total circumferential area of r , . By differentiating Eq. (74) with respect to Y', the resulting ratio, yV/rw, was found to no longer be a constant, i.e., (g)lI8or 9, but to depend on the heat flux, the operating temperature, and the thermodynamic properties of the working fluid. Typical values for this ratio, are illustrated in Fig. 48 for a high temperature lithium heat pipe. Anand et aZ. (14) and Anand (96) employed advantageously, Eq. (36) as derived by Cotter. The total length Z was replaced by (Z, Z,) and the condenser length 1, was replaced by Qe/C where C is a condenser parameter defined by

+

C = 2?Tkw(Tv- Tw)/ln(rP/yv) 0.86

0.82

--LITHIUM z ~ 2 cm 0 z c = 40 cm :

--rm=0.6cm

o,,

'JrW:

(75)

516

RADIAL HEAT FLUX (W/cm2)

FIG.48. Optimized mlationship between vapor radius and wall radius for lithium heat pipe. p.E. Hampel and R. P. Koopman, Reactivity Self-Control on Power and Temperature in Reactors Cooled by Heat Pipes. Lawrence Radiation Lab., Univ. of California, Livermore, UCRL-71198, Nov. 1, 1968.1

THEHEATPIPE

297

This procedure led to a quadratic term in Q with the solution Qopt =

[)Cm,3hga

+

~(#pvpi~/pvpib)1’2

&‘2~e2]’/2

- 4Cle

(76)

Equation (76) illustrates how a restraint on the axial heat transport capability is imposed by the radial heat flow in the condenser section. The condenser parameter C may be varied by wick flooding, introduction of noncondensable gases, or by manually changing the surface area. Equation (76) reveals the possible applicability of these techniques for heat pipe control. Brosens (110) started with Eq. (19) and considered both the liquid and the vapor flows as laminar, steady and incompressible, i.e., Poiseuille flow. The wick structure was thought to consist of n cylindrical capillary tubes and pressure equality was assumed at the far end of the condenser. For the case of zero gravity and perfect wetting of the wick, the expression for the maximum heat transfer was found to be

The same result was independently arrived at by Schlinder and Wassner (124). Brosens optimized Eq. (77) with respect to the capillary radius and obtained Tc,opt = c & / w ) ~ ’ ~ (78) from which resulted Qmax.opt = 3~~v4htg/161vvrc,opt

(79)

The values predicted by Eqs. (77) and (79) tend to be much larger than those measured experimentally. The deviations are caused by the many simplifying assumptions made during the derivation of these equations. Frank et al. (30) assumed Poiseuille flow for the liquid and a modified Poiseuille flow for the vapor and found the vapor pressure drop APv

= @v(8pv~~~v/~rv*pv)

(80)

where If is the effective flow length and extends from the mid-point of the evaporator to the midpoint of the condenser, and the function @ is given by @

-

v -

Notice that

QV

1’0.00494 ;

Re3l4;

Re Re

< 2200

> 2200

(81)

is a discontinuous function at Re = 2200 (by approxi-

298

E. R. F. WINTERAND W. 0. BARSCH

mately SO%, so some reservations as to its applicability in this range are indeed justified. Frank made use of Eq. (80) to derive an expression for the maximum heat transfer capability of a horizontal heat pipe: Qmax = &(acF~/32)(D?/rh)N

where and

+ (rilya))~ 0 6s [I + @v(~v/~l)(Dl/2rv)a (&/nrva)I-'

a = (1

FL

=

(82) (83)

(84)

D, in the preceding expression is the hydraulic diameter of the capillary pores. Frank (26) next discussed the application of Eq. (82) for a grooved heat pipe and by assuming perfect wetting, i.e., a = 1, and by replacing E by ALIA,, obtained for maximum heat flow Qmax = F L D ~ ' A L N / ~ ~ ~ V

(85)

Referring to Fig. 49, a number of auxiliary variables are defined. T h e mean radius of the grooves is Y,,, = yW - S/2 = r,

+ S/2

(86)

while the dimensionless pitch of the grooves is given by

p

= (w

+ wu')lw

(87)

and the number of grooves is n

= 2~rm/lgw

(88)

T h e aspect ratio of the grooves is defined by 01

= sjw

(89)

FIG.49. Sketch illustrating design variables in grooved heat pipe. [S. Frank el al., Heat pipe design manual. Martin Nuclear Report MND-3288, Martin Marietta Corp., Baltimore, Maryland, Feb. 1967.1

THEHEATPIPE

299

and finally a depth ratio is given as

t+lJ

(90)

= @/rm

By inserting Eqs. (86)-(90) into (85) the following expression is obtained: Qmax/rw3 = (8C/(l

where

+ S ) ) . (mt+lJ'/(l + 201)' (1 + 4 ~ ) ~ )

C

and

(92)

= nN/Plf

s = (l/FL)

(91)

-1

(93)

Equation (91) was next extremized to yield the optimum value of and if a bar is used to denote the optimum value, then the result is

QmaX/yw3

Qjmaxlrw' =

C(p/(1

+ $)'),[(lo$ - 1 - ?)/( 1 + 7$)]

(94)

where $ is given implicitly by: [(I

-

$I5

(2 - $)(I

+ 7$)/@(10$

-

1

-

?I2]

=

(16/P)(vv/vdGv

(95)

and & is given by

cu

=

(3

+ $)(1 + $)/2(10$

-

1 - p)

(96)

In addition, since the optimum heat flow generally occurs when the vapor flow is turbulent, Ov may be written as @v = 0.00494 Re3l4 = O.O0494[(2/nk,,clv)(Qmax/r,)(

1

+ $)/( 1 - #)I3//"

(97)

T h e solutions of Eqs. (94)-(97) were obtained by Frank and are reproduced in Fig. 50-53, respectively. T h e procedure for optimizing the ratio Qmax/rw3is an iterative one. First, the effective flow length If the operating temperature, the working fluid, and the pitch p are selected. T h e pitch has a minimum value of unity and it is advisable to make it as small as possible. I n general, the minimum value of the pitch will be imposed by machining and strength requirements. Once these values have been selected, an initial value of Qjv = 1 is assumed and $ may be found from Fig. 52. Using this value of $, the optimum value of Qmax/rw3can be taken from Fig. 50 and Ov from Fig. 53. If QjV I, the vapor flow is laminar and no further iterations are required. If, on the other hand, Qjv > 1 , then the prevailing vapor flow regime is turbulent and the procedure should be repeated using the new value of Ov to enter Fig. 52. T h e optimum value of OT can be found from




+

(102)

Notice that for Re, 1, McKinney worked with the Poiseuille flow assumption for the vapor in the adiabatic pipe section whereas Haskin assumed fully developed turbulent flow for the evaluation of the pressure drop. For intermediate axial Reynolds numbers, therefore, these two techniques may both be employed to encompass the actual pressure drop and the resultant heat transfer limit. McKinney with the aid of Eqs. (98) and (100) developed a computer program for the graphical display of Qmax versus the ratio rv/y, for various adiabatic lengths, permeabilities, wetting angles, and temperatures. I n all cases, the maximum heat transfer tended to maximize in the neighborhood of yV/yw = 0.3. T h e reason for the large deviation of his value from the values obtained by Cotter and Werner, i.e., and $, respectively, cannot be explained without a more detailed study of the problem. In all of the preceding analyses the pressure drop in the vapor space in one way or another was considered. Since for low temperature heat pipes this pressure drop may generally be assumed negligible, a quick estimate of the capillary limited heat flux can be obtained by regarding only the pressure drop in the liquid. Several authors whose results are discussed below have made use of this simplification. Phillips (29) and Phillips and Hindermann (53)applied Darcy's law to the liquid flow and 21, I,) and obtained arrived at Eq. (25) where I is replaced by (I, for 1, = 0:

(g)1/2

+

+

Qmax = 4 [ ~ h r g / v l l ( K ~ w / r c ~ ) 2[plhfg/vll

KAW g cos P

(103)

In addition, they recommended the use of the following expression for heat pipes which employ a bypass or arterial type of wick:

+ +

4 = ( 8 ~ ~ Q / A a ~ h g ) [ l 2 kla &] (nvl(rv 4-rw)'Q/16Krv(rw - yv) h g ) ( ( &

+

+ le)/(& - re))

(104)

T h e first term in Eq. (104) represents the pressure drop in the artery while the second term pertains to the pressure drop associated with the flow of liquid to and from the artery in the circumferential direction. It is noteworthy that the effective length utilized in the pressure drop &. and that I, = 0, calculation Eq. (25), is essentially +le 1, which reduces it to half of the total heat pipe length. Feldman (8) and

+ +

304

E. R. F. WINTER AND W. 0. BARSCH

Streckert and Chato (125, 126) have used the entire heat pipe length in their pressure drop calculations; consequently their resultant maximum heat transfer rate is one half of that calculated with Eq. (103). A similar analysis has been performed by Neal (28). Langston and Kunz (13,55) considered mass, momentum, and energy balances based on an elemental thickness of wick in the condenser. By assuming an infinite meniscus radius at the far end of the condenser and a minimum value at the condense-evaporator interface, they obtained the limiting heat flux

where the minimum meniscus radius has been evaluated from rmin = (2gou/p1ghmax)wa

(105)

(106)

T h e subscript WR refers to the temperature conditions at which the maximum wicking rise is measured. Implicit in the derivation of Eq. (105) is the assumption that E = 1. Cosgrove et al. (37, 38) extended the analysis of Langston and Kunz to include the effect caused by an adiabatic heat pipe section and the temperature variation of the fluid properties. The maximum heat transfer is then given by

where T, is the temperature of the adiabatic section and is equal to the saturation temperature of the fluid, and T , is the condenser temperature and is taken as the average of the saturation and sink temperatures. Notice that for the special case of I, = 0, Eq. (107) reduces to Eq. (105). The above discussion has dealt solely with the capillary limit to heat pipe operation. As mentioned earlier this type of limit is especially important for low temperature applications where relatively low vapor velocities and heat fluxes prevail. For high vapor velocities, on the other hand, the sonic and entrainment limits become important. Levy (127) performed a one-dimensional compressible vapor flow analysis on a control volume basis restricted to the vapor space. Two models were used to relate the thermodynamic properties in the vapor. First by treating the vapor as a perfect gas, the sonic limiting heat transfer rate was found to be

+

Qmax = pv77yv2i7ah~g/(2(K

(108)

THEHEATPIPE

305

This condition is reached when the vapor flow chokes at the downstream side of the evaporator. The second model described a single-component equilibrium two-phase saturated vapor and the analysis to which it was applied yielded a complex trancendental equation for the limiting heat transfer rate. Equation (108) therefore may be used to obtain the theoretical limiting curves which were illustrated earlier in Fig. 25. Levy compared the limiting heat transfer rates predicted with both models for a particular sodium heat pipe. These limiting rates are illustrated in Fig. 54 as a function of temperature. Curve A was obtained with Eq. (108) while curves B and C represent the two phase model solutions. Curve B was calculated using the temperature at the upstream end of the evaporator and curve C was obtained using the temperature at the downstream end. Also displayed are the wicking limit derived from Cotter’s fundamental equations and experimental data provided by Kemme (85). Relatively good agreement is discernible between the sonic limiting curves and the experimental data for temperatures less than 600°C. Above that temperature the sonic limit curve greatly overpredicts the measured maximum heat transfer rates and, in fact,

-5 6000 3

9 W

I-

a

a a

v,

z

a

a !-

tW X

t

4000-

-

2 2000-

r

3

2

P

1,

I

0‘500

1

1

I

700 800 TEMPERATURE.T (“C)

600

I

900

FIG. 54. Comparison of perfect gas model and two phase model for sonic limit to experimental data. Sodium heat pipe: (-) data from Reference 85, (----) incomplete perfect gas model,(----)two phase rnodel;d= 1.5cm, theory,wick limit (85), (---) le = 8 cm, It = 30 cm. [E. K. Levy, Theoretical investigation of heat pipes operating at low vapor pressures. “Aviation and Space: Progress and Prospects Annual Aviation and Space Conference, June 1968,” pp. 671-6.1

306

E. R. F. WINTERAND W. 0.BARSCH

the measured rates were probably limited by the pumping ability of the capillary system. It becomes also apparent from the agreement between curves A and B that the perfect gas model (Eq. (108)) is useful to estimate the heat transfer rate required to achieve choking within the evaporator section. The theory of the entrainment limit has received little attention, the reason being the dependency of this limit on the details of the geometry and the interfacial shear stress distribution. Cotter (19) and Kemme (86) claim that, for Weber numbers greater than unity, the possibility of entrainment exists. The Weber number is the ratio of inertial force to surface tension force written as Weber number

= pvVv21'/a

(109)

where I' is a characteristic dimension associated with the wick surface. An estimate of the entrainment limited heat flux may be established by equating the Weber number to unity. This assumption, together with the energy equation (Eq. (19)), yields Kemme claimed that for screen wicks, the characteristic length, Z', is very nearly equal to the screen wire diameter and that it probably depends to some extent on the wire spacing. The heat transfer limit associated with boiling within the wick did not receive much attention either. This type of limit is difficult to predict since it requires, among other properties, a thorough knowledge of the cavity dimensions in the wick and of the effective thermal conductivity of the saturated wick. The boiling limit was illustrated in Fig. 7 and has to be considered as qualitative in nature according to Deverall (18). Notice that the limiting heat flux decreases with increasing temperature. Neal (28) took the superheat which is necessary for the incipience of nucleation in the wicking into consideration and related the superheat to the temperature difference existing across the wick in the evaporator and obtained for the boiling limited heat flux: Qmax =

~

2771ekw UTgat 2 cos 8, ln rw/rv pvhfg -

[x

(111)

Marcus (128) contributed yet another relation for the boiling heat transfer limit

THEHEATPIPE

307

He further recommended to evaluate the effective wick conductivity with kw = 4 (1 - E) k,,,, (113)

+

Obviously the above expressions are quite different from the qualitative limit given by Deverall since both Neal and Marcus conclude that the limiting heat transfer increases with increasing vapor temperature. Hence the boiling limit curve, depicted in Fig. 7, should have a positive slope instead of a negative one. A great deal more experimental and theoretical effort must be expended before the boiling limit can be treated with sufficient confidence. The foregoing discussions pertained to the numerous predictions of the maximum heat transport capability of heat pipes in view of the wicking, sonic, entrainment, and boiling limit, respectively. Only a few analyses have been undertaken on other aspects of heat pipe technology. Lyman and Huang (129) studied the problem of two-dimensional liquid flow and heat conduction within the wick near the condenser entrance. Assuming constant pressure and a constant rate of condensation in the condenser they computed the temperature distribution in the wick. The results of the analysis are displayed in Fig. 55 and 56. The numbers 0

2

3

isotherms and FIG. 55. Isothermal and adiabatic curves in condenser wick: (-) - -) adiabats. [F. A. Lyman and Y. S. Huang, Analysis of temperature distributions in heat pipe wicks. Presented at the ASME-AIChE Heat Transfer Conf., Minneapolis, Minnesota, August 3-6, 1969, Paper No. 69-HT-23.1 (-

on the isotherms represent the temperature above the coolant temperature T , in units of (Qlbk,) while the numbers on the adiabatic curves give the fraction of the heat flow Q which passes through the portion of the wick to the left of the curve. Figure 56 illustrates the dimensionless temperature distribution in the midplane and at the surface of the wick. For wick matrices of low conductivity large temperature gradients are possible in the wick at the junction between the adiabatic section and

308

E. R. F. WINTERAND W. 0. BARSCH

the condenser. Such temperature gradients have been observed qualitatively in several experiments with low temperature water heat pipes. Unfortunately in general, the thermocouples were not placed sufficiently close to accurately verify the steepness of the gradient. It should be emphasized that this analytical solution is applicable only to condensers with a fixed temperature boundary condition, e.g., calorimeter cooling. It is also evident that major condensation and the associated heat flow into the wick occur very near the entrance to the condenser section.

TEMPERATURE AT SURFACE OF WICK

TEMPERATURE AT MIDPLANE OF WICK

02

X

-2

-1

0

2

3

'W

FIG. 56. Temperature distributions in midplane and surface of wick. [F. A. Lyman and Y. S. Huang, Analysis of temperature distributions in heat pipe wicks. Presented at the ASME-AIChe Heat Transfer Conf., Minneapolis, Minnesota, August 3-6, 1969, Paper No. 69-HT-23.1

The effective wick conductivity was calculated by assuming parallel heat conduction through the liquid and wick material as described by Eq. (113). Gorring and Churchill (130) and Nissan et al. (131) have suggested various techniques for the measurement and the computation of the effective thermal conductivity of other wick materials. Bressler and Wyatt (132)solved the differential equation for the velocity during the transient capillary rise of a liquid in grooves of various geometries. T h e mean velocity is plotted in terms of dimensionless groups shown in Fig. 57. The constant C equals [2 cscs(p/2) - 2 csc2(p/2)], [2], and [S(n - 2 ) / 4 and D is [-csc2(p/2)], [2], and [2], for triangular,

THEHEATPIPE

309

semicircular, and square cross section grooves, respectively. For a given groove geometry, fluid properties, and temperature difference between the wall and liquid surface, ( T , - TB),the mean velocity V may be determined from Fig. 57. T h e total heat flux at steady state can then be calculated from

Q

= p1cALhg

(114)

Bressler and Wyatt used the method to investigate the effect of groove geometry on the maximum heat transfer. They found that a vertex angle of approximately 30" led to the highest heat transfer rates among

FIG. 57. Calculated mean velocity in triangular, semicircular, and square grooves. [R. G . Bressler and P. W. Wyatt, Surface wetting through capillary grooves. Presented at the ASME-AIChE Heat Transfer Conf., Minneapolis, Minnesota, August 3-6, 1969. AIChE, Preprint 19.1

triangular grooves. Furthermore, it was found that square grooves are characterized by the highest heat transfer rates per unit groove width when all grooves have an optimized depth. T h e results were different, however, when all grooves were compared at the same depth. Many more comparisons of groove characteristics may be made depending on the specific requirements to be evaluated. I n another study, Galowin and Barker (133)employed a two parameter, fourth order velocity profile in the Karman-Pohlhausen boundary layer integral method to determine velocity and pressure fields in a twodimensional heat pipe. T h e vapor was assumed to be incompressible while the injection and suction velocities at the wick surface were considered as small. For the case of I, = E, = L and uniform injection

E. R. F. WINTERAND W. 0. BARSCH

310

and suction rates with a velocity V , , the velocity and pressure distributions were found to be: For 0 ,< z

< L: VItl(Z)

For L

= (3 VwI2)(~/R)(4L)

(115)

P(z) - P(0) = ( -3pvVw/2R)(L/R)2 ( z / L ) ~

(116)

V m ( 4 = (3Vw/2)(L/R)(2 - z/L)

(117)

< z < 2L:

P(z) - P(0) = (3pvVw/2R)(L/R)'[(zlL)' - 4(x/L)

+ 21

(118)

Approximate solutions were also obtained for the case where the injection velocity obeys a ramp function. Miller and Holm (134) considered the possibility of using model heat pipes to predict the performance of different prototype heat pipes. A material preservation scheme was employed in which the same working fluid and wick material was used in both the model and the prototype. This implies that the thermal conductivities of the wall and wick, the emittance of the condenser surface, and the permeability of the wick have to be preserved. With a starred quantity representing the model to prototype ratio of that quantity, the modeling equations are (T, - To)* = q*/z*

(119)

and Experimental verification of the above equations showed that prototype thermal behavior could be predicted from the model behavior to within 10°F over a temperature range of 140-330°F for a pair of water heat pipes. Modeling equations for another scheme which preserves the heat flux from model to prototype were also presented but an experimental verification was not attempted.

V. Summary A comprehensive literature collection composed of publications, papers presented at meetings and reports of varying nature, which appeared during the period from 1964 through midyear 1970 on heat pipe technology and on related topics, is classified and evaluated.

THEHEATPIPE

31 1

Although it is to be expected that Russian heat pipe publications may well exist, none were found in the common literature indexing systems, Generally the six interdependent processes which are assumed to effect the functioning of the heat pipe in the so-called heat pipe regime are in agreement with most qualitative observations. However, the evaporation mechanism, in particular, which is commonly regarded to take place at the liquid-wick vapor interface is the subject of some controversy. Additional investigations utilizing everted, or perhaps coplanar heat pipes with partly transparent containers are recommended for direct observation of evaporation. Moreover, the modes of energy transport in the liquid-wick matrix in both the condenser and evaporator are uncertain for low thermal conductivity fluids and further experimental evidence is needed to clarify whether conduction in the wick structure is indeed the dominant mode of energy transfer. T h e possible contribution of convective currents within the pores to the total heat transfer should be examined. Since the efficiency of the heat pipe depends largely upon the shape of the meniscus in the evaporator, the functional relationship between the heat flux and the meniscus shape together with its position in the wick matrix must be investigated. T h e operating conditions of a heat pipe are determined by the type of boundary conditions imposed on the surface of the condenser and evaporator. T h e simplified presentation of the influence which these constraints exert upon the operation of a heat pipe can be considerably expanded by employing more realistic assumptions. T h e case of floating temperature conditions for both the sink and source, especially needs exploration. Discrepancies between theoretical predictions and experimental observations of the four operating limits were found. T h e deviation was notably severe for the case of the boiling limit. Further work on theoretical predictions and experimental verification of the operating limits is required for the successful design of heat pipes. T h e influence of noncondensable gases on heat pipe operation has been discussed at length and must be dealt with as it occurs in a given heat pipe. A theoretical study of the transient behavior of heat pipes supported by transient experiments would enhance the understanding of their startup behavior. An abundance of data on heat pipe materials including working fluid properties, wick-fluid interactions, and material compatibility was generated in the course of a sizeable number of research programs; but little effort was spent on a systematic classification and evaluation

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E. R. F. WINTERAND W. 0. BARSCH

of materials in view of their potential for heat pipe application. In order to avoid further duplication of effort and expense, a coordinated materials research program is essential. Because the functioning of heat pipes is effected by the surface tension characteristics of fluids, attempts should be made to alter favorably these characteristics in fluids for use in the intermediate temperature range (200-600"C). Little is known about the applicability of fluid mixtures for heat pipe use. Separation of the working fluid into its components may occur and should be investigated. Some of the heat pipe applications may have been suggested in view of eventual government funding of extensive research programs; nevertheless, the heat pipe promises potential solutions to problems of temperature control, passive heat transfer, heat flux conversion and variable thermal conductance. Cotter's original analysis stimulated a variety of modifications and extensions to his one-dimensional heat pipe analysis and likewise to his prediction of the capillary limit. Very few multidimensional analyses have been attempted to describe fundamental heat pipe operation. A two-dimensional analysis supported by experiments using a coplanar device should eventually be followed by a comprehensive three-dimensional analysis for the conventional pipe geometry. Considerable effort has been spent on the prediction of the capillary limit, but only a marginal effort on the sonic and boiling limits. The entrainment limit requires even more attention because virtually no attempt has been made to formulate an analytical model.

NOMENCLATURE BASICSYMBOLS channel shape factor (Eq. (49)) dimensionless constant (Eq. (49)) area of artery (Eq. (104)) area of boiling heat transfer surface area available for liquid flow area available for vapor flow area of wick normal to flow dimensionless constant (Eq. (13)), width of 2-D wick (Fig. 51) specific heat screen mesh opening half width groove depth (Fig. 52) screen wire radius dimension parameter hydraulic diameter

g

gravitational acceleration gravitational constant Q/Ab E his elevation difference between evaporator and condenser (Fig. 12) film boiling coefficient latent heat of vaporization maximum wicking height net rate of heat addition per unit length permeability correction for channel shape (Eqs. (481, (54)) ratio of specific heats (Eq. (108)) thermal conductivity length of heat pipe

THEHEAT PIPE characteristic dimension of wick surface mass flow rate mach number molecular weight number of channels liquid transport factor pressure vapor pressure of liquid capillary pumping pressure heat flux radial heat flux heat transfer rate radius meniscus radius minimum effective radius pore radius effective channel radius radius of bubble nucleus mean groove radius (Eq. (86)) r l , rg meniscus radii of curvature in evaporator channel half depth (Eq. 115) R gas constant constant (Eq. (16)) S time t wick thickness tn temperature T velocity V groove width (Eq. (87)) W land width between grooves W’ (Eq. (87)) wick length (Fig. 10) X axial coordinate z OL

wick inclination (Fig. 10) aspect ratio of grooves (Eq. (89)) accommodation coefficient

313

inclination from vertical dimensionless pitch of grooves (Eq. (87)) groove depth (Eq. (86)) dimensionless constant (Eq. ( 5 1)) wick porosity wetting angle viscosity, dynamic viscosity, kinematic density surface tension vapor blanket thickness (Fig. 20) depth ratio of grooves

NONDIMENSIONAL GROUPS Np pressure number = plug/P2gc Pr Prandtl number = pc/k Re Reynolds number = rV/v Rer radial Reynolds number = r,Vr/vY St Stanton number = hr/CG

SUEISCRIPTS a

sonic condenser evaporator e effective eff effective flow f liquid 1 midchannel m condenser exterior 0 heat pipe container P liquid-vapor surface S sat saturation vapor V wick matrix, wall, or interface W 1 , 2 locations (Fig. 10)

C

REFERENCES 1 R. S. Gaugler, Heat Transfer Device. U. S . Patent 2, 350, 348, June 6, 1944. 2. L. Trefethen, On the Surface-Tension Pumping of Liquids, or, a Possible Role of the Candlewick in Space Exploration. General Electric Tech. Inform. Ser., No. 61SD114, Feb. 1962. 3. G. M. Grover et al., Structures of very high thermal conductance. J. Appl. Phys. 35, 199(rl (1964). 4. T. P. Cotter, Theory of Heat Pipes. Los Alamos Sci. Lab., LA-3246-MS, Feb. 1965. 5. S. Katzoff, Heat Pipes and Vapor Chambers for Thermal Control of Spacecraft. AIAA Thermophysics Specialist Conf., AIAA 67-3 10, April, 1967.

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6. V. H. Gray, The rotating heat pipe-a wickless, hollow shaft for transferring high heat fluxes. NASA Tech. Mem. X-52540. 7. E. C. Conway and M. J. Kelley, A continuous heat pipe for spacecraft thermal control.” Aviation and Space: Progress and Prospects-Annual Aviation and Space Conference, June 1968,” pp. 655-8. 8. K.T. Feldman, Jr., Heat Pipe Design and Analysis. Northrop Corp. Lab., NCL 68-llR, Feb. 29, 1968. 9. F. T. Feldman, Jr., and G. H. Whiting, Applications of the heat pipe. Mech. Eng., 48-53 (1968). 10. W. Harbaugh, Heat Pipe Applications. Presented at Heat Pipe Technology and Manned Space Station Applications Technical Interchange, Huntsvile, Alabama, May 27, 1969. 11. A. Basiulis and J. C. Dixon, Heat Pipe Design for Electron Tube Cooling. Presented at the ASME-AIChE Heat Transfer Conference, Minneapolis, Minnesota., Aug. 3-6, 1969, Paper No. 69-HT-25. 12. A. T. Forrester and F. A. Barcatta, Surface tension storage and feed systems for ion engines. J. Spacecraft Rockets 3, No. 7, 1080-85 (1966). 13. L. Langston and H. R. Kunz, Vapor Chamber Fin Studies. NAS 3-7622, lst, 2nd, and 3rd quart. rep. 14. D. K. Anand et al., Effects of condenser parameters on heat pipe optimization. J. Spacecraft Rockets 4, No. 5 , 695-6 (1967). 15. T. P. Cotter, et al., Status report on theory and experiments on heat pipes at Los Alamos. European Nuclear Energy Agency and Institution of Electrical Engineers, Intern. Conf. Thermionic Electrical Power Generation, London, Sept. 20-25, 1965. 16. G. B. Andeen, et al., The Heat Pipe. AEC Contract A T (30-1)-3496: Progress Rep., June 30, 1965. 17. J. E. Kemme, Ultimate heat pipe performance. “IEEE Conference Record of 1968 Thermionic Conversion Specialist Conference, Oct. 21-23, 1968,” pp. 266-71. 18. J. E. Deverall, Capability of heat pipes. Presented at Heat Pipe Technology and Manned Space Station Applications Technical Interchange, Huntsville, Alabama, May 27, 1969. 19. T. P. Cotter, Heat pipe startup dynamics. Thermionic Conversion Specialist Conference, Oct. 30-Nov. 1, 1967,” pp. 344-8. 20. G. Y. Eastman, The heat pipe. Sci. Am. 218, No. 5, 38-46 (1968). 21. K. T. Feldman, Jr. and G. H. Whiting, The heat pipe. Mech. Eng. 30-33 (1967). 22. K. T. Feldman, Jr., The heat pipe, an interesting heat transfer device. Mech. Eng. 4, NO. 2, 24-27. 23. J. E. Deverall, and J. E. Kemme, High Thermal Conductance Devices Utilizing the Boiling of Lithium or Silver. LA-3211, L o s Alamos Sci. Lab., April 9, 1965. 24. G. H. Parker and J. P. Hanson, Heat pipe analysis. Adwan. Energy Conven. Eng. 847-57 ( I 967). 25. W. J. Haskin, Cryogenic Heat Pipe. AFFDL-TR-66-228, June 1967. 26. S. Frank, Optimization of a grooved heat pipe. “Intersociety Energy Conversion Engineering Conference, Aug. 13-17, 1967,” pp. 833-45. 27. N. P. Jeffries and R. S. Zerkle, Honeywell Heat Pipe Study. Rept. 1 and 2. 28. L. G. Neal, Analytical and Experimental Study of Heat Pipes. TRW Rept. 9990061 14-R000, Jan. 1967. 29. E. C. Phillips, Low temperature heat pipe research program. NASA CR-66792. 30. S. Frank et al., Heat pipe design manual. Martin Nuclear Report MND-3288, Martin Marietta Corp., Baltimore, Maryland, Feb. 1967.

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31. European Atomic Energy Community, Ispra, Italy, Liquid Metals for Heat Pipes, Properties, Plots and Data Sheets. Rept. No. EUR-3653 E., No. N68-14750, Nov. 1967. 32. V. E. Hampel and R. P. Koopman, Reactivity Self-control on Power and Temperature in Reactors Cooled by Heat Pipes. Lawrency Radiation Lab., Univ. of California, Livermore, UCRL-71198, Nov. 1, 1968. 33. A. Basiulis and J.C. Dixon, HeatPipe Design for ElectronTubeCooling.Private comm. 34. A. E. Scheidegger, “The Physics of Flow Through Porous Media.” MacMillan, New York, 1960. 35. B. G . McKinney, An Experimental and Analytical Study of Water Heat Pipes for Moderate Temperature Ranges. Ph.D. Dissertation, Univ. of Alabama, 1969. 36. A. Carnesale, et al., Operating limits of the heat pipe. “Proceedings of Joint AEC/ Sandia Laboratories Heat Pipe Conference,” Vol. 1, No. Sc-M-66-623, pp. 27-44, Oct. 1966. 37. J. H. Cosgrove et al., Operating characteristics of capillary limited heat pipes. J. Nucl. Energy 21, No. 7, 547-58 (1967). 38. J. H. Cosgrove, Engineering Design of a Heat Pipe. Ph.D. Thesis, North Carolina State Univ. 1966. 39. J. K. Ferrell and A. Carnesale, A Study of the Operating Characteristics of the Heat Pipe. TID-23503, 5th. quart. prog. rep., Oct. 1, 1966. 40. J. K. Ferrell and A. Carnesale, A Study of the Operating Characteristics of the Heat Pipe. Quarterly Progress Reps: 5th-1 lth quart. prog. reps. ORO-3411-5-11. 41. J. K. Ferrell and J. Alleavitch, Vaporization Heat Transfer in Capillary Wick Structures. Presented at the ASME-AIChE Heat Transfer Conf., Minneapolis, Minnesota, Aug. 3-6, 1969, AIChE. Reprint 6. 42. A. P. Shlosinger, Heat pipes for Space Suit Temperature Control. Aviation and Space: Progress and Prospects-Annual Aviation and Space Conference, June 1968,” pp. 644-48. 43. A. P. Shlosinger, Heat Pipe Devices for Space Suit Temperature Control. TRW Systems Rept. No. 06462-6005-RO-00, Nov. 1968. 44. A. P. Shlosinger, et al., Technology Study of Passive Control of Humidity in Space Suits, N66-14556. 45. TRW Systems: Heat Pipe Experience and Technology. A promotional booklet. 46. J. Bohdansky, et al., The Use of a New Heat Removal System in Space Thermionic Power Supplies. European Atomic Energy Community, EUR 2229.3, 1965. 47. C. A. Busse, et al., Performance studies on heat pipes. European Nuclear Energy Agency and Institution of Electrical Engineers, Intern. Conf. on Thermionic Electrical Power Generation, London, Sept. 20-25, 1965. 48. W. A. Ranken and J. E. Kemme, Survey of Los Alamos and Euratom heat pipe investigations. Thermionic Conversion Specialist Conference, IEEE, Oct. 1965, pp. 325-36. 49. A. T. Calimbas and R. H. Hulett, An Avionic Heat Pipe. Presented at the ASMEAIChE Heat Transfer Conference, Paper No. 69-HT-16, Minneapolis, Minn., August 3-6, 1969. 50. T. I. McSweeney, The Performance of a Sodium Heat Pipe. Presented at the ASMEAIChE Heat Transfer Conf., Minneapolis, Minnesota, Aug. 3-6, 1969, AIChE preprint 7. 51. R. C. Turner and W. E. Harbaugh, “Design of a 50,000-watt heat pipe space radiator.” Aviation and Space: Progress and ProspectsAnn. Aviation and Space Conf., June 1968,” pp. 639-43.

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52. R. C. Turner, The Configuration Pumped Heat Pipe-An Analysis and Evaluation. Private comm., Feb. 1969. 53. E. C. Phillips and J. D. Hindermann, Determination of properties of capillary media useful in heat pipe design. Presented at the ASME-AIChE Heat Transfer Conf., Minneapolis, Minnesota, Aug. 3-6, Pap. No. 69-HT-18. 54. D. M. Ernst, Evaluation of theoretical heat pipe performance. Thermionic Conversion Specialist Conference, Oct. 30-Nov. 1, 1967,” pp. 349-54. 55. L. S. Langston and H. R. Kunz, Liquid transport properties of some heat pipe wicking materials. Presented at the ASME-AIChE Heat Transfer Conference, Minneapolis, Minnesota, August 3-6, 1969, Paper No. 69-HT-17. 56. K. Ginwala e t al., Engineering Study of Vapor Cycle Cooling Components for Space Vehicles. ASD-TDR-63-582, Sept. 1963, pp. 12&80. 57. R. A. Farran and K. E. Starner, Determining wicking properties of compressible materials for heat pipe application. “Aviation and Space: Progress and ProspectsAnnual Aviation and Space Conference, June 1968,” pp. 659-70. 58. J. Schwartz, Performance map of the water heat pipe and the phenomenon of noncondensible gas generation. Presented at the ASME-AIChE Heat Transfer Conf Minneapolis, Minnesota, Aug. 3-6, 1969, Paper No. 69-HT-15. 59. G. M. Grover, Theory and recent advances. Presented at Heat Pipe Technology and Manned Space Station Applications Technical Interchange, Huntsville, Alabama, May 27, 1969. 60. J. E. Deverall and J. E. Kemme, Satellite Heat Pipe. Los Alamos Sci. Lab., LA3278-MS, April 20, 1965. 61. G. M. Grover, Heat Pipe Systems. Post Conference Rep. Intern. Conf. on Thermionic Electrical Power Generation, London, pp. 12-16. 62. C. A. Busse et al., Prototypes of heat pipe thermionic converters for space reactors. European Nuclear Energy Agency and Institution of Electrical Engineers, Intern. Conf. on Thermionic Electrical Power Generation, London, Sept. 20-25, 1965. 63. C. A. Busse et al., Heat pipe life tests at 1600°C and 1000°C. “IEEE Conf. Record of 1966 Thermionic Conversion Specialist Conf.,” pp. 149-1 58. 64. B. I. Leefer, “Nuclear thermionic energy converter. “Proceeding of the 20th Annual Power Sources Conf., May 1966, pp. 172-75. 65. W. E. Harbaugh, and R. W. Longsderff, The development of an insulated thermionic-converter/heat pipe assembly. “IEEE Conf. Record of 1966. Thermionic Conversion Specialist Conf.,” pp. 139-48. 66. RCA, Heat pipe sweats to harness nuclear reactor heat. Electromch. Design 11, 20 (1967). 67. P. K. Shefsiek, Thermal measurements of a thermionic converter/heat pipe system. IEEE Conf. Record of 1966. Thermionic Conversion Specialist Conf., pp. 169-74. 68. J. F. Judge, RCA tests thermal energy pipe. Missiles Rockets 18, 36-38 (1966). 69. D. M. Ernst, et al., Heat pipe studies at Thermo Electron Corporation. “IEEE Conf. Record of 1968 Thermionic Conversion Specialist Conf.,” pp. 254-57. 70. D. Ernst, Heat pipe developments in thermionics. Presented at Heat Pipe Technology and Manned Space Station Applications Technical Interchange, Huntsville, Alabama, May 27, 1969. 71. D. M. Ernst and G. Y. Eastman, Thermionic two-piece heat pipe converter. Proceedings of the 21st Annual Power Sources Conf., 1967. 72. G. D. Johnson, “Compatibility of various high-temperature heat pipe alloys with working fluids.” IEEE Conf. Record of 1968 Thermionic Conversion Specialist Conf., pp. 258-65.

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73. K. F. Bainton, Experimental Heat Pipes. U. K. Atomic Energy Authority, AEREM1610, 1965. 74. W. B. Hall, Heat pipe experiments. “IEEE Conf. Record 1965. Thermionic Conversion Specialist Conf.,” pp. 337-40. 75. B. R. Bowman and R. W. Crain, Jr., An Ambient Temperature Water Heat Pipe. Private comm. 76. J. Bohdansky et al., Heat transfer measurements using a sodium heat pipe working at low vapor pressure. “Thermionic Conversion Specialist Conference, Houston, Texas, Nov. 1966,” pp. 144-8. 77. W. D. Allingham and J. A. McEntire, Determination of boiling film coefficient for a heated horizontal tube in water-saturated wick material. J. Heat Transfer, Pap. No. 60-HT-11, 1-5 (1960). 78. D. K. Anand, On the performance of a heat pipe. J. Spacecraft Rockets (Eng. Note) 3, NO. 5 , 763-65 (1966). 79. D. K. Anand et al., Heat Pipe Application for Spacecraft Thermal Control. Johns Hopkins Univ., Appl. Phys. Lab., AD 662241. 80. P. J. Marto and W. L. Mosteller, The effect of nucleate boiling on the operation of low temperature heat pipes. Presented at the ASME-AIChE Heat Transfer Conf., Minneapolis, Minnesota, Aug. 3-6, 1969, Pap. No. 69-HT-24. 81. R. A. Moss and A. J. Kelley, Neutron Radiographic Study of Limiting Planar Heat Pipe Performance. Private comm. 82. C. P. Costello and E. R. Redeker, Boiling heat transfer and maximum heat flux for a surface with coolant supplied by capillary wicking. Chem. Eng. Progr. Symp. Ser. 59, No. 41, 104-13 (1963). 83. G. S. Dzakowic et a/., Experimental study of vapor velocity limit in a sodium heat pipe. Presented at the ASME-AIChE Heat Transfer Conf. Minneapolis, Minnesota, Aug. 3-6, 1969. Paper No. 69-HT-21. 84. J. E. Deverall et al., Orbital Heat Pipe Experiment. N67-37590, June 22, 1967. 85. J. E. Kemme, Heat pipe capability experiments. Proceedings of Joint AECiSandia Labs. Heat Pipe Conf. 1, Sc-M-66-623, pp. 11-26, Oct. 1966. 86. J. E. Kemme, High performance heat pipes. “IEEE Conf. Record 1967. Thermionic Conversion Specialist Conf.,” pp. 355-358. 87. J. E. Deverall, et al., Heat pipe performance in a zero-g gravity field. J. Spacecraft Rockets 4, No. 11, 1556-7 (1967). 88. J. E. Deverall and E. W. Salmi, Heat pipe performance in a space environment. “IEEE C o d . Record 1967 Thermionic Conversion Specialist Conference,” pp. 359-62. 89. J. E. Deverall, The Effect of Vibration on Heat Pipe Performance. Los Alamos Sci. Lab., Rep. LA-3798, No TID-4500. 90. J. Bohdansky and H. E. J. Schins, Heat transfer of a heat pipe operating at emitter temperature. “Intern. Conf. on Thermionic Electrical Power Generation, London, Sept. 20-25, 1965.” European Nuclear Energy Agency and Institution of Electrical Engineers, London. 91. B. G. McKinney, An Experimental and Analytical Study of Water Heat Pipes for Moderate Temperature Ranges. N A S A T. M.-53849 (1969). 92. C. L. Tien, Two Component Heat Pipes. AIAA Pap. No. 69-631, June, 1969. 93. K. Dannenburg, Space station program. Presented at Heat Pipe Technology and Manned Space Station Applications Technical Interchange, Huntsville, Alabama, May 21, 1969. 94. J. Madsen, Spacecraft thermal modulation using heat pipes. Presented at the Heat

E. R. I;. WINTERAND W. 0. BARSCH Pipe Technology and Manned Space Station Applications Technical Interchange, Huntsville, Alabama, May 27, 1969. 95. W. Bienert, Heat pipes for electronic equipment and temperature control. Presented at the Heat Pipe Technology and Manned Space Station Applications Technical Interchange, Huntsville, Alabama, May 27, 1969. 96. D. K. Anand, Heat pipe application to a gravity-gradient satellite. “Aviation and Space: Progress and Prospects-Annual Aviation and Space Conf., June 1968,” pp. 634-38. 97. Johns Hopkins University, The GEOS-2 Heat Pipe System and Its Performance in Test and in Orbit, Rep. No S2P-3-25, NASA CR-94585, NASA N68-23540, April 29, 9968. 98. J. E. Deverall, Total hemispherical emissivity measurements by the heat pipe method. “Aviation and Space: Progress and Prospects-Annual Aviation and Space Conf., June 1968,” pp. 649-54. 99. K. Schretzmann, The effect of electromagnetic fields on the evaporation of metals. Phys. Letters 24A, No. 9, 478-79 (1967). 100. J. Bohdansky and H. E. J. Schins. New method for vapol--pressure measurements at high temperatures and high pressures. /. Appl. Phys. 36, No. 11, 3683-4 (1965). 101. C. A. Heath and E. Lantz, Nuclear thermionic space power system concept employing heat pipes. N A S A T N D-4299. 102. H. C. Haller and S. Lieblein, Feasibility studies of space radiators using vapor chamber fins. Proceedings of Joint AECiSandia Labs. Heat Pipe Conf. 1, No. SC-M-66-623, Oct. 1966, pp, 47-68. 103. H. C. Haller et al., Analysis of a low temperature direct condensing vapor-chamber fin and conducting fin radiators. N A S A TND-3103 (1965). 104. H. C. Haller, Analysis and evaluation of a vapor-chamber fin-tube radiator for high power Rankine Cycles. N A S A TND-2836 (1965). 105. R. W. Werner and G. A. Carlson, Heat Pipe Radiator for a 50-MWT Space Power Plant. Rept. No. UCRL-50294, June 30, 1967. 106. Los Alamos Scientific Laboratory, Quarterly Status Report on Advanced Reactor Technology (ART) for period ending July 31, 1965, LA-3370-MS, 1965, pp. 58-62. 107. RCA, Discussion of Heat Pipe Principles. Radio Corporation of America, Direct Energy Conversion Dept., Lancaster, Pennsylvania. 108. J. J. Roberts et ul., A Heat-Pipe-Cooled Fast-Reactor Space Power Supply. Argonne National Lab., ANL-7422, June 1968. 109. Ruhle et ul., Employment of Heat Pipes for Thermionic Reactors, Atomkernenergie 10, 399-404 (1965). 110. P. J. Brosens, Thermionic converter with heat pipe radiator. Advances in Energy Conversion Engineering Conf., Aug. 13-7, 1967, pp. 181-9. 111. C. A. Busse, Optimization of Heat Pipe Thermionic Converters for Space Power Supplies. European Atomic Energy Community, EUR 2534.q 1965. 112. J. Bohdansky, “Thermionic Converter and Its Use in a Reactor,” EUBU 5-4. 113. G. R. Frysinger and G. Y. Eastman, 3 kW flame heated thermionic energy converter. “Proceedings of the 20th Annual Power Sources Conference, May 1966,” pp. 169-71. 114. W. B. Hall and S. W. Kessler, Advances in heat pipe design. “Proceedings of the 20th Annual Power Sources Conference, May 1966,” pp. 166-69. 115. L. J. Lazarids and P. G. Pantazelos, Tests on flame heated thermionic diode.” “Proceedings of the 20thAnnual Power Sources Conferences, May 1966,”pp. 175-77.

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J. Bohdansky, et al., Integrate Cs-Graphite reservoir system in a heat pipe thermionic converter. “Thermionic Conversion Specialist Conf., Palo Alto, California, Oct. 30NOV. 1, 1967,” pp. 93-6. 117. P. Brosens, Heat Pipe Thermionic Converter Development. Final Rep. No. T E 4067-61-68, NASA CR-93664, NASA N68-19392, Dec. 1967. 118. R. W. Werner, The Generation and Recovery of Tritium in Thermonuclear Reactor Blankets Using Heat Pipes. Lawrence Radiation Lab., Univ. of California, Livermore, UCID-15390, Oct. 3, 1968. 119. T. WYATT,Controllable Heat Pipe Experiment. Applied Physics Laboratory, SCO-I 134, Johns Hopkins Univ., March 1965. 120. S. W. Yuan and A. B. Finkelstein, Laminar pipe flow with injection and suction through a porous wall. Tvuns. ASME 78, 719-24 (1956). 121. B. K. Knight and B. B. McInteer, Laminar Incompressible Flow in Channels with Porous Walls. LA-DC-5309, Los Alamos Sci. Lab., 1965. 122. W. E. Wageman and F. A. Guevara, FIuid flow through a porous channel. Phys. Fluids 3, No. 6, 878-81 (1960). 123. C. A. Busse, Pressure drop in the vapor phase of long heat pipes. “Thermionic Conversion Specialist Conference, Palo Alto, California, Oct. 3gNov. 1, 1967,” pp. 391-98. 124. M. Schindler and G. Wossner, Theoretical considerations on heat transfer in heat pipes. Atomkernenergie 10, 395-98 (1965). 125. J. H. Streckert and J. C. Chato, Development of a Versatile System for Detailed Studies on the Performance of Heat Pipes. Tech. Rept. No. ME-TR-64, University of Illinois, Urbana, Illinois, Dec., 1968. 126. J. C. Chato and J. H. Streckert, Performance of a wick-limited heat pipe. Presented at the ASME-AIChE Heat Transfer Conf., Minneapolis, Minn., August 3-6, 1969, Paper No. 69-HT-20. 127. E. K. Levy, Theoretical investigation of heat pipes operating at low vapor pressures. “Aviation and Space: Progress and Prospects Annual Aviation and Space Conference, June 1968,” pp. 671-6. *128. B. D. Marcus, On the Operation of Heat Pipes. TRW Space Techno]. Lab. Rept. No. 99900-611 4 - R 0 0 0 , May, 1965. 129. F. A. Lyman and Y. S. Huang, Analysis of temperature distributions in heat pipe wicks. Presented at the ASME-AIChE Heat Transfer Conf., Minneapolis, Minnesota, August 3-6, 1969, Paper No. 69-HT-23. 130. R. L. Gorring and S. W. Churchill, Thermal conductivity of heterogeneous materials. Chern. Eng. Progr. 57, No. 7, 53-9 (1961). 131. A. H. Nissan et al., Heat transfer in porous media containing a volatile liquid. Chem. Eng. Progr. Symp. Ser. 59 (1963). 132. R. G. Bressler and P. W. Wyatt, Surface wetting through capillary grooves, Presented at the ASME-AIChE Heat Transfer Conf., Minneapolis, Minnesota, August 3-6, 1969, AIChE Preprint 19. 133. L. S. Galowin and V. Barker, Heat pipe channel flow distributions. Presented at the ASME-AIChE Heat Transfer Conference, Minneapolis, Minnesota, August 3-6, 1969, Paper No. 69-HT-22. 134. P. L. Miller and F. W. Holm, Investigation of Constraints in Thermal Similitude. Tech. Rep. AFFOL-TR-69-91, Vols. I and 11. 116.

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E. R. F. WINTERAND W. 0. BARSCH Appendix: Recent and European Literature on Heat Pipes

Part of the articles in this appendix was presented at the Space Technology and Heat Transfer Conference, Los Angeles, California, June 21-24, 1970. Together with European publications they are cited as a supplement to the previously discussed literature so that the article is as comprehensive and up to date as possible. During a sabbatical leave spent in Europe one of the authors (E.R.F. Winter) in cooperation with P. Zimmermannl became acquainted with the European literature on heat pipes. T h e majority of these contributions has its origin at the Institut fur Kernenergetik, Technische Universitat Stuttgart in West Germany. T h e discussion of the most recent American presentations is followed by a brief evaluation of the European work relating to heat pipes. Deverall (235) found that the construction of mercury heat pipes for high heat transfer rates is feasible for operation between 200 and 360 "C. Previously encountered wetting difficulties with mercury were virtually eliminated by the additions of magnesium and titanium to the liquid metal. Schwartz (236) tested an ammonia-stainless steel heat pipe. T h e operating characteristics were compared to those obtained with a geometrically identical pipe employing water as a working fluid (58).It was established that the ammonia-filled pipe was more efficient in transporting thermal loads than the water filled device u p to an operating temperature of approximately 90 O F . Above this temperature the ammonia pipe's relative advantage vanished rapidly until dryout occurred, at which point the water pipe became superior and was able to transport 30% more energy. I n a similar study, Waters and King (137) tested the capability of an ammonia-filled heat pipe to function properly for extended periods of time without failure caused by fluid loss or by degradation of the energy transport mechanism. T h e heat pipe used was fitted with an aluminum container and with a stainless steel screen wicking structure. Accelerated time testing for both continuous heat pipe operation and alternating freeze-thaw cycles indicated no degradation in thermal performance. Subsequent metallurgical examination of the pipe revealed little material corrosion. T h e investigators concluded that such a heat pipe should have a useful operating life in excess of 20 years when operated at about 80 O F . Heat pipes in the cryogenic temperature range have been theoretically studied by Joy (138) who derived equations for optimum pore size, The authors are indebted to Dipl. Ing. Peter Zimmermann, Universitat Stuttgart Institut fur Kernenergetik, for providing them with material for the references (146-1 70).

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optimum wick thickness ratio, and maximum heat transport. T h e effect of gravity was found to play a major role and it must be taken into consideration when designing cryogenic heat pipes. An interpretation of his equations leads to the selection of oxygen as a working fluid and a channel wick for an optimum cryogenic heat pipe design for the temperature range from 77 to 90 O K . Chi and Cygnarowicz (139) also presented a theoretical analysis relating to cryogenic heat pipes. T h e influence of liquid property variations proved to be significant. T h e theoretical predictions compare favorably with the experimental results given by Haskin (25). Ferrell and Johnson (140) obtained experimental results for both the heat transfer coefficient and the critical heat flux through saturated beds of monel and glass beads. T h e liquid in the proximity of the heater was supplied only by capillary action. Various wick inclinations were employed in the tests. T h e conduction mechanism through a thin liquid-bead matrix in contact with the heating surface as proposed by Ferrell and Alleavitch (41) was confirmed to be substantially correct. Soliman et al. (141) measured the effective thermal conductivity of both dry and water-saturated sintered fiber-metal wicks. Correlations were derived for the effective thermal conductivity in terms of thermal conductivities of the solid and liquid phases and the wick porosity. Substantial differences in the effective conductivity were found when measured either along or across the fibers which were attributed to the importance of the contact resistance between the fibers. T h e effect of the working fluid, in either the liquid or vapor phase, within the reservoir of hot reservoir gas-controlled heat pipes was investigated by Marcus and Fleischman (142). They observed that the presence of liquid in the reservoir at start-up led to transient pressure and temperature variations in excess of design conditions. T h e installation of a perforated nonwetting plug at the reservoir entrance, however, eliminated this problem. Bliss et al. (143) tested a flexible heat pipe subject to varying degrees of deformation and to various transverse and longitudinal modes of vibration while in straight shape. It was discovered that flexible heat pipes art. feasible and that the amount of bending had little effect on operation. T h e vibrational environment, in general, tended to increase the heat transfer rate ; however, some critical longitudinal vibrational frequencies caused cessation of heat pipe operation. Bilenas and Harwell(Z44) discussed the development and construction of a set of heat pipes designed to minimize temperature gradients in structures of the Number 3-OAO spacecraft to be launched in 1970. Carlson and Hoffman (145) studied the influence of magnetic fields

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E. R. F. WINTERAND W. 0. BARSCH

on heat pipes. Such effects were found to be important when electrically conducting working fluids (such as liquid metals) are employed and the pipe axis is not aligned with the magnetic field. For such cases, the presence of a magnetic field always reduces the heat transfer capability of the device, However, the heat transfer rate obtained in absence of a magnetic field may be re-established by redesigning the pipe with a compound wick structure with a larger liquid flow passage and a proportionately smaller vapor flow passage than utilized in the nonmagnetic field design. T h e equations necessary for such a redesign are presented in their paper. The earliest German publications appear to be those by Schindler and Wossner (124) and by Ruehle et al. (146). Schindler and Wossner derived relations for optimum pressure and temperature differences and for maximum heat transfer rates in heat pipes. Their article includes diagramatic presentations suitable for the determination of maximum heat transfer rates and design parameters of sodium-filled heat pipes. Ruehle et al. showed convincingly how heat pipes could be employed in thermionic reactors. Their theoretical studies stimulated an extensive experimental and theoretical heat pipe research program, especially in Stuttgart at the Institut fur Kernenergetik. Dorner et al. (147) published a report relating to experimental investigations made on sodium-filled heat pipes, Of special interest was a longitudinally composite heat pipe which functioned satisfactorily. They report the application of X-ray diagnostic techniques and give local heat fluxes, maximum heat transfer rates and temperature profiles measured on sodium heat pipes. Pruschek et al. (148) in an article including a brief survey section on heat pipes also described experiments performed with a sodium-filled heat pipe and suggested possible applications of the then new device. Zimmermann (249) in this Diplomarbeit (master thesis) contributed a worthwhile theoretical study of heat pipes. In particular he investigated the influence of surface tension on heat pipe operation and elaborated on the possibility of nucleate boiling in wicks. Based on laminar liquid flow and laminar as well as turbulent vapor flow models, the work contains computations of fluid velocities, pressure drops, and heat flow rates. An appendix lists properties for sodium as a function of temperature. In view of subsequent investigations and later publications however, the report has lost some of its usefulness. Gammel and Waldmann (150) measured maximum heat fluxes in sodium and lithium heat pipes and Leonhardt (151) computed optimal radiator systems incorporating heat pipes. Dagbjartsson et al. (152) conducted a design study of a thermionic reactor to be employed as a

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power source for spacecraft into which he integrated a number of cylindrical heat pipes as structural and simultaneous heat transfer components. Subsequently Dagbjartsson et al. (153) improved the design of this low power thermionic reactor, again employing high temperature heat pipes for waste heat removal. Coining the term “heat pipes of the second generation,” i.e., heat pipes with arteries built into the wick structure, as compared to “heat pipes of the first generation,” i.e., pipes fitted with screen layers etc., Moritz and Pruschek (154) demonstrated that the 2nd-generation-type heat pipes are superior to the 1st-generation-type heat pipes. Concurrently they reported about conclusive measurements of heat fluxes in the evaporator zone of heat pipes provided with liquid flow arteries. Without citing any references Moritz (155) discribed successful experiments made with unique threaded wall-artery wick heat pipes. The paper also contains a number of construction details and design concepts. It is understood that patent negotiations are being conducted in order to secure eventual commercial profits for the inventor. Zimmermann (156) in a survey article covering 33 publications evaluated a fraction of the heat pipe literature and contributed his own well conceived supplementary studies. Lack of time did not permit a thorough evaluation of his contribution, although a brief inspection of the material stimulates one of the authors of this monograph (E. R. F. Winter) to advise the interested reader to examine Zimmermann’s report in detail. A cursery inspection of a Ph. D. thesis by Moritz (257) provided additional information on the threaded wall-artery wick heat pipes (Gewinde-Arterien Waermerohre). The thesis comprises lengthy discussions on surface evaporation and surface boiling with special emphasis on phase transformation in capillary structures and on grooved surfaces. He gives equations predicting the maximum heat transfer rate for a given heat pipe system in which his more efficient wick design is utilized, but the experimental results were afflicted with sufficient ambiguity, consequently requiring further studies before an affirmative statement can be made as to the validity of the predictions. Groll and Zimmermann (158) studied the qualification of working fluids for heat pipe operation and evaluated their degree of applicability in terms of dimensional groups (Kenngroessen). They included a short description of operating limits and displayed graphically some characteristic parameters as a function of temperature. I n a subsequent publication Groll and Zimmermann (259) optimized design features of various heat pipes in view of maximum heat transfer capabilities of the different systems; however, the analytical predictions are not substan-

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E. R. F. WINTERAND W. 0. BARSCH

tiated by experimental results in the article. T h e transient behavior of heat pipes under start-up conditions was described qualitatively in a further paper by the same investigators (260). Assuming convective cooling they also studied the stepwise and continuous variation of the heat load in several different heat pipes resulting in a number of graphically displayed predictions. Eventual applications of heat pipes in spacecraft were proposed by Zimmermann and Groll (261). T h e article reveals a high degree of candid optimism relating to the applicability of heat pipes in spacecraft operation. Concluding, the reader’s attention may be drawn upon a series of pending publications in Forsch. Ingenieurw. 2 (197 1) (162-1 70). T h e articles of which only title and abstract were available for perusal, should reflect the content of presentations and the ensuing discussions at a symposium on heat pipes held in Stuttgart in 1970 which was attended by a sizeable number of European researchers engaged in heat pipe work.

APPENDIXREFERENCES 135. J. E. Deverall, Mercury as a heat-pipe fluid. A S M E Space Technology and Heat Transfer Conf., Los Angeles, June 21-24, 1970. Paper No. 70-HTISpT-8. 136. J. Schwartz, Performance map of an ammonia (NH,) heat pipe. A S M E Space Technology and Heat Transfer Conf., Los Angeles, June 21-24, 1970. Paper No. 70-HT/SpT-5. 137. E. D. Waters and P. P. King, Campatibility evaluation of an ammonia-aluminumstainless steel heat pipe. A S M E Space Technology and Heat Transfer Conf., Los Angeles, June 21-24, 1970. Paper No. 70-HT/SpT-l5. 138. P. Joy, Optimum cryogenic heat-pipe design. A S M E Space Technology and Heat Transfer Conf., Los Angeles, June 21-24, 1970. Paper No. 70-HT/SpT-7. 139. S. W. Chi and T. A. Cygnarowiu, Theoretical analyses of cryogenic heat pipes. A S M E Space Technology and Heat Transfer Conf., Los Angeles, June 21-24, 1970. Paper No. 70-HT/SpT-6. 140. J. K. Ferrell and H.R. Johnson, The mechanism of heat transfer in the evaporator zone of a heat pipe. A S M E Space Technology and Heat Transfer Conf. Los Angeles, June 21-24, 1970. Paper No. 70-HT/SpT-12. 141. M. M. Soliman, D. W. Graumann, and P. J. Berenson, Effective thermal conductivity of dry and liquid-saturated sintered fiber metal wicks. A S M E Space Technology and Heat Transfer Conf., Los Angeles, June 21-24, 1970. Paper No. 70-HT/SpT-40. 142. B. D. Marcus and G . L. Fleischman, Steady-state and transient performance of hot reservoir gas-controlled heat pipes. A S M E Space Technology and Heat Transfer Conf.,Los Angeles, June 21-24, 1970. Paper No. 70-HTISpT-11. 143. F. E. Bliss, Jr., E. G. Clark, and B. Stein, Construction and test of a flexible heat pipe. A S M E Space Technology and Heat Transfer Conf., Los Angeles, June 21-24, 1970. Paper No. 70-HT/SpT-13.

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144. J. A. Bilenas and W. Harwell, Orbiting astronomical observatory heat pipesDesign, analysis, and testing. A S M E Space Technology and Heat Transfer Conf., Los Angeles, June 21-24, 1970. Paper No. 70-HT/SpT-9. 145. G. A. Carlson and M. A. Hoffman, Effect of magnetic fields on heat pipes. A S M E Space Technology and Heat Transfer Conf., Los Angeles, June 21-24, 1970. Paper NO. 7O-HT/SpT-lO. 146. R. Ruehle, G. Steiner, R. Fritz, and S. Dagbjartsson, Verwendung von Waermeuebertragungsrohren fuer Thermionikreaktoren. Atomkernenergie (9/ lo), 399-404 (1965). 147. S. Dorner, F. Reiss, and K. Schretzmann, Experimentelle Untersur-hungen an Natrium-gefuellten Heat-Pipes. Rep. KFK 5 12, Gesellschaft fuer Kernforschung M.B.H. Karlsruhe, Germany, January 1967. 148. R. Pruschek, M. Schindler, and K. Moritz, Das Waermerohr. Chem. Zng. Tech. 39 (I), 21-6 (1967). 149. P. Zimmermann, Theoretische Betrachtungen zum Waermerohr. Rep. No. 5-50d, IKE, Universitaet Stuttgart. October 1967. 150. G. Gammel and H. Waldmann, Messung des maximalen Leistungsdurchsatzes von Waermeleitrohren mit fluessigem Metall. BBC-Nachr. 49 (I), 34-8 (1967). 151. H. Leonhardt, Optimierung von Abstrahlersystemen mit Kollektor-Waermeleitrohren (heat pipes). BBC-Nachr. 49 (lo), 38-44 (1967). 152. S. Dagbjartsson, M. Groll, and P. Zimmermann, Ein Thermionikreaktor kleiner Leistung mit aussen angeordneten Konvertern und Kollektorkuehlung durch Waermerohre. Raumfahrtforschung 13 (1) (1969); Rep. No. 5-51, Universitaet Stuttgart, 1968. 153. S. Dagbjartsson, M. Groll, 0. Schloerb, and R. Pruschek, An improved out-of-core thermionic reactor for low power. Thermionic Conversion Specialist Conf., Framingham, Massachusetts, 1968; ZEEE Trans. Electron Devices ED-16(8), 713-17 (1969). 154. K. Moritz and R. Pruschek, Grenzen des Energietransports in Waermerohren. Chem. Zng. Tech. 41 (1/2), 30-7 (1969). 155. K. Moritz, Ein Waermerohr neuer Bauart-das Gewinde-Arterien-Waermerohr. Chem. Ing. Tech. 41 (1/2), 37-40 (1969). 156. P. Zimmermann, Das Waermerohr-Stand des Wissens. Rep. No. 5-65, IKE, Universitaet Stuttgart, 1969. 157. K. Moritz, Zum Einfluss der Kapillargeometrie auf die maximale Heizflaechenbelastung in Waermerohren. Dissertation, IKE, Universitaet Stuttgart, 1969. 158. M. Groll and P. Zimmermann, Kenngroessen zum Beurteilen von Waermetraegern fuer Waermerohre. Chem. Zng. Tech. 41 (24), 1294-1300 (1969). 159. M. Groll and P. Zimmermann, Das maximale Waermetransportvermoegen optimal ausgelegter Waermerohre. Chem. Zng. Tech. 42 ( 1 5), 977-81 (1970). 160. M. Groll and P. Zimmermann, Instationaeres Betriebsverhalten von Waermerohren. Chem. Zng. Tech. 42 (16), 1031-34 (1970). 161. P. Zimmermann and M. Groll, Waermerohre in der Satellitentechnik. Raumfahrtforschung 14, 1970. 162. M. Groll et al., Leistungsgrenzen, Technologie und Anwendungen von Waermerohren. Vortrag auf dem Symposium ueber Waermerohre, Stuttgart 1970. F O Y S C ~ . Ingenieurw. 2 (1971). 163. C. A. Busse, Werkstoffprobleme bei Hochtemperatur-Waermerohren. Vortrag auf dem Symposium ueber Waermerohre, Stuttgart, 1970. Forsch. Ingenieurn. 2 (1971) (to be published). 164. K. R. Schlitt, Temperaturstabilisierung durch Waermerohre. Vortrag auf dem

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Symposium ueber Waermerohre, Stuttgart, 1970. Forsch. Ingenieurw. 2 (1971) (to be published). 165. P. H. Pawlowski, Messung der axialen Leistungsdurchsaetze von Natrium- und Kalium- Heat Pipes. Vortrag auf dem Symposium ueber Waerrnerohre, Stuttgart, 1970. Forsch. Ingenieurw. 2 (1971) (to be published). 166. P. Zimmermann, Dynamisches Verhalten von Waerrnerohren. Vortrag auf dem Symposium ueber Waermerohre, Stuttgart, 1970. Forsch. Ingenieurw. 2 (1 971) (to be published). 167. H. Beer, Die dynamische Blasenbildung beim Sieden von Fluessigkeiten an Heizflaechen. Vortrag auf dem Symposium ueber Waermerohre, Stuttgart, 1970. Forsch. Ingenieurw. 2 (1971) (to be published). 168. F. Reiss and K. Schretzmann, Siedeversuche an offenen Rillenkapillarverdampfern. Vortrag auf dem Symposium ueber Waermerohre, Stuttgart, 1970. Forsch. Ingenieurw. 2 (1971). 169. A. Quast, Experimentelle Untersuchungen an einer Kapillar-Verdampfungskuehlung mit Wasser als Betriebsmittel. Vortrag auf dem Symposium ueber Waermerohre, Stuttgart, 1970. Forsch. Ingenieurw. 2 (1971) (to be published). 170. D. Quataert, Investigation of the corrosion mechanism in tantalum-lithium high temperature heat pipes by ion analysis. Vortrag auf dem Symposium ueber Waerrnerohre, Stuttgart, 1970. Forsch. Ingenieurw. 2 (1971) (to be published).

Film Cooling .

RICHARD J GOLDSTEIN Department of Mechanical Engineering. University of Minnesota. Minneapolis. Minnesota I . Introduction . . . . . . . . . . . . . . . . . . . . . . . . I1. Adiabatic Wall Temperature and Film Cooling Effectiveness . . .

321

. 326 A. Incompressible Flow . . . . . . . . . . . . . . . . . . . 326 B. High-speed Flow . . . . . . . . . . . . . . . . . . . . 327

C . Impermeable Wall Concentration . . . . . . . . . . . . . 111. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . A . General Remarks . . . . . . . . . . . . . . . . . . . . B. Two-Dimensional Incompressible Flow Film Cooling-Heat Sink Model . . . . . . . . . . . . . . . . . . . . . . . . . C . Energy Balance in the Boundary Layer . . . . . . . . . . . D . Two-Dimensional Incompressible Flow Film Cooling-Other . . . . . . . . . . . . . . . . . . . . . . . . Models E. Two-Dimensional Film Cooling in a High-speed Flow . . . . F. Injection through Discrete Holes-Three-Dimensional Film Cooling . . . . . . . . . . . . . . . . . . . . . . . . IV . Experimental Studies . . . . . . . . . . . . . . . . . . . . A . General Remarks . . . . . . . . . . . . . . . . . . . . B. Two-Dimensional Film Cooling-Incompressible Flow . . . . C . Two-Dimensional Film Cooling-Compressible Flow . . . . . D . Three-Dimensional Film Cooling . . . . . . . . . . . . . V. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

329 330 330 330 331 338 340 341 342 342 351 361 369 315 316 371

.

I Introduction The need to protect solid surfaces exposed to high-temperature environments is an old one. In general the high-temperature environment is gaseous. and it may be highly ionized as in the stream surrounding a vehicle reentering the atmosphere or in the constrictor of an electric arc or plasma jet . During the last twenty-five years. relatively sophisticated 321

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RICHARDJ. GOLDSTEIN

cooling methods have been used in rockets, reentering space vehicles, high performance gas turbines, and plasma jets. One method is to introduce a secondary fluid into the boundary layer on the surface to be protected. There are different means of introducing this secondary (or injected or coolant) fluid into the boundary layer including ablation, transpiration (or sweat or mass transfer), and film cooling. In ablation cooling, an added coating or “heat shield” decomposes, and by sublimation and other highly endothermic processes a significant quantity of gas enters the boundary layer. In transpiration cooling, the surface is usually a porous material, and the secondary fluid enters the boundary layer through this permeable surface. Both ablation and transpiration cooling are primarily designed to protect the region where the secondary fluid enters the boundary layer. They are highly effective in this regard as a considerable portion of the heat transferred toward the wall can be taken up by the injected coolant right where the heat transfer load is highest. In addition the gas entering the boundary layer effectively thickens it, decreasing the heat transfer rate. These two methods do, however, suffer from serious disadvantages which preclude their use in many applications. The ablating material is not in general renewable and so ablation cooling has been restricted to systems with high heat fluxes of short duration, such as reentering vehicles. This restriction does not apply to transpiration cooling since a coolant can be continually introduced through the porous surface. However, porous materials to date have not had the high strength required for certain applications (e.g., turbine rotor blades) and small pore size often leads to clogging and a resulting maldistribution of coolant flow. In addition, variation in the external pressure distribution can result in a nonoptimum secondary flow distribution through the permeable surface. Although a secondary fluid is also added to the boundary layer in film cooling there are considerable differences in operation and even in goals as compared with ablation and transpiration cooling. A key difference is that film cooling is not primarily intended as protection of the surface just at the location of coolant addition, but rather the protection of the region downstream of the injection location. Film cooling is thus the introduction of a secondary fluid (coolant or injected fluid) at one or more discrete locations along a surface exposed to a high temperature environment to protect that surface not only in the immediate region of injection but also in the downstream region. Eckert and Livingood ( I ) examined transpiration and film cooling (as well as internal convective cooling) to see how a given amount of fluid could be used most effectively. I n their comparison, however,

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the maximum rather than the average temperature of a film cooled wall downstream of the injection slot was considered. As might be expected, they found transpiration cooling more efficient in use of coolant. If a real conducting wall were considered, the average film cooled wall temperature would be more appropriate, and the difference in the effectiveness of the two methods would be greatly reduced. The geometry and flow field at the point of injection are significant variables in film cooling. In two-dimensional (including axisymmetric) film cooling not only is the external flow two-dimensional, but the secondary fluid is also introduced uniformly across the span as in Fig. I . Secondary fluid can enter through a porous region (Fig. la) or through a continuous slot at some angle to the wall surface and the mainstream (Figs. l b and lc). T h e flow downstream of a transpiration

( 0 )

4 Z F R MAINSTREAM

( C )

FIG. 1. Representative two-dimensional film cooling geometries: (a) porous slot,

(b) tangential injection-step

down slot, ( c ) slot angled to mainstream.

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RICHARDJ. GOLDSTEIN

cooled or ablation cooled region is similar to film cooling, the porous wall of the transpiration cooled section or coated wall section of ablation region acting as the slot or injection region. Similarly with liquid film cooling the cooling effect in the region downstream of the point where all the liquid is evaporated can be considered similar to gas-to-gas film cooling. Although the injection geometry can influence the film cooling performance in two-dimensional flow, the effect is usually of second order compared to geometrical effects in three-dimensional film cooling. In this latter flow (Fig. 2) the injection of secondary fluid is not uniform across the span, but rather occurs at isolated locations often through discrete holes in the surface. This can lead to the jets of secondary fluid being blown off the surface and the mainstream flow coming between and/or under the coolant jets decreasing the effectiveness of the film cooling process. Even so, for structural reasons it is usually impossible to have a truly continuous two-dimensional injection slot,

COOLANT

/-

FIG.2. Film cooling with injection through inclined tubes: (a) injection through single tube inclined at angle a to mainstream, (b) injection through single row of discrete tubes inclined at angle a to mainstream.

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and so interrupted slots and even rows of multiple holes have been used. Although film cooling has primarily been used to reduce the convective heat transfer rate from a hot gas stream to an exposed wall, it could also be used to shield a surface from thermal radiation if the radiation absorbtivity of the injectant is high. This can be effectively accomplished with gas particle suspensions or a liquid coolant. In this review, however, all the fluids are considered transparent to radiation, and the convective and radiation heat transfer are then independent and can be treated separately. Only the effect of film cooling on convective heat transfer will be considered. T h e introduction of secondary fluid into the boundary layer with film cooling may be considered to produce an insulating layer (film) between the wall to be protected and a gas stream flowing over it. Alternatively the injected fluid can be considered as a heat sink that effectively lowers the mean temperature in the boundary layer. As will be discussed, the secondary fluid usually serves both functions. T h e introduction of the secondary fluid into the boundary layer at a temperature lower than the mainstream and its resultant mixing with the fluid in the boundary layer reduces the temperature in the region downstream of injection. Note that there is usually considerable mixing of the injected fluid and the mainstream flow downstream of injection. T h u s the concept of a film of secondary fluid maintaining its structure for some distance downstream and isolating the solid surface from the hot mainstream is not strictly valid, especially with a gas coolant. Although a separate discrete insulating film is not produced, injection of the secondary gas can increase the boundary layer thickness and the mass of fluid entrained into the boundary layer from the free stream. T h e increased boundary layer thickness tends to decrease the heat transfer to the wall. However, the increased mainstream flow entrained in the boundary layer causes increased dilution of the secondary fluid with a resulting decrease in its effectiveness as a heat sink. T h e significance and relative importance of these two opposing effects will be discussed subsequently. This review is restricted to film cooling with both the mainstream fluid and the secondary fluid being gases, although not necessarily the same gas, and with a turbulent boundary layer downstream of injection. Both two-dimensional and three-dimensional secondary flow geometries are considered. I n addition, film cooling in compressible flows as well as in incompressible flows is examined. Although the emphasis is on adiabatic wall temperatures and uniform mainstreams, the effects of heat transfer and variable free stream velocity are discussed.

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J. GOLDSTEIN

11. Adiabatic Wall Temperature and Film Cooling Effectiveness

A. INCOMPRESSIBLE FLOW In most film cooling applications the heat transfer from the hot gas to the surface to be protected is not zero. There is usually some type of internal cooling, or, in a transient problem, the heat capacity of the wall material itself is used to take up the heat transferred. T h e general problem in film cooling is to predict or measure for a given geometry, mainstream, and secondary flows the relationship between the wall temperature distribution and heat transfer. Conversely, for a given mainstream and allowable wall heat transfer the requirement may be to predict the secondary flow needed to maintain the surface temperature below some critical value. With constant property flows the velocity distribution is independent of the temperature field and it is convenient to use the concept of a heat transfer coefficient. Thus = h AT = h(Tw -

Taat)

(1)

where T , is the local wall temperature. A question arises as to the datum (i.e., base or reference) temperature Tdat to use in Eq. (1). I n the limiting case of a perfectly insulated (i.e., adiabatic) surface the heat flux would be zero and the resulting surface temperature (distribution) is called the adiabatic wall temperature Taw. Thus the adiabatic wall temperature could be used as the datum temperature. T h e heat flux with film cooling would then be q = h(Tw - Taw)

(2)

Use of Eq. (2) yields a heat transfer coefficient that is independent of the temperature difference for a constant property flow. Note that in the absence of blowing, Tawwould be equivalent to the free stream temperature or in the case of high speed flow the recovery temperature. Most film cooling studies have treated the determination of the heat transfer coefficient and adiabatic wall temperature distribution separately with primary emphasis on the latter. Often the heat transfer coefficient is found to be relatively close to the value without secondary flow, i.e., dependent primarily on the mainstream boundary layer flow. On the other hand, the adiabatic wall temperature distribution can vary considerably and is thus harder (and more important) to predict. I n addition the adiabatic wall temperature is significant in that it is the limiting value of wall temperature that can be obtained without internal wall cooling. Primary emphasis is given to prediction and measurement of the

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adiabatic wall temperature distribution even when the assumption of a constant property flow is invalid, for example in high speed compressible flow. For this case the use of a reference temperature or reference enthalpy as with normal boundary layer flow in the absence of injection will be found useful in predicting heat transfer and will permit application of Eq. (2). T h e adiabatic wall temperature is not only a function of the geometry and the primary and secondary flow fields but also the temperatures of the two gas streams. T o eliminate this temperature dependence a dimensionless adiabatic wall temperature, q , called the film cooling effectiveness is used. For low speed, constant property flow the film cooling effectiveness is given by

’=

m

law

-

m

1,

T, - T,

(3)

where the temperatures of the secondary fluid T , and the mainstream fluid T , are assumed constant. Note that in general Taw< T , and T, < T , in a film cooling application. Since the constant property energy equation is linear in temperature the film cooling effectiveness is dependent only on the primary and secondary flows and the position on the surface. Note that the film cooling effectiveness usually varies from unity at the point of injection (where Taw= T,) to zero far downstream where, because of dilution of the secondary flow, the adiabatic wall temperature approaches the free stream temperature.

B. HIGH-SPEED FLOW For high-speed flow the film cooling effectiveness must be defined somewhat differently. At the point of injection the wall temperature T,, would be expected to be the recovery temperature of the secondary flow or possibly the total temperature of the secondary flow. Far downstream the wall temperature might be expected to approach the mainstream recovery temperature T , (evaluated in the absence of secondary flow). An expression often used for compressible flow film cooling is

Note that the film cooling effectiveness T,I~ reduces to Eq. (3) when compressibility effects can be neglected. An alternate convenient definition of effectiveness for high speed flows employs the isoenergetic-injection wall temperature distribution,

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328

The isoenergetic temperatures (Taw, and TW2,) are the temperatures on the adiabatic wall for the same mainstream conditions, and the same secondary flow rate, but with secondary flow stagnation temperature equal to the stagnation temperature of the mainstream (see Fig. 3). Note that TaWiis a function of position while the recovery temperature used in Eq. (4) can often be considered constant. The definition of effectiveness (Eq. 5) as in Eqs. (3) and (4) gives an effectiveness of unity at the point of injection and a zero value of effectiveness far downstream.

F Tozi

TWF-

"""""'T~

"

Tawi

FIG. 3. Temperatures with isoenergetic injection in high speed flow, injection equal to the mainstream stagnation temperature Tom. stagnation temperature To,,

The isoenergetic film cooling effectiveness is found to correlate experimental results better than an effectiveness based on the recovery temperature without secondary flow. The isoenergetic injection temperature distribution Tawiis obtained at the same blowing rate and free stream conditions as with film cooling. Assuming a constant property fluid, the flow fields are the same in the two cases and independent of the temperature of the secondary fluid. The viscous dissipation terms are also the same in the isoenergetic and normal film cooling runs and subtraction of the temperature distributions from the two runs eliminates the viscous dissipation effect. Thus the difference solution vis for the high-speed flow should be the same as for low-speed flows if the pertinent dimensionless variables describing the flow remain the same. For low-speed flow qis reduces to the usual incompressible flow effectiveness (Eq. 3). For these reasons it may be possible to predict the isoenergetic film cooling effectiveness using the results (theoretical or experimental) from low-speed flows. When using compressible-flow film cooling, results in a design problem not only qis but also the adiabatic wall temperature distribution for isoenergetic injection TaWimust be known. However, for many applications the difference between the recovery temperature without

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329

secondary flow, T,, and Tawiis much smaller than the difference between either of these and the expected temperature of the secondary fluid. Then the recovery temperature T , may be used as a reasonable approximation for TaWi. T h e enthalpy can also be used in defining the film cooling effectiveness in high-speed flows. This would be useful at large temperature differences. T h e proper enthalpies could then be substituted (for the temperatures) directly into the above expressions for effectiveness.

C. IMPERMEABLE WALLCONCENTRATION I t is often difficult to design test systems with walls that sufficiently approximate adiabatic surfaces. This is particularly apparent when the adiabatic wall temperature distribution has large gradients and with a very high temperature mainstream. I n such cases a mass transfer process can be used as an analogue to film cooling. Thus instead of injecting a gas at a different temperature from the mainstream, a gas of different composition would be injected isothermally. This might apply particularly in a study of the effects of three-dimensional film cooling with large density differences (between the primary and secondary flows). T h e mass transfer analogy is also useful for two-dimensional film cooling with large temperature (density) differences. T h e injected gas can be completely different in composition from the free stream, or only a tracer gas might be used in the secondary flow. If the secondary fluid is otherwise the same as the mainstream, the use of a tracer gives results comparable to low density (temperature) differences. T h e mass transfer process is analogous to the heat transfer process (neglecting thermal diffusion phenomena) if the equivalent dimensionless parameters of the flow are the same in the two cases and if the Lewis number is unity. (The Lewis number is the ratio of the Schmidt number for the mass transfer process to the Prandtl number for the corresponding heat transfer process.) T h e turbulent Lewis number as well as the molecular Lewis number should be unity for the analogy to hold. If the flow is sufficiently turbulent, variations in the molecular Lewis number from unity may not play an important role, but in all cases studied to date the value of the turbulent Lewis number should be considered. When using the mass transfer analogy with foreign gas injection, the quantity analogous to the adiabatic wall temperature is the concentration of the injected gas at an impermeable wall. Although there is some question as to the proper concentration to use, the mass fraction C is the most widely used. Equivalent to the film coolii g effectiveness

330

RICHARDJ. GOLDSTEIN

based on adiabatic wall temperature is an effectiveness based on the impermeable wall concentration

If the secondary fluid contains a single constituent not contained in the mainstream, then C, = 0, C, = 1 and Eq. (6) becomes v c = Ciw

(7)

In. Analysis

A. GENERAL REMARKS A number of theoretical correlations and predictions have been developed for the film cooling effectiveness. Since interest is chiefly in turbulent film cooling, the analyses are at least partly empirical yet often suggestive of the significant features of the flow. Much of the interest has centered on relatively simple heat sink models in which the added secondary flow is considered as a sink of heat at the point of injection reducing the temperature in the downstream boundary layer and thus the temperature of the wall. The models have been applied to two-dimensional incompressible and compressible flows and lately to three-dimensional film cooling. Other analyses use some of the recent numerical techniques for predicting two-dimensional turbulent boundary layers and separated flows to obtain predictions of film cooling effectiveness. In this section some of the theoretical analyses will be developed and differences between them discussed. Comparison with experimental results will be deferred till the next section in which the various experimental studies are described.

B. TWO-DIMENSIONAL INCOMPRESSIBLE FLOWFILM COOLING-HEAT SINKMODEL The first heat sink model for film cooling was given by Tribus and Klein (2) in a work primarily concerned with developing kernels to predict the heat transfer and temperature distribution along nonisothermal surfaces. At the suggestion of Eckert, they used Rubesin’s kernel for the wall temperature distribution with a turbulent boundary layer to predict the temperature on an adiabatic surface downstream from a line source of heat. The strength of the line source is determined

FILMCOOLING

33 1

- T,)I by the net enthalpy flow of the secondary fluid I p2U2CP2(T2 and the calculation ignores any effect of the injection on the mainstream flow. Integration of the kernel yields

77 = 5.76 Pr2/3Re~.2(p2/pm)o.z (Cp2/Cpm)(~/M~)-o.s

where

=

5.76 Pr2/3(C,2/C,,)

(8)

,!-O.*

,t = ( X / ~ S ) [ ( P ~ / CRe21-0.25 L~)

(9)

T h e dimensionless blowing rate parameter M is the ratio of the mass velocity of the injected fluid to the mainstream mass velocity. T h e distance x is measured downstream from the point of injection. Tribus and Klein compare their analysis to results for air injected into an air mainstream. For p2 = pm, C,, = C,, , and Pr M 0.72, 77 = 4.62 Rei.2(x/Ms)-o.s

(10)

T h e parameters in Eq. (8) appear in essentially all of the heat sink models and they are very useful for correlating data, particularly at low blowing rates. Equations (8) and (lo), however, predict higher values of effectiveness than have usually been found experimentally. This is apparently due to the assumption that the injected gas does not affect the velocity boundary layer. Figure 4, however, indicates that the boundary layer is considerably thickened by injection. This particular figure was obtained for secondary flow through a porous section ( 3 ) . Later heat sink analyses were made by Librizzi and Cresci (4 ), Kutateladze and Leont'ev (9, Stollery and El-Ehwany (6, 7), and Goldstein and Haji-Sheikh (8). These analyses have a great many similarities, and all proceed from an initial energy balance on the boundary layer.

C. ENERGYBALANCE IN

THE

BOUNDARY LAYER

T h e mass flowing within the boundary layer is considered to be composed of two different fluids from two different streams-the injected gas (ni2) and the mass which enters (is entrained into) the boundary layer from the mainstream (&). These gases are assumed to be well mixed in the boundary layer. At any position downstream of injection the mass flowing per unit time (ni) in the boundary layer is given by (see Fig. 5a)

RICHARDJ. GOLDSTEIN

332 c

E

.-

c

x - 1

m u) u)

!

W

z

I

Y

0

r

I-

I,

5

0.I

L

z 0

0.08 0.051

I

I

2

I I I I I I 6 810

4

40

20

60

80100

DISTANCE FROM TRIP WIRE (cml

FIG. 4. Effect of injection on boundary-layer momentum thickness for injection through a porous slot. ~~

~~

~

~

Symbol

M

&(gm/sec)

Um(m/sec)

X 0

0.0389 0.0263

22.55

56.1

0.01 62 0.00776

0

+

15.41 10.08 4.90

55.2 54.6 56.5

[R. J. Goldstein, G . Shavit, and T. S. Chen, J. Heat Transfer 87, 353 (1965).]

The mean temperature

T in the boundary layer is given by

Assume constant property ideal gases. The average specific heat for the boundary layer is given by,

Cu

=(hzC92

+ kmCDm)/(niz + k m )

Ca (13)

If the wall over which the fluid flows is adiabatic, application of the steady flow energy equation (see Fig. 5b) at any downstream position yields (ni2 ni,) CUT= k2Cu2T2 ?ilmCumT, (14)

+

+

Rearranging terms in Eq. (14) and using Eq. (13) yields

T - T-

1

(15)

,

FILMCOOLING

333

ENTRAINED MASS CONTROL A VOLUME m :A=

-1 --

--

-----

I

I I

--t"m=

m,+m,

FLOW RATE

CONTRF VOLUME

/kioocpmT~

ENTRAINED ENTHALPY Fu)w RATE

(b)

FIG. 5. Control volume when performing (a) mass and (b) energy balances.

I n References (4-6) the mean temperature T in the boundary layer is assumed equal to the adiabatic wall temperature. (Librizzi and Cresci (4) did consider the temperature variation across the boundary layer in a compressible flow model,) Equation (15) reduces to Eq. (3), and the film cooling effectiveness becomes

These analyses all use essentially the same method to predict m m . They assume a +th power turbulent velocity profile and a boundary layer thickness given by SIX' = 0.376 Re;?'5 (17) to predict the entrained flow rate & . The parameter x' is the distance from the point at which the boundary layer starts. Some of the original derivations use a slightly different value of the constant (0.376), but for

RICHARDJ. GOLDSTEIN

334

comparison purposes they have all been recalculated using Eq. (17). Equation (17) is valid in the absence of injection and the analyses assume that it is still valid with injection. T h e primary difference between the three analyses is the assumed location where the total mass flow in the boundary layer starts (i.e., where x' = 0). For a 3th power velocity profile starting at x' = 0, the mass in the boundary layer from the mainstream at some distance downstream is = 0.329pmUmx' Re;?'5

mm = pp,U,6

(18)

Librizzi and Cresci ( 4 ) assume that the boundary layer starts at the point of injection (x' = x) and at injection (x = 0) m = m,

+ mm = m,

(19)

Using fi, = pzUzs mm/& = 0.329(~/Ms)O.~ [Rez(p2/pm)]-0.2

(20)

Putting this into Eq. (16) and using Eq. (9)'

'

For C,, = C,,

1

+ 0.329(C,,/c,,)

1

'

50.'

1

=

I

+ 0.32950.'

Kutateladze and Leont'ev (5) assume the boundary layer downstream of injection grows as if it had started upstream of injection at some distance x". I n order to calculate the distance, x", they assume that the upstream (fictitious) boundary layer grows as a turbulent boundary layer which started sufficiently far upstream to have a net mass flow in it at the point of injection equal to the secondary mass flow rate. Calculation of the mass flow rate from Eqs. ( 1 1 ) and (18) using the above assumptions gives m,/&

= 0.329(4.01

Inserting this into Eq. (16) gives

'

=

and for C,, = C,,

1

+ [)""

1

-1

+ (C,,/C,,)[O.329(4.0I + no."- 11

'

1

= 1

+ 0.249[O.'

(23)

(24)

(25)

Leont'ev (9) extended the model to include the effect of film cooling

FILMCOOLING

335

on heat transfer and also the effect of rough surfaces on film cooling performance. Stollery and El-Ehwany (6, 7) assume the boundary layer starts at injection (x = x') and also that the total mass flow in the boundary layer is zero at the point of injection. Thus at x = 0, and for x

>0

m=0 m

=

gp,u,s

m,

=

ipP,U,6 - m2

Thus using Eq. (1 1)

Inserting this into Eq. (16) (assuming C,, = C P m )and noting m 2 = pzU2s, . - - yields 1 -8 _P Z_ UZ_ S rl = 7 p,U,6 7 p,U,6 -~ 8 m2 Using Eq. (17) 77 = 3.03(r/Ms)-0.8Re,(p2/pm)o.z or introducing Eq. (9)

77 = 3.03t-O.'

They also indicate in their analysis the effect of foreign gas injection (when C,, # CPm)and suggest an approach for determining the film cooling effectiveness with a variable free stream velocity. Since the above heat sink models assume complete mixing of the secondary fluid in the mainstream boundary layer, their validity would be expected to suffer when applied near the point of injection. Both the Tribus and Klein prediction (Eq. 10) and the Stollery and El-Ehwany prediction (Eq. 31) essentially assume no mass flow in the boundary layer at injection yet a finite heat source. Consequently they predict a value of infinity for the effectiveness at x = 0. T h e Librizzi and Cresci and the Kutateladze and Leont'ev correlations, by their assumption of rh = riz, at the point of injection yield an effectiveness of unity at x = 0. This is a convenience which should not be overlooked. Note that three of the correlations (Eqs. (22), (25), and (31)) approach the same prediction far downstream where t is large. It is of interest that these last three correlations (Eq. (22), (25), and (31)) are in better agreement with experimental data, as will be shown below, than the Tribus and Klein correlation (Eq. (8) or (lo)), even though they use the additional major assumption that the temperature in the boundary layer is constant. That this latter assumption is

336

RICHARDJ. GOLDSTEIN

not true was known from the earliest test results of Wieghardt (10) as is shown in Fig. 6. The temperature profile is observed to be similar for different positions downstream of injection. The reason the correlations (Eq. (22), (25), and (31)) work so well follows from the unwritten law that sometimes two invalid assumptions are better than one. Thus the assumption that the boundary layer is unaffected by the secondary flow would indicate less flow into the boundary layer from the mainstream (i.e., less dilution) than actually occurs, reducing m, in Eqs. (1 1) and (16), and thus predict a larger effectiveness than occurs. However, the mean temperature in reality is significantly different from the adiabatic wall temperature; T being between Tawand T, . Thus the assumption used to get Eq. (16) that

’

=

T - T, T,- T ,

gives a lower effectiveness than the true value

since I Taw - T , 1 > 1 T - T, I. The success of the correlations is apparently due to these two effects counterbalancing each other.

Y/8, FIG. 6. Dimensionless boundary layer temperature profiles at various positions downstream of injection: M = 0.74, s = 10 mm, Ta- T, = 28T,(-) e~p[-O.768(y/6~)’~/~], (- - -) e~p[-O.785(y/6~)~]. Distance from injection in meters: ( 0 )0.05, (+) 0.2, (a) 1.0, (v) 2.0, ( 0 ) 4.0. [K. Wieghardt, AAF Translation No. F-TS-919-RE (1946).]

FILMCOOLING

337

Goldstein and Haji-Sheikh (8) use a modified heat sink analysis as an attempt to correct the two assumptions mentioned above. An overall energy balance is performed yielding Eq. (15). T h e temperature variation through the boundary layer is considered, as is the effect of the injection on increasing the size of the boundary layer, and thus the mass flow entering the boundary layer from the mainstream. Assuming a power law velocity profile and a similar temperature profile (cf. Fig. 6 ) the mean temperature in the boundary layer is calculated from I' - T , = X(Ta, - T,) (34) Combining Eqs. (34) and (15),

where h depends on the temperature and velocity profiles. Using the profiles of Wieghardt (10) and extrapolating the results to zero blowing rate so they can be compared to Tribus and Klein results gives l / X = 1.9 Pr2/3

(36)

though the variation with Prandtl number would not be expected to hold over a large range. T h e ratio of the mass flow in the boundary layer with blowing m, , to the mass flow without blowing, mmo, is determined from experimental results of previous investigations. Figure 7 shows that the mass added to the boundary layer from the free stream increases with secondary flow rate and angle of injection (from the mainstream direction). This figure specifically refers to the flow when the secondary and primary gases are the same. For different gases km/km0 =1

+ 1.5 x lo-* Re2(p2W,/p,W2) sin a

(37)

is obtained where a: is the angle of injection (measured relative to the wall). Combining Eq. (37) with Eqs. (35) and (36) and using Eq. (18) to predict mmo(not i , directly),

' where

1.9 Pr2I3

1

+ O.329(CDm/C,,)['.'fi

B = 1 + 1.5 x lo-* R e , ( ~ W m / p , W 2sin ) a

(39)

338

RICHARDJ. GOLDSTEIN 6

-

5 -

-

4ma0 0

8

3-

-

2

-

.€

\

8

.€

I

0.9

I

I

2

I

3

4

5

6

7

8

910

FIG. 7. Ratio of boundary layer entrained mass flow rate with secondary injection kirm to entrained mass flow rate for zero injection +zmo as a function of secondary flow injection angle and flow rate. Data: (0, . , r , D ) , R . J.Goldstein,G.Shavit,andT.S.Chen,J.Heat Transfer87,353(1965); ( 0 , A), K. Wieghardt, AAF Translation No. F-TS-919-RE (1946); (v), J.P.Harnett, R. C. Birkebak, and E. R. G. Eckert, J.Heat Transfer 83,293 (1 961); (D), R. A. Seban and L. H. Back, J. Heat Transfer 84, 45 (1962). [R. J. Goldstein and A. Haji-Sheikh, in Japan SOC.Mech. Engr. 1967 Semi-Intern. Symp., 213-218, Tokyo (1967).]

D. TWO-DIMENSIONAL INCOMPRESSIBLE FLOWFILMCOOLINGOTHERMODELS

Hatch and Papell (ZZ) use a theoretical model for tangential injection, in which they envision that the injected gas remains in a separate film apart from the free stream and try to calculate the heat exchange between this film and a turbulent boundary layer atop it. I t would be expected that such a correlation would fit best very close to the region of injection. Saarlas (12) uses a boundary layer model to predict film cooling effectiveness. T h e analysis also permits an approximate calculation of the heat transfer with film cooling and the effect of a variable mainstream velocity.

FILMCOOLING

339

Seban and Back (13, 14, 25) use the similarity of the temperature profiles to predict film cooling effectiveness based on a uniform eddy diffusivity across the boundary layer. They divide the flow (tangential injection) into three regions: a wall-jet-like flow near the slot where the secondary flow is preserved, a mixing region, and finally a normal turbulent boundary layer region. They observe (for M < 0.8) that the < 56M1.5 (14). T h e model initial region of the flow is defined by XIS uses a linearized form of the energy equation and an upstream effective starting point of the boundary layer and predicts reasonable values of effectiveness for tangential secondary flow injection. T h e wall jet region is particularly significant at a velocity ratio, U , / U , , greater than unity. Note that it is apparently the velocity ratio that plays the key role in determining the approach to wall jet behavior rather than the mass velocity ratio or blowing rate M . T h e significance of the velocity ratio might be expected since it indicates (for tangential injection) whether the secondary fluid will tend to accelerate the mainstream ( U , > U,) or be accelerated by the mainstream ( U , < U,). Spalding (16) proposes relations for film cooling with a tangential slot through which a fluid similar to the mainstream fluid is injected. Although basically empirical the model reduces to a relation similar to the heat sink models at low blowing rates and behaves similar to what would be expected for a wall jet at large blowing rates. He predicts, for 6' < 7, 7)=l

and for

(40)

8' 3 7, ?1 = 715'

where

6'

= O . ~ ~ ( X / I M SRe,0.2 )~.~

+ 1.41{[1 - ( U J U J ]

(404

x/s}O.~

(40b)

All of the analyses described so far use models that employ considerable empirical input. I n an attempt to use a more analytical approach Whitelaw and co-workers (17, 18, 19) have tried to solve the turbulent flow boundary layer equations for film cooling with tangential injection. They use the Patankar and Spalding (20) approach in which a mixing length and effective turbulent Prandtl number distribution are assumed. Although this method has many difficulties and there are questions about its validity and accuracy, it does offer the hope of future solutions valid for the region close to injection point as well as predicting the heat transfer coefficient. Other references using this approach include (21)-(23).

RICHARDJ. GOLDSTEIN

340

E. TWO-DIMENSIONAL FILMCOOLING IN

A

HIGH-SPEEDFLOW

Film cooling in a two-dimensional high-speed flow has been analyzed (24, where the reference temperature (enthalpy) method was combined with some of the incompressible flow analyses to obtain the film cooling effectiveness. As mentioned above, an effectiveness using the isoenergetic wall temperature distribution as a reference appears to work best for compressible flows. T h e reference temperature used is T* = T,

+ 0.12(Tr

-

Tm)

(441)

and all properties in the boundary layer are evaluated at this temperature. Thus t* = (x/Ms)(Rez P ~ ~ P * ) (P*/P,) - ~ . ~ ~ (42) T h e local wall temperature T , or Tawwould be used in place of T , in Eq. (41) if very large temperature differences are encountered. Corresponding to the Kutateladze and Leont’ev model (Eq. 24) 71s = {I

+ (Cgm/C,2)[0.33(4.00 + E*)o’8

- 1]}-’.’

(43)

When the injected fluid is the same as the mainstream fluid the relations derived for high-speed flow are: Kutateladze and Leont’ev model; qis = (1

Librizzi and Cresci model; q,, = (1

+ 0.25&)-0*8

(44)

+ 0.3340;8)-1

(45)

Stollery and El-Ehwany model; qi, = 3.03(;0.8

(46)

and Goldstein and Haji-Sheikh model; qi, = 1.9 Pr2I3(1

where /3 = 1

+ 1.5 x

+ 0.33[0;8p)-1 Re,(pz/p*) sin 01

(47)

(48)

I n deriving these expressions for high-speed film cooling the constant used in the boundary layer growth equation comes from a best fit to experimental skin friction data proposed by Schlichting (25). Laganelli (26) uses a similar analysis based on reference properties to predict film cooling performances in supersonic flow. His results are similar to those given above. He also extends his results to an

FILMCOOLING

34 1

axisymmetric coordinate system. Librizzi and Cresci (4) have also considered film cooling in an axisymmetric supersonic flow. THROUGH DISCRETE HOLES-THREE-DIMENSIONAL FILM F. INJECTION IN G COOL

The heat sink concept has been applied to film cooling following injection through discrete holes (27). With such a geometry there is little hope of getting a relatively exact analytic description of the velocity and temperature distributions. At relatively low mass injection rates the mass addition through a single hole can be considered to act as a localized heat sink on the film-cooled surface. T h e transfer process in the boundary layer is approximated by the conduction equation, the problem being equivalent to determining the temperature distribution in a semi-infinite solid medium along whose surface a point heat source is moving in a straight line with constant velocity. The medium is the mainstream gas, the strength of the source is determined from the net enthalpy flow added through the hole and the velocity of the source in that of the free stream, though in the reverse direction. A major difficulty (and approximation) is to evaluate an effective thermal conductivity or thermal diffusivity of the mainstream. T h e resulting temperature distribution in the mainstream is,

(49)

Along the adiabatic surface Y = 0, rl(x' )'

=

+

IMUmD exp [--0.693 8r(X/D 0.5)

(T)2 ' ] 112

The reference coordinates used in Eq. (49)are shown in Fig. 2; Y is the vertical distance from the surface, 2 is the lateral distance from the center plane of the injection hole, and X is the distance downstream are distances at which the from injection. The distances Y1l2and 21/2 temperature difference drops to half its value along the centerline on the tunnel surface. Note that the effectiveness is now a function of lateral position from the hole centerline as well as of downstream location. T h e form of this relation is found to approximate the experimental data at low blowing rates and thus it has proven useful. Extension to higher injection rates faces formidable obstacles, particularly as the jet appears to leave the surface at large blowing rates.

342

RICHARDJ. GOLDSTEIN

Either Eq. (50) or the direct experimental results for single hole injection can be used to predict the film cooling performance of a row of holes using the principle of superposition. As long as the flows from the individual jets do not interact greatly, superposition appears to work reasonably well (28). At large blowing rates and far downstream the jets come together and superposition of single hole results to predict film cooling from a number of holes cannot be used.

IV. Experimental Studies A. GENERAL REMARKS A summary of some of the experimental studies of film cooling is presented in Table I. A brief description of the geometry of secondary injection as well as the range of pertinent experimental variables is included. Discussion of the results of the individual investigations is given below. In many of the tests, film “heating” rather than film cooling is employed for reasons of convenience. With small temperature differences the flow can be considered constant property. Then, if there is no radiant energy input, the dimensionless temperature distribution in the boundary layer (and thus the film cooling effectiveness) will be independent of whether the secondary gas is hotter or colder than the mainstream. This has been discussed previously in the section on film cooling effectiveness and is also applicable to the determination of the heat transfer coefficient when a heated or cooled wall is used. In some applications, however, there can be a considerable temperature difference between the mainstream and the coolant. Then a key parameter may be the ratio of the densities of the two fluids. This can, of course, be studied using real temperature differences, but attainment of adiabatic wall conditions is very difficult with large temperature differences, so several investigations have utilized the mass transfer analogy. The equivalent of the film cooling effectiveness is then the impermeable wall effectiveness as discussed above. This method is convenient even at small temperature (density) differences as there are always errors introduced due to thermal conduction in the wall when finite temperature differences are employed. This is particularly true in three-dimensional film cooling though the analogy has not been applied there as yet. One difficulty in using the mass transfer analogy is the simulation of a (heat) transfer process at the wall-gas interface. So far only impermeable surfaces have been used. There is also the lingering question of the direct equivalence of the analogy; Burns and Stollery (29),in particular, question whether the turbulent Lewis number is unity.

FILMCOOLING

343

B. TWO-DIMENSIONAL FILM COOLING-INCOMPRESSIBLE FLOW

1 . Injection of Air into Air-Constant

Mainstream Velocity

T h e first well-known study, Wieghardt ( l o ) ,on film cooling not only used a heated secondary gas, but was specifically applied to a film-heating problem, namely, the de-icing of airplane wings. Air was ejected through a slot (see Table I) inclined at an angle of about 30" to the wall surface. Note that as in most studies in which injection occurs through a slot the secondary flow was not fully developed nor was the temperature profile completely uniform at the slot exit. As previously noted, Wieghardt found that the temperature profiles were approximately similar and could be expressed (Fig. 6) in the form T, Taw

-

T, T,

-

e~p[-0.768(y/6,)~~/~]

Some distance downstream of injection, temperature distributions similar to this have been found in other film cooling studies indicating the relative insensitivity of the temperature profile to the secondary injection geometry. At high blowing rates and near the injection location, however, similarity of the temperature profile may not be assumed. Wieghardt found a maximum in film cooling effectiveness at a blowing rate of about unity (Fig. 8). He was able to correlate his adiabatic wall temperature distribution at low blowing rates ( M < 1) and some distance downstream (x/s > 100) with a simple relation 71 = 2 1 . 8 [ ~ / M ~ ] p O . ~

(52)

which, since his range of slot Reynolds number Re, was small, is not too different from some of the predicted results of the heat sink model. This simple equation, even today, is widely used to obtain initial estimates of film cooling performance. Other workers (30), using the same geometry as Wieghardt, found a relation similar to Eq. (52) although the numerical value of the constant was lower (16.9 instead of 21.8). Eckert and Birkebak (31) using the same geometry were able to correlate their results with Eq. (52). Others (32) studied film cooling with injection through both a normal and a tangential slot. For normal injection they found 7 = 2.2(x/Ms)-O.5

(53)

Seban (33) studying the film cooling effectiveness downstream of a stepdown slot (Fig. lb) correlated his data at low blowing rate ( M < 1) with the relation 71 = 2 5 M 0 . 4 ( ~ / M ~ ) - 0 . 8 (54)

TAR1,E I

Ref.

Injection gas"

Geometry

Density ratio, pJpm

0.78

Wieghardt (10)

to

Scesa (60)

Air

Air

Air

@m ?,;

(35) Papell, T r o u t

(39)

Hatch, Papell (11)

7 7

Air

-

to

0.22

0.046

to

to

1.9

0.092

0.81

1.09

0.2

0.025

0.91

to 1.23

0.81

1.09

to

to

+ wall jet

to 1.14

to

0.065

0.08

0.037

to

to

0.92

1.23

0.916

0.108

0.81

1.09 1.23

0.19 to 1.14

0.02s

0.92

to

0.83

0.85

0.25

0.056

1.17 0.52

to

to

He

1.1

1.2

to

Chin, Shirvin, Hayes, Silver

Blowing Free stream rate, M = T2/Tm pzUzlpmUm Mach No.

0.91

to

Seban, Chan Scesa (32)

Temp. ratio,

to

to

0.065

1.20

to

2.5

0.145 to

0.32

0.0

0.1 5

13.9

0.80

0.018 to

0.53

3.54

to 1.8

2.45

0.34

6.5

0.67

1.55

0.57

1.2

0.37

0.05

0.20

to

to

to

to

to

__.c

Papell (53)

Air

to

to

to

to

4.8

0.85

12.0

0.70

0.88

1.14

0.17

0.0045

AT ANGLES 90:80:45'

Seban (33)

/%-////

Air

to

u,_

Seban (34)

0.13

0.27

0.09

/KL//// Air

0.88

u,_ Chin, Skirvin, Hayes, Burggraf

1.15 to

1.13

(38)

1.0

to

to

20.8

to

1.14

0.76

0.87

0.0512

0.0822

0.887

1.026

0.152

to

to

to

MULTIPLE SLOTS

" Unless otherwise noted, mainstream is air. Tests also

to slot of 10.25 mm and 5.65 m m .

for single and double rows of holes.

' Values assume same as Seban (33).

EXPERIMENTAL STUDIES I N FILMCOOLING Velocity ratio,

Velocity

Urn

U2/U,

mlsec

0.246

15.8

to

to

2.44

32.0

0.286

9.7

to

20.4

0.097

13.0 to

1.06

37.0

0.286

9.7

to

to

1.35

20.4

0.26

18.9

to

2.85

to

to

425.0

0.36

3.175

1.2

0.22

3.175

1.3 3.8

3.175

2.7

\

to

105

\

to

lo5

0.33

K

lo5

0.205

'i

IW

0.216

1.0 x lo5 to 2.09 x 105

0.212

15.0 x 106 to 160.0 X 10'

0.94

6.35

0.193

1.52

1.59 3.175 6.35

0.4 x lo5

1.59 3.175 6.35

0.76

2.92

10.3

to

23.6

45.8

0.31

30.5

0.865

Estimated.

30.0 to

56.0

1.1

X

X

lo5 106

to

13.4 x lo6 Y

to

lo5

19.5 x 105

to

0.12 0.03 to

1.2

800

0.60

550

10

to

to

to

580

10

2600

130

q

0.7

550 to

10 to

1.0

1400

9

to

to

Taw

Taw

2.36

0.03

1200

1.o to 0.05

holes to

0.05

to

80

to

q Taw

to

slot

360

130

Taw

to

1.0 to

2500

6.35

to

395.0

1.13

0.222

Slot Reynolds Range of No.,Re, wls

0.095

1.59 3.175 6.35 12.7

2.55

to

to

Taw

147.0

0.036

0.057

to

Q

316.0

3.84

to

to

Y

to

Taw

to

Taw

256.0

to

0.16

Effecq/Taw/Cw tiveness

3.175 6.35 12.7

0.095 to

to

&*Is

15.0 .: lo5

2.9

los

Measured parameters,

8.0

54.0 168.0

Starting length Reynolds No., Re,,

5.0 10.0

to

1.35

to

Slot size, mm

4 Taw

4 Taw

Taw

to

0.095

to

2500

to

130

to

0.13

8200

233

1.o to

0.0

0 to 450,000

0 to 540

1.0

672

3.18

to

to

to

0.16

46,802

271

0.95

to 0.22

2,500

4

260,000

152

0.95

620

5

to

to

to

0.04

7950

0.75

760

to

to

'

to

to

300 2 to

300

0.05

4400

1.00

413

7.9

0.174

6100

177.8

to

to

to

Also considered injection through multiple rows of discrete holes. Holes equivalent

TABLE I

Ref.

Geometry

Density ratio, PslPm

Temp. ratio,

Blowing rate, M =

1.15 to 1.13

0.87

0.0861 to 1.25

0.0671 to 0.1712

0.78 to 0.98

1.02 to 1.27

0.265 to 0.288

0.145

Air

0.875 to 0.935

1.07 to 1.14

0.28 to 1.23

0.1 185

Air

0.90 to 0.97

1.03 to 1.11

4.95 12.6

0.00965 to 0.0149

0.88

1.13

O.Oh

0.04

Injection gas'

Burggraf, Chin, Hayes (70)

Free stream Ta/Tm PSUpIpmUm Mach NO.

MULTIPLE LOUVERS Hartnett, Birkebak, Eckert (30) Hartnett, Eckert Birkebak, (46) Seban, Back (15)

# 7 B .

..

.

/m&///d

Seban, Back (14)

Nishiwaki, Hirata. Tsuchida (41)

Air

&

Birkebak (31)

to 0.0975

0.0 17 to 0.086

0.87 to 1.00

1.0 to 1.15

0.2 to 0.9

0.083 to 0.1 10

0.87

1.15

0.19 to 0.93

0.14

Air

0.83 to 0.95

1.05 to 1.207

0.012 to 0.040

0.0961 to 0.1615

Air

1.1 I to 1.28

0.78 to 0.905

0.25 to 3.18

0.040 to 0.085

Air

3.4

0.8

0.0

3.01

/M-///L Air

r&7m

Samuel, Joubert (37)

Goldstein, Eckert, Tsou Haji-Sheikh (63)

~

~

Air

Shavit, Chen (3)

4%

to

0.095

0.0

&

Eckert,

to 0.70

1.21

0.825

~

Seban, Back (13)

to

Th

He

to

to

to

2.04

1.25

0.408

0.3

0.31 to 0.39

0.01 to 0.02

to

0.4

Velocity accelerated downstream of injection slot to 2.5 and 1.6 times initial values, measured velocitv data.

3.01

Values

EXPERIMENTAL STUDIES I N

Velocity Velocity urn ratio, U,/Um mbec

0.099 to 1.44

25.2 to

FILMC O O L I N G (Costinrted)

Slot Starting length size, Reynolds No., mm Re,,

1.62

52.0

0.294 to

50.0

&*is

8.62 x 105 to

18.0

3.12

Measured parameters,

'
0.13

0.2

0.055

12) 1.o

2.4 to

~

Metzger, Fletcher (45)

Psi, Whitelaw (48)

Air

@ %

Williams (58)

Hydrogen Arcton 12

0.069 to 4.17

1.0

Nitrogen'

2.38 to 3.07

0.33 to 0.42

0.85

1.18

0.75

0.07

0.021 to

6.85

0.03 to 0.06

0.308 to 2.99

0.04 to 2.5

~~

Goldstein, Ramsey Eriksen, Eckert, (28)

to

to

& &

Air

~

0.1 to 2.0

y

~~

0.088 to 0.176

* Value for S taken form Whitelaw (40). Based on maximum possible velocity. " Average heat gradients. Air-hydrogen combustion products form mainstream gas. Accelerated flow. y Hole

EXPERIMENTAL STUDIES IN FILMCOOLING(Continued) Velocity Velocity ratio, U, U,/Um m/sec

0.288 to

21.4

2.66 0.55 to

10.1 to

2.21

20.8

0.2

20.8

to

Slot size, mm

Starting length Reynolds No., Re,,

1.88 2.4 3.35 6.35, 12.7

to

6.1

to

Effecq/Taw/Cativeness

S*/s

0.0542 to

*

cw

6.26

2.17 x lo5

0.107

4.35 x

0.191

to

1.59

0.3

105

lo5

Y

to

to

0.236 to

Cw

17.4

0.51 to I .68

16.8

1.59

1.o

17.6

1.59

0.26

15.2 24.4

1.27 2.54

4

0.78

to

0.55

10.0

2.54

cw

to

20.7

0.127

38.8

to

I .09 0.118 to

30.5 to

2.36

61.0

0.588 to

30.5

25

44,OOO

150

I .o

70.8

2.5

0.005

14,250

212.5

0.97

1,500

12

0.20

18,400

210

1.O to

2,220

0

0.85

i

lo5

0.28

0.85

i

los

0.236

Cw

0.9 to 0.05

0.236 to 0.66

Cw

1.0 to 0.05

0.85

10'to

'~

3.03 Y lo5

0.3

"

0.635 1.522

Taw

9.7

lo5 to

17.0

9.7

11.8

\

Y

lo5

lo5

0.052 to

=

Taw

0.125

0.052

2.36

transfer over section from x diameter.

1 .O

to 0.01

0 to x = I was measured.

Taw

to

to

to

17,420 113 to 368

to

to

to

to

512 0 to 512

228 to 11,300

0 to 512

325

11s = 5.0

to

3500

70.0

70.8

0 to

to

14,400

212

3,100

12.7

0.16

24,100

138

0.85

5200

1.O to

11.8

to

to

Taw

to

2.21

745

to

Cw

Slot Reynolds Range of No., Re, XIS

to 0.15

to

4.00

to

0.95

Cw

0.366

2.54

2.4

0.53

lo5

Y

Measured parameters,

to

to

to 0.0

52000

3 to 80

0.4

12900

3

to 0.0

" Favorable

to

to 51800

to

80

and nonfavorable pressure

RICHARDJ. GOLDSTEIN

352

At large blowing rates several different empirical and semiempirical relations were used to approximate the wall-jet-like effects observed. Seban also investigated (34) the influence of mainstream boundary layer thickness at the point of injection through a tangential slot. Only a very slight decrease in film cooling effectiveness was found with increased boundary layer thickness at the point of injection [cf. References (29, 35, and 36)].

0.2

0 -4

08

0.8

I

2

FIG. 8. Film cooling effectiveness at I/S = 100 as a function of blowing rate M . [K. Wieghardt, AAF Translation No. F-TS-919-RE (1946).

Several investigators have used stepdown slots and correlated their data with (different) empirical relations (35, 37); others have also studied the effect of multiple slots (38). Papell and Trout (39) using tangential injection measured the film cooling effectiveness at very large temperature differences. Papell and Trout correlated their results with empirical and semiempirical correlations. Whitelaw (40) measured the impermeable wall concentration for air injection with helium as a tracer gas. The orders of magnitude of the results were found to be similar to previous film cooling effectiveness measurements. The possibility of turbulent Lewis numbers different from unity was suggested. This would reduce the value of a direct comparison, but

FILMCOOLING

353

would still permit impermeable wall tests to suggest trends and give relative results. Studies have been made of film cooling downstream of a porous section through which air was injected (3,41-43). Note that with normal blowing the velocity distribution near the porous section is severely affected (Fig. 9). T h e results of these different investigations agree quite well. A comparison with results for other geometries is shown in Fig. 10. I

FIG. 9. Effect of injection on boundary-layer velocity profiles with relatively large blowing rate through 35.6 nim porous section with a trip wire 11.2 cm upstream of injection. Data: m2 = 22.55 gmlsec, M = 0.0389, U = 56.0 m/sec. Distance from trip wire (cm): (A) 6.35, (B) 10.16, (C) 15.88, (D) 21.60, (E) 27.95, (F) 53.40, ( G ) 68.50. [R. J. Goldstein, G. Shavit, and T. S. Chen, /. Heat Transfer 87, 353 (1965).]

T h e film cooling results for porous injection have been found to agree relatively well with the analyses of Librizzi and Cresci ( 4 ) , Kutateladze and Leont’ev (5), and Stollery and El-Ehwany (6).Actually there is little difference between these three models, and the resulting equations approach the same value far downstream. Comparisons of the Kutateladze and Leont’ev relations with some of the data for tangential injection are shown in Fig. 1 1. Agreement is quite good. Figure 12 shows a comparison of some porous wall film cooling experiments with the predictions of Tribus and Klein (Eq. lo), Librizzi and Cresci (Eq. 22)’ and Goldstein and Haji-Sheikh (Eq. 38). Note the

RICHARDJ. GOLDSTEIN

3 54

relatively good agreement with the latter two analyses. In Fig. 13, Eq. (38) is observed to compare favorably with the film cooling effectivenesses obtained for tangential injection. The same relation (Eq. 38) when integrated to predict average values over the length of a wall (44), gives good agreement with the average film cooling effectiveness measured for injection through angled slots by Metzger and Fletcher (45).

I .02 10

I

I

20

40

I I 60 8ODO

200

400

1000

x/Ms

FIG. 10. Comparison of film cooling effectiveness as determined in various investigations. Data: (111//////), Goldstein et al., (++) Wieghardt 7 = 2 1 . 8 ( x / M ~ ) - ~ . ~ , (-.-) Hartnett et al. 7 = 16.9(x/M~)-~*~, (----) Nishiwaki et al. 7 = 1.77 (x/Ms)-O.~, (--..-) Scesa 7 = 2.20(x/M~)-~.~, (-.-) Seban 7 = 25.0M0.4(~/Ms)-0.8, (- -) Hatch et al. q = 1.31 exp(--0.229 Re&* (xh-' - l)/M). [R. J. Goldstein, G. Shavit, and T. S. Chen, J: Heat Transfer 87, 353 (1965).]

2. Variable Free Stream Velocity and Free Stream Turbulence Several investigators (14, 46) have reported studies of film cooling on surfaces with variable mainstream velocity (mainly accelerating). Little change in the boundary-layer temperature profiles was observed

FILMCOOLING

355

I .o

F u) u)

y

0.8 0.6

W

L

k-

0.4

LL LL

W

I3

z-I

0 0

0.2

0

5

LL

0.01

10

20

40

60

80 100

200

400

600

X

Ms

FIG. 11. Comparison of effectiveness for tangential slot geometry with analysis of Kutateladze and Leont’ev. (-) Eq. (25); ( 0 ) R. A. Seban, J. Heat Transfer 82, 303 (1960); ( v ) S. Papell and A. M. Trout, NASA Tech. Note TN D-9 (1959). [S. S. Kutateladze and A. I. Leont’ev, Thennal physics of high temperatures 1, No.2, 281-290 (1963).]

and the film cooling effectiveness could be found by multiplying the effectiveness predicted for uniform mainstream flow by a function of the local velocity (46). T h e relatively small change in effectiveness was attributed (14) to the thermal boundary layer being considerably thicker than the velocity boundary layer. Very strong acceleration ( U , increasing by a factor of 24 or 3) caused a slight decrease in effectiveness. Escudier and Whitelaw (47) measured the impermeable wall effectiveness for injection through a porous section with strong adverse pressure gradients. Little influence of pressure gradient on effectiveness was observed up to separation, agreeing with the earlier studies in a favorable pressure gradient (45,46). T h e small effect observed was an increase in effectiveness. Pai and Whitelaw (48) found little influence of a favorable pressure gradient on impermeable wall effectiveness unless the boundary layer ceases to be fully turbulent. Carlson and Talmor (49) report a large change (decrease) in film cooling effectiveness with acceleration of the free stream. I n their apparatus the test wall along which the secondary gas is injected is not flat; a substantial bend occurs at the point of injection, which may produce separated flow. They also indicate that increasing the free stream turbulence

356

RICHARDJ. GOLDSTEIN

9

F

iy -~

GEOMETRY BOTH REFERENCES

--__--

P .lo

TRIBUS AND KLEIN EON. 10 L l B R l t Z l AND CRESCI EQN. 22 GOLDSTIEN AND HAJI-SHEIKH

z 4 A4

V

LL

t

SOURCE

Re2

REF.42

982 81 6 4444 4361

REF.42 REF.42 REF.3

A 0

o

EQN. 38

M

0.0127 0.0155 0.0517 0.0400

*02

I

2

4

6

810

20

40

60 80 100

FIG. 12. Film cooling effectiveness with injection of air through a porous section including comparison with several analyses. [R. J. Goldstein and A. Haji-Sheikh, in Japan SOC.Mech. Engr. 1967 Semi-Intern. Symp., pp. 213-218, Tokyo (1967).]

at the slot location can significantly reduce film cooling performance due to the greater mixing of the secondary gas and the free stream. Going from a free stream turbulence intensity of 3 to 22% almost halves the effectiveness some distance downstream of injection. Kacker and Whitelaw (50) changed the turbulence intensity of the secondary gas in the injection slot from 5.5 to 9.5 yo and found no significant change in impermeable wall effectiveness.

3. Slot Geometry For tangential injection the ratio of lip thickness t to slot opening s can influence the film cooling effectiveness particularly when the velocity

FILMCOOLING

357

I .o .8 .6

-

I

GEOMETRY BOTH REFERENCES

.4

F ln

ln

w .2 2 W

> F

EOUATlON 38

U

w

h.10 w

w

E

0

-

.oe

-

RANWMLY

.06

SELECTED

SYMBOL

I

S (mm)

M

Re2

1.6

0.18

6.35

0.26 0.39 0.39 0.58

620 2420 1360 2120 3970 6220

0

4 .04

a

LL

D

1.6

0

3.175 3.175 3.17s

0 .o 2

.o I

I

2

DATA FROM REF13 ANQ REF33

4

6

0.18

8 1 0

20

40

60

80 100

FIG. 13. Film cooling effectiveness with tangential injection including comparison with analysis of Goldstein and Haji-Sheikh. [R. J. Goldstein and A. Haji-Sheikh, in Japan SOC.Mech. Engr. 1967 Semi-Intern. Symp., pp. 213-218, Tokyo (1967).]

ratio U,/U, is near unity. This has been demonstrated using the mass transfer analogy by Kacker and Whitelaw (50,51) and Burns and Stollery (29). Figure 14 shows the effectiveness for different values of t / s . For a lip thickness less than about 40% of the slot opening, the effects are small. T h e influence of lip thickness also diminishes as the velocity ratio U,l U , is decreased. Similar phenomena are reported by Sivasegaram and Whitelaw (52). T h e significant reduction of film cooling effectiveness that occurs for large lip thicknesses is probably due to the pronounced separated and reverse flow region at the lip edge. Under those conditions the simple heat sink models cannot be used directly, though Eq. (25) and

RICHARDJ. GOLDSTEIN

358 I.o

0.8 0.6

’

0.4

0.2 10

20

30 40 50

x /s

100

FIG. 14. Effect of increasing slot lip thickness on impermeable wall effectiveness for tangential injection at p 2 / p m m 1 and U,/U.Z= 1.07: t / s : (0) 0.126, ( A ) 0.38, ( v ) 0.63, ( 0 ) 0.89, (+) 1.14, ( 0 ) 1.90. [S. C. Kacker and J. H. Whitelaw, J. Mech. Engr. Sci. 11, 22(1969).]

(38) could possibly be modified to account for the role of geometry in the entrainment of mainstream flow into the boundary layer. Sivasegaram and Whitelaw (52) report the effect of injection angle on film cooling effectiveness. As expected [cf. Eq. (38)] the larger the angle the smaller the film cooling effectiveness due to the greater mixing of the coolant with the mainstream at the point of injection. Papell (53) and Metzger and co-workers (45,54) find a similar trend. 4. Effect of Large Temperature DajGerences Few experimental studies have used the extreme temperatures that might be encountered in film cooling applications. Large temperature differences can introduce significant errors in assumed boundary conditions and make accurate measurements difficult, particularly in getting adiabatic wall temperature distributions. Film cooling studies with large temperature differences include those by Papell and Trout (39) and Papell (53) on a flat plate. In their tests the temperature of the hot gas stream was as high as about 800K. Milford and Spiers (55) examined film cooling in a gas turbine combustion chamber at temperatures to 1950K. Lucas and Golladay (56, 57) measured film cooling performance in rocket nozzles and combustion chambers with free stream gas temperatures up to 3000K. Williams (58) studied film cooling in a rocket nozzle with a free stream temperature of about 870K. Other studies made at high temperatures are discussed in the section on compressible flow film cooling.

FILMCOOLING

359

5. Foreign Gas Injection There have been few studies of film cooling with the heated or cooled injection of a foreign gas into an air mainstream. Hatch and Papell (ZZ) injected helium through a near tangential slot into a hot air mainstream. Other workers (42) injected heated helium through a porous section. Burns and Stollery (29) find relatively close agreement between these data and a correlation similar to Eq. (31) though the constant is considerably larger, having been increased empirically to give the best fit with experimental data. T h e adiabatic wall temperature results for helium injection from (42) were found to be somewhat higher than the prediction of Eq. (21) and (24). However, Fig. 15 shows that Eq. (38) fits the data relatively well. Investigations with isothermal foreign gas injection have been performed to study the effect of a density difference between the injection gas and the mainstream using the mass transfer analogy. Nicoll and Whitelaw (18) and Burns and Stollery (29) used the mass transfer analogy, injecting foreign gases through tangential slots into an air mainstream and measuring the impermeable wall concentration. Figure 16 from Reference (29) shows the variation of impermeable wall concentration at a velocity ratio U J U , close to unity for different density ratios, p 2 / p m . I n this study the influence of boundary layer thickness on effectiveness is found to be small. T h e thickness of the slot lip plays a significant role near the slot for a relatively light injection gas. A decrease in effectiveness is found with increasing lip thickness which is attributed to increased mixing in the separated region immediately downstream of the lip. With the heaviest coolant, increasing the velocity ratio U,l U , increases the effectiveness, though past unity the increase is small. For helium injection the effectiveness continues to increase considerably even for velocity ratios U,/U, greater than unity. Pai and Whitelaw (59) measured the impermeable wall effectiveness downstream of a tangential slot through which hydrogen, air (with a helium tracer), argon or Arcton 12 (Refrigerant 12) were injected. With injection of a relatively dense gas the effectiveness reaches a plateau at a velocity ratio U,/U, of about unity (Fig. 17). For light gas injection increase of the velocity ratio above unity continues to yield further increases in effectiveness. For air injection, the effectiveness at most locations increases with blowing rate, finally reaching a plateau at a velocity ratio U,lU, of about unity. From unity to the highest velocity ratio used (-3.1) the effectiveness stays approximately constant.

360

RICHARDJ. GOLDSTEIN

SYMBOL

I

4.04U

.02

-

AT( C )

Tar C)

V

44.8

27.2

D 0

45.7 58.7

0

56.0

M

Po0

27.2 29.2

0.0022 2 0.00330 0.00442

I83 185 360

30.1

0.00683

364

-

WTA FROM REF. 4 2

.o I

1

I

I

1

I

I

I

FIG. 15. Film cooling effectiveness with injection of He through a porous section into a mainstream of air including comparison with analysis of Goldstein and Haji-Sheikh. [R. J. Goldstein and A. Haji-Sheikh, in Japan SOC.Mech. Engr. 1967 Semi-Intern. Symp., pp. 213-218, Tokyo (1967).]

It should be noted that the lip thickness t was about 60% of the slot opening in these tests. Their results were in reasonable agreement with calculations made using the turbulent boundary-layer equations.

6. Heat Transfer Measurements have been made of the heat transfer with film cooling on a surface over which a uniform mainstream flowed (30) and on a surface with a pressure gradient (46). Except at large blowing rates, they reported that shortly downstream of injection the heat transfer coefficient

FILMCOOLING

361

reduces to the heat transfer coefficient one would expect with no blowing (Fig. 18). Near the injection region the blowing usually causes a slight increase in heat transfer coefficient. T h e driving force in defining the heat transfer coefficient is the difference between the actual wall temperature and the adiabatic wall temperature. Scesa (60) and Seban and co-workers (24,32,33) found similar results in that the heat transfer coefficient was not significantly altered by blowing, although in these studies the heat transfer coefficient was sometimes found to be reduced slightly by the blowing. T h e difference in injection geometry used in (30) as compared to that used in (33) may account for this different trend. Metzger and co-workers (45,54) observed a slightly larger effect (increase) on heat transfer coefficient than the other studies, particularly at large blowing rates and close to the injection location.

FIG. 16. Effect of density ratio, p a / p m , at a velocity ratio U,/Um 1 on impermeable wall effectiveness for tangential injection. [W. K. Burns and J. L. Stollery, Intern. /. Heat Mass Transfer 12, 935 (1969).]

C. TWO-DIMENSIONAL FILMCOOLING COMPRESSIBLE FLOW Many applications of film cooling occur in high-speed flows. Although the incompressible flow results can often be used for compressible flow problems, this transformation must be checked experimentally. This is particularly true if the wall geometry is such as to produce shock interactions in the film cooled region. I n several reports (62-63) measurements have been made of the adiabatic wall temperature distribution downstream of a step-down slot in supersonic flow. Either air or helium could be injected tangentially into an air mainstream which had a Mach number of approximately

362

RICHARDJ. GOLDSTEIN

1.0

2.0

3 .Q

VELOCITY RATIO U2IUm

FIG. 17. Impermeable wall effectiveness for air injection with He tracer as a function of velocity ratio U , / U m . [S. C. Kacker and J. H. Whitelaw, J. Heat Transfer 90, 469 (1 968).]

three. Both heated and cooled secondary flows were used. Due to the flow over the edge of the splitter plate (separating the secondary and mainstream flows) there is an expansion fan, a lip shock, a separated region, and a reattachment shock, whose magnitudes are dependent on the rate of secondary mass addition. T h e effect of blowing rate on the flow field is shown by schlieren photography in Fig. 19a and 19b. At the larger secondary flow rates choking occurs in the injection slot. T h e results were correlated, using a film cooling effectiveness based on the isoenergetic flow conditions as described earlier. I n measuring the isoenergetic film cooling effectiveness two test runs are required

FILMCOOLING

363

2.0 I .8

GEOMETRY

I. 2

I. 0

I

I

I

at each blowing rate-one to obtain the isoenergetic wall temperature distribution and the other to obtain the film-cooled wall temperature distribution. Since the total temperature of the mainstream may be somewhat different in the two runs it is useful to normalize the temperatures in Eq. ( 5 ) by dividing them by the total temperature of the free stream for the test in which they are obtained. An empirical correlation of the results for air injection at low blowing rates (Fig. 20) is 771s =

where

550(&2.0

5

for

M

< 0.12

(55) (56)

=(x/~’)[(i/~)~o.4

Note that the step height h‘ (slot height plus lip thickness) is used, indicating the importance of the geometry. These test results are of the order of magnitude of the Tribus and Klein correlation indicating higher values of effectiveness than is found in most of the subsonic studies. At higher values of blowing rate ( M > 0.12) the effectiveness results are considerably higher than even the Tribus and Klein equation, the empirical correlation over the range of parameter studied being vis =

1 6 2 ( ~ / M h ’ ) - l . ~ for 0.12 < M

< 0.408

(57)

A comparison of these empirical correlations and some of the other predictions is shown in Fig. 21. Note that, compared to subsonic correlations, the supersonic results for injection through a tangential slot

M

-

0.136

FIG. 19a. Schlieren photographs for injection through a step down slot in supersonic flow for mainstream Mach number = 3.01, slot opening s = 4.62mm, step height h' = 6.07 mm. [R. J. Goldstein, E. R. G. Eckert, F. K. Tsou, and A. Haji-Sheikh, AIAA (Am.Inst. Aeron. Astronaut.) 1.4, 981 (1966).]

M 0.412 FIG.19b. Schlieren photographs for injection through a step down

slot in supersonic flow for mainstream Mach number = 3.01, slot opening s = 4.62 mm, step height h' = 6.07 mm. [R. J. Goldstein, E. R. G. Eckert, F. K. Tsou, and A. Haji-Sheikh, AZAA (Am.Inst. Aevon. Astronaut.) 1.4, 981 (1966).]

FIG.20. Correlation of film cooling effectiveness for supersonic mainstream flow with heated air injection at small blowing rates. [R.J. Goldstein, E. R. G.Eckert, F. K. Tsou, and A. Haji-Sheikh, Univ. of Minnesota, Heat Transfer Lab. Rept. H T L T R 60 (1965).]

FIG.21. Film cooling effectiveness as predicted by subsonic and supersonic correlations. [R. J. Goldstein, E. R. G. Eckert, F. K. Tsou, and A. Haji-Sheikh, Univ. of Minnesota, Heat Transfer Lab. Rept. HTL TR 60 (1965).]

FILMCOOLING

367

indicate a much more substantial length immediately downstream of the slot where the effectiveness is close to unity although further downstream the effectiveness diminishes more rapidly with distance than with subsonic film cooling. For helium injection through the tangential slot, the film cooling effectiveness based on the recovery temperature (obtained from tests with no blowing) was used (62,63). T h e results are correlated by the relation rlr = 10,000(x/h’)-2~0M0~8for 0.01 < M < 0.02 (58) Mukerjee and Martin (64) studied film cooling with injection or air into a Mach 1.5 to 1.7 air mainstream. Their system had approximately tangential injection though the relative lip thickness was much greater than in References (62) and (6.3). I n fact, the slot opening was only from 10 to 30% of the total step height h’. T h e secondary flow was apparently not increased to the point of choking. They report an empirical relation and compare their measured isoenergetic film cooling effectiveness with the results of Reference (63) and the equation of Tribus and Klein. At low blowing rates relatively good agreement is found with this latter equation though some of the results indicate considerable deviations. T h e measured effectiveness values are in qualitative accord with the earlier study (63). Differences between the two studies are possibly due to the difference in injection lip geometry. Interestingly, at high blowing rates a significantly shorter length along the wall for which the effectiveness is unity and then a more gradual diminishment of effectiveness with distance is found in Reference (64) as compared to References (62) and (63). Parthasarathy and Zakkay (65) conducted an extensive series of tests for film cooling with an axisymmetric Mach 6 air mainstream. Helium, hydrogen, argon, and air were employed as coolants with sonic injection in the upstream, normal and downstream directions. T h e boundary layer thickness at the injection location was much larger than in most other studies. For downstream injection they correlate their results with the relation rlo = K [ ( ~ ~/- 0~. 8 1)- 0 . 7 (59) where K = 155, 120, 35, and 30 for injection of hydrogen, helium, air, and argon, respectively. Normal injection gave less effective cooling than injection in the downstream direction, while no correlation could be obtained for injection in the upstream direction. It should be noted that the definition of effectiveness ?lo uses the free stream stagnation temperature as a reference rather than the recovery temperature or isoenergetic temperature.

368

RICHARDJ. GOLDSTEIN

Dannenberg (66) injected helium nearly tangentially on a hemisphere close to the stagnation point over which a Mach 10 airstream with a stagnation temperature of 4800K was flowing. The peak heating rate was reduced by a factor as large as 2.5 and immediately downstream of the injection point almost complete protection (7,rn 1) could be obtained. T h e most effective film cooling was found when the velocity of the injected coolant was matched to the velocity of the gas stream at the point of injection. Film cooling of a Mach 2.4 axisymmetric nozzle was studied by Lieu (67). T h e mainstream air was heated to 670K. Injection took place near the entrance (subsonic region) to the nozzle through a slot inclined at an angle of 10" to the main flow. Optimum film cooling performance was obtained when the free stream and coolant velocities were approximately equal. A modified version of the Hatch and Papell correlation (11)was used to correlate the results. Redeker and Miller (68) used film cooling in the stagnation region of a cylinder exposed to a Mach 16 crossflow. Nitrogen and helium could be injected either normal or tangential to the surface. Considerable reduction in heat transfer was found with injection. With normal injection the aerodynamic heating could be cut in half while with tangential injection it could be reduced to one-tenth of the non-filmcooled value. The downstream film cooling effects of nitrogen, helium, and argon injected through a transpiring flat plate into a Mach 8 airflow was studied by Woodruff and Lorenz (69). T h e reduction in the turbulent heat flux in the downstream region was found to be relatively independent of Mach number. Use of the blowing parameter eliminated the influence of Reynolds number and the nature of the coolant on the results. Studies have been made (24) of film cooling with a Mach 3 mainstream and injection of air through a narrow porous strip. Use of a reference temperature enabled them to correlate their adiabatic wall temperatures, in the form of isoenergetic film cooling effectiveness with modified subsonic incompressible flow relations (see section on analysis ). I t should be recalled that other film cooling results with compressible flow could not be correlated well with modified incompressible flow correlations, I n those, however, the flow geometry was much more complicated. Usually a step down slot of some type was used and the resulting flow pattern, including lip shock, separated region and reattachment shock could be such as to preclude simple correlations.

FILMCOOLING

369

D. THREE-DIMENSIONAL FILMCOOLING I n many applications of film cooling, design considerations prevent the use of continuous slots for introduction of the coolant. Discrete holes may be used for injection, or a slot with discontinuities (due to structural supports) may be used. If the mainstream is essentially two-dimensional in the absence of injection, blowing through discrete openings will result in a nonuniform flow across the span of the film cooled wall. This is the type of three-dimensional film cooling that will be discussed and reviewed in this section. T h e film cooling effectiveness for an adiabatic wall is still of interest, but now the effectiveness is a function of lateral position as well as downstream distance. T h e film cooling effectiveness for injection through discrete holes is usually considerably less than for slot injection at the same rate of secondary flow per unit span. I n addition, as the blowing rate M is increased past a relatively low value (perhaps M -. 0.5 for pz m pa), the effectiveness for injection through discrete holes falls off rapidly. These phenomena can be understood qualitatively by considering the interaction of a nontangential jet and a mainstream. There is usually ample room across the span for mainstream air to flow between the individual secondary flow entrances. At low blowing rates the jets entering the flow are quickly turned toward the surface by the mainstream. As the blowing rate is increased, the jets penetrate into the mainstream permitting mainstream gas to flow around and under the entering secondary flow jets. This separates the injected fluid from the wall and results in relatively low values of film cooling effectiveness. At still higher blowing rates the jets penetrate further and mix more with the mainstream. It should be noted that the dynamic head or dynamic pressure ratio (p2U22/pmU,2), rather than the blowing rate M is probably the parameter to use, for a given geometry, in predicting the secondary flow for which significant penetration of the jet (and reduced effectiveness) occurs. T h e dynamic head ratio would be important in predicting results for an application where the densities of the secondary and mainstream flows are quite different from test results for approximately constant density studies. Another important parameter would be the geometry of the hole through wbich the secondary fluid enters. A geometry which turns the secondary fluid (and thus the jet momentum) towards the wall as it enters the mainstream would be desirable in terms of optimizing the film cooling performance.

RICHARDJ. GOLDSTEIN

370

Wieghardt (10) covered his continuous slot with perforated sheets to study the effects of both a single row and two rows of holes running transversely across the wall. He studied only one blowing rate with this geometry. With two rows of holes the effectiveness was relatively uniform across the span although less by a factor of two than with the same air flow through a continuous slot. With injection through a single row of perforations he found very low values of effectiveness except directly downstream of the central region of each hole. Papell (53) measured film cooling with injection through multiple rows of discrete circular holes. Injection sections with either two rows or four rows of holes could be inserted in the wall. The holes were at an angle of 90" to the mainflow. Data were taken over a large range of injection rates and could be correlated using an empirical modification of the relation he used for film cooling through a continuous slot. Use of rows of punched crescent louvers to inject a film coolant has been reported (70). The louvers apparently turn the individual jets downstream so the problem of jet departure from the surface was not

TUBES

L

FIG. 22a. Injection section and coordinate system for a row of inclined jets. Detail and flow field are shown for only a single jet interacting with a mainstream. [R. J. Goldstein, E. R. G. Eckert, V. L. Eriksen, and J. W. Ramsey, Israel J. Technol. 8, 145 (1970).]

FILMCOOLING

37 1

encountered. The data was correlated using the same parameters as were used for film cooling through a number of two-dimensional slots (38). Far downstream the louvers were almost as effective as slots in protecting the surface. Several publications (28, 71, 72) have appeared from the University of Minnesota on measurements of film cooling with injection through circular tubes (ending flush to the surface) inclined at various angles to the main flow. Both single tubes and a transverse row of tubes were used. The general flow configuration is presented in Fig. 22, which shows qualitatively the flow of the jet entering the mainstream. Figure 23 shows the film cooling effectiveness downstream of a single hole through which air enters at an angle 01 of 35" to the main flow. Even along the hole centerline (2 = 0) the effectiveness is considerably less than what would be expected for injection through a continuous slot as shown by the top two curves (Eq. 52). Off centerline (2# 0)

UNNEL FLOOR

FIG.22b. Flow field and coordinate system associated with laterally inclined jet interacting with a mainstream. [R. J. Goldstein, E. R. G. Eckert, V. L. Eriksen, and J. W. Rarnsey, Isruel J. Technol. 8, 145 (1970).]

RICHARDJ. GOLDSTEIN

372 0.9

I

I

0.8

I

I 3 =21,8&-io:

I

I

I

I

I

ASSUMING 30 S W I N G ACROSS SPAN

6 0.7

2

w 2

0.6

6

0.5

B

0.4

w

(3

z

c

2 LL

0.3

0.2 0.I

0

0

5

10

I5 20 25 30 35 40 DIMENSIONLESS DISTANCE DOWNSTREAM, X/D

45

FIG. 23. Axial effectiveness distributions for injection through a single hole at an injection angle of 35" and M = 0.5. [R. J. Goldstein, E. R. G. Eckert, and J. W. Ramsey, J . Eng. Power 90, 384 (1968).]

the effectiveness is even less. T h e results shown in this figure are for M = 0.5, which is approximately the optimum blowing rate to maximize the film cooling through a single tube at an angle of 35". Figure 24 shows how the effectiveness varies with blowing rate at different downstream positions. Data for a single hole and a row of holes inclined at 35" to the mainstream are presented here. Note that for a single row of holes the effectiveness reaches a maximum at a blowing rate M 0.5. This could be interpreted as the blowing rate (for p z w pm) above which the jet is no longer turned by the mainstream to hug the wall along which it enters; above that value it increasingly penetrates into the main flow. At higher blowing rates not only is the effective protection per unit mass of coolant reduced, but the absolute value of effectiveness is reduced as well. At low blowing rates the flows of the individual jets from a row of holes appear to be independent of one another. T h e two-dimensional adiabatic wall temperature distribution can then be approximated by

373

FILMCOOLING 0.7 GEOMETRY

I

I

'$zD

c0.6

W

I = 0.22 105

- 0.124

z _ -

0.5

D -0.0

z I

w

5W 0.4

SINGLE

ROW0

lL

U W

0.3

z J 0 g 0.2 z

-I

C0.l

0

0

.5

1.5

1.0

BLOWING R A T E ,

2 .o

M

FIG. 24. Comparison of the centerline film cooling effectiveness for single hole and multiplehole injectionat an injection angle of 35"with the flow for various blowing rates M . [R. J. Goldstein, E. R. G. Eckert, V. L. Eriksen, and J. W. Ramsey, Israel I. Technol. 8, 145 (1970).]

superposition of the effect of a number of single film cooling jets (28). At larger blowing rates where the jets tend to blow off the wall the flows from the individual holes interact. This interaction of the jets results in effectively blocking part of the region where the mainstream might flow around the jets. The secondary flow is then more effectively turned toward the wall, giving considerably higher effectiveness at large M than would be given by a single jet or by superposition (cf. Fig. 24). At large blowing rates the effectiveness for injection through a row of holes increases with position downstream and then remains approximately constant for a considerable distance. If the injection tube is inclined laterally (Fig. 22b), the film cooling effect is spread out further across the span. At a given blowing rate the average effectiveness across the span can be higher for this geometry than for normal injection or injection through a tube inclined downstream (28). Figure 25 shows contours of film cooling effectiveness for injection through different inclined tubes. Lateral inclination seems to impede the penetration of the jet into the mainstream at moderate values of M.

RICHARDJ. GOLDSTEIN

374

GEOMETRY

-2 -I

~

-

I

'

o .I0

b .I5

I

d 25

C

0 I -

c

20

c

.30

Y'I.0

-

I

I

I

DIMENSIONLESS

AY

DISTANCE

I

-

I Q=W.

~

I

I

DOWNSTREAM

U.35'

X/D

FIG.25. Lines of constant film cooling effectiveness for single hole injection at

= 1.0 for various angles of injection. [R. J. Goldstein, E. R. G. Eckert, V. L. Eriksen, and J. W. Ramsey, Israel J. Technol. 8, 145 (1970).]

M

FILMCOOLING

375

It should be recalled, however, that M is based on the velocity within the injection tube. As the tube is inclined at a greater angle from the normal the elliptical exit hole area increases and the component of the injection velocity normal to the mainstream decreases, tending to decrease the penetration of the jet into the flow. The effect on penetration is apparently more significant when the other component of the injection velocity is in the lateral direction rather than the downstream direction. Metzger and Fletcher (45) measured the average film cooling effectiveness (in lateral and downstream directions) following injection through a row of holes inclined downstream. Their trends for the average film cooling effectiveness are similar to those from other studies (71).They also measure the average heat transfer downstream of the holes. Aside from the region close to injection it appears, at least for moderate blowing rates, that the average heat transfer coefficients can be approximated by the values determined without blowing, i.e., for a normal two-dimensional turbulent boundary layer.

V. Concluding Remarks Considerable understanding of film cooling processes has developed in the last twenty-five years. Recent important applications indicate that there are still significant advances to be made. Further work on numerical solutions to the equations for turbulent flows should enhance our ability to predict two-dimensional film cooling phenomena. Accurate predictions for film cooling injected at an angle to the mainstream with a relatively thick splitter plate, with high-speed flow or with large density differences may, however, prove elusive. For secondary flow through discrete holes or even interrupted slots, the difficulties in predicting film cooling performance are even greater. The resulting three-dimensional flow is not yet accessible to anything but simplified analysis. Much work must still be done experimentally to understand the effects of hole geometry, density differences, and the interaction of individual jets on the adiabatic wall temperature distribution. In addition, information on the effect of the mass addition on the local heat transfer is required.

ACKNOWLEDGMENT Several colleagues were of great aid during the preparation of Table I and in reviewing the manuscript for errors. Particular thanks are due to D. R. Pedersen, who also offered invaluable assistance in preparing the figures and text for publication.

RICHARDJ. GOLDSTEIN NOMENCLATURE C mass fraction of foreign gas Ctw mass fraction of foreign gas at an impermeable wall C, mass fraction of foreign gas present in secondary flow C m mass fraction of foreign gas present in mainstream C, specific heat average specific heat of gas in boundary layer, see Eq. (13) C,, specific heat of secondary fluid C,, specific heat of mainstream D diameter of injection tube h convective heat transfer coefficient h' step height, i.e., sum of slot height and lip thickness K empirical constant used in Eq. (59) Le Lewis number; ratio of Schmidt number to Prandtl number M blowing rate or blowing parameter

c,

pz U a l P m u m

Ma, injection Mach number Mam mainstream Mach number m mass flow rate per unit span in boundary layer at any point including both secondary fluid and fluid entrained from mainstream secondary fluid mass flow rate per ti2, unit span paU,s tibo mass flow rate per unit span in boundary layer of fluid entrained from mainstream m m o mass flow rate per unit span in boundary layer of entrained fluid with no secondary injection Pr Prandtl number heat flow per unit time and area q ReHD Reynolds number based on hydraulic diameter of tunnel Re. mainstream Reynolds number based on distance downstream of injection p m U m x / p m Re,, mainstream Reynolds number based on starting length Re,

pco U m x ' l p m

slot Reynolds number based on slot height p z U z s / p n

Stanton number with injection Stanton number without secondary injection injection slot height lip thickness of slot at injection adiabatic wall temperature adiabatic wall temperature with isoenergetic injection datum or reference temperature used in defining the heat transfer coefficient mainstream stagnation temperature stagnation temperature of secondary stream isoenergetic stagnation temperature of secondary stream (should equal Torn) wall recovery temperature in absence of secondary flow wall temperature wall temperature at point of injection, high-speed flow wall temperature at point of injection with isoenergetic injection temperature at a distance y from the surface temperature difference temperature of secondary fluid at injection mainstream temperature property reference temperature mean temperature in boundary layer, Eq. (12) velocity in boundary layer velocity of secondary fluid in injection slot (rica/p,s) mainstream velocity molecular weight of injection gas molecular weight of mainstream gas distance downstream from point of injection through hole (downstream edge) (Fig. 2) distance from point of injection (Fig. 1) distance from starting position of turbulent boundary layer starting length of Reference 5.

FILMCOOLING distance normal to adiabatic wall distance normal to surface in threedimensional film cooling studies (Fig. 2) Y l / 2 vertical position at which ( T W , Y, 0) - T m ) / ( T ( X0,O) , - Tm)

y Y

=B

lateral distance from centerline of injection (cf. Fig. 2) lateral position at which ( T W , 0 , Z ) - Tm ) / ( T( X,0,O)- Tm)

=&

angle of injection in YX-plane (Fig. 2) injection parameter of Reference 8, Eq. (39) boundary layer thickness boundary layer momentum thickness thermal boundary layer thickness boundary layer displacement thickness turbulent thermal diffusivity parameter defined in Eq. (56) film cooling effectiveness, lowspeed flow, Eq. (3)

377

impermeable wall effectiveness, based on concentration, Eq. (6) isoenergetic film cooling effectiveness, Eq.( 5 ) film cooling effectiveness based on total temperature of gas stream film cooling effectiveness based on recovery temperature, Eq. (4) dimensionless temperature parameter ( - T m ) / ( Taw - Tm) viscosity of secondary fluid viscosity at reference temperature T, viscosity of mainstream fluid dimensionless film cooling parameter defined in Eq. (9) dimensionless film cooling parameter defined in Eq. (40b) dimensionless film cooling parameter for high speed flow defined in Eq. (42) density of secondary fluid density at reference temperature T* density of mainstream fluid angle of injection in XZ-plane (Fig. 22)

REFERENCES I. 2. 3. 4. 5.

6. 7. 8. 9. 10.

11. 12. 13. 14. 15. 16.

E. R. G. Eckert and J. N. B. Livingood, NACA Rept. 1182 (1954). M. Tribus and J. Klein, Heat Transfer, Symp. Univ. Mich 1952, 21 1 (1953). R. J. Goldstein, G. Shavit and T. S. Chen, J. Heat Transfer 87, 353 (1965). J. Librizzi and R. J. Cresci, AIAA (Am. Inst. Aeron. Astronaut.) J. 2, 617 (1964). S. S. Kutateladze and A. 1. Leont’ev, Thermal physics of high temperutures 1, No. 2, 281-290 (1963). J. L. Stollery and A. A. M. El’Ehwany, Intern. /. Heat Mass Transfer 8, 55 (1965). J. L. Stollery and A. A. M. El-Ehwany, Intern. J. Heat Muss Transfer 10, 101 (1967). R. J. Goldstein and A. Haji-Sheikh, in Japan Soc. Mech. Engr. 1967 Semi-Intern. Symp., 213-218, Tokyo (1967). A. I. Leont’ev, “Advances in Heat Transfer” (T. F. Irvine, Jr. and J. P. Hartnett, eds.), Vol. 3, p. 33-100. Academic Press, New York, 1966. K. Wieghardt, AAF Translation No. F-TS-919-RE (1946). J. E. Hatch and S. S. Papell, NASA Tech. Note. 2“-130 (1959). M. Saarlas, Ph.D. Thesis, Univ. of Cincinnati (1967). R. A. Seban and L. H. Back, /. Heat Transfer 84, 45 (1962). R. A. Seban and L. H. Back, J. Heat Trunsfer 84, 235 (1962). R.A. Seban and L. H. Back, Intern. J. Heat Mass Transfer 3, 255 (1961). D. B. Spalding, AIAA (Am. Inst. Aeron. Astronaut.) J. 3, 965 (1965).

378

RICHARDJ. GOLDSTEIN

17. S. C. Kacker, B. R. Pai, and J. H. Whitelaw, “Progress in Heat and Mass Transfer” (T. F. Irvine, Jr., W. Ibele, J. P. Hartnett, and R. J. Goldstein, eds.), Vol. 2, p. 163-1 86. Macmillan (Pergamon), New York, 1969. 18. W. B. Nicoll and J. H. Whitelaw, Intern. J. Heat Mass Transfer 10, 623 (1967). 19. S. C. Kacker, W. B. Nicoll, and J. H. Whitelaw, Imperial College, Dept. of Mech. Engr. Rep. TWF/TN/30, London, 1967. 20. S. V. Patankar and D. B. Spalding, “Heat and Mass Transfer in Boundary Layers.” Morgan-Grampian Press, London, 1967. 21. M. Wolfshtein, Imperial College, Dept. of Mech. Engr. Rep. SF/TN/7, London, 1967. 22. E. H. Cole, D. B. Spalding and J. L. Stollery, Imperial College, Dept. of Mech. Engr. Rep. EHTITNI11, London, 1968. 23. B. R. Pai, Imperial College, Dept. of Mech. Engr. Rep. EHT/TN/9, London, 1968. 24. R. J. Goldstein, E. R. G. Eckert and D. J. Wilson, J. Eng. Ind. 90, 584 (1968). 25. H. Schlichting, “Boundary Layer Theory,” 6th ed., p. 600. McGraw-Hill, New York, 1968. 26. A. L. Laganelli, Intern. Heat Transfer Conf., 4th, Versailles/Paris, 1970 Pap. No. 69-IC-191 (to be presented). 27. J. W. Ramsey, R. J. Goldstein, and E. R. G. Eckert, Intern. Heat Transfer Conf., 4th, VersaillesiParis, 1970 Pap. No. 69-IC-136 (to be presented). 28. R. J. Goldstein, E. R. G. Eckert, V. L. Eriksen, and J. W. Ramsey, Israel J. Technol. 8, 145 (1970) (cf. NASA CR-72612;also Univ. of Minnesota, Heat Transfer Lab. Rept. H T L T R 91 (1969)). 29. W. K. Burns and J. L. Stollery, Intern. J. Heat Mass Transfer 12, 935 (1969). 30. J. P. Hartnett, R. C. Birkebak, and E. R. G. Eckert, J. Heat Transfer 83, 293 (1961). 3 1. E. R. G. Eckert and R. C. Birkebak, in “Heat Transfer, Thermodynamics and Education, Boelter Anniversary Volume” (H. A. Johnson, ed.), p. 150-163. McGraw-Hill, New York, 1964. 32. R. A. Seban, H. W. Chan and S. Scesa, Am. SOC.Mech. Engrs. Pap. 57-A-36 (1957). 33. R. A. Seban, J. Heat Transfer 82, 303 (1960). 34. R. A. Seban, J. Heat Transfer 82, 392 (1960). 35. J. H. Chin, S. C. Skirvin, L. E. Hayes, and A. H. Silver, Am. SOC.Mech. Engrs. Pap. 58-A-107 (1958). 36. S. C. Kacker and J. H. Whitelaw, Intern. J. Heat Mass Transfer 10, 1623 (1967). 37. A. E. Samuel and P. N. Joubert, Am. SOC.Mech. Engrs. Pap. 64-WAIHT-48 (1964). 38. J. H. Chin, S. C. Skirvin, L. E. Hayes, and F. Burggraf, J. Heat Transfer 83, 281 (1961). 39. S. Papell and A. M. Trout, Nasa Tech. Note TN D-9(1959). 40. J. H. WhiteIaw, Aeron. Research Council, London, Current Pap. No. 942, 1967. 41. N. Nishiwaki, M. Hirata, and A. Tsuchida, in “International Developments in Heat Transfer,” part IV,p. 675. ASME, New York (1961). 42. R. J. Goldstein, R. B. Rask, and E. R. G. Eckert, Intern. J. Heat Mass Transfer 9, 1341 (1966). 43. I. Mabuchi, JSME (Bulletin of Japanese Soc. of Mech. Engr.) 8, 406 (1965). 44. E. R. G. Eckert, R. J. Goldstein, and D. R. Pedersen, A Discussion of AIAA Pap. 69-523 by D. E. Metzger and D. D. Fletcher. (cf. Reference 45). 45. D. E. Metzger and D. D. Fletcher, AIAA ( A m . Inst. Aeron. Astronaut.) Paper 69523, to published in J. Aircraft (1969). 46. J. P. Hartnett, R. C. Birkebak, and E. R. G. Eckert, in “International Developments in Heat Transfer,” Part IV, p. 682. ASME, New York, 1961.

FILMCOOLING

379

47. M. P. Escudier and J. H. Whitelaw, Intern. J. Heat Mass Transfer 1 1 , 1289 (1968). 48. B. R. Pai and J. H. Whitelaw, Imperial College, Dept of Mech. Engr. Rep. E H T TN/A/l5, London, 1969. 49. L. W. Carlson and E. Talmor, Intern. J. Heat Mass Transfer 11, 1695 (1969). 50. S. C. Kacker and J. H. Whitelaw, J. Heat Transfer 90, 469 (1968). 51. S . C. Kacker and J. H. Whitelaw, Intern. J . Heat Mass Transfer 12, 1196 (1969). 52. S. Sivasegaram and J. H. Whitelaw, /. Mech. Engr. Sci. 11, 22 (1969). 53. S. S. Papell, N A S A Tech. Note TN D-299 (1960). 54. D. E. Metzger, H. J. Carper, and L. R. Swank, (1.Engr. Power) 90, 157 (1968). 55. C. M. Milford and D. M. Spiers, in “International Developments in Heat Transfer,” Part IV, p. 669. ASME, New York, 1961. 56. J. G. Lucas and R. L. Golladay, Nasa Tech. Note TN N-1988 (1963). 57. J. G. Lucas and R. L. Golladay, N A S A Tech. Note TN D-3836 (1967). 58. J. J. Williams, Ph. D. Thesis, Univ. of California, Davis, California, 1969. 59. B. R. Pai and J. H. Whitelaw, Aero. Research Council, London, Paper 29928, H.M.T. 182, 1967. Also Imperial College Dept. of Mech. Engr. EHT/TN/8, London, 1967. 60. S. Scesa, Ph.D. Thesis, Univ. of California (1954). 61. R. J. Goldstein, F. K. Tsou and E. R. G. Eckert, Univ. of Minnesota, Heat Transfer Lab. Rep., H T L T R 54, 1963. 62. R. J. Goldstein, E. R. G. Eckert, F. K. Tsou, and A. Haji-Sheikh, Univ. of Minnesota, Heat Transfer Lab. Rept. H T L T R 60 (1965). 63. R. J. Goldstein, E. R. G. Eckert, F. K. Tsou, and A. Haji-Sheikh, A I A A ( A m . Inst. Aeron. Astronaut.) J. 4, 981 (1966). 64. T. Mukerjee and B. W. Martin, in “Proceedings of the 1968 Heat Transfer and Fluid Mechanics Institute” (A. F. Emery and C. A. Depew, eds.), p. 221. Stanford Univ. Press, Stanford California, 1968. 65. K. Parthasarathy and V. Zakkay, Aerospace Research Lab. Tech. Rep., Contract F33615-68-C-1184 Project 7064, Wright Patterson Air Force Base, Ohio, 1968. 66. R. E. Dannenberg, N A S A Tech. Note TN D-1550 (1962). 67. B. H. Lieu, U. S . Naval Ordance Lab. NOLTR No. 224, White Oak, Maryland, 1964. 68. E. Redeker and D.S. Miller, in “Proceedings of the 1966 Heat Transfer and Fluid Mechanics Institute” (M. A. Saad and J. A. Miller, eds.), p. 387. Stanford Univ. Press, Stanford, California, 1966. 69. L. W. Woodruff and G. C. Lorenz, A I A A ( A m . Inst. Aeron. Astronaut) J . 4, 969 (1966). 70. F. Burggraf, J. H. Chin, and L. E. Hayes, J. Heat Transfer 83, 286 (1961). 71. R. J. Goldstein, E. R. G. Eckert, and J. W. Ramsey, J. Eng. Power 90,384 (1968). 72. R. J. Goldstein, E. R. G. Eckert, and J. W. Ramsey, N A S A CR-54604; Also Univ. of Minnesota, Heat Transfer Lab. Rep. H T L T R 82, 1968.

Author Index Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although his name is not cited in the text. Numbers in italics show the page on which the complete reference is listed. Bell, S., 190, 214 Berenson, P. J., 320b (141). 320e Bhatnagar, P. L., 171, 212 Bialokoz, J., 66 (46), 85 Bienert, W., 213 (95), 318 Bienkowski, G. K., 174, 212 Bilenas, J. A., 320a, 320f Bird, G. A., 198, 216 Bird, R. B., 10, 83 Birkebak, R. C . , 343 (30, 31), 346, 354 (46), 355 (46), 360 (30,46), 363, 378 Bishop, A.A., 50 Bliss, F. E., Jr., 320b, 320e Bohm, U., 98 (13), 160 Bogdanoff, S. M., 204 (134), 217 Bohdansky, J., 239, 250, 251, 271, 274 (46), 215, 275 (116), 289, 292, 315, 317,318, 319 Borishansky, V. M., 10, 83 Bourke, P. J., 40 (51), 42, 66, 67, 86 Bowman, B. R., 249, 317 Boylan, D. E., 204, 217 Bressler. R. G., 308, 309, 319 Bringer, R. P., 49,84 Brock, J. R., 180,213 Brodowicz, K., 66 (46), 85 Brosens, P. J., 275 (110, 1171, 297, 318, 319 Brown, C. K., 52 (29), 84 Brown, W. S., 145, 161 Brun, E. A., 193, 194 (80), 214 Brundin, C. L., 200, 201 Burggraf, F., 344,346, 352 (38), 370 (70). 371 (38), 378, 379 Burns, W. K., 342 (29), 350, 352 (29), 359, 361, 378

A Abadzic, E., 79, 79. 81. 86 Abarbanel, S., 190,214 Acrivos, A., 144 (53). I61 Agar, J. N., 90, 160 Alleavitch, J., 238 (41), 241, 242 (41), 258 (41), 260, 315, 320b, 320e Allingham, W. D., 254, 255, 317 Anand, D. K., 227 (14), 254, 256, 274, 216, 277, 296, 314, 317, 318 Andeen, G . B., 227 (16), 246, 314 Aoki, H., 129, 161 Arai, H., 129 (39), 161 Armstead, B. H., 52 (29). 84 Arpaci, V. S., 178 (41a), 212 Asada, K., 132 (42), 161 Atassi, H., 193, 194 (80), 214

B Back, L. H., 339, 346, 354 (14). 361 (14), 377 Bainton, K. F., 249, 317 Bakker, C. A.P., 116, 118, 160 Baldwin, L. V., 193, 194(79, 83), 200 (79), 214 Barcatta, F. A., 225 (12), 314 Barker, V., 309, 319 Bartz, J. A., 205, 217 Bassanini, P., 171, 172, 176, 178, 212 Basuilis, A., 223 ( I I), 224, 236, 250, 275 ( I I), 314, 315 Becker, G. H., 202, 216 Becker, M., 204,217 Beckwith, I. E., 202 (124). 216 Beer, H., 320e (167), 320g 381

AUTHOR INDEX

382

Bush, W. B., 197, 215 Busse, C. A., 239. 248, 248 (47, 62), 254, 265, 275 (1 1 l), 289, 290, 291, 292, 315, 316, 318, 319, 320e (163), 320f C

Calimbas, A. T., 240, 271, 275 (49), 315 Carden, W. H., 197, 215 Carlson, G. A,, 274, 294, 318, 348, 355, 379, 320a, 320f Carnesale, A.. 238 (36, 39, a), 241 (36, 39,40), 242 (36, 39,40), 315 Carper, H. J., 348, 358 (54), 361 (54), 379 Carver, J. R., 50 Cercignani, C., 171 (28, 29), 172, 172 (28,29), 175, 176 (29), 178 (29), 181, 212 Chahine, M. T., 191, 192 (73), 214 Chambre, P. L., 164 (2). 165 (2). 168, 187, 188, 190 (2). 203 (2). 211 Chan, H. W., 343 (32), 344, 361 (32), 378 Charwat, A. F., 206, 217 Chato, J. C., 304, 319 Chen, T. S., 191 (74), 192(74), 214, 331 (3), 332, 338, 346, 353, 353 (3), 354, 377 Cheng, A. L., 197, 216 Cheng, H. K., 197,215, 216 Chi, S. W., 320b, 320e Chilton, T. H., 103, 125, 160, 161 Chin, J. H., 344, 346, 352, 370(70), 371 (38), 378, 379 Chow, R., 197, 215 Christiansen, W. H., 200, 216 Churchill, S. W., 308, 319 Cipolla, J. W., 179, 213 Clapp, J. T., 119 (32), 161 Clark, E. G., 320b (143). 320e Colburn, A. P., 103, 160 Cole, E. H., 339, 378 Comings, E. W., 119, 161 Conway, E. C., 223, 247, 273 (7), 314 Corcoran, W. H., 52 (28), 84 Cosgrove, J. H., 238 (37, 38), 241 (37, 38), 242 (37, 38), 250, 252, 304 (37), 304, 315 Costello, C. P.. 260, 317 Cotter, T. P.,220, 227. 232, 233, 278, 284, 306, 313, 314

Crain, R. W., 249, 317 Creager, M. 0.. 189, 190(66), 200, 201 (66), 214 Cresci, R. J., 331, 333, 334, 341, 353, 377 Curtiss, C. F., 10, 83 Cybolski, R. J., 193, 194 (79), 200 (79), 214 Cygnarowicz, T. A., 320b, 320e

D Dagbjartsson, S., 32Oc (146, 152). 32M, 320f Dannenberg, R. E., 368, 379 Dannenburg, K., 273 (93), 317 Davies, R. T., 197, 215 Deissler, R. G., 52, 52 (31), 84, 85, 103, 160, 170 (19), 185, 211 Denton, E. B., 88 (2), 91 (2), 98 (2), 108 (2), 160 Denton, W. H.. 40 (51). 42, 66 (51), 67 (51), 86 Deverall, J. E., 230, 231, 235, 238, 247, 247 (23), 267, 268, 270, 274, 275 (23), 306, 314, 316, 317, 318, 320a, 320e Devienne, F. M., 168, 211 Dewey, C. F., 200, 201, 216 Dickinson, N . L., 49, 84 Dimopoulos, H. G.. 117. 120, 142 (30), 161 Dixon, J. C., 223 (Il), 224, 236, 250, 275 (1 I), 313, 315 Dobry, R., 116, 118, 160 Domin, G., 31 (15), 32, 33 (IS), 84 Dorner, S., 320c, 320f Doughty, D. L., 58, 85 Drake, R. M., 58, 85, 193, 202, 209, 214, 216,218 Draper, R., 60. 61, 76. 76 (44),77, 78, 79, 80, 81, 82 (44). 85 Drew, T. B., 125 (36), 161 Dubrovina, E. N., 57, 58, 60, 85 Dzakowic, G. S., 264, 264, 265, 317

E Eastman, G. Y., 235 (20). 248 (71), 273 (20), 274 (20), 275 (71, 113). 314, 316, 318

AUTHOR INDEX Eberly, D. K., 202, 204, 216 Eckert, E. R. G., 5 5 , 56, 57, 85, 322, 340 (24), 341 (27). 342 (28), 343 (30, 31), 346, 348, 350, 353 (42), 354 (44, 46), 355 (46), 359 (42), 360 (30, 46), 361 (30, 61, 62, 63), 363, 364, 365, 366, 367 (62. 63). 368 (24). 370, 371, 371 (28, 71, 72), 372, 373, 373 (28), 374, 375 (71), 377, 378, 379 Einarsson, A., 133 (44),161 Eisenberg. M.. 89 (6), 96 (6), 98 (6, 11). 130, 160, 161 Elberly, D. K., 203 El'Ehwany, A. A. M., 331, 333 (6), 335, 353,377 Ellinwood, J. W., 206, 207, 217, 218 Elliot, E., 206, 217 Endo, Y., 151 (50), 213 Eriksen, V. L., 342 (28). 350, 370, 371, 371 (28), 372, 373, 373 (28), 374, 378 Ernst, D. M., 241, 248, 265, 266, 275 (71), 285, 286, 287, 288, 316 Escudier, M. P., 348, 355, 379 Evans-Lutterodt, K., 26 (9), 37, 39, 40 (20), 42, 66, 67, 67 (9), 83, 84

F Farran, R. A., 244,245,316 Fay, J. A., 196, 197,214 Feldman, K. T., Jr., 223 (8, 9), 235 (21, 22), 236 (S), 246, 273 (8, 9), 274, 303, 314 Fench, E. J., 100,160 Ferrell, J. K., 238 (39), 241 (39; 41), 242, 258. 260, 315, 320b, 320e Ferri, A., 197, 215 Finkelstein, A. B., 279,319 Finn,R.K., 116,118,160 Fleischman, G. L., 320b, 320e Fletcher, D. D., 350, 354 (44). 354, 355 (49, 358 (45). 361 ( 4 9 , 375,378 Flugge-Lotz, I., 197, 215 Fonad, M. G., 98 (I 2). 160 Forrester, A. T., 225 (12), 314 Frank, S., 236, 236 (26), 270 (26), 297, 298,298,300,301,302,314

383

Frei, A. M., 98 (13), 160 Friend, W. L., 103,160 Fritsch, C. A., 66 (47), 85 Fritz, R., 320c (146), 320f Frysinger, G. R., 275 (1 13), 318 Fukuda, A., 128 (37), 161 Fukui, S., 129 (38), 161

G Galowin, L. S., 309,319 Gammel, G., 320c, 320f Gaskill, H. S., 88 (2), 91 (2), 98 (2), 108 (2), 160 Gaugler, R. S., 219, 313 Giedt, W. H., 197, 202,215, 216 Gill, L. E., 40(51), 42, 66(51), 67(51),

86

Ginwala, K., 241, 316 Goldman, K., 30, 53, 84, 85 Goldstein, R. J., 331 (3), 332, 337, 338, 340 (24), 341 (27), 342 (28), 346, 348, 350, 353 (3, 42), 354, 354 (44), 356, 357, 359 (42), 361 (61-63), 364, 365, 366, 367, 367 (62, 63), 368 (24), 370, 371, 371 (28, 71, 72), 372, 373, 373 (28), 374, 375 (71), 377, 378, 379 Golladay, R. L., 358, 379 Goodwin, G., 189, 190 (66), 200, 201 (66), 214 Goren. S. L., 133, 161 Gorring, R. L., 308, 319 Gosman, A. 0..132 (43), 161 Grad, H., 170 (20), 211 Graham, R. W., 53 ( 3 9 , 85 Grassmann, P., I 16, I 18, 120, 160 Graumann, D. W., 320b (141), 320e Gray, V. H., 223, 314 Greif, R., 177, 212 Griffith, P., 37, 40, 84 Grigull, U., 55 (37), 79, 81, 85, 86 Groll, M., 320c (152), 320d (153, 158, 159). 320e (160-162), 320f Grosh, R. J., 66 (47), 85 Gross, E. P., 171, 172, 212 Grove, A. S., 144 (53), 161 Grover, G. M., 220, 222, 227 (3), 234, 246, 247,247 (61), 274 (61), 313, 316 Guevara, F. A., 280, 319

384

AUTHOR INDEX H

Haji-Sheikh, A., 337, 338, 356, 357, 360, 361 (62, 63), 364, 365, 366, 367 (62, 63), 377, 379 Hall, W. B.,24, 31, 35, 50, 52(12), 54, 66, 66 (52), 68, 70 (53), 72 (53), 72, 83, 84, 86, 249, 275 (74, 114), 317, 318 Haller, H. C., 274 (102). 274, 318 Hamilton, R. M., 108, 109, 160 Hampel, V. E., 236 (32), 240, 275, 294, 296, 315 Hanratty, T. J., 88, 108, 110, 110 (25), 112, 114, 117, 120, 134, 134(46a), 140, 142 (30), 160, 161 Hanson, J. P., 235, 284, 285, 314 Harbaugh, W. F., 223, 223 (lo), 240 (51), 248 (10, 65), 273 (lo), 274(51), 275 (65), 314, 315, 316 Harbour, P . J., 204, 217 Harrington, S. A., 206, 218 Harriott, P., 108, 109, 160 Hartnett, J. P., 168, 191 (9,211, 343, 346, 351 (301, 354 (46), 355 (46), 360 (30,46), 361 (30), 363, 378 Harwell, W., 320a, 320f Hasegawa, S., 66 (48), 85 Haskin, W. J., 235 (25), 238, 271, 293, 314, 320h, 320e Hatch, J. E., 338, 344, 359, 368 ( I l ) , 377 Havekotte, J. C., 181 (47), 213 Haviland, J. K., 171, 174, 175, 212 Hayes, L. E., 344, 346, 352 (35), 370 (70), 371 (38), 378, 379 Hayes, W. D., 164 (3), 187 (3), 194 (3), 203 (3), 211 Heath, C. A., 274 (lo]), 318 Hendricks, R. C.. 53, 85 Herring, T. K., 197, 215 Hess, H. L., 53, 85 Hickman, R. S., 197, 215 Hilpert, R., I19 (33), I61 Hine, F., 132 (42), I61 Hindermann, J. D., 241, 243, 259, 261, 303, 316 Hiraoka, S., 122 (35), 161 Hirata, M., 34, 84, 346, 353 (41), 378 Hirschfelder, J. O., 10, 83

Ho, H. T., 197, 215 Hoffman, M. A,, 320a, 320f Holm, F. W., 310, 319 Hoti, E., 150, 161 Horstman, C . C . , 206, 208 (152), 218 Hoshizaki, H., 197, 216 Hsu, S. K., 176, 212 Hsu, Y. Y., 53 (39, 85 Huang, Y. S . , 307, 307, 308, 319 Hubbard, D. W.. 108, 109, 160 Hulett, R. H., 240, 271, 275 (49), 315 Hurlhut, F. C . , 167, 168, 173, 211

I Ibl, N., 88(2), 98, 116(28), 118 (28), 120 (28), 160 Ibusuki, A,, 122 (35). I61 Ilkovic, D., 88 (l), 160 Inman, R. M., 185, 186, 187, 213 Iribarne, A,, 132, 161 Ito, R., 108 (24), 110 (24), 122 (39, 128 (37), 135, 160, 161

J Jackson, 1. D., 24 (7), 31, 35, 37, 39, 40 (20), 42, 50, 52, 54 (12), 66, 66 (12, 52), 67, 68, 68 (7), 72, 83, 84,86 Jain, A. C., 204, 217 Jeffries, N. P., 236 (27), 247,277 (27), 314 Jerbens, R. H., 125 (36), I 6 I Johnson, G. D., 248, 316 Johnson, H. R., 320b, 320e Jolls, K. R., 134, 134 (46a), 161 Jonsson, V. K., 184, 184 (53), 191 (74), 192 (74), 213 Joubert, P. N., 346, 352 (37). 378 Joy, P., 320a, 320e Judge, J. F., 248 (68), 275 (68), 316 Jukoff, D., 190, 214

K Kacker, S.C., 339 (17, 19), 348, 350, 352 (36), 355, 256, 257, 358, 362, 378, 379 Kakarala, C. R., 50 Kao, H. C., 197, 112

AUTHORINDEX Kataoka, K., 128 (37), 161 Katzoff, S., 220, 223, 273, 276, 313 Kavanau, L. L., 193, 214 Kelley. A. J., 258, 317 Kelley. M. J., 223, 247. 273 (7). 314 Kemme, J. E., 230 (17), 235, 238, 240, 247, 247 (48), 260, 262, 263, 268, 269, 270, 271,272, 274 (23), 275 (23). 305, 306, 314,315, 316, 317 Kemp, N. H., 197,214,215 Kennard, E. H., 169 (lo), 170, 211 Kessler, S. W., 275 (114), 318 Kestin, J., 119, 161 Khan, S. A., 24 (7), 35, 68 (7), 83 King, P. P., 320a, 320e Kinney, R. B., 169, ZI/ Klebanoff, P. S., 142 (52), 153, 155, 161 Klein, J., 330, 377 Knapp, K. K., 59, 60, 85 Knight, B. K., 279, 319 Knudsen, J. G., 97 (7), 98 (7), 160 Koopman, R. P., 236(32), 240, 275, 294, 296, 315 Kopal, Z., 209, 218 Koppel, L. B., 50, 84 Koshmarov, Y. A,, 206(141), 208, 209, 217 Krasnoschekhov, E. A., 48 (22), 49 (22), 50,84 Kruger, Ch. H., 170 (22). 171 (22), 211 Kuhns, P. W., 200 (121), 216 Kunz, H. R., 53, 85, 226 (13), 236, 239, 241, 241 (13), 243, 243 (13), 254 (13), 274 (13), 274, 304, 304, (13), 314, 316 Kussoy, M. I., 206, 208 (152), 218 Kutateladze, S. S., 10 (5), 48, 50, 83, 331, 333 (5), 334, 353, 355, 377

L Laganelli, A. L., 340, 378 Lamb, D. E., 89, 160 Langston, L. S., 226 (13), 236, 239, 241, 243, 254, 274(13), 304(13, 155). 314, 316 Lantz, E., 274 (lo]), 318 Larsen, P. S., 178 (41a), 212 Larson, J. R., 45, 62, 63, 85 Laufer, J., 142 (50, 51), 161, 200, 216

385

Laurence, J. C., 194 (83), 214 Lavin, M. L., 171, 174, 175, 212 Lazarids, L. J., 275 (1 15), 318 Leefer, B. I., 248 (64), 275, 316 Lees, L., 171, 171 (23), 172, 173. 178, 178 (30), 180, 211, 212, 213 Lenard, M., 197, 214, 215 Leonhardt, H., 320c, 320f Leont’ev, A. I., 331, 333 (5). 334, 353, 355, 377 Leontiev, A. S., 48, 50, 84 Leppert, G., 145 (55), 161 Levy, E. K., 304, 305, 319 Lewis, J. H., 204. 217 Levinsky, E. S., 197, 215 Li, T. Y., 204, 217 Librizzi, J., 331, 333, 334, 341, 353, 377 Lieblein, S., 274 (102), 318 Lieu, B. H., 368, 379 Lightfoot, E. N., 108, 109, 160 Lighthill, M. J., 146, 161 Lin, C. S., 88, 91, 98, 103, 108, 110, 160 Lin, S. H., 185,186,213 Livingood, J. N. B., 322,377 Lokshin, V. A., 31 (16), 33.84 Longsderff, R. W., 248 (65). 275 (65), 316 Lorenz, G. C., 368, 379 Lucas, J. G., 358, 379 Lui, C. Y., 171, 172, 173, 178, 212 Lundgren, T. S., 184 (53), 191 (74), 192 (74), 213 Lyman, F. A., 307, 307, 308, 319

Mc McClellan, R., 200,216 McCroskey, W. J., 204,217 McDougall, J. G., 204 (134), 217 McEntire, J. A., 254,255,317 McFadden, P. W., 50 Mclnteer, B. B., 279,319 McKinney, B. G.,271,301,315,317 McSweeney, T. I., 240, 253, 266, 315

M Mabuchi, I., 348, 353 (43), 378 Madsen, J., 273 (94), 317 Maeder, P. F., 119, 161 Maise, G., 190, 214

386

AUTHOR INDEX

Mani, R. U. S., 133, 161 Manning, F. S.. 89 (5c, 5d). 160 Marcus, B. D., 306, 319, 320b, 320e Martin, B. W . , 367, 379 Maslach, G. J., 209, 218 Maslen, S. H., 184, 197, 213, 215 Maxwell, J. C.. 163, 210 Mederios, A. A., 53 (39, 85 Metzger, D. E., 348, 350, 354 (44), 354, 355 (49, 358, 361, 370, 375, 378, 379 Metzner. A. B., 103, 160 Michels, A,, 9 Mikami, H., 181, 213 Milford, C. M., 358, 379 Marto, P. J., 256, 256, 257, 317 Miller, J. T., 197, 215 Miller, D. L., 310, 319 Miller, P. S., 368, 379 Mirels, H., 206, 207, 217, 218 Miropolsky, 2. L., 30, 38 (21), 39, 49, 50, 66, 67, 84, 85 Mitchell, J. E., 88, 140, 160, 161 Mitsumura, H., 129 (38), 161 Mizushina, T., 108, 110, 122, 128, 160, 161 Moritz. K., 32Oc (148). 320d. 320f Morosova, N. A., 9 Morse, T. F., 176, 179, 212, 213 Moss, R. A., 258, 317 Mosteller, W. L., 256, 256, 257, 317 Moulton, R. W., 103 (IS), 108 (18), 110 (18), 160 Mukerjee, T., 367, 379 Mullin, T. E., 66 (49), 85 Muramoto, H., 108 (24), 110 (24), 160

N Nakajima, Y.,128 (37), 161 Neal, L. G., 236 (28), 237,239,249, 266 (28), 304,306,314 Newman, J., 90 (S), 160 Nicoll, W . B., 339 (18. 19). 348, 359 (IS), 378 Nikolayev, V. S., 206, 217 Nishiwaki, N., 34, 84, 346, 353 (41), 378 Nissan, A. H., 308, 319 Nohira. H., 129 (39). 161 Noordsij, P., 130, 161

Novikov, I. I., 10 (3,83 Nukiyama, S., 88, 160 0

Oberai, M. M., 197, 215 Ogino, F., 108 (24), 110 (24), 160 Oguchi, H., 204, 204 (129, 130), 216, 217 Okada, S., 132, I61 Oman, R. A.. 184 (57), 192,213 Oppenheim, A. K., 187, 209,214

P Pagani, C. D., 171 (28, 29). 172 (28, 29), 176 (29), 178 (29), 181, 212 Page, F., 52 (28), 84 Pai, B. R., 339 (17, 23), 350, 355, 359, 378, 379 Pan, Y.S., 204, 206, 208, 217 Pantazelos, P. G., 275 (1 1 9 , 318 Pappell, S. S., 338,344,352 (39). 355.358, 358 (53), 359, 368 (1 I), 370, 377 378, 379 Parker, G. H., 235, 284, 285, 314 Parker, J. D., 66 (49), 85 Parthasarathy, K., 367, 379 Patanker, S. V., 339, 378 Pawlowski, P. H., 320e (165). 320g Payne, H., 170 (21), 211 Pedersen, D. R., 354 (44), 378 Perlmutter, M., 175, 212 Petersen, H. L., 181, 213 Petukhov, B. S.. 48, 49, 50,84 Phillips, E. C., 236 (29), 241, 243, 259, 261, 303, 314, 316 Phillips, W. F., 178, 212 Picus, V. J., 39, 66 (SO), 67 (50), 86 Pikus, V . U., 38 (21). 84 Pitts, C. C., 145 ( 5 9 , 161 Potter, J. L., 197, 198, 215 Presler, A. F., 52 (31), 85 Probstein, R. F., 164 (3), 187, 187 (3), 194 (3), 195, 196 (64). 197, 198 (64), 199, 203 (3), 204, 205, 206 (64), 208, 211, 214, 215, 217 Protopopov, V. S., 48 (22), 49 (22), 50, 84 Pruschek, R., 32Oc (148), 320d (153, 154), 320f

AUTHORINDEX Pulling, D. J., 40 (51), 42, 66 (51), 67 (51), 86 Putnam, G. L., 88 (2). 91 (2), 98 (2), 103 (18), 108 (2, 18), 110(18), 160

Q Quast, A., 320c (169), 320g Quataert, D., 320c (170), 320g

R Ramsey, J. W., 341 (27), 341 (28), 348. 350, 370, 371, 371 (28, 71, 72). 372, 373, 373 (28), 374,375 (71), 378,379 Ranken, W. A., 240, 247(48), 271, 242, 315 Ranz, W. E., 88, 145, 160 Rask, R. B., 348, 353 (42), 359 (42), 378 Ratonyi, R., 169, 183 (12), 211 Rebont, J., 197, 215 Reddy, K. C., 192, 214 Redeker, E. R., 260, 317, 368, 379 Reiss, F., 32Oc (147), 320f Reiss, L. P., 110 (25), 160, 320e (168), 3208 Riddell, F. R., 196, 197, 214 Roberts, J. J., 275 (108), 318 Rott, N., 197. 214 Rotte, W., 130, 161 Rowlinson, J. S., 6, 6 (2), 7 (2), 10 (l), 83 Ruehle, R., 320c, 320f Ruhle, V. R., 275 (109), 318 S

Saarlas, M., 338, 377 Sabersky, R. H., 59, 60, 85 Sage, B. H., 52 (28). 84 Sakaguchi, I., 122 (359, I61 Salmi, E. W., 270 (88), 317 Samuel, A. E., 346, 352 (37), 378 Sandberg, R. O . , 50 Sandborn, V. A., 194 (83), 214 Sauer, F. M., 190, 193, 202 (128), 214, 216 Scesa, S., 343 (32), 344, 351, 361, 379 Schaaf, S. A., 164 (2), 165 (2), 168, 168, 187, 188, 190 (2), 190, 202 (128), 211, 216

387

Schamberg, R., 170 (17), 191, 192 (75, 76), 211, 214 Scheidegger, A. E., 236, 242 (34). 315 Scheuing, R. A., 184 (57), 192, 213 Schindler, M., 297, 319, 320c (148), 320f Schins, H. E. J., 2711(90), 274, 289, 317 Schlichting, H., 340, 378 Schlinger, W. G., 52 (28), 84 Schlitt, K. R., 32Oe (164), 320f Schloerb, O., 320d (153), 320e Schmidt, E., 5 5 , 85 Schmidt, K. R., 31 (14), 33, 84 Schoenhals, R. J., 45, 62, 63, 85 Schretzmann, K., 274, 318, 320c (147), 320e (168), 320f, 320g Schwartz, J., 246, 274, 294, 294, 316, 320a (58, 136), 320e Schutz, G., 101, 110, 160 Seban, R. A., 339, 343, 344, 346, 351 (32), 352, 354 (14), 355, 361, 377, 3 78 Sengers, J. V., 9 Serafimidis, K.,133 (44), 161 Shair. F. H., 144 (53). 161 Shavit, G. 331 (3), 332, 338, 346, 353, 353 (3), 354, 377 Shaw, P. V., 108, 110, 112, 160 Shefsiek, P. K., 248 (67), 316 Sheldon, D. B., 182 (51), 213 Shen, S. F., 170 (18), 211 Sherman, F. S., 182, 200, 213, 216 Shiralkar, B. S., 37, 40, 84 Shitsman, M. E., 30, 31 (13), 32, 33, 36, 37, 39, 49, 50, 66, 66 (50), 67, 67 (50), 84 Shlosinger, A. P., 238, 254, 266, 267, 277, 277 (44), 277, 315 Shorenstein, M. L., 204, 205, 217 Short, B. E., 52 (29), 84 Sibulkin, M., 146, 161 Silver, A. H., 352, 344, 378 Simon, H. A., 56, 57, 85 Sivasegaram, S., 357, 358, 379 Skirvin, S. C., 344, 352 (35). 371 (38), 378 Skripov, V. P., 57, 58, 60, 85 Sleicher, C. A., Jr., 52 (27), 84 Smith, F. G., 4, 83 Smith, J. M., 49, 84 Soliman, M. M., 320b, 320e

388

AUTHOR INDEX Townsend, A. A., 24, 83 Trefethen, L., 219, 313 Tribus, M.. 81, 81, 86, 330, 377 Trout, A. M., 344,352 (39). 355,358,378 Trub, J., 116 (28), I I8 (28), 120 (28), 160 Tsou, F. K., 346, 361 (61, 62, 63), 364, 365, 366, 367 (61, 62, 63), 379 Tsuchida, A., 346, 353 (41), 378 Turner, R. C., 240, 274(51), 315,316 Tzederberg, N. V., 9

Solomon, J. M., 206, 217 Son, J. S., 108, 110, 114, 160 Sonnernann, G., 53,85 Spalding, D. B., 132 (43), 161, 339, 377, 378 Spangenberg, W. G., 200 (122), 216 Sparrow, E. M., 169, 184, 185, 186, 191, 192, 211, 213, 214 Spiers, D. M., 358, 379 Springer, G. S., 169, 174, 180, 181 (47), 182 (36, 51), 183 (12). 211, 212, 213 Stalder, J . R., 189, 190, 200, 201, 214 Starner, K. E., 244, 245, 316 Stein, B., 320b (143), 320e Steiner, G., 320c (146), 320f Stewartson, K., 206, 217 Stine, H. A., 200 (123), 216 Stollery, J. L., 331, 333 (9,335, 342 (29), 350, 352 (29), 353, 359, 361, 377, 3 78 Streckert, J. H., 304, 319 Street, R. E., 204, 217 Su, C. L., 169, 178, 211 Sutey, A. M., 97, 98 (7), 160 Swank, L. R., 348, 358 (54), 361 (54), 379 Swensen, H. S., 50

Valensi, J., 197, 215 van der Hegge Zijnen, B. G., 118, 161 van Driest, E. R., 53, 85, 133, 161 Van Dyke, M., 187, 197, 214, 215 Vidal, R. J., 205, 217 Vikrev, Y. V., 31 (16). 33, 84 Vincenti, W. G., 170 (22), 171 (22), 211 Vogtlander, P. H., 116, 118, 160 Vrebalovich, T., 200, 216

T

W

Tachibana, F., 129 (38). 161 Takao, K., 193, 214 Takashima, Y., 181 (50), 213 Talmor, E., 348, 355, 379 Tanaka, H., 34, 84 Tanazawa, Y., 88 (5b), 160 Tanneberger, H., 8 (3), 83 Taylor, J. F., I t 9 (32), 161 Teagan, W. P., 174 (36), 182 (36), 212 Teilsch, H., 8 (3), 83 Tewfik, 0. E., 202, 216 Thomas, L. B., 168, 212 Tien, C. L., 273, 317 Ting, L., 197 (89, 103), 215 Tironi, G., 172, 175, 212 Tobias, C. W., 89 (6), 96 ( 6 ) , 98 (6, 1 I), 100, 130 (40), 160, 161 Tong, L. S., 50 Touba, R. F., 50 Touryan, K., 190, 214

U Uhlenbeck, G. E., 170(16), 171, 172, 211, 212 Urushiyama, S., 145, 161

V

Wachman, H. Y., 168, 169, 211 Wageman, W. E., 280, 319 Wagner, C., 98, 160 Wan, S. F., 169, 180, 211 Wang-Chang, C . S., 170 (la), 171, 172, 211,212 Waldron, H. F., 206. 218 Waldmann, H., 320c, 320f Waters, E. D., 320a, 320e Watson, A., 31, 50, 52 (12), 54 (12), 66 ( 1 2), 84 Welander, P., 170 (15), 17 1, 211 Welch, C. P., 49, 84 Weltmann, R. N., 200 (121), 216 Werner, R. W., 274, 276, 294 (1 18), 295, 318, 319 Westwater, J. W., 81, 81, 86 Whitelaw, J. H., 339, 348, 350, 352, 352 (36), 355, 356, 357, 358, 359, 378, 379

AUTHOR INDEX Whiting, G. H,, 223 (9), 235 (21), 273 (9), 274, 314 Wiederecht, D. A., 53, 85 Wieghardt, K., 336, 337 (lo), 338, 343, 344, 370, 377 Wilhelm, R. H., 89 (5c, Sd), 160 Wilke, C. R., 89, 96, 98, 98 (1 I), 130 (40), 160, 161 Wilkson, D. B., 206, 218 Williams, J. J., 350, 358, 379 Willis, D. R., 169, 171, 175, 177, 178, 179, 198, 211, 212, 213, 216 Wilson, D. J., 340 (24), 348, 368 (24), 3 78

Wilson, M. R., 197, 215 Winovich, W., 200 (123), 216 Wittliff, C. E., 197, 215 Wossner, G., 297, 319, 320c Wolfshtein, M., 339 (21), 378 Woodruff, L. W., 368, 379 Wragg, A. A., 133, 161

389

Wyatt, P. W., 308, 309, 319 Wyatt, T., 276, 319

Y Yasuhara, M., 206, 217 Yokoyama, S., 128 (37), 161 Yoshihara, H., 197 (96), 215 Yoshikata, K., 88 (5a), 160 Yoshizawa, S., 132 (42), 161 Yoskioka, K., 66 (48), 85 Yuan, S. W., 279, 319

Z Zakkay, V., 197(89, 102, 103), 215, 367, 3 79

Zerkle, R. S., 236 (27), 247, 277 (27), 314 Ziering, S., 171, 172, 212 Zimmermann, P., 320c (149, 152), 320d, 32Oe (160, 161, 166), 320j, 320g Zuber, N., 81, 81, 86

Subject Index heat transfer near, 1-86 molecular structure near, 8 physical properties near, 3-1 5 thermodynamic properties near, 3 transport properties near, 8

A Adiabatic wall temperature, 326 Accommodation coefficient, 165 B Boiling, 74 film, 78 nucleate, 76 pseudo, 87 Boundary layer flow, critical region, 17 Bulk compressibility, 7 isentropic, 7 isothermal, 7 Bulk temperature, 185 Buoyancy effects, 17, 20, 69

D Dissipation effects, 8, 21

E

C Channel flow, critical region, 19 Compressibility, bulk, 7 Corresponding states principle, 5 Critical isochor, 6 Critical isotherm, 4 Critical region boiling near, 74 boundary layer flow, 17 buoyancy effects near, 17, 20, 69 channel flow, 19 compressibility near, 7 defined, 2 energy equation, 15 equation of motion, 15 forced Convection near, 25 forced and free convection combined near, 66 free convection near, 55 heat capacity near, 6

390

Electrochemical method in transport phenomena, 87-1 62 application to mass transfer measurements, 94-136 i n free convection, 98-103 cylinder, 101 plate horizontal, 100 vertical, 98 sphere, 103 in forced convection, 103-136 local mass transfer fluctuation intensity, 125 transfer coeficient, local, 125 transfer factor, space-averaged, 124 artificially waved liquid layer, 133 concentric rotating cylinder annulus, 128 cross flow, 116 mass transfer coefficient local, 120 space-averaged, 1 18 falling liquid film, 132 jet flow, 135 packed beds, 134 rotating and vibrating bodies, 130

SUBJECT INDEX tube flow, 103 fluctuation of mass transfer rates, 112 laminar, 112 mass transfer coefficient, dynamic response of, I I6 turbulent entry region, 110 fully developed, 103 application to shear stress measurements, 136-144 in boundary layer, 140, 142 in well-developed flow, 138, 140 application to velocity measurements, 144-158 fluctuating velocity agitated vessel, 155 tube flow, 153 time-smoothed velocity boundary layer, 1 5 1 tube flow, 147 Enthalpy of evaporation, reduced, 6 Equation of state (van der Waals), 3

F Film boiling, 78 Film cooled wall temperature, 363 Film cooling, 321-379 analysis, 330-342 applications, 358 blowing rate parameter, 331, 339 correlations, 334-337 defined, 322 effect of injection angle on, 358, 373 effectiveness, 324, 326-333 high-speed flow, 327, 329 impermeable wall concentration, 329, 330, 355 incompressible flow, 327 isoenergetics, 328 experimental studies, 342-359 free stream turbulence effect, 354 heat sink model, 330, 337 heat transfer measurements, 360 high speed flow, 327, 340, 361 incompressible flow, 326, 338, 351 influence of boundary layer thickness, 359

391

injection of air into air, 351 through discrete holes, 341 isoenergetic, 328 large temperature difference effects, 358 mass transfer analogy, 329, 359 measurements, 371 porous injection in, 353, 355 slot geometry effects on, 351, 356 stagnation region, 368 surface effects on, 334 three-dimensional, 341, 369 two-dimensional, 323, 338, 340, 351, 361 compressible flow, 340, 361 incompressible flow, 351 variable free stream velocity effects in, 338, 354 Film heating, 342, 351 Forced convection agitated vessel, 122 artificially waved liquid layer, 133 concentric rotating annulus, 128 critical region, 25-55 cross flow, 116 falling liquid film, I32 jet flow, 135 packed beds, 134 tube flow, 103 Free convection critical region, 55-66 cylinder, 101 plate horizontal, I00 vertical, 98 sphere, 103 Free molecule flows, 164, 169, 184, 187 Free stream acceleration effects, 18

G Gas-to-gas film cooling, 324, 351 Grashof number, 15, 64

H Heat flux, turbulent, 22, 52 Heat pipe, 219 ammonia filled, 320a applications, 273

392

SUBJECT INDEX

control, 276 cryogenic, 320 a definition of, 220 description of, 220 flexible type, 320b fluids for, 235 functions of, 224 lithium-filled, 320c magnetic field effects on, 320b material test for, 235-249 compatibility of components, 246 life tests of components, 246 wicks, 236 working fluid, 235 mercury-filled, 320a operating characteristics of, 249-273 basic studies, 270 heat transfer limit investigations, 250 phenomenology, 220 rotating type, 223 sodium-filled, 320c surface tension effects, 3 2 0 ~ surveys, 234, 320d theory, 278, 320d threaded wall-artery wick, 320d transient behavior of, 320e types of, 220 vibrational environment effect on, 320 b wicks of, 221, 236 Heat transfer near critical point, 1-86 boiling, 74-82 film, 78 nucleate, 76 pseudo, 81 equation of motion and energy, 15-25 boundary layer flow, 17 buoyancy effects, 17 dissipation, 18 free stream acceletation effects, 18 channel flow, 19 acceleration effects, 21 buoyancy effects, 20 dissipation, 21 turbulent shear stress and heat flux, 22 effect of variable properties on, 22 forced convection, 25-55 correlation of experimental data, 43-51

acceleration, buoyancy, and dissipation effects, 44 existing correlations, 48 limiting form of correlations, 46-48 small temperature differences, 46 large temperature differences, 47 experimental data, 31-42 gaps in experimental data, 38 local heat transfer coefficient, 35 experimental measurements, 26-3 1 semipirical theories, 5 1-54 presentation of data in terms of dimensionless groups, 30 heat transfer coefficient, 29 local conditions, 27 free convection, 55-66 experimental results, 55 temperature differences, large and small, 56, 58 theoretical methods and correlations, 63 basic correlations, 64 theoretical correlations, 65 forced and free convection combined, 66-74 experimental results, 67 heat transfer deteriorations, 68 buoyancy effects on shear stress distribution, 69 influence of wall heat flux, 74 local deterioration in heat transfer coefficient, 71 shear stress distribution effects on turbulence, 70 physical properties near critical point, 3-1 5 molecular structure, 8 property variation effects on heat transfer, 10 effects of temperature difference,

11

limit as temperature difference tends to zero, 12, 14, 15 thermodynamic properties, 3 compressibility and velocity of sound, 7 heat capacity, 6 law of corresponding states, 5 van der Waalsp model, 3

SUBJECT INDEX transport properties, 8 Heat transfer in rarefied gases, 163-218 accommodation coefficients, 165 external flows, 187 free molecule flow, I87 ternpcrature jump (slip) regime, 192 transition regime (M I), 193, 194 cylinders, 200 cones, 206 flat plate, sharp leading edge, 202 spheres, 202 stagnation point, blunt body, 196 gas at rest, 169 free molecule conditions, 169 temperature jump approximation, 1 70 transition regime, 170 concentric cylinders, 178 parallel plates, 17I internal flows, 183 free molecule flow, 184 temperature jump (slip) regime, 184

I Isoenergetic temperature, 327, 328

K Knudsen number, 164

L Liquid film cooling, 324

M Mass transfer analogy, film cooling, 329, 359 Mass transfer measurements, 94 electrochemical method, 87 free convection, 98 forced convection, 103 Momentum accommodation coefficient, 166

N Nucleate boiling, 76 Nusselt number, 30, 65, 185

393 P

Physical properties, critical region, 3 Principle of corresponding states, 5 Pseudo boiling, 87

R Radiation and rarefaction interaction, 177 Rarefaction and radiation interaction, 177 Rarefied gases, heat transfer in, 163 external flows, 187 gas at rest, 169 internal flows, 183 stagnation region, 196 transition regime, 170, 193, I94 Recovery factor, 189 Recovery temperature, 189, 326, 327 Reduced enthalpy of evaporation, 6 Reduced isotherms, 6 Rotating heat pipe, 223

s Shear stress, turbulent, 22, 55, 70 Shear stress measurements, electrochemical method, 136 Slip velocity (thermal creep), 185 Sound velocity, 8 Stagnation point heat transfer, rarefied gases, 196 Stagnation temperature, 189 Stanton number, 189 Subcooled vapor, 4 Subcritical pressure, 2 Subcritical temperature, 8 Supercritical conditions, 2 Supercritical fluid, 2 Supercritical heat transfer, 2 Supercritical pressure, 2 Supercritical temperature, 8 Superheated liquid, 4

T Temperature dependence of thermal conductivity, 9 of viscosity, 9 Temperature jump approximation, 170

394

SUBJECT INDEX

Temperature jump (slip) regime, 184, 192 Thermal accommodation coefficient, 165 Thermal accommodation, incomplete, 172 Thermal conductivity, 8-1 3 near critical region, 13 temperature dependence of, 9 Thermal creep (slip velocity), 185 Thermal diffusivity, 14 Thermodynamic properties, critical region,

3

Transpiration cooling, 322, 323 Transport properties, critical region, 3 Turbulent flow, critical region, 22 heat flux, 22, 52 shear stress, 22, 55, 70

V Vapor chambers, heat pipe, 223 Velocity of sound, 8 Velocity measurements, electrochemical method, 144 Viscosity, 8-10 temperature dependence of, 9

W Wick inclination, heat pipe, 320b heat pipe, 221, 236 Wicks thermal conductivity, 320b