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

<0

is not satisfied, and the fluid would separate into two distinct phases. The regions bc and de represent, respectively, superheated liquid and subcooled vapor; the extent of these metastable regions is indicated by broken lines in Fig. 1. Equilibrium between the liquid and vapor phases (with a plane interface between them) is achieved with states marked b and e. (Note that unstable equilibrium between liquid and vapor can be achieved with a curved interface along bc and de. In these cases, surface tension forces at the bubble or droplet surface lead to a difference between the liquid and vapor pressures.) The isotherm marked o in Fig. 1 is known as the critical isotherm and

HEATTRANSFER NEAR

CRITICAL POINT

THE

5

passes through the critical point. It represents the isotherm for which the points bcd and e all coincide, thus giving a point of inflection at the critical point (CP on Fig. I), so that

(ap/aV)c,= 0 (aZpplaP2);

=0

2. The Law of Corresponding States The behavior of the critical isotherm, as embodied in Eqs. (2) and (3), can be used to eliminate the constants a and b in the van der Waals equation as follows: Equation (1) may be written as

p

= R T / ( P - b)

-

alpz

and, using Eqs. (2) and (3), -RTc 2a (*);= 0 = (VC b y t($)+ 2RTc -- 6a -

(P)S

(Be

(P)4

1 3 a/b2;

T e = 8a/27bR

- b)3

from which we find that

VC

= 3b;

=

Introducing the “reduced” quantities,

the van der Waals equation becomes

( p * + 3/(V*)’)(3V* - 1)

= 8T*

(4)

An interesting aspect of this equation is the fact that it involves only

p*, V*, and T* and not any quantities that are characteristic of a

particular substance. I n the above form it applies only to substances for which the van der Waals equation is true; however, the same principle may be stated in more general terms by asserting that there is a unique relationship between p*, V*, and T* for all substances. This is known as the princzple of corresponding states and is frequently stated in the form 2 = Z ( p * , T*)

(5)

6 where

W. B. HALL

z =~ P ~ R T

(For those substances for which van der Waal’s equation is true Zc = pcpe/RTc = 318,

and

Z

=

$P*V*/T*)

T h e “reduced” isotherms, p* = p*(V*), as determined by the principle of corresponding states, have the same shape for all substances; we may therefore conclude that for the same value of T*, all substances conforming to this principle must have the same reduced saturated vapor pressures and the same reduced specific volumes of saturated vapor and liquid. Further, the reduced enthalpy of evaporation, h,,/RTc must be the same function of T* for all substances. Thus h,,/RTC =f(T * ) (6) (This function tends to a constant value of approximately 10 at temperatures appreciably below the critical temperature.) T h e importance of the principle of corresponding states in the present context is that it provides a qualitatively accurate description of thermodynamic behavior near the critical point.

3 . Heat Capacities near the Critical Point Rowlinson (1) has reviewed the state of knowledge concerning singularities in the thermodynamic and transport properties near the critical point. While the specific heat at constant volume is always finite for a van der Waals gas, it has been shown experimentally that this model is inadequate at the critical point; it appears that cv does in fact become infinite but that the infinity is much weaker than that in cp , the specific heat at constant pressure. In most heat transfer problems we are more concerned with the value of cp ,and, in this case, the van der Waals model does predict a value of infinity at the critical point; this may be demonstrated as follows: T h e difference in the specific heats is given by the thermodynamic identity c, - c y =

w%J/w v12/(aP/av)T

and it may be shown (2) that the slope of the critical isochor ( 8 ~ j a T ) , ~ is equal to the slope of the vapor pressure curve at the critical point, which is finite. On the other hand, (@/8V),c is zero, so that cp - cv becomes infinite. T h e question of the precise nature of the singularities at the critical

HEATTRANSFER NEAR

THE

CRITICAL POINT

7

point is somewhat academic in most practical situations because of the extreme difficulty of achieving critical conditions precisely. In many heat transfer systems the pressure will be maintained somewhat above the critical value; under these circumstances the singularities will be avoided although the property variations may still be severe. For example, the peak in the specific heat is large even at pressures considerably greater than the critical, as may be seen from the data for CO, shown in Fig. 2. !I 18

-

16

U

.$.'4

- 12 7

P

-P

U

c

10

u

c

a

v)

6 4 2 20

30

40

50

Temperature ("C,

FIG. 2. Specific heat (at constant pressure) of carbon dioxide near the critical point.

4. Compressibility and the Velocity of Sound

T h e compressibility of the fluid may be defined for an isothermal or for a reversible adiabatic (isentropic) process as follows: Isothermal coefficient of bulk compressibility,

K~

=

-(aV/ap),/V

Isentropic coefficient of bulk compressibility,

K~

=

-(aV/ap),/ V

I t will be evident from what has been said in section 11, A, 1 that the isothermal coefficient, K ~ is, infinite at the critical point. It may be shown (2) that the two coefficients are related in the following manner: l/KS

= l/KT

+ Tv(ap/aT);/CV

(7)

W. B. HALL It may also be shown that the vapor pressure curve is continuous with the critical isochor beyond the critical point. At the critical point, therefore, (i?p/i?T),is equal to the limiting value of the slope of the vapor pressure curve, which is finite. Provided that the value of cy at the critical point is not zero, the isentropic compressibility will be finite. T h e velocity of sound, c, is given by From Eqs. (7) and (8) it will be clear that a maximum in c y will lead to a maximum in K, and a minimum in c near the critical point. Measurements in carbon dioxide at the critical temperature have indicated a minimum velocity of sound of 140 mjsec at a pressure about 0.5 atm higher than the critical pressure; at the critical pressure the measured value was 172 m/sec, compared with the calculated value of 155 mjsec (3).

B. MOLECULAR STRUCTURE NEAR

THE

CRITICALPOINT

T h e transition from a subcritical to a supercritical temperature at a slightly supercritical pressure does not, of course, involve a change of phase. While from a macroscopic standpoint the change of density is continuous, there is conclusive evidence that on a molecular scale the fluid is far from homogeneous. T h e phenomenon of “critical opalescence’’ indicates the presence of a structure large enough to produce scattering of light; moreover, X-ray diffraction patterns characteristic of the liquid have been detected at supercritical temperatures when the macroscopic density is much less than that of the liquid. An interesting survey of the information on structure has been made by Smith (4). It appears that as the temperature is increased (at a slightly supercritical pressure), the liquid structure gives way to liquidlike clusters in a matrix of gas; these reduce in size until the situation is virtually one of a gas with a high degree of association. I n most Auid flow and heat transfer problems it is probably reasonable to regard the fluid as a continuum because the dimensions of the system will usually be much greater than the scale of the molecular structure.

C. TRANSPORT PROPERTIES T h e pattern of variation of viscosity and thermal conductivity with temperature and pressure is illustrated in Figs. 3 and 4, which refer to carbon dioxide in the near-critical region. T h e theoretical basis for describing the variation of transport properties is less well-developed than that for the thermodynamic properties, and the problems of

HEATTRANSFER NEAR

c

20

THE CRITICAL POINT

30 40 Tern p e r a t u re ( "C I

9

50

FIG.3. Viscosity of carbon dioxide near the critical point.

I

20

I

I

30

40

Temperature

I

50

('C)

FIG.4. Thermal conductivity of carbon dioxide near the critical point: (a) data of N. V. Tzederberg and N. A. Morosova, Teplmergetika No. 1, 75 (1960); (b) data of J. V. Sengers and A. Michels, Progr. Int. Res. Thermodyn. Tramp. Prop., Pap. Symp. Thermophys. Prop. 2nd 1962 (1963).

W. B. HALL measurement are even more severe. I t appears that the thermal conductivity certainly becomes infinite near the critical point ( I ) , but there is less certainty about the viscosity. T h e effect of viscosity variations on fluid flow and heat transfer to low pressure gases is generally dealt with by using approximate expressions of the form in which T , is some reference temperature, and ps is the corresponding viscosity. It is quite clear that this technique will fail completely near the critical point, and one must seek a more general relationship between the transport properties and the temperature and pressure. One possible approach is to attempt to describe the transport in terms of “reduced” quantities in an analogous manner to the description of thermodynamic properties by the principle of corresponding states. Such an approach has been proposed by Borishansky et al. (5) and has been applied by them to the generalization of heat transfer processes at subcritical pressures. Hirschfelder, Curtiss, and Bird (6) have reviewed various methods by which the transport properties may be correlated. T h e most suitable, for our present purpose, relates the viscosity and thermal conductivity to the reduced temperature and reduced pressure in the following manner

(Pc )2131 - P*(P*, T*)

P* = P ( R p ) l / 6 / [ ~ 1 / 2

k*

= KM1’2(RTc)1’6/[R(pc)2’9] = A*@*, T*)

While the extension of thermodynamic similarity to heat transfer is certainly interesting, it seems likely that when it is taken together with the conditions for dynamical similarity, the resulting requirements for complete similarity will be extremely restrictive in all but the simplest cases. T h e matter is considered in more detail in Sections IV, C and V, B, in which the correlation of experimental data for forced and free convection is considered. T h e lack of accurate data on transport properties for most fluids near the critical point makes it essential in reporting experimental work to quote the raw data so that new theories or proposals for generalization may be tested against them.

D. THEIMPLICATIONS OF PHYSICAL PROPERTY VARIATION ON HEATTRANSFER T h e problem of physical property variation in an extreme form is a central feature of all near-critical heat transfer processes. I n many cases

HEATTRANSFER NEAR

THE

CRITICALPOINT

11

it produces a quantitative difference in behavior, and in some cases a phenomenon appears which at first sight has no parallel in constantproperty heat transfer. These effects will be considered in detail later; in this section we consider some of the more general implications of physical property variation. 1. The Effect of Temperature Diflerence

Heat transfer processes may be divided into two categories depending upon whether or not physical property variation forms an essential part of the process. Conduction and forced convection both may take place in the absence of variation in any property but the heat content of the substance involved. Boiling and condensation, on the other hand, involve phase changes with distinct properties for the separate phases; free convection involves a fluid flow pattern which is a direct result of density variation caused by heating or cooling. T h e existence of a meaningful constant property situation is frequently useful in analyzing forced convection data; the concept of a limiting value of the heat tranfer coefficient as the temperature difference tends to zero is often used to separate the effects of property variation from the inherent heat transfer and fluid flow processes. Provided that they are relatively small, property variations may be represented by the inclusion of an empirical function of temperature difference. From a mathematical standpoint, the constant property situation is important because the fluid flow and heat transfer problems are separable, and the energy equation is linear in temperature; complicated boundary conditions may therefore be built up by the superposition of solutions with simpler boundary conditions. In contrast to the case of forced convection, boiling is a process which necessarily involves variations in physical properties throughout the fluid. It is true that property variations within the separate liquid and vapor phases may be small, but the relative proportions in which these phases are present will, in general, depend upon the rate at which heat is added to the system, and therefore on the temperature difference between the fluid and the heating surface. There is, therefore, no meaningful limit of the heat transfer coefficient for boiling as the temperature difference tends to zero. T h e necessity for a simultaneous solution of the equations of motion and energy renders free convection a more difficult problem, at least from a mathematical standpoint, than forced convection. T h u s even when all properties (except density) are constant, superposition is prohibited by the nonlinearity of the equations; neither does the heat

12

W. B. HALL

transfer coefficient tend to a constant value as the temperature difference decreases. Nevertheless, it is possible to make a simplification when the property variations are small. This arises because the term in the momentum equation that links it (through temperature variation) with the energy equation involves the difference between the fluid density and the density that would be obtained if there were no heat transfer; this term is still important even when changes in density are entirely negligible in those other terms of the equation where it occurs as a factor. This simplification, which is implicit in most free convection theory, is of doubtful validity near the critical point at all but the smallest temperature differences.

2. The Limit as Temperature Di#erence Tends to Zero While this limit may be of little practical significance, it is often useful, as mentioned, as a device for separating the effects of property variation from heat transfer and fluid flow phenomena. It has been suggested in the preceding section that this limit is relevant only to the cases of conduction, forced convection, and, in a more restricted sense, to free convection; these cases are considered in more detail in the following. a. Conduction. Consider the problem of steady conduction through a slab of fluid at a slightly supercritical pressure, the two surfaces of the slab having temperatures which span the critical temperature. If the temperature difference is large, then there will be a thin layer of fluid in the interior of the slab in which the thermal conductivity exhibits a peak. If the thickness of the layer is small in relation to the thickness of the slab, the effect of the peak in conductivity will be negligible. If, however, the temperature difference is small, the whole of the fluid may have a conductivity equal to the peak value. Is it then possible that a reduction in temperature difference could, by increasing the average conductivity of the fluid, increase the heat flux through the slab ? T h e answer may be obtained as follows: From the definition of conductivity, k, =

--~(~)a~/ax

where q is the heat flux through the slab and x is the distance in the direction of heat flow. Thus

where b is the thickness of the slab, T I is the temperature of the cold surface, and Tz is the temperature of the hot surface. Equation (10)

HEATTRANSFER NEAR

THE

CRITICAL POINT

13

is illustrated in Fig. 5 , which is drawn for a fluid which has an infinite value of K at the critical temperature, but for which JK(T)dT through the critical temperature is finite.

'I

'2

Temperature, T

+

FIG.5. Illustration of conduction through a supercritical pressure fluid, Eq. (10).

(Note: if, following Rowlinson (I),we express K( T) near the critical point of the form k = C I T - TC (-0.2 then

1K

dT

=

C I T - TC1O.*/0.8

which remains finite as we pass through the critical point.) Referring to Fig. 5 and Eq. (lo), we see that for a slab of fixed thickness, the heat flux is proportional J k( T) dT. If TI is held constant, then q can only decrease as T, is decreased. On the other hand, if the temperature

14

W. B. HALL

difference is maintained at a constant small value, the heat flux will, of course, change as the temperatures traverse the range. T h e matter may be summarized by saying that it is not the fact that the conductivity may become infinite which is important, but that its integral through the critical temperature remains finite. Unsteady conduction depends upon the density and specific heat of the fluid as well as on the thermal conductivity. With constant properties it is possible to group these three quantities into a single parameter, klpc,, the thermal diffusivity, which governs the rate of transmission of temperature changes through the medium. With variable properties the parameters are k/pc, at some reference condition plus parameters which express the property variations with temperature. T h e question then arises as to whether the fact that k/pc, becomes zero at the critical point (because c, has a stronger infinity than that in k at the critical point) has any heat transfer significance. T h e matter may be resolved by referring to the unsteady conduction equation which, for a constant pressure system, may be written in the two identical forms

and

where qz ,qsl ,qz are the heat fluxes in the x-, y-, x-direction. T h e first equation contains quantities (c, and k) which become infinite at the critical point. However, it has been shown above that $ k dT remains finite through the critical temperature; the same is true of J’cP dT. T h e second equation therefore does not contain any singularities, and the only heat transfer effect will be that %/at will momentarily become zero as the temperature of the fluid passes through the critical temperature. A zero value of the thermal difiusivity does not imply in this case that the fluid is impervious to heat!

b. Forced Convection. Steady state forced convection is governed by equations of motion together with an energy equation which is formally similar to that for unsteady conduction, (see Section 111). As with conduction, therefore, the singularities in k and cp do not have the implications that might at first be attributed to them; i.e., the zero value of klpc, does not prevent the diffusion of heat into the flow. As the temperature difference tends to zero, convective heat transfer

HEATTRANSFER NEAR

THE

CRITICALPOINT

15

tends to a constant property process, and one might expect the usual correlations to apply, i.e., for a pipe flow or@

0.023 Reo.8(c,p/k)u.4

which, for a constant mass flow in the pipe gives a proportional to

kO .6,-Cp”. 4,

.4.

In other words, the heat transfer coefficient becomes infinite at the critical point as the temperature difference tends to zero. This fact has little practical significance again because the integrals of k and cp with respect to temperature remain finite through the critical temperature. Thus the heat flux will remain finite as the temperature difference tends to zero. c.

Free Convection. T h e remarks made about the singularities in k and

cp in connection with conduction and forced convection apply equally to

free convection. There is, however, an additional aspect which deserves mention. I t will be shown in Section I11 that one of the dimensionless parameters governing free convection, the Grashof Number, arises naturally from the basic equations in the form Gr

= gd3 dp/v2p

where d p is a characteristic density difference, usually that between the fluid at the heated surface and that outside the thermal layer. Because most normal fluids have a fairly constant expansion coefficient, /3, it is convenient to write the Grashof Number in the form Gr

= gd3,6A T / v 2

If this form is used at the critical point, then difficulties will arise because /3 becomes infinite. Reverting to the original expression for Gr, however, we see that the true Grashof number remains finite because dp (corresponding to a given value of A T ) remains finite as the system temperature traverses the critical temperature. 111. The Equations of Motion and Energy

Well established techniques exist for simplifying the basic equations when they are to be applied to constant property fluids in particular circumstances. Thus, for example we frequently use “boundary layer’’ forms of the equations and sometimes neglect the effects of viscous dissipation, buoyancy forces, and pressure gradients. We now consider

16

W. B. HALL

whether the same techniques may be applied to variable property flows. We begin with the equation of continuity, momentum, and energy for a two-dimensional, nonturbulent boundary layer flow (with the x-coordinate in the upward direction) Continuity:

= u - - - + Tay ax + per ax ay ay

ah pu -

Energy:

ap

ah

a4

au

(13)

I t is an advantage to use the above form of the shear stress and heat flux terms when the physical properties p and k vary, as they do, in an eccentric manner near the critical point. Thus, even though k may become infinite at the critical point, aT/ay will simultaneously become zero, the product q = --K 8Tjay remaining finite at some value between zero and the maximum value q,, at the wall. It is much easier to make reasonable estimates of q and T than of aT/ay and aujay. Similar equations may be written down for a turbulent flow. Putting u =B u', etc., for the mean and fluctuating components, we find that the equations become

+

-x -a@ dp pu-+pv-=--+--pg ax

ay

ax

a7

-

ay

where T and q are now given by

Our first concern will be to establish the conditions under which we may neglect the first and third terms on the right-hand sides of Eqs. (15) and (16) with respect to the second term in each equation. I n establishing criteria for neglecting these terms we shall employ constant property empirical relationships for such quantities as boundary layer thickness, 6, friction factor,f, and Stanton number, St. T h e criteria

HEATTRANSFER NEAR

THE

CRITICAL POINT

17

will therefore be very approximate and will certainly give no indication of the effect that the terms may have if they are large. We then investigate the relative importance of the turbulence terms in Eqs. (17) and (18), and attempt to express them in terms of time mean flow quantities. We shall consider a boundary layer and a “fully developed” channel flow. T h e latter type of flow frequently arises with constant property fluids but is less likely when properties change significantly in the direction of flow; the analysis must therefore be treated with reserve until the hypothesis can be tested adequately.

A. BOUNDARYLAYERFLOW T h e turbulent form of the equations will be used, since turbulent conditions are of the greatest practical interest. If Eq. (15) is applied to the (nonturbulent) free stream outside the boundary layer, we find -dWx

= P P , au,lax

+ p,g

(19)

which may be inserted in Eqs. (15 ) and (16), giving

-a@ pu-

ax

+ -ai pvay

au,

= Ps%%

aT ++ (Ps -P)g aY

T h e first term on the right-hand side of Eqs. (20) and (21) represents the effect of the acceleration of the free stream; the last term in Eq. (20) represents the effect of bouyancy forces; the last term in Eq. (21) represents the effect of viscous dissipation. We now consider the magnitudes of these terms in relation to & / ~ J J and aqjay. 1. Buoyancy Eflects

T h e magnitude of 1 I is of the order ~ ~ (where / 6 T~ is the wall shear stress and 6 the boundary layer thickness), and the buoyancy term therefore may be neglected if (Ps

-P)gS/70

<1

This will have a maximum value at the wall where criterion may therefore be written 2 Gr 6


P

= po

, and the

W. B. HALL

18 where Gr

Re = u,x/vs ;

= ( p S - p0)gx3/pSv,2;

if

= TO/pSus2

and x is the distance from the start of the boundary layer. Empirical data for a turbulent boundary layer with a uniform velocity and Qf = 0.0295 Re-0.2 free stream show that S/x = 0.037 (interpreting 8 as the momentum thickness). Thus the above criterion becomes, approximately, Gr/Re2

<< 1

2. Dissipation

T h e dissipation term in Eq. (21) may be neglected if

This will have a maximum value at the wall where so that the criterion may be written

where St

= qO/pSuS Ah;

7

&/ay = r o 2 / p 0 ,

E = u,'/Ah

and Ah is the enthalpy difference from surface to free stream. T h e group E is the Eckert number, ususally quoted for a constant property situation (at constant pressure) as u,"/c, d T . Using again empirical data for a turbulent boundary layer, the preceding equation becomes, approximately, E

< 1000

(25)

(The viscosity ratio has been omitted since it will always be between about 0.5 and 1.0.)

3 . The Effect of Free Stream Acceleration An acceleration of the free stream produces a pressure gradient which, acting uniformly on the boundary layer, causes it also to accelerate. T h e difference, psus &,/ax - P. &/ax, (Eq. (20)), is available for modifying the shear stress gradient, a ~ / a y and , is greatest at the wall where pU Z j a x = 0. Thus the change in the shear stress gradient at the wall can be neglected only if psu, au,/ax is much less than ~ ~ / 6 .

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19

Thus acceleration of the free stream is negligible if

i.e., if 2 s K-Re-
where

f

r

Using empirical data for a turbulent boundary layer, this reduces to the form KRe<1 (27) In a similar manner, the criterion for neglecting the acceleration term in Eq. (21) is found to be KKeE
B. CHANNELFLOW T h e terms containing the pressure gradient can also be eliminated when Eqs. (15) and (16) are applied to channel flow. Suppose we have a

W. B. HALL

20

channel formed by parallel planes a distance 2b apart, and that fully developed conditions are established (i.e., p T aii/ax independent of y ) . Integrating Eq. (15) from the wall to the center plane of the channel gives or

wheremis the mass flow through the channel per unit width. [Note that the assumption of uniform aii/ax across the channel is inconsistent with the acceleration resulting from a pressure gradient which is uniform across the channel; this would give uniform jiiaiijax. However we shall get a good estimate of the pressure gradient if we take the gradient of the mean velocity, diim/dx, in Eq. (29).] Substituting Eq. (29) into Eqs. (15) and (16) we get, noting that p. = 0,

As in the case of the boundary layer, we now consider the conditions under which dissipation, buoyancy forces, and acceleration may be neglected. 1. Buoyancy Effects

If we consider the case where dii,/dx

=

0 then Eq. (30) reduces to

+ ( a ~ / a+~ G) m -

0 = (~o/b)

And if we use the approximation -&/ay M T ~ / Z J (which is certainly true if (pm - p ) g is small), then T~ is indeterminate. The point is that when buoyancy effects are present, they are balanced by the change in &/ay from its initial value of ~ , / b .Thus, if the buoyancy force assists the pressure flow near the wall, then I a ~ / a Iy will be greater near the wall. Buoyancy forces will be significant, therefore, if they are able to y the value 7o/b. Thus once produce a significant change in a ~ / a from again the criterion becomes &Pm

-P)g/To

<< 1

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21

As in the boundary layer case, this may be expressed in terms of the Grashof and Reynolds numbers

Gr/(Re)l.s

< 0.1

(32)

where in this case

Buoyancy effects are frequently important in forced convection with supercritical fluids, and have rather large and unexpected results; these are discussed in Section VI. 2. Dissipation Following the same argument as that for a boundary layer we find

which, using empirical data for channel flow, becomes Reo.* E

< 100

where, in this case, Re = u,(4b)/vm and E

(33) =

urn2/&

3 . Acceleration Efects Whereas with the boundary layer the acceleration was imposed by applying a pressure gradient to the free stream, in the case of a channel of uniform cross section it will occur because of the expansion of the fluid as it is heated. T h e effect will be greatest at the wall where P. &/ax = 0 in Eq. (30), and the acceleration term will be negligible with respect to rO/bif m 2b dx

b T~

-

li2 du, 2r0 dx

T h e value of du,/dx may be assessed by calculating the rate of expansion of the fluid as it is heated by the specified heat flux q,, through the channel walls. As an approximation, this calculation is based on mean values of the variables taken across the channel (denoted by subscript m). We find that dii,,,ldx w e,,$ dhm/dx = 2 q o ~ m ~ m 1 ~ ~ , ,

22

W. B. HALL

T h e criterion then becomes

If we again assume that St m f / 2 this becomes

(Note that for a perfect gas this would become ATIT. T h e group 3/, dh,C,, may in fact have a value of order unity near the critical point, so that it will not normally be safe to neglect acceleration effects. However, we shall usually find that, unless the pipe diameter is very small, buoyancy effects are even more important.) Turning to the energy equation, Eq. (31), we find that the term k , / b (and the acceleration term (iifi/2b)(diiWJdx),which will usually be less than U T , / ~ except very close to the critical point where it may be of the same order) can be neglected if or (provided again that St w f / 2 )

C. THETURBULENT SHEAR STRESS AND HEAT FLUX Equations (17) and (18) define the turbulent shear stress and heat flux for a variable property fluid. If we are to draw on the large volume of data on turbulent diffusion in constant property fluids, we must make ( 18) and theircountersome estimate of the differences between (1 7) and parts for constant property fluids (i.e,, T~ = -pu'v' and q1 = ph'v').

1. The Effect of Variable Properties on T~ and qt T h e relative magnitude of the terms in Eqs. (17) and (18) may best be displayed by expressing the equations in dimensionless form. Thus, we define the dimensionless quantities

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23

and the subscript s refers to a reference condition, e.g., the free stream. Equations (17) and (18) then become 7t =

-

-

-@,2 U'V."- pau,U R'U'

qt =

-

Ah H'V'

-

p , ~ , 2R'U'V'

(36)

+ pI Ah 6 R H ' + p p , Ah R H ' V ' __

(37) T h e fluctuations, R' and H', may be expressed in terms of the dimensionless temperature fluctuations, 8', provided that we may assume that the turbulent pressure fluctuations are small compared with the mean pressure. Thus, if 6' = T'jAT, where AT = To - T , , AT R'=8'-Ps

( aapT )

P

-

-8'f-AT/3,

where

Ps

1 a=-(+ v

av a2

We now make the assumption that 8' m U', and that the correlations between 0' and other quantities are the same as between U' and those quantities. This is the crucial assumption and can be justified only by appealing to an analogy between the temperature and x-direction velocity fields. Thus the velocity fluctuations u', normalized by the overall velocity difference, us are compared with the temperature fluctuations T ' , normalized by the overall temperature difference; both are seen as the result of turbulent transport by the fluctuating velocity 71' in the transverse direction. While this argument is of doubtful validity with extreme property variations, it probably gives the correct order of magnitude of the terms. With these assumption, Eqs. (36) and (37) become -@82[U'V' - (ti/uS)/3 AT U'U' - /3 AT U'U'V'] ___ =~u~c~AT[UV'-(U/U~)B~T -/3ATU'U'V'] U'U' ~

T~

~

=

~

qt

(38) (39)

T h e following comments can be made about the quantities in these equations: I U'U' I because the correlation between u' and v' (i) I U'V' I is usually about 0.4

-

(ii)

u"v'
I U' I

<1

<

us (iii) for a boundary layer or channel flow 3 (iv) the product /3 AT, which for a perfect gas is ATiT, probably does not exceed unity except very close to the critical point.

W. B. HALL

24

On this basis, therefore, the last two terms in Eqs. (38) and (39) are probably negligible, and the expressions revert to the constant property form. There is one very important point to note, however: while the expressions for T~ and qt may remain the same, the magnitudes of the correlations &’ and h’v’, when expressed in terms of the mean flow quantities, will almost certainly be different. Thus, for example, Hall el a1 (7) have suggested that the effect of the relatively large expansion of the fluid undergoing turbulent diffusion may significantly affect the turbulence level. I t seems likely that such effects will outweigh any errors resulting from the omission of the lower order terms in Eqs. (38) and (39), and we therefore proceed using the constant property forms for T~ and q1 .

2.

r t and ql

in Terms of Time Mean Flow Quantities

There is, at present, insufficient data to assess accurately the effect of T h e approach variable properties on the correlations UTand adopted in Sections IV, V, and VI is to use a “mixing length” model of turbulent diffusion in which

m.

where I is the “mixing length,’) and E is the turbulent diffusixity. (Townsend (8) has shown that such a model may be derived from the equation governing the production, dissipation, convection, and diffusion of the kinetic energy of turbulence, under circumstances when the production and dissipation are in local equilibrium.) An assumption concerning the variation of I throughout the flow thus enables the turbulent shear stress and heat flux to be related to gradients of the time mean quantities ?i and h. A common assumption is that I = 0.4y, at any rate in the region close to the wall. From what has been said, this relationship might well be affected by variable property effects in supercritical fluids. An alternative method of specifying the level of turbulent diffusion is to establish a relationship between the turbulent diffusivity, E , and position in the flow. This is quite permissible, but one must remember

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that the expressions in common use are based on measurements in which the shear stress distribution in the flow has its normal constant property value. From the definition of E we see that it may be expressed as

and

where d is a characteristic dimension, and s refers to a reference condition such as the free stream. Thus, provided that the shear stress distribution, T / T ~is always the same, that ps/p M 1, and that 1 can be expressed in terms of y, then E / V , = f (Re, y/d). I n many supercritical heat transfer problems, however, buoyancy forces are large enough to change significantly the shear stress distribution; it is then important to avoid the use of an expression for E/V, which has in it an implicit assumption concerning T / T ~ . T h e matter is discussed in more detail in Section VI.

IV. Forced Convection Almost the whole of the data on forced convection near the critical point has been obtained using pipes or channels of uniform cross section, in most cases with a uniform heat flux boundary condition. This apparently simple situation has nevertheless yielded a diversity of experimental results that is matched only by the range of correlations produced to describe them! One is faced not only with discrepancies between correlations, but also between sets of data which have been obtained under apparently similar conditions. This interesting situation cannot be attributed solely to inadequate experimental techniques, and one must question whether there are some important factors which were not controlled in the experiments. We begin by discussing the manner in which forced convection heat transfer data are commonly presented and the difficulties that arise when physical property variations are severe. T h e apparent discrepancies between typical sets of experimental data and the deficiencies of current methods of correlation are then demonstrated. Finally, the prediction of heat transfer by numerical solution of the semiempirical equations of motion and energy is discussed.

26

W. B. HALL

A. METHODS OF PRESENTATION OF DATA Forced convection heat transfer data are frequently presented in a form in which neither the temperature of the heat transfer surface nor that of the fluid is given explicitly. T h e implication is that the heat flux is proportional to the temperature difference between surface and fluid, and that any effect due to the general level of temperature can be adequately expressed by evaluating the physical properties of the fluid at some characteristic temperature, e.g., the bulk mean fluid temperature. Proportionality of heat flux and temperature difference occurs with constant property fluids and is a consequence of the facts that the energy equation is linear in temperature, and the heat transfer process does not affect the flow process. Presentation of data in this form is inappropriate for fluids near their critical point, and confusion often results when data are forced into such a pattern. T h e matter is best illustrated by presenting the same data in a variety of forms; this is shown as follows for the case of carbon dioxide at a pressure of 75.8 bars flowing in a downward direction in a heated vertical pipe of diameter 1.90 cm (9). (The pressure of 75.8 bars is somewhat above the critical pressure (73.8 bars). It is usual to relate the behavior of the fluid to the "transposed critical temperature," Tpc, the temperature at which the specific heat reaches its maximum value. At the pressure of 75.8 bars this temperature is 32"CJrather than the critical temperature of 31.04"C.) Data were obtained for upward flow, but these gave anomalous results which will be discussed in Section VI.

1. The Experimental Measurements T h e quantities measured in the experiments were the mass flow of

CO, the fluid inlet temperature, the heat input (which was arranged to

give a very nearly uniform wall heat flux into the fluid), and the temperature of the pipe wall. T h e latter temperature was measured at intervals of one pipe diameter along the length of the test section. Typical results are given by the full lines in Fig. 6 , which shows the variation in wall temperature, T o , along the pipe for three different heat fluxes but with the same mass flow and inlet fluid temperature. T h e complete set of results for this mass flow involves three different fluid inlet temperatures each with five different heat flux levels, i.e., fifteen curves in all. Results in this form embody the whole of the information and should always be made available in tables if not also in graphs. Subsequent operations all involve either the input of physical property data (sometimes of doubtful accuracy) or the use of some simplifying hypothesis such as the elimination of an experimental variable or its combination with

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27

another. Difficulties are caused not by the well-intentioned attempt to find simpler means of expressing the results but by the rejection of information; it may be impossible to recover this from partly processed results. It is unfortunate that a great deal of the heat transfer data on supercritical fluids are of considerably less value than they should be because of such omissions.

; W a l l temperature

_----_----

-__--,------_--,---__----

Bulk ternperoture

\-/--___----*_ _ - &

*_--

-#

--

&&*-k==

0

20 40 60 a0 100 Distance from start of heatmg (diameters)

120

I

140

FIG. 6. Temperature distribution along a I .9-cm diameter vertical pipe for downward flow. Carbon dioxide at a pressure of 75.8 bars and a mass flow of I60 gmlsec. (a) wall heat flux, 3.09 W/cmZ;(b) wall heat flux, 4.05 W/cmz; (c) wall heat flux, 5.19 W/cm2.

2. Description in Terms of Local Conditions Only One of the most useful concepts in forced convective heat transfer is that of “fully developed” fluid flow and heat transfer conditions in a pipe. It is asserted that both the velocity and the temperature distributions across a pipe will become invariant aftes a certain distance from the pipe inlet. (Note that in the case of temperature it is only the shape of the distribution that remains unchanged since the bulk temperature must, of course, increase as heat is added.) There is ample evidence to show that such conditions are established in about 10 to 20 pipe diameters with a turbulent constant property fluid and a uniform cross section pipe; and if we therefore exclude the “entry region,” the heat transfer coefficient will be independent of position along the pipe. I n the case of a fluid whose properties change with temperature, and therefore with distance along the pipe, the hypothesis of a “fully developed” condition is less plausible and must be tested experimentally.

28

W. B. HALL

Figure 7 shows the results, of which Fig. 6 is a sample, presented in the form of heat flux against wall temperature, with the bulk fluid temperature T , as parameter. For this purpose it is necessary to calculate the bulk fluid temperature along the pipe (shown by dotted lines in Fig. 6); this may be done by applying a heat balance from the pipe inlet to the point in question and requires an accurate knowledge of enthalpy as a function of temperature. The hypothesis is then made that the heat transfer rate (expressed as heat flux for particular values of wall temperature and bulk temperature) is independent of the point along the pipe at which these particular conditions occur; the result is Fig. 7.While

w a i l temperature

("c)

FIG. 7. Heat flux versus wall temperature for various bulk fluid temperatures. Same data as for Fig. 6. Bulk fluid temperature: ( O ) , 19°C; (+), 22°C; (v), 25°C; ( x ) , 28°C; (O), 31°C.

there is a good deal more scatter of the experimental results than in the curves of Fig. 6 (in which the experimental points fall within 0.1"C of the smooth line drawn through them), the result does suggest that the heat transfer process is very largely governed by local conditions. The lower parts of the curves are drawn in broken lines because no results were obtained in this region; it is possible, however, to fix the point at which they intersect the To axis because at this point q = 0 and To = T , . Incidentally, the slopes of the curves as they cross the To axis give the limiting value of the heat transfer coefficient as the temperature difference tends to zero.

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29

3 . Presentation in Terms of a Heat Transfer Coejicient This form of presentation is shown in Fig. 8 in which the points are the same as those of Fig. 7. The heat transfer coefficient is by no means independent of either the wall temperature or the bulk temperature. This being so, one must question whether the concept of a heat transfer coefficient has any useful purpose to serve since the results could equally well be presented as in Fig. 7. It may perhaps be useful to see whether the results show a tendency towards a constant heat transfer coefficient in certain limiting conditions; for example, it appears that this may be so as the wall temperature increases, the bulk temperature still being below the transposed critical temperature. However, one might easily be misled by Fig. 8 into thinking that high heat fluxes could be achieved with small temperature differences whereas Fig. 7 shows that this is not the case. 0.5

" 0

N

5

-3\

0.4

C m .r c

" 0)

0 L

2 0 --

03 0

0

0

I

+ A x

0

0.2

I

I 20

I

I

40 Wall

temperature

I

I

60

I

J 80

("c)

FIG. 8. Heat transfer coefficient versus wall temperature for various bulk fluid temperatures. Same data as for Figs. 6 and 7. Bulk fluid temperature: (a),19°C; (+),22"C; (A), 25°C; ( x), 28°C; ( o ) ,31°C.

30

W. B. HALL

T h e usefulness of the heat transfer coefficient when applied to supercritical fluids has been questioned by Goldman (10); he has suggested that rather than expressing heat transfer results as a relationship between Nusselt number, Reynolds number, and Prandtl number, it is more appropriate to collect together all the terms in the dimensionless groups that are temperature dependent. Thus starting from the assumption that Nu

=c

Ren Prs

(42)

where c, n, and s are constants, he obtains qod'-"/(P)"

= f(T0 I

Tm)

(43)

This presentation is rather like that of Fig. 7 except that it postulates a specific form of variation with pipe diameter, d, and mass velocity, pu, whereas the data of Fig. 7 are for one pipe diameter and one mass velocity only. Equation (43) is, however, no more valid than Eq. (42) from which it was derived; there is no a priori reason to suppose that with a supercritical fluid Reynolds number and Prandtl number effects are adequately represented by an equation of the form of Eq. (43).

4. Presentation in Terms of Dimensionless Groups T h e data of Fig. 8 are shown in dimensionless form in Fig. 9. T h e form of correlation that has been used is due to Miropolsky and Shitsman (11) and is of the form Nu,

= c(Re,)"

(Prmjn)8

(44)

T h e physical properties in the Nusselt number and the Reynolds number are evaluated at bulk temperature, but the Prandtl number is the lower of the two values obtained by evaluating properties at the wall temperature and at the bulk temperature. T h e exponent n has been chosen as 1.4 to give the best fit to the results, and thus N u , , / ( R ~ , ) ~ . ~ has been plotted against Prmin. While the correlation appears at first sight to be reasonably good, it should be noted that in fact the scatter is a good deal more than in the original data or in the form of presentation shown in Fig. 7. Bearing in mind the restriction which must be placed on its use in conditions outside the range of the data on which it is based, the use of such a correlation is of doubtful value. An even more serious criticism is the fact that it is impossible to recover the original data from such a presentation of results.

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31

t

FIG. 9. Correlation of the data shown in Fig. 8. (Physical properties based on bulk temperature for N u and Re; Prmj,,is the lesser of the values at wall and bulk temperatures.)

B. EXPERIMENTAL DATA Existing data for forced convection have been reviewed by Hall, Jackson, and Watson (12). Most experimenters have used circular cross section pipes with a uniform heat flux boundary condition. I n spite of the very considerable amount of data that exists, the situation is still somewhat confused; it is not simply that one is unable to correlate the results in terms of the usual parameters, but rather that one suspects that there may be some important parameters that have not been controlled. T h e situation is made worse by the fact that in some cases experimental results have been presented in “correlated” form, and the physical property data used in the correlation has not been quoted. Little purpose would be served in presenting a detailed review of experimental data here; it is perhaps more useful to identify the important discrepancies and to consider what physical factors may have been responsible for them. Figures 10 and 1 1, which are based on uniform heat flux measurements in a circular pipe using water (2.3-16), illustrate some of the apparent discrepancies between experiments. In all cases the measurements were made in pipes of circular cross section with a uniform wall heat flux, 4, into the water. Provided that the entry conditions are similar, one would expect the wall temperature to be a function of the bulk enthalpy, the mass velocity of the water, the pipe diameter, and the wall heat flux. T h e latter three parameters are quoted on the figures, and the

W. B. HALL

32 580 560 540

5 20 500 4 80

u 460 .e

P

4 40

420 400

380

360 340

1600

1800

2000

2200

2400

2600

2800

Bulk e n l h o l p y (J/grn)

FIG. 10. Experimental wall temperature distributions as a function of local bulk enthalpy along a pipe. p* = 1.05 (W. B. Hall, J. D. Jackson, and A. Watson, “Symp, Heat Transfer and Fluid Dynamics of Near Critical Fluids.” Proc. Znst. Mech. Eng. 182. Part 31 (1968)). (a) Shitsman (13): q = 34 W/cm*, ni/A = 43 gm/sec cm2, d = 0.8 cm vertical. (b) Shitsman (13):q = 28.5 W/cm*, m/A = 43 gm/sec cmz, d = 0.8 cm vertical. (c) Shitsman (13):q = 28.0 W/cm*,& / A = 43 gm/sec cmB,d = 0.8 cm vertical. (d) Domin (IS):q = 72.5 W/cmz, &/A = 68.6 gm/sec cm*, d = 0.2 cm horizontal. ( e ) Domin (15): q = 72.5 W/cm2, m/A = 72.4 gm/sec cm2, d = 0.2 cm horizontal.

first is used as abscissa; Fig. 10 is for a pressure closer to the critical than Fig. 11. While it is not possible to make direct comparisons between the various experiments, it is difficult to believe that the sharp peaks of wall temperature in Shitsman’s data, the broader peaks of Domin, of Vikrov and Lokshin, and of Schmidt, and the sharp depression of wall temperature observed by Domin (Fig. l l ) , all form part of a single consistent pattern. Nevertheless one can identify certain trends, as follows: (i) I n all cases where the wall temperature behaves in an anomalous manner, it does so just before the bulk temperature reaches its critical value. (ii) There is a strong heat flux effect evident in all four sets of data; that is to say, the heat transfer coefficient is strongly dependent on heat flux. This is strikingly illustrated by Shitsman’s data (curves a, b, and c on Fig. 10). (iii) There is evidence both for a local improvement and also for a local deterioration in heat transfer when the critical temperature lies between the wall temperature and the bulk fluid temperature.

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Recent data have shown that one of the factors which is important in forced convection is the orientation of the pipe. Of the above sets of data, only Shitsman’s was obtained for a vertical pipe, the remainder being horizontal. This matter will be dealt with in more detail in Section VI. T h e experimental evidence for local increases and decreases in heat transfer coefficient are summarized in the following.

“ /

Bulk fluid temperalure

/ 1750

2000

2250

2500

Bulk enthalpy (J/gm)

FIG. 11. Experimental wall temperature distributions as a function of local bulk enthalpy along a pipe. p * = 1.15 (W. B. Hall, J. D. Jackson, and A. Watson, “Symp. Heat Transfer and Fluid Dynamics of Near Critical Fluids.” Proc. Inst. Mech. Eng, 182, Part 3 I (1968)). (a) Vikrev and Lokshin (16): q = 69.9 W/cm2, rit/A = 100 gm/sec cma, d = 0.8 cm horizontal. (b) Vikrev and Lokshin (16): q = 69.9 W/cm2,1h/A= 40gm/seccm2, d = 0.8 cm horizontal. (c) Schmidt ( 1 4 ) : q = 58 W/cmz,rit/A = 61 gm/sec cm2,d = 0.5 cm horizontal.(d) Schmidt (14): q = 82 W/cm2,rit/A = 61 gm/sec cm2,d= 0.5 cm horizontal. (e)Domin (15):q = 91 W/cma,rh/A = 101 gm/sec cm2,d = 0.2 cm horizontal.(f) Shitsman (13):q = 39.6 W/cm2, rh/A = 44.9 gm/sec cm2, d = 0.8 cm vertical.

1. Local Increases in Heat Transfer Coefficient One example of this has already been mentioned (Fig. 11, curve e ) . A similar effect has also been found in experiments with CO,, three examples of which follow: (a) T h e data of Figs. 7 and 8 clearly show an enhancement in the heat transfer coefficient for conditions in which the heat flux is small and the critical temperature lies between the wall and bulk fluid temperatures.

W. B. HALL

34

As the heat flux is increased the enhancement becomes less marked. It is important to note that these data were obtained with a downward flow in a 1.905-cm diameter vertical tube; results for upward flow can be quite different, as will be shown later. 2 .o

TPC 0

0

I

O I

0 O1

0

28

30 Bulk

32

34

temperature

36

38

("C)

FIG. 12. Variation of the heat transfer coefficient with bulk temperature for forced convection in a heated pipe. Data of H. Tanaka, N. Nishiwaki, and M. Hirata, Turbulent heat transfer to super-critical carbon dioxide. Nippon Kikai Gakkai Rombunshu ( 1 967). (Carbon dioxide at a pressure of 78.5 bars flowing upwards in a 1.0-cm diameter vertical pipe.) 0 Theory; ( A ) exp.: G = 140 f 4.4kg/hr, q = 1.44 W/cma. @ theory; ( X ) exp.: G = 140 f 3.1 kg/hr, q = 2.73 W/crne. @ theory; ( 0 )exp.: G = 280 f 5.6 kg/hr, q = 3.32 W/cm2. @ theory; ( 0 ) exp.: G = 280 If 7.8 kg/hr, q = 5.20 W/cm2.

(b) T h e data of Tanaka, Nishiwaki, and Hirata (17) are shown in Fig. 12. Peaks in heat transfer coefficient occur when the bulk temperature is slightly below the transposed critical temperature, the peaks being more marked when the heat flux is low, i.e., when the wall temperature is also close to (but slightly above) the transposed critical tem-

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35

Hot wall ternDeroture ("C)

FIG.13. Variation of heat flux through a 1 .O cm high channel (formed by horizontal planes) with the temperature of the upper (heated) wall. (The lower (cooled) wall temperature is kept constant at 28.25"C.) (Carbon dioxide at a pressure of 75.8 bars; mass velocity 37 gm/scc cm2). Data ofS. A. Khan, Ph.D. Thesis, University of Manchester, 1965.

perature. These results were obtained in a 1-cm diamater vertical tube with upward flow. (c) Hall, Jackson, and Khan (7) measured the overall heat transfer coefficient for a flat duct 1 x 18 cm in cross section with one of the longer sides heated and the other cooled. This arrangement produces a situation which is basically simpler than that in a tube because it is possible to arrange that the fluid temperature does not change in the direction of flow. There is thus no convection, and the experiment is a direct measure of the diffusive power of the turbulent stream. A sample of the results is shown in Fig. 13 in which the heat flux is plotted against the temperature of the heated wall, with the temperature of the cooled wall as parameter. Again, there is a sharp increase in heat flux as the heated wall passes through the transposed critical temperature (32°C in this case), and the increase becomes larger as the cooled wall approaches the transposed critical temperature. T h e flow in these experiments was horizontal with the upper surface of the duct heated.

2. Local Decreases in Heat Transfer Coeflcient There appear to be two distinct types of situations in which a local reduction in the heat transfer coefficient occurs, both of which are

t

p = 2 4 5 bars

p - 2 4 5 bars

515

-

35 25

0 \

*

45-

8b

55-

7b

65 -

6b

75

5b

85 -

95 100

0

I

100

\ \a

200

4 4,h \ \

300

1

400

tw , t b ( O C )

FIG. 14. Wall temperature, t , , and bulk temperature, te , as a function of distance ( x / d ) along a vertical heated pipe (1.6-cm diameter). Upward flow of water at a pressure of 245 bars. Data of M. E. Shitsman, “Symp. Heat Transfer and Fluid Dynamics of Near Critical Fluids.” Proc. Inst. Mech. Eng. 182, Part 31 (1968). (1) m/A = 382 gm/sec cm2, q = 27 W/cm2. (2) m/A = 382 gm/sec cm2, q = 37 W/cm2. (3) &/A = 400 gm/sec cmz, q = 45 W/cmP. (4) m/A = 375 gm/sec cm2, q = 52 W/cm2.

FIG. 15. Wall temperature, t , , and bulk temperature, t B , as a function of distance ( x / d ) along a vertical heated pipe (1.6-cm diameter). Downward flow of water at a pressure of 245 bars. Data of M. E. Shitsman, “Symp. Heat Transfer and Fluid Dynamics of Near Critical Fluids.” Proc. Inst. Mech. Eng. 182, Part 31 (1968). (5) m/A = 400 gm/sec cma, q = 27 W/cmz. (6) A / A = 400 gm/sec cm2, q = 36 W/cmz. (7) m/A = 393 gm/sec cm2, q = 43 W/cme. (8) m/A = 381 gm/sec cm2, q = 50 W/cm2

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37

illustrated in Figs. 10 and 11. T h e data of Domin, Schmidt, and Vikrov and Lokshin, all of which are for horizontal pipes, show rather broad peaks of temperature at higher heat fluxes; similar peaks have been reported by Griffiths and Shiralkar (18). Shitsman’s data, on the other hand, show sharp peaks of temperature when the flow is upwards in a vertical pipe. (a) Figs. 14 and 15 show some of Shitsman’s data (19) in more detail. Wall and bulk temperature are plotted against the dimensionless distance x / d from the pipe inlet, for several (uniform) heat fluxes. Fig. 14 is for upward flow and Fig. 15 for downward flow. I t is seen that while there is no anomalous behavior for downward flow, the wall temperature for upward flow rises to a sharp peak once a particular value of heat flux is exceeded. (b) A very similar behavior has been found by the author’s colleagues, J. D. Jackson and K. Evans-Lutterodt (20) using carbon dioxide (Fig. 16). These results form part of the same series as those presented in Fig. 6 but in this case the flow is in the upward direction. T h e pipe, of diameter 1.905 cm, had some 200 thermocouples distributed along its length, so that the shape of the temperature peaks could be accurately

100 110

90 -

-

-

nu

80

-

70

-

2 2 60 ?

; c

3

50

-

40 30

-

20

-

10

‘

I

0

I

20

I

40

Dlstance from

I

60

I

80

I

I00

I

I20

start of heating ( d i a m e t e r s )

FIG. 16. Temperature distribution along a 1.9-cm diameter vertical pipe for upward flow. Carbon dioxide at a pressure of 75.8 bars and a mass flow of 160gm/sec. Wall heat flux (a) 3.09 W/cma, (b) 4.05 W/cm2,(c) 5.19 W/cm2,(d) 5.67 W/cm*.

38

W. B. HALL

determined. Again, there is a sharp deterioration in heat transfer once a particular heat flux is exceeded with the flow in the upward direction. There is one interesting difference between these results and those of Shitsman; while it seems that the deteriorations occur in the CO, results only after the wall temperature has passed through the transposed critical temperature, they occur in Shitsman’s experiments even when it is substantially below. They do however also occur in some of Shitsman’s experiments when the wall temperature has passed the transposed critical temperature. (c) T h e data of Tanaka et al. (Fig. 12) was obtained under conditions rather similar to those described in (b) above, and yet no localized deteriorations in heat transfer were found. T h e heat flux, mass velocity, and fluid temperatures are all similar to those used by Jackson and Evans-Lutterodt, but the tube diameter was 1 cm rather than 1.905 cm; it will be shown later in Section VI that this difference can account for the differences in heat transfer behavior. T h e deteriorations that occur in horizontal tubes are generally less localized than those in vertical tubes with upflow. Only in a few cases has the temperature distribution around the tube been measured; Fig. 17 shows the temperature distribution along the upper and lower surfaces of a heated horizontal pipe carrying a flow of supercritical pressure water (21);there is a very considerable difference in temperature, corresponding to a reduction in the heat transfer coefficient for the upper surface, when compared with the lower, by a factor of about four. While such temperature variations may have been suppressed in other experiments by conduction around the pipe wall, there could then have been large variations in heat flux around the circumference of the pipe. I t is probably not worth attempting detailed comparisons between sets of data for horizontal pipes until the question of circumferential variations has received much more attention.

3 . Gaps in the Experimental Data I t has been clear for some time that many of the apparent discrepancies between different sets of heat transfer data are the result of differences in the experimental arrangement (sometimes wrongly assumed to be of no importance and therefore inadequately described). As mentioned, there is strong evidence to suggest that sharp local reductions in heat transfer occur with upward flow only and that even then they can be suppressed by a reduction in heat flux or tube diameter. Also confusion has been caused by the presentation of results in dimensionless form without giving the raw measurements or the property data used.

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39

FIG. 17. Temperature distributions as a function of local bulk enthalpy along heated vertical and horizontal pipes (1.6-cm diameter). Data of Z. L. Miropolsky, V. J. Picus, and M. E. Shitsman, Proc. Imt. Heat Tvansj'er Conf., 3rd, Chicago, 1966 Vol. 11, Paper No. 50 (1967). Water at a pressure of 245 bars, I ~ / = A 60 gm/sec cm2; q = 52 W/cm'. (1) Horizontal pipe, upper surface. (2) Horizontal pipe, lower surface. (3) Vertical pipe, upward flow. (4) Fluid temperature.

In spite of this rather confused situation, the writer believes that the basic forced convection problem of steady flow in vertical tubes with uniform heating could be resolved by a relatively small amount of experimental work. Once it is recognized that buoyancy forces are responsible for large differences between upward and downward flow (see Section VI), the data begin to make sense. Because of difficulties in generalizing physical property data it is still necessary to carry out experiments with a wide range of fluids; there is already good data for CO, , less detailed but reasonably satisfactory data for water, and rather inadequate data for most other fluids. Experiments must cover upward and downward flow, and should preferably involve a range of pipe sizes. It would be helpful if the experiments could be planned so that it is possible to determine the limiting situation as the temperature difference tends to zero at a range of fluid temperatures spanning the critical temperature. Very detailed pipe wall temperature measurements are required both in the axial and the circumferential directions; Jackson and Evans-Lutterodt (20),for example, use some 200 thermocouples on

40

W. B. HALL

a 1.9-cm diameter pipe of a length of 3 meters. Such experiments are not to be undertaken lightly since they make large demands on experimental skill if the results are to be reliable. It is to be hoped, therefore, that those engaged in the work will maintain closer liaison with each other than has been the case so far. More detailed work will certainly be required on horizontal pipes and also on the effect of heat transfer boundary conditions with both horizontal and vertical pipes. Note added in proof: A recent publication by B. Shiralkar and P. Griffith [The effect of swirl, inlet conditions, flow direction and tube diameter on the heat transfer to fluids at supercritical pressure, ASME Paper No. 69-WA/HTl (1969)] provides an opportunity to compare three sets of data on CO, with both upward and downward flow in a vertical pipe; the other two sets of data are those of references (20) and (51). T h e three sets of data differ only in the diameter of the test section on which they were obtained; moreover it is possible to choose conditions giving approximately the same Reynolds number in each case. In the absence of buoyancy and acceleration effects (see Sections 111, 3 and IV, C, 3, c) one would expect identical temperature distributions under these conditions, provided that the data are compared at similar values of qwd, where qw is the wall heat flux, and d is the pipe diameter. A study of the basic equations (Section 111, B) shows that the acceleration effect is the same for each pipe under the conditions imposed above; on the other hand, the buoyancy effect, characterized by the parameter Gr/(Re)1.8of equation (32), is certainly very different for the three cases. A comparison of the relationships between the wall temperature and the bulk temperature for the three sets of data might therefore be expected to yield differences that are attributable to buoyancy effects. T h e conditions for three approximately similar experiments, one taken from each of the three sets of data, are shown in Table I ; in each case results were obtained for upward and downward flow. Figure 17a shows the wall temperature as a function of the bulk enthalpy for the three cases shown in Table I. Shiralkar and Griffith’s data are represented by a single curve since they found that there was no significant difference between upward and downward flow in their tests. Very different results were obtained for upward and downward flow in the larger pipes. In both cases the sharp peaks mentioned in section IV, B, 2 were found with upward flow. T h e results for the two larger pipes are very similar in form; the differences in level may be accounted for by the fact that it was not possible to choose identical values of pressure, Reynolds number, and qwd for the comparison.

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41

TABLE I COMPARISON OF DATA FOR CO, Reference Shiralkar and Griffith Jackson and Evans-Lutterodt (20) Bourke et al. (54)

IN

VERTICAL PIPES

Ra

d (cm)

(W/cm2)

0.635 I .905

15.8 5.67

1.24

2.285

5.1

0.82

QW

1 .o

G“

$ q,d (W/cm)

Pressure (bars)

1 21

10.0 10.8

75.8 75.8

46.5

11.6

74.5

‘R = Reynolds number/Reynolds number (Shiralkar and Griffith); G = Grashof number/Grashof number (Shiralkar and Griffith).

Three extremely interesting points emerge from the comparison: (i) T h e sharp wall temperature peaks observed with the two larger pipes are present only in upward flow whereas the rather broad peak obtained in the small pipe is insensitive to flow direction. Except for the sharp peaks themselves, the wall temperatures (ii) for upward flow are everywhere less than those measured on the small pipe. (iii) T h e wall temperatures for downward flow in the two larger pipes show no sign of peaks and are everywhere considerably lower than those for the small pipe. T h e results illustrate, in a rather striking manner, the remarks made in Section IV, B, 2 concerning the two types of temperature peak that have been observed with supercritical fluids; they are undoubtedly of different origin (see also Fig. 10). Griffith and Shiralkar propose a mechanism for the deterioration in heat transfer which depends essentially on physical property variations across the pipe. Thus when the pipe wall passes through the critical temperature there appears at the wall a low conductivity “gaslike” layer, the core remaining in a “liquidlike” state moving with a relatively low velocity; the heat transfer coefficient is therefore reduced. As a greater proportion of the fluid is heated through the critical temperature, the flow velocity increases and the heat transfer coefficient is thereby restored to something like its initial value. While the above mechanism may be valid in the absence of buoyancy effects (i.e., at low values of Gr/Relas), it is radically modified when these effects are large. A mechanism for the effect of buoyancy is proposed in Section VI, where it is suggested that the shear stress distribution across the pipe, and hence the turbulence production, is drastically modified.

42

W. B. HALL

1 0

FIG. 17a. Comparison of the data of Shiralkar and Griffith, and J. D. Jackson and K. Evans-Lutterodt, Rept. N.E.Z., Simon Engineering Labs., University of Manchester, 1968, and P. J. Bourke, D. J. Pulling, L. E. Gill, and W. H. Denton, Atomic Energy Research Establishment, Harwell, Rept. No. AERE. R5952, for forced convection of carbon dioxide flowing upwards and downwards in vertical heated pipes. Key: (--.--.-) Shiralkar and Griffith; (-) Jackson and Evans-Lutterodt; (- - -) Bourke et al.

With upward flow the shear stress is rapidly reduced to zero in the core as the wall passes through the critical temperature and is then reversed, thus reestablishing turbulence production; the heat transfer coefficient thus passes through a minimum and then increases. For downward flow the effect of the buoyancy forces is always to increase the shear stress in the core of the flow and thus to improve heat transfer. The effect described by Shiralkar and Griffith may also be present, but at the higher values of Cr/Re1.8, it appears to be completely dominated by the buoyancy effect.]

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43

C . CORRELATION OF EXPERIMENTAL DATA T h e great virtue of dimensionless correlations is that, by grouping the variables, one is able to describe a particular situation by a smaller number of parameters. With constant property fluids there is, in this respect, no distinction between the physical properties and the other parameters governing the flow; thus we may achieve Reynolds number similarity between two systems by adjusting any of the component parameters, p, d, u , or p. T h e situation is very different when the property variations are large. Formally, one might introduce further dimensionless ratios to describe the property variations; for example the variation of viscosity with temperature could be expressed by a series of dimensionless ratios P(Tl)IP( Ts), P V 2 ) / P V J , P.(T J I P V s ) , etc. where T , is some reference temperature. Unless the variation is particularly simple one might require a large number of such groups and thus reach the rather ridiculous situation where the result of dimensional analysis was a series of groups far greater in number than the original quantities needed to specify the problem. I n this situation it is simpler to specify the fluid, the boundary conditions (flow and heat transfer) and accept the fact that there is little to be gained by comparisons between different fluids. This is perhaps an unduly pessimistic view of the situation as far as supercritical fluids are concerned, for it is possible that one might be able to make use of reduced coordinates (of temperature and pressure) to describe property variations in similar classes of fluids in the critical region. On the other hand it cannot reasonably be claimed that the use of dimensionless parameters has done much, so far, to clarify the situation. T h e problems of correlation can best be illustrated by referring to the momentum and energy equations developed in Section 111, (Eqs. (12), (13), (15), and (16)). I t will be recalled that this particular form of the equations, in which the shear stress and heat flux appear explicitly, was chosen in order to facilitate comparison of the terms on the right hand side of the equations. If we are to integrate the equations it will be necessary to express T in terms of the velocity gradient, and q in terms of the temperature (or enthalpy) gradient. Suitable relationships were developed in Section 111, C. We confine our attention to turbulent flow in channels since this is the case which has received the most attention experimentally. T h e appropriate momentum and energy equations are obtained by combining Eqs. (30), (31), (40), and (41) of Section 111. If all the terms in these

W. B. HALL

44

equations are retained, we find the trivial result that, for strict similarity, two systems must be identical in every respect! (That is, we cannot compare different sizes of systems using different fluids under different flow conditions.) We therefore restrict our arguments to cases where the dissipation, acceleration, and buoyancy terms are negligible although we shall examine separately the very important buoyancy effect in Section VI. 1. The Efects of Dissipation, Acceleration, and Buoyancy

In order to illustrate the nature of the above restrictions we shall estimate now the magnitude of the dissipation, acceleration, and buoyancy effects in the case of the results presented in Fig. 7. We take wall and bulk temperatures of 35°C and 29"C, respectively, and the wall heat flux will therefore be about 2.5 W/cm2. T h e other relevant quantities will be approximately as follows: po = 0.3

lit

=

gm/cm; pm

160 gm/sec; u,

= 0.7 M

gm/cm3;

70 cm/sec;

vm =

8.10-* cm2/sec; Re m lo5

cDm= 6

J/gm°C;

Prn = O.O38/OC.

(i) Referring to Section 111, B, 2, we see that dissipation may be neglected if ReO.8 E 100 or, in this case

<

E

< 106)/Re0-8

=

T h e enthalpy difference between the wall and bulk conditions is about 100 J/gm or loDcm2/sec2. Thus E = u,2/dh M 5 x T h e effects of dissipation on the energy equation are thus seen to be negligible. (ii) Referring to Section 111, B, 3, the criterion given by Eq. (34) is in this case PmAh/c,, = 0.64 and thus acceleration effects cannot safely be neglected. However, we shall find that bouyancy effects (which have a rather similar effect on &/ay near the wall when the flow is upwards) are even more important. (iii) Referring to Section 111, B, 1, the criterion for the neglect of buoyancy effects is Gr/(Re)1.8 0.1 I n the present case

<

HEATTRANSFER NEAR Thus

THE

CRITICAL POINT

45

Gr/(Re)1.8 = 5 x 10n/lOB= 5

This is greatly in excess of the value 0.1, and buoyancy effects cannot therefore be neglected. This is confirmed by the very large difference between the results for downward flow (Figs. 6 and 7) and upward flow (Fig. 16). There may also be an acceleration effect present, and, in the absence of buoyancy, this effect should be the same for upward as for downward flow. I t is conceivable that the rather broad temperature peaks mentioned in Section IV, B, 2 could be due to this rather than to buoyancy. They do, in fact, seem to arise with horizontal or with small diameter pipes and in both of these cases buoyancy effects would tend to be less important. Summarizing these very approximate calculations, we see that the effect of dissipation is certainly negligible, acceleration effects may be important, and buoyancy effects are certainly important, at any rate when the flow is vertically upwards. None of the correlations so far proposed have taken account of acceleration and buoyancy effects; indeed the problem of correlation quickly becomes intractable if they are included. We shall nevertheless proceed to deal with those cases where both effects are small, recognizing, however, that the range of applicability will be severely restricted.

2. Correlations in Which Buoyancy, Acceleration and Dissipation Are Neglected

Under these conditions Eqs. (30) and (31), together with Eqs. (40) and (41) reduce to: (45)

These may be put into dimensionless form by transforming the variables as follows: Y

= y/b,

X

= x/b,

H

= h/Ah,

U

= tl/U,,,

Equations (45) and (46) then become

f

4

a

0 =-f--[-(1 2 Re EJY -

_--

P

pm

1 12 au +-Y”-Re/-i)-] 4 b2 ay

aiJ

ay

(47)

46

W. B. HALL

(Note: we have assumed that ah/aT = cp in Eq. (46), which implies constant pressure. This is correct since the pressure is constant in the y-direction.) I n most forced convection systems the pressure variations in the x-direction will also be small compared with the absolute pressure, and we may therefore write ahlax = c, a q a X (Noting that for a supercritical fluid cp is a strong function of temperature.) Thus Eq. (48) may be rewritten in terms of B = ( T - Tv,)/ ( T o- T,) = ( T - T,)/dT

T h e solution of the above equation will yield the temperature gradient at the wall and, thus, the heat flux. T h e Nusselt number can then be obtained Nu = a4b/k, = q,,4b/KO AT = 4(a6/aY), Thus we may write the solution formally as

where it is implied that pip,, etc., are expressed as functions of temperature. Once again it is necessary to stress that the preceding arguments apply only when it is permissible to neglect buoyancy forces and accelerations. This is a severe restriction, and taken together with the necessity to specify the fluid (because of the property ratios in Eqs. (50) and (51)) it is questionable whether the attempt to find a general correlation in terms of dimensionless groups is worthwhile.

3 . Limiting Forms of Correlations a . Small Temperuture Dajferences. T h e limiting situation as the temperature difference AT tends to zero is one of constant physical properties, and one might therefore expect that in this limit any correlation should reduce the same form as those for constant property fluids. Measurements under such conditions tend to be somewhat inaccurate, and it is a good plan to determine the Nusselt number by measuring

HEATTRANSFER NEAR

47

THE CRITICAL P O I N T

the limiting slope of a curvc of heat flux against wall temperature as AT tends to zero (see, for example, Fig. 7). It should be noted that, as AT 0 and the fluid temperature everywhere becomes T I ) &it, is unnecessary to retain the property ratios, p , ’ ~ , , ~etc., , and the correlation then becomes ---f

Nu

=

Nu(Ke, Pr,n , X )

(52)

While it is certainly of interest to test whether any praposed correlation is consistent with constant property data, the result does not throw much light on the manner in which physical properties must be introduced into the correlation when d T is finite.

b. Large Temperature DqfJerence. T h e form of property variation in the region of the critical temperature is such that the greatest changes are restricted to a rather small temperature range spanning the critical temperature. I n particular, the rate of change with temperature soon becomes small as the temperature increases above the critical value. T h u s with turbulent flows in which the wall is heated and in which the wall and fluid temperatures span the critical temperature, it may be reasonable to treat the important region close to the wall as one having constant properties equal to the values at the wall. There will still, of course, be a layer of fluid in which the property variation is severe; however the thickness of this layer will decrease as the temperature difference increases. This suggests the use of a two-region model in which the property variation is characterized by two discrete vaIues (corresponding to the wall temperature and the fluid temperature, i.e., free stream temperature in a boundary layer or bulk temperature in a pipe) rather than by the relationships given in Eqs. (50) and (51). Introducing the ratios of properties at the wall temperature (subscript 0) to those at the bulk temperature (subscript m),Eq. (51) becomes

It will be seen later that this forms the basis of many correlations; while it may be made consistent with the constant property form as d T 0, the arguments that may be adduced in support of it are, as shown, very tenuous indeed. In particular, it should be noted that it is possible to achieve the same two values of specific heat cpoand cpmin different waysone in which the two temperatures span the critical temperature and one in which they do not. Such a form of correlation is unlikely to have any general validity, although, as previously shown, it may be a reasonable approximation when the temperature difference is large. --f

W. B. HALL

48

c. Correlation Based on a Similar “Reduced” Temperature Distribution. If we postulate that the physical properties are not significantly affected by the pressure variations in the fluid, then it may be possible (Section 11) to describe the properties in the following manner: Taking the case of viscosity as an example, (54)

P =Pmf(T*)/f(Tm*)

where pnLis the viscosity at the reduced temperature T,* and T , is the bulk mean temperature. Thus dpm

=.f((To* - Tm *) 0

=

T,/Tc,

+ Tm*)lf(Tm*)

where B = ( T - T,)/(To- T,,), the variable employed in Eq. (49). T h e property variations can then be characterized by the parameters To* and Tm*, the reduced wall and bulk temperatures. Equation (50) would then become Nu

= Nu(Re, Pr,

, X , To*,Tm*)

(5 5 )

This has the great advantage over Eq. (51) in that it may be used to generalize data for different fluids, albeit with the restriction that To* and T,* must have the same values for all cases. T h e correlation is still extremely restrictive, particularly when one recalls that, in addition to containing the parameters To* and Tm*, it also involves implicitly the assumption that dissipation, buoyancy, and acceleration effects are negligible. Nevertheless, it would be interesting to attempt a correlation between different fluids within the limits imposed by these restrictions. Suitable data for such a comparison is at present very limited.

4. Existing Correlations Table I1 shows some of the correlations that have been used to describe supercritical forced convection heat transfer. I t will be seen that most are of the form used for constant property fluids, expressing the Nusselt number as a simple function of the Reynolds number and the Prandtl number, with extra terms involving property ratios. T h e expressions by Petukhov et al. (22) and by Kutateladze and Leontiev (23) reduce to the constant property form in the limiting case of small temperature difference. I n all cases it is implicitly assumed that the Reynolds number, Prandtl number, and property ratio effects are separable, i.e., it is assumed that a change in Prandtl number or in the property ratios does not affect the functional relationship between

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49

TABLE I1 CORRELATIONS FOR VARIABLE PROPERTY FORCED CONVECTION Author/Fluid used (symbol from Fig. 18)

Correlation

Mirapolsky and Shitsman ( 0)/H2O

Nu, = 0.023 Re': Pr": where Prmtn is the lesser of Pr, and Pro

Petukhov, Krasnoschekov, and Protopopov (m)/CO,

NU, = N ~ o ( p ~ / p O )(km/kO)-0.a3 ~." (c^,/~,~)'.'s where Nu, = 0.1251 Re,Pr,/[12.7(5/8)1'2(Pr~'3 1 = 1/(1.82 log,, Re, - 1.64)2 and t, = (h, - h,,,)/(T,- Tm)

Kutateladze and Leontiev (A)

Nu,

Bishop, Sandberg, and

Nu, = 0.0069 Re': ~ r ~ 6 6 ( p 0 / p m )(1 0 ~ 4 s2.4/(L/D)) A where Pr, = t p p m / k m

Tong ( v ) / H 2 0

=

0.023 Re:'

Pr~4[2/((p,/po)"2

-

1)

+ 1.071

+ l)]* +

Swenson, Carver, and Kakarala (+)/H,O

Nu, = 0.00459 ~'813(p0/p,)0.231 A where Pro = i?9po/k,

Touba and McFadden ( x )/H,O

Nu,,

=

0.0068 Re:'

A

Pr, exp[2.19(hm/h,, - 0.801)]

Nusselt number and Reynolds number. There is no a priori justification for this, and it has not yet been tested adequately by experiment. T h e accuracy with which the correlations are claimed to fit their respective data is of the order of 1 1 5 % . Some authors have made comparisons with data other than their own: e.g., Petukhov, Krasnoschekhov, and Protopopov (22) report that data from the water experiments of Miropolsky and Shitsman (11) and Dickenson and Welch (24, and the carbon dioxide experiments of Bringer and Smith (25),lie within f20% of their correlation. Figure 18 shows a comparison of the correlations when applied to water at 254 bars ( p / p c = 1.15); the heat transfer coefficient is shown as a function of the wall temperature for a constant bulk temperature of 360°C. Agreement, in terms of the level of the heat transfer coefficient, is seen to be poor; the trend, as the wall temperature approaches the transposed critical value (387°C in this instance), is fairly consistent, however.

W. B. HALL

50

Distributions of the heat transfer coefficient, a , along a pipe reveal striking differences when compared with those for the constant property case. Koppel (26),using CO, , made detailed measurements of the axial variation of wall temperature; he observed that unusual variations of the 4.0r I

Y U

in

El 3.0 -

N

--.

7 L

C

‘? .u 2.0 e 0 W L

al +

g

e

1.0-

*

0

I

I

I

0 360 370 300

I

I

I

I

390 400 410 420 W a l l temperature (“CI

I

I

430

440

I I 450 460

FIG. 18. Heat transfer coefficient for water in a 0.8-cm diameter pipe as predicted

by various correlations (W. B. Hall, J. D. Jackson, and A. Watson, “Symp. Heat Transfer and Fluid Dynamics of Near Critical Fluids.” Proc. Inst. Mech. Eng. 182, Part 31 (1968))

( p * = 1.15; tiz/A = 200 gm/sec cm2; T, = 360°C). Key: (m) Z. L. Miropolsky and M. E. Shitsman, Zh. Tekhn.Fig. 27, No. 10 (1957); ( H) B. S.Petukhor,E.A. Krasnoschekhov, and V. S.Protopopov, Int. Develop. Heat Transfer, Proc. Heat Transfer Conf., 1961 (1963); ( A ) S. S. Kutateladze and A. I. Leontiev, “Turbulent Boundary Layers in Compressible Gases.” Arnold, London, 1964; (v) A. A. Bishop, R. 0. Sandberg, and L. S. Tong, Forced convection heat transfer to water at near-critical and super-critical pressures. A.I. Ch. E.-I. Chem. E. Symp. Ser. No. 2, 1965; (+) H. S. Swensen, J. R. Carver, and C. R. Kakarala, J . Heat Transfer, November (1965);( x ) R. F. Touba and P. W. McFadden, Combined Turbulent Convection Heat Transfer to Near Critical Water. Tech. Rept. No. 18 COO-1177-18, Purdue Res. Foundation, Indiana, 1966.

heat transfer coefficient occurred along the tube depending upon the proximity of the inlet bulk temperature to Tc and that local maxima and minima in 01 could be produced. H e pointed out that these axial variations could have contributed to the scatter among data based on values of 01 at some arbitrary point along a tube. For low and moderate heat fluxes, however, Koppel’s results were in agreement with the trend indicated by the correlations, i.e., towards improved heat transfer under near-critical temperatures.

HEATTRANSFER NEAR

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CRITICAL POINT

51

I t must be concluded that existing correlations are inadequate as a means of predicting heat transfer in the critical region. This is hardly surprising in view of the additional restrictions that are placed on similarity by large physical property variations. Some of the experimental data have been affected by parameters that are not even included in the correlation; the orientation of the tube in conjunction with buoyancy forces is noteworthy in this respect. One of the alternatives to the conventional dimensionless correlation is presented in the following section; this consists of solving the equations of motion and energy numerically using empirical data on turbulent diffusion. Unfortunately the technique of correlation focuses attention on the problem of finding suitable mathematical functions to fit an empirical result sometimes to the detriment of the physical understanding of the phenomenon. A numerical method does not suffer from this disadvantage.

D. SEMIEMPIRICAL THEORIES T h e difficulties encountered in any attempt to correlate empirical heat transfer data by means of the usual dimensionless parameters stem not so much from a lack of understanding of heat transfer mechanisms as from the interaction between these and the variation of physical properties. This being so, it seems profitable to approach the problem from a rather different angle and to attempt a numerical solution of the equations of motion and energy into which physical property data may be fed in tabular form. In the case of turbulent flow it is necessary also to feed in empirical information concerning the effect of turbulence on the diffusion of heat and momentum; sometimes this information is backed up by an hypothesis concerning the mechanism by which turbulence operates. Such calculations can be made for a range of conditions, and the results compared directly with experimental data. This technique has already proved useful with constant property fluids, and preliminary results for supercritical fluids are quite promising. T h e crucial assumption to be made in calculating heat transfer in a turbulent flow is that concerning turbulent diffusion, and, unfortunately, this has not yet been adequately tested in situations where the property variations are large. Some of the ways in which the description of turbulent diffusion may be affected have already been discussed in Section 111, C. T h e mixing length model forms the basis of most theories, and various methods have been used to allow for the effect of variable properties on the magnitude of the mixing length.

52

W. B. HALL

1. Specijication of the Turbulent Shear Stress and Heat Flux' As shown in Eqs. (40) and (41), the turbulent shear stress and heat flux may be written in terms of the turbulent diffusivity 6 as follows: rt = PE

aulay;

qt = p E ahlay

4t

aTPY

or, at constant pressure, PCP"

This result followed from the mixing length theory. A more general specification would allow the diffusivities in the two equations to take different values, eM and e H , and in fact there is some evidence which suggests that e H / e M= f (Re, Pr, y/a), and that eH is slightly greater than eM for the values of Pr with which we are concerned (1 < Pr < lo), (27)-(29).However, the dependence of eH/eM on Re, Pr, and y / a is not well established even for constant properties in this range of Pr, and its value under supercritical pressure conditions is likely to remain a matter of speculation for some time, T h e differences between the semiempirical theories are therefore centered around the manner in which they specify E ~ Deissler . (30) attempted to predict the heat transferred to supercritical water at a reduced pressure, pip" = 1.8; he then proposed an improved analysis (31) which was used for the case of common gases at normal pressure but with large temperature differences. It is the second of these analyses with which we are concerned. A two-region model was employed in which the following expressions for e M were developed from a dimensional analysis of the problem:

0 < y+ < y1';

y+

= y(T~/po)~/~/Vo ; U+ = u / ( T o / p ~ ) ~ / ~

= +u+y+[ 1 - exp( -u&u+y+/v)],

yl+ < y+

< a+;

EM _ -- K2(d~+/dy+)s vg

(56)

(57)

(d2u'/dy+"*

Values of n, K , and yl+ were selected so as to give the best possible agreement between predicted and measured distributions of u+ versus y+ for the case of unheated, fully developed turbulent pipe flow. Thus, n = 0.124, K = 0.36, and yl+ was chosen as the value of y+ at which E ~ / V= 1.92 ;these values were then assumed to apply also to the variable This section is based on reference (12), and its content is very largely the work of the author's colleague, J. D. Jackson.

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53

property case. Deissler put forward tentative arguments, based on the effect of viscosity on turbulent diffusivity near the wall, in support of the exponential damping terms in Eqs. (56) and (57). T h e equations give results which are in good agreement with velocity distributions and pressure drop-flow relationships for unheated pipe flow ; this is perhaps not surprising in view of the fact that the three constants n, K , and y,+ are available for adjustment. I t is well known that several different approaches (e.g., Van Driest (32))give equally good results when applied to unheated flows; this, of course, is no reason to expect the same methods to give good predictions for variable property conditions. We next consider a rather different approach which has been used by Wiederecht and Sonnemann (33) for liquids with large property variations, and later by Hess and Kunz (34)for supercritical pressure hydrogen ( pipc = 1.5). I t is an application of the unheated flow model of Van Driest (32), and is based on a modification of the Prandtl mixing length model to include wall damping. Thus Van Driest obtained the following expression for e M : cM/vo= K2(y+)2[1 - exp(-y+/A+)I2 I du+/dy+ I

(58)

Good agreement with experiment for unheated pipe flow is obtained with K = 0.4 and A+ = 28. Wiederecht and Sonnemann assumed that Eq. (58) could be used under variable property conditions, but Hess and Kunz suggested that the damped layer thickness (defined in terms of the relaxation distance A+)should be allowed to vary; the property determining the value of A+ was taken to be kinematic viscosity, which appears also in the damping terms of the Deissler expressions (Eqs. (56) and (57)). By applying Eq. (58) to the supercritical hydrogen data of Hendricks et al. (35),Hess and Kunz decided that A+ should vary according to the relationship A+ = 30.2 exp(--0.0285 vo/vnr).T h e Van Driest formulation is convenient because it can be put into the form of a single expression for E~ as a function of y+:

However, it will be apparent that the problem of deciding on the manner in which kinematic viscosity variations might cause a change in the damping of turbulence in the wall layer is a difficult one, and the application of the model under conditions of extreme property variation must be no less tentative than Deissler’s model. We turn finally to the model proposed by Goldmann (36) which involves a rather different method of taking account of the effect of

W. B. HALL

54

property variations on eM . Goldman defines “variable property” universal parameters for velocity and distance from the wall as follows:

and Hence and

du+/dy+ = (po/p)(dU++/dy++)

Goldman then assumes that u++ and y++ will be related under variable property conditions in exactly the same manner that uf and yf are related for the case of the unheated pipe flow; it follows that similar relationships exist also between du++/dy++ and y+ and between du+/dy+ and y+. The effect of Goldmann’s hypothesis can be illustrated by using Van Driest’s model; this gives -EM = - - [ 11 v- 1 1 + 4 7 [ K y vo 2 vo 70

+

( 1 - exp ( - ~ ) ) ] e / l ’ z ]

VO

(62)

which may be compared with Eq. (59). Unfortunately there are insufficient accurate data on velocity profiles under variable property conditions to check directly the basic assumption in Goldmann’s model.

2 . Application of the Theories to Supercritical Heat Transfer Almost all the calculations that have so far been performed using the discussed models of turbulent diffusion have assumed that the shear stress distribution across the channel is affected to only a minor extent by physical property variations. As we have seen, this constitutes a serious limitation to their usefulness; in Section VI we shall attempt to make an allowance for the effect of buoyancy forces in modifying the shear stress distribution. Comparisons between the various theories has been made difficult by the fact that they have been applied to widely differing situations. Hall et al. (12) have therefore obtained numerical solutions using three rather different theories and employing in each case the same sets of physical property data; the results of these calculations were then compared with experimental data. While it cannot be claimed that the degree of agree-

HEATTRANSFER NEAR

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55

ment is satisfactory, certain trends were reproduced which were reflected in the experimental data. Certainly, the agreement was no worse than that between the empirical correlations and the experimental data. I n spite of the rather limited success of the above methods at present, the general line of attack seems to be worth following. I t will certainly be necessary in future work to allow for buoyancy forces and possibly also for acceleration effects. V. Free Convection

Free convection systems were among the first in which the unusual heat transfer properties of a supercritical fluid were demonstrated. In 1939, Schmidt et al. (37) made measurements on a loop filled with ammonia at critical conditions; later measurements by Schmidt (38,39) using ammonia and CO, in a closed vertical pipe, the bottom end of which was heated and the upper end cooled, gave heat transfer rates several thousand times greater than would be achieved by conduction in a copper bar of the same dimensions. These effects were attributed to the abnormally large values of the expansion coefficient and the specific heat, and the relatively low viscosity of fluids near the critical point; thus a large Grashof number could be achieved in a relatively small system with a small temperature difference. By the same token, turbulent conditions could readily be achieved. In contrast to the case of forced convection, free convection data show a fairly unambiguous trend towards higher transfer coefficients as conditions approach the critical. It appears fairly certain, in fact, that the basic theoretical models established for constant property fluids are still adequate although their application to supercritical fluids poses a formidable problem which has not yet been satisfactorily solved for a wide range of conditions. Many of the difficulties encountered in predicting forced convection are present with free convection also; the same reservations must be made about the use of a heat transfer coefficient, and the problem of predicting the effect of property variations on turbulent diffusion is at least as great. I n the remainder of this section we shall refer only briefly to these common problems and shall concentrate on the points of difference.

A. EXPERIMENTAL RESULTS Most of the measurements of free convection with supercritical fluids have been made using horizontal wires or vertical plane surfaces. I n

56

W. B. HALL

some cases the temperature difference has been kept small so that conditions approach those of constant properties while in others the temperature difference is large and frequently spans the critical temperature. T h e two ranges of temperature difference are illustrated by the samples of data discussed in the following; as in the case of forced convection the data quoted are by no means exhaustive and are intended to illustrate the more important trends. 1. Small Temperature Dtyeerences Simon and Eckert (do), using an interferometer to measure density variations, experimented with a heated vertical plate in CO,; the sensitivity of their technique allowed them to employ extremely small temperature differences (0.001"C to 0.01"C). In addition to measuring the overall density difference from fluid to heated plate, they were able to determine the density gradient, and hence temperature gradient, at the plate surface; from these measurements, together with the heat flux, they were able to compute not only the heat transfer coefficient but also the thermal conductivity at the wall. They found that the heat transfer coefficient and also the thermal conductivity increased wih the heat flux when the fluid density was close to its critical value. T h e smooth curves which they fitted to their results have been replotted in Fig. 19 so that a comparison can be made between the relative increases in heat transfer coefficient and thermal conductivity. T h e full lines represent the heat transfer coefficient, a, normalized by dividing by its value at a fluid density of 0.4 gm/cm3, and a heat flux of 5.75 x 10-6 W/cm2; similar curves of the normalized thermal conductivity are shown by broken lines. I t will be seen that the increase in the heat transfer coefficient is generally somewhat greater than the increase in thermal conductivity; both are very significant, and show peaks near the critical density (0.468 gm/cm3). The striking aspect of these results (Fig. 19) is the fact that in all cases the temperature of the fluid is everywhere within 0.01"C of the value measured at some distance from the plate. This temperature range would normally be expected to result in quite negligible property variations (ens., a temperature difference of 0.01"C would give a density variation of about 0.2% under these conditions and the variations in conductivity and viscosity would be of the same order). If the measurements of thermal conductivity are to be believed, therefore, they must represent an effect of heat flux rather than temperature. Simon and Eckert suggest that the effect may be connected with the existence of clusters of molecules in the fluid at near-critical conditions. Thus they write, "one might agree that the formation and break up of such clusters

HEATTRANSFER NEAR THE CRITICAL POINT

57

3.0-

2.01 \

-r \ U

*

1.0-

0.4

0.45 Density (gm /cm3 )

0.5

0.55

FIG. 19. Replot of data of H. A. Simon and E. R. G. Eckert [Laminar free convection in carbon dioxide near its critical point. Intern. /. Heat Muss Trunsfer 6,681-690 (1963)] for free convection from a vertical plate to CO, . Heat transfer coefficient, a, and thermal conductivity, k, normalized with respect to the values at a fluid density of 0.4 gm/cmS W/cma (ar,k,). (a) heat flux = 2.11 x W/cm2; and a heat flux of 5.75 x (b) heat flux = 3.72 x W/cm2;(c) heat flux = 5.75 X 10-6W/cm2.Key:(-)a/a,; (-

- -) k/k, .

near the heated plate surface through the shear within the boundary layer is the cause of the dependence of the thermal conductivity on heat rate." No further light appears to have been thrown on the subject by subsequent work. While there does not appear to be any obvious reason to suspect the use of an optical measurement of density under these conditions, it would perhaps be useful to attempt to repeat the measurements of the heat transfer coefficient using direct measurements of the wall and fluid temperatures. Dubrovina and Skripov (41) have measured the heat transfer for a horizontal wire (29 x IO-O-cm diameter) to CO, in the region of the critical point. Most of their measurements were made at a temperature difference of 0.5"C; this is considerably larger than that used by Simon and Eckert, and the property variations throughout the fluid will certainly be significant (a temperature variation of 0.5"C around a temperature of 32°C and a pressure of 75.8 bars gives a density variation of about 20 to 30%). On the other hand, the small size of the wire reduces the Grashof number, for given fluid conditions, by a factor of about lo9 compared with that for Simon and Eckert's experiments. Their deter-

W. B. HALL

58

mination of the heat transfer coefficient as a function of pressure, for a number of fluid temperatures, is shown in Fig. 20. Unfortunately, it is not possible to make comparisons with Simon and Eckert’s data because of the difference in shape of the heat transfer surface. It is worth noting, however, that Dubrovina and Skripov find that for conditions close to the critical, the heat transfer coefficient increases as the temperature difference is decreased. I

12-

3

2

10 -

.*

N

E

:0 8 -3

-

-3

c

?

2

06-

0 W

L

W

4

t 04-

FIG. 20. Free convection heat transfer coefficient for a 2.9 x 10-%rn diameter wire in carbon dioxide as a function of pressure. Data of E. N. Dubrovina and V. P. Skripov, in “Heat and Mass Transfer” (A. V. Lykov and B. M. Smol’skii,eds.), Vol. I. Israel Program for Scientific Translations, 1967. (1) Fluid temperature 31.S”C; (2) Fluid temperature 32.0”C; (3) Fluid temperature 34.0”C; (4) Fluid temperature 37.0”C.

2. Large Temperature oisferences “Large” means, typically, greater than 1°C. I n such cases there may be variations in density by as much as a factor two from point to point in the fluid, and it is not relevant to make comparisons with constant property data. Doughty and Drake (42) made measurements of the free convection

HEATTRANSFER NEAR

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59

from a 0.025-cm diameter horizontal wire in Freon 12. For fluid close to the critical temperature and pressure, they found that the maximum heat transfer coefficient occurred with a temperature difference of about 5°C (An error of a degree or two in the temperature of fluid or wire would make a very great difference to the computed value of the heat transfer

FIG. 21. Heat transfer from a horizontal 0.025-cm diameter wire to carbon dioxide at a pressure of 89.6 bars. (Fluid temperature 9.5"C; wire temperature 154°C; heat flux, 46.5 W/cm2.) Data of K. K. Knapp and R. H. Sabersky, I n t . J. Heat Muss Transfer 9, 41-51 (1966). Magnification x 10 approx.

W. B. HALL

60

coefficient as the temperature difference decreases; it is possible that such errors may account for the apparent discrepancy between these results and those of Dubrovina and Skripov (41) which indicate a maximum heat transfer coefficient as A T -+ 0). Knapp and Sabersky (43) reported the appearance of a “bubble-like” flow in experiments with a heated wire in CO,; Fig. 21 illustrates this phenomenon at a pressure well above the critical, and with fluid and wire temperatures spanning the critical. It is very difficult to accept the idea of a sharp “phase boundary” under these conditions, and yet the “bubbles” beneath the wire do give such an impression. Short of a gross error in determining the fluid conditions, which is highly unlikely, the only other possibility seems to be contamination by a second component. I n this context, Draper (44) has reported the appearance of a separate phase in the form of droplets on a heated wire in sulphur hexaflouride; in this case it was identified as a separate component, but beyond the fact that it contained water, its composition was not established. Boiling and free convection from a heated horizontal wire immersed in sulphur hexaflouride have been studied by Draper, Figs. 22-24 show his results for supercritical conditions (results for subcritical pressures shown in Figs. 30-32). Each figure refers to one fluid temperature (the critical temperature is 456°C) and on each are plotted several

- 30

x

=

I

‘I00

Temperature difference, To -Ts (“C)

FIG. 22. Free convection heat transfer from a 0.01 I -cm diameter wire immersed in supercritical pressure sulphur hexaflouride.Fluid temperature 23.3”C. (Data of R. Draper, M.Sc. Thesis, University of Manchester, 1968.)

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61

FIG.23. Free convection heat transfer from a 0.01 I-cm diameter wire immersed in supercritical pressure sulphur hexaflouride. Fluid temperature 39.6"C.(Data of R. Draper, M.Sc. Thesis, University of Manchester, 1968.)

FIG.24. Free convection heat transfer from a 0.011-cm diameter wire immersed in supercritical pressure sulphur hexaflouride. Fluid temperature 43.1% (Data of R. Draper, M.Sc. Thesis, University of Manchester, 1968.)

W. B. HALL

62

curves of heat flux against temperature difference for a range of pressures (the critical pressure is 37.7 bars). It is of interest to note that the whole range of results (for different fluid temperatures and pressures) fall within & l o % of a single curve. I n all cases the fluid temperature is below the critical, and the measurements were not sufficiently sensitive to determine accurately any anomalous results as the wire temperature passed through the critical temperature. For the greater part of the range the temperature difference is sufficiently large to establish something like the "two-region" pattern described in Section IV. Reference to Figs. 3032, which are for subcritical pressures, shows that the results for film boiling are also very close to those for supercritical pressures.

U

0

10 Temperature

20

30

d i f f e r e n c e ("Cl

FIG. 25. Free convection from a 6-in high, 0.5-in wide vertical ribbon to water at a pressure of 223 bars. Data of J. R. Larson and R. J. Schoenhals, J. Heat Transfer, November, 407 (1966). Bulk temperature: (*) 374"C,( A ) 368.5"C,( x ) 376.8"C,( 0 )379"C, ( 0 ) 383"C, (+) 385.6"C. T,,.= 374.9"C.

Free convection data for a 15-cm high vertical flat plate in supercritical pressure water have been obtained by Larson and Schoenhals (45)' and are shown in Figs. 25 and 26. When comparing these data with Draper's data (Figs. 22-24), it should be noted that in Fig. 25 only two and on Fig. 26 only three of the curves refer to conditions in which the fluid temperature is below the transposed critical temperature (whereas all Draper's data were for this situation). For these curves there appears to be a stronger effect of fluid temperature than was obtained by Draper;

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63

however the fluid temperatures are relatively closer to the critical, and the range of temperature difference is much smaller. T h e conditions are also different in that Draper used a horizontal wire.

n

"0

I

10 Temperature

I

20

30

difference ("C)

FIG. 26. Free convection from a 6-in high, 0.5-in wide vertical ribbon to water at a pressure of 228 bars. Data of J. R. Larson and R. J. Schoenhals, J. Heat Transfer, November, 407 (1966). Bulk temperature: (+) 375.8"C, (0)376.4"C, (0)376.7"C, ( x ) 365.8"C, ( A ) 377.9"C, (*) 378.3"C, (0) 382.9"C. T,, = 376.5OC.

B. THEORETICAL METHODS AND CORRELATIONS Most of the free convection problems that arise in engineering are likely to give rise to turbulent boundary layers. This may be illustrated by considering the case of a vertical plane surface at a temperature, T o , of 400°C, immersed in water at 250 bars (critical pressure, 221.2 bars) at a temperature, T , , of 370°C (critical temperature, 374°C). T h e relevant physical properties are as follows: ps = 0.540 gm/cm3; p,$= 6.35

x

po = 0.166

gm/cm sec;

gm/cm3, Y, =

:.

( p s - po)/ps = 0.692

1.17 x 10-9

cm2/sec

W. B. HALL

64 Thus the Grashof number,

Gr -" 8 gx3 - 5.0 x 108x9 Ps

v2

If we assume that the criterion for transition to a turbulent boundary layer is still approximately the same as for fluids with small property variations, (i.e., G r M 2 x log), then this would occur in the above example with a plate 1.5 cm high.

1. The Basis of Correlations T h e dimensionless parameters governing turbulent free convection may be obtained from Eqs. (14), (20), and (21), together with Eqs. (40) and (41). We proceed in a similar manner to that employed in Section IV, C, 2, noting, however, that in this case there is no characteristic velocity which may be extracted as a parameter. Thus we employ the following change of variables: X

= x/d;

Y =y/d;

6' = ( T - T,)](To - T,)

=

H

=

(h - h,)/(ho - h,)

( T - Ts)/AT;

Ul

=

(h - hs)/Ah;

= ud/vs ;

Vl

= vd/vS

We also assume, as in Section IV, C, 2, that dissipation effects are negligible, and also that u, = 0. T h e following equations, analogous to Eqs. (47) and (48) for free convection, are then obtained:

pU,aR -ps

,iiaH

ax+p,=

where

If we again assume that pressure variations throughout the system are small compared with the absolute pressure, we obtain the equation analogous to Eq. (49):

HEATTRANSFER NEAR

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CRITICAL POINT

65

Thus, we see that the solution is formally similar to that for forced convection, except that the Grashof number replaces the Reynolds number. Thus

Similar arguments to those used in Sections IV, C, 3, a , b, and c can also be proposed. With reference to the limit as LIT --f 0, it has already been pointed out in Section 11, D, 2, c that the Grashof number does not become infinite at the critical point as has sometimes been suggested. Nevertheless, the problem remains more difficult than that of forced convection because of the interconnection between the momentum and energy equations (the term ( p , - p ) / ( p , - po), which is a function of temperature, links Eqs. (63) and (64)); thus the energy equation remains nonlinear in temperature even if the physical properties are constant. It would be interesting to follow up the possibility of using reduced coordinates to make comparisons between different fluids. As with forced convection, this would involve the comparison of data at the same values of reduced pressure, p * and the same values of reduced surface temperature To* and reduced fluid temperature, T,*. It is doubtful, however, whether sufficient data exist at the present time to make such a comparison.

2. Theoretical Calculations It seems likely that numerical integration of the basic equations will eventually prove the most profitable line of attack although the difficulties to be overcome are greater than with forced convection. As has been demonstrated previously, most practical problems are likely to involve turbulent conditions, and far less is known about turbulent diffusion in free convection than in forced convection boundary layers. For want of a better model, one might adopt similar descriptions of, for example, the mixing length, 1, in Eqs. (63)-(65) as those which are used in forced convection; this hypothesis has not yet been tested for constant property fluids, however. A number of attempts have been made to solve the free convection equations for nonturbulent conditions, usually by means of numerical methods. Some of these involve the use of the same similarity variables as are used in the constant property case, which seems questionable. Unfortunately it has not been possible adequately to test the calculations; this is not surprising when one considers the ease with which turbulence is produced, and the small temperature differences that are required if

66

W. B. HALL

it is to be avoided. Some of the calculations are described in references (46)-(49).

VI. Combined Forced and Free Convection

It has long been recognized that the very large density differences and small kinematic viscosities that occur with supercritical fluids produce ideal conditions for free convection. It is therefore surprising that more attention has not been directed towards free convection effects in forced flows. Most of the forced convection experiments have not, in fact, been designed to detect such effects; a change of flow direction in a vertical pipe, or measurements around the circumference of a horizontal pipe, are necessary for this purpose. One of the earliest investigations in which such effects were clearly present in a horizontal pipe was presented by Mirapolsky et al. (50). Results for upward and downward flow in vertical pipes have been reported more recently by Shitsman (19) using water, by Jackson and Evans-Lutterodt (20) and by Bourke et al. (51) using CO, . Shitsman clearly identified the sharp deteriorations in heat transfer coefficient which he observed for upward flow with a free convection effect; Hall et al. (12) proposed a mechanism for the effect in terms of the redistribution of the turbulent shear stress across the pipe. There is now little doubt that localized reductions in heat transfer in vertical pipes can be caused by buoyancy effects. I t was very difficult to understand, until a detailed mechanism was proposed, how free convection could combine with an upward forced convection to give a lower heat transfer coefficient than that measured with forced convection alone. This was particularly the case since the opposite effect had been predicted by a number of theoretical investigations of combined forced and free convection under nonturbulent conditions. (The model we shall describe refers to turbulent flows.) One interesting sidelight on the problem is the close connection between it and the phenomenon of “laminarization” of a turbulent boundary layer by means of an acceleration of the flow (52). Indeed, attempts were made to describe the supercritical heat transfer deteriorations in terms of an acceleration of the mean flow (following the rapid reduction in density as the fluid is heated through the critical temperature), but it soon became clear that buoyancy forces were more important than acceleration forces in this case. It seems likely that in both phenomena a reduction of turbulent diffusion follows from a modification of the shear stress distribution; this modification is produced in the one case by inertia effects in an accelerated flow and in the other by buoyancy forces in a flow with large density differences.

HEATTRANSFER NEAR

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67

Apart from the work of Miropolsky et nl. (50) there is very little detailed information on free convection effects in horizontal pipes. This is a very important area in which experimental data will be required before many practical applications can be adequately assessed. For the remainder of the present section, however, we shall restrict the discussion to vertical pipes.

A. EXPERIMENTAL RESULTS T h e most revealing comparison in the present context which can be made is that between upward and downward flow in a heated vertical pipe; this is illustrated in Figs. 6 and 16, both of which are based on the work of Jackson and Evans-Lutterodt (20). T h e curves for two particular heat fluxes are superimposed in Fig. 27 to ease comparison. It will be seen that at a low heat flux there is only a small difference between the results for the two flow directions, but as the heat flux is increased, a large peak occurs in the wall temperature for upward flow; as the heat flux is increased further, the peak sharpens and moves towards the inlet of the pipe. A very similar behavior has been obtained by Shitsman (19) and by Bourke et al. (51). T h e peak occurs, typically, when the surface temperature is above and the fluid temperature below the critical temperature, although Shitsman has observed similar effects when both temperatures, just before the peak, are below the critical value. T h e magnitude of the circumferential variation of temperature around the pipe, shown by vertical lines in Fig. 27, is seen to increase in the region of the peaks in upward Aow. T h e results presented in Figs. 6 , 16, and 27 form part of an extensive range of data in which the fluid inlet temperature, the mass flow, the pressure, and the heat flux were varied (9). T h e temperature peaks are present for upward flow at all pressures investigated (from the critical pressure to about 1.1 times the critical pressure) and for all Reynolds numbers (covering the range 2.5 x lo4 to lo5). T h e peaks generally disappear as the fluid temperature at the pipe inlet approaches the critical temperature and are replaced by a uniformly lower heat transfer coefficient (when compared with the corresponding downward flow results). There is also a tendency for an increase in pressure or mass flow to raise the heat flux at which the peaks begin to occur. Were it not for the fact that the temperature peaks occur only with upward flow, one would be tempted to explain the phenomenon in terms of a reduction in thermal conductivity in the viscous wall layer as the fluid in this region passes from a subcritical to a supercritical temperature. (This effect has, in fact, been shown not to result in a reduced

W. B. HALL

68

heat transfer coefficient (7) presumably because of the corresponding thinning of the viscous layer as the viscosity decreases). The same difficulty arises with any explanation based on the acceleration of the flow as it is heated; were this the case, it should presumably occur with downward as well as upward flow. 110

A

-

100 -

90 -

-:

80 -

0

70-

608

n

5

50-

: c

4030 20

I

20 Distance

I

40

I

60

I

80

I 100

I

I20

I

from start of heating (diameters)

FIG.27. Comparison of data of Figs. 6 and 16 for upward and downward flow. (a) Wall heat flux = 3.09 W/cma, (b) Wall heat flux = 5.67 W/cm2.(-) upward flow; (- - -) downward flow.

B. A PROPOSED MECHANISM FOR

THE

HEATTRANSFER DETERIORATIONS

The following analysis is substantially the same as that described by Hall and Jackson (52). I t may assist the reader to follow the argument if it is preceded by a brief statement of the more important steps. We begin by evaluating the approximate magnitude of the buoyancy forces in a typical case. It is found that even a very thin low density layer near the pipe wall is sufficient to generate forces which are of the same order as the turbulent shear force on the pipe wall. The shear stress distribution across the pipe is therefore drastically changed, and with it the amount of energy being fed into the turbulence (c.f. Section 111, C, 2.) Consequently the level of turbulence, and with it the turbulent diffusivity, decreases. At a later stage of development of the low density layer adjacent

HEAT TRANSFER NEAR

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69

to the wall, the shear stress is reversed, the energy input to the turbulence is restored, and with it the heat transfer coefficient. 1. The Effect of Buoyancy Forces in Modifying the Shear Stress Distribution T h e sharpness of the density change in passing from a subcritical to a supercritical temperature allows one to use, with a fair degree of accuracy, a two-region pattern of density when calculating the shear stress distribution across a pipe. We assume that there is a layer of fluid at a uniform low density pw adjacent to the wall, and that the core is at a uniform high density pe; we also neglect inertia effects in the wall layer. T h e result is a modification to the shear stress distribution as shown diagrammatically in Fig. 28 for three different thicknesses of wall layer. T h e particular thickness, A, for which the shear stress at the edge of the layer (and throughout the core) is zero, is calculated below. Typical values of pw and pe corresponding to carbon dioxide are Pipe

‘

E

l

i

i

1:

c , Y)

P ’ Y ) ’

o / r,

I

v)

I

, /

0/ / /

,

/

/

I

I

FIG. 28. Diagram showing the shear stress distribution across a channel for three thicknesses of the low density Iayer adjacent to the wall (upward flow).

W. B. HALL

70

0.3 gm/cms and 0.7 gm/cm3, respectively. (These values are appropriate to wall and bulk temperatures of about 35°C and 29°C.) If the shear stress is to be reduced to zero at y = A, then the buoyancy force per unit length of pipe, B , must balance the wall shear force per unit length of pipe, S. T h e latter force increases slightly as the wall layer thickness increases (53), but the value for a uniform density equal to that in the core is still a good approximation when the shear stress in the core is reduced to zero; at a Reynolds number of lo5 in a 1.9-cm pipe this gives a value for S of about 15 dyn/cm. Thus and

B

h

= ndh(p,

- pw)g

=

S

= S / d ( p , - pw)g 11

= 6.4

5 / ~x 1.9 x 0.4 x 981.

x lo-* cm

Thus a very thin layer of low density fluid at the wall is capable of reducing the shear stress in the core to zero. This will have a profound effect on the production of turbulence and, therefore, on the turbulent diffusivity, as shown in the following section.

2. The Effect of the Shear Stress Distribution on Turbulence T h e turbulence in the flow is maintained by an energy input which arises from the shearing of the turbulent fluid by the mean velocity gradient; this input is equal to r 1 afijay, and has its greatest value close to the wall. (Not at the wall, because the flow there is nonturbulent, and 71 = 0.) One might suspect, therefore, that the modifications to the shear stress shown in Fig. 28 reduce the level of turbulence and thus the turbulent diffusivity, e; this will certainly be the case in the core where both r d and ai?/ay become zero. I n the wall layer the net effect will depend upon the relative changes in T~ and &jay, and can be estimated by using a mixing length model of turbulent diffusion. T h e mixing length model asserts that rt =

-

p uw

= pi2

I aqay I aqay

T h e turbulent diffusivity is then given by = Tti,j aqay = i ( T t l , j ) l / z

Thus, provided that the mixing length 1 is not changed significantly by the density variation, a reduction in total shear stress will involve a reduction in T~ and an associated reduction in the turbulent diffusivtiy.

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As shown diagrammatically in Fig. 28, the shear stress very close to the wall is increased by the presence of the low density layer. However, this increase will usually be in a region in which turbulence is damped out by the wall and in which the turbulent shear stress is small. T h e increase will not, therefore, give rise to an increase in turbulence production. T h e above analysis is open to question on at least two grounds; firstly, it is by no means certain that the mixing length will be unaffected by the presence of the low density layer; secondly, and perhaps more fundamentally, the use of a mixing length model may not be justified in these circumstances. I t was mentioned in Section 111, C, 2, that a description of the distribution of turbulence based on the turbulent kinetic energy equation (which balances the production, dissipation, convection, and diffusion of the kinetic energy of turbulence) reduces to the mixing length model for an equilibrium boundary layer when the convection and diffusion terms in the equation are omitted; this implies local equilibrium between the production and dissipation of turbulence. I n the present application it is likely that the turbulence level will change quite rapidly in the direction of flow and that convection of turbulent kinetic energy will become important. Nevertheless, it is felt that the mechanism proposed above is qualitatively correct.

3 . A Mechanism for the Local Deteriorations in Heat Transfer Coejicient T h e consequence of the events described previously is that the turbulent diffusivity is reduced in upward flow when the low density wall layer becomes thick enough to reduce materially the shear stress in the region where energy is normally fed into the turbulence. This will, of course, reduce the diffusivity for heat, and, therefore, the heat transfer coefficient. As the process develops along the tube the wall temperature rises, the density difference becomes greater, and the buoyant layer thickens; both these effects accentuate the laminarization. It is possible that the low density layer eventually becomes sufficiently thick for the sign of the shear stress to be changed in the central region, as shown in Fig. 28. T h e wall layer will then exert an upward force on the core: the production of turbulence will be restored, and the turbulent diffusivity will increase. T h e model, therefore, accounts not only for the deterioration of the heat transfer coefficient but also for its subsequent improvement. This progression of events is illustrated in Fig. 29 which is based on an approximate theoretical model of the flow of supercritical pressure CO, between parallel planes 1.5 cm apart at a Reynolds number of

W. B. HALL

72

lo6 (53).It was assumed in the calculations (53) that the flow in the low density wall layer was nonturbulent, and the velocity distribution in the core was described by a “velocity defect” law in which the characteristic shear stress was taken as that at the interface between core and wall layer rather than that at the wall. Values of y+ at the interface are marked on the curves in Fig. 29 and show that the assumption of nonturbulent flow in the wall layer is a reasonable approximation. A comparison is made in (53)between this simple model and one that uses a van Driest formulation of the velocity distribution, which avoids the necessity for separate treatment of the wall layer and the core.

- 20 Shear stress

0

1

2

3

4

5

Distance

6

7

from wall

8

9

1

0

1

1

1

2

(10-3cm)

FIG.29. Calculated velocity and shear stress distribution near the wall of a channel with upward flow and three thicknesses of low density layer adjacent to the wall. (W. B. Hall and J. D. Jackson, Laminarisation of a turbulent pipe flow by buoyancy forces. 1l t h National Heat Transfer Conf., Special Session on Laminarization of Turbulent Flows, Minneapolis, Paper No. 69-HT-55, 1969. W. B. Hall, The Effect of Buoyancy Forces on Forced Convection Heat Transfer in a Vertical Pipe. Rept. N.E.1, Simon Engineering Labs., University of Manchester, 1968.)

It is interesting to note that when the shear stress falls to zero at the interface, (curve B of Fig. 29), the core is completely decoupled from the wall, and is moving upwards as a true “plug flow.” The wall layer, on the other hand, is entirely motivated by buoyancy forces, which are acting in precisely the region where they are required to overcome viscous shear (rather than being transmitted by a shear process from the core-a process which normally maintains the production of turbulence).

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If the wall layer is so thin that turbulence in it is supressed by the wall, then the whole flow could, if the situation persisted long enough, become nonturbulent even though the Reynolds number was well above the transition value. An approximate criterion for the conditions under which heat transfer may be impaired by buoyancy forces can be obtained by the methods that were developed in Section 111, €3, 1. In this case we apply the condition that the modification to the shear stress distribution is not merely significant (with respect to the initial distribution given by &lay = - ~ , / b ) , but large enough to reduce the shear stress to zero where it really matters, i.e., in the region y + w 30, where turbulence production is normally a maximum. If this is to happen (Fig. 28), &/ay = - ~ ~ / h , where

and

(pc - p w ) g must

h = ~OV,,/(T,/~,)~/*

be comparable in magnitude to ~ , / h i.e., ,

-

This may be expressed in the form Gr/Re2.7

where

1.2 x 10-4

(In obtaining Eq. (67) it has been assumed that the usual relationships between T~ and Re apply, i.e., T~ =f ip&u,Z and Q f= 0.023 Re-Oe2, and that vc/vzo 1). If we insert the data appropriate to Figs. 6 and 16 (i-e., that quoted in Section IV, C, I), we find that,

-

Gr/Re2.7m 0.7 x lo4

Thus it appears that the criterion established in Eq. (67)is approximately satisfied in the case of these results, (which certainly show buoyancy effects). The main difficulty in applying the criterion lies in the selection of suitable values of pw and pc . The above analysis is based on the assumption that a “two-region” model is adequate, as far as the density variation across the channel is concerned; the validity of this assumption depends very much on the wall heat flux level. This matter is discussed in the following.

74

W. B. HALL

4. The Injuence of Heat Flux The experimental results show that at low heat fluxes there is no sudden deterioration in heat transfer and that the wall temperature distribution for upward flow is not greatly different from that for downward flow. As the heat flux tends to zero, the temperature variation (and hence the density variation) across the flow becomes so small that a thick wall layer would be required in order to affect the shear stress significantly; the shear stress reduction would then take place too far from the wall to modify the turbulence production greatly. It seems, therefore, that in addition to the criterion expressed in Eq. (67), we must also stipulate a heat flux which is sufficiently high to produce a fairly sharp density change between wall and core regions. There will be a range of conditions where the “two-region” model of density is inadequate, e.g., a small density difference may be compensated by a low Reynolds number, and deteriorations may still occur. T h e analysis of such conditions is considerably more difficult; the most promising line appears to be the numerical solution of the flow and energy equations for a range of boundary conditions. Such work is in progress using a mixing length model of turbulent diffusion, and it is proposed to follow this by a more sophisticated model based on the turbulent kinetic energy equation. The most that can be said at the present time is that the criterion based on Eq. (67) must be regarded with caution when the heat flux is so low as to produce a temperature difference which does not span the region of rapid property variation.

VII. Boiling Boiling can occur only at subcritical pressures and is, thus, strictly outside the scope of this article. Nevertheless, it is of interest to touch briefly upon the characteristics of the boiling process at pressures approaching the critical pressure. In this respect the critical point may be regarded as the condition where the boiling and convection merge. T h e particular characteristics of the boiling process which set it apart from convection are, of course, (i) the coexistence of two separate phases each with its own density, enthalpy, and transport properties the phenomenon of surface tension at the interface between (ii) the phases; unless the interface is plane, this implies a pressure difference between the phases.

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At the critical point the distinction between the two phases disappears, as does the surface tension. As the critical point is approached, therefore, boiling is characterized by diminishing property differences between phases and diminishing surface tension. T h e ease with which a vapor bubble may be nucleated in a liquid depends upon the enthalpy change associated with the change of phase and upon the surface tension. The effect of surface tension is to increase the pressure in a vapor bubble (by an amount that increases as the bubble size decreases) and thus to make it necessary for the liquid to be superheated before the bubble can grow. For a given liquid and superheat there is a particular size of vapor bubble that can exist in equilibrium with the liquid. T h e equilibrium is unstable and a slight increase in bubble size will cause it to grow indefinitely, whereas a slight decrease will cause it to shrink and eventually disappear. For nucleation to occur it is necessary to create a bubble that exceeds this equilibrium size; associated with this is a critical value of enthalpy input for the nucleation process. It is found that nucleation takes place preferentially at a solid surface rather than in the body of the liquid; also that the superheat required is frequently much less than the theoretical value. This suggests that in practice the process must depend upon the stabilization of a relatively large nucleus (i.e., one containing many thousands of molecules), possibly at some cavity in the solid surface. T h e main effect of near-critical conditions on nucleation is the reduction in the superheat required to initiate bubble growth at a given size of cavity. I t will be shown later, however, that the range of heat fluxes over which nucleate boiling occurs is reduced as the critical point is approached, and film boiling occurs more readily. Surface tension has a large effect on the initial rate of growth of the vapor bubble; high surface tensions cause the equilibrium pressure in the bubble to be greater, and it is the excess of this pressure over that of the liquid that causes the rapid initial growth. This is one of the factors which causes the high heat transfer coefficient in nucleate boiling; the effect will be less marked at near-critical conditions because of the reduction in surface tension. At the same time the free convection that precedes nucleate boiling will generally be more effective as one approaches the critical point. These factors operate so as to produce a less marked change in the heat transfer coefficient when nucleate boiling begins; indeed it is sometimes difficult to be sure, from measurements of the heat transfer coefficient, that boiling has commenced. One of the most striking differences observed at near-critical conditions occurs in the case of film boiling. At low pressures the transition from nucleate to film boiling presents a rather confused visual appearance.

76

W. B. HALL

Large surface tension and buoyancy forces combine to cause the bubble departure from the surface to be a somewhat unstable and violent process. Near the critical point, however, both these factors are smaller, and film boiling presents a surprisingly orderly appearance. These effects are briefly illustrated, using the data of Draper (sulphur hexaflouride and a horizontal 0.1-mm diameter tungsten wire) and Grigull and Abadzic (CO, and Freon 13 and a horizontal 0.1-mm platinum wire).

BOILING A. NUCLEATE Figures 30-32 show Draper's results for SF, (critical pressure 37.7 bars and critical temperature 45.6"C) at three different fluid temperatures (44). On each figure the heat flux is plotted against the temperature difference between wire and fluid. It will be seen, from the saturated vapor pressure corresponding to each fluid temperature, that the liquid is, in all cases, subcooled. T h e surface heat flux was controlled, and therefore departure from nucleate boiling led directly to film boiling, and no points in the transition region could be obtained. It will be seen that the effectiveness of nucleate boiling in improving heat transfer rapidly diminishes as the pressure approaches its critical value. Figure 33 shows the wire under nucleate boiling conditions; the fluid is subcooled (at a temperature of 21.3"C compared with a saturation temperature of

Temperature difference, T,-T,

PC)

FIG. 30. Pool boiling from a 0.01I-cm diameter wire in sulphur hexaflouride. Fluid temperature 233°C. (Data of R. Draper, MSc. Thesis, University of Manchester, 1968.)

*

P

01

0

I 10

I 5

Temperature

I

15

I

1 25

20

difference ,To-Ts

('C)

FIG. 31. Pool boiling from a 0.011-cm diameter wire in sulphur hexaflouride. Fluid temperature 39.6"C. (Data of R. Draper, MSc. Thesis, University of Manchester, 1968.)

i

10

I -

U

/ *

I

/

Temperature difference ,To-Ts

("C)

FIG. 32. Pool boiling from a 0.01 1-cm diameter wire in sulphur hexaflouride. Fluid temperature 43.1"C. (Data of R. Draper, M.Sc. Thesis, University of Manchester, 1968.)

78

W. B. HALL

31.5"C). There appears to be little coalescence of bubbles, and their condensation as they leave the wire can be seen clearly. Figure 34 shows the results of Grigull and Abadzic for CO, boiling under saturated conditions (critical pressure 73.8 bars and critical temperature 31.1"C). I n this case hysteresis in the transition between nucleate and film boiling was observed by reducing the heat input under film boiling conditions until nucleate boiling was again established.

FIG. 33. Nucleate boiling from a 0.011-cm diameter wire in sulphur hexaflouride at a pressure of 27.6 bars. Data of R. Draper, MSc. Thesis, University of Manchester, 1968. Fluid temperature 21.3"C; wire temperature 34.4"C; heat flux, 23.8 W/crn*. Magnification x 25 approx.

T h e full lines at the lower left hand corner of the figure represent free convection conditions. Since the fluid was at its saturation temperature, and since the superheat required to initiate nucleate boiling was only 2.4"C at the pressure of 55.7 bars and 0.1"C at 71.2 bars, the free convection condition was difficult to achieve; the authors found that once nucleate boiling was established the superheat of the wire surface could be reduced below the value at which boiling first occurred, and that natural convection could only be reestablished by disconnecting the power supply for a few minutes. T h e results obtained for Freon 13 were generally similar to those for CO, .

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30

"c

FIG. 34. Heat flux as a function of temperature difference between a 0.01-cm diameter horizontal wire and carbon dioxide. Data of U. Grigull and E. Abadzic, "Symp. Heat Transfer and Fluid Dynamics of Near Critical Fluids." Proc. Inst. Mech. Eng. 182, Part 31 (1968). Key: ( A ) T = 18.7"C; ( A ) T = 18.9"C; ( 0 ) T = 20.1"C; ( V ) T = 20.7"C; ('I) T = 22.3"C; (11) T = 23.2"C; ( 0 ) T = 24.2"C; T = 25.4"C; ( 0 ) T = 26.5"C; (u) T = 27.6"C;(+) T = 29.5"C;(0) T = 30.7"C;( X ) T = 30.9"C.

(a)

B. FILMBOILING T h e film boiling process near the critical point exhibits some interesting characteristics. Movies of the process have been produced by Grigull(54) and by Draper (44);Figs. 35 and 36 are reproduced from that by Draper. T h e behavior of the vapor film is much more orderly than its is at low pressures, and exhibits instabilities which are very reminiscent of those proposed by Zuber in connection with departure from nucleate boiling on a horizontal plate. As conditions approach more closely to the critical point, the vapor layer tends firstly to form into tubes which subsequently break up into bubbles (Fig. 35) and then into thin sheets of vapor in the form of festoons rising from the wire (Fig. 36). Grigull and Abadzic (54) have produced some very beautiful pictures of these phenomena using

co, .

Draper has compared his data for departure from nucleate boiling

80

W. B. HALL

FIG. 35. Film boiling from a 0.011-cm diameter wire in sulphur hexaflouride at a pressure of 34.8 bars. Data of R. Draper, M.Sc. Thesis, University of Manchester, 1968. Fluid temperature 25.2"C; wire temperature 1658°C; heat flux 34.3 W/cme. Magnification x 25 approx.

FIG.36. Film boiling from a 0.011-cm diameter wire in sulphur hexaflouride at a pressure of 36.2 bars. Data of R. Draper, M.Sc. Thesis, University of Manchester, 1968. Fluid temperature 41°C; wire temperature 221.3"C; heat flux 42.9 W/cmZ.Magnification x 25 approx.

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under subcooled conditions with the method of prediction proposed by Zuber, Tribus and Westwater (57). Figure 37 show the comparison; the degree of agreement is encouraging. Grigull and Abadzic (54, however, found that their data for saturated conditions did not compare well with this type of prediction.

A

e-

I

Fluid temp

\

pi I

FIG. 37. Comparison between the results of R. Draper, M.Sc. Thesis, University of Manchester, 1968 and the theory of N. Zuber, M. Tribus, and J. W. Westwater, The hydrodynamic crisis in pool boiling of saturated and subcooled liquids. Int. Deoelop. Heat Transfer, Proc. Heat Transfer Conf., 1961 Pt 11, Paper No. 27 (1963) (shown by curves) for departure from nucleate boiling in a subcooled liquid. (Sulphur hexaflouride; critical pressure 37.7 bars, critical temperature 456°C.)It should be noted that while the experimental data are for a wire, the theory refers to a horizontal plane surface.

C. PSEUDOBOILING There have been many attempts to explain unusual heat transfer behavior of supercritical pressure fluids in terms of a “pseudo-boiling”

W. B. HALL phenomenon. Thus, an increase in heat transfer coefficient has been attributed to the occurrence of something like nucleate boiling, and a decrease to the onset of film boiling. It appears to the writer to be both irrational and unnecessary to introduce this concept at pressures which prohibit the existence of two distinct phases. It is true that the grouping of molecules into clusters is in some respects similar to a phase change; it seems likely, however, that the scale of this phenomenon will usually be small compared with the scale of the system, and that the fluid may therefore be treated as a continuum. The high level of turbulence that can be produced at a heated surface in a supercritical fluid has also been attributed to “pseudo boiling,” but it is not necessary to look further than the very large density changes, and consequently violent free convection, for an explanation. The one piece of evidence that is still difficult to explain in terms other than “pseudo boiling,” has already been mentioned in Section V, and is illustrated in Fig. 21. I t has not been possible to repeat this behavior using SF, under similar conditions (44,and until further evidence is produced the writer remains unconvinced that it is necessary to invoke the concept of boiling to explain heat transfer at supercritical pressures.

ACKNOWLEDGMENT I t is a pleasure to acknowledge the help of my colleagues and research students at the Simon Engineering Laboratories, University of Manchester. I wish particularly to thank J. D. Jackson whose pertinacity in our many arguments has forced me to think harder about the subject than I would otherwise have done.

NOMENCLATURE U

A b

B C

CII Ce

d E

f g

Gr

pipe radius; constant in Eq. ( I ) constant in Eq. (58) half width of channel; constant in Eq. (1) buoyancy force per unit length of Pipe velocity of sound specific heat at constant pressure specific heat at constant volume pipe diameter Eckert number (defined in Section 111, A, 2) friction factor (defined in Section 111, A, 2 ) gravitational acceleration Grashof number (defined in Section 111, A, 1)

h

H k K 1 rh

M n

Nu

P 9

R

Re

S

St

enthalpy ( h - ha)/(& - he) or ( h - M / ( h o- hm) thermal conductivity acceleration parameter (defined in Section 111, A, 3) mixing length (defined by Eqs. (40) and (41)) mass flow rate molecular weight constant in Eq. (56) Nusselt number pressure heat flux p / p , ; molar gas constant Reynolds number shear force per unit length of pipe Stanton number

HEATTRANSFER NEAR t

T Tllc

U

U UI V

V Vl X>Y, 2

X

Y

Z a

B 8 €

e KT K8

h

time temperature transposed critical temperature velocity in the x-direction

THE

CRITICAL POINT

p v

p T

4%

83

viscosity p / p , kinematic viscosity density shear stress (turbulent component 7,)

ud/v velocity in the y-direction v / u , ; specific volume (P = molar volume)

SUBSCRIPTS c

vdlv

m 0 s

Cartesian coordinates xjd or xjb Y/_d or Y / b

P V W

heat transfer coefficient coeflicient of expansion, (1 /v)(av/aT)” boundary layer thickness eddy diffusivity (c,, for heat, eM for momentum) ( T - T.)I(To- TJ or ( T - T,fl)/(To - T m ) - ( l j V ) ( a V / a p ) , ,isothermal compressibility -(I/V)(aV/ap), , isentropic compressibility thickness of wall layer for which T = 0 in core of pipe (Section VI, B, 1)

w

in the turbulent core bulk mean at a wall in the free stream, or, when stated, at some other reference condition in a wall layer

SUPERSCRIPTS c

* -

+ ++

value of parameter at the critical point reduced coordinate: e.g., P* = PIPC fluctuating component time mean value of a fluctuating quantity “universal” parameter (see Eq. 56) variable property “universal” parameter (see Eq. 60)

REFERENCES J. S. Rowlinson, Singularities in the thermodynamic and transport properties of a f lluid at its critical point, in “Symp. Heat Transfer and Fluid Dynamics of Near Criitical Fluids.” Proc. Inst. Mech. Eng. 182, Part 31 (1968). 2. J. S. Rowlinson, “Liquids and Liquid Mixtures.” Butterworth, London and Wrishington, D.C., 1959. 3. H. Teilsch and H. Tanneberger, Z. Phys. 137, 256 (1954). 4. F. 1G. Smith, “Review of Physico-Chemical Data on the State of Supercritical Fluids.” Eccm. Geol. 48, No. 1 (1953). 5. V. M. Borishansky, I. I. Novikov, and S. S. Kutateladze, Use of thermodynamic s i riilarity in generalising experimental data of heat transfer. Int. Dewelop. Heat Trdzmfer, Proc. Heat Transfer Conf, 1961, Pt. 11, Paper No. 56 (1963). 6 . J. 3. Hirschfelder, C. F. Curtiss, and R. B. Bird, “Molecular Theory of Gases and Licluids.” Wiley, New York, 1954. 7. W. B. Hall, J. D. Jackson, and S.A. Khan, Investigation of forced convection heat trainsfer to super-critical pressure CO, Proc. Znt. Heat Transfer Conf., 3rd, Chicago, 19ti6 (1967). 8. A . ,A. Townsend, Equilibrium layers and wall turbulence. J. Fluid Mech. 11.97 (1961). 9. K. Evans-Lutterodt, Ph.D. Thesis, University of Manchester. 1.

.

84

W. B. HALL

10. K. Goldmann, Heat transfer to super-critical water at 5000p.s.i. flowing at high mass flow rates through round tubes. Int. Develop. Heat Transfer, €’roc. Heat Transfer Conf., 1961, Pt. 11, Paper No. 66 (1963). 11. 2. L. Miropolsky and M. E. Shitsman, Heat transfer to water and steam with varying specific heat (in the near critical region). Zh. Tekhn. Fi2. 27, No. 10 (1957). 12. W. B. Hall, J. D. Jackson, and A. Watson, A. review of forced convection heat transfer to fluids at super-critical pressures, in “Symp. Heat Transfer and Fluid Dynamics of Near Critical Fluids.” Proc. Inst. Mech. Eng. 182, Part 31 (1968). 13. M. E. Shitsman, Impairment of the heat transmission at super-critical pressures. Teplofiz. Vys. Temp. 1, No. 2 (1963). 14. K. R. Schmidt, Thermal investigations with heavily loaded boiler heating surfaces. Mitt. Ver. Grosskesselbetr.No. 63, 391 (1959). 15. G. Domin, Warmeubergang in kritischen und iiberkritischen Bereichen von Wasser in Rohren (Heat transfer to water in pipes in the critical/super-critical region). Brennst.-Warme-Kraft 15, No. 11 (1963). 16. Y. V. Vikrev and V. A. Lokshin, An experimental study of temperature conditions in horizontal steam generating tubes at super-critical pressures. Teploenergetika 11, No. 12 (1964). 17. H. Tanaka, N. Nishiwaki, and M. Hirata, Turbulent heat transfer to super-critical carbon dioxide. Nippon Kikai Gakkai Rombunshu 127 (1967). 18. P. Griffith and B. S. Shiralkar, The Deterioration in Heat Transfer to Fluids at Super-critical Pressure and High Heat Fluxes. Dept. Mech. Eng., M.I.T. Rept. NO. 70332-51, 1968. 19. M. E. Shitsman, Natural convection effect on heat transfer to a turbulent water flow in intensively heated tubes at super-critical pressure, in “Symp. Heat Transfer and Fluid Dynamics of Near Critical Fluids.” Proc. Inst. Mech. Eng. 182, Part 31 (1968). 20. J. D. Jackson and K. Evans-Lutterodt, Impairment of Turbulent Forced Convection Heat Transfer to Super-critical Pressure CO, Caused by Buoyancy Forces. Rept. N.E.2, Simon Engineering Labs, University of Manchester, 1968. 21. 2. L. Miropolsky and V. U. Pikus, Heat transfer in super-critical flows through curvilinear channels, in “Symposium on Heat Transfer and Fluid Dynamics of Near Inst. Mech. Eng. 182, Part 31 (1968). Critical Fluids.” PYOC. 22. B. S. Petukhov, E. A. Krasnoschekhov, and V. S. Protopopov, An investigation of heat transfer to fluids flowing in pipes under super-critical conditions. Int. Develop. Heat Transfer, Proc. Heat Transfer Conf. 1961 (1963). 23. S. S. Kutateladze and A. I. Leontiev, “Turbulent Boundary Layers in Compressible Gases.” Arnold, London, 1964. 24. N. L. Dickinson and C. P. Welch, Heat transfer to super-critical water. Trans. ASME 80,746 (1958). 25. R. P. Bringer and J. M. Smith, Heat transfer in the critical region. Am. Inst. Chem. Engrs., Paper 3, No. 1 (1957). 26. L. B. Koppel, Heat Transfer and Thermodynamics in the Critical Region. Ph.D. Thesis in Chemical Engineering, Northwestern University, Illinois, 1960. 27. C. A. Sleicher, Jr., Experimental velocity and temperature profiles for air in turbulent pipe flow. Am. SOC.Mech. Engrs. Paper No. 57-HT-9 (1957). 28. W. H. Corcoran, F. Page, W. G. Schlinger, and B. H. Sage, Temperature gradients in turbulent gas streams. Ind. Eng. Chem. 44, Pts. 1-4, 410 (1952). 29. C. K. Brown, B. H. Amstead, and B. E. Short, The transfer of heat and momentum in a turbulent stream of mercury. Am. SOC.Mech. Engrs. Paper No. 55-A106 (1955). 30. R. G. Deissler, Heat transfer and fluid friction for fully developed turbulent flow

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CRITICAL POINT

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of air and super-critical water with variable fluid properties. Trans. ASME 76, January (1954). 31. R. G. Deissler and A. F. Presler, Computer reference temperatures for turbulent variable property heat transfer in a tube for several common gases. Int. Dewelop. Heat Transfer, Proc. Heat Transfer Conf., I961 (1963). 32. E. R. van Driest, On turbulent flow near a wall. 1.Aeron. Sci. November (1956). 33. D. A. Wiederecht and G. Sonnemann, Investigation of the nonisothermal friction factor in the turbulent flowof liquids. Am. SOC.Mech. Engrs. Paper No. 60-WA-82 (1960). 34. H. L. Hess and H. R. Kunz, A study of forced convection heat transfer to supercritical hydrogen. J. Heat Transfer, February (1965). 35. R. C. Hendricks, R. W. Graham, Y. Y. Hsu, and A. A. Mederios, Correlation of hydrogen heat transfer in boiling and super-critical states. A R S (Am. Rocket Soc.) 32, February (1962). 36. K. Goldmann, Heat transfer to super-critical water and other fluids with temperature dependent properties. Chem. Eng. Progr. Symp. Ser., Nucl. Eng. 50, Part I, No. 11 (1954). 37. E. Schmidt, E. Eckert, and U. Grigull, Heat transfer by liquids near the critical state. A.A.F. Trans. No. 527, Air Material Command, Wright Field, Dayton, Ohio. 38. E. Schmidt, Warmubertragung bei natiirlicher Konvektion, insbesondere durch Stoffe in der Nahe ihres kritischen Zustandes, in “Advances in Aeronautical Sciences” (Proc. 1st Int. Congress in the Aeronautical Sciences Madrid 1958), Vol. 1, 333-342. Pergamon Press, New York, 1959. 39. E. Schmidt, Heat transfer by natural convection. Int. Develop. Heat Transfer, Proc. Heat Transfer Conf., 1961 (1963). 40. H. A. Simon and E. R. G. Eckert, Laminar free convection in carbon dioxide near its critical point. Intern. J. Heat Mass Transfer 6, 681-690 (1963). 41. E. N. Dubrovina and V. P. Skripov, Convective heat transfer in the super-critical region of carbon dioxide, in “Heat and Mass Transfer” (A. V. Lykov and B. M. Smol’skii, eds.), Vol. I. Israel Program for Scientific Translations, Jerusalem, 1967. 42. D. L. Doughty and R. M. Drake, Free-convection heat transfer from a horizontal right circular cylinder to Freon 12 near the critical point. Trans. ASME 78, 1843 (1956). 43. K. K. Knapp and R. H. Sabersky, Free convection heat transfer to carbon dioxide near the critical point. Int. J. Heat Mass Transfer 9, 41-51 (1966). 44. R. Draper, M.Sc. Thesis, University of Manchester, 1968. 45. J. R. Larson and R. J. Schoenhals, Turbulent free convection in near-critical water. J, Heat Transfer, November, 407 ( 1 966). 46. K. Brodowicz and J. Bialokoz, Free convection heat transfer from a vertical plate to Freon 12 near the critical state. Arch. Budowy Masz. 10, 289 (1963). 47. C. A. Fritsch and R. J. Grosh, Free convective heat transfer to a super-cfitical fluid. Int. Develop. Heat Transfer, Proc. Heat Transfer Conf., 1961, Pt. 11, Paper No. 121 (1963). 48. S. Hasegawa and K. Yoskioka, An analysis for free convection heat transfer to supercritical fluids. Proc. Znt. Heat Transfer Conf.,3rd, Chicago 1966, Vol. 11, Paper No. 63 (1967). 49. J. D. Parker and T. E. Mullin, Natural convection in the super-critical region. Symp. Heat Transfer and Fluid Dynamics of Near Critical Fluids. Proc. Inst. Mech. Eng. 182, Part 31 (1968). 50. 2. L. Miropolsky, V. J. Picus, and M. E. Shitsman, Regimes of deteriorated heat

86

51.

52. 53. 54. 55.

56. 57.

W. B. HALL transfer at forced flow of fluids in curvilinear channels. Proc. Int. Heat Transfer Conf, 3rd, Chicago 1966, Vol. 11, Paper No. 50 (1967). P. J. Bourke, D. J. Pulling, L. E. Gill, and W. H. Denton, Forced Convective Heat Transfer to Turbulent COa in the Super-critical Region. Part I. Rept. No. AERE. R5952, Atomic Energy Res. Establishment, HarweII, England, 1969. W. B. Hall and J. D. Jackson, Laminarisation of a turbulent pipe flow by buoyancy forces. 1lth National Heat Transfer Conf., Special Session on Laminarization of Turbulent Flows, Minneapolis, Paper No. 69-HT-55, 1969. W. B. Hall, The Effect of Buoyancy Forces on Forced Convection Heat Transfer in a Vertical Pipe. Rept. N.E.1, Simon Engineering Labs., University of Manchester, 1968. U. Grigull and E. Abadzic, Heat transfer from a wire in the critical region, in “Symp. Heat Transfer and Fluid Dynamics of Near Critical Fluids.” Proc. Znst. Mech. Eng. 182, Part 31 (1968). N. V. Tzederberg and N. A. Morosova, Heat conductivity of carbon dioxide at pressures 1-200 kg/cma and temperatures up to 1200°C. TepZoenergetika No. 1, 75 (1960). S. A. Khan, Ph.D. Thesis, University of Manchester, 1965. N. Zuber, M. Tribus, and J. W. Westwater, The hydrodynamic crisis in pool boiling of saturated and subcookd liquids. Int. Develop. Heat Transfer, Proc. Heat Transfer Cunj., 1961, Pt. 11, Paper No. 27 (1963).

The Electrochemical Method in Transport Phenomena

.

T MIZUSHINA Department of Chemical Engineering. Kyoto University. Kyoto. Japan

I. Introduction . . . . . . . . . . . . . . . . . . . . . . I1. Electrochemical Reaction under the Diffusion-Controlling Condition

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

I11. Application to Mass Transfer Measurements . . . . . . . . A . Principle and Method of Measurements . . . . . . . . .

B. Free Convection . . . . . . . . . . . . . . . . . . . C . Forced Convection . . . . . . . . . . . . . . . . . . IV . Application to Shear Stress Measurements . . . . . . . . . A Principle and Method of Measurements . . . . . . . . . B. Shear Stress in a Well-Developed Flow . . . . . . . . . C . Shear Stress in the Boundary Layer . . . . . . . . . . . V Application to Fluid Velocity Measurements . . . . . . . . . A . Principle and Method of Measurements . . . . . . . . . B Time-Smoothed Velocity . . . . . . . . . . . . . . . C . Fluctuating Velocity . . . . . . . . . . . . . . . . . Symbols . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

87 89 94 94 98 103 136 136 140 142 144 144 147 153 159 160

.

I Introduction

The electrochemical method cannot be applied directly to heat transfer measurements . Nevertheless. many investigators of heat transfer have adopted the electrochemical method since the knowledge of the mass transfer obtained by this method can be correlated with heat transfer by the analogy between both transport phenomena . Especially. the chemical engineers who are mostly interested in operations involving liquids make use more frequently of the method of electrochemical reaction under the diffusion-controlling condition 87

88

T.MIZUSHINA

because this method enables them to measure not only the average value but also the local value of the mass transfer rate and shearing stress at a liquid-solid interface. From the limiting current density at the cathode, the transfer rate of mass or momentum can be calculated. Since the response of the cathode is very fast, we can obtain the fluctuation value as well as the time-smoothed value; further, a hot wire-type device to measure the velocity of liquid can be made from such cathodes of thin wire. Though the diffusion-controlling electrochemical reaction is a very strong weapon to attack transport phenomena in liquids, it should be noted that there are several limits in using this method. First, it is limited to liquids; accordingly, the data of mass transfer are limited for high Schmidt numbers. Second, only certain kinds of liquid mixtures can be used, i.e., those in which a diffusion-controlling electrolytic reaction occurs. Furthermore, we cannot use this method for a velocity larger than a critical flow rate at which the reaction resistance at the cathode becomes relatively significant compared with decreasing resistance of diffusion. For practical purposes, it is better to choose the condition in which the limiting current is reached before the potential becomes larger than the hydrogen overvoltage. Otherwise, the discharge current of hydrogen ions is added and the limiting current cannot be found. First, the electrochemists took up the problem of concentration polarization in analyzing the reaction in polarography (I)and electrolytic cells (2). Not very many years ago, the diffusion-controlled electrode reaction also began to be applied to the measurements of transport phenomena. Lin et al. (3) made a systematic study of the transfer rates of ions and other reacting species in electrochemical reactions in several kinds of mixtures and measured the mass transfer coefficients in laminar and turbulent flows by this method. Ranz (4) discussed the problems in applying this method to the measurement of the velocity of liquids and suggested that a long development and considerable study of chemical mechanisms are necessary in order to make practical instruments. Mitchell and Hanratty ( 5 ) developed the shear-stress meter and used this for a study of turbulence at a wall. In the following chapter, the method of the diffusion-controlling electrochemical reaction and its applications in the study of transport phenomena will be discussed. At the end of introduction, however, it seems necessary to mention the electroconductivity method, one of the electrochemical methods other than those described above. Nukiyama et al. (5a, b) developed the electroconductivity method to simulate complex problems in thermal con-

THEELECTROCHEMICAL METHOD

89

ductivity. Lamb et al. (5c) made an electrical conductivity probe to measure liquid-phase concentration fluctuations, and Manning and Wilhelm (5d) applied this method to measurement of turbulent characteristics in an agitated vessel.

II. Electrochemical Reaction under the Diffusion-Controlling Condition In electrode reactions, there exist concentration polarization and chemical polarization in series, i.e., the ions move from the bulk of the solution to the surface of the electrode where chemical and physical changes occur. I n measuring the rates of mass transfer by the use of electrochemical reactions, it is better and more usual to make the chemical polarization negligible because the mass transfer coefficients are most easily obtained from the limiting current when the concentration at the liquid-solid interface can be assumed to be zero. The ions are transferred from the bulk of the solution to the surface of the electrode principally by (a) migration due to the potential field, (b) diffusion due to the concentration gradient, and (c) convection by the flow. Assuming that the transfer is steady and unidirectional in the y-direction perpendicular to the surface of the electrode, the rate of transfer of a reacting species is expressed as NA

=

t9

+

€Y')

+

cAtntdRT)(ay/Ylay)- (9

cD>taCA/@)

+

OCA

tl)

and the current density at the electrode is expressed as i/An,F

=N A

(2)

The three terms on the right of Eq. (1) represent the contributions of migration, diffusion, and convection, respectively. The last term for convection vanishes in the redox processes because there is no net bulk flow in the y-direction. But it does not vanish in the processes of metal depositing on the electrode because there is a net bulk flow. However, its effect is very small at ordinary conditions and usually negligible. For example, Wilke et al. (6) showed that the error from neglecting this effect was never larger than 0.3 yofor the maximum flux of deposit in their experiments. Consequently, it may be assumed that the . zero. last term of Eq. (1) is Next, a simplification can be achieved concerning the migration term by adding a large excess of an unreactive electrolyte to the solution. If such electrolytes, which do not react at the electrode, exist in the solution in relatively high concentrations and have high conductivity

90

T. MIZUSHINA

compared with the reacting species of ions, there should be no sharp potential gradient near the electrode, i.e., a!P/ay may be assumed to be zero. The migration current then becomes negligible and almost all of the electrolysis current arises from the reaction of ions which reach the electrode surface by diffusion. I n this case the migration current was estimated to be of the order of 1 yo of the total current1 (8). Thus, the migration term in Eq. (1) can be neglected and only the diffusion term remains to give NA = 4 9 9)ac,/?Y (3)

+

I n the general case, the integration of Eq. (3) gives the following expression for the rate of mass transfer:

NA = k(Cb - C i )

(4)

Substituting Eq. (2) into Eq. (4), one obtains i/An,F

= k(c, - ci)

(5)

If the effect of ionic migration in the potential field is eliminated as mentioned above, the analogy between heat and mass transfer is valid as discussed by Agar (9). Thus the results of experiments by the electrochemical method may be correlated with the knowledge of heat transfer at a wall of constant temperature. At the limiting current, the concentration at the surface of the electrode, ci , becomes zero, and Eq. ( 5 ) is simplified to i/An,F = kcb

(6)

The limiting current can be seen easily from the potential-current curve as shown in Fig. 1. As the negative potential of the cathode is increased, the positive ions migrate, to a certain extent, from the bulk of the solution into the double layer. The accumulation of the ions at the surface of the cathode is equivalent to the charging current. T o keep a finite current between two electrodes in an electrolytic cell, the applied potential difference must exceed the equilibrium one by a finite voltage of concentration overpotential. As the applied potential is made higher, the current increases exponentially and thereafter approaches a constant value, i.e., a limiting current, asymptotically. Under the condition of a limiting current, the ions transferred to the electrode surface react very soon and the increasing potential does not result in an increase in the If the indifferent electrolyte is not added to the solution, the current due to the migration is of the order of 10% of the total current (8).

THEELECTROCHEMICAL METHOD

91

rate of the desired reaction. Lin et al. (3) indicated that the limiting currents observed under the same flow condition were exactly proportional to the bulk concentration of the ions which reacted at the cathode. This fact proves Eq. (6) to be valid. However, it should be noted that this theoretical limiting current density cannot be perfectly achieved. T h e polarization developed at the cathode is approximately equal to the electromotive force of a concentration cell of two solutions of the reacting ion at concentrations cb and ci ,

E

= (RT/n,F)In yb/yi

(7)

(RT/n,F)In cb/ci

Therefore, the value of ci decreases exponentially toward zero with the increase of the cathode potential, but it never quite reaches zero.

-

0.6

.

0.4

-

0.2

-

#

N

E

u

\

2

Y z 9

t .-

v)

C

a

’0 +

C

E! 3

u

0

I

I

FIG. 1.

I

I

Limiting current.

I n Fig. 1, it is seen that a further increase of the potential over the limiting current region causes a steep increase in current density due to the discharge by a secondary reaction such as hydrogen evolution on the cathode. As the diffusion rate of ions is made to increase, e.g., by increasing the flow rate, under the same conditions of electrolysis, the value of the limiting current is raised and finally the flat portion of the polarization curves disappears above a certain upper limit of the flow rate as shown in Fig. 2. I n such situations the limiting current is no longer indicated since the reaction is too slow to remove all ions reaching the electrode surface. This upper limit of flow rate is called the “critical flow rate.” T h e higher

92

T. MIZUSHINA

the reaction rate is, the higher the critical flow rate obtained. Since the diffusivity is an approximately linear function of temperature while the reaction rate constants vary exponentially with the temperature, the critical flow rate is very sensitive to temperature. The data above the critical flow rate should not be used in correlations since the concentration at the electrode surface is not zero.

c

40-

N

5

\

a

E

30-

x

c

0, C

0 c

20-

g, =I

u

IOuc 0

0

-0.2

Ucrlllcol I

I

I

-04 -0.6 -0.8 Cathode potential ( V )

-

FIG. 2. Critical flow rate.

The electrolytic solutions to be used to measure the mass transfer rates are those of redox couples such as ferrocyanide and ferricyanide ions in NaOH as an unreactive electrolyte and those for reduction of metal ion such as Cu2+deposition on Cu cathode in H2S0, solution. The reactions of these systems are: (1)

+

Fe(CN)e3- e + Fe(CN),4Fe(CN)e4- + Fe(CN),3-

(2) Cu*+

+ 2e

Cu

+

-+

Cu Cus+

+ 2e

+e

at cathode at anode

at cathode at anode

I n the former solutions, the presence of dissolved oxygen influences the mass transfer measurement. It is better to remove oxygen from the test solution and to seal the surface of the solution with an inert gas. When the latter solutions are used, it should be noted that the conditions of the cathode surface are changing due to metal deposition during electrolysis.

THEELECTROCHEMICAL METHOD

93

It is recommended that a metal which has as high a hydrogen overvoltage as possible and is inert in acidic solutions be selected as the cathode. In the case of CuSO, in H,SO, solution, H,SO, reacts with H,O to form oxonium ions, H,O+, and sulfate ions, SO,2-. It is desirable that the potential applied be sufficient for the reduction of Cu2+ ions to Cu metal at the cathode but not for hydrogen evolution by reduction of H,O+ ions. If the hydrogen overvoltage on that cathode is lower than the potential for the limiting current, the polarization curve obtained may not have a flat part just as that obtained when the flow rate is over the critical one. However, in this case the polarization curve can be corrected theoretically with the information on the hydrogen overvoltage on that cathode, and hence it is possible to estimate the “true” limiting current. Suppose that simultaneous discharges of Cu2+and H,O+ on a copper cathode surface are occurring and that the hydrogen discharge current is superimposed on the limiting current due to Cu2+ ion discharge. The overvoltage for the former process seems to be mainly caused by the charge transfer overvoltage while that for the latter is evidently by the diffusion overvoltage. Therefore, it may be assumed that these processes do not interfere with each other and that these current densities can be added. The hydrogen overvoltage is expressed as a function of the current density of hydrogen ions by Tafel’s experimental equation TH = a

+Pin 5~

(8)

The constants 01 and j3 in Eq. (8) are dependent on the combination of cathode and electrolyte. For example, for 2N-H2S0, on Cu cathode, f H = 10-7-10-3 A/cm2 at 20°C, 01 = 0.78 and fl = 0.043. On the other hand, the cathode potential, E , at the given current density is expressed as follows: E, = (RT/F)In YH - TH (9) From Eqs. (8) and (9), the current density by hydrogen discharge under the given potential can be calculated as

5br ,=exp[(l/@)((RT/F)In

- EC

-

(10)

Subtracting this calculated current density of hydrogen ions from the observed total current density at that potential, one can obtain the “true” limiting current for copper deposition as shown in Fig. 3. When this electrochemical method is applied to measurements of transport phenomena, in general an anode of larger surface area and a cathode of smaller surface area are coupled. This arrangement of

94

T. MIZUSHINA

Electrode potential ( V )

FIG.3.

“True” limiting current.

electrodes makes the current density at the anode smaller than that at the cathode, and the process at the anode has no noticeable effect on the shape of the applied potential-current curve. However, to obtain more accurate data, it is recommended that a standard electrode be used to measure the liquid potential. The addition of an unreactive electrolyte to the solution for eliminating the migration contributions has an advantage in practical operations. It makes the ohmic resistance of the solution negligibly small.

III. Application to Mass Transfer Measurements

A. PRINCIPLE AND METHOD OF MEASUREMENTS Utilizing the electrochemical method, three kinds of mass transfer coefficients (i,e., a space-time-averaged value, a local time-smoothed value, and a local fluctuating value at a solid-liquid interface in many different flow systems) can be easily measured. If one chooses an electrolytic system in which only one kind of reaction occurs at the electrode to be used as a mass transfer surface, the limiting current is reached provided that the opposite electrode does not limit the rate of the reaction. These conditions are satisfied when the opposite electrode is

THEELECTROCHEMICAL METHOD

95

large enough and when the reaction occurring at the measuring electrode is sufficiently rapid. And then the limiting current is a direct measure of the mass flux of a specific species of ion at the measuring surface. If the concentration of this species of ion in the bulk of the solution, c, , is known, the mass transfer coefficient can be calculated from Eq. (6). The time-smoothed value of the limiting current on the whole electrode surface gives the space-time-averaged value of the mass transfer coefficient, K. On the other hand, point electrodes which are electrically isolated from the surrounding electrode surface as shown in Fig. 4 and

Polyvinyl chloride

Isolated cathode

Cothode

FIG. 4. Isolated point cathode.

held at the same potential as other parts of the electrode are used to obtain the distribution of local time-smoothed values, k, . Furthermore, the instantaneous fluctuating values of these point electrodes give the local fluctuating values, A,’, from which information concerning the turbulent behavior of diffusion processes is obtained. The electrochemical method to measure the local values of the mass transfer coefficients gives more accurate results than those obtained by the ordinary measurement of mass or heat since the electrical insulation of an isolated cathode is an easy method. Figure 5 shows the electrical circuit of an experimental apparatus to measure the mass transfer rate in an agitated vessel. A potential applied between the anode and cathode is adjusted with a rheostat and the current in the circuit is measured. After the flow conditions are set, current is passed through the cell and increased in small increments at intervals of some time until the limiting current is reached and usually increased further to the hydrogen evolution point. Approximately one minute is recommended as the time interval at each rheostat setting because this is long enough to make the potential and current reach a steady state. Thus, the polarization curve can be drawn.

96

T. MIZUSHINA

On the other hand, Wilke et al. (6) in their experiments on free convection mass transfer at a plate studied various ways of obtaining current-potential curves with changes in the time of duration of electrolysis and intervals for potential measurement following each current setting. They concluded that the limiting current density was independent of the initial shape of the current-potential curve of each different time program. T h e maximum deviation from the average value was only 3.2%.

FIG. 5. Agitated vessel to measure mass transfer coefficients.

For metal depositing processes, especially, it is desirable to repeat the limiting current determinations three or four times to ensure reproducibility. When fluctuating values of the mass transfer coefficient are measured, the instantaneous limiting current of point cathodes are recorded to obtain the turbulent characteristics, such as intensity of turbulence, from the fluctuation curve. As discussed in Section 11, the cathode potential is measured against the saturated calomel electrode through a capillary in the illustrated circuit. The effect of location of capillary junctions along the cathode surface proved to be negligible on the observed limiting current. The shape and dimensions of the cathode are designed to form the transfer surface to be investigated, and the test sections should be electrically insulated from the rest of the experimental apparatus, such as

THEELECTROCHEMICAL METHOD

97

the pumping system, by means of rubber gaskets. T h e whole surface of electrodes including point cathodes should be carefully machined and polished to eliminate any roughness or discontinuity on the surface. For ferri-ferro redox systems, platinum or nickel is generally used as the electrode material. However, it has been reported that a platinum electrode containing a small percentage of rhodium showed a downward drift in the limiting current. Before each run, the electrodes should be cleaned with CC1, and buffed with soft paper. It is recommended that the electrodes be cleaned cathodically in 5% NaOH solution to make sure of eliminating chemical polarization for redox system. After each run a check must be made for corrosion deposits formed on the electrode surface. For Cu2f reduction system, copper metal is usually used for electrodes. T o obtain reproducible quality after machining, the electrodes must be polished by emery paper No. 400, then washed and degreased and used immediately after preparation. It should be noted again that the redox system should be kept under nitrogen to avoid oxidation of the ferrocyanide and reaction of the dissolved oxygen at the electrode although this effect is very small. Furthermore, it is recommended that nitrogen be bubbled through the storage tank to remove any oxygen prior to the runs. In addition, it is better to keep the test solution away from light. T h e light decomposes potassium ferrocynaide slowly to hydrogen cyanide, which will poison the electrodes. In fact, it is better to use the solutions which have been prepared just before the experiment, using specially treated distilled water, since colloidal ferrihydroxide is formed and the color of the solution becomes brown after a few days. It is important to know the actual effect of dissolved oxygen in a ferro-ferricyanide redox couple when the presence of air is unavoidable, as in two-phase air-liquid flow systems. However, there is uncertainty concerning this effect since the effects of oxygen in contaminating the solution and the electrodes are complicated. The actual electrochemical reaction involving oxygen may not occur at such a relatively low voltage as applied in this method since oxygen activation polarization is generally high. But mixed potential and oxide films on electrodes might have some influence. Anyway, absorption of oxygen by the solution and contamination of electrodes depend upon time. Therefore, suitable measurements may be finished before such an effect of contamination becomes controlling. Sutey et al. (7) indicated that the mass transfer coefficients measured at oxygen saturations below 70 % were within 5 % error for an operating time of 275 min. Finally, precautions must be taken to keep the temperature of the test

T. MIZUSHINA

98

section constant since the physical variables and, accordingly, the mass transfer rate depend on the temperature. Lin et al. (3) studied the following two systems as well as redox systems in measuring mass transfer rate in annular flow. (1) Reduction of quinone on a silver cathode in a strongly buffered solution. C,H402

+ 2Hf + 2e

-f

C,H,(OH),

(2) Reduction of oxygen on a silver cathode in a NaOH solution. 0,

+ 4e + 2H,O

-f

40H-

I n these two systems the electrode reactions are rapid enough, and the electrode materials are inert to the solutions. In the case of reduction of quinone, hydrogen ion is also a reacting material. In the presence of a strong buffer, however, the concentration of hydrogen ion is essentially constant across the boundary layer because the buffer reaction rate is more rapid than the diffusion rate of quinone. With regard to the reduction of oxygen, the situation is not complex since water always exists in large excess. It was found that for reduction of quinone, the critical flow rates were comparatively low, whereas the limiting current was still obtained at Re = 17,500 with reduction of oxygen and up to Re = 29,800 with ferricyanide reduction. Consequently, the ferro-ferricyanide redox couple may be best since in alkaline solutions it is stable and the chemical polarization on the cathode is so small that the critical flow rate is very high. An additional advantage is that the bulk composition of the solution is constant since the same reaction (but in opposite directions) occurs at cathode and anode.

B. FREECONVECTION 1. Free Convection Mass Transfer at a Plate a. Free Convection Mass Transfer at a Vertical Plate. The free convection mass transfer at a vertical plate has been studied by Wagner (lo), Wilke et al. (6),(ZZ), and Ibl et al. (22, 13). Wagner measured the limiting current by the deposition of copper from an acidic solution of copper sulphate, Wilke et al. by the deposition of copper and silver, and Ibl et al. by the deposition of copper and the reduction of ferricyanide in a solution of K,Fe( CN)6-K,Fe( CN),NaOH, respectively. The apparatus used by Wilke et al. is made of an electrolytic cell containing a solution of CuSO, and equipment for measuring cathode potential relative to the bulk solution in the cell and total current.

THEELECTROCHEMICAL METHOD

99

Fifteen different solutions of CuSO, concentration ranging from 0.01 to 0.74 mole/liter and that of H,SO, ranging from 1.38 to 1.57 moles/liter were used. I n Fig. 6 the results of Wilke et al. are plotted in the range of Rayleigh numbers = 4 x lo6 6 x loll. T h e correlating equation of these results is N

Sh

= 0.671

(Gr S C ) O . ~ ~

(1 1)

T h e Sherwood number or the Nusselt number for diffusion is given by Sh

=

KxX,/B

(12)

where k is the averaged mass transfer coefficient over x, x is the vertical height of electrode surface, and Xr is the logarithmic mean volume fraction of nondiffusing species.

1o3 ul

102

Gr Sc Wilke FIG. 6. Free convection mass transfer coefficients on a vertical plate. Key: (0)

et al.; ( 0 ) Ibl et al.; ( X ) Wagner.

T h e Grashof number is given by the following equation since the free convection in the ion transfer is caused by density differences due to concentration change between bulk solution and electrode surface: Gr

= g(Pb - Pi) PX3/P2

(13)

where p b and pi are the fluid densities in bulk solution and at the electrode surface, respectively. Equation (1 1) is in close agreement with the equation which correlates the data of the experiment on solid dissolution. This is an encouraging indication of the general validity of the electrochemical method. However, these experimental results of the mass transfer disagree with the predic-

T.MIZUSHINA

100

tions of Eq. (14), which is obtained from the modification of the SchmidtPohlhausen-Beckmann theory for heat transfer to air. Sh

= 0.525

(Sc * Gr)l/4

(14)

The disagreement may be expected since the numerical coefficient of

Eq. (14) is applicable only to systems with Pr or Sc

0.7.

Wagner measured the mass transfer rate from three electrodes of heights 1,4, and 16 cm in a solution of 0.1 mole of CuSO, and 1.0 mole of H,SO, per liter. I n his experiment it was found that deviations from the vertical position of the electrode surface up to 15" had no appreciable effect. Sherwood numbers calculated from his results are plotted in Fig. 6. It is seen that they are in good agreement with the results of Wilke et al. Using a solution of 0.2-0.025 mole of ferricyanide, 0.1 mole of ferrocyanide, and 2 moles of NaOH per liter, Ibl et al. obtained the mass transfer data not only in the laminar range, i.e., 3 x los < Sc . Gr < 4 x lo8 but also in the transition range, i.e., 1 x 1012 < Sc Gr < 1 x 10l6, while Wilke et al. and Wagner covered only the laminar region. Their data in the laminar region are correlated by Eq. (11) when the effect of the existence of a diaphragm is negligible. As shown in Fig. 6, the data of Ibl et al. for the transition region fall on an extension of Eq. (11) and are correlated by Sh = 0.31 (Sc * Gr)0*28 (15) Wagner divided a cathode into five electrically isolated sections, the heights of which are, respectively, 1, 3, 1, 7,and 1 cm from the bottom, and measured the current density of each section separately. The results are plotted in Fig. 7, which indicates the distribution of local mass transfer coefficients. b. Free Convection Mass Transfer at a Horizontal Plate. Fench and Tobias (14) studied the free convection mass transfer at a horizontal plate using the cathodic reduction of Cua+ to Cu from CuSO, solutions. Making use of standard electrodes, they measured the concentration polarization directly instead of measuring the applied potential. The anode was separated from the cathode by a $in. thick ceramic diaphragm so that convection currents generated by the dissolving anode would not influence convection in the cathode compartment. I n the process of the experiment, an interesting problem was presented. It is desirable to make the electrolysis period as short as possible to avoid decreasing the bulk concentration notably, especially in such a case as 0.01 moIe/liter solutions, and roughening the cathode surface excessively. On the other hand, the duration of electrolysis must be long

THEELECTROCHEMICAL METHOD

101

enough to allow natural convection to reach a steady state. As a result of the preliminary runs, 3-5 minutes were used for reaching the limiting current. The experimental data of Fench et al. are plotted in Fig. 8. The correlating equation is Sh

= 0.19

(Sc * Gr)”a

(16)

where the vertical distance between cathode and diaphragm was chosen as the characteristic dimension,

H

10.0

2.5

-0 k,

(cm/sec)

FIG. 7. Distribution of the local coefficients of free convection mass transfer on a vertical plate.

2. Free Convection Mass Transfer at a Cylinder Free convection mass transfer at a horizontal cylinder was studied by Schutz (15). In his experiments, the limiting currents were measured by the electrolysis of CuSO, in H,SO, as the supporting electrolyte. The experimental results of the space-time-averaged mass transfer coefficients are shown in Fig. 9. All the data for Gr Sc < loQ come together on a line of the gradient of 0.25. The correlation is expressed as

-

Sh

= 0.53 (Gr

-

(17)

T. MIZUSHINA

102

10’ L

Y,

lo2

.-

in‘

10’

10’

do

I 013

10”

Gr- Sc

FIG.8. Free convection mass transfer coefficients on a horizontal plate. Key:( 0 ) without glycerol; ( 0 ) with glycerol. Sh = 0.190 (Gr *

loj

cn

lo2

10 lo7

108 Gr-Sc

1o9

100

FIG.9. Free convection mass transfer coefficients on a cylinder.

THEELECTROCHEMICAL METHOD

103

This equation is in good agreement with the ordinary equation obtained for free convection heat transfer from horizontal cylinders. The data for Gr Sc > 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)

<0

The concentration profile becomes fully developed for very large values of x+, and ac+/ax+, at any value of y+, becomes constant. In such a region a mass balance gives -ac+/ax+ g -ac,+/axt

= 4Km+/Re

(21)

where K,+ = k,/u,. This term is very small because Km+is of the order of lo-*, while Re is of the order of 104. Taking account of the smallness of ac+/ax+, one integrates Eq. (19) as

Accordingly,

The fully developed mass transfer rate K,+ depends on the distribution of the eddy diffusivity, cD/u, near the wall, especially at high Schmidt numbers. The function describing eD/u near the wall may be expanded in a Taylor series, and usually only the leading term in the series €&

=

C(y+)"

(24)

is used. Substituting Eq. (24) into Eq. (23), one obtains K,+

= ( n / r )C1/lasin(r/n)(Sc)-(n-l)'n

(25)

I n general, the coefficient C and the exponent n in Eq. (24) could be a function of the Schmidt number and n may change slightly with y+.

THEELECTROCHEMICAL METHOD

107

Arguments can be presented to show that the exponent n should be three or greater. As described in Section 111, A, use of electrochemical methods offers a number of important advantages, the most significant of which is that surface roughness can be kept very small compared with the dissolving wall method. This is particularly important at high Schmidt numbers because the surface roughness influences the mass transfer rate significantly in this case. Typical experimental apparatus is given schematically in Fig. 14.

v Recorder

Potentiostat

Reference electrode -Luggin

-

capillary

I

Anode

Pump

U

Electrolyte reservoi r

Manometer

FIG. 14. Experimental apparatus for forced convection mass transfer in a tube flow.

T h e mass transfer coefficient is calculated by Eq. ( 6 ) from the experimental results of the electrochemical method provided that the necessary conditions are satisfied. From Eq. (19) the mass transfer rate is a function of Sc and x+, K,+

=

K,+(SC,

Xf)

(26)

At very large values of xf, K,+ = K,f. T h e average mass transfer rate over the length of the mass transfer section L f is expressed as K+

=

(l/L+)

's:

K,+ dxf

T. MIZUSHINA

108

If the length of the mass transfer section is very large, the entrance effects are negligible, i.e., L++m lim

K+

= K,+

(28)

Therefore, it is necessary to check that the measured value of the spaceaveraged mass transfer coefficient is a true asymptotic one to use as the fully developed mass transfer coefficient. Rather than such a method to measure K,+,however, use of an isolated cathode may be recommended at a downstream position where the concentration profile has been fully developed to measure the local value of K,+ directly. Lin et al. (3) measured the mass transfer rate between flow in the annulus of a double tube and the wall surface of the inner tube over a range of Schmidt numbers from 300 to 3000 and Reynolds numbers from 260 to 30,000, using four different systems of electrochemical reactions as mentioned in Section 11. Results for laminar flow are presented in Section 111, C, 1, d. Shaw and Hanratty (20) measured the fully developed mass transfer rate at Sc = 2400 and Re = 8000-50,OOO and found that K,+ is independent of Reynolds number. The maximum deviation of twenty nine measurements from the average was 4%. The process chosen was the reduction of ferricyanide ions at a nickel cathode in the presence of a large excess of sodium hydroxide (0.001 mole of K,Fe(CN), and K,Fe(CN), , respectively, and 2 moles of NaOH per liter). Hubbard and Lightfoot (21) used the diffusion-controlled reduction of potassium ferricyanide in excess caustic (0.005 mole of K,Fe( CN), and K,Fe(CN), , respectively, and 1 4 moles of NaOH/liter) in their measurement. Experiments were made at Sc = 1700-30,000 and Re = 7000-60,OOO in a rectangular channel. The data of Lin et al. (3), Shaw and Hanratty (20), Hubbard and Lightfoot (21) and unpublished data of Mizushina et at., along with the results of Harriott and Hamilton (22) which were obtained by the dissolving-wall method, are summarized in Fig. 15 and compared with predictions calculated from equations of Son and Hanratty (23), Lin et al. ( l a ) ,and Mizushina et al. (24) for eddy diffusivity. The broken line in Fig. 15 is given by Son et al. (23) and expressed as

K,f

SC-9'4

(29)

Re ( f / 2 ) l I 2Sc1I4

(30)

= 0.121

This equation is equivalent to Sh,

= 0.121

THEELECTROCHEMICAL METHOD

109

On the other hand, Hubbard and Lightfoot (21) and Harriott and Hamilton (22) have obtained Sh,

=

K

Ref

(31)

Sc1f3

Figure 15 indicates that the result of Son and Hanratty is not really conclusive as to the Schmidt number dependence of K,+. An equation containing S r 2 I 3rather than Sc3I4 seems to represent the data better. Accordingly, Eq. (31) seems to be a better correlation than Eq. (30).

I

lo2

I

I

I I l l 1 1

I

I

I

I

I I I I I

1 o3

104

1

I

I

I I l l l l

I

lo5

sc

FIG. 15. Variation of Km+with Sc. Key: (0)Hubbard et al.; ( A ) Lin et al.; (V) Harriott et al.; ( 0 ) Shaw et al.; (0) Mizushina et al.; (-) Mizushina et al.; (---) Son et al.; (--.--.-) Lin et al.

By comparing Eq. (29) with Eq. (25) and by assuming that the coefficient C is independent of Schmidt number, Son and Hanratty (23) obtained the following relation: ED/V =

0.00032(~+)~

(32)

However, the calculated curves based on Eq. (23) and the following equations for eddy diffusivity distributions represent the data better than Eq. (32).

T. MIZUSHINA

110 Lin et al. (18): Mizushina et al. (24):

F{ 1 - exp(-1IF)}

= (E&)

E&

F= EM/v

1

0.0344(~M/~)~/' SC

+ 0.240(~M/~)'/* Scl"

= 4.16

x 10-4(y+)9

(34) (344 (34b)

b. Turbulent Flow Mass Transfer in the Mass Transfer Entry Region. When a fluid in fully developed turbulent flow enters a mass transfer section, the local value of the mass transfer coefficient decreases from infinity at the inlet to a minimum value downstream. If the transfer section is long enough, that minimum will be the fully developed value. Accordingly, the space-averaged mass transfer coefficients decrease with an increase of the mass transfer section. Though the heat transfer measurements in such an entry region have been reported by several investigators, such studies are complicated because thermal isolation of a small transfer section from the remaining part of the pipe is very difficult. In this respect, the mass transfer experiments by electrochemical methods are superior for entry region investigations because electrically isolated cathodes are easily made. Shaw et al. (25) used the reduction of ferricyanide in determining the effect of the length of mass transfer section on the average rate of transfer, i.e., the relation between K+ and L+. Ten lengths of the mass transfer section, from 0.0177 to 4.31 diameters were used. T h e Schmidt number was constant at about 2400, while the Reynolds number was varied from 1000 to 75,000. The electrolyte concentration was 2 moles of NaOH and 0.01-0.0001 mole of K,Fe(CN), and K,Fe(CN), per liter. In Fig. 16 the results are plotted and compared with curves of predictions calculated numerically by substituting Eqs. (32), (33), and (34), respectively, into Eq. (19). Schutz (26) measured the local transfer rate K,+ in the mass transfer entry region. The Reynolds number was varied from 20,000 to 50,000 and the Schmidt number was 2170. T h e used solution contains 0.025 mole of K,Fe(CN), , 0.025 mole of K,Fe(CN), and 2 moles of NaOH per liter. I n Fig. 17 the results are plotted and compared with calculated curves based on the eddy diffusivities which are predicted by the equations of Lin et al. (18), Mizushina et al. (24), and Son and Hanratty (23).

THEELECTROCHEMICAL METHOD

111

+

Y

lo-&

1 o2

L’

(-

FIG. 16. Variation of K+ with L+. Key: ( 0 ) Son et al.; (-) Son at al.; (--.--.-) Lin et ul. Sc = 2400.

- -)

Mizushina et al.;

* Yx

X*

(-

FIG. 17. Variation of K.+ with x+. Key: (0)Schutz; (-) - -) Son et ul.; (--.--.-) Lin et al. Sc = 2170.

Mizushina et al.;

112

T. MIZUSHINA

c. Fluctuations of Mass Transfer Rates. The small eddies in the region of the viscous sublayer of a turbulent flow are known to play an important role in determining the transfer rates to the wall, especially in the case of high Schmidt or Prandtl numbers. Since the flow of these small eddies is unsteady, the local mass transfer rates to the wall are fluctuating. For studying the unsteady flow, optical methods are not useful because the quantitative data are difficult to obtain and hot-wire anemometer measurements are limited in that the size of the probe is so big relative to the region being studied that the flow may be disturbed by the probe. To overcome these difficulties an electrochemical method to measure the fluctuations in the local rate of mass transfer to the wall has been developed by Shaw and Hanratty (20). The fluctuating values of the local mass transfer rate can be calculated directly from the fluctuations of the electrical current to the isolated cathodes surrounded by the active cathode surface with the following equation: k,’ = i,‘/(A,n,Fc,) (35) The root-mean-square, the frequency spectra, and the scale of fluctuations in the local mass transfer coefficients are calculated from the measured values of fluctuation. Shaw and Hanratty measured the instantaneous rate of transfer in a fully developed boundary layer at Reynolds numbers ranging from 10,OOO to 60,000 and a Schmidt number of about 2400. As shown in Fig. 18, the size of the isolated cathode affects the measurements of the mass transfer fluctuations remarkably because the fluctuations of the small circumferential scale tend to be averaged over the cathode surface and larger cathodes give smaller signals. However, the correction by assuming a circumferential scale of As+ = 2.1 and an - all data into exponential form for the correlation coefficient brought agreement and indicated that the true local value of [(km’)a]1/2/ikm 0.48. Maximum deviation of the data from this value was about 10%. The axial correlation coefficients obtained with two isolated cathodes of 0.65-mm diameter which are located at a distance of 5 to 30 mm (center to center) in the axial direction are shown in Fig. 19. By integrating the curve in Fig. 19 graphically, longitudinal integral scales of the fluctuation are determined as Az+ E 350, which is about lo2 times as large as the circumferential scale. The frequency spectra are plotted against the dimensionless frequency ++ = + Y / ( v * ) ~ in Fig. 20.

d. Mass Transfer of Laminar Flow. A theoretical equation for the heat transfer in fully developed laminar flow with constant wall tempera-

1 o4

1o5

Re

FIG. 18. Cathode size effects on fluctuation measurements. Electrode diameter: (1) 0.0157 in.; (2) 0.0259 in.; (3) 0.0640 in.; (4) 0.1250 in.

0.0

I

0

I

2

I

1

4

6 (x;

1

8

- x; 1

Y

I

4

10

12

16'

FIG. 19. Axial correlation coefficients.

I

14

16

114

T. MIZUSHINA

ture was given by Graetz. For short tube lengths, the Graetz solution reduces asymptotically to LCvCque's equation, which is modified for mass transfer as Sh = 1.62(Re Sc * D/L)Il3 (36) This is equivalent to the following equation for a circular pipe:

(37)

K+ = 0.81(L+)-'/S ( S C ) - ~ ' ~

The mass transfer rates measured by Lin et al. (3) in laminar flow in an annulus are plotted in Fig. 21, and compared with Eq. (36). Their data are correlated well with Eq. (36). 1.0 I

I

0.9

-

0.8

-

0.7

-

0.6

-

0.b

-

0.3

-

0.2

-

0.1

-

0.5

0.0

I

0

I

2

I

1

4

I

I

6

I

I

8 @*B

1

I

10

I

I

12

I

I

14

I

I

I

16

18

104

FIG.20. Frequency spectra of the mass transfer fluctuations. Key: ( 0 ) Re = 8700;

( A ) Re = 23,150; ( 0 ) Re = 50,700.

Son and Hanratty (23) measured the mass transfer rate of laminar flow in a circular pipe. Their data, along with Eq. (37), were used to calculate the Schmidt numbers. The results for lengths ranging from 0.0174 to 1.91 pipe diameters and Reynolds numbers from 335 to 2200 are plotted in Fig. 22. The agreement of their results with Eq. (37) means

THEELECTROCHEMICAL METHOD I

r Ill

115

I

1

-

lo2

ReScDIL

FIG. 21. Mass transfer coefficients in laminar flow.System: (0)oxygen; ( 0 )quinone;

(0) ferricyanide ion; ( A ) ferrocyanideion.

sc = 2,LOO

FIG.22. Variations of K + with Lf in laminar flow.

T. MIZUSHINA

116

that the exponent for L+ in Eq. (37) and thus that for LCvCque’s equation are confirmed experimentally.

e. Dynamic Response of the Mass Transfer Coe@cient. I n Kyoto University the dynamic response of the mass transfer coefficient itself is being studied by Mizushina et al. using the flow of solution of ferriferrocyanide redox system in a circular tube and a rectangular duct. In Fig. 23, an experimental result of a step response of the space-averaged mass transfer coefficient is compared with a theoretical calculation.

Shear stress at the wall

1.0

0.5 t

1.5

2.0

(secl

FIG.23. Step response of the mass transfer coefficient.

2. Forced Convection Mass Transfer from Cross Flow Many investigators have studied the transfer of heat from a cylinder to fluids in cross flow. Only a few corresponding studies of the mass transfer have been made, however. A knowledge of the diffusional transport of mass in cross flow is of more than academic interest. For example, it may be of some practical importance in heterogeneous catalytic processes. An electrochemical method based on a diffusion-controlled electrode reaction seems suitable for studying mass transfer to a cylinder in a liquid. This enables us to obtain a lot of valuable data which are very difficult to get by any other method. Dobry and Finn (27) measured the space-time-averaged mass transfer coefficients to very fine wires at Reynolds numbers ranging from 0.08 to 10. Grassmann et al. (28) made an experiment at Reynolds numbers from 120 to 12,000 and a Schmidt number of 2780. They measured the local and the average mass transfer coefficients to cylinders and confirmed the analogy between heat and mass transfer. Vogtlander and Bakker (29) also obtained mass transfer data which agree with heat transfer results in the Reynolds number range between 5 and 100.

THEELECTROCHEMICAL METHOD

117

Dimopoulos and Hanratty (30) used electrochemical techniques in studies of flow around cylinders to measure the velocity gradient in the boundary layer. They also measured local mass transfer rates to cylinders and confirmed good agreement with calculated values from measured velocity gradients for Re = 60-360. These investigators used reduction of ferricyanide on the test cylinders in the system composed of ferricyanide, ferrocyanide, and sodium hydroxide as supporting electrolyte. Their experiments were carried out in different Reynolds number ranges and their correlating curves cannot be linked in a single curve, probably due to the effect of different turbulent intensities in the approaching flow. In construction of the experimental apparatus, it is necessary to satisfy the following requirements: (1) The ratio between the diameter of the test cylinder and that of the channel in which they are mounted, i.e., the blockage ratio, is small enough. (2) The approaching flow has a flat velocity profile and its fluctuations are as small as possible. For this purpose, a head tank or a water tunnel which has a convergent nozzle and calming section containing honeycombs and screens is used. I n addition, it is necessary to measure the turbulence intensity in the approaching flow. An experimental apparatus which is being used in Kyoto University is shown schematically in Fig. 24. A test cylinder of 1-cm diameter is 16'

JU

'4

screen

'61

i:Test cylinder ( l c m * cathode1

30' 30' I

k

!

Heat Electric heater exchanger NI Orifice

Storage tank

Frc. 24.

Experimental apparatus for mass transfer in cross flow.

118

T. MIZUSHINA

mounted horizontally in a channel of 8 x 16 cm cross section and has a platinum cathode of 0.5-mm diameter, which is embedded in, but isolated electrically, from the main cathode covering around the middle part of the cylinder surface in width of 2 cm. The cylinder can be rotated during the experiments so as to make the position of the isolated cathode relative to the front stagnation point vary. The anode, which is a large nickel plate of 20 x 10 cm, is located downstream. The circulating solution contains 0.01 mole of K,Fe(CN), and K,Fe(CN), ,respectively, and 2 moles of KOH per liter, and is kept at 30°C. T h e oxygen in the solution is purged with nitrogen before each experimental run. In order to make the velocity profile as flat as possible, five screens are installed in series in the duct before the test cylinder. The turbulent intensities are varied from 0.8 to 2.3 yo by changing the screens located 21 cm upstream from the test cylinder. Reynolds numbers are calculated from the velocity based on the corrected free cross section and vary from 3900 to 10,400.

a. Space-Averaged Mass Transfev Coe$&ents. In Fig. 25 the results of Dobry and Finn (27), Grassmann et al. (28), and Vogtlander and Bakker (29) are plotted together. Compiling - the results of many investigators, mainly on heat transfer, van der Hegge Zijnen (31) obtained the following correlation:

+

Sh = 0.38 Sc0u2 (0.56 Re0.5 10'

+ 0.001 Re) S C O . ~ ~

.

100

10.'

1o2

t

I

lo-'

1

I oo

I

I

10'

10'

1o3

I'0

lo5

Re

FIG.25. Mass transfer coefficients in cross flow. Key: ( 0 ) Dobry et al.; (0)Vogtliinder et al.; ( A , V, 0 ) Mizushina et al., 100(i/i)l/a/ub = 0.8, 1.2, and 2.1, respectively.

THEELECTROCHEMICAL METHOD

119

This equation, which is represented for Re = 1-500 and Sc = 1000 in Fig. 25, correlates the data by Vogtlander and Bakker for the middle range of Reynolds numbers quite well while it predicts j factor values which are somewhat higher than the data by Dobry and Finn for low Reynolds numbers. On the other hand, the results of Grassmann et al. for Reynolds numbers which are greater than 160 appear to be rather larger than predictions by Eq. (38). Perhaps this can be explained by two effects, i.e., the effect of blockage and turbulence in approaching flow. Since the ratio between the diameter of the test cylinders and the duct in which they were mounted was not very small in the experiments of Grassmann et al., the liquid might be accelerated near the cylinders. When Re is based on the mean velocity in the smallest cross section instead of in approaching flow, the results in Fig. 25 go down about 10yo. T h e unpublished results of Mizushina and Ueda of Kyoto University are also plotted in Fig. 25. T h e experiments were carried out with various levels of turbulent intensity in the approaching flow, namely 0.8, 1.2, and 2.1 %. T h e data for turbulent intensities of 1.2 and 2.1 yo are in good agreement with the heat transfer results by Comings et al. (32) and the corrected values of Grassmann et al., and the data for the intensity of turbulence (0.8%) agree with Hilpert’s heat transfer data for the intensity (0.3 %) (33). T h e effect of turbulence was studied by Kestin and Maeder (34) in heat transfer. Their results together with the results of Mizushina and Ueda are shown in Fig. 26. From this, the analogy between heat and mass transfer is confirmed even in the effect of turbulence. I

I

I

1.0

I

2.0

100J;r;/ur

FIG. 26. Effect of turbulent intensity on the mass transfer coefficients in cross flow.

120

T.MIZUSHINA

b. Local Mass Transfer Coeficients. T h e distribution of the local mass transfer coefficients obtained by Grassmann et al. (28) are shown for various Reynolds numbers in Fig. 27. It is shown that the concentration boundary layer starts at the front stagnation point and develops, as it proceeds around the cylinder, and reaches the separation region.

'*0°

k

800

i lJY

.40°

t 30

60

90

120

150

100

8 (deg)

FIG.27. Distribution of the local mass transfer coefficients in cross flow. Key: ( 0 ) Re = 10,380; (0)Re = 5210; (0)Re = 7240; ( A ) Re = 3870. 100(u'a)l/a/ub = 2.3.

Dimopoulos and Hanratty (30) measured the local mass transfer rates and velocity gradients on the wall in the range of Reynolds numbers from 115 to 356 and compared their mass transfer data with the prediction based on the measured velocity gradients. Except in the case of Re = 115, the predicted mass transfer rates are in good agreement with the measured values as shown in Fig. 28. For the higher Reynolds number region the local mass transfer coefficients are also sensitive to turbulence in the approaching flow. As seen in Fig. 27, there is a minimum at the front stagnation point. This may be due to the effect of turbulence. The fluctuations of the local mass transfer coefficients near the front stagnation point are plotted in Fig. 29 from which it is seen that the fluctuation is negligible at the 2.5", and decreases again to front stagnation point, increases to 0 9 = 5". Such a behavior of fluctuation corresponds to the distribution of the mass transfer coefficients near the front stagnation point in Fig. 27.

THEELECTROCHEMICAL METHOD I

I

I

I

1

1

I

121 I

Calculated line

1.0 -

-

0.8-

A A ”

-

A A A

-

0.2 0

A

I

I

I

I 10

I

1

I

I

1 30

I

20

I

1 40

I

L

50

t (sec) FIG. 29. Fluctuations of the local mass transfer in cross flow at various angles.

122

T. MIZUSHINA

In Kyoto University, Mizushina and Ueda are studying the effect of turbulence on the local mass transfer coefficients. The experiments are limited to the laminar boundary layers at the front half of the cylinder for the subcritical Reynolds number region from 7240 to 10,380. Since data for zero turbulent intensity are not available, those values were predicted by Dienemann's approximate method based on the static pressure distribution around the cylinder. In Fig. 30, Sh,/( Rell2 Sc1/3) are plotted against 8, the angle from the front stagnation point, and different curves were obtained for different turbulent intensity. From these curves the ratios of measured Sherwood number to those for zero intensity and then (Sh/Sh, - 1) are calculated and plotted against 0 on a semilog scale as Fig. 3 1. Different straight lines were obtained for each turbulent intensity. With increase of the angle 8, this ratio increases, i.e., the effect of turbulence increases. Since it becomes very large near the separation point, it is expected that the turbulence influences the separation of the boundary layer. With respect to the measurement of the separation point, two electrochemical methods, i.e., the shear-stress meter and the measurement of the local mass transfer fluctuation, are used. The former will be described in Section IV. I n the experiments of Mizushina and Ueda the latter was used. Since large vortices are shed at the separation point, the local mass transfer fluctuations have a sharp maximum there. T h e separation points at Re = 10,OOO were determined experimentally to be at 6 = 81", 82", and 84" in the flow, the turbulent intensities there were 2.3, 1.2, and 0.8 yo,respectively.

3. Forced Convection Mass Transfer in an Agitated Vessel The space-time-averaged and the local time-smoothed values and the local fluctuation of the mass transfer coefficients at the wall of an agitated vessel with paddle-type impellers were measured by Mizushina et al. (35) using an aqueous solution of 0.001 mole of CuSO, and 2 moles of H2S04per liter. As shown in Fig. 5, the cylindrical wall of the vessel is the cathode for measuring the average transfer coefficients. Nine isolated cathodes of 1.7-mm diameter for measuring the local transfer coefficients and of 0.3-mm diameter for measuring the fluctuation intensity are mounted at 1-cm intervals in the vertical direction on the cylindrical wall while the bottom plate is the anode. Both electrodes are made of copper. Since the limiting current was not found distinctly, owing to the discharge current of hydrogen ions in this case, it was determined by the calculation described in Section 11.

1

i

i

40

20

60

i

80

0 (deg 1

FIG.30. Turbulence effects on the local mass transfer coefficients in cross flow.

1.0

0.8 0.6 c

I

0.2

(

0.1

1

0

FIG.31. Turbulence

20

1

40

8 ldeg)

1

60

effects on the local mass transfer coefficients in cross flow.

T. MIZUSHINA

124

From the results of measurements of the mass transfer coefficients,

j factors are calculated by the following equation:

where V Bis the fluid velocity outside the laminar film and calculated from Vo = 1.52Rd (40) Equation (40) was obtained by measuring the time lag of the fluctuations at two horizontally separated electrodes. This equation was examined and found to be valid in the range of z / H = 0.1-0.9 for b/H = 0.7 and a / H = 0 . 3 4 . 7 for b/H = 0.2. Hence, it is assumed that Eq. (40) is valid in the whole range of z.

a. Space-Averaged Transfer Factors. T h e values of j , are plotted against Re, in Fig. 32. On the other hand, the friction factors in the to-‘

-k(

.t o 2

._

0

10

1o2

lo3

I 0‘

Red

FIG. 32. j factors of the mass transfer and the friction factors at the wall of an agitated vessel. Key: ( 0 ) mass, (0)momentum, b/H = 0.2; ( A ) mass, ( A ) momentum, b / H = 0.4; ( W) mass, (0) momentum, b / H = 0.5; (+) mass, (0) momentum, b/H = 0.7. j o = f / 2 = 0.15

tangential direction were measured in another agitated vessel which was designed for this purpose, and the measured values of f/2 are also plotted in Fig. 32. Both data are correlated with the same equation as follows:

THEELECTROCHEMICAL METHOD

125

Chilton et al. (36) obtained the following equation for heat transfer in agitated vessels with paddle-type impellers: Nu

= 0.36

Re:/3 Pr1’3(p/pt)0*14

(42)

Neglecting the correction term for viscosity, substituting Eq. (40) into Eq. (42) and taking into account that d / D = 0.66 in this experiment gives

jH

= 0.156

Re&”3

(43)

Thus, it is recognized that the average factors of transfer of momentum, mass, and heat are correlated with similar equations. This implies that the heat and mass transfer at the wall are caused mainly by tangential motion of the fluid and that those by vertical motion of fluid are very small.

b. Local Transfer Coe8cients. The vertical distributions of the mass transfer coefficients and local tangential friction factors which are obtained in a specially designed experimental apparatus are shown in Figs. 33 and 34, respectively. It may be recognized again that both transfer phenomena are quite similar. The distribution curves of fz/f and K,/k have peaks at the height of the impeller. The peak becomes lower with an increase of the width of the impeller and separates into two at a certain value of this width. The curves become flatter also with an increase of Reynolds number. The peak seems to be caused by the jet flow issuing from the impeller. Thus the two peaks seem to be made by the two separate jet flows issuing from the upper and lower edges of the impeller. On the other hand, the flat parts of the curves are attributed to the rotating motion of the fluid in the vessel. In Figs. 35 and 36 the local values of fz/2 and jDz at z / H = 0.5 are plotted against Re for b/H = 0.7 and 0.2, respectively. For the large value of b/H, the momentum and mass transfer factors are well correlated by a single curve. However, the plot for the small value of b/H shows that (jD)z,N=0.5 is a little bit larger than (f )z,a=o.5/2.This means that there is a vertical flow at this point by the jet flow from the impeller, and this flow increases the mass transfer coefficients to some extent. c. Local Mass Transfer Fluctuation Intensity. The electrical currents accompanying the fluctuations which were caused by the turbulent flow were measured with an AC amplifier and a square circuit. The vertical distribution of the fluctuation intensity is shown in Fig. 37. For the small value of b / H , this distribution curve has a peak

0.5

Red=16500

b 1

l 5 0

Y

0.5

y

N

I j

1

1

0

1

1

-

FIG. 33. Distribution of the local mass transfer coefficients at the wall of an agitated vessel.

1

. I -

x

3

kIk

0

0

1

-

0

0

1

i--.

1-

0 1

I

F

6

-

I

FIG.34. Distribution of the local friction factors at the wall of an agitated vessel.

T. MIZUSHINA

0 1.0

126

N

I

\

1 1 0

m=0.7

I B

- 3I

n 2

5

HlZ

I

1

Red =7OOO

p-++---

I

2

-i b/H=O/.

- OO

L: l':

1.0

127

THEELECTROCHEMICAL METHOD

Red

FIG.35. Variations of

jo, and fJ2 at zlH = 0.5 with Red for b / H = 0.7. Key:

( 0 ) mass; ( 0 ) momentum.

FIG.36. Variations of j D z and fJ2 at z l H

( 0 ) mass; ( 0 ) momentum.

=

0.5 with Rea for blH

=

0.2. Key:

1 -1 i

T. MIZUSHINA

128 1c

I

; 0.5

~/H=0.7

blH =0.5

b/H =OX

b/H=02

I n

"0

L

01

Q20

01

0

I

01

0

01

FIG.37. Distributions of the local mass transfer fluctuation intensities at the wall of an agitated vessel.

at the height of the impeller whereas for the large value of b/H, it has no such peak as seen in the local mass transfer coefficients distribution.

4.Forced Convection Mass Transfer from Rotating or Vibrating Bodies a. Mass Transfer in the Annulus of Concentric Rotating Cylinders. Many investigators have carried out theoretical analyses of instability and experimental analyses of transport phenomena of tangential flow in the annulus of concentric rotating cylinders. But special difficulties are met in measuring the local values because the wake behind any measuring probe, such as a thermocouple or pitot tube, which is inserted into the annulus between the cylinders is carried round with fluid flow. Mizushina et al. (37) have successfully employed the diffusion-controlled electrolytic mass transfer technique in studies of transport phenomena in fluid flowing tangentially between concentric rotating cylinders. T h e mass transfer rate to the inside surface of a stationary outer cylinder was measured by a depositing reaction of Cu2+ions. Figure 38 shows their experimental apparatus. Two kinds of inner rotors (44 and 58 mm in diameter) and one stationary outer cylinder (94 mm in diameter) are made of copper. The outer cylinder was used as a cathode for measuring the space-time-averaged mass transfer rate, and the inner one was used as the anode, the electrical circuit to which was connected by means of a mercury well. T h e electrolytic

THEELECTROCHEMICAL METHOD Inside surface of outer cylinder

129

I ~

Point cathodes detail

/ couple ' Copper anode

FIG.38. Apparatus for the mass transfer of the flow in an annulus of concentric rotating cylinders.

solutions contained 0.0005-0.0075 mole of CuSO, and 2 moles of H,SO, per liter. To change the viscosity of the solution, 0-55 weight per cent of glycerin was added. The experimental conditions were: Number of revolutions = 13-425 rpm, Taylor number = 59-19,500, and Schmidt number = 3 x 103-7.7x lo5. Figure 39 shows the experimental results of the space-time-averaged mass transfer coefficients together with correlations of the heat transfer coefficients presented by other investigators (38,39). It is seen that there exists an analogy between the heat and mass transfer in this case. In order to measure the local rates of mass transfer, the inside surface of the outer cylinder was equipped with 71 point cathodes which were embedded axially at intervals of 4 mm and angularly at intervals of 45", respectively, and insulated electrically from the surrounding main cathode (See Fig. 40). I n tangential flow between concentric rotating cylinders, a wide range of transition from laminar to turbulent flow occurs in the form of toroidal vortices, i.e., Taylor vortices spaced regularly along the axis. In Fig. 41, an example of an axial distribution of Sherwood numbers and a corresponding pattern of Taylor vortices are indicated. T h e timesmoothed mass transfer rates do not change significantly in the transition

T. MIZUSHINA

130

range, but the fluctuation of the instantaneous local mass transfer rates is very sensitive to the change of flow pattern. The oscillograms of the fluctuating limiting currents to embedded point cathodes and the axial distributions of the time-smoothed, local mass transfer coefficients are shown in Fig. 42. The former distinguishes the steps of transition from laminar to turbulent flow more precisely than the latter.

t

U '

3

m

FIG. 39. Transfer coefficients of the flow in an annulus of concentric rotating cylinders. Key: (1) Nu = 0.88 Talls Pro*s (Aoki et ul.); (2) N u = 0.84 Tall* Pr1/' (Tachibana et ul.); (3) Sh = 0.74 Tal/aS C ' / ~(Mizushina et d.).

Eisenberg et al. (40) obtained the space-time-averaged mass transfer rate to the inner rotating cylinder of an experimental apparatus similar to that shown in Fig. 38 using the redox system of ferrocyanide and ferricyanide. Their correlation is j D = 0.0791

for lo00

(44)

< Re, < 100,OOO.

b. Mass Transfer to Other Rotating or Vibrating Bodies. Noordsij and Rotte (41) applied an electrolytic redox reaction method to a study of the average mass transfer coefficients on a rotating and a vibrating sphere. I n their experiments a nickel sphere (25.3 mm in diameter), to which mass transferred, was used as the cathode.

THEELECTROCHEMICAL METHOD

FIG. 40.

131

Location of the isolated point cathodes in the experimental apparatus

M: 5 points at 4-mm intervals. L: 36 points at 4-mm intervals.

FIG. 41. Taylor vortices and the distribution of Sherwood numbers at the wall of the outer cylinder.

Their results can be summarized as follows: Rotating sphere:

k D / 9 = 10

+0 . 4 3 ( ~ D ~ / v ) ~ / ~

(v/.9)ll3

for 0.8 x lo3 < wD2/v < 27

(45)

Vibrating sphere:

+

k D / B = 2 0.24(+Da/~)l/~ for lo2 < +Da/v < 16 x lo2, 0.03

x lo3

< a / D < 0.063

(46)

T. MIZUSHINA

132 JJA

sh/s50.41

t

1 ' 1

0

1

2

la)

1

3 +

min

FIG.42. Transition from laminar to turbulent flow in an annulus of concentric rotating cylinders. (a) laminar flow, Ta < 41.2; (b) laminar vortex flow, 41.2 < Ta < 800; ( c ) transition flow, Ta g 1000; (d) turbulent vortex flow, 2000 < Ta < 10,000-15,000; (e) turbulent flow, Ta > 15,000.

Okada et al. (42) studied average rates of mass transfer on a rotating disk by means of electrolytic redox reactions. In a cylindrical cell, the distance between a rotating circular disk (60 mm in diameter) and a stationary disk (61 mm in diameter) was varied in the range of 0.2 to 2 cm. Their result is: k D / 9 = 0 . 6 1 ( ~ D ~ / v( v) /~9') ~l 1 3

for 4 x lo3 < wD2/v < 2 x lo* (47)

5 . Forced Convection Mass Transfer from Falling Film, Flow in a Packed Red, Jet Flow, etc.

a. Mass Transfer from Falling Liquid Film. Iribarne et al. (43) reported the results of a theoretical and experimental study on mass transfer between a falling liquid film and a wall, employing a diffusioncontrolled electrolytic reaction in aqueous solutions of 0.005 mole of K,Fe(CN), and K,Fe(CN), , respectively, and 0.5-6 moles of NaOH per liter, which are flowing on the outer surface of a vertical cylinder.

THEELECTROCHEMICAL METHOD

133

The nickel electrode on the surface of the cylinder was divided into fifteen different lengths electrically insulated from each other. These sections were used as cathodes and anodes. It was taken into account that the anode was larger than the cathode. In addition, since the ohmic drop in potential which occurs in a thin film is a serious problem, the distance between cathode and anode should be as short as possible. Their experimental results of the space-time-averaged mass transfer rates for Re < 700 are in good agreement with the predictions by a LkvvCque-type solution, and those for Re > 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
O<x
(53)

a,

T h e diffusion in the flow direction has been neglected in Eq. (52) since the condition of s L z / 9 > 5000 is satisfied in the ordinary case owing to the very small values of the diffusion coefficients of the ions. But it must be taken into account in the flow near the separation point. T h e velocity and the concentration are expressed as the sum of timesmoothed and fluctuating components. u=ii+u‘,

Z)=V+Z)’,

c=c+c’

(54)

Since the velocity gradient can be assumed to be linear in the very thin concentration boundary layer, ii is given by

u = sy

(55)

From the continuity equation, E is calculated as

where s is the time-smoothed velocity gradient at the wall and is a function of x. In this region, the fluctuating velocity can be represented by

where s‘ is the fluctuation in the velocity gradient at the wall and is a function of t and x. Using the above-mentioned equations of velocity distribution and neglecting the second-order terms in fluctuating equations, give the equations of time-smoothed and fluctuating concentration fields as follows: For time smoothed concentration,

138

T. MIZUSHINA

Boundary condition: E=O

at

y=O,

C=c, E=c,

at

y=00 x=O

at

O<x
(60)

For fluctuating concentration,

Boundary condition: c' = 0

at

y

c'=O

at

x=O

=

00

and y = 0,

(62)

These equations can be simplified in each case and solved to relate the mass transfer coefficient to the velocity gradient at the wall.

1 . Shear Stress in a Fully Developed Flow a . Time-Smoothed Velocity Gradient. For a fully developed flow, 0. Therefore, from Eq. (56), the second term of Eq. (59) becomes zero. Hence, the equation of time-smoothed concentration is simplified as follows: sy aqax = 9 a z E / a p (63) Boundary condition: =

t=cb

at

E=c,

at y = c o

s=O

at y = O ,

x=O

O<x
T h e solution of Eqs. (63) and (64) is

where

-77 = y(s/99x)1/3

T h e average mass transfer coefficient over the electrode surface is

THEELECTROCHEMICAL METHOD therefore s =

139

1.90 k 3 L / 9

When circular electrodes are used instead of rectangular ones, the effective length in the direction of flow is calculated by L,

(69)

= 0.81360

b. Velocity Gradient Fluctuation. as follows:

In pipe flow, Eq. (61) is simplified

Boundary condition: c'=O

at y =

c' = 0

at x = 0, at y = O ,

c'=O

00,

(71) O<x
Analyzing the fluctuations of the mass transfer coefficient and the velocity gradient into harmonic oscillations, one solves Eq. (70) for the oscillation of frequency, 4, to give &'/s = 3fl

+ O . O ~ ( ~ T C # J L ~)/ }~ / W&'/k~ S ~ / ~ 1'2

(72)

where f,' and L,' are the amplitudes of oscillations of frequency, 4,in the velocity gradient and the mass transfer coefficient. When the time constant or L2/9sa is very small as a result of making the electrode size and the Schmidt number small, the capacitance effect is negligible except for very large values of 4. T h e velocity gradient fluctuation intensity is calculated from the mass transfer coefficient fluctuation intensity and the spectral distribution function of the mass transfer fluctuations, W, by the following equation: )

((S')2)ll2/s

[SrnW4(l + 0.06(2m$L2/3/91/3~2/3)2f 41"' (73)

3[((K')")1/2/k]

+

0

Since W, for large values of is much smaller than that for small values of 4,Eq. (73)practically becomes ((s')2)'/2/S

=

3(0")'/"k

(74)

Since Eqs. ( 5 5 ) and (57) are assumed near the wall, ((u')2)'/2/u = ((s')2)1/2/s

(75)

140

T. MIZUSHINA

Problems which are similar to those for a hot-wire anemometer are encountered in using this method, i.e., the time response, the nonuniformity of the flow over the cathode, and the effect of nonlinearities. T h e time response has been already taken into account in the calculations. Since nonuniformities in the structure of turbulence are much greater in the circumferential direction than in the direction of flow, the measurements must be corrected, if necessary, by the use of measured values of the circumferential correlation coefficient. T h e assumption of the linearities is not a serious problem since it was found to give an error of less than 3 yo.

2. Shear Stress in the Boundary Layer Solving Eq. (59) by using a similarity transformation gives the concentration gradient at a position on the electrode surface.

Except in the region close to the front stagnation point (0 < 5") in the case of cross flow, the variation of s over the small length of cathode is negligible. Assuming that s is constant, one integrates Eq. (76) from 0 to L to give the space-averaged value of the mass transfer coefficient, k, over the cathode k L / 9 = (l/cb)

&/8y 0

I

dx = 0 . 8 0 7 ( ~ L ~ / 9 ) ~ / ~

1/-0

(77)

therefore s = 1.90(ksL/92)

B. SHEARSTRESSIN

A

WELL-DEVELOPED FLOW

Mitchell and Hanratty (48) measured the shear stress and the velocity fluctuations in a fully developed turbulent flow using measuring cathodes embedded in the wall of I-in. i.d. Lucite pipe. A fore-flow section of 180 diameters in length preceded the test section. T h e electrolyte solution contained 0.01 mole of K,Fe(CN), , 0.01 mole of K,Fe(CN),, and 2 moles of NaOH per liter. T h e test electrodes consisted of several nickel sheets which has a length of 0.003-0.021 in. and a width of 0.020 to 0.062 in. On the other hand, Reiss and Hanratty (49) used three sizes of nickel wire of 0.0398-0.1636 cm in diameter as cathodes which were arranged so as to measure longitudinal and circumferential correlations.

THEELECTROCHEMICAL METHOD

141

Different configurations of the rectangular cathodes such as shown in Fig, 45(a)-(d) were used for different purposes, i.e., (a) for the longitudinal velocity intensity, (b) for the circumferential correlation coefficient, (c) for the longitudinal correlation coefficient, and (d) for the circumferential velocity intensity. T o check the usefulness of this technique, the following experiments were carried out.

FIG.45.

Configurations of rectangular cathodes for various measurements.

A momentum balance in fully developed turbulent flow in a circular tube gives s = Um2f/2V

(79)

From Eqs. (67) and (79)

f

=

3.80(k/u,,J3 ( S C ) (L/D)(Re) ~

(80)

Thus the values of the friction factor can be calculated from the measurement of the mass transfer coefficients. On the other hand, the friction factors are calculated from Blasius' equation f

= 0.079

(81)

I n Fig. 46, the values of the friction factors calculated from Eqs. (80) and (81) are compared. T h e data obtained in Kyoto University in a circular tube are plotted also. T h e agreements of both predictions indicate that this technique is useful. Mitchell and Hanratty measured longitudinal and circumferential correlation coefficients also, and found that the circumferential integral scale was only about one-thirtieth of the longitudinal one.

T. MIZUSHINA

142

Using Eqs. (74) and (75), they calculated the velocity fluctuation intensity near the wall and obtained its value as 0.32, independent of Reynolds number. This result agrees with the measurements by other investigators (50)-(52) with the hot-wire anemometer.

f x lo3 (Blasius’ equation)

FIG.46. Comparison of the friction factors calculated from measurements and LID = 0.003; ( 0 )LID = 0.005; ( 0 ) LID = 0.007;( A ) Blasius’ equation. Key: (0) L/D = 0.0021 (Hanrattyet d.); ( 0 ) LID = 0.0424, ( r ) L / D= 0.0410 (Mizushinaet d).

C. SHEARSTRESSIN

THE

BOUNDARYLAYER

Dimopoulos and Hanratty (30) studied flow crossing cylinders and measured the velocity gradient in the boundary layer as described in Section 111, C, 2. Direct application of Eq. (77) to this case causes an error because diffusion in the x-direction and natural convection by density difference in the concentration boundary layer are neglected in deriving Eq. (52). Assuming that the interaction of the effects of natural convection and diffusion in the flow direction are negligible because both effects are small, one adds the correction terms for each effect to Eq. (77) as in Eq. (82). kL/B = 0.807(~L~/B)~’~ 0.19(~L~/9)-~’~

+

f 0.253(Gr,/Re,)(S~/sL~/B)~’~ sin B

(82)

THEELECTROCHEMICAL METHOD

143

where the plus sign is for aiding Aows and the minus sign is for opposing flows. These correction terms, however, are important only at very low Reynolds numbers and near the separation point. The test cylinder of 1-in. diameter was mounted in a 1 x 1 ft duct as the axis of the cylinder was perpendicular to the flow direction. T h e test section was connected to a gravity flow tunnel. Two kinds of test cylinders were used, one for velocity gradient measurements and another for mass transfer studies. The former was equipped with a platinum cathode of 0.020 x 0.500 in., and the latter with an isolated round platinum cathode of 0.020-in. diameter which was insulated electrically from the main cathode covering the whole cylinder surface. Both cylinders were rotated so as to change the position of the test cathode with respect to the front stagnation point. The mass transfer measurements have been described in Section 111, C, 2. A result of the measurements of the velocity gradient at the wall for Re = 151 is plotted in Fig. 47, and compared with the boundary layer calculation for Re = 174. T h e measurments agree with the calculations fairly well.

0 deg FIG. 41. Comparison between the velocity gradient measurements and the boundary Re = 174 layer calculation in cross flow. Key: (0)Re = 151 (measurements); (-) (boundary layer calculations).

T. MIZUSHINA

144

T h e separation angles determined by finding the position where the velocity gradient became zero are plotted against Reynolds number in Fig. 48 and are in reasonable agreement with the results of other investigators (53). I60

- 140 --

-

I

-

I

I

-

-

0

-

U

120-

-

O D

0 0 0

O g

rooI

I

al

Q

-

I

Re

FIG. 48. Measurements of the separation angles from the front stagnation point of the cylinders. Key: (0) 4-in. splitter plate; (0) without splitter plate.

V. Application to Fluid Velocity Measurements

A.

PRINCIPLE AND

METHODOF MEASUREMENTS

For measuring the velocity of air flow and its fluctuation, there are many devices. For liquid flow, however, convenient and reliable measuring techniques have not been developed. T h e pitot tube has a defect in response to the velocity fluctuations, and the hot-wire anemometer needs compensations for phase shift and amplitude and has limits in mechanical strength and operating temperature. Furthermore, it has a limit in the length of wire because the shorter wire causes the larger error due to the cooling effect of the supports. A diffusion-controlled electrochemical process on an extremely small cathode was found applicable to these purposes, and a device and technique for this process have been developed. T h e compensations for phase shift and amplitude attenuation of a fluctuating signal is not a serious problem because the measuring probe has practically no capacity for the transferred quantity. I n addition, this probe is extremely sensitive to low velocities and can measure the value of a few mmlsec. T h e maximum velocity which can be measured by this method was discussed in Section 11. O n the other hand, a capacitance effect of the concentration

THEELECTROCHEMICAL METHOD

145

boundary layer over the cathode limits the frequency range of the velocity fluctuations to be measured. Ranz (4) first suggested the usefulness of this method in measuring the liquid flow velocity and measured it in several kinds of electrolytic solutions. From the current-voltage curves obtained, he discussed the chemical reaction mechanism. The measuring probe which he used was a cylindrical type as shown in Fig. 49(a). Several types of probe other than this have been developed. Those are also shown in Fig. 49. A spherical type probe (b) was devised by Ito and Urushiyama (54) to determine the flow direction. A blunt-nose type (c) and hot-wire type (d) probes are used by Mizushina et al. in Kyoto University. The mass transfer coefficients calculated from the limiting currents to those electrodes are related to the liquid velocity. Since the electrodes in this case are so small, the boundary layers on these electrode surfaces are always laminar. The relations between Sh and Re for cylindrical electrodes was described in Section 111, C, 2. When the electrode is mounted at the front stagnation point of a sphere, the relations given experimentally by Brown et al. (55) are Sh

=

s

1.29 Re0.5S C O * ~ ~

(83)

,Stainless steel support

Glass

surface coating (a )

,Glass fusing

tfBare lcl

platinum surface

Glass fusing support platinum surface (b)

f t 1

Plat1 num wire

(d)

FIG.49. Various types of measuring probes. (a) cylindrical type, (b) spherical type, (c) blunt-nose type, (d) hot-wire type.

T. MIZUSHINA

146

and analytically by Sibulkin (56) are Sh

=

1.32 Re0.5Sco**

(84)

I n general, the mass transfer rate to a laminar boundary layer is proportional to Re1/2, so the limiting current is proportional to u1/2,i.e.,

where 0: is the natural convection term and /3 is the laminar convection term. For practical use, LY. and /3 are determined by experimental calibrations. As described in Section IV, A, this method is analogous to the hot-wire anemometer, not only in its characteristics but also in the limits of application. As a result, considerations similar to those for hot-wire anemometers must be given to the application of the electrochemical method. Lighthill (57) solved the linearized problem of the response of skin friction and heat transfer to the fluctuation in the velocity of incompressible laminar two-dimensional flow. He calculated the amplitude ratio relative to the quasi-steady amplitude and the phase lag in two-dimensional stagnation flow for Pr = 0.7. This calculation procedure was applied to axisymmetrical flows. I n this calculation it is assumed that the velocity at the outer edge of the boundary layer is equal to K X , where K is a constant. The results of the amplitude ratio for Pr or Sc = 0.7 and 2431 are represented in Fig. 50. It is seen that the amplitude ratio decreases more rapidly in the case of a higher Schmidt number and that the response of the mass transfer to liquid velocity fluctuation is worse than that of the heat transfer to air. For the comparison of the response between hot-wire anemometers and the electrochemical method, the critical frequencies, +crit , at which the amplitude ratios become 0.9 when the velocity measuring probes are placed in the stream of u = 100 cm/sec, are given as follows: Pr

= 0.7,

SC = 2431,

wire (D = 0.001 cm),

$crit =

wire ( D = 0.001 cm),

$crtt = 6.50

sphere (D = 0.001 cm),

$crlt

=

4.59 x lo4 Hz

x 10s Hz

5.16 x lo3 Hz

In the use of the electrochemical method in measuring the fluctuating velocity, this limitation in the ability of response must be always taken into account.

147

THEELECTROCHEMICAL METHOD 1.0

0.8

0.2

0. L

0.6

0.8

1.0

W I P

FIG. 50. Amplitude ratio in response to the fluctuations.

B. TIME-SMOOTHED VELOCITY 1. Time-Smoothed Velocity in a Tube Flow

To approach a wall in measuring the velocity of fluid flow, it is necessary to make the probe as small as possible. However, a small pitot tube is inadequate for measuring the fluctuating velocity because it needs a very long response time, especially in liquid flow. For this purpose the electrochemical method is very useful because the probe can be made very small to measure the velocity of bulk flow from the center to the vicinity of the wall and another probe embedded in the wall can measure the velocity gradient in the laminar sublayer at the wall as described in Section IV, and, as a result, the velocity profile in the whole cross section of flow can be obtained easily. I n addition, this method is conveniently used to determine the direction of flow. A hot-wire type probe designed for this purpose can be constructed smaller than a hotwire anemometer, which has a lower limit in wire length owing to the

148

T. MIZUSHINA

cooling effect of the supports. A blunt-nose type probe has an appreciable advantage. Flowing dirt particles hardly cling to the probe during an experiment, and the output signal is most reliable. These two types of probes, which have been developed and used in Kyoto University, are shown in Fig. 49. The hot-wire type probe has a platinum wire cathode of 0.003-cm diameter at the tips of the prongs, which are coated with glass. The blunt-nose type probe has a platinum wire cathode of 0.01-cm diameter which is embedded in the blunt glass nose and smoothed flush with nose surface so that only the tip of the wire is exposed. Before the experimental run, the probe must be calibrated with the electrolytic solution which will be used in the experiment. I n Kyoto University, a redox system containing 0.01 mole of K,Fe(CN)6 , 0.01 mole of K,Fe(CN), , and 2.0 moles of KOH/l. is used. The apparatus for calibration in Kyoto University consists of an arm of brass which is fixed perpendicular to a rotating shaft and a plastic basin which is of doughnut shape and placed around the shaft. The probe attached at the end of the arm is dipped in the electrolytic solution contained in the basin. By changing the rotating speed of the shaft, the traveling speed of the probe can be changed from 5 cm/sec to 5 m/sec. T h e probe must be attached to its place very carefully to meet the direction of flow. I n the basin a narrow passage for the probe is made with two parallel screens which depress the wake flow of the probe. In addition, the diameter of the circle of the passage is 90 cm, which is large enough to decay the wake within the period of one cycle. A typical plot of velocity versus limiting current for the calibration is shown in Fig. 51. In the calibration of a hot-wire type probe, its angular property is investigated by directing the axis of wire at various angles to its moving direction. The rate of mass transfer is determined mainly by the velocity component perpendicular to the wire axis. The effect of the velocity component parallel to the wire becomes noticeable only when the normal velocity component is very small. Thus, if the wire makes an angle t,b with the direction of flow, the mass transfer is accomplished essentially by the component usin#, and onIy at small values of $ does the component u cos t,b also assume importance. Consequently, the effective value of velocity is obtained approximately from the relation

where the factor K has a value between 0.1 and 0.4 and increases with decreasing velocity. A typical calibration curve for angular property is shown in Fig. 52, from which K is determined to be 0.39. However, for

THEELECTROCHEMICAL METHOD

149

the range of practical use, i.e., 30” < i+h 5 go”, it seems enough to consider the effective value of velocity to be u sinn 4. Thus, the following equation is obtained instead of Eq. (85):

i

=

01

+ p(sinn

t,/~ * ~ 4 ) ~ ’ ~

where a,8, and n are determined by calibration.

fi

(cm”2/sec1’2 )

FIG. 51. Calibration of the measuring probe.

I

90

I

70

1

I

1

50

30

JI ( d e g 1

I

FIG. 52. Calibration of the angular property of the hot-wire type probe.

10

150

T. MIZUSHINA

T h e essential part of the experimental apparatus in Kyoto University to measure the time-smoothed velocity profile in tube flow is a polyvinyl chloride plastic tube of 2-in. diameter, through which an oxygen-free redox system is circulated at controlled rates of flow. T h e test tube has a sharp leading edge with a trip wire from which the turbulent boundary layer develops. At various cross sections in the fully developed turbulent region and in “the entrance region” from the leading edge to the cross section where the boundary layer thickness becomes equal to the radius of tube, measurements of velocity profiles were made with a blunt-nose type probe of platinum cathode of 0.01-cm diameter. T h e anode is located at the tube wall downstream from the measuring cross section. T h e measurements of velocity were also carried out by a total pressure tube of 0.6-mm 0.d. T o achieve a flat velocity profile and a small turbulence level, a Hori type (58) water tunnel was constructed. T h e liquid from the convergent nozzle is passed into the leading edge of the test tube where the flow close to the wall of the nozzle is cut off to the bypass. Thus, a fairly flat velocity profile at the leading edge was obtained. On the other hand, the velocity gradients near the wall were measured by the electrochemical method described in Section IV. By combining the results of both experiments, the velocity distributions in fully developed tube flow were obtained. T h e result is represented by an ordinary equation of universal profile as shown in Fig. 53.

FIG. 53. Velocity distribution in a tube flow. Key: (- - -) fully developed region; entrance region; (o)x/D= 5.92; ( 0 ) x/D= 12.30;(0) x / D = 15.47;(A) x / D = 28.83; ( 0 )x/D= 73.40. (-)

151

THEELECTROCHEMICAL METHOD 2. Time-Smoothed Velocity in the Boundary Layer

T h e velocity distributions in the boundary layer in the entrance region of the same experimental apparatus were plotted in Fig. 53. As seen from Fig. 53, the similarity in velocity distributions at various cross sections may be assumed to be expressed in the form Uf =

C(y+)l’m

(88)

The value of m is a characteristic value in the entrance region and a little smaller than that in the fully developed region as follows. m

= 6.6,

C

=

7.6

at

Re,,, = 5 x lo4

m

=

6.7,

C

=

7.9

at

Re,,,

=

lo5

Using Eq. (88) and these experimental values, one can calculate the displacement thickness Sl and the momentum thickness 6, in the entrance region of the tube from the following equations:

jD’’

= (2/O)

(u/ub)(l

-r)

- u/ub)(D/2

dy

(90)

The shape parameters defined as P = 6J6, are calculated and plotted in Fig. 54.This indicates that the turbulent boundary layer develops from the leading edge owing to the existence of a trip wire. The macroscopic mass balance for the circular tube flow leads =

0

10

(D/4)(1

(91)

- um/ub)

20

30

40

X I D

FIG. 54. Shape parameter in the entrance region of a tube flow. Key: (0)Re, 5 x lo4; (0)Re, = lo6.

=

152

T. MIZUSHINA

Substituting Eq. (89) into this equation, one can obtain the experimental values of u&,. On the other hand, the momentum equation written in terms of P and 6, is given as

o'oal 0.06

xlD

FIG.55. Development of the momentum thickness in the entrance region of a tube theory, Re, = 5 x lo4; (---) Re, = lo6.

flow. Key: (-*-) Re, = 5 x 10';

theory, Re, = 10'; (0)exp.,

( 0 ) exp., 1

1

I

/'

/-

1.20-

/'

1

1

1.25-

0

/'

/

/'

1

0

.HO

,/'

-

/94

1.15-

'6

-

/',)

/;/'

1.101.05-

<;.'

4

&

-

0

/

-

p'

1.00

1

-

B

)I(

1

I

I

I

I

1

THEELECTROCHEMICAL METHOD

153

Solving this equation by the increment method using an electric computer gives the values of 6, , 6, , and u&, as functions of x. The experimental and predicted values of 26,/D and ub/umare compared in Fig. 55 and Fig. 56, respectively.

C. FLUCTUATING VELOCITY 1. Fluctuating Velocity in a Tube Flow The experimental apparatus is the same as described in Section V, B. The electric circuit used to measure the velocity fluctuation is shown in Fig. 57. The electric potential of the probe was held at -0.200V relative to the mercury oxide reference electrode with a potentiostat. Thus Cathode

Flow

-

DC component

- DC amp.

T I I \ r n n o

-Potentio- d e stat -

rms of fluctuation

c-

Power spectrum a nalvr er

FIG. 57. Electrical circuit for measuring the fluctuating velocities.

limiting currents through the circuit were ensured. The fluctuation of current was amplified 106 times, and then root-mean-squares and power spectrum densities of output signals were measured. The measured values of turbulent intensity in flow direction, ((u')2)1/2/u, at x / D = 4.638, 12.05, and 29.77 downstream from the leading edge are shown in Fig. 58. It may be noticed that the similarity in the distribution of turbulent intensity has been established even in the entrance region of tube flow. The relative turbulent intensity of free stream was roughly 0.7%. They are also compared with Klebanoff's results (59) measured in flow along a flat plate with no pressure gradient. It is shown that these results agree. The frequency spectrum densities at various positions in a cross section in the entrance region are shown in Fig. 59. The measurements

"i \

0 6

Flow along flat p l a t e

4

2

0.5

1.0

1.5

2.0

YJb

FIG.58. Distribution of turbulent intensities in the boundary layer in the entrance region of a tube flow. Key: ( A ) x/D = 4.638;(0) x / D = 12.05; (0)x / D = 29.77.

FIG. 59. Power spectrum density of u' in the entrance region of a tube flow. Key:

( A ) y / s = 1.086; ( 0 ) y/6 = 0.136; Mizushinaetal. (x/D = 12.05); (---) (-) y/S = 0.20 Klebanoff (with zero-pressure gradient).

y/8

=

1.00,

THEELECTROCHEMICAL METHOD

155

by Klebanoff (59) in a boundary layer on a flat plate with zero pressure gradient are also represented. There is a remarkable similarity in these measurements. T h e discrepancy between these measurements at very low frequency seems due to the difference in the mechanism of intermittent flow in the region of the outer side of the boundary layer. As described in Section V, A, the electrochemical method has a limit in the frequency which can be measured. This limit corresponds to cr 20 in Fig. 59. Accordingly, the measurements up to u = 20 are reliable. T h e correlation of the plots has two extensive regions. One is the inertial subrange where the spectrum varies in proportion to c r 5 I 3 . T h e other is the largest wave number region where viscosity plays an important role and the spectrum is proportional to 0-'. At the end of this section it should be mentioned that there was no difference in spectra obtained with different applied electrode potentials. This indicates that the reaction kinetics at the electrode surface does not influence the high frequency components of the fluctuations of mass transfer measurements and that the concentration of the diffusing ion is always regarded as zero at the surface of electrodes.

2. Fluctuating Velocity in an Agitated Vessel I n Kyoto University, time-smoothed and fluctuating velocities in an agitated vessel are measured with a hot-wire type electrode. T h e experiments have been carried out in cross sections where the vertical velocity component does not have a significant efiect on the measurements of the tangential and the radial velocity component. T h e experimental results are shown in Figs. 60-63. Figure 60 shows the equiturbulent intensity curves from which we may guess the movement of the turbulent eddies. I n Fig. 61, the radial distributions of the time-smoothed tangential and radial velocities, turbulent intensity in tangential and radial directions, and Reynolds stress and longitudinal spatial micro scale in the cross section at the center of impeller blades are plotted. All measured values have been reduced to a nondimensional form by dividing by a characteristic length d or a velocity Od where d is the impeller diameter and Q is the rotation speed, respectively. T h e Reynolds stress was calculated from the measured values of turbulent intensities in the direction of *45" to the tangential by the following equation:

- ((-)1/2 is the turbulent intensity in $45" where ( ( ~ - ' ) z ) l / ~is the turbulent intensity in -45" direction.

direction and

T. MIZUSHINA

156

The spatial micro scale in the flow direction, A,, was obtained from the time correlation curve of the fluctuating velocity and the Taylor hypothesis. Figure 62 shows the energy balance in the jet flow issuing from the impeller. From this, the following conclusions may be obtained: (1) The energy gain by convection balances the work by pressure, and the work by shear stress balances the conversion to turbulent energy. (2) The energy of the time-smoothed velocity first converts into turbulent energy which then dissipates into heat but does not directly convert into heat.

3.0 cm/sec

zt

0.5

0

0

0.5

I .o

r+

FIG.60. Equiturbulent intensity curves in an agitated vessel.

Figure 63 shows the turbulent energy balance in the same jet flow. In this diagram the loss by dissipation was calculated by assuming the isotropic turbulence theory in this flow. The gain by production is the same as the loss by conversion to turbulent energy in Fig. 62. From the total turbulent energy balance, the loss by turbulent diffusion was calculated and represented by a dotted curve. This result indicates that the local production is not equal to the local dissipation because of advection and diffusion effects and that the turbulent energy which is produced near the tip of the impeller is transferred toward the wall of the vessel.

THEELECTROCHEMICAL METHOD

157

0.10 '

0.05

0-

I .o

0.5

rt G.

61.

Distribution of the velocities and turbulent characters in an agitated vessel.

24

1.0 16 .-C

C

.-Q

rd

13

8

0

0

111 I 0

0.5

-8

UI

0"

4

-1 6

-0.5

1.o

0.5

r*

r+ FIG. 62. Energy balance in an agitated vessel. convection by mean motion; ( 0 ) work by Key: (0) pressure; ( x ) work by shear stress; (+) conversion to turbulent energy.

1.0

0.5 FIG. 63.

Turbulent energy balance in an agitated vessel. Key: (0)production; ( 0 ) dissipation; ( x ) advection, (- - -) turbulent diffusion.

THEELECTROCHEMICAL METHOD

159

ACKNOWLEDGMENTS This survey has been prepared in cooperation with Professor R. Ito of Osaka University, Assistant Professor F. Ogino, Messrs H. Ueda and Y. Hirata of Kyoto University, Mr. S. Hiraoka of Nagoya Institute of Technology, and Mr. K. Kataoka of Kobe University. The author acknowledges their great help. He also wishes to express his thanks to Professor P. Grassmann of E. T. H., Zurich, who first stimulated the interest of the author and his co-workers in the electrochemical method.

SYMBOLS A A, a

6

C

C

D

9 d E

F

f

g Gr

H 1

iD

in

K K+

k L m

N

no n

P

Pr

R

R18

Re

sc Sh

surface area of mass transfer surface area of isolated cathode at x amplitude width of agitating impeller experimental constant concentration diameter moiecular diffusivity diameter of agitating impeller electromotive force Faraday constant friction factor gravitational acceleration Grashof number depth of liquid in agitated vessel electric current j factor of mass transfer J’ factor of heat transfer experimental constant nondimensional mass transfer rate = k/o* mass transfer coefficient or its space-time-averaged value length of mass transfer section exponent mass transfer rate valence charge of an ion exponent shape parameter = &/A, Prandtl number gas constant correlation coefficient of mass transfer fluctuations Reynolds number Schmidt number Sherwood number or Nusselt number of mass transfer

velocity gradient near wall absolute temperature Ta Taylor number t time u velocity in x- or tangential direction w velocity in y- or radial direction w* friction velocity W spectral distribution function w velocity in z-direction x, y , z coordinates s

T

experimental constant experimental constant activity displacement thickness of boundary layer momentum thickness of boundary layer eddy diffusivity of mass eddy diffusivity of momentum angle from front stagnation point experimental constant integral scale of fluctuation spatial microscale in fiow direction viscosity kinematic viscosity current density by hydrogen ion hydrogen overvoltage density number of wave frequency electrostatic potential angle of wire with the direction of flow number of rotations per unit time angular velocity

T. MIZUSHINA SUPERSCRIPTS

+

fluctuating nondimensional

-

b

i

m

OVERLINES

,.

SU~SCRIPTS

amplitude time-smoothed

x

z co

bulk interface mean local value at x local value at z fully developed region

REFERENCES 1. D. Ilkovic, CoZlection Czech. Chem. Commun. 6, 498 (1934). 2. N. Ibl, Chem. Zng. Tech. 35, 353 (1963). 3. C. S. Lin, E. B. Denton, H. S. Gaskill, and G. L. Putnam, Ind. Eng. Chem. 43, 2136 (1951). 4. W. E. Ranz, A.Z.Ch.E. (Am. Inst. Chem. Engrs.) J. 4, 338 (1958). 5. J. E. Mitchell and T. J. Hanratty, J. Fluid Mech. 26, 199 (1966). 5a. S. Nukiyama and K. Yoshikata, J. /up. SOC. Mech. Engrs. 33, 77 (1930). 5b. S. Nukiyama and Y. Tanazawa, J. Jap. SOC. Mech. Engrs. 33,434, 595 (1930). 5c. D. E. Lamb, F. S. Manning, and R. H. Wilhelm, A.Z.Ch.E. (Am. Znst. Chem. Engrs.) J. 6, 682 (1960). 5d. F. S. Manning and R. H. Wilhelm, A.Z.Ch.E. (A m. Znst. Chem. Engrs.) J. 9, 12 (1963). 6. C. R. Wilke, M. Eisenberg, and C. W. Tobias, J. Electrochem. SOC.100, 513 (1953). 7. A. M. Sutey and J. G. Knudsen, Znd. Eng. Chem. Fundamentals 6, 132 (1967). 8. J. Newman, Znd. Eng. Chem. Fundamentals 5, 525 (1966). 9. J. N. Agar, Trans. Faruduy SOC.1, 31 (1947). 10. C. Wagner, J. Electrochem. SOC.95, 161 (1949). 11. C. R. Wilke, W. Tobias, and M. Eisenberg, Chem. Eng. Progr. 49, 663 (1953). 12. M. G. Fonad and N. Ibl, Electrochim. Acta 3, 233 (1960). 13. U. Bohm, N. Ibl, and A. M. Frei, Electrochim. Acta 11, 421 (1966). 14. E. J. Fench and C. W. Tobias, Electrochim. Acta 2, 311 (1960). 15. G. Schutz, Intern. J. Heat Mass Transfer 6, 873 (1963). 16. T. H. Chilton and A. P. Colburn, Znd. Eng. Chem. 26, 1183 (1934). 17. W. L. Friend and A. B. Metzner, A.Z.Ch.E. (Am. Inst. Chem. Engrs.) J. 4, 393 (1958). 18. C. S. Lin, R. W. Moulton, and G. L. Putnam, Znd. Eng. Chem. 45, 636 (1953). 19. R. G. Deissler, N A C A Rept. No. 1210 (1955). 20. P. V. Shaw and T. J. Hanratty, A.I.Ch.E. (Am. Znst. Chem. Engrs.) J. 10, 475 (1964). 21. D. W. Hubbard and E. N. Lightfoot, Znd. Eng. Chem. Fundamentals 5, 370 (1966). 22. P. Harriott and R. M. Hamilton, Chem. Eng. Sci. 20, 1073 (1965). 23. J. S. Son and T. J. Hanratty, A.I.Ch.E. (Am. Inst. Chem. Engrs.) J. 13, 689 (1967). 24. T. Mizushina, R. Ito, F. Ogino, and H. Muramoto, Mem. Fuc. Eng. Kyoto Uniw. 31, 169 (1969). 25. P. V. Shaw, L. P. Reiss, and T. J. Hanratty, A.Z.Ch.E. (Am . Znst. Chem. Engrs.) J. 9, 362 (1963). 26. G. Schutz, Intern. J. Heat Mass Transfer 7, 1077 (1964). 27. R. Dobry and R. K. Finn, Ind. Eng. Chem. 48, 1540 (1956). 28. P. Grassmann, N. Ibl, and J. Triib, Chem. Zng. Tech. 33, 529 (1961). 29. P. H. Vogtlander and C. A. P. Bakker, Chem. Eng. Sci. 18, 583 (1963).

THEELECTROCHEMICAL METHOD 30. 31. 32. 33. 34. 35.

161

H. G. Dimopoulos and T. J. Hanratty, J. Fluid Mech. 33, 303 (1968). B. G. van der Hegge Zijnen, Appl. Sci. Res., Sect. A 7, 205 (1958). E. W. Comings, J. T. Clapp, and J. F. Taylor, Znd. Eng. Chem. 40, 1076 (1948). R. Hilpert, Forsch. Gebiete Ingenieurw. 4, 215 (1933). J. Kestin and P. F. Maeder, N A C A T N No. 4018 (1957). T. Mizushina, R. Ito, S. Hiraoka, A. Ibusuki, and I. Sakaguchi, J. Chem. Eng. Ja9un.

2, 89 (1969). 36. T. H. Chilton, T. B. Drew, and R. H. Jerbens, Ind. Eng. Chem. 36,510 (1944). 37. T. Mizushina, R. Ito, K. Kataoka, S. Yokoyama, Y. Nakajima, and A. Fukuda, Kagaku Kogaku 32, 795 (1968). 38. F. Tachibana, S. Fukui, and H. Mitsumura, Trans. Japan SOC. Mech. Engrs. 25, 788 (1959). 39. H. Aoki, H. Nohira, and H. Arai, Trans. Jupun SOC.Mech. Engrs. 32, 1541 (1966). 40. M. Eisenberg, C. W. Tobias, and C. R. Wilke, Chem. Eng. Prog., Symp. Ser. 51, No. 16, 1 (1955). 41. P. Noordsij and W. Rotte, Chem. Eng. Sci. 22, 1475 (1967). 42. S. Okada, S. Yoshizawa, F. Hine, and K. Asada, Denki Kugaku 27, 179 (1959). 43. A.Iribarne,A. 0.Gosman, and D. B. Spalding, Intern. J. Heat Muss Trmfer 10, 1661 (1967). 43a. E. R. van Driest, J. Aeron. Sci. 23, 1007 (1956). 44. A, A. Wragg, R. Serafimidis, and A. Einarsson, Intern. J. Heat Mass Transfer 11, 1287 (1968). 45. S. L. Goren and R. U. S. Mani, A.I.Ch.E. (Am. Inst. Chem. Engrs.) J. 14, 57 (1968). 46. K. R. Jolls and T. J. Hanratty, Chem. Eng. Sci. 21, 1185 (1966). 46a. K. R. Jolls and T. J. Hanratty, A.1.Ch.E. (Am. Znst. Chem. Engrs.) J. 15, 199 (1969). 47. E. Weder, Chem. Zng. Tech. 39, 914 (1967). 48. J. E. Mitchell and T. J. Hanratty, J. Fluid Mech. 26, 199 (1966). 49. L. P. Reiss and T. J. Hanratty, A.Z.Ch.E. (Am. Inst. Chem. Engrs.) J. 9, 154 (1963). 50. J. Laufer, N A C A TR No. 1053 (1951). 51. J. Laufer, N A C A TR No. 1175 (1954). 52. P. S. Klebanoff, N A C A TN No. 3187 (1954). 53. A. S . Grove, F. H. Shair, E. E. Petersen, and A. Acrivos, J. Fluid Mech. 19, 60 (1964). 54. S . Ito and S. Urushiyama, Kagaku Kogaku 32, 267 (1968). 55. W. S. Brown, C. C. Pitts, and G. Leppert, J. Heat Trans. 84, 133 (1962). 56. M. Sibulkin, J. Aeron. Sci. 19, 570 (1952). 57. M. J. Lighthill, Proc. Roy. SOC.(London), Ser. A 224, 1 (1954). 58. E. Hori, J. Japan SOC.Aerot. Space Sci. 11, 229 (1963). 59. P. S. Klebanoff, N A C A Rep. No. 1247 (1955).

Heat Transfer in Rarefied Gases

.

GEORGE S SPRINGER Department of Mechanical Engineering. University of Michigan. Ann Arbor. Michigan I . Introduction . . . . . . . . . . . . . . . . . . . . . . 11. Accommodation Coefficients. . . . . . . . . . . . . . . . 111. Gas at Rest . . . . . . . . . . . . . . . . . . . . . . . A Free Molecule Conditions . . . . . . . . . . . . . . . B. Temperature Jump Approximation . . . . . . . . . . . C Transition Regime . . . . . . . . . . . . . . . . . . D Discussions . . . . . . . . . . . . . . . . . . . . . IV Internal Flows . . . . . . . . . . . . . . . . . . . . . A Free Molecule Flow . . . . . . . . . . . . . . . . . . B Temperature Jump (Slip) Regime . . . . . . . . . . . . . V . External Flows . . . . . . . . . . . . . . . . . . . . . A Free Molecule Flow . . . . . . . . . . . . . . . . . . B Temperature Jump (Slip) Regime . . . . . . . . . . . . C Transition Regime (M < 1) . . . . . . . . . . . . . . D. TransitionRegime(M > 1) . . . . . . . . . . . . . . VI Concluding Remarks . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . .

163 165 169 169 170 170 181 183 184 184 187 187 192 193 194 209 209 210

.

I Introduction

Convective heat transfer to bodies submerged in a rarefied gas has been the subject of many investigations since the studies of Maxwell (I). Most of the early investigators were concerned with heat transfer through gases at rest or flowing at a very low speed . During the last few years interest in these problems has arisen once again because many of these problems are simple enough for analytical solutions. In addition to these studies. attention has also been given to heat transfer in high-speed rarefied gas flows. as in the flight of bodies at high altitudes . T h e purpose of this paper is to survey the results of recent analytical and experimental investigations of heat transfer in rarefied gases. 163

164

GEORGE S. SPRINGER

The term “rarefied” means that the molecular mean free path A is not small compared to a characteristic dimension 1. The parameter which describes the degree of rarefaction is the Knudsen number (2)

The Knudsen number is related to the Mach number and the Reynolds number by Eq. (I). In defining the Knudsen number, it is important to select the appropriate characteristic mean free path and length. The choice for h and 1 is not always obvious and some of the problems arising in the selection of these parameters will be discussed later. When the Knudsen number is very small, then, in the vicinity of the body, the number of collisions between the molecules is large compared to the number of collisions between the molecules and the body. In this case the usual continuum concepts are applicable and the Navier-Stokes equations and the Fourier heat conduction law are valid. In continuum flows the flow is characterized by the Reynolds number and the Mach number only; the Knudsen number will not enter the problem explicitly since it has been already taken to be very small (3). When the Knudsen number becomes sufficiently large, then the continuum concepts must be modified for calculating the heat transfer. At very high Knudsen numbers (i.e., at the other end of the rarefaction scale from the continuum) where the number of collisions between the molecules and the wall is much larger than the number of collisions between the molecules, the flow is termed “free molecule.” Between the continuum and free molecule regimes there is a wide range of rarefied flow situations, and it is convenient to divide rarefied gas dynamics problems into the following regimes (2); free molecule (Kn > lo), transition (10 > Kn > O.l), temperature jump (slip) (0.1 > Kn > 0.01), and continuum (Kn < 0.01). The indicated Knudsen number ranges are approximate only, and one of the problems confronting us is determining the Knudsen numbers which limit the various regimes. We also wish to identify the different types of heat transfer phenomena and theoretical approaches in each of these regimes. The problems of heat transfer through rarefied gases could be discussed according to the density regimes just outlined. It is more convenient, however, to classify the problems according to whether the flow is related to “internal” geometries, such as pipes and ducts, or whether the flow is external to the body as in the flight of bodies in the atmosphere. I n the first group, the flow is generally at low speed (M < 1) and the problems are often amenable to theoretical analysis. For problems in the

HEATTRANSFER IN RAREFIED GASES

165

second group the flow is generally at high speed (M > 1) and, with the exception of free molecule or near continuum flows, most of the results are empirical. T h e following survey is divided into groups based on the geometry of the problem since such an approach was found to be helpful in comparing the various analytical and experimental results. In the figures presented, the parameter indicating the degree of rarefaction (Reynolds number or inverse Knudsen number) is always plotted along the horizontal axis so that free molecule conditions lie to the left and continuum to the right hand side of the graphs. It is noted that heat transfer through electrically neutral, nonreacting gases only is included in the present survey. The effects of ionization, dissociation, etc., which may arise in high-speed high temperature gas flows are not considered.

LI. Accommodation Coefficients I n order to determine the heat transfer to a body in a rarefied gas, it is necessary to know the flux of energy and momentum carried by the molecules impinging on and reemitted from the surface. Although considerable efforts have been made to describe the interaction between gas molecules and a surface, the detailed mechanism of the gas-surface interaction is not yet fully understood. In spite of the lack of information on the exact nature of the gas-surface interaction, it is possible to perform heat transfer calculations by the introduction of average empirical parameters called accommodation coefficients. The thermal accommodation coefficient is defined by (2) a

(Ein - Ere)/(Ein - E W )

(2)

E , , and E,, are the incident, and reflected energy fluxes from the surface, and E, is the energy flux the molecular stream leaving the surface would have if this stream were to carry the same mean energy per molecule as a stream issuing from a gas in equilibrium at the surface temperature T, . The thermal accommodation coefficient may vary between 1 (complete accommodation, diffuse reemission) and 0 (specular reemission). In the definition stated by Eq. (2) no distinction has been made between the accommodations of the energies associated with the different molecular degrees of freedom. Although it is possible to introduce separate accommodation coefficients for the translational, rotational, and vibrational energy components of the gas molecules, the limited experimental data presently available do not warrant such a distinction.

166

GEORGE S. SPRINGER

Thermal accommodation coefficients have been measured experimentally over a wide range of conditions and some typical values are presented in Table I. Both from the data and from theoretical analyses of gas-surface interaction it is evident that the value of 01 strongly depends on the composition and temperature of the gas and the surface, and on the composition of the adsorped gas layer on the surface. I n experiments it is difficult to ensure that the surface is free of contaminants, and, therefore, reported values of a may range from 0.01 to nearly 1.0, depending on the condition of the surface. T h e surfaces most commonly used in engineering practice are not clean but contain contaminant gas layers. For such surfaces the accommodation coefficient (for air) is expected to be between 0.8 and 0.98 (see Table I). It is noted, however, that the condition of such surfaces (and consequently a) may change when the surface is exposed to vacuum, and, therefore, the thermal accommodation coefficient values must be selected with caution. The earliest definition of the momentum accommodation coefficient was introduced by Maxwell (4). This definition assumes that a fraction F of the molecules impinging on the surface emerges with a Maxwellian velocity distribution, with a temperature possibly different from that of the surface, while a fraction of the molecules (1 - F) is “specularly” reflected. During specular reflection, there is a reversal in normal velocity components while the tangential velocity components remain unchanged. TABLE I

SOME TYPICALTHERMAL ACCOMMODATION COBPFICIENTS~ Gas

Surface

Air

Bronze Cast Iron Aluminum W Pt Glass

Indeterminate Indeterminate Indeterminate

Pt. Bright Pt, Black Pt

Indeterminate Indeterminate Saturated

Pt

Saturated

N,

OX

co,

w

Surface condition (absorbed gas)

Temp (“C)

0.88-0.95 0.87496 0.87-0.97

Indeterminate Indeterminate Indeterminate

cox

a

32 -170

0.624 0.5 0.38

0.81 0.93 30

0.74

32 30

0.990 0.76

Table continued

HEATTRANSFER IN RAREFIED GASES

167

TABLE I (continued) Gas

H,

He

Surface Pt, bright Pt, black Pt Glass Glass

Indeterminate Indeterminate Saturated Indeterminate Indeterminate

W, flashed W, not flashed W W W W W W W

Indeterminate Indeterminate Clean Clean Clean K on Ha H,on K 0 on K K on 0 Saturated Clean Clean Clean Indeterminate

W W

Clean Clean Clean

Pt K Na Glass Glass Ne

w

W W Pt K Na Glass

A

Xe

He

Na Saturated Clean Clean Indeterminate

W W W

Clean Clean Clean Saturated Clean Clean

W W W W W

Clean Clean Clean

Pt K Na

Kr

Surface condition (absorbed gas)

w

Clean Clean Clean

Temp ("C) -

30 25 -170 20 20 30 -30 -190 25 25 25 25 30 25 25 29 25 30 -30 -196 22 -194 30 25 25 25 30 -30 -196 30 25 25 30 -30 -196 30 -30 -183

01

0.32 0.74 0.220 0.29 1.O 0.17 0.53 0.0169 0.0153 0.0151 0.106 0.096 0.22 0.12 0.238 0.0826 0.0895 0.31 0.35 0.0412 0.0395 0.0495 0.17

0.32 0.57 0.1987 0.19 0.7 0.272 0.294 0.549 0.89 0.444 0.459 0.462 0.498 0.926 0.773 0.804 0.942

F. C. Hurlbut, Particle Surface Interactions in Hyper-velocity Flight, an Annotated Bibliography Memo RM 4885 PR, Rand Corp., Santa Monica, California, 1967.

GEORGE S. SPRINGER

168

Analogously to Eq. (2) Schaaf and Chambre (2) introduced the following tangential and normal momentum accommodation coefficients: 0

= (Tin - Tre)/(Tin - T,)

and

0’

= (pin

- pre)/(pin

- pw)

(3)

and p are tangential and normal components of the momentum. T h e subscripts in and re refer to the incident and reflected momentum fluxes. T h e subscript w denotes the momentum flux which the reemitted molecules would have if they were reemitted with a Maxwellian distribution of velocities corresponding to the surface temperature T, . For a coordinate system attached to the surface T, = 0. For completely diffuse reflection u = 0’ = 1, while for completely specular reflection u = u’ = 0. Very little data are available for momentum accommodation coefficients, and some of the earlier data are listed in Table 11. These, and also the newer data obtained by molecular beam experiments, T

TABLE I1 SOMETYPICAL TANGENTIAL ACCOMMODATION COEFFICIENTS~ Gas Air Air Air Air Air NZ COZ

coz COZ

H Z

He

Surface Machined brass Old shellac Fresh shellac Oil Glass Glass Machined brass Old shellac Oil Oil Oil

U

1 .oo 1 .oo 0.79 0.9 0.9 0.95 1 .oo 1 .oo 0.92 0.93 0.87

a S. A. Schaaf and P. L. Chambre, Flow of rarefied gases, in “Fundamentals of Gasdynamics” (H. W. Emmons, ed.), Vol. 111. Princeton Univ. Press, Princeton, New Jersey, 1958.

indicate that for engineering surfaces cr is in the range 0.8 to 1. T h e available information on a and u thus suggests that in many practical problems the molecules are reemitted diffusely from the surface. Until further evidence is generated, of may also be assumed to be near unity. T h e subject of accommodation coefficients is a large one and, consequently, could be discussed here only briefly. For further information on the subject the reader is referred to the excellent reviews by Hartnett (9,Wachman (6),Hurlbut (7), Thomas (8),and Devienne (9).

HEATTRANSFER IN RAREFIED GASES

I69

III. Gas at Rest We shall consider a stationary gas (M = 0) contained between two flat plates, two concentric cylinders, or two concentric spheres. T h e temperatures of the two surfaces are denoted by T, and T , ( T , > T,) and the thermal accommodation coefficients by a, and 0 1 , . In the case of cylinders and spheres TI and T , are the temperatures of the surfaces at radii R, and R , , respectively (R,> R,). Unless stated otherwise, the discussion and formulas are restricted to linearized problems, i.e., to problems in which the temperature difference between the surfaces is - 11 1. I n this case the mean free path may be small [(T1/T2) evaluated at either surface temperature (A, = A2 = A).

<

A. FREEMOLECULE CONDITIONS When h is large compared to the characteristic length, the heat conducted per unit time from a unit area at R, may be approximated well by Knudsen’s formula (ZO)

where R* = R,/R2and b is a constant equal to 0 for parallel plates, 1 for concentric cylinders, and 2 for concentric spheres; P is the pressure of a Maxwellian gas at the same density as the gas between the plates at a temperature T . For small temperature differences T may be replaced by ( T , 4-T,)/2. A? is the molecular weight of the gas, R is the molar gas constant, and C, is the molar heat capacity at constant volume. Equation (4) is based on the assumption of diffuse reflection at the surfaces (u = u’ = I). For parallel plates Sparrow and Kinney (ZI) derived an expression for u and u‘ different from unity. In the case of the parallel plate geometry, the appropriate characteristic length is the separation between the plates L. For concentric cylinders and spheres the choice of the characteristic length is not so obvious. Using experimental data, Wachman (6) argues that the important characteristic length is the inner radius R, . This conclusion is supported by the analytical results of Springer and Ratonyi (12),Springer and Wan (Z3),and Willis and Su (24). T h e characteristic length is of interest because it is needed in defining the Knudsen numbers for which the free molecule expression is applicable. The Knudsen number for which Eq. (4) may be used are discussed in Section 111, D (Fig. 8).

170

GEORGE S. SPRINGER

B. TEMPERATURE JUMP APPROXIMATION When h is small compared to the characteristic length, Smoluchowski’s temperature jump boundary condition (10) TIC - T w = g (aT/?Y)

(5)

can be applied to Fourier’s heat conduction equation. T, is the wall temperature, and Tk is the temperature the gas would have if the temperature gradient normal to the wall a T/aywould continue right up to the wall. T h e temperature jump distance, g, may be expressed as (10) 2-a

2y

A

g = a y f l P r

where Pr is the Prandtl number and y is the ratio of specific heats. Rearranging the results presented in Kennard (10) one obtains the following expression:

I n Eq. (7), G = L for parallel plates, G = R, In R,/R, for concentric cylinders and G = R,(l - R*) for concentric spheres; B = 1.0 for a monatomic gas and B = 45/38 for a diatomic gas. Similarly as in the case of free molecule heat conduction, for cylindrical and spherical problems the suitable characteristic length is R, (see SectionIII, D and Fig. 8). Equation (7) is based on the assumption that the molecular flux arriving at the surface can be characterized by the distribution function that exists in the gas away from the surface (10). Collisions between the incident and reflected molecules disturb this distribution function. Corrections for this effect have been made (15-21). The more refined analyses result in equations having the same form as Eq. (7) but the expression in the square bracket becomes more complex.

REGIME C. TRANSITION When h is of the order of the characteristic length the simple formulas given cannot be applied, and the heat flux is to be calculated from (22) m Qi

= ( 4 9 JJJ(Cf --m

- %)(C

-~

) z . f ~ ~ e

(8)

HEATTRANSFER IN RAREFIEDGASES

171

where c is the absolute velocity of the molecules of mass m,and u is the macroscopic gas velocity; dVc is an elemental volume in velocity space. For a stationary gas (M = 0), u = 0. For a gas in which the range of intermolecular forces is small compared to the molecular spacing, which in turn is small compared to A, the molecular distribution function f(ci, x i , t ) can be obtained from a solution of the Maxwell-Boltzmann equation (22). I n a Cartesian coordinate system, and in the absence of external forces this equation is (for other orthogonal coordinate systems see (23))

c and 5 and c' and 5' are the velocities of two colliding molecules before and after the collision, respectively; dPc is the differential collision cross section and depends on the nature of the intermolecular force field. Because of the complexity of Eq. (9), solutions for it have been assuming generally that the gas is composed of monatomic molecules either obeying Maxwell's inverse fifth power force law of repulsion (Maxwellian molecules) or behaving as elastic spheres (hard sphere molecules). Equation (9) can be simplified considerably by replacing the collision integral on the right hand side by a relaxation term suggested by Bhatnagar et al. (24) and by Welander (15) awt

+ ci a j p x i

=

.(f

-j e )

(10)

where v is a velocity dependent collision frequency and f e is the equilibrium value of the distribution function. Equation (10) (BKG model equation) has been used in many recent investigations of rarefied gas dynamics problems.

1. Parallel Plates Approximate solutions to Eqs. (9) and (10) have been obtained for the case of heat transfer through a rarefied monatomic gas contained between two parallel plates at unequal temperatures. This problem is one of the simplest for which solutions can be found, and this may explain the number of solutions available. Unfortunately, because of the simplicity of the geometry, this problem does not provide a good test for the analytical techniques employed in the solutions. Analytic results for the parallel plate geometry were reported by Wang-Chang and Uhlenbeck (25), Gross and Ziering (26,27), Bassanini et al. (28,29), Lees and Liu (30,32), Lavin and Haviland (32), Willis (33),

172

GEORGE S. SPRINGER

and Cercignani and Tironi (34). Wang-Chang and Uhlenbeck, Gross and Ziering, and Bassanini ei al. treat the linearized problem. The analyses of Lees and Liu, Lavin and Haviland, Willis, and Cercignani and Tironi also apply to large temperature differences. First, results for the linearized problem are presented. Wang-Chang and Uhlenbeck (25) utilized the polynomial expansion method for solving the linearized Boltzmann equation. In this method the distribution function is assumed to be a local Maxwellian function multiplied by a series of polynomials in velocity space with space dependent coefficients. Wang-Chang and Uhlenheck expanded the distribution function in full range polynomials and obtained results for successive approximations for a gas composed of Maxwellian molecules. I n the Gross and Ziering analyses (26,27),the distribution function is assumed to be a Maxwellian at the plate centerplane conditions multiplied by a series of polynomials in velocity space with space dependent coefficients. I n the “eight-moment method” there are eight such space dependent coefficients which are determined by taking four half range moments of the linearized Boltzmann equation. I n the “four-moment method” there are four coefficients which are determined by taking four full range moments of the linearized Boltzmann equation. Using the “eight-moment method,” Gross and Ziering obtained solutions both for hard sphere (26) and for Maxwellian molecules ( 2 7 ) . For the “fourmoment method” they presented results for hard sphere molecules only (26). I n the analyses of Gross and Ziering the boundary condition was that a fraction F of the molecules impinging on the plates emerge with a Maxwellian distribution characteristic of the plate temperature while a fraction of the molecules (1 - F) is specularly reflected. In comparing Gross and Ziering’s results to the data, the same numerical value was used for F as for the thermal accommodation coefficient a (Figs. 1,2). Bassanini et al. (28,29) transformed the linearized BGK model equation into a couple of integral equations for density and temperature, and solved these both by numerical and by variational methods for a = 1 (28).The results of the two methods agree within 0.5%. Using the variational method, Bassanini et al. also obtained solutions for incomplete thermal accommodation (29). None of the foregoing analyses yield closed form solutions for the heat transfer. Lees and Liu (30,31) proposed to evaluate the heat conduction between two plates by dividing the gas molecules (in velocity space) into two groups. Each of these groups is characterized by a Maxwellian distribution function containing four unknown spatial functions. These

HEATTRANSFER IN RAREFIED GASES

0

2

6 8 10 12 14 16 INVERSE KNUDSEN NUMBER L / A 4

18

173

20

FIG. 1. Conductive heat transfer through a monatomic gas at rest contained between L Y ~= 1; TJTp -I 1.

<

two parallel plates. Complete thermal accommodation: a, =

Key: 0 Hard sphere, four-moment (Gross and Ziering); @ Hard sphere, eight-moment (Gross and Ziering); @ Maxwell molecule, eight-moment (Gross and Ziering); @ Maxwell molecule, four-moment (Lees); @ Maxwell molecule, second approximation (WangChang and Uhlenbeck); @ Maxwell molecule, fourth approximation (Wang-Chang and Uhlenbeck); 0Hard sphere, four-moment (nonlinear, TITz= 1.2) (Lavin and Haviland); @ BGK model, numerical integration, variational method (Bassanini, Cercignani, and Pagani); @ Interpolation formula (Sherman); @ Temperature jump approximation.

unknowns are determined by sustituting the chosen distribution functions into the Maxwell integral equations of transfer. By restricting the analysis to the case of small temperature differences, Liu and Lees (31) obtained a closed form solution for the heat conduction between parallel plates for Maxwellian molecules and for complete thermal accommodation at the surfaces. Extension of the analysis to arbitrary values of the thermal accommodation coefficients, similarly as done by Hurlbut for concentric cylinders (35),yields (for a monatomic gas) Q/QFM

=

[I

+ (4/15)(L/X) m i a d ( m ~+

a2

-O

L~~I-~

(1 1)

GEORGE S. SPRINGER

174

Note that this expression is identical to the one given by the temperature jump approximation (Eq. 7) for all values of A/L. This result is one indication that the parallel plate geometry does not provide a strong test for the analyses. 1.o

0.9

e o

0.8

0.7 0.6

w in lL f 0.5

+LL

+ 0.4

9 I

0.3 0.2 0.1

0

0

2

4

6

8

10

12

14

46

18

INVERSE KNUDSEN NUMBER L/A

FIG. 2. Conductive heat transfer through a monatomic gas at rest contained between

<

01 = a1 = ap ; TJTP - 1 1. Key: 0 Hard sphere, four-moment (Gross and Ziering); @ Hard sphere, eight-moment (Gross and Ziering); @ Maxwell molecule, eight-moment (Gross and Ziering); @ Maxwell molecule, second approximation (Wang-Chang and Uhlenbeck); @ Maxwell molecule, four-moment (Lees); @ BGK model, variational method (Bassanini, Cercignani, and Pagani); @) Interpolation formula (Sherman); @ Temperature jump approximation. Data points: o Teagan and Springer; 0 Bienkowski.

two parallel plates.

Lees’ moment method has been applied by Lavin and Haviland (32) to the problem of heat transfer between parallel plates at large temperature differences for 01 = 1. Their analyses differed from Lees and Liu’s inasmuch as they presented a four-moment solution for hard sphere molecules and a six-moment solution for Maxwellian molecules. A comparison between the various results for the linearized problem (monatomic gases only) is presented in Fig. 1 (a = 1) and Fig. 2 (al = a2 = a # 1). In Fig. 2 the experimental data of Teagan and Springer (36) and Bienkowski (37) are shown also. For Knudsen numbers corresponding to transition and temperature jump conditions, the

HEATTRANSFER IN RAREFIED GASES

175

measured heat conduction values for argon agree with the linearized four-moment results of Gross, Ziering, and Lees, with the results of Bassanini, Cercignani, and Pagani, and with the second approximation of Wang-Chang and Uhlenbeck. Closest agreement is between experiment and the analyses of Lees, and Gross and Ziering. Also, for helium the data agree well with the results of Lees’ four-moment method (Eq. 11). Only the second approximation of Wang-Chang and Uhlenbeck’s analysis was compared to the data because it is extremely difficult to compute numerical results for the higher approximations when 01 # 1. A comparison between the second and fourth approximations computed for 01 = 1 (Fig. 1) indicates that the fourth approximation corrects the results in the proper direction. On the basis of this, one would expect the results of the fourth approximation to agree even better with the data in Fig. 2 than those of the second approximation. Note that the variational solution of the BGK model equation underestimates the heat transfer rate by a few percent. The lack of agreement between the data and the eight-moment method is somewhat surprising and thus far a satisfactory explanation has not been given for this. Solutions for a problem with a high temperature ratio ( T l / T z= 4) are shown in Fig. 3. Willis (33) and Cercignani and Tironi (34) found solutions to the BGK model equation by a numerical integration method. Using Monte Carlo techniques, Haviland and Levin (38) and Perlmutter (39)obtained heat transfer values at selected Knudsen numbers for hard sphere and Maxwellian molecules. All these results are for 01 = 1. A comparison between the results must be made with caution since at large temperature differences, certain parameters such as the mean free path cannot be evaluated at arbitrary positions, Again it appears that taking additional moments of the Maxwell-Boltzmann equation beyond the required minimum number does not improve the results. The linearized four-moment results of Lees and Liu agree best with the Monte Carlo solutions, at least for the 4 : 1 temperature ratio. Although temperature ratios of 4 : 1 or higher could be achieved in practice only with great difficulty, the results of such experiments would be helpful in evaluating the limits of linearized theories. Willis (33) also obtained solutions for the nonlinear problem using a Knudsen iteration technique. In this technique the distribution function is expanded in a series around the free molecule distribution function. Solutions found by using the first two terms of this series are referred to as the first iterate solutions. For high Knudsen numbers the first iterate and numerical integration results agree closely up to T,/T - 100 2.and, therefore, the former results are not shown separately in Flg. 3.

GEORGES. SPRINGER

176 1 .O

0.9

0.0 0.7

B

?

B

0.6

ul LL

2 + a IL tW

I

0.5

0.4

0.3

0.2

0.1

0 INVERSE KNUDSEN NUMBER, L / X

FIG. 3. Conductive heat transfer through a monatomic gas at rest contained between two parallel plates at large temperature differences: TJT2= 4. Complete thermal accommodation q = a8 = 1. Key: 0 Hard sphere, four-moment (Levin and Haviland); @ Maxwell molecule, six-moment (Levin and Haviland); @ BGK, numerical integration (Cercignani and Tironi); @ BGK, numerical integration (Willis); @ Maxwell molecule, four-moment (linearized) (Lees and Liu); @ Interpolation formula (Sherman). A Hard sphere, Monte Carlo (Perlmutter); ( 0 )Maxwell molecules, Monte Carlo (Haviland and Levin); ( 0 ) Hard sphere, Monte Carlo (Haviland and Levin).

Less information is available on the heat transfer through diatomic gases. Using a variational method, Hsu and Morse (40) obtained solutions to the BGK model equation for small temperature differences. Their results are compared to Springer and Teagan’s data in Fig. 4, and the agreement between theory and data is good. However, the data also agree well with the results of Bassanini et al. (29) for a monatomic gas. The data also compare favorably with the result of Lees’ moment

HEATTRANSFER IN RAREFIED GAS=

177

4 .O 0.9

0.8

0.7

z

? 0 W [L

z a

0.6

0.5

cr

+ l-

0.4

a

w I

0.3 0.2

O.!

C

1 2

I

1

6

I

I

1

I

, I

1

1

I

1

1

I

l

l

14 18 22 26 30 34 10 INVERSE KNUDSEN NUMBER. L / X

I

I

38

1

42

FIG.4. Conductive heat transfer through a diatomic gas at rest contained between two parallel plates. 01 = 0 1 ~= ; T,/T, - 1 1. Key: @ Maxwell molecules, fourmoment (“modified” Lees); @ BGK, variational method (Hsu and Morse); @ BGK, variational method (Bassanini, Cercignani, and Pagani); @ Interpolation I formula (Sherman). Data points, (0) N , (Teagan and Springer).

<

method, when the constant, 1514, in Eq. (I 1) is modified in such a way that Eq. (11) yields the proper free molecule limit for a diatomic gas. In all the foregoing analyses radiation effects were neglected. Greif and Willis (41) considered heat transfer between parallel plates including the effects of both rarefaction and radiation. The analysis is based on the approximation of a gray gas with radiative emission corresponding to local thermodynamic equilibrium. The conduction was treated by the BGK model equation and the Lees moment method, and the radiative transfer equations were formulated by the Milne-Eddington moment method. The solutions found for small temperature differences indicate that the heat transfer thus calculated agrees generally within about 5 %

GEORGES. SPRINGER

178

with the results computed by neglecting the interaction between the conductive and radiative heat fluxes.'

2. Concentric Cylinders T h e solutions available for the concentric cylinder geometry are fewer in number and are more limited by assumptions than those for the parallel plate case. Lees and Liu (42) applied Lees' four-moment method (30)to the heat conduction from a fine wire suspended coaxially in a cylinder ( R , R,) for complete thermal accommodation at the surfaces. Hurlbut's (35) extension of this analysis in which he includes arbitrary accommodation coefficient at the wire surface (but not at R,) gives (for a monatomic gas)

<

Q/QFM

=

C1 f (4/15)(Ri/h) 011 In R 2 / W 1

(12)

This method yields the proper free molecule limit. However, for large Knudsen numbers it does not yield accurate results for perturbations from free molecular quantities (43). Willis (43) calculated the heat transfer at near free molecule conditions by applying the Knudsen iteration technique to the BGK model equation. T h e first iterate solution gives (for a monatomic gas) Q/QFM =

1 - (7.rralR1/12X)[-ln(sin d)

+ +(d/sin

-( 4 8 )

+ d cot A]

(13)

where s i n d = R*. Recently Su and Willis (14) transformed the linearized BGK model equation into two coupled linear integral equations for the temperature and density and solved these equations by an asymptotic analysis for the case when the mean free path is much larger than the inner radius but is of the same order as the outer radius. I n the analysis they also assumed that a1 = 01, = 1. Comparison between the results of Lees and Liu, Willis, and Su (Fig. 5 ) shows that the heat transfer rates given by the three different analyses are within 1 yo when R,/h > 20. Bassanini et al. (29) also applied the variational method to the concentric cylinder geometry when the temperature difference is small The effects of rarefaction, optical depth, and ratio of conduction to radiation on the heat transfer through a stagnant diatomic gas between parallel plates have been investigated recently by Phillips et al. (41a). This analysis is based on the assumptions of (1) three-fluid formulation (translator-rotator-photon at low temperatures and translator-rotator and vibrator-photon at high temperatures), (2) continuous distribution functions, (3) molecule-molecule elastic collisions, (4) molecule-photon inelastic collisions, ( 5 ) gray gas, and (6) local equilibrium.

HEAT TRANSFER IN RAREFIED GASES

179

INVERSE KNUDSEN NUMBER, R,/

FIG. 5. Conductive heat transfer through a monatomic gas at rest contained between two concentric cylinders. Comparison between different analytical results at near free molecule conditions. Complete thermal accommodation. TJT,- 1 1. Key: (-) BGK, asymptotic solution (Su and Willis); (-------) BGK, first iteration (WiIIis); (- - -) Maxwell molecules, four-moment (Lees and Liu).

<

between the wire and the outer cylinder. Their results and the Lees-Liu solution (Eq. 12) are compared to the data of Dybbs and Springer (44) in Fig. 6. T h e data agree with the four-moment results within about 2.5 yo and with the results of Bassanini et al. within about 5 % . However, in the case of concentric cylinders the BGK model equation together with the variational method solution overestimates the heat flux rather than underestimating it as in the parallel plate case. T h e heat transfer through a polyatomic gas contained between two concentric cylinders was analyzed by Cipolla and Morse (45). Assuming small temperature differences between the cylinders they obtained a solution to the BGK model equation by using the Knudsen iteration technique similar to the one suggested by Willis for monatomic gases (43). T h e first iterate of this solution is long, but it is in closed form for the heat transfer from the inner cylinder. This solution is expected to be reasonable for Knudsen numbers larger than unity (45). Experimental verification of this analysis is not yet available.

GEORGE S. SPRINGER

180 1...0 ,

az

0.9

-

0.8

-

0.7

-

0.6 -

i3 Lf

k?

8

0.5 -

II-

$

0.4 -

0.3 0.2

0.1

0

-

a,.

-

-

2

4 6 8 10 INVERSE KNUDSEN NUMBER. R,/A

42

1

FIG. 6. Conductive heat transfer through a monatomic gas at rest contained between two concentric cylinders. Incomplete thermal accommodation at the surface of the inner cylinder; R,/R1 w 330, TJT, - 1 1. Key: - BGK, variational method (Bassanini, Cercignani, and Pagani); (- - -) Maxwell molecule, four-moment (Lees and Liu), and Interpolationformula(Sherman).Data fromDybbs andspringer: ( 0 )Helium(or, = 0.2831); ( 0 ) Neon (a1 = 0.4670); ( A ) Argon (or1 = 0.7328).

<

3. Concentric Spheres Lees (23) applied the four-moment solution to heat transfer between concentric spheres assuming complete thermal accommodation at the surfaces. Springer and Wan (13) included in the analysis arbitrary values of a at R, resulting in the expression (for a monatomic gas) Q/QFM

= (1

+ (4/15)(Ri/X) 4

1 - (RJQ1-l

(14)

Assuming complete thermal accommodation at the surfaces, Brock (46)

HEATTRANSFER I N RAREFIED GASES

181

calculated the heat transfer using the BGK model equation and the Knudsen iteration technique. For arbitrary thermal accommodation coefficients at the surface of the inner cylinder, the first iterate solution yields (47) (for a monatomic gas) Q/QFM

=1

-(7~R,~1/144h)

x (1

- R*

+ [ ( 1 / ~ * 3 ) ( 1 / ~ * ) 1 [ 1 - (1 -

~*2)1/212)

(15)

Equation (15) is expected to be valid in the same Knudsen number range as the similar results (Eq. 13) developed for concentric cylinders, i.e., when AIRl > 20. Cercignani and Pagani (48) analyzed the problem of heat transfer from a sphere submersed in a gas extending to infinity. Using the BGK model equation they obtained solutions by a variational method for a1 = 1 at R, . Again, as in the case of cylindrical geometry, this method gives higher heat flux values than the Lees moment method (Fig. 7). I n Fig. 7 a comparison is made between the results of Lees’ moment method and Petersen’s data (49),and the agreement between theory and data is good. Heat transfer from spheres surrounded by hydrogen-nitrogen and helium-nitrogen mixtures was measured by Mikami et al. (50). T h e experimental data indicate that the heat transfer through the gas can be calculated with reasonable accuracy by the Lees moment method (Eq. 14) when the thermal accommodation coefficient is replaced by =

1 i

XjCd3M:’2/F

where Xiis the mole fraction and component gas.

XjMy

(16)

Miis the molecular weight of t h e j t h

D. DISCUSSIONS T h e survey presented in the foregoing shows that in the transition regime only the results of Lees’ four-moment method and Willis’ Knudsen iteration technique can be expressed in closed form. T h e moment method is applicable for all Knudsen numbers while the Knudsen iteration technique may be used only at high Knudsen numbers. Although these results are approximate only, on the basis of available experimental evidence they appear to be adequate for describing the heat transfer between parallel plates, concentric cylinders and concentric spheres. T h e taking of additional moments of the Maxwell-Boltzmann integral equation beyond the minimum number required for a solution

GEORGE S. SPRINGER

182

INVERSE KNUDSEN NUMBER R,/ A

FIG.7. Conductive heat transfer through a monatomic gas at rest contained between two concentric spheres. Incomplete thermal accommodation at the surface of the inner sphere; T,/T,- 1 1 . Key: (-) Maxwell molecule, four-moment (Lees), and Interpolation formula (Sherman); (- - -) BGK, variational method (Cercignani and Pagani). Data points from Petersen, (0) (He).

<

does not seem to improve the accuracy of the results. Analyses requiring numerical computations, although of interest from a theoretical point of view, do not offer significant improvements in accuracies over the simpler closed form solutions. In addition to describing the heat transfer, Lees' four-moment method also approximates well the density distribution between two parallel plates (36) and the equilibrium temperature of a free molecule probe placed between two concentric cylinders (51). These results lend further confidence to this analysis. It is interesting to note that the simple interpolation formula suggested by Sherman (52) QIQFM = 11 +&?FM/~c)]" (17) is identical to the expressions given by Lees' four-moment method for

183

HEATTRANSFER IN RAREFIEDGASES

all three geometries discussed (Eqs. (1 I), (12), (14)). In Eq. (17), Qc is 0). the heat transfer in the continuum limit (Kn The approximate Knudsen number ranges can now be established in which the free molecule theory, temperature jump approximation, and the Fourier heat conduction equation give reasonable results. The Knudsen numbers limiting the free molecule, temperature jump, and continuum regimes are arbitrarily defined as those where the results of Lees' four-moment method agree within 1 % of the free molecule, temperature jump, and continuum (Fourier law) solutions (12). T h e results of these calculations are shown in Fig. 8. --j.

TRANSITION

0.04

1.0

0.1

10

400

4 000

INVERSE KNUDSEN NUMBER, l / K n

Classification of the density regimes. For flat plates concentric cylinders and spheres = and 0 1 ~= 1. (Y

(Y

= a1 = cx2.

For

( Y ~

IV. Internal Flows The mass flow rates of rarefied gases through tubes, nozzles, and parallel plates have been investigated widely for the special case of isothermal flow. The energy transport has received much less attention,

184

GEORGE S. SPRINGER

and the few available results are restricted (a) to gases with low mean velocities ( M Q 1) and (b) to density levels corresponding to either free molecule or to temperature jump conditions.

A. FREEMOLECULEFLOW Energy transport through tubes and parallel plate channels was analyzed in a series of papers by Sparrow and his co-workers (53-56). These analyses utilize the similarities between the free molecule energy transport and thermal radiation. Neglecting radiation effects, Sparrow et aZ(53) obtained an integral equation for the adiabatic wall temperature distribution along a circular tube and presented numerical solutions for selected conditions. Sparrow and Jonsson (54-55) studied the combined convective and radiative energy transfer through cylindrical tubes and between parallel plates. Assuming the gas to be transparent they derived a general expression for the heat transfer, which must be solved numerically for any given set of boundary conditions. T h e authors obtained numerical solutions for the special cases of uniform wall temperature, uniform wall heat flux, and adiabatic wall. Unfortunately, due to the large number of parameters describing the problem, these solutions cannot be presented in a concise form. T h e important conclusion indicated by these sample solutions is that at room temperatures or above, the molecular energy transport is negligible compared to the radiative energy transport, i.e., the results for the combined convectiveradiative transport differ little from those for pure radiation, provided that the thermal accommodation coefficient is nearly equal to the radiation absorption coefficient and that both coefficients are close to unity. A similar conclusion was reached by Sparrow and Jonsson (56) for the free molecule flow of gases through tapered tubes or conical nozzles.

B. TEMPERATURE JUMP (SLIP) REGIME When the gas is only slightly rarefied, then the energy transfer can be calculated using the Navier-Stokes equation and the energy equation together with the Fourier heat conduction law. T h e continuum boundary conditions have to be modified, however, to include a velocity slip and temperature jump at the wall. When the gas is moving, Smoluchowski’s temperature jump boundary condition as given by Eqs. ( 5 ) and (6) must be modified (57.) Furthermore, as pointed out by Maslen (58) the work done by the shear forces must also be considered when there is a slip velocity at the surface. Also, consideration must be given to the change in

HEATTRANSFER IN RAREFIED GASES

185

slip velocity (thermal creep) arising due to temperature gradients along the surface (10). Since these effects are presumably unimportant in low speed flows (M Q l), they were neglected in the derivations of the following results. Laminar heat transfer in circular tubes under slip flow conditions was analyzed by Sparrow and Lin (59),Inman (60, 61) and Deissler (19). For fully developed flow, and for uniform wall heat flux the Nusselt number is (59,60)

where d is the tube diameter, K is the thermal conductivity of the gas, q is the heat flux, T, is the wall temperature and Tb the bulk temperature of the gas Tb =

I

dl2

0

27rRTu d R / r 1 2 2rRu dR 0

T h e temperature jump distance g is given by Eq. (6). T h e ratio of slip velocity to average velocity is

where F is the Maxwell reflection coefficient discussed in Section 11. Sparrow and Lin (59) and Inman (60) also indicated the changes that would have to be made in Eq. (18) to include the effects of modified temperature jump, shear work, and thermal creep at the wall. Deissler (19) modified these results by including second order terms in the expressions for the velocity slip and the temperature jump at the wall. T h e maximum difference between Deissler's results and those given is about 15 yo at Knudsen numbers (Kn = 2h/d) of 0.2 and 01 = 1. At Knudsen numbers below 0.1 the difference becomes less than 0.5 yo for all values of a. When the uniform wall temperature is specified along the tube, then the Nusselt number depends on both the temperature jump and slip coefficient in a complex manner. Sparrow and Lin (59) and Inman (61) evaluated Nu numerically for selected conditions and the results are shown in Fig. 9. T h e problem of fully developed, laminar slip flow heat transfer between

186

GEORGE S. SPRINGER

-

4.0

I

-

0.8

-

0.6 -

-

-

5 W

cn 3

z

--

0.4 0.2

-

d

-

-

1

1

1

1

1

1

1

5

I

1

1

40

1

1

1

50

1

-

1 1 .

too

FIG. 9. Fully developed laminar flow in tubes under slip flow conditions. Variation of Nusselt number with inverse Knudsen number for uniform wall temperature. Pr = 0.7, y = 1.4, Nu, = 3.657, and F = 1.0. [E. M. Sparrow and S. H. Lin, Laminar heat transfer in tubes under slip flow conditions. J. Heat Transfer 84, No. 4, 363-369 (1962).]

parallel plates was treated by Inman (6042). For the case of uniform heat flux the Nusselt numbers are (60) NU =

4

8L _ T, - Tb K Z -

-

140 17

13

1 --1+-(1_) 3 u 1 u +-35g 26 ii 78 13 2L

for (Z = 2) (21)

for (Z = 1) (22)

where L is the distance between the plates and 2 is a parameter indicating the number of heating surfaces (i.e,, for Z = 1 one of the walls is insulated). Tb is the bulk temperature of the gas, defined analogously as for tubes (Eq. (19)). T h e temperature jump distance g is again given by Eq. (6). When the plates are at specified temperatures, the Nusselt numbers are equal at both plates even if the two plates are at unequal

HEATTRANSFER IN RAREFIEDGASES

187

temperatures provided the flow is fully developed (62).For this case the Nusselt number derived by Inman (62) is Nu

= 4/u

+kL11

(24)

Inman ( 6 0 6 2 ) also investigated the heat transfer in the entrance region of tubes and channels where the flow is not yet fully developed. T h e numerical results obtained are too lengthy to be presented here, but they indicate that T , - Tb generally approaches within 5 % of its fully developed value when the dimensionless axial distance [(xl2L)/Re Pr] is less than 0.1 (for tubes 2L is replaced by d ) . This distance decreases with increasing Knudsen number and decreasing a. Experimental results have not yet been reported on heat transfer in rarefied gas flows in pipes and ducts. Furthermore, theoretical analyses are unavailable for the transition regime. Thus, one cannot determine the Knudsen number ranges where the foregoing free molecule and temperature jump results may be applied with reasonable confidence. However, a direct interpolation should not be attempted between the free molecule and temperature jump results. It is well known that the mass flux for isothermal flow through a long tube has a minimum value in the transition regime and the heat transfer to the wall may behave similarly. V. External Flows T h e problem of convective heat transfer to bodies moving through rarefied gases is very relevant to the flight of bodies in the upper atmosphere. For this reason there has been considerable interest in this problem during the last two decades and summaries, emphasizing various aspects of the many investigations, have been given previously by Oppenheim (63), Schaaf and Chambre (2), Probstein (64) and Van Dyke (65). A. FREEMOLECULE FLOW Convective heat transfer per unit time to a differential surface element dA (Fig. 10) of a convex body submerged in the steady, uniform, free molecule flow of a perfect gas can be calculated from the following expression derived in detail in ( 2 , 3 )

x [exp(-+z)

+ & V ( l + erf +)] -

exp(-Q)l

2

dA

(25)

GEORGE S. SPRINGER

188

1

I

I

I

I

I

I

-

-

i

’-

n

w

Ir_

n

-

0

z

0

I

I

I

I

I

I

I

I

I

-

%

kL O

t

38 8

3

a

0.8 0.6

FIG. 10. Modified Stanton number and modified recovery factor for plates, cylinders, and spheres in free molecule flow. Key: (1) Flat plate = ~ / 2 (2) ; Flat plate /3 = 0; (3) Cylinder /3 = n / 2 ; (4) Sphere. [S. A. Schaaf and P. L. Chambre, Flow of rarefied gases in “Fundamentals of Gasdynamics” (H. W. Emmons, ed.), Vol. 111. Princeton Univ. Press, Princeton, New Jersey, 1958.1

where 4 = S sin 8, and 0 is the local angle of attack as shown in Fig. 10. The temperature of the wall is T,. The subscript 00 refers to free stream conditions. The speed ratio, S, is defined by (2,3) S = U,/(2WTm)1’a= (y/2)’ta M,

(26)

where U , is the free stream velocity. Equation (25) could be integrated, at least in principle, for any convex body shape. However, as discussed

HEATTRANSFER IN RAREFIED GASES

189

by Hayes and Probstein ( 3 , 6 4 ) one can make the following general observations without performing the integrations. At very high Mach 1 at every point), Eq. (25) reduces to numbers ( S sin 8

>

dQFM

=

1 - apmUmS sin 0 1 2

(

-

lY+l TUJ1 2 y - 1 T, dA

a)

~

(27)

From Eq. (27) the adiabatic recovery temperature can be readily calculated as TAT, = M Y

-

l)/(Y

+ 111 s2

(28)

Thus, if the heat transfer is due to convection only, then T, is independent of both a and 8. For high values of S , the stagnation temperature of the flow is given by ( 2 , 3 ) To = T , ( I

+

* Y

Sz)

T,

( e Y

From Eqs. (28) and (29) we find that in high speed free molecule flow the recovery temperature is higher than the free stream stagnation temperature. An explanation for this interesting phenomena was first given by Stalder, Goodwin, and Craeger (66). When S s i n 8 is large compared both to unity and to ( Tw/T,)112 (highly cooled surface), Eq. (27) becomes dQFM= gap, Urn3sin 9 dA (30) Equations (27) and (30) show the Mach number independence principle (3,64),i.e., in high speed rarefied flows the heat transfer depends only on U , ,p a ; , T, , and a. Using Eq. (25) heat transfer rates were calculated for some typical body shapes by taking a and T, to be constant over the surface. T h e results are generally expressed in terms of the modified recovery factor and modified Stanton number

For a flat plate at angle of attack

(Fig. 10) (2)

+ 1 - [l + T W ( N + erf I,!J) exp($2)]-'> St' = [1/(4m1w)][exp( -Q)+ + erf +)I Y'

= (l/S2){2S2

~1/2$(N

(32a) (32b)

GEORGE S. SPRINGER

190

where I/J = S sin 8. When the front and rear surfaces are isolated, A is the surface of one side only and N = 1. For the rear surface /3 is replaced by -/3. When the front and rear surfaces are in thermal contact, then A is the total surface area and N = 0. For a circular cylinder of surface area A at angle of attack /3 (Fig. lo), Talbot gives (67)

For a sphere with surface area A , Sauer’s (68) results may be rearranged to yield (2)

+ 1)[1 + (l/S)ierf S] + (2S2 l)(erf S)/(2S2) Se[1 + (1IS) ierf S ] + (erf S)/(2Sz) St‘ = [l/(SSz)][S2 + S ierfc S + (erf S)/2] Y

, = (2S2

-

(344 (34b)

where I,, and I, are modified Bessel functions, and ierfc(S) is the integrated complementary error function (ierfc(S) = ( 2 / ~ l / x~ ) $, dx Jiexp(-m2) dw). Some of the above results are shown in graphical form in Fig. (10). As can be seen, with the exception of a flat plate at fl = 0, the heat transfer (St’) becomes independent of the Mach number at a speed ratio of about 3, illustrating the Mach number independence principle. For a flat plate at fl = 0, St’ approaches zero asymptotically. The foregoing results must be modified for a nonuniform gas in which the distribution function is different from the Maxwellian distribution because of the presence of viscous stresses and heat flow. Expressions for heat transfer to cylinders and spheres for the free molecule flow of a nonuniform gas were derived by Bell and Schaaf (69) and Touryan and Maise (70). Combined radiative and convective energy transfer were considered by Stalder and Jukoff ( 7 4 , Stalder et al. (66) and Abarbanel (72). Stalder and his co-workers developed a general method for calculating the energy transport and the surface temperature of a body, and applied this method to flat plates (71) and concentric cylinders (66).Numerical solutions for flat plates have been presented (71) which indicate the effect of solar radiation on the surface temperature of bodies in high altitude flight. Abarbanel(72) derived generalized results for the surface temperature of bodies in free molecule flow when both convective and radiative

HEATTRANSFER IN RAREFIED GASES

191

transport are significant. H e presented solutions for flat plates, spheres, cylinders, and cones when the surface is a perfect conductor and outlined the calculating procedure for adiabatic surfaces. Although the lengthy results cannot be presented here in detail, one of the conclusions shown by Abarbanel’s calculations is that for high speed flows (S2 1, S sin B > 1) the adiabatic recovery temperature of a surface element with local angle of attack 6’(Fig. 10) is

>

where K is the Boltzmann constant, rn is the molecular mass, e is the emissivity and absorptivity of the surface, and u is the Stefan-Boltzmann constant. T h e previous results apply only when the surface is everywhere convex to the flow, i.e., when the reemitted molecules cannot strike another part of the surface. Concave shapes in free molecule flow were studied by Chanine (73), Sparrow et al. (74, and Schamberg (75, 76) for the assumptions of (a) hyperthermal flow ( S 2 1) with uniform incident velocity of molecules U , , (b) constant surface temperature T , , and (c) reemitted particles are distributed angularly according to Lambert’s cosine law and have a Maxwellian distribution of velocities. Chanine (73) obtained integral equations for a flow over a generalized concave surface when the incident molecular stream is not shielded by < w , Fig. 11). H e presented closed form any part of the surface solutions for spherical and infinite cylindrical surfaces, and gave numerical results for the special case of zero angle of incidence and S2 = 20. These results show ( 5 ) (a) that for 01 = 0 and 01 = 1 the concavity, w, does not have any effect on the heat transfer, (b) that for 0 < 01 < 1 the convective heat transfer to concave bodies is greater than

>

(a

FIG. 11. Schematic of the physical system for a concave surface in free molecule flow.

192

GEORGE S. SPRINGER

to convex bodies of equal cross sectional area normal to the flow and (c) that 01 has more effect for spherical surfaces than for the infinitely long circular cylinder. Sparrow et al. (74) performed an analysis to determine local and overall heat transfer rates and adiabatic recovery temperatures for flows incident on concave cylindrical surfaces without restricting the angle of attack (Q >( w ) . For the adiabatic recovery temperature the simple result was (74) T,

=

UmSI(Cg

+ CJ

(36)

which shows that T, is the same at all locations at the surface, and is independent of the angles SZ and ,6 and the thermal accommodation coefficient a. Sparrow et al. also derived closed form expressions for the local and overall heat transfer. Sample numerical solutions indicate similar effects of w on the heat transfer as do Chahine's results. Although the results presented in (73) and (74) are in closed form, they require numerical solution for a particular problem. Schamberg (75, 76) suggested the following simple approximate formula for the overall heat transfer to the surface:

where A is the projected area normal to the free stream, and a is 3 for cylinders and 2 for spheres. Equation (37) yields values that agree within 1 and 6 yo,respectively, with the values computed from the algebraically complex equations of Sparrow et al. and of Chahine.

B. TEMPERATURE JUMP (SLIP)REGIME When there is a velocity slip at the wall, then the work done on the wall due to shear must be included in the heat transfer (58), i.e., the heat transfer to the wall is

+

4 = ~[W?Y cL%(~~/aY)lY=o

(38)

where p is the viscosity and y is the coordinate normal to the wall. Using Eq. (38) together with the usual continuum boundary layer equations and Smoluchowski's temperature jump boundary condition, Oman and Scheuing (57) obtained lengthy but closed form expressions for the recovery factor and for the heat transfer in laminar flow over a flat plate. A similar problem was investigated by Reddy (77) by neglecting the shear work done by the slipping fluid.

HEATTRANSFER IN RAREFIED GASES

193

Sauer and Drake (78) analyzed the problem of forced convection heat transfer to cylinders in laminar slip flow on the basis of a continuum formulation of the energy equation and temperature jump boundary conditions at the surface. By neglecting the second term in Eq. (38) and by assuming a simple free stream velocity distribution around the cylinder, Sauer and Drake obtained approximate values for the average heat transfer coefficient. There are additional results available in the temperature jump regime, but these will be presented in the next two sections so as to compare these results directly with those in the transition regime. REGIME( M C. TRANSITION

< 1)

Heat transfer to cylinders and spheres in subsonic flow has been investigated experimentally over wide ranges of Mach numbers and Reynolds numbers. T h e results are generally presented as a Nusselt number versus Reynolds number plot with Mach number as a parameter. Although such plots give Nusselt number values for a large range of experimental conditions, they are limited in their use because the heat transfer (and hence the Nusselt number) depends on the thermal accommodation coefficient, which varied substantially in the reported experiments. For subsonic flows past cylinders the heat transfer data of Cybulski and Baldwin (79) indicate an a value of 0.56, while in Atassi and Brun’s experiments (80) a! was about 0.85. I n Kavanau’s (81) and Takao’s (82) experiments with spheres a appears to be close to unity. In order to eliminate the dependence of the results on the thermal accommodation coefficient and on the Mach number, the data of these investigators (79-82)were replotted in Fig. 12 as St/StFM versus S t F M / S t c . Here St is the Stanton number calculated from the data, SFM is the limit of St as Re -+0, and St, is the limit of St as M 40. Such a correlation was first suggested by Sherman, who applied it to Kavanau’s data (52). Generally, St,, can be calculated from theory such as given in Section V, A, while Stcmay be obtained either from theory or from experiments. Values for St, are given by Cybulski and Baldwin (79)and Baldwin et al. (83) for cylinders, and by Kavanau (80) and Sherman (52) for spheres. I n Fig. 12, we also show the interpolation formula suggested by Sherman (52), St/StFM = [I (stFM/stc)]-’ (39)

+

I n all the measurements shown in Fig. 12 the temperature difference was small between the free stream and the model. For such a condition Eq. (39) yields heat transfer rates which are within accuracies generally

GEORGE S. SPRINGER

0.01

I

I

a

8

0

,

,

1 1

1

L

STANTON NUMBER RATIO, StFM/Sls

Fro. 12. Heat transfer correlation for cylinders and spheres in subsonic flow. Key: M, = 0.1-0.69 (Kavanau); ( 0 ) Sphere, Ma = 0.0 (Takao); ( 0 ) Cylinder (select data), Ma = 0.1-0.9 (Cybulski and Baldwin); ( X ) Cylinder (select data) M, = 0.14-0.41 (Atassi and Brun). (0) Sphere,

adequate in engineering problems. It is noted that Eq.( 17) (SectionIII, D) is a special form of Eq. (39). Recovery factors have not yet been reported for spheres in subsonic flow. For cylinders recovery factors are given in (79),(80),and (83),but, because of the large spread in the data, firm conclusions cannot be reached about the variation of the recovery factor with the Mach and Reynolds numbers.

D. TRANSITION REGIME(M

> 1)

I n supersonic flows at sufficiently high Reynolds numbers a shock wave separates the undisturbed uniform free stream flow from the disturbed region behind the shock (Fig. 13). Due to the shock, the local mean free path around the body may be considerably smaller than in the free stream. Since it is this local mean free path that appropriately defines the Knudsen number, continuum type analyses often may be extended with success to high speed rarefied gas flows. In exploring the transition from continuum to free molecule flow it is convenient to divide the flow into the following regimes (3):(a) boundary

HEATTRANSFER IN RAREFIED GASES

195

DECREASING REYNOLDS NUMBER

*

SHOCK WAVE

MERGED LAYER

VISCOUS LAYER

BOUNDARY LAYER

la)

L-

CONTINUUM FLOW (b)

FIG.13. Density regimes in hypersonic rarefied flow past (a) a blunt body and

(b) a semi-infinite sharp leading edge flat plate. [R. F. Probstein, Heat transfer in rarefied gas flow, in “Research in Heat Transfer” (J. A. Clark, ed.), 33-60. Macmillan (Pergamon), New York, 1963.1.

layer, (b) viscous layer, (c) incipient merged layer, (d) merged layer, (e) transition, (f) first collision, and (g) free molecule. Groups (a) to (d) are continuum type flows inasmuch as the NavierStokes equations and the Fourier heat conduction law are used in describing the flow. I n the boundary layer regime the flow field between the body and the shock wave is divided into an inviscid and a viscous part; Prandtl’s boundary layer theory is applied in the viscous region near the body; Hugoniot relations are used across the shock. In the viscous layer regime the shock wave is treated as a discontinuity but the region between the body and the shock is fully viscous. In incipient merged

GEORGE S. SPRINGER

196

layer type flows the shock is approximated by a thin discontinuity obeying conservation laws but not the Hugoniot relations. For the merged layer class the shock is no longer thin. The various continuum type flow regimes are illustrated for a blunt body and a flat plate in Fig. 13. A complete description of the flow in the transition regime cannot be based on the above continuum type analyses but would require a solution of the Maxwell-Boltzmann equation (Eq. (9)). Such a solution has not yet been found. In the first order collision regime the local Knudsen number is large but not large enough to guarantee the validity of the free molecule results. Analyses of such problems take into account a single collision between a free stream molecule incident on and a molecule reemitted from the surface. Free molecule flow problems were discussed in Section V, A. Although high speed rarefied gas flows could be grouped in different ways, the foregoing classification will be of help in the next discussions. As will be shown, most of the theoretical results for the heat transfer are for the near continuum and for the near free molecule regimes. Experimental data do exist, however, spanning the entire range between the continuum and free molecule conditions. The data available are for a variety of body shapes but only for moderate temperature differences between the model and the free stream.

1. Stagnation Point of a Blunt Body At high Mach numbers, and at Reynolds numbers for which the boundary layer theory is valid, the heat transfer to the stagnation point of a highly cooled body (T,/T, 1) may be approximated by

<

qBL qFM

(-)7 -+1 1

m 1.072(Re0)l/2 y

I14 (-)112 2

(Pr)-0*6

y + l

(40)

1, i.e., qFM= 0.5 , U , ,the “nose” radius R, , and the viscosity p, evaluated at stagnation conditions.

where

qFM

is the free molecule heat transfer for

01

=

pmUrn3.Re, is a “mixed” Reynolds number based on pa

Equation (40) has been derived (64) from the hypersonic compressible boundary layer calculations of Fay and Riddell (84) by using an inviscid stagnation point velocity gradient for hypersonic flow, assuming a cold by one. body and replacing the ratio (pwpw/popo)o.l

HEATTRANSFER IN RAREFIED GASES

197

At decreasing Reynolds numbers secondary effects become important, resulting from (85) (a) the influence of the displacement thickness on the external flow, (b) the curvature terms in the Navier-Stokes equations neglected in classical boundary layer theory, (c) the temperature jump and velocity slip at the wall, and (d) the external vorticity due to gradients of entropy or total enthalpy. Analyses of stagnation point heat transfer including some or all of the above secondary effects have been reported, among others, by Rott and Lenard (85), Kemp (86), Herring (87), Probstein and Kemp (88), Ferri et al. (89), Cheng (90, 9 4 , Ho and Probstein (92), Lenard (93), Van Dyke (65,94,95), Levinsky and Yoshihara (96), Maslen (97),Oberai (98),Chow (99), Davis and Flugge (ZOO), and Bush (101). In Fig. 14 a comparison between some of the foregoing analyses and the experimental data of Ferri et al. (89,102, 103), Valensi and Rebont (104, Hickman and Giedt (105), Wittliff and Wilson (106), Potter and Miller (107)and Carden (108) is presented. I n Fig. 14, which has been adopted from Potter (109), qBL has been calculated according to the theory of Fay and Riddell (84) except for the theoretical results of Chow (99), Cheng (90),and Van Dyke (65) since these were already given in the form q/qBL. In Fig. 14, the subscript y refers to conditions downstream, immediately behind the shock. As can be seen from this figure there is qualitative agreement among the various analytical results and also between these results and the data. T h e spread in data is too large, however, to allow one to evaluate quantitatively the accuracies of the various analyses. One difficulty is that the results are rather sensitive to the density ratio across the shock, E = pm/p,, which varied significantly during the experiments. Also, Van Dyke, Cheng, and Chow neglect temperature jump and velocity slip effects at the wall. It is expected (84, 110) that neither the slip nor the temperature jump will affect the overall heat transfer rate (although they may be 1 and X,/Rw (T,,/7J1’2. present at the wall) as long as h,/R, Cheng (90) and Van Dyke (65) also assumed a linear viscosity-temperature relation. T h e calculations of Cheng and Cheng (111)show that the assumption of such a linear relationship does not introduce a significant error in the overall heat transfer results. T h e results of a third order boundary layer approximation obtained by Kao (112, 113) are also shown in Fig. 14. This approximation appears to be valid in the range of intermediate to high Reynolds numbers (Re, > 100, say) but fails at low Reynolds numbers. It is interesting to note that the result of Cheng for the incipient merged layer regime actually yields the free molecule solution when extended to low Reynolds numbers. T h e solutions referred to are based on continuum type analyses. I n

<

<<

198

A

GEORGE S. SPRINGER

t-

1

J

1.61

m

FREE MOL.

-

ITTLIFF AND WILSON (1962) DATA 6 = 0.15

0.4

'

I

10

1

1

1

'

I

'

" I '

lo3

102

REYNOLDS NUMBER, Re,=

urn Pv R,

'

I

'

io4

" ' &

105

P L V

FIG. 14. Heat transfer to the stagnation point of a blunt body in hypersonic flow. Key to theories: (-) Cheng(1961) c = 0.15; (-0-) VanDyke (1962) e = 0.16; (-0 0-) Chow (1963); (- - -) Kao (1964). [J. L. Potter, The transitional rarefied flow regime, in "Rarefied Gas Dynamics" (C. L. Brundin, ed.), Vol. 11, 881-938. Academic Press, New York, 1966.1

the first collision regime (near free molecule flow) the stagnation point heat transfer for highly cooled bodies may be approximated by (64)

+

According to Willis (214) F = 0.38 2.26 (uw/Ua)where u, is the mean velocity of the particles reemitted from the surface at a temperature corresponding to T, . It is worth noting that in the first coIlision-regime a Monte Carlo solution obtained by Bird (215) indicates an overshoot of the heat transfer beyond the free molecule value. T h e data in Fig. 14 are not adequate to confirm this effect. The results shown in Fig. 14 do not demonstrate clearly the trend of the stagnation point heat transfer in the various density regimes. In order to emphasize some of the characteristic features of the heat transfer, analytical results for a particular set of flow conditions are

HEATTRANSFER IN RAREFIED GASES

199

presented in Fig. 15 using coordinates somewhat different from those in Fig. 14. T h e independent variable in Fig. 15 is Re, , and the dependent which for hypersonic speeds is approximately variable is q Re,/p,, equal to the ratio Nu,/Pr. From Fig. 15 it can be seen that both the viscous layer and incipient merged layer results rise above the boundary layer solutions. T h e viscous layer model becomes invalid when (c Re,)-l is not small compared to unity ( 3 ) ,i.e., for the conditions in Fig. 15 it 100. One can also observe that the incipient breaks down at about Re, merged layer analysis fares smoothly into the free molecule result at low Reynolds numbers. Futhermore, the Nusselt number varies continuously with Reon, with the exponent n varying from +1 at low Reynolds numbers to for high Reynolds numbers. From Fig. 14 one may estimate the transition regime to lie between about Re, 1 and 300. For these Re, values and using a linear relationship between viscosity and temperature, we find Re, to be 0.5 and 150 for M, = 2, and 0.05 and 15 for M, = 10. One can now evaluate the free stream Knudsen numbers corresponding to these Reynolds numbers. Substitution of the above Reynolds numbers into Eqs. (1) and (41) yields Knudsen numbers of about 0.005 and 1 for M, = 2, and 0.05

-

VISCOUS LAYER

INCIPIENT MERGED LAYER (CHENG 1963)

--

0.4

t

10

REYNOLDS NUMBER, Reo =

PmU,Rw ~

100

PO

FIG. 15. Heat transfer to the stagnation point of a blunt body in hypersonic flow. 1.22, s. = 0.1, Pr = 0.71, M, >> 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<

%

FIG. 22. Stanton number for hypersonic flow past cones at zero angle of attack. Comparison of experimental data. Key: ( x x x x) He, 0, = 3", M, = 41, Tw/To 0.35, (Horstman and Kussoy); (111) Air, 8, = 5, 10,20°, M, = 1P-21, TWIT, 0.1(Waldron); (llll)Air, 0, = 6.3, 9",Mm = 14, 21, Tw/To= 0.1 (Wilkinson and Harrington).

all the data can be correlated along a straight line when the rarefaction parameter does not include C, . Koshmarov's experimental results exhibit a similar trend although from the information given in (141) this author was unable to transform Koshmarov's results to the coordinates of Fig. 22. I t has been observed (152) that the pressure distributions along the surface of the cone also correlate better when C , is excluded from the rarefaction parameter. These results are somewhat surprising since analyses imply that the correlation should include C, . This point would merit further consideration. The second point Fig. 22 illustrates is that at some values the data start to deviate from the straight line variation and tend towards the free molecule value. Contrary to this, in Koshmarov's experiments (141) the Stanton number appears to overshoot the free molecule value at near free molecule conditions (Mm/(Remz)l/PM 0.5-1) and at small cone angles (6, < 10'). As mentioned previously, this is in agreement with the flat plate solutions of Pan and Probstein (133), and Charwat (141). Unfortunately, one cannot evaluate how any of the cone data referred to approach the free molecule value of the heat transfer because the measurements do not extend to sufficiently large rarefaction parameters. Due to the scarcity of data, heat transfer in the near free molecule regime should receive further attention.

209

HEATTRANSFER I N RAREFIEDGASES

Recovery factors were measured by Drake and Maslach (153) and Koshmarov (141). Based on his experimental results in the ranges of M, = 2.5-10, Re,, = 25-5500, and 6, = 5"-30",Koshmarov recommends the following empirical relationship for the recovery factor

+

+

(y - 1)/2]MI2), J = (6.3 - 6,)(l 100,); M, is where T = T,,/{[l the Mach number on the cone surface in nonviscous flow as determined from tables (e.g., Koval (IN)),and TFM is the free molecule recovery factor as given, for example, by Oppenheim (63). T h e recovery factors reported by Drake and Maslach fall about 5% below the values given by Eq. (48).

VI. Concluding Remarks

T h e foregoing account, by no means complete, may indicate the strong activity in recent years in the field of heat transfer through rarefied gases. As this survey shows, our knowledge of the subject has been expanded considerably within the last two decades. However, in spite of the many excellent experimental and analytical studies, there are still areas that are in need of further exploration, such as problems where nonlinear effects are not negligible and the transition regime where neither the free molecule nor the continuum type solutions provide an understanding of the details of the heat transfer process. Until more information becomes available, the engineer will have to rely on the results of linearized theories in calculating heat transfer in nonlinear problems and on interpolation between free molecule and continuum solutions for computing the heat transfer in the transition regime.

ACKNOWLEDGMENT This work was supported by the National Science Foundation under Grant number

GK-1745.

NOMENCLATURE A b Cm

area constant (see Eq. 4) proportionality constant in temperature viscosity law, 1

PwTdPmTw

c absolute velocity of molecules C,, C, specific heats

D

d dVc

cylinder or sphere diameter diameter of circular tubes volume element in velocity space

210

GEORGE S. SPRINGER

energy flux Maxwell’s reflection coefficient velocity distribution function f parameter describing geometry G (see Eq. (7)) temperature jump distance (see g Eq. (6)) total enthalpy H overall heat transfer coefficient h I modified Bessel function thermal conductivity K Knudsen number (see Eq. (1)) Kn L distance between parallel plates 1 characteristic length M Mach number molecular weight m mass of molecule Nu Nusselt number P pressure normal component of momenP tum Pr Prandtl number heat transfer per unit time Q heat transfer per unit area per unit 4 time R radius R* radius ratio, RJR2 Re Reynolds number gas constant r, r‘ recovery factor, and modified recovery factor (see Eq. (31a)) S speed ratio (see Eq. (26)) St,St’ Stanton number, and modified Stanton number (see Eq. (31b)) T temperature t time U free stream velocity u macroscopic (mean) flow velocity x coordinate y coordinate normal to surface

boundary layer thickness (see Fig. 13) density ratio across shock recovery ratio, T,/ T, normalized recovery ratio, (7 - T M ~ F -M vJ local angle of attack (see Fig. 10) cone semivertex angle (see Fig. 21) mean free path viscosity molecular velocity density tangential and normal momentum accommodation coefficients (see Eq. (3)) tangential component of momentum modified speed ratio, S sin 0 modified speed ratio, S sin fl angles for concave surface (see Fig. 11)

E F

w

a

a

fl y

thermal accommodation coefficient (see Eq. (2)) angle of attack (see Fig. 10) specific heat ratio

SURSCRI PTS

BL b C

FM i in 0 I

re s

TJ W

Y 03

132

quantity given from boundary layer solution refers to bulk temperature of flow (see Eq. (19)) evaluated in continuum evaluated in free molecule flow coordinate direction (i = 1, 2, 3) incident on surface stagnation condition corresponding to recovery temperature reflected from surface evaluated with slip at the wall evaluated with temperature jump at the wall wall conditions conditions downstream immediately behind shock free stream conditions denotes two different surfaces; 1 for inner radius. 2 for outer radius

REFERENCES 1 . J. C. Maxwell, “The Scientific Papers of James Clark Maxwell” Vol. 2. Cambridge University Press, London and New York, 1890.

HEATTRANSFER I N RAREFIEDGASES

21 1

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The

Pipe

E. R. F. WINTER AND W. 0. BARSCH School of Mechanical Engineering, Purdue University, Lafayette, Indiana

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 11. Heat Pipe Phenomenology . . . . . . . . . . . . . . . . . A. Description and Types of Heat Pipes . . . . . . . . . . . B. Functioning of the Heat Pipe . . . . . . . . . . . . . . 111. Literature Survey . . . . . . . . . . . . . . . . . . . . . . A. General Literature . . . . . . . . . . . . . . . . . . . . B. Material Tests . . . . . . . . . . . . . . . . . . . . . C. Operating Characteristics of Heat Pipes . . . . . . . . . . D. Heat Pipe Applications . . . . . . . . . . . . . . . . . E. Heat Pipe Control . . . . . . . . . . . . . . . . . . . . F. Heat Pipe Theory . . . . . . . . . . . . . . . . . . . . 1V.Summary . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature. . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Recent and European Literature on Heat Pipes. . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

219 220 220 . 224 234 234 235 . 249 . 273 276 278 310 312 313 . 320a 320e

. .

I. Introduction

I n recent years it has become increasingly important to develop methods for the efficient transport of thermal energy from one location to another. Moreover the advent of the space age has stimulated research for heat transfer devices which are light weight and have relatively long life expectancies. Such a device, although not for space application, was first proposed by Gaugler (I) of the General Motors Corporation in 1944. Unfortunately for Gaugler, the thermal transport problems of that time could be solved using more conventional heat transfer methods and devices, thus effectively concealing the true potential of his invention for some 20 years. In 1962 Trefethen (2) submitted a report to the General Electric Company in which he suggested the possible use of a passive thermal device for spacecraft applications. 219

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

This device was to consist of a hollow tube with a porous liner covering the inside surface. Energy would be transferred from one end to the other by means of a capillary induced, continuous mass cycling. No experimental verification of this concept was attempted, however, and the suggestion was quietly buried in company files. In 1964 Grover et al. (3) of the Los Alamos Scientific Laboratory independently rediscovered a device similar to Gaugler’s and coined the name “heat pipe” to describe it. Grover and his co-workers were working in the area of spacecraft power generation at the time, but they immediately recognized the potential of the heat pipe in other areas. A heat pipe is defined as a closed structure containing some working fluid which transfers thermal energy from one part of the structure to another part by means of vaporization of a liquid, transport and condensation of the vapor, and the subsequent return of the condensate from the condenser by capillary action to the evaporator. Because energy is transferred by the flow of a pure saturated vapor, a heat pipe is usually very nearly isothermal. If the fluid is either accidentally contaminated or a second fluid is introduced intentionally as discussed by Cotter (4) and Katzoff (5), then the heat pipe may lose its isothermal nature. Since 1964, a growing interest in the heat pipe has encouraged numerous groups and individuals associated with universities and industries to initiate research and development programs of their own. As a result of these investigations well over 100 articles have appeared in the open literature in recent years. The articles to be discussed include those published up to approximately March, 1970. I n addition to these, about twelve papers to be presented at the 1970 ASME Space Technology and Heat Transfer Conference, Los Angeles, California, June 21-24, will be referenced in an appendix to this chapter.

II. Heat Pipe Phenomenology A. DESCRIPTION AND TYPES OF HEATPIPES As evident from the definition given in the introduction, all heat pipes have a number of common features. First, all heat pipes incorporate what is usually referred to as an evaporator. This is the part of the heat pipe through which thermal energy from some external source is introduced into its walls and from there subsequently transferred to the working fluid. Second, all heat pipes include a condenser section. T h e working fluid condenses here and ultimately transfers its heat of condensation t o an external sink. Many heat pipes contain also an

22 1

THEHEATPIPE

adiabatic section located between evaporator and condenser. T h e adiabatic section, besides providing a passage for the fluid, serves no function other than separating the heat source and heat sink to make the heat pipe compatible with any given external geometric requirement. I n addition to the longitudinal sections, i.e., evaporator, condenser, and adiabatic section, a heat pipe may also be subdivided for the purpose of discussion into three radial components. T h e outermost shell is usually referred to simply as the “container.” T h e container’s sole mechanical purpose is to enclose the functioning parts of the heat pipe and to lend it structural rigidity. Since the internal pressure is often different from the environmental pressure, the container must be capable of withstanding pressure differences without bulging or bursting. This constraint, along with cost and manufacturing considerations, has led to the wide use of cylindrical “pipes” as containing structures. I n addition to fluid and “pressure” containment, the container also acts as an important part of the heat flow path from the source to the sink (see Fig. I). Hence the container walls should be thin to minimize their thermal resistance. This feature is in direct opposition to the thick wall requirement for pressure containment and hence an opportunity for an optimization presents itself. T h e next radial element is usually referred to as the wick. For ease of the present discussion, this may be regarded simply as a porous material filled with small random interconnected capillary channels. Various types of wicks and their properties are discussed in greater detail later in this study. T h e wick returns the liquid from the condenser to the evaporator utilizing the surface tension forces of the liquid. Although it is not a requirement, the wick is usually firmly attached to or pressed against the inside wall of the container. Since the wick is in general saturated with a low conductivity working fluid (except in the case of liquid metals) the wick-fluid matrix represents usually the major resistance along the heat flow path. It is therefore I

I I

I

----

-F.

----+--A I VAPOR CORE I HEAT FLOW PATH

‘

:ONDENSER HEAT SINK)

I

FIG. 1.

ADIABATIC SECTION

II

EVAPORATOR EVA P ~ ATO R R (HEAT SOURCE )

Heat pipe components. components.

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

necessary to consider thermal properties as well as liquid transport properties when selecting a suitable wick. The interior space of the heat pipe is usually referred to as the vapor core which provides a passage for the vapor as it flows from the evaporator to the condenser. It should now be evident that the heat pipe definition entails no geometric constraints in regard to its structure; and, in fact, a large number of heat pipes of many different shapes have been built and tested. Several conventional and unconventional heat pipes are depicted in Figs. 2-4. A heat pipe, as conceived by Grover et al., is schematically shown in Figure 2A (3). This particular geometry exhibits two features which early investigators felt were important for efficient heat pipe operation, i.e., a relatively large length to diameter ratio and a porous wick material which covers the inside surface of the structure. Figures 2B and 2C illustrate two typical heat pipe configurations which also have a large length to diameter ratio, but which provide for capillary transport of the liquid in grooves and crevices forming an integral part

FIG.2. Typical heat pipe geometries.

THEHEATPIPE

223

of the containing structure, contrary to the porous wicking material sketched in Fig. 2A, which is only held against the inside wall. T h e heat pipe shown in Fig. 2D has a very small length to diameter ratio. Heat pipes having such proportions are often called “vapor chambers” or “vapor chamber fins.” T h e device shown in Fig. 2E also fits our definition of a heat pipe although the liquid and vapor flow paths are separated mechanically whereas in the more conventional heat pipe only the liquid-vapor interface separates the flow paths. Finally, the device depicted in Figure 2 F has recently been introduced (6) as a “rotating heat pipe.” Here the liquid return is caused by centrifugal forces of the rotating contrivance. Although the rotating device appears promising for many thermal transport problems, it does not fit our definition of a heat pipe and hence will not be discussed in detail. Evidently the variations in heat pipes hapes are unlimited; for example both Katzof€ (5) and Conway and Kelley (7) have considered a doughnut shaped heat pipe. Several investigators ( 5 , 8 , 9 ) have proposed and designed flexible heat pipes. RCA has built and operated heat pipes with a variety of geometries. Among these are the two illustrated in Figs. 3A and 3B (ZO). The configuration displayed in Fig. 3A will

FIG. 3. Various heat pipe geometries. [W. Harbaugh, Heat Pipe Applications, Presented at Heat Pipe Technology and Manned Space Station Applications Technical Interchange, Huntsville, Alabama, May 27, 1969.1

effectively transport heat around a 90” bend. The five-pronged device in Fig. 3B allows the use of any combination of prongs as evaporators and of the remaining prongs as condensers. For ease of manufacturing, both types have cylindrical cross sections although this is not a requirement. In addition to the above geometries, several investigators (10, IZ) have built and tested a so-called radial heat pipe. As illustrated in Fig. 4, the radial heat pipe provides for thermal energy transport from a heat source to a concentric heat sink. The wick lining the inner walls of the annulus in this case is complimented by spokes consisting of addi-

224

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

!/

EVAPORATOR

FIG. 4. Radial heat pipe. [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-25.1

tional wick materials. Here, as with most other heat pipe geometries, the relative positions of the condenser and evaporator may be interchanged in order to accommodate any particular thermal transport problem. The variety of geometries depicted above are by no means inclusive of all possible configurations and are presented only to illustrate the extreme versatility of the heat pipe for heat transfer problems.

B. FUNCTIONING OF THE HEATPIPE At first glance, the operation of a heat pipe appears exceedingly simple. Thermal energy is transferred from the evaporator to the condenser by continuous mass cycling and phase change of a suitable working fluid. The mechanism of phase change with the accompanying absorption or release of the latent heat of transformation has long been recognized as an efficient heat transfer process. Many gadgets, e.g., the coffee percolator and the reflux condenser, combine heat transfer by phase change with a gravity induced mass cycling. Boilers in many cases utilize a mechanical pump to continuously circulate and replenish the working fluid, I n a heat pipe, however, the working fluid is continuously cycled by the surface tension forces of the fluid itself. It is this unique method of mass transfer which has both stimulated a growing interest in the heat pipe and has also proved to be one of the major impediments for a successful heat pipe operation. T o better understand the functioning and the limitations of heat pipes let us consider in more detail the physical effects occurring in a heat pipe.

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225

T h e steady state operation of a heat pipe may be represented schematically as shown in Fig. 5. T h e inside wall of the container is lined with a porous capillary structure which is saturated with some working fluid. A sufficient amount of fluid must be supplied in the container in order to fill (saturate) all the pores of the capillary structure. T h e penalty for having a slight excess of fluid is small compared to the possibility of heat pipe failure which might arise from a deficiency of fluid. T h e vapor in the core of the pipe is essentially at the saturation pressure corresponding to the liquid surface temperature. I n actuality the saturation pressure of a vapor in equilibrium with a liquid surface

___--EVAPORATOR

- - - ---

qobr

FIG. 5.

(HEAT S I N K )

Heat pipe schematic diagram.

depends also on the radius of curvature of the surface. T h e vapor pressure is greater than that acting on a plane surface if the liquid surface is convex, and less if the meniscus is concave. T h e effect is usually too small to warrant consideration and is not significant until the meniscus radius is of the order of one micron (12).Since the typical capillary pores in most heat pipes are larger than one micron, no noticeable error is introduced by neglecting this effect. T h e heat transfer from the source to the sink is effected mainly by six simultaneous and interdependent processes: ( 1 ) heat transfer from the source through the container wall and wick-liquid matrix to the liquid-vapor interface; (2) evaporation of the liquid at the liquid-vapor interface in the evaporator; (3) transport of the vapor in the core from the evaporator to the condenser; (4)condensation of the vapor on the liquid-vapor interface in the condenser; ( 5 ) heat transfer from the

226

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

liquid-vapor interface through the wick-liquid matrix and container wall to the sink; and (6) return ffow of the condensate from the condenser to the evaporator caused by capillary action in the wick. Let us now consider each of these processes separately and in more detail. Heat transfer from the source to the liquid-vapor interface in the evaporator is essentially a conduction process. For low conductivity fluids, e.g., water or alcohols, the thermal energy is conducted through the wick-liquid matrix almost entirely by the porous wick material since the wick has a higher thermal conductivity than the fluid. For high conductivity liquid metals, however, the heat is conducted both through the wick structure and by the liquid in the pores. Heat transfer by convection is very small because the pores are too small for any significant convection currents to develop. The temperature drop associated with conduction across the wick-liquid matrix depends on the working fluid, wick materials, wick thickness, and the net radial heat flux. This temperature drop may range from a few tenths to several hundred degrees Fahrenheit and is one of the major temperature gradients along the heat path. Once the thermal energy has been transferred to the vicinity of the liquid-vapor interface, evaporation of the liquid can take place. As the liquid evaporates, the net mass flow away from the surface causes the liquid-vapor interface to recede into the wicking structure. T h e resulting concave shape of the meniscus, shown in Fig. 5 , is responsible for the functioning of the heat pipe. A simple force balance on a single pore shows that for a spherical interface the pressure of the vapor exceeds the liquid pressure by an amount equal to twice the surface tension divided by the meniscus radius. This pressure difference is the basic driving force for both the liquid and vapor flows. It is opposed mainly by the gravitational and viscous forces acting on the liquid during circulation. The assumed form of the liquid-vapor interface sketched in Fig. 5 is probably quite realistic for relatively low heat fluxes. As the heat flux increases, however, the meniscus recedes even further into the wick and assumes a more complex shape (13) which may eventually interfere with the liquid flow in the capillaries. Once the liquid has absorbed the latent heat of vaporization and is evaporated, the vapor begins to move through the core of the pipe towards the condenser. The flow is caused by a small pressure difference prevailing in the vapor core. This pressure difference is caused by the slightly higher temperature (saturation pressure) in the evaporator as compared to the temperature (and hence lower saturation pressure) in the condenser. This temperature drop is often used as a criterion for successful heat pipe operation, and if the difference is less than 1 or 2"F, the heat pipe is often said to be

THEHEATPIPE

227

operating in the “heat pipe regime,” i.e., isothermally (3, 14, 15). As the vapor flows toward the condenser, additional mass is added from the downstream portions of the evaporator and consequently the mass flow rate and velocity in the axial direction continue to increase throughout the evaporator. Inverse conditions prevail in the condenser section of the heat pipe. The vapor flow in the evaporator and condenser of a heat pipe is dynamically identical to pipe flow with injection or suction respectively through a porous wall. T h e flow may be either laminar or turbulent depending on the operating conditions of the heat pipe. As the vapor flows through the evaporator (and the adiabatic section) the pressure continues to decrease due to both viscous and acceleration effects. Once the condenser section is reached and the vapor begins to condense on the liquid-wick surface, a partial dynamic recovery in the decelerating flow tends to increase the pressure in the direction of fluid motion. It should be mentioned that the driving pressure in the vapor core is somewhat smaller than the vapor pressure difference of the fluid in the evaporator and condenser. This is so because the vapor pressure of the liquid in the evaporator must exceed the pressure in the adjacent vapor in order to maintain a continued evaporation process. Likewise, the pressure of the condensing vapor must exceed the vapor pressure of the adjacent liquid in order to maintain continued condensation. As the vapor condenses, the liquid saturates the pores in the condenser. The meniscus has a very large radius of curvature, and, in fact, it may be considered essentially infinite. Any excess working fluid in the pipe collects on the condenser surface thus virtually insuring a plane interface. The heat of condensation is conducted through the wick-liquid matrix and container wall to the heat sink. If excess liquid is present, the temperature drop from the interface to the outside of the container will be larger than the corresponding temperature drop in the evaporator. In fact, some investigators (14, 16) feel that the thermal resistance in the condenser is one of the major parameters to be considered in heat pipe design. Finally, the condensate is “pumped” through the wick to the evaporator by capillary action. T h e liquid flow is generally regarded to be laminar and assumed to be dominated by viscous forces. The pressure along the liquid flow path decreases due to both the viscous losses and the increase in elevation if the heat pipe is operated in a gravity field. Operation of the heat pipe in presence of gravity with the condenser above the evaporator actually defeats the purpose of the wick since gravity can be used to return the condensate along the inside of the container wall with less viscous loss than liquid flow in the wick

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

would cause. I n this mode of operation the heat pipe is said to have degenerated into a reflux condenser or thermosyphon. Therefore, in this chapter, the pressure loss due to gravity will always be considered greater than zero (evaporator above condenser) or equal to zero (horizontal orientation of heat pipe simulating a gravity free environment). Because the vapor temperature, or operating temperature as it is sometimes called, of the heat pipe is essentially determined by the coupling of the heat pipe to the heat source and the heat sink, a brief discussion of possible source-sink combinations and their effect on the heat pipe operation is warranted. T h e vapor temperature adjusts itself in such a way that the temperature drop across the wick-liquid matrix and the container wall in evaporator and condenser is adequate to transfer the given heat flow from heat source to heat sink. In other words, the absolute vapor temperature is established in response to the temperatures imposed on both the evaporator and condenser by the source and sink. T h e temperatures at the outside wall of the heat pipe may be either “fixed” or “floating” depending on the type of constraints imposed by source or sink. At the evaporator, a “floating” temperature is usually the result of forcing some sort of heat flux boundary condition upon the heat source. This is easily accomplished by employing resistance heaters, induction coils, rf coils or radiative heating for the heat source. At the condenser, a “floating” temperature is commonly effected by radiative cooling. A fixed temperature can be maintained at either end of the heat pipe with constant temperature baths or by utilizing the heat of evaporation or condensation of a secondary working fluid for heat addition or removal respectively, at constant temperatures. Let us now consider possible qualitative temperature profiles along the heat flow path (see Fig. 1) for several source-sink combinations. Figure 6A depicts the temperature profile which would be obtained if both the source and the sink were of the constant “fixed” temperature type. For such a situation, only one axial heat flow rate is possible. T h e vapor temperature in the heat pipe is quite close to the average of the source and sink temperatures and probably tends to be somewhat closer to the source temperature since the thermal resistance in the liquid-wick matrix is larger in the condenser than in the evaporator. Figure 6B illustrates the temperature profile which results if the sink temperature is fixed and the source temperature is allowed to float. This particular combination of source and sink is commonly found in laboratory testing of low temperature heat pipes (resistance heating and water jacket cooling). As shown by profiles a and c, the source temperature and the vapor temperature increase with increasing heat

THEHEATPIPE

SOURCE TEMP (FIXED1

229

HEAT F m w : g

I

I

I

I

SINK TEMP. (FIXED)

EVAPORATOR LIQUID VAPOR LIQUID VAPOR CONDENSER SURFACE INTERFACE INTERFACE SURFACE

SOURCE TEMP (FLOATING)

SINK TEMP. (FLOATING)

FIG. 6. Qualitative temperature profiles along heat flow path.

flux. Conversely, if the heat flux is maintained constant and the sink temperature is increased as in profiles a and b, the vapor temperature and the source temperature again increase, but now the temperature gradients in the evaporator and condenser remain the same. Figure 6C describes the profiles which are obtained if the source temperature is fixed and the sink temperature is allowed to float. Here we see from profiles a and b that the vapor temperature must drop in order to accommodate a larger heat flux for a fixed source temperature and, conversely, it must rise for a given heat flux if the source temperature is increased. A significant omission from Fig. 6 is the case where both the source and the sink temperature are allowed to float, a situation often encountered during the laboratory testing of high temperature liquid metal heat pipes (induction or rf coil heating and radiative cooling). For this case the operating temperature of the heat pipe adjusts itself to a value at which the total heat input equals the total heat rejection. Since this self-adjustment depends on the exact type

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

of source and sink used, and possibly on certain properties of the container wall itself, e.g., electrical resistivity or emissivity, an exact statement about the operating temperature cannot be made. In general, increased heat fluxes cause an increase in the operating temperature of the heat pipe. The proper functioning of heat pipes depends also upon a continuous circulation of the working fluid, consequently it is not surprising that virtually all of the limitations (limits) for successful heat pipe operation are associated in one way or another with the interruption of this mass circulation. T h e limit discussed and analyzed most often in the literature is the so-called “wicking limit.” This condition is reached when a given heat flux causes the liquid in the liquid-wick matrix to evaporate faster than it can be supplied by capillary pumping in the wick (wicking action). Once this condition occurs, the liquid-vapor meniscus continues to recede into the wick until all of the liquid has been depleted. T h e wick in the evaporator becomes dry and the container temperature increases without bound until a “burnout” condition is reached which usually results in destruction of the pipe. Of course, the burnout condition can be reached only if the source is of the floating temperature type. For a fixed temperature source, the heat pipe would simply cease functioning once the capillary limit is achieved, and no mechanical damage would occur. Two additional limits of heat pipe operation are commonly associated with heat pipe “startup” and low operating temperature conditions. One of these is called “sonic limit,” a condition found in heat pipes in which the source temperature is kept constant while the sink temperature is lowered. The vapor density decreases and the vapor velocity increases correspondingly until the velocity becomes sonic. The vapor flow “chokes” at the evaporator exit, just as it does when the sonic condition is reached at the throat of a convergent nozzle (17). Once choking occurs, a further decrease of the sink temperature, in analogy to a reduction of the exit pressure in a nozzle, does not result any longer in an increase of the total heat flow. A so-called “entrainment limit” is reached when the vapor velocity is high enough and the vapor stream shears off droplets from the liquid interface entraining and carrying them to the condenser. Quite frequently the droplets can be heard as they impinge upon the end cap of the heat pipe (18). The premature depletion of the working fluid from the wick means that less liquid can reach the evaporator where it is needed for successful heat pipe operation. The entrainment limit depends to a large extent on the surface pore size of the wick material and also on the surface tension of the working fluid. T h e use of small pore sizes and

THEHEATPIPE

23 1

fluids having large surface tensions is perhaps the most effective way of avoiding liquid entrainment. The relative position of the operating limits in a heat flow Q versus temperature T plot, is illustrated in Fig. 7. Here Q represents the total axial heat transfer rate and T is the average vapor temperature in the heat pipe. Successful heat pipe operation is possible only under conditions existing below the curve ABCDE. The shape of the area under the curve may vary drastically depending on the wick material and working fluid used, however, the basic shape of each limit curve should remain as shown. Numerical examples for these various limits will be presented later in the text of this study. SONIC

ENTRAINMENT

FIG. 7. Limits to heat pipe operation. [J. E. Devetall, Capability of heat pipes. Presented at Heat Pipe Technology and Manned Space Station Applications Technical Interchange, Huntsville, Alabama, May 27, 1969.1

The presence of a noncondensable gas in a heat pipe may have a detrimental effect on heat pipe performance. T h e noncondensable gas can be added intentionally for the purpose of control, or it can be the result of improper filling procedures, container leaks, or chemical reactions between the working fluid and the container or wick material. Neglecting the control aspect for the present, the most common noncondensabIes are air (from leaks) and hydrogen (from chemical reactions). During heat pipe operation the noncondensable gas is swept to the condenser and forms a stagnant gas layer. The temperature in this zone adjusts itself in such a way that the total pressure in the vapor core remains approximately constant throughout. Heat is transferred through this zone to the liquid-wick surface primarily by conduction. Because this mode of heat transfer is extremely slow compared to that taking place in a normal condensation process, the zone containing the stagnant gas is effectively eliminated as a functioning part of the heat pipe. The result is an effective shortening of the pipe, thus reducing

232

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

its total axial heat transfer capability. T h e length of the noncondensable gas zone depends on the operating temperature and pressure in the system. For increased axial heat fluxes and the corresponding increased pressures, the zone will contract and allow more of the condenser to become operative again. Conversely, for decreased operating pressures the gas zone expands and reduces the area available for condensation. If the quantity of noncondensable gas or the operating pressures are such that the entire condenser section comes to lie within the stagnant gas zone, heat pipe operation ceases. The possibility of using a noncondensable gas for the variation of the effective condenser area as a control technique will be expanded upon later. The preceding discussion has dealt exclusively with steady-state heat pipe behavior. Of equal importance for the practical use of these devices is an understanding of the transient operating conditions through which a heat pipe passes during startup. Cotter (19) conceived three basic modes of startup which may be recognized by the shape of the developing temperature profiles. The three modes are experienced when the evaporator is heated uniformly over its entire length at constant heat flux which may, however, be varied with time. The condenser is cooled uniformly either by radiation or heat conduction to a sink kept at uniform temperature. The various modes of startup are illustrated in Fig. 8, the abscissa of which represents the distance along the heat pipe axis and the ordinate the vapor temperature. The uniform startup in Fig. 8A takes place when the vapor density is high at the ambient temperature so that the working fluid begins to reflux throughout the pipe immediately in response to an increase in the heat flux. This type of startup procedure may be accomplished very rapidly without detrimental effects to the pipe. The frontal startup in Fig. 8B is encountered when the vapor density is very low at ambient temperature, a case often observed when starting liquid metal heat pipes from room temperature. In this case the vapor density is so low that the molecule mean free path exceeds the vapor core diameter. As the heat flux is increased, the vapor density in the evaporator section rises and the molecule mean free path becomes small compared to the vapor core diameter. The vapor in the evaporator section enters the continuum flow regime while the vapor in the condenser remains in the free molecule flow regime with, of course, a transition region located in between. This mode of startup is further complicated by compressible flow effects since transonic vapor velocities are achieved. Finally, the vapor may condense into liquid droplets in the vapor core since the vapor is nearly saturated in the evaporator but subcooled in its expanding flow toward the condenser. The frontal startup in Fig. 8C illustrates a situation

THEHEATPIPE

TIME

233

I I ,

m TIME

I I

I

I

I

I

HEAT

EVAWRATOR

'

PIPE

I

CONDENSER

FIG. 8. Transient temperature profiles of heat pipes during various startup modes: (A) uniform startup (vapor density high at ambient temperature), (B) frontal startup

(vapor density low at ambient temperature), (C) frontal startup (noncondensable gas present), (D) startup failure (runaway hot spot in evaporator).[T. P. Cotter, Heat pipe startup dynamics. Therrnionic Conversion Specialist Conference, Oct. 30-Nov. 1, 1967, pp. 344-8.1

which can be expected if a significant amount of noncondensable gas is present, a case in which the evaporator heats up relatively uniformly. As the vapor temperature and hence pressure increase the noncondensable gas is moved toward the condenser where it collects in a fairly well defined zone. T h e temperature in this zone adjusts itself so that the total pressure in the vapor core is approximately constant. As the heat flux is increased and the vapor pressure and temperature increase, the noncondensable zone is further compressed thus causing the temperature profiles displayed in Fig. 8C. This mode of startup may also be accomplished very rapidly. T h e startup modes described above are somewhat idealized and various intermediate modes may be observed depending on the vapor density at initial temperature and

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

the amount of noncondensable gas present. Figure 8D illustrates a common failure encountered during startup. A hot spot is formed in the evaporator and the temperature increases without bound until failure results. The failure is usually a consequence of either the attainment of the wicking or boiling limit during the startup sequence of heat pipes having a floating temperature source.

III. Literature Survey A. GENERAL LITERATURE A very extensive research effort has been devoted to heat pipes since

1964 when Grover and his co-workers (3) at the Los Alamos Scientific

Laboratory, Los Alamos, New Mexico, first reported the successful operation of a heat pipe. A 347 stainless steel container lined with five layers of 100-mesh 304 stainless steel screen saturated with 40 gm of sodium became the prototype of all subsequent heat pipes. Five chromelalumel thermocouples were welded along the 9-cm-long pipe. The temperature distribution was measured for various input power levels ranging from 50 to 600 W. Of particular interest among the experimental results reproduced in Fig. 9 are the constant temperature plateaus extending from the heated end of the pipe revealing the zones which were refluxing. The temperature drops occurring at the unheated end were attributed to stagnant hydrogen gas formed by the impure sodium at elevated temperatures. The measured temperature gradient in the refluxing region amounted to less than O.OS"K/cm. If the heat pipe were considered a solid rod, it would have an effective thermal conductivity

w

LL

700

3

a 500 a t-

200

100

300 0

20

40 60 80 DISTANCE (cm)

FIG.9. Temperature profile along heat pipe. [G. M. Grover et al., Structures of very high thermal conductance. J. Appl. Phys. 35, 1990-1 (1964).]

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in excess of 24,000 Btu/hr ft"F. I t is not surprising that a device capable of such performance stimulated considerable interest among large numbers of researchers. Subsequently, research and development programs were initiated simultaneously in many university and industrial laboratories leading to considerable duplication of the research efforts. Much of this effort was focused on the determination of basic material properties, especially wicking properties. Because of the large number of papers published in a relatively short period of time, it is very difficult to present a chronological discussion of the references. Instead, the literature will be dealt with by subject in the following order: material properties, operating characteristics of heat pipes, heat pipe applications, heat pipe control, and heat pipe theory. It should not be overlooked that several very useful review articles (20,21,22) have appeared in the literature, and while they did not add new experimental or theoretical information, they probably encouraged further research and hence contributed to the growing field of heat pipe technology.

TJSTS B. MATERIAL 1. Working Fluids The choice of a working fluid for a heat pipe application is dictated to a large degree by several physical properties of the fluid and by the chemical compatibility of the fluid with the container and the wick. Deverall and Kemme (23) were the first investigators to formulate the requirements for suitable heat pipe fluids: (1) high latent heat of vaporization, (2) high thermal conductivity, (3) low viscosity, (4) high surface tension, (5) high wetting ability, and (6) suitable boiling point. Parker and Hanson(24) showed that the vapor pressure curve dictates the temperature range of applicability for a given fluid. In general, a fluid should be used in a steeply sloped region of its vapor pressure-temperature curve so that the temperature change associated with the given pressure drop is minimized. In addition, the vapor pressure should be reasonably high since a low vapor pressure would result in low vapor densities and high pressure drops in the vapor flow. A wide variety of fluids ranging from cryogenic fluids to liquid metals have been employed by various heat pipe investigators (23,25). In experiments performed with the vast majority of heat pipes, either liquid metals were employed in high temperature studies or relatively common liquids, such as water or the alcohols, for low and moderate temperature experiments. In the intermediate temperature range from 200-350°C no proper working fluids have been found as yet, although Deverall(Z8) has recently suggested as a suitable fluid in this temperature

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

range the use of mercury to which a small amount of titanium and manganese is added to improve the wetting characteristics. T h e relative merit of various working fluids within a particular temperature range is usually established by comparing the values of a dimensional fluid property group for each fluid. The property group has been referred to by different authors as the fluid property group, dimensional liquid parameter, liquid transport factor, capillary pumping parameter, or the figure of merit; it has the dimensions of a heat flux and is defined as

N

= plhfg~/CLl

(1)

Later it will be demonstrated that this property group is a direct measure for the effectiveness of a substance as a working fluid in a heat pipe. Numerous authors have presented plots of the fluid property groups for various fluids as a function of temperature (8,26-30). I n addition, Frank et al. (30) have published extensive property group plots for water, sodium, and cesium while references (31, 32) yield values of liquid metals for possible heat pipe application. Langston and Kunz (13) presented a table comparing the value of N for several low temperature fluids including the freons, alcohols, and glycols. Basiulis and Dixon (33) have assembled some property data on potential working fluids which are electrically insulating. T h e choice of a particular fluid, of course, depends on the specific application; however, a few general conclusions can be drawn. For high temperature, high heat flux heat pipes, the liquid metals are definitely superior to nonmetallic liquids due to their vapor pressure characteristics, high surface tension, and high latent heat. T h e outstanding fluid for low temperature work appears to be water due to primarily its high surface tension and latent heat, No valid statements can be made at this time regarding the suitability of cryogenic fluids for low temperature heat pipe applications.

2. Wicks Prior to the recent interest in heat pipes, the majority of work on flow through porous materials came from such diversified fields as soil mechanics, petroleum engineering, water purification, and ceramic engineering. T h e particular type of flow in porous bodies under study was usually either a gravity induced flow or a forced or pressurized flow. Many publications in these fields as well as the fundamental theories on flow through porous media are presented and discussed in a book by Scheidegger (34). Instead of gravity or mechanical work,

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the heat pipe utilizes capillary induced fluid flow for its operation. This particular feature, along with several other prerequisites to be discussed subsequently, has led to considerable research efforts aimed at developing or finding existing wick materials suitable for employment in heat pipes. As mentioned earlier, the primary requisite for a heat pipe wick is that it acts as an effective capillary pump. That is, the surface tension forces developed between the fluid and the wick structure must be sufficient to overcome all viscous and other pressure drops in the pipe and still maintain the required fluid circulation. Because the heat pipe may often be required to operate in a gravity field with the evaporator located above the condenser, the wick should be capable of lifting the working fluid to heights equal to or greater than the maximum difference in elevation between the evaporator and condenser. T h e requirements are of opposing nature since on one hand large pore sizes are called for to minimize the viscous loss in the wick and on the other hand small pore sizes are needed to provide for sufficient capillary pumping and maximum lift height. As a result, some sort of pore size optimization procedure appears warranted and in fact numerous authors have addressed themselves to this problem and their work will be treated later. T h e property data on potential wicking materials, which have been accumulated to date, originate primarily from wicking height measurements and the measurement of the permeability, which is defined as a proportionality constant between the flow rate and the pressure drop in a porous body. T h e techniques used and the values obtained for these properties will be discussed later. I n addition to the operating characteristics, several mechanical features must be considered when examining potential heat pipe wicks. Of special importance is the reproducibility of a wick structure so that future heat pipe investigators may rely on data generated during earlier investigations. T h e wick should be mechanically stable and should be rigid enough so that its flow properties do not change in response to wick sagging or stretching. T h e ease of wick fabrication and the cost are also important and it is conceivable that these considerations could someday be the major criteria for wick selection if heat pipes are ever mass-produced. A wide variety of wicks have been successfully employed in heat pipes. As mentioned earlier, the first wick, and also probably the most widely used to date, consists of several layers of fine mesh screen. Various methods have been used to guarantee mechanical contact between the screen and the container. Neal (28) rolled the screen on a mandril and upon insertion into the pipe removed the mandril. T h e screen

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was held against the heat pipe wall by its own resilience, but Neal found that the resilience varied from screen to screen and hence the performance of the pipe was not reproducible. Deverall and Kemme (23) forced a steel ball through a heat pipe, after the screen layers had been inserted, and apparently achieved good contact between the wick and the wall. However, no attempt was made to check the reproducibility of the wick structures. Kemme (17) has constructed rigid screen wicks using the following procedure: Several layers of stainless steel screen were wrapped around a copper tube. The structure was placed into another copper tube and drawn through a die to compress the screen layers; the copper was removed chemically. The screen tube was then heated to 1000°C in a vacuum oven to bond the structure. Finally the baked, rigid screen tube was inserted into a heat pipe where the screen and the crescent annulus between screen and wall formed an effective wick structure with reproducible qualities. Numerous other techniques for screen wick preparation have either been used or suggested. McKinney (35)has employed a coiled spring to hold a screen wick firmly against the heat pipe wall. Katzoff (4) constructed wicks in which a single layer of screen is metallically bonded to the wall. One construction technique consisted of electroplating a thin coating of In or Sn onto both the screen and the wall, pressing the two firmly together and baking them in an oven (200°C for In and 275°C for Sn). Excellent results were also obtained by diffusion bonding stainless steel screens to stainless steel plates. The bonding was effected by pressing the screen against the plate with a pressure of 15 psi while baking them at 1100°C for 2 hr in a vacuum oven. In addition to using screens as wicks, several investigators have employed screens only as a retaining structure. Heat pipes constructed at North Carolina State University (36-41) have utilized wicks consisting of various types of beads packed in an annulus between a retaining screen and the heat pipe wall. Wicks of this type have been successfully constructed using beads of monel, glass, and stainless steel of various diameters. Several different textile fabrics have also been employed as wicks. Haskin (25) used a rayon cloth as wick for a nitrogen heat pipe. T h e cloth was held firmly against the heat pipe wall by sliding together two halves of a slotted, diagonally cut retaining tube. Shlosinger et al. ( 4 2 4 4 ) selected a commercially available quartz fiber cloth as a wick for their experiments with flexible heat pipes, in which the cloth was pressed against the wall with springs, Attempts to bond the cloth to the heat pipe wall met only with limited success. It was found that ordinary rubber cement produced acceptable bonds for operating temperatures

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up to 50°C if the rubber was allowed to cure and become “tacky” before the wick was pressed against the wall. In experiments with higher temperature, silicon rubber failed because low molecular weight silicon compounds were released in the curing process and effectively waterproofed the wick. Good bonding was achieved by applying heat sealable film materials, such as thermosetting and thermoplastic sheets of the polyester resin and polyethylenetype. Other investigators experimented with commercially available porous metals as potential heat pipe wicks. Neal (28) constructed a heat pipe using a sintered Cu fiber wick. Unfortunately the wick was not bonded to the heat pipe wall and poor results were obtained. Porous metal wicks are extremely difficult to machine and normal cutting techniques such as band-sawing, shearing, and grinding tend to close the surface pores along the cut. T h e use of a filler material which could be removed after cutting is not generally recommended since it is difficult to completely remove the filler, which changes the wetting characteristics of the porous material. Langston and Kunz (13) microscopically examined porous metals which were cut by electrodischarge machining (EDM) and electrochemical (ECM) machining techniques. They found that EDM tended to erode the surface whereas ECM cut the porous material very cleanly with a minimum of pore distruction. A technique to avoid the machining of porous metals completely was recently developed (45). T h e technique entails the application of a mixture of particular matter, binder, and solvent to a surface. As the solvent evaporates, the surface tension of the fluid draws the particles together, compacting them, yet leaving open pores. Of special significance is the fact that with this technique wicking structures can be applied to geometrically complex surfaces. Wick structures have been successfully fabricated from powders of A1,0,, S i c , Al, Cu, and Ni. With the exception of screen wicks, the wicking structures which have received the most attention are the so-called low resistance and composite wicks. Bohdansky et al. (46) first suggested the possibility of using channels, cut into the interior surface, running axially the length of the tube. Busse and his co-workers (47) constructed several heat pipes employing such an integrated type of wick and found that the structure was very stable and the pore size was easily controlled. Kemme (19) has advocated the use of “composite” wicks with a fine pore size at the liquid-vapor interface to provide good capillary pumping and a larger pore size underneath for the return flow of the liquid. He fabricated such a composite wick from several layers of screen of different mesh sizes with the finer screens installed on the inside to provide capillary pumping forces and the coarse screen located in the

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

annulus between the fine screen and the wall to serve as flow passage. Another type of composite wicks made of axially cut grooves or channels covered with a fine mesh screen has been successfully tested by Kemme (27) and Hempel and Koopman (32). Katzoff (4) constructed a low resistance wick by forming a single cylindrical passage or artery out of the same sheet of screen material which covered the interior of the pipe. A variety of other schemes have been devised for the construction of low resistance wicks. Ranken and Kemme (48) have employed a slotted corrugated stainless steel sheet which was formed into a cylinder and inserted into a heat pipe. T h e triangular passages formed between successive corrugations served as low resistance fluid return paths. Calimbas and Hulett (49) modified the basic screen wick by placing nickel ribbon spacers between the layers of screen to create a series of concentric annuli. McSweeney (50) made a wick by using a series of in. diameter rods wrapped with 10 mil wire to space them apart. T h e rods were held against the pipe wall by a coarse screen. Turner et al. (51,52) attempted to construct a noncircular heat pipe such that the corners of the tube would supply the capillary pumping force. Their “configuration pumped heat pipes” were not very successful due to structural deficiencies causing the pipes to bulge and distort under even minor pressure differences between the interior and the surroundings. A wick property of major importance is the maximum height to which a wick lifts a given working fluid. Two reasons must be cited to underline its importance for heat pipe operation. First, the maximum lift height places a constraint on the dimensions of a heat pipe if it is to be operated in a gravity field with the evaporator located above the condenser. Second, the measurement of the maximum lift height represents an efficient method for the evaluation of the capillary pumping capability of a potential wick. T h e capillary pumping pressure must be greater than the sum of all viscous pressure losses and gravity losses if the pipe is to function at all. Once the maximum lift height has been determined, the minimum effective radius of the capillary structure can be calculated with Eq. (2) hmax ==

(20 cos e)/(pi gymin)

(2)

Equation (2) is obtained from a simple static force balance performed on a meniscus located a distance h above a free surface in a cylindrical tube. After rnlin has been determined, the capillary pumping pressure is obtained from Eq. (3) APC

=2~lrmin

(3)

THEHEATPIPE

24 1

which is derived from a simple force balance. T h e idea of designating a pore radius or diameter is of course an idealized approach because in general capillary pores consist of irregular noncircular channels. This irregularity in channel size has prompted investigators at North Carolina State University (36-41) to define both a “rising” and a ‘‘falling’’ capillary equilibrium height. I n general the height to which a liquid will rise in a wick (rising height) is less than the height to which the liquid will fall in the same wick after it has first been completely soaked (falling height). Katzoff ( 4 ) has indicated that the difference between the two levels is typically of the order of 25% and attributes this to wetting difficulties associated with the rising fluid. Several investigators have experimentally measured capillary equilibrium levels in various wicks. Ferrell and his co-workers (40) measured the falling equilibrium height of water in packed beds consisting of stainless steel particles (40-100 mesh) and glass beads (80-100 mesh). Further work (41) indicated that the equilibrium height as a function of particle diameter could be closely predicted by assuming that the beads were arranged in a cubic array. Phillips and Hinderman (53) measured the maximum capillary pressure for a 200-mesh screen of stainless steel, bronze, and nickel using water, methanol and benzene, respectively. I n addition, several metal “foams” and “felts” were tested with these fluids. (The porosities of the samples ranged from 89-96%.) Capillary pressures were determined by the standard technique of measuring’the equilibrium height and the pressure necessary to force an air bubble through a saturated wick. Both methods gave comparable results and the latter approach proved to be easier and less time consuming, For the screens, capillary pressures ranged from 0.12-0.36 psi and for the porous metals from 0.02-0.13 psi. Katzoff ( 4 ) measured the lift capability of six screens and found the values of the minimum effective meniscus radius ranging between 0.75 and 0.90 times the spacing of the wires in the screen. Ernst (54) studied Katzoff’s data and concluded that the effective meniscus radius can be expressed by (d dl), where d is the mesh opening half width and dl is the radius of the wire in the mesh. Langston and Kunz (13,55) measured the equilibrium height of water and Freon-I 13 in 23 wick samples. T h e samples consisted of three classes of porous materials: sintered metal screens, sintered metal powders, and sintered metal fibers (the porosity of the samples varied from 47.7-91.8 yo.)Equilibrium heights greater than 16 in, were found for several of the sintered nickel fiber samples with water as test fluid. I n a study not directly concerned with heat pipes Ginwala et al. (56) examined 178 potential wicking materials which included cellular types, textile and synthetic fibers, filter papers,

+

242

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

inorganic fibers, porous ceramic and refractory products, porous and fibrous metals, etc. Equilibrium heights were measured for the fifteen most promising materials. Maximum rise heights were obtained with the Silica Vitreous fibers and filter papers while acceptable heights were observed in Viscous Rayon. A property of great importance when selecting a wick is its permeability. Permeability (K) has the dimensions of length squared and is defined by Darcy's law given by the following relationship: mi = (KAwpi/plx)[(Pl- Pz)- p i g x

sin a]

(4)

T h e permeability is dependent upon the dimensions and the geometry of the passages in the wicking material and can be determined experimentally by passing a liquid through a wicking material and measuring the pressure drop in the direction of the flow (see Fig. 10). The pressure drop and the measured liquid mass flow rate along with the area normal to the flow and the fluid properties are then used in Eq. (4) to evaluate the permeability. The flow through the porous body may be either forced or gravity induced since static pressure variations due to gravity are accounted for in the second term in brackets. Unfortunately, no correlations between permeability and more easily measured wick properties have been found which apply to general wick configurations. Attempts to correlate permeability with porosity have been unsuccessful (34) since porosity in no way accounts for the coupling of one pore to another. For wick materials whose geometry is easily identified, as for example spheres in a cubic array, correlations can readily be found as demonstrated by Ferrell and his co-workers (36,41) who measured the permeability of several beds of packed spheres of different diameter. For comparable packing the porosity of all beds was 40% and a correlation between permeability and particle diameter was obtained.

FIG. 10. Measurement of permeability.

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243

Numerous investigators have measured the permeability of more complex wicks for which no correlations exist. Ginwala et al. (56) measured the flow rate of distilled water flowing through various types of felts, fibrous materials, and cellular materials for three different pressure heads. Their experiments indicated that the flow rate under constant pressure head decreased with increasing time for all wicking materials. This behavior was thought to be caused by an accumulation of gases and microscopic particles in the wick initially dissolved and suspended in the liquid. Langston and Kunz (13, 55) experimentally determined the permeabilities for a number of sintered metallic materials fabricated from felted fibers, powders, and screens. Special care was taken to degas the fluids before their use. T h e permeability was found to be independent of the nature of the fluid, time, flow rate, and fluid temperature. A special test in which water was intentionally aerated before being passed through the sample was performed to evaluate the effect of dissolved gases in the fluid. T h e results showed that, with a high degree of air saturation, the permeability of the wicks decreased by about 18% in about 50 hr of operation. Using both equilibrium height and permeability data, values for a capillary pumping parameter defined as the product of maximum lift height and permeability were then evaluated for all samples. T h e magnitude of this pumping parameter is a direct measure of the efficiency with which a material might function as a pipe wick. T h e data attested that sintered metallic fibers, as a group, make the best heat pipe wicks while sintered powders were the next best performers and screens were the worst. Phillips et al. (29,53) measured the permeability of sintered metal screens, fibers, and foams using forced flow, gravity flow, and condenser flow in an operating heat pipe. For the force flow test the permeability was found to decrease for increasing flow rates. This is contrary to Langston and Kunz’s results which demonstrate that permeability is independent of the fluid flow rate. T h e disparity may perhaps be explained by the fact that for comparable samples Phillips et al. used a much wider range of flow rates than Langston and Kunz did (by a factor of almost four). Their data showed for a 96% porosity nickel foam wick a variation in permeability of approximately 23 yo over the entire range of flow rates. Gravity induced flows were used to measure the permeability of very thin wicks (such as one or two layers of screen or sintered metal samples less than 0.050 in. thick) as a function of meniscus radius and flow rate. T h e meniscus radius was maintained constant along a sample by adjusting the flow rate or angle of inclination in such a way that the viscous pressure drop was exactly countered by the increase in static head. This procedure assured pressure constancy throughout the liquid.

244

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

T h e meniscus radius was varied by changing the gas pressure on the vapor side of the vapor-liquid interface. For thin wicks the data revealed the permeability as a strong function of meniscus radius and for some samples (especially single layers of screen) also as a strong function of the liquid flow rate. Permeability values measured by forced and gravity induced flow techniques were in general not in agreement. T h e gravity induced flow technique yielded permeability values up to 2* times those measured by the forced flow method, T h e disagreement was attributed to the manner in which the samples were mounted. For the forced flow tests, the wicks were completely enclosed and the measured permeability was based on flow solely in the porous structure. In case of the gravity flow tests a fluid fillet formed along the edge of the wick which allowed some of the liquid to bypass the wick, leading to an erroneous and high value for the permeability. Several attempts were also made to measure permeability of a wick in an operating heat pipe. T h e pressure distribution in the wick was measured at five locations in the condenser and the fluid flow rate was determined from the product of the measured heat transfer rate and the known latent heat of vaporization of the liquid. Unfortunately the results were erratic and consequently their reproducibility for identical experiments was very poor. Farran and Starner (57) measured the capillary pressure and the permeability of a compressible wick of braided SiO, fibers. They noticed that the capillary pressure (referring to Fig. 10) is defined as, APc = P v l - P

(5)

could be evaluated from (P,- Pz)if the lower end of the wick was submerged and the meniscus had an infinite radius of curvature thus equalizing the pressure in both phases, P, = P,, (it is assumed that P,, = PY2).Equation (4) transforms to 1 = (dPtpiAwK/pi)[1/x]

-

(pi2gAwKsin ol/p1)

(6)

or inserting the relation for the mass flow rate in terms of the fluid velocity leads to dxldt

= (K~

p ~ / p ~ ) [I /(pl x ]gli: sin 4 p 1 )

(7)

T w o techniques were selected to evaluate the pressure difference A P , and the permeability. T h e first technique (displayed schematically in Fig. 11A) consisted of the measurement of the steady state mass flow in the wick. T h e fluid was removed from the top of the wick by evaporation. T h e second technique involved the measurement of the transient rise of a liquid in a previously unsaturated wick (displayed schematically

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in Fig. 11B). Inspection of Eqs. (6) and (7) then reveals that if m (or dx/dt) were plotted as a function of l/x, the resulting plot (Fig. 11C) should be a straight line, and the permeability and the capillary pressure could be found from the intercept and the slope, respectively. Data obtained by both methods failed to yield the expected linear relationship. For near horizontal wicks (sin a ‘v 0) the capillary pumping pressure and the permeability turned out to be constant and the two methods yielded values which differed by about 10%. For sin a # 0, both methods indicated that d P , was proportional to (x sin a ) and the reciprocal of the permeability was approximately proportional to

ADVANCING LIQUID FRONT (A)

- - _ WATER

(B)

h

FIG. 11. Determination of the capillary pumping pressure and permeability. K is found from the intercept in (C), Ape is found from the slope. [R. A. Farran and K. E. Starner, Determining wicking properties of compressible materials for heat pipe application. Aviation and Space:Progress and Prospects-Annual Aviation and Space Conference, June 1968, pp. 659-70.1

E. R. F. WINTERAND W. 0.BARSCH (xsin This behavior is perhaps best explained in terms of the existence of an optimum capillary radius corresponding to an optimum capillary flow area through which a maximum mass flow rate is pumped by capillary action. Farran and Starner showed an inverse variation of this radius with x sin a ; furthermore, they observed that if the optimum radius exists along the entire wick, d P , should be proportional to (x sin a) and 1/K to (x sin They hypothesized that if the pore size distribution is large enough at any particular position x, enough optimum sized pores would be available to dominate the flow. However, a determination of the pore size distribution was not made for the test samples thus leaving the hypothesis unchecked. Finally it is quite noteworthy that Feldman (8) has collected and presented a table summarizing permeability values for a variety of wicks including the sintered metals and some compressible materials.

3. Compatibility of Components and Life Tests T h e choice of suitable materials for heat pipe construction is dictated by a compatability criterion of the different materials. Many of the problems associated with long term heat pipe operation are a direct consequence of material incompatability which usually manifests itself in chemical reactions. I n general, improper selection of components results in a gradual appearance of noncondensable gases. For high temperature liquid metal heat pipes, improper material selection furthermore accelerates corrosion and dissolution of the wick structure. Grover and his associates (3) were the first to encounter and describe the generation of noncondensable gases. In their experiment, which was described earlier in this chapter, the temperature profile along the heat pipe dropped suddenly at the condenser (see Fig. 9) which was attributed to a pocket of hydrogen gas produced from impure sodium in the $Hz . Andeen et al. (16) tested a water-brass reaction, NaH + Na heat pipe and experienced severe problems with noncondensable gas. No attempt however, was made to determine the source of the gas. Schwartz (58) noticed the occurrence of a noncondensable gas in several water-stainless steel heat pipes. Samples of the gas were withdrawn from one of the pipes and their composition analyzed with a mass spectrometer. T h e results of the chemical analysis indicated that the noncondensable gas was composed of over 97 % hydrogen. Schwartz hypothesized that the hydrogen was formed as a result of a chemical reaction between the iron in the stainless steel and water. H e suggested that the problem of noncondensable gas generation could be avoided, either by choosing a heat pipe whose metal components range below

+

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247

hydrogen in the electromotive series or if the metals have an electrochemical potential above hydrogen, by using a nonreacting fluid other than water. T o test this concept, an ammonia-stainless steel heat pipe was built and operated continuously for three months with no measurable sign of noncondensable gas formation. Conway and Kelley (7) were also troubled by noncondensable gases in a water-stainless steel heat pipe. Although no tests were made to determine the origin of the gas, it was most likely that hydrogen was produced in the same chemical reaction which plagued Schwartz. Grover (59) has suggested a possible solution to the problems caused by hydrogen formation in low and moderate temperature heat pipes. He recommended to fabricate the condenser end cap out of palladium allowing hydrogen to diffuse to the outside while retaining the working fluids. T h e prohibitive cost of palladium, however, limits its use to only the most amply funded experimental programs and as yet, the concept has not been tested. Deverall and Kemme (60) have reported the successful operation of a water-stainless steel heat pipe for over 3000 hr without accumulation of noncondensable gas. T h e stainless steel tube and screen were first degreased in acetone and then bright-dipped to guarantee clean surfaces which were subsequently degassed at 600°C in a high vacuum oven. Since the other investigators who encountered hydrogen formation in water-stainless steel heat pipes reported no extensive cleaning procedures, it may be speculated that for the particular case of water-stainless steel, hydrogen formation is more a function of the techniques used in processing the materials than of the materials themselves. Jeffries and Zerkle (27) have commented on work done by Lyons who tested several fluids in capsules of aluminum alloy 6061 at temperatures from 155-322°F for durations in excess of 500 hr. Strong evidence of corrosion was found with methanol and ethanol; n-butane (1 55°F) and Monsanto Cp-34 (321 O F ) showed moderate corrosion; whereas no corrosion was evident with n-pentane (310"F), benzene (3 10"F), heptane (320"F), toluene (322"F), ammonia ( I 59"F), Freon-1 1 (156"F), and Freon-1 13 (155°F). I n addition, a capsule of 321 stainless steel was tested with water at 320°F. Definite signs of gas evolution were evident during the test, which proved in a qualitative mass spectrometric analysis to be hydrogen. T h e most severe compatibility problems are encountered when heat pipes are operated at elevated temperatures. A considerable effort was expended by workers at Los Alamos (15,23,48,59,61)to study this problem. A heat pipe was constructed selecting tantalum as container and wick material, and silver as the working fluid. T h e pipe was operated at 1900°C for 100 hr. Examination of the sectioned pipe revealed that

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

the wick in the condenser had almost disappeared while the wick in the evaporator was clogged with tantalum. This behavior was attributed to the dissolution of the wick in the working fluid. I n this case, small amounts of structural material were dissolved by the working fluid in the condenser section, transported to the evaporator section, and deposited there as the working fluid evaporated. T h e amount of tantalum deposited indicated a solubility of tantalum in silver of approximately 10 ppm. Dissolution and transport of the wick material also occurred in an In-W heat pipe which was operated at 1900°C for 75 hr. Recent tests have attested to the potential of a Ag-W heat pipe for high temperature operation. Such a pipe has been operated for 1000 hr with no noticeable deterioration in performance. Busse and his coworkers (47,62,63)have conducted an extensive experimental program to determine material compatability at the temperatures employed in thermionic converters, i.e., 1000°C for the collector and 1600-1800°C for the emitter. Special care was taken to select materials for which no known intermetallic compounds exist. For many material combinations, they found that the dissolution of the container or wick led to eventual heat pipe failure due to clogging of the wick in the evaporator. Other material combinations produced ultimate heat pipe failure due to a weakening of the wall caused by intergranular corrosion and wall penetration. It was also noticed for a particular thermionic converterheat pipe system that the lithium working fluid diffused through the wall at rates sufficient to significantly lower the power output of the converter. Three systems, however, proved to be promising for operations longer than 1000 hr at 1600°C: W/Li, W/Pb, and SGS-Ta/T1. I n the 1000°C temperature range, Na and Cs have operated for 1000 hr in a Nb-1Zr container with no significant corrosion. Workers at RCA (10,6448) have also investigated material compatibility at thermionic temperatures. Examination of a Li-TZM alloy heat pipe employing a Mo wire wick revealed about 1 0 p p m of the alloy in the Li working fluid after 600 hr of continuous undegraded operation. Another Li-TZM heat pipe has been operated successfully for 1400 hr while a K-Ni heat pipe has supposedly been working continuously for 26,000 hr. At lower temperatures, a water-copper heat pipe has accumulated 8500 hr of operating time with no degradation in performance. Ernst et al. (69-71) have reported the successful operation of a LCTZM heat pipe for u p to 5000 hr. Some deposits were noted in the evaporator and it was speculated that it was probably titanium oxide; however, no attempt was made to determine the composition of the substance. Johnson (72) investigated the compatibility at thermionic temperatures of the working fluids Ag, Ba, Ca, In, Li, Pb, and TI with containers

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made of Cb-IZr, Ta-lOW, and T Z M . T h e fluids (metals) were placed into reflux capsules which were heated for times up to 1000 hr. T h e capsules were then sectioned and examined by X-ray diffraction analysis. Results indicated that indium is not suitable for long term heat pipe operation. Only minor intergranular attack was observed when calcium was used as working fluid, while other material combinations showed varying degrees of attack and corrosion. I n summary, for low temperature heat pipes, care must be taken when selecting components to avoid any combination for which a possible chemical reaction exists which could lead to the formation of noncondensable gases. Cleanliness and material treatment techniques also seem to govern to a large extent the ultimate compatibility of heat pipe materials. For high temperature heat pipes, due consideration must be given to the formation of possible intermetallic compounds, solubility of one metal in another, and diffusion effects. Obviously a great deal of systematic research is called for in high temperature applications since very little is known about the crucial properties of metals at these temperatures. C. OPERATING CHARACTERISTICS OF HEATPIPES 1. General

I n addition to the experiments concerned with basic investigations of potential heat pipe materials, many investigators performed experiments to determine the operating characteristics of heat pipes. I n early investigations, often little more than the successful operation of a heat pipe was reported. Shortly after the work performed at Los Alamos had been published, Bainton (73) reported the successful operation of two sodium-stainless steel heat pipes. Temperature uniformity over most of the length of the pipe was verified by infrared photography. Workers at RCA (66) related the successful operation of a lithium heat pipe for over 9000 hr claiming that, as a thermal energy transfer device, this heat pipe operated 1000-10,000 times more efficiently than the thermal conduction process in an equivalent rod of metals such as copper or silver. Hall (74) tested a lithium-TZM heat pipe and verified that fluxes sufficient in magnitude to operate a thermionic converter could be obtained. Fluxes on the order of 40 W/cm2 were achieved for both a lithium and a sodium heat pipe. Bowman and Crain (75) operated a water-copper heat pipe at near ambient temperatures. T h e temperature profile along the axis of the pipe was measured to confirm heat pipe operation. An electrically insulating fluid was

250

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

employed by Basiulis and Dixon (11) in a radial heat pipe which could successfully dissipate 900 W regardless of its orientation with respect to gravity. Although no mention was made concerning the uniformity of the flux across the condenser surface (see Fig. 4),it is most probable that the flux around the circumference of the condenser was a function of its orientation in regard to gravity. 2. Investigations of Heat Transfer Limits Several investigators have experimentally determined the wicking limit of heat pipes as a function of vapor temperature and/or geometric parameters. Bohdansky et al. (76) measured the maximum possible heat flow in a Na-Nb heat pipe in the temperature interval from 500-800°C. T h e pipe was 50 cm long and had a 2-cm inner diameter. T h e capillary system consisted of 85 grooves of rectangular cross section with a width of 0.4 mm and a depth of 0.46 mm. T h e pipe was heated with an rf coil and cooled through a variable resistance helium thermal bridge with cooling water on the other end. In subsequent experiments the inclination of the pipe was varied in order to change the lift height h between evaporator and condenser. Upon each variation in orientation the power input was increased until a hot spot appeared at the far end of the evaporator indicating that the wick was no longer capable of supplying sufficient fluid to this part of the evaporator. T h e temperature variation was measured with thermocouples which were mounted at the outside wall of the pipe. Bohdansky et al. (76) plotted the heat flow rate versus operating temperature shown in Fig. 12, where the detrimental effect of lift height on the maximum heat transfer can be recognized. Occurrence of maximum heat flows for each experiment performed at a constant lift height is attributed by Bohdansky et al. to the decrease of the surface tension with increasing temperature. They even further illustrated the effect caused by elevation (lift) by replotting the maximum heat transfer values as a function of height as shown in Fig. 13. Neal (28) in a qualitatively similar representation for a water-stainless steel heat pipe fitted with four layers of 105-mesh screen as a wick structure confirmed the results of Bohdansky et al. It should be noticed that the shape of the curves in Fig. 12 corresponds to the form of the qualitative wicking limit curve displayed in Fig. 7. T h e curves can be compared qualitatively because the wall temperature and vapor temperature are closely related for heat pipes operating under steady state conditions. Cosgrove et al. (37) investigated two water-brass heat pipes in which the wick structures consisted of packed monel beads which were held

THEHEATPIPE

25 I

in an annulus between a retaining screen and the wall. For a particular wick structure and pipe orientation, the maximum heat transfer was considered reached when a hot spot began to form in the evaporator

500

600

700

800

900

T (“CI

FIG.12. Heat flow in a Na-Nb heat pipe as a function of temperature and inclination. [I. 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. 144-8.1

H Icml

FIG. 13. Maximum heat flow at 850°C as a function of inclination. [J. Bohdansky al., Heat transfer measurements using a sodium heat pipe working at low vapor pressure. “Thermionic Conversion Specialist Conference, Houston, Texas, Nov. 1966,” pp. 144-8.1 et

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

which was concurrently detected with thermocouples installed in the packed beads. T h e vapor temperature remained relatively constant in all experiments in which the primary variables were the pipe orientation and the diameter of the monel beads. Figure 14 illustrates the effect of particle diameter, and consequently pore size, on the maximum heat transfer as a function of inclination. For a given particle diameter, the maximum heat transfer decreases with increasing elevation once more reaffirming the results of Bohdansky and coworkers depicted in Fig. 13. Notice, that for a selected pipe inclination the maximum heat transfer increases with decreasing particle diameter. This effect is caused by the increased capillary pressure resulting from the smaller pore sizes. Cosgrove could not explain why the curve with the smallest particle diameter intersected with the other curves. If the pore sizes were decreased indefinitely however, eventually an optimum pore size would be obtained at which the viscous drag in the capillaries would become dominant and the maximum heat flow should decrease again. T h e results obtained by Bohdansky and Cosgrove and their coworkers suggest that the capillary limiting curve depicted in Fig. 7

I2O

t cos 4

0

FIG. 14. Maximum heat flow as a function of ischation and particle diameter. Particle diameter in units of ft: ( 0 ) 1.720, ( A ) 1.218, ( 0 )0.920, (v) 0.650, (0) 0.346. [J. H. Cosgrove et al., Operating characteristicsof capillary limited heat pipes. J . Nucl. Erzergy 21, No. 7, 547-58 (1967).]

THEHEATPIPE

253

is in reality a family of curves, each depending on the pipe orientation in the gravitational field. Moreover, the entire family of curves depends on the geometry of the heat pipe, and in particular on the characteristics of the wick structure. While Bohdansky et al. and Cosgrove et al. installed thermocouples in the wall and wick to signal the formation of a hot spot, McSweeney (50) has found that the vapor temperature is a much more sensitive indicator of wick dryout. He experimented with a sodium-stainless steel heat pipe and monitored both the vapor and the wall temperature as a function it is evident 750 L 700

-

TEMPERATURE

650 I

450

650

700 750 800 850 900

POWER TRANSFERRED (W)

950 (000

FIG. 15. Heat pipe dryout. [T. I. McSweeney, T h e Performance of a Sodium Heat Pipe. Presented at the ASME-AIChE Heat Transfer Conf., Minneapolis, Minnesota, Aug. 3-6, 1969, AIChE Preprint 7.1

that the vapor temperature at dryout changes more rapidly in response to power increase than the wall temperature. T h e vapor temperature decrease under dryout condition is somewhat puzzling to say th.e least. Unfortunately the location of the thermocouple probe in the vapor space was not mentioned. However, it appears most probable that the probe was located in the proximity of the condenser. Possibly the termination of the heat flow caused by the dryout of the wick resulted in considerable temperature variations within the now stagnant vapor core with rising temperatures in the evaporator and decreasing temperatures in the condenser which was still cooled. It is entirely possible that this type of behavior occurs only in heat pipes with a fixed temperature boundary condition imposed on the condenser. If a floating temperature sink were employed the entire vapor temperature would most likely increase once the dryout condition is reached.

254

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

In related studies, Shlosinger (43) found that for compressible wicks specifically, the manner in which the wick is retained against the heat pipe wall may significantly affect its wicking characteristics. For instance, if a helical spring is used to retain the wick, the fluid may have to travel by capillary action over a much longer spiral path from the condenser to the evaporator than otherwise would be necessary with a different type of retaining structure. This effect, of course, would not be present for more rigid wicking materials or for the commonly used axial slots, For instance, Busse et al. (63) found that heat pipes with axial grooves serving as a wick often formed a hot spot on the top side of the evaporator when the pipe was operated in the horizontal position, They attributed this behavior to the missing interconnections between the parallel grooves, thus preventing any cross flow between grooves in the evaporator. I n the condenser, however, the excess liquid tended to reside at the lower side of the pipe thus making the lower grooves the preferred paths for liquid transport. Their explanation was verified experimentally by shielding the lower side of the condenser of a W-Pb heat pipe. T h e shield forced the vapor to condense on the upper side of the pipe and subsequently no hot spot was formed. I n a heat pipe in which the grooves are interconnected by circumferential grooves for instance, local overheating should not occur in the horizontal mode of operation. A few investigators have studied the problem of heat transfer and boiling in wicks. I n a research project not directly related to heat pipes, Allingham and McEntire (77) measured boiling film heat transfer coefficients on a horizontal copper tube which was surrounded by a ceramic wick and immersed in a pool of water. For lower heat fluxes they measured values for the boiling film heat transfer coefficient in excess of those established under similar conditions in conventional pool boiling. T h e higher values were attributed to an increase in effective heat transfer surface area and also to an increase in active nucleation sites. At higher heat fluxes, however, the trend reversed itself and the values of boiling film heat transfer coefficients decreased for wick boiling below the corresponding values of normal pool boiling. T h e reason for the decrease appears to be twofold. First the very presence of the wick prevents agitation of the liquid otherwise so common in pool boiling. Second the vapor escaping through the pores impedes the liquid counterflow directed towards the heating surface. T h e data were reduced and an implicit function of the boiling heat transfer coefficient as a function of the radial Reynolds number, based on four times the hydraulic radius of the capillary pores is plotted in Fig. 16. I n a similar study, Anand et al. (78, 79) tested a water-stainless

THEHEATPIPE

255

KJ

1

Re-

, I

De G' CL

:

10-3

FIG. 16. Wick boiling heat transfer correlation. (St) P P = (0.072 Re)-O.". Boiling temperatures: (0)120"F, ( A ) 140"F, ( 0 ) 160°F. [W. D. Allingham and J. A. McEntire, Determination of boiling film coefficient for a heated horizontal tube in water-saturated wick material. 1. Heat Transfer, Pap. No. 60-HT-11, 1-5, (1960).]

steel heat pipe which had a 100-mesh stainless steel screen wick. T h e wall and vapor temperatures were recorded for different axial heat flows and the wick boiling heat transfer coefficient h, was calculated. Anand claims that the data can be correlated by St

= 0.0051 P

P NPO.~Re-1.43

(8)

However, the data were plotted using St = 0.00051

NP-O.~

(9)

Moreover, the ordinate on their graph was in error by a factor of ten. Allingham and McEntire corrected Anand's graph and presented it in Fig. 17 along with their own correlation. Both correlations show the same trend and are in relatively good agreement considering that one represents wick boiling on the outside of a tube, and the other one wick boiling in the interior of a heat pipe. T h e validity of Anand's results, however, may be somewhat questionable in view of the mistake made in their graphical presentation.

256

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

Marto and Mosteller (80) studied the problem of wick boiling using a so-called everted heat pipe. A sectional view of their heat pipe is given in Fig. 18. T h e unique feature of this pipe is a vapor space enclosed in an annulus between an interior tube and a confining envelope. T h e wick consisted of four layers of 100-mesh stainless steel screen attached to the outside of the inner pipe. Heat addition and removal

fo-4

to-3

REYNOLDS NUMBER

FIG. 17. Wick boiling heat transfer correlation. [D. K. Anand, On the performance of a heat pipe. J. Spacecraft Rockets (Eng. Note) 3, No. 5 , 763-65 (1966); D. K. Anand et al., Heat Pipe Application for Space-craft Thermal Control. Johns Hopkins Univ., Appl. Phys. Lab., AD 662241.1

VAPOR SPACE

ACCESS

FIG. 18. Sectional view of everted heat pipe. Thermocouple locations and number are indicated by ( 0 ) . [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-24.1

THEHEATPIPE

257

were accomplished using a resistance heater and a tap water cooling system both installed within the inner tube. T h e outer envelope was made of glass to facilitate visual observation of wick boiling. T h e results obtained with water as working fluid are demonstrated in Fig. 19, from which it becomes apparent that lower superheats were required under boiling conditions in a wick than in conventional pool boiling.

- I LL

2

A'

k ,b

;5 O ; ;5 ;O ;5 EVAPORATOR HEAT FLUX, O/A (W/IN2)

40

FIG. 19. Observed superheat versus radial heat flux. Constant pressure datap = 15.0 in. Hg vac (7.35 psia) including error limits. Runs made on different days in following order: ( 0 ) first, ( U) second, ( A ) third, and (v) fourth. Open symbols indicate no boiling, filled symbols indicate boiling. [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-24.1

As the radial heat flux was increased, dryout of the wick occurred at the same flux value whether or not boiling was observed. T h e authors concluded that wick boiling could exist in a heat pipe with no detrimental effect on its operation. T h e system pressure was also varied for both water and ethyl alcohol and it was found that for a given heat flux, the superheat decreased as the absolute pressure increased. I n addition, for a given superheat the heat flux in the case of water was almost one order of magnitude larger than that obtained with ethyl alcohol. This disparity in heat fluxes is expected in view of the higher surface tension and latent heat of water compared to the same properties of the alcohols. I n a study involving much larger superheats, Langston and Kunz (13) measured the heat flux through several wick samples as a function of the superheat. They found that for superheats of the order of 15°F and larger the flux through the sample wicks became much smaller than for boiling on a flat plate, thus confirming the results reported by Allingham and McEntire. T h e samples used by Langston and Kunz included sintered nickel powders and sintered nickel screens.

258

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

In another series of experiments on boiling in heat pipes, Moss and Kelly (81) employed a neutron radiographic technique to measure the liquid content (i.e., liquid thickness) in the wick of the evaporator in a coplanar heat pipe. T h e wick was made of sintered stainless steel screen ($in. thick) and the working fluid was water. Measurements proved that only under conditions of zero heat transfer did the wick in the evaporator remain completely saturated. As soon as heat was supplied to the evaporator, the liquid interface receded into the wick reducing the extent of saturation in the wick. I n addition, the data demonstrated that under normal operating conditions the degree of saturation of the wick in the evaporator was inversely proportional to the heat flux. The authors concluded that a vapor blanket formed at the base of the wick and that the existence of this blanket manifested itself in the reduced saturation of the wick. Two analytical models were formulated in an attempt to describe the heat transfer characteristics of the partially saturated wick. In a conventional model it was assumed that vaporization takes place at the liquid-vapor interface. I n the other model, however, the formation of a vapor blanket at the base of the wick was assumed. T h e vapor blanket thickness as a function of the heat transfer rate and contact angle is shown in Fig. 20. For heat fluxes smaller than 15,000 Btu/hr-ft2 the second model more closely predicts the measured values than the more conventional first model. T h e results led to some allegations that the study did not pertain to heat pipe operation since it is generally believed that vaporization should take place at the liquid vapor interface during successful heat pipe operation. T h e authors countered these allegations by insisting that their measurements indicated beyond any doubt that the previously accepted idea of heat transfer through wicks is in error. At the present time, however, there are more data available supporting the model of conduction through the wick, with vaporization taking place at the liquid vapor interface, than results sustaining the observations made by Moss and Kelly. For example, Ferrell and Alleavitch (42) measured the heat flux through packed beads saturated with water. Their data for 3-40-mesh monel beads are displayed in Fig. 21. I t is seen that the data fall very close to the line predicted by assuming pure conduction through the saturated bed. T h e curve obtained from the conduction model was calculated under the assumption that heat flows by conduction through a thin liquid-bead layer in contact with the. heating surface. T h e thickness of the liquid layer was determined by the location of the minimum pore diameter in the bead configuration. It is somewhat surprising that such good agreement was found between their theory

THEHEATPIPE 0.25C

,i I

8.600

I

- 0.20c -E

259

Is=oo

m

v)

w

5

0.15C

0

I

F F W

5a

0.1oc

1

m

E 0

a

a

'0.05C

/ 0

I

I

I

I

10 20 30 40 q ,HEAT FLUX (BTU/HR - F T ' x I O - ~ I

FIG. 20. Vapor blanket thickness versus heat transfer rate: (- - -) analytical (vapor evaporated from wick surface, model I), (-) analytical (vapor released from sides of wick, model 2), (A) data with, untreated by hydrogen annealing, (0)data with wick hydrogen annealed. [R. A. Moss and A. J. Kelley, Neutron Radiographic Study of Limiting Planar Heat Pipe Performance. Private comm.]

and experiments since in the experiments the bed of beads was completely flooded to a level well above the upper surface of the bed; hence, no such thin liquid-bead layer existed in actuality. Recent work by the same authors has included heat transfer measurements in a similar apparatus, except that now the bed was not flooded and the liquid was drawn to the evaporator section by capillary action identical in manner to an operating heat pipe. T h e more recent data showed excellent agreement with the conduction mechanism discussed above. I n a similar type of experiment Phillips et al. (29,53) measured the heat flux through a composite wick of nickel foam and stainless steel screen using water as a working fluid. T h e fluid was supplied through an artery and moved by capillary forces. A typical sample of their data is illustrated in Fig. 22. It is again obvious that conduction was the mode of heat transfer for low values of AT. T h e wick exhibited a

260

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

hysteresis effect after nucleate boiling was first observed. In addition, the maximum heat flux decreased with decreasing chamber pressure. The reduction was attributed to the significantly increased size of the vapor bubbles formed during nucleate boiling at decreased pressure, hence causing a premature burnout due to vapor blockage in the wick. T h e blockage occurred because the vapor was forced to vent through the top of the wick by purposely sealing its sides. The same effect was observed in experiments performed by Costello and Redeker (82). They concluded that proper venting of the vapor was necessary if the full capabilities of the capillary supply system were to be utilized. It is interesting to note that at the present time, no consideration is given to proper vapor venting of the wick in heat pipe design.

CURVE PREDICTED BY CONDUCTION MODEL FOR 30-40-MESH BEADS

1

1

FIG.21. Experimental results for surface covered with 30-40 mesh monel beads: Bed depth in inches: ( 0 ) Q , ( A ) , (0)1, ( 0 ) 14 [J. K. Ferrell and J. Alleavitch, Vaporization Heat Transfer in Capillary Wick Structures. Presented at the ASMEAIChE Heat Transfer Conf., Minneapolis, Minnesota, Aug. 3-6, 1969,AIChE Reprint 6.1

.

In addition to the capillary and boiling limits which restrict the heat transfer capability of all heat pipes, those transporting heat of the order of kilowatts instead of watts, are often limited by the sonic and/or entrainment limits. These limits are often encountered during startup procedures from near ambient conditions where the initial vapor pressure is very low and the resulting velocities in the vapor core are consequently very high. Kemme (17) investigated the sonic limit using several different liquid metals as working fluids. The heat pipes were

THEHEATPIPE 6000

-

26 1 NUCLEATE BOILING IS FIRST OBSERVED

CALCULATED ASSUMING HEAT CONDUCTION THROUGH A FILM OF WATER 0.056 IN. THICK (AVERAGE THICKNESS OF WICK ~0.056 IN.)

0

5

t0

15

20

25

30

35

FIG.22. Experimental results for distilled water and nicker foam. The liquid is distilled water. The wick material is 220-5 nicker foam and one layer of stainless steel screen. The wick thickness is 0.056 in. The artery is a stainless steel tube and two layers of nicker foam under tube. ~

Experiment No.

Chamber pressure (Psi4

Liquid head (in.)

Max. heat flux (Btu/hr-fta)

40 41 42

4.1 3.1 3.3

0.0 -0.5 -1.0

6900 3100

2000

[E. C. Phillips, Low temperature heat pipe research program. NASA CR-66792;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, 1969,Pap. No. 69-HT-18.1

heated by an induction coil and cooled through a gas gap with a water calorimeter. T h e use of different mixtures of argon and helium in the gap allowed heat pipe temperature variations at a constant heat input, or heat input variations at a constant heat pipe temperature. Figure 23 illustrates data obtained during the startup of a sodium heat pipe. T h e dashed line indicates the sonic limiting curve based on the vapor temperature existing in the evaporator exit. T h e heat flow was increased in discrete steps and the pipe was allowed to reach steady state before the heat flow and wall temperature measurements were made. T h e evaporator exit temperature followed the sonic curve until it reached 560°C. For temperatures lower than 560°C, the flow in the condenser

262

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

section consisted of continuum flow at the entrance and free molecule flow at the far end of the condenser. Hence, only that part of the condenser in which continuum flow existed contributed significantly to the heat removal from the pipe. As the heat flux increased, eventually the entire condenser region was in the continuum flow regime and once this occurred, the heat removal area of the system remained fixed, so that a further increase in heat input now resulted in a larger temperature rise at the evaporator exit than was previously possible. T h e

3.0

-

I

I

- -2

2.5

I

- 0 450

500

I

I

I

ox

0

a

0 %

5% 600 650 700 TEMPERATURE (“C)

FIG.23. Startup behavior for sodium heat pipe: ( x ) maximum evaporator temperature 1 and ( 0 ) evaporator exit temperature 2. [J. E. Kemme, Ultimate heat pipe performance. “IEEE Conference Record of 1968 Thermionic Conversion Specialist Conference, act. 21-23, 1968,” pp. 268-71.1

subsequent vapor density increase allowed the velocity at the evaporator exit to become subsonic. Further experiments indicated the occurrence of supersonic velocities at the condenser entrance. Figure 24 shows results obtained from steady state measurements at a constant heat input of 6.5 kW. T h e condenser temperature was adjusted by varying the concentration of the gas mixture confined in the gap described earlier. Curve A describes a condition in which the vapor velocity remained subsonic throughout the heat pipe. As the pressure in the condenser was decreased, the velocity became sonic at the evaporator exit (curve B). T h e existence of subsonic flow between curves A and B is evidenced by the changes in the condenser temperature and pressure which were transmitted to the evaporator where corresponding changes occurred. When the pressure in the condenser was further decreased,

THEHEATPIPE

263 ?

23 4

. -.EVAP

5

.-_,.

1

6 I

-.__.._--..,

7

0 1

CONDENSER

FIG.24. Transonic conditions in sodium heat pipe. Heat input = 6,5 kW. [J. E. Kemme, Ultimate heat pipe performance. “IEEE Conference Record of 1968 Thermionic Conversion Specialist Conference, Oct. 2 1-23, 1968,” pp. 266-71 .I

as shown by curves C and D, the evaporator pressure conditions remained constant. T h e vapor velocity did not decrease immediately upon entering the condenser but continued to expand and became finally supersonic followed by a more abrupt pressure recovery than was evidenced for subsonic flow. Figure 25 demonstrates the effect of different fluids on the sonic limit. Good agreement between experimental results and theoretical predictions was achieved, T h e sonic

FIG.25. Comparison of sonic limits in sodium, potassium, and cesium heat pipes:

experimental. [J. E. Kemme, Ultimate heat pipe performance. “IEEE Conference Record of 1968 Thermionic Conversion Specialist Conference, Oct. 21-23, 1968,” pp. 266-71.1 (- - -) calculated and (-)

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

264

limiting curve (see Fig. 7) is highly dependent on the working fluid and is also dependent on the pressure and temperature at which the heat pipe is operating. In a similar study, Dzakowic et al. (83)confirmed the results obtained by Kemme. A sodium-stainless steel heat pipe with five layers of 60-mesh screen serving as wick was employed to study the vapor velocity limit in heat pipe operation. Figure 26 is a plot of the axial temperature profiles obtained for two different heat inputs. I t is interesting to see that with increasing heat flow the transition from

I

fT/C

I

LOCATIONS

It!::: I

.O 800

f, 400 a

z

p

200

AXIAL POSITION

I

”

TRANSITIONAL OR FREE MOLECULAR FLOW

FIG.26. Axial temperature profiles for sodium heat pipe: (0-0) Q = 1440 W and ( O - - - o ) Q , = 2020 W. [G. S. Dzakowic et al., 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. Pap. No. 69-HT-21.1

supersonic to subsonic flow at the condenser entrance took place at nearly the same heat flow rate and temperature as measured by Kemme (see Fig. 23). This emphasizes again the dependency of the sonic limit on the selection of the working fluid and operating conditions of the pipe and deemphasizes its dependence on a particular pipe geometry. Various startup tests were also conducted. T h e data are presented in Fig. 27 along with the calculated capillary (or AP) and sonic limits. Note the data of Dzakowic et al. fall to the left of the sonic limit curve while Kemme’s data (Fig. 25) fall consistently to the right of the curve. An explanation for this discrepancy may be derived from

THEHEATPIPE

265

4000 t

-3 3000 -n w

a:

LL W

2 2000 a

+ Lz

c

a W

I

1000

400

500

600

TEMPERATURE MEASURE AT MIDPOINT OF ADIABATIC SECTION PC)

FIG. 27. Heat transfer rate versus temperature at the adiabatic section: ( A ) heat pipe center line inclined 22" from horizontal evaporator down (first startup), ( 0 ) heat pipe heat pipe center line inclined 22" from horizontal evaporator down (second startup), (0) horizontal (third startup). [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. Pap. No. 69-NT-21.3

the fact that Kemme plotted his data as a function of the maximum evaporator temperature while Dzakowic et al. used the temperature measured at the midpoint of the adiabatic section. It appears that perhaps the use of the evaporator exit temperature would move both sets of data closer to the predicted curve. Dzakowic et al. attributed their observed temperature discrepancy to possible supersaturation of the vapor and to a lesser degree, to the uncertainty in values of sonic velocity and specific heat of the vapor. T h e sonic limit is not the only factor which contributes to startup problems of heat pipes. Obviously the starting technique is also of major importance. Busse et al. (47) found it impossible to start a magnesium heat pipe unless the entire pipe was first preheated. Ernst et al. (69)experienced that the method of orienting the heat pipe during startup had a significant effect on its final operating condition. Figure 28 depicts the evaporator temperature drop as a function of difference in elevation between the evaporator and the condenser. For height

266

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

differences of less than 4.4in., no effect depending on pipe orientation was evident. Once the pumping limit was reached, however, the temperature drop was greatly influenced by the method used to attain the elevation difference. T h e upper curve represents the situation where the pipe was first placed into the desired position and subsequently heated, while the lower curve represents the case where the pipe was first heated horizontally and then subjected to the desired orientation. No satisfactory explanation was given for the difference in heat pipe behavior. McSweeney (50) measured the temperature in the vapor space of a sodium heat pipe which during the startup transient

FIG. 28. Temperature drop in evaporator versus opposing gravity head for 580°C sodium heat pipe operating at 350 W. [D. M. Ernst et al., Heat pipe studies at Thermo Electron Corporation. “IEEE Conf. Record of 1968 Thermionic Conversion Specialist Conf.” pp. 254-57.1

period unexpectedly oscillated with an amplitude of nearly 50°C and a period of approximately 10 sec. Oscillations were also observed for steady state operation of the pipe when the heat removal was highly concentrated by localized calorimeter cooling. As long as cooling was effected by free convection and radiation, no such oscillations were observed and the vapor temperature remained nearly uniform throughout the vapor space. The oscillatory behavior was attributed to either nonlinear wick characteristics or the presence of noncondensable gases. Neal (28) and Shlosinger (43) studied the startup behavior of low temperature heat pipes with the working fluid initially frozen. T h e transient temperature profile along a water heat pipe in response to

THEHEATPIPE

267

50 W of heat input is illustrated in Fig. 29. Even such a moderate heat input of only 50 W caused wick dryout and subsequent overheating in the evaporator section before the entire pipe thawed out and could begin to operate in its normal mode. With a heat input of 15 W, however, the pipe thawed without wick dryout and normal operation began after approximately one hour. In an attempt to shorten the transient period, an auxilliary heat pipe which employed a working fluid with a lower melting point was bonded to the primary heat pipe. 200 L t = l ! m i n

e g 400 V

I-

a

a W

0

i 5 I-

0 -25)

0

EVAPORATOR

10

1

20

I

30

I

40

CENTIMETERS ALONG PIPE

I

50

A 60 CONDENSER

FIG. 29. Transient temperature profile along partially frozen heat pipe. Notes: tubular stainless steel heat pipe, stainless steel screen wick, heat addition to evaporator started at time “0,”water is working fluid. [A. P. Shlosinger, Heat Pipe Devices for Space Suit Temperature Control. TRW Systems Rept. No. 06462-6005-RO-00, Nov. 1968.1

T h e use of the auxilliary pipe greatly enhanced the thawing of the primary heat pipe without local overheating and the transient period of time was reduced by a factor one half. Initially the pipe was frozen only in the condenser section and liquid was present in the rest of the pipe. Deverall et al. (84) investigated the startup behavior of a water heat pipe which was initially frozen over its entire length. T e n watts of power were supplied and the transient temperature profile, shown in Fig. 30, was measured. From the shape of the developing temperature profiles, Deverall deduced the following sequence of events: First heat was transferred in association with vapor flow along the full length of the pipe. T h e vapor was formed by sublimation in the evaporator. Since the wick was still frozen, no liquid returned to the evaporator to replace the sublimated fluid. T h e lack of return fluid resulted in a

268

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

rapid rise in evaporator temperature until it (see stations 1, 2, and 3) rose above the melting point. Water was then wicked into the heater area, and the temperature rise was temporarily arrested. Since most of the fluid was still frozen, not enough liquid was supplied to the heater. Dryout occurred and the temperature of the evaporator increased rapidly to above 100°C. When the entire working fluid was finally melted, sufficient liquid was wicked into the evaporator and the temperature fell suddenly to 44°C. Normal operation began and the pipe became isothermal.

FIG.30. Transient temperature profile along completely frozen heat pipe. [J. E. Deverall et al., Orbital Heat Pipe Experiment. N67-37590, June 22, 1967.1

Startup data were also obtained for a (nonfrozen) water heat pipe placed in earth orbit. Telemetry data indicated a frontal startup of the type illustrated earlier in Fig. 8B. T h e startup procedure and steady state operation of the heat pipe in orbit were similar to those experienced in laboratory tests. Phenomena associated with the startup behavior of high performance heat pipes have been studied by Kemme (85,86). He found that it was virtually impossible to start several sodium heat pipes with wicks consisting of axial channels (grooves) running the length of the pipe.

THEHEATPIPE

269

T h e difficulty was attributed to the attainment of the entrainment limit. As mentioned earlier, this limit is reached when a high velocity vapor stream shears liquid out of the wick and impedes a continuous supply of fluid to the evaporator. T h e entrainment limit is dependent primarily on the characteristics of the wick surface. Figure 31 shows test data

;

h

I

E4 /

W

a a

+ 2

+ a W

x n 500 600 700 000

TEMPERATURE (“C)

FIG.31. Heat transfer limit versus temperature for different wick surface configurations. [J. E. Kemme, Heat pipe capability experiments. Proceedings of Joint AEC/ Sandia Labs. Heat Pipe Conf. 1, Sc-M-66-623, pp. 11-26, Oct. 1966; J. E. Kemme, High performance heat pipes. “IEEE Conf. Record 1967 Thermionic Conversion Specialist Conf.,” pp. 355-358.1 Test number Heat pipe structure Material System length, cm Evaporator length, cm Condenser length, cm Channel width, mm Channel depth, mm Screen pore size, Screen wire diam, Tube o.d., cm Vapor flow area, cms (cross section) ~~

1 Ni

30 8 22 0.16 0.40 1.82 1.70

2

3

4

Ni 30 8 22 0.16 0.40

S.S. 85 8 77 0.75 0.75 I50 I15 1.82 1.48

S.S. 85 8 71 0.75 0.75 10 46 I .79 1.46

150 115

1.82 1.58

~

obtained for three-wick surface configurations. The maximum heat transfer was significantly improved by covering the channels with a layer of screens. T h e experimental curve labeled 4 is a good example of the entrainment limit and its shape confirms the curve initially depicted in Fig. 7. In another experiment, the effect of “screen fit” was studied and is shown in Fig. 32. T h e mechanical fit of the screen was very important in establishing the heat transfer limit of the heat pipe.

270

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

The higher limit with the loose fitting screen was probably caused by the extra fluid paths available in the annulus between the screen and the outer wall. The curve calculated for the contacting screen in Fig. 32 was based on the capillary limit and not the entrainment limit. T h e capillary and entrainment limits are both dependent on the wick structure and it is impossible to theoretically or experimentally change one without also altering the other. This dependency associated with the fact that the entrainment limit is generally attained only in a relatively narrow temperature range (between B and C in Fig. 7) has somewhat curtailed the investigation of this particular limit of heat transfer in heat pipes.

-

-3 2 5 0 0 I

5 2000 w

[L

fiGSE

i

FITTING SCREEN

FIG. 32. The effect of screen fit on axial heat flow in a sodium heat pipe with channels, Kemme [85, 861. Potassium is the working fluid, channels are 0.16 x 0.40 mm. (-) experimental, (- - -) calculated for contacting screen, [J. E. Kemme, Heat pipe capability experiments. Proceedings of Joint AEC/Sandia Labs. Heat Pipe Conf. 1, Sc-M-66-623, pp. 11-26, Oct. 1966; J. E. Kemme, High performance heat pipes. “IEEE Conf. Record 1967 Thermionic Conversion Specialist Conf.,” pp. 355-358.1

3. Basic Studies The investigations discussed in the preceding section have dealt primarily with the limits to heat pipe operation and startup characteristics. A number of additional studies have yielded information contributing to the rapidly growing heat pipe technology. Deverall et al. (60, 84, 87, 88) operated a water-stainless steel heat pipe in an earth orbit and its performance was monitored by telemetry at several tracking stations during fourteen revolutions. Results indicated that there was no degradation of the heat pipe performance in a zero gravity field. I n further tests (26,89) a similar heat pipe was subjected to various sinusoidal and random vibrations to determine the influence

THEHEATPIPE

27 1

of a vibratory environment on heat pipe performance. T h e experiments proved that vibration was not detrimental to heat pipe operation. On the contrary, vibration enhanced the wetting of the wick, forcing liquid into all parts of the wick structure, and thus actually improved heat pipe performance. Calimbas and Hulett (49) confirmed these results in a series of vibrational tests performed with a water-stainless steel heat pipe. Haskin (25) measured the total radial temperature drop in the evaporator and condenser in a low temperature nitrogen heat pipe. He found that for low heat loads (less than 36 W) the temperature drop in the condenser was larger than in the evaporator. T h e difference was assumed to be caused by the extra liquid which accumulated in the condenser during operation. For larger heat loads, the temperature gradient in the evaporator became greater. This was attributed to partial drying of the wick and to the formation of a superheated vapor film on the inner metal tube surface. Similar measurements were made by Schwartz (58) with a water-stainless steel heat pipe. For the same range of heat loads, but much higher temperatures obtained with different fluids he found that the temperature drop across the wall and wick in the evaporator was consistently higher than in the condenser. Hence, a higher condenser temperature drop is not the rule, but instead, the relation between condenser and evaporator temperature drops is dependent on the given geometry and the nature of working fluids. I n yet another study, Ranken and Kemme (48) measured the temperature variation along the length of a lithium heat pipe operating at about 850°C as shown in Fig. 33. T h e measured points have been fitted with a smooth curve which was corrected for the temperature drops arising from radial heat flows. Vapor pressure values associated with measured temperatures and with the temperature minimum are also displayed in Fig. 33. T h e temperature reaches a minimum between the evaporator and condenser. In addition a pressure recovery of 0.5 has occurred which compares favorably with theoretical predictions to be discussed in a later section. Ranken and Kemme compared data, obtained by Bohdansky et al. (90) using a lead-tantalum heat pipe, with predicted values for both laminar and turbulent flow (Fig. 34). Blasius’ turbulent flow equation gave considerably better results than the laminar flow equation, which was expected, because the axial flow Reynolds’ numbers were well over 1000 for this experiment. McKinney (35, 91) conducted extensive tests on a series of water heat pipes in a moderate temperature range (temperatures up to 400°F). Among other conclusions, which have already been amply discussed, he found that the magnitude of the radial Reynolds number had little or no effect on heat pipe operation.

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

272

CONDENSER 29 W/cmz Re,“ - 5

5

0

10 15 20 A X I A L POSITION ( c m )

25

FIG.33. Experimental observation of pressure recovery in lithium heat pipe. [W. A. Ranken and J. E. Kernme, Survey of Los Alamos and Euratom heat pipe investigations. “Thermionic Conversion Specialist Conference, IEEE, Oct. 1965,” pp. 325-36.1

CALCULATED HEAT FLUX LAMINAR FLOW

4000

/

’

3000

/

X

.

/

/ X

MEASURED HEAT FLUX

CALCULATED HEAT FLUX TURBULENT FLOW

4

IOOO 1700

I BOO

I900 T(”K)

2000

FIG.34. Results of Bohdansky heat transfer determinations compared to theoretical.

[W. A. Ranken and J. E. Kemme, Survey of Los Alamos and Euratom heat pipe investiga-

tions. “Thermionic Conversion Specialist Conference, IEEE, Oct. 1965,” pp. 325-36.1

THEHEATPIPE

273

Tien (92) measured the axial temperature distribution along the outside of a water-ethanol heat pipe. He determined the pressure inside of the pipe and compared this with the pressure which had to prevail if pure ethanol occupied the condenser and pure water occupied the evaporator. He concluded that separation into pure components in a heat pipe is extremely difficult, if not impossible, to achieve. Instead he found that if the initial composition was rich in ethanol, the data attested to the existence of a water-ethanol mixture in the evaporator while nearly pure ethanol (i.e., the azeotropic mixture) occupied the condenser section. All of Tien’s data, however, were obtained with the pipe operating vertically with the evaporator below the condenser. If the pipe contains excess liquid such an arrangement is usually referred to as a reflux condenser because gravity forces, instead of capillary forces, can always return the condensate to the evaporator. I n such cases results would have to be viewed with some caution because the wick structure may or may not significantly alter the liquid-vapor equilibrium conditions.

D. HEATPIPE APPLICATIONS Theoretically the heat pipe may be applied to almost a limitless number of thermal transport problems, which in general, can be subdivided into four broad topical categories (8-10,20) depending on the particular feature of a heat pipe which is to be exploited. These areas of possible application are: (1) Temperature Flattening, (2) SourceSink Separation, (3) Heat Flux Transformation, and (4) Constant Flux Production. T h e temperature equalizing feature of the heat pipe has prompted numerous suggestions and actual uses for the maintenance of a desired constant temperature environment. Much of the emphasis has been, and still is, focused on the problems of thermal control of spacecraft. It is well known that large temperature variations may occur on the surface of a spacecraft resulting from nonuniform heating of the craft. These temperature variations can cause a host of problems including undesirable thermal stresses. Katzoff ( 5 ) together with several other investigators (7, 93-95) recommended the use of long heat pipes, wrapped around the circumference of a spacecraft, to accomplish the necessary equalization of the temperature distribution. Naturally the evaporator sections of the circular pipes would have to face the sun while the radiation cooled condensers have to remain in the shadow of the craft. According to Katzoff, such an arrangement of heat pipes would reduce the temperature variations around a spacecraft with a

274

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

circumference of ten meters from 275 to 44°K. Anand (96,97)reported the successful employment of two Freon-1 1 heat pipes which reduced the temperature differences between transponders located in different parts of a Geos I1 spacecraft. Deverall utilized the isothermal walls of a heat pipe to measure the total hemispherical emissivity of variously prepared surfaces (23,98).Several heat pipe containers were plated or sprayed with different materials, of which emissivities were determined over a wide range of temperature with an estimated accuracy of 1 2 % . Schretzmann (99) employed an isothermal surface of a heat pipe as a metal source in a study of the effect of electromagnetic fields on the evaporation of metals. Bohdansky and Schins (100) used a heat pipe for determining pressure-temperature relations of metal vapors at high temperatures and pressures. Feldman and Whiting (9) suggested the construction of an isothermal flat plate for the installation of electronic components. They conceived a sandwhich-type plate, the interior filled with many interconnecting honeycombed heat pipes, which would rapidly distribute any localized heat flow and maintain the plate at a uniform temperature. Another use of the heat pipe would allow separation of the heat source from the heat sink. Again, possible spacecraft applications of heat pipes have received considerable attention since the heat source, for instance electronic components, is often located in the interior of the craft and the waste heat must often be transferred over some distance for ultimate rejection to outer space. Moreover, any sizeable temperature drop between the source and the radiator may induce a significant weight penalty because a large radiator area will be required at lower temperatures to dissipate the same amount of energy. Since the heat pipe is both light weight and nearly isothermal it appears to provide ideal solutions to many thermal dissipation problems encountered in spacecraft. Many investigators (13,46,61, 101-104) have considered its application to such energy dissipation systems and have cited numerous advantages including greater heat transfer per unit weight, and some degree of meteor protection when used in parallel arrangement. Werner and Carlson (105) have reported that heat pipes can operate sixty times more effectively as radiators than solid rods based on heat transfer rates per unit weight. Deverall (23) lends support to this claim by reporting that a silver heat pipe is 520 times more effective than an equivalent solid tantalum rod. A team of investigators at RCA (20,51) has designed, constructed and successfully tested a space radiator composed of one hundred individual heat pipes. T h e system which weighs less than twelve pounds is capable of rejecting 50,000 W of thermal energy at a temperature of 1420°F. McKinney (35) recommended the installation of cryogenic heat pipes around cryogenic

THEHEATPIPE

27 5

storage tanks. If the heat pipes contained a working fluid having a lower boiling point than the stored fluid it should be feasible to transfer the heat, leaked from the immediate surroundings into the storage vessel, to a remote sink for dissipation. Researchers at Los Alamos Scientific Laboratory explored a unique scheme in which the construction of a heat pipe plasma oven was proposed. T h e disposal of waste heat given up by electronic components at remote locations has already been mentioned several times (11, 49). Another feature of the heat pipe, which has generated much enthusiasm, is its ability for thermal flux transformation. Heat addition to and heat removal from a heat pipe are feasible across heat flow areas of different size. This potential for heat flux transformation has stimulated thermionic specialists to consider the conversion of low heat fluxes generated by radioactive isotopes, for instance, into a sufficiently high heat flux which is required for the operation of a thermionic converter. Leefer (64) achieved a flux transformation with a flux concentration ratio of approximately ten to one equivalent to an output flux density of 250 W/cm2 which is more than adequate to meet the requirements of a thermionic converter. Other investigators also reported measured flux conversions of a ratio of ten to one (107) and even twelve to one (68, 74). A number of papers have appeared in the literature (23,64,65,71,108117) extolling the virtues of heat pipes when used in conjunction with thermionic converters. Numerous heat pipe-thermionic converter assemblies have been built and tested. T h e results have generated much optimism and it is believed that the heat pipe applied to thermionic converters will reach technical maturity in the near future. Heat pipes may also be used to “flatten” flux variations supplied by an unsteady heat source. Researchers at RCA (107) have developed a “classified” heat pipe which supposedly maintains a constant thermal output flux independent of variations in thermal input flux up to a factor of eight or more. For instance, decaying radioisotopes could be used to provide a constant heat flux for at least three half lives. A qualified thermodynamicist might want to study these more exotic schemes in view of their compatibility with the requirements of the Second Law of Thermodynamics. Two particular potential heat pipe applications were proposed by investigators at the Lawrence Radiation Lab. Hampel and Koopman (32) suggested the utilization of the heat pipe for the control of small, fast-spectrum, high temperature reactors. His scheme is based on the capability of the heat pipe to respond to a sudden increase in heat flux with an increase in evaporation rate. If the evaporator ends of a sufficient number of heat pipes, filled with special working fluids having a negative void coefficient, were installed in the reactor core, an increase in core

276

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

heat flux would result in a decrease of the total mass contained in the core (evaporators). Such a hypothetical mass transfer device may eventually be used to provide some sort of reactor safety control. Werner (118) proposed to employ lithium heat pipes as tritium producers in the blanket structure of a reactor. T h e scheme calls for the transport of the tritium within the heat pipe to an accessible processing point outside of the blanket where it could be removed by diffusion or equivalent means and then be used to replenish the tritium consumed in the core.

E. HEATPIPE CONTROL Inasmuch as the heat pipe transfers energy between two points utilizing a continuous mass circulation, it is apparent that some degree of heat pipe control may be exerted by controlling the mass flow. Katzoff ( 5 ) suggested several concepts to accomplish thermal control with the heat pipe serving as a variable thermal conductor. One technique involves the intentional introduction of a noncondensable gas into the vapor space. As discussed earlier, the gas tends to collect in the condenser where it forms a relatively stagnant gas zone which effectively eliminates any working fluid condensation. T h e length of the gas zone, of course, depends on the working pressure in the pipe; the zone length decreasing with increasing pressures. T h e gas layer can be exploited in several ways. For example, suppose it is desired to always furnish solar energy to an instrument inside of a spacecraft regardless of the crafts orientation with respect to the sun. T h e task could be accomplished by mounting the instrument in the center section of a heat pipe containing some noncondensable gas. If the amount of gas was such that it filled slightly less than half of the pipe, then the instrument would always receive energy from the sun. T h e end of the heat pipe on the shaded side of the spacecraft should remain inoperative due to the blocking effect of inert gas and thus minimizing heat losses. T h e opposite problem could be solved by using a partially dry heat pipe. Now the dry part of the heat pipe, and hence the inoperative part, could always face the sun, and the instrument which might generate heat could reject this heat by radiation from the shaded side of the spacecraft to outer space. Besides Katzoff, Wyatt (119) and Anand et al. (79) have suggested the use of thermostatically controlled valves and/or bellows to supply or withdraw the noncondensable gas. Such an arrangement would allow the effective condenser area to be varied independently of the operating pressure prevailing in the pipe. Katzoff has also recommended a control

THEHEATPIPE

277

technique which involves the interruption of the liquid flow in the wick. This control technique would be practical only in wicking structures in which the bulk of the flow takes place in an artery. A thermostatically controlled valve could be used to impede, or stop the flow in the artery, in response to the control requirements. Anand et al. (79) and Shlosinger et al. (42-44) also considered a similar arrangement for controlling the vapor flow. Shlosinger incorporated this concept into the design of a variable conductance space suit. His design is illustrated in Fig. 35. BELLOWS OPERATED VAPOR CONTROL

Drr,

rPf,,

,r\

TrcrLcL I I V I

RADIATION TO SPACE

VACUUM INSULATION

EVAPORATII CHAMBER WICK

SUIT PRESSURE INCLUDING HEAT PIPE IN ALL CAVITIES

FIG.35. Schematic cross section of variable conductance space suit shell. [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; A. P. Shlosinger, Heat Pipe Devices for Space Suit Temperature Control. TRW Systems Rept. No. 06462-6005-RO-00, Nov. 1968; A. P. Shlosinger et al., Technology Study of Passive Control of Humidity in Space Suits, N66-14556.1

T h e design concept provides for both insulation when the valve is closed and for heat rejection when the valve is open during periods of high metabolic heat generation of the space suit wearer, Experimental results were interpreted to consider such a technique as feasible, but calling for much more research. Anand et al. (79) also recommended a control technique involving the use of two fluids whose pressuretemperature curve intersect at the desired operating temperature. I n anticipation of the difficulty of finding such fluids, this method of control has received only minor attention. Workers at Honeywell (27) have studied the concept of bellow controlled feeder wicks which are either in contact with the heated surface thus providing a path for the condensate return, or the contact is interrupted, thus preventing fluid return and hereby shutting off the heat pipe action. Heat pipes acting as “thermal switches” were built primarily in an effort to develop

278

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

variable conductance walls. Conductivity ratios of 150 have been reported depending on the position of the feeder wick. Obviously a great many possibilities for heat pipe control exist; however, most of these concepts remain still in the dreaming stage, and few controllable heat pipes have actually been built to date. Much more research will have to be devoted to this particular area of heat pipe application so that eventually some thermal problems can be solved with controllable heat pipes.

F. HEATPIPETHEORY T h e heat pipe theory developed up to date is due, in large part,

to a stimulus provided in a theoretical study performed by Cotter (4).

H e formulated the governing equations describing the processes taking place in the heat pipe and also developed a model to predict the capillary limit of heat pipe operation. His results have been used by other investigators who have often simplified or modified his equations to suit their particular assumptions and/or geometries. Because this has been done so extensively, Cotter’s analysis will be presented in necessary detail. T h e description of the analysis will be followed by a number of modified theories and by several other analytical approaches, all of which are concerned with the capillary limit. Derivations for the predictions of the other limits of heat pipe operation are given subsequently and finally several analyses which deal with specific problems conclude the analysis section. Cotter studied a heat pipe as shown in Fig. 36. The capillary structure is assumed to have a pore radius r0 and to be completely saturated with a working fluid. T h e radius of curvature of the meniscus surface

FIG. 36. Cylindrical heat pipe structure. [T. P. Cotter, Theory of heat pipes. Los Alamos Sci. Lab., LA-3246-MS, Feb. 1965.1

THEHEATPIPE

279

is thought to be dependent on the distance x and the pressure difference across a surface as a function of position is given by Pv(z) - Pl(Z) = 2a/r(z) = 2a cos B/r,

(10)

where ~ ( xis) the local meniscus radius of curvature. In Eq. (lo), of course, it is assumed that the meniscus is represented by one radius of curvature only. If, in fact, a complex meniscus shape were formed, the term (2/r(z)) in Eq. (10) should be replaced by (( 1 /rl) (1 / r z ) ) where rl and r2 are the two radii of curvature necessary to describe a three-dimensional surface. If a heat pipe is to be operated in a gravity field, then the maximum length z , ~ ,of~ the pipe is restricted by the lifting ability of the combined wick and liquid system. T h e maximum length is given by the well-known relation

+

Zmax

cos p =

(2a cos elpl g r C )

(1 1)

Cotter next considered the steady state operation of a heat pipe and applied the conservation of mass principle to arrive at the relation fh&)

+

fhi(Z) = 0

(12)

where the two mass flow rates are both positive in the plus z-direction. T h e pressure gradient in the liquid was determined from the NavierStokes equation for steady, incompressible, constant viscosity flow. By neglecting the inertial term and by modifying the viscous term with a dimensionless constant to account for different wicks, the pressure gradient in the z-direction was found to be dPl/dZ

p1 g cos /3

-

(bplrizl(z)/n(rW2- rv') p l E Y,')

(13)

where the dimensionless constant b is defined as crC2/Kand has a value of approximately 8 for nonconnected parallel cylindrical pores, and 10-20 for more realistic capillary structures with tortuous and interconnected pores. T h e radial pressure gradient has been assumed negligible as will be the case for long thin pipes, i.e., for those pipes for which rvl 3 rW2. T h e pressure gradient in the vapor was found by employing the results of Yuan and Finkelstein (120) and Knight and McInteer (121). These authors assumed incompressible laminar flow and uniform injection or suction at the vapor space boundary. T h e applicability of these results is based on the value of the radial Reynolds number Re, which is defined by =

-(pvrvvr/pv)

= (1 /(2vpv))d ~ v / d Z

(14)

280

E. R. F.

WINTER AND W.

0. BARSCH

It is positive for evaporation and negative for condensation. For I Re, I 1, the vapor flow is dominated by viscous forces and the

<

velocity profile approximates the usual parabolic shape for Poiseuille flow. For this case the vapor pressure gradient is given by dPvjdx

>

=

-(8pv&/npvrv4)(1

+ (3/4)Re, - (11127) Re? +

..a)

(15)

For I Re, I 1, however, the flow is qualitatively different in evaporator and condenser. For high evaporation rates, the velocity profile is not parabolic but is proportional to C O S ( T / ~ ) ( Y /which Y ~ ) ~ , was verified experimentally by Wageman and Guevara (122), and the pressure decreases in the direction of flow. For high condensation rates, on the other hand, the velocity profile is nearly constant across the vapor space with the transition to zero velocity occurring in a thin layer near the wall, and the pressure increases in the direction of motion due to a partial dynamic recovery in the decelerating flow. For I Re, I -+ co, the pressure gradient is found to be

where S = 1 for evaporation and S = 419 for condensation. Furthermore, Cotter hypothesized that for some situations where the Re, = 0 and the average vapor velocity is high, as might exist in heat pipes with a long adiabatic section, fully developed turbulent flow may occur. For such a case, Cotter recommended using

dPvldz = -0.0655pV2 Re7f4/pvrv3

(17)

in lieu of Eq. (15). The relation between vapor and liquid pressures and the mass flow rates is next given by a formula supplied by the kinetic theory. dmlldx = -dmv/dz = arv(Pv - P v a p ) / ( R T / 2 ~ M ) 1 / 2

(18)

where 01 is a numerical factor which includes the probability of condensation of a vapor molecule and the surface roughness of the meniscus. T h e value of a is very nearly unity. Cotter next considered the conservation principle of energy and by neglecting radiative and conductive contributions he arrived at

Finally, Cotter coupled the heat pipe to the surrounding environ-

28 1

THEHEATPIPE

ment by expressions accounting for heat fluxes or imposed temperature conditions, respectively,

where Eq. (21) relates the temperature on the outer surface of the heat pipe to the temperature at the liquid vapor meniscus. Equations (10)-(21) are generalized heat pipe relations as presented by Cotter. These equations can be solved if the vapor has a nearly uniform temperature To throughout the vapor space. If Eq. (21) is solved for T,,[T(z,rp)], then H in Eq. (20) may be expressed as a function of length z , temperature distribution in the vapor T,[T(z,r,)], and the heat flow Q. Also, if the heat flow through the pipe ends is denoted by Fo(T,) and F,(T,), then the average vapor temperature To and the corresponding axial heat flux distribution Qo(z)are determined from the relations dQoldz = H ( z , To Qo@) = Fo(To)

Qo(1)

9

Qd

= *I(

TO)

(22)

With this approximation for the heat flux, the mass flow rates may be determined from Eqs. (12) and (19). T h e vapor mass flow rate can then be used in the appropriate equation, (I 5 ) , (16), or (17), to determine the pressure distribution in the vapor. Also Eq. (13) may be integrated to yield the axial pressure distribution in the liquid. Finally, Eq. (18) can be solved for the vapor pressure of the liquid to within an additive constant. Cotter recommends using P(To) for this constant. Since the vapor pressure is a known function of liquid surface temperature, T J x ) may be determined. T h e consistency of this procedure can be verified if the axial variation of T, is small compared to T o . Cotter used this procedure for the special case of constant heat addition along the evaporator and constant heat removal along the condenser. Thus he obtained

where Qe is the total heat input to the evaporator. By employing Eqs. (23) in (15) and (16) and neglecting the term Re: in the former,

282

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

and assuming the vapor density pv to be constant in both, a straightforward integration yields

Likewise, integration of Eq. (13) yields APl = Pl(1) - Pl(0)

= pi gz cos

p

+ bp1Qez/2v(rw2- rva)pi

E r&ig

(25)

The temperature difference in the vapor space can be obtained from Eq. (1 8) leading to

APvap

= APv

-

IQe(RTo/2~M)”’/le(l- Ze) hfgarv

(26)

And finally by considering the vapor as an ideal gas, application of the Clapeyron-Clausius equation and neglecting the volume of the liquid phase results in (27)

AT, = RTo2APv,/MhigP(T,)

The preceding equations were applied to a horizontal sodium heat pipe and the results of those calculations are listed below. The left-hand column indicates the pertinent values used and the right-hand column shows the calculated parameters. Qe =500 W To= 920vK I = 9 0 em Ze = 13 cm rv = 0.64 cm rw = 0.80 em I , = 0.012 cm

niv(Ze) = 0.1

gm/sec

Pvap(T0) = 50 mm Hg AP, = -0.2 mmHg APvLp= -0.5 mm Hg 4 4 = 2 mmHg AT, = -7vK

An inspection of the computed values explains why the heat pipe has generated so much enthusiasm among thermal engineers. A relatively large energy transport is accomplished with an almost negligible temperature drop. Cotter next considered the maximum heat transport which is possible in a heat pipe which is limited only by the pumping ability of a wick in association with a given working fluid. The maximum pressure

THEHEATPIPE

283

difference which can be supported between the liquid and vapor is achieved when the meniscus radius in the evaporator achieves a minimum value, i.e., equal to r , . Therefore Eq. (10) becomes Pv(z)- Pl(X)

< 20 cos e/rc

(28)

which must be satisfied for all positions x. I n particular, for x = 0, Eq. (28) may be rewritten with the aid of Eqs. (24) and (25) to give the capillary limiting condition for the total axial heat flux

+ p 1 gz'OS

where Pv(0) - Pl(0) = Pv(1) - APV

'

bp1Q.d

+ 2n(rw2 - rv2)

+ dP1 - S ( Z )

==

E

rc2hfg

AP1 - APV

(30)

has been used to simplify the left-hand side of (28) prior to the substitution of Eqs. (24) and (25). Cotter next determined the optimum capillary pore radius, Y , in terms of Qe by considering Eq. (29) where the equality sign is used and the first term is simply --dP,. Now since AP, is directly proportional to Qe (or Qe2), the heat transported is a maximum if dP, is a maximum. T h u s Eq. (29) becomes APv = pllg cos fl

+ bplZQe/2~(r,~-

YV')

pl E

hfgrc2

(31)

With Eq. (31), the pressure drop AP, becomes a maximum when rc

= bpllQe/2n(rW2- rv2)p1 E hfgo cos 0

(32)

T h e maximum heat transport is determined by converting Eq. (28) to the form AP1 - APV - 20 cos 9/rc = 0 (33)

<

Now for Re, 1, and with no hydrostatic contribution to LIP, , Eq. (33) is used in conjunction with Eqs. (24), (25), and (32) to solve for Qe , leading to

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

284

Obviously 9, becomes a maximum when the term rV2(rw2- rv2) assumes a maximum. Simple calculus shows that this situation is achieved when rv/rw = 213

(35)

Substituting Eq. (35) into (34) finally yields

Q~ = (rrrw3hfgucos ei31)(2pvpl€/3bCLvpl)i/z

(36)

Inserting Eqs. (36) and (35) into (32) yields for the optimum pore size 1 for Re,

<

rc = rW[bp~~V/6~1pv~11~z

(37)

The analogous expressions to Eqs. (36) and (37) for Re$ 1 are Qe = + ~ w ~ h & v p pCOS’ ~ ~ d~/ ( r 2 - 4) bZp,)”3

and rc

=

[8pvb2plV/(,2 - 4) p12E% cos e]ll3

(38) (39)

Cotter’s model for Re,> 1 predicts that the maximum extent of pressure recovery is fixed at 4/m2 corresponding to 40.5% of the drop that occurs in the evaporator regardless of the amount of heat transferred. Therefore, the profile illustrated in Fig. 37 represents the vapor pressure distribution for any case which is dominated by inertial forces. It should be recalled that the profile in Fig. 37 is dependent on all the assumptions made by Cotter in his analysis and in particular, on the assumption that the vapor is incompressible. Parker and Hanson (24) wrote a digital computer code in which they consider the vapor to be compressible, and consequently, treat the vapor density as a variable. A

HEAT Ih

HEAT OUT

FIG. 37. Theoretical axial profile of pressure in vapor. [T. P. Cotter, Theory of heat pipes. Los Alamos Sci. Lab., LA-3246-MS, Feb. 1965.1

THEHEATPIPE

285

comparison of their results with Cotter’s predictions demonstrates excellent agreement for the pressure profile in the evaporator section. However, a significant departure from Cotter’s prediction was found to exist in the condenser section. Here, the extent of pressure recovery was dependent on the heat transported by the pipe. Figure 38 illustrates the predicted pressure recovery as a function of the heat transported for a particular sodium heat pipe. The extent of pressure recovery is seen to be considerably larger than that predicted by Cotter for large heat transport rates and hence the net end to end pressure drop in the heat pipe will be smaller. When compressibility effects are important, Cotter’s analysis underpredicts the maximum heat transfer capability of the heat pipe, and it may be used to provide a conservative estimate of the capability of the pipe. Of course, an even more conservative estimate could be obtained by neglecting pressure recovery altogether.

Y

2?

HEAT TRANSPORTED ( k W )

FIG. 38. Condenser pressure recovery versus heat transported. [G. H. Parker and J. P. Hanson, Heat pipe analysis. Adwan. Energy Conwers. Eng. 847-57 (1967).]

Ernst (54) has disputed the validity of the meniscus boundary conditions employed by Cotter for the special case of large radial Reynolds numbers, zero gravity, and incompressible vapor flow. I t should be recalled that Cotter assumed a meniscus profile of the type shown in Fig. 39. Using this profile, he went on to optimize the capillary radius and heat transfer rate and arrived at Eqs. (29), (38), and (39). Ernst has calculated the pressure profiles based on these equations for the special case of a sodium heat pipe operated at 700°C with rw = 1 cm, I, = 50 cm, and I, = 50 cm. His profiles are illustrated in Fig. 40 by the lower set of curves. Figure 40 shows that at the condition of maximum heat transfer predicted by Cotter, the pressure gradient in the liquid

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

286

--__ -EVAPORATOR

------CONDENSER

*

N

E 122

-

d I

-

0 4 8

L

12 16 20 24 28 32 36 40 44 48 LENGTH(cm)

FIG. 40. Pressure profile for 700°C sodium heat pipe. [D. M. Ernst, Evaluation of theoretical heat pipe performance. “Thermionic Conversion Specialist Conference, Oct. 3CNov. 1, 1967,” PP. 349-54.1

is smaller than that in the vapor. Consequently, the pressure of the liquid in the condenser is greater than the pressure in the vapor. Ernst has concluded that for such a situation, a meniscus profile of the type illustrated in Fig. 41 must exist. However, he regarded this meniscus profile as unrealistic and instead proposed the profile illustrated in Fig. 42 for the nonoptimized capillary case. In this revised profile, pressure equality between the liquid and the vapor is assumed to occur at the interface between the condenser and the evaporator instead

n------

I

EVAPORATOR -CONDENSER

n

FIG. 41. Interpretation by Ernst of Cotter’s liquid profile for optimized capillary at maximum Q. [D. M. Ernst, Evaluation of theoretical heat pipe performance. “Thermionic Conversion Specialist Conference, Oct. 3&Nov. 1, 1967,” pp. 349-54.1

THEHEATPIPE

--_-

-

--

EVAPORATOR

7

287 ---r

CONDENSER

-

of at the condenser end as assumed by Cotter. For the special case of a wire screen capillary structure, Ernst has derived the following equations for the pressure balance in the evaporator and condenser, the optimum mesh opening half-width, and the maximum heat transfer capability, respectively. Evaporator:

Optimum mesh opening half-width:

Maximum heat transfer: 1

(1

+

+ K ) 2((7774) +

,

vple02

cos2

bpdc

1

e

113

where Y, = d( 1 K) and K = d,/d has been used in deriving Eqs. (40)(43)and rv2 = -:rw2 has been applied to the derivation of Eqs. (42)and (43).Ernst also applied Eqs. (40-(43) to the same 700°C sodium heat pipe discussed earlier. His pressure profiles are illustrated by the upper set of curves in Fig. 40. The pressure equality between the liquid and vapor in the condenser implies a meniscus profile of the type shown in Fig. 43. A comparison of the revised maximum heat transport Eq. (43),with the expression derived by Cotter, Eq. (38), lead Ernst to believe that Cotter’s expression tends to overpredict the maximum heat transfer capability of the heat pipe. In particular, for the 700°C sodium heat pipe for which the pressure profiles in Fig. 40 were calcu-

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

288

FIG. 43. Revised liquid profile for maximum Q , optimized capillary. [D. M. Ernst, Evaluation of theoretical heat pipe performance. “Thermionic Conversion Specialist Conference, Oct. 3CNov. 1, 1967,” pp. 349-54.1

lated; this overprediction is approximately 40%. T h e extent of overprediction, of course, will vary for different heat pipe geometries. Ernst also claimed that the maximum heat transfer capability of a heat pipe can be increased if the wick in the evaporator is different from the wick in the condenser. If the optimum mesh half-width in the evaporator and condenser are, respectively, and

[( 1

+ K) 8b2pVp12Ze2/n2p12&cos 8]1/3

de

=

dc

= [4xbpiLpv~w~heg/3pi E Qm]1’2

(4.4) (45)

then the expression for the maximum heat transfer, given previously by Eq. (43), becomes Qm

= &Yw2heg[(

+

1/d( 1 K)’) . (pvpla2 COS’ 8/bp,le)]’13

A comparison of Eqs. (43) and (46) for a typical case where

(46) E

= 0.8

and 1, = 1, accompanied by some calculations shows that the use of different wicks in the evaporator and condenser may increase the maximum heat transport capability of heat pipes by over 15%. T h e percentage increase varies from pipe to pipe depending on the evaporator and condenser lengths, and on the porosity of the capillary structure. Many investigators have suggested various alternative expressions for the pressure drop in the gas and liquid for different heat pipe configurations which may be used in Eq. (29) instead of the pressure drops predicted by Cotter as given by Eqs. (24), (25). Bohdansky et al. (46), for instance, recommended the following expressions for a heat pipe which employs axial channels for the liquid return and for which zero g, laminar, incompressible flow is assumed in both the liquid and the vapor: and where K is a channel shape correction term which is approximately 1.3

THEHEATPIPE

289

for channels of rectangular cross section and a depth equal to twice the width. Equations (47) and (48) were used to derive the following relations for the maximum heat transfer rate: Qmax = (7.;3hf9/6~l)(K4A/2yV2y1)1’3

(49)

where a is a channel shape factor which has a value of about 2 for the channel discussed above, and A is another dimensionless number defined by n = Arv/rch (50) Notice that A is dependent on the number and spacing of the axial channels and has a maximum value of 7. Equation (49) is of limited importance to the design engineer because of the large number of dimensionless constants which must be employed, and the lack of a method for the determination of these constants. Busse et al. (47)modified Eq. (49) so that all the dimensionless parameters are incorporated into one constant. Their equation for the maximum heat flux is qmax = (0.13 Sr,&,

cos O/Zeff)

- (1/ v ~ ’ ~ v ! ’ ~ )

(51)

where S is this dimensionless constant. If the channel depth is I, , the channel width is I,, and the dimension between channels is then 6 is approximately 0.9 €or I, = 21, = 21, and 6 is approximately 0.5 for I , = I, = I,. The term Zeff is defined by Busse as leff = (1/qmax)

I

q(2) d z

(52)

and is the so-called effective length of the heat pipe where g(z) is the average axial heat flux at any position z averaged over the pipe cross section. The inclusion of Eq. (52) into Eq. (51) would, however, cause the term of interest, i.e., qmal, to cancel. A more straightforward approach is to simply replace leffin (51) by 1, the total heat pipe length. Bohdansky and Schins (90)recommended the following expressions for the vapor and liquid pressure drop for turbulent flow: and

APv

= (0.065/~~’”(Q/hig)’’~ (p~’4Zeff/pv~~Q’4)

(53)

API

= (0.065/1~~/~)(Q/nhf~)”~ (p~’4Z,ff/~l(Kreh)’Q’4)

(54)

Equation (53) is obtained by a straightforward integration of Eq. (17). T h e expression for AP, is obtained in a similar manner but has been modified to account for channel shape by the dimensionless K term.

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

290

As suggested above, leftshould perhaps be replaced by the total heat pipe length. Equation (54)is very likely of limited value because the liquid flow will generally be laminar even for very high heat transfer rates. Busse (123) considered laminar vapor flow in a cylindrical heat pipe which had an adiabatic section separating the evaporator from the condenser. For the case of constant heat addition and removal the vapor flow was described by the Navier-Stokes equation which was solved by approximating the axial velocity profile by a fourth power polynomial of the radius. The analysis furnished a velocity profile which was relatively constant along the evaporator; i.e., it approached the Poiseiulle profile in the adiabatic section, and deviated considerably from the Poiseiulle profile in the condenser. The pressure drop in the evaporator is given by

where Vm is the axial velocity averaged over the cross section in the evaporator, and A is given by

Equation ( 5 5 ) may be approximated with an error of less than 1 yo by PV(4

-

W O ) = -4cLvvmze r"2

[l

+ 0.61 Re, +

0.61 Re, 3.6 Rer

+

I 1,8 z2

(57)

The pressure drop in the adiabatic section is given by

where Re is the axial Reynolds number defined by Re = 2rvVmpv/pv and a is a correction to the Poiseiulle velocity profile defined by a exp( -22475) = A exp( - 2 2 4 7 5 ) exp( -144z/5rvRe)

(59)

Typical values of a are illustrated in Fig. 44 for several radial Reynolds numbers. Equation (58) can be approximated with an error of about 1 % by

PV(4

- PV(0)

=

"1

0.106 Rer 1 - exp(-3Oz/r R ) 18 5 Re, x/rvRe

+

(60)

THEHEATPIPE

-

0.75

t

0

-0.50

I

I

II

-

0.50

-0.25

1

I

Re,=al

29 1

Re: = 0

I

I I I I I

I I SHIELDED ZONE

HEATING ZONE

\ - 1.67

1

COOLING ZONE+

I

FIG.44. Velocity profile correction versus position in heat pipe. [C. A. Busse, Pressure drop in the vapor phase of long heat pipes. “Thermionic Conversion Specialist Conference, Palo Alto, California, Oct. 3CNov. 1 , 1967,” pp. 391-98.1

The pressure distribution in the condenser section is given by

where now a is found by the solution of

18

Here a, is the velocity profile correction at the beginning of the condenser 0.655. Busse plotted section. From Fig. 44 it is seen that 0 a, the dimensionless pressure distribution in the condenser section as a function of dimensionless length as illustrated in Fig. 45. Figure 45 represents the special case where the adiabatic section is so long that the flow at the beginning of the condenser has assumed a Poiseuille velocity profile, i.e., a, = 0. It also shows that up to a certain radial Reynolds number a pressure minimum occurs at the end of the condenser and that increasing condensation rates tend to move this minimum toward the beginning of the condenser. The dislocation of this minimum

< <

292

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

t

Z/P,

-

FIG.45. Pressure profiles in condenser. [C. A. Busse, Pressure drop in the vapor phase of long heat pipes. “Thermionic Conversion Specialist Conference, Palo Alto, California, Oct. JO-Nov. 1, 1967,” pp. 391-98.1

indicates a partial pressure recovery. The pressure drop of the vapor which should be employed in an overall pressure balance equation, such as Eq. (29), depends on the net heat transfer of the heat pipe. Busse has mentioned that for strong heating and cooling rates a pressure equality exists in the vapor and the liquid phase in the proximity of the beginning of the condenser. The observation was substantiated by Bohdansky et al. (76); recall that Ernst also had noted this equality in pressures. For such a situation, only the vapor pressure drop in the evaporator and adiabatic section has to be taken into account for the calculation of the maximum heat flow. Thus a combination of Eqs. ( 5 5 ) and ( 5 8 ) yields the total pressure drop in the vapor expressed by Busse as d p v = (4pvJ7m/rva)[441

where F = - 7- > +8a9 27

+F Re,) + 2 4 23aO2 405

(64)

THEHEATPIPE

293

T h e factor F can be approximated with an error of less than I

% by

and has values ranging from 0.61 to 0.81. On the other hand, for low heating and cooling rates, i.e., small Re,, the pressure equality between the vapor and liquid will generally exist at the far end of the condenser. For this case knowledge of the pressure drop in the entire vapor space is required and Busse combined Eqs. (55), (58), and (61) to arrive at the simple result APv

= (4pvVm17v2)(1e

+ 21, + lc)

(67)

The relation was previously obtained by Cotter, Eq. (24), for a heat pipe with no adiabatic section, i.e., I, = 0, where the bracketed lengths in equation (67) were replaced by the total heat pipe length. It is interesting to note that the length of the adiabatic section has more influence on the pressure drop than the contributions caused by the lengths of the evaporator or condenser. Haskins (25) has dealt with a heat pipe containing an adiabatic section. He transformed Eq. (23) into

I

ZQelze ;

Q o ( z ) = hr&v(z) = Qe ; (I - z)Qe/lc ;

O
< < < <

(68)

T h e pressure drop in the evaporator and condenser was obtained by substituting Eq. (68) into Eq. (15) and its subsequent integration. The pressure drop in the adiabatic section was found by assuming the existence of Poiseuille flow. The net end to end pressure drop for 1 is given by the sum of the individual contributions Re,

<

APv

=

-4pvQ)e(l

+ la)/npvrv'hg

(69)

This expression is identical to Eq. (67) which was derived by Busse. For the case of Re, 1, Haskin used Eqs. (68) and (16) for the condenser and evaporator and applied Eq. (17) to the adiabatic section and obtained

>

APv =

( )"*

-( 1 - 4/7r2)Qe2 - 0.O655p+''la Q 8Pv7v4h;g W V 3 "h&

(70)

Notice that both Eqs. (69) and (70) reduce to Cotter's results, Eq. (24), for the special case of I, = 0, The equation for the pressure drop in the liquid was found to be identical to that obtained by Cotter, Eq. (25),

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

294

with the exception that the length 1 in the viscous term is replaced by (1 1,). Schwartz (58) has used Cotter’s analysis and its extension by Haskin to display the liquid and vapor velocity and the vapor mass flow rate as a function of the heat load for a water-stainless steel heat pipe. The wick consisted of two layers of 100-mesh screen and the pertinent dimensions of the pipe were; 1, = 3 in., I, = 3 in., 1, = 8.5 in., and I, = 0.189 in. The results are presented in Fig. 46, and are typical for a low temperature heat pipe. The vapor velocity decreases with increasing heat load due to the large vapor density variation in the temperature range of interest.

+

0

.

2

5

150

7

- 20x10-3

0.20 -

0.15

- !5~10-~ P 2 -.>-

-

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

<

300

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

FIG. 50. Optimum value of Qmax/rna. Qmax/*raC =

[$'/(I

+ $la] [(lo$-

1 - &/(I

+ 7$)1

[S. Frank et al., Heat pipe design manual. Martin Nuclear Report MND-3288, Martin

Marietta Corp., Baltimore, Maryland, Feb. 1967.1

-

Ji

FIG. 51. Optimum value of a. = +(3

+ $)(1 + $)/(lo$-

1 - $9

[S. Frank et al., Heat pipe design manual, Martin Nuclear Report MND-3288, Martin Marietta Corp., Baltimore, Maryland, Feb. 1967.1

THEHEATPIPE

30 1

Fig. 51 once the iteration procedure has converged to constant values of QV. Frank et al. claim that this convergence is rapid and is generally accomplished by approximately three iterations. Completion of this procedure finally yields the optimum values of Qmsx/~,S and the aspect ratio, a. This implies that for a given heat pipe radius the maximum heat flow may be found, or conversely, for a given heat flow, the minimum pipe radius may be determined and, moreover, for both situations, the optimum groove shape (as given by the aspect ratio). Only the capillary limit to the maximum heat flow has been taken into account in the iteration procedure and other limits, such as the entrainment limit which is especially important for open grooves, may well pose additional restrictions and warrants further consideration. McKinney (35,92) discussed a heat pipe having an adiabatic section. He found for Re, Q 1 the pressure drop in the vapor to be identical

c-

0.2

0.1

-

0.4

0.3

J,

FIG.52. Optimum value of (I6//3)(.vh)

=

4. (1 - @(2

- $)(I

+ 7$)/@(10$

-1

- 4'9'

[S. Frank et al., Heat pipe design manual. Martin Nuclear Report MND-3288. Martin Marietta Corp., Baltimore, Maryland, Feb. 1967.1

302

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

4

I

0.008

q/Rw’ 1 kW/cm 0.004 0.10

0.45

FIG. 53. Graph for determining @V

0.20

dJ

0.25

0.30

0.35

.

@ . , ( h ~ ~ )= ~ / 0.00352[(q/Rv)((1 *

+ 4)/(1 - 4))13’4

[S. Frank et al., Heat pipe design manual. Martin Nuclear Report MND-3288, Martin Marietta Corp., Baltimore, Maryland, Feb. 1967.1

to that reported by Haskin, Eq. (69). The resulting expression for the maximum heat transport rate was given as

>

For the case of Re, 1, McKinney employed Cotter’s results for the evaporator and condenser and assumed Poiseuille flow in the adiabatic section finding the heat flow

THEHEATPIPE where

303

D =

(1 - 4 / 4 8P""gY,4

E=

8~vla ~ 1 ( 1 la) .rrPvhrgrv4 + 2Kplhfgn(rw2 - T"')

and

>

+

(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

3 12

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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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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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 . . .

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. 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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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,

FILMCOOLING

323

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.

324

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.

FILMCOOLING

325

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.

326

RICHARD

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

FILMCOOLING

327

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,

RICHARDJ. GOLDSTEIN

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

FILMCOOLING

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,

'<

Taw

105

6.1 x 105

0.244

0.333 0.31

42.0

to

3.11

4.97

Effecq/Taw/Cw tiveness

lo5

0.2

Taw 9

Taw

1.37 3.4

5.1 to

to

14.0

5.2

0.0

15.8

1.59 3.175 6.35

0.795

29.5

1.59 3.175 6.35

0.0 to

6.0

5.0

0.118

30.0

50.0

0.20

29.0

A

to

to

to

1.59

1.03

38.0

0.218

50.5

3.2

to

to

0.41 < lo5 to

0.76 x lo5

0.3 * to

1.2

9

Taw

Taw

20.0

3.175 6.35

to

Taw Q

3.3 x 105 to

0.24

Taw

to

16.8 x lo5

1.36

6.5 x. 105

0.2

T.,

1.07 0.013

30.5

0.042

55.0

0.23

15.2

2.48

30.5

to

to

to

to

1050

I050

35.6

0.59 ~-.lo5

0.0359

lo5

0.0403

to

1.06

x

to

3.175 6.35 9.525

Taw

Taw

1.63 3.12 4.62

4.0

1.63

4.0 x 105

Y

105

0.045 to 0.127

Tow

0.127

Taw

taken from Seban (33). ' Based on total step height.

' Based

Slot Reynolds No., Re,

Range of r/s

0.999

530

6.6

0.139

7700

390.7

0.85

1510

4

0.125

2880

140

0.96

2,200

6 to

to

to

to

to

to

to

to

0.19

to 10,OOO

1.0

3530

17

0.2

6960

300

0.95

0

0.06

6600

10 to

0.7

0

1.4

0.035

2200

180

0.95 to

lo00

5

0.06

7000

280

1.0

to

to

to

to

to

to

to

198

to

150

to

to

to 0.10

1800

2

8500

275

0.85

850

1.5

0.05

5800

17.44

1.0 to

1420

3.6

0.2

22,900

275.0

1.0 to 0.1

0

o j

11,100

73

I .o

0

0.1

23 I

0 ' to

to

to

to

to

to

to

to

to

to

to

to

13

on tunnel dimension rather than

TABLE I

Geometry

Ref.

Mabuchi (43)

yTgr/ I I-

Goldstein, Rask, Eckert (42) Nicoll, Whitelaw (18)

Metzger, Carper, Swank (54)

pJpm

Temp. Blowing Free ratio, rate, M = stream T2/Tm p z U z / p m U m Mach No.

0.788 to 0.878

1.14 to 1.27

0.02 to 0.146

0.02 to 0.03 1

Air

0.84 to 0.88

1.12 to 1.20

0.013 to 0.052

0.095 to

He

0. i 2

1. I2 to

0.0022 to 0.0076

0.095 to 0.16

Air with

1.20

e-m k

T -

*&&

Talmor (49)

0.16

~~

1.o

1.o

0.96

1.05

He tracer Air

Whitelaw (40)

Density ratio,

Air

' k r m e$j)3

Kacker, Whitelaw (36)

Carlson,

Injection gas"

0.47 to 2.26

0.06

0.25

0.04

to

1.o

to

1.49

0.07

0.3

0.06

Air with He tracer

1 .o

Air with

1.o

I .o

0.47 to 2.24

0.06

2.76

0.363

0.5 to 1.98

0.1

0.563 to 2.66

0.115 to

I .o

0.023 to 0.074

0.0702

2.9

0.08 to 0.17

2.1 to

He tracer

NP NP

2.76

0.363

Escudier, Whitelaw (47)

1 .O

Goldstein, Eckert, Wilson (24)

0.33 to 0.48

0.77 1.14

0.0085 to 0.0223

Goldstein, Eckert, Ramsey, (71) and (72)

0.84

1.2

0.10

to

to 2.0

to

0.5

0.242

______

* From Whitelaw (40). As temperatures are presented for only one case, assumed same for all

given boundary thickness.

Nitrogen used as mainstream gas.

Laminar boundary layer ahead of

EXPERIMENTAL STUDIFSIN FILMCOOLING (Continued) Velocity ratio,

Velocity

U,/Um

m/sec

0.26 to

0.185

Urn

7.0

to

10.0

Slot size, mm

15.1 to 51.0

0.015 to 33.2 to 0.0605 55.0

2.54

0.018 to 33.2 to 0.063 55.0

2.54

0.47

21.4

Starting length Reynolds No., Re,,

8.0

’ lo5

to

22.0

/

lo”

S*/s

0.011

Effecq/Tax/Cwtiveness

Taw

to

to

24.4

0.907 2.54

0.3

21.4

1.87

to

2. I

0.47 to

21.2

2.24 0.17

6.425 7.25

8.0 lo” to 14.0 lo5

0.037 to 0.046

Taw

0.80to

150to

0.78 to 31.0

0.95

4,035

4

0.20

19,500

218

312

l/s

2420

70

0.7

730

50

0. I

5000

200

1 .o to

4,035

4

0.26

19,500

218

1.0

10,350

2.4

A

lo”

0.107

C,

to

Q ” Taw

1.8

to

lo5

0.3

c w

to

11.0 . 105

1.28

2.4

0.095

f

lo5

to

Cx

Taw

Taw

24.4

cx

25.4

0.074 1025

0.12

30.5

2.38

61.0

to

12.6

23.5

6.0

6.0

lo5

to

21.0

lo5 lo”

0.026

0.033 to

0.058

runs. ’“ Average heat transfer over section from s injection. ‘I Hole diameter. ‘ From kk’hitelaw (40).

=

Taw

to

to

to

=

to

1.0

13,300

to

to

to

0.1 1

368,000

I .o

958

0.625

0.1

3000

20.6

0.9

I200

0.25

0.12

3700

8.0

0.85

10,000

2.0

0.005

100.000

40.0

to

to

35

to

565,000

to

Taw

to

0.1 1

to

0.90

300

to

to

0.107

1.59

to

to

0.03

to

0.19

to

35

0.78 to 31.0

0.67

0.023

1320

1 to

700 to 4000

1.59

to

0.06

to

0.80 to 0.04

2.26 15.2

358

Taw

/

to 1.55

x/s

0.037 to 0.046

to

0.26

Range of

0.95 to

0.092

Slot Reynolds No., Re,

8.0 / lo5to 14.0 lo5 /

6.425

Measured parameters,

to

to

0 to x = I was measured.

to

to

to

to

“ Calculated

from

TABLE I

Ref.

Geometry

@ m

Kacker, Whitelaw (50)

Pai Whitelaw (59)

Kacker, Whitelaw (51)

Injection gas"

k r m

-

/,m, urn_

Burns, Stollery (29)

pr/p~

Temp. Blowing Free stream ratio, rate, M = T,/Tm prUp/pmVmMach No.

Air with He tracer

I .o

I .o

Air H Air with He tracer Argor (Refrig.

0.07 to 4.17

1.0

Air with He tracer

1.o

Arcton 12 (Refrig. 12)

4.17

1.0

2.21 to 16.7

0.017 to 0.050

0.14

1.0

0.071 to 0.236

0.050

4.17 to 0.14

1.0

0.14 to 4.17

0.050

0.96

1.05

0.25

0.04

He

E;ror

Density ratio,

Air-Arcton

0.288 to 2.66

0.06

0.021 to 6.87

>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