Effect of a stepwise distribution of heat transfer on the supersonic flow over a flat plate

''''I In ~l' , '"

.J ,lii :\1

El

FT

~ r---~~.IO~UVJlnliU~~·~~~I~,---~ tz:j :t> 1--3 l2l ~ lA)

TRAINING CENTER FOR EXPERIMENT AL AERODYNAMICS

TECHNICAL NOTE 13

EFFECT OF A STEPWISE DISTRIBUTION OF HEAT TRANSFER

ON THE SUPERSONIC FLOW OVER A FLAT PLATE

by .

W.J. McCroskey

RHODE-SAINT-GENESE, BELGIUM

TRAINING CENTER FOR EXPERIMENTAL AERODYNAMICS

TN 13

EFFECT OF A STEPWISE DISTRIBUTION OF HEAT TRANSFER ON THE SUPERSONIC FLOW OVER A FLAT PLATE

by

W • J. McCroskey

FOREWORD

The work described herein was done in partial fulfilment of the requirements for receiving the Dip~oma of the Training Center for Experimental Aerodynamics. Mr McCroskey, an American student, obtained a Distinction grade, and was co-winner of the Belgian Government Prlze, awarded annually to the student ranked second in the

graduating class.

The subject of this investigation was suggested by Dr J.J. GinouK, as a side line from research sponsored under E.O.A.R. grant. Some of the equipment used by

ACKNOWLEDGEMENTS

The au thor gratefully acknowledges the assistance and encouragement given by his project advisor, Dr J.J. Ginoux. Discussions with Dr J.J. Smolderen and the generous technical assistance of Mr F. Thiry were also especia~ly beneficial.

The author also gratefully acknowledges the

financial support of the General Motors Research Corporation, whose fellowship made possible the present research.

SUMMARY SYMBOLS CONTENTS I. INTRODUCTION 0 II. THEORY 0 0 . 0 ' .

· .

. .

.

.

.

0 0 . 0 • 0 • • • • Integral analysis • 0 • • • • •

Restrictions and assumptions Longitudinal gradients

Higher order terms

. . .

• • • 0 •

· . .

• e • 0 • •

Prandtl number and viscosity • •

Similar profiles • • 0 • • • • •

l I l . EXPERIMENT o • 0 • o • • 0

Model design o • • . 0 . • • • • 0

Experimental technique • • 0 • 0 • •

Test procedure

. .

.

.

.

o • •

Results and discussion • • • 0 0 0

Araldite model 0 • • • 0 • 0 • •

Foam model • • 0 0' 0 • • • •

Conduction error •

.

. .

• • 0 •

IV. CONCLUS IONS • • • 0 • • • •

. .

Suggestions for further study

·

.

. .

.

REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D o • • • 0 • 0 o • 0 • 0 • •

•

Differ.ential equations • • • 0 • Integral equations • Numerical examples 0 Conduction error 0 •

· . .

.

.

.

·

.

.

.

. .

·

.

.. .~

.

.

APPENDIX E Determination of thermal

1 3

5

13

13

1,6

18

20 21 21 23

25

26

26

27 27

28

'29 32

35

39 42

46

conductivity • • • o . 50

SUMMARY

The effect of a stepwise distribution of heat transfer on a flat plate has been studied by considering J the ratio of the heat transfer coefficients h for a step

q

in heat flux to the heat transfer coefficient h for a uniformly heated plate. The ratio h/h was found to be a

q

simple function of the ratio of the distance x along the plate to the length ~ of the unheated portion, independent of Mach, Reynolds and Prandtl numbers. The theoretical predictions were verified experimentally in air at a Mach number of two, for various Reynolds numbers and values of

xl

~.

A A a B b C Cl c p c cf d f g h h K k ~ L M o SYMBOLS area

constant, defined as used constant

constant constant constant

Chapman-Rubesin viscosity constant specific heat at constant pressure constant

skin friction coefficient constant

Blasius dependent variable x-dependence of ti T

heat transfer coefficient stagnation enthalpy

constant

thermal conductivity

length of the unheated portion arbitrary reference length rvla eh number

N pr p Q q Re r S T u v

w

x

x y y " -dependence of temperature Prandtl number

fraction of total völume of foam plastic occupied by cavities

solution for T - T when ~

o

w aw heat flux

Reynolds number radial di~tance

Sutherland constant for viscosity temperature

velocity parallel to the wall

velocity perpendicular to the wall exact solution of the energy equation longitudinal coordinate

longitudinal coordinate

transformed normal coordinate

constant wedge angle constant

ratio of specific heats

tJ.

T transformed thermal layer thickness

ó boundary layer thickness

cS T thermal layer thickness

small length

n Blasius independent variable

e

angle in polar coordinates

k(T-T ) 2

a the quantity

e

viscosity

v ratio of thermal conductivities

the variabIe ng(x) p density T shearing stress the quantity p u C (T -T ) co (X) p w aw Pr stream function Subscripts a adiabatic value

m property of the model

n index

o stagnation value

q uniform heat flux case

w eonditions at the wall

eonditions outside the boundary layer

When used without subseripts, Tand p refer to

EFFECT OF A STEPWISE DISTRIBUTION OF HEAT TRANSFER

ON THE SUPERSONIC FLOW OVER A FLAT PLATE

Io INTRODUCTION

A class of fundamental problems in fluid mechanics which at present remain unsolved is that of boundary layer flows which develop in a conventional manner on the first portion of a body but are subject to discontinuous boundary conditions farther downstreamo The present investigation

concerns a simple problem of this type: the laminar, two-dimensional flow over a flat plate at zero incidence with an adiabatic front section folYowed by a portion with uniform flux of heat from the plate to the boundary layer.

The fundamental difficulties that arise due to discontinuous boundary conditions are not present in the

problems for which exact solutions exist, such as ~he familiar isothermal flat plate or a plate with uniform heat flux. The singularity at the discontinuity has been largely ignored by the previous investigators who have considered stepwise

distributions in wall temperature or heat flux (references 6-14)0 These analyses have all be~n approximate, in that longitudinal temperature gradients are neglected, have been restricted to incompressible flow, and for the most part

2.

laminar investigation i t was hoped that the diffieulty due

to a step in heat transfer would be less than for a step in

wall temperaturej and that as aresult both theory and

experiment would be simplified. However~ a solution to the

full problem was not obtained. On the other hand, the

ineom-pressible restrietion has been relaxed.

The present study is primarily eoneerned with the variation of the heat transfer coefficient hp defined by

h

T _w - T _aw

(where q is the heat flux to the boundary layer, T i~ the

, w w

wall temperature, and T is the adiabatic wall temperature)

aw

as eompared to the heat transfer coefficient h for a flat

q

plate uniformly heated everywhere. To this end an approximate

solution was obtained to the boundary layer equations giving

the ratio h/h _q as a simple function of _-', the length of the

unheated portion and the distance along the plate p independent

of Mach, Reynolds, and Prandtl numbers o Also, experimen~s

were eonducted in air at supersonie speeds on symmetrie wedge

models, and the results obtained on one of these models

agreed well with the theoretical predietions o Tests with the

other model we re not satisfactoryo

The present investigation is limited in seope to

•

steady, laminar boundary layer flow with no pressure gradient o

A perfect gas with constan~ specifie heat and Prandtl number

obtained o The results are not valid at the point of the discontinuity in heat flux. Reynolds number was varied

in the experiment$ but only one value of Mach number and

Prandtl number was possible o

Ilo THEORY

3.

The continuum flow field in the vicinity of asolid

body immersed in a general fluid can normally be described by the familiar boundary layer equations o However, the normal boundary layer approximations are not always valid when

conditions at the wall change discontinuously, such as the

step in heat transfer that is considered in the present investigation o In an exact analysis$ therefore, the full Navier-Stokes equations must be considered o

It is possible to obtain useful information about

the heat transfer coefficient and wall temperature on a flat

plate at zero incidence with a step in heat flux by an integral analysis of the boundary layer equations$ ignoring the

singu-larity at the point of the discontinuityo In the present

analysis the differential boundary layer equations are first transformed to the classical incompressible form, so that the approximations involved in the integral technique are on the same level as those normally encountered in the numerous incompressible analyses ava~lable in the literature.

4.

Before presenting the integral analysis, i t is instructive to consider the classical boundary layer assump-tions. Normally, gradients in the flow direction are neglected in comparison to those perpendicular to the flow. Chapman and Rubesin (1) have pointed out that the boundary layer continuity and momentum equations for a flat plate

(1 )

oU

.1.!!.

pu ox + pv oy _0_ _oy (~ lE..) _oy (2)

are not affected by strong temperature gradients, whereas the energy equa tion

Pu c _{P ox} oT

+

_P v c _P oT _oy is valid on1y if _ [Tw - TooJ (1.1.)

e

ox w - Ó or lesso

The present integral analysis gives the re sult

1 2

--(T - T ) ~ (1 _ l:.):? • (dT)

w aw x ' dx w ~ (1 _ 1:.) x 3

Thus it is cl ear that

(~)

is not neg1igible near the point

dx w

x

=

2., and the energy equation must be wri tten

(l.!!)2 + _o_(k 11.) + _o_(k 11.)

~ oy oy oy ox ox

(4)

a aT

The appearance of the new term ~(k ax) changes the nature of the energy equation from parabolic to elliptic, and the usual boundary layer solutions that are functions of ~

I;:

cannot be expected to holde No solutions to equation

(4)

appear to be available. Numerous investigators have treated problems involving abrupt changes in wall temperature or heat flux using equation

(3).

In the light of the above remarks, such solutions can only be considered approximate. However, this is not a serious drawback for most practical applications, because heat conduction within the body will always tend to smooth out the discontinuous boundary condition.

As the strong longitudinal temperature gradients are alocal phenomena, the regions away from the discontinuity aan normally be described by equation

(3).

In a following section the limits of applicability of equation (3) will be considered.

INTEGRAL ANALYSIS

The equations and boundary conditions describing

the compressible flow over a flat plate at zero incidence

with an insulated front portion followed by a uniformly heated portion, figure 1, are

apu + ~

6., PU au + pv ~ ax ay aT aT put: + p v e -p ax p ay u(y u(y T(y

=

ö)

=

0 00)

=

u 00)

=

T q (x

<

.

q

=

0 w Ol> Ol> q (x >R.) = const w J...( E ) ay IJ ay (2 )

(4)

assuming e

=

constant. Equations

(2),

(3)

and

(4)

ean be p

combined to g1ve

-aa (p uh )+-:-aa (pvh

)::--L

a (UT )+..L(kll)+_a (k.a1.)

x 0 y 0 y ay ay ax ax (5) au where T = j.l ay and ho 2 u

=

c pT + 2.

It is now assumed that the influenee of the heat flux for

x > R. is fe 1 t only in a "therma 1" layer of thi cknes s eST whi eh

develops downstream of x

=

R.. Upstream of x = ,R. the boundary layer is assumed to develop 1n the usual manner for flow over an adiabat1e flat plate. This behaviour is indieated qualita-tively in f1gure 1. More will be said about eST later; here i t 1s suffieient to suppose that some sort of "thermal" layer develops w1th1n the normal thermal boundary layer.

The next step is to integrate equations (1) and

(5)

w1th respect to y from the wall to eST. Comb1n1ng the results gives

7.

o

0

1

T:" (puh

o )dy - ho

(OTl

T

.~

(pu)dy "

=

~k

_{(aT) +lOT _a_( kl! )dY+}

o

[ T u+k aT]

w a Y él x ax a Y <5

o T

(6)

It is shown in Appendix B that if ho

=

const for y > oT' then

• = • -k (":\) aT

w oy w

L

OT OT

d él aT

-d _x (uh -uh )p dY1

-,,-(k~)dy

0 o~ oX oX

o 0

This condition for h is satisfied either by Pr

o

M « 1.

(7 )

1 or

by

Further progress requires additional assumptions regarding the fluid properties or the flow fièld properties. If the contribution of longitudinal temperature gradients,

l

OT él aT o • x (k

ax )

dy ,

q

=

w

is neglected in equation

(7),

then

d

dJ~PUO

(T - T )dy

x p a

o

(8)

where T (y) is the temperature that would exist if q 0

a w

everywhere.

Ai

this point attention is drawn to the fact that

a simple transformatioh

pdy p dY

00

relates the compressible boundary layer equations

(1),

(2)

aüd (3) in the (x,y) plane to classical incompressible

8.

the assumption that ~ - Tand Pr co·rist.

The details of the relationship between the com-pressible problem and the corresponding ficticious incompress-ible problem are given in Appendix A. The main point here is that the momentum equation in the transformed plane is inde-pendent of the energy equation, and that velocity profiles

are functions only of the variable

yl

Ix.

This means that

similar profiles exist, of the form

where u - = fn u _co (2..)

/i

ó

/::, =1

-1L dy o p ex> y

=

fn (-) /::,

i.e., /::, is the boundary layer thickness in the transformed plane.

The difficulty in obtaining an exact solution to the present problem is due to the fact that temperature

profiles are not similar, even in the transformed plane,

except in the case of 1

=

O. This special case has been ana~:.

lyzed by Ginoux (2), whose results show the temperature

pro-1

files to be similar in the parameter (q _w

lp

u c T )Re2 and to

e x > c o p c o X

be function of

YI

I~.

The foregoing remarks suggest that an approximate

solution could be obtained for 1 > 0 by assuming the

temperature profiles in the transformed plane, where the

9.

to be of the form T - T

(1-)

a fn T _w - T _aw 6 T where 6 T

=jT

î dy Poo 0

In order to integrate equation

(8),

i t is necessary

u

to speeify the funetional relationship of - - and

u

00 T - T _a

-T----T...:.:..- .' whi eh at the moment nepresent unknowns. One of the w aw

most general ways would be to represent the profiles by eonverging power series

The 00 u . T - T a T _w - T _aw eoeffieients a 00

L

a n=o n 00

~

b n n==o a 0 b 0 n = ==

=

a n

=

00

L

n=o and b n 1 0 0 1 b n

ean be arbitrary, ,exeept that

10.

al - related to skin friction coefficient

a

2

=

0 from Blasius'solution

b

2 0 from Ginoux's solution

If equation

(8)

is now transformed by equation

(9)

and the above series inserted for u and T, straightforward

' ±ntegration is possible, yielding

-d { p u . C (T

-dx co 00 P w

+ B ( ' : ) 4 + C (

'~

) 5 + ••• ] }

(10)

where A"B, C, 0 0 0 are numerical constants whose ~alues depend

upon the values of the coefficients a and b 0

n n

Now A > B > C, etc, and -t. T < 1 by assumption, 50

t.

that as a first approximation only the first term need be

consideredo If i t is postulated that t.T/t.

=

0 at x =~

,

then

integration with respect to x yields

But t.2 q

(x-~)

= Ap u c (T -T

)~

w 00 00 p w aw t. q = -w k w (dT) dy w (T - T ) w aw

(11)

(12)

11.

for the assumed temperature profile. Combining equations (11) and (12) gives T -T w aw

=

d P 1/3 2 öPr P (x-R.) 00

Now the c1assica1 incompressib1e boundary 1ayer theory gives

and therefore 2 2 1 1 _{iRa-Pr 3 3} (_c_) 3 qw x _(~) PoolJ oo ₍₁

_!.)

3 _{(13 )} P 00 U_oo c'

-Ab2 x 1 p w w T -T _w aw

This resu1t may be compared" with the approximate incompressib1e

solution of Smith and Shaw (3)

2.4 q

IR~-pr2/3

1 w x

(1 _

~ )3 P u ti x 00 p T -T _w 00

=

(14)

and with the exact solution obtained by Ginoux for R. = 0

T -T

w aw

G1noux's factor w(o) represents the numerical va1ue of the exact solution to equat10n (3) at the wal1 for a part1cu1ar va1ue of Pro Numer1ca1 ca1cu1ations indicate that the

..

w(o)

~

1.090 pr-l / 3

is valid to better than 1

%

accuracy in the range 0.5 < Pr < 1.

Thus the three analyses predict the same dependence upon the parameters q

/p

u c , Re , and Pro

w cocop x

c

In equation (13), the constant

(---2)

depends only

Ab l

upon the choice of the coefficients a and b that define the

n n

shape of the velocity and temperature profiles and upon the defini ti on of t. 0 S ome numerica 1 example s are gi Vlan in

Appendix C. Since an exact solution for ~= 0 exists, the

integral analysis need only relate the problem ot a step in

heat flux to the uniformly heated problem. Thus the heat transfer coefficients

T -T

w aw

are related by the simple expression

h h q -1/3

=

(1 - .&.) x

where the subscript q denotes the uniformly heated case.

Further, Ginoux has demonstrated that h is directly

propor-q

tional to h

T, the heat transfer coefficient for the case of an

13.

thereby relating the present solution to the wellknown exact

solution for T = constant.

be

w

RESTRICTIONS AND ASSUMPTIONS

The effect of a step in heat flux has been found to

~

-1/3

desoribed by a single factor (1 - -) ; however, a

x

number of simplifying assumptions we re made in the derivation. The significanee of these assumptions will now be discussed.

Longitudinal Gradients. The term that was previously neglected

oT

1

_d (k

l1.)

dX dX dy

o

becomes a dominant term i n the vicinity of x = ~; in that

region the results obtained cannot be expected to holde The extent of the domain in which longitudinal gradients are important can be obtained by considering the simplified case of a fluid with constant density, viscosity, and thermal

conductiv~tyo In this case equation

(7)

becomes

q

=

pc

~f~U(T_T

)dy -w P dx a o

2L

o

T

~

₂ (T-T)d _{- a y} dx 0

14.

u u 00 00 n=o T-T a T -T w aw 00

then proceeding as before yie1ds a differentia1 equation of the form where

a _

[k(T(awlf

a

=

numerical constant

s

=

numerical constant pu x Pr (_00) IJ (16)

3/2

The numerical factors depend upon the choice of the coefficients a and b • The main point here is that inc1usion of the

n n

de

longitudina1 gradients produces a new term a~ and changes the

equation for T from a1gebraic to different~a1.

w

Equation (16) can bè norma1ized by setting

1 1

[k (T -T )

-j

2

e

=

a

-2 sC2

w aw qw

1 1

15.

giving

d'e - 3/2

-- è ' -6 + x = 0

dx

There are two main points to be derived from equations (16) and

(17).

The first point is that either the similar profile assumption or the assumption that 0

=

0 at x

=

t i.e. 6(0)

=

0,

T

is in error. Otherwise, equation

(17)

predicts T _w <

T

_aw for x

=

t + E, where Eis small with respect to x and to This is another expression of the fact that the problem is fundamen~ ,

tally more difficult in the region near x

=

t .

Secondly, the previous result, which was equivalent to S6 3/2 =

x~(x-t),

is recovered by setting

dÊi

Considering the value of dx obtained by the flrst approximation

ë

=

x

2/3 , then the condttlon

o

dë

o « x -di'

-requires x » 1 or x - t t d

where d is a numerlcal constant, again dependent up on the coefficients a and b • For the examples cited in Appendix C,

n n

16 •.

be considered valid provided

Higher Order Terms o The expression for wall

tempera-ture, equation (13), was obtained by considering only the term

2 .

of order (~T/~ ) in equation (10). The small error so

intro-duced will now be considered by means of an illustration.

One of the examples considered in Appendix C is a quartic representation of velocity and temperature profiles,

4 _(I.) 1 4 u _{(I. )}

=

u 3 ~ 3 ~ co and T-T 4 (...L) 1 _(L)4 a 1 T -T =

-

3 + w aw öT 3 ~T

then equation (10) becomes

J}

Also, for this case

17.

The term

(~T/~)5

wi11 be retained; otherwise the procedure is the same as in the general case, and gives, af ter some

rearranging: where A B

c

~

=

1 - Q, x p u c (T -T ) 00 00 p w- aw

lRe-

q Pr x w

Equation (18) can be solved direct1y to give

On1y the roots associated with the negative sign are of interest here, i.e.

~ 3 =

c

+

B

+ • 0 0

(18 )

From this expression i t fo11ows that 1 A 2A2

--

1 + + - - + h (1 - .&) 3 B2 " 4 h

=

x I B 2A2C2 q AC 1 + B2 + B4 or 1

(1

+

]

--h (1 -

!)

3 1 A R,

'"

h x _{3 B2} x q

and for ,the examp1e considered

h h q '" (1 -1 ..&.)

3"

1

₁ + .0013 x

l"

Pr 1 3 o 0 0 + o • •

Thus the correction due to higher order terms is inverse1y

1

proportional to Xl and is 1ess than

4

%

for the examp1e consideredo

Prandt1 Number and Viscosity. An essential assump-tion in the transformation of the differential boundary 1ayer equations from compressib1e to incompressib1e form is that viscosity is linear1y proportiona1 to temperature. However, this is not a serious restriction, because the proportiona1ity constant

can be adjusted to yield nearly correct values for the

viscosity in the important region near the wall. Chapman and

Rubesin suggest matching Cl with the Sutherland viscosity law,

giving

Therefore Pw~w is a function of x, and a correction factor

arises giving h h q

-1

(1 _ .1:) 3 x T aw T + Q aw 1 T + Q

I-Re--

(1- ..8.)3 aw x x + Q

lR'ë-

+ S x /.--- 2. 1/3 l'Re (1 - -) x x + S 2 3

where Q

IRë-

represents Ginoux's exact solution for

x

1

3

T - T

w aw

when 2. = 0, equation (15). The correction is almost negligible

except at very large Q

11te--.

x

The effect of Prandtl number appears in the variation

of stagnation enthalpy outside the "thermal" layer. For Pr :/: 1,

equation (11) must be replaced by

t:J.2

q _w (x- 2.)= p u c (T -T )-l-+

~ 00 p w aw u

t:J.

p _GO

I

(uh -uh _{0 0} )dY

00

t:J. T

From Pohlhausens' classical solution of the flat plate thermometer pr oblem (4)

IJ.

1

(uh -uh )dY > 0 for _{IJ. T >} 0

o 0

00 IJ.

T

and Pr < 1

so that the wall temperature is slightly reduced when Pr < 1. However, .no general conclusions can be drawn about the effect

on the ratio h/h • q

Similar Profiles. Fund~mental assumptions in the foregoing analysis are that the velocity profiles are similar in the transformed hydrodynamic layer and that the temperature profiles are similar in the transformed "thermal" layero The former assumption is true provided ~ ~ Tc The latter

assumption, while not strictly satisfying the differential equations, appears reasonable in view of the results obtained and in comparison with the exact analysis o It has been used

by many authorsj the important point brought out in the present analysis is that only the magnitude and not the functional

dependence of the wall temperature depends upon the choice of the profile c Any physically acceptable similar profile that can be represented by a power series will give as a first approximation

h h

210

IIIo EXPERIMENT

In the preeeeding section eertain diffieulties of

the approximate theory were avoided and the results were

generalized by considering primarily the ratios of heat

trans-fer eoefficients h/hq and h/hT' relying upon exaet solutions for hq and h

To Similarly, certain dlfficultles and errors in

the experimental lnvestlgation were avoided by eonsidering the measurement of h/h and lts variation with x and t as the

q

primary test obJeetive o

The measurements were made under steady conditions at supersonic speeds on symmetrie w~dge modelsp shown in

figure 2, which gave uniform flow at M

=

2005 along their upper and lower surfaees o The models were tested in TCEA supersonic

wind tunnel S~l at free stream Reynolds numbers per meter of about 10

6

x 10

6

and

3

x 10

6

0

MODEL DESIGN

A major portion of the experimental program consisted of attempting to minimize two inherent difficulties assoeiated

with the present problem o The first of these diffieulties is due to eonduction of heat within the modelo The discontinuous

step in heat flux through the surface of the model to the boundary layer is smoothed out in pr~ctiee because the strong

longitudinal temperature gradient produced by the boundary layer

drives heat through the model fr om the heated portion to the

unheated portion o Thus the ·heat conduction is coupled in a

complicated~ nonlinear way to the boundary layer heat convectiono

An attempt to minimize the conduction error was made

by using models with small wedge angles and low heat

conductiv-itieso An approximate analysis in Appendix D illustrates the

importance of these two factors o The total wedge angle employed

was 10°0 Geometrically identical models were made from two

different insulating materialso The first model~ previously

used by Ginoux (2), was cast from an epoxy resin known as

"Araldite Type D" with a conductivity k~ about 00 17 kcal/hr m °Co

The second model was cast from a foam plastic resin~ known as

Thiakol "Rigithane" with k ~ 0006 kcal/hr m °C.for the bulk

*

material 0 As a reference~ i t may be recalled that for air at

room temperature k ~ 00022 kcal/hr m °c and for steel k ~ 40

kcal/hr m °Co

The second difficulty is that of obtaining a uniform

heat flux on the heated portion of the model o The basic technique

employed was the silver plating technique developed by Ginoux

(2)~ consisting of heat dissipated by Joule effect in a thin

layer of silver chemically deposited on the model surfaceo The

details of the plating process are des cri bed in reference 2~

*

The determination of the conductivity of these materials is

essentially i t is the process used in silvering mirrorsp and

ean be used with a variety of materialso In practicep howeverp

the silver is not deposited uniformly in thickness o Thus the

resistancep and hence the heat fluxp is not uniformly

distributed over the entire surface of the model o

Normally the plating process produced a layer with

variations in resistance of 10

%

or less when the surface had

had careful preparationp suah as thorough cleaning 'followed

by spraying with liquid araldite followed by scouring with

fine sandpaper o In cases where the variations in resistance

exceeded about 10

%

or were primarily localized₉ an improvement

could of ten be made by physiaally removing some of the silver

where it was excessively deposited o This was most easily done

by rubbing the silver layer with a soft rubber eraser in the

regions of lowest resistance o In generalp howev@r9 i t was

found preferable to replate the surface then to compensate for

large variations by removing some of the silver layer o The

data in figure 3 i llustrate scatter that ean be attributed

primarily to variations in resistance o

EXPERIMENTAL TECHNIQUE

A sketch of the models used in t he experiment is

shown in figure

30

The rear porti ons of both models were cast

of araldite and were at tached to a longitudinal sting o Both the

upper and the lower surfaces of the modells were plated and were

means of copper electrodes and silver circuit pAint, as shown

in figure

3.

The length of the unheated portion was varied

from 0 to 65 mm by scoring the silver layers with a series of

spanwise lines and isolating the portion upstream of the

appropriate lines, maintaining symmetry on the top and bottom

surfaces. Four independent AC power supplies were used for the four surfaoesg upper front, lower front, upper rear, and lower rear. The heat fluxes per unit area per unit time were

determined fr om measured voltages and currents and the total

areas of the heated elements. The voltage drop across each

heating element was measured directly at the electrodes to

avoid errors due to the power leads. The rated full~scale

I

acouracies of the voltmeter and ammeter were 2

4

%

and I ~

%,

respectively. However, these errors were largely cancelled by

considering the ratio of heat transfer coefficients

The temperature distribution at the surface of the

models was mea~ured by flush fitting copper-constantan

thermo-c~uple Junctions at the locations shown in figure

3.

Each

thermocouple was connected to an individual cold junction

placed i~ an individual mercury reservoir. The cold junctions

were maintained at 00

C in an ice bath. The output voltages

of the thermocouple cirouits were measured with a calibrated

thermo-couples used in the tests were found accurate to within 2

%

at 220

Co This error was partially cancelled by considering

the ratio h/h 0

q

TE8T PROCEDURE

The models were tested in the TCEA continuous wind

tunnel 8-10 Model alignment was -~erified by observing the

shock wave pattern with the schlieren system of tunnel 8-1.

Tunnel stagnatioR pressure was measured with mercury manometers.

8tagnation temperature was measured using a copper-constantan

thermocouple in the settling chamber and a recording

self-balancing potentiometer o

8teady state conditions were achieved for both

power-off and power-on conditions af ter about one hour of

running time between changeso For each set of readings with

heating~ a set of adiabatic temperature readings was recorded

so as to provide (T - T ) at each q ~ Re and to Allowance

w aw w x

was made for the gradual rise in stagnation temperature during

the transient intervalo Data were recorded for t = 0 for each

model and each Reynolds number per centimeter so as to provide

a reference value of h _q at each test condition which would be

RESULTS AND DISCUSSION

Araldite Model. Preliminary tests w~re made using

the araldite model with heater film resistanoe varying as

shown in figure

4.

The results of the measurements of h/h are q shown in figure

6.

The data have been oorreotedy based on

re~istanoe measurementsp by a factor expressing the ohange in

the ratio of looal heat flux to average heat fluxy as 1 was

varied.

The resistanoe of the s11ver layer was improved,as shown in figure 5_i and the tests were repeatedy giving the

results shown in figure

7.

It is interesting to note that the soatter in the data for h/h was substantially reduoed by

q

improving the heater resistanoey even though the absolute

value of h _q was not improved_i as shown in figures

4

and

5.

This illustrates the mutual oanoellation of some of the experimental errors as a result of measuning hand hunder

q nearly identioal oond1tions.

There appears to be no signifioant differenoe in the data reoorded at different values of 1 and Reynolds

number per meter *.The data in figure

7

a~ree wi~h theory to within

5

%

although the line (1 -

~)-.30

fits the data somewhat

x

better. No oorreotion for oonduotion error has been made in figures

6

and

7.

* The two values of Reynolds number oorrespond to tunnel stagnation pressures of 100 and 200 mm Hg.

Foam Model. Tests with the foam model were unsuc-cessful for the purpose of comparison with theorYj due t o

experimental difficulties attribut ed to the thermocouple circuits. However~ the data did allow a comparison of the conduction error of the foam model with that of the araldite model. This comparison is di~~ussed in the following sectiono

Conduction Errors. The data in figures 6 and 7 have not been corrected for the effect of conduction within the

model. However , the effect is believed to be almost negligible x

for ï greater than ab out 102, because of the good agreement with theory and because the data were not changed by varying ~ and free stream Reynolds number. If the conduction had been significant, a Reynolds number effect should have been evident. It should be mentioned , howevèr, that the apprpx1mate analysis in Appendix D gives an error of about 15

%

for the most severe test conditions at

T

x

=

102.

The error iS j of coursej largest in the region near

~ • Some results with the foam model and the araldite model are shown in figure 80 Despite the lower bulk conductivity of the

foam, the results are approximately the same. This seems to indicate that conduction withift the model was much stronger near the surface where the foam is composed more of solid

material and less of air bubbles and hence where the conduct~

vity i~' approximately that of the solid materia11 not the

measured bulk value. This effect may have been magnified by the thin layer of liquid araldite that was sprayed on the foam model for silver-plating purposes1 and by the silver layer itselfo

At any rate, no significant improvement in conduction error

was realized with the foam plastic model.

IV. CONCLUSIONS

In the preceeding sections the theoretical and

experimental investigations were described and the results

of each we re compared. The main conclusions of the

investiga-tion may be summarized as followsg

10 The ratio of heat transfer coefficients h/h _q for a flat

plate with a stepwise distribution of heat flux is a simple

x .'

function of

ï

₉ independent of Mach, Reynolds and Prandtl

numbers.

2. The present theory aan be used with good accuracy to

predict h/h provided

q

~ > 1 + 200

pr~

Re3 / 4

t

For ~ >

4,

h/h ,differs from unity less than 10

%,

so that

q

for most practical applications Ginouxvs exact solution for

h can be used downstream of x ::: 4t.

q

3.

The scatter of the experimental data of h/h was

signifi-q

aantly reduced by improving the uniformity of res1s~ance of

the silver layer, even though the absolute value of h was

not improved o The uniformity of the si~ver layer ean be controlled somewhat by careful preparation of the model surface and by physically removing some of the silver in regions where the resistance is lowesto In general the

former is preferable to the lattero

40

From the st~n~point of the conduction error associated

with the present problem, the foam plast ic is not signifi

-cantly better than araldite o However, the bulk thermal

conductivity of the foam is significantly less than that

of araldite o From the point of view of model construction

foam plastic seems to be more difficult to use than araldite.

Suggestions for further studyo It is suggested

that further investigation of lamina~, constant pressure

flow be directed to the region in the vicinity of the point of discontinuity. Downstream of x

=

i , it was found in the

present analysis that compressi~ity effects did not alter

the expression for h/h 0 However, the longitudinal gradient

q

term

which is important near x

=

i , is affected by variable fluid

properties o Therefore it is suggested that a full analysis be first restricted to an incompressible fluid o

as a first approximation in the analysis of the full energy

aT

equation, the term v

ay

could be neglected in comparison with the other terms. From an experimental point of view, the

~iscontinuous boundary condition wopld be more nearly realized using water rather than air, because of the large difference in Prandtl number and thermal conductlvity for these two fluids and their importance wi th respect to cond'uction error 0

The silver layer could ,be electrically insulated by spraying the model with a thin layer of aralditeo

For incompressible turbulent flows, various

investigations have shown h/hT and h/hq to be given by a simple expression of the form

where a and bare constants. It would be interesting to see whether these results could be extended to the compressible cas e, whether by the simple trans forma t i on p dy

=

p 00 dY or by some other treatment. The silver plating technique is also suitable for turbulent experiments, although the determination of h _q might be more difficult than in the laminar case. _..

Finally, i t would be' interesting to see to wha t extent the results of the present investigation are influenced by pressure gradientso For a flat plate, the temperature

profile can be taken to be

T - T _a

T _w - T _aw

=

1

-u u

without a1tering the resu1ts of the approximate ana1ysi~,

whether the wa11 is isotherma1 or uniform1y heated downstream

of x = t *0 Cohen and Reshotko (15) have shown the temperature

field to be directly related to the velocity field and the ratio of 100al stagnation enthalpy to enthalpy at the wall,

for an isothermal wa11 and for a olass of problems with pressure gradients and similar velocity solutions o These considerations sugge~t that progress might be made using an approximate~

integral teohnique a1png the lines of the present investigation o Experimenta1 verifioation of the results would, of course, be neoessaryo The silver plating technique is also suitable for tests of this sort.

*

It should be kept in mind, however, that the argument of

y y

T - T is , whereas the argument of u is 0

a 6

32.

REFERENCES

1. Chapman, DoR. and Rubesin, M.W.: Temperature velocity profiles in the compressib1e 1aminar boundary 1ayer with arbitrary d1stribution of surface

temperature.

J. Aero. Sci. vol 16, n° 10, P. 547, 1949.

2. Ginoux, J.J.: An exact soluti~n to the compressible

laminar boundary layer equations for the flat plate with constant heat flux.

T.C.E.A. TN 11, May 19630

3. Smith, A.G. and Shah, VoL.: ~eat transfer in the

incompress1ble boundary layer with arbitrary heat flux.

J. Aero. SCi., vol 28, n° 7, P. 738, 19610

4. pohlhausen, E.: Der Wärmeaustausch zwischen festen Körpern und Flüssigkeiten mit kleiner Reibung und kleiner Wärmeleitung.

Z.AoM.M., Bd 1, p. 115, 1921.

5. Lord Rayleigh: On the influence of obstacles arranged in rect~'ngular order .upon the properties of a medium.

Philosophical Magazine, vol XXXIV, 5th Series, po 481, 1892.

33.

6 0 Rubesin~ MoW0 8 An analytical investigation of the heat

transfer between a fluid and a flat plate parallel

to the direction of flow having a stepwise discon~

tinuous surface temperature.

MoS o Thesis~ University of Californiap Berkeley~

19450

70 Rubesin~ _MoW08 The effect of an arbitrary surface temper~

ature distribution on the convective heat transfer

in an incompressible turbulent boundary layer.

NoAoCoAo TN 2345~ 19510

80 Eckert~ EoRoGoand Drake~ RoM08 Introduction to the transfer

of heat and mass o

McGraw~Hill Book Co~ New Yorkp 19500

90 Reynolds _{9 WoC op Kaysp WoMo and Kline a} SoJ08 Heat transfer

in the turbulent incompressible boundary layer. Part II ~ Step wall temperature distribution o

NoAoSoAo Memo l2~2-58W9 Dec o 1958 0

100 Spaldingp DoB08 Heat transfer to a t urbulent stream from

a surface with a stepwise d1scontinuity in wall

temperatureo

International Developments in Heat Transfer~

110 Smith, AoG o and Shah, VoLog The calculation of wall and fluid temperature for the incompresslble turbulent boundary layer wlth arbitrary

dlstribution of wall heat flux.

International Jnl of Heat and Mass Transfer, vol 5, p. 1179, Dec. 1962.

12. Conti, Paul J.g Heat transfer measurements at a Mach number of 2 in the turbulent boundary layer on a flat plate having a stepwlse temperature distribution.

NoAoS.Ao TN D-159, Nov o 19590

13. Johnson, DoSog Veloclty and temperature fluctuation measurements in a turbulent boundary layer downstream of a stepwise discontinuity in wall temperature.

Jlo Applied Mech., p. 325, Sept o 19590

14. Meyers, GoE., Schauer, JoJ., and Eustis, R.H.g The

plane turbulent wall Jet o Part II - Heat transfer. Stanford Univ., Dept. Mech. En~r., TR n° 2,

Dec. 1961.

15. Cohen, eoBo and Reshotko, E.g Similar solutions for the compressible laminar boundary layer with heat transfer and pressure gradient o

350

ApPENDIX A

Differential equations o

With the assumption that c and Pr are constant p

p

the conservation equations for a steadyp constant pressure

boundary layer flow are

~ ₊ ~ _{= 0} ax a y au

au

a _{(~ ~)} pu + pv = ax ay _al" ay (A.2) a:r aT 1 a _{( ~T ) J:!... (~)} 2 pu _~x + pv =

- -

+ ay Pr ay ~ ay

c

ay p

Introducing a stream function defined by

u

=

o

9 v == ~ _{p ax}

p ay

the equations can be transformed from (xpy) into independent

variables (x,lP)

l !

₌

~ _a_ _{(p lA u}

_ll)

ax _{p-~. a1jJ} a1jJ (A.4)

aT 1 _{-L .a1.)} p ~u

(

~)

2

,

(p~ u

=

..

f

ax _{p2 Pr}

alP a1jJ 0 p2 atjJ

Using a linear viscosity law

Equation Ao4 ean be redueed to an ordinary differential equation by the transformation

n

=

~

The resultant momentum equation is

ff" +

ril'

o

(Ao

8)

whieh is the familiar Blasius equationo The veloeity is given by

u

=

~ f'(n)

The energy equation beo9mes

fil

T aT aT

+ Pr f - 2Pr f' ~

an

2

an

ax

Pr ( 1 )M2 _{T f"}

- TY-

ex> ex>

(AolO) The momentum equation is seen to be in exactly the same form as if the problem were ineompressible o The only differenee is

in the similarity variable

n ,

defined in the ineompressible

37.

n

yV

₂

"00

_{p, u}

~

,

00 00

Thus the compressible problem, regardless of the temperature field, is related to a corresponding incompressible problem

in a (X,Y) ,plane defined by

x

x

Y dy

The energy equation A.10 is linear in T, so that solutions may

be added in the (x,n) plane. A particular solution satisfying

the boundary condition of an adiabatic wall has been obtained

by Chapman and Rubesin

(

1

)

.

The remaining homogeneous equation

a2_{T "'T "'T}

+ Pr f _0 _ _ 2Pr f i X _0_

dX

o

(A.ll)

has to be solved for t he particular boundary condit lon of the

proble~, using the Blasius solut ion f or fen) and fU(n).

Solution by the method of separat ion of variables, i.e.

T

=

X(x) N(n), gives 1 (N" + PrfNU )

fTN

2Pr x Xi X

=

K

*

(A.12)

This method, however, does not satisfy the boundary conditions

*

Primes denote differentlation with respect to the variable of

of a step in heat flux, which are 2 k p w w = 0 for x < R. constant for x > R.

except in the limit of R. ~

o.

The assumption of similar profiles in the transformed "thermal" layer, made in the integral analysis of Section II, )

) is equivalent to T

=

X(x) N(E;) where E; n g(x) This gives 1

[N"

(t") + ..:...P.:;..r.:;..f

~:-:-:-:-...l..:I...L.j

f ' (n ) N ( E; ) .. 2Pr x [Xl (x) + g' (x) Nt (E: ) ] g2(x) X(x) N(E;)

which is not equivalent to equation A.12. The conclusion is that simi~ar temperature profiles do not exist for the present problem.

APPENDIX B

Integral equations

For a constant pressure boundary layer flow with strong longitudinal temperature gradient, the conservation equations can be written in the form

390

_0_ _{(puh )} +

....2.-

(pvh ) =

L

(u T) +

..L

(kL!:) + . .L (k~)

ox

0 ay 0 ay oy oy.

ox

-

ax (B.2) where 2 h = C T + u ₂

(

0 _P T = ~ ~ _oy

Integrating with respect to Y1 equation Bol gives

a

o x (pu) dy (B.3)

o o +

k aT]

Ö

ay

~

i

ÖT~

aT

+ (k-)dy

a

ax

o (B.4) (B.5)

where örepresents the outer edge of the boundary layer and Ö

T represents the outer limit of the influence of the heat flut

through the wall. Implicit in equation B.5 is the assumption

a

ax

for y > ÖT

Combining equation B.3 and B.5 gives

{ Ö

r

- h (Ö T)

_a

(pu)dy o

ax

o Ö

-1

-1.. ( p uh ) dy ê

ax

0 T (B.6)

Combining B.3, B.4 and B.6 gives

ê

aT

{T a

aT

= -k (--) + --(k-)dy w

ay

w .

ax

ax

o Ö -dd

r

puh dy

~Jö

0 T

Thus if h o Otherwise constant for y > 0 T' q • _{w -}

=.

Or

= dd

r

(h _ h )p u dy x

Jo

0 0a> OT

l

aaT

+ - ( k - ) d y o

a x ax

{ OT (h -h )p udy + o 0a> o

a

aT

- ( k - ) d y

_ax

_ax

+

1

0 d + -dx 0 T (h -h )pudy o 0 a> (h -h )p udy o 0 a> ...L(k 2-!)d

a

x

a

x

y

41. (BoB) (B.9)

APPENDIX C

NUMERICAL EXAMPLES

In Section II, velocity and temperature profiles were assumed of the form

'"

~a

(y) n u

=

u _{n=o n} fj, '"

'"

T-T

=~

Y n a T -T b ( - ) w aw n=o n

Lir

and the magnitude of the wa11 temperature was shown to depend

upon the choice of the coefficients a and b • Simp1e examp1es

n n

which are sometimes found in the 11terature will now be

considered. For simplicity i t is assumed that Cl = 1.

A. Cubic Profiles u u 00 T-T _a T -T w aw

/::, =

4.64 x

Equation 10 of Section 11 becomes

Equation 12 of Section 11 becomes

lJ p c q

=

_3 w w P (T -T ) w 2 prpoo/::'T w aw Equation 13 becomes p u c 00 00 p T -T _{w aw} t 1/3 (1 - ~)

and the skin friction coefficient is given by

00646

The numerical factors 2.40 and 00646 are to be compared with the exact values of 2.18 and 0.6640

Eo Quartic Profiles u T-T a T -T 4 Y 1 Y 4 -(-)

-

-

(-) 3 /::, 3 /::, w aw

44.

The boundary layer thickness is

x Equation 10 becomes Equation

13

becomes T -T w aw R.

1/3

(1 - -) x

and the skin friction coefficient is giv~n by

Co Sine Profiles

u

T-T _a

T -T w aw

x

{Re

x Equation 10 becomes d [ 211 [ 112 T I1T ]} q

= --

p u c (T -T )-- 1-(1+ )cos(~ --) w dx 00 00 p w a w 1T 11 2 -11 2 2 11 T = dd {p u c (T -T )211

r

t8

2- 1 ) (I1.T) 2 x 00 00 p w aw 1T

L

ti Equation

13

becomes T - T w aw

46.

APPENDIX D

Conduetion error

Sinee the boundary layer temperature at the wall is a funetion of x, temperature gradients exist within the model and some of the heat that should enter the boundary layer is

lo~t by eonduetion within the model. As the geometry of the model and the heating system are symmetrie, there is no heat flux aerossthe plane of symmetry of the model. Thus the steady state temperature distribution ean be determined from the

Laplaee equation in polar eoordinates

a2_T

l1'.

1 a2_T 0 r + + = ar2 ar r ae2

(D.l)

and the boundary eonditions

(11.)

₌

0 for e 0

ae

T(r,a) T (x)

where T (x) is the temperature distribution at y

w

°

in the

boundary layer coordinates.

An exact s'olution for T(r,e) can be obtained by

separation of variables if T (x) is assumed to be the boundary

w

layer solution to the constant heat flux problem witht = Oe

This solution is T(r,e) 2e18 q pr2/3 = T + w aw p u ei 00 00 p

e

COS 2 cos 2

and the ratio of heat conducted through the surface of the

model to the heat ideally convected away by the boundary layer

is

for small CL

However~ this solution is not valid for t > 0, i.eo

,',

for a step in heat flux o An approximate solution~ which agrees

w1th equation D.3 in the limit of t ~ 0, w111 now be der1ved.

For a th1n element assumed to have a temperature

4S.

qm

~

dq c qc

---...

',," ~ qc + - - 6 x _dx

~

..I

6y 6 x

the steady-state energy balance is dq

c

q 6 x + - -6 x6y 0

m dx

if no heat is transferred across the lower surface of the

elemental volume. In terms of the temperature gradient

flY

_m

If the wall temperature is assumed to be the

approximate boundary layer solution given by equation 13 in

d2T

Section II, then can be calculated at any point. The

dx2

ratio of heat conducted through the surface of the model to

the heat ideally convected away by the boundary layer is

6y m x -2/3 .&) x + 1 0

-5/3

2 2 J.

(1 _

~) + - ~(l 3 x x 9 x2

-S/3J

.l.) x

(D

.4)

If the thin element is interpreted as being a segment of the wedge model, then

=

tan Ct

Therefore equation Do4 agrees with equation Do3 for small Ct

and ~

=

0, and the correction factor due to t > 0 is the

quantity L

-l

(1

-~

-2/3 4

~

~

-5/3 8

~

)

- -

(1 - )

+ -x 3 x x

9

x₂ -8/3

J

(1 - ~) x

It should be noted that equation D.4 is not a valid solution to equation Dol. Also, the factor L is probably much_

,.

too large for all but infinitesimal wedge angles since equation D •'4 imp icitly contains the assumption that 1

as

aT

=

0 t roug ou • h h t And furthermore, equation D.4 was der~ved assuming T to be

w

given as a fi~st approximation by the solution of the boundary layer equations. This assumption is not true for large q /q •

m w

Thus equation D.4 should not be applied in the immediate

vicinity of ~. In spi te of these difficulties, however, certain conclusions can be drawn from equation D.4. First, the

conduction error is proportional to the wedge angle. Second, the conduction error is directly proportional to the conducti-vity of the model and inversely proportional to the conducticonducti-vity of the gas in the free. stream. Third, the error is very large in the neighbourhood of x

=

t, and can be expected to smooth out the ideal discontinuous boundary condition.

50.

APPENDIX E

Determination of thermal conductivity.

A simple experiment was conducted to measure the thermal conductivity of araldite and of the foam plastic, as no data appeared to be available for the foam. The test

consisted of measuring the heat flux and temperature difference across a thin slab of the test material. Samples 10 cm x 10 cm x 0.5 cm thick were sandwiched in the manner indicated below, with the inner faces silver plated

Plexiglas Silver

plating

and heated by Joule effect, The edges were sealed with strips of plexiglas, forming a box-like structure which was immersed in a well-stirred ice bath. Thermoco~ples were fitted on the inner faces, as shown below.

Circuit paint Test sample

51.

Thermocouple leads Power leads

The power to the two samples was adjusted 50 as to

have the same temperature on each inner face. From the power supplied and the temperature difference across the sample the thermal conductivity was computed according to the linear

conduction relation

k

where Q is the heat supplied per unit time. Ö x is the thickness

of the sample~ A is the area of the heated portion of the

sample~ and öT is the temperature difference across the sample.

The araldite samples were tested seven _times and the foam plastic samples five times, at various heating and temperature levels. The results were av~raged, giving the following dat ag

52.

Specific gravity k _hrmoC Kcal Maxodeviation

Araldite .155 +10% -9%

Foam Sample 1 040 0065 + 2% -4%

Foam Sample 2 037 .051 + 7% -5%

The manufacturer of'araldite has stated ~hat its

conductivity is about 0017 kcal/hr mO C, which is less than

10 % higher than the measu~ed value. The conductivity of the

foam plastic was found to be about 1/3 that of araldite 'and

about 3 times that of air o

Rayleigh (5) has suggested a method of calculating _,

the conductivity of asolid which contains spherical cavities arranged in a rectangular order o Calculations were made based on Rayleigh's expression ~2 + v) \ 2p ~ 1

-

\) ~

-k k solid ~2 + \)) (1

-

\)) + p mat'l

where \) is the ratio of the conductivity of the material in the cavity to that of the solid material and p is the fraction of the total volume that is occupied by the cavities o The

cavities were assumed to contain air, and p was found to be

about 006 to 008 for typical samples o Assuming a value of k for

53 •

• 20 kca1jhr moe, then Ray1eigh's formula 'gives 'k '" 0~07 kcal/ hr moe, compared with the measured v~lues of .051 and

54.

APPENDIX F

Remarks on the use of foam plastic.

The foam plastic used in the present program was a light weight, non-porous foaming resin, known as "Rigithane". The components we re

1.06 parts by weight Foaming Resin Rigithane 365 1.00 parts by weight Curing Resin C 365-R2HH

obtainable from

Thiokal Chemical Corporation 780 North Clinton Avenue Trenton, New Jersey, USA.

Test samples and the final model were cast in

aluminium molds which were thoroughly coated~ith vacuum grease

to prevent the foam from.,stick·-ing to the mold. The process consisted of pouring the curing resin into the foaming resin, followed by vigorous mixing for about 15 seconds. The mixture was poured into the mold and allowed to set for about 30 minutes.

The density of the finished product aan be varied

from about 0.1 gm/cc to about

.8

gm/cc by varying the mass

of mixture poured into a fixed volume mold. It is advisable to have a nearly air-tight mold except for a small vent hole

at the topo It is also advisable to preheat the mould to about 25°C to insure proper foaming action.

Cne of the most important factors in the process is thorough mixing of the two componentso·To this end an electric mixing or stirring machine is recommendedo It is also

recommended to cool both components to abo~t 15°C before mixing to allow more thorough mixing before the foaming action startso

t

aa

:J . -~ LU ...J al

o

0::

a..

...J <! U Cf) > ::t:

a..

LU J: t-lL..

o

J: U

.-LU ~ Cf) c)

--IJ..

>~~~~~---~---circuit paint unheated portion Thermocouple No Dlstance FOAM from leadlng edge mm ARAL A B C 16 26 31 8.5 13 19 D 36 23 silver-plated foam or arald1te E F G H 41 46 51 56 29 34 39 49 FIGURE 3 100 mm I J K L M 61 71 81 91 101 59 67 78 89 99 N 0 111 121 109 118 silver-plated araldite electrodes p Q 131 141 129 139