THE EFFECT OF HEAT I'-!SULATION ON THE

COOLD\G REQUIREMENTS OF THE INTERNAL

STRUCTURE OF HIGH-SPEED VEHICLES

by

John Noble Perkins

Thesis submitted to the Graduate Faculty of the

Vir~inia Polytechnic Institute

:in candidacy for the degree of

MASTF.R OF SCIID·lCE

in

Aeronautical Engineering

APPROVED: APPROVED:

Director of Graduate Studies Head of Department

Dean of Engineering Supervisor or Ma,ior Professor

April 29, 1958

Blacksburg, Virginia

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TABLE OF CONT~!TS

Page

1. INTRODUCTIOn • • • • • • • • • • • • • • • • • • • 5

2. DETERMINATION OF TIWJSIEHT SKrn T:FJ.WERATURES • • • 9

2.1 Governing Equations • • • • •••••• 2.2 Determination of Parameters ....... 2.3 Ru.rige-Kutta Hethod of Numerical.

9

14

Integration • • • • • • • • • • • • • • 20

2.4 E:xa.rnple Problem • • • • • • • • • • • • 23

3. DErERI·!INATIOIJ OF TRANSIFl·!T Tm.IPmATURE VARIATIOH

IT! INSULATI0rJ • • • • • • • • • • • • • • • • • • Z7

3.1 Introduction • • • • • • • • • • • • • • Z7

Heat Transfer in Insulation •••••• Governint: Equations • • • • • • • • • • •

Initial and Boundary Conditi.ons • •••

Finite-Difference l-!ethod of Solution ••

E:.lcar.ple Problem • • • • • • • • • • • •

DISCUSSIOli OF RESULTS • • • • • • • • • • •

4.1 AssUlned Trajectories • • • • • • •

• • •

• • • Co~pressibility Effects • • • • • • • •

Comparison of Two- and Three-Dimensional

Solutions • • • • • • • • • • • • • • •

Effect of Skin Thickness • • • • • • • •

28

29

33

35

37

42

l.i.2

42

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Jla~ts •••• • ••••••••

Iuulation • • • • • • • • • • • • • • 4.7 lacoimtan4&tions ••• • • • •••••

; • CONCLUSIONS • • • • • • • • • • • • • • • • • •

6. SUMMARY • • • • • • • • • • • • • • • • • • • •

7• A~ • • • • • • • • • • • • • • • •

8. BISLIOOKAPHY • • • • • • • • • • • • • • • • • •

9." VITA •• • • • • • • • • • • • • • • • • •" • • •

46

47 49

51 5, 54

;a

FIGURE 1

FIGURE 2

FIGURE 3

FIGURE 4

FIGURE 5

FIGURE 6

FIG!J'RE 7

FIG1JRE t1

FIGURE 9

FIGURE 10

FIGURE 11

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LIST OF FIGURES

Stagnation-Temperature Rise for Variable Cp • • • • • • • • • •

Mach Humber and Altitude Dia[;ram (Trajectory !Jo. 1) •••••••

1:Ia.ch ;<Jumber and Altitude Diagram (Trajectory i:o. 2) • • • • • • •

Mach Number and Altitude Diagram (Trajectory t.ro. 3) •••••••

Time History of Skin Temperature (Trajectory No. 1) • • • ••••

Time History of Skin Temperature (Tra~iectory No. 2) •••••••

Time HistoriJ of Skin Temperature (Trajectory No. 3) •••••••

• • • • • •

••••••

••••••

• • • • • •

• • • • • •

• • • • • •

• • • • • •

Co2parison of Two-Dimensional Solution for Skin Temperatures with Three-Dj_mensional Solution (Trajectory No. 3) ••••••••

Temperature Variation in Insulation (Trajectory ri!o. 1) • • • • • • • • • • • • • Temperature Varj_ation in Insulation (TrajectOr1J No. 2) • • • • • • • • • • • • •

Temperature Variation j_n Insulation (Trajectory lJo. 3) • • • • • • • • • • • • •

Page

59

60

61

62

63

64

65

66

67

68

69

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1. 1'.·lTRODUCTIO':

As actual and proro~ed speeds a.c:d alt~tudes of flit:ht

:Lncrease, the rrohlett of :na:inta:i-:1in~~ the ternr-erature of the

inter:1al structure of high-speed vehicles W:.thin structurally

perr:-,:i_ssable vnlues becor:ies :increas:ir::~ly d::fficult. Due to

aerodyn.a::ic heating, the internal terr:pera.tures of an aircra;."'t

travell r:{~ throu"'.h the atnoaphere at hi ch I:nch nm.ibers r;iay

becor;e excess:i ve unless a sufficient ar,i;mnt of cool inc is

supplied. As an example of one of the d:ifficult1es caused ry

aerodY'1a;.i.:ic heat:i ng, j t is known that sone of the Ger1ran A-4

r:~iss:1les experienced sufficient internal heating tc cause

exi:1csion of the fuel tru:ks rrior to iwract.

In attem:ptinc to solve the therrial prollem, it is first

necessary to determir1e the rate a.t which heat enters the

surface, and second, to provide some nea:·it3 wMch will prevent

the heat fron becord.ng excessive in the internal structure of

the vehicle.

The problem of deter;1d.ni';;~ the theoretical heat-transfer

cha.racter5 sties date back to the work of Pohlhausen27 and

L. Crocco28• In 1935 and a[c~ain :i.n 1938 von Yarman treated the

29 JO sulject of heat transfer for la.'Tdnar t-0undar::r layers ' ·•

Lore recently, the "WOrks by Huston, ,.,a.rfi.eld, and Stone31 , by

6 2 1 Lo , ty Van Driest , and by Truitt have all extended the

-6-

knowledee of heat-transfer characteristics for both laminar

and turbulent boundar'J layers. The nethod presented in this

thesis for determining the skin temperatures will be based

ma.inly on the work of these authors.

In order to un.derstand the fundamentals of aerodynamic

heating, it is first necessary to introduce some fundamental

concepts. As its na.m.e implies, aerodynamic heating is the

heating of an object as a result of the flow of air at high

speed about that object. Friction between the fluid filaments

as they stream along the body and compression at and near the

stagnation regions of forward surfaces convert the kinetic

energy of notion into heat within a thin layer of air

surrounding the body. The temperature of this layer increases

with the square of the speed so that, already at a Ha.ch number

of 5, the boundary layer temperature attains a vnlue of

approximately 3000 °R. Since this temperature j_s concentrated

j_n the air at the surf ace of the aircraft, heat will flow

readily from the boundary layer to the aircra~, the ease with

which it flows increasing also with the speed. Therefore, the

problem of aerodynamic heating tends to increase jn severity as

the speed increases. This fact :i.s aptly pointed out jn the

following quotation from reference 21

''Because of the increase of heat transfer with speed, it

arrears that a ••thermal barrier•' exists much as it

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appeared in the past that a ''sonic barrier'' existed. The

problems differ somewhat in that the sonic barrier existed

over a narrow band of Mach number, whereas the thermal

barrier does not occur over a limited range of Mach number

but rather tends to increase in severjty as the }fa.ch number

is increased. t' vihile, as in the case of the sonic barrier, proper design

can alleviate some of the problems arising from the thermal

barrier, it is obvious that as flight speeds become very large

(say, of the order of Mach numbers of 15 and greater) 1 the

surface temperatures w:ill become very high under any design

condition. Thus, it appears that the surfaces of high-speed

vehicles will have to be made of some material, such as ceramics,

which can withstand the high temperatures to which they will be

subjected.

Assuning, then, that the surface temperature will be very

large, the problem of keeping the temperature of the internal

structure from becorrQng excessive arises. This problem has

several possible solutions. Certainly, for flights of long

duration at very high speeds, some form of internal cooling will

be required. However, since weight is always of prime iinFOrtance

in the desir,n of any aircraft, the weight of a cooling system,

capable of handling the large temperatures encountered at flight

speeds of Mach nw:ibers of 15 or greater, may prove to be excessjve.

_g_

With this in mind, it was suggested by Dr. J. F. Vandrey,

Advanced Design Section, The Martin Company, that heat insulation

be used to prevent a portion of the heat from reaching the

internal structure of the vehicle. It is expected that, while

this will not eliminate the need for internal cooling, it will

make possible the use of a n1uch smaller, and thus lighter,

cooling system.

It was, therefore, the purpose of this thesis to perform a

theoretical analysis to determine the skin temperatures obtained

by a missile throughout several trajectories, and then to

ascertain the amount of heat transferred through a blanket of

insulation placed between the skin and the internal structure

of the vehicle.

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2. DE.'Tm.HI>lATIO: l OF TRASSIN:T SKI\l TFl:PEllATURES

2.1. Governinp: :§guations. F'or aerodynamic heating

problems, the local rate of heat transfer iEto (or out of) an

element of s1dn is equal to the local aerodynamic heat-transfer

rate at the element 1:d.nus the heat-transfer rate on the eler:ient

that is lost by radiation plus the equipment heat-release rate

rlus the heat-transfer rate into the element due to solar

radiation. This statement nay be expressed symbolically as

(2.1)

where Qlocal is the local rate of heat-transfer, in :BTU/sec,

at the sk5n elerner:t 6 A. This, :i.n turn, nay l.e rewritten as

wbere

dT 6 A c ts w ......!'! ,

dt

') ~ A = local element of surface area, (ft ... )

(2.2)

c = speci.fic heat of skin material, (BTU/lb-0 R)

ts = skin th:ickness, (ft)

w = srecific we1[>,ht of skin r:iaterial, (lb/ft3)

T = outer surface skin terr~perature, (0 R) w t = tine, (sec) •

The riua.:1t:it:y qa is the aerodynarrd.c heat-transfer rate, ir:

I?TU/sec, which may be wrjtten as

-10-

q = .6 A h (T - T ) ·a aw w , (2.3)

where h = film coefficient of heat tra.'1sfer,

(BTU/sec-ft2-°R)

Taw = adiabatic wall temperature, (0 R) •

The term ~ is the heat transfer rate lost by radiation from the

elemental surface, .6 A, in BTU/sec, and may be written as

where

(2.4)

€ = total hemispherical emissivity of the surface,

(dimensionless)

o = Stefan-Boltzmann radiation constant,

(BTU/sec-~2-0R) •

Usually in a continuum. flow the solar radiation term can

be neglected1 (i.e., q = O). Since the need for cooling s equipment is not known until a preliminary estimation of skin

temperature is made, the equipment heat release term is also

omitted. Therefore, the governing equation for the heat transfer

process is, from Eq. (2.1),

~ 4 6. A c t w --.Jal: == .6 A h (Ta;w- - Tw) - .6 A € o Tw • s dt (2.5)

Defining G = c ts w, as the heat absorption capacity of the skin

material, Eq. (2.5) can be restated as

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dT...,. h €<1 Tw4 - = - (T - Tw) - , dt G aw G

(2.6)

wherein the aerodynamic heat-transfer rate (Eq. (2.3)), per

unit area, in °R/sec, is

~ = ~ (T - T ) G G aw w • (2.7)

!~ote that Eq. (2.7) says that the direction of heat flow at the

skin surface does not depend on the difference between the wall

temperature, T...,., and the local free-stream temperature, T00, but

on the difference between the wall temperature and the adiabat:i.c

wall ter1perature, Ta.w•

The film coefficient of heat transfer, h, can be expressed

in terms of the Stanton nmnber2, Crb' as

where p == fluid mass density, (slugs/rt3)

V == velocity, (ft/sec)

(2.8)

Cp = specific heat at constant pressure, (ft2!°R-sec2)

and ( )00 refers to local free stream condition just outside

the boundary layer.

Substituting Eq. (2.8) into Eq. (2.7) gives the aerodyna.rr~c heat-

transfer rate in terms of Stanton nll!Ilber, or,

-12-

(2.9)

where J = the mechanical equivalent of heat (ft-lb/BTU).

Note that Eq. (2.9) is equal to

~ "" CHo:, Poo Vj [Taw - Tw • G JG 2 T aw - T00

(2.10)

Defining the recovery factor, R, as a measure of the fraction of

the local free-stream dynamic-temr~rature rise recovered at the

wall3:

R = Taw - Ten ' To - Too ro

where T0 = stagnation temperature, (0 R) , co

then, assumine the simple energy equation is valid

v 2 00

2

it follows that

T0 - T 00 00

(2.11)

(2.12)

(2.13)

For the case of turbulent flow the recover-f factor t'lay be expressed

at least approximately as4

R = (Pr)l/3 J (2.14)

-13-

where Pr = Prandtl nUJI1ber, a dimensionless parameter.

Substituting F.qs. {2.14), (2.13) and (2.11) into Eq. (2.10)

yields

{2.15)

an expression for the aerodynamic heat transfer involving the

recovery factor.

The Stanton nunber for turbulent flow is related to the

local skin-friction coefficient, (Crco), by the modified Reynolds

analogyl

{2.16)

Using Eq. (2.16) in Eq. (2.15) yields

' (2.17)

or in terms of the local free-stream Mach number 1

where Clo:> = local speed of sound just outside the boundary layer

(ft/sec).

On substituting F.qs. (2.18), (2.7) and (2.6) into the

governing heat transfer equation for turbulent flow, there

results the following expression:

-lh-

R Sa? o T 4 ro r1 3 - ___ w_ 00 •

2 G

(2.19) Equation (2.19) is a nonlinear differential equation which is

extrenely difficult to integrate explicitly, but which can be

integrated by numerical methods. One such method, the Runge-

Kutta method of numerical integration, which will be described

subsequently, is employed to obtain the solution of this equation.

2.2 Determination of Pararieters. For a given flight

trajectory and given body, certain parameters are either known

or must be calculated before inter.ration. The time history of

the flight path (velocity in ft/sec and the altitude in ft.

versus time in seconds) is usually known in advance. Therefore,

the time histoi;.r of the ambient (or true free-stream) temperature

and density can be found from standard atoosphere tables. Thus,

fron the time history of the flight path, the ambient speed of

sound and density car: be plotted versus time. J[nowing the time

histo!"J of the velocity and speed of sound, a f~ot of free-

stream Hach nur.tber versus time can be made. Using the potential

flow solution for the body" geometry to be investigated, the

temporal values for the '•local' ' free-stream l.!Ia.ch number, ter:i.per-

ature, de~1sity, and speed of sound can be calculated. If more than

one point on the body is to be investigated, a separate plot of

these variables is required for each such locati.on.

-15-

For a given body and skin material, the specific weight, w,

and the skin thickness, ts, are known constants for any point

on the body. However, the specific heat of the skin material, c,

being a function of the skin temperature, requires a plot of

instantar.eous specific heat values of the skin material as a

function of temperature. Therefore, the heat absorption capacity,

G1 is a function of time. E:lalrd.nation of Eq. (2.19) reveals

that a method for obtaining five of the parameters in the

governing equation, namely, 1'\:ot Ta,, p001 B.ro. and G has been out-

lined thus far.

From a study of radj_ation exchange measurements, Stefan5

found that the energy radiated by a body could be expressed as

proportional to the fourth power of the absolute temperature,

this relationship later being confirmed theoretically by

Boltzmarm6. The constant of portionality, o, is known as the

''Stefan-Boltzmann constant 11; its value having been found to be

4.8 x 10-l3 BTU/sec-~2-0R4.

The dimensionless coefficient € is termed the total hemi-

spherical emissivity to indicate that it applies to the total

radiation of all wave lengths emitted in all directions from

the element of surf ace over the entire solid angle of a hemi-

sphere. It represents the efficiency of the surface as an

emitter of thermal radiation, i.e., it is the ratio of the

entlssive power of the actual surface to that of a ''black body''•

-16-

Investigations have disclosed that the spectral hemispherical

e~dssivity cf actual surface frequently varies with wave length 7.

Since the range of wave len/:,ths er1itted varjes with tenperature,

the total hemispherical emJssivity of actual surfaces must be

regarded as a coefficient which r;iay vaT"'J with temperature.

Thus, an investigation of the variation of the emissivity of the

skin material is required. In eeneral, howe11er, tha endssivity

hejng only a weak function of temperature may be regarded as a

constant1 •

The adiabatic wall temperature, Taw' is a function of both

the local free-stream velocity, V00, and the specific heat, Cp•

For the actual case, the specific heat is a function of temper-

ature. In order to take into account variable specific heat,

it is necessary to i:nteGra.te the c;eneral energy equation

J g Cp dT = 0 • (2.20)

Integration of Eq. (2.20) gives

-= v 2

00 • (2.21)

2

Using Eq. (2.12), a set of staenation tenperature curves for

various altitudes and Hach nu.i;ll:;ers is obtained. In reference 8,

however, it was shown that by plotting the stagnation temperature

rise (T000 - T00 ) versus free-strearr, velocity, Vro, with Teo as a

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paral'"leter, the set of curves, for all practical FUrposes, fall

:' nto a si ~1gle curve for aJ.1 values of T00 ran::;:i ng fror" 392 °R

to 630 °R. This ter1perature range co:i."responds to the n:in:_r:m:~

and riax.ir:run a.r,ltient tenperatures fron sea-level to nbout 370,000

feet. Th~ s curve j_s shoun as Fi,c;-ure 1; its use 5.s expla:tried as

follows:

For turbulent flow, Eq. (2.11) becor:ies, us:in['" Eq. (2.11+),

T - T = Pz-1/3 (TrL - T ) aw ro --w ro (2.22)

where the Prandtl numl:er is based upon the wall tenperature.

Substituting .Eq. (2.22) i.nto Eq• (2.19) eives

• (2.23)

Thus, for each instant of time in the given trajector;rr, the

velocity, V00, is known and hence (T0co - To:) can be determined for

variable CP from the plot of (TOoo - T00 ) versus V00• Since

insta0taneous values of Teo and Pr1/J are known, then the

adiabatic wall temperature, Taw' can be calculated by Eq. (2.22).

Except for the local skin friction coefficient, Cr00, and

the skin temperature, Tw, the means for deterrdning the

instantaneous values of all of the parameters in Eo. (2.19)

have now been discussed. Since the skin temperature, Tw, is

the quantity to be determined by the jntegration, it remains only

to deternine the instantaneous values of Cfco •

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A semianalytical method of obtaining the local skin friction

coeffjcient for turbulent flow is formed from an extension of

von Karman' a mixing-length incompressible-now theory to take into

account variations of denslty and viscosity with temperature.

This result is91 lO

0.242 ( . -1 sin

Cr 1/2 ( _Y_-_1 H 2)1/2 c:o 2 00

(2.24)

where

A2 = ( y ; 1 1'6 2)/(Tw/Tc:o)

B = [ (1 + y ~ 1 1'1a, 2)/(Tw/Tro)] - 1

and in which f(n) is a function of the exponent n in the power

viscosity law ~L = const. Tn. It has been found that f(n)

depends upon whether the Prandtl or von Karman law for the

mixine length 1 is assumed, the choice being left to experiment2•

In reference 2, the solution to Eq. (2.24) is presented in the

form of a nomograph giving the Stanton nwnber as a function of

local Reynolds number, local Hach number, and ratio of wall to

local free-stream temperature. Thus, the local free-stream

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Reynolds number, Rea,, nrust be calculated for the point to be

investigated at each instant of time. Since the local free-

stream Mach number, 1''61 is known at each instant, it only

remains to know the value of Tw/Tro in order to determine the

value of the skin friction coefficient. At first, this appears

to be im~-0ssible since the skin temperature, Tw, is not known.

However, the numerical integration method of Runge and Kutta

(to be described subsequently) allows an iteration method of

finding the correct skin temperature for short time intervals.

It should be noted that the above analysis for deterrrdning

the skin frj_ction coefficient does not take into account the

phenomena of interaction between shock waves and the boundary

layer at hypersonic speeds. In general, the above analysis

predicts a decrease in the skin friction coefficient with an

increase in free-strearn Mach number; whereas, present theories

which take ieto account the shock :i..nteraction effects indicate

an increase in Cfoo with free-stream Mach numbez-1.1• Failure to

take into account interaction effects at hypersonic Mach numbers

can result in underestimating the skin friction by as much as

twenty percent12• Underestimating the skin friction coefficient

would obviously result in the underestimation of the wall

temperature. Thus, it is probable that at high Mach numbers the

skin temperatures (as calculated in the present thesis) are in

error. However, since the time history of the skin temperatures

-20-

was used only as a starting point :in estimating the effect of

the insulation on internal cooling requirements, it :is felt that

any error incurred by neglecting interaction effects will not

detract from the present analysis.

2.J Runge-Kutta Method of Nu.'llerical Integration. Equation

(2.23) is a nonlinear equation with variable coefficients of

the first order but the fourth degree. Since Eq. (2.23) can be

expressed symbolically as

' (2.25)

where o.6 f =- ' (2.26)

JG G

it is found that Eq. (2.25) can be solved by the Runge-Kutta

method of numerical integration13.

The problem may be stated as follows: To tabulate the

solution of the differential equation (2.25), which reduces to

Tw = T0 when t = t 0 , the tabular interval being .6 t. Then

Runge-Kutta•s method is carried out in the following self-

explanatory scheme:

-21-

t Tw f f(6t)

to To ql l/2(ql + ~)

t 0 + 1/26 t To + 1/2 ql q2 q2 + ~ (2.27) t 0 + 1/2.6. t T0 + 1/2 q2 (Eq. (2.36)) q3 SUM

t 0 + .6. t To+ q3 q4 q • 1/3 sum

t 1 = t 0 +6t Tl = To + q

The work is repeated with (t1, T1 ) as the pair of initial values,

giving T2 corresponding to t 2 = t 0 + 2 ~ t, and so on.

The Runge-Kutta formula given in the table above may be

considered as a special case of the recurrence formula14

Ts = Ts _ p + ph (weighted average of f values) (2.2.8)

where Ts is the approximate solution at T • Ts and h is the

(uniform) spacing .6 t m ts - l• Equation (2.28) can be

interpreted as requiring that Ts and Ts _ p be end points of a

chord whose slope is some weighted average of slopes of tangents

to curves which are solutions to Eq. (2.25).

The development of F.q. (2.27) neglects terms of the order

of 6 t5 and higher13. The error arising from neglecting these

terms is known as the truncation error and is 0( 6 t5). An

-22-

estimate of the truncation error, when .6. t is reasonably small,

may be made by repeating the evaluation of Eq. (2.27) using the

double interval 2 6. t. Let e be the error in T1, so that

approximately

e = c 6. t 5 , (2.29)

where c is a constant. Then the error in T2, calculated in two

stages, is 2e = 2c .6 t5. On the other hand the error in T2,

calculated in one stage, is

e• = 32c .6.. t5 J (2.30)

and therefore 2e = 1/15 (et - 2e) ' (2.31)

or • (2 • .32)

where T2 is the value determined by two stages, and T2• the

value determined in a single stage. Equation (2.32) is not the

total solution error since the round-off error and the inherited

error have been neglected. However, it does indicate that the

magnitude of the error is, in part, a function of the interval

size, At.

Referring to Eq. (2.28) it is seen that the curvature of the

solution to Eq. (2.25) also limits the size of .6. t. If the

curvature of the solution is small then the slopes of the

tangents to the solution at various points along the solution

curve are approximately equal. Thus, a large interval size,

-23-

£::::,. t, rnay be chosen without inducing any apprec1able error.

However, :i.f the curvature is large, the slopes of the tangents

are changing rapidly and therefore, the interval size must be

chosen sufficiently sir.all so that the weighted average of the

slopes of the tangents is a close approximation to the slope of

the solution over the chosen interval.

It is apparent then, that the choice of ~t depends upon

the particular system being integrated and the nature of the

result. Note, however, that no difficulty is incurred when

altering the size of L:::.. t during the integration. Therefore,

if the curvature is changing appreciably over a portion of the

solution, a small value of ~ t may be used, while a large value

may be used over the portion of the solution where the radius of

curvature is large.

2.4 Example Problem. Using the method of solution outlined,

the wall temperature, Tw, is calculated for the three trajectories

shown in :ngures 2, 3 and 4. For each trajectory the body

investigated is assumed to be a cone-cylinder configurat:on, and

the point to be investigated is located on the cylindrical

portion of the body. To simplify the analysis, the pressure co-

efficient, Cp, at the point in question, is assumed to be zero.

This is equivalent to stating that local flow parameters are

equivalent to true free-stream values.

-24-

The boundary layer is assumed to be turbulent throughout

a11 three trajectories. The treatment of turbulent boundary

layers on bodies of revolution can be carried out by the use of

the cone rule presented by Van Driest in reference 15. However,

Eckert, in reference 16, investigated the sinple case of a

circular cylinder aligned parallel to the flow, the longitudinal

gradient being zero. The results of this investigation showed

that for values of o/r (where 6 is the boundary layer thickness

and r is the radius of the cylinder) of the order of 0.01, there

were virtually no deviations from the flat-plate result. Even

for values of o/r = 0.10, the deviations from two-dimensional

results were found to be negligible for most practical purposes.

To illustrate the difference between the two solutions, a

comparison of the flat-plate solution with the three-dimensional

solution (for the trajec~ory shown in Figure 4) is presented

in Fie;ure 8. It is seen that the :maximum difference between

the two solutions is approximately 5 percent, with the flat-plate

solution predicting the higher temperatures. Without exper:'...l'.lental

data neither solution can be assumed to take preference over the

other. Iiowever, sjnce the difference between the two results js

srnall,, it would appear that the choice of the method of solution

is not too critical. Therefore, to further simplify the analysis,

and also to avoid 1mder-estima.ting the skin temperatures, the skin-

friction coefficients are here evaluated on the basis of a two-

dimensional analysis for all trajectories.

-25-

Since the Mach number and altitude is known from the

trajectories, then, using standard atmosphere tables, the true

free-stream conditions are ascertained. With the assumption of

zero pressure coefficient, the local free-stream conditions are

exactly the same as the true free-stream conditions. Application

of the Kutta-Runge method of integration requires that data be

available for twice the number of intervals as chosen for

inter,ration. For the trajectories shown in Figures 3 and 4,

which are equivalent to single-stage rockets, a choice of a two-

second integration interval was found to be sufficiently

accurate. Therefore, the data is calculated for every second.

For the trajectory shown in Fig. 21 which is equivalent to that

of a three-stage rocket, it was found that, in the regions where

the acceleration was changing rapidly (say, at the end (or

start) of one of the stages), it was necessary to use an

integration interval of one second and, therefore, the data is

calculated for every half-second in these regions. An explanation

of this is given in section 2.3.

The values of specific weight, specif5c heat, and emissivity

in Eq. (2.23) are determined by the choice of skin material. In

all three trajectories it is assumed that the skin material is

oxidized Inconel X of thickness (ts) = 0.005 inch, specific weight

(w) = 530 lbs/rt3 and emissivity ( € ) = 0.9 5• The choice of

such a small value of skj_n thickness is due to a desire to make

-26-

the radiation term in Eq. {2.3) as large as possible, and to be

able to assume that the temperature throughout the th1ckness of

the skin is constant at any instant of time. The necessity for

beinz able to n:i.ake this assUIDption is discussed in a later section

of this thesis.

The above data is used in the numerical integration process

to determine the wall temperature, Tw, for the three trajectories.

The integrated results are shown in Figures 5, 6 and 7. Note that

the effect of increasing the rnar:;rdtude of the radiation by

decreasing the skin thickness is quite evident toward the upper

end of the second trajectory (Figure 6). Although the Mach

number was increasing rapidly, the skin was cooling instead of

heating. This is a result of the radiation term (in Eq. (2.23))

overpowering the aerodynamic heating term.

In the next section the governing equation for one-dimensional

heat-transfer through solids is developed. Then, assuming the

skin temperatures obtained in thts section to be the temperature

variati.ou of one surface of a layer of insulation, the temperature

throughout the insulation is determined as a function of time.

-27-

3. DETERMI.!JATION OF TRA:·.ISIENT TEMPERATURE

VARIATIOIJ IF INSULATION

J.l Introduction. Transmission of heat energy is due to

the peculiar property of matter called temperature, and to the

second law of thermodynamics. This law states that free

migration of heat energy is always in the direction from a region

at higher temperature to a re3ion at lower temperature. Of

fundamental importance are the mechanisn~ for the transfer of

heat energy; where fo general, the modes of heat transmissiori

are separated into the familiar processes of conduction, con-,

vection, and radiation.

A survey of the literature reveals that it has been cor.unon

practice to treat the mechanisms of heat transfer separately,

with little or no attent:ion being given to the case where more

thar one kind of heat transfer }s present. In particular, the

case of heat transfer in a nonhomogeneous :rri.aterial, such as

insulation, has been igr.ored. w'hether this has been due to a

r:iisunderstanding of the problem or to the coHplexity of the

probler1 is r:ot known. However, it is felt that an understandi.ng

of the rr.anner in wrdch heat is transferred through an amorphous

material is important in the study of heat transrnission. There-

fore, a brief discussion of heat transfer through heat insulation

and a simple, althout3h approximate, riethod for cotlputing the

transi e:nt temperature varj_ation is presented in this section.

-28-

3.2 Heat Transfer in Insulation. host heat insulators

consist of solids containing pores, such as insulating bricks

and cork-board, or oi' M .. ghly porous materials having a continuous-

air type of arrangement, like powers and nineral wool. Heat is

transferred across an air space, such as a pore, by radiatj.on,

conduction, and/or convection. If the pore is small, and not

connected to other pores, the transfer of heat is by radiation

across the pore and by conduction of the gas within the pore. In

this process, with relatively srnall temperature drops across the

pore, the transfer of heat through the whole mass may be con-

sidered as conduction if a large munber of pores are uniformly

distributed through the material.

In a J,!2.terial having large pores, radiation becoI!leS an

inportant factor, and, with greater tertperature differences

across the pores, the heat transferred by radiation js neither

pro:r--'-Ortional to the temperature difference nor inversely

proportional to the thickness as in the case of conduction. In

lightweight conthmous-air type of insulat:ion, such as mineral

wool, the structure of the material is sufficiently open to

perm:it convection currents which can materially increase the rate

of heat transfer. Furthermore, some insulators are somewhat

trar:sparent to infared radiation so that a part of the heat may be

transferred directly through the material by radiation.

-29-

If radiation or convection becomes an important factor in the

rate of heat transmission through an insulating material, a

coefficient of thermal conductivity cannot correctly represent

its relative conductivity. In this event it would be better to

call the coefficient of heat transfer an apparent k or kappl7

(where kapp is found experimentally and takes into account all

modes of heat transfer present).

If the concept of an apparent coefficient of thermal con-

ductivity is utilized, the rate of heat transfer through the

insulation can be calculated, approximately, by treat]ng the

problem as though the only mechanism of heat transfer present is

that of conduction. Thus, the problem can be solved by the basic

law of heat conduction.

3.3 Governing Equations. The fact that materials differ

in their abilities to conduct heat has long been known. Although

Biot19 introduced the basic law of conduction for one direction

as

where

dQ = - k A !!!_ dt dx

Q = heat flow, (BTU)

' (3.1)

A = cross sectional area normal to the heat flow, (ft2)

dT/dx = temperature gradient, (0 R/ft)

t = time, (hr.)

and k = thermal conductivity, (BTU/hr-ft-0 R).

-JU-

this expressjor~ 1.a generally attrjbuted to F'cur1er20. It

appears that this lias come about sj.:nce Fourier used it as n

fundane!1tal eouation jr: h:i s analytic theory of heat. The x:iinus

s:ign follows from the second law of thermodynamics accordjng to

wh:ich heat naturally flows from regions of hit.her temperature

to ree:im•s of lower temperature. Thus, if dx is taken as

:pos:i.tive :h the direction of a positive heat now, then the

temperature difference, dT, ooat be negative. These conditions

are cor:sjster:t with each other only if the minus sip;n ia used.

Applying E;q. (3.1) in three mutually perpenmcular directions

leads to the differential equation of heat conduction. If for

a pa.rallelp:lped of infin:ltesjrnal size (Figure 3.1) a. heat balance

z

Figure J.l lleat Cor;ductjor; Through a Parnllelp1ped.

is descr:il:.ed, which is val.id lr; the differential of t:ime dt,

ther· the heat enterine from the left can be expressed as

-31-

d Q1 = - k ( dy • dz) aT dt , ,x ax (3.2)

and the heat leaving at the right side is

a aT dQ2,x = - k {dy • dz) ~ (T + ~ dx) dt 1 ax ax (3.3)

or aT a2T dO- = - k ( dy • dz) ( - + - dx) dt • """.c:,x ax ax2 (3.4)

Siirilar equations exist for the heat quantities d~,y' dQ2,y,

dQ_ 1 and dQ2 , which are conducted in the directions y and z. ·1, z ,z The total heat entering the parallelpiped in the time dt

can be written as

(3.5)

and the heat energy leaving can be expressed as

(3.6)

The heat energy stored in the body is

dQ'l = WC (dx • dy • dz) .2! dt ~ at ' (3.7)

where w = specific weight of the substance, (lh/ft3)

c = specific heat of the substance, (BTU/lb-0 R) •

Assuming that there is no heat source present in the body,

then, from the law of conservation of energy,

-32-

(J.8)

and substituting dQl' dQ2, and dQ3 from the equations above,

• (3 .9)

Equation (3.9) is the differential equation of conduction of

heat in a solid in which no heat is generated. It expresses the

conditions which govern the flow of heat in a body and con-

sequently any solution to a particular problem must satisfy this

equation.

During steady-state heat conduction, the only property of a

substance which determines the temperature distribution is its

thermal conductivity. However, when the temperature changes

with time, the thermal storage capacity (the product of the

specific weight and the specific heat) of a substance, :in

addition to its thermal conductivity, influences the temperature

variation. In this case, Eq. (3.9) shows that the behavior of

different substances will vary as the ratio of the thermal

conductivity to the thermal capacUy. It is therefore convenient

to define this combination of properties as

k -; a • (3 .10)

This single property, a, is called the thermal diffusivity of the

substaP..ce3.

-33-

Using Eq. (J.10) in Eq. {3.9) yields

dT • {3.11) -=

dt

Introducing the symLol \7 2 (del squared) to indicate seco:1d

partial differentiation, Eq. (3 .11) becomes

aT 2 - = a '\7 T • at

(3.12)

i:ote that Eq. (3.12) is that governing a potential field written

in terms of temperature. ''To some extent it resembles in form

the wave-equation. But there is the fundamental difference

that in the wave-equation the second differential coefficient

with respect to time occurs, whereas here it is the first

differential coefficient. This corresponds with the irreversibility

of the process of heat conduction, which excludes the possibility

of reversing the time, whereas this is possible in the case of

vibration phenomena.•rt

3.4 Initial and Eoundary Conditions. Before proceeding to

the r,"Jathernatical solution of Eq. {3.12) 1 it is necessary to

deterrrdr;e the ini t:ial and boundary conditions which the temperature

must satisfy. The body jnvestigated is assumed to be a semi-

k. Quoted from Planck, THEORY OF HEAT, P• 143, (Brose), 1-:acmj_llan,

'.{ew York, 1949.

-34-

infinite flat plate; its thermal conductivity, k, and diffusivity,

a:, are assumed to be cor.stant. Crate that for the insulation, k,

will be replaced by the apparent thermal conductivity, kapp' of

the insulation). The initial temperature throughout the plate is

a known function of x, where the surface temperature varies

continuously with t:ime. Under these conditions the heat flow

will l:e one-dimensional and the temperature history, T(x, t),

must satisfy the equation

.2! = a a2T for o <. x < L, t ";:P o at a-2 • (3.13)

In addition, the following in:i.tial and boundary conditions must

be satisfied:

and

where

T = g(x)

T == ¢(t)

for O < x .,.. L, t "" 0

for x = o, t ~ O ' L = thickness of plate, (ft.)

'

•

(J.14a) (J .14b)

If the variation of the surface temperature, ¢(t), is

known, Eq. (J.13) may be solved by one of several methods, e.g.,

Greens function in the TheorJ of Potentials, Duhamel' s method,

or by Laplace transformation21. However, in i:iany cases, even

though the time history of the surface temperature is known,

the mathematical expression for the function ¢(t) is so compli-

cated that the previously mentioned methods for obtaining the

solution can become exceedingly difficult or even ir.1possible.

-35-

The method of finite differences (to be described subsequently),

however, provides a simple and expedjent means of solving the

unsteady heat-conduction problem.

3.5 Finite-Difference Nethod of Solution. The method

based on the calculus of finite differences was indicated first

by Birider22 , then developed anew and more in detail by E. Schmidt23,

and further improved and extended by I'.essi and !Jis0lle24. The

following presentation of the subject is based mainly on the

wor~ of these authors.

In section 3.4 it was shown that for the flat plate the

temperature history must satisfy the equatfon,

-= aT for 0 < x < L, t >- 0 • (J.19) at

An equation corresponding to Eq. (J.19) written in finite

differences i1~stead of differentials is

.6 t T 6. x2 T ~~~- ~ ~~~~-

6. t ( 6. x ) 2 (J.20)

where the subscripts t and x indicate that the temperature

differences, 6. T and .6. 2T, are to be taken wjth varying t

and x, respectively.

The quantities 6. t and 6. x are the increnental chanees

in time and distance used in the step-wise procedure of the

method of finite differences. now, if Tm,n denotes the temperature

-36-

at a distance m • L:::,. x (measured from any starting point) and

at time n • 6 t (counted from any zero value of time); then in

the notation of finite differences

(3.2la)

L:::,. x T = Tm,n - T~ - 11 n 1 (J.2lb)

and .6 2 x T = Tm + 1, n - 2 Tm,n + Tm - 11 n • (3 .2lc)

Substituting Eqs. (J.2la), (J.2lb), and (J.2lc) in Eq. (J.20)

gives the recursion fornrula

Tm. ,n + 1 - Tm,n = o: 6 t IT 1 - 2T + T 1 1 • . ( 6 x)2 L m + ,n m,n m - ,nj (3.22)

Equation (J.22) can be used to calculate the temperature at the

point x = m • .6 x for the instant of time t = ( n + 1) L:::,. t

if the distribution close to the point is known at the previous

instant of time, t = n • 6 t.

Note however, that if values of 6. x and .6. t are chosen so

that

( 6. x)2 ----=2, a L:::,. t

then T n is eliminated and Eq. (J.22) reduces to n,

T = Tm + 1, n + Tm - 1, n m,n + 1 •

2

(3 .23)

(J.24)

-37-

Equation (3 .24) states that when A x and 6 t are selected

so as to satisfy Eq. (J.23), the temperature at the point

m • 6. x, at time (n + 1) 6. t, is the arithmetic mean of the

temperatures at the points (m + 1) .6 x and (m - 1) A x at

time n 6 t.

With the foregoing solution of the djfferential equation of

heat conduction, it is only a matter of routj_ne work to find the

temperature distribution through a semi-infinite plate at any

time t. For these calculations the therwal diffusivity (a) of

the rraterial must be known. In fact, it has been shown {see

references 18, 19 and 25) that the accuracy of the solution is

largely dependent upon the choice of a. A search of the

literature revealed no experimental data with which to compare

the present results. Therefore, the solutions obtained may be

somewhat in error due to the use of an incorrect value of a.

For this reasor. the results are presented in terms of the therna.l

diffusivny. Thus, if future experiments show that the value of

a used in the present work is in error, the results obtained in

this thesis rria.y be modified with only a small amount of additfonal

work. For illustratjve purposes, an example problem using rock

wool as the insulating material will be presented next.

J.6 Example Problem. A layer of insulation is assumed to

be placed between the .skin and the ir.ternal structure of a.

missile. Then, using the finite-difference method of solution

-38-

outlined above, the temperature distribution through the

insulation j_s computed for the three trajectories shown in

Figures 2 1 3 arid 4.

The surface temperature variation of the insulation is

assumed to be identical to that of the time history for the

skin. It is recognized that this asswnption is not quite

correct and becomes less so as the intimacy of contact between

the skin and the insulation decreases. 7\1ot even a so-called

••mirror finish'' surface is perfectly smooth in the microscopic

sense. As a result, when two surfaces are placed together, they

actually touch only at a limited number of points, the surr1 of

whkh is usually only a sr:ta.11 fraction of the total surface

area. In general, most of the heat flows through the actual

contact points. This means that at relatively great distances

fron the interface the area for heat flow is much greater than

at the interface. In effect this introduces a large resistance

of very short length in the heat flow path. The length of the

resistance is so short that an apparent temperature discontinuity

exists. The temperature drop depends on the jntimacy of contact

of the two surfaces and on the heat flow. Usually the temperature

drop is of such a sr.ia.11 value as to be negljgible and, since it

is usually not possible to calculate it accurately, it is con-

ventior:al procedure to overlook the effect of contact resistance3.

Note also that, since the effect of contact resistance is to

-39-

cause a drop in the temperature between the skin and the

insulation, the error incurred in neglecting this effect would

result i~: over-predicting the temperature distribution through

the insulation. Thus, the assumption of zero contact resistance

represents, to some extent, a safety factor, since the effect

of this assumption would be to over-estimate the amount of

insulation required.

Y.nowing the surface temperature variation of the insulation,

the next step is to deterrnine the type of insulation to be used.

An inspection of Eqs. (3.23) and (3.24) reveals that the solution

is largely independent of the type of insulation material used.

Although Eq. (3.23) must be satisfied if Eq. (.124) is to be

valid, note that when 6 t is fixed, it is possible for a

(and hence the type of insulation) to take on a range of values,

the effect beinr; to change the size of 6 x. Obviously there

is a restriction on the choice of CI since a large value may

result in 6 x becoming of such a magnitude that the accuracy

of the solution is destroyed. However, since the thermal

diffusivity, a:, of most fosulating materials js of the order of

magnitude of 0.01 ft2 per hr.25, the solution obta:ined from

Eq. (3.24) is sufficiently accurate for most insulating materials.

Thus, the temperature distribution through the insulation may be

found before deciding upon the type of insulation to be used.

-4G-

Before Eq. {3.24) can be used, it is necessary to decide

upon the size of the time foterval 6. t. A choice of 6. t

equal to 0.5 seconds was found to give sufficiently accurate

results; this requires a knowledge of the surface temperature

for every half-second of the trajectory. The temperature

through the slab of insulation is assumed to be a constant at

time equal to zero; then using Eq. (J.24) the temperature through

the slab is calculated at time equal to 0.5 seconds. Using this

temperature distribution, the temperature through the plate is

calculated at time 1.0 second, and so forth. Knowing the

temperature distribution through the slab at any time n 6 t,

it is now necessary to decide upon the type of insulatjon to be

used. In all three trajectories it is assun~ed that the insulating

material is rock wool. The thermal diffusivity, ex, is assumed

to be constant over the temperature range and its value is

taken to be 0.05 ft2/hr25 •

This value corresponds to a mean temperature of 1960 °R.

In reference 26 the thermal diffusivity of rock wool corresponding

to a mean temperature of 610 °R is eiven as 0.019 ft 2/hr. Thus,

as in the case of most materials, the thermal diffusivity of rock

wool increases with increasing temperature. However, due to the

lack of sufficient data, the manner in which the diffusjvity

varies with the temperature could not be ascertained. Therefore, in

-41-

order to be as conservative as possible, the largest known

value of a was chosen.

Substituting for a and C:::. t in Eq. (3.23), the value of

6. x was found to be 0.0448 inch. Thus, tre temperature at

every 0.044.8 inch of the insulation for every half-second of the

trajectory is kriown. Hance, the time hi.story of the temperature

at ar:y point in the insulation is known. These results for

several thicknesses of insulation for each tra,jector.r are shown

in Figures 9, 10 and ll. ;.1ote that the tl.licknesses are given

in terms of a, thereby ma.kins the choice of insulation and/or

the choice of a arbitrary. For rock wool, (a= c.05 ft2/hr)

the values for the thickness of the insulation are: O, 0.10,

0.25, and 0.50 inches. It is interesting to note that for each

trajectory, for a thickness of 0.50 inch of insulation, the

maxir:iun temperature rise through the insulation was 100 °H

although the skin temperature increased as much as 2000 °R.

In the next section a disucssion of the results obtained

fr; this thesis will be presented along w:Lth recommendations for

further investigation of the subject.

-42-

4. DISCUSSION OF RESULTS

4.1 Assumed Trajectories. Three arbitrary trajectories

are assmned as shown in Figures 2, 3 and 4. The first tra-

jectory (Figure 2) is equivalent to that of a three-stage

rocket. The trajectory is assumed linear and is at an angle of

inclinat:i.on of 53.13° with the horizon. The second trajectory

(Fieure 3) is equivalent to a sinrJ.e-stage rocket which is

fired vertically. The thj_rd trajectory (Figure 4) is equivalent

to a sfogle-stage rocket which follows a circular arc path.

1'1hile these trajectories are not the only types which could be

investigated, they are representative for the type of missile

considered.

4.2 Compressibility Effects. The skin temperature

variations with time for trajectories one, two, and three are

shown in Figures 5, 6 and 7, respectively. From these figures

it is seen that no appreciable rise in surface temperature is

obtained before a Mach nurriber of approximately one is reached.

Therefore, the problem of aerodynamic heating is negli.gible until

conpressibility effects become pronounced. The effects of

compressibility on the skin temperatures can be seen readily

in Figure 5, where for altitudes up to 75,000 feet the skin

temperature increases during periods of acceleration and decreases

during periods of deceleration. Thus, an increase in Mach nunber

-43-

results ir. an increase in skin temperature, whereas a decrease

in r'.ach r..umber results in a decrease in skin temperature.

For all three trajectories, the ~(in te~perature reaches a

maxi:r::wn value at an altitude of approxir:iately 80,000 feet,

indicating that the atmosphere becor:tes sufficiently rarefied at

this altitude to make the compressibility effects neglie;ible.

Althour,h the curves are not extended beyond 120,000 feet, it

appears that at altitudes of approxiw.a.tely 1001 000 feet and

M.e;her, the atmosphere becomes rarefied to such an extent that

the problem of aerodynamic heating is much less severe that at

lower altitudes. In fact, it has been shown in reference 1,

that for altitudes of 100 miles and greater, the aerodynamic

heating is negligible and the predominant heating problem is

due to solar radiation.

Note that irrespective of the type of trajector'J,

approximately the same maximum temperature of 2200 °R is

obtained. Thus, for the altitude and J:ach number range con-

sidered here, the type of trajectorJ is relatively unimportant

with re.c;ards to the r..aximum skin temperature obtained. Further

investir;ations would have to be ma.de before the above statement

could be generalized to include all Nach number and altitude

ranges.

4.3 Comparison of Two- and Three Dimensional Solutions. A

conparison of the two-dimensional solution with the three-

dimensional solution for the skin temperatures for trajectory

-44-

number three (Figure 4) is presented in Figure 8. The two-

dimensional solution corresponds to a flat-plate and the three-

dimensional solution corresponds to a cone, and :is computed by

the use of the cone rule presented by Van Driest in reference 13.

It is seen from Figure S that the max:Lmum difference between the

two solutions is approximately 5 percent. Hote, however, that

the present results are for a point on the body where the pressure

coeffid ent is zero, and are not to be construed to mean that the

percent difference between the two solutions is of such small

magnitude for all cases.

From Figure 8 it js seen that the flat-plate solution

predicts higher values of the skjn temperatures than does the

three-dimensional solution (the explanation for this is given in

referer:ce 2). Without experimental data for comparison purposes,

neither solution can be assumed to be correct. However, :in

order to be on the conservative side it is recommended that the

nat-rlate solutfon be used for a cylindrical afterbody.

4.4 Effect of Skin Thickness. In Figure 8, the skin

temperature variation for trajectory number three is calculated

us:l.ng a skin thickness of 0.010 inch. In Figure ?, the skin

te:r.-tperature variation for the same trajectory is presented for

a skin thickness of 0.005 inch. Comparing the two-dimensional

solution in Figure g with Figure 7, it :;_s seen that the effect

of doubling the skin thickness is to increase the skin temperatures

-45-

by approx:irrately 5 percent. This is, in part, due to the fact

that when the skin thickness is increased, the heat absorption

capacity of the skin is increased. A complete understanding of

the effect of skin thickness on the skb temperature variation

would require considerably more jnvestigation than has been

presented here; however, the present results indicate that the

skin ter.lperatures may be decreased by decreasine; the skin

thickness.

4.5 Effect of Insulation on Cooling Reguirements. The

temperature variations with time for three thicknesses of

insulation are shown for trajectories one, two and three in

Figures 9, 10 and 11, respectively. As was discussed in a

previous section, the thicknesses of the insulation are

presented in terms of o:. The data is presented in this form in

order to make the solution independent of the choice of

j_nsulation and/or the value of the thermal diffusivity (o:).

To determine the thickness of the insulation corresponding to

the curves shown, it is only necessary to multiply the square

root of o: by the proper constant as indicated on each curve.

For illustrative purposes, it is assumed that the insulating

material is rock wool (o: == O.C5C ft2/hr). For this value of the

thermal diffusivity, the thicknesses of insulation are found to

be C.10, 0.25, C.50 inches. Referring to Figures 9, 10 and 11

it is seen that if the insulating material has a thickness of

-46-

one-half inch, the artlount of heat transferred through the

insulation is negligible for all three trajectories. Furthermore,

when the thickness is reduced to one-fourth inch, the maximum

terr.perature rise through the insulation is less than 500 °R,

whereas the maximum skin temperature rise is approximately 1700 °R.

It should be noted that since the insulation only retards the

heat flow, trajectories lasting for longer periods of time would

require greater amounts of insulation than are jndicated in the

present results. However, for trajectories of short duration

(say, a minute or less), the use of insulation to prevent the

heat from reachine; the internal structure of a missile is very

effective.

4.6 V{eight PenalHjes Due to Addition of Insulation. Since

weight is always of prime importance in the design of any

aircraft, it is of interest to investigate the weie;ht penalities

resulting frorr, the addition of the insulation. Again, the

:insulating material will be assumed to be rock wool and the

specif:ic weight of the material is taken to be 7 lbs/ft3• Then,

if the thickness of the insulation is chosen to be one half inch,

one pound of insulation will cover J.43 square feet of area.

Thus, if a missile has a total surf<:..ce area of 100 square feet,

approximately 30 r:ou..Y!ds of insulation will be required. Con-

sidering the weight of present day missiles, this is a small

penalty to pay to prevent the temperatures of the internal structure

-47-

fror.1 becorrd.n(~ rnore than. 5 percent of the surface ter..perature.

In fact, jt js possible that the gross we:i.£'.;ht of the missile

will be less with the insulation thm-1 without, since the

internal structure w:Ul not have to be desic;ned to withstand

the high temperatures to which it would be subjected if the

iumlation was rot er:1ployed.

4.7 Recom.mendat:ions. It is the op).nior: of the author that

before the er.gineer can cor;fidently incorp:>rate heat jneulation

into the desigr, of a M.gh speed vehfole, the following areas of

research in heat trans~dssior: should be explored fUrtherl

1. A study of the them.al pro:perties of heat insulation at

hit~h temperatures should be rr.ade, w:ith particular attention

given to the var:ation of the thermal diffusivity wjth

ten:perature.

2. Investigations which would result in a clear understandfog

of the rnanner :in which heat is transferred through a cellular

er porous r.mi-homogeneous ~olid should be made. Particular

attent:ion should he g5ve~ to the a.mount of hea.t transferred

tJ radiation from surface to surface of the indiv::iduul

cells cf the ::>oHd.

-48-

3. The validity of usi11g an apparent coefficient of

thermal conductivity should be investigated, and the

limitations of this concept clearly stated.

4. Sound methods for calculating the total heat flow for

the case in which heat is transferred through a porous

solid by the combined mechanisms of conduction, radiation,

and/or convection should be developed.

-49-

5 • CONCLUSIONS

From the aforegoing results of the investigation, the

following conclusions were drawnl

1. The problem of aerodynamic heating is negligible until

compressibility effects become pronounced.

2. At an altitude of approxinately 80 1 000 feet, the

atmosphere becomes sufficiently rarefied to make the

compressibility effects on aerodynamic heating negligible.

J. ·For the Mach number and altitude range investigated,

the type of trajectory is relatively unimportant with

regards to the rraximur:i skin temperature obtained.

4. For large Reynolds numbers, there is less than 5 percent

difference in the values of surf ace temperatures predicted

by the two- and three-dimensional solutions.

5. The amount of heat lost by the skin due to radiation

ca:-1 be increased by decreasing the skin thickness.

6. The additional weight arising from the use of insulation

is negligible compared to the total weight of present day

missiles.

-50-

7. For trajectories of short duration (say, a minute or

less), the temperature of the internal structure of a

high-speed vehicle can be held to less than 5 percent of

the surface temperature by the use of a reasonable amount

of heat insulation placed between the skin and internal

structure of the vehicle.

-51-

6. SUMMARY

The present thesis project consisted of two parts. First,

a general method for deterrri.ining the transient skjn temperatures

of bodies during high-speed flight was developed. The governing

differential equation was presented for this purp:>se, giving the

fundamental relations between the transient skin temperature and

flir;ht history. The determination of all pertinent parameters in

the equation was discussed, and the Runge-Kutta numerical method

of integration was used to obtain the solution. The method was

employed to compute the time history of the skin temperatures for

several hypothetical flight plans, and the results presented in

the form of graphs. For the ~.iach number and altitude range

investigated, the maximum skin temperature obtained was approxi-

mately 2200 °R and was found to be largely independent of the type

of trajectory.

The second portion of the project consisted of determining

the effect of heat insulation on the cooling requirements of the

internal structure of a high-speed vehicle. The r.:soverning

equation for heat conduction through an isotropic solid was

developed, and then modified to account for nonhomogeneous

materials. The initial and boundary conditions for the governing

equation were specified, and the equation solved by the method of

finite-differences. The temperatures obtained,the first portion

-52-

the thesis, were used as the outer surface temperature variation

of the insulation, and the time history of the inner surface

ter~perature of the insulation (for several thicknesses) was

calculated. To mal:e the problem as general as possible, the

results were presented in terms of the thermal diffusivity of

the insulating material. For illustrative purposes, an example

problem was worked using rock wool as the insulating material.

It was found that, by using one-half inch of this insulating material,

the maximum temperature obtained by the internal structure was

less that 5 percent of the skin temperature. Thus, it was

concluded that the increase of the temperature of the internal

structure of a high-speed vehicle during a limited time of

flir:ht, can be held to structurally perrrissable values by the

use of heat insulation placed between the skin and the internal

structure of the vehicle.

-53-

7. ACK~10WLEDGl!lIE'ITS

It is the desire of the author to express his sincere

appreciation to Dr. Robert w. Truitt and

of the Aeronautjcal Engineering Department at the Virginia

Polytechnic Institute, for their patient assistance in the

successful completion of this project.

-54-

8. BIBLIOGRAPHY

1 Truitt, R. w., Fundamentals of AerosJ,mamic Heatin.g. The

Glenn L. Martin Co., March 1957.

2

3

4

5

Van Driest, E. R., The Problem of Aerodynamic Heatine•

Aeronautical Engineering Review• October, 1956.

Giedt, w. H., Princip1es of Fl!gineering Heat Transfer. lat.

F.d., D. Van Nostrand Company, Inc., Princeton, N. J ., 1957.

Ackermann, G., Plate Thermometer in High Velocity Flow with

Turbulent Boundary Layer. Forschung auf dem Gebiete des

Ingenieurwesens, Vol. 13, P• 226, 1942.

Stefan, J., Sitzungsber. d. Kais. Akad. d. Wiss. Wien, Ma.th. -

Naturwiss. Vol. 79, (p. 391) 1879.

6 Boltzmann, L., Wiedemanns A:lllilen, Vol. 22, (p. 291) 1884.

7 O'Sullivan, w. J., and Wade, kl. R., Theor:y and A't;paratus for

Measurement of :&nissivity for Radiactive Cooling of Ilypersonic

Aircraft with Data for Ineonel and Inconel x. NA.CA TN 41.21, 1957•

8 Lo, Hsu, Determination of Transient Skin Temperature of

Conical Bodies During Short-Time, High-Speed Fligl1t. HACA TN

1725, October, 194.8.

-55-

9 Van Driest, E. R., On the Boundary Layer with Variable Prandtl

Number, 1954 Jahrbuch der Wissenchaftlichen Gesellschaft

10

11

12

13

fur Luftfahrt e. v. (WGL), PP• 65-75; Friedr. Vieweg and

Sohn, Verlag, Braunschweig.

Van Driest, E. R., The Turbulent Boundary La.yer with

Variable Prandtl Number, 50 Jahre Grenzschichtforschung,

PP• 257-211; Friedr. Vieweg and Sohn, Verlag, Braunschweig.

Li, T:l.ng-Yi, and Nagamatsla., H. T., Shock-Wave Effects on

the Laminar Skin Friction of an Insulated Flat Plate at

Hzyersonic Sf!eds. Jour. Aero. Sci.,, Vol. 20, No. 5,

May 1953.

Bertram, ¥.d.tchel H., Boundary-Layer Displacement Effects

in Air at Mach tlumbers of 6.8 and 9.6. NACA TN 413.3, 1958.

Ince, E. L., Ordiparz Differential Equations.

Publications (New York), 1944.

Dover

l4 Crandall, s. H., llhgineering Analysis. McGraw-Hill Book

Co., Inc., New York, 1956.

15 Van Driest, E. R., Turbulent Boundary Layer on a Cone in a

Supersonic Flow at Zero Ang1e of Attack. Jour. Aero. Sci.,

Vol. 19, No. 1 P• 55 (1952).

16

17

lS

19

2C

21

22

24

-56-

Eckert, Ii. v., S:i.m.plified Treatment of the Turbulent Boundary

Layer Along a Cylinder in Cowressible Flow. Jour. Aero.

Sci., Vol. 19, No. 11 P• 23 (1952)

Finck, Hat. Bur. Standards, Research Paper 243; J. Research,

(1930).

r-icAdams, w. H., lieat Transmission, 3rd. ed., NcGraw-Hill

Book Co., Inc., New York, 1954.

Diot, J. B., Dibliothegue Britannigue, Vol. 'Z7 1 1S04.

Fourier, J. B. J., "Theorie analytique de le. chaleur• ',

Gauthier-Villars, Paris, 1822; English translation by Freer.nan,

Carabridge, uns.

Carslaw, H. s. and Jaeger, J. c., Conduction of Heat in

Solids, Oxford University Press, London, 1948.

Binder, L., Dissertation, Teclm. Hochschule 1-iuenehen, w. Knapp, Halle a. s., 1911.

Schmidt, E., Zeitschr. d. Ver. deutsch. Ipg., Vol. 70, 1947.

Nessi, A., and Nissolle, L., Methods graphigues p:>ur l•etude

des installations de chauf faf;;e et de refrigeration en

re;~ime discontinu, Dunod, Paris, 1929.

25

26

28

29

30

31

-57-

~!ilkes, G. B., Beat Insulation, John \1iley and Sons, Inc.,

New York,, 1950.

Wilkes and Wood, The 5J?!i?Cific Heat of Thermal Insulators,

Trans. A.S.H.V.E., Vol. 42, 1942.

Pohlhausen, E., Der Warmeaustausch Zwischen Frestem Korrnr

Und Flussirfseiten Mit Kleiner Reibune Und Kleiner

Warrneleitung. z. F. Angew. Math. u. Mech. Vol. 1, No. 21

1921, PP• 115-121.

Crocco, Luigi, Transmission of Heat from a Flat Plate to a

Fluid Flowing at a !-li&h i[elocit;z, i·JACA TH, l'~o. 690, 1932.

Von Karman, Th., The Problem of Resistance :in Compressible

Fluids. Royal Acade:rqy of Italy, Rome 1936.

Von Karman, Th. and Tsien, H. s., Boundary La.Y!;Zr in

Compressible Fluids. Jour. Aero. Sci., Vol. 51 No. 6,

April 1938,, PP• 2Z7-232.

Huston, w. B.,Warfield, C. ~:., and Stone, A. z., A Study of

Skin Temperatures of Conical Bodies in Supersonic Flight.

NACA Ti1 1724, October, 1948.

The vita has been removed from the scanned document

-59-

12000

10~00

I I

9000 ~· I I

I 7~00 ~- ·r

I I

I I

1000

To - Tm Q)

4SOO

ISOO

0 .....:~~~...1---------J...---------L--~--~...__~~~-L...~~ 0 2~00 sooo 7500

V00 (FT/SEC)

10000 12SOO 14000

FIG.I. STAGNATION - TEMPERATURE RISE FOR VARIABLE Cp·

Mach number

12 100 Altitude

9 75

hx 10-1

Mo (Feet)

~ 6 50

·~-~--·+- 25

--0 ...-.....::=...;;~....i...--------'--------....1.--------'---------..__------------~~..______. 0 0 4 8 12 16 20 24 28 30

t (Seconds)

FIG. 2. MACH NUMBER AND ALTITUDE DIAGRAM (TRAJECTORY NO. I)

- Mach number

12 100

-- Altitude I I

I I

9 75 /

hx 10-3 / Mo / (Feet) / I

CJ'.. I-'

6 50 I

/ /

/ /

3 / 25·

/ /

/ -__ _,,

0 0 0 3 6 9 12 15 18 20

f (Seconds)

f1G. 3. MACH NUMBER AND ALTITUDE DIAGRAM (TRAJECTORY No, 2 )

Mach number 12

Altitude

9 75

h x 10-5

Mo (Feet) !--

6 50 I\) I

0 L-~--.-:;iiiE::;..=;::::__~--l.~~~---1~~~~..l-~~~-l....~~~-..l.~~----l 0 0 5 10 15 20 30 34

t (Seconds)

FIG. 4. MACH NUMBER AND ALTITUDE DIAGRAM (TRAJECTORY NO. 3)

2500

2000

1500

; v - / /" -

TW ( 0 R)

!; 1000 '

' / I I I _/ ! i ._

I

I ~00

- I

0 0 4 8 12 16 20 24 28 30

t (Seconds)

FIG. 5 TIME HISTORY OF SKIN TEMPERATURE (TRAJECTORY NO. I)

2500

/ "' 2000 /

v 1~00

/

1000

___,/ v

~00

0 0 3 6 9 12 15 18 20

t (Seconds)

FIG. 6. TIME HISTORY OF SKIN TEMPERATURE (TRAJECTORY N0.2)

I r

2000 t-------+----~---- -r---2 ~ 0 0 o------+-

I ----l-----t-----

IOO O -------+

I J

500 t-------- _ ___._ ____ - ---- ------l-- - --- ------i I

0 L-~~~--J.~~~~-"-~~~~--~~~~'--~~~-'-~~~~--~~--0 5 10 15 20 25 30 34

t (Seconds)

FIG. 7. TIME HISTORY OF SKIN TEMPERATURE (TRAJECTORY No. 3 )

Flat plate -2000

Cone

Note: t1= 0. 01 inch

1000~-- -;-·-----, I

0 0 5 10 15 20 25 30 34

f (Seconds) FIG. 8. COMPARISON OF 2-DIMENSIONAL SOLUTION FOR SKIN TEMPERATURES WITH

3- DIMENSIONAL SOLUTION (TRAJECTORY NO. 3)

Note: t. ii the I

2000 thickness of

insulation Cf t.).

0 0 4 8 12 16 20 24 28

t (Seconds)

FIG. 9 TEMPERATURE VARIATION IN INSULATION (TRAJECTORY NO. I)

s I

30

Note: t. is the I

2000 thickness of Insulation (ft.).

T (0 R)

t; = 2.236../0

0 0 3 6 9 12 15 18 20

t (Seconds)

FIG. 10. TEMPERATURE VARIATION IN INSULATION (TRAJECTORY NO. 2)

2000

1500

T (0 R)

0

Note: t, : s the

thickness of

insulation (ft.)

I -- ----------,--··---+-·-----1 I , I I . I

·-·~· _J_~-·--· __ ____.___ _ __..__ _________ __ 0 5 10 15 20 2S 30 34

t (Seconds)

FIG. 11 TEMPERATURE VARIATION IN INSULATION (TRAJECTORY NO. 3)

I O" 'f