AN APPROXIMATE METHOD OF CALCUIATING THE WEIGHT N,

OF THE TWO-INSULATION--TWO-COOLANT

THERMAL PROTECTION SYSTEM

By

John Grady ~avis, Jr.

Thesis submitted to the Graduate Faculty of the

Virginia Polytechnic Institute

in candidacy for the degree of

MASTER OF SCIENCE

in

MECHANICAL ENGINEERING

Blacksburg, Virginia

" .• 1 ," ~.

, 'f .

- 2 -

II . TABLE OF CONTENTS

CHAPrER

I. TITLE . . . . . · · · · · II. TABLE OF CONTENTS · · · · ·

+:II. LIST OF FIGURES · · . IV. LIST OF SYMBOLS ·

V. INTRODUCTION

VI. THE REVIEW OF LITERATURE

Insulating Slab · . · · · · Insulated Structure . . . . . · Insulated and Cooled Structure · · · · Modified Insulated and Cooled Structure ·

VII. ANALYSIS . . . · · . . · · . VIII. RESULTS AND DISCUSSION

Discussion of Equations .

Material properties, X/L unrestricted.

Material properties, X/L restricted.

Outer coolant location

Comparison With Numerical Solutions

IX. CONCLUSIONS. .

X. ACKNOWLEDGMENTS

XI. REFERENCES

PAGE

1

2

· · · · · · 4

· · · · 5

8

12

· · · 12

· · · · · 17

· · · · · 19

22

· · · 25

· · · · · 32

32

· · · · 33

34

36

· · · · 44

· · · · 46

47

- 3 -

C:HAP.rER PAGE

XII. APPENDICES · · • 50

A. THERMAL CONDUCTIVITY DENSITY ANALYSIS 50

B. DIGITAL COMPUTER PROGRAM • . • · · 53

XIII. VITA . . . . . . . . . . . . . . . · · · 54

- 4 -

III. LIST OF FIGURES

FIGURE PAGE

1. Nose cap radiation equilibrium temperature versus

altitude and velocity • • • • • • • • • • • • • 9

2. Two-insulation-two-coolant thermal protection system. 10

(a) Vehicle wall 10

10

13

(b) Mathematical model of wall

3. Thermal protection systems ••••

(a) Insulating slab • • • • • • • • · • • •• 13

(b) Insulated structure • . . . . . . . . (c) Insulated and cooled structure

(d) Modified insulated and cooled structure •

. . . . . . 13

14

14

4. Effect of outer coolant location on 6lieff • • • • • 35

5. Temperature histories ••••••••• 37

(a) ~To = 2930° R, T = 0.25 hour . . · . 37

(b) ~To = 29300 R, T = 0.50 hour • . . . . • . . • . 37

(c) ~To = 29300 H, T = 0.75 hour • 37

Cd) ~To = 1400° H, T = 0.25 hour • 37

6. Percent difference between weights predicted by approximate

and numerical solutions • • • • • • • 39

7. Comparison of the sum of insulation and coolant weights

predicted by approximate and numerical solutions · . 41

- :; -IV. LIST OF SYMBOLS

A function defined by equation (18)

B constant in equation (Al), Btu-f't2 /hr .. ~-lb

C density corresponding to a minimum thermal conductivity-

density product in equation (Al), Ib/ft3

D constant ~in equation (AI), Btu-Ib/hr-ft4.Da

E constant in equation (A2), Btu-ft2 jhr-OR-lb

F density corresponding to' a minimum thermal conductivity-

density product in equation (A2), Ib/ft3

FE emissivity factor

G constant in equation (A2), Btu-lb/hr ... ft4-~

H enthalpy, Btu/lb

h convective heat transfer coefficient, Btu/hr_rt2_~.

k thermal conductivity, Btu/hr-ft-oR

L insulation thickness, ft

n integer

~TH dimensionless parameter defined by equation (24)

Q heat flux, Btu/ft2

q heating rate, Btu/hr-ft2

r nose cap radius, ft

T temperature, oa '1' average temperature, ~

u dummy variable of integration

W weight, lb/rt2

X outer insulation thickness, ft

y

a.

€

e

¢

p

T

- 6 -

inner insulation thickness, ft

thermal diffusi vi ty, ft2/hr

emissivity

Stefan-Boltzmann constant, 0.1714 • (10-8) Btu/hr .. ft2_~4

defined by equation (A5), .+b ... hr-oR/Btu

defined by equation (A4~, lb-hr-oR/Btu

density, lb/rt3

time, hr

dimensionless temperature ratio defined by equation (49)

Subscripts:

a

approx

aw

b

c

cond

cw

CH

eff

eq

f

fg

GH

g

apparent

approximate

adiabatic wall

back surface

coolant

conduction

cold wall

characteristic

effective

equivalent

final

vaporization

conditions during the ground-hold period for an aerospace

vehicle

gas

- 7 -

i initial

ins insulation

M mean

max maximum

min minimum

num numerical

o outer surface

opt optimum

r radiation cooling

s structure

ss steady state

vb vapor barrier

T function of time

1 refers to outer insulation or coolant

2 refers to inner insulation or coolant

- 8 -

V. INTRODUCTION

One item of major concern for aerospace vehicles is structural weight.

A significant portion of the total structure weight is in the thermal pro­

tection system necessitated by high-speed flight and possibly the use of

cryogenic fuel. The boundary layer surrounding a vehicle in hypervelocity

flight contains high-temperature air as a result of the conversion of

kinetic energy into internal energy. An example of the temperatures

encountered in high-speed flight can be seen in figure 1. Equation (10)

of reference 1 is used to obtain the curves shown in figure 1.

Two types of thermal protection systems are generally considered for

aerospace vehicles. They are radiation cooled or absorption type systems.

A major portion of the aerodynamic heat load is radiated from the surface

of the radiation cooled type system, while only a small percentage of the

heat load is conducted into the system. Examples of this type system are

insulated heat shields and high-temperature ablators. As the name implies,

most of the aerodynamic heat load is absorbed by the absorption type sys­

tem, while only a small percentage of the heat load is radiated from the

surface. Examples of this type system are low-temperature ablators and

heat sinks. Most systems considered to date have consisted of one or more

insulations and one coolant. However, the use of cryogenic fuels and the

long flight time~ envisioned for future aerospace vehicles will generate

a need for more sophisticated systems. For example, see figure 2.

In the preliminary design of an aerospace vehicle, the selection of

a thermal protection system is of great importance. Usually, this requires

investigating several different systems and various materiaJ. combinations

;-0

X

+J ~ ... Q)

'"0 ::l +J

+J

«

30

20

10

o

- 9 -

10

r = I ft E = .8

Note: Curves are computed

from eq. (10) ref. I

20

Velocity, ft/sec x 10-3

Figure 1.- Nose cap radiation equilibrium temperature versus altitude and velocity.

30

- 10 -

cooled structure

coolant passage

(coolant I)

Q

1

(a) Vehicle wall

I / /QCO~d X

L insulation I

coolant

1 y insulation 2

L coolant 2

(b) Mathematical model of wall

ryogenic propellant tank

pellant

(coolant 2)

insulation 2

insulation I

skin (outer surface)

outer surface

Figure 2. - Two-insulation-two-coolant thermal protection system.

- 11 -

for each. When elaborate computational procedures and equipment are

utilized, the selection process becomes very time consuming.

The problem to be considered in this paper is the development of an

approximate solution for estimating the weight of thermal protection sys­

tems. While the more simple thermal protection systems have been analyzed,

systems consisting of one or more insulations and two coolants require

study. The attention of this paper is devoted primarily to radiation

cooled nonablating type systems containing two coolants. A comparison of

weights calculated by the procedure developed in this paper and weights

obtained with the aid of an electronic computer are given. The effects of

material properties and coolant location on thermal protection weight are

discussed.

- 12 -

VI • THE REVIEW OF LITERATURE

A review of the literature indicates that approximate and exact

methods for calculating the weight of the thermal protection systems

shown in figure 3 have been developed. A discussion of the methods that

are suitable for use in obtaining preliminary weight estimates follows.

Insulating Slab

The thermal protection system shown in figure 3(a) is probably the

simplest one encountered. It consists of a single layer of insulation.

A convective heat load, Q, is imposed on the outer surface. Most of the

heat load is radiated from the surface, while a small percentage is con-

ducted into the slab. Heat losses from the back surface of the slab are

neglected. In order to estimate the weight of this system, a relationship

between insulation thickness, L, outer surface temperature, To, and the

back surface temperature rise, ~b' must be known. This relationship

has been presented in references 2 and 3.

An exact solution

1 - (1)

to the problem is given in reference 2 for the case where the outer sur-

face temperature does not vary during the heating period. A graphical

representation of equation (1) that may be used to aid in the rapid calcu-

lation of insulation thickness is included in reference 2. To use the

graph, the value of (6Tb/~o) is computed and the corresponding value of

r L

L

- 1; -

(a) Insulating slab.

(b) Insulated structure.

Figure 3.- Thermal protection systems.

--T o

vapor barrier

L

L

T

- 14 -

insulation /

/

(c) Insulated and cooled structure.

/

(d) Modified insulated and cooled structure.

Figure 3. - Concluded.

--T o

- 15 -

(uT/L2 ) is read from the graph. Knowing the thermal properties of the

insulation and the heating time, L can then be calculated and used to

determine the weight, (pL).

An approximate solution to the problem that includes the effect of a

variable outer surface temperature is presented in reference 3. The solu-

tion is based on the hypothesis that the actual problem (time varying

outer surface temperature) may be simulated by an equivalent problem

involving a constant surface temperature which produces the same transient

history at the back surface. A summary of the analysis of reference 3

follows. The actual heating rate, ~w, is replaced by an equivalent

heating rate,

where

and

~w(max)

Tqcw(max)

q' , which starts at an initial time, Ti, defined by cw

T

2 J qcw(max) qcw dT

o

~w(max) (2)

is the maximum heating rate achieved over the trajectory

is the corresponding time. A condition placed on the

transformation is

~w dT

and the variable free-stream enthalpy, Hg , is replaced by its time ,

average, Hgo

(4)

- 16 ..

Making use of equations (2) through (4), a heat balance at the outer sur-

face of the slab is written

Convective heat load

Radiation cooling

Heat conducted into slab

To obtain a solution to equation (5), an iteration procedure is required

since there are essentially two uriknowns, the surface temperature, To,

and the insulation weight, W. For the first iteration, WCp 6T.M and I

Hg/Ho are assumed equal to zero. This yields the maximum possible value

for To.

(6)

Knowing T~ and Tb , the ratio of mb/m~ is formed. Making use of

figure 3 in reference 3 which is a plot of mb/~~ and mM/m~ versus

kp(Tf - Ti) f" it WI . 2 ,the insulation weight for the lrst eration, ,lS CpW

computed. Note this is the maximum possible value for W. ,

Using To

and a known relationship between He and To, a hot-wall correction may

be applied to ~w. This represents the minimum possible hot-wall heat

load. The values obtained for WI , ~,and H~

equation, (5) • " Solving for To yields

are substituted into

- 17 ..

Equation (7) gives the minimum possible value for To' The iteration

/.

If procedure is repeated by forming a new value of ~b ~o and computing

a value for the insulation weight, W" • This process is continued until

suitable convergence is obtained. A comparison of weights obtained by the

method of reference 3 with weights obtained by a numerical solution

indicates a difference of less than 6 percent for the environment selected

in reference 3. Sample calculations are shown in reference 4.

Insulated Structure

The insulated structure system is shown in figure 3(b). The system

consists of an insulating layer and a load carrying structure. Boundary

conditions at the outer surface of the insulation are the same as for the

insulating slab previously discussed. All heat that passes through the

insulation is absorbed by the structure. This system has been studied

and five approximate solutions that are suitable for obtaining preliminary

weight estimates are presented in the literature.

One approximate solution that might be used is found in reference 5.

The solution

L -----:--1__ tan _1[; (To TSi "

In To ~ TSf

(8)

is applicable for the case where the outer surface temperature remains

constant during the heating period. Equation (8) is derived by adjusting

the coefficient of the first term in the exact solution given in refer-

ence 5 to satisfy the initial conditions of the problem, Other terms in

the exact solution are neglected. A graphical representation of

- 18 -

equation (8) that may be used to aid in the rapid calculation of insulation

weight (PinsL) is given in reference 6.

Two approximate solutions that are applicable for the case where

radiation cooling may be neglected and the convective heat transfer

coefficient, h, and adiabatic wall temperature, Taw' remain constant

are presented in reference 7. Both solutions are based on the assumption

that a linear temperature gradient exists through the insulation. The

first solution

TSf - TSi

Taw - TSi

omits the effect of the heat capacity of the insulation. The second

solution

TSf - TSi

Taw - TSi = 1 -

-~nsT exp ----------------~------------~

( k.) ( CPi PinsL) W C L 1 - ~ 1 + ns s Ps hL 2WsCps

(10)

attempts to account for the heat capacity of the insulation. Equations (9)

and (10) differ by less than 5 percent from the exact solution when the

criteria listed in reference 8 are satisfied. The results of the thermal

analysis presented in reference 7 are applied to develop an optimum design

procedure for insulated tension plates in that paper and for insulated

compression plates in reference 9.

The analysis of reference 7 is extended to include the effect of a

varying adiabatic wall temperature in reference 8. The solution is

- 19 -

(11)

where u is a dummy variable of integration.

An approximate solution that accounts for the heat capacity of the

insulation and is applicable to the case where the outer surface tempera-

ture varies is given in reference 3. The solution is

(12)

where W* is the weight of an insulating slab designed for the same back

surface temperature as the structure. The value of W* is computed by

the iteration procedure given in reference 3 for determining the weight of

an insulating slab. Sample calculations are shown in reference 4.

Insulated and Cooled structure

The insulated and cooled structure system shown in figure S(c) con-

sists of an insulating layer, a load carrying structure, and the coolant.

The structure and coolant are assumed to operate at the same temperature

and all heat that passes through the insulation is absorbed by the coolant.

- 20 -

Boundary conditions at the outer surface of the insulation are the same as

for the insulating slab. This system has been analyzed and various

approaches appear in the literature for estimating its weight.

An approximate solution to the problem that is applicable for the

case where the outer surface temperature remains constant

W = 2 (kp)ins(To - TC)T

Hfg

is presented in reference 10. In the derivation of equation (13) steady-

state conditions were assumed. A result of the analysis presented in

reference 10 is that minimum system weight is obtained when the insulation

and coolant weights are equal, or

(14)

The case where the outer surface temperature varies with time is

examined in reference 11. The solution presented is

w = 2 (

In the derivation of equation (15), it is assumed that a linear tempera-

ture gradient exists through the insulation and the heat capacity of the

insulation is neglected.

An approximate solution that accounts for the heat capacity of the

insulation and a varying outer surface temperature is given in reference 3.

- 21 -

A summary of the analysis presented follows. A new heating rate and time

are computed f'rom equations (2) and (3) of this paper. A heat balance at

the outer surface is written

, (H~) 4 _ _ k(dT) qcw 1 - -- - cr€To - dX_

Ho x~O

'-V--/~~ Convective Radia- Conduction heating tion into

cooling insulation

The minimum total weight of insulation and coolant is obtained from

where

(16)

(18)

and (Tss - Ti) is the time required for the insulation under transient

conditions to achieve the same mean temperature, TM, as exists during

steady state. Since To and A are unknown, an iteration procedure is

required to solve equation (17). For a first approximation, the right­

hand side of equation (16) is set equal to zero and T~ is computed from

the remaining portion of the equation- Knowing T~ and Tc ' the ratio'

~TM/ m~ is formed and the value of A' can be determined from figure 3

in reference 3. Figure 5 in reference 3 may then be used to determine the

value of Wins corresponding to T~ and A'. Solving for the steady-.

state conduction flux, qss = k.tns(T~ - Tc)/L, a second approximation for

- 22 -

To is computed from equation (16). This procedure is repeated by

forming a new ratio of ~M/6T~ and solving for the corresponding values

of A" and W" ins· The process is continued until suitable convergence

of the solution is obtained. Sample calculations are shown in refer-

ence 4. A comparison of weights obtained by the method of reference 3

with weights obtained by a numerical solution indicates a difference of

less than 10 percent for the environment selected in that paper.

Modified Insulated and Cooled Structure

The modified insulated and cooled structure system is shown schemat-

ically in figure 3(d). This system consists of an insulating layer, vent

space, the coolant, and a load carrying structure. The boundary condi-

tions at the outer surface are the same as for the three systems pre-

viously discussed. Heat passes through the insulation to the vapor

barrier where it is radiated to the coolant. All heat that crosses the

vent space is absorbed by the coolant since the coolant and structure

are assumed to operate at the same temperature. Attention is devoted

to this system in references 12 and where methods are given for

obtaining weight estimates.

The analysis presented in both reference 12 and reference 13 is

based on steady-state conditions. In addition, the assumption that

minimum system weight will be obtained when insulation and coolant weight

are equal is made in reference 12. A summary of the analysis of refer-

ence 12 follows. A value for the vapor barrier temperature, Tvb, is

selected and then the heat flux across the vent space is calculated from

- 23 -

The coolant weight is obtained from

(20)

and the insulation weight is given by

(21)

Weights obtained from equations (20) and (21) are compared. If the weights

differ, a new value for TVb is selected and the calculation process is

repeated. This process is continued until the insulation and coolant

weights obtained from equations (20) and (21) approach the same value.

As indicated in reference , the analysis of reference 12 can lead

to appreciable error in certain instances. However, for the environment

selected in reference 12, the assumption that coolant and insulation

weights are equal appears reasonable.

A more refined method of computing insulation and coolant weight is

given in reference The results of the analysis performed in refer-

ence 13 are as follows: Coolant weight is calculated from

2 1/4

(Wc) 3 -1/4(Wc ) - +4'P -WCH TH WCH

= 1 (22)

where

- 24 -

and

(24)

Insulation weight is given by

When the value of PTH is less than or equal to one-half, the insulation

weight equals zero. As the value of PTH becomes large, the value of

(Wc/WCH) approaches one. Hence, for large values of PTH, the insulation

and coolant weights are approximately the same value.

While weight estimates may be readily obtained for the thermal protec-

tion systems shown in figure 3, the estimation of weight for thermal

protection systems containing two insulations and two coolants appears to

still be a problem. The primary attention of the remaining portion of

this paper will be devoted to the development of an approximate method

for calculating the weight of two-insulation--two-coolant system.

- 25 -

VII. ANALYSIS

In this section of the paper, approximate weight equations are

developed for the two-insulation--two-coolant thermal protection

system. An example of this type system is shown in figure 2(a). The

system consists of an outer skin that forms the aerodynamic surface,

an insulating layer, the cooled structure, and the insulated cryogenic

propellant tank.

The mathematical model for this system used in the derivation of

weight equations is shown in figure 2(b). Note, the outer skin,

structure, and propellant tank wall are omitted from the model since

the temperature drop through these items will be negligible for most

aerospace applications.

Having defined the mathematical model, the next problem is to

determine the total weight of insulation and coolant required to

sustain a given thermal environment. The following assumptions are

made to simplify the analysis:

(1) The heating rate is uniform over the outer surface which

results in one-dimensional heat flow.

(2) The difference between the radiation equilibrium temperature

and the actual outer surface temperature is neglected because it is

small compared to the temperature drop across the outer insulation.

(3) Material properties are independent of temperature.

(4) A linear temperature profile exists through each insulation.

(5) The temperature gradients through the coolants are negligibly

small.

.. 26 ..

(6) Each coolant remains at a fixed temperature during the entire

heating period. For a noncirculating system such as the one shown in

figure 2, each coolant is considered to remain at its boiling temperature.

As a result of assumption (2) the outer surface temperature for the

system is given by .

T ~ 4{Ci; O,T \J~

Making use of assumption (4), the actual outer surface temperature,

TO,T' may be replaced by its time average value, To' in the suc­

ceeding derivation.

The thermal protection weight of the system may be divided into

two parts, insulation weight and coolant weight, or

Expressing each insulation and coolant weight in terms of material

properties and system geometry yields

in which the symbol 6H represents the amount of heat absorbed by

(26)

(28)

1 pound of coolant before it is removed from the system. (Usually the

heat required to raise the coolant from its initial temperature to its

- 27-

boiling temperature plus the heat of vaporization.) Equation (29) con-

tains two independent variables, the two insulation thicknesses, X

and Y. Applying the procedure given in reference 14 for determining

the minimum value of a function indicates that a minimum value for W

is obtained when

and

X kl(To - Tl)T

Pl6!il

Substituting equations (30) and (31) into equation (29) gives

W = 2

which is the minimum total weight of insulations and coolants. The

insulation weights that yield a minimum total weight of insulations

and coolants are obtained from

and

Wins2

(kP)l(To - Tl)T

Ml

The coolant weights that yield a minimum total weight of insulations

and coolants are given by

(30)

(31)

(32)

and

- 28 -

(kp)l(To - Tl)T

Ml

(kP)2(Tl - T2)T

(~2 -~J

1 (kP)2(Tl - T2)T =-~ (~2 -~l)

In certain cases, the two insulation thicknesses may not be inde-

pendent variables. For example, if the outer coolant is water and the . .

inner coolant is a cryogenic propellant, an upper limit would be placed

on values of X/L where

L=X+Y

This would be necessary because the water would freeze if it were

located at a position in the insulations where the steady-state tempera­

ture is below 320 F during. the ground-hold period for an aerospace

vehicle. The maximum permissible value of X/L is given by

in which To,GH is the outer surface temperature during the ground- .

hold period, Tl,GH is the freezing temperature of the outer coolant

and T2, GH is the temperature of the inner coolant during the gr?Und­

hold period. Note, the optimum value of X/L is given by

- 29 -

When the value of X/L obtained from equation (38) is greater than the

value obtained from equation (39), equation (32) may be used to calcu-

late the minimum system weight.

When the outer coolant location, X/L, differs from (X/L) opt ,

the equations presented in the following derivation may be used to cal-

culate the minimum total weight of insulations and coolants. Substi-

tuting equation (37) into equation (29) leads to

(40)

Differentiating equation (40) with respect to the total insulation

thickness, L, and setting the derivative equal to zero yields

L ::

(41)

for the 9ptimum value of the total insulation thickness. Substituting

equation (41) into equation (40) gives

(42)

which is the minimum total weight of insulations and coolants for any

given value of the ratio X/L. The insulation weights that yield a

minimum total weight of insulations and coolants are obtained from

- 30 -

and

(44)

The coolant weights that yield a minimum total weight of insulations and

coolants are given by

and

(46)

The value of L obtained from equation (41) is used in equations (43)

through (46).

An alternate form of equation (42) that is useful in determining

the effect of the outer coolant location on the total weight of insula-

tions and coolants is

where

in which

W = 2 (kP)I(To - T2)T

Alleff (47)

(48)

- 31 -

A discussion of equation (32), which is used to calculate the

minimum weight of the system when the insulation thicknesses are inde­

pendent variables, equation (42) which predicts the minimum weight when

the ratio of the outer insulation thickness to the total insulation

thickness is fixed, and equations (47) and (48) which are used to

detennine the effect of the outer coolant location on thennal protec­

tion weight is presented in the following section.

- 32 -

VIII. RESUIJrS AND DISCUSSION

Discussion of Equations

Material properties, X/L unrestricted.- First consideration is

given to eqUations (32) through (36) which are for the case where the

value of X/L is not restricted. It is apparent from equation (32)

that a minimum thermal conductivity density product (kp), is desirable

for both insulations used in the thermal protection system. Also, a

large enthalpy change (usually heat of vaporization) is desirable for

each coolant.

Adding equations (33) and (34) yields

Wins = (50)

Adding equations (35) and (36), then simplifying the resulting equation

produces

(kp)l(To - Tl)T +

tilil

Since the right-hand side of equations (50) and (51) are identical,

minimum system weight is obtained when the total insulation and total

coolant weights are equal.

- 33 -

Material properties, X/L restricted.- Now consideration is given

to equations (42) through (46) which are for the case where the value of

X/L is restricted. An examination of equation (42) indicates that a

large value for the enthaply change of each coolant is desirable. Note,

since the thermal, conductivity and density of most insulations are

related, the optimum value of these properties for use in equation (42)

is not readily apparent. It can be shown that equation (42) reduces to

equation (32) when the optimum value of X/L is substituted into

equation (42). Hence, for the case where the ratio X/L is optimum,

it is desirable to use minimum (kp) product insulations in the thermal

protection system.

When values other than (X/L)opt are substituted into equa-

tion (42), it is shown in appendix B that a mi~um (kp) product is not

the optimum choice of insulation properties_ When the ratio X/L is

less than (XjL)opt' a reduced system weight is obtained when the den­

sity of the inner insulation is less than its density at minimum

(pk)2 and the density of the outer insulation is greater than its

'density 'at minimum (pk)l- The reason for this apparent paradox is

the reduced dependence of the system weight on the thermal conductivity

of the inner insulation when X/L is less than (X/L)Opt1 because the

temper,ature gradient between coolants is reduced. Also, the thermal'

conductivity of the outer insulation is reduced, which lowers the heat

load to the outer coolant. When the ratio X/L is greater than,

(X/L)opt 1 a reduced system weight is obtained when the density of the

inner insulation is greater than: its density at minimum (pk)2 and

- 34 -

the density of the outer insulation is less than its density at

minimum (pk)l.

Substituting equation (41) into equations (43) and (44), then

adding the two resulting equations yields

Wins =

Substituting equation (41) into equations (45) and (46), then adding

the two resulting equations produces

Since the right-hand side of equations (52) and (53) are identical,

minimum system weight is obtained when the total insulation and total

coolant weights are equal. This result agrees with the result pre-

viously shown for the case where the optimum value of X/L is used.

Outer coolant location.- From equation (47) it can be seen that

minimum system weight is obtained when the value of Lilleff is maximum.

In order to gain insight on the effect that the outer coolant location,

X/L, has on the value of Lilleff, plots of Lilieff versus X/L are

shown in figure 4. An examination of figure 4 reveals the following

results. The value of Lilieff may vary considera.bly over the range

- 35 -

1.0

kl 1.0

AHI 5.0

k2 E""R2 =

PI 1.0

T 1-T2 = j.L = .8 P2 To-T2

.6 ~--------~--------~--------4-------~~------~ locus of max,

0.05

.4~~------~--------~~------~--------4-~~----~

== 0.25

....... --. ....... -- ,

-- -- ....... --. ....... '\ ....... \ , "---. --.' \

a .2 .4 .6 .8 I • a

x ["

Figure 4.- Effect of outer coolant ,location on AHeff -

- 36 -

X/L equal 0.0 to 1.0. As the value of X/L approaches zero, the

value of AHeff decreases rapidly. This result occurs because the

rate of vaporization of the outer coolant increases as X/L decreases.

When the outer coolant is located too near the inner coolant (in the

region where the curves are shown as dashed lines), the outer coolant

will not boil and should be omitted from the system. Or, the outer

coolant should be omitted from the system when the value of AHeff/(pk)l

is less than .6.:H2/(pk)a. As the boiling temperature of the outer

coolant approaches the boiling temperature of the inner coolant, the

value of AHeff increases sharply over the range of values of X/L

where the outer coolant should be included in the system.

Comparison With Numerical Solutions

This portion of the paper is concerned with the accuracy of the

approximate equations developed in this paper. The gross effects of

the assumptions made in the derivation of the approximate weight

equations will be demonstrated by comparison of the approximate solu­

tion with numerical solutions.

In order to make a comparison of solutions, temperature histories

had to be selected. Since a plot of the actual temperature versus time

for a point on the surface of an aerospace vehicle would probably be a

very complex shaped curve, the temperature histories shown in figure 5

were selected for use in the comparison of solutions. The thermal

environments shown in figure 5(a), 5 (b), and 5(c) might be considered

to be representative of the range of environments considered for

aerospace vehicles. (The heating periods, difference between the

0:: 0 ..

G) C-::s ..., D C-I) a. E I) ..., I) () D

If-C-::s fIJ

C-., ..., ::s 0

- 37 -

3500r-------------r-------------.-------------.------------.

2500

/

1500 / /

500 0

0

0

(a)

" (d)i~

• 125

and (d)

.250

( b)

.375

( c)

Time, hr

Figure 5.- Temperature histories.

(b), (c)

.250

.500

.750

- 38 -

maximum and minimum value of the outer surface temperature, the rate

of change of temperature with respect to time, and the heat load are

in the range considered for aerospace vehicles.) The thermal

environment shown in figure 5(d) is included so that the effect of the

difference (To,max - To,min) on the accuracy of the approximate

solution can be examined.

A plot of the percent difference between the weights predicted

by the approximate solution and numerical solutions versus the change

in outer surface temperature, (To max - To min), is shown in , , figure 6 for the temperature histories shown in figure 5. The

calculated data points are indicated in figure 6 by symbols. The

curve shown in figure 6 is obtained by fairing a line through the

data points. Examination of figure 6 indicates that the difference

between solutions decreases as the temperature difference decreases

and the heating time increases. Also the difference between

solutions is less than one percent for the environment and material

properties selected for the comparison. The material properties

used in the comparison are listed on figure 6.

The data used to construct figure 6 are obtained in the following

manner. Since the boiling temperature of the outer coolant was

selected to be 4000 R, a value of 0.364 was selected for X/L. An

examination of the material properties listed on figure 6 and the

boundary conditions during the ground-hold period

o o

c 3:

- 39 -

3.0 r---------~----------~----------~----------~

2.0

I .0

o

.03 Btu/hr-ft-OR

.0171 Btu/hr-ft-OR

p = 10 Ib/ft 3 2

c = C = .25 Btu/lb-oR PI P2

AHI = 1000 Btu/lb

AH2 200 Btu/lb

2000

° To(max) - To(min)' R

-1:' = .25 hr 1:' .50 hr

= .75 hr

4000

Figure 6.- Percent difference between weights predicted by approximate and numerical solutions.

- 40 -

indicates that a value of X/L equal to 0.364 will permit the outer

coolant to remain at its boiling temperature during the entire

heating period. Or, assumption (6) has no effect on the approximate

weights calculated for the comparison shown in figure 6. The total

weight of insulations and coolants (Wapprox) is calculated from

equation (42). The thickness of the outer insulation corresponding

to the value of Wapprox obtained from equation (42) is calculated

from equation (43). Using this insulation thickness, the outer

coolant weight is computed with the aid of a digital computer program

described in appendix B. The computer program is used to calculate

the temperature profile through the outer insulation versus time.

Once this is known, the net heat transfer rate to the outer coolant

may be calcu~ated from

~D.T) q-k -6X back surface of

outer insulation

The inner coolant weights obtained from the approximate and numerical

solutions are equal since a steaqy state temperature profile exists

through the inner insulation.

The comparison shown in figure 6 does not include the effect of,

assumption (6) or admit to the possibility that minimum system weight

might be obtained with an insulation thickness different from the,

value predicted by the approximate solution. Hence, a plot of the

weights obtained from the approxtmateand numerical solutions versus

total insulation thickness is shown in figure 7 for the case where the

- 41 -

2.0r---------------------~1--------------------~

1.0 -

o

o-minimum point

To

TI

T2 K I ,

PI

Cp I

aH I

=

= = =

apprOXimate_~

numeri~al~

1995 oR

660 oR T I' = 323 oR I,

40 oR

K2 = .03 Btu/hr-ft-OR

P2 = 10.0 Ib/ft3

C .25 ° = = Btu/lb- R P2

= 1000 Btu/lb = 677 Btu/lb

AH2 = 200 Btu/lb

.05

Insulation thickness, ft

.10

Figure 7. - Comparison of the sum of insulation and coolant weights predicted by approximate and numerical solutions.

- 42 -

outer coolant is initially a subcooled liquid which is raised to its

boiling temperature and then vaporized during the heating period. Note,

the difference in minimum weights is only about 0.12 Ib/ft2 (8 per

cent) and the corresponding difference in total thickness is only

about 0.005 ft (7 per cent). Material properties selected for use in

constructing figure 7 are listed on figure 7. The temperature history

utilized in calculating the weights shown in figure 7 is shown in

figure 5(a).

Figure 7 was constructed in the following manner. Theupper

curve (Wapprox) is a plot of equation (40). A value of x/L equal

to 0.423 was used in calculating the value of Wapprox versus

insulation thickness. This is the optimum value predicted by

equation (39). The lower curve (weights obtained from numerical

solutions) shown in figure 7 is constructed by fairing a line through

a series of data points. The data points (weights) were obtained in

the following manner. Using a value of 0.423 for the ratio of X/L,

a value for the total insulation thickness is selected and the coolant

weights are computed with the aid of the digital computer program

described in appendix B.

Since only one set of material properties for the insulations was

used in comparing the approximate solution with numerical solutions,

no precise statement can be made as to the effect of material

properties on accuracy. However, it would appear that low values of

thermal diffusivity for each insulation would tend to decrease the

accuracy of the approximate solution when the variation in outer

- 43 -

surface temperature is large and the heating time is short. This is

because the ass'umption that a linear temperature profile exists through

each insulation is incorrect except for steady state conditions.

- 44 -

IX. CONCLUSIONS

1. A comparison of the weights predicted by the approximate solu­

tion and numerical solutions indicates good agreement for the range of

variables examined in this paper. When the outer coolant remains at its

boiling temperature during the entire heating period, a difference of

less than 1.0 percent is indicated in figure 6. When the outer coolant

is initially a subcooled liquid which is raised to its boiling tempera­

ture and then vaporized during the heating period, a difference of about

8.0 percent is indicated in figure 7.

2. While numerical solutions do not readily reveal the effects of

material properties or the outer coolant location within the insulations

on system weight, an examination of the approximate weight equations for

the two-insulation--two-coolant thermal protection system indicates the

system weight is:

(a) Reduced as the enthalpy change (usually heat of vaporiza­

tion) of each coolant increases and the outer coolant should be omitted

from the system when the value of 6lieff/(pk)l is less than ~2/(pk)a.

(b) Reduced sharply as the boiling temperature of the outer

coolant approaches the boiling temperature of the inner coolant.

(c) Minimum when the outer coolant is at its optimum location,

each insulation is selected at its minimum thermal conductivity-density

product, and the total weight of insulations and total weight of coolants

are equal.

- 45 -

(d) Is reduced by selecting each insulation at a nonminimum

thermal conductivity-density product when an off-optimum value of X/L

is used.

(e) Significantly affected by the outer coolant location, X/L.

- 46 -

X. ACKNOWLEDGMENTS

The author wishes to express his appreciation to the National

Aeronautics and Space Administration for the opportunity to write this

thesis while in their employment.

He also wishes to thank Dr. H. L. Wood of the Virginia Polytechnic

Institute staff for his advice and supervision during the preparation of

this paper, and Mr. E. E. Mathauser and Mr. L. R. Jackson for their

criticism which proved very helpful in writing this paper.

- 47 -

XI. REFERENCES

1. Jackson, Charlie M.; and Harris, Roy V., Jr.: Effects of Peak

Deceleration on Range Sensitivity for Modulated-Lift Reentry at

Supercircular Speeds. NASA TN D-1955, September 1963.

2. Carslaw, H. S.; and Jaeger, J. C.: Conduction of Heat in Solids.

Second ed., Oxford Press, Longon, 1959, pp. 100-102.

3. Brown, John D.; and Shukis, Francis A.: An Approximate Method for

Design of Thermal Protection Systems. Paper presented at Thirtieth

Annual Meeting, lAS. (New York, N. Y.), January 22-24, 1962.

4. Hurwicz, Henryk; and Mascola, Robert: Thermal Protection Systems -

Application Research of Materials Properties and Structural

Concepts. ML-TDR-64-82, Air Force Systems Command, January 1965.

5. Dukes, W. H.; Goldberg, M. A.; and Brull, M. A.: Insulation.

Section IV of Structural Design for Aerodynamic Heating. Part II -

Analytical Studies, W. H. Dukes and A. Schnitt, eds., WADC Tech.

Rep. 55-305, pt II, u.S. Air Force, October 1955, pp. 89-128.

6. Dukes, W. H.; Goldberg, M.A.; and Brull, M. A.: Insulation Design.

Section 6.0 of Structural Design for Aerodynamic Heating. Part I -

DeSign Information, W. H. Dukes and A. Schnitt, eds., WADC Tech.

Rep. 55-305, pt I, U.S. Air Force, October 1955.

7. Davidson, John R.: Optimum Design of Insulated Tension Plates in

Aerodynamically Heated Structures. M.S. thesis, Virginia Polytechnic

Institute, August 1958.

- 48 -

8. Harris, Robert S., Jr.; and Davidson, John R.: An Analysis of Exact

and Approximate Equations for the Temperature Distribution in an

Insulated Thick Skin Subjected to Aerodynamic Heating. NASA

TN D-5l9, January 1961.

9. Davidson, John R.; and Dalby, James F.: optimum Design of Insulated

Compression Plates Subjected to Aerodynamic Heating. NASA TN D-520,

January 1961.

10. Schnitt, A.; Brull, M. A.; and Wolco, H. S.: Selection of Comparison

Parameters. Section VIII of Structural Design for Aerodynamic

Heating. Part II - Analytical Studies, W. H. Dukes and A. Schnitt,

eds., WADC Tech. Rep. 55-305, Pt II, u.S. Air Force, October 1955,

pp. 243-262.

11. Harris, Robert S., Jr.; and Davidson, John R.: Methods for Deter­

mining the Optimum Design of structures Protected From Aerodynamic

Heating and Application to Typical Boost-Glide or Reentry Flight

Paths. NASA TN D-990, March 1962.

12. Bridges, J. H.; and Richmond, F. D.: Design Considerations for a

Reentry Vehicle Thermal Protection System. Vol. X of Progress in

Astronautics and Aeronautics, sec. E, Clifford I. Cummings and

Harold R. Lawrence, eds., Academic Press (New York, N. Y.), 1963,

pp. 761-782 •

• Haviland, J. K.: The Optimization of an Actively Cooled Heat Shield

System. Report 00.256, Astronautics Div., Chance Vought Corp.,

July 17, 1963.

- 49 -

14. Sokolnikoff, I. So; and Redheffer, R. M.: Mathematics of Physics

and Modern Engineering. McGraw-Hill Book Company, Inc., 1958,

pp. 247-248.

- 50 -

XII . APPENDICES

APPENDIX A

THERMAL CONDUCTIVITY-DENSITY ANALYSIS

The thermal conductivity of most insulations is a function of the

insulation density. Since the optimum values of thermal conductivity

and density for use in equation (42) in the body of this paper are not

apparent, the following analysis is presented. For the purpose of the

analysis, it is assillued that

and

B(PI - C)2 + D kl = --~--------­

PI

=

2 E(P2 - E) + G

(Al)

(A2)

Substituting equations (AI) and (A2) into equation ( ) and rearranging

the resulting equation yields

(W)2 [ 2 ~ (X) [B(PI - C)2 + Dlrl ( X) 2 = B(PI - C) + D~~ L + PI J~P2 1 - L

(A3)

in which

(A4)

- 51 -

and

Differentiating equation (A3) with respect to PI produces

and similar, differentiating equation (A3) with respect to P2 yields

An examination of equations (Al) and (A2) indicates that a minimum value

for the thermal conductivity-density product, (kp), is obtained when

(AB)

and

If the 'optimum values of thermal conductivity and density for use in

equation (A3) are the values corresponding to (kp)l(min) and

(kp)2(min)' a necessary condition is that

(AIO)

- 52 -

However, substituting equations (AS) and (A9) into equations (A6) and

(A7) yields

- ~ ¢F (1 - ~) + Q. e(~) ~ 0 C2 L F L , (All)

(A12)

Thus, it is readily apparent from equations (All) and (A12) that the

values of thermal conductivity and density corresponding to (kp)min

are not always an optimum choice of values.

- 53 -

APPENDIX B

DIGITAL COMPUTER PROGRAM

A digital computer program, that may be used to compute the tempera­

ture profile through a series of insulations with varying boundary con­

ditions at the outer surface of the outer insulation, is available in

the central data facility of the Langley Research Center. The program

is for use on an IBM 7094 digital computer and is designated as program

number (p-57l).

- 54 -

XIII. VITA

The author was born in Durham, North Carolina, on November 23, 1938.

He attended public schools in Rocky Mount, North Carolina, and was

graduated from Rocky Mount Senior High School in June of 1957. He entered

North Carolina State College in September of 1958 and received the degree

of Bachelor of Science in Mechanical Engineering in June 1962. From that

time to the present, the author has been employed by the National Aero­

nautics and Space Administration at Langley Air Force Base, Hampton,

Virginia.

AN APPROXIMATE METHOD OF CALCULATING THE WEIGHT

OF THE TWO-INSULATION--TWO-COOLANT

THERMAL PROTECTION SYSTEM

By

John Grady Davis, Jr.

ABSTRACT

An approximate method of calculating the minimum total weight of

the two-insulation--two-coolant thermal protection system is developed.

The equations derived in the development of the approximate method

enable insight into the parameters that control the system weight. Two

cases are considered! the case where the outer coolant location is

unrestricted within the insulating wall and the case where the outer

coolant location is restricted within the insulating wall. The effects

on system weight of material properties and the outer coolant location

within the insulating wall are discussed. A comparison of weights pre­

dicted by the approx~ate method and numerical solutions is shown.