August 2012 

NASA/TP–2012-217595 

Analysis and Sizing for Transient Thermal 

Heating of Insulated Aerospace Vehicle 

Structures

Max L. Blosser 

Langley Research Center, Hampton, Virginia 
NASA STI Program . . . in Profile

Since its founding, NASA has been dedicated to the 

advancement of aeronautics and space science. The 

NASA scientific and technical information (STI) 

program plays a key part in helping NASA maintain 

this important role. 

The NASA STI program operates under the 

auspices of the Agency Chief Information Officer. 

It collects, organizes, provides for archiving, and 

disseminates NASA’s STI. The NASA STI 

program provides access to the NASA Aeronautics 

and Space Database and its public interface, the 

NASA Technical Report Server, thus providing one 

of the largest collections of aeronautical and space 

science STI in the world. Results are published in 

both non-NASA channels and by NASA in the 

NASA STI Report Series, which includes the 

following report types: 

 TECHNICAL PUBLICATION. Reports of 

completed research or a major significant phase 

of research that present the results of NASA 

Programs and include extensive data or 

theoretical analysis. Includes compilations of 

significant scientific and technical data and 

information deemed to be of continuing 

reference value. NASA counterpart of peerreviewed formal professional papers, but 

having less stringent limitations on manuscript 

length and extent of graphic presentations. 

 TECHNICAL MEMORANDUM. Scientific 

and technical findings that are preliminary or of 

specialized interest, e.g., quick release reports, 

working papers, and bibliographies that contain 

minimal annotation. Does not contain extensive 

analysis. 

 CONTRACTOR REPORT. Scientific and 

technical findings by NASA-sponsored 

contractors and grantees. 

 CONFERENCE PUBLICATION. 

Collected papers from scientific and 

technical conferences, symposia, seminars, 

or other meetings sponsored or cosponsored by NASA. 

 SPECIAL PUBLICATION. Scientific, 

technical, or historical information from 

NASA programs, projects, and missions, 

often concerned with subjects having 

substantial public interest. 

 TECHNICAL TRANSLATION. 

English-language translations of foreign 

scientific and technical material pertinent to 

NASA’s mission. 

Specialized services also include organizing 

and publishing research results, distributing 

specialized research announcements and feeds, 

providing information desk and personal search 

support, and enabling data exchange services. 

For more information about the NASA STI 

program, see the following: 

 Access the NASA STI program home page 

at http://www.sti.nasa.gov

 E-mail your question to help@sti.nasa.gov

 Fax your question to the NASA STI 

Information Desk at 443-757-5803 

 Phone the NASA STI Information Desk at 

443-757-5802 

 Write to: 

 STI Information Desk 

 NASA Center for AeroSpace Information 

 7115 Standard Drive 

 Hanover, MD 21076-1320
National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton, Virginia 23681-2199 

 

August 2012 

NASA/TP–2012-217595 

Analysis and Sizing for Transient Thermal 

Heating of Insulated Aerospace Vehicle 

Structures

Max L. Blosser 

Langley Research Center, Hampton, Virginia 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 

Acknowledgments 

The author would like to thank NASA Langley Research Center for the opportunity to 

perform this work under the Floyd Thompson Fellowship. Fellowship advisors, 

Professors Raphael Haftka and Bhavani Sankar from the University of Florida and Dr. 

William Garver from Lockheed Martin provided much helpful advice and 

encouragement. Professor Haftka contributed essential guidance for nondimensionalizing 

the governing equations and boundary conditions. Dr. Garver initially indicated the 

strong need for the type of simple design tool developed in this paper. Colleagues, Dr. 

Kamran Daryabeigi and Carl Poteet provided valuable review and discussion of 

the work as it progressed. 

The use of trademarks or names of manufacturers in this report is for accurate reporting and does not constitute an 

official endorsement, either expressed or implied, of such products or manufacturers by the National Aeronautics 

and Space Administration. 
Abstract

An analytical solution was derived for the transient response of an insulated

structure subjected to a simplified heat pulse. The solution is solely a function

of two nondimensional parameters. Simpler functions of these two parameters were

developed to approximate the maximum structural temperature over a wide range of

parameter values. Techniques were developed to choose constant, effective thermal

properties to represent the relevant temperature and pressure-dependent properties

for the insulator and structure. A technique was also developed to map a timevarying surface temperature history to an equivalent square heat pulse. Equations

were also developed for the minimum mass required to maintain the inner, unheated

surface below a specified temperature. In the course of the derivation, two figures of

merit were identified. Required insulation masses calculated using the approximate

equation were shown to typically agree with finite element results within 10%-20%

over the relevant range of parameters studied.

Nomenclature

a1, a2 Coefficients of approximate equation for maximum structural temperature

bn Coefficients of series solution for τ > τh

cm Coefficients of series solution for 0 < τ ≤ τh

cpe Effective insulator specific heat capacity

cps Structural specific heat capacity

de Insulator thickness

ds Structural thickness

fthr Fraction of the surface temperature range

ke Effective insulator thermal conductivity

m Mass per unit area

me Insulation mass per unit area

ms Structural mass per unit area

mopt Minimum total mass per unit area

mso Structural mass/area at which total mass/area is minimum

Pavg Average ambient pressure during equivalent square surface temperature pulse

T Temperature

t Time

1
Th Applied surface temperature rise of insulator

th Duration of heating pulse

Ti Initial temperature

Tm Maximum structural temperature rise

Tce Temperature to use for calculating effective insulation specific heat capacity

Tcs Temperature to use for calculating effective structural specific heat capacity

Tha Applied surface temperature of insulator

Tke Temperature to use for calculating effective insulation conductivity

Tma Maximum structural temperature

Tmx Maximum surface temperature for a surface temperature history

Tthr Threshold temperature for truncating insulator temperature rise integral

x Nondimensionalized spatial variable

x

0 Spatial variable, position through insulator thickness

β Ratio of insulator conductance/area to insulator heat capacity/area

βs Figure of merit for thermal performance of structure

δ Dirac delta function

 Error of approximate equation

γ Ratio of insulator to structural heat capacity/area

κe Figure of merit for thermal performance of insulator

λ Eigenvalue

ρe Effective insulator density

ρs Structural density

τ Nondimensionalized time

τh Nondimensionalized duration of heat pulse

τm Nondimensionalized time of maximum structural temperature

θ Nondimensionalized temperature

2
1 Introduction

Thermal protection systems are a critical component of hypersonic and atmospheric entry vehicles. The reusable ceramic tiles and blankets of the Space Shuttle

Orbiter work well as thermal insulators, but result in a fragile, high maintenance

exterior surface. An intriguing approach to this problem is to build the thermal insulation into the exterior vehicle wall. This deceptively simple idea will be difficult

to achieve because it requires a flight weight aerospace vehicle skin to not only carry

the required mechanical loads, but also to accommodate severe transient heating

with the corresponding hot outer surface and large temperature gradients through

its thickness.

A thermally insulating structural panel will likely be of a sandwich construction

as a result of both thermal and structural considerations. The outer, heated face

sheet of the sandwich will typically be a thin layer of non-insulating material that

will contribute little to the thermal response of the inner, unheated face sheet. Thus,

ignoring edge closeouts and joints, the transient thermal response of an insulating

sandwich panel will be similar in character to the thermal response of an insulated

structure like the external tiles and blankets covering the aluminum structure of the

Space Shuttle Orbiter. For a non-homogeneous sandwich core, it may be possible

to calculate effective thermal properties using a rule of mixtures to approximate its

thermal performance.

In this paper, a simplified transient thermal problem was investigated in an

attempt to gain basic insight that will be required to develop optimum sandwich

panels that can simultaneously insulate and carry structural loads. An analytical solution was derived for the transient response of an insulated structure to a

simplified heat pulse. The solution is a function of two nondimensional quantities.

Simpler functions of these two parameters were developed to approximate the maximum structural temperature over a wide range of parameter values. Techniques

were developed to choose constant, effective properties to represent the relevant

temperature and time-dependent thermal properties for the insulator and structure.

A technique was also developed to map a time-varying surface temperature history

to an equivalent square heat pulse. Using these techniques, the maximum structural

temperature rise was calculated using the analytical solutions and compared with

finite element simulations over a wide range of parameters. Equations were also

developed for the minimum mass for an insulated structure required to maintain

the inner, unheated surface below a specified temperature. A figure of merit which

correlates to the effectiveness of the heat capacity of the underlying structural material in reducing the amount of required insulation was developed. A second figure

of merit was identified for the combination of insulator thermal properties that minimizes the mass of an insulator required to limit the maximum temperature of an

underlying structure subjected to a transient heating pulse. Insulation was sized for

a number of surface heating histories, insulators, and structural materials by iterating a one-dimensional, finite element analysis and by using the simple equations

derived in this paper. Finite element results were correlated to the two figures of

merit developed in this paper.

3
Figure 1. Illustration of simplified problem

2 Problem Definition

The simplified problem investigated in this paper is illustrated in Fig. 1. A thermal insulator with thickness, de, density, ρe, specific heat capacity, cpe, and thermal

conductivity, ke, covers a structure (inner face sheet) of thickness, ds, density, ρs,

and specific heat capacity, cps. The inner surface of the structure is assumed to be

perfectly insulated to simplify the mathematics of the problem and because it is a

commonly used conservative assumption for sizing thermal protection systems. To

further simplify the problem, the structure is treated as a lumped heat capacitance

and the outer face sheet of the insulating sandwich panel is neglected. For this

solution, the material properties are assumed to be constant, so effective, averaged

properties would have to be used to approximate more complex material behavior.

A simple transient heat pulse is assumed. Initially the insulator and structure

are assumed to be at a uniform temperature of 0. The outer surface of the insulator

is assumed to instantaneously rise to a temperature Th at t = 0 and maintain that

temperature until t = th at which time it instantaneously returns to 0.

2.1 Mathematical Description

Using a nondimensional spatial variable, x =

x

0

de

, the governing differential equation for heat conduction through the insulator can be written as

∂T

∂t = β

∂

2T

∂

2x

(1)

where

β =

ke

ρecped

2

e

(2)

4
The boundary condition at x = 1 is defined as

∂T(1, t)

∂t = −βγ ∂T(1, t)∂x (3)

where

γ =

ρecpede

ρscpsds

(4)

The boundary condition at x = 0 is defined as

T(0, t) = 

Th for 0 < t ≤ th

0 for t > th

(5)

Finally, the initial condition is a uniform temperature of 0.

T(x, 0) = 0 (6)

The mathematical problem defined by Eqs. 1 through 6 can be completely nondimensionalized [1] by defining a nondimensional time, τ = βt and a nondimensional

temperature, θ =

T

Th

. The nondimensionalized differential equation becomes

∂θ

∂τ =

∂

2

θ

∂

2x

(7)

The boundary condition at x = 1 becomes

∂θ(1, τ )

∂τ = −γ

∂θ(1, τ )

∂x (8)

The boundary condition at x = 0 becomes

θ(0, τ ) = 

1 for 0 < τ ≤ τh

0 for τ > τh

(9)

where

τh =

ke

ρecped

2

e

th (10)

The initial condition becomes

θ(x, 0) = 0 (11)

2.2 Nondimensional Governing Parameters

Inspection of the mathematical problem defined by Eqs. 7 through 11 reveals that

any solution will be completely governed by two nondimensional parameters, γ and

τh. Each of these parameters has clear physical significance. The first parameter,

γ, is defined by Eq. 4 which shows it to be the ratio of the heat capacity per unit

area of the insulator to the heat capacity per unit area of the underlying structure.

Equation 10 defining the nondimensional heat pulse duration, τh, can be slightly

rewritten to show that it is the ratio of insulator conductance per unit area to heat

capacity per unit area times the duration of the heat pulse. The same ratio of

5
insulator conductance per unit area to heat capacity per unit area, β, is used to

nondimensionalize time.

τh =

ke

de

ρecpede

th (12)

3 Analytical Solution

An analytical solution for the problem illustrated in Fig. 1 can be obtained by

combining two existing solutions with modifications. The solution of the first part

of the problem, 0 < t ≤ th, can be found in the classic heat transfer textbook by

Carslaw and Yeager [2]. Converting the solution to the nomenclature used in this

paper and nondimensionalizing produces

θ(x, τ ) = T(x, τ )

Th

= 1 −

X∞

m=1

cm sin (λmx)e

−λ

2mτ

(13)

where

cm =

2(λ

2

m + γ

2

)

λm(λ2

m + γ

2 + γ)

(14)

The values for λm can be found by solving the equation

λm tan λm = γ (15)

Solving the second part of the problem, t > th, requires more effort. There

is an existing solution for a similar problem in which the insulator and lumped

mass are initially at a uniform temperature, but at t = 0 the temperature of the

insulator outer surface is instantaneously reduced to 0. A solution to this similar

problem, including its derivation, is presented in Reference [3]. A derivation of the

solution to the second part of the current problem is presented in Appendix A.

The derivation closely follows the approach used by De Chant, however, instead

of a uniform temperature, the solution to Eq. 13 at time th is used as the initial

temperature distribution.

So, for t > th or τ > τh the solution is

θ(x, τ ) = T(x, τ )

Th

=

X∞

n=1

bn sin (λnx)e

−λ

2

n(τ−τh)

(16)

where

bn = cn(1 − e

−λ

2

nτh ) −

X∞

m=1

m6=n





sin (λm−λn)

(λm−λn) −

sin (λm+λn)

(λm+λn) +

2 sin λm sin λn

γ

1 −

sin (2λn)

2λn

+

2 sin2 λn

γ



 cme

(−λ

2mτh)

(17)

6
The quantities cm and cn can be calculated using Eq. 14 and values for λm and

λn are obtained by solving Eq. 15. In Eq. 17 the summation excludes the term

m = n because the contribution of that term is captured by the exponential term

preceding the summation.

4 Numerical Example for Series Solution

The series solution described by Eqs. 13 and 16 can be applied to a wide range

of physical situations. However, the motivation for deriving the solution was to gain

insight into the thermal response of an insulated structure of an aerospace vehicle

subjected to a transient aerodynamic heating pulse. Therefore a numerical example

was chosen to represent a typical location on the Space Shuttle Orbiter at which the

aluminum structure is protected from aerodynamic heating by an LI-900 ceramic tile.

The material properties, obtained from Reference [4], are listed in Table 1 along with

dimensions, heating duration and the corresponding nondimensional parameters.

Thermal properties are temperature dependent, so the properties for the aluminum

structure are for 200◦F and the properties for the LI-900 tile are for 1250◦F. The

LI-900 thermal conductivity is also a strong function of pressure, so the conductivity

is chosen for a pressure of 0.01 atm (an arbitrary value in the mid-range of a typical

ambient pressure history for an atmospheric entry trajectory).

Table 1. Numerical values for initial example

Variable Value Units

ds 0.125 (0.003175) in(m)

ρs 175 (2800) lbm/f t3(kg/m3)

cps 0.216 (904) Btu/lbm/◦F(J/kg/K)

de 2,3,4 (0.051,0.076, 0.102) in(m)

ρe 9 (144) lbm/f t3(kg/m3)

cpe 0.296 (1238) Btu/lbm/◦F(J/kg/K)

ke 0.0492 (0.0851) Btu/f t/hr/◦F(W/m/K)

th 25 (1500) min(s)

A computer program was written using Version 2.7 of the Python programming

language to implement the series solution described by Eqs. 13 and 16. Routines

from Version 0.9 of the SciPy programming library [5] were used for numerical

solutions of nonlinear equations and fitting coefficients to nonlinear equations. For

times just after an instantaneous change in surface temperature, many terms of

the series solution were required for an accurate solution. However, for the results

shown in Figs. 2 through 5, eight terms were used to calculate the coefficients bn and

three terms to calculate cm. Use of additional terms produced negligible changes

in the results. To generate results for the wide range of parameter values shown in

Figs. 6, 7, 9, and 10, twelve terms each were used to calculate bn and cm to reduce

the chance of truncation error.

7
Three different tile thicknesses, de in Table 1, are considered to gain understanding of the solution described by Eqs. 13 and 16. The nondimensional temperature

distributions through the thickness of the insulator at several times are shown in

Figs. 2, 3, and 4 for LI-900 tile thicknesses of 2, 3, and 4 in. (0.051,0.076, 0.102m) respectively. In each figure, the solid blue line represents the temperature distribution

halfway through the heat pulse and the solid green line represents the temperature

at the end of the heat pulse, τ = τh. The dashed lines represent temperature distributions after the heat pulse has ended, τ > τh. As expected the tile interior and

underlying structure heats up more quickly and reaches a higher temperature for

thinner tiles. Fig. 2 shows that the structure at x = 1 has already started heating

up midway through the heating pulse, τ =

τh

2 = 0.14 for a tile thickness of 2 inches.

In contrast, Fig. 4 shows that the structure has not even started to heat up at the

end of the heat pulse, τ = τh = 0.07 for a tile thickness of 4 in. This behavior is

expected because it should take much longer for the heat to diffuse through twice

the thickness of tile.

Figure 2. Temperature distributions for 2 inch thick tile

After the tile surface temperature drops back to the initial temperature, the

heat stored in the tile interior begins to be conducted back out of the cooled outer

tile surface. However, the underlying structure, x = 1, continues to increase in

temperature until its temperature matches that of the tile material in contact with

it, then it begins to cool. This behavior is consistent with the assumption that the

structure is perfectly insulated on its inner surface.

For sizing of thermal protection systems, the temperature of the underlying

structure is of primary concern. Further insight into the solution can be obtained

by calculating the structural temperature as a function of time. Figure 5 shows a

8
Figure 3. Temperature distributions for 3 inch thick tile

Figure 4. Temperature distributions for 4 inch thick tile

number of temperature histories for different combinations of the governing nondimensional parameters. The ranges of the two parameters were chosen to bound

the values calculated for the numerical examples illustrated in Figs. 2 through 4. In

Fig. 5, the blue lines represent the structural temperature histories for τh = 0.05, the

9
Figure 5. Structural temperature histories as a function of governing parameters

green for τh = 0.15, and the red for τh = 0.3. The vertical dotted lines represent the

end of the heat pulse for the respective values of τh. The solid, dashed and dash-dot

lines represent values of γ of 1, 2, and 3 respectively. Because the nondimensionalized time axis makes the results more difficult to interpret physically, it is helpful to

consider the case of fixed material properties and heating duration. The remaining

free parameters would be the insulator thickness, de, and the structural thickness,

ds. The parameter τh varies as 1

d

2

e

, so smaller values of τh imply larger values of de.

Choosing a value of τh fixes the insulator thickness, so for a specified value of τh, γ

can only be varied by changing the structural thickness. The parameter γ varies as

1

ds

so larger values of γ imply smaller structural thicknesses. In Fig. 5 the structural

temperature histories for τh = 0.05 stay much cooler, which is consistent with the

expected behavior for thicker insulation. Conversely the structural temperatures

reach much higher values for τh = 0.3, which is consistent with the expected behavior for thinner insulation. For each value of τh the curve for γ = 1 reaches the lowest

maximum temperature, as expected for the correspondingly highest structural heat

capacity. Higher γ’s result in higher maximum structural temperatures as expected

for the associated reduction in structural heat capacity. All of the structural temperature history curves in Fig. 5 exhibit similar behavior. The temperature of the

structure continues to rise well after completion of the heating pulse, reaches a maximum value, and then begins to decrease. The maximum temperature is a primary

design driver for sizing of thermal protection systems, so it is desirable to be able

to readily calculate the maximum structural temperature.

10
5 Maximum Structural Temperature

All of the temperature histories, away from the heated surface, generated using

Eq. 16 would appear to heat up, reach a maximum temperature, and then begin

to cool. The time at which the maximum temperature occurs can be found by

differentiating Eq. 16 with respect to time, setting it equal to zero, and solving for

time. This results in the following equation to be solved for τm

0 = −

X∞

n=1

λ

2

n

bn sin (λnx)e

−λ

2

n(τm−τh)

(18)

Equation 18 can be solved for the time of the maximum temperature for any

location through the thickness of the insulator. However, for the current problem,

the maximum structural temperature at x = 1 is of greatest interest. The equation

therefore becomes

0 = −

X∞

n=1

λ

2

n

bn sin (λn)e

−λ

2

n(τm−τh)

(19)

Although Eq. 19 cannot be readily solved for τm in closed form, it can be solved

numerically. The nondimensional times at which the maximum structural temperature occurs for a range of the governing parameters are shown in Fig. 6.

Figure 6. Times at which maximum structural temperatures occur

The resulting τm can be substituted into Eq. 16 to obtain the maximum structural temperature rise. Following this procedure, the maximum structural temperature rise was calculated over a range of the governing nondimensional parameters

11
(0.2 ≤ γ ≤ 5 and 0.02 ≤ τh ≤ 0.5). Figure 7 shows a surface plot of the maximum

structural temperature rises calculated over a 49 by 49 point grid. The color contours from dark blue to red indicate increasing maximum structural temperature

rises. To gain insight into the physical implications of this plot it is again helpful to

consider the case of fixed material properties. For fixed properties, small values of

τh imply a short heat pulse and/or thick insulator and large values imply a long heat

pulse and/or thin insulator. It is logical to expect that as the heat pulse duration

goes to zero, τh → 0, the maximum structural temperature rise will also go to zero.

The parameter, γ, is the ratio of insulation to structural heat capacity per unit

area, so small values imply that the structural heat capacity per unit area is much

larger than that of the insulation and large values imply a relatively small amount

of structural heat capacity. Therefore, as the structural heat sink increases towards

infinity, γ will approach 0 and, for a finite heating duration, the maximum structural temperature rise will also go to zero. Also, for this problem, the maximum

structural temperature cannot exceed the applied surface temperature, so Tm

Th

≤ 1.

Figure 7. Maximum structural temperatures

The maximum structural temperature rise plot in Fig. 7 is useful to illustrate how

the maximum structural temperature rise varies with the two governing parameters,

but it was generated using a complicated series of numerical solutions. For quick

calculations, it would be much more useful to have a relatively simple algebraic

equation to approximate this surface. The approximate solution should go to zero if

either of the governing parameters goes to zero, and it should approach 1 as either

parameter goes to infinity. A number of candidate equations were evaluated and the

following approximate solution was chosen as a good compromise between simplicity

and accuracy.

12


Tm

Th



a

= 1 − e

(a1γ

a2 τh

2a2 )

(20)

where the subscript a indicates approximate.

A “least squares” routine was used to find the values of the coefficients in Eq. 20

that best approximate the surface shown in Fig. 7. The coefficient values are given

in Table 2.

Table 2. Coefficient values for Eq. 20

Coefficient Value

a1 -0.72058

a2 0.53649

Inspection of Table 2 reveals that a1 ≈ √−1

2

and a2 ≈

1

2

. Substituting these

coefficients into Eq. 20 simplifies the equation to



Tm

Th



a

= 1 − e

−τh

√γ

2 (21)

The approximate maximum structural temperature rises calculated using Eq. 20

with the coefficient values listed in Table 2 are shown in Fig. 8. The plot is similar

to that in Fig. 7.

Figure 8. Approximate maximum structural temperatures

13
The values from Fig. 7 were subtracted from those of Fig. 8 to calculate the

absolute errors resulting from the approximation.

a =



Tm

Th



a

−

Tm

Th

(22)

Figure 9 shows the distribution of error over the previously chosen range of

governing parameters. Over much of the surface errors are within ±1% of the

applied surface temperature rise, Th. Highest errors occur for small values of γ and

large values of τh.

Figure 9. Errors using approximate maximum structural temperature equation

It may be more instructive to consider the relative error distribution. The relative errors between the approximation of Eq. 20 and the series solution were calculated using the following equation

r =



Tm

Th



a

−

Tm

Th

Tm

Th

(23)

Figure 10 shows the distribution of the relative error over the previously chosen

range of governing parameters. Over most of the parameter space, the relative error

is within ±10%. For small values of γ, however, the relative error rises precipitately.

Therefore it would be prudent to check the values of γ for any calculations made

using Eq. 20 to avoid the inaccurate region. This inaccurate region of the approximate solution corresponds to a physical situation where the structural skin is much

heavier than the insulator.

14
Figure 10. Relative errors using approximate maximum structural temperature

equation

6 Comparison of Analytical and Numerical Solutions

The problem illustrated in Fig. 1 is considerably simpler than a typical numerical simulation of aerodynamic heating of an insulated aerospace vehicle structure.

Material properties, which can vary with temperature and pressure, are treated as

constant. The applied surface temperature is a simple square pulse, rather than a

more realistic transient profile. A numerical model was developed to: 1) demonstrate that the series solution, Eqs. 13 and 16, produces the expected results, 2)

verify that the governing nondimensional parameters (γ and τh), rather than individual material property values, determine the calculated temperatures, 3) develop

techniques for best using the analytical solution to approximate a realistic numerical

simulation, and 4) estimate typical errors involved in using the analytical solution

to approximate a realistic numerical simulation.

6.1 Numerical Model

A one-dimensional finite element model was developed using the DOLFIN [6]

finite element library for the Python programming language. Using the DOLFIN library, linear one dimensional elements were used to discretize the spatial dimension

and an implicit Crank-Nicolson time marching scheme was used to solve the weak

formulation of the diffusion equation. The model consisted of a layer of insulator

material in perfect contact with a layer of structural material similar to the configuration shown in Fig. 1, except that the structure in the finite element model was

15
not treated as a lumped heat capacitance. For the results shown in this paper, the

model consisted of 50 elements through the thickness of the insulator and 2 elements

through the thickness of the structure. Although a careful convergence study was

not performed, a model with half the number of elements was shown to produce

nearly identical results. The boundary condition on the outer surface of the insulator consisted of an imposed surface temperature that could be varied arbitrarily

with time and updated at each time step of the solution. The inner surface of the

structure was adiabatic. The material properties could be arbitrary functions of

temperature and ambient pressure. Material property values for each finite element

were updated at each time step of the analysis for the average temperature of each

element from the previous time step. Property values could be different for each

element, but did not vary spatially within an element. Time steps between 1 and 5

seconds were used to calculate the results presented in this paper.

When comparing results from the finite element and analytical solutions it is

important to understand and correctly handle a subtle difference in how the two solutions treat temperature. Because material properties are constant, the analytical

solution deals strictly with temperature rise from a uniform initial temperature (assumed to be zero for simplicity in the derivation). Therefore, Th and Tm have been

defined for the analytical solution as maximum temperature rise for the insulator

surface and structure, respectively. For the numerical solutions, however, absolute

temperatures must be used so that the solution is able to look up the correct temperature dependent material properties. Therefore, it is helpful to define two more

variables to clarify this distinction in subsequent discussions.

Tma = Tm + Ti

Tha = Th + Ti

(24)

6.2 Simplified Problem

A finely discretized finite element model of the problem illustrated in Fig. 1

should give exactly the same answer as the analytical series solution defined by

Eqs. 13 and 16. Both solution techniques solve the same differential equation with

the same boundary and initial conditions.

A numerical example, using the properties, dimensions, and heating values given

in Table 1, was studied with the previously described finite element model and the

series solution. A thickness of 3 inches for the LI-900 insulation, an initial temperature of 60◦F and an applied surface temperature, Tha, of 2000◦F were chosen to

complete the problem definition. Calculated temperature distributions at several

times are shown in Fig. 11. This solution is the same as that shown in nondimensional form in Fig. 3. The symbols in Fig. 11 represent the temperatures calculated

at each node of the finite element model. The solid lines represent the series solution defined by Eqs. 13 and 16. The blue and green circles represent distributions

halfway through and at the end of the heating pulse, respectively. The triangular

symbols represent temperature distributions at several times after the heat pulse has

16
ended. The finite element and series solutions are essentially in exact agreement, as

expected.

Figure 11. Temperature distributions from finite element model vs series solution

The primary concern for thermal protection system sizing is the maximum structural temperature. Figure 12 shows the calculated temperature history of the structure. Again the finite element analysis (temperature gradients in the structure were

negligible) and the series solution are essentially in exact agreement, as expected.

The red square shown in Fig. 12 was obtained by solving Eq. 19 for the time of maximum structural temperature and substituting that value into Eq. 16 to calculate

the maximum structural temperature. The green circle represents the approximate

maximum structural temperature calculated using Eq. 20, which is a temperature

rise 6% lower than the exact answer.

As shown earlier in the mathematical formulation of the problem, any solution

should depend only on the values of the nondimensional governing parameters and

not the individual parameter values. A parametric study was performed using the

finite element model to confirm that the numerical solution also depends only on the

value of the nondimensional parameters. Results of the study are shown in Table 3.

Again, the baseline properties are from Table 1 and were also used to generate

Figs. 11 and 12. Numerical values of the governing nondimensional parameters and

the maximum structural temperatures shown in Fig. 12 are presented in Table 3.

For the parametric study, selected combinations of material properties, dimensions,

and/or heating time were scaled so that the nondimensional parameter values remain

unchanged. Even though individual parameter values were doubled or halved, the

calculated maximum structural temperature was the same within 6 significant digits

because the nondimensional parameter values were unchanged.

17
Figure 12. Structural temperature history from finite element model vs series solution

6.3 Material Property Variation

Constant material properties were assumed to obtain the analytical solution.

However, the material properties can vary significantly during a transient heating

pulse. The specific heat capacities of the structure and insulator, cps and cpe, can

vary significantly with temperature. The thermal conductivity of the insulator, ke,

can vary greatly with both temperature and ambient pressure. To investigate the effect of variable material properties, the temperature dependent specific heats of the

aluminum structure and LI-900 insulation and the temperature and pressure dependent thermal conductivity of LI-900, Tables B1-B3 in Appendix B, were incorporated

into the one-dimensional finite element model. In addition, the thermal model for

heat transfer through flexible fibrous insulation from Reference [7] was incorporated

into a Python computer program and used to predict the effective thermal conductivity of Saffil
R

insulation in air. The objective of the finite element analysis was to

develop a rationale for choosing equivalent constant material properties for the analytical solution. The equivalent properties are chosen to approximate the effect of

temperature- and pressure-dependent properties on the calculated maximum structural temperature. The effect of each variable material property was investigated

separately with all other material properties held constant.

6.3.1 Structural Specific Heat Capacity

The structural specific heat capacity, cps, was investigated first because it was

expected to be the easiest to understand. The structure undergoes the smallest

18
Table 3. Numerical study of nondimensional governing parameters

Governing Parameters

γ = 1.69032, τh = 0.123318

Baseline maximum structural temperature, K

Series Eq. 20 Finite element

398.898 392.325 398.908

Parametric Study

Scaled parameters Scale factor Max. str. temp., K

cps, cpe,ke 2.0 398.908

ρs, cpe, ke 2.0 398.908

cps, ρe, th 0.5 398.908

ρs, de, ke, th 0.5 398.908

temperature change, is most isolated from the sharp temperature transients on the

surface of the insulator, and undergoes relatively moderate changes in specific heat

capacity as temperature varies. The obvious approach would be to use the specific

heat capacity at the average of the maximum and initial structural temperatures.

The previously described one dimensional finite element model was used to investigate the effect of temperature dependent cps on the calculated maximum structural

temperature. The finite element analysis was the same as that used for the baseline

results in Table 3, except that the constant value of cps was replaced by the temperature dependent values from Table B1. The results of the finite element analysis

with temperature dependent cps are shown in Table 4. The second column shows the

initial temperature, the average structural temperature, and the maximum structural temperature from the finite element analysis. The third column shows the

value of the temperature dependent structural specific heat capacity for the initial,

average, and maximum structural temperatures, respectively. The corresponding

γ and τh are shown for each value of cps. These values of γ and τh were used to

calculate the associated maximum structural temperatures using the series (Eq. 16)

and approximate (Eq. 20) solutions. Comparing the quantities in bold faced type

in Table 4, it is obvious that using cps for the average structural temperature in

the series solution gives almost exactly (within 0.1%) the same answer as the finite

element analysis with temperature dependent cps.

6.3.2 Insulator Specific Heat Capacity

The effect of temperature dependency of the insulator specific heat capacity,

cpe, on maximum structural temperature is a little more complicated. The insulator

undergoes much larger temperature excursions than the structure and experiences

large, transient temperature gradients whereas the structure has negligible gradients. It is not obvious which value of cpe should be used in the analytical solutions

to best match the finite element solution with temperature dependent cpe. Again,

19
Table 4. Effect of temperature dependent structural specific heat capacity

Structural Finite Element cps(T) γ τh Tma(Series) Tma(Eq. 20)

Temperature T, K 

J

kgK 

K K

Initial 288.71 842.7 1.8134 0.12332 403.01 396.10

Average 344.33 886.7 1.7234 0.12332 400.03 393.35

Maximum 399.95 924.4 1.6530 0.12332 397.60 391.15

th = 1500s Tha =1366K

the same finite element model was used as that for the baseline results in Table 3,

except that the constant value of cpe was replaced by the temperature dependent

values from Table B2. Finite element simulations were performed for two different

values of maximum surface temperature and three different values of heating time.

Results are presented in Table 5. For each simulation cpe values for the insulator were

calculated at three different temperatures: initial temperature, maximum structural

temperature, and the average between the initial and maximum insulator temperatures. The associated values of the governing nondimensional parameters are shown

for each value of cpe along with the corresponding maximum structural temperatures

calculated using the series and approximate solutions. Using the cpe values at the

maximum structural temperature in the analytical solutions most closely matched

the finite element result, as indicated by the bold faced temperatures shown in Table 5 for each simulation. The series solution results are within ±8% of the structural

temperature rise predicted by the finite element analysis of this limited set of cases.

Therefore, using cpe(Tma) as a constant value in the analytical solutions may be an

acceptable approach for approximating the effect of temperature dependent cpe on

maximum structural temperature.

6.3.3 Insulator Thermal Conductivity

Thermal conductivity of the insulator can be a strong function of both temperature and ambient pressure. For simplicity, the effects of pressure and temperature

were considered separately.

High aerodynamic heating is generally associated with high speed flight at high

altitudes with correspondingly low ambient pressure. Pressure increases rapidly

to one atmosphere as the vehicle slows and descends for landing. For an adiabatic

structure, the maximum temperature may occur long after the heating pulse is completed. Therefore, a simplified pressure history was investigated. The pressure was

assumed to have a uniform value, P1, during the heating pulse and a different uniform value, P2, after the heating pulse. The one-dimensional finite element model

was used to investigate the effect of this simplified pressure profile on the calculated

maximum structural temperature. Again, the material property values and heating

parameters were the same as those used for the baseline results in Table 3, except

20
Table 5. Effect of temperature dependent insulator specific heat capacity

Insulator Finite Element cpe(T) γ τh Tma(Series) Tma(Eq. 20)

Temperature T, K 

J

kgK 

K K

Ti 288.71 688.0 0.9393 0.22191 426.96 428.13

Tma 418.44 909.6 1.2419 0.16784 414.04 409.87

Tha+Ti

2

827.59 1209.8 1.6518 0.12620 400.15 393.55

th = 1500s, Tha =1366K

Ti 288.71 688.0 0.9393 0.11095 358.66 357.39

Tma 350.88 800.4 1.0929 0.09537 355.88 352.19

Tha+Ti

2

827.59 1209.8 1.6518 0.06310 344.74 339.88

th = 750s, Tha =1366K

Ti 288.71 688.0 0.9393 0.44382 554.38 561.19

Tma 531.14 1052.3 1.4368 0.29016 518.86 511.89

Tha+Ti

2

827.59 1209.8 1.6518 0.25239 506.19 497.47

th =3000s, Tha =1366K

Ti 288.71 688.0 0.9393 0.22191 357.84 358.42

Tma 355.42 808.7 1.1041 0.18879 354.16 352.99

Tha+Ti

2

558.15 1072.3 1.4640 0.14238 347.37 344.45

th =1500s, Tha =828K

that the constant value of ke was replaced by the temperature and pressure dependent values from Table B3. The values of P1 and P2 were systematically varied

between values of 0.0001, 0.001, 0.01, 0.1, and 1.0 atm. The resulting structural

temperature histories are shown in Fig. 13. Different color lines represent different

values of P1. Different line patterns represent different values of P2: solid is 1.0 atm,

dashed is 0.1 atm, and dotted is 0.01 atm. The curves in Fig. 13 exhibit a consistent pattern. The maximum structural temperature depends only on P1. Different

values of P2, cause the time of maximum temperature to vary, but the maximum

temperature itself varies less than 0.2K over this wide range of P2 values. Similar

results were obtained for Saffil insulation in air.

Thermal conductivity of the insulator is also generally a strong function of temperature. A simple approach for choosing an effective constant value of ke would be

to use the conductivity for a pressure, P1, at a temperature, Tke, that is the average of Tha and Ti

. A slightly more general approach would be to choose a thermal

conductivity at a temperature defined by

Tke = Ti + fkeTh (25)

where fke is the fraction (0 to 1) of the insulator temperature range. A value of 0.5

would result in a simple average of Tha and Ti.

A parametric numerical study was performed to determine if there was a sin21
Figure 13. Structural temperature histories for different ambient pressures

gle value of fke that could be used to predict accurately the maximum structural

temperature. Results for LI-900 are shown in Table 6, and for Saffil insulation in

Table 7. A range of parameters were varied, including P1, Th, th, de, and ρe. All

other material properties, except ke, were held constant. In Table 7 the default

value of de is 3 in. (0.0762 m) and the default value of ρe is 4.5 lbm

f t3 (72 kg

m3 ). For

each combination of parameters, the one-dimensional finite element model, with

ke(P1, T), was used to calculate the maximum structural temperature (column 2 of

Tables 6 and 7). The series solution, Eqs. 16 and 19, was solved iteratively to find

the fixed value of ke that would produce the same maximum structural temperature

as the finite element solution. The associated values of Tke (column 3 of Tables 6

and 7) and fke (column 4 of Tables 6 and 7) were also calculated. For the majority

of the results shown, fke ranged between 0.57 and 0.62. A value of 0.60 was chosen for fke, and the corresponding values of Tke, ke(P1, Tke), and τh (column 5 of

Tables 6 and 7) were calculated. The resulting approximate maximum structural

temperatures were calculated using the series solution (column 6 of Tables 6 and 7)

and the approximate solution (column 7 of Tables 6 and 7). For fke = 0.6, Tables 6

and 7 show good agreement between the finite element and series solutions for maximum structural temperature. Over most of the parameter ranges the approximate

solution is close to the series solution, but starts to diverge for small values of γ

(Table 7), as expected from the errors shown in Figs. 9 and 10.

22
Table 6. Maximum structural temperature calculated using effective LI-900 thermal

conductivity

P1 Finite Element Tke fke τh Tma(Series) Tma(Eq. 20)

atm Tma, K K K K

0.0001 358.72 954.0 0.617 0.07525 356.48 351.14

0.001 368.45 955.9 0.619 0.08577 365.86 360.21

0.01 399.02 950.6 0.614 0.12071 396.79 390.27

0.1 446.95 936.5 0.601 0.17853 447.03 439.33

γ = 1.6903, th = 1500s, Tha =1367K fke = 0.6, Tke = 935.6K

0.0001 422.26 1142.1 0.629 0.10789 410.38 402.56

0.001 429.54 1126.7 0.618 0.11867 422.34 414.20

0.01 469.68 1116.0 0.610 0.15824 465.80 456.62

0.1 541.09 1117.0 0.611 0.22374 535.89 525.19

γ = 1.6903, th = 1500s, Tha =1644K fke = 0.6, Tke = 1102.2K

0.0001 323.89 756.6 0.585 0.05398 324.93 321.53

0.001 330.85 756.9 0.585 0.06443 331.88 328.19

0.01 351.52 779.1 0.613 0.09315 350.89 346.55

0.1 381.23 756.1 0.584 0.14216 382.93 377.75

γ = 1.6903, th = 1500s, Tha =1089K fke = 0.6, Tke = 768.9K

0.0001 304.69 592.2 0.581 0.03755 305.27 303.42

0.001 309.04 587.3 0.571 0.04795 309.80 307.70

0.01 321.12 599.3 0.583 0.07527 321.65 319.06

0.1 336.38 591.6 0.580 0.11168 337.31 334.26

γ = 1.6903, th = 1500s, Tha =811K fke = 0.6, Tke = 602.3K

0.0001 293.96 439.2 0.615 0.02550 294.10 293.40

0.001 295.79 438.8 0.613 0.03445 295.93 295.10

0.01 301.10 442.2 0.627 0.06018 301.16 300.07

0.1 306.23 440.0 0.618 0.08544 306.28 305.00

γ = 1.6903, th = 1500s, Tha =533K fke = 0.6, Tke = 435.6K

0.0001 335.18 951.5 0.615 0.05017 334.03 329.61

0.001 341.67 953.2 0.616 0.05718 340.32 335.61

0.01 361.98 945.1 0.609 0.08047 361.14 355.64

0.1 393.83 926.2 0.591 0.11902 395.30 388.82

γ = 1.6903, th = 1000s, Tha =1367K fke = 0.6, Tke = 935.6K

0.0001 382.33 956.2 0.619 0.10034 378.80 372.77

0.001 395.35 958.9 0.622 0.11436 391.19 384.82

0.01 436.15 955.6 0.619 0.16095 431.90 424.54

0.1 499.87 947.6 0.611 0.23804 497.16 488.41

γ = 1.6903, th = 2000s, Tha =1367K fke = 0.6, Tke = 935.6K

23
Table 7. Maximum structural temperature calculated using effective Saffil thermal

conductivity (th = 1500s)

P1 Finite Element Tke fke τh Tma(Series) Tma(Eq. 20)

atm Tma, K K Tma, K Tma, K

0.0001 385.23 931.5 0.596 0.16694 386.50 387.99

0.001 405.99 941.7 0.606 0.19850 404.57 407.07

0.01 457.55 935.0 0.599 0.29376 457.92 463.63

0.1 482.63 932.8 0.597 0.34120 483.89 491.02

ρe = 72 kg

m3 , de = 0.0762m, γ = 0.8452, Tha =1367K fke = 0.6, Tke = 935.6K

0.0001 329.76 745.7 0.571 0.10155 333.20 332.72

0.001 344.56 754.8 0.582 0.13288 346.74 347.07

0.01 379.25 753.7 0.581 0.21580 382.05 384.33

0.1 393.60 752.7 0.580 0.25156 396.98 400.16

ρe = 72 kg

m3 , de = 0.0762m, γ = 0.8452, Tha =1089K fke = 0.6, Tke = 768.9K

0.0001 313.96 588.3 0.573 0.12546 315.57 317.88

0.001 328.84 592.0 0.581 0.19465 330.10 334.55

0.01 359.31 587.4 0.572 0.34894 361.62 371.04

0.1 370.14 587.8 0.572 0.40521 372.71 383.95

ρe = 72 kg

m3 , de = 0.0508m, γ = 0.5634 Tha =811K fke = 0.6, Tke = 602.2K

0.0001 301.90 425.9 0.561 0.24284 302.68 307.68

0.001 316.59 428.9 0.573 0.50759 317.22 328.42

0.01 339.90 430.0 0.577 0.96849 340.69 361.57

0.1 346.37 431.3 0.582 1.10387 347.12 370.46

ρe = 72 kg

m3 , de = 0.0254m, γ = 0.2817, Tha =533K fke = 0.6, Tke = 435.6K

0.0001 358.43 756.42 0.584 0.22445 361.50 369.79

0.001 379.61 763.0 0.593 0.28701 381.81 392.56

0.01 416.06 760.1 0.589 0.40934 418.58 435.96

0.1 427.75 760.5 0.590 0.44890 430.34 449.61

ρe = 48 kg

m3 , de = 0.0762m, γ = 0.5634, Tha =1089K fke = 0.6, Tke = 768.9K

24
6.4 Heating History Profile

For hypersonic aerospace vehicles, surface heating histories can vary greatly with

the vehicle mission, trajectory, configuration, and location on the vehicle. Surface

heating histories are often provided to a TPS designer in the form of radiation

equilibrium heating rates, radiation equilibrium temperatures, convective heating

coefficients with total enthalpies or recovery temperatures, or a series of heating

rates at constant wall temperatures that bracket the expected surface temperature

range. For simplicity, the current study will only consider radiation equilibrium

temperature histories, which are suitable for well insulated surfaces that have relatively low thermal mass at the surface. Radiation equilibrium temperature histories

can also be readily calculated from the other forms of heating histories.

Figure 14. Surface temperature histories

For the current study, surface temperature histories were chosen from Space

Shuttle Orbiter entry flight data [8, 9] and from previous studies [10, 11] of single

stage-to-orbit reusable launch vehicles (RLV’s). A total of seven surface temperature

histories are shown in Fig. 14. Surface temperature histories from four different body

points (BP9740, BP9678, BP9591, and BP9489) during the same atmospheric entry

flight of the Shuttle Orbiter were chosen to illustrate a range of different profile

shapes. The flight temperature data was smoothed to reduce noise from the crude

discretization produced by the flight instrumentation. Two heating histories from

an earlier TPS parametric weight study [10], body point A on the windward center

line of a winged cylindrical vehicle for the Access to Space study (ATSpA), and body

point A on windward centerline of a lifting body reusable launch vehicle (RLVpA)

are also shown in Fig. 14. The seventh surface temperature history, also studied in

25
Figure 15. Ambient pressure histories

Reference [11], was for a point on the windward center line of a slightly different

variation of the lifting body vehicle (RLV3c). The corresponding ambient pressure

histories are shown in Fig. 15.

The challenge was to find an equivalent square heating pulse for each of the

heating histories that would enable the analytical solution to calculate an accurate

estimate of the maximum structural temperature. The first step in idealizing the

heating histories was to simplify the temperature histories in Fig. 14 by truncating

the variable low temperature portions of each history at the beginning and end.

The histories were truncated by eliminating any portions below a threshold surface

temperature defined by

Tthr = Ti + fthr(Tmx − Ti) (26)

where Tmx is the maximum surface temperature for a surface temperature history,

and fthr is a fraction of the surface temperature range.

Several quantities, including integrated heat load, integrated absolute surface

temperature, and integrated surface temperature rise from the initial temperature,

were integrated over the time of the truncated temperature histories.

The one-dimensional finite element model with fixed material properties from

Table 1 was used to calculate the maximum structural temperature for a structural

thickness of 0.125 inch and insulator thicknesses of 1, 2, 3, and 4 inches for each

heating history. The resulting maximum structural temperatures were plotted as a

function of various integral values, such as integrated heat load, integrated absolute

temperature, and integrated surface temperature rise, in hopes of identifying a clear

correlation. A linear correlation, Fig. 16, was discovered between the maximum

26
structural temperature rise, Tm, and the integrated surface temperature rise from

the initial temperature, which is defined as

IT =

Z t2

t1

(T − Ti)dt (27)

where t1 and t2 are the beginning and ending times of the truncated temperature

history.

Figure 16. Maximum structural temperature vs integrated surface temperature rise

In Fig. 16, the maximum structural temperature rise is shown as a function

of the integral defined in Eq. 27. Each marker shape represents one of the seven

surface heating histories from Fig. 14. The triangular symbols represent the four

shuttle histories and the other symbols represent the RLV histories. The dashed

lines represent the best fit line for all the maximum structural temperatures calculated using the finite element model at each insulator thickness. The lines were

constrained to pass through the origin. The figure implies that for each configuration with fixed material properties, the maximum structural temperature rise is

directly proportional to the integral defined in Eq. 27. For the simple square heat

pulse, that integral is simply

IT = Thth (28)

To determine a reasonable equivalent square heat pulse it is helpful to bound the

ranges of the two parameters that define it. The maximum possible value of th would

be the time span of the truncated surface temperature history, t2−t1, and the largest

that Th could be is the maximum surface temperature rise, Tmx − Ti. Therefore,

27
the smallest that th could be is IT /(Tmx − Ti). For fixed material properties, the

series solution was used to calculate Tm by varying combinations of th and Th for

fixed values of IT . The calculated value of Tm was almost totally a function of IT

and showed little sensitivity to the particular combinations of th and Th over their

feasible ranges. However, for variable material properties, it is important to chose a

combination of th and Th that will enable calculation of constant effective properties

and average pressure during the heating pulse, P1. A simple approach is to use an

average of the bounding values of th.

th =

(t2 − t1) + IT

Tmx−Ti

2

(29)

Then Th can be readily found from Eq. 28. The resulting square heat pulse can

then be centered in the truncated time range between t1 and t2 for the purposes of

calculating an average value of P1, Pavg.

Figure 17. Simplified heating and pressure histories for BP9740

This approach for simplifying the surface heating history of Shuttle Orbiter

body point 9740 is illustrated in Fig. 17. The solid black line represents the surface

temperature history for BP9470. The horizontal blue dotted line represents the

threshold temperature defined in Eq. 26 for fthr = 0.15. The vertical blue dotted

lines, t1 and t2, bound the truncated heating pulse. The shaded area illustrates the

integral defined by Eq. 27. The heavy black dashed line represents the equivalent

square heating pulse with th calculated from Eq. 29. The square heat pulse has been

centered between t1 and t2 for the purpose of calculating an average pressure to use

as P1. The ambient pressure over the duration of the square heat pulse is indicated

28
by a solid green line. The heavy green dot-dash line represents a simple average of

the pressure over the interval of the square heat pulse.

Selecting the threshold temperature, or more specifically fthr, for truncating

the surface temperature histories is not straightforward. The intent of truncating

the temperature histories is to capture the main heating pulse without any spurious complications from variations in convective cooling effects before landing or the

missing initial portion of the heating history (RLV histories). The heating history

simplifications illustrated in Fig. 17 were applied to all seven heating histories for

three different values of fthr: 0.10, 0.15, and 0.20. Table 8 lists the resulting calculated quantities. The second column shows the maximum surface temperature rise.

The third column shows the threshold temperature for the given value of fthr and

the fourth and fifth columns show the times bounding the truncated heating interval. Columns 6 and 8 show the maximum and minimum possible durations for the

equivalent square heat pulse and column 9 is their average. The last three columns

contain the values that would be used for further calculations using the analytical solution. The integral in column 7 changes little over this range of fthr, which

indicates that there should be little effect on the calculated maximum structural

temperature for fixed material properties.

Table 8. Simplified surface temperature histories

Profile Tmx − Ti Tthr t1 t2 t2 − t1 IT

IT

Tmx−Ti

th Th Pavg

◦C K s s s 103s◦C s s

◦C Pa

fthr = 0.10

BP9470 976.4 386.3 77.2 1575.0 1497.8 1073.5 1099.4 1298.6 826.6 122.7

BP9678 914.0 380.1 76.7 1593.8 1517.1 1055.4 1154.7 1335.9 790.0 163.3

BP9591 969.3 385.6 126.4 1583.4 1457.0 975.5 1006.4 1231.7 792.0 122.2

BP9489 687.5 357.5 99.3 1516.8 1417.5 689.0 1002.1 1209.8 569.5 64.7

ATSpA 990.9 387.8 0.0 2166.3 2166.3 1670.4 1685.7 1926.0 867.3 1756.6

RLVpA 969.9 385.7 0.0 2297.9 2297.9 1763.4 1818.0 2058.0 856.9 1025.5

RLV3c 814.1 370.1 0.0 2160.2 2160.2 1399.1 1718.5 1939.4 721.4 1401.4

fthr = 0.15

BP9470 976.4 435.2 91.4 1544.7 1453.3 1068.1 1093.9 1273.6 838.6 98.0

BP9678 914.0 425.8 88.1 1565.9 1477.8 1051.0 1149.8 1313.8 799.9 133.0

BP9591 969.3 434.1 141.7 1554.2 1412.5 970.1 1000.8 1206.6 804.0 99.7

BP9489 687.5 391.8 114.2 1498.6 1384.3 686.1 998.0 1191.2 576.0 59.1

ATSpA 990.9 437.3 0.0 2145.7 2145.7 1667.8 1683.1 1914.4 871.2 1756.6

RLVpA 969.9 434.2 0.0 2269.7 2269.7 1760.0 1814.5 2042.1 861.8 1025.5

RLV3c 814.1 410.8 0.0 2125.1 2125.1 1395.5 1714.2 1919.6 727.0 1364.6

fthr = 0.20

BP9470 976.4 484.0 103.5 1523.2 1419.6 1062.3 1088.0 1253.8 847.3 84.0

BP9678 914.0 471.5 95.7 1538.6 1442.9 1045.4 1143.7 1293.3 808.3 107.3

BP9591 969.3 482.6 154.5 1529.6 1375.1 963.7 994.2 1184.7 813.5 82.7

BP9489 687.5 426.2 127.9 1480.9 1353.0 682.4 992.6 1172.8 581.9 54.2

ATSpA 990.9 486.9 0.0 2125.2 2125.2 1664.2 1679.5 1902.4 874.8 1732.7

RLVpA 969.9 482.7 0.0 2244.6 2244.6 1755.8 1810.2 2027.4 866.0 974.2

RLV3c 814.1 451.5 0.0 2091.9 2091.9 1390.8 1708.4 1900.2 732.0 1331.4

1 2 3 4 5 6 7 8 9 10 11

The equivalent square heat pulses, defined by the values of th and Th in Table 8,

and the fixed material properties from Table 1 were substituted into the series

29
solution to calculate the maximum structural temperature rise. The temperature

was predicted for a structural thickness of 0.125 in. and insulator thicknesses of 1, 2,

3, and 4 in. for each equivalent square heat pulse and compared to the finite element

results shown in Fig. 16. The largest errors in maximum structural temperature

rise were -5.4%, -5.9%, and -6.5% for fthr values of 0.10, 0.15, and 0.20 respectively.

Therefore, for fixed material properties, the solution is not sensitive to the threshold

temperature over the range considered. However, variable material properties are

sensitive to both Th and Pavg, which vary somewhat with fthr, as shown in Table 8.

With these considerations in mind, a value of fthr = 0.15 was chosen for further

calculations.

6.5 Approximation of Realistic Simulations

The key remaining question is: how accurately can the analytical solution, with

the previously described effective material properties and simplified square heat

pulse, predict the maximum structural temperature calculated using a finite element

solution with variable material properties and a time-accurate surface temperature

history? The one-dimensional finite element model with temperature-dependent

structural properties from Table B1 and insulator properties from Tables B2 and

B3 was used to calculate the maximum structural temperature for a structural

thickness of 0.125 in. and LI-900 insulator thicknesses of 1, 2, 3, and 4 in. for each

heating history.

For comparison, the maximum structural temperature was calculated for each

surface temperature history and geometry using the series and approximate analytical solutions, with the following approximations.

cps = cps(Tcs), where Tcs = Ti +

Tm

2

, (30)

cpe = cpe(Tce), where Tce = Ti + Tm, and (31)

ke = ke(Tke, Pavg), where Tke is defined in Eq. 25 (32)

Values for th, Th, and Pavg are taken from Table 8 for fthr = 0.15.

One difficulty is that the maximum structural temperature rise, Tm, must be

known to calculate the effective constant properties. This difficulty can be readily

overcome by making an initial guess and iterating to a converged solution. The

series solution and both approximate analytical solutions, Eqs. 20 and 21 were each

iterated independently to arrive at their respective converged solutions.

The finite element and analytical solutions for maximum structural temperature

are compared in Fig. 18. The figure shows the maximum structural temperature

rise as a function of LI-900 tile thickness. The solid circles represent the finite

element solutions, the solid lines represent the series solution, the dashed lines represent the approximate solution given by Eq. 20, and the dotted lines represent the

approximate solution given by Eq. 21. The colors correspond to the applied surface temperature histories: black - Shuttle orbiter body point 9470, green - Shuttle

30
Figure 18. Maximum structural temperature rise vs insulation thickness

orbiter body point 9489, and red - ATS reusable launch vehicle point A (windward

centerline). The series solution agrees with the finite element solution to within

-0.9% and 18.7% for the predicted maximum structural temperature rise of the

cases shown on Fig. 18. The close agreement between the finite element results and

the series solution tends to validate the methodology used to calculate the effective material properties and to map the transient surface temperature history to an

equivalent square pulse. The two approximate equations track the series solution

and the finite element solution well for most of the LI-900 thickness range, but start

to diverge for small thickness values.

Table 9 gives the detailed numerical comparisons between the finite element

calculations and results using the equations developed in this paper. The table is

divided into seven sections with each section consisting of results for four different

insulator thicknesses for a surface temperature history identified in bold print. For

each insulation thickness, values of the two governing nondimensional parameters

(associated with the series solution) are given, based on the effective constant material properties and equivalent square heating pulse defined previously. The finite

element prediction of the maximum structural temperature rise is given for variable

material properties and a time-accurate surface heating history. The series and approximate solutions for the maximum structural temperature rise are also shown,

along with their errors relative to the finite element solution. The best assessment

of the accuracy of using the material property simplifications combined with the

surface temperature history simplifications is error associated with the series solution. The approximate solution has the same errors as the series solution combined

with the approximation errors illustrated in Fig. 10. The errors for the series so31
lution range from 5.6% to 21.3%. The series solution is remarkably close to the

finite element solution, considering the complexity of the numerical simulation and

the large variation of material properties that occurs with time. For the much simpler approximate equations, Eq. 20 has errors that range from 1.5% to 38.2% and

Eq. 21 has errors that range from 9.6% to 46.0%. The errors for the approximate

solutions start to get large for relatively small values of γ, as expected from the

errors shown in Fig. 10. The series solution errors for the reusable launch vehicles

(ATSpA, RLVpA, and RLV3c) are less than 10% for all calculated cases.

7 Mass of Insulated Structure

For aerospace vehicles reducing mass is of utmost importance. Therefore, it is

helpful to develop further insight into the interplay between the mass of structure

and the mass of insulator required to limit the structural temperature. The mass

per unit area of this simplified, insulated structure can be written simply as

m = me + ms = ρede + ρsds (33)

The relationship between de and ds that is required to limit the maximum structural temperature to a specified value is given by Eq. 21. By substituting Eqs. 4

and 10 into Eq. 21 and rearranging terms, the following expression can be obtained.

dsd

3

e =

(keth)

2

2(ρscps)(ρecpe)



−ln(1 −

Tm

Th

)

2

(34)

The equation can be further manipulated to give an explicit expression for de.

This provides a useful equation to calculate the insulator thickness required to limit

the maximum structural temperature to a specified value.

de =





(keth)

2

2(ρscpsds)(ρecpe)



−ln(1 −

Tm

Th

)

2





1

3

(35)

Substituting Eq. 35 into Eq. 33 produces an expression for mass per unit area

as a function of structural thickness.

m =







√

ρeke

cpe 2

t

2

h

2ρsdscps 

−ln(1 −

Tm

Th

)

2





1

3

+ ρsds (36)

The first term in Eq. 36 represents the mass of insulator required to limit the

structure to the specified maximum temperature and the second term represents

structural mass. Eq. 36 can be written as a function of ms.

32
Table 9. Maximum structural temperature rise for LI900 insulation

de γ τh FE Series error Eq. 20 error Eq. 21 error

m ◦C

◦C % ◦C % ◦C %

BP9470

0.0254 0.4367 0.6536 164.86 174.10 5.6 207.58 25.9 214.64 30.2

0.0508 0.7858 0.1889 74.84 81.58 9.0 83.99 12.2 92.40 23.5

0.0762 1.1128 0.0902 44.74 49.42 10.5 47.22 5.5 54.23 21.2

0.1016 1.4400 0.0527 30.33 33.74 11.2 30.79 1.5 36.51 20.4

BP9678

0.0254 0.4364 0.6857 152.83 173.38 13.4 206.89 35.4 213.33 39.6

0.0508 0.7857 0.1980 68.76 81.49 18.5 84.03 22.2 92.15 34.0

0.0762 1.1128 0.0946 41.05 49.40 20.3 47.30 15.2 54.14 31.9

0.1016 1.4400 0.0552 27.81 33.73 21.3 30.85 10.9 36.47 31.1

BP9591

0.0254 0.4288 0.6144 139.86 155.54 11.2 186.49 33.3 193.71 38.5

0.0508 0.7735 0.1767 63.32 72.47 14.4 74.65 17.9 82.61 30.5

0.0762 1.1010 0.0839 37.78 43.76 15.8 41.70 10.4 48.20 27.6

0.1016 1.4290 0.0488 25.58 29.83 16.6 27.10 5.9 32.35 26.5

BP9489

0.0254 0.3969 0.5056 77.47 87.58 13.0 107.09 38.2 113.09 46.0

0.0508 0.7285 0.1408 33.92 39.79 17.3 41.10 21.2 46.49 37.1

0.0762 1.0588 0.0651 20.08 23.75 18.3 22.41 11.6 26.52 32.1

0.1016 1.3900 0.0373 13.55 16.09 18.7 14.39 6.1 17.60 29.8

ATSpA

0.0254 0.4826 1.3822 369.70 366.53 -0.9 426.90 15.5 423.03 14.4

0.0508 0.8856 0.4003 184.05 188.32 2.3 193.76 5.3 201.79 9.6

0.0762 1.2413 0.1957 112.23 117.62 4.8 114.58 2.1 123.82 10.3

0.1016 1.5723 0.1178 76.97 81.86 6.3 77.45 0.6 85.93 11.6

RLVpA

0.0254 0.4803 1.3174 358.76 348.28 -2.9 405.92 13.1 403.15 12.4

0.0508 0.8764 0.3826 175.04 177.50 1.4 182.78 4.4 190.99 9.1

0.0762 1.2327 0.1863 106.19 110.50 4.1 107.55 1.3 116.67 9.9

0.1016 1.5589 0.1122 72.75 76.85 5.6 72.59 -0.2 80.88 11.2

RLV3c

0.0254 0.4710 1.2408 270.03 276.63 2.4 324.47 20.2 323.31 19.7

0.0508 0.8446 0.3626 131.45 139.07 5.8 143.92 9.5 151.17 15.0

0.0762 1.1880 0.1760 79.73 86.22 8.1 84.07 5.4 91.61 14.9

0.1016 1.5121 0.1050 54.48 59.67 9.5 56.23 3.2 63.12 15.9

33
m =







√

ρeke

cpe 2

t

2

h

2cps −ln(1 −

Tm

Th

)

2





1

3

m

− 1

3

s + ms (37)

7.1 Material Properties and Insulator Mass

The first term in Eq. 37 represents the mass of insulator required to limit the

structure to the specified maximum temperature. Inspection of this equation illustrates the effect of key parameters on required insulator mass.

me =







√

ρeke

cpe 2

cps −ln(1 −

Tm

Th

)

2





1

3 

t

2

h

2ms

1

3

(38)

From Eq. 38, it is obvious that a longer heat pulse (increased th) will result in

a larger required insulator mass. Similarly, a larger structural mass will provide

more structural heat sink capacity and require less insulator mass. For a given

structural mass and a particular transient heating pulse, required insulation mass

will be determined by the thermal properties of the insulator and structure.

The specific heat capacities of insulators and structural materials vary significantly with temperature and the thermal conductivity of low density insulators is a

function of both temperature and pressure. Therefore, it is not immediately obvious

which material property values to use in the preceding equations to obtain results

that are meaningful and useful. However, the previously derived expressions for

equivalent constant thermal properties, Eqs. 30, 31, and 32, can be used to calculate values of cps, cpe, and ke for each surface temperature history. The quantities

defining the equivalent square heating pulses, th, Th and Pavg, are listed in Table 8

for the seven heating histories previously studied. The first four temperature histories were measured at four different locations on the windward surface of a Shuttle

Orbiter during a single atmospheric entry flight [8, 9]. It is interesting to note the

variation in the equivalent square heating pulses that occur over just the windward surface in a single flight. The last three heating histories were predicted for

a single point on the windward surface of three different proposed reusable launch

vehicles [10, 11].

Inspection of Eqs. 37 and 38 reveals that the thermal properties of the structure

and insulator can each be grouped separately. These grouped properties have the

potential to be useful for comparing the thermal effectiveness of both the structural

and insulator materials.

The specific heat capacity and maximum allowable temperature of the structure

directly affect the required insulation mass. However, the thermal properties of the

structure are coupled to the amplitude of the equivalent square heating pulse, Th.

The following expression is a candidate figure of merit for the thermal effectiveness

of the structural material.

34
βs = cps 

−ln(1 −

Tm

Th

)

2

(39)

A larger value of this parameter will lead to a smaller required mass of insulation. Therefore a structure with a high specific heat capacity and high maximum

temperature limit will tend to require less insulation.

For steady state heat conduction, minimizing the product of insulator density

and thermal conductivity, ρeke, minimizes the mass of required insulator [11]. However, a similar figure of merit for minimum mass of insulation subjected to transient

heating has not been available. The grouping of insulator properties in Eq. 38 suggests a candidate figure of merit for minimum mass insulation for a transient heating

pulse (or at least for the simplified heat pulse in this derivation).

κe =

ρeke

√cpe

(40)

A smaller value of this parameter will lead to a smaller required mass of insulation. Therefore, a low mass insulator for the transient heat pulse should have a

combination of low ρeke and high cpe.

7.2 Minimum Mass of Insulated Structure

The mass of the insulated structure, calculated using Eq. 37, is obviously a

function of the mass of the structural material. Increasing the amount of structural

mass linearly adds to the total mass, but results in a corresponding decrease in the

mass of required insulation. Therefore, in the absence of any overriding structural

requirements, there should be a structural mass for which the total mass is minimum.

The structural mass for which the total mass is minimum can be readily calculated

by taking the first derivative of Eq. 37 with respect to ms and setting it equal to

zero. The result can be solved for ms to produce

mso = 54(− 1

4

)√

th







√

ρeke

cpe 2

cps −ln 1 −

Tm

Th

2





1

4

(41)

Equation 41 is potentially useful for sizing insulated structure. The structure

could first be sized to carry the design structural loads. The resulting equivalent

structural mass can then be compared to that calculated using Eq. 41. If the result

of Eq. 41 is greater, then increasing the structural mass to that value will result in

the lowest combined mass of structure and insulator. However, if the result of Eq. 41

is less, then adding additional structural mass will only increase the combined mass.

Substituting Eq. 41 back into Eq. 37 produces the following expression for the

minimum mass of an insulated structure.

mopt = (3 + 1) mso (42)

35
mopt = 2 

2

3

(

3

4

)

√

th







√

ρeke

cpe 2

cps −ln 1 −

Tm

Th

2





1

4

(43)

The first term in parentheses in Eq. 42 indicates the relative contributions of the

insulation and structure, respectively, to the minimum total mass. For minimum

total mass, Eq. 42 predicts that the mass of the insulation will be three times the

mass of the structure. Eq. 43 defines the lower bound for the mass of an insulated

structure subjected to a transient heat pulse. The magnitude of the minimum total

mass is solely a function of the duration of the equivalent square heating pulse, th,

and the figures of merit defined in Eqs. 39 and 40.

8 Analytical Predictions and Numerical Simulations

Several numerical studies were performed to assess how the simple approximate

solutions derived in this paper compare to more accurate numerical simulations of

Earth atmospheric entry heating on reusable launch vehicles. The objectives of the

numerical studies were: 1) to assess the accuracy of using Eqs. 35 and 38 to size

insulation for a realistic transient heating pulse, 2) to determine the applicability

of the figures of merit defined by Eqs. 39 and 40 to results of atmospheric entry

simulations, and 3) to investigate the interplay between the mass of the structure

and the corresponding mass of insulation required to limit the maximum structural

temperature.

The finite element model and heating histories previously described were used for

all of these numerical studies. However, the finite element simulation was iterated to

determine the insulation thickness required to limit the structural temperature rise

to the specified value. Each transient simulation was continued until the structural

temperature had reached its maximum value.

8.1 Thermal Properties of Structural Material

Obviously, raising the structural temperature limit will decrease the amount of

required insulation. Less commonly considered is the effect of the structural specific

heat capacity on the required insulation. Equation 39 defines a candidate figure

of merit that combines the effects of structural temperature limit and specific heat

capacity on the amount of required insulation.

A numerical study was performed to assess how well the results of a series of full

numerical simulations would correlate to the parameter βs. To compare the effects of

the structural thermal properties, it was necessary to fix all other parameters in the

problem and vary only the choice of structural material. LI-900 tile was chosen as the

insulator and the structural mass per unit area was fixed at 6.10 kg

m2 (1.25 lbm

f t2 ). Two

different heating histories were considered, but results can only be directly compared

for a particular heating history. Table 10 shows thermal properties for four structural

36
Table 10. Thermal Properties of Four Structural Materials

Ti + Tm cps(Tcs) βs(BP7940) βs(AT SpA)

K(

◦F)

J

kgK  Btulbm◦R



J

kgK  Btulbm◦R



J

kgK  Btulbm◦R



Aluminum 2024 ρs =2803 kg

m3 (0.10 lbm

in3 )

422(300) 900(0.215) 27(0.006) 25(0.006)

450(350) 910(0.217) 41(0.010) 38(0.009)

Graphite/epoxy ρs =1576 kg

m3 (0.057 lbm

in3 )

366(200) 913(0.218) 9(0.002) 8(0.002)

422(300) 984(0.235) 30(0.007) 27(0.006)

Beryllium Aluminum ρs =2098 kg

m3 (0.076 lbm

in3 )

505(450) 1690(0.404) 151(0.036) 138(0.033)

561(550) 1735(0.415) 267(0.064) 244(0.058)

Titanium 6Al-4V ρs =4437 kg

m3 (0.16 lbm

in3 )

533(500) 574(0.137) 68(0.016) 62(0.015)

700(800) 594(0.142) 270(0.064) 242(0.058)

materials chosen for the study. The temperature dependent properties for these

materials are tabulated in Reference [11]. The maximum allowable temperature for

each material is application dependent, so an approximate upper and lower bound

was chosen for each material. For each limiting temperature, the structural specific

heat capacity (Eq. 30) and βs for each of the two heating histories are shown in

Table 10.

Iterative finite element analysis was used to size insulation for 16 cases (4 structural materials, 2 temperature limits, and 2 heating histories). The resulting insulation masses are shown in Fig. 19 and indicated by the square symbols. For the

ATSpA heating history, each particular structural material and temperature limit is

identified in the figure. The square symbols are not labeled for the BP7490 heating

history to avoid cluttering the figure. The approximate insulation masses, calculated

using Eq. 38, are indicated by circles corresponding to each of the 16 cases sized by

the finite element solution. For the Space Shuttle heating history, BP7490, half of

the approximate solutions are within 10% of the finite element solutions and all of

the approximate solutions are within 25%. For the reusable launch vehicle heating

history, ATSpA, half of the approximate solutions are within 10% of the finite element solutions and all of the approximate solutions are within 20%. Because the

value of cpe used in Eq. 38 is a function of the maximum structural temperature

limit (Eq. 31), the approximate insulation masses are not solely a function of βs.

The solid lines in Fig. 19 represent the approximate solution using a value of cpe

that is an average of the 8 cases for each temperature history. The close agreement

between the solid lines and circles indicates that, for LI-900 insulation, the variation

37
Figure 19. Effect of structural thermal properties on required insulator mass

of cpe with Tm has little effect on the mass of required insulation. The solid lines

clearly indicate the effect of βs on the required insulation mass predicted by the approximate solution, Eq. 38. Remarkably, the finite element solutions closely follow

this trend, even with four different structural materials. Therefore, the parameter βs

appears to be a useful indicator of the thermal effectiveness of a structural material.

8.2 Thermal Properties of Insulator

To better understand the significance of the parameter defined in Eq. 40, it is

helpful to consider a range of candidate insulator materials that could be used for

thermal protection on a reusable launch vehicle. A numerical study was performed

to investigate the relative mass efficiency of a range of insulators for two different

surface temperature histories (BP7490 and ATSpA) from Table 8. Five rigid ceramic

tile materials (LI-900, LI-2200, FRCI-12, AETB-8, AETB-16) and four flexible insulations (Saffil-3, Saffil-6, Qfiber-3, and Qfiber-6) were considered. The temperature

and pressure dependent thermal properties of the three Space Shuttle Orbiter tile

materials ((LI-900, LI-2200, and FRCI-12) were obtained from published tables [4].

The thermal properties for AETB-8 and AETB-16 were calculated using the model

developed by Daryabeigi [12] and the flexible insulation properties were calculated

using the models also developed by Daryabeigi [7]. For this study, aluminum with

a maximum temperature limit of 450K (350◦F) was chosen as the underlying structure. The structural mass per unit area was fixed at 6.10 kg

m2 (1.25 lbm

f t2 ).

Table 11 shows the effective cpe (Eq. 31) and ke (Eq. 32) of each insulation for

the two surface heating histories. The insulation figure of merit, calculated using

38
Eq. 40, is shown for each set of material properties. As shown in Table 11, the

insulation effective material properties and figure of merit, κe, for a given insulation

can be significantly different for different surface temperature histories. Therefore,

insulations should be directly compared only for a given surface temperature history,

structural material, and structural mass.

Table 11. Insulation figure of merit values

for aerospace insulations

Temp. History cpe ke κe = √

ρeke

cpe

W

mK

J

kgK

kg2

m3s2

√

K

LI-900 ρs =144 kg

m3 (9 lbmft3 )

BP7490 949 0.03399 0.1589

ATSpA 949 0.06060 0.2832

LI-2200 ρs =352 kg

m3 (22 lbmft3 )

BP7490 949 0.06046 0.6907

ATSpA 949 0.08326 0.9512

FRCI-12 ρs =192 kg

m3 (12 lbmft3 )

BP7490 949 0.04039 0.2517

ATSpA 949 0.07212 0.4494

AETB-8 ρs =128 kg

m3 (8 lbmft3 )

BP7490 975 0.05702 0.2338

ATSpA 975 0.08728 0.3578

AETB-16 ρs =256 kg

m3 (16 lbmft3 )

BP7490 975 0.05927 0.4860

ATSpA 975 0.08975 0.7360

Saffil-3 ρs =48 kg

m3 (3 lbmft3 )

BP7490 975 0.04352 0.0669

ATSpA 975 0.07302 0.1123

Saffil-6 ρs =96 kg

m3 (6 lbmft3 )

BP7490 975 0.02420 0.0744

ATSpA 975 0.05489 0.1688

Qfiber-3 ρs =48 kg

m3 (3 lbmft3 )

BP7490 932 0.04106 0.0646

ATSpA 932 0.07328 0.1152

Qfiber-6 ρs =96 kg

m3 (6 lbmft3 )

BP7490 932 0.02158 0.0679

ATSpA 932 0.04975 0.1564

The iterative finite element analysis,

previously described, was used to calculate the required insulation mass for

each insulation and surface heating history listed in Table 11. Fig. 20 shows

the required insulation masses versus

the insulation figure of merit, κe. Each

symbol shape represents the finite element solution for each of the insulations

represented in Table 11 and the solid

line represents the approximate solution

calculated using Eq. 38. The red line

and symbols correspond to the BP7490

temperature history and the black line

and symbols correspond to the ATSpA

temperature history. For each temperature history, the finite element results

for this wide range of insulators clearly

follow the trend predicted by the approximate solution.

For both surface temperature histories the flexible insulation materials are

more mass efficient than the rigid tile

materials. For the Shuttle Orbiter surface temperature history (BP7490), the

bulk of the heating takes place at such a

low pressure that gas conduction is not

significant and radiation dominates the

heat transfer through the insulation. Therefore the density of the flexible insulation

(affects the length of gas conduction path) has little effect on the thermal performance, so that all four flexible insulations have nearly the same mass efficiency.

Conversely, the bulk of the heating for the reusable launch vehicle surface temperature history (ATSpA) occurs at a higher pressure at which gas conduction through

the insulation is significant. Therefore, the lower density flexible insulations are

more mass efficient than the higher density flexible insulations. The lower density

insulations are more efficient because at a given mass they are thicker and thus provide a longer path for gas conduction. For both surface temperature histories, lower

density ceramic tiles tend to be more mass efficient than higher density tiles because

heat transfer due to solid conduction becomes more significant with increased density. However, the denser LI-900 tile is more mass efficient than the AETB-8 tile.

39
The required insulation masses predicted by the approximate equation are within

15% of the finite element solutions.

Figure 20. Required insulation mass for several insulators

8.3 Interplay Between Mass of Insulator and Structure

An additional numerical study was performed to gain a better understanding of

the interplay between the mass of the structure and the mass of insulator required

to limit the structural temperature. LI-900 ceramic tile was used as the insulator.

Aluminum, with a maximum temperature limit of 450K (350◦F), was chosen as the

underlying structure. The structural mass per unit area was varied between 0.49

and 14.6 kg

m2 (0.1 and 3.0 lbm

f t2 ). For each of the two different surface temperature

histories (BP7490 and ATSpA) from Table 8, iterative finite element analyses were

performed for thirteen different values of structural mass per unit area to calculate

the required insulation mass.

Results of this numerical study are shown in Fig. 21. The red symbols and

lines correspond to the BP7490 heating history and the black to the ATSpA heating

history. The blue line is the mass of the structure, which was the same for both

heating histories. Required insulation masses calculated using the finite element

analyses are represented by the solid circles. Dashed lines represent the corresponding approximate insulation masses calculated using Eq. 38 with properties defined

by Eqs. 30-32. The solid squares represent the sum of the finite element solutions

for insulation mass and the structural mass. The solid lines represent the sum of

the approximate solution for insulation mass and the structural mass. The solid

triangles represent the minimum total mass calculated using Eqs. 41 and 43.

40
Figure 21. Mass of insulated structure as a function of structural mass

Comparison of the required insulation masses calculated using the finite element

analysis and using Eq. 38 (solid circles vs dashed lines in Fig. 21) shows remarkably

good agreement. Over the middle range of structural masses studied the approximate solution is within 15% of the finite element solution for both heating histories.

However, for values of structural mass below 1 kg

m2 , the approximate solution begins

to sharply diverge from the finite element solution. Also, at the high end of the

range of structural masses, the approximate solution for the BP7490 heating history is beginning to diverge from the finite element solution. This indicates that the

approximate solution may become much less accurate when the ratio of insulation

mass to structural mass becomes very high or very low. The total mass is obtained

by simply adding the structural mass to the insulation mass, so the accuracy comparison between the finite element and approximate solutions is similar to that for

the required insulation masses.

As the structural mass increases, the required insulation mass decreases, so there

is the potential for a combination of structure and insulator that produces the minimum total mass. For the finite element results, the total mass minimum occurs at a

structural mass value of about 1 kg

m2 (0.2 lbm

f t2 ) for both heating histories. Equation 41

predicts a minimum at a structural mass of 2.1 kg

m2 for the BP7490 heating history

and 3.6 kg

m2 for the ATSpA heating history. Although Eq. 41 does not accurately

predict the structural mass at which the total mass is a minimum, the values of the

minimum total mass predicted using Eq. 43 are within 10% of the finite element

calculations for both heating histories. The three to one ratio of insulator mass to

structural mass predicted to occur (Eq. 42) at the minimum mass appears to be an

41
artifact of error in the approximation.

The total mass curves in Fig. 21 are relatively flat over a significant range of

structural masses. Therefore, the mass (and thickness) of the insulator can be

reduced by increasing the structural mass, for a relatively modest total mass penalty.

Thinner vehicle walls, insulator plus structure, could improve vehicle packaging

efficiency, perhaps resulting in a net mass savings.

9 Conclusions

A simplified transient thermal problem was investigated in an attempt to gain

basic insight into the thermal response of an insulated structure. A one-dimensional

problem, consisting of a homogeneous insulator in perfect contact with an underlying, perfectly insulated structure was defined. From an initial uniform temperature,

the outer surface of the insulator is instantaneously raised to an elevated temperature, held at that temperature for finite time, and then instantaneously returned to

the initial temperature.

An analytical solution was derived for the transient response of this simplified

transient problem. Although the solution is a rather unwieldy infinite series, the

thermal response is completely governed by two nondimensional parameters with

physical significance. Numerical examples were presented using properties and a

heating duration representative of ceramic tiles on the Space Shuttle Orbiter.

The analytical series solution was used to calculate the maximum structural

temperatures over a range of the two governing parameters. A simple function of

the two governing parameters was constructed and used to approximate the maximum structural temperature over the selected range of the parameters. From this

function, two approximate equations were developed for predicting the maximum

structural temperature rise of an insulated structure.

Techniques were developed to choose a constant effective value for each of the

temperature and pressure dependent material properties of the insulator and structural materials. A technique was also developed for defining an equivalent square

heating pulse for a wide range of surface temperature histories associated with atmospheric entry. Analytical solutions for maximum structural temperature rise, using

these constant effective material properties and simplified equivalent square heating

pulses were compared to finite element solutions with variable material properties

and time-accurate surface temperature histories for a range of insulator thicknesses.

Results for the analytical series solution were typically within 10% to 20% of the

finite element solutions. The approximate analytical solution had similar accuracy

for many of the cases studied, but began to lose accuracy as one of the nondimensional governing parameters, the ratio of insulation heat capacity to structure heat

capacity, became small.

Approximate analytical equations were developed for sizing the insulator thickness and mass required to maintain the insulated structural skin of hypersonic

aerospace vehicles below a specified temperature. Manipulation of the equations

revealed that the thermal properties of the insulator and of the structure could each

be collected into a single term. These terms could be used as figures of merit to

42
indicate the effect of the choice of insulator or structural material on the required

insulator mass. An equation was also developed for the minimum total mass of an

insulated structure sized to stay below the maximum structural temperature limit,

ignoring other design considerations.

A one-dimensional, finite element, transient thermal analysis was used to perform

numerical studies for comparison with the results of approximate equations developed in this paper. The finite element analysis included the effects of temperature

and pressure dependent material properties. Time-accurate surface temperature

histories and ambient pressure histories were also incorporated into the numerical

models.

A numerical study was performed to compare the mass of insulation required

to protect four different structural materials. Two different maximum structural

temperatures were considered for each structural material. The insulation material

and the structural mass were kept the same. Calculations were performed for two

different surface temperature histories. The approximate solution predicted the

required insulation masses to within 10% of the finite element results for more than

half of the cases and to within 25% for all of the cases considered. The finite element

results exhibited a close correlation to the figure of merit, βs, derived in this paper.

A second numerical study was performed to compare the mass efficiency of nine

different aerospace insulators, five ceramic tile materials and four flexible insulations. The structural material, mass and maximum temperature limit were held

constant while the required insulation masses were calculated for the same two surface temperature histories. For the insulations considered, the required insulation

masses predicted by the approximate equation were within 15% of the finite element solutions. The finite element results were closely correlated to the insulation

efficiency figure of merit, κe, derived in this paper.

A third numerical study was performed to investigate the effect of structural

mass on the mass of required insulation and to determine the minimum mass for an

insulated structure designed with only thermal constraints. The structural material

and maximum temperature, as well as the insulation material, were held constant

and the required insulation masses were calculated for a range of structural masses

and for the same two surface temperature histories. For very small or very large

values of structural mass the approximate solution diverges from the finite element

solutions, however, away from these extremes the approximate solution was within

15% of the finite element solution. The approximate equation for minimum total

mass was within 10% of the finite element solution, however, the approximate equation predicted the minimum total mass at a much larger value of the structural mass

than the finite element solution.

The approximate equations developed in this paper were shown to predict the

results of much more complex finite element calculations with surprising accuracy

for the wide range of cases considered. Achieving these accurate results requires

carefully following the techniques developed for mapping the transient surface temperature histories to equivalent square temperature pulses and for calculating the

effective property values to use in the approximate equations. These simple approximate equations are useful for the preliminary investigation of a wide range of design

43
space to identify attractive regions for more detailed study. The figures of merit

developed in this paper can also be helpful for choosing between available insulation

and structural materials as well as providing guidance for developing more efficient

materials. The successful development of constant, effective values for the temperature and pressure dependent thermal properties of monolithic insulators raises the

intriguing possibility of developing effective properties for more complex sandwich

cores or composite insulations for use in the simple approximations.

References

1. Gogu, C.; Haftka, R. T.; Bapanapalli, S. K.; and et. al.: Dimensionality Reduction Approach for Response Surface Approximations: Application to Thermal

Design. AIAA Journal, vol. 47, no. 7, 2009, pp. 1700–1708.

2. Carslaw, H. S.; and Jaeger, J. C.: Conduction of Heat in Solids. Oxford University Press, Oxford, 1959. Second edition.

3. De Chant, L. J.: Analytical Solutions for Diffusive Finite Reservoir Problems

Using a Modified Orthogonal Expansion Method. Mathl. Comput. Modelling,

vol. 28, no. 11, 1998, pp. 73–86.

4. Williams, S. D.; and Curry, D. M.: Thermal Protection Materials: Thermophysical Property Data. NASA RP–1289, Dec. 1992.

5. Jones, E.; Oliphant, T.; Peterson, P.; and et.al: SciPy: Open Source Scientific

Tools for Python. URL: http://www.scipy.org, 2001-2012.

6. Logg, A.; and Wells, G.: DOLFIN: Automated finite element computing. ACM

Transactions on Mathematical Software, vol. 37, no. 2, 2010, p. 28.

7. Daryabeigi, K.; Cunnington, G. R.; Miller, S. D.; and Knutson, J. R.: Combined

Heat Transfer in High-Porosity High-Temperature Fibrous Insulations: Theory

and Experimental Validation. AIAA Paper 2010-4660, June 2010.

8. Gibson, D. M.; and et. al.: HYTHIRM Radiance Modeling and Image Analyses

in Support of STS-119, STS-125 and STS-128 Space Shuttle Hypersonic Reentries. AIAA Paper 2010-244, Jan. 2010.

9. Zalameda, J. N.; and et. al.: Application of a Near Infrared Imaging System

for Thermographic Imaging of the Space Shuttle during Hypersonic Re-Entry.

AIAA Paper 2010-245, Jan. 2010.

10. Myers, D. E.; Martin, C. J.; and Blosser, M. L.: Parametric Weight Comparsion

of Current and Proposed Thermal Protection System (TPS) Concepts. AIAA

Paper 99-3459, June 1999.

11. Blosser, M. L.: Advanced Metallic Thermal Protection Systems for Reusable

Launch Vehicles. Ph.D. dissertation, University of Virginia, Charlottesville, VA,

May 2000.

44
12. Daryabeigi, K.; Cunnington, G. R.; and Knutson, J. R.: Heat Transfer Modeling

for Rigid High-Temperature Fibrous Insulation. AIAA Paper 2012-3004, June

2012.

45
Appendix A

Derivation of Series Solution

In Reference [3] De Chant derives the solution to a problem that is similar to the

second portion of the current problem, t > th. De Chant’s solution is for a uniform

initial temperature distribution with the outer surface temperature of the insulator

instantaneously reduced to zero. A solution to the same problem is presented in

Reference [2], but it appears to be incorrect – as noted by De Chant and verified in

the current effort.

In Reference [3] De Chant defines the following problem (converted to the nomenclature of the current paper) to be solved.

∂T

∂t = β

∂

2T

∂

2x

(A-1)

with the boundary condition at x = 1 as

∂T(1, t)

∂t = −βγ ∂T(1, t)∂x (A-2)

the boundary condition at x = 0 as

T(0, t) = 0 (A-3)

and the initial condition as

T(x, 0) = Tio (A-4)

The dimension de appears, indirectly through β, in Eqs. A-1 and A-2 as a result

of nondimensionalizing the spatial variable x.

De Chant then separates variables to get a first order ordinary differential equation in time that can be readily solved to obtain an exponential decay term and an

eigenvalue problem in the spatial variable.

φ

00(x) + λ2φ0

(x) = 0 (A-5)

The eigenfunctions that solve Eq. A-5 are given by

φn(x) = sin (λnx) (A-6)

and the eigenvalues can be obtained by solving Eq. 15.

The boundary condition at x = 1 causes some difficulty in proceeding with the

solution. De Chant overcomes this difficulty by developing a “weighting” function.

σ(x) = 1 + 1

γ

δ(x − 1) (A-7)

where δ denotes the Dirac delta function.

46
The weighting function is then used to ensure orthogonality of the eigenfunctions.

Z 1

0

φi(x)φj (x)σ(x)dx = 0 (A-8)

The next step in the solution is to satisfy the initial condition by using an

eigenvalue expansion.

To(x)

Th

=

X∞

n=1

bnφn(x) (A-9)

However, rather than using a uniform temperature distribution for the initial

condition of the second part of the problem, the temperature distribution from

Eq. 13 at time τ = τh is used as the initial condition.

To(x)

Th

= 1 −

X∞

m=1

cm sin (λmx)e

−λ

2mτh (A-10)

Application of the orthogonality relationship, Eq. A-8, yields

bn =

R 1

0

To(x)

Th

φn(x)σ(x)dx

R 1

0

φ2

n

(x)σ(x)dx

(A-11)

Substituting Eqs. A-6 and A-7 into Eq. A-11 and applying the Dirac delta function produces

bn =

R 1

0

To(x)

Th

sin(λnx)dx +

1

γ

To(1)

Th

sin λn

R 1

0

sin2(λnx)dx +

sin2 λn

γ

(A-12)

Further substitution of Eq. A-10 into Eq. A-12 and rearranging terms gives

bn =

R 1

0

sin (λnx)dx +

sin λn

γ



−

P∞

m=1 R 1

0

cm sin (λmx) sin (λnx)dx + cm sin λm sin λn



e

−λ

2mτh

R 1

0

sin2(λnx)dx +

sin2 λn

γ

(A-13)

Integrating Eq. A-13 produces

bn =



1−cos λn

λn

+

sin λn

γ



−

P∞

m=1 

λn sin λm cos λn−λm sin λn cos λm

λ2m−λ2

n

+

sin λm sin λn

γ



cme

−λ

2mτh

1

2 −

sin (2λn)

4λn

+

sin2 λn

γ

(A-14)

The first term in the numerator represents the solution for the uniform initial

temperature problem. It can be shown to be the same as the coefficients defined in

Eq. 14. Using this observation and trigonometrical identities for the product of sine

and cosine, Eq. A-14 can be rewritten as

47
bn = cn −

P∞

m=1 

sin (λn−λm)

λn−λm

−

sin (λn+λm)

λn+λm

+

2 sin λm sin λn

γ



cme

−λ

2mτh

1 −

sin (2λn)

2λn

+

2 sin2 λn

γ

(A-15)

Inspection of Eq. A-15 reveals a potential numerical problem when m = n. The

first term in the large parentheses of Eq. A-15 approaches 1 in the limit as λm → λn.

Therefore, for n = m the quantity in the large parentheses becomes equivalent to

the denominator and cancels out. Eq. A-15 can thus be rewritten as

bn = cn(1 − e

−λ

2

nτh ) −

X∞

m=1

m6=n





sin (λm−λn)

(λm−λn) −

sin (λm+λn)

(λm+λn) +

2 sin λm sin λn

γ

1 −

sin (2λn)

2λn

+

2 sin2 λn

γ



 cme

(−λ

2mτh)

(A-16)

which is the same as Eq. 17. A simple offset in the time variable completes the

solution shown in Eq. 16.

48
Appendix B

Material Properties

Table B1. Properties for aluminum 2024 (Ref. [4])

Temperature cp k

◦R(K) Btu/lbm/◦R(J/kg/K) Btu/f t/hr/◦R(W/m/K)

160 (89) 0.177 (741) 61.2 (106)

260 (144) 0.147 (615) 69.6 (120)

460 (256) 0.195 (816) 84.0 (145)

660 (367) 0.216 (904) 95.0 (164)

760 (422) 0.224 (937) 99.0 (171)

860 (478) 0.233 (975) 102.5 (177)

1060 (589) 0.250 (1046) 104.5 (181)

Table B2. Specific heat capacity for LI-900 (Ref. [4])

Temperature cp

◦R(K) Btu/lbm/◦R(J/kg/K)

460 (256) 0.150 (628)

710 (394) 0.210 (879)

960 (533) 0.252 (1054)

1210 (672 0.275 (1151)

1460 (811) 0.288 (1205)

1710 (950) 0.296 (1238)

1960 (1089) 0.300 (1255)

2210 (1228) 0.303 (1268)

2460 (1367) 0.303 (1268)

3460 (1922) 0.303 (1268)

49
Table B3. Thermal conductivity for LI-900 (Ref. [4])

T, ◦R

k, Btu/(ft hr ◦R)

Pressure, atm

0 0.0001 0.001 0.01 0.1 1.0

460 0.0075 0.0075 0.0100 0.0183 0.0250 0.0275

710 0.0092 0.0092 0.0125 0.0225 0.0316 0.0341

960 0.0125 0.0125 0.0167 0.0276 0.0400 0.0433

1210 0.0175 0.0175 0.0216 0.0325 0.0492 0.0534

1460 0.0233 0.0233 0.0275 0.0392 0.0600 0.0658

1710 0.0308 0.0308 0.0350 0.0492 0.0725 0.0782

1960 0.0416 0.0416 0.0459 0.0617 0.0875 0.0942

2210 0.0567 0.0567 0.0610 0.0767 0.1060 0.1130

2460 0.0734 0.0734 0.0782 0.0942 0.1270 0.1360

2760 0.0966 0.0966 0.1020 0.1160 0.1550 0.1670

2960 0.1660 0.1660 0.1230 0.1390 0.1790 0.1940

3260 0.1540 0.1540 0.1620 0.1800 0.2220 0.2420

3460 0.1900 0.1900 0.1960 0.2190 0.2620 0.2900

50
REPORT DOCUMENTATION PAGE Form Approved

OMB No. 0704-0188

2. REPORT TYPE 

Technical Publication

 4. TITLE AND SUBTITLE

Analysis and Sizing for Transient Thermal Heating of Insulated Aerospace 

Vehicle Structures

5a. CONTRACT NUMBER

 6. AUTHOR(S)

Blosser, Max L.

 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-2199

 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

 8. PERFORMING ORGANIZATION

 REPORT NUMBER

L-20165

10. SPONSOR/MONITOR'S ACRONYM(S)

NASA

13. SUPPLEMENTARY NOTES

12. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified - Unlimited

Subject Category 34

Availability: NASA CASI (443) 757-5802

19a. NAME OF RESPONSIBLE PERSON

STI Help Desk (email: help@sti.nasa.gov)

14. ABSTRACT

An analytical solution was derived for the transient response of an insulated structure subjected to a simplified heat pulse. The 

solution is solely a function of two nondimensional parameters. Simpler functions of these two parameters were developed to 

approximate the maximum structural temperature over a wide range of parameter values. Techniques were developed to 

choose constant, effective thermal properties to represent the relevant temperature and pressure-dependent properties for the 

insulator and structure. A technique was also developed to map a time-varying surface temperature history to an equivalent 

square heat pulse. Equations were also developed for the minimum mass required to maintain the inner, unheated surface 

below a specified temperature. In the course of the derivation, two figures of merit were identified. Required insulation masses 

calculated using the approximateequation were shown to typically agree with finite element results within 10%-20% over the 

relevant range of parameters studied.

15. SUBJECT TERMS

Insulated structure; Insulation; Thermal analysis; Thermal protection system

18. NUMBER

 OF 

 PAGES

55

19b. TELEPHONE NUMBER (Include area code)

(443) 757-5802

a. REPORT

U

c. THIS PAGE

U

b. ABSTRACT

U

17. LIMITATION OF 

 ABSTRACT

UU

Prescribed by ANSI Std. Z39.18

Standard Form 298 (Rev. 8-98)

3. DATES COVERED (From - To)

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

736466.01.08.07.09.02 

11. SPONSOR/MONITOR'S REPORT

 NUMBER(S)

NASA/TP-2012-217595

16. SECURITY CLASSIFICATION OF:

The public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, 

gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this 

collection of information, including suggestions for reducing this burden, to Department of Defense, Washington Headquarters Services, Directorate for Information Operations and 

Reports (0704-0188), 1215 Jefferson Davis Highway, Suite 1204, Arlington, VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person 

shall be subject to any penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number.

PLEASE DO NOT RETURN YOUR FORM TO THE ABOVE ADDRESS.

1. REPORT DATE (DD-MM-YYYY)

01- 08 - 2012