Closed-form solutions of the one-dimensional heat-conduction equations for the flow of heat into a plate with a laminar boundary layer have been obtained for a configuration entering a planetary atmosphere with constant velocity and negative entry angle. The atmospheric density was assumed to obey an exponential law and the temperature was assumed constant initially. The solution is in the form of a Fourier series expansion which, for most practical applications, can be approximated by retaining only one term of the expression. The solution applies to the initial part of the entry before the maximum heating conditions are encountered

McLellan, Charles. H.

Wells, William R.

English

NASA Technical Reports Server

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NASA TN D-1476
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TECHNICAL NOTE
D-1476
ONE-DIMENSIONAL HEAT CONDUCTION THROUGH THE SKIN OF
A VEHICLE UPON ENTERING A PLANETARY ATMOSPHERE AT
CONSTANT VELOCITY AND ENTRY ANGLE
By William R. Wells and Charles H. McLellan
Langley Research Center
Langley Station, Hampton, Va.
C4
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
WASHINGTON October 1962
https://ntrs.nasa.gov/search.jsp?R=19620006467 2020-03-11T13:27:36+00:00Z

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
TECHNICAL NOTE D-1476
ONE-DIMENSIONALHEAT CONDUCTION THROUGH THE SKIN OF
A VEHICLE UPON ENTERING A PLANETARY ATMOSPHERE AT
CONSTANT VELOCITY AND ENTRY ANGLE
By William R. Wells and Charles H. McLellan
SUMMARY
Closed-form solutions of the one-dimensional heat-conduction equa-
tions for the flow of heat into a plate with a laminar boundary layer
have been obtained for a configuration entering a planetary atmosphere
with constant velocity and negative entry angle. The atmospheric density
was assumed to obey an exponential law and the temperature was assumed
constant initially. The solution is in the form of a Fourier series
expansion which, for most practical applications, can be approximated by
retaining only one term of the expression. The solution applies to the
initial part of the entry before the maximum heating conditions are
encountered.
INTRODUCTION
One of the serious problems experienced during entry of a vehicle
into a planetary atmosphere is high convective aerodynamic heat-transfer
rates. In some instances when the heat is absorbed in the skin, large
thermal gradients can develop through the skin. Normally the heat con-
duction through a material with a time-dependent heat rate, such as that
encountered during entry of a vehicle into the atmosphere, is difficult
to analyze and can best be handled by step-by-step numerical processes
which are generally time consuming or require the use of automatic com-
puters. (See refs. i and 2.) A few simple conduction analyses have
been performed for special types of entry in which the boundary condi-
tions make it possible to obtain closed analytical expressions, such as
those given in reference 3- The temperature time history of the skin
of a vehicle entering an exponential atmosphere has been analyzed in
reference 4; in this analysis a constant entry angle and laminar flow
are assumed. The obtained solution for the skin of arbitrary thickness,
however, contains terms which generally have to be treated by numerical
integrations. The additional assumption of a constant velocity during
the initial entry phase has been assumed in the present analysis. From
these assumptions a simple but useful closed-form algebraic expression
2has been obtained for the temperature time history of the initial phase
of the entry which generally does not include the phase in which the
maximumheat rate occurs (about elght-tenths of the entry velocity
according to ref. 5). This analysis will however be applicable to a
significant part of manyentry trajectories.
SYMBOLS
= _kl=__=__Ce__hnt V_. _m OF/ft
B
C dimensional constant, 17,600 Btu/ft3/2-sec
Cm _ 1 ft2-secl/2-°F
_Km_mC m Btu
cm specific heat of material_ Btu/lb-°F
g local value of gravitational acceleration, ft/sec 2
h height of entry above surface of planet, ft
i imaginary unit,
j integer
Km thermal conductivity, Btu/sec-ft-°F
kI ratio of local heat flux to stagnation-point heat flux
m,n dimensionless constants, 3.15 and 0.50, respectively
Nm -
q
r
cosh_(T - x)
sinh_T
heat flux, Btu/ft2-sec
distance from planet center, ft
Rs
T
L/D
¥
V
w/c 
radius of curvature at a stagnation point, ft
variable in Laplace transform
temperature, OF
lift-drag ratio
transformed temperature function
ratio of velocity component normal to radius vector to circular
orbital velocity
resultant velocity, ft/sec
ballistic parameter, lb/ft 2
W = kl-_Ce-_hn I V 1TM Btu
# l_I ft- sec3/2
x distance normal to surface, ft
y altitude, ft
Y = _-(_nV sin y)t secl/2
-_nV sin 7
z complex variable
am thermal diffusivity, ft2/sec
atmospheric density decay parameter, --_ ft -I
23500
t time, see
p atmospheric density, slug/ft 3
Pm density of material, lb/ft3
T thickness of plate, ft
7 flight-path angle relative to local horizontal direction,
positive for climbing flight and negative for descent
= -_nV sin 7 sec-i
Subscripts:
o reference value
free stream (ambient atmosphere)
i initial condition
s stagnation point
ANALYSIS
Assumptions and Definitions
For the present analysis, the following assumptions are made:
(i) The atmospherevaries exponentially. (See ref. 5.)
(2) The flow is laminar and the heat rate of any station on the
body is proportional to the value at a stagnation point. (See refs. 5
and 6.)
(3) The velocity is sufficiently high so that the heating rate can
be considered independent of the surface temperature.
(4) The skin (being analyzed) has a zero heat rate on the back
(unexposed) side.
(5) Specific heats for the material are constant.
(6) The vehicle descends through the atmosphere at a constant
velocity and flight-path (or entry) angle.
(7) The flow is one dimensional with radiation neglected.
Assumptions (i) to (3) allowed Chapman (ref. 5) to write the fol-
lowing equations for p_ and q:
= poe- y (1)
n m
with
_=V cos y
Assumption (6) allows y to be written as
y = Vt sin 7 + h (3)
The constants kl, C, m, n, and _ are given in references 5 and 7.
Substitution of equations (i) and (3) into equation (2) gives
klC e-!3hnF V Ime-_nVt sin y
-
For convenience the following definitions are made:
B -- 5)
m = -_nV sin 7 6)
Equation (4) therefore can be expressed as
q = _BKm emt (7)
The applicability of these assumptions is shown in figures i to
for several entry conditions. Numerical integration of the equations of
motion has been used for the calculation of the time history of velocity,
entry angle 3 and altitude. These values in turn have been used with equa-
tion (2) and the values of the 1959 ARI_ atmosphere (ref. 8) in determining
the heating rate qs shown as solid-line curves in figures i to _. The
main test for the validity of the assumptions is whether the velocity and
entry angles are constant and how the deviations affect the heating rate.
Values of qs obtained from equation (7) (dash-line curves) are therefore
included in the figures. For initial values of velocity of 20,000 ft/sec,
entry angle of -20 ° at 400,000 feet, and W/CDA of 120 ib/ft 2, it can be
seen from figure i that the assumptions are good for the first 40 seconds
which include altitudes down to 120,000 feet. Increasing the value of
W/CDA to 1,000 ib/ft 2 in figure 2 increases the time to about 42 seconds
and brings the altitude down to 90,000 feet.
6In order to illustrate a condition in which these assumptions can-
not be expected to apply_ the same calculations have been made for a very
shallow entry angle. It can be seen in figure 3 that small changes in
the entry angle can represent large percentage changes and consequently
appreciable changes in the altitude and heating rate. This discrepancy
results from assumption 6 which assumes that the flight-path angle remains
constant throughout the analysis. The usefulness of the analysis can be
extended to these shallow entry angles by using a mean value of the angle
for the part of the entry that is being considered. To illustrate this
approach, curves have been included in figure 3 for 7 = -1.92o. The
heat rate as. computed from equation (7) is shown as dash-line curves.
An improvement in the agreement with the solid-line curve can be obtained
by changing the mean value of 7 to -2.10 ° which gives from equation (3)
an altitude which agrees with the corresponding solid-line curves for
altitude at t = 200 seconds. This mean value of 7 when substituted
into equation (7) gives the improved curve appropriately marked in
figure 3.
Transformation and Solution of Basic Equations
The transient temperatures of the surface are governed by the
Fourier equations:
a2T
C_n _x 2 - _t
and
(8)
aT
q = -Km_ (9)
From the previous assumptions the boundary conditions for the Fourier
heat-conduction equation are
_(x_t)
8x x=O
_(x't)1 = o
_X X= T
T(x,O) = T i
°
 :cr0 lo:
: j\:o/k::s
(lO)
These conditions are illustrated by the following sketch which shows T
as a function of x:
_)x ---q
T
=0
0 'I"
X
Equations (8) and (9) with boundary conditions (i0) specify the
solution to the heat-flow problem. This boundary valued problem can be
solved efficiently by the use of operational mathematics, the method of
the Laplace transform in particular. (See ref. 9-) If the Laplace
transform of T(x,t) with respect to t is denoted by
_0 °°
_(X, s) = e-StT(x, t)dt (ll)
then equation (8) and the transformed boundary conditions become,
respectively_
sT = -T i
am dx2
(12)
dY)x _ B
=0 S - 60
and
d¥)x = o (14)
8The solution to the total linear differential equation (eq. (12))
is simply
- Ti
¥(x,s): cle + c2e +- (15)s
The constants CI and C2
to be
are determined by conditions (13) and (14)
-e _(_n _-_
C1 -
2 sinh T]
B
s - LD
(16)
C2 : s - _ (17)
2 sinh(_ T)
Then,
A series solution, which converges quite rapidly for large values
of t, for the inverse transform of equation (18) can be found from the
inversion integral from the theory of complex variables. From this the
temperature is given as
_¢+i_T(x,t)- i lim eZt_(x,z)dz (19)
2_i @__ _ _-i@
where both _ and W are any real positive numbers. Equation (19)
can be evaluated by simply taking the sum of the residues of the expres-
sion
eZt< r_ _z ___
j2_2
The poles of this expression are at z = 0, to_ and -am T2
j = l, 2_ .... The residues of these poles are, then,
BamTi +_
tOT
xI
and
-_mJ 2_ 2t
e COS
T
.2 2
_mJ _
to +
T2
(j = i, 2, 3,
where
.)
Therefore 3
T(x,t) = T i + _ -
COT
B_--_t c°sh_( T- xI
-_j2_2t
2Bol_. _ e T2 cos(_)+ _ (20)
T c%nj2_ 2
j=l co +
T2
In general, most entry conditions are such that the last term
(series on the end) of equation (20) and the term B_m contribute very
_T
little to the temperature of the skin for values of t of i second or
more and can be neglected. This fact can be verified by simply evaluating
a few of the terms for any particular entry condition. The resulting
expression for the temperature is
i0
T(x,t) : Ti (21)
Working Charts
The expression for the temperature rise T - Ti from equation (21)
is
T- Ti-=_: (22)
This expression can be put into a fo_n which is convenient for applica-
tion by making the following definitions:
AT = T - Ti - WYNmC m (23)
where
y _ _
Nm _
e_ t e-(_nV sin 7)t
V-_nV sin 7
cosmical_xI
(24)
(25)
Cm = (26)
and
klC
W = -BK m - (27)
ii
In equation (23), only Y and Nm depend on the position and time
variables of the problem; W and Cm are entirely dependent on the
entry conditions V and 7 and on the material under consideration.
If plots are made of Y as a function of _ for various values of t
and of Nm as a function of TI_
x
for various values of
--, rapid pre-
T
diction of ZXT could be performed by use of these plots and a knowledge
of the entry conditions and the properties of the material. Such charts
are shown in figures 4 and 5-
Several significant facts can be obtained from figure 4:
(i) At values of T_ less than about 0.3_ the gradients are very
small through the skin (or plate) and the skin can be considered as a
thin plate.
(2) Above a value of T_ of about 2, the effect of thickness of
the plate on the surface temperature is negligible.
(3) The back-side temperature decreases very rapidly as T_
increases above i; at T_ = 3 the back-side temperature rise is only
i0 percent of the heated-side temperature rise and decreases to only
0.i percent at _ = 7.6.
v_m
CONCLUDING REMARKS
Closed-form expressions as approximate solutions of the one-
dimensional heat-conduction equations have been derived for a configura-
tion initially at constant temperatures entering a planetary atmosphere
with constant velocity and entry angle. This solution is for the part
of the entry before the maximum heat-transfer rates are reached. The
atmospheric density is assumed to be exponential and the boundary layer
to be laminar at entry. A series expression exists for the temperature
history. It is noted_ however_ that in general most entry conditions
are such that the only terms in the expression that contribute are the
initial temperature and the term involving the hyperbolic functions.
For this type of entry the expression was written in such form that
12
convenient working charts could be presented. An example of the use of
these charts is presented also.
Langley Research Center,
National Aeronautics and SpaceAdministration,
Langley Station, Hampton,Va., July ii_ 1962.
13
APPENDIX
ILLUSTRATION OF USE OF WORKING CHARTS
In order to illustrate the application of the expressions developed
in this report, a typical entry heating condition is considered. It is
assumed that the temperature distribution through copper skins of dif-
ferent thicknesses is required for a vehicle entering the atmosphere at
an altitude of 400,000 feet, a velocity of 20,000 feet per second, and
an entry angle of -20 ° . The skin is assumed to be electrolytic copper
which has a thermal conductivity of 0.0548 Btu/ft-sec-°F at 1,000 ° F.
The radius of the blunt nose is assumed to be i foot. The constants of
the problem are as follows:
V = 20,000 ft/sec
h = 400_000 ft
= _20 °
R =Ift
Km = 0.0548 Btu/ft-sec-°F
Dm = 521 lb/ft 3
Po = 0"00238 slug/ft3
C = 17,600 (from ref. 5)
m = 3.15 (from ref. 5)
n = 1/2 (from ref. 5)
= 23,500 ft (from ref. 5)
cm = 0.104 Btu/ib-°F
Cm = 0.580 ft2-secl/2-°F
Btu
am = 0.001011 ft2/sec
14
g = 32.2 ft/sec 2
r = 21,300,000 ft
This example treats the heating of two stations on the body: sta-
tion 1 which is at the stagnation point and station 2 which is at a point
that has a heat flux of i/i0 that at the stagnation point.
Substitution of the values of kl, C, Km, R, _, h,
g, and r into equations (5) and (27) and the values of _,
and y into equation (6) gives the following values for B,
n, m, V_
n_ V,
_, and W:
B : -28.2°F/ft
= 0.1455 see -I
W = 1.546 Btu/ft-sec 3/2
From equation (23) AT is given as
AT = WYNmC m = 0.897YN m
Then, depending on the particular time and station, ZkT can be deter-
mined once Nm and Y are obtained from figures 4 and 5, respectively.
Figures 6 to 8 show the results of these calculations. Figures 6
and 7 show the temperature-rise variation with time for specific values
of X/T for the two body stations under consideration and two plate
thicknesses. These curves are valid for times up to about 40 and
43 seconds for vehicles having values of W/CDA of 120 and 1, O00 lb/ft 2,
respectively, as seen from figures 1 and 2. Figure 8 gives the varia-
tion of the temperature ratio ZkT with x/T for several plate
thicknesses. It is interesting to note that these curves do not depend
on time.
15
REFERENCES
i. McAdams, William H.: Heat Transmission. Second ed., McGraw-Hill
Book Co.. Inc., 1942.
2. Hill, P. R.: A Method of Computing the Transient Temperature of
Thick Walls From Arbitrary Variation of Adiabatic-Wall Temperature
and Heat-Transfer Coefficient. NACA Rep. 1372, 1958. (Supersedes
NACA T_ 4105.)
3- Bergles, Arthur E., and Kaye, Joseph: Solutions to the Heat-
Conduction Equation With Time-Dependent Boundary Conditions.
Aerospace Sci. (Readers' Forum), vol. 28, no. 3, Mar. 1961,
pp. 251-252.
J our.
4. Sutton, George W.: The Temperature History in a Thick Skin Sub-
jected to Laminar Heating During Entry Into the Atmosphere. Jet
Propulsion, vol. 28, no. i, Jan. 1958, pp. 40-4D.
5- Chapman, Dean R.: An Approximate Analytical Method for Studying
Entry Into Planetary Atmospheres. NASA TR R-If, 1959. (Supersedes
NACA TN 4276.)
6. Lees, Lester: Laminar Heat Transfer Over Blunt-Nosed Bodies at
Hypersonic Flight Speeds. Jet Propulsion, vol. 26, no. 4, Apr.
1956, pp. 259-269, 274.
7. Detra, R. W., Kemp, N. H., and Riddeli, F. R.: Addendum to 'Heat
Transfer to Satellite Vehicles Re-Entering the Atmosphere.' Jet
Propulsion, vol. 27, no. 12, Dec. 1957, PP. 1256-1257.
8. Minzner, R. A., Champion, K. S. W., and Pond_ H. L.: The ARDC Model
Atmosphere, 1959. Air Force Surveys in Geophysics No. 115
(AFCRC-TR-59-267), Air Force Cambridge Res. Center, Aug. 1959.
9. Churchill, Ruel V.: Operational Mathematics. Second ed., McGraw-Hill
Book Co., Inc., 1958.
16
v I L
V = 20,no0 ft/sec7 = _ 20o
h = 400,000 ftL/D= o
k W/CDA : 120 ib/ft 2
o . f
_.-._..__ _
50O O
qs _ -BKme _t
24,000
-- 16,000
Y
-20 _- ----1
_00 _ _N_ 400,000
-_0 X
g 3_0 _ X X 30o,ooo
m /
. \
200 \ 200,000
loo / k I loo,ooo
0 _J 0
0 20 &O 60 80 i00 120 iAO
t_ see
V, ft/sec
-- 8,000
y,ft
Figure I.- Time history of velocity, entry angle, stagnation heat-transfer
rate_ and altitude for a high initial flight-path angle and a low
ballistic parameter.
17
0 _
V = 20,000 ft/sec
7 = - 20 °
h = 400 000 ft
L/D=O
w/c_ = ioooib/ft2
V
-2O
1,600 _ [
qs = -BKm e_t _'-_
-40 \ I
_ _,2oo_ _ ,
-60 "
° 800 \ //
//
40 _/y
400 F'x//
0 I0 20 30 40 50
24,000
t_ sec
-I00
16,000
V, ft/sec
_._ 8,000
--....
400,000
300,000
y, ft
200,000
i00,000
60 70
Figure 2.- Time history of velocity, entry angle, stagnation heat-transfer
rate, and altitude for a high initial flight-path angle and high bal-
listic parameter.
18
320
240
vqI
¢q
I
V = 25,380 ft./sec
_' =-1.92 °
h = 400,000 ft
L/_) = 0
0
W/CDA = 120 ib/ft 2
k
-2.0 _ _ 8,0O0
\
-4.0 0
\<
\
32,O00
m 24,000
V, ft/sec
-- 16,000
\
qs = -BKm ec''t - \
-8.o _ ,_.. _._ % %
_-8o "'_--- _, \
t / _ 3OO ,000
S 7=-1.920 "_"
80 - 12.0 _). _ 100,000
o _ _"_" _
0 50 lo0 150 200 250 300 350 400 450
t_ se¢
Figure 3-- Time history of velocity, entry angle, stagnation heat-transfer
rate, and altitude for a low initial flight-path angle and low ballis-
tic parameter.
19
10.0
8.0
6.0
4.0
3,0
.2 .3 .4 .6 .8 1.O 2.0 3.0 4.0 6.0 8.0 10.0
Figure 4.- Values of Nm for different values of x/T and the
parameter "r_.
2O
i0.0
1.0
.i
.01
.001
•0001 0 4 6 8
T
Figure 4.- Concluded.
10 12
21
100,000
i0,000
1,000
Y, sec I/2
i00
i0
Figure 5.- Variation of Y with _ for various values of t.
22
2,800
2,400
h = 400,000 ft
V = 20,000 ft/sec
7 = -20 °
R =l.O ft
Mat er lal _Copper
Km = 0-0548 Btu/ft-sec -°F
_m 52Z lb/ft 3
AT,
2,000
1,600
1,200
800
x- 0
.4
i.o
q=qs ,I
'I//,
,/
,)'
I
i
/'/
S °/ _-_°_qs-- S
0 8 16 24 32 40 /+8
t, sec
Figure 6.- Temperature-rise variation with time for a plate thickness
T of i inch.
23
i, 400
1,200
I,O00
8OO
AT, OF
600
40O
2OO
i
h = 400,000 ft
V = 20,000 ft/sec
= _20 °
R = 1.0 ft
Materialj Copper
Km = 0.0548 Btu/ft-see-°F
_m 52i ib/ft_
x= 0
.4
1.0
/
q 0"lqs_ /
q=qs
/ /\
/ \'%"
0 -- --
0 8 16 24 32 40 18
t, sec
Figure 7.- Temperature-rise variation with time for a plate thickness
of 4 inches.
24
h = 400,000 ft
V = 20,000 ft /sec
7 = -20 °
R= l.Oft
Material, copper
Km = O. 0548 Btu/ft- sec-°F
Pm = 521 ib /ft 3
AT
(AT)x=0
1.0 ____
\
.6
.4
.2
\
\
_=0
i/2',
I"
Z,"
0
0 .2 .4 .6 .8 1.0
x/_
Figure 8.- Variation of ratio of temperature rise at any point in plate
to that at the surface of plate with x/T.
NASA-Langley, 1962 L- 2029




One-Dimensional Heat Conduction Through the Skin of a Vehicle upon Entering a Planetary Atmosphere at Constant Velocity and Entry Angle

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