This document discusses the simulation of a missile's surface temperature over time. It provides governing equations that model the temperature change based on convection from the surrounding air and thermal radiation. Parameters like the missile's velocity, ambient air properties at different altitudes, and the missile's material properties are defined. The equations are solved numerically over a 10 second period to generate a temperature history profile for the missile's skin.

1. MathCAD - Missile Skin
Temperature.xmcd
Missile Skin Temperature History
by Julio C. Banks, MSME, P.E.
Reference
1. B. E. Gatewood, Ph.D., "Thermal Stresses with Applications to Airplanes, Missiles, Turbines, and
Nuclear Reactors". McGraw-Hill Publications. Pages 29 through 42.
2. Robert D. Blevins, "Applied Fluid Dynamics Handbook". ISBN 0-89464-717-2. Pages 385, 390.
1.0 Governing equations, and parameters (see the appendix for the units used in this paper)
C
d Ts(t)
d
 qC Ts t ( ) t    qR Ts t ( )    = Ts 0 ( ) Tsi
t
= Zi  z  Zf
Air Properties Altitude, kft: Z  ( 50 80 )T where kft  103ft
Density: ρz ( 203.3 48.34 )T 10 6   slug  32.174 lb
Heat Transfer Coefficient: hz  ( 39.10 12.39 )T
Missile velocity: V  3
Missile height at a given time: z t ( ) zi
 Z0
zi  Vt

Air density at a height given by time: ρ(t)  linterpZρzz(t)
Heat transfer coefficient at a height given by time: h(t)  linterpZhzz(t)
Check: τ  5.0 z(τ)  65 ρ(τ) 125.8 10 6   h(τ)  25.7
τ  10.0 z(τ)  80 ρ(τ) 48.34 10  6   h(τ)  12.4
Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 1 of 14

2. MathCAD - Missile Skin
Temperature.xmcd
Gravitational conversion, lbf
lb
ft
sec2

:
gc  32.2
Surface properties
Surface emissivity (dimensionless): εs  0.5
Metal weight density, lbf
ft3
: γs  484 ρs
γs
gc

Metal specific heat, BTU
lbf F
:
Cps  0.16
Wall thickness, ft : δ 8.3 10 3  
Thermal capacitance of the wall, BTU
ft2F
:
C  γsCpsδ3600 ( 3600sec = 1hr )
Ambient temperature* at altitude, R : Tas  1050 50kft  z(t)  80kft
Radiation sink effect of outer space, BTU
hrft2
:
G  20
Stefan-Boltzmann constant, BTU
hrft2R4
:
σ 1.714 10 9   
Convection, qCTst  h(t)Tas  Ts BTU
hrft2
:
Thermal Radiation, BTU/hr, BTU
hrft2
:
4  G  
qRTs εs σ Ts

 
* This is the recovery temperature of the air being adiabatically accelerated to the speed of the missile's
surface.
Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 2 of 14

3. MathCAD - Missile Skin
Temperature.xmcd
2.0 Spatial response parameters
Initial condition Tsi  700
Event duration T  10
Time step Δt  1
Number of time steps: Nt round
T
Δt


  1  11
3.0 Solve the governing set of equations
Note: ORIGIN  0 due to current MathCAD 2001i specifications. Future release
will not have this limitation. This constrain only applies to odesolve-function.
Given
C
d Ts(t)
d
 qC Ts t ( ) t    qR Ts t ( )    = Ts 0 ( ) Tsi
t
=
Φ  OdesolvetTNt
time  0  10
0 1 2 3 4 5 6 7 8 9 10
740
735
730
725
720
715
710
705
700
Missile Surface-Temperature, Ts, History
Φ(time)
time
Φ(time)
700.0
705.6
710.7
715.3
719.5
723.2
726.6
729.5
732.0
734.2
735.9

Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 3 of 14

4. MathCAD - Missile Skin
Temperature.xmcd
APPENDIX
This appendix describes the units being utilized in this simulation of the surface temperature history of a
missile. It should be noted that the missile altitude range corresponds to the tropopause (11 km to 20
km). Apropos, the tropopause is the top layer of the lower atmosphere, and it acts as a ceiling on
weather systems and pollutants because its constant temperature discourages vertical convection [2]
F  R
C
d Ts(t)
d
 qC Ts t ( ) t    qR Ts t ( )    = Ts 0 ( ) Tsi
T
= Zi  z  Zf
Air Properties Altitude: Z  ( 50 80 )Tkft where kft  103ft
Density: slug  32.174 lb ρz ( 203.3 48.34 )T 10 6  slug
ft3
 
Heat Transfer Coefficient: hz ( 39.10 12.39 )T BTU
hrft2F
 
Missile velocity: V 3
kft
sec
 
Missile height at a given time: z t ( ) zi
 Z0
zi  Vt

Air density at a height given by time: ρ(t)  linterpZρzz(t)
Heat transfer coefficient at a height given by time: h(t)  linterpZhzz(t)
Check: t  5.0sec z(t)  65kft ρ(t) 1.258 10 4  slug
  h(t) 25.7
ft3
BTU
hrft2F
 
t  10sec z(t)  80kft ρ(t) 4.834 10 5  slug
  h(t) 12.4
ft3
BTU
hrft2F
 
Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 4 of 14

5. MathCAD - Missile Skin
Temperature.xmcd
Gravitational conversion: gc
lbf
lb
 g
Surface properties
Metal weight density: γs 484
lbf
ft3
  ρs
γs
gc

Metal specific heat: Cps 0.16
BTU
lbf F
 
Wall thickness: δ 8.3 10 3   ft
Surface emissivity (dimensionless): εs  0.5
C  γsCpsδ
Time constant at t  0sec : τc
C
h(t)
 (for units, and magnitude checking purpose only)
Ambient temperature* at altitude: Tas  1050R 50kft  z(t)  80kft
Radiation sink effect of outer space: G 20
BTU
hrft2
 
Stefan-Boltzmann constant: σ 1.714 10 9  BTU
hrft2R4
 
Convection**: qCTst  h(t)Tas  Ts
4  G  
Thermal Radiation**: qRTs εs σ Ts

 
Check units: C 0.643
BTU
ft2F
  τc  59.2sec (units are OK)
* This is the recovery temperature of the air being adiabatic ally accelerated to the speed of the missile's
surface.
** Same units as the G-parameter (the stellar, solar, or interstellar radiation)
Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 5 of 14

6. MathCAD - Missile Skin
Temperature.xmcd
Characterize the Air Properties of the Tropopause
by Julio C. Banks, MSME, P.E.
The tropopause is the portion of the Earth's atmosphere extending from 11 km to 20 km. The tropopause
is the top of the lower atmosphere (the tropopause--sea level to 11 km), and it acts as a ceiling on
weather systems and pollutants because its constant temperature discourages vertical convection [1].
This paper will characterized the ideal (U.S. Standard) portion of the atmosphere referred to as the
tropopause.
Reference
1. Robert D. Blevins, "Applied Fluid Dynamics Handbook". ISBN 0-89464-717-2. Pages 385, 390.
Define:
Constant Parameters: γHg 0.4912
lbf
in3
  RU 53.35
ftlbf
Rlb
 
Units: kft  103ft psf
lbf
ft2
 F  R
U.S. Standard Atmosphere, 1962, in U.S. Customary Units.
Altitude Temperature Pressure Kinematic Viscosity
z
50
52
54
56
60
65
70
75
80


 kft TF
69.700
69.700
69.700
69.700
69.700
69.700
67.977
64.699
61.977


  F hHg
3.44440
3.13019
2.84468
2.58527
2.13537
1.68162
1.32521
1.04615
0.827295
 
 in ν
8.1587 10 4 
8.977 10 4 
9.8787 10 4 
1.0870 10 3 
1.3160 10 3 
1.6711 10 3 
2.1434 10 3 
2.7498 10 3 
3.5213 10 3 



ft2
sec
 
Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 6 of 14

7. MathCAD - Missile Skin
Temperature.xmcd
Process the atmosphere properties for heat transfer calculations
Pressure: P  γHghHg
Temperature: TR  TF  459.7R
Density: ρ
1
RU

P
TR


 

Dynamic Viscosity: μ 
(ρν)
TR
390.0
390.0
390.0
390.0
390.0
390.0
391.7
395.0
397.7


 R P
243.6
221.4
201.2
182.9
151.0
118.9
93.7
74.0
58.5


 psf ρ
11.709
10.641
9.671
8.789
7.259
5.717
4.485
3.511
2.758


10 3 lb
   μ
ft3
9.553
9.553
9.553
9.553
9.553
9.553
9.614
9.656
9.711


10 6 lb
ft sec
  
Notice that
ρ
363.94
330.74
300.57
273.16
225.63
177.68
139.41
109.14
85.72


10 6 slug
   and μ
ft3
2.969
2.969
2.969
2.969
2.969
2.969
2.988
3.001
3.018


10 7 slug
   where slug  32.174 lb
ft sec
Also note that Tavg  meanTR μavg  mean(μ)
Tavg  391.6R μavg 9.589 10 6  lb
ft sec
 
Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 7 of 14

8. MathCAD - Missile Skin
Temperature.xmcd
Transport Properties of Air per NBS Report No. 564
by Julio C. Banks, P.E.
Specific Heat
φ = ln(T)
Cp(φ)
a0
1
T
  a  1.0971462 0.46376559 0.066270566 3.1803792  10 3 3
i
ai  i φ




b0
1
3
i
bi  i φ






kJ
kgK

T
b 
 1 417.37692  10 3 58.990768  10 3 2.8067639  10 3 T  400K φ ln
T
K


 Cp(φ) 1.014
kJ
kgK
 
Dynamic Viscosity
μ T ( ) c0
 c1T
c2
T
 c3
ln(T)
T
  c4 e T  


10 5  kg
msec
 
c 4.6288478 0.0018081162 99.266534 126.70816 6.91712511041 T

Thermal Conductivity k(T)
α0
1
2
i
αi Ti  



β0
1
2
i
βi Ti   



 watt
mK


  μ
T
K


2.287 10 5  kg
msec
 
α 30.153311 10 6   100.83792 10 6  22.309819 10 9   T
 k
T
K


3.330 10 2  watt
mK
 
T
β 
 1 339.56524  10  6 192.5766  10 9 Prandtl Number Pr(T)
Cp(ln(T))μ(T)
 Pr
k(T)
T
K


 0.696
Specific Heat Ratio γ(T)
1
 γ
1
Rair
Cp(ln(T))

T
K


 1.395
Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 8 of 14

9. MathCAD - Missile Skin
Temperature.xmcd
Units & constants: kJ  103joule Rair 53.35
ftlbf
Rlb
 
Numerical evaluation of the transport properties of air
SI Units
T  400K ( T  720R ) φ ln
T
K



Cp(φ) 1.014
kJ
kgK
  μ
T
K


2.287 10 5  kg
  k
msec
T
K


3.330 10 2  watt
  Pr
mK
T
K


 0.696
γ
T
K


 1.395
US Customary System of Units
Convert any temperature given in degrees-F to degrees-R. Notice one degree change in F is the same
as one degree change in R, i.e., F  R
TFR(T)  T  460R Ti_F  990F Ti_R  TFRTi_F ΦRTR ln
TR
K



ΦF(T) ln
TFR(T)
K


 Ti_R  1450R φF  ΦFTi_F
CpφF 0.263
BTU
lbF
  μ
Ti_R
K


24.46 10 6  lb
  k
ftsec
Ti_R
K


3.360 10 2  BTU
hrftF
 
Pr
Ti_R
K


 0.6888 γ
Ti_R
K


 1.353
Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 9 of 14

10. MathCAD - Missile Skin
Temperature.xmcd
Example
Calculate the heat transfer coefficient at the surface of a missile traveling at a speed of 3,000 fps at a
5,000 ft altitude (that is in the tropopause). The measured surface temperature at 1 foot downstream
from the leading edge is 700 oR
Solution
The Eckert reference temperature, TER, must be found for use in this problem due to aerodynamic
heating.
The temperature in the tropopause is constant average: Tf  391.6R
Density Correlation of the Tropopause
Since only the density changes appreciably in the tropopause, then only a least-squares fit must be
found for this function.
ρ(z) e ABz lb
  where A  2.0447799 and B  0.048036945
ft3
Therefore, μatm μavg ν(z)
μatm
ρ(z)

Check: z  60 ρ(z) 7.248 10 3  lb
 ρ(z) 2.253 10 4  slug
ft3
ft3
 
ν(z) 1.323 10  3  ft2
sec
 
External Re: ReVxzTER Vxρ(z)
μ
TER
K



t  5sec V 3
kft
sec
  x 1 ft   zi 50  Z t ( ) zi
V
kft
  t z  Z(t)
Assume turbulent flow, i.e., the recovery factor is:
rVxzTER Pr
TER
K


ReVxzTER Tf
TER


if   2105
Pr
TER
K

 1
3
otherwise

Joule's constant: J 778
ftlbf
BTU
 
Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 10 of 14

11. MathCAD - Missile Skin
Temperature.xmcd
Assume: z  60 Tw  700R TER  Tw Taw  0.9TER ρ(z) 2.253 10 4  slug
ft3
 
Adiabatic Wall Temperature: TawVxzTER Tf
rVxzTERV2
2JCpΦRTER
 
Calculate the Eckert reference temperature
Given
TER = 0.5Tw  Tf   0.22TawVxzTER  Tf 
TER  FindTER
TER  691.0R ReVxzTER Tf
TER


  8.258  105 Pr
TER
K


 0.698
CpΦRTER 0.2416
BTU
lbF
  μ
TER
K


1.492 10 5  lb
sft

 ReVxzTER Tf
TER
K
rVxzTf   0.900 h 0.029 k


TER





0.8

rVxzTf 
x
 
h 26.27
BTU
hrft2F
 
If the wall temperature is changing with time such as in transient temperature response, then the
Eckert reference-temperate is itself a function of such surface temperature history. Therefore, express
the previous solution for the Eckert temperature as a function of the wall temperature, and height, z, as
Given
TER = 0.5Tw  Tf   0.22TawVxzTER  Tf 
TERTwz  FindTER
TERTwz  691.0R which is essentially the same result as before !
Also note that the heat transfer coefficient then becomes,
 ReVxzTERTwz Tf
hTwz 0.029 k
TERTwz
K


TERTwz





0.8

rVxzTf 
x
 
Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 11 of 14

12. MathCAD - Missile Skin
Temperature.xmcd
Redifine the convective coefficient to account for the Eckert reference temperature being a function of the
surface history.
Convection: qCTsz hTsz
 Tas  Ts
BTU
hrft2


Non dimensionalize all equations and parameters
Thermal capacitance: C
C
BTU
ft2F
 3600
Radiation:
qRTw qRTw
BTU
hrft2



2.0 Spatial response parameters
Initial condition Tsi  700
Event duration T  10
Time step Δt  1 Nτ round
T
Δt


 1  11 3.0 Solve the governing set of equations
Note: ORIGIN  0 due to current MathCAD 2001i specifications. Future release will not have this
limitation. This constrain only applies to odesolve-function.
Given
C
d Ts(τ)
d
 qC Ts τ ( ) R  Z t ( )    qR Ts τ ( ) R     = Ts 0 ( ) Tsi
τ
=
Tsurface Odesolve τT Nτ    
Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 12 of 14

13. MathCAD - Missile Skin
Temperature.xmcd
t  0  10
0 1 2 3 4 5 6 7 8 9 10
740
735
730
725
720
715
710
705
700
Ts = F(Tref)
Ts = G(Tre(Ts))
Missile Surface-Temperature, Ts, History
Φ(time)
Tsurface(t)
timet
Tref is the Eckert reference temperature
t
 Z(tsec)
0
1
2
3
4
5
6
7
8
9
10
50
53
56
59
62
65
68
71
74
77
80
 hTsurface(t)RZ(tsec)
38.6
34.3
30.6
27.2
24.2
21.6
19.2
17.1
15.2
13.5
12.0
BTU
hrft2F

 Tsurface(t)
700.0
703.2
706.3
709.4
712.5
715.5
718.5
721.5
724.5
727.4
730.2
 Φ(time)

700.0
705.6
710.7
715.3
719.5
723.2
726.6
729.5
732.0
734.2
735.9
Show the results at τ  10 Z(τsec)  80 TS  Tsurface(τ)
Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 13 of 14

14. MathCAD - Missile Skin
Temperature.xmcd
t
 Z(tsec)
0
1
2
3
4
5
6
7
8
9
10
50
53
56
59
62
65
68
71
74
77
80
 TERTsurface(t)RZ(tsec)
691.0
692.6
694.2
695.7
697.2
698.7
700.2
701.6
703.1
704.5
705.9
R
 TawVxZ(tsec) Tsurface(t)R
1051.1
1050.9
1050.7
1050.5
1050.2
1050.0
1049.8
1049.6
1049.4
1049.1
1048.9
R

Check results at 60 kft, 70 kft, and 80 kft: z  ( 60 70 80 )T
Initial guess of the time that the missile takes to go from 50 kft to 60 kft: τ  3sec
Define a function that finds the time at which a given (known) altitude is reached. Although our current
assumed trajectory is both vertical, and have a constant velocity (i.e., altitude function is a linear function
of time, it is best to use a block solver to allow a more general (and possibly complex) altitude function
of time.
Given
Z(τ) = Altitude
t(Altitude)  Find(τ)
i  0  2 ti  tzi Twi
Tsurface
ti
sec


 R
h Twi
zi  

 ti
26.2
17.8
12.0
BTU
hrft2F

 zi
3.33
6.67
10.00
s
 Twi
60
70
80
 TER Twi
710.5
720.5
730.2
R
zi  

696.2
701.1
705.9
R
 Taw Vxzi Twi
 

1050
1050
1049
R

Julio C. Banks, MSME, P.E. Sell-A-Vision@Outlook.com page 14 of 14