TRANSIENT THERMAL SENSITIVITY ANALYSIS OF A
SMALL SATELLITE
Hojjat B. Khaniki
Applied Science and Research Association (ASRA)
Now-Bonyad Sq., Tehran, Iran
Tele-Fax: +98212814323 , hbkhaniki@yahoo.com
ABSTRACT
In this paper, transient thermal sensitivity of a small satellite, that has a major role in the
satellite thermal design, is studied. Temperature distribution of the satellite components
is affected by parameters such as panels thermal conductivity, power dissipation of
dissipating packages, and surface emittance/absorbtance. Using numerical methods,
energy and sensitivity equations are solved to predict the tempreture distribution and
thermal sensitivity of satellite components, respectively. Comparisions shows that
battery temperature is sensitive to its and other packages power dissipation,
respectively. Battery is less sensitive to radiation/conduction couplings, but the effect of
radiation coupling is much more from conduction coupling.
1.INTRODUCTION
Thermal design of a system that its components must be maintained in a narrow
temperature range, needs to more accurate thermal analysis. Design parameters may be
consist of thermo-physical specifications of thermal surfaces, thermal properties such
as specific heat and thermal conductivity of materials, components power dissipations,
and thermal boundary conditions. Knowing how this parameters affects temperature
distribution of the satellite, may help designer to optimize the design.
Sensitivity analysis is the best assistance for a designer to design satellite thermal
control sub-system.
Here transient thermal sesitivity of a satellite has been studied. Math model as a
system of coupled nonlinear-differential equations, and numerical methods that may be
used to solve this equations are presented.
2.ENERGY EQUATION
For illustration the temperature distribution of spacecraft subsystems, it is needed to
solve transient energy equation for each subsystem. Because the spacecraft subsystems
are thermally coupled, a system of equations must be solved to finding the temperature
distribution of each subsystem. Thermal boundary conditions are also time dependent.
Eenergy equation consists of transient, conduction, and radiation terms plus boundary
conditions (Solar, Albedo, Earth radiation) as a source term. This equation can be
written as (2-1)[1]:
(mc p )i
n
dTi
= − Cij (Ti − T j ) −
dt
j =1
∑
n
∑ σ R (T
ij
4
i
− T j4 ) + QSun + QAlbedo + QEarth + QDissipation − σ ε i Ai Ti 4
(2-1)
j =1
In (2-1), conduction and radiation couplings are presented as Cij and Rij, respectively.
The sub-system specifications that are needed to solving (2-1) are thermal properties of
materials (Cp, k), thermo-physical properties of surfaces (α, ε), mass, dimensions, and
assembly/integration data.
3.SENSITIVITY EQUATION
Sensitivity function may be defined as (3-1)[3,4].
Si =
dTi
dα
(3-1)
Where S is sensitivity function, Ti is the temperature of nnd node (component), and α is a
design parameter such as battery power dissipation. Equation (3-1) explains the
variation of component temperature due to variation of design parameter. In other word,
when α differentially varies (dα), tempetature of ind component varies equal to
dTi = Si.dα.
Using this defenition, the sensitivity will be as (3-2).
( mc p ) i
+
n
dT dT j
d dTi
= − Cij ( i −
)−
dt dα
dα dα
j =1
∑
n
∑ σ R (4.T
ij
j =1
i
3
dT j
dTi
− 4.T j3
)
dα
dα
(3-2)
dQSun (i ) dQ Albedo (i ) dQEarth (i ) dQDissipation (i )
dT
+
+
+
− 4σ ε i Ai Ti 3 i
dα
dα
dα
dα
dα
Using (3-1), (3-3) will be obtained.
( mc p ) i
+
n
dS i
= − Cij ( S i − S j ) −
dt
j =1
∑
n
∑σ R
ij ( 4.Ti
3
S i − 4.T j3 S j )
j =1
(3-3)
dQSun (i ) dQ Albedo (i ) dQEarth (i ) dQDissipation (i )
+
+
+
− 4σ ε i Ai Ti 3 S i
dα
dα
dα
dα
Equation (3-3) is linear and does not need to linerized. Energy equation can be
solved coupled or un-coupled with sensitivity equation. Sensitivity equation also
consists of transient, conduction, and radiation terms plus boundary conditions
(Solar,Albedo, Earth radiation) as a source term.
4. BOUNDARY CONDITIONS
Thermal boundary conditions are consisting of Sun, Albedo, and Earth radiation [2].
Solar radiation may be written as (4-1).
Qsun = AP × α s × S
(4-1)
Where AP is projected area, αs is solar absorptance of external surfaces, and S is solar
costant.
Albedo is given according to (4-2).
QAlbedo = ( AP . × FSat − Earth ) × α s × f a × S × Cosθ
(4-2)
Where, fa is albedo factor and can vary for different surfaces of the earth. Here albedo
factor is assumed 0.34, and θ is the angle of satellite position (satellite-earth line)
respect to Zenith.
For earth radiation we have (4-3).
QEarth = ( AP . × FSat − Earth ).ε . G
(4-3)
Where ε is emittance of external surfaces, and G is earth radiation flux.
Here, solar constant is considered 1350 W/m2 and can vary ±3.5%, and earth radiation
flux is also considered 232 W/m2 [1].
5. NUMERICAL SOLUTION AND RESULTS
In order to finding in-orbit temperature and sensitivity distribution of satellite
component, equation
(2-1) and (3-3) must be solved numerically using avilable
methods [3]. Here, these equations are solved according to the Lumped Heat Capacity
method. As mentioned later, the non-linear term must be linerizad and transient term is
descritized using finite difference method. Therefore, there will be a system of linearalgebric equations that is solved by Gaus-Seidel iterative method.
Figure 5-1 illustrates temperature distribution of battery and its mounting during an
orbit, and figure 5-2 shows sensitivity distribution of baterry related to satellite
components power dissipations. Figure 5-3 also illustrates sensitivity distribution of
battery related to conduction coupling between battery and its mounting panel, and
finally figure 5-4 illustrates sensitivity distribution of battery related to radiation
coupling between battery and its enlcosure surfaces.
According to results, battery temperature is more sensitive to its power dissipation
and other components power dissipations. For batterry temperature control, variation of
radiation and conduction coupling between battery and its enclosure components will be
useful, respectively.
6. REFERENCES
1) Satellite Thermal Control for System Engineering, By Robert D. Karam, 1998
2) Satellite Thermal Control HandBook, By David G. Gilmore, 1994
3) Haftka, R. T., “Techniques for Thermal Sensitivity Analysis,” International Journal of Numerical
Methods in Engineering, Vol. 17, No. 1, 1981, pp. 71-80
4) Haftka, R. T., “Sensitivity Calculations for Iteratively Solved Problems,” International Journal of
Numerical Methods in Engineering, Vol. 21, No. 8, 1985, pp. 1535-1546
5) Haftka, R. T., and Malkus, D. S., “Calculation of Sensitivity Derivatives in Thermal Problems by
Finite Differences,” International Journal of Numerical Methods in Engineering, Vol. 17, No. 12,
1981, pp. 1811-1821
6) Suresha, S., and Gupta, S. C., and Katti, R. A., “Thermal Sensitivity Analysis of Spacecraft Battery,”
Journal of Spacecraft and Rockets, Vol. 34, No. 3, 1997, pp. 384-390
25
24
23
22
21
20
19
18
17
16
15
1.6
NODE 30 (BATTERY)
NODE 12 (PANEL)
SENSITIVITY FUNCTION
TEMPERATURE (°C)
7) Suresha, S., and Gupta, S. C., “Transient Thermal Sensitivity Analysis During Solar Eclipse with
Discontinuous Heat Load,” Journal of Spacecraft, Vol. 36, No. 6, 1999, pp. 916-918
8) Suresha, S., and Gupta, S. C., “Calculation of Sensitivity in Thermal Control Systems with Nonlinear
Inequality Constrraints,” Journal of Spaccecraft and Rockets, Vol. 35, No. 4, 1998, pp. 552-558.
9) Hoffmann K. A., Chiang S. T., “Computational Fluid Dynamics Engineers”, Engineering Education
System, Vol. 2, 1993
1.2
NODE 30 (BATTERY)
NODE 31
NODE 38
NODE 45
1
0.8
0.6
0.4
0.2
0
2000
4000
ORBIT TIME (Sec)
0
6000
Figure 5-1) Temperature distribution of battery
and its mounting during an orbit
2000
4000
ORBIT TIME (Sec)
6000
Figure 5-2) Sensitivity distribution of baterry
related to satellite components power
dissipations
SENSITIVITY FUNCTION *1E-8
0.18
SENSITIVITY FUNCTION
1.4
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
4
2
0
-2
-4
-6
-8
NODE 1
NODE 8
NODE 22
-10
0.00
NODE 5
NODE 12
NODE 28
-12
0
2000
4000
ORBIT TIME (Sec)
Figure 5-3) Sensitivity distribution of battery
related to conduction coupling between battery
and its mounting panel
6000
0
2000
4000
ORBIT TIME (Sec)
Figure 5-4) Sensitivity distribution of battery
related to radiation coupling between battery
and its enlcosure surfaces
6000