The document provides an overview of thermal control systems for spacecraft. It discusses passive and active thermal control systems, with passive relying on components like surface coatings and active using powered mechanisms. The document covers thermal analysis methods like conduction, convection, and radiation. It describes modeling the thermal environment from heat sources like solar radiation, planetary radiation, and albedo. Design of passive thermal control uses appropriate surface coatings to control absorptance and emittance.

1. Chapter 2. Thermal Control System
Dr Minkwan Kim
Semester 1

2. Outline and Contents
1. Overview of the Thermal Control System
2. Thermal Design
Passive
Active
3. Thermal Modelling and Testing
[Online Quiz] by 1/Nov (Monday)

3. Learning outcomes on Ch2
• Able to answer the following questions
⎻ What are the differences between active and passive systems?
⎻ What are the physical parameters affecting the temperature of S/C?
⎻ How does the orbit of a spacecraft affect on the Thermal control
system design?
⎻ What are the external sources of heat?
• Able to estimate the temperature of S/C.
• Able to select materials which can maintain the
temperature of S/C in acceptable level.
• Able to determine the required power of heater/cooler

4. 1. Introduction of Thermal Control System
• Functions
⎻ Monitors temperatures of key components.
⎻ Maintains the temperature of these components within
acceptable limits
⎻ Control the temperature of ALL individual components
throughout the Entire mission.
• Design Considerations:
⎻ Controlling the average spacecraft temperature requires a
balance of heat absorbed, generated and radiated
⎻ Once the average temperature is controlled effectively, the
temperature if individual components within the spacecraft can
be controlled via a wide variety of measures
⎻ Control system can be active, passive or a combination of the
two

5. 1. Introduction of Thermal Control System
• Typical spacecraft component temperature limits
Component / Subsystem
Operating
temperature (℃)
Survival
temperature (℃)
General electronics -10 ~ 45 -30 ~ 60
Batteries 0 ~ 10 -5 to 20
Motors 0 ~ 50 -20 ~ 70
Solar panels -100 ~ 125 -100 ~ 125

6. 2. Thermal Analysis
• All thermal analysis begins with the first law of
thermodynamics
Q = heat added to the system
W = rate of work production by the system
dU/dt = change in the internal energy U of the system
=
Thermal Analysis
𝑄𝑄 − 𝑊𝑊 =
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
𝐴𝐴 𝑑𝑑𝑑𝑑 𝜌𝜌 𝐶𝐶𝑝𝑝
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑
For a uniform solid with cross-section area A and length dx

7. 2. Thermal Analysis
• Assume:
⎻ spacecraft is in thermal equilibrium, and balance the heat
emitted with heat absorbed
Spacecraft Thermal Balance
Qin Qout
Qnet = Qin - Qout

8. 2. Thermal Analysis
Conductive Heat Transfer:
⎻ heat moves through a solid
⎻ microscopic diffusion and collision of particles
⎻ Fourier’s law:
Conduction
where = heat flux, W/m2
= material conductivity
𝑄𝑄𝑥𝑥 = −𝜅𝜅𝜅𝜅
𝑑𝑑𝑑𝑑
𝑑𝑑𝑑𝑑

9. 2. Thermal Analysis
Convection
Convective Heat Transfer:
⎻ heat transfer by the movement of fluids
⎻ dominant form of heat transfer in liquids and gases
⎻ Newton’s cooling law: (set of differential equation given by Fourier’s law)
where = convection coefficient
= surface temperature
𝑄𝑄𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = ℎ 𝑇𝑇𝑤𝑤 − 𝑇𝑇𝑓𝑓𝑓𝑓 ⋅ 𝐴𝐴

10. 2. Thermal Analysis
Radiation
Radiative Heat Transfer:
⎻ heat transfer vehicle in space and its external environment
⎻ transport of energy by electromagnetic waves emitted by all bodies
⎻ Stefan-Boltzmann law: where = emissivity
= Stefan-Boltzmann constant
𝑞𝑞𝑏𝑏 = 𝜎𝜎𝑇𝑇4

11. 2. Thermal Analysis
• Black body radiation
⎻ All bodies emit radiation due to their temperature. A black body is
an ideal emitter. A black body at temperature T emits radiation
with power per unit area of its surface given by the Stefan-
Boltzmann Law:
⎻ Absorptance (α)
• The ratio of radiant energy absorbed by a body to that incident on it
⎻ Emittance (ε)
• The ratio of energy emitted by a body to that emitted by a black body at the
same temperature.
Radiation
𝑞𝑞 = 𝜎𝜎 𝑇𝑇4
W
m2 where 𝜎𝜎 = 5.67 × 10−8
𝑊𝑊
𝑚𝑚2 ⋅ 𝐾𝐾4

12. 2. Thermal Analysis
Radiation properties
𝛼𝛼 + 𝑅𝑅𝜌𝜌 + 𝜏𝜏 = 1
Incident radiation Reflection, R𝜌𝜌
Transmission, 𝜏𝜏
Absorption, 𝛼𝛼

13. • Amount of heat radiating from a real surface is
• For bodies in thermal equilibrium at the same temperature,
2. Thermal Analysis
Radiation properties
𝑄𝑄𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 = 𝜀𝜀 𝑄𝑄𝑏𝑏
Energy being absorbed = Energy emitted

14. 2. Thermal Analysis
• Spacecraft thermal emission
Spacecraft Heat Emission
𝑄𝑄𝑠𝑠𝑠𝑠 = 𝜀𝜀 ⋅ 𝜎𝜎T4 ⋅ 𝐴𝐴𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 [W]
NOTE: Internal dissipation, Qdis
𝑄𝑄𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑃𝑃 [W]

15. 2. Thermal Analysis
i. Solar Radiation
ii. Planetary Radiation
iii. Albedo
A. Solar Radiation
⎻ Intensity Js of solar radiation at distance D:
External Heat Sources
Sun
𝐽𝐽𝑠𝑠 =
𝑃𝑃
4𝜋𝜋𝐷𝐷2
W
m2
where 𝑃𝑃 = 3.8 × 1026
𝑊𝑊
𝑄𝑄𝑠𝑠 = 𝐽𝐽𝑠𝑠 ⋅ 𝐴𝐴𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝−𝑆𝑆𝑆𝑆𝑆𝑆 ⋅ 𝛼𝛼𝑠𝑠 W

16. 2. Thermal Analysis
B. Planetary Radiation
⎻ All planets have temperatures above 0K so they emit
radiation
External Heat Sources
𝑄𝑄𝑝𝑝 = 𝐽𝐽𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ⋅ 𝐴𝐴𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝−𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ⋅ 𝐹𝐹𝑠𝑠−𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ⋅ 𝜀𝜀𝐼𝐼𝐼𝐼 W
S/C

17. 2. Thermal Analysis
C. Albedo
⎻ Albedo is the solar radiation that is reflected from a planet,
which is generally much more significant than planetary
radiation
External Heat Sources
v
S/C
Js
𝜙𝜙
𝑄𝑄𝑎𝑎 = 𝜌𝜌𝑎𝑎 ⋅ 𝐽𝐽𝑠𝑠 ⋅ 𝐴𝐴𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝−𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ⋅ 𝐹𝐹𝑠𝑠−𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ⋅ cos 𝜙𝜙 ⋅ 𝛼𝛼𝑠𝑠 W

18. 2. Thermal Analysis
C. Albedo
External Heat Sources
S/C
𝜙𝜙2
𝜙𝜙1

19. 2. Thermal Analysis
C. Albedo
External Heat Sources
𝑄𝑄𝑎𝑎 = 𝜌𝜌𝑎𝑎 ⋅ 𝐽𝐽𝑠𝑠 ⋅ 𝐴𝐴𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝−𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ⋅ 𝐹𝐹𝑠𝑠−𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ⋅ cos 𝜙𝜙 ⋅ 𝛼𝛼𝑠𝑠 W
𝑄𝑄𝑎𝑎 = 𝜌𝜌𝑎𝑎 ⋅ 𝐽𝐽𝑠𝑠 ⋅ 𝐴𝐴𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝−𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ⋅ 𝐹𝐹𝑠𝑠−𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 ⋅ cos 𝜙𝜙1 ⋅ cos 𝜙𝜙2 ⋅ 𝛼𝛼𝑠𝑠 W

20. 3. Thermal Design
• Principal trade-off in thermal control design
Passive and Active Systems
Passive:
No power requirement
No moving parts
Simple (reliable)
Low cost
 Inflexible
 Low heat transfer rates
 Performance variability
(e.g. Surface coatings)
Active:
Flexible and adaptive
High heat transfer rate
 Power required
 Mechanisms / moving
components (reliability)
 Mass
 High(er) cost
(e.g. fluid loop systems)

21. 3. Thermal Design
• Passive thermal control design has been proven on
Earth-orbiting missions (with ‘average’ power
requirements).
• Active systems are required for:
⎻ high (variable) power dissipation missions (e.g. high power
comms or military S/C).
⎻ S/C encountering extreme variations in thermal environment
(e.g. interplanetary missions).
⎻ precise thermal control (e.g. crewed missions).
• Cost drives use of passive methods wherever
possible.
Passive and Active Systems

22. 3. Thermal Design
i. Surface Coating
A. Passive Technique

23. 3. Thermal Design
i. Surface Coating
⎻ Temperature control by choice of surface coatings.
⎻ Use appropriate absorptance and emittance
A. Passive Technique

24. 3. Thermal Design
i. Surface Coating
⎻ Recall thermal balance equation: Qin+ Qdis = Qout
A. Passive Technique
𝑄𝑄𝑖𝑖𝑖𝑖 = 𝑄𝑄𝑠𝑠 + 𝑄𝑄𝑎𝑎 + 𝑄𝑄𝑝𝑝 =
T =
𝐽𝐽𝑠𝑠
𝜎𝜎
⋅
𝛼𝛼
𝜀𝜀
⋅
𝐴𝐴𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝐴𝐴𝑠𝑠/𝑐𝑐
1
4
+ ⋯
𝑄𝑄𝑜𝑜𝑜𝑜𝑜𝑜
𝑄𝑄𝑑𝑑𝑑𝑑𝑑𝑑
+

25. 3. Thermal Design
i. Surface Coating
- There are four basic thermal material categories
A. Passive Technique
a. Solar absorber
• High 𝛼𝛼, low 𝜀𝜀
• Example:
- gold: 𝛼𝛼 = 0.3 and 𝜀𝜀 = 0.02
- Teq ~ + 380 ◦C
High
𝛼𝛼
𝜀𝜀
> 1

26. 3. Thermal Design
i. Surface Coating
A. Passive Technique
b. Solar reflector
• Low 𝛼𝛼, high 𝜀𝜀
• Example:
- White paint: 𝛼𝛼 = 0.15 and 𝜀𝜀 = 0.9
- Teq ~ +380 ◦C
Low
𝛼𝛼
𝜀𝜀
< 1

27. 3. Thermal Design
i. Surface Coating
A. Passive Technique
c. Flat absorber
• high 𝛼𝛼, high 𝜀𝜀
• Example:
- black paint: 𝛼𝛼 = 0.9 and 𝜀𝜀 = 0.9
- Teq ~ +60 ◦C
d. Flat reflector
• low 𝛼𝛼, low 𝜀𝜀
• Example:
- Aluminium paint: 𝛼𝛼 = 0.3 and 𝜀𝜀 =
0.3
- Teq ~ +60 ◦C
𝛼𝛼
𝜀𝜀
≈ 1
𝛼𝛼
𝜀𝜀
≈ 1

28. 3. Thermal Design
i. Surface Coating
Example:
Evaluate the equilibrium temperature, Teq, of the given solar
array for a 3-axis stabilised GEO satellite in equinox conditions
(a) at local noon
(b) at local midnight
A. Passive Technique
𝑅𝑅𝐸𝐸 = 6378 𝑘𝑘𝑘𝑘
𝑅𝑅𝐺𝐺𝐺𝐺𝐺𝐺 = 42164 𝑘𝑘𝑘𝑘
𝐽𝐽𝑠𝑠 = 1400 𝑊𝑊/𝑚𝑚2
𝐽𝐽𝐸𝐸 = 240 𝑊𝑊/𝑚𝑚2
Albedo = 0.34
𝛼𝛼𝑠𝑠,𝐹𝐹 = ?
𝜀𝜀𝐹𝐹 = 0.8
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 = 6 𝑚𝑚2
𝛼𝛼𝑠𝑠,𝐵𝐵 = 0.7
𝜀𝜀𝐵𝐵 = 0.7
Solar cell efficiency = 0.14
Solar cell packing efficiency = 0.95
Average solar cell array absorptance = 0.8

29. 3. Thermal Design
ii. Bimetallic fins
• Fins provide:
⎻ an increase in spacecraft surface area
⎻ a change in effective 𝛼𝛼/𝜀𝜀 ratio
• Can be made an ‘active’ device in combination with a
heater.
A. Passive Technique

30. 3. Thermal Design
iii. Multi-layer insulation (MLI)
• Used to reduce heat loss by thermal radiation
⎻ Kapton is often used for inner and outer layers of a mylar MLI
blanket.
⎻ Number of layers: typically 20 to 25 per cm.
A. Passive Technique

31. 3. Thermal Design
iii. Multi-layer insulation (MLI)
• Applications
A. Passive Technique

32. 3. Thermal Design
iii. Multi-layer insulation (MLI)
A. Passive Technique
Effective venting of MLI blanket is necessary to prevent failure.
Assumes:
• No contact of layers
• Vacuum between sheets
• No edge leakages
ε =
εi
N + 1
where
𝜀𝜀𝑖𝑖 = original emittance
N = number of layers

33. 3. Thermal Design
iv. Heat Pipes
A. Passive Technique

34. 3. Thermal Design
iv. Heat Pipes
⎻ Heat pipes in space applications
A. Passive Technique

35. 3. Thermal Design
v. Passive Radiators
a. 3-Axis Stabilised:
A. Passive Technique

36. 3. Thermal Design
v. Passive Radiators
b. Dual-spin Stabilised:
A. Passive Technique

37. 3. Thermal Design
v. Passive Radiators
A. Passive Technique

38. 3. Thermal Design
i. Heater elements
B. Active Technique
• Particularly for eclipse operation:
─ e.g. hydrazine tanks / pipes / valves, thrusters, payload, battery environment
• Kapton laminate with etched wiring element
• Controlled automatically (by thermostats) or by ground command.

39. 3. Thermal Design
ii. Louver systems
B. Active Technique

40. 3. Thermal Design
iii. Thermoelectric System
B. Active Technique

41. 4. Thermal Modelling and Testing
• A Comprehensive Thermal Mathematical Model (CTMM) is
constructed in early Phase C/D:
- Model incorporates full S/C configuration details
- Typically 400 to 600 nodes
- Typically 10,000 to 15,000 thermal couplings
A. Modelling

42. 4. Thermal Modelling and Testing
Thermal-vacuum testing
B. Testing

43. 4. Thermal Modelling and Testing
• Conservation of energy:
•
For transparent materials with NO INTERNAL DISSIPATION
Thermal Analysis
𝑄𝑄𝑖𝑖𝑖𝑖 − 𝑄𝑄𝑜𝑜𝑜𝑜𝑜𝑜 + 𝑄𝑄𝑑𝑑𝑑𝑑𝑑𝑑 =
𝜕𝜕𝐸𝐸𝑖𝑖𝑖𝑖𝑖𝑖
𝜕𝜕𝜕𝜕
𝑄𝑄𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 𝑄𝑄𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 + 𝑄𝑄𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 + 𝑄𝑄𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡

44. 4. Thermal Modelling and Testing
• Assume:
⎻ Only has a Solar radiation
⎻ No internal energy dissipation
Steady State Temperature of Insulated Surfaces
𝑄𝑄𝑖𝑖𝑖𝑖 =
𝑄𝑄𝑜𝑜𝑜𝑜𝑜𝑜=
𝐽𝐽𝑠𝑠 𝐴𝐴𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝,𝑠𝑠 𝛼𝛼𝑠𝑠
𝜀𝜀 𝜎𝜎 𝑇𝑇4 𝐴𝐴𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
∴ 𝑇𝑇= 𝐽𝐽𝑠𝑠 𝛼𝛼𝑠𝑠
𝜀𝜀 𝜎𝜎
𝐴𝐴𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝐴𝐴𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
1
4

45. 4. Thermal Modelling and Testing
• Assume:
⎻ Only has a Solar radiation
⎻ No internal energy dissipation
Space Radiators
𝑄𝑄𝑖𝑖𝑖𝑖 =
𝑄𝑄𝑜𝑜𝑜𝑜𝑜𝑜=
𝐽𝐽𝑠𝑠 𝐴𝐴𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝,𝑠𝑠 𝛼𝛼𝑠𝑠
𝜀𝜀 𝜎𝜎 𝑇𝑇4
𝐴𝐴𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
∴ 𝑄𝑄𝑊𝑊= 𝜀𝜀 𝜎𝜎 𝑇𝑇4 𝐴𝐴𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 − 𝐽𝐽𝑠𝑠 𝐴𝐴𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝,𝑠𝑠 𝛼𝛼𝑠𝑠
+ 𝑄𝑄𝑊𝑊
𝐽𝐽𝑠𝑠 𝐴𝐴𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝,𝑠𝑠 𝛼𝛼𝑠𝑠 + 𝑄𝑄𝑊𝑊 − 𝜀𝜀 𝜎𝜎 𝑇𝑇4
𝐴𝐴𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 0

46. 4. Thermal Modelling and Testing
Solar Array/Plat Plate Min/Max Temperatures
Earth
sun 𝜙𝜙
ℎ
𝜃𝜃
Solar array/plat plate, A
𝑄𝑄𝑖𝑖𝑖𝑖 =
𝑄𝑄𝑜𝑜𝑜𝑜𝑜𝑜=
𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠 + 𝑄𝑄𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐼𝐼𝐼𝐼 + 𝑄𝑄𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
𝑄𝑄𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + 𝑄𝑄𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
• Assume:
⎻ No internal energy dissipation
⎻ Solar Array / plat plat is VERY thin
⎻ Different materials on front and back

47. 4. Thermal Modelling and Testing
Solar Array/Plat Plate Min/Max Temperatures
Earth
sun
ℎ
𝜃𝜃
Solar array/plat plate, A
𝑄𝑄𝑖𝑖𝑖𝑖 = 𝑄𝑄𝑠𝑠𝑠𝑠𝑠𝑠 + 𝑄𝑄𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐼𝐼𝐼𝐼 + 𝑄𝑄𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
𝐽𝐽𝑠𝑠 𝛼𝛼𝑠𝑠,𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝐽𝐽𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐴𝐴
𝑅𝑅𝐸𝐸
𝑅𝑅𝐸𝐸 + ℎ
2
𝜀𝜀𝐼𝐼𝐼𝐼, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
𝜙𝜙
𝐽𝐽𝑠𝑠 𝜌𝜌𝑎𝑎 𝐴𝐴
𝑅𝑅𝐸𝐸
𝑅𝑅𝐸𝐸 + ℎ
2
cos 𝜙𝜙 𝛼𝛼𝑠𝑠, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
= 𝐽𝐽𝑠𝑠 𝜌𝜌𝑎𝑎 𝐴𝐴 sin2 𝜌𝜌 cos 𝜙𝜙 𝛼𝛼𝑠𝑠, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏

48. 4. Thermal Modelling and Testing
Solar Array/Plat Plate Min/Max Temperatures
Earth
sun
𝜌𝜌
ℎ
𝜃𝜃
Solar array/plat plate, A
𝑄𝑄𝑜𝑜𝑜𝑜𝑜𝑜 = emitted energy
= 𝜎𝜎 𝜀𝜀𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝐴𝐴 𝑇𝑇4 + 𝜎𝜎 𝜀𝜀𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝐴𝐴 𝑇𝑇4
𝜙𝜙
𝑄𝑄𝑖𝑖𝑖𝑖 − 𝑄𝑄𝑜𝑜𝑜𝑜𝑜𝑜 = 0
−𝑄𝑄𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔
𝐽𝐽𝑠𝑠𝐴𝐴 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝜂𝜂

49. 4. Thermal Modelling and Testing
Solar Array/Plat Plate Min/Max Temperatures
Earth
sun
𝜌𝜌
ℎ
𝜃𝜃
Solar array/plat plate, A
𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 =
𝐽𝐽𝑠𝑠 𝛼𝛼𝑠𝑠,𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + 𝐽𝐽𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 sin2 𝜌𝜌 𝜀𝜀𝐼𝐼𝐼𝐼,𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 + 𝐽𝐽𝑠𝑠 𝜌𝜌𝑎𝑎 sin2 𝜌𝜌 𝛼𝛼𝑠𝑠, 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 − 𝐽𝐽𝑠𝑠𝜂𝜂
𝜎𝜎 𝜀𝜀𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + 𝜀𝜀𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
1
4
𝜙𝜙
𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 =
𝐽𝐽𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 sin2 𝜌𝜌 𝜀𝜀𝐼𝐼𝐼𝐼,𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
𝜎𝜎 𝜀𝜀𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 + 𝜀𝜀𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
1
4

50. 4. Thermal Modelling and Testing
Spherical Satellite Max/Min Temperatures
Earth
sun
𝜌𝜌
ℎ
Satellite
𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 =
𝐴𝐴𝑐𝑐 𝐽𝐽𝑠𝑠 𝛼𝛼𝑠𝑠 + 𝐴𝐴 𝐽𝐽𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝐹𝐹 𝜀𝜀 + 𝐴𝐴 𝐽𝐽𝑠𝑠 𝜌𝜌𝑎𝑎 𝐹𝐹 𝛼𝛼𝑠𝑠 + 𝑄𝑄𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝐴𝐴 𝜎𝜎 𝜀𝜀
1
4
𝜙𝜙
𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 =
𝐴𝐴 𝐽𝐽𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐹𝐹 𝜀𝜀 + 𝑄𝑄𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
𝐴𝐴 𝜎𝜎 𝜀𝜀
1
4