The objective of this assignment is to demonstrate an
understanding of thermal subsystem theory, components and design
1. Answer the following basic qualitative thermal subsystem
questions. They can all be answered with a few sentences at most:
a) What is the primary function of a satellite thermal
To maintain all spacecraft components
within their necessary temperature ranges.
b) What is the difference between an operating
temperature range and a survival temperature range.
If you are not using a component, you
may exceed the operating range but not the survival range. The
operating range specifies acceptable temperatures for immediate
functionality of the component. The survival range specifies the
range that - if ever exceeded - may lead to improper functionality at
any point in the future, no matter what the temperature is at that
c) What are the three primary heat transfer
mechanisms? Which one is often not applicable for certain
spacecraft applications/analyses - and, give a specific example of when
this specific mechanism IS a critical part of the thermal analysis for a
Conduction, radiation, convection
convection - examples would be
whenever there are liquids/gases, like manned space flight systems,
propulsion systems, etc.
d) Briefly describe the terms reflectivity,
transmissivity, absorptivity, and emissivity.
As light strikes an object, portions
of the light can be reflected back, transmitted through, or absorbed as
heat by the object. The fractions by which this occurs are
properties of the material known as reflectivity, transmissivity and
absorptivity, respectively. At a given temperature, the object
also emits energy; the amount emitted compared to an idealized blackbody
radiator at the same temperature is known as the emissivity.
From the perspective of thermal design, what is a “selective”
surface (or surface material, or surface coating, or surface
characteristic), and how is it useful?
A selective surface
has a distinct difference in the values of absorptivity and emissivity
in the visible vs in the infrared ranges, such that the ratio of
(visible) absorptivity to (infrared) emissivity is not 1. This
is useful since the choice of surface can be used to dramatically affect
the amount of heat coming into or going out of the system.
For grading, needed some type of definition relating to a and e,
and for addressing usefulness, needed to at least cite thermal control
or mention of heat balance.
f) How is the amount of radiated heat flux from an idealized
black-body related to the temperature of the body?
proportional to the 4th power of the temperature
g) What are the three primary environmental (i.e. external) thermal inputs for LEO
Direct solar, albedo (solar reflected off of
Earth), and Earth IR
h) What is the Beta angle of an orbit, and why is it
important for thermal design?
It is the angle btw the orbit plan and
the sun vector. Eclipse durations and albedo are a strong function
of - and can be parameterized by - the beta angle, so this is why it is
useful to know for thermal design.
i) Why are many thermal tests performed in a vacuum?
eliminate air that leads to convective heat transfer, which we typically
don't have in space
j) How do second surface reflectors achieve such dramatic
absorptivity to emissivity ratios when facing the sun?
SSRs use two materials to do this. The
first material has good high IR emissivity in order to dump heat and
since it is not facing the Earth, little heat is absorbed. This
material is also transparent to solar visible light which goes through
the material and gets mostly reflected back into space by the back
material, which is highly polished and reflective. Although the back
material absorbs some solar energy, it conducts this energy to the outer
layer which emits it out given its high emissivity.
k) Why is conductance through a joint problematic?
Actual contact only occurs over a small
fraction of the interface area, and conduction is dependent on a wide
variety of factors such as smoothness, pressure, etc.
l) Explain how you could "manage your power loads" (meaning
that you could turn components on and off) in order to influence (and
thereby control) the temperature at different points in your
satellite. Note that this might be considered a non-traditional
approach to thermal management, but it is often exploited on small
spacecraft and/or during contingency operations when anomalies exist.
Turning components on and off alters power
dissipation internal to the satellite. You could do this with
components placed "near" (meaning a good conduction or
radiation path) points in the satellite that need to have their
temperature altered. In effect, you are using standard components
2. Answer the following basic quantitative thermal subsystem
questions. Mind your units!
a) Consider a spherical cryogenic tank that is cooled
to 30ºK. The tank has a diameter of 0.25m. For the purposes of this
problem, the thickness of the tank wall is negligible; however, the
exterior of the tank is covered with an insulation blanket that is 2 cm
thick. The exterior of the blanket is 300ºK by virtue of the external
thermal environment. The thermal conductivity of the insulation is 0.003
W/mK. Because it affects the design of the cryogenic cooling system,
you must estimate how much heat flows from the exterior to the interior
(in W). Provide the equation you will use to do this as well as the
numeric solution (in W).
deltaT = 270 K; Ri=0.125m; Ro=0.145m; k=0.003
Q = (4*pi*k*Ri*Ro*deltaT) / (Ro-Ri) =
9.22W from outside to inside
b) Consider a 0.5m x 0.5m flat isothermal radiator
that "sees" only deep space and which has an IR emissivity of 0.8. This
radiator is used to reject heat from a transmitter, which is bolted to
the radiator with a metal-to-metal contact area that is 8 cm x 8 cm and
with a joint conductance of 1200 W/m2K. Assume that all other
sides of the transmitter are heavily insulated such that that all of the
100W of heat dissipated by the transmitter is rejected to space by the
radiator, analyze this system by finding the radiator temperature and
the transmitter temperature.
100W flows through conductive
interface and is emitted by the radiator, so the thermal path is:
conductance)---> Radiator --->(radiation)---> Space
So, first, find radiator temperature:
e = 0.8, sigma = 5.67 x 10^-8
W/(m^2K^4), A=0.25 m^2,
Q = e * sigma * Tr4 * A;
Tr = (100/(e*sigma*A))1/4=306.4K
Now, find transmitter temp given that
100W flows from it to the end of a conductor that is at 306.4K:
hc = 1200, A=0.0064, Tr=306.4, Q=100,
Q=hc*A*(Txmtr-Tr); Txmtr=100 / (hc*A)
+Tr = 319.4K
c) Imagine a spherical satellite orbiting the sun. For any
relative orientation to the sun, the exposed area to the sun is constant
(it is always pi*R2 where R is the satellite's radius).
Accordingly, if the sphere's surface has a constant absorptivity, then
there is no variation in absorbed solar energy as a function of the
orientation of the satellite.
Now, consider the same situation for a cube-shaped satellite (assume
the satellite's surface has constant absorptivity). Determine the
relative cube-sun orientations for minimum solar absorption and maximum
solar absorption. Describe these orientations. Find the
ratio of maximum absorption to minimum absorption. Is this value
significant enough that you think you would have to consider relative
orientation to the sun for a cubic spacecraft?
Min orientation: 1 face pointed directly at the
Max orientation: 1 vertex pointed directly at
Ratio: Max absorption / Min absorption = Amax/Amin
= sqrt(3) * s / s = sqrt(3) = 1.73 where s is the length of a side of
the cube. This problem boils down to spatial geometry (which is a
critical element of thermal analysis) - the projected area of a cube in
the direction of its vertex is equivalent
to 6 equilateral triangles with height of .707.
Significant? You betcha! It is a 73% increase
in absorbed energy!
d) A 50W payload (which can be turned completely on or
completely off) must be kept in a temperature range of -10 to +10ºC.
One side of this payload is a radiator that has an emissivity of 0.9 and
which is positioned to emit to space; the radiator and payload have the
same temperature and are perfectly coupled in a thermal sense. The
payload continuously absorbs 10W of constant heat transfer from
neighboring devices within the spacecraft. How big should the radiator
be in order to keep the payload below its maximum temperature? What
maximum heater power is necessary to keep the payload at its minimum
Hot case (T=10C): Qint+Qabs=60W=e*sigma*T4*A;
A=60/(e*sigma*T4) = 0.18m2
Cold case (T=-10C): Qint+Qabs+Qhtr=10+Qhtr=e*sigma*T4*A;
Qhtr=e*sigma*T4*A - 10 = 33.9W
3. Perform a series of simple thermal balances similar to the ones
shown on the thermal lecture slides 39-44. For this case, consider
a flat plate that is directly between the sun and the Earth, with one
flat side pointing perfectly at the sun and the other flat side pointing
perfectly at the Earth. We will assume that the edges of the plate
are perfectly insulated such that they neither receive nor release any
energy. The plate is above the Earth at an altitude of 800 km,
and the albedo factor a=0.3. The area of each side of the plate is
m2. The plate is not internally dissipating any energy.
a) What is the value of Jsun? 1358 W/m2
b) What is the value of Jalbedo? Note that
this is heavily dependent on the value of F that you use, which is a
bit difficult to read/estimate from the appropriate plot. Be
sure to specifically state the value of F that you are using.
= (0.3)*( 1 )*(1358W/m2) = 407W
note that F=1 from the plot of slide
29; you have to eyeball this, which is tough; I'll accept answers
where F was selected anywhere btw 0.8 and 1.3, which would give a
Jalbedo value in the range of 325 to 530
c) What is the value of Jplanet? 237*[ Re / (Re+h)
]2 = 237*(6378 / 7178)2 = 187W
d) What are Asun, Aalb, Aplanet, and Asurf?
Asun = Aalb = Aplanet = 1.75 m2, and
e) For (absorptivity/emissivity) =1, what is the steady state
temperature of the plate?
T4 = [187 / (2*sigma) ] +
(1)[ 1.75*(Jsun + Jalb) / (3.5*sigma)]
T4 = (1.65*109) +
for (a/e)=1, T=362.2 K
f) If the plate is dissipating 250W internally and is coated with
white enamel paint (#18 on the table shown on slide #40), what will the
for e=0.853 and (a/e)= 0.3, T4 =
(1.65*109) + [ 1.48*109 ] +(.3)(1.56*1010
g) Now assume that the plate is no longer dissipating any energy. If
you desired the temperature to be 30C, what (a/e) value would you like
to have for your surface properties?
Note that 30C=303K
T4 = (1.65*109) +
(a/e) = 0.435
h) Now assume that the plate is rotated by 30 degrees. Would you
expect the steady state temperature to go up or down? Why?
T will drop since the exposed areas to the
incoming Q is lowered but the exposed area for emitting energy stays the
4. A cubic LEO satellite with 1m sides orbits at 1,000
km altitude with a beta angle of 90º. As the satellite orbits,
the satellite's -z face points to the Earth, the satellite's +x face
points in the direction of orbital travel, and the satellite's +y face
always faces the sun. This problem attempts to walk you through
the process of finding the orbit-averaged heat flux contributions from
direct solar and Earth IR for each face of the satellite (given the
orbit, we will neglect albedo), with an emphasis on using the charts in
the slide package to determine view factors. Use 237 W/m2
as the Earthshine value and 1358 W/m2 for the direct solar
input; in addition, let solar absorptivity and infrared emissivity both
a) What is the Earth IR flux (in W/m2) at
this orbit? Determine this in 2 ways.
First, note that a simple approximation for the heat
contribution from the Earth infrared source is to use a simple inverse
square law. This approximation assumes that a given surface (or
its effective area, such as a spherical cross-section) is pointed
directly at the Earth. For example, on Slide 30, you are given
an inverse square law equation for determining the IR radiation on a
surface at a given orbit altitude. Compute Jir using this
Conversely, consider the geometric curves on Slides
21-24. For a square cross-sectional area in orbit above the
Earth, specifically consider the curves on Slide 24. The x-axis of
this chart quantifies the ratio of altitude to the radius of the Earth
(figure this out for the given problem). If the square surface is
pointing directly at the Earth, then lambda is 0 (so, use that
specific curve). Given these values, look up the View Factor and
compare to what you get for the inverse square law.
= 1000/6384 = 0.156
~ 0.75 or so
= 237 * .75 = 177.75
b) To perform the full analysis for the satellite,
fill out the table below. You will need to visualize the
geometric scenario described in the first paragraph of this problem.
First, for each side of the satellite (+z, -z, +y,
-y, +x, -x), determine the amount of direct solar energy on each face,
and enter these numbers in the first row of the table.
Second, consider how much IR heating from the Earth
impacts each face. To do this, you need to look up the
appropriate view factors using slide 24 given the geometric situation.
One of the faces directly faces the Earth, matching the scenario you
considered in part (a) of this problem. Another face, doesn't
'see' the Earth at all, and it therefore will have a view factor of 0.
The other faces are perpendicular to the Earth's surface - you can use
Slide 24 to determine the view factor for these sides, given a lambda
value of 90 degrees. Enter the view factors for each side on the
second row of the table, and then using these values use the third row
of the table to compute the amount of Earth IR energy incident on each
On the fourth row, add together the solar and IR
inputs to get the total input for each face.
|Solar input (W)
|Earth view factor
|IR input (W)
|Total input (W)
c) What is the total combined input for all faces?
This should be the sum of the
bottom-line entries: 1724.6 W
d) If the satellite internally dissipated 250W, what
would be the approximate steady-state temperature of the satellite
(we'll assume a high thermal mass, a single node, etc.)?
e*sigma*T4*Asurf = Q =
T = (1974.6 / (e*sigma*Asurf) )1/4