MECH 372 / ENGR 372
Space Systems Design and Engineering II
Assignment #3 - Thermal SubSystem Solutions
Objective: The objective of this assignment is to demonstrate an understanding of thermal subsystem theory, components and design processes. 

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 subsystem?

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 future time.

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 space system?

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.

e) 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 spacecraft?

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?

To 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 as heaters.

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:

     Transmitter  --->(joint 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 sun

Max orientation: 1 vertex pointed directly at the sun

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 temperature?

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 1.75 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.

Jalbedo=aFJsun = (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 Asurf=3.5 m2

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) + (1.56*1010 )

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 temperature be?

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)(1.56*1010 )

(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 same.


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 equal 1.  

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 technique.

Jir = 237*(6378/7378)2=177W/m2

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.

h/R = 1000/6384 = 0.156

F ~ 0.75 or so

Jir = 237 * .75 = 177.75

Darn close!

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 side.

On the fourth row, add together the solar and IR inputs to get the total input for each face.  

  +z -z +y -y +x -x
Solar input (W) 0 0 1358 0 0 0
Earth view factor 0 0.75 .2 .2 .2 .2
IR input (W) 0 177 47.4 47.4 47.4 47.4
Total input (W) 0 177 1405.4 47.4 47.4 47.4

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 = 250+1724.6 W

T = (1974.6 / (e*sigma*Asurf) )1/4 ~ 276ºK=3ºC