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<span class="style6">ESMD Course Material : Fundamentals of Lunar and
Systems Engineering for Senior Project Teams, with Application to a
Lunar Excavator</span><br />
<br />
Contact: David Beale, dbeale@eng.auburn.edu</div>
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<p align="center"><b>Thermal Considerations of Lunar Based Systems<a name="Chapter7"></a><o:p></o:p></b></p>
<p align="center"><b>Daniel K. Harris<o:p></o:p></b></p>
<p><b>1. Scope<o:p></o:p></b></p>
<p>The scope and breadth of this chapter will cover the least
rudimentary components of thermal design and analysis as applied to
lunar based systems. This chapter is as self-reliant and all
inclusive as could be reasonably expected while minimizing the
volume of this chapter. It is intended to give only the briefest of
outlines of the thermal science behind the engineering
described. The interested reader is guided to references that will
give more detail in the physics behind the heat<a href="Chapter7.htm#_ftn1" name="_ftnref1" title="">[1]</a> transfer
modes and mechanisms used here. The reader of this section is
assumed to have a baccalaureate in engineering or physics and is not
familiar at all with heat transfer and thermal design. The reader
is assumed to be interested in only the most general enveloping
analysis to guide his or her design of a mechanical structure
located and based on the lunar surface for extended time
periods. The design methodology is presented in an easy to mimic
style. The unique lunar environment and the challenges it presents
are offered in a simple to understand approach. The chapter
concludes with a case study giving the reader some final guidance,
and hopefully confidence in designing a thermal management approach
for whatever they are intending to deploy on the surface of our
moon. Radiation</p>
<p><b>2. Purpose<o:p></o:p></b></p>
<p>The objective of this chapter is to introduce the skill of
calculating temperatures and heat flux values for given boundary
conditions on the lunar surface. This skill will primarily consist
of adequately modeling radiation surface interchange phenomena in
both the UV and IR portions of the radiation spectrum. The methods
taught here will involve “hand” calculations using spreadsheets or
other easily available computer tools. More sophisticated
techniques requiring commercially available software are referenced
but not described in detail. Since the lunar diurnal cycle has a
period of approximately 28 Earth days, steady state calculations are
the most appropriate and are therefore stressed throughout this
chapter. The engineering design methodology applied here is to
identify worst case hot and cold conditions. Once these two worst
case extremes have been identified the engineer has to see how his
or her system can operate and survive under these two
conditions. Thermal remediation techniques and options will also be
taught in this chapter. It will then be up to the engineer to
“engineer” the system within the defined lunar conditions. </p>
<p><o:p> </o:p></p>
<p><b>1. Introduction<o:p></o:p></b></p>
<p>You may not be aware of just how large a role our atmosphere
plays in everyday life as a medium to transport heat and mass. Not
only does the atmosphere around us transport almost all of the
thermal energy to and from all entities, but it also plays a vital
role in moderating temperatures and diffusing (smearing out) large
discrepancies in our thermal environment. Air cooks our food, runs
our cars, heats and cools are homes, cools our computers, helps our
bodies regulate its temperature, and on and on. Needless to say
without an atmosphere thermal environments become very difficult for
anybody or anything to function within. Thermal design and
management is challenging enough in an atmosphere. But, in the
absence of an atmosphere (vacuum) it becomes more critical and a
main component of any design. Further, because of our everyday
experiences are from a perspective of being immersed in an
atmosphere, most aspects of a space thermal environment are
difficult to predict and in many cases non-intuitive. Therefore, we
cannot rely on our good judgment or everyday experiences when
designing a system for operation in space, or worse yet, on the
surface of the Moon. </p>
<p><b>a. </b><u>Heat conduction</u><a name="heatconduction"></a> is
the mechanism for heat transfer through bulk solids and fluids. It
is a gradient transport type of phenomena and is described by
Fourier’s empirical Law, which in its differential form is as shown
here. </p>
<p><img alt="" height="36" src="SiteImageFiles/Chapter7/image001.png" width="92" /></p>
<p>Once taken to its integral form it takes on the more familiar
appearance as,</p>
<p><img alt="" height="36" src="SiteImageFiles/Chapter7/image002.png" width="93" /></p>
<p>where k is the bulk material thermal conductivity. This heat
transfer mechanism will not be significant in most situations where
an external surface is interacting with the lunar
environment. However, once the system boundary conditions are
defined by the interactions with the external lunar thermal
environment, conduction heat transfer will be a very significant
heat transfer mechanism. </p>
<p><o:p> </o:p></p>
<p><b>b. </b><u>Radiation heat transfer</u><a name="radiationheattransfer"></a> phenomenon
is the dominant heat transfer mode on the lunar surface. Unlike
conduction heat transfer, which occurs inside a bulk solid or fluid,
radiation heat transfer is a surface phenomenon. Radiation heat
transfer occurs over the long wave length, infra-red, part of the
electromagnetic wave spectrum. All matter rejects and receives
thermal energy via thermal radiation. The radiating power of any
matter at a non-zero temperature is given by the Stefan-Boltzmann
equation,</p>
<p>
<img alt="" height="20" src="SiteImageFiles/Chapter7/image003.png" width="90" /></p>
<p>where A is the surface area, e is the emissivity, T is the
absolute temperature, and s is the Stefan-Boltzmann constant (5.670
51 x 10<sup>-8</sup> W/m K<sup>4</sup>). The thermo-optical
property of the surface called emissivity determines the ability of
the surface to both transmit and absorb (under the gray body
assumption) thermal radiation and lies between zero and unity. A
perfect thermal radiating surface will have an emissivity of one and
is termed a ‘blackbody’. While the physics behind this phenomenon
is quite complex, simple averaged values are used for the
emissivity. Surface emissivity can be both wave-length (spectrally)
and directionally dependent. Therefore, for the purposes of
engineering calculations values for the surface emissivity are used
that have been averaged (integrated) over all possible wavelengths
over all directions of a hemisphere. Bodies with an emissivity that
is spectrally independent are called ‘gray bodies’, where the
infra-red emissivity is assumed equal to the infra-red
absorbtivity. This is known as the gray body assumption and is
quite adequate for most common surfaces. </p>
<p>The interchange between two or more surfaces requires an optical
analysis to determine the view factor between the two surfaces. The
view factor between two surfaces (say surface i and surface j) is
identified by F<i><sub>ij</sub></i>. It also has a value between
zero and unity. It represents the percentage of view one surface
has to another surface. It is quite difficult to determine the view
factor even between two simple geometries. Therefore, a catalog of
common surface geometries is usually needed and the analyst can
simplify the calculations in order to estimate the view factor. See
for example [Modest, 1993], pp. 780-794 for some of the most common
shapes encountered. Once the view factor is determined then the
(net) radiation interchange between the surfaces at their respective
temperatures (<i>T<sub>i</sub></i>and <i>T<sub>j</sub></i>) is given
by the following equation. <o:p></o:p></p>
<p>
<img height="23" src="SiteImageFiles/Chapter7/image004.png" width="169" /></p>
<p>As an example we’ll compute the radiation heat transfer between a
sphere of radius r and a plane rectangle of dimensions 2l<sub>1</sub> by
2l<sub>2</sub>, separated by a distance d (measured from the
sphere’s center to the plane rectangle) and centered directly
‘under’ the sphere. </p>
<p align="center"><v:shapetype id="_x0000_t202" coordsize="21600,21600" o:spt="202" path="m,l,21600r21600,l21600,xe"><v:stroke joinstyle="miter"><v:path gradientshapeok="t" o:connecttype="rect"></v:path></v:stroke></v:shapetype><v:shape id="_x0000_s1094" type="#_x0000_t202" stroked="f" style="width: 204.9pt; height: 206.5pt; ">
<v:textbox>
<table cellpadding="0" cellspacing="0" width="100%">
<tr>
<td>
<div>
<p align="center">
<img height="177" src="SiteImageFiles/Chapter7/image005.png" width="228" /></p>
<p align="center">Figure 00: Geometry under analysis,
where only ¼ of plane is shown due to symmetry.)</p>
</div>
</td>
</tr>
</table>
</v:textbox><w:wrap type="none"><w:anchorlock></w:anchorlock></w:wrap></v:shape>
</p>
<p>As found in [Modest 1993] on page 792, item 47 shows the view
factor for this geometry under consideration as follows:</p>
<p>
<img height="76" src="SiteImageFiles/Chapter7/image007.png" width="459" /></p>
<p>In this example we’ll let the dimensions of the rectangle plane
be 2<i>d</i> by 2<i>d</i>. Therefore, both <i>l<sub>1</sub></i> and <i>l<sub>2</sub></i> are
equal to d, giving <i>D<sub>1</sub></i> and D<sub>2</sub> both equal
to unity. This gives <i>F</i><sub>1-2</sub> a value of,<o:p></o:p></p>
<p>
<img height="59" src="SiteImageFiles/Chapter7/image008.png" width="199" /></p>
<p>This gives the view factor of the sphere to one-quarter of the
rectangle; refer to Figure 00. The view factor of the sphere to the
entire plane rectangle is 4 times as much, or 0.1024. This can be
interpreted as saying that 10.24% of all the radiant energy leaving
the sphere will be incident onto the plane rectangle. Now there
exists a handy relationship for view factors between two surfaces
that allows the direct determination of the view factor from the
plane rectangle to the sphere (which is necessarily not the same
value). This is known as the <i>reciprocity relation</i>.<o:p></o:p></p>
<p>
<img height="22" src="SiteImageFiles/Chapter7/image009.png" width="80" /></p>
<p>This means that the view factor from the plane rectangle to the
sphere is found as<o:p></o:p></p>
<p>
<img height="20" src="SiteImageFiles/Chapter7/image010.png" width="107" /></p>
<p>
<img height="20" src="SiteImageFiles/Chapter7/image011.png" width="171" /></p>
<p>Now, if we let <i>d=2r</i>, then <i>F</i><sub>2-1</sub> becomes
equal to 0.0804. Only roughly 8% of the energy leaving the plane
rectangle is incident upon the sphere. </p>
<p>As an extension to more than two surfaces an additional
relationship is useful. The sum of all view factors for each
surface comprising a multi-surface enclosure is unity. This means
that for all surfaces comprising the enclosure, the following
relationship holds for all <i>N</i> surfaces.</p>
<p>
<img height="54" src="SiteImageFiles/Chapter7/image012.png" width="66" /></p>
<p>This is known as the <i>summation rule</i>. In fact, for an
enclosure comprised of <i>N</i> surfaces a total of <i>N</i><sup>2</sup> view
factors is needed. However, only <i>N(N</i>-1)/2 view factors need
to be determined directly by invoking the reciprocity relation and
the summation rule. </p>
<p>Carrying this example calculation forward, let’s consider values
for the temperatures and emissivities. This will allow for the last
component of a radiative heat transfer calculation to be
explained. The heat transferred between these two surfaces will be
found using the concept of radiosity. Radiosity is essentially the
concept whereby a surface can have thermal energy that it radiated
reflected back onto its surface. It is a concept needed for thermal
radiation calculations of multiple surfaces. Without going into too
much detail only the basics of radiosity network analyses (also
known as Oppenheim networks) is described here. </p>
<p>For gray and diffuse (directionally independent) surfaces the
emissivity e is assumed to be equal to the infra-red
absorbtivity a (not to be confused with the a representing the UV
absorbtivity), and therefore the reflectivity r of the surface is
equal to (1-a) or (1-e). The radiosity of a surface <i>J<sub>i</sub></i> can
be expressed as the following where <i>G<sub>i</sub> </i>is the
incident radiation onto the surface and <i>E<sub>bi</sub></i> is the
blackbody radiating power of the surface at its temperature, given
as <i>e<sub>i</sub>sT</i><sub>i</sub><sup>4</sup>.</p>
<p>
<img height="20" src="SiteImageFiles/Chapter7/image013.png" width="139" /></p>
<p>Therefore, the net radiative heat transfer rate from a surface is
given as follows. </p>
<p>
<img height="39" src="SiteImageFiles/Chapter7/image014.png" width="113" /></p>
<p>Hence, if the emissive power that a surface would have if it were
black exceeds its radiosity, there is net radiation heat transfer
from the surface; if the inverse is true the net transfer is to the
surface. For a network of surfaces that can view one another the
radiation balance for each surface can be represented as follows.</p>
<p>
<img height="54" src="SiteImageFiles/Chapter7/image015.png" width="111" /></p>
<p>Using this relationship for each surface a network can be
generated for all the surfaces and a non-linear system of equations
is produced that can be numerically solved. For a simple two
surface interchange the net radiation between the two surfaces is
found to be as follows. </p>
<p>
<img height="55" src="SiteImageFiles/Chapter7/image016.png" width="265" /></p>
<p>Since the objective of this chapter is to describe analyzing a
lunar thermal environment, we have to leave the radiation heat
transfer discussion and return to the application of these
principles described. Most design calculations that try to define
both the worst case hot and cold environment analyses will usually
involve only simple surface geometries interacting with other nearby
surfaces including the lunar surface, and space.</p>
<p><b>c. </b><u>Ultraviolet absorption</u><a name="uvabsorption"></a> as
a result of direct or reflected sunlight incident on external
surfaces solar absorption is another lunar surface phenomenon that
needs attention. Just as was the case for infra-red radiation heat
transfer discussed in the previous section only the rudiments are
covered here for design purposes. Any surface when exposed to
ultraviolet (uv) radiation will either reflect (<i>r</i>), absorb (<i>a</i>)or
transmit (<i>t</i>)the energy: note that <i>r+a+t=1</i>. For most
cases on the lunar surface (except for windows) the transmitted uv
energy will be zero. The energy will be either reflected or
absorbed and converted to heat. The amount of heat generated on a
surface due to absorbed uv energy (i.e. sunlight) is simply the
incident radiation flux onto that surface times the uv absorption
coefficient a, </p>
<p>
<img height="20" src="SiteImageFiles/Chapter7/image017.png" width="109" /></p>
<p>where <i>A </i>is the surface area, a is the absorbtivity, and <i>G</i><sub>solar</sub> is
the incident solar flux (ca. 1353 W/m<sup>2</sup> on the lunar
surface). Recall that the emissivity covers the infra-red region of
the electromagnetic spectrum while the absorbtivity discussed here
is for the ultra violet region. This is not to be confused with the
gray body assumption discussed previously where a surface’s
emissivity can be assumed equal to its absorbtivity over the
infra-red region. It is imperative that these two absorbtivities
not be used interchangeably. An example will help demonstrate the
point.</p>
<p>As an example let’s compute the resulting temperature of a clear
anodized aluminum surface exposed to direct sunlight on the lunar
surface and viewing only deep space, (i.e. no view to the lunar
surface). A simple heat balance applied to the surface gives that
the absorbed uv energy must equate to the infra-red energy radiated
away from the surface. </p>
<p>
<img height="20" src="SiteImageFiles/Chapter7/image018.png" width="116" /></p>
<p>
<img height="36" src="SiteImageFiles/Chapter7/image020.png" width="82" /></p>
<p>Using values of a=0.35, e=0.76 and <i>G</i><sub>solar</sub>=1353
W/m<sup>2</sup> gives a temperature of 323.8 K. As an interesting
aside the equilibrium surface temperature exposed to direct sunlight
with an a/e of unity is 393 K, and with an a/e = 0.25 is 277.9
K. This technique of computing equilibrium temperatures will be
expounded upon further when discussing thermal analysis of lunar
based devices. <o:p></o:p></p>
<p><u>Solar Entrapment</u> - Finally, a word of caution about solar
entrapment should be included here. Solar entrapment occurs when
direct solar energy is incident upon an open enclosure or
cavity. What happens is that the solar energy is almost all trapped
due to multiple reflections inside the cavity while the view factor
of the inner surface (which is heating up from the absorbed solar
energy) to the outside environment (the sink) is very low. In other
words, the inner surface of the cavity sees mostly itself and
therefore has very low emissive power to the outside
environment. The result of this entrapped solar energy without any
way of being effectively radiated away is for the surface to become
extremely hot. Therefore, it is always advisable to avoid designing
any cavity like structure that can be exposed to direct (or even
reflected) sunlight.</p>
<p><b>2. Lunar Thermal Environment<o:p></o:p></b></p>
<p>Quite a lot has already been discussed regarding the lunar
environment in an earlier chapter. This section will collect the
pertinent information needed to perform a complete analysis. While
redundant, it is intended to pull together in all one place the
tables and properties needed for lunar specific thermal
analyses. Essentially, there are three areas of interests in
defining the lunar thermal environment: the length of lunar nights,
the direct and reflected solar flux, lunar soil temperatures, and
associated lunar surface IR energy.</p>
<p><b>a. </b><u>Lunar day cycles</u><a name="Lunardaycycles"></a> are
quite long when compared to our own. In fact, at the equator the
lunar day lasts 14.75 Earth days out of the 29.5 Earth days in the
lunar day. Also, this daylight length period is a weak function of
latitude due to the very slight angle (1.533<sup>o</sup>) of the
Moon’s polar axis (spin axis) off the solar ecliptic. Comparing that
with the Earth’s orbit inclination of 23.5° we can understand why
the Moon has no annual seasons. Therefore, only at or very near
the Moon’s poles (within 5 to 7 degrees of the pole) are there
significantly long daylight and dark periods. At the poles the
daylight periods will last 180 Earth days. The Figure below shows
the length of daylight periods as a function of latitude for the
northern hemisphere as taken from [Eckart 2006].</p>
<p align="center"><v:shape id="_x0000_s1092" type="#_x0000_t202" stroked="f" style="width: 479.4pt; height: 276.45pt; ">
<v:textbox>
<table cellpadding="0" cellspacing="0" width="100%">
<tr>
<td>
<div>
<p align="center">
<img height="60%" src="SiteImageFiles/Chapter7/image019.png" width="60%" /></p>
<p align="center">Figure 01: Lunar Diurnal Cycle as
taken from [Eckart, 2006], pg. 118<a name="lunardaycyclechart"></a>.</p>
</div>
</td>
</tr>
</table>
</v:textbox><w:wrap type="none"><w:anchorlock></w:anchorlock></w:wrap></v:shape>
</p>
<p><b>b. </b><u>Solar light</u><a name="Solarlight"></a> on
the lunar surface is quite different in nature than that which we
are used to on the Earth. This is due to the lack of any lunar
atmosphere which scatters sunlight before reaching the
surface. Therefore, the solar energy incident on any lunar surface
is highly directional and therefore depends more strongly on
latitude and lunar hour. For a zenith facing surface the percentage
of full solar flux incident (solar flux fraction) is the product of
the sine of the local lunar time angle and the cosine of the
latitude, where the equator is 0 degrees latitude. The local lunar
time angle is defined as 0 degrees at sunrise, 90 degrees at noon
(sub-solar), and 180 degrees at sunset. The resulting solar flux
fraction for a zenith surface is shown here in Figure
02. Additionally, for an east oriented surface (leading face) the
percentage of full solar flux incident (solar flux fraction) is the
product of the cosine of the local lunar time angle and the cosine
of the latitude, where (again) the equator is 0 degrees latitude is
the equator. The solar flux fraction is a maximum value at sunrise
and vanishes at local noon for the remainder of the orbit. The
solar flux fraction is shown below in Figure 03. Consequentially,
for a west oriented face (trailing face) the solar flux fraction is
the opposite where the flux starts at noon and peaks at
sunset. This is shown for both faces in Figure 03. </p>
<p align="center"><v:shape id="_x0000_s1090" type="#_x0000_t202" stroked="f" style="width: 333.75pt; height: 256.2pt; ">
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<table cellpadding="0" cellspacing="0" width="100%">
<tr>
<td>
<div>
<p><img height="360" src="SiteImageFiles/Chapter7/image021.png" width="529" /></p>
<p align="center">Figure 02: Incident Solar Fraction on
a Zenith Oriented Surface dependence upon both Latitude
and Local Lunar Time. The Lunar Local Time is defined
as sunrise at 6:00 and sunset at 18:00, which represents
14.75 Earth days. </p>
</div>
</td>
</tr>
</table>
</v:textbox><w:wrap type="none"><w:anchorlock></w:anchorlock></w:wrap></v:shape>
</p>
<p>Finally, for a north oriented face in the southern hemisphere, or
equivalently, a south oriented face in the northern hemisphere, the
percentage of full solar flux incident on the surface is similar in
form to Figure 02 for the zenith face only the latitude dependence
is reversed. The resulting solar flux fraction graph is shown in
Figure 04. </p>
<p align="center"><o:p> </o:p></p>
<p align="center"><v:shape id="_x0000_s1086" type="#_x0000_t202" stroked="f" style="width: 333.75pt; height: 334.95pt; ">
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<tr>
<td>
<div>
<p><img height="360" src="SiteImageFiles/Chapter7/image022.png" width="529" /></p>
<p align="center">Figure 04: Incident Solar Fraction on
a North Oriented Surface in the southern hemisphere
dependence upon both Latitude and Local Lunar
Time. Note: this graph is identical to that for a South
Oriented Surface in the northern hemisphere. The Lunar
Local Time is defined as sunrise at 6:00 and sunset at
18:00, which represents 14.75 Earth days.</p>
</div>
</td>
</tr>
</table>
</v:textbox><w:wrap type="none"><w:anchorlock></w:anchorlock></w:wrap></v:shape>
</p>
<p><b>c. </b><u>Albedo</u><a name="albedo"></a> (the
reflected solar energy from the surface) on the lunar surface is
only around 7% of the solar flux incident onto the lunar
surface. That is, the lunar surface absorbs almost 93% of all
incident solar energy it receives. This compares to the Earth’s
orbital average albedo of 37%. Therefore, all surfaces exposed to
lunar surfaces during daylight periods will be exposed to the lunar
albedo, which be assumed as diffuse (directionally
independent). Recall that the absorbed albedo (heat) will be the
product of any incident albedo and the absorbtivity. The incident
albedo will also be a function of both the latitude and the local
lunar time. For analysis purposes the solar flux fraction for a
zenith oriented face shown in Figure 02 above can be used to find
the local albedo by multiplying the fraction found in the figure by
94.7 W/m<sup>2</sup> (i.e. 7% of 1353 W/m<sup>2</sup>). The view
factor from the surface to the lunar surface will also be needed in
addition to the solar absorbtivity. Therefore, the total heat
generated on a surface by both direct solar and albedo absorption
will be as follows.</p>
<p>
<img height="36" src="SiteImageFiles/Chapter7/image023.png" width="273" /></p>
<p>In the equation above the <i>F<sub>solar</sub></i> factor is
found in Figure 00 and the geometrical view factor, <i>F<sub>view</sub>,</i> will
have to be determined based upon the orientation of the surface
normal to the lunar surface.<o:p></o:p></p>
<p><b>d. </b><v:shape id="_x0000_s1030" type="#_x0000_t202" stroked="f" style="position: absolute; left: 0px; text-align: left; margin-left: 0px; margin-top: 76.2pt; width: 455.4pt; height: 140.3pt; z-index: 19; "><v:textbox><table cellpadding="0" cellspacing="0" width="100%">
<tr>
<td>
<div>
<p align="center">Table 01: Lunar Soil Monthly Averages
and Ranges, taken from [Eckart, 2006], pg. 143.</p>
<table border="1" cellpadding="0" cellspacing="0" width="570">
<tr>
<td width="94">
<p align="center"><b>Temperature<o:p></o:p></b></p>
</td>
<td width="80">
<p align="center"><b>Shadowed<o:p></o:p></b></p>
<p align="center"><b>Polar<o:p></o:p></b></p>
<p align="center"><b>Craters<o:p></o:p></b></p>
</td>
<td width="78">
<p align="center"><b>Other<o:p></o:p></b></p>
<p align="center"><b>Polar<o:p></o:p></b></p>
<p align="center"><b>Areas<o:p></o:p></b></p>
</td>
<td width="78">
<p align="center"><b>Front<o:p></o:p></b></p>
<p align="center"><b>Equatorial<o:p></o:p></b></p>
</td>
<td width="78">
<p align="center"><b>Back<o:p></o:p></b></p>
<p align="center"><b>Equatorial<o:p></o:p></b></p>
</td>
<td width="78">
<p align="center"><b>Limb<o:p></o:p></b></p>
<p align="center"><b>Equatorial<o:p></o:p></b></p>
</td>
<td width="84">
<p align="center"><b>Typical<o:p></o:p></b></p>
<p align="center"><b>Mid-Latitudes<o:p></o:p></b></p>
</td>
</tr>
<tr>
<td width="94">
<p align="center"><b>Average<o:p></o:p></b></p>
</td>
<td width="80">
<p align="center">40 K<o:p></o:p></p>
</td>
<td width="78">
<p align="center">220 K<o:p></o:p></p>
</td>
<td width="78">
<p align="center">254 K<o:p></o:p></p>
</td>
<td width="78">
<p align="center">256 K<o:p></o:p></p>
</td>
<td width="78">
<p align="center">255 K<o:p></o:p></p>
</td>
<td width="84">
<p align="center">220 – 255 K<o:p></o:p></p>
</td>
</tr>
<tr>
<td width="94">
<p align="center"><b>Monthly Range<o:p></o:p></b></p>
</td>
<td width="80">
<p align="center">none<o:p></o:p></p>
</td>
<td width="78">
<p align="center">10 K<o:p></o:p></p>
</td>
<td width="78">
<p align="center">140 K<o:p></o:p></p>
</td>
<td width="78">
<p align="center">140 K<o:p></o:p></p>
</td>
<td width="78">
<p align="center">140 K<o:p></o:p></p>
</td>
<td width="84">
<p align="center">110 K<o:p></o:p></p>
</td>
</tr>
</table>
<p><o:p> </o:p></p>
</div>
</td>
</tr>
</table>
</v:textbox><w:wrap type="square"></w:wrap></v:shape><u>The lunar
soil temperature</u><a name="lunarsoiltemperature"></a> is also
needed for lunar thermal environment analyses. The surface soil
(regolith and rocks, etc.) on the Moon has a very high diurnal
temperature swing between lunar noon and lunar night. In fact, the
temperature on the back equatorial has a range of 140 K to 256
K. Table 01 below taken from [Eckart 2006] and shows estimated
lunar surface temperatures</p>
<p>A first-order estimate of off equatorial variation of soil
temperatures as a function of the latitude angle (b) is also given
by [Eckart 2006], pg. 143, as follows.</p>
<p>
<img height="24" src="SiteImageFiles/Chapter7/image025.png" width="137" /></p>
<p>As an alternative, the lunar surface profile temperatures as a
function of latitude and local lunar time is shown in Figure 05, as
taken from [Eckart 2006], pg. 30]. </p>
<p>At the Apollo sites the mean temperature 35 cm below the surface
are 40 to 45 K above those at the surface. Figure 06 below shows
the temperature and soil thermal conductivity as a function of depth
at the Apollo 15 and Apollo 17 landing sites [Eckart, 2006]. Table
02, following gives some important soil thermal parameters and their
respective ranges. In Table 02, g is the thermal inertia of the
soil. </p>
<p>These influences include heat conduction to and from any surface
structure, and the infra-red radiation heat transfer interactions
with any structures that have an optical view of the
surface. Whereby, the latter is usually the most dominant mode of
heat transfer to and from a structure on the lunar surface. Knowing
the lunar thermal environment gives the designer the information
necessary to find the environmental thermal loads his or her lunar
device will be exposed to on the lunar surface. </p>
<p><v:shape id="_x0000_s1084" type="#_x0000_t202" stroked="f" style="width: 561pt; height: 280.2pt; ">
<v:textbox>
<table cellpadding="0" cellspacing="0" width="100%">
<tr>
<td>
<div>
<p align="center"><img height="312" src="SiteImageFiles/Chapter7/image026.png" width="328" /></p>
<p align="center">Figure 05: Lunar Surface Temperature
Profile at Different Latitudes, taken from [Eckart,
2006], pg. 301.</p>
</div>
</td>
</tr>
</table>
</v:textbox><w:wrap type="none"><w:anchorlock></w:anchorlock></w:wrap></v:shape>
</p>
<p align="center"><v:shape id="_x0000_s1082" type="#_x0000_t202" stroked="f" style="width: 561pt; height: 250.95pt; ">
<v:textbox>
<table cellpadding="0" cellspacing="0" width="100%">
<tr>
<td>
<div>
<p><img height="278" src="SiteImageFiles/Chapter7/image027.png" width="680" /></p>
<p align="center">Figure 06: Lunar Soil Temperature and
Thermal Conductivity as a Function of Depth at the
Apollo 15 and Apollo 17 Landing Sites, as taken from
[Eckart, 2006], pg. 140. </p>
</div>
</td>
</tr>
</table>
</v:textbox><w:wrap type="none"><w:anchorlock></w:anchorlock></w:wrap></v:shape>
</p>
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<tr>
<td>
<div>
<p align="center">Table 02 Lunar Soil Temperature and
Thermal Conductivity as a Function of Depth at the
Apollo 15 and Apollo 17 Landing Sites, as taken from
[Eckart, 2006], pg. 141. </p>
<p align="center"><img height="291" src="SiteImageFiles/Chapter7/image028.png" width="646" /></p>
</div>
</td>
</tr>
</table>
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</p>
<p><b>3. Thermal Analysis & Approach</b> - The
first step needed to design a system capable of operating and
surviving on the lunar surface is to define the effective
environmental temperatures expected from the defined environmental
loads. The design can then mature to define the exterior surface
finishes needed, heat rejection radiator sizing, heater power
requirements, etc. Before an analysis that can direct a thermal
design can commence the lunar location, thermal requirements (both
operating and survival), and the system power requirements and
budget need to be known. These are the constraints that the
mechanical design of any lunar device will have to operate and
survive under. The thermal design approach is shown graphically in
the following flow diagram. <b><o:p></o:p></b></p>
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<p>
<img height="325" src="SiteImageFiles/Chapter7/image029.png" width="378" /></p>
<p align="center">Thermal Design Flow Sequence<a name="thermaldesignsequence"></a>.</p>
</div>
</td>
</tr>
</table>
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</p>
<p><b>a. </b><u>The thermal requirements</u> (supplied
or derived) that are needed in developing a thermal design include
the following list:</p>
<p>· <v:shape id="_x0000_s1029" type="#_x0000_t202" stroked="f" style="position: absolute; left: 0px; text-align: left; margin-left: 0px; margin-top: 71.1pt; width: 561pt; height: 161pt; text-indent: 0px; z-index: 18; "><v:textbox><table cellpadding="0" cellspacing="0" width="100%">
<tr>
<td>
<div>
<p align="center">Table 03: Some Typical Components and
Operational Temperature Ranges, as adapted from [Wertz &
Larson, 1999], pg. 428.</p>
<div align="center">
<table border="1" cellpadding="0" cellspacing="0">
<tr>
<td rowspan="2" width="157">
<p align="center"><b>Component</b><o:p></o:p></p>
</td>
<td colspan="2" valign="top" width="246">
<p align="center"><b>Typical Temperature
Ranges (°C)</b><o:p></o:p></p>
</td>
</tr>
<tr>
<td valign="top" width="126">
<p align="center"><b>Operational</b><o:p></o:p></p>
</td>
<td valign="top" width="120">
<p align="center"><b>Survival</b><o:p></o:p></p>
</td>
</tr>
<tr>
<td valign="top" width="157">
<p>Batteries<o:p></o:p></p>
</td>
<td valign="top" width="126">
<p align="center">0 to 15<o:p></o:p></p>
</td>
<td valign="top" width="120">
<p align="center">-10 to 25<o:p></o:p></p>
</td>
</tr>
<tr>
<td valign="top" width="157">
<p>Power Box Baseplates<o:p></o:p></p>
</td>
<td valign="top" width="126">
<p align="center">-10 to 50<o:p></o:p></p>
</td>
<td valign="top" width="120">
<p align="center">-20 to 60<o:p></o:p></p>
</td>
</tr>
<tr>
<td valign="top" width="157">
<p>C&DH Box Baseplates<o:p></o:p></p>
</td>
<td valign="top" width="126">
<p align="center">-20 to 60<o:p></o:p></p>
</td>
<td valign="top" width="120">
<p align="center">-40 to 75<o:p></o:p></p>
</td>
</tr>
<tr>
<td valign="top" width="157">
<p>Antenna Gimbals<o:p></o:p></p>
</td>
<td valign="top" width="126">
<p align="center">-40 to 80<o:p></o:p></p>
</td>
<td valign="top" width="120">
<p align="center">-50 to 90<o:p></o:p></p>
</td>
</tr>
<tr>
<td valign="top" width="157">
<p>Antennas<o:p></o:p></p>
</td>
<td valign="top" width="126">
<p align="center">-100 to 100<o:p></o:p></p>
</td>
<td valign="top" width="120">
<p align="center">-120 to 120<o:p></o:p></p>
</td>
</tr>
<tr>
<td valign="top" width="157">
<p>Solar Panels<o:p></o:p></p>
</td>
<td valign="top" width="126">
<p align="center">-150 to 110<o:p></o:p></p>
</td>
<td valign="top" width="120">
<p align="center">-200 to 130<o:p></o:p></p>
</td>
</tr>
</table>
</div>
<p><o:p> </o:p></p>
</div>
</td>
</tr>
</table>
</v:textbox><w:wrap type="square"></w:wrap></v:shape>Temperature
limits, both operational and survival, for all system
components. Often the temperature of heat rejecting surfaces
(called radiators) and other surfaces exposed to the external lunar
environment will be defined by upper level system requirements. An
example of typical component survival and operational temperatures
is shown in Table 03. </p>
<p>· Equipment power dissipations and operating
modes. Usually the system will define several operational modes
such as nominal, stressed, and failure modes. Also, many components
can have operational loads that are dependent upon the health of the
system and other components. A complete description of all these
operating modes is necessary to indentify the worst case extremes. </p>
<p>· Location on lunar surface (especially the
latitude). The lunar location will define the direct and indirect
solar loads during lunar day along with the soil temperatures during
the lunar day and lunar night. There are no seasonal variations on
the Moon. </p>
<p>· Thermal s distortion budgets (e.g. antenna
surfaces). Also, thermal induced stresses in critical design
features, such as bearings, need to be identified. </p>
<p>· Interfaces with other sub-systems. </p>
<p>· Any other special thermal control requirements unique
to the design and mission.</p>
<p>It should be completely apparent from this list that a thermal
design is never created in a vacuum. The thermal design has to
support and yet lead the design at various stages along the design
maturation cycle. Communication between engineers during all design
iterations is crucial. Space engineering folklore is replete with
examples of simple communication lapses becoming major engineering
disasters during launch or in orbit. </p>
<p><b>b. </b><u>Worst Case Hot and Cold Assessments</u> must
be determined to be sure the final thermal control system (TCS)
designed will envelope all possible mission scenarios. The
parameters used are given in Table 04 below where it is recognized
that material thermo-optical properties change over time in space. </p>
<p><v:shape id="_x0000_s1026" type="#_x0000_t202" stroked="f" style="position: absolute; margin-left: 0px; margin-top: 0px; width: 561pt; height: 259.2pt; z-index: 16; ">
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<p align="center">Table 04: Thermal Parameter Variation
for Hot and Cold Assessments as taken and amended from
[Wertz & Larson, 1999], pg. 455. <b><o:p></o:p></b></p>
<div align="center">
<table border="1" cellpadding="0" cellspacing="0">
<tr>
<td valign="top" width="174">
<p align="center"><b>Parameter<o:p></o:p></b></p>
</td>
<td valign="top" width="138">
<p align="center"><b>Hot Case<o:p></o:p></b></p>
</td>
<td valign="top" width="125">
<p align="center"><b>Cold Case<o:p></o:p></b></p>
</td>
</tr>
<tr>
<td valign="top" width="174">
<p align="center"><b>Solar Constant<o:p></o:p></b></p>
</td>
<td valign="top" width="138">
<p align="center">1353 W/m<sup>2</sup></p>
</td>
<td valign="top" width="125">
<p align="center">0</p>
</td>
</tr>
<tr>
<td valign="top" width="174">
<p align="center"><b>Albedo<o:p></o:p></b></p>
</td>
<td valign="top" width="138">
<p align="center">97.7 W/m<sup>2</sup></p>
</td>
<td valign="top" width="125">
<p align="center">0</p>
</td>
</tr>
<tr>
<td valign="top" width="174">
<p align="center"><b>IR (equatorial)<o:p></o:p></b></p>
</td>
<td valign="top" width="138">
<p align="center">1,312 W/m<sup>2</sup> (390
K)</p>
</td>
<td valign="top" width="125">
<p align="center">3.7 W/m<sup>2</sup> (90 K)</p>
</td>
</tr>
<tr>
<td valign="top" width="174">
<p><b>Radiators:<o:p></o:p></b></p>
<p align="center"><b>Solar Absorptance<o:p></o:p></b></p>
<p align="center"><b>IR Emittance<o:p></o:p></b></p>
</td>
<td valign="top" width="138">
<p align="center"><o:p> </o:p></p>
<p align="center">Maximum</p>
<p align="center">Minimum</p>
</td>
<td valign="top" width="125">
<p align="center"><o:p> </o:p></p>
<p align="center">Minimum</p>
<p align="center">Maximum</p>
</td>
</tr>
<tr>
<td valign="top" width="174">
<p><b>MLI:<o:p></o:p></b></p>
<p align="center"><b>Solar Absorptance<o:p></o:p></b></p>
<p align="center"><b>IR Emissivity<o:p></o:p></b></p>
<p align="center"><b>Effectiveness<o:p></o:p></b></p>
</td>
<td valign="top" width="138">
<p align="center"><o:p> </o:p></p>
<p align="center">0.55</p>
<p align="center">.67</p>
<p align="center">0.01 cold side</p>
<p align="center">0.03 Sun side</p>
</td>
<td valign="top" width="125">
<p align="center"><o:p> </o:p></p>
<p align="center">0.35</p>
<p align="center">0.75</p>
<p align="center">0.03 cold side</p>
<p align="center">0.01 Sun side</p>
</td>
</tr>
<tr>
<td valign="top" width="174">
<p align="center"><b>Power Dissipation
Levels<o:p></o:p></b></p>
</td>
<td valign="top" width="138">
<p align="center">Maximum</p>
</td>
<td valign="top" width="125">
<p align="center">Minimum</p>
</td>
</tr>