-
A Modelica Library for Spacecraft Thermal Analysis
Tobias Posielek1
1Institute of System Dynamics and Control, DLR German Aerospace
Center, Oberpfaffenhofen, [email protected]
AbstractIn spacecraft missions it is vital to maintain all
space-craft components within their required temperaturelimits.
Thus, a model incorporating all main heatfluxes acting on the
spacecraft is necessary to allowfor the design of a thermal control
subsystem. Thispaper introduces the thermal space systems
librarywhich implements common models of radiation andthermal
components of a spacecraft. Special effort isput into the
calculations of the angles describing theorientation of the
spacecraft with respect to sun andearth. Issues occurring due to
the recalculation of theangles in each time step are shown and
methods fortheir determinations are given.Keywords: space modeling,
thermal modeling, angledetermination
1 IntroductionIn spacecraft engineering, it is essential to
ensurethat all components operate in their appropriate tem-perature
range to avoid malfunction and equipmentbreakage. Therefore, an
analysis of the thermal dy-namics is a necessity to design the
required thermalcontrol (Gilmore and Bello 1994), (Meseguer,
Pérez-Grande, and Sanz-Andrés 2012), (Fortescue, Swinerd,and Stark
2011). A rigorous description of the ther-mal system is difficult
as it has to incorporate theorbit and the orientation of the
spacecraft during themission, as well as the sun’s position and the
dissi-pated energy within the spacecraft. The modellingof the
thermal system is a present topic of interest(Ruan, Hu, and Sun
2017) (Lefeng et al. 2017) (Qianet al. 2015). Approaches of various
complexity ex-ist to design the thermal control. Simple design
ap-proaches consider only static worst case scenarios toaccount for
degradation and orbit thermal dynamics(Larson and Wertz 1991).
Other methods use ana-lytical models to obtain the dynamic
evolution of thetemperature over the course of multiple orbits
(Tsai2004).
The proposed library allows the simulation of thecomplete
spacecraft system including the thermal sys-tem as well as the
electric and mechanical systemproviding the dissipated energy and
spacecraft ori-entation dependent on the spacecraft mission.
Thelibrary is proposed in view of simple analytical mod-
els. Generally, a spacecraft is modelled by a hugenumber of
nodes with different heat fluxes acting oneach. We will only model
the most important nodese.g. each surface may be modelled as a node
for acuboid spacecraft. For each of these nodes the tem-perature
dynamic is determined by the dynamic ofits adjacent nodes and the
four main heat flows dueto the environment. One main point which
will beilluminated is the calculation of the angle betweenthe
spacecraft surfaces and its surroundings. As theattitude of
spacecraft is usually not known a prioriand determined online,
suitable methods to calculatethis angle are proposed. The library
is created inview of earth orbiting spacecraft. However, the
li-brary can also be used for simulations of spacecraftleaving
earth orbit as long as modifications regardingthe coordinate
systems and approximations, such asshadow calculations, are made.
The proposed libraryuses the other Modelica-based libraries of the
Insti-tute of System Dynamics and Control at the DLRGerman
Aerospace Center such as the Environmentlibrary (Briese, Klöckner,
and M. Reiner 2017) andSpaceSystems library (M. J. Reiner and Bals
2014).The library is created as an in-house library as a partof the
design of an energy management for spacecraft.Section 2 introduces
the essential fundamentals forthe thermal dynamics. Section 3 gives
details to theModelica implementation and in Section 4 an exam-ple
scenario is simulated to show the functionality ofthis library.
2 FundamentalsThis section introduces the coordinate systems,
heatfluxes, solar angles, form factor and shadow functionnecessary
to simulate the spacecraft thermal system.
2.1 Coordinate SystemsThe Earth-Centered Inertial (ECI) Frame is
definedsuch that the xI-axis points in direction of vernalequinox,
this is the intersection between the equa-tor and the sun’s
apparent orbit during spring. ThezI-axis is parallel to the mean
Earth’s rotation axisand towards the North Pole and the yI-axis
completesright handed coordinate system. For all following
ref-erence frames the rotation matrix to ECI coordinatesis given by
their coordinate axes. Each coordinatesystem will be denoted with a
superscript which will
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s
rξ
nφ
(a) Main angles influencing the temperature evolutionin a
spacecraft. The solar zenith angle ξ is definedas the angle between
the vector to the spacecraft rand the vector pointing to the sun s.
The normalsolar angle φ describes the angle between the normalof a
spacecraft surface n and the vector pointing fromspacecraft to the
sun. The dotted line describes thespacecraft orbit.
(s> xO )x
O +(s> zO )z
Oθβ
ξ
s
r
(b) Angles describing the influence of the solar zenithangle ξ.
The solar noon angle θ describes the anglebetween the vector
pointing to the spacecraft r andthe solar noon, i.e. the vector
pointing to the sunprojected on the orbit plane (xOs)xO +(zOs)zO.
Thebeta angle β is defined as the angle between the orbitplane and
the vector pointing to the sun s.
Figure 1. Solar Angles
be used for the notation of their coordinate axes androtation
matrices. We denote T S,I =
[xS yS zS
]>as the transformation from ECI coordinates to an ar-bitrary
coordinate system with superscript S. So forrI in ECI coordinates
the transformed vector rS iscalculated via
rS = T S,IrI . (1)The orbit frame is defined for a spacecraft in
an el-
liptical orbit with position rI(t) and velocity vI(t) ininertial
coordinates by the yO-axis which is normal tothe orbit plane in
direction of negative angular mo-mentum, the zO-axis which points
to geocentric nadirand the xO-axis which completes the right handed
co-ordinate system and is for circular orbits in directionof
velocity. We omit the time argument on the righthand side and
obtain the transformation from ECIcoordinates to orbit coordinates
as
TO,I(t) =[rI×(rI×vI)‖rI×(rI×vI)‖ −
rI×vI‖rI×vI‖ −
rI(t)‖rI(t)‖
]>. (2)
2.2 Environmental Heat FluxesMainly four environmental heat
fluxes are actingon a spacecraft surface, namely the heat flux
dueto direct solar irradiation, the solar radiation re-flected by
the earth, the radiation of the earth emit-ted in the infra-red
spectrum and the radiation ofthe spacecraft emitted to deep space
(Larson andWertz 1991),(Meseguer, Pérez-Grande, and Sanz-Andrés
2012). Each of these fluxes and its calculationis introduced in
this section.
2.2.1 Direct SolarThe solar radiation is the main factor
influencing tem-perature changes of the spacecraft. A solar
constantGs0 is defined as in (Meseguer, Pérez-Grande,
andSanz-Andrés 2012) which gives the mean solar irradi-ance acting
on a unit area perpendicular to the solarrays in a distance of 1ua
where ua denotes the astro-nomical unit. As the amount of
irradiance crossingspherical surfaces with different radii is
assumed tobe constant, the solar irradiation Gs scales with
dis-tance as
Gs(d) =Gs0d0d
2
where d is the distance in astronomical units andd0 = 1ua. The
solar energy is mostly distributedin visual and short wavelength
infra-red (Larson andWertz 1991). This allows for surfaces which
are veryreflective in the solar spectrum but highly emissiveto long
wavelength infra-red. A simple analyticalmodel incorporates the
angle φ = φ(n,r,s) ∈ [0,π]between the surface normal and the sun
and theshadow of the earth described by the shadow coef-ficient ν =
ν(r,s) ∈ [0,1] introduced in Section 2.5.Then the acting solar flux
reads
Qsun ={αGs
(‖s−r‖1ua
)Acos
(φ)ν if 0< φ < π2
0 if π2 < φ < π(3)
where α denotes the solar absorptance of the surfaceand A the
area of the surface.
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2.2.2 AlbedoBy albedo we denote the part of the solar
radiationwhich is reflected by the earth or scattered by theplanet
surface and atmosphere. Combining the simplemodels from (Larson and
Wertz 1991) and (Meseguer,Pérez-Grande, and Sanz-Andrés 2012) we
obtain
Qalb ={ρalbαGs(d)AFform cos
(ξ)
if 0< ξ < π20 if π2 < ξ < π
(4)
where ρalbedo ∈ [0,1] is the albedo coefficient. Thiscoefficient
can vary over the course of an orbit anddepends on the orbits
inclination. This model incor-porates the solar zenith angle
ξ(s,r)∈ [0,π], the anglebetween the sun and the spacecraft, and a
form factorFform(r,n) defined in Section 2.4 to describe the partof
the radiation that actually strikes the spacecraftsurface.2.2.3
Planetary RadiationAs planetary radiation we denote the thermal
radia-tion which is emitted by the planet as long
wavelengthinfra-red radiation. The emitted radiation can be
cal-culated as the absorbed solar radiation of the planetminus the
radiation emitted via albedo. Then, by as-suming the planet to be a
black body we obtain theplanetary infra-red thermal heat acting on
a sufraceof a spacecraft as in (Meseguer, Pérez-Grande,
andSanz-Andrés 2012)
Qplanet(r,n) = εAFform(r,n)σT 4p (5)
where Tp denotes the black body temperature of theearth, σ the
Stefan-Boltzmann constant and ε theinfra-red emissivity of the
surface. Instead of usingthe black body temperature of the earth
(5), it isoften written as
Qplanet(r,n) = εAFform(r,n)IIR (6)
where IIR is the intensity of earth infra-red flux toaccount for
the variation of Qplanet. Note that IIRis actually not a constant
but also varying over thecourse of an orbit. However, the variation
of IIR issmall in comparison to the albedo variation.2.2.4
Radiation to Deep SpaceThe outer surfaces of a spacecraft are
radiatively cou-pled to space. The energy of the reradiation to
spaceis usually in the long wave infra-red spectrum and canbe
described by
Qds = εAσT 4 (7)
where T denotes the temperature of the surface.These four heat
fluxes are the main environmen-
tal heat fluxes acting on the spacecraft. Other fluxesdues to
the environment exist but are neglected in the
analysis due to their minor influence on most space-craft.
Numerical values for the parameters describingthe solar
absorptivity and infra-red emissivity of dif-ferent surface can be
found in the literature such as(Larson and Wertz 1991). Hot and
cold case scenarioparameters for ρalb, IIR and Gs dependent on the
or-bit and can be found in (Larson and Wertz 1991).Formula to
describe the solar angles φ, ζ, the formfactor Fform and the shadow
coefficient ν are intro-duced in the following sections.
2.3 Thermal AnglesFor the calculation of solar, albedo and
infra-red ir-radiation, different angles describing the position
ofthe sun and the attitude of the surface are of inter-est. These
angles are visualized in Figure 1. Figure1a shows the zenith angle
ξ and normal solar angle φwhich are the angles influencing the
generation of heatand Figure 1b the solar noon angle θ and beta
angleβ which can describe the influence of the solar zenithangle as
will be explained later. In this section, wedefine these angles,
show the relations between themand introduce two different ways to
calculate theseangles.
Problem FormulationUsually two vectors v1 and v2 are given in a
referencecoordinate system. In order to calculate the anglebetween
these two vectors, it may seem advantageousto use the scalar
product as in formula (8) as youdo not need any other rotations or
coordinates sys-tem and use only the property of the scalar
product.This however, gives only angles between [0,π] whichis
sufficient for many calculations which use unevenfunctions but it
leads to undesired results when ro-tations are considered as
illustrated in Figure 3. Inthis figure the angle between the
vectors v1 and v2(t)is displayed on the left hand side over the
course ofa full uniform planar rotation illustrated in the mid-dle.
The angle moves between 0 and π which makesno unique identification
of the position of v2 fromthe angle possible. Note that in the
control context,the uniqueness issue can be solved by using
quater-nions which use the normal axes of the rotation asadditional
information. On the right hand side, thedesired angle evolution is
displayed which ensures thebijection between position and angle in
a single rota-tion. In order to achieve this angle definition
between(−π,π], we use the properties of cylinder coordinates.Such a
definition is simple if a coordinate system isconstructed which x−
y-plane describes the rotationplane or if the reference frame can
be rotated on therotation plane. Note however, that only a
minimumof information about the rotation is known and onecan only
rely on the current value v1(t) and v2(t) butnot on a closed
description of the functions v1(·) andv2(·). This information has
to be used to construct
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v1
v1 +v2
v1 −v2
x
y
(a) Angle definition using(8)
v1
v1 +v2
x
y
v1 −v2(b) Angle definition usingcoordinate system (9)
Figure 2. Different angle definitions
the same coordinate system at every time step overthe course of
the rotation. Problems using intuitivecoordinate systems
definitions are illustrated in Fig-ure 5 and 4. Thus, an additional
vector is necessaryto construct this coordinate system. In the case
of or-bit rotations this vector comes by the cross productof
velocity and position.
Angle Definitions
The most intuitive definition defines the angle θ ∈[0,π] as the
smaller positive angle between the twovectors v1,v2 ∈ R3 as
θ = ∠(v1,v2) := cos-1(
v>1 v2‖v1‖‖v2‖
)(8)
where cos-1 denotes the inverse of the cosine with adomain of
[0,π]. This definition is sufficient for mostpurposes especially if
only the cosine of an angle is ofinterest. However, as a result the
angle between v1and αv1 +v2 is the same as between v1 and
αv1−v2with v>1 v2 = 0 and α ∈R which is undesirable in viewof
planar rotations as illustrated in Figure 2a. Thisflaw can be
overcome by using a cartesian coordinatesystem and its polar
coordinate representation.
For cartesian coordinate system with axes x,yand z represented
by its transformation matrix T =[x y z
]so that Tv1 and Tv2 are in the x-y-plane,
we define the angle θ ∈ (−π,π] by
θ = ∠x(v1,v2) := atan2(e2Tv2,e1Tv2)−atan2(e2Tv1,e1Tv1) .
(9)
where ei denotes the i-th unit vector in R3 and atan2
the extension of the atan function as
atan2(b,a) :=
atan(ba
)if a > 0
atan(ba
)+π if a < 0 ∧ b≥ 0
atan(ba
)−π if a < 0 ∧ b < 0
π2 if a= 0 ∧ b > 0−π2 if a= 0 ∧ b < 0undefined if a= 0 ∧
b= 0
.
We use the superscript x in ∠x to reference to thecorresponding
x− y− z-coordinate system which de-scribes T . This definition
gives for planar rotationsangles the results as desired and is
illustrated in Fig-ure 2b. The angle between between v1 and v1 +
v2and v1 and v1−v2 have different signs in comparisonto Figure
2a.
With this definition we can describe the angles forplanar
rotations by using an at the beginning estab-lished coordinate
system with the mentioned prop-erties. However, as the desired
reference coordinatesystems for the calculations of the angles are
subjectto slow changes, it is necessary to redefine the coor-dinate
system at every point of time. This means thecoordinate axes have
to be constantly recalculated.Clearly, it is desirable to obtain
continuous axes thatdo not experience a change of sign.
Furthermore, thecoordinate system shall be right handed and use
onlyinformation about the current point of time.
By the definition of the cross product, it is suffi-cient to use
only two vectors v1 and v2 to define acoordinate system via
x= v1‖v1‖, (10a)
y = z×x, (10b)
z = v1×v2‖v1×v2‖. (10c)
However, for a constant v1 but a rotating v2 they- and z-axis
change their direction when v1 and v2become parallel as can be seen
in Figure 4.
Consider the Gram Schmidt process as a way toconstruct the
coordinate system with
x= v1‖v1‖, (11a)
y = v2− (x>v2)x
‖v2− (x>v2)x‖, (11b)
z = v3− (x>v3)x− (y>v3)y
‖v3− (x>v3)x− (y>v3)y‖. (11c)
However, with this definition it cannot be guaranteedthat the
resulting coordinate system is right handedas illustrated in Figure
5.
We combine these two methods in order to obtaina continuous
right handed coordinate system.
-
v1
v2(t2)π
π
π
π v2(t4) v2(t3)
v2(t1)
θ(t2)
θ(t1)
θ(t4)
θ(t3)v2(0)t1 t2 t3 t4 t1 t2
t3
t4
0 0
Scalar product angle definition Planar angle definitionPlanar
rotation
Figure 3. Angle of a planar rotation described by two different
definitions
y
x
z
v1
v2
y
x
z
v1v2
Figure 4. Defined coordinate system using (10)
y
x
z
v1
v2
v3
y
x
z
v1v2
v3
Figure 5. Defined coordinate system using (11)
Let xGram,yGram,zGram be the coordinate axes asin (11). Then
define for the coordinate system T =[x y z
]as
x= xGram , (12a)y = z×x, (12b)z = zGram . (12c)
Thus (9) gives with (11) and (12) a method to cal-culate the
continuous angle between v1 and v2 usingan additional vector v3.
This method is introduced in
view of continuous rotations of v1 and v2 in a slowlychanging
v1-v2-plane. However, it must be ensuredthat the plane normal does
not get perpendicular tov3.2.3.1 The Solar Noon AngleThe solar noon
angle θ is the angle between the space-craft vector r and the sun
pointing vector s projectedon the orbit plane
θ = ∠xSN(r,(xOs)xO + (zOs)zO
), (13)
using (9) and {xSN,ySN,zSN} denoting the coordinatesystem
obtained with Equation (11) and (12) with thevectors v1 = (xOs)xO
+(zOs)zO, v2 = r and v3 =−yO.This definition gives for a single
orbit of a spacecraftan angle between (−π,π] with one discontinuity
atmost.2.3.2 The Beta AngleThe beta angle β ∈ [−π2 , π2 ] defined
as in (Meseguer,Pérez-Grande, and Sanz-Andrés 2012) describes
therelative orientation of the orbit with regard to thesun, and is
defined as the minimum angle betweenthe orbit plane and the solar
vector. The beta an-gle is defined as positive if the spacecraft
orbits in acounter clockwise direction and negative if it
revolvesclockwise with respect to the sun as
β ={
∠(s,(xos)xo+ (zos)zo
)if s>yo < 0
−∠(s,(xos)xo+ (zos)zo
)if s>yo ≥ 0 (14)
using the definition of the orbit frame from Section2.1 and
Equation (8) . Another way to calculate thebeta angle is to use the
normal of the orbit plane andparameterise the vectors by the
orbital elements de-scribing the movement of the sun and the
satellite.Consider the sun as a satellite of the earth with the
-
inclination is and the sum of the argument of periap-sis and
true anomaly ωs+νs, i.e. the opliquity of theecliptic and the true
solar longitude of the ecliptic.Then the vector to the sun s and
the vector orthogo-nal to the plane yO can be written in ECI
coordinatesas:
s= cos(ωs+νs)xI + sin(ωs+νs)cos(is)yI
+ sin(ωs+νs)sin(is)zI ,yo = sin(Ω)sin(i)xI− cos(Ω)sin(i)yI +
cos(i)zI .
Instead of calculating the angle to the projection wecalculate
the angle to the orbit normal as
sin(β) =−cos(β+ π2 ) =−s>yo
⇒ β = sin-1(cos(ωs+νs)sin(Ω)sin(i)−
sin(ωs+νs)cos(is)cos(Ω)sin(i)+ sin(ωs+νs)sin(is)cos(i)
).
(15)
This description emphasises the dependence of the βangle from
the orbit inclination and longitude of theascending node.2.3.3 The
Solar Zenith AngleThe solar zenith angle is defined as in
(Meseguer,Pérez-Grande, and Sanz-Andrés 2012) to describe
theportion of the illuminated planet which is seen by
thespacecraft. The solar zenith angle ξ is defined as theangle
between the spacecraft vector r and the sunpointing vector s as
ξ = ∠(r,s) (16)
using (8). In order to enable a thermal analysis de-pendent of
the orbit attitude, the influence of this an-gle can be described
by the slowly time varying betaangle β and the periodic solar noon
angle θ.
For the solar zenith angle ξ, the beta angle β andthe solar noon
angle θ holds
cosξ = cosβ cosθ . (17)
As can be seen in Equation (4) the solar zenith an-gle
influences the acting heat significantly. By using(17) we have
introduced two different angles which al-low analysing the impact
of the chosen satellite orbit.The satellite orbit can be described
by the six orbitalelements a, ε, i, Ω, ω andM0. If the orbiting
object isonly influenced by a gravitation field described by
aspherical symmetric planet these orbital elements areconstant. In
many applications, orbits are chosen tobe circular sun synchronous
orbits. Thus, a uniformmovement is obtained and the solar noon
angle can bedescribed as θ= ωot, where ωo is the angular
rotationrate dependent on the semimajor axis a. However,the beta
angle is determined by the inclination of the
orbit i and the of the longitude of the ascending nodeΩ as can
be seen in (15). Therefore, the choice of Ωinfluences the heat
acting on the satellite due to thesun significantly.2.3.4 The
Normal Solar AngleThe normal solar angle φ is defined between the
nor-mal of a spacecraft surface n and the vector pointingto the sun
s− r as
φ= ∠(s− r,n
)≈ ∠
(s,n). (18)
This approximation holds because the distance be-tween earth
origin and spacecraft is negligible com-pared to the distance
between sun and spacecraft inlow earth orbits.
2.4 Form FactorFor the form factor described in the previous
sectionit is sufficient to assume the spacecraft surface to bea
infinitesimally small plate and the earth to be asphere. Then we
can use the results from (Juul 1979)and obtain the form factor as a
function of distanceto the plate and angle ζ = ∠(r,n), the angle
betweenthe normal of the plate n and vector between earthand plate
which is approximately the vector betweenearth and spacecraft r.
Let H = ‖r‖r⊕ where r ∈ R
3 isthe spacecraft position and r⊕ the radius of the earth,then
the form factor is
Fform =
cos(ζ)H2 ζ <
π2 − sin-1
( 1H
)Fform,2
π2 − sin-1
( 1H
)< ζ < π2 + sin-1
( 1H
)0 ζ > π2 + sin-1
( 1H
)(19)
with
Fform,2 =12 −
1π
sin-1(√
H2−1H sin(ζ)
)
+ 1πH2
(cos(ζ)cos-1
(−√H2−1cot(ζ)
)−√H2−1
√1−H2 cos(ζ)2
).
2.5 Shadow FunctionThe shadow function gives the occultation of
thesatellite due to the earth. We use cylindrical shad-ows as
illustrated in Figure 6. The distance betweenearth and sun is way
higher than the difference oftheir radii and the distance between
earth and space-craft, which is why it is sufficient to assume
cylindricalshadows instead of conic ones. We construct an
or-thonormal basis {x,y,z} ⊂ R3 with x = s‖s‖ then theshadow
coefficient ν = ν(r,s) is calculated as
ν ={
1 if r>x < 0∧‖r>y+ r>z‖< r⊕0 otherwise
. (20)
-
r
xshysh
zsh
β θ
Figure 6. Cylindrical Shadow Model
Note that instead of taking an arbitrary normal ba-sis we can
define a coordinate system ·sh using thedefined solar noon and beta
angle via
T sh,I =[xsh ysh zsh
]>=Ry(−π2 )Rx(β)Ry(θ)Ry(π)T o,I ,
where
Rx(θ) =
cos(θ) sin(θ) 0−sin(θ) cos(θ) 00 0 1
and
Ry(θ) =
cos(θ) 0 sin(θ)0 1 0−sin(θ) 0 cos(θ)
.We can use this coordinate system to parameterise
r asr
‖r‖ = cos(θ)cos(β)xsh + cos(θ)sin(β)ysh− sin(θ)zsh .
Then Equation (20) reads
ν ={
1 if |θ|> π2 ∧√
cos(θ)2 sin(β)2 + sin(θ)2 < r⊕‖r‖0 otherwise
.
(21)
Other methods divide the earth’s shadow into um-bra and
penumbra. The shadow coefficient ν ∈ (0,1)in penumbra is then
determined by the overlapping oftwo circular disks. A detailed
derivation can be foundin (Montenbruck and Gill 2011).
3 Modelica ImplementationThe implementation of the Thermal Space
library isan extension of the DLR Space Systems library from(M. J.
Reiner and Bals 2014) and uses gravity andsun models of the DLR
Environment Library (Briese,Klöckner, and M. Reiner 2017). The
implementedmodels are based on the Modelica Standard Library.
3.1 Heat Fluxes ImplementationEach of the solar radiation,
albedo radiation, infra-red radiation and deep space radiation is
imple-mented. We will discuss only the implementation ofthe Albedo
radiation in detail as all other radiationsfollow the same
implementation concept. The albedomodel is shown in Figure 7. The
user may providethe material specific solar absorptance parameter
αas well as the area of the surface A and the normal ofthe surface
nB in body coordinates. Additionally, theaverage solar flux
constant Gs0 and the albedo coeffi-cient ρalb may be provided.
Standard values for theseparameters exist, however it is often
desired to simu-late special hot and cold case scenarios which
makesan adaption of these parameters as implemented a de-sirable
feature. The model has two ports, a frame anda heat port connector.
As the spacecraft is usuallymodelled as a rigid body using the
Modelica Multi-Body Library (Otter, Elmqvist, and Mattsson
2003),the frame connector has to be connected to the bodymodelling
the spacecraft. Like this the orientation ofthe frame can be
accessed to provide the position rand orientation of the spacecraft
TB. Additionally,the outer world model is used to obtain the
positionof the sun s. Then Equations (8) and (16) are used
todetermine the solar zenith angle ξ. The orientation ofthe
spacecraft is used to transform the normal vectorin body
coordinates nB into ECI coordinates n usingEquation (1). Then the
position of the spacecraft rand the normal of the surface n are
used to deter-mine the form factor with Equation (19). Finally
thealbedo heat flow Qalb is calculated using (4) and fedto the heat
port as can be seen in Figure 7. This heatport can then be
connected to other sources and sinksof heat to model the thermal
dynamics. Instead ofusing (16), Equation (17) can be used with (9),
(12),(13) and (14) to describe the influence of the solarangle.
This gives the same results but uses the betaangle β instead of the
solar zenith angle ξ which maybe easier to parameterise with
respect to the satellitesorbit as can be seen in (15). The other
radiations have
-
Figure 7. Albedo Model Diagram
Figure 8. Spacecraft Surface Model Diagram
the same structure but use the Equations (3), (5) and(7),
respectively, with the angle defined in (18) andthe shadow function
(20).
3.2 Thermal Space ComponentsThe thermal model of a spacecraft
surface can be seenin Figure 8. The thermal dynamics are described
bythe differential equation
CṪ =Qalb +Qsun +Qplanet−εAσT 4 +Qr (22)
where C is the thermal capacitance of the surface andQr
describes all other heat fluxes which are acting onthe heat port.
This includes foremostly the internalpower dissipation of the
satellite. The capacitanceis implemented as a conditional
component. Thismodel offers the opportunity to remove the
thermalcapacitance if only the steady state calculations areof
interest. Additionally, a desired temperature of thesurface may be
given to obtain the necessary dissipa-tive power which have to be
for example produced byheaters to maintain this temperature.
Since many small satellites have the form of acuboid, a model
with six spacecraft surfaces with aninfinite resistance between
them is implemented. Thiscan be used to simulate the heat evolution
at eachspacecraft surface as in Section 4. In order to accountfor
the different satellite modes, attitude specific sur-face
configurations are implemented as for e.g. earthpointing mode in
which the attitude of the satellite is
fixed. Satellite components are modelled as a thermalcapacitor
which is connected to a spacecraft surface,usually a radiator. For
each of these components theparameters already discussed may be
provided to sim-ulate different scenarios of interest.
3.3 ArchitecturesThere are three thermal concepts commonly used
formicro- and nano satellites as described in (Baturkin2005) -
autonomous concept, centralized concept andcombined concept. Each
of these structures is imple-mented modelling the thermal coupling
between eachthermal component and the external heat exchange.
4 Example ScenarioIn order to show the functionality of the
library, thethermal dynamics of a cuboid earth pointing space-craft
are simulated. The cuboid is modelled by sixsurfaces having the
properties of a radiator. The sur-faces have the same area A= 1m2
and thermal prop-erties α = 0.25 and ε = 0.88. The spacecraft is
ina sun synchronous orbit with an altitude of 600kmand 10 : 30h
longitude of the ascending node simu-lated 2018-02-10 at 10 : 00h.
The earth’s gravitationfield is approximated up to the second zonal
coeffi-cient (Markley and Crassidis 2014). No dissipativeheat is
simulated and the parameter are chosen asGs = 1361Wm−2, ρalb = 0.3
and Tp = 255K. Onecomplete orbit, which takes about 5800s ≈ 97min,
issimulated. The satellite is earth pointing over thewhole orbit,
i.e. the spacecraft body axes which areperpendicular to the cuboid
surfaces are aligned withthe orbit frame.
Figure 9 shows the visualisation of the describedscenario. The
spacecraft itself is visualised as a sim-ple grey cuboid. The heat
flows, the sum of solar,albedo and infra-red radiation, acting on
each surfaceare visualised using head up displays from the
Visu-alization library (Bellmann 2009). It can be seen thatall but
the zenith direction are influenced by a con-stant heat flow due to
the earth’s infra-red radiation.Furthermore, it can be seen that
the spacecraft is inthe sunlight after approximately 1100s up to
4990sand that the transition between shadow and sunlitis
discontinuous. The nadir direction is mainly in-fluenced by the
infra-red and albedo irradiation as itsview to sun is mostly
blocked by the earth. Due to thelow solar absorptance of the
surface the change of theacting heat flow is comparatively small.
The zenithdirection however is mostly influenced by the
solarradiation. No albedo and infra-red radiation reachesthis
surface. The surface perpendicular to the orbitplane and in sun
direction is foremostly influenced bythe solar radiation as well.
However, due to the smallchange of the angle between this surface
and the di-rection to the sun, this heat flow is almost
piecewiseconstant. On the contrary, the surface perpendicular
-
Figure 9. Heat flows acting on the surfaces of a cuboid earth
pointing spacecraft
to the orbit plane and in anti sun direction is only in-fluenced
by the infra-red and albedo radiation. Dueto the small solar
absorptance, the albedo radiationinfluence is comparatively small
and this surface hasthe smallest heat flow changes. The surfaces in
veloc-ity and in anti velocity direction are mirrored with re-spect
to the solar noon of the spacecraft. Albedo andinfra-red radiation
are acting continuously on thesesurfaces while the influence of the
solar radiation canbe seen in the sudden discontinuity of the heat
flow.All in all, it can be observed that all surfaces aresubject to
high heat flux changes especially when thespacecraft enters and
exits the eclipse. The smallestvariation and overall incident heat
flux acts on thesurface orthogonal to the orbit plane in anti sun
di-rection making it suitable as a surface with radiator.
5 ConclusionsWe have presented a Modelica library suitable for
thedevelopment of a thermal spacecraft model. The mainacting
environmental and spacecraft heat flows are in-troduced and their
dependence on different angles isgiven as in the literature. Issues
regarding the de-termination of these angles have been described
andnovel methods for their calculation are given and dis-cussed. An
application example of the proposed li-brary is given to
demonstrate the usefulness and flex-ibility of the Modelica
implementation.
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IntroductionFundamentalsCoordinate SystemsEnvironmental Heat
FluxesDirect SolarAlbedoPlanetary RadiationRadiation to Deep
Space
Thermal AnglesThe Solar Noon AngleThe Beta AngleThe Solar Zenith
AngleThe Normal Solar Angle
Form FactorShadow Function
Modelica ImplementationHeat Fluxes ImplementationThermal Space
ComponentsArchitectures
Example ScenarioConclusions