Applied Thermal Engineering 193 (2021) 117039
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Applied Thermal Engineering
journal homepage: www.elsevier.com/locate/ate
Research paper
Simulation of a CubeSat with internal heat transfer using Finite Volume
Method
Edemar Morsch Filho a ,∗, Laio Oriel Seman b , Vicente de Paulo Nicolau a
a
b
Department of Mechanical Engineering, Federal University of Santa Catarina, Florianópolis, SC, 88040-900, Brazil
Graduate Program in Applied Computer Science, University of Vale do Itajaí, Itajaí, SC, 88302-901, Brazil
ARTICLE
INFO
Keywords:
CubeSat
Finite Volume Method
Temperature simulation
Internal heat transfer
Obstruction
Gebhart method
ABSTRACT
Estimating temperature is essential both to design reliable CubeSats and to keep it under maximum operating
efficiency. This paper presents the transient thermal simulation of a CubeSat 1U, where the heat transfer by
conduction and radiation (external and internal) are solved considering a typical CubeSat mission launched
from the International Space Station. The objective is to obtain the temperature field based on the Finite
Volume Method (FVM), with inner heat transfer by radiation. The Gebhart method computes the internal heat
exchange through successive reflections, and an obstruction model supports the calculation of view factors.
Three boundary conditions for the inner side of the CubeSat are tested: without internal heat transfer by
radiation, or zero emissivity (𝜖 = 0.0), with intermediate internal heat transfer by radiation (𝜖 = 0.5), and
maximum internal heat transfer by radiation (𝜖 = 1.0). The results show a significant impact of the internal
heat transfer by radiation in the temperature field of both inner and outer parts of the satellite, and therefore
it should not be ignored. Good agreement of transient temperature is found between the FVM and another
more straightforward formulation based on the Lumped Method, although the three-dimensional effects are
significant and cannot be obtained with such a simplified model.
1. Introduction
Before a satellite leaves the ground, the scenarios it will experience
in orbit must be tested to minimize the occurrence of failures or inefficient operation [1]. This practice is also essential for CubeSat projects,
a satellite’s category based on the dimensions of 10 × 10 × 10 cm
for the standard model 1U. CubeSats are generally of low cost, with
extensive use of Commercial Off-The-Shelf (COTS) components, short
schedule, short operational lifetime, limited redundancy, and extensive
testing focused on the system-level [2]. The CubeSat concept was
developed in 1999 [3], and since then, more than 1300 CubeSat were
already launched,1 most of them in Low Earth Orbit (LEO), but also
in successful missions to Mars [4]. The increasing number of electronic
operations to be performed during the mission, the complex integration
of the subsystems, and the power dissipation make thermal control in
space a challenging task [5].
The majority of CubeSats in LEO are below 600 km, has a nearcircular orbit, with a period of around 100 min, whose passages behind
the Earth’s shadow, also known as eclipse, may last around 40 min,
at maximum. Usually, the CubeSats are covered with photovoltaic
panels (PV) to generate power for the subsystems and recharge the
batteries, making it possible to operate during the eclipse [6]. In terms
of temperature, both solar panels and batteries are critical components
that require special attention because when exposed to temperature
levels beyond the recommended values, they may be permanently
damaged. Also, the PV has a Maximum Power Point (MPP) tracked
by the Electrical Power Systems (EPS) to generate power at maximum efficiency, which depends on the temperature levels [7,8]. Other
CubeSat’s examples with temperature dependence include power harvesting with active thermal control [9], thermo-mechanical structural
failure [10], thermoelectric generators [11], and biological experiments
in orbit [12]. Therefore, thermal simulation is a crucial tool to project
reliable and efficient CubeSat missions.
Analytical formulations are typical for the initial stages of a CubeSat’s development, where a quick estimation of the overall temperature
field is obtained and updated repeatedly as more information about the
inputs is available. Inevitably, idealizations make it a cheap and fast
approach at the expense of accuracy. However, nonlinear terms make
it difficult to obtain the analytical solution, and several linearization
techniques have been proposed [13]. Intrinsically to analytical solutions is using single nodes, as the Lumped Parameter Method (LPM)
∗ Corresponding author.
E-mail addresses: edemar@labcet.ufsc.br (E. Morsch Filho), laio@univali.br (L.O. Seman), vicente.nicolau@ufsc.br (V. de Paulo Nicolau).
1
nanosats.eu.
https://doi.org/10.1016/j.applthermaleng.2021.117039
Received 27 January 2021; Received in revised form 6 April 2021; Accepted 26 April 2021
Available online 30 April 2021
1359-4311/© 2021 Elsevier Ltd. All rights reserved.
Applied Thermal Engineering 193 (2021) 117039
E. Morsch Filho et al.
approach, to represent the satellite’s entire subsystems and reduce the
complexity of heat transfer estimation. Consequently, the conductive
and radiative heat transfer among the parts and their interaction are
oversimplified. Examples of thermal studies involving CubeSat’s heat
transfer with analytical approaches are Anh et al. [13], Tsai [14], Bulut
and Sozbir [15] and Pérez-Grande et al. [16].
On the other hand, numerical simulations provide further insights
into the satellite details by quite accurate solutions obtained under
higher computational cost and more complex algorithms [14]. Three
of the most known numerical techniques are the Finite Difference
Method (FDM), Finite Element Method, and Finite Volume Method
(FVM) [17]. Reyes et al. [18] present an example of CubeSat’s thermal
simulation where they develop an algorithm in MATLAB® based on
the FDM and compare the results with the commercial software Thermal Desktop. Kovács and Józsa [19] conduct a thermal analysis of a
nanosatellite through a thermal network approach and the commercial
software ANSYS® Workbench built on FEM. Bonnici et al. [20] has a
thermal model based on LPM to study the UoMBSat −1 PocketQube,
whose results are compared with the commercial software Ansys®
formulated as FEM. Escobar et al. [21] also use FEM to explore the
best thermal control configuration for a CubeSat. Corpino et al. [22]
is another example of FEM simulation, implemented by the authors in
MATLAB® , where they assess the worst hot and cold orbits. The authors
use the Absorption Factors Method for the internal boundary condition,
also known as the Gebhart Method [23]. Morsch Filho et al. [24]
simulate different attitudes in the commercial software CFX/ANSYS
built on FVM for the worst cold and hot scenarios of a CubeSat 1U.
From these thermal simulations, only Corpino et al. [22] considered
the radiation heat transfer in internal surfaces of the CubeSat. However,
in their in-house code, the Printed Circuit Boards (PCBs) and the battery
were modeled as lumped nodes; therefore, they did not account for
thermal gradients in these parts. Internally, the information about
the components’ dimensions and position is essential for estimating
obstructing surfaces and, consequently, calculating the heat transfer.
Except for the previous work of Morsch Filho et al. [24], none of
the mentioned papers have solved CubeSat’s temperature based on the
FVM. One of the main attractions of the FVM is the integration of
the governing equation over finite control volumes and its conversion
into a system of algebraic equations that still follows the conservative
balance [25]. These initial observations reveal a gap in the thermal
simulation of CubeSats.
The impact of internal boundary conditions on the CubeSat’s temperature field outlines the objective of this paper. This work’s first
contribution is to build a code based on the Finite Volume Method
for temperature estimation of a CubeSat 1U. There are no studies of
CubeSats’ simulations with detailed formulation regarding the internal
boundary conditions coupled with obstructing surfaces, which is the
second major contribution of our work.
The next Section 2 starts with the methodology of this paper,
Section 3 has the simulation setup of this work, and then at Section 4
there are results about the impact of the internal boundary condition
in the temperature field, as well as a comparison of the FVM and LPM
results. Section 5 brings the conclusions over the main findings.
Fig. 1. Domain for the LPM.
internal surfaces, and the heat transfer by conduction exists through
the CubeSat’s components, according to the equation:
𝜚𝑐
𝜕𝑇
− 𝜅∇. (∇𝑇 ) − 𝑆 = 0
𝜕𝑡
(1)
where 𝜚 is the density [kg/m3 ], 𝑐 is the heat capacity [J/kg K], 𝑇
is the temperature [K], 𝑡 is the time [s], 𝜅 is the thermal conduction
coefficient [W/m K] and 𝑆 is the source term [W/m3 ].
2.1. Lumped parameter method (LPM)
The methodology presented in this section is dedicated to solving
the thermal field of the satellite. In the LPM formulation, single points
represent each main parts of the CubeSat, as shown in Fig. 1.
The following assumptions are necessary to develop the equations:
• The CubeSat has seven main parts, each one represented by a
subindex 𝑤. Solar panels are represented from 1 to 6, while 7
is the battery;
• A photovoltaic panel covers each external side of the CubeSat;
• The battery is suspended in the center of the CubeSat;
• Each panel exchanges heat with the battery, proportional to
the temperature difference between these parts and inversely
proportional to the thermal resistance 𝑍 [K/W];
• Thermal resistance 𝑍 among the solar panels is infinite;
• Thermal resistance 𝑍 between the solar panel and battery is
unique and constant;
• Material and surface properties are constant;
• None internal heat generation.
Over the domain of Fig. 1, the classical energy balance (Eq. (1)) is
performed along the CubeSat’s orbit, for each node 𝑤 of the model,
resulting in:
𝑑𝑇𝑤
=0
(2)
𝑑𝑡
where 𝑄𝑟𝑤 is the heat transfer by radiation [W], 𝑄𝑍𝑤 the net heat
transfer between the solar panel and battery [W], and 𝑉 is the volume
[m3 ] of the part. The battery is inside the CubeSat, therefore 𝑄𝑟7 = 0.
The net heat transfer between parts may occur by conduction and
radiation, but the following equation will simplify this term.
𝑄𝑟𝑤 + 𝑄𝑍𝑤 − 𝜚𝑤 𝑉𝑤 𝑐𝑤
2. Methodology
𝑄𝑍𝑤
CubeSat’s temperature results from the radiation loads that it faces
in orbit, therefore it is a function of orbital parameters, attitude, the
geometry of the satellite, material, and surface properties. In this
section, the equations to perform the thermal simulation of a CubeSat
will be presented. The focus here is to obtain the CubeSat’s transient
temperature field through an in-house code based on the FVM, valid
for a typical CubeSat 1U. For thermal problems of orbiting spacecrafts,
a good approximation is a perfect vacuum, so the energy balance does
not have the convective term. In this work, the thermodynamic properties are constant, the radiation heat transfer occurs on external and
⎧
⎪ 𝑇7 − 𝑇𝑤
⎪ 𝑍
= ⎨∑
6
𝑇𝑤 − 𝑇7
⎪
⎪𝑤=1
𝑍
⎩
if 𝑤 ≤ 6
(3)
if 𝑤 = 7
The time derivative term will be evaluated according to the following finite difference scheme, where 𝛥𝑡 is the timestep and ‘‘it’’ the
iteration.
𝑑𝑇𝑤
𝑇 (𝑖𝑡) − 𝑇𝑤 (𝑖𝑡 − 1)
≈ 𝑤
(4)
𝑑𝑡
𝛥𝑡
The Newton–Raphson method [26] solves the nonlinear system
summarized by Eq. (2). The user only needs to inform the initial
condition and the balance convergence criteria of each part.
2
Applied Thermal Engineering 193 (2021) 117039
E. Morsch Filho et al.
The description of this term as a linear equation improves the convergence of the model [27].
Substituting these previous equations in Eq. (5) and rearranging the
terms:
(
)
𝜅𝑒 𝐴𝑒 𝜅𝑤 𝐴𝑤 𝜅𝑛 𝐴𝑛 𝜅𝑠 𝐴𝑠 𝜅𝑡 𝐴𝑡 𝜅𝑏 𝐴𝑏
𝛥𝑉
+
+
+
+
+
+ 𝜚𝑐
− 𝑆𝑃 𝑇𝑃
𝛿𝐸𝑃
𝛿𝑃 𝑊
𝛿𝑁𝑃
𝛿𝑃 𝑆
𝛿𝑇 𝑃
𝛿𝑃 𝐵
𝛥𝑡
𝜅𝑒 𝐴𝑒
𝜅𝑤 𝐴𝑤
𝜅𝑛 𝐴𝑛
𝜅𝑠 𝐴𝑠
𝜅𝑡 𝐴𝑡
𝜅 𝐴
=
𝑇 +
𝑇 +
𝑇 +
𝑇 +
𝑇 + 𝑏 𝑏 𝑇𝐵
𝛿𝐸𝑃 𝐸
𝛿𝑃 𝑊 𝑊
𝛿𝑁𝑃 𝑁
𝛿𝑃 𝑆 𝑆
𝛿𝑇 𝑃 𝑇
𝛿𝑃 𝐵
𝛥𝑉 𝑜
+𝜚𝑐
𝑇 + 𝑆𝑢
(9)
𝛥𝑡 𝑃
Finally, the previous differential Eq. (5) transforms into the next
discrete algebraic equation:
∑
𝑎𝑃 𝑇𝑃 −
𝑎nb 𝑇nb − 𝑎𝑜𝑃 𝑇𝑃𝑜 − 𝑆𝑢 = 0
(10)
Fig. 2. Control volume for application of conservation equations.
Source: Adapted from Versteeg and Malalasekera [25].
nb
where ‘‘nb’’ refers to the neighbor volumes (N, S, E, W, T, B), and the
coefficients 𝑎nb are the diffusive flux terms of the neighbors, e.g. 𝑎𝑊 =
∑
𝜅𝑤 𝐴𝑤 ∕𝛿𝑃 𝑊 . The term 𝑎𝑜𝑃 = 𝜚𝑐𝛥𝑉 ∕𝛥𝑡 and 𝑎𝑃 = nb 𝑎nb +𝑎𝑜𝑃 −𝑆𝑃 . For further details about the FVM, please refer to Versteeg and Malalasekera
[25].
The main advantage of FVM over the LPM formulation is the conservative nature of the integral solution since the flux entering a given
volume is identical to leaving the adjacent volume [17]. The iterative
method to solve Eq. (10) of this work is the TDMA (Tri-diagonal Matrix
Algorithm) [25]. In the FVM, the domain’s discretization into small
finite volumes results in three-dimensional solutions rather than single
values obtained for each part of the LPM’s formulation.
The external boundary condition discussed in the previous section
is identical for LPM and FVM. On the other hand, the internal radiative
heat transfer in the FVM will assume a conservative formulation instead
of the simplified radiative–conductive thermal resistance (𝑍) of the
LPM.
2.2. The finite volume method (FVM)
In the FVM, the terms of the energy conservation (Eq. (1)) are
integrated over time and over a finite number of Control Volumes (CV)
adjacent to each other, which results in conservative three-dimensional
solutions rather than single values, in contrast to the LPM’s formulation.
With the assistance of the divergence theorem, which transforms the
volume integration into spatial flux integration across the section area
(𝐴), the energy equation becomes:
(
)
𝜕
⃗ −
𝜚𝑐𝑇 𝑑𝑉 𝑑𝑡 −
𝑆𝑑𝑉 𝑑𝑡 = 0 (5)
(𝜅∇𝑇 ) .𝑑 𝐴𝑑𝑡
∫𝛥𝑡 𝜕𝑡 ∫𝐶𝑉
∫𝛥𝑡 ∫𝐴
∫𝛥𝑡 ∫𝐶𝑉
The numerical procedure to solve this problem requires the distribution of nodes in different positions of the domain, where the dependent
variable will be evaluated through a set of algebraic equations that
couple each node to its neighbors. Therefore, instead of an exact
solution obtained from the partial differential equation, the user will
have the discrete answer in specific points. This group of nodes forms
the mesh, also referred to grid [17].
Fig. 2 shows a control volume used to explain the discretization of
the equations of this work.
In this three-dimensional orthogonal volume there are six surfaces
identified by north (n), south (s), east (e), west (w), top (t), and bottom
(b). The variable of interest (temperature) is located at the center of this
volume and represented by P. The neighbor points N, S, E, W, T, B are
in the north, south, east, west, top, and bottom, respectively. In this
work, the mesh is structured, so each volume has the same quantity
of neighbors, and their organization follows a natural order [27]. The
volumes are hexahedral, and they do not overlap.
The transient term, which represents the time variation of temperature in Eq. (5), can be written as:
(
)
(
)
𝜕
𝜚𝑐𝑇 𝑑𝑉 𝑑𝑡 = 𝜚𝑐 𝑇𝑃 − 𝑇𝑃𝑜 𝛥𝑉
(6)
∫𝛥𝑡 𝜕𝑡 ∫𝐶𝑉
2.2.1. Internal heat transfer
The internal heat transfer by radiation will use the Gebhart Method
[23,28], where the following assumptions are necessary: (a) Each discretized surface is isothermal; (b) Gray and diffuse surfaces; (c) The
energy leaving a surface is uniformly distributed, so the view factor
between two surfaces is constant. The diffuse surface is an ideal assumption, necessary for the Gebhart Method, that differs from actual
surface properties used in the satellite’s projects, where absorptivity
and emissivity have directional dependence. To handle non-diffuse
cases, the reader can consult the Monte Carlo Ray Tracing method [29–
31]. The heat transfer by radiation on the surface 𝑖 is:
𝑞𝑖 = 𝐴𝑖 𝜖𝑖
∫𝛥𝑡 ∫𝐶𝑉
(11)
The term 𝐺𝑖−𝑗 is the absorption factor and represents the fraction of
energy emitted by surface 𝐴𝑖 that hits 𝐴𝑗 and there it is absorbed,
including direct (emission) and indirect (reflection) ways, from all
the surfaces 𝑀. The absorption factor has the reciprocity relation
∑
(𝜖𝑖 𝐴𝑖 𝐺𝑖−𝑗 = 𝜖𝑗 𝐴𝑗 𝐺𝑗−𝑖 ), follows the summation rule ( 𝑀
𝑗=1 𝐺𝑖−𝑗 = 1), and
it is calculated as:
(12a)
ℎ𝐺 = 𝑓
⎡ 𝐺1−1
⎢
𝐺
𝐺 = ⎢ 2−1
⎢ ⋮
⎢𝐺
⎣ 𝑀−1
(7)
where 𝛿 is the distance between two points.
The following linear equation describes the source term for the fully
implicit scheme:
(
)
̄
𝑆𝑑𝑉 𝑑𝑡 = 𝑆𝛥𝑉
= 𝑆𝑢 + 𝑆𝑃 𝑇𝑃 𝛥𝑡
(
)
𝐺𝑖−𝑗 𝜎 𝑇𝑖4 − 𝑇𝑗4
𝑗=1
The superscript ‘‘𝑜 ’’ is dedicated to a variable at time 𝑡, while the term
without it means a variable at time 𝑡 + 𝛥𝑡.
By using a central differencing approach for the face fluxes and fully
implicit method, the integration over the diffusive term result in the
following equation:
⃗
(𝜅∇𝑇 ) .𝑑 𝐴𝑑𝑡
∫𝛥𝑡 ∫𝐴
) 𝜅 𝐴 (
) 𝜅 𝐴 (
)
𝜅𝐴 (
= 𝑒 𝑒 𝑇𝐸 − 𝑇𝑃 + 𝑤 𝑤 𝑇𝑃 − 𝑇𝑊 + 𝑛 𝑛 𝑇𝑁 − 𝑇𝑃
𝛿𝐸𝑃
𝛿𝑃 𝑊
𝛿𝑁𝑃
) 𝜅𝐴 (
) 𝜅 𝐴 (
)
𝜅𝐴 (
+ 𝑠 𝑠 𝑇𝑃 − 𝑇𝑆 + 𝑡 𝑡 𝑇𝑇 − 𝑇𝑃 + 𝑏 𝑏 𝑇𝑃 − 𝑇𝐵
𝛿𝑃 𝑆
𝛿𝑇 𝑃
𝛿𝑃 𝐵
𝑀
∑
𝐺1−2
𝐺2−2
⋮
𝐺𝑀−2
⎡1 − 𝜌 𝐹̂
1 1−1
⎢
⎢ −𝜌1 𝐹̂2−1
ℎ=⎢
⋮
⎢
⎢ −𝜌 𝐹̂
⎣ 1 𝑀−1
(8)
3
⋯
⋯
⋱
⋯
𝐺1−𝑀 ⎤
⎥
𝐺2−𝑀 ⎥
⋮ ⎥
𝐺𝑀−𝑀 ⎥⎦
−𝜌2 𝐹̂1−2
1 − 𝜌1 𝐹̂1−1
⋯
⋮
−𝜌2 𝐹̂𝑀−2
⋱
⋯
⋯
(12b)
−𝜌𝑀 𝐹̂1−𝑀 ⎤
⎥
−𝜌𝑀 𝐹̂2−𝑀 ⎥
⎥
⋮
⎥
1 − 𝜌𝑀 𝐹̂𝑀−𝑀 ⎥⎦
(12c)
Applied Thermal Engineering 193 (2021) 117039
E. Morsch Filho et al.
Fig. 3. Test of the dot product to estimate the obstruction. Left: 𝑂(𝑖, 𝑗) = −1; Middle: 𝑂(𝑖, 𝑗) = −1; Right: 𝑂(𝑖, 𝑗) = 1.
⎡ 𝐹̂ 𝜖
⎢ 1−1 1
⎢ 𝐹̂ 𝜖
𝑓 = ⎢ 2−1 1
⎢ ⋮
⎢𝐹̂
⎣ 𝑀−1 𝜖1
𝐹̂1−2 𝜖2
𝐹̂2−2 𝜖2
⋯
⋮
⋱
𝐹̂𝑀−2 𝜖2
⋯
⋯
𝐹̂1−𝑀 𝜖𝑀 ⎤
⎥
𝐹̂2−𝑀 𝜖𝑀 ⎥
⎥
⋮
⎥
𝐹̂𝑀−𝑀 𝜖𝑀 ⎥⎦
(12d)
The term 𝜌 is the reflectivity [-] and 𝐹̂ [-] is the view factor. This view
∑
̂
factor must satisfy the summation rule ( 𝑀
𝑗=1 𝐹𝑖−𝑗 = 1).
By assuming that the radiative properties are independent of temperature, the Gebhart factors’ matrix is solved before the FVM, and the
values are recovered from a file in the memory. The computation of
view factors with potential obstructing requires considerable CPU time,
and for this reason, this process divides in two parts: determination of
obstructions and then the view factor itself.
Fig. 4. Test of the cone to estimate the obstruction.
2.2.2. Obstruction
Internally, depending on the components’ geometry and position,
there may have surfaces that cannot see each other. For such a case,
an algorithm is necessary to verify the obstruction, here based on
the work of [32]. The verification of obstructions is computationally
expensive, so tests for the obstruction are arranged to confirm it as early
as possible. For the pair of surfaces 𝑖 and 𝑗 with obstructing views, the
value 𝑂 (𝑖, 𝑗) = −1 will be attributed, while for the unobstructed pairs
it will be 𝑂 (𝑖, 𝑗) = 1.
The first test, shown in Fig. 3, uses the dot product between the
normal vector of surface 𝐴𝑖 , located at the center of that surface, and
the vertices of surface 𝐴𝑗 . If the result is negative for all of them, it
means that 𝐴𝑗 is completely behind 𝐴𝑖 , and there is an obstruction
(𝑂 (𝑖, 𝑗) = −1). The same happens if the dot product is 0, a condition
where the surfaces are coplanar. If the test is partially negative, there is
some view between the surfaces, but this work assumes as obstructed.
This condition will impact the estimation of view factors, as will be seen
in the next section. They entirely see each other and assume 𝑂(𝑖, 𝑗) = 1
if the dot product is entirely positive.
From this test, only the pairs with 𝑂 (𝑖, 𝑗) = 1 will continue to
the next. Although surfaces 𝑖 and 𝑗 are in front of each other, there
may have a third surface 𝑘 blocking their views. This procedure is
illustrated in Fig. 4. A radius circumscribing the surface 𝑘 (𝑅𝑘 ) and the
centroid of that surface (𝑥𝑘 , 𝑦𝑘 , 𝑧𝑘 ) are determined. An unitary vector
connecting the centroid of 𝑖 and 𝑗 is determined (𝐮𝑖,𝑗 ), as well as a
vector connecting the centroid of 𝑖 and 𝑘 (𝑉⃗𝑖,𝑘 ). The magnitude of the
dot product of these vectors (|𝑉⃗𝑖,𝑘 ⋅ 𝐮𝑖,𝑗 |) gives the projection of 𝑉⃗𝑖,𝑘 into
the direction of 𝐮𝑖,𝑗 . The radius circumscribing surface 𝑖 (𝑅𝑖 ) and 𝑗
(𝑅𝑗 ) are also determined. Therefore, a representative radius 𝑅𝑖𝑗𝑘 , at the
height of 𝑘, from the cone englobing both surfaces 𝑖 and 𝑗 is:
𝑅𝑖𝑗𝑘
|𝑉⃗𝑖,𝑘 ⋅ 𝐮𝑖,𝑗 | (
)
= 𝑅𝑖 +
𝑅𝑗 − 𝑅𝑖
|𝑉⃗𝑗 − 𝑉⃗𝑖 |
𝐷 = |𝑉⃗𝑖,𝑘 × 𝐮𝑖,𝑗 |. Therefore, the surface 𝑘 will not obstruct the view of 𝑖
and 𝑗 when:
{
(
)2
1,
𝐷2 > 𝑅𝑖𝑗𝑘 + 𝑅𝑘
𝑂 (𝑖, 𝑗) =
(14)
next test, otherwise
If the previous condition is not satisfied, there are still more tests to
check whether 𝑘 really blocks 𝑖 and 𝑗, done through the dot product.
The surfaces 𝑖 and 𝑗 are not blocked (𝑂 (𝑖, 𝑗) = 1) according to the cases
illustrated in Fig. 5.
If none of the above conditions are satisfied, they cannot see each
other. Further actions may improve the resolution of obstructing views,
for example, accounting for partial shadowing, but no further action
will be performed.
2.2.3. View factor
After determining the unobstructed surfaces, the view factor between them can be computed by several different techniques available
in the literature. Details about the mesh will be discussed later, but
only parallel and perpendicular surfaces exist in this work. For both
conditions of orientations shown in Fig. 6, the equation to compute the
view factor from surface 1 to 2 is [33]:
∑∑∑∑
(
)
1
𝐹1−2 = (
(−1)(𝑝+𝑞+𝑟+𝑠) 𝑈 𝑥𝑝 , 𝑦𝑞 , 𝜂𝑟 , 𝜉𝑠
)(
)×
𝑥2 − 𝑥1 𝑦2 − 𝑦1
𝑠=1 𝑟=1 𝑞=1 𝑝=1
2
2
2
2
(15)
For parallel and perpendicular surfaces, the function 𝑈 is given
by Eqs. (16) and (17), based on Figs. 6a and 6b, respectively.
𝑈 =
(13)
⎛
⎛
⎞
[
]1∕2
𝑦−𝜂
1 ⎜
⎟
tan−1 ⎜ [
(𝑦 − 𝜂) (𝑥 − 𝜉)2 + 𝑧2
]
1∕2
⎜ (𝑥 − 𝜉)2 + 𝑧2
⎟
2𝜋 ⎜
⎝
⎝
⎠
⎡
⎤
[
]1∕2
𝑥−𝜉
⎥
+ (𝑥 − 𝜉) (𝑦 − 𝜂)2 + 𝑧2
tan−1 ⎢ [
]
⎢ (𝑦 − 𝜂)2 + 𝑧2 1∕2 ⎥
⎦
⎣
The distance from the center of 𝑘 up to the line connecting the center
of 𝑖 and 𝑗 will be obtained by the magnitude of the cross product
4
Applied Thermal Engineering 193 (2021) 117039
E. Morsch Filho et al.
Fig. 5. Final test of obstruction.
Fig. 6. Diagrams to estimate the view factors.
−
]⎞
𝑧2 [
ln (𝑥 − 𝜉)2 + (𝑦 − 𝜂)2 + 𝑧2 ⎟
⎟
2
⎠
•
•
•
•
•
(16)
(
[
(
)1∕2
) (
)
1
1 ( 2
tan−1 (𝐵) −
𝑥 + 𝜉 2 ln 1 + 𝐵 2
(𝑦 − 𝜂) 𝑥2 + 𝜉 2
2𝜋
4
(
)])
1
− (𝑦 − 𝜂)2 ln 1 +
(17)
𝐵2
( 2
)
0.5
where 𝐵 = (𝑦 − 𝜂) ∕ 𝑥 + 𝜉 2
.
Regardless of the surface’s shape and orientation, the summation
rule of view factors must be satisfied in an enclosure [34]. This condition would not be satisfied here because the obstruction computation
assumes entirely blocked surfaces even when just a small fraction is
under the shadow. For this reason, a normalization is realized by
dividing the analytical view factor obtained with Eq. (15) by the sum
of view factors obtained with surface 𝑖, as follows:
𝑈=
𝐹𝑖−𝑗
𝐹̂𝑖−𝑗 = ∑𝑀
𝑗=1 𝐹𝑖−𝑗
The heat transfer by radiation (𝑄𝑟 ) on each external side 𝑤 of the
CubeSat is on Eq. (20), where the time dependency is omitted.
𝑄𝑟𝑤 = 𝑄sun𝑤 + 𝑄alb𝑤 + 𝑄𝑒𝑤 − 𝑄out𝑤
𝐹̂𝑖−𝑗 = 1
(20)
Further details about each term are in the following sections.
2.3.1. Solar radiation
Solar radiation is the primary source of heating for satellites in
LEO. At the distance of 1 AU (astronomical unit), the solar rays are
parallel, constant, resulting in an average solar flux of 𝑄′′
𝑠 = 1367
W/m2 , although there are small variations due to the 11-year solar
cycle and the elliptic orbit of the Earth (1322–1414 W/m2 ) [36]. This
radiation is short-wave, with peak of spectral emission around 0.5 μm,
and energy distribution of 7% in the ultraviolet, 46% in the visible,
and 47% in the near-infrared [37]. In this work, the solar irradiance
reaching the external surfaces 𝑤 of the CubeSat will be:
(18)
Thus, 𝐹̂𝑖−𝑗 satisfies the summation rule:
𝑀
∑
Orbit around the Earth;
Non-circular and non-equatorial orbits;
Orbit up to 800 km;
Constant ballistic coefficient;
CubeSat geometry of any size, without deployable parts.
(19)
𝑗=1
𝑄sun𝑤 = 𝑄′′
𝑠 𝐴𝑤 𝐹𝑤→sun 𝜓
The above models summarize the procedure to introduce the internal heat transfer by radiation with obstruction in the FVM. The next
section contains the formulation for the boundary conditions of surfaces
exposed to outer space, valid for both LPM and FVM.
(21)
The parameter 𝐴𝑤 is the area of the surface 𝑤, 𝐹𝑤→sun is the view
factor of surface 𝑤 towards the Sun, and 𝜓 is a variable to express the
shadow of the Earth.
2.3.2. Albedo radiation
As a consequence of the solar flux, there is a radiation load called
albedo heat flux, which is the solar radiation reflected by the planet’s
surface. For this case, the albedo coefficient 𝑏 is introduced, which is
the amount of reflected solar radiation over the total incoming. The
Earth’s albedo depends on a great variety of parameters, as atmospheric
conditions, clouds, and ground surfaces. The global annual average,
2.3. Boundary condition on external surfaces
The formulation of incoming radiation for the external sides of
the CubeSat relies on a previous work [35], where the authors developed an algorithm for radiation dedicated to CubeSat’s missions,
which includes the impact of orbital mechanics, orbit perturbations,
and attitude, valid for the following scenarios:
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Applied Thermal Engineering 193 (2021) 117039
E. Morsch Filho et al.
Fig. 7. The mesh of FVM simulation.
widely accepted for satellites in LEO, is 𝑏 = 0.3 [36,38–40]. Another
characteristic of the albedo heat flux is its dependency with the cosine
of the angle from the subsolar point; therefore, the albedo heat flux
is maximum at midday in the Equator’s line and vanishes in the line
separating day and night [37]. The albedo’s spectral distribution has
significant variation within 0.29 to 5.00 μm [38,41].
The equation for the albedo radiation will be:
𝑄alb𝑤 = 𝑄′′
𝑠 𝐴𝑤 𝐹𝑤→e cos (𝜒) 𝑏
(22)
where 𝐹𝑘→e is the view factor of surface 𝑤 towards the Earth [42], 𝜒 is
the angle between the solar ray and the satellite’s position, and 𝑏 = 0.3
is the albedo coefficient.
Fig. 8. Diagram for the internal view.
2.3.3. Earth’s emission
The third main thermal radiation source for a CubeSat in LEO is
the infrared radiation emitted from the Earth, a long-wave radiation.
This source is not constant, with warmer surfaces emitting more than
colder and clouds blocking it, but usually an average and constant value
2
assumed in the literature is a flux of 𝑄′′
𝑒 = 237 W/m [37,38]. The
Earth’s emission will be calculated as:
𝑄𝑒𝑤 = 𝑄′′
𝑒 𝐴𝑤 𝐹𝑤→𝑒
(23)
2.3.4. CubeSat’s emission
The satellite also emits radiation to space according to [34]:
(
)
4
𝑄out𝑤 = 𝜖𝑤 𝜎𝐴𝑤 𝑇𝑤4 − 𝑇∞
(24)
3.2. The domain of FVM simulation
The domain is discretized into a structured multi-block grid composed of a set of small hexahedral volumes, where the temperature
and the heat fluxes are evaluated at the centroid and surfaces of the
volume, respectively. The geometry represents a CubeSat 1U, according
to Fig. 7. It has the main external dimension of 10.0 × 10.0 × 10.0 cm,
with six solar panels covering the external sides of the CubeSat (10.0 ×
10.0 × 0.2 cm, in red), an internal structure (with a cross-section of
0.5 × 0.5 cm, in green), a battery (6.0 × 6.0 × 0.9 cm, in magenta),
four PCBs representing generic payloads (9.0 × 9.0 × 0.2 cm each, in
light-blue), and bolts to connect the PCBs and the structure (with a
cross-section of 0.5 × 0.5 cm each, in blue).
where 𝜖𝑤 is the emissivity, 𝜎 is the Stefan–Boltzmann constant, 𝑇𝑤 is the
satellite’s surface temperature, and 𝑇∞ is the outer space temperature,
corresponding to black-body radiation at 2.7 K.
3. Simulation setup
Each side of the CubeSat receives a number for its identification.
Notice that the solar panels cover the CubeSat’s external surfaces,
and the bolts provide conductive thermal path between sides 5 and
6 of the CubeSat. The battery is attached to the top of PCB2 . Fig. 8
shows a schematic view of the internal parts, without the structure and
bolts. One major cavity is formed by the solar panels, while the PCBs
and battery act as obstructions inside the cavity, also absorbing and
emitting radiation.
This section starts by delineating the study cases. Later, the simulation’s parameters are defined, including the geometry, material
properties, incoming radiation, convergence criteria, and the mesh
independence test.
3.1. Cases of study
To assess the impact of inner boundary conditions, the following
cases will be tested:
Any cabling, connectors, and electronics details were not included
in this work because the interest is in a generic simulation of a CubeSat
1U. Bolts connecting the printed circuit boards and the sides 5 to 6 are
in the simulation because a similar configuration is expected in a real
mission. Their inclusion is important because they form a significant
path for heat transfer. Wires and cables also offer conductive ways
of heat transfer; however, the polymer parts associated with them
usually have low thermal conductivity, and the cross-section of the
wires are small compared to the bolt’s area, which increases the thermal
resistance of this set. These parts make it difficult to heat exchange
by conduction, although they can contribute to the CubeSat’s thermal
inertia. The consequence of greater thermal inertia is to approach the
upper and lower extreme temperatures.
• E-0: Surfaces in the internal side of the CubeSat has emissivity
equal to 0.0;
• E-1/2: Surfaces in the internal side of the CubeSat has emissivity
equal to 0.5;
• E-1: Surfaces in the internal side of the CubeSat has emissivity
equal to 1.0;
The extreme conditions without internal radiation and maximum
internal radiation are designated by E-0 and E-1, respectively. An
intermediate configuration between these two limits will be simulated
in the case E-1/2, where the internal surfaces have emissivity equal to
0.5.
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Applied Thermal Engineering 193 (2021) 117039
E. Morsch Filho et al.
Table 1
Thermal and surface properties of CubeSat’s main parts.
Part
Thermal property
Surface property
𝜚 [kg/m3 ]
𝑐 [J/kg K]
𝜅 [W/m K]
𝜖 [–]
𝛼 [–]
Solar panel
1840 [20]
2810 [24]
800 [24]
862 [19]
1150 [20]
1600 [18]
1.03 [24]
0.60 [15]
0.85 [18]
0.68 [15]
0.85 [19]
0.91 [18]
Structure
2810 [24]
936 [21,43]
130 [24]
0.08 [15,18]
960 [18,24]
150
[21,43]
0.79–0.88 [44]
1840 [20]
2400 [24]
800 [24]
1150 [20]
0.25 [22]
1.03 [24]
0.22 [20]
0.85 [19]
2122 [45]
2180 [20]
2440 [46]
933 [45]
960 [20]
1200 [19]
1210 [46]
1250 [24]
12.5 [46]
21 [45]
36 [20]
0.7 [20]
–
PCB
Battery
0.37
[15,18]
0.25–0.91
[44]
Table 2
Thermal properties for the standard case.
Part
Solar panel
Structure
PCB
Battery
Thermal property
Surface property (external side only)
𝜚
𝑐 [J/kg
[kg/m3 ] K]
𝜅 [W/m
K]
𝜖 [–]
2325
2810
2120
2247
1.03
140
0.64
23
0.72
0.77
Defined according to the case study
Defined according to the case study
Defined according to the case study
1103
948
975
1110
𝛼 [–]
3.3. Material properties
The literature regarding thermal simulations of CubeSats and satellites presents a wide range for the thermal and surface properties of the
main parts, as shown in Table 1. While the emissivity is for long-wave
radiation, the absorptivity is for solar-wave radiation.
The average values from the previous table, shown in Table 2, are
used to compose the simulation’s thermal and surface properties. Only
the solar panels are exposed to the external radiations, with emissivity
and absorptivity according to the values of Table 2. The values for this
work’s surface properties have the ideal behavior of being independent
of the emission temperature and indifferent to the spectrum of the heat
source or their directions.
Fig. 9. Incoming thermal radiation on each side of the CubeSat.
smaller than 10−2 in the balance of each surface exposed to internal or
external heat transfer by radiation. The external boundary condition is
cyclic, and the interest is in results that are independent of the initial
condition. For this reason, an extra condition for the convergence of
the simulation is a temperature difference between the field on the last
instant of the orbit and that obtained in the first, within a margin of
±1 K.
The simulations were implemented in MATLAB. They were run in a
Ubuntu environment, in a computer with an Intel Core Xeon E5-2665
Processor (2.40 GHz), with 8 cores of 2 threads and 64 GB of RAM.
The overall features obtained with three meshes’ refinements are
indicated in Table 3. These results highlight the strong relation of
computational cost with the grid’s size. While Mesh 1 is around six
times smaller than Mesh 2, the total time to compute the obstruction is
27 times faster for Mesh 1 than Mesh 2 and 15 times faster to solve
the temperature field. With Mesh 3, the total time to compute the
obstruction is 16 times slower than Mesh 2 and five times slower to
solve the temperature field than Mesh 2.
In terms of the temperature of each main CubeSat’s part, the maximum deviation from Mesh 3 to Mesh 2 was 0.8 K, while for Mesh 1
and Mesh 2, it was 4.3 K. This difference is obtained by comparing the
central node of each main part of the CubeSat (solar panels, PCBs, and
battery), so the average values of all parts are even below it. For these
reasons, Mesh 2 will be used in the following results.
3.4. Attitude and orbit
The orbit scenario is coherent with a launch from the International
Space Station (ISS), so the satellite has an altitude of 431 km and an
orbit inclination of 51.6◦ . The orbit’s ascending node is set to 0◦ for a
maximum duration of the eclipse; consequently, the CubeSat will suffer
substantial variation of the incoming radiation. The CubeSat has an
attitude called nadir, where the normal vector of surface 2 continuously
faces the Earth’s surface. As a result of this attitude, the solar panels 1,
2, 3, and 4 are exposed to the Sun; however, only 3 and 4 will have
maximum projection towards it because of the combination of orbit
inclination, ascending node, and attitude. Fig. 9 shows the incoming
thermal radiation on each side of the CubeSat. The gap in the middle
of the plot results from the eclipse condition, where only the Earth’s
emission still occurs. The sides 5 and 6 are opposed, and for this orbit
and attitude, they have identical incoming radiation.
3.5. Convergence criteria and mesh independence test
For each timestep, set to 10 s, the convergence criteria is a normalized global residual smaller than 10−5 for the volumes of a mesh,
10−6 for the volumes in contact with a neighbor mesh, and a difference
7
Applied Thermal Engineering 193 (2021) 117039
E. Morsch Filho et al.
Fig. 10. Temperature field at𝑡 = 1720 s, for case E-1/2.
Table 3
Mesh independence study.
Parameter
Number
Number
Time to
Time to
An approximate view of the temperature field along the entire orbit
may be obtained by averaging the spatial distribution values on each
main part, as shown in Fig. 11, valid for the case E-1/2. As cited above,
at 1720 s occurs the maximum temperature gradient, which is when
the eclipse starts, while the minimum gradient is in the last instant
of the eclipse, at 3880 s. The peak at 1720 s is around 340 K, which
is 20 K colder than the observed in Fig. 10a because of the spatial
averaging. As observed in Fig. 9, side 1 receives more radiation in the
beginning of the orbit, resulting in maximum temperature for this side
in that instant. However, the maximum overall temperature is achieved
by side 4 because it is already warmer when the solar radiation starts
to raise its temperature. In comparison, side 3 has similar incoming
radiation to side 4, but it is colder than side 4 when the solar flux heats
it. Side 2 stays warmer than other sides near the end of the eclipse
because it receives more radiation from the Earth due to its projection
towards that source. The minimum temperature occurs on solar panel
1 because it does not receive any radiation from the Sun or the Earth
during the eclipse, even after entering the eclipse hotter than sides 3,
5, and 6. Sides 5 and 6 have identical behavior because their projection
towards the radiation sources is the same. These two surfaces do not
have significant variations because they do not receive solar radiation,
only albedo, and emission from the Earth.
As observed in Fig. 7a, the bolts passing through the PCBs connect
solar panels 5 to 6, serving as a thermally conductive path for inner
parts. The temperature of internal components shown in Fig. 11b
oscillates less than the parts exposed to external radiation sources,
and they are hotter than sides 5 and 6. The bottom PCB1 and top
PCB4 have similar curves, always colder than the intermediate PCB2
and PCB3 . These parts’ location can be one reason for it because the
influence of heat transfer by radiation from sides 1, 2, 3, and 4 is
greater in the intermediate PCB2 and PCB3 . The peak of temperature
occurs first at PCB2 , while the peak for PCB1 and PCB4 are the last,
Mesh 1 Mesh 2 Mesh 3
of volumes
829
of faces with internal heat transfer by radiation 982
compute obstruction [h]
0.2
solve the cyclic temperature field [h]
1.1
5212
2844
5.5
16.2
19 098
7174
90.4
89.3
4. Results
This section brings the results and discussions about the three cases
of internal radiation simulated with FVM: E-0 (internal emissivity is
0.0); E-1/2 (internal emissivity is 0.5); E-1 (internal emissivity is 1.0).
Before these findings, the discussion starts with the three-dimensional
field obtained with FVM and the values of the LPM.
4.1. Temperature field: FVM and LPM
The first result in Fig. 10 illustrates the importance of simulating
three-dimensional domains. These temperature fields are for𝑡 = 1720 s,
a maximum temperature gradient condition in the satellite, as will be
seen later. Notice that the temperature range for each part adjusts for
better visualization. In this case, solar panel 4 receives more radiation
than any other side (see Fig. 9). Due to the solar panel’s low thermal
conduction, there is a significant temperature gradient on this panel,
resulting in a peak temperature in the center and minimum values at
the border. This distribution is explained by the fact that the internal
structure, made of aluminum, is in contact with solar panel 4 only near
its borders. For all of these fields, it is evident that a single point of
temperature cannot represent the entire CubeSat’s temperature field,
neither of a single main part.
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Applied Thermal Engineering 193 (2021) 117039
E. Morsch Filho et al.
Fig. 11. The average temperature of the main parts of the CubeSat for case E-1/2.
Fig. 12. Comparison of LPM and FVM (E-1/2 case).
few seconds after PCB3 . These shifts increase among their minimum
temperatures, but the order of occurrence remains. The hypothesis for
the shift between them comes from their different exposition towards
the satellite’s sides. The temperature shift between PCB2 and PCB3
results from the larger thermal inertia of the battery plus the PCB2 .
Interesting to notice an elevation of the battery’s temperature after
several minutes of the eclipse beginning. The battery’s temperature is
strongly related to the heat transfer by conduction with PCB2 , so the
battery’s temperature keeps rising as long as the temperature of PCB2
is higher than the battery’s. The same conclusion is evident when the
battery’s temperature keeps falling after the eclipse and only raises if
the temperature of PCB2 is greater than the battery’s. The small swing
in the battery’s temperature also can be explained by its greater thermal
inertia (𝜌𝑐) than the PCBs.
Fig. 12 presents further temperature results of each side and battery
obtained with the FVM for case E-1/2, as well as from the LPM. The
temperature at the center of a part is given by a unique point (Tp ),
9
Applied Thermal Engineering 193 (2021) 117039
E. Morsch Filho et al.
Fig. 13. Temperature range and average temperature for the cases: E-1/2 (with internal heat transfer by radiation: 𝜖 = 0.5); E-0 (without internal heat transfer by radiation: 𝜖 = 0).
while the average temperature of a part (Tave ) is the sum of all temperature points weighted by the corresponding discretized volume and
divided by the total volume of the part. The temperature range (Trange )
comprehends the extreme maximum and minimum values of the part at
each instant. The temperature Tp , Tave and Trange are obtained from the
FVM simulation. In this same figure, the curves obtained with the LPM
formulation are plotted for different thermal resistances (parameter
in Eq. (3)), identified by the letter 𝑍 followed by the thermal resistance
value used in the simulation. The LPM uses the same thermal properties
of Table 2, dimensions of the FVM geometry, but discards PCBs and
bolts.
This Fig. 12 illustrates that Tp , Tave and Trange from the FVM are
different temperature views of the same part, and they together are
more informative rather than a single curve. By analyzing the curves
from the LPM, they are within or very close to the extreme values
obtained with FVM, except for 𝑍 = 0.00, where there is no heat
exchange between the solar panels and battery. With 𝑍 = 0.05, the
curves approximate to the values found in the center of the component
(Tp ), while 𝑍 = 0.10 approaches to the average value (Tave ). The temperature range (Trange ) of the FVM is somehow reproduced by the LPM
curves with 𝑍 = 0.05 and 𝑍 = 1E4. Interesting that these LPM curves
sometimes touch the upper bound and sometimes the lower bounds of
the FVM. With 𝑍 = 0.00, the battery does not exchange heat and, for
this reason, maintains its initial temperature of 273 K. The other values
of 𝑍 resulted in greater temperature oscillations at the battery, and
they did not follow the behavior found in the FVM results. The LPM
formulation with seven points can still follow the temperature tendency
obtained with the FVM, even with the oversimplification of the heat
transfer by conduction and radiation in the term 𝑍. Nevertheless, the
three-dimensional effect is essential to fully understand what is going
on in the satellite, which is a major drawback of this formulation when
compared to the FVM.
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Applied Thermal Engineering 193 (2021) 117039
E. Morsch Filho et al.
Fig. 14. Temperature range and average temperature for the cases: E-1/2 (with internal heat transfer by radiation: 𝜖 = 0.5); E-1 (with maximum internal heat transfer by radiation:
𝜖 = 1).
minimum inside the eclipse. On the other hand, inside the eclipse,
4.2. Zero internal radiation (E-0) and intermediate internal radiation (E1/2)
the zero emissivity case (E-0) underestimates the maximum levels of
PCB2 and PCB3 . Outside, it overpredicts and underpredicts the extreme
In Fig. 13 are the results for each main part of the CubeSat,
considering emissivity equal to zero in the internal surfaces (E-0), and
those with intermediate value of 0.5 (E-1/2), for both average (Tave )
and temperature range (Trange ). These figures show that around ±15
K and ±20 K of temperature difference exists in the average value
and temperature range of the component, respectively, by changing
from zero emissivity to 0.5. The values obtained without internal heat
transfer were overestimated in the hottest regions and underestimated
in the coldest areas compared to the case with the internal transfer.
For those solar panels with moderate temperature, for example, sides
2, 5, and 6, the effect from changing the internal surface properties
is less evident, although the zero emissivity results in lower minimum
temperatures.
Considering the PCBs, the temperature ranges are quite close between E-0 and E-1/2 for the hottest levels outside the eclipse and the
values of PCB1 and PCB4 , and almost always underestimation lower
limits of PCB2 and PCB3 . The introduction of internal radiation increase
the average temperature of PCB2 , PCB3 and battery, also the variation
of temperature range of PCB1 and PCB4 . It was expected because more
heat could arrive on these internal parts when there is internal heat
transfer by radiation, especially when the walls’ temperature is hot.
For the inside parts, the cavity temperature is more important than a
single wall, which explains the peak of temperature in the PCBs around
1100 s, before the CubeSat’s maximum absolute temperature at 1720 s
(solar panel 4). For both scenarios of internal boundary conditions, the
temperature gradient in the battery is narrow, explained by its greater
thermal conductivity (𝜅) when compared to the other parts.
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Fig. 15. Temperature distribution of the main parts of the CubeSat for each case. The average value is the pink diamond and the red line is the median.
panels 1, 3, and 4. When the internal components are assessed, the
opposite happens. The zero emissivity condition reduces temperature
variations, while the upper and lower values move away from each
other as increasing the emissivity enhances the internal heat transfer.
The difference by assuming maximum emissivity (𝜖 = 1) and minimum
(𝜖 = 0) may reach up to 10 K for the internal parts and around 20 K
for the externals.
4.3. Maximum internal radiation (E-1) and intermediate internal radiation
(E-1/2)
Fig. 14 shows the results of the case with maximum internal emissivity (E-1) together with the intermediate condition of emissivity (E-1/2).
In general, the curves of E-1/2 and E-1 are closer than in the pair E1/2 and E-0. The temperature range of E-1 has an opposite behavior
than case E-0, with extreme hot values being lower and extreme cold
being greater than the results obtained with E-1/2. Interesting to notice
an additional increase of temperature in PCB2 and PCB3 around 1700
s and 5000 s. The probable cause of that is the greater influence of
solar panels 4 and side 3, respectively, caused by the greater emissivity.
The maximum temperature of PCBs in the eclipse with maximum emissivity (E-1) is generally below the maximum levels obtained with the
intermediate emissivity (E-1/2). Both cases reproduce similar minimum
levels in the PCBs outside the eclipse, except near after the eclipse. Even
increasing the emissivity value, the battery’s temperature still presents
a low spatial and temporal temperature gradient, which evidences the
strong superiority of its heat transfer by conduction with PCB2 than by
radiative processes.
5. Conclusions
This work has simulated the heat transfer of a CubeSat 1U, with
solar panels covering the external sides, an internal structure of aluminum, four PCBs, and a battery without heat generation. The solution
was obtained through a Finite Volume Method (FVM) algorithm accounting for the internal heat transfer by radiation with obstructed
views, solved by the Gebhart Method. The results highlighted the
importance of including the internal radiation in thermal problems of
CubeSats.
The temperature field obtained with the FVM case was compared
to a Lumped Parameter Method (LPM). The LPM formulation is a fast
way to have an idea about the CubeSat’s heat transfer; however, as
the results have shown, there are appreciable threedimensional fields
that cannot be reproduced with this simple system of equations. The
attribution of a single point for each main part of the satellite in the
LPM was able to capture the overall behavior of the FVM results, and
an adjustment in the thermal resistance coefficient (𝑍) was able to
adhere to the range, central point, or average values obtained with
FVM simulations. For 𝑍 = 0.00, valid for none heat transfer among
the solar panels and the battery, the temperature levels are beyond
the extreme values of the FVM. In contrast, 𝑍 = 0.05 and 𝑍 = 1E4,
this last representing high heat transfer from solar panels towards the
battery, reproduce quite well the temperature range of the FVM, with
both values of curves defining the superior and inferior extreme levels,
depending on the instant of the orbit. With 𝑍 = 0.05 and 𝑍 = 0.10
the LPM approaches the central-point and average values of the FVM
formulation, respectively.
4.4. Statistical distribution of results
Fig. 15 summarizes the entire orbit’s average fields for each part
and case into boxplot and average values (pink diamond). The nonsymmetrical temperature distribution is more evident for solar panels
with the most significant temperature gradients (solar panels 1, 3, and
4) than the remainings. Since the quartile below the median is shorter
than the above, the CubeSat spends most of the time closer to its minimum temperature value than its maximum. As already observed, the
case without internal heat transfer by radiation (E-0) has the greatest
temperature differences, while the case of maximum internal emissivity
(E-1) has the narrowest ranges. However, the average temperatures are
essentially constant for the solar panels, with sides 5 and 6 having
the minimum levels. The medians are lower than the average values
at the panels with the most significant temperature gradients, namely
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E. Morsch Filho et al.
Fig. 16. Temperature data from CubeSats in orbit.
Source: Adapted from J. Firedel [47], Kramer [48]
and Kramer [49].
The boundary condition of inner surfaces varied from zero emissivity (E-0) to an intermediate case with 𝜖 = 0.5 on each internal part
(E-1/2), and maximum emissivity equal to one for all the internal surfaces (E-1). The inner boundary conditions impacted the temperature
field of both internal and external parts. For the components exposed
to outer space, the case without internal heat transfer by radiation
reproduced greater maximum levels outside the eclipse and colder
minimum values in the eclipse than the case with the internal emissivity
of 0.5. On the other hand, the internal parts receive less heat in the zero
emissivity case and reproduce lower temperatures for most of the time.
Opposite to it, the greatest internal emissivity resulted in the shortest
temperature peaks and gradients of the external parts, an expected
result because the solar panels could exchange heat through their
two opposite surfaces. Internally, the gradient was increased, but the
average values decreased. By statistically comparing the temperature
distribution in each case, there is no significant variation among the
entire orbit’s average values, only in their upper and bounder limits.
The conservative internal radiation boundary condition presented in
this paper can help other formulations, not only FVM. The inclusion
of it may assist in validating passive thermal controls centered on
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Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgment
This work was supported by CNPq, National Council for Scientific
and Technological Development - Brazil, via grant number 141276/
2018-5.
Appendix
Some examples of temperature behavior in orbit are shown in
Fig. 16. The oscillatory but nearly cyclic pattern comes from exposition
to the Sun followed by the Earth’s eclipse, resulting in an orbit period of
around 90 min for Low Earth Orbit (LEO) and eclipse of approximately
35 min for all the cases. Under the shadow of the Earth, these CubeSats
reach the lowest temperatures due to the lack of solar radiation. These
figures show significant transient and temperature gradients in orbit
and different temperature levels for the same satellite. Launched in
April 2007, CubeSat CP3 had a Sun-synchronous orbit, an altitude of
700 km, but no information was available regarding its spin. It can
be observed that its limits of temperature for one complete orbit are
more compressed a few weeks after the launch (left) than the sample
valid for one year later (right). CubeSat CP3 was below 300 K in the
beginning of the mission and reached more than 320 K twelve months
later. A similar trend is observed in the results from two complete orbits
of SwissCube, with around 15 K of temperature increase between the
figures that are 15 months apart. In both situations, the temperature
gradient expanded between these plots and could be associated with
different thermal radiation scenarios. In fact, in Fig. 16c the CubeSat
was rotating faster than 600 degrees per second, while less than 5
degrees per second in Fig. 16d, which created different exposition of
its surfaces to the irradiation fluxes. The internal temperature of some
parts of SwissCube and Zacube are in Figs. 16e and 16f, and show that
the internal parts of these satellites are less susceptible to temperature
variations.
The trend of these curves and their minimum values around 240
K for the external parts are quite similar to the numerical results.
However, the maximum levels for the numerical simulation are hotter
than these data. This discrepancy could be caused by the CubeSat’s
design, its materials, orbit, attitude, or electrical operation modes, but
there is not more information about these parameters.
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