### A.A. 2017/2018

### DIPARTIMENTO DI INGEGNERIA CIVILE ED INDUSTRIALE

### Corso di Laurea in Ingegneria Aerospaziale

## THERMAL MATHEMATICAL MODEL OF A MODULAR

## SATELLITE AND PRELIMINARY DESIGN OF A HIGH

## THERMAL PERFORMANCE COMPONENT

## RELATORI:

## Prof. Ing. Sauro Filippeschi

DESTEC, Università di Pisa## Dr. Ing. Mauro Mameli

DESTEC, Università di Pisa## Dipl.-Ing. Thomas Schervan

SLA, RWTH Aachen University## CANDIDATO:

## Stefano Piacquadio

## Dedications

Before starting the description of this work I would like to thank those who mostly contributed to help me get to it.

The first thank goes to my family, my parents, my little sister (a.k.a. Puffetta) and especially to my grandparents. Without them and their support I think I could not even be able to start studying at the university.

I would like to address a special thank to my supervisors, Prof. Sauro Filippeschi and Dott. Mauro Mameli. I started following them five years ago, when I was still in my Bachelor. They taught me a lot, but actually, when I think about what I got from them, the main thought is related not only to what I learned on the technical point of view, but on how they helped me to build-up my person. The teachings I will always remember are the life-related ones. Associated to them I want to thank all the people who worked in the PHOS and U-PHOS projects. Most of them were colleagues at first, but became great friends during these years.

I want to thank the people who helped me in this work at the RWTH Aachen University. My supervisor Ing. Thomas Schervan for his suggestions and for following through all the process. A particular thank goes to my friend Tobias Meinert. He gave me the possibility to perform this work, and helped me to face a new reality where I had to learn a new language and start understanding the German culture. I arrived in Aachen almost as a stranger to him, but, as I left, it was more like a farewell from family. I would like to thank the people who were put on my side in this period, especially Sebastian and Dominik. Their friendship helped me endure the fatigue of living in a foreign country and its wonderful weather.

My girlfriend Aliaksandra deserves a great thank. She helped me while reviewing this work, but especially walked with me while facing new emerging challenges.

Last, but not least, I want to thank the friends I’ve grown up with, for being the solid structure I can lean on, and the friends I made in these university years, for sharing with me this meaningful piece of my life. Among these, a special thought goes also to the friends I’ve made in my high-school, who are still with me, and to Prof. Ciletti, who stimulated in me a great interest in studying through his passion at teaching.

I can say I’m a lucky person because I almost always met people who care about me. We don’t simply get along together, but we share a path, we share life with most of them. This is the best gift a person could ever receive.

## Abstract

This work has the goal of describing the realisation of a Thermal Mathematical Model for the German joint research project on the "iBOSS" satellite. iBOSS is a research project carried on under a multilater agreement of the main German universities and the Deutsches Zentrum für Luft- und Raumfahrt (DLR). The target is the design, realisation and In Orbit Demonstration (IOD) of a completely modular satellite platform capable of assembling directly on orbit and performing robotic maintenance. The research in this kind of systems is by now a worldwide interesting topic related to the On Orbit Servicing (OOS) paradigm. OOS represents a set of technique capable of significantly extending satellites life and/or allowing a safe disposal of the satellite at its End of Life, helping to reduce hazards posed by phenomena like Space Debris. This work has been realised at the SLA - Institut für Strukturmechanik und Leichtbau of the RWTH Aachen University, major partner of the iBOSS project. In the work performed the in-put is the actual state of design of the satellite. The main components were already designed, but no thermal analysis with the target of verifying the operative conditions on the thermal point of view, had been performed, yet.

An extensive literature review helped understanding the process to accurately determine the en-vironmental heat loads on the satellite. Via a MATLAB script the algorithm to perform such calculations is implemented for variable inputs relative to the satellite orbit. The obtained val-ues are used as border conditions for the thermal model realised on the satellite geometry. The challenge faced in realising the model is the necessary modularity of the model itself. This because to object of the model is a modular satellite. This can mean frequent updates and variations of the mission scenario. Thus, it is necessary to have an easily manageable tool to simulate different, rapidly changing, scenarios due to multiple causes, i.e. coupling of several modules, detachment of obsolete ones or orbital plane changes. Although some limitations ex-ist, the SIMULINK thermal toolbox and its Graphic User Interface (GUI) constitute powerful instruments to realise such model. A thorough analysis of the modular satellite components properties helps understand the best approach for the model set-up, which is extensively de-scribed.

A reference mission scenario is then chosen in order to simulate a meaningful and short-term realistic application. The model allows to detect the critical elements from the thermal point of view. A following of the thesis focuses on analysing possible solutions to one of the spe-cific problems evidenced. A preliminary design of a heat transfer device is proposed to try addressing such problem. An updated model is run to verify whether the new design respects the thermal requirements. The performed design consists in a Flat Plate Pulsating Heat Pipe (FP-PHP) to be embedded in a pre-existing core-component which has the goal of distributing heat from and to temperature sensitive components. The goal of the FP-PHP is of aiding in

ii Abstract

the homogenization of temperatures on its surface, improving its heat transfer capability. This preliminary design takes into consideration also manufacturing aspects and proposes to use an Additive Manufacturing (AM) process. Literature about the application of such method for the production of FP-PHP suggests that promising conditions on the channels diameter shape and surface roughness can be obtained. This, according to literature, shall help improving the fluid motion activation with respect to FP-PHPs realised with more common and proven methodolo-gies.

From the model verification it is possible to understand that the results obtained by such meas-ure are promising and shall help improving the heat transfer capability of 200% with respect to the conditions not including this measure.

### Keywords

Thermal Mathematical Model, Modular satellites, On Orbit Servicing, Pulsating Heat Pipe, Flate Plate-Pulsating Heat Pipe, Additive Manufacturing, 3D printing.

## Contents

1 Introduction 3

1.1 iBOSS concept . . . 4

1.2 Spacecraft Thermal Control Systems and heat transfer. . . 6

1.3 Spacecraft thermal modelling . . . 24

2 iBOSS Thermal Mathematical Model 31 2.1 iBOSS Thermal architecture . . . 34

2.2 Simulation. . . 51

3 Thermal performance enhancement 73 3.1 Isothermalisation of the thermal junction . . . 73

3.2 Pulsating Heat Pipes working principles and space applications . . . 77

3.3 Preliminary design . . . 84

3.4 Model verification. . . 96

3.5 Design conclusions . . . 100

4 Conclusions 103 4.1 Limitations of the TCS and future work . . . 103

Bibliography 112 A Appendix - Thermal model schematics 113 A.1 Environmental heat loads . . . 113

A.2 Blocks description . . . 113

2 Contents

ABBREVIATIONS

iBOSS: intelligent Building blocks for On orbit Satellite Servicing TCS: Thermal Control System

LEO: Low Earth Orbit

GEO: Geostationary Earth Orbit FDM: Finite Difference Method FEM: Finite Element Method EOL: End Of Life

BOL: Beginning of Life

TMM: Thermal Mathematical Model PCM: Phase Change Material

CHF: Critical Heat Flux PHP: Pulsating Heat Pipe OHP: Oscillating Heat Pipe

CCHP: Constant Conductance Heat Pipe TRL: Technology Readiness Level

NOMENCLATURE T: Temperature [K,◦C]

k: thermal conductivity [W/(mK)] h: heat transfer coefficient [W/m2K] A: Area[m2]

cp: specific heat [J/(kg K)]

C: Thermal capacity [J/ K] ˙q: heat flux [W/m2]

σ=5, 67037321 ˙10−8W/(m2K4)Stefan-Boltzmann constant α absorptivity of a grey body

ε emissivity of a grey body M: emittance, exitance [W/m2]

E: eccentric anomaly [radians] ν:anomaly[radians]

RAAN Ω: Right Ascension of Ascending Node [radians] i: inclination [radians]

a: semimajor axis [km]

e: eccentricity [pure number 0-1] ω: perigeum argument [radians]

## 1 Introduction

In the last two decades space industry has rapidly grown. More countries, agencies and com-panies are capable of designing, realising and launching satellites for any kind of purpose. This increased the presence of satellite platforms in Earth orbits, causing concerns related to overcrowded orbital slots. The constant increase of abandoned satellites poses hazards to new spacecrafts, leading to the occupation of unique and economically valuable orbital slots that could be otherwise recycled for other uses. On Orbit Servicing (OOS) of satellites represents a key tool to create those architectures capable of maximizing functionality and sustainability of orbiting space assets. The Space field research is more and more oriented in finding tools to face performance degradation issues occurring to satellites due to the intrinsic difficulty to perform maintenance during their operative life. However, it is a paradigm which is experiencing a slow ascent due to the unreadiness of many crucial key components which are not flight-proven yet. Recently, the interest and the fundings for the realization of such systems has rapidly grown. OOS defines a series of activities performed in space comprising inspection, repairing, refuel-ling, refurbishment and update of existing orbital vehicles. Unmanned OOS is perceived as a very efficient and cost-effective way to extend the operational life of satellites orbiting the Earth.

Fig. 1.1: Artistic representation of iBOSS system performing On Orbit Servicing [1].

Satellites architecture can be split in payload and satellite-bus. The generic term bus includes integrative and functional subsystems of the satellite. The bus, which represents the supply unit of the satellite, allows the operation of the carried payload. In the OOS perspective the idea of modular satellites is gaining ground everyday more because of its feasibility and high possible revenues, especially with respect to other techniques such as direct human assisted

4 1 Introduction

maintenance, as happened for the Hubble Space Telescope. The underlying approach is the separation and subsequent modularization of the satellite bus on subsystems and component level. The individual satellite buses are integrated into distinct standardized modules. The modules are subsequently connected to each other, via a detachable interface, building up the full satellite bus.

### 1.1 iBOSS concept

The German joint research project iBOSS (Intelligent building blocks for On-Orbit Satellite Servicing) aims to design and create such a modular satellite system. iBOSS examines and develops potential concepts for standardization and modularization of future satellite systems enabling autonomous maintenance and servicing using robotic systems. This concept focuses on performing sustainable OOS operations such as maintenance, repairs, module upgrades or on-orbit-assembly with the aim to increase the lifespan and the flexibility of a space system. The modular design has positive effects on the overall mission effectiveness. This includes a great flexibility in satellite development and production. The iBOSS approach does not only fo-cus on the hardware aspects of modular space systems but also on the development of dedicated software such as computer aided satellite design, reconfiguration planning and detailed space-craft simulations which helps to reduce development time and costs. The positive economic effects resulting from increased sustainability of space missions are also accompanied by other benefits, such as the reduction of space debris. This furthermore provides a great motivation for the iBOSS project.

The concept centres around two key elements: modules, consisting of both structural and func-tional elements, and a standardized 4-in-1 interface. The multifuncfunc-tional interface provides mechanical and thermal coupling, power and data transfer. It enables countless block combina-tions due its androgynous and symmetric design. This approach enables excellent opportunities for industrial manufacturing and therefore increased economic efficiency. Using prequalified, off-the-shelf modules with interfaces the modular iBOSS concept also allows for rapid and on-demand development.

A single module of an iBOSS satellite, called iBLOCK, is represented by a cubic box with 40cm lateral size. A CAD image of a complete iBLOCK with its multifunctional interfaces is shown in Fig.1.2. By assembling via mating a variable number of iBLOCKs a complete satellite can be realized. Each iBLOCK has different customized purposes. The resulting structure concept is shown in Fig.1.3.

1.1 iBOSS concept 5

Fig. 1.2: Rendering of an iBLOCK assembly, which is a single module of an iBOSS satellite [1].

Fig. 1.3: Artistic representation of a complete iBOSS satellite system, realized by mating several iBLOCKs [1].

6 1 Introduction

As any kind of satellite, iBOSS needs to operate in a harsh environment where no convection is possible due to the absence of atmosphere. Typically, significant temperature variations occur when passing from the sunlit part of the orbit, where components can reach very high tem-peratures, to eclipse. To extend durability and prevent critical damage to components, thermal control systems are necessary. In order to size such systems, thermal models need to be de-veloped.

### 1.2 Spacecraft Thermal Control Systems and heat transfer

The main goal of spacecraft Thermal Control Systems (TCS) is to prevent overheating and un-dercooling in every part of equipment, for any phase of the spacecraft mission. The typical solution adopted in TCS to avoid overheating that causes permanent damage, is to use outer coverings with thermo-optical properties which allow to keep the system inside its operative range of temperature. To compensate the eventual undercooling, common at eclipses, the use of distributed electrical heaters is a common practice. Undercooling usually does not cause permanent damage but just a dormant non-operational state, which, however, may be critical to the mission. Some components, i.e. batteries and mechanisms, may instead be affected on future operation even after temperature is brought back inside the operative range. Some over-sizing is always applied to cover contingencies, as also suggested by the thermal control and thermal analysis normatives ECSS-E-ST-31C [2] and ECSS-E-HB-31-03A [3]. The problem with this solution is that electrical power availability is generally low in spacecrafts and more during eclipse periods, when no solar power can be generated. This closely connects the TCS system with the power management system in regard of battery capacity sizing. This explains why the choice of appropriate thermo-optical properties of external surfaces is never a trivial problem. The selection of too high emissivities and too low absorptivities causes a consistent risk of undercooling, requiring high power demands by heaters. Thus, it is necessary to select appropriate values which allow to minimize power consumption.

The TCS of a spacecraft is the group of components that allow the maintenance of critical components in their operative temperature range, which is listed in a general form in Tab.1.1. Classical TCS are based on radiative energy emission from the spacecraft envelop usually con-centrated on some surfaces specifically designed for the purpose of heat rejection (radiators), with some metal conduction along cold plates from equipment inside. In TCS, however, two-phase technologies have become the standard tool for spacecraft thermal control: heat pipes and loop heat pipes, micro electromechanic (MEMS) two-phase fluid loops, Phase-Change Materi-als (PCM), heat pumps, cryogenics, etc.

Thermal control technologies can be either passive or active. The first make use especially of the modulation of radiative properties of surfaces in order to control the absorbed and emitted energy via radiation. Special treatings and coatings can be applied. Moreover passive devices such as heat pipes can be used in order to transfer heat without the necessity of additional work. Active devices are normally used when it is not possible to reject or generate heat other-wise. The most common use of active devices consists in using survival heaters for temperature

1.2 Spacecraft Thermal Control Systems and heat transfer 7

Component Operative T [◦C] Storage T [◦C]

Batteries -5 to 50 -10 to 60

Solar panels -100 to 125 -100to 125

Momentum wheels 0 to 50 -20 to 70

Solid-state particle detectors -35 to 0 -35 to 35

IR detectors -269 to –173 -269 to 35

Analog electronics 0 to 40 -20 to 70

Digital electronics 0 to 50 -20 to 70

Table 1.1: Temperature range of basic spacecraft components [8].

sensible components which need to stay into a strict temperature range during operation, i.e. batteries. Thermo-electric coolers can be used when it is necessary to cool-down a component at temperatures lower than the steady-state average value of the satellite, normally for antennas which need to stay at very low temperatures to increase the signal to noise ratio.

Among the standard techonologies there are all of the typical radiative panels which, with a modulation of surface finish, can absorb and/or dissipate a certain amount of heat at a defined temperature. Solar reflectors, as second-surface mirrors, white paints and silver-aluminum backed Teflon are used to minimize absorbed solar energy, but emitting (and absorbing) en-ergy in the IR wavelength almost as a blackbody would. Black paints are instead used in the inside in order to maximize the heat transfer among internal components which don’t need to be isolated. Components which need to be thermally isolated from environmental sources and/or need to prevent heat loss are generally insulated with Multi Layer Insulation (MLI). MLI is composed of several layers of low-emittance films made out of Mylar sheets, each one with a vacuum deposited aluminum finish on one side.

Because of their very high thermal conductivity and lightweight, usually heat pipes are used as heat exchangers in order to connect a cold surface, i.e. radiator, to a heat source. Heat pipes use a closed two-phase liquid-flow cycle with an evaporator and a condenser to transport relat-ively large quantities of heat from one location to another without electrical power. A heat pipe can function in various ways as a thermal-control device. One-way, diode, heat pipes have been tested and flown, as have variable-conductance heat pipes (VCHPs), which maintain a constant-temperature evaporator surface under varying load conditions. Several new kinds of heat pipes under research are starting to be implemented in flight systems, i.e. Loop Heat Pipes (LHP), Ca-pillary Pumped Loops (CPL) and, lastly, Pulsating (or Oscillating) Heat Pipes (P/OHP). Many advantages of CPLs and LHPs are only truly exploited when these devices are considered early in the design phase, rather than treated as replacements for existing heat-pipe-based designs. LHP and a CPL theoretically only require wick material in the active evaporator zone; the re-mainder of the LHP is wickless tubing, condensers can use various designs, but they do not need to contain any wicks. Pulsating Heat Pipes were invented in a cost reduction perspective

8 1 Introduction

by Akachi [16] and basically consist of a wickless tube arranged in a serpentine shape both at evaporator and condenser sections. Their use in space applications is attractive, but still under research investigation.

In order to design a TCS, a thermal model needs to be realized. The thermal modelling of a spacecraft is highly related to the environmental heat loads acting on it. A thorough under-standing of common heat sources in the space environment is thus necessary to approach the problem.

1.2.1 Space Thermal Environment

The principal external heat sources are: direct sunlight, sunlight reflected off Earth surface (Al-bedo) and infra-red emitted energy from Earth. Free molecular heating is another source of heating that takes place during ascent phase of launch or for very low orbits, where friction is not negligible. The TCS is usually sized concentrating on the first three listed. The goal is to balance the IR natural emission to Space of the spacecraft surfaces with the external and internal heat sources.

In Low Earth Orbit (LEO), a space vehicle altitude is small compared to the diameter of Earth. This means that a satellite views only a small portion of the full globe at any given time. The satellite motion along the orbit, therefore, exposes it to rapidly changing environmental condi-tions as it passes over regions having different combinacondi-tions of land, ocean, snow, and cloud cover. These short-duration swings in environmental conditions are not of much concern to massive, well insulated spacecraft components. Infact, thermal inertia of spacecraft parts acts as a low-pass filter which averages most of the thermal response to rapidly changing scenarios. For instance, a polar-LEO satellite may be exposed to big changes in radiation input when going from the Equator to the Poles in less than 25 minutes (96 minutes polar orbit): local Earth Albedo factor may change from a=0.05 over uncloudy regions of tropical oceans to a=0.90 over the Antarctic, and Earth IR emission too, with surface temperatures over 300K over tropical lands to under 200K over Antarctica. These big changes have little influence on the thermal balance of the satellite, and only the most lightweight and decoupled spacecraft elements may follow such swift changes. Exposed lightweight components such as solar arrays and deployable radiators will respond to the extreme environments that are encountered for short time periods, so one must consider those environments in the design process. The shorter the thermal time constant a particular component has, the wider the range of environments that must be considered.

Direct Sunlight Sunlight is the most important source of environmental heating incident on spacecrafts in Earth orbit. The Sun is a very stable energy source and even the 11-year solar cycle has very little effect on the radiation emitted from the Sun, which remains constant within a fraction of 1% at all times. However, because of Earth’s orbit eccentricity, the intensity of sunlight reaching Earth varies during a year. At summer solstice, Earth is farthest from the Sun, and the received radiation heat flux is at its minimum value of 1322W/m2; at winter solstice, it

1.2 Spacecraft Thermal Control Systems and heat transfer 9

is at its maximum of 1414W/m2. The power-flux of sunlight at Earth’s mean distance from the
sun (1 AU) is known as the solar constant and is equal to 1367W/m2_{.}

Albedo Albedo is an optical phenomena that consists of sunlight reflected off a planet or moon surface. A very famous experience of it is the visibility of the dark side of the Moon before or after the new moon.

In a first approximation, Albedo heat load is expressed as: ˙q=a· I · F_{earth−sat}where a represents
the albedo factor, which is a pure number ranging from 0 to 1 usually about the value of 0,3, and
I is the Solar heat flux. However, Earth Albedo is highly variable. Reflectivity is greater over
continental regions than oceanic regions and generally increases with decreasing local
solar-elevation angles and increasing cloud coverage. Because of greater snow and ice coverage,
decreasing solar-elevation angle, and increasing cloud coverage, Albedo also tends to increase
with latitude. These variations make selection of the best Albedo value for a particular thermal
analysis uncertain. Another important point is that the Albedo heat flux reaching a spacecraft
will decrease as the spacecraft moves along its orbit and away from the subsolar point, which is
the point on Earth or another planet where the Sun is at the Zenith, even if the Albedo constant
remains the same. This happens because the albedo factor is a reflectivity, not a flux. As the
spacecraft moves away from the subsolar point it is over regions of Earth’s surface where the
local incident solar energy per square meter is decreasing with the cosine of the angle from
the subsolar point. The Albedo heat load on the spacecraft will therefore approach 0 near
the dividing line between the sunlit and dark sides of a planet, so-called terminator, even if
the albedo factor is 1.0. This geometric effect is accounted for by the analysis codes used to
perform spacecraft thermal analysis [9].

Earth-IR All incident sunlight not reflected as Albedo is absorbed by Earth and eventually
re-emitted as IR energy. While this balance is maintained fairly well on a global annual average
basis, the intensity of IR energy emitted at any given time from a particular point on Earth can
vary considerably depending on factors such as the local temperature of Earth’s surface and
the amount of cloud cover. A warmer surface region will emit more radiation than a colder
area. Generally, highest values of Earth-emitted IR will occur in tropical and desert regions, as
these are the regions of the globe receiving the maximum solar heating, and will decrease with
latitude. Increasing cloud cover tends to lower Earth-emitted IR. These localized variations in
Earth-emitted IR, while significant, are much less severe than the variations in albedo. This is
why, usually, contrary to Albedo calculation codes, no correction factor is applied to the
cal-culations of Earth-IR emission. In general an average value of Earth-IR heat load experienced
by satellites in LEO is extimated as ∼ 230W/m2_{. It is useful to remark that for GEO satellites}

it is common practice to neglect both Earth-IR and Albedo heat loads because, since the high distance between Earth and satellite makes view factors with Earth drop, they reach negligible values with respect to direct sunlight.

Unlike short-wavelength solar energy, Earth IR loads incident on a spacecraft cannot be reflec-ted away from radiator surfaces with special coatings since the same coatings would prevent the

10 1 Introduction

radiation of waste heat away from the spacecraft. Because of this, Earth-emitted IR energy can present a particularly heavy backload on spacecraft radiators in low-altitude orbits. The concept of Earth-emitted IR can be confusing, since the spacecraft is usually warmer than the effective Earth temperature, and the net heat transfer is from spacecraft to Earth. However, for analysis, a convenient practice is to ignore Earth when calculating view factors from the spacecraft to space and to assume that Earth does not block the view to space. Then the difference in IR energy is added back in as an "incoming" heat rate called Earth-emitted IR. The spacecraft ex-changes heat with deep space at a temperature nominally of 2.7K. Obviously this heat exchange is considered only as heat output, since the heat input from a "surface" at 2.7K is negligible at normal spacecraft temperatures.

Orbital parameters The orbit in which a spacecraft is placed has a great influence on its thermal response. A brief description of orbital parameters and their influence on the thermal analysis follows, focusing on Earth centred orbits.

Equatorial plane: plane of Earth’s Equator, which is perpendicular to Earth’s spin axis. Ecliptic plane: plane of Earth’s orbit around the Sun.

Sun day angle: position angle of the Sun in ecliptic plane, measured from vernal equinox. At vernal equinox it is 0°, at summer solstice it is 90° and so on.

The position of a satellite can be uniquely defined by six parameters: semimajor axis, eccentri-city, inclination, perigee argument, right ascension of the ascending node. The most important parameter for thermal analysis is given by the Orbit Beta Angle (β), which represents the min-imum angle between the orbit plane and the solar vector. It can vary between −90° and 90° degrees and is obtained by the formula [9]:

β=s−1(s(λe)s(i)s(Ω)− s(λe)c(incl)c(Ω)s(i) +s(λe)s(incl)c(i)) (1.1)

where s is used as notation for indicating the sine function while c is used for the cosine. λeis

the Sun longitude along the ecliptic, i is the orbit inclination, Ω is the satellite right ascension of the ascending node and incl is the inclination angle of the Earth axis with respect to the ecliptic.

Fig. 1.4: Graphic representation of the orbit β angle, useful for analysis of direct sunlight heat load [4].

The β angle is particularly useful in the LEO orbits thermal analysis since it helps understand the eclipse period.

1.2 Spacecraft Thermal Control Systems and heat transfer 11

As viewed from the Sun, an orbit with β equal to 0 deg appears edgewise. A satellite in such an orbit passes over the subsolar point on Earth where albedo loads are the highest, but it also has the longest eclipse time because of shadowing by the full diameter of Earth. As β increases, the satellite passes over areas of Earth further from the subsolar point, thereby reducing albedo loads; however, the satellite is also in the Sun for a larger percentage of each orbit as a result of decreasing eclipse times. At some point, which varies depending on the altitude of the orbit, eclipse time drops to 0. With β equal to 90°, a circular orbit appears as a circle as seen from the sun; no eclipses exist, no matter what the altitude, and Albedo loads are near 0. Note that β angles are often expressed as positive or negative; positive if the satellite appears to be going counter-clockwise around the orbit as seen from the sun, negative if clockwise. For any given satellite, β will vary continuously with time because of the orbit nodal regression (RAAN) and the season change. If the nodal regression of an orbit proceeds eastward at exactly the rate at which the Sun’s right ascension changes over the year, so "following" the Sun, the orbit is called sun-synchronous.

Parameters introduced above influence directly or indirectly the thermal behaviour of the satel-lite. In order to obtain quantitative data regarding heat fluxes and temperatures, it is necessary to properly understand and model the heat transfer phenomena. In the following, a brief de-scription of the main heat transfer phenomena in space and of common thermal modelling tools is provided.

1.2.2 Heat transfer in space

Because of the absence of air, the only heat transfer phenomena that can take place in a space-craft are radiation and conduction. The spacespace-craft receives heat from outer heat radiation sources and rejects heat only via radiation towards space. Other heat sources are represented by internal components Joule or friction dissipations. However, convection must be considered when analysing in detail components involving fluids, i.e. Heat Pipes or Phase Change Materi-als, or mission phases involving atmosphere interaction with the spacecraft.

In most heat-transfer problems, it is undesirable to ascribe a single average temperature to the
system, and thus a local formulation must be used, defining the heat flow-rate density, or simply
heat flux, as ˙q ≡ Q_{A}. According to the corresponding physical transport phenomena, heat flux
can be related to the local temperature gradient or to the temperature difference between the
sys-tem wall Twand the environment, far from the wall, T∞, because, at the wall, local equilibrium

implies T=Tw. In the classical three distinct modes of heat transfer, conduction, convection, and

radiation, the following models are respectively used [5]:

~˙q=−k∇T(x, y, z) (1.2)

˙q=h(T_{w}− T∞) (1.3)

12 1 Introduction

where k represents the material thermal conductivity, h is the heat transfer coefficient among the moving fluid and the bathed surface, σ is the universal Boltzmann constant σ= 5, 67 · 10−8W/(m2K4), and ε, pure number 0 < ε < 1 is the so-called emissivity factor which accounts for the fact that real bodies are not a blackbody.

These three heat-flux models are named as: heat transfer within materials, Fourier’s law, heat transfer at fluid-bathed walls, Newton’s law of cooling, and heat transfer through empty space, Stefan-Boltzmann’s law. An important point to notice is the non-linear temperature-dependence of radiation heat transfer, which forces the use of absolute values for temperature in any equation with radiation effects [6]. Since waste heat is usually dissipated via radiators, heat radiation is an important heat transfer mean to analyse. A deeper description of the main quantities is proposed.

Heat radiation

An explanation of the main quantities involved is given, afterwards focusing on the assumptions related to the case considered in this work.

Irradiance, I [W/m2], is defined as the radiant energy flowing per unit time and unit surface, normal to the propagation direction, if not otherwise stated. Irradiance is also the radiation power, φ, impinging on a unitary surface directly from a source or through intermediate reflec-tions, I=dφ/dA. Irradiance is measured by the effects of the incoming radiation on a detector. For one-directional radiation (like sunlight), irradiance depends on surface inclination in the way I=I0cos β, where I0 is the radiated heat flux if the impinging surface and the source were

directly facing. In general only a fraction of the irradiance is absorbed, the rest is reflected and, in the case of semi-transparent materials, transmitted. It is possible to define two quantities accounting for it, the Exitance and the Emittance. For a certain distributed source, the total power per unit surface exiting from that surface is termed exitance, M [W/m2], and it accounts for three different effects:

M=εσT4+ρI+τIrear (1.5)

where, εσT4accounts for the own emission of the body being ε the emissitivity and T the sur-face temperature, ρI is the part reflected from irradiance falling on it being ρ the reflectivity. The part accounting for transmission from the back, τIrear, being τ the transmissivity and Irear

the incoming radiation from the back of the surface, is absent in opaque objects. The emittance is the power emitted per unit surface area without accounting for other body inputs and is re-ferred with the same letter M used for exitance [6].

To study radiation heat transfer it is necessary to account for the fact that the heat exchange is directly depending on relative position and geometry of the facing objects which exchange heat. This is accounted by the so-called view-factors.

1.2 Spacecraft Thermal Control Systems and heat transfer 13

View factor The view factor F12 is the fraction of energy exiting an isothermal, opaque, and

diffuse surface 1, that impinges on surface 2. View factors depend only on geometry of the surfaces and their relative distance. With reference to Fig.1.5, consider two infinitesimal surface

Fig. 1.5: Geometry for view-factor definition [6].

patches, dA1and dA2, in arbitrary position and orientation, defined by their separation distance

S, and their respective tilting relative to the line of centres, θ1 and θ2, with 0 ≤ θ1 ≤

π

2 and

0 ≤ θ2≤

π

2, so seeing each other. The expression for the differential view factor dF12, where the differential symbol "d" is used to match infinitesimal orders of magnitude since the fraction of the radiation from surface 1 that reaches surface 2 is proportional to dA2, in terms of these

geometrical parameters, is as follows. The radiation power, d2Φ12 intercepted by surface dA2

coming directly from a diffuse surface dA1 is the product of its radiance L1=M1/π, times its

perpendicular area dA1⊥, times the solid angle, dΩ12, subtended by dA2[6]:

d2Φ12=L1dA1⊥dΩ12 =

L1dA1cos(θ1)· dA2cos(θ2)

S2 (1.6)

The differential view factor is then:

dF12=

d2Φ12

M_{1}dA_{1} =

cos(θ1)cos(θ2)

πS2 dA2 (1.7)

where, according to previous definitions, M1represents the exitance, or emittance, of surface 1.

When finite surfaces are involved, computing view factors is just a problem of mathematical integration, in general not a trivial one. The view factor from a patch dA1to a finite surface A2,

is the sum of elementary terms, whereas for a finite source, A1, the total view factor, being a

fraction, is the average of the elementary terms. The view factor between two finite surfaces A1

and A2is given by the following integral, averaged on A1:

F12 =
1
A_{1}
Z
A1
Z
A2
cos(θ1)cos(θ2)
πS2 dA2
dA1 (1.8)

14 1 Introduction

Analytical solutions for the view factor of two finite surfaces facing exist for particular geomet-ric configurations. One of them is the case of a large sphere radiating to a small patch. This is the case that can be considered in order to evaluate, in an approximate way considering Earth as spherical, the view factor between Earth and each face of a cubic satellite, i.e. iBLOCK. This allows to compute the Earth-IR and Albedo heat loads for the implemented model, as shown in Fig.1.7.

Consider the radiation from a large spherical surface of radius R to a small tilted, by an angle β, patch of area dA1, separated by a distance H, as shown in Fig.1.6.

Fig. 1.6: Schematic illustration of the geometrical case of a large sphere radiation to a small patch.

Assuming the sphere as isothermal, i.e. uniform emittance, it is possible to directly compute the power received by the patch dA1. Two cases must be considered, delimited by the semi-angle

subtended by the tangent to the sphere from the patch centre, βt=arcsin

R

R+H

[6]:

1. The plane containing the patch dA1does not cut the sphere, so the patch tilting angle, β,

is small, with β <π

2− βt. In this case, the view factor can be found geometrically by the projections method. It is the area of the projection on the patch-plane of the projection of the radiating sphere on the unit hemisphere centred at dA1, and divided by π:

F12 = Z A2 cos(θ1)cos(θ2) πS2 dA2= cos(β) h2 (1.9) where h=H/R.

2. The plane containing the patch dA1cuts the sphere, so the patch tilting angle, β, is large,

with π

2− βt < β < π

2+βt. In this case, the view factor is [6]:

F_{12}= 1

πh2

cos(β)arccos(y)− x sin(β)
p
1 − y2_{+}1
πarctan
sin(β)p1 − y2
x
!
(1.10)

1.2 Spacecraft Thermal Control Systems and heat transfer 15

For the analysed case of the cubic elementary module iBLOCK, these analytical formulas can be used to obtain the view factors of each face with Earth, assuming that one of the faces is al-ways pointing Earth, i.e. Earth observation satellite. In such case R is the Earth average radius, H is the satellite position with respect to centre of Earth and β is assumed to be 0 for the face pointing Earth and π

2 for the lateral faces. This calculation is implemented in an algorithm use-ful for the thermal mathematical model successively described. As an example, the following results of the variation of view factors during an elliptic orbit are shown in Fig.1.7. Such view

Fig. 1.7: View factors of the various faces of the cube with Earth, for a LEO orbit with e=0,1. Zero value of view factor represents the absence of view.

factors are employed in order to measure the incoming Earth-IR and Albedo heat loads on the satellite outer surface. It is possible to state that, since these loads are directly proportional to view factors, they decrease as the satellite approaches the apogee, reaching a minimum in such point, and reach a maximum at the perigee. For a circular orbit they would be constant. The zero value of view factor, while not delivering any information, represents the absence of view.

Heat radiation is necessary to evaluate the environmental heat loads acting on every space-craft. Usually the analysis of the radiation heat transfer in an enclosure, comprehending the Sun, the Earth and, eventually, the Moon, is performed. Software packages such as ESARAD or TRASYS are employed for such purpose. However, as the environmental heat loads are commonly considered as incoming heat inputs in the thermal balance equation of a spacecraft surface, it is possible to estimate the quantitative values of these inputs via analytical coupled problems solution. In the following a simplified approach to do that without the use of specific software packages is proposed. The algorithm described is then implemented in a MATLAB script used for the TMM developed.

16 1 Introduction

1.2.3 Environmental heat loads calculation

In order to properly evaluate the input heat loads due to the space environment, two coupled orbital mechanics problems need to be solved. One that gives in output the position of the satellite with respect to Earth, another that allows to obtain the sun-satellite mutual visibility and the orientation of each face with respect to Sun radiation. In the following a set of algorithms for the measurement of such data is explained, focusing on specific models which allow to simplify the problem.

Sun radiation evaluation

In order to estimate the Sun and Albedo loads, it is necessary to understand the position of the satellite and the Sun with respect to Earth and each other.

Sun position and β angle The first important parameter to evaluate is, once obtained in input the orbital parameters initial conditions, the variation of the RAAN ( ˙Ω ) over a year. The evaluation of ˙Ω is important because it influences the variation of the β angle during the year. Ω varies in time because of a precession motion due to the non-spherical nature of Earth, which creates a non-uniform gravitational field. The variation is expressed as [10]:

˙
Ω=−3
2J2n
R_{earth}
p
2
cos(i) (1.11)

where J2 is the oblateness perturbation, Rearth is the average Earth radius, p is the semi-latus

rectum, i is the orbit inclination and n is the mean angular speed:

n=n
"
1+3
2J2
R_{earth}
p
2
p
1 − e2
1 −3
2sin(i)
2
#
(1.12)

Sun position during the year The Sun position can be computed with an approximated

algorithm [11]. First input is the Julian starting date in universal time, i.e. in the example JD=2459205 corresponds the Winter solstice of year 2020.

JD_{ut} =2459205 (1.13)

day=JDut− 2451545 (1.14)

Then, all of the Sun position parameters are calculated with the following equations, for the indicated date.

λmsun=280.4606184+0.9856474 · day (1.15)

1.2 Spacecraft Thermal Control Systems and heat transfer 17

λe=λmsun+1.914666471 · sin(Msun) +0.019994643 · sin(2Msun) (1.17)

where λe is the Sun longitude along the ecliptic, λmsun the average sun longitude and Msun the

average anomaly of the sun.

r_{sun}=1AU · 1.000140612 − 0.016708617 · cos(M_{sun})− 0.000139589 cos(2Msun) (1.18)

where AU is the Astronomic Unit, corresponding to the average Sun-Earth distance.

incl=23.439291 − 0.0000004 · day (1.19)

where the variable incl stands for the inclination of the Earth axis with respect to the ecliptic plane.

~rsungeo =rsun

cos(λe)

cos(incl)sin(λe)

sin(incl)sin(λe)

(1.20)

Particular attention must be given to the angle units as the algorithm gives the output angles in degrees. Iterating the calculation it is possible to obtain the sun position in geocentrical equatorial coordinates during a year. The β angle is influenced by both λe(season change) and

Ω variation. Infact, as previously stated, it is given by the formula [9]:

β=s−1(c(λe)s(i)s(Ω)− s(λe)c(incl)c(Ω)s(i) +s(λe)s(incl)c(i)) (1.21)

where c stands for the cosine function and s stands for sine. An example of the variation of β during a year is shown in Fig.1.8. From the obtained data, the minimum value of β and its cor-responding date, and thus Ω, can be used for computing the "cold case" heat loads, since, when β is minimum, the eclipse duration is maximum. Viceversa, β maximum value, and corresping date and Ω, are used for the hot case.

Fig. 1.8: β angle variation during a year, for an orbit with semi-major axis a=8000km and eccent-ricity e=0.1

18 1 Introduction

Geocentric equatorial satellite position over one orbit

The satellite position is calculated solving the Inverse Kepler Problem via Newton method, taking as input the orbital parameters (semimajor axis "a", eccentricity "e", orbit inclination "i", perigeum argument "ω" and right ascension of ascending node previously calculated"Ω") and obtaining the eccentric anomaly, "E", of the satellite at a certain point. This operation is iterated for all eccentric anomaly angles from 0 to 2π. The satellite position in geocentric perifocal coordinates is in this way obtained. Via a rotation matrix calculation, T, based on the orbital parameters, the satellite position can expressed in geocentric equatorial coordinates [10].

ν=2 arctan
r
1+e
1 − etan
E
2
!
(1.22)
~rsatperi =
a(1 − e2)
1+ecos(ν)
cos(ν)
sin(ν)
0
(1.23)
T =
c(Ω)c(ω)− s(Ω)s(ω)c(i) s(Ω)c(ω) +c(Ω)s(ω)c(i) s(ω)s(i)
−c(Ω)s(ω)− s(Ω)c(ω)c(i) −s(Ω)s(ω) +c(Ω)c(ω)c(i) c(ω)c(i)
s(Ω)s(i) −c(Ω)s(i) c(i)
(1.24)
~rsatgeo =T
0_{·~r}
satperi (1.25)
Mutual visibility

The mutual visibility among Sun and satellite can be a non-trivial mathematical problem, de-pending on the satellite orbit altitude. The main difference resides in the shadow model to be adopted when analysing the eclipse period of the satellite. Two models exist: "cylindrical shadow model" and "conical shadow model". The second one is also the most accurate, since it takes into account the actual conical shape of the projected shadow and the penumbra. It is used mainly to study eclipse periods for orbits which have a not negligible penumbra periods, i.e. GEO and some MEO orbits. On the other hand, the cylindrical shadow model is less accurate, but easier to handle and analytically simple. It doesn’t take into account the presence of pen-umbra and the projected shadow shape is cylindrical. This model is commonly used to analyse the eclipse conditions in LEO orbits, since for such orbits the penumbra period is typically very short with respect to the complete umbra one [4].

In the following, a description of the mutual visibility problem assuming a cylindrical shadow model is proposed. Such simplifying assumption is justified by the fact that the primary goal of the mission considered in this work is related to the insertion of the iBOSS satellite in a LEO for an In-Orbit Demonstration (IOD). In this case, the mutual visibility among Sun and satellite is easily estimated once the Sun-satellite angle is known. Infact, if the Sun-satellite angle φ, as

1.2 Spacecraft Thermal Control Systems and heat transfer 19

defined in Fig.1.9, is such that cos(φ)and the distance of the normal vector to the Sun-satellite vector, from centre of Earth are lower than 0, then the satellite is in eclipse condition [12].

Fig. 1.9: Sun-satellite angle,φ, orbit β angle and angle from orbit noon, θ. [12].

The measure of the Sun-satellite angle and the consequent eclipse condition can be computed by the following formulas:

cos(φ) =

~r_{sat}_{geo}·~r_{sun}_{geo}

k~r_{sat}_{geo}k · k~r_{sun}_{geo}k (1.26)

dist =k~rsatgeok ·

q

1 − cos(φ2)− Rearth (1.27)

where the various position vectors shown have been explained in the previous paragraphs, φ is the Sun-satellite angle and the value dist represents the distance between the vector joining the satellite and the Sun and the Earth surface. The use of the previous equations allows, at each time step, j, to know if the satellite is sunlit or in eclipse, since for this last one to happen it must be both cos(φ(j))< 0 and dist(j)< 0.

When the satellite is in eclipse, it obviously does not experience any solar radiation. When it is sunlit, the various faces experience a different exposition to Solar radiation due to faces inclination, which depends on the Attitude Determination and Control System (ADCS) and the stabilization method employed. Most satellites, as the considered case of iBOSS, use three-axis stabilized systems because of their cubic structure. With the reasonable simplifying assumption that the face pointing Earth is always the same, i.e. antennas or observation instruments located on such face, it is possible to estimate the inclination angles with respect to Solar radiation of each face. Fig. 1.10 can help understand. With reference to Fig.1.10 and Fig.1.9, the Nadir pointing face is subjected to a heat load which varies with − cos(β)cos(θ), where θ is the orbit angle at noon. The opposite happens for the Zenith pointing face. An example is shown in Fig. 1.11. The faces pointing North and South will experience constant Solar heat flux, when sunlit, since their face inclination is always cos(β). Obviously, if β < 0, the only sunlit face is the North pointing one, while the opposite happens if β > 0, as shown in Fig.1.12.

20 1 Introduction

Fig. 1.10: Different orientation for faces: 1-Nadir pointing, 2-Direction opposite to velocity vector, 3-South pointing, 4-direction of the velocity vector, 5-North pointing, 6- Zenith pointing. Earth-IR and Albedo

Earth-IR and Albedo loads are given by:

Q_{Albedo}=AIαvisaKFEarth−satcos(φ) (1.28)

Q_{IR}=AI_{IR}αIRFEarth−sat (1.29)

where A is the exposed area, I is the Solar irradiance [W/m2], a is the albedo factor, K is the correction term for albedo, FEarth−sat is the view factor of the considered face with the

Earth surface, φ is the Solar Zenith Angle and IIR is the Earth infra-red emission considered

as constant. This last value is not exactly constant. When passing over colder areas (North or South-poles) it is clearly reduced, while the opposite happens when the satellite is facing hotter areas of Earth surface as a desert. But, in a first approximation, it can be considered as constant, since the variation of this value is very rapid and the main elements of a spacecraft are not particularly affected by this variation, as previously described. However, according to Gilmore [4], it is necessary to variate this value as the inclination of the orbit is varying, as shown in Tab.1.2.

Albedo correction factor The albedo correction factor, K, can be evaluated by an approxim-ated polynomial expression. The correction term derived from four months of data restricted to the -30 to +30 latitude band, was verified by testing another four months of data to wider latitude bands. This removes the Solar Zenith Angle dependence to within ± 0.04%. The correction is [9]:

Albedo(φ) =Albedo(φ=0) +C4·(φ)4+C3·(φ)3+C2·(φ)2+C1·(φ)

1.2 Spacecraft Thermal Control Systems and heat transfer 21

inclination Hot case Cold case

0 < i <π
6 a=0.21, IR=264W/m
2 _{a=0,16, IR=232 W/m}2
π
6 < i <
π
3 a=0,24, IR=248W/m
2 _{a=0,19, IR=221W/m}2
π
3 < i <
π
2 a=0,24, IR=233W/m
2 _{a=0,2, IR=224W/m}2

Table 1.2: Variation of average Earth-IR and average Albedo factor a depending on inclination for a time of 6h [4].

where φ is the Solar Zenith Angle in degrees and the Albedo is expressed as a fraction and C4= +4.9115 · 10−9, C3= +6.0372 · 10−8, C2=−2.1793 · 10−5, C1= +1.3798 · 10−3. The

results of the Sun, Earth-IR and Albedo heat loads evaluation algorithms are shown as example in Figg.1.11 and1.12 for a cubic satellite placed in an orbit with semimajor axis a=8000km and eccentricity e=0.1. Notice that the scale of the Figures is kept constant even for low values of the heat loads, as in Fig.1.12, in order to show the order of magnitude difference.

Once the understanding of the proper enviromental heat loads is performed, a study of the influence of such external and the internal additional heat loads on the spacecraft components temperature must be performed. In order to do that thermal models are developed with several methods. In the following a description of the process for the realisation of thermal models is provided. Further, after a brief literature review of the various methods, a trade-off study for the choice of the proper one for the case studied performed, is proposed.

22 1 Introduction

(a) Zenith heat load

(b) Nadir heat load

Fig. 1.11: Environmental Heat loads for Nadir and Zenith pointing faces over one orbit of a satellite in LEO, for hot case (β=46.7°).

1.2 Spacecraft Thermal Control Systems and heat transfer 23

(a) Face pointing South

(b) Face pointing North

Fig. 1.12: Heat loads for the two North and South pointing faces, at same β angle, over one orbit of a satellite in LEO.

24 1 Introduction

### 1.3 Spacecraft thermal modelling

Heat-transfer problems of a spacecraft geometry are generally too complicated for analytical study, and one has to resort to numerical simulation, with space and time discretisation along the following steps:

1. Spacecraft geometry definition.

2. Geometry is discretised, dividing the system into small pieces or lumps which, in the Finite Difference Method (FDM) are considered isothermal and represented by just one material point, the node, and in the Finite Element Method (FEM) are considered having a linear temperature field and represented by a few corner nodes.

3. The energy balance equation for each node is established, with the thermal capacity, heat dissipation and background loads ascribed to the node, and with the appropriate heat transfer couplings with the other nodes.

4. Time discretization provides a step-by-step updating temperature matrix, in terms of some initial conditions, which can be difficult to know, and the boundary conditions applied; a case study, including trajectory and operations, must be specified. Boundary conditions are changing all the time, so, only representative situations are studied. The most import-ant conditions are the worst hot case, which represents a case of maximum power and heat fluxes at End Of Life (EOL), and worst cold case, with minimum power and heat fluxes at Beginning Of Life (BOL), must be studied.

5. Assignment of particular power-dissipation profiles to each node, although these could be depending on eclipse timing and unknown operations.

6. Ascribe thermal-connection properties to node pairs: conductance factors to adjacent nodes, radiation factors to field-of-view nodes, and convection coefficients to internal fluid media, if any. This task is independent of spacecraft trajectory for fixed-geometry spacecraft, but it is coupled to orbit and attitude motion when there are some deployed or pointing elements with relative motion to the spacecraft body.

1.3 Spacecraft thermal modelling 25

1.3.1 Energy and thermal balance

For the purpose of thermal control, it is often assumed that the mass of the system is invariant,
thus propellant flow rates are not considered in thermal studies so that the energy balance is that
of a closed system, dE_{dt} =W˙ +Q. Internal energy variation in time is basically due to temperat-˙
ure change,dE_{dt} =m· c ·dT_{dt}, even though other types of energy may be important, e.g. electrical
energy in batteries and condensers, thermal energy in PCMs, or other physical-chemical or
nuc-lear energies. The open-system energy balance, regarding TCS sizing, is commonly necessary
only for re-entry analysis, to study thermal protection systems based on ablation, where mass is
lost.

As most spacecrafts incorporate solar cells, it is worth considering the following energy balance applicable to the whole spacecraft, with thermal, electrical, and electromagnetic energy terms. This equation could result unfamiliar with respect to the well known thermal balance equation for spacecrafts TCS, however, it is necessary to consider the global equation and then justify how and under which conditions, the de-coupled thermal balance is obtained:

dE

dt =

dE_{th}

dt +

dE_{el}

dt =W˙EM,net+W˙el,net+Q˙cond,net+Q˙rad,net+Q˙conv,net (1.31)

where only two types of internal energy are considered: thermal, Eth, and electrical, Eel and

only two types of power are considered: W˙EM,net, accounting for electromagnetic radiation

such as solar radiation, laser, or microwave radio-link, but not infra-red, which is accounted
as a thermal power, and ˙W_{el,net}, accounting for electrical currents. Finally, the three classical
thermal power types are considered: conduction through solids ˙Q_{cond,net}, convection ˙Q_{conv,net},
and radiation ˙Q_{rad,net}. Notice that solar radiation is included in the work term.The intent is to
restrict the analysis of radiation heat transfer to thermal sources in the far infra-red band of the
EM-spectrum, leaving thermal sources in the visible and near-IR, such as the Sun radiation, as
work exchanges independent on the system temperature.

Electrical energy depends on the State Of Charge (SOC) of the batteries. Several approximated measurement methods exist. If the battery efficiency is known, the electrical energy variation in time can be approximated asdEel

dt ∼ V

dQ

dt where V is the nominal voltage and Q is the capacity of the battery.

The total energy balance, in equation1.31, may be split into an electrical and a thermal balance, although some power source and sink term must be introduced in the single equations in order to take into account that only total energy is conservative, while thermal and electrical alone are not. The obtained result is thus:

VdQ

dt = (W˙EM,in− ˙WEM,out− ˙WEM,diss) + (W˙el,in− ˙Wel,out− ˙Wel,diss) CdT

dt =W˙EM,diss+W˙el,diss+Q˙cond+Q˙conv+Q˙rad

26 1 Introduction

where ˙WEM,dissis the dissipated electromagnetic radiation, and ˙Wel,dissis the dissipated electrical

power, both contributing to heating, and which are traditionally quoted as dissipated "heat", ˙

Q_{EM,diss} and ˙Q_{el,diss}, but recall that what enters to a resistor is electrical work, not heat. In
equation1.32, C represents the overall thermal capacity, which, if no phase change is involved,
can be expressed as the sum of all of the involved components mass, times their specific heat:
C=Σmc. The terms in the thermal balance equation are usually one order of magnitude bigger
than electrical terms. This is why the energy balance is generally reduced for TCS analysis, in
a first approximation, to equation1.32.

Substituting the proper environmental heat loads previously described in the terms of the thermal balance equation it is possible to obtain a one node model of the spacecraft temperature vari-ation study. The balance is generically expressed by an equvari-ation of the form as:

mcdT

dt =Q˙sun+Q˙Earth−IR+Q˙albedo+Q˙int− ˙Q∞=

αvisIsunAf rontal+αIRεEarthAFEarth−satσT4+αvisAFEarth−satKacos(φ) +Q˙int− εIRσT4 (1.33)

where ˙Q_{sun}is the incoming solar radiation, ˙Q_{Earth−IR}is the infra-red Earth emission and ˙Q_{albedo}
is the Albedo reflected solar radiation. Their evaluation procedure is explained in previous
sec-tions, while here a generic expression is given. The term ˙Qint accounts for internally dissipated

power, i.e. the electrical power dissipated by Joule effect. The term ˙Q∞is the output radiation

related to the thermo-optical properties of the covering materials of the spacecraft and their temperature.

Equation1.33 is useful to have a first understanding of the quantities involved in the thermal balance and can help in a preliminary sizing of thermo-optical properties of the surfaces. How-ever, in order to obtain a proper understanding of the internal components temperatures under operative conditions, it is necessary to develop a multi-nodal model. For simplicity a two-nodes model is reported: m1c1 dT1

dt =Q˙sun1+Q˙Earth−IR1+Q˙albedo1+Q˙int1+Q˙cond2,1+Q˙rad2,1− ˙Q∞1

m2c2

dT2

dt =Q˙sun2+Q˙Earth−IR2+Q˙albedo2+Q˙int2− ˙Qcond2,1− ˙Qrad2,1− ˙Q∞2

(1.34)

where the conductive and radiative heat exchange terms depend on temperatures, geometry and material properties, and are usually formulated as:

˙

Q_{cond}_{2,1} =G2,1(T2− T1)

˙

Q_{rad}_{2,1} =σR2,1(T24− T14) (1.35)

with G2,1thermal conductance and R2,1radiative coupling. The last one may depend in general

on the temperatures of the considered bodies and of the surrounding and must be computed by the exitance method, or by the Monte Carlo ray tracing method.

Usually, models with thousands of nodes are required to obtain a reliable understanding of the temperature and heat fluxes variation of the components.

1.3 Spacecraft thermal modelling 27

Multi-node models When analysing multi-nodal models it is necessary to perform a
discret-isation in order to be able to deal numerically with the problem. Usually a nodal equation is set
for the i-th node, where i=1, ..., n and n is the total number of nodes representing the spacecraft
geometry.
C_{i}T
+
i − Ti
∆t =
˙

Q_{sun}_{i}+Q˙_{Earth−IR}_{i}+Q˙_{albedo}_{i}+Q˙_{int}_{i}+

n

## ∑

j=16=i ˙ Q_{cond}

_{j,i}+ n

## ∑

j=16=i ˙ Q_{rad}

_{j,i}− ˙Q

_{∞}

_{i}(1.36)

where Ci is the thermal capacity of the node i, T_{i}+ and Ti are the node temperature after and

before advancing a ∆t in time. The node conductive coupling can be identified as:

n

## ∑

j=16=i ˙ Q_{cond}

_{j,i}= n

## ∑

j=16=i k_{i, j}

_{e f f}A

_{i, j}

_{e f f}Li, je f f (T

_{j}− Ti) (1.37)

and the radiative coupling is:

n

## ∑

j=16=i ˙ Q_{rad}

_{j,i}= n

## ∑

j=16=i σRi, j(Tj4− Ti4) (1.38)where ki, je f f is the effective conductivity of the materials implied, Ai, je f f is an effective

heat-flow area between the two nodes, Li, je f f is the effective distance between the two nodes and Ri, j

coincides with the view factor times area in the case of all node surfaces being blackbodies, but it must be obtained by solving the radiosities from the network model.

In order to solve the equation 1.36, several softwares have been developed to aid the thermal analysis numerical implementation.

1.3.2 Numerical methods

For the numerical implementation of equation 1.36 two methods can commonly be adopted: Finite Difference Method (FDM) and Finite Element Method (FEM).

• FDM is a method based on the direct approximation of Partial Differential Equations (PDEs). Such approximation is obtained substituting finite differences in the problem domain to the continuous partial derivatives. Such method is commonly employed for the solution of the heat equation:

δT

δt =α

δ2T

δx2 (1.39)

The FDM is used to discretize in time and space the equation. The discretization is performed expanding the derivatives via Taylor expansion, obtaining, with reference to equation1.39, a linear expression for temperature at a certain time step. Several methods exist in order to approach to the solution and translate the PDE problem into and ODE one. The main methods applied for the solution of the heat equation usually are:

28 1 Introduction

– Backwards Differences Method (Euler backward); – Crank Nicholson Method.

The difference among the various methods mainly relies in the conditions for the conver-gence of the method. Crank-Nicholson method is an unconditionately stable method, the Euler backward is also stable, even though for this method is advisable that the time steps are much smaller than the space intervals. For the Euler forward, instead exist several necessary conditions for convergence. An example of this last one method is shown in the following, however, not always it is chosen as solving method as further explained.

T_{i}τ+1− Tτ
i
∆t =α
Tτ+1
i+1 − 2Tiτ+Ti−1τ
∆x2 (1.40)

where τ is the n-th timestep and ranges from 0 to tend/∆t and i is the i-th node considered

in the lumped parameters model. In this equation Tτ

i is known from the previous timestep,

while Tτ+1

i is the equation unknown. The Courant-Friedrich-Lewy necessary condition

for convergence [13] of the method requires:

C=α ∆t

∆x2 ≤ 0.5 (1.41)

If the condition in equation1.41is satisfied for each i-th node, the FDM converges and it is possible to build a temperature matrix for all the nodes involved.

If the system is a nonstiff ODE system, the choice usually resides in an explicit method. Explicit methods require less computational effort than implicit ones, if other simulation characteristics are fixed. To find a solution for each time step, an explicit solver uses a formula based on the local gradient of the ODE system. If the system is stiff, the use of an implicit solver is recommended. Though an explicit solver may require less computational effort, for stiff problems an implicit solver is more accurate and often essential to obtain a solution as it is unconditionally stable. Implicit solvers require per-step iterations within the simulated time per-steps. With some implicit solvers, you can limit or fix these iterations. An implicit solver starts with the solution at the current step and iteratively solves for the solution at the next time step with an algebraic solver. An implicit algorithm does more work per simulation step, but can take fewer, larger steps. The underlying approach of an implicit method, i.e. the shown Euler Backward Method, still with reference to the heat equation is to evaluate difference approximations to derivatives at the next time step tτ+1and not the current time step tτwe are solving for:

T_{i}τ+1− Tτ
i
∆t =α
Tτ+1
i+1 − 2T
τ+1
i +T
τ+1
i−1
∆x2 (1.42)

Descriptions of implicit and explicit methods can be easily found in literature and they are employed by common built-in solvers of mathematical tools, such as MATLAB®and its ODE solvers environment [14]. This work in fact employs the ode23t implicit solver.

1.3 Spacecraft thermal modelling 29

• FEM: Contrarily with respect to FDM, which divides the domain in a reticle of points (nodes), the FEM splits the domain into the union of several sub-domains with element-ary shape. In such case the PDEs are not altered, while the domain is discretized. In a continuum problem, the field variable, i.e. pressure, deformation, temperature, is a function of each point of the domain. Thus, the problem has an infinite number of un-knowns. The finite elements discretization procedure reduces the number of unknowns to a finite number by sub-dividing the domain and expressing the unknown field in terms of approximating functions, defined inside the domain of each element. The approximat-ing functions, i.e. polynomial or exponential functions, also called shape functions, are defined by the values assumed by the dependent variable in certain points (nodes). The nodes are usually placed on the border of the elements, common to two or more elements. The values assumed by the field variable in the nodes define unequivocally the behaviour inside the element. In the FEM representation the nodal values of the field variable rep-resent the new unknowns. The approximation accuracy depends both on the order of the used polynomial and on the size of the dividing interval, i.e. with a linear polynomial the error is reduced as the nodes get closer. It is clear that in presence of high gradients of the field function, it is necessary to thicken the nodes only in the interested area and not in the whole domain. This flexibility is one of the advantages of FEM with respect to FDM. FEMs have the advantage of allowing the use of the same program for thermal analysis and for structural analysis. However, heat transfer, particularly in radiation, is highly non-linear and thermal analysts tend to use as few thermal elements (nodes) as possible to reduce computing effort and cost. Such limitations do not apply to structural FEMs, and thus thermal and struc-tural models are often not compatible for joint analysis.

Spacecraft thermal modelling often employs FDMs since such methods allow to properly evalu-ate the radiative heat exchange. Monte Carlo ray-tracing is used to determine the parameters for the radiative heat transfer between different parts of the spacecraft and between the spacecraft and the environment. These parameters are then added to a mathematical model representing the conductive heat transfer and iterative finite difference solvers are used to calculate temperatures within the spacecraft. The most common software adopting such methods are: ESATAN and SINDA. Such softwares include dedicated packages for the evaluation of radiation exchange

(ESARAD and TRASYS) [6].

This work has been realized via the implementation of a code in MATLAB®for the analysis of environmental heat loads and the definition of initial and border conditions. The SIMULINK® Simscape thermal toolbox is then used to discretize the domain and solve the heat equation for each node. This toolbox employs the MATLAB® ODE solver ode23t [14]. This solver is a variable step continuous solver and it employs an implicit method. This is preferrable to an explicit one in the case of a stiff ODE system, as implicit methods require fewer time steps. Actually, the main reason for this choice is due to the impossibility to access one of the two soft-ware packages earlier cited since the licenses were not available during the developing of the work. The main issues in using this package arise because of the difficulty in implementing the

30 1 Introduction

radiative coupling among the internal components and between the spacecraft and the environ-ment. In order to face these issues several simplifying assumptions, explained in the following chapter, have been performed. The option of using a FEM solver has also been considered. However, this model has the main goal of understanding the capability of the iBOSS system to transfer heat among several modules. The modelling of the coupled thermal interfaces requires assumptions which are easy to make only when employing a lumped parameter network. The number of coupled modules can vary, therefore a method employing a fixed geometry (CAD) is also inconvenient and inaccurate. Moreover the internal geometry of the iBLOCKs is not fixed and is customable depending on the payload requirements. These factors directly excluded the use of a FEM to perform the iBOSS thermal modelling, leaving no option but the use of the cited system.