3
Contents
1. Introduction and state-of-the-art ... 13
1.1 The importance of knowing temperature in space ... 13
1.2 Requirements and constraints of the project ... 17
1.3 Overview on spacecraft architectures ... 19
1.4 Structure and objectives of the project ... 22
2. Heat transfer mechanisms ... 24
2.1 Conduction ... 24
2.2 Convection ... 27
2.3 Radiation ... 29
2.3.1 Definitions and laws ... 29
2.3.2 Grey-body factors and view factors ... 32
3. External heat loads ... 36
3.1 The space environment in Earth orbits... 36
3.2 Sun heat loads ... 40
3.2.1 Cylindrical eclipse model ... 43
3.2.2 Conical eclipse model ... 45
3.2.3. Comparison between the models ... 48
3.3 Earth heat loads ... 50
3.4 Albedo heat loads ... 53
3.4.1 Sinusoidal albedo model ... 55
3.4.2 Refined albedo model ... 59
3.4.3 Comparison between the models ... 62
3.5 Thermal balance of a satellite ... 66
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3.5.1 Equilibrium temperature ... 67
3.5.2 Effect of thermal inertia ... 69
4. Attitude laws ... 72
4.1 Attitude determination ... 72
4.2 Kinematic model ... 75
4.3 Target selection and modelling ... 77
4.3.1 Fixed point in space ... 78
4.3.2 Satellite flying along another Earth orbit ... 83
4.3.3 Fixed point on Earth surface ... 85
4.3.4 Earth centre ... 88
5. Geometry and materials ... 90
5.1 From geometry to GMM ... 90
5.2 Thermo-optical model for materials ... 96
6. The thermal network ... 100
6.1 Pre-processing ... 100
6.2 The nodal structure ... 101
6.3 The nodal equations... 106
6.3.1 Calculus of conductive conductances ... 109
6.3.2 Calculus of radiative conductances ... 114
6.4 Post-processing and static tests ... 115
6.4.1 Effect of optical properties ... 115
6.4.2 Effect of thermal conductivity ... 116
6.4.3 Effect of a compartment at constant T ... 119
6.4.4 Effect of a compartment generating a constant Q ... 120
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6.4.5 Effect of facets at constant T ... 121
6.4.6 Mesh size effects ... 123
7. Integration and solution ... 128
7.1 Orbits and attitude integration ... 128
7.2 Advancement of solution in time ... 134
7.2.1 Forward-explicit method ... 136
7.2.2 Backward-implicit method ... 137
7.2.3 Comparison between the methods ... 138
7.3 Quantitative validation ... 140
8. Conclusions and future work... 146
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List of Figures
Figure 1.1 - Spacecraft model reduced at subsystem/component level ... 16
Figure 1.2 - Mesh model detail ... 18
Figure 1.3 - SpaceX's Dragon V2 Capsule ... 20
Figure 1.4 - Voyager 1 and its Golden Record ... 21
Figure 1.5 - Main components of SBIRS GEO satellite ... 22
Figure 1.6 - Work flow-chart ... 23
Figure 2.1 - Heat conduction through contacting points and relative temperature leap at the interface ... 26
Figure 2.2 - Schematics of natural (left) and forced (right) convection ... 27
Figure 2.3 - Thermal radiation in the electromagnetic spectrum ... 30
Figure 2.4 - Comparison of hemispherical monochromatic emission for a black, a grey (ε = 0.6), and a real surface ... 31
Figure 2.5 - Geometry for view-factor computation between two finite surfaces. θ1 and θ2 are the line-of-sight inclination angles, while r12 is the distance between the centroids of the surfaces ... 34
Figure 2.6 - Lambert's law and the dependence of hemispherical emittance on direction . 35 Figure 3.1 - Topics of Chapters 3 and 4 (yellow box) ... 36
Figure 3.2 - Schematics of the space environment for an Earth-orbiting satellite, with values of heat fluxes for the three main sources ... 37
Figure 3.3 - The six classical orbital elements in GRF ... 38
Figure 3.4 - Structure of external_power_inputs.m ... 39
Figure 3.5 - Comparison between measured solar spectral emittance and an equivalent one for a blackbody at T = 5800 K... 41
Figure 3.6 - a) Sun as a point-source in cylindrical eclipse model; b) Sun as a finite-size sphere in conical eclipse model ... 42
Figure 3.7 - Cylindrical eclipse model (despite the image, the solar disk is a point-source) ... 43
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Figure 3.8 - Geometry illustrating the role of β ... 44
Figure 3.9 – Square-wave shape of fday-night, cylindrical eclipse ... 45
Figure 3.10 - Conical eclipse model ... 46
Figure 3.11 - Geometry of the penumbra (vertex P) and umbra (vertex U) cones ... 46
Figure 3.12 - Solar disk occluded by Earth and graphical definition of A and aSun ... 48
Figure 3.13 - Sketch of the orbit with orbital elements listed above ... 49
Figure 3.14 - Solar power input on a MEO satellite, cylindrical eclipse model ... 49
Figure 3.15 - Solar power input on a MEO satellite, conical eclipse model ... 50
Figure 3.16 - Comparison between measured Earth’s spectral emittance and possible approximating blackbody curves ... 51
Figure 3.17 – Earth power input on a MEO satellite ... 53
Figure 3.18 - Definition of θs angle ... 56
Figure 3.19 – Sinusoidal shape of falb ... 57
Figure 3.20 - Sketch of the orbit with orbital elements listed above ... 58
Figure 3.21 - Albedo power input on a MEO satellite, sinusoidal albedo model ... 58
Figure 3.22 - Satellite's horizon ... 59
Figure 3.23 - Geometry for refined albedo model ... 60
Figure 3.24 - Albedo heat flux variation with altitude ... 62
Figure 3.25 - Concentric noon-midnight orbits ... 63
Figure 3.26 – Albedo models comparison, noon-midnight orbit at h = 600 km ... 64
Figure 3.27 – Albedo models comparison, noon-midnight orbit at h = 3000 km ... 64
Figure 3.28 – Albedo power input comparison for orbit in Figure 3.20 ... 65
Figure 3.29 - Albedo power input comparison for a dawn-dusk orbit, h = 600 km ... 65
Figure 3.30 - Dawn-dusk and noon-midnight orbits, h = 240 km ... 68
Figure 3.31 - Noon-midnight orbit comparison ... 69
Figure 3.32 - Dawn-dusk orbit comparison ... 69
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Figure 3.33 - Sketch of the orbit with orbital parameter as above ... 70
Figure 3.34 - Temperature profiles for different τ, solution at regime ... 71
Figure 4.1 – Yaw-pitch-roll sequence transforming BF into SFF ... 73
Figure 4.2 - Structure of spacecraft_attitude.m ... 74
Figure 4.3 - Flat plate attitude at initial time ... 75
Figure 4.4 - Minimum-angle rotation ... 77
Figure 4.5 - Fixed point in space ... 78
Figure 4.6 - Elliptic Earth orbit in HEF... 79
Figure 4.7 - Sketch of Earth’s orbit and target position ... 80
Figure 4.8 - Yaw-pitch-roll angles over one year ... 81
Figure 4.9 – Components of ω over one year ... 82
Figure 4.10 - Components of ω for the first 5 orbits ... 82
Figure 4.11 - ω0 over one year ... 83
Figure 4.12 - Orbits of target and follower ... 84
Figure 4.13 - Yaw angle along one orbit ... 84
Figure 4.14 - ωz along one orbit ... 85
Figure 4.15 - GEO orbit and target at time t0 ... 87
Figure 4.16 - Plot of yaw angle and ωz ... 88
Figure 4.17 - Comparison between OF and BF ... 89
Figure 5.1 - Topic of the chapter (yellow box) ... 90
Figure 5.2 - Mesher's main menu ... 91
Figure 5.3 - Internal view of the Reference Satellite ... 91
Figure 5.4 - Overall dimensions (mm) of the Reference Satellite ... 92
Figure 5.5 - Reference Satellite mesh example ... 93
Figure 5.6 - Corner and edge errors due to material modelling ... 96
Figure 5.7 - Thermo-optical model of materials ... 97
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Figure 5.8 - Excerpt and explanation of materials.txt ... 98
Figure 5.9 - Schematics of file management ... 99
Figure 6.1 - Topic of Chapters 6 and 7 (yellow box) ... 100
Figure 6.2 - Representation of a generic node (subscript x) in the generalized thermal network ... 102
Figure 6.3 - Position of nodes for Reference Satellite ... 104
Figure 6.4- Comparison between FEM and FDM meshes ... 106
Figure 6.5 - Schematics of a) conductive, b) convective and c) radiative conductances . 107 Figure 6.6 - Example of conduction through a wall made of a three-layer material ... 110
Figure 6.7 - Conduction between a compartment-node and its walls ... 111
Figure 6.8 - Conduction between two adjacent compartment-nodes ... 112
Figure 6.9 - Geometry for parallel conduction among adjacent facets ... 113
Figure 6.10 - Effect of optical properties ... 116
Figure 6.11 - Effect of thermal conductivity - aluminum alloy ... 117
Figure 6.12 - Effect of thermal conductivity - nickel alloy ... 118
Figure 6.13 - Effect of a compartment at constant T ... 119
Figure 6.14 - Battery mesh scheme, separated from the Reference Satellite ... 120
Figure 6.15 - Comparison of temperature distributions on the battery... 120
Figure 6.16 - Effect of facets at constant T ... 122
Figure 6.17 - Comparison with and without constrained facets - conductive material ... 122
Figure 6.18 - Comparison with and without constrained facets - insulating material ... 123
Figure 6.19 - Reference Satellite configuration to evaluate mesh size effects ... 124
Figure 6.20 - Global mesh reduction - 3592 nodes ... 124
Figure 6.21 - Strong local mesh reduction – 3501 nodes ... 125
Figure 6.22 - Panel-body interface, separated from the Reference Satellite ... 126
Figure 6.23 - Example of "good" mesh for the Reference Satellite - 5951 nodes ... 127
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Figure 7.1 - Orbit and attitude at t0... 129
Figure 7.2 - Instants at which the equilibrium solution is evaluated ... 130
Figure 7.3 – Temperature field at t1 and t9 – side to Sun... 131
Figure 7.4 - Temperature field at t1 and t9 – side to Earth ... 131
Figure 7.5 - Temperature field at t2 and t8 – side to Sun ... 132
Figure 7.6 - Temperature field at t2 and t8 – side to Earth ... 132
Figure 7.7 - Temperature field at t3 and t7 – side to Sun ... 133
Figure 7.8 - Temperature field at t3 and t7 – side in shadow ... 133
Figure 7.9 - Temperature field at t4, t5 and t6 ... 134
Figure 7.10 - Explicit and implicit derivatives ... 135
Figure 7.11 - Flat plate data ... 139
Figure 7.12 - Comparison between explicit and implicit methods ... 139
Figure 7.13 - a) Cube mesh; b) orbit GEO used in quantitative validation ... 141
Figure 7.14 - SYSTEMA vs ThermoCAS ... 142
Figure 7.15 - Nodal temperatures - ThermoCAS model ... 143
Figure 7.16 – Temperatures of central nodes – Cube ... 143
Figure 7.17 - Cube vs one-node satellite ... 145
Figure 8.1 - VIRAF software architecture ... 146
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List of Tables
Table 1.1 - Operating temperature requirements for common spacecraft hardware ... 14
Table 1.2 - Common TCS components ... 15
Table 2.1 - Values of k at various T for common materials of space application ... 25
Table 2.2 - Mean values of cp at 298 K for common materials in space application ... 26
Table 2.3 - Values of hav for common fluids ... 28
Table 2.4 - Optical properties of common spacecraft materials ... 32
Table 3.1 - Common values of albedo coefficients for various surfaces ... 54
Table 3.2 – Equilibrium temperatures for a satellite in LEO ... 67
Table 3.3 – Values of Tsat,0 for the selected orbits ... 68
Table 6.1 - Thermal-electrical system analogy ... 101
Table 6.2 - Example of nodal structure ... 105
Table 7.1 - Features of SYSTEMA model ... 140
Table 7.2 - Features of ThermoCAS model ... 141
