by navin chari - university of toronto t-space · navin chari master of applied science graduate...

SPACECRAFT THERMAL DESIGN OPTIMIZATION

by

Navin Chari

A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science

Graduate Department of Aerospace Science and EngineeringUniversity of Toronto

Copyright c©MMIX by Navin Chari

Abstract

Spacecraft Thermal Design Optimization

Navin Chari

Master of Applied Science

Graduate Department of Aerospace Science and Engineering

University of Toronto

MMIX

Spacecraft thermal design is an inverse problem that requires one to determine the choice of

surface properties that yield a desired temperature distribution within a satellite. The current

techniques for spacecraft thermal design are very much in the frame of trial and error. The goal

of this work is to move away from that procedure, and have the thermal design solely dependent

on heat transfer parameters. It will be shown that the only relevant parameters to attain this are

ones which pertain to radiation. In particular, these parameters are absorptivity and emissivity.

We intend to utilize an optimal/analytical approach, and obtain a solution via optimization. As

mentioned in the motivation, having a purely passive thermal system will greatly reduce costs,

and our optimization solution will enable that. This topic involves heat transfer (conduction and

radiation), spacecraft thermal network models, numerical optimization, and materials selection.

ii

Acknowledgements

First and foremost, I would like to express my appreciation to my thesis adviser, Dr. Chris J.

Damaren. His assistance and guidance has been pivotal in my research as well as my personal

growth. I would also like to express gratitude toward my family and friends for their support.

Finally, I would like to thank my colleagues in the Spacecraft Dynamics and Control Labo-

ratory, the members of my Research Assessment Committees, and the University of Toronto

Institute for Aerospace Studies community.

iii

Contents

1 Introduction 1

1.1 Current Thermal Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Nonlinear Thermal Network Models . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 Thermal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.5 Thermal Control Coatings (TCCs) . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Spacecraft Thermal Analysis 4

2.1 Initial Spacecraft Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Temperature Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Fourier’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Conduction Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.5 Radiation Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.6 Black Body View Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.7 Gebhart Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.8 Calculating Gebhart Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.9 Radiation Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.10 External Heat Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.11 Balance Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.12 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

iv

2.13 Forward Problem Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 The Inverse Problem 24

3.1 Solving the Inverse SQP Problem . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2 Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2.1 Next Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Gradient-Free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.1 Nelder-Mead Simplex . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.2 Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Spacecraft Model Modification 40

4.1 Shape Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2 Tray Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4 Orbital Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.5 External Flux Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.6 Multi-Scenario Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.6.1 Proof of Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.7 Surface Property Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5 Conclusion and Future Work 67

5.1 More Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2 Thermal Passivity at Lagrangian Points . . . . . . . . . . . . . . . . . . . . . 69

5.3 Passive Thermal Control Components . . . . . . . . . . . . . . . . . . . . . . 69

5.3.1 Phase Change Devices . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3.2 Multi Layer Insulation . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3.3 Thermal Doublers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.4 Thermal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

v

5.4.1 Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.4.2 Organic Dyes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Bibliography 73

vi

List of Tables

3.1 SQP Optimization Results (αs,0,i = 0.50, εout,0,i = 0.50, εin,0,i = 0.50) . . . . . 26



3.4 SQP Optimization Result Summary at Different Starting Points . . . . . . . . . 30

3.5 SQP Optimization Result Summary of Varying Discretization Rotated [π4

0 π4]T 30

3.6 SQP Optimization Simplified Results of 5× 5× 5 Cube . . . . . . . . . . . . 31

3.7 SQP Optimization Simplified Results of 10× 10× 10 Cube . . . . . . . . . . 32

3.8 SQP Optimization Simplified Results of 15× 15× 15 Cube . . . . . . . . . . 32

3.9 SQP Optimization Simplified Result Summary of Different Nodes . . . . . . . 32

3.10 Nelder-Mead Results (αs,0,i = 0.25, εout,0,i = 0.25, εin,0,i = 0.25) . . . . . . . . 35






3.16 Nelder-Mead Optimization Result Summary at Different Starting Points . . . . 37

4.1 Modified Shape: SQP Optimization Results . . . . . . . . . . . . . . . . . . . 41

4.2 3 Trays: SQP Optimization Results . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 Results Summary of 15× 15× 25 Spacecraft with Different Modifications . . 48

4.4 Summary of Values in Circular Orbit with ϕΦ = [0 0 0]T . . . . . . . . . . . . 57

vii

4.5 Summary of Values in Circular Orbit with ϕΦ = [0 i⊕ 0]T . . . . . . . . . . . 57

4.6 Summary of Values in Circular Orbit with ϕΦ = [π4i⊕

π4]T . . . . . . . . . . . 57

4.7 Summary of Values in Eccentric Orbit with ϕΦ = [π4i⊕

π4]T . . . . . . . . . . 58

4.8 Summary of Values in all Scenarios . . . . . . . . . . . . . . . . . . . . . . . 58

4.9 Results Summary of Spacecraft in Orbit at Different Altitudes (Td,i = 300K) . 61

4.10 Optimization Results at R = 6500 km, Td,i = 300K . . . . . . . . . . . . . . 62


4.12 Results Summary of Spacecraft in Orbit at Different Altitudes (Td,i = 400K) . 62




4.16 Summary of Values in Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.17 Results Summary at R = 8000 km for Different Td . . . . . . . . . . . . . . 65

4.18 Results Summary at R = 8000 km for Different Td . . . . . . . . . . . . . . 65

4.19 TCC Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

viii

List of Figures

2.1 Discretized Cube and Coordinate System . . . . . . . . . . . . . . . . . . . . 4

2.2 Full Eclipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Opposite Face Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Adjacent Face Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.5 Wetted Area of Axes Rotated by [π4

0 π4]T . . . . . . . . . . . . . . . . . . . . 7

2.6 Conduction Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.7 Conduction Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.8 Radiation View Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.9 Radiation View Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.10 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.11 Results For Full Eclipse (Scenario 1) . . . . . . . . . . . . . . . . . . . . . . . 21

2.12 Results For Full Eclipse (Scenario 1), ϕΦ = [π4

0 π4]T . . . . . . . . . . . . . . 21

2.13 Results For Opposite Face Flux (Scenario 2) . . . . . . . . . . . . . . . . . . . 22

2.14 Results For Opposite Face Flux (Scenario 2), ϕΦ = [π4

0 π4]T . . . . . . . . . . 22

2.15 Results For Adjacent Face Flux (Scenario 3) . . . . . . . . . . . . . . . . . . . 23

2.16 Results For Adjacent Face Flux (Scenario 3), ϕΦ = [π4

0 π4]T . . . . . . . . . . 23

3.1 Flow Chart of Network Correction . . . . . . . . . . . . . . . . . . . . . . . . 34

4.1 Inverse Problem: Modified Shape (15× 15× 25) - View 1 . . . . . . . . . . . 42

4.2 Inverse Problem: Modified Shape (15× 15× 25) - View 2 . . . . . . . . . . . 43

ix

4.3 Example of a Spacecraft with 3 Trays . . . . . . . . . . . . . . . . . . . . . . 44

4.4 Inverse Problem: 15× 15× 25 with 3 Trays - View 1 . . . . . . . . . . . . . . 46

4.5 Inverse Problem: 15× 15× 25 with 3 Trays - View 2 . . . . . . . . . . . . . . 47

4.6 θ = 0.0000 for Circular Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . 53





4.11 θ = −2.3562 for Circular Orbit . . . . . . . . . . . . . . . . . . . . . . . . . 55



4.14 100-Point Composite Image - View 1 . . . . . . . . . . . . . . . . . . . . . . . 59

4.15 100-Point Composite Image - View 2 . . . . . . . . . . . . . . . . . . . . . . . 60

x

Chapter 1

Introduction

1.1 Current Thermal Design

The current procedure for a spacecraft thermal design is to start with a passive system and add

other components as needed. An active system is used when there’s only a few degrees of

tolerance in the required temperature or if several kilowatts need to be dissipated. For large

mass spacecraft (i.e. greater than 1000 kg), one typically designs for the coldest permissible

range (cold bias) and uses heaters to trim [1].

1.2 Motivation

With respect to space applications, the installation and utilization of an active heating/cooling

system on a spacecraft entails increased mass and energy consumption, which translate into

very high costs. A passive system will reduce these costs, and also help mitigate the probability

of mechanical/electronic failures. Active thermal systems are generally 3− 4% of the total dry

weight of the spacecraft as well as 3− 4% of the total cost of the spacecraft [1].

1

CHAPTER 1. INTRODUCTION 2

1.3 Nonlinear Thermal Network Models

In the realm of thermal modelling and heat transfer analysis, numerous techniques have been

developed, and can be divided into two main categories: finite element and finite difference

techniques [2]. A widely used discretization method for modelling thermal systems, especially

in the aerospace thermal engineering community, is the thermal network approach. It is derived

from energy balance equations and is equivalent to a particular finite difference discretization

of the underlying heat transfer equation [3]. The thermal network model is based on having

a series of nodes similar to a circuit consisting of thermal resistances, capacitances and heat

sources. Here, the currents correspond to heat flow and the nodal potentials to temperatures.

In this approach, each node represents a particular isothermal component of the system, and

provides the basic advantage of fast computation time of complex systems.

1.4 Thermal Control

The thermal control subsystem (TCS) ensures that all components in a spacecraft stay within

their temperature bounds. It is a fundamental part of every spacecraft, but it must be designed

specifically for each one, catering to the mission constraints, mission objectives, the physical

design, and the encroaching thermal energies (solar radiation, Albedo, Earth’s terrestrial ra-

diation, and heat generated by onboard equipment). As mentioned in the Motivation, passive

and active are the two broad types of TCS. A passive system relies on conductive and radia-

tive heat paths and has no moving parts or electrical power input, whereas an active system

relies on pumps, thermostats, and heaters; uses moving parts; and requires electrical power [1].

Common passive components include: phase change devices, thermal control coatings (TCCs),

multi-layer insulation (MLI), and thermal doublers; common active components include: heat

pipes, louvers, Maltese crosses, second-surface mirrors, cold plates, thermal switches, electri-

cal heaters, pumped-loop systems, water evaporators, heat exchangers, and radiators. We will

be primarily focusing on TCCs.

CHAPTER 1. INTRODUCTION 3

1.5 Thermal Control Coatings (TCCs)

The spacecraft will either absorb or emit radiant energy based on the material surfaces. There

are three types of TCCs: solar reflectors, flat coatings, and solar absorbers. Solar reflectors have

a low solar absorptivity, but a high emissivity. They are useful in a solar or Albedo environment

as they reflect much of the impinging energy while retaining the high IR emissivity needed for

efficient rejection of the spacecraft waste heat [4]. These include white paints and optical solar

reflectors (OSRs), such as Teflon, glass, and quartz mirrors. Flat coatings reflect and absorb

nearly equally in the solar and IR spectra, and are widely used inside satellite canisters, and

exteriors of electronic covers [4]. They are primarily used to enhance radiative heat transfer

inside the satellite, and commonly include black and metallic paints. Solar absorbers have high

and unpredictable temperatures associated with them. These most commonly include silverized

finishes and vaporized deposited aluminum (VDA). These materials can also be combined with

each other in order in different patterns such as striped, dotted or checker-board. However, it

should be noted that all of these surface coatings can fade, deplete, corrode, and erode over

time; with the major instigators being contamination from debris/volatiles, and corruption from

exposure to UV radiation/charge particles.

Chapter 2

Spacecraft Thermal Analysis

2.1 Initial Spacecraft Model

Initially, the spacecraft will be modelled as a 1 m × 1 m × 1 m aluminum hollow cube, with

a thickness of 1 mm, and a thermal conductivity of 155 W/(m K). There will be a further

intention to expand to other shapes. This geometry, the location of each face and designated

coordinate system, is shown below in Figure 2.1.

TT11 TT22 TT33 TT44 TT55

T i th t f h

zTTxx

Ti is the centre of each area for i...N, x = N/6

x

y

Figure 2.1: Discretized Cube and Coordinate System

4

CHAPTER 2. SPACECRAFT THERMAL ANALYSIS 5

The cube will be discretized into distinct surface elements, with an inherent coordinate system.

Each component will be treated as a separate surface area, Ai, and treated as isothermal with

temperature, Ti. By placing Ti at the centre of each nodal element (on all three axes), we will

elect to neglect the temperature variation between the external and internal surfaces.

2.2 Temperature Distribution

Our initial goal is to model the forward problem of determining the temperature distribution of

the spacecraft (Ti, i = 1, . . . , N ) ensuring that the following three cases involving the Sun, the

Earth, and the spacecraft are examined:

1. Full Eclipse (Figure 2.2): In this scenario, the spacecraft will only receive Albedo and

Earth flux on Face 4, as the Earth is blocking the solar flux coming from the Sun.

Albedo

Earth FluxSolar Flux

Figure 2.2: Full Eclipse

2. Opposite Face Flux (Figure 2.3): In this scenario, the spacecraft will receive solar flux

on Face 4, and will also receive Albedo and Earth flux on Face 2, which is the direction

directly opposite to it.


AlbedoSolar Flux

Earth Flux

Solar Flux

Figure 2.3: Opposite Face Flux

3. Adjacent Face Flux (Figure 2.4): In this scenario, the spacecraft will receive solar flux

on Face 4, and will also receive Albedo and Earth flux on Face 3, which is the direction

adjacent to it.

Solar Flux

EarthFluxAlbedo Flux

Figure 2.4: Adjacent Face Flux

The modelled spacecraft will also have the ability to be rotated within each scenario, via the

rotational flux vector (ϕΦ), and receive radiation from any direction. This will enable the


analysis for the scenarios where the incident fluxes reach multiple faces at once. The worst

case scenario will arise when ϕΦ = [π4

0 π4]T , resulting in a corner of the cube being normal

to the incident fluxes. The aforementioned is considered to be the worst case scenario, because

the cross-sectional area exposed to incident fluxes, also known as the wetted area, is at its

maximum. This is shown in Figure 2.5:

Figure 2.5: Wetted Area of Axes Rotated by [π4

0 π4]T

2.3 Fourier’s Law

The time rate of conductive heat transfer through a material is proportional to the negative

gradient in the temperature and to the area at right angles, to that gradient, through which the

heat is flowing:

qC = −k∮S

∇T · dS (2.1)

where qC is rate of heat transferred, t is the time taken, k is the material’s conductivity, dS is

the surface through which the heat is flowing, and T is the temperature. When there is a tem-

perature gradient within a body, heat energy will flow from the region of high temperature to

the region of low temperature. The negative sign ensures that heat flows down the temperature

gradient. Fourier’s law can be further simplified to:

QC = −k∂T∂n

(2.2)

where QC is the conduction heat rate per unit area, which is proportional to the gradient of


the temperature in a direction n normal to the area [4]. Using Fourier’s Law, analysis will be

performed on each discretized element with its neighbour, as shown in Figure 2.6.

Figure 2.6: Conduction Analysis

By generalizing Eq. (2.2) for each node, we arrive at:

qC =−kAc,ijLij

(Ti − Tj) (2.3)

where Ac is the cross-sectional area, Lij is the length between the nodes, Tj is the temperature

of the neighbouring nodes, and Ti is the temperature of the current node. However, since the

geometry of the shapes are squares with equivalent dimensions, this can be further reduced to:

qC = kt(Tj − Ti) (2.4)

where t is the thickness of the conductive surface, since Ac,ij = Lijt. We can now introduce a

new term called Cij which represents the conduction coefficients. With this we can reduce Eq.

(2.4) to the following:

qC = Cij(Tj − Ti) (2.5)


2.4 Conduction Results

As previously mentioned, initial investigation into the representation of the spacecraft will

begin by modelling it as a cube, and neglecting the effects of radiation. Several simulations

were run and an instance of the temperature distribution is shown below. For the following

result, Scenario 2 was employed and a boundary condition of 300K was set at the middle of

the bottom face.

7250 7309 7329 7309 72507250 7309 7329 7309 7250

7277 7347 7369 7347 7277

7355 7431 7456 7431 7355

7488 7565 7592 7565 7488

7664 7750 7782 7750 7664

7161 7156 7226 7367 7575 7841 7989 8037 7989 7841 7575 7367 7226 7156 7161 7252 7310 7330 7310 72527161 7156 7226 7367 7575 7841 7989 8037 7989 7841 7575 7367 7226 7156 7161 7252 7310 7330 7310 7252

6988 6961 7025 7180 7429 7788 7979 8038 7979 7788 7429 7180 7025 6961 6988 7109 7174 7193 7174 7109

6729 6676 6733 6899 7172 7555 7750 7807 7750 7555 7172 6899 6733 6676 6729 6848 6906 6920 6906 6848

6362 6293 6330 6512 6803 7163 7310 7343 7310 7163 6803 6512 6330 6293 6362 6484 6505 6498 6505 6484

5952 5804 5783 6015 6366 6632 6637 6593 6637 6632 6366 6015 5783 5804 5952 6043 5958 5886 5958 6043

6015 5661 5409 5661 60156015 5661 5409 5661 6015

5399 4584 3719 4584 5399

4982 3558 300 3558 4982

5188 4366 3497 4366 5188

5601 5220 4956 5220 5601

Figure 2.7: Conduction Results

Although the results themselves are not viable nor realistic, the important conclusion that can

be garnered here is that it is not possible to attain a reasonable solution to the forward problem

by analyzing only the conductive heat transfer and neglecting the impact of radiation. The

consequence of ignoring the effects of radiation in the system produces multiple solutions and

reaches a singularity. However, this validated the construction of the conduction coefficient

matrix Cij .


2.5 Radiation Properties

There are four important radiation properties that we must consider within our analysis:

• Absorptivity (α): A surface property that measures the fraction of incident radiation that

is absorbed by the body.

• Reflectivity (ρ): A surface property that measures the fraction of incident radiation that

is reflected by the body.

• Transmissivity (τ ): A surface property that measures the fraction of incident radiation

that is transmitted through the body.

• Emissivity (ε): The ratio of the radiation emitted by a surface to the radiation emitted by

a black body at the same temperature.

The first three properties (absorptivity, reflectivity, and transmissivity) all depend on the tem-

perature of the body, the wavelength of the incident radiation, as well as its direction. The

following equation demonstrates how all three can be related to each other:

α + ρ+ τ = 1 (2.6)

However, when material is opaque, such as an aluminum spacecraft, the transmissivity value is

equal to 0 (τ = 0). This brings us to:

α + ρ = 1 (2.7)

The fourth important radiation property to consider is emissivity, which is defined as follows:

ε =WGB

WBB

(2.8)

where WGB and WBB are the respective power densities of a gray and a black body. When the

object is a black body, this implies that all of the energy is absorbed, resulting in ε = 1.


2.6 Black Body View Factors

Only a portion of the energy striking surface Ai reaches Aj . To account for this, we define the

view factor, Fij as the fraction of energy leaving Ai reaching Aj . This is shown in Figure 2.8

below.

Figure 2.8: Radiation View Factors

The rays hit every point on the surfaceAi, from every angle, and whatever is not absorbed exits

equally in all directions. A portion of rays leaving Ai will strike Aj . This leads to the idea of a

view factor, which can be calculated with the following equation:

FijAi =

∫Ai

∫Aj

cos θ1 cos θ2

πs2dAidAj (2.9)

where Fij is the view factor, dAi, dAj are differential areas, θi, θj are the respective angles of

incidence to dAi, dAj , and s is the distance between the two surfaces. This event is illustrated

in Figure 2.8.


Figure 2.9: Radiation View Factors

Integrating over all points on surface Ai and averaging gives the view factor Fij . It can also be

shown that:

AiFij = AjFji (2.10)

For gray bodies, we need Gebhart factors to account for the emissivity and reflectivity. It can

be shown that a similar reciprocity equation exists for gray bodies:

εiBijAi = εjBjiAj (2.11)

where Bij , the absorption factor, is the fraction of radiant energy arriving at surface Aj that is

emitted by surface Ai.


2.7 Gebhart Factors

For the purposes of this analysis, it can be assumed that the surfaces of the spacecraft are

opaque, which is also referred to as an adiabatic surface. As per Gebhart [5], it has been

proved that the temperature of a surface in radiant balance is independent of its surface prop-

erties. Even highly non-gray, specular surfaces are often essentially gray and diffuse under

the conditions of actual use as a result of surface coatings, corrosion, erosion, or other types

of surface alteration [5]. The gray body assumption, which assumes that the emissivity is a

constant, is an assumption that a surface’s spectral emissivity and absorptivity do not depend

on wavelength [5]. This leads us to Gebhart’s equation of radiant balance of an opaque surface:

qi = WjAj −N∑i=1

BijAi (2.12)

where WA is the rate at which energy is radiated from a surface. For an opaque surface

W = εσT 4 and Aj = Aa.

qa = εaAaσ(T 4a −

N∑i=1

BaiT4i ) = 0 (2.13)

where Bij , the absorption factor, is the fraction of radiant energy arriving at surface Aj . If a

given area in radiant balance is in an enclosure and is subject to highly non-uniform irradiation,

its temperature will be far from uniform. More accurate estimates of the temperature achieved

by such as surface may be obtained by subdividing it into separate zones for which individual

temperatures are calculated [5].


2.8 Calculating Gebhart Factors

For diffuse radiation and reflection, the absorption factors are given by [5]:

(F11ρ1 − 1)B1j + F12ρ2B2j + · · ·+ F1NρNBNj + F1jεj = 0

F21ρ1B1j + (F22ρ2 − 1)B2j + · · ·+ F2NρNBNj + F2jεj = 0

...

FN1ρ1B1j + FN2ρ2B2j + · · ·+ (FNNρN − 1)BNj + Fijεj = 0

(2.14)

where i is the node of departing energy, j is the node of destination energy, ρj is the transmis-

sivity, εj is the emissivity, Fij are the view factors, and Bij are the Gebhart absorption factors.

The absorption factor relations may be written more compactly as:

N∑k=1

(Fikρk − δik)Bkj + Fijεj = 0, i, j = 1, . . . , N (2.15)

δik =

1, if i = k

0, if i 6= k

where δik is Kronecker’s delta. For each individual wavelength, we can apply Kirchhoff’s

law, which states that at thermal equilibrium, the emissivity of a body (or surface) equals its

absorptivity for that wavelength:

εi = αi (2.16)

We can now apply Kirchhoff’s law (Eq. (2.16)) to Eq. (2.7) to reduce Eq. (2.15):

N∑k=1

(Fik(1− εk)− δik)Bkj + Fijεj = 0, i, j = 1, . . . , N (2.17)


This matrix can be expressed as a matrix equation to isolate the absorption factors:

F11(1− ε1)− 1 F12(1− ε2) . . . F1N (1− εN )

F21(1− ε1) F22(1− ε2)− 1 . . . F2N (1− εN )

...

FN1(1− ε1) FN2(1− ε2) . . . FNN (1− εN )− 1

B11 . . . B1N

B21 . . . B2N

...

BN1 . . . BNN

= −

F11ε1 . . . F1N εN

F21ε1 . . . F2N εN

...

FN1ε1 . . . FNN εN

(2.18)

This can be reduced to the form:

FεB = −Fε (2.19)

Finally, we can solve for the absorption factors by:

B = −Fε−1Fε (2.20)

2.9 Radiation Heat Transfer

The definition of emissivity, εi, leads to the Stefan-Boltzmann Law, stating that:

P = εσAT 4 (2.21)

where P is the rate at which energy is radiated for an object, and σ is the Stefan-Boltzmann

constant. The heat transfer rate between two black bodies is given by:

qR,2BB = σAiFij(T4i − T 4

j ) (2.22)

Again, just like the conduction analysis, since the elements are squares of the same size, we

can state that Ai = A. The heat transfer rate out from one gray body is given by:

qR,out = σεout,iAi(T4i − T 4

j ) (2.23)


where εout,i is the emissivity value of the spacecraft’s exterior at each node. The heat transfer

rate into one gray body is given by:

qR,in = σεin,iAi(T4i − T 4

j ) (2.24)

where εin,i is the emissivity value of the spacecraft’s interior at each node. Similarly, the heat

transfer rate between two gray bodies is given by:

qR,2GB = σεin,iAiBij(T4i − T 4

j ) (2.25)

The net heat transfer rate of radiation in a system is given by:

qR = qR,out + qR,in − qR,2GB (2.26)

With this we can reduce Eq. (2.26) to the following:

qR = σRij(T4j − T 4

i ) (2.27)

where Rij are the radiation coefficients represented as the summation of the qR,out, qR,in, and

qR,2GB terms, which consist of A,B, εout, and εin.


2.10 External Heat Sources

The incident rate of solar flux qs is generated by:

qs,i = Ai Φs αs,i cos(θs,i) (2.28)

where Φs is the solar flux, αs is the solar absorptivity, and θs is the angle between the vec-

tor normal to the face and the solar flux vector. We can put this into the following matrix

representation:

qs = A Φs αs θs (2.29)

where qs = colqs,i, αs = diagαs,i, and θs = colcos(θs,i). The incident rate of Albedo

qa, is generated by:

qa,i = Ai Φs f αs,i cos(θe,i) (2.30)

where f is the Albedo factor, and θe is the angle between the vector normal to the face and

the Earth flux vector. Using the same convention as Eq. (2.29), the previous equation can be

rewritten as:

qa = A Φs f αs θe (2.31)

where qa = colqa,i, αs = diagαs,i, and θe = colcos(θe,i). The incident rate of Earth

flux qe, is generated by:

qe,i = Ai Φe εout,i cos(θe,i) (2.32)

where Φe is the Earth flux, and εout is the emissivity. Using the same convention as Eq. (2.29),

the prior equation can be rewritten as:

qe = A Φe εout θe (2.33)

where qe = colqe,i, εout = diagεout,i, and θe = colcos(θe,i). The rate of internal power


dissipation qp, is generated by:

qp = Ai P (2.34)

where qp = colqp,i, and P = colPi. The net heat transfer rate of all external heat sources

on the system is given by:

q = qs + qa + qe + qp (2.35)

However, for this work, we will neglect the effects of power dissipation to remain purely pas-

sive. This reduces Eq. (2.35) to:

q = qs + qa + qe (2.36)

2.11 Balance Equation

For each surface element with temperature Ti, we have the following energy balance:

N∑j=1

Cij(ΩC)(Tj − Ti) +N∑j=1

σRij(ΩR)(T 4j − T 4

i ) = qi, i = 1, . . . , N (2.37)

where Cij is the conduction exchange factor, ΩC are the conduction parameters, Rij is the

radiation exchange factor, ΩR are the radiation parameters, and qi is the net heat transfer rate

(Eq. (2.35)) [6]. This can be reduced to the form:

F (T ) = CT +RT 4 − q = 0 (2.38)

where q = colqi,C = matrixCij,R = matrixRij, T = colTi, and T 4 = colT 4i .

Taking the partial derivative with respect to temperature gives the Jacobian:

∂F

∂T T= J(T ) = C + 4RD(T ) (2.39)

whereD(T ) = diag[T 31 , ..., T

3N ], and J(T ) is the Jacobian.


2.12 Newton’s Method

The scalar form of Newton’s Method for solving f(x) = 0, for x is shown below:

xn+1 = xn −f(xn)

f ′(xn), n = 0, 1, 2, . . . (2.40)

This method will usually converge, provided the initial guess is close enough to the unknown

zero, and that f ′(x0) 6= 0. For a nonlinear system of equations (Eq. (2.38)), Newton’s Method

is of the following form:

J(xn)(xn+1 − xn) = F (xn) (2.41)

Rearranging and solving for xn+1

xn+1 = xn − J−1(xn)F (xn) (2.42)

Solving in terms of temperature:

Tn+1 = Tn − J−1(Tn)F (Tn) (2.43)


2.13 Forward Problem Results

Given the validation of the conduction matrix, it is time to design the radiation matrixR, with

the most intensive component being the correct calculation of the view factors and subsequently

the Gebhart factors (Eq. (2.20)). This enables the solution to the forward heat transfer problem

(solving for Ti, i = 1, . . . , N ) utilizing Newton’s Method. Upon its completion, the next step

is to set up the balance equation (Eq. (2.38)), calculate its Jacobian (Eq. (2.39)), take its

inverse, and apply Newton’s Method to iterate our previously derived equation (Eq. (2.43)).

This course of action enables us to converge to an array of temperatures, corresponding to each

node on the cube. We discretized a cube according to 25 × 25 × 25 while setting values for

αs,i = 0.90, εin,i = 0.50, and εout,i = 0.10. Using the coordinate system shown in Figure 2.10,

results were found for all six scenarios, and are subsequently shown.

Figure 2.10: Coordinate System


234 226 217 209 201 193 186 180 174 168 162 157 153 148 144 141 137 134 131 128 126 124 122 121 120

237 227 218 209 201 194 186 180 174 168 163 157 153 148 144 141 137 134 131 128 126 124 122 121 119

240 229 219 210 202 194 187 180 174 168 163 158 153 149 144 141 137 134 131 128 126 124 122 120 119

242 231 220 211 202 194 187 180 174 168 163 158 153 149 144 141 137 134 131 128 126 124 122 120 119

245 233 222 212 203 195 188 181 174 168 163 158 153 149 145 141 137 134 131 128 126 124 122 120 118

247 234 223 213 204 196 188 181 175 169 163 158 153 149 145 141 137 134 131 128 126 124 122 120 118

249 236 224 214 205 196 188 181 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 120 118

251 237 225 215 205 197 189 182 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 119 118

252 238 226 216 206 197 189 182 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 119 118

253 239 227 216 206 198 189 182 175 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

253 240 227 216 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

254 240 228 217 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

254 240 228 217 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

254 240 228 217 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

253 240 227 216 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

253 239 227 216 206 198 189 182 175 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

252 238 226 216 206 197 189 182 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 119 118

251 237 225 215 205 197 189 182 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 119 118

249 236 224 214 205 196 188 181 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 120 118

247 234 223 213 204 196 188 181 175 169 163 158 153 149 145 141 137 134 131 128 126 124 122 120 118

245 233 222 212 203 195 188 181 174 168 163 158 153 149 145 141 137 134 131 128 126 124 122 120 118

242 231 220 211 202 194 187 180 174 168 163 158 153 149 144 141 137 134 131 128 126 124 122 120 119

240 229 219 210 202 194 187 180 174 168 163 158 153 149 144 141 137 134 131 128 126 124 122 120 119

237 227 218 209 201 194 186 180 174 168 163 157 153 148 144 141 137 134 131 128 126 124 122 121 119

234 226 217 209 201 193 186 180 174 168 162 157 153 148 144 141 137 134 131 128 126 124 122 121 120

242 247 252 256 259 262 264 266 267 268 269 269 269 269 269 268 267 266 264 262 259 256 252 247 242 234 226 217 209 201 193 186 180 174 168 162 157 153 148 144 141 137 134 131 128 126 124 122 121 120 119 118 118 118 117 117 116 116 116 116 116 116 116 116 116 116 116 116 116 117 117 118 118 118 119 120 121 122 124 126 128 131 134 137 141 144 148 153 157 162 168 174 180 186 193 201 209 217 226 234

247 255 261 266 270 274 276 278 280 281 282 282 282 282 282 281 280 278 276 274 270 266 261 255 247 237 227 218 209 201 194 186 180 174 168 163 157 153 148 144 141 137 134 131 128 126 124 122 121 119 118 118 117 116 116 116 115 115 115 114 114 114 114 114 114 114 115 115 115 116 116 116 117 118 118 119 121 122 124 126 128 131 134 137 141 144 148 153 157 163 168 174 180 186 194 201 209 218 227 237

252 261 269 275 279 283 286 288 290 291 292 293 293 293 292 291 290 288 286 283 279 275 269 261 252 240 229 219 210 202 194 187 180 174 168 163 158 153 149 144 141 137 134 131 128 126 124 122 120 119 118 117 116 116 115 115 114 114 113 113 113 113 113 113 113 113 113 114 114 115 115 116 116 117 118 119 120 122 124 126 128 131 134 137 141 144 149 153 158 163 168 174 180 187 194 202 210 219 229 240

256 266 275 281 286 291 294 296 298 300 301 301 301 301 301 300 298 296 294 291 286 281 275 266 256 242 231 220 211 202 194 187 180 174 168 163 158 153 149 144 141 137 134 131 128 126 124 122 120 119 118 116 116 115 114 114 113 113 112 112 112 112 112 112 112 112 112 113 113 114 114 115 116 116 118 119 120 122 124 126 128 131 134 137 141 144 149 153 158 163 168 174 180 187 194 202 211 220 231 242

259 270 279 286 292 297 300 303 305 306 307 308 308 308 307 306 305 303 300 297 292 286 279 270 259 245 233 222 212 203 195 188 181 174 168 163 158 153 149 145 141 137 134 131 128 126 124 122 120 118 117 116 115 114 113 113 112 112 112 111 111 111 111 111 111 111 112 112 112 113 113 114 115 116 117 118 120 122 124 126 128 131 134 137 141 145 149 153 158 163 168 174 181 188 195 203 212 222 233 245

262 274 283 291 297 301 305 308 310 312 313 313 313 313 313 312 310 308 305 301 297 291 283 274 262 247 234 223 213 204 196 188 181 175 169 163 158 153 149 145 141 137 134 131 128 126 124 122 120 118 117 116 115 114 113 112 112 111 111 110 110 110 110 110 110 110 111 111 112 112 113 114 115 116 117 118 120 122 124 126 128 131 134 137 141 145 149 153 158 163 169 175 181 188 196 204 213 223 234 247

264 276 286 294 300 305 309 312 314 316 317 317 318 317 317 316 314 312 309 305 300 294 286 276 264 249 236 224 214 205 196 188 181 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 120 118 116 115 114 113 112 112 111 111 110 110 110 109 109 109 110 110 110 111 111 112 112 113 114 115 116 118 120 121 123 126 128 131 134 137 141 145 149 153 158 163 169 175 181 188 196 205 214 224 236 249

266 278 288 296 303 308 312 315 317 319 320 321 321 321 320 319 317 315 312 308 303 296 288 278 266 251 237 225 215 205 197 189 182 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 119 118 116 115 114 113 112 111 111 110 110 109 109 109 109 109 109 109 110 110 111 111 112 113 114 115 116 118 119 121 123 126 128 131 134 137 141 145 149 153 158 163 169 175 182 189 197 205 215 225 237 251

267 280 290 298 305 310 314 317 320 321 323 323 323 323 323 321 320 317 314 310 305 298 290 280 267 252 238 226 216 206 197 189 182 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 119 118 116 115 113 112 112 111 110 110 109 109 109 108 108 108 109 109 109 110 110 111 112 112 113 115 116 118 119 121 123 126 128 131 134 137 141 145 149 153 158 163 169 175 182 189 197 206 216 226 238 252

268 281 291 300 306 312 316 319 321 323 324 325 325 325 324 323 321 319 316 312 306 300 291 281 268 253 239 227 216 206 198 189 182 175 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117 116 114 113 112 111 110 110 109 109 108 108 108 108 108 108 108 109 109 110 110 111 112 113 114 116 117 119 121 123 126 128 131 134 137 141 145 149 153 158 164 169 175 182 189 198 206 216 227 239 253

269 282 292 301 307 313 317 320 323 324 326 326 326 326 326 324 323 320 317 313 307 301 292 282 269 253 240 227 216 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117 116 114 113 112 111 110 110 109 109 108 108 108 108 108 108 108 109 109 110 110 111 112 113 114 116 117 119 121 123 126 128 131 134 137 141 145 149 153 158 164 169 176 182 190 198 207 216 227 240 253

269 282 293 301 308 313 317 321 323 325 326 327 327 327 326 325 323 321 317 313 308 301 293 282 269 254 240 228 217 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117 116 114 113 112 111 110 109 109 108 108 108 108 108 108 108 108 108 109 109 110 111 112 113 114 116 117 119 121 123 126 128 131 134 137 141 145 149 153 158 164 169 176 182 190 198 207 217 228 240 254

269 282 293 301 308 313 318 321 323 325 326 327 327 327 326 325 323 321 318 313 308 301 293 282 269 254 240 228 217 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117 116 114 113 112 111 110 109 109 108 108 108 108 108 108 108 108 108 109 109 110 111 112 113 114 116 117 119 121 123 126 128 131 134 137 141 145 149 153 158 164 169 176 182 190 198 207 217 228 240 254

269 282 293 301 308 313 317 321 323 325 326 327 327 327 326 325 323 321 317 313 308 301 293 282 269 254 240 228 217 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117 116 114 113 112 111 110 109 109 108 108 108 108 108 108 108 108 108 109 109 110 111 112 113 114 116 117 119 121 123 126 128 131 134 137 141 145 149 153 158 164 169 176 182 190 198 207 217 228 240 254

269 282 292 301 307 313 317 320 323 324 326 326 326 326 326 324 323 320 317 313 307 301 292 282 269 253 240 227 216 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117 116 114 113 112 111 110 110 109 109 108 108 108 108 108 108 108 109 109 110 110 111 112 113 114 116 117 119 121 123 126 128 131 134 137 141 145 149 153 158 164 169 176 182 190 198 207 216 227 240 253

268 281 291 300 306 312 316 319 321 323 324 325 325 325 324 323 321 319 316 312 306 300 291 281 268 253 239 227 216 206 198 189 182 175 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117 116 114 113 112 111 110 110 109 109 108 108 108 108 108 108 108 109 109 110 110 111 112 113 114 116 117 119 121 123 126 128 131 134 137 141 145 149 153 158 164 169 175 182 189 198 206 216 227 239 253

267 280 290 298 305 310 314 317 320 321 323 323 323 323 323 321 320 317 314 310 305 298 290 280 267 252 238 226 216 206 197 189 182 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 119 118 116 115 113 112 112 111 110 110 109 109 109 108 108 108 109 109 109 110 110 111 112 112 113 115 116 118 119 121 123 126 128 131 134 137 141 145 149 153 158 163 169 175 182 189 197 206 216 226 238 252

266 278 288 296 303 308 312 315 317 319 320 321 321 321 320 319 317 315 312 308 303 296 288 278 266 251 237 225 215 205 197 189 182 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 119 118 116 115 114 113 112 111 111 110 110 109 109 109 109 109 109 109 110 110 111 111 112 113 114 115 116 118 119 121 123 126 128 131 134 137 141 145 149 153 158 163 169 175 182 189 197 205 215 225 237 251

264 276 286 294 300 305 309 312 314 316 317 317 318 317 317 316 314 312 309 305 300 294 286 276 264 249 236 224 214 205 196 188 181 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 120 118 116 115 114 113 112 112 111 111 110 110 110 109 109 109 110 110 110 111 111 112 112 113 114 115 116 118 120 121 123 126 128 131 134 137 141 145 149 153 158 163 169 175 181 188 196 205 214 224 236 249

262 274 283 291 297 301 305 308 310 312 313 313 313 313 313 312 310 308 305 301 297 291 283 274 262 247 234 223 213 204 196 188 181 175 169 163 158 153 149 145 141 137 134 131 128 126 124 122 120 118 117 116 115 114 113 112 112 111 111 110 110 110 110 110 110 110 111 111 112 112 113 114 115 116 117 118 120 122 124 126 128 131 134 137 141 145 149 153 158 163 169 175 181 188 196 204 213 223 234 247

259 270 279 286 292 297 300 303 305 306 307 308 308 308 307 306 305 303 300 297 292 286 279 270 259 245 233 222 212 203 195 188 181 174 168 163 158 153 149 145 141 137 134 131 128 126 124 122 120 118 117 116 115 114 113 113 112 112 112 111 111 111 111 111 111 111 112 112 112 113 113 114 115 116 117 118 120 122 124 126 128 131 134 137 141 145 149 153 158 163 168 174 181 188 195 203 212 222 233 245

256 266 275 281 286 291 294 296 298 300 301 301 301 301 301 300 298 296 294 291 286 281 275 266 256 242 231 220 211 202 194 187 180 174 168 163 158 153 149 144 141 137 134 131 128 126 124 122 120 119 118 116 116 115 114 114 113 113 112 112 112 112 112 112 112 112 112 113 113 114 114 115 116 116 118 119 120 122 124 126 128 131 134 137 141 144 149 153 158 163 168 174 180 187 194 202 211 220 231 242

252 261 269 275 279 283 286 288 290 291 292 293 293 293 292 291 290 288 286 283 279 275 269 261 252 240 229 219 210 202 194 187 180 174 168 163 158 153 149 144 141 137 134 131 128 126 124 122 120 119 118 117 116 116 115 115 114 114 113 113 113 113 113 113 113 113 113 114 114 115 115 116 116 117 118 119 120 122 124 126 128 131 134 137 141 144 149 153 158 163 168 174 180 187 194 202 210 219 229 240

247 255 261 266 270 274 276 278 280 281 282 282 282 282 282 281 280 278 276 274 270 266 261 255 247 237 227 218 209 201 194 186 180 174 168 163 157 153 148 144 141 137 134 131 128 126 124 122 121 119 118 118 117 116 116 116 115 115 115 114 114 114 114 114 114 114 115 115 115 116 116 116 117 118 118 119 121 122 124 126 128 131 134 137 141 144 148 153 157 163 168 174 180 186 194 201 209 218 227 237

242 247 252 256 259 262 264 266 267 268 269 269 269 269 269 268 267 266 264 262 259 256 252 247 242 234 226 217 209 201 193 186 180 174 168 162 157 153 148 144 141 137 134 131 128 126 124 122 121 120 119 118 118 118 117 117 116 116 116 116 116 116 116 116 116 116 116 116 116 117 117 118 118 118 119 120 121 122 124 126 128 131 134 137 141 144 148 153 157 162 168 174 180 186 193 201 209 217 226 234

234 226 217 209 201 193 186 180 174 168 162 157 153 148 144 141 137 134 131 128 126 124 122 121 120

237 227 218 209 201 194 186 180 174 168 163 157 153 148 144 141 137 134 131 128 126 124 122 121 119

240 229 219 210 202 194 187 180 174 168 163 158 153 149 144 141 137 134 131 128 126 124 122 120 119

242 231 220 211 202 194 187 180 174 168 163 158 153 149 144 141 137 134 131 128 126 124 122 120 119

245 233 222 212 203 195 188 181 174 168 163 158 153 149 145 141 137 134 131 128 126 124 122 120 118

247 234 223 213 204 196 188 181 175 169 163 158 153 149 145 141 137 134 131 128 126 124 122 120 118

249 236 224 214 205 196 188 181 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 120 118

251 237 225 215 205 197 189 182 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 119 118

252 238 226 216 206 197 189 182 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 119 118

253 239 227 216 206 198 189 182 175 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

253 240 227 216 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

254 240 228 217 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

254 240 228 217 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

254 240 228 217 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

253 240 227 216 207 198 190 182 176 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

253 239 227 216 206 198 189 182 175 169 164 158 153 149 145 141 137 134 131 128 126 123 121 119 117

252 238 226 216 206 197 189 182 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 119 118

251 237 225 215 205 197 189 182 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 119 118

249 236 224 214 205 196 188 181 175 169 163 158 153 149 145 141 137 134 131 128 126 123 121 120 118

247 234 223 213 204 196 188 181 175 169 163 158 153 149 145 141 137 134 131 128 126 124 122 120 118

245 233 222 212 203 195 188 181 174 168 163 158 153 149 145 141 137 134 131 128 126 124 122 120 118

242 231 220 211 202 194 187 180 174 168 163 158 153 149 144 141 137 134 131 128 126 124 122 120 119

240 229 219 210 202 194 187 180 174 168 163 158 153 149 144 141 137 134 131 128 126 124 122 120 119

237 227 218 209 201 194 186 180 174 168 163 157 153 148 144 141 137 134 131 128 126 124 122 121 119

234 226 217 209 201 193 186 180 174 168 162 157 153 148 144 141 137 134 131 128 126 124 122 121 120

Figure 2.11: Results For Full Eclipse (Scenario 1)

256 252 248 244 241 237 235 232 230 227 225 223 221 219 217 215 213 211 209 207 205 203 201 198 197

254 248 243 238 235 231 228 225 222 220 217 215 213 211 209 207 205 204 202 200 198 196 194 193 192

252 245 239 234 229 225 222 219 216 213 211 208 206 204 202 200 198 197 195 193 191 190 188 187 186

250 242 236 230 225 221 217 213 210 207 205 202 200 198 196 194 192 190 189 187 185 184 183 182 181

248 240 233 227 221 217 212 209 205 202 199 197 194 192 190 188 186 185 183 182 180 179 178 177 177

247 238 230 224 218 213 209 204 201 198 195 192 189 187 185 183 181 180 178 177 175 174 173 173 172

245 236 228 221 215 210 205 201 197 194 190 188 185 183 181 178 177 175 173 172 171 170 169 168 168

244 234 226 219 213 207 202 198 194 190 187 184 181 179 176 174 173 171 169 168 167 166 165 164 164

243 233 224 217 210 204 199 195 190 187 183 180 178 175 173 171 169 167 166 164 163 162 161 161 160

241 231 223 215 208 202 197 192 188 184 180 177 174 172 169 167 165 164 162 161 160 159 158 157 157

240 230 221 213 206 200 195 190 185 181 178 174 171 169 166 164 162 161 159 158 157 156 155 154 154

239 229 220 212 205 198 193 188 183 179 175 172 169 166 164 162 160 158 156 155 154 153 152 152 151

237 227 218 210 203 196 191 186 181 177 173 170 167 164 161 159 157 155 154 152 151 150 150 149 149

236 226 217 208 201 195 189 184 179 175 171 168 164 162 159 157 155 153 152 150 149 148 147 147 146

235 224 215 207 200 193 187 182 177 173 169 166 163 160 157 155 153 151 150 148 147 146 145 145 144

233 223 214 206 198 192 186 181 176 172 168 164 161 158 155 153 151 149 148 146 145 144 143 143 142

232 221 212 204 197 190 184 179 174 170 166 163 159 157 154 152 150 148 146 145 144 143 142 141 141

230 220 211 203 195 189 183 178 173 169 165 161 158 155 153 150 148 146 145 143 142 141 140 140 139

228 218 209 201 194 188 182 177 172 168 164 160 157 154 151 149 147 145 143 142 141 140 139 138 138

226 216 208 200 193 187 181 176 171 167 163 159 156 153 150 148 146 144 142 141 140 139 138 137 137

224 214 206 198 192 186 180 175 170 166 162 158 155 152 150 147 145 143 142 140 139 138 137 137 136

221 212 204 197 191 185 179 174 169 165 161 158 154 152 149 147 144 143 141 139 138 137 136 136 135

219 211 203 196 190 184 178 173 169 165 161 157 154 151 148 146 144 142 140 139 138 137 136 135 135

216 209 202 195 189 183 178 173 168 164 160 157 154 151 148 146 144 142 140 139 137 136 135 135 135

214 208 201 195 189 183 178 173 168 164 160 157 153 150 148 145 143 141 140 138 137 136 135 135 134

261 261 260 259 258 257 256 255 254 253 252 251 249 248 247 245 243 242 239 237 234 231 228 224 220 214 208 201 195 189 183 178 173 168 164 160 157 153 150 148 145 143 141 140 138 137 136 135 135 134 134 134 135 135 136 137 138 139 141 142 144 146 148 151 154 157 160 164 168 172 177 181 186 192 197 201 204 208 211 213 216 218 220 222 224 226 228 230 232 234 236 238 240 243 245 248 251 254 257 260

266 267 267 267 266 266 265 264 263 262 261 260 259 258 257 255 253 251 249 246 243 240 236 230 224 216 209 202 195 189 183 178 173 168 164 160 157 154 151 148 146 144 142 140 139 137 136 135 135 135 134 135 135 135 136 137 138 139 141 142 144 146 149 151 154 157 160 164 168 172 177 182 187 192 198 204 209 213 217 220 223 226 228 230 232 234 236 238 240 241 243 245 247 249 252 254 257 259 262 264

271 272 273 273 273 273 272 272 271 270 270 269 268 266 265 263 262 259 257 254 251 247 242 236 228 219 211 203 196 190 184 178 173 169 165 161 157 154 151 148 146 144 142 140 139 138 137 136 135 135 135 135 135 136 136 137 138 139 141 143 144 146 149 151 154 157 161 164 168 173 177 182 188 194 200 207 213 218 222 226 229 232 235 237 239 241 243 244 246 248 250 251 253 255 257 259 262 264 266 268

275 277 278 278 279 279 278 278 278 277 276 275 274 273 272 270 268 266 263 260 257 252 247 240 231 221 212 204 197 191 185 179 174 169 165 161 158 154 152 149 147 144 143 141 139 138 137 136 136 135 135 135 136 136 137 138 139 140 141 143 145 147 149 152 155 158 161 165 169 173 178 183 189 195 202 210 217 222 227 231 234 237 240 242 244 246 248 250 252 253 255 257 258 260 262 264 266 268 270 272

279 280 282 283 283 283 283 283 283 282 282 281 280 279 278 276 274 272 269 266 261 257 251 243 234 224 214 206 198 192 186 180 175 170 166 162 158 155 152 150 147 145 143 142 140 139 138 137 137 136 136 136 136 137 137 138 139 140 142 144 145 147 150 152 155 158 162 166 170 174 179 184 190 197 204 213 220 226 231 235 239 242 245 247 249 251 253 255 256 258 259 261 263 264 266 268 270 272 274 276

282 284 285 286 287 287 288 288 287 287 286 286 285 284 282 281 279 276 273 270 266 260 254 246 237 226 216 208 200 193 187 181 176 171 167 163 159 156 153 150 148 146 144 142 141 140 139 138 137 137 137 137 137 138 138 139 140 141 143 144 146 148 151 153 156 159 163 166 171 175 180 186 192 199 206 215 222 229 234 239 242 246 249 251 253 255 257 259 260 262 263 265 266 268 269 271 273 275 277 279

284 287 288 289 290 291 291 291 291 291 290 289 288 287 286 284 282 280 277 273 269 263 257 249 239 228 218 209 201 194 188 182 177 172 168 164 160 157 154 151 149 147 145 143 142 141 140 139 138 138 138 138 138 139 139 140 141 142 144 145 147 149 152 154 157 160 164 167 172 176 181 187 193 200 208 217 225 231 237 241 245 249 252 254 257 259 260 262 264 265 266 268 269 271 272 274 275 277 279 282

287 289 291 292 293 293 294 294 294 294 293 292 292 291 289 287 285 283 280 276 272 266 259 251 242 230 220 211 203 195 189 183 178 173 169 165 161 158 155 153 150 148 146 145 143 142 141 140 140 139 139 139 139 140 140 141 142 143 145 147 148 150 153 155 158 161 165 169 173 178 183 188 195 202 210 219 227 234 239 244 248 252 255 257 259 261 263 265 266 268 269 270 272 273 275 276 278 280 282 284

289 291 293 294 295 296 296 296 296 296 296 295 294 293 292 290 288 285 282 279 274 268 262 253 243 232 221 212 204 197 190 184 179 174 170 166 163 159 157 154 152 150 148 146 145 144 143 142 141 141 141 141 141 141 142 143 144 145 146 148 150 152 154 157 159 163 166 170 174 179 184 190 196 204 212 221 229 236 241 246 250 254 257 260 262 264 266 267 269 270 271 273 274 275 277 278 280 281 284 286

290 292 294 296 297 297 298 298 298 298 297 297 296 295 294 292 290 287 284 281 276 270 263 255 245 233 223 214 206 198 192 186 181 176 172 168 164 161 158 155 153 151 149 148 146 145 144 143 143 142 142 142 143 143 144 144 145 147 148 149 151 153 156 158 161 164 168 171 176 180 186 191 198 205 213 223 231 237 243 248 252 256 259 262 264 266 268 269 271 272 273 274 276 277 278 280 281 283 285 287

291 294 296 297 298 299 299 299 300 299 299 299 298 297 295 294 292 289 286 282 278 272 265 257 247 235 224 215 207 200 193 187 182 177 173 169 166 163 160 157 155 153 151 150 148 147 146 145 145 144 144 144 144 145 145 146 147 148 150 151 153 155 157 160 163 166 169 173 177 182 187 193 199 207 215 224 232 239 245 250 254 258 261 263 266 268 269 271 272 274 275 276 277 278 280 281 283 284 286 289

293 295 297 298 299 300 300 301 301 301 300 300 299 298 297 295 293 291 287 284 279 273 266 258 248 236 226 217 208 201 195 189 184 179 175 171 168 164 162 159 157 155 153 152 150 149 148 147 147 146 146 146 146 147 147 148 149 150 152 153 155 157 159 162 165 168 171 175 179 184 189 195 201 208 217 226 234 241 246 251 256 259 262 265 267 269 271 272 274 275 276 277 278 280 281 282 284 285 287 290

293 296 297 299 300 301 301 302 302 302 301 301 300 299 298 296 294 292 288 285 280 274 268 259 249 237 227 218 210 203 196 191 186 181 177 173 170 167 164 161 159 157 155 154 152 151 150 150 149 149 148 149 149 149 150 151 152 153 154 156 157 159 161 164 167 170 173 177 181 185 191 196 203 210 218 227 235 242 248 253 257 261 264 266 268 270 272 273 275 276 277 278 279 281 282 283 285 286 288 291

294 296 298 299 301 301 302 302 302 302 302 301 301 300 299 297 295 292 289 286 281 275 269 260 251 239 229 220 212 205 198 193 188 183 179 175 172 169 166 164 162 160 158 156 155 154 153 152 152 151 151 151 151 152 152 153 154 155 157 158 160 162 164 166 169 172 175 179 183 187 192 198 204 212 220 229 237 243 249 254 258 262 265 267 269 271 273 274 276 277 278 279 280 281 283 284 285 287 289 291

295 297 299 300 301 302 302 303 303 303 302 302 301 300 299 297 296 293 290 286 282 276 270 261 252 240 230 221 213 206 200 195 190 185 181 178 174 171 169 166 164 162 161 159 158 157 156 155 154 154 154 154 154 155 155 156 157 158 159 161 163 165 167 169 172 174 178 181 185 190 194 200 206 213 221 230 238 245 250 255 259 263 266 268 270 272 274 275 276 278 279 280 281 282 283 285 286 288 290 292

295 297 299 300 301 302 303 303 303 303 303 302 302 301 299 298 296 294 291 287 282 277 270 262 253 241 231 223 215 208 202 197 192 188 184 180 177 174 172 169 167 165 164 162 161 160 159 158 157 157 157 157 157 158 158 159 160 161 163 164 166 168 170 172 174 177 180 184 188 192 197 202 208 215 223 232 239 246 252 256 260 264 267 269 271 273 274 276 277 278 279 280 282 283 284 285 286 288 290 292

295 297 299 300 301 302 303 303 303 303 303 302 302 301 300 298 296 294 291 287 283 278 271 263 254 243 233 224 217 210 204 199 195 190 187 183 180 178 175 173 171 169 167 166 164 163 162 161 161 160 160 160 161 161 162 163 164 165 166 168 169 171 173 175 178 180 183 187 190 194 199 204 210 217 225 233 241 247 253 257 261 264 267 270 272 274 275 276 278 279 280 281 282 283 284 285 287 288 290 292

295 297 299 300 301 302 303 303 303 303 303 302 302 301 299 298 296 294 291 288 283 278 272 264 255 244 234 226 219 213 207 202 198 194 190 187 184 181 179 176 174 173 171 169 168 167 166 165 164 164 164 164 164 165 166 166 167 169 170 171 173 175 177 179 181 184 187 190 193 197 202 207 212 219 226 235 242 248 254 258 262 265 268 270 272 274 276 277 278 279 280 281 282 283 284 286 287 289 290 293

295 297 299 300 301 302 302 303 303 303 302 302 301 300 299 298 296 294 291 288 283 278 272 265 256 245 236 228 221 215 210 205 201 197 194 190 188 185 183 181 178 177 175 173 172 171 170 169 168 168 168 168 168 169 170 171 172 173 174 176 177 179 181 183 185 188 190 193 197 200 205 209 215 221 228 236 244 250 255 259 263 266 269 271 273 275 276 277 279 280 281 282 283 284 285 286 287 289 290 293

295 297 299 300 301 301 302 302 302 302 302 301 301 300 299 297 296 293 291 287 283 279 273 266 257 247 238 230 224 218 213 209 204 201 198 195 192 189 187 185 183 181 180 178 177 175 174 173 173 172 172 172 173 173 174 175 176 178 179 180 182 184 185 187 190 192 194 197 200 204 208 212 217 223 230 238 245 251 256 260 263 266 269 271 273 275 276 278 279 280 281 282 283 284 285 286 287 289 290 292

295 297 298 299 300 301 301 301 301 301 301 301 300 299 298 297 295 293 290 287 283 279 273 266 258 248 240 233 227 221 217 212 209 205 202 199 197 194 192 190 188 186 185 183 182 180 179 178 177 177 177 177 177 178 179 180 181 183 184 186 187 189 191 192 194 197 199 202 205 208 212 216 220 226 232 240 246 252 257 261 264 267 270 272 274 275 277 278 279 280 281 282 283 284 285 286 287 289 290 292

294 296 297 298 299 300 300 300 300 300 300 299 299 298 297 296 294 292 289 286 283 278 273 267 259 250 242 236 230 225 221 217 213 210 207 205 202 200 198 196 194 192 190 189 187 185 184 183 182 181 181 182 182 183 184 186 187 188 190 191 193 195 196 198 200 202 204 207 209 212 216 219 224 229 235 242 248 253 257 261 265 267 270 272 274 275 277 278 279 280 281 282 283 284 285 286 287 288 290 292

294 295 296 297 298 299 299 299 299 299 299 298 297 297 296 294 293 291 288 285 282 278 273 267 260 252 245 239 234 229 225 222 219 216 213 211 208 206 204 202 200 198 197 195 193 191 190 188 187 186 186 187 188 189 190 192 193 195 196 198 199 201 203 204 206 208 210 212 215 217 220 224 227 232 237 243 249 254 258 262 265 268 270 272 274 276 277 278 279 280 281 282 283 284 285 286 287 288 290 291

293 294 295 296 297 297 297 297 297 297 297 296 296 295 294 292 291 289 287 284 280 277 272 267 261 254 248 243 238 235 231 228 225 222 220 217 215 213 211 209 207 205 204 202 200 198 196 194 193 192 192 192 194 195 197 199 200 202 204 205 207 208 210 212 213 215 217 219 221 223 226 229 232 235 240 245 250 255 259 262 265 268 270 272 274 276 277 278 279 280 281 282 283 284 285 286 287 288 289 291

292 293 294 294 295 295 295 295 295 295 295 294 293 293 291 290 289 287 284 282 279 275 271 266 261 256 252 248 244 241 237 235 232 230 227 225 223 221 219 217 215 213 211 209 207 205 203 201 198 197 197 198 200 202 204 206 208 210 212 213 215 217 218 220 221 223 225 226 228 230 232 235 237 240 243 246 251 255 259 262 265 268 270 272 274 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290

260 257 254 251 248 245 243 240 238 236 234 232 230 228 226 224 222 220 218 216 213 211 208 204 201

264 262 259 257 254 252 249 247 245 243 241 240 238 236 234 232 230 228 226 223 220 217 213 209 204

268 266 264 262 259 257 255 253 251 250 248 246 244 243 241 239 237 235 232 229 226 222 218 213 207

272 270 268 266 264 262 260 258 257 255 253 252 250 248 246 244 242 240 237 234 231 227 222 217 210

276 274 272 270 268 266 264 263 261 259 258 256 255 253 251 249 247 245 242 239 235 231 226 220 213

279 277 275 273 271 269 268 266 265 263 262 260 259 257 255 253 251 249 246 242 239 234 229 222 215

282 279 277 275 274 272 271 269 268 266 265 264 262 260 259 257 254 252 249 245 241 237 231 225 217

284 282 280 278 276 275 273 272 270 269 268 266 265 263 261 259 257 255 252 248 244 239 234 227 219

286 284 281 280 278 277 275 274 273 271 270 269 267 266 264 262 260 257 254 250 246 241 236 229 221

287 285 283 281 280 278 277 276 274 273 272 271 269 268 266 264 262 259 256 252 248 243 237 231 223

289 286 284 283 281 280 278 277 276 275 274 272 271 269 268 266 263 261 258 254 250 245 239 232 224

290 287 285 284 282 281 280 278 277 276 275 274 272 271 269 267 265 262 259 256 251 246 241 234 226

291 288 286 285 283 282 281 279 278 277 276 275 273 272 270 268 266 264 261 257 253 248 242 235 227

291 289 287 285 284 283 281 280 279 278 277 276 274 273 271 269 267 265 262 258 254 249 243 237 229

292 290 288 286 285 283 282 281 280 279 278 276 275 274 272 270 268 266 263 259 255 250 245 238 230

292 290 288 286 285 284 283 282 280 279 278 277 276 274 273 271 269 267 264 260 256 252 246 239 232

292 290 288 287 285 284 283 282 281 280 279 278 276 275 274 272 270 267 264 261 257 253 247 241 233

293 290 289 287 286 284 283 282 281 280 279 278 277 276 274 272 270 268 265 262 258 254 248 242 235

293 290 289 287 286 285 284 283 282 281 280 279 277 276 275 273 271 269 266 263 259 255 250 244 236

292 290 289 287 286 285 284 283 282 281 280 279 278 276 275 273 271 269 266 263 260 256 251 245 238

292 290 289 287 286 285 284 283 282 281 280 279 278 277 275 274 272 270 267 264 261 257 252 246 240

292 290 288 287 286 285 284 283 282 281 280 279 278 277 275 274 272 270 267 265 261 257 253 248 242

291 290 288 287 286 285 284 283 282 281 280 279 278 277 276 274 272 270 268 265 262 258 254 249 243

291 289 288 287 286 285 284 283 282 281 280 279 278 277 276 274 272 270 268 265 262 259 255 250 245

290 289 288 287 286 285 284 283 282 281 280 279 278 277 276 274 272 270 268 265 262 259 255 251 246

Figure 2.12: Results For Full Eclipse (Scenario 1), ϕΦ = [π4

0 π4]T


296 281 266 253 241 231 222 214 207 201 197 193 190 189 188 188 190 192 195 199 203 209 216 223 230

301 283 268 254 242 231 222 214 207 201 197 193 191 189 188 188 190 192 195 199 204 210 216 224 232

305 286 269 255 243 232 222 214 207 202 197 193 191 189 188 189 190 192 195 199 204 210 217 226 235

310 289 271 256 244 232 223 215 208 202 197 194 191 189 189 189 190 192 196 200 205 211 219 228 238

313 292 273 258 245 233 223 215 208 202 197 194 191 189 189 189 190 193 196 200 206 212 220 230 240

316 294 275 259 246 234 224 216 208 202 198 194 191 190 189 189 191 193 196 201 207 213 222 231 243

318 296 277 260 247 235 225 216 209 203 198 194 192 190 189 190 191 193 197 202 207 214 223 233 244

320 297 278 261 247 235 225 216 209 203 198 194 192 190 189 190 191 194 197 202 208 215 224 234 246

322 298 279 262 248 236 226 217 209 203 198 195 192 190 190 190 192 194 198 202 208 216 225 235 247

323 299 280 263 249 236 226 217 210 203 199 195 192 190 190 190 192 194 198 203 209 216 225 236 248

323 300 280 263 249 237 226 217 210 204 199 195 192 191 190 190 192 195 198 203 209 217 226 236 249

324 300 280 263 249 237 226 217 210 204 199 195 192 191 190 190 192 195 198 203 209 217 226 236 249

324 300 280 264 249 237 226 217 210 204 199 195 192 191 190 191 192 195 198 203 209 217 226 237 249

324 300 280 263 249 237 226 217 210 204 199 195 192 191 190 190 192 195 198 203 209 217 226 236 249

323 300 280 263 249 237 226 217 210 204 199 195 192 191 190 190 192 195 198 203 209 217 226 236 249

323 299 280 263 249 236 226 217 210 203 199 195 192 190 190 190 192 194 198 203 209 216 225 236 248

322 298 279 262 248 236 226 217 209 203 198 195 192 190 190 190 192 194 198 202 208 216 225 235 247

320 297 278 261 247 235 225 216 209 203 198 194 192 190 189 190 191 194 197 202 208 215 224 234 246

318 296 277 260 247 235 225 216 209 203 198 194 192 190 189 190 191 193 197 202 207 214 223 233 244

316 294 275 259 246 234 224 216 208 202 198 194 191 190 189 189 191 193 196 201 207 213 222 231 243

313 292 273 258 245 233 223 215 208 202 197 194 191 189 189 189 190 193 196 200 206 212 220 230 240

310 289 271 256 244 232 223 215 208 202 197 194 191 189 189 189 190 192 196 200 205 211 219 228 238

305 286 269 255 243 232 222 214 207 202 197 193 191 189 188 189 190 192 195 199 204 210 217 226 235

301 283 268 254 242 231 222 214 207 201 197 193 191 189 188 188 190 192 195 199 204 210 216 224 232

296 281 266 253 241 231 222 214 207 201 197 193 190 189 188 188 190 192 195 199 203 209 216 223 230

311 321 329 335 340 343 346 348 350 351 351 352 352 352 351 351 350 348 346 343 340 335 329 321 311 296 281 266 253 241 231 222 214 207 201 197 193 190 189 188 188 190 192 195 199 203 209 216 223 230 237 242 247 250 254 256 258 260 261 262 263 264 264 264 263 262 261 260 258 256 254 250 247 242 237 230 223 216 209 203 199 195 192 190 188 188 189 190 193 197 201 207 214 222 231 241 253 266 281 296

321 336 346 354 359 364 367 369 371 372 373 373 373 373 373 372 371 369 367 364 359 354 346 336 321 301 283 268 254 242 231 222 214 207 201 197 193 191 189 188 188 190 192 195 199 204 210 216 224 232 242 250 256 260 264 267 270 272 273 275 275 276 276 276 275 275 273 272 270 267 264 260 256 250 242 232 224 216 210 204 199 195 192 190 188 188 189 191 193 197 201 207 214 222 231 242 254 268 283 301

329 346 358 367 374 379 382 385 387 388 389 389 389 389 389 388 387 385 382 379 374 367 358 346 329 305 286 269 255 243 232 222 214 207 202 197 193 191 189 188 189 190 192 195 199 204 210 217 226 235 247 256 263 268 273 276 279 281 283 284 285 286 286 286 285 284 283 281 279 276 273 268 263 256 247 235 226 217 210 204 199 195 192 190 189 188 189 191 193 197 202 207 214 222 232 243 255 269 286 305

335 354 367 377 385 390 394 396 398 400 400 401 401 401 400 400 398 396 394 390 385 377 367 354 335 310 289 271 256 244 232 223 215 208 202 197 194 191 189 189 189 190 192 196 200 205 211 219 228 238 250 260 268 275 280 284 287 289 291 292 293 294 294 294 293 292 291 289 287 284 280 275 268 260 250 238 228 219 211 205 200 196 192 190 189 189 189 191 194 197 202 208 215 223 232 244 256 271 289 310

340 359 374 385 392 398 402 405 407 408 409 410 410 410 409 408 407 405 402 398 392 385 374 359 340 313 292 273 258 245 233 223 215 208 202 197 194 191 189 189 189 190 193 196 200 206 212 220 230 240 254 264 273 280 285 289 293 295 297 299 300 300 300 300 300 299 297 295 293 289 285 280 273 264 254 240 230 220 212 206 200 196 193 190 189 189 189 191 194 197 202 208 215 223 233 245 258 273 292 313

343 364 379 390 398 404 408 411 413 414 415 416 416 416 415 414 413 411 408 404 398 390 379 364 343 316 294 275 259 246 234 224 216 208 202 198 194 191 190 189 189 191 193 196 201 207 213 222 231 243 256 267 276 284 289 294 297 300 302 304 305 305 306 305 305 304 302 300 297 294 289 284 276 267 256 243 231 222 213 207 201 196 193 191 189 189 190 191 194 198 202 208 216 224 234 246 259 275 294 316

346 367 382 394 402 408 412 415 417 419 420 420 420 420 420 419 417 415 412 408 402 394 382 367 346 318 296 277 260 247 235 225 216 209 203 198 194 192 190 189 190 191 193 197 202 207 214 223 233 244 258 270 279 287 293 297 301 304 306 308 309 310 310 310 309 308 306 304 301 297 293 287 279 270 258 244 233 223 214 207 202 197 193 191 190 189 190 192 194 198 203 209 216 225 235 247 260 277 296 318

348 369 385 396 405 411 415 418 420 422 423 423 424 423 423 422 420 418 415 411 405 396 385 369 348 320 297 278 261 247 235 225 216 209 203 198 194 192 190 189 190 191 194 197 202 208 215 224 234 246 260 272 281 289 295 300 304 307 309 311 312 313 313 313 312 311 309 307 304 300 295 289 281 272 260 246 234 224 215 208 202 197 194 191 190 189 190 192 194 198 203 209 216 225 235 247 261 278 297 320

350 371 387 398 407 413 417 420 423 424 425 426 426 426 425 424 423 420 417 413 407 398 387 371 350 322 298 279 262 248 236 226 217 209 203 198 195 192 190 190 190 192 194 198 202 208 216 225 235 247 261 273 283 291 297 302 306 309 312 313 314 315 315 315 314 313 312 309 306 302 297 291 283 273 261 247 235 225 216 208 202 198 194 192 190 190 190 192 195 198 203 209 217 226 236 248 262 279 298 322

351 372 388 400 408 414 419 422 424 426 427 427 427 427 427 426 424 422 419 414 408 400 388 372 351 323 299 280 263 249 236 226 217 210 203 199 195 192 190 190 190 192 194 198 203 209 216 225 236 248 262 275 284 292 299 304 308 311 313 315 316 317 317 317 316 315 313 311 308 304 299 292 284 275 262 248 236 225 216 209 203 198 194 192 190 190 190 192 195 199 203 210 217 226 236 249 263 280 299 323

351 373 389 400 409 415 420 423 425 427 428 428 428 428 428 427 425 423 420 415 409 400 389 373 351 323 300 280 263 249 237 226 217 210 204 199 195 192 191 190 190 192 195 198 203 209 217 226 236 249 263 275 285 293 300 305 309 312 314 316 317 318 318 318 317 316 314 312 309 305 300 293 285 275 263 249 236 226 217 209 203 198 195 192 190 190 191 192 195 199 204 210 217 226 237 249 263 280 300 323

352 373 389 401 410 416 420 423 426 427 428 429 429 429 428 427 426 423 420 416 410 401 389 373 352 324 300 280 263 249 237 226 217 210 204 199 195 192 191 190 190 192 195 198 203 209 217 226 236 249 264 276 286 294 300 305 310 313 315 317 318 319 319 319 318 317 315 313 310 305 300 294 286 276 264 249 236 226 217 209 203 198 195 192 190 190 191 192 195 199 204 210 217 226 237 249 263 280 300 324

352 373 389 401 410 416 420 424 426 427 428 429 429 429 428 427 426 424 420 416 410 401 389 373 352 324 300 280 264 249 237 226 217 210 204 199 195 192 191 190 191 192 195 198 203 209 217 226 237 249 264 276 286 294 300 306 310 313 315 317 318 319 319 319 318 317 315 313 310 306 300 294 286 276 264 249 237 226 217 209 203 198 195 192 191 190 191 192 195 199 204 210 217 226 237 249 264 280 300 324

352 373 389 401 410 416 420 423 426 427 428 429 429 429 428 427 426 423 420 416 410 401 389 373 352 324 300 280 263 249 237 226 217 210 204 199 195 192 191 190 190 192 195 198 203 209 217 226 236 249 264 276 286 294 300 305 310 313 315 317 318 319 319 319 318 317 315 313 310 305 300 294 286 276 264 249 236 226 217 209 203 198 195 192 190 190 191 192 195 199 204 210 217 226 237 249 263 280 300 324

351 373 389 400 409 415 420 423 425 427 428 428 428 428 428 427 425 423 420 415 409 400 389 373 351 323 300 280 263 249 237 226 217 210 204 199 195 192 191 190 190 192 195 198 203 209 217 226 236 249 263 275 285 293 300 305 309 312 314 316 317 318 318 318 317 316 314 312 309 305 300 293 285 275 263 249 236 226 217 209 203 198 195 192 190 190 191 192 195 199 204 210 217 226 237 249 263 280 300 323

351 372 388 400 408 414 419 422 424 426 427 427 427 427 427 426 424 422 419 414 408 400 388 372 351 323 299 280 263 249 236 226 217 210 203 199 195 192 190 190 190 192 194 198 203 209 216 225 236 248 262 275 284 292 299 304 308 311 313 315 316 317 317 317 316 315 313 311 308 304 299 292 284 275 262 248 236 225 216 209 203 198 194 192 190 190 190 192 195 199 203 210 217 226 236 249 263 280 299 323

350 371 387 398 407 413 417 420 423 424 425 426 426 426 425 424 423 420 417 413 407 398 387 371 350 322 298 279 262 248 236 226 217 209 203 198 195 192 190 190 190 192 194 198 202 208 216 225 235 247 261 273 283 291 297 302 306 309 312 313 314 315 315 315 314 313 312 309 306 302 297 291 283 273 261 247 235 225 216 208 202 198 194 192 190 190 190 192 195 198 203 209 217 226 236 248 262 279 298 322

348 369 385 396 405 411 415 418 420 422 423 423 424 423 423 422 420 418 415 411 405 396 385 369 348 320 297 278 261 247 235 225 216 209 203 198 194 192 190 189 190 191 194 197 202 208 215 224 234 246 260 272 281 289 295 300 304 307 309 311 312 313 313 313 312 311 309 307 304 300 295 289 281 272 260 246 234 224 215 208 202 197 194 191 190 189 190 192 194 198 203 209 216 225 235 247 261 278 297 320

346 367 382 394 402 408 412 415 417 419 420 420 420 420 420 419 417 415 412 408 402 394 382 367 346 318 296 277 260 247 235 225 216 209 203 198 194 192 190 189 190 191 193 197 202 207 214 223 233 244 258 270 279 287 293 297 301 304 306 308 309 310 310 310 309 308 306 304 301 297 293 287 279 270 258 244 233 223 214 207 202 197 193 191 190 189 190 192 194 198 203 209 216 225 235 247 260 277 296 318

343 364 379 390 398 404 408 411 413 414 415 416 416 416 415 414 413 411 408 404 398 390 379 364 343 316 294 275 259 246 234 224 216 208 202 198 194 191 190 189 189 191 193 196 201 207 213 222 231 243 256 267 276 284 289 294 297 300 302 304 305 305 306 305 305 304 302 300 297 294 289 284 276 267 256 243 231 222 213 207 201 196 193 191 189 189 190 191 194 198 202 208 216 224 234 246 259 275 294 316

340 359 374 385 392 398 402 405 407 408 409 410 410 410 409 408 407 405 402 398 392 385 374 359 340 313 292 273 258 245 233 223 215 208 202 197 194 191 189 189 189 190 193 196 200 206 212 220 230 240 254 264 273 280 285 289 293 295 297 299 300 300 300 300 300 299 297 295 293 289 285 280 273 264 254 240 230 220 212 206 200 196 193 190 189 189 189 191 194 197 202 208 215 223 233 245 258 273 292 313

335 354 367 377 385 390 394 396 398 400 400 401 401 401 400 400 398 396 394 390 385 377 367 354 335 310 289 271 256 244 232 223 215 208 202 197 194 191 189 189 189 190 192 196 200 205 211 219 228 238 250 260 268 275 280 284 287 289 291 292 293 294 294 294 293 292 291 289 287 284 280 275 268 260 250 238 228 219 211 205 200 196 192 190 189 189 189 191 194 197 202 208 215 223 232 244 256 271 289 310

329 346 358 367 374 379 382 385 387 388 389 389 389 389 389 388 387 385 382 379 374 367 358 346 329 305 286 269 255 243 232 222 214 207 202 197 193 191 189 188 189 190 192 195 199 204 210 217 226 235 247 256 263 268 273 276 279 281 283 284 285 286 286 286 285 284 283 281 279 276 273 268 263 256 247 235 226 217 210 204 199 195 192 190 189 188 189 191 193 197 202 207 214 222 232 243 255 269 286 305

321 336 346 354 359 364 367 369 371 372 373 373 373 373 373 372 371 369 367 364 359 354 346 336 321 301 283 268 254 242 231 222 214 207 201 197 193 191 189 188 188 190 192 195 199 204 210 216 224 232 242 250 256 260 264 267 270 272 273 275 275 276 276 276 275 275 273 272 270 267 264 260 256 250 242 232 224 216 210 204 199 195 192 190 188 188 189 191 193 197 201 207 214 222 231 242 254 268 283 301

311 321 329 335 340 343 346 348 350 351 351 352 352 352 351 351 350 348 346 343 340 335 329 321 311 296 281 266 253 241 231 222 214 207 201 197 193 190 189 188 188 190 192 195 199 203 209 216 223 230 237 242 247 250 254 256 258 260 261 262 263 264 264 264 263 262 261 260 258 256 254 250 247 242 237 230 223 216 209 203 199 195 192 190 188 188 189 190 193 197 201 207 214 222 231 241 253 266 281 296

296 281 266 253 241 231 222 214 207 201 197 193 190 189 188 188 190 192 195 199 203 209 216 223 230

301 283 268 254 242 231 222 214 207 201 197 193 191 189 188 188 190 192 195 199 204 210 216 224 232

305 286 269 255 243 232 222 214 207 202 197 193 191 189 188 189 190 192 195 199 204 210 217 226 235

310 289 271 256 244 232 223 215 208 202 197 194 191 189 189 189 190 192 196 200 205 211 219 228 238

313 292 273 258 245 233 223 215 208 202 197 194 191 189 189 189 190 193 196 200 206 212 220 230 240

316 294 275 259 246 234 224 216 208 202 198 194 191 190 189 189 191 193 196 201 207 213 222 231 243

318 296 277 260 247 235 225 216 209 203 198 194 192 190 189 190 191 193 197 202 207 214 223 233 244

320 297 278 261 247 235 225 216 209 203 198 194 192 190 189 190 191 194 197 202 208 215 224 234 246

322 298 279 262 248 236 226 217 209 203 198 195 192 190 190 190 192 194 198 202 208 216 225 235 247

323 299 280 263 249 236 226 217 210 203 199 195 192 190 190 190 192 194 198 203 209 216 225 236 248

323 300 280 263 249 237 226 217 210 204 199 195 192 191 190 190 192 195 198 203 209 217 226 236 249

324 300 280 263 249 237 226 217 210 204 199 195 192 191 190 190 192 195 198 203 209 217 226 236 249

324 300 280 264 249 237 226 217 210 204 199 195 192 191 190 191 192 195 198 203 209 217 226 237 249

324 300 280 263 249 237 226 217 210 204 199 195 192 191 190 190 192 195 198 203 209 217 226 236 249

323 300 280 263 249 237 226 217 210 204 199 195 192 191 190 190 192 195 198 203 209 217 226 236 249

323 299 280 263 249 236 226 217 210 203 199 195 192 190 190 190 192 194 198 203 209 216 225 236 248

322 298 279 262 248 236 226 217 209 203 198 195 192 190 190 190 192 194 198 202 208 216 225 235 247

320 297 278 261 247 235 225 216 209 203 198 194 192 190 189 190 191 194 197 202 208 215 224 234 246

318 296 277 260 247 235 225 216 209 203 198 194 192 190 189 190 191 193 197 202 207 214 223 233 244

316 294 275 259 246 234 224 216 208 202 198 194 191 190 189 189 191 193 196 201 207 213 222 231 243

313 292 273 258 245 233 223 215 208 202 197 194 191 189 189 189 190 193 196 200 206 212 220 230 240

310 289 271 256 244 232 223 215 208 202 197 194 191 189 189 189 190 192 196 200 205 211 219 228 238

305 286 269 255 243 232 222 214 207 202 197 193 191 189 188 189 190 192 195 199 204 210 217 226 235

301 283 268 254 242 231 222 214 207 201 197 193 191 189 188 188 190 192 195 199 204 210 216 224 232

296 281 266 253 241 231 222 214 207 201 197 193 190 189 188 188 190 192 195 199 203 209 216 223 230

Figure 2.13: Results For Opposite Face Flux (Scenario 2)

340 335 331 327 324 322 319 318 316 315 314 313 313 312 312 311 311 310 310 309 309 308 307 306 306

338 331 326 322 318 315 313 311 309 308 307 306 305 305 304 304 303 303 303 302 302 302 301 302 302

337 329 322 317 313 310 308 306 304 302 301 300 299 299 298 298 297 297 297 297 297 297 297 297 299

336 327 320 314 310 306 303 301 299 298 296 295 295 294 293 293 293 292 292 292 292 293 293 294 295

335 326 318 312 307 303 300 298 296 294 293 291 291 290 289 289 289 288 288 289 289 289 290 291 293

335 325 316 310 305 301 297 295 293 291 289 288 287 287 286 286 285 285 285 286 286 287 287 289 290

334 324 315 309 303 299 295 292 290 288 287 286 285 284 283 283 283 283 283 283 283 284 285 287 288

334 323 314 307 302 297 294 291 288 286 285 283 282 282 281 281 280 280 281 281 281 282 283 285 287

333 322 313 306 301 296 292 289 287 285 283 282 281 280 279 279 279 279 279 279 280 281 282 283 285

333 322 313 306 300 295 291 288 285 283 281 280 279 278 278 277 277 277 277 278 278 279 281 282 284

332 321 312 305 299 294 290 287 284 282 280 279 278 277 276 276 276 276 276 276 277 278 279 281 283

332 321 312 304 298 293 289 286 283 281 279 278 277 276 275 275 275 275 275 275 276 277 278 280 282

331 320 311 304 298 293 288 285 282 280 278 277 276 275 274 274 274 274 274 275 275 276 278 279 281

331 320 311 303 297 292 288 284 282 279 278 276 275 274 274 273 273 273 273 274 275 276 277 279 281

330 319 310 303 296 291 287 284 281 279 277 276 274 274 273 273 272 272 273 273 274 275 276 278 280

330 319 309 302 296 291 287 283 281 278 276 275 274 273 272 272 272 272 272 273 273 274 276 277 279

329 318 309 301 295 290 286 283 280 278 276 275 273 273 272 272 271 271 272 272 273 274 275 277 279

328 317 308 301 295 290 286 282 280 277 276 274 273 272 272 271 271 271 271 272 272 273 275 276 278

327 316 307 300 294 289 285 282 279 277 275 274 273 272 271 271 271 271 271 271 272 273 274 276 278

326 315 306 299 294 289 285 282 279 277 275 273 272 271 271 270 270 270 270 271 272 272 274 275 277

324 314 305 299 293 288 284 281 279 276 275 273 272 271 271 270 270 270 270 271 271 272 273 275 276

322 312 304 298 292 288 284 281 278 276 274 273 272 271 270 270 270 270 270 270 271 272 273 274 276

320 311 303 297 292 287 284 281 278 276 274 273 272 271 270 270 269 269 270 270 270 271 272 273 275

317 309 302 296 291 287 283 280 278 276 274 273 271 271 270 269 269 269 269 270 270 271 272 273 274

315 308 301 296 291 287 283 280 278 275 274 272 271 270 270 269 269 269 269 269 270 271 271 272 273

347 348 348 348 348 348 347 347 347 347 346 346 346 345 345 344 343 342 341 339 337 335 332 327 322 315 308 301 296 291 287 283 280 278 275 274 272 271 270 270 269 269 269 269 269 270 271 271 272 273 275 276 277 278 279 280 280 281 281 282 283 283 284 285 286 287 288 289 291 293 295 298 301 304 307 311 313 315 316 317 318 318 319 320 320 321 321 322 323 323 324 325 326 328 330 332 334 337 341 344

353 355 357 357 358 358 358 358 358 357 357 357 357 356 356 355 354 353 352 350 347 344 340 335 327 317 309 302 296 291 287 283 280 278 276 274 273 271 271 270 269 269 269 269 270 270 271 272 273 274 276 278 279 280 281 282 282 283 284 284 285 286 286 287 288 289 290 292 293 295 297 300 303 306 310 314 318 320 322 324 325 326 326 327 328 328 329 329 330 331 331 332 333 335 336 338 340 343 346 349

358 361 363 364 365 365 366 366 366 366 366 365 365 365 364 363 362 361 360 358 355 351 347 340 332 320 311 303 297 292 287 284 281 278 276 274 273 272 271 270 270 269 269 270 270 270 271 272 273 275 277 279 280 282 283 284 284 285 286 286 287 288 288 289 290 291 292 293 295 297 299 302 304 308 312 318 322 325 327 329 330 331 332 333 333 334 334 335 336 336 337 338 339 340 342 343 345 348 351 354

362 366 368 369 370 371 372 372 372 372 372 372 371 371 370 370 369 368 366 364 361 357 351 344 335 322 312 304 298 292 288 284 281 278 276 274 273 272 271 270 270 270 270 270 270 271 272 273 274 276 278 280 282 283 284 285 286 287 287 288 289 289 290 291 292 293 294 295 297 298 301 303 306 310 314 320 325 328 331 333 334 335 336 337 338 338 339 340 340 341 341 342 343 344 346 347 349 351 354 358

365 369 372 373 375 375 376 376 377 377 377 377 376 376 375 375 374 372 370 368 365 361 355 347 337 324 314 305 299 293 288 284 281 279 276 275 273 272 271 271 270 270 270 270 271 271 272 273 275 276 279 281 283 284 285 286 287 288 289 289 290 291 291 292 293 294 295 296 298 300 302 305 308 312 316 322 327 331 334 336 337 339 340 341 341 342 343 343 344 344 345 346 346 348 349 350 352 354 357 361

368 372 374 376 378 379 379 380 380 380 380 380 380 380 379 378 377 376 374 371 368 364 358 350 339 326 315 306 299 294 289 285 282 279 277 275 273 272 271 271 270 270 270 270 271 272 272 274 275 277 280 282 284 285 286 287 288 289 290 291 291 292 293 293 294 295 296 298 299 301 303 306 309 313 318 324 329 333 336 338 340 341 343 343 344 345 345 346 346 347 348 348 349 350 351 353 354 357 360 363

370 374 377 379 380 381 382 382 383 383 383 383 383 382 382 381 380 378 376 374 370 366 360 352 341 327 316 307 300 294 289 285 282 279 277 275 274 273 272 271 271 271 271 271 271 272 273 274 276 278 280 282 284 286 287 288 289 290 291 292 292 293 294 294 295 296 297 299 300 302 304 307 310 314 319 326 331 335 338 340 342 343 345 346 346 347 347 348 349 349 350 350 351 352 353 355 356 359 361 365

372 375 378 380 382 383 384 384 385 385 385 385 385 384 384 383 382 380 378 376 372 368 361 353 342 328 317 308 301 295 290 286 282 280 277 276 274 273 272 272 271 271 271 271 272 272 273 275 276 278 281 283 285 287 288 289 290 291 292 292 293 294 294 295 296 297 298 299 301 303 305 308 311 315 320 327 332 336 339 342 344 345 346 347 348 349 349 350 350 351 351 352 353 354 355 356 358 360 363 367

373 377 379 382 383 384 385 386 386 386 386 386 386 386 385 384 383 382 380 377 374 369 362 354 343 329 318 309 301 295 290 286 283 280 278 276 275 273 273 272 272 271 271 272 272 273 274 275 277 279 281 284 286 287 289 290 291 292 292 293 294 295 295 296 297 298 299 300 302 304 306 309 312 316 321 328 333 337 340 343 345 346 348 348 349 350 350 351 351 352 353 353 354 355 356 357 359 361 364 368

374 377 380 383 384 385 386 387 387 387 387 387 387 387 386 386 384 383 381 378 375 370 363 355 344 330 319 309 302 296 291 287 283 281 278 276 275 274 273 272 272 272 272 272 273 273 274 276 277 279 282 284 286 288 289 291 292 292 293 294 295 295 296 297 298 298 300 301 302 304 307 309 313 317 322 329 334 338 341 344 346 347 348 349 350 351 351 352 352 353 353 354 355 356 357 358 360 362 365 369

374 378 381 383 385 386 387 388 388 388 388 388 388 388 387 386 385 384 382 379 375 370 364 356 345 330 319 310 303 296 291 287 284 281 279 277 276 274 274 273 273 272 272 273 273 274 275 276 278 280 283 285 287 289 290 291 292 293 294 295 295 296 297 297 298 299 300 302 303 305 307 310 313 317 323 329 334 339 342 344 346 348 349 350 351 352 352 353 353 354 354 355 356 356 357 359 361 363 366 369

375 379 382 384 385 387 387 388 388 389 389 389 389 388 388 387 386 384 382 380 376 371 365 356 345 331 320 311 303 297 292 288 284 282 279 278 276 275 274 274 273 273 273 273 274 275 276 277 279 281 283 286 288 289 291 292 293 294 295 295 296 297 297 298 299 300 301 302 304 306 308 311 314 318 323 330 335 339 342 345 347 349 350 351 352 352 353 353 354 354 355 355 356 357 358 359 361 363 366 370

375 379 382 384 386 387 388 388 389 389 389 389 389 389 388 387 386 385 383 380 376 371 365 357 346 331 320 311 304 298 293 288 285 282 280 278 277 276 275 274 274 274 274 274 275 275 276 278 279 281 284 286 288 290 291 293 294 294 295 296 297 297 298 299 300 300 301 303 304 306 308 311 314 319 324 330 335 340 343 345 347 349 350 351 352 353 353 354 354 355 355 356 356 357 358 360 361 364 366 370

375 379 382 384 386 387 388 389 389 389 389 389 389 389 388 387 386 385 383 380 377 372 365 357 346 332 321 312 304 298 293 289 286 283 281 279 278 277 276 275 275 275 275 275 275 276 277 278 280 282 285 287 289 291 292 293 294 295 296 297 297 298 299 299 300 301 302 303 305 307 309 312 315 319 324 331 336 340 343 346 348 349 351 352 352 353 354 354 355 355 356 356 357 358 359 360 362 364 367 370

375 379 382 384 386 387 388 389 389 389 389 389 389 389 388 387 386 385 383 380 377 372 366 357 346 332 321 312 305 299 294 290 287 284 282 280 279 278 277 276 276 276 276 276 276 277 278 279 281 283 286 288 290 292 293 294 295 296 297 298 298 299 300 300 301 302 303 304 306 307 310 312 315 320 325 331 336 340 344 346 348 350 351 352 353 353 354 354 355 355 356 356 357 358 359 360 362 364 367 370

375 379 382 384 386 387 388 389 389 389 389 389 389 389 388 387 386 385 383 380 377 372 366 357 347 333 322 313 306 300 295 291 288 285 283 281 280 279 278 278 277 277 277 277 278 278 279 281 282 284 287 289 291 293 294 295 296 297 298 298 299 300 300 301 302 303 304 305 306 308 310 313 316 320 325 331 337 341 344 346 348 350 351 352 353 354 354 355 355 355 356 356 357 358 359 360 362 364 367 370

375 379 382 384 386 387 388 388 389 389 389 389 389 388 388 387 386 385 383 380 377 372 366 358 347 333 322 313 306 301 296 292 289 287 285 283 282 281 280 279 279 279 279 279 279 280 281 282 283 285 288 290 292 294 295 296 297 298 299 300 300 301 301 302 303 304 305 306 307 309 311 314 317 321 326 332 337 341 344 347 349 350 351 352 353 354 354 355 355 356 356 357 357 358 359 360 362 364 367 370

375 379 382 384 385 386 387 388 388 389 389 389 388 388 388 387 386 384 382 380 376 372 366 358 347 334 323 314 307 302 297 294 291 288 286 285 283 282 282 281 281 280 280 281 281 281 282 283 285 287 289 292 293 295 296 298 299 299 300 301 302 302 303 303 304 305 306 307 308 310 312 314 317 321 326 332 337 341 344 347 349 350 351 352 353 354 354 355 355 356 356 357 357 358 359 360 362 364 367 370

375 378 381 383 385 386 387 387 388 388 388 388 388 387 387 386 385 384 382 379 376 372 366 358 347 334 324 315 309 303 299 295 292 290 288 287 286 285 284 283 283 283 283 283 283 283 284 285 287 288 291 293 295 297 298 299 300 301 302 302 303 304 304 305 306 306 307 308 310 311 313 315 318 322 327 333 338 342 345 347 349 350 352 352 353 354 354 355 355 356 356 357 357 358 359 360 362 364 366 370

374 378 381 383 384 385 386 386 387 387 387 387 387 387 386 385 384 383 381 379 375 371 365 358 348 335 325 316 310 305 301 297 295 293 291 289 288 287 287 286 286 285 285 285 286 286 287 287 289 290 293 295 297 298 300 301 302 303 304 304 305 306 306 307 307 308 309 310 311 312 314 316 319 323 327 333 338 342 345 347 349 351 352 353 353 354 354 355 355 356 356 357 357 358 359 360 361 363 366 369

373 377 380 382 383 384 385 385 386 386 386 386 386 385 385 384 383 382 380 378 375 370 365 358 348 335 326 318 312 307 303 300 298 296 294 293 291 291 290 289 289 289 288 288 289 289 289 290 291 293 295 297 299 301 302 303 304 305 306 307 307 308 308 309 310 310 311 312 313 314 316 318 321 324 328 334 339 342 345 347 349 351 352 353 353 354 354 355 355 356 356 357 357 358 359 360 361 363 366 369

373 376 378 380 382 383 383 384 384 384 384 384 384 384 383 383 382 380 379 376 373 369 364 357 348 336 327 320 314 310 306 303 301 299 298 296 295 295 294 293 293 293 292 292 292 292 293 293 294 295 298 300 302 303 305 306 307 308 309 309 310 311 311 312 312 313 314 314 315 316 318 320 322 325 329 335 339 343 345 348 349 351 352 353 353 354 354 355 355 356 356 356 357 358 359 360 361 363 365 368

371 375 377 378 380 381 381 382 382 382 382 382 382 382 381 380 379 378 377 374 372 368 363 357 348 337 329 322 317 313 310 308 306 304 302 301 300 299 299 298 298 297 297 297 297 297 297 297 297 299 301 303 304 306 308 309 310 311 312 313 313 314 314 315 315 316 317 317 318 319 321 322 324 327 330 335 340 343 345 348 349 351 352 352 353 354 354 355 355 355 356 356 357 358 358 359 361 362 364 367

370 373 375 376 377 378 378 379 379 379 379 379 379 379 378 377 377 375 374 372 369 366 361 355 348 338 331 326 322 318 315 313 311 309 308 307 306 305 305 304 304 303 303 303 302 302 302 301 302 302 304 306 308 310 312 313 314 315 316 317 317 318 319 319 320 320 321 321 322 323 324 325 327 329 332 336 340 343 346 348 349 351 352 352 353 354 354 355 355 355 356 356 357 357 358 359 360 362 364 366

368 370 371 373 373 374 375 375 375 375 375 375 375 375 374 374 373 372 370 368 365 362 358 353 347 340 335 331 327 324 322 319 318 316 315 314 313 313 312 312 311 311 310 310 309 309 308 307 306 306 307 310 312 314 316 318 319 320 321 322 323 323 324 324 325 325 326 326 327 327 328 329 330 332 334 337 340 343 346 348 349 350 351 352 353 354 354 354 355 355 356 356 357 357 358 359 360 362 363 365

344 341 337 334 332 330 328 326 325 324 323 323 322 321 321 320 320 319 318 318 317 316 315 313 311

349 346 343 340 338 336 335 333 332 331 331 330 329 329 328 328 327 326 326 325 324 322 320 318 314

354 351 348 345 343 342 340 339 338 337 336 336 335 334 334 333 333 332 331 330 329 327 325 322 318

358 354 351 349 347 346 344 343 342 341 341 340 340 339 338 338 337 336 335 334 333 331 328 325 320

361 357 354 352 350 349 348 346 346 345 344 344 343 343 342 341 341 340 339 337 336 334 331 327 322

363 360 357 354 353 351 350 349 348 348 347 346 346 345 345 344 343 343 341 340 338 336 333 329 324

365 361 359 356 355 353 352 351 350 350 349 349 348 347 347 346 346 345 343 342 340 338 335 331 326

367 363 360 358 356 355 354 353 352 351 351 350 350 349 349 348 347 346 345 344 342 339 336 332 327

368 364 361 359 357 356 355 354 353 353 352 351 351 350 350 349 348 348 346 345 343 340 337 333 328

369 365 362 360 358 357 356 355 354 353 353 352 352 351 351 350 349 348 347 346 344 341 338 334 329

369 366 363 361 359 357 356 356 355 354 354 353 353 352 352 351 350 349 348 346 344 342 339 334 329

370 366 363 361 359 358 357 356 355 355 354 354 353 353 352 352 351 350 349 347 345 342 339 335 330

370 366 364 361 360 358 357 356 356 355 355 354 354 353 353 352 351 350 349 347 345 343 340 335 330

370 367 364 362 360 359 358 357 356 356 355 355 354 354 353 352 352 351 349 348 346 343 340 336 331

370 367 364 362 360 359 358 357 356 356 355 355 354 354 353 353 352 351 350 348 346 344 340 336 331

370 367 364 362 360 359 358 357 356 356 355 355 355 354 354 353 352 351 350 348 346 344 341 337 331

370 367 364 362 360 359 358 357 357 356 356 355 355 354 354 353 352 351 350 349 347 344 341 337 332

370 367 364 362 360 359 358 357 357 356 356 355 355 354 354 353 352 351 350 349 347 344 341 337 332

370 366 364 362 360 359 358 357 357 356 356 355 355 354 354 353 352 352 350 349 347 345 342 338 333

369 366 363 361 360 359 358 357 357 356 356 355 355 354 354 353 353 352 351 349 347 345 342 338 333

369 366 363 361 360 359 358 357 357 356 356 355 355 354 354 353 353 352 351 349 347 345 342 339 334

368 365 363 361 360 359 358 357 356 356 356 355 355 354 354 353 353 352 351 349 348 345 343 339 335

367 364 362 361 359 358 358 357 356 356 355 355 355 354 354 353 352 352 351 349 348 345 343 340 335

366 364 362 360 359 358 357 357 356 356 355 355 355 354 354 353 352 352 351 349 348 346 343 340 336

365 363 362 360 359 358 357 357 356 356 355 355 354 354 354 353 352 351 350 349 348 346 343 340 337

Figure 2.14: Results For Opposite Face Flux (Scenario 2), ϕΦ = [π4

0 π4]T


335 323 313 304 296 289 283 278 273 269 265 262 258 255 252 249 246 243 239 236 232 228 224 220 216

333 319 306 296 287 280 273 267 262 257 253 249 246 242 239 236 233 229 226 223 219 216 213 210 207

333 315 301 290 280 272 264 258 252 247 243 238 234 231 227 224 221 218 215 211 208 205 203 200 198

332 313 298 285 274 265 257 250 244 238 233 229 225 221 217 214 211 207 204 201 198 196 193 191 189

331 311 295 281 269 259 251 243 237 231 225 221 216 212 208 205 201 198 195 192 189 187 184 182 181

330 309 292 278 265 255 246 238 231 224 219 213 209 204 200 197 193 190 187 184 181 179 176 174 173

330 308 290 275 262 251 241 233 225 218 212 207 202 197 193 189 186 182 179 176 174 171 169 167 165

329 307 288 272 259 247 237 228 220 213 207 201 196 191 187 183 179 176 172 169 167 164 162 160 158

328 305 286 270 256 244 234 224 216 209 202 196 191 186 181 177 173 169 166 163 160 158 155 153 151

327 304 285 268 254 242 231 221 212 205 198 192 186 181 176 172 168 164 160 157 154 152 149 147 145

326 303 283 267 252 239 228 218 209 201 194 187 182 176 171 167 162 159 155 152 149 146 144 142 140

326 302 282 265 250 237 225 215 206 198 190 184 178 172 167 162 158 154 150 147 144 141 139 137 134

325 301 281 263 248 235 223 213 203 195 187 180 174 168 163 158 154 150 146 143 140 137 134 132 130

323 300 279 262 246 233 221 210 201 192 184 177 171 165 159 155 150 146 142 139 135 132 130 127 125

322 298 278 260 245 231 219 208 198 190 182 174 168 162 156 151 147 142 138 135 132 129 126 124 121

321 297 276 259 243 229 217 206 196 187 179 172 165 159 153 148 143 139 135 131 128 125 122 120 118

319 295 275 257 241 228 215 204 194 185 177 170 163 156 151 145 141 136 132 128 125 122 119 117 114

317 293 273 255 240 226 213 202 192 183 175 167 160 154 148 143 138 133 129 126 122 119 116 114 112

315 291 271 253 238 224 212 201 191 181 173 165 158 152 146 141 136 131 127 123 120 117 114 111 109

312 289 269 251 236 222 210 199 189 180 171 164 156 150 144 138 133 129 125 121 117 114 112 109 107

309 286 266 249 234 221 209 197 187 178 170 162 155 148 142 137 131 127 123 119 115 112 110 107 105

304 283 264 247 232 219 207 196 186 177 168 160 153 147 140 135 130 125 121 117 114 110 108 105 103

300 279 261 245 231 218 206 195 185 175 167 159 152 145 139 133 128 123 119 115 112 109 106 104 102

294 276 259 244 229 217 205 194 184 174 166 158 150 144 138 132 127 122 118 114 111 108 105 103 101

290 273 257 242 228 216 204 193 183 173 165 157 149 143 136 131 125 121 116 113 109 106 104 102 100

349 353 355 356 356 356 356 356 356 355 354 354 353 352 351 349 348 346 343 340 336 331 324 316 305 290 273 257 242 228 215 203 192 182 173 164 156 148 142 135 130 124 120 115 112 108 105 103 101 100 99 100 100 102 103 105 107 110 113 116 119 123 128 133 138 144 150 157 164 172 180 188 197 207 216 224 230 236 241 246 250 254 258 261 264 267 270 273 276 279 283 287 291 296 301 308 315 323 332 341

360 366 371 373 375 376 376 376 376 376 376 375 374 374 373 371 370 367 365 361 356 350 342 331 316 294 275 258 242 228 215 203 192 182 172 163 155 148 141 135 129 123 119 114 111 107 104 102 100 99 98 99 99 100 102 103 105 108 111 114 118 122 126 131 137 142 149 156 163 171 179 188 198 208 218 230 239 246 252 258 262 266 270 273 277 280 282 285 288 291 294 298 302 306 311 317 323 331 339 349

368 377 382 386 388 390 391 391 392 392 391 391 390 390 389 387 386 384 381 377 371 364 355 342 324 299 278 260 243 229 215 203 192 181 172 163 155 147 140 134 128 123 118 114 110 107 104 101 100 98 98 98 98 99 100 102 104 107 109 113 116 121 125 130 136 141 148 155 162 170 179 188 198 209 221 234 245 254 261 267 272 276 280 284 287 290 292 295 298 301 304 307 310 314 319 324 331 338 346 356

375 385 391 395 398 400 402 402 403 403 403 403 402 401 401 399 398 395 392 388 382 375 364 350 330 303 281 262 245 229 216 203 192 181 171 162 154 147 140 133 127 122 117 113 109 106 103 101 99 98 97 97 97 98 99 101 103 105 108 111 115 119 124 129 135 141 147 154 162 170 179 189 199 211 224 238 250 260 268 274 280 284 288 292 295 298 300 303 306 308 311 314 317 321 325 330 336 343 352 362

381 390 397 402 406 408 409 410 411 411 411 411 411 410 409 408 406 404 401 396 390 382 371 355 335 307 284 264 246 230 216 203 192 181 171 162 154 146 139 133 127 121 117 112 109 105 102 100 98 97 96 96 96 97 98 100 102 104 107 110 114 118 123 128 134 140 146 154 162 170 179 189 200 212 226 241 254 265 273 280 286 291 295 298 301 304 307 309 312 314 317 320 323 326 330 335 341 348 356 367

385 395 402 407 411 413 415 416 417 417 417 417 417 416 415 414 413 410 407 402 396 388 376 360 338 310 286 265 247 231 217 204 192 181 171 162 154 146 139 132 126 121 116 112 108 105 102 99 98 96 95 95 95 96 97 99 101 103 106 110 113 117 122 127 133 139 146 153 161 170 180 190 201 214 228 244 257 268 277 285 291 296 300 304 307 309 312 314 317 319 321 324 327 330 334 339 345 352 360 371

388 398 405 411 415 417 419 420 421 421 422 422 421 421 420 419 417 415 411 407 400 391 379 363 341 312 288 267 248 232 217 204 192 181 171 162 153 146 138 132 126 121 116 111 108 104 101 99 97 96 95 95 95 95 97 98 100 103 105 109 113 117 122 127 133 139 146 153 161 170 180 191 202 215 230 246 260 271 281 288 294 300 304 308 311 313 316 318 320 323 325 328 330 334 338 342 348 354 363 374

390 400 408 413 417 420 422 423 424 425 425 425 424 424 423 422 420 418 414 410 403 394 382 365 343 314 289 268 249 233 218 205 192 181 171 162 153 145 138 132 126 120 115 111 107 104 101 99 97 95 94 94 94 95 96 97 99 102 105 108 112 116 121 126 132 138 145 153 161 170 180 191 203 217 231 248 262 274 283 291 297 303 307 311 314 317 319 321 323 326 328 330 333 336 340 344 350 357 365 376

392 402 410 415 419 422 424 425 426 427 427 427 427 426 425 424 422 420 417 412 405 396 384 367 345 315 290 269 250 233 218 205 193 181 171 162 153 145 138 131 125 120 115 111 107 103 101 98 96 95 94 94 94 94 95 97 99 101 104 108 111 116 121 126 132 138 145 153 161 171 181 192 204 217 232 250 264 275 285 293 300 305 309 313 316 319 321 324 326 328 330 332 335 338 342 346 352 358 367 378

393 403 411 417 420 423 425 427 428 428 428 428 428 428 427 426 424 422 418 413 407 398 385 368 346 316 291 270 251 234 219 205 193 181 171 162 153 145 138 131 125 120 115 110 107 103 100 98 96 95 94 93 93 94 95 97 99 101 104 107 111 115 120 126 132 138 145 153 161 171 181 192 204 218 233 251 265 277 287 295 301 307 311 315 318 321 323 325 327 329 332 334 336 340 343 347 353 360 368 379

394 404 412 417 421 424 426 428 428 429 429 429 429 429 428 427 425 423 419 414 408 399 386 369 347 317 292 270 251 234 219 205 193 182 171 162 153 145 138 131 125 120 115 110 106 103 100 98 96 94 93 93 93 94 95 96 98 101 104 107 111 115 120 125 131 138 145 153 161 171 181 192 205 219 234 251 266 278 288 296 302 308 312 316 319 322 324 326 328 330 333 335 337 340 344 348 354 360 369 380

394 405 412 418 422 425 427 428 429 430 430 430 430 429 428 427 425 423 420 415 408 399 387 370 347 317 292 270 251 234 219 205 193 182 171 162 153 145 138 131 125 120 115 110 106 103 100 98 96 94 93 93 93 94 95 96 98 101 103 107 111 115 120 125 131 138 145 153 161 171 181 193 205 219 234 252 266 278 288 296 303 308 313 317 320 322 325 327 329 331 333 335 338 341 345 349 354 361 369 380

394 405 412 418 422 425 427 428 429 430 430 430 430 429 429 427 426 423 420 415 408 399 387 370 347 318 292 271 251 234 219 205 193 182 171 162 153 145 138 131 125 119 115 110 106 103 100 98 96 94 93 93 93 94 95 96 98 100 103 107 111 115 120 125 131 138 145 153 162 171 181 193 205 219 234 252 266 278 288 296 303 309 313 317 320 323 325 327 329 331 333 336 338 341 345 349 354 361 369 380

394 405 412 418 422 425 427 428 429 430 430 430 430 429 428 427 425 423 420 415 408 399 387 370 347 317 292 270 251 234 219 205 193 182 171 162 153 145 138 131 125 120 115 110 106 103 100 98 96 94 93 93 93 94 95 96 98 101 103 107 111 115 120 125 131 138 145 153 161 171 181 193 205 219 234 252 266 278 288 296 303 308 313 317 320 322 325 327 329 331 333 335 338 341 345 349 354 361 369 380

394 404 412 417 421 424 426 428 428 429 429 429 429 429 428 427 425 423 419 414 408 399 386 369 347 317 292 270 251 234 219 205 193 182 171 162 153 145 138 131 125 120 115 110 106 103 100 98 96 94 93 93 93 94 95 96 98 101 104 107 111 115 120 125 131 138 145 153 161 171 181 192 205 219 234 251 266 278 288 296 302 308 312 316 319 322 324 326 328 330 333 335 337 340 344 348 354 360 369 380

393 403 411 417 420 423 425 427 428 428 428 428 428 428 427 426 424 422 418 413 407 398 385 368 346 316 291 270 251 234 219 205 193 181 171 162 153 145 138 131 125 120 115 110 107 103 100 98 96 95 94 93 93 94 95 97 99 101 104 107 111 115 120 126 132 138 145 153 161 171 181 192 204 218 233 251 265 277 287 295 301 307 311 315 318 321 323 325 327 329 332 334 336 340 343 347 353 360 368 379

392 402 410 415 419 422 424 425 426 427 427 427 427 426 425 424 422 420 417 412 405 396 384 367 345 315 290 269 250 233 218 205 193 181 171 162 153 145 138 131 125 120 115 111 107 103 101 98 96 95 94 94 94 94 95 97 99 101 104 108 111 116 121 126 132 138 145 153 161 171 181 192 204 217 232 250 264 275 285 293 300 305 309 313 316 319 321 324 326 328 330 332 335 338 342 346 352 358 367 378

390 400 408 413 417 420 422 423 424 425 425 425 424 424 423 422 420 418 414 410 403 394 382 365 343 314 289 268 249 233 218 205 192 181 171 162 153 145 138 132 126 120 115 111 107 104 101 99 97 95 94 94 94 95 96 97 99 102 105 108 112 116 121 126 132 138 145 153 161 170 180 191 203 217 231 248 262 274 283 291 297 303 307 311 314 317 319 321 323 326 328 330 333 336 340 344 350 357 365 376

388 398 405 411 415 417 419 420 421 421 422 422 421 421 420 419 417 415 411 407 400 391 379 363 341 312 288 267 248 232 217 204 192 181 171 162 153 146 138 132 126 121 116 111 108 104 101 99 97 96 95 95 95 95 97 98 100 103 105 109 113 117 122 127 133 139 146 153 161 170 180 191 202 215 230 246 260 271 281 288 294 300 304 308 311 313 316 318 320 323 325 328 330 334 338 342 348 354 363 374

385 395 402 407 411 413 415 416 417 417 417 417 417 416 415 414 413 410 407 402 396 388 376 360 338 310 286 265 247 231 217 204 192 181 171 162 154 146 139 132 126 121 116 112 108 105 102 99 98 96 95 95 95 96 97 99 101 103 106 110 113 117 122 127 133 139 146 153 161 170 180 190 201 214 228 244 257 268 277 285 291 296 300 304 307 309 312 314 317 319 321 324 327 330 334 339 345 352 360 371

381 390 397 402 406 408 409 410 411 411 411 411 411 410 409 408 406 404 401 396 390 382 371 355 335 307 284 264 246 230 216 203 192 181 171 162 154 146 139 133 127 121 117 112 109 105 102 100 98 97 96 96 96 97 98 100 102 104 107 110 114 118 123 128 134 140 146 154 162 170 179 189 200 212 226 241 254 265 273 280 286 291 295 298 301 304 307 309 312 314 317 320 323 326 330 335 341 348 356 367

375 385 391 395 398 400 402 402 403 403 403 403 402 401 401 399 398 395 392 388 382 375 364 350 330 303 281 262 245 229 216 203 192 181 171 162 154 147 140 133 127 122 117 113 109 106 103 101 99 98 97 97 97 98 99 101 103 105 108 111 115 119 124 129 135 141 147 154 162 170 179 189 199 211 224 238 250 260 268 274 280 284 288 292 295 298 300 303 306 308 311 314 317 321 325 330 336 343 352 362

368 377 382 386 388 390 391 391 392 392 391 391 390 390 389 387 386 384 381 377 371 364 355 342 324 299 278 260 243 229 215 203 192 181 172 163 155 147 140 134 128 123 118 114 110 107 104 101 100 98 98 98 98 99 100 102 104 107 109 113 116 121 125 130 136 141 148 155 162 170 179 188 198 209 221 234 245 254 261 267 272 276 280 284 287 290 292 295 298 301 304 307 310 314 319 324 331 338 346 356

360 366 371 373 375 376 376 376 376 376 376 375 374 374 373 371 370 367 365 361 356 350 342 331 316 294 275 258 242 228 215 203 192 182 172 163 155 148 141 135 129 123 119 114 111 107 104 102 100 99 98 99 99 100 102 103 105 108 111 114 118 122 126 131 137 142 149 156 163 171 179 188 198 208 218 230 239 246 252 258 262 266 270 273 277 280 282 285 288 291 294 298 302 306 311 317 323 331 339 349

349 353 355 356 356 356 356 356 356 355 354 354 353 352 351 349 348 346 343 340 336 331 324 316 305 290 273 257 242 228 215 203 192 182 173 164 156 148 142 135 130 124 120 115 112 108 105 103 101 100 99 100 100 102 103 105 107 110 113 116 119 123 128 133 138 144 150 157 164 172 180 188 197 207 216 224 230 236 241 246 250 254 258 261 264 267 270 273 276 279 283 287 291 296 301 308 315 323 332 341

290 273 257 242 228 216 204 193 183 173 165 157 149 143 136 131 125 121 116 113 109 106 104 102 100

294 276 259 244 229 217 205 194 184 174 166 158 150 144 138 132 127 122 118 114 111 108 105 103 101

300 279 261 245 231 218 206 195 185 175 167 159 152 145 139 133 128 123 119 115 112 109 106 104 102

304 283 264 247 232 219 207 196 186 177 168 160 153 147 140 135 130 125 121 117 114 110 108 105 103

309 286 266 249 234 221 209 197 187 178 170 162 155 148 142 137 131 127 123 119 115 112 110 107 105

312 289 269 251 236 222 210 199 189 180 171 164 156 150 144 138 133 129 125 121 117 114 112 109 107

315 291 271 253 238 224 212 201 191 181 173 165 158 152 146 141 136 131 127 123 120 117 114 111 109

317 293 273 255 240 226 213 202 192 183 175 167 160 154 148 143 138 133 129 126 122 119 116 114 112

319 295 275 257 241 228 215 204 194 185 177 170 163 156 151 145 141 136 132 128 125 122 119 117 114

321 297 276 259 243 229 217 206 196 187 179 172 165 159 153 148 143 139 135 131 128 125 122 120 118

322 298 278 260 245 231 219 208 198 190 182 174 168 162 156 151 147 142 138 135 132 129 126 124 121

323 300 279 262 246 233 221 210 201 192 184 177 171 165 159 155 150 146 142 139 135 132 130 127 125

325 301 281 263 248 235 223 213 203 195 187 180 174 168 163 158 154 150 146 143 140 137 134 132 130

326 302 282 265 250 237 225 215 206 198 190 184 178 172 167 162 158 154 150 147 144 141 139 137 134

326 303 283 267 252 239 228 218 209 201 194 187 182 176 171 167 162 159 155 152 149 146 144 142 140

327 304 285 268 254 242 231 221 212 205 198 192 186 181 176 172 168 164 160 157 154 152 149 147 145

328 305 286 270 256 244 234 224 216 209 202 196 191 186 181 177 173 169 166 163 160 158 155 153 151

329 307 288 272 259 247 237 228 220 213 207 201 196 191 187 183 179 176 172 169 167 164 162 160 158

330 308 290 275 262 251 241 233 225 218 212 207 202 197 193 189 186 182 179 176 174 171 169 167 165

330 309 292 278 265 255 246 238 231 224 219 213 209 204 200 197 193 190 187 184 181 179 176 174 173

331 311 295 281 269 259 251 243 237 231 225 221 216 212 208 205 201 198 195 192 189 187 184 182 181

332 313 298 285 274 265 257 250 244 238 233 229 225 221 217 214 211 207 204 201 198 196 193 191 189

333 315 301 290 280 272 264 258 252 247 243 238 234 231 227 224 221 218 215 211 208 205 203 200 198

333 319 306 296 287 280 273 267 262 257 253 249 246 242 239 236 233 229 226 223 219 216 213 210 207

335 323 313 304 296 289 283 278 273 269 265 262 258 255 252 249 246 243 239 236 232 228 224 220 216

Figure 2.15: Results For Adjacent Face Flux (Scenario 3)

332 325 320 315 311 307 304 302 300 298 297 296 295 294 293 293 293 293 292 292 293 293 293 293 295

325 315 307 301 296 292 288 285 282 280 279 277 276 275 275 274 274 274 275 275 276 277 279 281 285

320 307 298 290 284 279 274 271 268 265 263 262 260 260 259 259 259 259 260 261 262 264 267 271 276

315 301 290 281 274 268 263 259 255 252 250 248 247 246 245 245 245 246 247 249 251 253 257 262 268

311 296 284 274 266 259 253 248 245 241 239 237 235 234 234 234 234 235 236 238 241 244 248 254 261

308 292 278 268 259 251 245 240 235 232 229 227 225 224 224 223 224 225 227 229 232 236 241 247 255

304 288 274 262 253 245 238 232 227 224 220 218 216 215 215 215 215 217 218 221 224 229 234 241 249

302 284 270 258 248 239 232 226 221 216 213 211 209 207 207 207 208 209 211 214 218 223 228 235 244

299 281 267 254 243 234 226 220 215 210 207 204 202 201 200 200 201 202 205 208 212 217 223 230 239

297 279 263 250 239 230 222 215 209 205 201 198 196 195 194 194 195 197 199 202 207 212 218 226 235

295 277 261 247 236 226 218 211 205 200 196 193 191 189 189 189 190 192 194 198 202 208 214 222 231

293 274 258 245 233 223 214 207 201 196 192 188 186 185 184 184 185 187 190 193 198 203 210 218 228

291 272 256 242 230 220 211 203 197 192 188 184 182 180 180 180 181 183 186 189 194 200 207 215 225

290 270 254 240 227 217 208 200 194 189 184 181 178 177 176 176 177 179 182 186 191 196 203 212 222

288 268 252 237 225 214 205 198 191 186 181 178 175 174 173 173 174 176 179 183 187 193 200 209 219

286 266 250 235 223 212 203 195 188 183 178 175 172 171 170 170 171 173 176 180 184 190 197 206 216

284 264 248 233 221 210 201 193 186 181 176 172 170 168 167 167 168 170 173 177 182 188 195 203 213

281 262 245 231 219 208 199 191 184 178 174 170 168 166 165 165 166 168 171 175 179 185 192 200 210

279 260 243 229 217 206 197 189 182 177 172 168 166 164 163 163 164 166 169 172 177 183 190 198 207

276 257 241 227 215 204 195 187 181 175 170 167 164 162 161 161 162 164 167 170 175 181 187 195 204

272 254 238 225 213 203 194 186 179 174 169 165 163 161 160 160 161 162 165 169 173 178 185 193 202

268 251 236 223 211 201 192 185 178 172 168 164 161 160 159 159 159 161 164 167 171 177 183 190 199

264 248 234 221 210 200 191 184 177 172 167 163 161 159 158 158 158 160 163 166 170 175 181 188 195

260 245 232 220 209 199 190 183 176 171 166 163 160 158 157 157 158 159 162 165 169 174 179 186 193

256 243 230 219 208 198 190 182 176 170 166 162 160 158 157 157 157 159 161 164 168 173 178 184 190

342 339 336 333 330 327 325 323 321 319 317 316 314 312 311 309 307 304 302 298 294 289 283 276 267 256 243 230 219 208 198 190 182 176 170 166 162 160 158 157 157 157 159 161 164 168 173 178 184 190 196 200 205 209 212 216 219 222 225 227 230 233 236 239 243 246 250 254 259 264 270 277 284 291 299 305 308 310 312 313 313 314 314 315 315 315 316 317 317 318 320 321 323 325 327 330 333 336 339 342

352 351 349 347 345 343 341 339 338 336 335 334 332 331 329 327 325 323 319 316 311 305 298 289 276 260 245 232 220 209 199 190 183 176 171 166 163 160 158 157 157 158 159 162 165 169 174 179 186 193 200 207 212 217 221 225 228 232 235 237 240 243 246 249 252 255 259 263 267 272 278 284 290 298 306 314 320 324 326 328 329 330 331 332 332 333 333 334 335 336 337 338 339 341 342 344 346 348 350 352

360 360 359 358 356 355 353 352 351 350 349 347 346 345 343 341 339 337 333 329 324 318 309 298 283 264 248 234 221 210 200 191 184 177 172 167 163 161 159 158 158 158 160 163 166 170 175 181 188 195 205 212 219 224 229 233 237 240 243 246 249 251 254 257 260 263 267 270 275 279 284 290 297 304 312 322 329 334 338 340 342 343 344 345 346 346 347 347 348 349 350 351 352 353 354 355 357 358 359 360

367 367 366 366 365 364 363 362 361 360 359 358 357 356 354 352 350 348 344 340 334 327 318 305 289 268 251 236 223 211 201 192 185 178 172 168 164 161 160 159 159 159 161 164 167 171 177 183 190 199 209 217 224 230 235 240 244 247 250 253 256 259 261 264 267 270 273 277 281 285 290 295 302 309 318 329 336 342 346 349 352 353 354 355 356 357 357 358 358 359 360 360 361 362 363 364 365 366 366 367

373 373 372 372 371 371 370 369 369 368 367 366 365 364 362 361 359 356 352 348 342 334 324 311 294 272 254 238 225 213 203 194 186 179 174 169 165 163 161 160 160 161 162 165 169 173 178 185 193 202 212 221 229 235 241 245 249 253 256 259 262 265 267 270 273 275 278 282 286 290 295 300 306 314 323 334 342 348 353 356 359 361 362 363 364 365 365 366 366 367 367 368 368 369 370 370 371 372 372 372

377 377 377 377 376 376 375 375 374 374 373 372 371 370 369 367 365 362 359 354 348 340 329 316 298 276 257 241 227 215 204 195 187 181 175 170 167 164 162 161 161 162 164 167 170 175 181 187 195 204 216 225 233 240 245 250 254 258 261 264 267 270 272 275 277 280 283 286 290 294 298 304 310 318 327 338 347 353 358 362 364 366 368 369 370 371 371 372 372 373 373 373 374 374 375 375 376 376 376 377

380 380 380 380 380 380 380 379 379 378 378 377 376 375 374 372 370 367 363 359 352 344 333 319 302 279 260 243 229 217 206 197 189 182 177 172 168 166 164 163 163 164 166 169 172 177 183 190 198 207 219 228 237 244 249 254 259 262 266 269 271 274 276 279 281 284 287 290 293 297 302 307 313 321 330 341 350 357 362 366 369 371 372 373 374 375 376 376 377 377 377 378 378 378 379 379 379 380 380 380

383 383 383 383 383 383 383 382 382 381 381 380 380 379 377 376 373 371 367 362 356 348 337 323 304 281 262 245 231 219 208 199 191 184 178 174 170 168 166 165 165 166 168 171 175 179 185 192 200 210 222 232 240 247 253 258 262 266 269 272 275 278 280 282 285 287 290 293 296 300 305 310 316 324 333 344 353 360 365 369 372 374 376 377 378 379 379 380 380 380 381 381 381 381 382 382 382 382 382 382

385 385 385 385 385 385 385 385 384 384 384 383 382 381 380 378 376 373 370 365 359 350 339 325 307 284 264 248 233 221 210 201 193 186 181 176 172 170 168 167 167 168 170 173 177 182 188 195 203 213 225 235 243 250 256 261 266 269 273 276 278 281 283 285 288 290 293 296 299 303 307 312 318 326 335 346 355 362 367 371 374 377 378 379 380 381 382 382 382 383 383 383 383 384 384 384 384 384 384 384

386 386 387 387 387 387 387 386 386 386 385 385 384 383 382 380 378 376 372 367 361 352 341 327 309 286 266 250 235 223 212 203 195 188 183 178 175 172 171 170 170 171 173 176 180 184 190 197 206 216 227 237 246 253 259 264 269 272 276 279 281 283 286 288 290 292 295 298 301 305 309 314 320 328 337 348 357 364 369 373 376 378 380 381 382 383 384 384 384 385 385 385 385 385 385 386 386 386 386 386

387 388 388 388 388 388 388 388 388 387 387 386 386 385 384 382 380 377 374 369 362 354 343 329 311 288 268 252 237 225 214 205 198 191 186 181 178 175 174 173 173 174 176 179 183 187 193 200 209 219 230 240 249 256 262 267 271 275 278 281 284 286 288 290 292 295 297 300 303 306 311 316 322 329 338 350 359 365 371 375 378 380 381 383 384 384 385 385 386 386 386 386 386 387 387 387 387 387 387 387

388 388 389 389 389 389 389 389 389 388 388 387 387 386 385 383 381 379 375 370 364 356 345 331 312 290 270 254 240 227 217 208 200 194 189 184 181 178 177 176 176 177 179 182 186 191 196 203 212 222 233 243 251 259 265 270 274 278 281 283 286 288 290 292 294 297 299 302 305 308 312 317 323 330 340 351 360 367 372 376 379 381 383 384 385 385 386 386 387 387 387 387 387 388 388 388 388 388 388 388

389 389 389 389 390 390 390 389 389 389 389 388 388 387 386 384 382 380 376 371 365 357 346 332 314 291 272 256 242 230 220 211 203 197 192 188 184 182 180 180 180 181 183 186 189 194 200 207 215 225 236 246 254 261 267 272 276 280 283 286 288 290 292 294 296 298 301 303 306 310 314 318 324 332 341 352 361 368 373 377 380 382 383 385 386 386 387 387 387 388 388 388 388 388 388 388 388 388 388 388

389 389 390 390 390 390 390 390 390 390 389 389 388 387 386 385 383 380 377 372 366 358 347 334 316 293 274 258 245 233 223 214 207 201 196 192 188 186 185 184 184 185 187 190 193 198 203 210 218 228 239 249 257 264 270 275 279 282 285 288 290 292 294 296 298 300 302 305 308 311 315 320 325 333 342 353 362 368 373 377 380 382 384 385 386 387 387 388 388 388 388 389 389 389 389 389 389 389 389 389

389 390 390 390 390 390 390 390 390 390 390 389 389 388 387 385 384 381 378 373 367 359 349 335 317 295 277 261 247 236 226 218 211 205 200 196 193 191 189 189 189 190 192 194 198 202 208 214 222 231 243 252 260 267 273 277 281 285 288 290 292 294 296 298 300 302 304 306 309 312 316 321 327 334 342 354 362 369 374 378 381 383 385 386 387 387 388 388 388 389 389 389 389 389 389 389 389 389 389 389

390 390 390 390 391 391 391 391 391 390 390 390 389 388 387 386 384 381 378 374 368 360 350 336 319 297 279 263 250 239 230 222 215 209 205 201 198 196 195 194 194 195 197 199 202 207 212 218 226 235 246 255 263 270 275 280 284 287 290 292 295 297 298 300 302 304 306 308 311 314 318 322 328 335 343 354 363 370 375 378 381 383 385 386 387 388 388 388 389 389 389 389 389 389 389 389 389 389 389 389

390 390 390 391 391 391 391 391 391 391 390 390 389 389 388 386 384 382 379 374 369 361 351 338 321 299 281 267 254 243 234 226 220 215 210 207 204 202 201 200 200 201 202 205 208 212 217 223 230 239 250 259 267 273 278 283 287 290 293 295 297 299 301 302 304 306 308 310 312 315 319 323 329 336 344 355 364 370 375 379 381 384 385 386 387 388 388 389 389 389 389 389 389 389 389 389 389 389 389 390

390 390 390 391 391 391 391 391 391 391 390 390 389 389 388 386 385 382 379 375 369 362 352 339 323 302 284 270 258 248 239 232 226 221 216 213 211 209 207 207 207 208 209 211 214 218 223 228 235 244 254 263 270 277 282 286 290 293 296 298 300 302 303 305 306 308 310 312 314 317 321 325 330 337 345 356 364 371 375 379 382 384 385 387 387 388 388 389 389 389 389 389 390 390 390 390 390 390 390 390

390 390 390 391 391 391 391 391 391 391 390 390 390 389 388 387 385 383 380 375 370 363 353 341 325 304 288 274 262 253 245 238 232 227 224 220 218 216 215 215 215 215 217 218 221 224 229 234 241 249 259 267 275 281 286 290 293 296 299 301 303 305 306 308 309 311 312 314 317 319 322 327 332 338 346 357 365 371 376 379 382 384 386 387 387 388 389 389 389 389 389 390 390 390 390 390 390 390 390 390

390 390 390 391 391 391 391 391 391 391 390 390 390 389 388 387 385 383 380 376 371 364 355 343 327 308 292 278 268 259 251 245 240 235 232 229 227 225 224 224 223 224 225 227 229 232 236 241 247 255 264 272 279 285 290 294 297 300 303 305 306 308 310 311 312 314 315 317 319 322 325 329 333 339 347 358 366 372 376 380 382 384 386 387 388 388 389 389 389 389 389 390 390 390 390 390 390 390 390 390

390 390 390 391 391 391 391 391 391 391 390 390 390 389 388 387 385 383 380 376 371 365 356 345 330 311 296 284 274 266 259 253 248 245 241 239 237 235 234 234 234 234 235 236 238 241 244 248 254 261 270 278 284 290 295 298 302 305 307 309 311 312 314 315 316 318 319 321 322 325 328 331 336 341 349 359 366 372 377 380 382 384 386 387 388 388 389 389 389 389 389 390 390 390 390 390 389 389 389 389

389 390 390 390 391 391 391 391 391 390 390 390 389 389 388 387 385 383 380 377 372 366 358 347 333 315 301 290 281 274 268 263 259 255 252 250 248 247 246 245 245 245 246 247 249 251 253 257 262 268 277 284 290 295 300 304 307 310 312 314 316 317 318 320 321 322 323 325 327 329 331 334 338 343 350 360 367 373 377 380 383 384 386 387 388 388 389 389 389 389 389 389 389 389 389 389 389 389 389 389

389 390 390 390 390 390 390 390 390 390 390 390 389 389 388 387 385 383 380 377 372 366 359 349 336 320 307 298 290 284 279 274 271 268 265 263 262 260 260 259 259 259 259 260 261 262 264 267 271 276 284 290 297 302 306 310 313 316 318 320 322 323 324 325 327 328 329 330 332 333 336 338 342 346 353 361 368 373 377 380 383 384 386 387 388 388 389 389 389 389 389 389 389 389 389 389 389 389 389 389

389 389 390 390 390 390 390 390 390 390 390 389 389 388 388 386 385 383 380 377 373 367 360 351 339 325 315 307 301 296 292 288 285 282 280 279 277 276 275 275 274 274 274 275 275 276 277 279 281 285 291 298 304 309 314 318 321 324 326 328 329 330 332 333 334 335 336 337 338 339 341 343 346 350 355 363 369 374 377 380 383 384 386 387 387 388 388 389 389 389 389 389 389 389 389 389 389 389 389 389

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342 339 336 333 330 327 325 323 321 320 318 317 317 316 315 315 315 314 314 313 313 312 310 308 305

352 350 348 346 344 342 341 339 338 337 336 335 334 333 333 332 332 331 330 329 328 326 324 320 314

360 359 358 357 355 354 353 352 351 350 349 348 347 347 346 346 345 344 343 342 340 338 334 329 322

367 366 366 365 364 363 362 361 360 360 359 358 358 357 357 356 355 354 353 352 349 346 342 336 329

372 372 372 371 370 370 369 368 368 367 367 366 366 365 365 364 363 362 361 359 356 353 348 342 334

377 376 376 376 375 375 374 374 373 373 373 372 372 371 371 370 369 368 366 364 362 358 353 347 338

380 380 380 379 379 379 378 378 378 377 377 377 376 376 375 374 373 372 371 369 366 362 357 350 341

382 382 382 382 382 382 381 381 381 381 380 380 380 379 379 378 377 376 374 372 369 365 360 353 344

384 384 384 384 384 384 384 383 383 383 383 382 382 382 381 380 379 378 377 374 371 367 362 355 346

386 386 386 386 386 385 385 385 385 385 385 384 384 384 383 382 381 380 378 376 373 369 364 357 348

387 387 387 387 387 387 387 386 386 386 386 386 385 385 384 384 383 381 380 378 375 371 365 359 350

388 388 388 388 388 388 388 387 387 387 387 387 386 386 385 385 384 383 381 379 376 372 367 360 351

388 388 388 388 388 388 388 388 388 388 388 387 387 387 386 386 385 383 382 380 377 373 368 361 352

389 389 389 389 389 389 389 389 389 388 388 388 388 387 387 386 385 384 382 380 377 373 368 362 353

389 389 389 389 389 389 389 389 389 389 389 388 388 388 387 387 386 385 383 381 378 374 369 362 354

389 389 389 389 389 389 389 389 389 389 389 389 388 388 388 387 386 385 383 381 378 375 370 363 354

390 389 389 389 389 389 389 389 389 389 389 389 389 388 388 387 386 385 384 381 379 375 370 364 355

390 390 390 390 390 390 390 390 389 389 389 389 389 388 388 387 387 385 384 382 379 375 371 364 356

390 390 390 390 390 390 390 390 390 389 389 389 389 389 388 387 387 386 384 382 379 376 371 365 357

390 390 390 390 390 390 390 390 390 389 389 389 389 389 388 388 387 386 384 382 380 376 372 366 358

389 389 389 389 390 390 390 390 390 389 389 389 389 389 388 388 387 386 384 382 380 377 372 366 359

389 389 389 389 389 389 389 389 389 389 389 389 389 389 388 388 387 386 384 383 380 377 373 367 360

389 389 389 389 389 389 389 389 389 389 389 389 389 389 388 388 387 386 384 383 380 377 373 368 361

389 389 389 389 389 389 389 389 389 389 389 389 389 388 388 387 387 386 384 383 380 377 374 369 363

388 388 389 389 389 389 389 389 389 389 389 389 389 388 388 387 387 386 384 382 380 377 374 369 364

Figure 2.16: Results For Adjacent Face Flux (Scenario 3), ϕΦ = [π4

0 π4]T

Chapter 3

The Inverse Problem

Upon the completion of the forward problem, the task at hand, was to analyze the results of the

temperature distribution to attain a solution to the inverse heat transfer problem (Solving ΩR)

via optimization. The equation will be formatted as a nonlinear, least-squares problem with

inequality constraints [7]. The optimization will be performed by minimizing the following

cost function:

Ψ(ΩR) =N∑i=1

ζi(Ti − Td,i)2 (3.1)

subject to the following constraints:

0 ≤ αs,i ≤ 1, i = 1, . . . , N

0 ≤ εout,i ≤ 1, i = 1, . . . , N

0 ≤ εin,i ≤ 1, i = 1, . . . , N

(3.2)

where ΩR are the set of optimized radiation parameters (αs, εout, εin), N is the total number

of discretized nodes on all six faces, ζi is the weighting at each node, Ti is the temperature at

each node, and Td,i is the desired temperature at each node. Conduction parameters ΩC do not

play a role within the optimization, because their properties are dependent on the material of

the spacecraft, which is outside the scope of this work.

24

CHAPTER 3. THE INVERSE PROBLEM 25

Traditionally this has been formulated as a least-squares problem with the constraints having

been replaced by penalty items in the objective function; however the penalty terms include

weighting factors whose values can only be specified empirically [2]. Advanced techniques

require the evaluation of the derivatives of the objective function. Based on this, the primary

method of optimization that we will be reviewing is Sequential Quadratic Programming (SQP),

as it is considered one of the most effective and reliable methods for nonlinear optimization.

3.1 Solving the Inverse SQP Problem

We set the desired temperature Td,i with the results of the forward problem, maintained an

equal nodal weighting (ζi = 1), and compared its proximity to that of the initial design param-

eters. Similar to the method employed by Papalexandris and Milman [2], an SQP algorithm has

been utilized. The iterates are evaluated along a descent direction for |F (T )|2, and the stepsize

is determined via a back-tracking line-search algorithm, based on a restricted stepsize Newton

method. It is important to note that that the MATLAB SQP routine uses a finite differencing

procedure to calculate its derivatives.

Tables (3.1), (3.2), and (3.3), demonstrate the results for a 3 × 3 × 3 cube attained via the

optimization that has been completed using MATLAB’s SQP algorithm. The first column, the

desired temperature (Td), is the solution to the forward problem of an unrotated spacecraft

(ϕΦ = [0 0 0]T ) with the following characteristics: αs,i = 0.50, εout,i = 0.75, εin,i = 1.00.

The next 3 columns are the parametric results that we arrive at after solving the problem from

different starting points. The fourth column, the evaluated temperature Teval, is the temperature

at each node re-evaluated via the forward problem using the new design parameters, and finally

the last column, ∆Teval, is the modulus of the difference between Td and Teval.


Table 3.1: SQP Optimization Results (αs,0,i = 0.50, εout,0,i = 0.50, εin,0,i = 0.50)

Td αs εout εin Teval ∆Teval

267.9016 0.5000 0.7035 0.7113 267.9044 0.0028250.4788 0.5000 0.6755 0.6648 250.4774 0.0014257.3816 0.5000 0.6616 0.6538 257.3837 0.0021266.9154 0.5000 0.6072 0.6842 266.9167 0.0013250.2797 0.5000 0.6632 0.6800 250.2780 0.0017256.6289 0.5000 0.5910 0.6276 256.6294 0.0005267.9016 0.5000 0.6655 0.7082 267.9004 0.0012250.4788 0.5000 0.6836 0.6730 250.4789 0.0001257.3816 0.5000 0.6900 0.6723 257.3798 0.0018288.6466 0.4779 0.8462 0.5778 288.6476 0.0010290.2408 0.4055 0.7204 0.6261 290.2403 0.0005288.6466 0.4716 0.8398 0.6051 288.6452 0.0014290.2408 0.3817 0.6033 0.5579 290.2417 0.0009292.5267 0.3876 0.6368 0.5289 292.5248 0.0019290.2408 0.4046 0.6915 0.6047 290.2404 0.0004288.6466 0.4626 0.8555 0.6194 288.6465 0.0001290.2408 0.3809 0.7022 0.6627 290.2377 0.0031288.6466 0.4633 0.8763 0.6353 288.6467 0.0001257.3816 0.5000 0.6997 0.6806 257.3791 0.0025250.4788 0.5000 0.6507 0.6578 250.4796 0.0008267.9016 0.5000 0.6507 0.7087 267.9029 0.0013256.6289 0.5000 0.6361 0.6598 256.6257 0.0032250.2797 0.5000 0.6354 0.6671 250.2802 0.0005266.9154 0.5000 0.5717 0.6762 266.9168 0.0014257.3816 0.5000 0.6451 0.6529 257.3795 0.0021250.4788 0.5000 0.6572 0.6586 250.4779 0.0009267.9016 0.5000 0.6236 0.6846 267.9015 0.0001313.0091 0.4737 0.8718 0.7065 313.0102 0.0011314.5805 0.4034 0.7091 0.7270 314.5798 0.0007313.0091 0.4661 0.8280 0.5308 313.0136 0.0045314.5805 0.4303 0.7781 0.7040 314.5818 0.0013316.7360 0.4078 0.7415 0.7848 316.7409 0.0049314.5805 0.4410 0.7976 0.6144 314.5832 0.0027313.0091 0.4667 0.8630 0.6986 313.0081 0.0010314.5805 0.4258 0.7662 0.6867 314.5766 0.0039313.0091 0.4542 0.8155 0.6718 313.0061 0.0030267.9016 0.5000 0.6868 0.7249 267.9028 0.0012250.4788 0.5000 0.6845 0.6741 250.4766 0.0022257.3816 0.5000 0.7264 0.6931 257.3807 0.0009266.9154 0.5000 0.6302 0.7104 266.9157 0.0003250.2797 0.5000 0.6223 0.6579 250.2810 0.0013256.6289 0.5000 0.6343 0.6591 256.6280 0.0009267.9016 0.5000 0.6996 0.7088 267.9061 0.0045250.4788 0.5000 0.6525 0.6517 250.4817 0.0029257.3816 0.5000 0.7232 0.6811 257.3835 0.0019267.9016 0.5000 0.6812 0.7153 267.8995 0.0021250.4788 0.5000 0.6590 0.6599 250.4784 0.0004257.3816 0.5000 0.6681 0.6615 257.3833 0.0017266.9154 0.5000 0.5607 0.6666 266.9165 0.0011250.2797 0.5000 0.6408 0.6701 250.2807 0.0010256.6289 0.5000 0.5947 0.6424 256.6307 0.0018267.9016 0.5000 0.6156 0.6814 267.9014 0.0002250.4788 0.5000 0.6501 0.6559 250.4790 0.0002257.3816 0.5000 0.6952 0.6775 257.3802 0.0014




267.9016 0.7500 0.7186 0.6906 267.8985 0.0031250.4788 0.7500 0.5905 0.5721 250.4777 0.0011257.3816 0.7500 0.7312 0.6635 257.3839 0.0023266.9154 0.7500 0.6799 0.6843 266.9150 0.0004250.2797 0.7500 0.5847 0.5937 250.2808 0.0011256.6289 0.7500 0.6164 0.6090 256.6305 0.0016267.9016 0.7500 0.7287 0.6969 267.8980 0.0036250.4788 0.7500 0.6063 0.5802 250.4780 0.0008257.3816 0.7500 0.7524 0.6764 257.3819 0.0003288.6466 0.4280 0.8088 0.5963 288.6459 0.0007290.2408 0.4158 0.7627 0.5458 290.2412 0.0004288.6466 0.4150 0.7926 0.6199 288.6477 0.0011290.2408 0.4122 0.7542 0.5452 290.2415 0.0007292.5267 0.4085 0.7690 0.5314 292.5284 0.0017290.2408 0.4207 0.7635 0.5359 290.2389 0.0019288.6466 0.4225 0.8066 0.6097 288.6478 0.0012290.2408 0.4077 0.7586 0.5580 290.2402 0.0006288.6466 0.4187 0.8046 0.6188 288.6451 0.0015257.3816 0.7500 0.7510 0.6785 257.3821 0.0005250.4788 0.7500 0.6172 0.5865 250.4786 0.0002267.9016 0.7500 0.7446 0.7068 267.9000 0.0016256.6289 0.7500 0.6309 0.6168 256.6277 0.0012250.2797 0.7500 0.5733 0.5868 250.2789 0.0008266.9154 0.7500 0.7179 0.7070 266.9178 0.0024257.3816 0.7500 0.7536 0.6789 257.3801 0.0015250.4788 0.7500 0.5758 0.5656 250.4796 0.0008267.9016 0.7500 0.6839 0.6778 267.9068 0.0052313.0091 0.4214 0.7338 0.5633 313.0063 0.0028314.5805 0.4322 0.7968 0.4886 314.5763 0.0042313.0091 0.4360 0.7714 0.5333 313.0042 0.0049314.5805 0.4243 0.7722 0.5233 314.5837 0.0032316.7360 0.3851 0.7048 0.6467 316.7406 0.0046314.5805 0.4054 0.7283 0.5001 314.5776 0.0029313.0091 0.4293 0.7598 0.5793 313.0160 0.0069314.5805 0.4266 0.7838 0.4903 314.5789 0.0016313.0091 0.4359 0.7702 0.5393 313.0055 0.0036267.9016 0.7500 0.7059 0.6904 267.8981 0.0035250.4788 0.7500 0.5814 0.5685 250.4781 0.0007257.3816 0.7500 0.7563 0.6815 257.3799 0.0017266.9154 0.7500 0.6855 0.6870 266.9119 0.0035250.2797 0.7500 0.5721 0.5862 250.2771 0.0026256.6289 0.7500 0.6179 0.6100 256.6296 0.0007267.9016 0.7500 0.7206 0.6910 267.8991 0.0025250.4788 0.7500 0.6122 0.5830 250.4775 0.0013257.3816 0.7500 0.7435 0.6684 257.3829 0.0013267.9016 0.7500 0.7264 0.6959 267.8994 0.0022250.4788 0.7500 0.5858 0.5690 250.4777 0.0011257.3816 0.7500 0.7391 0.6705 257.3827 0.0011266.9154 0.7500 0.6599 0.6728 266.9144 0.0010250.2797 0.7500 0.5903 0.5972 250.2804 0.0007256.6289 0.7500 0.5762 0.5864 256.6283 0.0006267.9016 0.7500 0.7217 0.6951 267.9017 0.0001250.4788 0.7500 0.6360 0.5978 250.4774 0.0014257.3816 0.7500 0.7433 0.6747 257.3812 0.0004




267.9016 1.0000 0.6731 0.7148 267.9005 0.0011250.4788 1.0000 0.6962 0.6995 250.4775 0.0013257.3816 1.0000 0.7126 0.7033 257.3816 0.0000266.9154 1.0000 0.6757 0.7413 266.9161 0.0007250.2797 1.0000 0.6735 0.7039 250.2771 0.0026256.6289 1.0000 0.7380 0.7298 256.6290 0.0001267.9016 1.0000 0.6887 0.7194 267.8985 0.0031250.4788 1.0000 0.7090 0.7058 250.4753 0.0035257.3816 1.0000 0.7111 0.7040 257.3803 0.0013288.6466 0.4678 0.8325 0.6317 288.6460 0.0006290.2408 0.4700 0.7968 0.5774 290.2396 0.0012288.6466 0.4831 0.8415 0.6019 288.6480 0.0014290.2408 0.4685 0.7928 0.5822 290.2413 0.0005292.5267 0.4605 0.8069 0.5730 292.5267 0.0000290.2408 0.4607 0.7874 0.5988 290.2413 0.0005288.6466 0.4651 0.8376 0.6407 288.6440 0.0026290.2408 0.4573 0.7862 0.6084 290.2402 0.0006288.6466 0.4589 0.8417 0.6570 288.6484 0.0018257.3816 1.0000 0.6975 0.6910 257.3807 0.0009250.4788 1.0000 0.7127 0.7066 250.4760 0.0028267.9016 1.0000 0.7335 0.7459 267.9020 0.0004256.6289 1.0000 0.7303 0.7268 256.6281 0.0008250.2797 1.0000 0.7102 0.7225 250.2782 0.0015266.9154 1.0000 0.6821 0.7444 266.9175 0.0021257.3816 1.0000 0.7038 0.7029 257.3809 0.0007250.4788 1.0000 0.7118 0.7080 250.4767 0.0021267.9016 1.0000 0.7030 0.7275 267.9025 0.0009313.0091 0.4881 0.8645 0.5275 313.0080 0.0011314.5805 0.4526 0.8069 0.6675 314.5764 0.0041313.0091 0.4790 0.8455 0.5336 313.0078 0.0013314.5805 0.4580 0.8210 0.6666 314.5795 0.0010316.7360 0.4599 0.8468 0.6866 316.7356 0.0004314.5805 0.4595 0.8258 0.6656 314.5848 0.0043313.0091 0.4808 0.8508 0.5427 313.0125 0.0034314.5805 0.4662 0.8412 0.6708 314.5783 0.0022313.0091 0.4844 0.8604 0.5226 313.0114 0.0023267.9016 1.0000 0.7422 0.7499 267.9009 0.0007250.4788 1.0000 0.7009 0.7016 250.4765 0.0023257.3816 1.0000 0.6996 0.6924 257.3839 0.0023266.9154 1.0000 0.6932 0.7501 266.9166 0.0012250.2797 1.0000 0.7153 0.7248 250.2800 0.0003256.6289 1.0000 0.7220 0.7202 256.6314 0.0025267.9016 1.0000 0.7477 0.7472 267.9025 0.0009250.4788 1.0000 0.7111 0.7056 250.4802 0.0014257.3816 1.0000 0.6888 0.6924 257.3827 0.0011267.9016 1.0000 0.7178 0.7320 267.8985 0.0031250.4788 1.0000 0.6987 0.7004 250.4754 0.0034257.3816 1.0000 0.6795 0.6894 257.3808 0.0008266.9154 1.0000 0.7020 0.7551 266.9138 0.0016250.2797 1.0000 0.7066 0.7212 250.2790 0.0007256.6289 1.0000 0.7285 0.7280 256.6292 0.0003267.9016 1.0000 0.7022 0.7273 267.8999 0.0017250.4788 1.0000 0.7079 0.7065 250.4780 0.0008257.3816 1.0000 0.6816 0.6929 257.3837 0.0021


The following quantities are going to be used throughout this work:

• T d: The average value of desired temperatures

• T eval: The average value of evaluated temperatures

• Ψ: The value of the cost function which is calculated by Eq. (3.1)

• ∆T d: The RMS difference of Td and Teval defined by

∆T d(T ) =

√(Td − Teval)T (Td − Teval)

N(3.3)

As it can be seen, multiple trials were performed, and all of them managed to converge. How-

ever as seen in Table (3.4), αs,0,i = εout,0,i = εin,0,i = 0.00 and αs,0,i = εout,0,i = εin,0,i = 0.25

show extremely poor results when satisfying the cost function. On the other hand, all of

the other iterations provided different yet reasonable results. Surprisingly αs,0,i = εout,0,i =

εin,0,i = 1.00 (Table (3.3)) yielded the lowest value to the cost function, even though αs,0,i =

0.45, εout,0,i = 0.70, εin,0,i = 0.95 had the starting point closest to the expected solution. It

is also interesting that multiple solutions exist for the problem, and none of these trials, with

the exception of αs,0,i = 0.50, εout,0,i = 0.75, εin,0,i = 1.00 actually converged to the expected

result of αs,i = 0.50, εout,i = 0.75, εin,i = 1.00.


Table (3.4) details a summary of the aforementioned quantities, with T d = 272.8993K.

Table 3.4: SQP Optimization Result Summary at Different Starting Points

αs,0 εout,0 εin,0 T eval Ψ ∆T d Time(s) Iter0.0000 0.0000 0.0000 251.6070 5.1764E+04 30.9612 855.67 10.2500 0.2500 0.2500 300.9662 7.3371E+04 36.8609 1423.18 20.5000 0.5000 0.5000 272.8994 2.0446E-04 0.0019 1075.35 890.7500 0.7500 0.7500 272.8989 2.9568E-04 0.0023 930.72 811.0000 1.0000 1.0000 272.8989 1.8692E-04 0.0019 942.03 790.4000 0.6500 0.9000 272.8997 2.2286E-04 0.0020 874.14 760.4500 0.7000 0.9500 272.8987 2.4656E-04 0.0021 814.04 70

Based on the results from the prior section, it is safe to assume that the best starting point

going forward is to have the initial values to be αs,0,i = 1.00, εout,0,i = 1.00, εin,0,i = 1.00.

Moreover, for the next set of results we will be setting Td,i = 300K, ϕΦ = [π4

0 π4]T , and

we will vary the number of nodes by having different sizes of each discretized elements. The

following table highlights this with our previously defined metrics as the number of nodes (N )

and design parameters (Ω) on the spacecraft are increased.

Table 3.5: SQP Optimization Result Summary of Varying Discretization Rotated [π4

0 π4]T

Dimensions N Ω T eval Ψ ∆T d Time(s) Iter1× 1× 1 6 18 300.0014 4.0500E-05 0.0026 4.11 322× 2× 2 24 72 300.0004 1.9144E-04 0.0028 72.56 583× 3× 3 54 162 300.0002 1.2572E-04 0.0015 983.37 864× 4× 4 96 288 300.0005 3.0073E-04 0.0018 9299.25 1355× 5× 5 150 450 299.9997 1.7861E-04 0.0011 44304.31 187

As it can be seen from the prior table, the average deviation tends to decrease as the number of

parameters are increased, however the time to compute this also increases exponentially. This

can be attributed to the computation of the forward problem in every iteration.


3.2 Simplification

By allowing every node to have their own set of design parameters, there appears to be too

many design variables, and thus degrees of freedom that the algorithm tries to optimize. When

engineering simulations are computationally expensive, optimization must often resort to ap-

proximation strategies in order to reduce the number of function evaluations required and also

to reduce the computational expense of each function evaluation [8]. The code has been modi-

fied so that each face will be constrained to only be optimized for one set of design parameters.

Therefore, regardless of the amount of nodes there will only be 18 design variables in to-

tal. Not only is this more complex from an analytical/mathematical perspective, it also closer

reflects reality with regards to the final application (surface realization). With very minus-

cule discretizations, it may not be feasible to apply a particular solution to each node, yet we

clearly want to have a large number of nodes to attain a more accurate solution. To compare

these approaches, three cubes with successively finer discretizations will be optimized using

this method and the results will be compared with the results from the previous section. All

three cubes will share the following properties: αs,0,i = 1.00, εout,0,i = 1.00, εin,0,i = 1.00,

Td,i = 300K

Table 3.6: SQP Optimization Simplified Results of 5× 5× 5 Cube

Face αs εout εin T evalFace 1 1.0000 0.5919 0.0870 300.0764Face 2 0.9013 1.0000 0.0000 299.9543Face 3 0.4523 1.0000 1.0000 300.0701Face 4 0.4583 1.0000 0.0000 299.9597Face 5 1.0000 0.6781 0.0000 299.9613Face 6 0.6481 1.0000 0.0000 299.9551

Average 299.9962




Average 299.9992



Average 300.0002

Table 3.9: SQP Optimization Simplified Result Summary of Different Nodes

Dimensions n Ω T eval Ψ ∆T d Time(s) IterUnsimplified 5 150 450 299.9997 1.7861E-04 1.0002 44304.31 187

Simplified 5 150 18 299.9962 7.2610E+01 0.6958 1139.83 122Simplified 10 600 18 299.9992 1.0346E+02 0.4152 19695.59 127Simplified 15 1350 18 300.0002 1.0987E+02 0.2853 178049.23 201

The primary conclusion from the prior set of results is that by decreasing the number of param-

eters, even though the number of nodes have been increased, the cost rises as well. However

the two most important quantities, the time to compute the results and the RMS difference,

both decrease drastically. Based on these findings, the evident approach to take is to simplify

our solution for all future analysis.


3.2.1 Next Steps

We will compare our current solution with two other optimization techniques that will deal

with our inability to find the global minimum. They will both be gradient-free technique and

should help us achieve this goal - or at least enable us to find a better starting point.

3.3 Gradient-Free

As we saw in the previous analysis, the solution to the problem was heavily dependent on

the location of the starting point, leading to local solutions, with no guarantee that they are

the optimal (global) solution. Increased computational power has increased the interest for

global optimization methods [9], where the main concern is not only to find a locally improved

solution, but a globally improved one [10]. There are two gradient-free methods that we will

examine: The Nelder-Mead Simplex and Genetic Algorithms.

3.3.1 Nelder-Mead Simplex

The simplex method of Nelder and Mead performs a search in n-dimensional space using

heuristic ideas. Its main strengths are that it requires no derivatives to be computed and that

it does not require the objective function to be smooth. The weakness of this method is that

it is not very efficient, particularly for problems with more than about ten design variables,

a number above which convergence becomes increasingly difficult. However, there has been

some work done in the past by Narayana [11] in the field of satellite thermal network correction

and Luersen et al. in engineering optimization [12], both achieving favourable results. The

Nelder-Mead algorithm starts with a simplex (n + 1 sets of design variables x) and then modifies

the simplex at each iteration using the following operations: reflection, expansion, outside

contraction, inside contraction and shrinking. The sequence of operations performed in one

iteration depends on the value of the objective at the new point relative to the other key points.

The flowchart in Figure 3.1 was the method identified by Narayana [11]:


SATELLITE THERMAL NETWORKS 271

where X is a parameter, L a n d U are lower and upper limit values, and N is the total number of parameters. Depending on the type of parameters and the associated uncertainty involved in their estimation, different bound values can be considered for each parameter. Thermal network correction using the optimization method consists of the minimization of Eq. (2) subject to the constraints (3). Many possible optimization methods have been cited in the l i t e r a t ~ r e ' ~ , ' ~ . Due to the difficulties involved in the evaluation of the gradient of the objective function for the problem under consideration, the search method should not require derivatives. Of the several possible search methods the simplex method of Nelder and Meadi4 is used in the present investigation.

When using the optimization method, the correction of parameters is done purely on the basis of the reduction in the objective function value (OFV). However, in some cases, a reduction in OFV alone may not result in better parameter value^'^. This is

SET OF INITIAL VALUES

OF PARAMETERS

OBJECTIVE FEdCTION VALUE

1 THEORETICAL EXPERIMENTAL

TVALUATION'OF STATES

wrr OF STATES

a

TECMNIPUE WITH THE GIVEN CONSTRAINTS

ON PARAMETER

1 NEW SET OF 7 cowcrm PARAMETER 1 VALUES

Figure 1 Flow chart or network correction by the optimization method.

Downloaded By: [University of Toronto] At: 19:19 15 January 2009

Figure 3.1: Flow Chart of Network Correction


The basic Nelder-Mead simplex algorithm is unconstrained; however this modification has

been made to satisfy our problem. Similar to the SQP approach, the starting point plays an

important role in attaining the solution to the problem - both in computational time and satis-

fying the cost function. The following tables are the results to the problem at different starting

points.

Table 3.10: Nelder-Mead Results (αs,0,i = 0.25, εout,0,i = 0.25, εin,0,i = 0.25)


Average 299.9922



Average 300.0031



Average 299.9912




Average 300.0390



Average 299.9556



Average 299.8927

One of the key observations that can be made here, especially in comparison with the SQP

method, is the diversity of the design parameter solution (ΩR), observed by comparing Table

(3.6) with Table (3.15). In Table (3.6) many of the parameters have their values close to the

constraints, whereas in Table (3.15) the answers appear random. Moreover, on top of the

diversity, we also have different answers for the different starting points. This leads us to

conclude that there is no unique global solution to the network model. Our key indicators in


evaluating the effectiveness of this solution is by comparing the values of the cost function

as well as the time taken to achieve the solution. Table (3.16) below show the results for the

Nelder-Mead Simplex method:

Table 3.16: Nelder-Mead Optimization Result Summary at Different Starting Points

ΩR,0 T eval Ψ ∆T d Time(s)0.0000 - - - 10544.950.2500 299.9922 5.1725E+02 1.8570 1606.420.3333 300.0031 3.3846E+02 1.5021 1606.220.5000 299.9912 3.1315E+02 1.4449 1607.560.6667 300.0390 3.3195E+02 1.4876 1585.210.7500 299.9556 4.0682E+02 1.6469 1603.651.0000 299.8927 5.7532E+02 1.9584 1597.63

We can see that the optimal solution with this method occurs when the starting location (ΩR,0)

is near 0.50, as it has the lowest cost function (Ψ) and RMS difference (∆T d). Moreover when

ΩR,0 = 0.00 a solution cannot be attained. By comparing Table (3.9) with Table (3.16) we

see that the cost function value of the SQP method is an order of magnitude lower, suggesting

that it is a better algorithm to solve the thermal network model. The time to solve via the SQP

method is also approximately 1.5 times faster.

3.3.2 Genetic Algorithms

In the last twenty years, a considerable number of global methods have been developed, with

most being based on trying to copy the efficiency and simplicity of self-optimized processes in

nature [10]. Genetic algorithms (GAs) for optimization were inspired by the process of natural

evolution of organisms and are based on reproduction processes and “survival of the fittest”.

Taking the previous cost function (Eq. (3.1)), and placing it within the context of GAs we will

call each design variable, ΩR a population member, and the value of the objective function,

Ψ(ΩR) is termed the fitness.


Genetic algorithms are radically different from the gradient-based methods we have covered

so far. Instead of looking at one point at a time and stepping to a new point for each itera-

tion, a whole population of solutions is iterated towards the optimum at the same time. Using

a population lets us explore multiple “buckets” (local minima) simultaneously, increasing the

likelihood of finding the global optimum [13]. The main advantages of a GA are: The popula-

tion can cover a large range of the design space and is less likely than gradient-based methods

to “get stuck” in local minima; as with other gradient-free methods, it can handle noisy ob-

jective functions; the implementation is straightforward; and it can be used for multi-objective

optimization. However, GAs are typically expensive when compared to gradient-based meth-

ods, especially for problems with a large number of design variables. Since this number has

been reduced, we can now see how well it will perform here.

As previously mentioned, all three design variables (αs, εout, εin), must be constrained to

be real numbers between 0 and 1. We also require at least 3 digits of accuracy within our

solution. To achieve these results, each creature will be modelled as a 10-digit binary value,

which allows 1024 integer values for each creature. When converting to real numbers the bi-

nary values are then divided by 1023, applying the required constraints. As per Martins [13],

the following are steps to formulate a genetic algorithm:

1. Initialize a Population: Each creature (population member) is a constrained design point

with an associated fitness value. The population size (number of creatures per genera-

tion) should be between 15-20 times the number of design variables.

2. Determine Mating Pool: Within the mating pool, the fitness method is utilized, where

better members reproduce more frequently. However, with this method there could be

values in a weak (unfit) creature that would be useful in finding a better optimum.


3. Generate Offspring: Offspring (next generation of creatures) are created via the crossover

operation. In the scheme employed here, the procedure is to randomly generate a crossover

probability between 65-75% and a crossover value. If the crossover value is less than the

crossover probability, then mating occurs based on a randomly generated creature value.

4. Mutation: Diversity is maintained by randomly generating a mutation probability be-

tween 0.5-10% and a mutation value. If the mutation value is less than the mutation

probability, then the creature is mutated.

5. Compute Offspring’s Fitness: Creatures are decoded from binary values into real num-

bers and their respective fitnesses are calculated.

6. Tournament Again: Here, the new generation of creatures (offspring) replace the old

(parents), and we return to Step 2 to determine the mating pool.

7. Identify the Best Creature: The best creature is identified as the one having the best fit-

ness value. There were 3 values which were stored: The best one in the final generation,

the best one in last 10 generations, and the best one in all generations. The best crea-

ture is identified in this manner, because the fitness values kept fluctuating and an ideal

convergence could not be established.

The major difference between the GA and the other methods that we had utilized thus far is that

the starting points and iteration processes are random; hence with every simulation a different

solution is reached. What was discovered was that these solutions all ended with poor cost

function values and took a tremendous amount of time to compute. Moreover, the failure to

converge towards a particular design solution further validates the statement that there is no

unique global solution to this problem. From what has been witnessed here, it is quite evident

that although the Nelder-Mead Simplex method provides somewhat reasonable results, neither

gradient-free method is an ideal solution method for this problem.

Chapter 4

Spacecraft Model Modification

4.1 Shape Modification

Let us now investigate solutions to the minimization problem with a modification to the space-

craft’s shape. The primary change was to modify the original cubic structure to become a

generic rectangular prism, which could have any size along all 3 axes. Previously, as a cube,

the element length as well as each element area were easy to calculate as the total lengths in

all 3 axes were the same. The size of the area was inherently square, and there were an equal

number of squares on all 6 faces. However, in this model, as the lengths may not be the same,

either the size of each element or the number of them on each face would have to be adjusted.

The approach that was taken was to keep the elements as square but create a different amount

on each face. Assessing the dimensions of these elements was the next problem, but this was

resolved by employing the following algorithm:

1. Establish the length or the largest dimension

2. Determine the largest dimension

3. Define the element length as the length/largest dimension

4. Element area = element length squared

40

CHAPTER 4. SPACECRAFT MODEL MODIFICATION 41

As laid out in the aforementioned algorithm, here is how it is applied to a 15 × 15 × 25, or

0.6 m× 0.6 m× 1.0 m spacecraft:

1. Largest Length = 1.0 m

2. Largest Dimension = 25 Elements

3. Element Length = 40 mm

4. Element Area = 1.6 mm2

The average temperature on each face as the solution to the inverse problem in Scenario 2

(Figure 2.3), with no rotation, are shown for the described spacecraft with Td,i = 300K.

Table 4.1: Modified Shape: SQP Optimization Results


Average 299.9948

A graphical representation to the solution to the inverse problem to the same scenario is shown

in Figures 4.1 and 4.2:


0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

287.584

288.843

290.101

291.359

292.617

293.875

295.133

296.391

297.650

298.908

Figure 4.1: Inverse Problem: Modified Shape (15× 15× 25) - View 1


0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

287.584

288.843

290.101

291.359

292.617

293.875

295.133

296.391

297.650

298.908

Figure 4.2: Inverse Problem: Modified Shape (15× 15× 25) - View 2


4.2 Tray Expansion

As the original design only accounted for a hollow cube, the next major step in this was to

include trays within the spacecraft. From this point forward, the term face will also refer to the

added trays.

Figure 4.3: Example of a Spacecraft with 3 Trays

The major challenges with the trays were to:

1. Place them in locations that would align with the nodes on the perpendicular faces

2. Modify the conduction logic to account for the interaction that would take place between

the nodes on the faces and the trays

3. Modify the radiation logic, specifically the Gebhart factor analysis to account for the

“chambers” that would be created/contained within these frame.


For a 15 × 15 × 25 prism with 3 trays, the average temperature on each face as the solution

to the inverse problem in Scenario 2 (Figure 2.3), with no rotation, is shown for the described

spacecraft with Td,i = 300K.

Table 4.2: 3 Trays: SQP Optimization Results

Face αs εout εin T evalFace 1 1.0000 0.0762 0.9251 300.0689Face 2 0.4093 1.0000 0.9377 300.0759Face 3 1.0000 0.0762 0.9251 300.0689Face 4 0.2959 1.0000 0.9377 300.0761Face 5 1.0000 0.2263 1.0000 300.0597Face 6 1.0000 0.2263 1.0000 300.0597Tray 1 1.0000 0.7906 0.8914 299.8377Tray 2 1.0000 1.0000 0.9177 299.7464Tray 3 1.0000 0.7906 0.8914 299.8377

Average 300.0021

A graphical representation to the solution to the inverse problem to the same scenario is shown

in Figures 4.4 and 4.5:


0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

264.158

276.798

289.438

302.078

314.718

327.358

339.998

352.637

365.277

377.917

Figure 4.4: Inverse Problem: 15× 15× 25 with 3 Trays - View 1


0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

264.158

276.798

289.438

302.078

314.718

327.358

339.998

352.637

365.277

377.917

Figure 4.5: Inverse Problem: 15× 15× 25 with 3 Trays - View 2


4.3 Comparison

In conclusion, by comparing the effects of the two different modifications made to the space-

craft (changing the shape, and adding trays), we see that in Table (4.3), that there is a slight

increase in the average temperature.

Table 4.3: Results Summary of 15× 15× 25 Spacecraft with Different Modifications

Tray N Ω T eval Ψ ∆T d Time(s) Iter0 1950 18 299.9948 9.1389E+01 0.2165 355540.55 1733 2625 27 300.0021 1.0086E+03 0.6199 1212971.15 175

Although, the comparative results are not drastic, by knowing the effects of modifying a space-

craft we can use this to our advantage in future designs.

4.4 Orbital Dynamics

To get a more accurate grasp of the spacecraft’s incident flux, we need to track its position

with respect to both the Earth and the Sun. To achieve this, we need to create a Heliocentric

coordinate system, and via the utilization of orbital dynamics we need to track the spacecraft’s

position. By taking R0 as the components of the position vector of an object in a body-fixed

frame we arrive at:

~R0 = ~FT

0R0 (4.1)

Then we can re-write the component equation (Eq. (4.1)) as:

~R0 = ~FT

b Cb0[0 0 r]T (4.2)

where r is the position of the body, calculated via the following orbital dynamics. This analysis

will be restricted to circular and elliptical orbits (0 ≤ e < 1). The semi-major axis (a) dictates

the orbital size, while the eccentricity (e) dictates the orbital shape. Knowing the periapsis

(rp), the closest point in the orbit, and the apoapsis (ra), the farthest point in the orbit, we can


calculate the semi-major axis of an ellipse using:

a =1

2(rp + ra) (4.3)

With the knowledge of a, we can calculate the eccentricity:

e =raa− 1 (4.4)

We can now evaluate the semi-latus rectum (p), the period (T ), and the mean anomaly (M ):

p = a(1− e2) (4.5)

T = 2π

√a3

µ(4.6)

M = 2πt− t0T

(4.7)

where t is the current orbital time, and t0 is the time of periapsis passage. The mean anomaly

(M ), which is the angle covered if the body moved constantly in orbit at an average rate, can

be used to calculate the eccentric anomaly (E), using Kepler’s equation:

M = E − esin(E) (4.8)

which can be written as:

f(E) = E − esin(E)−M = 0 (4.9)

Kepler’s Equation can be solved via Newton’s Method shown below:

En+1 = En −f(En)

f ′(En), n = 0, 1, 2, . . . (4.10)


Now, the true anomaly (θ), the angle between the orbital position and direction of perigee

measured from the focus, can be calculated:

θ = 2 arctan

(√1 + e

1− etan

(E

2

))(4.11)

Finally, we can calculate the magnitude r of the body’s position.

r =p

1 + e cos(θ)(4.12)

Using the calculated value of r, we can determine R0 = [0 0 r]T . Furthermore, we can obtain

r using:

r =

r cos(θ)

r sin(θ)

0

(4.13)

We will be using the following subscripts to identify the 3 major bodies described in our anal-

ysis: the Sun (), Earth (⊕), and the spacecraft (). Using the standard values for the Earth

(a⊕ = 1.496×1011m, e⊕ = 0.0167), we can solve Eq. (4.12), providing us with the magnitude

of the Earth’s distance from the Sun (∣∣r(,⊕)

∣∣). Assuming that the Sun and Earth both lie on

the same plane along the same axis, we can state that the Earth’s coordinates with respect to

the Sun’s are:

r(,⊕) =

r(,⊕)

0

0

(4.14)

Similarly, using values of µ⊕ = 3.986 × 1014m3/s2), we can solve Eqs. (4.12) and (4.13),

providing us with the magnitude of the spacecraft’s distance from the Earth (r(⊕,)), and its

coordinates (r(⊕,)).


By taking the relative distances from the Earth (Geocentric Frame), and putting it into a He-

liocentric one, we can then obtain the relative distance between the Sun and the spacecraft

(r(,)).

r(,) = r(,⊕) − r(⊕,) (4.15)

One of the requirements that were outlined was the ability to rotate the spacecraft at each

measurement. This can be achieved by creating the following rotation matrix:

C21 =

cos θ2 cos θ3 cos θ2 sin θ3 − sin θ2

sin θ1 sin θ2 cos θ3 − cos θ1 sin θ3 sin θ1 sin θ2 sin θ3 + cos θ1 cos θ3 sin θ1 cos θ2

cos θ1 sin θ2 cos θ3 + sin θ1 sin θ3 cos θ1 sin θ2 sin θ3 − sin θ1 cos θ3 cos θ1 cos θ2

(4.16)

By multiplying the 3-2-1 attitude sequence rotation matrix (Eq. (4.16)) with r(,) and r(⊕,),

we can account for the rotation of the spacecraft as well as the axial tilt of the Earth.

r(,) = C21r(,) (4.17)

r(⊕,) = C21r(⊕,) (4.18)

where the Earth’s axial tilt of i⊕ = 23.44 will be applied to θ2. Knowing the position of the

spacecraft, and our coordinate system, we can measure the angles of all incident fluxes (solar,

Earth, Albedo) and determine how they interact with the spacecraft.


4.5 External Flux Calculation

Now that we have established the angles of all incident fluxes (solar, Earth, Albedo) to the

spacecraft, we can determine the magnitude of the solar and Earth fluxes based on the assump-

tion that they are both black bodies. For the Sun,

Φs =σT 4R

2∣∣~r(,)

∣∣2 (4.19)

where T = 5778K is the average effective surface temperature of the Sun, and R =

695 500 km is the average radius of the Sun. Similarly, we can utilize the same procedure

for the Earth:

Φe =σT 4⊕R

2⊕∣∣~r(⊕,)

∣∣2 (4.20)

where T⊕ = 254K is the average effective surface temperature of the Earth, and R⊕ =

6378.1 km is the average radius of the Earth:

4.5.1 Results

By calculating the spacecraft’s orbital path around Earth, we are able to create a series of evenly

spaced points to determine how the incident fluxes will vary at each point, and when they will

impinge the spacecraft. We are further able to analyze the temperature distribution that occurs

at each point. In the following analysis we can see the temperature distribution at 8 different

points along the orbit. Using the same design parameters that we have been using thus far,

(αs,i = 0.50, εout,i = 0.75, εin,i = 1.00), for a 15 × 15 × 25 prism with 3 trays at an altitude

of (rp, = 8000 × 103 km, ra, = 8000 × 103 km), the following results are obtained for the

forward problem.


0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

251.920

258.951

265.981

273.011

280.041

287.072

294.102

301.132

308.163

315.193

Figure 4.6: θ = 0.0000 for Circular Orbit

0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

255.582

262.934

270.287

277.639

284.991

292.343

299.695

307.047

314.399

321.752



0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

239.321

248.114

256.907

265.701

274.494

283.288

292.081

300.875

309.668

318.462


0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

203.761

209.267

214.772

220.277

225.783

231.288

236.794

242.299

247.805

253.310



0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

188.196

194.580

200.965

207.349

213.734

220.118

226.502

232.887

239.271

245.656


0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

203.761

209.267

214.772

220.277

225.783

231.288

236.794

242.299

247.805

253.310

Figure 4.11: θ = −2.3562 for Circular Orbit


0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

188.199

194.584

200.968

207.353

213.738

220.122

226.507

232.892

239.276

245.661


0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

255.582

262.934

270.287

277.639

284.991

292.343

299.695

307.047

314.399

321.752



Table 4.4: Summary of Values in Circular Orbit with ϕΦ = [0 0 0]T

θ R,X R,Y Tmax T Tmin Φs Φe

0.0000 -8000000 0 315.1929 275.9242 251.9202 1413.0300 150.02070.7854 -5656854 -5656854 321.7515 285.0043 255.5824 1412.9850 150.02071.5708 0 -8000000 318.4616 274.5514 239.3206 1412.8763 150.02072.3562 5656854 -5656854 253.3101 227.1346 203.7611 1412.7676 150.02073.1416 8000000 0 245.6557 207.1989 188.1960 1412.7226 150.0207-2.3562 5656854 5656854 253.3101 227.1346 203.7611 1412.7676 150.0207-1.5708 0 8000000 245.6610 207.2026 188.1989 1412.8763 150.0207-0.7854 -5656854 5656854 321.7515 285.0043 255.5824 1412.9849 150.0207

321.7515 248.6444 188.1960 1412.8763 150.0207

Table 4.5: Summary of Values in Circular Orbit with ϕΦ = [0 i⊕ 0]T


0.0000 -8000000 0 316.3488 286.0882 264.4616 1413.0300 150.02070.7854 -5656854 -5656854 322.4312 293.4753 264.5813 1412.9850 150.02071.5708 0 -8000000 318.9925 281.5589 239.7569 1412.8763 150.02072.3562 5656854 -5656854 255.1915 231.3594 204.2855 1412.7676 150.02073.1416 8000000 0 248.3131 214.7265 189.5688 1412.7226 150.0207-2.3562 5656854 5656854 255.1915 231.3594 204.2855 1412.7676 150.0207-1.5708 0 8000000 245.6610 207.2026 188.1989 1412.8763 150.0207-0.7854 -5656854 5656854 322.4312 293.4753 264.5813 1412.9849 150.0207

322.4312 254.9057 188.1989 1412.8763 150.0207

Table 4.6: Summary of Values in Circular Orbit with ϕΦ = [π4i⊕

π4]T


0.0000 -8000000 0 319.4621 298.7581 280.2670 1413.0300 150.02070.7854 -5656854 -5656854 321.5395 298.4811 266.4849 1412.9850 150.02071.5708 0 -8000000 332.8659 300.4864 257.0342 1412.8763 150.02072.3562 5656854 -5656854 248.6654 212.7391 184.5071 1412.7676 150.02073.1416 8000000 0 254.1911 224.1250 194.5904 1412.7226 150.0207-2.3562 5656854 5656854 254.8175 224.8053 199.1732 1412.7676 150.0207-1.5708 0 8000000 255.1169 231.4835 204.1984 1412.8763 150.0207-0.7854 -5656854 5656854 318.2664 293.8292 266.3655 1412.9849 150.0207

332.8659 260.5885 184.5071 1412.8763 150.0207


Table 4.7: Summary of Values in Eccentric Orbit with ϕΦ = [π4i⊕

π4]T


0.0000 -6500000 0 321.0299 302.9669 285.9911 1413.0012 227.25031.0975 -3241414 -6329245 325.0749 300.8370 262.2130 1412.9386 189.87671.9374 2966255 -7725004 252.7425 226.8071 198.4744 1412.8193 140.21792.5816 7775061 -4874211 247.8370 215.6849 187.5669 1412.7269 114.01713.1416 9500000 0 247.2855 218.6519 190.6912 1412.6938 106.3859-2.5816 7775061 4874211 245.4451 215.7460 190.6664 1412.7269 114.0171-1.9374 2966255 7725004 252.7896 225.3447 202.0921 1412.8193 140.2179-1.0975 -3241414 6329245 326.5270 300.7868 271.8988 1412.9385 189.8767

326.5270 250.8532 187.5669 1412.8331 152.7325

A summary of all of the gathered data regarding the pertinent scenarios determining the ideal

orbital shape, orbital size, and orientation are presented in Table (4.8). Our analysis of it, will

enable us to ascertain the best way to achieve our spacecraft’s desired temperatures and dictate

how to proceed.

Table 4.8: Summary of Values in all Scenarios

rp (km) ra (km) ϕΦ Tmax T Tmin Φs Φe

8000 8000 [0 0 0]T 321.7515 248.6444 188.1960 1412.8763 150.02078000 8000 [0 i⊕ 0]T 322.4312 254.9057 188.1989 1412.8763 150.02078000 8000 [π

40 π

4]T 334.4261 261.1535 184.5071 1412.8763 150.0207

8000 8000 [π4i⊕

π4]T 332.8659 260.5885 184.5071 1412.8763 150.0207

6500 9500 [0 i⊕ 0]T 323.7972 248.0908 185.7212 1412.8331 152.73256500 9500 [π

4i⊕

π4]T 326.5270 250.8532 187.5669 1412.8331 152.7325

From what we have seen in Table (4.8), the best scenario to use is the circular orbit with the

spacecraft rotated [π4

0 π4]T , however we will incorporate the axial tilt for the sake of realism.

By collecting the data of a 100-point orbit, and then averaging the temperature at each node,

we arrive at an average composite image. This process took 184.3 mins, and two views of the

composite image are shown in Figures 4.14 and 4.15


0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

239.392

244.086

248.779

253.473

258.166

262.860

267.553

272.247

276.940

281.634

Figure 4.14: 100-Point Composite Image - View 1


0.10.2

0.30.4

0.5

0.10.2

0.30.4

0.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Temperature (K)

239.392

244.086

248.779

253.473

258.166

262.860

267.553

272.247

276.940

281.634

Figure 4.15: 100-Point Composite Image - View 2


4.6 Multi-Scenario Optimization

In Chapter 2, three specific scenarios (full eclipse, opposite face flux, adjacent face flux) were

highlighted. The goal of this was to capture all of the scenarios (best/worst case) that would

interface with the spacecraft. However, upon further analysis, instead of creating a weighted

optimization of the three, the approach taken was to optimize over the span of the spacecraft’s

orbit by wrapping the different scenarios together within a new cost function:

Ψ(ΩR) =Ns∑j=1

N∑i=1

ζi(Tij − Td,ij)2 (4.21)

where N is the total number of discretized nodes, Ns is the total number of orbital points, ζi

is the weighting at each node, Tij and Td,ij are the temperature and the desired temperature

respectively for each node i and each orbit j. Here, we will optimize over all the points in the

orbit to come up with a set of design parameters and consequently compare it to our desired

values. Again, we will be aware that there will be an equal weighting (ζi = 1) of all nodes.

4.6.1 Proof of Concept

To demonstrate the feasibility of this optimization method, I have elected to take a simplified

5× 5× 5 cube rotated [π4i⊕

π4]T over 20 points in orbit at altitudes of 6500 km and 8000 km.

Again, the desired temperature will be 300K at all nodes.

Table 4.9: Results Summary of Spacecraft in Orbit at Different Altitudes (Td,i = 300K)

R T eval Ψ ∆T d Time(s) Iter6500 254.7405 3.3627E+05 47.3473 3681.30 1518000 253.2345 3.6223E+05 49.1411 3529.75 144

As it can be seen, the ∆T d is fairly large which indicates that the optimization failed to get to

a value close to our desired value of 300K. The average temperature on each face, as well as

the overall average is shown in Tables (4.10) and (4.11) for both altitudes respectively:


Table 4.10: Optimization Results at R = 6500 km, Td,i = 300K

Face αs εout εin TevalFace 1 0.8554 1.0000 1.0000 253.9902Face 2 1.0000 0.5358 0.0000 250.8543Face 3 1.0000 0.8609 0.0940 265.5492Face 4 0.4995 1.0000 0.0000 265.3594Face 5 0.8085 1.0000 0.0000 228.7639Face 6 0.4636 1.0000 0.0000 263.9260

Average 254.7405



Average 253.2345

From the two prior tables, it appears as though there is no solution for a desired temperature

greater than 270K. This would imply that this approach of a purely passive thermal design is

not feasible for human-travel, or where there is a requirement for the spacecraft to be warmer.

However, if we set the desired temperature to 400K at all nodes, we attain the following results:

Table 4.12: Results Summary of Spacecraft in Orbit at Different Altitudes (Td,i = 400K)

R T eval Ψ ∆T d Time(s) Iter6500 332.6428 7.0253E+05 68.4362 4638.59 1388000 335.7124 6.4467E+05 65.5575 4083.49 145

Surprisingly, the cost functions for both altitudes, and hence the ∆T d, are much higher than

the optimization for the lower desired temperature. Moreover, when we examine the average

temperature on each face, as well as the overall average, we see that we indeed are able to

break our prior assumed upper threshold of 270K. The average temperature on each face, as

well as the overall average is shown in Tables (4.13) and (4.14) for both altitudes respectively:




Average 332.6428



Average 335.7124

There are a couple interesting conclusions that we can learn from Table (4.12). The first being

that if we set the desired temperature to be 400K, we can achieve an average temperature of

approximately 330K, which may be too hot, but convinces us that the approach of a purely

passive thermal design is feasible for human-travel. What could be useful here is either an

electrical appliance to dissipate heat or a mechanism to store heat/energy and redistribute it

when in an eclipse situation. But based purely on optimization, a logical next step would be

to try the optimization again with the desired temperature set to 350K. The second interesting

conclusion is that the design variables for both altitudes are very close to one another. Because

of this we will try our next simulation with αs,0,i = 1.00, εout,0,i = 0.50, εin,0,i = 0.00 at an

altitude of 8000 km. The results for this are shown in Table (4.15).




Average 298.8118

Interestingly enough, the prior simulation gets us an average temperature very close to the

300K we initially desired, and also cuts down the processing time and number of iterations

considerably. Table (4.16) shows the orbital temperature statistics of Table (4.15) highlighting

the temperature fluctuation that the spacecraft endures.

Table 4.16: Summary of Values in Orbit


0.0000 -8000000 0 381.2795 350.6992 321.8663 1413.0300 150.02070.3142 -7608452 -2472136 400.5414 347.1268 282.7316 1413.0225 150.02070.6283 -6472136 -4702282 410.9575 348.7188 269.1594 1413.0006 150.02070.9425 -4702282 -6472136 412.7024 350.6740 268.3168 1412.9666 150.02071.2566 -2472136 -7608452 404.5076 350.6616 266.8236 1412.9238 150.02071.5708 0 -8000000 406.9047 350.7168 265.9138 1412.8763 150.02071.8850 2472136 -7608452 339.2291 263.9807 180.0674 1412.8288 150.02072.1991 4702282 -6472136 332.3082 252.8953 180.1754 1412.7860 150.02072.5133 6472136 -4702282 321.2708 249.7845 202.8677 1412.7520 150.02072.8274 7608452 -2472136 303.1542 255.0961 208.5903 1412.7301 150.02073.1416 8000000 0 284.1628 251.9119 207.0827 1412.7226 150.0207-2.8274 7608452 2472136 297.6060 239.7462 200.0946 1412.7301 150.0207-2.5133 6472136 4702282 305.6071 241.7399 190.6808 1412.7520 150.0207-2.1991 4702282 6472136 310.2138 244.4751 185.7828 1412.7860 150.0207-1.885 2472136 7608452 335.9961 244.0819 170.6266 1412.8288 150.0207-1.5708 0 8000000 355.4360 251.0368 172.2773 1412.8763 150.0207-1.2566 -2472136 7608452 381.1297 345.1581 273.9682 1412.9238 150.0207-0.9425 -4702282 6472136 377.8884 342.8655 274.3369 1412.9666 150.0207-0.6283 -6472136 4702282 367.4104 344.8775 303.4564 1413.0006 150.0207-0.3142 -7608452 2472136 354.2030 349.9892 346.8069 1413.0224 150.0207

412.7024 298.8118 170.6266 1412.8763 150.0207


Table (4.17) summarizes the important findings attained from the simulations at our three de-

sired temperatures.

Table 4.17: Results Summary at R = 8000 km for Different Td

Td T eval Ψ ∆T d Time(s) Iter300 253.2345 3.6223E+05 49.1411 3529.75 144350 298.8118 4.0251E+05 51.8018 2829.83 108400 335.7124 6.4467E+05 65.5575 4083.49 145

The most apparent outcome here is that as the desired temperature (Td) increases, so do Ψ,

∆T d, and σT . By performing a similar simulation with the full parameter set (each node has

its own set of design parameters), we arrive at the following results:

Table 4.18: Results Summary at R = 8000 km for Different Td

Td T eval Ψ ∆T d Time(s) Iter300 260.4735 2.4030E+05 40.0253 103463.90 181350 300.9526 3.6522E+05 49.3435 115497.02 184400 342.3582 5.0102E+05 57.7937 116292.77 189

The starting points in the prior simulation (Table (4.17)) would not converge, so the following

had to be used in this simulation (Table (4.18)): αs,0,i = 0.50, εout,0,i = 0.50, εin,0,i = 0.50.

From this we can see that overall outcomes are fairly similar except that in Table (4.18) Ψ, and

σT have slightly smaller values, and a marginally warmer T eval. Moreover, as expected, the

time to perform the computation for the full parameter set is greatly longer. It is also interesting

to note that when Td = 350K in both simulations, the results from the two tables are closest

to each other. Future steps could look at focusing on a 15 × 15 × 25 prism with 3 trays, and

optimizing for a desired temperature of 350K to achieve our actual goal of 300K.


4.7 Surface Property Realization

Based on the results of the optimization, we will attain a particular emissivity (ε) and solar

absorptivity (αs) for each node. By taking the ratio of these values, we will define ε = αs

ε

Based on the prior distribution, we will be able to determine the ideal set of TCCs, which have

been categorized as follows [4]:

• Solar Reflectors (SR): Have a ratio of ε 1

• Flat Coatings (FC): Have a ratio of ε ≈ 1

• Solar Absorbers (SA): Have a ratio of ε 1

Taking our results from the orbital optimization we can determine values of ε and select appro-

priate TCC category type for each face.

Table 4.19: TCC Selection

Face Teval αs εout εin εout εin TCCout TCCinFace 1 286.4880 1.0000 0.3042 0.0000 3.2873 ∞ SA SAFace 2 294.4852 1.0000 0.2484 0.0000 4.0258 ∞ SA SAFace 3 309.9032 1.0000 0.5022 0.0000 1.9912 ∞ SA SAFace 4 306.9810 0.9256 1.0000 0.0000 0.9256 ∞ FC SAFace 5 290.3720 1.0000 0.9250 1.0000 1.0811 1.0000 FC FCFace 6 300.3226 0.9335 1.0000 0.1499 0.9335 6.2275 FC SA

Chapter 5

Conclusion and Future Work

Based on previous conventional wisdom, it was expected that the rejection of heat into space

will be of greater concern than the requirement of getting heat into the spacecraft. What was

discovered was that in general the opposite is true, and that we struggled to attain a reasonable

climate within the spacecraft. More importantly, the idea that we can sustain a particular

temperature or a temperature range throughout an orbit is difficult to support. As seen in

Table (4.16), the spacecraft reaches a maximum temperature of 412.7024K and a minimum

temperature of 170.6266K, even though the average temperature is 298.8118K. We have also

noticed that by making modifications to the spacecraft such as adding trays, slightly increases

the average temperature. There are several venues to build upon the conducted research thus

far, and the principle behind all of these approaches will be to remain energy neutral. These

forays will deal with temperature regulation, location, orbit, and materials.

67

CHAPTER 5. CONCLUSION AND FUTURE WORK 68

5.1 More Optimization

In this work we failed to find a global optimum; however the quest to find whether one exists

is still valuable. A gradient-free method that we did not use, but could prove effective is the

particle swarm method. It is similar to the genetic algorithm, but its algorithm tackles the

problem in a different manner. Also, throughout this work we have used cubes and rectangular

prisms as the shape of our spacecraft. It would be meaningful to perform an optimization of the

spacecraft’s geometry to see what the ideal shape and size are to attain the maximum energy

within a particular orbit. We have found that a circular orbit allows higher temperature than a

particular elliptical orbit (e = 0.6842). What we did not determine was the ideal orbital shape

for the spacecraft. We can achieve this by performing an optimization of the spacecraft’s orbit.

The answer will most probably be a circular orbit as close to the Earth as possible, however

this could prove to be an interesting investigation. Additionally, as we have determined that

the threshold problem is a reality, we can modify our equation to contain a series of inequality

constraints to handle the temperature threshold at each node. Moreover, we can make use of

the rate of internal power dissipation, qp (Eq. (2.34)) to place heat sources in the spacecraft

to complement the calculated temperature at each node, and even optimize for different power

dissipation scenarios.


5.2 Thermal Passivity at Lagrangian Points

NASA’s Space/Earth Science Enterprise missions are either within the Earth’s orbit, at a La-

grangian point, or they travel to another planet. Many of these proposed future missions involve

dealing with deep cryogenic temperatures, and very tight and stable temperature control. Due

to this, there is a compelling necessity for new thermal control technology that will minimize

mass/power and create a more reliable and robust space science missions to explore deep space

and other planets [14]. From the onset of this work we have always considered the situa-

tion where the spacecraft will have varying quantities of heat flux, and have had to account

particularly for the eclipse scenario. It is my belief that research can commence on placing

a spacecraft or space structure at a stable Lagrangian Point and determine whether it can be

thermally passive at this location. This will allow the sustainability of large structures, such as

hotels or mines, and be cost effective without an expensive thermal system.

5.3 Passive Thermal Control Components

As we know, the spacecraft’s temperature varies dramatically with its orbit. To ensure that

we have our desired temperature at all points in the orbit, the idea here is to add components

within our spacecraft which will store and release heat based on the external temperature. We

have already talked about thermal control coatings, some of the other ones as mentioned in the

introduction are: phase change devices, multi layer insulation, and thermal doublers.


5.3.1 Phase Change Devices

Phase change systems use the dimensional changes (expansion and contraction) that occur in

materials as they undergo changes between phases (solid, liquid and gas). These mechanisms

absorb thermal energy by changing from a solid to a liquid, and then re-solidify as the temper-

ature decreases, emitting energy. A common type of this is an aluminum container filled with

wax [1]. Downsides to this type of apparatus are that it will be unable to absorb anymore heat

after melting, and there are also concerns over the liquid leaking.

5.3.2 Multi Layer Insulation

By reducing the rate of heat flux between two boundary surfaces, this type of protection re-

duces thermal control requirements, and also prevents a large heat influx. Typically MLIs are

closely spaced layers of aluminized Mylar or Kapton alternated with a course net material [1].

Equipment within the spacecraft can be wrapped with MLIs to thermally isolate them, or the

entire spacecraft’s interior can be wrapped to prevent heat loss.

5.3.3 Thermal Doublers

By having highly conductive materials placed in thermal contact with a component, thermal

doublers act as a heat sink when the internal temperature rises. As the temperature cools,

the heat is dispersed by radiation or conduction. This is another useful tool in regulating the

variance of the spacecraft’s temperature.


5.4 Thermal Control

Thermochromism is the ability of substances to change colour due to a change in temperature.

According to Day [15] “Thermochromism is defined operationally as an easily noticeable re-

versible colour change in the temperature range limited by the boiling point of each liquid,

the boiling point of the solvent in the case of solution or the melting point for solids.” To

date, the most important thermochromic materials that are applied to colour textiles and other

commercial products involve either liquid crystals, or organic dyes [16].

5.4.1 Liquid Crystals

Liquid crystals exhibit a phase of matter that has properties between a conventional liquid, and

a solid crystal. There are three phases: crystalline (low-temperature), cholesteric, and isotropic

liquid (high-temperature), however only the cholesteric has thermochromic properties. Colours

will change from black to red, orange, yellow, green, blue, violet, and again to black. They

are used in precision applications, since they can be engineered to accurate temperatures, and

operate between 30−120C. Some common applications are fish-tank thermometers and mood

rings.

5.4.2 Organic Dyes

Organic dyes are mixtures of leuco dyes with other chemicals, displaying a colour change,

usually from colourless to coloured, based on temperature. They allow a much wider range

of colours than the liquid crystal, but are more difficult to accurately control. Hyper-colour

T-shirts and heat-sensitive fax paper are some applications.


5.4.3 Applications

Thermochromic paint involves the use of liquid crystal or leuco dye technology, and can be

implemented via different materials such as inks, dyes, paint, papers, etc. Inks, for example,

temporarily change colour with exposure to heat, whereas paints change the way they absorb

and emit light at a different wavelengths after absorbing a certain amount of heat. As mentioned

by White and Leblanc [16], thermochromic compounds have become increasingly important

in recent years in the study and production of coatings that respond to their environment (smart

coatings). Incorporated into a window system, a thermochromic material could be used to

control the flux of solar energy by changing the transmission of visible light in response to

the heating effect of sunlight. A potential application would be to apply a smart coating to

the spacecraft, to control its temperature by adjusting its colour and radiation properties, based

on the incoming heat flux. However it has also been mentioned that only organic compounds

have reversible colour changes (colouring and fading reactions) that are due to the effect of

temperature, exclusively [17]. This is why more research and expertise are needed in this field.

Bibliography

[1] M. Fischer, ed., 10 Thermal Control. Texas Space Grant Consortium, 1995.

[2] M. V. Papalexandris and M. H. Milman, “Active control and parameter updating tech-

niques for nonlinear thermal network models,” Computational Mechanics, vol. 27,

pp. 11–22, 2001.

[3] M. Milman and W. Petrick, “A note on the solution to a common thermal network problem

encountered in heat-transfer analysis of spacecraft,” Applied Mathematical Modelling,

vol. 24, pp. 861–879, July-August 2000.

[4] R. D. Karam, Satellite Thermal Control for Systems Engineers (Progress in Astronautics

and Aeronautics). AIAA (American Institute of Aeronautics & Astronautics, 1998.

[5] B. Gebhart, “Surface temperature calculations in radiant surroundings of arbitrary com-

plexity for gray, diffuse radiation,” Int. J. Heat Mass Transfer, vol. 3, pp. 341–346, 1961.

[6] C. K. Krishnaprakas, “Application of accelerated iterative methods for solution of thermal

models of spacecraft,” Journal of Spacecraft and Rockets, vol. 32, pp. 341–346, July-

August 1995.

[7] S. Suresha and S. C. Gupta, “Calculation of sensitivities in thermal control systems with

nonlinear inequality constraints,” Journal of Spacecraft and Rockets, vol. 35, pp. 552–

558, July-August 1998.

73

BIBLIOGRAPHY 74

[8] M. S. Eldred, “Optimization strategies for complex engineering applications,” tech. rep.,

Sandia National Laboratories, 1998.

[9] P. M. Pardalos, H. E. Romeijn, and R. Horst, Handbook of Global Optimization. Springer,

1995.

[10] R. L. Galski, F. L. de Sousa, F. M. Ramos, and I. Muraoka, “Spacecraft thermal design

with the generalized external optimization algorithm,” in Inverse Problems, Design and

Optimization Symposium, (Rio de Janeiro), 2004.

[11] K. B. Narayana, “Satellite thermal network correction,” Engineering Optimization,

vol. 12, no. 4, pp. 269–280, 1987.

[12] M. Luersen, R. L. Riche, and F. Guyon, “A constrained, globalized, and bounded nelder-

mead method for engineering optimization,” Structural and Multidisciplinary Optimiza-

tion, vol. 27, pp. 43–54, May 2004.

[13] J. R. R. A. Martins, “Optimization concepts and applications.” AER1415, University of

Toronto, 2009.

[14] T. D. Swanson and G. C. Birur, “Nasa thermal control technologies for robotic space-

craft,” Applied Thermal Engineering, vol. 23, no. 9, pp. 1055 – 1065, 2003. 12th Interna-

tional Heat Pipe Conference, 2002.

[15] J. H. Day, “Thermochromism,” Chem. Rev., vol. 63, pp. 65 – 80, 1963.

[16] M. A. White and M. LeBlanc, “Thermochromism in commercial products,” Journal of

Chemical Education, vol. 76, pp. 1201–1205, September 1999.

[17] A. Samat and V. Lokshin, Organic Photochromic and Thermochromic Compounds: Vol-

ume 2: Physicochemical Studies, Biological Applications, and Thermochromism (Topics

in Applied Chemistry). Springer, 2002.

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