numerical methods for computing the temperature

PROJECTE FINAL DE CARRERA

NUMERICAL METHODS FOR COMPUTING

THE TEMPERATURE DISTRIBUTION IN

SATELLITE SYSTEMS

Estudis: Enginyeria Electrònica Autor: Francisco José Gómez-Valadés Maturano Director/a: Martin Altenburg, Werner Grimm

Any:2012

i

i

Author’s Statement

Hereby I confirm that I did the thesis on my own except the advice of my supervisor,

using only the sources and aids mentioned in the report.

Furthermore, I certify that I know and accept that I have no right to exploit the results

of my Master Thesis by any means without the written permission of the “Institut für

Flugmechanik und Flugregelung”.

Stuttgart, 22nd March 2012

........................................ ..........................................

(place and date) (signature)

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TABLE OF CONTENTS

AUTHOR’S STATEMENT ................................ ................................................... I

TABLE OF CONTENTS ................................. .................................................... II

RESUM DEL PROJECTE ................................ ................................................. IV

RESUMEN DEL PROYECTO ............................................................................ V

ABSTRACT .......................................... ............................................................. VI

LIST OF FIGURES AND TABLES ........................ ........................................... VII

NOMENCLATURE ...................................... ...................................................... IX

INTRODUCTION ................................................................................................ 1

1. Context of the work ............................... ............................................................................ 1

2. The LISA mission................................... ............................................................................ 1

3. Contents of the thesis ............................ ........................................................................... 2

CHAPTER 1. THERMAL ANALYSIS ....................... ......................................... 3

1.0. Introduction ...................................... .................................................................................. 3

1.1. Thermal control systems ........................... ....................................................................... 3

1.2. TCS modeling philosophy ........................... ..................................................................... 5

1.3. The heat equation ................................. ............................................................................. 5

CHAPTER 2. THERMAL MATHEMATICAL MODEL ............. ........................... 9

2.1. Introduction ...................................... .................................................................................. 9

2.2. Partial differential equations .................... ........................................................................ 9

2.3. Methods to solve PDEs ............................. ........................................................................ 9

2.3.1. Analytical methods to solve PDEs .................. ................................................................ 9

2.3.2. Numerical methods to solve PDEs ................... ............................................................. 12

2.4. The finite methods ................................ ........................................................................... 16

iii

2.4.1. Finite difference method .......................... ....................................................................... 16

2.4.2. Finite element method ............................. ....................................................................... 17

2.4.3. Finite volume method (FVM) ........................ .................................................................. 17

2.4.4. What scheme to use ................................ ........................................................................ 18

2.5. The lumped parameter method ....................... ............................................................... 19

2.5.1. The lumped parameter equations .................... .............................................................. 20

2.5.2. LP conductivities on a rectangular grid ........... ............................................................. 20

2.5.3. Derivation of Lumped Parameter equations by Finite Differencing ........................... 21

2.6. Matrix notation for heat balance .................. .................................................................. 23

CHAPTER 3. PROCEDURE TO PERFORM TEMPERATURE DISTRIB UTION CALCULATIONS ...................................... ....................................................... 27

3.1. Introduction ...................................... ................................................................................ 27

3.2. Methodology ....................................... ............................................................................. 27

3.3. MATLAB algorithm .................................. ........................................................................ 30

3.4. ESATAN algorithm................................... ........................................................................ 32

CHAPTER 4. MODEL SIMULATIONS, ANALYSIS AND DATA VERIFICATION ................................................................................................ 33

4.1. Introduction ...................................... ................................................................................ 33

4.2. TMS.R3 with zero capacities ....................... ................................................................... 33

4.2.1. TMS.R3 with zero capacities ....................... ................................................................... 33

4.2.2. TMS.R3 vs. 10.2 with capacities ................... .................................................................. 34

4.3. Analysis of a simple case: the heating bar ........ ........................................................... 34

4.4. Analysis of a complex case: the LISA model ........ ....................................................... 36

4.4.1. The node 12120 .................................... ............................................................................ 38

4.4.2. The node 12090 .................................... ............................................................................ 41

4.4.3. Boundary nodes .................................... .......................................................................... 44

CHAPTER 5. CONCLUSIONS AND FUTURE WORK ............ ........................ 47

BIBLIOGRAPHY AND REFERENCES ....................... ..................................... 48

iv

Resum del projecte

La present tesi s’ha fet a l’empresa ASTRIUM per buscar nous mètodes dels quals

obtenir distribucions de temperatura. Els paquets de software actuals tals com ESATAN

o ESARAD no només proporcionen excel·lents solucions d’anàlisi tèrmica, encara que a

un preu molt alt doncs consumeixen molt temps, sino que també permeten realizar

simulacions radiatives en condicions d’òrbita. Degut a que les llicències d’aquest

producte estan normalment limitades per l’ús de molts enginyers, és important

proporcionar noves eines per fer aquests càlculs. En conseqüència, s’investiga una

aproximació diferent a l’anàlisi tèrmica per mitjà de MATLAB, ja que és un software

d’ús assequible per a una gran varietat d’enginyers.

Com la metodologia requereix dades previament generades pel paquet professional

esmentat anteriorment, va ser necessari una formació introductòria d’aquesta eina per

desenvolupar una primera solució. Després, com el treball considera el problema de

resoldre l’equació de la calor com una PDE, es van haver d’investigar tècniques

analítiques i numèriques per trobar la millor estratègia que la resolgués.

El context de la tesi es basa en la missió LISA, les característiques de la qual es

detallen en la secció introductòria. El model LISA s’ha utilitzat per realitzar l’anàlisi

tèrmica i avaluar els resultats de la simulació. Anteriorment, un model simple com una

barra escalfada es va utilitzat per posar en pràctica la fiabilitat del mètode.

Finalment, com a combinació de les eines ESATAN i MATLAB, el procediment va ser

establert i provat donant excel·lents resultats en termes de temps d’execució i bons

valors en termes d’error relatiu. Com a resultat, una metodología ha estat

desenvolupada proporcionant bones distribucions de temperatura per a xarxes

tèrmiques, servint de gran utiitat als enginyers per estudiar l’impacte de pertorbacions

externes en un sistema tèrmic.

v

Resumen del proyecto

La presente tesis se ha hecho en la empresa ASTRIUM para buscar nuevos métodos

de los que obtener distribuciones de temperatura. Los paquetes de software actuales

tales como ESATAN o ESARAD no solo proporcionan excelentes soluciones de análisis

térmico, aunque a un precio muy alto ya que consumen mucho tiempo, sino que

también permiten obtener simulaciones radiativas en condiciones de órbita. Debido a

que las licencias de este producto están normalmente limitadas para el uso de muchos

ingenieros, es importante proporcionar nuevas herramientas para hacer estos cálculos.

Como resultado, se investiga una aproximación diferente al análisis térmico por medio

de MATLAB, ya que es un software de uso asequible para una gran variedad de

ingenieros.

Como la metodología requiere datos previamente generados por el paquete

profesional mencionado anteriormente, fue necesaria una formación introductoria a

esta herramienta para llevar a cabo una primera solución. Después, como el trabajo

considera el problema de resolver la ecuación del calor en forma de PDE, se tuvieron

que investigar técnicas analíticas y numéricas para encontrar la mejor estrategia que la

resolviera.

El contexto de la tesis se basa en la misión LISA, las características de la cual están

detalladas en la sección introductoria. El modelo LISA se ha utilizado para realizar el

análisis térmico y evaluar los resultados de la simulación. Anteriormente, un modelo

sencillo como una barra calentada fue utilizado para poner en práctica la fiabilidad del

método.

Finalmente, por una combinación de las herramientas ESATAN y MATLAB, el

procedimiento fue establecido y probado dando excelentes resultados en términos de

tiempo de ejecución y buenos valores en términos de error relativo. Como resultado,

una metodología ha sido desarrollada proporcionando buenas distribuciones de

temperatura para redes térmicas, sirviendo de gran utilidad a los ingenieros para

estudiar el impacto de perturbaciones externas en un sistema térmico.

vi

Abstract

The present thesis has been done at ASTRIUM company to find new methods to

obtain temperature distributions. Current software packages such as ESATAN or

ESARAD provide not only excellent thermal analysis solutions, at a high price as they

are very time consuming though, but also radiative simulations in orbit scenarios. Since

licenses of this product are usually limited for the use of many engineers, it is

important to provide new tools to do these calculations. In consequence, a different

approach for thermal analysis, by means of a MATLAB program, is searched since this

software is very affordable in use for different kinds of engineers.

As the method requires data previously generated by the professional packages

mentioned above, an introductory training to these tools was needed to perform a

first solution. Afterwards, as the work considers the problem of solving the heat

equation as a PDE, analytical and numerical techniques were investigated to find the

best strategy to solve it.

The context of the thesis is based on the LISA mission, whose characteristics are

detailed at the introductory section. The LISA model is used to perform thermal

analysis and evaluate the simulated results. Previously, a simple model like a heating

bar was used to put into practice the reliability of the method.

Finally, by a combination of ESATAN and MATLAB tools, the procedure was

established and tested giving excellent results in terms of time execution and good

values in terms of relative error. As a result, a methodology has been developed

providing good temperature distributions for thermal networks, hence, being very

useful for engineers to study the impact of external perturbations in a thermal system.

vii

List of figures and tables

Figures

Figure 0. 1: Artist view of the three spacecraft forming the mission [15] ....................... 1

Figure 1. 1: Heat balance for a satellite [4] ...................................................................... 6

Figure 2. 1: Disposition of the nodes in a mesh for FEM and FDM [10] ........................ 17

Figure 2. 2: Rectangular grid with cells positioned non-centrally [10] .......................... 20

Figure 3. 1: Flow chart representing the implemented procedure ................................ 27

Figure 4. 1: Geometrical model of a bar composed by 10 nodes designed with ESATAN

................................................................................................................................ 35

Figure 4. 2: Transient solutions at node 1 of the heating bar, in blue by MATLAB, in

green by ESATAN .................................................................................................... 35

Figure 4. 3: Absolute relative error (%) ESATAN results with respect to MATLAB.

|(TE-TM)/TE| with node 1 in blue, 3 in green and 8 in red ...................................... 36

Figure 4. 4: Geometrical model of the LISA system represented with ESATAN ............ 37

Figure 4. 5: Optical bench where heat diodes are represented by nodes ‘12120’ and

‘12090’ [17] ............................................................................................................. 37

Figure 4. 6: Plot of temperature distribution of TE (green) and TM (blue) over time ..... 38

Figure 4. 7: Plot of the temperature absolute difference distribution between TE (blue)

and TM (red) ............................................................................................................ 39

Figure 4. 8: Plot of the temperature distribution for TE (blue) and TM (red).................. 39

Figure 4. 9: Absolute differences |TE-TM| at node ‘12120’ ............................................ 40

Figure 4. 10: Absolute relative error (%) |(TE-TM)/TE| at node ‘12120’ ........................ 40

Figure 4. 11: Plot of the temperature distribution of TE (green) and TM (blue) over time

................................................................................................................................ 41

Figure 4. 12: Plot of the temperature absolute difference distribution between TE

(blue) and TM (red) .................................................................................................. 42

viii

Figure 4. 13: Plot of the temperature distribution for TE (blue) and TM (red)................ 42

Figure 4. 14: Absolute difference |TE-TM| at node ‘12090’ ........................................... 43

Figure 4. 15: Absolute relative error |(TE-TM)/TE| at node ‘12090’ ............................... 43

Figure 4. 16: Radiative and linear coupled boundary envelope [17] ............................. 44

Figure 4. 17: Temperature distribution in Matlab (red) and Esatan (blue). ................... 45

Figure 4. 18: Temperature difference distribution of LISA model. ................................ 45

Figure 4. 19: Absolute relative error |(TE-TM)/TE| after 217000 seconds. ..................... 46

Figure 4. 20: Difference between final temperature and initial temperature. .............. 46

Tables

Table 1. 1: Typical device temperature ranges [12] ......................................................... 4

Table 3. 1: File name extensions provided by ESATAN [11] ........................................... 29

Table 4. 1: Maximum differences performed by TMS.R3 between temperatures with

and without zero capacities ................................................................................... 33

Table 4. 2: Comparative table between TMS.R3 and 10.2 versions with capacities

simulation ............................................................................................................... 34

ix

Nomenclature

Acronyms

CFD: Computational fluid dynamics

FD(M): Finite difference (method)

FE(M): Finite element (method)

FV(M): Finite colume (method)

GMM: Geometric mathematical model

LISA: Laser interferometer space antenna

ODE: Ordinary differential equation

PDE: Partial differential equation

RELXCA: Relaxation constant

SW: Software

TC(S): Thermal control (system)

TDMA: Tridiagonal-matrix algorithm

TMM: Thermal mathematical model

Greek symbols

α: Thermal diffusivity

δn: Angular phase

ε: Emissivity of a material

ρ: Heat density

σ: Boltzmann constant

ωn: Angular frequency

Physical and Mathematical Variables

Bij: Radiative factor

C: Thermal capacity

c: Specific heat

F: Linear conductivity matrix

Fij: Line of sight

GL: Linear conductivity matrix

GR: Radiative conductivity matrix

K: Radiative matrix

k: Thermal conductivity

Q: Heat energy

q: Heat flux density

S: Bar section

TE: Temperature obtained from ESATAN

TM: Temperature obtained from MATLAB

ut: time derivative of the temperature

uxx: second derivative of the temperature with respect to the x-position

NUMERICAL METHODS FOR COMPUTING THE TEMPERATURE DISTRIBUTION IN SATELLITE SYSTEMS 1

INTRODUCTION

1. Context of the work The target of the thesis is to develop new methods for thermal analysis. The current

software to obtain temperature distributions from a thermal network is the ESATAN

package, whose license is provided by the ALSTOM company. The main drawback of

this tool is the large execution time (from tens to hundred hours) needed to generate

accurate results. The project is based in a line of work taken by ASTRIUM to find new

alternative methodologies to perform temperature analysis in a more reasonable

execution time with the best accuracy. In the present document, a tool implemented

in MATLAB is shown.

As the project is located in the context of a science space mission called LISA and some

tests of the work will simulate its thermal model, a brief introduction to it is required.

2. The LISA mission LISA is a space mission whose main goal is measuring gravitational waves at low

frequencies and characterizing its sources. The measurement principle implements a

laser interferometry system built up in three identical spacecraft. The three satellites,

whose positions mark the vertices of an equilateral triangle of five million km side, are

in orbit around the Sun. The spacecraft separation sets the measurement bandwidth

ranges from 0.1 mHz up to 1 Hz. This range was chosen to reveal some of the most

interesting sources: mergers of massive black holes, ultra compact binaries, and the

inspirals of stellar-mass black holes into massive black holes [13].

Figure 0. 1: Artist view of the three spacecraft fo rming the mission [15]

2 NUMERICAL METHODS FOR COMPUTING THE TEMPERATURE DISTRIBUTION IN SATELLITE SYSTEMS

3. Contents of the thesis The current section presents a guideline of how the thesis is organized. Chapter 1

introduces to the reader a definition of thermal systems and the philosophy that lies

underneath of thermal modelling in the context of satellite systems. Moreover, some

problematic questions that thermal engineers may find are mentioned as well as

measures and procedures to solve them. As a preliminary step to calculate

temperature distributions, the heat equation and its derivative over time and space is

also shown.

In chapter 2 the heat equation is analysed as a PDE. Besides, analytical and numerical

methods to solve PDEs are presented. Since the software (ESATAN) implemented for

temperature distributions is based on numerical methods, a more detailed description

of finite methods is given. In a final section, the heat equation has been analysed as a

set of algebraic equations which give the base for the method developed in the next

chapter.

Chapter 3 presents a methodology combining data obtained from ESATAN software

and a tool developed with MATLAB to compute temperature distributions. In the

sections one can find the procedure applied to two examples, a simple one consisting

of the analysis of a heating bar and the complex case of the LISA model, which is the

context of the mission.

Chapter 4 reports the simulations run during the work. A comparison of the results of

the implemented procedure with respect to reference results (ESATAN data) is

presented.

Finally, there is a section with conclusions, where an evaluation of the comparative

analysis is done. Then, an introduction to future worklines to be carried out is stated.


CHAPTER 1. Thermal analysis

1.0. Introduction The content of this chapter is intended to be an introductory overview of some of the

more important terminology and concepts that one may come across in thermal

control, although it cannot supply a detailed introduction to the whole domain of

thermal management.

1.1. Thermal control systems [12] The aim of the TCS is to maintain all the devices of a spacecraft within their

allowed temperature limits during all mission phases using the minimum resources.

These mission phases shall be represented by a coherent set of thermal design cases to

be proposed by the TCS, covering the extreme range of conditions that a device will

experience during its lifetime, thus a worst cold and hot temperature shall be defined.

The adverse conditions of space have to be taken into account to maintain the

operating ability of a spacecraft. In the absence of a protecting atmosphere, satellites

and their components are exposed to both direct solar radiation and the extreme cold

of space. Thus, sophisticated thermal management is critical.

Some of the thermal problems to be solved by the TCS are the following:

1) The on-board equipment can be disabled or damaged when exposed out of the

narrow range of acceptable operational temperature values that electronic

systems have. Besides, the space environment is extreme on thermal loads

causing sudden changes.

2) During the entrance or exit of eclipses lightweight deployed parts suffer some

100 ºC temperature jumps, thermal stresses being very high.

3) Unwanted optical deflections and structural deformations can be produced by

thermal expansion, which is due to temperature gradients. Some fine

instruments demand temperature stability of the order of millikelvins, such

stability is achieved by fine thermal control of outer shells and thermal

insulation of the inner core.

4) Some equipment must be kept at cryogenic temperatures, like cryostats and

infrared detectors (to increase the signal-to-noise ratio).

The mission of a satellite highly influences its setup. Not only the equipment, but also

the thermal layout is heavily affected by the task of the satellite. Moreover, there

exists dissipation of internal components (e.g. batteries). The shape and type of the

orbit determine the major conditions, such as the incident loads, and therefore also


determine if temperatures on the upper or lower bound of the specifications should be

expected. Finally, the launch scenario may also be of importance, e.g. if the fairing is

jettisoned early and friction comes into account.

In addition to the points mentioned above and just focusing on traditional thermal

control systems, there is a multitude of thermal problems in space. As a result, the

most critical spacecraft components requirements for TCS (regarding LISA

specifications) are:

• Batteries (often tight and low temperature range)

• Instrument detectors & sensors (dependent on operation)

• Star sensors

• Optical benches and mirrors of telescopes (low temperature gradients & high

temperature stability)

• Mechanisms (low temperature gradient over bearings)

• Traveling wave tubes (high heat loads, high heat fluxes)

• Chemical propulsion systems and components

In the following table a list of typical design temperature ranges are presented.

Device Operating temperature Electronics -10°C – +50°C

Solar generator -150°C – +120°C

Batteries -5°C – +20°C

Harness -15°C – +55°C

Mechanisms -45°C – +65°C

Hydrazine +7°C – +35°C

Table 1. 1: Typical device temperature ranges [12]

Temperature control is mainly realized by sizing thermal diffusivity, heat fluxes, optical

properties and radiative surfaces, concepts that will be later presented, and by

analyzing the global steady state temperature. Therefore, taking these parameters into

account, some of the features and measures required for TCS are:

• Insulate spacecraft to minimize influence of environment

• Minimize absorbed external heat fluxes

• Balance internal heat dissipation with heat loss

• Define dedicated radiator areas for waste heat rejection to space

• Distribute waste heat to the radiators

• Install TC Hardware for cold operation cases and safe modes

• Analyze TCS

• Verify TCS by TB/TV test (hotter/colder case and thermal vacuum respectively)


1.2. TCS modeling philosophy In thermal engineering, TCS follow a set of steps from the concept to their operation.

The following scheme represents the modeling philosophy used as guideline for

engineers to develop thermal analysis [12].

Configuration and Mission requirements

o Graphical SW package:

• GMM set-up

• geometry visualization and check

• spacecraft orbit and altitude check

o Radiative SW

• view factor calculation

• radiative couplings calculation

• environment fluxes calculation

Configuration, Design cases, Boundary conditions

o Thermal SW

• TMM set-up

• temperature and heat fluxes prediction

• control system definition, power budget definition

• thermo-hydraulic parameters definition

o Graphical SW package

• visualization and check of

• temperatures

• fluxes

• pressures

1.3. The heat equation [2] [9] In thermodynamics, heat is a form of energy transferred typically from a hotter

to a colder body by virtue of the temperature difference, following a direction and

changing at a particular rate, which is the temperature gradient.

Thermal energy transfer takes place via three major mechanisms which are radiation,

conduction and convection. The first mechanism takes place via electromagnetic

waves, the second in fluids and solids in the absence of fluid motion, the third in a

flowing fluid and between the fluid and a solid wall (in the absence of an atmosphere,

which is our case, there is no convection). Although the behavior of each of these

mechanisms may be described by different physical laws, real systems often exhibit a


complicated combination, which are often described by several complex mathematical

methods.

The discipline of heat transfer deals with specific methods applied, by which thermal

energy in a system is generated, converted or transferred to another one. Although

the definition of heat implicitly means the transfer of energy, the term heat transfer,

that is typically considered an aspect of mechanical engineering, covers its usage in

many engineering branches.

For algebraic purposes, the total amount of energy transferred as heat is written as Q.

Heat delivered by a system to its surroundings is by convention a negative quantity

(Q < 0); when a system absorbs heat from its surroundings, it is positive (Q > 0). Heat

transfer rate, or heat flow per unit time, is denoted by

dt

dQQ =

•

(1.1)

Furthermore, heat flux density (Eq. 1.2) is defined as rate of heat transfer per unit

cross-section area, resulting in the unit watts per square meter.

An object's temperature is obtained by balancing the incoming and outgoing heat

fluxes. In the outer space, incoming heat fluxes are direct solar, indirect solar (albedo)

and planetary infrared. Outgoing heat fluxes are mainly between the spacecraft and

space. Only small transfers are made between the spacecraft and planet. Additionally,

there are internal heat fluxes between components as a result of temperature

gradients between them.

Figure 1. 1: Heat balance for a satellite [4]

Derivation in one dimension

[2][9] The heat equation is derived from Fourier's law and conservation of energy. By

Fourier's law, the flow rate of heat energy through a surface is proportional to the

negative temperature gradient across the surface,

ukq ∇−= (1.2)


where q is the heat flux density, k is the thermal conductivity and u is the temperature.

In one dimension, the gradient is an ordinary spatial derivative, and so Fourier's law is

xkuq −= (1.3)

where

x

uux ∂

∂= (1.4)

In the absence of work done, a change ΔQ in internal energy per unit volume in the

body is proportional to the change in temperature Δu. That is,

ucQ p ∆=∆ ρ (1.5)

where cp is the specific heat capacity and ρ is the mass density of the material. (In this

section Δ is the ordinary difference operator). Choosing zero energy at absolute zero

temperature, this can be rewritten as

ucQ p ρ= (1.6)

The increase in internal energy in a small spatial region of the material

xxxx ∆+≤≤∆− ξ

over the time period

tttt ∆+≤≤∆− τ

is given by

( ) ( )[ ] τξτ

ρξξξρ ddu

cdttuttuctt

tt

xx

xxp

xx

xxp ∫ ∫∫∆+

∆−

∆+

∆−

∆+

∆− ∂∂=∆−−∆+ ,,

The change in internal energy in the interval [x-Δx, x+Δx] is balanced by the flux of heat

across the boundaries when no work is done and there are neither heat sources nor

sinks. By Fourier's law this is

( ) ( ) τξξ

τττ ddu

kdxxx

uxx

x

uk

tt

tt

xx

xx

tt

tt ∫ ∫∫∆+

∆−

∆+

∆−

∆+

∆− ∂∂=

∆−∂∂−∆+

∂∂

2

2

,,

Therefore, by conservation of energy

[ ] 0=−∫ ∫∆+

∆−

∆+

∆−τξρ ξξτ ddkuuc

tt

tt

xx

xx p


This is true for any rectangle [t−Δt, t+Δt] × [x−Δx, x+Δx]. Thus, the integrand is

identically zero:

0=− xxtp kuuc ρ

This can be rewritten as:

xxp

t uc

ku

ρ= (1.7)

Or

∂∂=

∂∂

2

2

x

u

c

k

t

u

pρ

This equation, also known as the diffusion equation, models the flow of heat through a

rod by modeling the temperature along the rod at time t, where the coefficient (often

denoted α)

ρα

pc

k= (1.8)

is called the thermal diffusivity.


CHAPTER 2. Thermal mathematical model

2.1. Introduction This section deals with the problem of solving the heat equation to obtain the

temperature distribution and also shows the typical mathematical methods used to

solve it and to perform a software solution. Finally, an introduction to the software

used in the work is explained as well as the mathematical method it implements.

2.2. Partial differential equations

The heat conduction equation is commonly solved for one dimension, considering the

distribution of heat in a rod along x, we get

xxt uu ⋅= α , (2.1)

where the symbol ut is the partial derivative with respect to the time variable t, and

similarly uxx is the second partial derivative with respect to x. Hence, the heat equation

is a type of second-order PDE, which is governed by the heat transfer dynamic

containing multivariable functions and their partial derivatives (for this reason, PDEs

can often model multidimensional systems).

The heat equation (Eq. 2.1) roughly says that the temperature at a given time and

point will change at a rate proportionally to the difference between its temperature

and the average temperature around the point. The amount uxx evaluates the

temperature difference with respect to the mean value, which is a property of

harmonic functions and will be later taken into account.

2.3. Methods to solve PDEs

PDEs can be solved analytically but some cases like non-homogenous or nonlinear

PDEs require numerical methods. In the upcoming sections the methods for solving the

heat equation are showed as well as some of their advantages and disadvantages.

2.3.1. Analytical methods to solve PDEs

This section illustrates the application of the separation of variables method to

determine analytical solutions for steady-state and transient linear heat conduction

problems [8] [9].


In order to solve the heat equation we must give the problem some initial temperature

conditions. So regarding the temperature along a rod, we must first define an initial

temperature at every point at time t = 0, which we do with the function

u(x, 0) = u0(x) for 0 ≤ x ≤ L .

This function is known as the initial temperature distribution. Many heat conduction

problems found in engineering applications involve time as the independent variable.

The goal of analysis is to determine the variation of the temperature as a function of

time and position u (x, t) within the heat conducting body.

Since heat can only enter or exit the rod at its boundaries, we must define some

“boundary conditions”. Analytical solutions to boundary value problems in linear

ordinary differential equations are usually obtained producing a general solution by

simple integration. Incorporation of boundary conditions subsequently defines the

values of the integration constants. Therefore, we need to define the conditions of the

rod at its boundaries (0, L).

u(0, t) = T0, u(L, t) = TL for all t > 0

These are known as Dirichlet conditions. Combining the heat equation with the initial

conditions and boundary conditions, the problem is to find the u(x, t) distribution. A

physical interpretation of the problem is as follows: after fixing the temperature on the

boundaries according to the given specifications, heat flows until a stationary state is

reached in which the temperature at each point of the domain doesn't change any

more. The temperature distribution along the body will then be given by the solution

u(x, t) that satisfies both the initial and boundary conditions.

The first consideration is to say that the operators in the heat equation are linear

operators, such that x∂∂ and 22 x∂∂ are linear. Since the heat conduction equation

is linear we can find a linear combination of two solutions to reach another solution.

The heat equation with boundary and initial conditions can be transformed into two

equivalent but simpler problems, a steady state non homogeneous problem and a

transient homogeneous problem. First we introduce new functions = us(x) and v(x, t)

such that the temperature in the rod is

u(x, t) = us(x) + v(x, t) . (2.2)

Once the steady state us(x) has been reached, as 0=∂∂

tu because x

u∂

∂ is time

invariant, we can use Fourier’s law;

Sk

Q

x

u

⋅−=

∂∂

•

(2.3)


Then, by integrating along the rod

∫∫ ⋅⋅

−=•

x

L

u

T

dxSk

Qdu

0

one can find that

0)()( TxLSk

Qxu s +−⋅

⋅=

•

. (2.4)

The remaining part is the transitory solution whose homogenous boundary conditions

make finding a solution much easier. This characteristic allows using separation of

variables because the method requires that the problem be homogeneous in order to

find a solution. Therefore, we want to find product solutions with separated variables

of the form

v(x, t) = X(x)T(t). (2.5)

The method will lead us to two ODEs for T and X. By differentiating separately with

respect to t and x we obtain the following two equations;

0)()( 2

2

2

=+∂

∂xX

x

xXnω

0)()( 2 =⋅+

∂∂

tTt

tTnωα

These differential equations are similar to the harmonic functions, for which nω is the

angular frequency. The harmonic functions are all analytic within the domain where

the equation is satisfied. If any two functions (or any linear homogeneous differential

equation) are solutions to Laplace's equation, their sum (or any linear combination) is

also a solution. This property, called the principle of superposition is here useful. So

the solutions are of the form:

)sin()( nnn xaxX δω +⋅=

)exp()0()( 2tTtT nαω−⋅= ,

where α and nδ are the thermal diffusivity and angular phase, respectively. Therefore,

the temperature distribution would become

∑∞

=

•

+−+−⋅⋅

+=1

20 )sin()exp()(),(

nnnnn xtaxL

Sk

QTtxu δωαω . (2.6)


Finally, we would need to find the coefficients an that satisfy the initial conditions for a

particular case given.

2.3.2. Numerical methods to solve PDEs

Prior to the description of numerical methods, first let's introduce the mathematical

term 'well-posed problem', defined by Jacques Hadamard [9]. He believed that

mathematical models of physical phenomena should have the properties that

1. a solution exists,

2. the solution is unique, and

3. the solution depends continuously on the data, in some reasonable topology.

An example of archetypal well-posed problem we use in our work is the Dirichlet

problem for Laplace's equation and the heat equation with specified initial conditions,

in which there are physical processes that solve these problems.

Such continuous problems must often be discretized in order to obtain a numerical

solution. While in terms of functional analysis such problems are typically continuous,

they may suffer from numerical instability when solved with finite precision. If the

problem is well-posed, then there is a good chance to solve the problem on a

computer using a stable algorithm.

The Discretization Concept

[10] The numerical solution of a differential equation is considered to be a set of

numbers that can build the distribution of the dependent variable, let’s call it φ. For

instance, let us represent the variation of φ by a polynomial in x such as

m

m xaxaxaa ++++= ......2210ϕ (2.7)

and use a numerical method to find the finite number of coefficients a0,a1,a2…am. This

permits us to evaluate φ at any location of x by substituting the x and a’s values into

(Eq. 2.7). However, this procedure is not convenient when we are interested in

obtaining the value of φ at several locations. Hence, we should think of a numerical

method which treats the values of the dependent variable at a finite number of

locations, which are called the grid points, as its unknowns in the calculation domain.

Methodology

Once the continuous information contained in the exact solution of the differential

equation is replaced with discrete values, then the attention is focused on the values

at the grid points. Thus, the distribution of φ has been discretized so it is appropriate

to refer to this class of numerical methods as discretization methods. The algebraic

equations (now named the discretization equations), which involve the unknown


values of φ at chosen grid points, are derived from the differential equation governing

φ.

Some assumptions about how φ varies between the grid points have to be taken into

account. Although the profile of φ could be chosen such that a single algebraic

expression is enough for the whole calculation domain, it is often more practical to use

a profile defined piecewise such that a given segment describes the variation of φ,

over a small region in terms of φ values at the grid points, within and around that

region. Therefore, it is common to subdivide the calculation domain into a number of

subdomains.

It is this discretization of the space and dependent variables in the continuous

calculation that makes it possible to replace the governing differential equation with

simple algebraic equations, which can be solved with relative ease.

The Structure of the Discretization Equation

A discretization equation is an algebraic relation coupling the values of φ for a group of

grid points. Such an equation is derived from the differential equation governing φ and

thus expresses the same physical information as the differential equation.

The value of φ at a grid point thereby influences the distribution of φ only in its

immediate neighborhood. As the number of grid points becomes very large, the

solution of the discretization equations is expected to approach the exact solution of

the corresponding differential equation. This comes from the consideration that, as

the grid points get closer together, the change in φ between neighboring grid points

becomes small.

Deriving the discretization equations

For a given differential equation, the required discretization equations can be derived

in many ways and Taylor-Series formulation is one of them. The usual procedure for

deriving finite-difference equations consists of approximating the derivatives in the

differential equation via a truncated Taylor series. Assuming the function whose

derivatives are to be approximated is sufficiently smooth, by Taylor's theorem,

( ) ( ) ( ) ( ) ( ))(

!.....

!2!10

)(20

)2(0

'

00 xRhn

xfh

xfh

xfxfhxf n

nn

+⋅++⋅+⋅+=+ (2.8)

where n! denotes the factorial of n and Rn(x) is a remainder term, denoting the

difference between the Taylor polynomial of degree n and the original function. Again,

using the first derivative of the function f as an example, by Taylor's theorem,

f(x0 + h) = f(x0) + f'(x0)h + R1(x),

Setting x0=a and (x-a) =h we have,


f(a + h) = f(a) + f'(a)h + R1(x),

Dividing by h gives:

( ) ( ) ( )

h

xRaf

h

af

h

haf )(1' ++=+

Solving for f’(a):

( ) ( ) ( )h

xR

h

afhafaf

)(1' −−+=

so that for R1(x) sufficiently small,

( ) ( ) ( )h

afhafaf

−+≈'

(2.9)

Solutions to Algebraic Equations

For a linear problem, which requires the solution of algebraic equations only once, a

direct method (i.e., those requiring no iteration) may be acceptable; but in nonlinear

problems, since the equations have to be solved repeatedly with updated coefficients,

the use of a direct method usually is not resource-saving. The alternative method then

is an iterative method for the solution of algebraic equations, which starts from an

initial guess value of T (the dependent variable). Successive iterations of the algorithm

finally lead to a solution that is sufficiently close to the correct solution of the algebraic

equations.

There are many iterative methods for solving algebraic equations such as the Jacobi,

Gauss-Seidel or successive over-relaxation methods. Jacobi and Gauss-Seidel iterative

methods are easy to implement in simple computer programs, but they can be slow to

converge when the system of equations is large. Hence, they are not considered

suitable for rapidly solving tri-diagonal systems which is best done with Thomas

algorithm or the tri-diagonal matrix algorithm.

The TDMA is actually a direct method for one-dimensional problems, although it can

be applied iteratively, in a line-by-line fashion, to solve multi-dimensional problems. It

is computationally inexpensive and has the advantage that it requires a minimum

amount of storage, because only non-zero coefficients of the equations are stored in

core memory.

The Tri-diagonal Matrix Algorithm

[9] In numerical linear algebra, the tri-diagonal matrix algorithm is a simplified form of

Gauss elimination that can be used to solve tri-diagonal systems of equations. Let’s

suppose that the points ‘i’ in the grid were numbered 1, 2, 3 ... N with points 1 and N

denoting the boundary points. A tri-diagonal system may be written as


iiiiiii dTcTbTa =++ +− 11 (2.10)

Thus, the temperature Ti is related to the neighboring temperatures Ti+1 and Ti-1.

Additionally, TDMA refers to the fact that all the nonzero coefficients align themselves

along three diagonals of the matrix, for which format the system is written as

=

−

nnnn

n

d

d

d

d

x

x

x

x

ba

c

ba

cba

cb

MMOO

O 3

2

1

3

2

1

1

33

222

11

0

0

(2.11)

For such systems, the solution can be obtained in O(n) operations instead of O(n3)

required by Gauss elimination. A first sweep eliminates the ai, and then a backward

substitution produces the solution. Such matrices commonly arise from the

discretization of 1D Poisson equation used for the diffusion problem, which our case is

based on.

We have set a1=0 and cn =0 considering the form of the boundary-point equations, so

that the temperatures T0 and TN+1 will not have any meaningful role to play. When the

boundary temperatures are given, these boundary-point equations take a rather trivial

form.

The first step, called the forward sweep, consists of modifying the coefficients as

follows, denoting the new modified coefficients with primes:

−=′−

==′

−

1,...,3,2;

1;

1

niacb

c

ib

c

c

iii

i

i

i

i

and

−=′−′−

==′

−

− 1,...,3,2;

1;

1

1 niacb

add

ib

d

d

iii

iii

i

i

i

The solution is then obtained by backward substitution process:

1,...,2,1;1 −−=′−′=′=

+ nniTcdT

dT

iiii

nn


The equation for i=2 is a relation between T1, T2, and T3. Since T1 can be expressed in

terms of T2 and T2 can be expressed in terms of T3, this leads to a relation between T2

and T3. This process of substitution can be continued until Tn is formally expressed in

terms of Tn+1. By backward substitution Tn-1 is obtained from Tn and so on, which is the

essence of TDMA.

The discretization equations for the grid points along a chosen line contain the

temperature at the grid points along the neighboring lines. If these temperatures are

substituted by their previous values, the equation for the grid points along the chosen

line would look like one-dimensional equations and could be solved by the TDMA.

The convergence of the line-by-line method is very fast, because the boundary-

condition information from the ends of the line is transmitted at once to the interior of

the domain, no matter how many grid points lie in one line.

2.4. The finite methods There are three widely used numerical methods to solve PDEs: the finite element,

finite volume and finite difference methods [9] [4] [3]. These methods are based on

the idea of first discretizing the heat equation and then solving the resulting algebraic

problem. Discretization is implemented considering the medium as a collection of cells

or volumes of finite size. Nodes are usually associated with each cell producing a mesh

of points. The separation between each two nodes is the mesh spacing. Temperature

at each cell is then represented by the temperature at the corresponding nodal

location. A program is then used to solve the resulting algebraic equations.

2.4.1. Finite difference method

The FDM has historical importance as is one of the first applications of digital

computers to numerical solutions of physical systems, the so-called finite-difference

approach, already explained in the section before. The principle of FDM is that

derivatives of PDEs are approximated by linear combinations of function values at the

grid points, for which a mesh size has been previously defined locating the nodes at

the center of the mesh.

Depending on the remaining term of the truncation made in the Taylor’s series

expansion, the approximations of the derivatives can be of first order, second order or

higher order. The higher the order the better the resolution, a reasonable accuracy is

obtained on coarse grids. By contrast, for more grid points there is a considerable

overhead cost. Geometrically, the approximation can be reached by forward,

backward or central difference interpolation methods, which corresponds to explicit,

implicit or Crank-Nicolson methods, respectively. Central differences are popular, in

particular, which are accurate up to second order.


2.4.2. Finite element method

The FEM, whose practical application is also known as finite element analysis, is a

technique for finding approximate solutions of PDEs as well as integral equations.

Eliminating completely the differential equation or just representing the PDE by an

approximating system of ODEs are the usual solution approaches. A numerical

integration using standard techniques such as Euler’s method or Runge-Kutta is then

required.

When solving the PDE, the first challenge is to create an equation that approximates it.

The basic idea is to replace the infinite dimensional linear problem with a finite

dimensional version, typically represented by piecewise polynomial functions which

can be triangles, squares or curvilinear elements, placed at the corners of the

elements. The solution has to be numerically stable, meaning that errors in the input

and intermediate calculations do not accumulate and cause the resulting output to be

useless, which can be achieved by many ways. Normally, one has an algorithm for

taking a given mesh and subdividing it by a smaller mesh, for increasing precision, by

increasing the degree of the polynomials or combining both methods.

The FEM is used in structural analysis of solids, but is also applicable to fluids. Because

of the characteristics mentioned above, FEM is a good choice for solving PDEs over

complicated domains, for example, when the domain changes (as during a solid state

reaction with a moving boundary), when the desired precision varies over the entire

domain or when the solution lacks smoothness. Its formulation requires special care to

ensure a conservative solution, however. Despite this, it is much more stable than

other approaches like FVM, although it can require more memory.

Figure 2. 1 : Disposition of the nodes in a mesh for FEM and FDM [10]

2.4.3. Finite volume method (FVM)


In the FVM, values are calculated at discrete places on a meshed geometry, similar to

the FEM or FDM. The method is based on the control volume formulation of analytical

fluid dynamics, where the domain is divided into a number of control volumes (cells or

elements) in which the variable of interest is located at the center of the control

volume. Thus, "finite volume" refers to the small volume surrounding each node point

on a mesh.

The method is based on the principle that the divergence term, that frequently occurs

in differential equations governing each control volume, can be rewritten as a surface

integral using the divergence theorem. The problem then simplifies to evaluating

fluxes at the surfaces of each finite volume using different interpolation techniques,

like the ones mentioned for FDM, from discrete data, which are the control variables.

Since the flux entering a given volume is the same as that leaving the contiguous one,

the method is considered to be conservative. Even a coarse grid solution exhibits exact

integral balances. This makes the FVM stable, flexible and relatively easy to implement,

the main reasons why the FVM is commonly implemented in commercial CFD solvers.

2.4.4. What scheme to use

Before introducing the software used in the work and after presenting the existing

methods handled to perform temperature distribution calculations, which is one of the

purposes of the work, an analysis of the typically used software packages for this

object is required.

One of the important distinctions between structural analysis software packages and

some thermal analysis packages is that most of the first use the FEM while some

thermal packages such as ESATAN, which is the one used for the work, implement the

FDM. The main reason for this difference is that structural and thermal software

packages were originally developed independently of each another – the engineers in

their respective fields chose the method that suited them best. However, both can be

used for either application.

While FEM is based on the use of 1D, 2D or 3D elements, one of the real strengths over

FDM are the mesh-generation schemes. These techniques can easily handle irregular

surface shapes and interfaces between two different mesh schemes.

There are also mesh-generation packages, such as MSC.Patran or Patran which has

excellent pre and post processing abilities. FEM mesh generating schemes are still used

in most thermal software packages to develop and post process FD temperature

results. It is to be noted that there are some FDM packages available with comparable

capabilities to those of the FEM packages, however, their number is limited.

For a thermal analysis, FE models will always be larger than necessary because there is

a requirement for each element face to share a complete interface with another


element. There is also a lack of information about the error associated with

calculations; hence, a finer mesh will most likely be necessary. Additionally, curved

surfaces require far more finite elements to describe the shape than a comparable FD

model.

The software used as a reference to perform temperature distributions is ESATAN,

which obtains solutions to lumped parameter thermal models, whose definition will be

introduced. An ESATAN input deck contains the specification of a model and

instructions for its solution. The model must be generated by the user, though

extensions to ESATAN provide limited facilities for automatic model generation for

some geometry. The "Lumped Parameter" or "Network Analogy" method involves

modeling a continuous medium as a discrete network of nodes representing the

capacity of the system linked by "conductors" representing its conductivity. Therefore,

a fourth numerical method is presented in the next section.

2.5. The lumped parameter method [10] The lumped parameter method has its conceptual origins in the thermal/electric

analogy in which temperature corresponds to voltage and heat flow to current flow. In

the days before digital computers were available at affordable prices, it was common

to analyze a thermal problem by constructing an analogous electrical experimental

model. In the 1930s discrete models using networks of resistors and capacitors came

into use. Discrete systems do not often occur in real thermal applications, but are the

rule rather than the exception in circuit design. In experimental work it was mainly the

ease of construction which led to continuous thermal systems being approximated by

discrete analogies.

The discrete approximation has advantages in numerical as well as experimental work,

and lends itself to simple hand calculations as well as to computerized solution. A

continuous system obeys a differential equation, whereas a networked model obeys a

system of algebraic equations in a finite number of variables which can be solved by

standard techniques. Numerical methods based upon lumped networks were

developed in the 1940s. Therefore, ESATAN can be seen as a direct descendant of

these methods.

Mathematically, the lumped parameter method can be seen as a way of deriving a

first-order finite-difference approximation to obtain the governing differential

equation. One could think of it as a non-elegant method of obtaining a numerical

solution to a differential equation. Its major advantage, however, is its intuitive

perception and physical simplicity. In the construction of a thermal model the first

level of approximation will be physical. The lumped parameter method is well suited to

this, e.g. a complicated piece of mechanism may be replaced by a single node

representing its heat capacity, and conductors linking it to the rest of the model, when

the detailed temperature distribution within the mechanism is not of interest.


2.5.1. The lumped parameter equations Here we introduce the general Lumped Parameter equation. We first suppose n nodes

N1, N2, ..., Nn, for which Ti is the temperature of node Ni, Ci the heat capacity, and Qi an

internal heat source. The linear conductivity (which may represent conduction,

convection, or other linear process) between node Ni and node Nj is Kij and the

radiative exchange constant is Rij. Applying a heat balance to node i we have

( ) ( ) iij

ijijij

ijiji

i qTTRTTKdt

dTC +−+−= ∑∑

≠≠

44· (2.12)

which for the steady state case reduces to

( ) ( ) i

ijijij

ijijij qTTRTTK +−+−= ∑∑

≠≠

440 (2.13)

In the steady state case.

The above equations assume that the lumped parameters may be derived, but do not

specify how to derive them. A wide variety of methods may be used depending on the

geometry and other factors.

2.5.2. LP conductivities on a rectangular grid

Figure 2. 2: Rectangular grid with cells positioned non-centrall y [10]

The procedure for discretizing a solid body by means of a rectangular grid is of

particular importance. In the case of an irregularly spaced rectangular grid, the

following generalization of the method described in the above example is commonly

assumed. The extent to which the method can be mathematically justified is discussed

below. Let’s consider a thin plate of thickness d with thermal capacity k, specific heat c


and heat density ρ as defined in the example. Note that in this example there is no

object to transfer radiative conduction so only linear conductivity is considered. Figure

2.2 shows a cell of dimensions sx

by sy

with the node N0 at temperature T0

not

necessarily placed centrally. The distances to the four nearest nodes in the +x and +y

directions, N1, N2, N3, N4 are s1, s

2, s

3, s

4 and their temperatures are T

1, T

2, T

3, T

4. The

heat balance is then

( ) ( ) ( ) ( )044,0033,0022,0011,00

0 · TTKTTKTTKTTKdt

dTC −+−+−+−=

(2.14)

where

44,0

33,0

22,0

11,00 ,,,,

s

skdK

s

skdK

s

skdK

s

skdKsdscC xyxy

yx ===== ρ (2.15)

2.5.3. Derivation of Lumped Parameter equations by Finite Differencing

It will now be shown that, subject to certain constraints, the lumped parameter

method for a rectangular grid described above is equivalent to a first order finite

difference approximation to the heat conduction equation. We again consider the

problem of a thin plate of thickness d, where heat transfer is by conduction only, and

there are no internal heat sources. The extension to more complex cases follows the

same method. For the heat conduction equation we consider

2

2

2

222

yxandT

c

k

t

T

p ∂∂+

∂∂=∇∇⋅=

∂∂

ρ (2.16)

We now refer to figure (2.2), in which nodes N0, N1, N2, N3, N4 have temperatures T0,

T1, T2, T3, T4, respectively, at some instant of time. Then we expand T as a Taylor series

about the point N0, to obtain expressions for T0, T1, T2, T3, T4.

)(62

41

31

3

321

2

2

101 sOs

x

Ts

x

Ts

x

TTT +

∂∂+

∂∂+

∂∂+=

(2.17)

)(62

42

32

3

322

2

2

202 sOs

y

Ts

y

Ts

y

TTT +

∂∂+

∂∂+

∂∂+=

(2.18)

)(62

43

33

3

323

2

2

303 sOs

x

Ts

x

Ts

x

TTT +

∂∂+

∂∂+

∂∂+=

(2.19)


)(62

44

34

3

324

2

2

404 sOs

y

Ts

y

Ts

y

TTT +

∂∂+

∂∂+

∂∂+=

(2.20)

We can regard equations (2.17) and (2.19) as simultaneous equations in unknowns

xT ∂∂ and 22 xT ∂∂ and equations (2.18) and (2.20) as simultaneous equations in

unknowns and yT ∂∂ and 22 yT ∂∂ . Solving for

22 xT ∂∂ and 22 yT ∂∂ we obtain

( ) ( )

( ) ( ).....

3)(

2

)(

2

.....3)(

2

)(

2

3

342

424

04

422

022

2

3

331

313

03

311

012

2

+∂∂−

++−

++−

=∂∂

+∂∂−

++−

++−

=∂∂

y

Tss

sss

TT

sss

TT

y

T

x

Tss

sss

TT

sss

TT

x

T

Adding and dropping terms in 33 xT ∂∂ and higher, we obtain a finite difference

approximation to T2∇ . Evaluating at N0 the heat equation then becomes

+−

++

−+

+−

++

−≈

=∇⋅=∂∂

)()()()(2

,

424

04

313

03

422

02

311

010

2

0

0

sss

TT

sss

TT

sss

TT

sss

TT

dt

dT

c

kwhereT

c

k

t

T

pN

pN

α

ρα

ρ

(2.21)

which provides a spatially discretized finite difference approximation to the heat

conduction equation. For steady-state solutions, and if we apply (2.21) at each internal

node, and a suitable boundary equation at each boundary node, we obtain a set of

algebraic equations which can be solved for the Ti. To obtain transient solutions, we

must apply a finite difference scheme to the time variable. Let’s consider, for instance,

a forward differencing scheme. Ti,t is the temperature at node i at time t . We choose

time steps t1, t2 , . . . , not necessarily equally spaced, with Δtn=tn+1-tn. Then

1,0,00

+==∆−

≈ nnn

ab tbandtawheret

TT

dt

dT

And we have

+−

++

−+

+−

++

−⋅∆+≈

+ )()()()(2

424

,0,4

313

,0,3

422

,0,2

311

,0,1,0,0 1 sss

TT

sss

TT

sss

TT

sss

TTtTT nnnnnnnn

nn

ttttttttntt α

(2.22)

This is a finite difference approximation to the heat equation discretized in both space

and time. For suitable initial temperatures, temperatures at successive time steps can

be calculated.


One can see that the temperatures Ti are not average temperatures over a cell, but

approximations to the temperatures at the nodal points. These finite difference

equations can be given a physical interpretation by casting them in the form of the

lumped parameter method. If we now introduce the idea of a cell surrounding the

point N0, and the quantity A0 = (s1+s3)( s2+s4)/4 is identified as the area of the cell, then

V0 = A0d is the volume of the cell, and C0=c·ρ·V0 is the heat capacity of the cell.

Equation (2.21) can then be written as

( )∑=

−≈4

10,0

00

iii TTK

dt

dTC

(2.23)

where

4

314,0

3

423,0

2

312,0

1

421,0

22

22

s

sskdK

s

sskdK

s

sskdK

s

sskdK

+=+=

+=+=

Provided the node boundaries are equidistant between the nodes, these expressions

correspond to the physical conductors calculated by the lumped parameter method

(Eq. 2.15), and (Eq. 2.23) corresponds exactly to the lumped parameter heat balance

(Eq. 2.14). Similarly, (Eq. 2.22) can be rewritten as

( )∑=

−∆+≈4

1,0,,0,0,0

iaaiinab TTKtTT

(2.24)

One can observe that if s1=s3 and s2=s4, i.e. for a regular grid, the terms in third order

cancel. Otherwise, the approximation is good only to second order. In addition, the

expression A0 = (s1+s3)·(s2+s4)/4 is only true if the cell boundaries are equidistant from

the node points on each side. In the case of a regular grid, this is equivalent to placing

the node points at the centers of the cells, otherwise it is not. If the boundaries are not

placed as stated, then the lumped parameter expressions will not correspond to those

here derived, and in general a second-order approximation will not be achieved.

2.6. Matrix notation for heat balance [13] The purpose of this section is to define a thermal network by analyzing the matrix

notation of the heat equation. The system exposed here is discretized in a number of n

nodes. Hence, the general heat balance equation in case of presence of conduction

and radiation only is given in (Eq. 2.25) for a certain node i of the thermal network.

( )[ ] ( )[ ] i

n

jjiijji

n

jiijji

ii qTTGRTTGL

dt

dTC +−+−= ∑∑

=≠=≠ 1,

44,

1,, (2.25)


In (Eq. 2.25) qi is considered an internal heat source while Ci is the thermal capacity or

heat capacity at node ‘i’. The thermal capacity C is the ability of a body to store

thermal energy and is calculated from the expression

cVC ··ρ= (2.26)

where ρ is the heat density, V the volume and c the specific heat.

The specific heat or so-called specific heat capacity, is defined as the amount of energy

that has to be transferred to or from one unit of mass to change the system

temperature by one degree. Specific heat is a physical property, which means that it

depends on the material under consideration and its state as specified by its

properties.

When heat is expected to flow between nodes, it is said that they are coupled by

conductors. A conductivity is usually calculated by assuming that the heat flows one-

dimensionally from one section to the contiguous parallel to a line between the nodes

(which is certainly the case here). Assuming this, the conductivity GL is given by the

formula

ji

jijiji l

SkGL

,

,,,

·= , (2.27)

where k is the thermal conductivity of the material, S is the cross-section area of the

bar and l the distance between the nodes i and j. The term L on the conductivity

reflects the linear dependence on temperature of the heat flux qL along the conductor,

consequently

TGLqL ∆= · (2.28)

where ΔT is the temperature difference between the nodes.

Now, the net heat flux transferred by radiation between two nodes i and j at

temperatures Ti and T

j respectively, is given by

( )44jiijiiR TTBAq −= σε (2.29)

where σ is the Stefan-Boltzmann constant, εi the emissivity of node i, Ai the surface

area of i, and Bij the radiative exchange factor (REF) between i and j.

The REF is defined as the fraction of the energy emitted by i which is finally absorbed

by j, this energy arriving at j either directly or via reflection or transmission by other


nodes in the model, where there is a direct line of sight. There holds Bij=αj·Fij in which

αj is the absorptivity of node j and Fij is the view factor between i and j. By definition

the heat exchange is independent of the direction between nodes i and j, i.e. GLi,j = GLj,i

and GLi,i = 0. The quantity

jiiiji BAGR ,, ···εσ= (2.30)

is known as the radiative conductivity.

(Eq. 2.25) is applied to a three-node model (for operational simplicity), which leads to

(Eq. 2.31-2.33)

( ) ( )[ ]( ) ( )[ ] 1

41

4313

41

4212

1313121211

··

···

qTTGRTTGR

TTGLTTGLTC

+−+−+

−+−=•

(2.31)

( ) ( )[ ]( ) ( )[ ] 2

42

4323

42

4112

2323211222

··

···

qTTGRTTGR

TTGLTTGLTC

+−+−+

−+−=•

(2.32)

( ) ( )[ ]( ) ( )[ ] 3

43

4223

43

4113

3223311333

··

···

qTTGRTTGR

TTGLTTGLTC

+−+−+

−+−=•

(2.33)

which can be further developed to (Eq. 2.34), in which a system of linear and radiative

conductivity matrices have been denoted by GL and GR, respectively.

−−−−

−−

+

−−−−

−−

+

=

43

42

41

23132313

23231212

13121312

3

2

1

23132313

23231212

13121312

3

2

1

·

3

·

2

·

1

3

2

1

·

·

·

00

00

00

T

T

T

GRGRGRGR

GRGRGRGR

GRGRGRGR

T

T

T

GLGLGLGL

GLGLGLGL

GLGLGLGL

q

q

q

T

T

T

C

C

C

(2.34)


As the reader can observe from the system of equations, GL and GR are symmetric

matrices with the main diagonal being the sum of the other line elements multiplied by

-1. Single values GLi,j , GRi,j are found on the secondary diagonals. In addition, the heat

capacity matrix is presented as an identity matrix but with capacity terms in the

diagonal. Subsequently, the three node model is substituted now by a general form

with n nodes, (Eq. 2.35).

•

•

−••

•••

••−••

•••

••−

+

•

•

−••

•••

••−••

•••

••−

+

•

•=

•

•

•

•

∑

∑

∑

∑

∑

∑

=−

−

=

=

=−

−

=

=

4

4

41

1,1,1,

,1,

1,

,11,2

,12,11

,1

1

1,1,1,

,1,

1,

,1,2

,12,11

,1

1

·

·

·

11

·

·

·

0

n

k

n

jjnnnn

nnji

n

jjk

j

n

n

jj

n

k

n

jjnnnn

nnji

n

jjk

ji

n

n

jj

n

k

n

k

n

k

T

T

T

GRGRGR

GRGR

GR

GRGR

GRGRGR

T

T

T

GLGLGL

GLGL

GL

GLGL

GLGLGL

q

q

q

T

T

T

C

C

C

(2.35)

Thus, one obtains a more compact system (Eq. 2.36) in general matrix notation, in

which the linear conductivity and radiation coefficient matrices will be denoted by GL

and GR, respectively. The symbol ‘[ ]’ indicates matrices and vectors while diagonal

matrices are marked by ‘diag’.

[ ] [ ] [ ] [ ] [ ] [ ]qTGRTGLdt

dTCdiag ++=

4··· (2.36)


CHAPTER 3. Procedure to perform temperature distribution calculations

3.1. Introduction As seen in the previous chapter, the software used to obtain temperature distributions

is ESATAN. In this section, an alternative procedure combining ESATAN and a MATLAB

tool is shown. Subsequently, the algorithms that lie underneath each program are also

explained. It is crucial to simulate the behavior of the thermal model, not only with the

highest possible accuracy but also in a reasonable time. With this goal in mind, the

methodology presented here will be applied to practical examples in the upcoming

chapter.

3.2. Methodology The methodology implemented follows some of the guidelines described in section 1.2

of chapter 1. Figure 3.1 represents the sequence followed to perform temperature

distributions. In a final stage, the accuracy of the procedure with respect to the

reference software ESATAN is compared.

Figure 3. 1: Flow chart representing the implemented procedure

TMM set up; in a first phase the thermal mathematical model is defined. By using the

lumped parameter method, a first step is to divide the model into sections and

associate a discrete node with each section. All the properties of a section (thermal


capacitance, temperature, heat flux, etc.) are then considered to apply at the node,

hence the term lumped parameter.

So now we have discretized our model into a node network; in general, the more

nodes we use the more accurate the solution will be. In theory boundary nodes have

half the volume of the others; this is because the boundary conditions are to be

applied at the end surfaces of the bar rather than within its interior. At this point we

will have a network of nodes and conductors. To complete the lumped-parameter

model we must identify the boundary conditions and fix temperatures.

The next step will be to create a model file. Having thus defined our TMM, we can now

represent it in an ESATAN file, which is programmed in MORTRAN code (extended

FORTRAN) [11]. The model file is by custom called <model>.d where <model> is the

same name given on the $MODEL line of the file. Additionally, we may need a

geometric model for running radiative analysis and other system calculations under

orbit conditions.

Processing the model; once we have the file ready we can run it with ESATAN and

compute the required steady-state or transient temperature distributions. ESATAN will

accept models of components or systems presented in thermal network terms,

together with instructions for their solution. All models are checked during input for

self-consistency and translated into a database which is held permanently on the

computer storage system.

There are two principal steps in processing the model, preprocessing and solution.

Preprocessing involves parsing the input and constructing a machine-readable model

database. In the solution step, a FORTRAN program is generated, compiled and

executed which reads the model database and calls the appropriate library subroutines

to carry out the analysis and provide the required output.

Besides the model database, the preprocessor always produces a log file which echoes

the input and reports any errors or potential errors found, as well as giving a summary

of the structure of the model (the number of nodes, the number of conductors, etc.). If

ESATAN fails to preprocess the model one should look in the log file to see why.

Further errors may be detected during the FORTRAN generation, in which case a

second log file is produced.

Files generated by ESATAN are given the base name <MODEL>, i.e. the model name in

upper case, truncated for historical reasons to 8 characters. The following table lists

the main file-name extensions used.


File Extension

Model database .MDB

Preprocessor log file .log

FORTRAN generation log file .lgf

Solution progress monitoring .MON

Standard solution output .out

Table 3. 1: File name extensions provided by ESATAN [11]

In addition to the usual files generated by ESATAN, some lines of MORTRAN code have

been programmed to export model data in a ‘.csv’ format, easier to manage later with

MATLAB.

A first simulation of the steady-state will provide the initial values for temperatures in

equilibrium. Then, our interest will be to change some heat fluxes representing heat

dissipation for a selected number of nodes. The purpose of this test is to do an

evaluation of the impact that thermal perturbations can produce in our system. As a

result, a new set of temperatures will be achieved, this time for a transient state, and

stored in order to compare the temperature variation over time for different solvers

such as ESATAN and a MATLAB routine.

Processing thermal network matrices; once the temperature distribution is performed

by ESATAN, the next step is to prepare the MATLAB solver. The method is based on the

algebraic equations (Eq. 2.35-2.36) seen in chapter 2, hence we will first need to

arrange the network matrices for solving the heat equation. There are three main ‘.csv’

files used for this phase; ‘.gr’ and ‘.gl’ files containing the radiative and conductive

matrices specified by (Eq. 2.36) and ‘.nl’ that provides temperatures, capacities and

heat sources. At this point, the matrices [GR], [GL] and [C] detailed at (Eq. 2.35-2.36)

have been defined to proceed solving the heat equation.

Heat equation solver; at this phase of the procedure the heat equation has been

solved by an iterative calculation programmed with MATLAB. The programmed solver

is based on the Crank-Nicolson method and computes the transient solution for a

given time value. The temperature variation is iterated over time while spatial

variation is obtained as the result of algebraic calculation. The previously defined

matrices relate the conductive and radiative behavior of the model, considering also

boundary conditions. The solver stores the temperature variation over time in an

output file.

Finally, a comparison between the output files obtained from the two solvers will be

done so that differences in execution time and data accuracy can be examined.


3.3. MATLAB algorithm The routine programed in MATLAB is intended to calculate temperature distributions,

what has been stated along the document. This kind of calculation depends on initial

and boundary values as well as the evolution of temperature variable along time and

space. Considering all this things, the method typically used to solve such problems is

the Crank-Nicolson method, also called central difference method.

The purpose is then to solve a set of algebraic equations defining a thermal network.

Our network has been obtained by combination of several mathematical

considerations explained in chapter 2 such as lumped parameter, TDMA and finite

difference methods. With this goal in mind, the temperature variation over time can

be arranged from (Eq. 2.36) as

[ ] [ ] [ ] [ ] [ ] [ ]( )qTGRTGLCdiagdt

dT ++=

− 41 ··· , (3.1)

where GL and GR matrices have been described at (Eq. 2.35), C is a diagonal matrix

with heat capacities and T and q are vectors defining the initial temperatures and heat

sources of the model, respectively.

Once the thermal model has been processed with ESATAN, we can get information

such as C, T and q vectors as well as GLi,j and GRi,j values from the generated database.

This information will be properly arranged to implement our algorithm.

The initial step will be to build the required matrices. As the capacity matrix is full of

zeros and we need to invert every single term, we will need to change those values by

nonzero values to avoid divergences, as ∞=01 . Those zeros are first replaced by very

small values, which once have been inverted are easy to detect and replace by a zero

after the inversion has been made.

Afterwards, GR and GL matrices are built using GLi,j and GRi,j values. As coefficients

with i=j are zero, there are neither radiation nor conductivity between a node and

itself, these positions must be replaced by a negative addition of the rest of the terms

of the corresponding line/column (Eq. 2.35). The meaning of this operation is to define

the coupling relations between nodes, those being diffusive or boundary, as observed

in (Eq. 2.31-2.33).

Once the matrices are prepared, we need to give initial conditions to the equation. The

temperature vector is loaded from the equilibrium vector, what means no

perturbation in the system, obtained as a steady state solution. An important

consideration is to convert grades ºC to kelvins, for which adding the term 273.15 is

compulsory. Furthermore, we also initialize the heat source vector q which will be later

perturbed at any node.


In a final stage, the temperature variation is calculated by iteration over time. To show

how this variation is obtained, let’s analyze the process sequentially. First of all, a

counter for time length and storing data are defined.

Counters t1 = 0, 1 ..., tf and t2 = 0, dt , ..., dt−1

Prior to any calculation, the number of nodes needs to be defined so that we are able

to operate the temperature vector. We then settle the vector ‘i’.

i = [1, 2, ..., 1395]

The next step is to obtain the temperature vector to the power of four in order to

operate the radiative equation.

(TR)i = Ti4 (3.2)

Once the radiative terms are obtained we can calculate the temperature variation by

operating temperature vectors and matrices built in MATLAB with the data extracted

from ESATAN.

ΔTi = dt·( Ti ·[GL] i,j + (TR)i ·[GR] i,j + [q] i)·diag[C]i,j -1 (3.3)

In (Eq. 3.3) the temperature variation ΔTi is calculated as the product of the inverted

capacity matrix diag[C]i,j -1

, the time step dt and the contribution of the heat power

obtained from the heat source q, linear terms Ti ·[GL] i,j and radiative terms (TR)i ·[GR] i,j.

The reader must also consider that indexes ‘i’ and ‘j’ have the same length. Moreover,

if we do a dimensional analysis we can verify that dimensions match:

[Kelvin] = [seconds]·[Watt]·[Kelvin/Joule], where [Joule] = [seconds]·[Watt]

Once the temperature variation is calculated we can iterate the temperature over the

counter t2, which sets a loop for a period of one second.

Ta= Ti + ΔTi , where a is t2 (3.4)

Finally, temperature is stored at a rate of one second, so we can have the temperature

distribution over the total execution time.

Tb = Ta +275.13, where b is t1 (3.5)

At (Eq. 3.5) vector Tt1 has dimensions time t1 and number of nodes ‘i’, besides it

converts Kelvin units to Centigrade by adding the term 273.15.


3.4. ESATAN algorithm [10] ESATAN software provides different types of solvers depending on how large the

thermal model is. For instance, for a steady-state solution with small to medium-size

models SOLVFM, which implements a forward method, is recommended by the

software specifications. SOLVFM takes few iterations to converge but has high memory

consumption.

The iterative routines are most frequently used as they are the most suitable for

nonlinear problems, which is certainly the case when solving the radiative equation. An

example is the Crank-Nicolson transient solver SLCRNC which calculates the

temperature change rate with respect to time at the start and end of each time step.

Subsequently, the temperature change is averaged over time step.

For most transient analyses SLCRNC is recommended, since it is stable for large

timesteps, although it should be remembered that accuracy is inevitably reduced when

using very large timesteps. SLFWBK is an alternative solver that has been used which

combines both Crank-Nicolson and forward-backward difference methods. This

method is centered on the midpoint of the time interval, is numerically stable

whatever the interval size is and the time discretization is second order. As an implicit

scheme, SLFWBK gives simultaneous solution of temperatures at each time step.

In order to compute a solution, various parameters are needed to specify, for instance,

what level of convergence is required. Convergence on temperature is controlled by

RELXCA (the smaller this value, the better the convergence) and the number of

iterations the solver will perform before giving up is specified by NLOOP. When setting

the convergence criteria RELXCA for the iterative solution schemes, we must consider

the temperature accuracy required.


CHAPTER 4. Model simulations, analysis and data verification

4.1. Introduction At the beginning of this chapter a study of LISA model performed by ESATAN TMS.R3

and 10.2 versions, under different conditions, has been carried out. The purpose of the

study was to estimate possible mismatches between these versions when simulating

steady state.

Then, the strategy implementing the methodology exposed in chapter 3 has been

followed to first analyze a simple case so that a first approach was obtained. Finally,

once the procedure was operating properly a more complex model was tested. At the

end of the chapter an analysis and verification of data extracted from the simulations

presented is also done.

4.2. TMS.R3 with zero capacities At the beginning of the study, a steady state simulation workbench was developed for

the two ESATAN versions. The two principal parameters changed were the relaxation

factor ‘RELXCA’ defined in the NASTRAN model file and the zero capacities. So on the

one hand, our interest is first to compare results by using zero capacities while solving

the heat equation. On the other hand, we also want to evaluate time execution and

data accuracy while tuning the ‘RELXCA’ variable.

4.2.1. TMS.R3 with zero capacities In the steady state analysis capacities are not required, as there is no dependency on

them in the equation (Eq. 2.5). One interesting point is to check the influence of using

zero matrices instead of the capacity values defined by the system.

RELXCA Exec. Time with

zero cap

Exec. Time without

zero cap

Max. Temp. Difference °C

1.0E-1 51 sec 60 sec 0

1.0E-3 66 sec 64 sec 0

1.0E-5 76 sec 75 sec 5.1220e-006

1.0E-7 85 sec 85 sec 0

1.0E-9 199 sec 193 sec 0

Table 4. 1: Maximum differences performed by TMS.R3 between temperatures with and without zero capacities


Table 4.1 shows how small differences have been found in terms of execution time. By

comparing temperature distributions, there is also a big similarity. Differences of the

order of few microkelvins are an excellent result. One may also observe that the

smaller the relaxation constant is, the longer the simulation. That is because the solver

requires more iterations, which is time consuming.

4.2.2. TMS.R3 vs. 10.2 with capacities At this point, a comparison between results performed by different versions of

ESATAN is shown in table 4.2.

RELXCA Exec. Time TMS.R3 Exec. Time 10.2 Max. Difference °C

1.0E-1 51 sec 40 sec 5.1860e-006

1.0E-3 66 sec 59 sec 5.1860e-006

1.0E-5 76 sec 77 sec 5.1220e-006

1.0E-7 85 sec 102 sec 0

1.0E-9 199 sec 119 sec 5.1220e-006

Table 4. 2: Comparative table between TMS.R3 and 10 .2 versions with capacities simulation

This time there is a slight discrepancy in the execution time, although it is not

significant regarding the time duration. In terms of accuracy difference the

approximation is still good, considering microkelvins as an excellent accuracy indicator.

According to these results, one can extract that both ESATAN TMS.R3 and 10.2

versions offer similar features when providing thermal analysis.

4.3. Analysis of a simple case: the heating bar We begin with a very simple example [11] to introduce some modeling phases such as

the geometrical model and the mathematical thermal methods. The following case

considers a heating bar.

Let’s consider a rectangular metal bar, insulated along its length, with one end held at

a constant 20 °C while a heat flux of 100 W is applied to the other. The metal has a

thermal conductivity of 240 W/m °C, a specific heat of 900 J/kg °C and a density of

2700 kg/m3. The bar is 45 cm × 4 cm × 4 cm. Finally, the environment is fixed to 20°C.

We want to know the temperature distribution in the bar when it has reached the

steady state, i.e. when it has been left long enough for the temperature at each point

to be constant in time.

The first step to simulate the model is to build a Nastran file with the description of it.

In appendix A one can find the ‘.d’ and ‘.erg’ files that have been built detailing the bar

TMM and GMM, respectively. Then we run the ESATAN software and load the model


file. In figure 4.1 there is a geometrical view of the model with 10 volumes

representing each active node.

Figure 4. 1: Geometrical model of a bar composed by 10 nodes designed with E SATAN

As a goal for this case, we want to do a transient simulation for the nodes defined. It is

convenient to run the model with an appropriate value for the convergence criterion.

Although a big value could give a faster solution, computationally this means fewer

iterations, accuracy could be lost. This trade-off between speed and accuracy is an

important consideration when defining our model.

Figure 4. 2: Transient solutions at node 1 of the heating bar, i n blue by MATLAB, in green by ESATAN


After running a simulation of 10000 seconds, data from the two solvers has been

compared. The relaxation constant has been chosen at a reasonable value for a good

accuracy. During the start-up the difference between the two programs is maximum,

and then the solution converges to a better approximation. As seen in figure 4.3, node

1 (where the heating source is applied) presents a higher absolute relative error than

the others, while the error diminishes when moving away from node 1. In this

calculation TE is the temperature obtained from ESATAN and TM the one obtained from

MATLAB.

Figure 4. 3: Absolute relative error (%) ESATAN results with respect to MA TLAB. |(T E-TM)/TE| with node 1 in blue, 3 in green and 8 in red

An execution time of 242 seconds was reported for ESATAN while MATLAB took 3.628

seconds for 10000 seconds iteration. The maximum difference was also found at node

1, what can be observed from Figure 4.3; here this value is 57.2 m°C. This temperature

difference can be seen as a reasonable value, depending on how critical the system is.

What cannot be ignored is the excellent execution time exhibited by our tool.

4.4. Analysis of a complex case: the LISA model Since the context of the project is based on the LISA mission, the simulation of its

model is required. As the thermal network is a complex system to be described in this

work, a model file has been provided by ASTRIUM; for this reason the descriptive files

are not included in the work. In figure 4.4, a view of the geometrical model is shown.


Figure 4. 4: Geometrical model of the LISA system r epresented with ESATAN

We have first run a steady state simulation with ESATAN to drive the system to an

equilibrium point. Afterwards, we have used the generated data as the initial

conditions for a transient simulation. Once again, we are generating a workbench,

based on different configurations of our model, to evaluate several situations where

the network is perturbed by a step input of 0.2W. Then the same conditions will be

performed in MATLAB so that a database can be stored for further comparison. With

this goal in mind, three tests have been carried out and are now shown.

Figure 4.5 shows an optical bench, which takes place at the telescope module, and two

nodes representing two diodes that will be perturbed by a dissipating source.

Figure 4. 5: Optical bench where heat diodes are re presented by nodes ‘12120’ and ‘12090’ [17]


4.4.1. The node 12120 First, a perturbation on the node ‘12120’ is produced. This node is interesting because

it represents a diode from the optical bench of the telescope (one of the warmest

parts) and has a visual contact with several components, thus providing radiative

coupling.

In figure 4.6 one can see a plot of the temperature distribution at node ‘12120’ for a

transient simulation of 218000 seconds. The temperature has been stored at a rate of

1 sample every 100 seconds that is why the plot represents a bit more than 2000

samples. Initially, the temperature difference is of the order of microcentigrades, but

as time increases the difference gets higher and becomes visible in the graph.

Figure 4. 6: Plot of temperature distribution of T E (green) and T M (blue) over time

In figure 4.7 one can see the absolute difference value of the temperature distribution

over the nodes for a steady state simulation. This plot shows how the nodes change

over time, so depending on the couplings one can see that there are nodes that barely

change, while others change at a a maximum of near 60mºC. The model also exhibits a

geometrical symmetry, i.e. there are two telescopes built with identical structure, so it

is remarkable to observe how the temperature distribution follows a symmetrical

behavior. In the left side of the figure, where the node has been perturbed, the

distribution performed is the same as on the right side, but in the latter, temperatures

are smaller because are not directly affected.


Figure 4. 7: Plot of the temperature absolute diffe rence distribution between T E (blue) and TM (red)

Figure 4.8 shows the temperature distribution. At first sight one cannot appreciate

differences as they are of the order of milli/micro ºC with respect to the temperature

range of the plot. Here the symmetry of the system is obvious.

Figure 4. 8: Plot of the temperature distribution f or T E (blue) and T M (red)

Finally, figures 4.9 and 4.10 show the temperature absolute difference and relative

error, respectively for node ‘12120’. The most remarkable feature of the graphs are

the increasing error over time, however, the curve smooths while a stable value is


expected. The main reason for this behavior is the effect of nonlinearities caused by

the radiative terms.

Figure 4. 9: Absolute differences |T E-TM| at node ‘12120’

Figure 4. 10: Absolute relative error (%) |(T E-TM)/TE| at node ‘12120’


While ESATAN spent 71h 22m MATLAB took 2h 34m. The model processing takes

almost 30 times in ESATAN while MATLAB presents a relative error of around 0.014%

as shown in figure 4.10.

4.4.2. The node 12090 Now, a perturbation on the node ‘12090’ has been produced. This node is near node

‘12120’ and belongs to the optical bench, too. Once again, dissipation at this node is

interesting to check the impact it causes in the rest of the model.

Figure 4.11 shows a plot of the temperature distribution at node ‘12090’ simulated

with ESATAN and MATLAB. The transient simulation performs 250000 seconds at a

rate of 1 sample every 100 seconds, what gives 2500 samples. Here the behavior is

similar to the one seen in the test at node ‘12120’.

Figure 4. 11: Plot of the temperature distribution of T E (green) and T M (blue) over time

Analogously to the ‘12120’ node case, we here obtain a similar temperature difference

as shown in figure 4.12.


Figure 4. 12: Plot of the temperature absolute diff erence distribution between T E (blue) and T M (red)

Figure 4. 13: Plot of the temperature distribution for T E (blue) and T M (red)

The temperature distributions of the model are quite similar when compared in figure

4.13, here some red points from MATLAB simulation is observable. The absolute

difference exhibited at figure 4.14 shows once more a very small difference with

respect to the value ranges of the system. Figure 4.15 shows the relative error with

respect to the ESATAN calculation, results are very precise.


Figure 4. 14: Absolute difference |T E-TM| at node ‘12090’

Figure 4. 15: Absolute relative error |(T E-TM)/TE| at node ‘12090’

This test has taken 3h09min in MATLAB, while ESATAN took 91h19min.


4.4.3. Boundary nodes

The nodes pictured with blue color in figure 4.16 represent radiatively inactive nodes

which are linearly coupled to an internal black MLI layer. All nodes pictured in red

represent radiatively active nodes, which are interacting with internal nodes. The

structure shows the boundary envelope, which is formed by boundary nodes, some of

which will be also perturbed during the test.

Figure 4. 16: Radiative and linear coupled boundary envelope [17]

In the last test the nodes changed were the boundaries ‘16010’, ‘16020’, ‘16030’,

‘16040’, ‘16050’ and ‘16060’. Boundary nodes are fixed in a system, so it is interesting

to see the influence of these nodes when changing. Figure 4.17 shows the distribution

of the nodes for a simulation time considered to be the steady state with ESATAN and

MATLAB.


Figure 4. 17: Temperature distribution in Matlab (red) and Esatan (blue).

The temperatures represented in the plots are always in grades centigrade. Actually,

one of the most frequent mistakes made at the beginning of the simulations was to

use ºC for every calculation in MATLAB, when for radiative calculations equations must

consider kelvins instead.

Figure 4. 18: Temperature difference distribution o f LISA model.

Figure 4.18 represents the temperature difference distribution between TE and TM.

Once again the difference is of the order of tens of milli ºC, but this time there is a

higher dispersion, more nodes are near the maximum difference.


Figure 4. 19: Absolute relative error |(T E-TM)/TE| after 217000 seconds.

Observing figure 4.19 one can see that the distribution of the error has the same

aspect as in figure 4.12, but this time the relative error is ten times higher than before.

The dispersion due to the dissipation in several nodes has increased the error when

calculating with MATLAB.

Figure 4. 20: Difference between final temperature and initial temperature.

This simulation took 71h55min with ESATAN and 2h40min with MATLAB.


CHAPTER 5. Conclusions and future work

Once the end of the work has been reached several ideas can be extracted. First of all,

a methodology to obtain temperature distributions has been developed. This

methodology has been based on the algebaric equations extracted from the finite

difference method, which has been described in chapter 2. The algebraic equations

have been built by a routine programmed in MATLAB with a model database obtained

from ESATAN software. In the end, the heat equation has been solved to achieve a

temperature distribution.

Secondly, results of the programmed MATLAB tool have exhibited very good values, in

terms of time execution and data accuracy, when comparing to those simulated with

ESATAN. MATLAB is thirty times faster and a relative error from 0.01% to 0.6% is found

in our simulations. As a result, an approach to perform temperature distributions for

thermal networks has been achieved.

In order to improve the thermal tool, several things need to be further developed.

First, a revision of the effect of nonlinearities when solving the radiative equation, this

might be a cause of the increasing error as time increases. Secondly, an approach with

Simulink should be done, as it provides a more intuitive interface for anybody not used

to program in MATLAB, at a first attempt during this work, the way Simulink manages

importing ‘.csv’ files was not found. Then one could compare if the new results are

improved by Simulink. Finally, an input bench to set different types of perturbation can

also be implemented in order to provide simulations in an easier way.


BIBLIOGRAPHY AND REFERENCES

[1] Matlab und Simulink, Josef Hoffman. Addison-Wesley, 1999.

[2] Introduction to Heat transfer, Frank P. Incropera. John Wiles & Sons, 1996.

[3] Finite-Difference Approximations to the Heat Equation, Gerald W. Recktenwald,

2011.

[4] Numerical Techniques for Solving the One-Dimensional Heat Equation Michael A.

Chupa, 1998.

[5] http://www.qrg.northwestern.edu/projects/vss/docs/thermal/2-what-is-heat-

balance.html

[6] http://www.mathworks.co.uk/help/

[7] http://www.scribd.com/doc/2071569/Partial-Differential-Equations-in-MATLAB

[8]http://online.redwoods.cc.ca.us/instruct/darnold/DEProj/sp02/AbeRichards/paper.

pdf

[9] http://en.wikipedia.org/wiki/Main_Page

[10] ESATAN-TMS Thermal Engineering Manual

[11] ESATAN-TMS Thermal Training Manual

[12] Thermal Control, Tasks and Processes for Thermal Engineering, EADS ASTRIUM

GmbH.

[13] Application of Linear Control Methods to Satellite Thermal Analysis, Martin

Altenburg and Johannes Burkhardt, EADS Astrium GmbH, 2008.

[14] Mission Design Description, LISA Mission Formulation, EADS ASTRIUM 2008.

[15] Coupled Thermal Analysis for MDR Baseline, LISA Mission Formulation, EADS

ASTRIUM 2008.

[16] LISA: Probing the Universe with Gravitational Waves, EADS ASTRIUM 2007.

[17] Thermal Design & Analysis for MDR Baseline, EADS ASTRIUM 2008.

numerical methods for computing the temperature

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