brno university of technology - vysoké …lab4.fme.vutbr.cz/pohanka/pdf/phdthesispohankam.pdf ·...

BRNO UNIVERSITY OF TECHNOLOGY

FACULTY OF MECHANICAL ENGINEERING

HEAT TRANSFER AND FLUID FLOW LABORATORY

Ph.D. THESIS

(Disertační práce)

Study Branch

DESIGN AND PROCESS ENGINEERING

Topic:

Technical Experiment Based Inverse Tasks in Mechanics

(Inverzní úlohy mechaniky s vazbou na technický experiment)

Author: Ing. Michal Pohanka

Supervisor: Doc. Ing. Miroslav Raudenský, CSc.

2006

Acknowledgments

All who know Miroslav Raudenský acknowledge his unique contribution. The topics which are

presented in this work could have hardly been conceived without his initial insight. I have been

fortunate to have the continuing encouragement of Jaroslav Horský, Keith A. Woodbury, and Miloslav

Druckmüller. I would also like to acknowledge the contribution of colleagues Petr Kotrbáček,

Jonathan Woolley, and Antonín Dobšák. Special thanks are due to my wife, my son, and my mother for

their great support.

i

ABSTRACT

In this work, complex inverse heat conduction problems are studied. Thermal numerical

models for design and control purposes in metallurgical industry require a precise description

of heat transfer phenomena. Comprehensive quantitative information on the heat transfer

phenomena is not available for cooling of hot moving surfaces. In this work, attention is

focused on the search for boundary conditions describing the heat transfer in engineering

applications of spray cooling of metal surfaces and cooling of molds during casting. First,

a finite difference method is extended to cover phase changes and temperature dependent

material properties. Behavior of the iteration process during numerical solution of heat

transfer partial differential equations is investigated and a stopping criterion based on the

geometric convergence of the computed results is proposed. Methods for discretization error

estimation are presented and tested for the simplification of the object, its boundary

conditions, and the time domain when the numerical approach is applied. These methods are

then used for discretization optimization. For solution of inverse tasks, such as determination

of boundary conditions, temperature dependent thermal material properties, and model

calibration, combined optimization methods or methods of artificial intelligence are used to

achieve a higher accuracy of results and to speed up the computational process. Finally,

a method for correction of measured pressure distribution of a spraying water nozzle is

presented to allow comparison of cooling intensity and real pressure distribution.

ii

Contents

CONTENTS

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

2 PROBLEM FORMULATION AND GOALS.........................................................................3

3 PRESENT STATE OF KNOWLEDGE – OVERVIEW............................................................6

3.1 Fundamentals of Heat Transfer Phenomena.....................................................................6

3.1.1 Heat Transfer............................................................................................................6

3.1.2 Boundary and Initial Conditions...............................................................................7

3.2 Analytical Solution of Heat Equation...............................................................................8

3.2.1 The Semi-Infinite Solid............................................................................................9

Constant boundary conditions......................................................................................................9

3.2.2 The Plane with Convection.....................................................................................10

3.3 Numerical Methods for Solution of Heat Transfer Partial Differential Equations.........10

3.3.1 Diffusive Initial Value Problems.............................................................................11

3.3.2 Diffusion Equation in Multi-dimensions................................................................13

3.3.3 Finite Difference Method (FDM)...........................................................................14

3.3.4 Finite Volume Method (FVM)................................................................................14

3.3.5 Finite Element Method (FEM)...............................................................................15

3.3.6 FDM, FVM, and FEM Similarity...........................................................................17

3.3.7 Finite-Volume-Unstructed-Mesh (FV-UM)............................................................18

Cell-centered triangular method................................................................................................18

Vertex-centered quadrilateral and triangular method in local co-ordinates.............................20

Comparison of RMS error for various approaches....................................................................21

3.3.8 Boundary Element Method (BEM).........................................................................22

3.3.9 Transmission Line Matrix (TLM)...........................................................................22

3.3.10 Phase Change Implementation..............................................................................24

Apparent heat capacity...............................................................................................................26

Total enthalpy.............................................................................................................................26

A general form of phase change implementation.......................................................................27

3.3.11 Solution of Linear Algebraic Equations................................................................27

3.3.12 Non-linear Set of Equations..................................................................................28

Under-relaxation........................................................................................................................29

3.4 Inverse Heat Conduction Problem..................................................................................30

iii

Contents

3.4.1 IHCP by Beck.........................................................................................................31

IHCP in multi-dimensions by Beck............................................................................................32

3.4.2 Comparison of Inverse Heat Conduction Methods by Raynaud.............................33

Test cases....................................................................................................................................36

Comparison................................................................................................................................36

3.4.3 Application of Beck’s Algorithm............................................................................37

Numerical simulation of casting of aluminum............................................................................37

Investigation of transient heat transfer coefficient in quenching experiments...........................39

A modified sequential approach.................................................................................................40

A two-dimensional IHCP...........................................................................................................41

3.4.4 Application of BEM in IHCP.................................................................................43

3.4.5 Gaussian Least Square Differential Correction scheme.........................................44

3.4.6 Conjugate Gradient Method in IHCP.....................................................................45

3.4.7 Methods Based on Artificial Intelligence...............................................................46

Artificial neural network............................................................................................................46

Genetic algorithm.......................................................................................................................49

3.5 Data Filtration.................................................................................................................49

3.5.1 Filtration in Time Domain......................................................................................50

3.5.2 Filtration in Frequency Domain..............................................................................50

Filtration of measured data........................................................................................................52

3.5.3 B-Splines................................................................................................................52

4 PROBLEM SOLUTION..............................................................................................54

4.1 Numerical Heat Conduction – Derivation of Discretization Equations.........................55

4.1.1 Cartesian Coordinates – One-dimensional Heat Conduction.................................56

Temperature dependent heat conductivity..................................................................................57

Temperature dependent mass density and specific heat ............................................................58

Source-term linearization...........................................................................................................61

Set of equations for all internal grid points in the general form................................................61

4.1.2 Cylindrical Coordinates – One-dimensional Heat Conduction..............................62

Equations for internal grid points in the general form for 1D cylindrical coordinates.............64

4.1.3 Treatment of Boundary Conditions.........................................................................64

Surface temperature....................................................................................................................65

Surface heat flux.........................................................................................................................65

Convection..................................................................................................................................66

Radiation....................................................................................................................................66

iv

Contents

4.2 Solution of Discretization Equations..............................................................................67

4.2.1 Error Estimation of Iteration Algorithm.................................................................70

4.3 Discretization Error Estimation......................................................................................71

4.3.1 Boundary Conditions..............................................................................................72

4.3.2 Mesh Discretization Error.......................................................................................73

4.3.3 Time Discretization Error.......................................................................................75

4.4 Optimization of Discretization.......................................................................................77

4.4.1 Refinement of Boundary Conditions in Time Domain...........................................77

4.4.2 Mesh Optimization.................................................................................................78

4.4.3 Time Step Refinement............................................................................................79

4.5 Multi-level Filtration of Measured Data.........................................................................79

4.6 Solutions of IHCP..........................................................................................................82

4.6.1 Computation of Boundary Conditions....................................................................83

Usage of optimization method....................................................................................................84

Usage of feed-forward neural network......................................................................................87

4.6.2 Computation of 2D Boundary Condition...............................................................89

4.6.3 Determination of Temperature Dependent Material Properties..............................91

4.6.4 Sensor Calibration..................................................................................................94

4.7 Data Focusing.................................................................................................................97

5 SOLUTION PRESENTATION AND ANALYSIS................................................................101

5.1 Direct Solutions of Heat Conduction...........................................................................101

5.1.1 Test of Iteration Algorithm of Non-linear Heat Conduction Problem..................101

Two-dimensional model with annular copper..........................................................................102

5.1.2 Implementation of Phase Change and Heat Generation.......................................103

Two-dimensional model of a heater unit..................................................................................103

5.1.3 Verification of Discretization Error Estimation....................................................105

Boundary Conditions and Time Discretization Error...............................................................105

Mesh Discretization Error........................................................................................................109

5.2 Filtration of Real Measured Data..................................................................................111

5.3 Inverse Heat Conduction Problems..............................................................................114

5.3.1 Boundary Conditions Computation......................................................................114

Beck's approach using filtered data..........................................................................................114

Modified downhill simplex optimization method......................................................................118

Feed-forward neural network...................................................................................................119

v

Contents

5.3.2 Calibration of Computational Model of Measuring Sensor.................................120

5.3.3 Comparison of Two-dimensional Boundary Conditions Computations using

An Optimization Method and Sequential Beck’s Approach.....................................123

5.3.4 Usage of Optimization Method for Determination of Temperature Dependent

Material Properties....................................................................................................126

5.4 Pressure Impact – 2D Field Correction........................................................................128

6 PREVIEW OF NEXT RESEARCH...............................................................................133

7 CONCLUSION......................................................................................................134

8 LITERATURE.......................................................................................................136

9 AUTHOR’S REFERENCES.......................................................................................141

10 APPENDIX A – CONDUCTION AND HEAT DIFFUSION EQUATION....................................I

10.1 Rate equation...................................................................................................................I

10.2 Heat Diffusion Equation................................................................................................II

11 APPENDIX B – CONVECTION..................................................................................V

12 APPENDIX C – RADIATION ...................................................................................VI

13 APPENDIX D – SOLUTION OF LINEAR ALGEBRAIC EQUATIONS................................VIII

13.1 Gauss-Jordan elimination..........................................................................................VIII

13.2 LU decomposition......................................................................................................VIII

13.3 Tridiagonal systems of equations.................................................................................IX

13.4 Band diagonal systems of equations..............................................................................X

13.5 Iterative improvement of a solution of linear equations...............................................XI

13.6 Conjugate gradient method..........................................................................................XI

14 APPENDIX E – COMPARISON OF FDM, FVM, AND FEM....................................XIII

14.1 Constant Heat Flux....................................................................................................XIV

14.2 Constant Heat Transfer Coefficient...........................................................................XVI

14.3 Time-dependent Heat Flux........................................................................................XIX

15 APPENDIX F – MULTIDIMENSIONAL HEAT CONDUCTION.......................................XXII

15.1 2D Heat Conduction.................................................................................................XXII

15.1.1 Cartesian Coordinates.......................................................................................XXII

15.1.2 Cylindrical Coordinates...................................................................................XXIII

15.2 3D Heat Conduction................................................................................................XXV

vi

Contents

15.2.1 Cartesian Coordinates......................................................................................XXV

15.2.2 Cylindrical Coordinates..................................................................................XXVI

16 APPENDIX G – PHOTO DOCUMENTATION..........................................................XXIX

16.1 Linear Test Bench...................................................................................................XXIX

16.1.1 Sensor Calibration............................................................................................XXX

16.2 Mold Casting..........................................................................................................XXXI

16.2.1 Alloy Casting Experiment...............................................................................XXXI

16.2.2 Experiment for Determination of Mold Material Properties.........................XXXII

16.3 Pressure Impact Measurement.............................................................................XXXIII

17 SUMMARY ...........................................................................................................A

18 RESÜMEE............................................................................................................B

19 RESUMÉ..............................................................................................................C

vii

Nomenclature

NOMENCLATURE

A............area, m2

A............matrix of coefficients

a.............coefficients in matrix A

Bi...........Biot number

b.............column vector of quantities, bias

vectors of the feed-forward neural

network

C............capacity, F

c.............thermal capacity, J/kg.K

cA...........apparent thermal capacity, J/m3.K

cp............thermal capacity at constant pressure,

J/kg.K

cvol..........volume-averaged apparent thermal

capacity, J/m3.K

E............thermal internal energy, J; Emissive

power, W/m2

ög...........rate of energy generation, W

öin..........rate of energy transfer into a control

volume, W

öout.........rate of energy transfer out of control

volume, W

öst...........rate of increase in energy stored

within a control volume, W

f.............frequency, s-1, m-1

f.............N-dimensional function

G............irradiation, W/m2

g.............volume fraction, 0..1

H............total volumetric enthalpy, J/m3

δH..........difference between solid and liquid

enthalpy, J/m3

h.............convection heat transfer coefficient,

W/m2.K

i, j..........unit vectors

k.............thermal conductivity, W/m.K

L............characteristic length, m; latent heat,

J/kg

L............lower triangular matrix

M...........measured data in frequency domain

m............measured data

N............shape function

n.............number of elements

n............normal vector

P............filter function in frequency domain

p.............filter function

Q............generalized heat source

q............heat transfer rate, W

q............rate of energy generation per unit

volume, heat source, W/m3

qa, qb.....coefficients of linearized heat source,

W/m3, W/m3.K

q.............heat flux, W/m2

R............resistance, Ω

r, ϕ, z.....cylindrical coordinates, m, rad

r, ϕ, Θ....spherical coordinates, m, rad

RMS.......root-mean-square error

s.............phase velocity, m/s

SSE........sum square error

T............temperature, K

T *..........measured temperature, K

T............average temperature, K

viii

Nomenclature

T............filtered temperature, K

T............initial guess of temperature, K

t.............time, s

U............upper triangular matrix

u, v.........local co-ordinates

V............volume, m3

W...........weight matrixes of the feed-forward

neural network

w............weighting function

x, y, z......rectangular coordinates, m

x.............vector of unknowns

Z............impedance, Ω

α............thermal diffusivity, m2/s

β, δ, γ.....substitution parameters

ε.............emissivity, 0..1

............safety coefficient

µ............position, m

............general boundary condition

ϕ............azimuthal angle, rad

............unknown variable

ψ............variation of a function within a cell

ρ............mass density, kg/m3

σ............Stefan-Boltzmann constant,

5.67×10-8 W/m2.K4

............under-relaxation factor, 0..1

............estimated error

Θ............zenith angle, rad

............sensitivity coefficient

Subscripts

b.............blackbody

c.............column index

d.............solid phase

f.............future time step index

i.............general index

J, K, P....boundary indexes along x-axis,

y-axis, and z-axis, respectively

j, k, p......node indexes along x-axis, y-axis,

and z-axis, respectively

L............left side

l.............liquid phase

R............right side

r.............row index

rad.........radiation

s.............surface condition

sur.........surroundings

w............temperature interval index

x.............local conditions on a surface

∞............free stream conditions

Superscripts

m............time index

i.............iteration index

ix

Introduction

1 1 II NTRODUCTIONNTRODUCTION

Efficient, accurate and stable numerical methods for solving fluid flow problems, heat

and mass transfer processes, chemical reactions, and turbulent phenomena are of great

importance in many industrial applications. It is nowadays generally recognized that computer

analysis of complex problems may provide a cost-effective, quick and sufficiently reliable

method in many cases. Sometimes, the computational methods may also be an alternative or

a complement to experimental investigations. Despite the tremendous developments and

achievements in methodologies, computer capacity and range of applications during last few

decades, still research on many topics is needed.

Many industries and companies worldwide use commercially available so-called

CFD-codes (CFD = Computational Fluid Dynamics) for simulation of flow and heat transfer

topics in heat exchangers, for enhancement of heat transfer, turbulent combustion in gas

turbines and combustion engines, electronics cooling and gas turbine blade, etc. Among these

codes are: ANSYS, FLUENT, CFX, STAR-CD, PHONECIS, CFD2000, FIDAP, ADINA and

others. However, to successfully apply such codes and to interpret the computed results, it is

necessary to understand the fundamental methods of such computations.

Although computation of heat transfer has reached a certain level and it can be

significantly helpful in many engineering and industrial applications, still comprehensive

research is needed in, e.g., heat transfer, handling of complex geometries, modeling of

two-phase convective flow and turbulence modeling. In this work, attention is focused on the

search for boundary conditions describing the heat transfer in engineering applications of

spray cooling of metal surfaces (up to 1200 oC). The cooling in casting, descaling, cooling of

products in hot rolling, and cooling of rolls in rolling technology are the typical industrial

applications. Thermal numerical models for design and control purposes in metallurgical

industry require a precise description of heat transfer phenomena. Comprehensive quantitative

information regarding the heat transfer phenomena is not available for quenching of hot

moving surfaces.

An almost all-engineering analysis is based on a model that has nice smooth

mathematical properties. On the other hand, any measured data has some noise. Combining

measurement with an inverse analysis often results in an ill-conditioned problem.

1

Introduction

Mathematically, such problems are ill-posed and have to be overcome by developing new

numerical methods, introduction of new objective functionals into the optimal control

algorithm, new regularization techniques, new experimental procedures, etc.

The inverse problem is one type of an ill-conditioned or ill-posed problem. In a typical

direct problem, one is given a model, the initial conditions of the state variables, and the

forcing terms, and is asked to produce a solution. Solving the direct problem, an error in input

data is usually suppressed. In the general inverse problem, one is given a model and

measurements of some state variables, and is asked to estimate the initial conditions, the

forcing terms, and the rest of state variables. Unlike the direct problem, an error in input data

is extremely amplified in the inverse problem. The given results are often unusable, if any

results are obtained. The noise significantly decreases the stability of the computational

algorithm.

An improvement of computational algorithms and new designs of experimental

measurements are needed. The major attention is focused on solving inverse heat conduction

problems (IHCP), such as determination of the fast changing boundary conditions,

suppression of influence of an error in measured data, and calibration of computational

models. Along with these problems, an improvement in the self-adaptive design of

computational models and knowledge of accuracy of computed results using a numerical

method are necessary.

New approaches, such as usage of various optimization methods and data filtration

could be useful to overcome the above specified problems. Nowadays, an artificial

intelligence could play an important role. Approaches like feed-forward neural network or

genetic algorithm may be considered.

The contemporary revolution in computer technology is producing larger, faster, and

inexpensive computers with almost no limits in sight. This technology has made it possible

for engineers and scientists to construct more realistic mathematical models of physical

processes. An important area of engineering that will become even more significant in the

future is the combining of measurements with engineering models.

2

Problem Formulation and Goals

2 2 PPROBLEMROBLEM F FORMULATIONORMULATION ANDAND G GOALSOALS

Many papers that describe solution of various problems using both direct and inverse

heat conduction, already exist. Reviewing these papers, it can be easily found that most of

them neglect errors. Some of them describe conditioned stability of the solution with respect

to the noise in the measured data. A few of the papers describe some sensitivity of computed

coefficients and other sensitivity analysis. No wonder that only a few papers consider errors in

computed data. Even wide spread professional computational software products do not

support error estimation for unsteady or dynamic conditions.

Performing various heat transfer experimental analyses, a very important problem

arises: How precise are the computed data? An answer to this question has a much wider

range of applicability than could seem. The knowledge of precision of the computed results

can be very useful for uncovering the weakest points during a complicated and interconnected

heat transfer experimental analysis. During the analysis, an experiment and a computational

part are performed. In both the experiment and the computational part, a number of various

errors are accumulated, such as activation error, error in measured data, assumption error,

error due to the computational model simplification, and various numerical errors (iterations,

rounding, etc.).

Investigating boundary conditions for applications like casting, descaling [A6, A10],

cooling of products in hot rolling, cooling of rolls in rolling technology [A4, A14], etc., many

various problems are encountered. Usually, it is almost impossible or cost inefficient to make

direct measurement in a plant, and therefore an experiment is designed. One of the first

problems is how to design such an experiment and the experimental apparatus. Various

additional experiments are made to calibrate temperature sensors used for measurements. To

evaluate the measured data, an ill-posed inverse heat conduction problem (IHCP) has to be

solved. Adaptively generated models are very efficient, but only a few papers dealing with the

unsteady heat conduction problem have been published. Problems with stability of the inverse

task and with precision of the computed results still remain. In addition, the precision of the

computed results is hardly known. These problems increase for experiments with rapid

changes in boundary conditions.

3


The main goal of this work is to improve the inverse heat conduction method for

applications in casting, descaling, cooling in hot rolling, etc. This includes stabilization of

computational methods and increase in precision of the computed results. Attention will also

be focused on error estimation of the computed results.

Along with the main goals, minor problems will have to be solved. For complex

geometries, there are no exact analytical solutions and thus numerical methods must be used.

The numerical methods simplify the reality – e.g. the temperature profile is piecewise

linearized. This simplification leads to a computational error which increases for highly

transient problems. A method for an adaptively generated mesh of the computational model on

the basis of the estimated error will have to be developed to describe the real problem as

accurately as needed. During computation, time dependent boundary conditions are also

simplified. An automatic time-step refinement for unsteady problems will be needed to

achieve the desired accuracy. The material properties change with temperature. Typically, the

temperature increases with increasing thermal energy of the object and most of the

computational methods are based on this phenomenon. In some cases, there is a change in the

internal structure of the material called the phase change. During this change, the temperature

may remain constant with increasing thermal energy. A modification of the computational

method will have to be made to include phase changes.

To determine boundary conditions, an inverse task is used that requires some measured

data. The data are measured using a sensor but none of the sensors are exactly the same. The

position of the thermocouple may differ as well as surface contact conductance. To calibrate

the computational model, an experiment will have to be designed. To achieve the desired

accuracy of the computational model, an adaptively generated model and calibration of the

computational model using experimental data will also be needed. To compute boundary

conditions from the measured temperature, an inverse heat conduction problem has to be

solved. Some methods have already been developed but none of them is universal. They have

strict limitations of usage and thus they can be used only for a certain sort of problems. It will

be necessary to design a proper inverse heat conduction computational method to solve IHCP.

The input information for the IHCP is usually blurred along the time axis. To use an adequate

time interval, the solution method should be able to determine adequate time interval for the

inverse method.

4


Some applications also require description of other than thermal boundary conditions.

Knowledge of water spray impact is often used for comparison of thermal and non-thermal

boundary conditions. Usually, a water pressure impact field is measured. For such a kind of

measurement, a real finite-size pressure sensor is used. In some cases, the impact distribution

is very small or at least very narrow with respect to the dimensions of the measuring sensor.

The distribution is being averaged during the measurement. The shape of the obtained spray

distribution is larger and the measured maximum is lower than the real one. A method will be

needed to obtain a real distribution from the measured data.

The laboratory experiments in continuous casting, descaling, and cooling of rolls have

already started and experimental data, which can be used for testing the new methods, are

already available. However, an improvement in the design of experiments is also needed.

5

Present State of Knowledge – Overview

3 3 PPRESENTRESENT S STATETATE OFOF K KNOWLEDGENOWLEDGE – O – OVERVIEWVERVIEW

3.1 3.1 Fundamentals of Heat Transfer PhenomenaFundamentals of Heat Transfer Phenomena

3.1.1 Heat Transfer

Whenever a temperature difference exists in a medium or between media, heat transfer

must occur. A simple definition [1] provides a sufficient description of heat transfer:

Heat transfer is energy in transit due to a temperature difference.

Three different types of heat transfer processes are known (see Figure 3.1). When

a temperature gradient exists in a stationary medium, which may be a solid or a fluid, we use

the term conduction for the heat transfer that occurs across the medium (see Appendix A for

more details). The term convection refers to heat transfer that occurs between a surface and

a moving fluid when they are at different temperatures (see Appendix B for more details). The

third kind of heat transfer is termed thermal radiation. All surfaces of finite temperature emit

energy in the form of electromagnetic waves. Owing to radiation between two surfaces at

different temperatures when there is an absence of an intervening medium a net heat transfer

occurs (see Appendix C for more details).

Figure 3.1 – Conduction, convection, and radiation heat transfer modes.

6

T1

Conduction througha solid or a stationary fluid

Convection from a surface to a moving fluid

Net radiation heat exchange between two

surfaces

T1 > T

2T ST∞

T2

TS

Moving fluid T∞

Surface, T1

Surface, T2

q

q q1

q2


A major objective in a direct conduction analysis is to determine the temperature field

(distribution) in a medium resulting from conditions imposed on its boundaries. Considering

a homogeneous medium within which there is no bulk motion, the temperature is expressed in

Cartesian coordinates as T(x, y, z). The general form of the heat diffusion equation [1], in

Cartesian coordinates, also known as the heat equation, is

∂∂ x k

∂T∂ x

∂∂ y k

∂T∂ y

∂∂ zk

∂T∂ z q=⋅cp

∂T∂ t

. (3.1.1)

In the cylindrical coordinates, the heat diffusion equation is

1r∂∂ r kr

∂T∂ r

1

r 2

∂∂ k

∂T∂

∂∂ zk

∂T∂ z q=⋅cp

∂T∂ t

. (3.1.2)

In heat transfer analysis, the ratio of the thermal conductivity to the heat capacity is

an important property termed the thermal diffusivity α, which has units of m2/s:

=kcp

. (3.1.3)

It measures the capability of a material to conduct thermal energy relatively to its capability to

store thermal energy. Materials with large α will respond quickly to changes in their thermal

environment, while materials with small α will respond more slowly, taking longer to reach

a new equilibrium condition.

3.1.2 Boundary and Initial Conditions

The solution of the heat equation depends on the physical conditions existing at the

boundaries of the medium and, if the situation is time dependent (unsteady), on conditions

existing in the medium at some initial time. The three kinds of boundary conditions, which

commonly occur in heat transfer, are summarized in Table 3.1. They are specified at the

surface x = 0 for a one-dimensional model.

The first condition corresponds to a situation for which the surface is maintained at

a known temperature (e.g. when the surface is in contact with boiling liquid or the surface

temperature is measured). It is commonly termed a Dirichlet condition, or a boundary of the

first kind. The second condition corresponds to the existence of a known heat flux at the

surface. This heat flux is related to the temperature gradient at the surface according to

Fourier’s law. It is termed a Neumann condition, or a boundary condition of the second kind.

7


This condition can be measured using a special sensor usually equipped with two

thermocouples. A special case of this condition corresponds to the perfectly insulated, or

adiabatic, surface for which ∂T /∂ x∣x=0=0. The boundary condition of the third kind

corresponds to the existence of convection cooling (or heating) at the surface and is obtained

from the surface energy balance. In addition, boundary condition of the third kind may be used

for radiation heat transfer, however, the h coefficient depends strongly on temperature. The

boundary condition of the third kind has a wide range of usage and is frequently used in our

analysis – the water cooling intensity distribution described by h( T ).

Table 3.1 – Boundary conditions for the heat diffusion equation at the surface (x=0)

1. Surface temperature

T 0,t =TS(3.1.4)

TS

T(x, t)

x

2. Surface heat fluxa) Finite heat flux

k∂T∂ x∣x=0=qs (3.1.5) T(x, t)

x

qs

b) Adiabatic or insulated surface

∂T∂ x∣x=0=0 (3.1.6)

T(x, t)

x

qs=0

3. Convection surface condition

k∂T∂ x∣x=0=h [T∞T 0,t ] (3.1.7)

T(0, t)

T(x, t)

x

T∞, h

3.2 3.2 Analytical Solution of Heat EquationAnalytical Solution of Heat Equation

Analytical methods may be used, in certain cases, for exact mathematical solutions of

conduction problems. These solutions have been obtained for many simplified geometries and

boundary conditions and are well documented in the literature [1, 2, 3]. However, more often

than not, geometries and boundary conditions preclude such a solution. In these cases, the best

8


alternative is the one using a numerical technique. However, the analytical solution is very

useful for testing numerical methods and their accuracy. Hence, some of the most important

cases to be used for testing accuracy of the numerical models are reviewed here.

3.2.1 The Semi-Infinite Solid

A simple geometry for which analytical solution may be obtained is the semi-infinite

solid. It may also be used to approximate the transient response of a finite solid such a thick

slab. For this situation, it would be reasonable to use the approximation only for a short time

period, during which temperatures in the slab interior are not influenced by the change in

surface conditions. A constant starting temperature T0 is assumed.

Constant boundary conditions

A sudden change in boundary conditions is assumed. Analytical solutions are

summarized as follows [1].

Constant surface temperature

T x , tT s

T0T s

=erf x

2 ⋅t (3.2.1)

qst =k T sT0

⋅⋅t(3.2.2)

Constant surface heat flux

T x , t=2 qs⋅t

k⋅exp x2

4 ⋅t qs⋅x

k⋅erfc x

2 ⋅t T0(3.2.3)

or in the form of dimensionless temperature solution for a unit step increase in surface heat

flux, but valid only for t0.1, the solution is [4]

T x , t=t1 3x

x2

2

2

2 ∑k=1

∞1

k2 ek2 2 t cosk x (3.2.4)

Constant surface convection

T x , tT0

T∞T0

=erfc x2 ⋅t exph⋅xk

h2 ⋅⋅t

k2 ⋅erfc x2 ⋅t

h⋅t

k (3.2.5)

9


3.2.2 The Plane with Convection

In comparison with the previous case, this one can be used also for a long time period.

We consider the plane wall of thickness 2L. The thickness must be small with regard to the

width and height of the wall. In this case, we can assume that the conduction occurs only in

the x direction. If the wall is initially at uniform temperature T0 and is suddenly exposed to

a constant heat transfer coefficient and constant ambient temperature on both sides, an exact

solution can be expressed as [1]

T x , tT∞

T0T∞

=∑n=1

∞ 4sinn

2nsin2n⋅expn

2 ⋅⋅t

L2 ⋅cosn⋅x (3.2.6)

where the discrete values of n are positive roots of the transcendental equation

n tann=h⋅Lk

. (3.2.7)

Since the convection conditions are the same on both surfaces, the temperature distribution is

symmetrical about the mid-plane where x = 0.

3.3 3.3 Numerical Methods for Solution of Heat Transfer PartialNumerical Methods for Solution of Heat Transfer Partial

Differential EquationsDifferential Equations

For situations where no analytical solution is available, the numerical method can be

used. Nowadays there are several methods that enable us to solve numerically the governing

equations of heat transfer problems. These include: the finite difference method (FDM), finite

volume method (FVM), finite element method (FEM), and boundary element method (BEM),

and others.

The numerical treatment of partial differential equations (PDEs) is a broad subject.

Partial differential equations play one of the most important role of computer analyzes or

simulations of continuous physical systems, such as heat conduction, mechanics, fluids,

electromagnetic, etc. In most mathematics books, PDEs are classified into the three

categories: hyperbolic, parabolic, and elliptic [5]. For the heat conduction problem, the

parabolic equation is used. The prototypical parabolic equation is the diffusion equation

10


∂T∂ t=∂∂ x ∂T

∂ x (3.3.1)

where the α is the diffusion coefficient (thermal diffusivity, Eq. (3.1.3)). The equation

describes how T (x, t) propagates itself forward in time (see Figure 3.2). In other words,

Eq. (3.3.1) describes time evolution.

Figure 3.2 – An initial value problem - initial values are given onone time slice, and it is desired to advance the solution in time.

The goal of a numerical code should be to track time evolution with some desired accuracy. In

general, it is not possible to „integrate forward in time” stably using an initial value. The goal

of a numerical code is to converge to the correct solution. An important point is the stability of

the algorithm. Many reasonable-looking algorithms for initial value problems just do not

work – they are numerically unstable. For example, the simplest problem on a 100 x 100

spatial grid would involve 10 000 unknowns. This would result in a 10 000 x 10 000 matrix

containing 108 elements.

3.3.1 Diffusive Initial Value Problems

The diffusion equation in one space dimension is Eq. (3.3.1). Let us assume ≥1,

otherwise Eq. (3.3.1) has a physically unstable solution [5]. A small disturbance leads to

increasing concentration of energy instead of to its dispersion. Consider first the case when α

is constant. Then Eq. (3.3.1) can be differenced in the obvious way

T jm1T j

m

t=[T j1

m 2T jmT j1

m

x2 ] . (3.3.2)

11

initial values

boundaryconditions


This is the Forward Time Centered Space (FTCS) or explicit differencing scheme [5]. The

stability criterion is

2 t

x2≤1 . (3.3.3)

The physical interpretation of the restriction is that the maximum allowed time-step is, up to

a numerical factor, the diffusion time across a cell of width x .

The fully implicit differencing scheme is only first-order accurate in time for the scales

that we are interested in:

T jm1T j

m

t=[T j1

m12T jm1T j1

m1

x2 ] . (3.3.4)

Schemes with this character are called fully implicit or backward time, by contrast with FTCS

(which is called explicit). To solve Eq. (3.3.4) one has to solve a set of simultaneous linear

equations at each time-step. Fortunately, this is a simple problem for one-dimension because

the system is tridiagonal. The scheme is unconditionally stable. The details of the small-scale

evolution from the initial conditions are obviously inaccurate for large t . But the correct

equilibrium solution is obtained. This is the characteristic feature of implicit methods.

Combining the stability of an implicit method with the accuracy of a method that is

second-order in both space and time yields to the average of the explicit and implicit FTCS

schemes which is the Crank-Nicholson differencing scheme

T jm1T j

m

t=2 [ T j1

m12T jm1T j1

m1T j1m 2T j

mT j1m

x2 ] (3.3.5)

which is second-order accurate in time. Both the left and right-hand sides are centered at

time-step n+1/2. The Crank-Nicholson scheme should be stable for any t , but unfortunately

not in case of large t and temperature dependent thermal diffusivity α.

t or m

(a) (c)(b)Explicit (FTCS) Fully implicit Crank-Nicholsonx or j

Figure 3.3 – Three differencing schemes for diffusive problems.

12


3.3.2 Diffusion Equation in Multi-dimensions

The methods for problems in 1 + 1 dimension (one space and one time dimension) can

easily be generalized to N + 1 dimensions [5]. However, the computing power necessary to

solve the resulting equations is enormous. Having solved a one-dimensional problem with 100

spatial grid points, solving the two-dimensional version with 100 x 100 mesh points requires

at least 100 times as much computing. Some papers recommend going to higher-order

methods to improve accuracy. However, this should not be used unless the solution is known

to be smooth, and the high-order method is known to be extremely stable.

Let us consider the two-dimensional diffusion equation

∂T∂ t=∂

2 T

∂ x2∂2 T

∂ y2 . (3.3.6)

An explicit method (FTCS) can be generalized from the one-dimensional case. We implement

the Crank-Nicholson scheme in two dimensions

T j , km1=T j , k

m 1 2 ⋅ t

2 x2 T j , k

m1x2 T j , k

m y2 T j , k

m1y2 T j , k

m (3.3.7)

where

≡ x= y , (3.3.8)

x2 T j , k

m1≡T j1,km1 2T j , k

m1T j1,km1

x2 T j , k

m ≡T j1,km 2T j , k

m T j1,km

y2 T j , k

m1≡T j , k1m1 2T j , k

m1T j , k1m1

y2 T j , k

m ≡T j , k1m 2T j , k

m T j , k1m

.

(3.3.9)

A problem arises when solving the coupled linear equations. In one space dimension the

system was tridiagonal, which is no longer true, though the matrix is still very sparse. One

possibility is a slightly different way of generalizing the Crank-Nicholson algorithm. It is still

second-order accurate in time and space. It uses powerful concept of operator splitting or time

splitting and is called the alternating-direction implicit method (ADI). The idea is to divide

each time-step into two steps of size t /2 . In each sub-step, a different dimension is treated

implicitly

T j , km1/2=T j , k

m 1 2 ⋅ t

2 x2 T j , k

m1/2y2 T j , k

m , (3.3.10)

13


T j , km1=T j , k

m1/21 2 ⋅ t

2 x2 T j , k

m1/2y2 T j , k

m1 . (3.3.11)

The advantage of this method is that each sub-step requires only the solution of a simple

tridiagonal system. This method is usually stable but not always. It can exhibit unrealistic

behavior when time-step t exceeds a certain value. Hence, the diffusive problems are

usually best treated implicitly as will be described later. The implicit method is the only one

that allows us to use any value of time-step t without producing physically unrealistic

results [6]. It is true that, for smaller t , the Crank-Nicholson method can be more accurate

than fully implicit method. However, the guarantee of producing realistic results for any value

of time-step t is considered to be often more important.

3.3.3 Finite Difference Method (FDM)

The finite difference method is the oldest method and also the easiest one to apply to

problems with simple geometries.

The computational domain is covered by a grid. Taylor series expansion or polynomial

fitting is used to approximate the derivatives of the variables with respect to coordinates at

each grid point – exact differential equation of the heat equation is reduced to an approximate

algebraic equation. Algebraic equations are achieved at each grid point and the resulting set of

equations may then be solved simultaneously for the temperature at each node. This method is

also called lumped capacitance because all of the control volume thermal capacitance is

lumped at the center node point for all interior control volumes. Further information is

available in [7].

For the numerical solution of an unsteady situation, time is discretized as well into the

time-steps t . There is a number of ways of deriving a discretization equation. Three already

mentioned methods are explicit, Crank-Nicholson, and fully implicit.

3.3.4 Finite Volume Method (FVM)

In the finite volume method [6, 8], the domain is subdivided into a number of so-called

control volumes of arbitrary and finite size as in Figure 3.4. The integral form of the

conversion equations is applied to each control volume. Conservation of energy for a solid

requires that the sum of the rate of heat flow across the control volume boundary and of the

time rate of change in energy within the control volume be equal to the rate at which energy is

14


produced within the control volume. In equation form, the conservation of energy can be

written as

∬A

q⋅dA∭V

q⋅dV=∭V

⋅c∂T∂ t

dV (3.3.12)

where A and V are the closed surface area and volume, respectively. The q parameter is the

heat flux vector leaving the control volume surface, q is the energy producing rate per unit

volume, and ⋅c mass density and specific heat.

dA=ndA

Control volumeboundary

Region of interest

Figure 3.4 – Finite control volume.

Each element has its own thermal properties and an element boundary can also

correspond to a material interface. Algebraic equations are obtained for each control volume.

In these equations, values of the variables for neighboring control volumes appear.

The FVM is very suitable for complex geometries and gives more accurate results than

the finite difference method, however, this capability demands more computation. This

method is conservative as long as surface integrals are the same for the control volumes

sharing the boundary.

3.3.5 Finite Element Method (FEM)

The finite element method [9, 10, 11] was mainly applied to problems in solid

mechanics. However, it has been developed to a general tool for solving partial differential

equations. It has become popular also for fluid flow and heat transfer problems.

Many methods can be used to develop finite element type systems of differential

equations. The Garlekin-type method of weighted residuals is one of the simpler approaches

and requires that the integral weighted average of the partial differential equation be zero over

some domain. There are similarities between the FEM and FVM. The domain is divided into

15


w(x)

x j1 x j x j1

x

1.0

0

Figure 3.5 – Weighting function for Garlekin-type method of weighted residuals.

finite elements (volumes). The distinguished feature of the FEM is that the conservation

equations are multiplied by a weight function before integration over the domain. For

example, for a slab

∫0

L

wx∂T∂ t

∂2 T

∂ x2 dx=0 (3.3.13)

where L is the thickness of the slab and w(x) is a weighting function to be specified. There is

an infinite number of possible weighting functions. A weighting function that gives good

results should be chosen. A popular weighting function is the so-called tent function which is

shown in Figure 3.5:

wx=w j x=xx j1

x jx j1

, x j1≤x≤x j

w j x=x j1x

x j1x j

, x j≤x≤x j1

w j x=0, xx j1∧x j1x. (3.3.14)

Since the weighting function wj(x) is zero over a large portion of the domain 0≤x≤L ,

Eq. (3.3.14) can be written as

∫x j1

x j1

w j x∂T∂ t

∂2 T

∂ x2 dx=0 . (3.3.15)

Using the linear element temperature profile assumption and Eq. (3.3.14) for wj(x), the finite

element equation for an arbitrary interior node j can be written as

kT jT j1

xk

T jT j1

xc

x2

ddt 13T j1

23

T jc

x2

ddt 23 T j

13

T j1=0 ; j=2, 3,..., N1.(3.3.16)

16


The only difference between FVM and FEM is the different weighting coefficients on the

thermal capacitance terms. The FVM gives greater weighting to the center temperature Tj. The

FEM equations for the surface nodes are

kT1T2

xqt c

x2

ddt 23 T1

13

T2 =0 , (3.3.17)

qN t kT N1T N

xqt c

x2

ddt 13 T N1

23

T N=0 . (3.3.18)

It can also be demonstrated that if the weighting function w(x) is chosen to be the Dirac delta

function

w j x=xx j = 1, x=x j

0, all otherx (3.3.19)

then the FEM results for interior nodes are identical to the FDM.

3.3.6 FDM, FVM, and FEM Similarity

All of the FDM, FVM, and FEM equations can be put in a similar form:

ddtT1T2=

x2T1T 2

qt c x , (3.3.20)

ddtT j12T jT j1=

x2 T jT j1

x2 T j1T j , (3.3.21)

ddtT N1T N =

x2 T N1T N qN t

c x(3.3.22)

where β and γ have the values listed in Table 3.2. Equations (3.3.20-3.3.22) are restricted to

temperature-independent thermal properties but the concepts can be extended to T-variable

cases.

Table 3.2 – Values of the β and γ of Eq. (3.3.20-3.3.22)

β γ β + γFDM 1/2 0 1/2

FVM 3/8 1/8 1/2

FEM 2/6 1/6 1/2

17


3.3.7 Finite-Volume-Unstructed-Mesh (FV-UM)

The FVM can be modified to better fit in complex geometries. In a two-dimensional

case, the domain is divided into triangular or quadrilateral elements [12, 13].

Cell-centered triangular method

The method for transient heat transfer conduction is presented by Duda [12] in which

an unstructured cell-centered finite volume method is used. The method is appropriate for

complex shape bodies and the temperature-dependent material properties (k, c and ρ) are used

in this method. The equation governing the transient heat conduction problem is given by heat

equation 3.1.1. The material thermal properties: specific heat, thermal conductivity and

density are the known functions of temperature. The heat conduction equation may be

integrated over the volume V and surface area A to obtain

∫V

∇∇∇∇ q⋅dV=∫V

⋅c∂T∂ t

dV . (3.3.23)

Using the mean value theorem for integrals on the right and the divergence theorem on the left

gives

∫A

q⋅⋅⋅⋅n dA=V T c T d Tdt

(3.3.24)

where the bar indicates a suitable average value over the region.

Figure 3.6 – An irregularly shapeddomain and its discretization into

triangular elements.Figure 3.7 – A typical triangular

element.

The domain is divided into triangular elements. A sample domain discretization is shown in

Figure 3.6. In this discretization scheme, curved boundaries are approximated by

18

b

j =1

x

y

j =2

j =3

c

a

o

noc

noa

β2 β

1

α2α

1


piecewise-linear curves. Then the centroids of the elements are connected with the midpoints

of the corresponding sides (see Figure 3.7): this creates polygonal control volumes around

each node in the calculation domain. An integral formulation can be obtained by applying the

conservation principle for energy to a control volume V1aoc

∫a

o

q⋅n dA∫o

c

q⋅n dA=V 1aoc⋅T1⋅cT1⋅dT1

dt(3.3.25)

where n is a unit outward vector normal to the differential area element dA. A linear

temperature variation is assumed throughout the triangular element. In each element heat flux

q can be expressed in terms of its components in the x and y directions

q=qx iqy j=[k T ∂T∂ x ] i[k T ∂T

∂ y ] j (3.3.26)

where i, j are unit vectors in the x and y directions, respectively. The normal vectors noa and noc

needed to evaluate the integrals in Eq (3.3.25) are given by

noa=cos1⋅icos2⋅ j=ya

xa2ya

2⋅i

xa

xa2ya

2⋅ j , (3.3.27)

noc=cos1⋅icos2⋅ j=yc

xc2yc

2⋅i

xc

xc2yc

2⋅ j . (3.3.28)

Using the local x, y co-ordinate system shown in Figure 3.7, the integrals in Eq. (3.3.25) that

represent the transport of energy by conduction can be approximated as

∫a

o

q⋅n dA=[qx , qy]⋅[ya , xa]

xa2ya

2 xa2ya

2, (3.3.29)

∫o

c

q⋅n dA=[qx , qy]⋅[yc , xc]

xc2 yc

2 xc2yc

2. (3.3.30)

Writing the heat balance equations for all control volumes, the system of N non-linear

ordinary equations is obtained whose solutions are the temperature changes in the central

nodes of the control volumes. The number of equations is equal to the number of the control

volumes N. The initial condition gives the starting values of the nodal temperatures.

19


Nodes

Sub-Control Volume

Control Volume (cell)

Figure 3.8 – Finite-Volume unstructed-mesh vertex based discretization.

Vertex-centered quadrilateral and triangular method in local co-ordinates

The nodes are located at vertices and this is different from the cell-centered method.

This makes it straightforward to ensure that internal boundaries are coincident with vertices.

The variables describing the energy are stored at the nodes, only the material type is stored

within a control volume also called cell (see Figure 3.8). The control volume for each node is

made up from the sum of sub-control volumes from each of the neighboring nodes.

The conversion equations in integral form are applied to the control volumes as outlined

for the FVM. The equations are treated in local co-ordinates so that each element may be

treated identically irrespective of how distorted any cell may actually be in global

Figure 3.9 – Sub-control volume definition.

co-ordinates. Each sub-control volume (SCV) contributes two surfaces to the control volume

boundary. Integrating over the control volume boundary, approximation is made at the

mid-point (IP) of every cell surface (SS) (see Figure 3.9).

20

IP1

IP2

SS1

SS2

SCVIP1

IP2

SS1SS2

SCV


Comparison of RMS error for various approaches

Several computational approaches were compared in [13]. The structured mesh

approach has proved to be both accurate and efficient in computer time. The structured mesh

was compared with an unstructured mesh (cell-vertex and cell-centered procedure). The

unstructured mesh can better describe a complicated shape but the algorithm is difficult to

converge and can give inaccurate results. The accuracy of the results is comparable to

structured mesh (Table 3.3). The unstructured mesh algorithm also requires longer

computation time (Figure 3.10).

Table 3.3 – Comparison of RMS error values for several mesh refinement

Approach 11 x 11 21 x 21 31 x 31 41 x 41

Structured mesh FVSM 5.18 2.63 2.40 2.30

Vertex-centered quadrilateral FVUMVQ 4.34 2.49 2.48 2.70

Vertex-centered triangular FVUMVT 4.59 2.80 2.52 2.60

Cell-centered quadrilateral FVUMCQ 5.18 2.63 2.40 2.30

Cell-centered triangular FVUMCT 3.43 2.26 2.30 2.15

0 250 500 750 1000 1250 1500 1750 20000

2500

5000

7500

10000

12500

15000

17500

20000

22500

25000

27500

Number of nodes

Tim

e [s

]

FVUMVT

FVUMVQ

FVUMCTFVUMCQ

FVSM

Figure 3.10 – Comparison of computation time of various methods [13].

21


3.3.8 Boundary Element Method (BEM)

The boundary integral method usually transforms the governing equations into boundary

integrals that are solved numerically [14]. For simple heat conduction problems it is well

suited but becomes more complicated for convective flow and heat transfer. The first step in

solving it numerically (FDM, FVM, FEM) is to discretize the solution domain using finite

differences, volumes, or elements. On the other hand, when using the BEM only the boundary

needs to be discretized and this shortens computation time and reduces storage requirements.

In addition, in the BEM no domain discretization is needed and so the location of internal

points, where the temperature is measured, can be chosen in a quite arbitrary way. Unlike

other numerical methods, the BEM gives a straightforward result of the unknown surface

temperature and the heat flux. Although the surface heat flux is more difficult to calculate

accurately than the surface temperature, their direct determination avoids the additional finite

differencing required in the conventional finite element approach. The disadvantage is that

BEM requires analytical solution of the internal heat conduction which is available only for

a limited set of simplified models. Lesnic describes in his study [15] a solution of

a one-dimensional, unsteady linear heat conduction equation in a slab geometry with constant

physical properties. Han has also investigated one-dimensional unsteady heat conduction [16]

using the time-dependent fundamental solution of the heat equation. This fundamental

solution is transformed into the boundary integral equation.

Ochiai has proposed an improved multiple-reciprocity BEM for steady heat conduction

[17]. Using this method, a highly accurate solution may be obtained solely using the

fundamental lower-order solution and simplifying the task of preparing data. The domain

integral of each step of this method is divided into point, line and area integrals.

Based on the improved multiple-reciprocity [17], Ochiai has presented triple-reciprocity

BEM [18]. He applied this method to unsteady heat conduction problems. In this method, heat

generation and the initial temperature distribution are interpolated using the boundary integral

equation and thin plate spline. A numerical example of unsteady heat conduction is shown

with an arbitrary initial temperature distribution on 2D surface with heat generation.

3.3.9 Transmission Line Matrix (TLM)

The appropriateness of this method is in application where high-power and short-pulse

are used, such as pulsed lasers in micro-electromechanical systems in the trimming resistors.

22


The material surface experiences a very high temperature gradient from the pulse. The

duration of the pulse is less than few picoseconds. Under these conditions, the finite speed of

heat propagation becomes important and a wave type propagation of heat, instead of diffusion,

has been proposed. The TLM method was first developed by Johns and Beurle [19] to treat

electromagnetic propagation. Transmission line matrix is a discrete numerical time-domain

modeling technique that can frequently be reduced to a solution of the diffusion equation.

The space is discretized into a network of finite cells with node points at the center of

each cell (see Figure 3.11). The following analogy of the electromagnetic propagation with

heat conduction is used [20]: in the equivalent TLM network, the resistors remain clustered

around the nodes and represent the thermal resistance of the material. The capacitor / inductor

combinations are replaced by impedance Z, which connect each node to its neighbors and

carry voltage pulses between the nodes in a finite time ∆t. According to the fundamental

transmission line theory, the impedance is

Z= LC=

a⋅ tC

=L

a⋅ t. (3.3.31)

Rb

R

R

R

R

Z

Z

Z

Z

Zb

ZR

Zb=0 R

f=-1

Sinking Boundary

Zb=∞ R

f=1

Insulating Boundary

Zb=Z R

f=0

AbsorbingBoundary

RadiativeConvectiveBoundary

Figure 3.11 – Different TLM boundary conditions around one cell.

Since the capacitor models the heat capacity of the material, the TLM parameters R and Z can

be evaluated from

2R=1

k⋅ x, (3.3.32)

23


Z=a⋅ t

c x3 (3.3.33)

where the parameter a represents the number of link lines in each elemental volume.

Figure 3.12 – Space-time history during three iterations of one-dimensionalscatter of a single input.

A TLM solution is obtained by repeatedly considering delta voltage (temperature) pulses

to be applied simultaneously on all parts of all nodes. These pulses are scattered

instantaneously into pulses which, during the time-step ∆t, travel along link transmission lines

to appear at neighboring nodes (see Figure 3.12). The TLM routine operates on the traveling,

scattering, and connecting of these pulses in the network. The transmission lines in the model

act as delay lines.

Transmission lines introduce a time delay, so that TLM-based models are explicitly

discretized in both space and time. Although this method is best known in electromagnetic

applications, the technique is also suitable for the numerical simulation of diffusion-based

phenomena such as heat transfer. However, this method is reasonable for problems where

high-power and short-pulse are used, such as pulsed lasers. The duration of the pulse is

usually less than few picoseconds.

3.3.10 Phase Change Implementation

Physical processes, such as solid/liquid and solid state transformations, involve phase

changes. The numerical treatment of this non-linear phenomenon involves many problems.

24

=R

RZ

Input

22

Spatial position

k = 0

Time index

k = 1

k = 2

k = 3


Methods for solving the phase change usually use a total enthalpy H, an apparent specific heat

coefficient cA, or a heat source q.

The nature of a solidification phase change can take many forms. The classification is

based on the matter in the phase change region. The most common cases follow:

a) Distinct : The phase change region consists of solid and liquid phases separated by a

smooth continuous front – freezing of water or rapid solidification of pure metal.

b) Alloy : The phase change region has a crystalline structure consisting of grains and

solid/liquid interface has a complex shape – most metal alloys.

c) Continuous : The liquid and solid phases are fully dispersed throughout the phase

change region and there is no distinct interface between the solid and liquid phase –

polymers or glasses.

In a distinct phase change, the state is characterized by the position of the interface. In such

cases the class of the so called front tracking methods are usually used. However, in cases b)

and c) the models uses the phase fraction.

The phase change process can be described by a single enthalpy equation

∂H∂ t∇⋅gd H d sdgl H l sl =∇⋅k∇T (3.3.34)

where g is the phase volume fraction, s is the phase velocity, and subscript d and l refer to

solid and liquid phases, respectively [21]. The k is (in this case) a mixture conductivity defined

as

k=gd kdgl k l (3.3.35)

and H is the mixture enthalpy

H=gs∫T ref

T

d cd dTgl∫T ref

T

l cl dTl gl L (3.3.36)

where Tref is an arbitrary reference temperature. To overcome the problem of the non-linear

(discontinuity) coefficient of a specific heat a non-linear source term is used. The term

∂H /∂ t can be expanded as

∂H∂ t=cvol

∂T∂ tH

∂ gl

∂ t. (3.3.37)

Neglecting convection effects in Eq. (3.3.34) and substituting Eq. (3.3.37) results in

25


cvol

∂T∂ t=∇⋅k∇T q (3.3.38)

where

q=H∂ gl

∂ t. (3.3.39)

Eq. (3.3.34) is non-linear and it contains two related but unknown variables H and T. It

is convenient to reformulate this equation in terms of a single unknown variable with

non-linear latent heat.

Apparent heat capacity

The apparent specific heat [22, 23] can be defined as

cA=dHdT=cvolH

dgl

dT(3.3.40)

where

cvol=gdd cdgl l cl

H=∫T ref

T

l cld cd dTl L .(3.3.41)

Neglecting convection effects and substituting into Eq. (3.3.34) yields apparent heat capacity

equation

cA

∂T∂ t=∇⋅k∇T . (3.3.42)

Total enthalpy

From Eq. (3.3.36) it can be written

∇T=∇ H /cvolH ∇ gl /cvol . (3.3.43)

Substitution in Eq. (3.3.34) will result in a total enthalpy equation

∂H∂ t=∇⋅ k

cvol

∇ H ∇⋅ kcvol

H ∇ gl . (3.3.44)

26


A general form of phase change implementation

All the governing equations suitable for a fixed grid numerical solution can be written in

the general form

c∂ ∂ t=∇⋅∇ Q (3.3.45)

where is the unknown variable, c is a specific heat, is a diffusion coefficient and Q is

a heat source [21]. The appropriate values are provided in the Table 3.4.

Table 3.4 - Possible values for the general form of the phase change equation

1. Basic equation

=H ; c=1; =0 ; Q=∇⋅k∇T

2. Apparent heat capacity

=T ; c=cA ; =k ; Q=0

3. Source

=T ; c=cvol ; =k ; Q=q=H ∂ gl /∂ t

4. Total enthalpy

=H ; c=1; =k /cvol ; Q=∇⋅k /cvolH ∇ gl

Note: Mixture volumetric enthalpy H=gd H dgl H l

Mixture conductivity k=gd kdgl k l

Mixture volumetric specific heat cvol=gdd cdgl l cl

3.3.11 Solution of Linear Algebraic Equations

Numerical methods usually build sets of algebraic equations that can be linear or

nonlinear. A general set of linear algebraic equations [5] look like

a11 x1a12 x2⋯a1N xN=b1

a21 x1a22 x2⋯a2N xN=b2

⋮aM1 x1aM2 x2⋯aMN xN=bM (3.3.46)

where the N unknowns xc are related by M equations. The coefficients ar,c are known numbers

as well as the quantities br. The set of equations can be written in a matrix form as

A⋅x=b (3.3.47)

27


where

A=[a11 a12 a13 ⋯ a1N

a21 a22 a23 ⋯ a2N

⋮aM1 aM2 aM3 ⋯ aMN

] x=[x1

x2

x3

⋮xN

] b=[b1

b2

⋮bM] .

(3.3.48)

If N = M then there are as many equations as unknowns, and there is a good chance of

obtaining a unique solution of xc set. Analytically, it can fail if one or more of the M equations

is a linear combination of the others. This kind of equations is called a singular one.

Numerically, at least two further things can go wrong:

• Some of the equations may be so close to become linearly dependent. The roundoff

errors may yield them linearly dependent during the solution process. In this case,

the numerical procedure will fail.

• Accumulated roundoff errors during the solution process can produce a false

solution. This problem arises when N is too large. The numerical procedure does

not fail algorithmically. However, it returns a set of xc that are wrong, as can be

discovered by direct substitution back into the original equations. It is difficult to

give any guidelines but there is a rough idea: Linear sets with N=20⋯50 can be

solved in single precision (32 bit floating representations) without resorting to

sophisticated methods, if the equations are not close to singular. With double

precision (64 bits), this number can be extended to several hundreds.

The general solution of Eq. (3.3.47) can be obtained as follows

x=A1 ⋅b (3.3.49)

but for huge A matrixes it might be an unstable and time-consuming task to compute

an inverse matrix. Thus sophisticated methods have been published to compute the unknown

vector x. The most often used methods are summarized in Appendix D.

3.3.12 Non-linear Set of Equations

A non-linear set of equations may have no solutions at all or it may have more than one

solution. In vector notation, we want to find one or more N-dimensional solution vectors x

28


f x =0 (3.3.50)

where f is the N-dimensional function. Its components are the equations to be solved

simultaneously. For smoothly varying functions and appropriate algorithm, it will converge

when the initial guess is good. Generally, the problem of finding solution vectors x is very

complicated.

In the heat transfer problems the equations are usually only slightly non-linear. The set

of non-linear equations can be usually solved by iterating the method for solving the linear set

of equations. We start by guessing or estimating the temperature at all grid points. Treating

this as the known temperature field, we calculate tentative values of the coefficients in the

discretization equations. Then we solve the equations to obtain a new temperature field.

Taking this field as a better estimate for temperature, we recalculate the coefficients and solve

the equations again. This process is repeated until the solutions are identical for all subsequent

iterations. This unchanging solution is called the converged solution. In practice, the iteration

process is terminated when the change in the solution is smaller than a prescribed small

number called convergence criterion. Although we proceed through a series of linear

equations, the final converged solution is the correct solution of the non-linear problem.

Under-relaxation

If values of coefficients change very rapidly from iteration to iteration, the iterating

method can diverge. In a highly non-linear problem, it is convenient to slow down the changes

in temperature from iteration to iteration. This process is called under-relaxation [24].

To slow down the changes in the of computed unknown variables x from iteration to

iteration, the value of xi+1 for the next iteration is a combination of values from this iteration xi

and from the previous iteration xi-1

x i1= x i1 x i1 (3.3.51)

where i superscript is an index of iteration and the is the under-relaxation factor, which is

between 0 and 1. An optimum of the depends on the nature of the non-linearity, number of

unknown variables, and other factors. To find a suitable value, exploratory computations have

to be made. Although this is not very convenient, it is good to know, that the divergence can

be suppressed by using appropriate under-relaxation.

29


3.4 3.4 Inverse Heat Conduction ProblemInverse Heat Conduction Problem

If the boundary conditions (temperature, heat flux, or heat transfer coefficient histories)

at the surface of a solid are known as functions of time (we can measure them), then the

temperature distribution inside can be found. This is termed a direct problem, during which

the impulse is dispersing. The physical situation at the surface may be unsuitable for attaching

a sensor. Very often, it is easier to measure accurately the temperature history at an interior

location. If the boundary conditions of a solid must be determined from transient temperature

measurements at one or more interior locations (see Figure 3.13), it is an inverse heat

conduction problem (IHCP) during which the dispersed impulse must be found. There are

numerous other inverse problems in transient conduction and diffusion, but this particular

problem has been named so and is the main subject of this work. The IHCP is much more

difficult to solve than the direct problem. The boundary conditions can be positive or negative,

constant or abruptly changing, periodic or non-periodic, etc. In the IHCP, the surface boundary

conditions are a function of time and may require hundreds of individually estimated

components to define it adequately.

Figure 3.13 – Example of the IHCP.

The IHCP is one of many mathematically ill-posed problems. Such problems are

extremely sensitive to measurement errors. There is a number of procedures that have been

advanced for the solution of ill-posed problems in general. Tikhonov has introduced the

regularization method [25] to reduce the sensitivity of ill-posed problems to measurement

errors. The mathematical techniques for solving sets of ill-conditioned algebraic equations,

called single-value decomposition techniques, can also be used for the IHCP [26]. There were

extremely varied approaches to the IHCP. These included the use of Duhamel’s theorem (or

30

q(t), unknown heat flux

Temperature sensor 1

L

x1

x

known boundary condtion


convolution integral) which is restricted to linear problems [27]. Numerical procedures such

as finite differences [7, 28, 29] and finite elements [30] were also employed, due to their

inherent ability to treat non-linear problems. Exact solution techniques were proposed by

Burggaf [31], Imber and Khan [32], Langford [33], and others. Some techniques used Laplace

transforms but these are limited to linear cases [34]. The improvement in artificial intelligence

has brought new approaches, such as genetic algorithm [35] and neural networks [36, 37].

3.4.1 IHCP by Beck

One of the first approaches for solving IHCP using numerical methods was proposed by

Beck [11]. The main feature of Beck’s approach is sequential estimation of the time varying

boundary conditions. He demonstrated that function specification and regularization methods

could be implemented in a sequential manner and that they gave in some cases nearly the

same results as the whole domain estimation. Moreover the sequential approach is

computationally more efficient. Beck’s approach has been widely used to solve inverse heat

conduction problems to determine unknown boundary or material property information.

The method uses sequential estimation of the time varying boundary conditions and uses

future time steps data to stabilize the ill-posed problem. The heat transfer coefficient is found

after determining the heat flux at the surface using Eq (11.1). To determine the unknown

surface heat flux at the current time tm, the measured temperature responses T i* ,m, are

compared with the computed T jm from the forward solver (e.g. FDM, FVM, FEM, etc.), using

n f future times steps

SSE= ∑f =m1

mn f

∑j≡i ; i=1

nT *

T i* , fT j

f 2 . (3.4.1)

Using the linear minimization theory, the value of the surface heat flux that minimizes

Eq. (3.4.1) is

qm=∑

f =m1

mn f

∑j≡i ; i=1

nT *

T i* , fT j

f |qm=0if

∑f =m1

mn f

∑i=1

nT *

if 2

(3.4.2)

where T jf |qm=0 are the temperatures at the temperature sensor locations computed from the

forward solver using all the previously computed heat fluxes, but without the current one qm.

31


The if is the sensitivity of the i th temperature sensor at time t f to the heat flux pulse at time tm.

These sensitivity coefficients are mathematically the partial derivatives of the computed

temperature field to the heat flux pulse, but in this case they physically represent the rise in

temperature at the temperature sensor location for a unit heat flux at the surface. The

sensitivity coefficient of our interest is defined as

jm=∂T j

m

∂qm. (3.4.3)

Once the heat flux is found for the time tm, the corresponding surface temperature T0m

may be computed using the forward solver. When the surface heat flux qm and surface

temperature T0m are known, the heat transfer coefficient is computed from

hm=qm

T∞mT0

mT0m1/2

. (3.4.4)

This approach is limited to linear problems. However, it can be extended to nonlinear

cases. The modification of this procedure involves an outer iteration loop which continues

until the computed temperature field is unchanging. The nonlinearity requires iteration only to

determine the present value of the heat flux, while the computations to determine the surface

temperature and heat transfer coefficient need only be performed once for each time tm. The

sensitivity coefficients are also nonlinear, due to the dependence of the thermal properties on

the temperature field, and they must be computed for each iteration.

Once the heat transfer coefficient at the ‘‘present’’ time is computed, the time index m is

incremented by one, and the procedure is repeated for the next time step. For n measured time

steps only n – f can be computed owing to the use of future data as a regularizing approach.

IHCP in multi-dimensions by Beck

The sequential approach can also be used for multidimensional IHCP. The temperatures

in one-, two-, or three-dimensional objects with temperature independent thermal properties

can be obtained using

T=T |q=0q (3.4.5)

where

32


T=[T m

T m1⋮

T m f 1] , T i =[

T1i

T2i

⋮

T nT*

i ] , q=[qm

qm1

⋮qm f 1

] , qi =[q1

i

q2i

⋮

qnq*

i ] ,=[

12 1⋮ ⋱

f f 1 ⋯ 1] , i =[1,1i ⋯ 1,nq*

i

⋮ ⋱nT * ,1

i nT * , nq*i ] . (3.4.6)

The sequential approach then temporary assumes that q is independent on time. Then using

Z= I * where I *=[11 0

⋱0 1n

] where n=nq* (3.4.7)

the function to minimize is

SSE=T * ,mT m |q=0Z mqmT T * ,mT m |q=0Z mqm . (3.4.8)

The matrix derivative of Eq. (3.4.8) with respect to q gives the estimated heat fluxes

qm=[Z mT Z m]1Z mT T * , mT m|q=0 . (3.4.9)

After it is obtained, m is increased by one and the procedure is repeated for the next time step.

3.4.2 Comparison of Inverse Heat Conduction Methods by Raynaud

Beck [11] has stabilized the IHCP using several future time temperatures with the

least-squares method while Hensel and Hills used smoothing functions on the input data. Beck

and Murio have combined this method with a regularization one [38]. Regularization method

[25] has been presented to reduce the sensitivity to the measurement error.

The quantitative comparison of methods is difficult. Raynaud [4] shows two test

problems and demonstrates how to compare the results given by different inverse algorithms.

The ideal inverse method would not be sensitive to the measurement error and thus the

minimum deterministic bias and minimum sensitivity to measurement errors are required. The

following four methods are compared by Raynaud: Method A – D’Souza [39]; Method B –

Raynaud and Bransier [40]; Method C – Weber [41]; Method D – Beck [11]. They are

stabilized using different principles but all of them use future temperatures except Method A.

33


For the methods A, B, and C, the surface heat flux is calculated from the temperature

distribution at the surface node using the following energy balance

qm=T1

mT 2m

x x2

T1m1T1

m1

2 t. (3.4.10)

x

n + 6

n + 5

n + 4

n + 3

n + 2

n + 1

n

n – 1

1 2 j – 1 j j + 1

t

Method A

Method C

Method B

– known temperature – temperature to be calculated

Figure 3.14 – Computation molecules.

D’Souza – Method A

This method uses the backward difference approximation for the time derivative and

central difference approximation for the space derivative

T j1m 2T j

mT j1m

x2 =T j

mT jm1

t. (3.4.11)

The computation molecule is shown in Figure 3.14. It is evident that this method uses only

past and present temperatures to obtain the present temperature estimate at node j – 1.After

calculating the temperature at node j – 1, the temperature at node j – 2 can be calculated in

an explicit way using Eq. (3.4.11). This continues to the surface temperature.

34


Raynaud and Bransier – Method B

This method uses double approximation given by energy equation. The first energy

balance in the dimensionless form is

T1, j1m T1, j

m

x

T1, jm1T1, j1

m1

x= x

T1, jm1T1, j

m

t(3.4.12)

and the second is

T 2, j1m T 2, j

m

x

T 2, jm1T 2, j1

m1

x= x

T 2, jm T 2, j

m1

t. (3.4.13)

The calculation in the time direction is performed for a given spatial node j before proceeding

to the next node. The computation molecule is shown in Figure 3.14. The estimation of the

surface temperature at time m involves the measured temperature at time m + j - 1 where j is

the node of the temperature sensor. At each step, T j1m is approximated by the arithmetic

average of T1, j1m and T2, j1

m . Grid refining leads to an increasing number of future

temperatures and of steps required to reach the surface. More space steps lead to the more

crude approximation of the surface temperature.

Weber – Method C

This method uses the hyperbolic form of the heat conduction equation

∂2 T

∂ t2 ∂T∂ t=∂2 T

∂ x2 . (3.4.14)

where is the nonnegative constant. Central differences in both time and space are used to

approximate Eq. (3.4.14)

T j

m12 T jmT j

m1

t2 T j

m1T jm1

2 t=

T j1m 2 T j

mT j1m

x2 . (3.4.15)

The computation molecule is shown in Figure 3.14. The computation of the temperatures

proceeds toward the surface as for method B. The future temperature at time m + j - 1 is

involved in the estimation of the surface temperature at time m as in method B.

35


Beck – Method D

This method is a time-marching technique. At each time step the surface flux is

computed using a least-squares minimization procedure for the simulated temperature

difference as described in the previous IHCP by Beck chapter.

Test cases

The one-dimensional linear heat conduction problem is chosen as a test case. The

temperature is known as a discrete function of time at position x under the surface. The

boundary at x = L is insulated. It is desired to predict the transient flux history at x = 0.

The purpose of the first test case is to estimate the deterministic bias of inverse methods.

A heat flux impulse is considered that is constant and equal to unity over just one time step

and zero at other times. The impulse heat flux is the smallest temporal fluctuation that can be

estimated, and any time variation of the surface flux can be represented by superposition of

such basic fluxes due to the principle of superposition because we are assuming a linear

problem. The bias is defined as the square root of the sum of the squares of the deviations of

estimated fluxes from the true ones

B=∑m=1

n

qmqm2

. (3.4.16)

The purpose of the second test case is to study the sensitivity to measurement errors of

inverse algorithms. The test case considers input temperatures equal to zero, except at time t0

when the temperature is equal to unity. These input data can be considered as an error of

magnitude 1 at time t0. If the random errors are additive, uncorrelated, and have zero mean

and constant variance then the standard deviation of the heat flux components for linear

problems is

q=2 ∑m=1

n

qm2 . (3.4.17)

For stable algorithms, the qm approach zero.

Comparison

Tables 3.5 and 3.6 show the computed bias and standard deviation of the estimates,

respectively. Since Method A does not use future temperatures the heat flux impulse is not

estimated at correct time but as moved along the time axis in positive direction. For other

36


methods, when the stabilizing parameters are increased, the impulse is spread in time. As it

can be seen from the result tables 3.5–3.6 the best methods are B and D. Method B is a bit

more accurate and Method D is less sensitive to the measurement error.

Table 3.5 – Comparison of inverse methods – bias value [4]

t 0,005 0.01 0.05

Method A 1,85 1,65 1,43

Method B 0,79 0,81 0,8

Method C 0,76 0,84 1,03

Method D 0,82 0,84 0,91

Table 3.6 – Comparison of inverse methods – standard deviation [4]

t 0.005 0.01 0.05

Method A 35 832 33 02 78,2

Method B 573 63 5,4

Method C 883 103 10,8

Method D 442 53 3,1

3.4.3 Application of Beck’s Algorithm

Numerical simulation of casting of aluminum

The algorithm for solution of non-linear problems was verified using a numerical

experiments. The casting of aluminum in a sand mold was simulated using one-dimensional

model by Woodbury [42]. The temperature dependent material properties were used for silica

sand. The heat transfer coefficient (HTC) was taken as a triangular time function, starting at

zero, raising to the peak value at t = 5 s, going down to the half peak value at t = 20 s, and

then remaining at this level. Three different values of the peak were investigated: h = 10; 100;

1000 W/m2.K. The forward problem was simulated using FEM with a very fine mesh, both in

time and space, compared to the FEM used in the inverse method.

First, the noiseless data were used as the input to the inverse algorithm. The

computational grid was 300 elements and the time step was 0.1 s. The results were extremely

37


good and were in agreement with the expectations done by the sensitivity study. All results

were obtained using a stabilizing parameter, the number of forward time steps, set to 3.

A larger error for a larger HTC was expected, since the sensitivity coefficients were smaller

for a larger HTC. The errors for noiseless data are listed in Table 3.7.

Table 3.7 – Error summary for noiseless data [42]

HTC peak Maximum error RMS error

10 2.0 % 0.5 %

100 2.3 % 0.6 %

1000 4.5 % 2.4 %

Next, the Gaussian noise with a standard deviation of 0.5 °C was added to the simulated

temperature history. The estimated HTC seemed good for the HTC peak, but worse for the

constant HTC (see Figure 3.15). For a larger stabilizing parameter, the loss of accuracy is

present near the peak in the curve. Table 3.8 gives a summary of the error obtained for the

cases with noisy data.

Table 3.8 – Error summary for noisy data [42]

HTC peak Stabilizingparameter f

Maximumerror

RMS error

10 7 218 % 57 %

10 10 100 % 33 %

10 13 68 % 23 %

100 7 45 % 13 %

1000 7 56 % 15 %

1000 10 24 % 8.4 %

The used inverse algorithm gives very good results with errorless data. The error of

computed HTC increases with increasing magnitude of HTC. When errors are present in the

input data, a larger stabilizing parameter is required. Data errors are more problematic when

the response in the domain is small. The proposed inverse algorithm appears to give better

results when the HTC is changing with time.

38


A) B)

Figure 3.15 – Estimated heat transfer coefficients compared with exact values – noisy data;A) f = 7; B) f = 10; by Woodbury [42].

Investigation of transient heat transfer coefficient in quenching experiments

The transient heat transfer coefficient history was estimated from interior transient

temperature measurements during quenching by Osman [43]. A hollow hot copper sphere was

immersed into the cooling baths without boiling. Three thermocouples were embedded into

a sphere positioned 2.6 mm from the outer surface. The average value of these three

thermocouples (see Figure 3.16) was used as the input to the inverse code. After heating the

sphere to the prescribed uniform temperature, the data acquisition process started and the

sphere was immersed into the cooling bath. The time for complete immersion of the sphere

was about 0.2 s. The data were measured at a constant rate of 0.1 s during first 50 s. Two types

of bath were used: ethylene glycol (27.5 °C), and water (23 °C). The sphere temperature was

150 °C for ethylene glycol and 99 °C for the water bath.

Figure 3.16 – Measured temperature histories for ethylene glycol [43].

39


The stabilizing parameter, the number of forward time steps, was set to 2 for a time

shorter than 2.5 s and 12 for a longer time. The computed HTC history (see Figure 3.17

showed that during the important early time the HTC increased rapidly to a maximum value in

about 0.5 s after immersion. Then the HTC decreased with time with a distinct change of

slope at about 2 s. The results were very similar for ethylene glycol and water, only the

magnitude was different. The thermal field around the sphere is quite complicated due to

sudden immersion of a relatively large sphere. The HTC were compared with well-known

steady-state empirical equations for free convection (see Figure 3.18). The computed HTC

coefficients were about 100-120 % higher than the ones obtained using empirical equations

due to the sudden immersion. For later time, the obtained data match well the empirical ones.

Figure 3.17 – Estimated heat transfercoefficient for ethylene glycol [43].

Figure 3.18 – Comparison of the estimatedresults with empirical correlations

(ethylene glycol) [43].

A modified sequential approach

Using the modified algorithm it was tried to eliminate the leading error caused by

adding future information [44]. The method makes use of the standard Beck’s approach as

a preliminary estimation of the real heat flux. By reviewing the computed results, linear,

triangular, stepped, or parabolic slope functions are used as the main characteristic of the

searched boundary conditions. The proper function is then used in the sequential equations.

The modified approach was tested on a one-dimensional model with constant material

properties. The input data were prepared using numerical simulation for three types of

boundary conditions: triangular, stepped source, and parabolic slope. A 3 % random

40


measurement error was added to the input data. The fourth numerical experiment makes use

of the triangular shape of boundary conditions with a 10 % error in the input data.

The proposed method effectively reduces the average relative error in all four cases.

When a 10 % random measurement error was considered the error in the computed heat

source was reduced from 17.82 % to 2.65 % (see Figure 3.19). The disadvantage of this

method is the computation of large inverse matrices. The more the future time steps, the larger

the matrix. This approach can hardly be used for multidimensional models. The results will be

probably much worse for nonlinear problems.

Figure 3.19 – The comparison of estimation results for sequential approaches in example 4,10% random measurement error considered [44].

A two-dimensional IHCP

Woodbury [45] tested the multidimensional Beck’s approach on a two-dimensional

model using a numerical experiment. The unknown heat flux on the boundary is assumed as

a function of space and time. Functions are piecewise constant for space and time domains.

xInsulation

y, q

Sensor 1 Sensor 2 Sensor 3

Pulse 1

Pulse 2

Pulse 3

Figure 3.20 – Scheme of the numerical experiment of the 2D IHCP.

41


During a numerical experiment, the object (see Figure 3.20) was exposed to known

(time and space dependent) heat flux (see Figure 3.21). The computed temperature histories of

three sensors inside the body were recorded (see Figure 3.22). A random noise was added to

the recorded values and these data were used as an input for the IHCP algorithm:

T data=T puree×rand 1,1 . (3.4.18)

Figure 3.21 – Time history of the tree heat flux pulses.

The inverse algorithm uses stabilizing parameter f = 1 for the case with noiseless data

(see Figure 3.23 for computed results) and the data with additional noise e=0.0017 (see

Figure 3.24). Very good results were obtained. For the case with more noisy data, e=0.017,

the results are much worse (see Figure 3.25). The maximum error of the estimated heat fluxes

was almost 100 % at the beginning of the experiment. It would be interesting to compare the

results computed using the 2D inverse algorithm and 1D inverse algorithm for each sensor

separately.

Figure 3.22 – Artificial data for test case. Figure 3.23 – Calculated surface heat fluxwith errorless data.

42

q, H

eat f

lux

time0 1 2

Pulse 1Pulse 2 Pulse 3


Figure 3.24 – Calculated surface heat fluxwith random error e=0.0017.

Figure 3.25 – Calculated surface heat fluxwith random error e=0.017.

3.4.4 Application of BEM in IHCP

Lesnic used the BEM approach for solving the one-dimensional, linear, inverse,

unsteady IHCP problem [46]. He used the fundamental solution of the heat conduction

Eq. (3.1.1) in one-dimension of the form

F x , t , ,=1

4 texp[x

2

4t ]H t . (3.4.19)

For simplicity, he also assumed constant boundary elements, i.e. the temperature and the heat

flux constant over each element. A liner heat flux in the time domain on the boundary was

used in the numerical experiment.

Various methods have been tested for the solution of IHCP using BEM: direct method,

least square method, regularization method, and minimal energy method. The computed

results were compared with an analytical solution. It has been shown that the least square and

minimal energy methods give very inaccurate results at the end of the computed time interval

and are very unstable, while the regularization method gives good results also for noisy data.

The minimal energy method gives good results for a noisy data except at the end of the

computed time interval. Although the regularization method gives the best results, it is not

an easy task to determine the regularization parameter in practical computations. Similar

results were obtained by Han [16].

43


3.4.5 Gaussian Least Square Differential Correction scheme

The main goal of the described approach [47] is to simultaneously determine the

thermal conductivity and volumetric heat capacity (of sand mold used in castings) as

a function of temperature from a single experiment. The functions are assumed to be

piecewise linear with n nodes that serve as interpolation points

k T ≈∑i=1

n

N i T k i (3.4.20)

cT ≈∑i=1

n

N i T ci . (3.4.21)

To determine the unknown temperature dependent material properties, a least-square function

is used. The measured values of the temperature sensors are compared with the ones

computed using a numerical model. The result of this approach is the Gaussian Least Square

Differential Correction scheme

=T 1 T T *T (3.4.22)

where T * is the vector of measurements, T are the values computed using the numerical

model, is the sensitivity matrix, which expresses the temperature dependence on the

unknown parameters, and the is a correction vector of the unknown parameters

=[k1

k2

⋮kn

c1

⋮cn

] . (3.4.23)

A simple numerical experiment was used to test the approach. A one-dimensional

model, with temperature dependent thermal properties, was heated at one end by a constant

heat source while the other sides were insulated. The following thermal properties were used

k T =expT /5 (3.4.24)

cT =1sinT /10 (3.4.25)

44


within the temperature range 0≤T≤5. During the numerical simulation with a time step of

0.001 s, temperature histories were recorded in five locations.

The results obtained from the algorithm vary with the number of nodes used for

approximation and with the additive noise. The number of nodes varies from 3 to 8. The

errors are the highest at the end nodes and lower in the middle for the temperature range (see

Figure 3.26–3.27). This is caused by the lack of the information on one side of the end nodes.

The results have shown that with four or five nodes the distribution of errors is approximately

uniform while for six or more nodes, the errors on the end nodes become significant, while the

errors of the interior nodes are negligible. The estimated temperature dependent functions of

the material properties were good, even in the case of 1% noise in the input data.

Figure 3.26 – Errors in ki for errorless datawith increasing number of knots.

Figure 3.27 – Errors in ci for errorless datawith increasing number of knots.

3.4.6 Conjugate Gradient Method in IHCP

The variation of air-gap heat resistance was estimated by Huang [48] using the

conjugate gradient method. An inverse analysis of heat conduction involving a phase change

was used to determine the time dependent heat resistance of the gap formated between the

casting and mold during the casting process. The liquid region together with the mold is

initially at a uniform temperature. A convective cooling is applied on the outer mold surface.

During the solidification process, the liquid-solid interface moves into the center of the

casting and forms the gap.

45


A numerical experiment was used to illustrate the accuracy of the approach. A triangular

and a step constant contact conductance are examined. Constant material properties of the

mold were assumed. One temperature sensor was located on the outer surface of the mold and

the second one was inside the mold. A noise was added to the measured data.

It has been found that the conjugate gradient method converges faster and is less

sensitive to the measurement error than the least square method. The disadvantage of this

method is that the function of the gap heat resistance is estimated at once for the whole time

domain. Thus, it can be hardly used for experiments with a lot of input data.

3.4.7 Methods Based on Artificial Intelligence

In the last ten years, a novel approach started to be used in IHCP. The new approach

uses artificial neural networks and a genetic algorithm to determine boundary conditions using

the temperature history from one or more internal sensors.

Artificial neural network

Radial basis function training algorithm

One of the first tests was done with artificial neural network (ANN) trained using the

radial basis function algorithm. The radial basis function algorithm is more efficient for

training than the back propagation method. The goal was to develop an alternative method for

approximation of the inverse function using ANN. Such networks are capable of

approximation of nonlinear functions. The ANN maps a vector of nin real-valued inputs onto

a vector of nout real-valued outputs (see Figure 3.28). The precise function computed by the net

is determined by the network’s adjustable parameters – its weight coefficients. These are

usually determined during training. Training a network involves adjusting some or all of the

network’s weights to minimize the cost function. Once the training has been completed, the

network’s ability to approximate the desired function can be tested by applying data not

included in the training.

The data used to train the ANN are usually computed by a forward solver. Dumek [49]

has tested ANN on a one-dimensional object (with internal temperature sensors) that was

subjected to time dependent boundary conditions. The ANN was trained using the radial basis

function algorithm and 300 forward solutions. The input vector contained 50 temperatures and

the output vector had 13 values of the desired HTC. The hidden layer had 30 neurons.

46


Input vector

1 2 3 nin

. . .

1 2 nh

. . .

Input layer

Hidden layer

1 2 4 nout

. . .3 Output layer

Output vector

Figure 3.28 – Structure of artificial neural network.

After completion of the training process, the ANN was tested using the data obtained

from numerical experiments where time dependent HTC were used. These consist of

sinusoidal, constant and triangular functions. The network generates a roughly sinusoidal

output. The results were a bit better for triangular HTC but very inaccurate for constant HTC.

The error of the computed boundary conditions varies from 10 % to 30 %. However, the

advantage of the described approach is the computation speed of HTC after the ANN was

trained.

Back propagation

Sablani [50] tested the trained ANN using the back propagation algorithm on

determination of Biot number

Bi=h⋅Lk

. (3.4.26)

Numerical experiments of a cooled cube were realized – linear and nonlinear problems.

A simple ANN was used for the estimation of Biot number from the temperatures recorded

inside the cube. The neural network models were capable in generalizing the behavior of both

linear and nonlinear problems. The error of the computed results was usually up to 10 %,

however, it increases up to 30% for very small Biot number (Bi < 0.1).

Cascade correlation

The inverse heat conduction task, treated by Raudenský [51], looks simultaneously for

boundary conditions and the time constant of a temperature internal sensor on the basis of the

47


knowledge of temperature measurement from that sensor. The recorded temperatures can be

seriously influenced by thermal resistance between the surface of the temperature sensor and

the measured material. The error increases when very fast dynamic processes are studied.

The one-dimensional heat conduction problem with constant material parameters was

studied. The cascade correlation learning algorithm was used. It has two special features: the

cascade architecture, and the learning rule. The cascade architecture is shown in Figure 3.29.

At the beginning, the system works with no hidden unit. When the error stops decreasing

hidden units are added one by one to the net during the training process. The main advantage

is that learning is much faster and the hidden units are added dynamically to the learning

process. No fixed topology has to be chosen at the beginning.

Inpu

t vec

tor

Hidden neurons

Output vector

+1

Adjustable weights

Frozen weights

Figure 3.29 – Topology of the cascade correlation.

A set of 200 training vectors was used. Each pair of the training vectors consists of 12

values of HTC plus one value of time constant and 50 values of temperature in time instants 1

to 50. The neural net had 14 hidden units at the end of the training. The accuracy test of ANN

was done using the data obtained from numerical experiments computed using triangular and

parabolic functions of HTC which were not included in the training set. The average relative

error of HTC was 4.2 % and the error of the appropriate time constant response was 1.7 % for

the triangular HTC. The average relative error of HTC was 3.6 % and the error of the

appropriate time constant response was 8 % for the parabolic HTC. The results obtained using

48


cascade correlation topology are much better than the ones computed using classical ANN

trained using radial basis function training or the back propagation algorithm.

Genetic algorithm

Genetic algorithms (GA) are random search strategies which use partially random

methods to evolve a solution toward the optimum or near-optimal solutions. In each step, the

GA deals with a group of solutions called population. The algorithm keeps the best found

solutions and it determines a new search direction in the space of possible solutions on their

base. In each step, the search strategy tries to generate a new generation of population

representing a better solution. The new population depends on the previous one. Using the

selection procedure, crossover and mutation operators, a new population is generated. The

procedure of generating a new population is repeated until an acceptable solution is found.

Because of the simple mathematical background of GA, they can be easily used in many fields

such as IHCP. However, it is usually more efficient to use the classical approach. Only if the

classical approach is unstable a GA may be a good choice.

The GA was tested by Raudenský [52] on a data obtained from a quenching experiment.

The obtained results were compared with the data computed using the classical Beck’s

approach. The accuracy of the obtained results are comparable with that obtained using ANN.

However, data from a real experiment were used and thus it would be very interesting to

compare temperature histories computed by the forward solver using the HTC obtained using

GA and Beck’s approach.

Liu [53] tested intergeneration-projection genetic-algorithm (IP-GA) for IHCP. This

inverse procedure should speed up the process of finding the desired parameters.

Identification of the heat convection constants of a typical microelectronic package was

performed as an example. The performance was compared with standard GA. It has been

found that the reduced-basis method combined with the IP-GA significantly speeds up the

solution process.

3.5 3.5 Data FiltrationData Filtration

Any data obtained from real experimental measurements contain some noise which can

be negligible, however, more often than not, the noise is significant and thus must be

suppressed. The data filtration task becomes even more important for the data used as inputs

49


for inverse tasks, because the noise is gained. Various techniques are used to filter measured

data. Some use filtration in the time domain, some cut high frequencies in a frequency

domain, and others approximate the measured data using some function.

3.5.1 Filtration in Time Domain

One of the simplest and often used methods is averaging several values into one value.

This approach is inbuilt directly in measuring apparatuses when an integration measuring

method is used. However, this approach seriously limits the shortest possible time interval

between consecutive samples.

A similar approach can be used without stretching the time interval. One sample is

filtered using several neighboring samples multiplied by some weight coefficient

mt =1

∑

p∑

m⋅pt (3.5.1)

where m are measured data, and p is a filter. An arbitrary filter can be chosen. However, the

most often used one is constant for a few neighboring samples, tent function, and Gaussian

filter defined as

p=e2

22 . (3.5.2)

3.5.2 Filtration in Frequency Domain

Noise is usually most significant at high frequencies. To suppress noise, it is very

convenient to work in the frequency domain. A physical process can be described either in the

time domain, as a function mt of time t, or in the frequency domain [5], where the process is

specified by its amplitude M as a function of frequency f. It is useful to use mt and M f as

two different representations of the same function. The filtration in the time domain

(Eq. 3.5.1) can also be done in the frequency domain, even with sophisticated filters.

However, we must convert the measured data into the frequency domain and backwards.

Fourier transform

We can go between these two representations using Fourier transform equations

50


M f =∫∞

∞

mt e2 i f t dt , (3.5.3)

mt =∫∞

∞

M f e2 i f t dt . (3.5.4)

If t is measured in seconds, then f is in cycles per second. However, the equations work with

other units too.

With two functions mt and pt (where mt are the measured data and pt is a filter

function), and their corresponding Fourier transforms M f and P f , we can make

convolution of the two functions m∗p as

m∗p=∫∞

∞

m⋅pt d (3.5.5)

The functions m∗p is a simple transform pair called Convolution theorem

m∗p⇔M f ⋅P f . (3.5.6)

In other words, the Fourier transform of the convolution is just the product of the individual

Fourier transforms.

Fourier transform of discretely sampled data

In the most common situations, function mt is sampled at some time intervals. If the

time interval between consecutive samples is t, the sequence of sampled values is

mn=mn t n=...,3,2,0,1,2,3,... . (3.5.7)

The time interval t is called the sampling rate. If t is measured in seconds, then the

sampling rate is the number of samples recorded per second.

For any sampling interval t, there is also a special frequency fc, called Nyquist critical

frequency [5], given by

f c=1

2 t. (3.5.8)

The Nyquist critical frequency is important for two reasons. The first one is known as the

sampling theorem: A continuous function mt (sampled at an interval t) is the bandwidth

limited to frequencies smaller than fc. This theorem shows that the information content of

51


a bandwidth limited function is smaller than that of a general continuous function. It is bad

that all of the power spectral density that lies outside of the frequency range –fc < f < fc is

spuriously moved into that range. This phenomenon is called aliasing. Any frequency

component outside of the frequency range (–fc; fc) is aliased (falsely translated) into that range

(see Figure 3.30).

f

true Fourier transform

M( f )

1/2 t 1/2 t

aliased Fourier transform

Figure 3.30 – Fourier transform of sampled function.

Filtration of measured data

The measured data are first transformed into the frequency domain using Fourier

transform. Next, they are multiplied by a filter function. A common filter is the so called low

pass filter when high frequencies are simply set to zero. Having modified the measured data in

the frequency domain, they are transformed back into the time domain.

3.5.3 B-Splines

Another technique for data smoothing was used by Loulou [54]. He used B-splines to

smooth the measured data. The measured data T * t were replaced by the smooth function

T t in the sense of least squares. The B-splines function of the order of b=4 was used. The

smoothed function can be written as

T t =∑j=1

n

a j p jbt (3.5.9)

where p jbt are the known basis functions. An optimum number of parameters or basis

functions was chosen by using a visual approach. Loulou started with small number of

parameters and then he added one by one until the smooth profile become unstable. When he

52


observed that the smoothed curve is in agreement with the physical evolution of the profile

and when it become unstable he stopped increasing the number of parameters and chose the

last one as optimum.

53

Problem Solution

4 4 PPROBLEMROBLEM S SOLUTIONOLUTION

The solution of the inverse heat conduction problem (IHCP) is an extensive task that

involves many sub-problems. All reviewed IHCP methods incorporate a forward solver of

heat conduction. To solve accurately IHCP, an accurate forward solver is needed. Any noise in

the input data is gained by the inverse task. Thus, the processing of measured data is obvious.

And, finally, the design of a measuring apparatus is important. An unsuitably placed thermal

sensor can significantly reduce the accuracy of the computed results.

Several methods have been considered for the forward solver of heat conduction. For

problems under solution in this work, the most suitable ones are FDM, FVM, and FEM. The

accuracy of these methods was tested as shown in Appendix E. It can be seen that the accuracy

of these methods is very similar. Even the simplest one, FDM, gives best results at nodes of

control volumes for very fast changes in boundary conditions. The equations of FVM and

FEM methods become much complicated for models with temperature dependent material

properties in comparison with FDM. Other methods were also considered. However, the BEM

is not suitable for multidimensional models with temperature dependent material properties.

The FV-UM can better fit within complex geometries. However, the computation is much

more complicated and time consuming. It is probably more efficient to use FDM and a finer

mesh. The TLM method is suitable mainly for extremely fast changes in boundary conditions,

such as a time interval shorter than few picoseconds.

Dealing problems in the metallurgical industry, the studied objects are usually made of

steel. Most of the common steel materials have temperature dependent material properties.

Even more, the phase change occurs not only during solidifying or melting but also at a lower

temperature, when the internal structure of the solid changes. Thus the forward solver must be

able to deal with latent heat.

As the finite methods build an equation for each control volume, the set of equations

should be built in a smart manner to allow as faster a solution as possible. In the case of

one-dimensional problem, the set of algebraic equations can be built as a tridiagonal system of

equations which can be solved very effectively and fast. However, moving into the

multidimensional heat conduction the system of equations becomes much more complicated

and the created matrices are sparse but neither tridiagonal nor band diagonal. The matrices

54

Problem Solution

become extremely large and only much slower algorithms than the tridiagonal one can be

used, such as LU decomposition.

To solve the IHCP, we first derive discretization equations for the forward solver. They

must handle the temperature dependent material properties, phase change, and effectively also

the multidimensional problem. Next, automatic refinement will be proposed in both time and

space domains. Effectively measured input data filtration will be designed and compared with

others that are usually used. Having prepared the forward solver and input data, we will treat

the IHCP.

4.1 4.1 Numerical Heat Conduction – Derivation of DiscretizationNumerical Heat Conduction – Derivation of Discretization

EquationsEquations

The discretization equations obtained using the FDM have a clear physical meaning – it

is not simply a formal mathematical approximation. The derived equations represent

conservation principles (mainly energy conservation) for each control volume and the

resulting numerical solution correctly satisfies the conservation over the whole calculation

domain. During the derivation of the discretization equations, we will consider the following

conditions:

• Unsteady heat conduction – it is time-dependent.

• The temperature profile is piecewise-linear (it is linear from the control-volume

boundary to the center of the control volume).

• The node is located in the center of the control volume.

• The surface control volume has a zero thickness.

• The material properties are temperature-dependent but constant over the control

volume and related to the node of the control volume.

• The material properties may vary for each control volume.

We first derive the equations in Cartesian coordinates. Next, the equations will be

extended to cylindrical coordinates. Treatment of the three kinds of boundary conditions

(surface temperature, surface heat flux, and convection) will be applied. The equations will

also be extended to handle the phase change.

55

Problem Solution

4.1.1 Cartesian Coordinates – One-dimensional Heat Conduction

For unsteady one-dimensional (1D) heat conduction in Cartesian coordinates,

Eq. (3.1.1) can be written as

dqdxq=⋅c

dTdt

(4.1.1)

where

q=kdTdx

. (4.1.2)

The time domain is discretized into the time steps where the index of the current time

step is m and the time step is defined as

t=tmtm1≈dt . (4.1.3)

T, T

empe

ratu

re

x, position

Node

Controlvolume

boundary

j-1 j0 1 2 j+1

0 1 2 3 J+2J+1JJ-1

x j

2

x j

j-1j j+1

q0

qJ

qJ+1

Figure 4.1 – One-dimensional model for planar geometry showing controlvolume and its boundaries.

56

Problem Solution

The object domain is discretized into a number of control volumes (see Figure 4.1).

Some methods suggest to create the surface volume of zero thickness and others suggest the

use of surface volume of the half size of the interior volumes. We use zero thickness, because

we will not use constant size of the interior volumes. This enables us to create small volume

next to the zero-volume which gives us the possibility to compute surface temperature more

accurately.

For each control volume a conservation of energy is applied to derive a set of algebraic

equations. Integrating Eq. (4.1.1) over the j-th control volume, we can write

qJmqJ1

m x j⋅q j= x j⋅ j T jm⋅c j T j

m⋅T j

mT jm1

t(4.1.4)

where j T jm and c j T j

m are the temperature-dependent mass density and specific heat,

respectively, T jm1 is the temperature of the node in the previous time step and t is the

time step.

When there are temperature-dependent parameters in the equations (such as material

properties, heat source, or boundary conditions), the set of equations becomes non-linear. The

set of these non-linear equations is solved by iterating a method for the linear problem as will

be described later. For the first iteration, the initial guess Tm of the temperature Tm is equal

to T m1 . After first iteration, the T m is set to T i1,m , where i – 1 is the previous iteration

index. During the iteration process, the Tm becomes Tm .

Temperature dependent heat conductivity

Because the conductivities kj-1 and kj of the neighboring volumes may be different, there

may be a discontinuity of the slope (dT/dx) at the control-volume boundary (see Figure 4.1).

The heat flux can be expressed on the J-th boundary at the time step m as

qJm=

k j1T j1m

x j1

2

⋅T j1m T J

m=k j T j

m

x j

2

⋅T JmT j

m(4.1.5)

where j is the node index, k j T jm is temperature-dependent thermal conductivity, and the m

superscript represents the time index. The heat flux qJm of Eq. (4.1.4) can be evaluated as

57

Problem Solution

qJm=[

x j1

2

k j1T j1m

x j

2

k j T jm ]

1

⋅T j1m T j

m . (4.1.6)

The temperature on the J-th boundary T Jm can be evaluated as

T Jm=[

k j1T j1m

x j1

2

k j T j

m

x j

2 ]1

⋅ k j1T j1m

x j1

2

T j1m

k j T jm

x j

2

T jm . (4.1.7)

Whenever there is a need to get the temperature at a control-volume boundary (e.g. during

interpolation of temperature across the control volume), this equation should be used together

with the temperature at the node.

Temperature dependent mass density and specific heat

When the mass density ρ or specific heat c becomes more dependent on the temperature,

the results are more inaccurate. Special care must be paid to time integration: if the time step

is too large, the latent heat of the phase change may be neglected [A9]. Figure 4.2 shows the

difference between the correct amount of energy stored and the amount of energy stored in the

case of neglected latent heat. In the case of a phase change, the properties are highly

dependent on the temperature.

H

T

⋅c , H

T f

T f

⋅c H*

Figure 4.2 – Comparison of correct enthalpy H and enthalpy withphase change H* neglected.

58

Problem Solution

This problem can be eliminated by solving it using the enthalpy instead of specific heat

and mass density as was described in the Phase Change Implementation chapter. The

following approach was inspired by the Apparent Heat Capacity method. Enthalpy H

represents the energy stored and is defined as follows

H T =∫T ref

T

T ⋅cT dT . (4.1.8)

The derivative of this equation gives us

T ⋅cT =dH T

dT. (4.1.9)

Using discretization of this equation, we obtain

j T jm⋅c j T j

m≈H j T j

mH j T jm1

T jmT j

m1 . (4.1.10)

Substitution of this relation for j T jm⋅c j T j

m in Eq. (4.1.4) reduces the possibility of energy

loss. This substitution can be used when the denominator T jmT j

m1 is non-zero, otherwise

j T jm⋅c j T j

m should be used (e.g. for the first iteration when Tm=Tm1 ).

Enthalpy for piecewise linearized mass density and specific heat

The temperature dependent material properties are usually given in a table or the given

dependence can be easily piecewise linearized (see Figure 4.3). Thus it is very convenient to

T

, c

Tw1

wth temperature interval

c

Tw Tw1

Figure 4.3 – Linearized thermal material properties.

59

Problem Solution

express the enthalpy H using piecewise linearized mass density and specific heat. Equations

T =a ,wTb ,w , (4.1.11)

cT =ac ,wTbc ,w (4.1.12)

describe mass density and specific heat in one temperature interval w from temperature Tw–1 to

Tw. Applying Eq. (4.1.11–4.1.12) in the enthalpy Eq. (4.1.8) gives us the equation for

computing enthalpy H within one wth temperature interval ⟨Tw1 ;T w⟩

H T =∫Tw1

T

a ,wTb ,w⋅ac ,wTbc ,wdT=aH ,wT3bH ,wT2cH ,wTd H ,wH 0 (4.1.13)

where

aH ,w=1 3

aac , (4.1.14)

bH ,w=1 2 a ,wbc ,wb ,wac ,w , (4.1.15)

cH ,w=b ,wbc ,w , (4.1.16)

d H ,w=aH ,wTw13 bH ,wTw1

2 cH ,wTw1 , (4.1.17)

H 0= 0 ; for w=1 H Tw1 ; for w1 . (4.1.18)

The H0 parameter is used for connecting two adjoining intervals and is equal to H(Tw–1) that is

computed using the preceding temperature interval ⟨Tw2 ;T w1⟩ .

Phase change at constant temperature

The major problem in using the described method is the specification of an accurate

numerical approximation for the rapidly changing material properties – an isothermal phase

change. Let us consider an isothermal phase change occurring at temperature Tm. To avoid

problems associated with the change in internal energy at a constant temperature, a small

artificial temperature region is introduced. Within this region, the heat conductivity can be

approximated by the linear piecewise continuous curve, and the product of mass density and

specific heat is constant and can be computed as

60

Problem Solution

T ⋅cT =L

TwTw1

(4.1.19)

where L is the latent heat and the wth temperature interval is the introduced small artificial

temperature region.

Source-term linearization

When the source term q is temperature-dependent in Eq. (4.1.4), its value can be

obtained as

q j=q j T jm . (4.1.20)

In some cases the source term can be expressed also using the unknown temperature

T jm as

q j=q ja T j

mq jb T j

m⋅T jm . (4.1.21)

The simplest example is a linear heat source which can be expressed as

q j=q jaq j

b⋅T jm (4.1.22)

as well as using Eq. (4.1.20) as

q j T jm=q j

aq jb⋅T j

m . (4.1.23)

The usage of Eq. (4.1.22) gives the solution faster, but this requires modification of the set of

linear equations. Eq. (4.1.23) is easier to implement, however, it can diverge as the q jb term is

increasing in absolute value and the under-relaxation must be used.

Set of equations for all internal grid points in the general form

The equations for all internal points of 1D model can be written in a general form

a j , j1⋅T j1i ,ma j , j⋅T j

i ,ma j , j1⋅T j1i ,m=b j (4.1.24)

where the j subscript is the node index from 1 to N – 1. The i superscript is an index of

iteration and the m superscript is a time step. The solution of these equations give us the

temperatures for time step m. The a and b parameters can be obtained by substituting

Eqs. (4.1.6), (4.1.10), and (4.1.21) into Eq. (4.1.4). The result for the first iteration is

61

Problem Solution

a j , j1=2 ⋅ x j1

k j1T j1m1

x j

k j T jm1

1

,

a j , j=a j , j1a j , j1 x j⋅[q jbT j

m1 j T j

m1⋅c j T jm1

t ] ,

a j , j1=2 ⋅ x j

k j T jm1

x j1

k j1T j1m1

1

,

b j= x j⋅[q jaT j

m1 j T j

m1⋅c j T jm1

t⋅T j

m1] .

(4.1.25)

For all subsequent iterations T ji1,m substitutes for T j

m1 in aj, j-1 and aj, j+1. The aj, j and bj

parameters are

a j , j=a j , j1a j , j1 x j⋅q jbT j

i1,m ,

b j= x j⋅[q jaT j

i1,mH j T j

i1,mH j T jm1

t ] .

(4.1.26)

4.1.2 Cylindrical Coordinates – One-dimensional Hea t Conduction

The heat conduction in cylindrical coordinates is described in a one-dimensional space

by the equation

1 r

dqdrq=⋅c

dTdt

(4.1.27)

where

q=r⋅kdTdr

. (4.1.28)

The object domain is discretized into a number of control volumes as shown in

Figure 4.4. Integrating Eq. (4.1.27) over the j-th control volume, we may write

qJmqJ1

m r j⋅ r j⋅q j=r j⋅ r j⋅ j T jm⋅c j T j

m⋅T j

mT jm1

t(4.1.29)

62

Problem Solution

T, T

empe

ratu

re

r, radius

Node

Controlvolume

boundary

j-1 j0 1 2 j+10 1 2

3

J+2J+1

JJ-1

r j

2

r j

j-1j j+1

q0= 0

qJ

qJ+1

rJ

rj

Figure 4.4 – One-dimensional model for cylindrical geometry showing controlvolume and its boundaries.

where r is the radius (distance from the center). For the cylindrical coordinates, the heat flux

qJm of Eq. (4.1.29) can be evaluated as

qJm=r J⋅[

r j1

2

k j1T j1m

r j

2

k j T jm ]

1

⋅T j1m T j

m (4.1.30)

and the temperature on the J-th boundary T Jm can be evaluated as

T Jm=[

k j1T j1m

r j1

2

k j T j

m

r j

2 ]1

⋅ k j1T j1m

r j1

2

T j1m

k j T jm

r j

2

T jm . (4.1.31)

63

Problem Solution

Equations for internal grid points in the general form for 1D cylindrical coordinates

We now use the same form of equations (Eq. 4.1.24) for all internal points as for the 1D

model in Cartesian coordinates. The a and b parameters can be obtained by substituting

Eqs. (4.1.30), (4.1.10), and (4.1.21) into Eq. (4.1.29). The result for the first iteration is

a j , j1=2r J⋅ r j1

k j1T j1m1

r j

k j T jm1

1

,

a j , j=a j , j1a j , j1r j⋅ r j⋅[ q jbT j

m1 j T j

m1⋅c j T jm1

t ] ,

a j , j1=2r J1⋅ r j

k j T jm1

r j1

k j1T j1m1

1

,

b j , j=r j⋅ r j⋅[q jaT j

m1 j T j

m1⋅c j T jm1

t⋅T j

m1] .

(4.1.32)

For all subsequent iterations T ji1,m substitutes for T j

m1 in aj, j-1 and aj, j+1. The aj, j and bj,

parameters are

a j , j=a j , j1a j , j1r j⋅ r j⋅q jbT j

i1,m ,

b j=r j⋅ r j⋅[q jaT j

i1,mH j T j

i1,mH j T jm1

t ] .

(4.1.33)

Derivation of discretization equations for 2D and 3D heat conduction in Cartesian and

cylindrical coordinates can be found in the Appendix F.

4.1.3 Treatment of Boundary Conditions

Having described the interior control volumes, we need to derive equations for the

surface control volumes. The zero thickness of the surface control volumes is a smart choice,

because the heat transfer needs to be described in one direction only and the derived equations

are the same for Cartesian and cylindrical coordinates. In other directions the heat flux is zero

because the surface perpendicular to that direction has the zero dimension.

64

Problem Solution

0 1

0 1 2T

s

x1

2

x1

qs

NN-1

N N+1N-1T

s

xN

2

xN1

qs

-1

T∞ T∞

Figure 4.5 – Applying boundary conditions.

Surface temperature

The situation is very simple. We just know the temperature of the node 0

T0=T s . (4.1.34)

By rearranging this simple equation to the form of Eq. (4.1.24) we obtain

a j , j=1 ; a j , j1=0 ; b j=T s (4.1.35)

where j = 0. The parameters for the surface control volume N are

a j , j1=0 ; a j , j=1 ; b j=T s (4.1.36)

where j = N.

Surface heat flux

Knowing the surface heat flux we can write the equation for the surface control volume

qs=kT0T1

x1

2

. (4.1.37)

By rearranging this equation to the form of Eq. (4.1.24) we obtain

a j , j=a j , j1 ; a j , j1=k1T1

m

x1

2

; b j=qs (4.1.38)


65

Problem Solution

a j , j1=kN1T N1

m

xN1

2

; a j , j=a j , j1 ; b j=qs (4.1.39)

where j = N.

Convection

The surface heat flux can be computed as

qs=hT sT∞ . (4.1.40)

Therefore, we can write the equation for the surface control volume

hT0T∞=k1T1

m

x1

2

T1T0 . (4.1.41)

By rearranging this equation to the form of Eq. (4.1.24) we obtain

a j , j=ha j , j1 ; a j , j1=k1T1

m

x1

2

; b j=h⋅T∞ (4.1.42)


a j , j1=kN1T N1

m

xN1

2

; a j , j=ha j , j1 ; b j=h⋅T∞ (4.1.43)

where j = N.

Radiation

The heat flux due to the radiation from the surface is expressed in Eq. (12.3). It is

convenient to express the radiation heat flux in the form

qrad=h T sT sur (4.1.44)

where the radiation heat transfer coefficient h is

h=⋅ T sT sur⋅T s2T sur

2 . (4.1.45)

Therefore, the radiation can be simply integrated into the equations for convection where

66

Problem Solution

h=hconvh . (4.1.46)

Because this boundary condition is strongly dependent on surface temperature, it should be

updated for each iteration during solution of discretization equations.

4.2 4.2 Solution of Discretization EquationsSolution of Discretization Equations

This chapter describes an effective solution algorithm for the set of equations of 1D, 2D,

and 3D heat conduction. The set of Eq. (4.1.24) can be written in a form of Eq. (3.3.46) and

also in the matrix form of Eq. (3.3.47). The solution is quite simple for 1D heat conduction

because it is a tridiagonal system of equations – the A matrix has nonzero elements only on

the diagonal plus or minus one column (see Figure 4.7). It is not a linear system of equations,

however, we can use the solution algorithm described in the Appendix D – Solution of Linear

Algebraic Equations – Tridiagonal Systems of Equations. By iterating this algorithm for linear

equations, we are able to solve our set of non-linear equations. As can be seen, the

non-linearity is caused by the parameters which are dependent on the unknown temperature

T jm . For the first iteration, the initial guess T j

m of the temperature T jm is set to temperature

T jm1 which was computed for the previous time step m – 1. After the first iteration, T j

m is

set to T ji1,m , where i – 1 is the previous iteration index. This is repeated for all next

iterations. During the iteration process, T jm becomes T j

m . The number of required iterations

will be described later.

Moving into the 2D or 3D space, each internal control volume has four or six neighbors,

respectively, while in the 1D space each internal control volume has only two neighbors. This

would result in five or even seven unknown temperatures in one equation for computing the

temperature of one internal node [j, k] or [j, k, p] :

/ 2D model /

a j , k1⋅T j , k1i ,m a j1,k⋅T j1,k

i ,m a j , k⋅T j , ki ,ma j1,k⋅T j1,k

i ,m a j , k1⋅T j , k1i ,m =b j , k

, (4.2.1)

/ 3D model /

a j , k , p1⋅T j , k , p1i ,m a j , k1, p⋅T j , k1, p

i ,m a j1,k , p⋅T j1,k , pi ,m a j , k , p⋅T j , k , p

i ,m

a j1,k , p⋅T j1,k , pi ,m a j , k1, p⋅T j , k1, p

i ,m a j , k , p1⋅T j , k , p1i ,m =b j , k , p

. (4.2.2)

67

Problem Solution

x

j–1 j j+1

J+2J+1JJ–1

qJ

qJ+1

1D heat conduction

x

qJ, k

qJ+1,k

2D heat conduction

J+2J+1JJ–1

y

j–1, k+1 j, k+1 j+1, k+1

j–1, k j, k j+1, k

j–1, k–1 j, k–1 j+1, k–1K–1

K

K+1

K+2

qj,K+1

qj,K

A matrix

zeros

zeros

A matrix

zeros

zeros

zeros

zeros

3D heat conduction A matrix

qJ,k,p

qJ+1,k,p

qj,K+1,p

qj,K,p

j,k,p

x

yz q

j,k,P

qj,k,P+1

Figure 4.6 – Control volumes and A matrixes for 1D, 2D, and 3D heat conduction.

By arranging the set of equations into the matrix form, Eq. (3.3.47), we will get very

sparse matrixes A. Examples of these matrices are shown in Figure 4.6 for rectangular and

cubic geometries. Although A matrixes may seem to be band diagonal systems of equations

68

Problem Solution

they are not because the condition bandwidth << N is not fulfilled. This set of equations can

be effectively solved by Conjugate Gradient Method or LU Decomposition, but there is

a more efficient solution of this problem.

The key point to solution is the principle of superposition of heat conduction – two or

three-dimensional conduction may be expressed as a product of one-dimensional conductions.

To solve our multi-dimensional problem, we can split it into separate one-dimensional

problems along the main axes (Cartesian, cylindrical, or spherical coordinates). This approach

is also called line-by-line method. We can compute Eq. (4.2.1) and Eq. (4.2.2) within two and

three iterations, respectively, as follows:

/ 2D model /

a j1,k⋅T j1,ki ,m a j , k⋅T j , k

i ,ma j1,k⋅T j1,ki ,m =b j , ka j , k1⋅T j , k1

m a j , k1⋅T j , k1m , (4.2.3)

i1 i (increase iteration index)

a j , k1⋅T j , k1i ,m a j , k⋅T j , k

i ,ma j , k1⋅T j , k1i ,m =b j , ka j1,k⋅T j1,k

m a j1,k⋅T j1,km . (4.2.4)

/ 3D model /

a j1,k , p⋅T j1,k , pi ,m a j , k , p⋅T j , k , p

i ,m a j1,k , p⋅T j1,k , pi ,m

=b j , k , pa j , k , p1⋅T j , k , p1m a j , k1, p⋅T j , k1, p

m a j , k1, p⋅T j , k1, pm a j , k , p1⋅T j , k , p1

m ,


a j , k1, p⋅T j , k1, pi ,m a j , k , p⋅T j , k , p

i ,m a j , k1, p⋅T j , k1, pi ,m

=b j , k , pa j , k , p1⋅T j , k , p1m a j1,k , p⋅T j1,k , p

m a j1,k , p⋅T j1,k , pm a j , k , p1⋅T j , k , p1

m ,


a j , k , p1⋅T j , k , p1i ,m a j , k , p⋅T j , k , p

i ,m a j , k , p1⋅T j , k , p1i ,m

=b j , k , pa j , k1, p⋅T j , k1, pm a j1,k , p⋅T j1,k , p

m a j1,k , p⋅T j1,k , pm a j , k1, p⋅T j , k1, p

m . (4.2.5)

We have three unknown temperatures on the left-hand side and all parameters known on the

right-hand side of the equations. The equation for the first iteration, Eq. (4.2.3), represents k

sets of tridiagonal systems equations – one set for one line of nodes along X direction. Each

line of nodes along the X axis is solved as a 1D heat conduction. The next equation,

Eq. (4.2.4), represents j sets of tridiagonal systems equations – one set for one line of nodes in

69

Problem Solution

the Y direction. A similar situation arises for the 3D model. The approach splits a huge set of

equations (especially for 3D model) into a number of relatively small tridiagonal systems of

equations which can be solved very effectively.

A smart choice of temperature Tm can significantly reduce the number of required

iterations. So far, the temperature from the previous iteration T i1,m was used for temperature

Tm . However, more up-to-date values can be used. We can use the values of the current

iteration of already computed lines. To avoid the new values are being every time on the same

side of the line (e.g. at the bottom and on the left side for the 2D model in Figure 4.6), it is

recommended to invert the order of the computed lines after each iteration.

4.2.1 Error Estimation of Iteration Algorithm

The previous chapter describes the iteration process for the solution of the non-linear

equations. The iterative solution process gives us approximate temperatures of the control

volume nodes. The temperatures may converge to the solution or diverge during the iteration

process. If divergence occurs, an under-relaxation should be considered although it is not very

common and it occurs only in some special cases with a heat source.

The maximum estimated error is usually computed as

i≈∣T ji ,mT j

i1,m∣ (4.2.6)

where T ji ,m and T j

i1,m are the node temperatures computed from the last and next-to-last

iterations, respectively. This estimation is good for problems that converge fast. As you can

see in Figure 4.7, this estimation can be very inaccurate if the problem under solution

converges slowly [A9]. The estimated error after the i th iteration is i but the difference

between the last computed value and the value computed after an infinite number of iterations

is i . This difference is the sum of the estimated errors from all next iterations that have not

been solved yet.

A better estimate of the convergence error can be determined using a method based on

the observed geometric convergence of heat conduction problems. The difference i between

the last computed value and the value computed after an infinite number of iterations can be

estimated using the sum of a geometric series

70

Problem Solution

Figure 4.7 – Estimated and real errors after ith iteration.

i=i1i2⋯∞=p

b1p

b2p

b3⋯p

b∞= p∑

n=1

∞ 1

bn= p

1 b1

(4.2.7)

Parameters p and q can be computed as

b=∣T ji2T j

i1

T ji1T j

i ∣ , (4.2.8)

p=∣T ji1T j

i∣ (4.2.9)

where T ji2,m , T j

i1,m , and T ji ,m are the results computed in the last three subsequent

iterations.

The iteration process converges faster for a higher b parameter. For the cases where

b≥2, the estimated error is less than p and thus Eq. (4.2.6) can be used. On the other hand,

whenever the b parameter becomes closer to 1, the solution will converge slowly and

Eq. (4.2.7) should be used. This occurs mainly when highly conductive materials are used

inside materials with poor thermal conductivity.

4.3 4.3 Discretization Error EstimationDiscretization Error Estimation

The numerical approach used requires some simplification of continuous physical

systems. The simplification is used for the object, its boundary conditions, and time domain.

The stress energy used in mechanics can be applied in heat conduction. The heat energy will

be used for an accuracy analysis of heat conduction problems solved using the numerical

approach instead of stress energy.

71

Tem

pera

ture

number of iterationsi-3Tm

T i ,mT i1,m

T i2,m

T i3,m

i-2 i-1 i

i1

i

i

Problem Solution

4.3.1 Boundary Conditions

We are working with unsteady boundary conditions, but the boundary conditions are

assumed to be constant within one time step as can be seen from Eqs. (4.1.34), (4.1.37), and

(4.1.40). Therefore, the time dependent profile of boundary conditions must be replaced by

a discontinuous profile consisting of a number of small constant pieces. Their length is the

time step (see Figure 4.8).

Bou

ndar

y co

nditi

on

time

(A)

time step

real profile

measured value

piecewise constant profile

Bou

ndar

y co

nditi

on

time

(B)

piecewise linear profile

Figure 4.8 – Simplification of time-dependent boundary conditions.

In most cases we do not know the real profile of boundary conditions, we only know the

values in certain measured time instants (see Figure 4.8). We usually use a piecewise

linearized profile between the measured values. Knowing the boundary conditions in certain

measured time instants, we can simplify them in various ways. Two methods are shown in

Figure 4.9. The B method is often used, but the A method is more effective.

We can estimate the error of simplified boundary conditions for case A as a sum of

partial errors

=∑m=1

n

∣ mm1

4 ∣ (4.3.1)

where the m is the measured boundary condition at time step m and n is the number of

computed time steps. The estimated error for case B is twice higher.

72

Problem Solution

Bou

ndar

y co

nditi

on

time

(A)

time step Bou

ndar

y co

nditi

on

time

(B)

sum of partial errors

m m + 1 m + 2 m m + 1 m + 2

Figure 4.9 – Error estimation of boundary conditions.

4.3.2 Mesh Discretization Error

For rapid changes in the heat flux boundary conditions and near the heterogeneous

material, a fine mesh is required [A9]. This is because the smooth temperature profile has

been piecewise linearized. Figure 4.10 shows two meshes – a rough one and a finer one.

Before any estimation of mesh discretization error and mesh refinement can be done, the user

must create a fine mesh that reflects the expected temperature profile. As you can see in

Figure 4.10, if the user creates a rough mesh (case A), the analysis just skips the left

temperature peak. If the mesh is fine enough (case B), the mesh discretization error can be

computed, and based on this information the mesh refinement can be done.

T,

Tem

pera

ture

x, position

(A)

(B)

control volume size

control volume size

Figure 4.10 – Mesh design.

73

Problem Solution

We will investigate the mesh discretization error by comparing the heat energy stored

within the control volume. The heat energy of the control volume can be expressed as

an enthalpy

H j*=

1 x j

∫x=xJ

xJ1

∫T ref

T

⋅c dT dx. (4.3.2)

Because we are assuming constant material properties within one control volume, we can

simplify Eq. (4.3.2) as follows

H j*=⋅c⋅T j

* . (4.3.3)

This simplifies our search for the mesh discretization error. We can only investigate the

average temperature of the control volume and, if necessary, we can recalculate the average

temperature to heat energy using Eq. (4.3.3).

First, we must avoid a very rough mesh as shown in Figure 4.10, case A. Next, we can

assume two extreme conditions shown in Figure 4.11. Case A shows the situation shortly after

the rise in temperature of the control volume j – 1 has risen. The temperature on the boundary

T J is very high compared to the rest of the temperatures within the control volume j. Case B

shows the situation later after the change. The thermal shock propagates deeper inside the

control volume. Because of the piecewise linearized profile, we only know the temperature at

the center of the node and we can estimate temperature on its boundaries. We do not know

anything about the real temperature profile between the centers of the control volumes.

However, we can express the two extreme conditions shown in Figure 4.11. As you can see in

case A, the real average temperature of the j th control volume T j* is very close to the

temperature of the j th node T j . On the other hand, in case B, the real average temperature is

much closer to the average temperature computed from the temperatures of the node and on

the boundaries of the control volume as follows

T j=T J , jT j , J1

2=T JT j /2T jT J1/2

2. (4.3.4)

If we take a closer look at FDM and FVM, we find that the FDM assumes the

temperature of the node T j for the average temperature of the control volume, while the

FVM assumes T j expressed in Eq. (4.3.4).

74

Problem Solution

T, T

empe

ratu

re

x, position

Tj–1

Tj

Tj+1T j

FDM

TJ

TJ+1

Tj–1

Tj T

j+1

TJ

TJ+1

x, position

(A) (B)

T J , j=T JT j /2

T j=T J , jT j , J1

2

T J , j=T JT j /2

T jFVM

real temperature profile

T jFDM

T jFVM

T j , J1=T jT J1/2

T j=T J , jT j , J1

2

T, T

empe

ratu

re

T j*

T j*

Figure 4.11 – Temperature of a control volume in FDM, FVM, and real averagetemperature of the control volume.

Assuming the real average temperature is between the T jFDM and T j

FVM , we can

estimate the discretization error as the difference between the temperature at the center of the

node and the average temperature T j using Eq. (4.1.7) and (4.3.4) as

T=∣T jFDMT j

FVM∣=∣T jT j∣=∣T j

2

1 4

k j1T j1⋅ x j⋅T j1k j T j ⋅ x j1⋅T j

k j1T j1⋅ x jk j T j ⋅ x j1

1 4

k j T j ⋅ x j1⋅T jk j1T j1⋅ x j⋅T j1

k j T j ⋅ x j1k j1T j1⋅ x j

∣ . (4.3.5)

The same equation can be obtained for cylindrical coordinates using Eq. (4.1.31). The x is

only replaced with r.

4.3.3 Time Discretization Error

Dealing with the time discretization, we should compare the three main schemes for

temperature variation in time – explicit, Crank-Nicholson, and fully implicit shown in

Figure 4.12. The explicit scheme essentially assumes that the old temperature lasts through the

entire time step. The fully implicit scheme supposes that the temperature suddenly drops at the

beginning of the time step and stays over the whole time step, while the Crank-Nicholson

scheme assumes a linear variation of the temperature during the time step.

Working only with the unconditionally stable fully implicit scheme, we will derive

a method for estimating the error caused by time discretization. As you can see in Figure 4.13,

none method describes correctly the variation of temperature in time shortly after a change in

the boundary condition.

75

Problem Solution

Crank-Nicholson

T, T

emp

erat

ure

time

time step

mm – 1

Explicit

Fully Implicit

Figure 4.12 – Variation of temperature in time for three different schemes.

The trick for estimating the error caused by time discretization is based on comparing

the error of the temperature (hatched area in Figure 4.13) for the time step and the double time

step. As you can see, for the later change in boundary condition, the hatched area for the

double time step is about twice larger than for two steps of the normal time step. Knowing

that the error (the hatched area) is half the size of that for half time step and having computed

our problem under solution using the double time step and normal time step we can estimate

the error caused by time discretization as

t T j =∣T j//T j

/∣ (4.3.6)

where T j// is the temperature computed using the double time step and T j

/ is the temperature

computed using the normal time step.

T, T

emp

era

ture

time

time step

double time steptime

T,

Tem

pera

ture

Shortly after change in boundary condition

Longer after change in boundary condition

Real tempearatureReal tempearature

Fully implicit scheme Fully implicit scheme

time step

double time step

Figure 4.13 – Temperature variation and time step refinement.

The situation is slightly different for the case shortly after the change in boundary

condition, because the hatched area for the double time step is not twice larger than that for

two steps of the normal time step, but slightly less. No ratio lower than 1.5 was observed

76

Problem Solution

when testing the method; the lowest values did not drop under 1.7. Knowing this behavior and

applying this ratio in Eq. (4.2.7) describing the sum of a geometric series, we can estimate the

error caused by time discretization using modified Eq. (4.3.6) as

t T j =⋅∣T j//T j

/∣ (4.3.7)

where the coefficient varies from 1 to 2, depending on the problem under solution. The

=2 should be used to make sure that the estimation of the maximum error caused by time

discretization is correct (=2 for ratio 1.5 and =1 for ratio 2). For the infinite time step

(steady conditions), the implicit scheme describes the temperature variation in time almost

perfectly and there is no need to make such an estimation.

4.4 4.4 Optimization of DiscretizationOptimization of Discretization

Knowing the methods for discretization error estimation, we can use them for mesh and

time step optimization. The next chapters will describe the methods for setting the optimal

mesh and time steps, but first we will deal with refinement of boundary conditions measured

in discrete time instants.

4.4.1 Refinement of Boundary Conditions in Time Dom ain

Before mesh optimization can be done, appropriate boundary conditions should be

prepared. We will assume boundary conditions measured in discrete time instants and the

linearized profile between the measured values. However, the computation numerical method

assumes constant boundary conditions within one time step. The error caused by this

simplification is expressed by Eq. (4.3.1).

Figure 4.14 shows that the error is half in size for a half time step. This behavior can be

used for time step refinement

t= t*

trunc* /max1 (4.4.1)

where t is a new time step, t* is time discretization of measured boundary condition, *

is the sum of the partial error for the time step t* , max is the maximum allowed value, and

the trunc function returns only the integer part of the expression in brackets.

77

Problem Solution

Figure 4.14 – Boundary condition refinement.

4.4.2 Mesh Optimization

In the forward solver, an adaptive meshing scheme should be used to keep the number

of computational points at minimum whilst ensuring accuracy of the solution [A9]. The

linearized profile will give reasonable representation of the solution only if the grid is

sufficiently fine.

Before mesh optimization can be done, a starting one must be created. The starting mesh

should not be very rough as you can see in Figure 4.10 case A. The mesh must reflect the

expected temperature profile. It is better to create a finer mesh, which can be coarsened, than

to create a too rough one.

Eq. (4.3.5) will be used as a criterion equation for mesh refinement. If the T parameter

is higher than our acceptable value Tmax , the control volume should be divided into more

control volumes. If the center node of a control volume corresponds to a temperature sensor

location, the volume should be divided into odd number of volumes so that the position of the

center node remains unchanged. If the T parameter is lower for two neighboring control

volumes than the limit value Tmin , the control volumes can be merged. However, avoid

merging volumes corresponding to a temperature sensor location and volumes already

divided. The T parameters should be checked for all control volumes after each time step

and the computation along with mesh refinement should be repeated until no re-meshing is

necessary.

78

Bou

ndar

y co

nditi

on

time

(A)

time step Bou

ndar

y co

nditi

on

time

(B)

sum of partial errors

m m + 1 m m + 1 m + 2

half timestep

constant value

A

B=A /2

linearized value

Problem Solution

4.4.3 Time Step Refinement

The used fully implicit scheme does not precisely describe the temperature variation of

control volumes in time, which causes some inaccuracy. As shown in Figure 4.13, the finer

time step results in a higher accuracy of the computed result. The estimated error caused by

time discretization is expressed by Eq. (4.3.7). This error should be checked after

computation. If the error is higher than the acceptable value tmax , the problem should be

computed using the half time step. This approach allows us to use repeatedly Eq. (4.3.7).

4.5 4.5 MultiMulti --level Filtration of Measured Datalevel Filtration of Measured Data

Noise suppressing is important for data that are used as inputs for inverse tasks, because

the noise is mostly gained by the inverse task and may seriously degrade the computed results.

Many papers testing the noise problem in input data consider very simply generated random

noise [45, 48 ] or sophisticated Gaussian noise [42]. However, noise does not have such

a characteristic very often. The data are recorded using a measuring card that converts the

continuous value into the digital representation. The resolution of digitalized data is usually 8,

12, or 16 bits. More bits result in a higher accuracy but a lower measuring frequency and

a higher price of the measuring card. Now imagine that you need to measure within the

temperature range 0–2000°C. If you use a 8-bits card, the measuring range can be divided only

into 256 levels. Therefore, you can measure at a maximum accuracy of 8°C. The accuracy for

12-bits and 16-bits cards is 0.5°C and 0.03°C, respectively. An example of measured data

using the 12-bits card is shown in Figure 4.15. It can be seen that the noise in the shown data

is far from a random or Gaussian noise. The values are simply leaping on two or three certain

layers.

1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0Time [s]

901.0

901.5

902.0

902.5

903.0

903.5

904.0

Tem

pera

ture

[°C

]

Figure 4.15 – Measure data using 12-bits A/D card at 300 Hz

79

Problem Solution

Figure 4.16 – Gaussian filter function with krn = 3.

The proposed multi-level filtration procedure was designed especially for

a thermocouple sensor built in a stainless steel plate 1.5 mm under the cooled surface.

However, it can also be modified for similar cases. The temperature was measured using

a 12-bits card at 100 and 300 Hz. An example of measured data is shown in Figure 4.15. The

algorithm uses convolution (Eq. (3.5.5) and Eq. (3.5.1) for discrete data) with the varying

Gaussian filter function

pt =et2

2 krn/32 (4.5.1)

to filter the measured data, where krn represents the with of influenced area (see Figure 4.16).

First, several sets of filtered data are computed. Each set is computed using constant

filter Eq. (4.5.1). The krn varies from krnmin = f / 10 to krnmax = 2f / 10 with step krnstep = 1

which represents one sample. The filtration procedure starts at the beginning of the recorded

temperature history with the first measured sample for which the value from the set computed

using krnm=krnmin= f /10 is used where the m superscript represents the current time step.

For the next value, the set computed using krnm=krnm12 (stronger filtration) is used but

only if the following condition is fulfilled

T *T step≤ T≤T *T step∨

T * mT step≤ Tm

∧

Tm2≤T * m2T step∨ Tm1≤T * m1T step∨ T

m1≤T * m1T step∨ Tm2≤T * m2T step

∨ Tm≤T * mT step

∧

T * m2T step≤ Tm2∨T * m1T step≤ T

m1∨T * m1T step≤ Tm1∨T * m2T step≤ T

m2. (4.5.2)

80

-3.0 -2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0t

0.0

0.2

0.4

0.6

0.8

1.0p(

t )

Problem Solution

0.9 1.0 1.1 1.2 1.3Time [s]

910

911

912

913

914

915

Tem

pera

ture

[°C

]

1.4 1.5 1.6 1.7 1.8Time [s]

901.0

901.5

902.0

902.5

903.0

903.5

904.0

904.5

905.0

Tem

pera

ture

[°C

]

MeasuredFilteredCorrected

Figure 4.17 – Before the correction function is applied, the filtered data are below themeasured ones in the part where the data changes the slope.

If this condition is not fulfilled, smaller krnm=krnm1 is tested by Eq. (4.5.2). The krn is

decreasing until the condition Eq. (4.5.2) or

krnm=krnm12 is fulfilled. This enables us to

change adaptively the strength of the used Gaussian

filter. If the slope of the temperature history is almost

constant the strength of the filter is increasing. As the

filtration procedure is approaching a change in slope

of the temperature history the strength is decreasing.

Next, a soft Gaussian filter with krn=5 is

applied to the filtered data. This operation is required

because the filtered temperature can abruptly change

as the krn changes during the first stage of the

filtration.

The filtered data are now below or above the measured ones in parts where the

temperature history changes its slope (see Figure 4.17). Therefore, a correction function,

which represents the smoothed deviation of the filtered and measured temperature, is

computed as

d T t =1

∑

p∑

[ T T * ]⋅pt (4.5.3)

81

0.0 0.2 0.4 0.6 0.8 1.0x

0.0

0.2

0.4

0.6

0.8

1.0

y( x

)

Figure 4.18 – Non-linear correctionfunction.

Problem Solution

where pt is expressed in Eq. (4.5.1) and krn= f /10 is used. The correction function

Eq. (4.5.3) is modified by the non-linear correction function (see Figure 4.18) and then

applied as

T m= T m1 2 cos

d T m

d T∣max∣

1d T∣max∣ for m=0..n . (4.5.4)

The non-linear function shown in Figure 4.18 ensures that the correction function from

Eq. (4.5.3) is applied more intensively for a bigger deviation. When this correction procedure

is applied twice, the smoothed values are pushed back within the measured ones as can be

seen in Figure 4.17.

4.6 4.6 Solutions of IHCPSolutions of IHCP

In this chapter we focus on the determination of boundary conditions of heat conduction

problems, determination of thermal material properties [A5], and model calibration of

the measuring sensor [A12]. Typical applications are cooling by water or mist air-water

nozzles in casting [A13], descaling [A10], and hot rolling [A4]. The computed boundary

conditions are usually used for the description of the cooling efficiency while the material

properties and calibrated model are needed for computation of these boundary conditions.

As described in the previous sections, the inverse methods usually make use of the

knowledge of the direct solution of the problem. For non-linear problems, the Beck’s

algorithm estimates boundary conditions from the knowledge of internal temperatures using

some algorithm and then it verifies the estimation using the direct solution of the problem.

After comparing the direct solution computed using estimated boundary conditions with the

measured values, next estimation is performed. If the measured values match the computed

ones, the iteration process is ended.

Usage of a new alternative approach is described in the following chapters. The main

idea is the usage of the general optimization method for matching the measured temperatures

with the computed ones by setting the parameters we are looking for. This process is also

often called identification. The modified downhill simplex multidimensional optimization

method [A9] will be used. However, other multidimensional methods can be considered.

82

Problem Solution

4.6.1 Computation of Boundary Conditions

Reviewing the inverse methods, the sequential Beck’s approach has been found to be

very suitable for IHCP. The method is relatively fast and stable. The disadvantage of this

method is the decreasing accuracy for cases where a large stabilization parameter must be

used, e.g. for noisy input data, and where the boundary conditions are changing fast. However,

it can give a good idea about the main characteristic of the boundary condition.

The proposed approach makes use of the knowledge of the main characteristic of the

boundary condition. The time dependent boundary conditions are approximated by

an appropriate function within a certain time interval at which the boundary conditions are

abruptly changing [A3, A6, A1]. For time intervals where the boundary conditions are almost

constant, the fast and stable sequential Beck’s algorithm is used with a large stabilization

parameter.

Our goal is to find transient boundary conditions from the measured temperature history

inside the body. However, we only know the direct solution – computation of the temperature

from boundary conditions. First, the sequential Beck’s approach with a large stabilization

parameter should be used to have an idea about the shape of boundary conditions. Next,

an appropriate approximation function should be used. An example of such a function for the

water nozzle cooling application is shown in Figure 4.19. The optimized heat transfer

coefficient (HTC) function is described by the following equations

HTCx=⋅ex2

22

=L for xR for x≥

(4.6.1)

where the parameters δ and γ represent the maximum and minimum value of the HTC,

respectively. The parameter σ describes the shape in the x direction. The shape for x ,

where µ represents the nozzle position, is different from x≥ . Hence the parameter σ is

divided into the two parameters σL and σR for the left and right sides, respectively.

The method is based on the fact that the optimum approximation function of HTC

minimizes the sum square error between the measured temperatures T i* and the computed

temperatures T i where the sum square error function is defined as

83

Problem Solution

SSE=∑i

T i*T i

2. (4.6.2)

This equation is used as a criterion function for the optimization method. The measured

temperatures T i* at an interior location are known from the measurement and the computed

temperatures T i at the same interior location, but of the computational model, are computed

using tested boundary conditions.

Several methods can be used to determine unknown parameters of the approximation

function by minimizing the criterion function. In most cases, the approximation function has

more than one parameter to determine. The parameters must be thus determined using

methods which are able to compute multiple parameters at once. Usage of the downhill

simplex multidimensional optimization method and an artificial neural network will be

described in the following chapters.

Figure 4.19 – Approximation function for water nozzlecooling applications.

Usage of optimization method

The downhill simplex method [5] is suitable for a small number of optimized

parameters and does not require any derivatives of the optimized function. Although no

derivatives are required, this method uses information about the gradient of the optimized

function and thus converges quite fast.

The optimized space is searched using a moving simplex. A simplex is the geometrical

figure consisting, in n dimensions, of n + 1 points (or vertices) and all their interconnecting

line segments, polygonal faces, etc. The algorithm is supposed to make its own way downhill

84

Time [s]

HT

C [W

/(m

².K

)]

T [°

C]

HTCSurface temperature

2σL 2σR

Nozzle position

δ

γ

Problem Solution

through the unimaginable complexity of N-dimensional topography until it encounters

a minimum.

Figure 4.20 illustrates possible changes in the simplex size within one step and simplex

movement during three steps for a 2D problem. The simplex was expanded during the first

step. The mirrored simplex (dotted line) was thrown out. During the second step, the

expanded simplex was thrown out and the mirrored one was accepted. During the third step,

neither the mirrored simplex nor the contracted one (dashed line) was accepted. Therefore, the

half simplex was used.

Figure 4.20 – Vertex meaning within one step and moving simplex in 2D space.

The basic downhill simplex method was modified to allow optimization of a problem

with constrained parameters, and restart that helps overcome small local minimums is also

implemented [A9, A5]. The description of the modified downhill simplex is as follows:

1. Create a simplex object and compute the value for each vertex of the simplex.

2. Define α (reflection), β (contraction), and γ (expansion) parameters. Define the

maximum of optimization steps and maximum of restarts.

3. Define the limitation dimensions used in step 6. These limits can be different for

any parameter and should reflect the ability of the numerical method to compute

different results for two different input parameters. If the difference in input

parameters is smaller than the chosen limit, there might not be any difference in the

computed values or the difference is incorrect due to the rounding errors in the

numerical method. These limits help avoid simplex degradation.

85

sx

x ′′

x ′′ rx

mx

x

ex

x Half simplex

0

0 0

1 2

3

3

Step number

Maximum value

Minimum value

Centroid

Reflection vertex

Original simplex

Problem Solution

Find r where ( ) ( )jnjr xfxf 11min +≤≤=

Simplex nodes are defined as nnxx ℜ∈+11 ,,L

Create empty vector sr

Find s where ( ) ( )jsjnjs xfxf r∉+≤≤= ,11max

Compute centroid ∑+

≠=

=1

,1

1 n

sjjjx

nx

Compute reflection node ( )sxxxx −+= αˆ

x is within limits

( ) ( )xfxf r ˆ>

Compute expanded node ( )xxxxe −+= ˆγ

ex is within limits

and ( ) ( )exfxf >ˆ

se xx → sxx →ˆ

Attach s to vector s

r

3<− snn r

Create half simplex using

( )jrjj xxxx −+=2

1

Find m where ( ) ( ) sjxfxf jnjm ≠= +≤≤ 11max

( ) ( )xfxf m ˆ<

Find x′ where ( ) ( ) ( ) sxfxfxf ,ˆmin=′

Compute contraction node ( )xxxx −′+=′′ β

( ) ( )xfxf ′>′′

Create half simplex using

( )jrjj xxxx −+=2

1

sxx →ˆ

sxx →′′

End

Start

Figure 4.21 – One optimization step of the Downhill Simplex optimization method withparameter limitation.

86

Problem Solution

4. Define the smallest allowed dimension of the entire simplex during the

optimization process. The smallest simplex should reflect the desired precision of

the optimization process.

5. Make a simplex movement using one optimization step as shown in Figure 4.21.

6. If any simplex dimension is smaller than the appropriate limit chosen in step 3, go

to step 8 (restart).

7. If the entire simplex is smaller than the smallest simplex defined in step 4, go to

step 8 (restart).

8. (This step stands for restart) If the number of restarts is higher than the defined

maximum or if the position of this minimum is very near to the previous restarting

minimums, you have found the optimum (minimum). If not, choose the node with

a minimum value and create a new simplex around this vertex. This simplex must

be bigger than the limits defined in steps 3 and 4. Restarts help overcome small

local minimums. Go to step 5.

In our case, the optimized values are the parameters of the approximation function, e. g.

the δ, γ, µ, σL, and σR parameters for Eq. (4.6.1). The downhill simplex method minimizes the

criterion function defined by Eq. (4.6.2) by setting the parameters of the approximation

function. The lower value of the criterion function indicates that the computed temperature

history matches the measured history better and thereby the HTC described by the

approximation function is more accurate.

Usage of feed-forward neural network

Let us imagine so called black box that can compute directly the desired boundary

conditions from the temperature history measured inside the investigated body. No such

general theory is available. However, we can create and train an artificial neural network for

our particular problem [A2]. Such a network will serve us as a black box where the input

vector is a measured temperature history, and the output vector consists of the approximation

function parameters of boundary conditions (see Figure 4.19).

Structure

The basic feed-forward neural network will be used for explanation (see Figure 4.22).

However, the cascade correlation can also be considered. A two-layer network [55] is used as

the main model. The number of neurons in the output layer is equal to the number of the

87

Problem Solution

requested parameters. The number of hidden neurons should reflect the complexity of the

problem under solution.

Training

The training data must be available before the start of the training. A set of various

vectors of approximation function parameters, which should cover the expected range of the

parameters to be computed, is prepared. The real measurement is simulated on a computer for

each vector in the prepared set. Different parameters and the computed temperature history

represent the requested training data. However, the training data should be standardized (to the

range from 0 to 1). Having prepared the training data, the feed-forward neural network is

trained using fast back-propagation, i. e. Levenberg-Marquardt functions.

Figure 4.22 – Feed-forward neural network.

Computing boundary conditions

Computation using the learned neural networks is fast and simple. Use the measured

temperature history as an input data, and the neural network gives you the desired parameters

of the approximation function. However, it has been proved that not all of the learned

networks can be used. Some of them return confused results. Verification should thus be

made, e. g. direct heat conduction computation using the computed approximation function

88

Σ

T(2)

Σ b2(1)

Σ b1(2)

Σ b2(2)

Σ

b1(3)

Σ

b2(3) W1(n,3)

W2(3,3)

W1(1,1)W2(1,1)

b1(1)

T(3)

T(n)

T(1)

INPUT OUTPUT HIDDEN LAYER OUTPUT LAYER

δ

σL

σR

hyperbolic tangent sigmoid transfer function

linear transfer function

T(1) ... T(n)

b1, b2

W1, W2

- temperature history

- weight matrixes

- bias vectors

Problem Solution

and a comparison of the computed temperature history with the measured one. Note that the

learned neural network can be used only for that particular geometry for which the learning

process has been accomplished.

4.6.2 Computation of 2D Boundary Condition

In this section, an alternative computation of more than one boundary condition is

explained. The inverse method simultaneously computes two boundary conditions: the

thermal contact resistance (TCR) between an alloy casting and the silica mold, and the heat

transfer coefficient (HTC) between the mold and a bath [A9].

Problem description

The alloy-casting apparatus (see Figure 4.23 and 16.6–16.8 in Appendix G) consists of

a solidifying alloy, mold, and furnace with a molten metal bath. The furnace keeps the metal at

a prescribed temperature. This enables control of the speed of the solidification process. At the

beginning of the experiment the mold with liquid alloy is immersed into the tin bath. During

solidification, the temperature histories of four thermocouples are recorded. The thermocouple

T1 is placed into the center of the casting. Two thermocouples, T 2 and T3 , are placed into

the mold (see Figure 16.9) near the inside and outside surfaces of the mold, respectively. The

fourth thermocouple, T 4 , measures the temperature of the molten metal bath.

T2 T1 T3

Furnace Tin bath

Mold Alloy T4

Figure 4.23 – Alloy casting experimental apparatus.

The computational model consists of an alloy, mold, and gap between the alloy and the

mold. The 1D computational model is considered in micro gravity, that is, the process is

89

Problem Solution

considered as purely conductive, and as one object with spatially and temperature dependent

material properties. The gap between the alloy and the mold, with some thermal resistance and

no specific heat, enables us to simulate the effect of the TCR between an alloy casting and the

mold. During the computation, TCR is converted to the thermal conductivity kgap using

kgap

xgap

T=1

TCRT=h1 T (4.6.3)

where x is the width of the space and T is a difference between the temperatures at the

sides of the gap. If the space is very small, it does not affect the model. In our model, we fix

the distance xgap and vary the conductivity kgap to determine the TCR. Using Eq. (4.6.3) the

TCR can be converted into the thermal conductivity or h1 . On the outside surface of the

mold, a boundary condition of the third kind is considered. The HTC between the mold and

bath is marked as h2 . The temperature of the bath is considered as the ambient temperature.

Inverse task

A common solution of the inverse task uses Beck's sequential method [11] but any

optimization method can be used. In some cases, optimization methods can work better than

Beck's approach, especially for multiple heat fluxes with temperature-dependent material

properties where the problem under solution becomes nonlinear.

The downhill simplex method can be used to determine time-dependent TCR and HTC

( h1, h2 ). The measured temperature histories (temperature sensors T1 , T 2 , T3 , and T 4 )

are used as the input for the inverse task. The inverse task finds the boundary conditions by

minimizing the difference between the measured temperature histories and the temperature

histories computed using the direct heat conduction that uses the tested boundary conditions

and computational model of the problem under solution (a computational model of the

problem must be available to solve the direct heat conduction task). The optimal boundary

conditions minimize the sum square error between the measured temperatures T i* ,m and the

computed temperatures T im where the sum square error function is defined for this case as

SSE=∑i=1

nT

∑m=t

t f

T i* ,mT i

m2 (4.6.4)

90

Problem Solution

where the nT is the number of compared temperature sensors, and f is the number of future

time steps. This equation is used as a criterion function for the modified downhill simplex

optimization method [A9]. One value of the boundary condition is computed for each time

step. This means that the described procedure is repeated for each time step.

The computation starts with searching boundary conditions for the first time step. The

optimization process starts with some initial values of the boundary conditions. During the

optimization, the boundary conditions are found by minimizing the criterion function. The

boundary conditions usually remain constant for f future time steps in the criterion function.

However, you may also use time-dependent functions, such as the linear one. Having found

the optimal boundary conditions for the first time step, the direct heat conduction computation

is performed for one time step and the search for boundary conditions is repeated for the next

time step until the boundary conditions for the whole time history are found.

4.6.3 Determination of Temperature Dependent Materi al Properties

The modified downhill simplex method [A9] can also be used for the determination of

thermal temperature-dependent material properties. The inverse task of this kind is explained

on the determination of thermal material properties of the mold form transient and steady state

measurements of the temperature histories [A5] (see Figures 4.24–4.25).

Computer

Controlled power source

Data acquisition

Measured power

Measured temperatures

Heater

Mold

Figure 4.24 – Scheme of the experimental apparatus.

Measurement description

A special annular mold was made (see Figure 16.10 in Appendix G), similar to that used

for casting. This silica mold consists of two different layers: soft smooth inner layer and rough

outer backup layer. Two extra thin temperature sensors were placed on the surface on the

91

Problem Solution

opposite sides of the cavity in the mold (see Figure 16.11) and two were placed on the outer

surface. Four more temperature sensors were built-in inside the mold as shown in Figure 4.25.

A cylindrical heater was placed inside the cavity of the mold and attached to the controlled

power source. Another thermocouple was placed on the surface of the heater to monitor the

temperature of the heater. All thermocouples and power were measured and information was

sent from the data acquisition system to the personal computer (see Figure 4.24).

At the beginning of the experiment, the mold and the heater were held at uniform room

temperature. Afterwards, the heater was heated up a little bit and the temperature responses

including the measured power for the heater were recorded into the computer. After reaching

steady conditions with the heated heater, the power of the heater was increased. This was

repeated several times to cover the whole investigated temperature range.

Figure 4.25 – Cross-section of the mold in detail.

Thermal conductivity from steady data

Measured temperature histories allow computing thermal temperature dependent

material properties of the silica mold. Knowing the power of the heater under steady

conditions, the conductivity k of both kinds of layers can be solved using analytical equations

q=2 L k1 T aTb

ln r b/ r a(4.6.5)

for backup layers and

q=T4T3

ln r m/ r 4

2 L k2 L

ln r 3/ r m

2 L k1 L(4.6.6)

92

Problem Solution

for the smooth inner layer. The parameter q represents the heat rate generated by the heater.

Parameter L is the length of the mold. The indexes a and b represent any thermocouple T1, T2,

or T3 where the b thermocouple must be closer to the outer surface and different from the a

one. Parameters T and r represent the temperature and distance of the thermocouple from the

center of the mold, respectively. Similar equations can be obtained for the opposite side of the

mold where thermocouples T5, T6, T7, and T8 are placed.

There is some inaccuracy because we have to know the precise outgoing heat flux but

some unknown heat is lost on the top and bottom of the cylindrical mold. In addition, the

relative temperature for the computed conductivity is approximate and to compute the thermal

conductivity k2 we have to know conductivity k1 for the same relative temperature.

Computing simultaneously temperature - dependent thermal conductivity and specific heat

The basic idea for this algorithm is that the computational model must match the

experimental model. In other words, if the computational model matches the experimental

one, the computed temperature histories must be equal to the measured temperature histories.

The indicator SSE that represents the accuracy of the computational model is given by

SSE=∑i=1

8

∑m=1

n

T i* ,mT i

m2 . (4.6.7)

This sum is computed over all eight temperature sensors and over whole temperature

histories.

Two 1D models are used for computational models, for sides A and B. The boundary

conditions are very simple because we have measured all four surface temperatures. So we

know the surface temperatures on both sides for each model. On the other hand, the material

properties of the silica mold are unknown. And these have to be found in order to match the

computed temperature histories to the measured ones.

The temperature dependent material properties can be computed as piecewise linearized

ones. For each conductivity k1, k2, specific heat c1, and c2 several values are optimized. These

are conductivity and specific heat for node temperatures. The right material properties can be

found using an optimization method, e. g. the modified downhill simplex optimization

method [A9]. The optimization method minimizes the criterion function (given by Eq. 4.6.7)

by changing the material properties. When the minimum of the criterion function is reached

the desired material properties are found.

93

Problem Solution

4.6.4 Sensor Calibration

The heat transfer phenomenon is usually measured indirectly. For this purpose,

thermocouples (see Figure 16.5 in Appendix G) are placed inside the body that is under

investigation [A13]. An inverse task has to be used to compute HTC from the measured

temperature history. This inverse task is one of many ill-posed problems. It means that a small

error in the input data results in a big error in the output data. Errors in the input data in

computations are of three different kinds:

• Geometry and dimensions of the computational model

• Material properties of materials used

• Noise in measured temperature histories

To maximize the accuracy of the computed results by the inverse task, an accurate

computational model must be used. The calibration of the computational model is described in

the following. First, some experiments are performed to obtain data for the calibration. Next,

the model is calibrated using an optimization method [A12].

Figure 4.26 – Application of sensor, and its structure in detail.

Sensor description

To measure temperature inside the body, special sensors with built-in K-thermocouples

are used as shown in Figure 4.26. The main body of the sensor is made of stainless austenitic

steel. A hole of 1.1 mm in diameter for a thermocouple is made from the side of the sensor.

The axis of the hole is 1 mm under the investigated surface and is perpendicular to the

94

A

A

A-A

Main body

Gap

Thermocouple wires

Insulation

Shield

Problem Solution

expected heat flux, so that the most important part of the inserted thermocouple lies in one

isotherm. Inside the sensor, a shielded ungrounded K-thermocouple is placed. The gap

between the sensor and the thermocouple is filled with ceramic material that can be exposed

to a high temperature. During measurements that serve for computing HTC, the sensors are

placed in the steel object on which the HTC is investigated.

Calibration experiment

None of the sensors is exactly the same. The position of the junction point inside the

shielded thermocouples differs. Also the hole inside the sensor is a bit bigger than the

thermocouple so that its position may differ. As the thickness of the material in the gap differs,

the heat resistance does, too. These are the main reasons why the calibration experiment has to

be done for each sensor.

The main idea is to perform an experiment we know all about except the heat resistance

between the sensor and built in thermocouple. The sensor at room temperature is exposed to

a high temperature. The measured temperature history must match the computed history in

case the computational model is correct.

DAQ PC

Controlled Power Supply

Cu

Heater

High-conductive Grease

Tested Sensor

Switch

Thermocouples

Flat Thermocouple

Figure 4.27 – Experimental apparatus forsensor calibration.

Main body of the sensor

Heated copper

rod

High-conductive grease

Cross-section of the shielded thermocouple

High-conductive grease

Heated copper

Gap

Thermocouple

Insulation

Shield

Upper part

Lower part

Figure 4.28 – Computational 2D model.

The experimental apparatus (see Figure 4.27 and 16.4)) consists of an arm with

an attached sensor that can be slid down and up. The temperature is measured using four

thermocouples during the calibration process. One thermocouple is in the sensor, two are

inside the copper rod, and the last one measures temperature inside the high conductive

grease.

95

Problem Solution

At the beginning of the experiment the copper rod is heated up to a uniform temperature

and the sensor is held at room temperature. A high conductive grease is applied onto the

surface of the copper rod to ensure a full contact between the sensor and copper rod.

Afterwards, the sensor is stuck to the heated copper rod and the switch indicates the time

when the contact occurred. A 0.05 mm thin pad placed in the grease ensures every time the

same thickness of the grease between the sensor and the heated copper rod.

Computational Model

A 2D axis symmetric model was used as shown in Figure 4.28. The model consists of

the heated copper rod, high conductive grease, and the tested sensor. The model includes the

shielded thermocouple with all its parts. The thermocouple must be taken into account

because the homogeneity of material is disturbed by the inserted thermocouple, and thus the

temperature profile is also disturbed.

Model Calibration

The 2D computational model described in the previous section is modified to match the

computed temperature histories with the measured ones. Supposing the axis of the hole of

1.1 mm in diameter is precisely 1 mm under the investigated surface, the following parameters

can vary:

• Junction point inside the shielded ungrounded K-thermocouple

• Position of the thermocouple inside the hole of 1.1 mm in diameter, and the

thickness of the material in the gap between the sensor and thermocouple

• Material properties (thermal conductivity, specific heat, and mass density) of the

material that fills the gap

Computational experiments have showen that knowing the specific heat and mass

density of the material that fills the gap, the remaining unknown parameters can be eliminated

by setting only two parameters – thermal conductivity of the material that fills the gap in the

upper and lower parts. The dimensions of the model remain, only the material that fills the gap

is divided into the upper and lower part. Both parts can differ in thermal conductivity.

Changing independently the thermal conductivity allows compensation of the position error of

both the junction point and the thermocouple inside the hole. The modified downhill simplex

optimization method [A9] is used for finding the appropriate thermal conductivity of the

material that fills the gap in the upper and lower parts.

96

Problem Solution

Figure 4.29 – Data flow during the optimization process.

To calibrate the computational model, two parameters are optimized – thermal

conductivity of the material in the upper and lower parts of the gap, ku and kl, respectively (see

Figure 4.29). If the computational model matches the real sensor, the computed temperature

history must match the measured one. Therefore, the criterion function for the optimization

method is used as follows

SSE=∑m=1

n

T * ,mTm2 (4.6.8)

where T and T* are the measured and computed temperatures of the thermocouple inside the

sensor, respectively.

4.7 4.7 Data FocusingData Focusing

A comparison of a pressure impact field and heat transfer distribution of a spraying

nozzle on a cooled surface is often required in various applications. The precise description of

a pressure impact field can be mostly done only by measurement. Such a measurement is done

using a pressure sensor of a real finite size. In some cases, the shape of the distribution is very

small or at least very narrow [A12]. Measuring such distributions with a pressure sensor,

which has finite dimensions, results in incorrect data. The pressure is being averaged during

the measurement, and the shape of the obtained spray distribution is larger and the measured

pressure maximum is lower than the real one. The correction of the measured data is also one

of many ill-posed inverse problems. The following section describes how to correct the

measured spray distribution in a frequency domain [A11].

97

Experiment Initial

Conditions

Measured Temperature

History

Computational Model

Computed Temperature

History

ku & kl

Criterion Function

Downhill Simplex optimization method

Problem Solution

DAQ PC Nozzle

Impact area

Pressure Sensor Moving Plate

Pressure

Position

Figure 4.30 – Pressure distribution measurement.

Experimental measurement of pressure distribution

Measuring the pressure distribution (see Figure 16.12 in Appendix G), the nozzle sprays

on a moving plate (see Figure 4.30). This plate is equipped with a pressure sensor (see

Figure 16.13 in Appendix G) that may be of circular or rectangular shape. For a given nozzle

configuration, a pressure is measured as position dependent while the plate with the sensor is

moving under the spraying nozzle.

For nozzles with a very narrow spray spot and for a small distance of a nozzle from the

moving plate, the measured data doesn’t represent a real pressure distribution. The values are

averaged over a „large” pressure sensor. The precision of the measured data depends on the

size ratio of a nozzle spray spot and the sensor. As the ratio becomes smaller the precision of

the measured data is getting worse.

Fourier transform

A physical process can be described either in the time domain or else in the frequency

domain as shown in Eqs. (3.5.3–3.5.4). If m is a function of position x (in meters), M will be

a function of inverse wavelength (cycles per meter). A two-dimensional Fast Fourier

Transformation (FFT) [5] is used to convert both the measured data and pressure sensor to the

frequency domain. The correction of the measured spray distribution is done in the frequency

domain and then an inverse two-dimensional FFT is used to convert the data back to the space

domain. This allows suppressing noise in the measured data that would be extremely

amplified.

98

Problem Solution

Correction of measured data

A large sensor averages real values and one measured value is equal to

m=1 A∬

A

mx , ydx dy (4.7.1)

where A is the surface of the sensor. The whole measured distribution can be described using

the following convolution equation

m∗p=∬mx , y⋅pxX , yYdx dy (4.7.2)

where p is a filter function. This filter function describes how the sensor averages real values.

To obtain a real distribution from a measured one, a convolution equation can also be

used. In this case, the filter function is an inverse function to the sensor filter function. The

convolution, inverse function computation and noise reduction can be done more easily in the

frequency domain than in the space domain. The measured data and sensor filter function are

transformed from the space domain into the frequency domain using FFT.

As we have values transformed into the frequency domain we can easily compute the

inverse sensor filter using

P1f x , f y=1

P f x , f y. (4.7.3)

Having the inverse filter we can do the convolution, Eq (3.5.6), in the frequency domain

using this inverse filter and measured data to obtain a real pressure distribution (still in the

frequency domain). In our case, the convolution is described by

M f x , f y= M f x , f y⋅P1 f x , f y (4.7.4)

where M are the measured data, P1 is the inverse sensor function and M represents

a sharpened data. Transforming sharpened data from the frequency domain into the space

domain using inverse FFT we obtain a pressure distribution which should be very close to the

real pressure distribution. However, some noise is visible in the sharpened data (small waves).

Noise reduction and aliasing phenomenon

Noise is most significant at high frequencies and because we are working in the

frequency domain, noise can be suppressed. The sensor function consists of the main

frequency spectrum (in the middle) and higher harmonic frequencies. Cutting off the higher

99

Problem Solution

harmonic frequencies and making them equal to zero also in the inverse sensor filter, we get

a cut inverse sensor filter. Using this filter for convolution (Eq. 4.7.4) instead of the inverse

filter and transforming the sharpened data using inverse FFT, we get sharpened measured data

with suppressed noise. However, the computed maximum is still lower than the real one. This

is due to the aliasing effect that is described in the previous Data Filtration chapter.

The sampled continuous function that is not bandwidth limited to less than the Nyquist

critical frequency results in an incorrect frequency spectrum. Any frequency component that

lies outside of the range f c f f c is spuriously moved into that range (this phenomenon

is called aliasing). This effect is more significant for the sensor function because of sharp

edges of the sensor.

The aliasing can be avoided by using a low pass filter. The sensor function passes

through the Gaussian low pass filter in the space domain. Transforming this function into the

frequency domain, we get high frequencies equal to zero. This means there is no aliasing

effect. Using the cut smooth inverse sensor filter for convolution and transforming the

sharpened data using inverse FFT, we get sharpened measured data with a maximum which is

very close to the real maximum.

100

Solution Presentation and Analysis

5 5 SSOLUTIONOLUTION P PRESENTATIONRESENTATION ANDAND A ANALYSISNALYSIS

The iteration algorithm of the non-linear heat conduction problem is tested first to prove

the correctness of the iteration error estimation algorithm within one time step for slowly

converging models. Next, the enhanced method of the direct heat conduction is presented to

show its capabilities to solve 2D heat conduction with temperature dependent material

properties, to treat a phase change, and heat generation. Further, the error estimation schemes

are tested to prove their rightness. The estimation of the mesh discretization error is used in

the presented example of mesh optimization. Having prepared the method for direct heat

conduction, capabilities of the enhanced data filtration method are presented.

First, the classical sequential Beck’s approach is used with filtered data on the input.

Next, the usage of the modified multidimensional downhill simplex optimization method is

presented in the solution of several inverse heat conduction problems and compared with the

usage of the artificial neural network in one of them. Real sensor calibration is shown and

accuracy of its model is tested. The usage of the optimization method is also presented in

multidimensional IHCP and in identification of temperature dependent material properties.

The advantage of the usage of the approximation function is compared with the classical

sequential Beck’s approach. Finally, correction of an artificial and a real measured pressure

impact field of a spraying nozzle is presented.

5.1 5.1 Direct Solutions of Heat ConductionDirect Solutions of Heat Conduction

5.1.1 Test of Iteration Algorithm of Non-linear Hea t Conduction Problem

Solution of one-dimensional models and multi-dimensional models with almost uniform

material properties converges very fast within one time step. In most cases three iterations are

enough to prove the high accuracy of the computed values using Eqs. (4.2.7–4.2.9) and for the

simple error estimation Eq. (4.2.6) can used. However, when materials with totally different

heat conductivities are used, the number of required iterations may rapidly increase for

multidimensional models.

101


Two-dimensional model with annular copper

To show slow convergence, a two-dimensional model was tested. A square (4 ×4 mm)

stainless steel body with an inbuilt annular made of highly conductive copper filled by Al2O3

was cooled down only on one side from the initial temperature 300°C. The heat transfer

coefficient was 1000 W/m2.K and the surrounding temperature was 0°C. The other sides were

insulated. The iteration convergence was tested after 100 uniform time steps at a time of 0.1 s.

The temperature distribution is shown in Figure 5.1. The stainless steel is represented by gray

color, the copper by dark gold color, and Al2O3 by aqua color inside the annular.

Figure 5.1 – Temperature field at t = 0.1 s.

Figure 5.2 – Error after the third iteration.Figure 5.3 – Error after the fourth iteration.

Figure 5.2 and 5.3 show the error distribution after the third and fourth iteration,

respectively (zero error is assumed after an infinitive or very high number of iterations). The

error was computed for each node of the 2D model after each iteration and the maximum error

evolution in time is shown in Figure 5.4. The chart also shows the estimated error range

computed using Eqs. (4.2.7–4.2.9) and the simple error estimation computed using Eq. (4.2.6).

It is obvious that the simple error estimation cannot be used for slowly converging models.

102


Figure 5.4 – Maximum error and its estimation after each iteration.

Although 44 and 145 iterations were required to reach the 0.0001 K accuracy and final

solution, respectively, the used line-by-line method with tridiagonal systems of equations is

still much faster than the full system of equations solved by LU decomposition. The full

system has nx×ny=n=82 ×82=6 724 equations and the solution requires 1/3n3 operations.

Moreover, the system is nonlinear. Thus, the solution should be iterated at least three times to

know the accuracy of the computed solution. The total number of operations is

304 006 671 424 at least. In the case of the line-by-line method, the 6 724 equations have to

be also solved. However, these equations are arranged in nxny=82 82=164 sets of

tridiagonal equations. The solution of these nonlinear equations takes only

2 ×145 ×6 724=1 949 960 operations, which is 155 904 times faster than LU

decomposition. Also the memory requirements are n times smaller for the line-by-line method

than for LU decomposition.

5.1.2 Implementation of Phase Change and Heat Gener ation

The purpose of this computation is to show the ability to deal with the implementation

of phase change [A10] and heat generation in a computational model of the heat conduction

problem.

Two-dimensional model of a heater unit

Let us imagine a heater unit as shown in Figure 5.5. The unit consists of a closed tube

placed in an insulation. A heater is situated in the center of the tube and the rest of the space in

103

293.94

293.96

293.98

294.00

294.02

294.04

294.06

294.08

0 10 20 30 40 50Iteration within one time step

Tem

pera

ture

[°C

]Temperature

Upper range

Lower range

Simple upper range

Simple lower range


the tube is filled with a medium that will be melted. The problem is considered in micro

gravity to avoid heat conduction due to the convection of the melted medium. The heater unit

is at an initial temperature of 0°C. The phase change of the melting medium occurs at 100°C.

At the beginning, the heater is switched on. The adiabatic surface conditions are used between

the insulation and unit. The model computes only one quarter of the problem because of the

symmetry. Thus, the adiabatic surface conditions are also used on the symmetry lines.

Shield

HeaterMelted medium

Insulation

Heater

3D Scheme Computational Model

Sensor

Figure 5.5 – Heater unit.

Figure 5.6 – Temperature history for themelted medium 15 mm from the center.

Figure 5.7 – Temperature profile of the heaterunit model at time = 100 s.

During the computational experiment, a temperature history was recorded for the melted

medium at a distance of 15 mm from the center of the rotation symmetry. The recorded history

is shown in Figure 5.6. First, the whole system was heated and the temperature was smoothly

increasing. As the temperature of the melted medium reached the melting temperature, a first

edge appeared at a time of 60 s. The medium was melting and the increase in temperature was

104

0 20 40 60 80 100 120 140 160Time [s]

0

50

100

150

200

Tem

pera

ture

[°C

]


slower due to the latent heat. Figure 5.7 shows the temperature distribution at a time of 100 s

after the start of the heating. The sharp edge in the temperature history represents the

liquid / solid interface. When all medium was melted, which occurred at a time of 130 s after

the start of the heating, the temperature increased more rapidly.

A phase change occurs also in solid states, e. g. when carbon steel is heated or cooled.

Tests of a phase change and heat generation are important for rolling applications when

carbon steel is cooled down from high temperatures and where heat is generated during

deformation of rolled products.

5.1.3 Verification of Discretization Error Estimati on

Boundary Conditions and Time Discretization Error

To verify the method for the estimation of the time discretization error using Eq. (4.3.7)

a one-dimensional model was used. The model was assumed to be made of stainless steel. It

was 10 mm long for all tests except for the one with a constant heat transfer coefficient where

its length was 100 mm. One side of the model was assumed to be insulated while the other

one exposed to various boundary conditions. The starting temperature of the model was 0°C.

A constant heat flux of 100 000 W/m2 was applied to one boundary. The step duration

was 1 s and 10 s for the first and second experiment, respectively. The temperature profiles

computed using a very fine time step are represented in Figures 5.8–5.9 by the solid line with

crosses. These temperature profiles were compared with those computed using a single time

step of 1 s and 10 s duration for the first and second experiments, respectively. The absolute

values of computed differences are represented in Figures 5.8–5.9 by the thick solid black

line. To verify the use of Eq. (4.3.7) for the estimation of the time discretization error, the

temperature profiles were also computed using half time steps. The estimated errors were

computed for =1 ; 1.5; 2 and absolute values are shown in Figures 5.8–5.9. It is obvious

that the estimated errors computed using =2 and 1.5 well cover the error caused by time

discretization for the half time step. The only place where the computed error is higher than

the estimated one is where the error is very low. However, in most cases the important value is

the maximum error, which is correct.

105


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 1 2 3 4 5 6 7 8 9 10Position [mm]

Te

mp

era

ture

err

or

[°C]

0

2

4

6

8

10

12

14

16

Tem

per

atu

re [°

C]

Error for normal time step

Error for half time step

Estimated error

1.5*Estimated error

2.0*Estimated error

Temperature

Figure 5.8 – Temperature profile for a constant heat flux of 100 000 W/m2 after 1 s.

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6 7 8 9 10Position [mm]

Te

mp

era

ture

err

or

[°C]

0

10

20

30

40

50

60

Tem

per

atu

re [°

C]



Estimated error

1.5*Estimated error

2.0*Estimated error

Temperature

Figure 5.9 – Temperature profile for a constant heat flux of 100 000 W/m2 after 10 s.

The estimation algorithm was also verified for time dependent boundary conditions. The

sinusoid, triangular heat fluxes, and constant heat transfer coefficient were used as shown in

Figures 5.10, 5.12, and 5.14. The charts shows real boundary conditions and discretized ones

for normal time and half time steps. The computed temperature profiles, time discretization

errors, and estimated errors are shown in Figures 5.11, 5.13, and 5.15. It is obvious that the

estimated errors computed using =2 and 1.5 also well cover the error caused by the time

discretization as in the previous computational experiments. Moreover, =1 can be used for

the constant heat transfer coefficient.

The computational experiments have proved that Eq. (4.3.7) can be used for error

estimation caused by time discretization. The only place for which the equation should not be

used are those where the error is small (e.g. 10%) in comparison with the maximum one.

106


-150 000

-100 000

-50 000

0

50 000

100 000

150 000

0 2 4 6 8 10 12 14 16 18 20Time [mm]

Hea

t flu

x [W

/m²]

Real

Normal time step

Half time step

Figure 5.10 – Time dependent heat flux q = 100 000*sin(t).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6 7 8 9 10Position [mm]

Te

mp

era

ture

err

or

[°C]

-1

0

1

2

3

4

5

6

7

8

Tem

per

atu

re [°

C]



Estimated error

1.5*Estimated error

2.0*Estimated error

Temperature

Figure 5.11 – Temperature profile for heat flux q = 100 000*sin(t) after 20 s.

0

20 000

40 000

60 000

80 000

100 000

120 000

0 5 10 15 20 25 30 35 40Time [mm]

Hea

t flu

x [W

/m²]

Real

Normal time step

Half time step

Figure 5.12 – Time dependent triangular heat flux.

107


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1 2 3 4 5 6 7 8 9 10Position [mm]

Te

mp

era

ture

err

or

[°C]

45

46

47

48

49

50

51

52

53

54

55

Tem

per

atu

re [°

C]



Estimated error

1.5*Estimated error

2.0*Estimated error

Temperature

Figure 5.13 – Temperature profile for triangular heat flux after 40 s.

0

20 000

40 000

60 000

80 000

100 000

120 000

0 100 200 300 400 500 600 700 800 900 1000Time [mm]

Hea

t flu

x [W

/m²]

Real

Normal time step

Half time step

Figure 5.14 – Heat flux for constant heat transfer coefficient h = 1000 W/m2.K.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 10 20 30 40 50 60 70 80 90 100Position [mm]

Te

mp

era

ture

err

or

[°C]

0

10

20

30

40

50

60

70

80

90

100T

emp

erat

ure

[°C

]



Estimated error

1.5*Estimated error

2.0*Estimated error

Temperature

Figure 5.15 – Temperature profile for constant heat transfer coefficient h = 1000 W/m2.Kafter 1000 s.

108


Mesh Discretization Error

Two one-dimensional models were used to verify the method for error estimation caused

by space discretization. However, the method can also be used for multi-dimensional models.

The first model was assumed to be made of stainless steel and the second one was a sandwich

made of two pieces of stainless steel and asbestos between them. The thicknesses of steel were

30 mm and 50 mm, and that of asbestos was 20mm. The starting temperature was 0°C and the

model was exposed to the constant heat flux 100 000 W/m2 on the side with 300 mm steel.

The temperature profiles were computed using a very fine mesh (20000 nodes) and the

tested rough one with 20 nodes. The computed results are shown in Figures 5.16–5.20. The

temperature profile computed using 20 nodes was compared with the results computed using

a very fine mesh. The differences are marked as a real error in the Figures. Next, mesh

discretization error estimation was computed using Eq. (4.3.5) from the mesh of 20 nodes.

These values are marked as the max. estimated error. The real errors are lower than the

estimated maximum values except for very small errors in heterogeneous material and long

time step of 1000 s as can be seen in Figure 5.20. Also the magnitude of the real errors is

comparable with the estimated maximum values.

Eq. (4.3.5) should be used for the adaptive meshing to keep the accuracy of the

computational model at a reasonable level. This is mainly important for multi-dimensional

models where the number of nodes rapidly increases and the computation time is much longer.

0

0.5

1

1.5

2

2.5

3

0 10 20 30 40 50 60 70 80 90 100Position [mm]

Tem

pera

ture

err

or [°

C]

0

5

10

15

20

25

30

35

40

45T

empe

ratu

re [°

C]

Real error

Max. estimated error

Temperature

Figure 5.16 – Homogeneous model of steel after 10 s.

109


0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 10 20 30 40 50 60 70 80 90 100Position [mm]

Tem

pera

ture

err

or [°

C]

0

20

40

60

80

100

120

140

Tem

pera

ture

[°C

]

Real error


Temperature


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 10 20 30 40 50 60 70 80 90 100Position [mm]

Tem

pera

ture

err

or [°

C]

0

50

100

150

200

250

300

350

400

450

500

Tem

pera

ture

[°C

]

Real error


Temperature


0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 10 20 30 40 50 60 70 80 90 100Position [mm]

Tem

pera

ture

err

or [°

C]

0

20

40

60

80

100

120

140

160

Tem

pera

ture

[°C

]

Real error


Temperature

Figure 5.19 – Model of steel and asbestos after 100 s.

110


0

2

4

6

8

10

12

0 10 20 30 40 50 60 70 80 90 100Position [mm]

Tem

pera

ture

err

or [°

C]

0

100

200

300

400

500

600

700

800

900

1000

Tem

pera

ture

[°C

]

Real error


Temperature

Figure 5.20 – Model of steel and asbestos after 1000 s.

5.2 5.2 Filtration of Real Measured DataFiltration of Real Measured Data

Real measured data were used to compare filtration methods. Measuring frequencies of

100 Hz and 300 Hz were used for recording temperature histories. These measured data are

shown in Figures 5.21–5.24 using crosses. The filtration was done in the time domain as well

as in the frequency domain.

In the frequency domain, the low pass filter was used. The data were converted from the

time domain to the frequency domain using the Fast Fourier transform, the high frequencies

were set to zero and the data were converted back into the time domain using using the

inverse Fast Fourier transform.

In the time domain, the proposed multi-level filtration (described in the previous

Multi-level Filtration of Measured Data chapter) and a convolution using Gaussian function

were employed. For convolution, Eq. (4.5.1) was used as a filter where the parameter shown

in Figures 5.21–5.24 is s = k / 3. Two parameters for convolution were tested – a stronger one

(higher value) and a less stronger (lower value).

The filtered data for a frequency of 300 Hz are shown in Figures 5.21–5.22 and for

100 Hz in Figures 5.23–5.24. The data filtered using low pass filters are oscillating for both

strong and less stronger filtration. The oscillation is the most significant at the beginning and

at the end of the temperature history but the filtered data are oscillating over the whole range.

Such data are very unsuitable for inverse algorithms. Much better results were obtained using

the Gaussian filter. However, the data filter using a less stronger filter are oscillating in places

111


where the slope is mild, see Figure 5.24. The stronger filter works better in these places but it

fails in places where the temperature history changes the slope rapidly as shown in

Figures 5.22–5.23. The proposed multi-level filtration algorithm works well in mild slopes as

well as in places where the temperature changes the slope rapidly.

900

905

910

915

920

925

0 0.5 1 1.5 2 2.5 3Time [s]

Te

mp

era

ture

[°C

]

Measured

Low pass f=50

Low pass f=200

Multi-level filtration

Gaussian s=0.03

Gaussian s=0.06

Figure 5.21 – Filtration of data measured at 300 Hz.

917

917.5

918

918.5

919

919.5

920

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2Time [s]

Te

mp

era

ture

[°C

]

Measured

Low pass f=50

Low pass f=200


Gaussian s=0.03

Gaussian s=0.06

Figure 5.22 – Filtration of data measured at 300 Hz in detail.

112


857

857.5

858

858.5

859

859.5

860

11.3 11.35 11.4 11.45 11.5 11.55 11.6Time [s]

Te

mp

era

ture

[°C

]

Measured

Low pass f=1000


Gaussian s=0.01

Gaussian s=0.02

Figure 5.23 – Filtration of data measured at 100 Hz – sudden change of slope.

856.5

857

857.5

858

858.5

859

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10

Time [s]

Te

mp

era

ture

[°C

]

Measured

Low pass f=1000


Gaussian s=0.01

Gaussian s=0.02

Figure 5.24 – Filtration of data measured at 100 Hz – mild slope.

113


5.3 5.3 Inverse Heat Conduction ProblemsInverse Heat Conduction Problems

First, determination of the single boundary condition of the unsteady problem is

presented using several approaches. An optimized mesh and filtered data are used for classical

Beck's sequential approach. Alternative approaches, a modified downhill simplex

optimization method and feed-forward neural network, are used for single boundary

conditions determination. Next, the optimization method is used in IHCPs where two-

dimensional boundary conditions are computed, temperature dependent material properties are

determined, and the computational model is calibrated.

5.3.1 Boundary Conditions Computation

In this chapter, the classical Beck's sequential approach, modified downhill simplex

optimization method and feed-forward neural network are used for the solution of IHCPs

where single boundary conditions are determined for unsteady problems.

Beck's approach using filtered data

The cooling intensity of a water nozzle was studied on a linear test bench (see

Figure 5.25 and Figure 16.1 in Appendix G). During the experiment a heated stainless steel

plate was moving repeatedly under a spraying nozzle, and the measured temperature inside the

steel plate was recorded using measuring frequencies of 100 Hz and 300 Hz (see

Figures 5.21–5.24). The internal structure of the temperature sensor (see Figure 16.3) in the

steel plate is shown in Figure 4.26 and Figure 16.5 in Appendix G.

Figure 5.25 – Principal scheme of the linear test bench 1. cooling medium supply, 2.pressure gauge, 3. nozzle, 4. moving deflector, 5. manifold, 6. test plate, 7. moving trolley,

8. data logger, 9. roller, 10. electric motor, 11. hauling steel wire rope, 12. girder.

114


Figure 5.26 – Temperature profile of the optimized 2D model after 1s with applied heat fluxq = 100 000 W/m2 on a surface.

Before the IHCP was solved, the mesh of the 2D computational model (see Figure 4.28)

was optimized (see Figure 5.26). The estimated discretization error fields for heat conduction

in X and Y directions are shown in Figure 5.27.

The time dependent surface heat flux was computed from the measured temperature

histories using classical Beck’s sequential approach. The input data (temperature histories)

were used in several forms: raw data, data filtered using the Gaussian filter, and data filtered

using proposed multi-level filtration. The computed boundary conditions are shown in

Figures 5.28 and 5.29 for measuring frequencies of 100 Hz and 300 Hz, respectively. The

results computed using raw measured data are very noisy. The noise and measuring resolution

step were strongly magnified by the IHCP algorithm. The situation is getting better for data

filtered using the Gaussian filter in the time domain. The improvement is more significant for

Error [°C]

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008184

Figure 5.27 – Estimated errors in X (left chart) and Y (right chart) directions caused byspace discretization after mesh optimization.

115


a measuring frequency of 300 Hz. However, the maximum peek is lower in comparison with

the raw and multi-level filtered data. In case of the 100 Hz, the problem with temperature

resolution step (see Figure 5.24) was not suppressed. Thus the computed results are still

oscillating for low heat fluxes (Figure 5.28). The results computed from the data filtered using

multi-level filtration are very good for low heat fluxes as well as for peaks. The oscillation of

the heat flux was well suppressed for low values and the peak was not smoothed. The

maximum and width of the peak is comparable with that computed using raw data.

Solution of the IHCP can be well improved when a sophisticated filtration is used for

the input data. An improvement is described in next chapter, where an alternative optimization

method instead of sequential Beck's approach was used for IHCP.

-500000

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

0 5 10 15 20 25 30Time [s]

Hea

t flu

x [W

/m²]

Measured

Gaussian s=0.01


-500000

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

7.5 7.7 7.9 8.1 8.3 8.5 8.7 8.9 9.1 9.3 9.5Time [s]

Hea

t flu

x [W

/m²]

Measured

Gaussian s=0.01


Figure 5.28 – Heat flux history computed using 8 forward time steps from data measured at 100 Hz.

116


-100000

0

100000

200000

300000

400000

500000

600000

700000

800000

0 2 4 6 8 10 12Time [s]

Hea

t flu

x [W

/m²]

Measured

Gaussian s=0.03


200000

300000

400000

500000

600000

700000

800000

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9Time [s]

He

at f

lux

[W/m

²]

Measured

Gaussian s=0.03


-100000

-50000

0

50000

100000

150000

200000

5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7Time [s]

Hea

t flu

x [W

/m²]

Measured

Gaussian s=0.03


Figure 5.29 – Heat flux history computed using 25 forward time steps from data measured at 300 Hz.

117


Modified downhill simplex optimization method

The approach uses classical Beck’s sequential method in combination with a modified

downhill simplex optimization method. The results computed using Beck’s method with

several forward steps are smooth, unfortunately, even if the physical reality dictates sharp

peaks in the surface heat flux or heat transfer coefficient. The combined approach is useful for

very steep changes in the measured temperature and for the demand on precise results. The

hydraulic descaling process is a typical example of the process with rapid changes in boundary

conditions. Hydraulic descaling uses high energetic sprays with narrow spray spots to remove

oxides from the steel surface. The typical pressures used reach 500 bar, the surface is usually

at high temperature (about 1000°C) and runs under sprays at a velocity of several meters per

second.

Data from the experiment done on linear test bench, which was described in the

previous chapter (see Figure 5.25), were used as an input for the combined inverse algorithm

that is described in the Computation of Boundary Conditions chapter. For time intervals where

the boundary conditions were almost constant, the fast and stable sequential Beck’s algorithm

was used with a large stabilization parameter. Within a certain time interval, at which the

boundary conditions were abruptly changing, the time dependent boundary conditions were

approximated by the function for the water nozzle cooling application shown in Figure 4.19.

For the time when the HTC peak occurred, Eq. (4.6.1) was used as approximation functions.

The optimized values were the parameters of the approximation function, the δ, γ, µ, σL,

and σR parameters for Eq. (4.6.1). The downhill simplex method minimized the criterion

function defined by Eq. (4.6.2) by setting the parameters of the approximation function.

time [s]

HT

C [W

/m˛.

K]

HTC inv. 5

HTC inv. 15

HTC opt.

Figure 5.30 – Interpolation of HTC history by function around HTC maximum.

118


The real measured data were taken to compare classical Beck’s sequential approach with

the combined approach. Three different computations were made (see Figure 5.30). The first

and second computation used classical Beck’s approach with 5 and 15 forward steps,

respectively. The third one combined the classical Beck’s approach with an optimization

method. The classical approach with 5 forward steps matched the measured temperature

history almost perfectly. The RMS error, defined as:

RMS= 1 n∑m=1

n

T* mT computedm 2 (5.3.1)

where T is a measured or computed temperature and n is a number of values in the

temperature history, was only 0.093 K and the maximum error was 0.25 K. But the computed

HTC is very noisy because the HTC tries to follow all noise in the measured temperature

history.

The computation with 15 forward steps shows that the noise can be quite well

suppressed. But the computed temperature history does not follow the measured one very

well. The RMS error increased to 0.273 K and the maximum error was 0.96 K. High numbers

of forward steps limit the maximum slope and also the maximum value of the computed HTC.

The shape of the HTC is also deformed and the maximum is moved to the right on the time axis.

The combined method removed the noise in the computed HTC and matched very well

the measured temperature history. The RMS error was 0.203 K, which is very close to the

noise in the measured data. The maximum error was only 0.38 K, which is much lower than in

the case of the classical approach with 15 forward steps. The combination of two numerical

methods allows an increase in precision of inverse calculation. The new investigative

approach does not have the negative impact on the stability of the inverse task.

Feed-forward neural network

As a substitution for the modified downhill simplex optimization method the

feed-forward neural network was used to solve IHCP described in the previous chapter, but

only for the first peak. The training data were prepared using computational simulation of the

experiments for several different parameters of the approximation function, and the computed

temperature histories together with the parameters represent the requested training data.

A computation experiment has shown that it is enough to perform three simulations for each

parameter.

119


The feed-forward neural network (see Figure 4.22) was trained using fast

back-propagation (Levenberg-Marquardt functions). The γ and µ parameters were not

computed using the neural network but the position µ was taken from position information,

and γ was computed using Beck’s approach from the data far from the peak. The neural

network reached the sum-squared error goal (0.002) usually in less than 50 training epochs.

Computation using the learned neural networks has proved that not all of them can be used.

Some of them returned confused results. A computation experiment has shown that the

learned neural networks should fulfill the condition that the maximal absolute value of the

element of the matrixes W 1, b1, W 2, andb2 is less than 1.7 (Eq. 5.3.2). The neural network

with the lowest maximum, within the group with equal sum-squared error goals, represented

the best one. In that case, the maximal value was 1.19.

maxW 1, max,W 2,max,b1, max,b2,max1.7

W 1,max=maxm ,i∣wm ,i∣; W 1=wm , i for m=1,..n ; i=1,..3

W 2, max=maxi , j∣wi , j∣; W 2=wi , j for i=1,..3; j=1,..3

b1,max=maxi∣bi∣; b1=bi for i=1,..3

b2,max=maxi∣bi∣; b2=bi for i=1,..3. (5.3.2)

Training of the neural network is a quite time consuming operation and the obtained

results are not very accurate. The trained network often depends on the random initial

matrixes W 1, b1, W 2, andb2 . The computed results must be verified because some networks

give confused results for data not included in the training set.

5.3.2 Calibration of Computational Model of Measuri ng Sensor

In experiments described in the previous chapters, a special sensor (see Figure 4.26 and

Figure 16.3 in Appendix G) was used for measuring temperature inside the steel plate. As

mentioned in the Sensor Calibration chapter, the sensor geometry may vary and the

methodology of the computational model calibration was described in that chapter.

During the experiment, the sensor held at room temperature was exposed to a high

temperature (see Figure 16.4 for experimental apparatus). The sensor was stuck to the heated

120


copper rod. The recorded temperature histories are shown in Figure 5.31. The sensor was held

at room temperature and the copper was heated to a uniform temperature at the beginning. As

the sensor was stuck to the heated copper rod, the temperature of the high-conductive grease,

which was spread on the copper rod, suddenly dropped down and the sensor temperature

started to increase.

20

30

40

50

60

70

80

90

0 1 2 3 4 5t [s]

T [°

C]

Sensor

Copper1

Copper2

Grease

Figure 5.31 – Measured temperature histories during the calibration experiment.

After model calibration, three experiments, which differ in the thickness of the grease

between the sensor and heated copper, were made using the calibrated sensor. The measured

temperature histories and that computed using the calibrated model are almost identical (see

Figure 5.32). Perfectly overlapping computed and measured temperature histories have

confirmed the correctness of the calibrated model.

20

25

30

35

40

45

50

55

0 0.5 1 1.5 2t [s]

T [°

C]

Inceasing grease thickness

Measured Computed

Figure 5.32 – Perfectly overlapping measured and computed temperature histories for threedifferent thicknesses of high conductive grease and the same computational model of the

sensor (0.05 mm, 0.11 mm, and 0.17 mm)

121


Two computational experiments were made to show the importance of the usage of the

calibrated model (see Figure 4.28). The measured temperature history inside the sensor

(Tmeasured) and the computed temperature history with the calibrated model (Ttherm.) are shown in

Figure 5.33. Both curves are almost identical. Moreover, this figure shows the computed

temperature history for the model with no internal structure – homogeneous steel (Tsteel). These

values were computed using the same boundary conditions as the ones computed with the

calibrated model. It is obvious that the response of such a sensor would be somewhat faster. In

addition, HTC were computed (see Figure 5.33) on the heated side (see Figure 4.28) of the

sensor during the experiment. The HTC were computed using both the calibrated model and

the homogeneous steel model, HTCtherm. and HTCsteel, respectively, from real measured

temperature history. The heated copper and grease were supposed as the surrounding medium.

The ambient temperature was the temperature of the copper. It is obvious that the computation

with the homogeneous steel model failed at the beginning because there is no reason why

HTC should be the lowest. HTC must be the highest at the beginning because the temperature

gradient in the heated copper rod is the lowest. Therefore, the heated copper rod is the most

intensive source of heat. As the temperature gradient increases in the heated copper, the HTC

decreases because the colder part of the copper is becoming a temperature resistor through

which the heat must pass.

20

25

30

35

40

45

50

55

60

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

t [s]

T [°

C]

0

2000

4000

6000

8000

10000

12000

14000

16000

HT

C [W

/m².

K]

T measured

T therm.

T steel

HTC therm.

HTC steel

Figure 5.33 – Comparison of the results computed using two models – one made ofhomogeneous steel and a calibrated one with a built in thermocouple.

122


5.3.3 Comparison of Two-dimensional Boundary Condit ions Computations

using An Optimization Method and Sequential Beck’s Approach

In this chapter, two methods are compared to obtain both the thermal contact resistance

(TCR) between an alloy casting and the silica mold, and the heat transfer coefficient (HTC)

between the mold and a bath that was used to control the speed of the solidification process

(see Figure 16.7 in Appendix G). The modified downhill simplex optimization method and

Beck’s approach were used in the inverse task. The experiment has already been described in

the Computation of 2D Boundary Condition chapter.

The 1D computational model consisted of an alloy, silica mold, and gap between the

alloy and the mold. Thus, the computational model, which can handle the phase change, was

used. The process was considered as purely conductive (in micro gravity), and as one object

with spatially and temperature dependent material properties.

Al mold bath

T1 T T3

Spa

ce

2 Too

Figure 5.34 – Computational model of the mold casting experimental apparatus.

The modified downhill simplex optimization method searches for the 2D space (TCR

and HTC) and for a minimum SSE (Eq. 4.6.4). On the other hand, sequential Beck's approach

makes use of linear estimation to compute TCR and HTC. Because of the nonlinear problem,

several iterations are needed to get a converged result.

To compare downhill simplex and Beck's approach, the direct computation was made

first (see Figure 5.35). This direct computation simulated the real experimental measurement

of A356 alloy solidification. Temperature histories for all 3 thermocouples were recorded

during this computation. The starting temperature of the alloy was 700°C, the starting

temperature of the silica mold was 300°C and the bath temperature was 300°C. The

temperature histories (see Figure 5.36) were recorded for 5 minutes while the solidification

123


process lasted the first 2 minutes. The sharp rise of HTC indicates that the mold was

immersed into the molten bath 20 seconds after the start of the recording.

Figure 5.35 – Temperature profile of model consisting of alloy and mold during casting.

After direct computation, the inverse task was made using a modified downhill simplex

and sequential Beck's approach. These two approaches are compared in Tables 5.1 and 5.2

where the time steps are 1 s and 0.1 s, respectively. A minimum of the required forward steps

is the number when the inverse tasks give us realistic results. The RMS error is defined as

RMSHTC= 1 2

n∑i=1

2

∑m=1

n

HTCi* mHTCi

m2 (5.3.3)

where HTCi* m are the original heat transfer coefficients on both sides of the mold in the mth

time step that was used in direct computation, and HTCim are the heat transfer coefficients

computed using the inverse task. The Downhill Simplex optimization method was faster and

gives better results than Beck’s approach. This is because the problem under solution is

nonlinear, and Beck’s approach uses linear approximation. Comparing the results for time

steps 1 s and 0.1 s, it is obvious that the shorter time step gives better results. This is because

there is a fast change in boundary conditions when the mold is immersed into the tin bath. The

reason for why a shorter forward time is necessary for a smaller time step is that there are still

124


more forward time steps, and the rounding errors of the numerical part with several iterations

are averaged.

Figure 5.36 – Simulated temperature histories during A356 alloy solidification .

Table 5.1 – Comparison of downhill simplex and Beck’s approach for the 1 s time step.

Downhill simplex Beck’s approach

Minimum of required forward steps (MFS) 2 7

Computation time using MFS 15 s 57 s

RMS error of HTC using MFS 31 W/m2.K 123 W/m2.K

Computation time using 10 forward steps 67 s 123 s

RMS error of HTC using 10 forward steps 161 W/m2.K 180 W/m2.K

Table 5.2 – Comparison of downhill simplex and Beck’s approach for the 0.1 s time step.

Downhill simplex Beck’s approach

Minimum of required forward steps (MFS) 14 40

Computation time using MFS 20 s 324 s

RMS error of HTC using MFS 8.2 W/m2.K 21.6 W/m2.K

Computation time using 40 forward steps 58 s 324 s

RMS error of HTC using 40 forward steps 20.3 W/m2.K 21.6 W/m2.K

125

300

400

500

600

700

0 50 100 150 200 250 300time [s]

T [°

C]

0

500

1000

1500

2000

2500

3000

[W/m

².K

]

T1

T2

T3

TCR

HTC


5.3.4 Usage of Optimization Method for Determinatio n of Temperature

Dependent Material Properties

An experimental method together with an optimization method was used to obtain

thermal material properties of the mold described in the previous chapter. The mold, which

was made of fused silica, is usually used for casting aluminum alloys. The mold consisted of

several layers, where the first inner one had different properties.

The experiment described in the Determination of Temperature Dependent Material

Properties chapter started at uniform room temperature (see Figure 16.10 in Appendix G).

Afterwards, the heater was heated up a little bit and the temperature responses including the

measured power for the heater were recorded. After reaching steady conditions, the power of

the heater was increased. This was repeated 15 times. The maximum temperature attained

inside the mold was 1080 K. An example of the recorded temperature history is shown in

Figure 5.37.

The mass density was measured at room temperature because the dependence on

temperature covers the specific heat as the density and specific heat appear together as

a product in equations. A small block of mold was prepared. The average dimensions of this

block were 9.99*42.88*35.18 mm. It weighted 22.12 g, and therefore the computed mass

density of the backup layer of the mold was 1 468 kg/m3.

For both kinds of layers (noticeable in Figure 16.9), temperature-dependent thermal

conductivity and specific heat were computed using measured temperature histories. Two

600

650

700

750

800

0 20 40 60 80 100t [s]

T [

K]

0

500

1000

1500

2000

2500

P [W

]

T1

T2

T3

T4

Power

Steady Unsteady conditions

Figure 5.37 – Measured temperature histories and power for steady andunsteady conditions.

126


k1 = -1.184e-11*T4 + 2.957e-08*T3 - 2.604e-05*T2 + 9.803e-03*T - 6.626E-01

0.00

0.20

0.40

0.60

0.80

1.00

300 400 500 600 700 800 900 1000T [K]

k1 [W

/m.K

]

Figure 5.38 – Approximate conductivity of the silica mold computed usingan eligible heat source for the steady conditions.

approaches were used. One used measurement only from steady state conditions and was able

to compute only thermal conductivity. The obtained results are shown in Figure 5.38. The

other computed simultaneously temperature-dependent thermal conductivity and specific heat

using the modified downhill simplex method. The temperature-dependent material properties

were computed as piecewise linearized ones. For each conductivity, k1 and k2, specific heat,

c1 and c2, seven temperature values (300 K, 400 K, 500 K, 600 K, 700 K, 800 K, and 900 K)

were optimized at the beginning. It was altogether 28 optimized parameters. After finding the

first optimum, some parameters were omitted because of the shape of the function and

because of insensitivity of some parameters to the result of the criterion function.

0

0.5

1

1.5

2

300 400 500 600 700 800 900

T [K]

k [W

/m.K

]

0

200

400

600

800

1000

1200

1400

Cp

[J/k

g.K

] k1

k2

Cp1

Cp2

Figure 5.39 – Temperature-dependent material properties of the silicamold computed using the Downhill Simplex optimization method.

127


Having reduced the number of parameters, four values for k1, four values for k2, and two

values for c1 were computed. It has been found that the measured data do not contain enough

information to compute c2 as temperature-dependent. Hence specific heat c2 is assumed to be

constant. The results are shown in Figure 5.39. There are no values for k2 and c1 related to the

300 K temperature because any value gives the same value of the criteria equation. This

means that no information is involved in the measured data for this temperature. Comparing

the results computed only for steady conditions with the results obtained for unsteady

conditions, it is obvious that conductivity k1 is higher for steady conditions. This is due to the

improper assumption that the heat lost at the top and bottom of the mold can be neglected.

Comparing the results, the heat loss amounted to about 20 %.

Comparing the computed conductivity k1 of the backup mold with the values from

MAGMA database, the results differ by about 5% only. This can be due to the different

density of the mold. The thermal conductivity is also slightly increasing for a temperature

range of 600–900 K. The computed mass density is by about 3.5 % lower than that in the

MAGMA database. The computed specific heat is by about 5 % lower than that in the

MAGMA database. This is probably due to the lower mass density of our mold. Looking at

the rapidly decreasing thermal conductivity k2, a similar effect can be seen for the Al2O3 in

database or for sapphire and polycrystalline aluminum oxide.

5.4 5.4 Pressure Impact – 2D Field CorrectionPressure Impact – 2D Field Correction

To test the sharpening method described in the Data Focusing chapter, artificial data

were prepared first. Next, the method was tested using a real measured spray distribution. The

correction was done in the frequency domain and then converted back to the space domain.

First, the evolution of the sharpened data is illustrated using an artificial spray

distribution (shown in Figure 5.40a) computed from a simulated real distribution

(Figure 5.40b). The measured maximum is 77.8 MPa but the real one is 100 MPa and the

measured impact shape is wider than the real one. To be able to compute convolution in the

frequency domain, the measured distribution and the circular sensor (Ø12 mm) were

converted into the frequency domain as shown in Figure 5.41. The inverse sensor computed

using Eq. (4.7.3) is shown in Figure 5.42a. We can now compute the convolution of the

128


measured data and the inverse sensor. The computed result is shown in Figure 5.42b in the

frequency domain and in Figure 5.43a in the space domain. The computed pressure

distribution is close to the real pressure distribution, however, some noise is visible in the

sharpened data (small waves).

Noise can be partially suppressed as was described in the Data Focusing chapter. Cutting

off the higher harmonic frequencies and making them equal to zero also in the inverse sensor

filter, we get a cut inverse sensor filter (Figure 5.45a). Using this filter for convolution we

obtain sharpened measured data with suppressed noise. To suppress the aliasing phenomenon,

the measured data and the sensor were filtered using Gaussian low-pass filter in the space

domain. Transforming these functions into the frequency domain (see Figure 5.44), we get

high frequencies equal to zero. This means that the aliasing phenomenon was suppressed.

Using the cut smooth inverse sensor filter for convolution, we get sharpened measured data

(see Figure 5.45b) with a maximum of 98.9 MPa which is very close to the real maximum,

100 MPa. You can also notice that the noise was well suppressed, compared with the

computed result shown in Figure 5.43a.

Real measured data are shown in Figure 5.46a. The measurement was made for a high

pressure flat jet nozzle where the distance from the surface was 150 mm and water pressure

was 20 MPa. The measurement was made using a square sensor (3x3 mm). Some noise in

measured data is obvious. The maximum in the sharpened data (see Figure 5.46b) rose by

about 18 % (from 1534 to 1814 kPa).

Another measurement was made using a different measuring apparatus. A circular

sensor of 1.5 mm in diameter was used. Two overlapping flat nozzles were measured. The

measured data are almost noise free compared to the previous data measured using

a rectangular sensor (see Figure 5.47a). The measuring area was 30x160 mm. The maximum

of the sharpened data rose from 1527 to 1587 kPa (see Figure 5.47b). The increase is small

because a small pressure sensor was used.

129


(a) (b)

Figure 5.40 – An example of (a) simulated measured distribution and

(b) simulated real distribution using a circular sensor of 12 mm in diameter.

(a) (b)

Figure 5.41 – (a) Data measured in frequency domain;

(b) circular sensor in frequency domain.

(a) (b)

Figure 5.42 – (a) Inverse sensor function in frequency domain;

(b) convolution of measured data and inverse sensor function in frequency domain.

130


(a) (b)

Figure 5.43 – (a) sharpened measured data – convolution of measured data and inversesensor function in space domain; (b) real pressure distribution in frequency domain.

(a) (b)

Figure 5.44 – (a) Circular sensor function in space domain filtered using low-pass filter;

(b) circular sensor function in frequency domain with removed high frequency.

(a) (b)

Figure 5.45 – (a) cut inverse sensor function in frequency domain;

(b) sharpened measured data.

131


(a) (b)

Figure 5.46 – (a) Real data measured using rectangular sensor (3x3 mm);

(b) sharpened data.

(a) (b)

Figure 5.47 –(a) Real data measured using circular sensor (1.5 mm in diameter);

(b) sharpened data

132

Preview of Next Research

6 6 PPREVIEWREVIEW OFOF N NEXTEXT R RESEARCHESEARCH

Alternative methods for solution of inverse heat conduction problems have been

presented. They used optimized models, however, some improvement can be achieved.

Methods for estimation of errors of computational models for direct solution have been

presented. However, the mesh discretization error was not computed for dynamic boundary

conditions, as in the case of estimation of the time discretization error, but only for one time

step. Knowledge of treatment of the mesh discretization error for unsteady boundary

conditions, which are developing in time, would be helpful. This would significantly help to

increase the accuracy of the computed results of direct heat conduction problems and the

known accuracy could be used in further analysis.

In the case of inverse heat conduction problems, results were obtained using alternative

methods. They are performed in a shorter time and give more accurate results. In the next

research, other multidimensional methods could be tested, such as Powell's method, simulated

annealing methods, or conjugate gradient method. Some of them might be even more

powerful. However, we still do not know the accuracy of the computed results. The situation

becomes even more complicated for time dependent boundary conditions. In this case, it

would be necessary to compute a safety zone along the curve that represents the computed

boundary condition history. This problem is very complicated because a number of various

errors are accumulated in inverse heat transfer analysis. The analysis involves the experiment

and the computational part where various errors are accumulated such as the activation error,

error in measured data, assumption error, error due to the computational model simplification,

and various numerical errors. Knowledge of these errors would be useful for improvement of

inverse tasks.

133

Conclusion

7 7 CCONCLUSIONONCLUSION

Many of inverse problems are often dismissed as ill-conditioned because the solution is

extremely difficult or almost impossible using concurrent methods. If any results are obtained,

they are inaccurate and the precision is unknown. Typical inverse problems studied in this

work include determination of boundary conditions in applications such as cooling in casting,

descaling, cooling of products in hot rolling, and cooling of rolls in rolling technology.

Because the studied inverse methods use direct computation of heat conduction, several

direct methods were compared. The suitable ones were finite difference method (FDM), finite

volume method (FVM), finite element method (FEM), and finite volume unstructured

methods (cell-centered and vertex-centered). The unstructured methods were not used in

inverse tasks because the demand of the inverse tasks for the computational power is high and

these methods are very slow. The FDM was found to be most convenient when testing the

FDM, FVM, and FEM for a direct solver of heat conduction. The speed was several times

higher when temperature dependent material properties were used and the accuracy of

computed results in nodes was the best for fast changes in boundary conditions. The general

FDM method was modified to treat phase change and temperature dependent material

properties.

Attention was also focused on accuracy of results computed by the direct heat

conduction method. Convergence of the iteration method for numerical solution of heat

transfer partial differential equations was tested. The convergence error was estimated using

a method based on the observed geometric convergence of heat conduction problems.

Methods for discretization error estimation were proposed and tested for the simplification of

the object, its boundary conditions, and time domain when the numerical approach is applied.

These methods were then used for discretization optimization.

New approaches and a special multi-level filtration for input data were developed for

solution of inverse heat conduction problems (IHCPs). The new approaches (optimization

method and neural network) were compared with classical Beck’s sequential approach. Usage

of the modified downhill simplex multidimensional optimization method was tested in

one-dimensional and several multi-dimensional inverse problems where the input data were

taken from real measurement. These inverse problems included determination of time

134

Conclusion

dependent boundary conditions (one- and two-dimensional ones), determination of

temperature dependent material properties (multidimensional inverse problem), and

calibration of computational model. Very good results were obtained using the optimization

method in contrast with usage of the artificial neural networks. The developed approaches that

use the optimization method gave better results than classical Beck's approach, and the

computational times were even shorter.

To compare the intensity of cooling with pressure impact distribution of a spraying

water nozzle, a measured impact field was corrected using the developed method based on

data focusing in the frequency domain. Data measured by a relatively large pressure sensor

were corrected using this method to obtain a more accurate distribution. The increase in the

maximum pressure impact was even 18 % in the presented case.

Combining various methods and usage of newly presented methods could bring

a significant improvement in solving ill-posed problems. This improvement would be useful

for designing and optimizing cooling sections in applications like continuous casting,

descaling, and hot rolling technology.

135

Literature

8 8 LL ITERATUREITERATURE

[1] Incropera, F. P.; DeWitt, D. P. Fundamentals of Heat and Mass Transfer. 4th ed. New

York: Wiley, 1996. ISBN 0-471-30460-3.

[2] Kakac, S.; Yener, Y. Heat Conduction. New York: Hemisphere Publishing, 1985.

[3] Poulikakos, D. Conduction Heat Transfer. Englewood Cliffs, NJ: Prentice-Hall, 1994.

[4] Raynaud, M.; Beck, J. V. Methodology for comparison of inverse heat conduction

methods. J. Heat Transfer, 1988, Vol. 110, pp. 30-37.

[5] William, H. P.; Saul A. T.; William, T. V.; Brian, P. F. Numerical Recipes in C. 2nd ed.

1997. ISBN 0-521-43108-5.

[6] Patankar, S. V. Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing

Corporation, 1980. ISBN 0-891-16522-3.

[7] Smith, G. D. Numerical solution of partial differential equations. UK: Oxford

University Press, 1978. ISBN 0-198-59650-2.

[8] Versteeg, H.; Malalasekera, W. An introduction to Computational Fluid Dynamics – the

Finite Volume Method. Longman Sc. & Tech., 1995. ISBN 0-582-21884-5.

[9] Fletcher, C. A. Jr. Computational techniques for fluid dynamics. Berlin: Springer, 1991.

ISBN 0-387-18151-2.

[10] Reddy, J. N.; Gartling, D. K. The finite element method in heat transfer and fluid

dynamics. Boca Raton, FL: CRC Press, 1994. ISBN 0-849-39410-4.

[11] Beck, J. V.; Blackwell, B.; Charles, R. C. Inverse Heat Conduction: Ill-posed Problems.

New York: Wiley, 1985. ISBN 0-471-08319-4.

[12] Duda, P.; Taler, J Numerical method for the solution of non-linear two-dimensional

inverse heat conduction problem using unstructured meshes. Int. J. for Numerical

Methods in Engineering, 2000, Vol. 48, pp. 881-899.

[13] Chow, P.; Cross, M. An enthalpy Control-Volume-Unstructured-Mesh (CV-UM)

algorithm for solidification by conduction only. Int. J. for Numerical Methods in

Engineering, 1992, Vol. 35, pp. 1849-1870.

[14] Power, H.; Worbel, L. Boundary integral methods in fluid mechanics. UK: WIT Press

(Computational Mechanics Publications), 1995.

136

Literature

[15] Lesnic, D.; Elliott, L.; Ingham, D. B. Application of the boundary element method to

inverse heat conduction problems. Int. J. Heat Mass Transfer, 1996, Vol. 39, No. 7, pp.

1503-1517.

[16] Han, H.; Ingham, D. B.; Yuan, Y. The boundary elementh method for the solution of the

backward heat conduction equation. J. of Computational Physics, 1995, Vol. 116, pp.

292-299.

[17] Ochiai, Y.; Sekiya, T. Steady heat conduction analysis by improved multiple-reciprocity

boundary element method. Eng. Analysis with Boundary Elements, 1996, Vol. 18, pp.

111-117.

[18] Ochiai, Y. Two-dimensional unsteady heat conduction analysis with heat generation by

triple-reciprocity BEM. Int. J. for Numerical Methods in Engineering, 2001, Vol. 51, pp.

143-157.

[19] Johns, P. B.; Beurle, R. L. Numerical solution of two dimensional scattering problems

using a transmition line matrix. Proc. IEE, 1971, Vol. 119, pp. 1203-1208.

[20] Cogan, D.; Soulos, A. Inverse thermal modeling using TLM. Num. Heat Transfer, 1996,

Vol. 29, pp. 125-135.

[21] Voller, V. R.; Swaminathan, C. R. Fixed grid techniques for phase change problems: A

review. Int. J. for Numerical Methods in Engineering, 1990, Vol. 30, pp. 875-898.

[22] Song, R.; Dhatt, G.; Cheikh, A. B. Thermo-mechanical finite element model of casting

systems. Int. J. for Numerical Methods in Engineering, 1990, Vol. 30, pp. 579-599.

[23] Comini, G.; Giudice, S. D.; Saro, O. A conservative alghorithm for multidimensional

conduction phase change. Int. J. for Numerical Methods in Engineering, 1990, Vol. 30,

pp. 697-709.

[24] Patankar, S. V. Computation of Conduction and Duct Flow Heat Transfer. Innovative

Research, Inc., 1991.

[25] Tikhonov, A. N.; Arsenin, V. Y. Solution of Ill-Posed Problems. Washington, D.C.:

Winston, 1977. ISBN 0470991240.

[26] Mandrel, J. Use of the singular value decomposition in regression analysis. Am. Stat.,

1982, Vol. 36, pp. 15-24.

[27] Stloz, G. Jr. Numerical solutions to an inverse problem of heat conduction for simple

shapes. Int. J. Heat Transfer, 1960, Vol. 82, pp. 20-26.

[28] Beck, J. V. Nonlinear estimation applied to the nonlinear heat conduction problem. Int.

J. Heat and Mass Transfer, 1970, Vol. 13, pp. 703-716.

137

Literature

[29] Beck. J. V.; Litkouhi, B.; St. Clair, C. R. Jr. Efficient sequential solution of the nonlinear

inverse heat conduction problem. J. Numerical Heat Transfer, 1982, Vol. 5, pp. 275-286.

[30] Bass, B. R. Applications of the finite element to the inverse heat conduction problem

using Beck’s second method. J. Eng. Ind., 1980, Vol. 102, pp. 168-176.

[31] Burggraf, O. R. An exact solution of the inverse problem in heat conduction theory and

applications. Int. J. Heat Transfer, 1964, Vol. 86C, pp. 373-382.

[32] Imber, M.; Khan, J. Prediction of transient temperature distributions with embedded

thermocouples. Al AA J., 1972, Vol. 10, pp. 784-789.

[33] Lengford, D. New analytic solutions of the one-dimensional heat equation for

temperature and heat flow rate both prescribed at the same fixed boundary (with

applications to the phase change problem). Q. Appl. Math., 1967, Vol. 24, pp. 315-322.

[34] Grysa, K.; Cialkowski, M. J.; Kaminski, H. An inverse temperature filed problem of the

theory of thermal stresses. Nucl. Eng. Des., 1981, Vol. 64, pp. 169-184.

[35] Sláma, L.; Raudenský, M.; Horský, J.; Březina, T.; Krejsa, J. Evaluation of quenching

test of rotating roll with unknown time constant of sensor using genetic algorithm. Int.

Conf. Mendel, Brno, 1996.

[36] Dumek, V.; Grove, T.; Raudenský, M.; Krejsa, J. Novel approaches to the IHCP: Neural

networks. In Int. symposium on inverse problems - Inverse problems in Engineering

Mechanics, Paris, 1994, pp. 411-416.

[37] Krejsa, J.; Sláma, L.; Horský, J.; Raudenský, M.; Pátiková, B. The comparison of

traditional and non-classical methods solving the inverse heat conduction problem. In

Int. Conf. Advanced Computational Methods in Heat Transfer, Udine, July 1996, pp.

451-460.

[38] Beck, J. V.; Murio, D. A. Combined function specification regularization procedure for

solution of inverse heat conduction problem. AIAA J., 1986, Vol. 24, pp. 180-185.

[39] D'Souza, N. Numerical solution of one-dimensional inverse transient heat conduction

by finite difference method. ASME Paper No. 75-WA/HT-81, 1975.

[40] Raynaud, M., Bransier, J. A new finite difference method for nonlinear inverse heat

conduction problem. Num. heat Transfer, 1986, Vol. 9, No. 1, pp. 27-42.

[41] Weber, C. F. Analysis and solution of the ill-posed inverse heat conduction problem. Int.

J. Heat Mass Transfer, 1981, Vol. 24, No. 11, pp. 1783-1792.

[42] Woodbury, K. A.; Ke, Q. An inverse algorithm for direct determiniation of heat transfer

coefficients. In 34th National Heat Transfer Conference, 2000, pp. 663-669.

138

Literature

[43] Osman, A. M.; Beck, J. V. Investigation of transient heat transfer coefficient in

quenching experiments. J. Heat Transfer, 1990, Vol. 112, pp. 843-848.

[44] Lin, S. M.; Chen, Ch. K.; Yang, Y. T. A modified sequential approach for solving inverse

heat conduction problems. J. Heat and Mass Transfer, 2004, Vol. 47, pp. 2669-2680.

[45] Woodbury, K. A.; Thakur, S. K. A two-dimensional inverse heat conduction algorithm.

In Proc. 1st International Conference on Inverse Problems in Engineering : Theory and

Practice, 1993, pp. 219-226.

[46] Lesnic, D.; Elliott, L.; Ingham, D. B. Application of the boundary element method to

inverse heat conduction problems. Int. J. Heat Mass Transfer, 1996, Vol. 39, No. 7, pp.

1503-1517.

[47] Woodbury, K. A.; Boohaker, Ch. G. Simultaneous determination of temperature

dependent thermal conductivity and volumetric heat capacity by an inverse technique. In

Proceedings of the ASME Heat Transfer Division, HDT-Vol. 332, Volume 1, 1996, pp.

269-274.

[48] Huang, C. H.; Ozisik, M. N.; Sawaf, B. Conjugate gradient method for determining

unknown contact conductance during metal casting. Int. J. Heat Mass Transfer, 1992,

Vol. 35, No. 7, pp. 1779-1786.

[49] Dumek, V.; Grove, T. D.; Raudenský, M.; Krejsa, J. Novel approaches to IHCP: Neural

networks. In Inverse Problems in Engineering Mechanics, Rotterdam, 1994, pp. 411-

416. ISBN 90-5410-517-8.

[50] Sblani, S. S.; Kacimov, A.; Perret, J.; Mujumdar, A. S.; Campo, A. Non-iterative

estimation of heat transfer coefficients using artificial neural network models. Int. J

Heat Mass Transfer, 2004, Vol 48, pp. 665-679.

[51] Raudenský, M.; Horský, J.; Krejsa, J. Usage of neural network for coupled parameter

and function specification inverse heat conduction problem. Int. Comm. Heat Mass

Transfer, 1995, Vol. 22, No. 5, pp. 661-670.

[52] Raudenský, M.; Horský, J.; Krejsa, J.; Sláma, L. Non-classical approach to the inverse

problems. In MMSP 96, General Workshop, Davos, 1996, pp. 288-298.

[53] Liu, G. R.; Lee, J. H.; Patera, A. T.; Yang, Z. L.; Lam, K. Y. Inverse identification of

thermal parameters using reduced-basis method. Comput. Methods Appl. Mech. Engrg.,

2005, Vol. 194, pp. 3090-3107.

139

Literature

[54] Loulou, T.; Artyukhin, E. A., Bardon, J. P. Estimation of thermal contact resistance

during the first stages of metal solidification process: I-experiment principle and

modelisation. Int. J. Heat Mass Transfer, 1999, Vol. 42, pp. 2119-2127.

[55] Demuth, H.; Beale, B. Neural Network Toolbox User's Guide. Ver. 3 MathWork Inc.,

1998.

140

Author’s References

9 9 AAUTHORUTHOR ’’ SS R REFERENCESEFERENCES

[A1] Pohanka, M.; Raudenský, M.; Horský, J.; Druckmüller, M. How to Precisely Define

Computational Models of Heat Process with Experimental Setting under Marginal

Conditions (in Czech). In Inženýrská mechanika 99. Svratka (Czech Republic), 1999,

pp. 717–722. ISBN 80-214-1323-9.

[A2] Pohanka, M.; Raudenský, M.; Horský, J. Attainment of more precise parameters of

a mathematical model for cooling flat and cylindrical hot surfaces by nozzles. In

Advanced computational methods in heat transfer VI. Madrid: WIT Press, 2000, pp.

627–635. ISBN 1-85312-818-X.

[A3] Pohanka, M.; Raudenský, M.; Horský, J. Optimizing parameters of a mathematical

model for cooling hot surfaces by nozzles. In Engineering mechanics 2000. Svratka

(Czech Republic), 2000, pp. 267–272. ISBN 80-86246-03-5.

[A4] Raudenský, M.; Horský, J.; Pohanka, M. Optimal cooling of rolls in hot rolling.

Journal of material processing technology. 2002, Vol. 125–126, pp. 700–705.

[A5] Pohanka, M.; Woodbury, K. A.; Wolley, J. Obtaining temperature dependent thermal

properties of investment casting mold. In Proc. of the International Mechanical

Engineering Congress and Exposition. New Orleans (LA): ASME, November 2002,

2-20-5-1.

[A6] Raudenský, M.; Pohanka, M.; Horský, J. Combined inverse heat conduction method

for highly transient processes. In Advanced computational methods in heat transfer VII,

Halkidiki: WIT Press, 2002, pp. 35–42. ISBN 1-85312-9062.

[A7] Tošovský, J.; Pohanka, M.; Kotrbáček, P. Stress analysis of scale layer for descaling

process. In 40th int. conf. experimental stress analysis. Praha, 2002, pp. 247–252.

ISBN 80-01-02547-0.

[A8] Pohanka, M.; Raudenský, M. Determination of heat resistances between installed

thermocouple and body used for computing heat transfer coefficients. In Engineering

mechanics 2002, Svratka (Czech Republic), 2002, pp. 227–228. ISBN 80-214-2109-6.

[A9] Pohanka, M.; Woodbury, K. A. A Downhill Simplex method for computation of

interfacial heat transfer coefficients in alloy casting. Inverse problems in engineering,

October 2003, Vol. 11, No. 5, pp. 409–424.

141

Author’s References

[A10] Raudenský, M.; Horský, J.; Pohanka, M.; et al. Experimental Study of Parameters

Influencing Efficiency of Hydraulic Descaling. In 4th Int. Conf. Hydraulic Descaling.

London, 2003, pp. 29–39.

[A11] Pohanka, M. Two-dimensional correction of data measured using a large pressure

sensor. In Computational methods and experimental measurements XI. Halkidiki: WIT

Press, 2003, pp. 587–596. ISBN 1-85312-969-0.

[A12] Kotrbáček, P.; Horský, J.; Raudenský, M.; Pohanka, M. Influence of parameters of

hydraulic descaling on temperature losses and surface quality of rolled material. In

Metal Forming 2004. Kraków (Poland), 2004, pp. 367–370. ISBN 3-937057-08-0.

[A13] Horský, J.; Raudenský, M.; Pohanka, M. Experimental study of heat transfer in hot

rolling and continuous casting. In Material Science Forum. Switzerland: Trans Tech

Publication, 2005, Vols. 473–474, pp. 347–354. ISBN 0-87849-957-1.

[A14] Kotrbáček, P.; Horský, J.; Raudenský, M.; Pohanka, M. Experimental study of heat

transfer in hot rolling. In 26e Journées sidérurgiques internationales. Paris, 2005, pp.

42–43. ISBN 2-911212-05-3.

142

Appendix A – Conduction And Heat Diffusion Equation

10 10 AAPPENDIXPPENDIX A A – C – CONDUCTIONONDUCTION A ANDND H HEATEAT D DIFFUSIONIFFUSION E EQUATIONQUATION

The conduction (heat transfer by diffusion) refers to the transport of energy in a medium

due to a temperature gradient. The physical mechanism is based on random atomic or

molecular activity.

10.1 10.1 Rate equationRate equation

It is possible to quantify the heat transfer process in terms of appropriate rate equations.

These equations may be used to compute the amount of energy being transferred per unit time.

For heat conduction, the rate equation is known as Fourier’s law. For the one-dimensional

plane wall shown in Figure 10.1, having a temperature distribution T(x), the rate equation is

expressed as

qx=kdTdx

. (10.1.1)

Figure 10.1 – One-dimensional heat transfer by conduction (diffusion of energy).

The heat flux qx (W/m2) is the heat transfer rate in the x direction per unit area perpendicular

to the direction of transfer and it is proportional to the temperature gradient dT/dx in this

direction. The proportionality constant k is a transport property known as the thermal

conductivity (W/m.K). The minus sign is the consequence of heat transfer in the direction of

decreasing temperature. Under the steady-state conditions shown in Figure 10.1, where the

temperature distribution is linear, the temperature gradient may be expressed as

dTdx=

T2T1

L(10.1.2)

I

T

T1

T2

T(x)

L x

qx


and the heat flux is then

qx=kT2T1

L=k

TL

. (10.1.3)

This equation provides a heat flux – the rate of heat transfer per unit area. The heat transfer

rate by conduction, qx [W], through a plane wall of area A is then the product of the flux and

the area

qx=qx⋅A . (10.1.4)

10.2 10.2 Heat Diffusion EquationHeat Diffusion Equation

A major objective in a direct conduction analysis is to determine the temperature field

(distribution) in a medium resulting from conditions imposed on its boundaries. The approach

follows the methodology of applying the energy conservation requirement. We define

a differential control volume, identify the relevant energy transfer processes, and introduce the

appropriate rate equations [1].

Considering a homogeneous medium within which there is no bulk motion, the

temperature is expressed in Cartesian coordinates as T(x, y, z). We first define

an infinitesimally small (differential) control volume, dx⋅dy⋅dz, as shown in Figure 10.2. If

there are temperature gradients, then conduction heat transfer, indicated as a heat rate (qx, qy,

qz), will occur across each of the control surfaces. The conduction heat rates at the opposite

surfaces can then be expressed as a Taylor expansion, and neglecting higher order terms we

obtain

qxdx=qx∂qx

∂ xdx , (10.2.1)

qydy=qy∂qy

∂ ydy , (10.2.2)

qzdz=qz∂qz

∂ zdz . (10.2.3)

II


Figure 10.2 – Differential control volume for conduction analysis in Cartesian coordinates.

Within the medium there may also be an energy source term associated with the rate of

thermal energy generation. This term is expressed as

ö g=q⋅dx⋅dy⋅dz (10.2.4)

where q is the rate at which energy is generated per unit volume of the medium. In addition,

there may occur changes in the amount of the internal thermal energy stored by the material in

the control volume. The energy storage may be expressed as

ö st=⋅cp

∂T∂ t

dx⋅dy⋅dz (10.2.5)

where ⋅cp∂T /∂ t is the time rate of the change in thermal energy of the medium per unit

volume.

On a rate basis, the general form of the conservation of the energy requirement is

ö inöoutö g=ö st (10.2.6)

and substituting Eq. (10.2.1–10.2.3), we obtain

∂qx

∂ xdx

∂qy

∂ ydy

∂qz

∂ zdzq⋅dx⋅dy⋅dz=⋅cp

∂T∂ t

dx⋅dy⋅dz . (10.2.7)

The conduction heat rates may be evaluated from Fourier’s law, Eq. (10.1.1),

III

dx

dy

dzEg , Est

qzdz qydy

qxdxqx

qyqz


qx=k⋅dy⋅dz∂T∂ x

, (10.2.8)

qy=k⋅dx⋅dz∂T∂ y

, (10.2.9)

qz=k⋅dx⋅dy∂T∂ z . (10.2.10)

Substituting Eq. (10.1.4–10.1.3) into Eq. (10.1.2) and dividing out the dimensions of the

control volume dx⋅dy⋅dz, we obtain

∂∂ x k

∂T∂ x

∂∂ y k

∂T∂ y

∂∂ zk

∂T∂ z q=⋅cp

∂T∂ t

. (10.2.11)

This equation is the general form, in Cartesian coordinates, of the heat diffusion equation also

known as the heat equation. In the cylindrical coordinates, the heat diffusion equation [1] is

1r∂∂ r kr

∂T∂ r

1

r 2

∂∂ k

∂T∂

∂∂ zk

∂T∂ z q=⋅cp

∂T∂ t

. (10.2.12)

IV

Appendix B – Convection

11 11 AAPPENDIXPPENDIX B B – C – CONVECTIONONVECTION

The convection boundary condition is one of the most common cases in our problems

under solution and thus we will discuss it a bit more in detail. The convection heat transfer

mode needs some fluid and is comprised of two mechanisms. In addition to the energy

transfer due to the random molecular motion (diffusion), energy is also transferred by a bulk,

or macroscopic, motion of the fluid. Such a motion contributes to the heat transfer and the

total heat transfer is a superposition of energy transport by the random motion of molecules

and by the bulk motion of the fluid.

We speak of forced convection when the flow is caused by external forces, such as

spraying water or atmospheric winds. In contrast, for free (or natural) convection the flow is

induced by buoyancy forces that arise from density differences caused by temperature

variations in the fluid. In many cases, mixed forced and natural convection may exist.

However, there are convection processes for which there is a latent heat exchange. This latent

heat exchange is associated with the phase change e.g. between the liquid and vapor states of

the fluid (boiling and condensation).

Regardless of the particular nature of the convection heat transfer process, the

appropriate rate equation has the form

q=h T sT∞ (11.1)

where q is the convective heat flux. This expression is known as Newton’s law of cooling, and

the proportionality constant h is termed the convection heat transfer coefficient (HTC). For

typical problems, typical values of HTC are given in Table 11.1.

Table 11.1 – Typical values of the convection heat transfer coefficient [1]

Process h [W/m2.K]

Free convectionGases 2–25Liquids 50–1000

Forced convectionGases 25–250Liquids 50–20 000

Convection with phase changeBoiling or condensation 2 500–100 000

V

Appendix C – Radiation

12 12 AAPPENDIXPPENDIX C C – R – RADIATIONADIATION

The heat transfer due to the radiation becomes important for hot surfaces such as steel

plates in continues casting or descaling applications. Thermal radiation is the energy emitted

by the matter that is at a finite temperature [1]. The energy of the radiation field is transported

by electromagnetic waves (or, alternatively, photons) and the presence of material medium is

not required. The rate at which the energy is released per unit area is termed the surface

emissive power E. There is an upper limit to the emissive power that is prescribed by the

Stefan-Boltzmann law

Eb=T s4 (12.1)

where Ts is the absolute temperature [K] of the surface and σ is Stefan-Boltzmann constant.

Such a surface is called an ideal radiation or blackbody. The heat flux emitted by a real surface

is less than that of a black body and is given by

Eb=T s4 (12.2)

where ε is a radiative property of the surface termed the emissivity 0≤≤1.

Radiation may also be incident on a surface from its surroundings. Such radiation is

termed irradiation G. A portion, or all, of the irradiation may be absorbed by the surface. The

rate at which the radiant energy is absorbed may be evaluated by Gabs=G. The absorptivity

α may be within the range 0≤≤1. The radiation may be reflected or, in the case of

a semitransparent material, it may be transmitted. The irradiation may be approximated by

emission from a blackbody at Tsur (surrounding temperature), in which case G=⋅T sur4 . The

net rate of radiation heat transfer from the surface is

qrad=qA=⋅Eb T sG= T s

4T sur4 . (12.3)

There are many applications for which it is convenient to express the net radiation heat

exchange in the form

qrad=h⋅AT sT sur (12.4)

where, from Eq. (12.4), the radiation heat transfer coefficient h is

VI

Appendix C – Radiation

h=⋅ T sT sur⋅T s2T sur

2 . (12.5)

The surfaces may also simultaneously transfer heat by convection to an adjoining gas. For

example, a 5 m long slab heated to a temperature 800oC moving at a rate of 5 m/s at room

temperature gives

q=qconvqrad=hT sT∞ ⋅ T s4T sur

4

=14 ⋅7800.9⋅5.67×108 ⋅1.3×1013=10 92067 304=78 224 W/m2 . (12.6)

VII

Appendix D – Solution of Linear Algebraic Equations

13 13 AAPPENDIXPPENDIX D D – S – SOLUTIONOLUTION OFOF L L INEARINEAR A ALGEBRAICLGEBRAIC E EQUATIONSQUATIONS

13.1 13.1 Gauss-Jordan eliminationGauss-Jordan elimination

This method produces both the solution vector x and also the inverse matrix A-1.

However, it requires all the right-hand sides to be stored and manipulated at the same time,

and when the inverse matrix is not desired, it is three times slower than the best alternative

technique for solving a single linear set. Its strength is that it is as stable as other direct

methods, perhaps even a bit more stable when full pivoting is used. It is straightforward,

understandable, solid as a rock, and is good backup when something is going wrong.

Gaussian elimination with back-substitution stands between full elimination schemes

such as Gauss-Jordan and triangular decomposition schemes. Gaussian elimination reduces

a matrix only halfway to the identity matrix. The components on the diagonal and above

remain nontrivial. The advantage of Gaussian elimination with back-substitution over

Gauss-Jordan elimination is simply that it is faster in operations count. The innermost loops of

Gauss-Jordan elimination are executed M3 and M2N times. The corresponding loops in

Gaussian elimination are executed only 1/3M3 times (only half the matrix is reduced), and

1/2M2N times.

13.2 13.2 LU decompositionLU decomposition

This method uses the fact that the matrix A can be written as a product of two matrices

L⋅U=A (13.2.1)

where L is the lower triangular (has elements only on the diagonal and below) and U is the

upper triangular (it has elements only on the diagonal and above). The decomposition is used

to solve the linear equations

A⋅x=L⋅U ⋅x=L⋅U⋅x =b (13.2.2)

first by solving the vector y

L⋅y=b (13.2.3)

and then solving

VIII


U⋅x= y . (13.2.4)

The advantage is that the solution of a triangular set of equations is quite trivial. Eq. (13.2.3)

can be solved by forward substitution and Eq. (13.2.4) can be solved by back-substitution. To

solve L and U for given A, the Crout’s algorithm is used. The decomposition is „in place” and

does not require an additional memory. The pivoting is necessary for the stability of the

Crout’s algorithm. Only partial pivoting (interchange of rows) is implemented efficiently. The

matrix A is not usually decomposed into LU form, but only a row-wise permutation of A is

used. This is enough to make the method stable.

The LU decomposition requires about 1/3M3 executions of the inner loops (each with

one multiply and one add). This is 3 times better than the Gauss-Jordan routine, and 1.5 times

better than a Gauss-Jordan routine which does not compute the inverse matrix.

13.3 13.3 Tridiagonal systems of equationsTridiagonal systems of equations

This is a special case of a system of linear equations. It has nonzero elements only on

the diagonal plus or minus one column. The set of equations to be solved is

a1,1x1a1,2x2=b1 , (13.3.1)

ai , i1 xi1ai , i xiai , i1 xi1=bi for i=2⋯N≡M 1 , (13.3.2)

aN , N1 xN1aN , N xN=bN , (13.3.3)

and in the matrix form

[a1,1 a1,2 0 ⋯ 0 0 0 a2,1 a2,2 a2,3 ⋯ 0 0 0

⋮0 0 0 ⋯ aN1,N2 aN1,N1 aN1,N

0 0 0 ⋯ 0 aN , N1 aN , N

]⋅[x1

x2

⋮xN1

xN

]=[b1

b2

⋮bN1

bN

] . (13.3.4)

In the tridiagonal matrix algorithm [24], Eq. (13.3.1) can be written as

x1=P1 x2Q1 (13.3.5)

where

IX


P1=a1,2

a1,1

Q1=b1

a1,1

.(13.3.6)

This relation is substituted into Eq. (13.3.2) for i = 2. The result is that x2 is expressed using

x3. As the substitution process continues, it expresses each xi-1 using xi

xi1=Pi1 xiQi1 . (13.3.7)

By substituting this relation into Eq. (13.3.2), we get

ai , i xi=ai , i1 xi1ai , i1Pi1 xiQi1bi . (13.3.8)

The expressions for Pi and Qi can be written as

Pi=ai , i1

ai , iai , i1 Pi1

Qi=biai , i1Qi1

ai , iai , i1 Pi1

.(13.3.9)

The solution algorithm is as follows:

1. Calculate P1 and Q1.

2. Obtain Pi and Qi for i = 2, 3, ..., N by using the recursive relations.

3. Set xN = QN.

4. Substitute into Eq. (13.3.7) for i = N, N-1, ..., 3, 2 to obtain xN-1, xN-2, ..., x2, x1.

13.4 13.4 Band diagonal systems of equationsBand diagonal systems of equations

Band diagonal systems are slightly more general and have m1≥0 nonzero elements to

the left (below) of the diagonal and m2≥0 nonzero elements immediately to the right (above)

it. This classification is only useful if m1 and m2 are both <<N. The solution of the linear

system is obtained by modified LU decomposition which is much faster, and requires much

less storage than for the general N x N case. The precise definition of a band diagonal matrix

is

ar , c=0 for crm2 or rcm1 (13.4.1)

X


Band diagonal matrices are usually stored in a compact form. It is not possible to store the LU

decomposition of a band diagonal matrix A as compactly as the compact form of A itself

because the decomposition produces additional nonzero values.

13.5 13.5 Iterative improvement of aIterative improvement of a solution of linear equationssolution of linear equations

For large sets of linear equations, it is not always easy to obtain precision comparable to

the computer’s limit. In direct methods the roundoff errors accumulate and many significant

figures can be lost. This method restores almost the full machine precision. Suppose that

a vector x is the exact solution of Eq. (3.3.47). However, only the slightly wrong solution

x x is known, where x is the unknown error. When the slightly wrong solution is

multiplied by the matrix A, it gives a product slightly different from the desired b

A⋅x x =bb . (13.5.1)

Subtracting Eq. (3.3.47) from (13.5.1) gives

A⋅ x=b (13.5.2)

Eq. (13.5.1) can be solved by substituting Eq. (13.5.2) for b which gives

A⋅ x=A⋅x x b (13.5.3)

In this equation, the whole right-hand side is known. In the case of LU decomposition the A is

already available and the x can be easily and qickly computed by back-substitution.

13.6 13.6 Conjugate gradient methodConjugate gradient method

This method can be very efficient for a properly stored sparse matrix. The simplest,

„ordinary” conjugate gradient algorithm solves Eq. (3.3.47) only in case that A is symmetric

and positive definite. It is based on the idea of minimizing the function

f x =1 2

x⋅A⋅xb⋅x . (13.6.1)

This function is minimized when its gradient

∇ f =A⋅xb (13.6.2)

XI


is zero, which is equivalent to Eq. (3.3.47). To solve Eq. (3.3.47), an initial guess of x1 is

made for the solution. The residual is

r 1=bA⋅x1 (13.6.3)

where the superscripts are iteration indexes and the initial values are

r1=r 1

p1=r 1

p1=r

1 . (13.6.4)

Then the sequence of improved estimates is

x i1=x ii pi (13.6.5)

where

i= ri⋅r i

pi⋅A⋅pi

r i1=r ii A⋅pi

ri1=r

ii AT⋅pi

i=r i1⋅r i1

ri⋅r i

pi1=r i1i pi

pi1=r

i1ip

i .(13.6.6)

When rm+1 = 0, the xm+1 is the solution of Eq. (3.3.47). There is no guarantee that the whole

procedure will not break down or become unstable for general A. The iterative process should

be halted when some appropriate error criterion is met.

XII

Appendix E – Comparison of FDM, FVM, and FEM

14 14 AAPPENDIXPPENDIX E E – C – COMPARISONOMPARISON OFOF FDM, FVM, FDM, FVM, ANDAND FEM FEM

This part deals with the testing of accuracy of the FDM, FVM, and FEM methods. As

can be seen in Eq. (3.3.21), all the methods use a similar form of equations to describe heat

conduction inside the body. The parameters for the equations are listed in Table 3.2 for each

method. First, the methods are compared with analytical solutions (Eq. 3.2.3 and Eq. 3.2.6)

along with a very fine model. After confirmation of the very high accuracy of the very fine

model, the model is used as an etalon for computations where no analytical solution is

possible.

The one-dimensional model is used for testing. The testing boundary conditions are

applied on the left side. An insulated surface is assumed on the right side. The initial

temperature is 0 °C. The length of the object is 0.1 m and is divided into 10 internal control

volumes. The surface volume has zero thickness. Stainless steel of constant material

properties is used:

• mass density: 7900 kg/m3

• specific heat: 510 J/kg.K

• thermal conductivity: 14 W/m.K.

The time step is fine enough during computation. Further refinement of the time steps

does not lead to a significantly higher accuracy.

The computed results shown in the following chapters were analyzed separately for

internal nodes and separately for the surface node. however, the overall accuracy of the

computed results is very similar for all three methods. For internal nodes the best results are

computed by FDM and FVM. In some cases, FDM is more accurate while in the others FVM

is better. Generally, the FEM gives the worst results. On the other hand, FEM gives the best

results for the surface nodes.

FVM and FEM show a very inconvenient behavior for a short time interval. The

temperature of internal nodes decreases even when the material is only heated. FDM seems to

be the most convenient for IHCP because it does not show this behavior and the inaccuracy of

the surface node can be suppressed by dividing the volume next to the surface into two control

volumes. When all the three methods are modified to treat temperature dependent material

properties, the FDM requires a much less computational time than FVM or FEM does.

XIII


14.1 14.1 Constant Heat FluxConstant Heat Flux

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-3.986

0

5

10

15

20

25

30

35

38.154

Tem

pera

ture

[°C

]

Fine modelAnalyticalFDMFVMFEM

Figure 14.1 – Temperature distribution; q = 100 000 W/m2; t = 1 s

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-4.407

-2

0

2

4

6

8

10

12

14

16

18

20

23.127

dT [°

C]


Figure 14.2 – Accuracy; q = 100 000 W/m2; t = 1 s

XIV


0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-0.389

10

20

30

40

57.04T

empe

ratu

re [°

C]



0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-1.865

0

2

4

6

9.529

dT [°

C]



0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-0.001

50

100

152.969

Tem

pera

ture

[°C

]



0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-0.65

0

1

2

2.728

dT [°

C]



XV


14.2 14.2 Constant Heat Transfer CoefficientConstant Heat Transfer Coefficient

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0t [s]

86584.103

90000

95000

100000

q [W

/m²]

Figure 14.7 – Surface heat fllux; h = 1 000 W/m2.K; t = 1 s

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-2.988

0

5

10

15

20

25

27.629

Tem

pera

ture

[°C

]

Fine modelFDMFVMFEM

Figure 14.8 – Temperature distribution; h = 1 000 W/m2.K; t = 1 s

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-3.383

0

2

4

6

8

10

12

14.213

dT [°

C] Fine model

FDMFVMFEM

Figure 14.9 – Accuracy; h = 1 000 W/m2.K; t = 1 s

XVI


0 1 2 3 4 5 6 7 8 9 10t [s]

65852.238

70000

75000

80000

85000

90000

95000

100000q

[W/m

²]


0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-0.301

5

10

15

20

25

30

36.952

Tem

pera

ture

[°C

]



0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-1.498

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.8042

dT [°

C]



XVII


0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100t [s]

35149.688

50000

60000

70000

80000

90000

100000q

[W/m

²]


0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-0.001

10

20

30

40

50

64.851

Tem

pera

ture

[°C

]



0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-0.7009

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1392

dT [°

C]



XVIII


14.3 14.3 TimeTime--dependent Heat Fluxdependent Heat Flux

0 1 2 3 4 5 6 7 8 9 10t [s]

-99999.992

-50000

0

50000

100000

q [W

/m²]

Figure 14.16 – Surface heat fllux q = 100 000 sin(t) W/m2; t = 10 s

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-15.52

-10

-5

0

5

9.047

Tem

pera

ture

[°C

]

Fine modelFDMFVMFEM

Figure 14.17 – Temperature distribution; q = 100 000 sin(t) W/m2; t = 10 s

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-20.663

-15

-10

-5

0

5.351

dT [°

C] Fine model

FDMFVMFEM

Figure 14.18 – Accuracy; q = 100 000 sin(t) W/m2; t = 10 s

XIX


0 1 2 3 4 5 6 7 8 9 10t [s]

0

20000

40000

60000

80000

100000q

[W/m

²]

Figure 14.19 – Square surface heat fllux; t = 10 s

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-0.535

2

4

6

8

10

12

14

16

18.635

Tem

pera

ture

[°C

]

Fine modelFDMFVMFEM

Figure 14.20 – Temperature distribution; square surface heat fllux; t = 10 s

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-8.132

-7

-6

-5

-4

-3

-2

-1

0

1

2.349

dT [°

C] Fine model

FDMFVMFEM

Figure 14.21 – Accuracy; square surface heat fllux; t = 10 s

XX


0 1 2 3 4 5 6 7 8 9 10t [s]

0

20000

40000

60000

80000

100000q

[W/m

²]

Figure 14.22 – Triangular surface heat fllux; t = 10 s

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-1.069

2

4

6

8

10

12

14

16

18

21.387

Tem

pera

ture

[°C

]

Fine modelFDMFVMFEM

Figure 14.23 – Temperature distribution; triangular surface heat fllux; t = 10 s

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10x [m]

-10.69

-8

-6

-4

-2

0

2

3.321

dT [°

C] Fine model

FDMFVMFEM

Figure 14.24 – Accuracy; triangular surface heat fllux; t = 10 s

XXI

Appendix F – Multidimensional Heat Conduction

15 15 AAPPENDIXPPENDIX F F – – M MULTIDIMENSIONALULTIDIMENSIONAL H HEATEAT C CONDUCTIONONDUCTION

15.1 15.1 2D Heat Conduction2D Heat Conduction

15.1.1 Cartesian Coordinates

The unsteady two-dimensional heat conduction in Cartesian coordinates can be written

as

dqx

dx

dqy

dyq=⋅c

dTdt

(15.1.1)

where

qx=kdTdx

,

qy=kdTdy

.(15.1.2)

The discretization of the 2D model is shown in Figure 4.6. Integrating Eq. (15.1.1) over

a control volume gives

yk qJ , km qJ1,k

m x j q j , Km q j , K1

m x j⋅ yk⋅q j , k

= x j⋅ yk⋅ j , k T j , km ⋅c j , k T j , k

m ⋅T j , k

m T j , km1

t.

(15.1.3)

We can write Eqs. (15.1.3, 4.1.6, 4.1.10, and 4.1.21) in the form of Eqs. (4.2.3 and 4.2.4). The

a and b parameters are as follows:

a j , k1=2 ⋅ x j⋅ yk1

k j , k1T j , k1m1

yk

k j , k T j , km1

1

,

a j1,k=2 ⋅ yk⋅ x j1

k j1,k T j1,km1

x j

k j , k T j , km1

1

,

a j , k=a j , k1a j1,ka j1,ka j , k1

x j⋅ yk⋅[ q j , kb T j , k

m1 j , k T j , k

m1⋅c j , k T j , km1

t ] ,

XXII


a j1,k=2 ⋅ yk⋅ x j

k j , k T j , km1

x j1

k j1,k T j1,km1

1

,

a j , k1=2 ⋅ x j⋅ yk

k j , k T j , km1

yk1

k j , k1T j , k1m1

1

,

b j , k= x j⋅ yk⋅[q j , ka T j , k

m1 j , k T j , k


t⋅T j , k

m1] . (15.1.4)

For all subsequent iterations T j , ki1,m substitutes for T j , k

m1 in aj, k-1, aj-1, k, aj+1, k, and aj, k+1. The

aj, k and bj, k parameters are

a j , k=a j , k1a j1,ka j1,ka j , k1 x j⋅ yk⋅q j , kb T j , k

i1,m ,

b j , k= x j⋅ yk⋅[q j , ka T j , k

i1,mH j , k T j , k

i1,mH j , k T j , km1

t⋅T j , k

m1] . (15.1.5)

15.1.2 Cylindrical Coordinates

The unsteady two-dimensional heat conduction in cylindrical coordinates can be written

as

1 r

dqr

dr

1 r

dqd

q=⋅cdTdt

(15.1.6)

where

qr=r⋅kdTdr

,

q=kr

dTd

.(15.1.7)

Integrating Eq. (15.1.6) over the control volume results in

k qJ , km qJ1,k

m r j q j , Km q j , K1

m r j⋅ r j⋅k⋅q j , k

=r j⋅ r j⋅k⋅ j , k T j , km ⋅c j , k T j , k

m ⋅T j , k

m T j , km1

t.

(15.1.8)

For the cylindrical coordinates, heat flux qJ , km can be evaluated as

XXIII


qJ , km =r J⋅[

r j1

2

k j1,k T j1,km

r j

2

k j , k T j , km ]

1

⋅T j1,km T j , k

m , (15.1.9)

heat flux q j , Km can be evaluated as

q j , Km =

1 r j

⋅[k1

2

k j , k1T j , k1m

k

2

k j , k T j , km ]

1

⋅T j , k1m T j , k

m . (15.1.10)

We can write Eqs. (15.1.8, 15.1.9, 15.1.10, 4.1.10, and 4.1.21) in the form of Eqs. (4.2.3 and

4.2.4). The a and b parameters are as follows:

a j , k1=2 r j

r j k1

k j , k1T j , k1m1

k

k j , k T j , km1

1

,

a j1,k=2 ⋅r J⋅k⋅ r j1

k j1,k T j1,km1

r j

k j , k T j , km1

1

,

a j , k=a j , k1a j1,ka j1,ka j , k1

r j r j⋅k⋅[ q j , kb T j , k

m1 j , k T j , k


t ] ,

a j1,k=2 ⋅r J1⋅k⋅ r j

k j , k T j , km1

r j1

k j1,k T j1,km1

1

,

a j , k1=2 r j

r j k

k j , k T j , km1

k1

k j , k1T j , k1m1

1

,

b j , k=r j⋅ r j⋅k⋅[q j , ka T j , k

m1 j , k T j , k


t⋅T j , k

m1] . (15.1.11)

For all subsequent iterations T j , ki1,m substitutes for T j , k

m1 in aj, k-1, aj-1, k, aj+1, k, and aj, k+1. Theaj, k and bj, k parameters are

a j , k=a j , k1a j1,ka j1,ka j , k1r j⋅ r j⋅k⋅q j , kb T j , k

i1,m ,

b j , k=r j⋅ r j⋅k⋅[q j , ka T j , k

i1,mH j , k T j , k

i1,mH j , k T j , km1

t⋅T j , k

m1] . (15.1.12)

XXIV


15.2 15.2 3D Heat Conduction3D Heat Conduction

15.2.1 Cartesian Coordinates

The unsteady three-dimensional heat conduction in Cartesian coordinates can be written

as

dqx

dx

dqy

dy

dqz

dzq=⋅c

dTdt

(15.2.1)

where

qx=kdTdx

,

qy=kdTdy

,

qz=kdTdz

.(15.2.2)

The discretization of the 3D model is shown in Figure 4.6. Integrating Eq. (15.2.1) over the

control volume gives

yk⋅ zpqJ , k , pm qJ1,k , p

m x j⋅ zpq j , K , pm q j , K1, p

m x j⋅ yk q j , k , Pm q j , k , P1

m

x j⋅ yk⋅ zp⋅q j , k , p= x j⋅ yk⋅ zp⋅ j , k , pT j , k , pm ⋅c j , k , pT j , k , p

m ⋅T j , k , p

m T j , k , pm1

t.

(15.2.3)

We can write Eqs. (15.2.3, 4.1.6, 4.1.10, and 4.1.21) in the form of Eq. (4.2.5). The a and b

parameters are as follows:

a j , k , p1=2 ⋅ x j⋅ yk⋅ zp1

k j , k , p1T j , k , p1m1

zp

k j , k , pT j , k , pm1

1

,

a j , k1, p=2 ⋅ x j⋅ zp⋅ yk1

k j , k1, pT j , k1, pm1

yk

k j , k , pT j , k , pm1

1

,

a j1,k , p=2 ⋅ yk⋅ zp⋅ x j1

k j1,k , pT j1,k , pm1

x j

k j , k , pT j , k , pm1

1

,

a j , k , p=a j , k , p1a j , k1, pa j1,k , pa j1,k , pa j , k1, pa j , k , p1

XXV


x j⋅ yk⋅ zp⋅[q j , k , pb T j , k , p

m1 j , k , pT j , k , p

m1 ⋅c j , k , pT j , k , pm1

t ] ,

a j1,k , p=2 ⋅ yk⋅ zp⋅ x j

k j , k , pT j , k , pm1

x j1


1

,

a j , k1, p=2 ⋅ x j⋅ zp⋅ yk

k j , k , pT j , k , pm1

yk1

k j , k1, pT j , k1, pm1

1

,

a j , k , p1=2 ⋅ x j⋅ yk⋅ zp

k j , k , pT j , k , pm1

zp1

k j , k , p1T j , k , p1m1

1

,

b j , k , p= x j⋅ yk⋅ zp⋅[q j , k , pa T j , k , p

m1 j , k , pT j , k , p

m1 ⋅c j , k , pT j , k , pm1

t⋅T j , k , p

m1 ] . (15.2.4)

For all subsequent iterations T j , k , pi1,m substitutes for T j , k , p

m1 in aj, k, p-1, aj, k-1, p, aj-1, k, p, aj+1, k, p,

aj, k+1, p, and aj, k, p+1. The aj, k, p and bj, k, p parameters are


x j⋅ yk⋅ zp⋅q j , k , pb T j , k , p

m1 ,

b j , k , p= x j⋅ yk⋅ zp⋅[q j , k , pa T j , k , p

i1,mH j , k , pT j , k , p

i1,mH j , k , pT j , k , pm1

t⋅T j , k , p

m1 ] . (15.2.5)

15.2.2 Cylindrical Coordinates

The unsteady three-dimensional heat conduction in cylindrical coordinates can bewritten as

1 r

dqr

dr

1 r

dqd

dqz

dzq=⋅c

dTdt

(15.2.6)

where

qr=r⋅kdTdr

,

q=kr

dTd

,

dqz=kdTdz

.(15.2.7)

XXVI


Integrating Eq. (15.2.6) over the control volume results in

k⋅ zpqJ , k , pm qJ1,k , p

m r j⋅ zpq j , K , pm q j , K1, p

m r j⋅ r j⋅k q j , k , Pm q j , k , P1

m

r j⋅ r j⋅k⋅ zp⋅q j , k , p=r j⋅ r j⋅k⋅ zp⋅ j , k , pT j , k , pm ⋅c j , k , pT j , k , p

m ⋅T j , k , p

m T j , k , pm1

t.

(15.2.8)

We can write Eqs. (15.2.8, 15.1.9, 15.1.10, 4.1.6, 4.1.10, and 4.1.21) in the form of

Eq. (4.2.5). The a and b parameters are as follows:

a j , k , p1=2 ⋅r j⋅ r j⋅k⋅ zp1

k j , k , p1T j , k , p1m1

zp

k j , k T j , k , pm1

1

,

a j , k1, p=2 r j⋅ zp

r j k1

k j , k1, pT j , k1, pm1

k

k j , k , pT j , k , pm1

1

,

a j1,k , p=2 ⋅r J⋅k⋅ zp⋅ r j1


r j

k j , k , pT j , k , pm1

1

,


r j r j⋅k⋅ zp⋅[q j , k , pb T j , k , p

m1 j , k , pT j , k , p

m1 ⋅c j , k , pT j , k , pm1

t ] ,

a j1,k , p=2 ⋅r J1⋅k⋅ zp⋅ r j

k j , k , pT j , k , pm1

r j1


1

,

a j , k1, p=2 r j⋅ zp

r j k

k j , k , pT j , k , pm1

k1

k j , k1, pT j , k1, pm1

1

,

a j , k , p1=2 ⋅r j⋅ r j⋅k⋅ zp

k j , k , pT j , k , pm1

zp1

k j , k , p1T j , k , p1m1

1

,

b j , k , p=r j⋅ r j⋅k⋅ zp⋅[q j , k , pa T j , k , p

m1 j , k , pT j , k , p

m1 ⋅c j , k , pT j , k , pm1

t⋅T j , k , p

m1 ] . (15.2.9)

For all subsequent iterations T j , k , pi1,m substitutes for T j , k , p

m1 in aj, k, p-1, aj, k-1, p, aj-1, k, p, aj+1, k, p,

aj, k+1, p, and aj, k, p+1. The aj, k, p and bj, k, p parameters are

XXVII



r j⋅ r j⋅k⋅ zp⋅q j , k , pb T j , k , p

i1,m ,

b j , k , p=r j⋅ r j⋅k⋅ zp⋅[q j , k , pa T j , k , p

i1,mH j , k , pT j , k , p

i1,mH j , k , pT j , k , pm1

t⋅T j , k , p

m1 ] .

(15.2.10)

XXVIII

Appendix G – Photo Documentation

16 16 AAPPENDIXPPENDIX G – P G – PHOTOHOTO D DOCUMENTATIONOCUMENTATION

16.1 16.1 Linear Test BenchLinear Test Bench

Figure 16.1 – Linear test bench with moving trolley.

Figure 16.2 – Steel plate withtemperature sensors.

Figure 16.3 – Detail of temperature sensorwith inserted shielded thermocouple.

XXIX


16.1.1 Sensor Calibration

Figure 16.4 – Experimental apparatus forsensor calibration.

Figure 16.5 – Detailed structure oftemperature sensor.

XXX


16.2 16.2 Mold CastingMold Casting

16.2.1 Alloy Casting Experiment

Figure 16.6 – Burning of mold.

Figure 16.7 – Pouring alloy into mold before inserting

mold into bath.

Figure 16.8 – Outer surface of mold. Figure 16.9 – Determination ofthermocouple position inside mold.

XXXI


16.2.2 Experiment for Determination of Mold Materia l Properties

Figure 16.10 – Mold equipped with thermocouple sensorsand a heater inside.

Figure 16.11 – Surfacethermocouple on inner surface

of mold.

XXXII


16.3 16.3 Pressure Impact MeasurementPressure Impact Measurement

Figure 16.12 – Pressure impact measurement of narrow water stream.

Figure 16.13 – Circular pressure sensor.

XXXIII

17 17 SSUMMARYUMMARY

A computer analysis of complex problems may provide a cost-effective, quick and

sufficiently reliable method for many applications. Computational methods may also be

a complement to experimental investigations, especially for inverse tasks. Although computer

science development has reached a very high level and a wide range of applications is now

available, research on many topics is needed. Heat transfer computation can be significantly

helpful in many engineering and industrial applications but comprehensive research is still

needed. Thermal numerical models for design and control purposes in metallurgical industry

require a precise description of heat transfer phenomena. Comprehensive quantitative

information on the heat transfer phenomena is not available for quenching of hot moving

surfaces. In this work, attention is focused on the search for boundary conditions describing

the heat transfer in engineering applications of spray cooling of metal surfaces and cooling of

molds during casting. Determination of boundary conditions represents solving inverse heat

conduction problems using experimental data. Combining measurement with an inverse

analysis often results in an ill-posed problem. Such problems are extremely sensitive to

measurement errors. If any results are obtained, they are very inaccurate and the precision is

often unknown. This work is also aimed at estimation of errors caused by discretization in

both space and time domains. Inverse tasks, such as determination of boundary conditions and

temperature dependent thermal material properties, are computed using optimized models.

Combining optimization methods or methods of artificial intelligence with classical

approaches seems to be useful in solving such ill-posed problems. A higher accuracy is

achieved during solution of ill-posed inverse tasks.

A

18 18 RRESÜMEEESÜMEE

Die Computeranalyse technischer Probleme kann eine finanzgünstige, schnelle und

genug zuverlässige Methode bei der Lösung verschiedener Probleme darstellen. Die

Berechnungsmethoden können auch ein Komplement der Experimentalforschung bilden,

insbesondere bei der Lösung der Inversionsaufgaben. Obwohl die Entwicklung in der

Computerwissenschaft bereits ein hohes Niveau erreichen hat und eine Reihe von

Programmsystemen existiert, ist es immerhin notwendig, Forschungen in vielen Richtungen

vorzunehmen. Die Berechnung der Intensität der Wärmeübertragung kann in vielen Ingenieur-

und Industrieapplikationen nützlich sein, aber häufig ist es noch immer eine weitere

Forschung notwendig. Die für den Entwurf und die Steuerung in der metallurgischen Industrie

angewandten nummerischen Modelle der Thermoprozesse erfordern eine genaue

Beschreibung der Bedingungen der Wärmeübertragung und eine Kenntnis der

temperaturabhängigen Materialcharakteristiken. Zusammenfassende und quantitative

Informationen, die die Bedingungen bei der Wärmeübertragung zur Kühlung heißer, sich

bewegender Oberflächen betreffen, sind noch immer nicht verfügbar. In dieser Arbeit befaßt

man sich mit der Feststellung der Randbedingungen, die die Wärmeübertragung in den

Ingenieurapplikationen bei der Kühlung von Metalloberflächen durch Bespritzen und bei der

Kühlung der Formen beim Abgießen beschreiben. Die Bestimmung der Randbedingungen

stellt eine Lösung der Wärmeleitunginversionsaufgabe mit der Ausnutzung von experimentell

gewonnenen Daten dar. Die Kombination einer Messung mit einer Inversionsaufgabe führt oft

zur Lösung eines nicht recht bedingten Problems. Die Berechnungen sind dann sehr

meßdatenfehlerempfindlich. Wenn irgendwelche Ergebnisse gewonnen werden, sind sie sehr

ungenau und oft ist ihre Präzision nicht bekannt.Die Arbeit befaßt sich auch mit der

Berechnung des Maximalfehlers, entstanden durch die Diskretisierung eines kontinuierlichen

Problems sowohl in einer Raum-, als auch in einer Zeitdomäne. Mit Hilfe optimalisierter

Modelle werden dann Inversionsaufgaben zum Gewinn von Randbedingungen und zur

Bestimmung temperaturabhängiger Materialeigenschaften berechnet. Die Kombination der

Optimalisierungs- und der klassischen Methoden, beziehungsweise die Ausnutzung der

Kunstintelligenzmethoden, die in der letzten Zeit immer mehr ausgenutzt werden, scheint bei

der Lösung nicht recht bedingter Probleme sehr nützlich zu sein. So erreicht man höhere

Genauigkeiten bei der Lösung nicht recht bedingter Inversionsaufgaben.

B

19 19 RRESUMÉESUMÉ

Počítačová analýza technických problémů může poskytovat finančně výhodnou, rychlou

a dostatečně spolehlivou metodu při řešení různých problémů. Výpočtové metody mohou být

také doplňkem experimentálního výzkumu, zejména při řešení inverzních úloh. Ačkoliv vývoj

v počítačové vědě dosáhl již velmi vysoké úrovně a existuje řada programových systémů, je

stále zapotřebí provádět výzkum v mnoha směrech. Výpočet intenzity přenosu tepla může být

velmi užitečný v mnoha inženýrských a průmyslových aplikacích, ale často je stále ještě

zapotřebí další výzkum. Numerické modely tepelných procesů v metalurgickém průmyslu pro

návrh a řízení vyžadují přesný popis podmínek přenosu tepla a znalost teplotně závislých

materiálových charakteristik. Souhrnné a kvantitativní informace, které se týkají podmínek při

přenosu tepla pro ochlazování horkých pohybujících se povrchů, ještě stále nejsou k dispozici.

V této práci je zaměřena pozornost na zjišťování okrajových podmínek popisujících přenos

tepla v inženýrských aplikacích při chlazení kovových povrchů ostřikem a chlazení forem při

odlévání. Určení okrajových podmínek představuje řešení inverzní úlohy vedení tepla

s využitím experimentálně získaných dat. Kombinace měření s inverzní úlohou vede často

k řešení špatně podmíněného problému. Následné výpočty jsou pak velice citlivé na chyby

v naměřených datech. Pokud jsou vůbec nějaké výsledky získány, jsou velmi nepřesné a často

není známa chyba výsledku. Tato práce se také zaměřuje na výpočet maximální chyby

vzniklou diskretizací spojitého problému, a to jak v prostoru, tak v čase. Pomocí

optimalizovaných modelů jsou pak počítány inverzní úlohy pro získání okrajových podmínek

a pro určení teplotně závislých materiálových vlastností. Kombinace optimalizačních

a klasických metod, případně využití metod umělé inteligence, které jsou v poslední době

stále více využívány, se zdá být velmi užitečné při řešení špatně podmíněných problémů.

Dosahuje se tak vyšších přesností při řešení špatně podmíněných inverzních úloh.

C

brno university of technology - vysoké …lab4.fme.vutbr.cz/pohanka/pdf/phdthesispohankam.pdf ·...

Documents