Radiative Heat transfer and Applications for Glass Production Processes

Glass 1

Radiative Heat transfer and Applications for Glass Production Processes

Axel Klar and Norbert Siedow

Department of Mathematics, TU Kaiserslautern

Fraunhofer ITWM Abteilung Transport processes

Montecatini, 15. – 19. October 2008

Glass 2

Radiative Heat transfer and Applications for Glass Production Processes Planning of the Lectures

1. Models for fast radiative heat transfer simulation

2. Indirect Temperature Measurement of Hot Glasses

3. Parameter Identification Problems

Glass 3

Indirect Temperature Measurement of Hot Glasses

N. Siedow

Fraunhofer-Institute for Industrial Mathematics,

Kaiserslautern, Germany

Montecatini, 15. – 19. October 2008

Glass 4

Indirect Temperature Measurement of Hot GlassesOutline

1. Introduction

2. Some Basics of Inverse Problems

3. Spectral Remote Sensing

4. Reconstruction of the Initial Temperature

5. Impedance Tomography

6. Conclusions

Glass 5

Models for fast radiative heat transfer simulations 1. Introduction

Temperature is the most important parameter in all stages of glass production

Homogeneity of glass melt Drop temperature Thermal stress

To determine the temperature:

Measurement Simulation

Glass 6

Indirect Temperature Measurement of Hot Glasses 1. Introduction

With Radiation

Without Radiation

Temperature in °C

Conduct

ivit

y in W

/(K

m)

Radiation is for high

temperatures the dominant

process

Heat transfer on a

microscale

Heat radiation on a

macroscale

mm -

cm

nm

Glass 7


Heat transfer on a

microscale

Heat radiation on a

macroscale

mm -

cm

nm

( , ) ( ( ) ( , )) , ( , )m m t

Tc r t k r T r t r t D

t

)(Tqr

( , , ) ( ) ( , , ) ( ) ( ( , ), )I r I r B T r t

20

( , , )r

S

q I r d d

0( ,0) ( ),T r T r r D

+ boundary conditions

),())(1(),',()(),,( agg TBrIrI

Glass 8


Direct Measurement

• Thermocouples

Indirect Measurement

• Pyrometer(surface temperature)

• Spectral Remote Sensing

Glass 9


Glass is semitransparent

Inverse ProblemSpectrometerT(z)

[µm]10 2 3

0.8

4

1

0.6

0.4

0.2

0

Emissivity

[°C]

Depth [mm]10 2 3 4

1000

950

900

Spectral Remote Sensing

Glass 10

Inverse Problems are concerned with finding causes for an observed or a desired effect.

Control or Design, if one looks for a cause for an desired effect.

Identification or Reconstruction, if one looks for the cause for an observed effect.

Indirect Temperature Measurement of Hot Glasses 2. Some Basics of Inverse Problems

Glass 11

Example 1:

Black Box

Input Signal

Output Signal - Measurement

I gf

0

( ) ( ) ( ) ( )x

Af x I x t f t dt g x

Assume: 1I 0

( ) ( ) ( )x

Af x f t dt g x If: • Continuous differentiable

(0) 0g •

Solution: ( ) '( )f x g x


Glass 12

Example 1:

21( )

2g x x ( )f x x

We find:Given is:

• analytically

• exact


Glass 13

Example 1:

21( )

2( )

g x x

f x x

2

4 5 4

1( ) ( )

2

( ) 10 sin(10 ) 10

g x x x

x x

A small error in the measurement causes a big error in the reconstruction!


Glass 14

Example 1: Numerical Differentiation

In praxis the measured data are finite and not smooth

1( ), , 1, 2,...,i i i i ig g x h x x i n ig

( )i if f x 1 1

2i i

i h i

g gf D g

h


Glass 15

Example 1: Numerical Differentiation

1%, 0.1h

1%, 0.01h

• A finer discretization leads to a bigger error


Glass 16

Example 2: Parameter Identification

0 1( ) ( ) ( ), 0 , (0) (0) , ( )u u

a x x f x x l a g u l gx x x

Practical meaning: Heat transfer equation

( )u x Temperature ( )a x Thermal conductivity

Diffusion equation

( )u x Concentration ( )a x Diffusivity

„Black-Scholes“ equation

( )u x Option price ( )a x Stock price


Glass 17


0 1( ) ( ) ( ), 0 , (0) (0) , ( )u u

a x x f x x l a g u l gx x x

Practical meaning: Electrical potential equation

( )u x Electrical potential

( )a x Electrical conductivity

0

0

( )

( )

x

g f y dy

a xux

Knowing the potential find the conductivity

Elasticity equation

( )u x displacement ( )a x Youngs Modulus


Glass 18


( ) ( ) 4 2, 0 1, (0) (0) 4, ( ) 4u u

a x x x x a u lx x x

0

0

( )

( )

x

g f y dy

a xux

( )u x

0%

Exact Measurement


Glass 19

( ) ( ) 4 2, 0 1, (0) (0) 4, ( ) 4u u



0

0

( )

( )

x

g f y dy

a xux

( )u x

0.001% 0.001%

Noisy Measurement


Glass 20

( ) ( ) 4 2, 0 1, (0) (0) 4, ( ) 4u u



0

0

( )

( )

x

g f y dy

a xux

( )u x

0.01%

Noisy Measurement

0.01%


Glass 21

Example 4:

1 1.99999801 62 1.99 10 8

3 1.0 10 0x 1x

1 8f 2 4f 3 0f

( ) ( ) ( ), 0 ,

(0) (0) 0, ( ) ( ) ( )

ux x f x x l

x x

u ul l u l

x x

1

2

2

1 1 0 1

1 1.000001 0.000001 1

0 0.000001 1.000001 0

u

u

u

0.01

0.01

0

Reconstruction:

1

2

2

20001.03

20000.02

0.000002

u

u

u


Glass 22

A common property of a vast majority of Inverse Problems is their ill-posedness

A mathematical problem is well-posed, if

Hadamard (1865-1963)

1. For all data, there exists a solution of the problem.

2. For all data, the solution is unique.

3. The solution depends continuously on the data.

A problem is ill-posed if one of these three conditions is violated.


Glass 23

What is the reason for the ill-posedness?

Example 1: ( ) '( )f x g x

( ) ( ) sin( )g x g x nx

4 510 , 10n

( ) ( ) cos( )f x f x n nx n

A small error in measurement causes a big error in reconstruction


Glass 24


Example 1: ( )i h if x D g

2

''6h

hD g f f

h

optimalh

Step size must be taken with respect to the measurement error


Glass 25


Example 2:

0

0

( )

( )

x

g f y dy

a xux

Numerical differentiation of noisy data


Glass 26


Example 4:

1

2

3

1 1 0 1

1 1.000001 0.000001 1

0 0.000001 1.000001 0

u

u

u

Eigenvalues:

61 2 30.5 10 , 1.000001, 2.0000005

Condition number: 64 10


Glass 27


Example 4:

1

2

3

1 1 0 1

1 1.000001 0.000001 1

0 0.000001 1.000001 0

u

u

u

Eigenvalues:

61 2 30.5 10 , 1.000001, 2.0000005

1 2 3, ,v v vLet be the eigenvectors

31

1

( , )i i ii

u f v v

The solution can be written as:

A small error in f f 3 3

1 1

1 1

( , ) ( , )i i i i i ii i

u f v v v v

2

61

1020000

0.5 10


Glass 28

What can be done to overcome the ill-posedness?

Regularization

31

6

0.5

( , ) 0.5

0.5 10i i iu f v v

2i

Regularization Methods

1. Truncated Singular Value Decomposition

We skip the small eigenvalue (singular values)

identical to the minimization problem

2( ) min

LJ u Au f

and take the solution with minimum norm

2min

Lu

Replace the ill-posed problem by a family of neighboring well-posed problems


Glass 29


2. Tichonov (Lavrentiev) Regularization

We look for a problem which is near by the original and well-posed

We increase the eigenvalues

3

1

1( , )i i

i i

u f v v

How to choose ?


Glass 30

2Lu

* 0.01977

2LAu f


2. Tichonov (Lavrentiev) Regularization

3

1

1( , )i i

i i

u f v v

*

8

1.0004

0.0102

1 10

u

Take 0n

L-curve method


Glass 31


3. Landweber Iteration

* *A Au A fWe consider the normal equation

1 * 0( ),k ku u A Au f u givenUse a fixed point iteration to solve

Iteration number plays as regularization parameter

1

k

Stopping rule = discrepancy principle

*( , ) inf |k f k N Au f

Solution after 4 iterations: * 70.5 0.5 1 10Tku


Glass 32


4. Classical Tichonov Regularization

* *A Au Iu A f Regularization of the normal equation

Equivalent to the minimization problem

2 2

2 2( , ) min

L LJ u Au f u

Tichonov (1906-1993)

Dealing with an ill-posed problem means to find the right balance between stability and accuracy


Glass 33




Equivalent to the minimization problem 2 2

2 2( , ) min

L LJ u Au f u

L-curve

7

0.4965

0.4965

5 10

u

7

0.5

0.5

5 10

u

Discrepancy principle


Glass 34




Equivalent to the minimization problem 2 2

2 2( , ) min

L LJ u Au f u

2 2

2 2( , ) 1 min

L LJ u Au f u

To get a better solution we need to include more information!

Assume:2

21

Lu

9

1.00104

0.00104

9 10

u


Glass 35

Indirect Temperature Measurement of Hot Glasses 3. Spectral Remote Sensing

formal solution:

s

s

D

aag

dsdseeB

TBeTBnee

sDs

DD

D

0

2

0

122

22

'),(},])[,(),(

)],(),([)1{(1

1

)'(sTT T

)(mI

Non-linear, ill-posed integral equation of 1. kind

One-dimensional radiative transfer equation

),()(),,()(),,( TBzIzz

I

Glass 36

Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement

Formal solution:

s

s

D

aag

dsdseeB

TBeTBnee

sDs

DD

D

0

2

0

122

22

'),(},])[,(),(

)],(),([)1{(1

1

)'(sTT T

)(mI

mITG )(

Linearization: )]()[(*)'())((')(*)'( 1 km

kkkkk TGITGTTTGTG

)]()[(*)'()]()(')(*)'[( 1 km

kkkkk TGITGTTTGTG AkRegularization:

Glass 37


(Rosseland-Approximation)

0

)),,(()(

1

3

4

dtzT

T

Bkr),,( tz

z

Tkq rr

is FDA ofA

cqz rq

Temperature satisfies the radiative heat transfer equation

Radiative Flux

Iteratively regularized Gauss – Newton method:

)]())()((*)'[(])(')(*)'[( 11 skk

mkkkkk TTTGITGTGTGTT

Ak

k

How to choose ?

A?

Glass 38


Discrepancy principle

Iteratively regularized Gauss – Newton method:

)]())()((*)'[(])(')(*)'[( 11 skk

mkkkkk TTTGITGTGTGTT

Ak

k

How to choose ?

k? 0lim,1,01

k

kk

kk r

Stopping rule for k?

?

1,0)()(

nkITFITF kn

II m

Glass 39


Furnace Experiment

Furnace

Glass slab

Thermocouples

Glass 40


Drop Temperature

Glass 41


The Improved Eddington-Barbier-Approximation

If we assume that

D

0

)()),(()()1()( dsesTBI sm

D

sDsag

Dm dseesTBTBneI0

22

2 }])[),(()(),()1){(1()(

Glass 42


The Improved Eddington-Barbier-Approximation

If we assume that

D

0

)()),(()()1()( dsesTBI sm

)1)(1)(1(

),()1()()),((

)()(

22)(2

DD

agDm

ee

TBneIzTB

)1)(1)((

)1()()1)(1()()(

)()()(

DD

DDD

ee

eDeez

2/0 Dz

D

sDsag

Dm dseesTBTBneI0

22

2 }])[),(()(),()1){(1()(

Glass 43


A fast iterative solution of the integral equation

i. For :0l2

0),()(0 DzzTzT E Calculate using the IEB-method

iii. For :1ll Use to calculate

)(zT l

ii. Using some additional information continue to )(0 zT Dz 0

D

sDslag

Dl dseesTBTBneI0

22

2 }])[),(()(),()1){(1()(

Glass 44


A fast iterative solution of the integral equation

v. Calculate a new temperature profile in the IEB points using

)(1 zT l

20

Dz

)1)(1)(1(

)()()),(()),((

)()(1

DD

lmll

ee

IIzTBzTB

vi. Using the additional information continue to )(1 zT l Dz 0

and go back iii.

iv. If then STOP else continue with v. )()( lm II

Glass 45


Iteration 1

Glass 46


Iteration 2

Glass 47


Iteration 3

Glass 48


Iteration 4

Glass 49


Iteration 5

Glass 50


Iteration 10

Glass 51


Error 1%

Glass 52


Error 1%

Glass 53

Indirect Temperature Measurement of Hot Glasses 4. Reconstruction of Initial Condition

Initial condition

Boundary condition

Temperature

+ Additional Measurements of boundary temperature / heat flux

Is it possible to reconstruct the initial temperature distribution from boundary measurements?

?

Glass 54


*

0

0 (0, *) (0, )

(0, ) (0, )

( ,0) 0 (0, *)

( , ) 0 (0, *)

t xx t

x

x

u u in Q t

u x u on

u t t t

u t t t

Additional measurement at boundary

0Au y

( ,0)u t y

Problem is ill-posed:

• Solution exists

• Solution is unique

• Continuous dependence on right hand side is violated

Glass 55


Backward Heat Sideways Heat New Heat Problem

Given:

2 conditions at y2 Additional bc at yTemperature at T

Looking for:

bc at y1 Initial conditionInitial condition

Glass 56


2

00

( , ) ( , ) ( )n tn n

n

u t x u w e w x

0Au y


• Solution exists



1 0( )

2 cos( ) 0n

nw x

nx n

For2

( ) ( ) 2k k ty t y t k e

One obtains 0 0 2 cos( )ku u k kx

Glass 57


2

00

( , ) ( , ) ( )n tn n

n

u t x u w e w x

0Au y


• Solution exists



1 0( )

2 cos( ) 0n

nw x

nx n

For ( ) ( ) 0ky t y t

One obtains 0 0ku u

Glass 58

• Given data


hy Y

0Au y

Solve the system by Tichonov regularization

• While searching for the correct

Solve * *0,h h hA A I u A y

Adjust according to the parameter choice rule

with

hYy y

by some method

• is the regularized solution 0,1

N

n nn

u a u

Glass 59


*

0

0 (0, ) (0, )

(0, ) (0, )

( ,0) 0 (0, )

( , ) 0 (0, )

t xx t

x

x

u u in Q T

u x u on

u t t T

u t t T

L. Justen. An Inverse Heat Conduction Problem with Unknown Initial Condition. Diploma Thesis, TU KL, 2002

9( ) ( ,0) ty t u t e

Exact solution:

0 ( ) cos(3 )u x x

Tichonov regularization with Morozov‘s Discrepancy stopping rule

Glass 60


2D Example: Parabolic Profile – Four-Sided Measurement

Glass 61


Glass 62


S.S. Pereverzyev, R. Pinnau, N. Siedow. Proceedings of 5th Conf. on Inverse Problems in Engineering, 2005

S.S. Pereverzyev, R. Pinnau, N. Siedow. Inverse Problems, 22 (2006), 1-22

Initial Temperature Reconstruction for a Nonlinear Heat Equation:

Application to Radiative Heat Transfer PhD S.S. Pereverzyev

124

21

( , ) ( ), ( ) 4 2

( , ) ( ( , )),

( ,0) ( )

b b

T Tx t D T D T T Id

t x

Tx t B T x t

nT x u x

( ) : ( )D u D T : ( , ( ))Fu L u D u y

Glass 63


S.S. Pereverzyev, R. Pinnau, N. Siedow. Proceedings of 5th Conf. on Inverse Problems in Engineering, 2005

S.S. Pereverzyev, R. Pinnau, N. Siedow. Inverse Problems, 22 (2006), 1-22

: ( , ( )) ( ,0) (0, ( )) :Fu L u D u L u L D u Au Gu y

Decomposition of the non-linear equation:

Linear part non-linear

measurement

Fixed-point iteration: 1k kAu y Gu

Tichonov regularization:

* *1k kA A I u A y Gu

Glass 64


Parameter choice rule: Quasi-optimality criterion

• It does not depend on the noise level

1. Select a finite number of regularization parameters

0 10 ... m

which are part of a geometric sequence, i.e.

0 , 1ii q q

2. For each solve and obtaini * *1,( ) ( )

ii k kA A I u A y Gu 1, iku

3. Among 1, 0i

m

k iu

choose 1, jku such that

21 11, 1, 1, 1, (0, )min , 1,2,...,

j j i ik k k k L lu u u u i m

Glass 65


Glass 66


Reconstruction is dependent on

• noise level

• discretization

How to choice the right discretization parameters depending on the noise level?

Glass 67

Indirect Temperature Measurement of Hot Glasses 5. Impedance Tomography

The knowledge of the temperature of the glass melt is important to control the homogeneity of the glass

Glass melting in a glass tank

Glass 68


• Thermocouples at the bottom and the sides of the furnace

• Use of pyrometers is limited due to the atmosphere above the glass melt

Glass 69

Glass melt

Determine the temperature of the glass melt during the melting process using an impedance tomography approach

23.2))(lg(890353)(

xxT

( )x

applyElectric current

measure Voltage

Neutral wire

Experiment

electrode


Glass 70

Inverse Problems are concerned with finding causes for an observed or a desired effect.

A common property of a vast majority of Inverse Problems is their ill-posedness(Existence, Uniqueness, Stability)

To solve an ill-posed problem one has to use regularization techniques(Replace the ill-posed problem by a family of neighboring well-posed problems)

The regularization has to be taken in accordance with the problem one wants to solve

Indirect Temperature Measurement of Hot Glasses 6. Conclusions

Glass 71

Further Examples of Inverse Problems:

Computerized Tomography

0

ln ( , )n

L

R

Idt s w

I ( , , )f s w t

Inverse Scattering

2 2 ( ),s s i su k u k u u f 21 ( ),f n x

2

( ) ( ) ,4

ik x ys ik e

u x u y dy x Rx y

( )f y

0I

LI

iusu

( ) 1/ ( )n x c x


Glass 72

Example 3:

1.0 0x 1x

1f 2f 3f

0 1

( ) ( ) ( ), 0 ,

(0) (0) , ( ) ( )

ux x f x x l

x x

u ug l l g

x x

1

0

(1) (1) (0) (0) ( ) 0u u

f x dxx x

0 1

0

( ) ,l

f x g g If exists no solution

0 1

0

( ) ,l

f x g g If exist an infinite number of solutions

Gauge condition

1

1 0

0

( ) 0g g f x dx

Integration


Glass 73

Example 4:

1 1.99999801 62 1.99 10 8

3 1.0 10 0x 1x

1 8f 2 4f 3 0f

( ) ( ) ( ), 0 ,

(0) (0) 0, ( ) ( ) ( )

ux x f x x l

x x

u ul l u l

x x

1

2

2

1 1 0 1

1 1.000001 0.000001 1

0 0.000001 1.000001 0

u

u

u

Exact Solution:

1

2

2

1

0

0

u

u

u

1


Glass 74


Example 3:

1 0 12

2 2

3 1 3

1 1 0

1 2 1 0 22

0 1 1

u g fh

u h f

u g f

System is singular

1 0 12

2 0 1 2

3 0 1 1 2 3

1 1 0

0 1 1 22

0 0 0 2

u g fh

u h g f f

u g g f f f


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Radiative Heat transfer and Applications for Glass Production Processes

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