Radiative Heat transfer and Applications for Glass Production Processes

transcript

Glass 1

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

y in W

Radiation is for high

temperatures the dominant

process

Heat transfer on a

microscale

Heat radiation on a

macroscale

Glass 7

Heat transfer on a

microscale

Heat radiation on a

macroscale

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

Tc r t k r T r t r t D

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

( , , )r

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

Emissivity

Depth [mm]10 2 3 4

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

( ) ( ) ( ) ( )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:

2g x x ( )f x x

We find:Given is:

• analytically

• exact

Glass 13

Example 1:

1( ) ( )

( ) 10 sin(10 ) 10

g 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

g gf D g

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

g f y dy

Knowing the potential find the conductivity

Elasticity equation

( )u x displacement ( )a x Youngs Modulus

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

a x x x x a u lx x x

g f y dy

( )u x

Exact Measurement

Glass 19

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

g f y dy

( )u x

0.001% 0.001%

Noisy Measurement

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

g f y dy

( )u x

Noisy Measurement

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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

u ul l u l

1 1 0 1

1 1.000001 0.000001 1

0 0.000001 1.000001 0

Reconstruction:

20001.03

20000.02

0.000002

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

hD g f f

optimalh

Step size must be taken with respect to the measurement error

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Example 2:

g f y dy

Numerical differentiation of noisy data

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Example 4:

1 1 0 1

1 1.000001 0.000001 1

0 0.000001 1.000001 0

Eigenvalues:

61 2 30.5 10 , 1.000001, 2.0000005

Condition number: 64 10

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Example 4:

1 1 0 1

1 1.000001 0.000001 1

0 0.000001 1.000001 0

Eigenvalues:

61 2 30.5 10 , 1.000001, 2.0000005

1 2 3, ,v v vLet be the eigenvectors

( , )i i ii

u f v v

The solution can be written as:

A small error in f f 3 3

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

u f v v v v

1020000

0.5 10

Glass 28

What can be done to overcome the ill-posedness?

Regularization

( , ) 0.5

0.5 10i i iu f v v

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

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

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2. Tichonov (Lavrentiev) Regularization

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

We increase the eigenvalues

1( , )i i

u f v v

How to choose ?

Glass 30

* 0.01977

2LAu f

2. Tichonov (Lavrentiev) Regularization

1( , )i i

u f v v

1.0004

0.0102

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

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( , ) 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

0.4965

Discrepancy principle

Glass 34

Equivalent to the minimization problem 2 2

2 2( , ) min

L LJ u Au f u

2 2( , ) 1 min

L LJ u Au f u

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

Assume:2

1.00104

0.00104

Glass 35

Indirect Temperature Measurement of Hot Glasses 3. Spectral Remote Sensing

formal solution:

dsdseeB

TBeTBnee

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

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

)'(sTT T

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

One-dimensional radiative transfer equation

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

Glass 36

Indirect Temperature Measurement of Hot Glasses 3. Indirect temperature measurement

Formal solution:

dsdseeB

TBeTBnee

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

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

)'(sTT T

mITG )(

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

kkkkk TGITGTTTGTG

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

kkkkk TGITGTTTGTG AkRegularization:

Glass 37

(Rosseland-Approximation)

)),,(()(

Bkr),,( tz

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

How to choose ?

Glass 38

Discrepancy principle

Iteratively regularized Gauss – Newton method:

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

mkkkkk TTTGITGTGTGTT

How to choose ?

k? 0lim,1,01

Stopping rule for k?

1,0)()(

nkITFITF kn

Glass 39

Furnace Experiment

Furnace

Glass slab

Thermocouples

Glass 40

Drop Temperature

Glass 41

The Improved Eddington-Barbier-Approximation

If we assume that

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

Dm dseesTBTBneI0

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

Glass 42

The Improved Eddington-Barbier-Approximation

If we assume that

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

)1)(1)(1(

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

TBneIzTB

)1)(1)((

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

)()()(

2/0 Dz

Dm dseesTBTBneI0

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

sDslag

Dl dseesTBTBneI0

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

)1)(1)(1(

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

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

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Iteration 3

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Iteration 4

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Iteration 5

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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, *)

t xx t

u u in Q t

u x u on

u t t t

Additional measurement at boundary

( ,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

( , ) ( , ) ( )n tn n

u t x u w e w x

• Solution exists

1 0( )

2 cos( ) 0n

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

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

Glass 57

( , ) ( , ) ( )n tn n

u t x u w e w x

• Solution exists

1 0( )

2 cos( ) 0n

For ( ) ( ) 0ky t y t

One obtains 0 0ku u

Glass 58

• Given data

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

by some method

• is the regularized solution 0,1

Glass 59

0 (0, ) (0, )

(0, ) (0, )

( ,0) 0 (0, )

( , ) 0 (0, )

t xx t

u u in Q T

u x u on

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

( , ) ( ), ( ) 4 2

( , ) ( ( , )),

( ,0) ( )

T Tx t D T D T T Id

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

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)(

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

ln ( , )n

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

( ) ( ) ,4

ik x ys ik e

u x u y dy x Rx y

( )f y

( ) 1/ ( )n x c x

Glass 72

Example 3:

1.0 0x 1x

1f 2f 3f

( ) ( ) ( ), 0 ,

(0) (0) , ( ) ( )

ux x f x x l

u ug l l g

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

f x dxx x

( ) ,l

f x g g If exists no solution

( ) ,l

f x g g If exist an infinite number of solutions

Gauge condition

( ) 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

u ul l u l

1 1 0 1

1 1.000001 0.000001 1

0 0.000001 1.000001 0

Exact Solution:

Glass 74

Example 3:

1 0 12

1 2 1 0 22

u g fh

System is singular

1 0 12

2 0 1 2

3 0 1 1 2 3

0 1 1 22

0 0 0 2

u g fh

u h g f f

u g g f f f

Radiative Heat transfer and Applications for Glass Production Processes

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