glass 1 radiative heat transfer and applications for glass production processes axel klar and...

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

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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

Parameter Identification Problems

N. Siedow

Fraunhofer-Institute for Industrial Mathematics,

Kaiserslautern, Germany

Montecatini, 15. – 19. October 2008

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Parameter Identification ProblemsOutline

1. Introduction

2. Some Basics

3. Shape Optimization of Pipe Flanges

4. Impedance Tomography

5. Further Examples

6. Optimization of Thermal Stresses

7. Conclusions

Glass 5

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

Conductivity is unknown

Additional information: ( ) , 1, 2,...,i iu x u i n Measurement , 1, 2,...,

iu i n

Formally we can write ( )F a u or2

( ) minF a u

We have to calculate derivatives!

'( ) 2 ( ) 0J a F a u *'( )F a

Parameter Identification Problems 1. Some Basics

Glass 6

A very convenient way of calculating derivatives is the Adjoint Method

2minu u subject to ( ; ) 0e u a (Partial Differential Equation)

Lagrangian:2

( , , ) ( ; ),L a u p u u e u a p

Derivatives:

( ; ) 0L

e u ap

*

2( ) ( ; ) 0L e

u u u a pu u

*

( ; ) ...L e

u a pa a

*'( ) ( ) 0F a F a u

State Equation

Adjoint Equation

Parameter Equation


Glass 7

Example 2:


Lagrangian: 20 0

( , , ) ( ) ( ) ( ) ( ) ( )l l u

L a u p u x u x dx a x x p x dxx x

Derivatives:

( ) ( ) ( )u

a x x f xx x

( ) ( ) 2( ( ) ( ))p

a x x u x u xx x

( ) ( ) ( ) ( )new u pa x a x x x

x x

State Equation

Adjoint Equation

Parameter Equation

+ b. c.

+ b. c.


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Example 2:


1%


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Example 2:

subject to ( ; ) 0e u a (Partial Differential Equation)2

minu u 2u c

1%


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Electric heating to keep the glass at desired temperature

• Control temperature (e.g. to avoid solidification)

• Control the input current (hot spots, cold shocks)

Shape Optimization

Constraints:

Objective:

gT

Parameter Identification Problems 2. Shape Optimization of Pipe Flanges

Glass 11

min)(2

1 2 dxTxT

S

g

Find a surface shape of the flange so, that in a small section near the pipe boundary:

Under certain constraints:

• Equation of electrical potential (1)

• Heat transfer equation (2)

• . . .

S


Glass 12

xxe

,0)(

1

eaae

xxxCxn

,0)(,,)(1

xxTke

t

,

)()(

2

aaeggt xxTxTxTxTxn

Tk

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

(2)

(1)


3D Model

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• Flange is very thin compared to the other dimensions

• Assume: the shape of the flange is symmetric with respect to z

electrical potential: 01

2

2

2

2

2

2

zyx ee

,,, zzyyxx - small parameter 1


Asymptotic Approach

Glass 14

Electrical potential: ...),,(),(),,( 10 zyxyxzyx

),(,0),(),( 00 yxy

yxhyx

yxhx

+ boundary

conditions

),(,)(),(

),(),(2

000 yxk

yxh

y

Tyxh

yx

Tyxh

x et

+ boundary conditions

Heat transfer: ...),,(),(),,( 10 zyxTyxTzyxT

(3)

(4)


Asymptotic Approach

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Find of the flange so, that in near the pipe

Under certain constraints:

• Equation of electrical potential (3)


• . . .

S),( yxh

xdTyxTS

g

2

0 ),(2

1 minxdyxh

),(2

2

• Minimal material


Parameter Optimization

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Lagrangian method:

• Potential equation (3)


• Total Lagrangian

• Necessity condition

),...,,,,( 00 ThL p v

0L

• Adjoint potential equation

• Adjoint heat transfer equation


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Before optimization After optimization

Thickness


Example

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Temperature


Example

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Heat Flux


Example

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

Glass melting in a glass tank

Parameter Identification Problems 3. Impedance Tomography

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• Thermocouples at the bottom and the sides of the furnace

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


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Glass melt

Determine the temperature of the glass melt during the melting process

23.2))(lg(890353)(

x

xT

( )x

applyElectric current

measure Voltage

Neutral wire

Experiment

electrode


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The forward Problem

1

0

( ) ( ) 0,

( ) ( ) ,

( ) ( ) ,

( ) 0,

( ) ( ) 0,

j

j

j

j

j

j

j

x x x

Ix x x

n r

Ix x x

n r

x x

x x xn

1 0( )j j

0

( )

( )j

x

x

I

r

Glass 24


The forward Problem

Electric potential Electric current density

Glass 25


The inverse Problem

Find so that( )x 2, 2

1 1

1( ) ( ) ( ) min

2 2

NP NEXPj j mes

ii j

E x dx g x dx

under the constraints that

2. is solution of the (so-called) form equation( )x

1. is solution of the potential equation( )j x

0

( ) ( ),

( ) ( ),

x g x x

x x x

Looking for a smooth solution

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The inverse Problem

• Potential Equation

( ) ( ) ( ),

( ) 0,

j j

j

w x x v x x

w x x

• Adjoint Potential Equation

,

0

( ) 0,

( ) ( ) ,

( ) 0,

( ) 0,

j

jj j mes

i i

j

j

v x x

vx x x

n

v x x

vx x

n

0i

• Adjoint Form Equation

• New Form Function

( ) ( ) ( ) ( ) ,newg x g x g x w x x

Glass 27

)2

1()

2

1()3sin(25625 zzyyx

CxTC 1445)(1400

Example

CxTC 1445)(1401

99.25)(02.24 x26)(24 x

original

Reconstruction


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Heat Transfer Coefficient

0( , ) ( , ) , ( , ) ( ( , ) ( , )), ( ,0) ( )t au x t a u x t f an u x t u x t u x t u x u x

Dip Experiment

Parameter Identification Problems 4. Further Examples

Glass 29

Brinkmann, Siedow. „Heat Transfer between Glass and Mold During Hot Forming.“ In Krause, Loch: Mathematical Simulation in Glass Technology; Springer 2002

Heat Transfer Coefficient


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Initial condition

( , ) ( , ) , ( , ) ( ( , ) ( , )), ( ,0)t au x t a u x t f a u x t u x t u x t u x

0 ( )u x

Boundary condition

( , ) ( , ) , ( , ) ( ( , ) ), ( ,0) ( )tu x t a u x t f a u x t u x t u x u x

( , )au x t

Control Problem


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Wrong cooling of glass and glass products causes large thermal stresses

Undesired crack

Parameter Identification Problems 6. Optimization of Thermal Stresses

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Thermal tempering consists of:

1. Heating of the glass at a temperature higher the transition temperature

2. Very rapid cooling by an air jet

Better mechanical and thermal strengthening to the glass by way of the residual stresses generated along the thickness

N. Siedow, D. Lochegnies, T. Grosan, E. Romero, J. Am. Ceram. Soc., 88 [8] 2181-2187 (2005)


Glass 33

How to control the heating to achieve a predefined temperature profile inside the glass products?


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Linear Cooling Optimized Cooling

Minimize thermal stresses during the cooling process


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Used Cooling Thermal Stress

Minimize the cooling time with constraint that permanent thermal stress < 3.5 Mpa


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-24%691s

915s

Used Cooling Thermal Stress

Minimize the cooling time with constraint that permanent thermal stress < 3.5 Mpa


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Optimal Mould Design during Pressing

How to design the mould that after cooling the glass lens has the desired shape?

Sellier, Breitbach, Loch, Siedow.“An iterative algorithm for optimal mould design in high-precision compression moulding.“ JEM606, IMechE Vol.221,2007, 25-33


Glass 38

Deviation of the upper glass side from the desired shape


Deviation of the lower glass side from the desired shape

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Parameter Identification Problems are inverse problems

• Ill-posed Regularization

We have discussed different Parameter Identification Problems

For constraint optimization problems the Lagrangian approach is very convenient for calculating derivatives

Parameter Identification Problems 7. Conclusions

glass 1 radiative heat transfer and applications for glass production processes axel klar and...

Documents

potential equation

heat transfer equation

d model slide

pipe boundary

surface shape

z electrical potential

adjoint method subject

lagrangian method