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

ITWM Activities in GlassGlassmaking

Form of the gob(FPM)

Shape optimization of thermal-electrical flanges

Gob temperature(Spectral remote

sensing)

Coupling of glass tank with electrical

network

Temperature(Impedance

Tomography)PATENT

Glass 3

ITWM Activities in GlassGlassprocessing I

PressingTV panelsLenses

Floatglasswindow glassesdisplay glasses

BlowingBottles

Foaming

Fiberproduction

Interface Glass-Mould (Radiation)

Identification of the heat transfer coefficient

High precision forming

. . .Wavyness of thin display glasses

Minimization of thermal stresses

Fluid-Fiber-Interaction

Optimal shape of the furnace

Glass 4

ITWM Activities in GlassGlassprocessing II

Tempering of glass

Free cooling

Cooling in a furnace

Simulation of temperature field

Control of furnace temperature to minimize

the thermal stress

Glass 5

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 6

Models for fast radiative heat transfer simulations

N. Siedow

Fraunhofer-Institute for Industrial Mathematics,

Kaiserslautern, Germany

Montecatini, 15. – 19. October 2008

Glass 7

Models for fast radiative heat transfer simulationsOutline

1. Introduction

2. Numerical methods for radiative heat transfer

3. Grey Absorption

4. Application to flat glass tempering

5. Conclusions

Glass 8

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 9


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 10


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 11

Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer

Heat transfer on a

microscale

Heat radiation on a

macroscale

mm -

cm

nm Rosseland-Approximation

ITWM-Approximation-Method

PN-Approximation

Discrete-Ordinate-Method (FLUENT)

• Radiation = Correction of Conductivity

Glass 12


We study the optically thick case. To obtain the dimensionless form of the rte we introduce

1

ref refx

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

which is small in the optically thick – diffusion – regime.

and define the non-dimensional parameter

' / ' /ref refr r r

( ', , ) ( ', , ) ( , )'

I r I r B T

Glass 13


We rewrite the equation

And apply Neumann‘s series to (formally) invert the operator

E I B

12( )I E B E O B

0

4 1( , ) ( ) ( , )

3m m

T Bc r t k r T r t

t T

Rosseland-Approximation

Glass 14

• Treats radiation as a correction of heat conductivity

• Very fast and easy to implement into commercial software packages

• Only for optically thick glasses

• Problems near the boundary


0

4 1( , ) ( ) ( , )

3m m

T Bc r t k r T r t

t T

Rosseland-Approximation

BUT • Standard method in glass industry

Glass 15


Heat transfer on a

microscale

Heat radiation on a

macroscale

mm -

cm



PN-Approximation



• Spherical Harmonic Expansion

Glass 16


Larsen, E., Thömmes, G. and Klar, A., , Seaid, M. and Götz, T., J. Comp. Physics 183, p. 652-675 (2002).

Thömmes,G., Radiative Heat Transfer Equations for Glass Cooling Problems: Analysis and Numerics. PhD, University Kaiserslautern, 2002

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

optical thickness (small

parameter)

1

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

Neumann series

2 3 42 3 4

2 3 4( , , ) ( ) ( ) ( ) ... ( ( , ), )I r E B T r t

Glass 17


2 14

3G G B

SP1-Approximation O(4)

2 1( 2 ) 4

3G U G B

SP3-Approximation O(8)

2 9 2 8

35 5 5U U G B

identical to P1-Approximation

coupled system of equations

Glass 18


Example: Cooling of a glass plate

Parameters:

Density 2200 kg/m3 Specific heat 900 J/kgKConductivity 1 W/KmThickness 1.0 mSurroundings 300 K gray mediumAbsorption coefficient: 1/m

Glass 19


Heat transfer on a

microscale

Heat radiation on a

macroscale

mm -

cm



PN-Approximation




• Full-discretization method Klar

Glass 20


Heat transfer on a

microscale

Heat radiation on a

macroscale

mm -

cm



PN-Approximation




• Full-discretization method

Glass 21



),(

0

),( ))((),(),(xd

skk

xdb

kk dsesxTBexIxI kk Formal solution:

with

1

),,(),(k

k

dxIxI k

1

)),(())((k

k

dxTBxTBk

1.)( kkk const

Taylor Approximation with respect to

x

Glass 22



),(

0

),( ))((),(),(xd

skk

xdb


with

1

),,(),(k

k

dxIxI k

1

)),(())((k

k

dxTBxTBk

1.)( kkk const

)(111

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

dBedexTBexIxI

kd

kk

dkxdb

kk kkk

Glass 23



),(

0

),( ))((),(),(xd

skk

xdb


with

1

),,(),(k

k

dxIxI k

1

)),(())((k

k

dxTBxTBk

1.)( kkk const

1( , ) ( ( )) ( )

kk k

k

dBI x B T x T x

dT Rosseland:

Glass 24



),(

0

),( ))((),(),(xd

skk

xdb


with

1

),,(),(k

k

dxIxI k

1

)),(())((k

k

dxTBxTBk

1.)( kkk const

)(111

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

dBedexTBexIxI

kd

kk

dkxdb

kk kkk

Glass 25


2

( ) ( , )1 1 ( ) ( , )k d x T

S

A d x e d

dxTeTdT

dB

dexITBtxTAdT

dBxq

dMK

k S

k

xd

S

bkk

MK

kk

MK

k

kk

kr

k

k

)(1)(

),()(),(1

)(

1

),(

11

2

2

Improved Diffusion Approximation

Lentes, F. T., Siedow, N., Glastech. Ber. Glass Sci. Technol. 72 No.6 188-196 (1999).

• In opposite to Rosseland-Approximation all geometrical information is conserved

Glass 26


2

( ) ( , )1 1 ( ) ( , )k d x T

S

A d x e d

dxTeTdT

dB

dexITBtxTAdT

dBxq

dMK

k S

k

xd

S

bkk

MK

kk

MK

k

kk

kr

k

k

)(1)(

),()(),(1

)(

1

),(

11

2

2


Lentes, F. T., Siedow, N., Glastech. Ber. Glass Sci. Technol. 72 No.6 188-196 (1999).

• Correction to the heat conduction due to radiation with anisotropic diffusion tensor

Glass 27


2

( ) ( , )1 1 ( ) ( , )k d x T

S

A d x e d

dxTeTdT

dB

dexITBtxTAdT

dBxq

dMK

k S

k

xd

S

bkk

MK

kk

MK

k

kk

kr

k

k

)(1)(

),()(),(1

)(

1

),(

11

2

2


• Boundary conditions

• Convection term

Glass 28


1( , ) ( , ) ( ), ,

( , ) ( ) ( , ') (1 ( )) ( ), , 0b b a b

I r I r B T r G

I r I r B T r G n

Two Scale Asymptotic Analysis for the Improved Diffusion Approximation

Introduce ( , , ), ,I r y y G so that ( , ) ( , , )I r I r r

1( , , ) ( , , ) ( , ) ( )r yI r y I r y I r B T

Glass 29


Two Scale Asymptotic Analysis for the Improved Diffusion Approximation

Ansatz:0

1( , , ) ( , , )ii

i

I r y I r y

1( , , ) ( , , ) ( , ) ( )r yI r y I r y I r B T

Comparing the coefficients one obtains the Improved Diffusion Approximation

F. Zingsheim. Numerical solution methods for radiative heat transfer in semitransparent media. PhD, University of Kaiserslautern, 1999

Glass 30


)(111

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

dBedexTBexIxI

kd

kk

dkxdb

kk kkk

Alternatively we use the rte 2S

q B I d

Formal Solution Approximation

2 2

( , )

1 1

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

kMK MKd x dk k

r k b kk kS S

dBq x B T I x e d T d e T x d

dT

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

Glass 31


Example: Heating of a glass plate Parameters:

Density 2500 kg/m3 Specific heat 1250 J/kgKConductivity 1 W/KmThickness 0.005 m Semitransparent Region:0.01 µm – 7.0 µm

Absorption coefficient:0.4 /m … 7136 /m (8 bands)

Wall T=800°C

Wall T=600°C

Glass T0=200°C

Glass 32


Example: Heating of a glass

plate

Computational time for 3000 time steps

Exact 81.61 s

Ida 00.69 s

Fsa 00.69 s

Glass 33


Example: Cooling of a glass plate

Glass 34


Example:

adiabatic

T=1300 K

adiabatic

T=1800 K1 m

5 m

Radiation with diffusely reflecting gray walls in a gray material

gravity

Radiation and natural convection (FLUENT)

Glass 35


Example:

FLUENT-DOM ITWM-UDF

>5000 Iterations 86 Iterations

Diffusely reflecting gray walls in a gray material

m/40Radiation and natural convection (FLUENT)

Glass 36

Models for fast radiative heat transfer simulations 3. Grey Absorption

The numerical solution of the radiative transfer equation is very complex

Discretization: • 60 angular variables

• 10 wavelength bands

• 20,000 space points

• 12 million unknowns

Not suitable for optimization

Development of fast numerical methods

Reduce the number of unknowns „Grey Kappa“

(„Find a wavelength independend absorption coefficient?“)

Glass 37

Problem: • many frequency bands yield many equations

• Averaging the SPN equations over frequency is possible, yields nonlinear coefficients.

• POD approaches are possible as well.

Klar: Remark – Frequency averages


Glass 38

Typical absorption spectrum of glass


Glass 39

One-dimensional test example:

• Thickness 0.1m

• Refractive index 1.0001

Source term for heat transfer is the divergence of radiative flux vector


Glass 40

Values from literature:

Planck-mean absorption coefficient

Rosseland-mean absorption coefficient

0

0

( ) ( , )

( , )P

B T d

B T d

0

0

( , )

1( , )

( )

R

BT d

T

BT d

T


Glass 41

Values from literature:

Planck-mean absorption coefficient

Rosseland-mean absorption coefficient

0

0

( ) ( , )

( , )

MK

MKP

B T d

B T d

0

0

( , )

1( , )

( )

MK

MKR

BT d

T

BT d

T

155.4071P m 10.4202R m


Glass 42

Comparison between Planck-mean and Rosseland-mean

Good approximation for the boundary with Planck

Good approximation for the interior with Rosseland


Glass 43

The existence of the exact “Grey Kappa”

• We integrate the radiative transfer equation with respect to the wavelength

0 0 0

( , , ) ( ) ( , , ) ( ) ( ( ), )MK MK MK

I x d I x d B T x dx

• We define an ersatz (auxiliary) equation:

0

( , ) ( , ) ( , ) ( , ) ( ( )), ( ( )) ( ( ), )MKJ

x x J x x D T X D T X B T x dx

• If

0

0

( ) ( , , ) ( ( ), )

( , )

( , , ) ( ( ), )

MK

MK

I x B T x d

x

I x B T x d

then

0

( , ) ( , , )MK

J x I x d


Glass 44

The existence of the exact “Grey Kappa”

• The “Grey Kappa” is not depending on wavelength BUT on position and direction

• The “Grey Kappa” can be calculated, if we know the solution of the rte

How to approximate the intensity?

0

0

( ) ( , , ) ( ( ), )

( , )

( , , ) ( ( ), )

MK

MK

I x B T x d

x

I x B T x d

How to get rid of the direction? AND


Glass 45

How to approximate the intensity?

0

0

( ) ( , , ) ( ( ), )

( , )

( , , ) ( ( ), )

MK

MK

I x B T x d

x

I x B T x d

We use once more the formal solution

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

d x d x dT dBI x e B T d x e x T

dx dT

How to get rid of direction?0 / 2

( , ) ( )/ 2

x x ld x h x

l x l x l


Glass 46

1 2( , ) ( ) ( ) ( ) ( )P Rx T T G x T G x

New (approximated) „grey kappa“ can be formulated as

0

0

( ) ( )

1

( ) ( ) ( ) ( )

( ) ( , )

( )

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

MK

MK

h xref

h x h xP ref ref ref

B T e d

G xa dB

T B T e h x e T ddT

1 2

1 2

0 : ( ) 1 ( ) 0

: ( ) 0 ( ) 1

d G x G x

d G x G x

Planck-mean value

Rosseland-mean value

Planck-Rosseland-Superposition


Glass 47

Example of a 0.1m tick glass plate with initial temperature 1500°C


Glass 48

Example of a 0.1m tick glass plate with initial temperature 1500°C


Glass 49

Summary:

For the test examples the Planck-Rosseland-Superposition mean value gives the best results

For the optically thin case: PRS PlanckFor the optically thick case: PRS Rosseland

1 2( , ) ( ) ( ) ( ) ( )PRS P Rx T T G x T G x

Stored for different temperatures in a table

Calculated in advanced


Glass 50

Summary:

For the test examples the Planck-Rosseland-Superposition mean value gives the best results

For the optically thin case: PRS PlanckFor the optically thick case: PRS Rosseland

These are ideas! – Further research is needed!


Glass 51

Models for fast radiative heat transfer simulations 4. Application to flat glass tempering

Wrong cooling of glass and glass products causes large thermal stresses

Undesired crack

Glass 52

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


Glass 53

Cooling of the glass melt depends on the temperature distribution in time and space

Characteristically for glass:

• No fixed point where glass changes from fluid to solid state

• There exists a temperature range

• The essential property is the viscosity of the glass

temperature

low high

• high viscosity • low viscosity

• Linear-elastic material • Newtonian fluid


Glass 54

Viscosity changes the density depending on the temperature

Change in density (structural relaxation) influences the stress inside the glass

A numerical model for the calculation of transient and residual stresses in glass during cooling, including both structural relaxation and viscous stress relaxation, has been developed by Narayanaswamy und Tool

Commercial software packages like ANSYS and ABAQUS have implemented this model


Glass 55



• ITWM model gives the closest result for temperature

Glass 56



• Rosseland gives the worst surface and

mid-plan temperature difference

CPU time in s:

• ITWM model comparable with Rosseland and much faster thanexact solution model

Glass 57



• ITWM model gives the closest result for transient and residual stresses

Glass 58


• Production of bodies, like cubes, cylinders, angles („Kipferl“), ….• Special products by post- processing (grinding) of these simple geometrical pieces

• Deformation after cooling

Glass 59


Glass 60

Models for fast radiative heat transfer simulations 5. Conclusions

1. Temperature is one of the main parameters to make „good“ glasses

2. To simulate the temperature behavior of glass radiation must be taken into account

3. One needs good numerics to solve practical relevant radiative transfer problems - Improved Diffusion Approximation methods are alternative approaches for simulating the temperature behavior in glass

4. A grey absorption coefficient can save CPU time

5. The right temperature profile is necessary to simulate stresses during glass cooling

Glass 61

radiative heat transfer and applications for glass production processes

Documents