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 1
ITWM Activities in Glass
Glassmaking
Shape optimization of thermal-electrical flanges
Temperature
(Impedance Tomography)
PATENT
Coupling of glass tank
with electrical network
Gob temperature
(Spectral remote
sensing)
Form of the gob
(FPM)
Glass 2
ITWM Activities in Glass
Glassprocessing I
Pressing
TV panels
Lenses
Floatglass
window glasses
display glasses
Interface Glass-Mould (Radiation)
Identification of the heat transfer coefficient
High precision forming
Blowing
Bottles
Wavyness of thin display glasses
...
Foaming
Minimization of thermal stresses
Optimal shape of the furnace
Fiberproduction
Fluid-Fiber-Interaction
Glass 3
ITWM Activities in Glass
Glassprocessing II
Free cooling
Simulation of temperature
field
Cooling in a furnace
Control of furnace
temperature to minimize the
thermal stress
Tempering of glass
Glass 4
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 5
Models for fast radiative heat transfer simulations
N. Siedow
Fraunhofer-Institute
for Industrial Mathematics,
Kaiserslautern, Germany
Montecatini, 15. – 19. October 2008
Glass 6
Models for fast radiative heat transfer simulations
Outline
1.
Introduction
2.
Numerical methods for radiative heat transfer
3.
Grey Absorption
4.
Application to flat glass tempering
5.
Conclusions
Glass 7
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
To determine the temperature:
 Measurement
 Thermal stress
 Simulation
Glass 8
Models for fast radiative heat transfer simulations
1. Introduction
nm
mm - cm
Conductivity in W/(Km)
Heat transfer on a microscale
With Radiation
Without Radiation
Temperature in °C
Heat radiation on a macroscale
Radiation is for high temperatures the
dominant process
Glass 9
Models for fast radiative heat transfer simulations
1. Introduction
Heat transfer on a microscale
cm  m
nm
T
(r , t )    (k (r )T (r , t ))    qr (T ) , (r , t )  Dt
t
T (r ,0)  T0 (r ), r  D
+ boundary conditions
mm - cm

qr    I (r , ,  )d d 
0 S2
Heat radiation on a macroscale
 I (r , ,  )   ( ) I (r , ,  )   ( ) B(T (r , t ),  )
  

 
I (rg , ,  )   () I (rg , ' ,  )  (1   ()) B(Ta ,  )
Glass 10
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Heat transfer on a microscale
 Rosseland-Approximation
nm
•
Radiation = Correction of Conductivity
 PN-Approximation
mm - cm
 Discrete-Ordinate-Method (FLUENT)
Heat radiation on a macroscale
 ITWM-Approximation-Method
Glass 11
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Klar:
 I (r , ,  )   ( ) I (r , ,  )   ( ) B(T (r , t ),  )
We study the optically thick case. To obtain the dimensionless form of the rte we
introduce
r '  r / rref
 '   /  ref
and define the non-dimensional parameter

1
 ref xref
which is small in the optically thick – diffusion – regime.

 I (r ', ,  )  I (r ', ,  )  B(T ,  )
'
Glass 12
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
We rewrite the equation



E




I  B



And apply Neumann‘s series to (formally) invert the operator
1






I   E     B   E     O( 2 )  B






Rosseland-Approximation



T
4 1 B 
cm m
( r , t )     k ( r ) 
 T (r , t ) 

t
3 0  T 


Glass 13
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Rosseland-Approximation
BUT



T
4 1 B 
cm m
( r , t )     k ( r ) 
 T (r , t ) 

t
3 0  T 


•
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
• Standard method in glass industry
Glass 14
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Heat transfer on a microscale
 Rosseland-Approximation
nm
•
Radiation = Correction of Conductivity
 PN-Approximation
mm - cm
•
Spherical Harmonic Expansion
 Discrete-Ordinate-Method (FLUENT)
Heat radiation on a macroscale
 ITWM-Approximation-Method
Glass 15
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer



E




 I (r , ,  )  B(T (r , t ),  )



 optical thickness (small parameter)
1



I (r , ,  )   E     B(T (r , t ),  )



Neumann series



2
3
4
2
3
4
I (r , ,  )   E    2 ()  3 ()  4 ()  ...  B(T ( r , t ),  )







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
Glass 16
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
 SP1-Approximation
O(4)
 1

    G    G  4 B
 3

2
 SP3-Approximation
identical to P1-Approximation
O(8)
 1

    (G  2U )    G  4 B
 3

2
2
8
 9

 2  
U   U   G    B
5
5
 35

coupled system of equations
Glass 17
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Example: Cooling of a glass plate
Parameters:
Density
Specific heat
Conductivity
Thickness
Surroundings
2200 kg/m3
900 J/kgK
1 W/Km
1.0 m
300 K
gray medium
Absorption coefficient:
1/m
Glass 18
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Heat transfer on a microscale
 Rosseland-Approximation
nm
•
Radiation = Correction of Conductivity
 PN-Approximation
mm - cm
•
Spherical Harmonic Expansion
 Discrete-Ordinate-Method (FLUENT)
•
Heat radiation on a macroscale
Full-discretization method
Klar
 ITWM-Approximation-Method
Glass 19
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Heat transfer on a microscale
 Rosseland-Approximation
nm
•
Radiation = Correction of Conductivity
 PN-Approximation
mm - cm
•
Spherical Harmonic Expansion
 Discrete-Ordinate-Method (FLUENT)
•
Heat radiation on a macroscale
Full-discretization method
 ITWM-Approximation-Method
Glass 20
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
 ITWM-Approximation-Method
Formal solution:
I k ( x, )  I k ( xb , )e  k d ( x ,)   k
d ( x , )
 k s
k
B
(
T
(
x

s

))
e
ds

0
with
I k ( x, ) 
k 1
 I ( x, ,  )d
B k (T ( x)) 
k
k 1
 B(T ( x),  )d
k
 ( )   k  const. k    k 1
Taylor Approximation with respect to
x
Glass 21
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
 ITWM-Approximation-Method
Formal solution:
I k ( x, )  I k ( xb , )e  k d ( x ,)   k
d ( x , )
 k s
k
B
(
T
(
x

s

))
e
ds

0
with
I k ( x, ) 
k 1
 I ( x, ,  )d
B k (T ( x)) 
k
k 1
 B(T ( x),  )d
k
 ( )   k  const. k    k 1


I k ( x, )  I k ( xb , )e  k d ( x , )  B k (T ( x)) 1  e  k d 
1
k
1  1   d e 
 k d
k
dB k
  T ( x)
dT
Glass 22
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
 ITWM-Approximation-Method
Formal solution:
I k ( x, )  I k ( xb , )e  k d ( x ,)   k
d ( x , )
 k s
k
B
(
T
(
x

s

))
e
ds

0
with
I k ( x, ) 
k 1
 I ( x, ,  )d
B k (T ( x)) 
k
k 1
 B(T ( x),  )d
k
 ( )   k  const. k    k 1
Rosseland:
k
1
dB
I k ( x, )  B k (T ( x)) 
 T ( x)
 k dT
Glass 23
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
 ITWM-Approximation-Method
Formal solution:
I k ( x, )  I k ( xb , )e  k d ( x ,)   k
d ( x , )
 k s
k
B
(
T
(
x

s

))
e
ds

0
with
I k ( x, ) 
k 1
 I ( x, ,  )d
B k (T ( x)) 
k
k 1
 B(T ( x),  )d
k
 ( )   k  const. k    k 1


I k ( x, )  I k ( xb , )e  k d ( x , )  B k (T ( x)) 1  e  k d 
1
k
1  1   d e 
 k d
k
dB k
  T ( x)
dT
Glass 24
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Improved Diffusion Approximation
 MK 1 dB k k 
 MK
  qr ( x)     
A T ( x, t )    k  B k (T )  I k ( xb , ) e  k d ( x , ) d

 k 1  k dT
 k 1 S 2



dB k

(T )  1  e  k d   T ( x)d
k 1 dT
S2
MK
Ak 

 (  ) d ( x ,  )
T


1

1


(

)
d
(
x
,

)
e



d


2 

S
• In opposite to Rosseland-Approximation all geometrical information is conserved

Lentes, F. T., Siedow, N., Glastech. Ber. Glass Sci. Technol. 72 No.6 188-196 (1999).
Glass 25
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Improved Diffusion Approximation
 MK 1 dB k k 
 MK
  qr ( x)     
A T ( x, t )    k  B k (T )  I k ( xb , ) e  k d ( x , ) d

 k 1  k dT
 k 1 S 2



dB k

(T )  1  e  k d   T ( x)d
k 1 dT
S2
MK
Ak 

 (  ) d ( x ,  )
T


1

1


(

)
d
(
x
,

)
e



d


2 

S
• Correction to the heat conduction due to radiation
with anisotropic diffusion tensor

Lentes, F. T., Siedow, N., Glastech. Ber. Glass Sci. Technol. 72 No.6 188-196 (1999).
Glass 26
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Improved Diffusion Approximation
 MK 1 dB k k 
 MK
  qr ( x)     
A T ( x, t )    k  B k (T )  I k ( xb , ) e  k d ( x , ) d

 k 1  k dT
 k 1 S 2



dB k

(T )  1  e  k d   T ( x)d
k 1 dT
S2
MK
Ak 

 (  ) d ( x ,  )
T


1

1


(

)
d
(
x
,

)
e



d


2 

S
• Boundary conditions
• Convection term
Glass 27
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Two Scale Asymptotic Analysis for the Improved Diffusion Approximation
1

 I (r , )  I (r , )  B(T ), r  G,
I (rb , )   () I (rb ,  ')  (1   ()) B(Ta ), rb G, n    0
Introduce
1

I (r , y, ), y   G,
so that
I (r , )  I (r ,  r , )
  r I (r , y, )    y I (r , y, )  I (r , )  B(T )
Glass 28
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Two Scale Asymptotic Analysis for the Improved Diffusion Approximation
1

  r I (r , y, )    y I (r , y, )  I (r , )  B(T )

Ansatz:
I ( r , y , )  
i 0
1

I ( r , y , )
i i
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 29
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
I ( x, )  I ( xb , )e
k
k
 k d ( x , )

 B (T ( x)) 1  e
k
 k d
  1  1   d e 
1
 k d
k
k
dB k
  T ( x)
dT
  q     B  I d 
Alternatively we use the rte
S2
  qr ( x)    k   B (T )  I ( xb , )  e
MK
k
k 1
S2
k
 k d ( x ,  )
dB k
d  
(T )  1  (1   k d )e  k d   T ( x)d 
k 1 dT
S2
MK
Formal Solution Approximation

N. Siedow, D. Lochegnies, T. Grosan, E. Romero, J. Am. Ceram. Soc., 88 [8] 2181-2187 (2005)
Glass 30
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Example: Heating of a glass plate
Wall T=800°C
Glass T0=200°C
Wall T=600°C
Parameters:
Density
Specific heat
Conductivity
Thickness
2500 kg/m3
1250 J/kgK
1 W/Km
0.005 m
Semitransparent Region:
0.01 µm – 7.0 µm
Absorption coefficient:
0.4 /m … 7136 /m (8 bands)
Glass 31
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
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 32
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Example: Cooling of a glass plate
Glass 33
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Example:
Radiation and natural convection (FLUENT)
adiabatic
1m
T=1300 K
gravity
T=1800 K
adiabatic
5m
Radiation with diffusely reflecting gray walls in a gray material
Glass 34
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
Example:
Radiation and natural convection (FLUENT)
Diffusely reflecting gray walls in a gray material
FLUENT-DOM
>5000 Iterations
  40 / m
ITWM-UDF
86 Iterations
Glass 35
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 36
Models for fast radiative heat transfer simulations
3. Grey Absorption
Klar:
averages
Remark – Frequency
Problem:
• many frequency bands yield many equations
• Averaging the SPN equations over frequency is possible, yields
nonlinear coefficients.
• POD approaches are possible as well.
Glass 37
Models for fast radiative heat transfer simulations
3. Grey Absorption
Typical absorption spectrum of glass
Glass 38
Models for fast radiative heat transfer simulations
3. Grey Absorption
One-dimensional test example:
• Thickness 0.1m
• Refractive index 1.0001
Source term for heat transfer is
the divergence of radiative flux
vector
Glass 39
Models for fast radiative heat transfer simulations
3. Grey Absorption
Values from literature:
Planck-mean absorption coefficient
Rosseland-mean absorption coefficient

P 
  ( ) B(T ,  )d 
0

 B(T ,  )d 
0

R  
B
0 T (T ,  )d 
1 B
0  ( ) T (T ,  )d 
Glass 40
Models for fast radiative heat transfer simulations
3. Grey Absorption
Values from literature:
Planck-mean absorption coefficient
Rosseland-mean absorption coefficient
MK
MK
P 
  ( ) B(T ,  )d 
0
MK

B(T ,  )d 
0
 P  55.4071m1

R  
0
MK

0
B
(T ,  )d 
T
1 B
(T ,  )d 
 ( ) T
 R  0.4202m 1
Glass 41
Models for fast radiative heat transfer simulations
3. Grey Absorption
Comparison between Planck-mean and Rosseland-mean
Good approximation for the
boundary with Planck
Good approximation for the
interior with Rosseland
Glass 42
Models for fast radiative heat transfer simulations
3. Grey Absorption
The existence of the exact “Grey Kappa”
• We integrate the radiative transfer equation with respect to the wavelength


x
MK


I ( x,  ,  ) d  
0
MK
MK
  ( ) I ( x,  ,  )d     ( ) B(T ( x),  )d 
0
0
• We define an ersatz (auxiliary) equation:
J
 ( x,  )   ( x,  ) J ( x,  )   ( x,  ) D(T ( X )), D(T ( X )) 
x
MK
• If
 ( x,  ) 
  ( )  I ( x,  ,  )  B(T ( x),  ) d 
0
MK
  I ( x,  ,  )  B(T ( x),  ) d 
then
MK

B(T ( x),  )d 
0
J ( x,  ) 
MK


I ( x,  ,  ) d 
0
0
Glass 43
Models for fast radiative heat transfer simulations
3. Grey Absorption
MK
The existence of the exact “Grey Kappa”
 ( x,  ) 
  ( )  I ( x,  ,  )  B(T ( x),  ) d 
0
MK
  I ( x,  ,  )  B(T ( x),  ) d 
0
• 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?
AND
How to get rid of the direction?
Glass 44
Models for fast radiative heat transfer simulations
3. Grey Absorption
MK
 ( x,  ) 
  ( )  I ( x,  ,  )  B(T ( x),  ) d 
0
MK
  I ( x,  ,  )  B(T ( x),  ) d 
How to approximate the intensity?
0
We use once more the formal solution
I ( x,  ,  )  1  e (  ) d ( x ,  )  B(T ,  ) 
How to get rid of direction?

dT
dB
1  1   ( )d ( x,  )  e (  ) d ( x ,  ) 
( x)
(T ,  )  ...
 ( )
dx
dT
0 xl/2
 x
d ( x,  )  h ( x )  
l  x l / 2  x  l
Glass 45
Models for fast radiative heat transfer simulations
3. Grey Absorption
New (approximated) „grey kappa“ can be formulated as
 ( x, T )   P (T )G1 ( x)   R (T )G2 ( x)
Planck-Rosseland-Superposition
MK
  ( ) B(T
ref
G1 ( x) 
 P (Tref )
MK

0
,  )e  (  ) h ( x ) d 
0


a
dB
 (  ) h ( x )
1  1   ( )h( x)  e  (  ) h ( x ) 

(Tref ,  )  d 
 B(Tref ,  )e
 ( )
dT


  d  0 : G1 ( x)  1 G2 ( x)  0
  d   : G1 ( x)  0 G2 ( x)  1
Planck-mean value
Rosseland-mean value
Glass 46
Models for fast radiative heat transfer simulations
3. Grey Absorption
Example of a 0.1m tick glass plate with initial temperature 1500°C
Glass 47
Models for fast radiative heat transfer simulations
3. Grey Absorption
Example of a 0.1m tick glass plate with initial temperature 1500°C
Glass 48
Models for fast radiative heat transfer simulations
3. Grey Absorption
Summary:
 For the test examples the Planck-Rosseland-Superposition mean value gives the
best results
 For the optically thin case: PRS
For the optically thick case:PRS
Planck
Rosseland
 PRS ( x, T )   P (T )G1 ( x)   R (T )G2 ( x)
Stored for different
temperatures in a table
Calculated in advanced
Glass 49
Models for fast radiative heat transfer simulations
3. Grey Absorption
Summary:
 For the test examples the Planck-Rosseland-Superposition mean value gives the
best results
 For the optically thin case: PRS
For the optically thick case:PRS
Planck
Rosseland
 These are ideas! – Further research is needed!
Glass 50
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 51
Models for fast radiative heat transfer simulations
4. Application to flat glass tempering
 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 52
Models for fast radiative heat transfer simulations
4. Application to flat glass tempering
 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
low
high
temperature
• high viscosity
• low viscosity
• Linear-elastic material
• Newtonian fluid
Glass 53
Models for fast radiative heat transfer simulations
4. Application to flat glass tempering
 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 54
Models for fast radiative heat transfer simulations
4. Application to flat glass tempering

N. Siedow, D. Lochegnies, T. Grosan, E. Romero, J. Am. Ceram. Soc., 88 [8] 2181-2187 (2005)
• ITWM model gives the closest result for temperature
Glass 55
Models for fast radiative heat transfer simulations
4. Application to flat glass tempering

N. Siedow, D. Lochegnies, T. Grosan, E. Romero, J. Am. Ceram. Soc., 88 [8] 2181-2187 (2005)
CPU time in s:
• ITWM model comparable with
Rosseland and much faster than
exact solution model
• Rosseland gives the worst surface and
mid-plan temperature difference
Glass 56
Models for fast radiative heat transfer simulations
4. Application to flat glass tempering

N. Siedow, D. Lochegnies, T. Grosan, E. Romero, J. Am. Ceram. Soc., 88 [8] 2181-2187 (2005)
• ITWM model gives the closest result for transient and residual stresses
Glass 57
Models for fast radiative heat transfer simulations
4. Application to flat glass tempering
• Production of bodies, like cubes,
cylinders, angles („Kipferl“), ….
• Special products by postprocessing (grinding) of these
simple geometrical pieces
• Deformation after cooling
Glass 58
Models for fast radiative heat transfer simulations
5. Application to flat glass tempering
Glass 59
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 60
Glass 61