Strahlung bei Glasabkühlung 1. ProblemAxel Klar and Norbert Siedow
Department of Mathematics, TU Kaiserslautern
Fraunhofer ITWM Abteilung Transport processes
Montecatini, 15. – 19. October 2008
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Gob temperature
Temperature
High precision forming
Minimization of thermal stresses
1726.unknown
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Planning of the Lectures
Indirect Temperature Measurement of Hot Glasses
Parameter Identification Problems
N. Siedow
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Outline
Introduction
Grey Absorption
Conclusions
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1. Introduction
Temperature is the most important parameter in all stages of glass production
Homogeneity of glass melt
1. Introduction
With Radiation
Without Radiation
mm - cm
1. Introduction
+ boundary conditions
mm - cm
Rosseland-Approximation
ITWM-Approximation-Method
PN-Approximation
mm - cm
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
We study the optically thick case. To obtain the dimensionless form of the rte we introduce
Klar:
and define the non-dimensional parameter
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We rewrite the equation
And apply Neumann‘s series to (formally) invert the operator
Rosseland-Approximation
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Very fast and easy to implement into commercial software packages
Only for optically thick glasses
Problems near the boundary
Rosseland-Approximation
BUT
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Rosseland-Approximation
ITWM-Approximation-Method
PN-Approximation
mm - cm
Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
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 optical thickness (small parameter)
Neumann series
SP1-Approximation O(e4)
SP3-Approximation O(e8)
Example: Cooling of a glass plate
Parameters:
Rosseland-Approximation
ITWM-Approximation-Method
PN-Approximation
mm - cm
Rosseland-Approximation
ITWM-Approximation-Method
PN-Approximation
mm - cm
ITWM-Approximation-Method
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ITWM-Approximation-Method
ITWM-Approximation-Method
ITWM-Approximation-Method
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
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Improved Diffusion Approximation
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
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Improved Diffusion Approximation
Two Scale Asymptotic Analysis for the Improved Diffusion Approximation
Introduce
Two Scale Asymptotic Analysis for the Improved Diffusion Approximation
Ansatz:
F. Zingsheim. Numerical solution methods for radiative heat transfer in semitransparent media. PhD, University of Kaiserslautern, 1999
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Models for fast radiative heat transfer simulations
2. Numerical methods for radiative heat transfer
N. Siedow, D. Lochegnies, T. Grosan, E. Romero, J. Am. Ceram. Soc., 88 [8] 2181-2187 (2005)
Alternatively we use the rte
Formal Solution Approximation
Example: Heating of a glass plate
Parameters:
Wall T=800°C
Wall T=600°C
Glass T0=200°C
Example: Heating of a glass
plate
Exact 81.61 s
Ida 00.69 s
Fsa 00.69 s
Example: Cooling of a glass plate
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Example:
Radiation with diffusely reflecting gray walls in a gray material
gravity
adiabatic
Example:
FLUENT-DOM
ITWM-UDF
Radiation and natural convection (FLUENT)
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3. Grey Absorption
The numerical solution of the radiative transfer equation is very complex
Discretization:
„Grey Kappa“
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• Averaging the SPN equations over frequency is possible, yields nonlinear coefficients.
• POD approaches are possible as well.
Klar: Remark – Frequency averages
3. Grey Absorption
3. Grey Absorption
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3. Grey Absorption
One-dimensional test example:
Refractive index 1.0001
Source term for heat transfer is the divergence of radiative flux vector
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3. Grey Absorption
Values from literature:
Planck-mean absorption coefficient
Rosseland-mean absorption coefficient
3. Grey Absorption
Values from literature:
Planck-mean absorption coefficient
Rosseland-mean absorption coefficient
3. Grey Absorption
Good approximation for the boundary with Planck
Good approximation for the interior with Rosseland
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3. Grey Absorption
The existence of the exact “Grey Kappa”
We integrate the radiative transfer equation with respect to the wavelength
We define an ersatz (auxiliary) equation:
If
then
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3. Grey Absorption
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?
How to get rid of the direction?
AND
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3. Grey Absorption
We use once more the formal solution
How to get rid of direction?
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3. Grey Absorption
Planck-mean value
Rosseland-mean value
3. Grey Absorption
Example of a 0.1m tick glass plate with initial temperature 1500°C
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3. Grey Absorption
Example of a 0.1m tick glass plate with initial temperature 1500°C
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3. Grey Absorption
For the test examples the Planck-Rosseland-Superposition mean value gives the best results
For the optically thin case: PRS Planck
For the optically thick case: PRS Rosseland
Stored for different temperatures in a table
Calculated in advanced
3. Grey Absorption
For the test examples the Planck-Rosseland-Superposition mean value gives the best results
For the optically thin case: PRS Planck
For the optically thick case: PRS Rosseland
These are ideas! – Further research is needed!
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4. Application to flat glass tempering
Wrong cooling of glass and glass products causes large thermal stresses
Undesired crack
4. Application to flat glass tempering
Thermal tempering consists of:
Heating of the glass at a temperature higher the transition temperature
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)
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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
temperature
low
high
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
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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
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4. Application to flat glass tempering
N. Siedow, D. Lochegnies, T. Grosan, E. Romero, J. Am. Ceram. Soc., 88 [8] 2181-2187 (2005)
Rosseland gives the worst surface and
mid-plan temperature difference
exact solution model
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
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4. Application to flat glass tempering
Production of bodies, like cubes, cylinders, angles („Kipferl“), ….
Special products by post- processing (grinding) of these simple geometrical pieces
Deformation after cooling
5. Application to flat glass tempering
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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
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