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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. ITWM Activities in Glass Glassmaking.

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## Radiative Heat transfer and Applications for Glass Production Processes

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**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**ITWM Activities in GlassGlassmaking**Shape optimization of thermal-electrical flanges Gob temperature (Spectral remote sensing) Temperature(Impedance Tomography)PATENT Coupling of glass tank with electrical network Form of the gob(FPM)**ITWM Activities in GlassGlassprocessing I**PressingTV panels Lenses Interface Glass-Mould (Radiation) Identification of the heat transfer coefficient Floatglasswindow glasses display glasses High precision forming Blowing Bottles Wavyness of thin display glasses . . . Foaming Minimization of thermal stresses Optimal shape of the furnace Fiberproduction Fluid-Fiber-Interaction**ITWM Activities in GlassGlassprocessing II**Simulation of temperature field Free cooling Tempering of glass Control of furnace temperature to minimize the thermal stress Cooling in a furnace**Radiative Heat transfer and Applications for Glass**Production Processes Planning of the Lectures • Models for fast radiative heat transfer simulation • Indirect Temperature Measurement of Hot Glasses • Parameter Identification Problems**Models for fast radiative heat transfer simulations**N. Siedow Fraunhofer-Institute for Industrial Mathematics, Kaiserslautern, Germany Montecatini, 15. – 19. October 2008**Models for fast radiative heat transfer simulationsOutline**• Introduction • Numerical methods for radiative heat transfer • Grey Absorption • Application to flat glass tempering • Conclusions**To determine the temperature:**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 • Measurement • Simulation**Heat transfer on a microscale**nm Conductivity in W/(Km) With Radiation mm - cm Without Radiation Temperature in °C Heat radiation on a macroscale Models for fast radiative heat transfer simulations 1. Introduction Radiation is for high temperatures the dominant process**Heat transfer on a microscale**nm mm - cm Heat radiation on a macroscale Models for fast radiative heat transfer simulations 1. Introduction + boundary conditions**Heat transfer on a microscale**nm mm - cm Heat radiation on a macroscale Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • Rosseland-Approximation • Radiation = Correction of Conductivity • PN-Approximation • Discrete-Ordinate-Method (FLUENT) • ITWM-Approximation-Method**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer Klar: We study the optically thick case. To obtain the dimensionless form of the rte we introduce and define the non-dimensional parameter which is small in the optically thick – diffusion – regime.**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer We rewrite the equation And apply Neumann‘s series to (formally) invert the operator Rosseland-Approximation**BUT**• Standard method in glass industry Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Rosseland-Approximation • 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**Heat transfer on a microscale**nm mm - cm Heat radiation on a macroscale Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • Rosseland-Approximation • Radiation = Correction of Conductivity • PN-Approximation • Spherical Harmonic Expansion • Discrete-Ordinate-Method (FLUENT) • ITWM-Approximation-Method**e optical thickness (small parameter)**Neumann series 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**identical to P1-Approximation**• SP3-Approximation O(e8) coupled system of equations Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • SP1-Approximation O(e4)**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer Example: Cooling of a glass plate Parameters: Density 2200 kg/m3Specific heat 900 J/kgKConductivity 1 W/KmThickness 1.0 mSurroundings 300 Kgray mediumAbsorption coefficient: 1/m**Heat transfer on a microscale**nm mm - cm Heat radiation on a macroscale Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • Rosseland-Approximation • Radiation = Correction of Conductivity • PN-Approximation • Spherical Harmonic Expansion • Discrete-Ordinate-Method (FLUENT) • Full-discretization method Klar • ITWM-Approximation-Method**Heat transfer on a microscale**nm mm - cm Heat radiation on a macroscale Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • Rosseland-Approximation • Radiation = Correction of Conductivity • PN-Approximation • Spherical Harmonic Expansion • Discrete-Ordinate-Method (FLUENT) • Full-discretization method • ITWM-Approximation-Method**Taylor Approximation with respect to**Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer • ITWM-Approximation-Method Formal solution: with**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer • ITWM-Approximation-Method Formal solution: with**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer • ITWM-Approximation-Method Formal solution: with Rosseland:**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer • ITWM-Approximation-Method Formal solution: with**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer Improved Diffusion Approximation • 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).**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer Improved Diffusion Approximation • 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).**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer Improved Diffusion Approximation • Boundary conditions • Convection term**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer Two Scale Asymptotic Analysis for the Improved Diffusion Approximation so that Introduce**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer Two Scale Asymptotic Analysis for the Improved Diffusion Approximation Ansatz: 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**Alternatively we use the rte**Formal Solution Approximation 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)**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer Example: Heating of a glass plate Parameters: Density 2500 kg/m3Specific heat 1250 J/kgKConductivity 1 W/KmThickness 0.005 mSemitransparent Region:0.01 µm – 7.0 µm Absorption coefficient:0.4 /m … 7136 /m (8 bands) Wall T=800°C Glass T0=200°C Wall T=600°C**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**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer Example: Cooling of a glass plate**adiabatic**T=1800 K 1 m T=1300 K adiabatic 5 m Models for fast radiative heat transfer simulations 2. Numerical methods for radiative heat transfer Radiation and natural convection (FLUENT) Example: gravity Radiation with diffusely reflecting gray walls in a gray material**Models for fast radiative heat transfer simulations 2.**Numerical methods for radiative heat transfer Radiation and natural convection (FLUENT) Example: Diffusely reflecting gray walls in a gray material FLUENT-DOM ITWM-UDF >5000 Iterations 86 Iterations**„Grey Kappa“**• Reduce the number of unknowns („Find a wavelength independend absorption coefficient?“) 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 • 12 million unknowns • 20,000 space points Not suitable for optimization • Development of fast numerical methods**Problem: • many frequency bands yield many equations**• Averaging the SPN equations over frequency is possible, yields nonlinear coefficients. • POD approaches are possible as well. Models for fast radiative heat transfer simulations 3. Grey Absorption Klar: Remark – Frequency averages**Models for fast radiative heat transfer simulations 3. Grey**Absorption Typical absorption spectrum of glass**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**Models for fast radiative heat transfer simulations 3. Grey**Absorption Values from literature: Rosseland-mean absorption coefficient Planck-mean absorption coefficient**Models for fast radiative heat transfer simulations 3. Grey**Absorption Values from literature: Rosseland-mean absorption coefficient Planck-mean absorption coefficient**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**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 • We define an ersatz (auxiliary) equation: • If then**Models for fast radiative heat transfer simulations 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 AND How to get rid of the direction? How to approximate the intensity?**Models for fast radiative heat transfer simulations 3. Grey**Absorption How to approximate the intensity? We use once more the formal solution How to get rid of direction?**Models for fast radiative heat transfer simulations 3. Grey**Absorption New (approximated) „grey kappa“ can be formulated as Planck-Rosseland-Superposition Planck-mean value Rosseland-mean value**Models for fast radiative heat transfer simulations 3. Grey**Absorption Example of a 0.1m tick glass plate with initial temperature 1500°C**Models for fast radiative heat transfer simulations 3. Grey**Absorption Example of a 0.1m tick glass plate with initial temperature 1500°C**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 PlanckFor the optically thick case: PRS Rosseland Stored for different temperatures in a table Calculated in advanced**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 PlanckFor the optically thick case: PRS Rosseland • These are ideas! – Further research is needed!

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