Download - 7.2.1 Equation of Phonon Radiative Transfer
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School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
7.2.1 Equation of Phonon Radiative Transfer
Dongwoo, Shin
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School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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Contents
• Equation of Radiative Transfer (ERT)
• Equation of Phonon Radiative Transfer
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School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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Equation of Radiative Transfer
• Radiative transfer is the physical phenomenon of energy transfer in the form of electromagnetic radiation.
• The equation of radiative transfer describes the radiation intensity of mi-croscopic point of view.
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School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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Equation of Radiative Transfer
• : intensity in a participating medium• : velocity of light. (• : absorption coefficient• : scattering coefficient• : scattering phase function.
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School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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Equation of Phonon Radiative Transfer
• Problems exist using ERT at thin film or superlattice because local equilib-rium breaks down in the acoustically thin limit.
• EPRT can describe Heat conduction across layered structures.
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School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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Equation of Phonon Radiative Transfer
• The phonon BTE under the relaxation time approxima-tion.
• Make simple : 1-D , without internal source
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School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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Equation of Phonon Radiative Transfer
• is for equilibrium distribution.
• ERT ::
• corresponds to the inverse of the absorption coef-ficient of ERT.
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School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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Equation of Phonon Radiative Transfer
– Acoustically thick limit / macroscale regime.– Most phonons will collide with phonons or defects inside
the medium Local equilibrium situation.– Fourier’s law is applicable without at a very short time.
– Acoustically thin limit / microscale regime– Most phonons will collide with the boundaries.
The walls are different T even in steady states.– BTE is applicable.
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School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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Equation of Phonon Radiative Transfer
• Using Bose-Einstein statistics,
• Integrating over all frequencies The total intensity
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School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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Equation of Phonon Radiative Transfer
• When upper limit with
• Derive heat capacity from energy flux
High T : Low T :
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School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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Equation of Phonon Radiative Transfer
• Kinetic expression of thermal conductivity
• When , C is the volumetric heat capacity of all phonon modes.
• Near the Debye temperature, C is the fraction of the volumetric specific heat.
• Also , we must use the appropriate upper limit in the integral when applying the EPRT.
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School of Mechanical and Aerospace Engineering
Seoul National University Computer Aided Thermal
Design Lab
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Equation of Phonon Radiative Transfer
• The heat flux per unit frequency
• The Criterion for radiative equilibrium
• Based on the energy density
• Local equilibrium condition