7.2.1 Equation of Phonon Radiative Transfer

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

7.2.1 Equation of Phonon Radiative Transfer

Dongwoo, Shin

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

2

Contents

• Equation of Radiative Transfer (ERT)

• Equation of Phonon Radiative Transfer

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

3

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.

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

4

Equation of Radiative Transfer

• : intensity in a participating medium• : velocity of light. (• : absorption coefficient• : scattering coefficient• : scattering phase function.

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

5

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.

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

6

Equation of Phonon Radiative Transfer

• The phonon BTE under the relaxation time approxima-tion.

• Make simple : 1-D , without internal source

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

7

Equation of Phonon Radiative Transfer

• is for equilibrium distribution.

• ERT ::

• corresponds to the inverse of the absorption coef-ficient of ERT.

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

8

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.

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

9

Equation of Phonon Radiative Transfer

• Using Bose-Einstein statistics,

• Integrating over all frequencies The total intensity

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

10

Equation of Phonon Radiative Transfer

• When upper limit with

• Derive heat capacity from energy flux

High T : Low T :

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

11

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.

School of Mechanical and Aerospace Engineering

Seoul National University Computer Aided Thermal

Design Lab

12

Equation of Phonon Radiative Transfer

• The heat flux per unit frequency

• The Criterion for radiative equilibrium

• Based on the energy density

• Local equilibrium condition