# Multi-scale Heat Conduction Solution of the EPRT - PowerPoint PPT Presentation

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Dec. 6 th , 2011. Multi-scale Heat Conduction Solution of the EPRT. Hong goo, Kim 1 st year of M.S. course. Contents. Introduction Two-Flux Method Modeling of Thin Layer Solution Example 7-4 Thermal Resistance Network Method Scheme Thermal Resistance Network

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Multi-scale Heat Conduction Solution of the EPRT

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• Dec. 6th , 2011

## Multi-scale Heat ConductionSolution of the EPRT

Hong goo, Kim

1st year of M.S. course

### Contents

• Introduction

• Two-Flux Method

• Modeling of Thin Layer

• Solution

• Example 7-4

• Thermal Resistance Network Method

• Scheme

• Thermal Resistance Network

• Three-Layer Structure

• Two-Flux Method

Modeling of Thin Layer

• Assumption

Scheme

• Medium is gray

Emission

• Absorption coefficient is independent of phonon frequency

T1

T2

• Walls are diffuse and gray

ε1

ε2

• Absorption coefficient is independent of direction and phonon frequency

• Emission at wall is independent of direction

• Governing Equation

EPRT

Gray medium

; Positive direction

; Negative direction

x = L

x = 0

Two-Flux Method

• Modeling of Thin Layer

• Boundary Conditions

• Temperature at the walls

(7.46)

• Intensity at the walls

(7.47)

Emission

Emission

(7.48)

Reflection

Reflection

T1

T2

x = L

x = 0

Two-Flux Method

• Solution

• Derivation of the Solutions

• From the governing equation (EPRT)

(7.45a)

• Integrating from 0 to x, after multiplying on both sides

LHS

RHS

LHS

RHS

Two-Flux Method

• Solution

• Derivation of the Solutions (continued)

• For positive directions (from left to right)

(7.49)

(7.50)

Attenuation of the intensity originated from the left surface (x = 0)

Generation term

• For negative directions (from right to left)

Two-Flux Method

• Solution

• Spectral Net Heat Flux (in x-direction)

(7.37)

Appendix (1)

(7.51a)

• For a diffuse surface (x = 0, x = L)

Diffuse Surface

(7.51b)

Two-Flux Method

• Solution

• Energy Balance

• Differentiation of heat flux (diffuse surface) from (7.51b)

Appendix (2)

(7.52)

Two-Flux Method

• Solution

• Energy Balance

• Total blackbody emissive power = total radiosities at 1 and 2

• Blackbody emissive power

• Energy balance, from (7.52)

Two-Flux Method

• Example 7-4

• Objectives

• Heat flux

• Thermal conductivity

• Temperature distribution

• Assumptions

• Medium is gray

• Surfaces are diffuse and gray

• Radiative thicklimit : Kn =Λ/L<< 1

Two-Flux Method

• Example 7-4

• Spectral Heat Flux

• In the radiative thick limit : Λ << x, Λ << L − x

• Local equilibrium holds if location of x is not too close to either surfaces

• Flux originating from the left/right surfaces are attenuated to ‘0’

(7.51a)

Two-Flux Method

• Example 7-4

• Spectral Heat Flux (continued)

• Exponential terms are significant only in the neighbor of x

• Taylor series 1st order approximation is valid

(7.54)

Two-Flux Method

• Example 7-4

• T << θD

• Net heat flux obtained from integrating (7.54) over frequency

• Thermal conductivity

Thermal conductivity

(7.55a)

• From kinetic theory

• Integration of (7.55a) overx from 0 to L :

(7.55b)

LHS

RHS

(7.56a)

Net heat flux

Two-Flux Method

• Example 7-4

• T << θD(continued)

• Temperature distribution

• By comparing (7.55a) and (7.56a)

(7.56b)

• Thermal resistance

• By definition of the thermal resistance and (7.56a)

(7.57)

Two-Flux Method

• Example 7-4

• T > θD

• Spectral heat flux

(7.54)

(7.34)

• Total intensity

• Net heat flux

(7.58)

Two-Flux Method

• Example 7-4

• T > θD

• Thermal conductivity

• Net heat flux

• From kinetic theory,

(7.58)

(7.59)

• Assuming small temperature difference

• Thermal conductivity can be approximated as a constant

• Thermal resistance

• Temperature distribution

Two-Flux Method

• Example 7-4

• Temperature Profiles

• T

• (T > θD)

• (T << θD)

• x

Two-Flux Method

• Example 7-4

• Local equilibrium condition, gray medium

(7.40)

• is the average of and

• Assuming T1 > T2: net heat flux in positive x

• should be greater than

• Local equilibrium is not a stable state

• Heat flux in the radiative thin limit

(7.60)

Thermal Resistance Network

• Scheme

• Internal Thermal Resistance

• Diffusion process: classical Fourier law

• Boundary Thermal Resistance

• When medium is not in radiative thick limit

• Does not exist in classical Fourier law

• Temperature jump approaches to zero in the radiative thick limit ( Kn << 1 )

• Restrictions

• Applicable to one-dimensional problem

• Results in temperature jump at the boundaries

Thermal Resistance Network

• Energy Transport

• T << θD

• Heat flux in thermal network resistance

(7.61)

Bulk

• For blackbody walls ( ε1 = ε2= 1 )

• Temperature difference between T1 and T2 is small (T1, T2≈ T )

Effective

thermal conductivity

(7.63)

Bulk

thermal conductivity

(7.55b)

Thermal Resistance Network

• Energy Transport

• T > θD

• Heat flux in thermal network resistance

(7.64)

• Effective vs. bulk heat conductivity ratio is the same as in low temperature for blackbody walls

• Discussion

• Fourier law can be applied inside the medium

• Heat flux for thermal resistance network can be applied between the diffusion and ballistic extremes

Thermal Resistance Network

• Three Layer Structure

• Thermal Resistance Network

TH

TL

TH

T1a

T1b

T2a

T2b

T3a

T3b

TL

Thermal Resistance Network

• Three Layer Structure

• Internal Resistance

• Due to diffusion (Fourier’s law)

• Interface Resistance

• Transmission of phonon through the interface

Γij : transmissivity from ito j

• Boundary Resistance

• Transmission of phonon transport considered

Thermal Resistance Network

• Three Layer Structure

• Total Resistance

• Heat Flux

• Effective Thermal Conductivity

• Appendix (1)

• Appendix (2)