Multi scale heat conduction solution of the eprt
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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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Multi scale heat conduction solution of the eprt

  • Dec. 6th , 2011

Multi-scale Heat ConductionSolution of the EPRT

Hong goo, Kim

1st year of M.S. course


Contents

Contents

  • Introduction

  • Two-Flux Method

    • Modeling of Thin Layer

    • Solution

    • Example 7-4

  • Thermal Resistance Network Method

    • Scheme

    • Thermal Resistance Network

    • Three-Layer Structure


  • Multi scale heat conduction solution of the eprt

    Two-Flux Method

    Modeling of Thin Layer

    • Assumption

    Scheme

    • 1-D, Steady state

    • 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

    Steady-state

    EPRT

    Gray medium

    ; Positive direction

    ; Negative direction

    x = L

    x = 0


    Multi scale heat conduction solution of the eprt

    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

    Irradiation

    Irradiation

    x = L

    x = 0


    Multi scale heat conduction solution of the eprt

    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


    Multi scale heat conduction solution of the eprt

    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)


    Multi scale heat conduction solution of the eprt

    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)


    Multi scale heat conduction solution of the eprt

    Two-Flux Method

    • Solution

    • Energy Balance

    • For 1-D steady-state,

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

    Appendix (2)

    (7.52)


    Multi scale heat conduction solution of the eprt

    Two-Flux Method

    • Solution

    • Energy Balance

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

    • Blackbody emissive power

    • Total radiosity

    • Energy balance, from (7.52)

    → Radiative equilibrium condition


    Multi scale heat conduction solution of the eprt

    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


    Multi scale heat conduction solution of the eprt

    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)


    Multi scale heat conduction solution of the eprt

    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)


    Multi scale heat conduction solution of the eprt

    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


    Multi scale heat conduction solution of the eprt

    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)


    Multi scale heat conduction solution of the eprt

    Two-Flux Method

    • Example 7-4

    • T > θD

    • Spectral heat flux

    (7.54)

    (7.34)

    • Total intensity

    • Net heat flux

    (7.58)


    Multi scale heat conduction solution of the eprt

    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


    Multi scale heat conduction solution of the eprt

    Two-Flux Method

    • Example 7-4

    • Temperature Profiles

    • T

    • (T > θD)

    • (T << θD)

    • x


    Multi scale heat conduction solution of the eprt

    Two-Flux Method

    • Example 7-4

    • Radiative Equilibrium Conditions

    • 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)


    Multi scale heat conduction solution of the eprt

    Thermal Resistance Network

    • Scheme

    • Internal Thermal Resistance

    • Diffusion process: classical Fourier law

    • Boundary Thermal Resistance

    • When medium is not in radiative thick limit

    • Due to radiation slip

    • 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


    Multi scale heat conduction solution of the eprt

    Thermal Resistance Network

    • Energy Transport

    • T << θD

    • Heat flux in thermal network resistance

    (7.61)

    Bulk

    Radiation slip

    • 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)


    Multi scale heat conduction solution of the eprt

    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


    Multi scale heat conduction solution of the eprt

    Thermal Resistance Network

    • Three Layer Structure

    • Thermal Resistance Network

    TH

    TL

    TH

    T1a

    T1b

    T2a

    T2b

    T3a

    T3b

    TL


    Multi scale heat conduction solution of the eprt

    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


    Multi scale heat conduction solution of the eprt

    Thermal Resistance Network

    • Three Layer Structure

    • Total Resistance

    • Heat Flux

    • Effective Thermal Conductivity


    Multi scale heat conduction solution of the eprt

    • Appendix (1)


    Multi scale heat conduction solution of the eprt

    • Appendix (2)


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