Dual-Phase-Lag Model(Chap 7.1.2)

Yoon kichulDepartment of Mechanical EngineeringSeoul National University

Multi-scale Heat Conduction

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Contents

1. Introductory Explanation of the Chapter

2. Heat Flux Equation (Lagging Behavior)

4. Dual-Phase-Lag Model by Tzou

6. Simplified BTE for Phonon System

1) Gurtin and Pipkin

2) Joseph and Preziosi

5. Parallel or Coupled Heat Diffusion Process

3. Jeffrey Type Lagging Heat Equation

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1. Introductory Explanation of the Chapter

∙Title of this chapter “Dual-Phase-Lag Model”

- “Dual” : Two different phenomena (heat temp. gradient, and reverse)

- “Phase-Lag” : Lag in time phase lag by Fourier’s transform

∙The concept of this chapter

- Temperature gradient Heat flux

- Heat source Temperature gradientNot instantaneously

Lagging behavior b/w heat flux and temperature gradient

( , ) ( , )t i tT t T e d

r r lag in time (t) phase lag (iωt)

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2. Heat Equation

1) Gurtin and Pipkin

( , ) ( ) ( , )t

t K t t T t dt

q r r : kernal function( )K t t

- ( ) ( )K t t t t

- ( ) ( / ) exp[ ( ) / ]qK t t t t , exp ( , )t

q q

t tt T t dt

q r r

( , )( , ) ( , )q

tt T t

t

q rq r r : Cattaneo equation

( , )( , ) exp ( , ) ( , ) exp ( , )

t t

qq q q q

t t t t tt T t dt T t T t dt

t

q rq r r r r

( , ) ( ) ( , ) ( , ) ( , )t t

t t t T t dt T t dt T t

q r r r r

( , ) ( , )t T t q r r : Fourier’s law

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2. Heat Equation

2) Joseph and Preziosi

0 1( ) ( ) ( / ) exp[ ( ) / ]qK t t t t t t

- : effective conductivity, : elastic conductivity 0 0 1

With assumption

10( , ) ( ) ( , ) exp ( , )

t t

q q

t tt t t T t dt T t dt

q r r r

1) 2)

1) 0 ( , )T t r

2) 1 exp ( , )t

q q

t tT t dt

r

( , )tt

q r 0 ( , )T tt

r

1 exp ( , )t

q q

t tT t dt

t

r

1)

2)

By Leibniz integral rule( ) ( )

( ) ( )

( ) ( )( , ) ( ( ), ) ( ( ), ) ( , )

b b

a a

db daf x dx f b f a f x dx

d d

, ( ) , ( ) , , ( , ) exp ( , )q

t tt b t a x t f x T t

r

( , ) ( ) ( , )t

t K t t T t dt

q r r

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2. Heat Equation

2) 1 exp(0) ( , ) 0 exp ( , )t

q q

dt t tT t T t dt

dt t

r r

1 12

( , ) exp ( , )t

q q q

t tT t T t dt

r r

0q qT Tt t

q

q 0 1, : steady-state thermal conductivity

10( , ) ( , ) ( , ) exp ( , )

t

qq q

t tt t T t T t dt

t

q r q r r r

10 1( , ) ( , ) exp ( , )

t

qq q

t tT t T t T t dt

t

r r r

0 1 0( , ) ( , ) ( ) ( , ) ( , )q qt t T t T tt t

q r q r r r

1 10 2

( , ) ( , ) ( , ) exp ( , )t

q q q

t tt T t T t T t dt

t t

q r r r r

10( , ) ( , ) exp ( , )

t

q q

t tt T t T t dt

q r r r

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3. Jeffrey Type Lagging Heat Equation

With eqn. (7.3) p p

T Tq c q c

t t

q q

22 2

02p q q p q

T q Tq c c T T

t t t t

22 2

2

1 qT

T TT T

t t t

0 : retardation timeqT

: Jeffrey’s eqn.

0 0 1, T q q T

0

0 1

1 01 ( 0) T q

0 10 ( 0) q Tt

q

q

: Fourier’s law

: Cattaneo equation

Generally, Thermal process b/w Fourier’s law and Cattaneo equation 0 1

2 20q qT T

t t

q

qBy on both side,

0q qT Tt t

q

q

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4. Dual-Phase-Lag Model by Tzou

Extended from lagging concept

With assumption ( , ) ( , )q Tt T t q r r

Delay timeT

q: heat source temperature gradient

: temperature gradient heat flux

In certain cases such as short-purse laser heating, both and exist T q

By Taylor expansion

Only requirement : , 0T q (does not require )q T

However, DPL model produces a negative conductivity component

Generalized form of DPL needs to be considered

q TT Tt t

q

q

same as heat equation by Joseph and Preziosi with 0T q

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4. Dual-Phase-Lag Model by Tzou

( , ) exp ( , ) ( , )t

Tq q

t tt T t T t dt

t

q r r rGeneralized form of DPL

1) 0T ( , ) exp ( , ) t

q qq

t tt T t d Tt

t

q

r qq r

2) T q ( , ) exp ( , ) exp ( , )t t

qq q q q

t t t tt T t dt T t dt

t

q r r r

q q q

T TT T

t t t

q

q

T q

Here, is not defined by can be theoretically allowed 0 1 T q

∴ More general than Jeffrey’s equation

Can describe behavior of parallel heat conduction

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5. Parallel or Coupled Heat Diffusion Process

( ) , ( )s p s f p fC c C c : volumetric heat capacities

d : rod diameter, N: number of rods, D : inner diameter of pipe

P N D : total surface area per unit length2 2 2/ 4, ( / 4)( )c fA N d A D Nd : total cross-sectional areas of rods and fluld

f sκ κ

/ , c f f f cG hP A C C A /A

Solid-fluid heat exchanger AssumptionsFluid is stationary, pipe is insulated from outside

Rods are sufficiently thin use average temp. in a cross section

Heat transfer along x direction only

Average convection coefficient h

Cross-sectional area of the fluid is also sufficiently thin

Constant Cs, Cf, κs, κf

2

2( )s s

s s s fc

T T PC h T T

t x A

2 2

2 2( ) ( )f f f f f f

f f s f f f s ff c c c

T T A T A TP PC h T T C h T T

t x A A t A x A

fC G

G

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2

2( ), ( )fs s

s s s f f s f

TT TC G T T C G T T

t x t

2 2

2 2 f fs s s s s s

s s ff f

T TT T C T TC C

t x t t C t C x

1) 2)

By combining eqn. 1) and 2) to eliminate Tf obtain differential equation for Ts

2 2 2 2 2

2 2 2 2 2 f fs s s s s s s s s s s s

s sf f

T TT T T C T T T C T TC G G

t t x t t t G t G t x t C t C x

1)t

2 2 2 2 2

2 2 2 2 21 1f f f ss s s s s s s s s s s s

f f s f s

C C C CT C T C T T T C T T

C G t x C t G t x G t x C t G t

1) 2) 3)

1) f

T

C

G

2) 1

1f f f ss s

s f s s s

C C C CC C

C

3) 1

fs

f s s fs T s Tq

s s s s s f

CCC C C CC CG

G C C

5. Parallel or Coupled Heat Diffusion Process

s T s f q T qC C C

T/( ), /s s f fC C C G

q /( ) ,s T s f TC C C relaxation timeq

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2 2 2

2 2 2

1 qs s s sT

T T T T

x t x t t

Differential equation for Ts

- Solutions exhibit diffusion characteristics

5. Parallel or Coupled Heat Diffusion Process

- Equation describes a parallel or coupled heat diffusion process

In this example, Dual-Phase-Lag model can still be applied

Initial temperature difference b/w rods and fluid local equilibrium X at the beginning

2 2 2

2 2 21f f f ss s s s s

s f s

C C C CT T C T T

x G t x C t G t

T

1

1

q

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6. Simplified BTE for Phonon System

0 1

N

f f f ff f

t

vr

Callaway

- No acceleration term, simplified scattering terms (two-relaxation time approx.)

- : relaxation time for U process : Not-conserved total momentum after scattering

- : relaxation time for N process : Conserved total momentum after scatteringN

- f0 , f1 : equilibrium distributions

Guyer and Krumhansl solved the BTE derived following equation

22 2

2 2 2

9 3 3

5N

a a

T TT T

t t t

: average phonon speedav

When 2 9

, , 3 5a N

q T

v 2

2 22

1 q

T

T TT T

t t t

Same as Jeffrey’s equation

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6. Simplified BTE for Phonon System

When Energy transfer by wave propagation N T q

The scattering rate for U process is usually very high

N process contributes little to the heat conductionAt higher temperature

Heat transfer occurs by diffusion mechanism, rather than by wave-like motion

Only at low temperature

Mean free path of phonons in U process is longer than specimen size

Scattering rate of N process is high enough to dominate other scatterings

Heat transfer occurs by wave-like motion called second sound

, 9 / 5q T N