Post on 04-Feb-2016
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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