Upload
justin-wade
View
216
Download
1
Embed Size (px)
Citation preview
Hyperbolic Heat Equation
1-D BTE 0x
f ff fv
t x
0 0f ff T
x x T x
assume
Integrate over velocity space assuming that is an averaged relaxation time
2 00x x x x
fTv fd v d v f d v fd
t x T
201
3
fv d
T
(4.63a) q fv d
(4.64)
20 ,x x
fv d
T
x xq fv d
x
x x
q Tq
t x
20x x x
f fv v v f f
t x
or
or xx x
q Tq
t x
In 3-D q
( , )( , ) ( , )
q r tq r t T r t
t
Maxwell-Cattaneo equation
Energy equation without convection
( , )( , ) ( , )p
T r tc q r t q r t
t
q
qq T
t
2
qq q Tt
p
Tc q q
t t
2
2p
T qc q
t t t
2
q q q
pcq Tq T
t t
2qp
Tq c q T
t t
Cattaneo equation
22
2q q q
pp
cT q T qc T
t t t
2q q2
2
p pc cT T q qT
t t t
2q q2
2
1T T q qT
t t t
Without internal heat generation2
22 2tw
1 1,
T TT
v t t
Damped wave equation
twq
v
Decay in amplitude: q
expt
21
3 v gc v For an insulator:
21
3 gv tw3gvv
hyperbolic equation
speed of thermal wave
Hyperbolic heat equation : valid strictly when tp >
semi-infinite solid under a constant heat flux at the surface
propagation speed : vtw
pulse wavefront: x1 = vtwt1, x2 = vtwt2
short pulse long pulse
In the case of a short pulse, temperature pulse propagates and its height decays by dissipating its energy to the medium as it travels.
Entropyu
T s qt
Local energy balance
Local entropy balance gen
s qs
t T
Entropy production rate
gen q2 2
1 1 qs q T q q
T T t
Hyperbolic heat equation sometimes predicts a negative entropy generation. (Energy transferred from lower-temp. region to higher-temp. region)
Negative entropy generation is not a violation of the 2nd law of thermodynamicsBecause the concept of “temperature” in the hyperbolic heat equation cannot be interpreted in the conventional sense due to the lack of local thermal equilibrium
Dual-Phase-Lag Model
Finite buildup time after a temperature gradient is imposed on the specimen for the onset of a heat flow: on the order of the relaxation time Lag between temperature rise and heat flow
In analogy with the stress-strain relationships of viscoelastic materials with instantaneous elasticity
( , ) ( ) ( , )t
q r t K t t T r t dt
( ) ( ),K When
( , ) ( , )q r t T r t
q/
q
( )K e
When
→ Cattaneo eq.
electrons → lattice phonons → temperature rise
By assuming that p/1
0q
( ) ( )K e
10
q q
( , ) exp ( , )t t t
q r t T T r t dt
2
q 2T2
1,
T TT T
t t t
q q 0 ,q
q T Tt t
0 1
0T q
retardation time
0 : effective conductivity (heat diffusion)
1 : elastic conductivity (hyperbolic heat)
( , )( , ) ( , )p
T r tc q r t q r t
t
dual-phase-lag model
q T( , ) ( , ),q r t T r t
q q
( , )( , ) ( , )
q r tq r t q r t
t
T T( , ) ( , ) ( , )T r t T r t T r tt
q T
( , )( , ) ( , ) ( , )
q r tq r t T r t T r t
t t
First order approximation
q q 0
qq T T
t t
q 0 T
T q
T q, : intrinsic thermal properties of the bulk material
T q The requirement may cause .1 0
1T
q q
( , ) exp ( , ) ( , )t t t
q r t T r t T r t dtt
The heat flux depends not only on the history of the temperature gradient but also on the history of the time derivative of the temperature gradient.
→ Cattaneo eq. When
T 0
When
T q → Fourier law
Solid-fluid heat exchanger
immersed long thin solid rods in a fluid inside a sealed pipe insulated from the outside
rod diameter d, number of rods N
inner diameter of pipe D
total surface area per unit lengthtotal cross-sectional area of the rodstotal cross-sectional area of the fluid
2c / 4A N d
2 2 2
2 2f c4 4 4 4
D D N dA A D Nd
average convection heat transfer coefficient h P N d
ff s f
TC G T T
t
f ff
c c
, C AhP
G CA A
2
s ss s s f2
T TC G T T
t x
2f
f 2 0T
x
s f assume
2 2 2qs s s s
T2 2 2
1T T T T
x t x t t
s sfT q T T
s f s f
, , CC
C C G C C
Due to the initial temperature difference between the rod and the fluid , a local equilibrium is not established at any x inside the pipe until after a sufficiently long time.
Two-Temperature Model
electrons → electron-phonon interaction → phononsUnder the assumption that the electron and phonon systems are each at their own local equilibrium, but not in mutual equilibrium (valid when t > )
ss e s
TC G T T
t
e : electron, s : phononC : volumetric heat capacity = cp
G : electron-phonon coupling constant
ee e e s a
TC T G T T q
t
Assume the the lattice temperature is near or above the Debye temperature so that electron-electron scattering and electron-defects scattering are insignificant compared with electron-phonon scattering. 2 2
e Be e
F2
n kC T
2
B,e
F2v
k Tc R
(5.25) esT
2 2e B
e3
n k T
m
(5.55b)
2 2e B
ee3
n kT
m
eq
es
TT
eq : thermal conductivity at
e sT T2 2 2 2 2 2
eq eqe B e B e Beq s e e
e s e s e
, , 3 3 3
n k n k n kT T T
m T m T m
2 2e e a
s6
m n vG
T
or 24e a B
eq18
n v kG
1/ 3
aD a
B
3
4
nhv
k
(5.10)1/ 3
B Da
a
4
3
kv
h n
ee e e s a
TC T G T T q
t
ss e s
TC G T T
t
2
q2 2 a a e eTe T e 2
1q q T TT T
t t t t
2
q2 2 a s ss T s 2
1q T TT T
t t t
s e eT q T T
e s e s
, , C C C
C C G C C G
q : not the same as the relaxation time due to collisionThermalization time: thermal time constant for the electron system to reach an equilibrium with the phonon system
q T: 30 ~ 40 fs, : 0.5 ~ 0.8 ps, : 60 ~ 90 ps
Heat Conduction Across Layered Structures
Equation of Phonon Radiative Transfer (EPRT)
00
scatt ( , )
f ff f fv S
t r t T
S0: electron-phonon scattering
Phonon BTE in 1-D system
0x
f ff fv
t x
*BE
B
1
exp 1
f f
k T
*
x
f f f fv
t x
or
phonon Intensity: summation over the three phonon polarizations
g
1( , , ) ( )
4 P
I x t v f D
phonon intensity under non-equilibrium distribution function: energy transfer rate in a direction from a unit area, per unit frequency, per unit solid angle
( ) ( ) ( )x xq x v f x D d
g gcosxv v v
+ forward direction ( > 0)- forward direction ( < 0)
T1
T2
q
I
I
x
*
x
f f f fv
t x
*
g g
( )1
( , )
I I I T I
v t x v T
Equation of Phonon Radiative Transfer (EPRT)
*
4
1( )
4
I Ia I a I I d
c t x
Radiative Transfer Equation (RTE)
optical thickness of the medium ,L a L
neglecting scattering 0
*1 I Ia I I
c t x
acoustical thickness of the mediumg
L
L
v
g
1 1
v
corresponds toa
Equilibrium intensity, Bose-Einstein statistics
B B
2 3g*/ 3 /3 2
p
( , )e 1 8 e 12
k T k TP P
v k dkI T
d v
g
1( , , ) ( )
4 P
I x t v f D
B
2*
/ 2
1, ( )
2e 1k T
k dkf D
d
(5.33)
22
2p p p
,p k k kv v v
g
dv
dk
analogy to blackbody intensity
B B B
33 3
/ / /2 2 2 2
2 2 / 22( )
e 1 e 1 2 e 1b h k T k T k T
hI
c c c
integrating over all frequency → total intensity for all 3 phonon modes
3 4B
SB 3 2a40
k
v
: Phonon Stefan-Boltzmann constantva: average phase velocity of the 2 transverse and 1 longitudinal phonon modes
SB
44 4 3* * SBB
3 3 20 0a
3( ) ( , )
8 e 1x
Tk T x dxI T I T d
v
2
4 341 1
4/50 0 0 02
2 2
e 11b b xC T
C C T x dxI I d d T
Ce
2 2
1 0 0 2 0 B 0 B2 , / 2 /C hc c C hc k c k
2 55 2 401 B
44 3 22 00 B
2 22
15 6015 2 /
cC k
C cc k
At temperature higher than Debye temperature
high frequency limit because the shortest wavelength of the lattice wave should be on the order of atomic distances, the lattice constants
mD
B
,h
k
D m m
mB B
1h
xT k T k T
m3
* * m B3 20
p
( ) ( , )8
kI T I T d T
v
energy flux at phonon equilibrium: I* invariant with direction
* * *ˆ ˆ( ) ( ) cos ( )q I T nd I T d I T
particle flux
4N
nvJ (4.12a)
energy carried by a phonon = energy flux / particle fluxenergy density= energy carried by a phonon X number of phonons per unit volume
At low temperature
duC
dT
**
g g
( ) 4( ) ( )
/ 4
I Tu T n I T
nv v
* 4SB
g g
4 4( ) ( )u T I T T
v v
3C T
At high temperature3
* m B3 2
g g p
4 4( ) ( )
8
ku T I T T
v v v
Dulong-Petit law: 3vc R
0 0dq
qdx
radiative equilibrium
*14
dqI G
dx
2 1
4 0 0 1
ˆ( ) ( , ) 2G x I x d I d I d
1*
1
4 2GI I d
m m 1*
0 0 1
1 14 2I d I d d
total quantities
1*
1
1
2I I d
or
2 1
4 0 0 1
ˆ ˆ cos sin 2q I nd I d d I d
T1
T2
q
I
I
x
Solution to EPRT
steady state*
g
( ( ))
( , )
I I T x I
x v T
Two-flux method in planar structures
*
, 0 1I I I
x
when
*
, 1 0I I I
x
when
*1 1 1(0, ) ( ) 1 (0, )I I T I
boundary conditions: gray medium, diffuse and gray walls
*2 2 2( , ) ( ) 1 ( , )I L I T I L
Solutions to EPRT
*
0( , ) (0, )exp ( )exp
xx x dI x I I
*( , ) ( , )exp ( )expL
x
L x x dI x I L I
Heat Flux (spectral)
1
0
* *2 20
2 (0, )exp ( , )exp
2 ( ) 2 ( )x L
x
x L xq I I L d
x d x dI E I E
1 2
0( ) expn
n
xE x d
2 1
4 4 0 0 1
ˆ ˆ cos cos 2q I nd I d I d I d
*3 20
( ) 2 (0) ( ) ( ( )) ( )q I E I T E d
*
3 22 ( ) ( ) ( ( )) ( )L
L LI E I T E d
For diffuse surface
3 3
* *2 20
2 (0) 2 ( )
2 ( ) 2 ( )x L
x
x L xq I E I L E
x d x dI E I E
total heat flux
1
12xq q d I d d
Heat flux at very low temperature
4 4SB 1 SB 2
1 2
4
3 1 1 1 1 41
2 2 3
x
T Tq
L Kn
Heat flux between blackbodies with small T difference
beff4
13
x
T Tq
KnL L
Thermal conductivity
3SB
b
16
3
T
1
02 (0, )exp ( , )exp
x L xq I I L d
* *2 20
2 ( ) 2 ( )x L
x
x d x dI E I E
acoustically thick limit
** *( ) ( ) ( )
dII x I x
dx
x
z
Let*
204 ( )
Iq zE z dz
x
1 2 D ,T T when
3SB16
3x
T dTq
dx
3
b SB
16
3T
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Effective Conductivity of the Film
3 effb SB
b
16 143 1
3
TKn
KnL
eff
b
Kn
Thermal Resistance Network
1bT
2aT 2bT
3aT3bT
xq
HT
LT1 2 3
H L
totx
T Tq
R
H1R1R
12R2R
23R3R
3LR
effective thermal conductivity of the whole layered structuretot
efftot
L
R
Internal thermal resistance with Fourier’s law
ii
i
LR
Thermal resistance at the interface
4 1 1 4 1 1
3 2 3 2ji
iji ij j ji
R
:ij transmissivity
Thermal resistance at the boundaries
31H1 3L
1 1 3 3
4 1 1 4 1 1,
3 2 3 2R R
H1R1R
12R2R
23R3R
3LR
Thermal Boundary Resistance (TBR)
TH
11 22 33TL
Acoustic Mismatch Modelspecular reflection of phononssimilar to geometric opticsno scattering or diffusion
Diffuse Mismatch Modeldiffusive reflection of phononsno information (except for energy) retained after a scattering
Thermal Contact Resistancedue to incomplete contact between two materialsthermal resistance between two bodies, usually with very rough surfaces, rms > 0.5 m Thermal Boundary Resistance
due to the difference in acoustic properties of adjacent materials
1T 2T
22
11
1 1, v 2 2, v
Acoustic Mismatch Model (AMM)For longitudinal phonon modes with linear dispersion in a Debye crystal (equilibrium, isotropic, perfect contact)
g
1( , , ) ( )
4 P
I x t v f D
1 1 1 1
1( , , ) , ( )
4 lI x t v f T D
1 2 ,1 2 12 1cosq q d I d d
m 2 / 2
1 2 1 1 1 12 1 1 1 10 0 0
1( , ) ( )cos sin
4 lq v f T D d d d
1 2 1 2 1 2
12 21 2
1 1 2 2 2 1
4 cos cos
cos cosl l
l l
v v
v v
transmission coefficient
1 2
1 2
sin sin
l lv v
Snell’s
law
Let 12 1
12
1
cos
cos
d
d
: hemispherical transmission
2 / 2 / 2
12 1 1 1 1 12 1 1 10 0 0
1cos sin 2 cos sind d d
m 2 / 2
1 2 1 1 1 12 1 1 1 10 0 0
1( , ) ( )cos sin
4 lq v f T D d d d
m 2 / 2
1 1 1 12 1 1 1 10 0 0
1( , ) ( ) cos sin
4 lv f T D d d d
m / 2
1 1 1 12 1 1 10 0
1( , ) ( ) cos sin
2 lv f T D d d
m m 312 121 1 1 1 1 120 0
1
1( , ) ( ) ( , )
4 4l ll
v f T D d v f T dv
1 2
1 2
sin sin
l lv v
Snell’s law critical angle: C
1 22
1
sin
sinl
l
vv
C21 21
2 2 22 202 2
2 cos sinl l
dv v
C / 22 221 2 12 1 121 2 1 12 2 20 0
1 2 1 1 1
cos cos2 sin 2 sin
sin sinl l l
d dv v v
m 3121 2 1 1 12 0
1
1( , )
4 ll
q v f T dv
1 2 2 1xq q q
m 3 3121 1 1 2 2 22 0
1
1( , ) ( , ) ( )
4 l ll
v f T v f T D dv
B
2
/ 2
1, ( )
2e 1k T
k dkf D
d
(5.33)
22
2p p p
,p k k kv v v
g
dv
dk
2
g g 2
1 1( , ) ( ) ( , )
4 4 2
k dkv f T D v f T
d
B
2 3
3 2 /3 2p p
1( , )
8 8 e 1k Tf T
v v
For longitudinal phonon modes with linear dispersion in a Debye crystal (equilibrium, isotropic, perfect contact)
p l
dv v
dk k
B
3
/3 2
1( , ) ( )
4 8 e 1l k T
l
v f T Dv
m 3 3121 1 1 2 2 22 0
1
1( , ) ( , ) ( )
4x l ll
q v f T v f T D dv
B
3 3
3 2 /3 2
1 ( , )( , ) ( )
4 8 8 e 1l k T
l l
f Tv f T D
v v
33
2
( , )( , ) ( )
2l
f Tv f T D
m 3121 1 2 22 2 0
1
1( , ) ( , )
8xl
q f T f T dv
B( / , 1 2)jx k T j or
m1 m24 3 3
4 412 B1 22 2 3 0 0
1 8 1 1
x x
x xl
k x dx x dxT T
v e e
In the low temperature limit,
2 4
4 412 B1 22 3
1 120xl
kq T T
v
For all 3 phonon modes
3 4
0 1 15x
x dx
e
m1 m24 3 3
4 412 B1 22 2 3 0 0
1 8 1 1
x x
x x xl
k x dx x dxq T T
v e e
2 2 21 1 1j l t
j
v v v
2 4
4 4 2B1 2 12 13120x j
j
kq T T v
Thermal boundary resistance
31 2 1 2
b,AMM 2 4 4 4 2B 1 2 12 1
120 1
x jj
T T T TR
q k T T v
3
2 4 22 2B 12 11 2 1 2
120 1 1
jj
k vT T T T
1 2T T T
TBR by AMM is proportional to T-3.
AMM assumes perfect specular reflection which is valid only when the characteristic wavelength of the phonons is much larger than the surface roughness. (mp >> rms)→ DMM (Diffuse Mismatch Model)
When the temperature difference is small,
2 2 31 2 1 2 4T T T T T
3 3
b,AMM 2 4 2B 12 1
30
jj
TR
k v
Dmp a
T
a: lattice constant
Diffusion Mismatch Model (DMM)In DMM, all phonons striking the interface are scattered once, and are emitted into the adjoining substances elastically ( = 0) with a probability proportional to the phonon density of states (DOS) in the respective substances.
Distribution of the emitted phonons is independent of the incident phonon, whether it is from side 1 or 2, longitudinal or transverse.
,12 12 ,21 21 12 21( , ) ( ), ( , ) ( ), ( ) ( ) 1j j
integration over the solid angle12 21( ) ( ) 1
2 212 1 21 2j j
j j
v v 12 212 2
1 2l lv v
2 22 1
12 212 2 2 21 2 1 2
, j j
j j
j j j jj j j j
v v
v v v v
at low temperature
by DMM (12) with the assumption1 2T T T
2 4
4 4 2B1 2 12 13120x j
j
kq T T v
3 2 2 31 2
b,DMM 2 4 2 2B 1 2
30 j jj j
j jj j
v v T
Rk v v
3 3
b,AMM 2 4 2B 12 1
30
jj
TR
k v
2 21 2 12,AMM
b,DM
2b,AM 2
j jj j
jj
v vR
R v
Measured Data
TBR between indium and sapphire[2]. X, normal indium; ● super conducting indium. Data B are for roughened sapphire surface. A and C are for smooth sapphire surfaces with different indium thicknesses. AMM predicts a flat line at 20.4 cm2K4/W