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Temperature and heat conduction beyond Fourier law
Peter VánHAS, RIPNP, Department of Theoretical Physics
and BME, Department of Energy Engineering
1. Introduction2. Theories
− Cattaneo-Vernotte− Guyer-Krumhansl− Jeffreys type
3. Experiments
Common work with B. Czél, T. Fülöp , Gy. Gróf and J. Verhás
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Heat exchange experiment
http://remotelab.energia.bme.hu/index.php?page=thermocouple_remote_desc&lang=en
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)( 0TTlT
Model 1 (Newton, 3 parameters):
Estimate Std. Error
l 0.02295 0.00014
T0 52.52 0.02
Tini 27.33 0.09
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)( 0TTlT
Model 1 (Newton, 3 parameters):
Model 2 (Extended Newton, 5 parameters):
)( 0TTlTT
Estimate Std. Error
l 0.02295 0.00014
T0 52.52 0.02
Tini 27.33 0.09
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)( 0TTlT
Model 1 (Newton, 3 parameters):
Model 2 (Extended Newton, 5 parameters):
)( 0TTlTT
Estimate Std. Error
l 0.02295 0.00014
T0 52.52 0.02
Tini 27.33 0.09
Estimate Std. Error
l 0.0026 0.0009
T0 53.3 0.30
Tini 26.98 0.05
35.8 1.6
vTini 0.617 0.004
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Why?
− two step process
)(1
0TTTT
)(
)(
1
011
TTT
TTT
)( 0TTlTT
T0 T1 T
α β
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Oscillations in heat exchange:
− parameters model - values micro - interpretation
− macro-meso mechanism
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Fourier – local equilibrium (Eckart, 1940)
0
0
ii
ii
Js
qe
ji q
TJes
1,
Entropy production:
0,1
2 TT
T
L
TLq iiii
Constitutive equations (isotropy):
Fourier law
01111
Tqq
TTqq
TT
qe
de
dsJs iijiiiii
iiii
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Constitutive equations (isotropy):
Cattaneo-Vernotte
Cattaneo-Vernotte equation (Gyarmati, 1977, modified)
0
0
ii
ii
Js
qe
ji q
TJq
mes
1,
22
TT
Lqq
T
mLq
T
m
TLq iiiiii
2
1
Entropy production:
011
iii
iiiiiiii q
T
m
Tq
T
qqq
T
mq
TJs
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0 iiqe
Heat conduction constitutive equations
.
,
,
,
,
2
21
ijjii
iiii
ijjjijiii
iii
ii
qaTq
TlTqq
qaqaTqq
Tqq
Tq
Fourier (1822)
Cattaneo (1948), (Vernotte (1958))
Guyer and Krumhansl (1966)
Jeffreys type (Joseph and Preziosi, 1989))
Green-Naghdi type (1991)
Tkkc iiii ,21
there are more…
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0
0
ii
ii
Js
qe
jiji qBJ
mes
,
22
Thermodynamic approachvectorial internal variable and current multiplier (Nyíri 1990, Ván 2001)
Entropy production:
01
1
iijijiijijji
jijiiiiiii
T
mqB
TBq
qBT
mq
TJs
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Constitutive equations (isotropy):
.1
ˆ,ˆ
ˆ,ˆ
321
22221
1212121
ijkkijjiijij
ijiji
ijiji
qkqkqkT
B
T
mlllBl
T
mlllBlq
,),(,),(
,,,1
22
1231
2
112
2231
21
22
122
21
2
kl
lbkk
l
lbk
l
Lakk
l
La
Tl
l
Tl
L
l
ijjjijijjjijiiii qbqbqaqaTTqq 212121
0
,0,0
0ˆˆ
,0,0
3
21
211221
21
k
kk
llllL
ll
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1+1 D:
'.'''''
,0'
21 qbaqTTqq
qTc
'.'''ˆ''ˆ1TaTaTTT
''ˆ''ˆ
''ˆ''ˆ
''ˆ
''ˆ
TaTT
TaTTT
TTT
TT
Fourier
Cattaneo-Vernotte
Guyer and Krumhansl and Jeffreys type
Green-Naghdi type
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Calculations
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iii
iii
ii
qqqTqq
Tq
~ˆ~~~
ˆˆ
'ˆ')ˆ~
(
'ˆ'~
ˆ'ˆ'~~
TTqq
TTqqTTqq
1+1D:
a) Jeffreys-type equation – heat separation
Jeffreys
‘Meso’ models
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),(),( trTtrq ii
Taylor series:
TkTqq iiii 1
b) Jeffreys-type equation – dual phase lag
Jeffreys
This is unacceptable.
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''''1
)''()(''
12
12
111
112
12
2
212
2
21111
Tc
gT
c
cgc
Tcc
gT
c
gTT
c
gTgTgTc
1+1D:
)(
),(
2122
1
2111
TTgc
Tq
TTgqcii
ii
c) Jeffreys-type equation – two steps
Jeffreys
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Waves +… in heat conduction:
− thermodynamic framenonlocal hierarchy (length
scales)
− macro-meso mechanismsframe is satisfied
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Memory Nonlocality Objectivity Thermo-dynamics
Fourier no no research ok
Cattaneo-Vernotte yes no research ok
Guyer-KrumhanslGyarmati-Nyíri, (linearized Boltzmann)
yes yes ? ok
Jeffreys type1. internal variable, 2 .heat separation3. dual phase lag,
4. two steps
yes yes ? ok
Ballistic-diffusive(Boltzmann split)
yes yes ? ?
Heat conduction equations
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Experiments
Homogeneous inner structure – metals- typical relaxation times: = 10-13- 10-17 s- Cattaneo-Vernotte is accepted: ballistic phonons(nano- and microtechnology?)
Inhomogeneous inner structure- typical relaxation times: = 10-3- 100 s- experiments are not conclusive
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Resistance wire Thermocouple
Kaminski, 1990
Particulate materials: sand, glass balottini, ion exchanger
= 20-60 s
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Mitra-Kumar-Vedavarz-Moallemi, 1995
Processed frozen meat: = 20-60 s
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Scott-Tilahun-Vick, 2009
− repeating Kaminski and Mitra et. al.
− no effect
Herwig-Beckert, 2000
˗ sand, different setup˗ no effect
Roetzel-Putra-Das, 2003
− similar to Kaminski and Mitra et. al.
− small effect
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Summary and conclusions
− Inertial and gradient effects in heat conduction
− Internal variables versus substructures(macro – micro) black box – universality
− No experiments for gradient effects(supressed waves?)
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iiiu
ii qqquuuqu ˆ~,~ˆ,
1D:
,ˆ~,ˆ
ˆ
~~
0
0 fffff
f
ffff
f
uqq
uqu
iii
ii
ˆˆˆ
,~~~
Ballistic, analytic solution
Ballistic-diffusive equation (Chen, 2001)Boltzmann felbontás
diffusive
?