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

Heat exchange experiment

http://remotelab.energia.bme.hu/index.php?page=thermocouple_remote_desc&lang=en

)( 0TTlT

Model 1 (Newton, 3 parameters):

Estimate Std. Error

l 0.02295 0.00014

T0 52.52 0.02

Tini 27.33 0.09

)( 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

)( 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

Why?

− two step process

)(1

0TTTT

)(

)(

1

011

TTT

TTT

)( 0TTlTT

T0 T1 T

α β

Oscillations in heat exchange:

− parameters model - values micro - interpretation

− macro-meso mechanism

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

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

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…

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

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

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

Calculations

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

),(),( trTtrq ii

Taylor series:

TkTqq iiii 1

b) Jeffreys-type equation – dual phase lag

Jeffreys

This is unacceptable.

''''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

Waves +… in heat conduction:

− thermodynamic framenonlocal hierarchy (length

scales)

− macro-meso mechanismsframe is satisfied

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

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

Resistance wire Thermocouple

Kaminski, 1990

Particulate materials: sand, glass balottini, ion exchanger

= 20-60 s

Mitra-Kumar-Vedavarz-Moallemi, 1995

Processed frozen meat: = 20-60 s

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

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?)

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

?