Post on 21-Mar-2018
Ecole Thématique du CNRS en Thermoélectricité
Approche Thermodynamique de la Thermoélectricité
Christophe Goupil Laboratoire Interdisciplinaire des Energies de Demain, LIED
Université Paris Diderot
Part one:Electrons and
thermodynamics
Thermoelectricity.
PELTIER (1834) SEEBECK (1823)
“Coupling Ohm’s law and Fourier’s law”
1794 --- 1795: Letter to professor Antonio Maria Vassalli (accademia delle
scienze di torino ) "... I immersed for a mere 30 seconds the end of
such arc into boiling water, removed it and allowing no time for it to cool
down, resumed the experiment with two glasses of cold water. It was then
that the frog in the water started contracting, and it happened even two,
three, four times on repeating the experiment till one end of the iron
previously immersed into hot water did not cool down".
The Volta Story
Alessandro Volta
18 February 1745 – 5 March 1827)
Thermodynamics
What is a good system?
21 QQWP
)1(1
21
2
2
1
121
T
TQW
T
Q
T
QSS
02121
QQWQQ
Fully reversible Fully irreversible
Perfect system means:
conservative transport of the entropy.
Thermal & Electrical coupling.
ThTc1
2
34
• Reversible adiabatic transport of the carriers. (isentropic)
• The convective part contribute to the entropy transport…
…but not the conductive part .( leak)
• => Reduce the conductive part and increase the convective part .
• The electical conductivity shoud be large.
•=> Yes but by increasing mobility only.
• 1: Adiabatic.
• 2: Isothermal.
• 3: Adiabatic.
• 4: Isothermal.
P
Th
Tc
ap(T
h-T
c)
Simple Thermoelectric generator.
+ + + ++ + +
+ ++
N
Th
Tc
an(T
h-T
c)
- - - -- - -
- --
I
I
I
I
Gibbs free energy and entropy.
v1, N1 v2, N2
p
« Entropy per carrier »
Fluid & « Lattice » .
Purely electronic part: “Gas”
(transported entropy)Lattice contribution.
(“boiling walls” )
(Th , Sh)
(Tc , Sh)
( m, N)
Qh
Qc
WTF
Latt
ice
Electronic gas + Lattice Steam + boiling walls
1
1
12
a
el
latel
TZT
ZT = a2
(e + L)· T
Seebeck coefficientElectrical
conductivity
Thermal conductivity
1017 1018 1019 1020 1021
S
insu
lato
rs
met
als
S2
Carrier Concentration
Tota
l K
ZT
semiconductorsL
e
a
ZTmax
PGEC: « Phonon-Glass Electron-Cristal »
The « Graal » of the best TE material.
mH
mCm
En
erg
y
TCTH
eVS = mH-mC =-ea(TH-TC)
eVS
Thermal biasing
I=0
Th
Tc1
2
34
mH
mCm
En
erg
y
TCTH
Vapp=VS
Thermal biasing
I=0TH
eVS = mH-mC =-ea(TH-TC)
mH
mC
m
Energ
y
TCTH
eVS = mH-mC =-ea(TH-TC)
eVS
Vapp= VS +Vpol
eVpol
eVpol
I=0
Thermal & Electrical biasing
Different operating modes: generator of receptor
TH
Operating modes
Generator
(TEG)
Receptor
(Heater)
Receptor
(Cooler)
IRin
VS Vap
p
Ioffe basic TE model.
aITc
Th
Tc
I
a(T
h-T
c)
aITh
K(T
h-T
c)
1/2RI2
1/2RI2
TEG
aITh
I
aITc
K(T
h-T
c)
Tc
Th
1/2RI2
1/2RI2
TEC
I
aITh
aITc
K(T
h-T
c)
Th
Tc
1/2RI2
1/2RI2
TEH
Efficiency or Power for a TEG
ZT=1.5, 15, 30, 300,
Non endoreversible
endoreversible
Thermodynamicof the fluid.
Elastic coefficients
TC
S
C
C
N
TN
N
m2
1
Thermodynamic of the fluid.
Vining 1997TTE
0
2
0
1
a
TC
S
C
C
N
TN
N
m2
1
TT
T
Q
0
2
1
a
Finite time Thermodynamics:Onsager-Callen
Gibbs relation & state equation.
Gibbs relation.
The Fi are potentials of the system.
S(Xi) contains all the information of the system.
Potentials,
1/T is the conjugate of E.
mi /T is the conjugate of N.
Affinities, Generalized forces
2 subsystems exchanging Xi adn X’ i at constant X0i.
Total S is the sum of the entropy of the 2 subsystems.
Fi =0, S extremal
at equilibrium.
« The forces are given by the gradient of the Potentials ».
Potentials Affinities, forces
Fluxes, entropy and linear response.
Flux of the quantity Xi.
Entropy production.
Linear response.
Matrix of
kinetic coefficients.
Generalized
Force.
Coupled fluxes: the Onsager model.
Generalized
Forces.
• Linear & Stationary transport means dissipation.
Remark: Quasi-static thermodynamics
Time constants.
« Instantanate entropy »
Out of equilibrium description.
Yes, but:
• Quasi-staticStationary cond.
• Minimal entropy production.
• Electronic fluid description.
The thermoelectric cells should be large enough to consider:
• Irreversible processes (avoid microreversibility).
• Slow relaxation time compared to microscopic relaxation time.
Jin
JEin
Jout
JEout
T
QdS
QS
T
JJ
QS
What can we do with that?
Current and heat fluxes.
WQdE NeEQ JJJ
m
!
NJeJ
WQdE NeQE JJJ
m
!
Decoupled processes:
Isothermal case: Ohm’s law
J=0, or Dm=0 : Fourier’s law
)1(2 TZT elecJJTE a
Coupled processes.
J=0 : Seebeck effect Isothermal : Peltier effect
There is no « specific» Seebeck or Peltier, nor Thomson, coefficients.
These are only expression of the same underlying physic
under specific thermodynamic condtions, isothermal, no current…
The entropy per carrier is a measurement
of the quality of the electronic fluid
to carry the entropy (remove or add).
The « Entropy per carrier ».
Kinetic coefficients & transport.
===
Are there any more familiar expressions for this?
The conductivity matrix.
The Seebeck coefficient IS the coupling parameter.
Heat &Entropy fluxes
Heat transformation per unit volume.
Energy conservation:
Carriers conservation:
Heat transformation contributions.
Peltier-Thomson & Co.
Isothermal case: Peltier
Thermal gradient case: Thomson
Kelvin relations:
Entropy production per unit volume.
The Wiedeman-Franz Law.
Metal
Non Degenarate Semi-Conductor
Lorentz number:
22
11
22
0
2211
122
22
11
22
12
1
TeL
L
T
LL
L
TeL
L
T
LT
J
T
J
Lorentz number and ZT
The lattice contribution is pure loss!
Part twoSystem optimization
The CNCA engine
Perfect engine? Thot
Tcold
W
Qin
Qout
hot
cold
in
CT
T
Q
W 1
No power
BUT infinite time do produce W.
Why ? Reversible also means acausal. No defined startup conditions!
Solution ? Modification of the boundary conditions.
• J. Yvon, The saclay Reactor: Two Years of Experience in the Use of a Compresed gas as a Heat Transfer Agent, Proceedings of the International Conference on the Peaceful Uses of Atomic Energy (1955)
• P. Chambadal Les centrales nucléaires. Armand Colin, Paris, France, 4 1-58, (1957)
• I.I. Novikov, Efficiency of an Atomic Power Generation Installation, Atomic Energy 3 (1957)• F.L. Curzon & B. Ahlborn, Efficiency of a Carnot Engine at Maximum Power Output, Am. J. Phys. 43 (1975)
Endoreversible
Thot
Tcold
Work
P
ηηCA ηCarnot
Pmax
Finite Time Thermodynamics
FTT
Thot
Tcold
Power
Endoreversible
hot
cold
in
CT
T
Q
W 1
hot
cold
in
CAT
T
Q
W
1
Generator (TEG)Receptor (Heater) Receptor (Cooler)
I
I(V) response:
0
E
E
=>
ZT
I
II
CC
TEG 1)( 0
ZTE 10
General model: presentation
ZT
I
IKIK
CC
TEG 1)( 0
2
0
1
.. .Q hM cM
I VR R
I T TT TK
R R
a
a a
D
0. .( )Q hM cM
advectionconduction
I T I K T Ta
( )oc hM cMV T Ta
Y. Apertet, H. Ouerdane, O. Glavatskaya, C. Goupil et Ph. Lecoeur, EPL 97 (2012)
Effective thermal conductance !
Force-Flux :
2
0 ( )
adv
Q hM cM
load
K
TI K T T
R R
KTEG
a
General model: Onsager description
Conduction
Convection
Is a function
of Rload!
Thevenin model:
Y. Apertet, et al. EPL 97 (2012)
( )oc hM cMV T Ta
2
0 0
' '
contactoc
contact contact
K TV T I
K K K K
V Roc
aa D
General model: resulting picture
ZT
I
IKIK
CC
TEG 1)( 0
For givenThermal contacts
M. Freunek et al., J. Elec. Mat. 38 (2009)K. Yazawa et A. Shakouri, JAP 111 (2012)
See also:
Electric adaptation Thermal adaptation
0 1
1
contact
load
K K ZT
R R ZT
The thermal adapatation is fundamentalfor correct working conditions!
Power
Y. Apertet, et al. EPL 97 (2012)
Special thanks to
• Henni Ouerdane
• Yann Apertet
• Philippe Lecoeur
• Aurélie Michot
• Olga Glavatskaya
• Eckhard Müller
• Knud Zabrocki
• Wolfgang Seifert
• Jeffrey Snyder
• Cronin Vining
Dilbert 10-10-1993
1. Rowe, D.M., Ed. CRC Handbook of Thermoelectrics: Macro to Nano; RC:
Boca Raton, FL, USA, (2006).
2. Seebeck, T.J. Ueber den Magnetismus der galvanischen Kette. Technical
report for the Royal Prussian Academy of Science: Berlin, Germany, (1821).
3. Peltier, J.C.A. Nouvelles expériences sur la caloricité des courants électrique.
Annales de Chimie et de Physique, 56, 371---386 , (1834)
4. Thomson, W. On the Mechanical Theory of Thermo-electric Currents. Trans.
R. Soc. Edinburgh: Earth Sci. 3, 91---98, (1851)
5. Onsager, L. Reciprocal Relations in Irreversible Processes. I. Phys. Rev. 37,
405---426, (1931). Onsager, L. Reciprocal Relations in Irreversible Processes.
II. Phys. Rev. 38, 2265---2279, (1931).
6. Callen, H.B. The Application of Onsager's Reciprocal Relations to
Thermoelectric, Thermomagnetic, and Galvanomagnetic Effects. Phys. Rev.
1948, 73, 1349--1358. Callen, H.B. On the theory of irreversible processes.
PhD thesis, Massachusetts Institute of Technology - (M.I.T.), Cambridge, MA,
USA, (1947)
7. de Groot, S.R. Thermodynamics of Irreversible Processes; North-Holland
Publishing Company: Amstedam, The Netherlands, 1963.
General Bibliography
1. Müller E. , Zabrocki K. , Goupil C., Snyder G.J., and W. Seifert. Functionally
graded thermoelectric generator and cooler elements. In D.M. Rowe, editor,
CRC Handbook of Thermoelectrics: Thermoelectrics and Its Energy
Harvesting, Vol. 1, Chapter 4. CRC Press, Boca Raton, FL, 2012.
2. Vining, C.B. The thermoelectric process. In Materials Research Society
Symposium Proceedings: Thermoelectric Materials - New Directions and
Approaches; Tritt, T., Kanatzidis, M., Lyon, H.B., Jr., Mahan, G., Eds.;
Materials Research Society: Warrendale, PA, USA; pp. 3---13 (1997)
3. Snyder, G.J.; Ursell, T.S. Thermoelectric Efficiency and Compatibility. Phys.
Rev. Lett. 91, 148301, (2003)
4. Goupil, C. Thermodynamics of the thermoelectric potential. J. Appl. Phys.
106, 104907, (2009)
5. Ioffe, A. Semiconductor Thermoelements and Thermoelectric Cooling;
Infosearch, ltd.: London, UK, (1957)
6. Curzon, F.; Ahlborn, B. Efficiency of a Carnot engine at maximum power
output. Am. J. Phys. , 43, 22---24, (1975)
7. Andresen, et al. Thermodynamics in finite time: Extremals for imperfect heat
engines. J. Chem. Phys. 66, 1571---1577, (1977)
8. Apertet Y. et al. Physical Review E 85, 041144 (2012)
9. Apertet Y. et al. Europhysics Letters 97, 28001 (2012)
Specific Bibliography
Addendum
Mesoscopic version
CouplingY Apertet et al. EPL 97 (2012) N. Nakpathomkun et al. PRB 82
(2010)
Macroscopic Mesocopic
Macrososcopic
• One thermodynamic fluid: => ZT
• One engine
• Two heat exchangers
• Two reservoirs
• Strong coupling, or possible leakage
TcoldThot
Power
K0
Mesoscopic• One thermodynamic fluid: => ZT?
• One engine?
• Two heat exchangers?
• Two reservoirs?
• Strong coupling or possible leakage?
?
mH
mC
m
TCTH
eVpol
I=0
Thermal & Electrical biasing
1 2
Ok if one isolated level
TH
T(E)
Coupling & Broadening
• One thermodynamic fluid: => ZT? NO because ZT=f(E)
• One engine? NO
• Two heat exchangers?
• Two reservoirs?
• Strong, or not, coupling possible?
To be considered together
YES for the lattice and
broadening means dissipation