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Chaire de Physique de la Matière Condensée Antoine Georges Cycle 2012-2013 ``Enseigner la recherche en train de se faire’’ THERMOELECTRICITE: CONCEPTS, MATERIAUX ET ENJEUX ENERGETIQUES
Chaire de Physique de la Matière Condensée
Antoine Georges Cycle 2012-2013
``Enseigner la recherche en train de se faire’’
THERMOELECTRICITE: CONCEPTS, MATERIAUX
ET ENJEUX ENERGETIQUES
Second lecture – March 27, 2013
• Transport coefficients, linear response • Thermodynamics of energy conversion and
application to thermoelectrics [mostly on blackboard, detailed notes available on website]:
- Entropy current and entropy production rate - Efficiency of energy conversion: - Reminders on Carnot engines - Endoreversible engines and Chambadal-Novikov
efficiency - The efficiency vs. Power diagram
Excerpt from abstract:
Séminaire – 27 mars 2013
Transport equations: Linear response
and thermoelectric coefficients
Transport equations (linear response) : Grand-canonical potential (per unit volume):
Particle-number and Entropy (densities):
Particle and entropy currents: linear response
Positivity of entropy production (2nd principle of thermo.) implies that L is a positive semi-definite matrix (see later). This amounts to:
Onsager’s reciprocity relation (in the absence of an applied magnetic field):
Chemical potential: in fact `electrochemical’ potential’
In practice, electric field:
Electrical and heat currents:
Heat :
Electrical conductivity:
Thermal conductivity: (no particle current)
Seebeck and Peltier coefficients: 1. Seebeck effect: thermal gradient induces a voltage drop between the
two ends of a conductor
2. Peltier effect: electrical current induces heat current
Kelvin’s relation (consequence of Onsager):
The Seebeck coefficient measures the entropy per charge flow:
(eliminating µ)
Conductivity matrix:
Dimensionless figure of merit:
Note:
0.01 0.1 1 10 100 1000
Conductivité thermique (W/mK)
Métaux purs!
Pu!(5.2)!
Al!(237)!
Ag!(436)!
Alliages métalliques!
Solides non métalliques!
Oxydes!Glace!S!
(0.27)!
Fibres!Mousses!
Isolants!
Liquides!
Hg!(8.3)!
H20!(0.61)!Huiles!
H2!
(0.18)!
Gaz!
O2!
(0.03)!
Semi-conducteurs (faible et large gap)!
Ge!(60)!
Si!(150)!
Diamant!(2000)!
Bi2Te3!
(2.0)!
Pt!(72)!
Conductivité thermique : ordre de grandeur à T = 300 K!
Verre!
Matériaux cristallins ou amorphes!
From: B.Lenoir, GDR Thermoelectricite summer school 2012
Effet Seebeck : ordre de grandeur à 300 K!
Ag, Cu, Au!
I&I (V/K)$
Constantan (Cu – Ni)!
10-7!
10-6!
10-5!
10-4!
10-3!
10-2!
10-1!
Germanium, Silicium purs!
Bi2Te3!
Semi-conducteurs!
Semi-métaux!
Métaux!
Bismuth!
5!
5!
5!
5!
5!
5!
Nickel!
Isolants!
n! I&I$
From: B.Lenoir, GDR Thermoelectricite summer school 2012
Coupling constant characterizing energy conversion :
Entropy and heat production rates
The Kelvin effect
Heat: rate of heat change contains a REVERSIBLE term In addition to the irreversible Joule heating. Heat production or absorption depending on sign of current
: Thomson coefficient
EFFICIENCY OF ENERGY CONVERSION - General considerations
and application to Thermoelectrics -
Maximum theoretical efficiency: the Carnot reversible engine
Carnot efficiency:
Since it corresponds to a reversible, quasi-static and hence infinitely slow process,
a Carnot engine delivers ZERO POWER !
Carnot cycle :
A Carnot cycle maximizes efficiency… but delivers zero power !
A general cycle (non-Carnot): Carnot maximizes the ratio of the white to total area, subject to the constraints of the 2nd principle.
Sadi Carnot (1796-1832) - Officer and Physicist/Engineer -
[Not to be confused with President Sadi Carnot (1837- 1894)] Both are descendents of Lazare Carnot, great revolutionary, statesman,
also a mathematician and physicist, and one of the founders of Ecole Polytechnique1753-1823
Sadi Carnot, then a student at Ecole Polytechnique (painting by Louis Leoplod Boilly [Wikipedia])
Efficiency at maximum power of an `endoreversible’ engine:
the Chambadal-Novikov (Curzon-Ahlborn) efficiency
Efficiency at maximum power, according to:
Chambadal-Novikov (Curzon-Ahlborn),
Endoreversible engines, « Finite-Time Thermodynamics »
TC TH
TiH TiC
QC,KC QH,KH
W
An endoreversible engine
Chambadal-Novikov efficiency
0 0.2 0.4 0.6 0.8 1x = Force Ratio normalized to stopping condition
0
0.2
0.4
0.6
0.8
Rel
ativ
e Ef
ficie
ncy
ZT=1 , g2=1/2 ZT=2 , g2=2/3 ZT=4 , g2= 0.8ZT=10 , g2 = 0.91ZT=100 , g2 = 0.99
0 2 4 6 8 10Dimensionless Figure of Merit
0
0.2
0.4
0.6
0.8Maximum efficiencyForce ratio at max effEfficiency at max power
0 0.2 0.4 0.6 0.8 1Coupling g**2
0
0.2
0.4
0.6
0.8
1 Maximum EfficiencyForce ratio at max eff.Efficiency at Max Power
0 0.2 0.4 0.6 0.8 1P/Pmax
0
0.2
0.4
0.6
0.8R
elat
ive
Effic
ienc
y
