1. No category

When thermoelectric systems meet the Feynman ratchet:
harmonic response and feedback
Christophe Goupil, Yann Apertet, Henni Ouerdane, Philippe Lecoeur
Feynman ratchet model
Tcold
Thot
Energy taken
from hot side

   L 

N   exp 
 k BThot 
I Qhot    L  N eff
 


N   exp 
 k BTcold
 
escape
frequency

 



Energy delivered
to the cold side

torque
I Qcold   N eff

spring energy
 
N eff  N  N
P  I Qhot  I Qcold  L N eff
Strong coupling configuration
effective « current »
Velasco at al. J. Phys. D: Appl. Phys.34 (2001) 1000–1006
Linearization
   L 

N   1 
k BThot 

 
Tcold
Thot



N   1 
 k BTcold
 

N eff 
R
V  T  R I
V  Rload I
V0
Rload

k BThot
 T



L0  L 
 L  
 Tcold
 k BThot
L  L0 
L0 



T
Tcold
k BThot


N eff
: stopping force (open voltage)
L  0 : ( short circuit )
Apertet et al. PRE 2014
Power budget
Tcold
Thot

I Qhot    L  N eff
I Qcold


k BThot  2
Thot N eff 
N eff




Tcold N eff

I Qhot  Thot I  KT  RI 2
Rin
V0
 
 
 Tcold
 
 
 Tcold
Rload
I Qcold  Tcold I  KT  1   RI 2
dissipative
resistance
 
P  
 Tcold

k BThot 
T 
N


 
eff  N eff

P  T  RI I
Apertet et al. PRE 2014
Extension to non-linear model



I Qhot    L  N eff   FRThot N eff  L0  L  N eff
I Qcold
 
 
 Tcold




Tcold N eff   FRTcold N eff


P  L N eff   FR T N eff

L0  L   2

N

eff
N eff
Rdis 
lin
Rdis

L0  L

N eff
k BThot


L0

N eff
 Rload
Rdyn  
d L0  L 


d N eff
Rdyn shows the variation trend of the dissipation
d  L 

 Rload (t )
d N eff
Apertet et al. PRE 2014
Non-linear model & efficiency
Parameters :
 x ∶ Force
 I : Flux
• R dyn = −
dx
dI
• R dissip =
x0 − x
I
•
γ ∶ coefficient de répartition
de la chaleur dissipée
Linear System
In a non linear system these
parameters are functions of
the working point!
Supra-Linear
System
Maximum power :
Sub-Linear
System
Efficiency at maximum power :
After Wang et Tu, EPL, 98 (2012) 40001
Feynman & Meso
a) Feynman’s ratchet: Energy reference at the cold side
and not influenced by the load.
b) Mesoscopic thermoelectric: mean of the Fermi levels
of the reservoirs, and are modified by the load (biaising V).
Apertet et al. PRE 2014
Carnot, Yvon, Chambadal, Novikov, Curzon & Ahlborn
Thot
Thot
Finite Time
Thermodynamics
FTT
Work
Power
Endoreversible
Tcold
Tcold
P
Tcold
W
C 
 1
Qin
Thot

Pmax
Tcold
CA    1 
Thot
Q in
W
ηCA
ηCarnot
η
• 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)
From coupling to feedback
Y Apertet et al. EPL 97 (2012)
Macroscopic
N. Nakpathomkun et al. PRB 82 (2010)
Mesocopic
Thot
General model: Onsager description
 1
I  R
 I    T
 Q 
 R
Khot
ThTE
TE
TTE
K0
R
KTE(I)
TcTE
+
voc
L
O
A
D
VTE
I


 VTE 
R


 2T

T
 K 0   TE 
R

VOC ( I )  TTE ( I )
I Q ( I )  IT  K 0 TTE ( I )
  2T

IQ (I )  
 K 0  TTE ( I )
 R  Rload

Kcold
Tcold


I
K TE ( I )  K 0 1 
ZT 
 I CC

Convection
Conduction
Y. Apertet, H. Ouerdane, O. Glavatskaya, C. Goupil et Ph. Lecoeur, EPL 97 (2012)
General model: presentation
Thot
Khot
Qin
TTE
TE
Power
Qout
Kcold
Tcold
Thermal boundary conditions: mixed but two extremal configurations
• Neumann: TTE imposed  Khot and Kcold inifinite  Carnot…impossible!
• Dirichlet: Heat flux imposed  (Khot=Kcold=0)  Solar heating or gas burner
• Adiabatic (Khot=Kcold=0)  impedance spectroscopy (TTE =0).
General model: resulting picture
Thot
VOC ( I )  TTE ( I )
Khot
TEG
T
RTE
TTE K0
KTE(I)
TcTE
Kcold
Tcold
Y. Apertet, et al. EPL 97 (2012)

K c    2T 
  I 

VOC ( I )   T
Kc  K0   Kc  K0 

I
ThTE
R
+
v’oc
L
O
A
D
K hot K cold
Kc 
K hot  K cold
VTE
V 'OC
RTE
Thevenin model
• The feedback comes from Kc
• Adiabatic if Kc=0.
• No RTE if the coupling is perfect, Kc =>∞.
Power
Electric adaptation
Thermal adaptation
For given
Thermal contacts

 K contact K 0  ZT  1


 Rload R  ZT  1
See also:
M. Freunek et al., J. Elec. Mat. 38 (2009)
K. Yazawa et A. Shakouri, JAP 111 (2012)
The thermal adapatation is fundamental
for correct working conditions!
Y. Apertet, et al. EPL 97 (2012)
1rst order harmonic response
Thot
Chot
Khot
TEG
I
ThTE
TTE
RTE
Cth
K0
KTE(I)
TcTE
Kcold
Ccold
Tcold
R
+
v’oc
L
O
A
D
VTE
Small signal analysis
i(t)
R
dTTE
CS
K0
KC
iT
+
𝐾𝑇𝐸 (𝑖) =
𝛼𝑖𝑇
𝛿𝑇𝑇𝐸
L
O
A
D
CS  Cth  Chot  Ccold
dvTE
KC 
K hot K cold
K hot  K cold
dTTE
dTTE (i ) 
iT
jCS  K C  K 0
Small signal analysis
 2iT
i
dTTE 

jCTh
1
jCTh  K c  K 0

 2T
RTE
dTTE
RTE
i

CTh
1  j 2 RTE
 T
Total ouput impedance
RTE 
 2T
KC  K0
CS
CTE  2
 T
 TE  RTE CTE
Not a real electrical capacitance !!!
RTE
Z ( j )  R 
1  j TE
Maximal output power if impedance matching => Better @ non zero frequency?
Conclusion
• Feynman’s ratchet as a system model for thermoelectricity.
• The entropy per tooth generalizes the entropy per carrier concept.
• Dynamic resistance.
• Thermoelectric resistance: RTE
• Thermoelectric capacitance: CTE
• 1rst order dynamic response: TE =RTE CTE
Special thanks to
• Henni Ouerdane
• Yann Apertet
• Philippe Lecoeur
• Knud Zabrocki
• Wolfgang Seifert
• Cronin Vining
Dilbert 10-10-1993