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Multifunctional Materials Lab
The Effective Thermoelectric Properties of Composite Materials
Yang Yang and Jiangyu Li Department of Mechanical Engineering
University of Washington
3rd Thermoelectrics Applications Workshop 2012
Multifunctional Materials Lab
Thermoelectric Figure of Merit
Snyder and Toberer (2008)
phe
TZκκσα+
=2
HCCTE TTZT
ZT/111
++−+
=ηη
Multifunctional Materials Lab
Nanostructured Thermoelectrics
Li et al (2010)
Multifunctional Materials Lab
Intrisic Heterogeneity
Pichanusakorn and Prabhakar Bandaru (2010)
Multifunctional Materials Lab
Thermoelectric Composites
Can the effective thermoelectric figure of merit of a composite material be higher than all its constituents, excluding the effects of size and interface? and how can we take advantages of the size and interfacial effects of composite materials to further enhance their thermoelectric figure of merit?
Multifunctional Materials Lab
Effective Figure of Merit
Bergman et al (1991, 1999): The effective figure of merit of a composite can never exceed the largest value of figure of merit in any of its component, in the absence of size and interface effects.
Multifunctional Materials Lab
Current State of Art
Graeme Milton (2002): There is a lot of confusion, particularly in the composite materials community, as to appropriate form of the thermoelectric equations.
0
10
20
30
40
50
60
70
80
90
100
TE Composite
Homogenization
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Governing Equations
σ φ σα− = ∇ + ∇TJ2( )σα φ σα κ α κ= − ∇ − + ∇ = − ∇Q T T T T TJ J
φ= +U QJ J J
0∇⋅ =UJ0∇⋅ =J
φ∇ ⋅ = −∇ ⋅QJ J
Transport Equations
Field Equations
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Homogeneous Thermoelectric
2 2
2 σκ= −
d T Jdx
2 2
2
φ ασκ
=d Jdx
22
1 22JT x c x cσκ
= − + +
2 22
1 0 3[ ( ) ]2 2α αφ ασκ σκ σ
= − + + − +J J Jx T T x c
0 1 0 1( ) ( )ασ σ φ φ= − + −J T T
All the coefficients are assumed to be temperature independent
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One-dimensional Analysis
Bi2Te3
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Heterogeneous Thermoelectric
22
22
,02
, 12
σ κ
σ κ
− + + ≤ <= − + + < ≤
A AA A
B BB B
J x a x b x fT
J x a x b f x
22
22
( ) ,02
( ) , 12
α ασ κ σ
φα ασ κ σ
− + + ≤ <
= − + + < ≤
AA A A
A A A
BB B B
B B B
J Jx a x c x f
J Jx a x c f x
All the coefficients are assumed to be temperature independent
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One-dimensional Analysis
Bi2Te3-LAST
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Equivalency Principle
Under identical boundary conditions of temperature and electric potential, the composite and homogenized medium would have: (1) identical current density and (2) identical energy flux.
Yang et al (2012)
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Enhanced Effective Figure of Merit
Bi2Te3-LAST
Due to nonlinearity, the effective properties depend on boundary conditions
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Q 0 LT T>
Q
0T
LT LTlR
Q
p n
Conversion Efficiency Analysis
TE material
Q
Super conductor
0 LT T>
Q
0T
LT LTlR
Q κ α φ= − ∇ + +UJ T TJ J
0 0==∑U xE SJ
2
0
lL
I RHE
=
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Enhanced Conversion Efficiency
Z of constituent phases is fixed, but individual coefficients are allowed to vary
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Enhanced Conversion Efficiency
Snyder (2004) Bian and Shakouri (2006)
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Thermoelectric Composite
Multifunctional Materials Lab
One-dimensional Asymptotic Analysis
η
L
ξη
=x
ξ1
x
K
BK
AKx: slow or macroscopic variable
ξ: fast or microscopic variable
20 1 2( , ) ( , ) ( , ) ( , ) ...ξ ξ η ξ η ξ= + + +T x T x T x T x
20 1 2( , ) ( , ) ( , ) ( , ) ...φ ξ φ ξ ηφ ξ η φ ξ= + + +x x x x
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0( ) 0κξ ξ
∂∂=
∂ ∂T2η−
1η− 0 0 0φσ σαξ ξ
∂ ∂+ =
∂ ∂T
0 0 010( ) ( ) φκ α κ κ
ξ ξ ξ ξ∂ ∂ ∂∂∂ ∂
+ − + + =∂ ∂ ∂ ∂ ∂ ∂
T T TJT Jx x
0 01 1φ φσ σ σα σαξ ξ
∂ ∂∂ ∂− = + + +
∂ ∂ ∂ ∂T TJ
x x
0 1 1 20 1
0 1
( ) ( )
( )
α κ κ α κ κξ ξ ξ
φ φξ
∂ ∂ ∂ ∂∂ ∂− + + + − + +
∂ ∂ ∂ ∂ ∂ ∂∂ ∂
= +∂ ∂
T T T TJT JTx x x
Jx
0η
Asymptotic Analysis
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Homogenized Equations
20 1 2( , ) ( , ) ( , ) ( , ) ...ξ ξ η ξ η ξ= + + +T x T x T x T x 2
0 1 2( , ) ( , ) ( , ) ( , ) ...φ ξ φ ξ ηφ ξ η φ ξ= + + +x x x x
2 22 2 20
02
1 1 1( ) 0α ακ κ κ σ κ
− < > − < >< > + < >< > =d T J T Jdx
22
2
1 0σκ
+ =d T Jdx
2 22 10 0
2
23 1 2 2
0
1( )
1 1( ) 0
φ α ακ κ κ
α α α ακ κ κ κ σ κ
−
−
+ < > − < > < >
− < >< > − < > < > − < >< > =
d dTJdx dx
J T J
22
2 0φ ασκ
− =d Jdx
Homogenized
Homogeneous
Homogenized and homogeneous materials satisfy different type of governing equations!
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Homogenized Solution
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Concluding Remarks
Rigorous asymptotic analysis has been developed to predict the effective thermoelectric behaviors of composite materials, and preliminary results suggest that conversion efficiency higher than that of constituents is possible in composites
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