Thermal Radiaton: Planck’s Law
T
I
M
Perfectly
Basic Relations
,
Reflecting Wall
n
Frequency ν
e
o
at T
t
h
Angular Frequency ω=2πν
l
Inside the Cavity
C
a
Wavelength λ
EM Wave In
g rm
Wavevector
magnitude k=2π/λ
n
Equilibrium at
e
a
Wavevector
k=(k ,k
,k
Temperature T
k
h
G /T ) ion
©c = νλlar ωrs= ck = c k + k + k
t
e
h
o
ω(k): Dispersion relation (linear)
v
g
i
S
n
r ct
o
y
energy in the cavity?
p ire How much
C
o
C D Ur=g2y hωf (ω,T ) =
∑∑∑
e
λ 9λ7
λ 97
n
L = 9,2 ,..., n
,...
9
E
2.2 2π 2r 2. 2 cal 2∫ (2πdk/ 2L ) ∫ (2πdk/ 2L ) ∫ (2πdk/ 2L )hωf (ω,T )
k =n o
i
r
2
L
F ct
dk
dk
dk
e
=
2
hωf (ω , T )
l
∫
∫
∫
(2π / L ) (2π / L ) (2π / L )
Two
E polarization
x
y
z
2
x
∞
x
x
∞
∞
∞
∞
y
x
x
x
x
0
x
2
z
n x =1 n y =1 n z =1
x
x
x
∞
2
y
∞
y
0
∞
x −∞
z
0
∞
y
x
−∞
z
z
y
−∞
z
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Thermal Radiaton: Planck’s Law
T
I
• Energy density per
ω interval
M
∫ ∫ ∫ hωf (ω , T )dk dk dk
,
n
u (ω ) = hωf (ω ,e
T )D(ω ) o
2V
t
h
(
)
=
h
ω
f
ω
T
π
k
dk
,
4
l
hω C 1
8π ∫
a
=
Planck’s law
c
πg
⎛ hωm
⎞
n
exp⎜⎜ r ⎟⎟ − 1
2V
⎛ω ⎞ ⎛ω ⎞
e
a
=
hωf (ω , T )4π ⎜ ⎟ d ⎜ ⎟
⎝k T ⎠ n
∫
h
G
8π
⎝c⎠ ⎝c⎠
T
o
/
•
Intensity:
energy
flux per unit
i
r
©
s
ω
U
a angle
r
t olsolid
= ∫ hωf (ω , T )
dω
e
h
V
π c
g t S dAnv cu(ω ) hω
i
r
1
o
y
(
)
=
=
I
ω
c
= ∫ hωf (ω , T )D(ω )dω p
C
e
4π
4π c
⎛ hω ⎞
o
r
⎟⎟ −1
i
exp⎜⎜
y
C D rg
k
T
⎝
⎠
7 ne Solid Angle Per unit wavelength interval
= ∫ u (ω )d9
ω7
9
9
9
E
.
.
l
dA
I (ω )dω 4πch
1
2 r 2 ca
dΩ =
I (λ ) =
=
R
dλ
λ
i
⎛ 2πhc ⎞
o
D(ω)-density
of states
per
r
⎜⎜
⎟⎟ −1
exp
t
F
whole
space
c
unit volume per unit
⎝ k Tλ ⎠
e
Planck’s
law
4π
angular frequency
El interval
2V
U= 3
8π
∞ ∞ ∞
x
y
z
− ∞− ∞− ∞
∞
2
3
3
0
2 3
∞
2
B
3
0
∞
2
2 3
0
∞
p
3
3 2
0
∞
B
0
p
2
5
B
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Thermal Radiaton: Planck’s Law
T
I
M
Q&
Wien’s displacement law
,
n o
e
t
λ T = 2898 Kμmh
l
C
a
g rm
n
e
a
h
n
G
T
o
/
Emissive Power
i
r
©
s
t ola er
h
g t S nv
i
Q& (λ ) = AπI (λ )
r
o
y
c
p
C
1o
hω
e
r
=A
i
y
4π c
⎛C
hω ⎞
g
r
⎟⎟ − 1 D
exp⎜⎜
7 ⎝ k T 9⎠ 7 ne
9
9
9
E
.
.
l
2 r 2 ca
Total
i
o
r
tT
Q& = ∫F
Q& (λ )dλ = c
Aσ
e
l
E
2
EMISSIVE POWER (W/cm μm)
max
3
2 2
B
10
4
10
3
10
2
10
1
5600 K
2800 K
1500 K
10
0
∞
800 K
4
0
10
-1
0
2
4
6
WAVELENGTH (μm)
8
10
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
T
I
M
,
Introduction to Thermoelectricity
n o
e
t
h
l
C
a
g rm
Gang Chen
n
e
a
h
G /T ion
Mechanical Engineering
© lar Department
s
r
t
Massachusetts
Institute
of Technology
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
i
y
CURL: http://web.mit.edu/nanoengineering
g
D
r
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Seebeck Effect
T
I
Hot, M
n o
e
t
h
l
C
a
g rm
n
a
Conductor
1 he
G /T ion Conductor 2
© lar rs
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
Cold
C D i rg y
7 97 ne
9
9
9
E
Voltage
.
.
l
2 r 2 ca
i
Seebeck effect: Discovered in 1821
o
r
Thomas
Johann
Seebeck
t
F
Temperature difference generates voltage
1770-1831 c
e
El
http://www.sil.si.edu/silpublications/dibner-library-lectures/scientific-discoveries/text-lecture.htm
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Peltier Effect
T
I
M
,
Heatingnor Cooling
e to
h
l
C
a
g rm
n
e
a
h
G /T ion
r
©Conductor
s
Conductor 2
1
a
r
t
l
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
C D i rg y
A
7 97 ne
9
9
9
E
.
.
Jean
Charles
Athanase
Peltier
l
2
2 ca
1785-1845 r
Peltier Effect: Discovered in 1834
i
o
An electrical current creates a cooling or
F ctr
heating effect at the junction depending
e
l
E
on the direction of current flow.
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Thomson Effect
T
I
M q(x)
,
heat release/absorption
n o
e
t
h
l
C
a
m
T
T ng
r
e
a
h
n
G /T iocurrent
© lar rs
t
e
h
o
v
g
i
S
n
r
t
Thomson
effect predicted, 1855
o
y
c
p ire
C
o
y
C
g
r
William 7
Thomson7 D
e
n
9
9
(Lord
Kelvin)
9
9
E
.
.
l
1824
–
1907
2 r 2 ca
i
o
r
F ct
e
l
E
cold
hot
http://www.sil.si.edu/silpublications/dibner-library-lectures/scientific-discoveries/text-lecture.htm
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
An Intuitive Picture of Thermoelectric Effects
T
I
M
• Current Flow in an Isothermal ,Conductor
n o
e
t
h
l
C
a
g rm
n
e
a
h
G /T ion Electrical
© lar rs
Field [V/m]
t
e
h
o
v
g
i
S
nσε = ε / ρ Electrical
r
t
Current Density Jye = enc
o
v
=
en
μ
ε
=
Resistivity [Ωm]
p
C
e
o
r gy
i
C
D
Electron
Electron
Drift
r
Mobility
Electrical
e
7
7
Density
[1/m
]
Charge [C]
Velocity
[m/s]
[m /s.V]
n
Conductivity [1/Ωm]
9
9
9
9
E
2. r 2. cal
q
i
Heat Flux
Fo ctr J q = qnv = e J e = ΠJ e
e
l
Peltier Coefficient [J/A]
E Heat Per Charge [J]
•
•
•
•
•
•
•
•
•
• •
•
•
•
•
•
••
••
••
• •
•
• •
•
•
• • •
• • •
• •
•
•
3
•
•
•
•
•
•
• •
•
•
• •
•
•
••
• •
• •
•
• •
• •
•
•
•
•
•
2
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Peltier Effect
T
I
M
,
J,J n
e to
1
h
l
C
a
Q
Q
g
m
n er
a
G2 /Th ion
© lar J , Jrs
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
Qir(Peltier)
=
(Π 1−Π2)J
o
y
C D rg
7 97 ne
9
9
9
E
.
.
•
Heating
l
2 r 2 ca and cooling at junctions
• Reversible
with current direction
i
o
r
F ct
e
l
E
e
e
q1
q2
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Seebeck Effect
T
I
M
Built-In Potential
,
n o
e
t
h
l
• a
•• • • • • • • • • C
••••
V
V
•
•
g
•
•
•
•
m
•
r
• • •• •• •
n
•
T
T
•
e
a
••• •
•
• h•
•
G /T ion
© lar rs
t
Temp. Gradient
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
• Charge
diffusion
under
a temperature gradient
i
y
C
g
D
r
•7
Built-in potential
resisting
diffusion
e
7
n
9
9
9
9
E
l = -(V -V )/(T -T )
2. r 2.S = c-ΔV/ΔT
a
hot
cold
hot cold
i
o
F ctr
e
l
E S --- Seebeck Coefficient
cold
hot
cold
hot
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Thomson Effect
T
I
M
,
heat release/absorption q(x)
n o
e
t
h
l
C
a
• T
m
• • • • • • •ng•
•
r
•
•
T
•• • • • • a
•e
•••
•
•
•
•
h
•
• • • • • • • • G• • /T
• • ion
r
© current
s
a
r
t
l
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
i
1
dq
dT
y
C
g
D
τ=
r
Thomson
Coefficient
7 97 ne
I dx dx
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
t
F cRelations:
• Kelvin
Π = ST; τ = T dS/ dT
e
l
E
cold
hot
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Properties are Temperature Dependent
IT
M
,
n o
e
t
h
l
C
a
g rm
n
e
a
h
G /T ion
© lar rs
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
C D i rg y
e
7
7
n
9
9
9
9
E
2. r 2. cal
i
o
r
F ct
e
l
E
Images removed due to copyright restrictions.
Please see Fig. 2a,b in Poudel, Bed, et al. "High-Thermoelectric
Performance of Nanostructured Bismuth Antimony Telluride Bulk Alloys."
Science 320 (May 2, 2008): 634-638.
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Thermoelectric Devices
T
I
M
,
n o
e
t
h
l
C
a
g rm
n
a
Ne
I
h
G /T ioPn
© lar rs
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
C D i rg y
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
COLD SIDE
COLD SIDE
-
+
HOT SIDE
HOT SIDE
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
T
I
M
,
n o
e
t
h
l
C
a
g rm
n
e
a
h
n
G
T
o
Performance of Thermoelectric
Devices
/
i
r
©
s
t ola er
h
g t S nv
i
r
y ec Co
p
o
C Dir rgy
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Other Basic Relations: Heat Conduction
T
I
Th
M
• Fourier Law for heat conduction
,
n o
e
t
h
l
q =g−Ck∇Tma
n er
a
h
n
G ] Thermal
Heat Flux [W/m
Conductivity
[W/m.K]
T
o
/
i
© lar rs
t
e
h
o
v
g
L
i t S on heat conduction
•yrOne-dimensional
Area A
c
p
C
e
o
r
C Di rgy Th − Tc
7 97 Qne= Ak
= KΔT
9
9
9
E
.
.
L
l
2 r 2 ca
kA
i
o
r
F ct
Thermal Conductance : K =
Tc le
L
E
2
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Device Analysis: Cooling
T
I
• Ideal Devices
M
,
Q
T
n o
e
No Joule Heating,
t
h
l
C
No Heat Conduction
a
g rm
n
Qc = (Π
a
p-Πn)•Ιe
p
G /Th ion
n
• ©
Real Devices:
r
s
a
r
t
l
e
h
o
v
g
Joule
Heating
& Heat Conduction
i
S
n
r
t
o
y
c
p
2R/2 - K (T -T )
C
e
Q
=
(Π
-Π
)•Ι
−
I
o
r
c
p
n
h c
i
y
T
QC
g
D
r
e Resistance Thermal Conductance
7 97 Electrical
n
9
9
9
E
.
.
l
2 r 2 ca
L p ρ p Ln ρ n
k A
k A
R
=
+
i
+
K=
Fo ctr
Ap
An
L
L
e
l
E
c
c
h
h
p
p
p
n
n
n
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Refrigerator Performance
T
I
V = IR + ( S − S )(TM
−T )
Voltage Drop:
,
n o
e
t
Coefficient of Performance: h
l
C
a
1 2
g
m
S
S
IT
K
T
T
−
−
−
−
( H C) I R
n) C
r
Q C ( p an
2
e
φ=
=
G(Sp −/STnh) I ( THi−oTnC ) + I2R
W
T Q
© lar rs
t
e
h
T
o
v
g
+
−
1
ZT
i
S
n
r
T
T
t
o
y
φ
=
c
Optimize Current:p
e
T − TC
(
) 1 + ZT + 1
o
r
i
y
C D rg
T = 0.5(T +
T )
e
7
7
n
9
9
9
9
E
2
2. r 2. ca(Slp − Sn )2
S
S
−
(
p
n)
Zi =
=
o
r
KR
⎛ L p ρp L ρ ⎞⎛ k p A p k A ⎞
F ct
n n ⎟⎜
n n ⎟
+
+
⎜
e
⎜ Ap
A n ⎟⎜ L p
L
n
⎟
El
Tc Qc
p
h
n
h
c
h
M
C
max
H
H
C
C
M
M
⎝
⎠⎝
h
c
⎠
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Figure of Merit Z
T
I
M
,
⎛ L p ρp L ρ ⎞⎛ k p A p k A n⎞
KR = ⎜
+ n en ⎟
+ n n ⎟⎜
o
t
⎜ Ap
⎟⎜
⎟
h
A
L
L
n ⎠⎝
p
n ⎠ l
⎝
C
a
g rm
1/ 2
n
e
2
L n A p ⎛ ρp k n ⎞
a
n
=⎜
⎟
( KR )min = ( k pρp + k n ρn ) Gwhen/Th
o
L piA n ⎜⎝ ρn k p ⎟
r
©
⎠
s
a
r
t
l
e
h
o
v
g
i
S
2
n
r
t
o
y
Sp − Sn )
2
2
(
c
p
S
σS
C
e 2 yFor a single material: Z = =
Zmax =
o
r
i
Ck pρp + Dk n ρn ) rg
ρk
k
(
7 97 ne
9
9
9
E
.
.
l
pn a
pairs are used:
2 In a device,
2
r ric
o
(1)
F Areascoft each type of legs need to be optimized
e
(2) Two
types of legs should have comparable properties
l
E
(3) Current input to the device needs to be optimized
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Typical Number
T
I
M
,
n o
e
t
• Bi2Te3-based materials ~300 C
Kh
l
a
g
m
r
n
Power
Factor:
S
σ=48 μW/cm-K
S=220 μV/K
e
a
h Z=3.2x10
n
Figure
1/K
G of/TMerit:
σ=10 Sm
o
i
r
©
ZT=1
k=1.5 W/mK
s
t ola er
h
g t S nv
i
r
• Device Leg: p
1y
mm x 1cmm xC2omm
e y
o
r
i
C
g
D
−3
r
L
2 × 10
e
7
7
R
0.02 Ω
=
=
=
n
9
9
1
−
5
6
σA
9
9
10 ×l10E
.
.
2 r 2 ca
i
o
r
Legs
are
t electrically in series
F but thermally
c
in parallel
e
l
E
2
5
-1
-3
Image by michbich at Wikipedia.
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Two Tc
thermocouples
soldered into
drilled holes
T
I
M
,
n o
e
t
h
l
C
a
g rm
n
e
a
h
G /T ion
© lar rs
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
C D i rg y
• Match two leg size
7 97 ne
• Minimize contact
9
9
9
E
resistance
2. r 2. cal
i
o
• Optimize current
r
T = 100 °C
F ct
e
l
E
An Example
Temperature Difference (oC)
120
100
80
60
40
20
One Th
thermocouple
soldered into
drilled holes
Copper blocks (Th)
Experimental data
Theoretical prediction
Ceramics
plate for
electrical
insulation
h
0
0
2
4
6
8
10
–WARREN
Current
(A) M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Temperature Dependence of Properties
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Images removed due to copyright restrictions.
Please see Fig. 2a,b,d,e in Poudel, Bed, et al.
"High-Thermoelectric Performance of Nanostructured
Bismuth Antimony Telluride Bulk Alloys." Science 320 (May 2, 2008): 634-638.
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Differential Analysis
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–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Thermoelectric Power Generation
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–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Thermoelectric Refrigeration
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–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
T
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–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Commercial Thermoelectric Devices
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Images removed due to copyright restrictions.
Please see http://www.hi-z.com/index.php
http://www.marlow.com/thermoelectric-modules/
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Current Applications in Refrigeration
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C
a
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Images removed due to copyright restrictions.
Please see: http://www.roadtrucker.com/12-volt-coolers-accessories/
12-volt-coolers-products/igloo-40-quart-kool-mate-40-12-volt-thermo-electric-cooler-6402.jpg
http://image.made-in-china.com/2f0j00kvZEKWVPgtlu/Refrigerator-BC-65A-.jpg
http://www.newdavincis.com/images/wc-1682%2016%20bottles.jpg
http://www.rmtltd.ru/datasheets/TO812.4MD04116xx.pdf
http://www.medsystechnology.com/images/gem4000_w32a.jpg
http://amerigon.com/ccs_works.php
.
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Current Applications in Power Generation
T
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n o
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t
h
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Images removed due to copyright restrictions. Please see:
http://thermoelectrics.caltech.edu/images/mhw-rtg.gif
http://globalte.com/pdf/teg_5120_spec.pdf
http://www.roachman.com/thermic/thermic1.jpg
http://www.research.philips.com/newscenter/pictures/downloads/misc-sustainability_05-0_h.jpg
–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
System Consideration
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–WARREN M. ROHS
ENOW HEAT AND MASS TRANSFER LABORATORY, MIT
ROHSENOW
MIT
Heat to Electricity Recovery
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–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
Vehicle Systems
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M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
heat losses Coolant –WARREN Exhaust
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Images removed due to copyright restrictions. Please see
http://cdn-www.greencar.com/images/waste-exhaust-heat-generates-electricity-cars-efficient.php/bmw-teg-1.jpg
http://cache.gawkerassets.com/assets/images/12/2009/03/BMW_TEG.jpg
–WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
MIT OpenCourseWare
http://ocw.mit.edu
2.997 Direct Solar/Thermal to Electrical Energy Conversion Technologies
Fall 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.