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Thermal Radiaton: Planck’s Law
Inside the Cavity EM Wave In Equilibrium at Temperature T
Perfectly Reflecting Wall at T Frequency ν
Angular Frequency ω=2πν Wavelength λ Wavevector magnitude k=2π/λ
νλ=c
Wavevector k=(kx,ky,kz) 222 zyx kkkcck ++==ω
ω(k): Dispersion relation (linear)
k
x xx
x x
xx x
Lnk
nL
2 2
,...2
,...,2
2,2
π
λλλ
=
=
Basic Relations
How much energy in the cavity?
( )
( )
( )TfL
dk
L
dk
L
dk
TfL
dk
L
dk
L
dk
TfU
z
z
y
y
x
x
z
z
y
y
x
x
n n nx y z
,)/2()/2()/2(
2
,)2/2()2/2()2/2(
2
,2
000
1 1 1
ωωπππ
ωωπππ
ωω
h
h
h
∫∫∫
∫∫∫
∑∑∑
∞
∞−
∞
∞−
∞
∞−
∞∞∞
∞
=
∞
=
∞
=
=
==
Two polarization
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MITWARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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For 2.997 Direct Solar/T
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Thermal Radiaton: Planck’s Law ( )
( )
( )
( )
( ) ( )
( ) ωω
ωωωω
ωπ ωωω
ωωπωωπ
πωωπ
ωωπ
du
dDTf
d c
TfV
U
c d
c TfV
dkkTfV
dkdkdkTfVU zyx
∫
∫
∫
∫
∫
∫ ∫ ∫
∞
∞
∞
∞
∞
∞
∞−
∞
∞−
∞
∞−
=
=
=
⎟ ⎠
⎞⎜ ⎝
⎛⎟ ⎠
⎞⎜ ⎝
⎛ =
=
=
0
0
32
2
0
2
0 3
2
0 3
3
,
,
4,8 2
4,8 2
,8 2
h
h
h
h
h
D(ω)-density of states per unit volume per unit angular frequency interval
• Energy density per ω interval
( ) ( ) ( )
1exp
1 ,
32
3
−⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ =
=
Tk
c
DTfu
B
ωπ ω
ωωωω
h
h
h
Planck’s law
Solid Angle
dAp
2RdA
d p=Ω
whole space 4π
• Intensity: energy flux per unit solid angle
( ) ( )
exp
1 44 23
3
⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ ==
Tk
c
cuI
B
ωπ ω
π ωω
h
h
Per unit wavelength interval
( ) ( ) 2 exp
14 5
⎜⎜ ⎝
⎛ ==
πλ π
λ ωωλ
Tk
c
d
dII
B
h
h
Planck’s law
⎟ −1
c ⎞
λ ⎟⎠⎟ −1
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MITWARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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Thermal Radiaton: Planck’s Law
( ) ( )
1exp
1 4 22
3
−⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ =
=
Tk
c A
IAQ
B
ωπ ω
λπλ
h
h
&
Q&
Total
( ) 4
0
TAdQQ σλλ == ∫ ∞
&&
10-1
100
101
102
103
104
0 2 4 6 8 10
EMIS
SIVE
PO
WER
(W/c
m2 μm
)
WAVELENGTH (μm)
5600 K
2800 K
1500 K
800 K
Emissive Power
Wien’s displacement law
mK2898max μλ =T
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Introduction to Thermoelectricity
Gang Chen
Mechanical Engineering Department Massachusetts Institute of Technology
URL: http://web.mit.edu/nanoengineering
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97 Copyrig
.997 Dir
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2.9
ht © Gang Chen, M
IT
For 2
ect Solar/T
hermal to
Electrical Energy C
onversion
Seebeck Effect
Seebeck effect: Discovered in 1821 Temperature difference generates voltage
http://www.sil.si.edu/silpublications/dibner-library-lectures/scientific-discoveries/text-lecture.htm
Thomas Johann Seebeck 1770-1831
Conductor 1 Conductor 2
Hot
Cold
Voltage
Copy
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Peltier Effect
Jean Charles Athanase Peltier 1785-1845 Peltier Effect: Discovered in 1834
An electrical current creates a cooling or heating effect at the junction depending on the direction of current flow.
Conductor 1 Conductor 2
Heating or Cooling
A
yr
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IT
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Thomson Effect
http://www.sil.si.edu/silpublications/dibner-library-lectures/scientific-discoveries/text-lecture.htm
Thomson effect predicted, 1855
William Thomson (Lord Kelvin) 1824 – 1907
ThotTcold current
heat release/absorption q(x)
anghe
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IT
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Current Density
Heat Flux
An Intuitive Picture of Thermoelectric Effects
• Current Flow in an Isothermal Conductor
• • •
• • •
•
•
•• •
•
•
• • • •
•
• • •
•
• • • • • •
• •
• •
•
• •
• •
• • •
•
•
•
• •
•
•
• • • •
•
•
•
•
•
•
•
• ••
• •
•
•
•
•
• •
ρσμ /εεεvJ ==== enene
Electron Charge [C]
Electron Density [1/m3]
Drift Velocity [m/s]
Mobility [m2/s.V]
Electrical Conductivity [1/Ωm]
Electrical Field [V/m]
eeq e
qqn JJvJ Π===
Heat Per Charge [J] Peltier Coefficient [J/A]
Electrical Resistivity [Ωm]
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MITWARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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Peltier Effect
Q (Peltier) = (Π 1−Π2)J
1
2
Je, Jq1
Je, Jq2
• Heating and cooling at junctions • Reversible with current direction
ang er•••••••
•
•• •
•• •
•• ••• •
•
•• •
•• •
•• • •
•• •• •
•
••
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MITWARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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Chen, MIT
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mal to
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n
S = -ΔV/ΔT = -(Vhot-Vcold)/(Thot-Tcold)
S --- Seebeck Coefficient
Built-In Potential
Temp. Gradient
Vcold Tcold
Vhot Thot
Seebeck Effect
•• • ••• •
•
• • •
• • •
• • • • • •
•
• • •
• • •
• • • •
• • •• •
•
• •
• Charge diffusion under a temperature gradient • Built-in potential resisting diffusion
Ga he•••••••
•
•• •
•• •
•• ••• •
•
•• •
•• •
•• • •
•• •• •
•
•• ••
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MITWARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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IT
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rmal to
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onversion
Thomson Effect
ThotTcold
current
heat release/absorption q(x)
1 dq dT I dx dx
τ =Thomson Coefficient
• Kelvin Relations: Π = ST; T dS/ dTτ =• Kelvin Relations: Π = ST; T dS/ dTτ =
• • ••• •
•
• • •
• • •
• • • • •
•
• • •
• • •
• • • •
• • •• •
•
• •
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Properties are Temperature Dependent
Images removed due to copyright restrictions. Please see Fig. 2a,b in Poudel, Bed, et al. "High-ThermoelectricPerformance of Nanostructured Bismuth Antimony Telluride Bulk Alloys." Science 320 (May 2, 2008): 634-638.
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY,WARREN M. ROHS MITENOW HEAT AND MASS TRANSFER LABORATORY, MIT
997
For 2.997
trica
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2. Copyrig
ht © Gang Chen, M
IT
Direct Solar/T
hermal to
Elec l Energy C
onversion
Thermoelectric Devices
COLD SIDE
HOT SIDE
-
I
+
N P
HOT SIDE
COLD SIDE
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MITWARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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Gang Chen, MIT
For 2.997 Direct Solar/T
hermal to
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onversionPerformance of Thermoelectric Devices
97r
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ht © Gang Chen, MIT
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Electrical Energy Conversio
n
Other Basic Relations: Heat Conduction
• Fourier Law for heat conductionTh
Tc
Tk∇−=q Thermal Conductivity [W/m.K]Heat Flux [W/m2]
Area AL • One-dimensional heat conduction
TKL
TTAkQ ch Δ= −
=
L
kAK =:eConductancThermal
p
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hermal to
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onversion
Device Analysis: Cooling • Ideal Devices
No Joule Heating, No Heat Conduction
Qc = (Πp -Πn)•Ι
QcTc
QhTh
p n • Real Devices:
Joule Heating & Heat Conduction
Qc= (Πp -Πn)•Ι − I2R/2 - K (Th-Tc
Electrical Resistance Thermal Conductance
p
pp
L
Ak K +=
n
nn
p
pp
A
L
A
L R
ρρ +=
)
kn An
Ln
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hermal to
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Refrigerator Performance QcTc
QhTh
Coefficient of Performance:
( ) ( )
( ) ( )
2 p n C H CC
2 p n H C
1S S IT K T T I RQ 2 W S S I T T I R
− − − − φ = =
− − +
( )
HM
C C max
H C M
T1 ZTT T
T T 1 ZT 1
+ −
φ = − + +
( ) ( )2 2 p n p n
p p p pn n n n
p n p
S S S S Z
KR L k AL k A A A L
− − = =
⎛ ⎞⎛ρ ρ+ +⎜ ⎟⎜⎜ ⎟⎜
⎝ ⎠⎝
Optimize Current:
.0MT =
Voltage Drop: ))(( chnp TTSSIRV −−+=
5(Th Tc+ )
⎞⎟⎟Ln ⎠
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY,WARREN M. ROHS MITENOW HEAT AND MASS TRANSFER LABORATORY, MIT
C
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For 2.997 Direct Solar/T
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Figure of Merit Z
p p p pn n n n
p n p n
L k AL k AKR A A L L
⎛ ⎞⎛ ⎞ρ ρ = + +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟
⎝ ⎠⎝ ⎠
( ) ( )2 p p n nmin KR k k= ρ + ρ when
1/ 2 n p p n
p n n p
L A k L A k
⎛ ⎞ρ = ⎜⎜ ρ⎝
( ) ( )
2 p n
max 2 p p n n
S S Z
k k
− =
ρ + ρ For a single material: Z =
In a device, pn pairs are used:
(1) Areas of each type of legs need to be optimized (2) Two types of legs should have comparable properties (3) Current input to the device needs to be optimized
⎟⎟⎠
S2 σS2=
ρk k
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY,WARREN M. ROHS MITENOW HEAT AND MASS TRANSFER LABORATORY, MIT
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MITWARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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Typical Number
• Bi2Te3-based materials ~300 K
S=220 μV/K σ=105 Sm-1
k=1.5 W/mK
Power Factor: S2σ=48 μW/cm-K Figure of Merit: Z=3.2x10-3 1/K
ZT=1
• Device Leg: 1 mm x 1 mm x 2 mm
3
1 5 6 L 2 10R 0.02 A 10 10
−
−
× = = = Ω
σ ×
Legs are electrically in series but thermally in parallel
Image by michbich at Wikipedia.
––WARREN M. ROHSENOW HEAT AND WARREN M. ROHSENOW HEAT AND
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Electrical Energy Conve
Ceramics plate for electrical insulation
Two Tcthermocouples soldered intodrilled holes
Copper blocks (T )
One Ththermocouple soldered into drilled holes
MASS TRANSFER LABORATORY, MITMASS TRANSFER LABORATORY, MIT
er
rsion
Current (A) 0 2 4 6 8 10
Tem
pera
ture
Diff
eren
ce (o C
)
0
20
40
60
80
100
120 Experimental data Theoretical prediction
An Example
Th = 100 °C
hh
Ceramics plate for electrical insulation
Two Tc thermocouples soldered into drilled holes
Copper blocks (T )
One Th thermocouple soldered into drilled holes
• Match two leg size • Minimize contact
resistance • Optimize current
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MITWARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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Temperature Dependence of Properties
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, MITWARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
2.997 Copyright ©
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For 2.997 Direct Solar/T
hermal to
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Differential Analysis
x
2 2
d dT dS dTk JT J 0 dx dT dxdx
⎛ ⎞ − + ρ =⎜ ⎟⎝ ⎠
Thomson Effect, Usually Neglected Joule Heating
Boundary Conditions (Cooling):
O
L x=0: c dT dT q k J k ST dx dx
= − + Π = − +Tc given or Jc
.997 Copyright © Gang Chen, MI
or 2.997 Direct Solar/Thermal to
2
T
FElectri
cal Energy Conversion
Thermoelectric Power Generation
ALKALI METAL THERMAL TO
AL
RCIEL LLE
TH
TEC
0.1
0.2
0.3
0.4
CARNOT CYCLE 10
7
ZT m
1
TNAPLERWPOMER
SEINGENEVIOTOMAUT
SC
ALKALI METALTHERMALTOALKALI METALTHERMALTO
IED LES ALP NT
4
ERENONII C
RO
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
CARNOT CYCLE 10
7
2
T
50.
ZTm
SRTOARENEGERWPOCITRELECOMERTH
AERENGGNILRIST
SROTAGMERTH
ave
Constant Properties
0.60.0.0.666Efficiency
POW
ER G
ENER
ATI
ON
EFF
ICIE
NC
Y
POW
ERG
ENER
ATI
ON
EFFI
CIE
NC
Y
0.50.0.0.555⎛ T ⎞ 1 + ZT −1⎜⎜ c ave
Th ⎟⎟
Tc
η = 1 −⎝ ⎠ 1 + ZTave + 4THTHERMALERMAL
POPOWERWERPLPLANTANT
Th
Tc + ThT 2= DIESEL PLANT
2 STIRLINGELELECTRIC CELLSECTRIC CELLSGENERATORTPV 1
THERMIONIC AUTAUTOMOTIVEOMOTIVEGENERATORS 0.5 ENENGINESGINES
THERMOELECTRICPOWER GENERATORS
00001 2 3 4 5 6 7 8 9 111 2 3 4 5 6 7 8 9 10111000TEMPERATURE RATIO (T /T )TEMPERATURE RATIO (T /T )hot coldhot cold
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY,WARREN M. ROHS MITENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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Electrical Energy Conversio
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Thermoelectric Refrigeration
0
1
2
3
4
5
6
1 1.2 1.4 1.6 1.8
CO
EFFI
CIE
NT
OF
PER
FOR
MA
NC
E
TEMPERATURE RATIO (Thot
/Tcold
) 2.0
0.51 2 4 710 ZT
m
CARNOT CYCLE STIRLING REFRIGERATORS HOUSEHOLD REFRIGERATORS & AIR-CONDITIONERS
THERMOELECTRIC REFRIGERATORS
STIRLING CRYOCOOLERS
Coefficient of Performance
11 /1
++
−+
− =
ave
chave
ch
c
ZT
TTZT
TT
TCOP
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MITWARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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Gang Chen, MIT
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hermal to
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Current and Potential Applications
2.997 Copyright ©
Gang Chen, MIT
For 2.997 Direct Solar/T
hermal to
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onversion
Commercial Thermoelectric Devices
Power Generators from Hi-Z
Coolers from Marlow Industries
Images removed due to copyright restrictions. Please see http://www.hi-z.com/index.php http://www.marlow.com/thermoelectric-modules/
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY,WARREN M. ROHS MITENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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hermal to
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onversion
Current Applications in Refrigeration
.
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
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hermal to
Current Applications in Power Generation
http://www.research.philips.com/newscenter/pictures/downloads/misc-sustainability_05-0_h.jpg
Electrical Ener
Conversion
Images removed due to copyright restrictions. Please see: http://thermoelectrics.caltech.edu/images/mhw-rtg.gif
http://www.roachman.com/thermic/thermic1.jpg
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY,WARREN M. ROHS MITENOW HEAT AND MASS TRANSFER LABORATORY, MIT
http://globalte.com/pdf/teg_5120_spec.pdf
2.997 C
MIT
For 2.
Object being cooled
N NP PTE Element
Ceramicsubstrate
Carriers moving heat
Electricalinterconnect
+_
DC Power source
Heat sink
TobjectTcold
Thot
TambientT
Tte
Ele Figure by MIT OpenCourseWare.c
System Consideration
Sometimes, thermal systems more expansive ––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY,WARREN M. ROHS MITENOW HEAT AND MASS TRANSFER LABORATORY, MIT
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MITWARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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Heat to Electricity Recovery from Gas Stream
Hot Gas m, Th,i
Hot Gas Th,o
Heat Transfer Surfaces
Coolant
Insulation Hot Side TemperatureTH
Cold Side TemperatureTc
• For thermoelectric devices, TH higher is better • However, maximum heat intercepted from hot
gas stream, mcp(Th,i-TH), decreases with TH
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MITWARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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For 2.997 Direct Solar/Thermal to
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Vehicle Systems
Gasoline 100 kJ
10kJ 30kJ 35kJ
Parasitic heat losses Coolant Exhaust
9kJ 10kJ
6kJ Auxiliary
Driving
Mechanical losses
Main AC: TE Local cooling of seats: TE
Catalytic converter: TE
Radiator: TE
In US, transportation uses ~26% of total energy.
––WARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MITWARREN M. ROHSENOW HEAT AND MASS TRANSFER LABORATORY, MIT
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Prototypes
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2.997 Direct Solar/Thermal to Electrical Energy Conversion Technologies Fall 2009
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