97 Copyright © Gang Chen,

or 2.997 Direct Solar/Thermal to

Electrical Energy Conversio

n

Importance of Heat

2.9

MIT

F

Courtesy of Lawrence Livermore National Laboratory. Used with permission.

yright © Ga

Direct Solar/Ther

ergy Conversion

Gasoline100 kJ

10kJ 30kJ 35kJ

Parasiticheat losses Coolant Exhaust

9kJ10kJ

6kJ

Exhaust

.99

hen, MI

or 2.

o

27 Cop

ng C

T

F997

mal t

Electrical En

Vehicle Systems

• In US, transportation uses ~26% of total energy.

Coolant

Gasoline 100kJ

10kJ

30kJ 35kJ

9kJ

10kJ

6kJ Auxiliary

Driving

Mechanical losses

Parasitic heat losses Exhaust

Photo from Wikimedia Commons, http://commons.wikimedia.org

2.997 Copyright © Gang Chen, MIT

For 2.997 Direct Solar/Thermal to

Electrical Energy Conversio

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Heating

TE Recovery

PVElectricity

Oil or Nat’l Gas

Entropy Thermal Power

Electrical Power

Heating

TE Recovery

PVElectricity

Oil orNat’l Gas

Oil orNat’l Gas

EntropyThermal Power

Electrical Power

Co-Generation in Residential Buildings

In US, residential and commercial buildings consume ~35% energy supply

Photo by bunchofpants on Flickr.

Image removed due to copyright restrictions.Please see any photo of the Honda freewattMicro-CHP system, such as http://www.hondanews.com/thumbnails/2007/4/3/13644_preview.jpg

Refrigeration &Refrigeration &AppliancesAppliances

2.997 Copyright ©

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Ele ctrical Energy C

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Industrial Waste Heat

Fig. ES.1 in Hemrick, James G., et al. "Refractories for Industrial Processing:Opportunities for Improved Energy Efficiency." DOE-EERE Industrial TechnologiesProgram, January 2005.

Photos by arbyreed and toennesen on Flickr.

.

, MI

2 997 Copyright © Gang Chen

T

For 2.997 Direct Solar/Thermal to

Electrical Energy Conversio

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Renewable Heat Sources

Photos by Jon Sullivan at http://pdphoto.org/ and NASA.

2.997

ght © Gang Chen, M

IT

For 2.997 DSolar/T

hermal to

Electrical Energy C

onversion

Solar Thermal

http://www.treehugger.com/Solar-Thermal-Plant-photo.jpg

http://media.photobucket.com/

Images by Sandia National Laboratories and NREL.

Photos of solar hot water tubes removed due tocopyright restrictions. Please see, for example,http://image.made-in-china.com/2f0j00KeoavBGJycbN/

rpy iUnpressurized-Solar-Water-Heater-VERIOUS-.jpg

Co irect http://ns2.ugurpc.com/productsimages/solarevacuatedtube_202160.jpg

2.997 Copyright ©

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onversion

Direct Energy Conversion COLD SIDE

HOT SIDE

Thermoelectrics

Thermophotovoltaicshttp://www.keelynet.com/tpvcell.jpg

Photovoltaics http://www.solareis.anl.gov/images/photos/Nrel_flatPV15539.jpg

Image removed due to copyright restrictions. Please see http://web.archive.org/web/20071011185223

/www.eneco.com/images/science-new.jpg

Image by Nadine Y. Barclay, USAF. Courtesy of John Kassakian. Used with permission.

2.997 Copyright © Gang Chen, MIT

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Solar Spectrum

0 0.5 0

200

400

600

1800 Te

rrest

rial S

olar

Spe

ctru

m (W

/m2 μ

m)

AM1.5 Solar Spectrum Energy Usable for Silicon PV Cells

Bandgap of Silicon (1.1 μm)

1600

1400

1200

1000

800

1 1.5 2 2.5 3 Wavelength (μm)

D rior gi

Cyp

n ms oiar

C

T/reh

© Ggna

S nolt oCcgye

er

ah

h0

2.997

t

en, MIT

For 2.997

re

l to

Electrical En

v0 2 4 6 8 10 12

Irradiance From Emitter

0 2 4 6 8 10 120

00

0.5

1.0

1.5

Selective Absorber

Emitter

TPV Cell

Thermal Management

0 0.5 1.0 1.5 2.0 2.5 3.0 0

Optical Concentrator

Emis

sivi

ty A

bsor

ptan

ce

Wavelength (μm)

Wavelength (μm)

Pow

er (W

/m2 μ

m)

Pow

er (W

/m2 μ

m)

(d)

(b)

Solar Thermophotovoltaics

Theoretical maximum efficiency: 85.4%; comparable to that of infinite number of multi-junction cells, but with only a single junction PV cell. Key Challenges: Selective surfaces absorbing solar radiation but re-emitting only in a narrow spectrum near the bandgap of photovoltaic cells, working at high temperatures.

1500 Solar Insolation 1000

500

Wavelength (μm)(a)

1.5E4 Absorber

1.0E4 5 10 15

1.5 0.5E4 (c) Selective Emitter

1.0

0.5

•

•

©

2.997 Copyright

Gang Chen, MIT

For 2.997 Direct Solar/T

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Electrical Energy C

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Solar Thermoelectrics

0.0

5.0

10.0

15.0

20.0

25.0

30.0

0.5 1.0 1.5 2.0 2.5

EFFI

CIE

NC

Y (%

)

AVERAGE FIGURE OF MERIT ZT

700 C

400 C

150 C 200 C

T cold

=30 C

600 C 500 C

250 C

T hot

-T cold

(b)

• Low materials cost and low capital cost, potentially high efficiency. • Key Challenges: Develop materials with high thermoelectric figure of

merit; and selective surfaces that absorb solar radiation but do not re-radiative heat.

2.997 Copyright © Gang Chen, MIT

For 2.997 Direct Solar/Thermal to

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1st Law of Thermodynamics

System Q W

Environment

Boundary

WQdt dE

WQdE

WQEE

&& −=

−=

−=−

δδ 121212

State Properties: Process Independent

Process Dependent Quantities

...Energy)(Internal +++= UPEKEE

]m-J/Korkg,-[J/KHeatSpecific 3

dT

duC =

Closed System Open System

Closed:

2.997 Copyright ©

Gang Chen, MIT

For 2.997 Direct Solar/T

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Electrical Energy C

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2nd Law of Thermodynamics

)0(Sgen12 ≥+=− ∫ gen boundary

ST

QSS δ

∫ = 0dS

c

c

h

h

T

Q

T

Q −=0

Entropy Change State Properties

Entropy Transfer

Entropy Generation

Heat Reservoir Th

Heat Reservoir Tc

W

Qh

Qc

During a cycle:

No entropy generation

Maximum Efficiency (Carnot Efficiency)

h

c

h

ch

h T

T

Q

QQ

Q

W −=

− == 1η

Th=223 oC, Tc=23 oC, η=40% Th=5800 K, Tc=300 K, η=95% Thermal power plant η~40%, IC engines η~25%

2.997 Copyright ©

Gang Chen, MIT

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Microscopic Picture of Entropy

Ω= lnBkS Ω

= 1P

• For Isolated Systems • Microstate: a quantum mechanically allowed state

• A total of Ω microstate • Principle of equal probability:

each microstate is equally possible to be observed

kB=1.38x10-23 J/K ---Boltzmann constant

Boltzmann Principle

• Constant Temperature and Closed Systems

)/()( TkE BAeEP −=

• Constant Temperature But Open Systems

()( EAeEP −= μ --- chemical potential (driving force for mass diffusion); average energy needed to move a particle in/out off a system

Probability

−μ ) /(kBT)

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Maxwell distribution

A box of gas molecules

( )22 y

2 x vvv

2 1

zmE ++=

( ) ⎥ ⎥ ⎦

⎤

⎢ ⎢ ⎣

⎡ ++ =

Tk

mAP

B

z x 2

vvv exp)v,v,v(

22 y

2 x

zy

All Probability must normalize to one

( ) ⎥ ⎥ ⎦

⎤

⎢ ⎢ ⎣

⎡ ++ = ∫∫∫

∞

∞−

∞

∞−

∞

∞− Tk

mA

B

z

2 vvv

expdvdvdv1 22

y 2 x

zyx 2⎜⎜ ⎝

⎛ =A

( ) ⎢ ⎢ ⎣

⎡ ++ ⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ =

Tk

m

Tk

mP B

z

B x 2

vvv exp

2)v,v,v(

22 y

2 x

2/3

zy π Maxwell Distribution

⎞⎟⎟

3/ 2m

πkBT⎠

⎤⎥⎥⎦

2.997 Copyright ©

Gang Chen, MIT

For 2.997 Direct Solar/T

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Electrical Energy C

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One molecule

∫=E

( ) ( ) ⎥⎦⎢

⎢ ⎣

⎡ ++ ++= ∫∫∫

∞

∞−

∞

∞−

∞

∞− Tk

mAmE

B

z z 2

vvv expvvv

2 1dvdvdv

22 y

2 x22

y 2 xzyx

TkE B2 3

=

Equipartition Principle: every quardratic term in microscopic energy contributes kBT/2.

meV26 /106.1

105.14

5.14300KJ/K1038.1

19

21-

23

= ×

× =

=××=

−

−

eVJ

J

TkB

Oxygen Atom at 300 K

How much Is kBT at room temperature

1067.116 300/1038.133k v 27

23 B

××

××× == −

− KJ

m

T

⎤⎥

-2110 J×

K= 220 m/s

kg

2.997 Copyright ©

Gang Chen, MIT

For 2.997 Direct Solar/T

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Fermi-Dirac Distribution • From quantum mechanics

• Energy levels are quantized • Each quantum state can have

maximum one electron • Planck-Einstein Relation • Planck constant h=6.6x10-34 Js,

kh/p:Momentum :Energy

h==

==

λ

νhE

)2/( πh=h

• Consider one quantum state with an energy E at constant temperature T. The state can have zero electron (n=0) or one electron (n=1). What is the average number of electrons if one does many observations?

⎥ ⎦

⎤ ⎢ ⎣

⎡ ⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ −+⎟⎟

⎠

⎞ ⎜⎜ ⎝

⎛ == −−

= ∑ Tk

E

TkAAe

BB

TkE

n

B exp1exp1 )/()(

1,0

μμ

• Average number of electrons in the state

hω

r o97

t

2.9Copyr gi h © Gang mChen, M

IT

F2.997 Direct S

olar/Ther al to

Electrical Energy C

onversion

Fermi-Dirac Distribution

1exp

1)/()(

1,0 +⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ − == −−

= ∑

Tk E

Aenf

B

TkE

n

B

μ μ

• Average number of electrons in the state

Fermi-Dirac Distribution

0

0.2

0.4

0.6

0.8

1

-0.1 -0.05 0 0.05 0.1

FER

MI-D

IRA

C D

ISTR

IBU

TIO

N

E-μ (eV)

1000 K

300 K 100 K

At T=0K, μ is called Fermi level, Ef

F=1 for E<μ F=0 for E>μ

2.997 Copyright ©

Gang Chen, MIT

For 2.997 Direct Solar/T

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Electrical Energy C

onversion π )

Natural Frequency

1ν =2π

Energy of Mode E = ⎛⎜ n + 1 ⎞⎟hω n =

⎝ 2 ⎠ Basic vibrational energy quanta hν is called a phonon

Photons and Phonons From quantum mechanics

• EM waves are quantized, basic energy quanta is called a photon

• Photon has momentum • Planck-Einstein Relation • Each quantum state of photon (an

EM wave mode) can have only integral number of photons

h/p:Momentum :Energy h

==

=

λ

ωνhE

2/(Js;106.6 34 hh =×= − h

One Photon

Energy of a quantum state:

2 ⎠⎝ n 1 =⎟

⎞⎜⎛ += ωhnE

Zero point energyClassical Oscillator

M

Spring

M

K

•

=

hk

0,1,2...

•

0,1,2...

2.997 Copyright ©

Gang Chen, MIT

For 2.997 Direct Solar/T

hermal to

Electrical Energy C

onversion

Bose-Einstein Distribution

0

1

2

3

4

5

0 0.1 0.2 0.3 0.4 0.5

BO

SE-E

INST

EIN

DIS

TRIB

UTI

ON

FREQUENCY (X1014 Hz)

5000 K

1000 K

300 K

100 K

• Consider one quantum state in thermal equilibrium

)/()()( TkE n

BnAeEP μ−−=

1exp

1

−⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛ − =

Tk E

f

B

μ

Average number of photons/phonons in one mode (quantum state)

Usually μ=0

• Bose-Einstein Distribution

2.997 Copyright ©

Gang Chen, MIT

For 2.997 Direct Solar/T

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Electrical Energy C

onversion

Heat Transfer Modes

Heat Conduction

Thot Tcold

• Fourier Law

L

[W]dx

dTkAQ −=&

[ ]2W/mT)-k( ∇=−= dx

dTkq&

• Heat Flux

Thermal Conductivity [W/m-K] Materials Property

y

x

yy

ux

uy

Ta

uuu

FluidFluidFluid

TwTx

x

uy

xx

uyx

uy

TaTa

w

Convection

• Newton’s law of cooling

( )aw TThAQ −=&

Convective Heat Transfer Coefficient [W/m2K] Flow dependent

• Natural Convection • Forced Convection

Thermal Radiation

Thot Tcold

• Stefan-Boltzmann Law for Blackbody

4TAQ σ=&

Stefan-Boltzmann Constant σ =5.67x10-8 W/m2K4

• Heat transfer

( )44 coldhot TTAFQ −= εσ&

Emissivity of two surfaces

View factor F=1 for two parallel plates

Cross-Sectional Area

2.997 Copyright ©

Gang Chen, MIT

For 2.997 Direct Solar/T

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Electrical Energy C

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Heat Conduction

Heat Conduction

Thot Tcold

L

th

coldhotcoldhot

R

TT

L

TTkAQ −

= −

=&

1D, no heat generation

Thermal Resistance kA

LRth =

10-2

10-1

100

101

102

103

104

105

101 102 103

Ther

mal

con

duct

ivity

(W/m

K)

Temperature (K)

Quartz single crystal (// to c-axis)

Water (saturated)

Fused quartz

Ice

Steam (saturated)

Stainless steel (type 304)

Copper

Silicon

Diamond

Air (1 atm)

Helium (1 atm)

Thot Tcold Rth

Convection hA

Rth

1 =

Q&CurrentHeat

aG©

Cy

7 2.99 997

op right ng Chen, M

IT

Fo cr 2.

DirecgSt olar/T

hermal to

Eletric

al Ener y Conversio

n

Heat Conduction: Kinetic Picture

qx

x

Hot Cold

xvxτ

qx

x

Hot Cold

xvxτ

( ) ( ) τvxxτvxxx xx nEv

2 1nEv

2 1

+− −=

dx

dTk−=−=

−=)

=

dx dTC

3 τv

dx dT

dT dU

3 τv

dx d(Env-vq

2

2 x

xx τ

Λ= v 3 1 CkThermal Conductivity

• Energy per particle: E [J] • Number of particles per

unit volume: n [1/m3] • Average random

velocity of particles v • Average time between

collision of two particles τ---relaxation time

• Average distance travelled between collision Λ=vτ---Mean free path

• Volumetric specific heat

[ ]KdT

dUC 3m J=

cC ρ=

Density

q

Specific heat per unit mass

922.9

7 Copyright © Gang Chen, MIT

For .997 Direct Solar/Thermal to

Electrical Energy Conversio

n

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

2.997 Copyright ©

Gang Chen, MIT

For 2.997 Direct Solar/T

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Electrical Energy C

onversion

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

( ) ( ) 44 23

3

== c

cuI π ω

π ωω

h

Per unit wavelength interval

( ) ( ) 4 5 == λ π

λ ωωλ

c

d

dII h

Planck’s law

1

⎛ hω ⎞⎟⎟ −exp⎜⎜

⎝ 1

kBT⎠

1

exp⎜⎜⎛ 2πhc ⎟⎟

⎞−1

⎝ kBTλ ⎠

t

2.997 C Dopyright ©

oGang Chen, MIT

For 2.997irec S

lar/Thermal to

Electrical Energy Conversio

n

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 μ

m)

WAVELENGTH (μm)

5600 K

2800 K

1500 K

800 K

Emissive Power

Wien’s displacement law

mK2898max μλ =T

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2.997 Direct Solar/Thermal to Electrical Energy Conversion Technologies Fall 2009

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