Radiative heat transfer - MIT OpenCourseWare · Convective Heat Transfer Coefficient [W/m2K] Flow...

Tw

2.997 Copyright © Gang Chen, MIT

For 2.997 Direct Solar/Thermal to

Electrical Energy Conversio

n

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

xx

uy

xx

uyx

uy

TaTa

Tw

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




n

Thermal Radiation: Planck’s Law

Solid Angle

dAp

R

dAd p sin2 ==Ω

( )λI

• Planck’s law

Emissive power: power per unit surface area

π

λ

π

π λλ

ϕ

θ

Id

dIe

=

Ω=

∫∫

∫ 2/

0

2

0

2

cos

cos

θ

• Intensity: power per unit •solid angle in the direction of propagation

Power=Iλ dA⊥dΩdλ θ sinθdθ = πIλ

4πc2h 1=

λ5 ⎛ 2πhc ⎞ exp⎜⎜ ⎟⎟ −1

⎝ kBTλ ⎠

e( )λ = 4π 2c2h 1

λ5 ⎛ 2πhc ⎞θdθdϕ exp⎜⎜ ⎟⎟ − ⎝ kBTλ ⎠

1

t

2.997 Copyright ©

oGang Chen, MIT

For 2.997 Direc Slar/T

hermal to


n

Thermal Radiation: Planck’s Law Q&

Total

( )

4

4

0

Te

TAdQQ

b σ

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

Wien’s displacement law

mK2898max μλ =T

428 W/m1067.5 K−×=σ

Stefan-Boltzmann constant

.997 Copyright © Gang Chen, MI

or 2.99irect Solar/T

hermal to

2

T

F

7 D


n

Universal Blackbody Curve

) 12 exp

1

1

5

−⎟⎟ ⎠

⎞ ⎜⎜ ⎝

⎛

Tk c

T

Bλ π

λ

h

( )

( ) 1

2

5 1

0 5

1 4

0

λσ

π

λ λσ

λ

λ

λ

λ

λ

=

−⎟⎟⎞

⎜⎜ ⎝

⎛ ==

∫

∫ ∫

T

b

dC

d

T

C

T

de

F h

Fraction of Energy between [0,λ] Relative to Total BlackbodyFunction of λT only

4π 2c2 1hebλ =T 5 (λT )5 ⎛

⎜⎜ 2πhc ⎞

⎟⎟ − ⎝ ⎠

exp ck 1expB kBλT⎠C1 (λ )= T(λT⎛ 2πh ⎞⎜⎜ ⎟⎟ − ⎝

T c0 1expkBλT⎠

C1 λT 1 dx x

= ∫ 5 = ∫ f (x)dxσ 0 x exp⎜

⎛ C2 ⎟⎞ −1 0

⎝ x ⎠ = F(0 - λT)

Copyright ©

C

Direct

Thermal

r

version

I

or 2

l

2.997

Ganghen, M T

F.997

So ar/

to

Electrical Ene gy Con

Universal Function

Tλ (μmK)

f(λT)

Tλ (μmK) F(

0-λT

)

2.997 Copyri

For 2.997 DiSola

ectrical

Conv

Sun 147,300,000 km152,100,000 km

September equinoxSept 22/23

March equinoxMar 20/21

PerihelionJanuary 3

DecembersolsticeDec 21/22

June solsticeJun 21/22

AphelionJuly 4

erng Chen, MI

ermal to

ght © Ga

T

re

/Th

El

En

i n

Earth Orbital

32'

Sun

= 1.495 x 1011 m= 9.3 x 107 mi

1.7%

1.27 x 107 m7900 mi

Earth

Solar constant= 1367 W/m2

= 433 Btu/ft2 hr= 4.92 MJ/m2 hr{Gsc

{Distance is

1.39 x 109 m8.64 x 105 mi

Figure by MIT OpenCourseWare.


Image by Robert Simmon (NASA).

Earth Tilt

Chen,

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io ns©

rig S np CDir gy

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

http://commons.wikimedia.org/wiki/Main_Page

.997 Copyright © Gang Chen, M

or 2.997 Direct Solar/Thermal to

rical Energy Conversio

n

2

IT

FElect

Solar Spectral Outside Atmosphere

http://www.newport.com/images/webclickthru-EN/images/798.gif

Image by Robert A. Rohde/Global Warming Art.

7 Copyright © Gang Ch

r 2.997 Direct Solar/Thermal t

trical Energy Conversio

n

.

I

2 99

en, M T

Fo

o

Elec

Solar Going Through Atmosphere

Source: http://marine.rutgers.edu/mrs/education/class/yuri/erb.html

Image by NASA.

http://www.nasa.gov/audience/forstudents/5-8/features/F_The_Role_of_Clouds.html



hermal to

Solar spectra

2

T

F

7 D


n

Solar spectra are named by their air mass (AM), which is the ratio of the path length of air the rays travel through to the shortest possible pathlength (i.e. directly overhead). Thus the AM0 solar spectrum is measured above theearth’s atmosphere, AM1 occurs when the sun is directly overhead, and AM1.5 occurs when the sun’s rays travel through 50% more of theatmosphere than when the sun is directly overhead. Note 1/cos(48.2°) =1.5, so AM1.5 occurs when the solar zenith angle is 48.2°. AM0 is an average of measured data from many satellites, the space shuttle, high-altitude aircraft, sounding rockets, etc. AM1.5G and AM1.5 Direct + Circumsolar are calculated values based on atmospheric constituent and particle concentrations, humidity, ground surface albedo, the U.S. Standard Atmosphere, and various other parameters. They are defined as follows:

AM1.5 Direct + Circumsolar is only the solar radiation that comes from the sun and the cone of sky of half-angle 2.5 degrees surrounding the sun. The reference surface is normal to the sun, with an air mass of 1.5 (solar zenith 48.2°). (Defined in ASTM G173-03)

– AM1.5G accounts for radiation from the sun, the entire sky, and reflections off the ground. The solar zenith is 48.2°, but the panel is tilted at an angle of 37°. This results in an angle of incidence of the sun of 11.2°. (ASTM G173-03)

– AM1.5G is the spectrum used to calibrate solar cells. This spectrum gives highersolar cell efficiency than AM0, and higher power per square meter than AM1.5 Direct + Circumsolar. (ASTM E490-00a)

•

•

–




n

Air mass standards

48.1°

AM1.5

Earth




n

Air mass standards

48.1°

Earth Earth

11.1°

AM1.5 Direct + Circumsolar AM1.5 Global

Copyright © Gang Chen, MIT

2.997 Direct Solar/Thermal to

lectrical Energy Conversio

n

. 99 72 oF r

EImage by Robert A. Rohde/Global Warming Art.

Source: http://en.wikipedia.org/wiki/File:Atmospheric_Transmission.png

http://en.wikipedia.org/wiki/File:Atmospheric_Transmission.png




nHere the black body at 5777 K is scaled such that the total power is the solar constant at AM0: 1366 W/m2.

.997 Copyriht © Gang Chen, MI


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2

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Surface Properties: Emissivity Iλ(Ω)

Directional-spectral ( )

λ

λ λε

bIIT

Ω =)('

Hemispherical-spectral

Directional-total

λ

λ λε

beeT =)(

Hemispherical-total beeT =)(ε

Diffuse Emitter

λλ εε = '

Gray Emitter '' εελ =

Diffuse-Gray Emitter

( ) bI

IT Ω

=)('ε


or 2.99irec

olar/Thermal to

Surface Properties: Absorptivity

Iλ(Ω) Directional-spectral

absorbed

( )dΩ=

Ω

power

Iα ' (T )λ

o v r i nλ

t S n e s o

y CHemispherical-spectral αλ (T )

7 D e g

l En rDirectional-total α ' (T )

Hemispherical-total α (T )

Kirchoff’s Law

' = 'ε α

FE e t icl c r aλ λ




n

Surface Properties: Reflectivity&Transmissivity

Bi-directional Reflectivity

Diffuse Reflector

Gray Reflector

Diffuse Gray Reflector

( )ΩiI ,λ ( )' , ΩrIλ

( ) ( )Ω

Ω = r

I I , ' λ

λρ

Directional-spectral:

Hemispherical-spectral:

Directional-total:

Hemispherical-total:

Reflectivity:

Transmissivity:

"

λ ,i

'ρλ

ρλ

ρ '

ρ

" ' 'τ λ ,τ λ ,τ λ ,τ ,τ

' ' ' Energy Conservation ρλ +αλ +τ λ =1




n

View Factor

dAi

dAj

θ1

θ2

2

i

2

i

j

coscos I

cos cos

dAleavingpower dAreachingpower

ij

jji

i

ij

jj iii

dAdA

R

dA

dA

R

dAdAI

F ji

π

θθ

π

θθ

=

=

= −

ji A A ij

ji

i AA dAdA

RAF

i j

ji ∫ ∫= − 2

coscos1 π

θθ

Diffuse surface Radiation leaving Ai is uniform

Assumptions:



hermal to

2

T

F

7 D


n

View Factor Relations

jdA dA ij − AiAA AFAF

ji − =

kji AAAA FFF +− += )(Summation

Reciprocity

FdA dAi FdA=dA Ai j− −i j j

Aj Ai Ak− −i



hermal to

2

T

F

7 D


n

Earth-Sun Radiation

9 2

2

1067.54/ − − ×==

se

e e R

Dπ

2

2

79.64/ ×==

se

s s R

DF π

see FA − =

4

2 e

e

DA π

=

esss FAe −

Radiation Reaching Earth

Solar Radiation Per Unit Area Normal to Sun on Earth Outside Atmosphere

J

πD 2 sAs =

4

es AsFs−eFs e Fs e = = −s sAe

= 5.67 ×108 ×(5777)4 × 6.79×10−5

10−5 W1365−e = 2m

A Fs s−e Solar Constant: 1366 W/m2

32'

Sun

= 1.495 x 1011 m= 9.3 x 107 mi

1.7%

1.27 x 107 m7900 mi

Earth

Solar constant= 1367 W/m2

= 433 Btu/ft2 hr= 4.92 MJ/m2 hr{Gsc

{Distance is

1.39 x 109 m8.64 x 105 mi


Fo

yright © Gang Chen, MI

9irect Solar/T

hermal to

2.997 Cop

T

7 D


n

Maximum Efficiency of a Solar Thermal Engine

Jh

Jc

Engine W

Blackbody At Sun’s Temperature Ts

Blackbody Absorber at T

Heat Transferred to Absorber

( )44 TTQ sh −= σ

Thermal Efficiency

( ) 4

4

4

44

1 ss

s th T

T

T

TT −=

− =

σ ση

Carnot Efficiency

T

Ta−=1η

r 2.9Ta

alar/T

he

Conversion

2.997 Copyright © G

ng Chen, MIT

For 2.997 Direct So

rmal to

Electrical Energy

Maximum Efficiency of a Solar Thermal Engine

⎟ ⎠

⎞ T

Ta

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1000 2000 3000 4000 5000 6000

Series8 E

ffici

ency

Maximum: 85% @ T=2450K For Ta=300 K

⎛ T 4 ⎞ ⎜⎜ ⎟⎟

⎠ η =ηthηC 1−

⎛1−⎜=T 4

s ⎝⎝

Temperature (K)

h ol ve

.997 Co

ahen, MI

or 2.99ir

ermal to

2

pyrigt © G

T

F

7 Dect S

ar/T

Electrical Energy Con rsio

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How Hot a Surface Can Get By Solar Radiation?

Solar Radiation In Thermal Emission Out

Zero Emissivity on This Side

Absorptivity αλ

Emissivity ελ

Ambient Ta

( ) ( ) [ ] λελα λλλλλ dTeTedJC obb∫∫ ∞∞

−= 00

Concentration

ng Ch

opyright ©

.997 Direct Solar/T


n

2.97 C

hen, MIT

o

rmal to

Selective surface • Black body: emissivity = 1 for all wavelengths

g Ca e

• Selective surface: emissivity is high in the solar spectrum and low in the infrared. Ideally the transition would occurabruptly at the cutoff wavelength λc.

Gn h

9r 2

FE el c




n

Approximate the Sun as A Blackbody

( ) ( ) ( ) [ ] λλ λ

λλλ dTeTeAdTe obbsunbsurface ∫ −= 0

( ) ( ) [λ λ

λ

λ

λ eTeAdTeAF bsunbse ∫∫ −= −

00

4 )0(

sun

s se

TF

T

TFF σ

λ =

− −

One Sun C=1

λ

∫A Fsun sun−

0

]dλ( ) Tobλ

λ(0− ) F (0 − λTa )−4 4σT σTa

997 Copyright © Gang Chen,



n

IMaximum Temperature AM0

2.

M T

.997 Copyright © Gang Chen, MIT



n

2

opyright ©

.997 Direct Solar/T


n

2.97 C

hen, MIT

o

rmal to

Selective surface • Fix Temperature, choosing a

g Ca e

cut-off wavelength

Gn h

9r 2

FE el c

© G

F

t Solar/Th

lec

rgy Conversion

yrig

hen, MI

99i

l to

2.997 Copht

ang C

T

or 2.7 D rec

erma

E tric

al Ene

Efficiency of a Solar Thermal Engine: 1 SunSolar Js

Selective Absorber

Jh

Jc

Engine W

• Fix Temperature, choosing a cut-off wavelength

thη Thermal Efficiency




n

2

TThermal Efficiency for AM1.5G with No Concentration




n

2

IT System (Surface x Carnot) Efficiency for AM1.5G with No Concentration




n

System performance • Increasing the temperature of the

hot side of a heat engine increasesthe Carnot efficiency.

• However, increasing the temperature increases radiationlosses, which decreases theamount of heat delivered to the heat engine (surface efficiency).

• Example: Sunselect® surface with no concentration:




n

Sunselect absorber

2

IT

Optimal operation point



hermal to

2

T

F

7 D


n This same analysis can be performed to find the ideal absorber for various concentrations and operating temperatures. This is the absolute upper limit onsystem efficiency. We can compare a blackbody, the Sunselect absorber, and an ideal absorber

•

•




n

Blackbody absorber

2.

MIT




n

Sunselect absorber

2.

MIT




n

Ideal absorber

2.

MIT

t la

2.997 Copyrigh © Gang Chen, MIT

For 2.997 Direct Sor/T

hermal to


n

Heat Transfer on A Suspended Surface

Thermal Emission Out

Emissivity ελ2

Absorptivity αλ, Emissivity ελ1 Ambient Ta

( )[ b TedJ = ∫∫ ∞∞

ελα λλλλ 00

Suspension

Solar Radiation In

]dλ T −Ta− ( ) To +ebλ Rth

Thermal Resistance by Conduction

o

.997 C epyright © Gang Chen, MI

or 2.99ir ct Solar/T

rmal to

2

T

F

7 D

he


n

Heat Conduction

Heat Conduction

Thot Tcold

L

th

coldhotcoldhot

R

TT

L

TTkAQ −

= −

=&

1D, no heat generation

Thermal Resistance kA

LRth =

Thot Tcold Rth

Convection hA

Rth

1 =

Q&CurrentHeat

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.

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