Radiative heat transfer - MIT OpenCourseWare · Convective Heat Transfer Coefficient [W/m2K] Flow...
Transcript of 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
2.997 Copyright © Gang Chen, MIT
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
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
Electrical Energy Conversio
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
Electrical Energy Conversio
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.
Figure by MIT OpenCourseWare.
Image by Robert Simmon (NASA).
Earth Tilt
Chen,
ngGa
r/T hermal to
io ns©
rig S np CDir gy
7cal
ec
Ener y o vt
o9ri
C.9ct
97
For 2.92
T MI
ht ola er
EleImages from Wikimedia Commons, http://commons.wikimedia.org
.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.
.997 Copyright © Gang Chen, MI
or 2.99irect Solar/T
hermal to
Solar spectra
2
T
F
7 D
Electrical Energy Conversio
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)
•
•
–
2.997 Copyright © Gang Chen, MIT
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
n
Air mass standards
48.1°
AM1.5
Earth
2.997 Copyright © Gang Chen, MIT
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
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
2.997 Copyright © Gang Chen, MIT
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
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
or 2.99irect Solar/T
hermal to
2
g
T
F
7 D
Electrical Energy Conversio
n
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
2.997 Copyright © Gang Chen, MIT
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
2.997 Copyright © Gang Chen, MIT
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
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
2.997 Copyright © Gang Chen, MIT
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
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:
.997 Copyright © Gang Chen, MI
or 2.99irect Solar/T
hermal to
2
T
F
7 D
Electrical Energy Conversio
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
.997 Copyright © Gang Chen, MI
or 2.99irect Solar/T
hermal to
2
T
F
7 D
Electrical Energy Conversio
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
Figure by MIT OpenCourseWare.
Fo
yright © Gang Chen, MI
9irect Solar/T
hermal to
2.997 Cop
T
7 D
Electrical Energy Conversio
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
n
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
trical Energy Conversio
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
2.997 Copyright © Gang Chen, MIT
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
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,
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
n
IMaximum Temperature AM0
2.
M T
opyright ©
.997 Direct Solar/T
trical Energy Conversio
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
.997 Copyright © Gang Chen, MI
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
n
2
TThermal Efficiency for AM1.5G with No Concentration
.997 Copyright © Gang Chen, M
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
n
2
IT System (Surface x Carnot) Efficiency for AM1.5G with No Concentration
2.997 Copyright © Gang Chen, MIT
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
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:
.997 Copyright © Gang Chen, M
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
n
Sunselect absorber
2
IT
Optimal operation point
.997 Copyright © Gang Chen, MI
or 2.99irect Solar/T
hermal to
2
T
F
7 D
Electrical Energy Conversio
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
•
•
997 Copyright © Gang Chen,
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
n
Blackbody absorber
2.
MIT
997 Copyright © Gang Chen,
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
n
Sunselect absorber
2.
MIT
997 Copyright © Gang Chen,
For 2.997 Direct Solar/Thermal to
Electrical Energy Conversio
n
Ideal absorber
2.
MIT
t la
2.997 Copyrigh © Gang Chen, MIT
For 2.997 Direct Sor/T
hermal to
Electrical Energy Conversio
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
Electrical Energy Conversio
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.