Heat Transfer Modes
T
I
M
Thermal
Radiation
Convection
,
Heat Conduction
n o
e
t
h
l
C
a
T
T
T
g
T
m
r
n
e
a
L
G /Th ion
• Fourier Law
© lawlaofr coolingrs• Stefan-Boltzmann
• Newton’s
t
e
h
o
Law for Blackbody
dT
v
g
&
i
S
& =thA
Q = −kA
[W]
r
(
Q
T −oTn )
dx
y
& = AσT
c
Q
p
C
e
Cross- o
r
y
C Di Convective
Stefan-Boltzmann Constant
Thermal
Sectional
g
Heat
r
Conductivity 7
σ=5.67x10 W/m K
Area
e
Transfer
Coefficient
7
n
9
9
[W/m-K]
[W/m
K]
9
9
E
• Heat transfer
.
.
Materials Property
2 r 2 cal Flow dependent
)
Q& = AF εσ (T − T
i
o
r
• HeatF
Flux ct
• Natural Convection
Emissivity of
View factor
e
l
dT
&
E [ ] • Forced Convection F=1 for two two surfaces
Fluid Ta
y
cold
hot
uy
ux
x
cold
hot
Tw
w
4
a
-8
2
4
2
4
hot
q = −k
dx
(= -k∇T) W/m 2
parallel plates
4
cold
Thermal Radiation: Planck’s Law
T
I
M
,
•
Intensity: power per unit •
Emissive power:
power per
n
earea to
solid angle in the direction unit surface
h
l
of propagation
C
a
θdΩ
e =g∫ I cosm
n er
Power
a
h
I =
n
G
T
o
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i
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r
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s
t ola = ∫ deϕr∫ I cosθ sin θd
θ = πI
h
v
g
i
S
n
r
dA
t
o
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• Planck’s
law
c
p
C
e y
θ
o
r
i
1
C D rg I (λ ) = 4πc h
7 97 ne
λ
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9
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9
9
E
.
.
l
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2
2
a
r ric
Solid Angle
o
4π c h
1
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e(λ ) =
λ
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e
l
= sin θdθdϕ
dΩ =
⎟⎟ − 1
exp⎜⎜
E
λ
λ
2π
λ
2π
π /2
⊥
λ
0
0
p
2
5
B
2 2
R
p
2
5
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λ
Thermal Radiation: 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
G /T ion
Total
© lar rs
t
e
h
o
v
g
i
S
n
r
&
&
t
Q = ∫ Q(λ )dλ = AσT
o
y
c
p
C
e
o
r
i
y
C
g
D
e = σT
r
7 97 ne
9
9
9
E
.
.
l
Stefan-Boltzmann
constant
2 r 2 ca
o W/mtrKi
σ = 5.67F
× 10
c
e
El
2
EMISSIVE POWER (W/cm μm)
max
∞
4
0
4
b
−8
2
10
4
10
3
10
2
10
1
5600 K
2800 K
1500 K
10
0
4
800 K
10
-1
0
2
4
6
WAVELENGTH (μm)
8
10
Universal Blackbody Curve
T
I
M [0,λ]
Fraction of Energy
between
,
nBlackbody
Function of λT only Relative to Total
e
o
t
h
l
C
a
e
4π c h
1
g
e dλ r m
∫
n
=
C
dλ
e
a
T
(λT ) exp
⎛⎜ 2πhc ⎞⎟ − 1
F=
=∫ n
h
G
T
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o
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⎞⎟ − 1
/
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i
r
©
s
⎠
⎝
⎜ k λT
⎟
a
r
t
l
e
h
⎝
⎠
o
v
g
C
1
i
S
n
r
=
t
C
d (λT
)
1
o
y
c
(λT
) exp⎛⎜ 2oπp
hc ⎞
=
C
e
∫
⎟
−
1
r
σ
(λT
) exp
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i
y
C⎜⎝ k λT ⎟⎠D rg
⎜ k λT
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7 97 ne
⎝
⎠
9
9
9
E
.
.
C
dx
1
l
2 r 2 ca
=
= ∫ f (x)dx
∫
σ x
i
⎛C ⎞
o
r
exp
⎜ ⎟ −1
F ct
x ⎠
⎝
e
El
bλ
5
λ
2 2
bλ
5
λ
0
1
5
4
0
B
B
λT
1
5
1
5
0
B
B
λT
x
1
5
0
= F(0 - λT)
2
0
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
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9
9
λ
T9(μmK)l E
.
.
λT (μmK)
2 r 2 ca
i
o
r
F ct
e
l
E
F(0-λT)
f(λT)
Universal Function
T
I
M Orbital
Earth
,
n o
e
t
h
l
C
a
g rm
n
e
a
h
G /T i n
© la
r
t
e
h
o
v
g
i
S
n
r
o
y e
p
C
o
r
C Di
7 97 n
9
9
9
E
.
.
l
2 r 2 { ca
i
o
r
F { ct
e
l
E
Image by Robert Simmon (NASA).
March equinox
Mar 20/21
Sun
1.27 x 107 m
7900 mi
1.39 x 109 m
8.64 x 105 mi
Earth
32'
Solar constant
2
Gsc
Distance is
= 1367 W/m
= 433 Btu/ft2 hr
= 4.92 MJ/m2 hr
= 1.495 x 1011 m
= 9.3 x 107 mi
1.7%
June solstice
Jun 21/22
Aphelion
July 4
152,100,000 km
Sun
147,300,000 km
Perihelion
January 3
December
solstice
Dec 21/22
September equinox
Sept 22/23
Figure by MIT OpenCourseWare.
Figure by MIT OpenCourseWare.
Earth Tilt
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 Di rgy
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
Images from Wikimedia Commons, http://commons.wikimedia.org
Solar Spectral Outside Atmosphere
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
i
y
C
g
D
r
2 7
7 ne
9
9
.F9 2c.9t al E
e
l
r
oE ric
Image by Robert A. Rohde/Global Warming Art.
http://www.newport.com/images/webclickthru-EN/images/798.gif
Solar Going Through Atmosphere
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
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
Source:
E http://marine.rutgers.edu/mrs/education/class/yuri/erb.html
Image by NASA.
Solar spectra
T
I
M of the path
Solar spectra are named by their air mass (AM), which is the, ratio
length of air the rays travel through to the shortest possible
pathlength (i.e.
n
directly overhead). Thus the AM0 solar spectrum is e
measured
above the
o
t
h
earth’s atmosphere, AM1 occurs when the sun is directly overhead,
and
l
C
a
AM1.5 occurs when the sun’s rays travel through 50% more of the
g rm
atmosphere than when the sun is directlyn
overhead.
Note 1/cos(48.2°) =
1.5, so AM1.5 occurs when the solar zenith
anglee
is 48.2°.
a
h
n
Gfrom many
•
AM0 is an average of measured data
satellites,
the space
T
o
/
i
shuttle, high-altitude aircraft, sounding
rockets,
etc. s
r
©
a
r
t
l
•
AM1.5G and AM1.5 Directh
+ Circumsolar
are calculated
values based on
e
o
v
atmospheric constituentig
and particle
concentrations,
humidity, ground
S
n
r
surface albedo, the y
U.S. Standard
Atmosphere,
and various other
t
o
c
parameters. They
are defined
as follows:
p
C
e
o
r
–
AM1.5 Direct + Circumsolar
is only y
the solar radiation that comes from the sun
i
C
g
and the cone of skyD
of half-angle
2.5 degrees surrounding the sun. The
r
reference
surface
is
normal
to
the
sun, with an air mass of 1.5 (solar zenith
e
7
7
48.2°).
9
9
(Defined
in ASTM n
G173-03)
9
9
E
accounts
for
radiation from the sun, the entire sky, and reflections off
.
l
2–. AM1.5G
2
the ground.
The solar
zenith is 48.2°, but the panel is tilted at an angle of 37°.
a
r
This results inian
angle of incidence of the sun of 11.2°. (ASTM G173-03)
c
o
r
– FAM1.5G is t
the spectrum used to calibrate solar cells. This spectrum gives higher
solar cellc
efficiency than AM0, and higher power per square meter than AM1.5
Directle
E + Circumsolar. (ASTM E490-00a)
Air mass standards IT
M
,
n o
e
t
h
l
C
a
g rm
n
e
a
h
G /T ion
r
©
s
a
r
t
48.1° l
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
C D i rg y
AM1.5
e
7
7
n
9
9
9
9
E
Earth
2. r 2. cal
i
o
r
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e
l
E
Air mass standards IT
M
,
n o
e
t
h
l
C
a
g rm
n
e
a
AM1.5 Direct + Circumsolar
AM1.5 Global
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
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7 97 ne
9
9
9
11.1°
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.
.
l
2 r 2 ca
i
o
r
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Earth
e
l
E
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
i
y
C
g
D
r
7
e
7
n
9
9
9
9
E
2. r 2. cal
i
o
r
F ct
Ele
Image by Robert A. Rohde/Global Warming Art.
Source: http://en.wikipedia.org/wiki/File:Atmospheric_Transmission.png
T
I
M
,
n o
e
t
h
l
Here the black
body at a
5777 K is
C
scaled such
g that thermtotal power is the
n
solara
constant ateAM0: 1366 W/m .
G /Th 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
2
Surface Properties: Emissivity
T
I
M I (Ω )
I (Ω)
,
Directional-spectral en
ε λ (T )to= λ
h
Iλ
l
C
a
g rm
n
eλ
e
a
h
Hemispherical-spectral
G /T ioεnλ (T ) = e
© lar rs
λ
t
e
h
o
v
g
i
S
n
r
I (Ω )
t
o
y
Diffuse Emitter p
c
Directional-total
ε (T ) =
C
e
o
r
i
'
I
y
C
g
ελ = ελ
D
r
7 97 ne
9
9
E
Gray
Emitter.9
.
l
2 ' r 2' ca Hemispherical-total ε (T ) = e
e
εFλo= ε ctri
e
l
Diffuse-Gray
E Emitter
λ
'
b
b
'
b
b
Surface Properties: Absorptivity
T
I
M
Directional-spectral
I (Ω)
,
n o
e
t
absorbed
power
h
l
α λ (T )
=C
a
g rm
I
λ (Ω )dΩ
n
e
a
h
G /T i
on
© lar rs α λ (T )
Hemispherical-spectral
t
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
o
r
Kirchoff’s
rgy
α (T )
CLaw
Di Directional-total
e
7
7
n
'9
' 99
9
E
2.ε λ =r α2λ. cal
i
o
r
Hemispherical-total
α (T )
F ct
e
l
E
λ
'
'
Surface Properties: Reflectivity&Transmissivity
T
I
M
,
I (Ω )
n o
I (Ω')
Reflectivity:
e
t
h
l
C
I (Ω')
a
ρ
=
g Reflectivity
Bi-directional
m
r
I (Ω )
n
e
a
h
n
G
T
o
/
i
ρ
Directional-spectral:
r
©
s
t ola er
h
v
g Hemispherical-spectral:
i
ρ
S
n
r
t
y ec Co
p
Directional-total:
ρ
o
r
i
y
C D Hemispherical-total:
g
ρ
r
7 97 ne
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9
9
9
E
.
.
l
2Reflector
2 ca Transmissivity:
τ ,τ ,τ ,τ ,τ
Gray
r
i
o
r
t
F
c
Diffuse Grayle
Reflector
ρ + α +τ = 1
Energy Conservation
E
λ ,i
λ ,r
λ ,r
"
λ
λ ,i
'
λ
λ
'
"
'
λ
'
λ
λ
'
λ
'
λ
'
λ
View Factor
T
I
M
,
n
power reaching
dA
e
o
F
=
t
h
l
dA
power
leavingadA
C
g rm
cos θ dA
n
I dA coseθ
a
G= /Th ioRn
© lar πIrsdA
θ
t
e
h
o
v
g
i
S
θ cos θ dA
cosn
r
t
y ec C
= o
θ
p
πR
o
r
i
y
C D rg
7 97 ne
cos θ cos θ
1
9
= ∫∫
dA dA
F
dA .9
9
E
.
A
πR
2 r 2 ca l
i
o
r
t
F Assumptions:
c
Diffuse surface
e
l
E
j
dAi − dA j
j
i
i
i
j
2
ij
i
2
i
j
i
i
j
j
2
1
ij
i
i
j
Ai − A j
2
i Ai A j
i
ij
Radiation leaving Ai is uniform
j
View Factor RelationsIT
M
,
n o
e
t
h
Reciprocity
l
C
a
g rm
n
FdA −
dA
dAi =
FdA
− dA dAGj a
FeA − A n
Ai = FA −
A A j
h
T
o
/
i
r
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s
a
r
t
l
e
h
o
v
g
i
S
n
r
t
o
y
c
p
C
e
r
Summationo
C Di FA −r(gA y+ A ) = FA −
A
+ FA −
A
e
7
7
n
9
9
9
9
E
2. r 2. cal
i
o
r
F ct
e
l
E
i
j
j
i
i
i
j
k
i
j
j
j
i
k
i
Sun
1.27 x 107 m
7900 mi
Earth-Sun
T
I
Radiation
M
,
n o
{
e
t
h
Radiation
Reaching
Earth
l
{
C
a
g rm
n
es As Fs −e
e
a
G /Th ion
r
©
s
πD
Solar Radiation
Per Unit
a
r
t
π
D
l
e
h
A
=
o
A = ig
Area
v Normal to Sun on Earth
4
S
n
4
r
t
Outside Atmosphere
o
y
c
p
C
e
o
r
i
y
4
πD / C
e AF
g
D
=
e F
J =
F
= 7
= 57.67 ×10 er
A
R
n
9
9
9
9
E
.
.
= 5.67 ×10 × (5777 ) × 6.79×10
l
2 πDr 2/ 4 ca
i
W
=
6.79 ×10
F =Fo
r
t
=
1365
R ec
m
l
E
1.39 x 109 m
8.64 x 105 mi
Earth
32'
Solar constant
2
Gsc
Distance is
= 1367 W/m
= 433 Btu/ft2 hr
= 4.92 MJ/m2 hr
= 1.495 x 1011 m
= 9.3 x 107 mi
1.7%
Figure by MIT OpenCourseWare.
2
2
s
e
s
e
2
s −e
−9
e
2
s
s−e
s
8
2
e −
s
e
se
e
−
s
s
s
4
−5
s
2
se
As F
s−
e
= Ae Fe − s
2
Solar Constant: 1366 W/m2
−5
Maximum
Efficiency
Blackbody At
T
I
Sun’s
of
a
Solar
Thermal
Engine
M
Temperature T
,
n
e to
Heat Transferred tohAbsorber
l
C
a
g rm
Blackbody
n
Q =
σ (T h−eT )
a
Absorber at T
G /T ion
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t
Thermal
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h
o
g t S nv
i
r
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y ec Cσo(T − T ) T
p
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o
C Dir rgηy = σT = 1 − T
7 97 ne
9
J
9
9
E
.
.
l
2 r 2 ca Carnot Efficiency
i
r
TFo
t
c
Ta
e
l
η = 1−
E
s
h
4
4
4
4
s
h
4
s
th
4
s
c
a
T
4
s
Maximum Efficiency
T
I
of a Solar Thermal Engine
M
,
n o
e
t
h
l
C
a
⎛ T ⎞⎛ T ⎞
g
m
r
n
η = η η =
⎜⎜1
−
⎟⎟⎜
1−
⎟
e
a
h
T
T
n
⎠ G
⎠⎝
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o
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i
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s
t ola er
h
g t S nv
i
r
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p
o
C Dir rgy
Maximum:7
85% @ T=2450K
7 ne
9
9
For T.9
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l
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i
o
r
F ct
Temperature (K)
e
El
4
0.9
a
4
s
0.8
0.7
Efficiency
th C
0.6
0.5
0.4
0.3
Series8
0.2
0.1
a
0
0
1000
2000
3000
4000
5000
6000
How Hot a Surface Can Get
By Solar Radiation? MIT
,
n
e to
h
l
Solar Radiation In Thermal C
Emission Out
a
g rm
n
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e
a
h
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T
© laαr rs
Absorptivity
t
e
h
o
Emissivity
ε
v
g
i
S
n
r
t
o
y
c
p
C
Zero
Emissivity
on
This Side
e
o
r
C D i rg y
7 97 ne
9
9
9
E
.
.
l
2 r 2 Cc∫aα J dλ = ∫ ε [e (T ) − e (T )]dλ
i
o
r
F ct
e
l
E Concentration
a
λ
λ
∞
∞
λ
0
λ
λ
0
bλ
bλ
o
Selective surface
T
I
• Black body: emissivity = 1 for all wavelengths
M
,
nsolarospectrum • Selective surface: emissivity is high in the
e
t
and low in the infrared. Ideally the transition
would
occur
h
l
a
.
abruptly at the cutoff wavelength λcC
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
Approximate the Sun as A Blackbody
T
I
M
,
One Sun C=1 en to
h
l
C
a
g rm
n
e
a
h
(T )]dλ
e (T )dG
λ = ∫ A[T
e (T ) − e n
∫
A F
o
/
i
r
©
s
t ola er
h
g t S nv
i
r
o
y
(
)
(T ) − e (T
)]dλ
[
AF
e
T
d
λ
=
A
e
c
p
∫ o
∫
C
e
C Dir rgy
e
7
7
n
9
9
F
T
(
0
) F (0
−
λ
T )
F (0 − λT )
−
λ
9
9
E
.
.
2 r 2F cσaTl = σT
−
σT
i
o
r
F ct
e
l
E
λ
λ
sun
sun−
surface bλ
bλ
sun
bλ
0
0
λ
λ
e − s bλ
bλ
sun
0
bλ
o
0
a
s
e− s
4
sun
4
4
a
o
Maximum Temperature AM0
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
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
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
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
Selective surface
T
I
M
,
• Fix Temperature, choosing
a
n
e to
h
l
cut-off wavelength 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
Efficiency of a Solar
Solar Js
T
I
Thermal Engine: M
1 Sun
,
n
• Fix Temperature, choosing
a cut-off
e
o
t
h
l
wavelength
C
a
g rm
n
Selective
e
a
h
G /T ion
Absorber
r
©
s
a
r
t
l
J
e
h
o
v
g
i
S
n
r
t
o
y
c
Wp
C
e
o
r
i
y
Engine C
g
D
r
7 97 ne
9
9
E
J .9
.
l
2 r 2 ca
i
o
r
F ct
ηth Thermal Efficiency
e
l
E
h
c
Thermal Efficiency for AM1.5G with No Concentration
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
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
System (Surface x Carnot) Efficiency for AM1.5G with No Concentration
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
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
System performance
T
I
• Increasing the temperature of
the
M
,
hot side of a heat enginenincreases
the Carnot efficiency. e
o
t
h
• However, increasing
the al
C
g rradiation
m
temperature increases
n
e
losses, whichadecreases
the
h
n
G
T
amount of heat delivered
to
the
o
/
i
r
©
s
heat engine
(surface
efficiency).
a
r
t
l
e
h
o
v
g
• Example:
Sunselect®
surface with
i
S
n
r
t
o
no concentration:
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
Sunselect absorber
T
I
M
,
n o
e
t
h
l
C
a
g rm
n
e
a
h
G /T ion
© lar rs
Optimal
t
e
operation pointh
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
T
I
M
,
n o
e
t
h
l
C
a
•
This same analysis can be performed
to find
the ideal
g
m
r
n
absorber for various concentrations
and
operating
e
a
temperatures. This is the absolute
n on
G /Thupperiolimit
system efficiency.
© lar rs
t
e
h So the vSunselect
•
We can compare iagblackbody,
absorber,
n
r ct
and an ideal absorber
o
y
p ire
C
o
C D rg y
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
Blackbody absorber
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
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
Sunselect absorber
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
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
Ideal absorber
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
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
Heat Transfer on A Suspended Surface
T
I
M
,
nOut o
Solar Radiation In
Thermal Emission
e
t
h
l
C
a
g rm Ambient T
n
Absorptivity α , Emissivity
ε e
a
G /Th ion
© lar rs
t
Suspension
e
h
o
v
g
i
S
Emissivity
ε
n
r
t
o
y
c
p
C
e
o
r
C D i rg y
7 α 9J7dλ = nε e[e (T ) −
e
(T
)]dλ
+
T − T
9
9
9
∫
∫
E
.
.
R
l
2 r 2 ca
i
o
r
F ct
e
l
Thermal Resistance by Conduction
E
λ
a
λ1
λ2
∞
∞
λ
0
λ
λ
0
bλ
bλ
a
o
th
Heat Conduction
T
I
M
,
n
e
Heat Conduction
Thermal Resistance
o
R
t
h
l
C
a
&
g
Heat
Current
Q
m
n er
T
T
a
h
n
G
T
L
o
/
T i
T
r
©
s
R
a
r
t
l
e
h
o
1D, no heat generation
v
1
g
i
S
Convection
n
=
R
r
t
o
y
hA
T
T
T
T
−
−
c
&
p
Q = kA
=
C
e y
o
R ir
L
C D rg
7 97 ne
9
9
9
E
.
.
l
2 r 2 ca
i
o
r
F ct
e
l
E
th
cold
hot
hot
cold
th
th
hot
cold
hot
cold
th
=
L
kA
MIT OpenCourseWare
http://ocw.mit.edu
2.997 Direct Solar/Thermal to Electrical Energy Conversion Technologies
Fall 2009
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