radiative heat transfer - espci parisradiative heat transfer black body radiation p bb(t )= z 1 0 p...
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Radiative heat transfer
Radiative heat transfer
Black body radiation
Radiative heat transfer
Black body radiation
Spec
tral i
nten
sity
(W.m
-2 µ
m-1
)
Pe(�, T ) =2⇡hc2
�5
1
exp(hc/�kBT )� 1<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Power emitted per unit surface :
Radiative heat transfer
Black body radiation
Spec
tral i
nten
sity
(W.m
-2 µ
m-1
)
Pe(�, T ) =2⇡hc2
�5
1
exp(hc/�kBT )� 1<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Power emitted per unit surface :
�M (m) =2.88⇥ 10�3
T<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Wien’s displacement law
Radiative heat transfer
Black body radiation
PBB(T ) =
Z 1
0Pe(�, T )d� =
2⇡5k4B15 h3c2
T 4 = �T 4
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Stefan’s law
Spec
tral i
nten
sity
(W.m
-2 µ
m-1
)
Pe(�, T ) =2⇡hc2
�5
1
exp(hc/�kBT )� 1<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Power emitted per unit surface :
�M (m) =2.88⇥ 10�3
T<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Wien’s displacement law
Radiative heat transfer
Black body radiation
PBB(T ) =
Z 1
0Pe(�, T )d� =
2⇡5k4B15 h3c2
T 4 = �T 4
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Stefan’s law
Spec
tral i
nten
sity
(W.m
-2 µ
m-1
)
Pe(�, T ) =2⇡hc2
�5
1
exp(hc/�kBT )� 1<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Power emitted per unit surface :
Non black bodies
Emissivity ε = absorptivity a
P(λ,T) = Pe(λ,T) ε(λ)
Kirchoff ’s law
�M (m) =2.88⇥ 10�3
T<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Wien’s displacement law
Radiative heat transfer
Black body radiation
Grey bodies
Emissivity independent of λ
ε(λ) = a(λ) = C
PBB(T ) =
Z 1
0Pe(�, T )d� =
2⇡5k4B15 h3c2
T 4 = �T 4
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Stefan’s law
Spec
tral i
nten
sity
(W.m
-2 µ
m-1
)
Pe(�, T ) =2⇡hc2
�5
1
exp(hc/�kBT )� 1<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Power emitted per unit surface :
Non black bodies
Emissivity ε = absorptivity a
P(λ,T) = Pe(λ,T) ε(λ)
Kirchoff ’s law
�M (m) =2.88⇥ 10�3
T<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Wien’s displacement law
Diameter of the Sun: 1,4 106 kmSurface temperature of the Sun : 5800 KDistance to the Sun: 1,5 108 kmDiameter of the Earth: 12700 km
Effective albedo of the Earth: 0.31
The Donald Trump problem at zeroth order : show that with a completely transparent atmosphere, it would be difficult to grow lawn for a golf course on Earth.
Earth and Sun are assumed to be black bodies
-19°C
Diameter of the Sun: 1,4 106 kmSurface temperature of the Sun : 5800 KDistance to the Sun: 1,5 108 kmDiameter of the Earth: 12700 km
Effective albedo of the Earth: 0.31
The Donald Trump problem at zeroth order : show that with a completely transparent atmosphere, it would be difficult to grow lawn for a golf course on Earth.
Earth and Sun are assumed to be black bodies
Radiative flux from the sun, averaged over entire Earth surface : 342 W/m2
Flux adsorbed : 235 W/m2
Corresponding equilibrium temperature : 254°K
The Donald Trump problem at order one : show that with a one layer atmosphere adsorbing in the infrared, it would be easier to grow lawn for a golf course on Earth.
Incoming solar radiation
Reflected solar radiation
Atmosphereemissivity εTa
Te Earth surfaceemissivity 1
Write the energy flux balance for the earth + atmosphere system
Write the energy flux balance for the atmosphere at temperature Ta
Determine Ta and Te as a function of ε
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
emissivity
210
220
230
240
250
260
270
280
290
300
310
Temperature
TearthTatmos
Te =
(1� a)Js�(1� ✏/2)
�1/4
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
The Donald Trump problem at order 2 : Evaluate the temperature distribution in a stable « grey » non scattering atmosphere
Atmosphere clear to solar radiation ; all absorption takes place at the ground.Gray atmosphere with no clouds in the IR (no scattering of light)Layers of atmosphere are grey thermal radiatorsPlane parallel or stratified atmosphereEddington approximation for the IR emission : I(z,θ ) = I0(z) + I1(z) cos θ
z
z+dz θ
s
ds
I(s)
I(s+ds)
s
ds
I(s)
I(s+ds)Extinction (absorption, scattering)
dIext = �ext I ds<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
s
ds
I(s)
I(s+ds)Extinction (absorption, scattering)
dIext = �ext I ds<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Augmentation (emission, scattering)
dIaug = aug J ds<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
s
ds
I(s)
I(s+ds)
If no scattering, emission by thermal radiation
dIaug = ext�T 4
⇡= extB(T )
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Extinction (absorption, scattering)
dIext = �ext I ds<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Augmentation (emission, scattering)
dIaug = aug J ds<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
s
ds
I(s)
I(s+ds)
If no scattering, emission by thermal radiation
dIaug = ext�T 4
⇡= extB(T )
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Extinction (absorption, scattering)
dIext = �ext I ds<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Augmentation (emission, scattering)
dIaug = aug J ds<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
dI
ds= ext(�I +B)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
dI
ds= ext(�I +B)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
dI
ds= ext(�I +B)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
d⌧ = �ext ds<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical path length (dimensionless) τ
dI
ds= ext(�I +B)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
d⌧ = �ext ds<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical path length (dimensionless) τ
dI
d⌧= I �B
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Js (1-a)
Js (1-a)
Fdown(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Js (1-a)
Fdown(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fup(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Js (1-a)
Fdown(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fup(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⌧ =
Z 1
zext dz
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical path from atmosphere top
Atmosphere top τ = 0
Js (1-a)
Fdown(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fup(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⌧ =
Z 1
zext dz
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical path from atmosphere top
Atmosphere top τ = 0
Js (1-a) Fdown(0) = 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fdown(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fup(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⌧ =
Z 1
zext dz
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical path from atmosphere top
Ground τ = τ*
Atmosphere top τ = 0
Js (1-a) Fdown(0) = 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fdown(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fup(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⌧ =
Z 1
zext dz
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical path from atmosphere top
Ground τ = τ*
Atmosphere top τ = 0
Js (1-a) Fdown(0) = 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fdown(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fup(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⌧⇤ =
Z 1
0ext dz
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical depth of atmosphere
⌧ =
Z 1
zext dz
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical path from atmosphere top
Ground τ = τ*
Atmosphere top τ = 0
Js (1-a) Fdown(0) = 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fdown(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fup(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fup = Fup(⌧⇤) = ⇡B(Tg)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Radiation going up :
⌧⇤ =
Z 1
0ext dz
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical depth of atmosphere
⌧ =
Z 1
zext dz
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical path from atmosphere top
Ground τ = τ*
Atmosphere top τ = 0
Js (1-a) Fdown(0) = 0<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fdown(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fup(z)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Fup = Fup(⌧⇤) = ⇡B(Tg)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Radiation going up :
Jsa+ Fdown(⌧⇤)
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Radiation going down :
⌧⇤ =
Z 1
0ext dz
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical depth of atmosphere
⌧ =
Z 1
zext dz
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical path from atmosphere top
ext = 0 exp(�z/Hw)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Hw = 1.6 km (water vapor)
ext = 0 exp(�z/Hw)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Hw = 1.6 km (water vapor)
Ground Temp.
�T 4g = Js(1� a)
✓1 +
3⌧⇤
4
◆
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
ext = 0 exp(�z/Hw)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Hw = 1.6 km (water vapor)
Ground Temp.
�T 4g = Js(1� a)
✓1 +
3⌧⇤
4
◆
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⌧⇤ =
Z 1
0ext dz
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical depth of atmosphere
ext = 0 exp(�z/Hw)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Hw = 1.6 km (water vapor)
Ground Temp.
�T 4g = Js(1� a)
✓1 +
3⌧⇤
4
◆
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
⌧⇤ =
Z 1
0ext dz
<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Optical depth of atmosphere
ext = 0 exp(�z/Hw)<latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit><latexit sha1_base64="(null)">(null)</latexit>
Hw = 1.6 km (water vapor)
Steep temperature gradient leads to convective instability
390 W/m2