Radiation Heat Transfer - MECH14 -...
Transcript of Radiation Heat Transfer - MECH14 -...
Radiation Heat Transfer 1
Radiation Heat Transfer
Subhransu Roy
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Radiation Heat Transfer 2
Thermal Radiation – EM Wave
Wave character
• consisting of electromagnetic waves (EM waves), or
• consisting of photons
Each electromagnetic wave is identified either by its
• frequency, ν, (Hz), or
• wavelength, λ (µm), or
• wavenumber, η (cm1)
They are related by,
ν =c0nλ
=c0nη
where c0: velocity of light in vacuum (n = 1) and n: refractive index
Radiation (ν, λ, η) in medium with refractive index n becomes (ν,nλ,η/n) in vacuum
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Radiation Heat Transfer 3
Electromagnetic Wave Spectrum
Colour of light depends on ν
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Radiation Heat Transfer 4
Interaction with matter
Volume phenomenon
for gases & semitransparent liquid/solid
Surface phenomenon
for emission from solid or liquid surfaces
spectral distribution directional distribution
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Radiation Heat Transfer 5
Radiation Terminology
directional : evaluated in one direction
hemispherical : integrated over all directions in half-sphere
spectral : at one wavelength
total : integrated over all wavelengths
gray : independent of wavelength
diffuse : independent of direction
black : independent of direction & wavelength
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Radiation Heat Transfer 6
Comparison between radiation, conduction and convection
Attribute Conduction Convection Radiation
Transfer medium Yes Yes No
T dependence Linear Linear T 4
Volume for energy balance Infinitesimal Infinitesimal Entire volume under consideration
Type of equation Differential Differential Integral
Note:
1. Mean free path of molecular collision during conduction is 10−10m
2. Mean free path of photon collision during radiation is 10−8m for metals or 1010m for sun to earth
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Radiation Heat Transfer 7
Radiative Intensity
spectral intensity, Iλ ≡ radiative energy flow / time / area
normal to rays / solid angle / wavelength
total intensity, I ≡ radiative energy flow / time / area
normal to rays / solid angle
Projected Area
Ap = so · no dA
Solid Angle
dΩ = sin θ dψ dθ
concept of solid angle based on unit hemisphere
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Radiation Heat Transfer 8
Radiative Heat Flux
Radiative heat flux into the surface :(dQλ)in
dA= (qλ)in =
∫I
Iλ(si)si · ni dΩi
Radiative heat flux from surface :(dQλ)out
dA= (qλ)out =
∫
I
Iλ(so)so · no dΩo
Net heat flun in s - direction :
(qλ)net = qλ · n =
∫
4π
Iλ(s)s · n dΩ
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Radiation Heat Transfer 9
Surface Radiation
Emissive power : Eλ = (qλ)out =1
dA
∫
I
Iλ(so) so · no dA︸ ︷︷ ︸
projected
area
dΩ︸︷︷︸
solid
angle
for a diffuse surface : Eλ = πIλ,out
and, E = πIout
Irradiation, Hλ = (qλ)in =1
dA
∫I
Iλ(si) si · ni dA︸ ︷︷ ︸
projected
area
dΩ︸︷︷︸
solid
angle
for diffuse irradiation : Hλ = π Iλ,in
and, H = π Iin
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Radiation Heat Transfer 10
Calculation of Radiation Intensity & Emission
The sun having radius,Rs, and blackbody temperature Ts gives sunshine to earth from a distance of
SES .
The total heat rate leaving the sun is Qs = 4πR2sEb(Ts).
The solar irradiation on earth
Hearth =4πR2
sEb(Ts)
4πS2ES
=Ebπ
πR2s
S2ES
= Ib(Ts) ΩES =
∫
I
I(si) si · ni dΩ
At any distance S from the sun, the irradiation will be
H(S) = Ib(Ts)πR2
s
S2= Ib(Ts) ΩS
We can clearly see that the intensity always remains Ib(Ts)mech14.weebly.com
Radiation Heat Transfer 11
Blackbody Radiation - An Ideal Reference
• GRAY : A black body absorbs all incident radiation, regardless of wavelength and direction.
• PERFECT EMITTER: For a prescribed temperature and wavelength, no surface can emit more
energy than a blackbody.
• DIFFUSE : Although the radiation emitted by a blackbody is a function of wavelength and temperature,
it is independent of direction.
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Radiation Heat Transfer 12
Blackbody Radiation Spectrum
Fig: Plank distribution
Ebν dν = Ebλ dλ
Ebν(T, ν) =2πhν3n2
C0
[exp( hνkT )− 1
]W
m2Hz
Ebλ(T, λ) =2πhC2
0
n2λ5[exp( hC0
nkT )− 1]
W
m2 µm
Eb = n2σT 4Stefan-Boltzmann Law
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Radiation Heat Transfer 13
f(nλT ) =
Fraction of radiation energy in
the range 0 : λT
i.e.,∫ λ
0Ebλdλ/
∫∞
0Ebλdλ
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Radiation Heat Transfer 14
Radiative Properties
reflectivity, ρ ≡reflected part of incoming radiation
total incoming radiation
absorptivity, α ≡absorbed part of incoming radiation
total incoming radiation
transmissivity, α ≡transmitted part of incoming radiation
total incoming radiation
for an opaque surface, i.e. τ = 0, ρ+ α = 1
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Radiation Heat Transfer 15
emisivity(T ), ǫ ≡energy emitted from a surface at T
energy emitted by a black surface at T
spectral directional emisivity, ǫ′λ =Iλ(T, λ, so)
Ibλ(T, λ)
spectral directional absorptivity, α′
λ =H ′
λ,abs(λ, si)
H ′
λ(λ, si)
α′
λ(T, λ, si) = ǫ′λ(T, λ, so)
only for the special case of gray and diffuse irradiation, α = ǫ
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Radiation Heat Transfer 16
Fig: Normal spectral emissivity for selected
materials
Fig: Directional variation of surface emissivities
for several metals and non-metals
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Radiation Heat Transfer 17
Calculation of absorptivity
αλ =Hλabs
Hλ=
∫
I ǫλIλ(λ, Tsource, si)si · n dΩi dA∫
I Iλ(λ, Tsource, si)si · n dΩi dA
α =
∫ ∞
0
αλ dλ =Habs
H=
∫∞
0αλHλdλ
∫∞
0Hλdλ
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Radiation Heat Transfer 18
Radiative Exchange Between Two Surfaces — View Factor Calculation
dFdAi−dAj≡
diffuse energy leaving dAi intercepted by dAjdiffuse energy leaving dAi
=I cos θi dAi dΩ
∫
I I cos θi dAi dΩ
Based on the assumption that intensity leaving Ai (Radiosity, J ) does not vary across the surface, the
view factor between finite surfaces:
FAi−Aj=
1
Ai
∫
Ai
∫
Aj
cos θi cos θjπS2
dAj dAi
FAi−AjAi = FAj−Ai
Aj
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Radiation Heat Transfer 19
Radiation view factors can be determined by:
1. Direct integration:
• Surface integration by analytical or numerical method
• Contour integration by analytical or numerical method
2. Statistical determination:
• Monte Carlo method
3. Special methods:
• view factor algebra
• crossed-strings method
• inside sphere method
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Radiation Heat Transfer 20
View Factor Calculation – Algebraic Method
AiFij = AjFjiN∑
j=1
Fij = 1
(A1 +A2)F(1+2)−3 = A3F3−(1+2) = A3(F3−1 + F3−2)
= A1F1−3 +A2F2−3
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Radiation Heat Transfer 21
View Factor Calculations – Crossed-Strings Method
This is useful for arbitrary two-dimensional configurations
The ends of the two surfaces are indicated by labels A1 : a, b and A2 : c, d
Connect surface points a, b with tight string and also surface points c, d with tight string
Connect end points a, c with tight string and points b, d with tight string to form the sides of the rectangle
abdc
Connect diagonals a, d with string and diagonals b, c with tight string
FA1−A2=
(Abc +Aad)− (Aac +Abd)
2A1
=diagonals − sides
2× originating area
Note that A1 6= Aab
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Radiation Heat Transfer 22
View Factor calculation – Inside Sphere Method
θ1 = θ2
R
R
θ2
θ1
A2
A1
Fd1−2 =
∫
A2
cos θ1 cos θ2π(2R cos θ)2
dA2
=
∫
A2
dA2
4πR2
=A2
Asphere
i.,e., independent of dA1
∴ F1−2 =A2
Asphere
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Radiation Heat Transfer 23
Radiation Exchange Between Diffuse Gray Surfaces
Surface Radiosity, J = The total heat flux leaving a surface per unit area.
J = ǫEb︸︷︷︸
emission
+ ρH︸︷︷︸
reflection
since, both emission and reflection are diffuse the resulting intensity leaving the surface:
I(so) = J/π
If the temperature of the emitting surface is uniform and irradiation (H) on the surface is uniform then the
radiosity (J ) will be uniform over the surface.
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Radiation Heat Transfer 24
Fig: Qualitative spectral behaviour of radiosity for irradiation from an isothermal source
A gray surface being “colour blind” cannot distinguish between reflected and directly emitted component of
the irradiation from a heat source.
It treats all irradiation as coming from a surface with an effective emissive power J .
Therefore radiation analysis only requires balancing the net outgoing radiation J travelling directly
between surfaces.
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Radiation Heat Transfer 25
Radiation Exchange Between Diffuse Gray Surfaces in Enclosure
Net heat flux input (q) : q = ǫEb − αH = J −H
Elliminating irradiation (H) : q − αq = (ǫEb − αH)− α(J −H) = ǫEb − αJ
For Diffuse gray surface α = ǫ : q =ǫ
1− ǫ(Eb − J)
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Radiation Heat Transfer 26
Consider an enclosure containing i = 1 to N gray diffuse surfaces with uniform radiosity
qiAi =Aiǫi1− ǫi
[Ebi − Ji] (1)
JiAi = qiAi︸︷︷︸
heat in
+N∑
j=1
Jj(Fi−jAi)
︸ ︷︷ ︸
irradiation Hji
+HoiAi︸ ︷︷ ︸
outside irr.
(2)
JiAi = ǫiEbiAi︸ ︷︷ ︸
emission
+(1− ǫi)
N∑
j=1
JjFi−jAi
︸ ︷︷ ︸
reflection component (αHji)
+(1− ǫi)HoiAi (3)
if Ebi(Ti) is known then equation (3) is used and if qi is known then equation (2) is used to solve for Ji
for each surface in the enclosure.
then Eq.(1) is used to find the unknownEbi(Ti) or qi
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Radiation Heat Transfer 27
The following set of equations that do not contain Ji can also be solved directly
Ebi =1
ǫiqi +
N∑
j=1
(
Ebj −1− ǫjǫj
qj
)
Fi−j +Hoi (4)
N∑
j=1
(1− ǫjǫj
qj
)
Fi−j −1
ǫiqi =
N∑
j=1
(EbjFi−j)− Ebi +Hoi (5)
[Cij ]qi = [Dij ]Ebi+Hoi (6)
If temperature is known then Eb is known and q is solved for. Otherwise we solve for Eb.
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Radiation Heat Transfer 28
For building network the following equation is used
qiAi =
N∑
j=1
Ji(Fi−jAi)−
N∑
j=1
Jj(Fi−jAi) +HoiAi
Qi = qiAi =
N∑
j=1
[Ji − Jj ]
(Fi−jAi)−1
︸ ︷︷ ︸
Qij
+HoiAi =[Ebi − Ji]
1−ǫiǫiAi
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Radiation Heat Transfer 29
Radiation exchange – very small to large enclosing surface
A1: small surface,A2: very large surface enclosingA1 such thatA1/A2 ≪ 1.
Therefore F12 = 1
q1A1 =Eb1 − Eb2
1− ǫ1A1ǫ1
+1
A1F12+
1− ǫ2A2ǫ2
= −q2A2
=A1(Eb1 − Eb2)
1− ǫ1ǫ1
+ 1 +A1
A2
(1− ǫ2ǫ2
)
q1A1 = A1ǫ1(Eb1 − Eb2)
A1ǫ1(Eb1 − Eb,sky)
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Radiation Heat Transfer 30
Surface Radiation Transport – Integral Equation
x2, ξ2 = x2/h
y φ
dx1
dx2
h
w
A2: T,ǫ
A1: T,ǫx1, ξ1 =
x1
h
J = ǫEb︸︷︷︸
emission
+ ρH︸︷︷︸
reflection
and q =ǫ
1− ǫ[Eb − J ]
J(r) = ǫ(r)Eb(r) + ρ(r)
[∫
A
J(r′)dFdA−dA′ +H0(r)
]
J1(x1) = ǫσT 4 + (1− ǫ)
[∫ w
0
J2(x2)dFdA1−dA2
]
J2(x2) = ǫσT 4 + (1− ǫ)
[∫ w
0
J1(x1)dFdA2−dA1
]
dx1dFd1−d2 = dx2dFd2−d1 =1
2cosφ dφ dx1 =
1
2
h2 dx1 dx2
[h2 + (x1 − x2)2]3/2
if J =J
σT 4; W =
w
hthen J (ξ) = ǫ+ (1− ǫ)
[∫ W
0
J (ξ′)dξ′
[1− (ξ′ − ξ)2]3/2
]
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Radiation Heat Transfer 31
Radiation in Participating Media
enclosure filled with
• absorbing gas or semitransparent solid or liquid
• absorbing, scattering particles (bubbles)
Radiative Intensity in Participating Medium
1. As radiation travels at the speed of light, for most engineering problems radiative energy can be
assumed to arrive “instantaneously ” every where in the medium.
2. If the medium is non-participating then the radiation intensity will be constant along its path.
3. This property makes radiation intensity a most suitable quantity for the description of absorption,
emission and scattering of energy within a medium, because any changes in intensity along its path
must be due to these phenomena
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Radiation Heat Transfer 32
Absorption and Scattering
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Radiation Heat Transfer 33
Attenuation of Intensity by Absorption
Absorption over distance ds
(dIη)abs = −κηIη ds︸︷︷︸
=dX
= −κρηIη (ρ ds)︸ ︷︷ ︸
=dX
= −κpηIη (p ds)︸ ︷︷ ︸
=dX
κη : (linear) absorption coefficient
κρη : mass absorption coefficient
κpη : pressure absorption coefficient
integrating, Iη(s) = Iη(0) exp(−
∫ s
0
κη ds)
τ =∫ s
0κη ds is the optical thickness through which the beam has travelled, and Iη(0) is the intensity
entering the medium at s = 0. For gas layer of thickness s,
spectral absorptivity, αη =Iη(0)− Iη(s)
Iη(0)= 1− e−τη
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Radiation Heat Transfer 34
Attenuation of Intensity by Scattering
Scattering over distance ds:
(dIη)sca = −σsηIη ds = −σsρηIη (ρds) = −σspηIη (pds)
σsη : (linear) scattering coefficient
σsρη : mass scattering coefficient
σspη : pressure scattering coefficient
Total attenuation by absorption and scattering:
extinction coefficient, βη = κη + σsη
and, optical distance = τη =
∫ s
0
βη ds
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Radiation Heat Transfer 35
Augmentation of Intensity by Emission
Emission over distance ds:
(dIη)em = κηIbη ds
net emissiondIηs
= κη(Ibη − Iη)
Iη(s) = Iη(0)e−τη + Ibη(1− e−τη), τη =
∫ s
0
κη ds
if only net emission is considered i.e., Iη(0) = 0
emissivity ǫη =Iη(s)
Ibη= 1− e−τη = αη absorptivity
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Radiation Heat Transfer 36
Augmentation of Intensity by In-scattering
The total amount of energy scattered away from si is
σsηIη(si)(dAsi · s) dΩi dη
(ds
si · s
)
= σsηIη(si)dAdΩi ds.
Of this, the fractionΦη(si, s)dΩ
4πis scattered into the cone Ωaround s-direction.
The scattering function Φη(si, s) is the probability that a ray in one direction, si, will be scattered into a
certain direction, s. Therefore,∮
Φη(si, s)
4πdΩ = 1
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Radiation Heat Transfer 37
Equation of Radiative Transport
Attenuation :
Absorption : (dIη)abs = −κηIηds
Out-Scattering : (dIη)sca = −σsηIηds
Augmentation :
Emission : (dIη)em = κηIbηds
In-Scattering : (dIη)sca(s) = (σsη/4π)∫
4πIη(si)Φη(si, s) dΩi
∂Iη∂s
= κηIbη − κηIη − σsηIη +σsη4π
∫
4π
Iη(si)Φη(si, s) dΩi
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Radiation Heat Transfer 38
Introducing single scattering albedo
single scattering albedo : ωη ≡ση
κη + σsη=σsηβη
and, optical coordinates, τη =
∫ s
0
βη ds
leads to,∂Iη∂τη
= −Iη + (1− ωη)Ibη +ωη4π
∫
4π
Iη(si)Φη(si, s) dΩi︸ ︷︷ ︸
source function Sη(τη,s)
assumes a simple form∂Iη∂τη
+ Iη = Sη(τη, s)
The above equation is an integro-differential equation (in space, and in two directional coordinates with
local origin). Furthermore, the Plank function Ibη is generally not known and must be found by
considering the overall energy equation (adding derivatives in three space coordinates and integrations
over two more directional coordinates and η).
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Radiation Heat Transfer 39
Radiative Behaviour of Clean Gas
Fig: spectral absorptivity of an isothermal mixture of N2 and CO2
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Radiation Heat Transfer 40
Radiative Properties of Particles
size parameter x =2πa
λwhere a is the effective radius of particles.
There are three distinct behaviour regimes:
x≪ 1 : Rayleigh scattering, σs ∝ ν4 or 1/λ4
x = O(1) : Mie scattering
x≫ 1 : Geometric optics
Fig: polar plot of phase function
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Radiation Heat Transfer 41
Some Simplifications to Equation of Transfer
(a) Non-scattering medium σsη = 0, ωη=0
If the medium only absorbs and emits, the source function reduces to the local black body intensity, and
Iη(τη) = Iη(0)e−τη +
∫ τη
0
Ibη(τ′
η)e−(τη−τ
′
η) dτ ′η
This equation is an explicit expression for the radiation intensity if the temperature field is known. However,
generally the temperature is not known and must be found in conjunction with the overall conservation of
energy.
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Radiation Heat Transfer 42
(b) Cold medium
If the temperature of the medium is so low that the blackbody intensity at that temperature is small as
compared with the incident intensity, then the equation of transfer is decoupled from other modes of heat
transfer. However, the governing equation remains a third-order integral equation, namely,
Iη(τη, s) = Iη(0)e−τη +
∫ τη
0
ωη4π
∫
4π
Iη(τ′
η, si)Φη(si, s) dΩie−(τη−τ
′
η) dτ ′η
For isotropic scattering or Φ ≡ 1,
Iη(τη, s) = Iη(0)e−τη +
∫ τη
0
ωη4πGη(τ
′
η)e−(τη−τ
′
η) dτ ′η
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Radiation Heat Transfer 43
(c) Purely scattering medium
If the medium scatters radiation, but does not absorb or emit, then the radiative transfer is again
decoupled from other heat transfer modes. In this case ωη ≡ σsη/(κη + σsη) ≡ 1, and the equation of
transfer reduces to:
Iη(τη, s) = Iη(0)e−τη +
1
4π
∫ τη
0
∫
4π
Iη(τ′
η, si)Φη(si, s) dΩie−(τη−τ
′
η) dτ ′η
Again, for isotropic scattering it is further simplified to:
Iη(τη, s) = Iη(0)e−τη +
1
4π
∫ τη
0
Gη(τ′
η)e−(τη−τ
′
η) dτ ′η
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Radiation Heat Transfer 44
Boundary Conditions for Equation of transfer
Diffusely emitting and reflecting opaque surface
Iη(s) = Iη =Jηπ
=ǫηEηbπ
+ρηHη
π
= ǫηIηb(T ) +ρηπ
∫
I
Iη(s′)(s′ · n) dΩ′
︸ ︷︷ ︸
irradiation
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Radiation Heat Transfer 45
Radiation Intensity I(τ, θ) inside a grey participating medium slab of optical thickness τL bounded by
black diffuse isothermal plate.
τs
τ
I+(τ, θ)
τ ′
s
τ ′
θ
θ
Tw = T1
Tw = T2
I−(τ, θ)
τL
black, diffuse, isothermal plate
Grey medium
0 < θ < π/2
π/2 < θ < π
Iw = I1
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Radiation Heat Transfer 46
Radiative Heat Flux
For a hypothetical (i.e., totally transmissive) surface element placed arbitraryly inside an enclosure, the
spectral radiative heat flux on to the surface element:
qη · n =
∫
4π
Iη s · n dΩ
qη = qxη i+ qyη j + qzηk =
∫
4π
Iη(s)s dΩ
the total heat flux vector q =
∫ ∞
0
qη dη =
∫ ∞
0
∫
4π
Iη(s)s dΩ dη
Incident Radiation Function
total intensity impinging on a point from all directions:Gη ≡
∫
4π
Iη(s) dΩ
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Radiation Heat Transfer 47
Divergence of Radiative Heat Flux
for infinitesimal pencil of rays
∂Iη∂s
= s · ∇Iη κηIbη − βηIη(s) +σsη4π
∫
4π
Iη(si)Φη(si, s) dΩi
integrated over all directions
∇ ·
∫
4π
Iη(s)s dΩ = 4πκηIbη − βη
∫
4π
Iη(s) dΩ
+σsη4π
∫
4π
Iη(si)
(∫
4π
Φη(si, s) dΩ
)
︸ ︷︷ ︸
=4π
dΩi
∇ · qη = κη
(
4πIbη −
∫
4π
Iη(s) dΩ
)
= κη(4πIbη −Gη)
integrating over all wavelengths
∇ · q = ∇ ·
∫ ∞
0
qη dη =
∫ ∞
0
κη(4πIbη −Gη) dη
for a gray medium (κη =constant)
∇ · q = κ(4n2σT 4 −G)
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Radiation Heat Transfer 48
Isothermal Sphere (non-scattering)
Iη(τR, θ) =
∫ 2τR cos θ
0
Ibη(τ′
η)e−(τη−τ
′
η) dτ ′η
= Ibη(1− e−2τηR cos θ
)for constant Ibη
Knowing that e−x = 1− x/1! + x2/2!− x3/3! + ...
for τηR ≪ 1 : Ibη(1− e−2τηR cos θ
)= Ibη2τηR cos θ
For gray body
q =
∫ 2π
0
∫ π/2
0
Ib2τR cos θ cos θ sin θ dθ dψ =4π
3τRIb =
4
3τRn
2σT 4
Q = 4πR2q = 4κn2σT 4
(4
3πR3
)
= 4κn2σT 4Volume
for τR ≫ 1 : Ibη(1− e−2τR cos θ
)= Ibη
q = πIb = n2σT 4black body and Q = n2σT 4(4πR2)
2τR cos θ θτR
n
s
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Radiation Heat Transfer 49
τs =τ
cos θ=τ
µ
For grey non-scattering medium
S(τ ′s, θ) = Ib(τs)
For grey, Ψ = 1 and ∇ · qrad = 0
S(τ ′s, θ) = (1− ω)Ib(τs) +ω
4πG
G = 4πIb
S(τ ′s, θ) = = Ib(τs)
τs
τ
I+(τ, θ)
τ ′
s
τ ′
θ
θ
Tw = T1
Tw = T2
I−(τ, θ)
τL
black, diffuse, isothermal plate
Grey medium
0 < θ < π/2
π/2 < θ < π
Iw = I1
I(τs, θ) = Iwe−τs +
∫ τs
0
S(τ ′s, θ)e−(τs−τ
′
s)dτ ′s
S(τ ′s, θ) = (1− ω)Ib(τs) +ω
4π
∫ 2π
ψi=0
∫ π
θi=0
I(τs, θ)Φ(θ, ψ, θi, ψi) sin θidθidψi
mech14.weebly.com
Radiation Heat Transfer 50
The equation for radiative transfer in a isotropically scattering and grey plane parallel medium of optical
thickness τL bounded by gray diffuse isothermal flat plates 1 & 2 is
intensity I+(τ, µ) = I1(µ)e−τ/µ +
∫ τ
0
S(τ ′, µ)e−(τ−τ ′)/µ dτ′
µ,
0 < µ < 1
τ > τ ′
I−(τ, µ) = I2(µ)e(τL−τ)/µ −
∫ τL
τ
S(τ ′, µ)e(τ′−τ)/µ dτ
′
µ,
−1 < µ < 0
τ ′ > τ
G(τ) =
∫ 2π
ψ=0
∫ π
θ=0
I(τ, θ) sin θdθdψ = −
∫ 2π
ψ=0
∫ −1
cos θ=1
I(τ, θ)d(cos θ)dψ
= 2π
[∫ 0
−1
I−(τ, µ)dµ+
∫ 1
0
I+(τ, µ)dµ
]
q(τ) =
∫ 2π
ψ=0
∫ π
θ=0
I(τ, θ) cos θ sin θdθdψ = −
∫ 2π
ψ=0
∫ −1
cos θ=1
I(τ, θ) cos θ d(cos θ)dψ
= 2π
[∫ 0
−1
I−(τ, µ)µ dµ+
∫ 1
0
I+(τ, µ)µ dµ
]
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Radiation Heat Transfer 51
We introduce Elliptic integrals of order n as:
En(x) =
∫ ∞
1
e−xtdt
tn=
∫ 1
0
e−x/µ µn−2dµ
En(0) =1
n− 1d
dxEn(x) = −En−1(x)
En(x) =
∫ ∞
x
En−1(x) dx
En+1(x) =1
n[e−x − xEn(x)]
mech14.weebly.com
Radiation Heat Transfer 52
For S(τ, θ) = Ib and black plates with intensities Ib1 and Ib2 we get
G(τ) = 2π
[
Ib1E2(τ) + Ib2E2(τL − τ)
+
∫ τ
0
Ib(τ′)E1(τ − τ ′)dτ ′ +
∫ τL
τ
Ib(τ′)E1(τ
′ − τ)dτ ′]
q(τ) = 2π
[
Ib1E3(τ)− Ib2E3(τL − τ)
+
∫ τ
0
Ib(τ′)E2(τ − τ ′)dτ ′ −
∫ τL
τ
Ib(τ′)E2(τ
′ − τ)dτ ′]
mech14.weebly.com
Radiation Heat Transfer 53
For grey medium Ib = n2σT 4/π and in radiative equllibrium dq/dτ = 0, i.e. q is constant over the
thickness of the medium. Also G = 4πIb = 4n2σT 4. Therefore
Ib(τ) =1
2
[
Ib1E2(τ) + Ib2E2(τL − τ) +
∫ τL
0
Ib(τ′)E1(‖τ
′ − τ‖)dτ ′]
T 4 =1
2
[
T 41E2(τ) + T 4
2E2(τL − τ) +
∫ τL
0
T 4(τ ′)E1(‖τ′ − τ‖)dτ ′
]
q(τ) = 2n2σ
[
T 41E3(τ)− T 4
2E3(τL − τ)
+
∫ τ
0
T 4(τ ′)E2(τ′ − τ)dτ ′ −
∫ τL
τ
T 4(τ ′)E2(τ − τ ′)dτ ′]
q(τ) = q(0) = 2n2σ
[1
2T 41 − T 4
2E3(τL)−
∫ τL
0
T 4(τ ′)E2(τ′)dτ ′
]
mech14.weebly.com
Radiation Heat Transfer 54
Φb(τ) =T 4(τ)− T 4
2
T 41 − T 4
2
=1
2
[
E2(τ) +
∫ τL
0
Φb(τ′)E1(|τ − τ ′|)dτ ′
]
Ψb(τ) =q
n2σ(T 41 − T 4
2 )= 2
[
E3(τ) +
∫ τ
0
Φb(τ′)E2(τ − τ ′)dτ ′
−
∫ τL
τ
Φb(τ′)E2(τ
′ − τ)dτ ′]
= 1− 2
∫ τL
0
Φb(τ′)E2(τ
′)dτ ′
τL 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0
Ψb 1.0000 0.9157 0.8491 0.7934 0.7458 0.6672 0.6046 0.6046 0.5532
τL 1.5 2.0 2.5 3.0 5.0
Ψb 0.4572 0.3900 0.3401 0.3016 0.2077
for τL ≫ 1, Ψb =4/3
1.42089 + τL
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Radiation Heat Transfer 55
Radiative Properties of Molecular Gas
Emission or absorption of photon goes hand in hand with the change of rotational and/or vibrational
energy levels in molecules in the gas.
Rotational vibrational degrees of freedom
mech14.weebly.com
Radiation Heat Transfer 56
Rotational Transition: (rigid rod, no stretching, ∆j =
±1 or 0)
Vibrational Transitions: (Stretching or vibrations,
∆v = ±1,±2...)
Allowed transitions ∆v = ±1,∆j = ±1, 0 leads
to three branches:
P-branch (∆j = −1
R-branch (∆j = 0
Q-branch (∆j = 1
Typical spectrum of vibration rotation bands
For CO ∆v = ±1: ηo = 2143 cm−1 and ∆v = ±2: ηo = 4260 cm−1mech14.weebly.com
Radiation Heat Transfer 57
Line Radiation
Mechanism of broadening of spectral lines
collisionbroadening Lorentz profile
Dopplerbroadening Doppler profile
S ≡
∫
∆η−line
κη dη line-integrated absorption coefficient or line strength
mech14.weebly.com
Radiation Heat Transfer 58
for Lorentz profile:
κη =S
π
bL(η − η0)2 + b2L
bL = line half-width
bLbL0
=p
p0
√
T0T
p = total gas pressure, 0 = reference state
must use effective pressure = a function of partial pressure of radiating gases and non-radiating gas
species.
mech14.weebly.com
Radiation Heat Transfer 59
Radiation From an Isolated Line
•for an absorbing-emitting (not scattering) gas∂Iη∂s = κη(Ibη − Iη)
Iη(X) = Iη(0)e−κηX
︸ ︷︷ ︸
incoming intensity
+ Ibη(1− e−κηX)︸ ︷︷ ︸
emission andself−absorption
where,X = L or ρL or pL for linear, mass or pressure κη
difference between exiting and incoming intensities∫
∆η−line
[Iη(X)− Iη(0)] dη
︸ ︷︷ ︸
net emission
≈ [Ibη − Iη(0)]︸ ︷︷ ︸
max. net emissionevaluated at η0
∫
∆η−line
(1− e−κηX) dη
︸ ︷︷ ︸
line emissivity
for a Lorentz line
∫
∆η−line
(1− e−κηX) dη = SXe−x[I0(x) + I1(x)], where x =SX
2πbL
and I0(x) &I1(x) are modified Bessel functions
NOTE: τη =
∫X
0κη dX = κη X for an isothermal gas
mech14.weebly.com
Radiation Heat Transfer 60
thank you
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