Radiation Heat Transfer - MECH14 -...

Radiation Heat Transfer 1

Radiation Heat Transfer

Subhransu Roy

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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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Electromagnetic Wave Spectrum

Colour of light depends on ν

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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 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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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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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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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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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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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


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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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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f(nλT ) =

Fraction of radiation energy in

the range 0 : λT

i.e.,∫ λ

0Ebλdλ/

∫∞

0Ebλdλ

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Radiative Properties

reflectivity, ρ ≡reflected part of incoming radiation

total incoming radiation

absorptivity, α ≡absorbed part of incoming radiation


transmissivity, α ≡transmitted part of incoming radiation


for an opaque surface, i.e. τ = 0, ρ+ α = 1

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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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Fig: Normal spectral emissivity for selected

materials

Fig: Directional variation of surface emissivities

for several metals and non-metals

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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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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 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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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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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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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 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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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 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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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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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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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 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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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 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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Absorption and Scattering

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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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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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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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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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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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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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Radiative Behaviour of Clean Gas

Fig: spectral absorptivity of an isothermal mixture of N2 and CO2

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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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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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(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,


∫ τη

0

ωη4πGη(τ

′

η)e−(τη−τ

′

η) dτ ′η

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(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:


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:


1

4π

∫ τη

0

Gη(τ′

η)e−(τη−τ

′

η) dτ ′η

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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 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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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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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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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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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

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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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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)]

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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 ′]

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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 ′

]

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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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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

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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


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

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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.

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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

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thank you

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