Radiation Heat Transfer
MEL 725
Power-Plant Steam Generators (3-0-0)
Dr. Prabal Talukdar
Assistant Professor
Department of Mechanical Engineering
IIT Delhi
Contents
• Basic definitions and laws with radiation –
1 class
• Concept of View factor, radiation
exchange between surfaces – 1 class
• Radiation from flame – 1-2 class
Thermal Radiation
• Radiation heat transfer can take place in a vacuum. It
does not need a medium unlike conduction/convection
• Thermal radiation is the stream of electromagnetic
radiation emitted by a material entity on account of its
finite absolute temperature
• Infrared radiation from a common household radiator or
electric heater is an example of thermal radiation, as is
the light emitted by a glowing incandescent light bulb.
• Thermal radiation is generated when heat from the
movement of charged particles within atoms is converted
to electromagnetic radiation
• Dominant in high temperature applications
Spectrum of Electro-magnetic
Radiation
Thermal radiation falls in the range of 10-1-102 µm of the
Electro-magnetic spectrum.
Emission Process
Emission by a surface
Gray
Diffuse
Solid angle
Solid angle
• dω= dAn/r2 = (r2 sinθ dθ dΦ)/r2 =sinθ dθ dΦ
Solid Angle for a Hemisphere
2π π / 2
π/ 2
∫ dw = ∫ ∫ sin θdθdφ = 2π ∫ sin θdθ = 2π sr
h
0
0
0
Spectral Intensity
• Iλ,e(λ,θ,φ)=dq/(dA1cos θ.dω.dλ)
•
Iλ,e is the rate at which radiant energy is emitted at the wave length λ
in the (θ, φ) direction, per unit area of emitting surface normal to this
direction, per unit solid angle about this direction and per unit
wavelength interval dλ about λ.
Heat Flux
•
dqλ=dq/dλ—rate at which radiation of wavelength λ leaves dA1 and
passes through dAn (unit: W/µm)
• dqλ= Iλ,e(λ,θ, φ) dA1cos θ dω dλ
• Spectral radiation flux associated with dA1is
dq "λ = I λ ,e (λ, θ, φ) cos θ sin θdθdφ
• Spectral heat flux associated with emission into hypothetical
hemisphere above dA1 is
2π π / 2
q "λ (λ ) =
∫ ∫I
0
λ ,e
(λ, θ, φ) cos θ sin θdθdφ
0
• Total heat flux associated with emissions in all directions and
at all wavelengths is then
∞
q " = ∫ q "λ (λ)dλ
0
Emissive Power
• Emissive power is the amount of radiation
emitted per unit surface area
• Spectral , hemispherical emissive power
2π π / 2
•
∫ ∫I
Eλ(λ)(W/m2.µm)=
0
λ ,e
(λ, θ, φ) cosθ sin θdθdφ
0
– Eλ →based on actual surface area
– Iλ,e →based on projected surface area
• Total hemispherical Emissive power:
∞
E
=
∫
0
E
λ
( λ ) d λ
Relation between Emissive Power
and Intensity
2π π / 2
• Eλ=
∫ ∫I
0
λ ,e
(λ , θ, φ) cos θ sin θdθdφ
0
• For a diffuse surface, Iλ(λ,θ,φ)= Iλ(λ)
Eλ =
2π
π/2
0
0
I λ ,e ( λ ) ∫
∫ cos θ sin θ d θ d φ
Eλ= π Iλ,e(λ)
Spectral basis
E=πIe
Total basis
Irradiation
• Intensity of incident radiation: It can be defined as
the rate which radiant energy of wavelength λ is
incident from the (θ,φ) direction, per unit area of
the intercepting surface normal to the direction,
per unit solid angle about this direction, and per
unit wavelength interval dλ about λ
• Radiation incident from all directions gives the
irradiation
2π π / 2
• Spectral irradiation Gλ(λ) = ∫ ∫ Iλ,i (λ, θ, φ) cosθ sin θdθdφ
∞
0
0
• Total irradiation G = ∫ G λ (λ)dλ W / m 2
0
Radiosity
• Radiosity accounts for all the radiant energy
leaving a surface
– Emitted and reflected part
∞
J=∫
0
2π π / 2
∫ ∫I
0
= πI e + r
0
λ ,e + r
(λ, θ, φ) cos θ sin θdθdφdλ
For a surface which is diffuse emitter
and diffuse reflector
Blackbody Radiation
• A blackbody absorbs all incident radiation,
regardless of wavelength and direction
• For a prescribed temperature and
wavelength, no surface can emit more
energy than a black body
• Although the radiation emitted by a
blackbody is a function of wavelength and
temperature, it is independent of direction.
That is blackbody is a diffuse emitter
Planck Distribution
• The spectral distribution of blackbody emission is
given by Planck as
2hc 02
I λ , b (λ, T ) = 5
λ [exp(hc 0 / λkT) − 1]
where, Planck constant h = 6.6256x10-34J.s and
Boltzmann constants k = 1.3805x10-23 J/K
speed of light in vacuum c0=2.998x108 m/s
C1
E λ ,b (λ, T ) = πI λ ,b (λ, T ) = 5
λ [exp(C 2 / λT ) − 1]
Planck Distribution
The emitted radiation varies continuously
with wavelength
•At any wavelength the magnitude of the
emitted radiation increases
with increasing temperature
A significant fraction of the radiation emitted
by the sun, which may be
approximated as a blackbody at 5800K, is
in the visible region of the spectrum.
In contrast, for T<800K, emission is
predominantly in the infrared region
of the spectrum and is not visible to the eye
The spectral radiation in which the radiation
is concentrated depends on temperature,
with comparatively more radiation appearing
at shorter wavelengths as the temperature
increases
Discussion
• The emitted radiation varies continuously
with wavelength
Wien’s Displacement Law
C1
E λ ,b (λ, T ) = πI λ ,b (λ, T ) = 5
λ [exp(C 2 / λT) − 1]
• Differentiating the above equation with respect to λ and
setting the result equal to zero, we get
λmaxT = 2897.8 µm.K
• According to this law, the maximum spectral emissive
power is displaced to shorter wavelengths with increasing
temperature
• The emission is in the middle of the visible spectrum (λ =
0.5 µm) for solar radiation, since the sun emits
approximately as a blackbody as 5800K
Stefan-Boltzmann Law
C1
E λ , b (λ , T ) = 5
λ [exp(C 2 / λT) − 1]
• Total Emissive power
∞
C1
Eb = ∫ 5
dλ
λ [exp(C 2 / λT) − 1]
0
E b = σT 4
σ = 5.67x10-8 W/m2K4
Since this emission is diffuse, the total intensity
associated with blackbody emission Ib=Eb/π
Surface Emission
• Emissivity is the ratio of radiation emitted by the
surface to the radiation emitted by a blackbody
at the same temperature
• Spectral directional emissivity ελ,θ(λ,θ,φ,T) of a
surface at the temperature T is the ratio of the
intensity of the radiation emitted at the
wavelength λ and in the direction of θ and φ to
the intensity of the radiation emitted by a
blackbody at the same values of T and λ.
I λ ,e (λ, θ, φ, T)
Hence
ε λ ,θ (λ, θ, φ, T) ≡
I λ , b (λ , T )
Blackbody and Real Emission
Emissivity
• Total directional emissivity ε θ (θ, φ, T) ≡
• Spectral hemispherical
emissivity
I λ ,e (θ, φ, T)
I b (T )
E λ (λ, T )
ε λ (λ , T ) ≡
E λ , b (λ , T )
2 ππ / 2
=
∫ ∫I
λ ,e
(λ, θ, φ, T) cos θ sin θdθdφ
0 0
2 ππ / 2
∫ ∫I
λ ,b
(λ, T ) cos θ sin θdθdφ
0 0
π/ 2
= 2 ∫ ε λ ,θ (λ, θ, T) cos θ sin θdθ
0
Total Hemispherical Emissivity
• Total
hemispherical
emissivity
• Directional
distributions of
total diretctional
Emisivity
E (T )
ε (T ) =
E b (T )
Absorption, Reflection and
Transmission
• Gλ = Gλ,ref+ G λ,abs+ Gλ,tr
• When a surface is opaque, Gλ,tr = 0
• There is no net effect of the reflection
process on the medium, while absorption
has the effect of increasing the internal
energy of the medium
• Surface absorption and reflection are
responsible for our perception of color
Absorptivity
• The absorptivity is a property that determines the
fraction of the irradiation absorbed by a surface
G abs
α=
G
• Surface exhibits selective absorption with
respect to the wavelength and direction of the
incident radiation
• Does not depend much on surface temperature
Reflectivity
• It is a property that determines the fraction of the
incident radiation reflected by a surface
G abs
α=
G
• This property is inherently bi-directional
• In addition to depending on the direction of the
incident radiation, it also depends on the
direction of the reflected radiation
• Surface may be idealized as diffuse or specular
Transmissivity
• Total hemispherical transmissivity
G tr
α=
G
• For a semitransparent medium, ρ+α+τ = 1
Kirchoff’s law
• Consider a large isothermal surface of
surface temperature Ts within which
several small bodies are cofined
Ts
G
A1
A2
E1
E2
A3
E3
Kirchoff’s Law (cont’d)
• Regardless of its radiative properties, such a
surface forms a blackbody cavity
• Accordingly, regardless of its orientation, the
irradiation experienced by any body in the cavity
is diffuse and equal to emission from a
blackbody at Ts i.e. G = Eb(Ts)
• Under steady-state conditions,
T1 = T2 = T3 =---- = Ts
and net energy transfer to each surface is zero
Kirchoff’s Law (cont’d)
• Applying an energy balance to a control surface
about body 1,
α1GA1 – E1(Ts)A1 = 0
E1(Ts)/ α1= Eb(Ts)
Applying to all bodies,
E1(Ts)/ α1 = E1(Ts)/ α1= ------- = Eb(Ts)
ε1 ε 2
=
= − − −− = 1
α1 α 2
ε=α