Thermal Radiation-I - Basic properties and Laws

Lectures on Heat Transfer –
RADIATION-I:
Basic Properties and Laws
by
Dr. M. ThirumaleshwarDr. M. Thirumaleshwar
formerly:
Professor, Dept. of Mechanical Engineering,
St. Joseph Engg. College, Vamanjoor,
Mangalore
India

Preface:
• This file contains slides on RADIATION-I:
Basic Properties and Laws.
• The slides were prepared while teaching
Heat Transfer course to the M.Tech.
students in Mechanical Engineering Dept.
of St. Joseph Engineering College,
Vamanjoor, Mangalore, India, during Sept.
– Dec. 2010.
Aug. 2016 2MT/SJEC/M.Tech.

• It is hoped that these Slides will be useful
to teachers, students, researchers and
professionals working in this field.
• For students, it should be particularly
useful to study, quickly review the subject,useful to study, quickly review the subject,
and to prepare for the examinations.
•
Aug. 2016 3MT/SJEC/M.Tech.

References:
• 1. M. Thirumaleshwar: Fundamentals of Heat &
Mass Transfer, Pearson Edu., 2006
• https://books.google.co.in/books?id=b2238B-
AsqcC&printsec=frontcover&source=gbs_atb#v=onepage&q&f=false
• 2. Cengel Y. A. Heat Transfer: A Practical
Approach, 2nd Ed. McGraw Hill Co., 2003
Aug. 2016 MT/SJEC/M.Tech. 4
Approach, 2nd Ed. McGraw Hill Co., 2003
• 3. Cengel, Y. A. and Ghajar, A. J., Heat and
Mass Transfer - Fundamentals and Applications,
5th Ed., McGraw-Hill, New York, NY, 2014.

References… contd.
• 4. Incropera , Dewitt, Bergman, Lavine:
Fundamentals of Heat and Mass Transfer, 6th
Ed., Wiley Intl.
• 5. M. Thirumaleshwar: Software Solutions to• 5. M. Thirumaleshwar: Software Solutions to
Problems on Heat Transfer – Radiation-Part-I,
Bookboon, 2013
• http://bookboon.com/en/software-solutions-heat-transfer-
radiation-i-ebook
Aug. 2016 MT/SJEC/M.Tech. 5

Thermal Radiation – I
Basic properties and Laws:
Outline…
• Introduction – Applications –
Electromagnetic spectrum – Properties
Nov.2010 MT/SJEC/M.Tech. 6
Electromagnetic spectrum – Properties
and definitions – Laws of black body
radiation – Planck’s Law – Wein’s
displacement law – Stefan Boltzmann Law
– Radiation from a wave band – Emissivity
– Kirchoff’s Law- Problems

Introduction:
• In Radiation heat transfer, there is no need for a medium
to be present for heat transfer to occur.
• Net radiation heat transfer occurs from a higher
temperature level to a lower temperature level.
• Two theories concerning the radiation heat transfer:
• Two theories concerning the radiation heat transfer:
• (i) classical electromagnetic wave theory of Maxwell,
according to which energy is transferred during radiation
by electromagnetic waves, and
• (ii) the ‘Quantum theory’ of physics, according to which
energy is radiated in the form of successive, discrete
‘quanta’ of energy, called ‘photons’

Applications of radiation heat
transfer:
• industrial heating, such as in furnaces
• industrial air-conditioning, where the
effect of solar radiation has to be
considered in calculating the heat loads
considered in calculating the heat loads
• jet engine or gas turbine combustors
• industrial drying
• energy conversion with fossil fuel
combustion etc.

Some features of radiation:
– The electromagnetic magnetic waves are of
all wave lengths, traveling at the velocity of
light, i.e. c = 3 x 1010 cm/s
– Frequency (f) and wave length (λ) are
connected by the relation: c = λ.f, which
connected by the relation: c = λ.f, which
means that higher the frequency, lower the
wave length
– Smaller the wave length, more powerful is
the radiation, and also more damaging, e.g.
X – rays and Gamma rays

Wave lengths of different types
of Radiation:

Electromagnetic spectrum:

Properties and definitions:
• ‘Spectral’ means, dependence on wave length.
• Value of a quantity at a given wave length is called
‘monochromatic value’.
• Absorptivity, Reflectivity and Transmissivity:
• When radiant energy (Qo) is incident on a surface, part of it ay
be absorbed (Qa), part may be reflected (Qr) and part may be
transmitted (Qt) through the body:
transmitted (Qt) through the body:

Spectral and Spatial energy
distribution:
• Distribution of radiant energy is non-uniform with
respect to both wave length and direction, as
shown below:

Black body:
• A body which absorbs all the incident radiation is called
a ‘black body’.
• A perfect black body does not exist in nature; however, a
perfect black body an be approximated in the laboratory
by having a sphere coated black on the inside; This is
known as ‘Hohlraum’. See Fig. 13.3.
Sphere coated black on
Fig. 13.3 Simulation of a black body
in laboratory - ‘Hohlraum’
Q
Sphere coated black on
inside surface

Reflection:
• Reflection may be ‘specular’ (or mirror-like) or ‘diffuse’.
See Fig. below.

Emissive power (E):
• The ‘total (or hemispherical) emissive power’ is
the total thermal energy radiated by a surface
per unit time and per unit area, over all the wave
lengths and in all directions.
• Note, in particular, that only the original, emitted
energy is to be considered and the reflected
energy is not to be included.
• Total emissive power depends on the
temperature, material and the surface condition.

• Solid angle: “Solid angle’ is defined as a region of a
sphere, which is enclosed by a conical surface with the
vertex of the cone at the centre of the sphere. See Fig.
13.5.

• If there is a source of radiation of a small area at
the centre of the sphere O, then the radiation
passes through the area An on the surface of the
sphere and we say that the area An subtends a
solid angle ω when viewed from the centre of the
sphere.
• Note that with this definition, An is always normal
n
to the radius of the sphere.
• Mathematically, solid angle is expressed as:
ω
A n
r
2
steradians (sr)......(13.2)

• If a plane area A intercepts the line of
propagation of radiation such that the
normal to the surface makes an angle θ
with the line of propagation, then we
project the incident area normal to the line
of propagation, such that, the solid angle
is now defined as:
ω
A cos θ( ).
r
2
sr.....(13.3)
Note that, A.cos(θ) = An is the projected area of the incident surface,
normal to the line of propagation.

Intensity of radiation (Ib):
• Intensity of radiation for a black body, Ib is defined as
the energy radiated per unit time per unit solid angle per
unit area of the emitting surface projected normal to the
line of view of the receiver from the radiating surface.i.e.
I b
dQ b
dA cos θ( ).( ) dω.
W/(m2.sr).....(13.4)
I b
dA cos θ( ).( ) dω.
i.e. I b
dE b
cos θ( ) dω.
W/(m2.sr).....(13.5)
Note that Emissive power Eb of a black body refers to unit
surface area whereas Intensity Ib of a black surface refers to unit
projected area.

• Ibλ is the intensity of blackbody radiation for radiation of a given
wave length λ. And, Ib is the summation over all the wave lengths,
i.e.
I b
0
∞
λI bλ d W/(m2.sr).....(13.6)
Lambert’s cosine law:
Consider a small, black surface dA emitting radiation all over a
hemisphere above it. See Fig.13.6.
hemisphere above it. See Fig.13.6.

• In any direction θ from the normal, rate of energy
radiated is given by Lambert’s cosine law:
• “A diffuse surface radiates energy such that the rate of
energy radiated in a direction θ from the normal to the
surface is proportional to the cosine of the angle θ”. i.e.
Q θ Q n cos θ( ).
For a black surface, the intensity is the same in all
For a black surface, the intensity is the same in all
directions. Such a surface is known as ‘diffuse surface’.
For a diffuse, black surface, radiation intensity is
independent of direction and such surfaces are also
known as ‘Lambertonian surface’.

Laws of black body radiation:
• Planck’s Law for spectral distribution:
• Planck’s distribution Law relates to the spectral black body
emissive power, Ebλλλλ defined as ‘the amount of radiation energy
emitted by a black body at an absolute temperature T per unit time,
per unit surface area, per unit wave length about the wave length λ’.
• Units of Ebλ are: W/(m2.µm). The first subscript ‘b’ indicates black
body and the second subscript ‘λ’ stands for given wavelength, or
monochromatic.
• Planck’s distribution law is expressed as:
E bλ λ( )
C 1 λ
5
.
exp
C 2
λ T.
1
W/(m2.µm).....(13.7)
where, C 1 3.74210
8. W.µm4/m2
and, C 2 1.438710
4. µm.K

• Plots of Ebλ vs. λ for a few different temperatures are shown in Fig.
13.7:
1
10
100
1 10
3
1 10
4
1 10
5
1 10
6
1 10
7
1 10
8
1 10
9
Spectral Emissive Power of a Black body
Spectralemissivepower,W/(m^2.micron)
0.01 0.1 1 10 100 1 10
3
1 10
4
1 10
3
0.01
0.1
1
T = 100 K
T = 500 K
T = 1000 K
T = 5800 K = temp. of Sun
Wave length , microns
Spectralemissivepower,W/(m^2.micron)
Fig. 13.7 Planck's distribution law for a black body

• This is an important graph that tells us
quite a lot about the characteristics of
black body radiation:
• At a given absolute temperature T, a
black body emits radiation over all wave
lengths, ranging from 0 to ∞.
• Spectral emissive power curve varies
continuously with wave length.
• At a given wave length, as temperature
increases, emissive power also
increases.

• At a given temperature, emissive power curve
goes through a peak, and a major portion of
the energy radiated is concentrated around
this peak wave length λ max.
• A significant part of the energy radiated by
Sun (considered as a black body at a
temperature of 5800 K) is in the visible region
λ
(λ = 0.4 to 0.7 microns), whereas a major part
of the energy radiated by earth at 300 K falls in
the infra-red region.
• As temperature increases, the peak of the
curve shifts to the left, i.e. towards the shorter
wave lengths.

• Area under the curve between λ and (λ + dλ)
= Ebλ.dλ = radiant energy flux leaving the
surface within the range of wave length λ to (λ
+ dλ).
• Integrating over the entire range of wave
lengths:
∞
4.
E b
0
λE λ d σ T
4. ....(13.8)
Eb is the total emissive power (also known as ‘radiant
energy flux density’ ) per unit area radiated from a black
body, and σ is the Stefan – Boltzmann constant
= 5.67 x 10-8W/(m2.K4).

Corollaries of Planck’s law:
• For shorter wave lengths, (C2/λ.T) becomes very large,
and exp(C2/λ.T) >> 1. Then, Planck’s formula (eqn.
13.7) reduces to:
E bλ
C 1 λ
5
.
exp
C 2
λ T.
.....(13.9)
λ T.
This equation is known as ‘Wein’s law’ and is accurate
within 1 % for λ.T < 3000 µm.K.
• For longer wave lengths, the factor (C2/λ.T) becomes
very small,and exp(C2/λ.T) can be expanded in a series
as follows:

• And, Planck’s law becomes:
exp
C 2
λ T.
1
C 2
λ T.
1
2 !
C 2
λ T.
2
. ....
i.e. exp
C 2
λ T.
1
C 2
λ T.
...approx.
C λ
5
. C T.
E bλ
C 1 λ
5
.
1
C 2
λ T.
1
C 1 T.
C 2 λ
4
.
.....(13.10)
This is known as ‘Rayleigh – Jean’s law’ and is accurate within
1 % for λ.T > 8 x 105 µm.K. This law is useful in analyzing long wave
radiations such as Radio waves.

Wein’s displacement law:
• It is clear from Fig. 13.7 that the spectral distribution of
emissive power of a black body at a given absolute
temperature goes through a maximum.
• To find out the value of λmax, the wave length at which
this maximum occurs, differentiate Planck’s equation
w.r.t. λ and equate to zero.
w.r.t. λ and equate to zero.
• We get:

• Wein’s displacement law is stated as: “product of
absolute temperature and wave length at which emissive
power of a black body is a maximum, is constant”.
• Stefan-Boltzmann Law:
• Monochromatic emissive power of a black body is obtained
from the Planck’s Law.
• Then, the total emissive power of a black body over all the
entire wave length spectrum is obtained by integrating Ebλλλλ.
entire wave length spectrum is obtained by integrating Ebλλλλ.
Total emissive power (or, hemispherical total emissive power)
is denoted by Eb, and is given as:
E b
0
∞
λE bλ d
0
∞
λ
C 1 λ
5.
exp
C 2
λ T.
1
d

• Performing the integration, we get:
E b σ T
4. W/m2....(13.13)
where, σ 5.67 10
8. W/(m2.K4)
σ is known as ‘Stefan-Boltzmann constant’.
Eqn. (13.13) is the governing rate eqn. for radiation from a black body.
Eqn. (13.13) is the governing rate eqn. for radiation from a black body.
Its significance lies in the fact that just with a knowledge of the absolute
temperature of a surface, one can calculate the total amount of energy
radiated in all directions over the entire wave length range.
Net radiant energy exchange between two black bodies at temperatures
T1 and T2 is, therefore, given by:
Q net σ T 1
4
T 2
4. W/m2....(13.14)

Radiation from a wave band:
• This is expressed as a fraction of the total emissive
power and is written as Fλ1-λ2. Then, we can write:
F λ1_λ2
λ 1
λ 2
λE bλ d
∞
λE bλ d
1
σ T
4.
λ 1
λ 2
λE bλ d.
0
λE bλ d
i.e. F λ1_λ2
1
σ T
4. 0
λ 2
λE bλ d
0
λ 1
λE bλ d.
i.e. F λ1_λ2 F 0_λ2 F 0_λ1 ......(13.15)

• Express F0-λ as follows:
F 0_λ
0
λ
λE bλ d
σ T
4.
0
λ T.
λ T.( )E bλ d
σ T
5.
i.e. F 0_λ f λ T.( ) ....(13.16)
i.e. Now, F0-λ is expressed as a function of the product of wave
length and absolute temperature ( = λ.T) only.
Values of F0-λ vs. λ.T are tabulated in Table 13.2 and plotted in
Fig. 13.8.
Note that the units of product λ.T is (micron.Kelvin).

• Therefore,
F λ1_λ2 F λ1.T_λ2.T
1
σ
0
λ2 T.
λ T.( )
Ebλ
T
5
d
0
λ1 T.
λ T.( )
E bλ
T
5
d.
i.e. F λ1_λ2 F 0_λ2.T F 0_λ1.T ......(13.17)
Fig. 13.8 Fraction of Black body radiation in the
range (0 - Lambda.T)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
100 1000 10000 100000
Lambda.T (micron.K)
F0-Lambda

Relation between Radiation intensity and
Emissive power:
• Consider a differential black emitter dA1
radiating into a hemisphere of radius r, with the
centre of the hemisphere located at dA1.
• We first calculate the rate of energy falling on a
• We first calculate the rate of energy falling on a
differential area dA2 on the surface of the
hemisphere using the definition of intensity, then
calculate the rate of energy falling on the whole
of the hemisphere by integrating, and then
equate this amount to the rate of radiant energy
issuing from the black surface dA1.

• Let the rate of radiant energy falling on dA2 be dQ. Solid angle
subtended by dA2 at the centre of the sphere, dω = dA2/r2. Projected
area of dA1 on a plane perpendicular to the line joining dA1 and dA2
= dA1.cos(θ).
• Then, by definition, intensity of radiation is the rate of energy emitted
per unit projected area normal to the direction of propagation, per
unit solid angle, i.e.

• But, it is clear from Fig. 13.9 that differential area dA2 is equal to:
I b
dQ
dA 1 cos θ( ). dω.
i.e. I b
dQ
dA 1 cos θ( ).
dA 2
r
2
.
....(13.18)
dA 2 r dθ.( ) r sin θ( ). dφ.( ).
dA 2 r dθ( ) r sin θ( ) dφ( )
i.e. dA 2 r
2
sin θ( ). dθ. dφ. .....(13.19)
Then, from eqns. (13.18) and (13.19),
dQ I b dA 1
. sin θ( ). cos θ( ). dθ. dφ.

• Then, total rate of radiant energy falling on the hemisphere, Q, is
obtained by integrating this value of dQ over the entire
hemispherical surface.
• Noting that the whole of hemispherical surface is covered by taking
θ from 0 to (π/2) and, φ from 0 to (2.π), we write:

i.e. Total emissive power of a black (diffuse) surface
is equal to times the intensity of radiation.

Emissivity, Real surface and Grey surface:
• ‘Emissivity (εεεε)’ of a surface is defined as ‘the ratio of
radiation emitted by a surface to that emitted by a black
body at the same temperature’.
• Value of ε varies between 0 and 1. For a black body, ε =
1, and emissivity of a surface is a measure of how
closely that surface approaches a black body.
• ε depends on nature of the surface, temperature, wave
length, method of fabrication etc.
• ελ refers to the emissivity at a given wave length, λ, and
is known as spectral emissivity. When it is averaged
over all wave lengths, it is known as total emissivity.

ε T( )
E T( )
E b T( )
E T( )
σ T
4.
.....(13.22)
Total hemispherical emissivity (εεεε):
where E(T) is the emissive power of
the real surface.
Emissivity values for a few surfaces at
Emissivity values for a few surfaces at
room temperature are given in Table
13.3.
A surface is said to be grey if its
properties are independent of wave
length, and a surface is diffuse if its
properties are independent of direction.

Kirchoff’s Law:
• Kirchoff’s Law establishes a relation between the total,
hemispherical emissivity, ε of a surface and the total,
hemispherical absorptivity.
• This is a very useful equation in calculating the net
radiant heat loss from surfaces.
• Kirchoff’s Law states that the total hemispherical
emissivity, ε of a grey surface at a temperature T is
emissivity, ε of a grey surface at a temperature T is
equal to its absorptivity, α for black body radiation from a
source at the same temperature T. i.e.
ε(T) = α(T)…..(13.27)
• Similar to eqn. (13.27), we can write for monochromatic
radiation:
ελ(T) = αλ(T) ……(13.28)

See Table 13.2 in Slide 36.
9/10/2016 52

Thermal Radiation-I - Basic properties and Laws

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