thermal radiation-i - basic properties and laws

Lectures on Heat Transfer –RADIATION-I:Basic Properties and Laws

by

Dr. M. ThirumaleshwarDr. M. Thirumaleshwarformerly:Professor, Dept. of Mechanical Engineering,St. Joseph Engg. College, Vamanjoor,MangaloreIndia

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

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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’sdisplacement 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 bodyin 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


nto the radius of the sphere.

• Mathematically, solid angle is expressed as:

ωA n

r2steradians (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 θwith 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( ).

r2sr.....(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 bdQ b

dA cos θ( ).( ) dω.W/(m2.sr).....(13.4)


I bdA cos θ( ).( ) dω.

i.e. I bdE b

cos θ( ) dω.W/(m2.sr).....(13.5)

Note that Emissive power Eb of a black body refers to unitsurface area whereas Intensity Ib of a black surface refers to unitprojected 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 b0

∞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 isindependent of direction and such surfaces are alsoknown 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.

expC 2

λ T.1

W/(m2.µm).....(13.7)

where, C 1 3.742 108. W.µm4/m2

and, C 2 1.4387104. µm.K

• Plots of Ebλ vs. λ for a few different temperatures are shown in Fig.13.7:

1

10

100

1 103

1 104

1 105

1 106

1 107

1 108

1 109

Spectral Emissive Power of a Black body

Spec

tral

em

issi

ve p

ower

, W/(

m^2

.mic

ron)


0.01 0.1 1 10 100 1 1031 10 4

1 10 3

0.01

0.1

1

T = 100 KT = 500 KT = 1000 KT = 5800 K = temp. of Sun

Wave length , microns

Spec

tral

em

issi

ve p

ower

, W/(

m^2

.mic

ron)

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 b0

λE λ d σ T4. ....(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 bλC 1 λ 5.

expC 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) becomesvery small,and exp(C2/λ.T) can be expanded in a seriesas follows:

• And, Planck’s law becomes:

expC 2

λ T.1

C 2

λ T.1

2 !

C 2

λ T.

2

. ....

i.e. expC 2

λ T.1

C 2

λ T....approx.

C λ 5. C T.


E bλC 1 λ 5.

1C 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 b0

∞λE bλ d

0

∞

λC 1 λ 5.

expC 2

λ T.1

d

• Performing the integration, we get:

E b σ T4. 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 absolutetemperature of a surface, one can calculate the total amount of energyradiated in all directions over the entire wave length range.

Net radiant energy exchange between two black bodies at temperaturesT1 and T2 is, therefore, given by:

Q net σ T 14 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 λ 1_λ 2λ 1

λ 2λE bλ d

∞λE bλ d

1

σ T4. λ 1

λ 2λE bλ d.


0

λE bλ d

i.e. F λ 1_λ 21

σ T4. 0

λ 2λE bλ d

0

λ 1λE bλ d.

i.e. F λ 1_λ 2 F 0_λ 2 F 0_λ 1 ......(13.15)

• Express F0-λ as follows:

F 0_λ0

λλE bλ d

σ T4.

0

λ T.λ T.( )E bλ d

σ T5.

i.e. F 0_λ f λ T.( ) ....(13.16)


i.e. Now, F0-λ is expressed as a function of the product of wavelength 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 λ 1_λ 2 F λ 1.T_λ 2.T1

σ0

λ 2 T.

λ T.( )E bλ

T5d

0

λ 1 T.

λ T.( )E bλ

T5d.

i.e. F λ 1_λ 2 F 0_λ 2.T F 0_λ 1.T ......(13.17)

Fig. 13.8 Fraction of Black body radiation in the range (0 - Lambda.T)


00.10.20.30.40.50.60.70.80.9

1

100 1000 10000 100000

Lambda.T (micron.K)

F0-L

ambd

a

Relation between Radiation intensity and

Emissive power:

• Consider a differential black emitter dA1radiating 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 bdQ

dA 1 cos θ( ). dω.

i.e. I bdQ

dA 1 cos θ( ).dA 2

r2.

....(13.18)

dA 2 r dθ.( ) r sin θ( ). dφ.( ).


dA 2 r dθ( ) r sin θ( ) dφ( )

i.e. dA 2 r2 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) surfaceis 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 T4......(13.22)

Total hemispherical emissivity (εεεε):

where E(T) is the emissive power ofthe 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 itsproperties are independent of wavelength, and a surface is diffuse if itsproperties 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)

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See Table 13.2 in Slide 36.

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