This document discusses radiative heat transfer, covering key laws such as Planck's, Stefan-Boltzmann, and Wien's laws, which describe energy transfer via photon emission. It also explores the mathematical differences between radiation and other heat transfer modes, emphasizing the integral equations used in radiation problems. Additionally, it details various concepts like shape factors, net energy exchange, and provides an overview of recent technological developments in fields requiring advanced understanding of radiant heat transfer.

1. Indian Institute of Engineering Science and Technology, Shibpur
Department of Mechanical Engineering
Radiative Heat transfer

2. CONTENT
Topic Page Number
1. Introduction 1-2
2. Different Laws in Radiation
 Planck’s Law 2
 Wein’s law 3
 Rayleigh Jeans Law 3
 Stefan Boltzman Law 4
 Wein’s Displacement Law 5-6
 Kirchoff’s Law 6-7
 Lambert’s cosine Law 7
3. Shape factor 7-10
4. Net energy exchange by a body in radiation 10-11
5. References 11

3. 1 | P a g e
Radiative Heat Transfer
Introduction: -
Radiation is a method of energy transfer across a system boundary due to a temperature difference,
by the mechanism of photon emission or electromagnetic wave emission. As the mechanism of
energy transmission is photon emission (unlike conduction and convection), so no intermediate
matter is needed to enable the heat or energy transfer. We are well acquainted with a wide range
of electromagnetic phenomena in modern life. These phenomena are sometimes thought of as
wave phenomena and are, consequently, often described in terms of electromagnetic wave length.
Examples are given in terms of the wave distribution is shown in fig. 1:
The interaction of radiation upon other modes of heat transfer (i.e. conduction and convection) is
an area which has recently aroused considerable attention. Basically, such interaction effects may
be broken down into two main categories. The first involves radiation passing through an
absorbing-emitting medium such as water vapor or quartz, for which net radiant energy is
transferred to or from each element of the medium. Consequently, the conduction or convection
process may be thought of as one involving heat sources and sinks. Due to the fact that the
analytical expression for the source-sink effect must be formulated in terms of emissive power,
problems of this type are of course nonlinear. In addition, radiation to or from an element will
take place over paths of finite length resulting in an integral expression for the source-sink term,
and consequently the equation expressing conservation of energy will be an integrodifferential
Fig. 1: Radiation variation with wavelength
distribution

4. 2 | P a g e
equation. The second category involves radiation interaction through the boundary conditions of
the conduction or convection process. One example is transient conduction in a solid with radiation
at the surface as treated; while in convection processes the boundary condition along the heated or
cooled surface may be altered due to radiation exchange [1]. Recent technological developments
in hypersonic flight, gas cooled nuclear reactors, power plants for space exploration needs,
meteorology, fission and fusion reactions, a rocket engine have emphasized the need for a better
quantitative understanding of radiant heat transfer through absorbing and scattering media. Little
theoretical or experimental work of an engineering nature has been done in this area except for
those studies which have been concerned with furnaces and combustion chambers [2]. On the other
hand, considerable background information has been made available during the past sixty years by
astrophysicists because of their concern for radiation transfer in planetary atmospheres, the sun
nebulae, and galaxies [3]. There exists little similarity between radiant heat transfer and the other
modes of heat transfer; that is, conduction and convection. In a physical sense there are significant
differences because radiation is transported by electromagnetic waves while conduction and
convection involve contact between micro and macroscopic particles of matter. In a mathematical
sense, radiation problems must usually be formulated with integral equations. However,
conduction and convection problems are usually formulated with differential equations; only in
the case of very dilute gases are integral equations of importance. It can be said by way of pointing
out similarities, that both radiation and conduction are governed by special cases of the integral
equation which describes the most general transport processes [4].
Now-a-days, the simultaneous radiative and conductive heat transfer has become more popular.
Many researches have been done in this new field such as Viskanta and Grosh (in 1962), Anteby
et al. (in 2000), Siegel and Howell (in 2002) and others.
Different Laws in radiative heat transfer:
1. Planck’s Law:
The actual form of Planck’s law for a black body emission is,
2
b 5
2 1
( )
exp 1
c h
E
ch
k T




= 
 
− 
 
(1)
where, Eb = single wave length radiant energy emitted by the black body per unit time per unit
area i.e. monochromatic emissive power

5. 3 | P a g e
c = velocity of light in vacuum = 3 × 108
𝑚/𝑠,
h = Planck’s constant = 6.626 × 10−34
𝐽. 𝑠,
 = wavelength of the radiation emitted (m),
T = absolute temperature of the black body (K),
The simplified form of this equation is,
5
1
b
2
( )
exp 1
C
E
C
T



−
=
 
− 
 
(2)
where, C1 = 3.742 × 1016
𝑊. 𝑚2
, C2 = 1.4389 × 10−2
𝑚. 𝐾
2. Wein’s Law:
Wein’s law defines the Planck’s law for shorter wave length. For shorter wave length
2
1
C
T
 , so the emissive power of black body can further be written as:
5
1
b
2
( )
exp
C
E
C
T



−
=
 
 
 
(3)
3. Rayleigh-Jeans Law:
For longer wave length 2
1
C
T
 .
2
2 2 2 21
exp 1 1
2!
C C C C
T T T T   
   
 = + + + − − − − −  +   
   
So, Planck’s law reduces to,
5
1 1
4
2 21 1
b
C C T
E
C C
T



−
= =
+ −
(4)
This identity is known as Rayleigh-Jeans Law.

6. 4 | P a g e
4. Stefan-Boltzman Law:
This law states that the emissive power (radiant energy per unit time per unit area) of a
black surface is proportional to the fourth power of the absolute temperature of that body.
bE ∝ 𝑇4 4
b bE T = (5)
Proof:
From Planck’s law we know that for monochromatic radiation emissive power is,
5
1
b
2
( )
exp 1
C
E
C
T



−
=
 
− 
 
Now total emissive power would be,
5
1
b
0 0 2
( )
exp 1
b
C
E E d d
C
T


 

− 
= = 
 
− 
 
Let, 2 2
2
C C
y d
T y T


=  = −
5 5 4 3 4
3 11 2 1 1
5 y 2 4 y 4
0 0 02 2 2
4
3 2 31
4
02
( 1)
C (e 1) C (e 1)
( ...)
y
b
y y y
C T y C C T y C T
E dy dy y e dy
Ty C
C T
y e e e dy
C
  
−

− − −
 =  = = −  
− −
= + + +
We know that,   1
0
!
exp( )n
n
n
y ay dy
a

+
− =
( )
4
1
4 4 4 4
2
4 16
41
44 2
2
4 8 2 4
b
3! 3! 3!
.....
1 2 3
3.742 10
6.48 6.48
1.4389 10
where, 5.67 10
b
b b
C T
E
C
C T
T
C
E T W m K 
−
−
− − −
 
 = + + +  

=  =  

 = = 
If there are two bodies then net radiant heat flux, ( )4 4
1 2net bQ T T= − (6)

7. 5 | P a g e
5. Wein’s Displacement Law:
The product of absolute temperature and the wave length associated with the maximum
emissive power at that temperature is constant.
1 2 31 2 3. 2 .m m mIn the fig it is shown that T T T const   =  =  =
Proof:
For maximum rate of emission, b( ) 0
d
E
d


=
5
1
2
22
6 1 2
1 7
2
2
0
1
5 ( 1)
1 1
5
C
T
CC
T T
C
T
Cd
d e
C C
C e e
T
C
e
T

 






−
−
 
 = 
 
− 
 − − = −

 
 − = 
 
Fig. 2: Wein’s Displacement Law
1 2 31 2 3 .m m mT T T const   =  =  =

8. 6 | P a g e
2
max
4.965
0.0029
C
T
T m K


 =
  = − (7)
max denotes the wave length at which the emissive power is maximum at a certain temperature.
6. Kirchoff’s Law:
The emissivity and absorptivity are equal for a real is always equal for radiation of fixed
temperature and wave length.
Proof:
From, the definition of emissivity we know that,
( )
( )
( )b
Emissive power of real surface E
at same temperature
Emissive power of black body E
 =
In the fig. 3, let E be the radiant energy emitted by the non-black surface and falls upon the black
surface which is fully absorbed. Let Eb be the radiant energy emitted by the black surface of which
Eb portion is absorbed by the non-black surface and the rest portion (1-)Eb is reflected back by
the non-black surface. So, the total heat energy exchange of the non-black surface is 𝐸 − 𝛼𝐸 𝑏.
Fig. 3: Heat transfer between black and non-black surfaces by radiation

9. 7 | P a g e
If both surfaces are at same temperature i.e. T = Tb then the net interchange of heat will be zero.
0b
b
E E
E
E


 − =
 = (8)
7. Lambert’s cosine law:
The law that states that the intensity of perfectly diffusing, emitting, plane surfaces, known as
“Lambertian radiators,” “Lambertian sources,” or “Lambertian reflectors,” is dependent upon
the cosine of the angle between the viewing direction and the normal to the surface, is given by
the relation –
I = Io cos  (9)
where I is the radiance, i.e., the radiant intensity, usually expressed in watts per square meter
per steradian, i.e., W · m−2
· sr−1
, where Io is the radiance normal, i.e., perpendicular, to the
emitting surface, and  is the angle between the viewing direction and the normal to the surface
being viewed, i.e., the emitting surface, and is a maximum in the direction of the normal to the
surface and decreases in proportion to the cosine of the angle between the normal to the surface
and the viewing direction. [5]
➢ Shape Factor/View Factor:
The view factor of body-2 with respect to body-1 (F1-2) is defined as the fraction of energy
emitted from surface 1, which directly strikes surface 2.
Fig. 4: The area dA1 as seen from the prospective of a
viewer, situated at an angle θ from the normal to the surface

10. 8 | P a g e
I1 is the intensity of radiation leaving the surface-1 (dA1).
The radiant energy emitted from the elemental area dA1 striking the surface dA2 is
12 1 1 12 1
1 2 1 2 2 2
1 122 2
cos
cos cos cos
, solid angle
dQ I dA d
dAdA dA
I as d
S S

  
=   
 
=  =  
4
1
1
4
1
12
1 2 1 2
2
1 2
cos cos
b b
b
E T
I
T
Q
A A
dAdA
S

 


 
= =
 =   (11)
Using eq. (5), eq. (10) and eq. (11) we get,
1 2
12
1
1 2
2
direct radiation from surface to surface
F
total radiation emitting from surface
Q
Q
−
− −
=
−
= (10)
Fig. 5: Radiation emitting from surface-1, falling upon surface-2

11. 9 | P a g e
4
1
12 4
1 1
1
1 2 1 2
2
1 2
1 2 1 2
2
1 2
1
1
cos cos
cos cos
b
b
T
F
T A A A
A A A
dA dA
S
dA dA
S

 

 
 

 = 

=
 
  (12)
Rules for determining shape factor:
1) Summation/Conservation Rule:
In order that we might apply conservation of energy to the radiation process, we must account for
all energy leaving a surface. We imagine that the surrounding surfaces act as an enclosure about the
heat source which receive all emitted energy. Should there be an opening in this enclosure through
which energy might be lost, we place an imaginary surface across this opening to intercept this
portion of the emitted energy. For an n surfaced enclosure, we can then see that
1
1 [ ]
n
ij
j
F i
=
=  (13)
2) Reciprocity Theorem:
We may write the view factor from surface i to surface j as:
1 2
2
cos cos
i j i
i j
i j
F A
A A
dA dA
S
 
→

=   (14)
Similarly, between surface j and surface i we may write,
1 2
2
cos cos
j i j
i j
j i
F A
A A
dA dA
S
 
→

=   (15)
Comparing eq. (14) and eq. (15) we get,
i j i j i jF A F A→ →= This is known as reciprocity theorem. (16)
3) Associative Rule:
Consider the set of surfaces shown to the right: Clearly, from conservation of energy, the fraction of
energy leaving surface i and striking the combined surface 𝑗 + 𝑘 will equal to the fraction of energy
emitted from i and striking j plus the fraction leaving surface i and striking k.

12. 10 | P a g e
i j k i j i kF F F→ + → → = + (17)
➢ Net energy exchange by a body in radiation:
The net energy leaving a surface will be the difference between the energy leaving a surface and the
energy received by a surface:
1[ ]bq E G A = −  (18)
Combine this relationship with the definition of Radiosity to eliminate G, we get –
[ ]b
b
J E G
J E
G
 


= +
−
 = (19)
Using eq. (19) in eq. (18),
1 1[ ] [ ] [ 1]
1
b b
b b
J E J E
q E A E A
 
     
 
− −
= −  = −  + =
−
i
j
k
Fig. 6: Orientation of surfaces i, j and k
Fig. 7: Radiant, reflective and incident energy

13. 11 | P a g e
1
1
1
[ ' , ]
1
[ ]
1
[ ]
1
b
b
b
b
J E
E A By Kirchoff s law
A
E J
E J
Q
A

   





− 
= −  = − 
= −
−
−
 =
−
(20)
The term
1
1
A


−
in eq. (20) is known as the surface resistance.
Thus, we may develop an electrical analogy for radiation which is as followed –
Table-1: Comparison of electric circuit with the radiant energy exchange
Equivalent Current Equivalent Resistance Potential Difference
Ohms Law I R V
Net Energy Leaving
Surface
Q1
1
1
A


− Eb – J
Net Exchange in the
space in between
i jQ→
1 12
1
A F
(known as space resistance)
J1 – J2
References:
1. R. D. Cess (1962), Appl. Sci. Research Sect. A, 10, 430, pp.- 1-2.
2. R. Viskanta & R. J. Grosh (1962), J. Heat Trans, ASME.
3. R. Viskanta & R. J. Grosh (1962), Int. J. Heat Mass Transfer, vol. 5, pp.- 729-734.
4. L.S Wang & C. L. Tien (1967), Int. J. Heat Mass Transfer, vol. 10, pp.- 1327-1338.
5. Martin H. Weik (2017), “Lambert's cosine law”, doi: 10.1007/1-4020-0613-6_9901
6. George B. Rybicki (1996), J. Astrophys. Astr., vol. 17, pp.- 95-112.
Fig. 8: Electrical Analogical circuit of radiant heat exchange