Radiation Shape factor algebra make

the study of radiation exchange between

surfaces simple & easy.  Number of shape

factors is equal to the square of the number

of radiating surfaces.

Radiation exchange is considered between

black surfaces. It has also been considered

between real surfaces. If exchange is in excess,

then radiation shields will be considered to

reduce the radiations between surfaces. 

Definition of a Shape Factor

There is a factor called the SHAPE FACTOR (Fij) OR view factor or configuration factor or form factor or geometric factor. It accounts for the orientation and geometry of the surfaces exchanging radiations.

F12 =Shape factor=ratio of radiations intercepted by surface 2/radiations leaving surface 1.

 Symbol is Fij.  

 For example 140 units of energy emitted by body A.

35 units of energy are received by a body B.

Then FAB= 35/140=0.25.

This depends how the surfaces view each other.

Also, it depends how the surfaces radiate energy with respect to each other.


(i). Reciprocity law  AiFij = AjFji  OR A2 F23 =A3 F32

Imagine surface 1 is convex and is enclosed in another surface 2 (a small ball in a big ball),

then F12=1

from the law of reciprocity F21=A1/A2

(ii). Summation Law – Say for four bodies,1,2,3 and 4

Firstly F11+F12+F13+ F14 =1

Secondly F21+F22+F23+ F24 =1

Thirdly F31+F32+F33+ F34 =1

Fourthly F41+F42+F43+ F44 =1

(iii). Shape factor for a body itself 

Fii or F11 energy received by body 1 out of energy emitted by body 1

Thus, shape factor for a flat surface Fii =0

Further, for a convex surface Fii = 0 and hence F11 =F22=F33=0

For a concave surface Fii ≠ 0 and hence F11 ≠0, F22≠0, F33≠0

(iv). If there are ‘n’ surfaces, there are n2 shape factors

 example,  for three bodies, 9 shape factors are required.

HOW TO FIND THESE n2 shape factors

There are three steps.

(i)  Apply summation rule to get n equations to give n shape factors i.e. 3

(ii)  Use Reciprocity relation n (n—1)/2 times to get n (n—1)/2 shape factors i.e. 3

(iii)  Now it is required to find the remaining shape factors i.e. Remaining=  n2 –n– n (n—1)/2 shape factors i.e. 3 in this case.

Case of three surfaces

Remaining shape factors will be = 32–3–3(3—1)/2=9-3-3=03

These remaining shape factors are found from general observations like F11 = 0, F12=1, F21=A1/A2 etc.


Equations for shape factors are not available for complex shapes. Complex shapes are converted  into simpler shapes. Shape factors then for simpler shapes can be found from the shape factor algebra.  It is based on

(i). The definition of shape factor

(ii). Law of reciprocity

(iii). Summation law

There are two surfaces A1 and A2. Say receiving surface A2 is a  complex surface. It is divided further into two areas A3 and A4 in simple areas.

A1F12=A1F13 +A1F14

Of course, the shape factors can be found from standard graphs for the following cases.

(i).  Parallel surfaces (rectangular, equal circular, non-equal circular, etc)

(ii). Surfaces at right angles

Fig. Shape Factors for two Parallel Surfaces

Fig. Shape Factors Between Two Rectangular surfaces at right angles with a common edge

Radiation exchange between any numbers of black  surfaces depends on the followings:

(i). Orientation— position in relation to each other

(ii). Geometrical shape-(Flat, concave, convex)uses areas for shape factors

(iii). Temperature

(iv). Radiative properties like emissivity,  absorptivity,  transmissivity, reflectivity and  radio city


Radiation exchange from body 1 to body 2 =q1 = F12 A1q1= F12 A1 Eb1

Radiation exchange from body 2 to body 1, q2   = F21 A2q2 = F21 A2 Eb2

Net radiative exchange between 1 and 2,  q12  = q1

q12= F12 A1q1 — F21 A2q2= F12 A1 (q1 – q2) = F12 A1 σ(T14–T24)                    (since F12 A1= F21A2)


There are two methods to find radiation exchange between grey bodies.


Shape factor for grey bodies is also called the Configuration factor/interchange factor/equivalent emissivity. It is represented for grey bodies by f12 and for black bodies by F12.

Various equations for f12

(i). For two small grey bodies    f121 Є2

(ii). A small body 1 is enclosed in a large body 2,      f121

(iii). Between two parallel infinite surfaces                 f12 =1

(iv). Among two infinite long concentric cylinders/spheres

f12 =

(v). A large body 1 enclosed into another large body 2

f12 =

Q12= f12A1σb(T14—T24)=  A1σb(T14—T24)

PROOF: Radiation exchange for infinite parallel real surfaces

Fig. Radiation exchange between two parallel infinite grey surfaces

Assumptions used

(i). Configuration factor for either surface is unity i.e. f12=1 and f21=1

(ii). There is a non absorbing medium like air between these surfaces.

(iii). Properties like emissivity, reflectivity and absorptivity  is constant over the entire surfaces.

Let surface 1 emits E1 with α1, Є1 at temperature T1 and surface 2 emits E2 with α2, Є2 at temperature T2

After number of reflections

Q1net =E1–[ α1(1– α2) E1+ α1 (1—α1) (1—α2)2E1 + α1(1—α1)2 (1– α2)3 E1+…………………….]

After number of reflections,

it becomes  Q1net =E1— After number of reflections

Therefore,    Q1net =E1— α1(1– α2)E1[1 + (1—α1) (1—α2)+ (1—α1)2 (1– α2)2 +…………………………]

Taking (1—α1) (1—α2) as common

In order to simplify, Put  Z=(1—α1) (1—α2)

We get, Q1net =E1— α1(1– α2)E1[1 +Z+Z2 + Z3 + ………………………………]

As Z<1 the sum of infinite series 1+Z+Z2+Z3+…………….=1/(1—Z)

Thus, Q1net =E1— α1(1– α2)E1/(1—Z)

Substitute the value of Z

It gives, Q1net =E1 [1— α1(1– α2)/(1– (1—α1) (1—α2)]

As per Kirchhoff’s law α=Є i.e. α11  and α22

Then, Q1net =E1 [1— Є1(1– Є2)/(1– (1—Є1) (1—Є2)]

Opening the brackets,

Q1net =E1 [1—(1– Є1)(1– Є2)— Є1(1– Є2)]/[(1– (1—Є1) (1—Є2)]

Where Q1net =It is net energy going from body 1 towards body 2.

Similarly Q2net

It is net energy going from body 2 towards body 1.

Radiation Exchange between surfaces 1 and 2

Q12 = Q1 –Q2 =

Now E1 = σ T14       and E2= σ T24

Q12 =f12 σ (T14 —  T24 )

where f12 is interchange factor or configuration factor for grey bodies


ANALOGY METHOD – Applicable both for black as well as for grey bodies. This method is direct, more general, and much simpler. This method uses incident G and radiosity J (W/m2).

 G = radiations flux incident on the surface, W/m2

Radiosity J = (emitted + reflected ) radiation flux from a grey body

Eb1 emitted energy from  body 1 as a black body at a certain temperature T1

J1 total energy coming out from body 1 as a grey body at the temperature T1

J= ( emitted+ reflected) radiation flux

Difference in Eb1 and J1 is due to the surface resistance

J1 total energy coming out from body 1 as a grey body at temperature T1

J2 total energy coming out from body 2 as a grey body at temperature T2

There is a resistance between grey bodies J1 and J2 called the space resistance.

J2 total energy coming out from body 2 as a grey body at temperature T2

Eb2 emitted energy from body 2 as a black body at temperature T2

Difference in Eb2 and J2 is due to the surface resistance

Q12 = J1A1 F12 —J2A2 F21

It becomes  Q12 =A1F12 (J1—J2)  since A1 F12 = A2 F21

Q12 = (J1—J2)/(1/ A1F12)= (J1—J2)/resistance

Now 1/A1F12 is called the SPACE RESISTANCE between the two grey bodies. Because it is due to the geometry and distance between the two grey bodies.

Fig. Space and surface resistances between three bodies

Fig. Space & surface resistance between two bodies


Fig. Radiosity J = ρG+ ε E

( Radiosity  J = emitted +reflected) radiation flux

J=Eg +ρG

J=ЄEb+ ρG   where Eb is the emissive flux from a black body

Now α+ ρ+ ς=1

ς =0 since the surface assumed to be opaque.

Now α+ ρ=1


J=ЄEb+ (1—α )G

α=Є by Kirchhoff’s Law

J=ЄEb+ (1—Є )G

J– ЄEb=(1—Є )G

G = (J– ЄEb)/ (1—Є )


=A(J–(J– ЄEb)/ (1—Є ))

=A[(J(1—Є ))– (J– ЄEb)]/ (1—Є )

=A[(JЄ–ЄEb)]/ (1—Є )

=(Eb –J)/[ (1—Є )/ЄA)]

This equation can be represented in the form of an electric network

Qnet is rate of heat transfer like current which is flow of electric charge

(Eb –J) is like voltage difference

(1—Є )/ЄA) is like resistance.

This resistance In this case is called surface resistance . It is because (Eb –J) refers to difference in heat flux of a black body and a grey surface at the same temperature.

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