RADIATION EXCHANGE BETWEEN GREY AND BLACK SURFACES
RADIATION EXCHANGE
BETWEEN GREY AND
BLACK SURFACES
Radiation Shape factor algebra makes
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.
LAWS FOR SHAPE FACTORS
(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 IN SHAPE FACTORS
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
2. NET RADIATION EXCHANGE BETWEEN TWO BLACK SURFACES 1 AND 2 :
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)
METHODS TO FIND RADIATION EXCHANGE BETWEEN GREY SURFACES
There are two methods to find radiation exchange between grey bodies.
(a) BASED ON CONFIGURATION FACTOR/INTERCHANGE FACTOR/EQUIVALENT EMISSIVITY BETWEEN TWO 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 f12 =Є1 Є2
(ii). A small body 1 is enclosed in a large body 2, f12=Є1
(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. α1=Є1 and α2=Є2
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
(b) BASED ON ELECTRIC NETWORK
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
3. NET RADIATIVE EXCHANGE BETWEEN TWO GREY SURFACES USING RADIOSITY
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
ρ=1—α
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 )
Qnet=A(J—G)
=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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