7.4 Radiative heat exchanger between surfaces
Till now we have discussed fundamental aspects of various definitions and laws. Now we will study the heat exchange between two or more surfaces which is of practical importance. The two surfaces which are not in direct contact, exchanges the heat due to radiation phenomena. The factors those determine the rate of heat exchange between two bodies are the temperature of the individual surfaces, their emissivities, as well as how well one surface can see the other surface. The last factor is known as view factor, shape factor, angle factor or configuration factor.
7.4.1 View factor
In this section we would like to find the energy exchange between two black surfaces having area A1 and A2, respectively, and they are at different temperature and have arbitrary shape and orientation with respect to each other. In order to find the radiative heat exchange between the bodies we have to first define the view factor as
| F12 = fraction of the energy leavings surface 1 which reaches surface 2 | |
| F21 = fraction of the energy leaving surface 2 which reaches surface 1 or in general, | |
| Fmn = fraction of the energy leaving surface m which raches surface n |
Thus the energy leaving surface 1 and arriving at surface 2 is Eb1A1F12 and the energy leaving surface 2 and arriving at surface 1 is Eb2A2F21. All the incident radiation will be absorbed by the blackbody and the net energy exchange will be,
Q = Eb1A1F12 - Eb2A2F21
At thermal equilibrium between the surfaces Q12 = 0 and Eb1 = Eb2, thus
0 = Eb1 (A1F12 - A2F21)
A1F12 - A2F21 |
(7.19) |
Equation 7.19 is known as reciprocating relation, and it can be applied in general way for any blackbody surfaces.
AiFij - AjFji |
(7.20) |
Though the relation is valid for blackbody it may be applied to any surface as long as diffuse radiation is involved.
7.4.1.1 Relation between view factors
In this section we will develop some useful relation of view factor considering fig. 7.5
Fig. 7.5: Exchange of energy between area A1 and A2 (A is area of blackbody)
View factor for radiation from A1 to the combined area A2,
F12 = F13 + F14 |
(7.21) |
and using the reciprocating relations for surface 1 and 4,
A1F14 = A4F41 |
(7.22) |
Using eq. 7.21 and 7.22,
Thus the unknown view factor F14 can be estimated if the view factors F12 and F13, as well as their areas are (A1, A2) known.
Now, consider a flat plate (for eg.) which is emitting the radiation, it can be understood that the radiation of the flat plat cannot fall on its own surface (partly or fully). Such kind or surfaces are termed as “not able to see itself”. In such situations,
F11 = F22 = F33 = F44 = 0
However, if the surface can see itself like concave curved surfaces, which may thus see themselves and then the shape factor will not be zero in those cases.
Another property of the shape factor is that when the surface is enclosed, then the following relation holds,
|
(7.23) |
where, Fij is the fraction of the total energy leaving surface i which arrives at surface j.
In case of N-walled enclosure, some of the view factors may be evaluated from the knowledge of the rest and the total N2 view factors may be represented in square matrix form shown below,
