Shape factors, radiation

Consider two black surfaces Ai and A2, as shown in Fig. 8-8. We wish to obtain a general expression for the energy exchange between these surfaces when they are maintained at different temperatures. The problem becomes essentially one of determining the amount of energy which leaves one surface and reaches the other. To solve this problem the radiation shape factors are defined as 

- 2 = fraction of energy leaving surface 1 which reaches surface 2 

Other names for the radiation shape factor are view factor, angle factor, and configuration factor. The energy leaving surface 1 and arriving at surface 2 is 

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Radiation shape factor The angle factor representing the fraction of the angular field of view from which energy exchange is trading places. 

Flo-M Sketch showing area elements used in deriving radiation shape factor. 

Fig. 8-10 Spherical coordinate system used in derivation of radiation shape factor.

Fig. 8-12 Radiation shape factor for radiation between parallel rectangles...

By making use of these reciprocity relations, the radiation shape factor Fiy may be expressed by... 

This example illustrates how one may make use of clever geometric considerations to calculate the radiation shape factors. 

When two infinite parallel planes are considered, At and A2 are equal and the radiation shape factor is unity since all the radiation leaving one plane reaches the other. The network is the same as in Fig. 8-26, and the heat flow per unit area may be obtained from Eq. (8-40) by letting A, = A2 and Fn = 1.0. Thus... 

If the emissivities of all surfaces are equal, a rather simple relation may be derived for the heat transfer when the surfaces may be considered as infinite parallel planes. Let the number of shields be n. Considering the radiation network for the system, all the surface resistances would be the same since the emissivities are equal. There would be two of these resistances for each shield and one for each heat-transfer surface. There would be n + I space resistances, and these would all be unity since the radiation shape factors are unity for the infinite parallel planes. The total resistance in the network would thus be... 

In reality, the radiation shape factors F, 2, F( m, and F2 m are unity for this example, so that the expression for the heat flow could be simplified to some extent however, these shape factors are included in the network resistances for the sake of generality in the analysis. 

All the preceding discussions have considered radiation exchange between diffuse surfaces. In fact, the radiation shape factors defined by Eq. (8-21) hold only for diffuse radiation because the radiation was assumed to have no preferred direction in the derivation of this relation. In this section we extend the analysis to take into account some simple geometries containing surfaces that may have a specular type of reflection. No real surface is completely diffuse or completely specular. We shall assume, however, that all the surfaces to be considered emit radiation diffusely but that they may reflect radiation partly in a specular manner and partly in a diffuse manner. We therefore take the reflectivity to be the sum of a specular component and a diffuse component ... 

The radiation shape factor F2(3), is the one between surface 2(3) and surface I. The reflectivity is inserted because only this fraction of the radiation gets to 1. Of course, A2 - A2I ). We now have... 

Making use of Eqs. (8-99) and (8-102) gives the complete network for the system as shown in Fig. 8-61. Of course, all the radiation shape factors in the above network are unity, but they have been included for the sake of generality. In this network Jw refers to the diffuse radiosity on the left side of 2, while J 2D is the diffuse radiosity on the right side of this surface. 

Basic radiation shape factor reciprocity relation r n A p nm (8-18a)... 

Find the radiation shape factors F, for the situations shown. 

F - or F Radiation shape factor for radiation from surface m to surface n... 

Lenenberger, H. and Pearson, R. A., Compilation of Radiation Shape Factors for Cylindrical Assemblies, Paper 56-A-144, ASME, New York, 1956. 

B. T. F. Chung and P. S. Sumitra, Radiation Shape Factors from Plane Point Sources, J. Heat Transfer, 94(3), pp. 328-330, August, 1972. 

Figure 4.11-4. Area elements for radiation shape factor.

A very important aspect of radiative heat transfer is the system geometry. This is accounted for by using radiation shape factors, also called view factors, angle factors, or configuration factors and defined as follows... 

Radiation shape factors can be computed for different situations and then used with the equations... 

Determination of the radiation shape factor for a given system is a complex problem. Some results (see Figures 9-3, 9-4, and 9-5) have been published for straightforward cases. 

Figure 9-3. Radiation shape factors between adjacent surfaces (1).

Figure 9-5. Radiation shape factor between a plane and rows of tubes (1).

In practice, it is possible to find radiation shape factors between parts of a system. Consider, for example, the situation depicted in Figure 9-6 where we need the shape factor between areas 1 and 4. 

In order to solve this problem, we need the radiation shape factor between the surfaces. This involves the use of Figure 9-4. For the situation (no re-radiating walls) we select from curves 1-4. The geometry in this case is a 2 to 1 (i.e., 3.04/1.52) rectangle, which means that curve 3 is the appropriate one to be used. The ratio is... 

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