View factor summation rule

For a given geometry, view factors are related to each other, one example being the reciprocity relationship given in equation 9.126. Another important relationship is the summation rule which may be applied to the surfaces of a complete enclosure. In this case, all the radiation leaving one surface, say i, must arrive at all other surfaces in the enclosure so that, for n surfaces ... 

This means that the sum of the exchange areas associated with a surface in an enclosure must be same as the area of that surface. The principle of the summation rule may be extended to other geometries such as, for example, radiation from a vertical rectangle (area 1) to an adjacent horizontal rectangle (area 2), as shown in Figure 9.40iii, where they are joined to a second horizontal rectangle of the same width (area 3). In effect area 3 is an extension of area 2 but has a different view factor. 

Consider an enclosure shown above, consisting of three surfaces which are very long in the direction perpendicular to the plane of the figure. The summation rule for view factors gives... 

The conservation of energy principle requires that the entire radiation leaving any surface i of an enclosure be intercepted by the surfaces of the enclosure. Therefore, t/ie svru of the view factors from surface i of an enclosure to all surfaces of the enclositne, including to itself, must equal unity. This is known as the summation rule for an enclosure and is expressed as (Fig. 13-9)... 

Thc sumniation rule can be applied to each surface of an enclosure by varying i from 1 to /V. Therefore, the summation rule applied to each of the N surfaces of an enclosure gives N relations for the determination of the view factors. Also, the reciprocity rule gives N(N — 1) additional relations. Then the total number of view factors that need to be evaluated directly for an /V-suiface enclosure becomes... 

For example, for a six-surface enclosure, we need to determine only I X 6(6 - 1) = 15 of the 6 = 36 view factors directly. The remaining 21 view factors can be determined from the 21 equadons that aie obtained by applying the reciprocity and the summation rules. 

Actually it would be sufficient to determine only one of these view factors by inspection, since we could alway.s determine the other one from the summation rule applied to surface 1 as F -. + Pf - 1-The vievr factor h n is determined by applying the reciprocity relation to surfaces 1 and 2 ... 

Finally, the view factor F, is determined by applying the summation rule to surface 2 ... 

Sul first we need to evaluate the view factor F 3. After checking the view factor charts and tables, we realize that we cannot determine this vievr factor directly. However, v/e can determine the view factor from Fig. 13-5 to be., 2 = 0,2, and vre knew that F, =0 since surface 1 is a plane. Then applying the summation rule to surface 1 yields... 

The view factor from the base to the top surface is, from Fig. 13-7, ft = 0.38. Then the viev/ factor from the base to the side surface is determined by applying the summation rule to be... 

C What are the summation rule and Ihe superposition rule for view factors ... 

Consider an enclosure consisting of 12 surfaces. How many view factors does this geometry involve How many of these view factors can be determined by the application of Ihe reciprocity and the summation rules Ansr.ers 144. 78... 

A simple example for the application of the relationships (5.132) and (5.133) is provided by radiation in an enclosure formed by two spherical surfaces 1 and 2, Fig. 5.52. There are four view factors in this case, l) i. fqo, Toi and F22 The summation rule is applied to the inner sphere in order to calculate them ... 

The fourth view factor is found by applying the summation rule F21 + To2 = 1 to the outer sphere... 

View factors are not always as easy to find as for the simple geometry present in the example we have just looked at. Then the multiple integral in (5.130) has to be evaluated. However, not all the view factors have to be calculated in this manner. In an enclosure bounded by n surfaces there are n2 view factors in total. From these, n view factors can be found by the application of the summation rule (5.133) on each of the n surfaces. In addition to this, n(n — l)/2 view factors can be determined using the reciprocity rule (5.132). Therefore the number of view factors that have to be calculated from (5.130) is only... 

The view factor depends only upon the geometric arrangement of the surfaces, and satisfies the reciprocity relation A E j=AjFj. The view factor must be between 0 and 1. In an enclosure consisting of N surfaces, the summation rule gives... 

Determine the unknown potentials (emissive powers) and cunrents (heat fluxes) by the usual solution procedures of electrical circuitry. Use summation and reciprocity rules and choose the most convenient view factors. 

When dealing with realistic geometries, the accuracy of the computed view factors should be checked. For example, according to the enclosure rule (Chap. 7), the summation of view factors from an individual surface to the enclosure must equal unity in order to satisfy conservation of radiative energy. Sample predicted sums of the view factors from surface 1 (see Fig. 18.35) are shown in Table 18.6, along with the maximum error of A,F/ [171]. Monte Carlo-generated view factors are treated as the exact values, and only Monte Carlo-generated view factors whose estimated accuracy is better than 90 percent are used in the comparison exercise. 

Figure 9.41. View factors obtained by using the summation rule " ...

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