View factor reciprocity relation

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 relation is referred to as the reciprocity relation or the reciprocity rule, and it enables us to determine the counterpart of a view factor from a knowledge of the view factor itself and the areas of the two surfaces. When determining the pair of view factors Fj >j and it makes sense to evaluate first the easier one directly and then the more difficult one by applying the reciprocity relation. 

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... 

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 ... 

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... 

The scheme of interpolation followed in the iterative regression strategy explained above is not very simple from a practical point of view. The method of triangles requires at least four different equations, one for each established subspace. The use of a single equation to describe the retention behavior of a solute, in the whole variable space, seems to be more convenient to predict the retention of a solute in any mobile phase, with a minimum effort. Table 8.1 shows some of the models (equations) that have been considered, where the logarithm (eqs. 8.15-8.22), or the reciprocal of the retention factor (eqs. 8.23-8.30) are related to micelle concentration and volume fraction of organic modifier. 

Figure 4 Top view (A) and side view (B) of a lattice of long, linear molecules with bulky head groups. The molecules tilt in the (Nearest-Neighbour) direction, and the lattice is expanded in the Ur direction. (C) shows a top view of the corresponding reciprocal lattice and (D) is a side view of it, showing the plane of maximal structure factor that is orthogonal to the long molecular axes in (B). (E) is the resulting 2D powder pattern. (A) also illustrates the relation between the lattices (odh = dh, Vdh 120°) and (or 6r, yr = 90°). Adapted from ref. [16].

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