Outline of the problem & objectives

image279 Подпись: (1)

The reciprocity principle enunciated by Lambert in his paramount book Photometria, written in Latin (Lambert, 1760), yields the following well-known integral equation:

Подпись: Figure 1. The reciprocity principle and quantities' significance for surfaces Aj and A;

Relevant terms in equation (1) are depicted in figure 1.

From the times of Lambert to our days, researchers and scientists in the fields of Geomet­ric Optics and Radiative Transfer have striven to provide solutions for the canonical equation 1. This is no minor feat, since the said equation will lead in most cases to a quadruple integration and to be sure the fourth degree primitive of even simple mathemat­ical expressions implies lengthy calculations. For this reason, direct mathematical calcula­tion of circular emitters was avoided, and only expressions for some particular position with respect to the emitter were available. In this sense, a detailed catalogue of configura­tion factors is provided online in [1], but with respect to circular emitters, only specific ones where the receiving point location is restricted to an axis passing through the center of the emitting circle are included[2],[3].

Considering the importance of these emitters and its wide application in architectural design and engineering, the objectives of the research aimed at establishing precise mathematical expressions for the required configuration factors. Such procedure entails exact analytical solutions of the quadruple integral, in order to yield expressions that barely include geometric parameters.

That said, a circular source that emits with constant power has been considered; receiving points are located freely in any parallel or inclined plane. Starting from canonical equation (1) and following mathematical procedures, new configuration factors are developed for these surfaces.

Updated: August 16, 2015 — 6:05 am