# Innovative Tool to Determine Radiative Heat Transfer Inside Spherical Segments
## 2 Materials and Methods--General Departing Equation  
The canonical equation that regulates a general exchange of radiant energy for arbitrary and non-planar surfaces, as depicted in Figure 1, can be written in the following form (Equation (1)):  
\[\mathrm{d}\phi_{12}=(\mathrm{E}_{1}-\mathrm{E}_{2})\cos\theta_{1}\cos\theta_ {2}\frac{\mathrm{d}A_{1}\mathrm{d}A_{2}}{\pi r_{12}^{2}} \tag{1}\]  
Equation (1) derives from the reciprocity theorem first stated by [PERSON] [14; 15; 16] and it gives the rate of diffuse radiant energy, in W/m\({}^{2}\), that is transmitted into the void from each of the surfaces, namely, \(\mathrm{E}_{1}\) and \(\mathrm{E}_{2}\). The angles \(\theta_{1}\) and \(\theta_{2}\) are formed between the normal and the respective surfaces and the segment that links any arbitrary couple of points termed \(r_{\mathrm{ij}}\) (see nomenclature). Figure 1: Radiation interchanges for a set of arbitrary curved surfaces, called \(\mathrm{A}_{1}\) and \(\mathrm{A}_{2}\). Thorough solving of Equation (1) implies four rounds of integration [4]. The present author has been able to finish them with much difficulty for the following shapes: perpendicular rectangles with a common edge, parallel rectangles, parallel coaxial disks, and a parallel coaxial circle under a sphere [8]. The last two are possible since the polar coordinates coincide for the involved coaxial elements. Other authors have reported a similar solution but they have not fully demonstrated their integral procedures [16; 17; 18], and some of them seem inconclusive or incomplete although they are frequently quoted [19; 20; 21]. For other different positions of the surface sources, an exact solution has not been achieved yet [5]. Neither for triangles or irregular fragments of rectangles and circles or even for the familiar form of the sphere. Intermediate steps to proceed only with the first and the second round of integration have been proposed [22]. They usually end at a single point or elementary area that belongs to the receiving surface. Some of them are recorded under the misleading manner of nomograms at catalogues of radiation view factors [23; 24]. Such results are somewhat helpful but they are by no means general and can be confusing in many situations. Subsequently, if we try to apply the former calculations to the interior of a sphere, we can identify two important postulates for this domain. To begin with, suppose for instance that the fragment under study is a simple hemisphere (Figure 2), in this situation the half-sphere encloses a space made of two surfaces, that is, a cap and a circular base [25]. Their respective areas amount to A\({}_{1}\) = 2\(\pi\)R\({}^{2}\) (cap) and A\({}_{2}\) = \(\pi\)R\({}^{2}\) (disk). Under such circumstances, it is not feasible to extend double integration to the disk and the hemisphere at the same time, or at least not in the way that we would apply for parallel rectangles running only through coordinates x and y, because Cartesian coordinates are interchangeable and polar coordinates are not, since they need to originate from the same central point and that makes them unmanageable. However, by the principle of conservation of energy, we know that all radiant interchanges inside a closed volume must add to 100% or the unity [6]. Thus, the orderly array of form factor algebra for the two surfaces implies,  
\[\text{F}_{11}+\text{F}_{12} =1; \tag{2}\] \[\text{F}_{21}+\text{F}_{22} =1. \tag{3}\]  
However, the disk is planar, and in Equation (3) we know that F\({}_{22}\) equates to zero. Thus, F\({}_{21}\) must be one in order to fulfil the equation. Figure 2: Form factor calculation applied to a hemisphere, Surf. 1 (eq: x\({}^{2}+\) y\({}^{2}+\) z\({}^{2}=\) R\({}^{2}\), with origin at 0,0,0) with a circular base, Surf. 2 (eq: x\({}^{2}+\) y\({}^{2}=\) R\({}^{2}\)) at the same origin.