## Graphical visualization

To help visualize the results of this research, some formulas have been programmed by the authors in Matlab® computational language, which greatly enhances understanding of radiative exchange between emitting surfaces and receiving planes. 3D graphs have been produced for a generic semicircular emitter.

Figure 7 shows a generic semicircular emitter that gives energy to a perpendicular plane in its base. Thanks to this new configuration factor, several radiative properties for these shapes can be clarified. For instance, a semicircular emitter is not capable of transferring more than 50% of its energy to a perpendicular plane; this is particularly important in some engineering lighting applications, such as lighted vaults or tunnels. Figure 7...

## Configuration factor between a sphere and a plane

Extending the previous deduction to a finite rectangle located at a certain distance to the sphere in a random position (figure 6), a new unknown factor has been deducted:

 F = 1 A1-A2 4p

 (56)

 7-‘Jxf+yf+ -2

 -■Jx 2 + yj2 + z 2 Zy]xl + y2 + z 2

 z x22 + y22 + z2 Figure 6. Configuration factor between a sphere and a parallel plane ## Extension to three dimensional emitters – Configuration factor between a sphere and a differential element placed at a random position

After analyzing the previous form factors for the circle, a new question can be deducted. A sphere can be considered, in terms of radiative transfer as a circle , as the viewed area of the said sphere from a distant point equals always a circle, because only half of the emitter is visible. Let us consider as emitting source a sphere of radius r, and a differential element, placed randomly in space at a distance (x, y,z), referenced to the three coordinate directions as shown in figure 5:  Figure 5. Calculations parameters for the sphere and a differential element at a random position

 The differential element, as in former cases, is defined by its normal, and it is necessary to find the radiation vector Fr impinging on it...

## Resolution of the integral for the third coordinate plane  Being radiation a vector, the resolution for a third coordinate plane that obviously cuts the emitting circle in two halves is required; the outline of the integral in this case yields:

In this particular case, the limits of the integral cannot be extended to 2n, as the value would be nil. If (41) is integrated with respect to 0, in the numerator the derivatives of cos0, – sin0 could be found. Therefore by making this change:

t = cosQ dt = -sin0d0 (42)  Integral (41) can therefore be expressed as:  Figure 4. Calculation parameters for the semicircle

Taking out all the constants, and integrating with respect to r, the primitive is just the quotient of the numerator:

 -i

 -hr2

 1

 hr

 1

 1

 (44)

## Integration process for circular emitters

3.1. Direct integration for a differential element to a circular disk on a plane parallel to that of the element

Terms depicted in figure 2 are:

d: vertical distance between the center of the emitting circle and the plane XY.

b: horizontal distance between differential element dAj and the plane ZX that contains the said circle.

S: Distance between differential elements (in the canonical equation (1) of the configuration factor, it is denoted as r, but in order to differentiate it from radius of the disk (emitting surfa...

## Outline of the problem & objectives  The reciprocity principle enunciated by Lambert in his paramount book Photometria, written in Latin (Lambert, 1760), yields the following well-known integral equation: 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...

## New Computational Techniques for Solar Radiation in Architecture

Jose M. Cabeza-Lainez, Jesus A. Pulido Areas, Carlos Rubio Bellido, Manuel-Viggo Castilla and Luis Gonzalez-Boado

1. Introduction

Many architectural examples rank among masterpieces for its beautiful and harmonious use of solar radiation. However, their creation had to rely solely on intuition because they possessed a curvilinear nature. As the necessary tools required for evaluating shapes derived from the sphere or the circle were not available, such forms could not be assessed.

Circular emitters represent an important issue not merely in architecture but in the field of configuration factors calculation...