**New Computational Techniques for Solar Radiation in Architecture**

Jose M. Cabeza-Lainez, Jesus A. Pulido Arcas, Carlos Rubio Bellido, Manuel-Viggo Castilla, Luis Gonzalez-Boado and Benito Sanchez-Montanes Macias

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/60017

## **1. Introduction**

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[8] H. T. Lin, Ecology, Energy Saving, Waste Reduction and Health, National Cheng Kung University. Taipei: Architecture and Building Research Institute; 2011.

and Building Research Institute; 2010.

156 Solar Radiation Applications

and Building Research Institute; 2009.

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. The circle form is present in a variety of devices and emitters that find ample application in the realms of thermal engineering, daylighting in architecture and artificial light, amongst others. In the past, several factors have been found for specific positions of the unit area in relation to the sources of such surface, centered with respect to the circle, but not for a generic location whether parallel or inclined. In this respect, perpendicular semicircles have been totally disregarded. As a result, calculation for the said configuration factors was sustained by iterative methods, which do not provide the desired accuracy in every situation and also require considerable effort and time in terms of computational capacity.

In previous researches, new configuration factors have been devised for complex forms and shapes, such as the paraboloid, the ellipsoid, the sphere and the straight cone, which are ever present in architecture and engineering. What is more, several configurations of volumes that include similar elements could also be assessed by virtue of adroit mathematical deduction. As a result, researchers and designers were provided with new configuration factors, so that the design process is entirely freed from iterative methods.

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In this chapter, an exact analytical solution derived from complex double integration is presented. The expression obtained significantly soothes the calculation of the configuration factor between a circular emitter and a point that lies in a plane located at any position to the former, not only in an axis perpendicular to its center. Those results were checked against more conventional formulas. Based on such calculus procedures, an entirely new factor for the semicircle to a perpendicular plane that contains the straight edge has been deducted. Likewise, the solution has been converted into an original algorithm and programmed in simulation software developed by the authors so that interactive maps of the radiative field can be visualized in a consistent and accurate way. Thus, computer simulation techniques, engineering and image applications will be greatly enhanced and benefitted.

#### **2. 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:

$$\mathcal{D}\_{1-2} = \left(E\_{b1} - E\_{b2}\right) \int\_{A\_1 A\_2} \int \cos\theta\_1 \cos\theta\_2 \frac{dA\_1 dA\_2}{\pi r^2} \tag{1}$$

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

**Figure 1.** The reciprocity principle and quantities' significance for surfaces A1 and A2

From the times of Lambert to our days, researchers and scientists in the fields of Geomet‐ ric 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 mathemat‐ ical expressions implies lengthy calculations. For this reason, direct mathematical calcula‐ tion 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 configura‐ tion 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.
