*3.1. Fundamentals*

A variety of mathematical models has been employed to determine the potential of the physical components at the nave of the church and to simulate its daylighting fields. The basic operations involve a configuration factor algebra newly developed by the authors. Another important innovation of the method proposed is that it incorporates both the direct and reflected component of light for curved surfaces, as we will discuss in more detail below. This procedure, a further advance of the research of Yamauchi and the American engineers H. Higbie and Levine [26], completes Lambert's corollary of reciprocity [27], representing a more generalized version of the form factors concomitant with those used in heat transfer for radiometric systems [28].

The model extends the radiation properties of diffuse sources to luminous exitance of all kinds of building surfaces that are subsequently considered as radiative emitters. Once the initial intensity of each surface is known and the primary shape of the exchangers is fixed, successive interchanges are obtained until a balance of the required accuracy is achieved.

The configuration factor is a dimensionless fraction (varying from zero to one) that originated in the work of J. Lambert (1760). It expresses the amount of radiant flux *Eb* that arrives from a given surface to others directly exposed to the first, if both are perfectly diffuse emitters. It depends solely on the position, size and form of the given surfaces. Therefore, we could term it as geometrical and the corresponding domain of study of these matters is geometric optics [29]. Mathematically speaking, we can define it by the following integral equation whose terms appear in Figure 7.

$$F\_{12} = \frac{1}{A\_1} \left[ \int\_{A\_2} \int\_{A\_1} \frac{\cos \theta\_1 \cos \theta\_2}{\pi r^2} dA\_1 dA\_2 \right] \tag{1}$$

where *Ebi* is the radiant power emitted by the corresponding surface 1 or 2 (lumen/m2). *Ai* is the area of surface, *dAi* is the differential of area (m2); *r* is the distance radio-vector (m). *θi* is the angle between radio-vector at differential element *i* and the normal to the surface (radians), *dΦ*1–2 is the differential of radiant flux from surface 1 to 2 (lumen).

The previous expression states that radiant interchange for every given form depends on its shape and its relative position in the three-dimensional space. From the times of Lambert to our days, researchers and scientists in the fields of geometric optics and radiative transfer have sought to provide solutions to the canonical equation in the figure for a variety of forms [6]. This implies no minor feat, since the said equation leads, in most cases, to a quadruple integration and the fourth degree primitive of even simple mathematical expressions often entails lengthy calculations [4].

The configuration factors *Fij* possess the well-known property by virtue of which,

$$\sum\_{j=i}^{n} F\_{ij} = 1,\tag{2}$$

For any surface *Ai* and this implies within an enclosure of *n* surfaces,

$$F\_{11} + F\_{12} + F\_{13} + \dots + F\_{1n} = 1.0,\tag{3}$$

If only planar surfaces are considered *F*11 = 0 (a flat surface does not see itself). But in the particular case of baroque churches with many curved surfaces like spheres and cylinders, the factors with repeated sub-indexes tend to have values which have not been described in the literature in an exact formulation. Such is a significant part of the novelty and opportunity of our approach and simulation program, one of the main reasons behind this article.

**Figure 7.** Equation parameters. Source: Joseph Cabeza-Laïnez.

The authors have devoted more than a decade of their careers to solve the former integral equation in an exact manner and the main outcome is that both the accuracy and speediness of computational time with these procedures are radically enhanced.

The second fundamental property of configuration factors is lack of symmetry. Since, due to Lambert's law, *A*1*F*12 = *A*2*F*21 consequently, *F*12 = *F*21, unless both surfaces have the same size.

The authors have established based on these two principle axioms a complete algebra for the first time in the literature that includes not only addition but also scalar product. We judge that to delve on this product exceeds the scope of the present article. Suffice to say that it permits the analysis of concatenated curved surfaces (previously untreated).

For the particular case of rectangular windows, if we are discussing their effects in the daylighting performance of a building, we should be utterly aware of how and where and especially under which conditions the light reaches the inner spaces. With this objective in mind, expressions for the configuration factor between rectangular inclined surfaces have been integrated at several angles to be used for daylighting simulation (Figure 8).

*Sustainability* **2018**, *10*, 3352

The window is considered to be, a uniform diffuse luminous source (its luminance *L* in lumen is constant for a given instant of time). As a result, we can obtain the following and more general expression for its illuminance *E* (lumen/m2):

$$E = \frac{L}{2} \left[ \begin{array}{c} \frac{a \cos \varphi - y}{\sqrt{a^2 + y^2 - 2ay \cos \varphi}} \text{arctg} \frac{b}{\sqrt{a^2 + y^2 - 2ay \cos \varphi}} + \\ + \frac{b \cos \varphi}{\sqrt{b^2 + y^2 \sin^2 \varphi}} \text{arctg} \frac{a \sqrt{b^2 + y^2 \sin^2 \varphi}}{b^2 + y^2 - ay \cos \varphi} + \text{arctg} \frac{b}{y} \end{array} \right] \tag{4}$$

Angle *ϕ* defines the relation with the normal to the considered work-plane. If *ϕ* = 90◦, cosine is 0 and the expression equates the formula for vertical rectangles to a point-source.

$$E = \frac{L}{2} \left[ \text{arctg} \frac{b}{y} - \frac{y}{\sqrt{a^2 + y^2}} \text{arctg} \frac{b}{\sqrt{a^2 + y^2}} \right],\tag{5}$$

The above expression (4) alters if we are trying to find the illuminance of the perpendicular surface; consequently, the solving equation would be in this case,

$$E = \frac{L}{2} \left[ \begin{array}{c} \frac{a \sin \phi}{\sqrt{a^2 + y^2 - 2ay \cos \phi}} \text{arctg} \frac{b}{\sqrt{a^2 + y^2 - 2ay \cos \phi}} + \\ + \frac{b \sin \phi}{\sqrt{b^2 + y^2 \sin^2 \phi}} \text{arctg} \frac{a \sqrt{b^2 + y^2 \sin^2 \phi}}{b^2 + y^2 - ay \cos \phi} \end{array} \right],\tag{6}$$

It is easy to notice in this situation that if *ϕ* = 90◦, this would give cos*ϕ* = 0 and sin*ϕ* = 1, substituting these values in the former equation:

$$E = \frac{L}{2} \left[ \frac{a}{\sqrt{a^2 + y^2}} \text{arctg} \frac{b}{\sqrt{a^2 + y^2}} + \frac{b}{\sqrt{b^2 + y^2}} \text{arctg} \frac{a}{\sqrt{b^2 + y^2}} \right],\tag{7}$$

This expression represents the exchange between a horizontal rectangle and a point-source parallel surface.

**Figure 8.** Configuration factor between a rectangle and a point that belongs to a tilted plane by the angle *ϕ*. Source: authors.

### *3.2. The Projected Solid Angle Principle*

The former calculations allow us to find configuration factors that consist of geometric parameters or proportions. Thus, it seems reasonable that the desired values can also be determined through graphic procedures like those employed in geometry. In science, something that may sound reasonable is not always easy to prove but after some time researching we could arrive at the proper demonstrations.

The advantages for architects and researchers are obvious compared with formulas because if during the calculations doubts appear due to the difficulty of the lighting problem, this mathematical model provides fast and easy visualization and it is suitable for BIM protocols.

In order to obtain the flux transfer between a surface and a point, we could simply draw a cone whose vertex is the study-point and its base the surface. The intersection of the said cone with a sphere of unit radius (*r* = 1) is then projected onto the reference plane (horizontal, vertical, etc.). The area inside this projection, divided by the projected area of the whole sphere on the same plane (i.e., *π*), gives the value of the configuration factors much in the same way as with the analytical methods already described (Figure 9) [30].

To summarize, view factors are dimensionless quantities or ratios that we can expressed as a relation of areas. The first area is a projection and the second is the total surface area expressed in circular terms *π*.

**Figure 9.** The light-cone in this figure cuts the area *σ* from the unit sphere. Orthogonal projection of *σ* onto the illuminated or irradiated plane gives the area *σ*", this surface divided by *πr*<sup>2</sup> = *π* equates the value of the configuration factor. Source: authors.

The main consequence that we can draw from this fact is that the problem of finding the configuration factor will have a unique solution regardless of its complexity because the area of a projection always produces a definite value.

Moreover, the factors can be understood as projections and thus they possess the additive property, only recently we have defined the scalar product of configuration factors (see above). Addition is useful when dealing with several light sources, if we can add their effects or under special conditions multiply them. The average of this geometric proportion extended over the corresponding surface equates the configuration factor, which in some cases it is non-trivial to find with analytical expressions, due to calculus constraints.

The useful corollary is that we can alternately use graphic or analytical methods. In either way, we assume that we are able to solve one of the fundamental problems in radiative transfer by means of geometric procedures. This signifies that the geometric form is very important in this type of problem and not all building shapes may receive the same performance from a radiation standpoint.

In this regard, the centralized plan of Saint Louis' church contributes to increase the radiative transfers between the interior surfaces. This type of geometry with axial symmetry generates greater configuration factors than a parallelepiped surface. Theretofore, higher daylighting levels are obtained with the same reduced proportion of windows to total surface area.
