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. The radiative field generated by a half disk of radius 3m. Over the perpendicular plane that contains the straight edge of the disk in a grid of 10 by 10 m.

Such new configuration factors can also be employed in the analysis of the architectural heritage in terms of environmental values and specifically in natural lighting; as was stated in the beginning of this chapter, several of these paradigms of architecture feature a balanced treatment of natural lighting coming from the sun but, as no calculation methods were available, their designs were merely the result of intuition and happenstance.

These new configuration factors find application in bringing light to the understanding of the said designs. In this sense, the authors would like to present the simulation cases of two epitomes of ancient Roman architecture, whose accurate radiative performance was largely unknown: the Pantheon (Figures 8, 9) and its superb baroque evolution the Church of Sant’Andrea all Quirinale (Figures 10, 11, 12).


Figure 8. The Roman Pantheon illuminated by diffuse radiation of an intensity of 10000 lumen/m2 (lux). Typical situa­tion in autumn and spring. Scale 0 to 400 lux.


Figure 9. Simulations of the lighting field inside the Pantheon under clear sky conditions.

In Figures 8 and 9, luminous radiation is dimmed and constant for the lower spaces. It is outlined that the values for the Pantheon were not significant (sometimes, under 200 lux) and this fact may have led to the introduction of vertical windows in the drum of the cylinder by late Renaissance or Baroque epochs. Differences in the peak levels are remarkable (400 lux and 1000 lux) due to the amount of energy coming from the sun and the sky; also, in figure 9 the solar penetration inside the Pantheon can be distinctly noticed due to the reflection on the left side of the drawing. These simulations were only possible; thanks to the new configuration factors for circular emitters presented in this text.

Radiative performance does not show an acute seasonal variation, but allows for sunshine to reveal certain decorative details of the structure adding to the reputation of spiritual luminous atmosphere that encompass the work of Bernini (Figures 10, 11, 12).


Figure 10. Plan of Sant’Andrea all Quirinale’s Church by Bernini (Rome) illuminated by direct solar radiation in win­ter. Values in lux (0-800)


Figure 11. Sant’Andrea all Quirinale’s Church. Transversal Section under direct solar radiation in winter. Values in lux (0-1600)

The architect and sculptor of light, Gian Lorenzo Bernini completed this masterwork, consid­ered to be his own spiritual retreat (Figure 12) and paved the way for further illumination achievements by Guarino Guarini Figure 13).


Figure 12. Sant’Andrea all Quirinale’s Church. Longitudinal section under direct solar radiation in winter. Values in lux (0-1600)

SAN LORENZO CHURCH TORINO (ITALY) On-site measured data Apnl 23rd. 2012 12:00 PM


Figure 13. Values measured at Guarini’s church in Torino

In the same fashion of studying radiation due to circular emitters, a building currently under construction, the new railway station at the airport of Barcelona (Spain) is briefly presented in an effort to show how simulation can help in the design process and assessment (Figures 14, 15).


Figure 14. Section of the new railway station in Barcelona. Radiative performance design by the author. Project by the architects Cesar Portela and Antonio Barrionuevo. Values in lux (0-600)


Figure 15. Plan of the railway station in autumn. Values in lux

Changing the scale for the modern requirements of transportation spaces which have become the cathedrals of our time, the author proposes a lighting design in which the oculus reaches a diameter of 30 metres and the radiative energy is distributed by means of massive aluminium louvers with a height exceeding 3 metres in total. The simulations show good values in winter and summer and an acceptable raise of temperature levels at the glazed aperture due to the solar protection and the mild climate of Barcelona.


The final case to be introduced is the Rautatalo building of 1955, by the modern Finnish master Alvar Aalto. Originally a department store, it beckoned Helsinki’s citizens by its intelligent use of luminous radiation, enhanced by conical skylights subtly adapted to the solar path in this lively northern city. (Figures 16, 17)

The latter example, the Rautatalo building, brings the reader back to the efforts of the modern movement in architecture to control radiation. With 40 circular skylights it was subsequently

adapted to many projects around the world, which generally speaking fared less well than the original for climatic and economic circumstances.

3. Conclusions

In this chapter four new configuration factors related to circular emitters have been presented. They have been deducted via direct analytical work, solving the integral according to the canonical expression of the theory of configuration factors.

In this sense, the components of the radiative field for the three directions of the space with respect to a circular emitter have been found. It is important to stress that thanks to this new factor radiative field can be assessed in any point of the space. In this way, former restrictions regarding the position of the receiving point [8],[9] have been superseded by the new expressions.

It is suggested that several complex surfaces can be estimated in a similar manner, provided that they allow for some decomposition into clusters of tangent circular elements; to perform this operation only the direction of the normal vector at each point considered is needed. With the aid of CAD software and simulation programs, such procedure is readily facilitated.

Following mathematical deduction this factors can be extended to three-dimensional emitters; the case of a spherical source is remarkable, as the viewed portion of a sphere from a receiving differential element can be assimilated to a circular emitter. Extension of this factor for a finite receiving surface, that is, a rectangle, allows for more complex calculations.

After this mathematical deduction, advances in the practical application of these new factors have been presented, mainly in the field of lighting engineering, natural lighting in architecture and thermal engineering but also in human comfort and medicine areas. The architectural examples, a set of climate-responsive buildings would remind the reader that, in order to produce universal results there is the need to consider local weather parameters.

Such a meticulous task can only be achieved in the case of solar radiation with the help of scientific Approach that we believe to have greatly facilitated by the discovery of new expressions to regulate the transfer of energy due to circular and other curved emitters and by creating and diffusing powerful and simple computer programs that successfully implement the desired algorithms.


Jose M Cabeza is grateful to the extraordinary librarians at Kansai Gaidai University (Japan). Professors Junko and Tsubasa were always very helpful. The authors would like to thank Juan Manuel Bonilla Martinez for his excellent drawing of figure 6

[1] Note that values under 0.5 can also be found for this relationship in a sort of globular cap with an area bigger than the hemisphere.

Updated: August 18, 2015 — 5:10 pm