Angular Distribution of Sky Diffuse Radiance and Luminance

Jose Luis Torres and Luis Miguel Torres

1 Introduction

As a rule, available global irradiance data in weather stations are referred to the horizontal plane. On the other hand, the different solar systems are generally placed on sloping surfaces. Therefore, it is necessary to establish procedures for calculating the existing irradiance on tilted planes in which direct, diffuse and ground reflected irradiances are evaluated in separated ways. The importance of the diffuse fraction of the global radiation is commonly underestimated. However, it must be pointed that in latitudes from 40 to 60° N, this fraction may become 40% to 60% of the yearly radiation received on a horizontal plane. Although this fraction can be lower on a sloping plane, it still supposes an important percentage that should be calculated in the most exact way, especially, in climates with frequent covered skies.

Estimation of the diffuse component has been tackled in most cases by means of models that calculate the radiation on a tilted plane from radiation data on the hori­zontal plane. Existing models basically differ in the treatment each of them makes of the sky diffuse radiation. In fact, diffuse radiation is caused by a number of complex processes due to the interaction of solar radiation with the molecules and particles of the atmosphere, i. e., simple and multiple scattering and absorption phenomena, which may take place simultaneously or not at every wavelength, modify the inten­sity and the spectrum of the incident radiation in the high part of the atmosphere and redistribute the energy in different directions until it reaches the Earth surface. The physical bases of the aforementioned phenomena have been known for a long time since J. W. Strutt (later known as Lord Rayleigh) in 1871 set the physical laws that govern the light dispersion for very small particles and later in 1908, Mie proposed his theory for bigger spherical particles. All the described phenomena cause that the

Jose Luis Torres

Public University of Navarre, Pamplona, Spain, e-mail: jlte@unavarra. es Luis Miguel Torres

Public University of Navarre, Pamplona, Spain, e-mail: lmtorresgarcia@gmail. com diffuse radiation has an anisotropic nature, highly non-uniform, and that the energy received from the different areas of the sky vault may be different. The newest mod­els for calculating radiation on the tilted plane from radiation on the horizontal plane provide approximations for this physical reality. Three areas of brightness more or less differentiated depending on the insolation conditions are considered.

An alternative to this procedure consists of carrying out the calculation of the dif­fuse radiance on a sloping plane by integrating the sky radiance distribution coming from the part of the sky that is “seen” by the said plane. This alternative is particu­larly interesting for estimating the radiation on planes placed in urban environments or complex terrains. In these situations, the presence of obstacles more or less close to the plane of collection means that some parts of the sky vault may not be seen by the said plane at given moments of the day and, as a consequence, it is necessary to know the radiance corresponding to the hidden area in order to achieve a exact estimation of the available energy. The application of the described alternative is clear for the design of active and passive solar systems as well as for determining the thermal and energetic performance of buildings. In addition, the present interest for integrating photovoltaic systems gives importance to the calculation of the ex­isting radiation on vertical surfaces due to the fact that modules are usually placed on the vertical walls of large buildings exhibiting a relatively small surface for solar exploitation on their roofs.

Moreover, the sky radiance and luminance have the same origin and nature (Luminance is directly related to radiance through the luminous efficacy), and the relative distributions of both magnitudes are almost identical, being different their absolute values. As a consequence, there is a parallelism between the angular distri­bution models of radiance and luminance in the sky vault. Luminance models can also be used for designing illumination systems. In fact, different computer tools, as Radiance, take into consideration the distribution of luminance in their calculations of both outside and inside illumination.

The best way of knowing the radiance or luminance distribution in the sky is by measuring it. Nevertheless, there are very few places with the said existing mea­surements (or even registrations of illuminance on the horizontal plane) whereas, generally, there are irradiance data available. Therefore, the use of mathematical models of angular distribution of radiance or luminance becomes essential in most places and there is where their importance lies.