Radiative loss processes include both the net emitted radiation from receivers as a consequence of their temperature and the reflection of some of the incident concentrated solar radiation. Surfaces emit and absorb radiation as essentially independent processes, with the net energy transfer taking place being the combination of the two. Each surface in a receiver will emit radiation in proportion to the fourth power of temperature, at a rate given by the black body emissive power multiplied by its emissivity.
Some fraction of the incident radiation will be reflected from any surfaces on which it is incident. The fraction of this radiation that is lost from the receiver will depend on the geometry. If glass covers are involved in the receiver construction, they will also introduce reflection losses that may be mitigated by anti-reflective coatings. For cavity receivers in particular, radiation reflected or emitted from one part of a receiver surface is quite likely to be incident on other parts, so calculating the net absorbed radiation requires a simultaneous solution of the whole process. Ray-tracing software typically does this for reflected incident radiation (but not emitted radiation).
Many real surfaces can be modelled as grey bodies, meaning that they have a constant emissivity across all wavelengths of interest. An important exception is the selectively absorbing surface which, in solar thermal applications, is designed specifically to have a high absorptivity in the wavelength range of solar radiation and a low emissivity in the wavelength range associated with the (infrared) radiation emitted from hot receiver surfaces. Chapter 15 covers such surfaces in detail.
For diffusely reflecting surfaces, the methods for calculating the final distribution of incident solar radiation are the same as those required to determine how the diffuse emitted radiation from hot receiver surfaces is ultimately distributed.
The radiation leaving a surface will be partly intercepted by all the other surfaces in the field of view, in proportion to the view factor (also called radiation shape factors),
Fij = the fraction of radiation leaving surface i and
reaching surface j. [2.39]
If these other surfaces reflect and absorb various fractions, working out the final distribution of absorbed energy becomes a complex problem. General presentations of radiation heat transfer can be found in Bergman et al. (2011).
The radiative energy balance of a particular (diffusely reflecting) surface element is illustrated in Fig. 2.13. In this diagram, Gi is the total irradiance
2.13 Radiation energy balance on a diffusely emitting and reflecting surface.
incident on surface i, J is the radiosity of surface i, defined as the total radiative flux (reflected plus emitted) leaving that surface, Ftj is the view factor from surface i to surface j, and At is the area of surface i. Each wavelength range (e. g. solar vs. thermal wavelengths) can be considered independently in this way.
For each surface in a radiation exchange network, an equation for the energy balance is written. Together, these equations form a linear system in terms of radiosity, and so they can be readily solved to evaluate the net heat transfer at each surface. Boundary conditions are required at each surface; the surface temperature can be fixed and the equation solved for net heat transfer, or else vice versa.
For a solar concentrator receiver, the radiation exchange between surfaces within it can be solved in this manner. The starting point is that the amount of concentrated solar radiation coming in through the aperture and striking each surface needs to be known from the optical properties of the concentrator. The aperture itself can be regarded as a black body surface to all radiation incident on it from other internal surfaces. Boundary conditions for external convective losses can also be introduced, relating surface temperature and the externally convected flux.
Analysis of simple grey-surface radiation exchange can be achieved with the free open-source software View3D (Walton, 2002). More sophisticated models incorporating coupled radiation and convection heat transfer are described in Section 2.5.2.
In a simplified model, if the aperture is treated as a single surface at the average receiver temperature, interacting only with the environment, then the emitted radiation loss will be given by
Qrad = oAeFRs( – TL), [2.40]
where Frs is a simplified shape factor between receiver and surroundings. In a simplified model, reflection losses may be approximated using a single effective net absorptivity for the receiver aperture area. Thus
Qref = (1 – a)AQsol. [2.41]