Net radiation method for diffuse surfaces

Figure 1: Open-ended cylindrical enclosure closed with virtual surface A3.

Consider an enclosure as shown in Fig. 1. Such an enclosure is useful because it allows a mathematical description that makes use of the configuration factor summation rules, resulting in a set of final equations that can be solved for the required unknowns. The enclosure must be constructed so that any surface k views other enclosure surfaces regardless of the direction from k. This would seem restrictive for many problems; however, a virtual enclosure can always be constructed by replacing openings in a physical enclosure with a virtual surface that has the radiative behavior of the opening.

Consider the open-ended cylindrical enclosure shown in Figure 1. To make the system into a radiative enclosure, the opening at the left end is replaced with a virtual black surface A3 at T. This virtual surface has the same radiative behavior as the opening, because no radiation from the internal cylinder surfaces will reflect from the opening; the opening acts as a perfect absorber (i.e., it is black), and radiation entering the enclosure from the surroundings will be at T. Thus, the opening can be replaced with a virtual black surface at T3 = T. This strategy can almost always be used to construct a virtual enclosure from an open radiating system.

Contents

An individual surface k within the enclosure will have energy incident on it from all other surfaces in the enclosure. Let this incident radiation per unit area Ak in the wavelength interval dλ from all surfaces in the enclosure (possibly including from k itself if it is concave) be given the symbol Gλ,kdλ The G is called the irradiation on surface k. The emitted plus reflected radiation leaving the surface is called the radiosity of the surface.

Gray Surfaces

Because wavelength-dependent properties are often not available, the net radiation method is quite often applied under the further assumption that all surfaces in the enclosure are gray. In that case, the three basic equations can be integrated over wavelength to become

${J_k} = {\varepsilon _k}{E_{b,k}} + {\rho _k}{G_k} = {\varepsilon _k}\sigma T_k^4 + \left( {1 - {\varepsilon _k}} \right){G_k}$
q''k = JkGk
${G_k} = \sum\limits_{j = 1}^N {{J_j}} {F_{k - j}}$

See Main Article Gray Surfaces.

Nongray Surfaces

If the surfaces exchanging energy have wavelength-dependent properties, then the absorptivity and emissivity of the surfaces are not equal as for the gray surfaces. If all surfaces in the enclosure are close in temperature, then the total absorptivity and total emissivity of a given surface may not differ significantly; however, in systems such as solar concentrators, radiant furnaces with high-temperature heating elements, and systems with combustion sources, this will not be the case, and the wavelength dependence must be considered.

See Main Article Nongray Surfaces.

Surfaces with Varying Temperature, Radiative Flux, or Properties

One of the stipulations for use of the configuration factor is that the radiosity J must not vary across the surface described by the configuration factor. In many cases of finite surfaces, this requirement is not met; in particular, near the intersection of two surfaces, the irradiation G is likely to vary considerably, so that even if ε and T are constant across the surface, the radiosity $J = \varepsilon \sigma {T^4} + \rho G$ will vary.

See Main Article Surfaces with Varying Temperature, Radiative Flux, or Properties

References

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.