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. A portion of the irradiation striking surface k will be reflected from k. Let k be a diffuse surface. In that case, the reflected radiative flux from Ak will be $d{{q’’}_{\lambda, ref, k}} = {{\rho}_{\lambda, k}} {G_{\lambda, k}}d \lambda \qquad \qquad(1)$

The spectral radiative energy flux leaving Ak will be made up of this reflected flux plus the radiation emitted by the surface. The emitted plus reflected radiation leaving the surface is called the radiosity of the surface, and is given the symbol J, so that ${J_{\lambda ,k}} = {\rm{emitted + reflected flux = }}{\varepsilon _{\lambda ,k}}{E_{\lambda b,k}} + {\rho _{\lambda ,k}}{G_{\lambda ,k}}\qquad \qquad(2)$

Because the surfaces are assumed to be diffuse, both the emitted and reflected flux are independent of direction, and the use of configuration factors is justified in finding energy exchange among surfaces.

We are now in a position to perform a radiative energy balance on surface k. The factors we can use are shown in Fig. 1. The energy balance is then $d{q''_{\lambda ,k}} = {\rm{(outgoing - incoming radiation)}} = {J_{\lambda ,k}}d\lambda - {G_{\lambda ,k}}d\lambda \qquad \qquad(3)$

The dq''λ,k is the heat flux in wavelength interval dλ that must be provided to surface k in order to balance the difference between the outgoing and incoming radiation fluxes.

Equations (2) and (3) provide two equations involving the factors J,G,q" and Eb. It is required that either Eb or q" (effectively temperature or radiative heat flux) be specified as a boundary condition. Nevertheless, we are left with two equations in three unknowns for each surface k, so an additional relation is required. Here, we can invoke the concept of an enclosure to provide the additional relation. The factor Gλ,k is made up of the radiation incident on Ak from all N surfaces making up the enclosure. The radiation leaving each of these surfaces is made up of the emitted plus reflected radiation (the radiosity.) Thus, the irradiation on Ak is ${G_{\lambda ,k}}{A_k} = \sum\limits_{j = 1}^N {{J_{\lambda ,j}}} {A_j}{F_{j - k}}\qquad \qquad(4)$

or, invoking reciprocity ${F_{1 - 2}} = \int_{{A_1}} {{F_{d1 - 2}}} d{A_1} = \int_{{A_1}} {\left[ {\int_{{A_2}} {d{F_{d1 - d2}}} } \right]} d{A_1} = \int_{{A_1}} {\left[ {\int_{{A_2}} {\frac{{\cos {\theta _1}\cos {\theta _2}d{A_2}}}{{S_{1 - 2}^2}}} } \right]} d{A_1}$ <center>

to obtain:

<center> ${G_{\lambda ,k}} = \sum\limits_{j = 1}^N {{J_{\lambda ,j}}} {F_{k - j}}\qquad \qquad(5)$

The three equations (2), (3) and (5) are the basis of the net radiation method for treating radiative transfer.

Note the restrictions that are implicit in the method so far: all surfaces making up the enclosure are diffuse; otherwise the configuration factors could not be used in eqs. (4) and (5). Further, the values of Gλ,j and Eλb,j must be invariant across each surface j, so that Jλ,j will be uniform and the configuration factors can be used. This condition is difficult to meet exactly in enclosures with large surfaces, where in particular the irradiance G will probably vary across a large surface. In that case, surfaces must be subdivided until the requirement of uniform radiosity from each surface is met.

An alternative approach to radiative exchange in enclosures is the radiative network method (Oppenheim, 1956). This approach is often used in undergraduate texts (e.g., Incropera et al., 2007). It is based on the same fundamental assumptions and relations as in the net radiation approach used here, but is somewhat more difficult to extend to more than a few surfaces and to cases with wavelength dependence.

## References

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

Incropera, F.P, DeWitt, D.P., Bergman, T.L., and Lavine, A.S., 2007, Fundamentals of Heat and Mass Transfer, 6th ed. John Wiley & Sons, New York, NY.

Oppenheim, A.K., 1956, “Radiation Analysis by the Network Method, Trans. ASME, Vol. 27, pp. 725–736.