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. The net radiation equations for a small wavelength interval dλ were given in the following equations:

${J_{\lambda ,k}} = {\rm{emitted + reflected flux = }}{\varepsilon _{\lambda ,k}}{E_{\lambda b,k}} + {\rho _{\lambda ,k}}{G_{\lambda ,k}}\qquad \qquad(1)$
$d{q''_{\lambda ,k}} = {\rm{outgoing - incoming radiation}} = {J_{\lambda ,k}}d\lambda - {G_{\lambda ,k}}d\lambda \qquad \qquad(2)$
${G_{\lambda ,k}} = \sum\limits_{j = 1}^N {{J_{\lambda ,j}}} {F_{k - j}}\qquad \qquad(3)$

As for the simplified case of gray surfaces, these can be combined (retaining the spectral dependence) to give

${J_{\lambda ,k}}d\lambda = {E_{\lambda b}}d\lambda - \frac{{\left( {1 - {\varepsilon _{\lambda ,k}}} \right)}}{{{\varepsilon _{\lambda ,k}}}}d{q''_{\lambda ,k}}\qquad \qquad(4)$

And

${J_\lambda }_kd\lambda = d{q''_{\lambda ,k}} + \sum\limits_{j = 1}^N {{J_{\lambda ,j}}} {F_{k - j}}d\lambda \qquad \qquad(5)$

The configuration factors are not wavelength dependent, but their use does require that all surfaces be treated as diffuse. As for gray surfaces, the radiosity can be eliminated from the equation set, resulting in

$\sum\limits_{j = 1}^N {{F_{k - j}}({E_{\lambda b,k}} - } {E_{\lambda b,j}})d\lambda = \sum\limits_{j = 1}^N {\left[ {\frac{{{\delta _{kj}}}}{{{\varepsilon _{\lambda ,j}}}} - {F_{k - j}}\frac{{\left( {1 - {\varepsilon _{\lambda ,j}}} \right)}}{{{\varepsilon _{\lambda ,j}}}}} \right]} d{q''_{\lambda ,j}}\qquad \qquad(6)$

If all temperatures Tk are given as boundary conditions, then eq. (6) can be solved by dividing the spectrum into M finite wavelength intervals Δλm, and solving for the Δq"j, m in each interval. The total heat flux for each surface is then found from

${q''_{j}} = \sum\limits_{m = 1}^M \Delta {q''_{jm}} \qquad \qquad(7)$

For the case of all surface temperatures known, the problem is reduced to solving the gray problem in M wavelength intervals, and then summing the results.

Suppose now that, rather than specifying the temperatures of all surfaces, the total heat flux q"k is given as a boundary condition on one or more surfaces. Equations (1) through (7) still apply: Now, however, the left hand side of eq. (7) is a known value for the one or more surfaces with specified radiative heat flux. To solve for the unknown temperatures on these surfaces requires an iterative solution. The temperatures for the flux-specified surfaces must be guessed, and the M values of all heat fluxes are then found. The computed spectral flux values for the flux-specified surfaces are substituted into the right-hand side summation in eq. (7), and the result checked against the specified radiative flux for that surface. If it differs, then new temperatures for the flux-specified surfaces are assumed, and the solution is repeated until a converged result is reached.

References

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