Question 6.13: A cylindrical cavity open at one end has a specularly reflec...
A cylindrical cavity open at one end has a specularly reflecting cylindrical wall and base (Figure 6.28a). Determine the fraction of radiation from ring element dX_1 that reaches dX by means of one reflection from the base with reflectivity ρ_{s,1} and one reflection from the cylindrical wall with reflectivity ρ_{s,2} .
As shown in Figure 6.28b, for this geometry the reflected radiation from the base can be regarded as originating from an image of dX_1. The second reflection occurs from an element of width dX/2 midway between the image dX_1 and dX. The desired radiation fraction is given by the configuration factor from the image dX_1 to the dashed ring area dX/2 multiplied by the two reflectivities:
\rho_{s,1}\rho _{s,2}dF_{dX_1-dX}=\rho_{s,1}\rho _{s,2}\left\lgroup1-\frac{[(X+X_1)/2D]^3+3(X+X_1)/4D}{\left\{[\left(X+X_1\right)/2D ]^2+1\right\}^{3/2} } \right\rgroup\frac{dX}{2}
Another type of curved specular surface of practical importance is a paraboloidal mirror such as in a solar furnace. The mirror axis is aligned in the direction of the sun and a receiver is placed at the mirror focal plane. It is desired to estimate the receiver temperature. Information on concentrators for solar furnaces was provided by Cobble (1961) and Kamada (1965). An inverse analysis for designing a mirror to achieve a specified intensity distribution at the receiver is in Zakhidov (1989). In Maruyama (1991, 1993), the angular distribution of intensity emitted from systems of cylindrical black emitters with various reflector shapes (circular arc, parabolic, and involute) are found by ray tracing. The reflector surfaces are assumed to be either perfect or metallic reflecting surfaces with reflectivity predicted by electromagnetic theory. In Maruyama (1993) there is an experimental verification of predictions for circular arc and involute reflectors. The involute is effective for providing a uniformly distributed radiation source.
