Radiation heat transfer down an elongated spheroidal cavity

Abstract

AbstractEffects of elongation on the radiation heat transport down a spheroidal cavity, located in a conducting solid with a diffusely reflecting cavity–solid interface, are examined. An effective conductivity λe and a void radiation conductivity λr are obtained as a function of cavity eccentricity α; and surface emissivity ε. To facilitate the calculations and produce readily applicable equations, a rigorous variational principle is used. Exact solutions are generated in the neighborhood of the spherical cavity (α2 → 0) for any ε > 0, a long needle‐shaped void (α2 → 1) for any ε > 0, and a perfect reflector (ε → 0) for arbitrary elongation (0 ≤α2 ≤ 1). Significant differences arising from the shape change are observed. The α2 → 0 edge demonstrates a linear increase in λr with ε. At the opposite edge α2 → 1 and positive ε, λr is a horizontal line independent of ε, much like the long cylinder, whose conductivity is a factor of 32/(9π) (= 1.13) larger. In the neighborhood of ε 0, λr is always zero for any 0 ≤ α2 ≤ 1. The emissivity slope for ε → 0 starts from unity at α2 = 0 and increases monotonically with elongation to a singularity 3π[16(1 – α2)]‐1 as α2 → 1 for the long needle.