An iterative formula for the directional hemispherical apparent reflectivity of an opaque isothermal cavity with an arbitrary surface is derived on the basis of an integral equation representing the apparent reflectivity. Nested integrals in the formula are computed by the hit-or-miss Monte Carlo method. The formula is applied to right circular cylinders whose surfaces are diffuse and to cylindrical cones whose surfaces are characterized by a uniform specular-diffuse model. For the cylinders, a table is given from which one can easily calculate the apparent emissivities of the center of the bottoms for arbitrary surface reflectivities. An analytical recalculation of I2 of Quinn gives corrected values that agree with those produced by the Monte Carlo method. For the cylindrical cones the directional apparent emissivities of the bottoms are obtained by varying the apex angle and the fraction of specular component. We point out that angles of 120° and 70° have advantages even for cavities with moderate specular components. Results are analyzed by studying numerically the individual terms of the iterative formula. The validity of an approximation for the directional apparent emissivity is discussed in terms of the constant asymptotic distribution of impact points of rays in cavities.
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