An accurate evaluation of geometric view factors for modelling radiative heat transfer in randomly packed beds of equally sized spheres

This paper proposes an effective general numerical procedure for the accurate evaluation of the geometric view factors between equally sized spheres in a randomly packed assembly. The procedure is a novel combination of three key techniques: the Tanaka integral expression for the view factor between two equal spheres, the product form of the Fibonacci integration scheme for integrals on spheres, and the non-uniform variable scaling. The Tanaka integral not only provides a simple integral to numerically compute the exact value of the view factor between two unit spheres without a blockage, but also plays a fundamental role in improving the computed shaded view factor between two spheres subject to any blockage. The product Fibonacci integration scheme, essentially a ray-tracing method, appears to be a low cost but sufficiently accurate cubature for evaluating the dual-integral defining the view factor. The use of a particular non-uniform scaling function very favourably modulates the properties of the view factor integrals, and thus substantially improves the solution accuracy of the Fibonacci integration scheme. Several practical issues associated with the accuracy and efficiency of the procedure are discussed, and test cases are provided to illustrate the applicability of the procedure.