Evaluation of Approximate Expressions to Calculate the Area of the Intersection between a Sphere and a Cylindrical Plane

Cylindrical packed beds of spheres are found in many industrial and practical applications where heat and mass transfer and fluid flow occur. A proper understanding of the porous structure is imperative for the analysis and design of the performance of packed beds. The porosity of the packed bed and in particular the radial variation in porosity is of interest. When the positions and sizes of the spheres in the packed bed are known, the areas of the intersections between the spheres and selected cylindrical planes can be used to obtain the radial variation in porosity. The aim of the study is to evaluate the performance of approximate expressions that had been derived to calculate the intersection areas. Firstly, the ability of the approximate expressions to calculate the intersection area is evaluated by considering several typical sphere–cylindrical plane configurations. Secondly, the application of the approximate expressions to obtain the radial variation in porosity for a selection of cylindrical packed beds is evaluated. It is concluded that the approximate expressions should only be applied to packed beds with aspect ratios larger than 6 and for radial positions larger than 1.5 sphere diameters from the centre line of the cylindrical packed bed.