Optimum Size and Heat Transfer and Velocity Correlations of the Outer and Inner Surfaces of Vertical Hollow Polygonal Cylinders in Natural Convection

Laminar natural convection heat transfer from vertical hollow polygonal cylinders with a wide range of cross-sectional areas is investigated. The buoyancy-driven three-dimensional (3D) flow around hollow polygonal cylinders immersed in quiescent ambient air with equal outer and inner surface temperatures is analyzed. The governing equations are numerically solved in nondimensional variables using the finite volume method. The numerical solution is validated using available experimental and numerical data. Results of the mean Nusselt number for the outer $(Nu¯ho)$ and inner $(Nu¯hi)$ surfaces are obtained by varying a number of key parameters. These parameters are the Rayleigh number based on the cylinder height (Ra$h$⁠) in the range 10³$≤$ Ra$h$$≤$ 10⁷, the nondimensional cross-sectional area (A$C$⁠) in the range 0.006 $≤$ A$C$$≤$ 0.5, and the number of sides of the polygon (N) in the range 6 $≤$ N $≤$$∞$⁠. In all cases, a Prandtl number (Pr) of 0.7 has been assumed. The study shows that at a certain Rayleigh number and a certain number of sides, the heat transfer rate from the inner surface decreases (by as much as 79.8%) as the polygon area decreases (by as much as 83.32%), whereas the heat transfer rate on the outer surface increases (by as much as 133.3%) as the polygon area decreases (by as much as 83.32%). It has also been found that the behavior of the buoyancy-driven flow in the vicinity of the outer surface is fundamentally different than that near the inner surface. Additional details about this fundamental difference are presented in the Results and Discussion section of the paper. New correlations to calculate the average velocity at the exit surface of the cylinder inner core and the mean Nusselt number for both the outer and inner surfaces have also been developed. Also, correlations have been developed for selecting the optimal cross-sectional area for purposes of identifying the regions where the thermal and velocity boundary layers overlap within the inner core of the cylinder.