Jerod C. Day Mechanical and Energy Engineering, University of North Texas, Denton, TX 76203 Matthew K. Zemler Mechanical Engineering, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114 Matthew J. Traum Mechanical Engineering, Milwaukee School of Engineering, Milwaukee, WI 53202 Sandra K. S. Boetcher 1 Mechanical Engineering, Embry-Riddle Aeronautical University, Daytona Beach, FL 32114 e-mail: sandra.boetcher@erau.edu Laminar Natural Convection From Isothermal Vertical Cylinders: Revisting a Classical Subject Although an extensively studied classical subject, laminar natural convection heat transfer from the vertical surface of a cylinder has generated some recent interest in the literature. In this investigation, numerical experiments are performed to determine average Nusselt numbers for isothermal vertical cylinders (10 2 < Ra L < 10 9 ; 0:1 < L=D < 10, and Pr ¼ 0.7) situated on an adiabatic surface in a quiescent ambient environment. Average Nusselt numbers for various cases will be presented and compared with commonly used correlations. Using Nusselt numbers for isothermal tops to approximate Nusselt numbers for heated tops will also be examined. Furthermore, the limit for which the heat transfer results for a vertical flat plate may be used as an approximation for the heat transfer from a vertical cylinder will be investigated. [DOI: 10.1115/1.4007421] Keywords: laminar, natural convection, vertical cylinders, classic solutions, CFD Introduction Laminar natural convection heat transfer from the vertical surface of a cylinder is a classical subject, which has been studied extensively. When the boundary layer thickness d is small com- pared to the diameter of the cylinder, Nusselt numbers may be determined by approximating the curved vertical surface as a flat plate. However, when the boundary layer thickness is large com- pared to the diameter of the cylinder, effects of curvature must be taken into account. Many investigators have studied the curvature limits for which the flat-plate model can be applied to estimate Nusselt numbers for vertical cylinders. Furthermore, these investi- gators have presented Nusselt number correlations for isothermal vertical cylinders. In most heat transfer textbooks, including but not limited to Incropera et al. [1,2], Holman [3], Burmeister [4], and Gebhart et al. [5], the accepted limit for which the flat-plate solution can be used to approximate average Nusselt numbers for vertical cylinders (Pr ¼ 0.72) within 5% error is D L  35 Gr 0:25 L (1) where D is the diameter of the cylinder, L is the height of the cylinder, and Gr L is the Grashof number based on the height of the cylinder. This limit was derived by Sparrow and Gregg [6] in 1956 using a pseudosimilarity variable coordinate transformation and perturbation technique for solving the heat transfer and fluid flow adjacent to an isothermal vertical cylinder. They assumed the boundary layer thickness at the leading edge to be zero and they made use of the boundary layer approximation (all pressure gra- dients are zero and streamwise second derivatives are neglected) and Boussinesq approximation (density difference are small). In addition, Nusselt numbers for vertical cylinders (Pr ¼ 0.72 and 1; 0 < n < 1) are presented as a truncated series solution and plotted. The curvature parameter n arose from a coordinate transformation done by Sparrow and Gregg [6] and is defined as n ¼ 4L D Gr L 4  1=4 (2) Around the same time as Sparrow and Gregg, LeFevre and Ede [7,8] solved the governing equations using the same assumptions as [6] with an integral method to obtain a correlation for vertical cylinder average Nusselt numbers, which is shown below Nu L ¼ 4 3 7Gr L Pr 2 5ð20 þ 21PrÞ   1=4 þ 4ð272 þ 315PrÞL 35ð64 þ 63PrÞD (3) In this equation, Nu L is the average Nusselt number and Pr is the Prandtl number. In 1974, Cebeci [9] extended the work of Ref. [6] by numeri- cally solving the governing equations using the boundary layer approximation for 0:01  Pr  100 and 0 < n < 5. The results of Cebeci for the average Nusselt number for an isothermal vertical cylinder Pr ¼ 0.72 have been correlated in Popiel [10] with range of deviation from 0.34% 0.66% Nu L Nu L;fp ¼ 1 þ 0:3 32 0:5 Gr 0:25 L L D   0:909 (4) In this equation, Nu L,fp is the average Nusselt number for the isothermal flat plate. Typically, the value for Nu L,fp is taken from the Churchill and Chu [11] correlations for natural convection from a vertical flat plate. Nu L;fp ¼ 0:68 þ 0:670Ra 1=4 L ½1 þð0:492=PrÞ 9=16  4=9 (5) In this equation, Ra L is the Rayleigh number based on the height of the cylinder. 1 Corresponding author. Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received December 26, 2011; final manu- script received August 13, 2012; published online January 3, 2013. Assoc. Editor: Giulio Lorenzini. Journal of Heat Transfer FEBRUARY 2013, Vol. 135 / 022505-1 Copyright V C 2013 by ASME Downloaded 06 Feb 2013 to 155.31.130.193. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm