## Free convection IN a slender vertical cylinder

Hi,

I have run into a problem and can't find an answer. I am trying to estimate the heat transfer coefficient for water INSIDE a vertical cylinder exposed to uneven heating.

The cylinder is 1m high with a diameter of 0.035m. D/L is then 0.035.

I have calculated the Grashof number 417200 and the Rayleigh number 3950887. There is a correlation equation for D/L>=35/Gr^(1/4). But I have D/L=0.035 and 35/Gr^(1/4)=1.38, so the equation is not valid. Apparently the curvature of the cylinder walls affects the fluid flow, but how? Would a Nusselt number (hence the heat transfer coefficient) calculated with the invalid correlation equation be a minimum, assuming flow is more inhibited with increased curvature?

I have searched high and low but can't find anything for flow INSIDE a vertical pipe for this geometry. A paper by LeFevre and Ede (1956) got me excited, but it is for flow OUTSIDE the slender vertical pipe.

Any suggestions or help would be greatly appreciated.

LeFevre, E. J., and Ede, A. J. (1956). Laminar Free Convection from the Outer Surface of a Vertical Circular Cylinder, Proc. 9th International Congress on Applied Mechanics, Brussels,
Vol. 4, pp. 175–183.