Now I assume that heat transfer area in this case includes the top and botom of the vertical cylinder. So does my characteristic length become$$L_{c}=\frac{2\pi r^{2}+\pi DH}{2\pi r}$$

Or do I neglect the surface area of the top and bottom to have$$L_{c}=\frac{\pi DH}{2\pi r}=\frac{\pi DH}{\pi D}=H$$

There is no unique definition for characteristic length. One can define the length scale as the square root of the surface area, as the third root of the volume, as the volume-to-surface ratio, etc. It all depends what length scale you want identify. For your problem (convective heat transport of an object submerged in a liquid at elevated temperature) the characteristic length is the vertical size of the object. This is because the convection will take place in the vertical direction (cold water flowing down, hot water rising).

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If your object is a cylinder in vertical orientation, the convective length scale is the height H of the cylinder. If your object is a cylinder placed horizontally the characteristic length is determined by the diameter D of the cylinder.

The characteristic length is commonly defined as the volume of the body divided by the surface area of the body, i.e.

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$L_c = \frac{V_{body}}{A_{surface}}$

The surface area should include the top and bottom of the cylinder.

Both of the above answers are correct. The characteristic lengths for the following shapes are:Sphere = $Radius/3$ = $Diameter/6$Cylinder = $Radius/2$ = $Diameter/4$Plate (ie: flat cylinder) = $Length/2$Cube = $Length/6$

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