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I am trying to find the Nusselt number for a small element of the outside of a cylinder that has a height of $\Delta z$. I found the average Grashof number of a surface as

$$Gr_{L}=\frac{\beta \rho (T_{s}-T_{\infty})L_{c}^{3} }{\nu^{2}}$$

If I wanted to find $Gr_{x}$ of a certain element, would I use a characteristic length of $\Delta z$ or could I find the z-location of the elements centroid and use that as the $L_{c}$. For example, If the cell was located at a z coordinate of 1 m, would that be the characteristic length or would the characteristic length be the height of the differential element?

I was thinking that I could treat each element as a small isothermal cylinder and have the characteristic length of each element be $\Delta z$

Also, does anyone know the a Nusselt number correlation for a vertical cylinder with a Prandtl number of less than one? I have only found one for when it is greater than 1. Thank you

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