# Nusselt Number Correlation of long cylinder in axial flow

What is the Nusselt number correlation of long cylinder in axial (forced) laminar flow?

Assume gravity does matter.

I have seen numerous sources and lists for cylinders in crossflow which is not the same thing or what I want.

I am looking for either a correlation, evidence that a correlation isn't valid or there, or quantification of the error of when using the flat plate correlation.

• So the length of cylinder is parallel to the flow? I don't know if such a correlation exists. Perhaps you could approximate it as a flat plate – nluigi Dec 20 '17 at 8:42
• You are talking about an external flow, correct? – Chet Miller Dec 20 '17 at 12:26
• @ChesterMiller Yes, external flow – James Urban Dec 20 '17 at 16:48
• @nluigi Yes, the length of the cylinder is along the direction of the flow. I've done a pretty exhaustive literature review and seen no evidence that one exists (which seems very strange, as it seems very "classical"). Currently I'm using the flat plate. – James Urban Dec 20 '17 at 16:51

Update regarding justification of use of flat-plate:

In essence you need to evaluate the Blasius equation in cylindrical coordinates to determine the velocity profile in the momentum boundary layer. When you have determined the momentum boundary layer ($\delta_v$) it can be used to evaluate the thermal boundary layer ($\delta_T$) where in general you will find $\delta_T/\delta_v\sim Pr^{-\frac{1}{3}}$.

According to this article which i found by googling 'Blasius cylinder':

Tutty, O. R.; Price, W. G. & Parsons, A. T. Boundary layer flow on a long thin cylinder Physics of Fluids, 2002, 14, 628-637

[...] it may appear that the leading term in the boundary layer solution will be flate plate Blasius flow [...]. While this is true near the leading edge of the cylinder, further downstream this approximation breaks down. [...] Blasius (flat-plate) flow will not be the leading term in the solution when the boundary layer thickness becomes comparable with the cylinder radius.

This unfortunately somewhat disqualifies my previous answer that you can simply use (to engineering accuracy) the flat-plate solution for the Nusselt correlation as it depends on the ratio of the cylinder radius to length $R/L$. If $R/L\ge1$ you can use the flat-plate approximation else you should use the result from article and apply it to the heat equation to get the Nusselt correlation.

I will leave the actual calculation for OP as it is probably a very involved process and I don't have the time but hopefully the article gives a way to go forward.

Old answer valid only for $R/L\ge1$:

I am not familiar with such a correlation for an external flow along a cylinder (as oppossed to a cross-flow). However, I think (to engineering accuracy) the correlation for a flat plate will suffice given that most correlations: $$\bar{\mathrm{Nu}}=C\mathrm{Re_x}^{\frac{1}{2}}\mathrm{Pr}^{\frac{1}{3}}$$ and the proportionality constant is generally $C\sim1$. For example, for a flat plate $C=0.67$, for crossflow $C=0.62$. The assumptions here are roughly $\mathrm{Re\lt1}$ and $\mathrm{Pr>1}$.

Depending on the radius of your cylinder you may have some curvature effects which will result in a a small error. For larger radii, this error will become smaller and smaller.

• I don't disagree, but I'm looking for the correlation, proof/evidence that it is impossible, or quantification of the acceptableness of treating it as a flat plate. – James Urban Dec 23 '17 at 20:35
• @JamesUrban - see my edit with some good and bad news – nluigi Dec 24 '17 at 17:21
• Thank you, I thought I had searched all of the literature - apparently not! – James Urban Dec 24 '17 at 18:04