I've been trying to find a Nusselt number correlation for a sphere cooling in a forced gas cross flow where the sphere temperature is much higher than the free stream temperature.

I want something like this: $ Nu = F(Re, Pr, ...)$ that is applicable for a hot spherical body cooling in a forced crossflow of air at roughly $ 10< Re < 1000 $ and $ .5< Pr < .8 $. I'm interested in cases where the temperature differences are quite large (~$1000^oC$). The closest I've found (DOI 10.1002/aic.690180219) is correlations of the form: $$ Nu = F\left(Re, Pr, \frac{\mu_\text{free stream}}{\mu_\text{surface}} \right)$$ but in these correlations the viscosity ratio must be $>1$ which is consistent with a sphere being heated not cooled (for air the viscosity increases with temperature).

Other details:

  • I'm not interested in radiation for this (I'm keeping track of it separately)
  • I'm aware you can use a "film temperature" but this is typically for much smaller temperature differences.
  • It can be theoretically or empirically based - as long as it is applicable!
  • If there is a way to show that the effect of large temperature differences shouldn't matter that is equally useful.
  • $\begingroup$ Are you also going to include radiation? $\endgroup$ Sep 28, 2016 at 18:11
  • $\begingroup$ Does anyone know if there is a Nusselt number correlation. Correlation between what and what? $\endgroup$
    – Gert
    Sep 28, 2016 at 18:13
  • $\begingroup$ Here's a suggestion. For the low end of your Reynolds numbers, the flow is still probably laminar. So solve the coupled laminar flow and heat transfer equations numerically for different viscosity ratios (using CFD for example), and develop the viscosity ratio correction factor yourself for values of the ratio you are interested in. Then assume that this same correction factor applies (approximately) at higher Reynolds numbers. $\endgroup$ Jun 25, 2018 at 12:36

1 Answer 1


Here is an equation that you can use:

enter image description here

  • $\begingroup$ Unfortunately this is from the source (Whitaker) that I already found and the original source says that it is only valid when the viscosity ratio is greater than one, in my case the ratio is less than one. $\endgroup$ Sep 28, 2016 at 21:05
  • $\begingroup$ How accurate does your answer need to be? I would still be inclined to use the equation. $\endgroup$ Sep 28, 2016 at 21:18
  • 1
    $\begingroup$ That is always the question. Currently I plan to use this correlation if I can't find anything better, but I won't know how truly accurate this is until I have the "correct" answer. $\endgroup$ Sep 28, 2016 at 21:33

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