Stanton Number for Heat Transfer

There are two ways to make a convection coefficient dimensionless, and they answer different questions. Nusselt divides $h$ by conduction through a fluid layer of thickness $L$ and asks how much better than still fluid this surface is doing. Stanton divides $h$ by the heat-carrying capacity of the stream flowing past, $\mathrm{St} = h/(\rho v c_p)$, and asks what FRACTION of what goes by actually gets delivered to the wall. The second is the more readable of the two: a Stanton number of 0.004 says four parts in a thousand of the stream's thermal capacity is transferred per unit of surface, and that sentence needs no further translation.

Its structural advantage is that no characteristic length appears in it. Nusselt cannot be quoted without stating what $L$ was, and comparing two Nusselt numbers from different geometries means first arguing about characteristic dimensions — the same argument that makes the Grashof number so easy to misuse. Stanton sidesteps it entirely, which lets a finned coil be compared with a bare tube, or a plate-fin core with a shell-and-tube bundle, without the geometries being reconciled. That is why compact heat exchanger data are almost always published as Stanton or as the Colburn $j$-factor rather than as Nusselt.

Stanton is a composite, and the cancellation is worth doing once by hand: $\mathrm{St} = \mathrm{Nu}/(\mathrm{Re}\,\mathrm{Pr})$, because $\mathrm{Nu} = hL/k$ and $\mathrm{Re}\,\mathrm{Pr} = \rho v L c_p/k$, and the $L$ and the $k$ both divide out. The product $\mathrm{Re}\cdot\mathrm{Pr}$ is the thermal Péclet number, so Stanton is also $\mathrm{Nu}/\mathrm{Pe}$. This is where the Colburn analogy is genuinely written: $j_H = \mathrm{St}\,\mathrm{Pr}^{2/3} = f/2$, which says the fraction of the stream delivered to the wall is set by the same wall friction that sets the pressure drop, once corrected for the sublayer where heat must move by conduction alone. That also puts a ceiling on the number — since the FANNING friction factor in turbulent flow rarely exceeds about 0.02, Stanton numbers much above 0.01 are physically suspicious, and the usual culprit is that $h$ and $v$ were read from different cross-sections.

There are two Stanton numbers and they are not the same. This page is the heat-transfer one; the catalog also carries the mass-transfer Stanton number, $\mathrm{St}_D = k_c/u$, which is the identical idea with a mass-transfer coefficient in place of $h/(\rho c_p)$. They are exact counterparts — the Chilton–Colburn analogy states that $j_H$ and $j_D$ are both equal to $f/2$, which is the same claim made twice — and in a gas, where Prandtl and Schmidt are both near 0.7, the two numbers come out nearly equal. In a liquid they differ by a factor of ten or more. Neither should ever be quoted as simply "the Stanton number" without saying which, and a paper that does so has to be read carefully enough to work out which coefficient is in the numerator.