Schmidt Number

The Schmidt number asks a single question about a fluid: when you disturb it, does momentum spread through it faster than molecules do? It is the ratio of the kinematic viscosity to the mass diffusivity, $\mathrm{Sc} = \nu/D = \mu/(\rho D)$, and it belongs to the fluid alone. No velocity appears in it, no pipe, no geometry — which is exactly why it is the group every mass-transfer correlation is written around. Ernst Schmidt proposed it in the 1920s as the mass-transfer counterpart of the Prandtl number, and the parallel is exact: swap the thermal diffusivity for the mass diffusivity and one becomes the other.

The striking thing is how far apart gases and liquids sit. In a gas, momentum and molecules are carried by the very same molecular motion, so $\nu$ and $D$ come out nearly equal and Schmidt lands between roughly 0.2 and 3 for almost every pair you will meet. Better still, it is nearly independent of pressure, because both $\nu$ and $D$ scale inversely with density and the ratio survives unchanged. In a liquid the two mechanisms decouple completely: momentum passes along through molecules that stay put and jostle, while a solute must physically thread its way between them. Water at room temperature gives Schmidt numbers of several hundred to a few thousand, and a viscous oil can reach a hundred thousand.

That gap has a physical shape you can picture. Schmidt is the ratio of the velocity boundary layer's thickness to the concentration boundary layer's, raised to a power near one third. In air the two layers are roughly the same size, which is why the heat, momentum and mass analogies work so well there. In water the concentration layer is a thin skin buried deep inside the velocity layer — perhaps a tenth of its thickness — and all the resistance to transfer is crammed into that skin. This is the reason liquid-phase mass transfer is so much slower than watching the flow would ever suggest, and the reason correlations carry $\mathrm{Sc}$ to the one-third or one-half power rather than ignoring it.

Two practical cautions. The temperature dependence is steep and it runs the wrong way from intuition: heating a liquid drops the viscosity and raises the diffusivity at once, so Schmidt can fall by half over thirty degrees, and a value quoted without its temperature is not usable. And in a gas mixture the diffusivity is specific to a PAIR of species — oxygen in nitrogen is not oxygen in helium — so a Schmidt number always names two substances, even when the literature is careless enough to print only one.