Péclet Number for Mass Transfer

The Péclet number asks whether the stream carries a solute along faster than diffusion can smear it out: $\mathrm{Pe} = uL/D$, advection over diffusion. It is named for Jean Claude Eugène Péclet, and it is exactly the product of Reynolds and Schmidt — a useful identity, because it means a large Péclet number in a liquid may be saying nothing more interesting than that Schmidt is large.

Its main job is deciding what kind of reactor or vessel you are actually operating. A large Péclet number means material moves through in an orderly front with little back-mixing, which is plug flow. A small one means diffusion spreads solute along the flow direction as fast as the flow carries it, which is a well-mixed tank. Real vessels sit between, and the dispersion model uses Péclet as its single adjustable parameter, with values above roughly 100 usually treated as plug flow and below about 1 as fully mixed. The consequences are not cosmetic: for most reactions a plug-flow vessel needs considerably less volume than a stirred tank for the same conversion, and back-mixing in a vessel designed as plug flow shows up as a conversion shortfall nobody can explain from the chemistry.

The most consequential mistake with this group is putting the molecular diffusivity in the denominator. In any real pipe or packed bed, solute spreads along the flow direction thousands of times faster than molecules alone would manage, because the velocity profile carries the centreline ahead of the wall and turbulent eddies stir the difference — Taylor dispersion. The coefficient that belongs here is the axial DISPERSION coefficient, which is an empirical property of the geometry and flow, not a property of the fluid. Using molecular $D$ inflates Péclet by orders of magnitude and makes every vessel look like ideal plug flow, which is the flattering direction and therefore the dangerous one.

A note on the name, since it causes genuine confusion in the literature. There is a thermal Péclet number, $uL/\alpha$, built on the thermal diffusivity, and it is a completely different number for the same apparatus — in water, the two differ by a factor of several hundred. Papers frequently write "Péclet number" without saying which, and the only reliable way to tell is to find the diffusivity in the denominator and see what it is.