Chilton–Colburn Analogy for Mass Transfer

Here is an idea that should not work as well as it does: you can predict how fast a gas will absorb by measuring the pressure drop across the equipment. Thomas Chilton and Allan Colburn published it in 1934, building on Colburn's heat-transfer paper of the year before, and the reasoning is that momentum, heat and mass are all carried across a turbulent stream by the same eddies. If the eddies are vigorous enough to produce a certain friction, they are vigorous enough to produce a proportional amount of mass transfer. The result is written as a j-factor, $j_D = \mathrm{St}_D\,\mathrm{Sc}^{2/3} = f/2$, which rearranges into $\mathrm{Sh} = (f/2)\,\mathrm{Re}\,\mathrm{Sc}^{1/3}$.

The Schmidt exponent is where the physics hides. If the analogy were perfect the groups would simply be equal and no correction would be needed, and for gases at Schmidt near 1 that is very nearly true. But right at the wall the eddies die out, and a thin sublayer remains in which nothing but molecular diffusion operates. Momentum crosses that sublayer easily and a solute in a liquid crosses it with great difficulty, and the $\mathrm{Sc}^{2/3}$ factor is the empirical correction for the discrepancy. It is not derived from first principles; it is fitted, and it is the reason the analogy carries an error band rather than an equals sign.

That error band is usually quoted at 20 to 30 percent, which is honest and rather good for a method that needs no mass-transfer data at all. The validity range matters more than the scatter, though. The analogy was fitted to turbulent data, conventionally Reynolds above about 10,000, and Schmidt from roughly 0.6 to 2500. In laminar and transition flow the boundary layers stop resembling one another and the whole argument collapses; there a laminar result such as the Graetz–Lévêque solution is the right tool.

The single most expensive error made with this equation is the friction factor. The analogy is written on the FANNING friction factor, defined from the wall shear stress, and the Darcy–Weisbach factor used in every pipe-sizing calculation is four times larger. Entering Darcy where Fanning belongs overstates the transfer coefficient fourfold, and the answer is not absurd enough to catch the eye. A useful check: Fanning runs near 0.005 in a smooth turbulent pipe where Darcy runs near 0.02. There is a second restriction of the same kind — the analogy holds only for skin friction, so on packed beds, tube banks and anything else where form drag contributes to the measured pressure drop, $f$ is simply not the quantity the equation wants, and the prediction runs high.