Transfer Units for Dilute Absorption

The design equation for a packed absorber is an integral — $N_{OG} = \int dy/(y - y^*)$ up the column — and Colburn's 1939 paper did the integration for the case that matters most, a straight equilibrium line and a dilute system. The result looks almost identical to the Kremser stage equation, with the same logarithm on top, and differs only in the denominator: $(1 - 1/A)$ here where Kremser has $\ln A$. That small difference is the whole distinction between a transfer unit and an equilibrium stage.

A transfer unit is the height over which the concentration change equals the average driving force producing it — one e-fold of approach to equilibrium, in effect. An equilibrium stage is a discrete step to the equilibrium line and back. They are different objects, and the ratio between them is not a constant: it depends on $A$, converging only when $A$ approaches 1 and the two lines run parallel. For the same duty at $A = 2$, a column needs about 3.4 equilibrium stages or about 4.7 transfer units, and there is no fixed conversion between the two counts.

The most revealing feature of the equation is a floor that has no counterpart in stage counting. As $A$ grows without bound, $N_{OG}$ does not fall toward zero — it approaches $\ln(y_1/y_2)$, and no amount of solvent will take it below that. The reason is that even with the driving force everywhere at its maximum, the gas still has to be diluted by the factor $y_1/y_2$, and each transfer unit only ever accomplishes one e-fold of that dilution. A 99 percent removal therefore needs at least $\ln 100 = 4.6$ transfer units, however generous the solvent. Kremser stage counts, by contrast, fall toward zero as $A$ rises. Seeing that difference is the clearest way to understand that the two methods are not two scales for the same thing.

The dilute restriction is real and it is where the equation is most often abused. "Dilute" means the solute is a small enough fraction that the gas and liquid molar flows are effectively constant down the column and the equilibrium line is effectively straight — conventionally under about 5 to 10 mole percent. Above that, the carrier flows change as the solute leaves the gas, the operating line curves, and this closed form quietly overestimates the performance. Concentrated absorbers are worked in mole RATIOS rather than mole fractions, on solute-free flows, or numerically. As with Kremser, a solvent entering with solute in it is handled by replacing every $y$ with $(y - m x_2)$.