Kremser Equation for Absorption Stages

Before Kremser, sizing an absorber meant drawing an operating line and an equilibrium curve on graph paper and stepping off the staircase between them by hand. In 1930 A. Kremser published, in a trade weekly rather than a journal, the observation that if both lines are straight the staircase is a geometric series and can be summed in closed form. Souders and Brown extended it two years later, which is why the result carries all three names in some texts. It replaced an afternoon of drafting with a logarithm, and it is still what a process engineer reaches for first.

Everything in the equation is dominated by the absorption factor $A = L/(mV)$, and the reason is visible in the algebra: $A$ sits inside the logarithm and again in the denominator, so it acts on the answer twice. Physically, $A$ compares the liquid's capacity to carry solute away against the gas's capacity to deliver it. Above 1 the solvent always has room for more and each stage takes another bite; below 1 the solvent saturates and a fraction of the feed passes straight through no matter how many trays are stacked up. That is not a poor design but a hard wall, and it is why an absorber failing to meet its outlet specification is almost never fixed by adding trays.

The count that comes out is THEORETICAL stages, and no real column contains any. A real tray does not bring its two streams to equilibrium; dividing by an overall tray efficiency is what turns the answer into hardware. Absorber efficiencies are lower than distillation efficiencies and sometimes much lower — 70 percent is a comfortable case, and 20 to 50 percent is ordinary for a poorly soluble gas where the liquid film governs and contact time on the tray is short. A design that forgets the efficiency step is out by a factor of two to five in the direction that matters.

Two assumptions are baked in and both deserve a check. The equilibrium line is taken as straight across the whole concentration range, which is fair for a dilute system and poor for a concentrated one; where it curves, the honest approaches are to evaluate $m$ at an average composition, to break the column into sections with their own slopes, or to abandon the shortcut and integrate. And the solvent is assumed to enter free of solute, which a recycled and regenerated solvent never is. The correction is clean: replace every $y$ with $(y - m x_2)$, where $x_2$ is the solute in the entering liquid, and the equation stays exactly correct. Skipping it is optimistic, because a solvent returning dirty from the stripper raises the outlet the column can reach.