Stairs on a graph.

The still and the tower are the same machine.

Boil a mix and the vapour comes off richer in whatever boils more easily. Catch that vapour, cool it back to a liquid, and you have something stronger than what you started with. Boil it again and it is stronger still. Every boil-and-catch is one stage, and a stage is the unit everything below is counted in.

A pot still is one stage, maybe two, because the vapour rising through the neck condenses a little and runs back. That is why whisky is made in passes:

A column still is the same trick stacked. Forty stages, eighty, more, one on top of the next, and it takes the same wash to about 95% ABV in one continuous pass. The Scotch rules cap it just below 94.8% ABV precisely because a tall enough column would keep going until the spirit tasted of nothing. A refinery column separating propane from propylene has a hundred and fifty stages and stands over a hundred metres tall, and the arithmetic that sized it is the arithmetic on this page.

A note for the people who will check: the first vapour off a 7% ABV wash is in equilibrium at about 44% ABV, well above the “about 20” the still collects. A pot still is a batch: the wash gets weaker as it boils, the vapour gets weaker with it, and the low wines are the average of the whole run. The stairs below are drawn for a continuous column, where every stage sees a steady feed.

The graph.

Liquid composition along the bottom, vapour composition up the side, both as mole fraction ethanol. One curve says how much richer the vapour gets than the liquid it boiled from, at every composition. That is measured data, ethanol and water at one atmosphere, and it is the only physics on the graph. One straight line says what is flowing past each stage inside the column, which is a mass balance and depends on how much of the top product you send back down as reflux. The stairs are drawn between them from the top product down to the bottoms, and the number of steps is the number of theoretical stages.

Push the slider left and the stairs pinch: the operating line climbs towards the curve, each step buys less, and at minimum reflux the count goes to infinity. Push it right and the count falls, but every unit of reflux is vapour you have to boil and condense again, so the reboiler and the condenser grow. Textbook heuristic: run at 1.2 to 1.5 times the minimum. That is where this column sits.

The lie inside the method.

The straight line is only straight if every mole of vapour that condenses on a stage boils exactly one mole of liquid in its place. That is the constant molar overflow assumption, and every textbook admits it in the paragraph after it introduces it. It holds when the two molecules take about the same heat to boil. They usually do not.

So in an acetone and water column, condensing a mole of water vapour boils 1.4 moles of acetone, the vapour flow changes from stage to stage, and the operating line bends. The honest construction is the Ponchon Savarit diagram, which carries the enthalpy, or the thing every simulator does instead: write the mass balance and the heat balance for every tray and solve all of them at once, tray by tray, until the numbers stop moving. McCabe Thiele gets you the shape and the rough count in ten minutes. The solver gets you the answer.

Where theory becomes trays.

Every step on the graph is a perfect stage: the vapour leaving it is in equilibrium with the liquid leaving it. A real tray is a sheet of metal with holes in it and a weir, with vapour blowing up through a froth of liquid for a fraction of a second. It gets part of the way to equilibrium, and the part is the tray efficiency: about 0.5 to 0.7 for most columns, as low as 0.4 for viscous or very light systems and up near 0.9 for the cleanest ones. So you divide:

A count becomes a height.

Trays are spaced so the froth on one does not reach the one above and so a person can get in between them to clean: about 0.6 metres, two feet, is the standard pitch, tighter in small columns and wider in foaming services. Then a few metres at each end, for the vapour to disengage from the liquid at the top and for the reboiler and a liquid inventory at the bottom.

Sources: the McCabe Thiele construction, the q line and minimum reflux at the pinch are in LibreTexts, Distillation. The heuristics (1.2 to 1.5 times minimum reflux, efficiency 0.5 to 0.7, two feet per tray plus five to ten feet at each end) are from Douglas, Distillation Design. Whisky strengths from Whisky Advocate and the Scotch Whisky Regulations 2009. Latent heats from the NIST WebBook (acetone) and Engineering Toolbox (water). Ethanol and water equilibrium: Carey and Lewis (1932), as tabulated in Perry's.