Stripping Operating Line (McCabe–Thiele)

Below the feed the column has a different job. Above it, the object is to enrich the vapour going up; below it, the object is to strip the light component out of the liquid going down so the bottoms product is clean. The stripping operating line is the material balance around that end — the reboiler, the bottoms draw, and every stage below the cut.

Written with the boilup ratio, as on this page, the slope is $(V_B + 1)/V_B$, which is always greater than one. Its counterpart above the feed has slope $R/(R+1)$, always less than one. So the two operating lines approach the 45-degree diagonal from the same side but with opposite steepness, and both lie between the diagonal and the equilibrium curve — the vertical gap between line and curve at any composition is the driving force available to the stage sitting there.

The same line is written at least three ways in the literature and they are all the same line. In internal flows it is $y = (\bar{L}/\bar{V})x - (B/\bar{V})x_B$; in the boilup ratio it is the form here, since $\bar{L}/\bar{V} = (V_B+1)/V_B$. The boilup form is the one worth internalising, because $V_B$ is what the reboiler is actually asked to raise, and reboiler duty is where distillation's enormous energy footprint comes from. Steepening this line to sharpen the bottoms has an invoice attached to it that arrives the same month.

The detail that most often costs a tray is the reboiler. A reboiler is a vapour-liquid contacting device and it counts as one theoretical stage — the last step on the diagram is the reboiler, not a tray. Count it as a tray and you buy a column one tray taller than the calculation asked for. The mirror-image question at the top of the column has the opposite answer: a total condenser is not a stage, because it changes no composition, while a partial condenser taking vapour product is one.