Sizing a column by hand has been done the same way since the 1930s, and every step is a named paper. It works by bracketing the answer between two extremes that are cheap to compute and impossible to build.

Fenske gives the floor: the fewest stages, at total reflux, where nothing is drawn off at all. $N_{min} = \dfrac{\ln\left[\dfrac{x_D}{1 - x_D}\cdot\dfrac{1 - x_B}{x_B}\right]}{\ln \alpha}$, read aloud N-min equals the log of the separation, over the log of alpha. $N_{min}$ is the minimum theoretical stage count, including the reboiler as one; $x_D$ and $x_B$ are the light key mole fractions overhead and in the bottoms; $\alpha$ is the relative volatility from the first lesson. The bracket is a ratio of ratios, so it is bare before the logarithm sees it — and because both logarithms have the same base, it does not matter which base you use, so long as it is the same one twice.

Underwood gives the other limit, the least reflux, at infinite stages. Between those two sits Gilliland, who plotted the results of a great many rigorous calculations and drew a curve through the cloud. The chart's axes are $X = \dfrac{R - R_{min}}{R + 1}$ across and $Y = \dfrac{N - N_{min}}{N + 1}$ up. Read $Y$ off the curve at your $X$, and rearrange: $N = \dfrac{N_{min} + Y}{1 - Y}$. Two points to keep in your head, because they will check your reading: at $X = 0.5$ the published curve passes through $Y \approx 0.25$, and at $X = 0.2$ through $Y \approx 0.46$.

Say plainly what Gilliland is: a correlation, not a derivation. The individual points scatter by roughly ±10% about the curve, and more at the ends. It is good enough to size a shell and cost a project; it is not good enough to guarantee a purity with. And the old rule of thumb that a real column runs near twice $N_{min}$ is simply this curve, evaluated at the reflux everybody uses.

Last, the reality check. All of that counts theoretical stages, and no real column has any: a real tray never quite reaches equilibrium. $E_o = \dfrac{N_{theor}}{N_{actual}}$ is the overall efficiency, so $N_{actual} = N_{theor}/E_o$ — divide, then round up, because half a tray cannot be welded in. Distillation trays run 60–80%. Absorber trays run far worse, sometimes 20–30% on a poorly soluble gas, and that difference has sunk more than one column.