Three correlations, one design
The FUG method designs a column in three moves:
Fenske gives the minimum number of stages, Nmin, at total reflux — the theoretical floor set by the relative volatility α and your recovery targets.
Underwood gives the minimum reflux ratio, Rmin — the floor on reflux below which no finite column can make the split.
Gilliland bridges the two: pick an operating reflux as a multiple of Rmin (typically R/Rmin ≈ 1.1–1.5) and it estimates the actual number of stages N.
More reflux → fewer stages but higher reboiler duty. Fewer reflux → more stages. That is the central capital-vs-energy trade-off of distillation.
Nmin (Fenske), Rmin (Underwood), then N at your chosen R/Rmin (Gilliland).