Fenske, Underwood, Gilliland: The Shortcut Method for Sizing a Distillation Column

Why a Shortcut Is Still Worth Having

A rigorous column simulation solves mass, energy, and equilibrium equations stage by stage, and it needs a starting guess: how many stages, what reflux, where to feed. The Fenske-Underwood-Gilliland method, usually called FUG, supplies those guesses from a handful of equations. It is fast enough to screen dozens of design options by hand or in a spreadsheet, and process simulators include it as a shortcut column model for exactly that purpose.

The method works in terms of two key components. The light key is the more volatile component whose recovery in the bottoms is specified, and the heavy key is the less volatile component whose recovery in the distillate is specified. Everything lighter than the light key is assumed to go overhead, and everything heavier than the heavy key to the bottoms.

Fenske: Minimum Stages at Total Reflux

At total reflux, where everything condensed overhead is returned to the column and nothing is withdrawn, the column achieves the most separation per stage, so it needs the fewest stages. Merrell Fenske's 1932 equation gives that minimum directly: N_min = ln[(x_LK/x_HK)_D·(x_HK/x_LK)_B]/ln α_avg, where the compositions are the key component ratios in the distillate and bottoms and α_avg is the relative volatility of the light key to the heavy key.

The equation assumes constant relative volatility, so in practice α_avg is taken as the geometric mean of the values at the top and bottom of the column. The result counts the reboiler as a stage, a detail that is easy to forget. Because relative volatility sits inside a logarithm in the denominator, close-boiling pairs with α near 1 drive N_min up very quickly: a pair with α = 1.1 needs roughly seven times as many minimum stages as a pair with α = 2 for the same split.

Underwood: Minimum Reflux

The opposite limit is minimum reflux, the lowest reflux ratio at which the separation is possible at all, needing an infinite number of stages. A. J. V. Underwood's equations, published in their most-used form in 1948, find it in two steps. First, solve Σ α_i·z_i/(α_i − θ) = 1 − q for the root θ that lies between the relative volatilities of the two keys, where z_i is the feed composition and q describes the feed's thermal condition. Then compute R_min + 1 = Σ α_i·x_i,D/(α_i − θ) using the distillate composition.

Underwood's method assumes constant relative volatility and constant molar overflow. Within those assumptions it is exact, and it handles multicomponent mixtures properly, which a McCabe-Thiele diagram cannot. In real columns, varying volatility and non-constant molar flows usually make the true minimum reflux somewhat different, so the result is treated as an estimate.

Gilliland: Stages at the Actual Reflux

Real columns run between those two limits. Edwin Gilliland's 1940 correlation connects them, plotting (N − N_min)/(N + 1) against (R − R_min)/(R + 1) for a large set of stage-by-stage calculations. The data fell on a single curve, which lets you pick an actual reflux ratio and read off the stage count. Molokanov and coworkers later fitted the curve with an equation, Y = 1 − exp[((1 + 54.4X)/(11 + 117.2X))·((X − 1)/√X)], which is more convenient than reading the chart.

Long-standing design heuristics, such as those collected by Walas, put the economic reflux ratio at about 1.2 to 1.5 times the minimum, which usually corresponds to roughly twice the minimum number of stages. The feed stage location is then commonly estimated with Kirkbride's empirical equation. The whole FUG sequence is only as good as its assumptions, and for non-ideal mixtures or azeotropes it can mislead, but as a first estimate of column size it remains the standard tool.

References

1. Fenske, M. R. (1932). Fractionation of straight-run Pennsylvania gasoline. Industrial & Engineering Chemistry, 24(5), 482–485. doi:10.1021/ie50269a003
2. Underwood, A. J. V. (1948). Fractional distillation of multicomponent mixtures. Chemical Engineering Progress, 44(8), 603–614.
3. Gilliland, E. R. (1940). Multicomponent rectification: Estimation of the number of theoretical plates as a function of the reflux ratio. Industrial & Engineering Chemistry, 32(9), 1220–1223. doi:10.1021/ie50369a035
4. Molokanov, Y. K., Korablina, T. P., Mazurina, N. I., & Nikiforov, G. A. (1972). An approximate method for calculating the basic parameters of multicomponent fractionation. International Chemical Engineering, 12(2), 209–212.
5. Seader, J. D., Henley, E. J., & Roper, D. K. (2011). Separation Process Principles (3rd ed.), Chapter 9. Hoboken, NJ: Wiley.
6. Walas, S. M. (1988). Chemical Process Equipment: Selection and Design. Boston: Butterworths.