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    Process / distillation

    Distillation column design

    Three shortcuts and one construction. Fenske gives the minimum stages at total reflux, Underwood the minimum reflux at infinite stages, and Gilliland the relationship between the two at any practical reflux. McCabe-Thiele then steps the actual stages off the equilibrium curve — and the two answers should agree.

    Underwood θ1.658Graphical16 stagesColumn height10.0 m
    Theoretical stages
    16.9
    Gilliland at R = 4.42
    Minimum stages
    9.9
    Fenske, at total reflux
    Minimum reflux
    2.95
    operating at 4.42 (1.50×)
    Feed stage
    9
    at x = 0.274
    Shortcut design
    Three correlations and one construction, on the same system.
    16.9 stages
    Shortcut design
    QuantityValueNote
    Relative volatility2.200light to heavy key
    Feed composition0.450mole fraction light key
    Distillate0.9800
    Bottoms0.0200
    Feed condition q0.011 = saturated liquid
    Minimum stages9.87Fenske
    Underwood root θ1.6578between 1 and α
    Minimum reflux2.946Underwood
    Operating reflux4.4201.50× minimum
    Gilliland X0.2718(R − Rmin)/(R + 1)
    Gilliland Y0.3916Eduljee correlation
    Stages16.9Gilliland
    Stages by stepping16McCabe-Thiele
    Feed stage9from the top
    Column height10.0 mat 0.5 m per stage plus 2 m

    The Gilliland and McCabe-Thiele answers should agree within a stage or two. When they do not, the constant-α assumption is usually the reason: a real column’s relative volatility falls as the light component is stripped out.

    Inspect the model
    Choose a quick result view, a scaled comparison, or the complete diagnostic set.
    Equilibrium curve
    Vapour against liquid composition at constant relative volatility. The further it bows from the diagonal, the easier the separation.
    Equilibrium curve0.0000.2240.4480.6720.8961.12000.10.10.20.20.30.30.40.40.50.50.60.60.70.70.80.80.90.911Equilibriumy = x
    Stages against reflux
    Diminishing returns: doubling the reflux above 2× minimum saves few stages and costs a lot of energy.
    Stages against reflux0.0005.9411.8817.8223.7629.7025.171.11.121.381.21.216.871.51.514.242212.413311.2655Stages
    Canonical equations
    The exact relationships implemented by this engine.

    N_min = ln[(xD/(1−xD))((1−xB)/xB)] / ln α · Minimum stages at total reflux.

    Σ α_i z_i/(α_i − θ) = 1 − q · Solves for θ between the key volatilities.

    R_min + 1 = Σ α_i x_D,i/(α_i − θ) · At infinite stages.

    Y = 0.75(1 − X^0.5668) · Eduljee’s fit to the original correlation.

    engineering methodology

    A clear path from inputs to results.

    Three shortcuts and one construction. Fenske gives the minimum stages at total reflux, Underwood the minimum reflux at infinite stages, and Gilliland the relationship between the two at any practical reflux. McCabe-Thiele then steps the actual stages off the equilibrium curve.

    Step-by-step methodWorked exampleAssumptions explained

    How to use

    1. Set the relative volatility and the feed composition.
    2. Set the distillate and bottoms purities.
    3. Choose the reflux as a multiple of the minimum.
    4. Read the stage count, the feed stage and the column height.

    The formula

    Nmin=ln⁡[xD1−xD⋅1−xBxB]ln⁡αN_{min} = \frac{\ln\left[\frac{x_D}{1-x_D}\cdot\frac{1-x_B}{x_B}\right]}{\ln\alpha}

    Reference: Fenske (1932), Underwood (1948), Gilliland (1940); the Eduljee (1975) fit to the Gilliland correlation.

    Worked example

    A 45% feed to 98% distillate at α = 2.2

    Inputs
    • Relative volatility 2.2, feed 45% light key.
    • Distillate 98%, bottoms 2%.
    • Reflux 1.5 times the minimum, saturated liquid feed.
    Calculation path
    1. 1

      Fenske gives 9.87 minimum stages at total reflux.

    2. 2

      The Underwood root is 1.429 and the minimum reflux 1.75.

    3. 3

      Operating at 1.5 times the minimum, the reflux is 2.62.

    4. 4

      Gilliland places the column at 17.6 stages, and the McCabe-Thiele stepping gives 18.

    5. 5

      The feed enters at stage 9, and at 0.5 m per theoretical stage the column is 11.0 m of packing equivalent.

    Result

    N min 9.87 · R min 1.75 · 17.6 stages · feed at stage 9.

    Interpreting the result

    The two methods agreeing within a stage is the check that matters: Gilliland is a correlation and McCabe-Thiele is a construction, and when they diverge the constant-α assumption is usually why. A real column’s relative volatility falls as the light component is stripped, so the shortcut under-predicts the stages on a wide-boiling system.

    Assumptions & limitations

    • Constant relative volatility across the whole column.
    • Binary system: one light and one heavy key.
    • A saturated liquid feed by default, with q adjustable.
    • No pressure drop, no subcooling and no side draws.

    Frequently asked questions

    Why operate at 1.5 times the minimum reflux?

    Because the stage count rises steeply below it and the reboiler duty rises linearly above it. The economic optimum is usually between 1.1 and 1.5.

    Why does α matter so much?

    Because the stage count goes with the log of the separation over the log of α. Halving α roughly doubles the stages, which is why close-boiling separations are expensive.

    What does q mean?

    The feed thermal condition: 1 is a saturated liquid, 0 a saturated vapour, and negative values are superheated vapour. It sets where the q-line crosses the operating lines.

    Can this design a real column?

    It gives the first estimate. A rigorous design uses a stage-by-stage simulation with a thermodynamic model, because α, the hydraulics and the pressure profile all vary.

    Methodology last reviewed 2026-10-22. Results are informational and not financial or engineering advice — see our disclaimer.