Distillation Column Design Theory
The Fenske-Underwood-Gilliland (FUG) shortcut method for designing multi-component distillation columns with rigorous analytical equations.
Overview
The FUG method is a three-step shortcut procedure for preliminary distillation column design:
- Fenske Equation: Calculate minimum stages at total reflux
- Underwood Equations: Determine minimum reflux ratio
- Gilliland Correlation: Find actual stages for a given reflux ratio
Fenske Equation (Minimum Stages)
At total reflux (R → ∞), the minimum number of stages is:
Nmin = ln[(xD,LK/xD,HK) × (xB,HK/xB,LK)] / ln(αavg)
Where:
- Nmin = Minimum number of equilibrium stages
- xD,LK, xD,HK = Distillate mole fractions of light and heavy keys
- xB,LK, xB,HK = Bottoms mole fractions of light and heavy keys
- αavg = Average relative volatility between keys
Underwood Equations (Minimum Reflux)
The minimum reflux ratio requires solving two coupled equations:
First Equation (Find θ):
1 - q = Σ (αi × zi) / (αi - θ)
Second Equation (Find Rmin):
Rmin + 1 = Σ (αi × xD,i) / (αi - θ)
Where θ is the Underwood root (αHK < θ < αLK) and q is the feed quality.
Gilliland Correlation
Relates actual stages to minimum stages using dimensionless variables:
Define: X = (R - Rmin) / (R + 1)
Then: Y = (N - Nmin) / (N + 1)
Molokanov correlation (curve fit):
Y = 1 - exp[(1 + 54.4X) / (11 + 117.2X) × (X - 1) / √X]
Feed Quality (q)
The feed quality parameter q represents the thermal condition:
| q Value | Feed Condition |
|---|---|
| q > 1 | Subcooled liquid |
| q = 1 | Saturated liquid (bubble point) |
| 0 < q < 1 | Two-phase (partially vaporized) |
| q = 0 | Saturated vapor (dew point) |
| q < 0 | Superheated vapor |
Design Procedure
- Specify feed composition, flow rate, and thermal condition
- Select light key (LK) and heavy key (HK) components
- Specify desired recoveries or product compositions
- Calculate average relative volatility
- Use Fenske equation to find Nmin
- Use Underwood equations to find Rmin
- Select operating reflux (typically 1.2-1.5 × Rmin)
- Use Gilliland correlation to find actual stages N
- Determine feed stage location using Kirkbride correlation
Kirkbride Equation (Feed Stage)
Estimates the optimal feed stage location:
NR/NS = [(zHK/zLK) × (xB,LK/xD,HK)² × (B/D)]0.206
Where NR is stages in rectifying section and NS is stages in stripping section.
References
- Fenske, M.R. (1932). "Fractionation of Straight-Run Pennsylvania Gasoline". Ind. Eng. Chem.
- Underwood, A.J.V. (1948). "Fractional Distillation of Multicomponent Mixtures". Chem. Eng. Prog.
- Gilliland, E.R. (1940). "Multicomponent Rectification". Ind. Eng. Chem.
- Seader, J.D. and Henley, E.J. (2006). Separation Process Principles. Wiley.