Engineering task and calculation objective
The SCUT module designs distillation columns for multi-component mixtures using the shortcut method. Instead of a rigorous tray-by-tray calculation, the classical approximation methods of Fenske, Underwood and Gilliland are combined: they deliver the minimum number of theoretical trays Nmin at infinite reflux, the minimum reflux ratio Rmin at an infinite number of trays, and from these, for a chosen reflux ratio R, the number of theoretical trays N actually to be installed.
The separation task is defined via the light and heavy key components; their required split between top and bottom product determines the sharpness of the separation. In addition, the thermal state of the feed enters via the mole fraction of liquid in the feed, which influences the location of the feed tray and the Underwood evaluation.
Shortcut calculations of this kind are the standard tool of the concept and basic-engineering phase: they quickly deliver reliable starting values for tray count, reflux ratio and energy demand of a rectification column and narrow down the search space for the subsequent rigorous simulation.
Calculation workflow
- Define the separation task and key components: For the mixture with its number of components, the light and heavy key components are chosen and their required concentrations or recoveries in distillate and bottoms are specified; the thermal state of the feed is described by the mole fraction of liquid in the feed.
- Minimum number of trays per Fenske: At total reflux, the Fenske equation with the mean relative volatility of the key components yields the minimum number of theoretical trays Nmin; at the same time, the distribution of the non-key components can be estimated.
- Minimum reflux ratio per Underwood: The Underwood equations deliver, via the common root parameter, the minimum reflux ratio Rmin that just permits the separation at an infinite number of trays; the feed condition enters directly here.
- Choose the operating point: The operating reflux ratio R is chosen as a multiple of Rmin – factors of about 1.2 to 1.5 are customary as a compromise between capital cost (tray count) and energy cost (reboiler duty).
- Number of trays per Gilliland: The Gilliland correlation links (R − Rmin)/(R + 1) with (N − Nmin)/(N + 1) and delivers the number of theoretical trays N to be installed at the chosen reflux ratio; in addition, the location of the feed tray can be estimated.
Input quantities
| Quantity | Symbol | Unit |
|---|---|---|
| Zulauf | Zulauf | – |
| Zulauf | Zulauf | – |
| Zulauf | Zulauf | – |
| Zulauf | Zulauf | – |
| Zulauf | Zulauf | – |
| Zulauf | Zulauf | – |
| Zulauf | Zulauf | – |
| Zulauf | Zulauf | – |
| Zulauf | Zulauf | – |
| Zulauf | Zulauf | – |
| Kopf | Kopf | – |
| Kopf | Kopf | – |
| Kopf | Kopf | – |
| Kopf | Kopf | – |
| Kopf | Kopf | – |
| Kopf | Kopf | – |
| Kopf | Kopf | – |
| Kopf | Kopf | – |
| Kopf | Kopf | – |
| Kopf | Kopf | – |
| Sumpf | Sumpf | – |
| Sumpf | Sumpf | – |
| Sumpf | Sumpf | – |
| Sumpf | Sumpf | – |
Worked example
An equimolar binary mixture (50 mol% light component) is to be separated at boiling-point feed such that distillate and bottoms each reach 95 mol% purity. The mean relative volatility is α = 2.5. Find Nmin, Rmin and the theoretical number of trays N at R = 1.3 · Rmin.
Given values
| Feed concentration of light component x_F | 0.50 |
| Distillate x_D | 0.95 |
| Bottoms x_B | 0.05 |
| Mean relative volatility α | 2.5 |
| Feed condition | saturated liquid (q = 1) |
| Reflux ratio R | 1.3 · Rmin |
Solution
Minimum number of trays per Fenske
Nmin = ln[(xD/(1 − xD)) · ((1 − xB)/xB)] / ln α
Nmin = ln[(0.95/0.05) · (0.95/0.05)] / ln 2.5 = ln 361 / ln 2.5 = 5.889 / 0.916 = 6.43 theoretical stages (including the reboiler)
Minimum reflux ratio per Underwood
For a binary mixture with saturated liquid feed (q = 1):
Rmin = 1/(α − 1) · [xD/xF − α · (1 − xD)/(1 − xF)]
Rmin = 1/1.5 · [0.95/0.50 − 2.5 · 0.05/0.50] = 1/1.5 · (1.90 − 0.25) = 1.10
Operating reflux ratio and Gilliland parameter
R = 1.3 · Rmin = 1.3 · 1.10 = 1.43
X = (R − Rmin)/(R + 1) = (1.43 − 1.10)/(1.43 + 1) = 0.33/2.43 = 0.136
Number of trays per Gilliland (Molokanov equation)
Y = 1 − exp[ (1 + 54.4 X)/(11 + 117.2 X) · (X − 1)/√X ] = 1 − exp(−0.7307) = 0.518
With Y = (N − Nmin)/(N + 1) it follows that:
N = (Y + Nmin)/(1 − Y) = (0.518 + 6.43)/(1 − 0.518) = 14.4 theoretical stages
In practice, around 15 theoretical stages (including the reboiler) are therefore installed.
Result
| Minimum number of trays Nmin (Fenske) | 6.43 |
| Minimum reflux ratio Rmin (Underwood) | 1.10 |
| Operating reflux ratio R | 1.43 |
| Theoretical number of stages N (Gilliland) | ≈ 14.4 → 15 stages |
All values are illustrative. The applicable standard and project-specific boundary conditions remain authoritative.
Frequently asked questions
What are key components and how are they chosen?
The light key component is the heaviest component that should predominantly go into the top product; the heavy key component is the lightest that should predominantly remain in the bottoms. The separation cut runs between the two. Components substantially lighter or heavier than the keys distribute almost completely to the top or bottom, respectively; their distribution follows from the Fenske estimate.
How accurate is the shortcut method compared to rigorous simulation?
For nearly ideal mixtures with approximately constant relative volatility, the tray count typically lies within a few trays of the rigorous result. For strongly non-ideal systems, temperature-dependent volatility, azeotropes or side draws, the approach fails conceptually – the shortcut result then serves only as a starting estimate for the rigorous tray-by-tray calculation.
Why is R chosen significantly larger than Rmin?
At R = Rmin, an infinite number of trays would be required, and close to Rmin the tray count grows extremely steeply. With increasing R the tray count falls, but reboiler and condenser duties rise proportionally to (R + 1). Experience puts the economic optimum at about 1.2 to 1.5 times the minimum reflux ratio; the module makes this trade-off directly visible by juxtaposing Rmin, R, Nmin and N.
Does the reboiler count as a theoretical tray?
Yes, the bottoms reboiler acts as an equilibrium stage and is included in the stage count determined per Fenske/Gilliland; a total condenser, however, is not. In the transition to the real column, the theoretical tray count must additionally be converted into actual trays or packing height via the tray efficiency or the HETP value of a packing.