OpenSpecEngineering

Finds the minimum number of equilibrium stages a binary distillation needs to reach a given distillate and bottoms purity. That minimum applies at total reflux, when no product is drawn and every stage works hardest. Process engineers use it as the first step of a shortcut column design, before choosing a reflux ratio. It is a limit, not the stage count of a working column; that needs a McCabe-Thiele or Gilliland step at the operating reflux.

Equation

N_min = ln[(x_D/(1 − x_D))·((1 − x_B)/x_B)] / ln α

This is the Fenske equation [1], [2]. x_D is the mole fraction of the light (more volatile) component in the distillate, and x_B its mole fraction in the bottoms. α is the relative volatility of the light component to the heavy one: the ratio of their vapor-to-liquid ratios, K_light/K_heavy. N_min counts equilibrium stages, and includes a partial reboiler as one stage. The page also gives N_min − 1, the number of theoretical trays in the column itself.

Limits and common mistakes

• α must be roughly constant through the column. If it changes from top to bottom, use the geometric mean of the top and bottom values, √(α_top·α_bottom) [1].
• Both purities are light-component fractions. For a bottoms that is 98 % heavy component, enter x_B = 0.02, not 0.98.
• Close-boiling mixtures (α near 1) need very many stages, and the result rises sharply as α falls toward 1.
• The stages are ideal. Divide by the tray efficiency, often 50–80 %, to get real trays.
• A real column runs at a reflux ratio above the minimum, typically 1.2 to 1.5 times it. A common rule of thumb is that it then needs about twice N_min.

Worked example

A column separates benzene from toluene, with an average relative volatility of 2.4. The distillate must be 95 % benzene and the bottoms no more than 5 % benzene. What is the fewest stages that could do it?

1. Enter Relative Volatility (α) 2.4.
2. Enter Distillate Purity (xD, light-key mole fraction) 0.95 and Bottoms Purity (xB, light-key mole fraction) 0.05.
3. Press Solve. The Solution shows: Minimum Number of Equilibrium Stages (N_min): 6.72654
   Minimum Stages Excluding Reboiler: 5.72654

Check by hand: the separation factor is (0.95/0.05)² = 361, and ln 361/ln 2.4 = 5.889/0.8755 = 6.727. Round up: the column needs at least 7 equilibrium stages, or 6 theoretical trays plus the reboiler, before reflux and tray efficiency are allowed for.