ChemicalSeparationshardCourseworkPE / Advanced

Minimum Stages by the Fenske Equation

A binary distillation produces a distillate of $x_D = 0.95$ and a bottoms of $x_B = 0.05$ (mole fractions of the light key).

The relative volatility is constant at $\alpha = 2.5$ .

Find the minimum number of theoretical stages at total reflux.

Given

xD	0.95 -	Distillate composition
xB	0.05 -	Bottoms composition
α	2.5 -	Relative volatility

Hint 1

Fenske gives the minimum stages at total reflux for constant relative volatility.

Hint 2

The separation factor is a product of two ratios, one at each end of the column.

Hint 3

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

Worked solution — try the problem first

Fenske equation for minimum stages at total reflux:

$N_{min} = \frac{\ln\left[\left(\dfrac{x_D}{1-x_D}\right)\left(\dfrac{1-x_B}{x_B}\right)\right]}{\ln\alpha}$

Separation factor:

$\frac{x_D}{1-x_D} = \frac{0.95}{0.05} = 19, \qquad \frac{1-x_B}{x_B} = \frac{0.95}{0.05} = 19$

$\text{product} = 19 \times 19 = 361$

Evaluate:

$N_{min} = \frac{\ln(361)}{\ln(2.5)} = \frac{5.889}{0.9163} = 6.43$

$\boxed{N_{min} = 6.43 \text{ stages}}$

What this number means. It is a theoretical floor, achievable only at total reflux with no product withdrawn — of no use for production, but the standard benchmark. A real column at a practical reflux ratio typically needs about twice $N_{min}$ , so roughly 13 theoretical stages here. Divide by the tray efficiency (perhaps 0.7) to get actual trays, and add one for the reboiler.

Do not round down. 6.43 means 7 stages minimum; 6 would not achieve the specification.

Concepts:DistillationFenske equationMinimum stagesRelative volatility

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