CHE-04

Fenske minimum stages

Nmin = ln[(xD/(1−xD))((1−xB)/xB)] / ln α.

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DistillationFenske 1932McCabe–Thiele

Governing equation

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

$\alpha \in \left[1.05, 8\right]$

where

x_D: Distillate mole frac. (—)
x_B: Bottoms mole frac. (—)
\alpha: Relative volatility (—)
N_{\min}: Minimum stages (—)

The figure, part by part

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①

Governing law — red

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

$\alpha \in \left[1.05, 8\right]$

$\alpha = 1.05$ $N_{\min} = 120.7$

Along the curve

②

Domain of the variable — blue

\alpha_{\min} = 1.05,\quad \alpha_{\max} = 8

$\alpha \in \left[1.05, 8\right]$

$\alpha = 1.05$ $\alpha = 1.05$

Along the curve

③

Response curve — green

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

$\alpha \in \left[1.05, 8\right]$

$\alpha = 1.05$ $N_{\min} = 120.7$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}N_{\min}}{\mathrm{d}\alpha}

$\alpha \in \left[1.05, 8\right]$

$\alpha = 1.094$ $N_{\min}^{\prime} = -857.5$

Along the curve

Lecture brief

Historical brief

From CSTR/PFR mole balances and Arrhenius rates to McCabe–Thiele stages and NTU exchangers, chemical engineering is conservation plus equilibrium. The lab is that design arithmetic. This sheet (CHE-04 — Fenske minimum stages) is the form associated with Fenske 1932 · McCabe–Thiele. Working symbols: $x_D$ , $x_B$ , $\alpha$ $\rightarrow$ $N_{\min}$ . At total reflux McCabe–Thiele steps collapse to the Fenske logarithm of the relative volatility, the minimum number of equilibrium stages.

Purpose

Purpose: compute

N_{\min}

from

x_D

x_B

\alpha

in Chemical engineering via

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

Nmin = ln[(xD/(1−xD))((1−xB)/xB)] / ln α. Use it when a real chemical engineering question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

x_D = 0.950\,\mathrm{—}

x_B = 0.050\,\mathrm{—}

\alpha = 2.200\,\mathrm{—}

, the governing relation

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

yields

N_{\min} = 7.469\,\mathrm{—}

. Binary, constant α, total reflux. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Distillate mole frac. x_D0.95 —Bottoms mole frac. x_B0.05 —Relative volatility \alpha2.2 —

Outputs

Minimum stages N_{\min}7.469 —

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Narration of this film

Binary, constant α, total reflux.

At total reflux McCabe–Thiele steps collapse to the Fenske logarithm of the relative volatility, the minimum number of equilibrium stages.

Reading speed

Watch on YouTube