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Separation Processes

One flash (Chapter 3) buys you one stage of separation — often not enough. Stack many equilibrium stages, add reflux at the top and a reboiler at the bottom, and you get distillation: the workhorse that does most of the separation in the chemical industry. This chapter is how a column is modelled, how to read what it's doing, and how to size one fast.

5.1 A column is a stack of equilibrium stages

Picture the flash of Chapter 3 repeated on every tray: liquid falls down the column, vapor rises up it, and on each stage the two come to equilibrium — vapor enriching in the light component as it climbs, liquid enriching in the heavy component as it descends. A condenser at the top returns part of the overhead as reflux (ratio

), and a reboiler at the bottom boils part of the bottoms back up. The example below is the ethanol-water column from Chapter 1, now the object of study.

5.2 The MESH equations

Rigorously, every stage

is described by four kinds of equation — the MESH equations — written for each component :

$l_{i,j},\; v_{i,j}$

Liquid- and vapor-phase molar flow of component i leaving stage j (the unknowns the M equations balance).

$f_{i,j}$

Feed flow of component i entering stage j (zero on every stage except the feed tray).

$y_{i,j},\; x_{i,j}$

Vapor and liquid mole fraction of component i on stage j.

$K_{i,j}$

Equilibrium ratio of component i on stage j (the same K from Chapter 3's flash, evaluated at that stage's T, P, composition).

$L_j,\; V_j$

Total liquid and vapor molar flow leaving stage j.

$h^L_j,\; h^V_j$

Liquid- and vapor-phase molar enthalpy on stage j.

$Q_j$

Heat added to (or removed from) stage j — nonzero only at the condenser and reboiler for an ordinary column.

That's

coupled nonlinear equations for an -stage, -component column, solved simultaneously by a Newton method — the equation-oriented idea from Chapter 1, applied inside a single unit. MaximaLabs shows you the exact set on the column's Theory tab:

5.3 McCabe-Thiele: the picture behind the column

Before rigorous solvers existed, engineers sized binary columns graphically with the McCabe-Thiele construction — and it's still the clearest picture of what a column does. On the equilibrium

diagram from Chapter 2 you draw two operating lines (the mass balances above and below the feed),

$R$

Reflux ratio L/D — moles refluxed per mole of distillate.

$x_n,\; y_{n+1}$

Liquid composition leaving stage n and vapor composition entering it from below — successive points on the operating line.

$x_D$

Distillate composition (the line's fixed point: at x_n=x_D, y_{n+1}=x_D too).

and step off stages as a staircase between the operating lines and the equilibrium curve — each tread is one theoretical stage. MaximaLabs draws it for a solved column, actual stages and all:

5.4 Reading the column: stage profiles

A rigorous solve gives more than top and bottom products — it gives the full internal state, stage by stage: temperature, the liquid and vapor traffic (

), and the composition on every tray. These profiles are how you diagnose a column — spot a pinch (stages doing no separation), find the real feed-stage optimum, or see a temperature break:

5.5 Shortcut design (Fenske-Underwood-Gilliland)

A full MESH solve needs a column to already exist. To design one — how many stages, what reflux — engineers start with the FUG shortcut (Aspen's DSTWU). Fenske gives the minimum stages at total reflux from the relative volatility

and the desired split:

$N_{\min}$

Minimum theoretical stages, at total reflux (R=infinity) — the fewest stages any design at this split could use.

$\alpha_{LK,HK}$

Relative volatility of the light key over the heavy key.

$(x_{LK}/x_{HK})_D,\;(x_{HK}/x_{LK})_B$

Light-to-heavy-key ratio in the distillate, and heavy-to-light-key ratio in the bottoms — how sharp the desired split is.

Underwood gives the minimum reflux ratio — the other end of the design-space wall, where an infinite number of stages would be needed. For a multicomponent feed you first solve for the root(s)

of

$\alpha_i,\; x_{F,i}$

Relative volatility and feed mole fraction of component i.

$\theta$

The Underwood root(s) — a value strictly between the heavy- and light-key volatilities, solved for numerically.

$q$

Feed liquid fraction (thermal condition): q=1 saturated liquid, q=0 saturated vapor, in between a flashed feed.

where

is the feed's liquid fraction (thermal condition), then use in a second summation over the distillate to get :

$R_{\min}$

Minimum reflux ratio — the other design-space wall (infinite stages needed below this).

$x_{D,i}$

Distillate mole fraction of component i.

Gilliland then closes the loop: given

(Fenske) and (Underwood), pick an actual operating reflux and read off the actual stage count from an empirical correlation between the two dimensionless ratios:

$N$

Actual stage count at the chosen operating reflux R — what the correlation solves for.

$X$

The dimensionless reflux ratio (R-R_min)/(R+1) — Gilliland's correlating variable.

Fenske → Underwood → Gilliland is the classic one-shot sequence: three cheap algebraic equations instead of a full MESH solve, good enough to size a column before committing to the rigorous model. MaximaLabs exposes exactly this as the Shortcut column tool (Analysis & reports ▸ Engineering analysis) — enter the light/heavy key recoveries and it returns

, , and the Gilliland-estimated actual in one call, the same three equations above.

5.6 The wall: azeotropes

Distillation separates by relative volatility, so where volatility vanishes —

, the azeotrope — an ordinary column stalls. On the McCabe-Thiele picture the staircase can't step past the point where the equilibrium curve meets the diagonal; ethanol-water pinches at about 89 mol% ethanol no matter how many trays you add. Breaking that wall (pressure-swing, entrainer, extractive or azeotropic distillation) is its own craft — and exactly the sort of thing you'd ask the copilot to set up.

5.7 Try it

A column is just Chapter 3's flash, stacked and wired with reflux and reboil — and everything you've learned so far (thermodynamics, flash, the equation-oriented solve) is doing the work inside that one node.

5.8 Exercises