McCabe-Thiele Method
A graphical method for designing binary distillation columns, developed by Warren McCabe and Ernest Thiele in 1925.
Overview
The McCabe-Thiele method is a graphical technique for determining the number of theoretical stages (or trays) required for a binary distillation. It combines the vapor-liquid equilibrium (VLE) curve with operating lines to step off stages graphically.
Key Assumptions
- Constant molar overflow (CMO) - liquid and vapor flow rates are constant in each section
- Binary mixture - only two components
- Theoretical stages - each stage achieves equilibrium
- Adiabatic operation - no heat losses
- Steady-state operation
Vapor-Liquid Equilibrium
For an ideal mixture following Raoult's law, the equilibrium relationship can be expressed using relative volatility (α):
y = αx / (1 + (α-1)x)
Where α is the relative volatility of the more volatile component to the less volatile component.
Operating Lines
Rectifying Section
The operating line for the rectifying (enriching) section above the feed:
y = (R/(R+1))x + xD/(R+1)
Stripping Section
The operating line for the stripping section below the feed passes through (xB, xB) and the intersection of the rectifying line with the q-line.
q-Line (Feed Line)
The q-line represents the feed condition:
y = (q/(q-1))x - zF/(q-1)
Where q is the feed quality:
- q = 1: Saturated liquid feed
- q = 0: Saturated vapor feed
- 0 < q < 1: Two-phase feed
- q > 1: Subcooled liquid
- q < 0: Superheated vapor
Minimum Reflux Ratio
The minimum reflux ratio (Rmin) occurs when the operating line passes through the pinch point - where it intersects the equilibrium curve at the q-line. At minimum reflux, an infinite number of stages would be required.
Practical designs typically use R = 1.1 to 1.5 times Rmin.
Stepping Off Stages
Starting from the distillate composition (xD, xD) on the diagonal line:
- Draw horizontal line to equilibrium curve
- Draw vertical line down to operating line
- Repeat until reaching bottoms composition (xB)
- Count the number of steps = number of theoretical stages
The feed stage is where you switch from the rectifying to stripping operating line.
Design Parameters
| Symbol | Parameter | Description |
|---|---|---|
| α | Relative volatility | Measure of ease of separation |
| zF | Feed composition | Mole fraction of light component in feed |
| xD | Distillate composition | Desired purity of overhead product |
| xB | Bottoms composition | Desired purity of bottom product |
| R | Reflux ratio | L/D = liquid returned / distillate removed |
| q | Feed quality | Thermal condition of feed |
References
- McCabe, W.L. and Thiele, E.W. (1925). "Graphical Design of Fractionating Columns". Industrial & Engineering Chemistry.
- Seader, J.D. and Henley, E.J. (2006). Separation Process Principles. Wiley.
- Perry, R.H. and Green, D.W. (2008). Perry's Chemical Engineers' Handbook. McGraw-Hill.