McCabe-Thiele Method Fundamentals

1. Overview & Applications

The McCabe-Thiele method is a graphical technique for designing binary distillation columns developed by Warren McCabe and Ernest Thiele in 1925. It determines the theoretical equilibrium stages required for a specified separation based on vapor-liquid equilibrium (VLE) data, reflux ratio, and feed thermal condition.

Complete McCabe-Thiele diagram for propane/butane separation showing equilibrium curve (α=2.5), rectifying and stripping operating lines, vertical q-line at feed composition 0.50, and 10 stepped-off theoretical stages with feed at stage 5 — McCabe-Thiele construction for C3/C4 separation: equilibrium curve, operating lines intersecting at q-line, and 10 theoretical stages stepped off from distillate (x_D=0.95) to bottoms (x_B=0.05).

NGL Fractionation

Deethanizer, Depropanizer

Ethane/propane and propane/butane separations in gas plants.

Condensate Stabilization

Light Ends Removal

Remove C1-C2 from condensate to meet RVP specifications.

Debutanizer

LPG Production

C3/C4 overhead (LPG) and C5+ bottoms (natural gasoline).

Preliminary Design

Simulation Input

Quick tray count estimate before rigorous modeling.

Key Definitions

Theoretical stage: Equilibrium stage where vapor and liquid leaving are in thermodynamic equilibrium (Murphree efficiency = 100%)
Equilibrium curve: y vs. x plot from VLE data at column pressure; lies above 45° diagonal for light component enrichment
Operating line: Material balance relating passing vapor (y_n+1) and liquid (x_n) streams
Reflux ratio (R): L/D where L = reflux returned to column, D = distillate withdrawn
q-line: Feed condition line; intersects 45° line at (z_F, z_F)
Minimum reflux (R_min): Reflux ratio where operating line is tangent to equilibrium curve (pinch point)

Fundamental Assumptions

Binary or pseudo-binary: Two components or key component approximation
Constant molal overflow (CMO): L and V constant in each section; valid when latent heats are similar (ΔH_vap,A ≈ ΔH_vap,B)
Adiabatic operation: No heat loss through column walls
Total condenser: All vapor condensed; x_D = y₁ (partial condenser requires modification)
Negligible pressure drop: Constant pressure throughout (typically valid for low-pressure columns)

Engineering Value: McCabe-Thiele provides immediate insight into column behavior that simulators cannot—graphical understanding of how R, q, x_D, and x_B affect stage count. Essential for troubleshooting operating columns, optimizing feed location, and validating simulation results.

2. Equilibrium Curve Construction

The equilibrium curve plots vapor composition (y) versus liquid composition (x) at column pressure—the foundation of McCabe-Thiele design.

Y-X diagram comparing equilibrium curves at relative volatilities α=1.5, 2.0, 2.5, and 3.0, showing how higher α creates larger gap from 45-degree diagonal line indicating easier separation — Equilibrium curves at different relative volatilities: higher α moves curve away from 45° diagonal, reducing required stages. α=2.5 typical for C3/C4 separation.

Vapor-Liquid Equilibrium

Relative Volatility Equation: y = (α × x) / [1 + (α - 1) × x] Where: α = K_light / K_heavy = P°_A / P°_B (for ideal systems) x = liquid mole fraction of light component y = vapor mole fraction of light component Physical Interpretation: α > 1 → Light component enriches in vapor (separation possible) α = 1 → No separation (equilibrium = 45° line) α → ∞ → Complete separation in one stage Temperature/Pressure Effects: Higher pressure → lower α → harder separation α varies with composition for non-ideal systems

Typical Relative Volatilities

System	Relative Volatility (α)	Separation Difficulty	Typical Trays
Methane/Ethane (demethanizer)	4.0-6.0	Moderate (cryogenic)	30-50
Ethane/Propane (deethanizer)	3.5-4.5	Moderate	25-40
Propane/Butane (depropanizer)	2.5-3.0	Easy-moderate	15-25
Butane/Pentane (debutanizer)	2.0-2.5	Easy	10-20
Benzene/Toluene	2.3-2.5	Easy	15-25
n-Butane/i-Butane	1.2-1.3	Very difficult	80-120
Ethanol/Water (azeotrope)	1.0 @ 95.6% EtOH	Impossible (azeotrope)	N/A

Generating Equilibrium Data

Method 1: Constant α (most common) y = (α × x) / [1 + (α - 1) × x] Use when α varies < ±15% across column. Method 2: VLE Data Tables For non-ideal systems, use T-x-y data from: • Perry's Handbook Chapter 13 • DIPPR database • Process simulator (Aspen, HYSYS, ProMax) Method 3: K-Value Charts y = K_light × x where K = f(T, P) Use GPSA K-charts or DePriester charts.

Example: Propane/Butane at 200 psia

x (C3)	y (C3)
0.00	0.000
0.20	0.394
0.40	0.634
0.60	0.796
0.80	0.912
1.00	1.000

Based on α = 2.5 (average). Curve lies above 45° line—propane enriches in vapor.

3. Operating Lines & McCabe-Thiele Construction

Operating lines are straight lines representing material balances in the rectifying (above feed) and stripping (below feed) sections. Their slopes depend on internal liquid-to-vapor ratios.

McCabe-Thiele operating lines construction showing rectifying line from (x_D, x_D) with slope R/(R+1)=2/3, stripping line from (x_B, x_B), vertical q-line at x_F=0.50, and intersection point with formula box — Operating lines construction: rectifying line (slope = R/(R+1)) and stripping line intersect on the q-line at feed composition.

Rectifying Section Operating Line

Rectifying Operating Line: y = [R/(R+1)] × x + x_D/(R+1) Where: R = L/D = Reflux ratio x_D = Distillate composition (mole fraction light) Key Points: • Slope = R/(R+1) — increases with R, approaches 1 as R → ∞ • y-intercept = x_D/(R+1) • Passes through point (x_D, x_D) on 45° diagonal • Higher R → steeper line → more stages but better separation

Stripping Section Operating Line

Stripping Operating Line: Passes through (x_B, x_B) and q-line intersection point. Slope = L'/V' = (L + qF)/(V + (1-q)F) Where: L' = Liquid flow in stripping section V' = Vapor flow in stripping section q = Feed liquid fraction Key Points: • Slope typically > 1 (steeper than rectifying) • Always passes through (x_B, x_B) on 45° line • Intersects rectifying line ON the q-line

Step-by-Step Construction

McCabe-Thiele Step-Off Procedure: 1. Plot equilibrium curve and 45° diagonal 2. Mark x_D, x_F, x_B on x-axis 3. Draw rectifying line from (x_D, x_D) with slope R/(R+1) 4. Draw q-line from (z_F, z_F) with slope q/(q-1) 5. Draw stripping line through (x_B, x_B) and q-line intersection 6. Step off stages: • Start at (x_D, x_D) • Horizontal to equilibrium curve • Vertical down to operating line • Repeat until x ≤ x_B 7. Count steps = N_theoretical (including reboiler as one stage)

McCabe-Thiele stage step-off method showing stages 1-10 with horizontal steps to equilibrium curve and vertical steps to operating line, Stage 5 highlighted as feed stage in yellow, with column schematic inset — Stage step-off procedure: alternate horizontal (to equilibrium) and vertical (to operating line) steps. Feed stage (Stage 5) marks transition from rectifying to stripping section.

Example: C3/C4 Separation

Parameter	Value
Feed (z_F)	0.50
Distillate (x_D)	0.95
Bottoms (x_B)	0.05
α	2.5
q	1.0 (sat. liq.)
R	2.0

Result: ~10 theoretical stages, feed at stage 5 from top.

Design Trade-off: Higher R → steeper rectifying line → fewer stages needed BUT higher reboiler/condenser duties. Optimal R typically 1.2–1.5 × R_min, balancing capital (trays) vs. operating cost (energy).

4. Minimum Reflux Ratio Determination

Minimum reflux (R_min) is the lowest reflux ratio achieving the specified separation—requiring infinite stages. At R_min, the operating line touches the equilibrium curve at a "pinch point."

Minimum reflux vs operating reflux comparison showing pinch point where R_min operating line touches equilibrium curve, and steeper operating reflux line (1.3 × R_min) with operating margin gap — Minimum reflux (R_min) creates pinch point requiring infinite stages. Operating at 1.3 × R_min provides margin for finite stage design.

Graphical Method

Finding R_min: 1. Draw q-line from (z_F, z_F) 2. Find q-line / equilibrium curve intersection: (x_p, y_p) 3. Draw line from (x_D, x_D) through (x_p, y_p) 4. Slope = R_min/(R_min+1) 5. Solve: R_min = (x_D - y_p)/(y_p - x_p)

Underwood Method

Binary System (Perry's 9th Ed. Eq. 13-61/62): Step 1 — Find θ (where 1 < θ < α): (α × z_F)/(α - θ) + (1 - z_F)/(1 - θ) = 1 - q Step 2 — Calculate R_min: R_min + 1 = (α × x_D)/(α - θ) + (1 - x_D)/(1 - θ) Solve iteratively or use graphical method.

Reflux Ratio vs. Stages

R/R_min	Stages	Energy	Assessment
1.0	∞	Minimum	Infeasible
1.1	Very high	+10%	Marginal
1.2–1.3	Moderate	+20-30%	Typical optimum
1.5	Lower	+50%	Conservative
2.0	Low	+100%	High energy cost
∞	N_min	∞	No product

Economic Optimum

Total Cost Trade-off: Capital cost ∝ N_trays (column height, diameter) Operating cost ∝ R (reboiler + condenser duty) Optimum R = 1.2 to 1.5 × R_min (minimizes total annualized cost) Rule of thumb: R_opt ≈ 1.3 × R_min for most NGL fractionation Factors shifting optimum: - Energy cost (high → favor lower R) - Equipment cost (high → favor higher R, fewer trays) - Space constraints (limited height → higher R) - Existing column (fixed trays → adjust R to meet spec)

Fenske Minimum Stages: At total reflux (R → ∞), minimum stages N_min = ln[(x_D/(1-x_D)) × ((1-x_B)/x_B)] / ln(α). Use Gilliland correlation to relate N and R between these limits.

5. Feed Condition & Actual Trays

Feed thermal condition (q-value) determines q-line slope and affects stage distribution between rectifying and stripping sections.

Five q-line orientations radiating from feed point (z_F, z_F): subcooled liquid (q>1, steep positive), saturated liquid (q=1, vertical), two-phase (0<q<1, negative slope), saturated vapor (q=0, horizontal), superheated vapor (q<0, shallow positive) — q-line orientations depend on feed thermal condition: vertical for saturated liquid (q=1), horizontal for saturated vapor (q=0). Formula: y = [q/(q-1)]x - z_F/(q-1).

Feed Condition Parameter (q)

Definition: q = (H_sat.vapor - H_feed) / (H_sat.vapor - H_sat.liquid) q = fraction of feed that is liquid q-Line Equation: y = [q/(q-1)] × x - z_F/(q-1) All q-lines pass through (z_F, z_F) on 45° diagonal.

q-Value Reference

Feed Condition	q	q-Line
Subcooled liquid	> 1	Steep positive
Saturated liquid (bubble pt)	1.0	Vertical
Two-phase	0–1	Negative slope
Saturated vapor (dew pt)	0	Horizontal
Superheated vapor	< 0	Shallow positive

O'Connell Tray Efficiency

Converting Theoretical to Actual Trays: N_actual = N_theoretical / E_overall O'Connell Correlation (Perry's 9th Ed.): E_o = 0.492 × (α × μ_L)^(-0.245) Where: μ_L = Liquid viscosity (cP) at column conditions α = Average relative volatility Valid range: 0.1 < (α × μ_L) < 10 Typical Efficiencies: Valve/sieve trays: 60–80% Low α×μ (< 0.5): E ≈ 75-85% High α×μ (> 2): E ≈ 50-65%

Efficiency Example

Given: α = 2.5, μ_L = 0.3 cP α × μ = 0.75 (within valid range) E_o = 0.492 × (0.75)^(-0.245) = 0.492 × 1.07 = 0.53 ≈ 53% If N_theoretical = 10 stages: N_actual = 10 / 0.53 = 18.9 → 19 trays Add 10-20% margin: 21-23 actual trays installed.

Design Best Practices

Design margin: Add 10–20% extra trays for flexibility, fouling, and capacity growth
Feed location: Actual feed nozzle ±1–2 trays from calculated optimal location
Tray spacing: 18–24" typical; 24" for high-pressure or foaming systems
Pressure drop: 0.05–0.15 psi/tray; check reboiler temperature impact
Validate with simulation: Use Aspen HYSYS, ProMax, or Pro/II for final design with actual VLE and tray hydraulics

When to use McCabe-Thiele: Preliminary design, quick estimates, sensitivity analysis, troubleshooting, validating simulator results, and understanding column behavior. For final equipment specifications, always use rigorous simulation.

McCabe-Thiele Graphical Distillation Method

y = f(x) from VLE

Rectifying & stripping

R_min from pinch

Contents

Use this guide when you need to: