Ponchon-Savarit method and minimum reflux

Why it matters

McCabe–Thiele assumes constant molal overflow, which fails when the components have very different latent heats or large heats of mixing (ammonia–water, ethanol–water at high purity, aqueous acids). The Ponchon–Savarit method uses an enthalpy–concentration diagram, so it handles energy balances exactly and gives condenser and reboiler duties directly. Minimum reflux – whether found from a pinch on either diagram or from the Underwood equations – sets the lower bound on energy use for every column.

Key ideas

Enthalpy–concentration (H–x–y) diagram. For a binary at fixed pressure, plot molar (or mass) enthalpy against composition of the light component. Two curves appear: saturated vapour H_V(y) on top and saturated liquid H_L(x) below. Tie lines join a liquid point and the vapour in equilibrium with it (data from VLE). Mixtures and stream additions obey the lever rule: the mixture of two streams lies on the straight line joining them, divided inversely to their amounts.

Difference points. In the rectifying section the net upward flow of material, V_n+1 − L_n = D, is constant, and so is the net upward flow of enthalpy, which includes the condenser duty. The combination is represented by the point

• Δ_D at (x_D, Q′_D), with Q′_D = H_D + Q_C/D. Every line through Δ_D cuts the vapour curve at V_n+1 and the liquid curve at L_n: these are the operating lines of Ponchon–Savarit. Similarly, in the stripping section the net downward flow is W, represented by
• Δ_W at (x_W, Q′_W), with Q′_W = H_W − Q_B/W (lies far below the liquid curve). An overall balance shows that Δ_D, F and Δ_W lie on one straight line.

Stepping off stages. From the top: V₁ (= x_D for a total condenser) → tie line to L₁ → line from Δ_D through L₁ meets the vapour curve at V₂ → tie line to L₂ → … When a tie line crosses the line Δ_D–F–Δ_W, switch to Δ_W (optimum feed stage). Continue to x_W.

Reflux ratio on the diagram. By the lever rule at the top, R = L₀/D = (Q′_D − H_V1)/(H_V1 − H_L0). A higher reflux ratio raises Δ_D (and lowers Δ_W), making the operating lines steeper and needing fewer stages.

Limiting cases.

• Total reflux: D = 0, Q′_D → ∞; Δ points go to infinity, operating lines become vertical, giving the minimum number of stages.
• Minimum reflux: an operating line coincides with a tie line, creating a pinch. Extend the tie lines in the rectifying section to cut the vertical x = x_D; the highest intersection (often the tie line through the feed) gives Δ_D,min and hence R_min. Any real Δ_D must lie above it.

Relationship to McCabe–Thiele. If the H_V and H_L curves are parallel straight lines (equal molar latent heats, no heat of mixing), Ponchon–Savarit reduces exactly to McCabe–Thiele with straight operating lines.

Underwood equations (constant α). For ideal mixtures with constant relative volatilities, minimum reflux can be computed directly: first find the root θ that lies between the α values of the key components, then evaluate R_min. For a binary with a saturated-liquid feed it gives the same answer as the McCabe–Thiele pinch.

Practical reflux. Operating reflux is usually 1.2–1.5 R_min; below R_min the specification is unattainable with any number of stages.

Formulas

Difference points and duties: Q′_D = H_D + Q_C/D, Q′_W = H_W − Q_B/W Overall energy balance: F·H_F + Q_B = D·H_D + W·H_W + Q_C, equivalently F·H_F = D·Q′_D + W·Q′_W Reflux ratio (lever rule): R = (Q′_D − H_V1)/(H_V1 − H_L0) Total condenser duty: Q_C = V₁ (H_V1 − H_L0) = (R + 1)·D·(H_V1 − H_L0) (saturated reflux)

• H – molar enthalpy, kJ/kmol; Q′ – enthalpy coordinate of a difference point, kJ/kmol; Q_C, Q_B – condenser and reboiler duties, kJ/h; F, D, W – kmol/h.

Minimum reflux from a pinch (x′, y′) on the x–y diagram: R_min = (x_D − y′)/(y′ − x′)

Underwood (constant α, components i): Σ α_i·z_i/(α_i − θ) = 1 − q (solve for θ between the key α values) R_min + 1 = Σ α_i·x_D,i/(α_i − θ)

• α_i relative to the heavy key (or any reference), z_i feed mole fractions, x_D,i distillate mole fractions.

Worked examples

Example 1 – duties and difference points (standard). Given: F = 100 kmol/h, saturated liquid with H_F = 9 000 kJ/kmol; D = W = 50 kmol/h; total condenser with saturated reflux and distillate, H_L0 = H_D = 8 000 kJ/kmol; vapour to condenser H_V1 = 40 000 kJ/kmol; bottoms H_W = 10 000 kJ/kmol; R = 2.

1. Q′_D from the lever rule: Q′_D = H_V1 + R(H_V1 − H_L0) = 40 000 + 2 × 32 000 = 104 000 kJ/kmol.
2. Q_C = D(Q′_D − H_D) = 50 × 96 000 = 4.80 × 10⁶ kJ/h. Check: (R + 1)D(H_V1 − H_L0) = 3 × 50 × 32 000 = 4.80 × 10⁶. ✓
3. Q′_W = (F·H_F − D·Q′_D)/W = (900 000 − 5 200 000)/50 = −86 000 kJ/kmol.
4. Q_B = W(H_W − Q′_W) = 50 × (10 000 + 86 000) = 4.80 × 10⁶ kJ/h.
5. Energy check: Q_B = Q_C + D·H_D + W·H_W − F·H_F = 4.8 × 10⁶ + 400 000 + 500 000 − 900 000 = 4.8 × 10⁶. ✓

Q′_D = 104 000 kJ/kmol, Q′_W = −86 000 kJ/kmol, Q_C = Q_B = 4.80 × 10⁶ kJ/h (1.33 MW).

Example 2 – minimum reflux by Underwood and by pinch (GATE level). Given: binary, saturated-liquid feed (q = 1), z_F = 0.5, x_D = 0.95, α = 2.5 (light) and 1 (heavy).

1. Underwood: 2.5 × 0.5/(2.5 − θ) + 1 × 0.5/(1 − θ) = 1 − q = 0.
2. 1.25(1 − θ) + 0.5(2.5 − θ) = 0 → 2.5 − 1.75θ = 0 → θ = 1.4286 (between 1 and 2.5 ✓).
3. R_min + 1 = 2.5 × 0.95/(2.5 − 1.4286) + 1 × 0.05/(1 − 1.4286) = 2.2167 − 0.1167 = 2.100.
4. R_min = 1.10.
5. Pinch check: y′ at x′ = 0.5 is 2.5 × 0.5/1.75 = 0.7143; R_min = (0.95 − 0.7143)/(0.7143 − 0.5) = 1.10. ✓

R_min = 1.10 by both methods.

Common mistakes

• Placing Δ_W above the liquid curve – it lies below because the reboiler adds heat (Q′_W = H_W − Q_B/W).
• Forgetting that Δ_D, F and Δ_W must be collinear.
• Taking the minimum-reflux tie line as the one through the feed when another rectifying tie line extends higher.
• Choosing an Underwood root θ outside the range between the key volatilities.
• Assuming Ponchon–Savarit and McCabe–Thiele give different answers when latent heats are equal – they agree.
• Mixing mass-basis enthalpy diagrams with molar flows.

For GATE CH

Expect conceptual questions on difference points, the lever rule, collinearity of Δ_D–F–Δ_W, total and minimum reflux on the H–x–y diagram, and when Ponchon–Savarit is needed. Numericals typically ask for Q′_D, R from enthalpies, condenser or reboiler duty, or R_min from a pinch or the Underwood equation for a binary. Keep the energy balance as a check.

Quick check

1. Where are the difference points at total reflux?
2. Write the reflux ratio in terms of Q′_D, H_V1 and H_L0.
3. Under what condition does Ponchon–Savarit reduce to McCabe–Thiele?
4. Which three points are always collinear on the H–x–y diagram?

Answers: 1. at infinity (operating lines vertical); 2. R = (Q′_D − H_V1)/(H_V1 − H_L0); 3. parallel straight saturated-liquid and vapour enthalpy lines (constant molal overflow); 4. Δ_D, F and Δ_W.