Distillation: VLE, relative volatility and flash distillation

Vapour–liquid equilibrium from Raoult's and Dalton's laws, K-values, bubble and dew points, relative volatility and the constant-α curve, and binary and multicomponent flash calculations.

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Why it matters

Distillation is the workhorse separation of the chemical and petroleum industries, and every distillation calculation starts from vapour–liquid equilibrium (VLE). Relative volatility tells you at a glance whether a separation is easy, hard or impossible by distillation, and the flash calculation – one equilibrium stage – is the building block of every column simulation.

Key ideas

Vapour–liquid equilibrium. When a liquid mixture and its vapour are in contact long enough, temperature, pressure and the chemical potential of each component become equal in both phases. Compositions no longer change; the vapour is richer in the more volatile component. For a binary at fixed pressure, specifying one composition fixes the temperature and the other composition (phase rule: F = 2 − π + C = 2).

Ideal solutions. For chemically similar components (benzene–toluene, hexane–heptane) at low pressure:

  • Raoult's law: partial pressure of i = x_i·P_i^sat.
  • Dalton's law: partial pressure = y_i·P.
  • Hence y_i = x_i·P_i^sat/P and the K-value K_i = y_i/x_i = P_i^sat/P. Non-ideal liquids need an activity coefficient γ_i (from Margules, van Laar, Wilson, NRTL – data book), and strong deviations produce azeotropes.

Phase diagrams.

  • T–x–y diagram at constant P: the bubble-point curve (saturated liquid, lower) and the dew-point curve (saturated vapour, upper); between them two phases coexist.
  • x–y diagram: the equilibrium curve lies above the 45° line for the light component; the further above, the easier the separation. An azeotrope is where the curve crosses the diagonal.

Bubble and dew points. Bubble point: the temperature at which the first vapour bubble forms, found from Σ K_i·x_i = 1. Dew point: the first liquid drop forms, Σ y_i/K_i = 1.

Relative volatility. α_AB = K_A/K_B = (y_A/x_A)/(y_B/x_B), conventionally light over heavy so α ≥ 1. For ideal systems α = P_A^sat/P_B^sat, which varies only slowly with temperature, so a constant (average, often geometric-mean of top and bottom) α is a good approximation. α generally falls as temperature and pressure rise, so lower-pressure distillation often makes separation easier. α = 1 means no separation by distillation at that composition.

Flash (equilibrium) distillation. A feed is partly vaporised (by heating, or by reducing pressure across a valve) and the vapour and liquid, in equilibrium with each other, are separated in a drum. It is a single equilibrium stage – a cheap, rough split suitable for widely different volatilities (α large) or as a pre-separation. The vaporised fraction f = V/F is set by the heat input (or by the feed enthalpy for an adiabatic flash) and the drum pressure.

Formulas

Ideal VLE: y_i·P = x_i·P_i^sat (Raoult + Dalton); K_i = y_i/x_i = P_i^sat/P Binary bubble-point liquid composition at given T and P: x_A = (P − P_B^sat)/(P_A^sat − P_B^sat), y_A = x_A·P_A^sat/P

  • P, P^sat in kPa (any consistent unit); x, y – mole fractions.

Relative volatility: α_AB = (y_A/x_A)/(y_B/x_B) = P_A^sat/P_B^sat (ideal) Constant-α equilibrium curve: y = α·x / (1 + (α − 1)·x) or x = y / (α − (α − 1)·y)

Flash balances (F feed, V vapour, L liquid, f = V/F): F = V + L, F·z = V·y + L·x Flash operating line: y = −((1 − f)/f)·x + z/f (slope −L/V, passes through (z, z)) Multicomponent (Rachford–Rice): Σ z_i (K_i − 1)/(1 + f(K_i − 1)) = 0

Worked examples

Example 1 – VLE of an ideal binary (standard). Given: benzene (A) – toluene (B) at 95 °C and 101.325 kPa; P_A^sat = 155.7 kPa, P_B^sat = 63.3 kPa (from Antoine data). Find x_A, y_A and α.

  1. x_A = (P − P_B^sat)/(P_A^sat − P_B^sat) = (101.325 − 63.3)/(155.7 − 63.3) = 38.03/92.4 = 0.4115.
  2. y_A = x_A·P_A^sat/P = 0.4115 × 155.7/101.325 = 0.6324.
  3. α = 155.7/63.3 = 2.46. Check: y = 2.46 × 0.4115/(1 + 1.46 × 0.4115) = 0.632. ✓

x_A = 0.412, y_A = 0.632, α = 2.46.

Example 2 – flash with constant relative volatility (GATE level). Given: an equimolar binary feed (z = 0.5) is flashed so that 40 mol % vaporises (f = 0.4); α = 2.5. Find x, y and the fraction of the light component recovered in the vapour.

  1. Flash line: y = −(0.6/0.4)x + 0.5/0.4 = −1.5x + 1.25.
  2. Equilibrium: y = 2.5x/(1 + 1.5x).
  3. Equate: 2.5x = (−1.5x + 1.25)(1 + 1.5x) = 1.25 + 0.375x − 2.25x² → 2.25x² + 2.125x − 1.25 = 0.
  4. x = [−2.125 + √(2.125² + 4 × 2.25 × 1.25)]/(2 × 2.25) = (−2.125 + 3.9706)/4.5 = 0.4101.
  5. y = −1.5 × 0.4101 + 1.25 = 0.6348 (check: 2.5 × 0.4101/1.6152 = 0.6348 ✓).
  6. Recovery of light component in vapour = f·y/z = 0.4 × 0.6348/0.5 = 0.508.

x = 0.410, y = 0.635; 50.8 % of the light component leaves in the vapour. One stage gives only a modest enrichment, which is why columns use many stages.

Common mistakes

  • Using the feed composition z as if it were the liquid composition x in a flash.
  • Taking α = P_A^sat/P_B^sat for a non-ideal mixture (it needs activity coefficients).
  • Inverting α (heavy over light) and getting a curve below the diagonal.
  • Forgetting that the flash line has a negative slope −L/V and passes through (z, z).
  • Mixing up bubble point (Σ K·x = 1) and dew point (Σ y/K = 1).
  • Assuming α is independent of pressure.

For GATE CH

Expect bubble/dew point and K-value problems with given vapour pressures, y from x with constant α, α from equilibrium data, and binary flash calculations (find x, y or f). Multicomponent flash with given K-values (Rachford–Rice) and identification of azeotropes on T–x–y or x–y diagrams also appear. Practise solving the flash quadratic quickly.

Quick check

  1. Define relative volatility for a binary.
  2. For α = 3 and x = 0.5, find y.
  3. What are the slope and the fixed point of the flash operating line?
  4. What does α = 1 mean?

Answers: 1. α = (y_A/x_A)/(y_B/x_B); 2. y = 1.5/2 = 0.75; 3. slope −(1 − f)/f = −L/V, passing through (z, z); 4. no separation by distillation – vapour and liquid have the same composition (an azeotrope).

See it move

All Chemical animations
Distillation: VLE and Relative Volatility

Adjust the mole fraction of the more volatile component in the liquid phase and observe how the vapor phase composition changes. Notice how relative volatility affects separation.

Equations used
  • α = (y₁/x₁) / (y₂/x₂) — α: Relative volatility, y₁, y₂: Mole fractions in vapor, x₁, x₂: Mole fractions in liquid
  • y = αx / (1 + (α - 1)x) — y: Mole fraction in vapor, x: Mole fraction in liquid, α: Relative volatility

Interview questions

All Mass Transfer interview questions

Try answering each one aloud before you open it.

  1. 1.What is vapor-liquid equilibrium (VLE) in the context of distillation?Concept

    Vapor-liquid equilibrium (VLE) refers to the state where the rate of evaporation equals the rate of condensation in a distillation process. At this point, the composition of the vapor phase and the liquid phase remains constant. VLE is crucial for designing and analyzing distillation processes because it helps determine the concentration of components in each phase at a given temperature and pressure.

  2. 2.Explain the concept of relative volatility in distillation.Concept

    Relative volatility of A to B is the ratio of their K-values, α_AB = (y_A/x_A)/(y_B/x_B), conventionally light over heavy so that α ≥ 1. For an ideal mixture obeying Raoult's law it equals the ratio of pure-component vapour pressures, P_A^sat/P_B^sat; for non-ideal mixtures activity coefficients enter as well. It is dimensionless and changes only slowly with temperature, so a constant α gives the simple curve y = αx/(1 + (α − 1)x). The larger α, the fewer stages and the lower the reflux needed; α = 1 means distillation cannot separate the mixture.

  3. 3.What is flash distillation and how does it differ from other distillation methods?Concept

    Flash (equilibrium) distillation partly vaporises a feed, by heating it or by letting down its pressure across a valve, and separates the vapour and liquid in a drum after they have reached equilibrium. It is a single equilibrium stage, so x and y are linked by the equilibrium curve and by the material balance line of slope −L/V through (z, z). It is continuous, simple and cheap but gives only a modest enrichment, so it suits mixtures with large relative volatility or serves as a pre-separation. Multistage columns with reflux are needed for sharp splits.

  4. 4.Why is relative volatility important in the design of a distillation column?Application

    Relative volatility is important because it determines the number of theoretical stages required for a given separation. A higher relative volatility means fewer stages are needed, which can reduce the size and cost of the distillation column. It also affects the energy consumption of the process, as higher volatility typically requires less energy for separation.

  5. 5.What happens if the relative volatility of a mixture is close to 1?Application

    If the relative volatility of a mixture is close to 1, it means that the components have similar volatilities and are difficult to separate by distillation. In such cases, a large number of stages or alternative separation methods, such as azeotropic distillation or extractive distillation, may be required to achieve the desired separation.

  6. 6.How does temperature (or column pressure) affect vapour-liquid equilibrium in distillation?Application

    At a fixed pressure the equilibrium temperature is fixed by composition, so temperature is not an independent variable for a binary; what the designer chooses is the column pressure, which sets the temperatures. Raising the pressure raises boiling points and usually lowers the relative volatility, because the vapour pressures of the components approach each other at higher temperature. Lower-pressure (even vacuum) operation therefore often eases the separation and protects heat-sensitive materials, at the cost of larger vapour volumes and colder, more expensive condensing duty.

  7. 7.In a flash distillation process, what factors determine the fraction of the feed that is vaporized?Application

    The fraction of the feed that is vaporized in a flash distillation process is determined by the feed composition, the feed temperature, the pressure of the flash drum, and the enthalpy balance. The operating conditions must be carefully controlled to achieve the desired vaporization and separation.

  8. 8.Calculate the relative volatility of an ideal binary mixture where the vapour pressure of component A is 80 kPa and of component B is 40 kPa at the system temperature.Numerical

    For an ideal mixture obeying Raoult's law, α_AB = K_A/K_B = P_A^sat/P_B^sat = 80/40 = 2. A is the more volatile component, so at any liquid composition the vapour is enriched in A, e.g. at x_A = 0.5 the equilibrium vapour has y_A = 2 × 0.5/(1 + 0.5) = 0.667. For a non-ideal mixture activity coefficients would also be needed.

  9. 9.The liquid leaving a flash drum contains 60 mol % of component A. If the relative volatility of A to B is 2, what is the composition of the vapour leaving the drum?Numerical

    The vapour and liquid leaving a flash drum are in equilibrium, so y_A/(1 − y_A) = α·x_A/(1 − x_A) = 2 × 0.6/0.4 = 3. Hence y_A = 3/4 = 0.75, i.e. 75 mol % A. Note that the liquid composition, not the feed composition, must be used here; the feed composition lies between x and y on the flash operating line.

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