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Rayleigh Equation (Simple Batch Distillation)

Also known as rayleigh equation · differential distillation · simple batch distillation · pot still equation · batch still material balance · rayleigh distillation

ln⁡B0B1=1α−1ln⁡ ⁣[x0x1(1−x11−x0) ⁣α]\ln\frac{B_0}{B_1} = \frac{1}{\alpha - 1}\ln\!\left[\frac{x_0}{x_1}\left(\frac{1 - x_1}{1 - x_0}\right)^{\!\alpha}\right]

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See also
Relative Volatility (Binary) — Supplies the α this integral assumes constant, and the equilibrium y that drives the depletion.Fenske Equation (Minimum Stages) — The opposite extreme: a column at total reflux, where staging does the work instead of time.Column Material Balance (Distillate and Bottoms Split) — The continuous version of the same bookkeeping — a steady split rather than a drifting one.

Learning zone

This is the oldest equation in the shard and the simplest apparatus in the trade: a pot, a heat source, and a condenser. No column, no packing, no reflux. Vapour is drawn off as fast as it forms, and because the vapour is always richer in the light component than the liquid it left, what stays behind gets steadily poorer. Lord Rayleigh set the bookkeeping down in 1902 as a differential material balance, and it has not needed revising.

The derivation is a small, satisfying piece of calculus. Over an instant, a little vapour dBdB leaves at the equilibrium composition yy, and the light-component balance on the pot gives dB/B=dx/(y−x)dB/B = dx/(y - x). Integrating both sides gives ln⁡(B0/B1)=∫dx/(y−x)\ln(B_0/B_1) = \int dx/(y-x), which is the general Rayleigh equation and is valid for any mixture at all. The closed form on this page appears only when you substitute the constant-volatility equilibrium relation; for a real mixture the integral is evaluated numerically against measured equilibrium data, and that general version is what a batch chemist actually uses.

Two things about the distillate are commonly misread. It is not the composition that would be in equilibrium with anything in particular: it is a running average of everything collected, rich at the start of the batch and progressively poorer, and it is found from the overall balance B0x0=B1x1+DxˉDB_0x_0 = B_1x_1 + D\bar{x}_D rather than from this equation. And the separation obtained is genuinely poor — a single equilibrium stage, once, spread over time. That is why the practice of cutting a batch into fractions exists: the early cut is far better than the average, the late cut far worse, and keeping them apart recovers much of what a single collected pot loses.

The assumption that deserves scrutiny is no reflux. A pot still with a packed neck, a long column, or a cooled head is not a simple still at all — it is a small rectifier, and it will do considerably better than this predicts. That is precisely why those features are fitted. Treat the Rayleigh result as the worst case: the separation available with no rectification whatever, and therefore the floor that any real still with any column on it should beat.

Rayleigh Equation (Simple Batch Distillation)
ln⁡B0B1=1α−1ln⁡ ⁣[x0x1(1−x11−x0) ⁣α]\ln\frac{B_0}{B_1} = \frac{1}{\alpha - 1}\ln\!\left[\frac{x_0}{x_1}\left(\frac{1 - x_1}{1 - x_0}\right)^{\!\alpha}\right]
B0B1αx1
Where
  • B1B_1= Charge remaining (kmol)
  • B0B_0= Initial charge (kmol)
  • x0x_0= Initial still composition
  • x1x_1= Final still composition
  • α\alpha= Relative volatility
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