The Rayleigh model applies a single-stage batch distillation as an un-refluxed, equilibrium vaporization process. A liquid mixture is charged to a reboiler, heated, and continuously removed as vapor that is assumed to be in vapor-liquid equilibrium with the liquid remaining in the still. The model tracks how the liquid inventory and composition change as the more volatile component is preferentially vaporized.

Core takeaway: The Rayleigh equation connects the decrease in liquid holdup to the changing liquid and vapor compositions. For a binary mixture, the equilibrium relationship between liquid composition (x_B) and vapor composition (y), together with the Rayleigh balance, predicts residue composition, distillate accumulation, and batch separation performance.

How the Rayleigh Model Represents the Pilot Plant

Single-stage batch operation

The pilot plant is modeled as a Rayleigh still: a reboiler containing a well-mixed liquid charge, a heat source, and a condenser that collects vaporized material without returning liquid reflux.

There is no rectifying section and no internal liquid-vapor contacting sequence. The remaining reboiler liquid is therefore treated as the single liquid phase controlling the instantaneous vapor composition.

Differential vapor removal

As heating proceeds, vapor is continuously generated and removed. The liquid holdup decreases from its initial value (B_0) to a final value (B_1).

The more volatile component is removed preferentially, so its mole fraction in the remaining liquid, (x_B), generally decreases during the batch. The instantaneous vapor composition (y) is higher in the volatile component than the liquid composition when separation is possible.

Vapor-liquid equilibrium assumption

At every instant, the generated vapor is assumed to be in equilibrium with the liquid in the still:

[ y = y^*(x_B) ]

This is the defining equilibrium assumption of the ideal Rayleigh model. The equilibrium relationship may come from experimental VLE data, an activity-coefficient model, or a constant-relative-volatility approximation.

Core Governing Equations

Overall liquid balance

Let (B) be the total number of moles of liquid remaining in the still. The accumulated distillate is

[ H_a = B_0 - B_1 ]

where:

(B_0) is the initial liquid amount,
(B_1) is the remaining liquid amount,
(H_a) is the total condensed distillate.

This balance assumes that vapor holdup and material losses are negligible.

Differential Rayleigh equation

For a binary mixture, let (x_B) be the liquid mole fraction of the selected volatile component and (y) its equilibrium vapor mole fraction. A differential component balance gives

[ B,dx_B = (y-x_B),dB ]

or, equivalently,

[ \frac{dB}{B} = \frac{dx_B}{y-x_B} ]

Because both (B) and (x_B) decrease during normal operation, the differential changes are negative. The equation therefore remains consistent with a positive separation factor (y-x_B).

Integrated Rayleigh equation

Integrating between the initial state ((B_0,x_{B0})) and final state ((B_1,x_{B1})) gives

[ \ln\left(\frac{B_0}{B_1}\right)

\int_{x_{B1}}^{x_{B0}} \frac{dx_B}{y-x_B} ]

This is the principal governing equation used to calculate the final liquid composition or the remaining liquid quantity.

An equivalent form is

[ \ln\left(\frac{B_1}{B_0}\right)

\int_{x_{B0}}^{x_{B1}} \frac{dx_B}{y-x_B} ]

The two forms are identical; only the integration limits and logarithm direction differ.

Constant Relative Volatility Model

Equilibrium relationship

For a binary mixture with constant relative volatility (\alpha), the vapor composition is approximated by

[ y = \frac{\alpha x_B} {1+(\alpha-1)x_B} ]

where

[ \alpha = \frac{K_{\text{volatile}}}{K_{\text{less volatile}}} ]

For (\alpha>1), the selected component is more volatile and (y>x_B) over the composition range (0<x_B<1).

Substitution into the Rayleigh equation

Using the constant-(\alpha) equilibrium relation,

[ y-x_B = \frac{(\alpha-1)x_B(1-x_B)} {1+(\alpha-1)x_B} ]

Therefore,

[ \frac{1}{y-x_B}

\frac{1+(\alpha-1)x_B} {(\alpha-1)x_B(1-x_B)} ]

The integrated equation becomes

[ \ln\left(\frac{B_0}{B_1}\right)

\frac{1}{\alpha-1} \left[ \ln\left(\frac{x_{B0}}{x_{B1}}\right)

\alpha \ln\left( \frac{1-x_{B0}}{1-x_{B1}} \right) \right] ]

This form is useful for binary laboratory mixtures when relative volatility can reasonably be treated as constant.

Component-mole form

Define the moles of the volatile and less volatile components remaining in the still as

[ N_{\mathrm{v}}=Bx_B ]

and

[ N_{\mathrm{l}}=B(1-x_B) ]

The constant-relative-volatility result can also be written as

[ \ln\left(\frac{N_{\mathrm{v},1}}{N_{\mathrm{v},0}}\right)

\alpha \ln\left(\frac{N_{\mathrm{l},1}}{N_{\mathrm{l},0}}\right) ]

This expression shows that the volatile component inventory decreases more rapidly than the less volatile component inventory when (\alpha>1).

Adding Time and Heat Input

Vaporization rate

The Rayleigh equation describes the composition and holdup trajectory, but it does not by itself specify how quickly the batch proceeds. Time can be introduced through the vaporization rate.

If the heat input (Q) and molar latent heat of vaporization (\Delta H_{\mathrm{vap}}) are approximately constant, the molar vaporization rate is

[ \dot n_V = \frac{Q}{\Delta H_{\mathrm{vap}}} ]

Since liquid is removed as vapor,

[ \frac{dB}{dt}=-\dot n_V ]

For constant (Q) and (\Delta H_{\mathrm{vap}}),

[ B(t)=B_0-\dot n_V t ]

until the assumptions cease to apply or the batch is stopped.

Composition dynamics

Combining the material balance with the Rayleigh relation gives

[ \frac{dx_B}{dt}

\frac{\dot n_V}{B} (x_B-y) ]

Equivalently,

[ \frac{dx_B}{dt}

-\frac{\dot n_V}{B} (y-x_B) ]

Because (y>x_B) for the more volatile component, (dx_B/dt) is negative. The liquid becomes progressively depleted in that component.

Effect of heat input

Increasing the heat input generally increases (\dot n_V), reducing the time required to reach a specified liquid holdup. Under ideal assumptions, it changes the time scale of the batch but not the equilibrium relationship between (B) and (x_B).

In practice, excessive heat input can invalidate this separation between time and composition by causing entrainment, foaming, non-equilibrium vaporization, heat losses, or condenser limitations.

Applying the Model to Pilot-Plant Data

Required measurements

A typical experiment requires:

Initial liquid amount (B_0) or initial volume and composition.
Initial volatile-component mole fraction (x_{B0}).
Remaining liquid amount (B_1) at a selected time.
Residue composition (x_{B1}).
Distillate accumulation (H_a).
Vapor or distillate composition, when available.
Heating rate, heat input, or vaporization rate.

Volume measurements must be converted to moles using the mixture density, molecular weights, and composition. Direct mole measurements are preferable when available.

Predicting the residue

If (B_0), (x_{B0}), and (B_1) are known, the Rayleigh equation can be solved for (x_{B1}). With a constant-(\alpha) model, this is usually a nonlinear calculation.

If the residue composition is measured instead, the equation can be used to predict the remaining liquid amount:

[ B_1 = B_0 \exp\left[ -\int_{x_{B1}}^{x_{B0}} \frac{dx_B}{y-x_B} \right] ]

The predicted value can then be compared with the experimental liquid inventory.

Evaluating separation performance

The instantaneous vapor composition is calculated from the liquid composition:

[ y_1 = \frac{\alpha x_{B1}} {1+(\alpha-1)x_{B1}} ]

The average composition of the accumulated distillate follows from an overall component balance:

[ \bar y_D = \frac{B_0x_{B0}-B_1x_{B1}} {B_0-B_1} ]

This average distillate composition is not generally equal to the final instantaneous vapor composition. The vapor composition changes throughout the batch, so the accumulated product represents an integrated history of the operation.

Understanding the Trade-offs

Constant relative volatility is an approximation

The assumption that (\alpha) is constant may be unsuitable for strongly nonideal mixtures or broad composition ranges. Relative volatility can vary with temperature, pressure, and composition.

For higher-fidelity work, replace the constant-(\alpha) equation with measured or thermodynamically calculated VLE data and evaluate the Rayleigh integral numerically.

Equilibrium does not guarantee ideal pilot performance

The ideal model assumes that the vapor leaving the liquid is in equilibrium with it. Real equipment may exhibit superheating, entrainment, incomplete mixing, vapor holdup, heat losses, and delayed sampling responses.

These effects can make the measured vapor composition differ from the equilibrium value predicted from the instantaneous residue composition.

Heating rate affects duration, not always outcome

Under the ideal model, heat input changes the rate of holdup reduction but not the equilibrium trajectory. Real systems can depart from this behavior when higher heat rates cause foaming, entrainment, flooding of the condenser, or insufficient heat-transfer and mixing performance.

A single stage is not a distillation column

The Rayleigh model represents one equilibrium vaporization step repeated over time. It does not account for theoretical trays, reflux, column holdup, tray efficiency, minimum reflux, or plate-to-plate composition profiles.

Methods such as Fenske-Underwood-Gilliland analysis belong to staged-column design and should not be substituted for the single-stage Rayleigh balance unless the pilot plant includes a genuine contacting section.

How to Apply This to Your Project

The following workflow provides a practical model-validation sequence:

If your primary focus is residue composition: Use the integrated Rayleigh equation with the measured or assumed VLE relationship to solve for (x_{B1}) at the measured remaining liquid amount.
If your primary focus is batch duration: Estimate (\dot n_V=Q/\Delta H_{\mathrm{vap}}), integrate the holdup balance in time, and couple it to the Rayleigh composition relation.
If your primary focus is distillate recovery: Calculate (H_a=B_0-B_1) and determine accumulated distillate composition from the overall volatile-component balance.
If your primary focus is model validation: Compare predicted residue and distillate compositions against time-stamped samples, then investigate deviations through changing (\alpha), heat loss, entrainment, and non-equilibrium effects.
If your primary focus is process design: Use the Rayleigh model as a baseline for an un-refluxed batch, and use a staged dynamic model when the pilot plant includes trays, packing, reflux, or significant internal holdup.

The Rayleigh model provides the essential theoretical benchmark for understanding how a single-stage batch distillation separates a binary mixture as the reboiler liquid is progressively depleted.

Summary Table:

Topic	Key equation or principle	Application
Liquid balance	Hₐ = B₀ − B₁	Calculates accumulated distillate
Rayleigh equation	dB/B = dxᴮ/(y − xᴮ)	Relates liquid holdup to composition change
Integrated model	ln(B₀/B₁) = ∫ dxᴮ/(y − xᴮ)	Predicts residue composition or remaining liquid
VLE relationship	y = y*(xᴮ)	Determines instantaneous vapor composition
Constant relative volatility	y = αxᴮ/[1 + (α − 1)xᴮ]	Simplifies binary-mixture calculations
Vaporization rate	ṅᵥ = Q/ΔHᵥₐₚ	Connects heat input to batch duration
Average distillate composition	ȳᴰ = (B₀xᴮ₀ − B₁xᴮ₁)/(B₀ − B₁)	Evaluates overall separation performance
Model limitations	Assumes equilibrium, ideal mixing, and negligible losses	Identifies causes of pilot-plant deviations

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