Reference ID: MET-92C4 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The Rayleigh equation is a fundamental mathematical model in process engineering used to describe differential distillation. In a simple batch distillation process, a liquid mixture is charged into a still and heated. As vapor is generated, it is continuously removed from the system and condensed. Because the vapor is typically richer in the more volatile component (MVC) than the remaining liquid, the composition of the liquid in the pot changes over time.
This calculation is critical for determining the amount of residue remaining in the still after a desired degree of separation has been achieved. It is widely used in chemical and pharmaceutical industries for batch processing, solvent recovery, and small-scale purification where steady-state continuous distillation is not feasible.
Methodology & Formulas
The Rayleigh equation is derived from a material balance on the more volatile component and the assumption of instantaneous equilibrium between the liquid and the vapor phase. The general form of the equation is expressed as:
When the relative volatility (\(\alpha\)) can be assumed constant over the composition range, the integral can be solved analytically. The resulting formula used for the calculation is:
Well-mixed liquid, no vapor holdup, constant pressure
To calculate the change in composition over time during a simple batch distillation, follow these steps:
Determine the initial moles of the charge and the initial liquid composition.
Establish the equilibrium relationship between the vapor and liquid phases, typically using the relative volatility constant.
Integrate the Rayleigh Equation, which relates the change in the number of moles to the change in liquid composition.
Solve for the final amount of residue or the final composition based on your specific separation requirements.
The Rayleigh Equation assumes an ideal batch distillation scenario. Key assumptions include:
The liquid in the still is perfectly mixed at all times.
The vapor leaving the still is in thermodynamic equilibrium with the liquid remaining in the still.
There is no liquid holdup in the condenser or the vapor space.
The process operates under adiabatic conditions.
Assuming a constant relative volatility simplifies the integration of the Rayleigh Equation significantly. When \(\alpha\) is treated as a constant, the differential equation can be solved analytically to produce a closed-form expression. If the relative volatility varies significantly with temperature or composition, process engineers must use numerical integration methods, such as Simpson's rule, to achieve an accurate result.
Worked Example: Rayleigh Equation for Batch Distillation
Scenario: A batch still is charged with 1000 L of an ethanol–water mixture at 1 atm. The initial mole fraction of ethanol (the more volatile component) is 0.40. The mixture is distilled until the residue reaches a mole fraction of 0.10 ethanol. For the composition range, a constant relative volatility of 2.5 is assumed. Determine the final liquid volume in the still.
