The Rayleigh Equation Revisited What to Do When Alpha Isn't Constant

Abstract

Integration of the Rayleigh Equation for batch distillation in closed analytical form has heretofore required that relative volatility (α) be assumed constant. A new technique presented in this paper produces a continuous analytical function for the Rayleigh Equation integral whether α is constant or not. The newly developed equation reduces algebraically to the traditional function when α is absolutely constant.

Both the traditional expression at constant α and the new equation are evaluated against numerical integration for a number of ideal and non-ideal binary vapor-liquid equilibrium (VLE) systems. The new method has proven especially useful for the non-ideal systems investigated, in which α varies widely with composition.

The method is then utilized successfully in a computer-simulated batch distillation of an ethanol-water solution at 1 atm pressure, a popular student laboratory exercise and one of the original systems studied by Lord Rayleigh. Results compare well with expected values.

A summary of VLE relationships for ideal and non-ideal systems plus temperature-dependent methods for analysis of constituent composition specific to the ethanol-water system are discussed in separate appendices. These latter include, among others, refractive index measurements assembled from various sources and estimates of liquid density for ethanol-water mixtures extended beyond the range of published data.

The paper consists of new material supported by background information tutorial in nature.

Keywords. Alcohol “Proof” Levels; Boiling Point Curve; Density of Ethanol-Water Solutions; Ethanol-Water Flash Points; Ethanol-Water Refractive Index; Gas Chromatography (GC); Numerical Integration; Rayleigh Equation; Simpson's Rule; Systems: Acetone-Water, Benzene-Toluene, Ethanol-Water, Ethylene Dichloride (EDC)-Toluene; Vapor-Liquid Equilibrium.

Summary and Conclusions

• Batch, or differential, distillation without reflux is described by the Rayleigh Equation:

+ Relates composition and amount of material remaining in distilling flask

+ Other quantities determined by material balance

+ Illustrative example from the literature presented.

• Numerical integration required for the Rayleigh Equation in its basic form.

• Substitution of relative volatility (α) allows analytical integration for constant α.

• New equation derived here allows analytical integration whether α is constant or not.

• Analytical integration utilizing α has been evaluated here against numerical integration:

+ For constant or nearly constant α

+ For a number of real vapor-liquid systems, where α varies with composition.

• For ideal systems, with α not varying widely, use of an “average” α is adequate.

• For non-ideal systems, only the new equation follows the track of numerical integration.

• The new function was used in simulation of batch distillation of pure ethanol and water.

• Ethanol-water results compare favorably with expectations, for example:

+ Boiling point vs. composition curve

+ Still pot temperature vs. cumulative volume of distillate collected

+ Density of solution remaining in still pot.

• Methods to analyze liquid composition are discussed for the ethanol-water system:

+ Gas chromatography (GC)

+ Refractive index data compiled from various sources

+ Densities of ethanol-water solutions extended beyond range of published data

+ Qualitative ignition test / flash points of ethanol-water solutions.