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

AIChE Spring Meeting and Global Congress on Process Safety
April 4, 2012 - 12:00am


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.

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