Application to a Simplified Theory of Distillation

In his analysis of the open tube distillation column Westhaver (1942) goes into a detailed consideration of radial concentration gradients which is very similar to Taylor s approach. His final formula, however, is the same as if he had assumed a constant velocity profile and an effective diffusion coefficient (Dt + llU2r2l48Dt). This is just the diffusion coefficient that we have found for viscous flow in the presence of a film on the tube wall in which the solute concentration is infinitely greater than in the fluid. This is clearly the case for 

It is therefore suggested that provided the diffusion coefficient is rightly chosen the following simple theory of distillation will prove quite servicable. In the simplified theory we assume that the velocity of the vapour is everywhere constant and equal to v, that diffusion takes place in the direction of the vapour flow only with diffusion coefficient D and that concentrations are constant on planes perpendicular to the flow. If X is the mole fraction in the vapour phase of the more volatile component of a binary mixture and Y its mole fraction in the condensate, then at total reflux a mass balance over a section of a column gives 

But for a relative volatility a close to 1 (this is of interest since it applies to difficult separations) the equilibrium relationship between X and Y may be written 

I am indebted to M. J. E. Golay for allowing me to see his contribution to the Amsterdam Gas Chromatography Symposium whilst it was still in typescript, and to J. W. Westhaver for correspondence on his work. Fruitful discussions of the distillation problem were had with N. Macleod of the Chemical Engineering Department, Edinburgh University. The formulae in 3 were checked by Mr. Kung-You Lee, who also did the calculation for the figure. 

1958 Contribution to the Gas Chromatography Symposium, Amsterdam. Lighthill, M. J. 1958 Fourier analysis and generalized functions. Cambridge University Press. Taylor, Sir Geoffrey 1953 Proc. Roy. Soc. A, 219,186. 

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