Differential Separation of a Binary Mixture

Let the solution consist of two components, one of transits into the gaseous phase when the pressure above the surface of the mixture diminishes. The pressure is kept at constant value. Consider a thin layer of mixture with volume Vand height H. Assume that all parameters depend on time and on a single coordinate x whose direction coincides with the direction of gravity. The value x = 0 corresponds to the upper surface of the layer. The assumptions just made allow us to write Eq. (23.8) as 

By implication, dmi/dt is the rate of mass change for component 1 in the bubble, therefore 

Expressions for the diffusion flux for various cases were presented in the previous chapter. To solve Eq. (23.9), we must know the mass distribution of bubbles. Let N (xo,t) be the number of bubbles (germs) per unit volume per unit time formed at the depth Xq at the moment t. It is obvious that the number of germs decreases in time and with the reduction of depth xo as the mixture is being depleted. The flux of bubbles through a fixed unit cross section is conserved, therefore 

Appearing in the left-hand side is the flux through the cross section of a vertical column at the depth x at the moment t, and in the right-hand side -the same quantity at the depth Xq x. Suppose the bubble growth formula mi = mi (x, Xo, Mn), where m is the mass of a germ, is known to us. This formula tells us the mass that the bubble nucleated at the depth xo x acquires when it rises to the depth x. Eor a fixed x, we can write dmi = (dmi/dxo) dxo, in which case the expression (23.11) gives us 

Let the sizes and the masses of incipient bubbles be identical, constant in time and over the height of the layer. Then 

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