Differential Distillation The Rayleigh Equation

We start the derivation of the model by composing the unsteady integral mass balances for the system  

To eliminate the time variable, we resort to a favorite trick of ours, one that is frequently used and should be part of the analyst s tool box The two mass balances are divided, thereby eliminating not only time as a variable but also the unknown distillation rate D. Thus, 

Formal integration of this expression then leads to 

This expression is known as the Rayleigh equation and relates the amount W left in the still at any instant to its composition x.  

We start by noting, as we have done on other occasions, that any of the variables appearing in Equation 7.5g may be regarded as an xmknown. Thus, for a given boiler content W, the equation will yield the corresponding composition Xy, and conversely we can calculate die fraction to be distilled 1 - W/W°, by prescribing a desired enrichment Xy - Xy/)/Xyf. What is often overlooked is that the equilibrium relation/(x, ) can also be established from experimental vs. W data. A particularly simple case arises when the/(Xw) 

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