A complete analytical solution is developed for the number of equilibrium stages in separations processes for which operating and equilibrium lines are not necessarily straight. The resulting equations are solutions of a Riccati difference equation and are applied to liquid-liquid extraction and the distillation of a binary mixture where the liquid and vapor flows are not constant because of the variation of the saturated liquid and vapor enthalpies with concentration. A specific example is given for the separation of ethyl alcohol and water in which the equilibrium curve is highly nonlinear and where a large number of plates are required in a pinch region. Another example is given for the extraction of methylcyclopentane from its solution in n-hexane using aniline as a selective solvent
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