A $6$ mol$/$s saturated liquid mixture, containing $40$ mol$\%$ propane and $60$ mol$\%$ butane, is fed to a continuous tray fractional distillation column equipped with a partial condenser and a boiler, in order to obtain mixtures of saturated liquids. One mixture should be $92\%$ propane and the other $92\%$ butane.

a$)$ Determine the flows in the distillate and the tails

b$)$ Determine the minimum number of stages with total reflux

c$)$ Determine the minimum reflux ratio

d$)$ If the optimal flow is $1\mathrm{.}5$Rm$$n$,$How many stages will there be in the system and what will be the liquid and vapor flows in the rectification and depletion zone?