We want to design a continuous fractionation distillation column to separate a methanol$-$water mixture. $19,850$ kg$/$h are fed, containing $57\%$ methanol and $43\%$ water $(\%$ by weight$),$ in the distillate product $350$ kmol$/$h is obtained and in the residues they contain $95\%$ water $(\%$ by weight$).$ A reflux ratio of $3$ moles per mole of product will be used. The molar latent heats of boiling for methanol and water are $35\mathrm{.}7$ kJ$/$mol and $40\mathrm{.}656$ kJ$/$mol$.$ Assuming that this mixture forms an ideal system, use the relative volatility, for the relevant calculations, at $80\backslash$deg C$.$

The feed is saturated steam at the dew point

a$)$ Calculate the moles of distilled and residual products per hour.

b$)$ Determine the number of ideal plates and the position of the feed plate

c$)$ If water steam at an absolute pressure of $230$ kPa is used for heating, what amount of steam is required per hour, neglecting heat losses and assuming that the reflux is a saturated liquid?

d$)$ If the cooling water enters the condenser at $20\backslash$deg C and leaves at $50\backslash$deg C$,$ what amount of cooling water will be needed?

e$)$ What is the minimum reflux ratio?

f$)$ What is the minimum number of ideal plates?

g$)$ What is the number of real plates with an efficiency of $67\%?$

We have the following equilibrium data for the methanol$-$water system