A distillation column, shown schematically, is used to separate a mixture of met

Question

A distillation column, shown schematically, is used to separate a mixture of methanol and ethanol. The feed has equimolar amounts of methanol and ethanol. 95% of the methanol in the feed goes overhead. Pick a basis for the feed rate (i.e. do not consider it to be an unknown) Use the following nomenclature:Methanol = species 1 Ethanol = species 2 X_1, X_2 = mole fractions in the feed y_1, y_2 = mole fractions in the overhead z_1, z_2 = mole fractions in the bottoms a) How many degrees of freedom are there in this problem? b) Arbitrarily assign values to the required number of variables to solve the problem. Then solve the problem (Solving the problem means finding the flow rate and composition of the overhead and bottom streams.) Assume the column operates at steady state.

Explanation / Answer

let us assume a basis of 100 Moles/hr of feed to column, F=100 moles/hr

there are three streams,

feed, F = 100 moles/hr (X1 = 0.5, X2 = 0.5)

overhead, M = ? moles/hr (Y1, Y2=1-Y1)

bottom, E=? moles/hr (Z1, Z2=1-Z1)

unknowns are, M,X1,E and Z1

number of equations are = mass balances for species 1 and 2 , so 2 equations and an information is also provided as 95% of methanol in feed goes overhead, so there are total 3 equations.

degree of freedom = number of unknowns - number of equations = 4-3 = 1

so we have to assume one unknown to have solution.

assume 50 moles/hr in bottom, so E=50,

that's why, M = F-E = 100-50 = 50.

material balance on methanol,

methanol in feed is = 100*0.5 = 50 moles, because 95% of methanol goes overhead so,

M*Y1 = 50*0.95 = 47.5, so,

Y1 = 47.5/50 = 0.95, Y2=1-0.95 = 0.05

because 5% of methanol goes bottom, so

E*Z1 = 50*0.05 = 2.5

Z1 = 2.5/50 =0.05 so, Z2=1-0.05 =0.95

M=50 (Y1=0.95, Y2=0.05)

E=50 (Z1=0.05, Z2=0.95)

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