Question

A first order reaction is to be treated in a series of two mixed reactors. Show that the total volume of the two reactors is minimum when the reactors are equal in size.

Answer 1

let the volume of 1st reactor be V1 and that of second V2

for 1st order reaction

T= (CO- C1)/ KC1, T =space time = V1/VO, K =rate constant

V1/VO= (CO-C1)/ KC1, VO= Volumetric flow rate

V1= (1- C1/CO)*VO/K, C1= concentratino at the ouilet of 1st reactor and CO= initial concentration

for the second reactor, since the reactors are in series, Vo remains the same.

V2= (1- C2/C1)* VO/K , C2= Concentratino at the outlet of second reactor

let V = V1+V2 = (1-C1/CO)*VO/ K + (1-C2/C1)* VO/K =total volume of reactors

V has to be minimum with respect to intermediate concentration C1

dV/dC1 = -1/CO+C2/C₁²= 0

C₂/C₁²= 1/C_O

C₁²= C₂CO, C1/CO= C2/C1    (1)

V1= (1-C1/CO)*VO/K

V2= (1-C2/C1)* VO/K , from (1) V2= (1-C1/CO)*VO/K

hence V1= V2