Conversion Calculation for 50 Tanks in Series and Second Order Reaction 
Author's Data: Housam BINOUS
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Mean residence time in a tank is τ. V is the total volume of all tanks. F is the flow rate. Ntanks=50 tanks.

Ntanks=50;V=1;F=0.01;

τ = V/F/Ntanks

RTD from MacMullin and Weber's derivation (1935). 50 tanks case.

tbar=Ntanks τ

RTD[t_]:=1/tbar (t/tbar)^(49) 50^50/49! Exp[-50 t/tbar]

plt1 = 
      Plot[RTD[t], {t, 0, 500}, PlotStyle -> RGBColor[1, 0, 0], 
        PlotRange -> All]

Conversion using the RTD. Assume first order reaction with rate constant k=0.05.

k=0.05;

NIntegrate[RTD[t] 1/(1 +  k t), {t, 0, 1000}]

Computing conversion using the system of ordinary differential equations that governs the problem.

Rate constant

k=0.05;

Input concentration of component A

(Subscript[CA, 0])[t_] := 1

Governing Equations

i=2;

For[j = 2, 
      j < 51, {eq[i] = 
          D[(Subscript[CA, i])[t], {t, 
                1}] == ((Subscript[CA, i - 1])[t] - 
                    (Subscript[CA, i])[t])/τ -   k (Subscript[CA, i])[t]^2, 
        i = i + 1}, j++]

eq[1] = 
        D[(Subscript[CA, 1])[t], {t, 1}] == ((Subscript[CA, 0])[t] - (Subscript[CA, 1])[t])/τ - 
            k  (Subscript[CA, 1])[t]^2;

eqn=Table[eq[i],{i,2,50}];

Eq=Join[{eq[1]},eqn];

init = Table[(Subscript[CA, i])[0] == 0, {i, 1, 50}];

sys=Join[Eq,init];

Unknown = Table[(Subscript[CA, i]), {i, 1, 50}];

computing the exit concentration from each tank.

sol=NDSolve[sys,Unknown,{t,0,1000}];

Plot of the concentration profile at the exit of the last tank

plt2 = 
      Plot[(Subscript[CA, 50])[t] /. sol, {t, 0, 500}, 
        PlotStyle -> RGBColor[0, 1, 0], 
        PlotRange -> All]

Conversion is equal to the steady state value of the exit concentration of the last tank. Conversion obtained by solving the system of ODEs and by using the RTD are both equal to 0.17.

(Subscript[CA, 50])[t] /. sol /. t -> 500