Finding best reactor set-up
Author's Data: Housam BINOUS
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Acknowledgement :
This problem was solved graphically by O. Levenspiel in Chemical Reaction Engineering, 3rd Edition, Wiley, 1999.

Volumetric Flow

v=0.1

Inlet concentration (Subscript[C, A0]), outlet concentration (Subscript[C, A])and space time τ in a laboratory mixed flow reactor

(Subscript[C, A0]) = {2, 5, 6, 6, 11, 14, 16, 24};

(Subscript[C, A]) = {0.5, 3, 1, 2, 6, 10, 8, 4};

τ = {30, 1, 50, 8, 4, 20, 20, 4};

χ = N[(τ/((Subscript[C, A0]) - (Subscript[C, A])))]

\({Join[{\*""\!\(C\_A0\)""}, C\_A0], 
        Join[{\*""\!\(C\_A\)""}, C\_A], Join[{"τ"}, τ], 
        Join[{\*""-1/\!\(r\_A\)""}, χ]} // MatrixForm\)

Data interpolation of ((-1)/(Subscript[r, A]))versus (Subscript[C, A])

InvReactRate=Interpolation[Sort[Transpose[{(Subscript[C, A]), χ}]],InterpolationOrder -> 3]

PLT1 = 
      ListPlot[Sort[Transpose[{(Subscript[C, A]), χ}]], 
        PlotStyle -> RGBColor[1, 0, 0], PlotJoined -> True, 
        PlotRange -> {{0, 12}, {0, 12}}, 
        DisplayFunction -> Identity]

PLT2 = 
      Plot[InvReactRate[x], {x, 0.5, 10}, PlotRange -> All, 
        DisplayFunction -> Identity]

Show[PLT1,PLT2,DisplayFunction->$DisplayFunction]

<< Graphics`FilledPlot`

Volume of a plug flow rector with recycle = 1.48 (m^3).

sol1=FindRoot[{InvReactRate[ξ]==InvReactRate[ω],Integrate[InvReactRate[x],{x,1,ξ}]==Integrate[InvReactRate[ξ]-InvReactRate[x],{x,ξ,ω}]},{ξ,1.7},{ω,6}]

plt1=FilledPlot[{InvReactRate[x],InvReactRate[ξ]/.sol1},{x,1,ξ}/.sol1,PlotRange->{{0,12},{0,12}},DisplayFunction->Identity]

plt2=FilledPlot[{InvReactRate[x],InvReactRate[ξ]/.sol1},{x,ξ,ω}/.sol1,PlotRange->{{0,12},{0,12}},DisplayFunction->Identity]

plt3=FilledPlot[{InvReactRate[ξ]/.sol1},{x,1,10}/.sol1,PlotRange->{{0,12},{0,12}},Fills->{GrayLevel[.9]},
 Curves -> Front,DisplayFunction->Identity]

Show[plt1,plt3,plt2,PLT2,AspectRatio->1,DisplayFunction->$DisplayFunction]

A1=Integrate[InvReactRate[ξ]/.sol1,{x,1,10}]

V=v A1

Recycle ratio and recycle flowrate

R=(10-ω)/(ω-1)/.sol1

(Subscript[v, R]) = v R

Volume needed if one uses an arrangement of a plug flow reactor and a stirred tank reactor = 0.73 (m^3).

plt4=FilledPlot[{InvReactRate[x],0},{x,1,4},PlotRange->{{0,12},{0,12}},DisplayFunction->Identity]

plt5=FilledPlot[{InvReactRate[4],0},{x,4,10},PlotRange->{{0,12},{0,12}},Fills->{RGBColor[1,0.7,0.5]},
 Curves -> Front,DisplayFunction->Identity]

Show[plt4,plt5,PLT2,AspectRatio->1,DisplayFunction->$DisplayFunction]

A1=Integrate[InvReactRate[4],{x,4,10}]

A2=Integrate[InvReactRate[x],{x,1,4}]

V=v (A1+A2)

Volume of two stirred tanks = 2.07 (m^3).

sol2=FindRoot[InvReactRate'[ξ]==-(10-InvReactRate[ξ])/(10-ξ),{ξ,5.5}]

plt6=FilledPlot[{InvReactRate[1]},{x,1,ξ}/.sol2,PlotRange->{{0,12},{0,12}},Fills->{GrayLevel[.9]},
 Curves -> Front,DisplayFunction->Identity]

plt7=FilledPlot[{InvReactRate[ξ]/.sol2},{x,ξ,10}/.sol2,PlotRange->{{0,12},{0,12}},Fills->RGBColor[0.8,0.2,0.8],
 Curves -> Front,DisplayFunction->Identity]

y=(InvReactRate'[ξ]/.sol2) (x-ξ/.sol2)+InvReactRate[ξ]/.sol2

plt8=Plot[y,{x,1,4},PlotStyle->RGBColor[0,1,1],DisplayFunction->Identity]

Show[plt6,plt7,plt8,PLT2,Graphics[{RGBColor[0,1,1],Line[{{ξ,10}/.sol2,{10,InvReactRate[ξ]/.sol2}}]},{RGBColor[0,1,1],Line[{{ξ,10}/.sol2,{10,InvReactRate[ξ]/.sol2}}]}],AspectRatio->1,DisplayFunction->$DisplayFunction]

A1=Integrate[InvReactRate[1],{x,1,ξ}]/.sol2

A2=Integrate[InvReactRate[ξ],{x,ξ,10}]/.sol2

V=v (A1+A2)

Best reactor setup is the second arrangement (i.e. plug flow and stirred tank arrangement).