Adiabatic Reactors: Multiple Steady States
Author's Data: Housam BINOUS
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Acknowledgement :
Numerical values were obtained from The Engineering of Chemical Reactions, Lanny D. Schmidt, Oxford University Press, 1998.

Relationship between temperature and concentration for an adiabatic CSTR and an adiabatic PFTR

Sol1 = 
      Solve[ρ (Subscript[c, p]) (T - T0) + ((Subscript[C, A0]) - 
                    x) (-ΔH) == 0, T] // 
        FullSimplify

Sol2 = 
      Solve[(ρ (Subscript[c, p]) (T - 
                    T0) + (-ΔH) ((Subscript[C, A0]) - 
                    x)) == 0, x]

Numerical values

(Subscript[k, 0]) = 2.6 10^20;

Ea=15000 8.314;

ΔH = 20000;

T0=300;

R=8.314;

ρ = 20000/(Subscript[c, p]) 1/20. ;

(Subscript[C, A0]) = 2;

Relationship between temperature and concentration for an adiabatic CSTR and an adiabatic PFTR

T[a_]:=First[T/.Sol1/.x->a]

rate constant

k[x_] := (Subscript[k, 0]) Exp[-Ea/(R T[x])]

reaction rate

r[x_]:=k[x] x

Concentration versus residence time for an adiabatic CSTR

pp=Plot[(2-x)/r[x],{x,0,2},PlotStyle->RGBColor[0,1,0],DisplayFunction->Identity]

pp1=Show[pp/.x_Line:>Map[Reverse,x,{2}],DisplayFunction->$DisplayFunction];

Concentration versus residence time for an adiabatic PFTR

τ[C_]:=NIntegrate[1/r[x],{x,2,C}]

kk=Plot[-τ[x],{x,0,2},PlotStyle->RGBColor[1,0,0],DisplayFunction->Identity]

kk1=Show[kk/.x_Line:>Map[Reverse,x,{2}],DisplayFunction->$DisplayFunction];

Concentration versus residence time  for an adiabatic PFTR and an adibatic CSTR. We find that the CSTR require much smaller residence times than the PFTR.

Show[pp1,kk1,PlotRange->{{0,5},{0,2}}]

Relationship between concentration and temperature for an adiabatic CSTR and an adiabatic PFTR

x[a_]:=First[x/.Sol2]/.T->a

rate constant

k[T_] := (Subscript[k, 0]) Exp[-Ea/(R T)]

reaction rate

r[T_]:=k[T] x[T]

Tempearature versus residence time for an adiabatic CSTR

ll=Plot[(2-x[T])/r[T],{T,300,340},PlotStyle->RGBColor[0,1,0],DisplayFunction->Identity]

ll1=Show[ll/.x_Line:>Map[Reverse,x,{2}],DisplayFunction->$DisplayFunction];

Tempearature versus residence time for an adiabatic PFTR

mm=Plot[NIntegrate[1/r[a] 1/20,{a,300,T}],{T,300,340},PlotStyle->RGBColor[1,0,0],DisplayFunction->Identity]

mm1=Show[mm/.x_Line:>Map[Reverse,x,{2}],DisplayFunction->$DisplayFunction];

Tempearature versus residence time for an adiabatic PFTR and an adiabatic CSTR

Show[mm1,ll1,PlotRange->{{0,5},{300,340}}]

Inverse of reaction rate (1/( (Subscript[kC, A]))) versus concentration diffrence (Subscript[C, A0]) - (Subscript[C, A])

T[a_]:=First[T/.Sol1/.x->a]

k[x_] := (Subscript[k, 0]) Exp[-Ea/(R T[x])]

r[x_]:=k[x] x

jj=Plot[1/r[2-x],{x,0,1.99},PlotRange->{{0,3},{0,7}}]

In an adiabatic CSTR, multiple steady states are obtained for example for a residence time equal to one. Residence time is the colored area in the figures. The same area is obtained for three different values of the concentration and temperature. 
Conclusion: We obtain three steady states for certain values of the residence time in an adiabatic CSTR.

Off[General::spell1]

sol1=FindRoot[1/r[ξ] (2-ξ)==1,{ξ,0.1}]

sol2=FindRoot[1/r[ξ] (2-ξ)==1,{ξ,1}]

sol3=FindRoot[1/r[ξ] (2-ξ)==1,{ξ,2}]

<< Graphics`FilledPlot`

plt1=FilledPlot[{1/r[ξ]/.sol1},{x,0,2-ξ}/.sol1,PlotRange->All,Fills->{RGBColor[0.9,0.9,0]},
 Curves -> Front,DisplayFunction->Identity]

plt2=FilledPlot[{1/r[ξ]/.sol2},{x,0,2-ξ}/.sol2,PlotRange->All,Fills->{RGBColor[0.9,0.5,0.8]},
 Curves -> Front,DisplayFunction->Identity]

plt3=FilledPlot[{1/r[ξ]/.sol3},{x,0,2-ξ}/.sol3,PlotRange->All,Fills->{RGBColor[0.5,0.5,0.8]},
 Curves -> Front,DisplayFunction->Identity]

Show[plt1,jj,DisplayFunction->$DisplayFunction]

Show[plt2,jj,DisplayFunction->$DisplayFunction]

Show[plt3,jj,DisplayFunction->$DisplayFunction]