Singular Point Analysis
Author's Data: Housam Binous 
Department of Chemical Engineering
National Institute of Applied Sciences and Technology
Tunis, TUNISIA
Email: binoushousam@yahoo.com

Reference :

K. Sundmacher & Z. Qi, Importance of Reaction Kinetics for Catalytic Distillation, Reactive Distillation, Status and Future Directions, Edited by Kai Sundmacher & Achim Kienle, Wiley VCH 2002.

Case a: (Subscript[α, 13])=0.2 and (Subscript[α, 23])=3.0

Singular point curve

y3=1-y1-y2;x3=1-x1-x2;

y1=x1/(x1+(Subscript[α, 21]) x2+(Subscript[α, 31]) x3);y2=x2/((Subscript[α, 12])x1+x2+(Subscript[α, 32]) x3);

eq=(x1+x3) (1+y3)-(y1+y3) (1+x3);

(Subscript[α, 21])=3.0/0.2;(Subscript[α, 12])=0.2/3.0;(Subscript[α, 31])=1/0.2;(Subscript[α, 32])=1/3.0;

plt1=ContourPlot[eq==0,{x1,-1,2},{x2,-1,2},RegionFunction->Function[{x1,x2},Evaluate[Abs[eq]<1.0]]]

eq2=-(y2-x2)+(-1+x2) Da (x1 x2-x3/K);

Kinetic curve for Damköhler number Da=2.33

K=5;Da=2.33;

plt2=ContourPlot[eq2==0,{x1,-0.5,2},{x2,-0.2,2},RegionFunction->Function[{x1,x2},Evaluate[Abs[eq]<1.0]],ContourStyle-> Red]

sol=FindRoot[{eq==0,eq2==0},{x1,0.8},{x2,1.7}]

Show[plt2,plt1,Graphics[{{RGBColor[1,0,1],Line[{{0,0},{0,1}}]},{RGBColor[1,0,1],Line[{{0,0},{1,0}}]},{RGBColor[1,0,1],Line[{{0,1},{1,0}}]},{RGBColor[0,1,0],PointSize[0.025],Point[{x1/.sol,x2/.sol}]},{RGBColor[0,1,0],PointSize[0.025],Point[{1,0}]},{RGBColor[0,1,0],PointSize[0.025],Point[{0,1}]}}]]

Case a: (Subscript[α, 13])=5.0 and (Subscript[α, 23])=3.0

(Subscript[α, 21])=3.0/5.0;(Subscript[α, 12])=5.0/3.0;(Subscript[α, 31])=1/5.0;(Subscript[α, 32])=1/3.0;

Singular point curve

plt3=ContourPlot[eq==0,{x1,-1,2},{x2,-1,2},RegionFunction->Function[{x1,x2},Evaluate[Abs[eq]<1.0]]]

Kinetic curve for Damköhler number Da=1.5

K=5;Da=1.5;

eq2=-(y2-x2)+(-1+x2) Da (x1 x2-x3/K);

plt4=ContourPlot[eq2==0,{x1,-0.2,2},{x2,-0.2,2},RegionFunction->Function[{x1,x2},Evaluate[Abs[eq]<1]],ContourStyle-> Red]

sol1=FindRoot[{eq==0,eq2==0},{x1,0.8},{x2,0.7}]

sol2=FindRoot[{eq==0,eq2==0},{x1,0.08},{x2,0.3}]

Show[plt4,plt3,Graphics[{{RGBColor[1,0,1],Line[{{0,0},{0,1}}]},{RGBColor[1,0,1],Line[{{0,0},{1,0}}]},{RGBColor[1,0,1],Line[{{0,1},{1,0}}]},{RGBColor[0,1,0],PointSize[0.025],Point[{x1/.sol1,x2/.sol1}]},
{RGBColor[0,1,0],PointSize[0.025],Point[{x1/.sol2,x2/.sol2}]},{RGBColor[0,1,0],PointSize[0.025],Point[{1,0}]},{RGBColor[0,1,0],PointSize[0.025],Point[{0,1}]}}]]