Construction of Attainable Region 
for van de Vusse Kinetics

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

We obtain the extended convex attainable region for van de Vusse kinetics using CSTRs and PFRs with and without mixing and bypass.

Reference: Seider, Seader and Lewin, Product & Process Design Principles, 2nd Edition, Wiley, 2004.

Rate constants

k1=0.01;k2=5;k3=10;k4=100;

Trajectory for a PFR. This constitute a candidate AR. But it does not form a convex region.

sol=NDSolve[{D[CA[t],{t,1}]==-k1 CA[t]+k2 CB[t]-k4 CA[t]^2,
D[CB[t],{t,1}]==k1 CA[t]-k2 CB[t]-k3 CB[t],CA[0]==1,
CB[0]==0},{CA[t],CB[t]},{t,0,10}]

plt1=ParametricPlot[Evaluate[{CA[t],CB[t]}/.sol],{t,0,1},PlotStyle->{{RGBColor[0,1,0],Thickness[0.015]}},PlotRange->All,AspectRatio->1,PlotPoints->10^5]

Extending the AR with a line arc, which represents mixing between the PFR effluent and the feed stream. We form the convex hull.

tbl=Table[First[Evaluate[{CA[t],CB[t]}/.sol]],{t,0,10,0.0001}];

func=Interpolation[tbl];

D1=D[func[x],{x,1}]

FindRoot[D1 (x-1)==func[x],{x,0.3},
MaxIterations->10^8,AccuracyGoal->10^-30]

Show[plt1,Graphics[{Dashing[{0.025, 0.025}],RGBColor[0,0,1],Thickness[0.015],Line[{{1,0},{0.288747,func[0.288747]}}]}],AspectRatio->1]

Trajectory for a CSTR. Can we enlarge the AR?

τ = Table[0.0001 i, {i, 0, 10000}];

solution[i_]:=FindRoot[{1-cA==τ[[i]] (k1 cA-k2 cB+k4 cA^2),cB==τ[[i]] (k1 cA-k2 cB-k3 cB)},{cA,0.5},{cB,0.2}]

CSTR1=Table[{cA,cB}/.solution[i],{i,1,10000}];

τ=Table[i,{i,1,100}];

CSTR2=Table[{cA,cB}/.solution[i],{i,1,100}];

CSTR=Join[CSTR1,CSTR2];

plt2=ListPlot[CSTR,PlotStyle->{RGBColor[1,0,0],Thickness[0.01]},PlotRange->All,PlotJoined->True,AspectRatio->1]

Show[plt1,plt2]

Blue arc represents a CSTR with bypass stream

func=Interpolation[CSTR];

D1=D[func[x],{x,1}]

FindRoot[D1 (x-1)==func[x],{x,0.3}]

Show[plt1,plt2,Graphics[{Dashing[{0.025, 0.025}],RGBColor[0,0,1],Thickness[0.015],Line[{{1,0},{0.387288,func[0.387288]}}]}],AspectRatio->1]

Extending the previous AR to form an expanded convex candidate AR

sol2=NDSolve[{D[CA[t],{t,1}]==-k1 CA[t]+k2 CB[t]-k4 CA[t]^2,
D[CB[t],{t,1}]==k1 CA[t]-k2 CB[t]-k3 CB[t],CA[0]==0.387288,
CB[0]==func[0.387288]},{CA[t],CB[t]},{t,0,10}]

plt3=ParametricPlot[Evaluate[{CA[t],CB[t]}/.sol2],{t,0,1},PlotRange->All,PlotStyle->{{RGBColor[0,1,1],Thickness[0.015]}},AspectRatio->1]

Show[plt1,plt2,plt3,Graphics[{Dashing[{0.025, 0.025}],RGBColor[0,0,1],Thickness[0.015],Line[{{1,0},{0.387288,func[0.387288]}}]}],AspectRatio->1]