The synthesis of maleic anhydride,
C
4
H
2
O
3
, involves the following reactions in the presence of a vanadium pentoxide catalyst:​​
C
6
H
6
+
9
2
O
2

C
4
H
2
O
3
+2
CO
2
+2
H
2
O
(1)​
C
4
H
2
O
3
+3
O
2
→4
CO
2
+
H
2
O
(2)​
C
6
H
6
+
15
2
O
2
→6
CO
2
+3
H
2
O
(3)​If air is present in excess, the concentration of oxygen can be considered as constant and the reaction rates are given by
r
1
=
k
1
C
A
,
r
2
=
k
2
C
P
, and
r
3
=
k
3
C
A
, where
k
i
is the rate constant of reaction
i
(expressed in
3
m
-1
s
kg
catalyst),
C
A
is the concentration of benzene, and
C
P
is the concentration of maleic anhydride.​The temperature-dependent rate constants are given by
k
1
=4280
-12660
T
e
,
k
2
=70100
-15000
T
e
, and
k
3
=26
-10800
T
e
.​The governing equations at steady state, obtained by writing molar balances, for a packed-bed reactor modeled as a PFR (plug-flow reactor) are:​
v
0
d
C
A
dW
=-
k
1
C
A
-
k
3
C
A
, with
C
A
(0)=10mol/
3
m
the concentration of benzene in the feed stream (i.e. at the entrance of the reactor), and​
v
0
d
C
P
dW
=
k
1
C
A
-
k
2
C
P
, with
C
P
(0)=0
the concentration of maleic anhydride in the feed stream, where
v
0
=0.0025
3
m
/s
is the volumetric flow rate of the feed stream, which is composed mainly of air, and
W
is the weight of catalyst (expressed in kg catalyst).​The final/outlet concentrations of benzene (
C
A
) and maleic anhydride (
C
P
) are obtained by solving the steady-state equations numerically, with the weight of the catalyst ranging from 0 to 10,000 kg.​This notebook plots the attainable region for maleic anhydride synthesis for a temperature equal to 600 K. A profit function is defined by
f(
C
P
,
C
A
)=(α
C
P
-β
C
A
)/
C
A
(0)
, where
α
and
β
are user-set values of the prices of maleic anhydride and benzene. When this function is maximized, the optimal catalyst weight (shown in green) and the value
f
max
(shown in magenta) are determined. The optimal outlet values of the concentrations of benzene (
C
A
) and maleic anhydride (
C
P
) are identified on the plot by the green dot. The dashed red line represents the equation
f
max
=(α
C
P
-β
C
A
)/
C
A
(0)
and is always tangent to the attainable region.​Attainable regions define the achievable compositions that may be obtained from a network of chemical reactions. The attainable region in composition space was first presented by Horn in 1964, with more recent extensions by Glasser and coworkers (1987).​​Reference:​[1] F. J. M. Horn, “Attainable Regions in Chemical Reaction Technique,” Third European Symposium of Chemical Reaction Engineering, London: Pergamon Press, 1964.[2] D. Glasser, C. Crowe, and D. Hildebrandt, “A Geometric Approach to Steady Flow Reactors: The Attainable Region and Optimization in Concentration Space,” Industrial Engineering & Chemistry Research, 26(9), 1987 pp. 1803–1810.[3] W. D. Seider, J. D. Seader, and D. R. Lewin, Product and Process Design Principles: Synthesis, Analysis, and Evaluation, 2nd ed., New York: Wiley, 2004.​
In[]:=
T=600;(*temperature*)​​CostA=6;(*benzenecost=β *)​​CostP=6.5;(*maleicanhydridecost=α*)
In[]:=
v0=0.0025;cA0=10;k1=4280Exp
-12660
T
;k2=70100Exp
-15000
T
;k3=26Exp
-10800
T
;
In[]:=
sol=NDSolve[{v0D[cA[W],{W,1}]-k1cA[W]-k3cA[W],cA[0]cA0,​​v0D[cP[W],{W,1}]k1cA[W]-k2cP[W],cP[0]0},{cA[W],cP[W]},{W,0,10000}];​​
In[]:=
obj[W_]:=First[Evaluate[CostPcP[W]-CostAcA[W]/.sol]];
In[]:=
solopt=NMaximize[{obj[W],W≥0},W];
In[]:=
plt1=ParametricPlot{cA[W]/cA0,cP[W]/cA0}/.sol,{W,0,10000},PlotRange{{0,1},{0,1}},ImageSize{500,350},PlotStyleThick,FrameLabel"
C
A
C
A
[0]
","
C
P
C
A
[0]
",PlotPoints1000,FrameTrue,GridLinesAutomatic,PlotLabelRow[{Style[Row[{"profit = ",NumberForm[solopt[[1]],{5,2}]}],Darker@Magenta,16]," ",Style[Row[{"catalyst = ",NumberForm[W/.solopt[[2]],{5,2}]}],Darker@Green,16]}];plt2=ParametricPlot[{x,(solopt[[1]]/10+CostAx)/CostP}/.sol,{x,0,1.0},PlotStyle{Thick,Red,Dashed},PlotRange{{0,1},{0,1}}];
In[]:=
Show[plt1,plt2,Graphics[{Green,PointSize[0.025],Point[{cA[W]/cA0,cP[W]/cA0}/.sol/.solopt[[2]]]}]]
Out[]=