We plots the bifurcation diagram (pinch composition versus reflux ratio) for a mixture of benzene -component 1) and ethylenediamine (component 2) at 1 atm. We compare out data to those reported in Figure 4.40 page 171 of Ref. 1. The tangent pinch is shown as a black dot in the original diagram given in Ref. 1. For this particular binary mixture, the vapor-liquid equilibrium (VLE) relation is as follows:​​
y=
ax
1+(a-1)x
+bx(1-x)
, where
a=9
and
b=-0.6
.​The bifurcation diagram is obtained using arc-length continuation. Since the pinch point corresponds to the case where the operating line and the equilibrium curve meet, the implicit function relating the reflux ratio,
r
, and the pinch point location,
x
P
, is given by:​​
y
P
-
r
r+1
x
P
-
x
D
r+1
=0
, ​where
r
is the reflux ratio and
x
D
the distillate composition, and ​​
y
P
=
a
x
P
1+(a-1)
x
P
+b
x
P
(1-
x
P
)
​​​The tangent pinch (operating line is tangent to the equilibrium curve) obeys the following conditions: ​
dy
dx
=solveoftheequilibriumcurve=
r
r+1
=slopeofoperatingline
and
y-
r
r+1
x-
x
D
r+1
=0
.​As
x
D
decreases, the curve loses its distinctive S-shape. The critical point where this transition occurs is verified by the following relations:
2
d
y
d
2
x
=0,
dy
dx
=
r
r+1
, and
y-
r
r+1
x-
x
D
r+1
=0
.
Benzene (light component)
In[]:=
a=9;b=-0.6;
distillate composition (i.e., mole fraction of Benzene in the distillate stream) is treated as a parameter ranging from 0.84 to 0.99 with a constant step equal to 0.03
In[]:=
xd={0.99,0.96,0.93,0.90,0.87,0.84};
pinch point occurs when the equilibrium curve intersects the operating line
In[]:=
eq1[i_]:=
ax
1+(a-1)x
+bx(1-x)-
r
r+1
x-
xd[[i]]
r+1
0/.xx[s]/.rr[s]
The auxiliary equation
In[]:=
auxeq=x'[s]^2+r'[s]^21
Out[]=
2
′
r
[s]
+
2
′
x
[s]
1
Solution of the Differential Algebraic System of Equations using NDSolve
In[]:=
sol[i_]:=Quiet@NDSolve[{eq1[i],auxeq,x[0]0.75,r[0]0.2,x'[0]0.1,r'[0]0.1},{x,r},{s,-10000,10000},AccuracyGoal5]
In[]:=
tbl=Table[sol[i],{i,1,Length[xd]}];
Plotting the bifurcation diagram (pinch composition versus reflux ratio). Red triangles correspond to the Tangent pinch situation
In[]:=
plt[i_]:=ParametricPlot[{x[s],r[s]}/.tbl[[i]],{s,-100,100},FrameTrue,GridLinesAutomatic,AspectRatio0.8,PlotRange{{0,1},{0,5}},PlotPoints100,PlotStyle{Thickness[0.010],Hue[(xd[[i]]-0.84)5],Dashed},FrameLabel{Style["pinch liquid composiiton",14],Style["reflux ratio",14]}]
In[]:=
soltp[i_]:=QuietFindRoot
ax
1+(a-1)x
+bx(1-x)-
r
r+1
x-
xd[[i]]
r+1
0,D
ax
1+(a-1)x
+bx(1-x),{x,1}
r
r+1
,{r,0.5,1.0},{x,0.2,0.9},MaxIterations100
In[]:=
p1=Show[Table[plt[i],{i,1,Length[xd]}],Graphics[Table[Text[Style["▲",Red,18],{x,r}/.soltp[i]],{i,1,Length[xd]-2}]]]
Out[]=
Comparison of our data versus Fig. 4-40 of Ref. 1
In[]:=
IMG=
;
In[]:=
Show[p1,GridLinesAutomatic,FrameTrue,PlotRange{{0,1},{0.0,3.0}},AspectRatio0.75,ImageSize{600,500},Prolog{Raster[ImageData[IMG,DataReversedTrue],{Scaled[{0.0,0.0}],Scaled[{1,1}]}]}]
Out[]=
Illustration 1​
​
We get a pinch for xd=0.87 and r=0.4435 (the operating line in solid red intersects with the equilibrium curve shown in blue). The required number of stages becomes prohibitive and the separation is unfeasible. When we increase the value of the reflux ratio,
r
(e.g.,
r=1.5×0.4435=0.6653
) the operating line (show in dashed magenta) does not intersect the equilibrium curve anymore and the separation becomes feasible.
In[]:=
xd=0.87;xp=0.6;r=.;xf=0.5;
In[]:=
sol1=NSolve
axp
1+(a-1)xp
+bxp(1-xp)-
r
r+1
xp-
xd
r+1
0,r
Out[]=
{{r0.443584}}
In[]:=
r=First[r/.sol1]
Out[]=
0.443584
In[]:=
eqplt=Plot
ax
1+(a-1)x
+bx(1-x),{x,0,1},FrameTrue,GridLinesAutomatic,FrameLabel{Style["liquid mole fraction of benzene"],Style["vapor mole fraction of benzene"]},AspectRatio1,PlotStyleThick,Epilog{Green,Thick,Line[{{0,0},{1,1}}]}
Out[]=
In[]:=
opline=Plot
r
r+1
x+
xd
r+1
,{x,xp,xd},FrameTrue,GridLinesAutomatic,FrameLabel{Style["liquid mole fraction of benzene"],Style["vapor mole fraction of benzene"]},AspectRatio1,PlotStyle{Darker@Red,Thick};
In[]:=
rnew=1.5r
Out[]=
0.665376
In[]:=
oplinenew=Plot
rnew
rnew+1
x+
xd
rnew+1
,{x,xf,xd},FrameTrue,GridLinesAutomatic,FrameLabel{Style["liquid mole fraction of benzene"],Style["vapor mole fraction of benzene"]},AspectRatio1,PlotStyle{Darker@Magenta,Dashed,Thick};

Reference

◼
  • [1] M. F. Doherty and M. F. Malone, Conceptual Design of Distillation Systems, Boston: McGraw-Hill, 2001.

    • CITE THIS NOTEBOOK

    Pinch Composition versus Reflux Ratio for a Benzene-Ethylenediamine Mixture at 1 atm ​
    by Housam Binous and Ahmed Bellagi
    Wolfram Community, STAFF PICKS, January 28, 2024
    ​https://community.wolfram.com/groups/-/m/t/3111106