## Figure 8.30 (page 471):

### Inverse of reaction rate versus concentration; optimal sequence to achieve 95% conversion is PFR--CSTR--PFR. ## Code for Figure 8.30

### main.m

```%% Copyright (C) 2001, James B. Rawlings and John G. Ekerdt
%%
%% This program is free software; you can redistribute it and/or
%% modify it under the terms of the GNU General Public License as
%% published by the Free Software Foundation; either version 2, or (at
%% your option) any later version.
%%
%% This program is distributed in the hope that it will be useful, but
%% WITHOUT ANY WARRANTY; without even the implied warranty of
%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
%% General Public License for more details.
%%
%% You should have received a copy of the GNU General Public License
%% along with this program; see the file COPYING.  If not, write to
%% the Free Software Foundation, 59 Temple Place - Suite 330, Boston,
%% MA 02111-1307, USA.

global a b fixrate theta1 theta2 theta3 c0
a=5; b=0.05;
c0=5;
conv=0.95;
d  = sqrt((1-2*b)*(1-2*b) - 4*b*(b+1));
w1 = (1-2*b + d)/(2*b);
w2 = (1-2*b - d)/(2*b);
x1  = sqrt(w1/a);
x2  = sqrt(w2/a);
c=linspace(0.01,10,300)';
y = 1./rxrate(c);

%% find the outlet of the first PFR, c1, which satisfies r(c1)=1/x2
fixrate=rxrate(x2);
x0=c0;
[x, fval, info] = fsolve('solverate',x0);
info;
c1=x;

%% size the first PFR, theta1 is size such that c(theta1)=c1
t0=0;
ncs=10;
cout = linspace(c0,c1,ncs);
opts = odeset ('AbsTol', sqrt (eps), 'RelTol', sqrt (eps));
[dummy,tout] = ode15s(@pfr,cout,t0,opts);
theta1 = tout(ncs);

%% size the cstr, theta2= (c1-c2)/r(c2), c2=x2
c2=x2;
theta2 = (c1-c2)/rxrate(c2);

%% size the second PFR, c3, which satisfies c3=(1-conv)*c0
c3 = (1-conv)*c0;
t0=0;
ncs=10;
cout = linspace(c2,c3,ncs);
opts = odeset ('AbsTol', sqrt (eps), 'RelTol', sqrt (eps));
[dummy,tout] = ode15s(@pfr,cout,t0,opts);
theta3 = tout(ncs);
c0; c1; c2; c3;
theta1; theta2; theta3;

%% optimal reactor is PFR --->    CSTR --->  PFR
%%                    theta1      theta2     theta3

%% now compute segregated reactor conversion
npts = 50;
zseg = linspace(0,1,npts);
cseg0 = [c0;0];
[dummy,cseg] = ode15s(@seg,zseg,cseg0);
csegfin = cseg(npts,2);

%% solve for cinf, which seems to be the same as what would come out of
%% a cstr of theta2 with feed conc c0
x0=c0;
[x, fval, info] = fsolve('climit',x0);
info;
cinf=x;

%% now compute maximally mixed conversion
npts=50;
z=linspace(0,1,npts);
[dummy,cmm] = ode15s(@mm,z,cinf);
cmmfin = cmm(npts);
table = [c y];
extras = [c0 1/rxrate(c0) c1 1/rxrate(c1) c2 1/rxrate(c2) c3 1/rxrate(c3) ...
c1 1/rxrate(c1) ;
c0 0          c1 0          c2 0          c3 0          ...
c2 1/rxrate(c2)];
save -ascii leven_extras.dat extras;
plot (table(:,1), table(:,2), ...
extras(:,1), extras(:,2), ...
extras(:,3), extras(:,4), ...
extras(:,5), extras(:,6), ...
extras(:,7), extras(:,8), ...
extras(:,9), extras(:,10));
axis ([0,6,0,10])
title ('Figure 8.30')
```

### rxrate.m

```function rate = rxrate(c)
global a b fixrate
rate = c./(1+a*c.*c) + b*c;
```

### solverate.m

```function retval = solverate(c)
global a b fixrate
retval = fixrate - rxrate(c);
```

### pfr.m

```function dtdc = pfr(c,t)
dtdc = - 1/rxrate(c);
```

### seg.m

```function xdot = seg(z,x)
global theta1 theta2 theta3
c    = x(1);
ctot = x(2);
if (z < 1)
t = z/(1-z);
if (t <= theta1+theta3)
p=0;
else
p=exp(-(t-(theta1+theta3) )/theta2)/theta2;
end
xdot = [-rxrate(c)/(1-z)^2; p*c/(1-z)^2];
else
xdot = [0; 0];
end
```

### climit.m

```function retval = climit(c)
global theta2 c0
retval = c-c0+rxrate(c)*theta2;
```

### mm.m

```function dcdz = mm(z,c)
global theta1 theta2 theta3 c0
if (z == 0.)
dcdz = 0;
else
t = (1-z)/z;
if (t <= theta1+theta3)
f = 0;
else
f = 1/theta2;
end
dcdz = - (f*(c-c0) + rxrate(c) )/(z*z);
end
```