## Figure 6.14 (page 301):

### Eigenvalues of Jacobian matrix vs. reactor conversion. ## Code for Figure 6.14

### 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 k_m T_m E c_Af c_A C_ps T_f DeltaH_R U T_a
%
% multiplicity parameters
%
% units are kmol, min, kJ, K, m^3
%
%
k_m      = 0.001;
T_m      = 298;
E        = 8000;
c_Af     = 2;
C_p      = 4;
rho      = 1000;
C_ps     = rho*C_p;
T_f      = 298;
T_a      = T_f;
DeltaH_R = -3e5;
U        = 0;

x0=[1; T_f];
nc_As = 200;
clear tmp_table;
c_Avect = linspace(0.995*c_Af, .005*c_Af, nc_As);
for i = 1: nc_As
c_A = c_Avect(i);
opts = optimset ('MaxFunEvals', 2000*numel (x0), ...
'MaxIter', 500*numel (x0));
[x, fval, info] = fsolve('st_st_cA', x0, opts);
theta = x(1);
T     = x(2);
conv     = (c_Af - c_A) / c_Af;
%%
%% compute the eigenvalues of the Jacobian
%%
k = k_m*exp(-E*(1/T - 1/T_m));
Jac = [-1/theta - k,     -k*c_A*E/(T*T);
-k*DeltaH_R/C_ps, -U/C_ps - 1/theta - k*c_A*DeltaH_R/C_ps*E/(T*T)];
lambda = eig(Jac);
a=real(lambda(1)); b=imag(lambda(1)); c=real(lambda(2)); d=imag(lambda(2));
if (a >= c)
lamrow = [a b c d];
else
lamrow = [c d a b];
end
tmp_table(i,:) = [theta, T, conv, lamrow, info];
x0=x;
end
table = [tmp_table];

plot(table(:,3),[table(:,4),table(:,6)]);
axis([0,1,-1,1]);
title ('Figure 6.14')
```

### st_st_cA.m

```function retval = st_st_cA(x)
global k_m T_m E c_Af c_A C_ps T_f DeltaH_R U T_a
theta = x(1);
T     = x(2);
k         = k_m*exp(-E*(1/T - 1/T_m));
retval(1) = c_Af - (1+k*theta)*c_A;
retval(2) = U*theta*(T_a - T) + C_ps*(T_f - T) - k*theta*c_A*DeltaH_R;
```