Figure 6.23:

Eigenvalues of the Jacobian matrix versus reactor conversion in the region of steady-state multiplicity.

Code for Figure 6.23

Text of the GNU GPL.

main.m


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% 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 theta C_ps T_f DeltaH_R U T_a
%
% limit cycle parameters
%
% units are kmol, min, kJ, K, m^3
%
% 9/16/98, jbr
%
k_m      = 0.004;
T_m      = 298;
E        = 15000;
c_Af     = 2;
C_p      = 4;
rho      = 1000;
C_ps     = C_p*rho;
T_f      = 298;
T_a      = T_f;
DeltaH_R = -2.2e5;
U        = 340;
theta    = 35;
T_set    = 321.53;
c_set    = 0.48995;
T_fs     = T_f;
Kc       = 0;
gamma    = E/T_f;
B        = -DeltaH_R*c_Af*gamma/(C_ps*T_f);
beta     = U/C_ps*theta;
Da       = k_m*exp(-E*(1/T_f-1/T_m))*theta;
x2c      = (T_a-T_f)/T_f*gamma;

%
% building the lower branch;
% use (theta,T) as dependent and
% c_A as independent
%
x0=[1; T_f];
npts1 = 200;
xvect = logspace(log10(1e-4), log10(0.75), npts1);
c_Avect = (1-xvect)*c_Af;
clear tmp_table;
for i = 1: npts1
  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 = sort(real(eig(Jac)))';
  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];
%
% stuff in some points near the extinction point to resolve
% rapid change in the eigenvalues
%
npts2 =200;
xnow = conv;
xvect = linspace(xnow, .95, npts2);
c_Avect = (1-xvect)*c_Af;
clear tmp_table;
for i = 1: npts2
  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;
  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 = [table; tmp_table];

%
% after turning the corner on the upper branch; switch to (c_A,T) as
% dependent and theta as independent load x with current solution
%
x0=[c_A; T];
cortheta= theta;
npts3 = 200;
theta_vect = logspace(log10(cortheta), log10(20.7), npts3);
clear tmp_table;
for i = 1: npts3
  theta = theta_vect(i);
  [x, fval, info] = fsolve('st_st_theta', x0);
  c_A   = x(1);
  T     = x(2);
  conv     = (c_Af - c_A) / c_Af;
  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 = [table; tmp_table];
%
% jam a bunch of points in interval theta=[20.7,20.8] to pick up pair
% of conjugate eigenvalues branching from real axis theta as
% independent load x with current solution
%
x0=[c_A; T];
cortheta= theta;
npts4 = 100;
theta_vect = linspace(cortheta, 20.8, npts4);
clear tmp_table;
for i = 1: npts4
  theta = theta_vect(i);
  [x, fval, info] = fsolve('st_st_theta', x0);
  c_A   = x(1);
  T     = x(2);
  conv     = (c_Af - c_A) / c_Af;
  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 = [table; tmp_table];
%
% finish out to large theta theta as independent load x with current
% solution
%
x0=[c_A; T];
cortheta= theta;
npts5 = 200;
theta_vect = logspace(log10(cortheta), log10(500), npts5);
clear tmp_table;
for i = 1: npts5
  theta = theta_vect(i);
  [x, fval, info] = fsolve('st_st_theta', x0);
  c_A   = x(1);
  T     = x(2);
  conv     = (c_Af - c_A) / c_Af;
  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 = [table; tmp_table];
%
% for eigenvalue plot for s-part of curve, save rows 1:npts1+npts2
%
% for eigenvalue plot for limit cycle part of curve, save rows
% 441:npts1+npts2+npts3+npts4+npts5
%
% Obviously the above row numbers change depending on how the points
% were placed, so this part has to be redone if the problem parameters
% change. I didn't see a better solution without going to a full
% continuation approach, which seems like overkill for one or two
% problems of this type. jbr 9/16/98

tmp = table(1:npts1+npts2,:);
save -ascii st_st_osc_eigs.dat tmp;

if (~ strcmp (getenv ('OMIT_PLOTS'), 'true')) % PLOTTING
plot (tmp(:,3), tmp(:,[4,6]));
axis ([0, 1, -200, 400]);

figure()
plot (tmp(:,3), tmp(:,[4,6]));
axis ([0, 1, -1, 1]);
% TITLE
end % PLOTTING

st_st_cA.m


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function retval = st_st_cA(x)
global k_m T_m E c_Af c_A theta 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;

st_st_theta.m


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function retval = st_st_theta(x)
  global k_m T_m E c_Af c_A theta C_ps T_f DeltaH_R U T_a
  c_A   = 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;