Figure 6.19:

Steady-state temperature versus residence time.

Code for Figure 6.19

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
%
% 9/10/98, 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];
nc_As = 200;
%tmp_table(1,:) = [0 T_f 0 -Inf -Inf 0];
c_Avect = linspace(0.9999*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 = sort(real(eig(Jac)))';
  lambda = eig(Jac);
  lamrow = [real(lambda(1)) imag(lambda(1)) real(lambda(2)) ...
            imag(lambda(2))];
  tmp_table(i,:) = [theta, T, conv, lamrow, info];
  x0=x;
end
table = [tmp_table];

%
% 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;
nthetas = 100;
theta_vect = logspace(log10(cortheta), log10(500), nthetas);
clear tmp_table;
for i = 1: nthetas
  theta = theta_vect(i);
  opts = optimset ('MaxFunEvals', 2000*numel (x0), ...
                   'MaxIter', 500*numel (x0));
  [x, fval, info] = fsolve('st_st_theta', x0, opts);
  c_A   = x(1);
  T     = x(2);
  conv     = (c_Af - c_A) / c_Af;
  k = k_m*exp(-E*(1/T - 1/T_m));
  %
  % compute the eigenvalues of the Jacobian
  %
  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);
  lamrow = [real(lambda(1)) imag(lambda(1)) real(lambda(2)) ...
            imag(lambda(2))];
  tmp_table(i,:) = [theta, T, conv, lamrow, info];
  x0=x;
end

table = [table; tmp_table];
save -ascii st_st_osc.dat table;

if (~ strcmp (getenv ('OMIT_PLOTS'), 'true')) % PLOTTING
plot(table(:,1),table(:,3));
axis ([-5,100,0,1]);

figure()
semilogx(table(:,1),table(:,3));
axis([.001,1000,0,1]);

figure()
plot(table(:,1),table(:,2));
axis ([-5,100,280,420]);

figure()
semilogx(table(:,1),table(:,2));
axis([.001,1000,280,420]);
% 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;