Figure 6.18:

Steady-state conversion versus residence time.

Code for Figure 6.18

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.

%
% limit cycle parameters
%
% units are kmol, min, kJ, K, m^3
%
% 9/10/98, jbr
%
% Revised 8/13/2018

p.k_m      = 0.004;
p.T_m      = 298;
p.E        = 15000;
p.c_Af     = 2;
p.C_p      = 4;
p.rho      = 1000;
p.C_ps     = p.C_p*p.rho;
p.T_f      = 298;
p.T_a      = p.T_f;
p.DeltaH_R = -2.2e5;
p.U        = 340;
p.theta    = 35;
p.T_set    = 321.53;
p.c_set    = 0.48995;
p.T_fs     = p.T_f;
p.Kc       = 0;
p.gamma    = p.E/p.T_f;
p.B        = -p.DeltaH_R*p.c_Af*p.gamma/(p.C_ps*p.T_f);
p.beta     = p.U/p.C_ps*p.theta;
p.Da       = p.k_m*exp(-p.E*(1/p.T_f-1/p.T_m))*p.theta;
p.x2c      = (p.T_a-p.T_f)/p.T_f*p.gamma;


%
% building the lower branch;
% use (theta,T) as dependent and
% c_A as independent
%

x0=[1; p.T_f];
nc_As = 200;
%tmp_table(1,:) = [0 T_f 0 -Inf -Inf 0];
c_Avect = linspace(0.9999*p.c_Af, .005*p.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(@(x) st_st_cA(x, c_A, p), x0, opts);
    theta = x(1);
    T     = x(2);
    conv  = (p.c_Af - c_A)/p.c_Af;

    % compute the eigenvalues of the Jacobian
    k = p.k_m*exp(-p.E*(1/T - 1/p.T_m));
    Jac = [-1/theta - k,...
           -k*c_A*p.E/(T*T);
           -k*p.DeltaH_R/p.C_ps,...
           -p.U/p.C_ps - 1/theta - k*c_A*p.DeltaH_R/p.C_ps*p.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(@(x) st_st_theta(x, theta, p), x0, opts);
    c_A   = x(1);
    T     = x(2);
    conv     = (p.c_Af - c_A)/p.c_Af;
    k = p.k_m*exp(-p.E*(1/T - 1/p.T_m));

    % compute the eigenvalues of the Jacobian
    Jac = [-1/theta - k,...
           -k*c_A*p.E/(T*T);
           -k*p.DeltaH_R/p.C_ps,...
           -p.U/p.C_ps - 1/theta - k*c_A*p.DeltaH_R/p.C_ps*p.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, c_A, p)
    theta = x(1);
    T     = x(2);
    k         = p.k_m*exp(-p.E*(1/T - 1/p.T_m));
    retval(1) = p.c_Af - (1 + k*theta)*c_A;
    retval(2) = p.U*theta*(p.T_a - T) + p.C_ps*(p.T_f - T) -...
                    k*theta*c_A*p.DeltaH_R;

st_st_theta.m


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function retval = st_st_theta(x, theta, p)
    c_A   = x(1);
    T     = x(2);
    k         = p.k_m*exp(-p.E*(1/T - 1/p.T_m));
    retval(1) = p.c_Af - (1 + k*theta)*c_A;
    retval(2) = p.U*theta*(p.T_a - T) + p.C_ps*(p.T_f - T) -...
                    k*theta*c_A*p.DeltaH_R;