Figure 7.20:

Dimensionless concentration versus radius for the nonisothermal spherical pellet: lower (A), unstable middle (B), and upper (C) steady states.

Code for Figure 7.20

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.

%
% compute the pellet temperature and concentration profiles
% for the multiple steady state Weisz Hicks problem
%
% 7/18/01
%
% jbr
%
% Revised 8/14/2018

Gamma = 30;
beta  = 0.4;
Phiscale  = 0.01;
tol = 1e-6;


if (exist ('OCTAVE_VERSION')) % IF OCTAVE
    [value, ier, nfun, err] = quad(@(c) integrand(c, beta, Gamma), 0, 1, tol);

    if (ier ~= 0)
        fprintf ('darn, quad failed on integral\n nfun and err =\n');
        nfun, err
    end

else % ELSE
    [value, nfun] = quadl(@(c) integrand(c, beta, Gamma), 0, 1, tol);
end % ENDIF

intercept = sqrt(2*value);
Phi = intercept*Phiscale;



rout = linspace(0,3,100)';
%c0=1e-11;
c0vec = [1e-11; 0.5; 0.95];
nc = length(c0vec);
results(1:nc) = {[]};

for i = 1:length(c0vec)
    c0 = c0vec(i);
    x0 = [c0; 0; 1; 0];
    ode_opts = odeset();
    rel = ode_opts.RelTol;

    if (isempty (rel))
        rel = sqrt (eps);
    end

    ode_opts = odeset('AbsTol', rel*c0, 'RelTol', sqrt(eps));
    [rsolver,xout] = ode15s(@(r, x) pelletode(r, x, beta, Gamma, Phi),...
                        rout, x0, ode_opts);
    %
    % checked sensitivies by finite difference, done 7/18/01
    %
    ceinit = xout(length(rout),1);
    c0init = c0;
    y0     = [ceinit; c0init];
    ydot0  = [-1; -1/xout(length(rout), 3)];
    tsteps = linspace(0, ceinit - 1, 10)';
    dae_opts = odeset ('AbsTol', rel*c0init, 'RelTol', sqrt(eps));
    [tsolver,y]  = ode15i(@(t, y, ydot) continode(t, y, ydot, beta, Gamma, Phi, rout),...
                        tsteps, y0, ydot0, dae_opts);
    %
    % solve the pellet problem from correct c0
    %
    c0 = y(length(tsteps),2);
    x0 = [c0; 0; 1; 0];

    ode_opts = odeset('AbsTol', rel*c0, 'Reltol', sqrt(eps));
    [rsolver,xout] = ode15s(@(r, x) pelletode(r, x, beta, Gamma, Phi),...
                        rout, x0, ode_opts);

    conc = xout(:,1);
    temp = beta*(1 - conc);
    eta(i) = xout(length(rout),2)/Phi^2;
    results{i} = [rsolver, conc, temp];
end

save whmss.dat results;

if (~ strcmp (getenv ('OMIT_PLOTS'), 'true')) % PLOTTING
    subplot (2, 1, 1);
    hold on

    for i = 1:nc
        plot (results{i}(:,1), results{i}(:,2));
    end

    % TITLE whmss
    subplot (2, 1, 2);
    hold on

    for i = 1:nc
        plot (results{i}(:,1), results{i}(:,3));
    end

    % TITLE whmssT
    hold off
end % PLOTTING

integrand.m

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function retval = integrand(c, beta, Gamma)
    retval = c *exp(Gamma*beta*(1-c)/(1+beta*(1-c)));

pelletode.m

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function xdot = pelletode(r, x, beta, Gamma, Phi)
    ca    = x(1);
    dcadr = x(2);
    s1    = x(3);
    s2    = x(4);
    xdot = zeros (4, 1);
    xdot(1) = dcadr;
    if (r == 0)
        xdot(2) = 1/3* Phi^2*ca*exp(Gamma*beta*(1-ca)/(1+beta*(1-ca)));
        xdot(4) = 0;
    else
        tmp = Phi^2*ca*exp(Gamma*beta*(1-ca)/(1+beta*(1-ca)));
        xdot(2) = - 2/r*dcadr + tmp;
        xdot(4) = tmp*(1/ca - Gamma*beta/(1+beta*(1-ca))^2)*s1 - 2/r*s2;
    end
    xdot(3) = s2;

continode.m

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function res = continode(t, y, ydot, beta, Gamma, Phi, rout)
    ce = y(1);
    c0 = y(2);
    %
    % solve pellet problem from c0
    %
    x0=[c0; 0; 1; 0];
    ode_opts = odeset('AbsTol', 1e-5*c0, 'RelTol', sqrt(eps));
    [rsolver,xout] = ode15s(@(r, x) pelletode(r, x, beta, Gamma, Phi),...
                        rout, x0, ode_opts);
    %
    res = zeros (2, 1);
    res(1) = ydot(1) + 1;
    res(2) = ydot(2) + 1/xout(length(rout),3);