Figure 7.19 (page 390):

Effectiveness factor versus normalized Thiele modulus for a first-order reaction in nonisothermal spherical pellet.

Code for Figure 7.19

Text of the GNU GPL.


%% 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
%% 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.

%% Solve for the nonisothermal dimensionless pellet profile
%% first order reaction in a sphere
%% See Weisz and Hicks, 1962
%% 1/r^2  d/dr r^2 dc/dr  -Phi^2 c exp( gamma beta (1-c) / (1+ beta(1-c)) )
%% c(3)      = 1
%% dc/dr (0) = 0
%% goal is to not use a shooting method to avoid numerical problems
%% try it as a pure ODE
%% jbr, 7/11/01
%% first effort: consider dimensional problem
%% x1 = c_A
%% x2 = dc_A/dr
%% dx1/dr = x2
%% dx2/dr = -2/r x2 - p1 x1 exp(gamma beta (1-x1)/ 1 + beta(1-x1))
%% x1(0) = c0 
%% x2(0) = 0
%% in which
%% p1 = k / De
%% solve the IVP until x1=1, call this distance r1, then let
%% Phi^2 = p1*r1^2
%% Drawback is that we are solving the ODE a lot.
%% Based on Weisz, Hicks, ChE Sci, 17, 265-275, 1962.
%% Finally, normalize Thiele modulus so that eta,Phiscale curves have 
%% common asymptote for large Phi.

more off;
global p1 Gamma beta
tol = 1e-6;
%% March ODE solver out to ca=1 stopping condition for a range of Gamma beta
%% values
Gamma   = 30;
betavec = [0.6; 0.4; 0.3; 0.2; 0.1; 0;  -0.8];
% betavec = [0.6];
% betavec = [0.2];
% betavec = [0.4];
% betavec = [0.3];
% betavec = [0];
% betavec = [-0.2];
% betavec = [-0.8];
nbeta   = length(betavec);

%% store the data to plot some straight lines on the figure
pts = [0.001 1000 0.01 0.2;
       10 0.1  0.01 500 ];

loglog (pts(:,1), pts(:,2), pts(:,3), pts(:,4));
hold ('on');

t1 = logspace (-11, -3, 10);
t2 = logspace (-3, log10 (0.5), 15);
t3 = 1 - logspace (log10 (0.5), -3, 15);
for i = 1:nbeta
  beta = betavec(i);
  [value, nfun] = quadl (@integrand, 0, 1, tol);
  intercept = sqrt(2*value);
  switch beta
    case 0.6
      rstop = 3;
      c0vec = [1e-295, 1e-125, 1e-50, 1e-25, t1, t2, t3];
    case 0.4
      rstop = 10;
      c0vec = [1e-200, 1e-100, 1e-50, 1e-25, t1, t2, t3];
    case 0.3
      rstop = 12;
      c0vec = [1e-100, 1e-50, 1e-25, t1, t2,  t3];
    case 0.2
      rstop = 15;
      c0vec = [1e-75, 1e-50, 1e-25, t1, t2,  t3];
    case 0.1
      rstop = 50;
      c0vec = [1e-50, 1e-25, t1, t2, t3];
    case 0
      rstop = 100;
      c0vec = [t1, t2, t3];
    case -0.8
      rstop = 500;
      c0vec = [logspace(-3,log10(0.3),15), 1-logspace(log10(0.3),-4,15)];
      warning ('invalid value of beta: %f', beta);

  nc0   = length(c0vec);
  clear Phivec Phiscale etavec
  Phivec = zeros (nc0, 1);
  Phiscale = zeros (nc0, 1);
  etavec = zeros (nc0, 1);
  for j = 1: nc0
    c0 = c0vec(j);
    opts = odeset ('RelTol', 1e-9, 'AbsTol', 1e-5*c0, 'Events', @g);
%     r0=1e-10;
    rtotal= rstop*linspace(r0,1,100)';
%    rtotal=[logspace(-10,log10(0.5),10), (1-logspace(log10(0.5),-3,10)), 1]';
%    rtotal=rstop*[ logspace(-10,-3,10), linspace(.001,1,50) ]';
    [rout,x] = ode15s (@weiszhicksode, rsteps, x0, opts);
    if (length (rout) == npts)
      fprintf ('hey, did not reach ca=1, increase rstop\n');
    nout = length(rout);
    cbar = x(:,1);
    a    = rout(nout)/3;
    rbar = rout/a;
%    plot (rbar,cbar);
    Phivec(j) = sqrt(p1)*a;
    Phiscale(j) = Phivec(j)/intercept;
    etavec(j) = a*x(nout,2)/Phivec(j)^2;
  results = [Phivec Phiscale etavec];
  save ('-ascii', sprintf ('wh_%d.dat', i), 'results');
  axis([1e-4, 10, 0.1, 1e3])
  loglog (results(:,2), results(:,3));
title ('Figure 7.19')
hold ('off');


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


function xdot = weiszhicksode(t,x)
  global p1 Gamma beta
  ca    = x(1);
  dcadr = x(2);
  xdot = zeros (2, 1);
  xdot(1) = dcadr;
  if (t == 0)
    xdot(2) = 1/3* p1*ca*exp(Gamma*beta*(1-ca)/(1+beta*(1-ca)));
    xdot(2) = -2/t*dcadr + p1*ca*exp(Gamma*beta*(1-ca)/(1+beta*(1-ca))); 


function [stp, isterminal, direction] = g(t,x)
  ca = x(1);
  stp = 1 - ca;

  isterminal = 1;
  direction = 0;

Data files


 8.29808752e-01 1.71117668e-02 5.76136200e+01
 3.58267028e-01 7.38794551e-03 1.30905799e+02
 1.49743792e-01 3.08791737e-03 2.98066905e+02
 8.00200749e-02 1.65012102e-03 5.13928291e+02
 4.11103671e-02 8.47750779e-04 8.11637600e+02
 3.86860385e-02 7.97757879e-04 8.34358478e+02
 3.62819587e-02 7.48182536e-04 8.54977656e+02
 3.39070846e-02 6.99209455e-04 8.71540499e+02
 3.15758140e-02 6.51135535e-04 8.80889432e+02
 2.93126137e-02 6.04465318e-04 8.77882327e+02
 2.71621569e-02 5.60120020e-04 8.54096086e+02
 2.52141941e-02 5.19950421e-04 7.95841682e+02
 2.36744578e-02 4.88198998e-04 6.82123156e+02
 2.31127276e-02 4.76615371e-04 4.88389662e+02
 2.31127276e-02 4.76615371e-04 4.88389662e+02
 2.32571126e-02 4.79592782e-04 4.34942524e+02
 2.35645232e-02 4.85931999e-04 3.78270236e+02
 2.40911943e-02 4.96792662e-04 3.19451051e+02
 2.49187998e-02 5.13858994e-04 2.60094704e+02
 2.61681515e-02 5.39622298e-04 2.02381693e+02
 2.80202811e-02 5.77815689e-04 1.48967892e+02
 3.07456603e-02 6.34016656e-04 1.02662983e+02
 3.47315642e-02 7.16211326e-04 6.58210684e+01
 4.04564714e-02 8.34266575e-04 3.95449817e+01
 4.82645834e-02 9.95280292e-04 2.31110349e+01
 5.78146985e-02 1.19221644e-03 1.41781376e+01
 6.78398921e-02 1.39894935e-03 9.87914622e+00
 7.77674884e-02 1.60366967e-03 7.88545018e+00
 9.21204213e-02 1.89964635e-03 6.36270909e+00
 9.21204213e-02 1.89964635e-03 6.36270909e+00
 1.15172069e-01 2.37500217e-03 4.43032909e+00
 1.36687386e-01 2.81867681e-03 3.00796283e+00
 1.49128953e-01 3.07523856e-03 2.16373647e+00
 1.49612359e-01 3.08520703e-03 1.68762995e+00
 1.40153258e-01 2.89014772e-03 1.41558410e+00
 1.24785720e-01 2.57324853e-03 1.25605319e+00
 1.07211539e-01 2.21084540e-03 1.15984409e+00
 8.99166535e-02 1.85420171e-03 1.10066192e+00
 7.41926185e-02 1.52995107e-03 1.06389256e+00
 6.05857349e-02 1.24935892e-03 1.04057233e+00
 4.91348411e-02 1.01322617e-03 1.02575627e+00
 3.96730334e-02 8.18111035e-04 1.01647036e+00
 3.19423860e-02 6.58694741e-04 1.01062042e+00
 2.56706446e-02 5.29363040e-04 1.00686677e+00


 2.15850373e+00 1.45668593e-01 6.74658645e+00
 1.09914726e+00 7.41769554e-02 1.30231951e+01
 5.68215326e-01 3.83465294e-02 2.43503152e+01
 3.01818700e-01 2.03685102e-02 4.28743966e+01
 1.52416856e-01 1.02859905e-02 7.18043154e+01
 1.43002879e-01 9.65067971e-03 7.45573213e+01
 1.33627253e-01 9.01795705e-03 7.73419407e+01
 1.24308371e-01 8.38906376e-03 8.00589992e+01
 1.15075659e-01 7.76598578e-03 8.25315050e+01
 1.05978558e-01 7.15205966e-03 8.44451544e+01
 9.71054980e-02 6.55325308e-03 8.52391050e+01
 8.86281135e-02 5.98114906e-03 8.39061060e+01
 8.09177795e-02 5.46081014e-03 7.86540330e+01
 7.49019809e-02 5.05482848e-03 6.65524910e+01
 7.49019809e-02 5.05482848e-03 6.65524910e+01
 7.40198866e-02 4.99529955e-03 6.26794635e+01
 7.33758246e-02 4.95183444e-03 5.83050002e+01
 7.30336538e-02 4.92874273e-03 5.34338743e+01
 7.30775841e-02 4.93170740e-03 4.81033512e+01
 7.36186859e-02 4.96822415e-03 4.23963851e+01
 7.48024675e-02 5.04811273e-03 3.64538357e+01
 7.68160767e-02 5.18400299e-03 3.04803694e+01
 7.98907334e-02 5.39149900e-03 2.47365615e+01
 8.42927201e-02 5.68857108e-03 1.95052578e+01
 9.02978905e-02 6.09383549e-03 1.50307649e+01
 9.81773052e-02 6.62558497e-03 1.14440331e+01
 1.08351122e-01 7.31217428e-03 8.70166916e+00
 1.22177298e-01 8.24524638e-03 6.56228756e+00
 1.44589818e-01 9.75777572e-03 4.58063915e+00
 1.44589818e-01 9.75777572e-03 4.58063915e+00
 1.71196663e-01 1.15533629e-02 3.02771952e+00
 1.86170167e-01 1.25638635e-02 2.17053717e+00
 1.86544310e-01 1.25891129e-02 1.69191742e+00
 1.74723203e-01 1.17913547e-02 1.41855192e+00
 1.55600257e-01 1.05008253e-02 1.25795041e+00
 1.33726824e-01 9.02467665e-03 1.16106754e+00
 1.12184063e-01 7.57084381e-03 1.10153707e+00
 9.25990116e-02 6.24912874e-03 1.06441529e+00
 7.56331392e-02 5.10417137e-03 1.04099142e+00
 6.13520272e-02 4.14039750e-03 1.02621004e+00
 4.95446909e-02 3.34356865e-03 1.01675021e+00
 3.98916182e-02 2.69212223e-03 1.01061870e+00
 3.20576543e-02 2.16344003e-03 1.00668067e+00
 2.57308937e-02 1.73647283e-03 1.00414561e+00


 2.50043019e+00 3.32534364e-01 2.92180260e+00
 1.28968203e+00 1.71515923e-01 5.50730288e+00
 6.81835094e-01 9.06778365e-02 9.85770534e+00
 3.39895582e-01 4.52030063e-02 1.72607338e+01
 3.18208210e-01 4.23187839e-02 1.80552777e+01
 2.96557816e-01 3.94394794e-02 1.88980321e+01
 2.74967193e-01 3.65681238e-02 1.97799331e+01
 2.53473328e-01 3.37096362e-02 2.06797738e+01
 2.32138881e-01 3.08723497e-02 2.15531573e+01
 2.11076072e-01 2.80711887e-02 2.23106960e+01
 1.90501211e-01 2.53349202e-02 2.27739907e+01
 1.70873657e-01 2.27246348e-02 2.25870126e+01
 1.53298024e-01 2.03872362e-02 2.10578710e+01
 1.53298024e-01 2.03872362e-02 2.10578710e+01
 1.49983970e-01 1.99464974e-02 2.04442110e+01
 1.46942647e-01 1.95420292e-02 1.97004776e+01
 1.44240916e-01 1.91827236e-02 1.88141546e+01
 1.41964413e-01 1.88799695e-02 1.77757603e+01
 1.40222938e-01 1.86483693e-02 1.65816087e+01
 1.39156637e-01 1.85065610e-02 1.52372393e+01
 1.38943137e-01 1.84781674e-02 1.37616174e+01
 1.39804747e-01 1.85927537e-02 1.21901578e+01
 1.42017275e-01 1.88869998e-02 1.05751171e+01
 1.45926100e-01 1.94068377e-02 8.98022592e+00
 1.52000356e-01 2.02146583e-02 7.46551617e+00
 1.61017729e-01 2.14138864e-02 6.06308863e+00
 1.74631170e-01 2.32243497e-02 4.74429643e+00
 1.96496339e-01 2.61322173e-02 3.39230080e+00
 1.96496339e-01 2.61322173e-02 3.39230080e+00
 2.16981536e-01 2.88565613e-02 2.35620091e+00
 2.21166780e-01 2.94131605e-02 1.79632744e+00
 2.10292817e-01 2.79670228e-02 1.47968327e+00
 1.89462188e-01 2.51967395e-02 1.29463800e+00
 1.64184555e-01 2.18350454e-02 1.18363030e+00
 1.38522770e-01 1.84222625e-02 1.11549186e+00
 1.14779869e-01 1.52646736e-02 1.07311769e+00
 9.39798002e-02 1.24984546e-02 1.04656068e+00
 7.63581562e-02 1.01549370e-02 1.02969522e+00
 6.17202554e-02 8.20822993e-03 1.01898429e+00
 4.97305073e-02 6.61370299e-03 1.01219164e+00
 3.99867998e-02 5.31787895e-03 1.00789362e+00
 3.21073651e-02 4.26998613e-03 1.00491291e+00
 2.57562892e-02 3.42535108e-03 1.00307621e+00


 4.93845513e+00 1.34827468e+00 7.21205444e-01
 3.35327926e+00 9.15497136e-01 1.04775363e+00
 1.76204119e+00 4.81064514e-01 1.91606127e+00
 8.64158598e-01 2.35928672e-01 3.54845302e+00
 8.06856185e-01 2.20284226e-01 3.74548353e+00
 7.49528152e-01 2.04632786e-01 3.96313766e+00
 6.92190444e-01 1.88978704e-01 4.20367956e+00
 6.34872721e-01 1.73330078e-01 4.46888998e+00
 5.77630450e-01 1.57702053e-01 4.75899530e+00
 5.20570470e-01 1.42123795e-01 5.07018151e+00
 4.63909873e-01 1.26654575e-01 5.38871449e+00
 4.08126670e-01 1.11424897e-01 5.67661155e+00
 3.54392030e-01 9.67545089e-02 5.83627756e+00
 3.54392030e-01 9.67545089e-02 5.83627756e+00
 3.43249406e-01 9.37124002e-02 5.83504828e+00
 3.32389072e-01 9.07473609e-02 5.81395958e+00
 3.21880945e-01 8.78784797e-02 5.76869576e+00
 3.11813312e-01 8.51298600e-02 5.69434122e+00
 3.02298765e-01 8.25322413e-02 5.58548494e+00
 2.93480081e-01 8.01246042e-02 5.43646082e+00
 2.85538489e-01 7.79564267e-02 5.24178098e+00
 2.78703017e-01 7.60902369e-02 4.99675434e+00
 2.73264016e-01 7.46053056e-02 4.69815713e+00
 2.69594209e-01 7.36033916e-02 4.34460551e+00
 2.68186355e-01 7.32190256e-02 3.93577420e+00
 2.69711518e-01 7.36354186e-02 3.46893792e+00
 2.75001465e-01 7.50796560e-02 2.93122058e+00
 2.83523350e-01 7.74062625e-02 2.28909313e+00
 2.83523350e-01 7.74062625e-02 2.28909313e+00
 2.82829463e-01 7.72168207e-02 1.75966323e+00
 2.66066212e-01 7.26401937e-02 1.46061232e+00
 2.38363519e-01 6.50769298e-02 1.28433982e+00
 2.05931310e-01 5.62224346e-02 1.17790275e+00
 1.73428906e-01 4.73487753e-02 1.11207580e+00
 1.43537948e-01 3.91880814e-02 1.07111721e+00
 1.17465039e-01 3.20697737e-02 1.04536358e+00
 9.53866806e-02 2.60420401e-02 1.02889902e+00
 7.70887345e-02 2.10464176e-02 1.01849270e+00
 6.20992296e-02 1.69540508e-02 1.01183529e+00
 4.99244227e-02 1.36301400e-02 1.00760795e+00
 4.00842131e-02 1.09436105e-02 1.00489602e+00
 3.21557088e-02 8.77900616e-03 1.00306680e+00
 2.57804345e-02 7.03845761e-03 1.00187796e+00


 1.03686150e+01 5.73940079e+00 1.70188857e-01
 5.40773143e+00 2.99337356e+00 3.19150149e-01
 2.60139625e+00 1.43996625e+00 6.29465110e-01
 2.42142928e+00 1.34034806e+00 6.70967972e-01
 2.24109354e+00 1.24052576e+00 7.18303486e-01
 2.06034205e+00 1.14047332e+00 7.72761431e-01
 1.87912201e+00 1.04016152e+00 8.36017321e-01
 1.69737782e+00 9.39559580e-01 9.10264167e-01
 1.51506150e+00 8.38640953e-01 9.98368756e-01
 1.33216564e+00 7.37401522e-01 1.10398343e+00
 1.14882754e+00 6.35917303e-01 1.23129065e+00
 9.65675563e-01 5.34536104e-01 1.38298958e+00
 9.65675563e-01 5.34536104e-01 1.38298958e+00
 9.26139275e-01 5.12651349e-01 1.41877825e+00
 8.86743541e-01 4.90844396e-01 1.45523264e+00
 8.47541853e-01 4.69144854e-01 1.49200192e+00
 8.08602113e-01 4.47590309e-01 1.52857183e+00
 7.70011409e-01 4.26228968e-01 1.56419871e+00
 7.31876736e-01 4.05120057e-01 1.59783017e+00
 6.94327122e-01 3.84334997e-01 1.62798851e+00
 6.57506948e-01 3.63953709e-01 1.65262691e+00
 6.21563524e-01 3.44057733e-01 1.66890406e+00
 5.86598807e-01 3.24703506e-01 1.67288571e+00
 5.52552262e-01 3.05857521e-01 1.65900282e+00
 5.18857036e-01 2.87206003e-01 1.61916334e+00
 4.83280788e-01 2.67513272e-01 1.54087146e+00
 4.37066456e-01 2.41931980e-01 1.40426475e+00
 4.37066456e-01 2.41931980e-01 1.40426475e+00
 3.79919101e-01 2.10298867e-01 1.26035771e+00
 3.24332418e-01 1.79529641e-01 1.16666798e+00
 2.71722714e-01 1.50408280e-01 1.10657512e+00
 2.24366200e-01 1.24194750e-01 1.06813588e+00
 1.83369525e-01 1.01501617e-01 1.04359211e+00
 1.48819199e-01 8.23767713e-02 1.02794931e+00
 1.20217596e-01 6.65447566e-02 1.01787222e+00
 9.68211470e-02 5.35939820e-02 1.01141143e+00
 7.78279466e-02 4.30805635e-02 1.00731726e+00
 6.24817856e-02 3.45859123e-02 1.00472598e+00
 5.01199541e-02 2.77431946e-02 1.00305503e+00
 4.01823538e-02 2.22423759e-02 1.00195987e+00
 3.22040900e-02 1.78261204e-02 1.00122508e+00
 2.58040980e-02 1.42834949e-02 1.00075022e+00


 9.80089122e+00 9.80089122e+00 9.85613926e-02
 9.09367921e+00 9.09367921e+00 1.05935615e-01
 8.38436133e+00 8.38436133e+00 1.14527918e-01
 7.67254043e+00 7.67254043e+00 1.24672530e-01
 6.95769280e+00 6.95769280e+00 1.36840103e-01
 6.23910898e+00 6.23910898e+00 1.51716146e-01
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