Figure 7.11 (page 375):

Dimensionless concentration versus radius for zero-order reaction in a spherical pellet.

Code for Figure 7.11

Text of the GNU GPL.

main.m

%% 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.

%%
%% jbr modifications to run on telemark; seems to have some parameters
%% set awfully high. Probably works on hazel with some better precision
%% than telemark.
%%
%% 6/4/00
%%

%% The pellet problem - Explanation of the structure of this program
%% This program computes the effectiveness factors for different thiele
%% moduli , for any order of the reaction and for the cases of
%% slab,cylinder and sphere(q=0,1,2).

%% The orders can be can be categorised into order=1 ,order<1 and
%% order>1. 

%% For order<1 , the problem is solved in two stages
%% Stage 1. 
%% The same intitial value problem approach as explained abouve is used,
%% but  here we march upto t=large no.This would yield all values of 
%% thiele  modulus for which the concentration at the centre is positive
%% Stage 2
%% Here the discontinuity in the concentration profile is
%% accomodated. We slightly modify the above initial value problem
%% approach. In this we start at a radius t=start, with the initial
%% conditions as c0=0 and a very small derivative, then we march forward
%% upto t=start+tsteps. The profile is rescaled to satisfy the outer
%% boundary condition c((q+1)*(start+tsteps)=1). Unlike the previous
%% case, the values of thiele modulus decreases for increase in
%% tsteps.The required solution is [Stage1;Stage2]

%% Limitations:

%% At extremely high thiele values (i.e >10000) for  order >1, the k and
%% c0 values chosen here will have to be changed suitably so that the
%% sensitivity of concentration w.r.t tsteps, is within the limit of
%% ode solver tolerances.
%% If one needs extremely high thiele modulus for order < 1, then ,the
%% initial derivative needs to be reduced and also the tsteps in stage 2
%% need to be very small.
%% Order =1,q=1 will not be able to yield very high thiele values. This
%% is because the thiele modulus is only proportional to tsteps, but
%% overflows occur at high tsteps.


%% This derivative program cvsrnsmall.m calculates the concentration
%% profile for n<1 to show that an inner sphere of the pellet has
%% exactly zero concentration of A. That is a result of the order of the
%% reaction being less than one.

global order k q thiele

%% Choose the orders n, geometry (q=0,1,2 for slab,cylinder,sphere) and
%% the thiele modulus upto which you require eta values.(If you do not
%% have an order=1, q=1 case then you can choose values of thiele
%% modulus to be as high as 1000, else you will have to have thiele to
%% be not greater than 100.

order=0;
q=2;

%%
%% The thiele values are chosen to so that for given order and geometry,
%% first one has nonzero center concentration, second one has zero
%% concentration right at center, and remaining ones have zero
%% conentration at finite radius.
%%
thielevec=[0.4; 0.57735; 0.8; 10];
%%
%% These are constants chosen arbitrarily
%% k     rate constant/diffusivity
%% c0    initial concentration at the centre
%% der   initial derivative for the case of dry regions in order <1
%% start the radius from which we start to march in the case of order<1 dry

k=2;
c0=0.01;
der=1e-8;
start=1;
tab = [];
%%
%% Stopping Critierion 
%%
%%
%% ODE's
%%
%%
%% Main program
%%
%%
%% Stage 1
%%  
for i = 1:2
  thiele = thielevec(i);
  tout=[0;logspace(-2,5,400)'];
  x0=[c0;0];
  y=k*c0^order;
  opts = odeset ('Events', @g, 'AbsTol', sqrt (eps), 'RelTol', sqrt (eps));
  [tsteps,x] = ode15s (@f, tout, x0, opts);
  phi = sqrt((order+1)/2.*x(:,1).^(order-1)*k).*(tsteps)/(q+1);
  eta = x(:,2)./ (k*(tsteps).*x(:,1).^order) *(q+1);
  phi(1) =0;
  eta(1) = 1.0;
  %%
  %% scale radius and concentration and save results to a numbered file
  %%
  table1 = [ tsteps./tsteps(end)*(q+1) ...
	    x(:,1)./x(end,1)]; 
  save ('-ascii', sprintf ('cvsrnsmall_%d.dat', i), 'table1');
  plot (table1(:,1), table1(:,2));
  hold on;
end
%%
%% Stage 2
%%
for i = 3:4
  thiele = thielevec(i);
  t0=start;
  nfine = 200;
  ncoarse = 100;
  fine = logspace(log10(start),log10(start+1),nfine)';
  coarse = logspace(log10(start+2),log10(40),ncoarse)';
  tout2 = [fine;  coarse(2:ncoarse)];
  c0 = 0;
  x0=[c0;der];
  y= -q/t0*der+k*c0^order;
  opts = odeset ('Events', @g, 'AbsTol', sqrt (eps), 'RelTol', sqrt (eps));
  [tsteps2,x2] = ode15s (@f, tout2, x0, opts);
  phidry = sqrt((order+1)/2.*x2(:,1).^(order-1)*k).*(tsteps2)/(q+1);
  etadry = x2(:,2)./ (k*(tsteps2).*x2(:,1).^order) *(q+1);
  %%
  %% scale radius and concentration and save results to a numbered file
  %%
  table2 = [tsteps2./tsteps2(end)*(q+1) ...
	    x2(:,1)./x2(end,1)];
  save ('-ascii', sprintf ('cvsrnsmall_%d.dat', i), 'table2');
  plot (table2(:,1), table2(:,2));
title ('Figure 7.11')
end

hold off

g.m

function [stp, isterminal, direction] = g(t,x)
  global order k q thiele
  if (x(1) == 0)
    pelthiele = Inf;
  else
    pelthiele = sqrt((order+1)/2*x(1)^(order-1)*k)*t/(q+1);
  end
  %% stop at each desired thiele modulus
  stp = pelthiele-thiele;

  isterminal = 1;
  direction = 0;

f.m

function xdot=f(t,x)
  global order q k
  xdot = zeros (2, 1);
  xdot(1) = x(2);
  if t==0
    xdot(2) = k*x(1)^order;
  else
    xdot(2) = -q/t*x(2)+k*x(1)^order;
  end

Data files

cvsrnsmall_1.dat

 0.00000000e+00 5.19997418e-01
 1.80277116e-01 5.21733289e-01
 1.87708726e-01 5.21879142e-01
 1.95446691e-01 5.22037267e-01
 2.03503640e-01 5.22208698e-01
 2.11892722e-01 5.22394555e-01
 2.20627630e-01 5.22596050e-01
 2.29722620e-01 5.22814500e-01
 2.39192534e-01 5.23051332e-01
 2.49052829e-01 5.23308092e-01
 2.59319598e-01 5.23586458e-01
 2.70009596e-01 5.23888246e-01
 2.81140271e-01 5.24215429e-01
 2.92729788e-01 5.24570143e-01
 3.04797063e-01 5.24954705e-01
 3.17361790e-01 5.25371626e-01
 3.30444477e-01 5.25823629e-01
 3.44066474e-01 5.26313666e-01
 3.58250014e-01 5.26844937e-01
 3.73018246e-01 5.27420913e-01
 3.88395273e-01 5.28045354e-01
 4.04406191e-01 5.28722340e-01
 4.21077131e-01 5.29456292e-01
 4.38435301e-01 5.30252003e-01
 4.56509032e-01 5.31114669e-01
 4.75327821e-01 5.32049925e-01
 4.94922381e-01 5.33063879e-01
 5.15324693e-01 5.34163153e-01
 5.36568055e-01 5.35354926e-01
 5.58687137e-01 5.36646982e-01
 5.81718040e-01 5.38047759e-01
 6.05698352e-01 5.39566406e-01
 6.30667210e-01 5.41212840e-01
 6.56665366e-01 5.42997815e-01
 6.83735250e-01 5.44932989e-01
 7.11921044e-01 5.47030999e-01
 7.41268747e-01 5.49305548e-01
 7.71826258e-01 5.51771490e-01
 8.03643450e-01 5.54444932e-01
 8.36772250e-01 5.57343333e-01
 8.71266726e-01 5.60485622e-01
 9.07183178e-01 5.63892322e-01
 9.44580222e-01 5.67585681e-01
 9.83518894e-01 5.71589821e-01
 1.02406275e+00 5.75930894e-01
 1.06627795e+00 5.80637249e-01
 1.11023340e+00 5.85739624e-01
 1.15600083e+00 5.91271344e-01
 1.20365495e+00 5.97268534e-01
 1.25327353e+00 6.03770364e-01
 1.30493754e+00 6.10819295e-01
 1.35873131e+00 6.18461365e-01
 1.41474264e+00 6.26746483e-01
 1.47306294e+00 6.35728760e-01
 1.53378739e+00 6.45466858e-01
 1.59701510e+00 6.56024378e-01
 1.66284927e+00 6.67470268e-01
 1.73139733e+00 6.79879284e-01
 1.80277116e+00 6.93332466e-01
 1.87708726e+00 7.07917678e-01
 1.95446691e+00 7.23730176e-01
 2.03503640e+00 7.40873229e-01
 2.11892722e+00 7.59458800e-01
 2.20627630e+00 7.79608270e-01
 2.29722620e+00 8.01453234e-01
 2.39192534e+00 8.25136362e-01
 2.49052829e+00 8.50812327e-01
 2.59319598e+00 8.78648818e-01
 2.70009596e+00 9.08827635e-01
 2.81140271e+00 9.41545875e-01
 2.92729788e+00 9.77017220e-01
 3.00000000e+00 1.00000000e+00

cvsrnsmall_2.dat

 0.00000000e+00 9.32495140e-07
 1.67256929e-04 9.35608023e-07
 1.74151804e-04 9.35869576e-07
 1.81330909e-04 9.36153138e-07
 1.88805959e-04 9.36460560e-07
 1.96589155e-04 9.36793850e-07
 2.04693201e-04 9.37155185e-07
 2.13131321e-04 9.37546925e-07
 2.21917288e-04 9.37971628e-07
 2.31065441e-04 9.38432068e-07
 2.40590711e-04 9.38931252e-07
 2.50508643e-04 9.39472441e-07
 2.60835425e-04 9.40059168e-07
 2.71587911e-04 9.40695265e-07
 2.82783649e-04 9.41384887e-07
 2.94440911e-04 9.42132538e-07
 3.06578725e-04 9.42943101e-07
 3.19216898e-04 9.43821869e-07
 3.32376058e-04 9.44774582e-07
 3.46077681e-04 9.45807461e-07
 3.60344130e-04 9.46927253e-07
 3.75198688e-04 9.48141271e-07
 3.90665600e-04 9.49457443e-07
 4.06770108e-04 9.50884366e-07
 4.23538496e-04 9.52431359e-07
 4.40998132e-04 9.54108524e-07
 4.59177511e-04 9.55926816e-07
 4.78106303e-04 9.57898110e-07
 4.97815402e-04 9.60035280e-07
 5.18336973e-04 9.62352283e-07
 5.39704511e-04 9.64864253e-07
 5.61952888e-04 9.67587595e-07
 5.85118416e-04 9.70540094e-07
 6.09238902e-04 9.73741034e-07
 6.34353714e-04 9.77211320e-07
 6.60503839e-04 9.80973616e-07
 6.87731958e-04 9.85052494e-07
 7.16082509e-04 9.89474592e-07
 7.45601761e-04 9.94268791e-07
 7.76337894e-04 9.99466403e-07
 8.08341070e-04 1.00510137e-06
 8.41663520e-04 1.01121050e-06
 8.76359631e-04 1.01783369e-06
 9.12486029e-04 1.02501419e-06
 9.50101674e-04 1.03279890e-06
 9.89267958e-04 1.04123866e-06
 1.03004880e-03 1.05038859e-06
 1.07251077e-03 1.06030845e-06
 1.11672316e-03 1.07106302e-06
 1.16275812e-03 1.08272255e-06
 1.21069080e-03 1.09536318e-06
 1.26059941e-03 1.10906746e-06
 1.31256543e-03 1.12392491e-06
 1.36667365e-03 1.14003254e-06
 1.42301238e-03 1.15749557e-06
 1.48167359e-03 1.17642804e-06
 1.54275300e-03 1.19695360e-06
 1.60635030e-03 1.21920630e-06
 1.67256929e-03 1.24333147e-06
 1.74151804e-03 1.26948668e-06
 1.81330909e-03 1.29784273e-06
 1.88805959e-03 1.32858484e-06
 1.96589155e-03 1.36191376e-06
 2.04693201e-03 1.39804718e-06
 2.13131321e-03 1.43722107e-06
 2.21917288e-03 1.47969128e-06
 2.31065441e-03 1.52573519e-06
 2.40590711e-03 1.57565350e-06
 2.50508643e-03 1.62977223e-06
 2.60835425e-03 1.68844483e-06
 2.71587911e-03 1.75205449e-06
 2.82783649e-03 1.82101663e-06
 2.94440911e-03 1.89578165e-06
 3.06578725e-03 1.97683785e-06
 3.19216898e-03 2.06471458e-06
 3.32376058e-03 2.15998578e-06
 3.46077681e-03 2.26327366e-06
 3.60344130e-03 2.37525279e-06
 3.75198688e-03 2.49665449e-06
 3.90665600e-03 2.62827165e-06
 4.06770108e-03 2.77096385e-06
 4.23538496e-03 2.92566301e-06
 4.40998132e-03 3.09337946e-06
 4.59177511e-03 3.27520856e-06
 4.78106303e-03 3.47233783e-06
 4.97815402e-03 3.68605471e-06
 5.18336973e-03 3.91775498e-06
 5.39704511e-03 4.16895187e-06
 5.61952888e-03 4.44128595e-06
 5.85118416e-03 4.73653581e-06
 6.09238902e-03 5.05662972e-06
 6.34353714e-03 5.40365822e-06
 6.60503839e-03 5.77988774e-06
 6.87731958e-03 6.18777541e-06
 7.16082509e-03 6.62998514e-06
 7.45601761e-03 7.10940500e-06
 7.76337894e-03 7.62916606e-06
 8.08341070e-03 8.19266286e-06
 8.41663520e-03 8.80357559e-06
 8.76359631e-03 9.46589410e-06
 9.12486029e-03 1.01839440e-05
 9.50101674e-03 1.09624148e-05
 9.89267958e-03 1.18063907e-05
 1.03004880e-02 1.27213837e-05
 1.07251077e-02 1.37133696e-05
 1.11672316e-02 1.47888270e-05
 1.16275812e-02 1.59547797e-05
 1.21069080e-02 1.72188425e-05
 1.26059941e-02 1.85892709e-05
 1.31256543e-02 2.00750153e-05
 1.36667365e-02 2.16857789e-05
 1.42301238e-02 2.34320816e-05
 1.48167359e-02 2.53253284e-05
 1.54275300e-02 2.73778842e-05
 1.60635030e-02 2.96031540e-05
 1.67256929e-02 3.20156711e-05
 1.74151804e-02 3.46311916e-05
 1.81330909e-02 3.74667973e-05
 1.88805959e-02 4.05410075e-05
 1.96589155e-02 4.38738998e-05
 2.04693201e-02 4.74872412e-05
 2.13131321e-02 5.14046304e-05
 2.21917288e-02 5.56516517e-05
 2.31065441e-02 6.02560423e-05
 2.40590711e-02 6.52478733e-05
 2.50508643e-02 7.06597463e-05
 2.60835425e-02 7.65270060e-05
 2.71587911e-02 8.28879714e-05
 2.82783649e-02 8.97841858e-05
 2.94440911e-02 9.72606881e-05
 3.06578725e-02 1.05366307e-04
 3.19216898e-02 1.14153981e-04
 3.32376058e-02 1.23681101e-04
 3.46077681e-02 1.34009888e-04
 3.60344130e-02 1.45207801e-04
 3.75198688e-02 1.57347971e-04
 3.90665600e-02 1.70509687e-04
 4.06770108e-02 1.84778907e-04
 4.23538496e-02 2.00248823e-04
 4.40998132e-02 2.17020468e-04
 4.59177511e-02 2.35203378e-04
 4.78106303e-02 2.54916304e-04
 4.97815402e-02 2.76287992e-04
 5.18336973e-02 2.99458019e-04
 5.39704511e-02 3.24577708e-04
 5.61952888e-02 3.51811116e-04
 5.85118416e-02 3.81336102e-04
 6.09238902e-02 4.13345493e-04
 6.34353714e-02 4.48048343e-04
 6.60503839e-02 4.85671294e-04
 6.87731958e-02 5.26460061e-04
 7.16082509e-02 5.70681035e-04
 7.45601761e-02 6.18623020e-04
 7.76337894e-02 6.70599126e-04
 8.08341070e-02 7.26948806e-04
 8.41663520e-02 7.88040079e-04
 8.76359631e-02 8.54271930e-04
 9.12486029e-02 9.26076918e-04
 9.50101674e-02 1.00392400e-03
 9.89267958e-02 1.08832159e-03
 1.03004880e-01 1.17982089e-03
 1.07251077e-01 1.27901948e-03
 1.11672316e-01 1.38656522e-03
 1.16275812e-01 1.50316049e-03
 1.21069080e-01 1.62956677e-03
 1.26059941e-01 1.76660961e-03
 1.31256543e-01 1.91518405e-03
 1.36667365e-01 2.07626041e-03
 1.42301238e-01 2.25089068e-03
 1.48167359e-01 2.44021536e-03
 1.54275300e-01 2.64547094e-03
 1.60635030e-01 2.86799792e-03
 1.67256929e-01 3.10924963e-03
 1.74151804e-01 3.37080168e-03
 1.81330909e-01 3.65436225e-03
 1.88805959e-01 3.96178327e-03
 1.96589155e-01 4.29507250e-03
 2.04693201e-01 4.65640664e-03
 2.13131321e-01 5.04814556e-03
 2.21917288e-01 5.47284769e-03
 2.31065441e-01 5.93328674e-03
 2.40590711e-01 6.43246985e-03
 2.50508643e-01 6.97365715e-03
 2.60835425e-01 7.56038312e-03
 2.71587911e-01 8.19647966e-03
 2.82783649e-01 8.88610110e-03
 2.94440911e-01 9.63375133e-03
 3.06578725e-01 1.04443132e-02
 3.19216898e-01 1.13230806e-02
 3.32376058e-01 1.22757926e-02
 3.46077681e-01 1.33086713e-02
 3.60344130e-01 1.44284626e-02
 3.75198688e-01 1.56424797e-02
 3.90665600e-01 1.69586512e-02
 4.06770108e-01 1.83855732e-02
 4.23538496e-01 1.99325648e-02
 4.40998132e-01 2.16097293e-02
 4.59177511e-01 2.34280203e-02
 4.78106303e-01 2.53993129e-02
 4.97815402e-01 2.75364817e-02
 5.18336973e-01 2.98534844e-02
 5.39704511e-01 3.23654534e-02
 5.61952888e-01 3.50887941e-02
 5.85118416e-01 3.80412927e-02
 6.09238902e-01 4.12422319e-02
 6.34353714e-01 4.47125168e-02
 6.60503839e-01 4.84748119e-02
 6.87731958e-01 5.25536886e-02
 7.16082509e-01 5.69757860e-02
 7.45601761e-01 6.17699845e-02
 7.76337894e-01 6.69675951e-02
 8.08341070e-01 7.26025631e-02
 8.41663520e-01 7.87116904e-02
 8.76359631e-01 8.53348755e-02
 9.12486029e-01 9.25153743e-02
 9.50101674e-01 1.00300082e-01
 9.89267958e-01 1.08739842e-01
 1.03004880e+00 1.17889771e-01
 1.07251077e+00 1.27809630e-01
 1.11672316e+00 1.38564204e-01
 1.16275812e+00 1.50223731e-01
 1.21069080e+00 1.62864359e-01
 1.26059941e+00 1.76568644e-01
 1.31256543e+00 1.91426087e-01
 1.36667365e+00 2.07533723e-01
 1.42301238e+00 2.24996750e-01
 1.48167359e+00 2.43929219e-01
 1.54275300e+00 2.64454776e-01
 1.60635030e+00 2.86707475e-01
 1.67256929e+00 3.10832646e-01
 1.74151804e+00 3.36987850e-01
 1.81330909e+00 3.65343907e-01
 1.88805959e+00 3.96086010e-01
 1.96589155e+00 4.29414932e-01
 2.04693201e+00 4.65548347e-01
 2.13131321e+00 5.04722238e-01
 2.21917288e+00 5.47192451e-01
 2.31065441e+00 5.93236357e-01
 2.40590711e+00 6.43154667e-01
 2.50508643e+00 6.97273397e-01
 2.60835425e+00 7.55945995e-01
 2.71587911e+00 8.19555649e-01
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