Figure 7.9 (page 374):

Effectiveness factor versus Thiele modulus in a spherical pellet; reaction orders greater than unity.

Code for Figure 7.9

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 set up as an initial value problem. We
%% start from the centre of the pellet (t=0) with the inner boundary condition
%% (dc/dr=0) and some c0=constant, and start marching forward in the
%% radial direction upto t=tsteps. The profile is then rescaled to satisfy
%% the outer boundary condition (c((q+1)*tsteps)=1). For different
%% values of tsteps we have different thiele modulus. We use the ODE
%% solver with stopping condition 
%% (ex.. thiele modulus=100). The solver marches forward until the
%% stopping condition is satisfied.(The stopping condition helps to
%% avoid the problem of blowing up)

%% For order=1, q=0 or 2, the analytical solutions have been used
%% For order=1, q=1 the solution is computed just like the case of
%% order>1 (Analytical solution for this is actually available, but it
%% involves Modified Bessels function which limits the value of thiele
%% modulus that can be attained.

%% 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
%% the 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.
  
global order k q thiele

%% Choose the orders , 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.

ordervec=[1; 2; 5; 10];
q=2;
thiele=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-30;
der=1e-8;
start=1;

%% 
axis([0.1,thiele,1/thiele,1])
more off;
%%

%%
%% ODE's for order<1


%% Stopping Critierion for order>2


%% Main program

for j=1:size(ordervec,1)
  order=ordervec(j);

  if (order < 1)

    %% Stage 1

    tsteps=[0;logspace(-2,5,200)'];
    x0=[c0;0];
    ode_opts = odeset ('AbsTol', 1e-18, 'RelTol', sqrt (eps));
    [tsolver,x] = ode15s(@ivp,tsteps,x0,ode_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;

    %% Stage 2

%    tsteps=[linspace(start,start+2,3000)';linspace(start+2,10000000,10)'];
    tsteps=[linspace(start,start+2,30)';logspace(log10(start+2),log10(40),20)'];
    x0=[0;der];
    ode_opts = odeset ('AbsTol', 1e-18, 'RelTol', sqrt (eps));
    [tsolver,x] = ode15s(@ivp,tsteps,x0,ode_opts);
    phidry = sqrt((order+1)/2.*x(:,1).^(order-1)*k).*(tsteps+start)/(q+1);
    etadry = x(:,2)./ (k*(tsteps+start).*x(:,1).^order) *(q+1);
    tab =[phi , eta ;flipud(phidry(2:end)), ...
		 flipud(etadry(2:end))];
    
  elseif (order==1)
    phi=logspace(-2,2,250)';
    if q==0
      eta=tanh(phi)./phi;
    elseif q==1
      eta = besseli(2*phi,1)./phi./besseli(2*phi,1);
    elseif q==2
      eta=(1./phi).*((1./tanh(3*phi))-(1./(3*phi)));
    end
    
    tab=[phi eta];
    save ('-ascii', sprintf ('etaordern_large_%d.dat', j), 'tab');
    
  elseif (order>1)
    x_0=[c0;0];
    y=k*c0^order;
    xdot_0=[0;y];
    t_out=[0;logspace(-1,10,2000)'];

    ode_opts = odeset ('Events', @g, 'AbsTol', sqrt (eps), 'RelTol', sqrt (eps));
    [t,x] = ode15s (@ivp, t_out, x_0, ode_opts);
    
    phi = sqrt((order+1)/2.*x(:,1).^(order-1)*k).*t./(q+1);
    eta = x(:,2)./ (k*t.*x(:,1).^order) *(q+1);
    tab=[ phi(2:end,:) eta(2:end,:)];
    save ('-ascii', sprintf ('etaordern_large_%d.dat', j), 'tab');
  end
  
  hold ('on');
  loglog (tab(:,1), tab(:,2));
title ('Figure 7.9')

end

hold ('off');

ivp.m

function xdot = ivp(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

g.m

function [stp, isterminal, direction] = g(t,x)
  global order q k thiele
  stp=sqrt((order+1)/2*x(1)^(order-1)*k)*t/(q+1)-thiele;
  isterminal = 1;
  direction = 0;

Data files

etaordern_large_1.dat

 1.00000000e-02 9.99940005e-01
 1.03768194e-02 9.99935399e-01
 1.07678380e-02 9.99930439e-01
 1.11735910e-02 9.99925099e-01
 1.15946336e-02 9.99919348e-01
 1.20315418e-02 9.99913156e-01
 1.24849136e-02 9.99906489e-01
 1.29553694e-02 9.99899310e-01
 1.34435528e-02 9.99891579e-01
 1.39501319e-02 9.99883256e-01
 1.44757999e-02 9.99874293e-01
 1.50212760e-02 9.99864643e-01
 1.55873068e-02 9.99854252e-01
 1.61746667e-02 9.99843063e-01
 1.67841595e-02 9.99831016e-01
 1.74166192e-02 9.99818044e-01
 1.80729111e-02 9.99804077e-01
 1.87539334e-02 9.99789038e-01
 1.94606179e-02 9.99772844e-01
 2.01939317e-02 9.99755409e-01
 2.09548782e-02 9.99736635e-01
 2.17444986e-02 9.99716421e-01
 2.25638734e-02 9.99694656e-01
 2.34141239e-02 9.99671222e-01
 2.42964134e-02 9.99645990e-01
 2.52119493e-02 9.99618822e-01
 2.61619844e-02 9.99589571e-01
 2.71478187e-02 9.99558077e-01
 2.81708011e-02 9.99524167e-01
 2.92323314e-02 9.99487658e-01
 3.03338623e-02 9.99448349e-01
 3.14769010e-02 9.99406027e-01
 3.26630116e-02 9.99360461e-01
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 2.68979310e-01 9.59117634e-01
 2.79114972e-01 9.56173833e-01
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 1.22561315e+00 5.95056092e-01
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 1.58782711e+00 4.97670830e-01
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 1.84102735e+00 4.44845954e-01
 1.91040083e+00 4.32127945e-01
 1.98238843e+00 4.19628317e-01
 2.05708867e+00 4.07356003e-01
 2.13460376e+00 3.95318555e-01
 2.21503976e+00 3.83522232e-01
 2.29850675e+00 3.71972090e-01
 2.38511893e+00 3.60672070e-01
 2.47499484e+00 3.49625091e-01
 2.56825744e+00 3.38833135e-01
 2.66503435e+00 3.28297331e-01
 2.76545801e+00 3.18018041e-01
 2.86966582e+00 3.07994930e-01
 2.97780039e+00 2.98227037e-01
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 3.20644723e+00 2.79450347e-01
 3.32727237e+00 2.70437086e-01
 3.45265044e+00 2.61670224e-01
 3.58275299e+00 2.53146583e-01
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 1.08678736e+01 8.91921084e-02
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 1.26009012e+01 7.72600983e-02
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 1.35684463e+01 7.18898226e-02
 1.40797316e+01 6.93426076e-02
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 1.51608270e+01 6.45092457e-02
 1.57321163e+01 6.22174348e-02
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 1.69400880e+01 5.78699950e-02
 1.75784234e+01 5.58091781e-02
 1.82408124e+01 5.38202981e-02
 1.89281615e+01 5.19009495e-02
 1.96414113e+01 5.00487993e-02
 2.03815378e+01 4.82615856e-02
 2.11495536e+01 4.65371159e-02
 2.19465097e+01 4.48732654e-02
 2.27734967e+01 4.32679757e-02
 2.36316462e+01 4.17192528e-02
 2.45221324e+01 4.02251656e-02
 2.54461738e+01 3.87838447e-02
 2.64050350e+01 3.73934799e-02
 2.74000278e+01 3.60523197e-02
 2.84325140e+01 3.47586687e-02
 2.95039062e+01 3.35108868e-02
 3.06156705e+01 3.23073875e-02
 3.17693283e+01 3.11466359e-02
 3.29664581e+01 3.00271479e-02
 3.42086981e+01 2.89474884e-02
 3.54977481e+01 2.79062697e-02
 3.68353720e+01 2.69021507e-02
 3.82234002e+01 2.59338346e-02
 3.96637319e+01 2.50000685e-02
 4.11583382e+01 2.40996415e-02
 4.27092641e+01 2.32313835e-02
 4.43186319e+01 2.23941640e-02
 4.59886438e+01 2.15868908e-02
 4.77215850e+01 2.08085092e-02
 4.95198267e+01 2.00580001e-02
 5.13858297e+01 1.93343794e-02
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 6.18250760e+01 1.60874601e-02
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 6.65722404e+01 1.49460631e-02
 6.90808114e+01 1.44059503e-02
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 7.43850987e+01 1.33833097e-02
 7.71880733e+01 1.28994222e-02
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 9.28691533e+01 1.07291892e-02
 9.63686429e+01 1.03409266e-02
 1.00000000e+02 9.96666667e-03

etaordern_large_2.dat

 5.77360615e-03 9.99972860e-01
 5.84722884e-03 9.99972103e-01
 5.92179041e-03 9.99971319e-01
 5.99730282e-03 9.99970507e-01
 6.07377820e-03 9.99969666e-01
 6.15122884e-03 9.99968796e-01
 6.22966717e-03 9.99967895e-01
 6.30910582e-03 9.99967046e-01
 6.38955753e-03 9.99966181e-01
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 6.55355198e-03 9.99964373e-01
 6.63712105e-03 9.99963426e-01
 6.72175587e-03 9.99962450e-01
 6.80747003e-03 9.99961443e-01
 6.89427730e-03 9.99960403e-01
 6.98219161e-03 9.99959331e-01
 7.07122710e-03 9.99958223e-01
 7.16139807e-03 9.99957079e-01
 7.25271900e-03 9.99955898e-01
 7.34520456e-03 9.99954678e-01
 7.43886961e-03 9.99953417e-01
 7.53372920e-03 9.99952115e-01
 7.62979856e-03 9.99950768e-01
 7.72709314e-03 9.99949377e-01
 7.82562856e-03 9.99948026e-01
 7.92542066e-03 9.99946851e-01
 8.02648546e-03 9.99945660e-01
 8.12883922e-03 9.99944448e-01
 8.23249836e-03 9.99943216e-01
 8.33747955e-03 9.99941960e-01
 8.44379965e-03 9.99940680e-01
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 8.66052515e-03 9.99938037e-01
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 8.88281416e-03 9.99935270e-01
 8.99608950e-03 9.99933835e-01
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 1.02114249e-02 9.99917046e-01
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 1.04735288e-02 9.99913023e-01
 1.06070934e-02 9.99910910e-01
 1.07423617e-02 9.99908726e-01
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