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.


%% 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.

%% 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.


%% 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

tab = [];
%% Stopping Critierion 
%% ODE's
%% Main program
%% Stage 1
for i = 1:2
  thiele = thielevec(i);
  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) ...
  save ('-ascii', sprintf ('cvsrnsmall_%d.dat', i), 'table1');
  plot (table1(:,1), table1(:,2));
  hold on;
%% Stage 2
for i = 3:4
  thiele = thielevec(i);
  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;
  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) ...
  save ('-ascii', sprintf ('cvsrnsmall_%d.dat', i), 'table2');
  plot (table2(:,1), table2(:,2));
title ('Figure 7.11')

hold off


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

  isterminal = 1;
  direction = 0;


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;
    xdot(2) = -q/t*x(2)+k*x(1)^order;

Data files


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 1.80277116e-01 5.21733289e-01
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 1.95446691e-01 5.22037267e-01
 2.03503640e-01 5.22208698e-01
 2.11892722e-01 5.22394555e-01
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 2.29722620e-01 5.22814500e-01
 2.39192534e-01 5.23051332e-01
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 2.70009596e-01 5.23888246e-01
 2.81140271e-01 5.24215429e-01
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 3.04797063e-01 5.24954705e-01
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 3.00000000e+00 1.00000000e+00


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