Solving PDE with Matlab - matlab

Question

`sol = pdepe(m,#ParticleDiffusionpde,#ParticleDiffusionic,#ParticleDiffusionbc,x,t);
% Extract the first solution component as u.
u = sol(:,:,:);
function [c,f,s] = ParticleDiffusionpde(x,t,u,DuDx)
global Ds
c = 1/Ds;
f = DuDx;
s = 0;
function u0 = ParticleDiffusionic(x)
global qo
u0 = qo;
function [pl,ql,pr,qr] = ParticleDiffusionbc(xl,ul,xr,ur,t,x)
global Ds K n
global Amo Gc kf rhop
global uavg
global dr R nr
sum = 0;
for i = 1:1:nr-1
r1 = (i-1)*dr; % radius at i
r2 = i * dr; % radius at i+1
r1 = double(r1); % convert to double precision
r2 = double(r2);
sum = sum + (dr / 2 * (r1*ul+ r2*ur));
end;
uavg = 3/R^3 * sum;
ql = 1;
pl = 0;
qr = 1;
pr = -((kf/(Ds.*rhop)).*(Amo - Gc.*uavg - ((double(ur/K)).^2).^(n/2) ));`
dq(r,t)/dt = Ds( d2q(r,t)/dr2 + (2/r)*dq(r,t)/dr )
q(r, t=0) = 0
dq(r=0, t)/dr = 0
dq(r=dp/2, t)/dr = (kf/Ds*rhop) [C(t) - Cp(at r = dp/2)]
q = solid phase concentration of trace compound in a particle with radius dp/2
C = bulk liquid concentration of trace compound
Cp = trace compound concentration at particle surface
I want to solve the above pde with initial and boundary conditions given. Tried Matlab's pdepe, but does not work satisfactorily. Maybe the boundary conditions is creating problem for me. I also used this isotherm equation for equilibrium: q = K*Cp^(1/n). This is convection-diffusion equation but i could not find any write ups that addresses solving this type of equation properly.

Answer 1

There are two problems with the current implementation.
Incorrect Source Term
The PDE you are attempting to solve has the form
which has the equivalent form
where the last term arises due to the factor of 2 in the original PDE.
The last term needs to be incorporated into pdepe via a source term.
Calculation of q average
The current implementation attempts to calculate the average value of q using the left and right values of q passed to the boundary condition function.
This is incorrect.
The average value of q needs to be calculated from a vector of up-to-date values of the quantity.
However, we have the complication that the only function to receive all mesh values is ParticleDiffusionpde; however, the mesh values passed to that function are not guaranteed to be from the mesh we provided.
Solution: use events (as described in the pdepe documentation).
This is a hack since the event function is meant to detect zero-crossings, but it has the advantage that the function is given all values of q on the mesh we provide.
So, the working example below (you'll notice I set all of the parameters to 1 since I didn't know better) uses the events function to update a variable qStore that can be accessed by the boundary condition function (see here for an explanation), and the boundary condition function performs a vectorized trapezoidal integration for the average calculation.
Working Example
function [] = ParticleDiffusion()
% Parameters
Ds = 1;
q0 = 0;
K = 1;
n = 1;
Amo = 1;
Gc = 1;
kf = 1;
rhop = 1;
% Space
rMesh = linspace(0,1,10);
rMesh = rMesh(:) ;
dr = rMesh(2) - rMesh(1) ;
% Time
tSpan = linspace(0,1,10);
% Vector to store current u-value
qStore = zeros(size(rMesh));
options.Events = #(m,t,x,y) events(m,t,x,y);
% Solve
[sol,~,~,~,~] = pdepe(1,#ParticleDiffusionpde,#ParticleDiffusionic,#ParticleDiffusionbc,rMesh,tSpan,options);
% Use the events function to update qStore
function [value,isterminal,direction] = events(m,~,~,y)
qStore = y; % Value of q on rMesh
value = m; % Since m is constant, it will never be zero (no event detection)
isterminal = 0; % Continue integration
direction = 0; % Detect all zero crossings (not important)
end
function [c,f,s] = ParticleDiffusionpde(r,~,~,DqDr)
% Define the capacity, flux, and source
c = 1/Ds;
f = DqDr;
s = DqDr./r;
end
function u0 = ParticleDiffusionic(~)
u0 = q0;
end
function [pl,ql,pr,qr] = ParticleDiffusionbc(~,~,R,ur,~)
% Calculate average value of current solution
qL = qStore(1:end-1);
qR = qStore(2: end );
total = sum((qL.*rMesh(1:end-1) + qR.*rMesh(2:end))) * dr/2;
qavg = 3/R^3 * total;
% Left boundary
pl = 0;
ql = 1;
% Right boundary
qr = 1;
pr = -(kf/(Ds.*rhop)).*(Amo - Gc.*qavg - (ur/K).^n);
end
end