Question on implementing Danckwert's boundary condition

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Deepa Maheshvare
Deepa Maheshvare 2018-11-10
评论： Ngo Ich Son 2023-3-17
I have the following code, which solves for the change in concentration of a species along the length of a channel of length L . The equation is ,
$ \frac{\partial C}{\partial t} = -v\frac{\partial C}{\partial z} + D\frac{\partial^2C}{\partial z^2} - kC $
The spatial direction is discretized and the resulting form solved using ode solver.
I have used constant values of concentration at the right and left boundary of the cylindrical channel.
function Model()
Nodes = 10;
v = 10;D =50;k = 10;
%C0 contains the concentration of the species at time 0 at all the nodes
C0(1:10) = [2000;1996;1980;1973;1965;1950;1947;1938;1878;1800]';
C(1:8,1) = C0(2:9);
tspan = 1:10;
[t C] = ode15s(@(t,v) diff(t,v), tspan ,C)
function dCdot=diff(t,C)
Convection = zeros(8,1);
ConvectionB = zeros(8,1);
ConvectionB(8,1) = C0(10);%Value at the right boundary(z=L) goes here
Matrix = -1*diag(ones(1,8));
for i=1:7
Matrix(i,i+1) = 1;
end
Convection = v*(Matrix*C + ConvectionB);
%The second term on the LHS of the first equation is discretized using
%central difference and a tridiagonal matrix is formed
Diffusion = zeros(8,1);
DiffusionB = zeros(8,1);
DiffusionB(1,1) = C0(1);%Value at the left boundary(z=0) goes here
DiffucionB(8,1) = C0(10);%Value at the right boundary(z=L) goes here
Diffusion = D*(full(gallery('tridiag',8,1,-2,1))*C + DiffusionB);
Reaction = -k*C;
dCdot = Convection + Diffusion + Reaction;
end
end
I would like to implement Danckwert’s boundary condition given as follows,
At z = 0,
C (z =0) = C^{feed} + \frac{D}{v}\frac{\partial C}{\partial z}
At z = L,
\frac{\partial C}{\partial z} = 0
Could someone advise me on how the above boundary condition can be implemented in the code?

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