Solving a system of mass balance equations with implicit variables

Hi,

I want to solve a system of mass balance equations which have concentrations implicit in them. I have attached a photo of the mass balance equations above. 'n' = 1:15 and 'm' = 1:20. so it is a 15 by 20 column of cells. I want to apply the mass balance so that it can update the concentrations of the cell (m,n) but there is no equation for the concentration separately and concentrations cannot be written explicitly (they are also part of the 'tNa' and 'tA' terms, and they cant be explicitly written in those terms)

[darova]

I'm not good in chemisty. Is it kind of diff equations or what? Can you explain more (some picture)?

[Zain Shabeeb]

Hi darova,

Thanks for your comment.

I have attached a photo of the model. I also posted the complete model in my last post here: Solving discretized algebraic equations using nested FOR loops.

I know it seems like a very detailed model, but I just need to get an idea of how to go about modelling it i.e the modelling approach. At the moment I am trying to use 'for' loops as you can see in my last post. I think the main problem is that the concentrations are implicit in the 'mass balance' equations. So if I use 'for' loops, There are no equations that are explicit for the concentration terms. So the concentrations don't get updated after each loop iteration.

I would be grateful for any help as I am really struggling to get going with this.

[Torsten]

The equations you supply are discretized PDEs for the concentrations C_Na+ and C_A that can readily be solved using an ODE integrator. If you want to turn them into algebraic equations for C_Na+ and C_A, you will have to discretize the d/dt - terms.

[Zain Shabeeb]

Hi Torsten.

Thanks for your comment. I understand that these are discretized equations. But how do I write the program for them? Should I use 'for' loops? I have tried to write the code for the model in my previous post here: Solving discretized algebraic equations using nested FOR loops.

I can use ode15i once I know how to actually write the algebraic equations in MATLAB.

[Torsten]

But the equations are ordinary differential equations for C_A(n,m) and C_Na+(n,m) and are readily written to be used in an ODE integrator like ODE15S. So I don't understand your problem if you want to use them together with an ODE integrator like ODE15S.

From your previous post, I assumed that you can't use an ODE integrator because you want to implement the model in an optimization environment which only allows algebraic equations.

[Zain Shabeeb]

If it is not possible to discretize with time then I can just use ode15s (forget about optimization), but even for that I would have to use the 'for' loop for the algebraic equations.

In my previous post I have tried to code the program to discretize with time. But in that program there is no variation with time or with space i.e. all the arrays after t=0 are zeros. So my question is what is wrong with the code in my previous post?

[Zain Shabeeb]

Another problem I am having is that the 'C' terms are implicit in the mass balance equations and cannot be separared.'t' is a function of 'C' and so in the mass balance equations, 'C' cannot be made explicit. So if i write the mass balance equations in a loop, how will the 'C' terms get updated after each loop iteration?

[Torsten]

The C-terms get updated by the ODE-integrator because they are the solution variables. You specify dC(m,n)/dt (m = 1,...,20 ; n = 1,...,15), and the formula you provided is explicit for

dC(m,n)/dt .