Exchange Problem with PDEPE solver

I want to use the PDEPE (or whatever else) solver to describe an exchange model between two different layers. Within the layers I want diffusion to occurr.

I want to start with a "hot" sphere surrounded by a cold spherical shell. Between those two reservoirs I want an exchange of the form d/dt U(leftborder) = -k*(U(leftborder)-U(rightborder). Within the layers I want a propagation (of heat for example) via diffusion.

My problems are:

(1) how to i solve a problem of two partial differential equations that are only coupled at their border in matlab. As far as i understand i can only solve coupled equations that have the same domain, i.e. the same xmesh.

(2) how can I implement a time dependent boundary condition like the one stated above.

Thanks for any help!

Best regards.

Answer 1

[Torsten]

There is no ready-to-use solver in MATLAB for this kind of problem.

You will have to discretize the PDEs in space and solve the resulting system of ODEs for the temperatures in the grid points by a solver for ordinary differential equations (e.g.ODE15s).

What second transmission condition should apply at the interface between the two layers ? Continuity of heat flux ?

Best wishes

Torsten.

[Thomas]

Hello Torsten and thank you for your answer! I do not fully understand what you mean by second transmission condition. The model should contain heat transfer towards the border by diffusion and then (kind of a rude discontinuity in the diffusion) an exchange at the border. In the second layer this should propagate by diffusion again. So in principle is a diffuse, hop over barrier (and also backwards hop) then keep diffusing type of problem.

Thank you very much, your answer already helped me a lot!

[Torsten]

You have two unknowns at the interface: U(leftborder) and U(rightborder). So you need two equations to solve for them.

Usually, if e.g. material properties change at the interface, the transmission conditions are continuity of temperature (Tleft=Tright) and continuity of heat flux (Lambda_left*dT_left/dx=Lambda_right*dT_right/dx).

In your case, only one condition is stated.

Best wishes

Torsten.

Answer 2

[Bill Greene]

Torsten said:

"Usually, if e.g. material properties change at the interface, the transmission conditions are continuity of temperature (Tleft=Tright) and continuity of heat flux (Lambda_left*dT_left/dx=Lambda_right*dT_right/dx)."

If, in fact, these are the conditions at the material interface, solution using the pdepe function is straightforward. You have a single xmesh that spans both layers. In your function that calculates the pde coefficients, you use the value of the x-variable that is passed in to calculate the flux, i.e, if x < x_interface, f= k1*dudx, otherwise f = k2*dudx.

Bill

[Torsten]

There seems to be a jump in the solution variable at the interface because one of the conditions reads

U(leftborder) = -k*(U(leftborder)-U(rightborder))

Best wishes

Torsten.

[Thomas]

Thorsten is right, at the border Uleft is basically an exponential decay while at Uright there is an exponential rise. Both edges have to come together for very long times in the end to equilibrate the Temperature over the whole system. I have thought of using one xmesh with if statements to distinguish the separate layers but that seemed a little messy to me. I might still try it out.

It is probably just me but i also have a little troube with my boundary condition since it is a time dependent one and not a spatial gradient which the PDEPE solver seems to ask for. This still throws me off a litte, but that is probably just me.

[Torsten]

Well, I repeat that PDEPE is not suited for your problem, but maybe its worth for gaining experience with your problem.

Best wishes

Torsten.

[Thomas]

Hm, okay it looks like i will have to do a numeric solution then! Thanks for the help!