Convection-Diffusion with Heat Sink - Mesh Effect
 

Hello Again,
I try to model the temperature change (of water) through an insulated pipe in time. So I solve:

https://i.postimg.cc/59nmWr8r/2019-0...-docx-Word.png

The thermal resistance for the pre-insulated pipe is based on x:
https://i.postimg.cc/HxjkcjPX/2019-0...yses-Gener.png

The problem is that whenever I change the mesh size (&number) in space x while solving the convection diffusion equation (with heat sink) the overall thermal resistance Rt is changing so the water temperature profile through the pipe. For Example (According to steady-state calculations the plateau should be 68.07°C!):

https://i.postimg.cc/CxBCDHzL/2019-0...s-Figure-1.png

QUESTION: How to avoid the mesh size effect on the heat sink? Are my formulations true in principle?

It is by your definition depending on the mesh size...

Are you sure that your problem has the resistence depending on the mesh size and not on a physical lenght width?

If I base on the pipe length instead of basing the mesh size then the heat loss becomes too high, here is the result of the same:

https://i.postimg.cc/4N0Q2FcT/2019-0...s-Figure-1.png

I don't know the physics of your problem but I cannot understand this problem that shows this PDE with a forcing term depending on the mesh size. First of all, I would understand the meaning of the solution for dx->0, that is the exact solution you would compute in the continuous meaning.

It is a buried pre-insulated pipe. I try to solve the temperature propagation through pipe in time.

I could not find any resource explaining the convection-diffusion equation with heat sink. Could you suggest any?

Convection-Dominated Flow | Finite Difference | Sink Term as Heat Loss
 

I found the mistake in my formulation:

There is a need to correct the heat sink term in the convection-diffusion equation, solving the temperature propagation through an insulated pipe segment in time (no thermal inertia considered). This way the value of c (check the formulation below) becomes constant whatever the spatial mesh size is! However, now I have trouble in my numerical results (oscillation, abnormal results).

https://i.postimg.cc/tCzKhdc1/2019-0...-docx-Word.png

Questions:
1) My results are oscillating when I remove the sink term. So can you please check the approximations? Is using FTCS with Upwind Treatment correct for this PDE (Check Figure1)?
2) With the sink term, the result is not realistic! Especially I want to ask you 'How to approximate T itself (in the sink part)?'. Is it true to have the approximation of T as Ti,n? Check Figure2!

Thanks in advance.

Figure1:
https://i.postimg.cc/Ls6SwCcF/2019-0...s-Figure-1.png

Figure2:
https://i.postimg.cc/CL1jMSbR/2019-0...s-Figure-1.png

The scheme that use the Forward Time and the Upwind discretization is denoted FTUS while the FTCS stands for centred discretization in space.
The FTUS is monotone, therefore you have a bug in your code. Is the transport velocity positive everywhere?

Yes the velocity is positive through the pipe length.

If the FTUS solution (without the source term) shows numerical oscillations in the range of the numerical stability you have a bug in the code.
What about the numerical condition at the outlet?

It is Neumann boundary condition at the pipe outlet i.e. dT/dx=0

However, you must assess that the solution is monotone in space, not in time at the outlet