Discretization of diffusion equation of a fluid in movement

In summary: T_f=\frac{h_{f2}T_1+h_{f2}T_2}{2}$$In summary, the equation states that the thermal behaviour of a moving heat transfer fluid is affected by convective exchanges with the walls. The equation can be solved using a second order spatial discretization with decentred schemes at the extremities, but not at the center. This is likely due to the discretization of the mcp term. To correct the problem, the convection term should have a positive sign.
  • #1
DianeLR
7
0
Hello,

I want to model the thermal behaviour of a moving heat transfer fluid in 1D, with convective exchanges with the walls. I have obtained the following equation (1 on the figure). I have performed a second order spatial discretization with decentred schemes at the extremities (y = 0 and H). After spatial discretisation, equations (2 to 4) are obtained.

By scoring these equations in OpenModelica (a software with a DASSL time integrator), I obtain consistent results at the extremities but not at the centre. I think this is due to the discretization, especially the mcp term.

Do you have any idea how to correct this problem?
 

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  • #2
In your initial equation, the convection term should have a positive sign, not a negative.

Why did you choose this method to solve this problem? I can be done so much more simply using the method of characteristics.
 
  • #3
Chestermiller said:
In your initial equation, the convection term should have a positive sign, not a negative.

Why did you choose this method to solve this problem? I can be done so much more simply using the method of characteristics.
Should the convection term be positive even if element 2 (at T2) is to the right of the fluid (element 1 to the left)?
I didn't know about the characteristics method... So it's easier for me to use the finite element method.
 
  • #4
DianeLR said:
Should the convection term be positive even if element 2 (at T2) is to the right of the fluid (element 1 to the left)?
I didn't know about the characteristics method... So it's easier for me to use the finite element method.
Yes. Derive it yourself.

You should learn how to apply the method of characteristics to this problem.'

Please write out the PDE.
 
  • #5
Chestermiller said:
Yes. Derive it yourself.

You should learn how to apply the method of characteristics to this problem.'

Please write out the PDE.
Thank you for the correction.

I will look into the method of characteristics to apply it to my problem.

The PDE are written in the figure below.
 

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  • #6
DianeLR said:
Thank you for the correction.

I will look into the method of characteristics to apply it to my problem.

The PDE are written in the figure below.
So $$\rho C A\left[\frac{\partial T}{\partial t}+v\frac{\partial T}{\partial x}\right]=L(h_{f1}+h_{f2})(T^*-T_f)$$with $$T^*=\frac{h_{f1}T_1+h_{f1}T_2}{(h_{f1}+h_{f2})}$$
 

1. What is the diffusion equation of a fluid in movement?

The diffusion equation of a fluid in movement is a mathematical representation of how a fluid (such as air or water) flows and spreads out over time. It takes into account factors such as the fluid's velocity, density, and viscosity.

2. Why is discretization necessary for the diffusion equation?

Discretization is necessary for the diffusion equation because it allows us to break down the continuous fluid flow into smaller, discrete time and space intervals. This makes it easier to solve the equation numerically using computer simulations.

3. How is the diffusion equation discretized?

The diffusion equation is discretized by dividing the fluid domain into a grid of discrete cells and approximating the fluid properties (such as velocity and density) at each cell. This creates a system of equations that can be solved iteratively to simulate the fluid flow over time.

4. What are the benefits of discretizing the diffusion equation?

Discretizing the diffusion equation allows us to accurately model and predict the behavior of fluids in movement. It also enables us to study complex fluid flow phenomena that would be difficult or impossible to solve analytically.

5. Are there any limitations to discretizing the diffusion equation?

One limitation of discretizing the diffusion equation is that it requires a lot of computational resources, especially for complex fluid flow problems. It also relies on certain assumptions and simplifications, which may not always accurately reflect real-world scenarios.

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