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 $$T^*=\frac{h_{f1}T_1+h_{f1}T_2}{(h_{f1}+h_{f2})}$$
 

FAQ: Discretization of diffusion equation of a fluid in movement

What is the discretization of the diffusion equation?

Discretization of the diffusion equation involves transforming the continuous partial differential equation (PDE) into a set of algebraic equations that can be solved numerically. This process typically uses methods such as finite difference, finite element, or finite volume techniques to approximate the derivatives and solve for the fluid's behavior at discrete points in space and time.

Why is discretization important for solving the diffusion equation of a fluid in movement?

Discretization is crucial because it allows us to solve complex diffusion equations that describe fluid movement, which are often impossible to solve analytically. By breaking down the problem into smaller, manageable parts, we can use computational methods to simulate and analyze the fluid's behavior under various conditions.

What are the common methods used for discretizing the diffusion equation?

The most common methods for discretizing the diffusion equation include the finite difference method (FDM), finite element method (FEM), and finite volume method (FVM). Each method has its strengths and is chosen based on the specific requirements of the problem, such as accuracy, computational efficiency, and the nature of the domain.

How do stability and convergence affect the discretization of the diffusion equation?

Stability and convergence are critical factors in the discretization process. Stability ensures that the numerical solution remains bounded and behaves correctly over time, while convergence guarantees that the solution approaches the true solution as the discretization parameters (e.g., time step, grid spacing) are refined. Properly chosen discretization schemes and parameters are essential to achieve accurate and reliable results.

What are the challenges in discretizing the diffusion equation for a fluid in movement?

Challenges in discretizing the diffusion equation for a moving fluid include handling complex boundary conditions, ensuring numerical stability and accuracy, dealing with large computational domains, and capturing the effects of fluid movement and interaction with other physical processes. Advanced techniques and high-performance computing resources are often required to address these challenges effectively.

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