A Discretization of diffusion equation of a fluid in movement

AI Thread Summary
The discussion focuses on modeling the thermal behavior of a moving heat transfer fluid in 1D, addressing issues with spatial discretization and convection terms in the governing equations. The user encounters inconsistencies at the center of the model when using OpenModelica, suspecting the discretization of the mcp term as the cause. Participants suggest that the convection term should be positive and recommend the method of characteristics as a simpler alternative to the finite element method. The user acknowledges the correction and expresses intent to explore the method of characteristics further. Overall, the conversation emphasizes the importance of accurate term signs and effective numerical methods in thermal modeling.
DianeLR
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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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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.
 
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.
 
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.
 
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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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 $$T^*=\frac{h_{f1}T_1+h_{f1}T_2}{(h_{f1}+h_{f2})}$$
 
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