# Discretization of diffusion equation of a fluid in movement

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DianeLR
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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Mentor
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.

DianeLR
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.

Mentor
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.'

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