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I could use some help to validate (or invalidate) and improve my equations as I am trying to model 2D heat dispersion in a Spa.

Basicaly my model consist of some air inside a cabinet, the cabinet is framed by 2 different insulants each insulant has heat flux on boundaries (a hot source and a cold one).

I model the heat inside insulants by a simple Poisson's equation { [tex]\nabla[/tex](-c[tex]\nabla[/tex]T) = 0, no problem here.

But I am really not sure about the equations of air.

I figure I should take 2 things into account, the convection and the conduction.

Which would mean my temperature equation becomes :

{ u[tex]\nabla[/tex]T - c [tex]\Delta[/tex] T = 0 , where u is my velocity field, I am not too doubtful about this one.

Then I also need to add my velocity field equation. From what I found it should be something like that :

{ [tex]\rho[/tex]u[tex]\nabla[/tex]u + [tex]\nabla[/tex]P = [tex]\mu[/tex][tex]\Delta[/tex]u

here [tex]\rho[/tex] would be my air density, [tex]\mu[/tex] the air viscosity and P the air pressure.

So I have a few questions about that, which could use some physicists insight (yeah I am more the mathematician kind).

First you might have noticed I decided to operate independtly of time, so I ignored the time derivatives in both the heat equation and the velocity field. I am working with constant over time fluxes and looking for a balance state of dispersion. So I think that is the right call. What is your opinion about that ?

I should first explain a bit more the physical conditions of my model. The air is in a range of temperature of 16-39° C (289-312 K). I easely modeled the dispersion without convection. Now my convection is created by 2 gaps in the layers of insulant, one at the bottom (floor - side junction) and the other at the top (lid-side junction). Gap are around 1-2 centimeters wide but all around the spa which is not seen in the 2D model but the ratio is respected imo.

Then 2 questions about my velocity field

1) Concerning the pressure, I don't really know how it is acting inside the cabinet but I hardly believe it would have big fluctuations, meaning I could just ignore the derivative term of pressure. Do you think that's a good estimation?

2) The viscous diffusion term is dependent of [tex]\mu[/tex], which if I am correct is around 15e-6 m²/s in my range of temperature, against a value of [tex]\rho[/tex] around 1.15 kg/m[tex]^{3}[/tex] so I might also be able to ignore the viscous diffusion. Is it a correct thinking?

Again, all this is really seen in a mathematical pov. I am trying to achieve a computed estimation so I don't need a 0.1 degree preciseness, it's really about knowing where it is hot and where it is cold with a rought estimation of temperature range if possible. But like I said I am not a physicist and could overlook some important effects so if you have anymore suggestion or reflexion please share it, I am always willing to improve the model.

Thank you for your time and answers.

Have a good day.

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# Heat transfert + Air convection, setting proper equations

Can you offer guidance or do you also need help?

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