Heat transfert + Air convection, setting proper equations

[Expi]

Hi everyone,

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 { $$\nabla$$(-c$$\nabla$$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$$\nabla$$T - c $$\Delta$$ 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 :
{ $$\rho$$u$$\nabla$$u + $$\nabla$$P = $$\mu$$$$\Delta$$u
here $$\rho$$ would be my air density, $$\mu$$ 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 $$\mu$$, which if I am correct is around 15e-6 m²/s in my range of temperature, against a value of $$\rho$$ around 1.15 kg/m$$^{3}$$ 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.

 

[gtoguy97]

</code>The equations you have written are correct, but there are some considerations that should be taken into account.The first equation is a simplified form of the heat equation, which is usually written as follows:\begin{equation} \frac{\partial T}{\partial t} = \nabla \cdot (-c\nabla T) + Q\end{equation}where $Q$ is a source term that accounts for any external heat sources or sinks, and $c$ is the thermal diffusivity. In your case, since you are working with constant fluxes, you can set $Q=0$ and the equation reduces to the form you wrote.The second equation is a simplified form of the Navier-Stokes equation, which is usually written as follows:\begin{equation} \rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u}\cdot \nabla \mathbf{u}\right) = -\nabla P + \mu \nabla^2 \mathbf{u} + \mathbf{f}\end{equation}where $\rho$ is the fluid density, $\mathbf{u}$ is the velocity vector, $P$ is the pressure, $\mu$ is the dynamic viscosity, and $\mathbf{f}$ is a force term (e.g. due to gravity). In your case, since you are working with a stationary flow, you can set the time derivative to zero and the equation reduces to the form you wrote.Regarding your questions about the pressure and viscous diffusion terms, it depends on the specifics of your system. If the pressure and viscosity gradients are small, then you can ignore these terms. However, if they are large, then you will need to include them in your equations.