# Model of heat sink problem with fan

I am trying to solve this problem using advection-diffusion model and finite element method for the solution, due to the complex geometry.

Basically the problem i'm trying to solve using OpenFOAM is see how heat is distributed when fan is placed on a heated power source, and how much speed or how many fans should be placed.

First, i'm thinking first about the advection-diffusion model for this, is it the proper model ?

Second, how about stokes equations (they are hyperbolic) , how would heat be expressed ? (They're not suited for the heat problem, if i'm not mistaken)

• Hi @Warrior4just and welcome to scicomp! The issue of choosing the right equations to model your problem is more formally a physics question than a computational one and i would recommend asking it on the Physics SE site. – Paul Jan 18 '15 at 2:27
• Thanks Paul, i will post this question in physics, although i was thinking that computational scientists would know the proper applicable models. – Warrior4just Jan 18 '15 at 8:11

Maybe you should ask yourself some general questions considering the physics

1. How large is the Reynolds number $Re$? (Stokes flow is only applicable to $Re\ll 1$.)
2. Are effects of compressibility relevant? So what is your Mach number $Ma$? A rule of thumb: $Ma<0.3$ incompressible flow, $Ma>0.3$ compressible flow. You can also use the Eckert number for characterization. Be careful if $Ma\to 1$ you have discontinuities in your field variables.
3. There are also similar numbers for heat transfer like Nusselt number.

You can select the appropriate OpenFOAM-solvers from the list at http://www.openfoam.org/archive/2.3.0/docs/user/standard-solvers.php#dx13-89118 . Remember to choose steady-state or transient. (Often a steady-state doesn't exist because of turbulence, hence no convergence of steady-state solvers.)

Convection-diffusion equation is in my opinion only applicable to frozen velocity fields. I guess thermoFoam solves this case.

OpenFOAM is a finite volume solver not a finite element one.

• Thanks Sebastian, excellent insights. I am very aware of the Mach number, similar singularities happen when dealing with the Froude Number close to 1. – Warrior4just Jan 19 '15 at 22:55
• Yes that is right. If I remember right the Froude number is similar to the Mach number for incompressbile flows. – sebastian_g Jan 20 '15 at 8:23

You will probably need two "part" to model your problem.

1. The first one will deal with the motion of the air. You can start with the Stokes equations, but this will probably not be suitable for your problem (Stokes flow are usually used for very viscous fluids... and air is not particularly viscous). The next steps would be to consider the Navier-Stokes equations, turbulence models,... but all these should already be present in OpenFOAM.
2. The second part deals with the distribution of the temperature in the fluid. And for that, the advection diffusion equation that you mention should be fine.

You need of course to link both parts.

• The main coupling will probably consider that the air velocity transports the temperature, i.e. the velocity of the air is the advection field of the advection-diffusion equation for the temperature.
• A second coupling would consider that when the air gets warmer, it becomes lighter, so it goes up. However, since you consider the action of the fans, you can probably neglect this effect is the fans are strong enough.

There are two pages in the tutorials of the dealII library that explain very well the modeling and resolution of such a problem. You can find them here and here. For the CFD part (i.e. motion of the air), some works using OpenFOAM for the modelling of wind turbine have already been done. You should have a look at them, you will probably face similar problematics (look at this presentation for example).

• As a comment, if the finite element method is the one you have to use and you have the choice of library, I would rather use a finite element library like dealII instead of OpenFOAM, which has limited finite element functionalities (and documentation). – Dr_Sam Jan 19 '15 at 13:31
• @Warrior4just Please consider upvoting the answers that you find useful for your question. This not only rewards those who took some time to answer you, but also gives indications if someone has a similar question and helps this beta. – Dr_Sam Jan 20 '15 at 11:54
• Dr_Sam, i would have for sure, but i have to have 15 in reputation. – Warrior4just Jan 22 '15 at 21:19
• @Warrior4just I didn't paid attention to that, sorry! Already upvoted your question, so I can't give you more rep :-) – Dr_Sam Jan 23 '15 at 9:14