The key technical term you're looking for here is convective cooling. Either you allow the air to rise through the chimney on its own accord (natural convection cooling) or you pump it through with a fan (forced convection cooling). The second is obviously more effective at pulling heat out of the system.
The amount of heat removed will be a factor of the flow rate of air through the system as well as the heat transfer coefficient (HTC) between the air and the CPU. You can get a good estimate by grabbing an HTC function from engineering toolbox or a text book and using Newton's Law of Cooling:
q = hc A dT
where q is heat transfer (in watts), hc is the heat transfer coefficient (W/m^2 K), A is the surface area to be cooled (m^2), and dT is the temperature difference between the air and the surface (in K).
The HTC for air is (for v between 2-20 m/s):
hc = 10.45 - v + 10 v^1/2
where v is your air flow speed in m/s. The faster the air is moving, the higher your HTC (and the more heat you can pull out of the system).
Broad estimates of HTCs:
Free convection: 0.5-1000 W/(m^2 K)
Forced convection: 10-1000 W/(m^2 K)
You should know from the specs of the machine roughly how much heat is produced in watts. Then your air flow will dictate your HTC - and try to maximize it so that you can pull more heat out of the system.
The concept of convective cooling is pretty widely discussed; any general engineering thermodynamics textbook will have more detail ("Thermodynamics: An Engineering Approach" by Cengel or "Fundamentals of Heat and Mass Transfer" by Incropera are good resources). There's quite a lot of stuff about general engineering approaches online, too: where to locate the object to be cooled in your enclosure, what sort of fans to use, etc. These are the sorts of problems that can be analyzed at a really detailed level with computational fluid dynamics. To check the feasibility, I would start with Newton's Law of Cooling, though.