This illustration showing how Thermal Chimneys work is from the green builder reference but there is a better picture showing how the chimney would be constructed at that same web site.

Thermal Chimneys
A thermal chimney is a device which is used to cool down the inside area of the house in summer. This is used when the air is hot and humid.

The thermal chimney is built in the form of a narrow passage to the roof, like a chimney. it has a black metal absorber, which is on the inside the chimney, behind a glazed front that is on the outside of the house and is able to reach high temperatures. This surface can be insulated from the house. The chimney must be able to terminate above the roof level of the house so that air will be drawn out through it.

There is also a rotating metal scoop at the top which opens opposite the wind, this will allow heated air to exhaust without being too overcome by prevailing wind.
The cost of a Thermal Chimney depends on the elements of the chimney its self. There is a layer of insulation on the inside wall and a glazed or 'window' surface on the outer wall. The only things that will cost are the insulation and the glazing and the black metal of the chimney. We would be insulating anyway so the only extra cost is the glazing, the metal and the rotating turbine.

To understand why the metal would be painted black please read the post about absorption by different colours by HEAT Team A

To understand why hot air rises check out Dr Noot and Team B's post on the behaviour of air when heated.

A post on convection currents should be coming up soon!
www.greenbuilder.com/sourcebook/PassSolGuide3.html