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Actually to be clearer, copper has a higher thermal conductivity than aluminium at and around room temperature (by a fair margin, but aluminium doesn't do too badly itself).
Specific heat capacity I'm not so sure on, but in any case in an equilibrium position (ie when your computer has been running for 30 mins or so I guess) it is irrelevant. The thermal emissivity of the heatsink comes in to play then, and I would guess that Al and Cu heatsinks when both painted black would have similar levels. My thermodynamics and fluid mechanics is a little bit rusty, but I guess the mechanism here is radiation to the surrounding (forced-moving) air. So a high airflow past a black (high thermal emissivity) heatsink that conducts heat quickly away from the CPU (copper) is the best setup for air cooling.
Turbulent flow of the air past the heatsink is probably a good thing (Greebe knows about this I think) as the air is given more chance to mix around and pick up the heat. Some of the heat will not be radiated to the air from the HS but instead conducted to the surface flow - helped more by turbulence probably.
Please others jump in to correct me - I haven't looked at a textbook in about 7 years and haven't used the knowledge for about as long!
Should add: obviously the really important things are as high a surface area as possible on the heatsink and a large temperature differential between the air and the heatsink surface.
All of the above mean that the zalman copper flowers make a lot of sense on a CPU.
Material, Thermal Conductivity in Watts/cm*°C at ~25°C
Type I Diamond, 9 (diamond with nitrogen atoms in the crystal lattice)
Type IIa Diamond, 24 (very little nitrogen)
Copper, 4
Aluminum, 2.35
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