why is the volumetric air compressor need to be cooled between two phases?

Answer 1

Compressing any gas results in an increase in temperature and the compression process becomes less efficient and requires larger equipment and more power the higher the temperature. Thus it is important to provide cooling between stages of compression. In air compression it is particularly important that the discharge temperature is not too high because of the danger of explosion with lubricating oil mist or other combustible debris.

Answer 2

The answer is still the same:

By cooling the air between the two stages you get the benefit of a smaller cylinder for the second stage than would otherwise be needed.

You also reduce the discharge temperature of the second stage resulting in less maintenance because the materials are subjected to temperatures that are not as high.

Finally if you look at the energy required to compress the air, it is less with the cooling of the air between stages.

Answer 3

This basically explains it.
http://en.wikipedia.org/wiki/Gas_compressor
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Andre' B.