As with your common experience, heat is
transferred from areas of high concentration
(i.e. a hot object) to areas with less
concentration (i.e. anything that is less hot).
So these carbon nanotubes would transmit the
heat to the air around it, to the material
behind them, and also the heat would conduct
along the nanotubes.
Copper is very conductive, so it is a good
idea to transfer heat using copper. However,
heat is not going to flow from an area of low
concentration (i.e. low temperature) to an area
of high concentration (i.e. high temperature).
In the same way that water does not flow up
hill, heat will not flow from cold to hot.
Because the hot tub is always hotter than the
swimming pool, any sort of system that connects
the two will always heat the swimming pool, and
cool the hot tub.
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