"Heat Pipes" are used to transfer heat efficiently from a hot source
to a cold source.  They are used to dehumidify air, keep computer
parts cool, protect frozen tundra from the Alaska oil pipeline, etc. 
Some of these can transfer heat from heat below to cold above.  Others
can do the opposite, transfering from top to bottom. My question is,
for the latter type of heat pipe that transfers heat from the top of
the tube to the bottom of the tube, what is the practical limit as to
the length of the tube for the device to still function?  They operate
by capillary action, so is there some inherent limit to capillary
action?

An acceptable answer at least fully answers the second question while
giving background on the first.  A great answer directly answers the
first question fully.