If the tank is empty (of air), leaving the valve open won't hurt anything.
I just leave my compressor charged up (with air), and open the drain for a
second each time it runs. They sell kits to do this automatically, here is
one from HF:
Their website isn't working right now, but that's the link for it, anyway.
None of the numerous air compressors I have bought over the years
mentioned anything like that in the operating instructions. Operating
manuals are available for many brands on line, and that is a source
for info as well. Unless something like that has been published it is
safe to assume that an open drain doesn't buy you anything in between
uses. Just drain it often in the summertime hot humid weather and you
will be fine.
It won't make any difference if it is not running. Are you getting much
water out? I drain mine about ne a year and get very little, but that
depends on use, climate, etc. Our big (150 hp) compressor at work drains
every 20 minutes.
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