The only way I see this working off gravity is to make the main feed pipe come from the bottom of the first barel and over flowing to the second and then vice versa the only problem being that once the water fills the second barel it may not have enoph force to fill the third do to an equalibrium being established and thus major oveflow of the second barel. Imo it is a posibilty you may have to concider. As well as concidering a pump to help move things along just a thaught.
. i hope i don't have the problem you've mentioned
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