I use one of these small pumps in a builders bucket positioned under the lathe, powered by an old router plug in 12v adaptor.
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with a length hose to one of these, and a return hose to the bucket from the swarf tray.
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works really well and is capable of supplying more than enough coolant.
I drop the pump into a fine mesh sleeve to protect it from metal particles, and dangle a magnet into the bucket now and then to collect any settled swarf.
Has been working fine for the last year or so, – now I've said that it will probably pack up.