I have the same unit that you have (thanks for showing it!). For cooling the air, wouldn't you want the fan to be running while using the tools instead of just when the compressor comes on?
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The way I have mine plumbed the air exits the pump, enters the cooler to be cooled, exits the cooler and hits the water separator and on to the tank. So to answer your question, no, in my case the air has already been cooled and most moisture removed prior to going in the tank. (it will not pass through the cooler again, so the fan running will not make any diff) It leaves the tank cool and will pass through my multistage filter dryer for the final conditioning. I also have drops that do not pass through the dryer for general use, blow guns , tires, etc. You can plumb it so the air exits the tank then through the cooler and on to your filters, in which case you would want to run the fan while using tools. I wanted to get rid of most of the water prior to going to the tank, and I wanted the fan operation to more automated so that I do not have to deal with turning it on an off, it's up to you how you want it to operate.
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Does the charge in the tank drop in temp quickly enough to no be a problem after the cooler/water filter?
This cooler seems to be very efficient, you can not touch the inlet side after a couple cycles and the outlet side is cold. I want to get a hold of a temp gun so I can see the swing.
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Does having the fan on the cooler run at the time of compression enough to cool the charge to allow water removal at the filter right after the cooler?
Yes, I have watched my filter's auto drain spit out the water during the cycling. What ever uncondensed moisture makes it past the first sperator, will have to pass through another set of filters and then to the dryer which will remove any uncondensed moisture.
Hope that helps, post up some pics of your setup once you get it installed