I'm with Pancho on this.
People seem to think they get more volume & can do more by adding extra storage, when it's really all about cfm.
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I'm on this camp as well. Theoretically, if you have an infinitely large tank, you don't need a pump, right? But practically, adding a 30 gallon tank doesn't do anything, unless you are running the compressor continuously, which at that point the tank size is almost irrelevant.
For media blasting, needle scaler, die grinder, spray painting, DA sanding, "maybe" there's an advantage. Certainly not for filling tires and nailing.
You can argue both ways I suppose.
I've got a 30 gallon 7.5 HP with a Curtis ES-100 pump. Puts out something like 24CFM. I made it a while ago, but hardly ever use it, unless I'm using the HVLP gun.
I plumbed a pressure switch and a pilot valve so I can run it start-stop or continuous-run. For such a small tank and very short recovery time, start-stops would be an issue with electric motors, that's why I ran the continuous-run, but this is obviously an odd duck. I never have actually used the start-stop, as I have smaller compressors.
I recall about 10 years ago, it seemed that like EVERYBODY thought an air compressor with a huge tank was the way to go. Seemed like folks were paying attention to the "HP" rating on the tank and the size of the tank itself back then, not much attention to the CFM rating...until your pump wasn't able to keep up... Didn't Sears or some else have a 60 gallon model with a little 120V pump found on the 20/30 gallon units? Still had the same CFM rating as the little guys.
Duty cycles are typically 50% on homeowner units, but are higher for industrial units, if anyone cares. The only compressors that I can think of have a large tank with low CFM are the HVAC control compressors, but those things are designed to be run 24/7/365 forever! 30 gallon duplex with alternator setup. They are actually kind of cool.
OP never mentioned the compressor in question, aux. tank or what he's using it for, so we are just guessing here, LOL