The theory of operation is that the beginning of the rain fills up the tube with bird poop and road dust contaminated rainwater, once thats full a hard plastic ball floating in it hits a tapered stop and diverts the remaining clean water through the spout into the barrel.
You can't see it but the tapered stop is actually a 3" to 2" reducer glued inside the tube.
There is a small weep hole (1 mm) in the middle of the bottom cap that slowly empties out the tube between rains. To keep the weep hole from clogging I had to cover it with a mini-basket of stainless mesh filled with poly pillow fluff. Dunno how stuff gets into the barrel big enough to clog it, but it does.
I replaced the original crappy plastic mesh on the barrel with a 50 mesh stainless screen basket that goes down into the barrel about 5 inches. Before I installed stainless micromesh gutter guards the basket got full every single rainfall (pine tree flowers and needles). After the gutter guards the basket never gets anything in it.
The tee is actually a full size schedule 40 fitting instead of a DWV sized fitting, it let me insert the square downspout adapter inside it, which shortened it up. Because it was too big for the DWV pipe I had to make an adapter from another fitting to snug it up, you can see the adapter ring in the first photo.
Thought this might help somebody else out who's thinking of doing rain collection.

