I'm not sure one can brown a round barrel and get good results. I've browned many octagonal barrels, which let you draw file the flats to get the rough edge needed for a really nice chocolate brown. A perfecly smooth surface would not allow the rust to cling efficently. But how would you file a round barrel without making it "un-round" in places?
However, if you do decide to go ahead with that, the best browning solution is sold by Track of the Wolf. It's a true rust brown accomplished in alternating steps of application and steel wool. Using steel wool is called "carding." After each application, card the barrel lightly to smooth out the spots that got too much rust, so that it blends in. It'll look splotchy the first several applicatons, but soon you'll have an even finish. The best way to get that nice finish in a hurry is to build a long, slender wooden box, maybe four inches side to side, and top to top. Rig something inside the box from which the barrel can hang. Put a heat source in the box, like a very weak light bulb. Also, place a water source for the heat to draw from. A damp sponge, or a small container of water. This will enhance the rusting and expedite the whole process.
Consider the historical correctness of a browned buffalo rifle. I'm not at all sure they were browned.