Like the others, I assume that a blowout near the end is a certainty. I leave my barrels 1/4 to 1/2 inch long to avoid this. Don't drill all the way through, and then cut the un-drilled excess off the end. When I drill on the lathe, I start with a bit that's a couple of sizes (1/32" for me) smaller than my target size, and I don't baby it. Heat builds up whether you're boring into fresh material or just spinning your tires, so I try to drill quickly, and back out to dump chips every 1/4" or so. I use a spray lubricant to keep the heat and noise down. While this may sound like a bad idea (poor glue adhesion, etc.), I then redrill the hole with a bit 1/64th bigger with no lubricant, and finally move up another 64th of an inch to final diameter. These cuts remove just a hair of extra material and all the lubricant with it. Since there's little material to go through, the heat isn't bad. I can get what little extra material there is to drill out in 1 continuous pass with the larger bits, and this leaves me with a cleaner hole, no blowouts, and no melted and smeared acrylic on the walls of the hole.
When turning, I use a 1/2" round nose scraper almost exclusively. It is the cheap, non-HSS Harbor Freight tool. Oddly enough, I find that a slightly dull edge (1 or 2 previous pens since sharpening) works best! In any case, do not use a tool with a burr; the angle of the burr makes the acrylic try to pull the tool into it. I've turned through the whole barrel and hit brass before I knew what was happening!
I first knock off the corners with some rough sandpaper at low speed. This reduces chipping later. Then, the lathe gets turned up to max speed, and I begin making SLOW light passes with the round nose scraper. This gets the tool and blank warmed up, which makes turning go much more smoothly. Once I have a cylinder, I take light cuts and move a bit more quickly. My goal is to make shavings that look like plastic Easter grass, if that makes sense. If my shavings begin to wad up and melt on the scraper, I back off for a few seconds or move to another area. Done right, I frequently have to stop the lathe to rip off a bird's nest of shavings that has accumulated. The kids go crazy for these shavings. I say they're not right, the wife says I'm paying for college anyway. :tongue:
If my scraper is too broad to get the detail I need, I use a skew scraper-style and take light passes, using the toe (yes!) to dig slightly into the material, and then push it along with the toe leading the way. Using the skew any other way on acrylic produces chatter for me. Don't try this on wood!
Done right, the barrels are almost glossy before sandpaper ever hits them. I can go straight to 600 grit MM, used wet with the lathe on LOW or med-low speed. Up through the grits to a reasonable number, then One-Step plastic polish or #2 Novus scratch remover--interchangeable in my experience. If I'm feeling froggy, a quick buff on the buffer finishes it off, but I just as often stop after the One-Step/Novus.