You also may need new steering stops
As the other guy said, you need to replace any filters and do a complete flush. Based on the mechanic using a used part, which would be over 12 years old and probably sitting in a junkyard for 2 years, just does not make sense. Sure for a non wear item like a bumper or AC hose, or something interior part like a seat motor, radio, etc. a junkyard is OK, but not that item.
Why did the first one fail, could just be wear, but could be the clogged filters or so much sludge in the rack, or the rack is sticking, causing high pressure.
Do flush the rack you have to put the front wheels up and with the car off, turn the wheel to full lock side to side 5-10 times. If fluid is real bad, may to fill up with spec fluid, drive around for a few days and drain again. Most shops do not do that. Fluid, you need spec fluid, not sure if is pentosin 11s?
If the first pump broke into pieces, those could be in the system causing problems.
You may need to find a shop that specializes in BMW so they can try to determine the problem before replacing the pump. Also they will know which after market parts to use. There is a big difference in parts today, some are just junk. My shop that works on volvo, audi and BMW did a rack and pump on my 04 s60, they offer a 1 year warranty on parts and labor for leaks and failure.
Your first mistake is using parts from the scrap yard, Maynard. Figure it out.
I have a 1998 BMW 740i and I have had to replace a handful of power steering pumps in just over a year...after not having to replace any since I bought the car in 2007. The mechanic i took it too replaced the first one with a pump i got at a junkyard, but that one soon broke too almost right away. He replaced that too along with the tensioners. That pump lasted for a little while before breaking too a few months later. These pumps had plastic pullies. The mechanic i took it too replaced the pump with metal pullies and it's lasted until the other day. The other times the pumps broke, the belts just came loose, but this time the belt snapped too. Each time the pump broke, the bolts were still on the engine block He's not sure what could be causing this many pumps to break. He's thinking there is a possiblity it could be the rack and pinion. Could this be causing it or could it be something else?