Rob, The black particles that you saw coming from the jets on your whirlpool bath are the accumulated residue of previous baths, ie: soap, bath oils, even body oils.
One way to reduce this problem is as follows: Fill the tub with hot water to at least two inches above the highest jet.
Very carefully pour at least two cups of household laundry bleach (Clorox), and two cups of white vinegar into the water.
Start the pump and let the system run for about 10 minutes. This will do a decent job of cleaning the water line of the residue. To clean the air lines you must turn off the system and let the solution stand for at least thirty minutes. After you have turned off the system, the solution will fill the air lines, giving all important "contact time".
Drain the tub, fill again with water, run the system about 10 minutes, and drain.
This is about all that you can do to this tub. I did find a company (mtiwhirlpools.com) that offers new tubs with a self-cleaning system that eliminates the problem. Go to their website and click on Fill/Flush.