The average plane spotter might well know more about compressor washes than I do. All I know is what I have seen in a documentary on engine maintenance at Lufthansa Technik's on DVD. It shows a bloke lowering what I think is a compressor casing into a basin. Reading from a manual, he eplains they were applying some alkaline rust remover for 15 to 60 minutes to remove "Zunderpartikel" (cinder particles?). The part was washed and then immersed in potassium permanganate to turn oil carbon into manganese oxides.
From
I gather engines can be washed on-wing as well.
Sorry I can't tell you anything more profound.