if you are inclined to rush that much culture may I recommend a modification to your method? Instead of whipping 8 guys, whip only 4 and draft the other 4. You will get 8 unhappiness all the same, but each 20 turns you will eliminate both a whipping unhappiness and a draft unhappiness. Thus: only 80 turns.
edit>> I think I may need to clarify. The purpose of the drafts is to disband for shields. One draft provides one disband-shield for every 4 shields that it would have cost to build the unit. It's a nice trick to aid in those 100k culture victories.