On the actual yeilds of acadmies, let's take a look at what you can get. I realize that crossmr did something like this but the freedom finisher was not included and the +15% from sovereignty is a multiplied modifier, not additive. It's easy to make these kind of errors. I do it all the time . I will try to get the numbers right, however.
+50% from university
+50% from observatory
+50% from research lab
+50% from national college
So that gives a multiplier of 3x. Now sovereignty is applied on top of this AFAIK. So this gives 1.15 * 3 = 3.45
The freedom finisher is tricky. It seems to double base tile yeilds but not the addons. So it doubles the 6 but not the extra 2 from Korea or Scientific Theory. At least that's what I've noticed (others feel free to correct me if I'm wrong). So under this assumption, the new base yeild of a Korean academy post freedom finisher is:
6 * 2 + 2 + 2 = 16
This gives a max academy yeild of 16 * 3.45 = 55.2:c5beakers:/turn
Now to be fair, for most of the game you won't have the research lab bonus nor in most games the observatory. The freedom finisher can be obtained in Renaissance but late Renaissance. So let's say for argument sake you get just the university and NC for the first half of the academies lifetime and the other bonuses for the last.
That gives the early game academy an effective yeild of: 14.64/turn
It follows that, under the above assumptions, the average bonus over the life of the academy is: 34.92/turn
If it lasts 150 turns, that gives you 150 * 34.92 = 5238/academy
Of course this assumes getting a lot of policies to modify these outputs. It's certainly possible but the 50-50 assumption I made won't hold. It's too hard to analyze any other way though and you can tear through renaissance with academies so I thought it was fair. It also doesn't include the extra RA bonuses, the fact that you will get techs fast than not meaning earlier public schools and such increasing overall science faster.
Just something to think on.