Why would I use more? In this fantasy world of mine, I'm thinking I could change out 10 percent each week rather than changing out the full tub in 10 weeks. Same difference water wise...
You use more water with incremental changes in order to get to the same level of water dilution (freshness). If you replace all the water at once, then you get 100% fresh water. If you replace only 10% of it and do this 10 times, then the replacement water mixes with the older water so you lose some of this freshening effect each time as follows:
1st 10%: 10% fresh + 90% old = 90% old overall
2nd 10%: 10% fresh + 90% of 90% old = 90%*90% = 81% old overall
3rd 10%: 90%*81% = 72.9% old overall
4th 10%: 90%*72.9% = 65.61% old overall
5th 10%: 90%*65.61% = 59.049% old overall
6th 10%: 90%*59.049% = 53.1441% old overall
7th 10%: 90%*53.1441% = 47.82969% old overall
8th 10%: 90%*47.82969% = 43.046721% old overall
9th 10%: 90%*43.046721% = 38.7420489% old overall
10th 10%: 90%*38.7420489% = 34.86784401% old overall
The faster general formula for incremental water changes such as this one is (1 - x)^(1/x) where "x" is the fraction of water that is changed, so 0.10 in this case so (1 - 0.10)^(1/0.10) = 0.9^10 = 34.86784401%
If you were to do continual water dilution with perfect mixing, then after using one full volume of water you would have e^(-1) = 36.78794% old water. So diluting only 10% at a time is nearly as imperfect as continual water dilution.
If you were to dilute by 50% in two steps then you would have:
1st 50%: 50% fresh + 50% old = 50% old overall
2nd 50%: 50% fresh + 50% of 50% old = 25% old overall
So you can see how rapidly water dilution becomes less effective unless you do it all at once and that there is a limit if you do it continuously.
