Water Change Formula for Daily WC's
Ralph Cote February 11, 2003

I was wondering about how different water change percentages affect the water quality in the tank and how much builds up over time. The formula applies only to water soluble wastes and does not apply to CO2 (gas is removed by water action) nor ammonia and nitrites (which are converted by bacteria). It does apply to nitrates (in unplanted tanks) and any unknown wastes which are the likely cause of stunting and poor health. If you are not a math fan, you can just skip to the bottom.

If W is the total daily waste output by all the fish in the tank, and WC% is the percent of tank volume that is changed daily:

The waste in the tank on the evening (after a WC) of the first day = W(1-WC%)

Waste in the tank after second day (after WC) = W(1-WC%) + W(1-WC%)(1-WC%)

Waste on third evening = W(1-WC%) + W(1-WC%)(1-WC%) + W(1-WC%)(1-WC%)(1-WC%)

Etc., etc.

In English it means the fish are constantly putting out waste and it is being partially removed every evening with the water change, so the waste builds up over time. It is shaped like a learning curve where it is steep at first then gradually levels out to a nearly constant amount (the time to level out (flattening of the curve) varies with the WC% but is around one week for most of our situations). The actual curve if done by hours would actually form a sawtooth pattern but still in the general form of a learning curve.

Since the formula is a geometric series, the eventual amount of constant waste can be found using a simple formula:

S = a/(1-r) where "S" is the sum of the series (the eventual amount of waste in the tank), "a" is the initial starting value (a = 1 - WC%), and "r" is the ratio between successive terms (r = 1 - WC%).

So, if you are doing a 50% water change every evening: S = .5 / (1 - .5) = 1.

In other words, after about 5 days, you have approximately 1 day of waste in the tank at all times (it builds up during the day to almost 2 days of waste right before the water change).
  • A 33% water change gives you about 2 days worth of waste as a constant.

  • A 25% WC gives you about 3 days of waste.

  • A 10% WC gives you about 9 days of waste

  • A 1 - 2% WC (10% every week) gives you around 75 days of waste!

I hope that made sense to somebody, it would be better with a graph but...

There is not a huge difference between 50% and 33%, or even 25% (though maybe to the discus) but the build up is big by the time you reach 10% water changes per day. And 10% per week, don't even bother.

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