@beaslbob I'm basically illiterate in the matter by experience - was just sharing what I read.
If I misinterpreted the article by the good doctor, it's my mistake.
no misinterpretation.
My analysis is based on where the system winds up at. Or to what final point is it going too. As opposed to all the "advanced" math with expended power functions which define where the system is at it goes along the path to that end point.
At the end point the amount of stuff removed by the water change is equal to the total change between water changes.
So if you do a 1/4 water change, the system build up to 4 times the total change between water changes. 1/10 10 times, 1/20 20 times and so on.
Plus whatever is in the replacement water.
Of course that assumes the stuff is a linear measure unlike things like pH.
And that you have a constant change between water changes also.
So
amount before water change=(amount in replacement water)+(total build up between water changes)/(fraction of water changes)
If you do a spreadsheet on water changes and do a plot, you will find that the system will approach that point.
Also notice this does not matter where the system started at. You could start at any old point and with an infinite number of water changes the end point is the same.
Of course we don't have time to do infinite water changes, but it will be within test kit results after 10-20 or so water changes.
And obviously things don't stay linear forever. For instance, calcium can't decrease at 1ppm/day forever. You run out of calcium. And things like nitrate and phosphate react to algae/coral build ups in the system. and bioload changes and feeding also.
But I make those assumptions to get a handle on how effective water changes are.
any extended agreement with reality may be purely accidental. LOL