# How to Calculate Water Change Amount and Frequency



## solarz (Aug 31, 2010)

Assume:
x = amount in ppm of NO3 produced per _time interval_
z = % of water change done once every interval

let y = (1-z), and we can calculate the concentration of NO3 in water for successive time intervals T

T1: xy
T2: (xy + x)y
T3: ((xy + x)y + x)y
T4: ...

If we expand this and add another x to the result, we get the following infinite series:

x + xy + xy^2 + xy^3 + ... + xy^n => x(1 + y + y^2 + ... + y^n)

For those a bit rusty with high school math, the above is a http://en.wikipedia.org/wiki/Geometric_series, which means that:










replace r with y, and we get:

x(1/(1-y)) = x/(1-y) => *x/z*

It's that simple: if your tank produces 10ppm of NO3 per week, and you do a 30% water change per week, after a few weeks, your NO3 will stabilize at:
10 / 0.3 = 33.3 ppm. If you want to control it to 20ppm, you'll need a 50% water change per week.

The time interval can be anything: 1 week, 2 weeks, 1 month, etc.

The tricky part is measuring x. You can do that by measuring NO3 right *after* a water change, taking down the number, and then measuring again a week later, *before* doing another water change. However, with most home test kits, it's hard to get an exact number. The colors for 10ppm and 20ppm on my test kit is almost identical to my eyes, and since the next color is 40ppm, I can never tell if my NO3 is closer to 40ppm or to 20ppm.

Also note that the "stable value" I mentioned above is the NO3 value right before your schedule water change. So if you change 30% water every week, and your tank produces 10ppm each week, your NO3 will actually vary between 23.3 to 33.3 ppm over the week.

Let me know what you think, or if you think there's an error in my reasoning.


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## Zebrapl3co (Mar 29, 2006)

Holy crux! 
I don't think there is any thing wrong with your caculation.
But, I think your x variable constant is not really constant but a running constant. You can't gaurentee x will always be the same every week. It's dependent on the amount of food you feed, fish appetite (believe it or not), tank temperature and how clog is your filter.
But assuming that if x is always 0 after a water change ... x/z = 0/z = 0, then it's all good ... 
heh heh, just joking. It didn't matter. As long as you do a weekly water change, you should be fine, provided that the tap water is relatively nitrate free.

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## solarz (Aug 31, 2010)

Zebrapl3co said:


> Holy crux!
> I don't think there is any thing wrong with your caculation.
> But, I think your x variable constant is not really constant but a running constant. You can't gaurentee x will always be the same every week. It's dependent on the amount of food you feed, fish appetite (believe it or not), tank temperature and how clog is your filter.
> But assuming that if x is always 0 after a water change ... x/z = 0/z = 0, then it's all good ...
> heh heh, just joking. It didn't matter. As long as you do a weekly water change, you should be fine, provided that the tap water is relatively nitrate free.


Yeah, this is not meant to be precise by any means. It's more of an estimate for deciding how often and how much water change is needed.

It's also interesting because it's another way to demonstrate the effect of bioload on a tank. Greater bioload = more NO3 produced = more/bigger water changes required.

I think it's also a better estimate than the "1 inch of fish per gallon" rule, as x already takes into account the effect of plants and the types of fish.


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## Zebrapl3co (Mar 29, 2006)

Well, if you really really want to play around with math. I still remember reading a thread on another forum that calculate the difference between daily water change as compare to end of the week water changes (of equal volume of water change).
If I remember correctly, the math comes out the same. But when it was actually experiemented on, it turns out that daily water changes was better than the once a week (both at equally volume of water changes by the end of the week). I can't remember exactly the reason, but I think it has to do with the theory that at smaller volume of nitrate, it gets processed by the nitrate eating bacteria (yes, such bacteria exist, it's just that it's not fast enough to keep up with the fish tank) and hence didn't build up as bad and therefore always remain at a better quality of water. Don't quote me on this one, but I believe the result of the test was that daily water changes was better than weekly water changes.

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## solarz (Aug 31, 2010)

Zebrapl3co said:


> Well, if you really really want to play around with math. I still remember reading a thread on another forum that calculate the difference between daily water change as compare to end of the week water changes (of equal volume of water change).
> If I remember correctly, the math comes out the same. But when it was actually experiemented on, it turns out that daily water changes was better than the once a week (both at equally volume of water changes by the end of the week). I can't remember exactly the reason, but I think it has to do with the theory that at smaller volume of nitrate, it gets processed by the nitrate eating bacteria (yes, such bacteria exist, it's just that it's not fast enough to keep up with the fish tank) and hence didn't build up as bad and therefore always remain at a better quality of water. Don't quote me on this one, but I believe the result of the test was that daily water changes was better than weekly water changes.


That's very interesting, and shows how you can't take into account everything with a simple mathematical model. 

I know that nitrate-eating bacteria do so under anaerobic environments, but I've never found any information on how effective they are in FW tanks.


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## ubr0ke (Jul 12, 2011)

I don't know if you guys have ever used this or not but here's a cool calculator
http://ei.petalphile.com/


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## J-P (Feb 27, 2011)

that formula is a little off.

It doesn't take into account fish growth, feeding and some other factors.

The general thought id 50% weekly. That way you will know exactly what your bioload is, BUT that changes. An NO, Nitrates don't stabilize over time. I have seen tanks that were consistently over the 100ppm mark.

When your nitrates are 20+ it is time for a water change. You'll get into a rhythm and will eventually stop testing weekly and go monthly.

The exception is planted tank that will throw off your reading and should be done weekly regardless.

This was just posted on The Puffer Forum:
http://www.thepufferforum.com/forum/library/tnk-mgr/vi-water-change-math-by-robert-t-ricketts/


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## BillD (Jun 5, 2006)

solarz said:


> I know that nitrate-eating bacteria do so under anaerobic environments, but I've never found any information on how effective they are in FW tanks.


They can be very effective if using a plenum system. However, the plenum needs to be a certain depth as does the gravel (of the correct size) above it to allow the correct water flow between the anoxic area under the plenum and the rest of the tank. It won't work with a tank of diggers.


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## solarz (Aug 31, 2010)

J-P said:


> that formula is a little off.
> 
> It doesn't take into account fish growth, feeding and some other factors.
> 
> The general thought id 50% weekly. That way you will know exactly what your bioload is, BUT that changes. An NO, Nitrates don't stabilize over time. I have seen tanks that were consistently over the 100ppm mark.


Consistently over 100ppm is still stabilization. Non-stable means NO3 will keep increasing over time, with no upper limit. (It can also mean dropping, but that's not what concerns us here.)

Feeding is already taken into account, assuming the feeding is done regularly. Fish growth does change the bioload over time, but it's a slow enough change that you can just take another measure after a couple of months, assuming you have fast growing fish.

50% weekly is okay for hardy fish, but not for delicate species like CRS. If you ever wondered whether your 10% weekly water changes were doing anything for your shrimp tank NO3, now you can find out.

I find that large water changes are also not practical in winter. Your tap water will always be very cold, and if you do a large water change, you have to shut off your heaters and filters or else risk leaving them running dry. This means that you will be refilling very cold water into a tank that's normally 75F. In winter, I usually do 30% WC.


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## J-P (Feb 27, 2011)

solarz said:


> Consistently over 100ppm is still stabilization. Non-stable means NO3 will keep increasing over time, with no upper limit. (It can also mean dropping, but that's not what concerns us here.)
> 
> Feeding is already taken into account, assuming the feeding is done regularly. Fish growth does change the bioload over time, but it's a slow enough change that you can just take another measure after a couple of months, assuming you have fast growing fish.
> 
> ...


Over 100 ppm isn't stabilization.. hobby test kits can't read that high. You are correct about the water changes. How you reduce the nitrates in the tank is up to the owner. Several smaller partials are recommended because it is easier to match the temperatures and not shock the fish or critters. There is still a need for water changes. They have to be done regardless.


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## ubr0ke (Jul 12, 2011)

Im a little lost...why would nitrates stablize at 100ppm?..nitrates would continue to climb until a water change was done...


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## solarz (Aug 31, 2010)

ubr0ke said:


> Im a little lost...why would nitrates stablize at 100ppm?..nitrates would continue to climb until a water change was done...


No, I'm saying that as long as you do water changes regularly, your nitrates will stabilize at a certain concentration, even if it's over 100ppm.

So even if your tank produces 100ppm NO3 per week, and you only do 10% WC per week, NO3 will still stabilize at 100/0.1 = 1000ppm.

The point is that with the formula, you can figure out how much/often WC you need to do.


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