[GoofBoy]

Couple of posts regarding water changes and nitrates got me thinking so I figured I finally needed to work this out for myself. So, I got my math on.

I will assume you want to do the same water change routine every week. How much water should I change to keep the Nitrates below a certain value?

Figure your tank generates N nitrates per day.

In a week you get 7N Nitrates in your tank.

You do a P percent water change at the end of a week. Assume 0 Nitrates in the incoming water.

You have 7N (1-P) of the Nitrates left after the water change.

Next week 7N nitrates generated again added to the previous left over:

7N + 7N (1-P)

You do a water change and are left with:

(7N + 7N (1-P))(1-P) = 7N (1-P) + 7N (1-P)(1-P)

Next week you get before the water change you end up with

7N + 7N (1-P) + 7N (1-P)(1-P)

Well, it looks like we have a geometric series of the form:

a + ar + ar^2 + ar^3 + ar^4 = ...

If r is less than 1 (and for us it is) the convergence for a series of this form will be:

a / (1 -r)

and,

a = 7N
r = (1 -P)

so,

(7N)/(1 - (1-P)) = 7N/P

For Maximum Steady state Nitrate levels then it is simply:

(Nitrate Increase per time period / Percentage Waterchange per time period) = Nitrates in your tank just before your next water change.

21ppm Nitrate Increase per week / 75% waterchange = 28ppm Nitrates Max before next water change

21ppm Increase in Nitrates per week / 30% water change = 70ppm Nitrates Max before next water change

21ppm Increase in Nitrates per week / 10% water change = 210ppm Nitrates Max before next water change

Makes too much sense that the steady state you will be removing what is added each week as in 10% of 210ppm is 21ppm - which matches the water change.

These values are for final steady state - so if your nitrates are going up you are not removing as much with your water changes as is being created by the tank each week.

This calculation shows where the stopping point will be.

Hope this helps someone.

 

[Number6]

I used excel once upon a time... my math showed the exact same result... if you have a steady increase in nitrate, then your water change routine eventually reaches a balance point. As soon as I realized this, I made sure to do a very large water change on many systems every "x" duration in order to do a reset. 95% on some setups...

 

[dielikemoviestars]

Oh math, how I've missed you since college.

The one issue here is assuming your new water has 0ppm NO3. I don't think most tap water does. At least not around here. Either way, I'm sure there's a way to incorporate that variable in the mix relatively easy, since it's still a ratio of X gallons at Y ppm and A gallons at B ppm.

 

[Number6]

If there is nitrate in your source water, you simply add that into the formula and it still works out.

 

[GoofBoy]

If there is nitrate in your source water, you simply add that into the formula and it still works out.

Specifically, it will work out to be:

(Nitrate Increase per time period / Percentage Waterchange per time period) + Nitrates in water = Nitrates in your tank just before your next water change.

So in the cases I listed if you assume 10ppm Nitrates from tap water.

21ppm Nitrate Increase per week / 75% waterchange = (28ppm + 10ppm from tap water) Nitrates Max before next water change or 38ppm

21ppm Increase in Nitrates per week / 30% water change = (70ppm + 10ppm from tap water) Nitrates Max before next water change 80ppm

21ppm Increase in Nitrates per week / 10% water change = (210ppm + 10ppm from tap water) Nitrates Max before next water change 220ppm

Basically, just add the concentration in the tap water to the end amount.

 

[Number6]

GoofBoy
No... you don't add the two ppm numbers unless your source water and tank water were a 50/50 mix.

E.g. if you add 30% source water with 10ppm to 70% tank water at a higher ppm like 70ppm then the end result is between 10ppm and 70. I haven't finished my coffee this morning so I'm not going to attempt the math just yet.

The only math I can do right now is brain + coffee= normal; brain - coffee < normal.

 

[GoofBoy]

I beg to differ and stand by the offset, please see below.

Tap water percentage is T

So Initial concentration is T before anything.

After 1 week:

7N + T

And the water change on week one TP is amount added in with the tap water

(7N + T)(1-P) + TP =>

(7N + T)(1-P) + TP = 7N - 7NP + T - TP + TP = 7N(1-P) + T = T + 7N(1-P)

So you are at T + 7N(1-P) after the week one water change.

Second week Nitrate buildup gives,

T + 7N(1-P) + 7N = T + 7N + 7N(1-P)

Water change,

(T + 7N + 7N(1-P) )(1-P) + TP =>

(T + 7N + 7N(1-P))(1-P) + TP = (7N + 7N(1-P))(1-P) + T(1-P) + TP = (7N + 7N(1-P))(1-P) + T

= T + 7N(1-P) + 7N(1-P)(1-P)

Add nitrates for third week

= T + 7N + 7N(1-P) + 7N(1-P)(1-P)

The nitrates series now becomes => T + a + ar + ar^2 + ar^3 + ar^4 = T + a / (1 -r)

And T is just the offset for the converging sum.

No fair, I've had some coffee.

 

[marvo]

this is to funny welcome to college ego 101

 

[Rhinox]

Hey I did this a while back. Without all the series and what not, N2=N1-N1x+n, gives you the same answer a little easier, when you set N1=N2. N1 = current nitrate level before water change, N2 = next weeks nitrate level before water change, x = water change percent, n = weekly nitrate production. Yes, the steady state max nitrate level before water change N = n/x. This works for any time interval too, it doesn't have to be weekly. You just have to know how much Nitrate is produced during the interval you want to do your water changes.

{If your tap has nitrate, the equation is N2=N1-N1x+NTx+n, where everything is the same except NT is nitrate concentration out of the tap}

edit: fixed, I have NOT had coffee

edit2: That means with nitrate in the tap, when N2=N1=N, then N=n/x + NT, so yeah goofboy is right.

The really interesting thing is when you look at it in plot form as Nitrate Level before water change on the Y axis vs % weekly water change on the X axis, for a given "nitrate produced per week" level.

The Y axis is a factor - you multiply it by your tanks' individual nitrate production for any time interval you want to do a water change, and the x axis is the % water change you do consistently at the same time interval. This is for a 0 nitrate tap by the way.

What this tells me is that you get diminishing returns as you increase your water change %, for a constant nitrate production. If you're doing small water changes already (<50%, but more noticeably <35%), you can improve water quality substantially with a small increase in weekly water change %. If you're already doing large water changes (50% or more), you really won't see a big improvement if you increase the size of your water changes.

 

[GoofBoy]

this is to funny welcome to college ego 101

Sad to say, college was well over 20 years ago.

This topic comes up quite a bit as far as how big of a water change to do and the effects thereof - lots of people have argued over it with no math behind the arguments.

Ego has nothing to do with it, I'm too old to be that insecure - Fish Geek 101 I'll humbly embrace, I just want to find the answers so I can understand and explain when it comes up again.

Rhinox

I'm stupid and have a huge ego so I have learned (

) - I don't understand how to use the chart.

What determines Y?

Could you use the example above we have 21ppm per week.

What Y do I use?

Thanks

 

[Rhinox]

Could you use the example above we have 21ppm per week.

Yeah, just multiply the Y axis, every value, by 21. Then the chart will tell you what the nitrate level will top out at (Y axis) for any water change percent (X-axis).

So for the example, if you're doing 50% water changes, nitrate will top out at 42ppm. If you're doing 20% water changes, nitrate will top out at 105ppm. etc.

If for example you never want your nitrate to go above 50ppm, and you want to figure out how big of a water change you should do weekly if your bioload is 21ppm, then its just a linear relationship between bioload and water change, so water change = 21/50 = 42% per week. If you check that on the chart, at 42% water change, the factor is ~2.4, so 2.4 times 21 = 50.4ppm. Close enough.

edit: this comes from N=n/x - rearrange for x=n/N

 

[GoofBoy]

To restate for the way I think I understand it.

Multiply your PPM generated by the Y value of the point on the line that has your water change %.

That gives the max concentration of Nitrates you will stabilize at.

Got it, thanks.

The chart gives a great visual on why bigger is definitely better.

Ciao.

 

[Rhinox]

To restate for the way I think I understand it.

Multiply your PPM generated by the Y value of the point on the line that has your water change %.

That gives the max concentration of Nitrates you will stabilize at.

Got it, thanks.

The chart gives a great visual on why bigger is definitely better.

Ciao.

Yep thats exactly right :thumb:

 

[Number6]

Your math with nitrates in the source water isn't correct.

Tank has a nitrate level of x ppm.
You remove water. Your tank has a nitrate reading of?
still x.

If you add source water with a lower ppm level of y, your result (and this can be verified by a nitrate test kit) is... between x and y.

The end result is proportionate to the % of source vs tank water.

 

[GoofBoy]

Your math with nitrates in the source water isn't correct.

Tank has a nitrate level of x ppm.
You remove water. Your tank has a nitrate reading of?
still x.

I think there is a slight disconnect between concentration and actual amounts in the total volume that is causing the disjoint.

x is in parts per million as you said - lets call it 40ppm.

For simplicity can we just call it parts per gallon

Gallon=million parts.

So x equal 40ppg.

1 Gallon that has x = 40ppg contains a total of 40 parts
2 Gallons that have x = 40ppg contains a total of 80 parts
10 gallons that hold x = 40ppg contains a total of 400 parts

All have the exact same x= 40ppg reading.

That 10 gallons contains 400 parts and it reads 40ppg, if I take away 8 gallons and get a reading of 40ppg on the remaining 2 gallons. Those 2 gallons at 40 ppg contain a total of 80 parts

The end result is proportionate to the % of source vs tank water.

It is.

50 gallon tank
10 ppg Tap water
30 ppg added during the week.
25% waterchange

Fill tank with 10ppg tap water and age one week (+30ppg) total concentrate is 40 ppg.

Borrowing from above looking at total parts in the tank:

40 ppg * 50g = total of 2000 parts

Remove 25% or the water (12.5 gallons leaving 37.5 gallons)

40 ppg * 37.5 g = 1500 parts left

The concentration is the same 40 ppg, we now have one quarter less parts total left because we took away one quarter of the water.

Add 12.5 gallons at 10 ppg tap water so a total of 125 parts added.

Total parts in the full tank after water change

(40 ppg * 37.5g) + (10 ppg * 12.5 g) = 1625 parts

1625 parts / 50 gallon tank size = 32.5 ppg concentration

Looking at the formula same parameters:

7N = 30 ppg from aging
T = 10 ppg from tap water
P = .25 (water change %)

(7N + T)(1-P) + TP

(7N + T) = (your x) full concentration before water change
-times-
(1-P) = amount of water left in tank as a percent (proportial part)

-plus-

T = (your y)concentration of tap water
-times-
P = tap water added to tank as a percent (proportial part)

(30 + 10)(1 -.25) + (10)(.25)
40*.75 + 2.5 = 32.5 ppg

Same result.

The final value 32.5 ppg arrived at both ways is between your x (40 ppg)and y (10 ppg) and is proportionate to the water change size P.

You are correct.

The algebra shows the final series sum proportional to the water change with an offset of the concentration of the tap water, assuming I didn't screw it up - which makes sense to me as that is the baseline concentration in every drop of water going through the tank - you simply cannot remove or change it.

Does this make any sense?

Edit:

If you were referring to this:

Specifically, it will work out to be:

(Nitrate Increase per time period / Percentage Waterchange per time period) + Nitrates in water = Nitrates in your tank just before your next water change.

This is the steady state max level of nitrates you will end up at after some long period of time of doing the same size water changes each week and not a calculation of a single water change.

 

[Number6]

Edit:

If you were referring to this:

Specifically, it will work out to be:

(Nitrate Increase per time period / Percentage Waterchange per time period) + Nitrates in water = Nitrates in your tank just before your next water change.

This is the steady state max level of nitrates you will end up at after some long period of time of doing the same size water changes each week and not a calculation of a single water change.

Thanks! I had to reread the thread to see where I thought you were off and you are right... I misread the above.


The other thing that keeps throwing me off though is the first part of the formula:
(7N + T)(1-P) + TP

It reads as if 10ppm nitrate is being added, when that is actually the theoretical starting point of the tank and you are assuming that it will match tap water (which it does in your example).

For purely aesthetic reasons, I think it deserves it's own Variable name.

(7N+X)(1-P) +TP where X= Reading of Nitrate on Tank.

Apologies for misreading and thanks for the patient corrections! :thumb:

 

[dielikemoviestars]

If anyone had any doubt that fish people are nerds... :lol:

 

[GoofBoy]

Apologies for misreading and thanks for the patient corrections! :thumb:

Absolutely no worries.

The thing I wanted to understand was - If you start a tank and do the same size water change every week what ends up happening to the Nitrates based on water change schedule?

I've seen plenty of water change threads and could never feel truely confident on a hard answer.

I get it now, and appreciate the challenges to make sure I understand thoroughly by being able to explain it.

If anyone had any doubt that fish people are nerds...

Your point is...

Ciao,

EnginerdBoy

 

[johnnyo513]

I just test my water for nitrates, usually before and after cleaning. In all four of my tanks. If the amount of Nitrates overall continues to remain high (never goes beyond 20ppm) I just do a larger water change the next time.........no math to worry about.......

 

[zimmy]

Thanks GoofBoy for posting this very helpful information. Maybe the thread should be moved to a more general forum (Tank Setup or General Aquaria Discussion?). If I wasn't looking at Malawi species I might have missed this.