The math behind water changes

[Scapexghost]

The power just went out so it seems like as good a time as any to talk about water changes. First, let's determine if a 25% water change every week is equal to a 50% every other week. At first, some may think that they are the same, while others will say that the 50% change is greater. In the case of the 50% water change, 50% of the original water is replaced. In the case of the 2 25% water changes, 25% of the original water is replaced, and then 25% of the 75% remaining original water is replaced, meaning the 2 25% water changes is only equal to a 43.75% water change. However, there is more to it. This is because the nitrate levels in the tank are changing.

So, let's map this out. Let's take 2 tanks, both producing 10 nitrates per week. The first tank we'll do 25% weekly water changes while the second we'll do 50% bi-weekly changes. At week zero, both tanks have zero nitrates. At the end of week one, both gain 10 nitrates. In tank one, we remove 2.5 nitrates, leaving 7.5. In tank two, we remove no nitrates, leaving 10. At the end of week two, after the water changes, tank one now has 13.125 nitrates and tank two has 10. After week three, tank one haa 17.34375 nitrates and tank two has 20 nitrates. After week four, tank one has 20.5078125 nitrates and two has 15 nitrates.

Let's stop for a moment, as two things are becoming appearant. First, it shifts back and forth between the two tanks as to which has more nitrates. Second, both tanks are increasing in nitrate levels despite the water changes. This is concerning, as it looks like the nitrates will keep increasing forever regardless of how many water changes we do. But will they?

Let's see. In tank one, the nitrates are increasing by 10 every week, and decreasing by 25%. So, the change in nitrates is equal to 10-.25N, with N being the number of nitrates in the tank. If we set this equal to zero, we can see what the nitrate level will be once the nitrates no longer change. So, 0=10-.25N, and N equals 40. So, the max nitrates in this tank are 40, and the nitrates will never pass this amount. So, what about tank two? Well, the formula for this tank is 20-.5N, and N will once again equal 40. So, in terms of maximum nitrate levels, a 25% weekly water change is the same as a 50% bi-weekly water change.

We can also use this formula to determine the ammount of water changes we need to do in any tank. Lets say n is the ammount of nitrates produced per week, N is the max ammount of nitrates we wish to allow, and x is the % of water we need to change to achieve this. So, the equation is 0=n-Nx. Solving for x, we get x=n/N.

Hopefully this helps answer the age-old question "how much water should i change," and clear any confusion around more, smaller water changes versus fewer, larger water changes.

Edited May 22, 2022 by Scapexghost
Added graph, added paragraphs

[Zac]

On 5/22/2022 at 1:08 AM, Scapexghost said:

The power just went out so it seems like as good a time as any to talk about water changes. First, let's determine if a 25% water change every week is equal to a 50% every other week. At first, some may think that they are the same, while others will say that the 50% change is greater. In the case of the 50% water change, 50% of the original water is replaced. In the case of the 2 25% water changes, 25% of the original water is replaced, and then 25% of the 75% remaining original water is replaced, meaning the 2 25% water changes is only equal to a 43.75% water change. However, there is more to it. This is because the nitrate levels in the tank are changing. So, let's map this out. Let's take 2 tanks, both producing 10 nitrates per week. The first tank we'll do 25% weekly water changes while the second we'll do 50% bi-weekly changes. At week zero, both tanks have zero nitrates. At the end of week one, both gain 10 nitrates. In tank one, we remove 2.5 nitrates, leaving 7.5. In tank two, we remove no nitrates, leaving 10. At the end of week two, after the water changes, tank one now has 13.125 nitrates and tank two has 10. After week three, tank one haa 17.34375 nitrates and tank two has 20 nitrates. After week four, tank one has 20.5078125 nitrates and two has 15 nitrates. Let's stop for a moment, as two things are becoming appearant. First, it shifts back and forth between the two tanks as to which has more nitrates. Second, both tanks are increasing in nitrate levels despite the water changes. This is concerning, as it looks like the nitrates will keep increasing forever regardless of how many water changes we do. But will they? Let's see. In tank one, the nitrates are increasing by 10 every week, and decreasing by 25%. So, the change in nitrates is equal to 10-.25N, with N being the number of nitrates in the tank. If we set this equal to zero, we can see what the nitrate level will be once the nitrates no longer change. So, 0=10-.25N, and N equals 40. So, the max nitrates in this tank are 40, and the nitrates will never pass this amount. So, what about tank two? Well, the formula for this tank is 20-.5N, and N will once again equal 40. So, in terms of maximum nitrate levels, a 25% weekly water change is the same as a 50% bi-weekly water change. We can also use this formula to determine the ammount of water changes we need to do in any tank. Lets say n is the ammount of nitrates produced per week, N is the max ammount of nitrates we wish to allow, and x is the % of water we need to change to achieve this. So, the equation is 0=n-Nx. Solving for x, we get x=n/N.

Hopefully this helps answer the age-old question "how much water should i change," and clear any confusion around more, smaller water changes versus fewer, larger water changes.

My brain hurts

[Scapexghost]

On 5/21/2022 at 11:56 PM, Zac said:

My brain hurts

Would a graph help?

[Cinnebuns]

On 5/22/2022 at 12:08 AM, Scapexghost said:

The power just went out so it seems like as good a time as any to talk about water changes. First, let's determine if a 25% water change every week is equal to a 50% every other week. At first, some may think that they are the same, while others will say that the 50% change is greater. In the case of the 50% water change, 50% of the original water is replaced. In the case of the 2 25% water changes, 25% of the original water is replaced, and then 25% of the 75% remaining original water is replaced, meaning the 2 25% water changes is only equal to a 43.75% water change. However, there is more to it. This is because the nitrate levels in the tank are changing. So, let's map this out. Let's take 2 tanks, both producing 10 nitrates per week. The first tank we'll do 25% weekly water changes while the second we'll do 50% bi-weekly changes. At week zero, both tanks have zero nitrates. At the end of week one, both gain 10 nitrates. In tank one, we remove 2.5 nitrates, leaving 7.5. In tank two, we remove no nitrates, leaving 10. At the end of week two, after the water changes, tank one now has 13.125 nitrates and tank two has 10. After week three, tank one haa 17.34375 nitrates and tank two has 20 nitrates. After week four, tank one has 20.5078125 nitrates and two has 15 nitrates. Let's stop for a moment, as two things are becoming appearant. First, it shifts back and forth between the two tanks as to which has more nitrates. Second, both tanks are increasing in nitrate levels despite the water changes. This is concerning, as it looks like the nitrates will keep increasing forever regardless of how many water changes we do. But will they? Let's see. In tank one, the nitrates are increasing by 10 every week, and decreasing by 25%. So, the change in nitrates is equal to 10-.25N, with N being the number of nitrates in the tank. If we set this equal to zero, we can see what the nitrate level will be once the nitrates no longer change. So, 0=10-.25N, and N equals 40. So, the max nitrates in this tank are 40, and the nitrates will never pass this amount. So, what about tank two? Well, the formula for this tank is 20-.5N, and N will once again equal 40. So, in terms of maximum nitrate levels, a 25% weekly water change is the same as a 50% bi-weekly water change. We can also use this formula to determine the ammount of water changes we need to do in any tank. Lets say n is the ammount of nitrates produced per week, N is the max ammount of nitrates we wish to allow, and x is the % of water we need to change to achieve this. So, the equation is 0=n-Nx. Solving for x, we get x=n/N.

Hopefully this helps answer the age-old question "how much water should i change," and clear any confusion around more, smaller water changes versus fewer, larger water changes.

Idk why my ADHD didn't kick in and start skipping parts but I loved that. But I'm also the weirdo that took Calc 2 in college as a music major for fun. 

[nabokovfan87]

On 5/21/2022 at 11:16 PM, Cinnebuns said:

Idk why my ADHD didn't kick in and start skipping parts but I loved that. But I'm also the weirdo that took Calc 2 in college as a music major for fun. 

Calc 3 is definitely a LOT more fun than Calc 2.  Oddly.  Calc 2 (for me) got into a bunch of weirdness and down a rabbit hole I just got annoyed by.  Infinitely adding an axis to a coordinate system and then doing things over and over.

I wonder if this equation is similar to (or how similar to) a lot of the math on something like half life.  There's got to be some crossover there. The difference being, you're assuming with half-life the nitrates are your chemical and they never increase. Higher water changes means a shorter half life. Smaller water changes means a longer half life for your "tank issues".

Very interesting stuff.

[Scapexghost]

On 5/22/2022 at 2:48 AM, nabokovfan87 said:

Calc 3 is definitely a LOT more fun than Calc 2.  Oddly.  Calc 2 (for me) got into a bunch of weirdness and down a rabbit hole I just got annoyed by.  Infinitely adding an axis to a coordinate system and then doing things over and over.

I wonder if this equation is similar to (or how similar to) a lot of the math on something like half life.  There's got to be some crossover there. The difference being, you're assuming with half-life the nitrates are your chemical and they never increase. Higher water changes means a shorter half life. Smaller water changes means a longer half life for your "tank issues".

Very interesting stuff.

Calc 2 is one of the hardest math courses I've taken. In Calc 2 you basically learn all the calculus and in calc three you just apply that same math in 3 (or 4, or 5) dimensions.

The equation for the graph is y=N-Ne^(-wx) with w being the percentage of water changed as a decimal. Certainly similar to a radioactive decay formula.

[nabokovfan87]

On 5/22/2022 at 2:13 AM, Scapexghost said:

Calc 2 is one of the hardest math courses I've taken. In Calc 2 you basically learn all the calculus and in calc three you just apply that same math in 3 (or 4, or 5) dimensions.

Ah for me. I stopped with this one, had to look it up!

Introduction to Linear Algebra and Differential Equations

---

Introduction to the solutions of ordinary differential equations and their relationship to linear algebra. Topics include matrix algebra, systems of linear equations, vector spaces, linear independence, linear transformations and eigenvalues.
 

Edited May 22, 2022 by nabokovfan87

[Scapexghost]

On 5/22/2022 at 3:40 AM, nabokovfan87 said:

Ah for me. I stopped with this one, had to look it up!

Introduction to Linear Algebra and Differential Equations

---

Introduction to the solutions of ordinary differential equations and their relationship to linear algebra. Topics include matrix algebra, systems of linear equations, vector spaces, linear independence, linear transformations and eigenvalues.
 

Oh yeah that was rough. I didnt find linear algebra to bad but the ODEs were killer.

[modified lung]

You gotta break up the paragraph and use bullet points or something.

[Scapexghost]

On 5/22/2022 at 8:11 AM, modified lung said:

You gotta break up the paragraph and use bullet points or something.

You got it

[Wrencher_Scott]

All I want to know is if do I have 10ppm or 20ppm or 15maybe ? or,  Is there red in that??

according to that API test. 😉

[Zac]

On 5/22/2022 at 2:11 AM, Scapexghost said:

Would a graph help?

Lol that’s about all my bio major brain can comprehend. I tapped out of math after calc 1 

[Patrick_G]

@Scapexghost, you should make a little calculator in HTML!

[Cory]

This is essentially my job. An algebraic equation is too advanced for the public. Also people would do the math once and up not updating the equation.

The first hurdle is to convince people that a water change every week isn't magical. Even though "it's always worked for me, it's just what I do". Then the next hurdle is to get people to test water. "I never test and my do great". 

Over the years working in a store, writing for blogs, doing youtube. I found that those who want to learn will, and the easier you can make it, the more will be successful in learning. My last attempt was this infographic on water changes.

https://www.aquariumcoop.com/pages/water-changes

[xXInkedPhoenixX]

So @Cory according to this chart you have if you have 50 Nitrates or less (and provided all the other parameters are good)- zero water changes need to happen? (My water ALWAYS tests 40ppm)

[Cory]

I believe that to be correct. Assuming ph, hardness and buffer tested out fine.

[anewbie]

The problem with the math is you presume there is a function that well defines water condition over time and you can therefore compare weekly water changes at 25% to bi-weekly 50% in some fashion of effectiveness. At the very best you can only state that for your aquarium it changed some parameters by so much but even then that ignores both unmeasured parameters and parameters not easily measured. I know that when I change the water in one of my tanks the kuhli all come out and greet me (of course in the tank next to it the kuhli hide during the whole process - so go figure). Like wise when i add leaf extract to another tank the clown loaches come out to see what is going on (as long as i don't look at them). Anyway i'm trying to figure out now why my 40B nitrate were running at 40ppm despite weekly water changes (50%); BUT when i missed a water change it went to 5ppm.... go figure.

[modified lung]

On 5/22/2022 at 9:24 AM, Patrick_G said:

@Scapexghost, you should make a little calculator in HTML!

If you're interested in a google sheets calculator, I got one I can post.

[Patrick_G]

On 5/22/2022 at 10:35 AM, modified lung said:

If you're interested in a google sheets calculator, I got one I can post.

Absolutely 

[modified lung]

On 5/22/2022 at 10:41 AM, Patrick_G said:

Absolutely 

I'll post a bunch of them in a couple days and tag you. I want to write up some instructions for a few of them first.

[Scapexghost]

On 5/22/2022 at 9:44 AM, Wrencher_Scott said:

All I want to know is if do I have 10ppm or 20ppm or 15maybe ? or,  Is there red in that??

according to that API test. 😉

They should make an app that can determine the exact ppm based on the rgb of a picture of the test tube. Most of the time i just think "orangish good, redish bad"

[Scapexghost]

On 5/22/2022 at 11:34 AM, anewbie said:

The problem with the math is you presume there is a function that well defines water condition over time and you can therefore compare weekly water changes at 25% to bi-weekly 50% in some fashion of effectiveness. At the very best you can only state that for your aquarium it changed some parameters by so much but even then that ignores both unmeasured parameters and parameters not easily measured. I know that when I change the water in one of my tanks the kuhli all come out and greet me (of course in the tank next to it the kuhli hide during the whole process - so go figure). Like wise when i add leaf extract to another tank the clown loaches come out to see what is going on (as long as i don't look at them). Anyway i'm trying to figure out now why my 40B nitrate were running at 40ppm despite weekly water changes (50%); BUT when i missed a water change it went to 5ppm.... go figure.

Thats true, fish can breed and die, fish can grow, food can change, all affecting nitrate build up. Plants are even more complicated. A tank with few plants will process few nitrates, but a tank full of plants will process few nitrates bc there is no room to grow. So, a tank half full of plants will process the most nitrates. If the blue line is plant mass, the red line is nitrate consumption.

Still, it is simple to adjust water changes if you are keeping track of your nitrate build up. If nitrates are plateuing at 60 instead of 40, increase water changes by 50%. If nitrates are plateuing at 20 instead of 40, decrease water changes by 50%.

 

[Scapexghost]

On 5/22/2022 at 11:01 AM, Cory said:

This is essentially my job. An algebraic equation is too advanced for the public. Also people would do the math once and up not updating the equation.

The first hurdle is to convince people that a water change every week isn't magical. Even though "it's always worked for me, it's just what I do". Then the next hurdle is to get people to test water. "I never test and my do great". 

Over the years working in a store, writing for blogs, doing youtube. I found that those who want to learn will, and the easier you can make it, the more will be successful in learning. My last attempt was this infographic on water changes.

https://www.aquariumcoop.com/pages/water-changes

Thats all true. The biggest issue is that everyone wants an easy answer and in fishkeeping there rarely is. Often people will ask "How often should I change water," but that is something they have to determine themselves, all anyone can do is guide them through the process of finding the answer. Even if someone doesn't use my formula, the main takeaways I want people to get are:

-doubling water change frequency is the same as doubling water change volume, halving water change frequency is the same as halving water change volume, etc.

-assuming nitrate build up is constant, nitrate levels will plateu given a consistent water change schedule.

[lefty o]

math 😲 not for this guy. 1+1= ahh heck i just change water when its needed, not on a schedule.

[anewbie]

To be honest for most of my tanks i do 50% twice a week. Also I have quite a few fishes that prefer extremely clean water and they will frequently get very lively after a large water change. Having said this - there are fishes that are intolerable to change and for them you have to be more careful about water changes and focus on smaller (and sometime less frequent) changes. Then there are species that are intolerable to any impurity and change (many of them are black-water fishes) and they are the most finicky requiring frequent but very small water changes. There is an interesting story about on specific species where the owner talks about how it would pout every time he made the smallest change to water chemistry.  Anyway i try to keep nitrate below 10 in my tanks but it really come down to species - some species are more at home in dense water than others. I tend to find live bearers the most tolerant to thick water but this is a generality. Back in the days before the internet and we didn't know about water changes we kept a 20 long in a dorm with a pair of swordtails and by the year end we had a mile of floating plants and 30 or 40 fishes that looked extremely healthy after a year of no water changes.