Originally Posted by
pascal
Mathematical approach to waterchanges
There have been many articles about waterchanges but very few about an analytical approach. Articles have been published by Randy Holmes-Farley (1) in 2005 and later by David Boruchowitz (2), (3) in 2009. These articles were very interesting and describe the build-up of waste when applying a waterchange schedule. In these articles the author applied an iteration method simulating what happens in the tank. The result was reported under the form of tables and graphs.
Until today, I never read an analytical mathematical approach to the problem, resulting in a single formula describing the concentration of nitrates that is expected as a result of fish population density (or food administration quantity) combined with a waterchange schedule.
The principle of dynamic equilibrium
The approach applied to derive this formula is based on the principle of a dynamic equilibrium. This can be illustrated with an example.
We start a tank with a perfect running filter and pure water. Every day I feed the fish, which will result in a slight build-up of nitrates. Let’s forget the steps involving nitrite and ammonium, we want to keep it simple. After a week, I apply a water change. I will almost eliminate no nitrates, since the water is still very clean. I continue to apply my water change schedule. After two months, I will eliminate much more nitrates during the same waterchange, just because the waste concentration in the water is much higher.
At a certain moment, the absolute value (expressed in gram) of nitrates I will eliminate with my periodic waterchange will be exactly the same as the absolute amount of nitrates I added during this same period by feeding the fish. From then on, there will be a dynamic equilibrium and the situation will remain constant in time.
The concentration of nitrates can’t increase, because we eliminate as much as we added during the same period. It will not decrease, because we never eliminate more than we added in the same period. Between two waterchanges, there will be a slight increase, but this situation will remain constant indefinitely.
What is the relation between mass of food and nitrate concentration?
Food has mainly 4 components : water, proteins, lipids and carbohydrates. In this article, we will only consider dry food. I mean if I feed 10g of frozen artemia, containing 90% of moisture, I will only add 1g of dry food.
As a mean, we can say that fishfood contain 45% of proteins. Some contains more (artemia more than 60%, some less like daphnia), but most of the flakes and pellets contain this ratio of proteins. Let’s take that value. This means that 1 g of dry food contains 0.45 g of proteins.
Proteïns are build up by aminoacids. These molecules contain nitrogen and will result in the build up of nitrates. They contain as a mean value 16% of nitrogen, this value can be found on the internet. So 1 g of dry food will result in maximum 1 x 0.45 x 0.16 = 0.072g of nitrogen.
When this nitrogen has passed the nitrogen cycle, resulting in nitrate (NO3-), these 0.072g of nitrogen will give rise to 0.32g of nitrate. So now I know that 1g of dry food can result in an amount of 0.32g of nitrate when the nitrogen cycle has done his job.
Building up the equation
We have explained that after a certain period, a dynamic equilibrium will be established. That means that with the applied waterchanges, the amount of waste (say nitrates) that is eliminated is equal to the amount of waste (nitrates) that was accumulated during this same period. Only then, we have a dynamic equilibrium and a constant situation. Simply said, when the amount of waste (nitrates) that I eliminate with a waterchange is equal to the amount that was added (by feeding fish) during the same period, the amount of waste will remain constant in time.
Consider we feed every day D gram of dry food, this results in an equivalent situation where we would add (D x 0,32) gram nitrates daily to the tank. Expressed in mg, this value will be 1000 times higher : (D x 320) mg.
Consider I apply a regular waterchange since a long time (I’m at equilibrium) and every p days (period “p” is expressed in days and is the time between 2 waterchanges, once a week means p = 7 days) I perform a waterchange of v%, expressed in % of the total volume. The total volume of water in the tank is V, expressed in liter.
The amount of water I change is then (v/100 x V). Example, if I have a tank of 300 l (V) and I’m changing 25% (v), the amount of water I’m changing is indeed 25/100 x 300 = 75l.
Then, at the equilibrium, the amount of nitrates I will eliminate is equal to the amount of nitrates that I added since the last waterchange. If I add daily an amount of D x 320 mg nitrates by feeding the fish, I will have cumulated in p days (since my last waterchange) an amount of p x D x 320 mg. Indeed, I know that at equilibrium, I’m eliminating with a waterchange exactly the amount of nitrates that I added in that same period.
Example, if p = 7 days, D= 0.2g, then the amount of nitrates eliminated will be p x D x 320 = 7 x 0.2 x 320 = 448 mg of nitrates.
I can then calculate what was the concentration of nitrates that was present in the tank just before the waterchange, by dividing the amount of nitrates I know (mg) by the volume of water in which this amount was present (liter).
If there is (p x D x 320) mg nitrate present in a volume of (v/100 x V) liter, the concentration of nitrates (mg/l) at that moment is the amount of nitrate (p x D x 320), divided by the volume of water (v/100 x V) in which this amount of nitrates was present.
This gives us our first formula, expressing the maximum nitrate concentration as a function of food administration and waterchange schedule (amount and frequency) :
n = (p x D x 32000) / (v x V)
Where
n : maximum concentration of nitrates present in the tank, due to administration of food, expressed in mg/l
p : period between two waterchanges, expressed in days
v : percentage of waterchange, expressed as % of the total volume
D : Amount of dry food administrated, expressed in gram
V : Volume of water in liter. Make the correction for filter, decoration and sand.
An example :
I apply a weekly waterchange of 50% on a 300l tank and feed 1g of dry food daily.
p = 7 days
D = 1 gram
v = 50%
V = 300 l
The maximum nitrate concentration will be n = 15 mg/l. After the waterchange, concentration will be 15/2 = 7,5 mg/l.
If I apply a daily waterchange of 7 %, resulting in almost 50% per week :
p = 1 day
v = 7%
I obtain the same maximum concentration n = 15 mg/l before each waterchange daily. After the waterchange, concentration will be 15 x (100-7)/100 = 14 mg/l. That means that changing daily water is less efficient than equivalent large waterchanges. Less efficient doesn’t mean worse in my opinion. It is sufficient to change frequently a little bit more.
A tank can eliminate by itself a certain degree of nitrates by plants, anaerobic zones in sand, decoration, filtration substrate. The concentration of nitrates will be lower than expected. We express this by multiplying the formula with a correlation factor k, which will have a value between 0 and 1. We can estimate this value by measuring the nitrate concentration on a tank where the waterchange schedule is applied since a longer period on one hand and calculating the theoretical value on the other hand. I did the exercise for my tank and I estimated k equal to 0.4 – 0.6.
The formula becomes :
N = (k x p x D x 32000) / (v x V)
Where k is the correction factor for the tanks ability to reduce a part of the nitrates. If you want to overestimate the nitrateconcentration, use k = 1. N becomes the estimated concentration of nitrates expressed in mg/l.
Some of us use tapwater, having a startconcentration of nitrates. Since this concentration of nitrates is present at the start and also in the water that is used for waterchanges, we can add this value as a constant to the formula. Be N0 the startconcentration of nitrates of the tapwater expressed in mg/l. The final formula becomes :
N = [(k x p x D x 32000) / (v x V)] + N0
Where :
N : the estimated amount of nitrates present in the tank, due to administration of food, expressed in mg/l
p : period between two waterchanges, expressed in days
v : percentage of waterchange, expressed as % of the total volume
D : Amount of dry food administrated, expressed in gram
V : Volume of water, expressed in l.
N0 : the beginconcentration of nitrates in the water used for waterchanges, mg/l
k : a correlationfactor with a value 0 < k < 1.
I want to highlight that this is just an estimation of the waste content at equilibrium due to addition of food on one hand (bioload of the tank, more fish means more food) and waterchange schedule on the other hand. I hope it is usefull to somebody.
(1) Randy Holmes-Farley (2005), “Waterchanges in Reef Aquaria”, http:// reefkeeping.com, November 2005.
(2) - Boruchowitz, David E. (2009), “Time for a Change: A Mathematical Investigation of Water Changes – part I,” Tropical Fish Hobbyist, November 2009
(3) - Boruchowitz, David E. (2009), “Time for a Change: A Mathematical Investigation of Water Changes – part II,” Tropical Fish Hobbyist, December 2009