Continuous vs. batch water changes

I have been wondering lately about the effectiveness of building a continuous or automatic daily water change system for my reef aquarium. I really hate batch water changes, but they seem like they would be more effective. I decided to do the math and find out if that was true. A continuous water change can be modeled as a differential equation:

$y'(t) = (f_1)(\frac{y(t)}{V}) - (f_2)(\frac{y(t)}{V})$

where $y(t)$ is the amount of some dissolved substance at time $t$, $f$ is the flow rate in and out of the system, and $V$ is the volume of the tank. We can simplify the model by making some assumptions: a continuous water change will consist of a small volume over a long period of time, thus the flow rate will be low enough to assume complete mixing in the high water flows of a reef aquarium. Flow rates should be chosen to provide an equivalent reduction of dissolved material as a weekly batch water change, however the rate is not of interest here since we are purely interested in comparing the amount of dissolved material reduction with a batch water change, not how long it will take. It will be easiest to calculate volume with a flow rate of 1 liter per hour. We can also assume an input concentration of 0, since the water coming in should be from an RO/DI system. This gives:

$y'(t) = - \frac{y(t)}{284}$

noting that my aquarium $V$ is 75 gallons, or 284 liters. We will also set this up as an initial value problem, with $y(0)$ being an initial concentration of 50 mg/L of nitrate. The goal is reduction to 45 mg/L which we will compare to a batch water change later. For my 284L aquarium, the total concentration of nitrate would be $50 mg/L * 284L = 14200 mg$. This gives:

$y'(t) = - \frac{y(t)}{284}, y(0) = 14200$

The particular solution for this IVP is:

$y(t) = 14200\ e^{\frac{-t}{284}}$

Now with the solution we can solve for a nitrate reduction to 45 mg/L, which is a total concentration of 12780 mg. This happens at t = 29.9, which is 29.9 liters or 7.9 gallons of exchanged water. Now to compare to batch water changes.

Batch water changes use the formula for dilution:

$M_i V_i = M_f V_f$

Where M is the molarity and V is the volume, for initial and final states. Molarity is the moles of solute per liter of solution and nitrate ($NO_3^-$) has a molecular mass of 62.0049 g/mol, so the molarity of nitrate with a 50 mg/L concentration is 806 uM. Our goal of 45 mg/L has a molarity of 726 uM. Substituting these values into the formula:

$806\mu\!M * V_i = 726\mu\!M * 284L$

yields an initial volume requirement of 255.8 liters, or 67.6 gallons. That means the amount of water removed would be 7.4 gallons, pretty much exactly a 10% water change.

Now we can compare the two: a 10% batch water change requires 7.4 gallons to reduce nitrates from 50 mg/L to 45 mg/L, while a continuous water change requires 7.9 gallons. That is only half a gallon difference.

So we can readily conclude that continuous water changes, daily water changes, and weekly 10% water changes all have roughly the same effect on dissolved substances in the aquarium. I used a deliberately high concentration of nitrate for this study in order to accentuate the changes, I have no doubt that at typical levels of nitrate and phosphate the difference is even less discernible.
Since the model’s function is exponential, continuous water changes will have a reduced effect compared to batch water changes as the amount exchanged becomes larger. For example, a reduction of nitrate from 50 mg/l to 25 mg/L with a continuous water change requires 52 gallons, where a batch water change only requires 37.5 gallons.

Sunday, November 18th, 2012 Uncategorized

1 Comment to Continuous vs. batch water changes

• juan pablo says:

Extremely interesting article, thow i dont understand much about those maths. It impressed me that theres practically no difference from one system to the other. I have a continuous exchange system and i allways assumed that i need a lot more water per week in a continuous system, given that Im extracting water that is allready cleaner…

On other words, and if i exaggerate greatly, if i compare a 100% water exchange it takes ie only 100 liters and water, and if i try to achieve the same with a continuous system in theory it would be an infinite amount of water since all the time new water comes in, it is being diluted with the old polluted water… (i dont now if my logic makes any sense)

For the sake of comparing, the important factor here is that providing continuous exchange will produce stable conditions, which is paramount on fish health… i believe.

cheers