## Archive for November, 2012

### 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:

$y(t) = 14200\ e^{\frac{-t}{284}}$
Sunday, November 18th, 2012 Uncategorized 1 Comment