The problem

Consider an aquarium that has a total volume of water $V$, measured in gallons. Suppose an aquarium device pumps water from the aquarium through the device and back into the aquarium at a rate of $F_0$ gallons per minute. Because the water that circulates through the device is returned immediately to the tank, processed and unprocessed water get mixed together.

Many aquarium devices return water to the tank after processing, so that subsequent input to the device comes from a mixture of processed and unprocessed water. A cannister filter is a good example of such a device. In fact, for most types of aquarium filters, water enters the filtration system, passes through the filter, and then immediately returns to the tank. An ultra-violet sterilizer is another example of a device that functions this way. Water from the tank is taken into the sterilizer at a some rate $F_0$, then the water is returned to the tank where the sterilized water mixes with the unsterilized water.

Question: how will we know when essentially all the water has passed through the device? The answer is NOT as simple as calculating total minutes using the formula

$$T = V / F_0.$$ (1)

To see why formula (1) produces a poor estimate, notice that at any moment $t$ after the device is started, there is a fraction of the flow into the device that has never been processed, and a fraction of the flow that has already been processed. The longer we let the device run, the lower is the flow rate of unprocessed water. Therefore, as time moves forward from the point at which the device is initially turned on, $F_0$ becomes an extremely inaccurate estimate of the flow rate of unprocessed water into the device.

So how can time required to process most or all of the tank water through a device be estimated? The next section documents a formula for estimating ``turn-over" time, or the time it takes to process most all the tank water.