# 4.3: Residence Time

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The concept of the residence time of water in a lake is a natural one, because typically the water stays in a lake a long time, and velocity is much smaller than the source of inflow and outflow. The concept is not particularly interesting for rivers, though, because the velocity is about the same everywhere. The * residence time *of a lake is defined as

*the volume of the water in the lake divided by the discharge into or out of the lake*.

Letting the volume of water in the lake be *V *and the discharge to and from the lake be *Q*, The residence time *T**r *is

(A comment on the discharge here: this is easy to deal with only for stream discharge, not for groundwater discharge.) You can easily check that Equation 1 provides the right physical dimensions for *T**r*, namely time. In terms of the dimensions mass **M**, length **L**, and time **T**, the dimensions of *T**r *are by Equation 1

or, canceling out the **L **dimensions, just **T**, which is what we should have expected.

The residence time is important because it tells you something about how long it might take to clear pollutant from a lake. The trouble with it is that it *deals only with the average properties of the lak*e. A given drop of water or molecule of pollutant may actually reside in the lake for a very short time or a very long time.

Here’s an example of residence time in a lake. Look at a lake that’s one kilometer by one kilometer in surface area and ten meters deep, with a stream feeding the lake with an average discharge of 20 cubic meters per second. This is a rather small river, but it’s larger than just a stream. You could have this kind of discharge in a river with a mean velocity of about half a meter per second, a width of 20 meters and a depth of 2 meters. By Equation 1 the residence time then is

or a little less than one day.

On the other hand, very large lakes like Lake Superior or Lake Baikal can have residence times of many hundreds of years! Two important implications are that *it takes a lot longer to pollute large lakes but also a lot longer to clean them up*.