Who among us hasn’t watched a flowing mountain stream, and wondered at its flow rate? And who hasn’t wondered how that flow rate compares to the national consumption of say, milk, or gasoline? Indeed mankind has asked these questions since the time of the ancients.

Here is a picture of a stream I was fortunate enough to visit, not far from Yosemite National Park: How can we measure this stream’s flow rate? There are many ways. There are a number of devices used to measure flow. Some meters work on Bernoulli’s principle, working on pressure drops across orifices or venturis. Some use the number of rotations of a set of paddles, like a wind meter. Of course, we could lace the stream with radioactive isotopes and buy a suitable detector, or just divert the whole lot into a large tank for some time interval.

Isotopes are increasingly hard to find for the everyday consumer, as are giant tanks and earth-moving equipment. Fortunately another option exists: a volumetric flow rate can be calculated as the product of a flowing stream’s cross-sectional area and its velocity.

Q = Av

This makes sense from an intuitive point of view. Think about a large stream, like the Mississippi. It has a large cross-sectional area. Imagine pinching it down, until it became whitewater rapids. The volumetric flow rate would stay the same, with the loss in area being made up for by the increase in velocity. And the units check out (ft² * ft/s = ft³/s).

I set out to measure the stream’s cross-sectional area first. If the stream happened to have a simple geometry, like that of a rectangular drainage ditch, this would be a simple matter. We could measure width and depth, and multiply. However, most natural streams do not have simple geometries. We can use a bit of numerical integration to estimate area, however. That’s a fancy term for chunking our stream into a series of rectangles (or other, more intricate shapes, if we want greater accuracy) and adding the area of each individually.

To do this, we start by measuring depth at regular intervals. I did, and here’s what this stream looks like:

I was lucky in that I had a little foot bridge over the stream, and could use the boards to get a regular width at which to measure depth. What’s going on at measurement point 4, you ask? That’s a rock.

This is the measuring instrument used. The units are in feet.

Now I had to measure the width. Fortunately the boards of the bridge looked to be spaced fairly regularly:

Of course it was good practice to make sure. I measured four boards to get a representative sample, and they were all within 3/8 of an inch of one another. The average came out to be 0.47 ft (.14 m).

Multiplying this width by each of the individual depth measurements, and adding the results together, we get the area:

A = 5.31 ft² (0.493 m²)

Now to measure velocity. This is a bit more involved. I found a method for doing so on a US Environmental Protection Agency site. To use this method, you measure out a length of stream and throw in a floating object (not a human or animal that can’t swim. That’d be breaking the law). Then, use a stopwatch to record the time it takes to get from from beginning to end. I was lucky enough to have a load of pine cones lying around, and grabbed 15 to get a good estimate.

A stream will generally flow more quickly at its center than at its edges. I tried to introduce the pine cones at different points in the width of the stream to get a representative sample. Here are the data I got:

Note the missing points at runs four and fifteen. The pine cones got caught in eddies in those cases, greatly extending transit times. I decided to throw these data out, though I’m not sure how intellectually honest this was. Surely if the pine cone is held up, some of the water is held up as well. And if the water is held up, it’s not flowing forward, affecting flow rate. I’d be interested to hear how this is dealt with in practice, but for the time being, I punted.

Taking an average of the transit times, 4.54 seconds, and dividing it into the run length, 11.5 feet, I got the average velocity:

v = 2.54 ft/s (0.77 m/s)

We can now solve using the formula first outlined:

Q = Av

= 5.31 ft² * 2.54 ft/s

= 13.5 ft³/s

Q = 108 gallons per minute (409 l/min)

This is what the 108 gpm stream looks like:

So, answering the questions of our ancestors:

Americans consume 368.51 million gallons of gasoline per day (EIA). That’s 2370 of these streams.

How about milk? American production was 201 billion pounds in 2013. That’s 425 of these streams.

Lunch to whoever finds the product produced at a rate closest to this stream! Cheers!