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Tuesday, December 5, 2023

How Many Water Bottles Could a Filling Station Save?

Filling a water bottle at a public fountain is a pain. It never fits, so you have to tilt your bottle sideways—which makes it even harder to get the water into the narrow mouth. But there are new fixtures that solve this problem; you might have seen them in office buildings or airports. Along with a normal spout for drinking, there’s a special outlet just for bottles.

Well, here's the thing. At my university they recently upgraded our ancient hallway fountain to this new kind. Cool, right? This particular model has a digital counter at the top that says "Bottles saved." What? You know what this means? It means I can track the counter value over some significant period of time and see what happens.

I'll be honest. I really can't help myself in cases like this; I just love to collect data. I don't even know what I'm looking for. I’m just looking.

Fountain of Truth

OK, let's start with the numbers. Over several months, I usually checked the counter reading at least a couple times a day. So here is a plot of the counter value as a function of time. The units for time are in days, but I have converted the approximate hour into a fraction of a day.

So, what do you see here? These are my initial observations:

  • It's fairly linear. If it was completely linear, then humans would be walking by and drinking at regular intervals. Of course, humans are never that predictable.
  • Cleary I didn't gather much data on the weekends, so there are some blank areas. There's a longer gap around day 40—that was Thanksgiving break.
  • Still, it seems like there wasn't much fountain use on weekends. Friday afternoon and Monday morning values are pretty similar.
  • After about 60 days, the line flattens out. This was after final exams.

So let's get a rough estimate of the usage rate in "counts per day." This would be the slope of the data above. If I just fit the data as is (including the low use on weekends but deleting the vacation data points), I get an average bottle count rate of 22.29 per day. If that's really the number of disposable bottles saved each day, that's pretty great.

What about the count rate for each week? If I just fit a linear function to a set of data points for each week, I get the following plot:

You can see that during the semester, the rate is about 35 per day, with a slight increase just before finals and then a big drop-off when most of the students leave. But my takeaway is that with a constant population, the usage is pretty consistent over time.

Go With the Flow

But what about the actual water flow rate? The data above is just the bottle count, so what is the conversion factor between bottles and liters of water? Yes, I could probably get this from the fountain manufacturer, but where's the fun in that? Instead, I'm going to fill up a container with water and take some measurements.

Watching the counter, it takes 8.5 seconds to increase by one count. That means the count rate is 0.118 counts per second. I can also fill up a beaker to measure the volume of a "count"; that gives me 500 milliliters (16.9 ounces). That means the volume flow rate for the fountain is 59 ml/s.

You get the idea. There's no end of interesting things you can do with a data set. But I don't want to take all the fun. Here are some questions for you to carry the analysis further:

  • How long will it take to reach a million bottles saved?
  • If you kept the fountain running continuously, how long would it take to fill a backyard swimming pool?
  • Based on the counter rate, estimate the number of people in the building.
  • If the downward flowing stream of water is cylindrical with a diameter of 0.69 cm at the top (I measured it), how fast is the water moving?
  • As the water falls, it speeds up. Since the water mass rate stays constant, the water stream is thinner at the bottom (because it's moving faster). Estimate the percent change in stream diameter if the water falls 20 cm.

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