How to calculate the temp of falling water?
November 7, 2017 1:34 PM   Subscribe

I'm curious how much the temperature of hot water (say, 37ºC) drops as it falls through the air, like in a shower. I don't know enough physics or fluid dynamics to even know where to start googling the problem, but if you had a shower 2m off the ground, and assumed ideal large drops, how much would the water temp drop after falling say, a meter in height?

I'm mostly looking for the magnitude of change, not a super precise number, i.e., if it's more like 0.1ºC in change (there's not much surface area on a drop of water) or if it's like 1-2ºC.

I googled around for an hour trying to figure this out, but don't even know where to start. I even considered just using a bucket at the top and bottom of a shower but couldn't think of a simple repeatable way to measure water temp reliably, so I'm curious how the math plays out.
posted by mathowie to Science & Nature (21 answers total) 4 users marked this as a favorite
Can you set up an instant read or probe thermometer at the right height and perform the experiment yourself? Also, wouldn't it really depend on the air temperature?
posted by bondcliff at 1:38 PM on November 7, 2017

Assume room temps? I realized I didn't have a thermometer that could measure even 1-2º of difference.
posted by mathowie at 1:44 PM on November 7, 2017 [1 favorite]

The shower is essentially an evaporative cooler. I'm betting that the evaporation of the water drops is the biggest element evaporating the water.

Some links that might kick you off:
Evaporative cooler
Why does water get significantly colder while falling through the air?
Passive Downdraft Evaporative Cooling
And its use in CPU cooling setups known as 'bong coolers':
Nuclear Tower Water Cooling

posted by suedehead at 1:50 PM on November 7, 2017

Amazon sells all manner of IR thermometers that I think should work with enough precision for your purposes.

Here's a primer on heat transfer and dissipation.

Why do you want to know this information? Sounds like you're solving a larger problem.
posted by jeffamaphone at 1:54 PM on November 7, 2017

You get more evaporation than other cooling. You also get cooling as the water bleeds off heat into the surrounding air, but that doesn't happen fast enough to measure a meter or two's worth of change. It cools a lot more on touching the tub, and it's possible it cools a tiny bit just before that, from being next to something that's cooler, but again, not enough to measure without instruments that are definitely not normal bathroom equipment.

How much it changes is going to be based on the water temp and air temp. 37°C water in a 20° room is going to be different from 37° water in a 0° room. Still, water temp change from the air, and even evaporation (and now we're adding humidity and air pressure to the calculations) is not going to be usefully measurable.

Most of why a shower seems cooler at floor level than at the showerhead is dispersal - there's a lot more water per square inch at the top. And some of it bounces off the walls and/or tub, and that cools it a lot more.

If you really want to know, go round up some physics students and set it as an applied science puzzle: Given, 100 gallons of water at 37°, and 10 cubic meters of air at 20°, how fast will the water lose 1° of temperature? ... and so on.
posted by ErisLordFreedom at 1:58 PM on November 7, 2017 [1 favorite]

We are matthowie's bunch of physics students in this situation.
posted by maryr at 2:02 PM on November 7, 2017 [4 favorites]

Why do you want to know this information? Sounds like you're solving a larger problem.

Two reasons. One, my partner is 1 full foot shorter than I am, and regularly says various showers don't get hot enough, so I'm curious if that's personal preference or actual SCIENCE! to explain the difference in our perception of most showers.

The second reason is I tested out a weird vaporizing shower a couple months ago, which sorta misted you to death, but I assume due to the ultra-high surface area of teeny mist droplets, it gave the coldest showers I've ever felt and was incredibly uncomfortable, and I was looking for some SCIENCE! to back up why that one felt so terrible.

I know humans are super sensitive to temps around 100ºF (basically body temp), so I'm curious if the math can show it's a small difference or a large enough one to warrant differences in perception of heat.
posted by mathowie at 2:10 PM on November 7, 2017 [5 favorites]

If you have any relatively wide mouth thermos, hold one in the stream at the elevation you want to test until it's full then pour it out quickly (to temper the thermos) and let it fill again about halfway and measure the temp inside; then do the same at the other elevations that interest you.

The differences in temperature at the various levels will give a pretty good indicator of how much the temperature falls from one elevation to another.

I think a foot will easily be worth a degree or two.
posted by jamjam at 2:25 PM on November 7, 2017 [1 favorite]

I have an infrared thermometer. When I get home, I will run an...experiment.
posted by notsnot at 3:13 PM on November 7, 2017 [2 favorites]

It looks noticeable in this IR video of a shower, but I can't quite tell what's happening in the scene.
posted by ambrosen at 3:14 PM on November 7, 2017

How about solving this with an adjustable shower head that slides up and down a rod. My sister is barely 5 ft tall and her husband is 6'4", that's their solution.
posted by mareli at 3:16 PM on November 7, 2017 [3 favorites]

If you have any relatively wide mouth thermos,

I'd go the other way, and use like a dixie cup - which has almost no thermal mass on its own, will fill up super quick, and still holds enough water to measure the temperature of.

Really you could probably just get two dixie cups, fill them both simultaneously from different heights, and use your own body to feel if one's noticeably colder. But the point about the spread of the water is a good one - even if the water's effectively the same temperature, it's more spread out lower down.

A totally non quantitative experiment for problem #1 would be to get down on your knees to simulate being 18" shorter. Be sure to kneel on some flip flops or dock shoes or something so it looks like you have feet!
posted by aubilenon at 3:18 PM on November 7, 2017 [1 favorite]

I forgot I had an infrared thermometer for cooking reasons. So I did some experiments just now.

Filling up a plastic cup, dumping it out, and waiting until it was half full or fully full consistently gave a temp of 105ºF at the very top of the shower. Filling the cup on the floor of the shower was slower and more difficult due to the spread, so water had time to cool down to fill up the cup. I threw out those early checks and instead tried to get the least amount of water as quickly as possible to get a good temp close to actual. I ended up getting consistently around 99ºF. I redid the top of the shower check with smaller amounts of water and continued to get about 105º each time.

So! It looks like in six feet of drop, my home shower drops about 1ºF per foot of fall, which frankly was higher than I would have guessed. For some reason I felt like several degrees F of change was too much and it would be miniscule real differences.

Thanks all for the help at getting ideas on how to solve this, it's something I've wondered about for years.
posted by mathowie at 3:33 PM on November 7, 2017 [18 favorites]

You need a partial differential equation changing temp with either position or time to help you estimate the solution to this problem. Your upper boundary layer will be your starting temp and either zero time or zero position. So either make it to graduate level math (calc 5) OR measure it. Or just agree with your partner because that will bring you happiness.
posted by Kalmya at 4:12 PM on November 7, 2017 [3 favorites]

hot water (say, 37ºC)
regularly says various showers don't get hot enough

If you tried to tell me a 37ºC shower was hot, I'd tell you that too...
posted by HiroProtagonist at 7:04 PM on November 7, 2017 [3 favorites]

I tested out a weird vaporizing shower a couple months ago, which sorta misted you to death, but I assume due to the ultra-high surface area of teeny mist droplets, it gave the coldest showers I've ever felt and was incredibly uncomfortable, and I was looking for some SCIENCE! to back up why that one felt so terrible.

The relatively extreme droplet surface area to volume ratio for very small droplets will certainly promote more rapid cooling of droplets in transit from nozzle to skin, though this might be offset to some extent by the reduced ability of very lightweight droplets to entrain a cooling air stream.

Something else to consider would be that whether you feel warm or cold depends on the rate that heat is being delivered to or removed from your body, and that this rate will be proportional to the temperature difference between your body and the applied water. For water temperatures close to body temperature on application, a water temperature difference of "only" one degree could easily cause an apparently disproportionate difference in net heat transfer, perhaps even altering a net warming effect to net cooling.

The heat transfer rate will also be roughly proportional to the volume of water being applied to your skin per unit time, as will the perceived stability of that rate. This is another reason why low flow showers (of which, I imagine, your weird vaporizer was a rather extreme example) yield such consistently miserable experiences.

All that said: if you want to go all SCIENCE! on this then it seems to me that thermodynamic and fluid dynamics models of droplet behaviour under these conditions are likely to require so many simplifying assumptions as to render them quite unreliable, and that you'd get far more useful information from surveying the subjective experiences of your household members and instrumenting your shower with a sampling cup, a cheap bulb-and-column weather or medical thermometer, and a flow rate meter.

Ask yourself: what would asavage do?
posted by flabdablet at 10:22 PM on November 7, 2017

bondcliff: Also, wouldn't it really depend on the air temperature
I think you would need to add this to your considerations - partly because the falling water will be cooled less rapidly if it is falling through warmer air - and partly because the general perception of warmth will be less if the air is colder.

If your shower is a closed cubicle then the warm, steamy air will start to fill it up from top to bottom as you take a shower. So a shorter person might feel this warmth on their head a little later than somebody taller. The same thing applies if your shower is in a larger open space - except that this time the rate at which the room fills with steam will be slower. As a room fills with steam it is often possible to see the level descend from the ceiling until it reaches your observation point. Is your partner getting immersed in the warm steamy air too- or is it just you?
posted by rongorongo at 10:57 PM on November 7, 2017

And a couple of other 'fresh-from-shower' observations:
1. The highest degree of heat variation by person height - however you calculate it - comes with a vertically mounted shower head located above you. The lowest comes from a 45 degree directed shower head mounted in front of you. Can you try mounting your shower head like that?
2. The degree to which a jet of water feels cold will be partly dictated by the speed at which it moves not just water but also air. Try a high mist shower setting and you will notice not only that it feels cooler but also that you can feel the current of air it generates. The same thing applies to a more regular jet. In that case the lowest level of interaction with the air - and hence movement of the air - is going to be near the shower nozzle where each jet of water is distinct. As the area of water spreads out and the jets get less distinct - more air will be moved along adding to the cooling effect. Finally, towards the floor, the rate of air movement will slack off again because the overall area covered will be going up while the amount of water will be constant. At least, that is my intuition. If your partner's head is at the height of maximum air flow then that will contribute to a feeling of coolness that you may not feel.
posted by rongorongo at 3:35 AM on November 8, 2017

I wound up running the same experiment as Mathowie. A big cup of water taken from the shower head got me 111F. Two or so feet down was 105F. Water from right before it hit the tub was 95F. And the tub surface, after steady-state conditions, was 94F, which confirms my last "cup" value.
posted by notsnot at 4:51 AM on November 8, 2017

I'm surprised that mathowie's results were so little variation, I would have expected more like what rongorongo reported. One thing that might make the difference is that it's the smallest droplets that cool the fastest and fill the cup the least. The cup does a very good average of droplets of different temperatures, whereas that's not necessarily the way we perceive it. The last hotel I stayed at, for instance, had a water-saving high-pressure tiny droplet spray, and even at the level of hte top of my head there was a dramatic difference between the very fine drops on the edge of the cone (tiny drops, cold air, cold water) and the center of the spray (larger droplets, self-heating of the air around them). Even though the average temperature was fine, the presence of cold water in the mix was really distracting and unpleasant. And if I'd had a cup, it would have been easier to collect the warmer center part than the edges.
posted by aimedwander at 7:06 AM on November 8, 2017

Weird that you had to experimentally verify with a thermometer... am I the only person who, on especially hungover weekend mornings, starts the shower standing up, then ends up laying down in the tub under the stream, and inevitably has to crank up the hot water to achieve my desired temperature?
posted by Grither at 8:23 AM on November 8, 2017 [2 favorites]

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