The Faulty Fluid Dynamics of Hotel Environmentalism

i-c515e11e77187da962f223ebd7dc640f-westin_shower.jpgBoskone this past weekend was held at the Westin Waterfront in Boston, which has these funky double showerheads that they charmingly call the "Heavenly(R) Shower" (hype aside, they are very nice showers). The picture at right is courtesy of lannalee on Twitter, as I didn't bring a camera.

Why am I telling you this? Because there was a sign glued to the wall in the shower that read:

Refresh yourself, restore our world

One of your Heavenly(R) Shower heads has been turned off in an effort to minimize water usage and protect one of our most precious natural resources.

The smarmy enviroweenieness of this was undercut somewhat by the next paragraph, which explained that you could turn it back on by pushing a little button on the showerhead (you can see one side of it on the lower head in the picture). And also by the fact that it's a completely stupid statement.

Turning off one of the two showerheads does essentially nothing to reduce the water usage. The flow rate of water coming into the shower is determined by the pressure and cross-sectional area of the pipes. If you turn off one of the two showerheads, it just makes the water come out of the other one faster-- at twice the speed, in the ideal case, which means you use just as much water per second in the shower with one head as with two. This is why putting your thumb over the end of the garden hose makes the water spray out so much farther-- the same amount of water needs to pass through a much smaller opening, so it has to move much faster on the way out. The only way turning one showerhead off can reduce the water usage by making showering slightly less pleasant, and thus getting people to take shorter showers.

But that's the ideal case-- does it hold up in reality? And, more importantly, can we test this?

Of course we can test this-- we're physicists. Well, I am. Also, I'm enough of a dork to want to check this out experimentally.

So, here's the configuration for the simple test. I tilted the upper showerhead up as close to horizontal as it would go, and let the water hit the far wall of the shower, like so:


The showerhead was roughly 1.9 m above the floor (measured by counting the number of tiles on the wall, which were 20cm high according to the ruler on the inside back cover of Matter and Interactions (which I had with me so I could field student questions by email)). The far wall was roughly 1.2m away, measured by counting 40cm floor tiles, and I could measure the height on the wall where the water hit.

With water flowing through both showerheads, the maximum height of the spray from the top showerhead hit the wall 0.9m above the floor. With one of them closed, the water hit higher-- somewhere between 1.1 and 1.2 m above the floor. This clearly shows that the water comes through the single showerhead faster than when both are open. But is it really twice as fast, as it would need to be to completely undo the supposed conservation benefits?

Explaining that requires a bit of math, but nothing too horrible. The height at which water hits the far wall is determined from simple kinematics, using the following equations:


Here, xf is the 1.2m distance to the far wall, yi is the 1.9m height of the showerhead, and yf is the height of the water on the far wall. The angle θ is the downward angle of the showerhead (which wouldn't go all the way to the horizontal position-- I guesstimated it at around 30 degrees, as I'm not nearly dorky enough to carry a protractor with me), and g is the acceleration due to gravity, 9.8m/s2.

Δt is the time required for water to travel from the showerhead to the wall. We don't know this, of course, but we can solve the top equation to find Δt in terms of the distance and the speed, giving us the following expression for the distance the water falls in terms of the angle and the distance to the wall:


That looks scary, I know, but the important fact here is that we know everything in this equation except for the speed v. We can do a bit of algebra and find the following expression for the speed of the water, v:


This looks faintly horrible, but it's pretty easy to crank these numbers into a calculator or a spreadsheet, and get a value for the speed of the water.

Using the numbers given above, the two-showerhead speed comes out at 5.5 m/s, which is a reasonable enough speed. Putting in 1.1m for the height with one showerhead closed off gives the speed as 9.4 m/s, not quite twice the speed; 1.2 m/s would require a speed of 36 m/s. The real height was somewhere between those two-- splitting the difference (1.15m) gives a speed of 12.8m/s. If instead of faffing about with approximate heights, we assume that the speed is doubled, and predict the height, we get 1.13m, which is entirely consistent with my eyeball measurements.

So, at best the "shut off one showerhead" strategy might be saving 15% of the water usage (using the lower bound on the speed); the real savings is probably considerably less, and consistent with zero, as a simple analysis of the plumbing would suggest.

So, if the folks at the Westin really believe this saves water, they're being tripped up by their own lack of physics intuition. Of course, the really important question (from the Westin's point of view) is not so much whether turning off one showerhead really does save water, as whether turning off one showerhead makes the people who stay at the Westin think they're saving water, and gives them a warm fuzzy feeling about how "green" they are. The answer to that is probably "yes," which suggests that the sign is either canny marketing or a depressing indicator of the innumeracy of the general population.

For the rest of us, a little bit of physics confirms that there's no reason to feel guilty about turning the second showerhead back on. As long as you don't linger longer in the double shower, you're not wasting any water by making full use of the facilities. And given what they charge for the rooms, you might as well get your money's worth.

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Ah, the kind of classical physics (mechanics) i like most. Here in Europe at least they keep throwing these "water saving shower heads" at us. Naturally, they save water by reducing the flow rate from 8-15 litres per minute to 4-8 litres per minute. But they are a terrible experience, probably due to less (perceived?) skin pressure. So i would keep the second showerhead off, so much more satisfaction!

what i'd like to know is why showerheads in the states all seem to be bolted to the wall. once you've used a few that are mounted on a 1.5-2 meter hose and merely set in a hook near the top of the wall for when you don't have a hand free to hold it, you won't go back.

(my showerhead came with a flow-rate limiting disk in the handle, designed to narrow down the pipe enough to actually choke off water flow. i wasn't dorky enough to take measurements, but subjective impressions before and after i tossed the silly thing indicate it did actually reduce flow noticeably. the added frustration at not getting rinsed quickly enough made the savings not worth it, to me.)

By Nomen Nescio (not verified) on 16 Feb 2010 #permalink

Hey Genius...turn shower head on for a minute and catch the water in a 5 gallon bucket, then measure. Turn both on and put the water into the same bucket (after emptying naturally)....only physics needed is a ruler.

By Rockinscotty (not verified) on 16 Feb 2010 #permalink

I have to admit that if I really wanted to know such a thing, I'd have to go with Rockinscotty on how to find the most accurate, reliable test to measure the flow. I mean as an experiment goes, it is certainly repeatedly and accuracy is determined by the type of catch container and the precision of the measuring instrument. On the otherhand, that is some really beautiful math work. Being able to set that up and make a prediction is impressive as all get out. But, wouldn't science be to use the math to make the prediction, and then getting out the bucket to measure the reality?

By Mike Olson (not verified) on 16 Feb 2010 #permalink

Alas, we do not travel with five gallon buckets, not even when we are driving. And I think it would be difficult to explain to the hotel why we wanted one.

Nomen Nescio, our showerhead is not only on the end of a hose but set on a vertical pole so the height can be easily adjusted (critical when it is shared by a 6'6" person and a 5'3" person).

My intuition is that Johan #1 is right: turning off the second shower head leads people to think they are getting an adequate rinse in a shorter time, so the water savings results from shorter showers. I don't have the means to test this theory, but Westin Hotels does, and perhaps they have done the experiment.

So why would they bother putting that second shower head in there? My guess is that the hotel assumed that you and Kate, or at least enough guests like you and Kate, might want to shower together. On flights where I have been bored enough to look at the Skymall catalog, I have seen a similar double shower head advertised, for exactly that purpose.

By Eric Lund (not verified) on 16 Feb 2010 #permalink

Oh, and the second shower head gets you spray over a much bigger vertical area (e.g., all of your back), which is really really nice. I'm not sure it would actually be better for a multiple-person shower.

The little signs are new since last year.

So why would they bother putting that second shower head in there? My guess is that the hotel assumed that you and Kate, or at least enough guests like you and Kate, might want to shower together. On flights where I have been bored enough to look at the Skymall catalog, I have seen a similar double shower head advertised, for exactly that purpose.

It's not so much a variable height/ joint shower thing as a greater area thing. You get less pressure with both showerheads, but you get hot water hitting a greater fraction of your body at the same time, which is really nice.

And yes, measuring the time to fill a constant volume would be a more direct measurement of the same thing. But that wouldn't serve to illustrate anything about classical mechanics, which is what I'm currently teaching.

People adjust their shower to get a comfortable water pressure on their skin. This requires a lower flow rate using one shower head than it does from two.

I am with Rockinscotty #3. That is a much more accurate measure of the flow. I am skeptical that the flow through the shower head changes that much due to pressure in the pipes. If it did, then the shower flow would be very sensitive to other water flow, sink usage and toilet flushing.

Variable levels of air entrainment might have a significant effect on the water trajectory.

Many hotel rooms use plastic containers for trash, and line them with a plastic bag. Even those that do not use plastic containers,, still use plastic bags. One of those could be used.

The other approach is to isolate a single stream from one of the shower jets and collect that single stream in a glass.

If only I'd known this while I was still at that stupid hotel!

By Jo Walton (not verified) on 16 Feb 2010 #permalink

Doesn't your premise assume that the shower head's maximum flow rate is greater than or equal to the pipe that feeds it Chad? The bucket test is really the only way to be sure.

Does a failure to understand fluid mechanics count as innumeracy? Inphysicsacy, maybe ... Doesn't really flow off the tongue, though, does it? :)

daedalus2u, you need some fancy hydraulic engineering in your shower head to keep the flow rate from being directly proportional to the water pressure in the pipe, over any reasonable range of water pressure. Something that constricted or expanded the nozzle in relation to the pressure, etc. I find this highly doubtful. The reason you notice a drop in flow rate in your home shower when someone flushes the toilet but not in a hotel has nothing to do with the shower head--it's the size of the hotel's plumbing system (decreasing the net effect of a single toilet flush) and the details of water delivery in it (most hotels, and most especially large high-rises, do not depend solely on the pressure supplied by the municipal water system to move water around the building).

By Scott Simmons (not verified) on 16 Feb 2010 #permalink

I was struck by the geometry (same size pipes in and out, doubling the area but with a 90 deg turn that must generate turbulence, destroying any laminar flow analysis) and your failure to mention Poiseuille's law (because it sounds cool when you say it).

By CCPhysicist (not verified) on 16 Feb 2010 #permalink

"I guesstimated it at around 30 degrees, as I'm not nearly dorky enough to carry a protractor with me"

Who needs a protractor? You have a ruler and trigonometry, so you've already got all the tools you need!

First, measure the length of the shower head, and set that equal to h. Next, get gravity to do the work for you and let your ruler hang vertically...use this to measure the vertical projection of the shower head and set that equal to y. Finally, solve for theta: sin(theta) = y/h

By ASFalcon13 (not verified) on 16 Feb 2010 #permalink

I'll one up even Rockinscotty.

There is this thing called a 'bathtub'. Some hotels even have them. All you would need to do such a measurement is a stopped drain, a ruler and a stop watch.

I do think the treatment here is rather elegant and more illustrative of more general truths useful in a variety of other settings. Definitively the difference between the approaches of a physicist and an engineer.

I think this is much easier to reason if you realize maximum flow rate is without a shower head because then there would be no resistance. Everything is relative to this baseline since there are no upstream changes.

* A pipe cap would be infinite resistance so no flow.

* One show head is in between. One head with a 90-degree bend would be more resistance so less flow.

* Two shower heads is closer to the head-less setup (I think resistors in parallel).

15-second reasoning says one head would save water. Pull out the trigonometry to guesstimate how much is saved: not to reason.

[I rent an apartment so I cannot change my plumbing but my preference would be a flow control handle to reduce the flow while soaping up and high flow when rinsing my hair. I ****hate**** low flow when rinsing my hair because it takes so much longer to rinse.]

A good illustration of some practical physics but I don't agree that it wont save water. Just because you can get the same flow through one shower head as through two doesn't mean you would want to. It's the velocity of the water droplets hitting the skin that you respond to when adjusting the taps (faucets). If 5.5m/s was adequate from two shower heads it would be adequate from one.
Maybe its just my Australian perspective where water use is always an issue.
nice post though. keep up the good work.

i thought the coriolis effect caused water to flow up from the drain and into the showerhead when you were south of the equator, like in australia?

and wouldn't the double showerhead set up an water interference pattern where some spots of your body would stay dry and some would get twice as wet? unless the de Broglie wavelength of 5.5 m/s water was very short compared to the width of the jets coming out of the showerhead. then there wouldn't be an interference pattern!

If you have an iPhone, you could have used the level app to find the angle.

1.9 meters is the shorter I (1.8m tall) can stand a shower head. Do you just kneel to rinse you're hair?

By marciepooh (not verified) on 16 Feb 2010 #permalink

Intuitively, some people like less pressure from the water that is hitting them, so when you use only one shower head I believe there would be a tendency to lower the setting of the shower.

Author states "The flow rate of water coming into the shower is determined by the pressure and cross-sectional area of the pipes. If you turn off one of the two showerheads, it just makes the water come out of the other one faster-- at twice the speed..."

It boggles my mind that a physicist can do all that math but doesn't realize how a water saving shower head actually works. They have restricted orifices as well as internal fluid friction that reduce the flow below what the water system can provide. If you add another shower head in parallel then the area of the restricting orifice is functionally double. Of course if the cross section of the two restricting orifices is greater than that of the supply line, it may not save much water. For example, if the shower can supply 3 gallons per minute with no shower head and the shower head is rated at 2 gpm, then the difference between 1 and 2 shower heads is about 1 gpm.

It boggles my mind that a physicist can do all that math but doesn't realize how a water saving shower head actually works.

Read the article again.
These are not water saving shower heads. They're two relatively ordinary shower heads in parallel, with a cut-off switch on one of them.

These are not water saving shower heads. They're two relatively ordinary shower heads in parallel, with a cut-off switch on one of them.

Here is a schematic from the company that makes the shower head and it contains an autoflow device which limits the flow to 2.5gpm. If the plumbing can supply more than 2.5gpm then it is a flow reducing head.

Speakman manufactures Westin's shower heads, according to their website.

@Victor #26

I am fully aware they are 2 parallel shower heads. That's the whole point my post in #23. The shower heads resist flow. Two equal resistors in parallel have less resistance than a single resistor of the same resistance. The link to the schematic in post #25 shows that indeed there is a restriction in the shower head in question.

If you don't understand any of my technical jargon, i apologize. You can easily look up any of their meaning online very easily.

Also, calling people Jack offs shows me that you are less interested in having a debate and are more likely just trolling.

well, there is certainly lots of disagreement about the shower heads and pressure and water saving shower heads.

i think we need some clever name for this controversy.

commenters, i bring you "watergate."

Personally, as non-numbers person, I think it's cool that Chad is comfortable enough with the math to jump to that automatically (although I'd be worried about his ability to teach physics if he weren't comfortable with math) I also think it's cool that somebody else came up with another way to test the showerheads.

Paul is right, as long as those showerheads have significantly higher flow resistance then pipes their number significantly impacts flow.

But the equally important factor from water economy point of view is the behavior of people interacting with the showers. The effects like compensating for lower flow by showering longer or adjusting water pressure are critically important to the outcome. Even mentioning the water saving campaign alone may significantly impact water usage.

So the only way to tell if the campaign is working is for the hotel to compare average water usage over representative periods from before and during this campaign while controlling for other variables.

Prof. Orzel, is there no basis for the claims made by waterpik for the "EcoFlow" shower head? Your experimental results are consistent with your predictions, and are clearly a counterpoint to the claims of the hotel. However, are there relevant physical parameters which your, er, "streamlined" model may not include in falsifying the more general claims of water "savings" made by, among others, waterpik?

Many hotels now also include little cards that suggest that you reuse your towels to "save the planet"... When I read one of these, the phrase that forms in my mind is: "save us the cost of doing your laundry..."

I'm sure there is more "green" savings in not forcing them to clean my towel everyday, but frankly I can't stand to wipe off with a dirty towel - so screw 'em.

Hey guys, as pointed out above the restriction in the shower itself is about 3/16" , flow in pipe has nothing to do with it. Local feed Pipe is 1/2 to 3/4" . I know because i drilled my restriction out, to get a real shower. There is plenty of water in Boston, and plenty of gas to heat it, but hotel saves gas $ by cutting back your shower. Why the asinine sign. So, it is entirely possible two showers on at once, give 2x flow, without any CFD, if restriction is orifice and not pipe,which is the case in a hotel,which has 2" water pipes in the hall ceiling.

Measuring bath level over a fixed time is best way to know.

This reminds me exactly of misapplication of finite elements in welds, happens all the time, math and fancy CAD system is very impressive, perfect, --modeling of problem and understanding of it is zero.

Also reminds me of flush twice, it's a water saving toilet.

Prius guys buy them.

"I'm sure there is more "green" savings in not forcing them to clean my towel everyday, but frankly I can't stand to wipe off with a dirty towel - so screw 'em."

Why would your towel be dirty after a single day? It should be used to dry oneself not to clean oneself.

I really don't see that this is innumeracy, since even Orzel's calculation showed that it could be saving 15%, which is not insignificant.

Also, clearly deviation from the ideal is significant in common situations. Anyone who has put their thumb over a garden hose knows that it comes out faster, but also that you can significantly reduce the volume that way.

There is no way it could save zero, though it could save a small amount.

Claiming that restricting the flow of water merely results in a higer water-speed thus completely eliminating any savings, would be true only if water experienced zero friction, and no increase in friction at higher speeds, neither of which are true.

We all know this, restricting an opening so as to let less water trough actually WORKS. Yes, the water increases in speed, so halving the opening won't halve consumption, but neither will it leave consumption COMPLETELY unchanged.

By Gunnar Tveiten (not verified) on 22 Feb 2010 #permalink

Most shower heads are low-flow to some extent, it's one of those federal mandates along the lines of low-flush toilets. As such I wouldn't really be surprised if on occasion two heads used almost twice as much water if the feed piping is large enough. This particular one may not, but it's not a priori certain for all such showers.

It is interesting on how many people on this thread are arguing over the water usage of a shower head, but have not seen the mechanism in operation. It is like arguing over the paint color on the walls of a dark room.

I have to say, I was at BosKone for the weekend, and noticed the shower head in question. My wife and I also took notice of the cute little sign that was posted nearby, about the water usage. Taking from actual usage, there was no change in water volume, just the pressure, when the value to moved. In reference to Pascal's laws, the applied pressure and constant volume, the amount of water released from the shower head will be the same. The only difference will be the adjustable pressure given by the usage of the value.

By the way, Prof. Orzel, the Time Travel board was entertaining and enjoyable.