u201cWhich Tank will Will Fill up First? (example with 12 tanks, A to L, see question source for image)
This is a recent meme thatu2019s getting shared on facebook and elsewhere. Repeating the image from the question source:This is a question that has gone viral recently. Most people answer u201cGu201d.But look closely, as the question says. Many of the pipes are blocked - the line that blocks off D from C is not a mistake.To find the real answer (this is assuming a low flow rate, as after all it is shown as a drip in the diagram):From A to B to C is straightforward. None of them can fill before the next one.J is a bit more complex. But as you fill J, as soon as the water rises to the outlet to L it overflows to L. So it can never get any higher. Yes, its level also rises in the outlet tube leading to I, but it can never get high enough to overflow to I.So it flows to L, which in turn fills F.So which fills first, L or F?By the time F is full, L will only be partly full (with the water at the same level for both).So your F is the answer.This video shows the idea, an animation by Nick Rossi using a physics engine, AlgodooIt doesn't quite flow like a real fluid, as he says, but it's enough to get the answer and show how it works.Here is another animation Which will fill first? from THE FLOW... by CorneliaXaosWhich will fill first? by CorneliaXaosThat answers the question, since it shows a dripping tap at a slow flow rate. But letu2019s go off on a tangent.WHAT HAPPENS IF THE WATER IS POURED INTO A AT A FASTER FLOW RATEIf the flow is very fast then obviously A will fill first. However, could any of the others fill up first before F and before A?Itu2019su2024 governed by the Hagenu2013Poiseuille equation so long asthe flow is due to a pressure difference.the fluid is incompressible and Newtonian (water is, approximately).the flow is laminar (not turbulent) - it is with water if it flows slowly through a narrow pipe.through a pipe of constant circular cross-sectionthat is substantially longer than its diameter,and there is no acceleration of fluid in the pipe.All those conditions seem to apply. The pipes are substantially longer than their diameter which is one of the most important requirements. And they are narrow, and the fluid is water.Under those conditionsIf the outlet is above water, the flow rate is proportional to the height of the head of water above the inlet to the pipe. If the outlet is below water, itu2019s proportional to the difference in height between the water above the inlet and the water above the outlet.The difference in height of the water here is often called the u201cheadu201d of water.It is inversely proportional to the length of the pipe.Or in short, the flow rate for laminar flow, in a pipe significantly longer than its diameter, is proportional to the pressure difference, and so to the head of water, but it is also inversely proportional to the length of the pipe.(it also depends on the radius of the pipe and the viscosity of the water but those are the same for all our pipes).Techy details. The equation is:There L is the length of the pipe and R is its radius.Q is the flow rate (what we are looking for).u0394P is the difference in pressure between the two ends of the pipe, which for water is proportional to the difference in height of the inlet and the outlet.Finally u03bc is the dynamic viscosityAll of those are constant (the pipes are all the same radius, and the viscosity is constant) except for L, the length of the pipe, Q which we are interested in and u0394P.So our equation simplifies to Q = c u0394P / L, where c is a constant which is the same for all the pipes in our example because they are all the same radius.Double the length of the pipe and you halve the flow rate. Double the head and you double the flow rate.So now for instance, can L fill at any flow rate?Its outlet is a very long pipe. Even if L is nearly full of water ,the head of water in F will mean the difference in heads between L and F is quite small even if F is nearly full and L is likewise.Its inlet is a much shorter pipe. Whether L can fill will depend on whether we can get J to have a high head to increase the flow rate of L's inlet pipe to more than that of its outlet pipe. But, at least at first sight, it would seem that such a high flow rate could mean that one of the other tanks earlier in the chain could fill firstSo - itu2019s quite a finely balanced question, and hard to answer.A obviously can fill first with a very fast flow rate, just fill it faster than it can empty.Well we can actually try this out with a real world experiment :).Well we can actually try this out with a real world experiment :).Prozix has made a 3D printed version of the puzzle. If you have a 3D printer you can download it here and print it out and test it yourself: Answer to the question Which one fill First / water equisystem by prozixI donu2019t have a 3D printer but he has uploaded some videos.First this is what happens with a slow flow rateNote that at 22 seconds in, J nearly fills briefly.If you look closely, you see that a bubble forms in the outlet from J to L, which makes sense, itu2019s a downward pipe and air is buoyant. The bubble then gets pushed out into L and then bursts.This shows the bubble just before it bursts (you can show the video at 1080p from the Settings)So - if the pipes are very thin - or the flow is just right - that might lead to J filling right there, if you can arrange it to fill before the bubble disperses.So even at a slow rate we have something anomalous already, though its because of a bubble.But what happens at faster flow rates? I asked in a comment to the video, and Prozix was interested and answered with a new videoAt 28 seconds in, at one of the flow rates, then L and F fill at the same time.Here, it all makes sense up to J. J canu2019t fill (apart from that possibility due to the bubble) at this stage because the pipe from C to J has only a tiny head above its inlet. Itu2019s outlet is about twice as long as its inlet, perhaps more.Aside: If C was nearly full, J would start to fill, and if we could have the level of water stay below the outlet into L while J fills, then C with its shorter pipe could continue to fill J even when it is nearly full. But as it is now, there is no chance of J filling.So that makes sense. But how can F fill at the same time as L? That's more mysterious.The pipe from L to F is three times the length of the pipe from J to L. Meanwhile, in the situation shown here, the head from J to L is about double the head from L to F.So by the Hagen Pousseville equation again, the flow rate from L to F should be about two thirds of the flow rate from J to L in this situation where J is half full and both L and F are almost full.So you expect L to fill faster than F.So, I donu2019t think they can get into this situation at all, with a steady flow into L. There must be something going on that doesnu2019t fit our assumptions of laminar flow, or something else such as a bubble forming.Letu2019s look at what lead up to this. If you look at the video, L fills faster than F to start with, keeping nearly the same head from L to F as from J to L.L is clearly filling faster than F and is on track to beat it. There is no sign of any bubbles in the inlet to L.But then a little while later you get this (25 seconds in)Now F is filling faster than J. Something has happened to reduce the flow rate into L, which then permits the two levels between L and F to equalize.But the head going into L hasnu2019t changed. Also the input pipe to L is full and there are no bubbles. I think the only possible answer is turbulence.You can see waves forming in J so maybe that means thereu2019s a bit of turbulence impeding the flow from J to L, especially since the water level for J is exactly at the level for the outlet to L. What are your thoughts?This is what happened with a moderately fast flow rate:Here is the video starting at that point.All of A, B, C, J, L and F are just about full. B, L and F started to overflow first and I think L just about beat the other two though it was almost simultaneous. In this frame you can see L just about to overflow and the other two though they have the water raised above the level of the top, havenu2019t yet actually started to flow down the side.So how do we understand that as a possible state in terms of the flow rates? Back to our diagram againWith A, B, C, J, L and F all filled, then A to B to C to J all have the same length of pipe and same head (height difference of the water in the tanks above inlet and outlet) so have the same flow rate. J to L has around 2.5 times the head of C to J, and the pipe is around 2.5 times the length, so the flow out of J is about the same as the flow into it, and the difference in head between the top of J and the outlet to I is small. From L to F, the difference in head is about the same as for C to J (which we already know is about the same as the flow from J to L) but the pipe is far longer, so L shouldnu2019t be able to empty as fast as it fills, and the water flows out of J faster than it flows out of L, so L should fill before J.From L to F, the difference in head is about the same as for C to J but the pipe is far longer, so L shouldnu2019t be able to empty as fast as it fills, so it should fill long before J fills,So if the flow rate is high enough for J to fill like this, L should fill before J and F doesnu2019t get a look in.So how could it happen? Well it could be the bubble from J to L, slows down the flow out of J so that J fills first before L.As for F filling, how did that happen? Letu2019s look at it again:The head from J to L is far higher than from L to F and the pipe is shorter, so the flow into L should be a lot more than the flow out of L to F. So it seems impossible for F to fill like this. It's not the bubble - the two tanks fill up reasonably steadily at the same rate. You can watch the video at quarter speed to check. Click the Settings icon in the lower-right corner, then click the Speed selector.Perhaps at this flow rate, its the double kink in the pipe from J to L causing more turbulence and so slowing down the input to L? What do you think? That could also help explain why J fills at this flow rate, if the pipe from J to L, has a slower flow rate than youu2019d expect from its length and head. What do you think? Do say in the comments.Even K can fill, though it is pretty hard to do. This is with a very strong flow into A, and several of the others have been overflowing for some time. They have turned off the inlet pipe at this point.Amusingly, in the real world, E ends up half full too after some time of running it at a high flow rate with the water overflowing from A.Here is the complete videoSo far the only confirmed alternatives to F are A (obviously) and L (pretty sure it wins at the moderate flow rate).Thatu2019s just a start. There are many other things to tryVarying flow rate. Can you get, J, say, to fill first or even K by turning the flow rate up and down at critical points during the filling process? This could cause bubbles to form, as well as adjust the heads of the various tanks.What happens if you scale the whole model up, or scale it down to a very small size? Scaling it down could make the flow rates out of some of the pipes very slow. It could also mean that bubbles like the one from J to L take a long time to disperse too. Scaling up could lead to more possibility of turbulent flow through the pipes.Try adding sugar for viscosityWhat if it is really hot, and you use a slow flow rate so that the water evaporates quickly?What if it is really cold so that the water freezes? That would seem to be a way to fill even B first, if the water freezes by the time it gets to B to C but remains unfrozen as far as the flow from A to B.NOTEIf you see anything in this to correct, however small or important it is, please either suggest an edit for my answer or say in a comment. Thanks!
