# When is a water level no level?

We are all familiar with the ancient but much venerated and still in use today water level.

It's simple, right?

You fill a tube with water, make sure that all of the bubbles are excluded, and then the two water surfaces - one in each leg of the tube - will always settle at the same level.

Simple, and truly brilliant.

Except, that my 50 metre long 10mm bore clear plastic tube filled with water and food colour point blank refuses to balance.

I have spent literally hours walking backwards and forwards slowly searching for and coaxing bubbles along the tube to one of the ends where it is expelled to the atmosphere.

As far as I can tell, ALL bubbles were long ago tracked down and ejected and yet I still can't get it to balance.

The two water levels settle at about 100mm (give or take) apart.

This just simply shouldn't be happening and it is driving me insane.

What physical process could be accounting for this that I have overlooked?

• Is the water denser at one of the two ends? This could happen if they were at different temperatures, for example. – Peter Shor Sep 9 '16 at 17:13
• Are you sure the die coloring is perfectly even? If you put it in without stirring, maybe it isn't. By the way, a picture of this would be beautiful :) – Steeven Sep 9 '16 at 17:16
• This kind of 'trouble-shooting' is quite difficult without access to the equipment. Can you post some photographs? And/or provide more details such as : diameters of all tubes and connections, any differences in liquid on both sides, the presence of any valves, whether either tube is sealed at the top or open to the atmosphere, etc. – sammy gerbil Sep 9 '16 at 17:18
• Which answer, Peter's or Steevens or sammy' s. Credit where is credit due....I gotta know – user108787 Sep 9 '16 at 17:21
• I will try to verify this over the next few days and report my results back. – Badger Sep 9 '16 at 17:25

UPDATE POSTED AFTER THE ADDITIONAL DETAILS

Contrary to my expectations the colorant is significantly more dense than water. You poured 4 bottles into one end and 2 into the other, which diffused to colored lengths of approx. 4m and 2m respectively. One bottle may have contained a different colorant. Can these details explain your observation of a 100mm difference in water level?

One 38 ml bottle of food colorant contained 45.5g per 100ml of 'sugars' - the only substantial ingredient besides waster - ie 17.3 g per bottle.

Treehouse Test. Here "although no measurement was taken of the level disparity, things seemed about the same." The vertical limbs were approx. 6m long, so all of the dye (in the top 4m or 2m) was in these limbs. With 4 bottles on one side and 2 on the other, this means that 2x17.3=34.6g of dye must balance a 100mm column of water in a tube of diameter 10mm - ie 31.4g of water.

The closeness of these 2 values confirms that the weight of dye can account for the difference of about 100mm in water level. It also suggests that the 6th bottle was identical to the others, and that the difference in water level at tree height was actually 110mm.

'Trouble-shooting' is difficult without access to the equipment. Some photographs and/or more details of your apparatus would be a great help. Further details have now been supplied in comments. I have made use of these at the end of my answer.

Deck Magazine contains an article about making and using a DIY Water Level. It identifies and addresses the 3 main problems briefly :

• Uneven thermal expansion. If one limb is in the shade and the other in the baking sun, then the levels can differ by as much as 10mm for a narrow bore tube.

• Kinks and objects resting on the pipe, obstructing flow.

• Air Bubbles. Easily overlooked. Having dye throughout the tube will make them easier to find.

Density Effect (material/thermal)

I am skeptical that the difference in density of pure vs dyed water (as suggested by Steeven), or hot vs cold water (as suggested by Peter Shor), can account for the effect you are seeing.

If each limb was 25.0m long vertically, the density of a 24.9m column of dyed water would have to exceed that of a 25.0m column of pure water by 0.1/25=0.4% to produce a difference of 100mm in water level. While such a difference is realistic, a height of 25m is not. If the limbs are only about 2m high, the required difference in density would have to be about 0.1/2=5%. If the limbs are 1m high, the difference would have to be about 10%.

A 5% difference in density requires approx. 100$^{\circ}$C difference in temperature [source]. This is unreasonable as an explanation. A 10% difference due to temperature is impossible.

Liquid food colorant as sold in supermarkets is mostly water; its density is close to that of pure water. Gel or solid dyes are also available, but when mixed into water the density is unlikely to be much different since 'a little goes a long way.' For reference a saturated salt solution (NaCl) has a relative density of 1.2 at 25$^{\circ}$C, while the relative density of sea water is 1.03. A 5-10% difference in density is possible if the colorant is used concentrated in one limb and very dilute in the other. However, I think the dilution which you have used is not likely to be significantly different from pure water.

I have removed a claim that the density of undiluted colorant would have to be about 20x that of water. This was based on a faulty calculation.

So I think there must be some other explanation for your observation.

**Update 1 ** following comments by Floris (use of alcohol-based dye) and DJohnM/Whit3rd (capillary action) and answers by Whit3rd (air pressure) and Graham S (uneven ground or unequal tube lengths).

Acohol-based colorant (density effect)

Alcohol has a density of approx. 0.8x that of water, a difference of 20%. A 50cm column of pure alcohol would balance a 40cm column of pure water, giving the required difference in height. This is physically realistic but it assumes no mixing and does not take account of your description :

I have just a couple of metres or so of food colour at each end of the tube with one end being distinctly more coloured than the other.

Assuming the dye at one end remained pure alcohol (density 0.8) while the other end mixed equally with water (density 0.9 $\to$ dilution by 50%) the columns would need to be 90cm and 80cm, which is still reasonable. However, this would require at least 500ml of this dye, which is mostly used for painting onto icing on cakes, so it is sold in small amounts of about 10-20ml.

Capillary Action

My calculation in answer to Size of a glass capillary for noticable capillary action shows that this effect is smaller for plastic tubing than for glass. Achieving a height of 100mm would require an inner diameter of about 0.3mm for glass tubing or 0.01mm for plastic. Bearing in mind that the effect here depends on a difference in capillary action between the two limbs, the required diameters would have to be sub-micron. Again, physically impossible and clearly incompatible with your description.

Except for capillary action, a difference in diameters of the two limbs would not cause any difference in height.

Air Pressure/Uneven Ground

Whit3rd's suggestion of a difference in air pressure due to wind is not likely because of your statement that "the ends of the pipe were held together in the same location."

Graham S's suggestions, that (a) the ground is not level or (b) the ends of the tube are at different heights, would not cause a difference in water level.

What other explanations are there?

Viscous Resistance (aka Friction Head Loss).

Especially in a coiled or kinked tube which is long and narrow, it could take several seconds for the water levels to equalise in the two limbs. This time increases exponentially as the difference in water levels (the pressure 'head' which drives the flow) decreases. So the explanation could be that you have not allowed enough time for the water levels to equalise.

Using an online calculator I estimate that a head of 100mm of water produces a flow rate of 5.5ml/s in a straight, smooth tube of inner diameter of 10mm. If this flow rate were sustained (as I've pointed out, it will decrease) it would take at least 1s for the levels to equalise. If the tube were kinked throughout like corrugated metal sheet the flow rate would be 2.2ml/s, and the levels would take at least 3s to equalise. These are only order-of-magnitude calculations.

Viscous resistance exists whether the tubing is vertical or horizontal, but the 'levelling time' will be shorter when the tubing is straight, which is easier to achieve when it is vertical.

Against this explanation is your observation that

The two water levels settle at about 100mm (give or take) apart.

Which suggests that either the difference is no longer decreasing, or the rate at which it decreases suddenly drops below a noticeable amount.

Air Bubbles

Since pressure is transmitted equally through air as through water, trapped air will only make a difference in water level if it is in the vertical limbs (density effect). Air trapped in the horizontal section of the tube increases resistance to flow, and could contribute to a long settling time. It could even stop the flow completely - which agrees with your observation quoted above. [Source : Air in Water Pipes, middle paragraph of page 3.]

Temporary Conclusion

Differences in density of liquid in the 2 limbs, and viscous resistance, are two possible explanations, but they either require you to do unlikely things (use large amounts of highly concentrated food colorant) or do not quite fit with your description of the levelling process.

You seem to have taken great care to eliminate air bubbles, but this still seems to be a possible and quite likely explanation.

Without more details of your apparatus and your levelling procedure, some other design fault or human error are still possibilities.

• Food colorant may be alcohol based - in which case the density difference could be larger (also explains why it remains at the surface). But you make valid points about calculating the magnitude of the stipulated effects. – Floris Sep 9 '16 at 19:09
• After failing to achieve a 'level' at approximately 1 metre an assistant was sent up into a convenient nearby tree house holding both ends of the pipe to see if that improved matters. Primarily, this was in order to see if a more direct (no kinks or bends in the tube) line would help with any bubble clearance even though confidence was already high that all bubbles had been excluded. It didn't, although no measurement was taken of the level disparity things seemed about the same. The tree house plus height at which the assistant was holding the pipe ends was approximately 6 metres. – Badger Sep 9 '16 at 22:26
• @Badger Science treehouse! – hobbs Sep 10 '16 at 4:53
• Here are the results of my experiments this morning. I emptied the original contents of the pipe - consisting of water with very uneven mixes of food colouring of approximately 4 metres at one end and two metres at the other - into a bucket. Gave it a quick stir and then set the bucket aside for reuse of the contents later. I then refilled the pipe with pure tap water being sure to purge it for some time to ensure all traces of the food colouring and any bubbles were removed. Result: the liquid in the two ends of the pipe now aligned perfectly! – Badger Sep 11 '16 at 11:47
• Returning to the bucket I gave the contents another quick stir (although maybe still not quite enough - see below) and then, having emptied the pure water out of the pipe, refilled it with the now somewhat more homogeneous mix of water and food colouring. Purged for bubbles by walking along the pipe keeping a loop raised such that the air (and all bubbles picked up along the way) were transported to the other end. Result: the liquid levels aligned to within approximately 4mm - an enormous improvement over the original 100mm, but still not the 'spot on' result of the pure water. – Badger Sep 11 '16 at 11:49

A U-tube can be a level, but only if the atmospheric pressure is the same at both locations. It is also a manometer. If one location has some airflow (prevailing winds, local air ducting, etc.) there may be a Bernoulli effect causing local low air pressure. It can move the liquid level up by a few centimeters, or an airplane by a million centimeters.

To control for this, you could use a second U-tube alongside the first, with a different density of liquid (maybe salted water is good enough). It might be quicker to check both locations with a barometer, or two barometers.

• For testing purposes (prior to deploying the instrument in earnest) both ends of the pipe were held together in the same location. Wind speed was effectively zero. – Badger Sep 9 '16 at 22:33

There are capillary forces in the tubes, of course; is the wetting angle of the liquid in one branch different from the wetting angle of the other liquid, in the other branch?

I've seen an object suspended in water on a wire, where the weight of the object underwent an apparent few-hundred-micrograms change on adding a droplet of soapy water to its bath. Adding nonsoapy water did nothing.

100mm seems too much for this effect, though. Have you any jokesters in the lab who would salt the water to change its density?

It is very likely that either a) the ground is not exactly flat and one side is 100mm higher than the other (if you are outside for instance). Or b) the vertical lengths of your u-tube are not of equal length (assuming you are measuring the distance from the top of your tube to the water level).

Regardless, you should borrow some surveying gear and measure the elevation of the water levels relative to some nearby datum. I think you will find that the water levels are indeed identical, in terms of meters above sea level.

• Neither of your suggestions (a) and (b) could create a difference in water levels. The water levels have been compared side by side at the same location - see Badger's comment to Whit3rd below - so difference in ground level is not the explanation either. – sammy gerbil Sep 11 '16 at 4:37