I was trying to understand the Physics of how water gets to the leaves of a tree and found this article which claims some experiment disproved a 300 year old law of Physics. Sounds pretty unlikely... What exactly are they talking about though? They cite that water can only flow a maximum of 32 feet up a tube. Where does this number come from?

This is not a Biology question so stick with me.

I've looked up how water gets from the bottom of trees to the leaves before, and my understanding is that it basically is capillary action and a pressure difference. So, water evaporates from the top of the xylem (which is just a tall skinny tube) creating "negative pressure" and this acts with the capillary action to raise water up to 100 meters for very tall trees.

At least, that's the idea. I would like to show this mathematically as it doesn't seem to be all that difficult, but I've never learned anything related to these ideas so I'm not sure where to start.

I found this equation for the height a liquid will rise by capillary action: $$h=\frac{2\gamma cos\theta}{\rho g r}$$ where $\rho$ is density of the fluid, $r$ is radius of the tube, $\gamma$ is the air-surface tension, $g$ is acceleration due to gravity, and $\theta$ is the angle of the tube.

So, I found the diameter of a xylem is around 10 microns and that $\gamma_{water}=.0728\frac Nm$

Then, $$h=\frac{(2)(.0728\frac Nm)(cos(0))}{(1000 \frac{kg}{m^3})(9.81 \frac{m}{s^2})(5*10^-6 m)}=2.96 m$$

Clearly, I'm using this equation incorrectly, or this just isn't everything that needs to be considered.

So, if water were to flow 100 meters upward against gravity, and you are allowed to have a pressure lower than atmospheric pressure at the top (as described above) and you account for capillary action, what pressure will do the job?

If anyone happens to know the physics of how water can flow to the top of a tree and can work it out mathematically, that would be massively appreciated and extremely interesting because I haven't been able to find reliable physics on this subject anywhere in the vastness of the interweb.

Also, regarding that article I posted, they say that they use a salt solution which increases the density of water and allows it flow higher. According the above equation, that should decrease the height of the liquid in a column. So what's going on?

  • 2
    $\begingroup$ The thermodynamic limit on how high you can pump water is a few kilometers (depending on the assumptions you make about humidity and ambient temperature). You have to consider that you can extract work from the evaporation of water. How exactly trees manage to do this is not known in full detail, but it's certainly not against the laws of physics. $\endgroup$ – Count Iblis Sep 10 '15 at 21:39
  • $\begingroup$ Certainly not against the laws of physics. It seems that there can't be that many factors at play with moving water up a tree. Is it not understood from a Physics or a biology perspective? Or are the two connected here? $\endgroup$ – jheindel Sep 10 '15 at 22:36
  • $\begingroup$ Yep, this is called a measurement error, or, more fitting, a naive interpretation of what is happening. What has been neglected here is the chemical affinity between the tube wall and the water. Let me see them repeat this experiment with a teflon tube. As for trees: they could pump water into the stratosphere if we go by just the laws of physics and not those of biology (there are no limits to active pump mechanism like those at work in cellular organisms). Evolution just didn't care to select for such high trees. :-) $\endgroup$ – CuriousOne Sep 10 '15 at 23:35

The website and linked blogs are pretty incoherent, but what they seem to think they're proving is that water can be lifted more than you would predict by mere air pressure.

The experiment was to take a 6mm water-filled tube with both ends in water containers, and then lift the center of the tube up to the top of a cliff. There was a small amount of salt water involved, but it's not clear where the salt water was located. The conventional belief (or at least what they thought the conventional belief was) is that when the column of water was tall enough so that the pressure difference from gravity was greater than atmospheric pressure, the water would stop rising and the lifted center of the tube would become a vacuum (or whatever water vapor pressure was at that temperature). Apparently this didn't happen, and they were able to raise the center of the tube to the top of the cliff, 78 feet above the bottom.

Why did this happen? I can only guess that it involved the relatively small diameter of the tube, or some aspect of water being attracted to itself. But, the language used in the blog postings, and the lack of details on exactly what was done (again: that salt water was implied to be critical, but they were remarkably vague on where it was in the tube) gives me the strong suspicion that this is classic pseudo-science, where anti-establishment conclusions are trumpeted but the methods are unclear.


Atmospheric Pressure

I think this is what they meant:

On earth, fluids are also subject to the force of gravity, which acts vertically downward, and has a magnitude γ = ρg per unit volume, where g is the acceleration of gravity, approximately 981 cm/s2 or 32.15 ft/s2, ρ is the density, the mass per unit volume, expressed in g/cm3, kg/m3, or slug/ft3, and γ is the specific weight, measured in lb/in3, or lb/ft3(pcf). Gravitation is an example of a body force that disturbs the equality of pressure in a fluid. The presence of the gravitational body force causes the pressure to increase with depth, according to the equation dp = ρg dh, in order to support the water above. We call this relation the barometric equation, for when this equation is integrated, we find the variation of pressure with height or depth. If the fluid is incompressible, the equation can be integrated at once, and the pressure as a function of depth h is p = ρgh + p0. The density of water is about 1 g/cm3, or its specific weight is 62.4 pcf. We may ask what depth of water gives the normal sea-level atmospheric pressure of 14.7 psi, or 2117 psf. This is simply 2117 / 62.4 = 33.9 ft of water.

In particular: This is the maximum height to which water can be raised by a suction pump, or, more correctly, can be supported by atmospheric pressure.

From rereading your question, you might already know this, so apologies for that, and I will delete this answer if necessary, but I think this is the experimental evidence they based their theory on.

  • $\begingroup$ I did not know where that came from so your answer is appreciated but in light of that, I still wonder how water in a river can actually reach height of 100 meters in a tree. Or, removing the tree, how does water travel up a very skinny tube 100 meters? $\endgroup$ – jheindel Sep 10 '15 at 22:30
  • $\begingroup$ biology.stackexchange.com/questions/11044/… this might relate to it, apologies if this is off topic for your question $\endgroup$ – user81619 Sep 10 '15 at 22:36

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