## 196 - Tall trees need water

How does a tall tree get water to its upper levels? We are taught in school Physics that a pipe will only support 10m of water. That suggests that trees shouldn’t achieve more than ten metre height.

Is this a non-issue? If so, how was is to signally failed to be covered in class? I remember feeling that I’d been disruptive yet again in asking what to me seemed a proper, interested question. When this sort of thing happens in school, this is the moment when a teacher demonstrates good subject knowledge or has to recognise that there are good questions one should have been asking earlier – and some sort of promise to find out; I might know the answer but try to see what the class thought before offering a contribution. If I really didn’t know, I’d say so and we (all) would go find out for (a part of) homework. If learning is the important thing, then this seems to me to be consistent with that idea.

As ever, the devil is in the detail: the physics lesson doesn’t teach that—or it isn’t supposed to. The lesson you were supposed to participate in showed that a water barometer would have a column height of around ten metres. ¹  Such a pipe is closed at the top, where the so-called Torricelli vacuum is. Trees are not built like that.

For a start, the water in columns in trees are not closed; there is some pressure from below from the roots and there is a suction effect from transpiration at the leaves. Issues then are what sort of values are these pressures (clearly, big enough, since tall trees exist ²). The physics of columns of fluid (slow moving or stationary) gives us cavitation (the column breaks, there are gas bubbles occurring) above the P=πgh limit (can I call that the Torricelli limit? Apparently not). Wikipedia tells me that cavitation ³ occurs in the xylem of vascular plants when the tension of water within the xylem becomes so great that liquid water (or sap) vaporises locally and dissolved air within the water expands to fill either the vessel elements or tracheids. Plants are generally able to repair cavitated xylem in a number of ways.
The theory that biologists learn about is called the cohesion-tension theory. To pull water to the top of a sequoia requires (says the linked site) 1.9x10³kPa (isn’t that about 2 MPa?).

I am aware that water is close to incompressible but wondered what the limit was for water under tension. Note that the experiments to measure the tensile strength of water degrade rapidly with water impurity, so whatever process plants use covers that issue well, somehow not requiring extreme values of tensile strength. This article covers some of the issues, this one is more academic and tells us the lack of a working theoretical model (hint, opportunity) but doesn’t give a figure; this one does; -33.3±3MPa, but presumably that was very clean water. We need 2 MPa for the tallest tree, so the water itself is not an apparent source of difficulty. ⁴  The reason that impurities bring down the tensile strength is that the larger molecules of the impurity offer opportunity sites for cavitation to begin [I thought that obvious].

Incidental info:
•  Small bore tubes filled with water are very much stronger than the empty tube, akin to filling the tube with steel wire.
•  trees have mechanisms for repairing xylem suffering from cavitation, so it can occur.
•  root pressure can be very low and 200kPa would be rated as a high value. This suggests that transpiration is primary.

The cohesion theory suggests that transpiration pull from the leaves and root pressure from below combine to provide movement of water through the plant. I found that root pressure is usually a small component of this, making the transpiration ‘suck’ the major factor.

Do please read the technical explanation below. ⁵  It implies a good deal of valuable research being done and explains that the topmost hole, the stomata in the leaf, is a critical factor, and though a large bore pipe nearby may cause difficulty in continuity of the water column, the slow growth of trees does in a sense avoid some of the recognised difficulties. Thus a tree cannot start from (nor, presumably recover from) having no liquid in the xylem, but it can grow very tall provided the water column is never broken. See biology stack exchange.

There is a related question that should have been answered earlier, really: How do we pump water above 10 metres? That assumes again that the original physics lesson has been misunderstood. Ten metres is the limit we could lift using a vacuum, simply because it relies upon atmospheric pressure to do the lifting. If we pump, that would be positive pressure, there is no practical limit. Well, there are limits imposed by the plumbing and the pump design. But there is nothing preventing staged lifting; for example lifting in 70 metre units. ⁶  70 metres implies 100psi and under 30 floors of a building.

What strikes me is that the perfectly straightforward content of a physics lesson given early in secondary school is so thoroughly misunderstood. I accept that water is largely incompressible; that means it can be pumped at high pressures (if the pumping system can take it), which means water can be pumped to great height. In turn that means that the biological mechanism of solutions seen in trees are differently interesting. I confess to surprise that there is (so much) research left to do in this field and encourage readers to grab the opportunity.

DJS 20160614

http://biology.stackexchange.com/questions/11044/how-do-trees-lift-water-higher-than-10-meters

1  Pressure = height x density x g. Atmospheric pressure is 950-1050 millibars and the standard pressure is 101.3kPa. That figure, using water as 1 tonne/m³ and g as 9.81m/sec², suggests that the height would be 10.33m and the range of figures would be ±5%, i.e. 10.33±0.066m.

2 The tallest tree is 116m, a California redwood. http://www.livescience.com/14667-tall-trees-grow.html That article suggests an upper limit (of that theory) of 122-130m (Americans, 400 feet). Typical tall trees in the Amazon rainforest are around the 30m mark.

3 We usually discuss cavitation in relation to movement of things in water, such as propeller blades. The principle does not require movement, but the formation of bubbles. Perhaps there is an adjective missing to describe what occurs in trees: static cavitation, perhaps?

4 Dare I suggest that some experimentation is called for to establish the range of figures applying to trees, to investigate how tensile strength decays with impurity? It would seem to me that such experiments would lead to understanding how the 'larger impure molecule leads to cavitation' helps cavitation occur. Would that require a hydrogen bond to break? Does the -33.3MPa figure tell us what is required, or is that an upper bound (for that temperature)? What is special about glycol that gives is an even higher tensile strength?

I have written before about concrete being weak in tension and read it is 2-5 MPa. Wood might be 40, bone 130, steel over 400 (I’m quoting ultimate strength not the lesser yield strength, for those that know about this or want to use the figures). Yield strength seems to be a value that can be visited often, whereas the ultimate value breaks the material. In this context, toughness means the ability to tolerate imperfections and still perform as expected.

5 V. IMPLICATION FOR WATER TRANSPORT TO THE TOPS OF TALL TREES Most plant physiologists accept the “cohesion-tension theory” as the explanation for the ascent of sap.26 In this qualitative theory, the move of water depends upon three important physical-chemical properties of water, which actually correspond to capillary rise (cohesion), cavitation (tension), and the hydrated wall (low contact angle), respectively. In this section, we focus only on the implication of the general force balance and loop hysteresis in the capillary rise of a tall tree. The height to which water in a tree rises is dependent on the size of the transport conduits. If one cuts down a tree and looks inside, the capillary dimensions of the relatively large conduits (the xylem tube) are on the order of 100 μm. [27] As a result, the capillary rise is about 0.1 m. If capillary pressure alone were to explain the water rise to the treetop of a 100 m tall tree, such as the coastal redwoods of California, a capillary radius of about 100 nm is required. It was suggested that the relevant capillary dimension is the air−water interfaces in the cell walls of the uppermost leaves. The matrix of cellulose microfibrils is highly wettable, and the spacing among them yields effective pore diameters of about 10 nm. It has been pointed out that it is not necessary for the capillary to have a small bore throughout its length. Only the bore at the meniscus (i.e., in the uppermost leaf) is relevant. [27] This consequence has been proved in our general expression of force balance, eq 4. Note that a microchannel containing corners or cusps on its cross section is not considered in the derivation of eq 4. Liquid filaments extend to infinity in the corners or cusps. [28] Nonetheless, the elevation of the liquid column is still inversely proportional to the characteristic dimension of the cross section of the tube. For the solutions satisfying the force balance, nevertheless, there exists the issue of physical stability. A small bore at the uppermost leaf connected to a larger xylem conduit reveals the presence of a convergent microchannel. As a consequence, multiple stable heights are possible, as described in the aforementioned analyses. However, the final state depends on the initial condition. The liquid will rise to a stable height corresponding to the larger xylem conduit if the microchannel is initially empty. In other words, the liquid will not rise of its own accord to the stable height near the top of the convergent channel because it will not be able to traverse the larger conduit of the channel. This situation is, nonetheless, stable if the liquid is sucked up to the top and then the suction is removed. How does a tall tree acquire such large negative (suction) pressures from above? As demonstrated in our experiments, the gradual rise of an initially immersed cone is able to maintain the stability of the meniscus on the top of the truncated cone as long as the force balance is satisfied. Note that the contact angle in the vicinity of the small pore mouth can be tuned up to fulfil the force balance when the microchannel is not high enough. The slow growth of the tree can be regarded as a gradual rise of the convergent channel. As long as water transport to the pores on the uppermost leaves is not interrupted throughout the entire course of the tree’s growth to 100 m, this stable height can be achieved without resorting to suction.  Wang Z et al. (2012) Capillary Rise in a Microchannel of Arbitrary Shape and Wettability: Hysteresis Loop. Langmuir 28: 16917-16926

6 Should be 7 shouldn’t it? 70m represents 7 atmospheres, 7 bar, 100psi. Many choices. look up booster sets, such as here.  For context, in a domestic situation, maximum anticipated pressure is around 80psi. It is expected in Britain that the minimum value will give you 7 metres of ‘static head’, i.e into the roof tank of a three storey building.

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