The height limit of a siphon | Scientific Reports

Although the siphon has been used since ancient times, the means of operation has been a matter of controversy1,2,3,4,5,6. Two competing models have been put forward, one in which siphons are considered to operate through gravity and atmospheric pressure and another in which gravity and liquid cohesion are invoked. Key evidence for the atmospheric model is that the maximum height of a siphon is approximately equal to the height of a column of liquid that can be supported by the ambient barometric pressure. In this model, a siphon is considered to be two back-to-back barometers. Another piece of evidence in support of the atmospheric model is the fact that siphon flow can occur with an air bubble inside the tube so that there is no physical connection between the water molecules. Evidence in support of the gravity cohesion model is that siphons have been shown to operate under vacuum conditions7,8,9 and the model can explain a curious waterfall-like feature when a siphon is operating close to the barometric limit10.

Both siphon models–atmospheric and cohesion–predict that the maximum height of a siphon is dependent on the ambient barometric pressure. In the case of the atmospheric model, the pressure of the atmosphere is required to hold the column of water together. In the cohesion model, the limit is explained by the pressure at the top of the siphon falling below the vapour pressure of water, at the given temperature, so that cavitation occurs, i.e. the water starts to boil thereby breaking the column.

However, the cohesion model predicts that if cavitation can be prevented, the barometric height limit can be broken. The reason for the cohesion is that surfaces cost energy and the water/air surface is no different. For water, the surface energy is often referred to as surface tension. The surface energy of the water/air interface is 0.072 J/m2. It costs energy to make bubbles in water because of the energy of the bubble surface. For a bubble to be stable it must be supported either by internal pressure of a gas or by the equivalent tension (negative pressure) in the water. For gas in a bubble the pressure (P) is given by (1). This equation11 is exact for an ideal gas, but an approximation for a real gas.

where γ is the surface energy (J/m2 or N/m) and r (m) is the bubble radius. A good benchmark pressure is the atmospheric pressure which is = 1.013 × 105 Pa (N/m2). An internal pressure of one atmosphere (or equivalent tension in the water) could support a bubble of radius r where:

That is, an internal pressure of one atmosphere is generated by a bubble of 1.42 μm radius (a diameter of 2.8 μm). Equivalently, tension equal to the support of one atmosphere would occur for an empty bubble of diameter 2.8 μm. A smaller bubble would support greater water tension and a larger bubble a lesser water tension. A bubble of 2.8 nm diameter could tolerate water tension equal to 1000 atmospheres (100 MPa).

Many experiments have been performed to measure the tensile strength of water12,13,14,15,16,17,18,19,20and values as high as −150 MPa have been achieved21. All these experiments have been performed in static samples. In this paper we report, for the first time, a siphon operating at above the barometric limit at ambient atmospheric pressure. Thus we demonstrate the bulk flow of water under tension.

In an initial experiment, 60 ml of ordinary tap water with a 4 ml silicon oil-capping layer was held under a vacuum of <10−3 Pa for a period of more than three weeks. During the initial degassing process, significant volumes of gas were evolved from both the water and capping layers. This process is commonly attributed to boiling, but as qualified in subsequent sections, this effect is entirely due to dissolved gasses coming out of the water. A small amount of water (~2 ml) was evaporated from the initial volume, mainly due to the exposure of the surface of the water when large bubbles passed though the capping layer.

Once the water and capping layer were fully degassed, there was no further loss of either fluid. After allowing the vessel to return to atmospheric pressure for a short time, subsequent evacuations did not cause more gas to evolve from the water (video sequence 1). However, returning the container to the ambient air pressure for several hours did allow gas to be reabsorbed into the oil-capping layer and over a longer period, into the water underneath. This gas was released again when the container was re-evacuated.

In the next experiment, the cohesive strength of water was tested using a simple inverted U-tube with the base exposed to vacuum, in the manner of a barometer (Fig. 1). Initially the U-tube was set to below the level of the surface of the liquid, while the glass vessel was evacuated and all gases fully removed from above and within the liquids. When the partial pressure inside the vessel reduced to 7.5 ± 0.05 × 10−1 Pa the U-tube was raised by lifting the apex of the tube to a height of 300 mm above the surface of the oil. With a density marginally lower than that of water, the oil surface was assumed to be close to that of a hypothetical water vacuum interface. It was observed that the water formed a continuous column with no bubbles/cavities forming at the top of the tube (Fig. 2). The inverted U-tube was then held in this position for a period greater than four weeks. After this time, the U-tube was tilted further, so the apex was 400 mm above the surface, while reducing the partial pressure above the liquid to 5 ± 0.05 × 10−3 Pa. In this position the water column was observed to be stable with no bubbles seen to evolve in the U-tube even after several hours.

Figure 1

Upper picture: Experimental apparatus for the degasification of water; Right picture: Expanded view of McLeod Gauge; Lower diagram: The 100 ml graduated glass measuring cylinder is filled with 60 ml of water and capped with 5 ml of oil, which stands on a small Perspex tray above the turbomolecular pump. Pressure gauges are marked 1) APG-M-NW16, 2) AIM-S-NW25 and McLeod.

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Figure 2

Diagram of a water-filled U-tube barometer.

The lower figure shows the position during evacuation and degassing of the water with an oil-capping layer and the upper figure shows the U-tube tilted into position while the base is held under vacuum.

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To test the ability of water to maintain cohesion under conditions of flow, a glass siphon was constructed such that both reservoirs could be held under high vacuum (Fig. 3), in a manner similar to that performed previously by Noaks8. In this arrangement, during the degassing process with the U-tube set below the oil, the level of the liquid in both reservoirs was equal with half filling each. When the U-tube was then raised to a vertical position, an offset in position allowed one reservoir to rise further than the other leading to a small height difference. With the U-tube initially in the lower position, water was degassed to a partial pressure of 9.5 ± 0.05 × 10−1 Pa. The apex of the U-tube was the raised 300 mm and water observed to flow from the higher chamber to the lower via the siphon tube into the lower chamber (video sequence 2).

Figure 3

Photograph of U-tube barometer while under vacuum.

Pressure readings are in Pa and the height of the apex is 300 mm above the surface of the liquid.

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While the flow was initiated independently of atmospheric pressure within the siphon, it was noted that the movement of the reservoirs between the static and flowing conditions exposed surfaces that were previously covered with water. As this happened the pressure in the vacuum region was observed to rise above 103 Pa. Realising that this represented a fundamental flaw, in this and in previous attempts by others at producing a water siphon under vacuum conditions, it was deemed that a moderate length siphon could not conclusively discount the effects of vapour pressure on supporting the column.

In order to discount the effect of external pressure acting on the liquid column, a second siphon was constructed, operating under atmospheric conditions, with a height above the nominal barometric limit of 10 m, using water degassed using a vacuum desiccator (Fig. 4).

Figure 4

Diagram of a water siphon under vacuum.

The lower figure shows the position during evacuation and degassing of the water with oil capping layer and the upper figure shows the position of the siphon tilted with liquid flowing from the upper to the lower reservoir, while each reservoir is held under vacuum.

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The siphon height, defined as the vertical distance between the surface of the water in the upper reservoir and apex of the tube, started at 1498 ± 2 cm and increased to 1504 ± 2 cm (Fig. 5). The barometric pressure during the experiment was 99.8 ± 0.1 kPa. The experiment was repeated a number of times and an example is shown in the relevant supplementary video (video sequence 3). After opening both taps at the base of the pre-primed siphon, water was observed to flow out of only the lower of the two siphon legs (video sequence 4). Approximately 400 ml of water flowed from the upper to lower reservoir in 850 s corresponding to a flow of 4.7 ± 0.05 × 10−7 m3 s−1 and average velocity of 1.7 ± 0.05 × 10−2 m s−1.

Figure 5

Diagram of a siphon taller than the barometric limit with the reservoirs open to air.

Water in the upper reservoir is capped with a 5 mm layer of silicon oil. A pulley is used at the apex to support the length of tube and prevent kinks in the pipe.

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To measure the effects of capillary action in making any contribution to lifting the water within the siphon tube, one end of the empty siphon tube was immersed in the degassed water, which was open to air, while the other open end of the tube was held above the level of the liquid. As no differential was observed between the heights of the liquid inside the nylon tube and outside, capillary action was discounted as playing any significant role in the siphon process.

The ability to completely degas water has always represented a significant challenge in performing experiments investigating liquid tensile strength. It is widely known that the great variance observed both within and across different methods investigating the properties of water is due to the unpredictable nature of the gases dissolved within22. In water free of all dissolved gases, bubbles only form when the energy gained in forming a cavity is greater than the binding energy of the surrounding molecules.

Cavity formation in fully degassed water thus represents the limit of cohesion of the water molecules. Of the methods used, such as boiling, sonication, membrane degasification and freeze pump thaw, those where water is exposed to a vacuum are generally considered to be the most effective at removing all dissolved gases. This can be understood by extrapolating to the limit of Henry’s law

where C is the solubility of a gas at a fixed temperature in a particular solvent, k is Henry’s constant and Pgas the partial pressure of the gas above the liquid. Accordingly, at zero pressure the amount of dissolved gas should also equally be zero. However, due to practical constraints it is difficult to achieve pressures above the surface much below that of the vapour pressure, which for water at 20 °C is approximately 2.33 kPa and consequently some dissolved gases will always be present.

At temperatures above freezing and below the boiling point, the bonds between adjacent water molecules at the liquid air interface are continuously being broken and reformed. This constant exchange between molecules leaving and re-joining is generally at equilibrium at atmospheric pressure and room temperature, which is why we see liquid water so abundantly on earth. However, once the pressure above the interface is reduced, or the temperature of the liquid below increased, the equilibrium shifts and water molecules are on average lost from the bulk liquid.

A simple method to overcome water loss is to change the energy barrier at the surface of the water by applying a layer of immiscible liquid above the surface. By floating a liquid with low specific gravity and ultra-low vapour pressure over the water, molecules at the interface are unable to leave the water and migrate through the capping liquid to the surface. Thus evaporative loss that normally occurs below the water vapour pressure is considerably reduced, if not entirely negated.

After initially degassing the water, there was no further evaporative loss or cavitation within the bulk liquid, or at any interface when the ambient pressure was below 10−3 Pa. While it could be argued that the oil was applying a downward force on the water raising the pressure above that of the vapour point, with a capping layer of only 5 mm, the oil would contribute a downward pressure of less than 43 Pa.

It was also observed that with the surface of the water capped by oil during the degassing stage there was only a drop in temperature, measured on a mercury thermometer, when the water surface became exposed to the vacuum, as happened when large bubbles exploded at the surface. The temperature of the water would then gradually increase over time returning to the ambient temperature of the lab. This very slow temperature increase was attributed in part due to some radiant energy through the Perspex front of the chamber but predominantly from thermal conduction through the apparatus. Over the period of 3 weeks, when under vacuum, the temperature of the water was observed to remain steady at approximately 21 °C.

This surprising behaviour is explained by considering the dynamics of evaporation, where on average the most energetic molecules tend to leave the surface first. In this case, by increasing the energy barrier at the surface, no evaporation can occur, therefore there is little or no net loss of energy from the system leaving the temperature constant. Consequently, while the oil acts as an effective barrier to the evaporative loss of water, it does not prevent gas transport in either direction, or significantly change the pressure gradient within the liquid. Consequently these experiments show that while exposed water does evaporate under low partial pressures, as would be expected, internal cavitation or nucleated boiling does not occur at room temperature even under extremely low ambient pressures.

For a siphon with dissolved gases the maximum height (hm) of a siphon is

where P0 is the ambient atmospheric pressure, Pv is the vapour pressure of water, v the mean velocity of the water and the other symbols are as previously defined in this paper. The expression for the atmospheric model is the same as equation (3) except with no Pv term.

The siphon in the experiment described in this paper was clearly operating above the barometric limit, which, at the given barometric pressure was 10.18 ± 0.01 m for the atmospheric model and 9.94 ± 0.01 m for the cohesion model (ignoring the negligible velocity term). Therefore, it is evident that atmospheric pressure plays no part in carrying the water over the apex of the siphon tube. Therefore it is clear that a new equation for the maximum height of a siphon is required for situations where cavitation does not occur.

The new equation is much simpler and is

where TSw is the tensile strength of water. So for example if the tensile strength of a sample of water was 1 MPa, the maximum height of a siphon would be about 100 m. In the case of the siphon is this experiment we can say that the tensile strength of the water was greater than −0.15 MPa.

Extrapolating these results from even the most conservative experimental measurements of the tension under which cavitation occurs it is possible that the cohesive strength of fully degassed water is able to support a continuous vertical column greater than several hundred meters. While the experiment performed here did not reach anywhere near the absolute limit predicted it does shed light on the stability of flowing water under tensile stress and the possibility of constructing apparatus of suitable dimensions to test such a limit. These experiments also lend support to the cohesion-tension theory of sap ascent in trees. It would be interesting to perform further experiments to see if it is possible to operate a flowing siphon at above 100 m. If tensions as high as the transient tension of several 100 bar can be maintained at the apex of a siphon, then in principle a siphon should work up to a height of several km. However, it would be challenging to verify this experimentally, requiring a helicopter or UAV with a ceiling of several km capable of supporting several kg of water-filled tubing and cable supporting the siphon. It would also be interesting to repeat the experiment with a larger diameter tube. In view of the many anomalies of bulk water23, it would be interesting to explore the physical properties of water in the negative pressure regime of a siphon above 10 m.