The following is a purely verbal description of the numerical modelling which was undertaken in order to estimate the efficiency of a very large scale steam-foam-driven bubble pump. A more detailed description can be found in the PDF version of these pages.
The Equal Velocity Assumption
Lower density material rising in a fluid gives rise to a plume. Such plumes are well described by three equations describing conservation of mass, conservation of momentum and conservation of buoyancy first described by Morton, Taylor and Turner (1956). These equations describe how plumes widen and slow down as they rise due to the entrainment of surrounding fluid.
Developments based on the three conservation assumptions have lead to predictions of plume behaviour in close agreement with plumes observed experimentally.
In the present case one of the underlying assumptions, that of conservation of buoyancy, no longer holds. That is because the bubbles are bubbles of steam-foam rather than of air and such bubbles will tend to collapse as they lose heat to the surrounding water so that buoyancy is not conserved in this case. For this reason it is not possible to model such a plume in the manner of Morton et al. Some other assumptions must be made.
The large theoretical yields calculated in the previous section are entirely a consequence of the buoyancy of the steam-foam injected into the water column by the thin pipe. If this steam-foam were to be instantaneously perfectly mixed with the surrounding colder water the foam would immediately collapse and the yield would be negligibly small. In thermodynamic terms there would be a large increase in entropy entailed in the sudden loss of heat brought about by such mixing.
It follows that the best method of preserving the buoyancy of the foam is to inhibit entrainment and the resulting mixing by containing the plume inside a second pipe. This second pipe will be called here the "fat pipe" to distinguish it from the "thin pipe" which brings the superheated water up from the ocean floor. The cross-sectional area of the fat pipe is assumed to be considerably greater than that of the thin pipe. Apart from suppressing entrainment, the fat pipe also serves to create an ascending column of water around the steam-foam emitted by the thin pipe.
The assumption which we now make and which will allow modeling of the ascending plume to proceed, is that the vertical velocity of the cold water in the fat pipe is equal to the vertical velocity of the ascending steam-foam plume.
Instead of a free plume, we now have a steam-foam-driven bubble pump similar to the small bubble pumps used in motor-less refrigerator systems but on a much larger scale. Because the relative velocity of the plume and the surrounding water column is zero there will be no entrainment of cold water into the plume in the manner described by Morton et al.
The single most critical factor determining the viability of such a bubble pump is the lifetime of the steam-foam plume after it comes into contact with cold water.
Inside the foam bubble the conductivity is effectively infinite because liquid is in equilibrium with its own vapour at a fixed pressure. This is the same situation that occurs in a heat pump. If heat is introduced into any part of the mixture the pressure will increase throughout the bubble or vessel so causing condensation and latent heat release elsewhere in the vessel. Because of this heat is transferred extremely rapidly; the rate of heat transfer is dependent only on the speed of sound in the liquid-vapour mixture.
Outside the boundary of the bubble, heat is conducted according to Fourier's equation. The thermal diffusivity of water is very small so that heat diffuses very slowly through water in the absence of convection.
In order to gain an insight into what happens at the boundary between steam-foam and cold water, a simple one dimensional numerical model was set up in which steam-foam and water meet at an interface.
The results showed that for the rate of heat transfer fell very rapidly over the first half second or so then leveled out. More significantly, the heat from the steam-foam only penetrated a short distance into the water. Even after 5 seconds it had barely penetrated 3 mm from the interface.
A Rising Spherical Bubble
The account given above describes an idealized first order model. In order to achieve greater realism a numerical model of a three dimensional "bubble" of steam-foam was constructed. By "bubble" is meant a bubble of steam and water foam such as that emitted by the thin pipe, not the tiny bubbles of pure steam which make up the foam itself. It was assumed that the bubble is rising at the same rate as the surrounding water and is held by surface tension in the form of a sphere which is sufficiently small for the vertical pressure gradient across the sphere to be ignored.
From the time of its creation the bubble not only loses heat to its surroundings but is also expanding due to the pressure drop as it rises through the water column. As it rises it will also cool and this pressure-drop cooling and heat losses due to conduction must be modeled simultaneously. Likewise expansion due to pressure-drop and contraction due to heat loss must also be accounted for simultaneously.
Results from a number of model runs are shown in the figure.
The figure shows how the bubble diameter changes when the bubble ascends with a vertical velocity of 10 m/s. The ratio of bubble diameter to initial bubble diameter is plotted against depth for 3 different values of initial diameter. Obviously the diameter and hence the buoyancy of a bubble is a strong function of its initial diameter. Small bubbles lose buoyancy as they rise while large bubbles gain in buoyancy because expansion due to decreasing pressure outweighs contraction due to conductive heat losses. This effect is a direct result of the differing ratio of surface area to volume of different size bubbles.
Large bubbles behave very differently from small bubbles. The dynamics of steam-foam bubble plumes is strongly dependent on scale.
A Steady State Plume
The previous section described the behaviour of a single isolated bubble of steam-foam surrounded in 3 dimensions by much colder water which extracts energy from the bubble in the form of heat. This is a pessimistic view of the real situation.
In a real steam-foam bubble plume, any single bubble will be accompanied by many similar bubbles all losing heat to their surroundings. In this situation temperature gradients will be much less on average than for a single bubble in isolation.
At this stage we do not know on either experimental or theoretical grounds how a steam-foam plume will break up into individual bubbles although it seems likely that it will do so. One way of dealing with this and so accounting for the proximity of neighbouring bubbles, is to consider a single conical plume of steam-foam rising with the same vertical velocity as the surrounding cold water.
Another assumption, which may not be justified in practice, is that the rising plume is in a steady state, i.e. that it retains the same three dimensional shape over time.
We follow the behaviour of a disk as it rises through the plume.
Varying the velocity caused large changes in the volume of the plume as can be seen in the figure. This occurs because as the vertical velocity increase the residence time of each disk of steam-foam in the plume decreases in inverse proportion.
Calculating the Fat Pipe Yield
Once the volume and mass of the plume have been found the effective body force on the water in the fat pipe can be calculated. It is convenient to express this quantity as an equivalent head of water, i.e. as the difference in water level at the top of the pipe that would create the same gravitational force on the water in the pipe as does the presence of the plume. Expressing the buoyant force in this way allows conventional formulas describing flow in pipes to be used. Once a vertical velocity has been found in this way the volume flux or yield, , of the bubble pump can be determined. Yield as a function of fat pipe diameter is shown in the figure.
The yield for a 10 MW hydrothermal vent injecting steam-foam into a fat pipe at 400m depth is seen to increase rapidly with fat pipe diameter at small diameters and then to level out to a value of 220 cubic metres per second for larger diameters.
Based on potential energy considerations alone the maximum possible yield of a 10 MW HTV injecting at 400m was calculated in the previous section to be 1151 m3/s. No machine is perfectly efficient. There always losses due to entropy increases incurred by heat losses and friction. The ratio of the actual yield for a heat engine such as this is termed the thermodynamic efficiency.
The thermodynamic efficiency of this hydrothermal bubble pump is estimated according to this numerical model as 20 percent. This compares favorably with the thermodynamic efficiencies of other heat engines such as the steam locomotive (5% to 8%) and the steam turbine (34% to 38%).
Free energy is lost from the system in 3 ways, viz: as heat loss from the foam to the surrounding water, as friction with the pipe walls and in the kinetic energy of the rising water column.
Cold water brought to the surface in this way might be expected to sink back down again immediately after it leaves the fat pipe because it is more dense than the surrounding water. However it also has considerable kinetic energy. The rising water mass will "hit" the ocean surface forcing it to rise a little. The gravitational force on this bulge will dissipate the vertical momentum of the water but leave its kinetic energy unchanged. It will spread out radially beyond the mouth of the fat pipe. It is likely that a toroidal eddy will form.
There will be a considerable velocity gradient between the new water and the surrounding water and entrainment of the surrounding water will occur. As this happens the kinetic energy will cascade down to smaller scales and finish up as turbulent kinetic energy.
Obviously the situation is not straightforward and further modeling and experimentation is required. Nevertheless it appears likely that the KE of the rising water and the turbulent kinetic energy that it generates will cause rapid mixing with the surrounding water and so militate against its sinking back down to the depths whence it came.
The lay public and often scientists themselves tend to place too much credence in computer models. The real world must always be more complex than any computer model can encapsulate because real phenomena operate over a large range of scales in a manner that numerical models cannot yet replicate. This means that all numerical models require assumptions, approximations and parameterizations to be made in order for them to be able to work at all. Numerical models are certainly never oracular predictors of real world behaviour. Rather they are heuristic devices which enable us to gain insights into complex physical processes.
The present case is no exception. The underlying "assumption of equal velocity" would certainly not be valid in the real world. The buoyancy of the steam-foam near the centre of the fat pipe must somehow be communicated to the surrounding cold water. There will be a velocity gradient between the upward force of central buoyant fluid and the downward force of the friction in the pipe walls.
Despite these difficulties some important insights have been gained. These include
• Scale size is of fundamental importance,
• Under some circumstances the plume can cool faster than the halo so that the steam-foam becomes "self-insulating",
• Despite free energy losses there is a massive amplification of flow whereby a flow of a few kilograms per second of superheated water is converted to a flow of hundreds of tonnes of cold water. (Seen as a current amplifier, the "gain" is 220,000 / 6.67 or 33,000.) and
• Energy losses in the form of kinetic energy may assist the cold water plume to remain near the surface after it has left the fat pipe.
|All materials and images © 2007 Ecofluidics Pty Ltd. No unauthorised reproduction.|