Bubble Potential Energy Rate
Once the volume flux of sea foam into the fat pipe has been determined it becomes possible to work out how much cold deep water can be brought to the surface in a given time interval.
A stationary foam bubble located at a particular depth in the fat pipe has a particular potential energy associated with it. It is the work that would be done in taking the same bubble at the surface and pushing it downwards against buoyancy forces until it reaches the depth in question. For notational simplicity we will assume the bubble contains the mass of sea foam that would be delivered in one second from the thin pipe as computed above. The potential energy of this bubble is the bubble potential energy rate, BPER. It is the rate at which new potential energy is delivered to the fat pipe.
Specific Potential Energy
Similarly the potential energy required to move unit mass of deep cold water to the surface is the work done in lifting the mass from the initial depth against the force of gravity whilst taking buoyancy due to the surrounding fluid into account. Since we are dealing with unit mass we can call this the specific potential energy, SPE, of the cold deep water.
A Definition of Yield
The mass of cold water that can be brought to the surface from depth in one second, the yield, is the bubble potential energy rate divided by the specific potential energy of the cold deep water.
The calculation of both of these quantities is simply a matter of integrating the body forces over height while taking the change in buoyancy due to changes in volume into account.
For present purposes there are only two free parameters associated with the fat pipe, viz: the nozzle depth, ND, the depth at which the bubble is injected by the thin pipe, and the depth of the bottom of the fat pipe from which the cold nutrient rich water is drawn, termed here the reference depth, RD.
These quantities and their ratio, the yield, were computed for various runs of the numerical model for a selection of nozzle depths and reference depths.
Calculated yields were of the order of ½ million to 1 million Kg/s for a 10 MW HV at 2500m depth.
A Cautionary Note
These calculated yields are maximum theoretical yields. They are the yields which would be obtained in the absence of pipe friction and heat losses and under the assumption that parcels of water move so slowly that kinetic energy considerations can be ignored. In practice none of these things are true. The hydrothermal bubble pump described here is a type of heat engine and will experience energy losses due to all of these effects. A more realistic figure is determined in the section on bubble pump modeling.
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