Making the Turbines Spin - Dick LaRosa

Kinetic energy does it. The kinetic energy density in the stream, water in this case, is 0.5(density)(velocity squared). Multiply this by velocity to get the rate of kinetic energy transport. Multiply this by the area swept by the turbine blades and you get the power that the turbine could extract if only it could stop the water completely and somehow get it out of the way to make room for new water. But that's not possible. The power coefficient tells us what fraction of the undisturbed kinetic energy transport is extracted as turbine shaft power. People generally figure 0.45 for a good turbine.

But no matter how many turbines you put in the water channel, you don't deplete the kinetic energy transport as long as the water enters and exits the channel at the same rate. In a channel with a free top surface, the water surface slopes downward from inlet to outlet. The force pushing an element of water downstream is (density)(acceleration of gravity)(slope of surface) . Add more turbines and the slope increases as long as the driving hydraulic head is available.

Plots of velocity vs. depth at stations across the Florida Current show that the turbulent boundary layer extends from the surface all the way to the bottom. The velocity profile is linear near the bottom. Layers of water slide over each other. The shear friction force is due to eddy viscosity rather than molecular viscosity. In turbulent flow, parcels of water are displaced upward and downward. Parcels from a slow layer tend to slow down a faster layer and parcels from a fast layer tend to speed up a slower layer. The slope of the velocity vs. depth profile decreases as we go up from the bottom because the gravitational force along a layer replaces some of the shear force of the faster layer above. The velocity profile is roughly parabolic and at some stations across the stream there is a rounded nose on the velocity profile.

The velocity changes across the channel as well as with depth. The water is stratified, with the cooler denser water at the bottom. Isopycnal layers are layers of constant density. Horizontal eddy motion is along isopycnal surfaces, so these tend to be large excursions, whereas vertical eddy motion is restrained by buoyancy forces and is therefore smaller in amplitude. The horizontal eddy viscosity is much greater than the vertical eddy viscosity, which is in turn much larger than the molecular viscosity.

The power that can be extracted from a stream by a multiplicity of turbines is the drop in level from inlet to outlet times the acceleration of gravity times the water density times the volume transport rate times an efficiency factor. In a current system such as we have through the Yucatan Channel, Gulf of Mexico, Florida Straits, and Florida Current, turbines must be floated near the top where the velocity is greatest. The drag of floats, mooring cables, and power cables uses some of the available power. The remainder, multiplied by the power coefficient, is the shaft power of the turbines. I have made sporadic attempts to estimate the parasitic drag and I always throw up my hands and guess that 50% of the available hydraulic power comes out as turbine shaft power. How wrong can that be? It's right in the middle between 0 and 100. As the song says, "But Baby, you've sure got a long way to go."