FablabSU

As the upstream water's height (on the left) and the downstream one (on the right) are known, the pressures Pupstream and Pdownstream can be deduced. As the height of the estuary decreases, the height of the ocean remains the same since its surface is way bigger than the estuary's one. As a consequence, the pressure on the right of the turbine remains the same during the whole draining.

The water flow in the in the penstock is considered as a steady process, incompressible and homogeneous. Whence the Bernouilli's principle:

The estuary's surface S0 orthogonal to the water flow is way bigger than the conduit's surface Supstream orthogonal to the water flow. Whence Vo«Vupstream and therefore:

Using the first assumptions:

As water is considered as incompressible, the area of the penstock and velocity of water are held constant in the conduit. The kinetic energy is supposed constant between the times t and t+dt. Yet, as defined by the kinetic energy theorem:

Ek(t+dt) -Ek(t) = 0 as the kinetic energy is conserved.

From the forces upstream and downstream the turbine the work expression can be deduced. It is composed of the pressure work and the work of the turbine on the water. By dividing this expression by dt, we get the power's expression of the water on the turbine. Energy losses from friction are neglected:

Using the fact that S=Supstream=Sdownstream and Vupstream=Vdownstream

<note>See reference (2)</note>

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