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Water turbines - notes on calculations

Figure [*] represents a physical setting of a turbine casing with respect to a dam. Let us denote by $ h$ the head, i.e. the difference between heights of water levels in front of the dam and at the back of it. Vertical distance between the center of mass of the inlet surface of volute and the high water level is denoted by $ h_{In}$. Vertical distance between the center of mass of the outlet surface of draft tube and the low water level is denoted by $ h_{Out}$. Finally, vertical distance between the centres of mass of inlet and outlet surfaces is denoted by $ h_{IO}$.
sketch water turbine levels

Figure: General water turbine sketch. Physical setting of a turbine casing with respect to a dam. Water levels indicated.

We can see there are several simple relations among quantities just defined




To simplify calculations we usually considers water an incompressible fluid with no phase changes possible and we solve the well known equations, the conservation laws the fluid obeys.

Coservation of linear momentum is expressed by the Navier-Stokes equation. In this case (incompressible fluid, steady state) it can take the form


where $ \vec{U}$ is a velocity field, $ \nu$ kinematic viscosity, $ \varrho$ density, $ p$ pressure field and $ \vec{f}$ is a force density of a gravitational field, given by ([*]).

Conservation of mass is expressed by the continuity equation. In this case it takes the simple form


emains to discuss boundary conditions for the equations mentioned above. For we usually do not know the inlet velocity field, we are about to prescribe pressure field for both inlet and outlet surface.

At the inlet surface a total pressure is known and it is equal to hydrostatic one. With the use of formula ([*]) we can write


where $ \vec{r}_{0,H}$ is a position vector of an arbitrary point of the high water level (plane). Since total pressure is defined as a sum of static and dynamical pressure


we can, by means of this definition ([*]) and formula ([*]), express $ p$ at the inlet


At the outlet the situation is a little bit complicated. After the flow (with high total pressure) exits the draft tube it mixes with water (with low total pressure) surrounding the draft tube. We usually take simplifying steps and assume the static pressure of exiting flow equals the hydrostatic pressure of surrounding water, i.e. we write


where $ \vec{r}_{0,L}$ is a position vector of an arbitrary point of the low water level.

Described steps correspond to real physical setting. In a task like this we need to know not only the $ h$, but even $ h_{In}$ (optionally $ h_{Out}$), i.e. a position of the turbine with respect to the high or low water level. Quantity $ h_{IO}$ can be obtained from the given turbine casing geometry, of course.