Question 5.10: The turbine in Fig. 5–24 is used in a small hydroelectric pl......
The turbine in Fig. 5–24 is used in a small hydroelectric plant. If the discharge through the 0.3-m-diameter draft tube at B is 1.7 m³/s, determine the amount of power that is transferred from the water to the turbine blades. The frictional head loss through the penstock, turbine, and draft tube is 4 m.
Fluid Description. This is a case of steady flow. Here viscous friction losses occur within the fluid. We consider the water to be incompressible, where γ = 9810 N/m³.
Control Volume. A portion of the reservoir, along with water within the penstock, turbine, and draft tube, is selected to be the fixed control volume. The average velocity at B can be determined from the discharge.
Q = V_BA_B; 1.7 m^3/s = V_B[\pi(0.15 m^2)]
V_B = 24.05 m/s
Energy Equation. Applying the energy equation between A (in) and B (out), with the gravitational datum set at B, we have
0 + 0 + 60 m + 0 = 0 +\frac{(24.05 m/s)^2}{2(9.81 m/s^2)} + 0 + h_{turbine} + 4 m
h_{turbine} = 26.52 m
As expected, the result is positive, indicating that energy is supplied by the water (system) to the turbine.
Power. Using Eq. 5–17, the power transferred to the turbine is therefore
\dot{W_s} = Q\gamma h_s = (1.7 m^3/s)(9810 N/m^3)(26.52 m)
= 442 kW
By comparison, the power lost due to the effects of friction is
\dot{W_L} = Q\gamma h_L = (1.7 m^3/s)(9810 N/m^3)(4 m) = 66.7 kW