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

Question

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.

Accepted Answer

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.

[latex] \begin{aligned} Q=V_B A_B ; \quad 1.7 m ^3 / s & =V_B\left[\pi(0.15 m )^2 ight] \ V_B & =24.05 m / s \end{aligned} [/latex]

Energy Equation. Applying the energy equation between A (in) and B (out), with the gravitational datum set at B, we have

[latex] \begin{gathered} \frac{p_A}{\gamma}+\frac{V_A^2}{2 g}+z_A+h_{ ext {pump }}=\frac{p_B}{\gamma}+\frac{V_B^2}{2 g}+z_B+h_{ ext {turbine }}+h_L \ 0+0+60 m +0=0+\frac{(24.05 m / s )^2}{2\left(9.81 m / s ^2 ight)}+0+h_{ ext {turbine }}+4 m \ h_{ ext {turbine }}=26.52 m \end{gathered} [/latex]

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

[latex] \dot{W}_s=\dot{m} g h_s=Q \gamma h_s [/latex] (5-17)

[latex] \begin{aligned} \dot{W}_s & =Q \gamma h_s=\left(1.7 m ^3 / s ight)\left(9810 N / m ^3 ight)(26.52 m ) \ & =442 kW \end{aligned} [/latex]

By comparison, the power lost due to the effects of friction is

[latex] \dot{W}_L=Q \gamma h_L=\left(1.7 m ^3 / s ight)\left(9810 N / m ^3 ight)(4 m )=66.7 kW [/latex]

Question 5.10: The turbine in Fig. 5–24 is used in a small hydroelectric pl...

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