Reconsider Prob. 666. The maximum efficiency of the turbine occurs when = 180, but this is

Reconsider Prob. 6–66. The maximum efficiency of the turbine occurs when β = 180°, but this is not practical. Investigate the effect of β on the power generation by allowing it to vary from 0° to 180°. Do you think we are wasting a large fraction of power by using buckets with a β of 160°?

Data from Exercise 66

Pelton wheel turbines are commonly used in hydroelectric power plants to generate electric power. In these turbines, a high-speed jet at a velocity of V_j impinges on buckets, forcing the wheel to rotate. The buckets reverse the direction of the jet, and the jet leaves the bucket making an angle β with the direction of the jet, as shown in Fig. P6–66. Show that the power produced by a Pelton wheel of radius r rotating steadily at an angular velocity of ω is Ẇ_shaft = ρωrV̇. (V_j – ωr)(1 – cos β), where ρ is the density and V̇ is the volume flow rate of the fluid. Obtain the numerical value for ρ = 1000 kg/m³, r = 2 m, V̇ =10 m³/s, ṅ = 150 rpm, β = 160°, and V_j = 50 m/s.

FIGURE P6–66

Transcribed Image Text:

Nozzle W Shaft r00 V₁-rw