Problem

A Pelton impulse wheel, as shown in Fig. P5.136(a), typically found in hydroelectric power...

A Pelton impulse wheel, as shown in Fig. P5.136(a), typically found in hydroelectric power plants consists of a wheel at the periphery of which are attached a series of buckets. As shown in Fig. P5.136(b), water jets impinge on the buckets and cause the wheel to spin about its axis (labeled O). Let υw and be the speed and the mass flow rate of the water jets at the nozzles (the nozzles are stationary), respectively. As the wheel spins, a given water jet will impinge on a given bucket only for a very small portion of the bucket’s trajectory. This fact allows us to model the motion of a bucket relative to a given jet (during the time the bucket interacts with that jet) as essentially rectilinear and with constant relative speed, as was done in Example 5.17. Although each bucket moves away from the jet, the fact that they are arranged in a wheel is such that the effective mass flow rate experienced by the vanes is instead of the reduced mass flow rate computed in Eq. (6). With this in mind, consider a bucket, as shown in Fig. P5.136(c), that is moving with a speed υ0 horizontally away from a fixed nozzle but subject to a mass flow rate . The inside of the bucket is shaped so as to redirect the water jet laterally out (away from the plane of the wheel). The angle θ describes the orientation of the velocity of the fluid relative to the (moving) bucket at B, the point at which the water leaves the bucket. Determine θ and υ0 such that the power transmitted by the water to the wheel is maximum. Express υ0 in terms of υw.