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24   Chapter Tw o


              must be used. If the blade moves at speed u, then with respect to the
              blade,

                     F = ˙m [−(v 0 − u) − (v 0 − u)] =−2˙m(v 0 − u)  (2-40)

              The power that is delivered to the blade (and, therefore, the rotor) is:

                                 P = Fu = 2˙m (v 0 − u) u         (2-41)

              When u = 0, then force F is the largest, but power P is zero. When
              u = v 0 , the rotor blade is moving away at the same speed as the water
              jet. Therefore, as expected, the force is zero and power is zero. Highest
              power is delivered when u = v 0 /2.
                                             2
                                     P max = ˙mv /2               (2-42)
                                             0
              Therefore,themaximumpowerthatcanbeextractedbyaPeltonwheel
              rotor is equal to the kinetic energy in water, which implies that the the-
              oretical efficiency of the Pelton wheel water rotor is 1 when it operates
              at u = v 0 /2.
                 From a mechanical efficiency standpoint, considering friction and
              other losses, most water rotors operate at 90% or higher efficiency. In
              contrast, the maximum efficiency at which a wind rotor operates is
              50%. The reason for such a large discrepancy in efficiency is the me-
              chanics of energy transfer. With water, the direction of flow of almost
              all the particles is the same as the overall flow direction and, therefore,
              it is able to deliver most of the energy to the rotor. If air were used in
              a Pelton wheel, a significant amount of the energy would be used up
              to increase the randomness in velocity of air particles as opposed to
              imparting energy to the rotor.


        References
              1. Danish Wind Industry Association. Wind Turbines Deflect the Wind
               [Online] June, 2003. http://guidedtour.windpower.org/en/tour/wres/
               tube.htm.
              2. Okulov, V. L., and Sorensen, J. N. “Optimum operating regimes for the ideal
               wind rotor,” Journal of Physics: Conf. Ser., 75: 2007.
              3. Hansen, Martin O.L. Aerodynamics of Wind Turbines, Chapter 5: Shrouded
               rotors. 2nd Edition. Earthscan, Sterling, VA, 2008.
              4. Patel, Mukund R. “Wind speed and energy,” Wind and Solar Power Systems,
               CRC Press, Boca Raton, FL, 2006.
              5. Burton, T., Sharpe, D., Jenkins, N., and Bossanyi, E. Wind Energy Handbook,
               Wiley, Hoboken, NJ, 2001.
              6. Wagner,H-J.,Mathur,J.IntroductiontoWindEnergySystems,Springer,Berlin,
               2009.
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