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The formula for the maximum efficiency (\(\eta_{max}\)) of a Pelton wheel turbine is ______, where \(\phi\) is vane angle at the outlet.

  • A.\(\eta_{max} = \frac{1 + \cos\phi}{2}\)
  • B.\(\eta_{max} = \frac{1 + \sin\phi}{2}\)
  • C.\(\eta_{max} = \frac{\cos\phi}{2}\)
  • D.\(\eta_{max} = \frac{\sin\phi}{2}\)

Answer: A

For a Pelton wheel the hydraulic efficiency is maximum when the bucket speed is half the jet speed, and then

\[ \eta_{max} = \frac{1 + \cos\phi}{2} \]

where \(\phi\) is the vane angle at the outlet. For a smooth bucket that turns the jet right round (\(\phi = 0\)) this gives 100 percent. The options with sin phi or with cos phi alone (over 2) do not have this property: the second would give an efficiency of 50 percent when phi is zero, and the third and fourth would give 50 percent and zero.

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