gH g²H²
[root]{2gH + [alpha]²r²} = [alpha]r + -------- - ----------- ...
[alpha]r 2[alpha]³r³
then
[eta] = 1 - gH/2[alpha]r + ...
which increases towards the limit 1 as [alpha]r increases towards
infinity. Neglecting friction, therefore, the maximum efficiency is
reached when the wheel has an infinitely great velocity of rotation.
But this condition is impracticable to realize, and even, at
practicable but high velocities of rotation, the friction would
considerably reduce the efficiency. Experiment seems to show that the
best efficiency is reached when [alpha]r = [root](2gH). Then the
efficiency apart from friction is
[eta] = {[root](2[alpha]²r²) - [alpha]r} [alpha]r/gH
= 0.414 [alpha]²r²/gH = 0.828,
about 17% of the energy of the fall being carried away by the water
discharged. The actual efficiency realized appears to be about 60%, so
that about 21% of the energy of the fall is lost in friction, in
addition to the energy carried away by the water.
§ 184. _General Statement of Hydrodynamical Principles necessary for
the Theory of Turbines._
(a) When water flows through any pipe-shaped passage, such as the
passage between the vanes of a turbine wheel, the relation between the
changes of pressure and velocity is given by Bernoulli's theorem (§
29). Suppose that, at a section A of such a passage, h1 is the
pressure measured in feet of water, v1 the velocity, and z1 the
elevation above any horizontal datum plane, and that at a section B
the same quantities are denoted by h2, v2, z2. Then
h1 - h2 = (v2² - v1²)/2g + z2 - z1. (1)