| √ (2gH + α2r2) = αr + | gH | − | g2H2 | ... |
| αr | 2α3r3 |
then
η = 1 − gH / 2αr + ...
which increases towards the limit 1 as αr 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 αr = √ (2gH). Then the efficiency apart from friction is
η = {√ (2α2r2) − αr} αr / gH
= 0.414 α2r2 / 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)