V_i, V_o be the velocities of the wheel at the inlet and outlet
surfaces,
v_i, v_o the velocities of the water,
u_i, u_o the velocities of flow,
v_(ri), v_(ro) the relative velocities,
h_i, h_o the pressures, measured in feet of water,
r_i, r_o the radii of the wheel,
[alpha] the angular velocity of the wheel.
At any point in the path of a portion of water, at radius r, the
velocity v of the water may be resolved into a component V = [alpha]r
equal to the velocity at that point of the wheel, and a relative
component v_r. Hence the motion of the water may be considered to
consist of two parts:--(a) a motion identical with that in a forced
vortex of constant angular velocity [alpha]; (b) a flow along curves
parallel to the wheel vane curves. Taking the latter first, and using
Bernoulli's theorem, the change of pressure due to flow through the
wheel passages is given by the equation
h´_i + v_(ri)²/2g = h´_o + v_(ro)²/2g;
h´_i - h´_o = (v_(ro)² - v_(ri)²)/2g.