V_o = V_i r_o/r_i,
where r_o, r_i are the radii of the outlet and inlet surfaces.
In a radial inward flow turbine,
V_i = 0.66 [root](2gH),
V_o = V_i r_o/r_i.
If the wheel were stationary and the water flowed through it, the
water would follow paths parallel to the wheel vane curves, at least
when the vanes were so close that irregular motion was prevented.
Similarly, when the wheel is in motion, the water follows paths
relatively to the wheel, which are curves parallel to the wheel vanes.
Hence the relative component, v_r, of the water's motion at c is
tangential to a wheel vane curve drawn through the point c. Let v_o,
V_o, v_(ro) be the velocity of the water and its common and relative
components at the outlet surface of the wheel, and v_i, V_i, v_(ri) be
the same quantities at the inlet surface; and let [theta] and [phi] be
the angles the wheel vanes make with the inlet and outlet surfaces;
then
v_o² = [root](v_(ro)² + V_o² - 2V_o v_(ro) cos [phi])
v_i = [root](v_(ri)² + V_o² - 2V_i v_(ri) cos [theta]), (13)
equations which may be used to determine [phi] and [theta].
[Illustration: FIG. 195.]
§ 192. _Condition determining the Angle of the Vanes at the Outlet
Surface of the Wheel._--It has been shown that, when the water leaves
the wheel, it should have no tangential velocity, if the efficiency is
to be as great as possible; that is, w_o = 0. Hence, from (10), cos
[beta] = 0, [beta] = 90°, U_o = V_o, and the direction of the water's
motion is normal to the outlet surface of the wheel, radial in radial
flow, and axial in axial flow turbines.
Drawing v_o or u_o radial or axial as the case may be, and V_o
tangential to the direction of motion, v_(ro) can be found by the
parallelogram of velocities. From fig. 195,
tan [phi] = v_o/V_o = u_o/V_o; (14)
but [phi] is the angle which the wheel vane makes with the outlet
surface of the wheel, which is thus determined when the velocity of
flow u_o and velocity of the wheel V_o are known. When [phi] is thus
determined,
v_(ro) = U_o cosec [phi] = V_o [root](1 + u_o²/V_o²). (14a)
_Correction of the Angle [phi] to allow for Thickness of Vanes._--In
determining [phi], it is most convenient to calculate its value
approximately at first, from a value of u_o obtained by neglecting the
thickness of the vanes. As, however, this angle is the most important
angle in the turbine, the value should be afterwards corrected to
allow for the vane thickness.