r_o = 2r_i,
and
d_o = d_i or ½d_i
according as the disk is parallel-sided or coned. The water enters the
wheel radially with the velocity u_i, and
u_o = Q/2[pi]r_o d_o. (3)
[Illustration: FIG. 211.]
Fig. 211 shows the notation adopted for the velocities. Suppose the
water enters the wheel with the velocity v_i, while the velocity of
the wheel is V_i. Completing the parallelogram, v_(ri) is the relative
velocity of the water and wheel, and is the proper direction of the
wheel vanes. Also, by resolving, u_i and w_i are the component
velocities of flow and velocities of whir of the velocity v_i of the
water. At the outlet surface, v_o is the final velocity of discharge,
and the rest of the notation is similar to that for the inlet surface.
Usually the water flows equally in all directions in the eye of the
wheel, in that case v_i is radial. Then, in normal conditions of
working, at the inlet surface,
v_i = u_i \
w_i = 0 > (4)
tan[theta] = u_i/V_i |
v_(ri) = u_i cosec [theta] = [root](u_i² + V_i²) /
If the pump is raising less or more than its proper quantity, [theta]
will not satisfy the last condition, and there is then some loss of
head in shock.
At the outer circumference of the wheel or outlet surface,
v_(ro) = u_o cosec [phi] \
w_o = V_o - u_o cot [phi] > (5)
v_o = [root]{u_o² + (V - _o - u_o cot [phi])²} /
_Variation of Pressure in the Pump Disk._--Precisely as in the case of
turbines, it can be shown that the variation of pressure between the
inlet and outlet surfaces of the pump is
h_o - h_i = (V_o² - V_i²)/2g - (v_(ro)² - v_(ri)²)/2g.
Inserting the values of v_(ro), v_(ri) in (4) and (5), we get for
normal conditions of working
h_o -h_i = (V_o² - V_i²)/2g - u_o² cosec² [phi]/2g + (u_i² + V_i²)/2g
= V_o²/2g - u_o² cosec² [phi]/2g + u_i²/2g. (6)
_Hydraulic Efficiency of the Pump._--Neglecting disk friction, journal
friction, and leakage, the efficiency of the pump can be found in the
same way as that of turbines (§ 186). Let M be the moment of the
couple rotating the pump, and [alpha] its angular velocity; w_o, r_o
the tangential velocity of the water and radius at the outlet surface;
w_i, r_i the same quantities at the inlet surface. Q being the
discharge per second, the change of angular momentum per second is
(GQ/g)(w_o r_o - w_i r_i).