| vro = uo cosec φ wo = Vo − uo cot φ vo = √ {uo2 + (Vo − uo cot φ)2} |
(5)
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
ho − hi = (Vo2 − Vi2) / 2g − (vro2 − vri2) / 2g.
Inserting the values of vro, vri in (4) and (5), we get for normal conditions of working
ho − hi = (Vo2 − Vi2) / 2g − uo2 cosec2 φ / 2g + (ui2 + Vi2) / 2g
= Vo2 / 2g − uo2 cosec2 φ / 2g + ui2 / 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 α its angular velocity; wo, ro the tangential velocity of the water and radius at the outlet surface; wi, ri the same quantities at the inlet surface. Q being the discharge per second, the change of angular momentum per second is
(GQ/g) (woro − wiri).
Hence