cosec φ = cot φ;
and then
η = (Vo, + uo cosec φ) / 2Vo,
which may approach the value 1, as φ tends towards 0. Equation (8) shows that uo cosec φ cannot be greater than Vo. Putting uo = 0.25 √(2gH) we get the following numerical values of the efficiency and the circumferential velocity of the pump:—
| φ | η | Vo |
| 90° | 0.47 | 1.03 √2gH |
| 45° | 0.56 | 1.06 ” |
| 30° | 0.65 | 1.12 ” |
| 20° | 0.73 | 1.24 ” |
| 10° | 0.84 | 1.75 ” |
φ cannot practically be made less than 20°; and, allowing for the frictional losses neglected, the efficiency of a pump in which φ = 20° is found to be about .60.
§ 210. Case 2. Pump with a Whirlpool Chamber, as in fig. 210.—Professor James Thomson first suggested that the energy of the water after leaving the pump disk might be utilized, if a space were left in which a free vortex could be formed. In such a free vortex the velocity varies inversely as the radius. The gain of pressure in the vortex chamber is, putting ro, rw for the radii to the outlet surface of wheel and to outside of free vortex,
| vo2 | ( 1 − | ro2 | ) = | vo2 | ( 1 − k2 ), |
| 2g | rw2 | 2g |
if