Case 1. Axial Flow Turbines.—Vi = Vo; and the first term on the right, in equation 17, disappears. Adding, however, the work of gravity due to a fall of d ft. in passing through the wheel,
hi − ho = (vro2 − vri2) / 2g − d.
17a
Case 2. Outward Flow Turbines.—The inlet radius is less than the outlet radius, and (Vi2 − Vo2)/2g is negative. The centrifugal head diminishes the pressure at the inlet surface, and increases the velocity with which the water enters the wheel. This somewhat increases the frictional loss of head. Further, if the wheel varies in velocity from variations in the useful work done, the quantity (Vi2 − Vo2)/2g increases when the turbine speed increases, and vice versa. Consequently the flow into the turbine increases when the speed increases, and diminishes when the speed diminishes, and this again augments the variation of speed. The action of the centrifugal head in an outward flow turbine is therefore prejudicial to steadiness of motion. For this reason ro : ri is made small, generally about 5 : 4. Even then a governor is sometimes required to regulate the speed of the turbine.
Case 3. Inward Flow Turbines.—The inlet radius is greater than the outlet radius, and the centrifugal head diminishes the velocity of flow into the turbine. This tends to diminish the frictional losses, but it has a more important influence in securing steadiness of motion. Any increase of speed diminishes the flow into the turbine, and vice versa. Hence the variation of speed is less than the variation of resistance overcome. In the so-called centre vent wheels in America, the ratio ri : ro is about 5 : 4, and then the influence of the centrifugal head is not very important. Professor James Thomson first pointed out the advantage of a much greater difference of radii. By making ri : ro = 2 : 1, the centrifugal head balances about half the head in the supply chamber. Then the velocity through the guide-blades does not exceed the velocity due to half the fall, and the action of the centrifugal head in securing steadiness of speed is considerable.
Since the total head producing flow through the turbine is H − ɧ, of this hi − ho is expended in overcoming the pressure in the wheel, the velocity of flow into the wheel is
vi = cv √ {2g (H − ɧ − (Vi2 − Vo2 / 2g + (vro2 − vri2) / 2g) ],
(18)
where cv may be taken 0.96.
From (14a),