h´´_i - h´´_o = (V_i² - V_o²)/2g.
Consequently the whole difference of pressure at the inlet and outlet
surfaces of the wheel is
h_i - h_o = h´_i + h´´_i - h´_o - h´´_o
= (V_i² - V_o²)/2g + (v_(ro)² - v_(ri)²)/2g. (17)
_Case 1. Axial Flow Turbines._--V_i = V_o; 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,
h_i - h_o = (v_(ro)² - v_(ri)²)/2g - d. (17a)
_Case 2. Outward Flow Turbines._--The inlet radius is less than the
outlet radius, and (V_i² - V_o²)/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 (V_i² -
V_o²)/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 r_o : r_i 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 r_i : r_o 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 r_i : r_o = 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 -
[h], of this h_i - h_o is expended in overcoming the pressure
in the wheel, the velocity of flow into the wheel is
v_i = c_v[root]{2g(H - [h] - (V_i² - V_o²/2g + (v{r0}² - v_(ri)²)/2g)}, (18)
where c_v may be taken 0.96.
From (14a),
v{r0} = V_o [root](1 + u_o²/V_o²).