# Page:EB1911 - Volume 14.djvu/117

TURBINES]
105
HYDRAULICS

relative velocity v-u making an angle gr with the direction of the vane's motion. Combining this with the velocity u of the vane, the absolute velocity of the water leaving the vane will bew § Bc. The component of -w in the direction of motion of the vane is Ba=Bb4ab =u-(11-u) cos ~a.' Hence

if Q is the quantity of

water reaching 'the vane

per second the changehof momentum per second in

the direction of the vanc's B a 6 motion is (GS/g)[v-{uv-u)¢0S°1}l=(/g)(v-u) cos a). R a.=0°,

cos a.=I, and the 'change

of momentum per second,

which is eq-ual to the

effort driving the vane, is

/4* P=2(GQ/g)(z/-u). The

work done on the vane is

Pu=2(GQ/g)(v-u)u. If a

series of vanes are interposed

in succession, the

quantity of water impinging

on the vanes per second is the total discharge of the' nozzle, and the energy expended at the nozzle is GQv”/zg. Hence the efficiency of the arrangement is, when a =0°, neglecting friction, 1/=2P1¢/GQv'=4(v-"lu/v'.

which is a maximum and equal to unity if u=%v. In that case the whole energy of the jet is usefully expended in driving the series of vanes. In practice a cannot be quite zero or the water leaving one vane would strike the back of the negt advancing vane. Fig. 203 shows a Pelton vane. The water divides each way, and leaves the vane on each side in a direction nearly parallel to the direction of motion of the vane. The best velocity of the vane is very approximately half the velocity of the jet.

""';£:f?

H

FIG. 205.

U i I

Q

L .. f .

li A

~, "" 1351 . .... w

-~- {@» M( f., ,, -0,1

vs

7 R ..., . . . ., .. ., , ., .

I ° .-»I ..

=, -

E3.

nl cy

W .

~ .

```N.
```
```- '|'
```

ty j '&

/ i"

(EJ hai

I || *

m @! | .

4 'iiill — r R"””"

Flo. 206.

In one of the Niagara pen stocks there are 400 tons of water flowing at ro ft. per second, opposing enormous resistance to rapid h change of speed of How. The sluices of water turbines also are

necessarily large and heavy. Hence relay governors must be