ia 
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482 APPLIED MECHANICS 
A. 
Useful work per second = Py, = 7 — v, cos B)v,v cos B. 
. v wAv ; 
Energy of jet per second = wAv - aS apie: : F 
Avs 2 a 
Efficiency = “A- v, cos 8)v,v cos B + tne (0 - v, cos 8) cos B. 
2g 
For given sass of v and P the efficiency will be a maximum ¥ 
v ; BY 
v,(v—v,cos 8) is a maximum, that is, when 0, = ‘Zeos B’ Hence th 
pee ptt efficiency is 3, or 50 per cent. It is obvious that w ta : 
= F008 B? V, =, and, = 0. 
It is evident that unless Bi is a small angle a single vane of Ii 
length BC could only remain in action for a very short time, wig 
the vane is receiving water, that part of it upon which the jet is impir 
ing must be straight and parallel to BH in order that there may nal 
shock at entrance. 
For a succession of vanes, with CN perpendicular to CL, the tot 
pressure on all the vanes in action at one time is P=— vcos B, where V 
is now the total weight of water delivered by the jet per second, and is 
wAv? cos B 
equal to wAv. Therefore P= , and the useful work per se 
Avy? 
ei ihe - It would therefore seem that the useful woe 
creases indefinitely with v,, but if the vanes are driven by the jet, tl 
useful work cannot exceed the energy of the jet. Hence the maximul 
wAv? - » 
useful work = “ag? the efficiency is then unity, and v, = Zeos 8’ 
is Pv, = 
419. Action of a Jet on a Revolving Vane.—A case of great im \- 
portance in connection with turbines and certain forms of water wh eels 
is that in which the vane upon which the water impinges is a 
a revolving wheel. Referring to Fig. oO 
780, O is the axis of the wheel which Rr Ri 
is perpendicular to the plane of the foe 
figure ; the acting surface of the vane aX ‘\ i hetthe 
is also perpendicular to that plane. v we 
The inner and outer edges of the vane 
are at distances 7, and r, from the 
axis of the wheel. In what follows 
, LE" \ a 
fer Ne 
F . rs ij DB} , 
the wheel is assumed to be moving if bee ‘ 
with uniform angular velocity. The 
linear velocities of the inner and outer uy Sx 
edges of the vane are c, and ¢, re- 
spectively. Evidently c¢,/r, = ¢,/r,. 
At entrance the axis of the jet makes 
an angle 6, with c,. The absolute Fie. 780. 
velocity of the water at entrance is v,. Completing the parallelogram 
of velocities at B,, the relative velocity u, of the water and vane a 
