WATER WHEELS AND TURBINES 491 
Therefore vj — vj = 2(v,c, cos A, — ve¢q cos O,). 
But vt = c] + Wt + 2c,u, cos d,, 
and Uy = Cy + UZ — cot. COs po, 
also % cos A, =c, +% cos dy, 
and Vy COS A, = Cy —Uy COS do. 
Substituting these equivalents in the equation 
y Ui — vz = 2(v,c, Cos A, —VgCq cos 4), 
_ the result is “@—-wg=c-d. 
It is evident that the efficiency of the wheel will be greater the 
smaller v, is, and v, will be smaller the smaller ¢, is. But since there 
must be a sufficient area of passage between the vanes at exit, ¢. cannot 
__ be made indefinitely small. For a given value of ¢, the velocity Vg will 
have nearly its minimum value when vz is equal to cy, and if wu be made 
equal to c., this leads to very simple relations between the various 
quantities. For since uj — aj =cj — &, it follows that if ug=c,, then 
U =cy, and ¢,=20,. Hence c,= Tee B," and the angular speed of the 
The energy given to the wheel per second = 3° (04 — v3), but since 
rv, sin a 
Ug = Cg, Ve = 2c, sin 3 zy eae , therefore energy per second 
ry sin % ‘ 
I , and the horse-power of the wheel is this 
r, cos 0, 
Biccecivn divided by 550. ° 
The energy in the water per second as it enters the wheel is he 
ty sin #2 3 F 
Se 6," The efficiency is 
1 
therefore greater the smaller the angles ¢, si 6,. 
In practice the angle @, is generally between 20° and 25°, and ¢, 
is generally between 15° and 20°. The ratio of r, to r, is is generally 
between 1°15 and 1-25. 
Taking friction into account the efficiency is about 80 per cent., and 
the efficiency is not reduced by diminishing the sluice opening when 
there is a reduction in the load. 
The axis of the wheel may be either horizontal or vertical. 
427. Jet Reaction Wheels.—The simplest form of the jet reaction 
wheel is that generally known as Barker’s mill. Fig. 791 shows a 
Barker’s mill constructed of ordinary wrought-iron or steel tubing. The 
vertical central tube AB has jointed to it two horizontal tubular arms 
CD and EF, which are opposite to one another. These arms are closed 
: Hence the efficiency of the wheel is 1 sh 
