Overshot and high breast wheels work badly in backwater, and hence if
the tail-water level varies, it is better to reduce the diameter of
the wheel so that its greatest immersion in flood is not more than 1
ft. The depth d of the shrouds is about 10 to 16 in. The number of
buckets may be about
N = [pi]D/d.
Let v be the peripheral velocity of the wheel. Then the capacity of
that portion of the wheel which passes the sluice in one second is
Q1 = vb(Dd - d²)/D
= v b d nearly,
b being the breadth of the wheel between the shrouds. If, however,
this quantity of water were allowed to pass on to the wheel the
buckets would begin to spill their contents almost at the top of the
fall. To diminish the loss from spilling, it is not only necessary to
give the buckets a suitable form, but to restrict the water supply to
one-fourth or one-third of the gross bucket capacity. Let m be the
value of this ratio; then, Q being the supply of water per second,
Q = mQ1 = mb dv.
This gives the breadth of the wheel if the water supply is known. The
form of the buckets should be determined thus. The outer element of
the bucket should be in the direction of motion of the water entering
relatively to the wheel, so that the water may enter without splashing
or shock. The buckets should retain the water as long as possible, and
the width of opening of the buckets should be 2 or 3 in. greater than
the thickness of the sheet of water entering.
For a wooden bucket (fig. 180, A), take ab = distance between two
buckets on periphery of wheel. Make ed = ½ eb and bc = 6/5 to 5/4
ab. Join cd. For an iron bucket (fig. 180, B), take ed = 1/3 eb; bc =
6/5 ab. Draw cO making an angle of 10° to 15° with the radius at c.
On Oc take a centre giving a circular arc passing near d, and round
the curve into the radial part of the bucket de.
[Illustration: FIG. 180.]
There are two ways in which the power of a water wheel is given off to
the machinery driven. In wooden wheels and wheels with rigid arms, a
spur or bevil wheel keyed on the axle of the turbine will transmit the
power to the shafting. It is obvious that the whole turning moment due
to the weight of the water is then transmitted through the arms and axle
of the water wheel. When the water wheel is an iron one, it usually has
light iron suspension arms incapable of resisting the bending action due
to the transmission of the turning effort to the axle. In that case spur
segments are bolted to one of the shrouds, and the pinion to which the
power is transmitted is placed so that the teeth in gear are, as nearly
as may be, on the line of action of the resultant of the weight of the
water in the loaded arc of the wheel.
The largest high breast wheels ever constructed were probably the four
wheels, each 50 ft. in diameter, and of 125 h.p., erected by Sir W.
Fairbairn in 1825 at Catrine in Ayrshire. These wheels are still
working.
[Illustration: FIG. 181.]
§ 181. _Poncelet Water Wheel._--When the fall does not exceed 6 ft., the
best water motor to adopt in many cases is the Poncelet undershot water
wheel. In this the water acts very nearly in the same way as in a
turbine, and the Poncelet wheel, although slightly less efficient than
the best turbines, in normal conditions of working, is superior to most
of them when working with a reduced supply of water. A general notion of
the action of the water on a Poncelet wheel has already been given in §
159. Fig. 181 shows its construction. The water penned back between the
side walls of the wheel pit is allowed to flow to the wheel under a
movable sluice, at a velocity nearly equal to the velocity due to the
whole fall. The water is guided down a slope of 1 in 10, or a curved
race, and enters the wheel without shock. Gliding up the curved floats
it comes to rest, falls back, and acquires at the point of discharge a
backward velocity relative to the wheel nearly equal to the forward
velocity of the wheel. Consequently it leaves the wheel deprived of
nearly the whole of its original kinetic energy.
Taking the efficiency at 0.60, and putting H for the available fall,
h.p. for the horse-power, and Q for the water supply per second,
h.p. = 0.068 QH.
The diameter D of the wheel may be taken arbitrarily. It should not be
less than twice the fall and is more often four times the fall. For
ordinary cases the smallest convenient diameter is 14 ft. with a
straight, or 10 ft. with a curved, approach channel. The radial depth
of bucket should be at least half the fall, and radius of curvature of
buckets about half the radius of the wheel. The shrouds are usually of
cast iron with flanges to receive the buckets. The buckets may be of
iron 1/8 in. thick bolted to the flanges with 5/16 in. bolts.
Let H´ be the fall measured from the free surface of the head-water to
the point F where the mean layer enters the wheel; then the velocity
at which the water enters is v = [root](2gH´), and the best
circumferential velocity of the wheel is V = 0.55f to 0.6v. The number
of rotations of the wheel per second is N = V/[pi]D. The thickness
of the sheet of water entering the wheel is very important. The best
thickness according to experiment is 8 to 10 in. The maximum thickness
should not exceed 12 to 15 in., when there is a surplus water supply.
Let e be the thickness of the sheet of water entering the wheel, and b
its width; then
bev = Q; or b = Q/ev.