b = 6Q/H [root](2gH).
Allowing for the contraction of the stream, the area of opening
through the sluice may be 1.25 be to 1.3 be. The inside width of the
wheel is made about 4 in. greater than b.
Several constructions have been given for the floats of Poncelet
wheels. One of the simplest is that shown in figs. 181, 182.
Let OA (fig. 181) be the vertical radius of the wheel. Set off OB, OD
making angles of 15° with OA. Then BD may be the length of the close
breasting fitted to the wheel. Draw the bottom of the head face BC at
a slope of 1 in 10. Parallel to this, at distances ½e and e, draw EF
and GH. Then EF is the mean layer and GH the surface layer entering
the wheel. Join OF, and make OFK = 23°. Take FK = 0.5 to 0.7 H. Then K
is the centre from which the bucket curve is struck and KF is the
radius. The depth of the shrouds must be sufficient to prevent the
water from rising over the top of the float. It is ½H to 2/3 H. The
number of buckets is not very important. They are usually 1 ft. apart
on the circumference of the wheel.
The efficiency of a Poncelet wheel has been found in experiments to
reach 0.68. It is better to take it at 0.6 in estimating the power of
the wheel, so as to allow some margin.
[Illustration: FIG. 182.]
In fig. 182 v_i is the initial and v_o the final velocity of the
water, v_r parallel to the vane the relative velocity of the water and
wheel, and V the velocity of the wheel.
_Turbines._
§ 182. The name turbine was originally given in France to any water
motor which revolved in a horizontal plane, the axis being vertical. The
rapid development of this class of motors dates from 1827, when a prize
was offered by the Société d'Encouragement for a motor of this kind,
which should be an improvement on certain wheels then in use. The prize
was ultimately awarded to Benoît Fourneyron (1802-1867), whose turbine,
but little modified, is still constructed.
_Classification of Turbines._--In some turbines the whole available
energy of the water is converted into kinetic energy before the water
acts on the moving part of the turbine. Such turbines are termed
_Impulse or Action Turbines_, and they are distinguished by this that
the wheel passages are never entirely filled by the water. To ensure
this condition they must be placed a little above the tail water and
discharge into free air. Turbines in which part only of the available
energy is converted into kinetic energy before the water enters the
wheel are termed _Pressure or Reaction Turbines_. In these there is a
pressure which in some cases amounts to half the head in the clearance
space between the guide vanes and wheel vanes. The velocity with which
the water enters the wheel is due to the difference between the pressure
due to the head and the pressure in the clearance space. In pressure
turbines the wheel passages must be continuously filled with water for
good efficiency, and the wheel may be and generally is placed below the
tail water level.
Some turbines are designed to act normally as impulse turbines
discharging above the tail water level. But the passages are so designed
that they are just filled by the water. If the tail water rises and
drowns the turbine they become pressure turbines with a small clearance
pressure, but the efficiency is not much affected. Such turbines are
termed _Limit turbines_.
Next there is a difference of constructive arrangement of turbines,
which does not very essentially alter the mode of action of the water.
In axial flow or so-called parallel flow turbines, the water enters and
leaves the turbine in a direction parallel to the axis of rotation, and
the paths of the molecules lie on cylindrical surfaces concentric with
that axis. In radial outward and inward flow turbines, the water enters
and leaves the turbine in directions normal to the axis of rotation, and
the paths of the molecules lie exactly or nearly in planes normal to the
axis of rotation. In outward flow turbines the general direction of flow
is away from the axis, and in inward flow turbines towards the axis.
There are also mixed flow turbines in which the water enters normally
and is discharged parallel to the axis of rotation.
Another difference of construction is this, that the water may be
admitted equally to every part of the circumference of the turbine wheel
or to a portion of the circumference only. In the former case, the
condition of the wheel passages is always the same; they receive water
equally in all positions during rotation. In the latter case, they
receive water during a part of the rotation only. The former may be
termed turbines with complete admission, the latter turbines with
partial admission. A reaction turbine should always have complete
admission. An impulse turbine may have complete or partial admission.
When two turbine wheels similarly constructed are placed on the same
axis, in order to balance the pressures and diminish journal friction,
the arrangement may be termed a twin turbine.
If the water, having acted on one turbine wheel, is then passed through
a second on the same axis, the arrangement may be termed a compound
turbine. The object of such an arrangement would be to diminish the
speed of rotation.
Many forms of reaction turbine may be placed at any height not exceeding
30 ft. above the tail water. They then discharge into an air-tight
suction pipe. The weight of the column of water in this pipe balances
part of the atmospheric pressure, and the difference of pressure,
producing the flow through the turbine, is the same as if the turbine
were placed at the bottom of the fall.
I. Impulse Turbines. | II. Reaction Turbines.
|
(Wheel passages not filled, and | (Wheel passages filled, discha-
discharging above the tail | rging above or below the tail
water.) | water or into a suction-pipe.)
(a) Complete admission. (Rare.) | Always with complete admission.
(b) Partial admission. (Usual.) |
\_________________________________\/_______________________________/
Axial flow, outward flow, inward flow, or mixed flow.
\_________________________________\/_______________________________/
Simple turbines; twin turbines; compound turbines.
§ 183. _The Simple Reaction Wheel._--It has been shown, in § 162,
that, when water issues from a vessel, there is a reaction on the
vessel tending to cause motion in a direction opposite to that of the
jet. This principle was applied in a rotating water motor at a very
early period, and the Scotch turbine, at one time much used, differs
in no essential respect from the older form of reaction wheel.
[Illustration: FIG. 183.]
The old reaction wheel consisted of a vertical pipe balanced on a
vertical axis, and supplied with water (fig. 183). From the bottom of
the vertical pipe two or more hollow horizontal arms extended, at the
ends of which were orifices from which the water was discharged. The
reaction of the jets caused the rotation of the machine.
Let H be the available fall measured from the level of the water in
the vertical pipe to the centres cf the orifices, r the radius from
the axis of rotation to the centres of the orifices, v the velocity of
discharge through the jets, [alpha] the angular velocity of the
machine. When the machine is at rest the water issues from the
orifices with the velocity [root](2gH) (friction being neglected). But
when the machine rotates the water in the arms rotates also, and is in
the condition of a forced vortex, all the particles having the same
angular velocity. Consequently the pressure in the arms at the
orifices is H + [alpha]²r²/2g ft. of water, and the velocity of
discharge through the orifices is v = [root](2gH + [alpha]²r²). If the
total area of the orifices is [omega], the quantity discharged from
the wheel per second is
Q = [omega]v = [omega] [root](2gH + [alpha]²r²).
While the water passes through the orifices with the velocity v, the
orifices are moving in the opposite direction with the velocity
[alpha]r. The absolute velocity of the water is therefore
v - [alpha]r = [root](2gH + [alpha]²r²) - [alpha]r.
The momentum generated per second is (GQ/g)(v - [alpha]r), which is
numerically equal to the force driving the motor at the radius r. The
work done by the water in rotating the wheel is therefore
(GQ/g) (v - [alpha]r) ar foot-pounds per sec.
The work expended by the water fall is GQH foot-pounds per second.
Consequently the efficiency of the motor is
(v - [alpha]r) [alpha]r {[root]{2gH + [alpha]²r²]} - [alpha]r} [alpha]r
[eta] = ----------------------- = -----------------------------------------------.
gH gH