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 18 in. thick bolted to the flanges with 516 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 = √ (2gH′), and the best circumferential velocity of the wheel is V = 0.55v to 0.6v. The number of rotations of the wheel per second is N = V/πD. 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.
Grashof takes e = 16H, and then
b = 6Q/H √ (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 12e 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 12H to 23H. 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.
Fig. 182.
In fig. 182 vi is the initial and vo the final velocity of the water, vr 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, discharging |
| discharging above the tail water.) | above or below the tail 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. | |
 |
| Fig. 183. |
§ 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.
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 of the orifices, r the radius from the axis of rotation to the centres of the orifices, v the velocity of discharge through the jets, α the angular velocity of
