tions, it is usual to make the diameter of the wheel about double the fall.

164. The Depth of Shrouding ought to be sufficient to prevent the water which glides up the vanes from overflowing their upper edges; because in order to produce the best efficiency, the water should all glide down again, and glance off at the lower edges of the vanes. The best velocity of the water relatively to the vanes is about 04 of the velocity of supply v1; but to provide for the contingency of that velocity amounting to 07 v1, it is advisable to give the shrouding the depth due to 07 v1; that is to say, about half the depth from the topwater level in the penstock to the outlet of the sluice.

165. The Regulating Sluice is placed as close as possible to the wheel, and is consequently inclined. The co-efficient of contraction c of its outlet (as already stated, Article 140), is from 0.74 to 0.8; therefore, the depth of its opening is from four-thirds to five-fourths of the depth of the stream which issues from it.

The greatest depth of that stream should not exceed about onefifth of the depth of the shrouding; therefore, the depth of opening of the sluice for the maximum flow should be about one-fourth of the depth of the shrouding, or one-eighth of the depth of the centre of the orifice below the topwater level.

Let Q be the greatest flow to be used, in cubic feet per second; h', the depth of the middle of the orifice below topwater; d, the depth of the orifice;

l, the length of the orifice, or breadth of the opening of the sluice; then

all dimensions being in feet.

V1

166. The Wheel Race is designed as follows (see fig. 66):— Draw H F G a tangent to the wheel, with a declivity of one in ten. This declivity is to preserve the velocity of supply v1 undiminished. At the height c d (Article 165) above H F G, draw KL to represent the upper surface of the stream, meeting the circumference of the wheel at the point L. Then

make the section of the bot

tom of the wheel race from G to F an arc of a circle, equal to G L, and of the same radius; that is, the radius of the wheel.

C

K

Ꮮ

From G to E the wheel race is formed so as to clear the wheel by about 0.4 inch.

167. The Surface Velocity of the wheel for the greatest efficiency has already been stated, in Article 146, to be

In this expression is to be held to represent the mean angle which the stream makes with a tangent to the wheel, which is very nearly

168. Vanes or Floats.-As to the number of vanes, from two to three in the length of the arc L G are in general enough.

The determination of the proper form for those vanes, near their outer edges, has already been explained in Articles 145, 146. They are usually curved in a circular arc, so that their inner ends are tangents to radii of the wheel.

169. The Efficiency has been stated, in Article 148, to be about 0-6 when the wheel is not drowned, and 0.48 when it is drowned. At these rates, the energy of the available fall from the penstock to the tail race, for each effective horse-power, is

170. Wheel in an Open Current.-Wheels of this class are carried by boats moored in a rapid current. Their floats are usually plane and radial, and fixed at distances apart equal to their length in the direction of a radius.

According to the experiments of Poncelet, the following is the useful work per second of such a wheel; v, being the velocity of the current; u, that of the centre of a float; A, the area of a float in square feet; and D, the weight of a cubic foot of water:

According to this formula, the velocity of the centres of the floats for the greatest efficiency is half the velocity of the current; and the efficiency at that speed is 0·4, if A v1 be taken to represent the volume of water acting on the wheel in a second.

189

CHAPTER V1.

OF TURBINES.

SECTION 1.-General Principles.

171. Turbines Generally Described and Classed.-A turbine is a water wheel with a vertical axis, receiving and discharging water in various directions round its circumference. The wheel consists of a drum or annular passage, containing a set of suitably formed vanes, which are curved backwards in such a manner, that the water, after glancing off them, is left behind with as little energy as possible.

Turbines have the advantage of being of small bulk for their power, and equally efficient for the highest and the lowest falls.

The supply of water takes place either directly from a reservoir, in which case the wheel is placed close to a suitable opening at the bottom of the reservoir, or through a supply pipe and wheel case. The former method is the best suited to moderate falls, the latter to very high falls.

The opening through which the water is delivered to the wheel is in most cases furnished with guide blades, to make the water arrive at the wheel in the direction best suited to drive it efficiently.

Turbines may be divided into three classes, according to the direction in which the water moves before reaching the guide blades, and after leaving the wheel, viz. :—

I. Parallel Flow Turbines, in which the water is supplied and discharged in a current parallel to the axis.

II. Outward Flow Turbines, in which the water is supplied and discharged in currents radiating from the axis.

III. Inward Flow Turbines, in which the water is supplied and discharged in currents converging radially towards the axis.

Those three classes of turbines differ in certain details; but there are general principles which are applicable to them all, and general equations which are adapted to any one of them merely by assigning suitable values to certain symbols in them. The diagrams which will now be given show the general arrangement of the principal parts of each, the details of their construction being reserved until later. A is the supply;

Fig. 67 represents a parallel flow turbine.

chamber, being an annular passage through the bottom of the reservoir, which contains the guide blades; these are vertical at

their upper edges: the form and position of their lower edges, as shown by dotted lines, are such as to direct the water in several small streams or jets obliquely against all parts of the circumference of the wheel B. The wheel B consists also of an annular passage between two cylindrical drums, containing a series of vanes, resembling the guide blades in shape, but turned with their lower edges pointing backwards.

Fig. 68 shows a vertical section of a few of the guide blades C, and vanes D.

Fig. 69 is a horizontal section of part of an outward flow turbine; A is the supply chamber, being a vertical cylinder with a ring of openings round its lower end; C are the guide blades for directing the water obliquely forwards as it rushes out of these openings; B is the wheel surrounding the ring of openings, and consisting of a pair of crowns, or flat rings, with a series of curved vanes D between them; these vanes are radial at their inner edges, and directed obliquely backwards at their outer edges.

Fig. 70 represents a plan of one form of the reaction wheel-3 kind of outward flow turbine without guide blades. The water is conducted by a vertical supply pipe A into the centre of a rotating

hollow disc, provided with two or three hollow arms, which discharge the water through orifices directed backwards. In the figure, the hollow disc, and its two arms B B, are shown of such a form as to leave the largest possible space for the motion of the water from the centre of the disc towards the circumference, in order to avoid friction, and for other reasons which will afterwards appear. C, C, are the orifices. The circumferences of the arms B, B, here perform the functions of vanes.

Fig. 71 is a horizontal section of an inward flow turbine. A is the supply chamber; C, one of the guide blades, directing the water obliquely forwards against the wheel; B is the wheel, occupying a central space surrounded by the supply chamber, and discharging the water through openings in its centre; it consists of a pair of crowns with a set of curved vanes D between them: these vanes are radial at their outer ends, and are directed obliquely backwards at their inner ends.

In treating of the theory of the efficiency of turbines, it will be assumed that they are constructed of the forms and proportions, and worked in the manner most favourable to efficiency, according to rules which will presently be explained. The waste of power caused by deviations from those rules can afterwards be allowed for by means of empirically-found multipliers.

172. By Velocity of Flow is to be understood the velocity of that component of the motion of the water by which it is carried towards, through, and away from the wheel; that is, the component, whether parallel to the axis or radial, which is at right angles to the motion of the vanes.

Let A denote the total effective sectional area in square feet of the orifices through which the water passes, whether in the wheel, or amongst the guide blades, as measured upon a surface perpendicular to the direction of the flow; that is, in a parallel flow turbine, on a plane perpendicular to the axis, and in an outward or inward radial flow turbine, on a cylindrical surface described about the axis.

Let Q be the volume of water used in cubic feet Then

is the velocity of flow.

Q ÷ A........

per

second.

.....(1.)

Inasmuch as sudden changes in the velocity of a stream are accompanied with waste of energy, it is desirable that the velocity of flow should either be constant, or change slowly during the passage of the water through the wheel.

In parallel flow turbines, such as fig. 67, the velocity of flow would be made constant, if the vanes were insensibly thin, by making the drum, or annular case containing the vanes, simply