Page:EB1911 - Volume 14.djvu/116

sluice successively, any number of sluices can be opened or closed as desired. The turbine is of 48 horse power on 5.12 ft. fall, and the supply of water varies from 35 to 112 cub. ft. per second. The efficiency in normal working is given as 73%. The mean diameter of the wheel is 6 ft., and the speed 27.4 revolutions per minute.

Fig. 200.
 

Fig. 201.
 

Fig. 202.

As an example of a partial admission radial flow impulse turbine, a 100 h.p. turbine at Immenstadt may be taken. The fall varies from 538 to 570 ft. The external diameter of the wheel is 41/2 ft., and its internal diameter 3 ft. 10 in. Normal speed 400 revs. per minute. Water is discharged into the wheel by a single nozzle, shown in fig. 202 with its regulating apparatus and some of the vanes. The water enters the wheel at an angle of 22° with the direction of motion, and the final angle of the wheel vanes is 20°. The efficiency on trial was from 75 to 78%.

§ 199. Theory of the Impulse Turbine.—The theory of the impulse turbine does not essentially differ from that of the reaction turbine, except that there is no pressure in the wheel opposing the discharge from the guide-blades. Hence the velocity with which the water enters the wheel is simply

v_i = 0.96 √2g (H − ɧ),

where ɧ is the height of the top of the wheel above the tail water. If the hydropneumatic system is used, then ɧ = 0. Let Q_m be the maximum supply of water, r₁, r₂ the internal and external radii of the wheel at the inlet surface; then

u_i = Q_m / {π(r₂² − r₁²)}.

The value of u_i may be about 0.45 √2g (H − ɧ), whence r₁, r₂ can be determined.

The guide-blade angle is then given by the equation

sin γ = u_i / v_i = 0.45/0.94 = .48;

γ = 29°.

The value of u_i should, however, be corrected for the space occupied by the guide-blades.

The tangential velocity of the entering water is

w_i = v_i cos γ = 0.82 √2g (H − ɧ).

The circumferential velocity of the wheel may be (at mean radius)

V_i = 0.5 √2g (H − ɧ).

Hence the vane angle at inlet surface is given by the equation

cot θ = (w_i − V_i) / u_i = (0.82 − 0.5)/0.45 = .71;
θ = 55°.

The relative velocity of the water striking the vane at the inlet edge is v_ri = u_i cosec θ = 1.22u_i. This relative velocity remains unchanged during the passage of the water over the vane; consequently the relative velocity at the point of discharge is v_ro = 1.22u_i. Also in an axial flow turbine V_o = V_i.

If the final velocity of the water is axial, then

cos φ = V_o /v_ro = V_i /v_ri = 0.5/(1.22 × 0.45) = cos 24º 23′.

This should be corrected for the vane thickness. Neglecting this, u_o = v_ro sin φ = v_ri sin φ = u_i cosec θ sin φ = 0.5u_i. The discharging area of the wheel must therefore be greater than the inlet area in the ratio of at least 2 to 1. In some actual turbines the ratio is 7 to 3. This greater outlet area is obtained by splaying the wheel, as shown in the section (fig. 199).

Fig. 203.

§ 200. Pelton Wheel.—In the mining district of California about 1860 simple impulse wheels were used, termed hurdy-gurdy wheels. The wheels rotated in a vertical plane, being supported on a horizontal axis. Round the circumference were fixed flat vanes which were struck normally by a jet from a nozzle of size varying with the head and quantity of water. Such wheels have in fact long been used. They are not efficient, but they are very simply constructed. Then attempts were made to improve the efficiency, first by using hemispherical cup vanes, and then by using a double cup vane with a central dividing ridge, an arrangement invented by Pelton. In this last form the water from the nozzle passes half to each side of the wheel, just escaping clear of the backs of the advancing buckets. Fig. 203 shows a Pelton vane. Some small modifications have been made by other makers, but they are not of any great importance. Fig. 204 shows a complete Pelton wheel with frame and casing, supply pipe and nozzle. Pelton wheels have been very largely used in America and to some extent in Europe. They are extremely simple and easy to construct or repair and on falls of 100 ft. or more are very efficient. The jet strikes tangentially to the mean radius of the buckets, and the face of the buckets is not quite radial but at right angles to the direction of the jet at the point of first impact. For greatest efficiency the peripheral velocity of the wheel at the mean radius of the buckets should be a little less than half the velocity of the jet. As the radius of the wheel can be taken arbitrarily, the number of revolutions per minute can be accommodated to that of the machinery to be driven. Pelton wheels have been made as small as 4 in. diameter, for driving sewing machines, and as large as 24 ft. The efficiency on high falls is about 80%. When large power is required two or three nozzles are used delivering on one wheel. The width of the buckets should be not less than seven times the diameter of the jet.

Fig. 204.

At the Comstock mines, Nevada, there is a 36-in. Pelton wheel made of a solid steel disk with phosphor bronze buckets riveted to the rim. The head is 2100 ft. and the wheel makes 1150 revolutions per minute, the peripheral velocity being 180 ft. per sec. With a 1/2-in. nozzle the wheel uses 32 cub. ft. of water per minute and develops 100 h.p. At the Chollarshaft, Nevada, there are six Pelton wheels on a fall of 1680 ft. driving electrical generators. With 5/8-in. nozzles each develops 125 h.p.

Fig. 205

§ 201. Theory of the Pelton Wheel.—Suppose a jet with a velocity v strikes tangentially a curved vane AB (fig. 205) moving in the same direction with the velocity u. The water will flow over the vane with the relative velocity v−u and at B will have the tangential