Page:EB1911 - Volume 14.djvu/100

water, and let AD represent in magnitude and direction the velocity of the vane. Completing the parallelogram, DC or AE represents the direction in which the water is moving relatively to the vane. If the lip of the vane at A is tangential to AE, the water will not have its direction suddenly changed when it impinges on the vane, and will therefore have no tendency to spread laterally. On the contrary it will be so gradually deviated that it will glide up the vane in the direction AB. This is sometimes expressed by saying that the vane receives the water without shock.

Fig. 160.

§ 159. Floats of Poncelet Water Wheels.—Let AC (fig. 160) represent the direction of a thin horizontal stream of water having the velocity v. Let AB be a curved float moving horizontally with velocity u. The relative motion of water and float is then initially horizontal, and equal to v − u.

In order that the float may receive the water without shock, it is necessary and sufficient that the lip of the float at A should be tangential to the direction AC of relative motion. At the end of (v − u)/g seconds the float moving with the velocity u comes to the position A₁B₁, and during this time a particle of water received at A and gliding up the float with the relative velocity v − u, attains a height DE = (v − u)²/2g. At E the water comes to relative rest. It then descends along the float, and when after 2(v − u)/g seconds the float has come to A₂B₂ the water will again have reached the lip at A₂ and will quit it tangentially, that is, in the direction CA₂, with a relative velocity −(v − u) = −√ (2gDE) acquired under the influence of gravity. The absolute velocity of the water leaving the float is therefore u − (v − u) = 2u − v. If u = 1/2v, the water will drop off the bucket deprived of all energy of motion. The whole of the work of the jet must therefore have been expended in driving the float. The water will have been received without shock and discharged without velocity. This is the principle of the Poncelet wheel, but in that case the floats move over an arc of a large circle; the stream of water has considerable thickness (about 8 in.); in order to get the water into and out of the wheel, it is then necessary that the lip of the float should make a small angle (about 15°) with the direction of its motion. The water quits the wheel with a little of its energy of motion remaining.

§ 160. Pressure on a Curved Surface when the Water is deviated wholly in one Direction.—When a jet of water impinges on a curved surface in such a direction that it is received without shock, the pressure on the surface is due to its gradual deviation from its first direction. On any portion of the area the pressure is equal and opposite to the force required to cause the deviation of so much water as rests on that surface. In common language, it is equal to the centrifugal force of that quantity of water.

Fig. 161.

Case 1. Surface Cylindrical and Stationary.—Let AB (fig. 161) be the surface, having its axis at O and its radius = r. Let the water impinge at A tangentially, and quit the surface tangentially at B. Since the surface is at rest, v is both the absolute velocity of the water and the velocity relatively to the surface, and this remains unchanged during contact with the surface, because the deviating force is at each point perpendicular to the direction of motion. The water is deviated through an angle BCD = AOB = φ. Each particle of water of weight p exerts radially a centrifugal force pv²/rg. Let the thickness of the stream = t ft. Then the weight of water resting on unit of surface = Gt ℔; and the normal pressure per unit of surface = n = Gtv²/gr. The resultant of the radial pressures uniformly distributed from A to B will be a force acting in the direction OC bisecting AOB, and its magnitude will equal that of a force of intensity = n, acting on the projection of AB on a plane perpendicular to the direction OC. The length of the chord AB = 2r sin 1/2φ; let b = breadth of the surface perpendicular to the plane of the figure. The resultant pressure on surface

= R = 2rb sin φ/2 × Gt/g·v²/r = 2G/gbtv² sin φ/2,

which is independent of the radius of curvature. It may be inferred that the resultant pressure is the same for any curved surface of the same projected area, which deviates the water through the same angle.

Case 2. Cylindrical Surface moving in the Direction AC with Velocity u.—The relative velocity = v − u. The final velocity BF (fig. 162) is found by combining the relative velocity BD = v − u tangential to the surface with the velocity BE = u of the surface. The intensity of normal pressure, as in the last case, is (G/g)t(v − u)²/r. The resultant normal pressure R = 2(G/g)bt(v−u)² sin 1/2φ. This resultant pressure may be resolved into two components P and L, one parallel and the other perpendicular to the direction of the vane’s motion. The former is an effort doing work on the vane. The latter is a lateral force which does no work.

P = R sin 1/2φ = (G/g)bt(v−u)² (1−cos φ);
 L = R cos 1/2φ = (G/g)bt(v−u)² sin φ.

Fig. 162.

Fig. 163.

The work done by the jet on the vane is Pu = (G/g)btu(v − u)²(1 − cos φ), which is a maximum when u = 1/3v. This result can also be obtained by considering that the work done on the plane must be equal to the energy lost by the water, when friction is neglected.

If φ = 180°, cos φ = −1, 1 − cos φ = 2; then P = 2(G/g)bt(v − u)², the same result as for a concave cup.

§ 161. Position which a Movable Plane takes in Flowing Water.—When a rectangular plane, movable about an axis parallel to one of its sides, is placed in an indefinite current of fluid, it takes a position such that the resultant of the normal pressures on the two sides of the axis passes through the axis. If, therefore, planes pivoted so that the ratio a/b (fig. 163) is varied are placed in water, and the angle they make with the direction of the stream is observed, the position of the resultant of the pressures on the plane is determined for different angular positions. Experiments of this kind have been made by Hagen. Some of his results are given in the following table:—

	Larger plane.	Smaller Plane.
a/b = 1.0	φ = . . .	φ = 90°
0.9	75°	721/2°
0.8	60°	57°
0.7	48°	43°
0.6	25°	29°
0.5	13°	13°
0.4	8°	61/2°
0.3	61/2°	..
0.2	4°	..

§ 162. Direct Action distinguished from Reaction (Rankine, Steam Engine, § 147).

The pressure which a jet exerts on a vane can be distinguished into two parts, viz.:—

(1) The pressure arising from changing the direct component of the velocity of the water into the velocity of the vane. In fig. 153, § 154, ab cos bae is the direct component of the water’s velocity, or component in the direction of motion of vane. This is changed into the velocity ae of the vane. The pressure due to direct impulse is then

P₁ = GQ (ab cos bae − ae)/g.

For a flat vane moving normally, this direct action is the only action producing pressure on the vane.

(2) The term reaction is applied to the additional action due to the direction and velocity with which the water glances off the vane. It is this which is diminished by the friction between the water and the vane. In Case 2, § 160, the direct pressure is

P₁ = Gbt (v − u)²/g.

That due to reaction is

P₂ = −Gbt (v − u)² cos φ/g.

If φ < 90°, the direct component of the water’s motion is not wholly converted into the velocity of the vane, and the whole