(see Power Transmission). The development of electrical
methods of transmitting and distributing energy has led to the
utilization of many natural waterfalls so situated as to be useless
without such a means of transferring the power to points where
it can be conveniently applied. In some cases, as at Niagara, the
hydraulic power can only be economically developed in very
large units, and it can be most conveniently subdivided and
distributed by transformation into electrical energy. Partly
from the development of new industries such as paper-making
from wood pulp and electro-metallurgical processes, which
require large amounts of cheap power, partly from the facility
with which energy can now be transmitted to great distances
electrically, there has been a great increase in the utilization
of water-power in countries having natural waterfalls. According
to the twelfth census of the United States the total amount of
water-power reported as used in manufacturing establishments
in that country was 1,130,431 h.p. in 1870; 1,263,343 h.p.
in 1890; and 1,727,258 h.p. in 1900. The increase was 8.4%
in the decade 1870–1880, 3.1% in 1880–1890, and no less than
36.7% in 1890–1900. The increase is the more striking because
in this census the large amounts of hydraulic power which are
transmitted electrically are not included.
XII. IMPACT AND REACTION OF WATER
§ 153. When a stream of fluid in steady motion impinges on a solid surface, it presses on the surface with a force equal and opposite to that by which the velocity and direction of motion of the fluid are changed. Generally, in problems on the impact of fluids, it is necessary to neglect the effect of friction between the fluid and the surface on which it moves.
During Impact the Velocity of the Fluid relatively to the Surface on which it impinges remains unchanged in Magnitude.—Consider a mass of fluid flowing in contact with a solid surface also in motion, the motion of both fluid and solid being estimated relatively to the earth. Then the motion of the fluid may be resolved into two parts, one a motion equal to that of the solid, and in the same direction, the other a motion relatively to the solid. The motion which the fluid has in common with the solid cannot at all be influenced by the contact. The relative component of the motion of the fluid can only be altered in direction, but not in magnitude. The fluid moving in contact with the surface can only have a relative motion parallel to the surface, while the pressure between the fluid and solid, if friction is neglected, is normal to the surface. The pressure therefore can only deviate the fluid, without altering the magnitude of the relative velocity. The unchanged common component and, combined with it, the deviated relative component give the resultant final velocity, which may differ greatly in magnitude and direction from the initial velocity.
From the principle of momentum, the impulse of any mass of fluid reaching the surface in any given time is equal to the change of momentum estimated in the same direction. The pressure between the fluid and surface, in any direction, is equal to the change of momentum in that direction of so much fluid as reaches the surface in one second. If Pa is the pressure in any direction, m the mass of fluid impinging per second, va the change of velocity in the direction of Pa due to impact, then
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| Fig. 152. |
If v1 (fig. 152) is the velocity and direction of motion before impact, v2 that after impact, then v is the total change of motion due to impact. The resultant pressure of the fluid on the surface is in the direction of v, and is equal to v multiplied by the mass impinging per second. That is, putting P for the resultant pressure,
Let P be resolved into two components, N and T, normal and tangential to the direction of motion of the solid on which the fluid impinges. Then N is a lateral force producing a pressure on the supports of the solid, T is an effort which does work on the solid. If u is the velocity of the solid, Tu is the work done per second by the fluid in moving the solid surface.
Let Q be the volume, and GQ the weight of the fluid impinging per second, and let v1 be the initial velocity of the fluid before striking the surface. Then GQv12/2g is the original kinetic energy of Q cub. ft. of fluid, and the efficiency of the stream considered as an arrangement for moving the solid surface is
§ 154. Jet deviated entirely in one Direction.—Geometrical Solution (fig. 153).—Suppose a jet of water impinges on a surface ac with a velocity ab, and let it be wholly deviated in planes parallel to the figure. Also let ae be the velocity and direction of motion of the surface. Join eb; then the water moves with respect to the surface in the direction and with the velocity eb. As this relative velocity is unaltered by contact with the surface, take cd = eb, tangent to the surface at c, then cd is the relative motion of the water with respect to the surface at c. Take df equal and parallel to ae. Then fc (obtained by compounding the relative motion of water to surface and common velocity of water and surface) is the absolute velocity and direction of the water leaving the surface. Take ag equal and parallel to fc. Then, since ab is the initial and ag the final velocity and direction of motion, gb is the total change of motion of the water. The resultant pressure on the plane is in the direction gb. Join eg. In the triangle gae, ae is equal and parallel to df, and ag to fc. Hence eg is equal and parallel to cd. But cd = eb = relative motion of water and surface. Hence the change of motion of the water is represented in magnitude and direction by the third side of an isosceles triangle, of which the other sides are equal to the relative velocity of the water and surface, and parallel to the initial and final directions of relative motion.
Fig. 153.
Special Cases
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| Fig. 154. |
§ 155. (1) A Jet impinges on a plane surface at rest, in a direction normal to the plane (fig. 154).—Let a jet whose section is ω impinge with a velocity v on a plane surface at rest, in a direction normal to the plane. The particles approach the plane, are gradually deviated, and finally flow away parallel to the plane, having then no velocity in the original direction of the jet. The quantity of water impinging per second is ωv. The pressure on the plane, which is equal to the change of momentum per second, is P = (G/g) ωv2.
(2) If the plane is moving in the direction of the jet with the velocity ±u, the quantity impinging per second is ω(v ± u). The momentum of this quantity before impact is (G/g)ω(v ± u)v. After impact, the water still possesses the velocity ±u in the direction of the jet; and the momentum, in that direction, of so much water as impinges in one second, after impact, is ±(G/g)ω(v ± u)u. The pressure on the plane, which is the change of momentum per second, is the difference of these quantities or P = (G/g)ω(v ± u)2. This differs from the expression obtained in the previous case, in that the relative velocity of the water and plane v ± u is substituted for v. The expression may be written P = 2 × G × ω(v ± u)2/2g, where the last two terms are the volume of a prism of water whose section is the area of the jet and whose length is the head due to the relative velocity. The pressure on the plane is twice the weight of that prism of water. The work done when the plane
