HYDRAULIC MACHINES
ordinates in fig. 150, and on these vertical ordinates suppose the velocities set off horizontally at their proper depths. Thus, if v is the measured velocity at the depth h from the surface in fig. 149, on vertical marked III., then at III. in fig. 15o take cd=h and ac=v, Then d is a point in the vertical velocity curve for the vertical III., and, all the velocities for that ordinate being similarly set off, the curve can be drawn. Suppose all the vertical velocity curves I. .
V. (fig. 15o), thus drawn. On each of these figures draw verticals corresponding to velocities of x, 2X, 3X . . . ft. per second. Then for instance cd at III. (fig. 150) is the depth at which a velocity of 2X ft. per second existed on the vertical III. in fig. 149 and if cd is set off at III. in fig. 149 it gives a point in a curve passing through points of the section where the velocity was 2X ft. per second. Set off on each of the verticals in fig. 149 all the depths thus found in the corresponding diagram in fig. 150. Curves drawn through the corresponding points on the verticals are curves of equal velocity.
The discharge of the stream per second may be regarded as a solid having the cross section of the river (fig. 149) as a base, and cross
Left bank
§ 152. Hydraulic machines may be broadly divided into two classes: (1) Motors, in which water descending from a higher to a lower level, or from a higher to a lower pressure, gives up energy which is available for mechanical operations; (2) Pumps, in which the energy of a steam engine or other motor is expended in raising water from a lower to a higher level. A few machines such as the ram and jet pump combine the functions of motor
IT
V
bt FIG. 15o.
P
oi, hg
M iNt0 €
N, n
Wp a .'a
QiR WW W tar'
W W; 9.24 9.80 118212.30 14.4114.8016.92 17.30 1957 19.80 22.15 22.30
Discharge per Second = Q= 14.10870ab.m: Curves of equal velocity,
Transform.ation ratio 10:1 Height scale=1:1.5
2 9 4 S 6 7 8 9 10
4
sections normal to the plane of fig. 149 given by the diagrams in fig. 150. The curves of equal velocity may therefore be considered as contour lines of the solid whose volume is the discharge of the stream per second. Let Sto be the area of the cross section of the river, 521, 12:. . the areas contained by the successive curves of equal velocity, or, if these cut the surface of the stream, by the curves and that surface. Let x be the difference of velocity for which the successive curves are drawn, assumed above for simplicity at 1 ft. per second. Then the volume of the successive layers of the solid body whose volume represents the discharge, limited by successive planes passing through the contour curves, will be
zx(Sto+S21), 2x(121+522), and so on.
Consequently the discharge is
Q=x(z(Sto I Stn) I S2i=02+ ... f 52 –i}.
The areas 1k, S21 . are easily ascertained by means of the polar planimeter. A slight difficulty arises in the part of the solid lying above the last contour curve. This will have generally a height which is not exactly x, and a form more rounded than the other layers and less like a conical frustum. The volume of this may be estimated separately, and taken to be the area of its base (the area S2„) multiplied by a to 2 its height.
Fig. 151 shows the results of one of Harlacher's gaugings worked151.
and pump. It may be noted that constructively pumps are essentially reversed motors. The reciprocating pump is a reversed pressure engine, and the centrifugal pump a reversed turbine. Hydraulic machine tools are in principle motors combined with tools, and they now form an important special class.
Water under pressure conveyed in pipes is a convenient and economical means of transmitting energy and distributing it to many scattered working points. Hence large and important hydraulic systems are adopted in which at .a central station water is pumped at high pressure into distributing mains, which convey it to various points where it actuates hydraulic motors operating cranes, lifts, dock gates, and in some cases riveting and shearing machines. In this case the head driving the hydraulic machinery is artificially created, and it is the convenience of distributing power in an easily applied form to distant points which makes the system advantageous. As there is some unavoidable loss in creating an artificial head this system is most suitable for driving machines which work intermittently
(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 papermaking from wood pulp and electrometallurgical 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 waterpower in countries having natural waterfalls. According to the twelfth census of the United States the total amount of waterpower reported as used in manufacturing establishments in that country was 1,130,431 h.p. in 1870; 1,263,343 h.p. in 18go; and 1,727,258 h.p. in 19oo. The increase was 8.4% in the decade 18701880, 3.1% in 188o18go, and no less than 36.7 % in 18go19oo. The increase is the more striking because in this census the large amounts of hydraulic power which are transmitted electrically are not included.
§ 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 P, is the pressure in any direction, m the mass of fluid impinging per second, vs the change of velocity in the direction of P„ due to impact, then
Pa =mva.
If v, (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 `G V  P for the resultant pressure,
P = mv.
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 vi 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
+t=Tu/(GQv12/2g)•
§ 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 thesurface. 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.
End of Article: HYDRAULIC 

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