CHAPTER XXIII HYDRAULICS WHEN a force is applied to a solid body a certain definite change of shape (omitting cases of actual breakage or deformation) is first produced, and the body then moves as a rigid body, the particles of which it is composed maintaining their relative positions unchanged. Finally, on removing the force, the body returns to its original shape. In the case of liquids, which we are now about to consider, we have the exact opposite to the above; there is no definite amount of change of shape produced when a given force is applied, but the amount is anything we please, depending entirely on how long the force is applied. Also the small particles do not keep their relative positions but move freely about among one another. If then we attempt to determine the motion of a fluid or liquid by simple application of the principle of work and the laws of motion, we should have to inquire into the motions of these small particles. This is done to a certain extent in the branch of mechanics called Hydrodynamics. The results of hydrodynamics can be rarely made available for practical purposes, and hence we must call actual experiment to our aid. We then treat, not of the motion of each particle, but of the water or other liquid as a whole, the results sought for being obtained by a combination of experimental facts with theoretical reasoning. The science which treats the subject in this way we term Hydraulics. We shall, with one exception, confine ourselves exclusively to one liquid, viz. water, though the methods will be applicable to all. Effect of Gravity—Head.—We commence by considering the effect of gravity on water, since a large part of the flow of water takes place under gravity alone. In Fig. 346 ABCD is a reservoir, AB being h the water level. B D Fig. 346. Take now any point E in the reservoir h feet below the water level, then we know from hydrostatics that, due to the weight of the water above E, there is at E a pressure the magnitude of which on every square foot is equal to the weight of a column of water I sq. ft. in section h ft. high. If, then, P=pressure per sq. feet at E, w= weight of I c. ft. of water in lbs., The fact that E is h ft. below AB we express in hydraulics by saying there is a Head of h feet at, or over, E, and this is, we see in the case of still water, equivalent to saying there is a pressure of P or wh lbs. per sq. foot at E. The pressure is plainly present on a horizontal plane at E, and it can be shown both experimentally and theoretically that it is also present as a direct normal pressure on a plane passing through E in any direction; if the plane be not horizontal the intensity of the pressure varies from point to point, but at E is wh lbs. per sq. foot. [Notice that we use now lbs. per sq. ft., not per sq. inch.] We have taken E in the body of the reservoir, but this was not necessary. For let there be a long pipe as shown (Fig. 346), then at F there is a head h1, and a pressure produced by it wh; and similarly for any other point so long as there is a continuous fluid connection, and the water is at rest. The latter qualification is, as we shall see, of the first importance. The pressure produced by a given head depends on the density of the liquid; in the case of water we take I c. ft. of fresh water weighs 62.5 lbs. I c. ft. of sea water weighs 64 lbs. In some cases mercury is used to balance or measure a pressure, the head being then generally measured in inches, and then I inch of mercury=.49 lbs. per sq. inch. We have in speaking of the pressure at E omitted the atmospheric pressure, so that the real pressure at E is wh plus that of the atmosphere. The mean value of the atmospheric pressure may be taken as 14.7 lbs. per sq. inch, or 2136 lbs. per sq. ft., and it is thus equivalent to a head of 34 feet of fresh water, or 33 feet of salt water. We do not, however, include the atmospheric pressure when we speak of the head at E; by that term we shall always be understood to mean the depth below the water surface. Unresisted Flow under a given Head.—ABCD is now a reservoir, from which water is flowing through the open end of the pipe DE (Fig. 347), and we sup D C Fig. 347. pose either that the reservoir is so large that the water level remains practically constant, or that it is maintained constant by water entering from some other source. Let h=depth of E below AB in feet, then the head over E is h feet. The question we wish to answer is: Suppose there be no resistance offered to the motion, with what velocity will the water flow out at E? [We see now why it was necessary on page 463 to insist on the water being still, for there is now no pressure produced by the head, since the stream leaving at E is exposed to the atmosphere (see page 465), the head being now a source of velocity.] Let us consider the time during which 1 lb. of water, which was all originally at the water level, falls to E and out at E. Then, there being no source of effort but gravity, Energy exerted=h ft. -lbs., I lb. having been exerted through h ft. No work has been done, and consequently if v be the velocity at E, h=0+ ... v2=2gh, or v= √2gh. Velocity of Flow. The question now arisesWhat velocity does v really represent ? To understand the point of this question, let us proceed to consider how much water flows out at E per second, or to find the discharge. Let A sectional area at E in sq. ft. |