HYDRAULIC DATA.
Water is practically non-elastic. A pressure of 30,000 lbs. to the square inch has been applied and its contraction has been found to be less than one-twelfth. Experiment appears to show that for each atmosphere of pressure it is condensed 471⁄2 millionth of its bulk.
The mechanical properties of liquids are determined on the hypothesis that liquids are incompressible; according to known general principles this is found to be for all practical purposes true, yet liquids are more compressible than solids. If water be confined in a perfectly rigid cylindrical vessel, its compression would equal 1⁄300000 of its length for every pound per unit of area of the end pressure.
Fig. 84.
Water is nearly 100 times as compressible as steel, yet for almost all practical purposes, liquids may be considered as non-elastic bodies without involving sensible error.
The pressure upon the horizontal base of any vessel containing a fluid, is equal to the weight of a column of the fluid, found by multiplying the area of the base into the perpendicular height of the column, whatever be the shape of the vessel.
This follows, since here the distance of the center of gravity of the base from the surface of the fluid, is the same as the perpendicular height of the column. With a given base and height, therefore, the pressure is the same whether the vessel is larger or smaller above, whether its figure is regular or irregular, whether it rises to the given height in a broad open funnel, or is carried up in a slender tube.
Hence, any quantity of water, however small, may be made to balance any quantity, however great. This is called the hydrostatic paradox. The experiment is usually performed by means of a water-bellows, as represented in Fig. 84. When the pipe AD is filled with water, the pressure upon the surface of the bellows, and consequently the force with which it raises the weights laid on it, will be equal to the weight of a cylinder of water, whose base is the surface of the bellows, and height that of the column AD. Therefore, by making the tube small, and the bellows large, the power of a given quantity of water, however small, may be increased indefinitely. The pressure of the column of water in this case corresponds to the force applied by the piston in the hydrostatic press.
Fig. 85.
We have already seen that the pressure on the bottom of a vessel depends neither on the form of the vessel nor on the quantity of the liquid, but simply on the height of the liquid above the bottom. But the pressure thus exerted must not be confounded with the pressure which the vessel itself exerts on the body which supports it. The latter is always equal to the combined weight of the liquid and the vessel in which it is contained, while the former may be either smaller or greater than this weight, according to the form of the vessel. This fact is often termed the hydrostatic paradox, because at first sight it appears paradoxical.
CD (Fig. 85) is a vessel composed of two cylindrical parts of unequal diameters, and filled with water to a. From what has been said before, the bottom of the vessel CD supports the same pressure as if its diameter were everywhere the same as that of its lower part; and it would at first sight seem that the scale MN of the balance, in which the vessel CD is placed, ought to show the same weight as if there had been placed in it a cylindrical vessel having the same weight of water, and having the diameter of the part D. But the pressure exerted on the bottom of the vessel is not all transmitted to the scale MN; for the upward pressure upon the surface n o of the vessel is precisely equal to the weight of the extra quantity of water which a cylindrical vessel would contain, and balances an equal portion of the downward pressure on m. Consequently the pressure on the plate MN is simply equal to the weight of the vessel CD and of the water which it contains.
Pressure exerted anywhere upon a mass of liquid is transmitted undiminished in all directions, and acts with the same force on all equal surfaces, and in a direction at right angles to those surfaces.
To get a clearer idea of the truth of this principle, let us conceive a vessel of any given form in the sides of which are placed various cylindrical apertures, all of equal size, and closed by movable pistons. Let us, further, imagine this vessel to be filled with liquid and unaffected by the action of gravity; the pistons will, obviously, have no tendency to move. If now a weight of P pounds be placed upon the piston A (Fig. 86), which has a surface A, it will be pressed inwards, and the pressure will be transmitted to the internal faces of each of the pistons B, C, D, and E, which will each be forced outwards by a pressure P, their surfaces being equal to that of the first piston. Since each of the pistons undergoes a pressure, P, equal to that on A, let us suppose two of the pistons united so as to constitute a surface 2a; it will have to support a pressure 2P. Similarly, if the piston were equal to 3a, it would experience a pressure of 3P; and if its area were 100 or 1,000 times that of a, it would sustain a pressure of 100 or 1,000 times P. In other words, the pressure on any part of the internal walls of the vessel would be proportional to the surface.
The principle of the equality of pressure is assumed as a consequence of the constitution of fluids.
By the following experiment it can be shown that pressure is transmitted in all directions; a cylinder provided with a piston is fitted into a hollow sphere (Fig. 87). in which small cylindrical jets are placed perpendicular to the sides. The sphere and the cylinder being both filled with water, when the piston is moved the liquid spouts forth from all the orifices, and not merely from that which is opposite to the piston.
Fig. 86.
Fig. 87.
The reason why a satisfactory quantitative experimental demonstration of the principle of the equality of pressure cannot be given is that the influence of the weight of the liquid and of the friction of the pistons cannot be altogether eliminated.
Note.—The influence of the weight (or gravity) of water and its fractional resistance in practical use is so great upon all the processes of numbers and of the application of the natural laws governing the operation of fluids, as stated under the heading of Hydraulic Data, that separate pages will hereafter be found devoted to a more extended explanation of this subject of gravity and friction of water.
Yet an approximate verification may be effected by the experiment represented in Fig. 88. Two cylinders of different diameters are joined by a tube and filled with water. On the surface of the liquid are two pistons, P and p, which hermetically close the cylinders, but move without friction. Let the area of the large piston, P, be, for instance, thirty times that of the smaller one, p. That being assumed, let a weight, say of two pounds, be placed upon the small piston; this pressure will be transmitted to the water and to the large piston, and as this pressure amounts to two pounds on each portion of its surface equal to that of the small piston, the large piston must be exposed to an upward pressure thirty times as much, or of sixty pounds. If now, this weight be placed upon the large piston, both will remain in equilibrium; but if the weight is greater or less, this is no longer the case.
Fig. 88.
It is important to observe that in speaking of the transmission of pressure to the sides of the containing vessel, these pressures must always be supposed to be perpendicular to the sides.
Equilibrium or state of rest of superposed liquids. In order that there should be equilibrium when several heterogeneous liquids are superposed in the same vessel, each of them must satisfy the conditions necessary for a single liquid, and further there will be a stable state of rest only when the liquids are arranged in the order of their decreasing densities from the bottom upwards.
The last condition is experimentally demonstrated by means of the phial of four elements. This consists of a long narrow bottle containing mercury, water, colored red, saturated with carbonate of potash, alcohol, and petroleum. When the phial is shaken the liquids mix, but when it is allowed to rest they separate; the mercury sinks to the bottom, then comes the water, then the alcohol, and then the petroleum. This is the order of the decreasing densities of the bodies. The water is saturated with carbonate of potash to prevent its mixing with the alcohol.
This separation of the liquids is due to the same cause as that which enables solid bodies to float on the surface of a liquid of greater density than their own. It is also on this account that fresh water, at the mouths of rivers, floats for a long time on the denser salt water of the sea; and it is for the same reason that cream, which is lighter than milk, rises to the surface.
The pressure upon any particle of a fluid of uniform density is proportioned to its depth below the surface.
Fig. 89.
Example 1. Let the column of fluid ABCD Fig. (1) be perpendicular to the horizon. Take any points, x and y, at different depths, and conceive the column to be divided into a number of equal spaces by horizontal planes. Then, since the density of the fluid is uniform throughout, the pressure upon x and y, respectively, must be in proportion to the number of equal spaces above them, and consequently in proportion to their depths.
Example 2. Let the column be of the same perpendicular height as before, but inclined as is Fig. (2); then its quantity, and of course its weight, is increased in the same ratio as its length exceeds its height; but since the column is partly supported by the plane, like any other heavy body, the force of gravity acting upon it is diminished on this account in the same ratio as its length exceeds its height; therefore as much as the pressure on the base would be augmented by the increased length of the column, just so much it is lessened by the action of the inclined plane; and the pressure on any part of Cc will be, as before, proportioned to its perpendicular depth; and the pressure of the inclined column ACac will be the same as that of the perpendicular column ABCD.
Fluids rise to the same level in the opposite arms of a recurved tube.
Fig. 90.
Let ABC, (Fig. 90) be a recurved tube: if water be poured into one arm of the tube, it will rise to the same height in the other arm. For, the pressure acting upon the lowest part at B, in opposite directions, is proportioned to its depth below the surface of the fluid. Therefore, these depths must be equal, that is, the height of the two columns must be equal, in order that the fluid at B may be at rest; and unless this part is at rest, the other parts of the column cannot be at rest. Moreover, since the equilibrium depends on nothing else than the heights of the respective columns, therefore, the opposite columns may differ to any degree in quantity, shape, or inclination to the horizon. Thus, if vessels and tubes very diverse in shape and capacity, as in Fig. p. 84 be connected with a reservoir, and water be poured into any one of them, it will rise to the same level in them all.
The reason of this fact will be further understood from the application of the principle of equal momenta, for it will be seen that the velocity of the columns, when in motion, will be as much greater in the smaller than in the larger columns, as the quantity of matter is less; and hence the opposite momenta will be constantly equal.
Hence, water conveyed in aqueducts or running in natural channels, will rise just as high as its source. Between the place where the water of an aqueduct is delivered and the spring, the ground may rise into hills and descend into valleys, and the pipes which convey the water may follow all the undulations of the country, and the water will run freely, provided no pipe is laid higher than the spring.
Pressure of water due to its weight. The pressure on any particle of water is proportioned to its depth below the surface. The pressure of still water in pounds per square inch against the sides of any pipe, channel, or vessel of any shape whatever, is due solely to the “head” or height of the level surface of the water above the point at which the pressure is considered and is equal to ·43302 lbs. per square foot, every foot of head or 62·355 lbs. per square foot for every foot of head at 62° F.
The pressure per square inch is equal in all directions downwards, upwards or sideways and is independent of the shape or size of the containing vessel; for example, the pressure on a plug forced inward on a square inch of the surface of water is suddenly communicated to every square inch of the vessel’s surface, however great and to every inch of the surface of any body immersed in it.
It is this principle which operates with such astonishing effect in hydrostatic presses, of which familiar examples are found in the hydraulic pumps, by the use of which boilers are tested. By the mere weight of a man’s body when leaning on the extremity of a lever, a pressure may be produced of upwards of 20 tons; it is the simplest and most easily applicable of all contrivances for increasing human power, and it is only limited by want of materials of sufficient strength to utilize it.