356 SUBAQUEOUS OPERATIONS. The upward thrust of the air being equal to the weight of a cylindrical column of the water on an equal base, the weight of this bell must exceed the weight of water it displaces. The same principles are employed in sinking caissons for underwater foundations, as in the Forth Bridge (fig. 77); or in driving a tunnel under a river through muddy soft soil, as in the Hudson River and the Blackwall tunnels, now in progress. Air is forced in to equalize the pressure of the head of water, and to prevent its entrance, being retained by air locks through which the workmen and materials can pass; a slight diminution of air pressure allows the water to percolate sufficiently to loosen the ground, but an increase of pressure is apt to allow the air to blow out in a large bubble. This system, due to Mr. Greathead, has overcome the difficulties of subaqueous tunnelling; but if employed in the projected Channel Tunnel, a pressure of about 10 atmospheres would be required, to which the workmen are not yet accustomed. (The Diving Bell and Dress, J. W. Heinke, Proc. Inst. Civil Eng. XV.; Diving Apparatus, W. A. Gorman, Proc. Inst. Mechanical Engineers, 1882; The Forth Bridge, Engineering, Feb., 1890.) Examples. (1) Two thin cylindrical gasholders which will hold four times their weight of water, and one of which just fits over the other, will float mouth downwards half immersed in water. If the larger one is now placed over the smaller, determine the position of equilibrium. (2) Two equal cylindrical gasholders, of weight W and height a, float with a length ma occupied by gas, which at atmospheric pressure would occupy a length a. If a weight P is placed upon one of them, and gas is transferred to the other till the top of the first just reaches the water, prove that the other rises a height (3) Prove that gas of constant pressure, measured by a height h of water, can be delivered by a gasholder in the form of a truncated cone, whose sides are inclined at an angle a to the vertical, if the thickness of the sides is h sin a. (4) Coal gas, of density 0.6 of that of the air, is delivered to the pipes at a pressure of 2 ins of water; prove that 300 ft higher the pressure will be given by 38 ins; the temperature being 10°C. (5) Prove that the small vertical oscillations of a cylindrical solid, closed at the top and inverted over mercury in a wide basin, will synchronize with a pendulum of length where Mg denotes the weight of the body, a and Bcm2 the horizontal cross sections of the interior and of the material of the body, σ and h the density of mercury and the height of the barometer, V cm3 the volume of air in the cavity, and z cm the difference of level of the mercury inside and outside. (6) A caisson, closed at the top and divided in the middle by a horizontal diaphragm, whose weight is half that of the water it will contain, is floating over water. Prove that the draft of the caisson will be doubled when a hole is opened in the diaphragm. (7) A diving bell with a capacity of 125 ft3 is sunk in salt water to a depth of 100 ft. If the s.G. of salt water is 1025, and the height of the fresh water barometer 34 ft, find the volume of atmospheric air required to clear the bell of water. (8) Two cylindrical caissons closed at the top, of equal cross section and heights H and H, are placed in water so that the first is just submerged, and the second at a depth such that the air occupies the same volume in each; prove that h/2 is the depth of the water surface in the second. What will happen if communication is made by a pipe between the air spaces in the two caissons? (9) Find how deep a cylindrical diving bell of height a and radius c, with a hemispherical top, must be sunk so that the water rises inside to the base of the hemisphere; and prove that the volume of atmospheric air now required to clear the bell of water is times the volume of the bell. (10) Prove that if two equal cylindrical diving bells of height a, whose air spaces communicate by a pipe, are sunk so that their tops are at depths, and Z and if a volume of atmospheric air is forced in, which would occupy a length b of either bell, the 1 THE PUMP. surface of the water in each bell is lowered {√[{H+1(≈1+%2)}2+2H(2a+b)] − {√[{H+1(%1+%2)}2+4aH]; 359 (11) Determine the effect on the level of the water in a diving bell, on the pressure of the air, and on the tension of the chain, due to a floating body inside, according as it has come from the exterior, or has been detached from the interior; or due to a workman leaving his seat in the bell to work on the bottom of the water. Prove that if a bucket of water weighing Plb is drawn up into the bell, then (§ 254), (i) the fall of water level in the bell, (ii) the diminution of volume of the air, (iii) the increase of tension of the chain are respectively (i) PV H 0 Write down the values of these expressions for a cylindrical bell. 258. Pumps. The simplest form of water pump is the common syringe, consisting of a piston rod and piston, working in a cylinder, which is dipped into water. If in contact with the lower surface of the piston, the water will, in consequence of the atmospheric pressure, follow the piston to a height which is only limited by the barometric head of water; the cylinder thus becomes filled with water, which is ejected on reversing the motion of the piston; this is the earliest form of fireengine. By the addition of valves, as in fig. 8, p. 19, the cylinder may be fixed in position, and the piston with its packing may be replaced by a plunger working through a stuffing 360 THE FORCING PUMP box, as easier of manufacture and of adjustment in working, and now this machine is called a force pump (§ 12); it is used on a large scale in Cornish pumping engines (§ 23) for driving water to a high reservoir in water works, and in draining mines; the water lifted being often 30) times the weight of the coal raised. Two such force pumps, placed side by side, and worked in alternate opposite directions by a lever, constitute the modern manual fire engine, which does not, however, differ essentially from the machine invented by Ctesibius, described in Hero's IIvevμaTIKà, B.C. 120; the pumps discharge into an air vessel, in which the cushion of air preserves a steady continuous stream of water in the hose. In a steam fire engine the piston rod of the steam cylinder actuates the piston of a double acting force pump (fig. 78), by which the continuous stream is produced. |