(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 ft 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

(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 1 and Z2 and if a volume of atmospheric air is forced in, which would occupy a length b of either bell, the

THE PUMP.

surface of the water in each bell is lowered

{√/[{H+}(≈1+≈2)}2+2H(2a+b)]

− } √/[{H + }(≈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

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 IIvevμaтika, 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.

AND FIRE ENGINE.

361

The Worthington pumping engine is of similar design ; the ratio of the piston area of the steam cylinder to that of the pump being made somewhat greater than the inverse ratio of the steam and water pressures, according to the speed at which the pump is to be worked.

Water may be used instead of steam to actuate the pump; and now the product (Ah) of its head (h ft) and of the area of piston on which it acts (A ft2) must exceed the product (Bk) of the height to which the water is forced (k ft) and of the area of the pump plunger (B ft2); the delivery (Q ft3/sec) depending on this excess.

For, denoting the length of stroke by 7 ft, the moving force D(Ah-Bk) lb on the piston, acting through x ft suppose, will for 7-x ft, the remainder of the stroke, be changed to a resisting force of DBk lb; and therefore if we take W lb as the inertia of the piston and the moving water in the pump, and v f/s as the maximum velocity acquired,

Wv2/g=D(Ah- Bk)x = DBk(l—x).

The average velocity of the piston being v, the de

(glB3Dk/W), as Ah

which increases from zero to

increases from Bk to infinity.

If the water pressure is used to check the motion of

the piston, then

Wv2/g=D(Ah-Bk)x = D(Ah+Bk)(l—x) ;

and therefore, as before,

362

WORCESTER'S AND SAVERY'S ENGINE.

The adjustment of the valves must be very accurate to secure this motion without shock; or else a connecting rod, crank, and fly-wheel must be added, as in the ordinary steam engine.

In the inventions of the Marquis of Worcester (1663) and of Savery (1696) steam acted directly upon the surface of the water, without the intervention of pistons; and considerable waste by condensation took place.

This method is nevertheless employed nowadays in the Pulsator or Pulsometer pump (fig. 79) where economy is not of so much importance as rapidity and certainty of action; an automatic spherical valve C admits steam to act alternately on the surface of the water in the vessels A and B, by which the water is forced to the required level; entrance and exit valves being provided to each chamber, as in a pump.