DYNAMICS OF THE SIPHON.

273

This supposes that the mercury column divides where the pressure vanishes or becomes negative; but if, as in Mr. Worthington's experiments (§ 6), we suppose that the mercury column can support a tension of a certain amount, ok suppose, without breaking, the siphon can still work, so long as the height of B' above H does not exceed k.

191. In its dynamical action the siphon may be assimilated to a chain coiled up at A, and led over a pulley at B so that the end hangs at C; the preponderating length will set the chain in motion, so that the coil at A will become gradually transferred into a coil at C.

If x cm of chain have passed over in t seconds, and the moving part ABC, of length 7 cm suppose, has then acquired a velocity v cm/sec, if w denotes the weight in g/cm of the chain, and T g denotes the tension of the chain at A, the equation of motion is

the momentum in second-grammes generated per second; so that

= gz-v2; dt

..(1)

and the chain therefore starts with an initial acceleration gz/l, and tends to a terminal velocity

(gz), just like

a body falling under gravity in a medium in which the resistance varies as the square of the velocity.

In the siphon there will be no loss of energy at A due to the continuous series of impacts, so that we may halve the above value of T; and now the equation of motion in the siphon becomes

274

STARTING THE ACTION

1 d v = gz - { v2,

dt

so that the terminal velocity in the siphon is(2gz). By integration of equation (2),

.(2)

192. A leak in the tube ABC, if above the level of A, will admit air, and vitiate the action of the siphon, even to the extent of stopping the flow if the leak is sufficiently large; but liquid will escape from a leak in the tube below the level of A.

The large siphons or standpipes of waterworks are designed to reach the altitude of the service reservoirs, so that the water in passing through may be cleared of air, which tends to accumulate in the mains.

In the distiller's siphon (fig. 62, p. 272) the action is started by opening the stopcock s.C., closing the end C with the hand, and sucking the air out by the curved mouthpiece at C''; as soon as the spirit passes the highest point of the bend at B, the action of the siphon commences, and it can be stopped and restarted by closing and opening the stopcock.

The preceding methods are not desirable with noxious liquids, such as acids, which cannot be handled or tasted with impunity; the siphon is then started by first closing the stopcock s.c., and filling the branch BC

through one of two small funnels at B, the other funnel permitting the escape of air; on shutting off these funnels by plugs or stopcocks and opening s.c., the liquid is made to flow through the siphon.

193. If however the length of the longer branch BC is insufficient, the liquid will not be drawn up to the level of the bend B in the branch AB, but will rest at a lower level B' a certain vertical height x above A, and at a certain distance y from A if the branch AB is curved; and the siphon will not start.

The vertical height of the liquid CC′ left in the longer branch BC will also be x, in consequence of the equality of the pressures at A and C, and at B' and C'; and the pressures at A and C being due to a head h of the liquid, the pressure in B'C' will be due to a head h−x.

Denoting by a, b the lengths of the branches AB, BC, and supposing for simplicity that BC is inclined at an angle a to the vertical, the air which originally occupied the length a of the shorter arm AB now occupies the length a+b-y-x sec a.

Therefore by Boyle's Law (Chap. VII.), which asserts that the product of the volume and pressure of a given quantity of air remains the same at the same temperature, ah=(a+b−y−x sec a)(h—x),

In the critical case when the liquid just reaches the bend B, y=a, and x denotes the vertical height of B above A; so that

and a greater value of b will start the siphon.

276

SIPHONS ON A LARGE SCALE.

In the siphons of fig. 61 the branches are vertical; and now if a, b denote the lengths of the vertical branches, and c the length of the horizontal part, then in the critical case

(a+c)h=(b+c− a)(h− a),

or

b=

(a+c)h
h-a

+u-c.

194. As employed for drawing off water over an embankment, the siphon is shown in fig. 63; for example, over the reservoir dam of water works (fig. 20), or in draining a fen or inundation.

(Proc. Inst. Civil Engineers, XXII.) An automatic valve, opening inwards, is placed at A and a stop valve at C.

The siphon is filled either through a funnel by means of a hand pump, or else by exhausting the air by an air pump at B. On opening the stop valve C, the water flows through the siphon; and on closing the stop valve, the siphon remains filled for an indefinite time, the valve at A preventing the return of the water in AB.

In this, as in all other cases, the height of B above the upper level of the liquid must be kept below the head of liquid corresponding to the atmospheric pressure.

Sometimes the siphon is inverted, as required for carrying a water main across the bed of a river; and now there is no limitation of depth to its working.

A water main, or a pipe line for conveying oil, carried in an undulating line in the ground, may be considered as a series of erect and inverted siphons; and on an emergency, the pipe may be carried over an obstacle, which is higher than the supply source or hydraulic gradient by something under the atmospheric head of the liquid.

THE INTERMITTENT SIPHON.

277

195. An intermittent siphon is shown in fig. 64; the vessel is gradually filled up to the level of B, when the action of the siphon suddenly commences, and the vessel is rapidly emptied; and so the operation goes on periodically.

The Cup of Tantalus, invented by Hero, depends on this principle; and it is also used for securing an intermittent scouring flow of water. The action of natural intermittent springs and geysers is explained in this manner; and the underground flow of certain rivers, such as the Mole, by subterranean inverted siphons.

Examples.

(1) If a vessel contains liquids of various densities, will the action of the siphon be impeded?

Two equal cylindrical pails of horizontal section A are placed, one on the ground, and the other on a stand of height h; the former is empty, and the latter contains masses m, m2 of two different homogeneous liquids; a fine siphon tube of negligible volume has its two ends at the bottoms of the two pails and through it flows liquid until equilibrium is attained, a mass m, of density p remaining in the upper pail; prove that

m1+m2-2m3 = Ahp.