A numerical comparison can be made with a four-stage centrifugal pump, described in the Engineer, June 17, 1910, which pumps 110 gallons/minute to a head of 225 feet, at 1,440 revs/min, and B.H.P. 10.

The loss of energy, measured in ft-lb/sec, is

and of this, half is thrown away in the rotating energy of the wake; the other half is lost by shock on the blade.

This last energy can be recovered if the leading edge A' of the blade is given a zero angle of attack, so that the screw has a gaining pitch, from at A' through its mean value Р to some

u

n

final pitch p', and the blade A A' is cambered.

If the camber is parabolic, so that cot increases uniformly in the axial direction, the mean effective pitch p is the harmonic

u

mean of the initial pitch and final pitch p',

n

The system of gaining pitch is always adopted with a turbine, intended to run at a given speed n in a given current u; the guide blades may be taken as the equivalent of another screw fixed in front.

Two screws on the same shaft line were employed in one of the earliest screw steamers, so as to recover the rotational energy of the wake; the system has been brought forward lately by Colonel Rota, of the Italian Navy; the system seems applicable to the Gnome motor on a flying machine, when the shaft is made to carry a screw as well as the cylinders, and is allowed to revolve in the opposite direction.

In this way the revolutions of each screw are halved, while the relative motion of the axle and cylinder remains the same as is desirable in practice.

So long as the slip s is small, the energy recovered from the wake would not be worth the extra weight and complication of a second screw.

But with the two screws the slip s may be made as large as 50 per cent., s, and then each screw is pulling hardest for its weight, so that size can be reduced and weight economised.

With uniform pitch there would be no economy of efficiency, as the energy recovered from the wake is lost again in shock at the second screw; but with appropriate gaining pitch the theoretical efficiency can be made perfect.

Large slip is preferred at sea for driving against a head sea, and diminution of racing.

Racing of the screw is due chiefly to variation of axial flow; the variation of the longitudinal velocity of the water in wave motion has more influence than the accompanying vertical component.

With a fine pitch and small slip this velocity variation causes a rapid change in s and L, not so rapid when s is large.

LECTURE VI

PNEUMATICAL PRINCIPLES OF AN AIR SHIP

THE flying machine as a practical success is only some two or three years old; but it looks as if it will displace the air ship balloon, with a large gas bag to give the ascensional force.

The air ship lighter than air is, however, still on its trial, and so we proceed to discuss the pneumatical theory involved; for the detailed calculation a reference must be made to Chapter VIII. of my Hydrostatics.

The first practical balloon to make an ascent with a man dates from 1783, the hot air balloon of Montgolfier (Fig. 54).

The legend goes that as Madame Montgolfier's silk dress was airing before a fire, it became inflated and rose to the ceiling.

Montgolfier followed up the idea, on a small scale at first, on the impression that the hot air was some new kind of gas; and finally, as a paper manufacturer, he was able to make a fire balloon large enough to take up the first two real aeronauts, Pilâtre de Rozier and the Marquis d'Arlandes, in November, 1783, from the Château de la Muette in Paris, and so realise finally the dream of the poet and artist of antiquity.

The principle is seen in the ordinary toy hot air balloon; the air in the balloon is rarefied by heat to an extent such as to make the total weight of the balloon, car, and passengers, and of the hot air it contains, equal to or less than the weight of the external cold air displaced.

The experimental laws of pneumatics required in the theory are embodied in the gas equation

connecting the pressure p, lb/ft2, specific volume v, ft3/lb, and absolute temperature 0, which we take Centigrade, with P2, v2, 02, in another state of the same given quantity of a gas.

100 Feet

FIG. 54.

FIG. 55.

FIG. 56.

This equation expresses Boyle's law when the temperature is constant, and the law of Charles, when varies, and either p or v, one at a time.

Denote by W, lb, the weight of the balloon, car, and aeronauts, corrected for buoyancy of the air, as if weighed in a vacuum, and denote by W' lb the weight of atmospheric air they displace, so that WW' lb is the apparent weight when weighed in air; denote also by V, ft3, the capacity of the gas bag of the balloon, so that M = Vp lb denotes the weight of atmospheric air which fills the balloon, at a density p, lb/ft.3 When the air inside is raised in temperature from to ' degrees, absolute Centigrade, part of the air will flow out of the balloon, leaving the rest at the same pressure, p, but at density

H

The balloon will be floating in equilibrium when the weight of the balloon and hot air it contains is equal to the weight of surrounding cold air displaced; that is when

(3)

Ө M
Ꮎ'

o'

determining '0, the increase of temperature required to rise.

The balloon is now in unstable equilibrium, like a bubble of air compressed to the density of the surrounding water; and it will begin to rise, as it cannot descend.

The balloon will continue to rise and the hot air to escape till another stratum is reached, at height z ft, suppose, where the density is P, and absolute temperature ; and then the pressure p, is given by the gas equation (1)

Reference must be made to Hydrostatics, Chapter VIII., for the further theory of Montgolfier's hot air balloon; but the principle was soon abandoned in favour of the hydrogen balloon, invented by the chemist Charles a few months later, having an advantage that the lift can be obtained with a gas bag much smaller (Fig. 55).

This is of great importance for military use, where the balloon is tethered, and size must be kept down on account of the wind; also in the large air ship, intending to take up numerous passengers, and to keep up in the air and make a journey as long as possible.