(7) A piston, of weight A, in a closed vertical cylinder of height a and section A is in equilibrium at a height a/n from the base, the pressure of the air underneath it being p.

Prove that a small rise 7 C of the temperature of the air underneath will raise the piston through a height approximately equal to

(8) An air thermometer is made of a bulb and tube inverted vertically in a reservoir of mercury of depth b so that the tube rests on the bottom.

Prove that if the volume of the bulb and tube is equal to a length c of the tube, and h is the height of the barometer, the graduation for the temperature is at a height above the bottom of the tube

¿(h+b+c) — √/ { } (h + b −c)2+(h+b)c

Ꮎ

where 0 is the absolute temperature at which the enclosed air begins to escape.

(9) Assuming that the relative distribution of oxygen and nitrogen at different heights in an atmosphere in equilibrium follows the law that one is not affected by the other, find at what height in an isothermal atmosphere the proportion of oxygen would be reduced to half what it is at sea level, where the proportions by weight may be taken to be 80 parts of nitrogen to 20 of oxygen, and where the densities are in the ratio of 14 to 16.

CHAPTER VIII.

PNEUMATIC MACHINES

235. The Montgolfier Hot-Air Balloon.

This balloon, invented by the Montgolfiers in 1783, is historically interesting as the first employed by the aeronauts Pilâtre de Rozier and d'Arlandes to make an ascent in the atmosphere.

The principle is the same as that of the ordinary hotair toy balloon; the air in the balloon is rarefied by heat to such an extent that the total weight of the balloon, of the hot air it contains, of the car and of the aeronauts is equal to or less than the weight of the external cold air displaced, when the balloon begins to rise.

Denote by W lb the weight of the balloon, car, and aeronauts, as weighed in vacuo, or corrected for the buoyancy of the air; and denote by W'lb the weight of cold air they displace, so that W- W' lb is the apparent weight when weighed in air; denote also by V ft3 the capacity of the balloon, so that M= Vp lb denotes the weight of cold air which fills the balloon, p denoting the density, in lb/ft3, of the surrounding cold air.

Then when the air inside is raised in temperature from to' degrees absolute, part of the air will flow out,

328

THE MONTGOLFIER

leaving the remainder at the same pressure but at density pe/e', and therefore of weight

By Archimedes' principle the balloon will float in equilibrium when the weight of the balloon and the hot air it contains is equal to the weight of cold air displaced; that is, when

giving 0′— 0, the requisite increase of temperature.

The balloon will now be in unstable equilibrium, like a bubble of air in water, and will begin to rise, as it cannot descend.

236. The balloon will continue to rise and the hot air to escape, till another stratum of air is reached, of height suppose, and of density pz and absolute temperature 0, and therefore of pressure pz, given by

p denoting the pressure at the ground.

The pressure of the hot air in the balloon being also Pz, the quantity of hot air left in the balloon, supposed always at the absolute temperature ', is

Pz

ρ

W'+M(1−02/0′)*

But the barometer and thermometer carried by the aeronauts give P2 the pressure and 0, the absolute temperature, compared with Ρ and 0, the pressure and temperature at the ground; and by the Gaseous Laws

Pz P Oz W'+M(1—0z/0')'

.(2)

If e' denotes the temperature which is just sufficient for levitation, as given by equation (1), then in (2)

or

M-W+W'

W+" (1-8).

P-Pz
Pz

W

.(3)

so that with this temperature 'the balloon will not rise unless 02/0 is less than unity, or unless the temperature of the air diminishes as we ascend in the atmosphere.

Thus in an atmosphere in thermal equilibrium of uniform temperature, e' must be increased beyond the value given by (1) for the balloon to ascend.

In such an atmosphere it has been shown that, with

p ρ

where k denotes the height of the homogeneous atmosphere at the temperature ; so that taking this height at the freezing temperature as 26,214 ft,

Ꮎ

k=26214 ≈ 96 0.

330

THE HYDROGEN

The temperature e' of the hot air required to ascend to a height z ft is now given by equation (2),

in an atmosphere in Convective Equilibrium (§ 226). 237. The Hydrogen or Gas Balloon.

In this balloon the requisite levitation is secured by filling the balloon with hydrogen, as first carried out by Charles and Robert in 1783, a few months after the first ascent in the Montgolfier balloon; or nowadays with coal gas, which is specifically lighter than air.

With hydrogen the balloon can be made of much smaller dimensions; but this advantage is counterbalanced by the difficulty of the manufacture of the gas and its great speed of diffusion; so that a balloon is now generally made of larger dimensions and filled with coal gas from the nearest gasworks.

For military purposes, however, where the balloon is required to be held captive at a height of about 1000 ft, it is important to keep down the size, so as to reduce the effect of the wind; so that military balloons are now filled with hydrogen, carried highly compressed in steel flasks, at a pressure of about 100 atmospheres.

With the same notation as for the Montgolfier balloon, suppose the gas employed has a specific volume nv and a density p/n, n times and one-nth that of air at the same pressure and temperature,