CHAPTER XIII.

PNEUMATICS.

SECTION I.-Equilibrium of air and elastic fluids.

1. THE fundamental propositions that belong to hydrostatics are common to the compressible and the incompressible fluids, and need not, therefore, be repeated here.

2. Atmospheric air is the best known of the elastic fluids, and has been defined an elastic fluid, having weight, and resisting compression with forces that are directly as its density, or inversely as the spaces within which the same quantity of it is contained.

The correctness of this definition is confirmed by experi

ment.

The weight of air is known from the Torricellian experiment, or that of the barometer. The air presses on the orifice of the inverted tube with a force just equal to the weight of the column of mercury sustained in it.

A bottle, weighed when filled with air, is found heavier than after the air is extracted. The pressure of the atmosphere is at a mean about 14 lbs. on every square inch of the earth's surface. Hence the total pressure on the convex surface of the earth = 10,686,000,000 hundreds of millions of pounds.

The elastic force of the air is proved, by simply inverting a vessel full of air in water: the resistance it offers to farther immersion, and the height to which the water ascends within it, in proportion as it is farther immersed, are proofs of the elasticity of the air contained in it.*

When air is confined in a bent tube, and loaded with different weights of mercury, the spaces it is compressed into are

It is in virtue of this property, and ought to be known as extensively as possible, that a man's hat will serve in most cases as a temporary life-preserver, to persons in hazard of drowning, by attending to the following directions:-When a person finds himself in, or about to be in, the water, let him lay hold of his hat between his hands, laying the crown close under his chin, and the mouth under the water. By this means, the quantity of air contained in the cavity of the hat will be sufficient to keep the head above water for several hours, or until assistance can be rendered.

found to be inversely as those weights. But those weights are the measures of the elasticity; therefore the elasticities are inversely as the spaces which the air occupies.

The densities are also inversely as those spaces; therefore the elasticity of air is directly as its density. This law was first proved by Mariotte's experiments.

In all this, the temperature is supposed to remain unchanged. These properties seem to be common to all elastic fluids.

Air resists compression equally in all directions. No limit can be assigned to the space which a given quantity of air would occupy if all compression were removed.

3. In ascending from the surface of the earth, the density of the air necessarily diminishes: for each stratum of air is compressed only by the weight of those above it; the upper strata are therefore less compressed, and of course less dense than those below them.

4. Supposing the same temperature to be diffused through the atmosphere, if the heights from the surface be taken increasing in arithmetical progression, the densities of the strata of air will decrease in geometrical progression. Also, since the densities are as the compressing forces, that is, as the columns of mercury in the barometer, the heights from the surface being taken in arithmetical progression, the columns of mercury in the barometer at those heights will decrease in geometrical progression.

As logarithms have, relatively to the numbers which they represent, the same property, therefore if b be the column of mercury in the barometer at the surface, and B at any height h above the surface, taking m for a constant coefficient, to be determined by experiment, h = m (log b — log 3), or h = m log 1/2: - where m may be determined by finding trigonome

b

3

:

trically the value of h in any case, where b and ẞ have been already ascertained.

5. If b be the height of the mercury in the barometer at the lowest station, 3 at the highest, t and t' the temperatures of the air at those stations, f the fixed temperature at which no correction is required for the temperature of the air; and if q and q' be the temperatures of the quicksilver in the two barometers, and n the expansion of a column of quicksilver, of which the length is 1, for 1° of heat; h being the perpendicular height (in fathoms) of the one station above the other.

If the centigrade thermometer is used, because the beginning of the scale agrees with the temperature f, so that f = 0, the formula is more simple; and if the expansion for air and mercury be both adapted to the degrees of this scale,

h = 10000 (1 + ·00441

t

(+) log.

b

B (1 + 00018 (q — q') 6. The temperature of the air diminishes on ascending into the atmosphere, both on account of the greater distance from the earth, the principal source of its heat, and the greater power of absorbing heat that air acquires, by being less compressed.

7. Professor Leslie, in the notes on his Elements of Geometry, p. 495, edit. 2d, has given a formula for determining the temperature of any stratum of air when the height of the mercury in the barometer is given. The column of mercury at the lower of two stations being b, and at the upper 3, the diminu

tion of heat, in degrees of the centigrade, is

This seems to agree well with observation.

b

25.

8. If the atmosphere were reduced to a body of the same density which it has at the surface of the earth, and of the same temperature, the height to which it would extend is, in fathoms,

Hence if b be the height of the mercury in the barometer, reduced to the temperature t, the specific gravity of mercury

is to that of air, as b to 4343 (1+4),

1000

or the specific gra

The divisor 72 is introduced in consequence of b being expressed in inches.-(Playfair's Outlines.)

SECTION II.-Pumps.

1. Def. The term Pump is generally applied to a machine for raising water by means of the air's pressure.

D

2. The common suction-pump consists of two hollow cylinders, which have the same axis, and are joined in A c. The lower is partly immersed, perpendicularly in a spring or reservoir, and is called the suction-tube; the upper the body of the pump. At a c is a fixed sucker containing a valve which opens upwards, and is less than 34 feet from the surface of the water. In the body of the pump is a piston D made air-tight, moveable by a rod and handle, and containing a valve opening upwards. And a spout & is placed at a distance greater or less, as convenience may require, above the greatest elevation of D.

A

B

C

On

3. To explain the action of this pump. Suppose the moveable piston D at its lowest depression, the cylinders free from water, and the air in its natural state. raising this piston, the pressure of the air above it keeping its valve closed, the air in the lower cylinder A в forces open the valve at a c, and occupies a larger space, viz., between в the surface of the water, and D; its elastic force therefore being diminished, and no longer able to sustain the pressure of the external air, this latter forces up a portion of the water into the cylinder A B to restore the equilibrium. This continues till the piston has reached its greatest elevation, when the valve at a C closes. In its subsequent descent, the air below D becoming condensed, keeps the valve at A c closed, and escapes by forcing open that at D, till the piston has reached its greatest depression. In the following turns a similar effect is produced, till at length the water rising in the cylinder forces open the valve at A c, and enters the body of the pump; when, by the descent of D, the valve in a c is kept closed, and the water rises through that in D, which on re-ascending, carries it forward, and throws it out at the spout G.

4. Cor. 1. The greatest height to which the water can be raised in the common pump by a single sucker is when the column is in equilibrio with the weight of the atmosphere, that is, between 32 and 36 feet.

5. Cor. 2. The quantity of water discharged in a given time is determined by considering that at eath stroke of the piston

a quantity is discharged equal to a cylinder whose base is a section of the pump, and altitude the play of the piston.

=

6. To determine the force necessary to overcome the resistance experienced by the piston in ascending. Let h = the height HF of the surface of the water in the body of the pump above E F the level of the reservoir; and ao: the area of the section м N. Let h' the height of the column of water equivalent to the pressure of the atmosphere; and suppose the piston in ascending to arrive at any position m n which corresponds to the height I F. It is evident that the piston is acted upon downwards by the pressure of the atmosphere a h', and by = a2 the pressure of the column B m = a2 X HI; therefore the whole tendency of the piston to descenda2 (h' + H 1.)

=

But the piston is acted upon upwards by the pressure of the air on the external surface E F of the reservoir aa h'; part of which is destroyed by the weight of the column of water having for its base m n, and height FI;

a2 ×

the whole action upwards a2 x (h'-FI);

whence Fao. (h' + H 1)=a2. (h' — F 1) a2. F H = a2 h,

=

that is, the piston throughout its ascent is opposed by a force equal to the weight of a column of water having the same base as the piston, and an altitude equal to that of the surface of the water in the body of the pump above that in the reservoir. In order, therefore, to produce the upward motion of the piston, a force must be employed equal to that determined above, together with the weight of the piston and rod, and the resistance which the piston may experience in consequence of the friction against the inner surface of the tube.*

When the piston begins to descend, it will descend by its own weight; the only resistance it meets with being friction, and a slight impact against the water."

7. Cor. 1. If the water has not reached the piston, let its

Suppose the body of the pump to be 6 inches in diameter, and the greatest height to which the water is raised to be 30 feet; suppose, also, the weight of the piston and its rod to be 10lbs., and the friction one-fifth of the whole weight. Then, by the rule at p. 201, 1 of the square of the diameter gives the ale gallons in a yard in length of the cylinder, and an ale gallon, p. 290, weighs 10 lbs. There fore (62 × 10) + 1 62×10)= 360+7·4 = 367-4 lbs. weight of the opposing column of water. And 367·4+10+ (377·4)=452·9 lbs., whole opposing pres

sure.

30

10

If the piston rod be moved by a lever whose arms are as 10 to 1, this pressure will be balanced by a force of 45.29 lbs., and overcome by any greater force.