In the absence of any direct method of determining the general relation between the pressure and volume of common steam, empirical formulæ expressing it have been proposed by different mathematicians.

The late Professor Navier proposed the following: Let S express the volume of steam into which an unit of volume of water is converted under the pressure P, this pressure being expressed in kilogrammes per square mètre. Then the relation between S and P will be

where a

1000, b = 0.09, and m = 0.0000484.

This formula, however, does not agree with experiment at pressures less than an atmosphere. M. de Pambour, therefore, proposes the following changes in the values of its co-efficients :-Let P express the pressure in pounds per square foot; and let b = 0.4227 m = 0.00258, and the formula will be accurate for all pressures. For pressures above two atmospheres the following values give more accuracy to the calculation:

a = 10000

a = 10000 b=1.421 m = 0.0023.

In these investigations I shall adopt the following modified formula. The symbols S and P retaining their signification, we shall have

where

α

S=

b+ P

(10.)

and may

These values of a and b will be sufficiently accurate for practical be used in reference to lowpurposes for all pressures, pressure engines of every form, as well as for high-pressure engines which work expansively.

When the pressure is not less than 30 pounds per square inch, the following values of a and b will be more accurate :

On the Expansive Action of Steam.

The investigation of the effect of the expansion of steam which has been given in the text, is intended to convey to those who are not conversant with the principles and language of analysis, some notion of the nature of that mechanical effect to which the advantages attending the expansive principle are due. We shall now,

The dynamical effect produced by any mechanical agent is expressed by the product of the resistance overcome and the space through which that resistance is moved.

Let P

the pressure of steam expressed in pounds per square foot.

S the number of cubic feet of steam of that pressure produced by the evaporation of a cubic foot of water. E the mechanical effect produced by the evaporation of a cubic foot of water expressed in pounds raised one foot. Then we shall have E = PS; and if W be a volume of water evaporated under the pressure P, the mechanical effect produced by it will be WPS.

By (10.) we have

SP = a - bs.

Hence, for the mechanical effect of a cubic foot of water evaporated under the pressure P we have

Let a cubic foot of water be evaporated under the pressure P', and let it produce a volume of steam S' of that pressure. Let this steam afterwards be allowed to expand to the increased volume S and the diminished pressure P; and let it be required to determine the mechanical effect produced during the expansion of the steam from the volume S' to the volume S.

Let E'

the mechanical effect produced by the evaporation of the water under the pressure P' without expansion. E" the mechanical effect produced during the expansion of the steam.

E

= the mechanical effect which would be produced by the evaporation under the pressure P without expansion. E = the total mechanical effect produced by the evaporation under the pressure P' and subsequent expansion.

Thus we have

EEE”.

Lets be any volume of the steam during the process of expansion, p the corresponding pressure, and e" the mechanical effect produced by the expansion of the steam. We have then by (10.)

which, taken between the limits s = S' and s = S, becomes

Hence it appears that the mechanical effect of a cubic foot of water evaporated under the pressure P may be increased by the

quantity a log.

S
S"

if it be first evaporated under the greater pres

sure P ́, and subsequently expanded to the lesser pressure P. The logarithms in these formulæ are hyberbolic.

To apply these principles to the actual case of a double acting steam engine,

Let L

the stroke of the piston in feet.

A = the area of the piston in square feet.

n

the number of strokes of the piston per minute.

...2n AL the number of cubic feet of space through which

the piston moves per minute.

Let cLA = the clearage, or the space between the steam valve and the piston at each end of the stroke.

Let V

The volume of steam admitted through the steam valve at each stroke of the engine will be 2n AL (1 + c).

the mean speed of the piston in feet per minute,

• 2nL = V.

The volume of steam admitted to the cylinder per minute will therefore be VA (1 + c), the part of it employed in working the piston being VA.

Let W = the water in cubic feet admitted per minute in the form

S

of steam through the steam valve.

the number of cubic feet of steam produced by a cubic foot of water.

By which the pressure of steam in the cylinder will be known, when the effective evaporation, the diameter of the cylinder, and speed of the piston, are given.

If it be required to express the mechanical effect produced per minute by the action of steam on the piston, it is only necessary to multiply the pressure on the surface of the piston by the space per minute through which the piston moves. This will give

a
VAP W
=

1+ c

-VAb;

(17.)

which expresses the whole mechanical effect per minute in pounds raised one foot.

If the steam be worked expansively, let it be cut off after the piston has moved through a part of the stroke expressed by e.

The volume of steam of the undiminished pressure P' admitted per minute through the valve would then be

and the ratio of this

being expressed by S',

VA (e + c);

volume to that of the water producing it we should have

The final volume into which this steam is subsequently expanded being VA (1 + c), its ratio to that of the water will be

The pressure P', till the steam is cut off, will be

The mechanical effect E' produced per minute by the steam of full pressure will be

and the effect E" per minute produced by the expansion of the steam will by (12.) be

(20.)

If the engine work without expansion, e = 1;

as before; and the effect per minute gained by expansion will therefore be

which therefore represents the quantity of power gained by the expansive action, with a given evaporating power.

In these formulæ the total effect of the steam is considered without reference to the nature of the resistances which it has to overcome.

These resistances may be enumerated as follows:

1. The resistance produced by the load which the engine is required to move.

2. The resistance produced by the vapour which remains uncondensed if the engine be a condensing engine, or of the atmospheric pressure if the engine do not condense the steam. 3. The resistance of the engine and its machinery, consisting of the friction of the various moving parts, the resistances of the feed pump, the cold water pump, &c. A part of these resistances are of the same amount, whether the engine be loaded or not, and part are increased, in some proportion depending on the load.

When the engine is maintained in a state of uniform motion, the sum of all these resistances must always be equal to the whole effect produced by the steam on the piston. The power expended on the first alone is the useful effect.

Let R = the pressure per square foot of the piston surface, which balances the resistances produced by the load.

mR = the pressure per square foot, which balances that part of the friction of the engine which is proportional to the load.

r = the pressure per square foot, which balances the sum of all those resistances that are not proportional to the load.

The total resistance, therefore, being R+mR+r, which, when the mean motion of the piston is uniform, must be equal to the mean pressure on the piston. The total mechanical effect