Now, in all calculations for steam engines where expansion is used, it is clear that no one of the formulæ can be used so as to give correct results, for the pressure may vary from 4 or 5 atmospheres to considerably less than 1 atmosphere; besides, the calculation would be so complicated as to be of little practical use; this being the case, empirical methods have been used to give the pressure and volume in terms of each other.

Pambour* gives for condensing engines,

2=

10000 422700258 p

(a)

* The same weight of high-pressure steam will raise the temperature of a given weight of water in the same degree, whatever may be the pressure of the steam. This is commonly known as the law of Watt, and has been proved experimentally by M. Clement and others; and although the recent laborious experiments of M. V. Regnault show that it is not correct, it is, nevertheless, likely to maintain its position among practical men, on account of its simplicity and sufficient accuracy for ordinary purposes. The experiments of Pambour are said to confirm the law of Watt, inasmuch as steam, "under an absolute pressure, varying from 27 to 44 atmospheres, and escaping into the atmosphere under an actual pressure of from 14 to 1.03 atmosphere, presented at its issue exactly the same temperature as though it were saturated."

for non-condensing engines,

10000

u =

1·421 +·0023 p

(b)

where u is the relative volume, or the ratio of the volume of the steam to that occupied by the same weight of water, and p the pressure expressed in pounds per square

here P is the pressure in lbs. per square inch, and √ its relative volume compared with that of its constituent

water.

Mr. Pole observes, that this formula may be adopted without considerable error throughout the range generally required in the Cornish engines, viz. from 65 lbs. to 5 lbs. per square inch.

Now, if we suppose a volume of water E to be converted into steam at a pressure p, and if M be the volume of steam which can be produced by it, then

1

u =

M = E

a + B p

(d)

If, again, the same volume of water be converted into steam at a pressure p', and M' is the absolute volume of steam at that time, we shall have

therefore, between the absolute volumes of steam which correspond to the same weight of water, we shall obtain, by eliminating E, the relation

a and B are the constants in Pambour's formula.

Now, by proceeding in the same manner with Pole's formula, we obtain the ratio of the volumes

On the Work done by the Engine on the Piston, per minute.

Let E represent the number of cubic feet of water converted into steam per minute, ▲ the area of the piston in square feet, the whole length of stroke, a that part of the stroke before the steam is cut off, P the pressure in the boiler, p' the pressure in the cylinder before expansion, p the pressure at the xth foot of the stroke, c the total clearance, N the number of single strokes per minute, and U the work of steam on the piston per minute.

Now, each cubic foot of water converted into steam exists in the cylinder before expansion begins, under a pressure p'; it therefore occupies a relative space by equation (d) represented by

Now, the number of cubic feet of water which is evaporated in the boiler, and passes into the cylinder at every stroke of the engine in the form of steam, is represented

E

by; therefore, the space occupied in cubic feet in the

cylinder when the valve is closed and expansion begins, will be represented by

In like manner, the space occupied at the rth foot of the stroke when the pressure p' has become P, is repre

sented by

But the

space in the cylinder filled by the steam before expansion begins, taken in cubic feet, is evidently represented by

A (a + c)

and, the space in the cylinder occupied by it at the oth foot of the stroke, by

But the work done per square foot upon the piston after expansion begins, is represented by

substituting for p its value from equation (ƒ),

E

α

Sp. dx = .log(2+) - = (-a).

a

ΝΑ β

Also, since the pressure upon the piston before expansion begins is represented by p', and it describes a feet under this pressure, the work done on each square foot is represented by

therefore, the whole work done upon each square foot of the piston, is the sum of these two values, and

Multiplying by NA, we shall have the whole work done on the piston per minute, or

The engine having attained uniform motion, the work

developed by the power per minute must be equal to that