ever less than 2 lbs. on the square inch, or nearly double of the pressure of condensation.

The principal cause, however, of increased back pressure, is resistance to the escape of the steam from the cylinder, by which, in condensing engines, the mean back pressure is caused to be from 1 to 3 lbs. on the square inch greater than the pressure in the condenser. There is as yet no satisfactory theory of that resistance, so that it cannot be computed for any proposed engine by means of a general formula.

The back pressure, therefore, in proposed condensing engines, can for the present only be estimated roughly from the results of experience in particular cases. The following is a summary of

some such results:

There is a deficiency of precise experimental data on this subject, because of the frequent omission to observe the atmospheric barometer at the time when the indicator diagrams of steam engines are taken. The consequence of that omission is, that the diagrams show only the effective pressures of the steam, and not the absolute pressures, which are left to be roughly estimated by guessing the probable atmospheric pressure.

It is certain, that if sufficient experimental data existed, the back pressure would be found to vary with the speed of the engine, being greater at higher speeds, and also with the density of the steam at the commencement of the exhaust, and with the size of the exhaust port through which it escapes from the cylinder

In non-condensing locomotive engines, a great number of experimental data as to back pressure have been collected and arranged, and to a certain extent reduced to a system of laws, in Mr. D. K. Clark's work On Railway Machinery. That author finds, that the excess of the back pressure above the atmospheric pressure varies nearly

As the

square of the speed;

As the pressure of the steam at the instant of release; that is, of the commencement of the exhaust;

Inversely as the square of the area of the orifice of the blast pipe, through which the steam is blown into the chimney to produce a draught.

Mr. Clark also finds, that the excess of back pressure is less, the greater the ratio of expansion; that it is less, the longer the time

during which the eduction of the steam lasts; and that it is increased by the presence of liquid water amongst the steam, being in certain cases greater in unprotected than in protected cylinders in the ratio of 1.72 to 1.

As an example of specific results obtained by Mr. Clark, it may be stated, that "with a mean of 16 per cent. of release,"—that is, with the exhaust port opened when the piston had performed 0.84 of its forward stroke-"with an admission of half stroke,”—that is, with the ratio of expansion 2, nearly, "and with a speed of piston of 600 feet per minute;" the excess of the back pressure above the atmospheric pressure, in protected cylinders, was about 0.163 of the excess of the pressure of the steam at the instant of release above the atmospheric pressure.

It is probable, that the general results arrived at by Mr. Clark may be safely applied to all engines, whether condensing or noncondensing, to the following extent :

That in the same engine, going at the same speed, the excess of the mean back pressure above the pressure of condensation, varies nearly as the density of the steam at the end of the expansion;

And that in the same engine, with the same density of steam at the end of the forward stroke, that excess of back pressure varies nearly as the square of the speed.

281. Thermodynamic Function, and Adiabatic Curve, for Mixed Water and Steam.-When, as in the present investigation, the volume of a pound of water, and its variations, are treated as insensibly small, the value of the thermodynamic function consists simply of the first term of the expression in Article 246, equation 1; that is to say,

J hyp log ;

J denoting, as usual, Joule's equivalent, or the dynamical value of the specific heat of water. Suppose the pound of water to be raised from a fixed temperature to any given absolute temperature

and then to be either wholly or partially evaporated; and let u be the volume of the steam produced, which for total evaporation is equal to v, the volume of one pound of saturated steam at the given boiling point, and for partial evaporation, may have any value less than v. Then from Article 255, equation 1, it is evident, that to complete the thermodynamic function for the aggregate of water and steam, we must add to the expression already found for the water in the liquid state, the following quantity :

d p น ; dr

giving for the complete thermodynamic function for one lb. of water and steam

[The same expression may be made applicable to any other fluid by putting instead of J, J c, the dynamical specific heat of the fluid in question in the liquid state.]

The equation of an adiabatic curve is

constant.

This enables us to find the equation of the form of the curve BC in the diagram, fig. 109, Article 278, when that curve is adiabatic; that is, when the steam expands without receiving or giving out heat. Attending to the notation of Article 279, we have, in the present case, for the point B in the curve,

from which is easily deduced the following expression for the volume u occupied by one lb. of water and steam at any pressure

When common instead of hyperbolic logarithms are used in the calculation, for J = 772 is to be substituted,

J hyp log 10 = 772 × 2.3026 = 1777.6.

According to Article 255, equation 3,

by means of which formula, with the aid of equation 1 of Article

d P

237,

can be computed. ατ

The use of the equation 3 for computing the value of u may be much facilitated, by employing the values of L, the latent heat per cubic foot, which are given for steam in Table IV. (and for æther in Table V.); for according to Article 255, equation 2 (neglecting the volume of the liquid water),

A convenient modification of equations 3 and 5 is the following:

Let the weight of steam under consideration be D1

1

21

= so that its initial volume u1 is one cubic foot. Then, instead of u may be

put r (= “), the ratio in which the steam is expanded; so that we

have for the value of that ratio,

282. Approximate Formula for Adiabatic Curve. From the results of numerical calculations of the co-ordinates of adiabatic curves for steam, it has been deduced by trial, that for such pressures as usually occur in the working of steam engines, the relation between those co-ordinates is approximately expressed by the following statement:-the pressure varies nearly as the reciprocal of the tenth power of the ninth root of the space occupied, that is to say, in symbols

This formula belongs to the class already explained in Article 279, Method II.; the value of the exponents and co-efficients being

The preceding equation 1, and those deduced from it, are most expeditiously employed by the aid of a table of logarithms. In the absence of a table of logarithms, the ninth root of any ratio can be found by extracting the cube root of the cube root, either by the aid of a table of cube roots, or by ordinary arithmetic.

283. Liquefaction of Steam Working Expansively.-The volume

of one pound of saturated steam (neglecting the volume of the liquid water), according to Article 256, equation 1, is

H' being the latent heat of evaporation of one pound. It appears by computation, that the volume u given by equation 3 or equation 5 of Article 281 is less than v in all cases which occur in practice; from which it follows, that when steam expands in driving a piston, and receives no heat from without, a portion is liquefied.

To find under what conditions, and to what extent this condensation by expansive working will take place, we have for the proportion borne by the condensed steam to the whole mass of steam and water, the following expression:

The value of H' is given approximately in foot-lbs. per pound of steam by the formula

H' = a − b r = 1109550-540·4 r..............

T.............(3.)

For any other fluid, J c would have to be put instead of J, and for a and b their proper values, supposing them to have been ascertained.

It may be shown by an investigation, which it is unnecessary here to give in detail, that the expression (2) is always positive so long as

α

- is less than 3 (= 1437°-2 for steam = 461o-2+976°).

1

J c

The principle just stated, as to the liquefaction of vapours by expansive working, was arrived at contemporaneously and independently, by Professor Clausius and the Author of this work in 1849. Its accuracy was subsequently called in question, chiefly on the ground of experiments which show that steam, after being expanded by being "wire-drawn," that is to say, by being allowed to escape through a narrow orifice, is super-heated, or at a higher temperature than that of liquefaction at the reduced pressure. afterwards, however, Professor William Thomson*proved that those experiments are not relevant against the conclusion in question, by showing the difference between the free expansion of an elastic fluid, in which all the energy due to the expansion is expended in agitating the particles of the fluid, and is reconverted into heat,

"Now Lord Kelvin.

Soon