it receives and rejects heat. Thus a standard of comparison for every individual engine may be obtained with which to compare its actual performance. The standard of comparison generally adopted for this purpose is obtained by calculating the efficiency of an engine working according to the Rankine cycle. That is to say, expansion is adiabatic and is continued down to the back pressure which in, a non-condensing engine is 14·7 ℔ per square inch, since any back pressure above this amount is an imperfection which belongs to the actual engine. The back pressure is supposed to be uniform, and there is no compression.

Fig. 21.

Fig. 21 shows the pressure-volume diagram of the Rankine cycle for one pound of steam where the initial pressure is 175 ℔ per square inch by the gauge, equivalent to 190 ℔ per square inch absolute. In no case could an engine receiving steam at the temperature corresponding to this pressure and rejecting heat at 212°F. convert more heat into work than is

represented by the area of this diagram. The area of the diagram may be measured, but it is usually more convenient to calculate the number of B.Th.U. which the area represents from the following formula, which is expressed in terms of the absolute temperature T_{1} of the steam at the steam-pipe, and the temperature T_{2} = 461° + 212° = 673° absolute corresponding to the back pressure:—

With the initial pressure of 190 ℔ per square inch absolute it will be found from a steam table that T_{1}=838° absolute. Using this and the temperature 673° in the expression, it will be found that U = 185 B.Th.U. per pound of steam. If *h*_{2} is the water heat at the lower temperature, *h*_{1} the water heat at the higher temperature, and L_{1} the latent heat at the higher temperature, the heat supply per pound of steam is equal to *h*_{1} − *h*_{2}+L_{1}, which, from the steam tables, with the values of the temperatures given, is equal to 1013 B.Th.U. per pound. The thermal efficiency is therefore

That is to say, a perfect engine working between the limits of temperature assigned would convert only 18% of the total heat supply into work. This would be an ideal performance for an engine receiving steam at 190 ℔ initial pressure absolute, and rejecting steam at the back pressure assumed above, and could never be attained in practice. When the initial pressure is 100 ℔ per square inch by the gauge the thermal efficiency drops to about nearly 15% with the same back pressure. The way the thermal efficiency of the ideal engine increases with the pressure is exhibited in fig. 22 by the curve AB. The curve was drawn by calculating the thermal efficiency from the above expression for various values of the initial temperature, keeping the final temperature constant at 673°, and then plotting these efficiencies against the corresponding values of the gauge pressures.

Fig. 22.—Engine Efficiency Curves.

The actual thermal efficiencies observed in some of the cases cited in Table XXI. are plotted on the diagram, the reference numbers on which refer to the first column in the table. Thus the cross marked 3 in fig. 22 represents the thermal efficiency actually obtained in one of Adams and Pettigrew’s experiments; namely, 0·11, the pressure in the steam-pipe being 167 ℔ per square inch. From the diagram it will be seen that the corresponding efficiency of the ideal engine is about 0·18. The efficiency ratio is therefore 0·11/0·18 = 0·61. That is to say, the engine actually utilized 61% of the energy which it was possible to utilize by means of a perfect engine working with the same initial pressure against a back pressure equal to the atmosphere. Lines representing efficiency ratios of 0·6, 0·5 and 0·4 are plotted on the diagram, so that the efficiency ratios corresponding to the various experiments plotted may be readily read off. The initial temperature of the standard engine of comparison must be the temperature of the steam taken in the steam-pipe. For further information regarding the standard engine of comparison see the article Steam Engine and also the “Report of the Committee on the Thermal Efficiency of Steam Engines,” *Proc. Inst; C.E.* (1898).

§ 16. *Piston Speed.*—The expression for the indicated horse-power may be written

(27)

where *v* is the average piston speed in feet per second. For a stated value of the boiler pressure and the cut-off the mean pressure *p* is a function of the piston speed *v*. For the few cases where data are available—data, however, belonging to engines representing standard practice in their construction and in the design of cylinders and steam ports and passages—the law connecting *p* and *v* is approximately linear and of the form

(28)

where *b* and *c* are constants. (See W. E. Dalby, “The Economical Working of Locomotives,” *Proc. Inst. C.E.*, 1905–6, vol. 164.) Substituting this value of *p* in (27)

(29)

the form of which indicates that there is a certain piston speed for which the I.H.P. is a maximum. In a particular case where the boiler pressure was maintained constant at 130 ℔ per square inch, and the cut-off was approximately 20% of the stroke, the values *c* = 55 and *b* =0·031 were deduced, from which it will be found that the value of the piston speed corresponding to the maximum horsepower is 887 ft. per minute. The data from which this result is deduced will be found in Professor Goss’s paper quoted above in Table XXI. The point is further- illustrated by some curves published in the *American Engineer* (June 1901) by G. R. Henderson recording the tests of a freight locomotive made on the Chicago & North-Western railway. Any modification of the design which will reduce the resistance to the flow of steam through the steam passages at high speeds will increase, the piston speed for which the indicated horse-power is a maximum.

§ 17. *Compound Locomotives.*—The thermal efficiency of a steam-engine is in general increased by carrying out the expansion of the steam in two, three or even more stages in separate cylinders, notwithstanding the inevitable drop of pressure which must occur when the steam is transferred from one cylinder to the other during the process of expansion. Compound working permits of a greater range of expansion than is possible with a simple engine, and incidentally there is less range of pressure per cylinder, so that the pressures and temperatures per cylinder have not such a wide range of variation. In compound working the combined volumes of the low-pressure cylinders is a measure of the power of the engine, since this represents the final volume of the steam used per stroke. The volume of the high-pressure cylinder may be varied within, wide limits for the same low-pressure volume; the proportions adopted should, however, be such that there is an absence of excessive drop between them as the steam is transferred from one to the other. Compound locomotives have been built by various designers, but opinion is still uncertain whether any commercial economy is obtained by their use. The varying load against which a locomotive works, and the fact that a locomotive is non-condensing, are factors which reduce the margin of possible economy within narrow limits. Coal-saving can be shown, to the extent of about 14% in some cases, but, the saving depends upon the kind of service on which the engine is employed. The first true compound locomotive was constructed in 1876 from designs by A. M. Mallet, at the, Creusot works in Bayonne. The first true compound, locomotive. in England was constructed at Crewe works in 1878 by F. W. Webb. It was of the same type as Mallet’s engine, and was made by simply bushing one cylinder of an ordinary two-cylinder simple engine, the bushed cylinder being the high-pressure and, the other cylinder the low-pressure cylinder. Webb evolved the type of three cylinder compound with which his name is associated in 1882.