OF STEAM.] STEAM-ENGINE 485 perfect gas, and may be called "steam gas." It then follows the equation PV = 85'5r, and the specific heat at constant pressure, K,,, is 371 foot-pounds or 0'48 thermal unit. At very low temperatures steam approximates closely to the condition of a perfect gas when very slightly super- heated, and even when saturated ; at high temperatures a much greater amount of superheating is necessary to bring about an approach to the perfectly gaseous state. The total heat required for the production of superheated steam under any constant pressure, when the superheating is sufficient to bring the steam to the state of steam gas, may therefore be reckoned by taking the total heat of saturated steam at a low temperature and adding to it the product of K p into the excess of temperature above that. Thus Rankine, treating saturated steam at 32 F. as a gas, gives the formula H' = 1092 + 0-48('-32) to express the heat of formation (under any constant pressure) of superheated steam, at any temperature t' which is so much above the temperature of saturation corresponding to the actual pressure that the steam may be treated as a perfect gas. Calculated from its chemical composition, the density of steam gas should be 0'622 times that of air at the same pressure and temperature. The value of 7 or K ;) /K r for steam gas is 1 '3. These formulas, dealing as they do with steam which is so highly superheated as to be perfectly gaseous, fail to apply to high-pressure steam that is heated but little above its temperature of saturation. The relation of pressure to volume and temperature in the region which lies between the saturated and the perfectly gaseous states has been experimented on by Him. 1 Formulas which are applicable with more or less accuracy to steam in either the saturated or superheated condition have been devised by Him, Zcuner, 2 Hitter, 3 and others. 66. The expansion of volume which occurs during the conversion of water into steam under constant pressure the second stage of the process described in 55 is isothermal. From what has been already said it is obvious that steam, or any other saturated vapour, can be expanded or compressed isothermally only when wet, and that evaporation (in the one case) or condensation (in the other) must accompany the process. Isothermal lines for a working sub- stance which consists of a liquid and its vapour are straight lines of uniform pressure. 67. The form of adiabatic lines for substances of the same class depends not only on the particular fluid, but alsc- on the propor- tion of liquid to vapour in the mixture. In the case of steam, it has been shown by Rankine and Clausius that if steam initially dry be allowed to expand adiabatically it becomes wet, and if initially wet (unless very wet 4 ) it becomes wetter. A part of the steam is condensed by the process of adiabatic expansion, at first in the form of minute particles suspended throughout the mass. The temperature and pressure fall ; and, as that part of the sub- stance which remains uncondenscd is saturated, the relation of pressure to temperature throughout the expansion is that which holds for saturated steam. The following formula, proved by Rankine 8 and Clausius 6 (see 75), serves to calculate the extent to which condensation takes place during adiabatic expansion, and so allows the relation of pressure to volume to be determined. Before expansion, let the initial dryness of the steam be q l and its absolute temperature TJ. Then, if it expand adiabatically until its temperature falls to T, its dryness after expansion is i( +"*?) L! and L are the latent heats (in thermal units) of 1 ft of steam before and after expansion respectively. When the steam is dry to begin with, 5' 1 = 1. This formula is easily applied to the construction of the adiabatic curve when the initial pressure and the pressure after expansion are given, the corresponding values T and L being found from the table. It is less convenient if the data are the initial pressure and the initial and final volumes, or the initial pressure and the ratio of expansion r. An approximate formula more appropriate in that case is Pt- n = constant, or P/P : = (v/Vj) n r n . Here v and i>j denote the volume of 1 Ib of the mixture of steam and water before and after expansion respectively, and are to be distinguished from V and V-,, which we have already used to denote the volume of 1 ft of dry saturated steam at pressures P and Pj. The index n has a value which depends on the degree of initial dryness q^ i Theorie Mecanique de la Chaleur. " Ztschr. d. Vereins deutscher Innenieure, vol. xi. 3 Wied. Ann., 1878. For a discussion of several of these formulas, see a paper by H. Dyer, Trans. Inst. of Engineers and Shipbuilders in Scotland, 1S85. 4 Prof. Cotterill, in liis Treatise on the Steam-Engine, | 79, has calculated (using the equation which follows in the text) that, when a mixture of steam and water expands adiabatically, steam condenses if the proportion of steam be, roughly, over 50 per cent., but water is evaporated if the proportion of steam he less than about 50 per cent. The exact proportion depends on the initial pressure. s Steam-Engine, 281. 6 Mechanical Theory of Heat (tr. by W. R. Browne), chap. vi. 12. According to Zeuner, 7 ?t=l-035 + - l(jr 1 , so that for ?! = ! 0-95 0-9 0-85 0'8 075 07 i = 1-135 1-130 1-125 1-120 1-115 I'llO 1-105. Rankine gave for this index the value , which is too small if the steam be initially dry. He determined it by examining the expansion curves of indicator diagrams taken from working steam- engines ; but, as we shall see later, the expansion of steam in an actual engine is by no means adiabatic, on account of the transfer of heat which goes on between the working fluid and the metal of the cylinder and piston. When it is desired to draw an adiabatic curve for steam, that value of n must be chosen which rulers to^the degree of dryness at the beginning of the expansion. 68. We are now in a position to study the action of a heat-engine Carnot's employing steam as the working substance. To simplify the first cycle with consideration as far as possible, let it be supposed that we have, as steam for before, a long cylinder composed of non-conducting material except working at the base, and fitted with a non-conducting pistoii ; also a source of substance, heat A at some temperature TJ ; a receiver of heat, or, as we may now call it, a condenser C, at a lower temperature T, ; and a non-con- ducting cover B (as in 40). Then we can perform Carnot's cycle of operations as follows. To fix the ideas, suppose that there is 1 Ib of water in the cylinder to begin with, at the temperature T, : (1) Apply A, and allow the piston to rise. The water will take in heat and be converted into steam, expanding isothermally at constant pressure Pj. This part of the operation is shown by the line ab in fig. 14. (2) Remove A and apply B. Allow the expan- sion to continue adiabatically (be), with falling pressure, until the temperature falls to T 2 . The pressure will then be P 2 , corresponding (in Table II.)toT 2 . (3) Remove B, apply C, and compress. Steam is condensed by rejecting heat to C. The action is iso- thermal, and the pressure remains P 2 . Let this be continued until a certain point d is reached, after which adiabatic compression will complete the cycle. FIG. 14. Carnnt's Cycle with water and steam for working substance. (4) Remove C and apply B. Continue the compression, which is now adiabatic. If the point d has been rightly chosen, this will complete the cycle by restoring the working fluid to the state of water at temperature TJ. The indicator diagram for the cycle is given in fig. 14, as cal- culated by the help of the equations in 67 and of Table II. for a particular example, in which ^ = 90 Ib per square inch (rj = 781), and the expansion is continued down to the pressure of the atmo- sphere, 147 ft per square inch (r 2 =673). Since the process is reversible, and since heat is taken in only at r l and rejected only at T 2 , the efficiency is (T I - T 2 )/r 2 . The heat taken in per ft of the fluid is Lj, and the work done is L t (TJ - T 2 )/Tj, a result which may be used to check the calculation of the diagram. 69. If the action here described could be realized in practice, Efficiency we should have a thermodynamically perfect steam-engine using of a saturated steam. The fraction of the heat supplied to it which perfect such an engine would convert into work would depend simply on steam- the temperature, and therefore on the pressure, at which the engine, steam was produced and condensed. The temperature of con- Limits of densation is limited by the consideration that there must be an tempera- abundant supply of some substance to absorb the rejected heat ; ture, water is actually used for this purpose, so that r has for its lower limit the temperature of the available water-supply. To the higher temperature TJ and pressure Pj no limit can be set except such as is brought about in practice by the mechanical diffi- culties, with regard to strength and to lubrication, which attend the use of high-pressure steam. By a very special construction of engine and boiler Jlr Perkins has been able to use steam with a pressure as high as 500 ft per square inch ; with engines of the usual construction the value ranges from 190 ft downwards. If the temperature of condensation be taken as 60 F., as a lower limit, the efficiency of a perfect steam-engine, using saturated steam, would depend on the value of Pj, the absolute pressure of production of the steam, as follows : For perfect steam-engine, with condensation at 60 F., P. in ft per square inch being 40 80 120 160 Highest ideal efficiency = -284 '326 '350 '368 But it must not be supposed that these values of the efficiency are actually attained, or are even attainable. Many causes conspire 1 prevent steam-engines from being thermodynamically perfect, and some of the causes of imperfection cannot be removed. These num- bers will serve, however, as a standard of comparison in judging ot
