Since (1) itself cannot be measured in practice in any convenient manner, it is usually deduced for any substance from the expres sion found by experiment for H. Since dQ=dH—aVdP, by (5), taking dH from (44), we obtain, which is seen to be identical with (49) if we substitute for SC from (39). This may be integrated, as in the case of H, from o to T at zero pressure, with S=S0, and from o to P at constant T. Taking as an example the simple equation (42) for V, with SC from (43), we find the expression, in which A is the constant of integration determined in the usual manner.
Use of the Entropy.—It should be observed that the entropy is not an independent property of the substance, since it can be de duced from H and V, but it is most useful in practice for defining the process of frictionless adiabatic expansion, and for deducing the maximum work obtainable from a quantity of heat supplied to the working substance under specified conditions. As a simple ex ample we may take the case of the steam-turbine. Any part of the heat received (e.g., in the superheater) at the constant initial pres sure P' is represented in virtue of relation (3) by the increase DH of the total heat of the steam over the range considered. Let D4 represent the increase of entropy taken from the tables over the same range of temperature.. In the ideal cycle after adiabatic expansion the corresponding part of the heat rejected at the con stant temperature T" of the condenser will be T"D4. The differ ence DH—T"Dit. represents the work obtainable from this part DH of the heat supplied, and the ideal efficiency is given by I —T"Dc13/DH. Thus it becomes possible to estimate the thermal efficiencies of different stages of the heating system, as depending on the temperature T' of heat reception by the working fluid.
The equivalent of the work theoretically obtainable in any part of an ideal cycle included between two adiabatics may also be found by subtracting the heat-drop along the lower adiabatic from that along the higher adiabatic. This gives the same result as subtracting the heat rejected from the heat received, but saves trouble in calculation if tables of adiabatic heat-drop are available.
If the "mean effective temperature" T. of heat reception DH at constant pressure P is defined as being equal to DH/Dc1), the expression for the ideal efficiency of the cycle between the two adiabatics differing in entropy by Df reduces to the form I—T"/T., the same as that of a Carnot cycle in which all the heat is received at one temperature T.. As G. M. Clarke has pointed out, the mean effective temperature thus defined affords a convenient method of expression for the ideal efficiency of any cycle as compared with the Carnot cycle. But it does not super
sede the use of the entropy, which is required in order to be able to calculate Efficiency of Expansion in a Turbine.—The foregoing method of finding the efficiency of an ideal cycle, or part of a cycle, between two adiabatics, depends on assuming that the useful work done is equal to the excess of the heat received over that rejected by the working fluid, and is restricted to the case in which no losses are incurred in expansion, and all the heat is rejected at one temperature T", as in the condenser of a steam engine or turbine. The method gives the fraction of the heat re ceived in any part of the heating system which could be con verted into work by a perfect engine under the conditions imposed. On the other hand, in analysing the performance of an actual en gine or turbine, it is necessary to take account of losses incurred in the engine during the expansion, and to compare the actual performance with that obtainable in adiabatic expansion under ideal conditions over the same range. The general principle of the method by which this may be accomplished in the case of a turbine is as follows: The Case of Dry Steam.—As explained previously, and implied by equation (12), the drop of H in expansion through a turbine, when corrected for the minor losses Q and K, is the equivalent of the useful work done by the steam. So long as the steam is dry, the value of H at any point of the expansion can be de termined by observing the pressure and temperature, and the actual drop of H, denoted by DH, between any two points can be deduced. The work theoretically obtainable in frictionless expan sion can also be found from the adiabatic equation as shown in (i7). The ratio of the actual heat-drop DH, as corrected, to the adiabatic heat-drop for the same drop of pressure, gives the efficiency of any stage or section of the turbine. This simple method fails when the steam is wet, as usually happens towards the end of the expansion, because there is no satisfactory means of measuring the degree of wetness under these conditions at low pressures. But if the initial value cf. of the entropy is known, the adiabatic heat drop can still be found from the final temperature Tv', with the aid of the tables giving and for dry saturated steam, since for wet steam in the state H", cf." , at any temperature T" we have the simple relation, Equation (54) is also the appropriate equation to employ for finding the adiabatic heat-drop over the whole range of ex pansion in the turbine, from admission to exhaust, when the final state of the steam is wet, as is usually the case with a condensing engine. It is equally applicable to the case of a reciprocating engine, since the discontinuities involved in the operation of this type of engine are supposed to be absent from the ideal cycle. The heat-drop thus found may be compared with the work actually done per lb. of steam, as deduced from measurements of the feed and the power, for which see STEAM-ENGINE.