Thermodynamics and Heat Engines

Application to Gases.--In the case of ideal gases obeying the law aPV =RT, it follows directly from the general relations (3) that the difference S—s is constant and equal to R. It will be shown later that the specific heats of such gases cannot vary with pressure, though they may vary considerably with temperature, while the difference S—s remains constant, as in the case of hydrogen. (See HEAT.) On the other hand, vapours, like steam, though they usually approximate closely to the law aPV =RT at low pressures, show large deviations from the gas-laws at high pressures near'saturation, and the value of S shows a wide range of variation with pressure, especially near the critical point. Fortunately these variations can be deduced from the thermo dynamical relations given in a later section of this article.

Cycle or Cyclical Process.

The application of the first law, as expressed in (I), to the cycle of a steam-engine, in which the working fluid is restored to its initial state of water at each repetition of the cycle, is fully considered in the articles already referred to. Briefly stated, if E=E0 at the conclusion of the cycle 2:0, it follows from (I) that the algebraic sum ZQ of all the quantities of heat Q received and rejected by the working substance during the cycle must be equal to the net balance of work done by the engine per cycle, work done by the fluid in expansion being reckoned positive, and work done on the fluid in compression being reckoned negative. With this understanding the formula for the cycle may be expressed as follows, In an ideal reversible engine, in which no heat is lost and no work is wasted in friction, the area of the cycle on the PV dia gram, expressed by the integral of PdV taken round the cycle, will represent the maximum work obtainable from the cycle considered. In any actual engine some of the heat received by the working fluid is lost before it has contributed its full quota of work, and some of the work done is reconverted into heat by internal friction and is rejected in the form of heat. 'If the properties of the working fluid are known, such losses may be estimated in the reciprocating engine by comparing the ideal diagram, as calculated, with the actual diagram as observed with the indicator. (See STEAM-ENGINE.) In the case of a turbine, to which the indicator method is inapplicable, it is necessary to use a different kind of diagram, or the ideal output may be calculated from the properties of the working fluid, and compared with that actually realised.

Frictionless Adiabatic Expansion.

The term "adiabatic" implies that there is no gain or loss of heat by the working fluid. Putting Q=0 in (I) we see that in this case E—E. must be equal to —W/J, or the intrinsic energy of the working fluid is diminished by an amount equivalent to the work done. If no work is wasted in friction, this represents the most efficient method of conversion of heat into work, which is the ultimate aim of every heat-engine. To calculate the work done from the integral of PdV under this condition, it is necessary to know the form of the expansion curve on the PV diagram, or the adiabatic equation representing the relation between P and V for the working fluid employed, which requires an appeal to experiment. Watt made the first experiments of this kind with his indicator, but found the expansion curve for a slow speed engine using wet steam to be approximately PV = constant, the same as Boyle's law for the expansion of a gas at constant temperature, whereas the fall of pressure in adiabatic expansion should have been more rapid than that given by Boyle's law owing to the fall of temperature. Watt was well aware that this anomaly was due to partial condensation of the steam on admission by the cool walls of the cylinder, followed by re-evaporation towards the end of the stroke, which made the conditions far from adiabatic.

Laplace subsequently showed (see HEAT) that the rate of drop of pressure dP/dV in the adiabatic expansion of a gas must exceed that for the same state at constant temperature, in the ratio, S/s=7, of the specific heats. Assuming Boyle's law for a gas at constant temperature, and y =constant in adiabatic ex pansion, it followed that the adiabatic equation must be of the form, PVC' = constant for a gas.

The adiabatic equation of Laplace and Poisson for a gas, was established long before the first law of thermodynamics was formulated, and has proved to be the most convenient type of equation for the purpose. But with the assistance of the laws of thermodynamics, the scope of the adiabatic equation in this form may be considerably extended. Thus with the assistance of the first law, in the form in which it is usually employed for mathematical purposes, namely Experimental Verification of the Adiabatic.—The most direct way of testing the adiabatic equation is to use a cylinder containing a constant charge of gas or vapour which is alternately compressed and expanded by a reciprocating piston. Indicator cards give fairly accurate values of the actual volume and pres sure, if the clearance is carefully measured, the pressure scale calibrated, and everything in perfect adjustment. But in working over large ranges of pressure, as is necessary for a satisfactory verification, there are formidable difficulties due to the wide variation of heat exchanges between the charge and the walls of the cylinder at different points of the stroke. These effects are further complicated by accidental leakage past the piston, and, in the case of steam, by the risk of condensation, both of which affect V indirectly by reducing the apparent mass of the charge. The most complete method of eliminating these uncer tain sources of error, which profoundly affect the uniformity of V throughout the cylinder, is to observe, instead of the relation between P and V, the relation between T and P, as given by the adiabatic equation in the form (io), deduced from the second law. In this case the temperature observed in the middle of the cylinder will be practically unaffected by the action of the walls, and is determined solely by the actual pressure as observed with the indicator. We are no longer concerned with the theoretical pressure, corresponding to the compression ratio by volume, which might be realized in the absence of condensation or leak age or heat-loss to the walls. These effects will still reduce the observed pressure below the theoretical value, but will not affect the relation between pressure and temperature. The success of the thermometric method depends on the construction of a ther mometer sufficiently sensitive to follow the rapid variations of temperature without appreciable lag, and on obtaining simul taneous readings of the relative values of the pressure with the same order of accuracy at the maximum and minimum points of the cycle. It is easy to cover the range of temperature with a single thermometer, but, the pressure range being upwards of o/i, it is necessary to use a separate indicator with a light spring for the low pressures. Applied in this manner the method is particularly suited to give the best average value of the index over large ranges of pressure and temperature. The procedure may be varied when it is desired to obtain the value of the index at some particular point of the scale, e.g., at high or low tem peratures.