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The efficiency of any heat engine depends primarily on the range of temperature through which the working fluid can be cooled while converting its heat energy into work done in adiabatic expansion. It is im portant for this reason to be able to observe and calculate the cooling effect, which is most conveniently defined as the ratio of the drop of temperature dt to the drop of pressure dp under the condition that no heat is supplied to the working fluid, and that the expansion is frictionless and reversible. The heating effect in compression is the same coefficient with the signs of both dp and dt reversed, so that its value is the same as that of the cooling effect in expansion. To find a thermodynamic expression for the cooling effect under the condition dQ = o, we may put dQ/dt=o in the general expression (34) for the specific heat; in which case we must also take dp/dt in (34) under the same condition at constant Q, i.e., we must write
which is the reciprocal of the required cooling effect
Substituting as before from Clapeyron's relation (33), we obtain immediately,
(37)
which gives a general expression for the cooling effect, in terms of the specific heat and the coefficient of expansion, both at con stant pressure, and comparatively easy to determine by experi ment.
Thus if the specific heat of a liquid and its coefficient of expan sion are known, as is the case for most liquids, it is easy to cal culate the heating effect of a sudden compression. The effect is zero for water at its point of maximum density, but increases rapidly with the coefficient of expansion, since S is nearly con stant.
the case of a gas or vapour, which approximates to the equa tion aPV =RT at low pressures, since (dV/dt) approaches the limit R/aP, the expression for the cooling effect becomes RT/SP, corresponding to the adiabatic equation in the form (I 0), with S/R = 1. This shows that, if a vapour, like steam, obeys an
adiabatic equation of this type with 72 constant, the limiting value
of its specific heat at zero pressure must be constant, i.e., independent of the temperature. Conversely in the case of a gas, such as hydrogen, which follows the law aPV =RT very accurately, there will be no variation of S with pressure, but any variation of S with temperature requires a corresponding variation in the in dex 7/-1- I, and in the cooling effect, which is equal to T/(n+ In such a case, it is often much easier to measure the cooling effect directly with a thermometer of a well adapted type, than it is to measure the specific heat itself at high or low temperatures. The variation of the specific heat with temperature may then be deduced from observations of the cooling effect. This method was
applied by Makower (Phil. Meg., Feb. 1903) to verify the con stancy of S in the case of steam by observing the cooling effect in a jacketed vessel, at iio° C. By allowing the steam to expand suddenly from a pressure of 81 cm. to atmospheric and observing the corresponding drop of temperature, he found a value of the index 7 =1.304, agreeing closely with that deduced from the engine experiments at much higher temperatures and pressures, as previously described. Brinkworth (Proc. Roy. Soc., 1925) employed a similar method for verifying the variation of the specific heat of hydrogen at low temperatures. It will readily be understood that it is most important in all such cases to have an exact thermodynamical relation such as (37) between the coeffi cients to be measured. Otherwise the interpretation of the results is apt to be uncertain. Thus it is usually possible to allow for the small variations of dv/dt in reducing the results, and similarly for the variations of S with pressure.
This is the most im portant coefficient of its type, and is most easily measured in steady flow. It was first measured by Joule and Thomson (1852), and was defined by them as the ratio of the drop of temperature to the drop of pressure in a pure throttling process, in which the kinetic energy generated by the pressure-drop is completely re converted into heat without any external heat-loss. Under these conditions, as explained previously, the value of H is the same after passing the throttle as in the initial state, except that in any actual experiment at high temperatures it is necessary to make a small correction for heat-loss. Since H remains constant in a pure throttling process, the cooling-effect in throttling may be defined as the ratio of dT to dP at constant H and may be denoted by
provided that we are dealing with small changes of temperature. In the example previously cited we were concerned only with the fact that the value of H at the high initial pressure and temperature must be the same as that observed at atmos pheric pressure after throttling when corrected for heat-loss. The drop of temperature amounted to upwards of 300° C, but the cooling effect represented by the ratio, 300.3°/3789 lb., or 0.07925° per lb., does not enter explicitly into the calculation, though it might be described as the mean cooling effect over the given range. To find the limiting value of the cooling-effect at a particular temperature and pressure, it is of course necessary to make measurements over much smaller ranges of temperature.