Thermodynamics and Heat Engines

Discharge Through a Nozzle.

The function of a nozzle in a turbine is to convert as much as possible of the total heat of the steam into kinetic energy for turning the wheels. For this reason the nozzle is formed with a bell-mouth on the high pressure side tapering smoothly to a nearly parallel throat, so as to pro duce a uniform jet of high velocity with the least possible friction or turbulence. In this respect the nozzle is the opposite of the throttle which reconverts as much as possible of the kinetic energy into heat, leaving the total heat practically unaltered. In one respect it is like the throttle, in that it has no moving parts and does no external work in itself, so that IV =0 in equation (12), when applied to either nozzle or throttle. The heat-loss Q in the case of a nozzle is also very small compared with the heat-flow, which is usually large owing to the high velocity of flow. Neglecting W and Q on this account, the heat drop is the exact equivalent of the kinetic energy generated, thus equation (I2) becomes in which the constant Jg is required for reducing to heat units. It may be observed that equation (I s) remains true if any proportion of the kinetic energy generated is reconverted into heat by friction, since this has the effect of increasing H2 exactly as much as it reduces K2. As a rule the initial velocity is small and is almost negligible in comparison with K2. The velocity generated may then be calculated from the heat-drop by the formula If H is taken in calories C and U in feet per sec., the value of the constant is 300.2, or a heat-drop of I cal. C will correspond to a velocity of 30o ft./sec.; or ioo cal. to 3,002 ft./sec. A velocity of ioo ft./sec., or 7o miles per hour, though not unim portant in meteorology, may often be neglected in dealing with turbines, as it corresponds to only 9th of a calorie C. Other conditions being equal, the heat-drop and velocity will evidently be greatest in the absence of friction, or in frictionless adiabatic expansion as defined by putting dQ=o in equation (5). This gives the condition dH=aVdP for finding the heat-drop by integrating aV dP along the adiabatic (9) between the given limits of pressure P1 to P2. Neglecting b, this reduces to the simple form, as given by Carnot, and applied by him to the following simple cases: Clapeyron's Equation.—In the vaporisation of unit mass of steam in a boiler, the whole work of expansion W along the iso thermal at constant pressure and temperature, would evidently be equal to p(V–v), the product of the vapour-pressure p by the increase of volume from that of water v to that of steam V. Thus the .work obtainable in a cycle of range dt would be (V–v)dp/dt, where dp/dt is the rate of increase of the steam pressure per degree rise of temperature. The heat Q absorbed in the vaporisation of unit mass is the latent heat L. Substituting these symbols in (24) we obtain the result commonly known as Clapeyron's equation, which evidently represents the condition of equilibrium between liquid and vapour at any temperature, and is applicable to any other substance with the same value of F't at the same tempera ture, but with different values of L, V, and p, depending on the properties of the substance considered. Carnot employed this relation for calculating the value of his function F't at 100° C from the properties of steam, which were roughly known at this temperature. Taking L=540 cal. dp/dt = 27.2 mm. of mercury, or 37o kg./sq. metre, and V–v =1.67o cb.m./kg., we find F't= 1.135 kilogram metres per kilocalorie for the work obtainable in a cycle of I° range at i oo° C. We may remark in passing that the value found is equal to 427/373, being the mechanical equiv alent of the kilocalorie in kgm. divided by the absolute tempera ture. This implies that the whole of the heat could be converted into work if the range of the cycle were extended to the absolute zero with the same rate of production of work per degree fall. But Carnot, who had at that time no knowledge of the value of the mechanical equivalent, naturally failed to notice this remark able coincidence, though the result he obtained for F't was correct.

Lowering of the Freezing Point of Ice by Pressure.—The appli cation of Carnot's equation (25) to the rate of increase of vapour pressure dp/dt with temperature, or to the rise of the boiling point dt/dp with pressure (which are merely different ways of expressing the same property of the substance) is easily made in any case in which the latent heat L and volume V of the vapour are known, and affords in fact one of the most direct methods of verifying Carnot's principle by experiment. A more dramatic verification was that made by James Thomson (1851), who ob served that equation (25) must apply just as exactly to the equilibrium between liquid and solid at the freezing-point, as to that between liquid and vapour at the boiling-point. In the case of solid and liquid, L is the latent heat of fusion, and V–v is the increase of volume in melting, which is positive for a substance like wax which expands in melting, but negative for a substance like water which expands on freezing. Thus in the case of wax, the sign of dt/dp should be positive, or the melting point should be raised by pressure (as is always the case for the boiling-point), whereas in the case of water the sign should be negative, or the freezing-point should be lowered by pressure. Taking the known values of L and V–v for water and ice, he predicted that the freezing-point of water should be lowered 0.0075° C per atmos phere of pressure, a result which was immediately verified by his brother, Lord Kelvin, and materially assisted in the final estab lishment of the second law of thermodynamics.

Application to Gases, Absolute Scale of T.—The application of Carnot's equation (24) to the case of a gas obeying the law PV =RT, is equally simple. The whole work W done in isothermal expansion, is represented by the analytical expression RT In r, where ln r denotes the natural logarithm of the ratio of expan sion. It immediately follows, as Carnot remarks, that dW/dt=Rlnr, or is simply equal to W/T. Thus equation (24) reduces to the form, Carnot deduced from this relation that the ratio W/Q of the work done to the heat absorbed in isothermal expansion must be the same for all gases at the same temperature, and that if equal volumes of different gases were taken at the same temperature and pressure (or masses proportional to the molecular weights) the heat Q absorbed in isothermal expansion must be the same for all gases under similar conditions since the work W was the same for all. It followed as a special case that the difference of the specific heats at constant pressure and volume, being the heat absorbed in an isothermal expansion equal to V/T, must be the same for equal volumes of all gases. His endeavours to extract a value of F't from this relation gave results similar to those ob tained from steam and other va pours, but failed to give a de cisive conclusion owing to the uncertainty of the ratios of the specific heats as deduced from the velocity of sound. At a later date when Joule (1845) proved by direct experiment that the heat absorbed by a gas in isother mal expansion was approximately equivalent to the work done, it became evident, by substituting W =IQ in equation (26) that the value of F't must be nearly equal to J/T for all substances. But since according to Regnault's experiments (1847) the temperature scales of actual gases differed quite appreciably, different gases would give different values of T at the same temperature, and it was impossible to say which should be selected. The essential point of Carnot's principle being that the value of F't was the same for all substances at the same temperature, the most logical method was that proposed by Kelvin, to define absolute temperature T as being proportional to the reciprocal of Carnot's function. Exact consistency with Carnot's principle and with the law of conservation of energy could thus be secured, while leaving the deviation of the scale of any particular gas from the absolute scale thus defined to be determined by experiment in each case.