W/H = (t1 − t0) / (t1 + 273) = (T1 − T0) / T1
(4)
which, he observed, agrees in form with that found by Rankine.
21. The Absolute Scale of Temperature.—Since Carnot’s function is the same for all substances at the same temperature, and is a function of the temperature only, it supplies a means of measuring temperature independently of the properties of any particular substance. This proposal was first made by Lord Kelvin (Phil. Mag., 1848), who suggested that the degree of temperature should be chosen so that the efficiency of a perfect engine at any point of the scale should be the same, or that Carnot’s function F′t should be constant. This would give the simplest expression for the efficiency on the caloric theory, but the scale so obtained, when the values of Carnot’s function were calculated from Regnault’s observations on steam, was found to differ considerably from the scale of the mercury or air-thermometer. At a later date, when it became clear that the value of Carnot’s function was very nearly proportional to the reciprocal of the temperature T measured from the absolute zero of the gas thermometer, he proposed a simpler method (Phil. Trans., 1854), namely, to define absolute temperature θ as proportional to the reciprocal of Carnot’s function. On this definition of absolute temperature, the expression (θ1 − θ0) / θ1 for the efficiency of a Carnot cycle with limits θ1 and θ0 would be exact, and it became a most important problem to determine how far the temperature T by gas thermometer differed from the absolute temperature θ. With this object he devised a very delicate method, known as the “porous plug experiment” (see [Thermodynamics]) of testing the deviation of the gas thermometer from the absolute scale. The experiments were carried out in conjunction with Joule, and finally resulted in showing (Phil. Trans., 1862, “On the Thermal Effects of Fluids in Motion”) that the deviations of the air thermometer from the absolute scale as above defined are almost negligible, and that in the case of the gas hydrogen the deviations are so small that a thermometer containing this gas may be taken for all practical purposes as agreeing exactly with the absolute scale at all ordinary temperatures. For this reason the hydrogen thermometer has since been generally adopted as the standard.
22. Availability of Heat of Combustion.—Taking the value 1.13 kilogrammetres per kilo-calorie for 1° C. fall of temperature at 100° C., Carnot attempted to estimate the possible performance of a steam-engine receiving heat at 160° C. and rejecting it at 40° C. Assuming the performance to be simply proportional to the temperature fall, the work done for 120° fall would be 134 kilogrammetres per kilo-calorie. To make an accurate calculation required a knowledge of the variation of the function F′t with temperature. Taking the accurate formula of § 20, the work obtainable is 118 kilogrammetres per kilo-calorie, which is 28% of 426, the mechanical equivalent of the kilo-calorie in kilogrammetres. Carnot pointed out that the fall of 120° C. utilized in the steam-engine was only a small fraction of the whole temperature fall obtainable by combustion, and made an estimate of the total power available if the whole fall could be utilized, allowing for the probable diminution of the function F′t with rise of temperature. His estimate was 3.9 million kilogrammetres per kilogramme of coal. This was certainly an over-estimate, but was surprisingly close, considering the scanty data at his disposal.
In reality the fraction of the heat of combustion available, even in an ideal engine and apart from practical limitations, is much less than might be inferred from the efficiency formula of the Carnot cycle. In applying this formula to estimate the availability of the heat it is usual to take the temperature obtainable by the combustion of the fuel as the upper limit of temperature in the formula. For carbon burnt in air at constant pressure without any loss of heat, the products of combustion might be raised 2300° C. in temperature, assuming that the specific heats of the products were constant and that there was no dissociation. If all the heat could be supplied to the working fluid at this temperature, that of the condenser being 40° C., the possible efficiency by the formula of § 20 would be 89%. But the combustion obviously cannot maintain so high a temperature if heat is being continuously abstracted by a boiler. Suppose that θ′ is the maximum temperature of combustion as above estimated, θ” the temperature of the boiler, and θ0 that of the condenser. Of the whole heat supplied by combustion represented by the rise of temperature θ′ − θ0, the fraction (θ′ − θ″) / (θ′ − θ0) is the maximum that could be supplied to the boiler, the fraction (θ″ − θ0) / (θ′ − θ0) being carried away with the waste gases. Of the heat supplied to the boiler, the fraction (θ′ − θ0) / θ″ might theoretically be converted into work. The problem in the case of an engine using a separate working fluid, like a steam-engine, is to find what must be the temperature θ″ of the boiler in order to obtain the largest possible fraction of the heat of combustion in the form of work. It is easy to show that θ” must be the geometric mean of θ′ and θ0, or θ″ = √θ′θ0. Taking θ′ − θ0 = 2300° C., and θ0 = 313° Abs. as before, we find θ″ = 903° Abs. or 630° C. The heat supplied to the boiler is then 74.4% of the heat of combustion, and of this 65.3% is converted into work, giving a maximum possible efficiency of 49% in place of 89%. With the boiler at 160° C., the possible efficiency, calculated in a similar manner, would be 26.3%, which shows that the possible increase of efficiency by increasing the temperature range is not so great as is usually supposed. If the temperature of the boiler were raised to 300° C., corresponding to a pressure of 1260 ℔ per sq. in., which is occasionally surpassed in modern flash-boilers, the possible efficiency would be 40%. The waste heat from the boiler, supposed perfectly efficient, would be in this case 11%, of which less than a quarter could be utilized in the form of work. Carnot foresaw that in order to utilize a larger percentage of the heat of combustion it would be necessary to employ a series of working fluids, the waste heat from one boiler and condenser serving to supply the next in the series. This has actually been effected in a few cases, e.g. steam and SO2, when special circumstances exist to compensate for the extra complication. Improvements in the steam-engine since Carnot’s time have been mainly in the direction of reducing waste due to condensation and leakage by multiple expansion, superheating, &c. The gain by increased temperature range has been comparatively small owing to limitations of pressure, and the best modern steam-engines do not utilize more than 20% of the heat of combustion. This is in reality a very respectable fraction of the ideal limit of 40% above calculated on the assumption of 1260 ℔ initial pressure, with a perfectly efficient boiler and complete expansion, and with an ideal engine which does not waste available motive power by complete condensation of the steam before it is returned to the boiler.
23. Advantages of Internal Combustion.—As Carnot pointed out, the chief advantage of using atmospheric air as a working fluid in a heat-engine lies in the possibility of imparting heat to it directly by internal combustion. This avoids the limitation imposed by the use of a separate boiler, which as we have seen reduces the possible efficiency at least 50%. Even with internal combustion, however, the full range of temperature is not available, because the heat cannot conveniently in practice be communicated to the working fluid at constant temperature, owing to the large range of expansion at constant temperature required for the absorption of a sufficient quantity of heat. Air-engines of this type, such as Stirling’s or Ericsson’s, taking in heat at constant temperature, though theoretically the most perfect, are bulky and mechanically inefficient. In practical engines the heat is generated by the combustion of an explosive mixture at constant volume or at constant pressure. The heat is not all communicated at the highest temperature, but over a range of temperature from that of the mixture at the beginning of combustion to the maximum temperature. The earliest instance of this type of engine is the lycopodium engine of M. M. Niepce, discussed by Carnot, in which a combustible mixture of air and lycopodium powder at atmospheric pressure was ignited in a cylinder, and did work on a piston. The early gas-engines of E. Lenoir (1860) and N. Otto and E. Langen (1866), operated in a similar manner with illuminating gas in place of lycopodium. Combustion in this case is effected practically at constant volume, and the maximum efficiency theoretically obtainable is 1 − loger / (r − 1), where r is the ratio of the maximum temperature θ′ to the initial temperature θ0. In order to obtain this efficiency it would be necessary to follow Carnot’s rule, and expand the gas after ignition without loss or gain of heat from θ′ down to θ0, and then to compress it at θ0 to its initial volume. If the rise of temperature in combustion were 2300° C., and the initial temperature were 0° C. or 273° Abs., the theoretical efficiency would be 73.3%, which is much greater than that obtainable with a boiler. But in order to reach this value, it would be necessary to expand the mixture to about 270 times its initial volume, which is obviously impracticable. Owing to incomplete expansion and rapid cooling of the heated gases by the large surface exposed, the actual efficiency of the Lenoir engine was less than 5%, and of the Otto and Langen, with more rapid expansion, about 10%. Carnot foresaw that in order to render an engine of this type practically efficient, it would be necessary to compress the mixture before ignition. Compression is beneficial in three ways: (1) it permits a greater range of expansion after ignition; (2) it raises the mean effective pressure, and thus improves the mechanical efficiency and the power in proportion to size and weight; (3) it reduces the loss of heat during ignition by reducing the surface exposed to the hot gases. In the modern gas or petrol motor, compression is employed as in Carnot’s cycle, but the efficiency attainable is limited not so much by considerations of temperature as by limitations of volume. It is impracticable before combustion at constant volume to compress a rich mixture to much less than 1⁄5th of its initial volume, and, for mechanical simplicity, the range of expansion is made equal to that of compression. The cycle employed was patented in 1862 by Beau de Rochas (d. 1892), but was first successfully carried out by Otto (1876). It differs from the Carnot cycle in employing reception and rejection of heat at constant volume instead of at constant temperature. This cycle is not so efficient as the Carnot cycle for given limits of temperature, but, for the given limits of volume imposed, it gives a much higher efficiency than the Carnot cycle. The efficiency depends only on the range of temperature in expansion and compression, and is given by the formula (θ′ − θ″) / θ′, where θ′ is the maximum temperature, and θ″ the temperature at the end of expansion. The formula is the same as that for the Carnot cycle with the same range of temperature in expansion. The ratio θ′ / θ″ is rγ−1, where r is the given ratio of expansion or compression, and γ is the ratio of the specific heats of the working fluid. Assuming the working fluid to be a perfect gas with the same properties as air, we should have γ = 1.41. Taking r = 5, the formula gives 48% for the maximum possible efficiency. The actual products of combustion vary with the nature of the fuel employed, and have different properties from air, but the efficiency is found to vary with compression in the same manner as for air. For this reason a committee of the Institution of Civil Engineers in 1905 recommended the adoption of the air-standard for estimating the effects of varying the compression ratio, and defined the relative efficiency of an internal combustion engine as the ratio of its observed efficiency to that of a perfect air-engine with the same compression.
24. Effect of Dissociation, and Increase of Specific Heat.—One of the most important effects of heat is the decomposition or dissociation of compound molecules. Just as the molecules of a vapour combine with evolution of heat to form the more complicated molecules of the liquid, and as the liquid molecules require the addition of heat to effect their separation into molecules of vapour; so in the case of molecules of different kinds which combine with evolution of heat, the reversal of the process can be effected either by the agency of heat, or indirectly by supplying the requisite amount of energy by electrical or other methods. Just as the latent heat of vaporization diminishes with rise of temperature, and the pressure of the dissociated vapour molecules increases, so in the case of compound molecules in general the heat of combination diminishes with rise of temperature, and the pressure of the products of dissociation increases. There is evidence that the compound carbon dioxide, CO2, is partly dissociated into carbon monoxide and oxygen at high temperatures, and that the proportion dissociated increases with rise of temperature. There is a very close analogy between these phenomena and the vaporization of a liquid. The laws which govern dissociation are the same fundamental laws of thermodynamics, but the relations involved are necessarily more complex on account of the presence of different kinds of molecules, and present special difficulties for accurate investigation in the case where dissociation does not begin to be appreciable until a high temperature is reached. It is easy, however, to see that the general effect of dissociation must be to diminish the available temperature of combustion, and all experiments go to show that in ordinary combustible mixtures the rise of temperature actually attained is much less than that calculated as in § 22, on the assumption that the whole heat of combustion is developed and communicated to products of constant specific heat. The defect of temperature observed can be represented by supposing that the specific heat of the products of combustion increases with rise of temperature. This is the case for CO2 even at ordinary temperatures, according to Regnault, and probably also for air and steam at higher temperatures. Increase of specific heat is a necessary accompaniment of dissociation, and from some points of view may be regarded as merely another way of stating the facts. It is the most convenient method to adopt in the case of products of combustion consisting of a mixture of CO2 and steam with a large excess of inert gases, because the relations of equilibrium of dissociated molecules of so many different kinds would be too complex to permit of any other method of expression. It appears from the researches of Dugald Clerk, H. le Chatelier and others that the apparent specific heat of the products of combustion in a gas-engine may be taken as approximately .34 to .33 in place of .24 at working temperatures between 1000° C. and 1700° C., and that the ratio of the specific heats is about 1.29 in place of 1.41. This limits the availability of the heat of combustion by reducing the rise of temperature actually obtainable in combustion at constant volume by 30 or 40%, and also by reducing the range of temperature θ′ / θ″ for a given ratio of expansions r from r.41 to r.29. The formula given in § 21 is no longer quite exact, because the ratio of the specific heats of the mixture during compression is not the same as that of the products of combustion during expansion. But since the work done depends principally on the expansion curve, the ratio of the range of temperature in expansion (θ′ − θ″) to the maximum temperature θ′ will still give a very good approximation to the possible efficiency. Taking r = 5, as before, for the compression ratio, the possible efficiency is reduced from 48% to 38%, if γ = 1.29 instead of 1.41. A large gas-engine of the present day with r = 5 may actually realize as much as 34% indicated efficiency, which is 90% of the maximum possible, showing how perfectly all avoidable heat losses have been minimized.
It is often urged that the gas-engine is relatively less efficient than the steam-engine, because, although it has a much higher absolute efficiency, it does not utilize so large a fraction of its temperature range, reckoning that of the steam-engine from the temperature of the boiler to that of the condenser, and that of the gas-engine from the maximum temperature of combustion to that of the air. This is not quite fair, and has given rise to the mistaken notion that “there is an immense margin for improvement in the gas-engine,” which is not the case if the practical limitations of volume are rightly considered. If expansion could be carried out in accordance with Carnot’s principle of maximum efficiency, down to the lower limit of temperature θ0, with rejection of heat at θ0 during compression to the original volume V0, it would no doubt be possible to obtain an ideal efficiency of nearly 80%. But this would be quite impracticable, as it would require expansion to about 100 times v0, or 500 times the compression volume. Some advantage no doubt might be obtained by carrying the expansion beyond the original volume. This has been done, but is not found to be worth the extra complication. A more practical method, which has been applied by Diesel for liquid fuel, is to introduce the fuel at the end of compression, and adjust the supply in such a manner as to give combustion at nearly constant pressure. This makes it possible to employ higher compression, with a corresponding increase in the ratio of expansion and the theoretical efficiency. With a compression ratio of 14, an indicated efficiency of 40% has been obtained In this way, but owing to additional complications the brake efficiency was only 31%, which is hardly any improvement on the brake efficiency of 30% obtained with the ordinary type of gas-engine. Although Carnot’s principle makes it possible to calculate in every case what the limiting possible efficiency would be for any kind of cycle if all heat losses were abolished, it is very necessary, in applying the principle to practical cases, to take account of the possibility of avoiding the heat losses which are supposed to be absent, and of other practical limitations in the working of the actual engine. An immense amount of time and ingenuity has been wasted in striving to realize impossible margins of ideal efficiency, which a close study of the practical conditions would have shown to be illusory. As Carnot remarks at the conclusion of his essay: “Economy of fuel is only one of the conditions a heat-engine must satisfy; in many cases it is only secondary, and must often give way to considerations of safety, strength and wearing qualities of the machine, of smallness of space occupied, or of expense in erecting. To know how to appreciate justly in each case the considerations of convenience and economy, to be able to distinguish the essential from the accessory, to balance all fairly, and finally to arrive at the best result by the simplest means, such must be the principal talent of the man called on to direct and co-ordinate the work of his fellows for the attainment of a useful object of any kind.”
Transference of Heat