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 15th 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
