# Popular Science Monthly/Volume 17/October 1880/Possible Efficiency of Heat-Engines

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* POSSIBLE EFFICIENCY OF HEAT-ENGINES.
By Professor WILLIAM A. ANTHONY.
THE theory of thermodynamics, which asserts the equivalence of heat and mechanical work, has now been generally accepted by men of science for thirty years. The equivalence of heat and work is accepted as an established fact by engineers and mechanics, and the mechanical equivalent of heat, as determined by Joule, is made the basis of computations regarding the energy of fuel, etc., by practical men, without a question as to its correctness. But there is one conclusion to which the theory leads, of great practical importance as regards the theory of steam-engines, which does not seem to have been as generally accepted, and yet it is just as firmly established as the fundamental principle of the equivalence of heat and work. A pound of carbon by its combustion in oxygen yields 14,400 heat-units, equal to 14,400 $\times$ 772 $=$ 11,116,800 foot-pounds of energy. One horse-power should, therefore, be developed by the combustion of about one sixth of a pound of carbon per hour, while our best steam-engines require ​about two pounds of coal per hour per horse-power. Hence it is said that the steam-engine is a very inefficient machine, which the genius of the future inventor may improve till a pound of coal is made to yield six, eight, or ten times the power it now gives.
But, while the theory of thermodynamics asserts that the combustion of a pound of carbon yields an amount of heat equivalent to 11,116,800 foot-pounds of energy, it also asserts just as clearly and just as firmly that, under the conditions which exist upon this planet, heat can not be transformed into mechanical effect without wasting a considerable portion of the energy due to it. In other words, if a given quantity of heat be taken from a source, as a steam-boiler, and made to do work by means of any form of engine, a considerable portion of that heat must, from the very nature of things under existing conditions, be allowed to pass through the engine and raise the temperature of other bodies to no useful purpose. It is to the discussion upon which this conclusion is based that I propose to devote this article.
In a very valuable essay published in 1824, Carnot furnished us with a conception which is an exceedingly useful one in this investigation. It is that of an engine completely reversible in all its physical and mechanical agencies. A water-motor which could be driven backward by means of some source of power, and would then raise to a given level as much water as it would use from that level to develop the power required to run it backward, would be a reversible engine. It is plainly to be seen that such a motor would be a perfect motor, and also a perfect pump. A reversible heat-engine would be one which, running forward and performing a certain amount of work by means of a given amount of heat derived from a source, would, if run backward by the performance upon it of the same amount of work, restore to the source the same amount of heat.
A reversible engine in this sense is, of course, impossible in practice, but the theoretical deductions from the conception are in no way invalidated by this fact. Such a heat-engine would be a perfect engine in the sense that it would produce as much mechanical effect as could be produced by any heat-engine under the same conditions from the same quantity of heat. The proof of this proposition rests upon two assumptions that are supported by all past experience, and may, therefore, be regarded as physical axioms: 1. That a perpetual motion is impossible; 2. That it is impossible by means of inanimate material agency to derive mechanical effect from any portion of matter by cooling it below the temperature of surrounding objects.
A heat-engine when at work must carry heat from a body of high temperature (the source) to one of low temperature (the refrigerator), and experiment has proved that the amount of heat given to the refrigerator is always less than that taken from the source. Now, let there be two heat-engines, A and B, of which B is reversible, working between the same source and refrigerator. Let each take the same ​quantity, H, of heat from the source, and, if possible, let A derive from this heat more work than B. Let h be the quantity of heat carried to the refrigerator, and W' the mechanical effect developed by B when running forward. Let W' greater than W by hypothesis, be the mechanical effect developed by A. A may be used to run B backward, and, since B is perfectly reversible, will, in so doing, by the expenditure of the mechanical effect W, take from the refrigerator the amount h, and carry to the source the amount H of heat. The two engines so coupled would then develop the mechanical effect W'—W, while no heat would be lost by the source. If A carries to the refrigerator the same quantity of heat that B takes away, the mechanical effect W'—W is developed without any change in external objects, without any consumption of energy. This would constitute a perpetual motion, which, by the first axiom, is impossible. If A transfers to the refrigerator less heat than B takes away, the refrigerator will grow colder and colder, and, since for the purposes of this discussion all other bodies may be assumed to be at the same temperature as the source, this will present the case of a machine producing mechanical effect while taking heat from the coldest of surrounding bodies. This is contrary to the second axiom. Therefore, A can not do more work under the conditions named than B. The reversible engine, then, derives as much mechanical effect from a given amount of heat as can be derived by any heat-engine whatever working between the same temperatures.
It follows further that all reversible engines working between the same source and refrigerator, and taking from the source the same amount of heat, must yield the same mechanical effect; in other words, must have the same efficiency. No matter what the working substance, or in what way heat is made to yield mechanical effect, so long as the process is completely reversible, the same amount of mechanical effect will always be derived from the same heat taken from the source.
It may be well to emphasize a little the first conclusion, that no heat-engine whatever can be more efficient than a reversible engine. If the reasoning is correct, no form of heat-engine, whether using air, or gas, or a condensable vapor, or a liquid, or a solid, as the working substance, or using thermo-electric currents, or any other means of converting heat into mechanical effect, can be more efficient than any one of the reversible engines. There is no escaping this conclusion except through the perpetual motion, or the derivation of mechanical effect from the heat of a body already cold. It has been sometimes claimed that the latter alternative was no impossibility, that the expansion of a compressed gas, the expansion of a gas into a vacuum, or the diffusion of one gas into another, may perform work at the expense of its own heat, while being cooled down to a lower temperature than surrounding bodies. But to compress the gas, or produce ​the vacuum, or separate the gases preparatory to diffusion, requires an expenditure of energy at least equal to the mechanical effect to he derived.
Since the reversible engine is as efficient as any heat-engine, and since all reversible engines of whatever construction and whatever the working substance have the same efficiency, it is allowable, in discussing the question as to the amount of mechanical effect derivable under given conditions from a given amount of heat, to assume any form of reversible engine, using any working substance which may be most convenient. And it makes no difference whether the engine assumed be practically possible, so long as we know the properties of the working substance well enough to determine its action under the assumed conditions. Sir W. Thomson, before 1851, assuming Carnot's engine with air as the working substance, furnished us with a very complete discussion of this question. The properties of air in relation to heat are very simple. Heat expands and cold contracts it with great uniformity. Compression heats and expansion cools it according to a well-known law. The effects of any change of volume or of temperature in the cylinder of an engine can, therefore, be exactly predicted.
Suppose a given mass of air to be compressed and the heat developed by compression removed, so that its temperature remains constant. The pressure exerted by it will increase, as shown graphically in the annexed diagram, where O a, measured on the horizontal axisFig. 1. O x, represents the initial volume, and a e perpendicular to O x represents the pressure exerted at that volume. O b, O c, and O d, represent other volumes, and b f, c g, d h, the corresponding pressures. The curved line e f g h, drawn through the extremities of the perpendiculars, represents to the eye the relation between volume and pressure when temperature is constant. It is called an isothermal line. Now, suppose the air to be compressed without loss or gain of heat. It is warmed by compression, and the rise of temperature causes it to exert a greater pressure. If, then, the substance be at the same initial volume, pressure, and temperature as before, and it be compressed to the volume O b without loss of heat, the pressure exerted will be b m greater than b f. Similarly the pressure c n will correspond to the volume O c, and d o to d. The line l m n o, which shows the relation between volume and pressure when no heat enters nor escapes, is called an adiabatic line.
​Carnot's engine consists of a cylinder having no outlet nor inlet, with an air-tight piston inclosing a mass of air which changes in volume with the movements of the piston. The piston and sides of the cylinder are supposed to be perfect non-conductors, while the bottom of the cylinder is a perfect conductor of heat. The engine has a source of heat and a refrigerator, whose temperatures are supposed to remain absolutely constant whether parting with or receiving heat. There is also a non-conducting stand, on which, if the cylinder be placed, no heat can enter or escape from it, however much the air within it may change in temperature. In the working of the engine there are four operations, as follows:
First Operation.—The air is supposed to be at the temperature of the refrigerator, which may be designated by t, and to have a volume represented by O a, and pressure represented by a A (Fig. 2). The Fig. 2. cylinder is supposed to stand upon its non-conducting support. The piston is now depressed, and, since no heat can escape, the air rises in temperature. The compression continues till the temperature of the air becomes that of the source, which we designate by T. The rise in pressure will be represented by the adiabatic line A B. Let O b represent the volume and B b the pressure at the end of the operation. It is plain that work must have been done to compress the air, equal to the space swept through by the piston multiplied by the mean pressure; but this is represented by the area of the figure A B b a.
Second Operation.—The cylinder is placed upon the source of heat and the piston allowed to rise, being forced upward by the pressure of the air. The bottom of the cylinder being a perfect conductor, heat will enter so rapidly as to maintain the temperature of the air while it expands. The pressure therefore falls, as indicated by the isothermal line B C. Let this operation continue until an amount of heat H is taken from the source, and suppose O c to represent the volume and c C the pressure of the air at that time. It will be seen that during this operation work represented by the area B C c b will have been done by the air.
Third Operation.—The cylinder is returned to its non-conducting support. The upward stroke of the piston continues, and the air expands without receiving heat, until its temperature falls to that of the refrigerator, that is, to the temperature that it had at the beginning of the first operation. The fall of pressure is represented by the adiabatic C D, and work represented by the area C D d c is done by the air.
​Fourth Operation—. The cylinder is placed upon the refrigerator, the piston caused to descend, and the air compressed until its initial volume is reached. Since the bottom of the cylinder and the refrigerator are supposed to be perfect conductors, the heat generated by the compression will escape to the refrigerator, and the temperature of the air will remain constant. The air is now in the same condition as regards temperature, volume, and pressure, as at the beginning of the first operation. The isothermal line, which represents the rise of pressure during the last operation, must, therefore, pass through the starting-point A. During this operation work represented by the area A D d a must be done upon the air, and a certain amount of heat—all that generated by compressing the air—must be given up to the refrigerator.
It will be noticed that, during the second and third operations, work represented by the area B C D d b is done by the air, and during the first and fourth operations work represented by B A D d b is done upon the air. During the complete cycle of operations, therefore, mechanical effect is developed equivalent to the difference between these areas, or to the area B C D A. This figure is, in fact, the indicator diagram of the engine. During the second operation, heat represented by H was taken from the source, and during the fourth operation heat represented by h was given to the refrigerator. During the cycle of operations, heat equal to H—h has disappeared, and, since the working substance is at the end of the cycle in precisely the same condition as at the beginning, this heat must be the equivalent of the mechanical effect developed, and the efficiency of the engine is H—hH
But it is easily shown that this cycle of operations is a completely reversible cycle. For suppose the substance at its initial volume O a, pressure A a, and temperature t. Place the cylinder on the refrigerator, and allow the air to expand to the volume A d. The same isotherm A D that represented the rise in pressure in the reverse operation will now represent the fall, and the same heat h that was before given to the refrigerator will now be taken from it. Now let the cylinder be placed upon its non-conducting support and the piston descend till the volume becomes O c. Since no heat escapes, the rise of pressure will be represented by the adiabatic D C, and the temperature will rise by the same amount as it fell during the expansion from c to d, that is, from t, the temperature of the refrigerator, to T, that of the source. Now, let the cylinder be placed upon the source, and the descent of the piston continue till the volume of the air becomes O b, the temperature remains that of the source, the isotherm C B represents the rise in pressure, and heat is given to the source precisely equal to the amount taken from it during the expansion from b to c in the direct working of the engine. Now let the cylinder be placed upon its non-conducting support, and the piston rise till the volume becomes O a. ​Since no heat can enter, the temperature will fall to t, and the pressure to A a. It is easy to show that, during this reverse cycle, an amount of mechanical energy represented by A B C D has been expended, and it is seen that heat equal to h has been taken from the refrigerator, and heat equal to H given to the source. The engine is, therefore, a perfectly reversible engine in the sense before defined, and it has already been seen that no other heat-engine of whatever construction, steam, gas, hot air, thermo-electric, or whatever it may be, working between the same temperatures, could develop more mechanical effect from the heat H taken from the source. In other words, any heat-engine working between the temperatures T and t, and taking from the source the amount H of heat, must transfer to the refrigerator an amount of heat at least equal to h, the amount given up by our reversible engine under the same conditions. It remains to be seen what relation this bears to the heat taken from the source.
Experiment proves that the lower the temperature the smaller is h, and it is evident that if the temperature of the refrigerator had been lower the isotherm A D would have been O X. The area A B C D would then be greater, and, since this represents the work done by the engine in one revolution, it is seen that this is greater the lower the temperature of the refrigerator. It appears, then, that the proportion of the heat taken from the source which can be converted into mechanical effect, is greater as the temperature of the refrigerator is lower, and the question arises, how low must this temperature be in order that the whole of the heat may be so converted. Perhaps the best way of approaching this question is by Sir W. Thomson's absolute scale of temperature. This may be defined as a scale upon which the temperatures of any two bodies are to each other as the heat, received is to the heat rejected by a reversible heat-engine using one of the bodies as a source and the other as a refrigerator. That is, if T and t are the temperatures upon the absolute scale of our source and refrigerator, T: t:: H: h, or T$-$t: T$=$H$-$h: H.
Let T be the temperature of boiling water, and t that of melting ice, and let T$-$t$=$180°, as in the Fahrenheit scale. From the properties of air we know that if it is used as the working substance of a reversible engine, with a source at the temperature of boiling water and a refrigerator at the temperature of melting ice, H$-$h : H :: 100 : 373 nearly. Hence
| 180 : T | :: | 100 : 373 |
| T | $=$ | 671·4 |
| and t | $=$ | 491·4. |
Any other temperatures may be easily determined. Suppose B C (Fig. 2) be the isotherm corresponding to the temperature of boiling water, A D that corresponding to that of melting ice, and m n an isotherm corresponding to some intermediate temperature, that marked 100° on the Fahrenheit scale, for instance, whose temperature t' upon ​the absolute scale we wish to determine. We have as above T$-$t : T :: H$-$h : H, and T$-$t' : T :: H$-$h' : h', if h' is the heat rejected at the temperature t'. Hence T$-$t : T$-$t' :: H$-$h : H$-$h'. But H$-$h is the heat converted into work by an engine working between the temperatures T and t, and is proportional to the area B C D A. Also H$-$h' is the heat converted into work by an engine working between the temperatures T and t', and is proportional to the area B C m n. Therefore T$-$t : T$-$t' :: area B C B A : area B C m n; or 180° : T$-$t' :: area B C B A : area B C m n.
Having the data for constructing the isothermal and adiabatic lines, the areas B C B A and B C m n can be computed, and hence t' determined. The divisions of an absolute scale so constructed are found to correspond very closely with the divisions of the air-thermometer, and to differ but little from the divisions of the Fahrenheit scale. We are led, then, to the conclusion that to convert all the energy of a given amount of heat into mechanical effect, a refrigerator at a temperature of 491 Fahrenheit degrees below the melting-point of ice, or 459° below zero Fahr., is necessary.
Let us recapitulate briefly the points of this argument.
1. It is impossible for any heat-engine, of whatever construction, to convert into mechanical effect a larger proportion of the heat derived from a given source than can be done under the same conditions by a reversible engine. This proposition can not be denied without involving a denial of two physical axioms which are founded upon the results of all past experience, viz.: That the perpetual motion is impossible, and that "it is impossible by means of inanimate material agency to derive mechanical effect from any portion of matter by cooling it below the temperature of the coldest of surrounding objects."
2. That all reversible engines, whatever the working substance, have the same efficiency; that is, taking from the source the same quantity of heat H, they will transfer to the refrigerator the same quantity h, and convert into mechanical effect the same quantity H$-$h. Hence whatever results are derived from a discussion of any one form of reversible engine will be true of all others.
3. If a scale of temperature be constructed such that the temperature of the source is to the temperature of the refrigerator of a reversible engine as the heat derived from the source is to the heat given to the refrigerator, the scale divisions will differ but little from the divisions of the scales in common use. The efficiency of an engine
will then T$-$tT $=$ 1 $-$ tT
4. Upon such a scale, if there are, as in the Fahrenheit scale, 180 between the freezing and the boiling points of water, the former point would be numbered 491·4 and the latter 671·4. The ​possible efficiency of an engine working between these two temperatures
would therefore be 671·4$-$491·4671·4, 27 per cent, nearly.
Heat-engines are often spoken of as very inefficient machines, because they transform into mechanical effect but a small proportion of the heat used. The inefficiency is not so much the fault of the machine as of the conditions under which it is worked. Consider the case of a condensing engine with a boiler pressure of 45 pounds and a vacuum of 25 inches of mercury. The temperature of the source is here about 294° and of the refrigerator 140° Fahr. The possible efficiency under these conditions is about 20 per cent., that is, a perfect engine working between those temperatures could give in mechanical effect no more than one fifth the energy of the heat. The best steam-engines would, under these circumstances, give one-horse power for something less than two pounds of coal per hour. This is an efficiency of 10 or 12 per cent., or more than half the possible efficiency. The engine, as a machine, is not so very imperfect. In speaking of the engine, I of course include the boiler as a part of the machine. Any great improvement must come from an increased range of temperature between the source and refrigerator. The temperature of the refrigerator can not well be lower than the general temperature of surrounding objects, and there are great practical difficulties in the way of a very high temperature of the source. Suppose an engine could be worked with a source at a temperature of 1250° of the absolute scale, or nearly 800° Fahr., and a refrigerator at 500° of the absolute scale, or nearly 40° Fahr., the possible efficiency would be 1$-$5001250, or only 60 per cent. It appears, then, that there is not much hope that any large percentage of the energy of heat can, by any practical means, be converted into mechanical effect. But are we, for this reason, to continue wasting the energy of fuel as it is wasted now? Is there no other way in which the energy of chemical separation of carbon from oxygen can be converted into mechanical effect except by first converting it into heat? Why may not the union of carbon with oxygen be made to generate electric currents instead of heat? Electric energies have been made that convert into mechanical effect 60 to 70 per cent, of the energy of the electric current, and a much higher efficiency might, no doubt, be obtained. Already something has been done toward the generation of electric currents by the union of carbon and oxygen; but, so far, no means has been discovered by which such a union can be effected, except at a high temperature, and this involves a great waste of energy in the form of heat. A discovery that would enable us to convert the energy of fuel into electric currents directly and completely would revolutionize, not only the methods of obtaining power, but the methods of obtaining light and distributing heat as well. I have shown elsewhere that, if a Brayton oil-engine is used to ​drive a dynamo-electric machine producing the electric light, more than twice as much light will be developed as would be obtained if the oil that runs the engine were burned in the ordinary coal-oil lamps. How much greater would be the economy if the energy of the oil could be converted directly into the energy of the electric current!
For warming buildings, the furnace would become an electric generator, from which wires, instead of pipes for steam or hot air, would lead to the rooms to be heated, when, by interposing a suitable resistance, the energy of the current would be converted into heat. The probability of being able to convert the energy due to combustion of fuel into electric instead of heat energy may be very small; but it is at least a possibility; that is, there is no known reason in the nature of things why it can not be done, while it is demonstrated that the whole of the energy of heat can not be converted into mechanical effect, except by means of a refrigerator at a temperature of nearly 500° below that at which water freezes—a temperature which has never yet been reached, and which it is impossible to obtain with our present surroundings, except by an expenditure of energy equal to that which would be gained.