Theory of Heat Engines
23. A heat-engines acts by taking in heat, converting a part of the heat received into mechanical energy, which appears as the work done by the engine, and rejecting the remainder, still in the form of heat. The theory of heat-engines comprises the study of the amount of work done, in its relation to the heat supplied and to the heat rejected. The theory is based on the two laws of thermodynamics, which may be stated here as follows:---
LAW 1. When mechanical energy is produced from heat, 1 thermal unit of heat goes out of existence for every 772 foot-pounds of work done; and, conversely, when heat is produced by the expenditure of mechanical energy, 1 thermal unit of heat comes into existence for every 772 foot-pounds of work spent.
The "thermal unit" is the heat required to raise the temperature of 1_ of water 1 degree Fahr. when at its temperature of maximum density. The equivalent quantity of work, 772 foot-pounds, was determined by experiments of Joule, and is called Joule’s equivalent. Later researches by Joule and others have indicated that this number is probably too small; should perhaps be as much as 774 foot-pounds. Joule’s original value is still generally used by engineers; and as enters into many published tables it may conveniently be adhered to until its accuracy is more definitely disproved. Since a definite number of foot-pounds are equivalent to 1 thermal unit, we may, if please, express quantities of work in thermal units, of quantities of heat in foot-pounds; the latter practice will frequently be found useful.
LAW 2. It is impossible of a self-acting machine, unaided by any external agency, to convey heat from one body to anther at a higher temperature.
This is the form in which the second law has been started by Clausius. Another statement of it, different in form but similar in effect, has been given by Thomson. Its force may not be immediately obvious, but it will be shown below that it introduces a most important limitation of the power which any engine has of converting heat into work. So far as the first law shows, there is nothing to prevent the whole heat taken in by the engine from changing into mechanical energy. In consequence of the second law, however, no heat-engine converts, of can convert, more than a small fraction of the heat supplied to it into work; a large part is necessarily rejected as heat. The ratio
Heat converted into work
Heat taken in by the engine
is a fraction always much less than unity. This ratio is called the efficiency of the engine considered as a heat-engine.
24. In every heat-engine there is a working substance which takes in and rejects heat, thereby suffering changes of form, of more commonly of volume, and does work by overcoming resistance to these changes of form or volume. The working substance may be gaseous, liquid, or solid. We can, for example, imagine a heat-engine in which the working substance Is a long metallic rod, arranged to act as the pawl of a ratchet-wheel with fine teeth. Let the rod be heated so that it elongates sufficiently to drive the wheel forward through the space of one tooth. Then let the rod be cooled (say by applying cold water), the wheel being meanwhile held from returning by a separate click of detent. The rod, on cooling, will retract so as to engage itself with the next succeeding tooth, which may then be driven forward by heating the rod again, and so on. To make it evident that such an engine would do work, we have only to oppose that the ratchet-wheel carries round with it a drum by which a weight is wound up. We have, then, a complete heat-engine, in which the working substance is a solid rod, which receives heat by being brought into contact with some source of heat at a comparatively high temperature, transforms a small part of this heat into work, and reject the remainder to what we may call a receiver of heat, at a comparatively low temperature. The greater part of the heat may be said simply to pass through the engine, from the source to the receiver, becoming degraded as regards temperature as it goes. We shall see presently that this is typical of the action of all heat-engines; when they are doing work, the heat which they reject is rejected at a temperature lower than that at which it is taken in. They convert some heat into work only by letting down a much larger quantity of heat from a high to a relatively low temperature. The action is analogous to that of a water-wheel, which does work by letting down water form a high to a lower level, but with these important differences that in the transfer which occurs in heat-engines an amount of heat disappears which is equivalent to the work done.
25. In almost all actual heat-engines the working substance is a fluid. In some it is air, in some a mixture of several gases. In the steam-engine the working fluid is a mixture (in varying proportions) of water and steam. With a fluid for working substance, work is doe by changes of volume only; its amount depends solely on the relation of pressure to volume during the change, and not at all on the form of the vessels in which the change takes place. Let a diagram be drawn (fig. 9) in which the relation of the intensity of pressure to the volume of any supposed working substance is graphically exhibited by the line ABC, where AM, CN are pressure and AP, CQ are volumes, then the work done by the substance in expanding from A to C is the area of the figure MABCN. And similarly if the substance be compressed from C back to its original volume is such a manner that the line CDA represents the relation of pressure and volume during compression, the work done upon the substance is the figure NCDAM. Taking the two operations together, we find that the substance has done a net amount of work equal to the area of the shaded figure ABCDA, or &Mac186;PdV. This is an example of and a generalization of the method of representing work which Watt introduced by invention of the indicator; the figure ABCDA may be called the indicator diagram of the supposed action.
26. Generally in heat-engines the working substance returns Cycle of periodical to the same state of temperature, pressure, volume, operation and physical condition. When this has occurred the substance is of work said to have passed through a complete cycle of operation. For in sub-example, in a condensing steam-engine, water taken from the hot-stance well is pumped into the boiler; it then passes into the cylinder as steam, passes thence into the condenser, and thence again into the hot-well; it completes the cycle by returning to the same condition at first. In other less obvious cases, as in that of the non-condensing steam-engine, a little consideration will show that the cycle is completed, not indeed by the same position of working substance being returned to the boiler, but by an equal quantity of water being fed to it, while the steam which has been discharged into the atmosphere cools to the temperature of the feed. In the theory of heat-engines it is of the first importance to consider (as was first done by Carot in 1824) the cycle of operation performed by the working substance as a complete whole. If we stop short of the completion of a cycle matters are complicated by the fact that the substance is in a state different from its initial state, and may therefore have changed its stock of internal energy. After a complete cycle, on the other hand, we know at once that, since the condition is the same, the internal energy of the substance is the same as at first, and therefore---
Heat taken in = work done + heat rejected.
27. It will serve our purpose best to approach the theory of heat-engines by considering, in the first instance, the action of an engine in which the working substance is any one of the so-called permanent gases, or a mixture of them, such as air. The word permanent, as applied to a gas, can now_ be understood only as meaning that the gas is liquefied with difficulty---either by the use of extremely low temperature or extremely high pressure or both. So long as gases are under conditions of pressure and temperature widely different from those which produce liquefaction, they conform very approximately to certain simple laws---laws which may be regarded as rigorously applicable to ideal substances called perfect gases. After starting these laws briefly we shall examine the efficiency of a heat-engine using a gas in a certain manner as working substance, and then show that the conclusion so derived has a general application to all heat-engines whatsoever. In this procedure there is no sacrifice o generality, and a part of the process is of independent service in the discussion of actual air-engines.
28. The laws of the permanent gases are the following:---
LAW 1 (Boyle). The volume of a given mass of gas varies inversely as the pressure, the temperature being kept constant.
Thus, if V be the volume of 1 _ of a gas in cubic feet, and P the pressure in pounds per square foot, so long as the temperature is unchanged---
P oc V__, or PV = constant.
For air the value of the constant is 26220 when the temperature is 32° F.
29. LAW 2 (Charles), under constant pressure equal volumes of different gases increase equally for the same increment of temperature. Also, if a gas be heated under constant pressure, equal increments of its volume correspond very nearly to equal intervals of temperature as determined by the scale of a mercury thermometer.
Thus, let us take, say, 493 cubic inches of hydrogen, also of oxygen, of air, &c., all at 32° F., and, keeping each at a constant pressure (not necessarily the same for all), heat all so that their temperature rises 1° F. We shall find that each has expanded by sensibly the same amount and now occupies 494 cubic inches. And further, if we heat any one through another 1° F. to 34° F., we shall find that its volume is now 495 cubic inches, and so on. Thus for any gas, kept at constant pressure, if the volume was it would be
493 at 32° F.,
492 at 31° F.,
:
461 at 0° F.,
:
And finally 0 at - 461° F.,
Provided the same law were hold at indefinitely low temperatures. This we may assume to be the case with a perfect gas, although any actual gas would change its physical state long before so low a temperature were reached.
This result may be concisely expressed by saying that if we reckon temperature, not from the ordinary zero but from a point 461° below zero of Fahrenheit’s scale, the volume of a given quantity of a gas, kept at constant pressure, is proportional to the temperature reckoned from the zero. Temperatures so reckoned are called absolute temperature, and the point - 461° F. is called the absolute zero of temperature. Denoting any temperature according scale by t, and the corresponding absolute temperature by r, we have
r = t + 461 on the Fahrenheit scale,
and r = t + 274 on the Centigrade scale.
Charles’s law shows that if temperature be measured by thermometers in which the expanding substance is air, hydrogen, oxygen, or any other permanent gas, and, if those intervals of temperature be called equal which correspond to equal amounts of expansion, then the indications of these thermometers always agree very closely with each other, and also agree, though less closely, with the indications of a mercury thermometer. We shall see later that the theory of heat-engines affords a means of forming a thermometric scale which is independent of the properties, as to expansion, of any substance, and that this scale coincides with the scale of a perfect gas thermometer, a fact which justifies the use of the term absolute, a applied to temperature measured by the expansion of a gas.
30. Combining laws1 and 2, we have, for a given mass of any gas,
PV = cr,
which c is constant depending on the specific density of the gas and on the units in which P and V are measured. In what follows we shall assume that P is measured in pounds per square foot that V is the volume of 1 _ in cubic feet, and that r is the absolute temperature in Fahrenheit degrees. In air, with these units,
PV = 53&Mac250;18r.
31. LAW 3 (Regnault). The specific heat constant pressure is constant for any gas.
By specific heat at constant pressure is meant the heat taken in by 1 _ of a substance when its temperature rises 1° F., while the pressure remains unchanged---the volume of course increasing. The law states that this quantity is the same for any one gas, no matter what be the temperature, or what the constant pressure, at which the process of heating takes place.
Another important quantity in the theory of heat-engines is the specific heat at constant volume, that is, the heat taken in by 1 _ of the substance when its temperature rises 1° F. while the volume remains unchanged---the pressure of course increasing. We shall denote specific heat at constant pressure by Kp and specific heat at constant volume Kv. Let 1 _ of a gas be heated at constant pressure P from temperature r, to temperature r2 (absolute). Let V, be the volume at r, and V2 the volume at r2. Heat is taken in, and external work is done by expansion of the gas, namely---
Heat taken in = Kp (r2-r1).
Work done = P (V2-V1) = c (r2-r1).
The difference between these quantities, or (kp-c)(r2-r1), is the amount by which the stock of internal energy possessed by the gas has increased during the process. We shall see presently that this gain of internal energy would have been the same had the gas passed in any other manner from r1 to r2.
33. LAW 4 (Joule). When a gas expands without doing external work and without taking in or giving out heat, its temperature does not change.
To prove this, Joule connected a vessel containing compressed gas with another vessel that was empty, by means of a pipe with a closed stop-cock. Both vessels were immersed in a tub of water and were allowed to assume a uniform temperature. Then the stop-cock was opened, the gas expanded without doing external work, and finally the temperature of the water in the tub was found to have undergone no change. The temperature of the gas was unaltered, and no heat had been taken in or given out by it.
34. Since the gas had neither gained nor lost heat, and done no work, its internal energy was the same at the end as at the beginning of the experiment. The pressure and volume had changed, but the temperature had not. We must therefore conclude that the internal energy of a given mass of a gas depends only on its temperature, and not upon its pressure of volume; in other words, a change of pressure and volume not associated with a change of temperature does not alter the internal energy. Hence in any change of temperature the change of internal energy is independent of the relation of pressure to volume throughout the operation. Now we have seen the above that the quantity
(Kp – c)(r2 – r1)
measures the gain of Internal energy when 1 _ of a gas has its temperature changed from r1 to r2 in one particular way, namely, at constant pressure. Hence this same quantity also measures the gain of internal energy when 1 _ of a gas has its temperature changed from r1 to r2 in any manner whatsoever.
35. Next consider the case of 1 _ of a gas heated form r1 to r2 constant volume. The heat taken in is
Kv (r2– r1).
Since no work is done, this is all gain of internal energy, and is therefore (§ 34) equal to
(Kp-c)(r2-r1).
Hence in any gas
Kp-c=Kv.
The ratio Kp/Kv will be denoted by y; obviously Kv=c/(y-1). The following table of Kp, kv, c, and y will be found useful in dealing with air and gas engines.
Table I. --- Properties of Gases.
Kp Kv c. y.
Dry air............................. Foot-lbs
183·4 Foot-lbs
130&Mac250;2 Foot-lbs
53&Mac250;2
1&Mac250;409
Oxygen........................... 167&Mac250;9 119&Mac250;3 48&Mac250;1 1&Mac250;402
Nitrogen......................... 188&Mac250;2 133&Mac250;4 54&Mac250;8 1&Mac250;411
Hydrogen...................... 2632 1864 768 1&Mac250;412
Carbonic oxide................ 189&Mac250;1 133&Mac250;4 55.7 1&Mac250;418
Carbonic acid.................. 167&Mac250;4 132.6 34&Mac250;8 1&Mac250;261
Olefiant gas..................... 311&Mac250;9 257&Mac250;7 54&Mac250;2 1&Mac250;213
Steam gas, or highly}
Super-heated steam....... 371 285&Mac250;5 85&Mac250;5 1&Mac250;30
36. We shall now learn to the consideration of diagrams like that of § 25, which exhibit the action of working substance by curves showing the relation of P to V during expansion. In most of the instances which occur in the theory of heat-engines such curves may be exactly or approximately represented by equations of the form.
PVn = constant,
where the index n has various numerical values. Let AB, fig. 10, be a curve of expansion (of any substance in which PVn is constant, from pressure P1 and volume V1 at A pressure P2 and volume V2 at B. We have, by assumption,
P1V1n = P2V2n.
The work done is
V2 PdV = P1V1n&Mac186; V2dV = p1v1n(v21-n-vd11-n) .
V1 v1 Vn 1 – n
This may also be written
P1V1(1 – r1-n) ,
n – 1
where r is the ratio V2/v1, which may be called the ratio of expansion.
Still another form of the above expression for the work done is
P1V1 – P2V2 .
n – 1
37. Applying this result to the case of an expanding gas, we have
Work done = c(r1-r2) (n-1).
The loss of internal energy during expansion is, by § 34,
Kv(r1-r2), or c(r1-r2) / (y-1), by § 35.
Suppose now that the mode of expansion is such that the loss of internal energy is equal to the external work done, then
C(r1-r2) = c(r1-r2) , or n = y,
n – 1 y – 1
and the law of expansion is
PVy = constant.
The same formula applies when a gas is being compressed in such a manner that the work spent upon the gas is equal to the gain of internal energy by the gas.
38. This mode of expansion (or compression) is termed adiabatic. It occurs when the working substance is neither gaining nor losing heat by conduction or radiation or internal chemical expansion. It would be realized if we had a substance expanding or being compressed, without chemical change, in a cylinder which (with the piston) was a perfect non-conductor of heat. In adiabatic expansion the external work is done entirely at the expense of the substance’s stock of internal energy. Hence in the adiabatic expansion of a gas the temperature falls, and in adiabatic compression it rises. To fine the change of temperature in a gas when expanded or compressed adiabatically we have only to combine the equations
P1V1y = P2V2y and P2V2 = r2,
P1V1 r2
and we find r2 = r1 (V1/V2)y-1 .
It is clear from the above that if, during expansion, n is less than y the fluid is taking in heat, and if n is greater than y the fluid is rejecting heat.
39. Another very important mode of expansion or compression is that called isothermal, in which the temperature of the working substance is kept constant during the process.
In the case of a gas the curve of isothermal expansion is a rectangular hyperbola, having the equation
PV = constant = cr.
When a gas expands (or is compressed) isothermally a t temperature r from V1 to V2 the work done by (or upon) it (per _) is
V2 PdV = P1&Mac186;V2dV= PVlog _ r ,
V1 V1 V
where is the ratio V2/V1 as before.1
During isothermal expansion or a compression a gas suffers no change of internal energy (by § 34, since r is constant). Hence during isothermal expansion a gas must take in an amount of heat just equal to the work and during isothermal compression it must reject an amount of heat just equal to the work spent upon it. The expression crlog_r consequently measures, not only the work done by or upon the gas, but also the heat taken in during isothermal expansion or given out during isothermal compression. In the diagram, fig.11, the line AB is an example of a curve of isothermal expansion for a perfect gas, called for brevity an isothermal line, while AC is an adiabatic line starting from the same point A.
We shall now consider the action of an ideal engine in which the working substance is a perfect gas, and is caused to pass through a cycle of changes each of which is either isothermal or adiabatic. The cycle to be described was first examined by Carnot, and is spoken of as Carnot’s cycle of operations. Imagine a cylinder and piston composed of a perfectly non-conducting material, except as regards the bottom of the cylinder, which is a conductor. Imagine also a hot body or indefinitely capacious source of heat A, kept always at a temperature r1, a perfectly non-conducting cover B, and a cold body or indefinitely capacious receiver of heat C, kept always at a temperature r2, which is lower than r1. It is supposed that A, B, or C can be applied to the bottom of the cylinder. Let the cylinder contain 1 _ of a perfect gas, at temperature r1, volume Va, and pressure Pa to begin with. The suffixes refer to the points on the indicator diagram, fig. 12.
(1) Apply A, and allow the piston to rise. The gas expands isothermally at r1, taking heat form A and doing work. The pressure changes to Pb and the volume to Vb.
(2) Removes A and apply B. Allow the piston to go on rising. The gas expands adiabatically, doing work at the expense of its internal energy, and the temperature falls. Let this go on until the temperature is r2. The pressure is then Pc, and the volume Vc.
(3) Remove B and apply C. Force the piston down. The gas is compressed isothermally at r2, since the smallest increase of temperature above r2, causes heat to pass into C. Work is spent upon the gas, and heat is rejected to C. Let this be continued until a certain point d (fig. 12) is reached, such that the fourth operation will complete the cycle.
(4) Remove C and apply B. Continue the compression, which is now adiabatic. The pressure and temperature rise, and if the point d has been properly chosen. When the pressure is restored to its original value Pa, the temperature will also have risen to its original value r1. Heat is now rejected to A, and the cycle is completed by the curve ba.
In this process the engine is not doing work; on contrary, work is spent upon it equal to the area of the diagram, or c(r1-r2)log_r. Heat is rejected to A in the fourth operation, to the amount cr1log_r. In the first and third operations there is no transfer of heat.
The action is now in every respect the reverse of what it was before. The same work is now spent upon the engine as was formerly done by it. The same amount of heat is now given to the hot body A as was formerly taken from it. The same amount of heat is now taken from the cold body C as was formerly given to it, as will be seen by the following scheme:—
Carnot’s Cycle, Direct.
Work done by the engine = c(r1-r2)log_r;
Heat taken from A = cr1log_r
Heat rejected to C = cr2)log_r
Carnot’s cycle, Reversed.
Work spent upon the engine = c(r1-r2)log_r ;
Heat rejected to A = cr1log_r ;
Heat taken to C = cr2log_r ;
The reversal of the work has been accompanied by an exact reversal of each of the transfers of heat.
43. An engine in which this is possible is called, from the thermodynamic point of view, a reversible engine. In other words, a reversible heat-engine is one which, if forced to trace out its indicator diagram reversed in direction, so that the work which would be done by the engine, when running direct, is actually spent upon it, will reject to the source to heat the same quantity of heat as, when running direct, it would take from the source, and will take form the receiver of heat the same quantity as, when running direct, it would reject to the receiver. By "the source of heat" is meant the hot body which acts as source and by "the receiver" is meant the cold body which acts as receiver, when the engine is running direct. Carnot’s engine is one example of a reversible engine. The engine of thermodynamic reversibility is highly important, for the reason that not heat-engine can be more efficient than a reversible engine, if both take in and reject heat at the same pair of temperatures.
44. To prove this, let it be supposed that we have tow engines M and N, of which N is reversible in the above sense, and that we have a hot body A capable of acting as a source of heat, and a cold body C capable of acting as a receiver of heat. The engine M is set to work as heat-engine, taking heat from A and rejecting heat to C. To prove that M cannot be more efficient than the reversible engine N, we shall assume that it s more efficient, and trace the consequences of that assumption.
Let M, working direct, be coupled so as to work N reversed; if we suppose that the engines are without mechanical friction, and can be coupled up without loss of power, the work represented by the indicator diagram of M is spent on N, and N will therefore reject to A an amount of heat which we will call QA and take from B an amount of heat which we will call QB. Now, since N is reversible, if it worked direct, taking QA from A, it would do the same amount of work as, in the supposed circumstances, is spent upon it. Hence, if M is more efficient than N it is taking form A an amount of heat less than QA, and consequently also is giving to B an amount of heat correspondingly less than QB. The joint effect, therefore, of M working direct and N working reversed is that the heat taken form A by M is less than the heat given to A By N, while the heat given to B by M is less(to an equal extent) than the heat taken from B by N. the consequence is that the hot body A is gaining heat on the whole, and the cold body B is losing an equal amount of heat; in other words, with the continued action of the double system heat passes, in indefinitely large quantity, from a cold body to a hot body, by means of an agency which, it is to be observe, is purely self-acting, for if we suppose there is no mechanical friction the system requires no help from without. Now, this result is, by the second law of thermodynamics (§ 23), contrary to all experience; and we are forced to conclude that the assumption that M is more efficient than the reversible engine N, when both take in and reject heat at the same two temperatures, is false. Hence, with given temperatures of source and receiver of heat no engine is more efficient than a reversible engine
Hence, let M and N both the reversible and both work between the same limits, but be different in any other respect. Then by the foregoing argument M cannot be more efficient than N, neither can N be more efficient than M. Hence all reversible heat-engines taking in and rejecting heat at the same temperature are equally efficient.
45. These results imply that reversibility, in the thermodynamic sense, is the criterion of what may be called perfection in a heat-engine. A reversible engine is perfect in the sense that it cannot be improved on as regards efficiency: no other regards efficiency: no other engine, taking in and rejecting heat at the same temperatures, will convert into work a greater fraction of the heat which it takes in. Moreover, if this criterion in satisfied, it is as regards efficiency a matter of complete indifference what is the nature of working substance, or what, in other respects, is the mode of the engine’s action.
46. Further, since all engines that are reversible are equally efficient, provided, they work between the same temperatures, an expression for the efficiency of one will apply equally to all. Now, the engine whose efficiency we have found in § 41 is one example of a reversible engine. Hence its efficiency
(r1-r2)/r1
Is the efficiency of any reversible heat-engine whatsoever taking in heat at r1 and rejecting heat at r2. And, as no engine can be more efficient than one that is reversible, this expression is the measure of perfect efficiency. We have thus arrived at the immensely important conclusion that no heat-engine can convert into work a greater fraction of the heat which it receives than is expressed by the excess of the temperature of reception above that of rejection divided by the absolute temperature or reception.
47. Briefly, recapitulated, the steps of the argument by which this result
48.
has been reached are as follows. After stating the experimental laws to which gases conform, we examined the action of a heat-engines in which the working substance took in heat when at the temperature of the source and rejected heat when at the temperature of the receiver, the change of temperature from one to the other of these limits being accomplished by adiabatic expansion and adiabatic compression. Taking a special case in which this engine had for its working substance a perfect gas, we found that its efficiency was (r1-r2)/r1(§41). We also observed that it was, in the thermodynamic sense, a reversible engine (§ 43). Then we found, by an application of the second law of thermodynamics, that no heat-engines can have a higher efficiency than a reversible engine, when taking in and giving out heat at the same two temperature r1 and r2; this was shown by the fact that a contrary assumption leads to a violation of the second law (§ 43). Hence, we concluded that all reversible heat-engines receiving and rejection heat at the same temperature r1 and r2 respectively are equally efficient, and hence that the efficiency (r1-r2)/r1,already determined for one particular reversible engine, measure the efficiency of any reversible engine, and is a limit of efficiency which no engine whatever can exceed
48. The second law of thermodynamics, on which (along with the first law) this conclusion rests, has been given in many different forms. The statement of it in § 23 is that of Clansius, and is very similar to that of Sir W. Thomson. Rankine, to whom with Thomson and Clansius is due the development of the theory of heat-engines from the point at which Carnot left it, has stated the second law in a form which is neither easy to understand, nor obvious, as an experimental result, when understood. His statement runs:— "If the absolute temperature of any uniformly hot substance be divided into any number of equal parts, the effects of those parts in causing work to be performed are equal. "1
To make this intelligibility we may suppose that any quantity q of heat from a source at temperature r1 is taken by the first of a series of perfect heat-engines, and that engine rejects heat at a temperature r2 less than r1 by a certain interval _r. Let the heat so rejected by the first engine the heat supply of a second perfect engine working from r2 to r3 through an equal interval _r; let the heat which it in turn rejects form the heat-supply of a third engine working again through an equal interval from r3 to r4; and so on. The efficiencies of the several engines are (by § 46) _r, _r, _r, &c. The amounts of heat supplied to
R1 r2 r3
them are q, qr2, qr3 , &c. Hence, the amount of work done by
r1 r2
each engine is the same, namely, q_r. Thus Rankine’s statement is to be
r1
understood as meaning that each of the equal intervals into which any range of temperature may be divided is equally effective in allowing work to be produced from heat when heat is made to pass, doing work in the most efficient possible way, through all the intervals from the top to the bottom of the range.
49. A point of much theoretical interest may be noted in passing. In place of measuring temperature, as we have done, by the expansion of a perfect gas, a scale of temperature might be formed thus. Starting from any one temperature, let a series of intervals be taken such that a series of reversible engines, each working with one of the intervals for its range, in the manner described in § 48 (so that the heat rejected by the first forms the supply of the second, and so on), will each do the same amount of work; then call these intervals equal. This gives a scale of temperature (originally suggested by Sir W. Thomson) which is truly absolute in the sense of being independent of the properties of any substance; it coincides, as is evident from § 48, with the scale we have been using, in which equal intervals of temperature are defined as those corresponding to equal amounts of expansion of a perfect gas under constant pressure; and it coincides approximately with the scale of a mercury thermometer when that is graduated to read from the absolute zero by the addition of a suitable constant. (§ 29).
50. the availability of heat for transformation into work depends essentially on the range of temperature through which the heat is let down from the hot source to the cold body into which heat is rejected; it is only in virtue of a difference of temperature between bodies that conversion of any part of their heat into work becomes possible. If r1 and r2 are the highest and lowest temperatures of the range through which a heat-engines works, it is clear that the maximum of efficiency can be reached only when the engine takes in all heat at r1 and rejects at r2 all that is rejected. With respect to every portion of heat taken in and rejected the greatest idea efficiency is
=Temperature of reception - temperature of rejection .
Temperature of rejection
Any heat taken in at a temperature below r1 of rejected at a temperature above r2 will have less availability for conversion into work than if taken in at r1 and rejected at r2, and hence, with a given pair of limiting temperatures, it is essential to maximum efficiency that no heat be taken in by engine except at the top of the range. Further, as we have seen in § 45, when the temperature at which heat is received and rejected are assigned, as engine attains the maximum of efficiency if it be reversible.
51. It is therefore important to inquire more particularly what kinds of action are reversible in the thermodynamic sense. A little consideration will show that a transfer of heat from the source or to the receiver is reversible only when the working substance is at sensibly the same temperature as the source of the receiver, as the case may be, and an expansion is reversible only when it occurs by the graduate displacement of some part of the containing envelope in such a manner that the expanding fluid does external work on the envelope, and does not waste energy to any sensible extent in setting itself in motion. This excludes what termed free expansion, such as that of the gas in Joule’s experiment, § 33, and it excludes also what may be called imperfectly-resisted expansion, such as would occur if the fluid were allowed to expand into a closed chamber in which the pressure was less than that of the fluid, or if the piston in a cylinder rose so fast as to cause, through the inertia of the expanding fluid, local variations of pressure throughout the cylinder.
To make a heat-engine, working within given limits of temperature, as efficient as possible we must therefore strive—(1) to take in no heat except at the highest temperature, and to reject no heat except at the lowest temperature; (2) to secure that the working substance shall, when receiving heat, be at the temperature of the body form which the heat occurs, and that it shall, when giving up heat, be at the temperature of the body of which heat is given up; (3) to avoid free of imperfectly-resisted expansion. If these conditions are fulfilled the engine is a perfect heat-engine.
The first and second of these conditions are satisfied if in the action of the engine the working substance changes its temperature from r1 to r2 by adiabatic expansion, and from r2 to r1 by adiabatic compression, thereby being enabled to take in and reject heat at the ends of the range without taking in or rejecting any by the way. This is the action in Carnot’s engine (§ 40).
52. But if were can cause the working substance to deposit heat in some body within the engine while passing from r1 to r2 in such a manner that the transfer of heat form the substance to this body is reversible (satisfying the second condition above), then when we wish the working substance to pass from r2 to r1 we may reverse this transfer and so recover the heat that was deposited in this body. This alternate storing and restoring of heat may then take the place of adiabatic expansion and compression, in causing the temperature of the working substance to pass from r1 to r2 and from r2 to r1 respectively. The alternate storing and restoring is an action occurring wholly within the engine, and is therefore distinct from the taking in and rejecting of heat by the engine.
53. In 1827 Robert Stirling designed an apparatus, called a regenerator, by which this process of alternate storing and restoring of heat could be actually performed. For the present purpose it will suffice to describe the regenerator as a passage through which the working fluid can travel in either direction, whose walls have a very large capacity for heat, so that the amount alternately given to or taken from them by the working fluid causes no more than an insensible rise or fall in their temperature. The temperature of the walls at one end of the passage is r1, and this tapers continuously down to r2 at the other end. When the working fluid at temperature r1 enters the hot end and passes through, it comes out at the cold end at temperature r2, having stored in the walls of the regenerator an amount of heat which it will pick up again when passed through in the opposite direction. During the return journey its temperature rises form r2 to r1. The process is strictly reversible, or rather would be so if the regenerator had an unlimited capacity for heat, if no conduction of heat took place along its walls from the hot to the cold end, and if no less took place by conduction or radiation from its external surface.
54. Using air as the working substance, and employing his regenerator, Stirling made an engine (to be described later) which, allowing for practical imperfections, is the earliest example of a truly reversible engine. The cycle of operation In Stirling’s engine is substantially this:—
(1) Air (which has been heated to r1 by passing through the regenerator) is allowed to expand isothermally through a ratio r, taking in heat in from a furnace and raising a piston. Heat taken in (per _ of air) = cr1log_r.
(2) The air is caused to pass through the regenerator from the hot to the cold end, depositing heat and having its temperature lowered to r2, without change of volume. Heat stored in regenerator = Kp(r1-r2). The pressure of course falls.
(3) The air is them compressed isothermally to its original volume at r2 in contact with a refrigerator (or receiver of heat). Heat rejected = cr2log _r.
(4) The air is again passed through the regenerator from the cold to the hot end, taking up heat and having its temperature raised to r1. Heat stored by the regenerator = Kp(r1-r2).
cr1log_r - cr2log_r = r1 – r2
Efficiency = cr1log_r r1
The indicator diagram of the action is shown in fig. 13, and a diagram of the engine is given in chap.XIV. Stirling’s engine is important, not as a present-day heat-engines (though it has recently been revived in small forms after a long interval of disuse), but because it is typical of the only mode, other than Carnot’s plan of adiabatic expansion and compression, by which the action of a heat-engine can be made reversible. Valuable as the regenerator has proved in metallurgy and other industrial process, its actual application to heat-engines has hitherto been very limited. Another way of using it in air-engines was designed by Ericsson, and attempts have been made by C. W. Siemens and F. Jenkin to apply is to steam-engines and to gas-engines and to gas-engines. But almost all actual engines, in so far as they can be said to approach the condition of reversibility, do so, not by the use of the regenerative principle, but by more or less nearly adiabatic expansion and compression after a manner of Carnot’s ideal engine.
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