ARE THERE BETTER WAYS TO UNDERSTAND THE SECOND LAW OF THERMODYNAMICS AND THE CARNOT EFFICIENCY OF HEAT ENGINES?

Transcription

1 ARE THERE BETTER WAYS TO UNDERSTAND THE SECOND LAW OF THERMODYNAMICS AND THE CARNOT EFFICIENCY OF HEAT ENGINES? W. John Dartnall 979 reised Abstract The Second Law of Thermodynamics imposes the Carnot upper limiting efficiency on heat engines. Traditional engineering thermodynamics tetbooks introduce both the Second Law of Thermodynamics and the Carnot efficiency in elegantly general ways that are not easily related to the practical and physical causes of the Carnot limitation. In this paper two theoretical models of heat engines are presented that bridge the conceptual gap between the elegantly abstract (and general, also traditional) Second Law approaches and physical/practical (also general) principles goerning the limiting efficiency of heat engines.. INTRODUCTION The second law of thermodynamics imposes the Carnot upper limiting efficiency on all heat engines. This efficiency is more stringent on "low temperature" heat engines such as flat plate solar thermal engines, where the source temperature is not far aboe the enironmental sink temperature than for internal combustion engines where the source temperature is related to the relatiely high air/fuel combustion temperature. Traditional engineering thermodynamics tetbooks usually introduce the Second Law of Thermodynamics in a manner that leaes the reader wondering why it really is that heat engines efficiency is limited to the Carnot efficiency. The tet books usually hae the following format: - the law is stated, Carnot efficiency for a hypothetical engine using an ideal gas is deried, it is then shown that any engine

2 haing greater than Carnot efficiency, coupled to the reersed Carnot Engine, would imply the absurdity of heat energy flowing from the lower temperature sink to the higher temperature source (Rogers and Mahew, 967). The Carnot Efficiency is therefore taken to be the ultimate. The Carnot cycle is found to be not the only cycle for which: ( ) Efficiency T T u T u Other cycles, such as the Stirling cycle and the Ericsson cycle are shown to hae Carnot efficiency. The aboe situation has raised many questions in my mind. Some of them are: -. Is it possible to construct either a mechanical of a mathematical model, which will demonstrate, practically and directly, the Second Law as it applies to heat engines?. Why is it that Carnot efficiency is independent of the fluid or its state? 3. Is it possible to generate an entire family of cycles, based on ideal gas as the working fluid, haing Carnot efficiency? 4. The harnessing of heat energy inoles mechanical deices (positie displacement and rotary epanders). Surely then, the entire phenomenon must be "mechanically" eplainable in terms of the mechanical properties and structure of matter. 5. What are the full implications of the Thermodynamic Temperature Scale? 6. Is there an easier way to eplain the meaning of entropy? 7. If this elusie (more fundamental) approach is found, will it relate more intelligibly to engine practicalities such as shape, size, structure, work ratio and the fundamentals of thermodynamic fluids? 8. Can some general theory be deised relating practical efficiency (say the best yet attained) to the ultimate efficiency with respect to source temperature with sink temperature fied at, say, 3 C?

3 In attempting to find answers to these questions, I hae considered a number of ideas and hae deeloped alternatie techniques for deriing ultimate heat engine efficiency based on practical and physically meaningful initial ideas and models. It is the purpose of this article to present these ideas and the techniques. These ideas hae been considered oer a period of seeral years and some of the preliminary ideas that now follow are well known. Howeer, they are included in order to gie a contet to my own noel ideas. First, I will present some preliminary ideas and concepts in section to follow. In sections 3 to 5 I outline two approaches that I hae deeloped in seeking a better understanding of the Carnot efficiency.. PRELIMINARY IDEAS AND CONCEPTS. Fundamental Problem of Heat Engines A heat engine is a deice that harnesses heat energy from a relatiely high temperature source by raising the pressure (and consequently, in some cases, the elocity) of a thermodynamic fluid. The fundamental problem seems to be to efficiently harness the energy due to random motion (i.e., kinetic energy) of the molecules of the thermodynamic fluid as it receie heat energy from the source. The thermodynamic fluid acts as a medium in transforming heat energy to mechanical work. If a mechanical engineer is proided with a system haing ordered kinetic energy (e.g., a rotating flywheel, a falling weight or a flowing liquid), he/she has no hesitation in saying that, in theory, he/she could harness % of the aailable kinetic energy. By means of a well-designed mechanical or hydraulic deice, he/she could couple the energy to an output deice. Howeer, when the kinetic energy is in particles (molecules), which are irtually infinite in number and moing simultaneously in different directions within a common region in space, the situation is different. The design would inole an infinite number of infinitesimally small connecting deices - a practical impossibility! 3

4 The seeming randomness of the motion and the quantity of particles in the system seems to present a problem for our understanding.. The Table Tennis Bat and Ball Analogy Consider a perfectly elastic bat, ball and table suitably aligned as depicted in Fig. Raising and lowering the bat will ary the frequency of the ibration of the ball. Howeer, a complete "cycle" (compression and epansion to starting point) will not produce any net work (taken from the bat). Introduction of eternal energy to the ball at some stage will raise the frequency but on completion of the current cycle, all following cycles will be identical to each other unless further eternal energy is introduced. Epansion Bat Compression Table surface Ball ibrating between bat and table surface. Figure : Ball ibrating between bat and table A little thought reeals that the only way to continuously (or with high frequency) apply energy to the ball and harness it at the bat is by applying it in such a manner that the kinetic energy of the ball is "on the whole" higher during epansion than during compression. This will necessitate the introduction of eternal energy during epansion and rejection of energy during a compression part of the cycle. 4

5 . The Concept of a Positie Displacement Heat Engine The diagrammatic model of a heat engine shown in Fig was a widely published model used in older tetbooks on heat engines (Lewitt, 946) A heat engine is taken to be a deice that may be modeled by the piston IC engine model in which heat energy is alternately supplied to and rejected from the molecules of a working gas in the cylinder space. In thermodynamic cycle analysis many well-known practical details such as friction and time influences are oerlooked. Thermodynamic Gas Heat Energy Source Piston Crank Connecting Rod Heat Energy Sink Cylinder Figure : Model of a piston type (positie displacement) heat engine 3. THE VIBRATING PARTICLE MODEL OF A HEAT ENGINE My idea was to remoe some of the compleity from the thermodynamic models by replacing the gas by a single "elastic" particle oscillating in only one dimension. If I could show that this system is able to generate (by analogous behaiour) the processes, cycles and importantly the Carnot limitation of thermodynamic cycle analyses, then I reasoned that the Carnot limitation would hae been shown to be independent of the Second Law of Thermodynamics. 5

6 I now show how this can be done. 3. The Model Consider a perfectly elastic particle of mass ibrating between two perfectly aligned, perfectly elastic surfaces, one of which is fied and the other adjustable as indicated below in Fig 3. s S F m Figure 3: Model of heat engine with thermodynamic fluid represented by a single ibrating particle S relatiely fied surface. S moeable surface (ideally zero mass). This surface represents a piston. Graitational, magnetic and electrostatic fields are ideally zero. Also air resistance is assumed to be zero. Now imagine that the process is initiated with a particle elocity o and with o. Round trip time for the particle: δ t (The absolute alue of is used to aoid the negatie resulting from the return trip. The bar oer the is used to indicate the aerage speed for the round trip.) Aerage absolute alue of the momentum of the particle for the round trip: 6

7 momentum m δ ( m) change in momentum per rebound m By differentiating momentum with respect to time we get: Mean force on S δ ( m ) δ t m m For conenience drop all signs and work only in absolute alues. Signs will be shown as + or - if necessary. The aboe may be written:- m F now define: r ; r Substituting gies ke F r Further, for any : m F r ke r... 5 Now let force F push S to the left by displacement -δ. Work done on the particle from initial position by the moing surface S : m δw Fδ ( δ) For a larger compression, say from o to : m W Fd d (the negatie sign indicating "compressie" work on the particle) 7

8 But W ke ke m m So we need to sole the equation: m d m m... 6 Try setting up for a small adjustment δ. δke m( + δ ) m ( δ δ ) m + m( δ ) m But δke δw F( δ) ( δ) let m( δ ) m d After cancellation this yields the differential equation: d d for which ln ln is the general solution and is equialent to the equation: c Try this in. c... 7 c ( ) m d m c c Integrating the left-hand side yields the right hand side: 8

9 mc d mc 3 So equation 7 can now be accepted. Now c c ( ) Also ke m m ( ) Or substituting r / : ( r ) r r ke m mr ( ) r r Note: Both elocity and kinetic energy tend to as r during an isentropic compression. 3. Model of the Stirling Cycle: Now consider, by analogy, a Stirling Cycle carried out on the particle.. By analogy, an isothermal process would correspond to constant ke based on the kinetic theory of gases.. Constant olume heating (by analogy) would be where r is held constant and ke is increased by introduction of eternal energy (say at surface S ) to increase. Vice ersa for constant olume cooling. 9

10 s (capable of echanging energy with the particle.) S Fc m Fe c e Figure 4: Model of heat engine applied to Stirling Cycle Commence the cycle at e with particle elocity l and compress from e to c. Work of compression (C) F c d e c m l d m l ln c e Increase particle elocity to u. By analogy, this would correspond to regeneration. Energy of regeneration (R) m ( ) u l Useful work of epansion (W) e u F e d d c m m u ln e c Energy of degeneration (D) m ( ) u l

11 The physical description and interpretation of the analogy is left to the reader. Cycle efficiency W C η s W m u ln m e c u ln m e c l ln e c After cancellation, u l η s... 8 u The aboe formula, according to the kinetic theory of gases* is analogous to the Carnot formula: η s T u T T u l *Note: The kinetic theory of gases gies the following relationship between the absolute temperature of a gas and the mean elocity of the particles: T m 3 k where: T absolute temperature of the gas m molecular (particle) mass k the Boltzmann constant the mean square elocity of the particles Carnot Cycle Similarly the Carnot cycle η may be deried.

12 3.3 Discussion of the aboe analysis of the ibrating particle model The aboe ibrating particle model is a purely mechanical model. From this model, the Carnot cycle efficiency has been deried. This indicates that the limitations of the second law of thermodynamics are not mysterious and that the Carnot cycle efficiency limitations are, in fact, simply caused by the mechanical-physical mode of transformation of energy from heat to work ia a pulsating olume of thermodynamic fluid. The need to employ a cycle and the consequent need to reject some of the heat energy receied during some parts of the cycle, during the "compression" part of the cycle, relates to the fact that the thermodynamic fluid is gas. Because this gas is contained under pressure it therefore reacts against the piston during compression and requires work of compression while it simultaneously rejects the work of compression to the sink so as to aoid an increase in its "internal energy". I hae not oerlooked the fact that the case of the single particle, aboe and een the more widely representatie case of and ideal gas (goerned by the kinetic theory) based heat engine cycles are not generally representatie of all heat engine cycles. They do not, for eample represent two phase cycles or solid state heat engine cycles. What I do wish to focus on is the mechanism that the ibrating particle model reeals and the fact that this is clearly the goerning mechanism for the ideal gas based heat engine and is therefore likely to be more general. Later, other cases may be inestigated to see if the other cases are also goerned by the same mechanism. Since the body of knowledge contained in classical thermodynamics already shows the generality of the Carnot limitation to all heat engines (whether two phase or solid state etc.) we already know that further inestigation will support our case. Unfortunately, howeer the classical thermodynamics does not readily reeal our mechanism in a way that is meaningful to an engine designer. 4. IDEAL GAS CYCLE MODEL ANALYSED ON A ln T - ln ν DIAGRAM

13 I now present the second of my two models. This is a mathematical model that is illustrated on a ln-ln diagram. 4. Background to the ln T - ln ν DIAGRAM After haing reached the conclusions of the ibrating particle model of the preious section of this report, I reasoned that the traditional ideal gas cycle model, well known in engineering tetbooks effectiely supports the same ideas. Howeer the traditional presentation in tetbooks does not hae a design approach to it. I reasoned that an engineering designer would deelop a method with the following lines of reasoning and with the following questions in mind. Heat engines are modeled with ideal cycles for ideal gases on diagrams. Cycles may be manipulated and analysed mathematically. "Manipulation" might include breaking down a cycle into elemental components and reconstructing and equialent cycle or mathematical transformation. What type of diagram would be the best for the manipulation process? Then, can a method of manipulation be found which will lead to the "best" cycle. This may be the one with the highest efficiency. Such a process would be an optimization process. Traditional tetbooks simply present and analyse a number of standard cycles: Carnot, Diesel, Stirling etc. They discuss the practicalities, adantages and disadantages of each of these cycles. This traditional tetbook approach lacks any suggestion of optimization processes. With the aboe thoughts in mind I now outline my preliminary ideas and assumptions and then present my deriation. 4. Preliminary ideas and assumptions I include with my preliminary ideas some of the basic laws and theories from the tetbooks. 3

14 4.. Processes and the First Law For a heat engine as in section., consider the application of the First Law to a number of elementary reersible changes in the system. If the cylinder is always pressurized then, work is done when the piston moes to the right, whilst work must be rejected (done on the system), in order to moe it to the left. Heat energy may be introduced concurrently whilst work is being done by the engine. Then three possibilities arise:- (i) The heat energy may eceed the work, in which case the internal energy of the fluid is increased. dqsup du incr + dw (e.g., constant pressure process) (ii) The heat energy may just equal the work done: dq sup dw (this is the isothermal process) (iii) The heat energy supplied may be less than the work done: dq sup dw du decr (in this case internal energy is being gien up to contribute to the work done by the engine.) Conersely, whilst the piston is doing work on the gas, heat energy may be rejected. Then, once again, there are three possible outcomes:- (i) Work done on the gas is less than heat energy rejected: dw dq rej du decr (ii) Work done on the gas just equals heat energy rejected: dw dq rej (iii) Work done on the gas is greater than the rejected heat energy: dw dq + rej du decr 4

15 Other special cases:- (i) Isentropic: du dw (ii) Constant olume: dq du 4.. Constant olume and isothermal processes These processes are special in that, in the first, no work crosses the boundary, whilst in the second, one hundred percent of the supplied heat energy crosses the boundary as work done. An interesting account of this process is gien in Pierce, (97) pp Regenerators Part of the gross work of a cycle may be stored as energy in a flywheel and later returned to the fluid. Similarly, heat energy may be stored in a regenerator and returned proided the temperature of regenerator and fluid are suitable to cause energy flow Capacitor Analogy The regenerator may be considered to be a thermal capacitor and the fluid itself acts as a thermal capacitor. Then an engine may be represented in a diagrammatic form as shown below. HEAT ENERGY INPUT HEAT ENGINE MECHANICAL FLYWHEEL MECHANICAL WORK OUTPUT THERMAL CAPACITORS REJECTED HEAT ENERGY Figure 5: Model of a heat engine haing both mechanical flywheel and thermal capacitors 5

16 4..5 Zeroth Law of Thermodynamics (Cardwell, D. S. L., 97, p 8; Reynolds, W. C., 9, p6) 4..6 Temperature (Cardwell, D. S. L., p 8; Reynolds, W. C., pp 6-66) 4..7 Internal Energy: (Pierce, 97, Ch 3) 4..8 Equilibrium: (Reynolds, pp 59,6) 4..9 Entropy - history of the discoery of: (Cardwell, pp 6-76) 4.. Open and Closed Cycles: (Mahew and Rogers, 967, pp 43,44) 4.. Kinetic Theory of Gases: (Lewitt, 946, pp 435 to 467; Pierce, pp3 to 36, Reynolds, pp 6 to 3) 4.. Pressure: Generally in a heat engine pressure results from the rebounding of molecules on the walls of the engine etc. In the liquid state or solid state, matter may eert pressure as a result of repulsie forces between molecules and rebounding of ibrating molecules (Pierce, pp 4 to 4). Ideal gas pressure may be diminished by attractie forces between atoms and molecules. It is eident, howeer that both pressure and temperature increase with internal energy. (Pierce pp 3 to 34, 3, 3) 5. THE TECHNIQUE FOR DERIVING ULTIMATE HEAT ENGINE EFFICIENCY. The technique is now presented. It commences by considering an ideal gas as proposed by the Kinetic Theory. The Zeroth and First Laws are accepted. 5. Assumptions 6

17 It is accepted that for an engine to perform non-triial work, the thermodynamic fluid must pass through a cycle in the accepted sense haing positie area on a P or T diagram and for conenience, with clockwise sense. All processes to follow are assumed to be reersible as it can be shown reersible processes only detract from efficiency. Mathematically, the Kinetic Theory supports the Characteristic Equation for Ideal Gases. Ro P M T... (a) or P RT...(b) Where:- From which:- P absolute pressure of the gas specific olume R o uniersal gas constant M mean relatie molecular mass R specific gas constant T absolute temperature R P T...() (i.e. pressure is proportional to temperature for a particular specific olume.) A further useful consequence of the Kinetic Theory is that C is independent of or that u (internal energy) depends only on T ie: u u + C T... (3a) o or u C T...(3b) Equations () and (3a and b) are now used along with the concepts at the beginning of this section to derie "ultimate efficiency". 7

18 Suppose we start the procedure with an arbitrary cycle on a T diagram as illustrated in Figure 6. TU T Cycle TL Figure 6: Arbitary cycle on a T- diagram The intention is to "manipulate the cycle mathematically" until it becomes the ultimate cycle(s). 5. Partitioning of the cycle To do this it is conenient to partition the cycle by ertical lines (constant olume lines) and to approimate the cycle by a chain of elementary reersible processes (e.r.p.'s) as show in Fig 7. Clearly the "limiting case" approaches the desired cycle in all properties, state functions and energy transfers. 8

19 Figure 7: The cycle (approimated by a series of e.r.p's) on a T- diagram 5.3 The use and definition of Elementary Reersible Processes (e.r.p's) Seeral e.r.p.'s are conenient to use and they are now discussed. It is proposed to construct the manipubatable cycle entirely from these e.r.ps. Figure 8.: e.r.p. for constant olume heating In this process all heat energy supplied is applied to the molecules resulting in increase in their kinetic energy. No eternal work is done. δq δu C δt Figure 8.: Working e.r.p. (constant temperature epansion process) This is the isothermal e.r.p. The aerage kinetic energy of the molecules is unchanged. 9

20 All heat energy supplied is conerted to work (which may be used at the "shaft" or temporarily stored as energy in the flywheel). δ Q dw Pd Figure 8.3: Working Isentropic e.r.p. (isentropic epansion process) In this case work done is proided only by degeneration of the kinetic energy of the molecules (i.e. no eternal heat energy supplied). For this case to occur it is eident that the molecular k.e. must hae been proided prior (in the cycle) to the eent and that:- δqprior to the eent CδT Pδ Figure 8.4: Degeneratie e.r.p. (constant olume) In this process molecular k.e. degenerates and no work is done. It would appear that this represents a total loss of heat energy. Howeer, assuming a perfect regenerator may be deised, all such energy may be stored and used for corresponding regeneratie e.r.p.'s. δq C δt deg Figure 8.5: Compressie e.r.p (isothermal compression) In order to repeat the cycle it is eidently necessary to reduce (recompress the gas) and by the first law this requires that:

21 δ Q δw rej rej Pδ During this e.r.p., stored mechanical energy is transferred to the gas and since the internal energy of the gas is inariant, the energy is immediately transferred out as rejected heat energy! Tentatiely we may consider this heat energy as recoerable, but later we see that it is not recoerable at T L. Figure 8.6: Compressie Isentropic e.r.p. Here the flywheel energy is used to regenerate molecular k e. No eternal heat energy is supplied or rejected. For conenience the e.r.p.'s are now labelled as follows: e.r.p Regeneratie Working Isentropic working Degeneratie Compressie Isentropic compression label R W WI D C CI 5.4 Partitioning and constructing of a cycle using e.r.p's Now consider particular strips of a cycle (Figure 9). The strips are each identified at a particular specific olume, and strip mean upper and lower temperatures, Tu and Tl.

22 Figure 9: Eamples of elemental strips in cycle(s) For any strip, stationed at a particular olume, let P u and P l respectiely be representatie upper and lower strip mean pressures. It is eident that no matter how it is partitioned, for the cycle to repeatedly produce work the following must be true: () The mechanism driing the engine is centered on the fact that P u > P l. This is because during epansion the gas does greater work than that required for re-compressing it. The gas is the link between heat energy and work; and δw Pδ, for a gien δ, increases with P. () At the upper end of any elemental strip, heat energy, δq, must be eternally proided for each W ( P ) from Q( P ) δ uδ. In the isothermal e.r.p., the energy for doing the work comes δ uδ. In other cases, such as the isentropic e.r.p. (WI), the energy is not proided concurrently with the work done and may be considered to hae been proided by an earlier R. (3) Work must be rejected at the lower end on each strip. δw rej Pδ l Figure below shows a typical cycle made up of such elementary processes:-

23 Figure : Eamples of elementary reersible processes approimating an engine cycle For this cycle the following may be obsered to be alid: Eery W requires δq δw Eery WI requires δq δw from one or more R's matching the fall in T during the WI At best eery R should be proided by a D; ecept when R proides WI. Note: Since δq C δt is inariant with we can match D's with R's. Hence for eery strip, at best:- η strip net. work. done minimum.heat.supplied Puδ Plδ Pδ u Pu Pl P u...(4) Equations () and (4) 3

24 T T T η strip u l u...(5) If a source at T U is aailable and a sink at T L then:- TU T L maimum η strip T U...(6) Considering the entire cycle, an obious solution for which η cycle is maimised is that cycle for which all strips hae T u T U and T l T L, i.e. the Stirling cycle. Figure : The Stirling cycle as a cycle of ultimate efficiency The Stirling cycle may therefore be considered to be the most fundamental solution to the question of which cycle(s) hae ultimate efficiency. At first it may appear to be the only solution to the problem, as introduction of strips where T u <T U or T l >T L would seem to detract from efficiency. Note that for the Stirling cycle all D must (in theory) be used for R. 5.5 Transforming to a ln T - ln chart to coneniently find ALL cycles of Carnot efficiency For a further study to look for other cycles haing ultimate efficiency it becomes conenient to transform the problem to a ln T - ln chart. 4

25 Figure : Transformation of the Stirling cycle onto a lnt-ln diagram For which:- δq δw RTδ ln isothermal isothermal ( ln ln ) Q W RT isothermal isothermal ( ) Q. C T T const ol c compressed olume e epanded olume *A little thought reeals that any diagram on this chart for which the left side "copies" the right side and which contains isothermals at T U and T L will hae Carnot efficiency. For eample, see Fig 3 below 5

26 Figure 3: Other cycles haing Carnot efficiency For which intermediate C's may in theory be proided by flywheel energy stored from matching intermediate W's thereby eliminating the apparent need for eternal compression energy at the C's. In a similar manner heat the matching C's may (in theory) proide energy for the matching W's. 6. THE COMMON CYCLES AND SOME COMMENTS By consideration of a number of such eamples and by etension to any number of matching steps (een approaching an infinite number) one may arrie at the statement *. 6. The chart below summarizes the most common cycles haing η carnot. Figure 4: Common cycles haing Carnot efficiency 6

27 In the chart the three cycles are contained between T U & T L and e & c. 6. Stirling cycle has maimum work/cycle W cycle Rln E C ( T T ) U ( ) Q C T T regen U L L 6.3 Carnot cycle has zero regeneration. The molecular k.e. is raised by compression and reduced by epansion. Hence a low work ratio. 6.4 Charts as aboe may be of considerable practical alue as straight lines represent all fundamental processes. The charts are easily constructed. Quantities are easily read and sealed off. See the appendi Figure 5: Common processes become linear on a ln T- ln diagram 6.5 Etension of ln T - ln charts to real gases and thermodynamic fluids including liquid and apour phases may be profitable. Most lines are close to being linear. For eample: 7

28 Figure 5: Typical lines for real fluids on a lnt -ln diagram 6.6 Three-dimensional charts could be useful. Construction of P and u surfaces to the bases ln T s ln yields substantially flat surfaces compared to the traditional cured surfaces. These surfaces would seem to be of considerable aid in understanding the arious fluids and cycles. 7. FURTHER WORK TO BE CARRIED OUT: So far this paper has only considered ideal gases and it in no way eplains ultimate efficiency for engines using real fluids. Figure 5 suggests a breakdown in the reasoning, as a constant temperature process does not infer constant internal energy during the apour phase. Also during the apour phase T is not proportional to P when is fied. Howeer, in general terms it is eident that T and P increase together at constant and hence, if a real cycle is partitioned into strips, greater work potential will eist at the upper end of the strips than the lower end. Eidently higher temperature is caused by higher molecular k.e. and it follows that the higher molecular k.e. Will produce higher molecular momentum and hence higher pressure which means greater δw for a gien δ. The question of regeneration is not easily answerable and the simple relationships for an ideal gas do not hold for either work potential or regeneration. 8

29 Neertheless, it is suspected that the model may be modified to include real fluids and that some general statement(s) based on the energy relationships of molecules and liquids inoling attractie and repulsie molecular forces and the structure of matter will show that real substances can only at best produce the efficiency shown for an ideal gas. It is hoped that etension of the theory will produce results that greatly aid the understanding with regard to real cycles. 8. CONCLUSIONS 8. This paper has taken a fresh look at the kinetic theory model of an ideal gas by inestigating some consequences of modeling a gas by a single "spherical-elastic" particle. It has then applied this model to the simple well-known source-sink model of a heat engine and shown that this model produces the Carnot limiting efficiency. Thus the Carnot efficiency is demonstrated with only two assumptions: use of a Newtonian-mechanical model and that absolute temperature is related directly to the kinetic energy of the particle. The compleity of modeling the gas by using the mean properties of an immense number of molecules in a three dimensional container is shown to be not necessary to obtain the Carnot limitation. 8. The least benefit of 8. is that it proides an educationally beneficial way of iewing the Carnot limitation, particularly for engineers. 8.3 A possible greater benefit of 8. is that it may hae proed the Carnot limitation from a simple Newtonian-mechanical model (the Kinetic Theory of Gases). In which case, since the Carnot limitation is a corollary of the Second Law of Thermodynamics it will hae reduced the Second Law to a consequence of the Kinetic Theory of Gases without the need to introduce the statistically deried concept of entropy. This would be a startling discoery! 8.4 Haing achieed 8. to 8.3, I then obsere that the implications of the single particle model are effectiely etended to the entire Kinetic Theory of Gases with all its refinements. 8.5 I then consider heat engine cycle diagrams for ideal gases. The mathematical relationships producing these cycle diagrams are supported by the Kinetic Theory. 9

30 I construct a heat engine cycle from e. r. p's (infinitesimally small reersible processes) and mathematically manipulate this cycle to find all cycles haing Carnot efficiency. For conenience, a ln T-ln diagram is used and I show that on such a diagram, all these cycles hae upper and lower isothermals of equal length which are connected by a left and a right process or series of processes that are geometrically identical. 8.6 Finally, I now point out that the central cause of the Carnot limitation is that during a heat engine cycle the gas must be epanded and recompressed or ice ersa. To any incremental epansion there must be a corresponding incremental recompression in the same olume range. If one eamines such an incremental pair it is eident that for an ideal gas, the isothermal work output during the incremental epansion is proportional to the absolute temperature as is the isothermal energy rejected during the corresponding incremental compression. This is simply because of the relationship between absolute temperature and mean particle kinetic energy*, giing rise to proportionately greater work output than energy rejection. Clearly some energy rejection must occur unless the sink temperature is absolute zero. *as well as particle elocity and consequences such as frequency of rebound of particles on the work boundary giing rise to proportionately higher pressure. References: Bloch, Eugène, 94, The Kinetic Theory of Gases, Methuen and Co. Ltd. London Cardwell, D. S. L., 97, From Watt to Clausius - The rise of thermodynamics in the early industrial age, Heinemann London, ISBN: Rogers, G.F.C. & Mayhew, Y.R., (967), Engineering Thermodynamics - Work and Heat Transfer. Second Edition, (Longman) Lewitt, E.H., (946), Thermodynamics Applied to Heat Engines, Sith Edition, (Longman) Pierce, James B., (97), The Chemistry of Matter, (Houghton Mifflin Co., Boston) 3

31 Reynolds, William C., (9), Thermodynamics. Second Edition, (McGraw Hill) Resnick, Robert, & Halliday Daid, (966), Physics, (Wiley) Masterton, W. L., & Sloweinski, E.J., (973), Chemical Principles, (Saunders) 3