A Meeting between Robert Stirling and Sadi Carnot in 1824

Israel Urieli – presented at the 2014 International Stirling Engine Conference (Updated 10/27/2018)

Introduction

This meeting never occurred, however the timing is feasible. In 1824 Sadi Carnot published his famous theoretical treatise "Réflexions sur la Puissance Motrice du Feu"1 ( Reflections on the Motive Power of Fire). This was 8 years after Robert Stirling had published his famous patent on the Stirling engine2. Consider the possibility that Stirling was intrigued with Carnot's treatise and invited him to Glasgow, Scotland, mainly in order to obtain a theoretical validation of the importance of the regenerator in the Stirling engine, which no one seemed to understand. I believe that had this meeting occurred it could have resulted in a significant difference both in the future of Stirling engine development and acceptance, and in the teaching of thermodynamics.

Consider for example the famous Schmidt Analysis3 done in 1871, 55 years after Stirling's patent. There is no mention of a regenerator in the analysis, and apparently the Lehman machine on which it was based did not have a regenerator.

Furthermore, in all the thermodynamics textbooks that I am aware of, the main example of an ideal heat engine described is the so-called "Carnot cycle" engine, which is a completely impractical machine, even though the ideal Stirling engine with a perfect regenerator has the identical ideal thermal efficiency.

The Fictitious Discussion

Stirling: I found your "reflections" somewhat difficult to follow. You compare the motive power of heat to that of a waterfall - each having a maximum that we cannot exceed. You state that the motive power of the waterfall depends on its height and quantity of the liquid, and that of heat depends on the quantity of caloric used, and also on the difference of temperature between the warm and cold bodies between which the exchange of caloric is made. I am very confused - what heat engine are you referring to, and what is the value of this maximum motive power that it can produce?

Carnot: My "reflections" are purely a theoretical abstraction. There are (and will be in the future) a number of different practical heat engines in the world - the Steam engine, your Stirling engine, in about fifty years time Nikolaus Otto will develop the Otto cycle engine, and in seventy years Rudolph Diesel will develop the Diesel engine. All of these machines have the same purpose - to convert heat energy into mechanical energy. The purpose of my "reflections" is to determine what are the basic requirements for a heat engine to produce mechanical energy, what defines an ideal heat engine and what is the maximum thermal efficiency that one can obtain from this ideal machine.

Since this is a fictitious meeting I am taking the liberty of basing my explanations on the science of Thermodynamics which will be developed in the future beginning with James Joule who will present his research on The Mechanical Equivalent of Heat in around 21 years, and finally rejecting the notion of heat being a flow of caloric.

To begin, we define the thermal efficiency of a heat engine by ηth = (W/QH), where W is the net work out and QH is the heat supplied by the fire. Based on my "reflections" we can determine that the maximum thermal efficiency of a heat engine is given by ηth = (1- TL/TH), where TL and TH are the respective lower and upper absolute temperatures.

Stirling: I am still confused. I understand that because of mechanical friction a practical heat engine has a reduced efficiency, however surely the ideal theoretical heat engine can convert all the heat from the fire into mechanical energy and thus can have a 100% thermal efficiency?

Carnot: First of all I need to explain the abstract concept of a heat engine as shown in the following diagram:

 

Figure 1 - An Abstract Concept of a Heat Engine 8

Note that there must be two temperature reservoirs TH and TL, with TH > TL. Heat QH is extracted from the high temperature source TH, part of that heat is converted into work W done on the surroundings, and the rest of the heat QL is rejected to the low temperature sink TL.

Stirling: I understand the heat source at temperature TH, that is the heat from the fire. However none of my heat engines have a heat sink at a temperature TL. Instead, I have a "heat economiser" (in future it will be referred to as a regenerator). It is simply a bundle of wires which absorbs the heat that is being rejected by the air. This heat is then transferred back to the air and reused in a different part of the cycle. Please explain to me the concept of a heat sink.

Carnot: Je ne comprend pas- I don't understand the concept of a heat economiser (regenerator?) - you will have to explain it to me later. First of all, my favorite explanation of the heat sink requirement is the explanation that will be presented by William Thomson (aka Lord Kelvin). You don't know him since he was only born this year (1824) in Ireland, however in about 25 years time he will be a professor at the University of Glasgow, and will use one of your engines in his lectures. He will explain the need for two temperature reservoirs by the following statement that in the future will be referred to as the Kelvin-Planck Statement of the Second Law of Thermodynamics:

It is impossible to construct a device that operates on a cycle and produces no other effect than the transfer of heat from a single body in order to produce work.

Note that this statement cannot be proven, however it has never been found to be violated. I prefer to present a less formal description of this statement in terms of the impossibility of a boat extracting heat from the ocean in order to produce its required propulsion work, as shown in Figure 2 below.
In the case of your engines, I believe that the flame acts as the high temperature heat source TH and the piston cylinder acts as the low temperature heat sink TL - you would probably get better performance if you water-cooled the piston cylinder.

 

Figure 2 - A simple example of the Second Law regarding heat engines8

Stirling: Very interesting - I have noticed on a few occasions that after running my engine for a few hours I needed to stop it and let it cool down. Water cooling will certainly be a feature in my engines from now on4. Now explain to me what is your definition of an ideal heat engine.

Carnot: Trés bien - since you accept the Kelvin-Planck statement based on your own experience I will continue. Notice that this is a negative statement in that it only implies what is impossible to achieve. In order to determine the maximum performance available from an ideal heat engine we need to introduce the concept of Reversibility, including both mechanical and thermal reversibility.
I will attempt to clarify these concepts in terms of the following example of a reversible piston cylinder device in thermal equilibrium with the surroundings at temperature T0, and undergoing a cyclic compression/expansion process.

 

Figure 3 - Mechanical and Thermal Reversibility8

For mechanical reversibility we assume that the process is frictionless, however we also require that the process be a quasi-equilibrium one. In Figure 3 we notice that during compression the gas particles closest to the piston will be at a higher pressure than those farther away, thus the piston will be doing more compression work than it would do if we had waited for equilibrium conditions to occur after each incremental step. Similarly, thermal reversibility requires that all heat transfer processes are isothermal. Thus if there is an incremental rise in temperature due to compression then we need to wait until thermal equilibrium is established. During expansion the incremental fall in temperature will result in heat being transferred from the surroundings to the system until equilibrium is established.
In summary, there are three conditions required for reversible operation:
* All mechanical processes are frictionless.
* At each incremental step in the process thermal and pressure equilibrium conditions are established.
* All heat transfer processes are isothermal.
So now you can see why I keep stating that this is a theoretical exercise - all that I am trying to show are the conditions required for a heat engine to obtain its highest theoretical thermal efficiency as well as to evaluate that efficiency value.

We now consider the analysis of heat engines. Since work done is evaluated by integration: W = ∫PdV (P = pressure, V = volume), it is customary to describe a heat engine in terms of a PV diagram. Thus the net work done per cycle of a specific engine is indicated by the area enclosed in the PV diagram. In Figure 4 I have chosen a specific example of a simple heat engine using atmospheric air, which is defined by the four processes shown: two isothermal processes (1)→(2) and (3)→(4), and two constant volume processes (2)→(3) and (4)→(1).

 

Figure 4 - An ideal heat engine having 2 isothermal and 2 constant volume processes

Stirling: Excuse me for interrupting, however there is something I don't understand. Why have you included the skinny cycle (1)→(2c)→(3)→(4c)→(1) in the PV diagram?

Carnot: The "skinny" cycle represents the so-called "Carnot" heat engine that I will describe later. Unlike all future Thermodynamics textbooks that I am aware of, I have drawn the PV diagram to scale, since I wanted to enable a meaningful comparison of two different heat engines, both completely impractical machines since the external environment needs to be changed between a constant low temperature TL and a constant high temperature TH every cycle. The cycle (1)→(2)→(3)→(4)→(1) presented in Figure 4 represents the maximum work output available under the specific volume ratio and temperature range given, mais malheureusement...

Stirling: I must interrupt you at this point. The cycle (1)→(2)→(3)→(4)→(1) in the above PV diagram definitely can be obtained from a practical heat engine, in fact this is exactly how an ideal form of my Stirling engine operates. Rather than changing the environment from hot to cold every cycle (and then back again to hot) I use a displacer piston to move the gas at constant volume between two separate spaces - the hot space TH and the cold space TL resulting in an identical PV diagram with two isothermal processes (1)→(2) and (3)→(4), and two constant volume processes (2)→(3) and (4)→(1) as shown in Figure 5 below.

 

Figure 5 - An ideal Stirling engine having 2 isothermal and 2 constant volume processes

Carnot: Incroyable! I am truly impressed! I must point out however that the ideal Stirling engine will have a much lower thermal efficiency than the ideal "Carnot" heat engine that I want to describe next, since the heat supplied externally during the constant volume process (2)→(3) is usually significantly higher than that supplied during the isothermal expansion process (3)→(4). This is not obvious on the PV diagram, and Rudolf Clausius is only 2 years old (born 1822 as Rudolf Gottlieb), hence we will need to wait until 1865 (41 years) before he introduces the concept of entropy (Clausius coined this from the Greek word τροπε meaning "transformation" and purposely wanted it to rhyme with energy). Meanwhile we will only be able to obtain the maximum thermal efficiency given by ηth = (1- TL/TH) for a reversible heat engine in which all the heat transfer processes occur isothermally, such as in the "Carnot cycle" engine shown as the "skinny" cycle (1)→(2c)→(3)→(4c)→(1) in Figure 6 below.

 

Figure 6 - The Carnot heat engine having 2 isothermal and 2 adiabatic processes

Stirling: All these new terms - "entropy" and "adiabatic" are all very confusing. I have never heard of the concept "adiabatic" which is obviously the vital key to a reversible heat transfer process - please explain it to me.

Carnot: Je regrette - I forgot that according to the Shorter Oxford English Dictionary the word "adiabatic" will only be introduced into the English language in 1875. It comes from the Greek word αδιαβατοσ meaning "incapable of being crossed", referring to a process in which there is no heat transfer between the system and the surroundings.
So referring again to Figure 6 we see that the so-called "Carnot" heat engine is defined by two isothermal processes (1)→(2c) and (3)→(4c), and two adiabatic processes (2c)→(3) and (4c)→(1). Unfortunately we see that there is very little work done per cycle and the environment needs to be changed four times per cycle: isothermal at TL, adiabatic, isothermal at TH, and adiabatic - a completely impractical machine, however the only one that can perform at the maximum thermal efficiency. C'est la vie,,,

Stirling: I must contradict you. If you had read my patent of 1816 you would have noticed the significant importance that I place on the heat economiser (or regenerator matrix - a bundle of wires wrapped around the displacer). As you indicated earlier this is a fictitious meeting, so I will also take the liberty of explaining this using the future science of Thermodynamics. Let us go back again to Figure 5. Since the amount of heat transferred during the constant volume displacement process is dependent only on the temperature difference, the heat absorbed by the regenerator matrix during the displacement process (4)→(1) is re-absorbed by the gas during the displacement process (2)→(3). This is not a simple process since as you mentioned previously a significant amount of heat is transferred, requiring an infinite area of contact between the gas and the wires. Since this heat transfer takes place exclusively between the wires and the gas, as far as the outside world is concerned these two processes are externally adiabatic. Let me develop the performance equations for the ideal Stirling engine - luckily I will not require the concept of entropy (which I still do not understand) for this engine since I will be able to derive all the relations needed based only on the energy equation (First Law of Thermodynamics). To keep things simple we will use specific quantities throughout. We define the system as the constant mass of gas (m[kg] - in our case air) enclosed by the piston cylinder representing the heat engine.

Carnot: Mon Dieu! We must make sure that this important analysis will be presented in the 2014 ISEC7. I pray that in 200 years time the thermodynamics community will never again associate my theory with the ridiculous Ideal Adiabatic (so-called "Carnot") heat engine. Finally I have understood the importance of the regenerator and the ideal Stirling Engine. I am amazed that this important result can be derived based on the First Law (energy equation) alone without any consideration or mention of the Second Law. This should have a profound effect on the future teaching of thermodynamics.

Conclusions

Concerning the teaching of thermodynamics, a typical basic thermodynamics course (presented to all engineering and science majors) starts off with energy and the First Law, and ends with the Second Law and ideal heat engines and heat pumps. Using the ideal Stirling cycle machine as the prime example of both laws will be a significant boon to the understanding and appreciation of these machines and in particular the important role of the regenerator. The only attempt to introduce this approach that I am aware of is presented in the web textbook Engineering Thermodynamics - a Graphical Approach. The analysis that was presented above is introduced and developed in this web resource immediately after the introduction of the First Law in Chapter 3b.

Concerning the development and acceptance of the Stirling engine in practical applications, we need to consider its unique characteristics:

1. External heat source - this enables any type of heat source, including low temperature sources (refer for example to the Cool Energy website), however it requires extremely effective heat transfer to the working gas leading to both physical and theoretical complexities. Physically effective heat transfer requires an extremely large surface area and a working gas with a high heat transfer coefficient and minimal flow friction. In order of preference the working gasses of choice are hydrogen, helium and air. The oscillating closed cycle leads to nonsteady flow convection heat transfer processes which are extremely difficult to analyze and can only be done by sophisticated computer analysis (refer for example to the web resource Stirling Cycle Machine Analysis).

2. Regenerator - this vital component was ignored for more than 100 years, resulting in the development of engines having a significantly lower thermal efficiency than what could have been obtained6. We can only surmise the possible outcome had this not have been the case.

3. Mechanical complexity - the engine presented in Stirling's patent diagram has a complex linkage system in order to provide the correct phasing between the displacer and the piston (as indicated in the website by Robert Sier). Furthermore, since the output power of the engine is proportional to the mean pressure, added mechanical complexity is required to allow variations in pressure or piston displacement in engines that require variations in power, such as for vehicle operation (refer for example to the Ford Philips 4-215 engine which was developed in the 1970's5. Very high pressure hydrogen was used in this engine, resulting in complex sealing as well as hydrogen embrittlement problems). Various other mechanical systems were devised over the years, however the major breakthrough occurred in the 1960's, when William Beale invented the Free-Piston Stirling Engine, eliminating entirely the linkage system, and in 1974 he formed the company Sunpower. Since 1974 Sunpower has developed Free-Piston Stirling Engine/Generators ranging in power levels from 35We to 7.5kWe. Electrical output power is easily produced by simply adding permanent magnets to the power piston, and coils of copper wire to the cylinder. With no output shaft one can easily use any working gas (such as helium) at extremely high pressures. The 1kW engine/generator is currently manufactured by Microgen Engine Corporation (refer to their History and Engine web pages). The only alternative to the gasoline powered vehicle that I can envision is an electric vehicle with a Free Piston Stirling engine/generator used when needed for keeping the batteries charged.

4. Reversible operation - the Free Piston Stirling Engine can easily be operated in reverse by driving the power piston electrically. This results in cooling, and if helium is used as the working gas this will enable reaching cryogenic temperatures. (Refer for example to the Sunpower and the Stirling Ultracold websites).

Finally, my hope is that all future Thermodynamics textbooks will use the Ideal Stirling Cycle engine as the prime example of both the First and Second Laws of Thermodynamics, and that the free piston Stirling engine/generator will become a vital component of electric vehicles as well as of distributed flat plate solar powered home generators.

Bibliography

1. Carnot, S.N.L. (1824). Réflexions sur la Puissance Motrice du Feu et sur les machines propres à développer cette puissance. Bachelier (1824).
Translated and Edited by Thurston, R.H. and published by Macmillan and Co. (1890) and reprinted in Am. Soc. Mech. Engrs, (1943). Published with an Introduction, additional footnotes and new Appendix by Mendoza, E. (1960, 1988). Sadi Carnot- Reflections on the Motive Power of Fire, Dover Publications, Inc.

2. Sier, R. (1995). Rev Robert Stirling D.D., Inventor of the Heat Economiser and Stirling Cycle Engine, published by L A Mair, Chelmsford.

3. Schmidt, G. (1871). Theorie der Lehmann'schen calorischen Maschine. Zeit. des Vereins deutscher Ing., 15 part 1, 1-12 (January). English translation presented in a condensed form in Appendix A of Urieli, I., Berchowitz, D.M. (1984). Stirling Cycle Engine Analysis, Adam Hilger, Bristol.

4. Finkelstein, T., Organ, A. (2001). Air Engines, ASME Press. Chapter 2 - The Stirling Engine, page 26.

5. van Giesel, R., Reinink, F. (1977). Design of the 4-215 D. A. Automotive Stirling Engine. Society of Automotive Engineers SAE 770082.
See also: Urieli, I., Berchowitz, D.M. (1984). Stirling Cycle Engine Analysis, Adam Hilger, Bristol. Chapter 2.4 - Case study - The Ford Philips 4-215 engine.

6. C. Lyle Cummins Jr. (1976). Internal Fire, Carnot Press/Graphics Arts Center, Portland, Oregon. Chapter 2 - Air Engines, page 23.

7. Stirling International (2014) - 16th International Stirling Engine Conference

8. Urieli, I - Engineering Thermodynamics - a Graphical Approach

9. Smith, L., Nuel, B., Weaver, S. P., Berkower, S., Weaver, S. C., Gross, B. (2016). 25 kW Low-Temperature Stirling Engine for Heat Recovery, Solar, and Biomass Applications - Presented at the 17th International Stirling Engine Conference.

10. Berchowitz, D. (2018). A Personal History in the Development of the Modern Stirling Engine Presented at the 18th International Stirling Engine Conference.

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Engineering Thermodynamics by Israel Urieli is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License