This paper is of historical value as it represents a snapshot of the understanding of Thermodynamics in 1886. We have to understand that Thermodynamics was still in its infancy. Despite earlier contributions by Boyle, Mariotte, Gay-Lussac and Avogadro,  Nicolas Léonard Sadi Carnot must be seen as the founder of thermodynamics. He studied how to improve the efficiency of steam engines and published his findings in 1824 in "Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance" in which he describes a cycle of four reversible steps. This cycle is today named after him as the "Carnot-cycle" 

Carnot proved that the work of steam engines is proportional to the amount of heat that is transferred from the boiler to the condenser, i.e. from the high-temperature reservoir to the low-temperature reservoir. This is also true for gas engines. Carnot also proved that the efficiency increases with a larger temperature difference."

Dugald Clerk was a Scottish engineer who designed a two-stroke engine with one working cylinder and a second cylinder for compression. This improved the efficiency of earlier developed two-stroke engines, bringing its efficiency close to the Otto-four-stroke engine. However the two cylinders made the engine relatively heavy. It is worth noting that he formed with George Croydon Marks, the intellectual property firm called Marks & Clerk.




Clerk referred to an experiment made by Bunsen in 1866, using a veery small explosion vessel of only a few cubic meters capacity. Bunsen used a homogenous mixture of hydrogen with atmospheric air.  Based on the total heat produced, the amount of hydrogen used and the specific heat of the gas mixture Bunsen calculated the  temperature that would occur under the conditions

  • the heat evolved completely at the moment of the explosion
  • the gases entered into complete combination at once
  • no heat losses occurred due to cooling while the explosion progressed.

Not only were these assumptions not permissible, also the formula is wrong and so Bunsen ended up, after some corrections for the nitrogen in the air and the condensation of the steam in the combustion product with a temperature of 8,812°. (The maximum, theoretical temperature of hydrogen flame actually is 2254°C.) In a next step Bunsen calculated the maximum pressure that should occur in the combustion process, which was much higher than what could be measured. In similar experiments Hirn in Germany and Mallard and Le Chatelier in France came to the same observation that their observed pressures were much lower than calculated. Explanations were given as

  • The limiting casus is the cooling effect of the enclosing walls (Hirn)
  • the approach of the ignited gases to the dissociating point of the compounds (carbon dioxide and water)(Bunsen)
  • the enormous increase of the specific heat of oxygen, nitrogen, and the products of combustion, as the temperature increases towards 1,000°C and 2,000°C.

Clerk therefore decided to repeat these experiments "in order to obtain data requisite for the more thorough understanding of the gas-engine and also, if possible, to arrive at some conclusions as to the relative truth of the three distinct theories".  

Experimental setup

Clerk then describes his experimental setup: "The explosion-vessel is a strong cast-iron cylinder, the internal space being 7 inches in diameter and 81/4 inches long. It is closed completely by covers at the top and at the bottom, firmly bolted, and so arranged as to be easily and rapidly separated for cleaning the walls. Upon the upper cover is mounted a Richards' indicator, from which the ordinary reciprocating drum has been removed and a revolving one substituted, driven by a falling weight and suitable gear; a fan, moving at a high velocity, serves to render the motion uniform. The revolving drum is enamelled, and a soft black lead pencil, held by the indicator-motion, marks upon the drum a line caused by the movement of the indicator-piston. A pair of insulated points project through a plug in the bottom cover, and serve to ignite the mixture when a spark from a coil and battery is passed between them."

He continues to explain that the explosion vessel is emptied before each test and the mixture of the inflammable gas is measured in a graduated glass measuring vessel. To guarantee a homogenous mixture the contents are mixed thoroughly and the vessel is allowed to stand for at least half an hour before the explosion. Then "the drum was made to revolve at a known rated and the spark being passed, a line was traced upon it by the pencil ascending and descending, as the pressure rose and fell. The line shows that amount of rise of pressure, and the times of rise and fall in terms of revolutions of the drum. The tracing is precisely analogous to the indictor-diagram of an engine. It shows the pressure within the explosion vessel fromthe moment of the beginning of the explosion to its completion, and after, till the pressure again becomes equal to that of the exterior atmosphere. The rising line is due to the explosion ;  the falling line is due to the cooling action of the cold walls upon the hot gases. The tracing is, in fact, a record of the rapidity, intensity, and duration of an eplosion. Careful tracings were mad, which are reporduced in Plate 1, Figs. 1, 2, and 3"

The first set of experiments, depicted in Fig. 1, the top figure, shows the results with coal gas from Glascow in mixtures with air of 1:14, 1:12, 1:10, 1:8, and 1:6. The highest pressure of 0.66 MPa (6.6 bar) was obtained for a mixture of 1:6. In this case, where less oxygen was present that what was needed for a complete combustion,  the explosion occured 0.05 seconds after the ignition. Pressure and time of explosion diminish with dilution of the gas mixture. With a dilution of 1:14 the maximum pressure has fallen to 0.36 MPa (3.6 bar) and the explosion time had a delay of 0.28 seconds.

The second set of experiments, depicted in Fig. 2, the middle figure above, shows the results with coal gas from Oldham. The recording drum revolved slower, at 0.5 second for one revolution, so that the graphs in Fig. 2 are compressed by a factor of 6:10 in comparison to Fig. 1. Bas mixtures of 1:14, 1:13, 1:12, 1:11, 1:9, 1:7, 1:6 and 1:5, 1:4 were used (There seems to be an error in the table. These figures have been taken from the legend shonw for Fig. 2). The highest pressure of 627 kPa (6.3 bar) was obtained at a mixture of 1:5 with an explosion delay of 0.055 second. With a mixture of 1:14, the pressure was fallen to half of this value and the time of explosion was extended to 0.45 second. A mixture of 1:15 did not ignite. The most rapid explosion delay of 0.49 was obtained with a mixture of 1:7.

The second set of experiments, depicted in Fig. 3, the bottom figure above, shows the results with hydrogen. The drum was rotating with 1 revolution in 0.33 second, so that the diagrams cannot directly be compared in the x-axis. The tests were conducted with a gas mixture of 1:7, 1.5, and 2:7. With a large excess of air the hydrogen-air mixture of 1:6 the rise of pressure upon explosion is comparatively slow, 0.15 seconds with a pressure maximum of 283 kPa (2.8 bar). The explosion  of a mixture of 2:7 the explosion occured in just 0.01 second so that the sensor started oscillating.   

Clerk's conclusions

It is obvious the best mixture for producing power is that in which the greatest increase in pressure is attained with the smallest relative amount of gas. As all experiments were conducted with the same volume of gas mixture, Clerk divided the pressure by the volume of the coal gas. He found that the most economical mixture was for Glasgow coal gas a mixture of 1:11 for Oldham coal gas a mixture of 1:12.

He also argued, that the best result my not be the highest pressure, but that one has to look at the shape of the curve to find which mixture delivers the most work:

"The explosion must not only give the greatest pressure for a given volume of gas, but it must be prolonged sufficiently to allow of its utilization by the piston. If it gives a good pressure which vanishes instantly, then it is useless. No explosion does this, but some explosions are more prolonged than others. Suppose the time taken by the piston in moving through the working portion of its stroke to be 0.2 second, then a comparison of the relative losses of pressure during 0.2 second from the attainment of the maximum pressure will show which is the best mixture in this respect." 

For his other findings one can look at the paper, which is easy to read. In summary he drew the following conclusions:

  1. Messrs. Mallard and Le Chatelier's theory of increased specific heat of the gases, nitrogen, and oxygen at high temperatures, is erroneous.
  2. Dissociation probably occurs at the higher temperatures to a considerable extent, but it is not the sole cause imposing a limit to the pressure.
  3. Combustion is very similar to the chemical actions, the first part of the reaction occurring rapidly, and proceeding with increasing difficulty as the combination approaches completion.
  4. The explosion-vessel is entirely filled with flame before the combustion is complete, The spread of the flame and the combustion of the gases are distinct phenomena.
  5. The limiting causes act after the flame has spread completely.
  6. The limiting causes in weak mixtures are diminution in the rate of burning as the reaction approaches completion, and consequent limiting by the rate of cooling. The combustion may cause the evolution of heat at rates greater than, equal to, and less than, the rate of cooling.

Todays State of Research



Dissociation refers to the disintegration of combustion products which occurs at temperatures higher than 1000°C. Chemical products are broken down to compounds or radicals thereby consuming heat. Consequently a temperature drop pressure drop occurs causing along with the temperature drop a drop in pressure. 

For example carbon dioxide is broken down to carbon oxid and oxygen

2 CO2 ⇒ 2 CO + O2

and water vapour into hydrogen and oxygen

2 H20 ⇒ 2 H2 + O2

The highest temperature drop is about 300°C. [6]. It is indeed not as significant as it was assumed by Bunsen or Clerk. When the combustion is near to its end the temperature drops further some of the products recombine and release heat, but then it is already too late for the release heat and pressure increase to contribute to the pushing of the piston.  

Heat transfer and exhaust heat flow

Clerk had denied that heat transfer through the cylinder walls has a prominent effect. He correctly argues that bigger cylinders should have a lower heat loss than smaller cylinder with a higher surface to volume ratio. So far I have not found an explanation why the differences for a 300 cubic centimeter vessel compared to a 4,000 cubic centimeter vessel observed by Berthelot were so insignificant that Clerk denied the effect. However he admitted that he had observed that in vessels with a smaller capacity the pressure falls more rapid after the maximum pressure has been attained.

Fact is that heat transfer accounts for around 60% of the losses of the theoretical thermodynamic efficiency limit of an Otto engine, assuming a compression ratio of 1:10. For an Otto engine with a cylinder wall temperature of 177°C (450K) 16.4% of the available energy is lost through the cylinder walls and 43.2% is transferred out of the cylinder by the exhaust gases, which have a temperature of about 1,327°C (1600 K). Only 39.7% of the available energy can be transformed into mechanical work. This example is based on an Otto engine running at 2000 rpm. Some of the heat transfer occurs during the compression phase when the gas mixture is compressed and thereby increases its temperature.[3] As we can see from the diagram on the diagram the loss through the cylinder walls can be reduced by accepting higher cylinder wall temperatures. However this comes and higher costs for the making the cylinder suitable for this higher temperatures and most of the advantage is diminished by an increased heat transfer by the hotter exhaust gases.

Please note that the diagram shows the heat balance of the the energy made available by the combustion. It does not account for the cycle efficiency which is for an Otto cycle at a compression ratio of 1:10 approximately 60%. With respect to the heating value of the gasoline a modern Otto cycle motor runs at a total efficiency of 25%.   

In Clerk's experiments the heat transfer by the exhaust gases was probably relatively small, as he was performing only one explosion and did not allow the combustion gases to exhaust. Most of this heat would then have been transferred through the cylinder walls, although some would have still added some mechanical work. Diesel two-stroke marine engines, which are running at low speed (100 rpm) and have combustion chambers with low surface area to volume ratios, further using turbochargers and waste heat recovery devices thus achieve thermal efficiencies in the order of 55%. [4]       

Flame speed and flame propagation

Today we have more technical possibilities at hand. We can measure pressure with pressure transducers with a very high degree of precision. Quartz glass windows in the cylinder head or in the piston and high speed cameras allow us to look inside the combustion chamber and each frame of the photographic sequence can be associated with the crank angle of the engine [7]. Parameters, such as heat release and mass burned fraction that are not directly accessible by sensors are calculated using mathematical models using the accessible data.

The spark between the electrodes of the spark plug creates a cylindrical spark channel in the order of 30 µm. The gas temperature in the channel may be greater than 60,000K. The diameter of the spark channel increases with current and time. After 1 microsecond the kernel has grown to 1 mm in diameter whilst the temperature has fallen below 10,000K [14]. The combustion of the premixed gases eventually starts about 1ms after ignition. This delay has to be taken into account as a time advance so that at different engine speeds the combustion starts around Top Dead Center (TBC).

When the combustion starts a flame front develops. "A flame is the (confined) region within which the fuel oxidizing reactions and chemical energy release occur. Engine flames are normally thin: the actual reaction region is a fraction of a millimetre thick. In the spark-ignition engine, the fuel-vapor air charge is well mixed so the flame is a premixed flame. The speed at which such a premixed flame propagates into the unburned gas ahead of it is called the flame speed or burning velocity. In a laminar or well-ordered flow, this velocity SL is a characteristic property of the unburned mixture ahead of the flame: it depends on the composition of the mixture, the properties of the fuel, and the mixture temperature and pressure."  Premixed SI engine flames become turbulent as they develop from the spark discharge, and their propagation rate is then defined by their turbulent flame speed. If engine flames were not turbulent, with a turbulent flame speed several times the laminar flame speed, engines could not operate satisfactorily because combustion, and the pressure rise it produces, would be much too slow. Because engine turbulence scales with piston speed, the combustion rate also scales with piston speed [6]. This fact is of great practical importance because it means that even at high engine speeds the flame propagation may still be completed early in the expansion stroke [13]. 

The figure summarizes some aspects of what is happening in a cylinder during combustion. Pressure data had been measured from an experimental four cylinder engine running at 1200 rpm and a residual fraction of 5% [7]. Temperature and burnt masses had been calculated from a mathematical model based on the pressure data. I calculated the volume curve from an engine with a 84mm stroke, a rod length of 140mm and a compression ratio of 10.4. At 1200 rpm a crank shaft angle of 10° is equivalent to 1.39 ms.

Today engines are running at least ten times faster than the first Ott-four-cycle engine. Therfore the ignition has to occur already before the piston arrives at the top dead center (TDC) (0° crank angle) so that the maximum pressure can develop early in the down stroke phase of the piston. We also see from the diagram that the main fraction of 90% of the combustible gases is already burnt before the crankshaft arrives at a crank angle of 30° after the TDC. This means that the piston has only travelled down less than 9% of its full stroke. Taking into account the clearance volume (the volume enclosed by the cylinder head and the piston head, when the piston is on the TDC) this is equivalent to 17% of the total volume when the piston is at the bottom dead center (BTC). 

Immediately following the spark, there is a period during which no heat heat release due to combustion is observed. Following this period, combustion begins, not at a point, but throughout some critical volume called flame kernel. The flame propagation speed of the flame kernel is slightly at the level of the laminar flame speed and accelerates continuously. [7]

The period from ignition to the 10% mark of the total burned mass is termed flame development period. It takes about one third of the total combustion period.  

At the beginning of the compression cycle the cylinder walls, which are cooled by a coolant, are hotter then the admitted gas mixture. Consequently there is a heat transfer from the cylinder walls to the enclosed gas mixture. When the burnt gases hit the cylinder wall, there will be a heat transfer from the burnt gases to the cylinder wall, which drops locally the temperature of the flame front and extinguishes the flame front at the cylinder wall. We also see that the temperature, due to the compression of the gas mixture has already risen to 2400K even before the ignition starts. Actually, if the compression rate would be higher, the temperature raise would also be higher, and there would be the risk that the gasoline would ignite prematurely, an effect that is known as knocking. The mean temperature of the burning gases which are away from the cylinder wall, is about 2600K while the gases at the flame front are close to 2900K. Obviously, due to pressure drop caused by  the piston moving towards BTC and the heat transfer to the cylinder wall the temperature inside the cylinder will dramatically fall. When the exhaust valves opens the temperature of the gases has fallen to 900K.

Today, with experimental data and mathematical models, we can look into the cylinder and have more profound ideas what happens inside the cylinder during combustion [6], [7], [8], [9]. Town gas, that was used in the beginning for Otto's gas engine, has a laminar flame propagation speed of 0.9 m/s. Laminar is somewhat what Otto had imagined in his explanation that the gas after being admitted to the combustion chamber does not mix with the earlier admitted layers of combustion residues or the air. Once ignited, the mixture of combustible gas and air  will propagate at a relatively slow  speed, without being accelerated to cause an explosion. And indeed, if we look at Fig. X, between when 10% and 90% of the combustible mixture has been burned, the burn rate, i.e. the mass of gas burned by time unit is almost constant. As the turbulent flame propagation is at its highest value this period is also called the rapid burning period (RBP)

According to the Benson Ford Research Center a 1883 Otto Engine of one horse power had a stroke of 23cm (9 inch) and a bore diameter of 12.4 cm (4.875 inch), which gives a volume of about 1 liter. After ignition a laminar flame would propagate in a chemical reaction zone as a relatively thin flame front spherically towards the piston. In case of Otto's idea it would take a laminar flame 0.14s to propagate from the lower dead end to the upper dead end of the piston. This is just to give an idea how slowly a flame would propagate, if there was not the effect of turbulence.

Turbulence, known from fluid mechanics is the effect that occurs when the speed of fluid particles is above a certain threshold, which depends for example of the diameter of a pipe. The fluid particles then do not follow the same direction as their neighbour fluid particles but start to move partially chaotically with high speed fluctuations and create wrinkles and eddies. Although several theories exist [10], [11] and it is not yet completely clear which effect prevails one can observe that the reaction zone becomes thicker than in the laminar case and the flame front propagates faster. One explanation is that the wrinkles around the flame front create a larger surface where the reaction takes place and thus the mass burn rate is increase and with it the propagation speed. Whereas gasoline has a flame propagation speed of 0.35 m/s the average flame propagation velocity with a wide open throttle increases to around 20m/s at 2250 rpm and around 35m/s at 5500 rpm. Higher engine velocities promote higher turbulences, increasing flame propagation velocity. [9]



Efficiency as a function of the compression ratio

The Carnot efficiency is a theoretical model for a thermodynamic engine by which the engine extract the most work from a thermal cycle. It thus provides an upper limit on the efficiency that any classical thermodynamic engine can achieve during the conversion of heat into work. It depends only on the temperature for the heat added and the temperature of the heat reservoir the heat is rejected to. In order to compare the Carnot cycle efficiency I used the Otto engine model and calculated the maximum temperature that will occur in an instantaneous combustion of the fuel. The burnt gases in the Carnot engine is then given sufficient time to cool down to an environment temperature of 20°C (red line). As we will see, the average temperature of the burnt fuel in a Otto engine depends on the temperature. The higher the compression ratio the higher the average temperature that can be achieved by the same amount of fuel. Therefore in this specific example where we compare of the same temperatures applied to a Carnot engine and achieved by an ideal Otto engine the efficiency is shown as a function of the compression ratio of an Otto engine.

Otto engines are operated with exhaust temperatures of aroung 300-580°C. In order to show this difference in efficiency the blue line shows the efficiency of Carnot engine if it is allowed to cool down only to 600K. 

In theory the thermal efficiency of the Otto cycle is only limited by the compression ratio, represented in the diagram by a purple line. We see that the maximum efficiency of an Otto engine is much lower than the efficiency of a Carnot engine. This is the price to pay to have relatively simple engines running at 6000 rpm, or in racing cars up to 20.000 rpm.

However, the design criteria of an Otto engine is to burn the fuel relatively slowly so as to avoid the self destruction of the Otto engine. The green line in the diagram shows the efficiency if the fuel is burned over a crank shaft angle of 50°, starting at TDC.

In reality, due to unavoidable losses we can be happy if a Otto engine eventually performs at 50% of its thermal efficiency. We see that Otto's first four stroke gas engine , if we assume a compression ratio of 1:2.5 could have achieved a maximum thermal efficiency of 31%, or about 15% break power efficiency. With around 10% break power efficiency his first machine was not so far away from this limit.

Todays Otto engines in cars us a compression ration of about 1:10. Higher compression ratios would increase the efficiency but also increase the risk that the fuel ignites prematurely by the temperature raise due to compression, an effect that is purposely used in Diesel engines. However, in Otto engines that on the long run would damage the engine. There was a time when a lead compound was added to the gasoline to reduce the risk of knocking. As some of us will remember this had a bad effect on the environment and lead based additives were reduced from 1985 on.





Spark Advance for Optimal Efficiency

Otto's gas engine was running at about 2 rounds per second, or 1 four stroke cycle per second. At this speed the gas easily was burned after ignition. In fast running engines however, the delay time of 0.4 to 1.0 seconds for the ignition to start at a considerable speed, shifts the maximum pressure of the combustion process towards BDC, which lowers the power that can be extracted from the combustion.

In order to position the pressure development in the cylinder such that the combustion produces maximum work the pressure peak is targeted at around 20° after top dead center (TDC). This means that under normal driving conditions the mixture is ignited around 15° - 30° before the piston has reached TDC. If the timing of the ignition is too early, the pressure rise starts too early and counteracts the piston movement. [16] This causes also a strain to the engine, for example for the bearings. If the pressure increase comes too late work is lost during the expansion phase.


[1] Clerk, D., Forrest, J., Institution of Civil Engineers (Great Britain). (1882). "On the theory of the gas engine ..: With an abstract of the discussion upon the paper. London: The Institution.


[3] Caton, Jerald. (2017). Maximum efficiencies for internal combustion engines: Thermodynamic limitations. International Journal of Engine Research. 19. 146808741773770. 10.1177/1468087417737700. Link to ResearchGate

[4] Caton, Jerald "Maximum efficiencies for internal combustion engines: Thermodynamic limitations" in International Journal of Engine Research 1-19. DOI: 10.1177/I468087417737700.

[5] Edson, M., "The Influence of Compression Ratio and Dissociation on Ideal Otto Cycle Engine Thermal Efficiency," SAE Technical Paper 620557, 1962,

[6] John Heywood, "Internal Combustion Engine Fundamentals", 2018, 2nd edition, Mc Graw Hill education

[13] Lloyd Withrow and Gerald Rassweiler, "Studying Engine Combustion by Physical Methods - Areview, Journal of Applied Physics 9, 362 (1938); DOI 10.1063/1.1710431  

[7] G.P. Beretta, M. Rashidi, and J.C. Keck, "Turbulent Flame Propagation and Combustion in Spark Ignition Engines, 1883, Massachusetts Institute of Technology, Cambridge, MA 02139 in COMBUSTION AND FLAME 52, pages 217-245 available at Amanote

[8] Lavoie, G.A.; Heywood, J.B.; Keck, J.C., "Experimental and Theoretical Study of Nitric Oxide Formation in Internal Combustion Engines" in Combustion Science and Technology, Vol. 1, 313 (1970) available at James-Keck-Memorial-Collection

[9] Guilherme Bastos Machado, Sergio Leal Braga, Jose Eduardo Mautone Barros, Carlos Valois Maciel Braga, "Methodologies for Flame Propagation Velocity Determination in Spark Ignited Engines", SAE Technical Paper Series 2017-36-0193

[10] David R. Lancaster and Roger B. Krieger (Research Labs., General Motors Corp.), Spencer C. Sorenson and William L. Hull (Mechanical Engineering Department, University of Illinois); "Effects of Turbulence on Spark-Ignition Engine Combustion";  SAE Technical Paper 760160, 1976,

[11] Stefan Pischinger, John B. Heywood, "A model for flame kernel development in a spark-ignition engine", Symposium (International) on Combustion, Volume 23, Issue 1, 1991, Pages 1033-1040, ISSN 0082-0784,

[12] L. Cornolti, T. Lucchini, G. Montenegro and G. D’Errico, "A comprehensive Lagrangian flame-kernel model to predict ignition in SI engines"; Department of Energy, Politecnico di Milano, via Lambruschini 4, 20156 Milano, Italy PDF from Politechnico Milano


[14] R. Maly in Comments to [15] at the end of [15]

[15] Jerzy Chomiak, "Flame Development From An Ignition Kernel In Laminar And Turbulent Homogenous Mixtures" Cloloquium on Turbulent Combustion Interactions

 [16] Lars Eriksson, "Spark Advance for Optimal Efficiency", Electronic Engine Controls 1999: Neural Networks, Diagnostic and Electronic Hardaware, and Controls (SP-1419), reprinted in SAE Technical Paper Series, 1999-01-0548, Vol. 108, Section 3: JOURNAL OF ENGINES (1999), pp. 789-800 (12 pages).