Carnot cycles

Type of physical science: Classical physics

Field of study: Thermodynamics

A cyclic process is a change of a thermodynamic system wherein the variables of that system (temperature, pressure, and the like) ultimately return to their original values. If the process is arranged to consist of isothermal and adiabatic parts, it is a Carnot cycle, and the energy or work balance of the process can be investigated easily.

Overview

The Carnot cycle is a particular, four-step process in which a system (like a vapor confined in a cylinder-piston engine) expands and contracts (driving the piston), while the temperature rises and falls and heat is added or exhausted. The steps are envisioned to be arranged so that the system returns to its initial state (of temperature, pressure, and volume) after the four steps, and the quantities of work done by the engine and the heat added (by ideal reservoirs at high and low temperatures) can be directly calculated from thermodynamic definitions. The immediate advantage of this construct is that many working heat engines, such as steam engines, can be considered to be modeled roughly by the Carnot cycle, and so their idealized work-output versus heat-input ratio (the "efficiency") can be calculated. From more theoretical considerations involving the second law of thermodynamics, this ideal calculation can be shown to give the maximum possible efficiency attainable by any cyclic engine (operating between the same high- and low-temperature reservoirs). Even in cases where the actual engine does not accurately reproduce the steps in the Carnot cycle, the maximum achievable efficiency is still known through the Carnot model, and alternative schemes for work/heat engines can be usefully compared. The Carnot cycle can be employed to illustrate the various formulations of the second law of thermodynamics and to clarify the limitation on such consequences as the inevitable increase in entropy in a "natural" system.

The particular steps in the Carnot cycle are chosen for convenience of calculation, not for their realism in approximating existing devices. They are: An expansion by heating of the system such that the temperature is held constant by regulating how fast the cooling by expansion does work (say on the piston) to "balance" the added heat. This is the first "isothermal" stage. Second, continued expansion of the system with the source of heat shut off, thus allowing the temperature to drop. This is the first "adiabatic" stage. Third, a compression (say by having a camshaft reverse the direction of the piston) where the temperature is again held constant (perhaps by having a condenser cool the vapor). Finally, further compression where the heat is not allowed to escape. At the end of this process one can envision that the final temperature, pressure, and volume is restored. The first law of thermodynamics asserts that overall there was no change in the state variables of the system (including internal energy), so that the total work balanced the total heat exchange. There was no heat exchange during the two adiabatic stages, and the work done on the environment (piston) during the adiabatic expansion was exactly balanced by the work done by the environment during the adiabatic contraction. This leaves only the work done and heat added or subtracted during the two isothermal stages to be computed. If one examines a system composed of an ideal gas (one in which it is known ideally how much work is done in an isothermal expansion), one can discover that the heat added (from the high-temperature reservoir) divided by the temperature at which it was added will be quantitatively equal to the heat exhausted (to the low-temperature reservoir) divided by that low temperature. In other words, the entropy added in the first isothermal stage is equal to the entropy subtracted at the second stage. Since there was no heat exchanged, by definition, in the adiabatic stages, this "conservation of entropy" applies to the complete Carnot cycle. It should be noted here that entropy does not increase, as would be suggested by the second law of thermodynamics, because the processes are ideal and "reversible" (the engine running backward through the stages--as a "refrigerator"--would undergo the same numerical work-heat balances).

The immediate consequence of this result is that the efficiency of this ideal Carnot engine can be directly calculated as the difference in temperatures (of the reservoirs) divided by the high temperature.

Since the theoretical arguments leading to maximum efficiency arguments of ideal engines refer to reversible models, the entire discussion can be reversed: heat extracted from the low-temperature reservoir and exhausted (along with heat added from the agency supplying work) to the high-temperature reservoir. Such a device, a refrigerator, can be characterized by a "coefficient of performance" related to the "efficiency" of its reverse. The coefficient of performance is defined as the heat extracted from the low-temperature reservoir divided by the work done to extract it. Again from the arguments of the ideal Carnot cycle (that the heat transfer divided by the temperature is the same everywhere), the coefficient of performance can be computed simply as the low temperature divided by the temperature difference.

While in engineering thermodynamics, the Carnot cycle is not as "realistic" as other cyclic engine models (which may not have obvious isothermal stages and "reservoirs"), it has certain advantages in more abstract arguments. One immediate example is to note that the formula for efficiency seems to require the concept of a "lowest temperature." If the temperature of the low-temperature reservoir were taken to be negative, the efficiency of a Carnot engine utilizing that reservoir would be greater than one (100 percent). This would seem to violate the first law of thermodynamics, for one joule of heat would produce more than one joule of work.

This argument (along with a technical piece of the second law) forms the basis for introducing an "absolute temperature scale," wherein the lowest theoretical temperature conceivable is -273 degrees Celsius.

The introduction of the second law of thermodynamics further illustrates the theoretical utility of the notion of a Carnot cycle. One can imagine that any reversible cyclic engine can be duplicated by stringing together an infinite number of infinitesimal Carnot cycles (operating from reservoirs of temperatures minutely different from one another). The result of this notion is that the sum of each heat added (or subtracted) divided by the temperature at which it was added (or subtracted) is zero over the entire cycle. Thus, if two points can be visualized (particular pressures, temperatures, and volumes) in the cycle, the entropy (heat divided by temperature) expended by arriving from one point to the other will be "returned" by completing the cycle from the second point to the first. In other words, the entropy expended (or absorbed) in the process from the first point to the second is the same for either process ("path")--the original process from the first point to the second, or the reverse of the "return" path. Since the cycle considered here was arbitrarily chosen (it could be any cycle as long as it could be composed of Carnot cycles), it follows that the entropy change in any reversible process is independent of the "path" of the process. That is, the entropy change depends only on the initial and final points; the entropy of a system depends only on the state of the system, not on the "history" of how it got that way. This is not true of the notions of "work" or "heat," which have to expend different values to cause the system to arrive at the same state by different paths. This means that entropy is a "property" of a thermodynamic system, while "heat content" (for example) is not.

This abstract introduction, via the Carnot cycle, of entropy into the family of thermodynamic variables (properties of a system) has many consequences, since it has been achieved as a logical consequence of the second law of thermodynamics. One immediate result is from reconsidering the change in entropy in an irreversible system. Since the entropy is a state variable, its changes are independent of the process, irreversible or not. For a simple example, imagine an ice cube sitting on a hot plate. One statement of the second law of thermodynamics is that it is not the case that heat will "spontaneously" flow from the cold ice cube to the hot plate.

Rather, the heat will "naturally" flow into the ice. Note that the same heat that left the hot plate flowed into the ice cube (required by the first law of thermodynamics). The lower temperature of the ice would mean that it received a greater entropy than the hot plate gave up (since entropy is heat divided by temperature). Subtracting the lesser yield of the hot plate from the greater gain of the ice, it is obvious that the total entropy of the system has increased. Thus, another version of the second law is indicated: In any isolated system containing an irreversible process, total entropy increases. If there are only ideal, reversible processes, the entropy will be conserved. The "increasing entropy" statement of the second law receives much attention. For example, chemists can envision the theoretical final entropy of alternative end-states of a hypothetical reaction and choose the predicted path as that which maximizes entropy. In abstract, the entire universe can be considered a closed (adiabatic) system (without "external" sources of heat). As such, its entropy must be ever increasing, as it surely contains irreversible processes. In this case, that evolution toward equilibrium would involve a cooling of the hot stars and hot planets, and a heating of the cold interstellar matter (and radiation). The subsequent "heat death" of the universe would then involve a freezing, not a boiling of the earth's oceans.

There are other advantages to examining the heat and energy flow in systems by use of a cyclic Carnot process. The meaning of the potentials (internal energy, enthalpy, and the like) can be clarified in many immediate examples. Many laboratory situations involve cyclic or periodic behavior, and are thus candidates for an efficiency analysis. Thus, for both theoretical and practical reasons, the Carnot cycle is a most useful thermodynamic idealization.

Applications

Because it requires knowing only temperature, the calculation of efficiency for an ideal Carnot engine (one whose working substance is an ideal gas) is particularly simple and memorable. Consider some hypothetical steam engine, whose boilers operate at around the boiling temperature of water (about 373 Kelvins) and whose condensers operate around room temperature (300 Kelvins). Were this engine employing an ideal gas in reversible Carnot cycles, its efficiency would be 73/373 (the temperature difference divided by the high temperature; note that the temperatures must be expressed in the Kelvin scale), or about 0.20 (or 20 percent). This means that for every 100 joules of heat provided by the boiler, only 20 joules is delivered as work to the pistons and 80 joules is exhausted to the condenser. Note that the efficiency would be 1 (100 percent) only if the condenser were at absolute zero. This is an "ideal" case. Presumably an actual engine, using steam rather than an ideal gas and operating with some friction in the parts (hence, "irreversible") would certainly be less efficient. It is a consequence of the second law of thermodynamics that no other arrangement of stages in a reversible cycle can be more efficient than a Carnot cycle, so simply knowing the temperatures of the "reservoirs" immediately gives a maximum efficiency to any thermodynamic device proposed to extract work from temperature differences.

An example of a refrigerator may be a household freezer operating between an interior reservoir of below freezing (about 270 Kelvins) and room temperature (about 300 Kelvins). The coefficient of performance of an ideal Carnot refrigerator would then be 270/30, or about 9. This means that for every joule of work done on the refrigerator (by the electric motor attached), there will be 9 joules removed from the interior of the refrigerator. From the first law of thermodynamics, one can see that there will then be 10 joules exhausted to the high-temperature reservoir, the kitchen. It can be anticipated that the second law of thermodynamics will dictate that a nonideal, irreversible refrigerator will do considerably less well at cooling.

The definition of a Carnot cycle included two isothermal (constant-temperature) processes. In practice, isothermal processes are hard to arrange, so that engineers have preferred to use other versions of an idealized cyclic engine to model the theoretical efficiency of practical devices. For steam-piston engines, the Rankine cycle--pairs of adiabatic and isobaric (constant pressure) stages--is considered more convenient. The internal vaporizations and condensations of water (or other working substances, often Freon in a refrigerator) take place at constant pressure.

For internal combustion engines, the Otto cycle--pairs of adiabatic and isochoric (constant volume) stages--is often preferred as a model. The firing of the spark plug happens quickly before the compression begins. This is roughly isochoric (constant volume). While these rough approximations are inexact, the Otto cycle stages (adiabatic-isochoric) are more representative than the Carnot stages (isothermal-adiabatic) for processes in the internal combustion engine.

The diesel cycle--isobaric and isochoric stages between two adiabatic stages--is a popular alternative to describe engines whose high-temperature combustion is provided by highly compressing the vaporized fuel. For one thing, this means that spark plugs are not required to ignite the fuel. The higher temperatures required for this self-ignition produce both advantages and disadvantages in the practical engineering of diesel engines.

It might seem that such explosive processes are far from appropriate for description by ideal, reversible models. Yet, it is often argued that the sudden combustion stage of an engine is so rapid that there is no time for heat to enter or leave the system. This accounts for the adiabatic stages, and the isothermal and isochoric processes can be approximated by other mechanical features of the actual engine design. There is no thought of exceeding the efficiency of a Carnot cycle in these alternate models. Their purpose is to make the ideal efficiency of the model approach more reasonably the actual efficiency of the engine.

The system undergoing cyclic changes in variables is not restricted to mechanical engines. Biological organisms take in heat and do work in a repetitive fashion. The "ideal efficiency" of an organism converting heat (by "digestion") to work, and the energy required for temperature control are useful parts of a model. This is only a sample of the utility of the concept of the Carnot cycle.

Context

Nicolas-Leonard-Sadi Carnot initiated the notions behind the "Carnot cycle" in his epochal article of 1824, "Reflections on the Motive Force of Fire." At the time of writing, Carnot had envisioned heat in the form of an invisible "fluid" ("caloric") the manner fashioned by Antoine-Laurent Lavoisier at the end of the eighteenth century. He considered an ideal steam engine involving a high-temperature reservoir representing the boiler, and a low-temperature reservoir representing the condenser. Then, by assuming that the amount of caloric was conserved as it flowed from high to low temperature, Carnot calculated the "efficiency" of the engine--the amount of useful work done by the engine by the "falling" caloric. Carnot was probably following an idea initiated by his father, Lazare-Nicholas-Marguerite Carnot, who had proposed a similar idea for improving the efficiency of waterwheels (to maximize efficiency by minimizing "splashing"). Carnot concluded that higher efficiency would always result from a greater temperature difference between the hot and cold reservoirs.

Historically, even before Carnot's memoir, a notion of a cyclic engine was envisioned by a Scots minister, Robert Stirling. In 1816, Stirling patented the design for a hot-air engine that would convert some of the energy of coal-burning into work. An idealized Stirling engine operates with two isothermal and two isochoric (constant-volume) stages.

The practical consequence of these ideas were probably minimal in engineering engines, for the steam-engineers of the day favored trial-and-error improvements over mathematical calculations. Nevertheless, the conceptual method of Carnot became the basis for the more powerful theoretical arguments of Rudolf Clausius. By mid-nineteenth century, Hermann Ludwig Ferdinand von Helmholtz (and others) had devised the quantity "energy" to measure the mechanical work done by an engine, and Clausius himself had replaced the notion of "caloric" by the kinds of motions known now as heat. By reconsidering Carnot's argument in terms of "converting" heat to work, Clausius could incorporate Helmholtz' "conservation of energy" into the calculation of efficiency of an ideal engine. Clausius' generalized "engine" no longer had to represent a steam device, but could (ideally) represent any cyclic process where work and heat were exchanged. Clausius went on to formulate the second law of thermodynamics in terms of heat, work, and entropy, and thus unified the modern field of thermodynamics.

Clausius' student, Rudolph Diesel, was one of the first practical engineers to adapt the idea of the Carnot cycle to the consideration of a novel engine. Since the nineteenth century, many similar fruits of this idea have emerged, and the notion has spread to model systems far from simple mechanical engines.

Thus, the general science of thermodynamics developed around the concept of the ideal Carnot engine, and still employs that concept to illustrate its main features, both practical and abstract.

Principal terms

ADIABATIC PROCESS: a process wherein no heat is added or subtracted

CHANGE IN ENTROPY: the heat added to a system divided by the temperature at which it was added

EFFICIENCY: the useful work divided by the heat input for an engine

ISOBARIC PROCESS: a process at constant pressure

ISOCHORIC PROCESS: a process at constant volume

ISOTHERMAL PROCESS: a process at constant temperature

RESERVOIR: a source or repository of heat that does not appreciably change its temperature

Bibliography

Cardwell, D. S. L. FROM WATT TO CLAUSIUS: THE ROLE OF THERMODYNAMICS IN THE EARLY INDUSTRIAL AGE. London: Heinemann, 1971. This is one of only a few historical surveys of the early history of thermodynamics and emphasizes the historical applications of thermal engineering. Cardwell is an eminent historian and an expert on nineteenth century science.

Carnot, Sadi. REFLECTIONS ON THE MOTIVE POWER OF FIRE. Gloucester, England: Peter Smith, 1977. This translation and commentary on the original "founding document" of thermodynamics is quite accessible to the general reader. The prethermodynamic notion of heat may seem misleading to later notions of energy, but here it helps the reader's intuition of efficiency and work in a thermodynamic machine.

Fenn, J. B. ENGINES, ENERGY, AND ENTROPY. New York: W. H. Freeman, 1982. Largely nonmathematical, this volume has particularly excellent graphics. It is helpful in relating thermodynamics to its context of kinetic theory and statistical mechanics.

Roller, Duane. THE EARLY CONCEPTS OF TEMPERATURE AND HEAT: THE RISE AND DECLINE OF THE CALORIC THEORY. Cambridge, Mass.: Harvard University Press, 1956. The series of short monographs in the Harvard Case Histories in Experimental Science have become essential resources. They are constructed from original sources, hence this book requires no advanced mathematics and can be appreciated by a precollege audience as well as advanced scholars.

Thomas, Donald. DIESEL: TECHNOLOGY AND SOCIETY IN INDUSTRIAL GERMANY. Fayetteville: University of Alabama, 1987. The story of the successes and failures of Rudolph Diesel is a story involving much more than applied thermodynamics. As a student of Clausius, Diesel may have even misunderstood the subject, though Thomas explains the principles behind the engineering of Diesel's engines. The technical arguments are often too advanced for the general reader, but the historical narrative is easily appreciated.

The Behavior of Gases

Thermal Properties of Matter

Laws of Thermodynamics