Carnot cycle
A Carnot cycle is a theoretical ideal thermodynamic cycle proposed by French physicist Sadi Carnot in 1824 and expanded upon by others in the 1830s and 1840s. It provides an upper limit on the efficiency that any classical thermodynamic engine can achieve during the conversion of heat into work, or conversely, the efficiency of a refrigeration system in creating a temperature difference by the application of work to the system. It is not an actual thermodynamic cycle but is a theoretical construct. There are practical thermodynamic cycles such as a Rankine cycle but no one can get the  efficiency equal or higher than the Carnot cycle efficiency, by the Carnot's theorem. Every single thermodynamic system exists in a particular state. When a system is taken through a series of different states and finally returned to its initial state, a thermodynamic cycle is said to have occurred. In the process of going through this cycle, the system may perform work on its surroundings, for example by moving a piston, thereby acting as a heat engine. A system undergoing a Carnot cycle is called a Carnot heat engine, although such a "perfect" engine is only a theoretical construct and  cannot be built in practice. However, a microscopic Carnot heat engine has been designed and run. Essentially, there are two "heat reservoirs" forming part of the heat engine at temperatures formula_1 and formula_2 (hot and cold respectively). They have such large thermal capacity that their temperatures are practically unaffected by a single cycle. Since the cycle is theoretically reversible, there is no generation of entropy during the cycle; entropy is conserved. During the cycle, an arbitrary amount of entropy formula_3 is extracted from the hot reservoir, and deposited in the cold reservoir. Since there is no volume change in  either reservoir, they do no work, and during the cycle, an amount of heat energy formula_4 is extracted from the hot reservoir and a smaller amount of waste heat formula_5 is given off into the cold reservoir. The net energy formula_6 is equal to the work done by the engine per cycle. Carnot heat engine as an impractical macroscopic construct. A Carnot heat engine is a heat engine performing a Carnot cycle, and its realizations in a macroscopic scale is impractical. For example, for isothermal expansion as a part of the Carnot cycle, the following conditions are required to be  satisfied simultaneously at every step in the expansion to realize it. These (and other) "infinitesimal" requirements makes the Carnot cycle to be performed over an infinite amount of time. There are other practical requirements that make the Carnot cycle hard to be realised (e.g., fine control of gas thermal contact with the surroundings including high and low temperature reservoirs), so the Carnot engine should be thought as a theoretical limit of macroscopic scale heat engines rather than a practical achievement goal. Stages. A Carnot cycle as an idealized thermodynamic cycle performed by a heat engine (Carnot heat engine) consists of  the following steps. In this case, since it is a reversible thermodynamic cycle (no net change in the system and its surroundings per cycle) or, This is true as formula_9 and formula_10 are both smaller in magnitude and in fact are in the same ratio as formula_11. The pressure–volume graph. When a Carnot cycle is plotted on a pressure–volume diagram (), the isothermal stages follow the isotherm lines for the working fluid, the adiabatic stages move between isotherms, and the area bounded by the complete cycle path represents the total work that can be done during one cycle. From  point 1 to 2 and point 3 to 4 the temperature is constant (isothermal process). Heat transfer from point 4 to 1 and point 2 to 3 are equal to zero (adiabatic process). Properties and significance. The temperature–entropy diagram. The behavior of a Carnot engine or refrigerator is best understood by using a temperature–entropy diagram ("T"–"S" diagram), in which the thermodynamic state is specified by a point on a graph with entropy ("S") as the horizontal axis and temperature ("T") as the vertical axis (). For a simple closed system (control mass analysis), any point on the graph represents  a particular state of the system. A thermodynamic process is represented by a curve connecting an initial state (A) and a final state (B). The area under the curve is: which is the amount of thermal energy (heat) transferred in the process. If the process moves to greater entropy (e.g., moving from a lesser entropy point A to a greater entropy point B in the integral above), the area under the curve is the amount of heat absorbed by the system in that process. If the process moves towards lesser entropy, it is the amount of heat removed or leaving  from the system. For any cyclic process, there is an upper portion of the cycle and a lower portion. In "T"-"S" diagrams for a clockwise cycle, the area under the upper portion will be the thermal energy absorbed by the system during the cycle, while the area under the lower portion will be the thermal energy removed from the system during the cycle. The area inside the cycle is then the difference between the two (the absorbed net heat energy), but since the internal energy of the system must have returned to its initial value, this difference must be the  amount of work done by the system per cycle. Referring to , mathematically, for a reversible process, we may write the amount of work done over a cyclic process as: Since "dU" is an exact differential, its integral over any closed loop is zero and it follows that the area inside the loop on a "T"–"S" diagram is equal to the total work performed by the system on the surroundings if the loop is traversed in a clockwise direction, and is equal to the total work done on the system by the surroundings as the loop is traversed in a counterclockwise  direction. The Carnot cycle. Evaluation of the above integral is particularly simple for a Carnot cycle. The amount of energy transferred as work is The total amount of thermal energy (heat) transferred from the hot reservoir to the system (in the isothermal expansion) will be and the total amount of thermal energy (heat) transferred from the system to the cold reservoir (in the isothermal compression) will be Due to energy conservation, the net heat transferred, formula_15, is equal to the work performed The efficiency formula_17 is defined to be: where This is the Carnot heat engine working efficiency definition as  the fraction of the work done by the system to the thermal energy received by the system from the hot reservoir per cycle. This thermal energy is the cycle initiator. Reversed Carnot cycle. A Carnot heat-engine cycle described is a totally reversible cycle. That is, all the processes that compose it can be reversed, in which case it becomes the Carnot heat pump and refrigeration cycle. This time, the cycle remains exactly the same except that the directions of any heat and work interactions are reversed. Heat is absorbed from the low-temperature reservoir, heat is rejected to a  high-temperature reservoir, and a work input is required to accomplish all this. The "P"–"V" diagram of the reversed Carnot cycle is the same as for the Carnot heat-engine cycle except that the directions of the processes are reversed. Carnot's theorem. It can be seen from the above diagram that for any cycle operating between temperatures formula_1 and formula_2, none can exceed the efficiency of a Carnot cycle. Carnot's theorem is a formal statement of this fact: "No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between those same reservoirs ." Thus, Equation gives the maximum efficiency possible for any engine using the corresponding temperatures. A corollary to Carnot's theorem states that: "All reversible engines operating between the same heat reservoirs are equally efficient." Rearranging the right side of the equation gives what may be a more easily understood form of the equation, namely that the theoretical maximum efficiency of a heat engine equals the difference in temperature between the hot and cold reservoir divided by the absolute temperature of the hot reservoir. Looking at this formula an interesting fact becomes apparent: Lowering the temperature of the cold reservoir will  have more effect on the ceiling efficiency of a heat engine than raising the temperature of the hot reservoir by the same amount. In the real world, this may be difficult to achieve since the cold reservoir is often an existing ambient temperature. In other words, the maximum efficiency is achieved if and only if entropy does not change per cycle. A entropy change per cycle is made, for example, if there is friction leading to dissipation of work into heat. In that case, the cycle is not reversible and the Clausius theorem becomes an inequality rather than an equality . Otherwise, since entropy is a state function, the required dumping of heat into the environment to dispose of excess entropy leads to a (minimal) reduction in efficiency. So Equation gives the efficiency of any reversible heat engine. In mesoscopic heat engines, work per cycle of operation in general fluctuates due to thermal noise. If the cycle is performed quasi-statically, the fluctuations vanish even on the mesoscale. However, if the cycle is performed faster than the relaxation time of the working medium, the fluctuations of work are inevitable. Nevertheless, when work and heat fluctuations are counted, an exact equality relates  the exponential average of work performed by any heat engine to the heat transfer from the hotter heat bath. Efficiency of real heat engines. Carnot realized that in reality it is not possible to build a thermodynamically reversible engine, so real heat engines are even less efficient than indicated by Equation . In addition, real engines that operate along the Carnot cycle style (isothermal expansion / isentropic expansion / isothermal compression / isentropic compression) are rare. Nevertheless, Equation is extremely useful for determining the maximum efficiency that could ever be expected for a given set of thermal reservoirs. Although Carnot's cycle is an  idealization, Equation as the expression of the Carnot efficiency is still useful. Consider the average temperatures, at which the first integral is over a part of a cycle where heat goes into the system and the second integral is over a cycle part where heat goes out from the system. Then, replace "T" and "T" in Equation by 〈"T"〉 and 〈"T"〉, respectively, to estimate the efficiency a heat engine. For the Carnot cycle, or its equivalent, the average value 〈"T"〉 will equal the highest temperature available, namely "T", and 〈"T"〉 the lowest, namely "T". For other less efficient thermodynamic cycles , 〈"T"〉 will be lower than "T", and 〈"T"〉 will be higher than "T". This can help illustrate, for example, why a reheater or a regenerator can improve the thermal efficiency of steam power plants and why the thermal efficiency of combined-cycle power plants (which incorporate gas turbines operating at even higher temperatures) exceeds that of conventional steam plants. The first prototype of the diesel engine was based on the Carnot cycle.