Carnot+Efficiency

The **Carnot cycle** is a theoretical thermodynamic cycle proposed by [|Nicolas Léonard Sadi Carnot] in 1824 and expanded by [|Benoit Paul Émile Clapeyron] in the 1830s and 40s. It can be shown that it is the most efficient cycle for converting a given amount of thermal energy into work, or conversely, creating a temperature difference (e.g. refrigeration) by doing a given amount of work. Every thermodynamic system exists in a particular thermodynamic 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, 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 limit and cannot be built in practice. The **Carnot cycle** when acting as a heat engine consists of the following __steps__:
 * 1) **Reversible isothermal expansion of the gas at the "hot" temperature, //T////H// (isothermal heat addition or absorption).**
 * 2) During this __step__ (A to B on Figure 1, 1 to 2 in Figure 2) the expanding gas makes the piston work on the surroundings. The gas expansion is propelled by absorption of quantity Q1 of heat from the high temperature reservoir.
 * 3) ** [|Isentropic] ( [|reversible adiabatic] ) expansion of the gas (isentropic work output).**
 * 4) For this step (B to C on Figure 1, 2 to 3 in Figure 2) the piston and cylinder are assumed to be thermally insulated, thus they neither gain nor lose heat. The gas continues to expand, working on the surroundings. The gas expansion __causes__ it to cool to the "cold" temperature, //T////C//.
 * 5) **Reversible isothermal compression of the gas at the "cold" temperature, //T////C//. (isothermal heat rejection)**(C to D on Figure 1, 3 to 4 on Figure 2)
 * 6) Now the surroundings do work on the gas, causing quantity Q2 of heat to flow out of the gas to the low temperature reservoir.
 * 7) **Isentropic compression of the gas (isentropic work input).**(D to A on Figure 1, 4 to 1 in Figure 2)
 * 8) Once again the piston and cylinder are assumed to be thermally insulated. During this step, the surroundings do work on the gas, compressing it and causing the temperature to rise to //T////H//. At this point the gas is in the same state as at the __start__ of step 1.
 * 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 3 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. To find the absolute temperature in kelvin, add 273.15 degrees to the Celsius temperature. 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.
 * 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 3 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. To find the absolute temperature in kelvin, add 273.15 degrees to the Celsius temperature. 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.

Figure 1: A Carnot cycle acting as a heat engine, illustrated on a temperature-entropy diagram. The cycle takes place between a hot reservoir at temperature TH and a cold reservoir at temperature TC. The vertical axis is temperature, the horizontal axis is entropy. ||  Figure 2: A Carnot cycle acting as a heat engine, illustrated on a pressure-volume diagram to illustrate the work done. ||
 * [[image:http://upload.wikimedia.org/wikipedia/commons/thumb/b/b6/CarnotCycle1.png/400px-CarnotCycle1.png width="400" height="300" link="http://en.wikipedia.org/wiki/File:CarnotCycle1.png"]]