Carnot cycle
A Carnot cycle is an ideal thermodynamic cycle proposed by French physicist Sadi Carnot in 1824 and expanded upon by others in the 1830s and 1840s. By Carnot's theorem, it provides an upper limit on the efficiency of any classical thermodynamic engine during the conversion of heat into work, or conversely, the efficiency of a refrigeration system in creating a temperature difference through the application of work to the system.
In a Carnot cycle, a system or engine transfers energy in the form of heat between two thermal reservoirs at temperatures and , and a part of this transferred energy is converted to the work done by the system. The cycle is reversible, merely transferring thermal energy between the thermal reservoirs and the system without gain or loss. When work is applied to the system, heat moves from the cold to hot reservoir. When heat moves from the hot to the cold reservoir, the system applies work to the environment. The work done by the system or engine to the environment per Carnot cycle depends on the temperatures of the thermal reservoirs per cycle such as, where is heat transferred from the hot reservoir to the system per cycle.
Stages
A Carnot cycle is an idealized thermodynamic cycle performed by a Carnot heat engine, consisting of the following steps:In this case, since it is a reversible thermodynamic cycle
This is true as and are both smaller in magnitude and in fact are in the same ratio as.
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. Heat transfer from point 4 to 1 and point 2 to 3 are equal to zero.The temperature-entropy graph
A Carnot cycle plotted on a Temperature-entropy diagram is rather simple. Isothermic paths are horizontal, adiabatic paths are vertical. The area enclosed by the cycle is the amount of heat energy extracted from the hot reservoir but not delivered to the cold reservoir, which has been converted into work.The heat energy injected into the engine is QH=TH and the heat energy extracted from the engine is QC=TC
Efficiency
The efficiency is defined to be:where
- is the work done by the engine system,
- > 0 is the heat taken from the engine system,
- > 0 is the heat put into the engine system,
- is the absolute temperature of the cold reservoir, and
- is the absolute temperature of the hot reservoir.
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 and, 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 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 are rare. Nevertheless, Equation is extremely useful for determining the maximum efficiency that could ever be expected for a given set of thermal reservoirs.This can help illustrate, for example, why a reheater or a regenerator can improve the thermal efficiency of steam power plants by increasing the value of and why the thermal efficiency of combined-cycle power plants exceeds that of conventional steam plants. The first prototype of the diesel engine was based on the principles of the Carnot cycle.
As a macroscopic construct
The Carnot heat engine is, ultimately, a theoretical construct based on an idealized thermodynamic system. On a practical human-scale level the Carnot cycle has proven a valuable model, as in advancing the development of the diesel engine. However the requirement of perfect reversibility is unattainable in a practical sense, and, ultimately, a perfect Carnot engine is incapable of doing any work in a finite amount of time.As such, per Carnot's theorem, the Carnot engine may be thought as the theoretical limit of macroscopic scale heat engines rather than any practical device that could ever be built.