Carnot cycle

The 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.Isothermal Expansion. Heat is transferred reversibly from high temperature reservoir at constant temperature TH (isothermal heat addition or absorption). During this step (1 to 2 on Figure 1, A to B in Figure 2) the gas is allowed to expand, doing work on the surroundings by pushing up the piston (stage 1 figure, right). Although the pressure drops from points 1 to 2 (figure 1) the temperature of the gas does not change during the process because it is in thermal contact with the hot reservoir at Th, and thus the expansion is isothermal. Heat energy Q1 is absorbed from the high temperature reservoir resulting in an increase in the entropy of the gas by the amount Δ S 1 = Q 1 / T h {displaystyle Delta S_{1}=Q_{1}/T_{h}} .Isentropic (reversible adiabatic) expansion of the gas (isentropic work output). For this step (2 to 3 on Figure 1, B to C in Figure 2) the gas in the engine is thermally insulated from both the hot and cold reservoirs. Thus they neither gain nor lose heat, an 'adiabatic' process. The gas continues to expand by reduction of pressure, doing work on the surroundings (raising the piston; stage 2 figure, right), and losing an amount of internal energy equal to the work done. The gas expansion without heat input causes it to cool to the 'cold' temperature, Tc. The entropy remains unchanged.Isothermal Compression. Heat transferred reversibly to low temperature reservoir at constant temperature TC. (isothermal heat rejection) (3 to 4 on Figure 1, C to D on Figure 2) Now the gas in the engine is in thermal contact with the cold reservoir at temperature Tc. The surroundings do work on the gas, pushing the piston down (stage 3 figure, right), causing an amount of heat energy Q2 to leave the system to the low temperature reservoir and the entropy of the system to decrease by the amount Δ S 2 = Q 2 / T c {displaystyle Delta S_{2}=Q_{2}/T_{c}} . (This is the same amount of entropy absorbed in step 1, as can be seen from the Clausius inequality.)Adiabatic reversible compression. (4 to 1 on Figure 1, D to A on Figure 2) Once again the gas in the engine is thermally insulated from the hot and cold reservoirs, and the engine is assumed to be frictionless, hence reversible. During this step, the surroundings do work on the gas, pushing the piston down further (stage 4 figure, right), increasing its internal energy, compressing it, and causing its temperature to rise back to Th due solely to the work added to the system, but the entropy remains unchanged. At this point the gas is in the same state as at the start of step 1. Q = ∫ A B T d S {displaystyle Q=int _{A}^{B}T,dS} (1) W = ∮ ⁡ P d V = ∮ ⁡ ( d Q − d U ) = ∮ ⁡ ( T d S − d U ) = ∮ ⁡ T d S − ∮ ⁡ d U = ∮ ⁡ T d S {displaystyle W=oint PdV=oint (dQ-dU)=oint (TdS-dU)=oint TdS-oint dU=oint TdS} (2) η = W Q H = Q H − Q C Q H = 1 − T C T H {displaystyle eta ={frac {W}{Q_{H}}}={frac {Q_{H}-Q_{C}}{Q_{H}}}=1-{frac {T_{C}}{T_{H}}}quad quad quad quad quad quad quad quad } (3) The 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. 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 Th and Tc (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 ΔS 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 energy ThΔS is extracted from the hot reservoir and a smaller amount of energy TcΔS is deposited in the cold reservoir. The difference in the two energies (Th-Tc)ΔS is equal to the work done by the engine.