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: Reversible isothermal expansion of the gas at the "hot" temperature, TH .
Reversible isothermal compression of the gas at the "cold" temperature, TC. Now the surroundings do work on the gas, causing an amount of heat energy Q2 and of entropy to flow out of the gas to the low temperature reservoir.
The behaviour of a Carnot engine or refrigerator is best understood by using a temperature-entropy diagram , in which the thermodynamic state is specified by a point on a graph with entropy as the horizontal axis and temperature as the vertical axis.
For a clockwise cycle, the
area under the upper portion will be the thermal energy absorbed during
the cycle, while the area under the lower portion will be the thermal
energy removed during the cycle.
Referring to figure 1, 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 if the loop is traversed in a clockwise direction, and is equal to the total work done on the system as the loop is traversed in a counterclockwise direction.
The total amount of thermal energy
transferred between the hot reservoir and the system will be and the
total amount of thermal energy transferred between the system and the
cold reservoir will be is the work done by the system , is the heat put
into the system , is the absolute temperature of the cold reservoir, and
is the absolute temperature of the hot reservoir. is the maximum system
entropy is the minimum system entropy This efficiency makes sense for a
heat engine , since it is the fraction of the heat energy extracted
from the hot reservoir and converted to mechanical work.
Therefore, all the processes that comprised it can be reversed, in which case it becomes the Carnot 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 in 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 cycle, except that the directions of the processes are reversed.
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.
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.
For the case when work and heat fluctuations are counted, there is exact equality that relates average of exponents of work performed by any heat engine and the heat transfer from the hotter heat bath. This relation transforms the Carnot's inequality into exact equality that is applied to an arbitrary heat engine coupled to two heat reservoirs and operating at arbitrary rate.
Carnot realized that in reality it is not possible to build a thermodynamically reversible engine, so real heat engines are less efficient than indicated by Equation 3. In addition, real engines that operate along this cycle are rare. Nevertheless, Equation 3 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 idealisation, the expression of Carnot efficiency is still useful.
The Carnot cycle when acting as a heat engine consists of the following steps: Reversible isothermal expansion of the gas at the "hot" temperature, TH .
Reversible isothermal compression of the gas at the "cold" temperature, TC. Now the surroundings do work on the gas, causing an amount of heat energy Q2 and of entropy to flow out of the gas to the low temperature reservoir.
The behaviour of a Carnot engine or refrigerator is best understood by using a temperature-entropy diagram , in which the thermodynamic state is specified by a point on a graph with entropy as the horizontal axis and temperature as the vertical axis.
Referring to figure 1, 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 if the loop is traversed in a clockwise direction, and is equal to the total work done on the system as the loop is traversed in a counterclockwise direction.
Therefore, all the processes that comprised it can be reversed, in which case it becomes the Carnot 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 in 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 cycle, except that the directions of the processes are reversed.
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.
For the case when work and heat fluctuations are counted, there is exact equality that relates average of exponents of work performed by any heat engine and the heat transfer from the hotter heat bath. This relation transforms the Carnot's inequality into exact equality that is applied to an arbitrary heat engine coupled to two heat reservoirs and operating at arbitrary rate.
Carnot realized that in reality it is not possible to build a thermodynamically reversible engine, so real heat engines are less efficient than indicated by Equation 3. In addition, real engines that operate along this cycle are rare. Nevertheless, Equation 3 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 idealisation, the expression of Carnot efficiency is still useful.
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