Carnot Cycle

The Carnot Cycle

A Carnot cycle is shown in Figure 3.4. It has four processes. There are two adiabatic reversible legs and two isothermal reversible legs. We can construct a Carnot cycle with many different systems, but the concepts can be shown using a familiar working fluid, the ideal gas. The system can be regarded as a chamber enclosed by a piston and filled with this ideal gas.

clip_image001

Figure 3.4: Carnot cycle — thermodynamic diagram on left and schematic of the different stages in the cycle for a system composed of an ideal gas on the right

The four processes in the Carnot cycle are:

  1. The system is at temperature clip_image002at state clip_image003. It is brought in contact with a heat reservoir, which is just a liquid or solid mass of large enough extent such that its temperature does not change appreciably when some amount of heat is transferred to the system. In other words, the heat reservoir is a constant temperature source (or receiver) of heat. The system then undergoes an isothermal expansion from clip_image003[1]to clip_image004, with heat absorbed clip_image005.
  2. At state clip_image004[1], the system is thermally insulated (removed from contact with the heat reservoir) and then let expand to clip_image006. During this expansion the temperature decreases to clip_image007. The heat exchanged during this part of the cycle, clip_image008)
  3. At state clip_image006[1]the system is brought in contact with a heat reservoir at temperature clip_image007[1]. It is then compressed to state clip_image009, rejecting heat clip_image010in the process.
  4. Finally, the system is compressed adiabatically back to the initial state clip_image003[2]. The heat exchange clip_image011.

The thermal efficiency of the cycle is given by the definition

clip_image012

(3..4)

In this equation, there is a sign convention implied. The quantities clip_image013, clip_image014as defined are the magnitudes of the heat absorbed and rejected. The quantities clip_image010[1], clip_image005[1]on the other hand are defined with reference to heat received by the system. In this example, the former is negative and the latter is positive. The heat absorbed and rejected by the system takes place during isothermal processes and we already know what their values are from Eq. (3.1):

clip_image016

clip_image018

The efficiency can now be written in terms of the volumes at the different states as

clip_image019

(3..5)

The path from states clip_image004[2]to clip_image006[2]and from clip_image003[3]to clip_image009[1]are both adiabatic and reversible. For a reversible adiabatic process we know that clip_image020. Using the ideal gas equation of state, we have clip_image022. Along curve clip_image004[3]clip_image006[3] , therefore, clip_image023. Along the curve clip_image009[2]clip_image003[4] , clip_image024. Thus,

clip_image025

Comparing the expression for thermal efficiency Eq. (3.4) with Eq. (3.5) shows two consequences. First, the heats received and rejected are related to the temperatures of the isothermal parts of the cycle by

clip_image026

(3..6)

Second, the efficiency of a Carnot cycle is given compactly by

clip_image027

(3..7)

The efficiency can be 100% only if the temperature at which the heat is rejected is zero. The heat and work transfers to and from the system are shown schematically in Figure 3.5.

clip_image028

Figure 3.5: Work and heat transfers in a Carnot cycle between two heat reservoirs

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