The Carnot Engine and the Carnot Cycle

The Carnot cycle is the cycle of a heat engine developed by the French engineer Sadi Carnot. It is a hypothetical, idealized heat engine that could have a possible maximum efficiency that is consistent with the second law of thermodynamics. The aim is to solve how much efficiency a heat engine could have given two heat reservoirs at temperatures \(T_H\) and \(T_C\), if, based on the second law, it could not have a 100% heat engine. This is heat engine is known as the Carnot engine.

The concept about reversibility and its connection to the direction of the different thermodynamic processes are important to understand the Carnot cycle. To get the maximum possible efficiency of a heat engine, all irreversible processes must be avoided. These irreversible processes include:

  1. Conversion of work to heat. Since conversion of work to heat is an irreversible process, the heat engine is devised to partially reverse this process – converting heat to work with as great efficiency as possible.
  2. Heat flow through a finite temperature drop. Any heat transfer process in the Carnot cycle must be isothermal at either \(T_H\) or \(T_C\).
  3. Heat transfer between the engine and either of the reservoirs. Any process in which the temperature \(T\) of the working substance changes must be adiabatic.

Thus, all the process happened in the Carnot cycle must either be adiabatic or isothermal. To make each process completely reversible, there must always be maintenance of the thermal and mechanical equilibrium.

The figure below is a representation of a Carnot cycle in which the working substance used is an ideal gas in a cylinder with piston. The arrows in each piston indicates the direction of its motion in every process.

Figure

The Carnot cycle has two reversible isothermal processes and two reversible adiabatic processes. Here are the following steps in the Carnot cycle.

  1. At \(A \rightarrow B\), there is isothermal expansion of the ideal gas at temperature \(T_H\) in which heat, \(Q_H\) is also absorbed.
  2. At \(B \rightarrow C\), the gas expands adiabatically,thus no heat energy enters or leaves the system. The temperature of the gas drops from \(T_H\) to \(T_C\).
  3. At \(C \rightarrow D\), the gas is compressed isothermally at temperature \(T_C\). In this process, the gas rejects heat \(Q_C\).
  4. At \(D \rightarrow A\), there is an adiabatic compression of the gas going back to its initial state at temperature \(T_H\).

The thermal efficiency of an engine is given by the equation

\(e = 1 – {|Q_C| \over |Q_H|}\)          (1)

Note that in a Carnot cycle,

\({|Q_C| \over |Q_H|} = {T_C \over T_H}\)          (2)

Thus, to solve for the thermal efficiency of the Carnot engine, we use the formula

\(e_C = 1 – {|T_C| \over |T_H|}\)          (3)

Equation (3) implies that all Carnot engines which operates between two the same temperatures have the same efficiency. If \(T_C = T_H\), the efficiency of the Carnot engine is zero. If \(T_C\) is decreased and \(T_H\) is increased, the efficiency of the engine increases.