Derive the equation for efficiency of Carnot cycle.
A Carnot cycle is a hypothetical cycle consisting of four different processes: two isothermal processes and two adiabatic processes.
The essential elements and p-V diagram for a Carnot cycle shown in Figure.
Assumptions made in the working of the Carnot cycle are:
- Working fluid is a perfect gas.
- Piston cylinder arrangement is weightless and does not produce friction during motion.
- The walls of the cylinder and piston are considered perfectly insulated.
- Compression and expansion process are reversible.
- The transfer of heat does not change the temperature of sources or sink.
The Carnot cycle has the highest possible efficiency and it consists of four simple operations as below:
Isothermal expansion (1 – 2)
- The source of heat (H) is applied to the end of the cylinder and isothermal expansion occurs at a temperature .
- During this process amount of heat is supplied to the system.
- Heat supplied per kg of gas during isothermal expansion process (1 – 2),
Adiabatic expansion (2 – 3)
- Non conducting (insulating) cover (C) is applied to the end of the cylinder and the cylinder becomes perfectly insulated.
- The adiabatic cover is brought in contact with the cylinder head. Hence no heat transfer takes place. The fluid expands adiabatically and the temperature falls from to .
- For adiabatic expansion process (2 – 3),
Isothermal compression (3 – 4)
- The adiabatic cover is removed and sink (S) is applied to the end of the cylinder.
- The heat, is transferred isothermally at a temperature from the system to the sink (S).
- Heat rejected per kg of gas during isothermal compression process (3 – 4),
Adiabatic compression (4 – 1)
- The adiabatic cover is brought in contact with the cylinder head. This completes the cycle and system is returned to its original state at 1.
- During the process, the temperature of the system is raised from to .
- For adiabatic compression process (4 – 1),
The expression of efficiency of Carnot cycle:
- By comparing Eq. (2) and Eq. (4),
- Net work output from the cycle per kg of air,
- Thermal Efficiency,