Basic Mechanical Engineering (3110006)

Question-3b

BE | Semester-1 Summer-2019 | 04-06-2023

Q3) (b)

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 $T_{1} = T_{2} = T_{H}$ .
During this process $Q_{s}$ amount of heat is supplied to the system.
Heat supplied per kg of gas during isothermal expansion process (1 – 2),
$Q_{s} = Q_{1 - 2} = p_{1} V_{1} \ln \frac{V_{2}}{V_{1}} = {RT}_{H} \ln \frac{V_{2}}{V_{1}} (∵ T_{H} = T_{1} = T_{2}) - - - - - [1]$

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 $T_{H}$ to $T_{L}$ .
For adiabatic expansion process (2 – 3),
$\frac{T_{2}}{T_{3}} = \frac{T_{H}}{T_{L}} = {(\frac{V_{3}}{V_{2}})}^{γ - 1} - - - - - [2]$

Isothermal compression (3 – 4)

The adiabatic cover is removed and sink (S) is applied to the end of the cylinder.
The heat, $Q_{r}$ is transferred isothermally at a temperature $T_{L}$ from the system to the sink (S).
Heat rejected per kg of gas during isothermal compression process (3 – 4),
$Q_{r} = Q_{3 - 4} = p_{3} V_{3} \ln \frac{V_{3}}{V_{4}} = {RT}_{L} \ln \frac{V_{3}}{V_{4}} (∵ T_{L} = T_{3} = T_{4}) - - - - - [3]$

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 $T_{L}$ to $T_{H}$ .
For adiabatic compression process (4 – 1),
$\frac{T_{1}}{T_{4}} = \frac{T_{H}}{T_{L}} = {(\frac{V_{4}}{V_{1}})}^{γ - 1} - - - - - [4]$

The expression of efficiency of Carnot cycle: