Carnot Engine - Thermodynamic Engine

Carnot Cycle

A Carnot cycle is defined as an ideal reversible closed thermodynamic cycle. Four successive operations are involved: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression. During these operations, the expansion and compression of the substance can be done up to the desired point and back to the initial state.

Following are the four processes of the Carnot cycle:

• In (a), the process is reversible isothermal gas expansion. In this process, the amount of heat absorbed by the ideal gas is q_in from the heat source at a temperature of T_h. The gas expands and does work on the surroundings.
• In (b), the process is reversible adiabatic gas expansion. Here, the system is thermally insulated, and the gas continues to expand and work is done on the surroundings. Now the temperature is lower, T_l.
• In (c), the process is a reversible isothermal gas compression process. Here, the heat loss q_out occurs when the surroundings do the work at temperature T_l.
• In (d), the process is reversible adiabatic gas compression. Again the system is thermally insulated. The temperature again rises back to T_h as the surrounding continue to do their work on the gas.

Steps involved in a Carnot Cycle

For an ideal gas operating inside a Carnot cycle, the following are the steps involved:

Step 1:

Isothermal expansion: The gas is taken from P₁, V₁, T₁ to P₂, V₂, T₂. Heat Q₁ is absorbed from the reservoir at temperature T₁. Since the expansion is isothermal, the total change in internal energy is zero, and the heat absorbed by the gas is equal to the work done by the gas on the environment, which is given as:

\(\begin{array}{l}W_{1\rightarrow 2}=Q1=\mu \times R\times T_{1}\times ln\frac{v_{2}}{v_{1}}\end{array} \)

Step 2:

Adiabatic expansion: The gas expands adiabatically from P₂, V₂, T₁ to P₃, V₃, T₂.

Here, work done by the gas is given by:

\(\begin{array}{l}W_{2\rightarrow 3}=\frac{\mu R}{\gamma -1}(T_{1}-T_{2})\end{array} \)

Step 3:

Isothermal compression: The gas is compressed isothermally from the state (P₃, V₃, T₂) to (P₄, V₄, T₂).

Here, the work done on the gas by the environment is given by:

\(\begin{array}{l}W_{3\rightarrow 4}=\mu RT_{2}ln\frac{v_{3}}{v_{4}}\end{array} \)

Step 4:

Adiabatic compression: The gas is compressed adiabatically from the state (P₄, V₄, T₂) to (P₁, V₁, T₁).

Here, the work done on the gas by the environment is given by:

\(\begin{array}{l}W_{4\rightarrow 1}=\frac{\mu R}{\gamma -1}(T_{1}-T_{2})\end{array} \)

Hence, the total work done by the gas on the environment in one complete cycle is given by:

\(\begin{array}{l}W= W_{1\rightarrow 2}+W_{2\rightarrow 3}+W_{3\rightarrow 4}+W_{4\rightarrow 1}\\ \\ W=\mu \: RT_{1}\: ln\frac{v_{2}}{v_{1}}-\mu \: RT_{2}\: ln\frac{v_{3}}{v_{4}}\end{array} \)

\(\begin{array}{l}Net\; efficiency =\frac{Net\; workdone\; by\; the\;gas}{Heat\; absorbed\; by\; the\;gas}\end{array} \)

\(\begin{array}{l}Net\; efficiency =\frac{W}{Q_{1}}=\frac{Q_{1}-Q_{2}}{Q_{1}}=1-\frac{Q_{2}}{Q_{1}}=1-\frac{T_{2}}{T_{1}}\frac{ln\frac{v_{3}}{v_{4}}}{ln\frac{v_{2}}{v_{1}}}\end{array} \)

Since the step 2–>3 is an adiabatic process, we can write T₁V₂^Ƴ-1 = T₂V₃^Ƴ-1

Or,

\(\begin{array}{l}\frac{v_{2}}{v_{3}}=(\frac{T_{2}}{T_{1}})^{\frac{1}{\gamma -1}}\end{array} \)

Similarly, for the process 4–>1, we can write

\(\begin{array}{l}\frac{v_{1}}{v_{2}}=(\frac{T_{2}}{T_{1}})^{\frac{1}{\gamma -1}}\end{array} \)

This implies,

\(\begin{array}{l}\frac{v_{2}}{v_{3}}=\frac{v_{1}}{v_{2}}\end{array} \)

So, the expression for net efficiency of Carnot engine reduces to:

\(\begin{array}{l}Net\; efficiency=1-\frac{T_{2}}{T_{1}}\end{array} \)

Frequently Asked Questions – FAQs

Q1

What is a working fluid in Carnot’s cycle?

The working fluid in a Carnot’s cycle is an ideal gas.

Q2

What is a Carnot heat engine?

Carnot heat engine is a theoretical engine that operates on a reversible carnot cycle. It has maximum efficiency that a heat engine can possess.

Q3

Name the processes involved in the Carnot cycle?

It involves four process: isothermal expansion, adiabatic expansion, isothermal compression and adiabatic compression.

Q4

What happens in the isothermal expansion process?

In the isothermal expansion process, gas is taken from P₁, V₁, T₁ to P₂, V₂, T₂. Heat Q₁ is absorbed from the reservoir at temperature T₁. The total change in internal energy is zero and the heat absorbed by the gas is equal to the work done.

Q5

Is Carnot’s cycle reversible?

The Carnot heat-engine cycle described is a totally reversible cycle.

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