Topic: Heat Engines and Refrigerators

Detailed Notes from Toppers

1. The Carnot Cycle

• The Carnot cycle is a theoretical cycle that describes the most efficient possible heat engine.
  • It consists of four processes:
    • Isothermal expansion: Heat is added to the system from a high-temperature reservoir.
    • Adiabatic expansion: The system expands without any heat transfer.
    • Isothermal compression: Heat is rejected from the system to a low-temperature reservoir.
    • Adiabatic compression: The system is compressed without any heat transfer.
• The efficiency of a Carnot cycle is given by:

$$\eta = 1 - \frac{T_L}{T_H}$$

Where:

• $T_H$ is the temperature of the high-temperature reservoir
• $T_L$ is the temperature of the low-temperature reservoir

2. The Otto Cycle

• The Otto cycle is a model of the gasoline engine.

  • It consists of four processes:
    • Adiabatic compression: The air-fuel mixture is compressed.
    • Isochoric combustion: The air-fuel mixture is ignited, causing the temperature and pressure to increase.
    • Adiabatic expansion: The hot gases expand, driving the piston.
    • Isochoric heat rejection: The exhaust gases are cooled, causing the pressure and temperature to decrease.

• The efficiency of an Otto cycle is given by:

$$\eta = 1 - \frac{1}{r^{k-1}}$$

Where:

• $r$ is the compression ratio -$k$ is the ratio of specific heats of the air-fuel mixture

3. The Diesel Cycle

• The Diesel cycle is a model of the diesel engine.

  • It consists of four processes:
    • Adiabatic compression: The air is compressed.
    • Isobaric heat addition: Fuel is injected into the air, causing the temperature and pressure to increase.
    • Adiabatic expansion: The hot gases expand, driving the piston.
    • Isochoric heat rejection: The exhaust gases are cooled, causing the pressure and temperature to decrease.

• The efficiency of a Diesel cycle is given by:

$$η=1−\frac{1}{r^k}\left[\frac{k}{k−1}\left(1−(\frac{V_4}{V_3})^k\right)\right]$$

Where:

• $r$ is the compression ratio
• $k$ is the ratio of specific heats of the air
• $V_3$ is the volume at the end of the compression stroke
• $V_4$ is the volume at the end of the expansion stroke

4. Refrigerator Cycles

Refrigerators use a cycle of compression and expansion to cool objects.

• The most common type of refrigerator cycle is the vapor-compression cycle.
• It consists of four processes:
  • Isothermal compression: The refrigerant gas is compressed by a compressor.
  • Adiabatic compression: The hot refrigerant gas is cooled by passing it through a condenser.
  • Isothermal expansion: The refrigerant gas expands through an expansion valve.
  • Adiabatic expansion: The cold refrigerant gas evaporates, absorbing heat from the inside of the refrigerator.

5. Heat Pump Cycles

Heat pumps are used to heat objects. The most common type of heat pump cycle is the vapor-compression cycle. It consists of four processes:

• Isothermal compression: The refrigerant gas is compressed by a compressor.
• Adiabatic compression: The hot refrigerant gas is cooled by passing it through a condenser.
• Isothermal expansion: The refrigerant gas expands through an expansion valve.
• Adiabatic expansion: The cold refrigerant gas evaporates, absorbing heat from the outside air or ground.

6. Entropy

• Entropy is a measure of the disorder of a system.
• In thermodynamics, entropy is defined as the change in heat energy divided by the temperature:

$$S = \frac{Q}{T}$$

Where:

• $S$ is entropy
• $Q$ is the heat energy transferred
• $T$ is the temperature

7. The Second Law of Thermodynamics

• The second law of thermodynamics states that the entropy of an isolated system always increases over time.
• This law has implications for the efficiency of heat engines and refrigerators:
  • The efficiency of a heat engine cannot be 100%.
  • The coefficient of performance of a refrigerator cannot be infinite.

References:

• NCERT Physics Class 11, Chapter 12: Thermodynamics
• NCERT Physics Class 12, Chapter 6: The Dual Nature of Radiation and Matter
• NCERT Physics Class 12, Chapter 7: Alternating Current