Polytropic process

Polytropic

Polytropic process

A polytropic process is a type of thermodynamic process that follows the relation:

\[ PV^n = \text{constant} \]

where \( P \) is the pressure, \( V \) is the volume, and \( n \) is the polytropic index. This equation describes a wide range of processes, including isothermal processes, adiabatic processes, and isobaric processes, depending on the value of \( n \).

Polytropic Index

The polytropic index \( n \) determines the nature of the process:

• \( n = 0 \): Isobaric process (constant pressure)
• \( n = 1 \): Isothermal process (constant temperature)
• \( n = \gamma \): Adiabatic process (no heat exchange), where \( \gamma \) is the heat capacity ratio
• \( n \to \infty \): Isochoric process (constant volume)

Applications

Polytropic processes are used to model the behavior of real gases in various thermodynamic cycles, such as the Carnot cycle, Otto cycle, and Brayton cycle. They are also applicable in the study of stellar structure and astrophysics.

Mathematical Formulation

The general form of the polytropic process equation is:

\[ PV^n = C \]

where \( C \) is a constant. The work done \( W \) during a polytropic process can be calculated using:

\[ W = \frac{P_1 V_1 - P_2 V_2}{1 - n} \]

for \( n \neq 1 \), and

\[ W = P V \ln \left( \frac{V_2}{V_1} \right) \]

for \( n = 1 \).

Related Concepts

• Thermodynamics
• Heat transfer
• Ideal gas law
• Entropy
• Enthalpy

See Also

• Isothermal process
• Adiabatic process
• Isobaric process
• Isochoric process
• Thermodynamic cycle
• Carnot cycle
• Otto cycle
• Brayton cycle

References

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