Polytropic process
Polytropic Process
A polytropic process is a thermodynamic process that obeys the relation:
[ PV^n = \text{constant} ]
where ( P ) is the pressure, ( V ) is the volume, ( n ) is the polytropic index, and the constant depends on the specific properties of the gas involved in the process.
Understanding the Polytropic Index
The polytropic index ( n ) determines the nature of the process:
- ( n = 0 ): Isochoric process (constant volume)
- ( n = 1 ): Isobaric process (constant pressure)
- ( n = \gamma ) (ratio of specific heats, ( C_p/C_v )): Adiabatic process (no heat transfer)
- ( n = \infty ): Isothermal process (constant temperature)
Polytropic Process Equation
The general equation for a polytropic process can be written as:
[ PV^n = C ]
where ( C ) is a constant for a particular process.
Work Done in a Polytropic Process
The work done ( W ) during a polytropic process is given by:
[ W = \frac{P_1V_1 - P_2V_2}{1 - n} ]
where ( P_1 ) and ( V_1 ) are the initial pressure and volume, and ( P_2 ) and ( V_2 ) are the final pressure and volume, respectively.
Heat Transfer in a Polytropic Process
The heat transfer ( Q ) can be calculated using the first law of thermodynamics:
[ Q = \Delta U + W ]
where ( \Delta U ) is the change in internal energy.
Differences and Important Points
| Property | Isochoric | Isobaric | Adiabatic | Isothermal | Polytropic |
|---|---|---|---|---|---|
| Volume | Constant | Variable | Variable | Variable | Variable |
| Pressure | Variable | Constant | Variable | Variable | Variable |
| Temperature | Variable | Variable | Variable | Constant | Variable |
| Heat Transfer | Variable | Variable | Zero | Variable | Variable |
| Work Done | Zero | Variable | Variable | Variable | Variable |
| Polytropic Index | ( n = 0 ) | ( n = 1 ) | ( n = \gamma ) | ( n = \infty ) | Any value |
Examples
Example 1: Adiabatic Process
For an adiabatic process where ( n = \gamma ), the work done is:
[ W = \frac{P_1V_1 - P_2V_2}{1 - \gamma} ]
Since there is no heat transfer, ( Q = 0 ), and ( \Delta U = -W ).
Example 2: Isothermal Process
For an isothermal process where ( n = \infty ), the work done is:
[ W = P_1V_1 \ln\left(\frac{V_2}{V_1}\right) ]
Here, ( \Delta U = 0 ) since the temperature is constant, and ( Q = W ).
Example 3: Polytropic Process with a Specific Index
Consider a polytropic process with ( n = 1.2 ), ( P_1 = 100 ) kPa, ( V_1 = 1 ) m³, ( P_2 = 50 ) kPa, and ( V_2 = 2 ) m³. The work done is:
[ W = \frac{100 \times 1 - 50 \times 2}{1 - 1.2} = \frac{0}{-0.2} = 0 ]
This is a special case where the work done is zero, which can occur depending on the values of ( P ), ( V ), and ( n ).
Conclusion
The polytropic process is a versatile concept in thermodynamics that can describe a wide range of processes by varying the polytropic index ( n ). Understanding this process is crucial for solving problems in heat and thermodynamics, especially in engineering applications.