**Content**

- Polytropic Process
- Definition polytropic process
- Polytropic equation
- Polytropic Index
- Some important relations between Pressure [P], Volume [V] and temperature [T] in Polytropic process for Ideal Gas
- Polytropic work
- Polytropic Heat transfer
- Polytropic vs isentropic process
- Polytropic process vs adiabatic process
- Polytropic efficiency
- Polytropic head
- Polytropic process for air
- Polytropic process examples

**Polytropic Process**

**Definition Polytropic process**

“A polytropic process is a thermodynamic process that obeys the relation: PV

^{n}= C, where where p is the pressure, V is volume, n is the polytropic index, and C is a constant. The polytropic process equation can describe multiple expansion and compression processes which include heat transfer.”

**Polytropic Equation** | **Polytropic equation of state**

The polytropic process can be defined by the equation

the exponent **n **is called polytropic index. It depends upon the material and varies from 1.0 to 1.4. This is constant specific heat procedure, in which heat absorption of gas taken into consideration because of unit rise in temperature is fixed.

**Polytropic index**

**n < 0:**Negative Polytropic index denotes a process where Work and heat transfer occurs simultaneously through the boundaries of system. However, such spontaneous process violates the 2^{nd}law of thermo-dynamics. These special cases are used in thermal interaction for astrophysics and chemical energy.**n = 0: P = C:**Represents an isobaric process or constant pressure process.**n = 1: PV = C:**Under the Assumption of Ideal Gas Law, The PV = C represents the Constant Temperature or Isothermal Process.**1 < n < γ:**Under the assumption of ideal gas law, In these Process the heat and work flow move in opposite direction (K>0) Like in Vapor compression cycles, Heat lost to hot surrounding.**n = γ:**Under the assumption of ideal gas law, represents the Constant entropy or Isentropic Process or reversible adiabatic process.**γ < n < Infinity:**In this process it is assumed that heat and work flow move in same direction like in IC engine when some amount of generated heat is lost to the cylinder walls etc.**n = Infinity :**Represents an isochoric process or constant volume process

**Some important relations between Pressure [P], Volume [V] and temperature [T] in Polytropic process for an Ideal Gas**

Polytropic equation is,

………………………. Relations between Pressure [P] and Volume [V]

………………………. Relations between Volume [V] and Temperature [T]

………………………. Relations between Pressure [P] and Temperature [T]

**Polytropic Work**

Ideal gas equation for polytropic process is given by

**Polytropic Heat transfer**

According to 1^{st} law of thermodynamics,

dQ=dU+W

**Polytropic vs isentropic process**

Polytropic process is a thermodynamic process which follows the equation

PV^{n} = C

This process takes into consideration the frictional losses and irreversibility factor of a process. It is a real-life actual process followed by the gas under specific conditions.

Isentropic Process also known as reversible Adiabatic process is an ideal process in which no energy transfer or heat transfer takes place across the boundaries of the system. In this process system is assumed to have an insulated boundary. Since Heat transfer is zero. **dQ = 0**

According to first law of thermodynamics,

**Polytropic process vs adiabatic process**

Polytropic process is a thermodynamic process which follows the equation

PV^{n} = C

This process takes into consideration the frictional losses and irreversibility factor of a process. It is a real-life actual process followed by the gas under specific conditions.

Adiabatic process is a special and specific condition of polytropic process in which .

Similar to Isentropic process in this process too, no energy transfer or heat transfer takes place across the boundaries of the system. In this process system is assumed to have an insulated boundary.

**Polytropic efficiency**

“Polytropic Efficiency well-defined as the ratio of Ideal work of compression for a differential pressure change in a multi-stage Compressor, to the Actual work of compression for a differential pressure change in a multi-stage Compressor.”

In simple terms it is an isentropic efficiency of the process for an infinitesimally small stage in a multi-stage compressor.

Where, γ = Adiabatic index

P_{d} = Delivery Pressure

P_{s} = Suction Pressure

T_{d} = Delivery Temperature

T_{s} = Suction temperature

**Polytropic head**

Polytropic Head can be defined as the Pressure Head developed by a centrifugal compressor as the gas or air is being polytropically compressed. The amount of pressure developed depends upon the density of the gas is compressed and that varies with variation in density of gas.

Where,

γ= Adiabatic index

z_{avg} = Average compressibility factor

η = Polytropic efficiency

P_{d} = Delivery Pressure

P_{s} = Suction Pressure

S = Specific gravity of gas

T_{s} = Suction temperature

**Polytropic process for air** | **Polytropic process for an ideal gas**

Air is assumed to be an Ideal gas and thus the laws of ideal gas is applicable to air.

Polytropic equation is,

………………………. Relations between Pressure [P] and Volume [V]

………………………. Relations between Volume [V] and Temperature [T]

………………………. Relations between Pressure [P] and Temperature [T]

**Polytropic process examples**

## 1. Consider a polytropic process having polytropic index **n = (1.1)**. Initial conditions are: P_{1} = 0, V_{1} = 0 and ends with P_{2}= 600 kPa, V_{2} = 0.01 m^{3}. Evaluate the work done and Heat Transfer.

Answer: Work done by Polytropic process is given by

Heat Transfer is given by

## 2. A piston-cylinder contains Oxygen at 200 kPa, with volume of 0.1 m^{3} and at 200°C. Mass is added at such that the gas compresses with PV^{1.2} = constant to a final temperature of 400°C. Calculate the work done.

Ans: Polytropic work done is given by

## 3. Consider Argon at 600 kPa, 30°C is compressed to 90°C in a polytropic process with n = 1.33. Find the work done on the Gas.

Ans: Polytropic work done is given by

for Argon at 30°C is 208.1 J/kg. K

Assuming m = 1 kg

work done is

## 4. Assume mass of 10kg of Xenon is stored in a cylinder at 500 K, 2 MPa, expansion is a Polytropic process (n = 1.28) with final pressure 100 kPa. Calculate the work done. Consider the system has constant specific heat.

Ans: Polytropic work done is given by

We know that,

for Xenon at 30°C is 63.33 J/kg. K

Assuming m = 10 kg

work done is

## 5. Take into consideration a cylinder-piston having initial volume 0.3 containing 5kg methane gas at 200 kPa. The gas is compressed polytropically (n = 1.32) to a pressure of 1 MPa and volume 0.005 . Calculate the Heat transfer during the process.

Ans: Polytropic Heat Transfer is given by

## 6. Take into consideration a cylinder-piston containing 1kg methane gas at 500 kPa, 20°C. The gas is compressed polytropically to a pressure of 800 kPa. Calculate the Heat Transfer with exponent n = 1.15.

Ans: Polytropic Heat Transfer is given by

We know that, R for methane = 518.2 J/kg. K

## 7. 1 kg of Helium is stored in a piston – cylinder arrangement at 303 K, 200 kPa is compressed to 400K in a reversible polytropic process with exponent n = 1.24. Helium is an ideal gas characteristics so specific heat will be fixed. Find the work and Heat Transfer.

Ans: Polytropic work done is given by

R for Helium is 2077.1 J/kg

Polytropic Heat Transfer is given by

## 8.Assume air stored in a cylinder having volume of 0.3 Liters at 3 MPa, 2000K. Air expands following a reversible polytropic process with exponent, n = 1.7, a volume ratio is observed as 8:1 in this case. Calculate the polytropic work for the process and compare it with adiabatic work if the expansion process follows reversible adiabatic expansion.

Ans: We are given with

Relations between Pressure [P] and Volume [V]

Polytropic work done is given by

Adiabatic work done is given by

For expansion process the Work done through reversible adiabatic process is greater than the Work done through reversible Polytropic process.

## 9. A closed container contains 200L of gas at 35°C, 120 kPa. The gas is in compressng in a polytropic process till it reaches to 200°C, 800 kPa. Find the polytropic work done by the air for n = 1.29.

Ans: Relations between Pressure [P] and Volume [V]

Polytropic work done is given by

## 10. A mass of 12 kg methane gas at 150°C, 700 kPa, undergoes a polytropic expansion with n = 1.1, to a final temperature of 30°C. Find the Heat Transfer?

Ans: We know that, R for methane = 518.2 J/kg. K

Polytropic Heat Transfer is given by

## 11. A cylinder-piston Assembly contains R-134a at 10°C; the volume is 5 Liters. The Coolant is compressed to 100°C, 3 MPa Following a reversible polytropic process. calculate the work done and Heat Transfer?

Ans: We know that, R for R-134a = 81.49 J/kg. K

Polytropic work done is given by

Polytropic Heat Transfer is given by

## 12. Is a polytropic process isothermal in nature?

Ans: When **n becomes 1 **for a polytropic process: Under the Assumption of Ideal Gas Law, The PV = C represents the Constant Temperature or Isothermal Process.

## 13. Is a polytropic process reversible?

Ans: a polytropic processes are internally reversible. Some examples are:

**n = 0: P = C: ** Represents an isobaric process or constant pressure process.

**n = 1: PV = C:** Under the Assumption of Ideal Gas Law, The PV^{γ} = C represents the Constant Temperature or Isothermal Process.

**n = γ: **Under the assumption of ideal gas law, represents the Constant entropy or Isentropic Process or reversible adiabatic process.

**n = Infinity :** Represents an isochoric process or constant volume process.

## 14. Is adiabatic polytropic process?

Ans: when **n = γ: **Under the assumption of ideal gas law PV^{γ} = C, represents the Constant entropy or Isentropic Process or reversible adiabatic process.

## 14. What is Polytropic efficiency?

Ans: Polytropic Efficiency can be defined as the ratio of Ideal work of compression, to the Actual work of compression for a differential pressure change in a multi-stage Compressor. In simple terms it is an isentropic efficiency of the process for an infinitesimally small stage in a multi-stage compressor.

In simple terms it is an isentropic efficiency of the process for an infinitesimally small stage in a multi-stage compressor.

Where, γ = Adiabatic index

P_{d} = Delivery Pressure

P_{s} = Suction Pressure

T_{d} = Delivery Temperature

T_{s} = Suction temperature

## 15. What is Gamma in Polytropic process?

Ans: In a Polytropic process when **n = γ: **Under the assumption of ideal gas law PV^{γ} = C, represents the Constant entropy or Isentropic Process or reversible adiabatic process.

## 16. what is n in polytropic process?

Ans: The polytropic process can be defined by the equation,

PV^{n} = C

the exponent **n **is called polytropic index. It depends upon the material and varies from 1.0 to 1.4. It is also called as constant specific heat process, in which heat absorbed by the gas taken into consideration because of unit rise in temperature is constant.

## 17. What conclusions can be made for a polytropic process with n=1?

Ans: when **n = 1:** PV^{n} = C **:** Under the Assumption of Ideal Gas Law becomes The PV = C represents the Constant Temperature or Isothermal Process.

## 18. What is a non-polytropic process?

Ans: The polytropic process can be defined by the equation PV^{n} = C , the exponent **n **is called polytropic index. When,

**n < 0:**Negative Polytropic index denotes a process where Work and heat transfer occurs simultaneously through the boundaries of system. However, such spontaneous process violates the Second law of Thermodynamics. These special cases are used in thermal interaction for astrophysics and chemical energy.**n = 0: P = C:**Represents an isobaric process or constant pressure process.**n = 1: PV = C:**Under the Assumption of Ideal Gas Law, The PV = C represents the Constant Temperature or Isothermal Process.**1 < n < γ:**Under the assumption of ideal gas law, In these Process the heat and work flow move in opposite direction (K>0) Like in Vapor compression cycles, Heat lost to hot surrounding.**n = γ:**Under the assumption of ideal gas law, PV^{γ}= C represents the Constant entropy or Isentropic Process or reversible adiabatic process.**γ<****n****< Infinity :**In this process it is assumed that heat and work flow move in same direction like in IC engine when some amount of generated heat is lost to the cylinder walls etc.**n = Infinity :**Represents an isochoric process or constant volume process

## 19. Why is heat transfer negative in a polytropic process?

Ans: Polytropic Heat Transfer is given by

When **γ < n**

**< Infinity**

**:**In this process it is assumed that heat and work flow move in same direction. The change in temperature is due to change in internal energy rather than heat supplied. Thus, even though heat is added in a polytropic expansion the temperature of the gas decreases.

## 20. Why does the temperature decrease on heat addition in the polytropic process?

Ans: Polytropic Heat Transfer is given by

For the condition:** 1 < n < γ: **Under the assumption of ideal gas law, In these Process the heat and work flow move in opposite direction (K>0) Like in Vapor compression cycles, Heat lost to hot surrounding. The change in temperature is due to change in internal energy rather than heat supplied. The work produced exceeds the amount of heat supplied or added. Thus, even though heat is added in a polytropic expansion the temperature of the gas decreases.

## 21. In a polytropic process where PV^{n} =constant, is temperature constant as well?

Ans: In a polytropic process where PV^{n} =constant, the temperature remains constant only when the polytropic index n = 1. For **n = 1: PV = C:** Under the Assumption of Ideal Gas Law, The PV = C represents the Constant Temperature or Isothermal Process.

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