Typically the polytropic problems are based on the polytropic compression or expansion process and can be solved by using the following polytropic work equation:

W= (P_{2}V_{2} – P_{1}V_{1})/ (n-1)……………**Eq.1**

However, before attempting to solve any such problem you must understand that all the polytropic process are governed by the following equation:

PV^{n} = C…………………………………….**Eq.2**

Also, you will require the following ideal gas equation:

PV=mRT………………………………….**Eq.3**

Where:

W – Work done by the polytropic process

P_{1} – Initial pressure

V_{1} – Initial volume

P_{2} – Final pressure

V_{2} – Final volume

n – Polytropic index (a real number)

C – Constant

m – Mass of the gas

R – Gas constant

Now, as you know all the necessary equations, let’s try to apply the equations for solving a typical problem.

**Problem statement: ** Some amount of inert gas is entrapped in a piston cylinder. The volume of the inert gas is 0.05 cubic meters at the temperature of 140 degree centigrade and 125 KPa pressure. The pressure of the inert gas changed to 105 KPa at the temperature of 110 degree centigrade. Assume that the system is behaving polytropically.

You have to find out the work done by the system during the expansion of the entrapped gas.

**Solution**:

- Given data:

P_{1} = 125 KPa

V_{1}=0.05 Cubic Meter

T_{1}=140 Deg C.

P_{2}=105 KPa

T_{2}=110 Deg C.

- By converting the temperature from centigrade to Kelvin:

T_{1}=413 K

T_{2}=383 K

- By using the
**Eqn.3,**V_{2 }can be calculated as

V_{2}=0.0552 Cubic Meter

- By using the
**Eqn.2,****n**can be calculated as

n=1.762199

- Finally, the
**Eqn.1**is used for calculating the work done by the system as

W=0.59564 KJ

The methodology used by the above sample problem can be used for solving any similar polytropic work equation problem related to polytropic fluid compression or expansion process.

**References**

http://www.grc.nasa.gov/WWW/K-12/airplane/thermo.html

http://www.taftan.com/thermodynamics/POLYTROP.HTM** **

### Shibashis Ghosh

Disclaimer: I work for Altair. mechGuru.com is my personal blog. Although i have tried to put my neutral opinion while writing about different competitor's technologies, still i would like you to read the articles by keeping my background in mind.

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Nice working out, really easy to follow your steps and understand. (insert sarcasm here)

R u mad or what total calculation is wrong

Deepak,

If you think something is wrong, please do suggest corrections.

please can solve this:-

A mass of air occupying 0.5 m3 at 2 bar and 200oC is compressed reversibly and adiabatically to 5 bar and then it

undergoes isobaric expansion so that it gives out 45 kJ of work. If the system

is to be brought back to its initial state what should be the polytropic index

? calculate the network interaction of this cycle. Sketch the cycle on a p-V

diagram. Also compute the power developed if the number of cycles executed per

minute is 300.