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= (P2V2 – P1V1)/ (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:
PVn = C…………………………………….Eq.2
Also, you will require the following ideal gas equation:
W – Work done by the polytropic process
P1 – Initial pressure
V1 – Initial volume
P2 – Final pressure
V2 – 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.
- Given data:
P1 = 125 KPa
V1=0.05 Cubic Meter
T1=140 Deg C.
T2=110 Deg C.
- By converting the temperature from centigrade to Kelvin:
- By using the Eqn.3, V2 can be calculated as
V2=0.0552 Cubic Meter
- By using the Eqn.2, n can be calculated as
- Finally, the Eqn.1 is used for calculating the work done by the system as
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.
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