Isothermal and adiabatic air compression different expressions

[Ameen1985]

For an Isothermal compression process of air in a vessel with constant volume, I found the following expressions

Two of them give the same exact answer while the third one gives a different answer I don't know why.

For adiabatic air compression, I found these two expressions which give different answer as well I don't know why

Let's say we have a 11 kWh to be stored, P0 = 101325 Pa, P1 = 6.3 bar, P2 = 10 bar, and T0= 298 K. What would be the required air vessel volume?

 

[goutam_freelance]

The third isothermal equation does not consider the effect of constant pressure at tank outlet P_A which reduces the work available from the compressed air. The reduction is
Reduction of work=(V_A-V_B)P_A=(P_B-P_A)V_B.
The second term in RHS of first equation is due to this. Please note the first term in RHS is negative as work is done on the system and you are adding a positive quantity with this.


Engineers, think what we have done to the environment !

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[Ameen1985]

Many thanks @goutam for your reply.

As you said, when I add " - (P_B-P_A)V_B " to the third equation. It gives me the same answer as Equations 1 and 2.

So, should I don't consider the effect of constant pressure at tank outlet as equation 3 suggests or equation 1 and 2 represent the isothermal process well ??

Regarding adiabatic process in equation 4 and 5, which one describes the process better ?

Many thanks in advance

 

[goutam_freelance]

@Ameen1985
Sorry for the delay in reply.
For isentropic compression you may use my formula (the other formula I did not check. I have taken the formula from Wikipedia.
Regarding isentropic compression the below formula(as given by you) is correct for initial compression from P_0 to P_1 and then final compression P_1 to P_2.

Engineers, think what we have done to the environment !

https://www.linkedin.com/in/goutam-das-59743b30/