Thermodynamics polytropic processes and work for them

[mihalj]

Hi, something is a bit confusing in regards to polytropic processes.Can anyone explain why boundary work in closed system for compressor is calculated as W = ∫pdV and polytropic process after integration is generally W = -(p2V2 - p1V1)/(1-n)...(numbers behind p and V are subscripts)...
When we talk about control volume, internally reversible process (isn't above one also reversible?) then work is defined as W = -∫Vdp and for polytropic process for compressor work is defined as W = -n*(p2V2-p1V1)/(n-1).
When I am supposed to use one way of calculation and when other? The results are not same. Aren't they same type of group as control volume compressor should behave during the process of compression as closed system (there is no constant add or removal of mass during compression).
Thanks in advance!

 

[Valkarie]

A:A polytropic process is a thermodynamic process in which the pressure $P$ of a gas varies linearly with its volume $V$, while its temperature remains constant. The work performed by a polytropic process on the system, with initial and final pressures $P_1$ and $P_2$ respectively, and initial and final volumes $V_1$ and $V_2$ respectively, is given by $$ W=-\frac{n}{n-1}\left(P_2V_2-P_1V_1\right)$$ where the polytropic index $n$ is a measure of how closely the polytropic process approximates an isothermal process; it is a property of the thermodynamic system.For a closed system, the work is calculated as $$W=\int_{V_1}^{V_2}P \mathrm{d}V$$For a control volume, where there is a net flow of energy and matter into and out of the system, the work is calculated as $$W=-\int_{V_1}^{V_2}V \mathrm{d}P$$Note that these equations are derived from the first law of thermodynamics, which states that the change in the internal energy of a system is equal to the sum of the heat added to the system and the work done on the system.