How can I demonstrate that the integral of vdP is used to calculate work in flow machines?

[Emmanuel S]

I have this little doubt because in class, the professor said the work is equal to the integrate of vdP, but i don't know how to prove it, why it isn't PdV like those exercises in a cylinder-piston

 

[robphy]

Possibly useful:
https://physics.stackexchange.com/questions/52001/what-is-v-dp-work-and-when-do-i-use-it

 

[Chestermiller]

Is your system a flow system, such that fluid is flowing into the system and out of the system through inlets and outlets to a control volume? An example would be a nozzle.

 

[Emmanuel S]

Is your system a flow system, such that fluid is flowing into the system and out of the system through inlets and outlets to a control volume? An example would be a nozzle.

Indeed. In fact, the problem was with a reversible compressor; the process was polytropic, and after performing the mass and energy balance, when calculating the compressor's specific work, the professor established w = ∫ vdP

And he said that for flow machines, the work took that expression, not PdV.

 

[Emmanuel S]

Possibly useful:
https://physics.stackexchange.com/questions/52001/what-is-v-dp-work-and-when-do-i-use-it

Thank you very much, now I understand the concept, I'm working on how to do the balance in such a way that I can demonstrate it in this case.

 

[Emmanuel S]

from Moran, et al, Fundamentals of Engineering Thermodynamics I saw the balance explained

 

[Chestermiller]

Indeed. In fact, the problem was with a reversible compressor; the process was polytropic, and after performing the mass and energy balance, when calculating the compressor's specific work, the professor established w = ∫ vdP

And he said that for flow machines, the work took that expression, not PdV.

For a differential section of the compressor operating at steady state, the open system version of the first law of thermodynamics tells us the $$dW_s=dh=Tds-vdP$$ where h is the enthalpy per unit mass of gas, s is the entropy per unit mass, v is the volume per unit mass, and $W_s$ is the shaft work. For adiabatic reversible operation, ds = 0, so $$dW_s=vdP$$