# Ws for Open Systems on a PV Diagram

#### SISO, Internally Reversible, SS

Negligible changes in kinetic and potential energies

#### Polytropic

processes:
Or:
Therefore:
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### Ch 8, Lesson B, Page 12 - Ws for Open Systems on a PV Diagram

• Consider how we can express the work per unit mass as an area on a PV Diagram with LINEAR axes.
• The shaft work per unit mass of flowing fluid is the integral of Vhat dP.  But wait, I thought work was the integral of P dV.  What is going on here ?
• Well, for internally reversible processes in a closed system, the BOUNDARY work IS the integral of P dV.
• But, for open systems, the other or shaft work per unit mass of flowing fluid is the integral of Vhat dP.  This is the area to the LEFT of the process path on the PV Diagram.
• Get this straight here and now.  I cannot tell you how many students get confused by the similarity of these two equations.
• Once you are comfortable with that, we are ready to roll.
• For polytropic processes, PVhat to the δ is equal to a constant that we’ll call “C.”
• We can solve this path equation for Vhat and plug it into the integral in our modified energy balance equation at the top of this page.
• Then, because C is a constant, we can pull it out of the integral.
• That’s where we stop with our general solution because evaluating this integral depends on whether δ = 1 or not.
• First, let’s consider the case in which δ is not equal to 1.