# What Do Areas on TS Represent (Open Systems)

## SISO, Internally Reversible, Steady-State Processes

In Lesson 7B, we showed that the area under an internally reversible process path on a
TS Diagram was equal to the amount of heat transferred during the process.
What is the relationship between and the area under the process path on a TS Diagram for an open system?
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### Ch 8, Lesson B, Page 6 - What Do Areas on TS Represent (Open Systems)

• Let’s begin with our steady-state entropy balance equation for SISO systems.
• First, let’s assume our system is at a uniform temperature.  This lets us eliminate the summation in the heat transfer term.
• Let’s also limit our analysis to internally reversible processes.  This means that Sgen,dot is zero.
• These assumptions greatly simplify the equation.
• It may be easier to see how the area under the process path can be interpreted if we write the differential form of the entropy balance equation.
• That is, mdot times dShat is equal to dQdot over T.
• A little algebra and the integration of dQdot gives us the equation in the first yellow box.
• It is important that M-dot is a constant at steady-state or the equation in the yellow box would not be true.
• This equation tells us that the area under the process path on a TS Diagram is equal to the ratio of the heat transfer rate to the mass flow rate.
• The equation gets even simpler when the temperature of the system is a constant.
• But, remember, these equations only apply for internally reversible, SISO processes operating at steady-state.
• You may be wondering why these equations are important.
• When these equations are combined with the 1st Law and Gibbs 2nd Equation, we get an extremely useful equation for determining the shaft work for a process.
• You’ll see how this works in the next few pages.