Area on a T-S Diagram: Irreversible Process

Ch 7, Lesson C, Page 5 - Area on a T-S Diagram: Irreversible Process

• Consider an irreversible process that changes the state of a system from state 1 to state 2 as shown here on a TS Diagram.
• The area under the curve is equal to the integral of T dS evaluated from state 1 to state 2.
• But two pages back we showed that dS is equal to dQ over T PLUS δSgen.  This is the equation that defines entropy generation.
• Now, we can rearrange this equation and solve for dQ.  No problem, right ?
• Now, we just integrate each term in the equation from state 1 to state 2.
• The integral of dQ from state 1 to state 2 is simply Q, the heat transferred during the process.
• The first equation on this page tells us that the integral of T dS from state 1 to state 2 is the area under the process path on the TS Diagram.  This is the quantity that we are interested in right now.
• What can we do with the integral of T δSgen ?
• In general, not much.  Because Sgen is a path variable, we cannot directly evaluate this integral without detailed knowledge of the process path.
• However, we know that T is positive because we are dealing with absolute temperature.
• We also know that Sgen and δSgen are always positive as well for irreversible processes such as this one..
• Therefore, we can conclude that the integral of T dS from state 1 to state 2 is ALWAYS positive for irreversible processes.
• This means that the area under the irreversible process path on the TS Diagram is NOT equal to Q.
• This fact is VERY important.
• I hope you can see that the area is always GREATER than Q for irreversible processes.
• Unfortunately, the area under an irreversible process path on the TS Diagram is not very useful because the integral of T δSgen generally cannot be directly evaluated.
• So, don’t use this area incorrectly.
• Now, let’s shift gears a bit and finally show how this lesson got its title.
• Let’s talk about the principle of increasing entropy.