# Heat & Work in Int Rev, Polytropic Processes

the value of δ.
How much

#### boundary work

, in kJ/kg, does it take to increase the pressure on an

#### ideal gas

in a closed system from 200 kPa to 400 kPa in an

,

#### polytropic

process ? Assume the specific heats are constant.
Roll the mouse pointer over the big yellow circles to see more information about each process path.
= Area under the process path.

#### Heat Capacity Ratioγ = 1.3985

δ = 0.75

Internally reversible processes in which δ < 1 require even more work than an

#### isothermal

process because they must reject extra heat in order to cool the system down.
δ = 1.0 < γ

Internally reversible processes in which δ < γ require more work than an isentropic process and as a result must reject some heat.
: Isothermal
δ = 1.2 < γ

Internally reversible processes in which δ < γ require even more work than an

#### isentropic

process and as a result must reject some heat.
δ = γ = 1.3985

This path represents an

,

process. The

#### isentropic

process requires the least work of any process that is strictly a compression (no heat added).
: Isentropic
δ = 2 > γ

The increase in pressure is caused both by

#### heat

to the system.
δ = ∞ > γ

The increase in pressure is caused only by the addition of

#### heat

to the system.
: Isochoric
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### Ch 7, Lesson E, Page 19 - Heat & Work in Int Rev, Polytropic Processes

• Here we have a PV Diagram in which both axes are linear.
• So, it is safe to interpret the area under the process paths as the specific boundary work for an internally
reversible process operating on a closed system.
• The processes we will consider begin at state 1 at the bottom left in the diagram.
• Six process paths are represented by the heavy black curves and six final states are represented by the big black dots with gold circles around them.
• Roll your mouse pointer over each final state to see the value of delta for each process and some comments about that process path.
• Begin with the isochoric path.  Do you know which path that is ?
• The isochoric process requires ZERO work to raise the pressure of the system from 200 to 400 kPa.
• That’s great, but it requires a heat input of more than 216 kJ/kg.
• Now, roll over the next final state to the left of the isochoric process.  For this process path δ is equal to 2.
• This path requires an input of 36 KJ/kg of work, but only requires 54 kJ/kg.
• The next final state to the left is the isentropic process path.  In this case δ = γ = 1.402.
• This path requires more work than either path we have considered so far.  The boundary work input for the isentropic path is 48 kJ/kg.
• But the isentropic process is also adiabatic, so it requires ZERO heat input.
• The next path corresponds to δ = 1.2 which is less than gamma, but still greater than 1.
• This path requires an input of 53 kJ/kg of work but REJECTS 27 kJ/kg in the form of heat.
• When δ equals 1, the process is isothermal and all of the energy put in as work is removed as heat.  Consequently, the work requirement and heat rejected are large, 60.3 kJ/kg.
• δ = 0.75 for the last process path.  This path requires the largest work input, but rejects even MORE heat because the temperature of the system DROPS during the process.
• The work input is 71.8 kJ/kg and the heat rejected is 116.4 kJ/kg.
• So, you see it’s a tradeoff.  You can raise the pressure using heat or work.  If you put in just the right amount of work, no heat transfer is required.  If you put in too little work, you have to add even more heat to make the final pressure of 400 kPa.
• Of course if you put in extra work, the energy must be rejected in the form of heat.
• You’ll have a chance to see the detail of calculations I did to construct this diagram when you look at Example #3.
• That concludes this lesson about isentropic and polytropic processes.
• I’ve prepared 3 good example problems for you to study before you try the quiz because this is a very meaty and important lesson.
• It was tough to summarize this lesson in just one page, but I did my best.  Be sure to take a look.
• This also wraps up this chapter.
• Now, that you have a good grasp of the nature of entropy we will move on to using entropy to analyze open systems and flow processes in Chapter 8.
• In some sense, the purpose of the whole course so far was just to prepare you for Chapter 8.  It’s that important.