Second Law of Thermodynamics Entropy

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Second Law of Thermodynamics - Entropy

• Chapter 6

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Entropy

• What is Entropy?
• Its a quantitative measure of disorder.
• The unit of entropy is J/K

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Clausius Inequality
Two part system Reversible Cyclic Device (heat
engine) Piston Cylinder Device
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Consider an energy balance for the entire system
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If the heat engine is completely reversible then
This is based on Chapter 5, where it has been
shown that heat transfer in a reversible system
is proportional to the temperature.
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This means that we can rewrite the energy
balance, and solve for the net work (the work of
the combined system.
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That means that we can rewrite the energy
balance, and solve for the net work (the work of
the combined system.
To find the total net work we need to integrate
over time
Why is this a cyclic integral?
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But we know that it is impossible for a system to
exchange heat with only one reservoir, and
produce work!!
That means Wc cant be positive, but it can be
zero or negative!!
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Since TR is always positive
This is called the Clausius inequality. It is
equal to 0 for the reversible case, and is less
than 0 for the irreversible case
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To prove that it is 0 for the reversible case?
If the system is reversible, we can run it
backwards.
If it produces negative work going one way, it
will produce positive work going the other way
But we know you cant exchange heat with only one
reservoir, and produce work!!
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Therefore, for a reversible system, the cyclic
integral must be 0!!!
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What kind of properties have cyclic integrals
equal to 0?
The cyclic integral of a property is 0!!!
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Remember, only the cyclic integral is equal to 0
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Consider a cycle composed of two
processes Process 1 (from state 1 to state 2) is
reversible Process 2 (from state 2 back to state
1 is arbitrary
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So, the definition of a change in entropy
becomes.
Or
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The Entropy Change Between Two Specific States
The entropy change between two specific states is
the same whether the process is reversible or
irreversible
Entropy is a state function
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Consider an adiabatic reversible process
So DS0
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But what if the process isnt reversible?
Sgen depends on the process
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What kind of things cause a system to be
irreversible?

• Friction
• Unrestrained gas expansion
• Mixing of two fluids
• Etc
• These factors plus heat transfer all cause
  entropy to be generated

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Consider an isolated system
Because it is isolated, there is no heat transfer
or work
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The equality holds if all the processes inside
the system are reversible
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Now consider the universe

• It can be considered to be an isolated system!!

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You can consider any system to be isolated, if
you draw your system boundaries out far enough
Sgen is always greater than 0 for real systems
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• How do you calculate entropy changes for real
  systems?
• If its reversible adiabatic, DS0
• If its isothermal heat transfer
• If its not an ideal gas, just look it up on the
  property tables

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Entropy is a property
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The entropy of a pure substance is determined
from the tables, just as for any other property
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Some Remarks about Entropy

• Processes can occur in a certain direction only,
  not in any direction such that
• Entropy is a non-conserved property, and there is
  no such thing as the conservation of entropy
  principle. The entropy of the universe is
  continuously increasing.
• The performance of engineering systems is
  degraded by the presence of irreversibilities,
  and entropy generation is a measure of the
  magnitudes of the irreversibilities present
  during that process.

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Isentropic Processes

• The entropy of a fixed mass can be changed by
• Heat transfer
• Irreversibilities
• If the entropy does not change, it is isentropic

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Engineering Devices

• Many engineering devices are essentially adiabatic

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• They perform best when irreversibilities are
  eliminated
• Isentropic model serves as an idealization of a
  real process
• These devices work best when they are isentropic

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The assumption that a process is isentropic,
gives us a connection between the inlet and
outlet conditions just like assuming constant
volume, or constant pressure
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Property Diagrams Involving Entropy
This area has no meaning for irreversible
processes
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Consider some special cases of the Ts diagram
Isothermal Process
Isentropic Process
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Q0
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T-s Diagram for the Carnot Cycle
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Temperature
Entropy
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Mollier Diagrams (h-s Diagrams)
For adiabatic, steady flow devices, Dh is a
measure of work, and Ds is a measure of
irreversibilities
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Mollier Diagram for Steam, see Appendix
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Work Done during a Process

• In Chapter 4 we found the work done in a closed
  system due to moving boundaries and expressed it
  in terms of the fluid properties
• In a process, there are usually no moving
  boundaries

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Work Done During a Process

• It would be useful to be able to express the work
  done during a steady flow process, in terms of
  system properties
• Recall that steady flow systems work best when
  they have no irreversibilities

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Energy Balance for a steady flow device
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All we have to do now is integrate!!
In order to integrate, we need to know the
relationship between v and P
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For solids and liquids
v is constant
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Steady flow of a liquid through a pipe or a nozzle
There is no work!!
Bernoullis equation
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Steady Flow of a Liquid through a pump or a
turbine
Or..
Note that the work term is smallest when v is
small, so for a pump (which uses work) you want v
to be small. For a turbine (which produces work)
you want v to be big.
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Compressor Work
We integrated this equation for v constant,
which is good for liquids but what about gases?
Consider an ideal gas, at constant T
Remember, this is only true for the isothermal
case, for an ideal gas
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Compressor Work
Another special case is isentropic
We derived the isentropic relationships earlier
in this chapter
Rearrange to find v, plug in and integrate
Now its just algebra, to rearrange into a more
useful form
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Remember, this equation only applies to the
isentropic case, for an ideal gas, assuming
constant specific heats
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Compressor Work
Another special case is polytropic
Back in Chapter 3 we said that in a polytropic
process Pvn is a constant
This is exactly the same as the isentropic case,
but with n instead of k!!
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Pv Diagram for Isentropic, Polytropic and
Isothermal compression, for the same final and
initial pressures
The area to the left of each line represents the
work, vdP
Note, it takes less work for an isothermal process
You should compress isothermally, and you should
use an isentropic process in a turbine!!
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How do you keep a compression process isothermal?

• The gas will heat up as it is compressed, so it
  needs to be cooled
• Intercooling is difficult
• Instead, multistage compression is more common,
  with cooling between steps

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Two stage Compressor
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How do you decide how to break up the compression
load?

• You save the most work by intercooling, when each
  compressor carries the same load

Since you cool back to T1 between stages, the
only things that change in this equation are the
Ps
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For the work done by each stage to be equal, the
pressure ratio must be equal
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Isentropic Efficiencies of Steady Flow Devices

• Real devices are never really isentropic
• There are always irreversibilities that downgrade
  performance
• We should compare real devices to isentropic ones
• Second Law Efficiency

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Isentropic Efficiency

• Most steady flow devices are intended to operate
  under adiabatic conditions
• Lets compare how well real devices work to how
  well comparable isentropic devices work
• Same inlet conditions
• Same outlet conditions
• Turbine, Compressor and Nozzle

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Turbines
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Remember, the work done in a turbine can be found
from the energy balance
Do example 7-14, page 314
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Isentropic Efficiencies of Compressors and Pumps

• Ratio of the work required to raise the pressure
  of a gas to a specified value, in a isentropic
  manner, to the actual work

Note that this equation is arranged so that it is
always less than one!!
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Remember, the work done by a compressor can be
found from the energy balance
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Applies to both gases and liquids
Isentropic work for a liquid
Only applies to a liquid
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Sometimes compressors are cooled intentionally
Why?

• Cooling reduces the specific volume, resulting in
  less work required for compression
• For compressors that are intentionally cooled,
  the isothermal model is more realistic

Do example 7-15, page 316
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Isentropic Efficiency of Nozzles

• The objective of a nozzle is to increase the
  kinetic energy of the gas
• Usually, the inlet velocity is low enough that we
  can consider it to have zero kinetic energy

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