Entropy : A Measure of Disorder

1
Chapter 6
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder

• Entropy A Measure of Disorder

2
Chapter Summary
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder

• Introduction to entropy, Clausius inequality
• Entropy, Entropy change of isolated system
• Increase of entropy principle
• Isentropic process
• T-s and h-s diagram, T-ds relations
• Entropy changes of pure substance, solid, liquid
  and ideal gasses
• Reversible steady flow work
• Minimizing the compressor work and intercooling
• Isentropic efficiencies of steady flow devices
• Entropy balance

3
Introduction to Entropy
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder

• The second law of thermodynamics leads to the
  definition of a new property called entropy,
  which is a quantitative measure of microscopic
  disorder for a system
• The definition of entropy is based on the
  Clausius inequality, given bywhere the
  equality holds for internally or totally
  reversible processes and the inequality for
  irreversible processes

4
Proof of Clausius Inequality
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder
Let Applying energy balance to the combined
system Assuming that the engine is totally
reversible Net work done by the combined
system For cyclic devices, no net change in
energy, thus
ve Wc leads to violation of Kelvin-Planck
Statement, thus it must be negative, so The
equality holds for totally reversible cycle
5
Entropy A Thermodynamics Property
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder

• Any quantity whose cyclic integral is zero is a
  property, thus (?Q/T)int rev must be a
  thermodynamics property in differential form.
  This property is called entropy defined as
• Thus, the change of entropy of a
  system during a process is
  
• For internally reversible heat transfer
  process

6
Entropy Change of Isolated System
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder
A system and its surrounding form an isolated
system
Entropy is an extensive property thus, the
entropy change of an isolated system is the sum
of the entropy changes of its components, and is
never less than zero
During a heat transfer process, the net disorder
(entropy) of the combined system increases (the
increase in the disorder of the cold body more
than offsets the decrease in the disorder in the
hot body)
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Increase of Entropy Principle
Department of Mechanical Engineering
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Chap. 6 Entropy A Measure of Disorder
The entropy of an isolated system during a
process always increases or, in limiting case of
a reversible process, remains constant
Proof
consider cyclic process 1-2 reversible 2-1
irreversible
Thus for an isolated system
Combined system and its surrounding
8
Increase of Entropy Principle (cont..)
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder
Entropy change is caused by heat transfer, mass
flow, and irreversibilities
Effect of heat transfer on entropy
A process must proceed in a certain direction
that complies with the increase of entropy
principle and a process that violates this
principle is impossible


The effect of irreversibilities is always to
increase the entropy. As all real processes are
irreversible, entropy can be used as a measure of
performance
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What is Entropy ??
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder

• Measure of molecular disorder or molecular
  randomness
• In solid phase (low temperature), the molecules
  are fix and its locations are predictable gt low
  entropy
• In vapour phase (high temperature), the molecules
  are moving at random and its locations are very
  unpredictable gt high entropy
• Heat is a form of disorganized energy (low
  quality, high entropy). The higher the
  temperature, the higher the entropy
• In any actual process, the quantity of energy is
  always preserved (1st law), but the quality is
  bound to decrease (2nd law). The decrease of
  quality is accompanied by increase of entropy

10
Isentropic Process
Department of Mechanical Engineering
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Chap. 6 Entropy A Measure of Disorder
The entropy change in a process that is
internally reversible and adiabatic will be
zero A process with zero entropy change is
called isentropic
?S 0 or s2s1 kJ/kgK
11
The T-s Diagram
Department of Mechanical Engineering
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Chap. 6 Entropy A Measure of Disorder
For the reversible process, dS equation gives
Thus
Thus, for a reversible process, the area under
the curve on a T-S diagram represent the heat
transfer
To evaluate Qnet requires relationship between T
and S
12
The T-s Diagram of Carnot Cycle
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder
T
Recall the Carnot Cycle
1
2
TH
Process 1-2 Reversible isothermal heat addition
at high temperature, TH gt TL to the working fluid
in a piston--cylinder device which does some
boundary work. Process 2-3 Reversible adiabatic
expansion during which the system does work as
the working fluid temperature decreases from TH
to TL. Process 3-4 The system is brought in
contact with a heat reservoir at TL lt TH and a
reversible isothermal heat exchange takes place
while work of compression is done on the
system. Process 4-1 A reversible adiabatic
compression process increases the working fluid
temperature from TL to TH
TL
3
4
A
B
S
S1S4
S2S3
Q12 Area A12BA
Q23 0
Q34 -Area A43BA
Q41 0
Thus Qnet Wnet Area1234
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The h-s Diagram
Department of Mechanical Engineering
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Chap. 6 Entropy A Measure of Disorder
Recall that for single stream control volume, eg.
In turbines, compressors and nozzles, the
enthalpy, h is a primary property
In h-s diagram, the vertical distance, dh gives
the change in enthalpy and the horizontal
distance is the change in entropy which is a
measure of irreversibility i.e how efficient is
the device in doing the process
h-s diagram is also known as Mollier Diagram
E.g. in turbines
14
The T-ds Relations
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder

• Entropy is a property, and it can be expressed in
  terms of more familiar properties through the Tds
  relations, expressed as and
• The equation is also known as Gibbs Equation and
  valid for both reversible and irreversible
  processes for closed and open system
• change in entropy can be evaluated by integrating
  the equation.
• This requires relationship between du or dh and
  the temperature as well as the equation of
  state

Tds du Pdv
or
Tds dh - vdP
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The Entropy Change of Pure Substance
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder
The entropy of a pure substance is determined
from the tables, just as for any other property
T-s Diagram for water
h-s Diagram for water
In any process, ?S m ? s m(s2-s1)
Isentropic process, s2 s1
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The Entropy Change of Solid and Liquid
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder

• For incompressible substance such as liquid and
  solid, the specific volume remains constant, thus
  using Tds relation
• Hence the entropy change for solid and liquid is
  
• For isentropic process

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The Entropy Change of Ideal Gasses
Department of Mechanical Engineering
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Chap. 6 Entropy A Measure of Disorder

• Using Tds relation and the ideal gas equation of
  state, the entropy change can be evaluated using
  
• With constant specific heat (approximate
  analysis)

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The Entropy Change of Ideal Gasses (cont ..)
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder

• Approximate analysis (unit mole basis)
• Variable specific heat (Exact analysis)

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The Isentropic Processes of Ideal Gasses
Department of Mechanical Engineering
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Chap. 6 Entropy A Measure of Disorder

• Constant specific heat (approximate treatment)
  
• Variable specific heat (exact treatment)
• Relative pressure and relative spec vol.

or
20
Reversible Steady Flow Work
Department of Mechanical Engineering
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Chap. 6 Entropy A Measure of Disorder
For incompressible substances (v constant)
steady-flow work for a reversible process
simplifies to
The steady-flow work for a reversible process can
be expressed in terms of the fluid properties as
The work done during a steady-flow process is
proportional to the specific volume.
Therefore, v should be kept as small as possible
during a compression process to minimize the work
input and as large as possible during an
expansion process to maximize the work output.
21
Minimizing the Compressor Work
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder
The reversible work inputs to a compressor
compressing an ideal gas from T1, P1, to P2 in an
isentropic (Pvk constant), polytropic (Pvn
constant), or isothermal (Pv constant) manner,
are determined by integration for each case with
the following results
isentropic
polytropic
isothermal
The isothermal process (maximum cooling) requires
the least work
22
Multistage Stage Compression with Intercooling
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder

• The work input to a compressor can be reduced by
  using multistage compression with intercooling

• The gas is compressed in stages and cooled
  between each stage through a heat exchanger
  called intercooler

• For maximum savings from the work input, the
  pressure ratio across each stage of the
  compressor must be the same.

23
Isentropic Efficiencies of Steady Flow Devices
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder

• Most steady-flow devices operate under adiabatic
  conditions, and the ideal process for these
  devices is the isentropic process
• For Turbines

• The parameter that describes how efficiently a
  device approximates a corresponding isentropic
  device is called isentropic or adiabatic
  efficiency

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Isentropic Efficiencies of Steady Flow Devices (c)
Department of Mechanical Engineering
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Chap. 6 Entropy A Measure of Disorder
For compressors and pumps
For nozzles
25
Entropy Balance
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder

• The entropy balance for any system undergoing any
  process can be expressed in the general form as
  

Entropy balance

• Or in rate form

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Entropy Balance (continue)
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder

• Entropy change of a system, ?Ssystem

• Mechanism of entropy transfer, Sin and Sout

• Entropy transfer by heat transfer

• Entropy transfer by work

• Entropy transfer by mass flow

• Entropy generatiom, Sgen

Caused by irreversibilities friction, mixing,
chemical reaction, heat transfer through finite
temperature diff., unrestrained expansion,
non-quasi-equilibrium compression etc ...
27
Entropy Balance (continue)
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder

• Thus, the entropy balance for closed system
  simplifies to

• And the entropy balance for control volume

• For a general steady-flow process

28
Extra Examples (1)
Department of Mechanical Engineering
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Chap. 6 Entropy A Measure of Disorder
Example 1 For a particular power plant the heat
added and rejected both occur at constant
temperature and no other processes experience any
heat transfer. The heat is added in the amount
of 3150 kJ at 440 oC and is rejected in the
amount of 1950 kJ at 20 oC. Is the Clausius
Inequality satisfied and is the cycle reversible
or irreversible? Example 2 Find the total
entropy change, or entropy generation, for the
transfer of 1000 kJ of heat energy from a heat
reservoir at 1000 K to a heat reservoir at 500 K.
Example 3 Determine the entropy change of
water contained in a closed system as it changes
phase from saturated liquid to saturated vapor
when the pressure is 0.1MPa and constant. Why is
the entropy change positive for this process?
Example 4 Steam at 1 MPa, 600oC expands in a
turbine to 0.01 MPa. If the process is
isentropic, find the final temperature, final
enthalpy of the steam, and the turbine work.
29
Department of Mechanical Engineering
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Chap. 6 Entropy A Measure of Disorder
Extra Examples (2)
Example 5 Aluminum at 100oC is placed in a
large, insulated tank having 10 kg of water at a
temperature of 30oC. If the mass of the aluminum
is 0.5 kg, find the final equilibrium temperature
of the aluminum and water, the entropy change of
the aluminum and the water, and the total entropy
change of the universe because of this process.
Before we work the problem, what do you think the
answers ought to be? Are entropy changes going to
be positive or negative? What about the entropy
generated as the process takes place? Example 6
Carbon dioxide initially at 50 kPa, 400 K
undergoes a process in a closed system until its
pressure and temperature are 2 MPa and 800 K,
respectively. Assuming ideal gas behavior, find
the entropy change of the carbon dioxide by first
assuming constant specific heats and then
assuming variable specific heats. Example 7
Air, initially at 17oC, is compressed in an
isentropic process through a pressure ratio of
81. Find the final temperature assuming
constant specific heats and variable specific
heats.
30
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder
Extra Examples (3)
Example 8 Air initially at 0.1 MPa, 27oC is
compressed reversibly to a final state. (a) Find
the entropy change of the air when the final
state is 0.5 MPa, 227oC. (b) Find the entropy
change when the final state is 0.5 MPa,
180oC. (c) Find the temperature at 0.5 MPa that
makes the entropy change zero. Show the two
processes on a T-s diagram. Example 9 Nitrogen
expands isentropically in a piston cylinder
device from a temperature of 500 K while its
volume doubles. What is the final temperature of
the nitrogen, and how much work did the nitrogen
do against the piston, in kJ/kg? Example 10 A
Carnot engine has 1 kg of air as the working
fluid. Heat is supplied to the air at 800 K and
rejected by the air at 300 K. At the beginning
of the heat addition process the pressure is 0.8
MPa and during this process the volume triples.
(a) Calculate the net cycle work assuming air is
an ideal gas with constant specific heats. (b)
Calculate the amount of work done in the
isentropic expansion process. (c) Calculate the
entropy change during the heat rejection process.
31
Department of Mechanical Engineering
Dr Zamri

Chap. 6 Entropy A Measure of Disorder
Extra Examples (4)
Example 11 Saturated liquid water at 10 kPa
leaves the condenser of a steam power plant and
is pumped to the boiler pressure or 5 MPa.
Calculate the work for an isentropic pumping
process. Example 12 The isentropic work of the
turbine in Example 6-6 is 1152.2 kJ/kg. If the
isentropic efficiency of the turbine is 90,
calculate the actual work. Find the actual
turbine exit temperature or quality of the
steam. Example 13 Air enters a compressor and is
compressed adiabatically from 0.1 MPa, 27oC to a
final state of 0.5 MPa. Find the work done on
the air for a compressor isentropic efficiency of
80. Example 14 Nitrogen expands in a nozzle
from a temperature of 500 K while its pressure
decreases by factor of two. What is the exit
velocity of the nitrogen when the nozzle
isentropic efficiency is 95? Example 15 An
inventor claims to have developed a water mixing
device in which 10 kg/s of water at 25oC and 0.1
MPa and 0.5 kg/s of water at 100oC, 0.1 MPa are
mixed to produce 10.5 kg/s of water as a
saturated liquid at 0.1 MPa. If the surroundings
to this device are at 20oC, is this process
possible? If not, what temperature must the
surroundings have for the process to be
possible?