ME2121 – Engineering Thermodynamics

ME2121 – Engineering
Thermodynamics
Entropy – Part I
Prof Arun S Mujumdar
2005-2006
September 28,2005
© Copyright 2005 Prof. Arun S. Mujumdar.

Entropy-Introductory Remarks
• Basic concept
• Origin in 2 nd Law of Thermodynamics
• Review Chapter 5, Cengel and Boles
• Concepts of reversibility and irreversibility are
quantified via definition of entropy
• Note; Entropy is a thermodynamic property
• Review Chapter 6 , Cengel and Boles
• Note: Some additional slides are inserted this
presentation to provide quick refresher on some
basic concepts / terms used.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 2

Review Chapter 5-2 nd Law
Concepts/Terms Used in this lecture
• Thermal Energy reservoirs
• Heat engines-definition
• Energy conversion efficiency
• PMM1, PMM2
• Reversible/irreversible processes
• Carnot cycle- 2 isothermal, 2 adiabatic steps
• Carnot principles
• Absolute Kelvin scale of temperature
• Chapter 5-focuses on cyclic processes, Ch 6 applies 2 nd
Law to processes
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 3

Spontaneous Changes:
Why do some processes happen spontaneously whilst
others do not
•Why does a hot body get cooler when surrounded by
a cooler medium rather than to get hotter
•Why does a gas expand into all available volume of a
container rather than to contract
•Some law must determine the direction of
spontaneous change, that is the direction of
change that happens without intervention, i.e., that
does not require work.
This law is the second law of Thermodynamics.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction
(ASM) 4

Direction of Energy Conversion
• No process is possible in which the sole result is
the absorption of heat from a reservoir and its
complete conversion into work.
• Kinetic energy of a bouncing ball for example is
converted into thermal motion
• Energy is not accumulated in ball and thermal
motion is not directional
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction
(ASM) 5

Entropy, S, and the Second Law
• The driving force for spontaneous change is an increase
in the chaos of energy dispersal of the isolated system.
• The Entropy, S, a thermodynamic state function, is a
measure of molecular disorder and helps us to define the
direction of spontaneous change
• The Entropy of an isolated system increases in the
course of a spontaneous change
• ∆S system + ∆S surroundings = ∆S total > 0
• Hence: ∆S universe > 0
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction
(ASM) 6

Entropy and Disorder
• The Entropy of the System depends on order of the system,
e.g. going from solid to liquid to gas increases disorder at
the molecular level.
• Hence entropy of system increases as the material changes
phase from solid to liquid to gas (vapor) state
• So, why then does water freeze spontaneously on a cold
night
Note:
Liquid (water molecules able to move)
Solid (molecules fixed in crystal)
Should entropy not decrease
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction
(ASM) 7

Answer
No, as for ANY spontaneous process, even in this
case the entropy of the universe increases!
Why
Solidification of water is an exothermic process
with heat of fusion of -6.88 kJ/mol. Hence the
system (water) passes heat to the surroundings.
Thus entropy change of system decreases,
entropy of surrounding increases, thus change
of entropy of the universe is positive.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction
(ASM) 8

Entropy of Crystalline Substance
The entropy of a crystalline substance at
equilibrium approaches zero as absolute
zero of temperature is approached.
This is an outcome of the Third Law of
Thermodynamics
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction
(ASM) 9

SOME INTRODUCTORY COMMENTS
Earlier, second law of thermodynamics and
applied it to cycles and cyclic devices. Here, we
apply the second law to processes.
The first law of thermodynamics deals with the
property energy and the conservation of it. The
second law leads to the definition of a new
property called entropy.
Entropy is a somewhat abstract property and it is
difficult to give a physical description of it without
considering the microscopic state of the system.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 10

INTRODUCTION (Cont’d)
We start with a discussion of the Clausius
inequality, the basis for the definition of entropy.
Unlike energy, entropy is a nonconserved
property, and there is no such thing as a
conservation of entropy principle.
The entropy changes that take place during
processes for pure substances, incompressible
substances, and ideal gases will be discussed.
Finally, entropy balance concept is introduced
and applied to various systems.
Examples are included at the end- some for selfstudy
for lack of time during lectures. Additional
examples to be included in tutorials
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 11

Clausius Inequality
The Second Law of Thermodynamics often leads to
expressions that involve inequalities.
An irreversible refrigerator or a heat pump has a
lower coefficient of performance (COP) than a
reversible one operating between the same
temperature limits.
Clausius inequality was first stated by the German
physicist R.J.E. Clausius (1822-1888), one of the
founders of thermodynamics.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 12

Clausius Inequality (Cont’d)
Clausius inequality is expressed as:
Q
T
The cyclic integral of δQ/T is always less than or equal to
zero.
0
Integration is performed over whole cycle.
This inequality is valid for all cycles, reversible or irreversible.
The integration is to be performed over the entire cycle.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 13

What are Heat engines
Brief overview
Strictly, cyclic devices characterized by: receives
heat from higher temperature source, converts a
part to useful work( rotating shaft etc), rejects
some of the heat input to a lower temperature
reservoir and operates in a cycle using a suitable
working fluid as energy carrier.
Any work producing device e.g. auto engine,
turbine is often also called a heat engine
although they are not cyclic devices. The “
working fluid” is rejected continuously and not
cycled.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 14

Brief Review: Reversibilities etc
Reversible processes-ones that can be
reversed without leaving a trace on the
surroundings. Both system and
surroundings must revert to initial state at
end of process.
Implies net heat and net work exchange
between system and surroundings for
combined process (original and reverse) is
zero.
Such a process is an idealization- does not
exist!
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 15

Reversible & Irreversible processes
Causes: friction, unconstrained expansion,
mixing, heat transfer under finite
temperature differences.
Internally reversible- no irreversibilities
within system
Externally reversible- no irreversibilities
outside system boundaries
Totally reversible or simply reversible, if
both have no irreversibilities
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 16

Clausius Inequality (Cont’d)
To demonstrate the validity of the Clausius
inequality consider a system connected to a
thermal energy reservoir at a constant absolute
temperature of T R through a reversible cyclic
device (See Fig 6-1).
The cyclic device receives heat δQ R from the
reservoir and supplies heat δQ to the system
whose absolute temperature at that part of the
boundary is T (a variable) while producing work
δW rev .
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 17

System for Clausius Inequality Proof
Adapted from Fig:6-1 (C+B)
Thermal
Reservoir δQ R
Tg
Reversible
Cyclic System
δW rev
δQ
Combined
System
System
δW sys
© Copyright 2005 Prof. Arun S. Mujumdar.

Clausius Inequality (Cont’d)
The system produces work δW sys as a
result of this heat transfer. Applying the
energy balance to the combined system
identified by dashed lines yields:
δW c = δQ r – dE c
Where δW c is the total work of the
combined system (δW rev + δW sys ) and
dE c is the change in the total energy of
the combined system. The cyclic device
is a reversible one.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 19

Brief Review: Carnot Principle and
Kelvin Scale
Carnot Principle: All reversible heat
engines operating between the same two
reservoirs have the same efficiency.
This means:
Ratio heat transfer from hot and to cold
reservoirs is equal to the ratio of the
absolute temperatures of the two
reservoirs.
This also leads to Kelvin scale of absolute
temperature.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 20

Clausius Inequality (Cont’d)
Q
T
R
R
Q
T
Above eqn is Eqn.5.18 in textbook. for a reversible cycle.
Eliminating δQ R from the two relations above yields
W
c
T
R
Q
T
dE
Let the system undergo a cycle while the cyclic device
undergoes an integral number of cycles.
c
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 21

Clausius Inequality (Cont’d)
Integrating over a cycle:
W
c
T
R
Q
T
Since the cyclic integral of energy (the net change
in the energy, which is a property, during a
cycle) is zero.
Here W C is the cyclic integral of δW c , and it
represents the net work for the combined cycle.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 22

Clausius Inequality (Cont’d)
It thus appears that the combined system is
exchanging heat with a single thermal energy
reservoir while involving (producing or
consuming) work W c during a cycle.
The Kelvin-Planck statement of the second law
states that no system can produce a net amount
of work while operating in a cycle and exchanging
heat with a single thermal energy reservoir
Hence, we reason that W c cannot be a work output,
and thus it cannot be a positive quantity.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 23

Clausius Inequality (Cont’d)
We must have
Q
T
0
which is the Clausius inequality. This inequality is
valid for all thermodynamic cycles, reversible or
irreversible, including the refrigeration cycles.
In the reversed cycle cases, all the quantities will
have the same magnitude but the opposite
sign.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 24

Clausius Inequality (Cont’d)
Therefore, the work W c which could not be a
positive quantity in the regular case, cannot be
a negative quantity in the reversed case.
Then it follows that W C int rev =0 since it cannot be
positive or negative quantity and therefore
Q
T
intrev
Thus we conclude that the equality in the
Clausius inequality holds for totally or just
internally reversible cycles and the inequality
for the irreversible ones.
0
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 25

Entropy
Clausius chose to name this property entropy. It
is designated S and is defined as( eq. 6.4)
dS
Q
T
intrev
(kJ /
K)
Entropy is an extensive property of a system and
sometimes is referred to as total entropy.
Entropy per unit mass, designated s, is an
intensive property and has the unit kJ/kg.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 26

Entropy (Cont’d)
The term entropy is generally used to refer to
both total entropy and entropy per unit mass
since the context usually clarifies which one is
meant. Check units!
The entropy change of a system during a
process between the initial and the final
state Eqn. 6.5:
S
2
Q
S2 S1
T
1
int
rev
(kJ / K)
ME2121 - Entropy I - Introduction (ASM) 27
© Copyright 2005 Prof. Arun S. Mujumdar.

Entropy (Cont’d)
Absolute values of entropy are determined on the
basis of the third law of thermodynamics.
Engineers are usually concerned with the changes
in entropy.
The entropy of a substance can be assigned a zero
value at some arbitrarily selected reference state.
To perform the integration in Eq. 6-5 one needs to
know the relation between Q and T during a
process. For the majority of cases we have to rely
on tabulated data for entropy.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 28

Entropy (Cont’d)
Note that entropy is a property. It has fixed values
at fixed states.
The entropy change ΔS between two specified
states is the same no matter what path,
reversible or irreversible, is followed during a
process.
From definition of entropy:
Also note that the integral of δQ/T will give us the
value of entropy change only if the integration is
carried out along an internally reversible path
between the two states.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 29

Important points to note on Entropy
The integral of δQ/T along an irreversible
path is not a property and different values
will be obtained when the integration is
carried out along different irreversible
paths.
Even for irreversible processes, the entropy
change should be determined by carrying
out this integration along some convenient
imaginary internally reversible path
between the specified states.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 30

Internally Reversible Isothermal
Heat Transfer Processes
Entropy change of a system during an
internally reversible isothermal heat
transfer process can be determined by
S
2
1
Q
T
int
rev
2
Q
T
1 0
intrev
1
T
0
2
Q
1
intrev
which reduces to
S
Q
T
0
(kJ /
K)
ME2121 - Entropy I - Introduction (ASM) 31
© Copyright 2005 Prof. Arun S. Mujumdar.

Internally Reversible Isothermal Heat
Transfer Processes (Cont’d)
T 0 is the constant absolute temperature of the
system and Q is the heat transfer for the
internally reversible process.
Useful for determining the entropy changes of
thermal energy reservoirs that can absorb or
supply heat indefinitely at a constant
temperature.
The entropy change of a system during an
internally reversible isothermal process can be
positive or negative, depending on the direction
of heat transfer.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 32

Internally Reversible Isothermal Heat
Transfer Processes (Cont’d)
Heat transfer to a system will increase the
entropy of a system, whereas heat
transfer from a system will decrease it,
losing heat is the only way the entropy of
a system can be decreased.
No entropy is associated with work.
Higher entropy means greater disorder,
greater irreversibilities-undesirable for
efficiency!
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 33

Closing Remarks
In the following lecture we will derive equations
for calculation of entropy changes and entropy
generation during thermodynamic processes
involving ideal gas and incompressible liquids and
solids
Also a number of illustrative examples
demonstrating application of the equations
derived as well as thermodynamic property tables
for calculation of entropy changes and entropy
generation in various simple problems of
engineering interest
Note that it is often necessary to simultaneously
apply conservation of mass and energy equations
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 34

Closure
Study chapter 6 of textbook - suggested
sections. Conceptually difficult
PPT Notes are closely tied to textbook to
ease learning and preparation of your
personal Notes
Do study examples given in textbook to
ensure you understand application of basic
concepts.
Note often you need to use First Law to
determine thermodynamic states before
2 nd Law calculations can be made.
© Copyright 2005 Prof. Arun S. Mujumdar.
ME2121 - Entropy I - Introduction (ASM) 35