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CHAPTER 8
THERMODYNAMICS
Thermodynamics is the science that deals with interaction of energy and matter. The subject is based on the
primitive concepts that have been formulated into the fundamental laws and govern the principles of energy
conversion and feasibility of the processes. Since all natural processes involve interaction between energy and
matter, thermodynamics encompasses a very large area of application. The emphasis of the present context is on
understanding the fundamental concepts and on systematic formulation and solution of problems from the first
principles of thermodynamics.
8.1
8.1.1
BASIC CONCEPTS
of action of the molecules which can be simply
perceived by human senses, such as pressure,
temperature, volume. Such an approach is called
macroscopic approach.
Thermodynamic Approaches
Behavior of a matter can be studied at two levels of
approach:
For example, pressure, a macroscopic quantity,
is the average rate of change of momentum due to
all the molecular collisions made on a unit area.
The effects of pressure can be easily felt.
1. Microscopic Approach
Microscopic approach is
concerned with the behavior of each molecule that
cannot be perceived by human senses. Behavior
of the concerned matter is described by summing
up the behavior of its molecules, such as in
kinetic theory of gases. This approach is also called
statistical thermodynamics.
Macroscopic approach conveniently disregards
the atomic nature of a substance to view it as
a continuous and homogeneous matter. This is
called the concept of continuum1 ; the substance is
treated free from any kind of discontinuity. This
idealization permits properties to be treated as
point functions, varying continually in space.
Size of the engineering
2. Macroscopic Approach
systems is generally much larger than the mean
free path of the molecules, therefore, molecular
level analysis is not appropriate due to time
consuming demerits. In such situations, the only
interest of study is to know the overall effects
The concept of continuum can be explained for
density as a property. Consider a mass δm in a
small volume δv surrounding a point in a system.
1 The
concept of continuum is also used in fluid mechanics,
mechanics of materials and also in heat transfer studies.
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The variation of average density δm/δv can be
plotted against δv. The average density tends to
approach an asymptote as δv increases [Fig. 8.1].
Surrounding
System
δm/δv
Discontinuity
Continuum
Boundary
Molecular
effects
Figure 8.2
ρ
Asymptote
δv
Concept of continuum.
When δv reaches δv ′ , so small as to contain
relatively few molecules, the average density fluctuates substantially due to random motion of the
molecules. In such a situation, the value of quantity δm/δv cannot be predicted. This threshold
volume δv ′ can be regarded as a continuum beyond
which there is no effect of molecular motion on
the properties of the system. Thus, macroscopic
density ρ of the system is defined as
ρ = lim
δv→δv ′
8.1.2.2 System Exchanges Between a given system
and its surroundings, the following two types of exchanges can occur:
1. Energy exchange
δv ′
Figure 8.1
2. Mass exchange
Here, energy means both heat and work transfers.
Heat transfer can takes place through a diathermal
boundary only. An adiabatic boundary does not allow
heat exchange to take place.
8.1.2.3 Types of Systems Classification of thermodynamic systems is based on the types of exchanges and
depends on selection of a fixed mass or a fixed volume
in the space for study. A thermodynamic system can be
closed, open, or isolated, explained as follows:
1. Closed System A closed system consists of a fixed
mass (thus, also known as control mass) on which
only energy transfer can occur [Fig. 8.3].
δm
δv
Energy
The concept of continuum can be similarly applied
to other properties of the matter.
8.1.2
System, surrounding, and universe.
Closed
system
Thermodynamic Systems
Boundary
A system in thermodynamics is the collection of matter
or region in space chosen for study. Thermodynamic
analysis can be simplified by defining an appropriate
system which in turn leads a systematic study.
8.1.2.1 System, Surrounding and Universe Thermodynamic system is a three-dimensional region of
space bounded by one or more surfaces. The boundary
can be real or imaginary and can change its size,
shape, and location. The region of physical space that
lies outside the defined boundaries of a system is
called surrounding. Whenever a thermodynamic system
is defined, the complementary region (surrounding) gets
automatically defined. A system and its surroundings
together comprise a universe [Fig. 8.2].
Surrounding
Figure 8.3
Energy
Closed system.
2. Open System Both matter and energy cross the
boundary of an open system [Fig. 8.4]. Most of the
engineering devices involving mass flow, such as a
compressor, turbine, nozzle, are examples of open
system.
Based on steadiness2 of exchange rates, the
open systems can be of two types:
2 The
term ‘steady’ implies no change with time. The opposite of
steady is ‘unsteady’ or ‘transient’. The term uniform, however,
implies no change with location over a specified region.
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8.1 BASIC CONCEPTS
Energy
Surrounding
Open
system
Mass
Mass
A quantity of matter homogeneous in chemical composition and physical structure is called a phase. A
substance can exist in any one of the three phases,
namely, solid, liquid, and gas. A homogeneous system
consists of single phase. A system consisting of more
than one phase is called heterogeneous system.
Energy
Boundary
8.1.3
Figure 8.4
Open system.
(a) Steady Flow System When flow rates of mass
and energy remain constant, the system is
called steady flow system. Most of the engineering devices (e.g. turbine, pumps, heat exchangers, refrigerators) come under this category.
Properties of the working fluid can change
from point to point within the control volume,
but at any fixed point they remain the same
during the entire process.
(b) Unsteady Flow System
When flow rates of
mass and energy vary with time, the system
is called unsteady flow system. This condition
mainly occurs in starting stages of steady flow
systems.
Energy flow associated with a fluid stream is often
expressed in rate form by incorporating the mass
flow rate (ṁ), the amount of mass flowing through
a cross-section per unit time.
Thermodynamic analysis of open systems involves study of a certain fixed volume in space
surrounding the system, known as the control
volume. Thus, there is no difference between open
system and control volume. Control surface is the
imaginary or real boundary of the control volume,
which can be fixed or moveable. The contact
surface is shared by both the system and the
surrounding.
3. Isolated System
An isolated system does not
have any interaction of mass or energy with its
surroundings. Therefore, mass and energy inside
an isolated system remain constant [Fig. 8.5].
Surrounding
Isolated
system
Boundary
Figure 8.5
501
No interaction
Isolated system.
State Properties
Physical condition (state) of a system is described by
certain characteristics, such as mass, volume, temperature, pressure. These characteristics are called properties
of the system. Thermodynamics is concerned with the
properties that are macroscopic in nature and approach
[Section 8.1.1].
Based on the dependency on the mass or extensiveness, thermodynamic properties can be of two types:
1. Intensive Properties Intensive properties are independent of the mass in the system, such as
density, pressure, temperature. These properties
are generally used to compare systems in an
absolute manner, irrespective of the mass of the
systems.
Intensive properties are generally denoted by
lowercase letters (temperature T is the obvious
exception).
2. Extensive Properties Extensive properties depend
on the extent of the system, such as volume,
energy, momentum. Their magnitude increases
with increase in mass of the system. Extensive
properties are used to observe the scale of the
systems.
Extensive properties per unit mass are called
specific properties, which are in fact intensive
properties, such as specific volume, specific energy,
density.
Extensive properties are generally denoted by
uppercase letters (mass m is the obvious exception), whereas specific extensive properties (i.e.
intensive properties) are denoted by lowercase
letters.
Extensiveness of a property can be determined by
dividing the system into two equal parts with an
imaginary partition. Each part will have the same value
of intensive properties as the original system but half
the value of the extensive properties.
8.1.4
Thermodynamic Equilibrium
A system is said to exist in a state of thermodynamic
equilibrium if its isolation from the surrounding does not
cause a change in any of the macroscopic properties of
the system. This is possible if there exist no unbalance
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force, no chemical reaction, and no change in energy.
Hence, thermodynamic equilibrium is meant for mechanical, chemical, and thermal equilibrium of a system.
The concept of thermodynamic equilibrium is related
to the concept of quasi-static process, the basis of all
theoretical thermodynamic cycles. An isolated system
always reaches, in the course of time, a state of
thermodynamic equilibrium and can never depart from
it spontaneously.
8.1.5
✐
Two Property Rule
8.1.5.1 Constitutive Relation Definite values of all
the properties of a system indicate a definite state of the
system. However, all the properties of a system cannot
be varied independently since they are interrelated
through Constitutive relation of the following type:
given initial state goes through a number of different
changes in state (i.e. through various processes) and
finally returns to its initial state, the system undergoes
a cyclic process, or simply a cycle. Therefore, at the
conclusion of a cycle, all the properties have the same
value as at the beginning. For a cyclic process, the final
state is identical with the initial state; cyclic integral of
a property is always zero [Fig. 8.6].
Using the state postulate, the properties of a system
can be taken as the state coordinates to describe the
state of the system as a point on a two-dimensional
thermodynamic property diagram. Therefore, processes
and cycle of a system can be conveniently represented
on two-dimensional property diagrams [Fig. 8.6].
p
b
Process
b
f (p, v, T ) = 0
where p, v, T are some interrelated properties of a
system. Interestingly, a system can be perfectly defined
by knowing how many variables can be varied independently. Two properties are independent if one property
can be varied while the other one is held constant.
b
Cycle
Experiments have shown that once a sufficient number of properties are determined, the rest of the
properties assume definite values automatically using
the constitutive relations.
8.1.5.2 State Postulate The number of properties
required to fix the state of a system is given by the state
postulate or two property rule. According to this rule,
the state of a simple compressible system is completely
specified by two independent intensive properties.
8.1.5.3 Compressible System In state postulate, a
system is called simple compressible system in absence of
electrical, magnetic, gravitational, motion, and surface
tension effects. These effects are caused by external
forces, and are negligible in most engineering problems.
Otherwise, an additional property needs to be specified
for each effect that is significant. A simple system of
compressible substance (gas) can be described by, (p,
v), or (p, T ), or (T , v), or (p, u), or (u, v), but cannot
by (T , u) because u is not independent of T .
8.1.6
Processes and Cycle
A change in one or more properties of a system is called a
change in state. The succession of states passed through
during a change of state is called the path of the change
of state. When the path is completely specified, the
change of state is called a process3 . When a system in a
3 The
prefix iso- is often used to designate a process for which a
particular property remains constant [Section 8.8.7]. For example,
v
Figure 8.6
Processes and cycle.
For a given state, there is a definite value for each
property. The change in a property of a system is
independent of the path the system follows during
the change of state. Therefore, properties are point
functions4 .
8.1.7
Modes of Energy
Energy5 exists in numerous forms, such as thermal, mechanical, kinetic, potential, electrical, magnetic, chemical, nuclear, and all these forms constitute the total
energy of a system. All these forms are called different
modes of energy. The mode of energy significantly
affects the efficiency and type of energy exchange of
a thermodynamic system. For thermodynamic studies,
the modes of energy are grouped into macroscopic and
microscopic modes, discussed as follows:
The macroscopic
1. Macroscopic Modes of Energy
energy of a system is related to motion and
the influence of some external potentials, such
isothermal process, isobaric process, isochoric (or isometric)
process.
4 Differentials of point functions are exact or perfect differentials.
5 The term “energy” was coined in 1807 by Thomas Young, and
its use in thermodynamics was proposed in 1852 by Lord Kelvin.
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8.1 BASIC CONCEPTS
503
as gravity, magnetism, electricity, and surface
tension. A system possesses such forms of energy
always with respect to some external reference
frame. Thus, kinetic energy and potential energy
come under macroscopic mode of energy, explained
as follows:
negligible. Thus, the total energy of a system consists of
kinetic energy, potential energy, and internal energies,
and is expressed as
(a) Kinetic Energy
The energy that a system
possesses as a result of its motion relative to
some reference frame is called kinetic energy.
For example, when all the parts of a system,
having mass m, move with the same velocity
V with respect to some fixed reference frame,
the kinetic energy is expressed as
Closed system generally remains stationary during a
process and thus experience no change in their macroscopic energy. For such systems, referred to as stationary
systems, the change in total energy is identical to the
change in internal energy:
T =m
V2
2
(b) Potential Energy The energy that a system
possesses as a result of its elevation in a
gravitational field is called potential energy.
For example, when all parts of a system, having
mass m, are at elevation z relative to the center
of a potential field, say gravity g, the potential
energy of the system is equal to
E = U +m
V2
+ mgz
2
∆E = ∆U
In absence of motion and gravity,
E=U
Thermodynamics aims for devising the means for converting disorganized internal energy into useful or organized work, or sometimes interchange between the above
two modes of energy. Thermodynamics does not inquire
about the absolute value of the total energy but deals
only with the change in the total energy.
Ug = mgz
Both of these forms of energy are the organized
form of energy, as these can be readily converted
into work.
2. Microscopic Modes of Energy The molecules are
always in random motion and possess energy
in several forms, such as translational energy,
rotational energy, vibrational energy, electronic
energy, chemical energy, nuclear energy. These are
the microscopic forms of energy which are related
to the molecular structure of a system and the
degree of the molecular activity. These forms of
energy are independent of outside reference frame.
These are the disorganized forms of energy6 that
cannot be readily converted into work.
The sum of all microscopic forms of energy is
called the internal energy7 of the system and is
denoted by U . Since internal energy of a system is
independent of outside reference frame, therefore,
it is a property of the system.
In most of the thermodynamic systems, the effects
of magnetic, electrical and surface tension fields are
6 The
kinetic energy of an object is an organized form of energy
associated with the orderly motion of all molecules in one direction
in a straight path or around an axis. In contrast, the kinetic
energies of the molecules are completely random and highly
disorganized.
7 The term internal energy and its symbol U first appeared in the
works of Rudolph Clausius and William Rankine in the second
half of the nineteenth century.
8.1.8
Equilibrium in Processes
Thermodynamic processes are categorized on the basis
of maintaining thermodynamic equilibrium at each state
point in the process. As such, a process can be quasistatic or irreversible, described as follows:
1. Quasi-Static Processes Quasi-static means “likestatic”. Hence, infinite slowness is the characteristic feature of a quasi-static process. A quasi-static
process is, thus, a succession of infinite equilibrium
states. Such processes are also called reversible
processes because once having taken place, can be
reversed, and in so doing leave no change in either
the system or surroundings.
One way to make real processes approximate
reversible process is to carry out the process
in a series of small or infinitesimal steps. For
example, heat transfer can be considered reversible
if it occurs by virtue of very small temperature
difference between the system and its surrounding.
2. Irreversible Processes An irreversible process is a
process, if reversed, cannot return both the system
and the surroundings to their original states. All
of the natural processes are irreversible processes.
Practically, there exist no truly reversible processes in
this world; however, the term “reversible” is used to
make the analysis simpler, and to determine maximum
theoretical efficiencies.
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8.2
CHAPTER 8: THERMODYNAMICS
ZEROTH LAW OF
THERMODYNAMICS
Several properties of materials depend on temperature
in definite way. This fact forms the basis for accurate
temperature measurement. For example, the commonly
used mercury-in-glass thermometer is based on the
expansion of mercury with temperature.
two systems represent the potential of heat transfer
between the systems.
8.3.1.1 Sign Convention Heat transfer is a directional quantity, and thus, the complete description of
heat interaction requires the specification of both the
magnitude and direction. Heat flow into a system is
taken as positive, and heat flow out of a system is taken
as negative [Fig. 8.7].
The zeroth law of thermodynamics8 states that if two
bodies are in thermal equilibrium with a third body, they
are also in thermal equilibrium with each other. This
law serves as the basis for the validity of temperature
measurement.
Heat (+ve)
ENERGY TRANSFER
The forms of energy not stored in a system can be viewed
as the dynamic forms of energy or as energy interactions
which are recognized only at the system boundary.
Energy can cross the boundary of a closed system in
two distinct forms, heat and work. Therefore, the term
energy for closed systems is meant for ‘work’ and ‘heat’
both. These are the energies in transit and are identified
at the boundary only. An energy interaction is heat
transfer if its driving force is temperature difference,
otherwise it is work.
A quantity transferred to or from a system during
an interaction is not a property since the amount of
such quantity depends on more than just the state of
the system. In other words, the systems possess energy,
but not heat or work. Both forms of energy interactions
are associated with process, not with a state. Therefore,
heat and work are path functions; their magnitudes
depend on the path followed during a process as well
as the end states.
8.3.1
Heat Transfer
Heat transfer is defined as the energy interaction across
a boundary of a system by virtue of a temperature
difference. Thus, the temperature difference between
8 The
zeroth law of thermodynamics was first formulated and
labeled by R. H. Fowler in 1931. This was recognized as
a fundamental principle more than half a century after the
formulation of the first and the second laws of thermodynamics.
The law is named zeroth law since it should have preceded the
first and second laws of thermodynamics.
Surrounding
System
In temperature measurements, the third body is
replaced with a thermometer and zeroth law is restated
as “two bodies are in thermal equilibrium if both have
the same temperature reading even if they are not in
contact”.
8.3
✐
Heat (−ve)
Boundary
Figure 8.7
Heat transfer (sign convention).
8.3.1.2 Heat Transfer in a Process The amount of
heat transfer during the process between two states
(say 1 and 2) is denoted by Q12 or just Q. Sometimes,
knowledge of heat transfer rate is desired instead of
total heat transferred over some time interval. The heat
transfer rate is denoted by Q̇. When Q̇ varies with time
(t), the amount of heat transfer during a process is
determined by integrating Q̇ over the time interval of
the process, as
Z 2
Q=
Q̇dt
1
When Q̇ remains constant during a process, the above
relation reduces to
Q = Q̇ (t2 − t1 )
Being a path function, heat transfer in a process from
state 1 to state 2 can be represented as
Z 2
Q1−2 =
T dX
1
where T is the temperature at the point in the path and
X is another property9 of the system.
A process in which heat cannot cross the boundary
of the system is called adiabatic process10 . Thus, an
adiabatic process involves only work interaction. It
should not be confused with an isothermal process. Even
though there is no heat transfer during an adiabatic
process, the energy content and thus the temperature
of a system can still be changed by other means, such as
work.
9 The
quantity X is actually the entropy of the system [Section
8.5.14].
10 The word adiabatic comes from Greek word adiabatos, which
means not to be passed.
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8.3.2
Work Transfer
If the energy crossing the boundary of a closed system
is not heat, it must be work. Work is done by a force
if it causes a body to move in the direction of the
force. A rising piston, rotating shaft, and an electric wire
crossing the system boundaries, all are associated with
work interactions.
The work done during a process between two states 1
and 2 is denoted by W12 or just W . Work done per unit
time is called power, denoted by P . The unit of power
is kJ/s or kW.
In thermodynamics, the work is said to be done by a
system if the sole effect on things external to the system
can be reduced to the raising of a weight (cannot actually
but imaginary).
8.3.2.1 Sign Convention Work transfer is also a
directional quantity. Work is arbitrarily taken to be
positive when the system does work, and negative when
the work is done on the system [Fig. 8.8].
Work (−ve)
Surrounding
Figure 8.8
505
8.3.2.3 Flow Work The flow work, significant only
in flow process or an open system, represents the energy
transferred across the system boundary as a result of
the energy imparted to the fluid by a pump, blower,
or compressor to make the fluid flow across the control
volume. It is analogous to displacement work.
The flow work per unit mass is equal to pv, equivalent
to the work required to push the volume of mass from
zero to v under constant pressure p.
8.3.2.4 Work in Free Expansion Free expansion of a
gas against vacuum is not a quasi-static process. Since
vacuum does not offer any resistance to the expansion
of the gas, therefore, there is no work transfer involved
in the free expansion.
8.3.2.5 Electrical Work In an electrical field, electrons in a wire move under the effect of electromagnetic
forces. Thus, electrons crossing the system boundary do
electrical work on the system.
In general, potential difference V and the current I
can vary with time, therefore, the electrical work done
during a time interval from t1 to t2 is expressed as
Z t2
W =
V Idt
t1
System
8.4
Work (+ve)
Boundary
✐
Work transfer (sign convention).
8.3.2.2 Work Done in a Process When volume (v)
of a gas changes with pressure (p) in a quasi-static
process, the work done by the system (gas) is given by
dW = pdv
Z v2
W12 =
pdv
(8.1)
FIRST LAW OF
THERMODYNAMICS
Experiments have shown that by means of proper
apparatus, any form of energy can be converted into
other forms, and that during this process absolutely
no part of energy is lost. Heat energy and mechanical
energy are thus found inter-convertible. Since nothing is
lost in such conversions, a unit of one form of the energy
must always give certain number of units of another
form. The first law of thermodynamics is the application
of the conservation of energy principle in thermodynamic
processes.
v1
Therefore, work is a path function; dW is an inexact or
imperfect differential.
8.4.1
For irreversible processes, the path cannot be certain,
therefore,
The expressions of the first law of thermodynamics
are different for closed and open systems (steady and
unsteady), discussed as follows:
 Z

 = pdv
Z
W

 6= pdv
Reversible processes
Irreversible processes
If the work done on a gas is equal to the change in
potential energy (of mass), it results in a situation where
dv = 0 and yet dW is not equal to zero.
Expressions of First Law
1. Closed System If a closed system [Fig. 8.9] within
given time period takes heat dQ, works dW , and
change in its internal energy is dU , then, the first
law of thermodynamics is represented as
dQ = dW + dU
(8.2)
Therefore, for a closed system, heat transfer is the
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dQ
dQ
Surrounding
dU
Figure 8.9
dW
2. Cyclic Processes If a closed system in a part of
its cycle takes heat dQ and works dW , the cyclic
integral of the change in internal energy (dU as
a property) would be zero. Therefore, using Eq.
(8.2), one gets
I
I
I
dQ = dW + dU
I
I
dQ = dW
(8.3)
4. Unsteady Flow Systems In the present context,
following is the common expression for two unsteady flow processes: filling and emptying,
Q−W =
(m2 − m1 ) ef
|
{z
}
change in flow energy
where
u1
u2
p1 v 1
p2 v 2
= c v T1
= c v T2
= m1 RT1
= m2 RT2
In the above equation, Q and W represent heat
and work transfer, m1 and m2 represent the
mass in the reservoir before and after the process,
respectively, u denotes specific internal energy, and
the value of ef is defined as
cycle
Therefore, for a cyclic process, the total heat
transfer is equal to total work transfer.
ef =
3. Steady Flow Systems In a steady flow system with
unit mass flow rate, the first law of thermodynamics is written as
(h2 − h1 )
| {z }
m2 u 2 − m1 u 1
{z
}
|
change in internal energy
−
sents the cyclic integration. For discrete energy
exchanges, Eq. (8.3) takes following form:
X X Q
=
W
(8.4)
Q−W =
Steady flow system.
Figure 8.10
This is the first law of thermodynamics Ifor cyclic
repreprocesses on a closed system where
cycle
h2 , V 2
dW
First law of thermodynamics.
sum of work transfer and change in internal energy
of the system.
z2
Control
volume
h1 , V 1
System
Boundary
Surrounding
z1
(
c p Tr
c p T2
Filling processes
Emptying processes
where Tr is the temperature of reservoir used in
filling process.
Change in enthalpy
+
V2 2 − V1 2
2 }
| {z
Change in K.E.
+ g (z2 − z1 )
| {z }
Change in P.E.
where Q and W represent heat and work transfer
per unit mass, respectively, h, V , z represent
specific enthalpies, velocities and elevation above
datum levels with subscripts 1 and 2 for inlet and
outlet points, respectively [Fig. 8.10].
This equation is called steady flow energy equation [Section 8.4.5].
8.4.2
Energy - A Property
The interactions of heat and work cause a change in
the stored energy (E) of the system. During energy
transfer, the system undergoes a change from one state
to another. When a closed system undergoes a cyclic
process, the total heat transfer is equal to the total work
transfer. In other words, the cyclic integral of the change
in energy of the system is zero; energy is a point function,
therefore, a property of the system.
Consider a system undergoing cycles between state 1
to state 2 in two alternative paths A and B and returning
by a common path C. So, the system undergoes a cycle
A-B [Fig. 8.11].
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1
b
A
B
C
b
2
Figure 8.11
Energy - a property.
507
this process, pressure and volume play dominant roles.
Therefore, specific heat can be measured in two ways,
by keeping constant volume or constant pressure. The
specific heat measured by keeping volume constant is
called specific heat at constant volume (cv ). Similarly,
the specific heat measured by keeping pressure constant
is called specific heat at constant pressure (cp ). Their
relationship with internal energy and enthalpy can be
established as follows:
1. Constant Volume Process
process, dv = 0.
Z
(∆u)v =
For a constant volume
T2
cv dT
T1
Using the first law for two processes A and B [Eq.
(8.2)]:
and for a closed system of unit mass
dQv = du + pdv
= du
Z T2
=
cv dT
dQC = dEC + dWC
dQA = dEA + dWA
For a cycle consisting of processes C and A [Eq. (8.3)]:
I
I
dQ = dW
T1
Thus, heat transfer at constant volume changes
the internal energy of the system in equal amount:
∂Q
cv =
∂T v
dQC + dQA = dWC + dWA
dQC − dWC = − (dQA − dWA )
dEC = −dEA
Therefore, specific heat of a substance at constant
volume cv is the rate of change of internal energy
with respect to temperature:
∂u
cv =
∂T
Similarly, for a cycle consisting of processes C and B:
dEC = −dEB
This indicates that the change in energy (dE) between
two states of a system is the same, irrespective of
the path the system follows between the two states.
Therefore, energy of a system is a point function and
a property of the system.
8.4.3
2. Constant Pressure Process
(dp = 0),
d (pv) = pdv + vdp
= pdv
Enthalpy
Therefore,
dQp = du + pdv
= du + d(pv)
= d(u + pv)
= dh
Enthalpy of a system is the energy in terms of sum of
internal energy, and flow energy. For a system of unit
mass, the specific enthalpy (h) is given by
h = u + pv
Hence, heat transfer at constant pressure changes
the enthalpy of the system with equal amount.
Therefore, specific heat at constant pressure cp
is the rate of change of enthalpy with respect to
temperature11 :
∂h
cp =
∂T
where u is the specific internal energy and pv is the
specific flow energy. Enthalpy is an important form
of energy, having special relevance in thermodynamic
analysis of open systems.
8.4.4
In an isobaric process
Specific Heats
Specific heat is the amount of heat required by unit
mass of the system for unit rise of its temperature. In
11 For
an ideal gas,
pv = nRT
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A relevant term, heat capacity (C), is defined as the
amount of heat required by full mass of the system
for unit rise of its temperature at constant volume or
constant pressure. For a system having mass m,
Cv = mcv
Cp = mcp
The specific energy e and specific enthalpy h are
written, respectively, as
V2
+ zg + u
2
h = u + pv
e=
The equation of continuity is used to relate flow area,
specific volume and velocity at inlet and outlet points:
A1 V1
A2 V2
=
v1
v2
where Cv and Cp are the heat capacities at constant
volume and constant pressure, respectively.
8.4.5
✐
Steady Flow Systems
Most of the engineering devices work at constant rate of
flow of mass and energy through the control surface and
the control volume in course of time attains an invariant
state with time. Such a state is called steady flow state.
At the steady state of a system, any thermodynamic
property will have a fixed value at a particular location,
and will not alter with time (t) but can vary with space.
Consider a steady flow system in which one stream
of fluid enters at point 1 and leaves the control volume
at point 2. There is no accumulation of mass or energy
within the control volume [Fig. 8.12].
dQ
z2 , ṁ, A2
The shaft work Wx is the only external work done by
the system. Steady flow systems involve flow energy
(pv) at inlet and outlet points. Therefore, assuming no
accumulation of energy within the system,
dQ
dWx
= ṁ (e2 + p2 v2 ) +
dt
dt
dQ
dWx
e 1 + p1 v 1 +
= e 2 + p2 v 2 +
dm
dm
2
dQ
V2 2
dWx
V1
+ z1 g +
= h2 +
+ z2 g +
h1 +
2
dm
2
dm
ṁ (e1 + p1 v1 ) +
where dQ/dm and dWx /dm represent the rate of heat
and work transfer per unit mass, respectively. This
equation can also be written as
Q − Wx = (h2 − h1 ) +
V2 2 − V1 2
+ g (z2 − z1 )
2
h 2 , p2 , v 2 , V 2
z1 , ṁ, A1
h 1 , p1 , v 1 , V 1
Control volume
mcv , z
Flow out
Flow in
dW
Figure 8.12
Steady flow system.
The symbols, A, ṁ, p, v, u, V and z are used to
represent cross-sectional area, mass flow rate, pressure
(absolute), specific volume, specific internal energy,
velocity, and elevation above an arbitrary datum, respectively, and mcv represents the mass of control volume.
The net rates of heat transfer and work transfer through
the control surface are dQ/dt, dW /dt, respectively.
(8.5)
where Q and Wx refer to energy transfer per unit mass.
This equation is known as steady flow energy equation
(SFEE). The differential form of the above equation is
dQ − dWx = dh + V dV + gdz
(8.6)
The application of SFEE is explained in the following
devices:
1. Nozzle and Diffuser
Nozzles are used in turbomachines to convert pressure energy into the
kinetic energy of a fluid. A diffuser increases the
pressure of a fluid at the expense of kinetic energy
[Fig. 8.13].
1
Flow in
Throat
2
Flow out
b
b
and internal energy is a function of temperature only. Therefore,
enthalpy can be written as
b
Insulated
surface
h = u(T ) + nRT
= f (T )
It means that for an ideal gas, enthalpy is also the function of
temperature only.
Figure 8.13
Nozzle and diffuser.
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8.4 FIRST LAW OF THERMODYNAMICS
throttling14 . The drop in pressure is compensated
by corresponding change in density or internal
energy of the flowing fluid.
For a nozzle or diffuser, the flow is assumed to
be adiabatic (dQ = 0) and there is no work transfer
(dWx = 0). Using Eq. (8.5),
h1 +
V1 2
V2 2
= h2 +
2
2
q
V2 = 2 (h1 − h2 ) + V1 2
3. Turbine and Compressor
Turbine and engines
are used to extract power from the working
fluid, whereas compressor and pumps are used to
energize the fluid [Fig. 8.15].
This expression12 is used in determining the exit
velocity.
2
Turbine
Mach number, an important quantity in study
of compressible flow (such as in nozzle or diffuser),
is defined as the ratio of the velocity
√ of gas (V ) to
velocity of sound in the gas (a = γRT )
M=
b
ṁ
1
Figure 8.15
2
b
b
W
Turbine and compressor.
For a well-insulated turbine system (Q = 0)
and ignoring the changes in kinetic and potential
energies,
Flow out
h1 = h2 +
Insulated
surface
Figure 8.14
ṁ
Flow in
2. Throttling Device Throttling13 is the process of
passing a fluid through a constricted passage, resulting in an appreciable pressure drop. Throttling
Flow in
W
m
W
= h1 − h2
m
Throttling device.
valves are usually small devices [Fig. 8.14], and
the flow through them can be assumed to be
adiabatic since there is neither sufficient time nor
large enough area for any effective heat transfer to
take place. Also, there is no work done and change
in potential energy:
The change in enthalpy of the fluid is equal to the
amount of work transfer.
4. Heat Exchanger
A heat exchanger is used to
transfer heat from one fluid to another [Fig. 8.16].
b
dQ = dWx = dz = 0.
Flow in
b
Figure 8.16
This indicates that the enthalpy of the fluid before
throttling is equal to the enthalpy of the fluid after
ṁ
1
ṁ
V1 2
V2 2
= h2 +
2
2
h1 = h2
Q
Heat exchanger.
When the law of conservation of energy is
applied, the rate of change of enthalpy of one fluid
is equal to that of the other fluid15 .
12 In
using above equation, the units of h and V should be observed
without mistake, because, generally h is in kJ/kg and V 2 /2 is in
J. Therefore, (h1 − h2 ) must be divided by 1000.
13 The magnitude of the temperature drop or rise during a
throttling process is governed by a property called the JouleThomson coefficient.
Flow out
2
The variation in velocities (V1 , V2 ) at inlet and
outlet can also be assumed to be negligible.
Therefore, using Eq. (8.5),
h1 +
Flow out
b
V
a
1
509
14 This
observation is used in measurement of dryness fraction of
steam.
15 Heat exchangers are specifically studied in the subject of heat
transfer.
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8.4.6
✐
CHAPTER 8: THERMODYNAMICS
Unsteady Flow Systems
For a finite time interval, the above equation
becomes
In general, unsteady flow processes are difficult to
analyze because the properties of the mass at the inlets
and outlets can change during a process.
Consider a device through which a fluid is flowing
under unsteady state [Fig. 8.17].
dQ
z2 , ṁ2 , A2
h 2 , p2 , v 2 , V 2
z1 , ṁ1 , A1
h 1 , p1 , v 1 , V 1
Control volume
mcv , z
Flow out
Flow in
dW
Figure 8.17
Unsteady flow system.
It requires idealization that the fluid flow at any inlet
or outlet is uniform and steady, and thus, the fluid
properties do not change with time or position over the
cross of an inlet or outlet. If they do, they are averaged
and treated as constant for the entire process.
As in the steady flow system, the equations of
conservation of mass and energy are applied here:
1. Conservation of Mass
The rate at which the
mass of fluid within the control volume (mcv ) is
accumulated is equal to the net rate of mass flow
across the control surface, as given below
dmcv
= ṁ1 − ṁ2
dt
The change in mass inside the control volume over
any finite period of time:
∆Ecv = Q − Wx
Z V1 2
+ z1 g dm1
+
h1 +
2
Z V2 2
−
h2 +
+ z2 g dm2
2
This equation is known as unsteady flow energy equation
(USFEE). The application of USFEE can be seen in the
following processes:
1. Charging Process Consider a process in which gas
bottle is filled from a pipeline. In the beginning
the bottle contains gas of mass m1 , at state (p1 ,
T1 , v1 , h1 , u1 ). The valve is opened and gas flows
into the bottle till the mass of gas in the bottle is
m2 at state (p2 , T2 , v2 , h2 , u2 ). The supply to the
pipeline is very large so that the state of gas in the
pipeline (indicated by subscript p) is constant at
(pp , Tp , vp , hp , up ) and velocity of flow is Vp .
The change in internal energy of the control
volume is
∆Ecv = m2 u2 − m1 u1
(8.7)
Since addition of energy is from single side (1) only,
therefore,
Z Vp 2
∆Ecv = Q − W +
hp +
dm1
2
By putting value of Ecv from Eq. (8.7),
Q − W = m2 u 2 − m1 u 1
Vp 2
− hp +
(m2 − m1 )
2
∆mcv = ∆m1 − ∆m2
This is the equation of first law for unsteady
charging process.
2. Conservation of Energy
Energy of the system
within the control volume is written as
mcv V 2
+ mcv gz
Ecv = U +
2
cv
By ignoring Vp , the equation can be further
reduced to
Open flow systems involve flow energy (pv) at
inlet and outlet points. The rate of increase in this
energy is equal to the net rate of energy inflow:
dEcv
V1 2
= h1 +
+ z1 g ṁ1
dt
2
V2 2
− h2 +
+ z2 g ṁ2
2
dQ dWx
+
−
dt
dt
If the tank is empty before the start of charging
process (m1 = 0), and there is no energy transfer,
then
Q − W = m2 u2 − m1 u1 − hp (m2 − m1 )
h p m2 = m2 u 2
c p Tp = c v T2
T2 = γTp
Thus, the temperature of gas after charging (T2 )
will be equal to γ times the temperature of gas in
the pipe (Tp ).
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2. Discharging Process Consider a case of discharging
of a bottle in which the extraction is from single
side (2) only. Thus,
Z V2 2
∆Ecv = Q − W −
h2 +
dm2
2
Therefore,
Q − W = m2 u 2 − m1 u 1
V2 2
− h2 +
(m2 − m1 )
2
This is the equation of first law for unsteady
discharging process. By ignoring V2 , the equation
can be further reduced to
Q − W = m2 u2 − m1 u1 − h2 (m2 − m1 )
The above analysis leads to the following common
equation:
Q − W = m2 u2 − m1 u1 − (m2 − m1 ) ef
where
ef =
(
c p Tp
c p T2
Charging process
Discharging process
which represents the specific enthalpy of the working
fluid in motion, as in the case of
1. Charging Process cp Tp is the enthalpy of charging
fluid (at pipe), and
2. Discharging Process
cp T2 is the enthalpy of
discharging fluid (at 2).
8.5
SECOND LAW OF
THERMODYNAMICS
The first law of thermodynamics does not impose any
restriction on the direction of a process; satisfying the
first law does not ensure that the process can actually
occur. However, natural processes occur only in one
direction, for example, heat flows from higher to lower
temperatures, water flows downward, time flows in
the forward direction. The reverse of these phenomena
never happens spontaneously. The spontaneity of the
process is driven by a finite potential, such as gradients
of temperature, concentration, electric potential. So
important is this observation, that it is called the second
law of thermodynamics, which remedies the inadequacy
of the first law in identifying the feasibility of a process.
Mechanical energy can be simply converted into heat
energy. For example, heat is produced by friction of
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511
moving bodies and in other similar phenomena. The first
law can be used to state that, if a process occurs, the
net change in energy will be zero:
W =Q
However, the change in the opposite direction is by
far the most difficult task. The apparatus necessary
to convert heat into mechanical forms of energy is
complicated and does not even theoretically convert all
of the supplied heat energy:
−
→
Q>W
Therefore, work is considered as high grade energy
while heat as low grade energy. The conversion of low
grade energy (heat, Q) into high grade energy (work,
W ) is possible through a cyclic heat engine, but it is
incomplete16 .
8.5.1
Energy Reservoirs
In the development of the second law of thermodynamics, hypothetical bodies known as energy reservoirs,
facilitate understanding of thermodynamic cycles or
processes. Thermodynamic analysis involves two types
of energy reservoirs:
1. Thermal Energy Reservoir A thermal energy reservoir (TER) is defined as a large hypothetical
body of infinite heat capacity which is capable
of absorbing or rejecting an unlimited quantity
of heat without suffering appreciable changes
in its thermodynamic coordinates. Atmosphere,
large rivers, and a two-phase system can be
conveniently modeled as thermal energy reservoirs.
2. Mechanical Energy Reservoir A mechanical energy
reservoir (MER) is a large body enclosed by an
adiabatic impermeable wall capable of storing
work as potential energy (e.g. raised weight or
wound spring) or kinetic energy (e.g. flywheel).
8.5.2
Cyclic Heat Engine
A heat engine cycle involves net heat transfer and work
transfer to the system. Heat engine can be a closed
system (e.g. gas confined in a cylinder and piston) or
an open system (e.g. steam or gas power plant).
Let heat Q1 be transferred to the system and work W
be the net work done by the system. Heat Q2 is rejected
from the system and the system is brought back to its
initial condition [Fig. 8.18].
16 Sadi
Carnot, a French military engineer, first studied this aspect
of energy transformation.
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Heat source
T1
T1
Heat sink
Q1
Q1
E
R
W
W
Q2
Q2
T2
Heat source
T2
Heat sink
Cyclic heat engine.
Figure 8.18
Figure 8.19
of desired thermal effect and work input to the system:
Q2
W
Q2
=
Q1 − Q2
The direction of arrow of heat engine cycles is
clockwise. Net heat transfer and work transfer in the
cycle are
COPr =
Q = Q1 − Q2
8.5.4
Using the first law of thermodynamics for cyclic systems:
Q=W
The efficiency of a heat engine (ηe ) is defined as the ratio
of net work output and total heat input to the cycle:
(8.8)
Heat Pump
A heat pump (HP) operates in a cycle to maintain a body
at a temperature higher than that of the surrounding.
Consider a body losing heat Q1 to the surroundings.
The cyclic effects in a heat pump are similar to that of
a refrigerator [Fig. 8.20].
W
Q1
Q
=
Q1
Q1 − Q2
=
Q1
Q2
= 1−
Q1
ηe =
T1
Body
Q1
HP
W
Q2
T2
The experience shows that W < Q1 , therefore, ηe < 1;
all the heat input to the heat engine cannot be converted
into work17 .
Figure 8.20
8.5.3
Cyclic refrigerator.
Refrigerator
Heat source
Cyclic heat pump.
Coefficient of performance of a heat pump is defined
A refrigerator operates in a cycle to maintain a body at
a temperature lower than that of its surrounding. Thus,
the direction of a refrigeration cycle is opposite to that
of a refrigeration cycle [Fig. 8.19].
as
To measure the performance of a refrigerator, a
coefficient of performance (COPr ) is defined as the ratio
Using Eqs. (8.8) and (8.9), for heat engine, heat
pump, and refrigerator working between the same
temperature limits:
17 Present
day engines, such as petrol engine, diesel engine, steam
engines, are efficient upto a range of only 30-45%, and engineers
are working rigorously to improve the efficiency.
COPp =
Q1
W
COPp = COPr + 1 =
(8.9)
1
ηe
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8.5 SECOND LAW OF THERMODYNAMICS
8.5.5
Statements of Second Law
The second law of thermodynamics has two but equivalent forms of statements:
1. Kelvin Planck Statement Kelvin and Planck stated
that it is impossible for a heat engine to produce
work in a complete cycle if it exchanges heat only
with bodies at a single fixed temperature. Thus,
for a cyclic heat engine,
Q2 6= 0
A heat engine has to therefore exchange heat
with two thermal energy reservoirs at two different
temperatures to produce work in a complete cycle.
2. Clausius Statement Clausius18 stated that it is
impossible to construct a device which, operating
in a cycle, will produce no effect other than the
transfer of heat from a cooler to a hotter body.
Thus, for a heat pump or refrigerator,
W 6= 0
Heat, therefore, cannot flow on itself from a body
at a lower temperature to a body at a higher
temperature. Some work must be expended to
achieve this.
513
If there is equilibrium and no dissipative effects, all
the work done by the system during a process in one
direction can be returned to the system during the
reverse process. Such processes are reversible in nature.
8.5.7
Carnot Cycle
The most interesting cycle, both historically and thermodynamically, is the Carnot cycle. It could be carried
out with any material as working substance, but can be
simply investigated for the case of a perfect gas [Fig.
8.21].
p
3
b
T =c
s=c
b
b
2
4
s=c
T =c
b
1
v
Both statements of the second law of thermodynamics
are equivalent to each other. This can be shown by
proving that violation of one statement implies the
violation of second, and vice versa.
8.5.6
Reasons of Irreversibility
Any natural process carried out with a finite gradient (of
temperature, pressure, voltage, etc.) is an irreversible
process. All spontaneous processes are irreversible.
Irreversibility of a process can be due to numerous
factors, such as friction, unrestrained expansion, mixing
of two fluids, heat transfer across a finite temperature
difference, electrical resistance, inelastic deformation of
solids, and chemical reactions. These factors can be
grouped into two categories:
1. Lack of Equilibrium This includes heat transfer
through a finite temperature difference, lack of
pressure equilibrium, free expansion, throttling,
etc.
2. Dissipative Work This includes friction, paddle
wheel work transfer, transfer of electricity through
a resistor.
18 Rudolf Julius Emanuel Clausius (1822-1888), was a German
physicist and mathematician, one of the central founders of the
science of thermodynamics. In 1850, he first stated the basic ideas
of the second law of thermodynamics. In 1865, he introduced the
concept of entropy.
Figure 8.21
Carnot cycle.
The Carnot cycle comprises the following four reversible processes,
1. Isothermal process (1 → 2): Heat Q1 is added to
the system.
2. Adiabatic process (2 → 3): Work We is done by
the system.
3. Isothermal process (3 → 4): Heat Q2 is rejected
by the system.
4. Adiabatic process (4 → 1): Work Wc is done on
the the system.
The subscripts e and c in We and Wc , respectively, are
used to signify the expansion and the compression of the
system during the respective processes. The net work
done by the system and net heat transfer into the system
in a cycle is
W = We − Wc
Q = Q1 − Q2
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Efficiency of the Carnot cycle is
T1
Q1B
Q1A
= Q1B
W
Q
Q1 − Q2
=
Q1
Q2
= 1−
Q1
ηCarnot =
WA
EA
Q2A
∃B
WB
Q2B
T2
8.5.8
Carnot Principles
Based on the Kelvin-Planck and Clausius statements
of the second law of thermodynamics, the two Carnot
principles are stated as follows:
1. All heat engines operating between a given constant temperature source and a given constant
temperature sink, none of them having a higher
efficiency than the reversible engine:
ηrev > ηirr
2. The efficiencies of all the reversible engines operating between the same two reservoirs are the same:
ηrev,1 = ηrev,2
The first statement of the Carnot principles can be
proved analytically by considering two heat engines
EA (irreversible) and EB (reversible) operating between
source temperature T1 , and sink temperature T2 [Fig.
8.22].
T1
Q1B
Q1A
= Q1B
WA
EA
Q2A
EB
WB
Q2B
T2
Figure 8.22
Both EA and EB as heat engines.
If ηA > ηB , then for the same amount of heat input
(Q1A = Q1B ),
WA
WB
>
Q1A
Q1B
WA > WB
When EB is reversed to act as heat pump ∃B then
the heat Q1B discharged by ∃B is the heat input to EA
[Fig. 8.23].
Q2B − Q2A
Figure 8.23
Engine EA and heat pump ∃B .
The net effect is a cyclic heat engine producing net
work WA − WB , while exchanging heat from a single
reservoir T2 . This violates the Kelvin Planck statement.
Hence, the assumption ηA > ηB is incorrect, so
ηB ≥ ηA
The second Carnot principle can be proved by
replacing the irreversible engine by another reversible
engine having higher efficiency than the first one. After
reversing the first engine, the net effect will violate the
Clausius statement by producing work while exchanging
heat with single reservoir. This proves that both the
reversible engines would have equal efficiencies.
8.5.9
Celsius Scale
Temperature measuring instruments should have thermometric properties, for example, the length of a
mercury column in a capillary tube, the electrical
resistance of a wire, the pressure of a gas in a closed
vessel, the e.m.f. generated at the junction of two
dissimilar metal wires. To assign numerical values to
the thermal state of a given system, it is necessary to
establish a temperature scale on which temperature of
a system can be read. Therefore, the temperature scale
is read by assigning numerical values to certain easily
reproducible states. For this purpose, Celsius scale19
uses the following two points:
1. Ice Point
The equilibrium temperature of ice
with air saturated water at standard atmospheric
pressure which is assigned a value of 0◦ C.
2. Steam Point The equilibrium temperature of pure
water with its own vapor at standard atmospheric
pressure is assigned a value of 100◦ C.
19 This
scale is called the Celsius Scale named after Anders Celsius.
In 1742 he proposed the Celsius temperature scale. The scale was
later reversed in 1745 by Carl Linnaeus, one year after Celsius’
death.
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8.5 SECOND LAW OF THERMODYNAMICS
8.5.10
Perfect Gas Scale
An ideal gas with unit mole obeys the following constitutive relation20 :
pv = ℜT
where ℜ (= 8.314 J/mol K) is the universal gas constant.
The perfect gas temperature scale is based on the
observation that the temperature of a gas at constant
volume increases with increase in pressure.
The temperature at the triple point (Ttp ) of water
has been assigned a value of 273.16 K. Therefore,
temperature (T ) of an ideal gas varies proportionally
w.r.t. pressure (p):
Here, vtp is the volume of the gas at the triple point
of water and v is the volume of the gas at the system
temperature.
8.5.11
Absolute Temperature Scale
The Carnot principles state that the efficiency of all
engines working between the same temperature levels
is the same, and independent of working substance.
Therefore, for a reversible cycle (say, Carnot cycle)
receiving heat Q1 and rejecting heat Q2 , the efficiency
will solely depend upon the temperatures t1 and t2 at
which heat is transferred:
T
p
=
Ttp
ptp
T = 273.16 ×
Q2
Q1
= f (t1 , t2 )
ηe = 1 −
p
ptp
(8.10)
Let a series of measurements with different amounts of
gas in a bulb be made. The measured pressures at the
triple point as well as at the system temperature change
depending on the amount of gas in the bulb. A plot of
the temperature T , calculated from Eq. (8.10) is shown
in Fig. 8.24.
Q1
= F (t1 , t2 ) (say)
Q2
H2
(8.11)
Let a reversible engine E1 receive heat from source at
t1 and reject heat at t2 to another reversible engine E2
which, in turn, rejects heat to the sink at t3 :
Q1
= F (t1 , t2 )
Q2
Q2
= F (t2 , t3 )
Q3
Air
T
515
Another heat engine E3 can operate between t1 and t3 :
b
Q1
= F (t1 , t3 )
Q3
Q1 /Q3
Q1
=
Q2
Q2 /Q3
He
ptp
Therefore,
Figure 8.24
F (t1 , t2 ) =
T versus ptp .
When these curves are extrapolated to zero pressure,
all of them yield the same intercept. This behavior can
be expected since all gases behave like ideal gas when
their pressure approaches zero.
The correct temperature of the system can be obtained only when the gas behaves like an ideal gas, and
hence, the value is to be calculated in limit ptp → 0.
Therefore,
p
Tptp →0 = 273.16 ×
ptp
A constant pressure thermometer can also be used to
measure the temperature:
v
Tvtp →0 = 273.16 ×
vtp
F (t1 , t3 )
F (t2 , t3 )
The temperatures t1 , t2 , t3 can assume arbitrary value.
Since the ratio Q1 /Q2 depends only on t1 and t2 and
is independent of t3 , t3 is eliminated and the above
equation takes the following form:
Q1
φ (t1 )
=
Q2
φ (t2 )
Kelvin proposed the simplest form of function φ(t) = T ,
therefore,
Q1
T1
=
Q2
T2
This scale is the absolute temperature scale, and is better
known as Kelvin scale21 .
21 The
20 This
equation is only an approximation to the actual behavior
of the gases. The behavior of all gases approaches the ideal gas
limit at sufficiently low pressure.
efficiency of Carnot cycle can be formulated by analysis of
its cycle composed of reversible processes as
ηCarnot = 1 −
T2
T1
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The SI system uses the Kelvin scale for measuring temperature which is based on the concept of
absolute zero, the theoretical temperature at which
molecules would have zero kinetic energy. Absolute zero
(−273.16◦ C) is set at zero on the Kelvin scale. This
means that there is no temperature lower than zero
Kelvin, so there are no negative numbers on the Kelvin
scale.
1
b
Q=0
b
1
′
T =c
b
2′
Q=0
b
8.5.12
Consider a reversible cycle consisting of two reversible
processes (constant entropy, s = c) and an isothermal
process (constant temperature, T = c). Heat transfer
can take place in the isothermal process but not in the
reversible processes [Fig. 8.25].
Reversible paths.
Hence,
Thus, a reversible path can be replaced by a reversible
adiabatic path, followed by a reversible isotherm, and
then by another reversible adiabatic path, such that the
heat transfer during the isothermal process is the same
as that transferred during the original process.
W
s=c
s=c
b
T =c
b
Figure 8.25
Figure 8.26
Q1−2 = Q1−1′ −2′ −2
b
Q
Isotherm
Impossible cycle.
The net effect of the cycle will be production of
work without discharging heat, violating the second law
of thermodynamics. Thus, such a cycle is impossible.
Alternatively, two reversible adiabatic processes passing
through the same end points must coincide with each
other.
8.5.13
2
Reversible Adiabatic Paths
Clausius Theorem
Consider a system changing state from an initial equilibrium state 1 to final equilibrium state 2. Let two
reversible adiabatic paths 1 − 1′ and 2′ − 2 be drawn [Fig.
8.26].
A reversible isotherm 1′ − 2′ is drawn in such a way
that area under 1 − 1′ − 2′ − 2 is equal to that under 1 − 2:
W1−1′ −2′ −2 = W1−2
Using the first law of thermodynamics for closed system
in two alternative paths,
Q1−2 = U2 − U1 + W1−2
Q1−1′ −2′ −2 = U2 − U1 + W1−1′ −2′ −2
This finding proves that the ideal gas temperature and Kelvin
temperature are equivalent.
Reversible cycle
Figure 8.27
Adiabatics
Clausius theorem.
Dividing any reversible cyclic process into such transformations results in large number of Carnot cycles [Fig.
8.27] as
I
dQ
=0
(8.12)
R T
Therefore, for a reversible cyclic process, cyclic integral
of dQ/T is zero. This is known as Clausius theorem.
The Clausius theorem [Eq. (8.12)] is very important
for thermodynamic analysis of power cycles. For example, in Carnot cycle,
Q1 Q2
−
=0
T1 T2
where Q2 is taken negative because it is the heat
rejected.
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8.5 SECOND LAW OF THERMODYNAMICS
8.5.14
Entropy - A Property
In thermodynamic analyses, the quantity dQ/T quantity
is of very frequent use. As established by the Clausius
theorem, the cyclic integral dQ/T is zero. This quantity
is a point function (i.e. a property of the system). This
quantity is the change in entropy of the system22 . The
entropy is represented by S. It is an extensive property
of system and sometimes referred to as total entropy.
Entropy per unit mass is designated as s, an intensive
property, and has the unit of kJ/kgK. Therefore,
dQ = T dS
The entropy change of a system during a process 1-2 can
be determined as
Z 2
dQrev
= (S2 − S1 )
T
1
which is independent of the path.
Based on the definition of entropy, following equations
can be derived for a system of unit mass:
1. First Tds Equation Using the first law of thermodynamics for a closed system [Eq. (8.2)]:
T ds = du + pdv
(8.13)
between the same end points:
Z 2
dQ
≤ (S2 − S1 )irr
T
1
This equation is known as the Clausius inequality23
or the entropy principle which is valid for all thermodynamic cycles, reversible or irreversible, including
refrigeration cycles. The equality in this equation holds
for totally reversible cycles and the inequality for the
irreversible ones.
The Clausius inequality is used as an alternative form
of the second law of thermodynamics, which helps in
examining the feasibility of a process (i.e. whether a
process is possible or impossible). The inequality can
also be used in finding the condition for maximum work.
This can be demonstrated through processes discussed
as follows.
8.5.15.1 Heat Transfer Consider a case when transfer of heat Q between two bodies takes place at a finite
temperature difference T1 − T2 [Fig. 8.28].
Q
T1
By definition of enthalpy
h = u + pv
dh = du + d(pv)
= du + pdv + vdp
= T ds + pdv
T ds = dh − vdp
This is the second T ds equation which is obtained
by eliminating du from the first T ds equation.
(8.15)
The above two equations are combined together as
I
dQ
≤0
(8.16)
T
This is the first T ds equation, also known as Gibbs
equation.
2. Second Tds Equation
517
T2
Figure 8.28
Heat transfer.
The total entropy change is the sum of entropy
changes in both bodies:
Q Q
+
T1 T2
Q (T1 − T2 )
=
T1 T2
dS = −
8.5.15
Clausius Inequality
The Clausius theorem [Eq. (8.12)] for reversible processes cycle is written as
I
dQ
=0
(8.14)
R T
However, entropy change in an irreversible process is
always higher than entropy change in a reversible process
22 Rudolph Clausius (1822-1888) realized in 1865 that he had
discovered a new thermodynamic property, and he chose to name
this property entropy, originally entropie (on analogy of Energie)
from Greek entropia “a turning toward,” from en “in” + trope “a
turning”.
For a naturally possible process (dS > 0), T1 > T2 ,
otherwise the process is impossible. This means that
heat transfer from a lower body to higher body is
impossible (without add of work), thus returning to the
second law of thermodynamics.
8.5.15.2 Mixing of Fluids Consider mixing of two
fluids of equal heat capacity C at temperatures T1 , T2 .
The final temperature of mixture is Tf [Fig. 8.29].
23 Clausius
inequality was first stated by the German physicis R. J.
E. Clausius (1822-1888), one of the founders of thermodynamics.
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m1 , c 1 , T 1
m2 , c 2 , T 2
The work will be maximum when entropy change is zero:
p
Tf =
m1 + m2 , c , T f
Therefore, the maximum possible work is
Wmax = Cp [(T1 − Tf ) − (T2 − Tf )]
p
= Cp T1 − T2 − 2 T1 T2
Mixing of fluids.
Figure 8.29
The total entropy change in mixing is
Tf
Tf
dS = C ln
+ C ln
T1
T2
Tf
Tf
+ ln
= C ln
T1
T2
!
2
Tf
= C ln
T1 T2
8.5.15.4 Work from a Finite Body Consider a finite
body of heat capacity Cp at temperature T . To extract
work from this body a TER at T0 can be used as sink
[Fig. 8.31].
T1 → T0
Q
E
2
Since Tf > T1 T2 , dS > 0, so, it is a possible case.
Q
Wmax
T0
Figure 8.31
Work from a finite body.
The temperature of TER does not change during the
heat transfer. If heat Q is taken out from the finite body
and heat engine extracts work W , then the change in
entropy of TER is
E
∆Sr =
Q − Wmax
T2 → Tf
Figure 8.30
Tf
Tf
Cp ln
+ Cp ln
≥0
T
T2
1
Tf
Tf
Cp ln
+ ln
≥0
T1
T2
Tf2
≥0
Cp ln
T1 T2
p
Tf ≥ T1 T2
Q−W
T0
The heat can be extracted until the temperature of body
reaches to that of TER. Therefore, the condition for
possibility of this process is
Work from finite bodies.
The net change in entropy is
W
Q−W
8.5.15.3 Work from Finite Bodies Maximum work
obtainable from two finite bodies of heat capacity Cp
at temperatures T1 and T2 can be investigated. Let
the final temperature reached by both the bodies after
extraction of maximum obtainable work is Tf [Fig. 8.30].
T1 → Tf
T1 T2
Cp ln
T0 Q − W
+
≥0
T
T0
Therefore,
T
W ≤ Cp (T − T0 ) − T0 ln
T0
8.5.16
The Increase of Entropy Principle
Consider a cycle made of arbitrary (reversible or irreversible) processes 1-2, and an internally reversible 2-1.
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519
Using the Clausius inequality for this cycle:
I
dQ
≤0
T
Z 1
Z 2
dQ
dQ
+
≤0
T
T
2
1
Z 2
dQ
+ S1 − S2 ≤ 0
T
1
Z 2
dQ
S2 − S1 ≥
T
1
closed system if it undergoes an internally reversible and
adiabatic process.
where S2 − S1 is the change of entropy when the
system undergoes any reversible process from state
1 to 2. In the above expression, the equality holds
for an internally reversible process and the inequality
for an irreversible process. The expression can also be
presented in differential form:
Any device that violates the first or second law of
thermodynamics is called a perpetual motion machine
(PPM), which can be of two types:
dS ≥
dQ
T
Therefore, entropy change of a closed system during an
irreversible process is greater than the integral of dQ/T
(entropy transfer) evaluated for that process. This can
be taken as entropy generated during an irreversible
process, due to entirely the presence of irreversibilities.
Entropy generation is denoted by Sgen , therefore,
Z 2
dQ
S2 − S1 =
+ Sgen
T
1
Entropy generation Sgen is not a property of the system
because it depends on the process. It is either a positive
quantity or zero (for reversible processes).
Heat transfer is accompanied with entropy transfer,
while work transfer does not involve entropy transfer.
For an isolated system undergoing an irreversible
path, the entropy transfer is zero; the entropy change of
a system is equal to the entropy generation. Therefore,
∆Sisolated ≥ 0
Therefore, entropy of an isolated system during a process
always increases or in the limiting case of reversible
process remains constant. This is known as increase in
entropy principle.
8.5.17
An isentropic process can serve as an appropriate
model for actual processes, such as in pumps, turbine,
nozzles, diffusers. Therefore, in many applications, isentropic processes are used to define isentropic efficiencies
for the processes to compare the actual performance of
these devices.
8.5.18
Perpetual Motion Machines
1. PPM-I Perpetual motion machines of first class
does work without input energy, thus violate the
first law of thermodynamics.
2. PPM-II Perpetual motion machines of second
class is an engine without any heat rejection or
a refrigerator without work input, thus violate the
second law of thermodynamics.
Despite numerous attempts, no perpetual motion machine is known to have worked.
8.6
THIRD LAW OF
THERMODYNAMICS
Consider a hypothetical situation when enough engines
are placed in series such that the heat rejected from
the last engine is zero; absolute temperature of the
last sink is zero. However, the second law of thermodynamics proves its impossibility. Thus, it appears
that a definite zero exists on the absolute temperature
scale, which cannot be reached without violation of the
second law. In other words, attainable values of absolute
temperature are always greater than zero. This is also
known as the third law of thermodynamics. In terms of
the Fowler-Guggenheim statement, it is impossible by
any procedure, no matter how idealized, to reduce any
system to the absolute zero of temperature in a finite
number of operations.
Isentropic Processes
A process during which entropy remains constant is
called an isentropic process24 . This is possible for a
8.7
24 An
The second law of thermodynamics prohibits complete
conversion of a low grade energy (heat) into high grade
energy (shaft work). That part of low grade energy
which is available for conversion is referred as available
energy (AE), while the part, which, according to second
isentropic process is not necessarily a reversible adiabatic
process because entropy increase of a substance during a process as
a result of irreversibilities can be offset by a decrease in entropy due
to heat losses. However, the term isentropic process is customarily
used in thermodynamics to imply an internally reversible adiabatic
processes.
EXERGY AND IRREVERSIBILITY
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CHAPTER 8: THERMODYNAMICS
law, must be rejected, is known as unavailable energy
(UAE). In this context, a new term, availability25 , is
introduced in the study.
8.7.1
Wu = W − p0 (v2 − v1 )
The volume of a steady flow system does not change,
hence, maximum useful work would remain same.
Exergy
A system can deliver the maximum possible work when
it undergoes from the specified initial state to the state
of its environment (dead state) through a reversible
process. This represents the useful work potential of the
system at the specified initial state, and is called exergy.
It is also termed as availability26 .
Availability or exergy is only the potential of work,
not the actual work, that a system can deliver without
violating any of the thermodynamic laws. It depends
upon the state of both the system and its surrounding.
The concept can be examined in the following systems.
8.7.3.1 Carnot Cycle Consider a closed system undergoing a Carnot cycle [Fig. 8.21] in which heat
rejected is given by
Q2 = T2
25 Josiah
T0
W = Q1 1 −
T1
Q1
= Q1 − T0
T1
(8.17)
8.7.3.2 Closed System Consider a closed system
which is given input heat Q during which it passes
through a path 1-2 [Fig. 8.32].
T
b
1
dQ1
T1
Useful Work
The work produced by a device is not always entirely in
a usable form. For example, when a gas of unit mass in a
piston cylinder device expands from v1 to v2 , part of the
work done by the gas is used to push the atmospheric
air at constant pressure p0 . The difference between the
actual work W and the surrounding work is called useful
work, written as
8.7.3
Heat rejection can be minimized by reducing T2 up to
T0 . Hence, the availability of the system is
Dead State
A system is said to be in the dead state when it is in
equilibrium with its surrounding. Thus, a system at the
dead state is in chemical, thermal and mechanical equilibrium; the system is at the temperature and pressure of
its environment, and has no kinetic or potential energy
relative to the environment. The properties of a system
at dead state are denoted by subscript zero, for example,
p0 , T0 , h0 , u0 , and s0 for a system of unit mass.
8.7.2
✐
Q1
T1
Willard Gibbs is accredited with being the originator of
this concept of availability.
26 The term availability was introduced in US in the 1940s. Its
synonym exergy was introduced in Europe in the 1950s, which
has found global acceptance partly because it is shorter, it rhymes
with energy and entropy.
2
b
T0
dS
S
Work on T -S diagram.
Figure 8.32
Using Eq. (8.17), the available work for a small heat
transfer dQ is
dQ
dW = dQ −
T0
T1
The maximum available work (availability) is
W =
Z
2
dW
1
Z
2
Z
2
dQ
1
1 T1
= Q − T0 (S2 − S1 )
=
dQ − T0
Unavailable energy is
UAE = Q − W
= T0 (S2 − S1 )
8.7.3.3 Finite Heat Source Consider a finite source
of heat capacity C, undergoing a temperature change
from T1 to T0 . The available energy (availability) of the
source is
Z
T1
Z
T1
dT
C
T
T0
T0
T1
= C (T1 − T0 ) − T0 ln
T0
W =
CdT − T0
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8.8 PROPERTIES OF GASES
8.7.3.4 Effect of Temperature Consider heat loss
from a hot gas flowing through a conductive pipe [Fig.
8.33]. The process is associated with entropy increase
given by
dS =
dQ
T
By definition, if the system changes its state from
1 to 2, and respective availability functions are φ1 ,
φ2 , then, reversible work (availability) is simply
derived as
Wmax = φ1 − φ2
2. Steady Flow System
Flow systems involve flow
work pv and macroscopic energy, therefore, availability function of a steady flow system having unit
rate of mass flow [Section 8.4.5] is defined as
dT
= mc
T
dT
T
∝
dS
mc
ψ = h − T0 s +
T
b
1
b
Wmax = ψ1 − ψ2
b
Another term for an steady flow process, B is
defined as
B = h − T0 S
b
b
S
Figure 8.33
which is equal to the availability function when
there is no variation in kinetic energy and potential
energy of the system.
Work on T –S diagram.
Therefore, as temperature increases, slope of T –S
diagram increases; loss of available energy is more when
heat loss occurs at a higher temperature than when
the same heat loss occurs at a lower temperature. Eq.
(8.16) shows that maximum work is possible only in
reversible process because S2 -S1 is higher than dQ/T
in an irreversible process.
8.7.3.5 Steady Flow System The availability of a
steady flow system [Section 8.4.5] is
V1 2
dWmax = h1 − T0 s1 +
+ gz1
2
V2 2
+ gz2
− h2 − T 0 s 2 +
2
8.7.4
Availability Function
When a system changes its state tending towards that
of its surrounding, the work potential diminishes and
finally ceases to exist at dead state. Thus, an important
quantity, availability function, is used to represent the
available energy or potential of a system with respect to
the surrounding, determined as follows:
1. Closed System For a closed system of unit mass,
availability function φ is defined as
φ = u − T 0 s + p0 v
V2
+ gz
2
By definition, if working fluid changes its state
from 1 to 2 through the system, and respective
availability functions are ψ1 , ψ2 , then, the reversible work (availability) is simply derived as
b
2
521
8.7.5
Irreversibility
Irreversibility, denoted by I, is the difference between
actual work done by the system and ideal maximum
work possible:
I = W − Wmax
It can be related to entropy change of universe (∆suniv ),
and temperature of surrounding as
I = T0 ∆Suniv
This is applicable to both closed and open systems. The
change in entropy is calculated as
∆Suniv = ∆Ssys + ∆Ssur
Thus, irreversibility represents the amount of heat
required to add entropy into the universe at its absolute
temperature.
8.8
PROPERTIES OF GASES
Gas and vapor are often used as synonymous words.
The vapor phase of a substance is customarily called
a gas when it is above the critical temperature. Vapor
usually implies a gas that is not far from the state of
condensation. This section deals with state or constitutive relations between properties of gases.
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CHAPTER 8: THERMODYNAMICS
Pure Substance
A pure substance is defined as the one that is homogeneous and invariable in chemical composition throughout its mass, even when processed. This includes atmospheric air, steam-water mixture, combustion products
of a fuel. The mixture of air and liquid air is not
a pure substance since the relative proportions of
oxygen and nitrogen differ in the gas and liquid phases
in equilibrium. Water, nitrogen, helium, and carbon
dioxide, for example, are all pure substances.
A pure substance does not have to be of a single
chemical element or compound. A mixture of various
chemical elements or compounds, also qualifies as pure
substance as long as the mixture is homogeneous. For
example, a mixture of oil and water is not a pure
substance because oil is not soluble in water, it will
collect on top of the water, forming two chemically
dissimilar regions.
8.8.2
✐
Using ideal gas equation for two different states of an
ideal gas,
p2 v 2
p1 v 1
=
T1
T2
Ideal gas is a gas model that obeys the ideal gas equation. In the range of practical interest, many familiar
gases such as air, argon, helium, hydrogen, krypton,
neon, nitrogen, oxygen, and even heavier gases, such
as carbon dioxide, can be conveniently treated as ideal
gases. Other gases follow the ideal gas equation only in
the range of high temperatures and low pressures when
their density is low.
8.8.3
Compressibility Factor
The deviation from ideal gas behavior at a given
temperature and pressure can accurately be accounted
for by the introduction of a correction factor called
compressibility factor (z), which is defined as [Fig. 8.34]:
Ideal Gas Equation of State
z=
Any equation that relates the macroscopic properties of
a system, such as pressure, temperature, and specific
volume of a substance, is called an equation of state.
The equation of state represents the behavior of a pure
substance, thus, it is also called constitutive relation.
pv
ℜT
T1
z
T2
The simplest form of equation of state is the ideal gas
equation27 of state for unit mole, written as
pv = ℜT
(8.18)
T3
1
b
T increasing
where ℜ is the universal gas constant which is equal to
8.314 kJ/kmol-K. It is related to gas constant R as
ℜ
R=
M
where M is the molar mass of the gas, defined as the
mass of one mole of a substance in grams, or the mass
of one kmol in kilograms.
Molar mass of a substance has the same numerical
value in both unit systems because of the way it is
defined. For example, molar mass of oxygen (O2 ) is 32
which means the mass of 1 kmol of oxygen is 32 kg.
27 In 1662, Robert Boyle, an Englishman, observed during his
experiments within a vacuum chamber that the pressure of gases is
inversely proportional to their volume. In 1802, J. Charles and J.
Gay Lusaac, Frenchmen, experimentally determined that at low
pressures, the volume of gas is proportional to its temperature.
Ideal gas equation was first stated by Clapeyron in 1834 as a
combination of Boyle’s law and Charles’s law. The equation was
also derived from kinetic theory by August Kronig in 1856 and
Rudolf Clausius in 1857. Universal gas constant was discovered
and first introduced into the ideal gas law instead of a large number
of specific gas constants by Dmitri Mendeleev in 1874.
0
b
Figure 8.34
p
Compressibility factor chart.
The compressibility factor can be related to the
following points:
1. For an ideal gas, by their definition, z = 1 but for
real gases, z can be greater than or less than unity.
The farther away z is from unity, the more the gas
deviates from ideal gas behavior.
2. For given absolute temperature T and pressure
p, the actual volume of a real gas vactual and
volume of an ideal gas videal can be related to the
compressibility factor z as
pvactual = zℜT
pvideal = ℜT
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Therefore,
1
z=
Tr = 2.0
b
vactual
videal
.5
Tr
z
8.8.4
Principle of Corresponding States
Tr
The ideal gas equation is closely valid for real gases at
low pressure and high temperature, which can keep the
gases far away from saturation. Therefore, the pressure
or temperature of a substance is high or low relative to
its critical temperature or critical pressure. Experiments
have shown that gases behave differently at a given
temperature and pressure, but they behave very much
the same at temperatures and pressure normalized with
respect to their critical temperatures and pressures.
This normalization of properties is done by defining the
reduced properties, as
p
pc
T
Tr =
Tc
pr =
The compressibility factor is approximately the same for
all gases at the same reduced pressure and temperature.
This is called the principle of corresponding states which
can also be stated as at the same reduced pressure
and reduced temperature, the reduced volume of different
gases is approximately the same:
v
vc
zℜT /p
=
zc ℜTc /pc
z Tr
=
z c pr
= f (pr , Tr , z)
vr =
where zc is critical compressibility factor (≈ 0.2 − 0.3),
which can be taken as constant.
Thus, Tr is plotted as a function of reduced pressure
pr and z, generalized compressibility chart is found
satisfactory for great variety of substances [Fig. 8.35].
By definition, two different substances are considered
to be in corresponding states, if their pressure, volume
and temperature are of the same fractions of the
critical pressure, critical volume, critical temperature,
respectively.
8.8.5
Van der Waals Equation
The ideal gas equation is based on the postulates of the
kinetic theory of gases proposed by Clerk Maxwell. Van
523
=
=1
1.0
b
0
Figure 8.35
pr
Reduced compressibility factor.
der Waals28 intended to improve this equation by two
corrections:
1. Intermolecular attraction forces, by incorporating
a/v 2
2. Volume occupied by the molecules themselves, by
incorporating b
Thus, the Van der Waals equation of state has two
constants that are determined from the behavior of a
substance at the critical point. The equation is written
as
a
p + 2 (v − b) = ℜT
(8.19)
v
This law is followed by real gases particularly at
high pressure and low temperature. Rearranging this
equation,
pv 3 − (pb + ℜT ) v 2 + av − ab = 0
This equation has three roots of v with the following
characteristics [Fig. 8.36]:
1. Out of three, only one root needs to be real for low
temperature (T < Tc ) (i.e. liquid phase).
2. Three positive real roots exist for certain range of
pressure (i.e. liquid plus gas phase).
3. As temperature increases, at critical point all three
roots become equal to each other. Above critical
temperature (T > Tc ) only one real root exists for
all values of p (i.e. gas phase).
The determination of the constants a and b is based
on the observation that the critical isotherm on a pv diagram [Fig. 8.36] has a horizontal inflection point
28 Van
der Waals (1837-1923) was a Dutch theoretical physicist and
thermodynamicist famous for his work on an equation of state for
gases (proposed in 1873) and liquids. His name is also associated
with van der Waals forces, van der Waals molecules, and van der
Waals radii. He won the 1910 Nobel Prize in physics.
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1. Using the first T ds equation:
p
du pdv
+
T
T
dT
dv
= cv
+R
T
v
ds =
pc
b
The entropy change between two states 1 and 2 is
written as
Z 2
s2 − s1 =
ds
Tc
1
vc
b
= cv
v
Figure 8.36
(8.21)
Z
2
1
dT
+R
T
Z
2
1
dv
v
v2
T2
+ R ln
= cv ln
T1
v1
Critical properties.
(8.22)
2. Using the second T ds equation for ideal gases:
dh vdp
−
T
T
dT
dp
= cp
−R
T
p
at the critical point. Therefore, the first and second
derivatives of p with respect to v at the critical point
must be zero:
ds =
1. First derivative
−ℜTc
(vc − b)
∂p
∂v
2
+
The entropy change between two states 1 and 2 is
written as
Z 2
s2 − s1 =
ds
=0
Tc
2a
=0
vc 3
1
= cp
2. Second derivative
∂2p
∂v 2
2ℜTc
Z
2
1
dT
−R
T
Z
2
1
p2
T2
− R ln
= cp ln
T1
p1
=0
dp
p
(8.24)
Tc
6a
3 −v 4 =0
c
(vc − b)
These two conditions are solved for
vc
b=
3
a = 3pc vc 2
Therefore, the universal gas constant is derived as
8 pc v c
ℜ=
3 Tc
Eliminating the individual coefficients a, b, and ℜ from
Eq. (8.19) 3
pr + 2 (3vr − 2) = 8Tr
(8.20)
vr
This equation is called reduced equation of state.
8.8.6
(8.23)
Entropy Change of Ideal Gases
Using the T ds equations, entropy change in ideal gases
is derived as follows:
8.8.7
Reversible Processes
Following is the constitutive relation for one unit mass
of an ideal gas:
pv = RT
(8.25)
By the definition of specific heats [Section 8.4.4], the
changes in internal energy and enthalpy of an ideal gas
are as follows:
u2 − u1 = cv (T2 − T1 )
h2 − h1 = cp (T2 − T1 )
The basic formulas for energy interaction (work and heat
transfer) are as follows:
Z v2
W1→2 =
pdv
Q1→2 =
Z
v1
T2
T ds
T1
The entropy change can be calculated as
Z T2
dQ
s2 − s1 =
T
T1
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8.8 PROPERTIES OF GASES
During a process, a pure substance follows specific
rule or law, that can be called process relation. Such a
law is conveniently expressed in terms of pressure p, and
specific volume v.
8.8.7.1 Polytropic Process Polytropic process is the
most general process by use of which expressions for all
other processes can be easily derived.
Following is the relationship between the area on
the p − v diagram which represents the work done [Fig.
8.37]:
Z v2
Z p2
pdv + p1 v1 = −
vdp + p2 v2
1. Process Relation The process relation for polytropic process is written as
v1
p
b
p1 v 1 − p2 v 2 = −
Z
1
p
p1
p2
vdp −
p1
b
b
pdv
−
2
b
pdv
R p2
p1
(8.26)
vdp
v1
R p2
p1
v1
v2
n−1
=
p2
p1
n−1
n
p
dp
= −n
dv
v
2. Work Done Work done can be calculated as [Fig.
8.38]
Z v2
W1→2 =
pdv
v1
vdp.
Equation (8.26) is very helpful in thermodynamics
of
Rp
compressors as open system, where − p12 vdp is found
to be a very interesting quantity.
The relationship between R, cp and cv of an ideal gas
can be found as
R
cp − cv
=
cv
cv
= γ −1
R
cp − cv
=
cp
cp
γ −1
=
γ
The heat capacity ratio (γ) for an ideal gas can be
related to the degrees of freedom (n) of a molecule by:
2
n
This can be examined for monoatomic gases (n = 3) and
diatomic gases (n = 5), as follows:
(
5/3 For monoatomic gases
γ=
7/5 For diatomic gases
In the following
for energy transfer,
fer, work transfer,
reversible processes
=
p2 v 2
pdv and −
γ = 1+
b
v
R v2
T2
T1
2
v
Figure 8.37
By differentiating both sides of Eq. (8.27), the
slope of p − v curve [Fig. 8.38] is found as
b
p1 v 1
(8.27)
where n is the index. Along with Eq. (8.25),
v1
pv n = c
pv = c
v1
v2
1
n
R v2
Z
pv n = c
subsections, important expressions
such as internal energy, heat transare derived for different types of
on ideal gases.
p
b
1
b
T
pv n = c
pv n = c
R v2
b
v1
1
2
2
pdv
b
b
b
R s2
s1
T ds
s
v
Polytropic process.
Figure 8.38
Therefore,
W1→2 =
=
Z
Z
v2
v1
v2
c
dv
vn
cv −n dv
v1
v2
c
−n+1
=
v
dv
−n + 1
v1
h c
iv2
1
=
vdv
(−n + 1) v n
v1
p1 v 1 − p2 v 2
=
n−1
R (T2 − T1 )
=−
n−1
(8.28)
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3. Heat Transfer Heat transfer in the process is [Fig.
8.38]
Q = W + (u2 − u1 )
R (T2 − T1 )
+ cv (T2 − T1 )
=−
n−1
cp − cv
= −
+ cv (T2 − T1 )
n−1
n−γ
=
cv (T2 − T1 )
n−1
|
{z
}
w.r.t. γ:


 > γ, ds > 0
n = γ, ds = 0


< γ, ds < 0
Decrease in entropy of the system is seen in
working of centrifugal and axial compressors.
(8.29)
5. Change in Internal Energy
energy can be found by
The change in internal
u2 − u1 = cv (T2 − T1 )
cn
Therefore, equivalent specific heat for a polytropic
process can be written as
n−γ
cn =
cv
(8.30)
n−1
This simplifies the calculations of heat transfer and
entropy change.
Heat transfer Q can be represented in terms of
W by using Eq. (8.28) and Eq. (8.29), as
n−γ
Q=−
cv W
R
n−γ
=−
cv W
(γ − 1) cv
γ −n
W
(8.31)
=
γ −1
8.8.7.2 Isobaric Process Constant pressure processes
are called isobaric processes.
1. Process Relation The process relation for isobaric
process (n = 0) is written as
p=c
2. Work Done Work done in isobaric processes [Fig.
8.39] is expressed as
W1→2 = p (v2 − v1 )
p
1
p=c
b
b
2
p (v2 − v1 )
Work transfer W can related to heat transfer Q as
W =
γ −1
Q
γ −n
(8.32)
Equations (8.30) and (8.32) can be used to derive
expressions of specific heats and work done for
other processes as described in Table 8.1.
Table 8.1
b
v
Figure 8.39
Isobaric process.
3. Heat Transfer
The process occurs at constant
pressure, therefore, specific heat at constant pressure is involved in the heat transfer:
Reversible processes
Process
n
c
W
Q
Polytropic
Isobaric
Isochoric
Isothermal
Adiabatic
n
0
∞
1
γ
cn
cp
cv
∞
0
(γ − 1) Q/γ
0
Q
∆u
cp ∆T
cv ∆T
W
0
4. Change in Entropy Change in entropy is
T2
s2 − s1 = cn ln
T1
Q = cp (T2 − T1 )
4. Change in Entropy The change in entropy is
Z 2
cp dT
s2 − s1 =
T
1
T2
= cp ln
T1
Using the second T ds equation for isobaric processes (dp = 0):
(8.33)
This equation can be used to determine the sign
of entropy change (ds) by knowing the value of n
T ds = cp dT + vdp
dT
T
=
ds p
cp
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8.8 PROPERTIES OF GASES
This is the slope of isobaric path on T –s diagram.
5. Change in Internal Energy The change in internal
energy
u2 − u1 = cv (T2 − T1 )
8.8.7.3 Isochoric Process Constant volume processes
are also known as isochoric process.
8.8.7.4 Isothermal Process In isothermal processes,
the temperature of the system remains constant.
1. Process Relation The process relation for isothermal processes (n = 1) is written as
pv = c
Therefore,
1. Process Relation The process relation for isochoric
processes (n = ∞) is written as
pdv + vdp = 0
Z
Z
pdv = − vdp
v=c
2. Work Done The volume remains constant during
isochoric process [Fig. 8.40], hence the work transfer is zero.
W1→2 = 0
2
b
p
2. Work Done Work done in isothermal processes
[Fig. 8.41] can be expressed as
Z v2
W1→2 =
pdv
v1
Z v2
C
dv
=
v1 v
v
v=c
b
= [C ln v]v21
v2
= p1 v1 ln
v1
v2
= p2 v2 ln
v1
1
b
v
Figure 8.40
Isochoric process.
The same expression can also be derived by taking
n → 1 in polytropic pdv-work.
p
b
1
T
3. Heat Transfer Iscochoric processes occur at constant volume, therefore, heat transfer involves
specific heat at constant volume cv :
4. Change in Entropy The change in entropy is
Z 2
cv dT
s2 − s1 =
T
1
T2
= cv ln
T1
Using the first T ds equation for isochoric processes
(dv = 0):
T ds = cv dT + pdv
dT
T
=
ds v
cv
This is the slope of isochoric path on T –s diagram.
5. Change in Internal Energy The change in internal
energy:
u2 − u1 = cv (T2 − T1 )
The work transfer is zero in the process, the heat
transfer is equal to change in internal energy:
Q = u2 − u1
1
b
b
2
pv = c
Q = cv (T2 − T1 )
(8.34)
R2
1
b
T (s2 − s1 )
2
pdv
b
b
s
v
Figure 8.41
Isothermal process.
3. Change in Internal Energy
Isothermal processes
occur at constant temperature:
Z 2
u2 − u1 =
cv dT
1
=0
There is no change in internal energy in isothermal
processes.
4. Heat Transfer Using the first law, heat transfer is
equal to work done:
Q = W1→2 + (u2 − u1 )
= W1→2
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Change in entropy is deter-
p
b
1
T
1
γ
pv = c
R2
1
b
2
pdv
b
b
Q
T
W1→2
=
T
s2 − s1 =
v
s
Adiabatic process.
Figure 8.42
1. Process Relation Useful expressions can be derived
for isentropic processes (ds = 0):
p
T
(a) Using Eq. (8.22)
γ
T2
v2
+ R ln
=0
T1
v1
b
n=0
1
n=∞
γ
n>γ
b
T2
p2
− R ln
=0
T1
p1
s
v
Figure 8.43
(b) Using Eq. (8.24):
Reversible processes.
Figure 8.43 enables comparison of slopes of the reversible
processes on p-v and T -s diagrams. It is depicted that
heat transfer between two points is maximum for n = 1
(isothermal process).
R/cp
p2
T2
= ln
T1
p1
(γ−1)/γ
T2
p2
=
T1 s
p1
ln
-1
1
R/cv
T2
v1
= ln
T1
v2
γ−1
T2
v1
=
T1 s
v2
cp ln
∞
n=0
ln
2
b
8.8.7.5 Adiabatic Process Adiabatic process does
not permit heat transfer. Therefore, such processes are
isentropic processes.
cv ln
b
5. Change in Entropy
mined as
Table 8.2 readily summarizes the formulations of
work and heat transfers for all the reversible processes
discussed above.
Combining the above two equations, one obtains
T2
T1
=
s
v1
v2
γ−1
=
p2
p1
(γ−1)/γ
(8.35)
This is very useful equation for dealing with
isentropic processes of ideal gases. Using this, the
process relation for adiabatic processes can be
written as
pv γ = c
Thus, for adiabatic processes the polytropic index
n is equal to γ.
2. Work Done Using Eq. (8.28), work done in an
adiabatic process [Fig. 8.42] is obtained as
W1→2 =
p1 v 1 − p2 v 2
γ −1
3. Heat Transfer There is no heat transfer involved
in adiabatic process.
4. Change in Entropy As there is no heat transfer,
there is no change in entropy.
8.8.8
Properties of Gas Mixtures
The properties of gas mixtures are affected by the
properties and fraction of the non-reactive constituent
gases. To quantify this, mole fraction (xi ) is defined as
the fraction of moles of given constituent to the total
moles of the mixture:
xi =
ni
n
where ni represents the number of moles of ith constituent, and n is the total number of moles in the gas
mixture. The summation of all the mole fractions would
be equal to unity:
n
X
xi = 1
i=1
8.8.8.1 Dalton’s Law According to the Dalton’s law,
pressure of a mixture of ideal gases is equal to the sum
of the partial pressures of constituents of the mixture.
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529
Partial volume is defined as
Reversible processes on ideal gas
Table 8.2
c
Process
W
Vi =
Q
Polytropic
pv n = c
ni ℜT
= xi V
p
Amagat’s law can be represented as
n−γ
n−1
cv −
R (T2 − T1 ) c (T − T )
n
2
1
n−1
V =
n
X
Vi
i=1
Total volume of the mixture is given by
γ −1
Q
=
γ −n
γ −n
=
W
γ −1
Isobaric
p=c
cp
p (v2 − v1 )
cp (T2 − T1 )
cv
0
cv (T2 − T1 )
Isochoric
v=c
Isothermal
pv = c
∞
p2 v2 ln
v2
v1
W
V =n
8.8.8.3 Gibbs-Dalton Law Under the ideal gas approximation, the properties of a gas are not influenced
by the presence of other gases, and each gas component
in the mixture behaves as if it exists alone at the mixture
temperature and mixture volume. This principle is
known as Gibbs-Dalton law, which is an extension of
the Dalton’s law. In view of this, the formula for the
properties of gas mixture can be summarized as below
Pn
i=1 mi Ri
Gas constant, R = P
n
i=1 mi
n
X
Molecular weight, µ =
x i µi
i=1
Adiabatic
pv γ = c
ℜT
p
Density, ρ =
p1 v 1 − p2 v 2
γ −1
0
0
Internal energy, U =
Enthalpy, H =
Ideal gas equation for n moles of an ideal gas having
volume V at temperature T and pressure p is written as
Entropy, S =
pV = nℜT
Consider a mixture of ideal gases having total volume
V at temperature T and pressure p. If a particular
constituent i has ni moles in the mixture, then partial
pressure of this constituent is given by
pi =
ni ℜT
= xi p
V
Dalton’s law can be represented as
p=
n
X
pi
n
X
i=1
n
X
i=1
n
X
i=1
n
X
ρi
u i mi
h i mi
s i mi
i=1
8.9
GAS COMPRESSION
Compressor is a type of machine that elevates the pressure of a compressible gas. Reciprocating compressors
are used to produce compressed gas used for industrial
applications for like gas transmission pipelines, petrochemical plants, refineries, cleaning, pneumatic control
devices.
i=1
8.8.8.2 Amagat’s Law According to Amagat’s law,
volume of a mixture of ideal gases is equal to the sum
of the partial volumes of constituents of the mixture.
8.9.1
Shaft Work
Let the air get compressed in a polytropic process pv n =
c from p1 , v1 state (1) to p2 , v2 state (2) [Fig. 8.44].
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p
p2
p
2
2′
3′
b
b
b
1
v3
Compression on p-v plane.
Total shaft work
p1 v 1 − p2 v 2
− p2 v 2
n−1
n
(p2 v2 − p1 v1 )
n
−
Z 1
= − vdp
( )
(n−1)/n
n
p2
=−
p1 v 1
−1
n−1
p1
where n is the index of compression.
Volumetric Efficiency
When clearance is provided in the cylinder, this affects
the mass flow rate. So, volumetric efficiency for compressor is defined as the ratio of actual volume intake to
swept volume of the cylinder [Fig. 8.45]:
ηv =
b
1
v
v
8.9.2
p1
4
v1
=−
b
pv n = c
b
p1
Wc = p1 v1 +
2
pv n = c
T =c
Figure 8.44
p2
b
v1 − v4
v1 − v3
Thus,
v1 = vs + vc
v3 = vc
where vc is clearance volume and vs is swept volume.
Also,
1/n
p2
v4 = v3
p1
where n is the index of expansion. Therefore,
( )
1/n
vc
p2
ηv = 1 −
−1
vs
p1
v4
Figure 8.45
v1
Volumetric efficiency.
This expression is used to consider the effect of vc in
power calculation29 .
( )
(n−1)/n
n
p2
Wc = −
mRT1
−1
n−1
p1
( )
(n−1)/n
n
p2
=−
η v p1 v 1
−1
n−1
p1
8.9.3
Staged Compression
Staged compression is used to reduce the work requirement. This is accomplished by introducing an
intercooler between two stages of compression. The
intercooler cools the compressed gas to initial temperature. Thus, staged compression tends to bring the
compression process towards an isothermal process [Fig.
8.46].
If index of expansion is same for all stages and air is
cooled to initial temperature after each stage, then the
maximum work required to compress the air from p1 to
p2 in N stages is given by
( )
(n−1)/(nN )
p2
n
mRT1
−1
Wc = N ×
n−1
p1
and if N → ∞ (isothermal process)
v1
Wc → p1 v1 ln
v2
The objective is to increase the pressure (and not
to increase internal energy), so when temperature is
maintained to T1 (i.e. isothermal process), the work
is not increased but remains the same for the whole
29 In
this way, the volume flow rate is not used, but mass flow rate
is used.
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8.10 BRAYTON CYCLE
p
p2
Heat
exchanger
2
b
Saving in work
QS
Compressor
n
pv = c
pi
2′
b
b
1′
b
v1
Wc
Wt
Heat
exchanger
Staged compression.
Figure 8.47
But with imperfect cooling, the work done in higher
pressure stage is more than that for lower stage. If m
and n are indices of compression in the first and second
stages, respectively, then for perfect cooling, the ratio of
compression works is determined as
(m−1)/m
(n−1)/n
p2
=
pi
W1
m/ (m − 1)
=
W2
n/ (n − 1)
m (n − 1)
=
n (m − 1)
1
b
Isentropic
s
Brayton cycle.
Two important ratios, compression ratio (r) and pressure ratio (rp ), are defined, respectively, as
V1
V2
p2
rp =
p1
p3
=
p4
r=
In the process 1 → 2,
γ
p2
V1
=
p1
V2
γ
rp = r
T2
T1 = γ−1
r
In the process 3 → 4,
T4 =
BRAYTON CYCLE
Brayton cycle was first proposed by George Brayton for use
in the reciprocating oil burning engine that he developed around
1870.
4
p2 = p3
p1 = p4
=
30 The
b
For the Brayton cycle shown in Fig. 8.47,
Brayton cycle30 , also known as Joule cycle, is the
theoretical cycle for gas turbines. It is a modified
version of Carnot cycle in which isothermal processes are
replaced by isobaric processes. Thus, the cycle consists
of two isentropic processes and two constant pressure
processes [Fig. 8.47].
3
4
v
For two-staged compression, intermediate pressure pi
is geometrical mean of suction and delivery pressures,
irrespective of perfect cooling provided with same indices
of compression in both stages and minimum total work
of compression,
√
pi = p1 p2
8.10
1
QR
movement of piston. Pressure increases in an adiabatic
process, so the work requirement also increases.
pi
p1
2
pv = c
1
b
Isobaric
3
2
b
Figure 8.46
Turbine
n
T =c
p1
T, h
b
(γ−1)/γ
p4
T3
p3
T3
rp (γ−1)/γ
T3
= γ−1
r
8.10.1
Thermal Efficiency
Heat interactions of the cycle are as follows:
QS = mcp (T3 − T2 )
QR = mcp (T4 − T1 )
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Efficiency of the Brayton cycle is determined as
QR
QS
T4 − T1
= 1−
T3 − T2
T3 /rγ−1 − T2 /rγ−1
= 1−
T3 − T2
1
= 1 − (γ−1)
r
1
= 1 − (γ−1)/γ
rp
ηBrayton = 1 −
ends at Tmax , the Carnot efficiency is reached, rp has
the maximum value rp−max . The efficiency ηBrayton is
found to be maximum, and equal to
(ηBrayton )max = 1 −
when
rp max =
8.10.2
Using the above expression, the following points can be
deduced:
T1
T3
T1
T3
γ/(γ−1)
Maximum Work Output
Figure 8.49 shows the effect of rp on ηBrayton and Wnet .
In the limiting case when rp reaches to the maximum
value rp−max , the net work output becomes zero.
Wnet
1. The efficiency of a Brayton cycle depends upon the
compression ratio (r) and γ:
b
b
ηBrayton = f (r, γ)
2. For the same compression ratio (rp ) with same
working fluid, efficiency of Brayton cycle is equal
to that of Otto cycle31 :
ηBrayton
(rp )opt
ηBrayton = ηOtto
b
3. The lower limit of temperature T1 is limited by the
atmospheric temperature (Tmin say). The highest
temperature T3 is limited by the characteristics of
material available for burner and turbine construction (say upto Tmax ). This is evident in the T − s
plot for the Brayton cycle with different values of
compression ratios [Fig. 8.48].
T, h
3′′
Tmax
b
b
2
2
Tmin
b
b
b
2′′
′
3′
3
b
b
4′
4
4
b
b
1
Figure 8.48
s
Effect of rp .
Figure 8.49 shows the effect of rp on ηBrayton . As
the pressure ratio is increased, the efficiency steadily
increases. In the limit when the compression process
31 Discussed
in Chapter 9.
0
Figure 8.49
Wnet versus rp .
The work output of the Brayton cycle is
Wnet = cp {(T3 − T4 ) − (T2 − T1 )}
T3
γ−1
T1 − T1
= cp
T3 − γ−1 − r
r
1
γ−1
= cp T3 1 − γ−1 − T1 r
−1
r
= cp T3 1 − r−γ+1 − T1 rγ−1 − 1
For maxima of Wnet w.r.t. r,
b
′′
(rp )max
rp
dWnet
=0
dr
T3 r−γ − T1 rγ−2 = 0
rγ−2
T3
=
r−γ
T1
T3
r2(γ−1) =
T1
1/(2(γ−1))
T3
r̄ =
T1
γ/(2(γ−1))
T3
r̄p =
T1
√
= rp max
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8.10 BRAYTON CYCLE
The maximum work output at this pressure ratio is
written as
np
p o2
max (Wnet ) = cp
T3 − T1
533
Polytropic efficiencies of compressor and turbine are
defined, respectively, as
T2′
=
T1
Mass flow rate is
T2
=
T1
3600
kg/kWh
ṁ =
Wnet
p2
p1
p2
p1
(γ−1)/(γηpc )
(γ−1)ηpt /γ
where Wnet is in kW.
In gas turbine power plants, the ratio of the compressor work to the turbine work, called back pressure ratio,
is very high. Usually one-half of the turbine work output
is used to drive the compressor. Therefore, a power plant
with high back work ratio requires a larger turbine to
provide the additional requirement of the compressor.
Hence, the turbines used in gas turbine power plants are
larger than those used in steam turbine power plants of
the same net power output.
8.10.4
Regeneration
Regeneration in Brayton cycle is the heat addition at
higher temperature, resulting in increase in the mean
temperature of heat addition and decrease in the mean
temperature of heat rejection. Thus, efficiency of the
Brayton cycle is increased but work output remains
unchanged.
Wt − Wc
Compressor
8.10.3
Isentropic Efficiencies
Wc
Figure 8.50 shows the effect of machine efficiencies on
the cycle.
QS
Heater
2
2
1
T, h
Turbine
b
3
3
′
4
b
b
Regenerator
4′
Condenser
QR
2′
2
b
b
4′
T, h
3
b
b
b
4
s=c
p=c
1
b
b
2′
s
p=c
2
Figure 8.50
Effect of ηt and ηc .
b
b
b
s=c
b
4
4′
Qreg
b
When turbine efficiency (ηt ) and compressor efficiencies (ηc ) are involved, then the efficiency of Brayton cycle
is written as
wt ηt − wc /ηc
ηBrayton =
Q
where
dTactual
ηt =
dTisentropic
(T3 − T4′ )
=
(T3 − T4 )
dTisentropic
ηc =
dTactual
(T2 − T1 )
=
(T2′ − T1 )
1
s
Figure 8.51
Regeneration on T -s plane.
The ratio of the actual temperature rise of air to the
maximum possible temperature rise [Fig. 8.51] is called
the effectiveness of the regenerator (ǫ):
ǫ=
T 2′ − T 2
T4 − T2
In an ideal regenerative cycle, the compressed air
is heated to the turbine exhaust temperature in the
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CHAPTER 8: THERMODYNAMICS
regenerator so that
T2′ = T4
T2 = T4′
For the isentropic process 1 → 2,
T2
=
T1
p2
p1
(γ−1)/γ
= rp (γ−1)/γ
staged heat input called reheat at turbine [Fig. 8.52]. It
is was found that the efficiency of cycle actually reduces
by staging the compression and intercooling because
heat to be added is increased but there can be a net gain
in efficiency when intercooling is adopted in conjunction
with a regenerator. Same is true with reheat. When the
number of stages is large, then Brayton cycle tends
towards Ericsson cycle32 .
T, h
Intercooling
T, h
3
b
3
3′
b
b
For the isentropic process 3 → 4,
T4
=
T3
p2
p1
= rp
Therefore,
b
(γ−1)/γ
2
T1 (T2 /T1 − 1)
T3 (1 − T4 /T3 )
T1 T2 (1 − T1 /T2 )
= 1− × ×
T3 T1 (1 − T4 /T3 )
T1
= 1 − × rp (γ−1)/γ
T3
ηBrayton−reg = 1 −
For a fixed ratio T1 /T3 , the cycle efficiency drops with
increasing pressure ratio.
In practice, a regenerator is costly, heavy and bulky
and causes pressure losses which bring about a decrease
in cycle efficiency. Above certain pressure ratio (p2 /p1 ),
the addition of a regenerator causes a loss of cycle
efficiency, when compared to original cycle. In this
situation, the compressor discharge temperature T2 is
higher than the turbine exhaust gas temperature T5 .
The compressed air will thus be cooled in the regenerator
and the exhaust gas will be heated.
8.10.5
Intercooling and Reheat
Efficiency of a Brayton cycle can be increased by the use
of staged compression with intercooling or and by using
′
1
Reheat
s
s
Figure 8.52
8.10.6
b
4
b
b
b
(a) Reheat
In such a case, the efficiency of the cycle can be found
as
Replacing values of T4′ and T2′ ,
4
1
b
T2
T4
=
= rp (γ−1)/γ
T1
T3
mcp (T4′ − T1 )
= 1−
mcp (T3 − T2′ )
(T4′ − T1 )
= 1−
(T3 − T2′ )
T1 (T4′ /T1 − 1)
= 1−
T3 (1 − T2′ /T3 )
2
b
(γ−1)/γ
1
ηBrayton−reg
2′
b
b
4′
(b) Intercooling
Reheat and intercooling.
Comparison with Otto Cycle
The Brayton cycle can be compared with the Otto cycle
to deduce the following points:
1. For the same compression ratio and work capacity,
Brayton cycle handles a larger range of volume and
a smaller range of pressure and temperature.
2. A reciprocating engine cannot efficiently handle a
large volume flow of low pressure gas, for which
the engine size becomes larger, and the friction
losses also becomes more. So, the Otto cycle is
more suitable in the reciprocating field.
3. An internal combustion engine is exposed to the
highest temperature (after combustion of fuel)
only for a short while, and it gets time to become
cool in the other processes of the cycle. On the
other hand, gas turbine plant, a steady flow device,
is always exposed to the highest temperature used.
So to protect material, the maximum temperature
of gas that can be used in a gas turbine plant
cannot be as high as in an internal combustion
engine. In the steady flow machinery, it is more
difficult to carry out heat transfer at constant
volume (Otto cycle) than at constant pressure
(Brayton cycle).
32 Discussed
in Chapter 9.
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IMPORTANT FORMULAS
535
IMPORTANT FORMULAS
Basic Concepts
V2
E = U +m
+ mgz
2
Z 2
Q=
Q̇dt
Second Law of Thermodynamics
1. Cyclic heat engine
= Q̇ (t2 − t1 )
dW = pdv
Z v2
W12 =
pdv
COPr =
Reversible
Q2
Q2
=
W
Q1 − Q2
3. Heat pump
Irreversible
First Law of Thermodynamics
COPp =
(a) Closed system
ηrev > ηirr ,
(b) Steady flow system
1
ηe
I = W − Wmax
I = T0 ∆Suniv
∆Suniv = ∆Ssys + ∆Ssur
ηrev−1 = ηrev−2
Properties of Gas
V2 2 − V1 2
2
+g (z2 − z1 )
Q − W = (h2 − h1 ) +
5. Clausius theorem
I
dQ
=0
R T
1. Ideal gas equation
pv = ℜT
ℜ
R=
M
For Carnot cycle:
Q1 Q2
−
=0
T1 T2
2. Enthalpy
h = u + pv
2. Compressibility factor
6. Entropy
3. Specific heats
∂u
∂Q
cv =
=
∂T v
∂T
cp =
∂Q
∂T
=
p
V2
+ mgz
2
2. Irreversibility
4. Carnot principles
dQ = dW + dU
I
dQ = dW
φ = u − T 0 s + p0 v
ψ = h − T0 s + m
Q1
W
COPp = COPr + 1 =
1. First law equations
I
Wmax = φ1 − φ2
2. Refrigerator
v1
 Z

 = pdv
Z
W

 6= pdv
1. Availability function
W = Q1 − Q2
W
Q2
ηe =
= 1−
Q1
Q1
1
Exergy and Availability
∂h
∂T
4. Steady flow systems
dQ = T dS
Z
2
1
dQrev
= (S2 − S1 )
T
7. T ds equations
T ds = du + pdv
T ds = dh − vdp
(a) Nozzle and diffuser
q
V2 = 2 (h1 − h2 ) + V1 2
8. Clausius inequality
I
dQ
≤0
T
(b) Throttling
9. Increase of entropy principle
Z 2
dQ
S2 − S1 ≥
T
1
Z 2
dQ
=
+ Sgen
T
1
∆Sisolated ≥ 0
h1 = h2
(c) Turbine and compressor
W
= h1 − h2
m
z=
pv
vactual
=
ℜT
videal
3. Corresponding states
pr =
p
,
pc
Tr =
T
Tc
4. Van der Waals equation
a
p + 2 (v − b) = ℜT
v
3
pr + 2 (3vr − 2) = 8Tr
vr
5. Entropy change of ideal gases
v2
T2
+ R ln
T1
v1
T2
p2
= cp ln
− R ln
T1
p1
s2 − s1 = cv ln
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6. Reversible process
u2 − u1 = cv (T2 − T1 )
h2 − h1 = cp (T2 − T1 )
Z v2
W1→2 =
pdv
Q1→2 =
s2 − s1 =
Z
Z
T ds
T1
T2
T1
−
Z
Z
n
X
p=
n
X
vdp
Amagat’s law
pdv
V =
1
2
pv n = c
n−γ
cv
cn =
n−1
R (T2 − T1 )
W =−
n−1
γ −1
=
Q
γ −n
Q = cn (T2 − T1 )
γ −n
=
W
γ −1
pi
n
X
Vi
Other properties
Pn
i=1 mi Ri
R= P
n
i=1 mi
n
X
µ=
x i µi
ρ=
U=
H=
S=
(b) Isobaric process
W = p (v2 − v1 )
Q = cp (T2 − T1 )
(c) Isochoric process
W =0
Q = cc (T2 − T − 1)
(d) Isothermal process
i=1
n
X
i=1
n
X
i=1
n
X
i=1
n
X
ρi
u i mi
h i mi
s i mi
i=1
8. Gas compression
Z
Wc = − vdp
( n−1
)
np1 v1
p2 n
=−
−1
n−1
p1
v1 − v4
v1 − v3
( 1
)
p2 n
vc
−1
= 1−
vs
p1
ηv =
pv = c
v2
v1
n−1
nN
=Q
(e) Adiabatic process
pv γ = c
p1 v 1 − p2 v 2
W =
γ −1
Q=0
)
n−1
p2 n
−1
p1
( n−1
)
nηv p1 v1
p2 n
=−
−1
n−1
p1
nmRT1
Wc = −
n−1
−1
)
pi
p1
√
p1 p2
n−1
p2 n
=
pi
W1
m (n − 1)
=
W2
n (m − 1)
m−1
m
10. Brayton cycle
i=1
(a) Polytropic process
W = p2 v2 ln
pi =
i=1
1
p2
p1
xi = 1
Dalton’s law
2
(
Wc
nmRT1
=
N
n−1
i=1
dQ
T
γR
cp =
γ −1
9. Staged compression
ni
xi =
n
v1
T2
p1 v 1 − p2 v 2 = −
R
,
cv =
γ −1
7. Gas mixtures
Mole fraction
(
ηBrayton = 1 −
1
rp (
(ηBrayton )max = 1 −
γ−1
γ
)
T1
T3
when
rp max =
T1
T3
γ
γ−1
For maximum work
γ/2(γ−1)
T3
r̄p =
T1
√
= rp max
np
p o2
max (Wnet ) = cp
T3 − T1
Isentropic efficiencies
dTactual
(T3 − T4′ )
=
dTisentropic
(T3 − T4 )
dTisentropic
(T2 − T1 )
ηc =
=
dTactual
(T2′ − T1 )
ηt =
Polytropic efficiencies
T2′
=
T1
T2
=
T1
p2
p1
p2
p1
1
γ−1
γ η
pc
γ−1
γ ηpt
Regenerator effectiveness
ǫ=
T 2′ − T 2
T4 − T2
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