ME6301 ENGINEERING THERMODYNAMICS - LECTURE NOTES

1 Engineering Thermodynamics

ME 6301
ENGINEERING
THERMODYNAMICS
[FOR THIRD SEMESTER B.E MECHANICAL ENGINEERING STUDENTS]
COMPILED BY
BIBIN.C
ASSISTANT PROFESSOR
DEPARTMENT OF MECHANICAL ENGINEERING
RMK COLLEGE OF ENGINEERING AND TECHNOLOGY
PUDUVOYAL
GUMMIDIPOONDI TALUK
TIRUVALLUR DISTRICT

CHAPTER 1 Basic Concept & First law
What is Thermodynamics?
For a wide range of engineering plant like steam turbines, reciprocating engines, turbo-jets,
rockets, combustion systems etc., we are concerned with the transfer of heat and work. In many
cases the objective is to convert one form of energy to another. Thermodynamics is science of
energy and entropy. “Thermodynamics is the science that deals with heat and work and
those properties of substances that bear a relation to heat and work.”
Some keywords:
• Properties: density, temperature, pressure, …
• State: a collection of properties
• Process: a path between states
• Energy: heat, work, internal energy, enthalpy
• Entropy: degree of disorder
What will we learn?
• Identification of thermodynamic properties and states
• Basic laws
Zeroth law : equality of temperature (thermal equilibrium)
1st law : conservation of energy
2nd law : conservation of entropy
• System and control volume analysis
• Directionality of process
• Efficiency analysis
• Practical devices and cycles
All the above laws were derived from experience without mathematical proofs.

Before going into the analysis of such changes, it is necessary to introduce some basic
concepts.
System / Control Volume
For defining the object under study, we draw a boundary around what we wish to study.
A system is a region of space containing a quantity of matter whose behaviour is being
investigated. This quantity of matter is separated from the surroundings by a boundary, which
may be a physical boundary like walls of a vessel, or some imaginary surface enveloping the
region. The term surroundings is restricted to those portions of the matter external to the
system, which are affected by changes occurring within the system. Before any thermodynamic
analysis is attempted, it is necessary to define the boundary of the system because it is across
the boundary that work, heat and mass are said to be transferred.
Now let us see what is happening at the boundary? Can work, heat, mass cross the boundary?
This makes for different definitions of the systems:
Definition Work Heat Mass
Isolated system
No No No
Closed system
Also called
Control Mass
Yes Yes No
Open system
Also called
Control Volume
Yes Yes Yes

Fig. Examples of Closed System
When the same matter remains within the region throughout the process under investigation it is
called closed system. In this case, only heat and work cross the boundary.
An open system is a region in space defined by a boundary across which the matter may flow
in addition to work and heat.
Fig. Examples of Open System
A closed system with moving boundary

The isolated system is one in which there is no interaction
between the system and surroundings. There is no mass or
energy transfer across the system boundary.
Examples of open system⇒ flow nozzles, steam turbine, boiler
etc.
Examples of closed system⇒ mixer of water and steam in a
closed vessel, a gas expanding in a cylinder by displacing a
piston. Hence, for a closed system, boundary need not be fixed; it may contract or expand to
accommodate any change in volume undergone by a fixed quantity of fluid.
The processes undergone in a closed system ⇒ non-flow process.
The processes undergone in an open system ⇒ flow process.
State, Property, Path, Process
The idea of system is defined in the introduction. A closed system is fully defined when
the following details are known:
• The fluid, i.e. whether gas, water etc.
• The boundary between the fluid under consideration and its surroundings
• The mass of the fluid within the boundary.
Every system has certain characteristics by which its
physical condition may be described, e.g. volume,
temperature, pressure etc. Such characteristics are
called properties of the system. These are all
macroscopic in nature. When all the properties of a
system have definite values, the system is said to
exist at a definite state. Any operation in which one
or more properties of a system change is called a
change of 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 process, e.g. a constant pressure process.
Fig. Isolated systems
System
Surroundings

The value of property does not depend on process through which the fluid is passed. The
change in the value of property depends on the initial and final states of the system. Pressure,
specific volume and temperature are some examples of basic properties. Three more
properties- internal energy, enthalpy and entropy emerge as a consequence of First and
Second Laws of Thermodynamics. From these six properties, only two may be selected to
determine the state of a closed system in thermodynamic equilibrium and the remaining four
values are then fixed. Care must be taken to see that the two properties are independent of
each other, i.e. it must be possible to vary one of these properties without changing the other.
For example, when a liquid is in contact with its vapour in a closed vessel it is found that the
temperature at which the liquid and vapour in equilibrium is always associated with a particular
pressure and one cannot change one without the other. Pressure and temperature cannot be
used to determine the state of such systems. However, pressure and specific volume may be
used to define the state of such system. It follows that the initial and final states of any
closed system can be located as points on a diagram using two properties as
coordinates. Properties may be of two types. Intensive (Intrinsic) properties are independent
of mass of the system, e.g. pressure, temperature, etc. Extensive properties are related to the
mass of the system, e.g. volume, energy etc.
Equilibrium
By specifying the P, T, ρ and V, the state of the system is defined.
Two adjacent systems (or system and surroundings) left for a long time will reach
equilibrium.
A system is said to be in thermodynamic equilibrium if no further changes occur within it
when it is isolated from the surroundings in such a way that no heat and work can cross the
boundary. The properties must be uniform throughout the system when it is in equilibrium. Only
under conditions of equilibrium can a single values of pressure or temperature be ascribed to
the system, and thus be used to determine its state.
For a system to be in equilibrium, system must be in mechanical, thermal and chemical
equilibrium.
If the system is imagined to pass through a continuous series of equilibrium states during the
process, the intermediate states could be located on the diagram, and a line representing the
path of the process could be drawn through all the points. Such a process is called a reversible

process or quasi-static process. However, in all real processes, the system is not in
equilibrium in any of the intermediate states. These states cannot be located on the coordinate
diagram because the properties do not have single unique values throughout the system. Such
processes are called irreversible processes.
Mechanical Equilibrium: Force Balance
For a system to be in mechanical equilibrium, summation of all the forces acting on the body
should be zero i.e. ΣF = 0 if acceleration is zero (Newton’s second law)

− Quasi-equilibrium versus non-equilibrium
Quasi-equilibrium: if the process is slow enough, the system is considered
approximately in equilibrium at each time.
22
1
Reversible Process Irreversible Process
1

For example, remove weights one by one. Each time the system reaches equilibrium
instantaneously. This process is called Quasi-Equilibrium Process.
If you remove weights all at once, Process is difficult to describe. Then it is Non equilibrium
process.
How slow is slow?
Relative to the relaxation time scales for pressure (speed of sound), temperature
(molecular collision), etc. These time scales are usually very short, so the quasi-
equilibrium approximation is valid (even for reciprocating engines).
Q: Can we define a path for a quasi-equilibrium process?
Q: Can we define a path for a non-equilibrium process?
Q: Can we calculate anything that happened during a non-equilibrium process?
Thermal equilibrium
The property, which distinguishes thermodynamics from other sciences, is temperature.
Temperature is associated with the ability to distinguish between hot from cold. When two
bodies at different temperature are brought into contact, after some time they attain a common
temperature and are said to be in thermal equilibrium. Two systems are said to have equal
temperatures if there is no change in any of their observable characteristics when they are
brought into contact with one another.
As
show
n in
the
fig., if
a copper body at a temperature T1 is brought into contact of a steel body at temperature T2 such
that T1 > T2, then after some time both the bodies will be at a temperature T3. The temperature
T3 will be in between T1 and T2. Both the bodies then can be said to be in thermal equilibrium. If
Copper
T1
Steel
T2
Copper
T3
Steel
T3
State 1 State 2
Process

two bodies (A and B) are each equal in temperature to a third body (C), they (A and B) are
equal in temperature to each other. This is the principle of thermal equilibrium and is known as
zeroth law of thermodynamics. In other words, the zeroth law of thermodynamics states that if
TA = TC and TB = TC, then TA = TB.
The possibility of measuring the temperature rests upon this principle.
Temperature scale:
C (Celsius) K (Kelvin) K = C + 273.15
F (Fahrenheit) R (Rankine) R = F + 459.67
F = 32 + (9/5)C

WORK and HEAT
Summary:
We can say that the state of a system changes when heat and work cross the boundary.
Thermodynamics provides a means of relating the quantities of heat and work with the
change of state. The structure of thermodynamics rests on two important principles called
the First and Second Laws of Thermodynamics. These cannot be proved and are treated
as axioms.


We can now:
1. Distinguish the system from its surroundings.
2. Find the properties of the system to identify its state.
Our objective now is to find out how we can use it to produce Work and Heat that are useful in
our living.

Energy Transfer across the system boundary by Work and Heat
WORK
Work is said to be done when force acting upon a body moves through a distance in the
direction of the force. If part of the boundary of a system undergoes a displacement
under the action of a pressure, the work done, W, is the product of the force (pressure x
area), and the distance it moves in the direction of the force.
The basic unit of work is Newton metre (Nm), and also called the joule (J). Work is a quantity,
which is not a property of a system. It is a transient quantity, which only appears at the
boundary while a change of state taking place within a system.
Heat
Surroundings
Surroundings
Work

Hence, work is ‘something’ which appears at the boundary when a system
changes its state due to movement of a part of the boundary under the action of
force. Another definition is: work is said to be done by a system if the sole effect on
things external to the system can be reduced to raising of weight.
Examples of Work

Work crossing the system boundary whose sole effect on the surroundings could be raising of a
weight
Various types of Work
(i) pdV- work or Displacement work

Let us consider a closed system where a part of the boundary is allowed to move under
such conditions that the external restraining force is infinitesimally smaller than the force
produced by the pressure of the system. The area of the piston is A and the pressure of
the fluid at any instant is p. If p is assumed to be constant during an infinitesimal
movement of the piston over a distance dl, the work done by the fluid in moving the
external force pA through this distance is pA dl. But A.dl is dV, the infinitesimal
change of volume, therefore
dW = pdV
If the expansion occurs from
pressure p1 to a pressure p2 in such
a way that the restraining force is
changed continuously, then the total
work done can be found out by
summing up all the increments of
work pdV, i.e.
∫=
2
1
dVpW
(ii) Electrical work
When a current flows through a
resistor taken as a system, there is
work transfer into the system. This is because current drives the motor, the motor can
drive a pulley and the pulley can raise the weight.
The current flow, I, in amperes is given by
τ
=
d
dC
I

where C is the charge in coulombs and τ is time in seconds. Thus dC is the charge
crossing a boundary during time dτ. If E is the voltage potential, the work is
dW = E. dC = EI. dτ
or ∫ τ=
2
1
dIEW
The electrical power will be
IEW =&
(iii) Shaft work
If T is the torque applied to the shaft and dθ is the angular displacement, the shaft work
is
∫ θ=
2
1
dTW
and the shaft work is
∫ ω=
τ
θ
= T
d
d
TW&
where ω is the angular velocity.
HEAT
Heat is ‘something’ which appears at the boundary when a system changes its
state due to temperature difference between the system and surroundings. It is
denoted by Q and the unit of heat is joule (J).
Sign Convention for Heat and Work
Work done by the system on the
surroundings is positive; work done on the
system by the surroundings is negative.

Heat flow into the system from the surroundings is positive and heat flow from the
system to the surroundings is negative.
Hence,
Qin is taken as positive.
Wout is taken as positive.
PATH FUNCTION AND POINT FUNCTION
From Fig. it can be seen that a system can be taken from state 1 to state 2 along many
quasi-static paths, such as A, B or C. Since the area under each curve represents the
work for each process, the amount of work involved in each case is not a function of the
end states of the process, and it depends on the path of the system follows in going
from state 1 to state 2. For this reason, work is called a path function and dW is an
inexact or imperfect differential.
Thermodynamic properties are point
functions, since for a given state; there is a
definite value for each property. The
change in thermodynamic property of any
system in a change of state is independent
of path of the system follows during the
change of state, and only depends on the
initial and final states of the system. The
differentials of point functions are exact or
perfect differentials and integration is
simply
12
2
1
VVdV −=∫
However,

12
2
1
WWdW −≠∫
Rather
21
2
1
WdW =∫ or W1-2
Work , heat and reversibility
Let us consider a closed system where
a part of the boundary is allowed to
move under such conditions that the
external restraining force is
infinitesimally smaller than the force
produced by the pressure of the
system. The area of the piston is A and
the pressure of the fluid at any instant
is p. If p is assumed to be constant
during an infinitesimal movement of the
piston over a distance dl, the work
done by the fluid in moving the external
force pA through this distance is pA dl.
But A dl is dV, the infinitesimal change
of volume, therefore
dW = pdV
If the expansion occurs from pressure p1 to a pressure p2 in such a way that the
restraining force is changed continuously, then the total work done can be found out by
summing up all the increments of work pdV, i.e.
∫=
2
1
dVpW

The condition under which this expansion has been imagined to proceed is just those
necessary for the expansion to be reversible. There are no pressure gradients and the
properties are uniform throughout the system at all the times. The system passes
through a series of equilibrium states and the process may be represented by a full line
on the p-v diagram. The area under the curve is given by ∫
2
1
dVp and it represents the
work done per unit mass of fluid.
It may be noted that ∫pdV is not always equal to the work done per unit mass, and the
equality only holds when the system passes through a series of equilibrium states.
Consider a container divided
into two compartments by a
sliding partition. One part
contains a mass of gas at a
pressure p1 and specific
volume v1 and the other is
evacuated. When the partition is withdrawn, the gas undergoes free expansion and
settles down to a new equilibrium state with pressure p2 and specific volume v2.
The intermediate states are indeterminate
because no single values of p and v can be
ascribed to the system as a whole. The end
states could be located in the p-v diagram, but
the process could be indicated only by dotted
lines. By having several partitions, some
intermediate pressures and specific volumes
could be measured and the corresponding state
points plotted in the p-v diagram. If a dotted
curve is drawn through all such points, the area under the curve would be equal to
∫
2
1
dVp . Yet no work has been done by the system on the surroundings at all because
no external force has been moved through a distance.
Free expansion
(b)(a)
Irreversible process
p
p2
p1
V2
dV
V1

Thus it implies that the reversible expansion or compression is an infinitely slow
process. Since any real process must occur within a finite time, it is clear that the
reversible process is an ideal process.
Similarly, it can be shown that the transfer of energy due to the temperature difference
can only be carried out reversibly if the temperature difference is infinitesimally small.
This implies that the rate at which the transfer can proceed is infinitely slow.
Similarities between Heat and Work
1. Heat & work are phenomena that occur at the system boundary.
2. The system does not possess heat & work; it only possesses energy.
3. The consequence of heat & work crossing the system boundary is to change the system
state or its energy content.
4. Both heat and work are path dependent phenomena
First Law of Thermodynamics of a Closed System.
The structure of thermodynamics rests upon two fundamental laws. The First law, which
will be introduced here, is concerned with the principle of conservation of energy as
Summary
Neither heat nor work are properties of a system, but both are transient quantities, only
appearing at the boundary while a change in state occurs within the system. Although
they cannot be used to describe the state of a system, heat and work can be used to
describe the process undergone by the system during the change of state. Heat and
work are not only functions of the initial and final state of the system, but also of the
process undergone by the system.

applied to a closed system, which undergo changes of state due to transfers of work
and heat across the boundary. The First law cannot be proved; its validity rests upon
the fact that neither it nor its consequences have ever been contradicted.
THE CYCLE
A closed system is said to undergo a cyclic
process, or cycle, when it passes through a
series of states in such a way that its final
state is equal in all respect to its initial state.
This implies that all its properties have
regained their initial values. The system is
then in a position to be put through the same
cycles of events again, and the procedure
may be repeated indefinitely.
FIRST LAW OF THERMODYNAMICS FOR A CLOSED SYSTEM
When any closed system is taken through a cycle, the net work delivered to the
surroundings is proportional to the net heat taken from the surroundings. This may be
expressed as
(ΣδQ)cycle ∝ (ΣδW)cycle
or
(ΣδQ)cycle = J (ΣδW)cycle
J is called the mechanical equivalent of heat or Joule’s equivalent; it expresses the
number of work units, which are equivalent to one unit. For SI unit the value of J is unity.
If kJ and kcal are chosen as the units of work and heat respectively, then the value of J
is 4.19 kJ/kcal. Hence, J can be dropped from above equation, which now can be
written as
(ΣδQ)cycle - (ΣδW)cycle = 0
FIRST LAW OF THERMODYNAMICS FOR A CHANGE OF STATE
2
1
A cycle
1-2-1 A cycle

The expression (ΣδQ)cycle = (ΣδW)cycle applies to a system undergoing a cycle. But if the
system undergoes a change of state during which both heat and work transfers are
involved, the net energy transfer will be stored or accumulated within the system. If Q is
the amount of heat transferred to the system and W is the amount of work transferred
from the system during the process, the net energy transfer (Q-W) will be stored in the
system. This stored energy is known as internal energy or simply energy of the system.
Hence, the First Law for a change of state is the net energy transfer during a process
involving heat and work transfers is equal to the change in increase in energy of the
system.
That is
Q – W = ΔE
ENERGY IS A PROPERTY OF THE SYSTEM
Proof: Consider a system which changes its state from state 1 to state 2., following the
path A and returns from state 2 to state
1 following path B. So the system
undergoes a cycle.
Hence,
QA = WA +ΔEA
and QB = WB +ΔEB
The process A and B together forms a cycle and hence
QA + QB = WA + WB
or QA - WA = WB - QB = - (QB - WB)
or ΔEA = -ΔEB
Similarly if the system returns from state 2 to state 1 following path C instead of path B
then
ΔEA = -ΔEC
Hence, ΔEB = ΔEC

That is the change in energy between the two states of a system is same and
independent of path followed by the system. Therefore energy has a definite value for
every state of the system. Hence, it is a point function and a property of the system.
Different forms of energy
The symbol E refers to the total energy of the system. Basically there are two modes in
which the energy may be stored in the system:
• Macroscopic energy mode;
• Microscope energy mode.
Macroscopic energy mode includes kinetic energy and potential energy of the system.
Kinetic energy,
2
Vm
E
2
K =
Potential energy, EP = mgZ
Where, m is the mass, V is the velocity and Z is the elevation of the system.
Microscopic energy mode refers to the energy stored in the molecular and atomic
structure of the system, which is called internal energy, denoted by U.
Hence, E = EK + EP + U
For a closed system EK = 0 and EP also can be assumed to be zero.
Hence, for a closed system E = U
and Q – W = ΔU = U2 – U1
or dQ – dW = dU
COROLLARIES OF THE FIRST LAW
Corollary 1. There exists a property of a closed system such that a change in its value
is equal to the difference between the heat supplied and the work done during any
change of state.
If the property is denoted by U, the corollary can be expressed as

( )∑ −=δ−δ
2
1
12 UUWQ
or Q – W = U2 – U1
where Q and W are the net quantities of heat and work crossing the boundary during
the change of state. The property U is called the internal energy of the system and
equation is called the non-flow energy equation.
The internal energy, U, is sum of microscopic energy due to
• Molecular level – translation, rotation, vibration
• Chemical reaction – bond energy (heat of formation)
• Dissociation – ion energy
• Atomic – nuclear energy
Internal energy is a thermodynamic property and hence it is a State function and independent of
process. If two independent properties are given, U is uniquely determined.
Corollary 2. The internal energy of a closed system remains unchanged if the system is
isolated from its surroundings.
If the system is isolated from the surroundings,
Q and W are both zero and hence, ΔU must be
zero. The system represented in Fig. below is
an example of an isolated system. All that
happens in this case is a spontaneous
redistribution of energy between parts of the
system, which continues until a state of
equilibrium is reached; there is no change in the
total quantity of energy within the system during the process. Corollary 2 is often called
the Law of Conservation of Energy.
Corollary 3. A perpetual motion machine of the first kind (PMM I) is impossible.
Boundary
Isolated system
Hot
Cold

Definition: PMM1 is a device which delivers work continuously without any energy
input.
The perpetual machine was originally conceived
as a purely mechanical contrivance which, when
once set in motion, would continue to run forever.
Existence of such a machine is impossible
because of the presence of friction. What would be
immense value is a machine producing a
continuous supply of work without absorbing energy from the surroundings.
It is always possible to devise a machine to deliver a limited quantity of work without a
source of energy in the surroundings. For example, a gas compressed behind a piston
will expand and do the work at the expense of internal energy of the gas. Such a device
cannot produce work continuously and for this to happen the machine must be capable
of undergoing a succession of cyclic process. But as per First Law no net amount of
work can be delivered by the system if a net amount of heat is not supplied by the
surroundings during a cycle. Thus the First law implies that a perpetual machine of first
kind is impossible.
THE ENERGY EQUATION AND REVERSIBILITY
The energy equation for the non-flow process can be written as
Q – W = U2 – U1
When the system is in equilibrium all the properties are uniform throughout the system
and each unit mass of the system will have same internal energy. Thus if u represents
the internal energy per unit mass of the system, then it is possible to write U = mu. u is
termed as specific internal energy. It is often convenient to work with the specific
quantities throughout an analysis and then multiply the final result by the mass of the
system. Thus the non-flow energy equation is usually written as
Q – W = u2 – u1
implying that Q and W are the quantities of heat and work per unit mass.
Q
W

Engine
PMM I

For a reversible process, the system is imagined to pass through a continuous series of
equilibrium states. In such cases the equation may be applied to any infinitesimal part of
the process between the end state, and the energy equation can be written as
dQ – dW = du
For irreversible process the system does not pass through a series of equilibrium states.
Hence, it is not possible to write mu for the internal energy of the system in any
intermediate state. Each elemental mass δm may have a different specific internal
energy e, which may include kinetic and potential energy of the mass δm. When
considering the irreversible processes, the energy equation can only be used in the
integrated form and it is then sufficient to be able to assign the values of u to the end
states.
CONSTANT VOLUME PROCESS
For a constant volume process, work can only be done by some method of churning
the liquid. There cannot be any ∫pdV work as no external force has been moved through
a distance. Hence, W must be zero or negative. For a constant volume process, unless
otherwise stated, work done is taken as zero. Thus energy equation for a constant
volume process is usually written as
Q = u2 – u1
If, in addition to work being zero, it is stipulated that the heat is transferred by virtue of
an infinitesimally small temperature difference, then the process is reversible and the
equation can be written in differential form as
dQ = du
CONSTANT PRESSURE PROCESS
A closed system undergoing at constant process is shown in Fig. The fluid is enclosed
in a cylinder by a piston on which rests a constant weight. If heat
is supplied, the fluid expands and work is being done by the
system in overcoming the constant force; if the heat is extracted,

W
Q

the fluid contracts and work is done on the system by the constant force. In the general
case
Q – W = (u2 – u1)
If no paddle work is done on the system, and the process is reversible,
dQ – p dv = du
Since p is constant, this can be integrated to give
Q – p (v2 – v1) = (u2 – u1)
A further simplification is for constant pressure process if a new property is introduced.
Since p is constant, p dv is identical to d(pv). Thus energy equation becomes
dQ – d(pv) = du
or dQ = d(u + pv) = dh
where h = u + pv, known as enthalpy. Since enthalpy is a combination of properties u, p
and v, it itself is a property. Hence
h = u + pv
and for any mass of fluid m, in a state of equilibrium
H = U + pV
Using this derived property, the energy equation for a reversible constant pressure
process becomes
dQ = dh
or in the integrated form
Q = (h2 – h1)
Thus heat added in a reversible constant pressure process is equal to increase of
enthalpy, whereas it is equal to increase of internal energy in the reversible constant
volume process.
SPECIFIC HEAT

The specific heat at constant volume CV is defined as the rate of change of specific
internal energy with respect to temperature when volume is held constant, i.e.
V
V
T
u
C ⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
For constant volume process
Δu = CV ΔT
For a closed system with constant volume process
Q = Δu = CV ΔT
The specific heat at constant pressure CP is defined as the rate of change of specific
enthalpy with respect to temperature when pressure is held constant, i.e.
P
P
T
h
C ⎟
⎠
⎞
⎜
⎝
⎛
∂
∂
=
For constant pressure process
Δh = Cp ΔT
For a closed system with constant pressure process
Q = Δh = CP ΔT
For a perfect gas u = f(T)
Hence, du = CV dT or u2 – u1 = CV (T2 – T1)
h = u + pv
= u + RT
or dh = du + R dT = CV dT + R dT = (CV + R) dT
Now, dh = CP dT ( this is true only for a perfect gas)
Hence, CP = Cv + R
Also, γ=
V
P
C
C

For air: CP = 1.005 kJ/kg K
Cv = 0.718 kJ/kg K
R = 0.287 kJ/kg K
γ = 1.4
PROCESS IN WHICH p v = constant
∫=−
2
1
21 dVpW
Now pv = p1v1 = c or
v
vp
p 11
=
1
2
11
2
1
11
2
1
11
21 ln
v
v
vp
v
dv
vpdv
v
vp
W ===∴ ∫∫
2
1
11
1
2
1121 lnln
p
p
vp
v
v
vpW ==∴
POLYTROPIC PROCESS
The constant volume and constant pressure processes can be regarded as limiting
cases of a more general type of processes in which both volume and pressure change
in a certain specified way. In many real processes, it is found that the states during
expansion or compression can be expressed as pvn
= constant, where n is a constant
called the index of expansion or compression and p and v are average values of
pressure and specific volume for the system. This is called Polytropic process. When
n=0 the relation reduce to p = constant and when n = ∞ it reduces to v = constant.

For the reversible Polytropic
process, single values of p and v can truly define the state of a system, and dW = pdv.
For any intermediate state
nn
22
n
11 vpvpvp ==
Therefore
( )
n1
vvpvvp
n1
vvvp
v
dv
vpdvpW
n1
1
n
11
n1
2
n
22
2
1
n1
1
n1
2
n
11
n
n
11
2
1
−
−
=
−
−
===
−−−−
∫∫
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
−
=
−
−
=
−
n
1n
1
2111122
p
p
1
1n
vp
n1
vpvp
The integrated form of energy equation for a reversible polytropic process may therefore
be written as
( )12
1122
uu
n1
vpvp
Q −=
−
−
−

Relation between p,v,T
pv = RT
11
22
1
2
2
22
1
11
vp
vp
T
T
or
T
vp
T
vp
==
n
nn
p
p
v
v
orvpvpNow
1
1
2
1
2
2211
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
==
n1
1
2
n
1n
1
2
1
2
v
v
p
p
T
T
−
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=∴
Reversible Adiabatic Process (Isentropic Process)
The term adiabatic is used to describe any process during which heat is prevented
from crossing the boundary of the system. That is, an adiabatic process is one
undergone by a system, which is thermally insulated from its surroundings. For
adiabatic non-flow processes, the energy equation reduces to
- W = (u2 – u1)
From First Law
Q = du + W
For adiabatic process, this becomes
0 = CV dT + pdv
Also pv = RT (equation of state)
or pdv + vdp = R dT
Hence, ( ) 0dpv
R
C
dvp1
R
C
or0pdvdpvdvp
R
C VVV
=+⎟
⎠
⎞
⎜
⎝
⎛
+=++
Now CP = CV +R

0
p
dp
v
dv
C
C
or
0vdp
R
C
pdv
R
C
V
P
VP
=+
=+∴
Integrating we get
γ ln v + ln p = const
or pvγ
= const.
ISOTHERMAL PROCESS
When the quantities of heat and work are so proportioned during a process such that
the temperature of the fluid remains constant, the process is said to be isothermal.
Since temperature gradients are excluded by definition, the reversibility of the
isothermal process is implied.
Recommended Solution Procedure
STEP 1
Sketch the diagram including all mass, heat and work flows, and direction of forces (e.g.
external pressures and gravitation).
STEP 2
Define a system. Identify the initial and final states (label them 1 and 2), and write down all the
known conditions.
STEP 3
Write the applicable laws for the system.
STEP 4
Determine if the substance can be handled as an ideal gas or by the table.
STEP 5
What do you know about the process? Anything constant or zero? Relationships

between properties?
STEP 6
Find which quantities are known and which are unknown. Determine which two independent
properties should be found for any given state. Diagrams are helpful.
STEP 7
Determine the unknown quantities.

First Law applied to Flow Processes
STEADY FLOW ENERGY EQUATION
For any system and in any process, the first law can be written as
Q = ΔE + W
Where E represents all forms of stored energy in the system.
For a pure substance
E = EK + EP + U
Where
EK= Kinetic energy =
2
Cm 2
; where C is the velocity of fluid.
EP = Potential energy = mgz
U is the residual energy (internal energy) stored in the molecular structure of the
substance. Hence,
Q = ΔEK + ΔEP + ΔU + W
Let
Ai, Ae ⎯ Cross sectional area , m2
Wi, we ⎯ mass flow rate, kg/s
pi, pe ⎯ absolute pressure, N/m2
vi, ve ⎯ specific volume, m3
/kg
δQ
Steady-flow process
e
i
Ce
Ci δW
Shaft
Steady flow device
ze
zi

ui, ue ⎯ specific internal energy, J/kg
Ci, Ce ⎯ velocity, m/s
zi, ze ⎯ elevation from an arbitrary datum level, m
Subscript i and e refers to the inlet and exit sections.
The total energy of mass δm at the inlet is
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
++++ i
2
i
ii
2
ii gz
2
C
umorzgmCm
2
1
U δδδ
Similarly, the total energy of mass δm at the outlet is
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
++ e
2
e
e gz
2
C
umδ
Lastly we must consider how the system changes its state from i to e. We may assume
that δQ units of energy are transferred to the system as heat and that δW units of
energy are transferred to the surroundings as work via. a turbine shaft. δW is not only
work done so far as the system is concerned
because parts of its boundary move at
sections i and e. For the element δm to enter
the open system, the system must be
compressed, its volume decreasing by δm vi.
This is accompanied by a force piA moving a
distance l = δm v1/A, where A is the cross-
sectional area of the element. The work done by the surroundings on the system is
therefore δm pi vi. Similarly it can be shown that the work done by the system to the
surroundings is δm pe ve. The net work done by the system during the change is
therefore
δW + δm (pe ve – pi vi)
We may now write the energy equation for the open system as
δQ – {δW + δm (pe ve – pi vi)} = ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
++ e
2
e
e gz
2
C
umδ - ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
++ i
2
i
i gz
2
C
umδ
By writing h for (u + pv), the equation becomes

⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
++−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
++=− i
2
i
ie
2
e
e z
2
C
hmz
2
C
hmWQ δδδδ
The continuous steady-flow process consists of the sum total of all the elemental mass
transfers across sections 1 and 2. It may therefore be represented as
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
++−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
++=− ∑∑ ∑∑ i
2
i
ie
2
e
e z
2
C
hmz
2
C
hmWQ δδδδ
If we consider the properties to be uniform over the cross-section of flow at inlet and
outlet and write m for Σδm then the above equation becomes after simplifying
( ) ( ) ( )ie
2
i
2
eie zzgCC
2
1
hhWQ −+−+−=−
where Q and W are the heat and work transfers per unit mass flowing through the
system. The assumptions upon which the equation is based may be summarised as
follows:
1. The mass flow at the inlet is constant with
respect to time, and equal to mass flow at
outlet.
2. The properties at any point within the open
system do not vary with time.
3. The properties are constant over the cross-
section of the flow at inlet and outlet.
4. Any heat or work crossing the boundary does
so at a uniform rate.
Finally we must note an important equation which
follows directly from assumptions (1) and (3). It is known as the continuity equation
and expresses the principle of conservation of mass in steady flow. If m is the rate of
mass flow, we have
eeeiii
e
ee
i
ii
ACAC
v
CA
v
CA
m ρρ ====
FIRST LAW AS A RATE EQUATION

It expresses the instantaneous or average rate at which energy crosses the system
boundary as heat and work and the rate at which the energy of the system changes.
Conservation of Mass
Mass flow in – mass flow out = accumulation in the
control volume.
i.e. δmcv = mi - me
Dividing by δt we get
t
m
t
m
t
m eicv
δδδ
δ
−=
or ei
cv
mm
dt
dm
−=
where e
e
i
i
m
t
m
andm
t
m
==
δ
δ
δ
δ
Conservation of Energy
Consider a time interval δt during which an amount
of heat δQ crosses the system boundary, an
amount of work δW is done by the system, the
internal energy change is ΔU, the kinetic energy
change is ΔK and potential energy change is ΔPE.
Since the mass flow rates at the inlet im and outlet
em are not same, there will be a change of energy
stored in the system. If the change in energy stored
in the system is ΔEcv, then

First Law states
δQ = ΔU + ΔKE + ΔPE + δW + ΔEcv
Dividing by δt we get

( ) ( )
dt
dE
W
dt
PEd
dt
KEd
dt
dU
Qor
t
E
t
W
t
PE
t
KE
t
U
t
Q
cv
cv
++++=
Δ
++
Δ
+
Δ
+
Δ
=
δδ
δ
δδδδ
δ
where W
t
W
andQ
t
Q
==
δ
δ
δ
δ
Hence, the First Law becomes
∑∑ ++⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
++=⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+++
dt
dE
Wzg
2
C
hmzg
2
C
hmQ cv
e
2
e
eei
2
i
ii
i.e. ∑∑ ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
++−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+++−= e
2
e
eei
2
i
ii
cv
zg
2
C
hmzg
2
C
hmWQ
dt
dE
Simplifications
1. Steady State Flow
Fluid properties may vary in space but are constant at each location.
eiei hh;mm ≠≠
0
dt
dEcv
=
Hence,
∑∑ ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
++−⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+++−= e
2
e
eei
2
i
ii zg
2
C
hmzg
2
C
hmWQ0
2. Steady state flow with uniform mass flow rate
Continuity equation becomes
mmm ei ==
( ) ( ) ( )ie
2
i
2
e1ie zzgmCC
2
m
hhmWQ −+−+−=−∴
3. Negligible KE and PE
( )1ie hhmWQ −=− &&&

4. No heat loss, work with negligible KE and PE
hi = he
Examples of Steady State Devices

OPEN SYSTEM WITH STEADY FLOW
Boiler
In the boiler, the fluid entering as liquid, leaves as a vapour at a constant rate. In this
case no work is done on or by the fluid as it passes through the system. The velocities
are usually quite low, so that the difference between kinetic energies at the inlet and
outlet is negligible compared to the other terms of the equation.

Assumptions
• Steady (dECV/dt = 0)
• Single inlet/exit
• No shaft work
• KE, PE change negligible (usually)
Hence, Steady Flow Energy Equation (SFEE)
becomes
Q = he – hi kJ/kg
Nozzles and Diffusers
A nozzle is a duct of varying cross-sectional area so designed that a drop in pressure from inlet
to outlet accelerates the flow. The flow through a nozzle usually occurs at a very high speed,
and there is little time for the fluid to gain or lose energy by a flow of heat through the walls of
the nozzle as the fluid passes through it. The process is therefore always assumed to be
adiabatic. Also, no work crosses the boundary during the process. The function of diffuser is the
reverse of that of nozzle.
Nozzle – A device to increase velocity at the expense of
pressure.
Diffuser – A device to increase pressure at the
expense of velocity.
Assumptions
• No heat transfer (assuming adiabatic)
• No shaft work
• PE change negligible
Hence, SFEE becomes
CeCi
Nozzle, Ci < Ce
Ci Ce
Diffuser, Ci > Ce

2
C
h
2
C
h
2
e
e
2
i
i +=+
If Ce > Ci (nozzle) ⇒ he < hi (cools down)
If Ce < Ci (diffuser) ⇒ he > hi (warms up)
Turbine/Compressor
A turbine is a means of extracting work from a flow of fluid expanding from a high
pressure to a low pressure. The fluid is accelerated in a set of fixed nozzles and the
resulting high-speed jets of fluid then change their direction as they pass over a row of
curved blades attached to the rotor. As first approximation, the velocity at the inlet and
outlet of the turbine can be assumed equal. Since the velocity of flow through the
turbine is very high, the process can be assumed to
be adiabatic. The rotary compressor can be
regarded as a reverse turbine, work being done on
the fluid to raise the pressure.
Assumptions
• No heat transfer (usually, but not always)
• KE, PE change negligible (usually)
Hence, SFFF becomes
W = hi - he kJ/kg
Throttling Valves
A flow-restricting device that causes a significant pressure drop accompanied by a large
drop in Temperature (valves, porous plug, capillary tubes in refrigerators). The term
throttling is usually applied to relatively low-speed flow, i.e. low enough for any

difference between the kinetic energy at the inlet and outlet to be negligible. Any heat
transfer across the boundary can be neglected. Also no work crosses the boundary.
Assumptions
• No heat transfer and shaft work
• KE, PE change negligible
Hence, SFFF becomes
hi = he kJ/kg
ui + pivi = ue +peve
Hence, pv increases at the expense of u (temperature drops)
For real gases, h = h (T) only, i.e. Ti = Te
Heat Exchangers
Consider a double-tube type heat exchanger (tube and shell). In heat exchanger, the
change in potential energy and kinetic energy terms are very small and can be
neglected. Also, there is no external work.
Assumptions
• Multiple inlets/exits
• No heat transfer and shaft work
• KE, PE change negligible
Hence, SFEE becomes
∑∑ =
e
ee
i
ii hmhm
or 44223311 hmhmhmhm +=+

Power Plant
Boiler: 21525121b hhcesin,)hh(m)hh(mQQ =−=−==
Turbine: )hh(mWW 2332T −=−=
Condenser: )hh(mQQ 3443C −==−
Pump: )hh(mWW 4554P −=−=−
Adding we have
0WWQQ 54324321 =−−+
or ∑ ∑ =− 0WQ δδ

CHAPTER 2 Second Law Entrophy and Availability
____________________________________________________
In this chapter the idea of cycle efficiency in introduced and the Second Law is then
stated and distinguished from the First Law. Formal definition of a reversible process is
made and its implications both for non-flow and steady-flow processes are discussed.
CYCLE EFFICIENCY
When a closed system undergoes a series of non-flow processes, during which the fluid
passes through a cycle of thermodynamic states, the First Law can be expressed as
∑δQ = ∑δW
If Q1 is the heat supplies, Q2 the heat rejected and W the net work done by the system
during the cycle, the First Law becomes
Q1 – Q2 = W (6.1)
A system operating in a cycle and producing a net quantity of work from a supply of
heat is called a heat engine. The greater the proportion of the heat supply converted
into work, the better the engine is. Consequently the cycle efficiency of a heat engine is
defined as
1Q
W
pliedsupheat
workdone
==η (6.2)
Hence, from equations (6.1) and (6.2), we have
1
21
Q
QQ −
=η (6.3)
The efficiency of any heat engine operating on either a closed system or an open cycle,
consisting of steady-flow processes, can also be expressed by equation (6.3). In this
case Q1 and Q2 must be interpreted either as the heat supplied and rejected per unit
mass flowing round the circuit, or as the rates of heat supplied and rejected.

THE SECOND LAW OF THERMODYNAMICS
Consider the following isolated systems with well-defined initial states:
We have never observed the reverse to occur, but it does not violate the 1st
Law. However,
there is a directionality of a process. You can make the reverse happen, but it does not occur
SPONTANEOUSLY.
More observations
The direction of spontaneous change for a ball bouncing on a floor. On each bounce
some of its potential energy is degraded
into the thermal motion of the atoms of the
floor, and that energy disperses into the
atoms of the floor. The reverse has never
been observed to take place.
The reverse, if it occurs, does not violate the
1st Law as long as the energy is conserved.
Recall also that only a small amount of
thermal energy is required to make the ball
jump very high.

Hence, the first Law only states that the net work cannot be produced during a cycle without
some supply of heat. However, First Law never says that some proportions of heat supplied to
an engine must be rejected. Hence, as per the First Law, cycle efficiency can be unity, which is
impossible in practice. All that First Law states that net work cannot be produced during a cycle
without some supply of heat, i.e. that a perpetual motion machine of the first kind is impossible.
So, the 1st Law is not enough. Something is missing! What is missing? A law that can tell us
about the direction of spontaneous change. The Second Law of Thermodynamics tells us about
the directionality of the process. We need it to ensure that systems we design will work. As we
will see later, ENTROPY is a property that we have invented (like internal energy) that will allow
us to apply the 2nd Law quantitatively.
HEAT ENGINE, HEAT PUMP, REFRIGERATOR
Energy reservoirs: Thermal energy reservoirs (TER) is defined as a large 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.
Source: TER from which heat is transferred to the system
operating in a heat engine cycle.
Sink: TER in which heat is rejected from the cycle during a
cycle.
Heat Engine cycle is a thermodynamic cycle in which there is
a net heat transfer to the system and a net work transfer from the system. The system,
which operates in heat engine cycle is called heat engine.
Efficiency ,
( )
( )
11
sup 1
2
1
21
1
≤−=
−
===
Q
Q
Q
QQ
Q
W
payweEnergypliedheat
wantweEnergyworkdone
η
Refrigerator: A device which operating in a cycle maintains a body at a temperature
lower than the temperature of the surroundings.
Heat engine
Q2
Q1
W
(Q1-Q2)
Hot Reservoir
Cold Reservoir

COPRef =
( )
( ) 21
22
QQ
Q
payweEnergyW
wantweEnergyQ
−
=
Heat Pump: A device which operating in a cycle maintains a body at a temperature
higher than temperature of surroundings
COPHP =
( )
( ) 21
11
QQ
Q
payweEnergyW
wantweEnergyQ
−
= = COPRef + 1
Engineering Examples:
A. System (gas) – undergoes a cycle.

Refrigerator
Q2
Q1
W
(Q1-Q2)
Atmosphere, T1
Body, T2
Heat pump
Q2
Q1 (=Q2 + W)
W
Body, T1
Atmosphere, T2
T2 < T1 T1 > T2

From our experience, the reverse cycle does not occur. W < 0 and Q < 0 are possible but W > 0
and Q > 0 are not. However, neither violates the 1st Law.
B. Heat transfer from high T to low T: only one direction is possible, so it is impossible to
complete a cycle. This leads us to heat engines and refrigerators – devices that can do the
opposite. What are the limitations on these devices?
The Second Law states that some heat must be rejected during the cycle and hence,
the cycle efficiency is always less than unity.
Thus the First Law states that net work cannot be greater than heat supplied, while the
Second Law goes further and states that it must be less than heat supplied.
If energy is to be supplied to a system in the form of heat, the system must be in contact
with a reservoir whose temperature is higher than that of the fluid at some point in the
cycle. Similarly, if heat is to be rejected, the system must be at some time be in contact
with a reservoir of lower temperature than the fluid. Thus Second Law implies that if a
system is to undergo a cycle and produce work, it must operate between two reservoirs
of different temperatures. A machine which will work continuously, while exchanging
heat with only a single reservoir, is known as a perpetual motion machine of the second
kind (PMM II); such a machine contradicts Second Law.
It is now possible to see why a ship could not be driven by
an engine using the ocean as a source of heat, or why a
power station could not be run using the atmosphere as a
source of heat. They are impossible because there is no
natural sink of heat at a lower temperature than the
atmosphere or ocean, and they would therefore be PMM II.
It should be noted that Second Law does not restrict that
work cannot be continuously and completely converted to
heat. In fact, a fluid in a closed vessel may have work done on it and the heat thus
generated is allowed to cross the boundary. The rates of work and heat may be made
equal and the internal energy of the system remaining constant. An important
consequence of Second law is work is a more valuable form of energy transfer than
Q
W
Heat source
PMMII

heat as heat can never be transformed continuously and completely to work, whereas
work can always be transformed continuously and completely to heat.
The following statements summarise the obvious consequences of the Second Law:
a) If a system is taken through a cycle and produces work, it must be exchanging heat
with at least two reservoir at different temperatures,
b) If a system is taken through a cycle while exchanging heat with one reservoir, the
work done must be zero or negative,
c) Since heat can never be continuously and completely converted into work whereas
work can always be continuously and completely converted into heat, work is more
valuable form of energy transfer than heat.
THE KEIVIN-PLANK’S STATEMENT OF THE
SECOND LAW
It is impossible to construct a system,
which will operate in a cycle, extract heat
from a reservoir and do an equivalent
amount of work on the surroundings.
THE CLAUSIUS STATEMENT OF THE
SECOND LAW
It is impossible to construct a system, which
will operate in a cycle and transfer heat from
a cooler to a hotter body without work being
done on the system by the
surroundings.(Corollary 1)

EQUIVALENCE OF KELVIN PLANK’S AND CLAUSIUS STATEMENTS.
Proof: Suppose the converse of the Clausius’ proposition is true. The system can be
represented by a heat pump for which W = 0. If it takes Q units of heat from the cold
reservoir, it must deliver Q units to the hot reservoir to satisfy the First Law. A heat
engine could also be operated between the two reservoirs; let it be of such a size that it
delivers Q units of heat to the cold reservoir while performing W units of work. Then the
First Law states that the engine must be supplied with (W + Q) units of heat from the hot
reservoir. In the combined plant, the cold reservoir becomes superfluous because the
heat engine could reject its heat directly to the heat pump. The combined plant
represents a heat engine extracting (W + Q) – Q = W units of heat from a reservoir and
delivering an equivalent amount of work. This is impossible according to Kelvin-Plank’s
statement of Second Law. Hence converse of Clausius’ statement is not true and the
original proposition must be true.

Reversibility and irreversibility
The Second Law of Thermodynamics enables us to divide all processes into two
classes:
a) Reversible or ideal process.
b) Irreversible or natural process.
A reversible process is one, which is performed in such a way that at the end of the
process, both the system and the surroundings may be restored to their initial states,
without producing any changes in the rest of the universe. Let the state of a system be
represented by a, and let the system be taken to state b by following the path a-b. In the
reverse process, the system is taken from state b to a by following the same path b-a. If
the system and also the surroundings are restored to their initial states (a) and no
change in the universe is produced, then the process a-b will be a reversible process.
A reversible process is carried out infinitely slowly with an infinitesimal gradient, so that
every state passed through by the system is an equilibrium state.
Any natural process carried out with a finite gradient is an irreversible process. A
reversible process, which consists of a succession of equilibrium states, is an idealised
hypothetical process, approached only as a limit. All spontaneous processes are
irreversible.
A reversible process
Wb-a
Wa-b
ba
b
a

Causes of Irreversibility:
• Friction
• Unrestrained expansion
• Heat transfer through a finite temperature difference
• Mixing of two different substances
The lack of equilibrium (mechanical, thermal or chemical) between the system and its
surroundings, or between two systems, or two parts of the same system, causes a
spontaneous change, which is irreversible. Heat transfer through a finite temperature
difference, free expansion are some examples.
The irreversibility of a process may be due to the dissipative effects in which work is
done without producing equivalent increase in the kinetic or potential energy of the
system. Friction, transfer of electricity through a resistor are some examples.
Question: If all the heat cannot be converted into work, then what is the maximum possible
efficiency?
Maximum efficiency is achieved by reversible engines, which must
• Always be in equilibrium.
• Avoid friction.
• Avoid heat transfer. If we must have heat transfer it should be through infinitesimal
temperature differences ‘dT’ not finite ‘ΔT’.
CARNOT CYCLE
A Carnot cycle consists of
a) A reversible isothermal process in which heat Q1 enters the system reversibly from
a constant temperature source of T1
Q1 = U2 – U1 + W1-2
b) A reversible adiabatic process (isentropic process) in which work WT is done by
the system adiabatically and reversibly at the expense of its internal energy.
0 = U3 – U2 + W2-3

c) A reversible isothermal process in which heat Q2 is rejected from the system
reversibly from a constant temperature source of T2
-Q2 = U4 – U3 – W3-4
d) A reversible adiabatic process (isentropic process) in which work WP is done by
the system adiabatically and reversibly at the expense of its internal energy.
0 = U1 – U4 – W4-1
Net work done = Wnet = WT – WP = Q1 – Q2
REVERSIBLE ENGINES OPERATING BETWEEN ONLY TWO RESERVOIRS
Carnot’s Theorem: It is impossible to construct an engine operating between only two heat
reservoirs, which will have a higher efficiency than a reversible engine operating between the same
two reservoirs. (Corollary 2)
Carnot Engine
WP
WT
Q2
Q1
Heat Exchanger (A), T1
Heat Exchanger (B), T2

Proof. Assume the converse of this proposition is true. Let X be such an engine having an
efficiency ηX. Let it receive heat Q1 from the source, does work WX, and reject heat (Q1 – WX) to
the sink. Then it is assumed that ηX > ηR, where ηR is the efficiency of a reversible engine R
operating between the same two reservoirs (a). If the reversible engine also receives heat Q1
from the source, it will do work WR such that WR < WX and the heat rejected will be (Q1 – WR)
which is greater than (Q1 – WX). Let the reversible engine be reversed and act as heat pump
(b). It now receives heat (Q1 – WR) from the low temperature reservoir, receives work WR from
the surrounding and rejects heat Q1 to the high temperature reservoir. If the engine X is coupled
to the heat pump such that the engine X receives heat Q1 directly from the heat pump, then the
combined plant represents a heat engine receiving heat (Q1 – WR) - (Q1 – WX) = (WX - WR) from
the surroundings and delivering an equivalent amount of work. According to Second Law this is
impossible and the assumption ηX > ηR cannot be true. Consequently the original proposition
must be true.
Corollary 3: All reversible engines operating between the same two reservoirs have the
same efficiency.
Since all reversible engines operating between the same two reservoirs have the same
efficiency, this efficiency must depend upon the only feature, which is common to them all viz.
the temperature of the reservoirs. It is called Carnot efficiency.
(a)
WX
(Q1-WR)(Q1-WX)
WR
Q1Q1
(b)
Net
(W W )
(Q1-WR)(Q1-WX)
WR
Q1Q1

THERMODYNAMIC TEMPERATURE SCALE
Corollary 4. A scale of temperature can be defined which is independent of any particular
thermometric substance, and which provides an absolute zero of temperature.
Let us consider the reversible engine operating
between two reservoirs represented symbolically in
Figure. Corollary 3 implies that its efficiency depends
only on the temperature of the reservoirs, and
therefore that it is independent of the particular
properties of the working fluid and of the quantity of
heat Q0 supplied to the engine. But
0
1
0
10
Q
Q
1
Q
QQ
−=
−
=η
and hence it follows that Q1/Q0 is a function only of the temperatures of the reservoirs.
( )01
0
1
T,T
Q
Q
1 φ=−=η∴
Now ( )21
2
1
T,T
Q
Q
φ=
W
Q1
QO
Hot Reservoir
Cold Reservoir

( )32
3
2
T,T
Q
Q
φ=
( )31
3
1
T,T
Q
Q
φ=
Since
32
21
3
1
QQ
QQ
Q
Q
=
∴ φ(T1,T3) = φ(T1,T2) x φ(T2,T3)
Since, the left side of the above equation is a function of T1 and T3, the right hand side of the equation will be a function of T1 and T3.
Hence we can conclude that

( )
( )
( )
( )3
2
32
2
1
21
Tf
Tf
)T,T(and
Tf
Tf
)T,T( =φ=φ
( ) ( )
( )3
1
31
3
1
Tf
Tf
T,T
Q
Q
=φ=∴
In general
( )
( )H
L
H
L
Tf
Tf
Q
Q
=
Lord Kelvin proposed that
H
L
H
L
T
T
Q
Q
=
With temperature so defined,
H
L
H
L
T
T
1
Q
Q
1 −=−=η
If Q is measured for several sinks at
different temperatures and plotted against
the relation between Q and T, we obtain a
temperature scale, which is linear in Q. The
temperature scale so defined is called the
thermodynamic scale, because it depends
solely on the thermodynamic law and not
TOT
Q
QO

upon the properties of any particular substance.
The thermodynamic scale is an absolute scale because it presents us with the idea of an
absolute zero, i.e. T = 0 when Q = 0. The Second Law states that Q can in fact never be zero,
and we may therefore deduce that absolute zero is a conceptual limit and not a
temperature, which can ever be reached in practice. Thus it appears that a definite zero
point exists on absolute temperature scale but this point cannot be reached without violation of
the Second Law. Hence, the Third Law of Thermodynamics can be stated as: It is
impossible by any procedure to reduce the temperature of any system to absolute zero in
a finite number of operations.
What we have learned so far:
• The 2nd Law tells us the direction of spontaneous change.
• The 2nd Law gives us the maximum possible efficiency (coefficient of performance) of a
heat engine (refrigerator).
• A reversible process is one in which both the system and the surroundings can be
returned to their original state.
• To achieve the maximum possible efficiency, we must have a reversible engine
(frictionless, adiabatic, isothermal heat transfer, quasi-equilibrium)
• One realisation of such an engine is a Carnot engine.
• All reversible engines have the same efficiency.
• This maximum efficiency depends only on the temperature of the reservoirs – this yielded
the thermodynamic temperature scale (K).

Entropy- A Measure of disorder /Irreversibility
The Second Law says that spontaneous change will occur only in a certain direction.
These processes are called irreversible. There are many factors that make the process
irreversible, such as
• Friction
• Unrestrained expansion
• Heat transfer through a finite temperature difference
• Mixing of two different substances
Question: Is there any way to quantify the degree of irreversibility?
CLAUSIUS INEQUALITY
Whenever a system undergoes a cycle, ∫
δ
T
Q
is zero if the cycle is reversible and
negative if irreversible, i.e. in general .0
T
Q
≤
δ
∫
For a Reversible Engine
∫ −= LH QQQδ > 0, since W = QH - QL
While defining the absolute temperature scale, we
showed that
L
L
H
H
L
H
L
H
T
Q
T
Q
or
T
T
Q
Q
==

Which gives us ∫ =−= 0
T
Q
T
Q
T
Q
L
L
H
Hδ
For Irreversible Engine
Wirr < Wrev since W = QH - QL
Hence, QH - QL,irr < QH - QL,rev
i.e. QL,irr > QL,rev
Therefore, ∫ −=
L
irr,L
H
H
T
Q
T
Q
T
Qδ
< 0
Hence, for any cycle (reversible or irreversible) we get,
.0
T
Q
≤
δ
∫
Consequences of the Second Law for non-flow process
ENTROPY – A Property of the System
Corollary: There exists a property of a closed system such that a change in its value is
equal to ∫
2
1
T
Qδ
for any reversible process undergone by the system between state 1 and
state2.
Proof.

Let us assume that ∫
2
1
T
Qδ
depends upon the
reversible process as well as upon the states.
Consider two reversible processes A and C by
which the system can change from state 1 to state
2. The assumption is that
∫∫ ⎟
⎠
⎞
⎜
⎝
⎛
≠⎟
⎠
⎞
⎜
⎝
⎛
2
1 C
2
1 A T
Q
T
Q δδ
In each case, let the system returns to its original state by a third reversible
process B. For each of the complete reversible cycles AB and CB we have

∫ ∫∫ ⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
=⎟
⎠
⎞
⎜
⎝
⎛
1
2 B
2
1 AAB T
Q
T
Q
T
Q δδδ
and ∫ ∫∫ ⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
=⎟
⎠
⎞
⎜
⎝
⎛
1
2 B
2
1 CCB T
Q
T
Q
T
Q δδδ
If the assumption is true then it follows that
∫ ∫ ⎟
⎠
⎞
⎜
⎝
⎛
≠⎟
⎠
⎞
⎜
⎝
⎛
CBAB T
Q
T
Q δδ
But this cyclic integral must be equal since they are zero for a reversible cycle
(Claussius inequality). Hence, ∫
2
1
T
Qδ
is independent of the path of the reversible process
and represents the change in property of the system.
If the property is denoted by S, this corollary can be expressed mathematically as
12
rev
2
1
SS
T
Q
−=⎟
⎠
⎞
⎜
⎝
⎛
∫
δ
or in the differential form as
revT
Q
dS ⎟
⎠
⎞
⎜
⎝
⎛
=
δ
This property S is called entropy of the system.
Entropy change in a process
Let us assume A and B are reversible processes
and C is an irreversible process. For the reversible
cycle consisting of A and B
∫ ∫∫ ⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
=⎟
⎠
⎞
⎜
⎝
⎛
1
2 B
2
1 AAB T
Q
T
Q
T
Q δδδ
= 0
or ∫∫ ⎟
⎠
⎞
⎜
⎝
⎛
−=⎟
⎠
⎞
⎜
⎝
⎛
1
2 B
2
1 A T
Q
T
Q δδ
For the irreversible cycle consisting of B and C, by
the inequality of Clausius
∫ ∫∫ ⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
=⎟
⎠
⎞
⎜
⎝
⎛
1
2 B
2
1 CCB T
Q
T
Q
T
Q δδδ
< 0

Hence, ∫∫ ⎟
⎠
⎞
⎜
⎝
⎛
+⎟
⎠
⎞
⎜
⎝
⎛
−
2
1 C
2
1 A T
Q
T
Q δδ
< 0
i.e. ∫ ⎟
⎠
⎞
⎜
⎝
⎛
2
1 AT
Qδ
> ∫ ⎟
⎠
⎞
⎜
⎝
⎛
2
1 CT
Qδ
Since the process A is reversible
∫ ⎟
⎠
⎞
⎜
⎝
⎛
2
1 AT
Qδ
= ( )∫
2
1
AdS
Since entropy is a property, changes for the path A and C would be the same.
Therefore,
( )∫
2
1
AdS = ( )∫
2
1
CdS
Hence, ( )∫
2
1
CdS > ∫ ⎟
⎠
⎞
⎜
⎝
⎛
2
1 CT
Qδ
Thus for an irreversible process,
dS >
T
Qδ
whereas for a reversible process
revT
Q
dS ⎟
⎠
⎞
⎜
⎝
⎛
=
δ
Therefore in general
T
Q
dS
δ
≥
or ∫≥−
2
1
12
T
Q
SS
δ
For an isolated system which does not undergo any energy interchange with the
surroundings, δQ = 0.
Therefore for an isolated system dS ≥ 0.
For a reversible system, dS = 0 or S = constant.

Thus the entropy for an isolated system can never decrease. It always increases and
remains constant only when the process is reversible. This is known as entropy
principle.
Change of Entropy for special Reversible processes:
rev
12
rev T
Q
ssor
T
Q
dS ⎟
⎠
⎞
⎜
⎝
⎛
=−⎟
⎠
⎞
⎜
⎝
⎛
=
δδ
1) Adiabatic
δQ = 0 or 0
T
Q
dS
rev
=⎟
⎠
⎞
⎜
⎝
⎛
=
δ
Reversible adiabatic is known as isentropic process.
2) Isothermal
( ) ( )
( ) ( )12rev21
2
1 H
rev21
rev
H
12
ssTQor
T
Q
Q
T
1
ss
−=
==−
−
−
∫ δ
Summary
• Since entropy is a property, it does not matter how we get from state 1 to state 2. Thus,
we can choose the most convenient reversible path to compute the integral.
• S2 – S1 is independent of the process – once evaluated, it is the same for all processes.
• Entropy of all pure substances is taken to be zero at T = 0 K.
• In the absence of chemical reactions, an arbitrary reference may be selected as in the
case of internal energy and enthalpy. (e.g. sf = 0 at triple point for water).

ENTROPY OF PURE SUBSTANCE
In the saturated region:
s = (1-x) sf + x sg or s = sf + x sfg
where, s is specific entropy in kJ/kg K = S / m.
ENTROPY CHANGE OF AN IDEAL GAS
From First Law: δQ = du + δW = du + pdv
For a reversible process:
dsTQords
T
Q
rev
==⎟
⎠
⎞
⎜
⎝
⎛
δ
δ
(from Second Law)
Hence,
Tds = du + pdv
Again
dh = du + d(pv) = du + pdv + vdp
So
Tds = dh – vdp
Hence, the general property relations are
Tds = du + pdv

Tds = dh – vdp
For ideal gas,
du = Cv dT ; dh = CP dT and pv = RT
The entropy change between two states 1 and 2 can be computed as
v
dv
RdTC
dv
T
p
T
du
ds
v +=
+=
Hence,
1
2
1
2
v12
v
v
lnR
T
T
lnCss +=−
Again
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
==
1
2
2
1
1
2
2
22
1
11
T
T
p
p
v
v
or
T
vp
T
vp
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+=−∴
1
2
2
1
1
2
v12
T
T
p
p
lnR
T
T
lnCss
or ( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−+=−
2
1
1
2
v12
p
p
lnR
T
T
lnRCss
i.e. ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=−
2
1
1
2
P12
p
p
lnR
T
T
lnCss
Similarly, substituting for
1
2
T
T
we get
1
2
P
1
2
v12
v
v
lnC
p
p
lnCss +=−
For Constant Volume Process
1
2
v12
2
2
1
1
T
T
lnCssHence;
T
p
T
p
=−=
For Constant Pressure Process
1
2
P12
2
2
1
1
T
T
lnCssHence;
T
v
T
v
=−=

For Adiabatic Process
s2 = s1
For Isothermal Process
P1v1 = p2v2 ; hence
1
2
1
2
12
p
p
lnR
v
v
lnRss −==−

CHAPTER 3 Properties of a Pure Substance and steam
power cycle
A pure substance is one that has homogeneous and invariable chemical composition. It may
exist in more than one phase, but chemical composition is same for all the phases. Thus, water,
mixture of water and ice and water and steam are all examples of pure substance. Sometimes a
mixture of gases e.g. air is considered as pure substance.
We have seen that two independent properties are sufficient to determine
thermodynamic state of a fluid when it is in equilibrium. Any other thermodynamic
property is a function of the chosen pair of independent properties. We shall first
consider the relation between the primary properties p, v and T, the equation expressing
this relation for any particular fluid being called the equation of state or characteristic
equation of the fluid.
p-v-T DATA OF WATER

Since we have three variables to consider, the obvious procedure is to measure the
variation of one with another while the third is kept constant and repeat this for a range
of values of the third variable.
Imagine unit mass of ice below the
freezing point, enclosed in a
cylinder by a piston under a
constant load of 1 atmosphere (1
atm. = 1.01325 bar = 101.325
kPa). If heat is added to the
cylinder while keeping the pressure
constant, the temperature rises
and ice expands until a
temperature of 273.15 K (00
C) is
reached (AB) as shown in Fig.
Further heating does not raise the
temperature of ice but causes a change to the liquid phase (BC). The change of phase
occurs at a constant temperature and by reduction of specific volume. The heat required
for this process is known as latent heat of fusion. Further heating results in a rise of
temperature of liquid and a further contraction in volume until the temperature is about
40
C and subsequent expansion until a temperature of 373.15 K (1000
C) is reached
(point D). At this point a second phase change occurs at constant temperature with a
large increase in volume until the liquid has been vaporised (point E). The heat required
in this case is called the latent heat of vaporisation. When vaporisation is complete,
the temperature rises again on heating (line EF). The heat transferred to a substance
while the temperature changes is sometimes referred to as sensible heat. This
constant pressure lines are called isobars.
If the pressure is reduced, there is a slight rise in the melting point and also there is a
marked drop in the boiling point of water and a marked increase in the change in
volume, which accompanies evaporation. When the pressure is reduced to 0.006113
A
B
v
T(K)
Fusion
Vapourisation
F
D
E
C
373.15
277.15
273.15

bar (0.6113 kPa), the melting point and boiling point temperatures become equal and
change of phase, ice-water-steam, are represented by a single line. The temperature at
which this occurs has been accepted internationally as a fixed point for the absolute
temperature scale and is by definition 273.16 K. Only at this temperature and pressure
of 0.6112 kPa, can ice, water and steam coexists in thermodynamic equilibrium in a
closed vessel and is known as triple point. If the pressure is reduced further, the ice,
instead of melting, sublimes directly into steam.
Consider now the behaviour at pressure above atmospheric. The shape of the curve is
similar to that of the atmospheric isobar, but there is a marked reduction in the change
in volume accompanying evaporation. At a sufficiently high pressure, this change in
volume falls to zero and the horizontal portion of the curve reduces to a point of
inflexion. This is referred to critical point. The values pressure and temperature of water
at which critical point reached are
pc = 221.2 bar = 22.12 MPa ; Tc = 647.3 K ; vc = 0.00317 m3/kg.

The pressure at which liquid vaporises or condenses is called saturation pressure
corresponding to a given temperature. Alternately, the temperature at which this
phenomena occur is called saturation temperature corresponding to the given
pressure.
A vapour in a state lying along the saturated vapour line is also called dry saturated
vapour and the vapour lying right of this line is called superheated vapour.
IMPORTANT POINTS TO BE REMEMBERED
• Saturation temperature: Temperature at which a pure substance changes phase at a
given pressure.
• Saturation pressure: Pressure at which a pure substance changes phase at a given
temperature.
• Latent heat: The amount of energy absorbed or released during a phase-change.

Melting/freezing: Latent heat of fusion.
Evaporation/condensation: Latent heat of vaporization.
• Temperature at which water starts boiling depends on the pressure => if the pressure
is fixed, so is the boiling temperature.
TABLE OF PROPERTIES OF WATER
(i) Saturated Table
P ts vf vg uf ug hf hg sf sg
Now how many properties do we need to know to determine the thermodynamic state?
For a single phase we need to know only two independent properties like (P, T), (T, v), etc.
For a two-phase mixture, p and T are not independent. Hence, we require defining “quality”.
Quality or Dryness fraction:
lv
v
mm
m
x
+
=
Value of x varies from 0 to 1. For
saturated water, when water just
starts boiling, x = 0 and for
saturated vapour, when
vaporisation is complete, x = 1
and the vapour is called dry
saturated.
Let V be the total volume of a
liquid vapour mixture of quality x.
Vf the volume of saturated liquid
and Vg the volume of saturated vapour, the corresponding masses are m, mf, and mg
respectively.

Now m = mf + mg
and V = Vf + Vg
or m v = mf vf + mg vg = (m – mg) vf + mg vg
or
( )
g
g
f
g
v
m
m
v
m
mm
v +
−
=
or v = (1 – x) vf + x vg
Again, enthalpy H = Hf + Hg
or m h = mf hf + mg hg = (m – mg) hf + mg hg
or
( )
g
g
f
g
h
m
m
h
m
mm
h +
−
=
or h = (1 – x) hf + x hg
Similarly,
s = (1 – x) sf + x sg
where, h is the specific enthalpy an s the specific entropy. We shall discuss about these
properties in the later chapters.
Now
v = (1 – x) vf + x vg = vf + x (vg – vf) = vf + x vfg.
Similarly,
u = uf + x ufg
h = hf + x hfg
s = sf + x sfg
where, vfg = vg – vf ; ufg = ug – uf ; hfg = hg – hf etc.
ii) Superheated steam Table

p
(ts)
t 200 300 400 …………..
15
(198.3)
20
( )
v
u
h
s
v
u
h
s
IDEAL GAS (PERFECT GAS)
From experimental observations it has been established that the following relation can
express the p –v – T behaviour of gases at the low pressure
TRvp =
where, R is the universal gas constant, 8.3143 J/mol K and v is the molar specific
volume, m3
/gmol. Dividing the equation (5.14) by the molecular weight M, we get
p v = RT
where v is the specific volume, m3
/kg, and R is the characteristic gas constant. It follows
from the above equations that this equation of state can be written in terms of total
volume, V, of the gas as
TRnVp =
p V = m R T
where, n is the number of moles and m is the mass of the gas. The above equation can
be written for the two states of the gas as

2
22
1
11
T
vp
T
vp
=
The above equations s called the ideal gas equation of state. At very low pressure or
density, all gases and vapour approach ideal gas behaviour.
Vapour Power Cycles and Processes.
About 80% of the electrical power is generated by steam power plants in which working
fluid (normally water) undergoes a cyclic change in phase from liquid state to vapour
state and back to the original liquid state. This change in phase is brought about by
transfer of heat to and from the working fluid. The analysis of such plants is based on
idealized cycle that may be closely approximated to the working of the actual plant.
THERMAL EFFICIENCY
The thermal efficiency of a cycle is defined as the ratio of the energy output to the
energy input according to the relation
H
n
th
Q
W
=η
where Wn is the net work output of the cycle and QH is the heat supplied to the working
fluid from the high temperature source.
In many situations, however, we like to evaluate the thermal efficiency of a certain
process, e.g. the efficiency of compression process in a compressor or the efficiency of
an expansion process in a turbine.
Summary
Ideal Gas: When is it a good approximation?
- Low density ⇒ low pressure, high temperature

The compressor efficiency applied to a pump or compressor is defined as the ratio of
theoretical isothermal work WT (in case of reciprocating compressor) or isentropic work
Ws (in case of rotary compressor or pump) to the actual work, Wac
ac
T
c
W
W
=η (for reciprocating compressor)
or
ac
s
c
W
W
=η (for rotary compressor or pump).
The expansion efficiency applied to a turbine is defined as the ratio of the actual work
Wac to the isentropic work Ws
s
ac
T
W
W
=η
IDEALISED STEAM POWER CYCLE (CARNOT VAPOUR CYCLE)
The efficiency of the Carnot Cycle is maximum and it does not depend on the working
fluid. It is, therefore, natural to examine if a steam power plant can be operated on
Carnot cycle.
Compressor or Pump Turbine

1-2 → Isentropic compression in pump.
2-3 → Constant Pressure heat addition in
boiler.
3-4 → Isentropic expansion in turbine.
4-1 → Constant pressure heat removal in
condenser.
The efficiency is given by
H
L
H
L
H
LH
H
net
cycle
T
T
1
Q
Q
1
Q
QQ
Q
W
−=−=
−
==η
where QH = heat transferred to
working fluid
QL = heat transferred from working fluid
Practical Problems
The constant pressure heat addition process 2-3 can be achieved in boiler. However,
for process
I. 3-4 Turbine has to handle high moisture content. Liquid droplets can damage or
erode turbine blades.
II. 4-1 condensing process in condenser must be terminated at state 1, where the
working fluid is a mixture of liquid water and vapour. It is impossible to attain
such condensation.
III. 1-2 Pump that handles two-phase fluid is not practical.
Due to these inherent practical difficulties, Carnot Cycle remains an ideal one.

RANKINE CYCLE
The simplest way of overcoming the inherent practical difficulties of Carnot Cycle
without deviating too far from it is to keep processes 2-3 and 3-4 of Carnot Cycle
unchanged and to continue the process 4-1 in the condenser until all vapour is
converted to liquid water. Water is then pumped into Boiler till its pressure is raised to
the pressure corresponding to state 2 and the cycle is completed. Such a cycle is
known as Rankine Cycle.
Processes:
1-2 → Isentropic (reversible adiabatic) compression in pump.
2-3 → Constant Pressure heat addition in boiler.
3-4 → Isentropic expansion in turbine.
4-1 → Constant pressure heat removal in condenser.
s
T
6 5
4
32’
2
1

P2
P1

If changes in kinetic and potential energies are neglected, the area under the curve 2-3
i.e. area 2-2’-3-5-6-2 represents the heat transfer to the working fluid in Boiler, which is
equal to (h3 – h2) and area under the curve 1-2 i.e. area 1-4-5-6-1 represents the heat
transferred from the working fluid in condenser, which is equal to (h4 – h1). The
difference between the two areas, namely area 1-2-2’-3-4-1, represents the work
obtained from the cycle. The thermal efficiency of the cycle is given by
( ) ( )
( )
( ) ( )
( ) H
PT
23
1243
23
1423
H
net
th
Q
WW
hh
hhhh
hh
hhhh
2653'22area
143'221area
Q
W
−
=
−
−−−
=
−
−−−
=
−−−−−
−−−−−
==η
where WT and WP are the turbine work and pump work respectively per kg of steam flow
through the cycle and h1, h2, h3, h4 are the specific enthalpies of the working fluid.
We know that the efficiency of the Carnot cycle depends only on the temperature levels
of high and low temperature reservoirs. Efficiency o the Rankine cycle similarly depends
on the average temperature at which the heat is transferred to and from the working
fluid. Any change that increases the average temperature at which heat is transferred to
the working fluid will increase the efficiency of the Rankine cycle. Similarly, any change
that decreases the average temperature at which heat is transferred from the working
fluid will increase the efficiency of the Rankine cycle.
An advantage of the Rankine cycle over all other power cycles is its low back work ratio,
which is expressed as the ratio of the pump work to the turbine work, i.e.
Back work ratio =
T
P
W
W
METHODS OF IMPROVING THE EFFICIENCY OF RANKINE CYCLE
The Rankine cycle efficiency is given by
( ) ( )
( ) H
PT
23
1243
th
Q
WW
hh
hhhh −
=
−
−−−
=η

Since the pumping work is very small compared to the turbine work, it may be
neglected. Hence, the efficiency of the Rankine cycle can be approximated as
( )
( )
( )
( )23
s
23
43
th
hh
h
hh
hh
−
Δ
=
−
−
=η
It can be seen that the Rankine efficiency depends on three values, h2, h3 and turbine
expansion work (Δh)s. The enthalpy of the steam entering the turbine h3 is determined
by the pressure and temperature of the steam entering the turbine. The enthalpy of feed
water h2 is determined by the condenser pressure (as in this case h2 = h1 since the
pump work is negligible). The isentropic heat drop (Δh)s in the turbine is determined by
the pressure and temperature at the entrance of steam turbine and the pressure at the
end of expansion in the turbine. That means the Rankine efficiency depends on
pressures P1 ( i.e. P4 ) , P2 (i.e. P3 ) and temperature T3.
a) Effect of Lowering Condenser pressure
By lowering the condenser pressure we can
achieve higher efficiency as the enthalpy of
the steam leaving the turbine decreases
thereby increasing heat drop (Δh)s in the
turbine. However, there is a limit to which
the condenser pressure can be lowered and
this limit is the saturated pressure Psat
corresponding to the condenser
temperature.

b) Effect of Increasing Steam pressure in the Boiler
In this analysis the maximum temperature of the steam T3 as well as the exhaust
pressure P4 are held constant.
P2, P2’, P2’’ are the pressures of steam at the entrance of the turbine at temperature T3
such that P2’’ > P2’ > P2 . x4, x4’,x4’’ are the qualities of steam at he exhaust pressure P4
of the turbine, where x4>x4’>4’’ .It is evident that as pressure n the Boiler increases, the
isentropic heat drop Δh)s increases with the result that the Rankine cycle efficiency
increases.
An adverse effect resulting from increasing the steam pressure in the Boiler is the
greater amount of moisture in the steam at the end of expansion in the turbine. If the
moisture content in the turbine exceeds ~10%, the turbine blades also get eroded which
leads to serious wear of the turbine blades.
c) Effect of Superheating
of Steam in the Boiler
The moisture in the steam at the
end of the expansion can be
reduced by superheating and
x4’
x4”
x4(Δhs)″
(Δhs)′
(Δhs)
h
s
P2’’ P2’
T3
P4
P2

increasing the superheat temperature of steam, T3’. Hence, it is natural to avoid the
erosion of the turbine blades by an increase of boiler pressure accompanied by
superheating at a higher temperature. By superheating to a higher temperature, the
heat drop in the turbine is increased from (h3 – h4) to (h3’ – h4’), thereby increasing the
efficiency of the Rankine cycle. However, the maximum temperature to which the steam
can be superheated is limited by materials.
d) Effect of Superheating &Reheating of Steam in the Boiler
We have noted earlier that the efficiency of the Rankine cycle can be increased by
increasing the steam pressure in the boiler. But this increases the moisture content of
the steam in the lower stages of the turbine, which may lead to erosion of the turbine
blades. The reheat cycle has been developed to take advantage of the increased
pressure of boiler, avoiding the excessive moisture of the steam in the low pressure
stage.
In the reheat cycle, the high pressure superheated steam after expansion in the high
pressure turbine is reheated at constant pressure, usually to the entrance temperature
of the steam in the high pressure turbine. After this, it expands in the low pressure
s
P3 P4
P6
4
3 5
6
Constant temperatureh

turbine to the exhaust pressure. Reheating has a two-fold advantage. Firstly, it reduces
excessive moisture in the low pressure stages of turbine, and secondly, a large amount
of work may be obtained at the cost of additional consumption of heat required for
reheating the steam. The net effect is an improvement in the thermal efficiency of the
cycle. Thus with reheat cycle, the efficiency of the cycle is increased without increase in
the maximum pressure or maximum temperature of the cycle.
The efficiency of the reheat cycle is given by
( ) ( ) ( )
( ) ( )4523
126543
H
PT
th
hhhh
hhhhhh
Q
WW
−+−
−−−+−
=
−
=η
REAL RANKINE CYCLE: DEVIATION FROM IDEAL CYCLES
a) Turbine and pump losses
1-2s & 3-4s → Isentropic drop.
1-2 & 3-4 → irreversible drop.
b) Pump loses
a-b: friction loss → entropy increases
b-c: heat loss → entropy decreases.

Example
CRITERIA FOR THE COMPARISON OF CYCLES
The choice of power plant for a given purpose is determined largely by considerations of
operating cost and capital cost. The former is primarily a function of overall efficiency
of the plant, while the latter depends mainly on its size and complexity.
The Second Law tells us that even in the best power cycle, some heat must be rejected.
The best form of cycle is one in which (i) all the heat supplied is transferred while the
working fluid is at constant temperature TH, and all the heat rejected from the working

fluid is at constant temperature TL; (ii) all processes are reversible. The efficiency of the
such cycle is
H
LH
T
TT −
, which is known as ideal cycle efficiency or Carnot efficiency.
However, all real processes are irreversible and irreversibility reduces cycle efficiency.
Hence the ratio of actual cycle efficiency to ideal cycle efficiency, i.e. the efficiency ratio
is one measure of comparison. Some cycles are more sensitive to irreversibilities than
others. That is, two cycles may have the same ideal cycle efficiencies, but allowing for
the process efficiencies, their actual cycle efficiencies may be markedly different.
Hence, Work Ratio rw is a criterion, which indicates the cycle sensitiveness to the
irreversibilities. Any cycle consists of both positive (turbine work) and negative (pump
work) work. The work ratio rw is defined as the ratio of net work to the positive work
done in the cycle. That is
T
PT
w
W
wW
r
−
=
If rw is near unity, then the effect of irreversibility on the cycle efficiency is less.
However, if rw is slightly greater than zero, quite a small amount of component
inefficiencies is sufficient to reduce the network output to zero thereby reducing the
actual cycle efficiency to zero.
Hence, we can say that a high ideal cycle efficiency together with high work ratio
provides a reliable indication that a real power plant will have a good overall efficiency.
The next consideration is some criterion which will indicate the relative size of plant for a
given power output. In general, the size of component depends on the amount of
working fluid, which has to be passed through them. A direct indication of relative sizes
of steam power plant is therefore provided by the Specific Steam Consumption (ssc)
i.e. mass flow of steam required per unit power output. If W is the net work output in
kJ/kg, then ssc can be found from
⎥
⎦
⎤
⎢
⎣
⎡
=⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
=
kWh
kg
W
3600
h
s
3600x
kWs
kg
or
kJ
kg
W
1
ssc

Gas Power / Air Standard Cycles.
For small power plants, gas is ideally preferred as the working fluid. The gasoline
engines, diesel engines and gas turbines are common examples.
The analysis of the air-standard cycle is based on the assumptions that are far from
real. In actual internal combustion (IC) engines, chemical reaction occurs inside the
engine cylinder as a result of combustion of air-fuel mixture and. The IC engines are
actually operated on Open cycles in which the working fluid does not go through a
cycle. The accurate analysis of IC engine is very complicated. However, it is
advantageous to analyse the performance of an ideal closed cycle that closely
approximates the real cycle. One such approach is air-standard cycle, which is based
on certain assumptions. The assumptions for idealized air-standard cycles are:
1) The working fluid, air, is assumed to be an ideal gas. The equation of state is
given by the equation pv = RT and the specific heats are assumed to be
constant.
2) All processes that constitute the cycle are reversible.
3) No chemical reaction occurs during the cycle. Heat is supplied from a high
temperature reservoir (instead of chemical reaction) and some heat is rejected to
the low temperature reservoir during the cycle.
4) The mass of air within the system remains constant throughout the cycle.
5) Heat losses from the system to the atmosphere are assumed to be zero.
In this we shall discuss about the Brayton cycle, Otto cycle and Diesel cycle.
BRAYTON CYCLE
The Brayton cycle is widely used as the basis for the operation of Gas turbine.
A schematic diagram of a simple gas turbine (open cycle) and the corresponding p-v
and T-s diagrams are shown below.
Air is drawn from he atmosphere into compressor, where it is compressed reversibly
and adiabatically. The relative high pressure air is then used in burning the fuel in the
combustion chamber. The air-fuel ratio quite high (about 60:1) to limit the temperature
burnt gases entering the turbine. The gases then expand isentropically in the turbine. A

portion of the work obtained from the turbine is utilised to drive the compressor and the
auxiliary drive and the rest of the power output is the net power of the gas turbine plant.
Simple gas turbine Brayton cycle with closed cycle consists of
1 – 2 Isentropic compression in the compressor.
2 – 3 Constant pressure heat addition.
v
p
4
32
1
s
T
4
3
2
1

3 – 4 Isentropic expansion in the turbine.
4 – 1 Constant pressure heat rejection.
Assuming constant specific heats, the
thermal efficiency of the cycle
H
L
th
Q
Q
1−=η
or
23
14
th
hh
hh
1
−
−
−=η
( )
( )23P
14P
TTC
TTC
1
−
−
−=
The thermal efficiency can be written as
⎥
⎦
⎤
⎢
⎣
⎡
−
⎥
⎦
⎤
⎢
⎣
⎡
−
−=
−
−
−=
1
T
T
T
1
T
T
T
1
TT
TT
1
2
3
2
1
4
1
23
14
thη
Now
γ
γ 1
1
2
1
2
p
p
T
T
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
= and
γ
γ 1
4
3
4
3
p
p
T
T
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
Since, p2 = p3 and p1 = p4, it follows
2
3
1
4
4
3
1
2
T
T
T
T
or
T
T
T
T
==
Hence,
γ
γ
γ
γ
η
1
p
1
1
2
2
1
th
r
1
1
p
p
1
1
T
T
1
−
− ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=−=
Where rp is the pressure ratio.
It can be seen that increasing the pressure ratio can increase the efficiency of the
Brayton cycle.
s
T
3
2
1
4

Effect of Pressure Ratio
As mentioned above, the thermal efficiency of the Brayton cycle depends on the
pressure ratio and the ratio of specific heat. For air, γ = 1.4 and the efficiency vs.
pressure ratio plot is shown below.
The highest temperature of the
cycle occurs at the end of
combustion process (state 3) and
the maximum temperature that the
turbine blades can withstand limits
it. This also limits the pressure ratio
that can be used for the cycle. In
gas turbine cycle, T1 is the
temperature of the atmosphere and
T3 is the temperature of the burnt
gases entering the turbine. Both
these temperatures are fixed, first
by ambient conditions and second by metallurgical conditions. Between these two
extreme values of temperatures, there exists an optimum pressure ratio for which
the net work output of the cycle is maximum.
The net work output of the turbine is given by
( ) ( )12P43Pn TTCTTCW −−−=
⎥
⎦
⎤
⎢
⎣
⎡
−−⎥
⎦
⎤
⎢
⎣
⎡
−−= 1
T
T
TC1
T
T
TC
1
2
1P
3
4
3P
Now
2
1
3
4
2
3
1
4
T
T
T
T
or
T
T
T
T
==
Pressure ratio, rp

( )
( ) ⎥⎦
⎤
⎢⎣
⎡
−−
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−−=
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
⎥
⎦
⎤
⎢
⎣
⎡
−−⎥
⎦
⎤
⎢
⎣
⎡
−−=∴
−
−
−
−
1rTC1
r
1
TC
1
p
p
TC1
p
p
1
TC
1
T
T
TC1
T
T
TCW
1
P1P1
P
3P
1
1
2
1P1
1
2
3P
1
2
1P
2
1
3Pn
γ
γ
γ
γ
γ
γ
γ
γ
The optimum pressure ratio is obtained by differentiating the net work with respect to
pressure ratio, rP and putting the derivative as zero.
Let n
1
=
−
γ
γ
( )
( )[ ]1rTC1
r
1
TCW
n
P1Pn
P
3Pn −−⎥
⎦
⎤
⎢
⎣
⎡
−−=
or [ ] [ ]1n
P1P
1n
P3P
P
n
rnTCrnTC
dr
dW −−−
−−−=
or 1n
P1P
1n
P3P rnTCrnTC0 −−−
−=
or 1n
P1
1n
P3 rnTrnT −−−
=
or ( ) ( )
3
11n1n
P
T
T
r =−−−−
or
3
1n2
P
T
T
r =−
or
1
3n2
P
T
T
r =
i.e. ( )
n2
1
1
3
optimumP
T
T
r ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
or ( )
( )12
1
3
optimumP
T
T
r
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
γ
γ

The maximum work is given by
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−= 1
T
T
TC1
T
T
TCW
2
1
1
3
1P
2
1
3
1
3Pmax
( )
[ ]2
13P
12
1
313P
TTC
TTT2TC
−=
⎥
⎦
⎤
⎢
⎣
⎡
+−=
i.e. [ ]2
minmaxPmax TTCW −=
Effect of Reversibility
In an ideal gas turbine plant, the compression and expansion processes are isentropic
and there is no pressure drop in the combustion chamber and heat exchanger. But
because of irreversibilities associated with the compressor and the turbine and because
of the pressure drop in the actual flow passages and combustion chamber, the actual
gas turbine plant cycle differs from the ideal one.
Compressor efficiency
12
1'2
c
hh
hh
−
−
=η
Turbine efficiency
'43
43
T
hh
hh
−
−
=η
T
s
4’
4
3
2’
2
1

Regenerative Brayton Cycle
The temperature of the exhaust gases of simple gas turbine is higher than the
temperature of air after compression. If the heat energy in the exhaust gases instead of
getting dissipated in the atmosphere is used in heating air after compression, it will
reduce the energy requirement from the fuel, thereby increasing the efficiency of the
cycle.
Air is drawn from the atmosphere into
compressor and compressed isentropically
to state 2. It is then heated at constant
pressure in the regenerator to state x by hot
burnt gases from the turbine. Since the
temperature of air is increased before it
reaches the combustion chamber, less
amount of fuel will be required to attain the
designed turbine inlet temperature of the
products of combustion. After the combustion at constant pressure in the combustion
chamber, the gas enters the turbine at state 3 and expands to state 4 isentropically. It
then enters the counter-flow regenerator, where it gives up a portion of its heat energy
to the compressed air from the compressor and leaves the regenerator at state y. In the
y
Combustion
Chamber
3x
2
1
4
Fuel
Regenerator
TurbineCompressor
4
3
x
2
y
1

ideal cycle, the temperature of the compressed air leaving the regenerator is equal to
the temperature of the burnt gases leaving the turbine, i.e. Tx = T4. But in practice, the
temperature of the compressed air leaving the regenerator is less than Tx.
The effectiveness of the regenerator is given by the ratio of the increase in enthalpy of
the working fluid flowing through the regenerator to the maximum available enthalpy
difference.
The effectiveness of regenerator is given by
2x
2x
R
hh
hh
−
−
= ′
η
Assuming constant specific heat
2x
2x
R
TT
TT
−
−
= ′
η
The thermal efficiency of an ideal gas turbine cycle with regenerator is
x3
1y
x3
1y
H
L
th
TT
TT
1
hh
hh
1
Q
Q
1
−
−
−=
−
−
−=−=η
Since for ideal condition
Tx = T4 and Ty = T2
Therefore,
⎥
⎦
⎤
⎢
⎣
⎡
−
⎥
⎦
⎤
⎢
⎣
⎡
−
−=
−
−
=
3
4
3
1
2
1
43
12
th
T
T
1T
1
T
T
T
1
TT
TT
η
Since
γ
γ
γ
γ
γ
γ 1
2
1
1
3
4
3
4
1
1
2
1
2
p
p
p
p
T
T
and
p
p
T
T
−−−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
∴
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
−
−
γ
γ
γ
γ
η
1
1
2
1
1
2
3
1
th
p
p
1
1
1
p
p
T
T
1

γ
γ 1
1
2
3
1
p
p
T
T
1
−
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
So the thermal efficiency of an ideal regenerative gas turbine cycle depends not only on
the pressure ratio but also on the ratio of two extreme temperatures. In this case, lower
the pressure ratio, higher the efficiency, the maximum value being
3
13
T
TT −
when rP = 1.
This is the Carnot cycle efficiency based upon maximum and minimum temperatures of
the cycle.
Intercooling and Reheating
The addition of regenerator improves the ideal efficiency but does not improve the work
ratio. The latter may be reduced by reducing the compressor work or increasing the
turbine work.
Consider the compressor work first.
The curvature of the constant
pressure lines on T-s diagram is such
that the vertical distance between
them reduces as we go towards the
left (shown by the arrow). Therefore,
further to the left the compression
process 1-2 takes place, smaller is
the work required to drive the
compressor. State 1 is determined by
the atmospheric pressure and temperature. But if the compression is carried out in two
stages, 1-3 and 4-5 with the air is being cooled at constant intermediate pressure pi
between the stages; some reduction of compression work can be achieved. The sum of
temperature rises (T3 – T1) and (T5 – T4) will be clearly less than (T2 – T1). Ideally, it is
possible to cool the air to atmospheric condition i.e. T4 = T1, and in this case
Intercooling is complete.
pi
2
3
1
5
4
T
s

With isentropic compression and
complete Intercooling, the
compressor work is given by
( ) ( )45P13P TTCTTCW −−−−=
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
⎪
⎭
⎪
⎬
⎫
⎪
⎩
⎪
⎨
⎧
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−=
−
−
1
p
p
TC
1
p
p
TC
1
i
2
1P
1
1
i
1P
γ
γ
γ
γ
The saving in work will depend on
the choice of intermediate
Intercooling pressure pi. By equating
idp
dW to zero the condition for minimum work is
given by
21i ppp =
Hence, Pi
i
2
1
i
r
p
p
p
p
==
and P
1
2
Pi r
p
p
r ==
Thus for minimum compressor work,
the compression ratio and work inputs
for two stages are equal. The
compression work can further be
reduced by increasing the number of
stages and intercoolers. But the
additional complexity and cost make
more than three stages uneconomic.
Similarly, it can be shown that the
work output from the turbine is
increased by multi-stage expansion
HP CompressorIntercooler
4
3
1
5
Fuel
LP Compressor
HP Turbine
LP Turbine
Reheater
9
8
6 10
pi6
8
7
9
10
T
s

with reheating. The figure in the left illustrates the relevant part of the cycle showing
expansion in two stages with reheating to the metallurgical limit i.e. T9 = T6.
The turbine work is increased from W6-7 to W6-8 + W9-10 which is given by
Wnet = CP(T6 – T8)+CP(T9 – T10)
It is possible to show that with isentropic expansion, the optimum intermediate pressure,
this time for maximum work, is given by
PPi76i rrorppp ==
Reheating can also be extended to more than two stages, although this is seldom done
in practice and with open cycle plant a limit is set by the oxygen available for
combustion.
Although intercooler and reheaters improve the work ratio, these devices by themselves
can lead to decrease in ideal efficiency. This is because the heat supplied is increased
as well as net work output. The full advantage is only reaped if a regenerator is also
included in the plant. The additional heat required for the colder air leaving the
compressor can be obtained from the hot exhaust gases, and there is a gain in ideal
cycle efficiency as well as work ratio.
BRAYTON CYCLE WITH INTERCOOLING, REHEATING AND
REGENERATION
Regenerator
10
HPIntercoole
4
3
1
5
Fuel
LP
HP
LP Turbine
Reheater
8
7
6 9


The thermal efficiency is given by
( ) ( )
( ) ( )7856
32110
H
L
th
hhhh
hhhh
1
Q
Q
1
−+−
−+−
−=−=η
OTTO CYCLE
The air standard
Otto Cycle is an
idealised cycle which closely approximates the operation of Spark Ignition (SI) Engine.
Process 1-2 is reversible adiabatic (isentropic) compression of air when the piston
moves from crank-end dead centre (BDC) to cylinder head dead centre (TDC). During
T
s
7
5
9
86
4
3
2
1
10
P
v
3
2 4
1
Reversible adiabatic
v = constant
4
3
2
T
s
1

the process 2-3, heat is transferred reversibly to the system at constant volume (this
process corresponds to the spark ignition of the actual engine). The curve 3-4
represents the reversible adiabatic expansion process when piston moves from TDC to
BDC and the curve 4-1 represents the heat rejected by the system reversibly at
constant volume.
The thermal efficiency
added
rejectedadded
th
Q
QQ −
=η
added
rejected
Q
Q
1−=
23
14
uu
uu
1
−
−
−=
( )
( )23v
14v
TTC
TTC
1
−
−
−=
23
14
TT
TT
1
−
−
−=

Chapter 4 Ideal And Real Gases And
Thermodynamic Relations
GAS MIXTURES
Pure substance is defined as a substance which means that it is homogeneous and
unchanging in chemical composition.
COMPOSITION OF A GAS MIXTURE:
MASS AND MOLE FRACTIONS
To determine the properties of a mixture, we need to know the composition of the mixture as
well as the properties of the individual components. There are two ways to describe the
composition of a mixture: either by specifying the number of moles of each component, called
molar analysis, or by specifying the mass of each component, called gravimetric analysis.

Consider a gas mixture composed of k components. The mass of the mixture mm is the sum of
the masses of the individual components, and the mole number of the mixture Nm is the sum of
the mole numbers of the individual components.
...............(1)
The ratio of the mass of a component to the mass of the mixture is called the mass fraction mf,
and the ratio of the mole number of a component to the mole number of the mixture is called the
mole fraction
.................(2)
Dividing (1) by mm or (2) by Nm, we can easily show that the sum of the mass fractions or mole
fractions for a mixture is equal to 1
................(3)
The mass of a substance can be expressed in terms of the mole number N and molar mass M
of the substance as m NM. Then the apparent (oraverage) molar mass and the gas constant
of a mixture can be expressed as
...........(4)
The molar mass of a mixture can also be expressed as
...........(5)
Mass and mole fractions of a mixture are related by
...........(6)

P-v-T BEHAVIOR OF GAS MIXTURES:
IDEAL AND REAL GASES
An ideal gas is defined as a gas whose molecules are spaced far apart so that the
behavior of a molecule is not influenced by the presence of other molecules—a situation
encountered at low densities. We also mentioned that real gases approximate this behavior
closely when they are at a low pressure or high temperature relative to their critical-point values.
The P-v-T behavior of an ideal gas is expressed by the simple relation Pv RT, which is called
the ideal-gas equation of state. The P-v-T behavior of real gases is expressed by more complex
equations of state or by Pv ZRT, where Z is the compressibility factor.
When two or more ideal gases are mixed, the behavior of a molecule normally is not
influenced by the presence of other similar or dissimilar molecules, and therefore a nonreacting
mixture of ideal gases also behaves as an ideal gas. Air, for example, is conveniently treated as
an ideal gas in the range where nitrogen and oxygen behave as ideal gases. When a gas
mixture consists of real (nonideal) gases, however, the prediction of the P-v-T behavior of the
mixture becomes rather involved.
The prediction of the P-v-T behavior of gas mixtures is usually based on two models:
Dalton’s law of additive pressures and Amagat’s law of additive volumes. Both models are
described and discussed below.
Dalton’s law of additive pressures:
Dalton's law (also called Dalton's law of partial pressures) states that the total
pressure exerted by a gaseous mixture is equal to the sum of the partial pressures of
each individual component in a gas mixture. This empirical law was observed by John
Dalton in 1801 and is related to the ideal gas laws.
Mathematically, the pressure of a mixture of gases can be defined as the
summation
or
where represent the partial pressure of each component.

It is assumed that the gases do not react with each other.
where the mole fraction of the i-th component in the total mixture of m
components .
The relationship below provides a way to determine the volume based
concentration of any individual gaseous component.
where: is the concentration of the ith component expressed in ppm.
Dalton's law is not exactly followed by real gases. Those deviations are
considerably large at high pressures. In such conditions, the volume occupied by the
molecules can become significant compared to the free space between them.
Moreover, the short average distances between molecules raises the intensity of
intermolecular forces between gas molecules enough to substantially change the
pressure exerted by them. Neither of those effects are considered by the ideal gas
model.
Amagat’s law of additive volumes:
Amagat's law or the Law of Partial Volumes of 1880 describes the behaviour
and properties of mixtures of ideal (as well as some cases of non-ideal) gases. Of use
in chemistry and thermodynamics, Amagat's law states that the volume Vm of a gas
mixture is equal to the sum of volumes Vi of the K component gases, if the temperature
T and the pressure p remain the same: [1]
.

This is the experimental expression of volume as an extensive quantity. It is
named after Emile Amagat.
Both Amagat's and Dalton's Law predict the properties of gas mixtures. Their
predictions are the same for ideal gases. However, for real (non-ideal) gases, the
results differ.[2]
Dalton's Law of Partial Pressures assumes that the gases in the mixture
are non-interacting (with each other) and each gas independently applies its own
pressure, the sum of which is the total pressure. Amagat's Law assumes that the
volumes of each component gas (same temperature and pressure) are additive; the
interactions of the different gases are the same as the average interactions of the
components.

MIXING OF IDEAL GASES

Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving point
particles that interact only through elastic collisions. The ideal gas concept is useful
because it obeys the ideal gas law, a simplified equation of state, and is amenable to
analysis under statistical mechanics.
At normal ambient conditions such as standard temperature and pressure, most
real gases behave qualitatively like an ideal gas. Generally, deviation from an ideal gas
tends to decrease with higher temperature and lower density, as the work performed by
intermolecular forces becomes less significant compared with the particles' kinetic
energy, and the size of the molecules becomes less significant compared to the empty
space between them.
The ideal gas model tends to fail at lower temperatures or higher pressures,
when intermolecular forces and molecular size become important. At some point of low
temperature and high pressure, real gases undergo a phase transition, such as to a
liquid or a solid. The model of an ideal gas, however, does not describe or allow phase
transitions. These must be modeled by more complex equations of state.
The ideal gas model has been explored in both the Newtonian dynamics (as in
"kinetic theory") and in quantum mechanics (as a "gas in a box"). The Ideal Gas model
has also been used to model the behavior of electrons in a metal (in the Drude model
and the free electron model), and it is one of the most important models in statistical
mechanics.
Classical thermodynamic ideal gas
The thermodynamic properties of an ideal gas can be described by two equations : The
equation of state of a classical ideal gas is given by the ideal gas law.

The internal energy at constant volume of an ideal gas is given by:
where:
• is a constant dependent on temperature (e.g. equal to 3/2 for a monoatomic
gas for moderate temperatures)
• U is the internal energy
• P is the pressure
• V is the volume
• n is the chemical amount of the gas
• R is the gas constant (8.314 J·K−1
mol-1
in SI units)
• T is the absolute temperature
• N is the number of gas particles
• k is the Boltzmann constant (1.381×10−23
J·K−1
in SI units)
The probability distribution of particles by velocity or energy is given by the Boltzmann
distribution.
The ideal gas law is an extension of experimentally discovered gas laws. Real fluids at
low density and high temperature approximate the behavior of a classical ideal gas.
However, at lower temperatures or a higher density, a real fluid deviates strongly from
the behavior of an ideal gas, particularly as it condenses from a gas into a liquid or
solid. The deviation is expressed as a compressibility factor.
Heat capacity
The heat capacity at constant volume of an ideal gas is:

It is seen that the constant is just the dimensionless heat capacity at constant
volume. It is equal to half the number of degrees of freedom per particle. For moderate
temperatures, the constant for a monoatomic gas is while for a diatomic gas
it is . It is seen that macroscopic measurements on heat capacity provide
information on the microscopic structure of the molecules.
The heat capacity at constant pressure of an ideal gas is:
where H = U + pV is the enthalpy of the gas. It is seen that is also a constant and that
the dimensionless heat capacities are related by:
Entropy
Using the results of thermodynamics only, we can go a long way in determining the
expression for the entropy of an ideal gas. This is an important step since, according to
the theory of thermodynamic potentials, of which the internal energy U is one, if we can
express the entropy as a function of U and the volume V, then we will have a complete
statement of the thermodynamic behavior of the ideal gas. We will be able to derive
both the ideal gas law and the expression for internal energy from it.
Since the entropy is an exact differential, using the chain rule, the change in entropy
when going from a reference state 0 to some other state with entropy S may be written
as ΔS where:

where the reference variables may be functions of the number of particles N. Using the
definition of the heat capacity at constant volume for the first differential and the
appropriate Maxwell relation for the second we have:
Expressing CV in terms of as developed in the above section, differentiating the ideal
gas equation of state, and integrating yields:
where all constants have been incorporated into the logarithm as f(N) which is some
function of the particle number N having the same dimensions as in order that the
argument of the logarithm be dimensionless. We now impose the constraint that the
entropy be extensive. This will mean that when the extensive parameters (V and N) are
multiplied by a constant, the entropy will be multiplied by the same constant.
Mathematically:
From this we find an equation for the function f(N)
Differentiating this with respect to a, setting a equal to unity, and then solving the
differential equation yields f(N):
where φ is some constant with the dimensions of . Substituting into the
equation for the change in entropy:

This is about as far as we can go using thermodynamics alone. Note that the above
equation is flawed — as the temperature approaches zero, the entropy approaches
negative infinity, in contradiction to the third law of thermodynamics. In the above "ideal"
development, there is a critical point, not at absolute zero, at which the argument of the
logarithm becomes unity, and the entropy becomes zero. This is unphysical. The above
equation is a good approximation only when the argument of the logarithm is much
larger than unity — the concept of an ideal gas breaks down at low values of V/N.
Nevertheless, there will be a "best" value of the constant in the sense that the predicted
entropy is as close as possible to the actual entropy, given the flawed assumption of
ideality. It remained for quantum mechanics to introduce a reasonable value for the
value of φ which yields the Sackur-Tetrode equation for the entropy of an ideal gas. It
too suffers from a divergent entropy at absolute zero, but is a good approximation to an
ideal gas over a large range of densities.

Equation Table for an Ideal Gas
Real gas
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Real gas, as opposed to a Perfect or Ideal Gas, effects refers to an assumption base
where the following are taken into account:
• Compressibility effects
• Variable heat capacity
• Van der Waals forces
• Non-equilibrium thermodynamic effects
• Issues with molecular dissociation and elementary reactions with variable
composition.

For most applications, such a detailed analysis is "over-kill" and the ideal gas
approximation is used. Real-gas models have to be used near condensation point of
gases, near critical point, at very high pressures, and in several other less usual cases.
Van der Waals modelisation
Real gases are often modeled by taking into account their molar weight and molar
volume
Where P is the pressure, T is the temperature, R the ideal gas constant, and Vm
the molar volume. a and b are parameters that are determined empirically for
each gas, but are sometimes estimated from their critical temperature (Tc) and
critical pressure (Pc) using these relations:
Redlich–Kwong modelisation
The Redlich–Kwong equation is another two-parameters equation that is used to
modelize real gases. It is almost always more accurate than the Van der Waals
equation, and often more accurate than some equation with more than two parameters.
The equation is

where a and b two empirical parameters that are not the same parameters as in
the Van der Waals equation.
Berthelot and modified Berthelot modelisation
The Berthelot Equation is very rarely used,
but the modified version is somewhat more accurate
Dieterici modelisation
This modelisation fell out of usage in recent years
Clausius modelisation
The Clausius equation is a very simple three-parameter equation used to model gases.
where

Virial Modelisation
The Virial equation derives from a perturbative treatment of statistical mechanics.
or alternatively
where A, B, C, A′, B′, and C′ are temperature dependent constants.
Peng-Robinson Modelisation
This two parameter equation has the interesting property being useful in modeling some
liquids as well as real gases.
Wohl modelisation
The Wohl equation is formulated in terms of critial values, making it useful when real
gas constants are not available.

where
Beatte-Bridgeman Modelisation
The Beattie-Bridgeman equation
where d is the molal density and a, b, c, A, and B are empirical parameters.
Benedict-Webb-Rubin Modelisation
The BWR equation, sometimes referred to as the BWRS equation
Where d is the molal density and where a, b, c, A, B, C, α, and γ are empirical
constants.
THERMODYNAMICS RELATIONS

Fundamental Equation of Thermodynamics
•
Enthalpy
•
Helmholtz free energy
•
Gibbs free energy
•
Maxwell relations
Incremental processes
•
•

•
•
Compressibility at constant temperature
More relations
Other useful identities

Proof #1
An example using the above methods is:
;
Proof #2
Another example:

Clausius–Clapeyron relation
The Clausius–Clapeyron relation, named after Rudolf Clausius and Émile
Clapeyron, who defined it sometime after 1834, is a way of characterizing the phase
transition between two phases of matter, such as solid and liquid. On a pressure–
temperature (P–T) diagram, the line separating the two phases is known as the
coexistence curve. The Clausius–Clapeyron relation gives the slope of this curve.
Mathematically,
where dP / dT is the slope of the coexistence curve, L is the latent heat, T is the
temperature, and ΔV is the volume change of the phase transition.
Disambiguation
The generalized equation given in the opening of this article is sometimes called
the Clapeyron equation, while a less general form is sometimes called the Clausius–
Clapeyron equation. The less general form neglects the magnitude of the specific
volume of the liquid (or solid) state relative to that of the gas state and also
approximates the specific volume of the gas state via the ideal gas law.
Pressure Temperature Relations

A typical phase diagram. The dotted line gives the anomalous behavior of water. The
Clausius–Clapeyron relation can be used to (numerically) find the relationships between
pressure and temperature for the phase change boundaries. Entropy and volume changes (due
to phase change) are orthogonal to the plane of this drawing
Derivation
Using the state postulate, take the specific entropy, s, for a homogeneous
substance to be a function of specific volume, v, and temperature, T.
During a phase change, the temperature is constant, so
Using the appropriate Maxwell relation gives

Since temperature and pressure are constant during a phase change, the
derivative of pressure with respect to temperature is not a function of the specific
volume. Thus the partial derivative may be changed into a total derivative and be
factored out when taking an integral from one phase to another,
Δ is used as an operator to represent the change in the variable that follows it—final (2)
minus initial (1)
For a closed system undergoing an internally reversible process, the first law is
Using the definition of specific enthalpy, h, and the fact that the temperature and
pressure are constant, we have
After substitution of this result into the derivative of the pressure, one finds
where the shift to capital letters indicates a shift to extensive variables.
This last equation is called the Clausius–Clapeyron equation, though some
thermodynamics texts just call it the Clapeyron equation, possibly to distinguish it
from the approximation below.
When the transition is to a gas phase, the final specific volume can be many times the
size of the initial specific volume. A natural approximation would be to replace Δv with

v2. Furthermore, at low pressures, the gas phase may be approximated by the ideal gas
law, so that v2 = vgas = RT / P, where R is the mass specific gas constant (forcing h and
v to be mass specific). Thus,
This leads to a version of the Clausius–Clapeyron equation that is simpler to integrate:
C is a constant of integration.
These last equations are useful because they relate saturation pressure and
saturation temperature to the enthalpy of phase change, without requiring specific
volume data. Note that in this last equation, the subscripts 1 and 2 correspond to
different locations on the pressure versus temperature phase lines. In earlier equations,
they corresponded to different specific volumes and entropies at the same saturation
pressure and temperature.
Other derivation
Suppose two phases, I and II, are in contact and at equilibrium with each other.
Then the chemical potentials are related by μI = μII. Along the coexistence curve, we
also have dμI = dμII. We now use the Gibbs–Duhem relation dμ = − sdT + vdP, where s
and v are, respectively, the entropy and volume per particle, to obtain

Hence, rearranging, we have
From the relation between heat and change of entropy in a reversible process δQ =
T dS, we have that the quantity of heat added in the transformation is
Combining the last two equations we obtain the standard relation.
Applications
Chemistry and chemical engineering
The Clausius–Clapeyron equation for the liquid–vapor boundary may be used in either
of two equivalent forms.
where
• T1 and P1 are a corresponding temperature (in kelvin or other absolute temperature
units) and vapor pressure
• T2 and P2 are the corresponding temperature and pressure at another point
• ΔHvap is the molar enthalpy of vaporization
• R is the gas constant (8.314 J mol−1
K−1
)
This can be used to predict the temperature at a certain pressure, given the
temperature at another pressure, or vice versa. Alternatively, if the corresponding
temperature and pressure is known at two points, the enthalpy of vaporization can be
determined.

The equivalent formulation, in which the values associated with one P,T point are
combined into a constant (the constant of integration as above), is
For instance, if the p,T values are known for a series of data points along the phase
boundary, then the enthalpy of vaporization may be determined from a plot of lnP
against 1 / T.
Notes:
• As in the derivation above, the enthalpy of vaporization is assumed to be constant over
the pressure/temperature range considered
• Equivalent expressions for the solid–vapor boundary are found by replacing the molar
enthalpy of vaporization by the molar enthalpy of sublimation, ΔHsub
Clausius–Clapeyron equations is given for typical atmospheric conditions as
where:
• es is saturation water vapor pressure,
• T is a temperature,
• Lv is latent heat of evaporation,
• Rv is water vapor gas constant.
Example
One of the uses of this equation is to determine if a phase transition will occur in
a given situation. Consider the question of how much pressure is needed to melt ice at
a temperature ΔT below 0°C. Note that water is unusual in that its change in volume
upon melting is negative. We can assume

and substituting in
L = 3.34 × 105
J/kg (latent heat of water),
T = 273 K (absolute temperature), and
ΔV = −9.05 × 10−5
m³/kg (change in volume from solid to liquid),
we obtain
= −13.1 MPa/K.
To provide a rough example of how much pressure this is, to melt ice at −7 °C
(the temperature many ice skating rinks are set at) would require balancing a small car
(mass = 1000 kg) on a thimble (area = 1 cm²).
The Joule–Thomson (Kelvin) coefficient
The rate of change of temperature T with respect to pressure P in a Joule–
Thomson process (that is, at constant enthalpy H) is the Joule–Thomson (Kelvin)
coefficient μJT. This coefficient can be expressed in terms of the gas's volume V, its heat
capacity at constant pressure Cp, and its coefficient of thermal expansion α as
See the Appendix for the proof of this relation. The value of μJT is typically
expressed in °C/bar (SI units: K/Pa) and depends on the type of gas and on the
temperature and pressure of the gas before expansion.

All real gases have an inversion point at which the value of μJT changes sign.
The temperature of this point, the Joule–Thomson inversion temperature, depends on
the pressure of the gas before expansion.
In a gas expansion the pressure decreases, so the sign of is always negative.
With that in mind, the following table explains when the Joule–Thomson effect cools or
warms a real gas:
If the gas temperature is then μJT is since is thus must be so the gas
below the inversion temperature positive always negative negative cools
above the inversion temperature negative always negative positive warms
Helium and hydrogen are two gases whose Joule–Thomson inversion
temperatures at a pressure of one atmosphere are very low (e.g., about 51 K (−222 °C)
for helium). Thus, helium and hydrogen warm up when expanded at constant enthalpy
at typical room temperatures. On the other hand nitrogen and oxygen, the two most
abundant gases in air, have inversion temperatures of 621 K (348 °C) and 764 K (491
°C) respectively: these gases can be cooled from room temperature by the Joule–
Thomson effect.
For an ideal gas, μJT is always equal to zero: ideal gases neither warm nor cool
upon being expanded at constant enthalpy.

Applications
In practice, the Joule–Thomson effect is achieved by allowing the gas to expand
through a throttling device (usually a valve) which must be very well insulated to prevent
any heat transfer to or from the gas. No external work is extracted from the gas during
the expansion (the gas must not be expanded through a turbine, for example).
The effect is applied in the Linde technique as a standard process in the
petrochemical industry, where the cooling effect is used to liquefy gases, and also in
many cryogenic applications (e.g. for the production of liquid oxygen, nitrogen, and
argon). Only when the Joule–Thomson coefficient for the given gas at the given
temperature is greater than zero can the gas be liquefied at that temperature by the
Linde cycle. In other words, a gas must be below its inversion temperature to be
liquefied by the Linde cycle. For this reason, simple Linde cycle liquefiers cannot
normally be used to liquefy helium, hydrogen, or neon.
Proof that enthalpy remains constant in a Joule–Thomson process
In a Joule–Thomson process the enthalpy remains constant. To prove this, the
first step is to compute the net work done by the gas that moves through the plug.
Suppose that the gas has a volume of V1 in the region at pressure P1 (region 1) and a
volume of V2 when it appears in the region at pressure P2 (region 2). Then the work
done on the gas by the rest of the gas in region 1 is P1 V1. In region 2 the amount of
work done by the gas is P2 V2. So, the total work done by the gas is
The change in internal energy plus the work done by the gas is, by the first law of
thermodynamics, the total amount of heat absorbed by the gas (here it is assumed that
there is no change in kinetic energy). In the Joule–Thomson process the gas is kept
insulated, so no heat is absorbed. This means that

where E1 and E2 denote the internal energy of the gas in regions 1 and 2, respectively.
The above equation then implies that:
where H1 and H2 denote the enthalpy of the gas in regions 1 and 2, respectively.
Derivation of the Joule–Thomson (Kelvin) coefficent
A derivation of the formula
for the Joule–Thomson (Kelvin) coefficient.
The partial derivative of T with respect to P at constant H can be computed by
expressing the differential of the enthalpy dH in terms of dT and dP, and equating the
resulting expression to zero and solving for the ratio of dT and dP.
It follows from the fundamental thermodynamic relation that the differential of the
enthalpy is given by:
(here, S is the entropy of the gas).
Expressing dS in terms of dT and dP gives:
Using
(see Specific heat capacity), we can write:

The remaining partial derivative of S can be expressed in terms of the coefficient
of thermal expansion via a Maxwell relation as follows. From the fundamental
thermodynamic relation, it follows that the differential of the Gibbs energy is given by:
The symmetry of partial derivatives of G with respect to T and P implies that:
where α is the coefficient of thermal expansion. Using this relation, the differential
of H can be expressed as
Equating dH to zero and solving for dT/dP then gives:

Chapter 5 Psychrometry
psychrometry is a term used to describe the field of engineering concerned with
the determination of physical and thermodynamic properties of gas-vapor mixtures. The
term derives from the Greek psuchron meaning "cold" and metron meaning "means of
measurement"
Common applications
The principles of psychrometry apply to any physical system consisting of gas-
vapor mixtures. The most common system of interest, however, are mixtures of water
vapor and air because of its application in heating, ventilating, and air-conditioning and
meteorology.
Psychrometric ratio
The psychrometric ratio is the ratio of the heat transfer coefficient to the
product of mass transfer coefficient and humid heat at a wetted surface. It may be
evaluated with the following equation
where:
• r = Psychrometric ratio, dimensionless
• hc = convective heat transfer coefficient, W m-2
K-1
• ky = convective mass transfer coefficient, kg m-2
s-1
• cs = humid heat, J kg-1
K-1
Humid heat is the constant-pressure specific heat of moist air, per unit mass of dry air.

The psychrometric ratio is an important property in the area of psychrometrics as
it relates the absolute humidity and saturation humidity to the difference between the dry
bulb temperature and the adiabatic saturation temperature.
Mixtures of air and water vapor are the most common systems encountered in
psychrometry. The psychrometric ratio of air-water vapor mixtures is approximately
unity which implies that the difference between the adiabatic saturation temperature and
wet bulb temperature of air-water vapor mixtures is small. This property of air-water
vapor systems simplifies drying and cooling calculations often performed using
psychrometic relationships.
Psychrometric chart
A simple psychrometric chart.

A psychrometric chart is a graph of the physical properties of moist air at a constant
pressure (often equated to an elevation relative to sea level). The chart graphically
expresses how various properties relate to each other, and is thus a graphical equation
of state. The thermophysical properties found on most psychrometric charts are:
• Dry-bulb temperature (DBT) is that of an air sample, as determined by an
ordinary thermometer, the thermometer's bulb being dry. It is typically the x-axis,
the horizontal axis, of the graph. The SI units for temperature are Kelvin; other
units are Fahrenheit.
• Wet-bulb temperature (WBT) is that of an air sample after it has passed
through a constant-pressure, ideal, adiabatic saturation process, that is, after the
air has passed over a large surface of liquid water in an insulated channel. In
practice, this is the reading of a thermometer whose sensing bulb is covered with
a wet sock evaporating into a rapid stream of the sample air. The WBT is the
same as the DBT when the air sample is saturated with water. The slope of the
line of constant WBT reflects the heat of vaporization of the water required to
saturate the air of a given relative humidity.
• Dew point temperature (DPT) is that temperature at which a moist air sample at
the same pressure would reach water vapor saturation. At this saturation point,
water vapor would begin to condense into liquid water fog or (if below freezing)
solid hoarfrost, as heat is removed. The dewpoint temperature is measured
easily and provides useful information, but is normally not considered an
independent property. It duplicates information available via other humidity
properties and the saturation curve.
• Relative humidity (RH) is the ratio of the mole fraction of water vapor to the
mole fraction of saturated moist air at the same temperature and pressure. RH is
dimensionless, and is usually expressed as a percentage. Lines of constant RH
reflect the physics of air and water: they are determined via experimental
measurement. Note: the notion that air "holds" moisture, or that moisture
dissolves in dry air and saturates the solution at some proportion, is an
erroneous (albeit widespread) concept (see relative humidity for further details).

• Humidity ratio (also known as moisture content, mixing ratio, or specific
humidity) is the proportion of mass of water vapor per unit mass of dry air at the
given conditions (DBT, WBT, DPT, RH, etc.). It is typically the y-axis, the vertical
axis, of the graph. For a given DBT there will be a particular humidity ratio for
which the air sample is at 100% relative humidity: the relationship reflects the
physics of water and air and must be measured. Humidity ratio is dimensionless,
but is sometimes expressed as grams of water per kilogram of dry air or grains of
water per pound of air.
• Specific enthalpy symbolized by h, also called heat content per unit mass, is the
sum of the internal (heat) energy of the moist air in question, including the heat of
the air and water vapor within. In the approximation of ideal gases, lines of
constant enthalpy are parallel to lines of constant WBT. Enthalpy is given in (SI)
joules per kilogram of air or BTU per pound of dry air.
• Specific volume, also called inverse density, is the volume per unit mass of the
air sample. The SI units are cubic meters per kilogram of dry air; other units are
cubic feet per pound of dry air.
The versatility of the psychrometric chart lies in the fact that by knowing three
independent properties of some moist air (one of which is the pressure), the other
properties can be determined. Changes in state, such as when two air streams mix, can
be modeled easily and somewhat graphically using the correct psychrometric chart for
the location's air pressure or elevation relative to sea level. For locations at or below
2000 ft (600 m), a common assumption is to use the sea level psychrometric chart.
The relationship between DBT, WBT, and RH is given by the Mollier diagram
(pressure-enthalpy) for water in air, developed by Richard Mollier. Willis Carrier,
considered the 'father' of modern air-conditioning, rearranged the Mollier diagram for
moist air (its T-s chart) to allow such graphical solutions. Many variations and
improvements to the psychrometric charts have occurred since, and most charts do not
show the specific entropy (s) like the Mollier diagram. ASHRAE now publishes what are
considered the modern, standard psychrometric charts, in both I-P and SI units, for a
variety of elevations or air pressures.

How to read the chart
The most common chart used by practitioners and students alike is the "ω-t"
(omega-t) chart in which the dry bulb temperature (DBT) appears horizontally as the
abscissa and the humidity ratios (ω) appear as the ordinates.
In order to use a particular chart, for a given air pressure or elevation, at least
two of the six independent properties must be known (DBT, WBT, RH, humidity ratio,
specific enthalpy, and specific volume). This gives rise to possible
combinations.
DBT: This can be determined from the abscissa on the x-axis, the horizontal axis
DPT: Follow the horizontal line from the point where the line from the horizontal axis
arrives at 100% RH, also known as the saturation curve.
WBT: Line inclined to the horizontal and intersects saturation curve at DBT point.
RH: Hyperbolic lines drawn asymptotically with respect to the saturation curve which
corresponds to 100% RH.
Humidity ratio: Marked on the y-axis.
Specific enthalpy: lines of equal values, or hash marks for, slope from the upper left to
the lower right.
Specific volume: Equally spaced parallel family of lines.
Dry-bulb temperature
Common thermometers measure what is known as the dry-bulb temperature.
Electronic temperature measurement, via thermocouples, thermistors, and resistance

temperature devices (RTDs), for example, have been widely used too since they
became available.
Wet-bulb temperature
The thermodynamic wet-bulb temperature is a thermodynamic property of a
mixture of air and water vapor. The value indicated by a simple wet-bulb thermometer
often provides an adequate approximation of the thermodynamic wet-bulb temperature.
A wet-bulb thermometer is an instrument which may be used to infer the amount
of moisture in the air. If a moist cloth wick is placed over a thermometer bulb the
evaporation of moisture from the wick will lower the thermometer reading (temperature).
If the air surrounding a wet-bulb thermometer is dry, evaporation from the moist wick will
be more rapid than if the air is moist. When the air is saturated no water will evaporate
from the wick and the temperature of the wet-bulb thermometer will be the same as the
reading on the dry-bulb thermometer. However, if the air is not saturated water will
evaporate from the wick causing the temperature reading to be lower.
The accuracy of a simple wet-bulb thermometer depends on how fast air passes
over the bulb and how well the thermometer is shielded from the radiant temperature of
its surroundings. Speeds up to 5,000 ft/min (60 mph) are best but dangerous to move a
thermometer at that speed. Errors up to 15% can occur if the air movement is too slow
or if there is too much radiant heat present (sunlight, for example).
A wet bulb temperature taken with air moving at about 1-2 m/s is referred to as a
screen temperature, whereas a temperature taken with air moving about 3.5 m/s or
more is referred to as sling temperature.
A psychrometer is a device that includes both a dry-bulb and a wet-bulb
thermometer. A sling psychrometer requires manual operation to create the airflow over
the bulbs, but a powered psychrometer includes a fan for this function.

Specific humidity
Specific humidity is the ratio of water vapor to air (including water vapor and dry air) in a
particular mass. Specific humidity ratio is expressed as a ratio of kilograms of water
vapor, mw, per kilogram of air (including water vapor), mt .
That ratio can be shown as:
Specific humidity is related to mixing ratio (and vice versa) by:
Relative humidity
Relative humidity is a term used to describe the amount of water vapor that exists in a
gaseous mixture of air and water.
Definition
The relative humidity of an air-water mixture is defined as the ratio of the partial
pressure of water vapor in the mixture to the saturated vapor pressure of water at a
prescribed temperature. Relative humidity is normally expressed as a percentage and is
defined in the following manner[1]
:
where:

is the relative humidity of the mixture being considered;
is the partial pressure of water vapor in the mixture; and
is the saturated vapor pressure of water at the temperature of the mixture.
The international symbols U[2][3]
and Uw,[4]
expressed in per cent, are gaining
recognition.
Measurement
A hygrometer is a device that is used to measure humidity.
The relative humidity of an air-water vapor mixture can be estimated if both the
measurement temperature (Tm) and the dew point temperature (Td) of the mixture are
known. This approach to measuring humidity requires an equation expressing the
saturation vapor pressure as a function of temperature. There are many such
equations[5]
, ranging from the approximate Antoine equation, to the more detailed Goff-
Gratch equation. A reasonable compromise proposed by many is the Magnus-Teton
formulation:
where the saturated vapor pressure of water at any temperature T is estimated by:
Where T is expressed in degrees Celsius, various researchers have published fitted
values for the α,β and λ parameters, some of which are listed by Vömel[5]
.
A slightly more complex variant of this equation was proposed by Buck.
Whichever equation is chosen for , the Relative humidity is calculated as:

In practice both Tm and Td are readily estimated by using a sling psychrometer
and the relative humidity of the atmosphere can be calculated.
The Fred Cirrani Factor: this factor is known for knowing all that seems to be
relavent in the theory of relative humidity and air conditioning versus heat.
Adiabatic Saturation
Consider an unsaturated mixture entering a chamber. Suppose water was
sprayed into the stream, so that the humidity increases and it leaves as a saturated
mixture. This is accompanied by a loss of temperature due to heat being removed from
the air which is used for vaporization. If the water supplied is at the temperature of exit
of the stream, then there is no heat transfer from the water to the mixture. The final
temperature of the mixture is called adiabatic saturation temperature.
Sensible Heating Process
Sensible Heating Process is a process during which the dry-bulb temperature of air is

increased. The process occurs at constant moisture content. The air passes over a hot
and dry surface which might be pipe coil using steam or hot water, electrical resistance
or an air-to-air heat recovery unit. The load on the heater is:
Sensible Cooling Process
Sensible cooling process at constant moisture content is a process during which
the dry-bulb temperature of air is decreased. The air passes over a cooling coil which
uses chilled water or direct expansion of some refrigerant into the pipe coil. The load on
the cooling coil is
Sensible Cooling with Dehumidification

In this process the dry-bulb temperature and the moisture content of air are
decreased. The heat in the condensed water is normally very small relative to the total
cooling load therefore the load on the cooling coil can be approximated as:
Dehumidification can only occur, if the coil surface temperature is blew the dew-point
temperature of the entering air stream.
Apparatus dew-point temperature (ADP) is defined as the average coil
temperature at air condition C. Where C is the intersection of the saturation line and the
straight line through conditions A and B on psychrometric chart.

ME6301 ENGINEERING THERMODYNAMICS - LECTURE NOTES

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