3

1
3
FUNDAMENTAL LAWS
AND
BALANCE
EQUATIONS
FOR
MASS,
ENERGY, ENTROPY,
EXERGY
,
AND
MOMENTUM
Equation Section 3
The purpose of studying thermodynamics is
to predict the behavior
of
systems
in terms of their
states
as they respond to
interactions
with their
surroundings
. Classical thermodynamics is an axiomatic
science; that is, the behaviors of systems can be predicted by
deduction from a few basic
axioms
or laws, which are assumed to be
always true. A law is an abstraction of myriads of observations
summarized into concise statements that are self

evident and
certainly without any contradiction.
We have already come across
the Zeroth Law of t
hermodynamics, which introduced temperature, a
thermodynamic property
, as an arbiter of thermal equilibrium
between two objects.
In this chapter we will introduce the conservation of mass and
momentum principle, the First Law and Second Law of
thermodynami
cs
and the concept of exergy
. A uniform framework in
terms of balance equations will be developed. Each fundamental
principle will be translated into a balance equation of a particular
property. Just as equations of state are the starting point for a state
evaluation, analysis of en
gineering systems and processes
in the
future chapters will
begin with
the balance equations
. While the
balance equations are derived in this chapter, their applications to
closed and open systems are delegated to Chapter 4 and 5
respectively.
To gain a comprehensive insight into these equations
Chapters 3, 4 and
5, therefore,
should be iteratively studied.
Chapter
3

2
3
Fundamental Laws and the Mass, Energy, Entropy, Exergy and
Mom
entum Balance Equations
................................
.......................
3

1
3.1
Balance Equation
................................
............................
3

3
3.2
Reynold
s Transport Equation (RTE)
.............................
3

5
3.3
Classification of Systems
................................
................
3

7
3.3.1
Open vs. Closed Systems
................................
........
3

7
3.3.2
Steady vs. Unsteady Systems
................................
..
3

7
3.3.3
Instantaneous Rates vs. Process
..............................
3

8
3.3.4
System Tree
................................
...........................
3

10
3.4
M
ass Equation
................................
...............................
3

10
3.4.1
Forms of Mass Balance Equation
.........................
3

11
3.5
Energy Equation
................................
............................
3

12
3.5.1
Forms of Energy Balance Equation
......................
3

14
3.6
Entropy Balance Equation
................................
............
3

16
3.6.1
Forms of Entropy Balance Equation
.....................
3

19
3.7
Exergy Balance Equation
................................
..............
3

20
3.7.1
Forms of Exergy Balance Equation
......................
3

26
3.8
Momentum Balance Equation
................................
.......
3

28
3.9
Balance Equations Summary
................................
........
3

30
3.9.1
General Form
................................
........................
3

30
3.9.2
Closed Systems
................................
.....................
3

31
3.9.3
Closed Process
................................
......................
3

31
3.9.4
Closed Steady
................................
........................
3

32
3.9.5
Open Steady
................................
..........................
3

32
3.9.6
Open Process
................................
.........................
3

33
3.10
Summary
................................
................................
.......
3

43
3.11
Index
................................
................................
..............
3

43
3

3
3.1
Balance Equation
Each fundamental law that will be introduced in this chapter will be
shown to be associated with a certain
global
extensive p
roperty

mass
, energy
, entropy
, or momentum
,
or

of the system. To develop a unified
framework, we will represent
these extensive properties with the
generic
symbol
and the
corresponding specific property with
. For a uniform system
(Fig.
3.1)
, as stated in Eq. (
2.94
),
. For a non

uniform system,
however,
has to be summed or integrated over the ensemble of
local systems, each represented by a differential element
as shown
in
Fig. 3.
2
.
For a local system with a volume
, Eq. (1.12) has to be
written in a differential form.
(
3
.
1
)
Integrating
,
(
3
.
2
)
The integration is carried out over the entire syst
em
, open or closed
,
at a given instant
.
In Fig. 3.2, notice
how the boundary is carefully
drawn
to pass through the
inlet and exit
ports at right angles so that
two unique
uniform
surface states, State

i and State

e, can describe
the inlet and exit conditi
ons. Moreover, situated inside the ports
slightly away from the main body of the system
, these states
are
more likely to be uniform than if they were chosen exactly at the
openings. Assuming uniformity across the inlet and exit surfaces, t
he
flow rates of
the property
at the ports can be obtained from Eq.
2.94.
(
3
.
3
)
where,
(
3
.
4
)
Beside mass transfer, a property can b
e transferred across the
boundary through other interactions
–
energy, for instance, is carried
with heat and work.
As
will be stated shortly, entropy can not only
be transferred, but also generated spontaneously within a system.
Therefore,
can be expected to change with time, i.e.,
.
To use the image analogy introduced in section
1.3.3
, the snapshot of
the system taken with the
state camera
at time
provides us with the
distr
ibution of
throughout the system at that
thermodynamic
instant. The global property
, therefore, can be obtained by
Fig. 3.1. A u
niform
system does not have to
be closed.
Fig. 3.2. The local
system used inside the
integral of Eq.
(
3
.
2
)
.
3

4
simply
analyzing the picture. Similarly, a snapshot taken after an
interval
can be used to evaluate
, the global property
at time
. On the other hand, if the
causes
for
a
change in
are accounted for, the
change

can be deduced
entirely from a different angle. The equality between the two
expressed on a rate basis constitutes the
balance equation
.
EXAMPLE 3

1
Total Property
for a Non

Uniform System.
The temperature of air trapped in a vertical rigid tank of diameter 1
m and height 1 m increases linearly from 300 K at the bottom to
4
00
K at the top. Determine the total mass of the stratified air
if the
pressure inside can be a
ssumed uniform at 100 kPa
.
Use the perfect
gas model.
SOLUTION
The
global
properties of a
non

uniform system is to be
determined by treating it as an aggregate of uniform local systems.
Assumptions
A differential slice of air of thickness
(Fig. 3.3)
constitutes a local system
in LTE
.
Analysis
From Table C

1, obtain the necessary material properties
of air:
= 29 kg/kmol,
=1.005 kJ/kg
K. The gas cons
tant and
are calculated
(see Section 2.5.3.1.2)
as
=8.314/29=0.287
kJ/kg
K and
= 1.005

0.287 = 0.718 kJ/kg
K.
The variabl
e
temperature can be expressed as
with
K and
K/m.
Using the ideal gas equation,
,
Eq.
(
3
.
2
)
can be
simplified
as
follows.
(
3
.
5
)
For
the evaluation of mass of the system,
; t
herefore,
.
Substituting this and the linear
temperature
relation, Eq.
(
3
.
5
)
can be
integrated.
TEST Solution
TEST can be used only for uniform systems or
binary non

uniform systems, which are made of two uniform
Fig. 3.3. A slice of air
acts as a uniform local
system.
3

5
subsystems. Therefore, this problem, which involve
s
an infinite
number of local systems, can
not be solved
using TEST.
Discussion
The evaluation of other properties such as energy is more
complicated since
as a function of
will complicate the integrand
of Eq.
(
3
.
5
)
. For more complex systems, where variation can be in
all three directions and are not known in functional terms, integration
of Eq.
(
3
.
5
)
may be impossible. Fortunately, the
global
properties of
non

uniform systems are seldom necessary to evaluate. Examples of
property evaluation for uniform systems
, which are more common,
can be found in Ex. 2.19 and 2.20.
3.2
Reynold
s T
ransport Equation
(RTE
)
The fundamental laws are usually described
with
closed systems
in
mind
. For instance, Newton’s Second Law
which states
that the net
external force on a particle equals its rate of chan
ge of momentum,
implicitly assumes the particle, the system in our case, to be closed.
Similarly, the conservation of mass principle, and the First and
Second Law are also easier to state as applied to closed systems.
Each of these laws expresses the rate
of change of a particular
extensive property with respect to time in terms of other variables. In
other words, a generic format for these laws can be written as
(
3
.
6
)
T
he
superscript
reminds us that
this equation cannot be applied to
open system as is. The
right hand side
(RHS)
is prescribed by the
specific l
aw
s to be introduced shortly.
With the help of
RTE
the
fundamental laws, which are known in the closed system format of
Eq.
(
3
.
6
)
,
are
expanded
in
to
balance equat
ions applicable to
any kind
of system, open or closed
.
We begin
the development of
the
RTE
by considering
a very
general
open
system
at time
and
as sketched in Fig. 2.
4
.
The
minor restriction of a sin
gle inlet and exit will be lifted as the last
step of this derivation.
The system, defined by the dotted black
boundary, is allowed to have
all
possible interactions
–
mass, heat
and work
–
with its surroundings. As shown in the sketch, even the
shape of t
he system is allowed to change. As
the
working substance
passes through the system, we identify a
closed system
marked by
the red boundary
at time
, which occupies the entire open system
plus
a little
region
I
near the inlet
. The
closed system
becomes
deformed as it flows through the open system
. After a small period
,
it
still occup
ies
the entire o
pen system; however, the region

I
completely disappears
and a new region, region

III,
not necessarily
equal
in size to region

I,
appears
near the exit.
This is not a
coincidence since for any given
, the
region

I
is carefully chosen
3

6
so that the entire fluid inside that region flows into the system during
that interval. Of course,
has to be sufficiently small so as not
to
allow the closed system to
loose its identity through disintegration
,
and
regions
I and II
can be considered uniform so that
, and
(
3
.
7
)
Because
is an extensive property, an inventory of
for a system
can be obtained by combining contributions from different sub

regions comprising the system. Referring to Fig.
3.3
, the change in
for the closed system as it passes through the open system
(regi
on
II)
can be written as
No special superscript is necessary for the open system because it is
the system by default.
Rearranging and substituting Eq.
(
3
.
7
)
Dividing both side by
and taking a limit
The LHS and the first term on the RHS of this equation are clearly
derivatives of the extensive property
with respect to time for the
closed and open system respectively.
Also, a
s
,
,
and the last two terms approach
and
, where the
sup
erscript
is not necessary anymore since each term in this
instantaneous expression refers to time
. The above equation, thus,
reduces to
Generalizing for multiple inlets an
d exits, the
Reynolds
Transport
Equation
(RTE) or the
general balance equation
can be written as
(
3
.
8
)
It relates the rate of change of an extensive property
of an open
system at a given instant to that of a closed system
which happens to
pass
through with the bounda
ries
of the two systems
aligning on top
of each other at that particular instant.
Fig. 3.1. A very general
system at two
neighboring
macroscopic instants.
3

7
3.3
Classification of Systems
In practical applications, thermodynamic systems or their behavior
are restricted in certain ways. Therefore the
general template of the
balance equation, Eq.
(
3
.
8
)
, can be simplified when applied to
specific systems. For instance, if a system is closed, the mass
transfer terms on th
e RHS drops out. In this section we will discuss,
in general terms, patterns that repeat across
the entire spectrum of
thermodynamic devices and processes. Recognizing these patterns
will help us simplify a system, classify its behavior and reduce the
gove
rning set of balance equations into custom forms. This
systematic approach will be cultivated throughout this book in favor
of the hit

and

miss approach of matching balance equations to
specific systems that gives thermodynamics a bad name among the
uninit
iated.
3.3.1
Open vs. Closed Systems
Classification of any system begins with the question, “Is there any
mass transfer across the boun
dary?” If there is no mass transfer, the
system is called closed. Otherwise, by
default, is considered open.
Obviously a system can only be open or closed, there is no other
alternative. It should be stressed here that heat or work transfer has
nothing to do with whether a system is open or closed.
For a closed system,
the mass transf
er terms drop out of Eq.
(
3
.
8
)
.
(
3
.
9
)
The
open system equation, thus, reduces to the
fundamental laws
from which they are derived. The
usefulness of such an obvious
equation will become clear when we introduce the individual balance
equations
.
3.3.2
Steady vs. Unsteady Systems
A system, by default, is
unsteady
; that is, its global state can change
with time. When the global state of a system remains frozen in time,
it is said to be
in
steady state
. In terms of our image analogy, the
snapshot of a steady system does not change whether or not the
system interacts with its surroundings. Hot and pressurized steam
flowing into a steam turbine exits at a much lower pre
ssure and
temperature. Shaft work, flow work and even heat transfer
from the
turbine
may occur. Yet, the turbine is most likely to operate in a
steady state.
At steady state
all
global properties
, the total property
included,
must
remain constant
since t
he global image does not
change
. Therefore, the
time derivative of the
LHS of Eq.
(
3
.
8
)
Fig. 3.5. System
classifi
cation: Open vs.
Closed systems.
Fig. 3.6. System
classification: Steady vs.
Unsteady systems.
3

8
summarily drops out making the general balance equation an
algebraic one.
(
3
.
10
)
Obviously this simplification is applicable to both open and closed
syst
ems giving rise to four types of systems already.
A closed system
passing through a steady open system need not be steady. If you
follow a control mass of steam as a closed system entering the
turbine, it will surely undergo changes. That is why the last t
erm in
Eq.
(
3
.
10
)
, which tracks the changes in the closed system flowing
through, cannot be
set to zero.
In the classification process, the second question to ask is,
“Does
the
image
of the system taken with a state camera change with
time?” Although the answer is a simple yes or no, sometimes it
depend
s
on the resolution or precision with which one answers the
question. Inside a turbine (take a virtual tour of turbine in th
e TEST
web site) the rotors spins at a very high RPM. Therefore,
instantaneous snapshots at two different times cannot be identical.
However, if the
thermodynamic
instant
(see Section 1.3.2)
is
stretched by increasing the camera exposure to a few milliseco
nds,
the pictures at two different times will be almost identical as all the
fluctuations would average out in those few milliseconds. In a
similar way, a car engine can be considered steady, as long as the
time resolution is large enough for the piston to
execute several
cycles of strokes. On the other hand if we are interested in a single
stroke of the piston, the picture obviously changes and the system
must be considered
unsteady
.
3.3.3
Unsteady
Process
The time derivative of
is non

zero for an unsteady system.
T
he
LHS of the
balance equations cannot be simplified any further if
instantaneous rate of change of
is important. For example, if we
are
interested
in
the rate of change of temperature of a cup of coffee
at a specific instant
as it cools down
, we have a
n
instantaneous,
unsteady, closed
problem. The general balance equations, by default,
apply to
instantaneous
, unsteady, open systems.
Often,
in
unsteady systems, the change of system properties
over a finite interval is of greater interest than
an
instantaneous
rate
of change
. For instance, in the compression stroke of an automobile
engine cycle, we are interested in the state of the
gas mixtu
re
at the
beginning and end of the stroke rather than
at
any intermediate state.
Similarly, in the charging of a propane tank, another unsteady
phenomenon, the instantaneous rates
maybe of less significance than
the overall changes during the entire proces
s
. The balance equations
Fig. 3.7. As water flows
through the constriction,
its pressure changes.
However the open
system is a steady one if
the global picture does
not change.
Fig. 3.8. System
classification: Process
vs. Instantaneous rate.
3

9
under
such situations can be simplified by integrating with respect to
time.
An unsteady system is said to execute a
process
if
it
undergoes changes from
a
beginning
global state, called the
b

state
or begin

stat
e
,
to
a final global state, called the
f

state
or final

state
.
The begin and finish states are also known as the
anchor states
of a
process.
The anchor states must be in equilibrium for a proces
s;
however, as the system moves from the b

state to f

state it does not
have to pass through a succession of equilibrium for the balance
equations to be simplified. For system which is uniform at the
beginning and end of the process,
the anchor states can
be spotted on
the familiar
diagram as sketched for
a
compression
process
in
Fig.
3
.
9
.
Note that without a thorough knowledge of the process, we
cannot select a path between the anchor states.
To identify if an unsteady system i
s undergoing a process, the
appropriate question to
ask
is, “
Does the unsteady system move from
a
clear begin
ning
to a
clear finish state?” If the answer is yes, we
have a process.
The simplification f
or a process
is achieved by multiplying
Eq.
(
3
.
8
)
with
and integrating
from the b

state to the f

state.
For
an
op
en unsteady system
, the inlet and exit states are often
assumed to remain uniform across the cross

section and invariant
with time. The assumption, known as the
uniform state and
uniform flow
assumption; can considerably sim
plify the above
equation as
and
, being independent of time, can be pulled out
of the integrals. The general balance equation
for an
open process
reduces to
(
3
.
11
)
Fig. 3.9. In this closed
process, a gas is
compressed from a b

State
to a f

State.
Fig. 3.10. Inflating a tire
is an open process.
3

10
The equation still looks quite formidable with an integral of a
derivative as one of it
s term. However, when we discuss specific
balance equations, say, mass or energy equation, this term will
be
shown to simplify much farther.
3.3.4
System Tree
The classification
of systems introduced until now
can be
organized
in a tree structure as shown in Fig.
3.11
, called the
system tree
. The
next two chapters will be devot
e
d
exclusively
to the discussion of
closed and open systems respectively. Further classification of
closed process
and
open steady
syste
ms will be deferred until then.
In TEST start at the daemons page, by using the Daemons
link on the Task Bar, to classify a system. A simplification table
provides links to all possible branches one can follow depending on
the answer to the question posed
at the table header.
At any stage of
simplification,
a system schematic and the customized set of
balance
equations
appear below the simplification table
. Once you gain
expertise in this step

by

step procedure, you can use the Map,
arranged like the tree
of Fig.
3.11
and linked from the Task

Bar
in
TEST
, to jump to a specific category of systems by clicking on its
node.
We now begin the development of fundamental laws into
balance equations and customize these equations for different classes
of systems.
3.4
Mass Equation
The
conservation of mass principle
can be stated through the
following
simple postulate
.
Mass cannot be created or destroyed.
For a closed system
the total mass
must remain constant
;
therefore
,
the time derivative of
must be zero, i.e.,
(
3
.
12
)
Substitute Eq.
(
3
.
12
)
into the RTE, Eq.
(
3
.
8
)
, with
and
,
to formulate
the
mass balance equation
for an open unsteady
system.
(
3
.
13
)
Fig. 3.11 The system
classification tree. The
Map in TEST displays a
similar clickable tree.
Fig. 3.12 Fl
ow diagram
for the mass balance
equation..
3

11
The meaning of the three terms is explained with the help of a
flow
diagram
in Fig. 3.12. The difference between the inflow and
outflow is accumulated in th
e balloon. Similar
flow
diagrams will be
constructed for other balance equations.
3.4.1
Forms of Mass Balance Equation
The general form of the mass balance equation
can be simplified for
different categories of systems cla
ssified in Fig. 3.11.
Closed System Simplification
For a closed system the mass transfer
terms drop out. For both steady and unsteady closed systems,
therefore,
or,
(
3
.
14
)
This is almost a trivial result; therefore,
a constant mass can be
implicitly assumed
for a closed system
without having to
refer to this
equation.
Open Steady Simplification
As explained in section
3
.3.2, at steady
state
t
he total mass
,
like all other
global properties,
remains constant.
; or,
(
3
.
15
)
This form of mass conservation is often referred as “what goes in
comes out”. If there is a single flow, i.e., o
nly one inlet and one exit,
the equation can be further simplified using Eq.
(
3
.
4
)
.
; or,
, or
(
3
.
16
)
Open Process Simplification
For a process involving an open
system Eq.
(
3
.
13
)
can be integrated or, alternatively, Eq.
(
3
.
11
)
can
be used to produce
(
3
.
17
)
This form is further simplified if there is only a single inlet or a
single exit as in
the case of charging a propane tank or a whistling
pressure cooker. Discussion of such specific cases, however, is
postponed until
Chapter 5
.
Fig. 3.13 Flow diagram
for the mass balance
equation, open steady
system.
Fig. 3.14 Flow diagram
for the mass balance
equation, open process.
3

12
3.5
Energy Equation
The
conservation of energy principle
also known as the
First Law
of thermodynamics can be stated through the following postulates.
i)
The
internal
energy
of a system is a thermodynamic property.
ii)
Energy
cannot be created or destroyed, only
tr
an
sferred through heat or work. On a rate basis this can be
expressed as
(
3
.
18
)
where,
is the net rate of heat transfer into the system and
is
the net rate of work or power transfer out of the system.
Substituting
,
and
for
,
and
respectively
in the RTE
and using the
second postulate
(
3
.
19
)
where,
and
, evaluated based on the open system boundary, are
substituted for
and
respectively since the boundaries of the
closed and open systems
become
coincident as
.
The energy
flow rates at the inlet and exit can be also be expressed through the
symbol
, which is used in the flow diagram of Fig. 3.16.
Equation
(
3
.
19
)
is now completely decoupled from the original
closed system and will be labele
d the
conservative form
of the
energy equation.
Different modes of heat and work transfer, shown in the flow
diagram of Fig. 3.16, will be quantitatively discussed in the next
chapter. As explained in Section 1.2.2.2, the transfe
r of heat through
the ports can be neglected compared to the transfer through the rest
of the boundary. The same, however, is not true about work transfer
through the system ports, called the
flow work
. As explained in
Section 1.2.2.4 different types of wo
rk transfer can be classified into
two major categories, flow and external work, to distinguish open
and closed systems.
Fig. 3.16 Flow diagram
explaining various
modes of heat and work
transfer.
Fig. 3.15 Flow diagram
for the conser
vative
form of the energy
balance equation, open
unsteady system.
3

13
(
3
.
20
)
For a closed system
and there is no distinction between
and
.
To evaluate the flow work
,
consider the small fluid e
lement
of length
in the simplified system of Fig.3.17 that is pushed out
of the system by the pressure force from the left against the pressure
from the right. The pressure force
does a work of
(see
Section 1.2.2.3) in
. According to the sign convention, the
exit work must be positive since work is done by the system. In a
similar manner, as a fluid element is pushed into the system against
the resistance of the inlet pressu
re, a negative work transfer with a
magnitude of
takes place in time
at the inlet. As
,
the net flow work rate or flow power can be written with the help of
Eq.
(
3
.
16
)
as
(
3
.
21
)
A port
with a very small
area
still can have very large
and,
thus
,
transfer
a relatively
significant
amount of
flow work.
Equation
(
3
.
21
)
can be g
eneraliz
ed for multiple inlets and
exits.
(
3
.
22
)
Each term on the RHS
resembles
flow rate of properties discussed in
Section
2.8
. The flow work too, therefore, can be regarded as a
flow
property.
Substituting the above expression for flow work after
s
eparating
it from all other work terms, the conservative form of the
energy equation,
Eq.
(
3
.
19
)
, can be rewritten as
(
3
.
23
)
In this modified form the mass flow can be seen to carry a
combination property consisting of energy
and a term that
Fig. 3.18 The flow of
flow energy
is
equivalent to the flow of
energy
and the
transfer of flow work
acros
s a control
surface.
Fig. 3.17 A fluid
element at the exit being
expel
led by the system
against an external
pressure.
3

14
represents the flow work performed per unit mass of
the
flow. We
call this combination property the
speci
fic flow energy
and represent
it with the symbol
in the absence of any universally accepted
symbol for this important convenience property.
(
3
.
24
)
Substituting the symbol
for the
specific flow energy, we obtain
the
balance equation for energy
in its most general form.
(
3
.
25
)
The
energy
carried by the flow
in the conservative form
,
Eq.
(
3
.
19
)
, is replaced in this equation by the
flow energy
carried by
the flow
,
. The advantage of this form is that only the
readily
recognizable
external work appears in this equation and the
hidden
work of flow can be
completely ignored since it is already accounted
for
in
the use of the property
. It
may seem that
this form of energy
equation is meant only for open systems. To the contrary, if we
substitute
and
into
Eq
.
(
3
.
25
)
,
the
second postulate of the First Law
is immediately recovered making
Eq.
(
3
.
25
)
the most general form
from which all other forms sho
uld
be derived
. The meaning of various terms in this equation is
explained through the flow diagram of Fig. 3.18.
3.5.1
Forms of Energy Balance Equation
As we did with the mass balance equation, the energy equation
is
customized for the
particular classes of systems introduced in
the
system tree of Fig. 3.11.
Closed System Simplification
For a
closed system
the mass transfer
terms drop out and
as there is no
possibility of any
flow
work. The
energy balance equation, Eq.
(
3
.
25
)
, reduces to the second
postulate of the First Law.
(
3
.
26
)
Obviously, this forms suits any
instantaneous unsteady closed
system
.
There is no need for the superscript
c
anymore because we
Fig. 3.19 By using
specific flow energy
instead of specific
energy
, the
cumbersome flow work
can be forgotten.
Fig. 3.20 For a closed
system there is no flow
work; therefore,
.
3

15
are deriving a restr
icted form from a more general form applicable to
both open and closed systems.
Closed Process Simplification
For an
unsteady closed system
going
through a process, Eq.
(
3
.
26
)
can be integrated from the b

state to the
f

state as outlined in section
3.3.3
producing
(
3
.
27
)
This is an algebraic equation that relates two anchor states through
two process variables
and
.
Closed Steady Simplification
For a steady system,
open or closed,
the ti
me derivative of any global property must be zero
. The energy
equation, thus,
simplifies to
(
3
.
28
)
The net rate of heat transfer to a
steady closed system
must be
exactly equal to the net rate of work delivered by the system.
Open Steady Simplification
The time derivative of all global
properties of the system must be zero
at steady state as the global
picture remains frozen at steady state. The energy equation
simplifies to what is commonly called the
steady flow energy
equation
(SFEE).
(
3
.
29
)
By rearranging the equation, it can be shown that t
he sum total of the
rate of flow of
flow energy
and heat into a
s
teady open system
must
be equal to the rate at which energy leaves the system through flow
energy and external work
.
Like the steady state mass balance
equation, it expresses
what goes in, comes out
in terms of energy.
Open Process Simplification
For a pr
ocess involving an open
system Eq.
(
3
.
26
)
can be integrated from the begin to the finish state
as outlined in section
3.3.3 for a generic property
. Using the uniform
flow u
niform state assumption, the energy equation reduces to
(
3
.
30
)
Fig. 3.21 Energy flow
diagram for a closed
process.
Fig
. 3.22 Energy flow
diagram for a closed
steady system.
Fig. 3.23 Energy flow
diagram for an open
steady system.
Fig. 3.24 Energy flow
diagram for an open
process.
3

16
The mass
transfers in such a process has already been examined in
section
3.4.1
.
3.6
Entropy Balance Equation
The
Second Law
of thermodynamics
can be stated
through
the
following postulates.
i)
Entropy
is an extensive property
that measures the
degree of
disorder in a system
.
The specific entropy
is a thermodynamic
property.
ii)
Entropy can be transferred
across a boundary
through heat but
not through work. Th
e rate of entropy transfer
by
crossing a
boundary at a temperature
is given as
.
iii)
Entropy cannot be destroyed. It can be generated by natural
processe
s
,
i.e.,
.
iv)
An isolated system achieves thermodynamic equilibrium when the
entropy of the system reaches a maxima.
Let
us go
over these statements one at a time.
From our
experience of chaos, we would tend to agree with the first postula
te
that entropy
, being a measure of total amount of chaos or disorder in
a system,
is an extensive property; that is, doubling the size of a
uniform system will double its entropy.
Heat transfer to a system can be expected to increase the
molecular disor
der and
,
hence
,
entropy. If a
uniform
system is at a
high temperature and,
therefore
, pretty chaotic to start with, addition
of heat cannot be expected to add as much entropy to the system as
would be the case for a cooler, less chaotic system. This
provid
es
justification as to
why the boundary temperature,
which is same as
the
system temperature
for a local system,
occurs in the denominator
of
the entropy transfer term in
postulate

II. Observe that transfer of
work does not seem to affect entropy of a syst
em. Work involves
organized motion such as the rotation of a shaft, motion of a
boundary,
and, in the case of electricity,
directed movement of
electrons, etc. The chaotic motion of the system, therefore, remains
unaffected by the transfer of organized mot
ion.
The third postulate states that every system has a natural
tendency towards generating entropy. Because entropy cannot be
destroyed, the generated entropy is a permanent signature of the
process. When heat
radiates from the Sun to earth
, the coffee
in
the
stirred cup
gradually
comes to rest, electrons flow
across a voltage
difference, a drop of ink dissipates in a bucket of water,
rubbing
one
hand against another
make
them w
a
rm, na
tural gas burns in air
forming hot flames
, a volcano erupts
–
there is o
ne thing that is
Fig. 3.25 CARTOON
Are you saying that the
Second Law left those
footprints?
3

17
common in all these
apparently unrelated phenomena
;
they all tend
to destroy a gradient of some
kind while generating entropy as
dictated by postulate

II
.
In the next chapter we will devote an entire
section going after these sources of sp
ontaneous entropy generation.
For the time being,
we will refer to all these gradient destroying
natural phenomena as
generalized friction
.
Generalized friction leave an
indelible
footprint in
the form
of
entropy generation.
Any process involving generalized friction,
therefore, cannot be completely reversed and are called
irreversible
,
the degree of
irreversibility
being proportional
to
the
entropy
generation. Generalized friction
due to system surroundings
interactions sometimes extends beyond the
system into the
immediate surroundings. Depending on the location where the
entropy
is generated with respect to the system boundary, the
associate
d irreversibilities are called
internal
if within the system
and
external
if outside
or at the boundary
.
For instance, entropy is
generated inside and in the immediate surroundings of a turbine
operating in a steady state. The
system
’s universe
enclosed by the
outer boun
dary of Fig. 3.26
includes both the internal and external
generation of entropy.
In the limiting situation of no entropy
generated in the system’s universe as a result of a particular process,
the system can be completely restored back to its original stat
e
without leaving any clue that the original process ever took place.
The system or process is said to be
reversible
under that ideal
situation.
T
he concept of
entropy generation will
be linked in the
next chapter
with
the design of
more
efficient engines, refrigerators
and various
other thermal
devices.
The third postulate (not to be confused with the Third Law of
thermodynamics
to be
introduced
in Chapter

8)
has tremendous
implications in predicting equilibrium,
which
will be discussed
in
more details in Chapter
8 and 10
. For the time being, consider two
closed insulated systems, initially at two different temperature
s
,
brought in
diathermal
contact by removing insulations from two
walls and pressing the two blocks against each other on
their un

insulated faces. The entropy of the combined system will start
to
increase
as entropy is generated due to heat transfer from the hotter
block to the colder one. We know from our experience that at
equilibrium temperatures of the two blocks will b
ecome equal, at
which point entropy will cease to increase any further, all the
temperature gradient having been completely destroyed. Thus
entropy has been maximized as the isolated system, consisting of the
two blocks, comes to equilibrium. As a matter o
f fact, we will show
in Chapter

8
, that starting from the second law, the equality of
temperature at equilibrium can be predicted. Although this
may seem
like a
trivial exercise, the same principle will help us deduce in
Fig. 3.26 The
interactions between the
system and its
surroundings causes
entropy generation
inside and in the
immediate surroundings
of a system.
Fig. 3.26 Entropy is
generated in the shaded
area which extends
beyond the system
boundary.
3

18
Chapter
–
10
, the emissions from com
bustion
, something far from
trivial
.
Getting back to our task of translating the fundamental laws
into balance equations, the second postulate can be written as.
(
3
.
31
)
where,
is the rate of entropy generation within the closed
system boundary and
is the rate of heat t
ransfer into the closed
system of Fig.
3.4
. Substituting
and
for
and
respectively
in the RTE, we obtain the
general
entropy balance equation
.
(
3
.
32
)
As
mentioned before
, th
e boundary of the closed system passing
through the open system of Fig. 3.
4
is almost identical to that of the
open system as
goes to zero. Therefore,
and
.
The comments und
er each term are keyed to the open system of Fig.
3.
4
as th
is general entropy equation
completely stands on its own
without any reference to the closed system to which it owes its
origin.
The flow diagram of Fig. 3.27 also explains the
various
terms
of the
entropy equation. An arrow with dots inside is used to signify
the generation of entropy.
For most systems on earth, the heat interaction takes place
with the surrounding atmosphere. If the system boundary is carefully
drawn to pass through the surroundi
ng air, atmospheric temperature
can be used for
. Obviously the precise location of the boundary
does not affect
or
, which are flow rates of energy; however,
being a cumulat
ive quantity,
depends entirely on the selection
of boundary. The total rate of entropy generation in the turbine of
Fig. 3.26, for instance, can be expressed as the sum of the entropy
generation inside the system and in the imme
diate surroundings
external to the system.
(
3
.
33
)
where the subscript
univ
is used to signify the
system’s universe
.
Fig. 3.27 Entropy is
accumu
lated
due to
generation and transfer
through mass and
energy.
Fig. 3.28
includes all sources of
entropy generation
inside and outside the
system.
3

19
If
a system exchanges heat with
different segments of the
surroundings at different temperatures as shown in Fig. 3.28, the
boundary of the extended system can be made to pass through
segments each
at a
uniform temperature
. The entropy balance
equation for the system
’s universe
modifies as follows
(
3
.
34
)
The total entropy
, the mass flow rates
or
and the heat
transfer rate
are
assumed not affected
significantly
by
extending
the system to
include the thin layer of immediate surroundings. The
entropy generation, however, can be huge outside the system, even
in a very thin layer. This will be discussed with examples i
n the next
chapter
.
3.6.1
Forms of Entropy Balance Equation
As we did with the mass and energy balance equations, we will
customize the entropy equation in a similar manner for different
classes of systems.
Although the
following equations are written for
a system with a fixed boundary
temperature
, they can be modified
for an extended system by replacing
with
and
with
.
Closed System Simplification
For a
closed system
the mass transfer
terms drop out and Eq.
(
3
.
34
)
reduces to
(
3
.
35
)
Obviously, this form suits any
instantaneous unsteady closed system
.
Closed Process Simpli
fication
For an
unsteady closed system
going
through a process, Eq.
(
3
.
35
)
can be integrated from the b

state to the
f

state as outlined in section 3.3.3 producing
(
3
.
36
)
This is an algebraic equation that relates two anchor states through
two
process variables
and
. Obviously
since
Fig. 3.29 Flow diagram
of entropy for an
extended system with
surroundings at two
different temperatures.
Fig. 3.30 Entropy flow
diagram for a closed,
instantaneously
u
nsteady system.
Fig. 3.31 Entropy flow
diagram for a closed,
process.
3

20
Closed Steady Simplification
For a steady system, the time
derivative of any global property must be ze
ro. Eq.
(
3
.
35
)
simplifies
to
(
3
.
37
)
A number of Second Law statements can be deduced from this
equation in the next two chapter.
Open Steady Simplification
With the time derivative of
disappearing at steady state, the entropy equation, Eq
.
(
3
.
32
)
,
simplifies to a form similar to the steady flow energy equation.
(
3
.
38
)
Note that unlike the mass or energy equation, the entropy equation
cannot be rearranged in the what

goes

in

must

come

out format.
Because of entropy generation,
what comes out is often more that
what goes in.
Open Process Simplification
For a process involving an open
system, Eq.
(
3
.
32
)
can be integrated from the begin to the fin
ish state
as outlined in section 3.3.3. Using the
uniform flow uniform state
assumption, the entropy equation reduces to
(
3
.
39
)
The mass transfers in such a process has already been examined in
section
3.4.1
.
3.7
Exergy Balance Equation
S
tagnant air and wind both have energ
y
. Yet it is much easi
er to
extract useful work out of the wind than stagnant air at atmospheric
conditions. One of the major quests
for engineers
at
all
times
has
been deliver
y of
useful work
in the form of shaft or electrical power
out of any source of
available
energy

wind
, ocean waves, river
streams, geo

thermal reserves, solar radiation, fossil fuels, nuclear
materials, etc. With the help of the mass, energy and entropy
balance equation
we are about
to predict the maximum possible
useful work that can be extracted from a
system.
In fact
manipulation of these equations will be shown to lead to a new
Fig. 3.32 The default
direction of heat flow is
inconsistent with the
direction of entropy
transfer.
Fig. 3.33 What comes
out may be more than
what goes in sinc
e
.
Fig. 3.34 Entropy
diagram for an open
process.
Fig. 3.35 CARTOON
–
A ship stuck in an ocean
or a car stuck in a
desert. Energy Energy,
every where but not a
drop of exergy to drink.
3

21
bala
nce equation for a new property called
exergy
, which measures
the useful energy content of a system. The ocean or the atmosphere
may have tremendous amount o
f energy due to their huge mass
alone, but very little exergy; that is why, a ship or an airplane cannot
extract any useful work out of these reservoirs of energy. Exergy,
thus, can be looked upon as a measure of the quality of energy. An
exergy analysis n
ot only helps in comparing two alternative sources
of energy

or should we say exergy
–
but also in rating the
performance of any device that
consumes or delivers
useful work
.
Because exergy is a derived concept, we must develop the balance
equation first
before identifying the different mechanisms of its
storage, transfer and generation, if any. Only then
can we precisely
define
exergy as a property.
Consider the system schematic sketched in Fig.
3.35
. Like
Fig.
3.4
,
it
has only one inlet and one exit, a
restriction that will be
readily removed towards the end of our derivation. The
surroundings, however, is divided into two thermal energy reservoirs
or
TERS
–
a
thermal energy reservoir
is a large system whose
t
emperature is assumed not affected by heat transfer
–
reservoir 0,
which is atmospheric air
at a
standard
temperature
and reservoir
, say, a furnace
at a temperature
. Note th
at a TER is
sometime
called a
heat reservoir
, a
n oxymoron for a form of energy which
can only be transferred
and not stored
.
Obviously, the number of
TERS can be extended just the same way as the number of inlets or
exits
.
For the e
xtended system enclosed by the boundary of Fig.
3.35, t
he energy and entropy equations can be written as
(
3
.
40
)
(
3
.
41
)
By including the immed
iate surroundings, we capture the entire
amount of entropy generation
(see Eq.
(
3
.
33
)
)
due to the
interactions between the system and its surroun
dings; moreover, the
choice of the boundary temperature is simplified as it passes through
the TER’s with constant temperatures
and
.
Note that
variables,
,
,
,
,
,
,
,
, or
are unaffected by
the choice of the ext
ended system. If the mass (or
thermal capacity
)
of the added layer of immediate surroundings can be considered
small compared to the
internal
system,
and
also can be
assumed identical between a system
and its extended version. The
3

22
only difference, therefore, comes from the entropy generation
contributed by the immediate surroundings.
Multiplying
Eq.
(
3
.
41
)
by
,
a
constant, and
subtracting
the
resulting equation from Eq.
(
3
.
40
)
we get
(
3
.
42
)
Before we proceed further, let us take a closer look at the external
work transfer.
Different modes of work transfer ha
ve
been
introduced in a
qualitative manner in Section 1.2.2.4 and through a
flow chart in Fig. 3.16
, which is further modified in Fig. 3.37
. Most
components of the external work transfer, electrical, shaft or power
transfer through a piston into the crank shaft are readily usefu
l. The
only exception is the part of boundary work transfer that involve
s
the
atmosphere. The work done by the piston in pushing the atmospheric
air in Fig. 3.36 cannot be used for any practical purpose. Similarly if
the piston is pushed downward, the atmo
spheric work comes free
and does not cost anything as does any other form of useful work.
The external work term in Eq.
(
3
.
42
)
, therefore, can be separated into
an useful a
nd atmospheric components as shown by the flow
diagram of Fig. 3.37.
(
3
.
43
)
Although evaluation of boundary work is a topic for the next
chapter, we can evaluate the
atmospheric work
transfer due to the
displacement
of the piston in time
as shown
in Fig. 3.38 by
recalling the definition of work (see Section 1.2.2.3) and the sign
convention.
(
3
.
44
)
The
inner
negative sign
accounts for the fact that the atmospheric
force and the displacement of its point of application are in opposite
direction. The
outer
negative sign converts the
work done by the
atmosphere into wor
k done by the system. The expression derived
for a single piston can be shown to remain valid even if the entire
boundary of a system is non

rigid. The boundary, in that case, can be
divided into a
large
number of
discrete
pistons

cylinder
arrangements
and
the contributions
from individual elements when
integrated
can be shown to produce the same result as Eq.
(
3
.
44
)
.
Of
course, the expression derived for the atmospheric wor
k is valid for
both expansion or contraction of the system.
Fig. 3.37 Different
modes of external work
in the energy flow for
the system in Fig
. 3.36.
Fig. 3.38 Evaluation of
atmospheric work
transfer.
3

23
Using Eqs.
(
3
.
44
)
and
(
3
.
43
)
, Eq.
(
3
.
42
)
can be rearranged as
(
3
.
45
)
Before we manipulate this equation
any
further, let us introduce the
concept of the dead state. Work is generally extracted during
gradient destroying natural process
–
a hydraulic power plant needs a
d
ifference in water height, a wind turbine requires a velocity
difference between the wind and the rotor blades, a thermal power
plant requires a temperature difference between the
boiler
and the
atmosphere,
to name a few
. When a system comes to
thermodynam
ic
equilibrium with
quiescent
atmospheric air at sea level, there is no
mechanical, thermal or chemical
driving force left
for it
to
interact
with the surroundings
. In other words, there is
no juice or exergy left
in the system to extract useful work out o
f.
Such a state with
,
,
and
is
said to be at its
dead state
.
Getting back to the
Eq.
(
3
.
45
)
,
if the stream flowing through
the system is brought to equilibrium with the surrounding air at
and
,
the combination property
carried by the stream
reduces to
(
3
.
46
)
It should be emphasized tha
t
at
the dead state
the
working substance
that make up the
system
is at
and
with no
kinetic
or
potential
energies
. However,
other specific properties such as
,
, etc., are
properties of the working substance of the system,
and are
not
necessarily equal to that of the
surrounding air
.
With the help of Eq.
(
3
.
46
)
, the
t
erm B
of
Eq.
(
3
.
45
)
can be
modified
by
using the dead state
as a reference
state
for the
combination
property
.
(
3
.
47
)
The combination property
carried by the mass flows
and
in
this equation is called the
specific flow exergy
and is r
epresented by
Fig. 3.39 CARTOON
You got a dead battery,
Madam!
A system in equilibrium
with the quiescent
ambient atmosphere is
said to be at its dead
state.
3

24
the symbol
.
Defined in terms of intensive property
, and
specific properties
and
,
must be an
extrinsic specific property since
is an extrinsic property. It has a
zero value at the dead state and has the same unit as
,
i.e., kJ/
kg.
The physical meaning of flow exergy will be discussed shortly
and
plenty of examples
will be covered
in Chapter
6
. T
erm H
in the
above equation
can be
further
simplified by employing
Eq.
(
3
.
46
)
and
the mass balance equation, Eq.
(
3
.
13
)
.
Taking the term H into the LHS of Eq.
(
3
.
47
)
,
(
3
.
48
)
The
combin
ation
extensive
system property
that appear
s
inside the
time derivative is now referenced at the dead state, i.e., its value at
the dead state is zero. This is
called
the
exergy
of the system
and has
the same unit as
, i.e., kJ. The total
exergy is
represent
ed by the
symbol
(pronounced phi) and
the corresponding specific property
by
,
and are defined in Eq.
(
3
.
49
)
.
While
is a total extensive
property,
is an extrinsic property because it has extrinsic
components such as ke and pe.
Substituting Eq.
(
3
.
48
)
into Eq.
(
3
.
47
)
, and generalizing the
number of inlets, exits and TER’s,
we obtain the
general exergy
balance e
quation
.
3

25
(
3
.
49
)
Although
the
equation looks
formidable
at first sight, it lends
itself to interpretation just like all other balance equations derived so
far.
The LHS,
as is usual,
represents the time rate of increase of
an
extensive property, the system
exergy
in this case. Like any other
total property of a non

uniform system (see Eq.
(
3
.
2
)
) it can be
obtained by integrating
or summing up
the specific exergy
over
the local systems comprising the global system.
The
RHS
lists all possible ways that affect the stor
ed
exergy
of
a system
. Flow exergy, just like flow energy
, is carried by the
flow in and out of the system. H
eat transferred into the system from
a
TER
at
carries
a
fraction
,
, of itself as exergy.
Note that for reservoir
, i.e., the ambient atmosphere, this
fraction reduces to zero
.
That is, there is no exergy is transferred
through heat transfer between the system and the ambient
atmosphere.
The implication of a cold reservoir
, i.e.,
,
will be
discussed in Section
4
.3.3.
4
.
The exergy delivered by the sys
tem as useful
work
,
,
appears with a negative sign as it drains the system of its stored
exergy
.
Finally,
the entropy generation can be seen to produce a term
that must be always non

positive since
(S
econd Law). It
is called the
rate of exergy destruction
or
the
rate of irreversibility
and
is
represented by the symbol
.
Fig. 3.40 Flow diagram
of exergy f
or an
extended system.
3

26
Each term of this equation
sketched in the flow diagra
m of
Fig. 3.40 will be explained with plenty of examples in the next two
chapters. A comparison of the flow diagrams for energy (Fig. 3.15),
entropy
(Fig. 3.27) and exergy (3.40) can be helpful in understanding
the similarities and differences in the inven
tory of the three
properties in terms of a common framework.
3.7.1
Forms of Exergy Balance Equation
As we did with the mass, energy and entropy balance equations, we
will customize the e
xerg
y equation in a similar mann
er for different
classes of systems.
Closed System Simplification
For a
closed system
the mass transfer
terms drop out and
Eq
.
(
3
.
49
)
reduces to
(
3
.
50
)
Obviously, this form suits any
instantaneous unsteady closed system
.
Closed Process Simpl
ification
For an
unsteady closed system
going
through a process, Eq.
(
3
.
50
)
can be integrated from the b

state
to the
f

state as outlined in S
ection
3.3.3
producing
(
3
.
51
)
The simplified form of the exergy equation for a closed process can
be
used to
explore the physical meaning of
some of its term
s
. For
instance, when a
closed system, say, a
warm cup of coffee
cools
down
from a temperature
to
the
room temperature
by rejecting
amount of heat
,
no useful work is produced. However, the
exergy equation can be used to see if it is possible to construct a
clever device to extract useful work out of this cooling process. With
,
Eq
.
(
3
.
51
)
simplifies
as
(
3
.
52
)
Clearly it is possible to convert some of the exergy in a coffee mug
into useful work. If the final state is the dead state, i.e., the coffee
in
the mug reaches equilibrium with the environment
,
.
Being a
non

negative quantity,
the
irreversibility
can be seen to reduce the
useful
work output
. In
fact for a regular
coffee cup, the
exergy is
Fig. 3.43 Energy flow
diagram for Eq.
(
3
.
54
)
.
The direction of the heat
arrow is reversed since
(
is a
positive quantity).
Fig. 3.41 A smart
coffee mug that
produces electricity as
the coffee cools down to
room temperature.
Fig. 3.42 The exergy of
a warm coffee mug is
the maxi
mum possible
useful work that can be
extracted as the coffee
comes to equilibrium
with the surrounding air.
3

27
completely destroyed by
.
If the irreversibility can b
e eliminated

and the Second Law does permit
as a limiting ideal case

the work produced is maximized
.
(
3
.
53
)
The exergy of a system, therefore, has the simple interpretation of
the maximum possible useful work that can be extracted out of it by
transferring heat with only the atmospheric TER.
One may naturally ask, w
hy
cannot
we
use
an energy
analysis
instead
to predict the maximum work transfer
?
The next
chapter will be devoted to analysis such as this for closed system. As
a preview let us see what the energy and entropy equation predict
abou
t the system at hand. Using the solid/liquid model for the coffee,
the energy equation,
Eq.
(
3
.
27
)
, can be simplified as
(
3
.
54
)
By eliminating
completely it seems that the change in internal
energy can be completel
y converted into work, i.e.,
. The Second Law however has been completely
disregarded in arriving at this conclusion.
In fact, an entropy
equation
for the process
, Eq.
(
3
.
36
)
,
yields
(
3
.
55
)
The first term on the RHS
being negative, an elimination of
would result in a negative
, which is a direct violation of the
Second Law.
Any conclusions from the energy equation, therefore,
must be tested for compliance with the
Second Law.
Conclusions
derived from the exergy balance equation,
on the other hand,
do not
run
into these types of difficulty as the exergy equation is firmly
rooted in the combination of mass, energy and entropy equations.
Closed Steady Simplification
F
or a steady system, the time
derivative
of
, a global property, is set to zero and
Eq.
(
3
.
50
)
simplifies to
(
3
.
56
)
Fig. 3.44 The change in
and
according to
the solid/liquid model as
the tempera
ture goes
from
to
. Mass of
the cup is neglected in
these expressions.
3

28
Open Steady Simplification
The steady state
exergy equation
,
similarly, can be expressed in an algeb
raic form as the time
derivative drops out
.
(
3
.
57
)
The destruction of
exergy term makes it impossible to express this
equation in the what

come
s

in

must

go

out format.
To explore the physical meaning of flow exergy, consider a
steady stream of fluid flowing through a system which has heat
interactions with only the atmosphe
ric reservoir. The power
delivered
by this device can be obtained from Eq.
(
3
.
57
)
as
(
3
.
58
)
The useful work is maximized when the exergy destruction is
eliminated and the flow exits at its dead state.
(
3
.
59
)
The flow exergy, therefore, can be interpreted as the maximum
possible useful work delivered per unit ma
ss of the flow if the flow
is brought to dead state by exchanging he
at with the atmospheric
TER.
Complete analysis of open systems will be carried out in
Chapter 5 at which point this will be a simple exercise to show that a
First Law analysis alone cannot
be used for predicting the maximum
work transfer
since
the Second Law may be violated
.
Open Process Simplification
For a process involving an open
system
Eq.
(
3
.
49
)
can
be integrated from t
he begin to the finish state
.
Using the
uniform flow uniform state
assumption, the exergy
equation reduces to
(
3
.
60
)
where
many
of the
symbols
have been explained in connection with
the corresponding form of the
energy and
entropy equatio
ns
.
3.8
Momentum Balance Equation
The
momentum equation will not be used until chapter
7
, where we
will discuss modern jet engines. However, this is the appropriate
place to cast Newton’s law into our common framework of a balance
equation that applies to all systems, open or closed.
Fig. 3.45 Flow diagram
of exergy simplified for
an open steady system.
3

29
Newton’s
Second Law of Motion
for a closed system can be state
d
as
The rate of change of momentum of a closed system
is
equal
to
the
net external force applied on the system.
Because momentum and force are vectors, the momentum equa
tion
can be split into three independent equations along
,
, and
direction
s
in the Cartesian coordinates. Along the
direction,
Newton’s Second Law
can be written as
(
3
.
61
)
Observe that in this equation the unit of forc
e is kN to be consistent
with all other balance equations and the unit of pressure, a deviation
from the standard use of N in mechanics.
Substituting
and
for
and
respectively in
the RTE, Eq.
(
3
.
8
)
, we obtain the
general
momentum balance
equation
.
(
3
.
62
)
As in the energy and entropy equation, the superposition of the
closed and open system is
exploited to substitute
.
Like
any
other extensive property
, momentum can be transported in and out of
the system with mass. Like the entropy generation term in the
entropy equation, the net external force acts as a source of
mom
entum.
For closed systems, Newton’s law of motion is
recovered
.
(
3
.
63
)
where,
is the acceleration in the
direction.
For
an
open steady
system
Eq.
(
3
.
62
)
reduces to
(
3
.
64
)
Th
ese are
the only form
s
of the momentum equation that will be
used in Chapter 7 and 1
1, although other forms can be derived as
easily.
Fig. 3.46 An external
force is necessary to
balance the momentum
flow.
3

30
The momentum equation in the
or
d
irections can be written
by
simply changing the subscript
into
and
respectively.
3.9
Balance Equations Summary
The complete set of governing balance equations are summarized
below for selected categories of systems that will be frequently
encountered in the rest of the chapters. Although momen
tum
equation is also included, often the
MEEE equations

the mass,
energy, entropy and exergy
equations

constitute the core governing
balance equations in thermodynamic problems.
3.9.1
General Form
The following are the balance equations for open and unsteady
systems. All other forms can be derived from this equation set.
Mass
(Eq.
(
3
.
13
)
)
(
3
.
65
)
Energy
(Eq.
(
3
.
25
)
)
(
3
.
66
)
Entropy
(Eq.
(
3
.
32
)
)
(
3
.
67
)
Exergy
(Eq.
(
3
.
49
)
)
(
3
.
68
)
Momentum
(Eq.
(
3
.
62
)
)
Fig. 3.46.1 System
schematic to
accompany
Section 3.9.1.
3

31
(
3
.
69
)
3.9.2
Closed Systems
Considerable simplification results as the mass transfer terms are
dropped from the balance equations for
closed systems. Moreover,
flow work being completely absent,
.
Mass
(Eq.
(
3
.
13
)
)
(
3
.
70
)
Energy
(Eq.
(
3
.
25
)
)
(
3
.
71
)
Entropy
(Eq.
(
3
.
32
)
)
(
3
.
72
)
Exergy
(Eq.
(
3
.
49
)
)
(
3
.
73
)
Mome
ntum
(Eq.
(
3
.
62
)
)
(
3
.
74
)
3.9.3
Closed Process
When an unsteady closed system undergoes a change of state from a
begin

state to a finish

state, it is said to have executed a
closed
process
.
M
ass
(Eq.
(
3
.
14
)
)
Fig. 3.46.2 System
schematic to
accompany
Section 3.9.2.
3

32
(
3
.
75
)
Energy
(Eq.
(
3
.
27
)
)
(
3
.
76
)
Entropy
(Eq.
(
3
.
36
)
)
(
3
.
77
)
Exergy
(Eq.
(
3
.
51
)
)
(
3
.
78
)
3.9.4
Closed Steady
When the image of a
clo
sed system
taken with a
state camera
does
not change with time, the time derivative of all global properties
becomes zero and the system is said to be a
closed steady system
.
Closed cycles
, as will be shown in the next chapter, can be treated as
a special
case of a closed steady system.
Mass
(Eq.
(
3
.
14
)
)
(
3
.
79
)
Energy
(Eq.
(
3
.
28
)
)
(
3
.
80
)
Entropy
(Eq.
(
3
.
37
)
)
(
3
.
81
)
Exergy
(Eq.
(
3
.
56
)
)
(
3
.
82
)
3.9.5
Open Steady
When the image of an
open system
taken with a
state camera
does
not change with time, the time derivative of all global properties
becomes zero and the system is said to be an
open steady system
.
Mass
(Eq.
(
3
.
15
)
)
(
3
.
83
)
Energy
(Eq.
(
3
.
29
)
)
Fig. 3.46.3 System
schematic to
accompany
Section 3.9.3.
Fig. 3.46.4 System
schematic to
accompany
Section 3.9.4.
Fig. 3.46.5 System
schematic to
accompany
Section 3.9.5.
3

33
(
3
.
84
)
Entropy
(Eq.
(
3
.
38
)
)
(
3
.
85
)
Exergy
(Eq.
(
3
.
57
)
)
(
3
.
86
)
Momentum
(Eq.
(
3
.
64
)
)
(
3
.
87
)
3.9.6
Open Process
When an
unsteady open system
undergoes
a change of state from a
begin

state to a finish

state, it is said to have executed an
open
process
. The inlet and exit states are carefully chosen so that their
properties can be assumed to remain unchanged over time and over
the cross

sectional areas. T
his is known as the
uniform state
uniform
flow assumption
.
Mass
(Eq.
(
3
.
17
)
)
(
3
.
88
)
Energy
(Eq.
(
3
.
30
)
)
(
3
.
89
)
Entropy
(Eq.
(
3
.
39
)
)
(
3
.
90
)
Exergy
(Eq.
(
3
.
60
)
)
(
3
.
91
)
Fig. 3.46.6 System
schematic to
accompany
Section 3.9.6.
3

34
EXAMPLE
3

2
MEEE Equations for
a
Closed
P
rocess.
Develop the appropriate form of MEEE (mass, energy, entropy and
exergy) equations for the following problem.
Determine the amount of heat necessary to raise the temperature of 1
kg of aluminum from 30
to 100
?
SOLUTION
The customized form of balance equations for various
classes of systems have been already identified in this chapter.
Therefore, the task at hand is to simplify the problem with suitable
assumptions and choose the appro
priate block of equations from
S
ection
3.9
.
Simplification
The system, obviously closed, is uniform so that a
single state describes its state at a given time. The system is
obviously unsteady, its image taken with a
state camera
changing
with time. Howev
er, the problem description clearly indicates the
system travels from a b

state to a f

state, the hallmark of any
process. The block of equation
summarized
in
S
ection
3.9.3
,
therefore, describes the appropriate form of the balance equations.
The equations
can be further simplified by noting that changes in
and
are most likely negligible making
.
Mass
Energy
Entropy
Exergy
Simplification Using TEST
Starting at the Daemons page,
progressively navigate through Closed, Process, Generic and
Uniform pages.
A
system schematic
and the s
et of equations that
describe that system are displayed at the bottom of the page.
An
appropriate
material model
is
selected as the last step
before the
Closed Process daemon is launched
.
Discussion
The boundary temperature is unknown in this problem.
S
ince the body is be
ing heated to a temperature of
100
, at least
one of the heat sources must be at a temperature of 100
or more.
Also note that
the MEEE equations derived in this problem are
Fig. 3.47 Heating the
block from a b

s
tate to a
f

state constitutes a
closed process.
3

35
applicable
regardless of the model chosen.
Individual terms of the
balance equations will be discussed in the next two chapters. Notice
that the equations are derived here for the extended system. Also
observe that the balance equations in their current form are
ind
ependent of the
material model
.
EXAMPLE
3

3
MEEE Equations for a
Closed
P
rocess.
Develop the appropriate form of MEEE (mass, energy, entropy and
exergy) equations for the following problem.
A piston

cylinder device initially contains 20 g of saturated wa
ter
vapor at 300 kPa. A resistance heater is operated within the cylinder
with a current of 0.4 A from a 240 V source until the volume
doubles. At the same time a heat loss of 4 kJ occurs. Determine the
final temperature and the duration of the process.
SOLUTION
To develop a customized set of MEEE equations.
Simplification
The simplification carried out in Ex.
3

2
applies to
this problem as well. In addition to heat transfer, there are two
modes of work transfer, electrical and boundary work. The closed
process equations of
S
ection
3.93
can be simplified as follows.
Mass
Energy
Entropy
Exergy
Simplification
Using TEST
The procedure remains unchanged to
the one described in the last problem.
Discussion
Steam trapped in a piston

cylinder device apparently has
no similarity with the block of aluminum of the last example.
However, as far as the governing MEEE e
quations are concerned,
the only difference between the two systems is the presence of work
transfer in this problem. As in the previous problem, the balance
equations in their current form are independent of the
material
model
.
EXAMPLE
3

4
MEEE Equation
s for a Non

Mixing
Closed
P
rocess.
Fig. 3.48 Steam
undergoes a closed
process just like the
block in Fig. 3.47.
3

36
Develop the appropriate form of MEEE equations for the following
problem.
A 40 kg aluminum block at 100
is dropped into an insulated tank
that contains 0.5 m
3
of liquid water at 20
. Determine the entropy
generated in this process.
SOLUTION
To simplify the problem so that the balance equations
can be reduced to one of the customized forms discussed in this
chapter.
Simplification
Water and the block constitute a
non

uniform closed
system
going through a process in this problem. Two states, one for
the block and one for water, can be used to describe the composite
begin state. At the end of the process, even though the temperature is
unifor
m, the finish

state still requires a composite description as the
density is different for the two sub

systems. Designating the two
subsystems as A and B, and neglecting any changes in
and
,
the close
d process equations can be simplified as follows.
Mass
Energy
Entropy
Exergy
Simplification Using TEST
Navigate through the Systems, Closed,
Process, Generic, Non

Uniform, Non

Mixing, pages to display the
progressively simplified system schematic and balance equations.
Discussion
The subsystems are closed
themselves since there is no
mass transfer between them. In TEST such systems are called
non

mixing non

uniform
systems. In the following example, on the
other hand, the subsystems of a non

uniform system can be seen to
be
mixin
g
. As in the previous problem, the balance equations in their
current form are independent of the
material model
.
EXAMPLE
3

5
MEEE Equations for
a
Mixing
Closed
P
rocess.
Fig. 3.49 The composite
system goes through a
non

mixing closed
process.
Fig. 3.50
T h e
c o mp o s i t e c l o s e d s y s t e m
g o e s t h r o u g h a mi xi n g
p r o c e s s.
3

37
Develop the appropriate form of MEEE equations for the follow
ing
problem.
A 0.5 m
3
rigid tank containing hydrogen at 40
, 200 kPa is
connected to another 1 m
3
rigid tank containing hydrogen at 20
,
600 kPa. The valve is opened and the system is allowed to reach
the
rmal equilibrium with the surroundings at 15
. Determine the
irreversibility in this process. Assume variable
.
SOLUTION
To simplify the problem so that the balance equations
can be reduced to one of
the customized forms discussed in this
chapter.
Simplification
By drawing the system boundary as shown in the
accompanying figure, gases in the two tanks, each of which acts as
an open system during the process, behave like a closed system. In
the resulti
ng non

uniform system, two states, one for tank A and one
for tank B, must be used to describe the composite begin state. At the
end of the mixing process, the finish state is uniform and can be
represented by a single state. Neglecting any changes in
and
, the closed process equations can be simplified as follows.
Mass
Energy
Entropy
Exergy
Simplification Using TEST
Navigate through the Systems, Closed,
Process, Generic, Non

Uniform, Mixing, pages to display the
progressively simplified system schemati
c and balance
equations.
Discussion
An interpretation of different terms of the balance
equation is postponed until the next chapter.
If the valve is closed
before mixing is complete, the finish state must be expressed through
a composite state just like t
he begin state.
The balance equations, it
should be noted,
are independent of the
material model
.
3

38
EXAMPLE
3

6
MEEE Equations for
a
Closed Steady System
.
Develop the appropriate form of MEEE equations for the following
problem.
A10 m
2
brick wall separate
s two chambers at 500 K and 300 K
respectively. If the rate of heat transfer is 0.5 kW/m
2
, determine the
entropy generation rate and the rate of exergy destruction in the wall.
Assume the wall surface temperatures to be the same as the adjacent
chamber tem
peratures. Also assume steady state.
SOLUTION
To simplify the problem so that the balance equations
can be reduced to one of the customized forms discussed in this
chapter.
Simplification
The brick wall in this problem, obviously, constitutes
a closed sy
stem at steady state. Because the area of the wall at the
edges are negligible compared to the two main faces, heat transfer
through the end faces can be neglected. Also the time derivatives of
and
can
be assumed zero.
Mass
Energy
;
Entropy
Exergy
Simp
lification Using TEST
Navigate through the Systems, Closed,
Steady pages to display the progressively simplified system
schematic and balance equations .
Discussion
Once again we will defer interpretation of various terms
until the next chapter.
With
, t
he exergy equation can be
shown to reduce to entropy equation for this particular system.
Notice that the equations are derived here for the extended system.
EXAMPLE
3

7
MEEE Equations for
an
Open Steady System.
Develop the app
ropriate form of MEEE equations for the following
problem.
Carbon dioxide enters steadily a nozzle at 35 psia, 1400
, and 250
ft/s and exits at 12 psia and 1200
. Assuming the nozzle to be
Fig. 3.51 A closed
system at steady state.
Fig. 3.52 A nozzle
operating at steady state.
3

39
adiabatic and t
he surroundings to be at 14.7 psia, 65
, determine
(a) the exit velocity, and (b) the entropy generation rate by the device
and the surroundings.
SOLUTION
To simplify the problem so that the balance equations
can be reduced to on
e of the customized forms discussed in this
chapter.
Simplification
The image of the nozzle taken with a state camera
remains frozen even though the state of the fluid flowing through the
nozzle changes. Hence, a nozzle is an open steady device. Although
c
hange in
can be neglected, the purpose of a nozzle is to
accelerate a flow and, therefore, the change in
must be
considered significant. Because there is a
single flow
through the
nozzle, the summation over inlets and exits of the open, steady
equations of section
3.9.5
r
educe
to
Mass
Energy
Entropy
Exergy
Simplification Using TEST
Navigate through the Systems, Open,
Steady, Generic, and Single

Flow pages to display the progressively
simplified system schematic and balance equations.
Discussion
Individual
terms of the balance equations will be
discussed in the next two chapters. Notice that the equations are
derived here for the extended system. Also observe that the balance
equations in their current form are independent of the
material
model
.
EXAMPLE
3

8
MEEE Equations for
a
Mixing, Open Steady
System.
Develop the appropriate form of MEEE equations for the following
problem.
3

40
Liquid water at 100 kPa and 10
is heated by mixing it with an
unknown amount of steam at 100 kPa and
200
, and by heating
the mixing chamber with a resistance heater with a power rating of 5
kW. Liquid water enters the chamber at 1 kg/s, and the chamber
looses heat at a rate of 500 kJ/min with the ambient at 25
. If the
mixture leaves at 100 kPa and 50
, determine (a) the mass flow
rate of steam, and (b) the entropy generation rate during mixing.
SOLUTION
To simplify the problem so that the balance equations
can be reduced to on
e of the customized forms discussed in this
chapter.
Simplification
The mixing chamber can be assumed to operate at
steady state. Although heat is transferred from the electrical heating
elements to the working fluid, it is electrical power
that crosses
the boundary and, therefore, must appear in the energy and exergy
equations as
and
respectively. Two inlet states, i1

S
tate
and i
2

S
tate, and one exit state, e

state, are req
uired in this
multi
flow
mixing
configuration. The open, steady equations of section
3.9.5
reduce to
Mass
Energy
Entropy
Exergy
Simplification Using TEST
Navigate through the Systems, Open,
Steady, Generic, Multi

Flow

Mixed pages to display the
progressively simplified system schematic and balance equations .
Discussion
Individual terms of the b
alance equations will be
discussed in the next two chapters. Notice that the equations are
derived here for the extended system. Also observe
that the balance
equations in their current form are independent of the
material
model
.
EXAMPLE
3

9
MEEE Equatio
ns for a Non

Mixing, Open, Steady
System.
Develop the appropriate form of MEEE equations for the following
problem.
Fig. 3.53 A steady state
mixing chamber.
3

41
Steam enters a closed feed
water heater at 1.1 MPa and 200
and
leaves as saturated liquid at the same pressure. F
eedwater enters the
heater at 2.5 MPa and 50
and leaves 12
below the exit
temperature of steam. Neglecting any heat losses, determine (a) the
mass flow rate ratio and (b) the entropy generation rate of t
he device
and its surroundings. Assume surroundings to be at 20
.
SOLUTION
To simplify the problem so that the balance equations
can be reduced to one of the customized forms discussed in this
chapter.
Simplification
The closed f
eed water heater shown in the
accompanying figure is a heat exchanger, where the flow of water is
heated by the flow of steam. For this
non

mixing multi

flow
configuration, two inlet states, i1

and i2

states, and two exit st
ates,
e1

and e2

states, describe the two flows, flow

A from i1 to e1 and
flow B from i2 to e2. Clearly there is no external work transfer for
this passive device. The open, steady equations of section
3.9.5
simplify into
Mass
Energy
Entropy
Exergy
Simplification Using TEST
Navigate through the Systems, Open,
Steady, Generic, Multi

Flow Non

Mixing pages to display the
progressively simplified system schematic and balance equatio
ns .
Discussion
Individual terms of the balance equations will be
discussed in the next two chapters.
EXAMPLE
3

10
MEEE Equations for an Open Process.
Develop the appropriate form of MEEE equations for the following
problem.
Fig. 3.54 A closed
feed
water
heater used in a
steam power plant.
Fi
g. 3.55 The selection
of the inlet state on the
outer side of the valve
ensures that State

i
remains unchanged
during the open process.
3

42
An insulated rigid tank is in
itially evacuated. A valve is opened, and
air at 100 kPa 20
enters the tank until the pressure in the tank
reaches 100 kPa when the valve is closed. Determine the final
temperature of the air in the tank. Assume variable specific
heats.
SOLUTION
To simplify the problem so that the balance equations
can be reduced to one of the customized forms discussed in this
chapter.
Simplification
The tank, an open system, goes from a vacuum b

state to a filled f

state as air from the suppl
y line rushes in. If the i

state is located above the position of the valve, its
thermodynamic
state
at all times can be considered identical to that in the supply
line. In this
open

process
, there is no external work or heat
transfe
r. The open, process equations of section 2.9.5 simplify into
Mass
Energy
Entropy
Exergy
Simplification Using TEST
Navigate through the
Systems, Open,
Process pages to display the progressively simplified system
schematic and balance equations
.
Discussion
Individual terms of the balance equations will be
discussed in the next two chapters.
3

43
3.10
Summary
The fundamental governing equations for
the interactions between a
system and its surroundings are derived in a common format called
the balance equation in this chapter.
The goal is to express the
governing equations in a customized format for a given system.
The
Reynolds
transport equation
or
the RTE
relates the rate of change of
any total extensive property of an open system
at a given instant
with
that of
a closed system
passing through, which happens to occupy
the entire open system at that time.
With the help of RTE the
fundamental laws of
thermodynamics, postulated for a closed
system, are converted into balance equation for
a very general
system.
In Section 3.3 systems are classified into a tree structure with
different branches representing
groups of systems that show some
similar patte
rns.
Mass balance equation is derived and expressed in
different formats in Section 3.4. Similarly, energy, entropy, exergy,
and momentum equations are derived in Sections 3.5 through 3.8.
Finally, in Section 3.9 the complete set of equations, called the
M
EEE equations are summarized for important classes of systems
that are often encountered
in the practice of thermodynamics
.
The next two chapters are devoted to understanding the
various equations derived in this chapter through comprehensive
analysis of
various closed and open systems.
3.11
Index
anchor states, 3

9
atmospheric work, 3

22
axioms, 3

1
balance equation, 3

4
Balance Equation, 3

3
Balance Equations
Closed Process Form
Summary
,
3

31
Closed Steady Form
Summary
,
3

32
Open
Process Form
Summary
,
3

33
Open Steady Form
Summary
,
3

32
Balance Equations, Closed
Systems Summary
,
3

30
Balance Equations, General
Form Summary
,
3

30
begin

state, 3

9
Classification of Systems
,
3

6
Closed Systems
,
3

7
conservative form, 3

12
dead state,
3

23
Energy Balance
Different Forms
,
3

14
, 3

19
Entropy Balance Equation
,
3

15
exergy, 3

20, 3

24
Exergy Balance
Different Forms
,
3

26
Exergy Balance Equation
,
3

20
exergy destruction, 3

25
extended system, 3

17
final

state, 3

9
First Law, 3

11
flow diagra
m, 3

10
general balance equation, 3

6
3

44
general balance equation,
energy, 3

13
general balance equation,
entropy, 3

18
general balance equation,
exergy, 3

24
general balance equation,
momentum, 3

29
generalized friction, 3

16
heat reservoir, 3

21
irreversibi
lity, 3

17, 3

25
irreversible, 3

17
Mass Balance
Different Forms
,
3

11
mass balance equation, 3

10
Mass Balance Equation
,
3

10
,
3

11
MEEE equations, 3

29
mixing systems, 3

36
Momentum Balance
Equation
,
3

28
multi flow, 3

40
multi flow, non

mixing, 3

41
New
ton’s Second Law, 3

28
non

mixing systems, 3

36
non

uniform systems, 3

36
open process, 3

9, 3

42
Open Systems
,
3

7
process, 3

9
reversible, 3

17
Reynold
s Transport
Theorem
,
3

5
RTE, 3

5
Second Law, 3

15
single flow, 3

39
specific flow energy, 3

13
steady
flow energy equation,
3

15
steady state, 3

7
Steady Systems
,
3

7
System classification
,
3

10
System tree
,
3

10
TER, 3

21
thermal energy reservoir, 3

21
uniform and steady flow, 3

9
unsteady, 3

7
Unsteady Instantaneous
,
3

8
Unsteady Process
,
3

8
Unsteady Sy
stems
,
3

7
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