# chapter03-balance - The Expert System For Thermodynamics

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27 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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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

................................
..

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

................................
........................

3
-
32

3.9.5

................................
..........................

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

A system, by default, is
; 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
. 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

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

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
.

3.3.3

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,
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

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
, 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

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

-
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.

As explained in section
3
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
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

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
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
system
.

There is no need for the superscript
c

anymore because we
Fig. 3.19 By using
specific flow energy

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

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
.

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

must be
exactly equal to the net rate of work delivered by the system.

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
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

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

Fig. 3.23 Energy flow
diagram for an open

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
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

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
.
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

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

Fig. 3.31 Entropy flow
diagram for a closed,
process.

3
-
20

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.

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

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

.

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
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,

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

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

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.

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

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

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

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 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

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
.

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

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
.

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

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

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
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,
-
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
-
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,
-
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

Summary
,
3
-
32

Open
Process Form
Summary
,
3
-
33

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

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

flow energy equation,
3
-
15

-
7

,
3
-
7

System classification
,
3
-
10

System tree
,
3
-
10

TER, 3
-
21

thermal energy reservoir, 3
-
21

uniform and steady flow, 3
-
9

-
7

,
3
-
8