3 FUNDAMENTAL LAWS
AND BALANCE
EQUATIONS FOR MASS,
ENERGY, ENTROPY,
EXERGY, AND
MOMENTUM
Chapter
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 thermodynamics, 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
thermodynamics 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 engineering 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.
3-1
3
Fundamental Laws and the Mass, Energy, Entropy, Exergy and
Momentum Balance Equations ....................................................... 3-1
3.1
Balance Equation ............................................................ 3-3
3.2
Reynolds 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
Mass 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
3-2
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 property mass m , energy E , entropy S , or momentum mVx , mV y or mVz of the system. To develop a unified framework, we will represent
these extensive properties with the generic symbol B and the
corresponding specific property with b . For a uniform system (Fig.
3.1), as stated in Eq. (2.94), B  mb . For a non-uniform system,
however, B 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 dV , Eq. (1.12) has to be
written in a differential form.

b
unit of B 3 
(3.1)
dB  bdm  b dV  dV  unit of B =
m
v
kg  m3 /kg 

Integrating,
b
B   bdm   dV
v
 unit of B 
(3.2)
The integration is carried out over the entire system, 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 conditions. 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, the
flow rates of the property B at the ports can be obtained from Eq.
2.94.
 unit of B kg unit of B 
Bi  mi bi ; and
Be  mebe 
=
(3.3)
s
s
kg 

where,
mi  i AV
i i 
AV
AV
i i
; and me  e AeVe  e e
vi
ve
 kg 
 s  (3.4)
Beside mass transfer, a property can be 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, B can be expected to change with time, i.e., B  B  t  .
To use the image analogy introduced in section 1.3.3, the snapshot of
the system taken with the state camera at time t provides us with the
distribution of b throughout the system at that thermodynamic
instant. The global property B  t  , therefore, can be obtained by
3-3
Fig. 3.1. A uniform
system does not have to
be closed.
Fig. 3.2. The local
system used inside the
integral of Eq. (3.2).
simply analyzing the picture. Similarly, a snapshot taken after an
interval t can be used to evaluate B  t  t  , the global property B
at time t  t . On the other hand, if the causes for a change in B  t 
are accounted for, the change B  t  t  - B  t  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 400
K at the top. Determine the total mass of the stratified air if the
pressure inside can be assumed 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 dx (Fig. 3.3)
constitutes a local system in LTE.
Analysis From Table C-1, obtain the necessary material properties
of air: M = 29 kg/kmol, c p =1.005 kJ/kg  K. The gas constant and
cv are calculated (see Section 2.5.3.1.2) as R =8.314/29=0.287
kJ/kg  K and cv = 1.005- 0.287 = 0.718 kJ/kg  K. The variable
temperature can be expressed as T  c1  c2 x with c1  300 K and
c2  100 K/m.
Using the ideal gas equation, pv  RT , Eq. (3.2) can be simplified
as follows.
b
bp  D 2
 D 2 p bdx
B   bdm   dV  
dx 
dx
v
RT 4
4 R 0 T
0
L
L
(3.5)
For the evaluation of mass of the system, B  m ; therefore, b  1 .
Substituting this and the linear temperature relation, Eq. (3.5) can be
integrated.
m

 D2 p L
L
dx
 D2 p
dx

ln  c1  c2 x   0

4 R 0 c1  c2 x
4 Rc2
 D2 p
4 Rc2
ln
T2
 0.948 kg
T1
TEST Solution TEST can be used only for uniform systems or
binary non-uniform systems, which are made of two uniform
3-4
Fig. 3.3. A slice of air
acts as a uniform local
system.
subsystems. Therefore, this problem, which involves an infinite
number of local systems, cannot be solved using TEST.
Discussion The evaluation of other properties such as energy is more
complicated since b as a function of x 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 Reynolds Transport 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 change 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
dBc
 known (from the fundamental Law)
dt
 unit of B 


s
(3.6)
The superscript c reminds us that this equation cannot be applied to
open system as is. The right hand side (RHS) is prescribed by the
specific laws 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 into balance equations 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 t and t  t as sketched in Fig. 2.4. The
minor restriction of a single 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 the 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 t , 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
t , it still occupies the entire open 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 t , the region-I is carefully chosen
3-5
so that the entire fluid inside that region flows into the system during
that interval. Of course, t 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
BIt  mIt bit ,
t t t t
BIIIt t  mIII
be
and
(3.7)
Because B is an extensive property, an inventory of B for a system
can be obtained by combining contributions from different subregions comprising the system. Referring to Fig. 3.3, the change in
B for the closed system as it passes through the open system (region
II) can be written as
B c ,t t  B c ,t   B t t  BIIIt t    B t  BIt 
No special superscript is necessary for the open system because it is
the system by default. Rearranging and substituting Eq. (3.7)
t t t t
B c ,t t  B c ,t   B t t  B t   mIII
be  mIt bit
Dividing both side by t and taking a limit
t t t t
mIII
be
mIt bit
Bt t  Bt
B c ,t t  B c ,t
lim
 lim
 lim
 lim
t 0
t 0
t 0
t 0 t
t
t
t
The LHS and the first term on the RHS of this equation are clearly
derivatives of the extensive property B with respect to time for the
closed and open system respectively. Also, as t  0 , bet t  bet ,
and the last two terms approach mebe and mi bi , where the
superscript t is not necessary anymore since each term in this
instantaneous expression refers to time t . The above equation, thus,
reduces to
dB c dB

 mebe  mibi
dt
dt
Generalizing for multiple inlets and exits, the Reynolds Transport
Equation (RTE) or the general balance equation can be written as
dB
dt
Rate of increase of B
for an open system

mb
i i
i
Net flow rate of B
into the system

m b
e e
e
Net flow rate of B
out of the system

dB c
dt
(3.8)
Rate of increase of B
for a closed system.
It relates the rate of change of an extensive property B of an open
system at a given instant to that of a closed system which happens to
pass through with the boundaries of the two systems aligning on top
of each other at that particular instant.
3-6
Fig. 3.1. A very general
system at two
neighboring
macroscopic instants.
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 the 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
governing 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
uninitiated.
3.3.1 Open vs. Closed Systems
Classification of any system begins with the question, “Is there any
mass transfer across the boundary?” 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.
Fig. 3.5. System
classification: Open vs.
Closed systems.
For a closed system, the mass transfer terms drop out of Eq.
(3.8).
0
0
dB
dBc
  mi bi   me be 
;
dt
dt
i
e

dB dBc
(3.9)

dt
dt
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 pressure 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
B included, must remain constant since the global image does not
change. Therefore, the time derivative of the LHS of Eq. (3.8)
3-7
Fig. 3.6. System
classification: Steady vs.
Unsteady systems.
summarily drops out making the general balance equation an
algebraic one.
dB
dt
0, Steady State
  mi bi   mebe 
i
e
dBc
dt
(3.10)
Obviously this simplification is applicable to both open and closed
systems 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 term 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
depends on the resolution or precision with which one answers the
question. Inside a turbine (take a virtual tour of turbine in the 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 milliseconds,
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.
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.
3.3.3 Unsteady Process
The time derivative of B is non-zero for an unsteady system. The
LHS of the balance equations cannot be simplified any further if
instantaneous rate of change of B 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 an 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 mixture 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 process. The balance equations
3-8
Fig. 3.8. System
classification: Process
vs. Instantaneous rate.
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-state, 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 process;
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 p  v 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.
Fig. 3.9. In this closed
process, a gas is
compressed from a bState to a f-State.
To identify if an unsteady system is undergoing a process, the
appropriate question to ask is, “Does the unsteady system move from
a clear beginning to a clear finish state?” If the answer is yes, we
have a process.
The simplification for a process is achieved by multiplying
Eq. (3.8) with dt and integrating from the b-state to the f-state.
Fig. 3.10. Inflating a tire
is an open process.
dBc
dB   mibi dt   mebe dt 
dt
dt
i
e
finish
finish
finish
finish
finish
finish
finish
dB c
  dB    mi bi dt    mebe dt  
dt
dt
i begin
e begin
begin
begin
dBc
 B f  Bb    mi bi dt    mebe dt  
dt
dt
i begin
e begin
begin
For an open 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 simplify the above
equation as bi and be , being independent of time, can be pulled out
of the integrals. The general balance equation for an open process
reduces to
finish
dB c
B  B f  Bb   mi bi   mebe  
dt
dt
i
e
begin
finish
where, mi 

begin
finish
mi dt , and me 

(3.11)
me dt
begin
3-9
The equation still looks quite formidable with an integral of a
derivative as one of its 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 devoted exclusively to the discussion of
closed and open systems respectively. Further classification of
closed process and open steady systems will be deferred until then.
Fig. 3.11 The system
classification tree. The
Map in TEST displays a
similar clickable tree.
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.
Fig. 3.12 Flow diagram
for the mass balance
equation..
Mass cannot be created or destroyed.
For a closed system the total mass mc must remain constant;
therefore, the time derivative of mc must be zero, i.e.,
dmc
0
dt
(3.12)
Substitute Eq. (3.12) into the RTE, Eq. (3.8), with B  m and b  1 ,
to formulate the mass balance equation for an open unsteady
system.
dm
dt
Rate of increase of mass
for an open system.

m
i
i
Net mass flow rate
into the system.

m
e
e
 kg 
 s 
(3.13)
Net mass flow rate of
out of the system.
3-10
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 the 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 classified 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,
dm
 0 or,
dt
m  constant
(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 the total mass, like all other global properties, remains constant.
dm
dt
0, steady state
  mi   me ;
i
m  m
or,
i
e
i
e
e
 kg 
 s 
(3.15)
Fig. 3.13 Flow diagram
for the mass balance
equation, open steady
system.
This form of mass conservation is often referred as “what goes in
comes out”. If there is a single flow, i.e., only one inlet and one exit,
the equation can be further simplified using Eq. (3.4).
mi  me  m ; or, m  i AV
i i  e AeVe , or m 
AV
AV
i i
 e e
vi
ve
(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
m  m f  mb   mi   me
i
e
finish
where, mi 

begin
(3.17)
finish
mi dt , and me 

me dt
begin
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.
3-11
Fig. 3.14 Flow diagram
for the mass balance
equation, open process.
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 u of a system is a thermodynamic property.
ii) Energy E  U  KE  PE cannot be created or destroyed, only
transferred through heat or work. On a rate basis this can be
expressed as
 kJ

 s =kW 
dE c
 Qc  W c
dt
(3.18)
where, Qc is the net rate of heat transfer into the system and W c is
the net rate of work or power transfer out of the system.
Fig. 3.15 Flow diagram
for the conservative
form of the energy
balance equation, open
unsteady system.
Substituting E , e and E c for B , b and B c respectively in the RTE
and using the second postulate
dE
dt
Rate of increase
of E for an
open system.

m e
i i
i
Net energy flow rate
into the system.

m e
e e
e
Net energy flow
rate out of the system.

Q
Net Rate of
heat transfer
into the system.

W
 kW 
Net Rate of
heat transfer
into the system.
(3.19)
where, Q and W , evaluated based on the open system boundary, are
substituted for Qc and W c respectively since the boundaries of the
closed and open systems become coincident as t  0 . The energy
flow rates at the inlet and exit can be also be expressed through the
symbol E  me , 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 labeled 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 transfer 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 work transfer can be classified into
two major categories, flow and external work, to distinguish open
and closed systems.
3-12
Fig. 3.16 Flow diagram
explaining various
modes of heat and work
transfer.

W
Net Rate of
Work transfer
out of the system.
W
F ,e
i

W
F ,i
i
Net Flow
Work Out
Net Flow
Work In
 Wsh  Wel  WB
Shaft Work
Out
Electrical
Work Out
Boundary
Work Out
Other Work, WO
Flow Work, WF
(3.20)
 WF  WO  WB  WF  Wext
External Work, Wext
For a closed system WF  0 and there is no distinction between W
and Wext .
To evaluate the flow work, consider the small fluid element
of length xe 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 Fe  pe Ae does a work of Fe xe
(see Section 1.2.2.3) in t . 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 pressure, a negative work transfer with a
magnitude of Fi xi takes place in time t at the inlet. As t  0 ,
the net flow work rate or flow power can be written with the help of
Eq. (3.16) as
Fe xe Fi xi pe Ae xe pi Ai xi



t
t
t
t
(3.21)
AeVe
AV
i i
 pe AeVe  pi AV
pe ve 
pi vi  me pe ve  mi pi vi
i i 
ve
vi
Fig. 3.17 A fluid
element at the exit being
expelled by the system
against an external
pressure.
WF  WF ,e  WF ,i 
A port with a very small area still can have very large pv and, thus,
transfer a relatively significant amount of flow work.
Equation (3.21) can be generalized for multiple inlets and
exits.
WF   me pe ve   mi pi vi
e
flow energy J is
equivalent to the flow of
energy E and the
transfer of flow work
WF across a control
(3.22)
i
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
separating it from all other work terms, the conservative form of the
energy equation, Eq. (3.19), can be rewritten as
dE
  mi  ei  pi vi    me  ee  peve   Q  Wext
dt
i
e
Fig. 3.18 The flow of
(3.23)
In this modified form the mass flow can be seen to carry a
combination property consisting of energy e and a term that
3-13
surface.
represents the flow work performed per unit mass of the flow. We
call this combination property the specific flow energy and represent
it with the symbol j in the absence of any universally accepted
symbol for this important convenience property.
j  e  pv  u  ke  pe  pv  h  ke  pe
(3.24)
Substituting the symbol j for the specific flow energy, we obtain
the balance equation for energy in its most general form.
dE
dt
Rate of increase
of E for an
open system.

m j
i i
m

j
e e
i
e
Net flow rate of flow
energy into the system.
Net flow rate of flow
energy out of the system.
where, j  h  ke  pe  h 

Q
Net Rate of
heat transfer
into the system.

 kW 
Wext
Net Rate of
work transfer
into the system.
V2
gz

, and Wext  WB  WO
2000 1000
(3.25)
The energy carried by the flow E  me in the conservative form,
Eq. (3.19), is replaced in this equation by the flow energy carried by
the flow, J  mj . 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 j . It may seem that this form of energy
equation is meant only for open systems. To the contrary, if we
Fig. 3.19 By using
specific flow energy j
instead of specific
energy e , the
cumbersome flow work
can be forgotten.
0
substitute WF  0 and W  WF  Wext  Wext 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 should
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 Wext  W 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.
dE
 Q W
dt
(3.26)
Obviously, this forms suits any instantaneous unsteady closed
system. There is no need for the superscript c anymore because we
3-14
Fig. 3.20 For a closed
system there is no flow
work; therefore,
W  Wext .
are deriving a restricted 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
Fig. 3.21 Energy flow
diagram for a closed
process.
E  E f  Eb  Q  W
finish
where, Q 

finish
Qdt , and W 
begin

finish
WB dt 
begin

WO dt  WB  WO
(3.27)
begin
This is an algebraic equation that relates two anchor states through
two process variables Q and W .
Closed Steady Simplification For a steady system, open or closed,
the time derivative of any global property must be zero. The energy
equation, thus, simplifies to
Q W
(3.28)
Fig. 3.22 Energy flow
diagram for a closed
steady system.
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).
0   mi ji   me je  Q  Wext
i
(3.29)
Fig. 3.23 Energy flow
diagram for an open
steady system.
e
By rearranging the equation, it can be shown that the sum total of the
rate of flow of flow energy and heat into a steady 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 process 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 uniform state assumption, the energy equation reduces to
E  E f  Eb   mi ji   me je  Q  Wext
i
e
finish
where, Q 

begin
(3.30)
finish
Qdt , and Wext 

Wext dt
begin
3-15
Fig. 3.24 Energy flow
diagram for an open
process.
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 S is an extensive property that measures the degree of
disorder in a system. The specific entropy s is a thermodynamic
property.
ii) Entropy can be transferred across a boundary through heat but
not through work. The rate of entropy transfer by Q crossing a
boundary at a temperature TB is given as Q / TB .
iii) Entropy cannot be destroyed. It can be generated by natural
processes, i.e., Sgen  0 .
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 postulate
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 disorder 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 provides
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 system. 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 motion.
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 warm, natural gas burns in air
forming hot flames, a volcano erupts – there is one thing that is
3-16
Fig. 3.25 CARTOON
Are you saying that the
Second Law left those
footprints?
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 spontaneous 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
associated 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 boundary 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 state
without leaving any clue that the original process ever took place.
The system or process is said to be reversible under that ideal
situation. The 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 temperatures,
brought in diathermal contact by removing insulations from two
walls and pressing the two blocks against each other on their uninsulated 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 become 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 of 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
3-17
Fig. 3.26 Entropy is
generated in the shaded
area which extends
beyond the system
boundary.
Fig. 3.26 The
interactions between the
system and its
surroundings causes
entropy generation
inside and in the
immediate surroundings
of a system.
Chapter –10, the emissions from combustion, something far from
trivial.
Getting back to our task of translating the fundamental laws
into balance equations, the second postulate can be written as.
dS c Q c
c

 Sgen
;
dt
TB
 kW 
 K 
c
Sgen
0
(3.31)
where, Sgen is the rate of entropy generation within the closed
system boundary and Qc is the rate of heat transfer into the closed
system of Fig. 3.4. Substituting S and s for B and b respectively
in the RTE, we obtain the general entropy balance equation.
dS
dt
Rate of increase
of S for an
open system.

m s
i i
i
Net flow rate of
entropy into
the system.
where,

m s
e e
e
Net flow rate of
entropy out of
the system.

Q
TB
Net Rate of
entropy transfer
through heat.

Sgen
Net Rate of
generation of
entropy inside
the system.
 kW 
 K 
(3.32)
Sgen  0
As mentioned before, the boundary of the closed system passing
through the open system of Fig. 3.4 is almost identical to that of the
c
open system as t goes to zero. Therefore, Q  Qc and Sgen  Sgen
.
The comments under each term are keyed to the open system of Fig.
3.4 as this 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 surrounding air, atmospheric temperature
can be used for TB . Obviously the precise location of the boundary
does not affect Q or W , which are flow rates of energy; however,
being a cumulative quantity, Sgen 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 immediate surroundings
external to the system.
Sgen,univ  Sgen,int  Sgen,ext
 kW 
 K 
(3.33)
where the subscript univ is used to signify the system’s universe.
3-18
Fig. 3.27 Entropy is
accumulated due to
generation and transfer
through mass and
energy.
Fig. 3.28
Sgen,univ  Sgen  Sgen,ext
includes all sources of
entropy generation
inside and outside the
system.
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 k
segments each at a uniform temperature Tk . The entropy balance
equation for the system’s universe modifies as follows
Q
dS
 kW 
  mi si   me se   k  Sgen,univ 
dt
 K 
i
e
k Tk
Fig. 3.29 Flow diagram
of entropy for an
extended system with
surroundings at two
different temperatures.
(3.34)
The total entropy S , the mass flow rates mi or me and the heat
transfer rate Qk 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 in 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 TB , they can be modified
for an extended system by replacing Sgen with Sgen,univ and
Qk
T
k
Fig. 3.30 Entropy flow
diagram for a closed,
instantaneously
unsteady system.
Q
with
TB
.
k
Closed System Simplification For a closed system the mass transfer
terms drop out and Eq. (3.34) reduces to
dS Q
  Sgen
dt TB
(3.35)
Obviously, this form suits any instantaneous unsteady closed system.
Closed Process Simplification 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
S  S f  Sb 
Q
 Sgen
TB
finish
where, Q 

begin
(3.36)
finish
Qdt , and Sgen 

Sgen dt
begin
This is an algebraic equation that relates two anchor states through
two process variables Q and S gen . Obviously Sgen  0 since Sgen  0
3-19
Fig. 3.31 Entropy flow
diagram for a closed,
process.
Closed Steady Simplification For a steady system, the time
derivative of any global property must be zero. Eq. (3.35) simplifies
to
Q
 S gen
TB
0
(3.37)
Fig. 3.32 The default
direction of heat flow is
inconsistent with the
direction of entropy
transfer.
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 S
disappearing at steady state, the entropy equation, Eq. (3.32),
simplifies to a form similar to the steady flow energy equation.
0   mi si   me se 
i
e
Q
 Sgen
TB
(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 finish state
as outlined in section 3.3.3. Using the uniform flow uniform state
assumption, the entropy equation reduces to
S  S f  Sb   mi si   me se 
i
i
finish
where, Q 

Q
 Sgen
TB
finish
Qdt , and Sgen 
begin

Fig. 3.33 What comes
out may be more than
what goes in since
Sgen  0 .
Fig. 3.34 Entropy
diagram for an open
process.
(3.39)
Sgen dt
begin
The mass transfers in such a process has already been examined in
section 3.4.1.
3.7 Exergy Balance Equation
Stagnant air and wind both have energy. Yet it is much easier 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 delivery 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
3-20
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.
balance 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 of 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 not 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
temperature is assumed not affected by heat transfer –reservoir 0,
which is atmospheric air at a standard temperature T0 and reservoir
k , say, a furnace at a temperature Tk . Note that a TER is sometime
called a heat reservoir, an 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 extended system enclosed by the boundary of Fig.
3.35, the energy and entropy equations can be written as
dE
 mi ji  me je  Qo  Qk  Wext
dt
(3.40)
Q Q
dS
 mi si  me se  o  k  Sgen,univ
dt
To Tk
(3.41)
By including the immediate surroundings, we capture the entire
amount of entropy generation Sgen,univ (see Eq. (3.33))due to the
interactions between the system and its surroundings; moreover, the
choice of the boundary temperature is simplified as it passes through
the TER’s with constant temperatures T0 and Tk . Note that
variables, mi , me , ji , je , si , se , Qo , Qk , or Wext are unaffected by
the choice of the extended system. If the mass (or thermal capacity)
of the added layer of immediate surroundings can be considered
small compared to the internal system, E and S also can be
assumed identical between a system and its extended version. The
3-21
only difference, therefore, comes from the entropy generation
contributed by the immediate surroundings.
Multiplying Eq. (3.41) by T0 , a constant, and subtracting the
resulting equation from Eq. (3.40) we get
d  E  T0 S 
 mi  ji  T0 si   me  je  T0 se 
dt
 T 
 Qk 1  0   Wext  T0 Sgen,univ
 Tk 
(3.42)
Before we proceed further, let us take a closer look at the external
work transfer. Different modes of work transfer have 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 useful. The
only exception is the part of boundary work transfer that involves 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 atmospheric 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 and atmospheric components as shown by the flow
diagram of Fig. 3.37.
Wext  Wu  Watm
Fig. 3.37 Different
modes of external work
in the energy flow for
the system in Fig. 3.36.
(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 x of the piston in time t as shown in Fig. 3.38 by
recalling the definition of work (see Section 1.2.2.3) and the sign
convention.
Watm   lim
t 0
  p0 Ax   lim
t
t 0
p0 V d  p0 V 
dV

 p0
t
dt
dt
(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 work 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 work is valid for
both expansion or contraction of the system.
3-22
Fig. 3.38 Evaluation of
atmospheric work
transfer.
Using Eqs. (3.44) and (3.43), Eq. (3.42) can be rearranged as
d  E  T0 S  p0 V 
 mi  ji  T0 si   me  je  T0 se 
dt
B
A
 T 
 Qk 1  0  
 Tk 
 T0 Sgen,univ
Wu
D
kW 
(3.45)
E
C
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
difference 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 thermodynamic
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 of. Such a state with V  0 ,
z  0 , T  T0 and p  p0 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 p0
and T0 , the combination property j  T0 s carried by the stream
reduces to
 j  T0 s 0 
0
0
j0  T0 s0  h0  ke0  pe0  T0 s0  h0  T0 s0
(3.46)
It should be emphasized that at the dead state the working substance
that make up the system is at p0 and T0 with no kinetic or potential
energies. However, other specific properties such as h0 , s0 , 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 term B of Eq. (3.45) can be
modified by using the dead state as a reference state for the
combination property  j  T0 s  .
A  mi  ji  T0 si    j0  T0 s0    me  je  T0 se    j0  T0 s0  
F

 j0  T0 s0  mi  me 
CDE
kW 
G
(3.47)
H
The combination property carried by the mass flows mi and me in
this equation is called the specific flow exergy and is represented by
3-23
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.
the symbol  . Defined in terms of intensive property T0 , and
specific properties j and s ,    j  T0 si    j  T0 s0  must be an
extrinsic specific property since j is an extrinsic property. It has a
zero value at the dead state and has the same unit as j , i.e., kJ/kg.
The physical meaning of flow exergy will be discussed shortly and
plenty of examples will be covered in Chapter 6. Term H in the
above equation can be further simplified by employing Eq. (3.46)
and the mass balance equation, Eq. (3.13).
dm
dt
dH 0
dS
d
d
  mh0   T0  ms0  
 T0 0
dt
dt
dt
dt
d
d
  H 0  T0 S0   U 0  p0 V0  T0 S0 
dt
dt
d
  E0  T0 S0  p0 V 
dt
H   j0  T0 s0  mi  me    h0  T0 s0 
Taking the term H into the LHS of Eq. (3.47),
A-H 
d
 E  T0 S  p0 V    E0  T0 S0  p0 V0 
dt
(3.48)
The combination extensive system property that appears 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 E , i.e., kJ. The total exergy is represented 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 equation .
3-24
d
dt
 m

i
i

i
Rate of increase of
total exergy  of
an open system.
Net flow rate of
flow exergy into
the system.

Wu
m
e
e
Net flow rate of
flow exergy out of
the system.

Net Rate of exergy
transfer through heat.
 kW 
I
Rate of irreversiblities
or exergy destruction
due to entropy generation.
Rate of exergy transfer
through useful
external work.
where,  
 T 
  Qk 1  0 
k
 Tk 
e
  dV   E  T S  p V    E
0
0
0
 T0 S0  p0 V0 
sys
(3.49)
 U  U 0   T0  S  S0   p0  V  V0   KE  PE
   u  u0   T0  s  s0   p0  v  v0   ke  pe
   j  T0 s    j0  T0 s0 
V2
gz
  h  h0   T0  s  s0  

2000 1000
I  T0 Sgen,univ
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 stored exergy
of a system. Flow exergy, just like flow energy j , is carried by the
flow in and out of the system. Heat transferred into the system from
a TER at Tk  T0 carries a fraction, 1  T0 / Tk , of itself as exergy.
Note that for reservoir k  0 , 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., Tk  T0 , will be
discussed in Section 4.3.3.4.
The exergy delivered by the system as useful work, Wu ,
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 Sgen,univ  0 (Second Law). It
is called the rate of exergy destruction or the rate of irreversibility
and is represented by the symbol I .
3-25
Fig. 3.40 Flow diagram
of exergy for an
extended system.
Each term of this equation sketched in the flow diagram 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 inventory 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 exergy equation in a similar manner for different
classes of systems.
Closed System Simplification For a closed system the mass transfer
terms drop out and Eq. (3.49) reduces to
 T 
d
  Qk 1  0   Wu  I
dt
k
 Tk 
Fig. 3.41 A smart
coffee mug that
produces electricity as
the coffee cools down to
room temperature.
(3.50)
Obviously, this form suits any instantaneous unsteady closed system.
Closed Process Simplification 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 Section 3.3.3 producing
 T 
   f   b   Qk 1  0   Wu  I
k
 Tk 
finish
where, Qk 

finish
Qk dt , Wu 
begin

(3.51)
Wu dt and, I  T0 S gen,univ
begin
The simplified form of the exergy equation for a closed process can
be used to explore the physical meaning of some of its terms. For
instance, when a closed system, say, a warm cup of coffee cools
down from a temperature Tb to the room temperature T0 by rejecting
Qloss 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
Tk  T0 , Eq. (3.51) simplifies as
Wu   b   f  I
(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,  f  0 . Being a
non-negative quantity, the irreversibility I can be seen to reduce the
useful work output. In fact for a regular coffee cup, the exergy is
3-26
Fig. 3.42 The exergy of
a warm coffee mug is
the maximum possible
useful work that can be
extracted as the coffee
comes to equilibrium
with the surrounding air.
Fig. 3.43 Energy flow
diagram for Eq. (3.54).
The direction of the heat
arrow is reversed since
Q  Qloss ( Qloss is a
positive quantity).
completely destroyed by I . If the irreversibility can be eliminated and the Second Law does permit Sgen,univ  0 as a limiting ideal case
- the work produced is maximized.
Wu ,max   b   f
0
I
0
(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, why 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
about the system at hand. Using the solid/liquid model for the coffee,
the energy equation, Eq. (3.27), can be simplified as
E  U f  U b   KE  PE  Q  W   Qloss   W
0
0
 W  U b  U f   Qloss  mcv Tb  T0   Qloss
(3.54)
By eliminating Qloss completely it seems that the change in internal
energy can be completely converted into work, i.e.,
Wmax  mcv Tb  T0  . 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
S  S f  Sb 
 Sgen,univ
 Qloss   S
Q
 Sgen 
gen,univ
TB
T0
T Q
 mcv ln 0  loss
Tb
T0
(3.55)
The first term on the RHS being negative, an elimination of Qloss
would result in a negative S gen,univ , 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 For a steady system, the time
derivative of  , a global property, is set to zero and Eq. (3.50)
simplifies to
 T 
0   Qk 1  0   Wu  I
k
 Tk 
(3.56)
3-27
Fig. 3.44 The change in
U and S according to
the solid/liquid model as
the temperature goes
from Tb to T0 . Mass of
the cup is neglected in
these expressions.
Open Steady Simplification The steady state exergy equation,
similarly, can be expressed in an algebraic form as the time
derivative drops out.
 T 
0   mi i   me e   Qk 1  0   Wu  I
i
e
k
 Tk 
(3.57)
The destruction of exergy term makes it impossible to express this
equation in the what-comes-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 atmospheric reservoir. The power
delivered by this device can be obtained from Eq. (3.57) as
Wu  mi i  me e  I  mi  i  e   I
(3.58)
The useful work is maximized when the exergy destruction is
eliminated and the flow exits at its dead state.
0
0
Wu ,max  mi i  me  e  I  mi i  i
(3.59)
The flow exergy, therefore, can be interpreted as the maximum
possible useful work delivered per unit mass of the flow if the flow
is brought to dead state by exchanging heat 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 the begin to the finish state.
Using the uniform flow uniform state assumption, the exergy
equation reduces to
 T 
   f  b   mi i   me e   Qk 1  0   I
i
i
k
 Tk 
(3.60)
where many of the symbols have been explained in connection with
the corresponding form of the energy and entropy equations.
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.
3-28
Fig. 3.45 Flow diagram
of exergy simplified for
an open steady system.
Newton’s Second Law of Motion for a closed system can be stated
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 equation
can be split into three independent equations along x , y , and z
directions in the Cartesian coordinates. Along the x direction,
Newton’s Second Law can be written as
dM xc
  Fxc
dt
 kN;
where, M x 
mVx
1000
kN kg.m 

kN.s= N s3 s  (3.61)
Observe that in this equation the unit of force 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 M x and Vx /1000 for B and b respectively in
the RTE, Eq. (3.8), we obtain the general momentum balance
equation.
dM x
dt
Rate of increase
of x -momentum
of an open system.
 mV

i x ,i
i
/1000   meVx ,e /1000 +
Net x -momentum flow rate
into the system.
i
Net x -momentum flow rate
out of the system.
 F  kN (3.62)
x
Net Rate of
generation
of x -momentum.
As in the energy and entropy equation, the superposition of the
closed and open system is exploited to substitute Fx  Fxc . 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
momentum.
For closed systems, Newton’s law of motion is recovered.
d  mVx /1000 
dM x
max
  Fx or,
  Fx or,
  Fx (3.63)
dt
dt
1000
where, ax is the acceleration in the x direction.
For an open steady system Eq. (3.62) reduces to
0
1 

mV

i x ,i   meVx ,e    Fx

1000  i
i

(3.64)
These are the only forms of the momentum equation that will be
used in Chapter 7 and 11, although other forms can be derived as
easily.
3-29
Fig. 3.46 An external
force is necessary to
balance the momentum
flow.
The momentum equation in the y or z directions can be written by
simply changing the subscript x into y and z 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 momentum
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))
dm
  mi   me
dt
i
e
 kg 
 s 
(3.65)
Energy (Eq. (3.25))
dE
  mi ji   me je  Q  Wext
dt
i
e
where, j  h  ke  pe  h 
 kW 
(3.66)
2
V
gz

; Wext  WB  WO
2000 1000
Entropy (Eq. (3.32))
dS
Q
 kW 
  mi si   me se   Sgen 
dt
TB
 K 
i
e
(3.67)
Exergy (Eq. (3.49))
 T
d
  mi i   me e   Qk 1  0
dt
i
e
k
 Tk

  Wu  I

where,    E  T0 S  p0 V    E0  T0 S0  p0 V0 
 kW 
(3.68)
   j  T0 s    j0  T0 s0 
I  T0 Sgen,univ
Momentum (Eq. (3.62))
3-30
Fig. 3.46.1 System
schematic to accompany
Section 3.9.1.
dM x
1 


mV

i x ,i   meVx ,e    Fx

dt
1000  i
i

mVx
where, M x 
1000
 kN
kN kg.m 

 kN.s= N s 2 s 
(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, W  Wext .
Mass (Eq. (3.13))
dm
0
dt
 kg 
 s  ;
 m  constant  kg 
(3.70)
Energy (Eq. (3.25))
dE
 Q  W  Q  WB  WO 
dt
 kW ;
(3.71)
Entropy (Eq. (3.32))
dS Q
 kW 
  Sgen 
dt TB
 K 
(3.72)
Fig. 3.46.2 System
schematic to accompany
Section 3.9.2.
Exergy (Eq. (3.49))
 T 
d
  Qk 1  0   Wu  I
dt
k
 Tk 
kW
(3.73)
 M x  constant
(3.74)
Momentum (Eq. (3.62))
dM x
  Fx
dt
 kN ;
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.
Mass (Eq. (3.14))
3-31
m  constant
(3.75)
[kg]
Energy (Eq. (3.27))
E  E f  Eb  Q  W  Q  WB  WO 
Entropy (Eq. (3.36))
S  S f  Sb 
Q
 Sgen
TB
(3.76)
Fig. 3.46.3 System
schematic to accompany
Section 3.9.3.
(3.77)
Exergy (Eq. (3.51))
 T 
   f  b   Qk 1  0   Wu  I
k
 Tk 
(3.78)
3.9.4 Closed Steady
When the image of a closed 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))
m  constant
(3.79)
Energy (Eq. (3.28))
0  Q W
(3.80)
Entropy (Eq. (3.37))
0
Q
 Sgen
TB
 T 
0   Qk 1  0   Wu  I
k
 Tk 
Exergy (Eq. (3.56))
Fig. 3.46.4 System
schematic to accompany
Section 3.9.4.
(3.81)
(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))
0   mi   me
i
e
 kg 
 s 
(3.83)
Energy (Eq. (3.29))
3-32
Fig. 3.46.5 System
schematic to accompany
Section 3.9.5.
0   mi ji   me je  Q  Wext
i
[kW]
(3.84)
 kW 
 K 
(3.85)
kW
(3.86)
e
Entropy (Eq. (3.38))
0   mi si   me se 
i
e
Q
 Sgen
TB
Exergy (Eq. (3.57))
 T 
0   mi i   me e   Qk 1  0   Wu  I
i
e
k
 Tk 
Momentum (Eq. (3.64))
0
1 

mV

i x ,i   meVx ,e    Fx

1000  i
i

 kN
(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. This is known as the uniform state uniform
flow assumption.
Mass (Eq. (3.17))
m  m f  mb   mi   me ;
i
(3.88)
e
Energy (Eq. (3.30))
E  E f  Eb   mi ji   me je  Q  Wext
i
(3.89)
e
Entropy (Eq. (3.39))
S  S f  Sb   mi si   me se 
i
i
Q
 Sgen
TB
(3.90)
Exergy (Eq. (3.60))
 T 
   f  b   mi i   me e   Qk 1  0   I
i
i
k
 Tk 
(3.91)
3-33
Fig. 3.46.6 System
schematic to accompany
Section 3.9.6.
EXAMPLE 3-2 MEEE Equations for a Closed Process.
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 o C to 100 o C ?
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 appropriate block of equations from
Section 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. However, 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 Section 3.9.3,
therefore, describes the appropriate form of the balance equations.
The equations can be further simplified by noting that changes in
KE and PE are most likely negligible making E  U .
Mass
Energy
m  constant
0
0
E  U  KE  PE  Q  W
0
Or, U  m  u f  ub   Q
Entropy
S  S f  Sb  
k
Exergy
Qk
 Sgen,univ
Tk
 T 
0
   f  b   Qk 1  0   Wu  T0 Sgen,univ
k
 Tk 
Simplification Using TEST Starting at the Daemons page,
progressively navigate through Closed, Process, Generic and
Uniform pages. A system schematic and the set 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.
Since the body is being heated to a temperature of 100 o C , at least
one of the heat sources must be at a temperature of 100 o C or more.
Also note that the MEEE equations derived in this problem are
3-34
Fig. 3.47 Heating the
block from a b-state to a
f-state constitutes a
closed process.
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
independent of the material model.
EXAMPLE 3-3 MEEE Equations for a Closed Process.
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 water
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 Section 3.93 can be simplified as follows.
Mass
Energy
m  constant
0
0
E  U  KE  PE  Q  W  Q  WB  WO
Or, U  m  u f  ub   Q  WB  WO
Entropy
S  S f  Sb  
k
Exergy
Qk
 Sgen,univ
Tk
 T 
0
   f  b   Qk 1  0   Wu  T0 Sgen,univ
k
 Tk 
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 equations 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 Equations for a Non-Mixing Closed Process.
3-35
Fig. 3.48 Steam
undergoes a closed
process just like the
block in Fig. 3.47.
Develop the appropriate form of MEEE equations for the following
problem.
A 40 kg aluminum block at 100 o C is dropped into an insulated tank
that contains 0.5 m3 of liquid water at 20 o C . 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
uniform, 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 KE and PE ,
the closed process equations can be simplified as follows.
Mass
Energy
Fig. 3.49 The composite
system goes through a
non-mixing closed
process.
mA  constant; mB  constant;
0
0
0
E  U  KE  PE  Q  W
0
Or, U  U f  U b   mAu f , A  mBu f , B    mAub , A  mBub , B   0
Entropy
S  S f  Sb  
k
0
Qk
 Sgen,univ
Tk
Or, S  S f  Sb   mA s f , A  mB s f , B    mA sb , A  mB sb , B   S gen
0
0
T 
   f  b   Qk 1  0   Wu  T0 Sgen
k
 Tk 
Or,  f   b   mA f , A  mB f , B    mAb , A  mBb , B   T0 Sgen,univ
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 nonmixing non-uniform systems. In the following example, on the
other hand, the subsystems of a non-uniform system can be seen to
be mixing. 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 Process.
3-36
Fig. 3.50 The
composite closed system
goes through a mixing
process.
Develop the appropriate form of MEEE equations for the following
problem.
A 0.5 m3 rigid tank containing hydrogen at 40 o C , 200 kPa is
connected to another 1 m3 rigid tank containing hydrogen at 20 o C ,
600 kPa. The valve is opened and the system is allowed to reach
thermal equilibrium with the surroundings at 15 o C . Determine the
irreversibility in this process. Assume variable c p .
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 resulting 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 KE and
PE , the closed process equations can be simplified as follows.
mA  mB  constant;
Mass
0
0
Energy
E  U  KE  PE  Q  W
Entropy
S  S f  Sb  
0
Or, U  U f  U b   mA  mB  u f   mAub , A  mBub , B   Q
k

Qk
 Sgen
Tk
S  S f  Sb   mA  mB  s f   mA sb , A  mB sb , B  
Q
 S gen,univ
T0
0
Exergy

 T 
0
   f  b   Qk 1  0   Wu  T0 Sgen,univ
k
 Tk 
 f   b   mA  mB   f   mAb , A  mBb , B   T0 Sgen,univ
Simplification Using TEST Navigate through the Systems, Closed,
Process, Generic, Non-Uniform, Mixing, pages to display the
progressively simplified system schematic 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 the begin state. The balance equations, it
should be noted, are independent of the material model.
3-37
EXAMPLE 3-6 MEEE Equations for a Closed Steady System.
Develop the appropriate form of MEEE equations for the following
problem.
Fig. 3.51 A closed
system at steady state.
A10 m2 brick wall separates two chambers at 500 K and 300 K
respectively. If the rate of heat transfer is 0.5 kW/m2, 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 temperatures. 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 system 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
KE and PE can be assumed zero.
Mass
m  constant;
0
Energy
0  QH  QC  W ;  QH  QC
Entropy
0
k
Exergy
Qk
 Sgen,univ
Tk
 1
1 
 Sgen,univ  QH   
 TC TH 
0
 T 
0   Qk 1  0   Wu  T0 Sgen,univ
k
 Tk 
 T 
 T 
 0  QH 1  0   QC 1  0   T0 Sgen,univ
 TH 
 TC 
Simplification 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 QH  QC , the 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 appropriate form of MEEE equations for the following
problem.
Carbon dioxide enters steadily a nozzle at 35 psia, 1400 o F , and 250
ft/s and exits at 12 psia and 1200 o F . Assuming the nozzle to be
3-38
Fig. 3.52 A nozzle
operating at steady state.
adiabatic and the surroundings to be at 14.7 psia, 65 o F , 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 one 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
change in PE can be neglected, the purpose of a nozzle is to
accelerate a flow and, therefore, the change in KE 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 reduce to
Mass
mi  mi  m
Energy
0  m  ji  je   Q  Wext ;
0
0

ji  je
0
0  m  si  se   
Entropy
k

Exergy
se  si 
Qk
 Sgen,univ  m  si  se   Sgen,univ
Tk
Sgen,univ
m
0
0
T 
0  m  i  e    Qk 1  0   Wu  T0 Sgen,univ
k
 Tk 
TS
  e   i  0 gen,univ
m
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.
3-39
Liquid water at 100 kPa and 10 o C is heated by mixing it with an
unknown amount of steam at 100 kPa and 200 o C , 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 o C . If the
mixture leaves at 100 kPa and 50 o C , 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 one 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 Wel that crosses
the boundary and, therefore, must appear in the energy and exergy
equations as Wext and Wext,u respectively. Two inlet states, i1-State
and i2-State, and one exit state, e-state, are required in this multi
flow mixing configuration. The open, steady equations of section
3.9.5 reduce to
Mass
mi1  mi 2  me
Energy
0  mi1 ji1  mi 2 ji 2  me je  Q  Wel ;
Entropy
0  mi1si1  mi 2 si 2  me se 
Q
 Sgen,univ
T0
Exergy
0
T 
0  mi1 i1  mi 2 i 2  me e   Qk 1  0   Wu  T0 Sgen,univ
k
 Tk 
 0  mi1 i1  mi 2 i 2  me e  Wel  T0 Sgen,univ
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 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 3-9 MEEE Equations for a Non-Mixing, Open, Steady
System.
Develop the appropriate form of MEEE equations for the following
problem.
3-40
Fig. 3.53 A steady state
mixing chamber.
Steam enters a closed feedwater heater at 1.1 MPa and 200 o C and
leaves as saturated liquid at the same pressure. Feedwater enters the
heater at 2.5 MPa and 50 o C and leaves 12 o C below the exit
temperature of steam. Neglecting any heat losses, determine (a) the
mass flow rate ratio and (b) the entropy generation rate of the device
and its surroundings. Assume surroundings to be at 20 o C .
Fig. 3.54 A closed feed
water heater used in a
steam power plant.
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 feed 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 states,
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
mi1  me1  mA ;
mi 2  me 2  mB ;
Energy
0  mi1 ji1  mi 2 ji 2  me1 je1  me 2 je 2  Q  Wext ;
0

0
mA  ji1  je1   mB  je 2  ji 2 
0
Entropy

0  mi1si1  mi 2 si 2  me1se1  me 2 se 2 
Q
TB
 Sgen,univ
0  mA  si1  se1   mB  si 2  se 2   Sgen,univ
0  mi1 i1  mi 2 i 2  me1 e1  me 2 e 2
Exergy

0
0
T 
 Qk 1  0   Wu  T0 Sgen,univ
k
 Tk 
0  mA  i1  e1   mB  i 2  e 2   T0 Sgen,univ
Simplification Using TEST Navigate through the Systems, Open,
Steady, Generic, Multi-Flow Non-Mixing 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.
EXAMPLE 3-10 MEEE Equations for an Open Process.
Develop the appropriate form of MEEE equations for the following
problem.
3-41
Fig. 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.
An insulated rigid tank is initially evacuated. A valve is opened, and
air at 100 kPa 20 o C 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 bstate to a filled f-state as air from the supply line rushes in. If the istate 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
transfer. The open, process equations of section 2.9.5 simplify into
0
m  m f  mb  mi ; Or, m f  mi
Mass
Energy
0
0
0
E f  Eb  mi ji  Q  Wext ;

0

m f u f  ke  pe

u f  hi
0
  m  h  ke
i
i
0
 pe
0

Entropy
Q
0
S f  Sb  mi si 
0
TB
 Sgen,univ
 m f s f  mi si  Sgen,univ
 s f  si 
Sgen,univ
mf
Exergy
0
0
T
 f   b  mi i   Qk 1  0
k
 Tk
 m f  f  mi i  T0 Sgen,univ
  f  i 

0
  Wu  T0 Sgen,univ

T0 Sgen,univ
mf
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.
3-42
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 patterns. 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
MEEE 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, 319
Entropy Balance Equation,
3-15
exergy, 3-20, 3-24
Exergy Balance
Different Forms, 3-26
Exergy Balance Equation, 320
exergy destruction, 3-25
extended system, 3-17
final-state, 3-9
First Law, 3-11
flow diagram, 3-10
general balance equation, 3-6
3-43
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
irreversibility, 3-17, 3-25
irreversible, 3-17
Mass Balance
Different Forms, 3-11
mass balance equation, 3-10
Mass Balance Equation, 310, 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
Newton’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
Reynolds 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 Systems, 3-7
3-44