 General description of the conservation of
mass and energy equations for a general
control volume.
 Steady-flow process, which is the model
process for many engineering devices such
as turbines, compressors, heat exchangers
and flow through ducts and pipes.
Chapter 4
1
4.1.- TERMODYNAMIC ANALYSIS OF CONTROL
VOLUMES
A large number of engineering problems involve
mass flow in and out of a system and, therefore,
are modeled as CONTROL VOLUMES.
A water heater, a car radiator, a turbine, and a
compressor all involve mass flow and should be
analyzed as control volumes (OPEN SYSTEMS)
instead of as control mass (CLOSED SYSTEMS).
In general, any arbitrary region in space can be
selected as a control volume.
The boundaries of a control volume are called a
CONTROL SURFACE and they can be real or
imaginary.
HOT WATER
OUT
WATER
HEATER
CONTROL
SURFACE
Chapter 4
(CONTROL
VOLUME CV)
COLD WATER
IN
2
Real boundary
Imaginary
boudary
A NOZZLE
(CONTROL
VOLUME CV)
CV
Moving
boundary
The terms STEADY and UNIFORM are used
extensively in this chapter and thus it is important
to have a clear understanding of their meanings.
Chapter 4
3
The term STEADY implies “NO CHANGE WITH
TIME”. The opposite of steady is UNSTEADY, or
TRANSIENT.
The
term
UNIFORM,
however,
implies “NO CHANGE WITH LOCATION” over a
specified region.
CONSERVATION OF MASS PRINCIPLE.The conservation of mass is one of the most
fundamental principles in nature.
For closed systems, the conservation of mass
principle in implicitly used by requiring that the
mass of the system remain constant during a
process. For control volumes, however, mass can
cross the boundaries, and so we want must keep
track of the amount of the mass entering and
leaving the control volume. The CONSERVATION
OF MASS PRINCIPLE for a control volume (CV)
undergoing a process can be expressed as
Chapter 4
4
Total mass  Total mass   Net change 
 entering    Leaving   in mass within 

 
 


 
 

CV
CV
CV
m  m
 mcv
Inlet
Exit
i
e
2kg
mcv  5kg
7 kg
MASS AND VOLUME FLOW RATES.The amount of mass flowing through a cross
section per unit time is called the MASS FLOW

RATES and denoted
Chapter 4
m . As before, the dot over
5
a symbol is used to indicate a quantity per unit
time.
A liquid or gas flows in and out of a control
volume through pipes or ducts. The mass flow
rate of a fluid flowing in a pipe or duct is
proportional to the cross-sectional area A of the
pipe or duct, the density
,
and the velocity
 of
the fluid. The mass flow rate through a differential
area dA can be expressed as

d m  ndA
Where  n is the velocity component normal to dA.
The mass flow rate through the entire crosssectional area of the pipe or duct is obtained by
integration:

m    n dA
(kg / s)
A
Chapter 4
6

m  av A
(kg / s)
Where
 = density, kg/m3 (1/  )
 av = average fluid velocity normal to A, m/s
A= cross-sectional area normal to flow direction, m2
The volume of the fluid flowing through a crosssection per unit time is called the VOLUME FLOW

RATE V and is given by

V    n dA   av A
(m 3 / s)
A
The mass and volume flow rates are related by


m  V 
Chapter 4

V

7
CONSERVATION OF ENERGY PRINCIPLE.The change in the energy of a closed system
during a process is equal to the net heat and work
transfer across the system boundary. This was
expressed as
Q  W  E
For control volumes, however, an additional
mechanism can change the energy of a system:
MASS FLOW IN AND OUT OF THE CONTROL
VOLUME.
When mass enters a control volume, the energy of
th e
control
volume
increases
because
th e
entering mass carries some energy with it.
Likewise, when some mass leaves the control
volume, the energy contained within the control
volume decreases because the leaving mass
takes out some energy with it.
Chapter 4
8
The conservation of energy equation for a control
volume undergoing a process can be expressed
as
 Total energy  Total energy  Total energy  Net change 
cros sin g boundary    of mass    of mass    in energy 

 
 
 

 as heat and work   entering CV   leaving CV   of CV 
Q  W  Ein,mass  Eout ,mass  Ecv
MASS IN
W
(CLOSED
SYSTEM)
Q
W
(CONTROL
VOLUME
CV)
Q
MASS OUT
The energy required to push fluid into or out of a
control volume is called the FLOW WORK, or
FLOW ENERGY.
Chapter 4
9
FLOW WORK.Unlike closed systems, control volumes involve
mass flow across their boundaries, and some
work is required to push the mass into or out of
the control volume. This work is known as the
FLOW
WORK,
or
FLOW
ENERGY,
and
is
necessary for maintaining a continuous flow
through a control volume.
A
F
V
P
m
CV
F
L
F  PA
Wf low  FL  PAL  PV (kJ)
w f low  P (kJ / kg)
Chapter 4
10
TOTAL ENERGY OF A FLOWING FLUID.2
e  u  ke  pe  u 
 gz
2
(kJ / kg)
The fluid entering or leaving a control volume
possesses an additional form of energy-the FLOW
ENERGY P . Then the total energy of a flowing
fluid on a unit-mass basis (denoted  ) become
  P  e  P  (u  ke  pe)
But the combination
P  u has been previously
defined as enthalpy h. So the above relation
reduces to
2
  h  ke  pe  h 
 gz
2
(kJ / kg)
4.2.- THE STEADY-FLOW PROCESS.A large number of engineering devices such as
turbines, compressors, and nozzles operate for
Chapter 4
11
long periods of time under the same conditions,
and they are classified as STEADY-FLOW devices.
Processes involving steady-flow devices can be
represent
reasonably
well
by
a
somewhat
idealized process, called the STEADY-FLOW
PROCESS.
A steady-flow process can be defined as A
PROCESS DURING WHICH A FLUID FLOWS
THROUGH A CONTROL VOLUME STEADLY. That
is, the fluid properties can change from point it
point within the control volume, but at any fixed
point they remain the same during the entire
process.
(Remember,
STEADY
means
NO
CHANGE WITH TIME.) A steady-flow process is
characterized by the following:
1. No properties (intensive or extensive) within
the control volume change with time.
2. No properties change at the boundaries of the
control volume with time.
Chapter 4
12
3. The heat and work interactions between a
steady-flow system and its surrounding do
not change with time.
MASS IN
300ºC
250ºC
(CONTROL VOLUME)
225ºC
200ºC
150ºC
MASS OUT
Time: 1 p.m.
MASS IN
300ºC
250ºC
(CONTROL VOLUME)
225ºC
200ºC
150ºC
MASS OUT
Time: 3 p.m.
Chapter 4
13
During a steady-flow process fluid properties
within the control volume may change with
position, but no with time.
MASS IN
CONTROL
VOLUME
mCV=Const.
ECV=Const.
MASS OUT
Under steady-flow conditions, the mass and
energy contents of a control volume remain
constant.


m1
m2
h1
h2
CONTROL
VOLUME

m3
h3
Under steady-flow conditions, the fluid properties
at an inlet or exit remain constant (do not change
with time).
Chapter 4
14
CONSERVATION OF MASS.During a steady-flow process, the total amount of
mass contained within a control volume does not
change with time (mcv =constant).
When dealing with steady-flow processes, we are
not interested in the amount of mass that flows in
and out of the device over time; instead, we are
interested in the amount of mass flowing per unit

time, i.e., THE MASS FLOW RATE m . The
CONSERVATION OF MASS PRINCIPLE for a
general steady-flow system with multiple inlets
and exits can be expressed in the rate form as
 Total mass   Total mass 

 

entering
CV

leaving
CV

 

 per unit time   per unit time 

 



 mi  me
(kg / s)
For devices with only one inlet and one outlet


m1  m2
Chapter 4
(kg / s)
15
11A1  2 2 A 2
1
1
1A1   2 A 2
1
2
Where:
 = density, kg/m3
 = specific volume, m3/kg (= 1 )
 = average flow velocity in flow direction, m/s
A = cross-sectional area normal to flow direction, m2

m 2  2kg / s

V 2  0.8m3 / s
Air
Compressor
During a steadyflow process,
volume flow rates
are not necessarily
conserved

m1  2kg / s

V 1  1.4m3 / s
Chapter 4
16
CONSERVATION OF ENERGY.It was pointed out earlier that during a steady-flow
process the total energy content of a control
volume remains constant (ECV=constant).
That is, the change in the total energy of the
control volume during such a process is zero
(  ECV=0).Thu, the amount of energy entering a
control volume in all forms (heat, work, mass
transfer) must be equal to the amount of energy
leaving it for a steady-flow process.
THE CONSERVATION OF ENERGY PRINCIPLE for
a general steady-flow system with multiple inlets
and exists can be expressed verbally as
 Total energy
  Total energy   Total energy 
cros sin g boundary  transport out of  transport int o




 as heat and work   CV with mass  CV with mass 

 
 

 per unit time
  per unit time   per unit time 
or




Q W   me e   mi i
Chapter 4
17
2
2
 





e
i
Q W   me  he 
 gz e    mi  h1 
 gzi 
2
2







For each inlet
For each exit
For the case single-stream (one-inlet, one-outlet)
the mass flow rate through the entire control



volume remains constant ( m1  m 2 ) and denoted m .
Then the conservation of energy for SINGLESTREAM STADY FLOW-SYSTEMS becomes
2
2


 2  1
Q W  mh2  h1 
 gz 2  z1 
2







(kW )

Q W  mh  ke  pe
(kW )

Dividing by m
  1
q  w  h2  h1  2
 gz 2  z1 
2
2
2
q  w  h  ke  pe
Chapter 4
(kJ / kg)
(kJ / kg)
18
Where:

q
Q

(heat transfer per unit mass, kJ/kg)
m

w
W

(work done per unit mass, kJ/kg)
m
If ke  0, pe  o
q  w  h
(kJ / kg)
The various terms appearing in the above
equations are as follows:

Q = rate of heat transfer between the control
volume and its surroundings.

W = Power
h  hexit  hinlet , for ideal gases h  Cp,av e T2  T1 

ke 
2
2
 1
2

2
pe  gz2  z1 
Chapter 4
19
4.3.- SOME STEADY-FLOW ENGINEERING
DEVICES.1.- NOZZLES AND DIFFUSERS.Nozzles and diffusers are commonly utilized in jet
engines, rockets, spacecraft, and even garden
hoses. A NOZZLE is a device that INCREASES
THE VELOCITY OF A FLUID at the expense of
pressure.
A
DIFFUSER
is
a
device
that
INCREASES THE PRESSURE OF A FLUID by
slowing it down.
 2  1
 2  1
1
NOZZLE
2
1
DIFFUSER
2
The relative importance of the terms appearing in
the energy equation for nozzles and diffusers is
as follows:


Q  0 , W  0 , h  0 , ke  0 , pe  0
Chapter 4
20
0
0
0
q  w  h  ke  pe
0  h  ke
  1
0  h2  h1  2
2
2
(kJ / kg)
(kJ / kg)
2
(kJ / kg)
 2  1


0

C
T

T

p
2
1
For ideal gases
2
2
2
(kJ / kg)
2.- TURBINES AND COMPRESSORS.In steam, gas or hydroelectric power plants, the
device that drives the electric generator is the
TURBINE. As the fluid passes through the turbine,
work is done against the blades, which are
attached to the shaft. As a result, the shaft
rotates, and the turbine produces work. THE
WORK DONE IN A TURBINE IS POSITIVE since it
is done by the fluid.
Chapter 4
21
COMPRESSOR, as well as pumps and fans, are
devices used to increase the pressure of a fluid.
Work is supplied to these devices from an
external
source
through
a
rotating
shaft.
Therefore THE WORK TERM FOR COMPRESSOR
IS NEGATIVE since work is done on the fluid.
A COMPRESSOR is capable of compressing the
gas to very high pressures. PUMPS work very
much like compressor except that they handle
liquids instead of gases.
For turbines and compressors, the relative
magnitudes of the various terms appearing in the
energy equation are as follows:
Gas
Turbine
Chapter 4
Air
Compressor
22



Q  0 , Q  0 , W  0 , h  0 , ke  0 , pe  0
0
q  w  h  ke  pe 0
q  w  h
(kJ / kg)
(kJ / kg)
q  w  h2  h1
(kJ / kg)
For ideal gases q  w  Cp T2  T1 
(kJ / kg)
3.- THROTTLING VALVES.Throttling valves are ANY KIND OF FLOWRESTRICTING DEVICES that cause a significant
PRESSURE DROP IN THE FLUID. Some familiar
examples are ordinary adjustable valves, capillary
tubes, and porous plugs.
The
pressure
accompanied
drop
by
in
a
th e
fluid
LARGE
is
DROP
often
IN
TEMPERATURE, and for that reason THROTTLING
devices are commonly used in refrigeration and
air conditioning applications.
Chapter 4
23
But in the case of an ideal gas h=h(T), and thus
the temperature has to remain constant during a
throttling process.
(a) An adjustable valve
(b) A porous plug
(c) A capillary tube


Q  0 , W  0 , h  0 , ke  0 , pe  0
0
0
0
q  w  h  ke  pe
0  h
Chapter 4
0
(kJ / kg)
(kJ / kg)
24
h2  h1
(kJ / kg)
4a.- MIXTURE CHAMBERS.In engineering applications, mixing two streams
of fluids is not a rare occurrence. This section
where
th e
mixing
process
takes
place
is
commonly referred to as a MIXING CHAMBER
The mixing chamber does not have to be a
distinct “chamber”. An ordinary T-elbow or Yelbow in a shower, for sample, serves as the
mixing chamber for the cold-and hot-water
streams.

m1
T1

CV

m2
m3
T3
T2
Chapter 4
25


Q  0 , W  0 , h  0 , ke  0 , pe  0
20
20
 

0
0


e
i



Q  W   m e  he 
 gz e `   mi  h1 
 gzi 
2
2





0

0
For each exit

For each inlet

m h  m
i
e
1
he
4b.- HEAT EXCHANGERS.As the name implies, HEAT EXCHANGERS are
devices
where
two
moving
fluid
streams
exchange heat without mixing. Heat exchangers
are widely used in various industries, and they
come in various designs.
The simplest form of a heat exchanger is a
DOUBLE-TUBE (also called tube-and-shell) heat
exchanger.
Chapter 4
26
Fluid B
Heat
Fluid A
Heat
Under steady operation, the mass flow rate of
each
fl u i d
stream
flowing
through
a
heat
exchanger remains constant.


Q  0 , W  0 , h  0 , ke  0 , pe  0
20
20
 

0
0


e
i



Q  W   m e  he 
 gz e    mi  h1 
 gzi 
2
2





0
0

For each exit

For each inlet

m h  m
i
Chapter 4
1
e
he
27
Fluid B
CV boundary
Heat
Fluid A
Heat
(a) System: Entire heat exchanger (QCV=0)
Fluid B
CV boundary
Heat
Fluid A
Heat
(b) System: Fluid A (QCV  0)
Chapter 4
28
5.- PIPE AND DUCT FLOW.The transport of liquids or gases in pipes and
ducts is if great importance in many engineering
applications. Flow through a pipe or a duct
usually satisfies the steady-flow conditions and
thus can be analyzed as a steady-flow process.


Q
We

W sf h


Q  0 , W  0 , h  0 , ke  0 , pe  0 `



Q W  mh  ke  pe


0
(kW )

Q W  mh  pe
(kW )

q  w  h  pe
Chapter 4
(kJ / kg)
29