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CHAPTER 5
MASS AND ENERGY ANALYSIS OF CONTROL VOLUMES
Lecture slides by
Dr. Fawzi Elfghi
Thermodynamics: An Engineering Approach 8th Edition in SI Units
Yunus A. Çengel, Michael A. Boles
McGraw-Hill, 2015
2
Objectives
• Develop the conservation of mass principle.
• Apply the conservation of mass principle to various systems
including steady-flow control volumes.
• Apply the first law of thermodynamics as the statement of the
conservation of energy principle to control volumes.
• Identify the energy carried by a fluid stream crossing a control
surface as the sum of internal energy, flow work, kinetic energy,
and potential energy of the fluid and to relate the combination of
the internal energy and the flow work to the property enthalpy.
• Solve energy balance problems for common steady-flow devices
such as nozzles, compressors, turbines, throttling valves,
• Apply the energy balance to general unsteady-flow processes with
particular emphasis on the uniform-flow process as the model for
commonly encountered charging and discharging processes.
3
CONSERVATION OF MASS Conservation of mass: Mass, like energy, is a conserved property, and it cannot be created or destroyed during a process.
Closed systems: The mass of the system remain constant during a process.
Control volumes: Mass can cross the boundaries, and so we must keep track of the amount of mass entering and leaving the control volume.
Mass m and energy E can be
converted to each other
according to
where c is the speed of light in a
vacuum, which is c = 2.9979 108
m/s.
The mass change due to energy
change is negligible.
4
Conservation of Energy for Control volumes
The conservation of mass and the conservation of energy principles for open systems
or control volumes apply to systems having mass crossing the system boundary or
control surface. In addition to the heat transfer and work crossing the system
boundaries, mass carries energy with it as it crosses the system boundaries. Thus,
the mass and energy content of the open system may change when mass enters or
leaves the control volume.
Thermodynamic processes involving control volumes can be considered in two
groups: steady-flow processes and unsteady-flow processes. During a steady-flow
process, the fluid flows through the control volume steadily, experiencing no change
with time at a fixed position.
5
A pressurized water nuclear reactor steam power plant has many
examples of open system operation. Some of these are the
pressure vessel, steam generator, turbine, condenser, and pumps.
Photo courtesy of Progress Energy Carolinas, Inc.
6
A heat exchanger, the heater core from a 1966 289 V8
Mustang, is another example of an open system. Cooling
water flows into and out of the tubes and air is forced
through the fin sturucture.
7
A hair drier is an example of a one entrance, one exit open
system that receives electrical work input to drive the fan
and power the resistance heater.
8
Let’s review the concepts of mass flow rate and energy transport by mass.
One should study the development of the general conservation of mass presented in
the text. Here we present an overview of the concepts important to successful
problem solving techniques.
Mass Flow Rate
Mass flow through a cross-sectional area per unit time is called the mass flow rate .
Note the dot over the mass symbol indicates a time rate of change. It is expressed
as
m
where is the velocity normal to the cross-sectional flow area.
Vn
9
If the fluid density and velocity are constant over the flow cross-sectional area, the
mass flow rate is
aveave
V Am V A
v
where is the density, kg/m3 ( = 1/v), A is the cross-sectional area, m2; and is the
average fluid velocity normal to the area, m/s. aveV
Example 5-1
Refrigerant-134a at 200 kPa, 40% quality, flows through a 1.1-cm inside diameter, d,
tube with a velocity of 50 m/s. Find the mass flow rate of the refrigerant-134a.
At P = 200 kPa, x = 0.4 we determine the specific volume from table A12, Page 918
3
0.0007533 0.4(0.0999 0.0007533)
0.0404
f fgv v xv
m
kg
2
2
3
4
50 / (0.011 )
0.0404 / 4
0.117
ave aveV A V dm
v v
m s m
m kg
kg
s
10
The fluid volume flowing through a cross-section per unit time is called the volume
flow rate . The volume flow rate is given by integrating the product of the velocity
normal to the flow area and the differential flow area over the flow area. If the
velocity over the flow area is a constant, the volume flow rate is given by (note we are
dropping the “ave” subscript on the velocity)
V
( / )V VA m s
3
The mass and volume flow rate are related by
( / )m VV
vkg s
Example 5-2
Air at 100 kPa, 50oC, flows through a pipe with a volume flow rate of 40 m3/min. Find
the mass flow rate through the pipe, in kg/s.
Assume air to be an ideal gas, so From table A1, page 898 , we find the gas constant
(R)
vRT
P
kJ
kg K
K
kPa
m kPa
kJ
m
kg
0 287273
100
0 9270
3
3
.(50 )
.
11
/ min
. /
min
.
mV
v
m
m kg s
kg
s
40
0 9270
1
60
0 719
3
3
Conservation of Mass for General Control Volume
The conservation of mass principle for the open system or control volume is
expressed as
or
( / )m m m kg sin out system
Steady-State, Steady-Flow Processes
Most energy conversion devices operate steadily over long periods of time. The
rates of heat transfer and work crossing the control surface are constant with time.
The states of the mass streams crossing the control surface or boundary are constant
with time. Under these conditions the mass and energy content of the control volume
are constant with time.
12
0CVCV
dmm
dt
Steady-state, Steady-Flow Conservation of Mass:
Since the mass of the control volume is constant with time during the steady-state,
steady-flow process, the conservation of mass principle becomes
or
( / )m m kg sin out
13
Flow work and the energy of a flowing fluid
Energy flows into and from the control volume with the mass. The energy required to
push the mass into or out of the control volume is known as the flow work or flow
energy.
The fluid up steam of the control surface acts as a piston to push a unit of mass into
or out of the control volume. Consider the unit of mass entering the control volume
shown below.
As the fluid upstream pushes mass across the control surface, work done on that unit
of mass is
14
FLOW WORK AND THE ENERGY OF A FLOWING FLUID
Flow work, or flow energy: The work (or energy)
required to push the mass into or out of the control
volume. This work is necessary for maintaining a
continuous flow through a control volume.
15
Total Energy of a Flowing Fluid
The total energy consists of three parts for a nonflowing fluid and
four parts for a flowing fluid.
h = u + Pv
The flow energy is
automatically taken
care of by enthalpy.
In fact, this is the
main reason for
defining the property
enthalpy.
16
Energy Transport by Mass
When the kinetic and potential energies
of a fluid stream are negligible
When the properties of the mass at
each inlet or exit change with time
as well as over the cross section
17
ENERGY ANALYSIS OF STEADY-FLOW SYSTEMS
Steady-flow process: A process
during which a fluid flows through
a control volume steadily.
18
The term Pv is called the flow work done on the unit of mass as it crosses the control
surface.
The total energy of flowing fluid
The total energy carried by a unit of mass as it crosses the control surface is the sum
of the internal energy, flow work, potential energy, and kinetic energy.
u PvV
gz
hV
gz
2
2
2
2
Here we have used the definition of enthalpy, h = u + Pv.
Energy transport by mass
Amount of energy transport across a control surface:
2
(kJ)2
mass
VE m m h gz
19
Rate of energy transport across a control surface:
2
( )2
mass
VE m m h gz kW
Conservation of Energy for General Control Volume
The conservation of energy principle for the control volume or open system has the
same word definition as the first law for the closed system. Expressing the energy
transfers on a rate basis, the control volume first law is
or
E E E kWin out system
Rate of net energy transfer by heat, work, and mass
Rate change in internal, kinetic, potential, etc., energies
( )
Considering that energy flows into and from the control volume with the mass, energy
enters because net heat is transferred to the control volume, and energy leaves
because the control volume does net work on its surroundings, the open system, or
control volume, the first law becomes
20
where is the energy per unit mass flowing into or from the control volume. The
energy per unit mass, , flowing across the control surface that defines the control
volume is composed of four terms: the internal energy, the kinetic energy, the
potential energy, and the flow work.
The total energy carried by a unit of mass as it crosses the control surface is
u PvV
gz
hV
gz
2
2
2
2
Where the time rate change of the energy of the control volume has been written
as ECV
21
Steady-State, Steady-Flow Processes
Most energy conversion devices operate steadily over long periods of time. The
rates of heat transfer and work crossing the control surface are constant with time.
The states of the mass streams crossing the control surface or boundary are constant
with time. Under these conditions the mass and energy content of the control volume
are constant with time.
dm
dtm
dE
dtE
CVCV
CVCV
0
0
Steady-state, Steady-Flow Conservation of Mass:
( / )m m kg sin out Steady-state, steady-flow conservation of energy
Since the energy of the control volume is constant with time during the steady-state,
steady-flow process, the conservation of energy principle becomes
22
E E E kWin out system
Rate of net energy transfer by heat, work, and mass
Rate change in internal, kinetic, potential, etc., energies
( )
0 or
or
Considering that energy flows into and from the control volume with the mass, energy
enters because heat is transferred to the control volume, and energy leaves because
the control volume does work on its surroundings, the steady-state, steady-flow first
law becomes
23
Often this result is written as
where
Q Q Q
W W W
net in out
net out in
Steady-state, steady-flow for one entrance and one exit
A number of thermodynamic devices such as pumps, fans, compressors, turbines,
nozzles, diffusers, and heaters operate with one entrance and one exit. The steady-
state, steady-flow conservation of mass and first law of thermodynamics for these
systems reduce to
24
where the entrance to the control volume is state 1 and the exit is state 2 and is the
mass flow rate through the device.
When can we neglect the kinetic and potential energy terms in the first law?
Consider the kinetic and potential energies per unit mass.
m
keV
2
2
For V = 45m
s
V = 140m
s
kem s kJ kg
m s
kJ
kg
kem s kJ kg
m s
kJ
kg
( / ) /
/
( / ) /
/
45
2
1
10001
140
2
1
100010
2
2 2
2
2 2
pe gz
For z m pem
sm
kJ kg
m s
kJ
kg
z m pem
sm
kJ kg
m s
kJ
kg
100 9 8 1001
10000 98
1000 9 8 10001
10009 8
2 2 2
2 2 2
./
/.
./
/.
25
When compared to the enthalpy of steam (h 2000 to 3000 kJ/kg) and the enthalpy
of air (h 200 to 6000 kJ/kg), the kinetic and potential energies are often neglected.
When the kinetic and potential energies can be neglected, the conservation of energy
equation becomes
( ) ( )Q W m h h kW 2 1
We often write this last result per unit mass flow as
q w h h kJ kg ( ) ( / )2 1
m
w
W
m
where and .
Some Steady-Flow Engineering Devices
Below are some engineering devices that operate essentially as steady-state, steady-
flow control volumes.
26
We may be familiar with nozzles, diffusers, turbines,
compressors, compressors, heat exchangers, and mixing
devices. However, the throttle valve may be a new device to
many of us. The Throttle may be a simple as the expansion
tube used in automobile air conditioning systems to cause
the refrigerant pressure drop between the exit of the
condenser and the inlet to the evaporator.
27
SOME STEADY-FLOW ENGINEERING DEVICES
Many engineering devices operate essentially under the same conditions
for long periods of time. The components of a steam power plant (turbines,
compressors, heat exchangers, and pumps), for example, operate nonstop for
months before the system is shut down for maintenance. Therefore, these devices
can be conveniently analyzed as steady-flow devices.
A modern land-based gas turbine used for electric power
production. This is a General Electric LM5000 turbine. It
has a length of 6.2 m, it weighs 12.5 tons, and produces
55.2 MW at 3600 rpm with steam injection.
28
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.
The cross-sectional area of a nozzle
decreases in the flow direction for
subsonic flows and increases for
supersonic flows. The reverse is true
for diffusers.
Energy
balance for
a nozzle or
diffuser:
29
Deceleration
of Air in a
Diffuser
30
Acceleration
of Steam in a
Nozzle
= 2.8 kJ/kg
Steam
5 kg/s
1 1.8 MPa, 400C
0.02 m2
2 1.4 MPa
275 m/s
31
Turbines and
Compressors Turbine drives the electric generator In
steam, gas, or hydroelectric power plants.
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.
Compressors, 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.
A fan increases the pressure of a gas
slightly and is mainly used to mobilize a
gas.
A compressor is capable of compressing
the gas to very high pressures.
Pumps work very much like compressors
except that they handle liquids instead of
gases.
32
Compressing Air
by a Compressor
33
Power Generation
by a Steam
Turbine
34
Throttling
valves
Throttling valves are any kind of flow-restricting devices that
cause a significant pressure drop in the fluid.
What is the difference between a turbine and a throttling
valve?
The pressure drop in the fluid is often accompanied by a large
drop in temperature, and for that reason throttling devices are
commonly used in refrigeration and air-conditioning applications.
Energy
balance
35
Expansion of
Refrigerant-134a
in a Refrigerator