Chapter 5: Mass, Bernoulli, and
Energy Equations
Eric G. Paterson
Department of Mechanical and Nuclear Engineering
The Pennsylvania State University
Spring 2005
Note to Instructors
These slides were developed1 during the spring semester 2005, as a teaching aid
for the undergraduate Fluid Mechanics course (ME33: Fluid Flow) in the Department of
Mechanical and Nuclear Engineering at Penn State University. This course had two
sections, one taught by myself and one taught by Prof. John Cimbala. While we gave
common homework and exams, we independently developed lecture notes. This was
also the first semester that Fluid Mechanics: Fundamentals and Applications was
used at PSU. My section had 93 students and was held in a classroom with a computer,
projector, and blackboard. While slides have been developed for each chapter of Fluid
Mechanics: Fundamentals and Applications, I used a combination of blackboard and
electronic presentation. In the student evaluations of my course, there were both positive
and negative comments on the use of electronic presentation. Therefore, these slides
should only be integrated into your lectures with careful consideration of your teaching
style and course objectives.
Eric Paterson
Penn State, University Park
August 2005
1 These
slides were originally prepared using the LaTeX typesetting system (http://www.tug.org/)
and the beamer class (http://latex-beamer.sourceforge.net/), but were translated to PowerPoint for
wider dissemination by McGraw-Hill.
ME33 : Fluid Flow
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Chapter 5: Mass, Bernoulli, and Energy Equations
Introduction
This chapter deals with 3 equations
commonly used in fluid mechanics
The mass equation is an expression of the
conservation of mass principle.
The Bernoulli equation is concerned with the
conservation of kinetic, potential, and flow
energies of a fluid stream and their
conversion to each other.
The energy equation is a statement of the
conservation of energy principle.
ME33 : Fluid Flow
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Chapter 5: Mass, Bernoulli, and Energy Equations
Objectives
After completing this chapter, you should be able
to
Apply the mass equation to balance the incoming
and outgoing flow rates in a flow system.
Recognize various forms of mechanical energy, and
work with energy conversion efficiencies.
Understand the use and limitations of the Bernoulli
equation, and apply it to solve a variety of fluid flow
problems.
Work with the energy equation expressed in terms of
heads, and use it to determine turbine power output
and pumping power requirements.
ME33 : Fluid Flow
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Chapter 5: Mass, Bernoulli, and Energy Equations
Conservation of Mass
Conservation of mass principle is one of the
most fundamental principles in nature.
Mass, like energy, is a conserved property, and
it cannot be created or destroyed during a
process.
For closed systems mass conservation is implicit
since the mass of the system remains constant
during a process.
For control volumes, mass can cross the
boundaries which means that we must keep
track of the amount of mass entering and leaving
the control volume.
ME33 : Fluid Flow
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Chapter 5: Mass, Bernoulli, and Energy Equations
Mass and Volume Flow Rates
The amount of mass flowing
through a control surface per unit
time is called the mass flow rate
and is denoted m
The dot over a symbol is used to
indicate time rate of change.
Flow rate across the entire crosssectional area of a pipe or duct is
obtained by integration
m
  m   V
Ac
n
dAc
Ac
While this expression for m is
exact, it is not always convenient
for engineering analyses.
ME33 : Fluid Flow
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Chapter 5: Mass, Bernoulli, and Energy Equations
Average Velocity and Volume Flow Rate
Integral in m can be replaced with
average values of  and Vn
V avg 
1
Ac
V
n
dAc
Ac
For many flows variation of  is
very small: m   V avg Ac
Volume flow rate V is given by
V 
V
n
dAc  V avg Ac  VAc
Ac
Note: many textbooks use Q
instead of V for volume flow rate.
Mass and volume flow rates are
related by m   V
ME33 : Fluid Flow
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Chapter 5: Mass, Bernoulli, and Energy Equations
Conservation of Mass Principle
The conservation of
mass principle can be
expressed as
m in  m o u t 
dmCV
dt
Where m in and m o u t are
the total rates of mass
flow into and out of the
CV, and dmCV/dt is the
rate of change of mass
within the CV.
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Chapter 5: Mass, Bernoulli, and Energy Equations
Conservation of Mass Principle
For CV of arbitrary shape,
rate of change of mass within the CV
dm C V
dt
d

dt

 dV
CV
net mass flow rate
m net 
  m   V
CS
CS
n
dA 
  V n  dA
CS
Therefore, general conservation
of mass for a fixed CV is:
d
dt
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
CV
 dV 
   V n  dA  0
CS
Chapter 5: Mass, Bernoulli, and Energy Equations
Steady—Flow Processes
For steady flow, the total
amount of mass contained in
CV is constant.
Total amount of mass entering
must be equal to total amount
of mass leaving
mm
in
out
For incompressible flows,
V
in
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n
An 
V
n
An
out
Chapter 5: Mass, Bernoulli, and Energy Equations
Mechanical Energy
Mechanical energy can be defined as the form of
energy that can be converted to mechanical work
completely and directly by an ideal mechanical device
such as an ideal turbine.
Flow P/, kinetic V2/g, and potential gz energy are the
forms of mechanical energy emech= P/  V2/g + gz
Mechanical energy change of a fluid during
incompressible flow becomes
D e m ech 
P2  P1

V 2  V1
2

2
2
 g  z 2  z1 
In the absence of loses, Demech represents the work
supplied to the fluid (Demech>0) or extracted from the fluid
(Demech<0).
ME33 : Fluid Flow
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Chapter 5: Mass, Bernoulli, and Energy Equations
Efficiency
Transfer of emech is usually accomplished by a rotating
shaft: shaft work
Pump, fan, propulsion: receives shaft work (e.g., from
an electric motor) and transfers it to the fluid as
mechanical energy
Turbine: converts emech of a fluid to shaft work.
In the absence of irreversibilities (e.g., friction),
mechanical efficiency of a device or process can be
defined as
h m ech 
E m ech , out
E m ech , in
1
E m ech ,loss
E m ech ,in
If hmech < 100%, losses have occurred during conversion.
ME33 : Fluid Flow
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Chapter 5: Mass, Bernoulli, and Energy Equations
Pump and Turbine Efficiencies
In fluid systems, we are usually interested in
increasing the pressure, velocity, and/or
elevation of a fluid.
In these cases, efficiency is better defined as
the ratio of (supplied or extracted work) vs. rate
of increase in mechanical energy
D E m ech , fluid
h pum p 
W shaft , in
h turbine 
W shaft , out
D E m ech , fluid
Overall efficiency must include motor or
generator efficiency.
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Chapter 5: Mass, Bernoulli, and Energy Equations
General Energy Equation
One of the most fundamental laws in nature is the 1st
law of thermodynamics, which is also known as the
conservation of energy principle.
It states that energy can be neither created nor
destroyed during a process; it can only change forms
Falling rock, picks up speed
as PE is converted to KE.
If air resistance is neglected,
PE + KE = constant
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Chapter 5: Mass, Bernoulli, and Energy Equations
General Energy Equation
The energy content of a closed
system can be changed by two
mechanisms: heat transfer Q and
work transfer W.
Conservation of energy for a closed
system can be expressed in rate
form as
Q net , in  W net , in 
dE sys
dt
Net rate of heat transfer to the
system:
Q net , in  Q in  Q out
Net power input to the system:
W net , in  W in  W out
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Chapter 5: Mass, Bernoulli, and Energy Equations
General Energy Equation
Where does expression for pressure work
come from?
When piston moves down ds under the
influence of F=PA, the work done on the
system is Wboundary=PAds.
If we divide both sides by dt, we have
 W pressure   W boundary  P A
ds
dt
 P A V piston
For generalized control volumes:
 W pressure   PdAV n   PdA V  n 
Note sign conventions:
n is outward pointing normal
Negative sign ensures that work done is
positive when is done on the system.
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Chapter 5: Mass, Bernoulli, and Energy Equations
General Energy Equation
Recall general RTT
dB sys
dt

d

dt
CV
 bdV  
CS
 b  V r n  dA
“Derive” energy equation using B=E and b=e
dE sys
dt
 Q net , in  W net ,in 
d

dt
CV
 edV  
CS
 e  V r n  dA
Break power into rate of shaft and pressure work
W net , in  W shaft , net ,in  W pressure , net ,in  W shaft , net ,in 
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 P V n  dA
Chapter 5: Mass, Bernoulli, and Energy Equations
General Energy Equation
Moving integral for rate of pressure work
to RHS of energy equation results in:
Q net , in  W shaft , net , in 
d
dt
P

 edV     e  e V r  n dA


CS 


CV

Recall that P/ is the flow work, which is
the work associated with pushing a fluid
into or out of a CV per unit mass.
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Chapter 5: Mass, Bernoulli, and Energy Equations
General Energy Equation
As with the mass equation, practical analysis is
often facilitated as averages across inlets and
exits
Q net , in  W shaft , net ,in 
m 
  V
d
dt

 edV 

out
CV
P

m  e



in
P

m  e



 n dAc
AC
Since e=u+ke+pe = u+V2/2+gz
Q net , in  W shaft , net , in 
ME33 : Fluid Flow
d
dt

CV
 edV 

out
2
P

V
m u 
 gz  
2


19

in
2
P

V
m u 
 gz 
2


Chapter 5: Mass, Bernoulli, and Energy Equations
Energy Analysis of Steady Flows
Q net , in  W shaft , net ,in 

out
2


V
mh 
 gz  
2



in
2


V
mh 
 gz 
2


For steady flow, time rate of change of the
energy content of the CV is zero.
This equation states: the net rate of energy
transfer to a CV by heat and work transfers
during steady flow is equal to the difference
between the rates of outgoing and incoming
energy flows with mass.
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Chapter 5: Mass, Bernoulli, and Energy Equations
Energy Analysis of Steady Flows
For single-stream
devices, mass flow rate
is constant.
V 2  V1
2
q n et , in  w sh a ft , n et , in  h 2  h1 
P1
w sh a ft , n et , in 
P1
1
ME33 : Fluid Flow

21
V1
1

V1
2
2
 gz1 
2
2
 gz1  w p u m p 
P2
2
P2
2
2

V2
2
2
2
 g  z 2  z1 
 gz 2   u 2  u 1  q n et , in 
2

V2
2
 gz 2  w tu rb in e  e m ech , lo ss
Chapter 5: Mass, Bernoulli, and Energy Equations
Energy Analysis of Steady Flows
Divide by g to get each term in units of length
P1
1 g

V1
2
2g
 z1  h p u m p 
2
P2
2g

V2
2g
 z 2  htu rb in e  h L
Magnitude of each term is now expressed as an
equivalent column height of fluid, i.e., Head
ME33 : Fluid Flow
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Chapter 5: Mass, Bernoulli, and Energy Equations
The Bernoulli Equation
If we neglect piping losses, and have a system without
pumps or turbines
P1
1 g

V1
2
2g
 z1 
P2
2g
2

V2
2g
 z2
This is the Bernoulli equation
It can also be derived using Newton's second law of
motion (see text, p. 187).
3 terms correspond to: Static, dynamic, and hydrostatic
head (or pressure).
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Chapter 5: Mass, Bernoulli, and Energy Equations
The Bernoulli Equation
Limitations on the use of the Bernoulli Equation
Steady flow: d/dt = 0
Frictionless flow
No shaft work: wpump=wturbine=0
Incompressible flow:  = constant
No heat transfer: qnet,in=0
Applied along a streamline (except for irrotational
flow)
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Chapter 5: Mass, Bernoulli, and Energy Equations
HGL and EGL
It is often convenient
to plot mechanical
energy graphically
using heights.
Hydraulic Grade Line
HGL 
P
g
 z
Energy Grade Line
(or total energy)
EGL 
ME33 : Fluid Flow
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P
g

V
2
z
2g
Chapter 5: Mass, Bernoulli, and Energy Equations
The Bernoulli Equation
The Bernoulli equation
is an approximate relation
between pressure,
velocity, and elevation
and is valid in regions of
steady, incompressible
flow where net frictional
forces are negligible.
Equation is useful in flow
regions outside of
boundary layers and
wakes.
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Chapter 5: Mass, Bernoulli, and Energy Equations