Chapter 5: Conservation of Mass
and Bernoulli 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.
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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.
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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.
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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
• The dot over a symbol is used to
indicate time rate of change.
• Flow rate across the entire
cross-sectional area of a pipe or
duct is obtained by integration
• While this expression for
is
exact, it is not always convenient
for engineering analyses.
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Chapter 5: Mass, Bernoulli, and Energy Equations
Average Velocity and Volume Flow Rate
• Integral in can be replaced with
average values of ρ and Vn
• For many flows variation of ρ is
very small:
• Volume flow rate is given by
• Note: many textbooks use Q
instead of
for volume flow rate.
• Mass and volume flow rates are
related by
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Chapter 5: Mass, Bernoulli, and Energy Equations
Continuity Equation
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Chapter 5: Mass, Bernoulli, and Energy Equations
Continuity Equation
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Chapter 5: Mass, Bernoulli, and Energy Equations
Continuity Equation
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Chapter 5: Mass, Bernoulli, and Energy Equations
Continuity Equation
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Chapter 5: Mass, Bernoulli, and Energy Equations
Continuity Equation
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Chapter 5: Mass, Bernoulli, and Energy Equations
Continuity Equation
Then the Continuity Equation
can be written in vector form
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Chapter 5: Mass, Bernoulli, and Energy Equations
Continuity Equation
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Chapter 5: Mass, Bernoulli, and Energy Equations
Continuity Equation
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Chapter 5: Mass, Bernoulli, and Energy Equations
Continuity Equation
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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
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/ρ, kinetic V2/g, and potential gz energy are the
forms of mechanical energy emech= P/ρ + V2/g + gz
• Mechanical energy change of a fluid during
incompressible flow becomes
• In the absence of loses, Δemech represents the work
supplied to the fluid (Δemech>0) or extracted from the fluid
(Δemech<0).
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
The Bernoulli Equation
• Assumptions to derive Bernoulli Equation are:
• Fluid is non-viscous
• Fluid is homogeneous and incompressible,
hence ρ = constant
• The flow is steady and continuous
• The flow is along the streamline
• Velocity is uniform over the section
• The only forces acting on the fluid are the gravity
forces and the pressure forces.
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
This is famous Euler Equation of Fluid Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
(Bernoulli Equation from Euler Equation:)
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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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: ρ = 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
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Chapter 5: Mass, Bernoulli, and Energy Equations
Energy Analysis of Steady Flows
• Divide by g to get each term in units of length
Magnitude of each term is now expressed as an
equivalent column height of fluid, i.e., Head
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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
• 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
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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
• Energy Grade Line (or
total energy)
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Navier-Stokes Equations of Motion
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Chapter 5: Mass, Bernoulli, and Energy Equations
Steady, Laminar flow between Fixed Parallel Plates (Poiseuilli flow)
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Chapter 5: Mass, Bernoulli, and Energy Equations
Steady, Laminar flow between Fixed Parallel Plates (Poiseuilli flow)
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Chapter 5: Mass, Bernoulli, and Energy Equations
Steady, Laminar flow between Fixed Parallel Plates (Poiseuilli flow)
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Chapter 5: Mass, Bernoulli, and Energy Equations
Steady, Laminar flow between Fixed Parallel Plates (Poiseuilli flow)
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Chapter 5: Mass, Bernoulli, and Energy Equations
Steady, Laminar flow between Fixed Parallel Plates (Poiseuilli flow)
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Chapter 5: Mass, Bernoulli, and Energy Equations
Steady, Laminar flow between Fixed Parallel Plates (Poiseuilli flow)
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Chapter 5: Mass, Bernoulli, and Energy Equations
Steady, Laminar flow between Fixed Parallel Plates (Poiseuilli flow)
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Chapter 5: Mass, Bernoulli, and Energy Equations
Steady, Laminar flow between Fixed Parallel Plates (Poiseuilli flow)
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Chapter 5: Mass, Bernoulli, and Energy Equations
Steady, Laminar flow between Fixed Parallel Plates (Poiseuilli flow)
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Chapter 5: Mass, Bernoulli, and Energy Equations
Steady, Laminar flow between Fixed Parallel Plates (Poiseuilli flow)
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Chapter 5: Mass, Bernoulli, and Energy Equations
Steady, Laminar flow between Fixed Parallel Plates (Poiseuilli flow)
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations
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Chapter 5: Mass, Bernoulli, and Energy Equations