Pharos University

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• Pharos University
• ME 259 Fluid Mechanics for Electrical Students
• Application of Bernoulli Equation

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• Eulers equation of motion
• Bernoulli equation

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INTRODUCTION

• The three equations commonly used in fluid
  mechanics are
• the mass, Bernoulli, and energy equations.
• The mass equation is an expression of the
  conservation of mass principle.
• The Bernoulli equation Conservation of kinetic,
  potential, and flow energies ( viscous forces are
  negligible.)

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Fluid Motion

• Two ways to describe fluid motion
• Lagrangian
• Follow particles around
• Eularian
• Watch fluid pass by a point or an entire region
• Flow pattern
• Streamlines velocity is tangent to them

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STEADY AND UNSTEADY FLOW

• Steady flow the flow in which conditions at any
  point do not change with time is called steady
  flow.
• Then, etc.
• Unsteady flow the flow in which conditions at
  any point change with time, is called unsteady
  flow.
• Then, etc.

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UNIFORM AND NON-UNIFORM

• The flow in which the conditions at all points
  are the same at the same instant is uniform flow.
• The flow in which the conditions vary from point
  to point at the
• same instant is non-uniform flow.

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ACCELERATION

• Acceleration rate of change of velocity
• Components
• Normal changing direction
• Tangential changing speed

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ACCELERATION

• Cartesian coordinates
• In steady flow ?u/?t 0 , local acceleration is
  zero.
• In unsteady flow ?u/?t ? 0 local acceleration
  Occurs.

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Example

• Valve at C is opened slowly
• The flow at B is non uniform
• The flow at A is uniform

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Laminar vs Turbulent Flow

• Turbulent

• Laminar

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Flow Rate

• Volume rate of flow
• Constant velocity over cross-section
• Variable velocity
• Mass flow rate

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Flow Rate

• Only x-direction component of velocity (u)
  contributes to flow through cross-section

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CV Inflow Outflow

• Area vector always points outward from CV

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Examples

• Discharge in a 25-cm pipe is 0.03 m3/s. What is
  the average velocity?

• A pipe whose diameter is 8 cm transports air with
  a temp. of 20oC and pressure of 200 kPa abs. At
  20 m/s. What is the mass flow rate?

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Example The velocity distribution in a circular
duct is , where r is
the radial location in the duct, R is the duct
radius, and Vo is the velocity on the axis. Find
the ratio of the mean velocity to the velocity on
the axis.

• Find

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Example Air (? 1.2 kg/m3) enters the duct
shown

• Find
• V/10y/.5 ? V20y
• dA1dy

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Example In this flow passage the discharge is
varying with time according to the following
expression . At time t0.5
s, it is known that at section A-A the velocity
gradient in the S direction is 2m/s per meter.
Given that Qo, Q1, and to are constants with
values of 0.985 m3/s, 0.5 m3/s, and 1 s,
respectively, and assuming that one-dimensional
flow, answer the following questions for time
t0.5 s.a. What is the velocity a A-A?b. What
is the local acceleration at A-A?c. What is the
convective acceleration at A-A?
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Example
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Systems, Control Volume, and Control Surface

• System (sys)
• A fluid system contains the
• same fluid particles.
• Mass does not cross the
• system boundaries.
• Thus the mass of the system
• is constant.

Control Volume (C.V) A control volume
is a selected volumetric region in space. Its
shape and position may change with time.(Open
System)
Control Surface (C.S) The surface
enclosing the control volume is called the
control surface.
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Systems, Control Volume, and Control Surface
(continued )

• Consider the tank shown, assume
• the control volume is defined by the tank walls
  and the top of the liquid.
• The control surface that encloses the control
  volume is designated by the dashed line.
• The liquid in the tank at time t is elected as
  the system and is indicated by the solid line.
• At this instant in time, the system completely
  occupies the control volume and is contained by
  the control surface. Thus, at this time

During the same period some liquid has entered
the control volume from the left, the amount
being
After a time some liquid has flowed out of
the control volume to the right. The amount that
flowed out is
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Systems, Control Volume, and Control Surface
(continued )

• Now the system has been deformed as shown in Fig.
  (b).
• Part of the system is the liquid that has flowed
  out across the control surface.
• The system remaining in the control volume has
  been deformed by the mass that has flowed in
  across the control surface.
• Also, the height of the control volume has
  changed to accommodate the net flow into the
  tank.
• The mass of the system at time t ?t can be
  determined by taking the mass in the control
  volume, subtracting the mass that entered, and
  adding the mass that left.

Subtracting (1) from (2), we have
Dividing by ?t and taking the limit as ?t ?0
yields
The equation relates the rate of change of the
mass of the system to the rate of change of mass
in the control volume plus the net outflow
(efflux) across the control surface
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Systems, Control Volume, and Control Surface
(continued )

• By definition, the mass of the system is constant
  so
• 
  Lagrangian statement
• And Eq. (3) becomes
• The corresponding Eulerian statement and can be
  written as

This equation states that there is an increasing
mass in the control volume if there is a net mass
influx through the control surface and decreasing
mass if there is a net mass efflux. This is
identified as the continuity equation.
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Systems

• Laws of Mechanics
• Written for systems
• System arbitrary quantity of mass of fixed
  identity
• Fixed quantity of mass, m

• Conservation of Mass
• Mass is conserved and does not change

• Momentum
• If surroundings exert force on system, mass will
  accelerate

• Energy
• If heat is added to system or work is done by
  system, energy will change

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Control Volumes

• Solid Mechanics
• Follow the system, determine what happens to it
• Fluid Mechanics
• Consider the behavior in a specific region or
  Control Volume
• Convert System approach to CV approach
• Look at specific regions, rather than specific
  masses
• Reynolds Transport Theorem
• Relates time derivative of system properties to
  rate of change of property in CV

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CV Inflow Outflow
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Continuity Equation

• In the case of the continuity equation, the
  extensive property in the control volume equation
  is the mass of the system, Msys, and the
  corresponding intensive variable, b, is the mass
  per unit mass, or
• Substituting b equal to unity in the control
  volume equation yields the general form of the
  continuity equation.
• This is sometimes called the integral form of the
  continuity equation.

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Continuity Equation

• The term on the left is the rate of change of the
  mass of the system. However, by definition, the
  mass of a system is constant. Therefore the
  left-hand side of the equation is zero, and the
  equation can be written as
• This is the general form of the continuity
  equation. It states that the net rate of the
  outflow of mass from the control volume is equal
  to the rate of decrease of mass within the
  control volume.
• The continuity equation involving flow streams
  having a uniform velocity across the flow section
  is given as

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Example at a certain time rate of rising is 0.1
cm/s

• Continuity equation

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Example Both pistons are moving to the left, but
piston A has a speed twice as great as that of
piston B. Then the water level in the tank is a)
rising, b) not moving up or down, c) falling.

• Select a CV that moves up and down with the water
  surface
• Continuity Equation

h
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Euler Equation

• Fluid element accelerating in l direction acted
  on by pressure and weight forces only (no
  friction)
• Newtons 2nd Law

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Example 1

• Given Steady flow. Liquid is decelerating at a
  rate of 0.3g.
• Find Pressure gradient in flow direction in
  terms of specific weight.

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Example 2

• Given g 10 kN/m3, pB-pA12 kPa.
• Find Direction of fluid acceleration.

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Example 3

• Given Steady flow. Velocity varies linearly
  with distance through the nozzle.
• Find Pressure gradient ½-way through the nozzle

V1/2(8030)/2 ft/s 55 ft/s
dV/dx (80-30) ft/s /1 ft 50 ft/s/ft
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Bernoulli Equation

• Consider steady flow along streamline
• s is along streamline, and t is tangent to
  streamline

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An alternative form of the Bernoulli equation is
expressed in terms of heads as The sum of the
pressure, velocity, and elevation heads is
constant along a streamline.
The Bernoulli equation states that the sum of the
kinetic, potential, and flow energies of a fluid
particle is constant along a streamline during
steady flow.
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Example 4

• Given Velocity in outlet pipe from reservoir is
  6 m/s and h 15 m.
• Find Pressure at A.
• Solution Bernoulli equation

Point 1
Point A
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Example 5

• Given D30 in, d1 in, h4 ft
• Find VA
• Solution Bernoulli equation

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Static, Stagnation, Dynamic, and Total Pressure
Bernoulli Equation
Hydrostatic Pressure
Dynamic Pressure
Static Pressure
Static Pressure moves along the fluid static
to the motion.
Dynamic Pressure due to the mean flow going to
forced stagnation.
Hydrostatic Pressure potential energy due to
elevation changes.
Following a streamline
Follow a Streamline from point 1 to 2
0
0, no elevation
0, no elevation
Total Pressure Dynamic Pressure Static
Pressure
Note
H gt h
In this way we obtain a measurement of the
centerline flow with piezometer tube.
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• Bernoulli equation The sum of flow, kinetic, and
  potential energies of a fluid particle along a
  streamline is constant.
• Each term in this equation has pressure units,
  and thus each term represents Energy per unit
  volune
• P is the static pressure.
• ?V2/2 is the dynamic pressure.
• ? gz is the hydrostatic pressure
• The sum of the static, dynamic, and hydrostatic
  pressures is called the total pressure. Bernoulli
  equation states that the total pressure along a
  streamline is constant.
• The sum of the static and dynamic pressures is
  called the stagnation pressure, and it is
  expressed as

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• The static, dynamic, and stagnation pressures are
  shown.
• When static and stagnation pressures are measured
  at a specified location, the fluid velocity at
  that location can be calculated from

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Stagnation Tube
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Stagnation Tube in a Pipe
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Pitot Tube
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Example Venturi Tube

• Given Water 20oC, V12 m/s, p150 kPa, D6 cm,
  d3 cm
• Find p2 and p3
• Solution Continuity Eq.
• Bernoulli Eq.

Nozzle velocity increases, pressure decreases
Diffuser velocity decreases, pressure increases
Similarly for 2 ? 3, or 1 ? 3
Pressure drop is fully recovered, since we
assumed no frictional losses
Knowing the pressure drop 1 ? 2 and d/D, we can
calculate the velocity and flow rate
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Ex

• Given Velocity in circular duct 30 m/s, air
  density 1.2 kg/m3.
• Find Pressure change between circular and
  square section.
• Solution Continuity equation
• Bernoulli equation

Air conditioning ( 60 oF)
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Ex

• Given r 1000 kg/m3 V1 30 m/s, and A2/A10.5,
  gm20000 N/m3
• Find Dh
• Solution Continuity equation
• Bernoulli equation

Heating ( 170 oF)

• Manometer equation

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Pitot Tube Application