Transcript CE 150 Fluid Mechanics

CE 150
Fluid Mechanics
G.A. Kallio
Dept. of Mechanical Engineering,
Mechatronic Engineering &
Manufacturing Technology
California State University, Chico
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Elementary Fluid
Dynamics
Reading: Munson, et al.,
Chapter 3
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Inviscid Flow
• In this chapter we consider “ideal”
fluid motion known as inviscid flow;
this type of flow occurs when either
1)   0 (only valid for He near 0 K), or
2) viscous shearing stresses are negligible
• The inviscid flow assumption is often
valid in regions removed from solid
surfaces; it can be applied to many
problems involving flow through
pipes and flow over aerodynamic
shapes
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Newton’s 2nd Law for
a Fluid Particle
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Newton’s 2nd Law for
a Fluid Particle
• The equation of motion for a fluid
particle in a steady inviscid flow:





ma  F  Fp  Fg  Fext



dV
m
 Fp  Fg
dt
• We consider force components in
two directions: along a streamline (s)
and normal to a streamline (n):
dV
mV
 Fps  W sin 
ds
V2
m
 Fpn  W cos 

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Newton’s 2nd Law
Along a Streamline
• Noting that
dz
m  V , W sin   gV ,
ds
dp
and Fps   V ,
ds
we have:
dV
dz dp
V
  g 
ds
ds ds
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Newton’s 2nd Law
Along a Streamline
• Integrating along the streamline:
dp

 12 V 2  gz  constant
s
• If the fluid density  remains
constant
p

 12 V 2  gz  constant along a streamline
or
p  12 V 2  gz  constant along a streamline
• This is the Bernoulli equation
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Newton’s 2nd Law
Across a Streamline
• A similar analysis applied normal to
the streamline for a fluid of constant
density yields
V2
p    dn  gz  constant

n
• This equation is not as useful as the
Bernoulli equation because the
radius of curvature () of the
streamline is seldom known
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Physical Interpretation of
the Bernoulli Equation
p  12 V 2  gz  constant along a streamline
• Acceleration of a fluid particle is due
to an imbalance of pressure forces
and fluid weight
• Conservation equation involving
three energy processes:
– kinetic energy
– potential energy
– pressure work
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Alternate Form of the
Bernoulli Equation
p V2

 z  constant along a streamline
g 2 g
• Pressure head (p/g) - height of
fluid column needed to produce a
pressure p
• Velocity head (V2/2g) - vertical
distance required for fluid to fall
from rest and reach velocity V
• Elevation head (z) - actual elevation
of the fluid w.r.t. a datum
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Bernoulli Equation
Restrictions
• The following restrictions apply to the
use of the (simple) Bernoulli
equation:
1) fluid flow must be inviscid
2) fluid flow must be steady (i.e., flow
properties are not f(t) at a given location)
3) fluid density must be constant
4) equation must be applied along a
streamline (unless flow is irrotational)
5) no energy sources or sinks may exist
along streamline (e.g., pumps, turbines,
compressors, fans, etc.)
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Using the Bernoulli
Equation
• The Bernoulli equation can be
applied between any two points, (1)
and (2), along a streamline:
p1  12 V12  gz1  p2  12 V22  gz2
• Free jets - pressure at the surface is
atmospheric, or gage pressure is
zero; pressure inside jet is also zero
if streamlines are straight
• Confined flows - pressures cannot be
prescribed unless velocities and
elevations are known
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Mass and Volumetric
Flow Rates
• Mass flow rate: fluid mass
conveyed per unit time [kg/s]
 
m

A
Vn dA
where Vn = velocity normal to area [m/s]
 = fluid density [kg/m3]
A = cross-sectional area [m2]
– if  is uniform over the area A and the
average velocity V is used, then
  AV
m
• Volumetric flow rate [m3/s]:
Q  AV
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Conservation of Mass
• “Mass can neither be created nor
destroyed”
• For a control volume undergoing
steady fluid flow, the rate of mass
entering must equal the rate of mass
exiting:
m 1  m 2
1 A1V1   2 A2V2
• If  = constant, then
A1V1  A2V2 or Q1  Q2
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The Bernoulli Equation
in Terms of Pressure
• Each term of the Bernoulli equation
can be written to represent a
pressure:
p  12 V 2  gz  constant ( pT )
• pgh : this is known as the
hydrostatic pressure; while not a real
pressure, it represents the possible
pressure in the fluid due to changes
in elevation
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The Bernoulli Equation
in Terms of Pressure
• p : this is known as the static
pressure and represents the actual
thermodynamic pressure of the fluid
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The Bernoulli Equation
in Terms of Pressure
• The static pressure at (1) in Figure
3.4 can be measured from the liquid
level in the open tube as pgh
•
1
2
V 2 : this is known as the dynamic
pressure; it is the pressure measured
by the fluid level (pgH) in the
stagnation tube shown in Figure 3.4
minus the static pressure; thus, it is
the pressure due to the fluid velocity
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The Bernoulli Equation
in Terms of Pressure
• The stagnation pressure is the sum
of the static and dynamic pressures:
p2  p1  12 V12
– the stagnation pressure exists at a
stagnation point, where a fluid
streamline abruptly terminates at the
surface of a stationary body; here, the
velocity of the fluid must be zero
• Total pressure (pT) is the sum of the
static, dynamic, and hydrostatic
pressures
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Velocity and Flow
Measurement
• Pitot-static tube - utilizes the static
and stagnation pressures to measure
the velocity of a fluid flow (usually
gases):
V  2( p3  p4 ) / 
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Velocity and Flow
Measurement
• Orifice, Nozzle, and Venturi meters restriction devices that allow
measurement of flow rate in pipes:
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Velocity and Flow
Measurement
• Bernoulli equation analysis yields
the following equation for orifice,
nozzle, and venturi meters:
– Theoretical flowrate:
Qideal  A2
2( p1  p2 )
 [1  ( A2 / A1 )2 ]
– Actual flowrate:
Qactual  CQideal
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(C  1)
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Velocity and Flow
Measurement
• Sluice gates and weirs - restriction
devices that allow flow rate
measurement of open-channel flows:
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Velocity and Flow
Measurement
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