bernoulli chapter 3
TRANSCRIPT
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2/21/2014 9:57 AM FLUIDS 1
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Chapter 3: The Bernoulli Equation
Newton’s Second Law F=ma along a streamline F=ma normal to a streamline Physical interpretations Static, Stagnation, Dynamic and Total Pressure Examples of use of the Bernoulli Equation The energy line and the hydraulic grade line Restrictions of use of the Bernoulli Equation
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Newton’s 2nd Law
The net force acting on the fluid particle must equal its mass times its acceleration
For inviscid fluid, we are assuming that the fluid motion is governed by pressure and gravity forces only
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Newton’s 2nd Law (cont.)
Streamlines – lines that are tangent to the velocity vectors throughout the flow field
Along the streamline,
Normal to the streamline,
Note : For steady, inviscid, incompressible flow, the pressure variation across streamline is merely hydrostatic (because of gravity alone), even though the fluid is in motion
s
VVas
R
Van
2
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F=ma along a streamline The equation of motion along the streamline
direction is
◦ A change in fluid particle speed is accomplished by the appropriate combination of pressure and particle weight along the streamline
s
VV
s
p
sin
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F=ma along a streamline (cont.)
Rearranging and integrating the equation for inviscid, incompressible flow gives
zVp 2
2
1Constant along a streamline
Applicable to steady flows only
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Example 1
Some animals have learned to take advantage of the Bernoulli effect. For example, a typical prairie dog burrow contains two entrances – a flat front door and a mounded back door. When the wind blows with velocity Vo across the front door, the average velocity across the back door is greater than Vo because of the mound. Assume the air velocity across the back door is 1.07Vo. For a wind velocity of 6 m/s, what pressure difference, p1-p2, is generated to provide a fresh air flow within the burrow.
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F=ma normal to a streamline
For steady, inviscid, incompressible flow
zdnR
Vp
2
Constant across the streamline
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F=ma normal to a streamline (cont.)
When the fluid travels along a curved path, a net force directed towards the center of curvature is required, due to either gravity or pressure or both.
When the streamlines are straight, the centrifugal effect is negligible and the pressure variation across the streamline is due to gravity alone even though the fluid is in motion.
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Physical Interpretation
An equivalent form of the Bernoulli Equation
Hzg
Vp
2
2
Constant along a streamline
Elevation head -related to potential energy of the particle
Velocity head -vertical distance needed for the fluid to fall freely (neglecting friction) if it is to reach V from rest
Pressure head -height of the column of fluid that is needed to produce the pressure p
Total head
Pressure head + Elevation head = Piezometric head
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Static, stagnation, Dynamic and Total Pressure
TpzVp 2
2
1Constant along a streamline
Dynamic pressure
Hydrostatic pressure
Total pressure
Static pressure Actual thermodynamic pressure
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Static, stagnation, Dynamic and Total Pressure (cont.)
For 2 points at the same height with V2=0
2
1122
1Vpp
Figure 1 : Measurement of
static and stagnation
pressures
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Static, stagnation, Dynamic and Total Pressure (cont.)
Then, p2 is called the stagnation pressure
The pressure at stagnation point, p2, is greater than the static pressure, p1
There is a stagnation point on any stationary body that is placed into a flowing fluid
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Example 2
Air is drawn into a small open circuit wind tunnel as shown. Atmospheric pressure is 98.7 kPa (abs) and the temperature is 27C. If viscous effects are negligible, determine the pressure at the stagnation point on the nose of the airplane. Also determine the manometer reading, h, for the manometer attached to the static pressure tap within the test section of the wind tunnel if the air velocity within the test section is 60 m/s. Elevation of (1),(2) and (3) is assume the same.
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Pitot Static Tube
Fluid speed can be calculated if we know the values of the static and stagnation pressures in a fluid.
Figure 2 : The Pitot static tube
43
14
2
21
3
2 ppV
ppp
Vpp
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Pitot Static Tube
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Examples of use of the Bernoulli Equation
Free Jets
Assumptions ◦ z1=h, z2=0 ◦ Reservoir is large, V1=0 ◦ Reservoir is open to atmosphere, p1=0 gage ◦ Fluid leaves as a free jet, p2=0
◦ Once outside nozzle, the stream continues as a free jet, p5=0
Figure 3 : Vertical flow from a tank
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Examples of use of the Bernoulli Equation
Free Jets (cont.)
Figure 4 : Vertical flow from a tank
ghh
22v2
Hhg 2v5
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Example 3
For the system in the figure, h= 36 ft and the diameter of the side opening is 2 in. Find the
(a) Jet velocity in units of ft/s
(b) Volume flow rate in units of gallon per min. (gpm)
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Solution: Example 3
P1 + ½ ρV12 + γZ1 = P2 + ½ ρV2
2 + γz2 P1 = 0 V1 = 0 Z1 = h Z2 = 0 P2 = 0 d1 = 2 in = 2/12 = 0.167 ft
Note that: γ = ρg, thus: γ/ρ = g
γ Z1 = ½ ρV22
a)→ V = (2gh)½ = (2 x 32.2 ft/s x 36 ft)½ = 48.15 ft/s
0 0 0 0
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Solution: Example 3
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Example 4
A smooth plastic, 10-m long garden hose with an inside diameter of 15 mm is used to drain a wading pool as shown. If viscous effects are neglected, what is the flowrate from the pool?
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Solution: Example 4
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Examples of use of the Bernoulli Equation
Confined Flows ◦ In many cases, fluid is confined and its pressure cannot be prescribed a priori – need to use the concept of conservation of mass
Figure 5 : Steady flow into and out of a tank
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Examples of use of the Bernoulli Equation
Confined Flows (cont.) ◦ In such case, mass is conserved, i.e. inflow rate must equal to the outflow rate
◦ In general, following Bernoulli, an increase in velocity (could be due to reduction of flow area) is accompanied by a decrease in pressure
◦ For flows of liquids, this may result in cavitation, a potentially dangerous situation that results when liquid pressure is reduced to vapor pressure and the liquid “boils”.
ible)incompress(ifor 211222111 VAVAVAVA
AVmAVQ ,
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Cavitation
• Cavitation is the formation
and then immediate implosion
of cavities in a liquid – i.e.
small liquid-free zones
("bubbles") – that are the
consequence of forces acting
upon the liquid.
• It usually occurs when a
liquid is subjected to rapid
changes of pressure that cause
the formation of cavities where
the pressure is relatively low.
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Example 5
Water is siphoned from the tank as shown. The water barometer indicates a reading of 30.2 ft. Determine the maximum value of h allowed without cavitation occurring. Note that the pressure of the vapor in the closed end of the barometer equals the vapor pressure.
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Solution: Example 5
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Solution: Example 5
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Example 6
Water flows from a large tank as shown. Atmospheric pressure is 14.5 psia and the vapor pressure is 2.88 psia. If viscous effects are neglected, at what height, h, will cavitation begin?
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Solution: Example 6
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Solution: Example 6
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Examples of use of the Bernoulli Equation
Free Jets (cont.)
– If exit of tank is not
smooth, well
contoured nozzle, the
diameter of the jet will
be less than the
diameter of the hole –
vena contracta effect
– Contraction coef.,
Cc=Aj/Ah
hole jet
Figure 3 : Typical flow patterns and
contraction coef. for various round exit
configurations
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Examples of use of the Bernoulli Equation
Flowrate measurement ◦ Assumptions – steady, inviscid and incompressible
Figure 6 : Typical devices for measuring
flowrate in pipes
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Examples of use of the Bernoulli Equation
Flowrate measurement (cont.) ◦ Between points (1) and (2)
2
12
212
2211
2
221
2
2
121
1
1
)(2
hence
and
AA
ppAQ
VAVAQ
VpVp
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Examples of use of the Bernoulli Equation
Flowrate measurement (cont.) ◦ The actual measured flowrate, Qactual will be smaller than this theoretical results because of the assumptions made in deriving the Bernoulli Equation
◦ Other flowmeters based on Bernoulli equation are used to measure flowrates in open channels such as flumes and irrigation ditches.
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Orifice – Nozzle – Venturi Flow Meter
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The Energy Line and the Hydraulic Grade Line
Energy line is a line that represents the total head available to the fluid
Under the assumptions of the Bernoulli equation, the energy line is horizontal
If the fluid velocity changes along the streamline, the hydraulic grade line will not be horizontal
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The Energy Line and the Hydraulic Grade Line (cont.)
Figure 7 : Representation of the
energy line and the hydraulic
grade line
Measures the sum of
the pressure head and
the elevation head. The
sum is called
piezometric head
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The Energy Line and the Hydraulic Grade Line (cont.)
Figure 8 : Representation of the energy line and the hydraulic grade line for flow from a tank
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The Energy Line and the Hydraulic Grade Line (cont.)
The distance from the pipe to HGL in Fig. 7 indicates the pressure within the pipe. ◦ If the pipe lies below HGL the pressure within the pipe is positive
◦ If the pipe lies above HGL the pressure is negative
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Example 7
Draw the energy line and the hydraulic grade line for the flow of Example 6.
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Restrictions of use of the Bernoulli Equation
Assumptions involved in deriving the Bernoulli equation ◦ Fluid is incompressible – ok with liquids
◦ Flow is steady
◦ Inviscid flow In the absence of viscous effects, the total energy of the
system remains constant
◦ There are no mechanical devices in the system between the two points along the streamline to which the equation is applied