Upload
lynne-snow
View
232
Download
7
Tags:
Embed Size (px)
Citation preview
Chapter 6: Bernoulli and energy equations
Fall 2009
University of PalestineCollege of Engineering & Urban PlanningApplied Civil Engineering
Lecturer:Eng. Eman Al.Swaity
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 2
OBJECTIVES
EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Derive the Bernoulli (energy) equation.
Demonstrate practical uses of the Bernoulli and continuity equation in the analysis of flow.
understand the use of hydraulic and energy grade lines.
Apply Bernoulli Equation to solve fluid mechanics problems (e.g. flow measurement).
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 3EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Let us first derive the Bernoulli equation, which is one of the most well-known equations of motion in fluid mechanics, and yet is often misused. It is thus important to understand its limitations, and the assumptions made in the derivation.
The assumptions can be summarized as follows: Inviscid flow (ideal fluid, frictionless) Steady flow (unsteady Bernoulli equation will not be discussed
in this course) Applied along a streamline Constant density (incompressible flow) No shaft work or heat transfer
The Bernoulli Equation
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 4EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
The Bernoulli Equation Limitations
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 5EGGD3109 Fluid Mechanics Chapter 6: 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, where the fluid motion is governed by the combined effects of pressure and gravity forces.
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 6EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Acceleration of a Fluid Particle
Describe the motion of a particle in terms of its distance s along a streamline together with the radius of curvature along the streamline.The velocity of a particle along a streamline is V = V(s, t) = ds/dt The acceleration can be decomposed into two components: streamwise acceleration as along the streamline and normal acceleration an in the direction normal to the streamline, which is given as an = V2/R.
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 7EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Acceleration of a Fluid Particle
Note that streamwise acceleration is due to a change in speed along a streamline, and normal acceleration is due to a change in direction. The time rate change of velocity is the acceleration
In steady flow, the acceleration in the s direction becomes
(Proof on Blackboard)
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 8EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Derivation of the Bernoulli Equation
Applying Newton’s second law in the s-direction on a particle moving along a streamline in a steady flow field gives
The force balance in s direction gives
where
and
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 9EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Derivation of the Bernoulli Equation
Therefore,
Integrating steady flow along a streamline
Steady, Incompressible flow
This is the famous Bernoulli equation.
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 10EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Derivation of the Bernoulli Equation
This is the Bernoulli equation, consisting of three energy heads
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 11EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Derivation of the Bernoulli Equation
A head corresponds to energy per unit weight of flow and has dimensions of length.Piezometric head = pressure head + elevation head, which is the level registered by a piezometer connected to that point in a pipeline.Total head = piezometric head + velocity head.
Applying the Bernoulli equation to any two points on the same streamline, we have
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 12EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
The Bernoulli Equation
Without the consideration of any losses, two points on the same streamline satisfy
where P/ as flow energy, V2/2 as kinetic energy, and gz as potential energy, all per unit mass. The Bernoulli equation can be viewed as an expression of mechanical energy balanceWas first stated in words by the Swiss mathematician Daniel Bernoulli (1700–1782) in a text written in 1738.
2 21 1 2 2
1 21 22 2
P V P Vz z
g g g g
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 13EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Example 1
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 14EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Example 2
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 15EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Example 2
-29.9 kpa
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 16EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Example 3
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 17EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Example 3
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 18EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
The Energy Equation
The energy equation is more general than the Bernoulli equation, because it allows for (1) friction, (2) heat transfer, (3) shaft work, and (4) viscous work (another frictional effect).
Where Ws is the shaft work , hL, called the head loss,In the absence of these two terms, the energy equation is identical to the Bernoulli equationWe must remember however that the Bernoulli equation is a momentum equation applicable to a streamline and the energy equation above is applied between two sections of a flow
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 19EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
The Energy Equation
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 20EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
The Energy Equation
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 21EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Example 1
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 22EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Example 2
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 23EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Example 2
waterof 97.12
1081.9*2
64.70000
22
2
2
222
1
211
mH
H
Hhzg
V
g
pz
g
V
g
p
P
P
PL
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 24EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Example 3
Pump draws water from reservoir (A) and lifts it to a higher reservoir (B), as shown below, the head loss from A to the pump = 4v2/2g and the head loss from the pump to B = 7 v2/2g. compute the pressure head the pump must deliver. The pressure head at the inlet of pump = -6m.
A
B
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 25EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Example 3
mH
HV
Hhzg
V
g
pz
g
V
g
p
mV
V
VV
hzg
V
g
pz
g
V
g
p
P
P
PL
L
7281.9*2
26.61150
81.9*2
115400400
22
26.681.9*2
510
81.9*2
40
81.9*26400
22
2
2
3
233
1
211
2
22
2
222
1
211
12
3
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 26EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
HGL and EGL
It is often convenient to plot mechanical energy graphically using heights.
P/g is the pressure head; it represents the height of a fluid column that produces the static pressure P.V2/2g is the velocity head; it represents the elevation needed for a fluid to reach the velocity V during frictionless free fall.z is the elevation head; it represents the potential energy of the fluid.H is the total head.
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 27EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
HGL and EGL
Hydraulic Grade Line (HGL)
Energy Grade Line (EGL) (or total head)
PHGL z
g
2
2
P VEGL z
g g
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 28EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
HGL and EGL
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 29EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Something to know about HGL and EGL:
For stationary bodies such as reservoirs or lakes, the EGL and HGL coincide with the free surface of the liquid, since the velocity is zero and the static pressure (gage) is zero.The EGL is always a distance V2/2g above the HGL. In an idealized Bernoulli-type flow, EGL is horizontal and its height remains constant. This would also be the case for HGL when the flow velocity is constant .For open-channel flow, the HGL coincides with the free surface of the liquid, and the EGL is a distance V2/2g above the free surface.
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 30EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Something to know about HGL and EGL:
At a pipe exit, the pressure head is zero (atmospheric pressure) and thus the HGL coincides with the pipe outlet.The mechanical energy loss due to frictional effects (conversion to thermal energy) causes the EGL and HGL to slope downward in the direction of flow. A steep jump occurs in EGL and HGL whenever mechanical energy is added to the fluid. Likewise, a steep drop occurs in EGL and HGL whenever mechanical energy is removed from the fluid.
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 31EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Something to know about HGL and EGL:
The pressure (gage) of a fluid is zero at locations where the HGL intersects the fluid. The pressure in a flow section that lies above the HGL is negative, and the pressure in a section that lies below the HGL is positive.
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 32EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Something to know about HGL and EGL:
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 33EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Examples on HGL and EGL:
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 34EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Examples on HGL and EGL:
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 35EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Examples on HGL and EGL:
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 36EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Examples on HGL and EGL:
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 37EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Examples on HGL and EGL:
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 38EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Examples on HGL and EGL:
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 39EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Examples on HGL and EGL:
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 40EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Static, Dynamic, and Stagnation Pressures
P is the static pressure; it represents the actual thermodynamic pressure of the fluid. This is the same as the pressure used in thermodynamics and property tables.V2/2 is the dynamic pressure; it represents the pressure rise when the fluid in motion.gz is the hydrostatic pressure, depends on the reference level selected.
The Bernoulli equation
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 41EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Static, Dynamic, and Stagnation PressuresThe sum of the static, dynamic, and hydrostatic pressures is called the total pressure (a constant along a streamline).The sum of the static and dynamic pressures is called the stagnation pressure,
The fluid velocity at that location can be calculated from
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 42EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
APPLICATIONS OF BERNOULLI & MOMENTUM EQUATION
1)Pitot tube.2)Changes of pressure in a tapering pipe:3)Orifice and vena contracta.4)Venturi, nozzle and orifice meters.5)Force on a pipe nozzle.6)Force due to a two-dimensional jet hitting an
inclined plane.7)Flow past a pipe bend.
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 43EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
PITOT TUBE
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 44EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
PITOT TUBE
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 45EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
PITOT TUBE IN THE PIPE-(method1)
Two piezometers
for ideal flow
To account for real fluid effects, the equation can be modified into
where is the coefficient of velocity to be determined empirically.
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 46EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
PITOT TUBE IN THE PIPE-(method2)
Using a static pressure taping in the pipe
1
water
mercurymercurywater hh
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 47EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
PITOT TUBE IN THE PIPE-(method2)
Example
mh
hh
water
water
mercurymercurywater
45.211025
136002.0
1
95.0 if
pipe in the water of velocity actual theDetermine
kg/m 1025
200mh shown, figure In the3
water
mercury
VC
Sol.
smV
ghCV
actual
Vactual
/ 6.645.281.9295.0
2
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 48EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
PITOT TUBE IN THE PIPE-(method3)
Using combined Pitot static tube
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 49EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
PITOT TUBE IN THE PIPE-(method3)
Using combined Pitot static tube. In which the inner tube isused to measure the impact pressure while the outer sheathhas holes in its surface to measure the static pressure
The total pressure is know as the stagnation pressure (or total pressure)
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 50EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
PITOT TUBE IN THE PIPE-(method3)
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 51EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
CHANGES OF PRESSURE IN ATAPERINGPIPE.
Changes of velocity in a tapering pipe were determined by using the continuity equation.
Changes of velocity will accompanied by a changed in pressure, modified by any changed in elevation or energy loss, which can be determined by the use of Bernoulli's equation.
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 52EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
CHANGES OF PRESSURE IN ATAPERINGPIPE-Example
Find:¢ the pressure difference across the 2m length ignoring any losses ofenergy.¢ the difference in level that would be shown on a mercury manometer connected across this length.
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 53EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
CHANGES OF PRESSURE IN ATAPERINGPIPE-Example
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 54EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
CHANGES OF PRESSURE IN ATAPERINGPIPE-Example
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 55EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Orifice and vena contracta
We are to consider the flow from a tank through a hole in the side close to the base
Looking at the streamlines you can see how they contract after the orifice to a minimum cross section where they all become parallel, at this point, the velocity and pressure are uniform across the jet. This convergence is called the vena contracta (from the Latin 'contracted vein'). It is necessary to know the amount of contraction to allow us to calculate the flow.
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 56EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Orifice and vena contracta
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 57EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Orifice and vena contracta
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 58EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Venturi , nozzle and orifice meters
The Venturi, nozzle and orifice-meters are three similar types of devices for measuring discharge in a pipe.
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 59EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
A Venturi meter in laboratory
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 60EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Venturi Meter
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 61EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Venturi Meter
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 62EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Venturi Meter
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 63EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Venturi Meter
This is also the theoreticaldischarge in terms ofmanometer readings
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 64EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Venturi Meter
22
21
21
21
22
2
21
2
22
22
12
2
21
22
21
22
21
2221
2
222
1
211
.2
11
2
)(2 )(2
)(2 22
22
22
AA
AAghQ
AAg
Qh
A
Q
A
QhgVVhg
VVhgg
V
g
Vh
g
V
g
V
g
p
g
p
zg
V
g
pz
g
V
g
p
22
21
21.2AA
AAghCQ dactual
Horizontal Venturimeter
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 65EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Venturi Meter Example 1
mh
h
AA
AAghCQ dactual
756.0
00786.003141.0
00786.003141.081.9296.01000/30
.2
22
22
21
21
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 66EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Venturi Meter Example 2
mh
hh
mh
water
water
mercurymercurywater
mercury
15.311000
1360025.0
1
25.0
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 67EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Venturi Meter
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 68EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Venturi Meter Example 3
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 69EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
Venturi Meter
smQ
Q
AA
AAghCQ dactual
/ 138.0
0314.01257.0
0314.01257.0*92.081.9298.0
.2
3
22
22
21
21
mh
hh
mh
oil
oil
mercurymercuryoil
mercury
92.01700
1360005.0
1
05.0
Chapter 3: Pressure and Fluid StaticsESOE 505221 Fluid Mechanics 70EGGD3109 Fluid Mechanics Chapter 6: Bernoulli and energy equations
PIPE ORIFICES
¢ A similar effect as the venturi meter can be achieved by insertingan orifice plate¢ The orifice plate has an opening in it smaller than the internal pipediameter