ebvf4103 (chapter 4) fluid mechanics for civil engineering
TRANSCRIPT
Copyright © ODL Jan 2005 Open University Malaysia
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
FLUID MECHANICS FOR CIVIL ENGINEERING TUTORIAL 2 – UNIT 2: Fluid Dynamics and Behaviour of Real Fluid Chapter 4: Application of Bernoulli and
Momentum Equations
Assoc. Prof. Dr Othman A. KarimEBVF4103 Fluid Mechanics for Civil EngineeringJan 2005
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
SEQUENCE OF CHAPTER 4
Introduction
Objectives
4.1 Application of Bernoulli Equation
4.1.1 Pitot Tube
4.1.2 Pitot Static Tube
4.1.3 Venturi Meter
4.1.4 Sharp Edge Circular Orifice
4.1.5 Nozzles
4.1.6 Flow over Notches and Weirs
Summary
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Introduction
• The Bernoulli equation can be applied to a great many situations not just the pipe flow we have been considering up to now.
• In the following sections we will see some examples of its application to flow measurement from tanks, within pipes as well as in open channels.
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Objectives
• Acknowledge practical uses of the Bernoulli and momentum equation in the analysis of flow
• Understand how the momentum equation and principle of conservation of momentum is used to predict forces induced by flowing fluids
• Apply Bernoulli and Momentum Equations to solve fluid mechanics problems
Copyright © ODL Jan 2005 Open University Malaysia
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
4.1 Application of Bernoulli Equation
X0
Figure 4.1: Streamlines passing a non-rotating obstacle
4.1.1 Pitot tube A point in a fluid stream where the
velocity is reduced to zero is known as a stagnation point.
Any non-rotating obstacle placed in the stream produces a stagnation point next to its upstream surface.
The velocity at X is zero: X is a stagnation point.
By Bernoulli's equation the quantity p + ½V2 + gz is constant along a streamline for the steady frictionless flow of a fluid of constant density.
If the velocity V at a particular point is brought to zero the pressure there is increased from p to p + ½V2.
For a constant-density fluid the quantity p + ½V2 is therefore known as the stagnation pressure of that streamline while ½V2 – that part of the stagnation pressure due to the motion – is termed the dynamic pressure.
A manometer connected to the point X would record the stagnation pressure, and if the static pressure p were also known ½V2 could be obtained by subtraction, and hence V calculated.
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Figure 4.2: Simple Pitot Tube
A right-angled glass tube, large enough for capillary effects to be negligible, has one end (A) facing the flow. When equilibrium is attained the fluid at A is stationary and the pressure in the tube exceeds that of the surrounding stream by ½V2. The liquid is forced up the vertical part of the tube to a height :
h = p/g = ½V2/g = V2/2g
above the surrounding free surface. Measurement of h therefore enables V to be calculated.
(4.3)ghV 2
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Measurement of the static pressure may be made at the boundary of the flow, as illustrated in (a), provided that the axis of the piezometer is perpendicular to the boundary and the connection is smooth and that the streamlines adjacent to it are not curved
A tube projecting into the flow (Tube c) does not give a satisfactory reading because the fluid is accelerating round the end of the tube.
Figure 4.3: Piezometers connected to a pipe
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Two piezometers, one as normal and one as a Pitot tube within the pipe can be used in an arrangement shown below to measure velocity of flow.
From the expressions above,
2112 2
1Vpp
12
2112
2
2
1
hhgV
Vghgh
Figure 4.3 : A Piezometer and a Pitot tube
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
4.1.2 Pitot static tube The tubes recording static pressure
and stagnation pressure are frequently combined into one instrument known as a Pitot-static tube
The ‘static’ tube surrounds the ‘total head’ tube and two or more small holes are drilled radially through the outer wall into the annular space.
The position of these ‘static holes’ is important. This instrument, when connected to a suitable manometer, may be used to measure point velocities in pipes, channels and wind tunnels.
Figure 4.5: Pitot static tube
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Consider the pressures on the level of the centre line of the Pitot tube and using the theory of the manometer,
PA = P2 + gX
PB = P1 + g(X-h) + maxgh
PA = PB
P2 + gX = P2 + g(X – h) + mangh
We know that , substituting this in to the above gives
)(2 max
1
ghV
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
4.1.3 Venturi meter The Venturi meter is a device for
measuring discharge in a pipe.
It consists of a rapidly converging section, which increases the velocity of flow and hence reduces the pressure.
It then returns to the original dimensions of the pipe by a gently diverging ‘diffuser’ section. By measuring the pressure differences the discharge can be calculated.
This is a particularly accurate method of flow measurement as energy losses are very small.
Figure 4.6: A Venturi meter
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Applying Bernoulli Equation between (1) and (2), and using continuity equation to eliminate V2 will give :
(4.4)
and Qideal = V1A1
To get the actual discharge, taking into consideration of losses due to friction, a coefficient of discharge, Cd, is introduced.
Qactual = Cd Qideal = (4.5)
It can be shown that the discharge can also be expressed in terms of manometer reading :
Q actual = (4.6)
where man = density of manometer fluid
1
2
2
2
1
2121
1
A
A
ZZg
ppg
V
1
2
2
2
1
2121
1
AA
ZZg
ppg
ACd
1
12
2
2
1
1
A
A
gh
AC
man
d
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Example 4.1 A Venturi meter with an entrance diameter of 0.3
m and a throat diameter of 0.2 m is used to measure the volume of gas flowing through a pipe. The discharge coefficient of the meter is 0.96. Assuming the specific weight of the gas to be constant at 19.62 N/m3, calculate the volume flowing when the pressure difference between the entrance and the throat is measured as 0.06 m on a water U-tube manometer.
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
What we know from the problem statement :
gg = 19.62 N/m2
Cd = 0.96d1 = 0.3md2 = 0.2m
Calculate Q:
V1 = Q/0.0707 V2 = Q/0.0314
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
For the manometer :
--- (1)
For the Venturi meter :
--- (2)
Combining (1) and (2) :
PwPgg gRRzgPgzP )( 2211
2
222
1
211
22z
g
V
g
Pz
g
V
g
P
gg
221221 803.0)(62.19 VzzPP
smQCQ
smQ
smV
V
ideald
ideal
ideal
/816.085.096.0
/85.02
2.0047.27
/047.27
423.587803.0
3
32
2
22
423.587)(62.19 1221 zzPP
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
4.1.4 Sharp edge circular orifice
(1)
(2)
h
datum
streamline
Consider a large tank, containing an ideal fluid, having a small sharp-edged circular orifice in one side.
If the head, h, causing flow through the orifice of diameter d is constant (h>>d), Bernoulli equation may be applied between two points, (1) on the surface of the fluid in the tank and (2) in the jet of fluid just outside the orifice. Hence :
lossesg
V
g
Ph
g
V
g
P 0
22
222
211
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Now P1 = Patm and as the jet in unconfined, P2 = Patm. If the flow is steady, the surface in the tank remains stationary and V1 0 (z2=0, z1=h) and ignoring losses we get :
or the velocity through the orifice,
(4.7)
This result is known as Toricelli’s equation.Assuming no loses, ideal fluid, V constant across jet at (2), thedischarge through the orifice is
hzzg
V 21
22
2
ghV 22
20 VAQ
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
where A0 is the area of the orifice
For the flow of a real fluid, the velocity is less than that given by eq. 4.7 because of frictional effects and so the actual velocity V2a, is obtained by introducing a modifying coefficient, Cv, the coefficient of velocity:
Velocity, (4.8)
or (typically about 0.97)
gh2AQ 0
ghCV va 22
velocityltheoretica
velocityactualCv
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
As a real fluid cannot turn round a sharp bend, the jet continues to contract for a short distance downstream (about one half of the orifice diameter) and the flow becomes parallel at a point known as the vena contracta (Latin : contracting vein).
d0
approx. d0/2
Vena contracta
P = Patm
Figure 4.8: The formation of vena contracta
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
The area of discharge is thus less than the orifice area and a coefficient of contraction, Cc, must be introduced.
Hence the actual discharge is :
(typically about 0.65)
or introducing a coefficient of discharge, Cd, where :
(typically 0.63)
(4.9)
(4.10)
orificeofarea
contractavenaatjetofareaCc
gh2CACQ v0ca
dischargeltheoretica
dischargeactualCd
gh2ACQ 0da
Cd = Cc . Cv
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
4.1.5 Nozzles
In a nozzle, the flow contracts gradually to the outlet and hence the area of the jet is equal to the outlet area of the nozzle.
i.e. Cc = 1.0
therefore Cd = Cv
Taking a datum at the nozzle, Torricelli’s equation gives the total energy head in the system as it assumes an ideal fluid and hence no loss of energy, i.e. theoretical head :
(4.11)
dnozzle
contraction within nozzle
Figure 4.9: Contraction within a nozzle
g
Vht 2
22
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
But the actual velocity is :
V2a = Cv V2
and the actual energy in the jet is :
as P2 and z2 are zero
Therefore the actual energy head is :(4.12)
And the loss of energy head, hf , in the nozzle due to friction is :
(4.13)
g
Vh a
a 2
22
g
VCh v
a 2
22
2
g
VC
g
Vhhh v
atf 22
22
22
)1(2
22
2vC
g
V
tvf hCh )1( 2
theoretical actual
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
4.1.6 Flow over notches and weirsNotch is an opening in the side of a tank or reservoir, which
extends above the surface of the liquid. It is usually a device for measuring discharge. A weir is a notch on a larger scale – usually found in
rivers. It may be sharp crested but also may have a
substantial width in the direction of flow – it is used as both a flow measuring device and a device to raise water levels.
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Weir Assumptions assume that the velocity of the fluid approaching the
weir is small so that kinetic energy can be neglected.
assume that the velocity through any elemental strip depends only on the depth below the free surface.
These are acceptable assumptions for tanks with notches or reservoirs with weirs, but for flows where the velocity approaching the weir is substantial the kinetic energy must be taken into account (e.g. a fast moving river).
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
A General Weir EquationTo determine an expression for the theoretical flow through a notch we will consider a horizontal strip of width b and depth h below the free surface, as shown in the figurevelocity through the strip
V = discharge through the strip,
Integrating from the free surface, h = 0, to the weir crest, h = H gives the expression for the total theoretical discharge,
Qtheoretical =
gh2ghhbAVQ 2
dhbhgH
0
21
2
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Rectangular WeirFor a rectangular weir the width does not change with depth so there is no relationship between b and depth h. We have the equation, b = constant = B.
Substituting this with the general weir equation gives:
(4.14)
Figure 4.11 : A rectangular weir
dhhgBQH
O
ltheoretica 21
2
23
23
2HgB
To calculate the actual discharge we introduce a coefficient of discharge, Cd, which accounts for losses at the edges of the weir and contractions in the area of flow, giving :
(4.15)23
23
2HgBCQ dactual
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Example 4.2 Water flows over a sharp-crested weir 600 mm wide. The
measured head (relative to the crest) is 155 mm at a point where the cross-sectional area of the stream is 0.26 m2. Calculate the discharge, assuming that Cd = 0.61.
As first approximation,
H = 155 mm
Cross sectionalArea = 0.26 m2
23
23
2HgBCQ dactual
23
)155.0(/62.196.03
261.0 2 msmm
= 0.0660 m3/s
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Velocity of approach =
= 0.254 m/s
H + V12/2g = (0.155 + 0.00328) m = 0.1583 m
Second approximation:
Further refinement of the value could be obtained by a new calculation of V1 (0.0681 m3/s 0.26 m2), a new calculation of H + V1
2/2g and so on. One correction is usually sufficient, however, to give a value of Q acceptable to three significant figures.
2
3
26.0
/0660.0
m
sm
msm
sm
g
V 32
22
1028.3/62.19
)/254.0(
2
smsmx /0681.0/1583.06.062.1961.03
2 332/3
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Summary
• Chapter 4 emphasized basically on the application of Bernoulli equation in order to solve problems related to fluid mechanics and the application of momentum equation to solve type of flows problem.
• Students should concentrate more on the examples given in chapter 4 and try to relate the concept in the real scenario.
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Subject Matter Expert/Author: Assoc. Prof. Dr Othman A. Karim (OUM)
Faculty of Engineering andTechnical Studies
Copyright © ODL Jan 2005 Open University Malaysia
Thank You