EBVF4103 (Chapter 4) Fluid Mechanics for Civil Engineering

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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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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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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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


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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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Thank You

Documents

EBVF4103 (Chapter 4) Fluid Mechanics for Civil Engineering