ch7application on bernoulli equation
TRANSCRIPT
Ministry of Higher Education & Scientific ResearchFoundation of Technical Education Technical College of Basrah
CH7:Application on Bernoulli Equation
Training Package
in
Fluid Mechanics
Modular unit 7
Application on Bernoulli Equation
By
Risala A. MohammedM.Sc. Civil Engineering
Ass. Lect.Environmental & Pollution Engineering Department
2011
Target population
CH7:Application on Bernoulli Equation
For the students of second class in
Environmental engineering Department in
Technical College
Central Idea
CH7:Application on Bernoulli Equation
The main goal of this chapter is to know the
application of Bernoulli equation in the many
situations of fluid mechanics.
Instructions
CH7:Application on Bernoulli Equation
1 -Study over view thoroughly
2 -Identify the goal of this modular unit
3 -Do the Pretest and if you-: * Get 9 or more you do not need to proceed
* Get less than 9 you have to study this modular
4 -After studying the text of this modular unit , do the post test and if you-:
* Get 9 or more , so go on studying modular unit eight * Get less than 9 , go back and study the modular unit seven
Performance Objectives
CH7:Application on Bernoulli Equation
At the end of this modular unit the student will be able to :-
1. Describe the pitot tube and calculate the velocity of fluid during
flowing through pipes by using it.
2.Describe the Venturi Venturi meter and calculate the discharge of fluid
during flowing through pipes by using it
3.Calculate the discharge from tanks by using orifices
4.Calculate the hydraulic coefficients ( Cv, Cd and Cc)
5.Calculate the time required for empting tank through an orifices.
6.measuring discharge in open channel by using notches and weirs
Pre test
CH7:Application on Bernoulli Equation
Q1)) ( 5 mark)Explain briefly how the co-efficient of velocity of a jet "issuing through an orifice can be
experimentally determined. Find an expression for head loss in an orifice flow in terms of co-efficient of velocity and jet velocity, The head lost in flow through a 50 mm diameter orifice under a certain head is 160 mm of water and. the velocity of water in the jet is 7 m/s. If the co-efficient of discharge be 0.6 l, determine: (i) Head on the orifice causing flow; (ii) The co-efficient of velocity; iii) The diameter of the jet.
Q2)) ( 5 mark)A Venturi meter having a throat diameter of 150 mm is installed in a horizonta1300-mm-diameter water main. The coefficient of discharge is 0.982. Determine the difference in level of the mercury columns of the differential manometer attached to the Venturi meter if the discharge is 0.142 m3/s.
Not Check your answers in key answer page
Introduction
CH7:Application on Bernoulli Equation
• 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.
Pitot Tube
CH7:Application on Bernoulli Equation
X0
Streamlines passing a non-rotating obstacle
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.
Pitot Tube
CH7:Application on Bernoulli Equation
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.
ghV 2
Pitot Tube
CH7:Application on Bernoulli Equation
Two piezometers, one as normal and one as a Pitot tube within the pipe 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 can be used in an arrangement shown below to measure velocity of flow. flow.
From the expressions above,From the expressions above,
2112 2
1Vpp
12
2112
2
2
1
hhgV
Vghgh
Figure 4.3 : A Piezometer and a Pitot tube
Venturi meter
CH7:Application on Bernoulli Equation
The Venturi meter is a device for The Venturi meter is a device for measuring discharge in a pipe. measuring discharge in a pipe.
It consists of a rapidly converging It consists of a rapidly converging section, which increases the velocity section, which increases the velocity of flow and hence reduces the of flow and hence reduces the pressure. pressure.
It then returns to the original It then returns to the original dimensions of the pipe by a gently dimensions of the pipe by a gently diverging ‘diffuser’ section. By diverging ‘diffuser’ section. By measuring the pressure differences measuring the pressure differences the discharge can be calculated. the discharge can be calculated.
This is a particularly accurate method This is a particularly accurate method of flow measurement as energy losses of flow measurement as energy losses are very small.are very small.
Venturi meter
CH7:Application on Bernoulli Equation
Applying Bernoulli Equation between (1) and (2), and using continuity equation to eliminate V2 will give :
and Qideal = V1A1
To get the actual discharge, taking into consideration of losses due to friction, a coefficient of discharge, Cd, is introduced.
It can be shown that the discharge can also be expressed in terms of manometer reading :
where man = density of manometer fluid
1AA
ZZρg
pp2g
V 2
2
1
2121
1
1
2
2A
1A
2Z
1Z
ρg2
p1
p2g
1A
dC
idealQ
dC
actualQ
1AA
1ρ
ρ2gh
ACQ 2
2
1
man
1dactual
Example(1)
CH7:Application on Bernoulli Equation
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
Solution
V1 = Q/0.0707,V2 = Q/0.0314
For the manometer :
PwPgg gRRzgPgzP )( 2211
423.587)(62.19 1221 zzPP
Example(1)
CH7:Application on Bernoulli Equation
For the Venturi meter :
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
Example (2)
CH7:Application on Bernoulli Equation
Example(3)
CH7:Application on Bernoulli Equation
Example (4)
CH7:Application on Bernoulli Equation
Example (5)
CH7:Application on Bernoulli Equation
Sharp edge circular orifice
CH7:Application on Bernoulli Equation
(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>10d), 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
Sharp edge circular orifice
CH7:Application on Bernoulli Equation
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,
This result is known as Toricelli’s equation. Assuming no loses, ideal fluid, V constant across jet at (2), the discharge through the orifice is
hzzg
V 21
22
2
ghV 22
20 VAQ
Sharp edge circular orifice
CH7:Application on Bernoulli Equation
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,
gh2AQ 0
ghCV va 22
velocityltheoretica
velocityactualCv (typically about 0.97)
Sharp edge circular orifice
CH7:Application on Bernoulli Equation
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
Sharp edge circular orifice
CH7:Application on Bernoulli Equation
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)
orificeofarea
contractavenaatjetofareaCc
gh2CACQ v0ca
dischargeltheoretica
dischargeactualCd
gh2ACQ 0da
Cd = Cc . Cv
Experimental Determination of Hydraulic Coefficients
CH7:Application on Bernoulli Equation
Determination of Co efficient of Velocity of Velocity (Cv)
A Fig. shows a tank containing water at a constant level, maintained by a constant supply. Let the water flow out -of the tank through an orifice, fitted in one side of the tank. Let the section C-C represents the point of vena contracta. Consider a particle of water in the jet at P.
Experimental Determination of Hydraulic Coefficients
CH7:Application on Bernoulli Equation
Let, x = Horizontal distance travelled by the particle in time 't',
y= Vertical distance between C-C and P, V;:= Actual velocity of the jet at vena-contracta, and
H;:= Constant water head.
Then, horizontal distance, x= V x t
Example (6)
CH7:Application on Bernoulli Equation
An orifice 50mm in diameter is discharging water under a head of l0 meters. If Cd = 0.6 and Cv
= 0.97, find: (i) actual discharge, and ii) Actual velocity of the jet at vena contraeta
Example (7)
CH7:Application on Bernoulli Equation
A large tank has a sharp edged circular orifice of 930 mm2 area at a depth of 3 m below constant water level. The jet issues horizontally and in a horizontal distance of 2.4m ,
it falls by 0.53 m, the measured discharge is 4.3 liters\sec. Determine coefficients of velocity, contraction and discharge for the orifice.
Example (8)
CH7:Application on Bernoulli Equation
The head of water over the centre of an orifice of diameter 30 mm is 1.5m. The actual discharge through the orifice is 2.55 liters/sec. Find the co-efficient of discharge.
Example (9)
CH7:Application on Bernoulli Equation
Nozzles
CH7:Application on Bernoulli Equation
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 :
dnozzle
contraction within nozzle
g
Vht 2
22
Nozzles
CH7:Application on Bernoulli Equation
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 :
And the loss of energy head, hf , in the nozzle due to friction is :
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
Example (10)
CH7:Application on Bernoulli Equation
The nozzle in Fig. below throws a stream of water vertically upward so that the power available in the jet at point 2 is 3.50 hp. If the pressure at the base of the nozzle, point 1, is 36.0 psi, find the (a) theoretical height to which the jet will rise, (6) coefficient of velocity, (c) head loss between 1 and 2, and (4) theoretical diameter of the jet at a point 18 ft above point 2.
Time Required For Empting Tank Through An Orifices
CH7:Application on Bernoulli Equation
Time Required For Empting Tank Through An Orifices
CH7:Application on Bernoulli Equation
Example (11)
CH7:Application on Bernoulli Equation
A circular tank of diameter 3 m contains water up to a height of 4m. The tank is provided with an orifice of diameter 0.4 m at the bottom, Find the time taken by water; (i) to fall from 4 m to 2 m, and (ii) for completely emptying the tank
Example (12)
CH7:Application on Bernoulli Equation
A 1 m diameter circular tank contains water up to a height of 4 m. At the bottom . of tank an orifice of 40 mm is provided. Find the height of water above the orifice after 1.5 minutes. Take co-efficient of discharge for the orifice Cd= 0.6.
Example (12)
CH7:Application on Bernoulli Equation
Flow over notches and weirs
CH7:Application on Bernoulli Equation
• 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.
Notch
Flow over notches and weirs
CH7:Application on Bernoulli Equation
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).
Flow over notches and weirs
CH7:Application on Bernoulli Equation
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 figure
velocity through the strip
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,
ghV 2
ghhbAVQ 2
dhbhgQH
ltheoretica 0
21
2
Flow over notches and weirs
CH7:Application on Bernoulli Equation
Rectangular Weir
For 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:
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 :
23
23
2HgBCQ dactual
Example (13)
CH7:Application on Bernoulli Equation
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
Example (13)
CH7:Application on Bernoulli Equation
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
Post test
CH7:Application on Bernoulli Equation
Post test
Q1)) ( 5 mark)A tank has two identical orifices in one of its vertical sides, The upper orifice is 1.5 below the water surface and the lower one is 3 m below the water surface . Find the point, at which the two jets will intersect, if the co-efficient of velocity is 0.92 for both the orifices
Q2)) ( 5 mark)Oil is flowing upward through a Venturi. meter as shown in . Assume a discharge coefficient of 0.984 What is the rate of flow of the oil?
Key answer
CH7:Application on Bernoulli Equation
Pre testQ1((
Key answer
CH7:Application on Bernoulli Equation
Pre testQ2((
Key answer
CH7:Application on Bernoulli Equation
Post testQ1((
Key answer
CH7:Application on Bernoulli Equation
Post testQ2((
References
CH7:Application on Bernoulli Equation
1. Evett, J., B. and Liu, C. 1989 “2500 solved problems in fluid mechanics and hydraulics” Library of Congress Cataloging- in-Publication Data, (Schaum's solved problems series) ISBN 0-07-019783-0
2. Rajput, R.,K. 2000 “ A Text Book of Fluid Mechanics and Hydraulic Machines”. S.Chand & Company LTD.
3. White, F., M. 2000 “ Fluid Mechanics”. McGraw-Hill Series in Mechanical Engineering.
4. Wily, S., 1983 “ Fluid Mechanics”. McGraw-Hill Series in Mechanical Engineering.