EAA 206 - Fluids Mechanics (Manual)

1

SCHOOL OF CIVIL ENGINEERING UNIVERSITI SAINS MALAYSIA

ENGINEERING CAMPUS

EAA 206/2

STRUCTURAL, CONCRETE AND FLUID MECHANICS ENGINEERING LABORATORY

FLUID MECHANIC LABORATORY MANUAL

2

LIST OF EXPERIMENTS

Code of

Experiment

Name of Experiments

Page

H1

EXPERIMENT 1 : REYNOLDS

NUMBER TEST

EXPERIMENT 2 : FLOW THROUGH

ORIFICE

1-9

10-18

H2

EXPERIMENT 3 : CALIBRATION OF

RECTANGULAR AND TRIANGULAR

NOTCH

EXPERIMENT 4 : BOURDON

PRESSURE GAUGE

19-26

27-30

H3

EXPERIMENT 5 : FLOW THROUGH A

VENTURI METER

31-41

3


ENGINEERING CAMPUS

EXPERIMENT 1 : REYNOLDS NUMBER TEST

1.0 OBJECTIVES

To demonstrate laminar, transitional and turbulent flow.

To calculate Reynolds number for each flow.

1.1 INTRODUCTION

When a fluid flows next to a solid boundary the nature of the flow depends on the

velocity relative to that boundary. At low velocities the layers of fluid move smoothly

over one another and this is termed “laminar” flow. However, as the velocity is increased

small disturbances cause eddies which “mix-up” the layers of fluid and produce a

different pattern of flow which is termed “turbulent”. This change has a marked effect on

the forces acting between the fluid and the solid boundary and an understanding of the

behaviour is of fundamental importance in the study of hydraulics and fluid mechanics.

The nature of flow over an aircraft wing affects the drag and hence determines the power

required to propel the aircraft forwards. Similarly, when fluid flows along a pipe the

nature of the flow determines the pressure loss and hence the power required to pump the

fluid along the pipe.

Before the advent of high speed transport, the most important application of fluid

mechanics was in the study of flow in pipes. Many engineers and scientists investigated

the behaviour of flow in pipes but it was a British physicist named Osborne Reynolds

(1842-1912) who first identified the variables controlling the flow and produced a

4

rational means of predicting the nature of flow. Reynolds showed that the behaviour

depends on the balance between inertia and viscous forces in the fluid. This led to the

definition of a non-dimensional parameter, now called Reynolds Number, which

expresses the ratio of inertia to viscous, forces and can be used to identify the conditions

under which the flow changes from laminar to turbulent. By experiment it was found that

the change always occurred at a similar value of Reynolds Number irrespective of the

fluid and the size of pipe.

The Reynolds Number and Transitional Flow Demonstration Apparatus has been

designed to demonstrated the kind of experiment which was conducted to show the

dependence of flow on Reynolds Number. The apparatus enables the nature of the flow in

a pipe to be studied by observing the behaviour of a filament of dye injected into the

fluid. The flow rate can be varied and the changed or “transition” between laminar and

turbulent flow can be clearly demonstrated. The effect of viscosity on the behaviour can

be demonstrated by varying the temperature using an optional temperature control

module, or by using different fluids.

1.2 THEORY

Consider the case of a fluid moving along a fixed surface such as the wall of a

pipe. At some distance y from the surface the fluid has a velocity u relative to the surface.

The relative movement causes a shear stress which tends to slow down the motion so

that the velocity close to the wall is reduced below u. It can be shown that the shear stress

produces a velocity gradient du/dy which is proportional to the applied stress. The

constant of proportionality is the coefficient of viscosity and the equation is usually

written :-

dy

du …………………..(1)

The above equation is derived in most text books and represents a model of a

situation in which layers of fluid move smoothly over one another. This is termed

“viscous” or “laminar” flow. For such conditions experiments show that Equation (1) is

valid and that is a constant for a given fluid at a given temperature. It may be noted

that the shear stress and the velocity gradient have a fixed relationship which is

5

determined only by the viscosity of the fluid. However, experiments also show that this

only applies at low viscosities. If the velocity is increased above a certain value, small

disturbances produce eddies in the flows which cause mixing between the high energy

and low energy layers of fluid. This is called turbulent flow and under these conditions it

is found that the relationship between shear stress and velocity gradient varies depending

on many factors in addition to the viscosity of the fluid. The nature of the flow is entirely

different since the interchange of energy between the layers now depends on the strength

of the eddies (and thus on the inertia of the fluid) rather then simply on the viscosity.

Equation (1) still applies but the coefficient no longer represents the viscosity of the

fluid. It is now called the ‘Eddy Viscosity” and is no longer constant for a given fluid and

temperature. Its value depends on the upstream conditions in the flow and is much greater

than the coefficient of viscosity for the fluid. It may be noted that this implies an increase

in shear stress for a given velocity and so the losses in the flow are much greater than for

laminar conditions.

We have seen that laminar flow is the result of viscous forces and that turbulent

flow is in some way related to inertia forces. This was realized by Reynolds who

postulated that the nature of flow depends on the ratio of inertia to viscous forces. This

led to the derivation of a non-dimensional variable, now called Reynolds Number, Re

which expresses this ratio. On physical grounds we may say that inertia forces are

proportional to mass times velocity change divided by time. Since mass divided by time

is the mass flow rate and this is equal to density ρ times cross – sectional area times

velocity, ν we may write:-

Inertia forces α uud .2 ………………………………………(2)

Where d is the diameter of the pipe.

Similarly the viscous forces are given by shear stress times area so, using Equation (1),

we may write :-

Viscous forces α 2ddy

du

6

α 2dd

u …………………………………..(3)

Dividing the inertia forces by the viscous forces we obtain Reynolds Number as :-

du

d

ud

22

Re …………………………………….. (4)

The term

is called the kinematic viscosity ν and it is often convenient to write

Equation (4) as :-

ud

Re …………………………………………..(5)

It may be noted that the above equations can also be derived by dimensional analysis but

in either case it should be remembered that Re represents the ratio of inertia to viscous

forces.

1.3 PROCEDURE

1. The apparatus is free-standing and can be mounted on any suitable bench or

working surface. A water supply and drain are required so a convenient position

should be chosen where these services are available.

2. A bag of glass beads is supplied and this should be used to fill the lower part of

the constant head tank. The level of beads should be 10-15 mm below the top of

the bell-mouth and should be as flat as possible.

3. Fit the injector tube to the dye reservoir (if not already fitted) and position the

mounting plate on the top of the tank such that the injector tube is in the centre of

the bell-mouth.

4. Turn the dye control valve to off and pour suitable dye (e.g water soluble tank)

into the dye reservoir.

5. Stand the thermometer such that the bulb is resting on the stilling bed.

6. Turn on the water supply, and partly open the discharge valve at the base of the

apparatus.

7

7. Adjust the water supply until the level in the constant head tank is just above the

overflow pipe and is maintained at this level by a small flow down the overflow

pipe. This is the condition required for all tests and at different flow rates through

the tube – the supply will need to be adjusted to maintain it. At any given

condition the overflow should only be just sufficient to maintain a constant head

in the tank.

8. Open the adjust dye injector valve to obtain a fine filament of dye in the flow

down the glass tube. If the dye is dispersed in the tube, reduce the water flow rate

by closing the discharge valve and adjusting the supply as necessary to maintain

the constant head. A laminar flow condition should be achieved in which the

filament of dye passes down the complete length of the tube without disturbance.

9. Slowly increase the flow rate by opening the discharge valve until disturbance of

the dye filament. This can be regarded as the starting point of transition of

turbulent flow. Increase the water supply as required maintaining constant head

conditions.

10. Record the temperature of the water using the thermometer, then measure the

flow rate by timing the collection of a known quantity of water from the discharge

pipe.

11. Further increase the flow rate as described above until the disturbances increase

such that the dye filament becomes rapidly diffused. Small eddies will be noted

just above the point where the dye filament completely breaks down. This can be

regarded as the onset of fully turbulent flow. Record the temperature and flow rate

as in step 10.

12. Now decrease the flow rate slowly until the dye just returns to a steady filament

representing laminar flow and again record the temperature and flow rate.

8

Figure 1.1 : Schematic diagram of the Reynolds Number apparatus

9

Figure 1.2 : Typical flow patterns at various flow conditions

10

1.4 RESULT

Initial water temperature =

Final water temperature =

Mean water temperature =

Kinematic viscosity of water at above temperature, =

Diameter of pipe, dp =

Cross sectional area of pipe, Ap =

Figure 1.3: Variation of some properties of water with temperature

11

Table 1.1: Result for different type of flows

No. Type of flow

Time for 200 ml

(s)

Velocity, (u)

(m/s)

Flow rate (Q)

(x 10-6 m3/s)

Kinematic viscosity

(υ) (x 10-6 m2/s)

Re

1 Laminar 2 Transition 3 Turbulent 4 Laminar

1.5 PRESENTATION OF RESULTS

1. From Table 1.1, determine the Re for each type of flow.

QUESTION FOR FURTHER DISCUSSION

1. What suggestions do you have for improving the apparatus?

12


ENGINEERING CAMPUS

EXPERIMENT 2 : FLOW THROUGH ORIFICE

2.0 OBJECTIVE

To determine cd CC , and vC for orifice.

2.1 INTRODUCTION

It often happens that when a fluid passes through a constriction, such as through a

sharp-edged hole or over a weir, the discharge is considerably less than the amount

calculated on the assumption that the energy is conserved and that the flow through the

constriction is uniform and parallel. This reduction in flow is normally due to a

contraction of the stream which takes place through the restriction and continues for

some distance down-stream of it, rather than to a considerable energy loss.

In this experiment, arrangements are made to measure the extent to the reduction in flow,

contraction of the stream and energy loss, as water discharges into the atmosphere from a

sharp-edged orifice in the base of a tank (Figure 2.1).

13

2.2 THEORY

Figure 2.2 shows the essential features of flow through the orifice. The tank is

assumed to be sufficiently large for the velocity of flow in it to be negligibly small except

close to the orifice. In the vicinity of the orifice, the fluid accelerates towards the centre

of the hole, so that as the jet emerges it suffers a reduction of area due to the curvature of

the streamlines, as typified by the streamline MN indicated on the figure. The reduction

of area due to this local curvature may be taken to be complete at about half the orifice

diameter downstream of the plane of the orifice; the reduced section is usually referred to

as the vena contracta.

The pressure everywhere on the surface of the jet is atmospheric; but within the jet

pressure does not fall to atmospheric until the acceleration is complete, i.e. until the vena

contracta is reached.

Figure 2.1: Arrangement of apparatus

14

Consider now the total head of the water at points M and N of a typical stream-line, M

being in the surface and N being in the plane of the vena contracta.

From Bernoulli, the total head at M is mmm z

w

P

g

V

2

2

and at N is mnn z

w

P

g

V

2

2

so that, if the energy were conserved, ie., if there were no loss of total head:

nnn

mmm z

w

P

g

Vz

w

P

g

V

22

22

………(1)

In this equation, mP and nP are equal (both being atmospheric) and mU is negligibly

small according to our assumption. Moreover,

onm Hzz ………(2)

So that, from Equation (1) and (2), the ideal velocity at N is given by

Figure 2.2: Diagrammatic sketch of flow through orifice

15

on Hg

V

2

2

………(3)

This result applies to all points in the plane of the vena contracta, so changing the

notation to let oV be the ideal velocity in the plane of the vena contracta, which would

occur if there was no energy loss.

oo Hg

V

2

2

….……(4)

Because of the energy loss, which in fact takes place as the water passes down to the tank

and through the office, the actual velocity cV in the plane of the vena contracta will be

less than oV , and may be calculated from the Pilot tube reading by the equation :

cc Hg

V

2

2

..…….(5)

It is clear that co HH represent the energy loss. The ratio of actual velocity cV and

ideal velocity oV is often referred to as the coefficient of velocity vC of the orifice. From

the Equation (4) and (5), we obtain :

o

c

o

cv H

H

V

VC ..…….(6)

In a similar sense, the coefficient of contraction cC is defined as the ratio of cross-section

of the vena contracta cA , to the cross –section of the orifice oA ,

o

cc A

AC ……..(7)

16

Finally, the coefficient of discharge dC is defined as the ratio of the actual discharge to

that which would take place if the jet discharged at the ideal velocity without reduction of

area. The actual discharge Q is given y:

cc AVQ ………(8)

and if the jet discharged at the ideal velocity oV over the orifice area oA the discharge oQ

would be :

ooooo gHAAVQ 2 ……….(9)

So, from the definition of the coefficient of discharge,

oo

cc

od AV

AV

Q

QC .………(10)

or in terms of quantities measured experimentally,

oo

dgHA

QC

2 .……….(11)

From Equations (6), (7) and (10) it follows immediately :

cvd CCC ………..(12)

2.3 PROCEDURE

1. The experiment may be divided into two parts, firstly, the measurement of

,, vd CC and cC at a single constant value of oH , and secondly, measurement of

discharge at a number of different values of oH .

2. The equipment is set on the bench and leveled so that the base of the tank is

horizontal.

3. The flexible supply pipe from the bench control valve is connected to the inlet

pipe of the apparatus which is positioned to discharge directly back to the weigh

17

tank and the overflow of the apparatus is directed onto the bench top. To obtain

the steadiest readings the vertical position of the inlet pipe should be adjusted to

be just submerged. The diameter of the sharp-edged orifice is noted.

4. In the first part of the experiment, water is admitted to the tank to allow it to fill to

the height of the overflow pipe and the inflow is regulated so that a small steady

discharge is obtained from the overflow. This ensures that the level in the tank

remains constant while the measurements are made.

5. To measure dC the discharge is obtained by collection of a known weight of

water from the orifice in the weighing tank, and recording the value of head oH

on the orifice.

6. To measure vC , the Pitot tube is inserted into the emerging jet close to the

underside of the tank, and the values of Pitot head, cH and head, oH on the

orifice are noted.

7. To measure cC , it is necessary to find the diameter of the jet at the vena

contracta.

8. This is done by utilizing the sharp-edged blade attached to the head of the Pitot

tube, the plane of the blade being normal to the direction of traverse of the tube.

9. The blade is brought to each edge of the jet in turn, just below the tank, and the

positions of the tube as read on the lead screw and graduated nut read in each

case. The difference of the readings represents the diameter of the jet.

10. In the second part of the experiment the inflow to the tank is reduced to lower the

level in the tank in stages, the discharge from the orifice being measured at each

stage.

11. Care should be taken to allow the level to settle to a steady value after the inflow

to the tank has been changed, and it is advisable to read this level several times

while the discharge is being collected and to record the mean value over the time

interval.

12. About 6-8 different flow rates should be sufficient to establish the relationship

between discharge and head on the orifice.

18

2.4 RESULT

Diameter of Orifice, do = 0.013 m

Area of Orifice, Ao = 4

2d

= m2 dc = Diameter of jet Ac = Area of jet Cc = Coefficient of contraction

Table 2.1 : Data recorded and the calculations of flow rate, Q.

No.

Stop watch reading Volumetric tank reading Q=V/t x 410 (m3/s)

dj = dc

(mm)

Coefficient of

contraction,

o

cc A

AC

Initial (s)

End (s)

Time (s)

Initial (m3)

End (m3)

Volume (m3)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 1 2 3 4 5 6 7 8

Average Value of cC =

19

Table 2.2: Data recorded and the estimation of vC .

No. Head of tank Head of pitot tube

Coefficient of velocity,

o

cv

H

HC

oH

(m) oH

(m1/2)

cH

(m) cH

(m1/2)

(1) (2) (3) (4) (5) (6) 1 2 3 4 5 6 7 8

Average Value of vC =

Table 2.3 : Calculation of dC , cC and vC for orifice

No. Q

x10-4

(m3/s)

oH

(m1/2)

vC ooo gHAQ 2

x 10-4 (m3/s)

od Q

QC

v

dc C

CC

(1) (2) (3) (4) (5) (6) (7) 1 2 3 4 5 6 7 8

Average

Average Value of vC =

Average Value of dC =

Average Value of cC =

20


1. Plot graph Q versus oH . The flow rate through the orifice is stated as:

HKQ

where gACK d 2

The plotted gradient line will represent K value while dC can be computed using

the equation below:

gAKCd 2/

The above equation represent the dC value of the orifice.

2.6 QUESTIONS FOR FURTHER DISCUSSION

1. What suggestions have you for improving the apparatus?

21


ENGINEERING CAMPUS

EXPERIMENT 3 : CALIBRATION OF RECTANGULAR

AND TRIANGULAR NOTCH

3.0 OBJECTIVE

To calibrate the rectangular and triangular notch by using the hydraulic bench and

panel

3.1 INTRODUCTION

Weirs are commonly used to regulate flow in rivers and in other open channels.

The purpose is often to maintain water depth for some purpose such as navigation, but a

weir may also be used to measure the flow rate.

In many cases, the rate of flow over the weir depends solely on the water level

just upstream of the weir (the relationship between flow rate and water level being

sometimes known as the “rating curve”). However, the water level downstream of the

weir may rise sufficiently as to affect the conditions of flow, so that the flow rate now

becomes a function not only of the upstream water level but also of the water level

downstream. The weir is then referred to as being “suppressed” or “drowned”.

The cross-section of a weir is usually determined by considerations of strength

and stability in relation to the conditions of the site, and availability of materials. The

crest is frequently rounded or broad as shown in Figure 3.1 (a) and (b). For such weirs,

the flow usually remains attached to the downstream surface. In the case of the sharp

crested weir shown in Fig 3.1 (c), however, the flow separates at the crest to form a

curved jet which plunges into the downstream pool. In plan view, the weir may be

straight or curved to suit site condition, or on grounds of aesthetics. Frequently, the crest

22

level is not uniform along the whole of the length. For instance, just part of the whole

length may used to carry normal flow, the reminder of the weir having a higher crest, so

that it comes into use only at higher flow rates.

Fig 3.1 Examples of various weir profiles

A form of weir, particularly suitable for flow measurement, is the “notch”, so

called because it comprises a sharp edged notch cut out of a metal plate. The cut out may

of course be of any shape, but rectangular and V-shaped notches.

a) Round crested b) Broad crested

c) Sharp edged

23

3.2 THEORY

Rectangular Notch Panel

L H P

Figure 3.2 : Front and side view of a rectangular notch

The relationship for the flow rate (Q) of a rectangular notch is :

2/32/1 )10/()2(3/2 HnHLgCQ d …………….(1)

Where : n = number of end contractions (= 2 for this case)

L = width of the crest

H = head over the weir

If end contraction is negligible, then:

2/32/1)2()3/2( LHgCQ d ……………(2)

or

2/31HkQ …………....(3)

Equation (2) and (3) shows that the value of Q and H3/2 should be plotted in a graph as a

straight line. The gradient of the graph will represent ‘k1’ value. Cd can be obtained using

the equation below :

LgkCd ])2(2/[3 2/11 ……………(4)

Peak level

24

Equation (2) also can be expressed as :

2/32/11 )2( LHgkQ ……………(5)

)/(05.04.01 PHk

)]/(05.04.0[*)2/3( PHCd ……………(6)

Where :

P is the height of the weir crest above from the base of the tank. Equation 6 is only valid

for H/P values up to 10 as long as the weir is well ventilated.

Triangular Notch Panel

θ H P

Figure 3.3 : Front and side view of triangular notch (V-Notch) panel

The relationship for the flow rate (Q) of a triangular notch is :

2/52/1)2)(2/tan()15/8( HgCQ d ……………….(7)

or

2/52HkQ ……………….(8)

Peak level

25

The above equations shows that the value of Q and H5/2 should be plotted in a graph as a

straight line. The gradient of the graph will represent ‘k2’ value. Cd can be obtained using

the equation below :

)]2/tan()2)(8/[(15 2/12 gkCd ………………(9)

3.3 PROCEDURE

1. The apparatus is first connected to the supply and then leveled.

2. When the correct level has been obtained, the point gauge is also brought exactly

to the water surface, and the calibrated dial either read or, if adjustment is

possible, set to zero.

3. Sets of measurements of discharge rate and head are taken for each notch in turn,

the flow being regulated by the bench supply valve.

4. It is recommended that the first reading be taken at maximum flow rate, and

subsequent values with roughly equal decrements in head.

5. Reading should be discontinued when the level has fallen to the condition at

which the stream ceases to spring clear of the notch plate; this is likely to occur

when the head is reduced to about 10 mm for the rectangular notch and about 30

mm for V notch.

6. As it takes a little time for the water level to stabilize after changing the flow rate,

care should be taken to ensure that condition have settled completely before

starting to measure the discharge.

7. The rate of flow is found by timing the collection of a known amount of water in

the weighing tank.

8. About 6-8 experimental points for each notch should be sufficient.

26

3.4 RESULT

Rectangular Notch

Width of the crest, L (m) = 0.030 m

Height of the crest, P (m) =

Number of end contractions, n = 2

Table 3.1: Data recorded and the calculation of the flow rate, Q

Serial Number

Time taken Volume recorded Q =V/t (m3/s)

Initial (s)

End (s)

Time (s)

Start (m3)

End (m3)

Volume (m3)

(1) (2) (3) (4) (5) (6) (7) (8) 1 2 3 4 5 6 7 8

Table 3.2 : Reading of point gauge for water level and calculation for Cd

Serial Number

Discharge Q

(m3/s)

Reading of point gauge (m)

H3/2

(L-

0.2H) (m)

Discharge coefficient ‘Cd’

Crest Level Head, H

Eq. (1)

Eq. (4)

Eq. (6)

(1) (2) (3) (4) (5) (6) (7) (8) 1 2 3 4 5 6 7 8

27

Triangular Notch Angle of V notch, θ = 30º

Depth of V notch from base, P (m) =

Table 3.3 : Data recorded and the calculation of the flow rate, Q

Serial Number

Time taken Volume recorded Q =V/t (m3/s)

Initial (s)

End (s)

Time (s)

Start (m3)

End (m3)

Volume (m3)

(1) (2) (3) (4) (5) (6) (7) (8) 1 2 3 4 5 6 7 8

Table 3.4: Reading of point gauge for water level and calculation for Cd

Serial Number

Discharge Q

(m3/s)

Reading of point gauge (m) H5/2

Discharge coefficient ‘Cd’

Crest Level Head, H

Eq.(7) Eq.(9)

(1) (2) (3) (4) (5) (6) (7) (8) 1 2 3 4 5 6 7 8


1. From Table 3.2, plot graph Q versus H3/2 for rectangular notch. The plotted

gradient line will represent k1 value. dC can be computed using Equation (1),

Equation (4) and Equation (6).

28

2. From Table 3.4, plot graph Q versus H5/2 for triangular notch. The plotted

gradient line will represent k2 value. dC can be computed using Equation (7) and

Equation (9).

3.6 QUESTION FOR FURTHER DISCUSSION


2. Discuss the importance of weir or notch concept in water resources.

29


ENGINEERING CAMPUS

EXPERIMENT 4 : BOURDON PRESSURE GAUGE

4.0 OBJECTIVE

To calibrate the Bourdon Pressure Gauge

4.1 INTRODUCTION

The pressure gauge fitted to the tester is of a type known as the Bourdon Gauge,

which is used to a very great extent is engineering practice.

The mechanism of the gauge may be seen through the transparent dial of the

instrument (illustrated in Figure 4.1). A tube having a thin wall of oval cross-section is

Figure 4.1 : Bourdon pressure gauge

30

bent to a circular are encompassing about 270 degrees. It is rigidly held at one end where

the pressure is admitted to the tube and is free to move at the other end, which is sealed.

When pressured is admitted, the tube tends to straighten, and the movement at the free

end operates a mechanical system which moves a pointer round the graduated scale, the

movement of the pointer being proportional to the pressure applied. The sensitivity of

the gauge depends on the material and dimensions of the Bourdon tube; gauges with a

very wide selection of pressure ranges are commercially available.

When it is desired to check the accuracy of a Bourdon gauge, the usual procedure

is to load it with known pressures by a dead weight tester using oil to transmit the

pressure. The present experiment, however, works satisfactorily with water instead of

oil.

4.2 INSTALLATION PROCEDURE

Remove the piston from the unit. The piston is delivered lightly oiled and should

be wiped only when the unit is to be used. Fill the cylinder with water, and remove air

trapped in the transparent tube by tilting and gently tapping the unit. A small amount of

air left in the system will not affect the experiment. Top up with water and insert piston

into cylinder, allowing air and excess water to escape through the top hole in the side of

the cylinder. Allow the piston to settle.

4.3 EXPERIMENTAL PROCEDURE

1. Ensure the cylinder is vertical.

2. Masses are added in approximately eight increments up to a maximum of 5.2 kg.

On no account should more than the supplied masses be loaded.

3. Always load the masses gradually, do not drop them onto the platform.

4. The pressure gauge reading should be recorded at each increment of loading.

5. To prevent the piston sticking, rotate the piston gently as each mass is added.

6. Reverse the above procedure, taking readings as the masses are removed.

7. The cross-sectional area and the mass of the piston should be noted.

31

4.4 RESULT AND CALCULATIONS

The actual hydrostatic pressure (P) in the system due to mass of M kg (including the

piston mass) applied to the given by :-

P = M x 9.81 x 10-3 kN/m2

A Mass of piston, m = 1.0 kg Cross section area of piston, A = 315 mm2

Increasing Pressure Decreasing Pressure Mass

added to piston

Total mass on piston

(M)

Actual pressure

(P)

Gauge reading

Gauge error

Gauge reading

Gauge error

kg kg kN/m2 kN/m2 kN/m2 kN/m2 kN/m2 0 1.0

0.5 1.5 0.5 2.0 0.5 2.5 0.5 3.0 0.5 3.5 0.5 4.0 0.5 4.5 0.5 5.0 0.2 5.2

Table 4.1: Result for increasing and decreasing pressure


1. From Table 4.1, plot the graph of the gauge pressure against actual pressure and

gauge error against actual pressure.

4.6 QUESTIONS FOR FURTHER DISCUSSION


32


ENGINEERING CAMPUS

EXPERIMENT 5 : FLOW THROUGH A VENTURI METER

5.0 OBJECTIVES

To validate Bernoulli’s Theorem

To calculate the value of dC by using venturi meter

5.1 INTRODUCTION

The Venturi tube is a device which has been used over many years for measuring

the flow rate along a pipe. As seen from Figure 5.1, it consists of a tapering contraction

section, along which the fluid accelerates towards a short cylindrical throat, followed by a

section which diverges gently back to the original diameter. (The slowly diverging

section is frequently referred to as a diffuser). As the velocity increases from the inlet

section to the throat, there is a fall in pressure, the magnitude of which depends on the

flow rate. The flow rate may therefore be inferred from the difference in pressure,

measured by piezometers placed upstream and at the throat. Such a unit is referred to as a

Venturi flow meter.

Another way of metering the flow would be to insert a sharp edged orifice into the

pipe; whereby the differential pressure produced by flow through the orifice may

similarly be used to infer the flow rate. An orifice meter has the advantages of simplicity

and cheapness. In comparison with the Venturi meter however, an orifice causes a bigger

head loss than a corresponding venturi tube. This is because in the venture tube, much of

the head loss is recovered as the fluid decelerates in the diffuser. The differential

piezometric head from inlet to the throat can be several times greater than the total head

loss across the whole device.

33

Although piezometer tappings are needed only at the upstream section and at the

throat to infer the flow rate, it is instructive in a laboratory experiment to insert numerous

further tappings to show the distribution of piezometer head along the whole length of the

Venturi tube. Therefore, it is possible to calculate the distribution of pressure along the

tube. Comparison with measurements will then show where the losses occur in the unit.

5.2 THEORY

Bernoulli’s Theorem stated that

tconsZP

g

vtan

2

2

. ……………(1)

(velocity head) + (pressure head) + (elevation head) = constant

Equation (1) can be expressed as:

22

22

11

21

22Z

P

g

vZ

P

g

v

…………….(2)

Figure 5.1 : Arrangement of venturi meter apparatus

34

Consider flow of an incompressible, inviscid fluid through the convergent-divergent

Venturi tube shown in Figure 5.2. The cross sectional area at the upstream section 1 is

A1, at the throat section 2 is A2, and at any other arbitrary section n is An. Piezometer

tubes at these sections register h1, h2 and hn above the arbitrary datum shown. Note that,

although the tube may have any inclination, it is necessary for the datum to always be

horizontal. So, the elevation head, z = z1=z2=z3=………zn. Pressure can be expressed in

piezometer head form: h1 , h2 , …… hn where it can replace P1/ γ, P2 /γ ……..Pn / γ

Equation (2) can be expressed as:

2

22

1

21

22h

g

vh

g

v ….. ………..(3)

The continuity equation is:

nn AvAvAvAv .......................332211

or

Figure 5.2: Ideal conditions in a venturi meter

35

1

221 A

Avv

Substituting in Equation (3), gives :

2

12

212

)/(1

)(2

AA

hhgv

or

2

12

212

)/(1

)(2

AA

hhgAQ

…………...(4)

This is the ideal flow rate, obtained by assuming inviscid, one-dimensional flow. In

practice, there is some loss of head between sections 1 and 2. Also, the velocity is not

absolutely constant across either of these sections. As a result, the actual values of Q fall

a little smaller of those given by Equation (5).

2

12

212

)/(1

(2

AA

hhgACQ d

..……….(5)

In which dC known as the coefficient of flow rate in venturi meter. Its value, which

usually lies between 0.92 - 0.99 is established by experiment.

Equation (5) can be expressed as :

)21( hhKQ …………………(6)

The above equation shows that when the graph Q versus 21 hh is plotted, it will form

a straight line which represent the gradient of the graph, K. dC can be computed by the

equation below :

gA

AAKCd

2

)/(1

2

212

……………(7)

36

To validate the Bernoulli’s Theorem, Equation (3) can be expressed as:

22

221

22

1

2/ v

vv

gv

hh nn

……………...(8)

The continuity equation gives )/(/ 1221 AAvv and )/(/ 22 nn AAvv

Equation (8) can be replaced with

2

2

2

1

222

1

2/

n

n

A

A

A

A

gv

hh ………….(9)

Equation (9) gives the value of gv

hhn

2/22

1 for inviscid fluid.

For the existing apparatus, the ideal values are tabulated in Table 5.1. The values will be

plotted against the distance from the inlet. The results are compared with the value of

gv

hhn

2/22

1which is calculated from the observation.

By expressing piezometric changes 1hhn as a fraction of the velocity head g

v

2

22 at the

throat, results at different discharges become directly comparable, and it is seen that the

experimental values follow the ideal curve quite well up to the throat, after which a

steadily increasing loss of energy becomes apparent. The dimensions of the meter and the

position of the piezometer tappings are shown in Figure 5.3.

37

5.3 PROCEDURES

1. The apparatus is first levelled. This is done by opening both the bench supply

valve and the control valve downstream of the meter, so as to allow water to flow

for a few seconds to clear air pockets from the supply hose.

Diameter (mm)

Distance from entrance (mm)

Figure 5.3 : Dimensions of venturi meter and positions of piezometer tubes

(1) (2)

38

2. The control valve is then gradually closed, subjecting the venture tube to a

gradual increase in pressure, which causes water to rise up the tubes of the

manometer, thereby compressing the air contained in the manifold.

3. When the water levels have risen to a convenient height, the bench valve is also

closed gradually, so that when both valves are finally shut off, the meter is left

containing static water at moderate pressure, and the water level in the manometer

tubes stands at a convenient height. The adjusting screws are then operated to give

identical readings for all of the tubes across the whole width of the manometer

board. The board should also be reasonably vertical when viewed from the end.

4. To establish the meter coefficient, measurements are made of a set of differential

heads (h2-h1) and flow rates Q. The first reading should be taken with the

maximum possible value of (h2-h1), i.e with h1 close to the top of the scale and h2

near to the bottom. This condition is obtained by gradually opening both the

bench valve and the control valve in turn.

5. Note all piezometer reading (from A to L) in Table 5.3. The rate of flow is found

by timing the collection of a known amount of water in the weighing tank (Table

5.2), whilst values of h1 and h2 are noted from the manometer scale.

6. Readings are then taken over a series of reducing values of (h2-h1), equally spread

over the available range from 250 mm to zero (Table 5.5). About 7-8 readings

should be sufficient.

39

5.4 RESULT

Experiment 1 : To validate Bernoulli’s Theorem

Table 5.1 : The distribution of ideal pressure as a fraction of velocity head at throat

Piezometer

tube No. ‘n’

Diameter

nd

(mm)

ndd /2

22 )/( nAA

2

22

12 )/()/( nAAAA

(1) (2) (3) (4) (5) A (1)

B C

D(2) E F G H J K L

Table 5.2: Flow rate

Volume, V (liter) Time, t (s)

Flow rate, Q (m3/s) Initial Final Total

Data required

Flow rate (Q) = m3/s

Velocity at throat section 22 / AQv = m/s

Velocity head at throat )2/( 22 gv = m

40

Table 5.3 : Pressure along the venturi meter.

Serial Number

Piezometer nh

(mm)

( 1000/)1hhn

(m) gv

hhn

2/

1000/)(2

2

1

(1) (2) (3) (4) (5) 1 A (1) 2 B 3 C 4 D (2) 5 E 6 F 7 G 8 H 9 J

10 K 11 L

* Notes : n change from A to L

Table 5.4 : Comparison between ideal values and measured values

Ideal values

22

212 )/()/( nAAAA

Measured values

gv

hhn

2/

1000/)(2

2

1

41

Experiment 2 : To calibrate venturi meter and determine Cd value

Diameter at inlet (d1) = mm

Cross sectional area at inlet (A1) = m2

Diameter at throat (d2) = mm

Cross sectional area at throat (A2) = m2

Table 5.5 : Determination of Cd

S

V (liter)

t (s) Q = (V/10000t)

(m3/s)

h1

(mm)

h2

(mm)

h1-h2

(m)

(h1-h2)1/2

(m)1/2

Cd

(1) (2) (3) (4) (5) (6) (7) (8) (9) 1 2 3 4 5 6 7 8

Average of Cd

Notes : h1 = Reading at piezometer A

h2 = Reading at piezometer D


1. From Table 5.4, plot graph measured gv

hhn

2/

1000/)(2

2

1and ideal pressure

22

212 )/()/( nAAAA distribution along venturi meter versus distance from

inlet to contraction section.

2. From Table 5.5, plot graph Q versus (h1-h2)

1/2. The plotted gradient line will

represent K value. dC can be computed using Equation 7.

42

5.6 QUESTION FOR FURTHER DISCUSSION

1. What suggestions have you for improving the apparatus?

EAA 206 - Fluids Mechanics (Manual)

Documents

Transcript of EAA 206 - Fluids Mechanics (Manual)