ABSTRACT
This experiment was conducted in order to find the volumetric flow rates of the water, the time taken to collect 3L volume of water, the pressure difference at all manometer tube, velocity, dynam dynamic ic head head and and also also the the total total head head.. The The comb combin inat atio ion n of vent ventur urii meter meter compl complete ete with with manometer tube and hydraulic bench were used. Bernoulli’s Theorem experiment’s apparatus consists of a classical venturimeter. A series of wall tappin allow measurement of the static pressure distribution alon the converin duct, while a total head tube is provided to traverse alon the centre line of the test section. These tappin are connected to a manometer bank incorporatin a manifold with air bleed valve. !urin the experiment, water is fed throuh a hose connector and the flow rate can be ad"u ad"uste sted d at the the flow flow reu reulat lator or valv valvee at the the outl outlet et of the the test test secti section on.. The The vent venturi uri can can be demonstrated demonstrated as a means of flow measurement measurement and the dischare dischare coefficient coefficient can be determined. determined. The results show the readin of each manometer tubes increase when the pressure difference increases.
INTRODUCTION
Bernoulli#s $rinciple states that for an inviscid flow of a non%conductin fluid, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid#s potential enery. enery. Bernoulli#s $rinciple is named in honor honor of !aniel Bernoulli who published it in his his book &ydrodynamica &ydrodynamica in '(3). *uppose a fluid is movin in a hori+ontal direction and encounters a pressure difference. This pressure difference will result in a net force, which is by ewton’s ewton’s *econd Law will cause an acceleration of the fluid. Bernoulli’s $rinciple can be demonstrated by the Bernoulli e-uation. The Bernoulli e-uation is an approximate relation between pressure, velocity, and elevation. hile the /ontinuity e-uation relates the speed of a fluid that movin throuh a pipe to the cross sectional area of the pipe. 0t says that as a radius of the pipe decreases the speed of fluid flow must increase and vice%versa.
The Bernoulli e-uation1 kinetic enery 2 potential enery 2 flow enery constant
&owever, Bernoulli’s $rinciple can only be applied under certain conditions. The conditions to which Bernoulli’s e-uation applies are the fluid must be fric tionless and of constant density. The flow must be steady, and the relation holds in eneral for sinle streamlines. 0n eneral, frictional effects are always important very close to solid wall and directly downstream of bodies. Thus, the Bernoulli approximation is typically useful in flow reions outside of boundary layers and wakes, where the fluid motion is overned by the combined effects of pressure and ravity forces. Bernoulli#s principle can be explained in terms of the law of conservation of enery. As a fluid moves from a wider pipe into a narrower pipe or a constriction, a correspondin volume must move a reater distance forward in the narrower pipe and thus have a reater speed. At the same time, the work done by correspondin volumes in the wider and narrower pipes will be expressed by the product of the pressure and the volume. *ince the speed is reater in the narrower pipe, the kinetic enery of that volume is reater. Then, by the law of conservation of enery, this increase in kinetic enery must be balanced by a decrease in the pressure%volume product, or, since the volumes are e-ual, by a decrease in pressure.
OBJECTIVES
'4 To find the time taken to collect 3L of water, the volumetric flow rates of the water, the pressure difference at all manometer tube 5static head4, velocity, dynamic head and also the total head. 64 To investiate the validity of the Bernoulli e-uation when applied to the steady flow of water in a tapered duct.
THEORY
Bernoulli#s e-uation clearly stated that the assumptions made in derivin the Bernoulli’s e-uation are1 •
The li-uid is incompressible.
•
The li-uid is non%viscous.
•
The flow is steady and the velocity of the li-uid is less than the critical velocity for the li-uid.
•
There is no loss of enery due to friction.
Bernoulli’s e-uation may be written as7 V'6 6 g
+
P'
+
γ
z'
=
V66 6 g
+
P6 +
γ
z 6
*tartin from a fluid element alon a streamline derived the Bernoulli e-uation for steady one% dimensional flow of an incompressible, in viscid fluid7 6
V
g
6
+
P γ
+
z
P 0 γ
here7 γ
the specific weiht of the fluid
z the elevation, V the velocity on the centre streamline in the 8enturi tube, P and P 0 the static and stanation 5total4 pressure respectively
0f hori+ontal tube levelled correctly, then +'+6 and the Bernoulli’s e-uation is simplified as7
6
V'
6 g
V '6 6 g
+
P'
6
+
γ
z'
=
V6
6 g
+
P6 +
γ
6
V'
z 6
6 g
+
P'
6
=
γ
V6
6 g
V
=
6
6 g
8elocity head h v
+
+
P6 γ
P γ
Total head, hT hs 2 hv
Then derive the expression for the velocity V alon the streamline as function of γ, P and P 0. 9rom the continuity e-uation for steady incompressible flow, the mean velocity U at each cross%section of the 8enturi tube is1
U
Q A
Q is the volume flow rate, A is the cross%section area.
APPARATUS
8enturi meter, &ydraulic bench, *top watch, ater, ater tank e-uipped with valves water controller, ater host and tubes.
The air bleed screw $ad of manometer tubes
The control valve
venturi
PROCEDURES
:-uipments *et ;p '. The Bernoulli’s e-uation apparatus is first set up on the hydraulic bench so that the base is in the hori+ontal position. 6. The test section is ensured to have the '<% tapered section converin in the direction of the flow. 3. The ri outflow tube is positioned above the volumetric tank. <. The ri inlet is connected to the bench flow supply, the bench valve and the apparatus flow control are closed and then the pump is started. =. >radually, the bench valve is opened to fill the test ri with the water. ?. 0n order to bleed air pressure tappin point and the manometers, both the bench valves and the ri flow control valves are closed. Then, the air bleed screw is opened and the cap from the ad"acent air valve is removed. (. A lenth of small%bore tunin from the air valve is connected to the volumetric tank. ). The bench valve is opened and allowed to flow throuh the manometer to pure all air from them. @. After that, the air bleed screw is tihtened and both the bench valve and ri flow control valve are partly opened. '. ext, the air bleed is opened slihtly to allow the air to enter the top of the manometers. The screw is re%tihtened when the manometer reach a convenient heiht. Takin A *et f Cesults '. The h' D h= are set to be = ml usin air bleed screw. 6. After the specific volume of h' D h= is reached, the ball valve is closed and the time taken to 3. <. =. ?. (.
accumulate 3L of fluid in the tank is measured. *teps ' and 6 are repeated with the different level of h' D h=. Then, the test section is reversed to et the diverin flow. The test section is removed by unscrewin the two couplin and bein reversed. The couplins are tihtened. *teps ' until 3 are repeated for diverin section.
RESULTS
:xperiment '5flow rate1 fast4 8olume 5 m Time 5s4
3
4
3 x '%3
''.3)
6.?
3
m
9low Cate 5
s
4
Cross
Using Bernoulli eqution
Using Continuit!
section
E
Di""erence
eqution
hF h&
2× g ×
hi 8iB
5mm4
¿
¿
Ai
8ic
8iB % 8ic
Q av
5mGs4
√ ¿
5mm4
π
Di
2
A i
4
5mGs4 5m64
5mGs4
A
'@
'(=
'.('?
=.3' x ' %<
.<@(
'.6'@
B
')
'?
'.@)'
3.?? x ' %<
.(6'
'.6?
/
'(=
'6
3.6)=
6.' x ' %<
'.3'3
'.@(6
!
'?=
'<
6.6'=
3.'< x ' %<
.)<'
'.3(<
:
'?=
'<=
'.@)'
3.) x ' %<
.?@=
'.6)?
9
'?<
'==
'.363
=.3' x ' %<
.<@(
.)6?
:xperiment 65flow rate1 medium4 8olume 5 m
3 x '%3
3
4
Time 5s4
@.= 3.'?x ' %<
3
m
9low Cate 5
s
4
Cross
Using Bernoulli eqution
Using Continuit!
Di""erenc
eqution
e
section
E
hF h&
2× g ×
hi
¿ √ ¿
8iB 5mm4
¿
Ai
5mm4
π
Di
2
8ic
8iB % 8ic
Q av
5mGs4
A i
4
5mGs4 5m64
5mGs4
A
6'
')=
6.6'=
=.3' x ' %<
.=@=
'.?6
B
6(
'(
6.?@<
3.?? x ' %<
.)?3
'.)3'
/
6=
(
=.'<(
6.' x ' %<
'.=(6
3.=(=
!
'@
'3=
3.6)=
3.'< x ' %<
'.?
6.6(@
:
')=
'=
6.?6
3.) x ' %<
.)36
'.())
9
')=
'?<
6.3
=.3' x ' %<
.=@=
'.<3=
:xperiment 35flow rate1 slow4 8olume 5 m
3 x '%3
3
4
Time 5s4
(.)) 3.)(x ' %<
3
m
9low Cate 5
Cross
s
4
Using Bernoulli eqution
section
E
hF h&
2× g ×
hi 8iB
¿
√ ¿
¿
Using Continuit!
Di""erenc
eqution
e
Ai
8ic
8iB % 8ic
5mm4
5mm4
π
Di
2
4
Q av
5mGs4
A i
5mGs4 5m64
5mGs4
A
636
6
6.=?
=.3' x ' %<
.('(
'.()@
B
63
'@
3.(?
3.?? x ' %<
'.<
6.???
/
66<
''
<.(6@
6.' x ' %<
'.)@<
6.)3=
!
6=
'<=
3.<3'
3.'< x ' %<
'.6'6
6.6'@
:
6
'?
6.)'
3.) x ' %<
'.6
'.(@@
9
66
'(=
6.36
=.3' x ' %<
.('(
'.=)=
CALCULATIONS E#$eri%ent &'
3
9low rate of water
(3 L×
1m
1000 L
3
) ÷ 11.38 s 6.?< x ' %<
S%$le Clcultion (cross section A)' Bernoulli eqution' 2× g ×
¿
8iB
8iB
¿
√ ¿ 2× 9.81 × ¿ ¿ √ ¿
8iB '.('? mGs
Continuit! eqution'
Ai
π
D i
2
4
( 26 × 10− )
3 2
Ai π
4
Ai =.3' x ' %< m6
m s
Q av
8ic
A i −4 2.64 × 10
8ic
5.31 x 10
−4
8ic .<@( mGs
Thus, 8iB % 8ic '.('? mGs % .<@( mGs '.6'@ mGs
E#$eri%ent *'
3
9low rate of water
( 3 L×
1m
1000 L
3
) ÷ 9.5 s 3.'? x ' %<
S%$le Clcultion (cross section A)' Bernoulli eqution' 2× g ×
¿ √ ¿
8iB
8iB
¿
2× 9.81 × ¿ ¿ √ ¿
8iB 6.6'= mGs
Continuit! eqution'
Ai
π
D i
2
4
( 26 × 10− )
3 2
Ai π
4
Ai =.3' x ' %< m6 Q av
8ic
A i
m s
−4 3 .16 × 10
8ic
5.31 x 10
−4
8ic .=@= mGs
Thus, 8iB % 8ic 6.6'= mGs % .=@= mGs '.?6 mGs
E#$eri%ent +'
3
9low rate of water
( 3 L×
1m
1000 L
3
) ÷ 7.8 8 s 3.)( x ' %<
S%$le Clcultion (cross section A)' Bernoulli eqution' 2× g ×
¿ √ ¿
8iB
8iB
¿
2× 9.81 × ¿ ¿ √ ¿
8iB 6.=? mGs
Continuit! eqution'
Ai
π
D i
2
4
( 26 × 10− )
3 2
Ai π
4
Ai =.3' x ' %< m6 Q av
8ic
A i −4
3.807 × 10
8ic
5.31 x 10 −4
8ic .('( mGs
m s
Thus, 8iB % 8ic 6.=? mGs % .('( mGs '.()@ mGs
DISCUSSION
The ob"ectives of this experiment is to investiate the validity of the Bernoulli e-uation when applied to the steady flow of water in a tapered duct and to measure the flow rates and both static and total pressure heads in a riid converent and diverent tube of known eometry for a rane of steady flow rates. This experiment is based on the Bernoulli’s principle which relates between velocities with the pressure for an inviscid flow.
To achieve the ob"ectives of this experiment, Bernoulli’s theorem demonstration apparatus alon with the hydraulic bench were used. This instrument was combined with a venturi meter and the pad of manometer tubes which indicate the pressure of h'until h). A venturi is basically a converin%diverin section, typically placed between tube or duct sections with fixed cross% sectional area. The flow rates throuh the venturi meter can be related to pressure measurements by usin Bernoulli’s e-uation.
9rom the result obtained throuh this experiment, it is been observed that when the pressure difference increase, the flow rates of the water increase and thus the velocities also increase.. The result show a rise at each manometer tubes when the pressure difference increases. As fluid flows from a wider pipe to a narrower one, the velocity of the flowin fluid increases. This is shown in all the results tables, where the velocity of water that flows in the tapered duct increases as the duct area decreases, reardless of the pressure difference and type of flow of each result taken.
9rom the analysis of the results, it can be concluded that the velocity of water decrease as the water flow rate decrease. 9or slow flow rate, the velocity difference at cross section A for water flow rate is 5'.6'@mGs4, B 5'.6?mGs4, / 5'.@(6mGs4, ! 5'.3(
slow condition is 3.)(x ' %<, then for medium flow rate is 3.'?x ' %
There are few mistakes that have occurred durin the experiment. 9irst, parallax and +ero error occurs when takin the measurement of each data. The observer must have not read the level of static head properly because the eyes are not perpendicular to the water level on the manometer. Also, while takin the readin of the manometer, there miht be possibility that the eye position of the readers is not parallel to the scale. Therefore, there are some minor effects on the calculations due to the errors.
Therefore, it can be concluded that the Bernoulli’s e-uation is valid when applied to steady flow of water in tapered duct and absolute velocity values increase alon the same channel. Althouh the experiment proof that the Bernoulli’s e-uation is valid for both flow but the values obtain miht be slihtly different from the actual value.
CONCLUSION
The results show the readin of each manometer tubes increase when the pressure difference increases. 9rom the result obtained, we can conclude that the Bernoulli’s e-uation is valid for converent and diverent flow as both of it does obey the e-uation. 9or both flow, as the pressure difference increase, the time taken for 3L water collected increase and the flow rates of the water also increase. Thus, as the velocity of the same channel increase, the total head pressure also increase for both converent and diverent flow.
RECO,,ENDATION
'. Hake sure the trap bubbles are removed before startin the experiment by tappin the surface of the rubber pipe. 6. 0n order to et more accurate results, repeat the experiment several times to et the averae values. 3. To avoid parallax error, the position of the observer’s eyes must be parallel to the water meniscus when takin the readin at the manometers. <. Hake sure there is no leakae of water in the instrument when conductin the experiment. RE-ERENCE
'. Bernoulli’s Theorem !emonstration, 6( Auust 6', at http1GGwww.solution.com.myGpdfG9H6<5A<4.pdf 6. Bernoulli’s $rinciple, 6( Auust 6', at http1GGtheory.uwinnipe.caGmodItechGnode?).html 3. Bernoulli’s $rinciple D /omputer !ictionary !efinition, 6( Auust 6', at http1GGwww.yourdictionary.comGcomputerGbernoulli%s%principle
APPENDI.
*mall bubbles in the tube that needs to be
0nitial heiht of water at '=mm
clear off
9inal heiht of water
