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Table of Contents
Abs t r ac t / Summary................................................................................................................2
Introduction................................................................................................................................2
A i m s / Objectives....................................................................................................................3
Theor y ...........................................................................................................................................4
Apparatus....................................................................................................................................7
Expe r i m en t a l Procedure........................................................................................................8
Res u l t s ..........................................................................................................................................9
Sam pl e Calculations.............................................................................................................19
Discussions...............................................................................................................................20
Conclusions..............................................................................................................................21
Recommendations..................................................................................................................21
References.................................................................................................................................22
Appendices...............................................................................................................................22
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ABSTRACT / SUMMARY
The m a i n pu r pose o f t h i s expe r i m en t i s t o i nves t i ga t e t he va l i d i t y o f t he
Ber nou l l i equa t i on when app l i ed t o t he s t eady f l ow o f wa t e r i n a t ape r ed
duc t and t o m easu r e t he f l ow r a t e and bo t h s t a t i c and t o t a l p r e s su r e h eads
i n a r i g i d conve r gen t / d i ve r gen t t ube o f known geom et r y f o r a r ange o f
s t eady f l ow r a t e s . The appa r a t us u sed i s Be r nou l l i s Theor em
Dem ons t r a t i on Appar a t us , F1 - 15 . I n t h i s expe r i m en t , t he p r e s su r e
d i f f e r ence t aken i s f r om h1 - h 5 . The t i m e t o co l l ec t 3 L wa t e r i n t he t ank
was de t e r m i ned . Las t l y t he f l ow r a t e , ve l oc i t y , d ynam i c head , and t o t a l
head wer e ca l cu l a t ed us i ng t he r ead i ngs we go t f r om t he expe r i m en t andf r om t he da t a g i ven f o r bo t h conve r gen t and d i ve r gen t f l ow . Based on t he
r e su l t s t aken , i t ha s been ana l ysed t ha t t he ve l oc i t y o f conve r gen t f l ow i s
i nc r eas i ng , whe r eas t he ve l oc i t y o f d i ve r gen t f l ow i s t he oppos i t e ,
wher eby t he ve l oc i t y dec r eased , s i nce t he wa t e r f l ow f r om a na r r ow a r ea
t o a w i de r a r ea . The r e f o r e , Be r nou l l i s p r i nc i p l e i s va l i d f o r a s t eady f l ow
i n r i g i d conve r gen t and d i ve r gen t t ube o f known geom et r y f o r a r ange o f
s t eady f l ow r a t e s , and t he f l ow r a t e s , s t a t i c heads and t o t a l heads p r e s su r e
a r e a s we l l ca l cu l a t ed . The expe r i m en t was com pl e t ed and success f u l l yconduc t ed .
INTRODUCTION
I n f l u i d dynam i cs , Be r nou l l i s p r i nc i p l e i s bes t exp l a i ned i n t he
app l i ca t i on t ha t i nvo l ves i n v i sc i d f l ow , wher eby t he speed o f t he m ov i ng
f l u i d i s i nc r eased s i m u l t aneous l y whe t he r w i t h t he dep l e t i ng p r e s su r e o r
t he po t en t i a l ene r gy r e l evan t t o t he f l u i d i t s e l f . I n va r i ous t ypes o f f l u i d
f l ow , Be r nou l l i s p r i nc i p l e u sua l l y r e l a t e s t o Be r nou l l i s equa t i on .
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Techn i ca l l y , d i f f e r en t t ypes o f f l u i d f l ow i nvo l ve d i f f e r en t f o r m s o f
Be r nou l l i s equa t i on .
Ber nou l l i s p r i nc i p l e com pl i e s w i t h t he p r i nc i p l e o f conse r va t i on o f
ene r gy . I n a s t eady f l ow , a t a l l po i n t s o f t he s t r eam l i ne o f a f l owi ng f l u i d
i s t he sam e a s t he sum o f a l l f o r m s o f m echan i ca l ene r gy a l ong t he
s t r eam l i ne . I t c an be s i m p l i f i ed a s cons t an t p r ac t i ce s o f t he sum o f
po t en t i a l ene r gy a s we l l a s k in e t i c en e r g y .
F l u i d pa r t i c l e s co r e p r ope r t i e s a r e t he i r p r e s su r e and we i gh t . As a
m a t t e r o f f ac t , i f a f l u i d i s m ov i ng ho r i zon t a l l y a l ong a s t r eam l i ne , t he
i nc r ease i n speed can be exp l a i ned due t o t he f l u i d t ha t m oves f r o m a
r eg i on o f h i gh p r e s su r e t o a l ower p r e s su r e r eg i on and so w i t h t he i nve r se
cond i t i on w i t h t he dec r ease i n speed . I n t he ca se o f a f l u i d t ha t m oves
hor i zon t a l l y , t he h i ghes t speed i s t he one a t t he l owes t p r e s su r e , whe r eas
t he l owes t speed i s p r e sen t a t t he m os t h i ghes t p r e s su r e .
AIMS / OBJECTIVES
1 . To i nves t i ga t e t he va l i d i t y o f Be r nou l l i equa t i on when app l i ed t o a
s t eady f l ow o f wa t e r i n a t ape r ed duc t .
2 . To m easu r e f l ow r a t e and b o t h s t a t i c and t o t a l p r e s su r e heads i n a r i g i d
conve r gen t / d i ve r gen t t ube o f known geom et r y f o r a r ange o f s t eady f l ow
r a t e s .
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THEORY
The spec i f i c hyd r au l i c m ode l u s ed i n t h i s expe r i m en t i s Be r nou l l i s
Theor em Dem ons t r a t i on Appar a t us , F1 - 15 .
The t e s t s ec t i on , wh i ch i s p r ov i ded w i t h a num ber o f ho l e - s i ded
p r e s s u r e t ap in gs , conn ec ted to t he m anom ete r s hous ed on t he r ig , i s
i ndeed an accu r a t e l y m ach i ned c l ea r ac r y l i c duc t o f va r y i ng c i r cu l a r c r oss
sec t i on . The t ap i ngs a l l ow t he m easu r em en t o f s t a t i c p r e s su r e head
s i m ul t aneous l y .
A f l ow con t r o l va l ve i s i nco r po r a t ed downs t r eam o f t he t e s t s ec t i on .
F l ow r a t e and p r e s su r e i n t he appa r a t us m ay be va r i ed i ndependen t l y by
ad j us t m en t o f t he f l ow con t r o l va l ve , and t he bench supp l y con t r o l va l ve .
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Cons i de r a sys t em wher eby Cham ber A i s unde r p r e s su r e and i s
connec t ed t o Cham ber B , wh i ch i s a s we l l unde r p r e s su r e . The p r e s su r e i n
Cham ber A i s s t a t i c p r e s su r e o f 689 .48 kPa . The p r e s su r e a t som e po i n t , xa l ong t he connec t i ng t ube cons i s t s o f a ve l oc i t y p r e s su r e o f 68 .95 kPa
exe r t ed 10 ps i exe r t ed i n a d i r ec t i on pa r a l l e l t o t he l i ne o f f l ow , p l us t he
unused s t a t i c p r e s su r e o f 90 ps i , and ope r a t e s equa l l y i n a l l d i r ec t i ons . A s
t he f l u i d en t e r s cham ber B , i t i s s l owed down , and i t s ve l oc i t y i s
changed back t o p r e s su r e . The f o r ce r equ i r ed t o abso r b i t s i ne r t i a
equa l s t he f o r ce r equ i r ed t o s t a r t t he f l u i d m ov i ng o r i g i na l l y , so t ha t t he
s t a t i c p r e s su r e i n cham ber B i s equa l t o t ha t i n cham ber A .
F r om t he above i l l u s t r a t i on , Be r nou l l i s p r i nc i p l e r e l a t e s m uch w i t h
i ncom pr ess i b l e f l ow . Be l ow i s a com m on f o r m o f Be r nou l l i s equa t i on ,
wher e i t i s va l i d a t any a r b i t r a r y po i n t a l ong a s t r eam l i ne when g r av i t y i s
cons t an t .
. . . . . . . . . . . . . . . ( 1 )
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wher e :
i s t he f l u i d f l ow speed a t a po i n t on a s t r eam l i ne ,
i s t he acce l e r a t i on due t o g r av i t y ,i s t he e l eva t i on o f t he po i n t above a r e f e r en ce p l ane , w i t h t he
po s i t iv e z- d i r ec t i on po i n t i ng upwar d so i n t he d i r ec t i on o ppos i t e
t o t he g r av i t a t i ona l acce l e r a t i on ,
i s t he p r e s s u r e a t t he po i n t , and
i s t he dens i t y o f t he f l u i d a t a l l po i n t s i n t he f l u i d .
If eq ua ti on (1 ) is mu lt ip li ed wi th fl ui d de ns it y, , it can be rewritten as the
followings;
. . . . . . . . . . . ( 2 )
Or
. . . . . . . . ( 3 )
wher e :
i s dynam i c p r e s su r e,
i s t he piezometric head or hydr au l i c head ( t he sum o f t he
e l eva t i on z and t he p r e s s u r e head and
i s t he to ta l pressure ( t he sum o f t he s t a t i c
p r e s su r e p and dynam i c p r e s su r e q ) .
The above equa t i ons sugges t t he r e i s a f l ow speed a t wh i ch p r e s su r e
i s ze r o , and a t even h i ghe r speeds t he p r e s su r e i s nega t i ve . M os t o f t en ,
gases and l i qu i ds a r e no t capab l e o f nega t i ve abso l u t e p r e s su r e , o r even
ze r o p r e s su r e , so c l ea r l y Be r nou l l i ' s equa t i on ceases t o be va l i d be f o r e
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http://en.wikipedia.org/wiki/Earth's_gravityhttp://en.wikipedia.org/wiki/Elevationhttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Dynamic_pressurehttp://en.wikipedia.org/wiki/Piezometric_headhttp://en.wikipedia.org/wiki/Hydraulic_headhttp://en.wikipedia.org/wiki/Pressure_headhttp://en.wikipedia.org/wiki/Earth's_gravityhttp://en.wikipedia.org/wiki/Elevationhttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Dynamic_pressurehttp://en.wikipedia.org/wiki/Piezometric_headhttp://en.wikipedia.org/wiki/Hydraulic_headhttp://en.wikipedia.org/wiki/Pressure_head7/27/2019 Bernoulli's Theorem Distribution Experiment
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ze r o p r e s su r e i s r eached . I n l i qu i ds , when t he p r e s su r e becom es t oo l ow ,
cav i t a t i ons occu r . The above equa t i ons u se a l i nea r r e l a t i onsh i p be t w een
f l ow speed squa r ed and p r e s su r e .
Gene r a l l y i n m any app l i ca t i ons o f Be r no u l l i s equa t i ons , i t i scom m on t o neg l ec t t he va l ues o f gz t e r m , s i nce t he change i s so sm a l l
com par ed t o o t he r va l ues . Thus , t he p r ev i ous exp r ess i on can be s i m p l i f i ed
as t he f o l l owi ng ;
. . . . . . . ( 3 )
wher e p 0 i s ca l l ed t o t a l p r e s su r e , and q i s dynam i c p r e s su r e , whe r eas p
usua l l y r e f e r s a s s t a t i c p r e s su r e . Thus ,
To t a l p r e s su r e = s t a t i c p r e s su r e + dynam i c p r e s su r e . . . . . . . ( 4 )
However , a f ew assum pt i ons a r e t aken i n t o accoun t i n o r de r t o
ach i eve t he ob j ec t i ves o f expe r i m en t , wh i ch a r e a s t he f o l l owi ngs :
The f l u i d i nvo l ved i s i ncom pr ess i b l e
The f l ow i s s t eady
The f l ow i s f r i c t i on l e s s
APPARATUS
Vent u r i m e t e r
Pad o f m onom et e r t ubes
Pum p
St opwa t ch
W at e r
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W at e r t ank equ i pped w i t h va l ves wa t e r con t r o l l e r
W at e r hos t s and t ubes
EXPERIMENTAL PROCEDURE
1. The t e s t s ec t i on t ube i s s e t t o be conve r g i ng i n t he d i r ec t i on o f f l ow .
2. The pum p swi t ch i s opened . The f l ow con t r o l va l ve i s t hen opened and
t he bench va l ve i s ad j us t ed t o a l l ow t he f l ow t h r ough t he m anom et e r .
3. Th e a i r b leed s cr ew i s opene d and th e cap i s r em ov ed f ro m t he
a dj ac en t a ir v al ve u nt il t he s am e l ev el o f w at er i n m an om et er i s
r e a c h e d. T h e b e n c h v a l v e i s a d j u s te d u n t i l t h e h 1 h 5 h e a d d i f f e r e nc e
o f 50m m wa t e r i s ob t a i ned .
4. T h e b a l l v a l ve i s c lo s ed a n d t he t im e t ak e n t o a cc um ul at e a k n ow n
v o lu m e o f 3 L f l u i d i n t h e t a n k i s m e as u r ed t o d e te r m in e t h e v o l u me
f l ow r a t e .
5. The who l e p r ocess i s r epea t ed us i ng ( h 1 h 5 ) 100 and 150 m m wa t e r .
6.Nex t , t he expe r im en t i s r epe a ted f o r d ive r ge n t t e s t se c t io n tu be .
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RESULTS
Con ver gen t F low
Pr essu r e d i f f e r ence = 50 m m wa t e r
Vo l um e ( m 3 ) = 0 .003
Ti m e ( s ) = 46
F l ow r a t e ( m 3 / s ) = 6 .522x10 - 5
No
Pr essu r e
head , h
Di s t ancei n t o duc t
( m )
Duc ta r ea , A
( m 2 )
Ve l oc i t y
( m / s )
S t a t i chead
h , ( m )
Dynam i chead ,
( m )
To t a lhead
h o ( m )1 h 1 0 .00 490 .9
x10 - 60 .132 9 1 45x
10 - 30 . 0009 0 .145 9
2 h 2 0 . 0603 151 .7
x10 - 60 .429 9 13 5 x
10 - 30 . 0094 0 .144 4
3 h 3 0 . 0687 109 .4
x10 - 60 .596 1 12 5 x
10 - 30 . 0181 0 .143 1
4 h 4 0 . 0732 89 . 9
x10 - 60 .725 5 11 0 x
10 - 30 . 0268 0 .136 8
5 h 5 0 . 0811 78 . 5
x10 - 60 .830 8 9 5 x
10 - 30 . 0352 0 .130 2
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Pr essu r e d i f f e r ence = 100 m m wa t e r
Vo l um e ( m 3 ) = 0 .003
Ti m e ( s ) = 31
F l ow r a t e ( m 3 / s ) = 9 .677x10 - 5
No
P r e s s u r e
head , h
Di s t ance
i n t o duc t
( m )
Duc t
a r ea , A
( m 2 )
Ve l oc i t y
( m / s )
S t a t i c
head
h , ( m )
Dynam i c
head ,
( m)
To t a l
head
h o ( m )1 h 1 0 .00 49 0 .9
x1 0 - 60 . 1971 17 0 x
1 0 - 30 .00 20 0 .17 20
2 h 2 0 .060 3 15 1 .7
x1 0 - 60 . 6379 14 5 x
1 0 - 30 .02 07 0 .16 57
3 h 3 0 .068 7 10 9 .4
x1 0- 6
0 . 8846 12 5 x
1 0- 3
0 .03 99 0 .16 49
4 h 4 0 .073 2 89 .9
x1 0 - 61 . 0760 10 0 x
1 0 - 30 .05 90 0 .15 90
5 h 5 0 .081 1 7 8 .5
x1 0 - 61 . 2330 7 0 x
1 0 - 30 .07 75 0 .14 75
Pr essu r e d i f f e r ence = 150 m m wa t e r
Vo l um e ( m 3 ) = 0 .003
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Ti m e ( s ) = 25
F l ow r a t e ( m 3 / s ) = 1 .200x10 - 4
No
P r e s s u r e
head , h
Di s t ance
i n t o duc t
( m )
Duc t
a r ea , A
( m 2 )
Ve l oc i t y
( m / s )
S t a t i c
head
h , ( m )
Dynam i c
head ,
( m)
To t a l
head
h o ( m )1 h 1 0 .00 49 0 .9
x1 0 - 60 . 2444 19 0 x
1 0 - 30 .00 30 0 .19 30
2 h 2 0 .060 3 15 1 .7
x1 0 - 60 . 7910 16 0 x
1 0 - 30 .03 19 0 .19 19
3 h 3 0 .068 7 10 9 .4
x1 0 - 61 . 0970 12 5 x
1 0 - 30 .06 13 0 .18 63
4 h 4 0 .073 2 89 .9
x1 0 - 61 . 3350 9 0 x
1 0 - 30 .09 08 0 .18 08
5 h 5 0 .081 1 7 8 .5x1 0 - 6
1 . 5290 4 0 x1 0 - 3
0 .11 92 0 .15 92
Di ve rg en t Fl ow
Pr essu r e d i f f e r ence = 50m m wa t e r
Vo l um e ( m 3 ) = 0 .003
Ti m e ( s ) = 30
F l ow r a t e ( m 3 / s ) = 1 .000x10 - 4
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No
P r e s s u r e
head , h
Di s t ance
i n t o duc t
( m )
Duc t
a r ea , A
( m 2 )
Ve l oc i t y
( m / s )
S t a t i c
head
h , ( m )
Dynam i c
head ,
( m)
To t a l
head
h o ( m )1 h 1 0 .00 49 0 .9
x1 0 - 6
0 . 2037 15 5 x
1 0 - 3
0 .00 21 0 .15 71
2 h 2 0 .060 3 15 1 .7
x1 0 - 60 . 6592 13 0 x
1 0 - 30 .14 03 0 .27 03
3 h 3 0 .068 7 10 9 .4
x1 0 - 60 . 9141 12 0 x
1 0 - 30 .04 26 0 .16 26
4 h 4 0 .073 2 89 .9
x1 0 - 61 . 1120 11 5 x
1 0 - 30 .06 30 0 .17 80
5 h 5 0 .081 1 7 8 .5
x1 0 - 61 . 2740 10 5 x
1 0 - 30 .08 27 0 .18 77
Pr essu r e d i f f e r ence = 100 m m wa t e r
Vo l um e ( m 3 ) = 0 .003
Ti m e ( s ) = 23
F l ow r a t e ( m 3 / s ) = 1 .304x10 - 4
No
P r e s s u r e
head , h
Di s t ance
i n t o duc t
( m )
Duc t
a r ea , A
( m 2 )
Ve l oc i t y
( m / s )
S t a t i c
head
h , ( m )
Dynam i c
head ,
( m)
To t a l
head
h o ( m )
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1 h 1 0 .00 49 0 .9
x1 0 - 60 . 2657 17 5 x
1 0 - 30 .00 36 0 .17 86
2 h 2 0 .060 3 15 1 .7
x1 0 - 60 . 8596 13 5 x
1 0 - 30 .03 77 0 .17 27
3 h 3 0 .068 7 10 9 .4x1 0 - 6
1 . 1920 8 5 x1 0 - 3
0 .07 24 0 .15 74
4 h 4 0 .073 2 89 .9
x1 0 - 61 . 4510 8 0 x
1 0 - 30 .10 73 0 .18 73
5 h 5 0 .081 1 7 8 .5
x1 0 - 61 . 6610 7 5 x
1 0 - 30 .14 06 0 .21 56
Pr essu r e d i f f e r ence = 150 m m wa t e r
Vo l um e ( m 3 ) = 0 .003
Ti m e ( s ) = 20
F l ow r a t e ( m 3 / s ) = 1 .500x10 - 4
NoP r e s s u r ehead , h
Di s t ance
i n t o duc t( m )
Duc t
a r ea , A( m 2 )
Ve l oc i t y( m / s )
S t a t i c
headh , ( m )
Dynam i c
head ,( m)
To t a l
headh o ( m )
1 h 1 0 .00 49 0 .9
x1 0 - 60 . 3056 18 5 x
1 0 - 30 .00 48 0 .18 98
2 h 2 0 .060 3 15 1 .7
x1 0 - 60 . 9888 13 5 x
1 0 - 30 .04 98 0 .18 48
3 h 3 0 .068 7 10 9 .4 1 . 3711 5 5 x 0 .09 58 0 .15 08
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x1 0 - 6 1 0 - 3
4 h 4 0 .073 2 89 .9
x1 0 - 61 . 6685 4 5 x
1 0 - 30 .14 19 0 .18 69
5 h 5 0 .081 1 7 8 .5
x1 0
- 6
1 . 9108 3 5 x
1 0
- 3
0 .18 61 0 .22 11
Pr essu r e
Head
( conve r gen t
f l ow)
U si ng Be rn ou ll i s Eq ua ti on U si ng C on ti nu it y
Equa t i on
Di f f e r ence
To t a l
Head , h
( m )
S t a t i c
Head ,
h i ( m)
V a =
[ 2g( h -
h i ) ]
Duc t
Ar ea ,
Ax10 6
( m 2 )
Vb =
Fl ow
r a t e Q /
A
( V a - Vb )
/ Vb ,
%
h 1 0 . 1459 0 .1 45 0 .1 329 4 90 .9 0 . 1329 0
h 2 0 . 1444 0 .1 35 0 .4 295 1 51 .7 0 . 4299 - 0 .0 9
h 3 0 . 1431 0 .1 25 0 .5 959 1 09 .4 0 . 5961 - 0 .0 3
h 4 0 . 1368 0 .1 10 0 .7 251 8 9 .9 0 . 7255 - 0 .0 6
h 5 0 . 1302 0 .0 95 0 .8 310 7 8 .5 0 . 8308 0 .02
Pr essu r e D i f f e r ence = 50m m
Pr essu r e
Head
( conve r gen t
f l ow)
U si ng Be rn ou ll i s Eq ua ti on U si ng C on ti nu it y
Equa t i on
Di f f e r ence
To t a l
Head , h
( m )
S t a t i c
Head ,
h i ( m)
V a =
[ 2g( h -
h i ) ]
Duc t
Ar ea ,
Ax10 6
( m 2 )
Vb =
Fl ow
r a t e Q /
A
( V a - Vb )
/ Vb
%
h 1 0 . 1720 0 .1 70 0 .1 981 4 90 .9 0 . 1971 0 .51
h 2 0 . 1657 0 .1 45 0 .6 373 1 51 .7 0 . 6379 - 0 .0 9
h 3 0 . 1649 0 .1 25 0 .8 849 1 09 .4 0 . 8846 0 .04
h 4 0 . 1590 0 .1 00 1 .0 759 8 9 .9 1 . 0760 - 0 .0 09
h 5 0 . 1475 0 .0 70 1 .2 331 7 8 .5 1 . 2330 0 . 008
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Pr essu r e D i f f e r ence = 100m m
Pr essu r e
Head
( conve r gen t
f l ow)
U si ng Be rn ou ll i s Eq ua ti on U si ng C on ti nu it yEqua t i on
Di f f e r ence
To t a l
Head , h
( m )
S t a t i c
Head ,
h i ( m)
V a =
[ 2g( h -
h i ) ]
Duc t
Ar ea ,
Ax10 6
( m 2 )
Vb =
Fl ow
r a t e Q /
A
( V a - Vb )
/ Vb ,
%
h 1 0 . 1930 0 .1 90 0 .2 426 4 90 .9 0 . 2444 - 0 .7 4
h 2 0 . 1919 0 .1 60 0 .7 911 1 51 .7 0 . 7910 0 . 013h 3 0 . 1863 0 .1 25 1 .0 967 1 09 .4 1 . 0970 - 0 .0 3
h 4 0 . 1808 0 . 09 1 .3 347 8 9 .9 1 . 3350 - 0 .0 2
h 5 0 . 1592 0 . 04 1 .5 293 7 8 .5 1 . 5290 0 .02
Pr essu r e D i f f e r ence = 150m m
Pr essu r e
Head
( d i ve r gen t
f l ow)
U sin g B er no ul li s Eq ua ti on Us in g C on tin ui t y
Equa t i on
Di f f e r ence
To t a l
Head , h
( m)
S t a t i c
Head ,
h i ( m)
V a =
[ 2g( h -
h i ) ]
Duc t
Ar ea ,
Ax10 6
( m 2 )
Vb =
Fl ow
r a t e Q /
A
( V a - Vb )
/ Vb ,
%
h 1 0 .15 71 0 . 155 0 .203 0 49 0 .9 0 . 2037 - 0 .3 4
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h 2 0 .15 21 0 . 130 0 .658 5 15 1 .7 0 . 6592 - 0 .1 1
h 3 0 .16 26 0 . 120 0 .914 2 10 9 .4 0 . 9141 0 .01
h 4 0 .17 80 0 . 115 1 .111 8 8 9 .9 1 . 1120 - 0 .0 2
h 5 0 .18 77 0 . 105 1 .273 8 7 8 .5 1 . 2740 - 0 .0 2Pr essu r e D i f f e r ence = 50m m
Pr essu r e
Head
( d i ve r gen t
f l ow)
U si ng Be rn ou ll i s Eq ua ti on U si ng C on ti nu it y
Equa t i on
Di f f e r ence
To t a l
Head , h
( m )
S t a t i c
Head ,
h i ( m)
V a =
[ 2g( h -
h i ) ]
Duc t
Ar ea ,
Ax10 6
( m 2 )
Vb =
Fl ow
r a t e Q /A
( V a - Vb )
/ Vb ,
%
h 1 0 . 1786 0 .1 75 0 .2 658 4 90 .9 0 . 2657 0 .04
h 2 0 . 1727 0 .1 35 0 .8 600 1 51 .7 0 . 8596 0 .05
h 3 0 . 1574 0 .0 85 1 .1 918 1 09 .4 1 . 1920 - 0 .0 2
h 4 0 . 1873 0 .0 80 1 .4 509 8 9 .9 1 . 4510 - 0 .0 1
h 5 0 . 2156 0 .0 75 1 .6 609 7 8 .5 1 . 6610 - 0 .0 1
Pr essu r e D i f f e r ence = 100m m
Pr essu r e
Head
( d i ve r gen t
U sin g B er no ul li s Eq ua ti on Us in g C on tin ui t y
Equa t i on
Di f f e r ence
Tota l S ta t i c
Head ,
V a =
[ 2g( h -
Duc t
Ar ea ,
Vb =
Fl ow
( V a - Vb )
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490 .9 x 10 - 6 m 2
= 0 .2657 m / s
D yn am ic he ad = v 2
2g
= (0 .2 65 7 m/s ) 2
2 x 9 .81m / s 2
= 0 .0036 m
Tot a l head = S t a t i c head + Dynam i c head
= ( 0 .0036 + 1175x10 - 3 ) m
= 0 .1786 m
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DISCUSSION
Ref e r r i ng back t o t he ob j ec t i ves o f t he expe r i m en t , wh i ch a r e t oi nves t i ga t e t he va l i d i t y o f t he Be r nou l l i s equa t i on when app l i ed t o t he
s t eady f l ow o f wa t e r i n a t ape r e d duc t a s we l l a s t o m easu r e t he f l ow
r a t e and bo t h s t a t i c and t o t a l p r e s su r e heads i n a r i g i d conve r gen t and
d i ve r gen t t ube o f known geom et r y f o r a r ange o f s t eady f l ow r a t e s .
As f l u i d f l ows f r om a w i de r p i pe t o a na r r ower one , t he ve l oc i t y
o f t he f l owi ng f l u i d i nc r eases . Th i s i s shown i n a l l t he r e su l t s t ab l e s ,
wher e t he ve l oc i t y o f wa t e r t ha t f l ows i n t he t ape r ed duc t i nc r eases a s
t he duc t a r ea dec r eases , r ega r d l e s s o f t he p r e s su r e d i f f e r ence and t ypeo f f l ow o f each r e su l t t aken .
F r om t he ana l ys i s o f t he r e su l t s , we can conc l ude t ha t f o r bo t h
t ype o f f l ow , be i t conve r gen t o r d i ve r gen t , t he ve l oc i t y i nc r eases a s t he
p r e s s u r e d i f f e r en ce inc r eas es . Fo r i n s t anc e , the v e loc i t i e s a t p r e s su r e
h e a d h 5 a t p r e s su r e d i f f e r ence o f 50 m i l l i m e t r e s , 100 m i l l i m e t r e s and
150 m i l l i m e t r e s f o r conve r gen t f l ow a r e 0 .8308 m / s , 1 . 5290 m / s and
1 .2740 m / s r e spec t i ve l y , wh i ch a r e i nc r eas i ng . The sam e goes t od i ve r gen t f l ow , wher eby t he ve l oc i t i e s a r e dec r eas i ng when t he p r e s su r e
d i f f e r ence be t ween h 1 and h 5 i s i nc r eased . No t e t ha t f o r d i ve r gen t f l ow ,
t he wa t e r f l ows f o r m p r e ssu r e head h5 t o h 1 , wh i ch i s f r om na r r ow t ube
t o w i de r t ube .
Nex t , t he to t a l hea d va l ue f o r co nve r gen t f l ow i s ca lcu la t ed t o b e
t he h i ghes t a t p r e s su r e head h 1 and t he l owes t a t p r e s su r e head h 5 ,
whe r eas t he t o t a l head f o r d i ve r gen t f l ow i s i n a d i f f e r en t ca se w her e i t
i s ca l cu l a t ed t o be t he h i ghes t a t p r e s su r e head h 5 and t he l owes t a t
p r e s s u r e head h 1 .
The r e m us t be som e e r r o r o r weaknesses when t ak i ng t he
m easu r em en t o f each da t a . One o f t hem i s , t he obse r ve r m us t have no t
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r ead t he l eve l o f s t a t i c head p r ope r l y , whe r e t he eyes a r e no t
p e r p end i cu la r to the wa t e r l eve l on t he m anom ete r . The r e f o r e , t he r e a r e
som e m i nor e f f ec t s on t he ca l cu l a t i ons due t o t he e r r o r s .
CONCLUSION
Fr om t he expe r i m en t conduc t ed , t he t o t a l head p r e s su r e i nc r eases
f o r bo t h conve r gen t and d i ve r gen t f l ow . Th i s i s exac t l y f o l l owi ng t he
Ber nou l l i s p r i nc i p l e f o r a s t eady f l ow o f wa t e r and t he ve l oc i t y i s
i nc r eas i ng a l ong t he sam e channe l .
The second ob j ec t i ves , whe r e t he f l ow r a t e s and bo t h s t a t i c andt o t a l head p r e s su r e s i n a r i g i d conve r gen t / d i ve r gen t o f known
geom et r y f o r a r ange o f s t eady f l ow r a t e s a r e t o be ca l cu l a t ed , a r e a l so
ach i eved t h r ough t he expe r i m en t .
RECOMMENDATION
Repea t t he expe r i m en t seve r a l t i m es t o ge t t he ave r age va l ue . M ake su r e t he bubb l e s a r e f u l l y r em oved and no t l e f t i n t he
m anom et e r .
The eye o f t he obse r ve r shou l d be pa r a l l e l t o t he wa t e r l eve l
on t he m anom et e r .
The va l ve shou l d be con t r o l l ed s l owl y t o m a i n t a i n t he p r e s su r e
d i f f e r ence .
The va l ve and b l eed sc r ew shou l d r egu l a t e sm oo t h l y t o r educe
t he e r r o r s
M ake su r e t he r e i s no l eakage a l ong t he t ube t o avo i d t he
wa t e r f l owi ng ou t
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REFERENCES
B.R . M unson , D .F . Young , and T .H . Ok i i sh i , Fund ament a l s o f
F l u id Mech an i cs , 3 r d ed . , 1998 , W i l eyand Sons , New Yor k .
D o u g l a s . J . F . , G a s i o r e k . J . M . a n d S w a f f i e ld , F lu id Mechan ics ,
3 r d ed i t i on , ( 1995) , Longm ans S i ngapor e Pub l i she r .
G il es , R .V ., E ve tt , J .B . a nd C he n g L ui , Schaumm s Out l i ne
S er ie s The or y and Pro blems o f F lu i d Me chan ics and
H ydr au l i c , ( 1994) , M cGr aw- Hi l l i n t l .
APPENDICES
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