bernoulli's theorem distribution experiment

7/27/2019 Bernoulli's Theorem Distribution Experiment

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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

2 | P a g e
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_head


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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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http://en.wikipedia.org/wiki/Cavitationhttp://en.wikipedia.org/wiki/Cavitation


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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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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


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


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


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)


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

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

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)


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

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 )

1 | P a g e


17/21


18/21

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

1 | P a g e


20/21

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

2 | P a g e


21/21

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

2 | P a g e

bernoulli's theorem distribution experiment

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