39165346 Bernoulli s Theorem Distribution Experiment

7/29/2019 39165346 Bernoulli s Theorem Distribution Experiment

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1 | P a g e

Table of Contents

Ab s t r ac t / S umm ary ............................................................................................................... 2

In t ro d uc t i on ............................................................................................................................... 2

Ai ms / O bj e c t i ves .................................................................................................................... 3

T heor y .......................................................................................................................................... 4

Ap p ar a tu s .................................................................................................................................... 7

Ex per im en ta l P ro cedu re ....................................................................................................... 8

R esu l t s ......................................................................................................................................... 9

S ampl e C al c ul a t io ns ............................................................................................................. 19

Di s cuss i ons .............................................................................................................................. 20

C onc l us ions ............................................................................................................................. 21

R ecommend at i on s ................................................................................................................. 21

R ef e ren ces ................................................................................................................................ 22

Ap p en di ces ............................................................................................................................... 22


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2 | P a g e

ABSTRACT / SUM M ARY

The ma i n pur pos e o f t h i s expe r i men 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 meas ur 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 es s u r e heads

i n a r i g i d conver gen t / d i ve r gen t t ube o f known geomet 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

Demons t r a t i on Appar a t us , F1- 15 . I n t h i s exp e r i men t , t he p r es s u r e

d i f f e r ence t aken i s f r om h 1 - h 5 . The t i me t o co l l ec t 3 L wa t e r i n t he t ank

was de t e r mi ned . Las t l y t he f l ow r a t e , ve l oc i t y , dynami 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 exper i men t and

f r om t he da t a g i ven f o r bo t h conver gen t and d i ve r gen t f l ow. Bas ed on t he

r es u l t s t aken , i t has been ana l ys ed t ha t t he ve l oc i t y o f conver gen t f l ow i s

i nc r eas i ng , wher 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 eas ed , 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 conver gen t and d i ve r gen t t ube o f known geomet 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 es s u r ea r e a s we l l ca l cu l a t ed . The exper i men t was compl e t ed and s ucces s f u l l y

conduc t ed .

I N TRODUCT I ON

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 n ed i n t he

app l i ca t i on t ha t i nvo l ves i n v i s c i d f l ow, wher eby t he s peed o f t he movi ng

f l u i d i s i nc r eas ed s i mul t aneous l y whe t he r w i t h t he dep l e t i ng p r es s u r e o r

t he po t en t i a l ene r gy r e l evan t t o t h e 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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3 | P a g e

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 ms o f

Ber nou l l i s equa t i on .

Ber nou l l i s p r i nc i p l e compl i e s w i t h t he p r i nc i p l e o f cons e 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 eaml i ne o f a f l owi ng f l u i d

i s t he s ame as t he s um of a l l f o r ms o f mechan i ca l ene r gy a l ong t he

s t r eaml i ne . I t can be s i mpl i f i ed a s cons t an t p r ac t i ces o f t he s um of

p o t en t i a l ene rg y a s we l l a s k i ne t i c ener g y.

Fl 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 es s u r e and we i gh t . As amat t e r o f f ac t , i f a f l u i d i s mov i ng hor i zon t a l l y a l ong a s t r eaml i ne , t he

i nc r eas e i n s peed can be exp l a i ned due t o t he f l u i d t ha t moves f r om a

r eg i on o f h i gh p r es s u r e t o a l ower p r es s u r e r eg i on and s o wi t h t he i nve r s e

cond i t i on wi t h t he dec r eas e i n s pe ed . I n t he cas e o f a f l u i d t ha t moves

hor i zon t a l l y , t he h i ghes t s peed i s t he one a t t he l owes t p r es s u r e , wher eas

t he l owes t s peed i s p r es en t a t t he mos t h i ghes t p r es s u r e .

A I M S / OB J ECT I VES

1 . To i nves t i ga t e t he va l i d i t y o f Ber 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 meas ur e 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 es s u r e heads i n a r i g i d

conver gen t / d i ve r gen t t ube o f k nown geomet r y f o r a r ange o f s t eady f l ow

r a t es .


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4 | P a g e

THEORY

The spec i f i c hyd r au l i c m ode l u sed i n t h i s expe r i m en t i s Be r nou l l i s

Theor em Demons 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 wi t h a number o f ho l e - s i ded

p re ss ur e t ap in gs , co n ne c te d to t he m an om et e r s hous ed on t he r i g , i s

i ndeed an accur a t e l y mach 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 os s

s ec t i on . The t ap i ngs a l l ow t he meas ur emen t o f s t a t i c p r es s u r e head

s i mul t aneous l y .

A f l ow con t r o l va l ve i s i ncor por a t ed downs t r eam of t he t e s t s ec t i on .

F l ow r a t e and p r es s u r e i n t he appar a t us may be va r i ed i ndependen t l y by

ad j us t men t o f t he f l ow con t r o l va l ve , and t he bench s upp l y con t r o l va l ve .


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5 | P a g e

Cons i de r a s ys t em wher eby Chamber A i s under p r es s u r e and i s

connec t ed t o Chamber B , wh i ch i s a s we l l under p r es s u r e . The p r es s u r e i n

Chamber A i s s t a t i c p r es s u r e o f 689 .48 kP a . The p r es s u r e a t s ome po i n t , x

a 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 es s u r e o f 68 .95 kPa

exer 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

unus ed s t a t i c p r es s u 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 . As

t he f l u i d en t e r s chamber 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 es s u r e . The f o r ce r equ i r ed t o abs or 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 movi ng o r i g i na l l y , s o t ha t t he

s t a t i c p r es s u r e i n chamber B i s equa l t o t ha t i n chamber A .

Fr om t he above i l l u s t r a t i on , Ber 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 ncompr es s ib l e f l ow . Bel ow i s a c om m on f o r m o f Ber nou l l i s eq ua 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 eaml i ne when g r av i t y i s

cons t an t .

. . . . . . . . . . . . . . . ( 1 )


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6 | P a g e

wher e :

i s t he f l u i d f l ow s peed a t a po i n t on a s t r eaml 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 of t he po i n t above a r e f e r ence p l ane , w i t h t he

pos i t i ve z- d i r ec t i on po i n t i ng upwar d s o i n t he d i r ec t i on oppos 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 re ssu 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 .

I f equa t i on ( 1 ) i s mul t i p l i ed wi t h f l u i d dens i t y , , it can be rewritten as the

followings;

. . . . . . . . . . . ( 2 )

Or

. . . . . . . . ( 3 )

wher e :

i s dynami c p r es s u r e ,

i s t he p i ez om et r i c h ead o r hydr au l i c head ( t he s um of

t he e l eva t i on z and t he pr ess u re h ea d and

i s t he to ta l pressure ( t he s um of t he s t a t i c

p r ess u re p and dynami c p r es s u r e q ) .

The above equa t i ons s ugges t t he r e i s a f l ow s peed a t wh i ch p r es s u r e

i s ze r o , and a t ev en h i ghe r s peeds t he p r es s u r e i s nega t i ve . Mos t o f t en ,

gas es and l i qu i ds a r e no t capab l e o f nega t i ve abs o l u t e p r es s u r e , o r evenze r o p r es s u r e , s o c l ea r l y Ber nou l l i ' s equa t i on ceas es t o be v a l i d be f o r e

ze r o p r es s u r e i s r eached . I n l i qu i ds , when t he p r es s u r e becomes t oo l ow,

cav i t a t i ons occur . The above equa t i ons us e a l i nea r r e l a t i ons h i p be t ween

f l ow s peed s qua r ed and p r es s u r e .
http://en.wikipedia.org/wiki/Earth%27s_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/Cavitationhttp://en.wikipedia.org/wiki/Cavitationhttp://en.wikipedia.org/wiki/Pressure_headhttp://en.wikipedia.org/wiki/Hydraulic_headhttp://en.wikipedia.org/wiki/Piezometric_headhttp://en.wikipedia.org/wiki/Dynamic_pressurehttp://en.wikipedia.org/wiki/Densityhttp://en.wikipedia.org/wiki/Pressurehttp://en.wikipedia.org/wiki/Elevationhttp://en.wikipedia.org/wiki/Earth%27s_gravity


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7 | P a g e

Gener a l l y i n many app l i ca t i ons o f Ber nou l l i s equa t i ons , i t i s

common 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 s o s ma l l

compar ed t o o t he r va l ues . Thu s , t he p r ev i ous expr es s i on can be s i mpl 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 es s u r e , and q i s dynami c p r es s u r e , wher eas p

u s ua l l y r e f e r s a s s t a t i c p r es s u r e . Thus ,

To t a l p r es s u r e = s t a t i c p r es s u r e + dynami c p r es s u r e . . . . . . . ( 4 )

However , a f ew as s umpt 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 exper i men 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 ncompr es s 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 es s

APPARATUS

Vent u r i me t e r

Pad o f monomet e r t ubes

Pump

St opwat ch Wat er

Wat er t ank equ i pped wi t h va l ves wa t e r con t r o l l e r

Wat er hos t s and t ubes


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8 | P a g e

EXPERI M ENTAL PROCEDURE

1 . The t e s t s ec t i on t ube i s s e t t o be conver g i ng i n t he d i r ec t i on o f f l ow.

2 . The pump s wi 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 manomet e r .

3 . The a i r b l eed s c r ew i s opened and t he cap i s r emoved f r om t he

ad j acen t a i r va l ve un t i l t he s ame l eve l o f wa t e r i n manomet e r i s

r eached . The bench va l ve i s ad j us t ed un t i l t he h 1 h 5 head d i f f e r ence

of 50mm wat e r i s ob t a i ned .

4 . The ba l l va l ve i s c l os ed and t he t i me t aken t o accumul a t e a known

vo l ume o f 3L f l u i d i n t he t ank i s meas ur ed t o de t e r mi ne t he vo l ume

f l ow r a t e .

5 . The who l e p r oces s i s r epea t ed us i ng ( h 1 h 5 ) 100 and 150 mm wat e r .

6 . Nex t , th e ex per i m en t i s r epea t ed f o r d i v er ge n t t e s t se c t i on t ube .


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9 | P a g e

RESULTS

Conver gen t F l ow

Pr es s u r e d i f f e r ence = 50 mm wat e r

Vo l ume ( m3) = 0 .003

Ti me ( s ) = 46

F l ow r a t e ( m3/ s ) = 6 .522x10

- 5

No

Pr es s u r e

head , h

Di s t ance

i n t o duc t

( m)

Duc t

a r ea , A

( m2)

Ve l oc i t y

( m/ s )

S t a t i c

head

h , ( m)

Dynami c

head ,

( m)

Tot a l

head

ho

(m)

1 h 1 0 .00 490 .9

x 1 0- 6

0 .1329 145x

1 0- 3

0 .0009 0 .1459

2 h 2 0 .0603 151 .7

x 1 0- 6

0 .4299 135 x

1 0- 3

0 .0094 0 .1444

3 h 3 0 .0687 109 .4

x 1 0- 6

0 .5961 125 x

1 0- 3

0 .0181 0 .1431

4 h 4 0 .0732 89 .9

x 1 0- 6

0 .7255 110 x

1 0- 3

0 .0268 0 .1368

5 h 5 0 .0811 78 .5

x 1 0- 6

0 .8308 95 x

1 0- 3

0 .0352 0 .1302


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10 | P a g e


Vo l ume ( m3) = 0 .003

Ti me ( s ) = 31

F l ow r a t e ( m3/ s ) = 9 .677x10

- 5

No

Pr es s u r e

head , h

Di s t ance

i n t o duc t

(m )

Duc t

a r ea , A

( m2)

Ve l oc i t y

( m/ s )

S t a t i c

h e a d

h , ( m)

Dynami c

head ,

( m)

Tot a l

head

ho

(m)

1 h 1 0 .00 490 .9

x 1 0- 6

0 .1971 170 x

1 0- 3

0 .0020 0 .1720

2 h 2 0 .0603 151 .7

x 1 0- 6

0 .6379 145 x

1 0- 3

0 .0207 0 .1657

3 h 3 0 .0687 109 .4

x 1 0- 6

0 .8846 125 x

1 0- 3

0 .0399 0 .1649

4 h 4 0 .0732 89 .9

x 1 0- 6

1 .0760 100 x

1 0- 3

0 .0590 0 .1590

5 h 5 0 .0811 78 .5

x 1 0- 6

1 . 2 3 3 0 7 0 x

1 0- 3

0 .0775 0 .1475


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11 | P a g e


Vo l ume ( m3) = 0 .003

Ti me ( s ) = 25

F l ow r a t e ( m3

/ s ) = 1 .200x10- 4

No

Pr es s u r e

head , h

Di s t ance

i n t o duc t

(m )

Duc t

a r ea , A

( m2)

Ve l oc i t y

( m/ s )

S t a t i c

h e a d

h , ( m)

Dynami c

head ,

( m)

Tot a l

head

ho

(m)

1 h 1 0 .00 490 .9

x 1 0- 6

0 .2444 190 x

1 0- 3

0 .0030 0 .1930

2 h 2 0 .0603 151 .7

x 1 0- 6

0 .7910 160 x

1 0- 3

0 .0319 0 .1919

3 h 3 0 .0687 109 .4

x 1 0- 6

1 .0970 125 x

1 0- 3

0 .0613 0 .1863

4 h 4 0 .0732 89 .9

x 1 0- 6

1 . 3 3 5 0 9 0 x

1 0- 3

0 .0908 0 .1808

5 h 5 0 .0811 78 .5

x 1 0- 6

1 . 5 2 9 0 4 0 x

1 0- 3

0 .1192 0 .1592


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12 | P a g e

Di ver gen t F l ow

Pr es s u r e d i f f e r ence = 50mm wat e r

Vo l ume ( m3) = 0 .003

Ti me ( s ) = 30

F l ow r a t e ( m3/ s ) = 1 .000x10

- 4

No

Pr es s u r e

head , h

Di s t ance

i n t o duc t

(m )

Duc t

a r ea , A

( m2)

Ve l oc i t y

( m/ s )

S t a t i c

h e a d

h , ( m)

Dynami c

head ,

( m)

Tot a l

head

ho

(m)

1 h 1 0 .00 490 .9

x 1 0- 6

0 .2037 155 x

1 0- 3

0 .0021 0 .1571

2 h 2 0 .0603 151 .7

x 1 0- 6

0 .6592 130 x

1 0- 3

0 .1403 0 .2703

3 h 3 0 .0687 109 .4

x 1 0- 6

0 .9141 120 x

1 0- 3

0 .0426 0 .1626

4 h 4 0 .0732 89 .9

x 1 0- 6

1 .1120 115 x

1 0- 3

0 .0630 0 .1780

5 h 5 0 .0811 78 .5

x 1 0- 6

1 .2740 105 x

1 0- 3

0 .0827 0 .1877


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13 | P a g e


Vo l ume ( m3) = 0 .003

Ti me ( s ) = 23

F l ow r a t e ( m3

/ s ) = 1 .304x10- 4

No

Pr es s u r e

head , h

Di s t ance

i n t o duc t

(m )

Duc t

a r ea , A

( m2)

Ve l oc i t y

( m/ s )

S t a t i c

h e a d

h , ( m)

Dynami c

head ,

( m)

Tot a l

head

ho

(m)

1 h 1 0 .00 490 .9

x 1 0- 6

0 .2657 175 x

1 0- 3

0 .0036 0 .1786

2 h 2 0 .0603 151 .7

x 1 0- 6

0 .8596 135 x

1 0- 3

0 .0377 0 .1727

3 h 3 0 .0687 109 .4

x 1 0- 6

1 . 1 9 2 0 8 5 x

1 0- 3

0 .0724 0 .1574

4 h 4 0 .0732 89 .9

x 1 0- 6

1 . 4 5 1 0 8 0 x

1 0- 3

0 .1073 0 .1873

5 h 5 0 .0811 78 .5

x 1 0- 6

1 . 6 6 1 0 7 5 x

1 0- 3

0 .1406 0 .2156


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14 | P a g e


Vo l ume ( m3) = 0 .003

Ti me ( s ) = 20

F l ow r a t e ( m3/ s ) = 1 .500x10

- 4

No

Pr es s u r e

head , h

Di s t ance

i n t o duc t

(m )

Duc t

a r ea , A

( m2)

Ve l oc i t y

( m/ s )

S t a t i c

h e a d

h , ( m)

Dynami c

head ,

( m)

Tot a l

head

ho

(m)

1 h 1 0 .00 490 .9

x 1 0- 6

0 .3056 185 x

1 0- 3

0 .0048 0 .1898

2 h 2 0 .0603 151 .7

x 1 0- 6

0 .9888 135 x

1 0- 3

0 .0498 0 .1848

3 h 3 0 .0687 109 .4

x 1 0- 6

1 . 3 7 1 1 5 5 x

1 0- 3

0 .0958 0 .1508

4 h 4 0 .0732 89 .9

x 1 0- 6

1 . 6 6 8 5 4 5 x

1 0- 3

0 .1419 0 .1869

5 h 5 0 .0811 78 .5

x 1 0- 6

1 . 9 1 0 8 3 5 x

1 0- 3

0 .1861 0 .2211

Pr es s u r e

Head

( conver gen t

f l ow)

Usi ng Ber nou l l i s Equa t i on Us i ng Con t i nu i t y

Equa t i on

Di f f e r ence

Tot 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 ,

Ax106

(m2)

Vb =

Fl ow

r a t e Q /

A

( V a - Vb )

/ Vb ,

%

h 1 0 .1459 0 .145 0 .1329 490 .9 0 .1329 0

h 2 0 .1444 0 .135 0 .4295 151 .7 0 .4299 - 0 .09

h 3 0 .1431 0 .125 0 .5959 109 .4 0 .5961 - 0 .03

h 4 0 .1368 0 .110 0 .7251 89 .9 0 .7255 - 0 .06

h 5 0 .1302 0 .095 0 .8310 78 .5 0 .8308 0 .02

Pr es s u r e D i f f e r ence = 50mm


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15 | P a g e

Pr es s u r e

Head

( conver gen t

f l ow)


Equa t i on

Di f f e r ence

Tot 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 ,

Ax106

(m2)

Vb =

Fl ow

r a t e Q /

A

( V a - Vb )

/ Vb

%

h 1 0 .1720 0 .170 0 .1981 490 .9 0 .1971 0 .51

h 2 0 .1657 0 .145 0 .6373 151 .7 0 .6379 - 0 .09

h 3 0 .1649 0 .125 0 .8849 109 .4 0 .8846 0 .04

h4

0 .1590 0 .100 1 .0759 89 .9 1 .0760 - 0 .009

h 5 0 .1475 0 .070 1 .2331 78 .5 1 .2330 0 .008


Pr es s u r e

Head

( conver gen tf l ow)


Equa t i on

Di f f e r ence

Tot 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 ,

Ax106

(m2)

Vb =

Fl ow

r a t e Q /

A

( V a - Vb )

/ Vb ,

%

h 1 0 .1930 0 .190 0 .2426 490 .9 0 .2444 - 0 .74

h 2 0 .1919 0 .160 0 .7911 151 .7 0 .7910 0 .013

h 3 0 .1863 0 .125 1 .0967 109 .4 1 .0970 - 0 .03

h 4 0 .1808 0 .09 1 .3347 89 .9 1 .3350 - 0 .02

h 5 0 .1592 0 .04 1 .5293 78 .5 1 .5290 0 .02



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16 | P a g e

Pr es s u r e

Head

( d i ve r gen t

f l ow)


Equa t i on

Di f f e r ence

Tot 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 ,

Ax106

(m2)

Vb =

Fl ow

r a t e Q /

A

( V a - Vb )

/ Vb ,

%

h 1 0 .1571 0 .155 0 .2030 490 . 9 0 .2037 - 0 .34

h 2 0 .1521 0 .130 0 .6585 151 . 7 0 .6592 - 0 .11

h 3 0 .1626 0 .120 0 .9142 109 .4 0 .9141 0 .01

h 4 0 .1780 0 .115 1 .1118 89 .9 1 .1120 - 0 .02

h 5 0 .1877 0 .105 1 .2738 78 .5 1 .2740 - 0 .02


Pr es s u r e

Head

( d i ve r gen t

f l ow)


Equa t i on

Di f f e r ence

Tot 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 ,

Ax106

(m2)

Vb =

Fl ow

r a t e Q /

A

( V a - Vb )

/ Vb ,

%

h 1 0 .1786 0 .175 0 .2658 490 .9 0 .2657 0 .04

h 2 0 .1727 0 .135 0 .8600 151 .7 0 .8596 0 .05

h 3 0 .1574 0 .085 1 .1918 109 .4 1 .1920 - 0 .02

h 4 0 .1873 0 .080 1 .4509 89 .9 1 .4510 - 0 .01

h 5 0 .2156 0 .075 1 .6609 78 .5 1 .6610 - 0 .01



17/22

17 | P a g e

Pr es s u r e

Head

( d i ve r gen t

f l ow)


Equa t i on

Di f f e r ence

Tot 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 ,

Ax106

(m2)

Vb =

Fl ow

r a t e Q /

A

( V a - Vb )

/ Vb ,

%

h 1 0 .1898 0 .185 0 .3069 490 .9 0 .3056 0 .43

h 2 0 .1848 0 .135 0 .9885 151 . 7 0 .9888 - 0 .03

h 3 0 .1508 0 .055 1 .3710 109 . 4 1 .3711 - 0 .01

h 4 0 .1869 0 .045 1 .6686 89 .9 1 .6685 0 .01

h 5 0 .2211 0 .035 1 .9108 78 .5 1 .9108 0



18/22

18 | P a g e

Fi gur e 1 Graph of Tota l Head versus Pressure Head for Convergent

Flow

Fi gur e 2 Graph of Tota l Head versus Pressure Head for Divergent F low

0

0.05

0.1

0.15

0.2

0.25

50 mm pressure difference



h1 h2 h3 h4 h5

Pressure Head

TotalHead(m

)

Total Head versus Pressure Head for Convergent Flow

0

0.05

0.1

0.15

0.2

0.25




h1 h2 h3 h4 h5

Pressure Head

TotalHead(m)

Total Head versus Pressure Head for Divergent

Flow


19/22

19 | P a g e

SAM PLE CALCULAT I ONS

Di ver gen t F l ow

Pr es s u r e d i f f e r ence = h 1 - h 5 = 100 mm wat e r

F l ow r a t e = 0 .003 / 23

= 1 .304 10- 4

m3/ s

Ve l oc i t y , v = F l ow r a t e

Ar ea i n t o duc t

= 1 .304 10- 4

m3/ s

490 .9 x 10- 6

m2

= 0 .2657 m/ s

Dynami c head = v2

2 g

= ( 0 .2657 m/ s )2

2 x 9 .81m/ s2

= 0 .0036 m

Tot a l head = S t a t i c head + Dynami c head

= ( 0 .0036+ 1175x10

- 3) m

= 0 .1786 m


20/22

20 | P a g e

D I SCUSSI ON

Ref e r r i ng back t o t he ob j ec t i ves o f t he exp er i men t , wh i ch a r e 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 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 ap e r ed duc t a s we l l a s t o meas ur 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 es s u r e heads i n a r i g i d conver gen t and

d i ve r gen t t ube o f known geomet 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 wi 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 eas es . Th i s i s s hown i n a l l t he r e s u 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 ap e r ed duc t i nc r eas es a s

t he duc t a r ea dec r eas es , r ega r d l es s o f t he p r es s u r e d i f f e r ence and t ype

of f l ow o f each r e s u l t t aken .

F r om t he ana l ys i s o f t he r e s u l t s , we can con c l ude t ha t f o r bo t h t ype

of f l ow, be i t conver gen t o r d i ve r gen t , t h e ve l oc i t y i nc r eas es a s t he

p r ess u re d i f f e renc e in c re as es . For i n s t an ce , t he v e l oc i t i e s a t pr ess u re

head h 5 a t p r es s u r e d i f f e r ence o f 50 mi l l i me t r es , 100 mi l l i me t r es and

150 mi l l i me t r es f o r conver gen t f l ow a r e 0 .8308 m/ s , 1 .5290 m/ s and

1 .2740 m/ s r e s pec t i ve l y , wh i ch a r e i nc r eas i ng . The s ame goes t o

d 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 es s u r e

d i f f e r ence be t ween h 1 and h 5 i s i nc r eas ed . 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 es s u r e head h 5 t o h 1 , wh i ch i s f r om na r r ow t ube

t o wi de r t ube .

Nex t , th e to t a l h ead va lu e fo r con ve rgen t f l ow i s ca l cu l a t ed t o be

t he h i ghes t a t p r es s u r e head h 1 and t he l owes t a t p r es s u r e head h 5 ,

wher 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 cas e wher 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 es s u r e head h 5 and t he l owes t a t

p r ess u re h ea d h 1 .

The r e mus t be s ome e r r o r o r weaknes s es when t ak i ng t he

meas ur emen t o f each da t a . One o f t hem i s , t he obs e r ve r mus t have no t

r ead t he l eve l o f s t a t i c head p r oper l y , wher e t he eyes a r e no t


21/22

21 | P a g e

pe rp en d i cu la r t o t he wa t e r l ev el on the m an omet e r . T he re fo re , t h er e ar e

s ome mi 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 .

CONCLU SI ON

Fr om t he exper i men t conduc t ed , t he t o t a l h ead p r es s u r e i nc r eas es

f o r bo t h conver 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 s ame channe l .

The s econd ob j ec t i ves , wher e t he f l ow r a t e s a nd bo t h s t a t i c and t o t a l

head p r es s u r es i n a r i g i d conver gen t / d i ve r gen t o f known geomet 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 b e ca l cu l a t ed , a r e a l s o ach i eved

t h r ough t he exper i men t .

RECOMMENDAT I ON

Repea t t he exper i men t s eve r a l t i mes t o ge t t he ave r age va l ue .

Make s u r e t he bubb l es a r e f u l l y r emoved and no t l e f t i n t hemanomet e r .

The eye o f t he obs e r ve r s hou l d be pa r a l l e l t o t he wa t e r l eve l

on t he manomet e r .

The va l ve s hou l d be con t r o l l ed s l owl y t o ma i n t a i n t he p r es s u r e

d i f f e r ence .

The va l ve and b l eed s c r ew s hou l d r egu l a t e s moot h l y t o r educe

t he e r r o r s

Make s u 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


22/22

22 | P a g e

REFERENCES

B.R. Muns on , D .F . Young , and T .H . Ok i i s h i , Fu nda ment a l s o f

F lu id Me chan i cs , 3 r d ed . , 1998 , Wi l ey

and Sons , New Yor k .

Dougl as . J .F . , Gas i o r ek . J .M. and Swaf f i e l d , F l u i d Mechan i cs ,

3r d

ed i t i on , ( 1995) , Longmans S i ngapor e Pub l i s he r .

Gi l es , R .V . , Eve t t , J .B . and Cheng Lu i , Schaumms Out l i ne

Ser i e s Theor y and Pr ob l ems o f F l u i d Mechan i cs and

Hyd rau l i c , ( 1994) , McGr aw- Hi l l i n t l .

APPEND I CES

Documents

39165346 Bernoulli s Theorem Distribution Experiment