blt1110 hydraulics notes

5/26/2018 BLT1110 Hydraulics Notes

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Hydraulics BLT1110

Module Notes

John Pickering & Rob Eggleton


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Hydraulics Module Notes

TABLEOFCONTENTS

Hydralic!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#S$ %nit!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""'a!! and (eight""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""")*i+en!ion!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""")Linear, con!tant acceleration""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""")*i+en!ional Ho+ogeneity"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-.Pre!!re"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-.Frictional Shear Stre!!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-.Pro/ertie! o0 Flid!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""--Coe00icient o0 *yna+ic 1i!co!ity 2 345 +6""""""""""""""""""""""""""""""""""""""""""""""""""""""""--

5ine+atic 1i!co!ity345 n6""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""--

Flid Flo7"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""--$ntrodction to Flid 'echanic!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-8

The La7! o0 Hydro!tatic!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-8Pre!!re 'ea!re+ent""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-9

Ab!olte and 4age Pre!!re"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-9Pre!!re Head""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-9The 'ercry Baro+eter"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-9The Aneroid Baro+eter""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-:The *ead7eight Pi!ton 4age""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-:The Bordon 4age"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-:Pre!!re Tran!dcer!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-:

The Pie;o+eter Tbe"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-:The %a+/le! ? Pre!!re 'ea!re+ent""""""""""""""""""""""""""""""""""""""""""""""""""""""-@Ttorial e!tion! ? Pre!!re 'ea!re+ent""""""""""""""""""""""""""""""""""""""""""""""""""""""-)Stability and Boyancy o0 Floating Bodie!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""8.

Archi+ede! Princi/le""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""8.*eter+ination o0 'etacentric Height, '4""""""""""""""""""""""""""""""""""""""""""""""""""""""""8-

(orked E>a+/le! < Boyancy"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""88Ttorial e!tion! < Boyancy"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""89Hydro!tatic Force! on Sb+erged Sr0ace!"""""""""""""""""""""""""""""""""""""""""""""""""""""""8:

'o+ent o0 a Force, Rotation and Eilibri+"""""""""""""""""""""""""""""""""""""""""""""""""""""8=(orked E>a+/le! ? Hydro!tatic Force!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""8@Ttorial e!tion! ? Hydro!tatic Force!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""8)

Hydrodyna+ic! ? Flid! in 'otion"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9.Ty/e! o0 Flo7"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9.*eter+ination o0 La+inarTrblent Flo7 ? Reynold! N+ber, Re""""""""""""""""""""9.Analy!i! o0 Flid Flo7""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9-(orked E>a+/le! < Continity"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""98Ttorial e!tion! ? Continity """""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""98

(ork, Energy and E00iciency"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""99

(ater Energy"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""994raDitational or Potential Energy"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""99Energy Head""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9:

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5inetic Energy""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9:Pre!!re Energy""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9:Energy Lo!!e!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9:

Frictional Energy Lo!!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9:'inor Energy Lo!!e!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9=

The Energy BernolliG Eation""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9=(orked E>a+/le! ? Energy""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9@Ttorial e!tion! ? Energy""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9#Flo7 'ea!re+ent in /i/eline!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9'o+ent+ Eation""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9)

Jet $+/act on a Stationary Flat Plate"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9)Jet $+/act on a Stationary CrDed 1ane"""""""""""""""""""""""""""""""""""""""""""""""""""""""""9)Force! on a Pi/e Bend""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""":.Force! on a Pi/e Contraction or No;;le"""""""""""""""""""""""""""""""""""""""""""""""""""""""""":.Thr!t Block!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""":.

(orked E>a+/le! ? 'o+ent+"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""":-

Ttorial e!tion! ? 'o+ent+"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""":8Friction Lo!! in Pi/eline! """""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""":9

The *arcya+/le! ? %ni0or+ Channel Flo7""""""""""""""""""""""""""""""""""""""""""""""""""""""""@9Ttorial e!tion! ? %ni0or+ Channel Flo7""""""""""""""""""""""""""""""""""""""""""""""""""""""""@@Non


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Fnction! and Ty/e!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#8Po!itiDe *i!/lace+ent P+/!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#8Rotodyna+ic P+/!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#9Ty/e! o0 Rotodyna+ic P+/!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#9

Centri0gal P+/! lo7 , high HG"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#9

A>ial Flo7 P+/! high , lo7 HG""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#:'i>ed Flo7 P+/!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#:'lti!tage P+/!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#:

P+/ Per0or+ance Characteri!tic! CrDe!G""""""""""""""""""""""""""""""""""""""""""""""""""""""#=Centri0gal P+/ CrDe!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#=A>ial Flo7 P+/ CrDe!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#='i>ed Flo7 P+/ CrDe""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#=Other Characteri!tic CrDe!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#@

Hydro Po7er Trbine!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#@Franci! Trbine"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#@A>ial Flo7 Trbine""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""##

Pelton (heel Trbine"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#Se7age P+/ing Station Layot!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#)

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L$STOFF$4%RES

Figre - Flid Flo7""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""--Figre 8 Pre!!re 1ariation7ith *e/th""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-8Figre 9 Pre!!re in Connected Flid!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-8Figre : 4age Pre!!re Scale""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-9Figre = 'ercry Baro+eter""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-9Figre @ Aneroid Baro+eter""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-:Figre # *ead7eight Pi!ton 4age"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-:Figre Bordon 4age"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-:Figre ) Pre!!re Tran!dcer"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-:Figre -. aG Pie;o+eter bG 'ano+eter 7ith Secondary 4age Flid""""""""""""""-=Figre -- *i00erential 'ano+eter""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-=Figre -8 (orked P"'" E>a+/le -""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-@

Figre -9 Soltion 0or P"'" E>a+/le -""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-@Figre -: (orked P"'" E>a+/le 8""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-#Figre -= (orked P"'" E>a+/le 9""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-Figre -@ P"'" Ttorial e!tion! Figre!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""-)Figre -# Boyancy Force""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""8.Figre - Eilibri+ Condition!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""8-Figre -) Hydro!tatic Force on Sb+erged Sr0ace""""""""""""""""""""""""""""""""""""""""""""8:Figre 8. 'o+ent o0 Force"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""8=Figre 8- (orked E>a+/le H"F" -""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""8@Figre 88 (orked E>a+/le H"F" 8""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""8#Figre 89 (orked E>a+/le H"F" 9""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""8#

Figre 8: (orked E>a+/le H"F" 9 Soltion""""""""""""""""""""""""""""""""""""""""""""""""""""""""""8Figre 8= (orked E>a+/le H"F" 9 Soltion 'o+ent!G""""""""""""""""""""""""""""""""""""""""8Figre 8@ H"F" Ttorial e!tion! """"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""8)Figre 8# Flo7 Continity""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9-Figre 8 (orked E>a+/le! Continity""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""98Figre 8) (ork < *irected Force"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""99Figre 9. Pre!!re Energy""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9:Figre 9- Bernolli Energy *iagra+"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9=Figre 98 (orked E>a+/le! E" -""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9@Figre 99 (orked E>a+/le! E" 8""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""9#Figre 9: Jet $+/act on Stationary Flat Plate"""""""""""""""""""""""""""""""""""""""""""""""""""""""9)

Figre 9= Jet $+/act on a Stationary CrDed 1ane""""""""""""""""""""""""""""""""""""""""""""""9)Figre 9@ Force! on a Pi/e Bend"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""":.Figre 9# Force! on a Pi/e Contraction"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""":.Figre 9 (orked E>a+/le! '" -""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""":-Figre 9) (orked E>a+/le! '" 8""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""":8Figre :. (orked E>a+/le! *"E""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""::Figre :- Pi/eline *e!ign Table""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""":@Figre :8 Table o0 'inor Lo!!e!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""":Figre :9 Entry to a Pi/e""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""":)Figre :: Sdden $ncrea!e in Pi/e *ia+eter""""""""""""""""""""""""""""""""""""""""""""""""""""""":)Figre := Sdden *ecrea!e in Pi/e *ia+eter""""""""""""""""""""""""""""""""""""""""""""""""""""""=.Figre :@ Bend!, Elbo7!, and Tee Jnction!"""""""""""""""""""""""""""""""""""""""""""""""""""""""=.Figre :# *iDergent Ta/er or *i00!er"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""=-

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Figre : Partially Clo!ed 1alDe!""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""=8Figre :) E>it 0ro+ a Pi/e into a re!erDoir""""""""""""""""""""""""""""""""""""""""""""""""""""""""""=8Figre =. E>it 0ro+ a Pi/e to at+o!/here"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""=9Figre =- (orked E>a+/le! '"L" -""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""=:Figre =8 (orked E>a+/le! '"L" 8""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""==

Figre =9 (orked E>a+/le! '"L" 9""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""=@Figre =: %ni0or+ Flo7"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""=)Figre == Nona+/le! %"C"F" -"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""@9Figre @. (orked E>a+/le! %"C"F" 8"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""@9Figre @- (orked E>a+/le! %"C"F" 9"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""@:Figre @8 (orked E>a+/le! %"C"F" :"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""@=Figre @9 Hydralic J+/""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""@#

Figre @: J+/ Eation *iagra+""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""@Figre @= Broad Cre!ted (eir Figre @@ Rond Cre!ted (eir""""""""""""""""""""""""""@Figre @= Broad Cre!ted (eir Figre @@ Rond Cre!ted (eir""""""""""""""""""""""""""@Figre @# 1ial Flo7 P+/""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#:Figre # 'i>ed Flo7 P+/""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#:Figre #) Centri0gal P+/ Characteri!tic CrDe!"""""""""""""""""""""""""""""""""""""""""""""""#=Figre . A>ial Flo7 P+/ Characteri!tic CrDe!""""""""""""""""""""""""""""""""""""""""""""""""#=Figre - 'i>ed Flo7 P+/ Characteri!tic CrDe""""""""""""""""""""""""""""""""""""""""""""""""#=Figre 8 *e!ign Point CrDe"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#@Figre 9 Fa+ilie! o0 Characteri!tic CrDe!"""""""""""""""""""""""""""""""""""""""""""""""""""""""""#@Figre : Franci! Trbine"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#@

Figre = A>ial Flo7 Trbine"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""##Figre @ Pelton (heel Trbine""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""#Figre # Se7age P+/! < A>ial and 'i>ed Flo7""""""""""""""""""""""""""""""""""""""""""""""""#)Figre Se7age P+/! < Centri0gal and Sb+er!ible"""""""""""""""""""""""""""""""""""".

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Hydraulics

Hydralic! 30ro+ the 4reek 7ord 0or 7ater6 i! the !tdy o0 the /roble+! o0 the 0lo7

and !torage o0 7ater, bt i! o0ten a//lied to other liid!"

Flid +echanic! i! the !ection o0 a//lied +echanic! concerned 7ith the !tatic! anddyna+ic! o0 liid! and ga!e!"

A 0lid can o00er no re!i!tance to any 0orce ca!ing change o0 !ha/e"

Change o0 !ha/e i! ca!ed by !hearing 0orce!"

$0 a 0lid i! at re!t there are no !hear 0orce! acting"

Flid! +ay be either liid! or ga!e!"

A liid i! di00iclt to co+/re!! and occ/ie! a 0i>ed Dol+e"

A ga! i! ea!ily co+/re!!ed and e>/and! to 0ill any De!!el"

Pre/ared Jne 8..# Page # o0 .


8/80


SI Units

The Sy!te+e $nternational ha! !i> ba!ic nit!

length metre (m)

+a!! kilogramme (kg)

ti+e second (s or sec)

electric crrent ampere (A)

ab!olte te+/eratre kelvin (k)

l+ino! inten!ity candela (cd)

All other nit! are deriDed 0ro+ the!e 0nda+ental nit!"

The nit o0 Delocity i! obtained by diDiding the nit o0 di!tance, the +etre, by thenit o0 ti+e, the !econd, and i! th! +etre! /er !econd"

The relation!hi/ bet7een +a!! and 0orce i! deriDed 0ro+ Ne7ton! !econd la7+aking the con!tant o0 /ro/ortionality eal to oneG"

FORCE I 'ASS ACCELERAT$ON

The nit o0 0orce i! a /rodct o0 +a!! and acceleration"

kg3+!ec86 I Ne7ton

Other !y!te+! e>i!t

0ootPre/ared Se/te+ber 8..= Page o0 .


9/80


ass and !eig"t

The 7eight o0 a body i! the 0orce 7ith 7hich it i! attracted to7ard! the earth" $0 abody 0all! 0reely then, neglecting air re!i!tance, it! acceleration i! con!tant" Thi!acceleration i! denoted by Kg and i! called the Kacceleration de to graDity"

i"e" (eight ( I +g 7here + i! +a!!

N"B" $0 +a!! +G i! +ea!red in kg and g in +!8, then ( i! +ea!red in NNe7tonG"

#imensions

$n +echanic! all antitie! can be e>/re!!ed in ter+! o0 0nda+ental di+en!ion!

+a!!

length L

ti+e T

th! acceleration $ distance % time&

there0ore dimensions o' acceleration $ L % T&

and 'orce $ L % T&

Linear constant acceleration

e"g", acceleration de to graDity

$0 a body 7ith initial Delocity !/eedG K ndergoe! con!tant acceleration Ka 0or ti+eKt to bring it to a 0inal Delocity KD oDer a di!tance K! then

-G

Frther, the ( )vuspeedaverage &1 += and tspeedaveragetravelleddistance =

th! ( )tvus &1 += 8GSb!titting 0or D 0ro+ -G into 8G giDe!

9G

Pre/ared Jne 8..# Page ) o0 .

atuvt

uv

timeinc"ange

speedinc"angea +=

== or

( ) &&

1

&

1 atutstatuus +=++= or


10/80

=


Sb!titting 0or t 0ro+ -G into 8G giDe!

i"e", &asuvuv&as &&&& +== or :G

#imensional Homogeneity

$0 an eation i! to re/re!ent !o+ething real then ter+! on both !ide! +!t haDethe !a+e di+en!ion! !o that like i! co+/ared 7ith like"

e"g" v $ u at

di+en!ion! o0 D are LT*1

di+en!ion! o0 are LT*1

di+en!ion! o0 at are LT*&+ T $LT*1

Note that !o+e /ractical 0or+lae do not a//ear to be di+en!ionally correct" Therea!on i! that a coe00icient +ay not be a /re n+ber" E"g", g +ay be 7ritten a!)"- bt ha! the nit! LT*&"

,ressure

The 7eight o0 the block acting on an area KA o0 the !r0ace /rodce! a /re!!re K/on that !r0ace, 7here

Pre!!re

i"e", /re!!re i! 0orce nit area

-rictional S"ear Stress

Shear 0orce!, !ch a! 0riction, /rodce a !hear !tre!! on the area oDer 7hich

they act, 7here

!hear !tre!! 8+N

A

F

area

0orce!hear==

Pre/ared Jne 8..# Page -. o0 .

(

Area KA

/

F

Area KA

( ) ( ) auvvus&

1 +=


11/80


,roperties o' -luids

ASS #./SIT 345 rho6 i! +a!! /er nit Dol+e, nit! kg+9

S,.I-I !.I2HT7 i! the 7eight /er nit Dol+e, nit! N+9

7eight I +a!! > graDitational accelerationgw =

S,.I-I 23A4ITor relatiDe den!ity i! the ratio o0 the den!ity o0 a!b!tance to the den!ity o0 7ater"

N"B" den!ity o0 7ater, w I -... kg+9

den!ity o0 +ercry, m I -9@.. kg+9

den!ity o0 dry air, a I -"9 kg+9at NTP

oe''icient o' #ynamic 4iscosity 5 627 mu8

The coe00icient o0 dyna+ic Di!co!ity i! de0ined a! the s"ear 'orce % unit areareired to drag one layer o0 0lid 7ith nit Delocity /a!t another layer nit di!tancea7ay 0ro+ it in the 0lid"

skg%mms/unitsSI & or

7inematic 4iscosity 627 nu8The kine+atic Di!co!ity i! the ratio o0 dyna+ic Di!co!ity to +a!! den!ity"

C!ec+nit!8

=

1i!co!ity o0 liid! decrea!e! 7ith increa!e in te+/eratre"

1i!co!ity o0 ga!e! increa!e! 7ith increa!e in te+/eratre"

-luid -lo9

A! 0lid! 0lo7 along /i/e! or channel!, the 0lid /article!ne>t to the bondary /i/e 7all or channel bedG are held atre!t by 0riction" Thi! generate! internal !hearing 0rictionG0orce! bet7een the layer! o0 0lid a! they !lide oDer oneanother" The Di!co!ity o0 a 0lid deter+ine! the rate at7hich it! /article! can !lide oDer one another nder a giDen!hear !tre!!"

-igure 1 : -luid -lo9

i;e;

=

dy

dv

Th! a KDelocity gradient e>i!t!, /er/endiclar to the direction o0 0lo7"

Pre/ared Jne 8..# Page -- o0 .


12/80


Introduction to -luid ec"anics

T"e La9s o' Hydrostatics

-" *irection o0 Force at a Bondary

(hen a 0lid i! at re!t, no 0riction !hearG 0orce e>i!t! at it! bondarie!"Hence, any 0orce e>erted by the 0lid on a bondary or i++er!ed obect+!t act in a direction nor+al /er/endiclarG to that bondary and beinde/endent o0 Di!co!ity"

8" 1ariation o0 Pre!!re $nten!ity 7ith *irection

The /re!!re inten!ity at a /oint in a 0lid ha! the !a+e +agnitde in alldirection!"

9" 1ariation o0 Pre!!re $nten!ity 7ith *e/th

$t i! co++on kno7ledge that the /re!!re in a liidincrea!e! 7ith de/th" Con!ider the Derticaleilibri+ o0 the cylinder o0 0lid !ho7n, i0 the areao0 the end i! A then

-igure & : ,ressure 4ariation9it" #ept"

7here atI at+o!/heric /re!!re /!hing do7n on the !r0ace o0 the 7ater

No7 AhggvolumegmW ===

hence AatpghApA +=

i"e", atpghp +=

th! /re!!re increa!e! linearly 7ith de/th"

:" Pre!!re in Connected Flid!

The /re!!re along any hori;ontal line MM 7ill be the !a+e +agnitde in eachar+ o0 the De!!el /roDiding the 0lid i!the !a+e"

-igure < : ,ressure in onnected -luids

Pre/ared Jne 8..# Page -8 o0 .

0== AatpWpAyR


13/80


,ressure easurement

Absolute and Gauge Pressure

-igure = : 2auge ,ressure Scale

N"B" Ab!olte ;ero /re!!re I /er0ect Dac+G

Ab!olte Pre!!re /ab!I /at gh$n /ractice, /re!!re i! +ea!red relatiDe to at+o!/heric /re!!re, a! gage/re!!re"

4age Pre!!re / I /ab!? /atI gh

N"B, NegatiDe gage /re!!re I !b/re!!ed

in ter+! o0 the height h o0 a col+n o0 0lid den!ity, G reired to /rodce that/re!!re regardle!! o0 ho7 the /re!!re i!, in 0act, /rodcedG"

The nit o0 +ea!re+ent i! then +etre! +G and the /re!!re i! e>/re!!ed inter+! o0 K+etre! head o0 0lid" E"g", K+etre! head o0 7ater or K+etre! head o0+ercry"

The Mercury Barometer

A gla!! tbe abot -+ long i! 0illed 7ith +ercryand then inDerted in a bo7l o0 +ercry" AboDe the+eni!c! +ercry Da/or 0or+! at Da/or /re!!re

0or +ercry, 7hich i! al+o!t ab!olte ;ero /re!!re"The at+o!/heric /re!!re acting on the !r0ace o0the +ercry in the bo7l !//ort! a col+n o0+ercry in the tbe o0 height h" Pre!!re at A leDel7ith liid !r0aceG I /re!!re on liid !r0ace Iat+o!/heric /re!!re, /at" 1ertical eilibri+ o0the +ercry col+n reire! that

Pre!!re 0orce acting /7ard! at A I 7eight o0+ercry aboDe A

-igure > : ercury Barometer

Pre/ared Jne 8..# Page -9 o0 .


14/80


i"e", /at> a I +gah /atI +gh

7here a I bore o0 tbe +I den!ity o0 +ercry

At+o!/heric /re!!re Darie! abot a +ean Dale o0 /atI -.-kN+8" E>/re!!ing

thi! a! a /re!!re head it i! ."#@+ o0 +ercry or -."9=+ o0 7ater"

The Aneroid Barometer

The 0le>ible cha+ber i! eDacated, bt/reDented 0ro+ colla/!ing by the !/ring"1ariation! in at+o!/heric /re!!re ca!e thebo> to +oDe in or ot !o that the /ll o0 the!/ring I 0orce de to at+o!/heric /re!!re"The!e +oDe+ent! are a+/li0ied +echanicallyand +oDe a /ointer oDer a calibrated !cale"

-igure ? : Aneroid Barometer

The Deadweight Piston Gauge

The /re!!re at A in the 0lid i! +ea!red by balancing it7ith 7eight! acting on the /i!ton" Thi! a//arat! i! +ainly!ed to calibrate other ty/e! o0 gage!"

-igure @ : #ead9eig"t ,iston 2auge

The Bourdon Gauge

The crDed tbe o0 0lattened cro!! !ection i! held rigidly atone end 7ith the 0ree end connected to a /ointer" (hen

/re!!re i! ad+itted to the tbe it tend! to !traighten" The+oDe+ent o0 the 0ree end ca!e! the /ointer to rotate oDera gradated !cale"

-igure : Bourdon 2auge

Pressure Transducers

Flid i! ad+itted to the !+all cha+ber on one !ide o0 thedia/hrag+" Change! in /re!!re re!lt in change! in the!tre!! and con!eently the electrical re!i!tance o0 thedia/hrag+" The re!i!tance change i! +ea!red on a digitalDolt+eter and calibrated 7ith the /re!!re change"

-igure : ,ressure Transducer

N"B" The dead7eight /i!ton gage, Bordon gage and /re!!re tran!dcer!+ea!re Kgage /re!!re"

The Piezometer Tube

The /re!!re / in the /i/eline !//ort! a col+n o0 liid height h in the Derticaltran!/arent tbe" The /re!!re i! th! +ea!red directly in +etre! head

/ I h +etre! o0 liid I gh N+8, 7here I den!ity o0 liid kg+9G

Pre/ared Jne 8..# Page -: o0 .


15/80


-igure 10 : (a) ,ieCometer (D) anometer 9it" Secondary 2auge -luid

N"B" The /ie;o+eter i! o/en to the at+o!/here and +ea!re! gage /re!!re"Thi! i! tre 0or all +ano+eter!"

The U-tube Manometer

Thi! !e! a !econdary gage liid, den!er than the 0lid in the /i/eline" Thi! i!nece!!ary

iG 7hen the /i/eline 0lid i! a ga! 7hich 7old e!ca/e 0ro+ a /ie;o+eter and7old not 0or+ a +eni!c!G

iiG 7hen h 7old be too large 7ithot the introdction o0 a +ch den!er gage0lid, e"g" 7ater or alcohol 0or air, and +ercry 0or 7ater in the /i/eline"

The /re!!re at leDel > i! the !a+e in both tbe!" i"e", />I / Agy I Bgy

Hence / I Bgh < Agy N+8

The Differential Manometer

Thi! allo7! the /re!!re di00erence bet7een t7o/oint! in a /i/eline to be +ea!red"

N"B" B A" $0 BQ Athen the +ano+eter

+!t be inDerted" $t i! then kno7n a! an KinDerted%


16/80

!orked .+amples E ,ressure easurement

-" For the !itation! !ho7n in Figre -8"

aG $0 the /i/eline and +ano+eter 0lid den!itie! are [email protected]+9and +I [email protected]+

9re!/ectiDely, > I .")+and the /re!!re at i! ). kN+8, 0ind the di00erence in+eni!c! leDel!, h"

bG For the !a+e 0lid den!itie! a! in aG, bt 7ith > I-"9+ and h I -":+, 0ind the /re!!re at "

-igure 1& : !orked ,;; .+ample 1

Solution:

aG *ra7 a dat+ leDel M ? M throgh the lo7er+eni!c!" The /re!!re at leDel M ? M 7ill be the !a+e inboth li+b! o0 the % ? tbe +ano+eter" See Figre -9"

For the right


17/80

8" For the !itation !ho7n in Figre -:, i0 the /i/elineand +ano+eter 0lid den!itie! are I --..kg+9brineGand +I [email protected]+

9re!/ectiDely, > I ."#+ and the/re!!re at i! ? 9. kN+8, 0ind the di00erence in+eni!c! leDel!, h"

-igure 1= : !orked ,;; .+ample &

Solution:

Note that the right


18/80

9" For the !itation! !ho7n in Figre -=, i0the /i/eline and +ano+eter 0lid den!itie!are I -...kg+9and +I [email protected]+

9

re!/ectiDely, the /re!!re di00erencebet7een ' and N i! 9=kN+8, a I -".+ and

b I ."9+, 7hat i! the di00erence in +eni!c!leDel!, h

-igure 1> : !orked ,;; .+ample i! Dertical" *en!ity o0

!ea 7ater I -.8=kg+9

G"3'4 I < .":8.+W n!table6

8" A rectanglar /ontoon -8+ long by #"=+ 7ide by 9+ dee/ ha! a +a!! o0 ).+etric ton! and 0loat! in !ea 7ater I -.8=kg+9 G" The centre o0 graDity o0 the/ontoon i! at the geo+etrical centre" Find the +etacentric height" 39"#)9+6

9" A rectanglar /ontoon -9"=+ > =":+ > 9".+ high 0loat! in 7ater 7ith the 9".+!ide Dertical" The /ontoon 7eigh! :8.kN and it! centre o0 graDity i! -"8+ aboDe

the ba!e o0 the /ontoon" Calclate the +etacentric height o0 the /ontoon 7hen0loating in

a" 0re!h 7aterb" !ea 7ater I -.8=kg+9G

$n 7hich o0 the!e 0lid! doe! the /ontoon haDe greater !tability

3a" '4 I 9"898+W b" '4 I 9"98+W !ea 7ater6


24/80

>

Hydrostatic -orces on SuDmerged Sur'aces

So 0ar 7e haDe !een that the /re!!re in a 0lid increa!e! linearly 7ith de/thW

/ I gh

'ore !ally, 7hen de!igning hydralic !trctre! !ch a! da+!, lock gate!, 0la/DalDe! etc" it i! the re!ltant hydro!tatic 0orce F that i! reired"

Thi! 0orce F i! the !++ation o0 the /re!!re 0orce! acting on the !b+ergedarea" $t act! at right angle! to the !b+erged !r0ace"

-igure 1 : Hydrostatic -orce on SuDmerged Sur'ace

Strctre! +ay be /artially or 7holly !b+erged in the 0lid, ho7eDer there i! ageneral 0or+la 7hich can be !ed to deter+ine the 0orce FW

(here A i! the !b+erged area, i! the Dertical di!tance 0ro+ the 7ater !r0aceto the centroid o0 the !b+erged area"

$t i! al!o !e0l to kno7 7here the re!ltant 0orce act! in relation to the 7ater!r0ace" The /oint at 7hich the 0orce act! i! kno7n a! the centre o0 /re!!re"

Thi! di!tance >/i! +ea!red /arallel to the !b+erged !r0ace and can bedeter+ined 0ro+ the general 0or+la

(here cgi! the !econd +o+ent o0 area abot the centroid o0 the !b+erged area"

i! the di!tance 0ro+ the 7ater !r0ace to the centroid o0 the !b+erged area+ea!red /arallel to the !b+erged !r0ace"

(hen the !b+erged !r0ace i! Dertical X I ).Y, hence !in X I - and

hRgAF=

h

>

>A

>cg

/ +

=

!inX

h> =


25/80

oment o' a -orce 3otation and .GuiliDrium

A 0orce +ay ca!e not only linear +otion, bt al!orotation o0 a body" Thi! /rinci/le i! !ed in the ca!e o0a leDer !y!te+" The /er!on /!he! do7n 7ith 0orce F,

ca!ing a load ( to ri!e by rotation abot the /iDot P"

-igure &0 : oment o' -orce

The +agnitde o0 F reired to ca!e rotation de/end! on ( and on > and y, theK/er/endiclar di!tance! 0ro+ the 0orce! to the /iDot P" Each 0orce i! !aid to/rodce a K+o+ent abot P"

i"e", the K+o+ent o0 the 0orce F abot P I F > anticlock7i!eGand the K+o+ent o0 the 0orce ( abot P I ( y clock7i!eGFor anticlock7i!e rotation to take /lace, F > ( y"

For clock7i!e rotation to take /lace, ( y F >"For Kbalance, Keilibri+ or Kno rotation, F > I ( y"


26/80

@.Y-"8+

h

>>

/

a

2

,

-"=+

9".+

!orked .+amples E Hydrostatic -orces

-" A rectanglar /late -"=+ > 9".+ i! !b+erged in 7ater and +ake! an angle [email protected] 7ith the hori;ontal, the -"=+ !ide! being hori;ontal" Calclate the +agnitde o0

the 0orce on the /late and the location o0 the /oint o0 a//lication o0 the 0orce, 7ithre0erence to the to/ edge o0 the /late, 7hen the to/ edge o0 the /late i! -"8+belo7 the 7ater !r0ace"

Solution:

m88625138615160

21x ....

sin

.=+=+

=

m499260886260xh .sin.sin ===

51349928191000Ahg- ... ==

N92109N1092109- 3 .. ==

x12

dx

x!d

1

12

!dx

xA

$xx

23,

. +=+=+=

m146326008862886212

3

8862x

2

. ..... =+=+=

Fro+ the to/ edge o0 the /late, m760138611463a ... ==

-igure &1 : !orked .+ample H;-; 1


27/80

8" A Dertical blkhead :+ 7ide diDide! a !torage tank" On one !ide o0 theblkhead /etrol S"4" I ."#G i! !tored to a de/th o0 8"-+ and on the other !ide7ater i! !tored to a de/th o0 -"8+" *eter+ine the re!ltant 0orce on the blkheadand the /o!ition 7here it act!"

Solution:

!gh!h2

hgAhg- 2

21 ===

N567N412819780- 221

1 ... ==

N2528N4218191000- 221

2 ... ==

-igure && : !orked .+ample H;-; &

Hence the re!ltant 0orce

N25392528567--- 21/ ... ===

( ) h6

h

2

h

2h!h

1

12

!h

2

h

xA

$xx

32

3,

. =+=+=+=

Fro+ the diagra+, x = h 2/3h =1/3h

Hence, x1= 2.1 / 3 = 0.7mand x2= 1.2 / 3 = 0.4m

Taking +o+ent! abot O, FRZ>RI F-Z>-? F8Z>8

i"e" 9)"8= >RI @#"= ."# ? 8"8= .": and hence >RI .")-@+

9" A hinged, circlar gate #=.++ india+eter i! !ed to clo!e the o/eningin a !lo/ing !ide o0 a tank, a! !ho7nin the diagra+ in Figre 89" The gatei! ke/t clo!ed again!t 7ater /re!!re/artly by it! o7n 7eight and /artly bya 7eight on the leDer ar+" Find the+a!! ' reired to allo7 the gate tobegin to o/en 7hen the 7ater leDel i!=..++ aboDe the to/ o0 the gate"The +a!! o0 the gate i! @. kg"Neglect the 7eight o0 the leDer ar+"G

-igure &< : !orked .+ample H;-; @..++ rectanglar 0lat /late 7ith the @..++ !idehori;ontalG i! !ed to clo!e the o/ening in a !lo/ing !ide o0 a tank, a! !ho7n

in Figre 8@*iagra+ 8" The gate i! ke/t clo!ed again!t 7ater /re!!re/artly by it! o7n 7eight and /artly by a 7eight on the leDer ar+" Find the+a!! ' reired to allo7 the gate to co++ence to o/en 7hen the 7aterleDel i! :..++ aboDe the to/ o0 the gate" Neglect the 7eight o0 the leDerar+G" 3898 kg6

-igure &? : H;-; Tutorial Fuestions

#iagram 1 #iagram &


30/80

Hydrodynamics E -luids in otion(hil!t hydro!tatic! i! a relatiDely /reci!e !bect, hydrodyna+ic! can only beanaly!ed 7ith a//ro>i+ate accracy" The rea!on 0or thi! i! that the +otion o0 0lid/article! i! o0ten rando+n/redictable 7hich create! inaccracie! in analy!i!"

The +aor e00ect i! 0riction bet7een the /i/e 7all and the 0lid 7hich create! aDelocity Dariation acro!! the /i/e dia+eter" Ho7eDer to !i+/li0y +atter! in thi! nitall Delocitie! con!idered 7ill be aDerage Delocitie!, i"e" D I A"

Types o' -lo9

Steady 'lo9condition! at a /oint in the 0lo7 !y!te+ re+ain con!tant7ith re!/ect to ti+e"

Unsteady 'lo9 condition! change 7ith re!/ect to ti+e"

Uni'orm 'lo9 0lo7 condition! re+ain con!tant 7ith re!/ect to di!tance"

/on*uni'orm 'lo9 condition! change 7ith re!/ect to di!tance"

Laminar 'lo90lid /article! retain !a+e relatiDe /o!ition! at!cce!!iDe cro!! !ection!"

TurDulent 'lo9 0lid /article! 0ollo7 rando+, di!organi!ed /ath!"

/;B; >K o' real li'e 'lo9s are turDulent;

#etermination o' Laminar%TurDulent -lo9 E 3eynolds/umDer 3e

Reynold! recogni!ed that Di!co!ity, aDerage Delocity and the dia+eter o0 the /i/edeter+ined the 0lo7 /attern" He deter+ined the Reynold! N+ber to /redict 0lo7ty/e, 7here

I aDerage Delocity d I /i/e dia+eter

2 I coe00icient o0 dyna+ic Di!co!ity I 0lid den!ity

Re :... and 0lo7 i! trblent

8... Q ReQ :... and 0lo7 i! tran!itional

ReQ 8... and 0lo7 i! la+inar

The +aor di00erence bet7een the t7o 0lo7 /attern! i! that greater 0rictional lo!!e!

occr in trblent 0lo7"

-

8

9

= vd

/&


31/80

Analysis o' -luid -lo9

The three ba!ic eation! are

-G Continity Eation

8G Energy Eation BernolliG9G 'o+ent+ Eation

-G Continity Eation

For continity, the 0lo7 rate entering !ection - +!t be eal to the 0lo7 rateleaDing !ection 8" i"e", -I 8I and

-I Dol+e!ec I length!ec > area i"e", -I 1-A-and 8I 18A8

th! I 1-A-I 18A8

-igure &@ : -lo9 ontinuity


32/80

!orked .+amples * ontinuity

A 9..++ dia+eter /i/eline diDide! into t7o !+aller /i/eline!, one being -=.++ india+eter and the other being 88=++ in dia+eter" $0 the Delocity in the -=.++ /i/e

i! ."=+! and that in the 88=++ /i/e i! ."+!, calclate the 0lo7rate in litre!!and the Delocity in +! in the 9..++ /i/e

Solution:

*m10848504

150vA

332

150150150=

== ..

.

*m10831804

2250vA

332

225225225=

== ..

.

( ) *'t&*6440*m10831848 33

225150300 ... =+=+=

*m575030

1064404

430

A

v

2

3

2

300

300

300300 .

.

.

.=

=

==

-igure & : !orked .+amples ontinuity

Tutorial Fuestions E ontinuity

A 8..++ dia+eter /i/eline diDide! into t7o !+aller /i/eline!, one being -..++ india+eter and the other being -=.++ in dia+eter" $0 the Delocity in the -..++ /i/e

i! ."9+! and that in the -=.++ /i/e i! ."@+!, calclate the 0lo7rate in litre!!and the Delocity in +! in the 8..++ /i/e" 3-9 l!, .":-9+!6

1>0mm

&&>mm


33/80

!ork .nergy and .''iciency(hen the /oint at 7hich the0orce act! +oDe!, the 0orce i!!aid to haDe done 7ork" The+an +oDing the block ha!done 7ork eal to thecon!tantG 0orce a//lied ti+e!the di!tance he ha! +oDed theblock in the direction o0 thea//lied 0orce"

-igure & : !ork * #irected -orce

i"e", (ork done I P! I 0orce > di!tance +oDed K(ork done i! a !calar antity"The nit! are N+ or ole! i"e", - N+ I - oleG"

A body 7hich ha! the ca/acity to 7ork i! !aid to /o!!e!! energy" A! a body doe!7ork it e>/end! energy e"g", the +an +oDing the blockG" Th!, nit! o0 energy arethe !a+e a! tho!e o0 7ork ole!G"

There are +any 0or+! o0 energyW e"g", electrical energy, che+ical energy, heatenergy, nclear energy, +echanical energy, etc" ObDio!ly, they all haDe the!a+e nit!" A 0nda+ental la7 o0 /hy!ic! !tate! that energy can be neithercreated nor de!troyed bt can change 0ro+ one 0or+ to another"

The a+ont o0 energy e>/ended by the +an +oDing the block I (ork done energy 7a!ted" The energy 7a!ted i! +ainly heat generated +oDing the block i!

a lot o0 7ork[G %100')pt)&gytpt%

''&)y =

i"e", the e00iciency o0 the +an! body in +oDing the block i!

!ater .nergy

(ater in a re!erDoir /o!!e!!e! energy !ince it can be !ed to do the 7ork o0driDing a trbine !ally to /rodce electrical energyG" (ater energy i! a+echanical energy and +ay be held by the 7aterG in three 0or+!

4raDitational or /otential energy

5inetic energyPre!!re energy

2ravitational or ,otential .nergy

$0 a body e"g", o0 7aterG o0 +a!! + i! at a height ; aboDe !o+e arbitraryG dat+,e"g", !ea leDel, then it i! !aid to /o!!e! a Potential energy I +g; ole!G relatiDeto the dat+" Thi! i! the 7ork 7hich +!t be done to rai!e a +a!! + o0 0lid by aheight o0 ; +etre!"

g&)&at&d&)&gyh&at*.

*

+


34/80

.nergy Head

Since 0lid! +oDe a! a contin+ rather than a! a di!crete +a!! +, it i!conDenient to con!ider their Kenergy /er nit 7eight or Kenergy head"

i"e", Potential energy head I +g; +gG I ;

Hence, energy head ha! the di+en!ion o0 length and i! +ea!red in +etre!"

7inetic .nergy

A body e"g", o0 7aterG +ay /o!!e!! energy de to it! +otion! a! 7ell a! it!/o!ition" The kinetic energy o0 a body +ay be de0ined a! the a+ont o0 7ork7hich +!t be done by a 0orce in order to bring it to re!t"

Let a body o0 +a!! + +oDing 7ith !/eed D be broght to re!t 7ith ni0or+retardation di!tance +oDed!ec

-igure


35/80

Minor #nergy $osses

Other energy lo!!e! occr at /articlar /lace! in the /i/e or channel 7hen theni0or+ 0lid 0lo7 i! interr/ted by bend!, DalDe!, change! in /i/e dia+eter or7hen the 7ater 0lo7! into or or o0 a re!erDoir" The!e energy lo!!e!, ca!ed bythe 0lid !7irling abot and generating heat, are +ch !+aller than the 0rictionalenergy lo!!e!" Hence they are called K+inor head lo!!e!, hL"

T"e .nergy (Bernoulli) .Guation

The Ktotal energy head H o0 the 0lid i! 0ond by adding the three ty/e! o0+echanical energy /o!!e!!ed by the 0lid at that /oint"

A! 7ater 0lo7! bet7een t7o /oint!, or !ection! o0 a /i/eline or channel, no energy

can be created or de!troyed 0nda+ental la7 o0 /hy!ic! concerning con!erDationo0 energyG" Ho7eDer, i0 any +echanical energy i! conDerted into !ay heat energythrogh 0riction then it i! lo!t to the +echanical !y!te+ and the Ktotal energy headH i! redced"

The energy head! inDolDed are +ea!red in +etre! and can be re/re!ented a!Dertical di!tance! height!G on an energy diagra+"

Con!idering 0lo7 bet7een 8 /oint! labelled - and 8

or

-igure


36/80

!orked .+amples E .nergy

-" Oil S"4" I ."=G at a /re!!re o0 -=. 5N+8 0lo7! at a rate o0 9+9+in throgh/i/e A -..++ dia+eterG" A ta/ered !ection, =+ long, connect! /i/e A to /i/e B

-=.++ dia+eterG !itated at a di!tance h belo7 /i/e A" $0 the /re!!re in /i/e Bi! -. 5N+8 and the energy lo!! de to 0riction throgh the ta/ered !ection i!."-8++ length, deter+ine the di!tance h"

Solution:

*m3766010

34

410

603

4d

v

222A

AA .

..=

=

=

=

*m83260150

34

4d

v

22+

+

+

..

=

=

=

;+

2++

A

2AA h

g2

v

g

p

g2

v

g

p+++

=++

-igure


37/80

8" A =..+ long /i/eline !lo/e! /7ard! at - in =. and change! 0ro+ :=.++ india+eter to 9..++ in dia+eter 9..+ 0ro+ it! lo7er end" $0 the 0rictional headlo!!e! in the /i/e! are -". and #".+k+ length re!/ectiDely and the /re!!re atthe //er end i! -8. 5N+8, 0ind the /re!!re at the lo7er end 7hen the 0lo7 ratein the /i/eline i! -.. l!"

Solution:

*m630450

104

4450

10

4d

v

2221

11 .

.

.

.

.=

=

=

=

*m42130

104

4d

v

22

2

22 .

.

.=

=

=

-igure


38/80

-lo9 easurement in pipelines

There are t7o deDice! 7hich are co++only !ed to +ea!re 0lo7 in /i/e!

-" The Dentri+eter

8" The ori0ice +eter

The Dentri+eter con!i!t! e!!entially o0 a conDergence in a /i/eline 0ollo7ed by a!hort /arallel !ided Kthroat and then a diDergence"

Continity reire! a greater Delocity at the throat than the inlet, there i!con!eentially a di00erence o0 /re!!re bet7een the inlet and throat and+ea!re+ent o0 thi! /re!!re di00erence allo7! the 0lo7 rate to be calclated"

The ori0ice +eter 7ork! on a !i+ilar /rinci/al to the Dentri+eter and i! chea/erand !i+/ler to in!tall bt it generate! larger energy lo!!e!" $t con!i!t! o0 a circlardi!c, 7ith an ori0ice +achined in it! centre, 7hich i! ea!ily bolted bet7een /i/e

length!"

The 0lo7 rate throgh a 1entri and Ori0ice +eter can be 0ond 0ro+ the eation

g:2aa

aa=

22

21

21d

=

7here the cro!!


39/80

omentum .Guation

(hen a 0orce act! on a 0lid in +otion, there i! a rate o0 change o0 +o+ent+ inthe 0lid in the direction o0 the 0orce, in accordance 7ith Ne7ton! Second La7 o0

'otion"i"e",

Thi! eation i! a//lied in Dario! !itation!, inclding the i+/act o0 0lid et! on/late! and Dane! trbine! and /+/!G and the 0orce! occrring on /i/e no;;le!,

contraction! and bend!"

%et &mact on a 'tationary "lat Plate

A//lying the +o+ent+ eation /er/endiclar to

the /late, F>I D>8?D>-G

No7, D>8I . ?all 0lo7 a0ter i+/act i! /arallel to the y a>i!"

D>-I D-co!

Hence F>I


40/80

th! ^ I tan


41/80

!orked .+amples E omentum

-" A ta/ered !ection in a hori;ontal /i/eline redce! the dia+eter 0ro+ @..++ to:=.++ in the direction o0 0lo7" $0 the 0lo7rate i! #=. l! and the /!trea+ /re!!re

i! 9.. 5N+8

, calclateaG the do7n!trea+ /re!!re

bG the +agnitde and direction o0 the 0orce on the ta/er

Solution:

A//lying Ne7ton! 8ndla7 o0 +otion, the re!ltant 0orce on the 7ater in the ta/er,

( )122211x vvAp-Ap/ == " F i! the 0orce e>erted by the ta/er on the 7ater"

Hence, ( )122211 vvApAp- =*m6532

60

7504

4d

v

22

1

11 .

.

.=

=

=

-igure


42/80

8" A :=.++ dia+eter /i/eline conDeying -". +9! o0 7ater contain! a 88"=Y bendin the hori;ontal /lane" $0 the /re!!re in the bend i! 8=. 5N+8, calclate the+agnitde and direction o0 the 0orce on the bend"

Solution:

*m296450

014

4d

v

22 ..

.=

=

=

( )( )+= 5221vpA-x .cos

( )N522129601104

45010250 3

23

+= .cos...

( )( ) ( ) N535221054652212967639 ..cos..cos.. ==+=

-igure D

D

88"=Y


43/80

gd2

;v

h

2

=

+=

Re.

.log

512

d73

2 *

-riction Loss in ,ipelines

T"e #arcy*!eisDac" .Guation

So 0ar 7e haDe e>/re!!ed 0rictional head lo!! in +etre!+etre length 7hen!b!titting in Bernolli! Eation" The 0ollo7ing detail! a +ore reali!tic 7ay o0a!!e!!ing /i/e 0riction lo!!e!"

TurDulent -lo9s

Trblent 0lo7 in /i/e! i! highly co+/le> and cannot be !olDed by +ath! alone"For trblent 0lo7! Ne7ton! La7! o0 1i!co!ity do not a//ly" Reynold!e>/eri+ent! and 7ork by *arcy (ei!bach !ho7ed that 0rictional head lo!! 7a!/ro/ortional to the !are o0 the Delocity" The 0ollo7ing eation 7a! deDelo/ed

(here h0i! the 0riction head lo!!, L i! the /i/e length, i! the 0lo7 Delocity, d i! the

/i/e dia+eter, i! the *arcy 0riction 0actor" Note i! di+en!ionle!!" The

eation i! !o+eti+e! 7ritten 7ith :0 re/lacing , thi! i! the A+erican Der!ion"

N"B" The head lo!! de to 0riction /er nit length o0 /i/e h0LG i! the hydralic

gradient iG" Th! i I h0L I 88dg" Thi! 7ill be the gradient o0 the TEL and H4L"

Th!, 0or a giDen 0lo7 rate, h0-d=, i"e" halDing the !i;e o0 /i/e to be !ed 7ill

increa!e 0rictional head lo!!e! 98 ti+e!" The head lo!! de to 0riction i! a +lti/le

LdG o0 the kinetic D88gG, i"e" h0I LdGD88gG"

Originally it 7a! thoght that 7a! con!tant, ho7eDer, e>/eri+ent! !ho7ed that it

7a! related to Reynold! N+ber and /i/e roghne!!"

oleDrook*!"ite -ormula 'or

The 0ollo7ing eation 0or 7a! deriDed by carrying ot laboratory e>/eri+ent! on

co++ercial /i/e!" The eation i!

(here k!i! height o0 /i/e !r0ace roghne!!, d i! the /i/e dia+eter and Re i! theReynold! N+ber o0 the 0lo7"

$n thi! 0or+at it i! di00iclt to 0ind directly, +odi0ication! 7ere +ade by Barr

+=

890* 12865

d73

2

1.

Re

.

.log

2

22

d

4

dg2

;h

d

4

A

===

?

@

?

henceandAl!o,

5

2

5

2

2d112

;h

d

;

g2

16h

.

@@

?=

=


44/80

!orked .+amples E #arcy*!eisDac" .Guation

A /i/eline connecting t7o re!erDoir! haDing a di00erence o0 leDel o0 @+ i! #8.+long, and ri!e! to a height o0 9+ aboDe the //er re!erDoir at a di!tance o0 8:.+0ro+ the entrance be0ore 0alling to the lo7er re!erDoir" $0 the /i/e i! -8+ in

dia+eter and the 0rictional coe00icient, $ .".:, 7hat 7ill be the di!charge and the/re!!re at the highe!t /oint o0 the /i/eline

-igure =0 : !orked .+amples #;.;

Solution:Figre :.!ho7! the layot" Since no in0or+ation i! giDen that e>it or entry to the/i/e i! !har/ no !hock lo!! need be con!idered" A//ly Bernolli! eation to Aand B on the 0ree !r0ace o0 the re!erDoir, 7here the Delocitie! are ;ero and the/re!!re! at+o!/heric" Take B a! dat+ leDel"

Total energy /er nit 7t at A $ Total energy /er nit 7t at B 0rictional lo!!

g2

v

d

;

g2

v

g

p0

g2

v

g

p:

22++

2AA ++

+=+

+

/AI /BI . and DAI DBI ., !o thatg2

v

d

;:

2=

Ptting H $ @+, $ .".:, L $ #8.+, d$ -"8+

g2

v

21

7200406

2

.

. = i"e" 924

720040

8192216v2 .

.

..=

= *m222v .=

*i!charge I area > Delocity I m5122224

21 2..

.=

9! An!7erG

To 0ind the /re!!re /Cat C a//ly Bernolli! theore+ to A and C, taking dat+leDel at A and DAI ."

g2

v

d

;

g2

vh

g

p

g

p 212

=A +++

=

+= d

;

1g2

v

hg

p

g

p 12

A=

Ptting /AI at+o!/heric /re!!re I ., h I 9+, I .".:, L- I 8:.+, 1 I 8"88+,dI -"8+

Bat&m26521

2400401

8192

22230

g

p 2= ..

.

.

.=

+

=

2

= mN6512658191000p /... == An!7erG


45/80

Tutorial Fuestions E #arcy*!eisDac" .Guation

-" *eter+ine the lo!! o0 head de to 0riction in a ca!t iron /i/e :..+ long and-=.++ in dia+eter 7hich carrie! :9 l!" %!e the *arcy eation 7ith I

.".8" 3-@".)+68" (hat !i;e o0 /i/e !hold be !ed to deliDer 9. l! o0 7ater throgh a

di!tance o0 -9..+ i0 the dro/ in the hydralic grade line i! not to e>ceed8=+" Take I .".8=" 3-=#++ ? ie" -#=++6

9" (ater i! di!charged 0ro+ a re!erDoir throgh a /i/e -=..+ long 7hich i!.":+ dia+eter 0or #=.+ and ."8=+ dia+eter 0or the re+aining di!tance"Calclate the 0lo7rate, taking only 0rictional lo!!e! into accont, i0 the otletleDel o0 the /i/e i! 9.+ belo7 the re!erDoir leDel" I .".-@ 0or the .":+/i/e and .".8: 0or the ."8=+ /i/e" 3-9# l!6


46/80

TaDles 'or t"e Hydraulic #esign o' ,ipes and Se9ers

.+planation o' TaDle Layout

The!e 7ere deDelo/ed a! an aid to /i/eline de!ign" They are ba!ed on the *arcya+/le o0 thi! i! a!hort /+/ing +ain 7here +inor lo!!e! +ay accont 0or / to a third o0 the totalenergy lo!!"

There are t7o +ethod! o0 analy!ing /i/eline! 7hich inclde +inor lo!!e!" The 0ir!ti! to !i+/ly add each +inor lo!! into the Bernolli EationW

AlternatiDely, they +ay be e>/re!!ed in ter+! o0 an eiDalent length o0 !traight/i/e 7hich 7old giDe the !a+e 0rictional energy lo!!" The length l i! !allye>/re!!ed a! a +lti/le o0 the /i/e dia+eter d !ee tableG" Th!W l I nd" Th!actal LG and eiDalent /i/e length! lG are added together to giDeW

g2

Ch

2;

;

=

2;1;2

222

1

211 hhh1 : !orked .+amples ;L; 1

Solution:A//ly Bernolli! theore+ to A and B Figre =-G, a!!+ing Delocity at A i! ;eroand that /A $ /B$ at+o!/heric /re!!re I ." ;BI . and ;AI H"

Total energy at A $ Total energy at B lo!! at entry 0rictional lo!!"

g2

v

d

;

g2

v

2

1

g2

v:

222 ++=

Ptting H $ -8+, $ .".:, L $ :=.+, d$ -..++ $ ."-+

g2v5181

1045004051

g2v12

22

=

+= .

...

i"e 315181

819212v2 .

.

. == *m141v /.=

*i!charge I area > Delocity I m1096814110 32

41 = ... 9! An!7erG


55/80

8" (ater i! di!charged 0ro+ a re!erDoir into the at+o!/here throgh a /i/e 9)+long" There i! a !har/ entrance to the /i/e and the dia+eter i! =.++ 0or -=+ 0ro+the entrance" The /i/e then enlarge! !ddenly to #=++ in dia+eter 0or there+ainder o0 it! length" Taking into accont the lo!! o0 head at entry and at theenlarge+ent, calclate the di00erence o0 leDel bet7een the !r0ace o0 the re!erDoir

and the /i/e e>it 7hich 7ill +aintain a 0lo7 o0 8" l!" Take a! .".-)8 0or the=.++ /i/e and .".89- 0or the #=++ /i/e"

-igure >& : !orked .+amples ;L; &

Solution:Figre =8!ho7! the arrange+ent" $0 QI di!charge, then

22

241

12

141 vdvd ==

4261050

10824

d

4v

2

3

21

1 ..

.=

=

=

+! and 63400750

10824

d

4v

2

3

22

2 ..

.=

=

=

+!

A//lying Bernolli! theore+ to A and * at 7hich /AI /*I at+o!/heric /re!!reand DAI .,

Total energy at A I Total energy at * !hock lo!! at B 0rictional lo!! in BC !hock lo!! at C 0rictional lo!! in C*"

There i! no !hock lo!! at * a! di!charge i! to at+o!/here"

( )

g2

v

d

;

g2

vv

g2

v

d

;

g2

v

2

1

g2

v:

2

2

2

22

2

21

2

1

1

1121

22 +

+

++=

( )

g2

vv

d

;1

g2

v

d

;

2

1

g2

v 2

21

1

1122

1

1121 +

++

+=

( )

g2

7920

0750

24023201

g2

6340

050

1501920

2

1

g2

4261 222 .

.

..

.

..+

++

+=

I ."@:# ."-#9 .".98

*i00erence o0 leDel I HI ."=8+ o0 7ater" An!7erG


56/80

9" T7o re!erDoir! are connected by a /i/eline 7hich i! -=.++ in dia+eter 0or the0ir!t @+ and 88=++ in dia+eter 0or the re+aining -=+" The entrance and e>it are!har/ and the change o0 !ection i! !dden" The 7ater !r0ace in the //erre!erDoir i! @+ aboDe that in the lo7er" Tablate the lo!!e! o0 head 7hich occrand calclate the rate o0 0lo7 in +9!" Friction coe00icient i! .".: 0or both /i/e!"

*ra7 al!o the hydralic gradient and the total energy gradient"

-igure >< : !orked .+amples ;L; ."9## I 9".=:+Enlarge+ent I -"=@ > ."9## I ."=+Friction in /i/e8 I 8"@# > ."9## I -"..#+E>it I -".. > ."9## I ."9##+

The total head lo!! i!, o0 cor!e, H I @+"

The total energy at any /oint can be re/re!ented gra/hically, a! !ho7n in Fig"53.At the entry to the /i/eline there 7ill be an entry lo!! ab; ne>t 0ollo7! the 0rictionallo!! in the 0ir!t /i/e, 7hich giDe! the !lo/ing !traight line bc. The lo!! at the!dden enlarge+ent i! !ho7n a! cd. Fro+ d to e the 0rictional lo!! in the larger/i/e ca!e! the total energy to 0all !lo7ly and the !hock lo!! at e>it efbring! thetotal energy gradient to the leDel o0 the !r0ace in the lo7er re!erDoir"

The hydralic gradient i! obtained by /lotting the !+ o0 the /otential and /re!!re

energy and 7ill there0ore be a di!tance eal to the Delocity head in the /i/e belo7the total energy gradient" The hydralic gradient !ho7! the leDel to 7hich the liidin the /i/e 7old ri!e i0 a Dertical !tand /i/e 7a! in!erted in the /i/eline at the/oint nder con!ideration"


58/80

Tutorial Fuestions E inor Losses

-" (ater 0lo7! 0ro+ a !torage tank along a 8=.++ dia+eter /i/e, =.+ long ata rate o0 ."-+9!" The /i/e inclde! t7o ).Y bend! kLI ."#G, t7o :=Y bend!

kLI ."8G, a 0lly o/en gate DalDe kLI ."8=G and a /artially !ht DalDe kLI@G" $0 the /i/e entrance i! !har/ and the 0lo7 di!charge! to at+o!/here atthe end o0 the /i/e, calclate the nece!!ary di00erence in leDel bet7een thetank 7ater leDel and the otlet" Take the /i/e 0riction 0actor, I .".:"39"#-9+6

8" (ater i! di!charged 0ro+ a re!erDoir into the at+o!/here throgh a /i/e=.+ long" There i! a !har/ entrance to the /i/e and the dia+eter i! #=++0or 8.+ 0ro+ the entrance" The /i/e then enlarge! !ddenly to -..++ india+eter 0or the re+ainder o0 it! length" A DalDe in the #=++ /i/e ha! a kLI-"= 7hen 0lly o/en" Taking into accont the lo!! o0 head at entry, at theDalDe and at the enlarge+ent, calclate the di00erence o0 leDel bet7een the!r0ace o0 the re!erDoir and the /i/e e>it 7hich 7ill +aintain a 0lo7 o0 -. l !"Take a! .".8 0or the #=++ /i/e and .".8= 0or the -..++ /i/e" 38"=9#+6

9" T7o re!erDoir! are connected by a /i/eline 7hich i! 88=++ in dia+eter 0orthe 0ir!t -8+ and -=.++ in dia+eter 0or the re+aining 8+" The entranceand e>it are !har/ and the change o0 !ection i! !dden, 7ith CcI ."@" The7ater !r0ace in the //er re!erDoir i! #"=+ aboDe that in the lo7er"Calclate the rate o0 0lo7 in +9! and tablate the lo!!e! o0 head 7hichoccr" Friction coe00icient i! .".9 0or the 88=++ /i/e and .".: 0or the-=.++ /i/e" 3.".@)+9!6


59/80

pen "annel -lo9

Flo7 in an o/en channel or dct in 7hich the liid ha! a 0ree !r0ace di00er! 0ro+

0lo7 in /i/e! in !o 0ar a! the /re!!re at the 0ree !r0ace i! con!tant nor+allyat+o!/hericG and doe! not Dary 0ro+ /oint to /oint in the direction o0 0lo7, a! the/re!!re can do in a /i/eline" A 0rther di00erence i! that the area o0 cro!! !ectioni! not controlled by 0i>ed bondarie!, !ince de/th can Dary 0ro+ !ection to !ection7ithot re!traint"

O/en channel 0lo7 i! generally +ore co+/licated than /i/e 0lo7" O/en channel!haDe !ection! Darying 0ro+ !i+/le geo+etric !ha/e! to irreglar !ection! o0 natral!trea+!" The !tate o0 the bondary !r0ace! +ay Dary 7idely 0ro+ !+oothconcrete to neDen bed! o0 riDer!"

The choice o0 a 0riction 0actor 0or an o/en channel i! th! likely to be +ore

ncertain that a !i+ilar choice 0or a /i/e"

Types o' "annel -lo9

'o!t 0lo7! in channel! are trblent" The 0lo7 i! !aid to be ni0or+ or non= : Uni'orm -lo9

+on-uniform "low or *aried "low

-igure >> : /on*uni'orm (or 4aried -lo9)

1elocity o0 0lo7 i! con!tant along the

channel ? occr! 7hen a cro!!

con!tant de/th

0lid !r0ace i! /arallel 7ithchannel bed

Flo7 Delocity i! not con!tant alongthe channel

de/th Darie!

0lid !r0ace i! not /arallel7ith channel bed


60/80

'teady "low

1elocity and hence de/th re+ain con!tant 7ith re!/ect to ti+e"

Unsteady "low

1elocity and de/th Dary 7ith re!/ect to ti+e at any !ection o0 the channel"

For analy!i! /r/o!e!, it i! !al to con!ider the 0lo7 in o/en channel! to be!teady ni0or+ 0lo7, ho7eDer +o!t real li0e 0lo7! are non? : "annel -lo9 #iagram

Ter+ *e0inition

y 0lo7 de/th I Dertical di!tance 0ro+ 0ree !r0ace to channel bed

kinetic or Delocity energy head

!/eci0ic energy, E

B channel 7idth I 7idth o0 channel at 0ree !r0ace

A area o0 0lo7 nor+al to direction o0 0lo7

P 7etted /eri+eter I length o0 7etted !r0ace +ea!red nor+al todirection o0 0lo7

R hydralic radi! I AP

So !lo/e o0 channel

Fr Frode N+ber Igy

0lo7 rate /er nit 7idth I B0or rectanglar channel! onlyG

g2

v2

yg2

v2+


61/80

.Guations

Continity, Energy and 'o+ent+ eation! +ay be a//lied in the !a+e 7ay a!/i/e 0lo7

-igure >@ : -lo9 #iagram 'or .Guations

Con!ider /oint M, 7here the Delocity i! D, at a de/th h, belo7 the 0ree !r0ace

Fro+ Bernolli 7e get ( )hyg2

v

g

p:

2

+++=D

/ I hydro!tatic /re!!re at de/th h belo7 0ree !r0ace

hg

p=

i"e"g2

vy:

2

++= I Total Energy Head -G

'ecific #nergy

$n channel 0lo7 ho7eDer, it i! !al to re0er to the channel bed, rather than ahori;ontal dat+" Hence, eation -G beco+e!

g2

vy

2

+ I Energy Head I E, !/eci0ic energy

The hydralic grade line coincide! 7ith the 0ree !r0ace" The total energy line lie!D88g aboDe thi!"

For a giDen Dale o0 !/eci0ic energy E the 0lo7 can haDe t7o de/th!" The 0ir!t i!7here the 0lo7 i! dee/ and !lo7 +oDing and i! kno7n a! tranil or !bcritical0lo7" The !econd i! 7here the 0lo7 i! !hallo7 and 0a!t +oDing and i! kno7n a!ra/id or !/ercritical 0lo7"


62/80

(ritical Deth

-igure > : ritical #ept" 2rap"

For a giDen Dale o0 S/eci0ic Energy, it can be !een 0ro+ the diagra+ aboDe, therei! one de/th corre!/onding to +a>i++ 0lo7 rate" Thi! i! called the critical de/th,yc"

Certain 0lo7 +ea!ring deDice! e"g" 7eir! and 0l+e! generate critical de/th 0lo7in their o/eration"

The eation 0or critical de/th i!g

E3y

2

=

The Frode N+ber, Fri! !ed to cla!!i0y the 0lo7

Fr -, 0lo7 i! !/ercritical ra/id, !hallo7G

FrI -, 0lo7 i! critical

FrQ -, 0lo7 i! !bcritical !lo7, dee/G

#uations for 'teady Uniform "low in .en (hannels

$nitial e>/eri+ental 7ork 7a! done by Che;y, 7hich led to

7here C I Che;y! coe00icient ? related to channel geo+etry and channel

roghne!!"

$t 7a! thoght that Che;y! C 7a! con!tant 0or all channel!, bt it 7a! 0ond to beDariable !ally =. ? #.G" So 'anning deDelo/ed an eation ba!ed on Che;y

7here n I 'anning! n+ber ? a roghne!! coe00icient"

21

F3

2

G/)

1v= 2

1

F3

2

G/)

A =or

'anning! Eation(very 9idely used in practice)

F/G=v = Che;y! EationF/GA==or


63/80

Ty/ical Dale! o0 'anning! Kn

!+ooth concrete .".--rogh concrete .".-9brick7ork .".-:!teel !heet /ile! .".-!+ooth earth .".88rogh rock .".9=

!orked .+amples E Uni'orm "annel -lo9

-" Find the gradient nece!!ary 0or a rectanglar 0l+e -"=+ 7ide and ."#=+ dee/to deliDer 9"=+9! o0 7ater 7hen rnning 0ll" Take C I #8"

Solution:2m125175051!yA ... ===

m3750251y2! =+=+= ..

Hence, m37503

1251

A/ .

.===

-igure > : !orked .+amples U;;-; 1

52061

721251

53

3750

1

A=

/

1G/GA=

22

00 ..

.

.=

=

==

8" Arectanglar o/en channel ha! a botto+ 7idth o0 8":+ and a !r0aceroghne!! corre!/onding to 'anning n I .".-=" $0 the !lo/e o0 the bed i! ."..-and the de/th o0 0lo7 i! -"8+, 7hat i! the di!charge nder condition! o0 ni0or+!teady 0lo7

Solution:2

m8822142!yA ... ===

m8421242y2! ... =+=+=

-igure ?0 : !orked .+amples U;;-; &

Hence, m6084

882

.

A/ .

.

.===

*m3240010600150

882G/

)

A

30

21

32

21

32

....

.===

y I ."#=+

b I -"=+

y I -"8+

b I 8":+


64/80

9" A channel o0 tra/e;oidal !ection, 7ith !ide !lo/e! at @.Y to the hori;ontal ba!eo0 the channel, conDey! 7ater at a de/th o0 .")+" Find the 7idth o0 the ba!e andthe gradient o0 the bed to di!charge -"#+9! 7ith a +ean Delocity o0 0lo7 o0 ."@+!"Take 'anning n I .".8="

Solution:

833260

71

v

AAv .

.

. ====

-igure ?1 : !orked .+amples U;;-;

t

b

y I .")+ @.Y


65/80

:" A canal ha! a botto+ 7idth o0 :+ and !ide! 7ith a !lo/e o0 - Dertical to -"=hori;ontal" The de/th o0 7ater i! -".+ 7hen the di!charge i! : +9!"

aG Calclate the !lo/e o0 the channel bed !ing the 'anning 0or+la 7ithn I .".88"

bG Calclate the di!charge in +9! 7hen the de/th o0 0lo7 i! -"8+"

Solution:

aG ( )[ ] 2

m5501514y512!!2

01A ....

.=+=++=

m672530124511y2!. 22

.... =+=++=

m724067

55

.

A/ .

.

.===

-igure ?& : !orked .+amples U;;-; =

25401

724055

40220

A/

)GG/

)

A

22

003

23

2

21

32

=

=

==..

.

bG ( ) 2m966215142151421A ...... =+=+=

m32782532124. ... =+=

m83603278

966

.

A/ .

.

.===

*m5752540

183600220966G/

)A 30

21

322132 .... ===

>

b

y I -".+ -

-"=


66/80

Tutorial Fuestions E Uni'orm "annel -lo9

-" Find the gradient nece!!ary 0or a rectanglar 0l+e -"8+ 7ide and ."@+dee/ to deliDer 8"8=+9! o0 7ater 7hen rnning 0ll" Take C I #." 3- in -=-6

8" Arectanglar o/en channel ha! a botto+ 7idth o0 8":+ and a !r0aceroghne!! corre!/onding to 'anning n I .".-=" $0 the !lo/e o0 the bed i!."..- and the de/th o0 0lo7 i! -"8+, 7hat i! the di!charge nder condition!o0 ni0or+ !teady 0lo7 3:"98 +9!6

9" A channel o0 tra/e;oidal !ection, 7ith !ide !lo/e! at :=Y to the hori;ontalba!e o0 the channel, conDey! 7ater at a de/th o0 ."#=+" Find the 7idth o0the ba!e and the gradient o0 the bed to di!charge -"8#+9! 7ith a +eanDelocity o0 0lo7 o0 ."#+!" Take Che;y C I @@" 3-":8+W -99--6

:" A canal ha! a botto+ 7idth o0 9+ and !ide! 7ith a !lo/e o0 - Dertical to 8hori;ontal" The de/th o0 7ater i! -"8+ 7hen the di!charge i! 9"=-+9!"

%!ing the 'anning 0or+la 7ith n I .".88

a" Calclate the !lo/e o0 the channel bed" 3-=...6

b" Calclate the di!charge in +9! 7hen the de/th o0 0lo7 i! -"9+"3:"-8+9!6


67/80

/on*uni'orm pen "annel -lo9

There are, broadly !/eaking, t7o kind! o0 !teady non


68/80

The eation o0 the hydralic +/ i!

-igure ?= : ump .Guation #iagram

-lo9 easuring #evices in "annels

There are three +ain ty/e!

-" Flo7 +ea!ring !trctre!, /er+anent, e>/en!iDe, a00ect! 0lo7 condition!G"

8" *eter+ination o0 0lo7 rate 0ro+ Delocity, chea/, te+/orary, accrateG"

9" Che+ical +ean!, !ed 0or !/ecial condition!, !/eciali!ed, accrateG"

/0 "low Measuring 'tructures

Ba!ically, an ob!trction i! /laced in the channel to create critical 0lo7 de/th y cG"There are t7o +ain kind! o0 !trctre! 7eir! and 0l+e!"

!eirs

-igure ?> : Broad rested !eir -igure ?? : 3ound rested !eir

b I 7idth o0 channelCd? 0ond 0ro+ BS 9@.

Broad cre!ted 7eir! are e!!entially lo7 da+! acro!! the entire channel, 7hich to/rodce critical de/th oDer the 7eir, +!t haDe a 7idth in the direction o0 channel0lo7, o0 at lea!t three ti+e! the +a>i++ height o0 7ater aboDe the 7eir" Th!

they are +a!!iDe and e>/en!iDe !trctre!, 7hich inDariably collect !ilt again!t

( )1-812

yy

22

21 +=

( )1-812y

y

2

1

1

2 +=

23

data !:71= .=


69/80

their /!trea+ 0ace! and reire annal +aintenance" A! +ea!ring !y!te+! theytend to be inaccrate at lo7 rate! o0 0lo7"

Cr+/ and 0lat


70/80

-lumes

-igure ? : -lume ,lan -igure @0 : -lume .levation

b I 7idth o0 channelCdCD? 0ond 0ro+ BS 9@.

Fl+e! do not !00er 0ro+ !iltation /roble+!, !ince any !ilt i! 7a!hed throgh the!trctre" Ho7eDer their length and th! co!t o0 con!trction are higher than 0ornor+al 7eir!" To +ini+i!e the!e co!t!, a large n+ber o0 0l+e! haDe beencon!trcted by driDing !heet /ile! into the channel bed and back0illing to theoriginal riDer bank 7ith !itable 0ill +aterial!"

Another adDantage o0 0l+e! i! the Dery li+ited backing / o0 the /!trea+ 7aterleDel it /rodce!" $n 0lat area! thi! i! a con!iderable adDantage"

-igure @1 : #ept" easurement

10 "low 2ate from *elocity

%!e! I 1A ? i"e" +ea!ring 1 and A" 1elocity +ea!red 7ith crrent +eter ?Dale! recorded at /oint! acro!! the !ection and aDerage Delocity taken"

-igure @& : 4ariation o' 4elocity

23

vdata !:==71 .=


71/80

-igure @< : Irregular Sections * 3ivers%Streams

-" Find area o0 each 7idth by !e o0 +ea!ring !ta00 I A

8" Record aDerage Delocity 0or each 7idth I 1

9" Flo7 rate 0or each 7idth I 1A I :" Total 0lo7 rate I I - 8 9f" n

Thi! +ethod i! kno7n a! UDelocity area gagingV" (ith care the accracy 0or the0lo7 rate eal! ="

30 (hemical Means

%!ed 0or rogh, highly di!trbed 0lo7, the +ethod inDolDe! the introdction o0 akno7n antity o0 a che+ical at a /re!cribed rate at an /!trea+ location" Thetrblent natre o0 the 0lo7 en!re! thorogh +i>ing o0 the che+ical" Sa+/le! o0the 0lid are taken at a do7n!trea+ location and by che+ical analy!i!, the

concentration o0 che+ical in the 0lid can be 0ond" Thi! degree o0 concentrationcan be related to 0lo7 rate" The che+ical diltion +ethod giDe! an accracy to="

So+e +odern deDelo/! inclde electroche+ical +ethod!, la!er !canning andltra!onic deDice!"


72/80

Hydraulic ac"ines

-unctions and Types

Hydralic +achine! are /o7er or energy conDerter!" They conDert bet7een+echanical and hydralic 0or+! o0 /o7er" There are t7o ty/e! /+/! andtrbine!"

P+/! A +otor conDert! electrical /o7er into +echanical /o7er" The /+/then conDert! the +echanical /o7er into hydralic /o7er"

Trbine! The trbine conDert! hydralic /o7er into +echanical /o7er" Agenerator then conDert! the +echanical /o7er into electrical /o7er"

The e00ect on the hydralic !y!te+ +ay be !ho7n on the total energy diagra+"

-igure @= : Total .nergy #iagram 'or ,ump and TurDine

- ? $nlet8 ? Otlet

? 0lo7 Delocity +SG

? /re!!re N+8G

H ? total head acro!! the +achine +G

'o!t /+/! and trbine! /roDide a contino! 0lo7 rate throgh a rotodyna+icaction" Ho7eDer, /o!itiDe di!/lace+ent /+/! are !ed in !o+e !itation!"

,ositive #isplacement ,umps

The!e 0nction a! a re!lt o0 Dol+etric change!7ithin the /+/" E"g" the reci/rocating or /i!ton/+/!f"

The /i!ton di!/lace! a 0i>ed Dol+e o0 0lid ateach !troke" The action re!lt! in an n!teadydi!contino!G 0lo7 rate"

The!e /+/! are !ed 0or de7ateringe>caDation!, tran!0erring !ldge at !e7agetreat+ent 7ork! and inecting che+ical! into the

0lo7 at 7ater treat+ent 7ork!"-igure @> : ,ositive #isplacement ,ump


73/80

3otodynamic ,umps

The +echanical /o7er i! tran!+itted throgh a rotating !ha0t P'I 8NT 7att!

7here N I rotational !/eed o0 !ha0t reD!G

T I tore in the !ha0t N+G$+/eller blade!, attached to the !ha0t, tran!0er /o7er energy to the 0lid" The

hydralic /o7er PHI gH 7att!

7here I di!charge 0lo7 rate throgh the /+/ +9!G

H I increa!e in total energy head acro!! the /+/ +G

i"e" H i! the ri!e in the T"E"L" in the direction o0 0lo7"

The e00iciency o0 the /+/ %% 100.

.100

pB&')pt

pB&tpt

"

: ==

Types o' 3otodynamic ,umps

Fro+ the eation PHI gH it i! eDident that t7o +achine! /rodcing the !a+e

/o7er +ay deliDer di00erent 0lo7 rate! at di00erent head!" The relatiDe Dale! o0 and H 7ill de/end on the !ha/e o0 the i+/eller, /rodcing di00erent ty/e! o0+achine! 7hich are !ited to di00erent ta!k!"

(entrifugal Pums 4low 5! high H6

-igure @? : entri'ugal ,umps

Flo7 /a!t the i+/eller i! entirely radial" The Dolte ca!ing i! de!igned to 7ith!tandthe high /re!!re! generated" The centri/etal action /rodce! a relatiDely high/re!!re di00erence"

The!e ty/e! o0 /+/! are !ed to e>tract 7ater 0ro+ borehole!, in 7ater !//ly/i/eline! and !e7age /+/ing throgh long ri!ing +ain!"


74/80

A7ial "low Pums 4high 5! low H6

-igure @@ : A+ial -lo9 ,ump

Flo7 /a!t the i+/eller i! entirely a>ial" The i+/eller, 7hich i! !ha/ed like a !hi/!/ro/eller, allo7! relatiDely high 0lo7!, bt the head di00erence 7hich can be+aintained acro!! it i! lo7" The t7i!t 7hich i! i+/arted to the 7ater i! /artiallyre+oDed !ing gide Dane! do7n!trea+, th! redcing the energy lo!!"

The!e ty/e! o0 /+/! are !ed in land drainage !che+e!, and to retrn actiDated!ldge in large !e7age 7ork!"

Mi7ed "low Pums

-igure @ : i+ed -lo9 ,ump

Flo7 /a!t the i+/eller i! /artially radial and /artial a>ial" The !ha/e o0 the i+/ellerand ca!ing de/end! on the characteri!tic! reired" So+e +i>ed 0lo7 /+/!

re!e+ble centri0gal /+/!, bt 7ith a thickened Dolte ca!ingW other! re!e+blea>ial 0lo7 /+/!, bt 7ith a Kblb or Kbo7l ca!ing arond the i+/eller"

The!e ty/e! o0 /+/! are !ed to /+/ !tor+ 7ater throgh !hort ri!ing +ain!,and in !+all land drainage !che+e!"

Multistage Pums

The!e are 0or+ed by bilding a n+ber o0 identical centri0gal i+/eller! onto a!ingle driDing !ha0t in !erie!G" The !a+e 0lo7 rate /a!!e! each i+/eller in trn,the 7ater receiDing an acc+lating head li0t at each !tage"

The!e /+/! are !ally in!talled 7ith the !ha0t Dertical in borehole! and dee/7ell!"


75/80

,ump ,er'ormance "aracteristics (urves)

'athe+atical eation! relating the Dariable! H, , N, P and are long and

co+/licated" $t i! +ore conDenient to hold the in0or+ation in the 0or+ o0 gra/h!,!ing data obtained 0ro+ /er0or+ance te!t!"

P+/! are !ally te!ted at con!tant !/eed N, and the other Dariable! /lottedagain!t di!charge " The te!t! are re/eated 0or di00erent !/eed!, /rodcing a0a+ily o0 characteri!tic crDe!"

(entrifugal Pum (ur)es

A K!table characteri!tic !ho7! a head 7hich 0all!contino!ly 7ith 0lo7 rate" High eddy lo!!e! in the/+/ +ay ca!e an Kn!table characteri!tic, allo7ingKhnting bet7een t7o 0lo7 rate! at a giDen head"

A Knon


76/80

.ther (haracteristic (ur)es

-igure & : #esign ,oint urve

-igure < : -amilies o' "aracteristic urves

Hydro ,o9er TurDines

There are +any hydroelectric /o7er in!tallation! throghot the 7orld" TheDario! +odern 7ater trbine de!ign! are ca/able o0 coDering a 7ide range o0o/erating condition!" Hydroelectric !che+e! +ay no7 be 0ond in +ontain!,

riDer or e!tarine enDiron+ent!"

For large !cale /o7er generation, three ty/e! o0 trbine! are do+inant"

-" Radical 0lo7 or Franci! trbine

8" A>ial 0lo7 trbine

9" Pelton (heel trbine

The 0ir!t t7o are kno7n a! reaction trbine!, the third i! called an i+/l!e trbine"

"rancis Turbine

The Franci! trbine +ay be regarded a! a centri0gal /+/ o/erating in KreDer!e"

-igure = : -rancis TurDine


77/80

The 7ater enter! the oter /art o0 the ca!ing and 0lo7! throgh the gide Dane!7hich direct the 0lo7 radially in7ard! throgh the rotating ele+ent called thernnerG, tran!0erring energy to the rnner" The 7ater then drain! ot o0 the ca!ingthrogh the central otlet /a!!age"

A7ial "low TurbineThe a>ial 0lo7 trbine +ay be regarded a! an a>ial 0lo7 /+/ o/erating inKreDer!e" The gide Dane! direct the 0lo7 !o that it +eet! the rnner in an a>ialdirection"

-igure > : A+ial -lo9 TurDine

There are t7o +aor di00erence! bet7een the trbine and the /+/

aG $n the trbine, the rotating ele+ent i! !rronded by gide Dane!, !o thatthe Delocity and direction o0 the 7ater can be ad!ted" Thi! /roDide! 0oro/ti++ 0lo7 condition! oDer a 7ide range o0 di!charge and /o7er ot/t"

N"B" electrical reire+ent! Dary oDer a 8: hor /eriod"GbG The 7ater leaDing the trbine /a!!e! throgh a dra0t tbe, 7hich i! ta/ered

!o a! to +ini+i!e the lo!! o0 kinetic energy" Thi! en!re! that the+a>i++ energy i! aDailable at the trbine"

1irtally all +odern nit! incor/orate +echanical +ean! o0 Darying the /ro/ellerblade angle o0 the rnner, !o that high rnner e00iciency +ay be +aintained oDerthe !/eci0ied range o0 di!charge and /o7er ot/t"

1ariable angle de!ign! are called K5a/lan trbine!" $n recent year!, a n+ber o0/+/ed !torage !che+e! haDe tili;ed a !/ecial ty/e o0 5a/lan nit" Thi! nito/erate! a! a trbine dring /eriod! 7hen the de+and 0or electricity i! high"Ho7eDer, de+and i! lo7 dring the night" The blade i! then reDer!ed and the niti! +otor driDen !o a! to act a! a /+/"

Both the 5a/lan and Franci! nit! rn 0ll o0 7ater" There i! there0ore acontino! 0lo7 o0 7ater bet7een the re!erDoir and the tail7ater"

4enerally, trbine! are Dery e00icient, i"e" hydralic and +echanical energy lo!!e!are lo7" They co++only attain e00iciencie! in e>ce!! o0 )."

5a/lan nit! are generally a//lied 7here there i! a large di!charge at a lo7/re!!re, 7herea! the Franci! nit i! a//ro/riate 0or +oderate di!charge! at+oderate head 8=. ? 9.. +G"


78/80

Pelton 8heel Turbine

The Pelton (heel trbine i! a//ro/riate 7here a high /re!!re 7ater !//ly i!aDailable" $n contra!t to the other trbine!, the ca!ing doe! not rn 0ll o0 7ater"

-igure ? : ,elton !"eel TurDine

E!!entially, the rnner co+/ri!e! a di!c, 7ith a !erie! o0 0lo7 de0lector! orKbcket! arond it! /eri/hery" The 7ater enter! the ca!ing throgh one or +oreno;;le! a! a high Delocity et" A! the rnner rotate!, the et i+/inge! /on eachbcket in trn" The 7ater i! de0lected by the bcket!, and the change in+o+ent+ tran!0er! a 0orce to the bcket, and hence a tore to the driDe !ha0t"

The et incor/orate! a !trea+lined control DalDe 7hich i! !ed to reglate thedi!charge, and hence the /o7er ot/t"

The gro!! head 0or the Pelton (heel i! de only to the height o0 the re!erDoir

aboDe the no;;le!"


79/80

Se9age ,umping Station Layouts

-igure @ : Se9age ,umps * A+ial and i+ed -lo9


80/80

-igure : Se9age ,umps * entri'ugal and SuDmersiDle

blt1110 hydraulics notes

Documents

flo7 ea

o ent o0

o0 hydro

aial flo7 p crde

ied flo7 p crde

o0 rotodyna ic p

age pre

lace ent p