Ir~FLUENCE OF SPi!F !<lU\L f\!~D CYUNURICI\L.

C!WSULES

ON PFzESSUr~E TRPJiS! ElfrS I! ~ HY DR!\ULI C PIPF.LH:FS

INFLUENC:t:: OF SrHERIC!\L. NHJ CYUtHWIC/\L

Ci\PSULES

ON PRESSURE TRANSIENTS IN HYDRAULIC PIPELIHLS

BY

RAJBIR SINGH S~~RA

A Thesis

Submitted to the Facul'cy of Graduate SturJ'ies

for the Dor:rce

Haster of En~rin:::erin9

McMaster University I·!Jli ::.nb·~ r·, i 9 I 0

Fd':ASTfT urn VEI<Sn Y Hamilton, Ontario

TITLE: lnflue:n.:.c of Spher·ic0l and Cylind!~ical Cvpsul8s

on Pressure Tnn1sients ·in Hyd!'aulic Pipelines

f\UTHOH:

SUPERVISO?: Dr. G. F. Rou;;d

NUf'-inE~ Or P;~GES: viii, 72

SCOPE Af!D CONTENTS:

The influence> of carsule:s on pressure tr·ansients in hydrc<ulic

p·ipel·itiC'S hc:s b:::2n r-cpurLcd ;n this dissertation. Th·is experimr~rrl:a"i

invcstiqation was carried out.for flow velocities of 1 to 4ft/sec.

for both cylindr·Ical and sph2r·ica·t c.:;psuh:s. Shock \'laves v:cre

c;:;psules of dian:etct" 0.87!: in. end U.95 in. The splie;·icnl capsulEs

we~c of diam2ter 0.75 in. and 0.875 in.

fl tllf'vr<.::tical an<:t'lysis for this type of waterharcme~- pi'ob1crn

using a per·tu·,botion t<:~c!nrique has been obtc-dncd. The thc=:on;t·lce;l

solution was fol!ild t(l be in c:.~wce:n2nt with the e>:p~rimcntal

observat'i ons for· cyl i ndd cc. 1 CC'lpsul es.

i j

The author ex:wcssos Ids dec~; sense of g;·at"itude v.nd ind::btness

·to D1·. G. F. Rouml, the Ch~irman of the clcrartme:nt of r':echanica.l

throughout this ~ork.

The resea\'clt for tfri s "Lh".:S is v1<:1~~ supported ·itt pzt~~t by the

Defence r~cseatch Botil'd of Canada, Gnmt No. 95SO--tiC: a.nd in pD.t::

CHAPTER

1

2

3

4

5

1.1

2.1

2.2

2.3

2.5

3.'1

3.2

3.3

3.4

TABLE OF CONTENTS

LIST OF FT GURES

NO!,!ENCU\TURE

INTHODUCTIOil

THEORY

Ass w·npt ions

Basic Equations

Non-Dimen3iona1 Form

Perturbation Equations

Analysis of Pressure

APP!\Rf\TUS MW EXPEi~H1EtlTf'1 PROCEDUHE

General Requircm2nts

Descdpt"ion of /\pparv.tus

Experimental Procedure

Error Anclysis

RESULTS N!D OISCUSSI0l'l

CONCLUS IOi~S

I~E F!::HENCFS

Capsule-·Frec Pipeline fJt' Va.lve Clostwc

Ccwtplctcd in less Than 2L/a Seconds.

iv

v

vii

l

6

6

6

9

10

18

21

21

21

33

35

37

48

66

69

69

LIST Or FI GURES

FIG. NO .

Schematic Diag ram of Apparatus

2 Details of Pipe Loop

3 Details of Test Sect ion

4( a) Overall V~ ew of Apparatus

(b) Ovcral"i Vi e\·t of In stnonentation

5 Pressure Cell Mounting

6 Schematic Diagram of Pressure Recording

System

7 Typi ca 1 Traces f tom Vi s ·i cordcr-Osc·i ll oqrc1ph

8 Traces fo r Transducer T3;

Ve 1 ocity == 2 ft / sec.

9 Traces for Transducer T3

;

Ve l ocity = 1 ft/scc .

10 Traces for Transdu cer T1;

Veloci(y = 2 ft / sec .

11 Traces for Tr-ansducer T 1;

Ve loc i ty= 1 ft/sec .

12 Traces for Transducer T3;

Vel ocity:.: 1 f t / sec .

13 Traces for Tr-ansducer T3,T2;

Velocity::: 2 f t / sec .

14 Traces for Transduce r T3;

Ve 1 oci ty ::: 3 ft/s ec

v

P/\GE

25

2F

27

28

30

31

3?

50

51

52

53

54

55

56

15 Pressure at T3 versus Fl mv Velocity

16 Pressure at r3

versus Hum er of Capsu l es -

0. 95 i n. Di;:,r,,eter Cy li nders

17 Pressu re at T1 versus Numb el~ of Capsu l es

0.95 in. o·i i:'meter Cyli nder

18 Pressure at T 3 v cr'SUS Nu r:1be r of Caps ul es ··

0. 875 ·i n. Dian,c t er Cylinders

19 Pressure at T 1 vers us flurnbe r of Caps ul es -

0. 875 in. Di ame~cr Cyli nders

20 Pressure at T 3

versus l•lttmbe }' of Capsu l cs -

0. 875 in. and 0.7 5 i n. o · ar,Jeter Spheres

21 Pressure at r1

versus N tir;- ,b~ r of Cupsu l es -

0. 875 i n . Di umcter Sph e r es

22 Hea ci Lrss versus DistdlCE· (steaJy st i.lte

co ncJ i t-i ons )

23 Fl01·1 /\reil versus D·istctnce /\lon ~t the Pip eli ne

2 ~ Pressure Transient History for Valv e Cl osure

l ess than 2L / il Seconds

Vi

57

58

59

60

61

62

63

64

65

72

t·;OJ·iEHCL ATU f<E

w* fluid velocity

p density

Pm m'2an density

P pressure

t time

a speed of sound in the flu·id

K i sothenna 1 bulk comrwess ion modt•l us

x axial co-ordinate

u axial component of vc~loclty

A al"ea of ci~oss-section

L len~Jth bch:ccn solenoid value and the pr·im(lry

refl ect·i on site

A0

area of cross-section of the pip~: at the solcr.o·icl

valve

Q disdli'lr~K~ at stetldy-state cond-ition

~1 ~lach number

B(x) function of capsule size and capsule train length

E perturbation parameter

a. an arbitrary constant

P0

,P1,P2 zero-or·der, fit~st-order, second-order per~urbation prc:ssur·e

----------------------------··------·--··---·------

* qm<ntities \·Jith a b<w are physical qzh;.ntitics h:.:viw1 d·imensions.

vii

q0

,q 1.q2 zero-·o:'dcr, fil'St-onier' and sccond-ol'dc·r pcl'turbntion

di sch(ii'ge

fl,f2,f3

cl • c2' c3,c4

at·bi trary funci::i ons of tin:;:;

functions of the 92o:netry o"f the sys tern

initial pn:ssur·e

zero-order pressure in a capsule-containing pipeline

zer-o··order ~wessure in a Cf;psu.le·-ftee p·ipt:1inc~

viii

1. INTRODUCTION

The influence of capsules on pressure tr·ansients in a

hydraul'ic pipeline (\':atel'ilstliimer' \'laves) hns been stud·;ed iri this

invcst·iqation. Th~· transpm'tv.tion of Ci:lp:;ulc~s in (Yip2l·1n.:;:s hus

aroused interest in this ared. The wo;-d 'capsule' in this context

The cc~psule may he hol1u.1 or solid, cast or extruch::d, coated or

uncoated, l'ig·id m~ non--ri~:rid and cylindtica1 ot spherical in shape.

The Research Cou~cil of Albe~ta has been largely responsible

for v.ror·k that has been do;1e on cD.psul c pi pel i ni ng. In a sE:l·'i Gs

patt (1-·9)[8··16] the empl1as·is has been on tho cit~terrnim<tion of

capsule velocity Oi' veloc-ity ratio (capsule/free streairl) and pt0ssure

drop~: entailed c<S a. funct·lon of avol·u~r~: ve'iccity~ d·i\;;n:;ttl' ratio

and capsule/liqu'ici cle;1shy 1At1Uo. The capsu·ies us<~d in the

exper'irnental tests have been cyl"indr·:c.-:11 ol~ sp!icr·ical, hollo·,,, or

solid and of a \·!ide ran~_;e of diDm~ter ratio; the cylindrical ones

hav·J ng e vm'i ety of l c~n9ths.

Part l of Uris sedes descr·ibr:s the emer92nce of the concept

of the pipE~line trc::n:;port of solids by a stream of capsules [8];

part 2 pr·esent!:. a s impl i f·i ed theory of the fl 0\v of l 0119 coi:centri c

cylindr·ical capsules [9], and part 3 d·iscusses an e;:pcdm:::ntal

investig!Jtion of the tn-~nsport by ~Jatel' of cylinders and spher·es

of equal density [10]. Parts 4 and 5 ai e concc:rned vrith

1

2

experhn·:::ntal results rc1aJcin0 to cylindr-ical c:n::i spher·ical capsules

of d;:TIS Hy gteatcr t!v:n \'mtc~i·' [ 11, P]. Pal·t 6 presents a numeri cu. 1

an<Jlys·is of som2 vada.bks affecting the flo~·; of cylinders [13].

Jn pu.r-t 7 exper·imtnts are l'elX~titcd usin9 vintcr VJith equal density

capsules in v.n oil carT·ier [ltr] .. Par·t 8 is an cxpc~i~·imental

invest'igatio;·, liith cylinder;; and sp!icr·.::?s cJc•nsf:i· thnn an oil

carl"ier· [15]. Part 9 ptese:nts a theor-etical stud_y• of a free flo·,;·in~J

infinite long capsule [16].

Hc;·;cvu·~ to d.:..te,;w signH:ic~:.nt t·:or·k lws been l'epotted on

the inf1uence of capsulrs on t:H::: pressure ttansients in hydraul·ic

pipel"inc~s. Tlie prop~\gc.•tion of pri..ssur·e tY'a.nsient by the

clisturbi:nce of steaciy concl'i"clons~ either intent'ion£~11y Ol' by n

system fi;i"fute such as a purnp stoppuSJe, i~; a major c.ons·idr:l'ation

in the des ·i £F' of p·i p:~ 1·i nc systems.

The bt~sic co;1c:epts cmd equot·ioliS of \wter-hamnic:r analysis

\ve~·e d2vc·1 opc:d clnd conf·i rm2d experh1c~ntally by Joukv,:sky [ 1] and

A 11 i ev·i [2] at the bc:girnri n9 of trl'i s century. It should be noted

that all of the rncthods used in vaterh<:n:met analysis stem from the

s0.me basic equat-ions of JoukcMsky and filliev·i. These bc:sic

relations havG been mnplified but st"l11 rernnin thr~ foundation of

all vwterham:n~~~ stuc!'ic:s. It has been definHely established that

the sa.m'::! basic equat-ions apply to the surge computntions for any

fluid in any type of closed COiiduH provided the physical factor·s

are knu.m.

NurM.'ri ca 1 v.nd 9n"cphi ca 1 methods have been usHl few the

3

solution of these equat-ions with fr·iction omitted from the theory,

but included in lUi;ip2d for;,l in the 9raphica1 methods. Smn2

analytical solutions have been obtained by linearizing the equations

\·lith f\niction ·iiiCluded [3,4,~)].

Hatet'he:nn:::r, as app1iccl to sl·igiltly cornpr·ess·ible liqu-ids in

elastic condu·its, nwy be clc.:;.ss·ified ·into h:a categor-ies: the

fluid ttansient ca.se ·in 1:h·ich the condit·ions change fr·om one

steo.dy-state situation to VJ1ot.her stetidy-stu.te situation and the

steady osd1latin9 case, in 1·1hich trw s<:;r:'12 cycle is rcpeatc,d

pel··iod·tcally. The f'it'St case solt1t'ion by the method of

char·<:1cter·lstics hu.s bce11 found to be mol C? convenient 'in that "it

takes mm-)inenrity ·into account and y·lelds simrle r<::lu.Uonsh·ips

for handling the most complex boundary concl'it'ions; in the second

case is more conveniently handled by tl1e h1pedence rn2thocl ·in v;ldch

fr'ict'ion ·is l'ine<:ri?.ed and the boundu.ry conditions <:;re satisf-ied

by takin9 a hiwmonic analysis of the e};cit·ing val"ic:J>le, such as

head or disdlta'92s then sunming tile solutions for each lnrmonic.

In recent years, Str'~cter and Hy1ie [6] have developed

computer solutions of hydraul-ic system transients. They first

took up the charactel'istics rn2thod and applied it to several situations

vurying from a simple pip2 to a pipe net\':ork. Pump pov:er· failure

\'lith the resulting prcs<.>ttre tt'ansient have also been discussed by

Strt~etcr and Hyl i e. The im~;edence nv::thr>d \·ms next dGveloped for

computer solut-ion <:.nd appl-ied to piping systems excited by

reciprocating pumps or by waves on a reservoir and to self-excited

systems.

4

The mc:-:thod of cha.r<;cter·istics hD.s be2n foLmd to be successful

for most of the prob 1 Cli'JS and m:my phy:; ·i ca 1 phtnomc:na can be fu ny

explain::::d by t!ris rn<'~Lhod. But it is importunt to note thil.t ·Jr. Uris

method Oi'1C! studies the flo·.-~ by div·id·inc; the flm·i field into a net\'tor-k

of char-&ctedstics (alon9 wh·ich certain flow properties remain

constc:nt) Glid th::>n sat'isf,:ring the boundc:u·y conditions 0t the ends.

The full equat·iollS <'ll'E: never solved. The method of che;;ractr::r··ist"ics

disturb::mcr:s in the flm·t f'ield~ such as hav·ing c0psu1es in a

pipe1'ine. In the rm:.:sent ·invc:sti9cd:!on, a m~:thcmaticu1 0pprov.ch

us·ing the pc:rtud.H1t'ion te:d;rdquc: ln1.s been used for such a problem.

inviscid, unsteady f"im: is governed by tv:o equaUons ·· cont·inuity

and 1nomsntum. ThQ variation in c!e:n3Hy ·is srnan and the cquution

of state relntes the velocity or sow1d to the COl!:prc:;s·ib·il"lty of

water. In the governin9 cquElt:·Jons) vc~locity has bc:<::n replaced by

dischr-JT~le as it is convenient to dc=al in tenns of dischar~Je for

a vcn-y-ing area sect'ion. The goverrrin9 equutions are non-dimensiono.li7Ed.

A perturbation pa1·ameter, c, exists and it ·is propol'tional to the:

square of appl·opt··iate ~1ach nwnllc~t'. /\symptot·ic expans·ions for the

flo'"' varic1bles (pressw'e and dischar0e) can be \'lritten as fo1lm·1s.

If f denotes th2 flow variable, then

suhsti tuting such expans"icms in the govenl'ing equations and coilPct'ing)

5

in tUi'n)tcr1:1s ·inc.L.:pcndent of c and fir·st po·,,'ei'S of c, w>::: 9et the

zero ord.:;r equa'dons cJ.lld the f'irst ord2·r' pr~rturba.tion equations.

The system of equations can be solvt:d tmd the integra.tion constnnts

can be evaluatf;d fron1 the bounca}'..\' cu:·;ditions. Asswnh1~J an

exponential funct·:on fol' v.::;lve closure, s·imp1e exptcssions fOi'

pressure trans·i ents may be obU:d ned.

The analysis in(Hcates i..hat qualitatively, ir1tc::rnc:1 ar'r'a

changc~s affect the !T:D£Jni tude of the pressure tr·ans·i ents and tht:Se

chctn~!2S result in litC\~;;·li f'i cat'i on of the \'Wtl;i·:·i<:<n.;rer wave i r1 a

pipel·in:~ conta"irdn9 cv.psule~; as ccr:.parec! to wavr:s in a cap:;ule-frec:

pipelin:::. f'IU.~Jnificb'..:ion of tile prc~.sm··c in the pn:sr:;1ce of

capsules nri~!ht se<~i~~ to be surprish1D hut this tws b en also

obsetvcd expcr-imenti:\ 1ly.

2. Tm:or~Y

2.1. 1\SSU!lij)Cion:'. --~-- -·-~------··-·-.----·-·--·--·-

1. The fluid (wat~r) is inviscid.

3. The f1trid (VIil.L:r·) is slightJy cor,!press·ible~ C!nd var·iat:lon~;

in dc:rsity arc: snwll.

4. Convective acceleP~tion is negligible as compared to the

local acceleration.

5. The: f1ct<I is unst1:i1dy.

7. Ther2 are no cxt~:nnl forces.

The fh/d of a sl·ightly comprcss·ible l·iquid is c\xnplet.cly

descl'ibcd by the follu.rit1~! eqi.-iat1ot!S [lB].

Ol'

Eu 1 er:

- D~ p _..:.~ ... - Vp

Dt

Continuity:

v· (;;\.:,) + -6~- = o at

Equation of State:

- a r) K= p-:

Clp

6

(1)

(2)

( 3)

7

v:h e fl..!: -\;/ is fltrid ve·loc·ity

(> is dens ·j ty

t is t·im2

-p is pressure -a is speed of sound ·in tile f1 tri d' and

K is i.:hc: ·!sotlieli"lll\1 bulk COlilfW•:S~:i on riiodu·t us.

reasonable vc1oc'itics (sc:.y less than ~0 ft/sE:c.) and pressure pulses

of the or·d:::r· of 1000 p:>"i. l·!·ith these assum;)·t"ions tl12 Eul C:'l' cq~w.t·i on

-dU

-· p . --·-Ill at

( 4)

-u is the ax"ii.:l co;11ponents of velocity.

S"irlCC onc-dh;Jension:::l (plane) fiO'<I hos been as:.u:nc:d r o.nd u are fuilcti ons only of x. and L

Averaginq equat-ion (4) acr-oss c.lf1 <1tbHr·ary cro::.s--sccc-ion of

aret-t A( x) thus gives

~:: pfil _il9. - ----- ( 5)

ax A(x) at

where - Ju dA q :::: (6)

A

1\ver.;:~~rin~: the contim!ity NJUc,tion (2) over a cross-sect·ion of

the pipeline c·nd utilizing the equr;t·ion of stv.tc~ (3) to elhninate

p gives

(7)

- -· x and q are positivr: 1·:hen both arc n:~a:>urcd in the same

direcUon; th<1t is) fl'O:n resuvoir to the ~:ate ('in Uds n:.se fl'O:n

the reservoh· R, to the: so1cnuid valve n (c. f. r>igs. l vnd !:)) .

movement 0t tilt~ dm:ns treu:n end, ·it is mm·e convc:ni c:r,t to express

the cJ·istancc~ to a section of the pire fro:-il the? lm't:T end of the

pipe as a posH"i\:0 distance, since h\'iJcial clistur~)c:nccs in the

flm·: occur' first at the dm·mstream Pnd of the p·ipf:l·inc: and then

move upstream.

Taking): as positivE: from tlle valve1 1.·rith -x = 0 dcnotin9

the sol cnoi d Vc< 1 ve Dnd x '" L denoting the r·efi ecti ng bend, the

governing equations becoiiK:

~i:: ~~ -~.9-ax J\(x) at

( 8)

(9)

For non-dimensionalisationJthe standard length, discharg~ and

cross-section have been t~ken as:

fo 11 O'o:s:

9

reflecticm s·ite

A = area of cross-·section of the pipe at th2 valv2 0

Let us dc:fin:':2 the: non-diPiC~n::dontil varit:tblcs as

-X

X :: [

-a ~ FJl7f]- :: f~

0

1 = - (say) £

('IO.v)

(lO.b)

(10.c)

(lO.d)

(10.c)

(lO.f)

(lO.g)

a perturbrtion par0m~ter

Also

£ :: o(r}) « 1

ClX ax

dx

dx

the qovcTn·l n~J equot·i 011s as fo 11 o;·:s:

d p 1 }51 -,ix == 1-\ CJ t

2 A. Zcro-Otdsl~ ar1f~ F·i l"S t-·Or·d::~r Pcl'tui bat·i Oil Equat·i ons ---~---~------·----------~----·--··----··--~·-·-· ~---·--~--~----~~'-·-~-···--·--·.,-,---··---·- --------·-

10

("iO.h)

( 11}

( 12}

( 13)

Lf.:t us clefi :10 asymptotic expansions for pressure and

discharge as folloi'Js:

where

p(x,t;c} = p0(x,t) (l4.a)

p0

,p1 ,p2 denott:s the zero--otdcr, first-order and second-ordct·

perttwbation p~'cssurc, and

11

Subst'itut'ing the tbO\'e exp;)nsions in cquut·ion ('12) and

( 1 3) h'C 9C t

(15.a)

and

(Jq __ Q_ + ()X

(15.b)

Collcct·in~:~ ·in turn, terms inclcpcnclent of c; and first Jhi'\·Jd' of c

we have

<lPo 1 -· 7i ()X

<lqo 0 -

()X

3pl 1 ----- A 8X

<lql = A --·--

(JX

Equat'itm

()(1 '0

Clt

<lql -a-r·

<lpo -at-

(l6.b)

(16.0.)

(16.b)

(lG.c)

(16.d)

·i1nplics that q is not a funct·ion of x. 0

dq q

0 == q

0(t) ilnd \'.'2 c:C:tn \'ll~·it~ -d--=to instead of a pai'tia1

de~ivative in (16.a).

q :;: q (t) 0 0 (17.a)

12

Solving equ~tion (16.a)

intc~n-atin9 \·Jith rc::.pect to :<

r o < x , t l , :~" J r,f~r ' f, < t l

It is ~" ite rcas a nab 1 e to I'll" He J 711fr = c1 J: in thi c. case

o.s the vz<r·iat'ion of /\ \vHh n:sp;;ct to x is only sicJnif·icant for

a Slil0.11 i11tci·val of x. c 1 is a function of the ~wo~nehj/ of the

t 1(t) is so1r::" funct'ion oft ancJ should be rvalu~:tcd fror:1 the

bou;·1d0ry wnditi ons. The: cnndi ti ons <Jt x 0" 0 Play be used fur

i.e. the total rise in pressure at th2 valve is just equivalent

to the zero-order pressure. This is quite reasonable as the first-

order· pr:TtUl"bat·ion pressure p1 ·is not siqnif'icant at the valve.

p(O_,t) = 0 + f 1(t)

so

(17.b)

pres;.ure and is cqutil to the sum of the ptessure at the valve plus

have

13

' uue

dq [. ~~ 0 · A -ere

ciU 0 x-,-:·-·­

ot rate of chanqe of tao,1:enturn pc·t'

un'it c:Teo].

Substitwc·in~J equE\t:ion (rl.b) ·into equ::1Uon (lG.d) 1·1e

Clp ::: [\ ( _____ Q_ )

(lt

ql = ~.2~Q fc-lx dt-

? 2 d-o x '0 dn(O t) ::: c - - - + v ._! .• elf'---· + f2( t) 2 2-- d?'

It is rcusonnblo to tel:e jc1 >:dx " c2 -{ in this Ci5e, nnd c2

is tho function of· th~ ~g:o::H::tly of tile: system.

c2 -- 1 for a capsule-free p·ipel·ine

> 1 for a capsule-containing pipeline

Jl\dx - V, is the vn 1 ume.

The firsi:-ordet~ perturbation velocity v:ill be zel'O 11.t

the valve (x = 0). For evaluat'ino f (t) we cvn use the boundary ·' 2

condition that pcrtul~bat'ion is zem at x "'0. Also at x == 0,

V ~ 0. Therefore,

f (t) = 0 2

vie huve

(17.c)

substHutin9 in equat·ion {l6.c) \'Je have an Qxpress·ion for the

ap1 l aql -a>:- = N>iT at

Integrati n~1 vrl th rcspt:ct to x

l\s variation of fl. \rith rcsp2ct to x ·is for a. sl,m't int£TVD.1 of x,

and

f* dx = c;;

c3 and c4 are functions of the geom2try of the system and

c3 = 1 for a capsule--free pipeline

> 1 for a capsule-contuining pipeline.

Also t 3(t) be ev<Juav::d fr·om the boundc<ry condition that

at the valve thf:~ first-order pl'CSSUl"e is zeto.

15

(17.d)

~Je Ira V(; the expressions for zr•r'O··ordcr and first-order

pel"tur-lntion cpn:iltH'ics [Equtt"ions 17(a)- (d)] in terms of

If \·te clcfhlc q (t) and p(O,t) by some 0

appropdatE:~ e>:pre:ss·ions, thE:n it is poss,ible to obtcdn q0 , p0 , ql

and p.1. t·Jc C·:<~J have c:1 mnr~~J2r of exprc::;sions for q0(t) and p(O,t)

In the anal,ysis to follm·t, p(O,t) is related to the

The eff(:ct or area dwr,ge can be seen frorn these expressio:iS.

The: prcssur'E' val'iation at the vo.lvr~ can be~ reasoivbl.v

es timatc:d fnyn the: v:cll···l~no·.ni tesul t for a waterhmn:n;::r· wave and

this equation is

-dH a dV ::: - --

9 ( le)

dH denotes a ri sc in the head; Vis the velocity of fl01v. Also

p=pgH (19)

By using equation (19) the equation (18) is modified as

dp = - a p dV

Averaging over the cross-section -dJ) m dp == -· -----

A(x) do

(20.a)

(20.b)

do cl p := - 7\rl' ( 21 )

tim·1 at the valve, i.e. x 0, tile pn:.:sslwe :-: p(O,t); /\ = 1 and

q(O,t) = q . c:s ql -·} 0 at X -· 0 o'

thcrofor'e

dp(O~t) 1 do -· - M '0

J\1 so \'!e can 1·1l'"i te

Thet·efore

~l~(~)_?.,!} l cl,l

0 dt - - H dt

To find p(O,t) vie inteqri;tc (;(JUi'ition (22.<;)

P qo

Jdn(O t) :: - _l Jliu I . ' t•1 ;

p. Q 1 . 1

o1. and 0. are the inititl nressure and flow rate . '1 ~-

p and q at·c the pn-:ssure and d·ischar~le at anytime 0

dtn·i ng va 1 vc~ closure.

p(O, t) - P.i

(22.a)

(22.b)

(22.c)

(23)

17

Non-dintcnsi(:nc;l 0; - 1. Suk:titutin0 in r:qtwt'ions (17 a···d) v:e.

get

o0(x)t) ·- 00 ( t)

p0

(x,i:) X dqo + .. c /i ·en-1

x2 12 q.

1(x,t) c?.

c qo .. -2- ·---2·-

clt'

3 13 c:~x ( 0

p 1 (x~t) '0 .. --r:s ___

·-d~~r

1 ("I H ·-

v - n

c4x - ·A'rf-

qo) +

do '0

(ff'

12 ( 0 '0 d_t_2_ ·-

P;

( 2 " \ .-'r.C.}

(24.b)

( 2t1. c)

(24.d)

The funct·i on q0

( t) n.'fcrs tc the: vad D. ti Ot1 of th<! fl CJ\'i

rCJ.te vrith respect to i.:irK' i:'1lld tlds d::pc:nds Oil tlie> ts 1<; of tlw

be such tiv;t:

fltt-0, q mu:~t b2 ec1tltil to the full vz·,lu:: of f'lo;·: 0

and for large t; q0 ~ 0.

The funct:i on shoul c! be cont i nuuus and an~•lyt-i c

throughout the range of interest. If the function or its derivatives

are discontinuous, then the analysis becomes invalid. It is

reasonable to u.ssume an exponcnt·ial function for the zero--order

dischurqr::: for valve closur·c, takin9

q (t) = Q e o:t (25.a) 0

when.: o: is an al'bit·fiHY constant; Q == 1 (non-·dirm:r,sional)

q ( t) -· ·o

.. o:t e (25.b)

It is possible to assume any otl:e:t' type of cxprc~ssion

18

for' q0(t), but this exponc;rUai fot:n \'li"ll ·ilh~str·(Ji:e: the 1wocedure

in a sirnple Victy. !\s po·lnted out ear1iel', the dlo·ice of the function

depends o:1 the valve c!Jcn·a:.::terist·ics c.nd phys·ical d·im:·11s·ions. For

q0

::: e o.t th2 equal.i()i;', (24 a--ci) (H'C n:odified a:; fol"iCi'\'/S:

- ut q = e 0

q.1(x,t) --

pl (X, t) ·-

u

...

(·.1 ! o: ~ at-.~ l ('I /1 "' t1

- at) c + P· )

ut --(:

2 -a e

o:c v "l ,,

A

2 v, ( o;

X c _2 ___ + f11 2

3 o:t X (

Ci c3 >~ c:/;

7\ ·--···6·--·- + .. ,!) !'t

~ at+ l (l _~at) + JJ. M 1

(2G.a)

(2G.b)

(?G. c)

(26.<1)

(2G.e)

It is i'CcSOilable to assur:e. th::.t the zel·o··C1l·del" rwcssure

then -x - - a t

and using non-dim~nsionalisation

X = - _!: t-1

Th·is express·imi (equv.tion 27.b) is appl·lcable only

( 27. a)

(27.b)

after valve closure hCts started~ as prop0gation of the wave is

related to valve closure. Substitution of equation (27.b) into

19

equation (26.b) gives

1 - at Clat po=-g[l+e (-A---1)] +pi (28)

For a capsule-free pipeline Cl'oss-section is constant (A= l)

throughout the length of the pipeline. But for a capsule-containing

pipeline the flow area is reduced for the portion equivalent to

the length of the capsule train. Let the capsule area be

denoted by B(x). Then, for a capsule-containing pipeline,

A= 1- B(x). In the present investigation, variation in area

is only for a small interval of x; say x = 0.01, i.e. capsules

are present over one percent of the total length of the pipeline.

Let the pressure transient for a capsule-free pipeline be

denoted by pf and for a capsule-containing pipeline be Pc·

For a capsule-free pipeline B(x) = 0 and c1 = 1, but for a

capsule-containing pipeline B(x) is positive for some interval of

x and c1 > 1. Then

1 t atc1 P~ = H [1 + ~ a (1 - B(x) - 1)] + pi

Therefore, the change in pressure due to capsules is:

cl ( -~-;\- 1)

1 - 8\XJ

( 29. a)

(29.b)

(30)

The above expression is al\'1ays positive for a capsule-containing

pipeline, c1 and B(x) depend on the size of the capsules and lenqth

of the capsule train.

20

The above analysis indiu.tes quulitat'ively that cap~;ules

present in a [ripclillf~ v!ill pt·oduce l:l()~ti1"if'ication of the \'Jater·hc.n:ftiE:r

VHlVC. The aniotmt of mugnif·icat·ion \Fill depend on the capsule size

and train len~Jth.

The f"irst-ord~··r pel'turbi!t'ion pi"essurc~ 1·Jil"l also be

slightly affected by the capsules but for shm~t lene1ths, the

curnmulative effect 1·rill be nc~(li~rible.

It is ir::portant to rer·:c:r·tbcr thi;t the above analysis

is true: fol' an inviscicl, sli~!Lt'ly coi,;pn::sslble nuid and the flm1

is one-dir::ensiona·l dnc: unstt•ady.

3. APPARATUS AND EXPERIMENTAL PROCEDURE

3.1 . Gener.11 PC(jlli rc-:m~nts -------------,~~-· ---·-~··-·· ·-----·-

The system w~s designed nnd instrumented to meet tl1e

(1) prov·ision fu· g2nerv.tion of Vio.tethu.m;ner \tJaves, i.e. a fast-clos·in~i

va lvc ot the do·.ms tn:~t":n1 end of the 1 i ne.

(2) the pip2linc is to be sufficiently long so that valve closure in

relation to U1c test section could be considered as instant closure.

This mGans thet valve closure time should have to be less than

2 L/a.

L = distance bet\;een test sect·ion vnd the primal')'

reflection site ·

a == velocity of sound in fluid used (watel~)

(3) provis·ion made so that the number of capsules mounted,

their spacin~J, and capsu·lc/p~ipe die1m~:ter rvtio could be vaded

('l) response tir112 rm· the ~·L··essurc tl'an::;du:..:el' system to be

srna 11

(5) recording instrumf:'!nt should be able to recotd pressure

transients continuovsly

(6) extraneous pressure variation should be minimum

(7) the system should not leak durino pr~ssure wave propagation.

The apparatus ~~ich WdS used in the experiments is shown

schematically in Figure 1. The flow direction from the reservoir,R,

to the soleilo·id vu1rc is sLc~;n in F·i9urc 1. The resr.:rvoiY' \"las u

21

£..(_

rectanguli'.r' tank 3G ·in. x 36 in.>: 30 in. f'itted with a 2 1/2 in.

diam2t.er- pipe lead·ing to th(:! pump; a 1 in. diameter· druin line wr:s

provided. The pump \·tas rated at 3 H.P. , del-ivering up to 100 I.G.P.f:i.

at 50 ft. \'Jatc>.:· head. The> pu:;ip wa~; directly couplec: to an electric

motot~ {3600 r.p.m.). lhe reset'Voir fucilitated the removal of

A 2 in. diamr.:ter discharge line connected the pu;np to the

SU\'9e ch<unber assembly. The surge ch&:1;ber had an effect':ve volurn2

of 90!J. cu. in. and \'!C<S connected to a nitrogen bottle through a

1 in. olobe v&lvr. J

lhe 2 in. dian~ter line was cornected to 1 in. diameter

loop via. a 90° e"lbo·,.;. The total length of 1 in. diameter line \•Jas

245ft. In onier that this l-ine could be: accvn:moduted in a

laborotory approximc:tely 50 ft. long~ the pipe \;~as bu'ilt in thl~ee

loops contain·in9 strai~Jht lengths of 37 ft. and semidr·cular be;1ds

each hfving a r~dius of curvtture of 1.5 ft. (c.f. Figure 2).

Swaffield [17] has indicated that if the ratio of bend radius to

pipe inside dic:metet' is greater· than six then there will be

practica"lly cor;1plcte transnrlssion of a pl'essure ttans·ient at thr

bend. In the present instance, care \'!as taken to maintain a bend

radius » 6, thus avoid·ing reflection at 1ntE:rmediate points.

The flo'// rate \'las controlled by the throttl·ing valve nem·

the junction of 1 in. and 2 in. diam2tet pip2. A rotameter,

covering the range 0- 10 I.G.P.M., indicated the flow rate.

The line \'las mounted on ang1e i1·on fram2s, v;hich \'Jere bolted

23

to the ground. C1·oss--bars \·!el'e provided to t•educe the vibl~ations

of the fl'uiil2. /\ lthough the vibrations of the 1 i ne \'H?re red•,;ced b_y

firmly holdir.9 it trith U clc.·mps, Hds source of extrc:ti<20t'S pressure

r·educed to nn ac:c0·ptr:JJ1 e 1 eve 1.

The shock \·nwc: v:as gcnc.ratcd by a solencrid--v.ctuated,

pr·essurf! oper·atcd fast-·closin£1 valve located at the er1d of pipe loop.

The valve selected had a closing time of less than l/15 sec. and it

ensured instantc;ncous c1osut'e cor,dit'ion at the test section. Six

pressure points were provided u~ the test section a~d one near the

primal~ reflection site. r1,T2 and r3 in Figures 1 and 2 indicate

the posit·ion of the pressure ti'Rnsducers. T1 indicc:tc:d tlie prcsstn~c

history at the r~C~flection bend. T2 o.nrl T3 indicated the pr·essure

history at tv:o point-:; on trn: test s;::ct'ion; on cithc:r- side of the

capsule train. The distance between T1 and T3 was 231 ft.

The pressm'e cells mounted huve the fo'tlo\:ring sped fications

and tr~~e identification.

Position Pressure Cell Ratinq Kg/ cm2 /P. f;~-- ·-----···P-SifP. F.

Tr~ce Identification

T3 54.5 774 upper trace

T2 4-0.0 568 middle tt"CiCC:

Tl 30.5 443 lower trace

A 6 ft. long, 1 in. diarnete·1· lucite tu!:F.: \·ns used <ts a

test section, located nem' the final dischat·ge end of the pipe loop.

The tc.:; t sccti on could be eEls i ly comwcted or detaclwd by rnc:ans of

Johns~Jn dresser coup1-inns. A nt!mbcT of holc~s \'.'<:'l"e provided in

the test sccdcm for 1ocatin9 the cc-.psu1cs. The cr:qJ:;;u·les could b~'

posit'ion Ly SliliJl"l bolts.

The C<lf":S ul cs u::.c·cl \'lc l"e:

(a) solid cy1indc~rs, le;·,~r\:.h 5 in., diiti'1C:tc'r 7/8 ·in. and 0.95 in.

(l.J) r·J·l::-.e·•·;c·· f'"I"J'l'~l'"'><' c··",· 7/ 0 .. ,.!. ;'!tlrl 'Jjfi ,·1- a'·,···l'j'f'!c''l-n·' C\-'":JLI. ''t'""' l. ... ") . 0 I, Q ... "J r lc C. -..~t .. -._i

transducca"S connected to turd i·1q p'l uqs ~ \':h·i ch i 11 turn 1·:er2 connc;ctcd

d·iaphr0.m and <1 f·ixl.''d clc~ctr·o(_1. This capz,citancc fonncd par·t of

a rc:>CJiE.:lt circu-it th.:::t contr·ol'ied the fre;qu<::ncy of an u:>ci1lntur·.

a.n analog D.C. volto.~:e. lienee, a chimqc: ·in v. mech<micc~l d·isplctccment

chf!ngc 1·:hich, in turn, could be converted into a D.C. voHc;g::~ in

a reactance convertor.

a pr·~::ssur-e channel few rn2asur··ing lri9!i pn~ssurc variations. It is

irnportcH!t to have a short pn'ssur.; chr.mnel for IK~astwing th€~

pressun: cell flush vrlth the \''i:.llls of test scctioil.

.-:.-C) >·-< f--­t.) L1 . ..! .. J (J_,

IJ...J t~~

c-r: 1--fo··(

Vl

! ~- . t:

y

[\_

Cl C) _J

"'-· ·~-I .-.J I cl t-· ;~-1· ••• .!

?~; :~:

Cl •·~ , ....... :n :--· .U,.I f") i· ~ l.! . .J

V;

c<;--------­-·~·---M-----------

,. "

(})

60.

() "~~:

f··

) If~·.: '··

I IJ.I

' lJ)

j_

·-:-_:.::-:::::J l:J % .. ' •• J

f< !·.I I , ~' l . : Ll

t~-:

l)li ·L:--j

~>"'

( ; f" , .. , (/) !

L' f.; (,t')

I., ··,) L)

•· J {)._

U.~ C;

1,/) ··-l '""·" c.:-r"' .::: IU C::l

. C'J ........ l.L

LLI [J)

CY :~ ll.J ~--f--LLJ lU ;;: 1--<:I; ,_, ...... u Q :.:J

_J _J <:I; ~ z (.,)

0:: i'-1 w ;<: 1- 1-:z:: .....

c:: . •r-s::: ··- "'-''

' ,.... r·-

,--~ ---·-r I I

I I I

L,i ::¢ ~··1

r-··L~ I : .. _

,I

r-ei! ,, I

... _ I

I t-[1 .._

I

I

II

I

I

'I I

I

I .. __

l' I I I

y ___ ,. i ____ :

-([)

_ _l

27

;::: 0 ·-· 1-u l;J Vl

1--Vl w

·-w :::c 1-

LL. 0

Vl _J

·--~ <:I; ~-w

Vl C.:l lLJ ... J C) :c

(Y) . C.LJ ,__, LL.

28

Vl ~ t­c:( 0::: c:( Q,. Q,. c:(

LL 0

3 LU -> ...J

<t 0::: LU > 0

[\ .. pipe loop

B so 1 enoi d- va 1 \'t'

c

R reserved t

s

D throt.tlin~1 v<:1lve

so 1 c: no i cJ ··· v i:i 1 v c)

T 2

txansdue:cr· at the tcs t section; ups tl't:t:m fr·o::1 tht::

C., 1' r- I I 1 '\ '' ( 1 ('\ ') (,t),, d. t.::. ... {._ •i il • fr(J:n tiH• solc.:no·ld--valve)

N .. ,r·lr.,f'l.) l'cactar:c:c convc;rtcts I t... ,1

E v·i s·i c:ord:-r···osci 11 v;r·ar:1

F multich2nncl D.C. amplifier

'J(' l. :J

30

ELEVt\T 10l,J

__ f J~ -··~-.!_-~-I] __ I I I

I I I /\rt-,·,-. l'(J·'--cf·-,,, I I / -.- -.... I I ,}, L ; '.' ·. ,I./I,~;.~. ~-~;;i '' >,: I v I " I l:(;~J..il '-'~1. h:ol! ,L·'!t: ill ~· 1 t 101( /'.D,c\eitY liLIUI -- a.;:--~i/+}9 -------· I t y I I " /I I ,_ _.-- I

-'-----t-:--1-

riC. 5

-·' ..... !

(.) (.r)

C..J

----1" " -·- --

--·-·------

~~ t::,::: c-: (,lJ

C)

··-' •... .l t-- -~

( . .' VJ C)

n: l,J,_I

Cl ft:. C'l t...l

;:;-... ;?~ J -·- -----~----

(,!) ... .,.. __ 0 I ,._, c< ci Cl !.!l n:

3?

33

d·isp1ayed on an osc'i'lloscepe ot n;c\.Jfdul on cl tri~1h spr::cd dw.rt

recor·c!el~. A 11ml::y,;eil 2WG visicordc~r--oscillogr'aph in conjunct-ion

circuitry protcctiCJn \li.?.S usc:d for· l't::cord·ing the pi'essur·e history of

the \·J<;tcr·:i c1l;:;nc\' Hc.vc•s.

adjustment to be tr!tldes e.g. one ver·Vica1 d·lvi~don (:= 1 itl.)

to 100 volts dc:v.:•di!l~l on the se1ect:or p:;sH·ion.

was a VERSA T.G.G. 2701.

The ~wcssure ce1ls \!l::l'L' u:.l'ilni1td using a clead-\·Jc·i~Jht

tester·. The t•oti:nnctC:l' \las cal·lbrated by VJ(::-igiling amounts of \·iatei·

collectt'd in a fixed tin:o.

The converter unHs Here s~·Jitchc:cl on to warm up fOl"' at 1 u;st

one hom· before tht.~ ~'tai't of et,ch expcr·irm:n\:. Thi~; helped in

The reser·voit· Has f'ilh:d v:itli \:ater to l/2 cilpvcity and

the Pl'n1J1 stuff·ln£.J box n•Jts (2 off) \!(';re cu:ljustcd so that thon:: wns

slight drip of water as the driving shaft was rotated slowly

by hand. This indicated that the pump was well primed.

34

Tlw capsules (cyl·lnd~:rs o1· spheres) \•Jer2 local:ed at the

desir·ed pos·ition ·in the test sect"iOi!. The test section was

connectPd in the line by means of the Johnson dresser couplings.

The va 1 ve at the june-Vi on of 1 ·in. ;:nid 2 ·j n. d·i mn-::tei~ 1"i nes \·tc<S

closed. fJ,fter the pump 1·1as st<trtecl tiH~ valve was opened slo'.'!ly and

at full openin9 vatter vw.s c-irculated for· 15 minutes, so that rdr

was sv1ept from the system. The flo\'/ v1a.s adjusted to the t'cquired

velocity.

l~e reactance converter for the appropriate transducer was

adjus t9d so that the output voltage was zero. For a patti cul ar

flow velocity, the scale on the wrrplif·ier v1as selected. Irnm~:d·iately

after the visicorder-osci1lograph drive was started at a selected

pupet speed, the soleno'ic!--operated valve v;as closed. The v1aterhan:mer

\·Jave thus gene1·atcd propa£Jatec: thl'OU]h the line and pressure s·ignals

v1ere recorded from all the transducel~s. Typical output s·igncl.ls,

for three traces, were recorded indicating the transient pressure

h·istory at three points 11, T2 and T3 on the pipel-ine. There\'Jas some

time ia~1 for the s·ignal from the transducer T1. The time lag was

detennined accurately' by running at lligher·chm·t speeds.

The pl·'ocedun~ dt~scribecl above was follv;vecl for the entire

exp•::dmcntal p;~o~n?m, i .(~. foi~ varjdng f'iow velocities, capsule

diameters, shapes, nurnbet· and pos i t'i on of capsules.

3!.J

3.4. Error 1\nalvsis ----·-~··---·- -·--···-·----·-··-- ·--· .'l>-~---·~···---

The- fo11m·ring c\lic:.lysis ·is bc:t::.ed on estimates of error due

(l) tl·ans"icnt p;··cssurc

(;~) flml velocity

(3) capsu1c c:nd p·ipc dircc:;:s·ions

a·ir bubb'le~. in thz:: systciil.

e s t i H1 a tul c 1 Tv:· '"

(b) calHH'i':tion enor. Fol' culihr-ation 'the: dcud w2·i~1ht tester

(c) the con:hi nr;:d e;TOl' due to th~.:· el cctri CC:l.l sys tc:rn ( pl"C:SSUl'e

cell··tun·inq p1ug-oscillator-t·cclctance converter·) rn~J.Y bf~

estim0ted ~ ±2.5%

Totu.l errot in ptc~~~.:ul't: ··· 4 + 0.5 + 2.5

-±?'i

It is n::;t pos~dblc.: to ht1ve cl.il estimate of el'r"I:W' introduced)

due to the air· bubh.IC<s in the system. Hm·:c~vel', tlrls ~our't::e of

36

air.

Eti'O\' in FlcM Vel r.city ----~ ~- -----·-···-·-- -·--- _________________ ._} __

reading = ±0.05 ft/St(.

(L) Ci.1.1·ib;·c:tUon E:i'T0r'

rn(:>:h;'!l!m f:iTor· in \·Jcight ol' \'Jatf::J· coliccl.cd- :1_0.25 1bs.

Total etTOI' ·in flu.: velocity·- ~i + 0.5 + 0.5

±6%

error in 3/4 in. = ±.001 in.

test section length Vli\S r:12c!sur·cd by il rn~ter rod i:lnd en--or

in ft. length"" ±1/8 in.

es ti mated er·rol~ -· ± l. 07~

4. RESULTS AND DISCUSSION

The experiments Here conducted using the f0l1mdr,9 series of

cupsu1es all rrw.Je of lucite:

0.875 in. diam2ter, 5 in. long cylh,del'S

0.% in. d)t.uneter~ 5 in. 1ong cylinders

0.75 in. diameter· spheres

0.8/5 in. di0m2te;~ sphctes

The pressur'e hansient history W:ls recoi·dcd by the transducers

mounted at r 1, T2 and T3

. Initially a set of experim2nts \·iaS made

in a capsule-free pipeline to investigz;.te the effect of varying

the flm·t velocity and to have the data avi.ril&b1c for cor1parison with

the capsule located pi[.•elhJc. Typical traces are shovm in Fi~;ut'e 7.

A waterhail'inlel" Have generated by H12 solenoid valve reached

point T3

first and then travel"led 231 ft. before rec;ch"ir;g point T1

.

The t'irne h19 bch:ccn s i g:1c\l s i n(J"i cated the time tu.ken by the v;ave

to travel the d·istance be·L~·:e.::n T3

and T1

• The chMt paper \'JO.S

run at higher speed (16 in/sec.) and the trac~~s were obtained fo¥'

measuring the time lart accurately. The speed of waterhamrner wave

\v~1.s found to be 231/0.05 = ti62G ft/sec. a.nd t·crnai ned cor.s tr<~lt for

all f1o1·1 velocities.

The ex peri m2nta 1 data ott a:; ned for· a caps u1 e- free pipe 1 i nr;;

agn::ed very well Hith the theon:tical v&lues obtained fi·om the

\'lell-knnm l·tate:'harnm:::i' cqu:.rtion dH ~, -~~ dV. The excellent agl'Eem;?nt g

(c.f. Figure 15) bttween tile two results assm·ed that:

(1) lhe instrumentation was working well.

(2) The valve closu'te could be cons·id~red to be inst0ntcu1;:ous.

37

:.m

A large numbet· of pressurt• trnces \<Jere obta·ined fOI" various

comiJ·inat·ions of:

(a) flm·i v0locity (l--4 ft/scc.)

(b) capsule stwp2 and d·imnc'Ler

(i) spheres 3/~ in. and 7/8 in. di~meter

(ii) cylim:lets 5 in. lonq, 7/8 in. and 0.95 in. diameter

(c) capsule tndn length and ccmfiguration

(l)train of capsules with no gap between adjacent capsules

(i) 2,4,6, and 8 cy1inders,c!'ivretcr 7/8 in.

(ii) 1,2 and 3 cylinders,d·iuilleter 0.95 in.

( 2)tr<:ti n of capsules vri th a gnp bet',ieen adjacerd: capsules

(i) 2,3,4, and 6 cy1indc:rs, c!imnotcr 7/8 'ir:. and gap 5 in.

(ii) 2,3,4,5 and 6 cylinders, diar112ter 7/8 in. and gap 2 1/2 in.

(-iii) 2 cylind~1·s, d·iank~ter 0.9:) in., gap 5 in. and 2 l/2 in.

(iv) ?,4,6,8 and 1·2 spheres, diameter 3/4 in. and centre to

centre distance 1 l/4 in.

(v) 2,4,6,8 and 12 spheres, diam2ter 7/8 in. and centre to

centre distance 1 l/4 in.

It has been observed that the pressure transient was

slightly affected by the capsules of 0.875 diameter ratio.

(capsule/pipe). But for 0.95 in. diametet' cylinclc:r·s the change in

magnitude was significant. Hm1ever the t1·ain len9U1 for such a

diameter ratio \'JCJS kept snwll because of the drop of f1o\'l velocity

with the addition of each cylindt:t and co:·responcling pressure

increase. Figures 8 - 14 have been selected to demonstrate different

39

effects for different combinations of para~eters.

Fi 9ures 7 - 1 r: shm'l the r-ise in pressure dur·i nf! the trans i E2nt

wave propa9at·ion. The tt·ans·ient pn~ssure recorded is the pressure

above the init·ial steo.ciy state static pr'essur··e. M'te1· valve

closun;, nw.xir;;Lnl pressure is ut the first p2ak and the rna~Jil"itl'de,

in the subsec!ucnt l'cf"le:ctions, clr·ops dm·:n; finally attaininr~

the steady state situ&tion. The initial steady state pressure is

different for different flow situations, and in Fi0ures 7 - 14

the base pressure is d-ifferent for all the tnlccs. Fi~;Ul'e 22

i."11ustratc:s tlds point cleiJrly.

Fi~urc'~ 22 shO\'tS the initial stec<dy state pressure (prcssun~

befon; any ttansient viave has been ~tencrateci in the systG'1) for

three flow situations.

(1) FOI' a capsule-free pipeline and fl 0\'! velocity - ft/sec.

(2) For a cars ul c:--frec p·ipeli11c (lllc.i nn\"/ ve·l oci ty :: 4 ft/sec.

(3) FOi' a caps~lc-containing p·ip~::line t:nd f"! 0\!

velocity = 1 ft/sec.

The drop in pressure along the pipeline is due to two main

factors.

(1) Friction in the pipeline.

(2) Throttlin0 at tli2 valve.

The follmdn0 calculat-ions can be used to obtain the heaci

loss due to friction:

Head Loss

40

where 1 = length of the pipe

0 = di2mcter cf the pipe

u = flo·.1 velocity

f = friction factor

Friction factor depends en the pipe roughness and Reynolds

number.

Heynolds number uD \)

u = 12 X 0.1217 X l0- 4

= 6,85.0 u

The flov; velocity vades from 1 - 4ft/sec. Reynolds number is

in the ran~e of 6,850 to 27~4no so the flm: is turbulent in the ranre

of interest.

Friction factor for the pipe used in this range of

Reynolds number= 0.07.

length betv.:een the throttling valve anc! discharge

en9 = 250 ft.

Diameter of the pipe = in. Therefore

_ flu 2 hf - D2 ---

9

2 = 1. 41 u psi

The follm:iw1 tabh~ i~; for' flmi velocities l - t: ft/scc.

Fl o1·1 Vc 1 oci ty in ft/scc.

1

2

3

4

Head Loss in lbs/sq.in.

1.41

5.6(

12.69

2? .IJG

41

Pl~cssun: at tl:c: c:-isc!Jar~JC enc: is atn:ospher·ic pressure and 30 ps·iq

just upstrec'rl thr throttlh:s' valve for all flm·; s·ituut"ions. There

is loss of head at tiH' do\mstrec:n;~ e::nd of the? solenoid vv.lve and

fl.s pointed out earlier, the drc1p ·in pressure fro~·il 30 ps·i9

(urstrec;n of the throttlin~~ valve) to l.!.i psi~l (at the solenoid

valve) is either due to the thi'Ottlin~~ valve Ol' due to fr·ict"ion

in the p·ipe.

In Figun:~ 22 the pl~essure vt u and g is alv:ays the

sa1::e and does not ~at'Y with di ffercnt fl O';J s i tua ti ons.

At a pressure = 30 psig

At g pressure- 1.5 psig

For flm·1 velocity l ft/sec. the loss of head due to fr'·iction

in the pipe is 1.41 Ds·i. Therefore) just c1 m·:nstn~c~n the thlAottlinq

vulve presstwe is 2.91 ps·ig and upstre<1r11 the thl'otU·in~J valve prt!Ssure

is 30 psig. The pressure drop across the valve is about 27 rsig.

In Fiqure 22 it is represented by po·ints a-b··C·ti-e·-·!-f-g.

42

For a flovr velocity ~· ft/sec. loss of head due to friction is

22.46 psi. So pressure just do\'rnstream of the throttling valve is

23.96 psi. The drop in pressure across the throttling valve is

about 6 psi. In Figure 22 it is represented by a-b-c-d-2-g.

In a capsule-containing pipeline there is pressure drop across

the capsule and in such a case the initial steady state pressure may

be represented by points a-b-c-3-h-f-g (c.f. Figure 22).

The initial steady state pressure is different for different

flov1 situations. The final steady state pressure (when the solenoid

valve is fully closed) is indicated by the dotted line a-4-j in

Figure 22. It is clear from Figure 22 that the pressure rise,

when the system goes from the initial steady state to final steady

state, depends on the fl m·.J s i tua ti on.

Next, consider the pres~ure history at point T, which

corresponds to point T1 in the present experimental set-up. For

flow velocity l.ft/sec. rise in pressure is from points 1 to 4 and

is about 27 psi. For 4 ft/sec. the rise in pressure is from points

2 to 4 about 7 psi. Similarily different amounts of pressure rise

for· different combinations of capsules as the system moves from

the initial steady state situation to the final steady state

situation.

In Figures 7 - 14 the pressure rise has been r.1easUl'ed by the

transducers T1, T2 and T3. The rise in pressure during transient

wave ptopagation is not affected by the final steady state or

the initial steady state pressure. Also, it is important to note

that the maximum pressure is the pressure of the primary wave

9enerated by the solcitCid va·lve. If the system is sufc for th·is

maximum pressure it \ri 1·1 be safe for the subsequc:nt pressure \'taves.

So tile d2signei~ need to have a look only at the maqnituclc of the

The number of reflections is different for different flow

situaUOiiS. Hlis n:ay Le cluc- to rt:sonance in tile rdp·inq systeLl.

The P1a0rli tude ~1ets attentuated in the subsec;uent refl ecU ons.

This aspect is not investigated ·in tile: prcsh1t \'mrk. Since the

111a~Jni tude ~Jets clttentuc~ted it \vi 11 not present any serious pr'ob 1 E'lri

in a pipe 1 i ne netv:or-1:.

Pl'CSSUl'e transient history in o. capsule fr·ee pipeline: for

valve closure C0!1iplctod in a finite til'H:: (less than 2L/a scccJJds)

is presented in the Appen~ix.

Figures 8 - 13 ·indicate the effect of 0.95 in. d·icnnetcr

cyl i ncicrs on the \'lt:1. tertlii!i!l:er \:ave.

Fi~ures 8 and 9 shq•.·J that at pos·itic•n l.., the pn::ssur·e \':ave ~)

bec:ar:le rno.gnific:d in tl1e pl't?sence of cyl incJrical capsules in

the pipeline.

Pn:ssurc at L \'Jas the same as 2t L for most cases, but i j

reached a lov:er value for a lal~0el' cc:J.psule diameter, ratio (Fiqurc

13.)

FiQures 10 and 11 indicate thc:t thei'e v1as attentuat·ion

system.

Fi ~Jure 14 sho•t~s heM the pres sure \·:ave, at T 3 bcca:•1c n'oclifi N1

be noted that the overan effect is small t,ut thef'e is slight

ma~n1ification of the fir~.t peale

It can be se':'n that aftel' valve clo:.ure there is a pl'irnvxy

pe0.k anc.! mony scconc:ary rc;;lcctions. The first p(~il.k has t!ie

greatest PiJ9nHuclc: and is nost si~ndf·icant fr'u:<; the desi~;n point

of vievt. The f-i t'st peak pressur-e for each set of cxperip;e.ntc..l

data has been plotted a~1ainst num0er of capsules for flov1

velocities 1 - ~ ft/sec.

Figure 15 shoh'S the pressLH'e T3 ver·sus flo\1' velocity for a

capsule-free p-ipeline. Thete is s1ooC: i19l'ec:rn·2nt bet\-'e(;n the

expcrir~ental data and the theoret-ical prediction.

In Figul·es lG and H~ the pressun:~ at T3

vet'sus the nur1bcr

of cylinders hbs been rl otted. The rwesstwe bcco;ilt::s f11Cl~!nifi eel

in the presence of cylindr·ical capsules and Iil~~:rlification derends

on the c!iar:1eter of the capsule and len9th of the capsule tr-ain.

This has been predicted tlleoreticCll1.Y i11 Chapt.et' 2.

In Figures 17 und 19 the pressure c:t T1 ver·sus the number

of cylinders has been plotted. The pressure at T1 becomes

attentuated in a capsule located line.

Figure 20 indicates the effect of spherical capsules on

the pressure at T3

. The effect is not very significant but the:

pressure v1ave becomes at'centuatc·d. Figure 21 shO\'IS the effect

of spheres of diameter 7/f; in. at T1. In 9t:'ner·al, there is

little effect of 7/8 in. diar;1etel' srhei'es and aln:ost no effect for

3/4 in. diameter spheres on the ptessut't:' v1ave. r}ut it is si~nificant

to note that the sphericu-1 capsule's vlitf: ~wrs in br:ti·JeE:n indiv·iduc:l

45

spheres produced attentuaticn of the ~tJaves and cylindrical

capsules produced magnification of the wave at T3 and attentuation

at T 1.

Attentuation of the ~t:aterhammer wave may be due to the

disturbance and friction introduced by the capsules. During valve

closure the pressure wave seems to be affected by a higher velocity*

change at capsule located section.

The pressure transient in a capsule-containing pipeline

seems to be affected as follows:

(1) area change alone is responsible for magnification of

the ~twve

{2) the added obstruction and the accompanying friction

produce attentuation of the wave

For a large diameter cylindrical capsule magnification due

to area change is greater than the attentuation due to friction

and the net eff~ct is one of magnification. But, for spherical

capsules the magnification produced seems to be less than the

attentuation and the net effect has been observed as attentuation.

In the present investigation, the capsules are present for

a short interval of length of the pipeline as shown in Figure 23(a).

Capsule train length is 0.5 to 1.5% of the total pipeline length.

This may not be the case in a rea 1 situation. In a rea 1 case,

the capsule train length ~ay be 50 - 100% of the length of the

* ' i.e. for a capsule of 0.95 diameter ratio, the pipe area covered is 90%, thus increasin9 the fluid velocity by a factor of ten.

pipeline.

If a pipel·ine is 100~; filled Hitl1 the capsules as shmvn in

Figun~ 23(b), the pressul"e tr-ansient J:'ay be analysed as follm·1s:

let

Takinq basis for conpar·ison as a capsule-free pipeline,

Flm'i velocity in D capsule-fl~ee pipeline'·· v1

Pipe area of cross-section = A1 Len0,th of the p·i pc '-' 1

1

Loss of head due to friction =

av ·f g

and ·let the lerlc!th, discliar~JL' and at'ea of cr'oss-scct·ion be the

S2111e in the h!o Cilses. If the p·ipc"linc is loo;; fil1ec1 v:Hh the

capsules, then ·it ·is equiv.:dent to a pipe of sn:a'llcr vrea of

c l'O s s - s c c ti o n .

Let the fl ov1 area !1 2 == n Jl,1

n c~cpenc!s on the qeor:1etry of the syster.1. If the Cr'oss-scction area

ratio (capsulc/pipe.line) is 0.9, then n = O.l. P.ssu:;:·ing the loss

of head due

the r:li-1>-:imum

to friction --

rr·essun:

a 9

t'i sc

Lf

at

av ~ av 1 (_ -- - ----·- ·-

---n

and f1 0'1'1 velocity v2 - v /n and i -~~

2 l

the valve closure = 'U (:, '2' therefore

47

The effective t'ise ·in pn:ssure v1ill be the sum of the pressure

dli 2 ancl Lf2

(pressure increase clue to a1·ea d1anc:e awJ pressure drop

due to ft'ict-ion). I11 ·~i;e S2l:it: 1·:ay, a pipeline filled vrith any

5. CONCLUSIONS

On the basis of the analyses and experimental data presented

above, the following conclusions can be made:

{i} The waterhammer wave is slightly affected by the capsules

located in a pipeline up to a diameter ratio 0.875. Only for

long trains of capsules of large diameter ratio (say 0.95) is

the effect significant.

{ii} In general, at the location of capsules there is a magnification

of the waterhammer wave in the presence of cylindrical capsules and

attentuation in the case of spherical capsules.

{iii) Pressure waves past the capsules are affected as follows:

{a} In the immediate neighbor.hood of the capsule, the pressure

is the same as at the location or just before the capsule.

{b} At some distance upstream from the capsule, the wave becomes

attentuated in a pipeline containing capsules.

(iv) Magnification or attentuation dep2nds on the flow velocity,

capsule size and train length.

{v} Pressure wave transient velocity re1~1ains constant for all

flow velocity and capsule located line. -

(vi} For a capsule-free pipel·ine, the magnitude of the pri_mary • - • • • • • • 4

wave can be obtained from the well-known ~xpression, dH = - a/g dv

for waterhammet~.

{vii} Gaps bebwen the adj.acent cylindrical capsules have no appreciable

effect on the waterhammer \'lave.

48

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------------------------LEGEND

0 EXPERH1tlllF\L POli'lTS

280 TllEORETI CN ... CURVt

260

22(1

200

lUi ..

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GO

40

20

0 --·---·-1- _______ ,,___ ______ L_ __ ,

1 2 3 4

VELOCITY IN ft/scc.

FIG'• lr:; p,-,,~ .. s~~v-: !•T ·r V',·IIS!'- Fl (1'.' \1'-Lr"y··v ..., t\L..) Lii\L • 3 Lr. J:-, ,.vd l. \),_ ll

58

LEGEND u in ft/sec.

[J 4

320 0 3

6 2 0 1

300

2GO

•·--i Vl CL 200 ;;....-_:: ,.._,t

p) lBO ~--

100

80

60

40 ·---~--·----L ------1 __ .1 2------- __ _l -~---- -------

NUMBER OF CYLINDERS . -

0

Fl G.

...------------------·--- ·-------------LEGEND

160

[J I!

150 0 3

6 2 lliO 'rl 0 1 ~

130

120

\ 110

t-c tf)

100 p_ -z l--4

·-t-· 90 -

80

70

6()

50

40

30 ---~

20 . --o--

__ .J_ ____ . .. ·~-J

0 2 3

F·c· 17 fl'l·--ss···lJ'''': ~.-r -- \I[D~u~ f}'l'.">':n OF c·~rF~Ul cs· l l. • Lt. ' L h ll 1\:-, 2l II Li! ") t f\ . .f" " ,_ L

(0 g r:~ ll~r-·~,..r··r, C"LI''r·,~·-·r-) • ::; Ill , l, '.I',L. t. i\ i I; ;;:_ t()

320

300

2f\O

260

2/fO

220

200 .. _, V) CL

;:.:_: 180 ,_, (•")

1-

160

140

1?.0

100

80

60

40

LtGEND

[J 4

0 3 6 2

0

U ·L, II• ft/sec.

---0-----D---

60

-----o-----0>--- --o------·0----

--~----·--1·-- I 0 2 __ 4 __ , ______ _1 __ · ----

8 l·!Uf.!8EH OF CYLINDERS

FIG. 18 · ' • ··\1...) rLL.:.J P!!ESSU:<E l1l T3

VEHSUS NLWSEP O·F c·1·<oc·tl· vr

(0. 875 in, DIM·ETEf' CY' nrr1r:·n···) l. l I • ~ ~.~ l \~)

160

1 so

140

130

120

110

100

...... 90 (/) o_

z ..... .-- 80

1--

70

60

50

40

30

20

-Lf GHlll ill ft/ SC:C.

[] 4

0 3

6 2

0 1

---D-·--··-o----o------o-_

-'---------- 2

FIC. 19

·r ·- J OF CYL HiDERS NUI·dt E _ .

'"· :" OF CJWSULtS VERSUS tiLL JEri rJR,,'r:.·s·sLJHE AT T.l )

r - . 'r· [)r n::, .. -·rE:R CYU·i.t." . DI p,,·;t. , (0.875 1n.

320

300

2no

{'()0

21W

220

200

lBO .. ,..>(

VJ 0. ..

.... p

"'- 1GO • ~·-1

CY) 1--

HO

120

100

go

co

11,0

-------------

LEGEi'iD

.875

0 \7 6 0

u in ft/src.

.75 [J 4

------'-----____ j___ ___ _.___ __

0 ~ 8 12

tl!Jf-18 U\ 0 t~ SPfiE RES

FIG. 20 PPESSUf{f: 1\f T3

VEF<SUS flUi-:JEP OF C1WSULES

C875 :111. a.nd 0. 7!i in. DIN•1ETEP SPHEPES)

63

LEC;EtW ·u in ft/scc

rl _.J 4

lGO 0 3

6 2

150 0 1

140

130

120

110 r--, ------··--u·-------0------o-- 0 ---- --·-...-·

100

90 ... ~ (/) ()_

80 z -----i':J-----6---------t..."::s--------6----··-·-~of

...... 1-

70

60 --o-----o--------o---------0--·

50

40

30

20 --------'------'-0 lJ

L--------1:----~ 8 '12

NUI·!llER Of SFlltFES

FIG. 21 PPESSUf:E !IT T l VEPSUS r·HJ:,;;;[p OF: Cf.P:;ULES

(0.875 in. DIA~ETER SPHERES)

(-C:

...----.---------- -----------------------·- I <J CJ ~· ·-l

>-.:- {./) l.!J C~'

.L

~r- --_c, I _______ !_ ___________________________ _

·o

-'·-' , _____ _L ___ _

C>

"' LO N

I .

Cl C'-.J

·--I ~r

OJ

'(, ::-J ( J

Vl

(f) •:_)

: J ~ -' .. ' r--­:-::~) .-. ~{ (/J

c

c •;-

t.;,. T"· ( '

: ( -:..~ ; . ,. ----r

l\ r::_; . ( '

' -·~

,,, l I

r-. r

' : i

I :

I i ( ,_\_

I< . -f- ,,,

. -·· C .. 1· ~ ~- . - .. ~~ :. j :./J C·. <-.:

C:.

;

~

C• t)

v:

t (/'

C•

'· •')

--' I ,-,

I

v. t/)

' /

I

i. i

·! ~ (-) (., . ~-.: (> ·~"'

-~-: ·!· ' <i,)

(!.::. (:!

'),, ;.::: (_; •r~

t ~- I """' . (:j

(..) ~:· . , ·,

r..:J.

(4_:_~ :·)

~~-) ::~j

cr_ l'-..1

------· c.:.

i./} ()

''

·:...:.:. ()

) .. I ...

.,.-i.J ..

(") ~;,_ .. ~; ~ ·:

•,·· ()

·~:~ +) " . ,-.. ,··-.: .

[L

\l'.l

(J 6 -!) , ..

' •'

l __ __l_ ... ' ...... '----- -- l~l ~- r') ('J

L-:; Ul •

C> I- ~,.:; C) c·:)

l !..~

... :;.._

•·-< __ J

LJ

' ' (l..

!•J

'-~ ' (~) .. -_ ,. '

f. V"'~

'

. -~ ' J f'.

'·

l'

t ·: ..

REFERENCES

1. Jouko;•Jsky, N.E., 11 Haterhammer", (translated by 0. Simis),

Proc. Am. Haten·:orks Assoc., p. 335 ( 1904).

2. Allievi, L.,t''!heo:-y of ~·!atethammel~", (translated hy

E. E. 1-1 a 1 mo s ) , ( 192 b) .

3. Hood, F.H.,"The /\pp1ication of Hcaviside Operational

Calculus to the Solution of 1·/aterhammer Problems'', T1·ans.

ASt-'iE, Vol . ~-· Pc:q)er !lyd-59--15, Novembel' ( 1937).

4. R·ich, G.R. ,"Haterhc.:mmer Annlysis by Laplace f··1ellin

Transfot·mationn, Trans. ASi<E (1905).

5. Rich, G.R.,"Hydl'aulic Trans·ient", f·1c-Gra'I'J Hi.ll Book Co.,

Ne~'/ York City, N.Y. ( 1951).

6. Strcetel', V.L. and i·lylie,E.B., 11 HydrcJUlic Trcmsients",

t·k-Gt·m·J Hill Book Co., Nei·t York City, N.Y. (1967).

7. Panna!:·ian, J.,"l·Jater·han~mer J\nalysis", Pre;1tice·-Hclll, Inc.s

~levt York (l%5).

8. Hodgson, G.I·J. and Charles, t~l.E. ,"The Pipeline Flov1 of

Capsules; Part 1: The Concept of Cupsule PipcliniwJ",

Can. J. Chern. Eng., Vol . .1.1_, 43-4!1, (1963).

9. Char·les, 1<1.E., 11 1he Pipeline Fio:: of Capsules; Part 2:

Theoretical Anu.,lys·is of th2 Concentric Flm; of Cylindrical

Forms", Can. J. Cheri1. Eng., Vol . .1_1_ ( 2), 46-51 ( 1963).

10. Ellis, H.S., "The Pipeline Flm·J of Capsules; Part 3: An

Experimental Investigation of the Trans~ort by Water of Single

CylindY'ical and Spherical Capsules with Density Equal to that

of Uater", Can. J. Chern. Eng., Vol. 42_(1), 1-·8 (l9GLi).

66

67

11. Ellis, H.S.,"The Pipeline Flm1 of C:1psules; Part 4: An

Experimental Investigation of the Transport in Water of

Sin~le Cylindt"ical Capsules with Density Greater than that

of Hater". Can. J. Chern. Eng., VoL _:+2(2), 69-76 (1964).

12. El"l-is, H.S.,''The Pipeline Flo':! of Capsules: Patt 5: J\n

Exper·imental Jnvest·igat·ion of the Transpm·t by l·!ater of

Single Sphedcal·Capsules vrith Density Gl'eater than tllat of

Hater", Can. J. Chern. Eng., Vol. 4~(4), 155-161 (1964).

13. Ne\·rton, R., Redbel'9Ci~, P.J. and Round, G.F.,"The Pipeline

Flw of Capsules; Part 6: Nu:n2rical Jl.nalysis of Sorw~

Variables Determinin0 Fre(~ Flo\·J", Can. J. Chern. Eng.,

Vol. ~g_(tl), 168-113 (1964).

14. Ell"is, H.S, Bolt, L.H.,"The Pipel·ine Flow of Capsules;

Pat·t 7: fl.n E>:pet'irnental Invcstigution of the Transpoi't by

Tv:o Oils of Single Tyl"indr·ical and Sphel'"iccil Capsules with

Density Equal to that of the Oil", Can. J. Chern. Eng.,

Vol. ~~(5), 201-210 (1964).

15. Round, G.F. and Bolt, L.H.,"The P·ipeline Fl01v of Capsules;

Part 8: An Expetn·imenta1 Investigat-ion of the Transport in

Oil of Single, Denset-than-·oil Spher-ical and Cylindrical

Capsules", Can. lJ. Chem. Eng., Vol. ~-:}_(4), 197-205 (1965).

16. Kruyer, Jun. Redberget·~ P.J. and Ell-is, I:.S.,"The Pipeline

FlovJ of Capsul::s; Par·t 9: J. Fluid r1echanics, Vol. ~Q(3),

513-531 ( 196/).

68

17. S\·Jaffield, J.A.,''Thc Influence of Bei1ds on Flu-id Transients

Propagated ·in Incompressible Pipe Flovt", Proc., Inst. r~1ech.

En9., Vol..l_?3(29L 603-613 (1968).

18. fvl"il ne-Tho.ns on, L. 1'1. , "Til co ret i cal Hyd r'odynanri cs", 4th ed.

The f·1acf·1i"llan Co., New York, (19GO).

19. Tarantine, F.lJ. and Roule(J, \.-J.T.;"l-Jatci'lJG.mii12t f'1ttentuation

\'lith a Tapered Lj n2", Trans. ASt·~E Papel' No. 68-IH\FE-6,

(1968), (1-ll).

1\PPEtlDI X

Pf:ESSUrH: Tft.I!SIEI:"i HISTOPY IN{' U.PSULE-Ff:EE PIPFLiriE FOP Vf\LVE

Cl(1Slll-H ((-1! 1P'L!:--E-·r; 1-~: LESS "flif'.f·i ''[/~ <:t:CO~'!-!S _ , ... • \ L .·, . . I t . . • 1. 1. c L .. a ...... L . ·'' , . ~ •

a reflected 1·wvc cannot n::tun1 to th~ soleno·id val'/2 bofol'e the

v0lvc rnot"ion is ccr:~p1etcd) max·i1~:um presstwe chc-:noe occurl'iw! at

the valve is thr: same as if the valve closed instantancou:,ly. For~

any valve closure, v:hich takes plr1c2 in lc?ss thc;n 21-ia secc\nds,

olita"irJec; as follc1·:s. If x2 is the distt'ncc fron the l·kitiw: point

to the p·i 1::a ry ref'! cc t"i c•n s itc, th r \113 vc- triwc l t"iP:c -rn,::1 the

solcnoiC: valve to the l"i:dtir~~ rcint is (L -- x2)/e seccnds. The

tir:~e t'ccwirecJ for the pressLwc \'ave to travE:l ftCi'l the solrrwid

valve to the prir1ary rcflcct·ion s·ite ancr bad to tht: l"ir'itin(l

point is (L + x2);a seconos. If thr total va·lve closure is

conq:leted in thlE T, the clasped tiue fl'Oi:1 the stat't of the

valve mover::ent to the instant of arr'ival of the final incre::1ent,:ll

pressure chanqe at the limitin~r point is T + (L - x2)/a.

Therefore, the limitins: point is located alcH1SJ the ripe \'!here the

direct \i!cve is rnet by the rdlected v:ave. Thtd: is

L - X?. = T + _____ ....:_ a

~-----·---

a

69

In the present investig2tion

1 T = 15 seconc:s

L = 250 ·ft.

a = ~620 ft/sec.

Therefore

x2 -· T5 X 46?0 __ T ___

:: 154 ft.

Pictol'·ial represent2tion of the !·.cad c!JiHl~'CS 2low• the pipelinE.'

fot' valve closm-c cc\:1r•lctcd in less '~:fE·~ ?L/c; scconcls is sl1o1m in

Fiqures 24(a - h). In Figure 2A the ris2 or drop in pressure tlbove

the steady state is inci·icateJ by the dottec: l"fr.c. Initial stoc:.cy

state ve·locity is n~prcscnted by v. 2nc~ the vc·locHy curing the 1

transient v:ave propagotion is rept'esr.ntcd by v. vis ah1ays less

than v. for a valve closina in finite ti~e. 1 -

The pressure rise takes place in a series of cmpl'essicn

v1avcs. As the valve closure sttn·ts the head of i..hc co:npl'ess·ion

~1ave starts prora9ating fl'Om the soleno·icJ valve to the fWir·:ar·y

reflection site.

Fi~JUi'es 2Ll, (a b) shm: the coinprcssion v1ave tl'<we"!lin0.

up to the reflection site and the valvt, up to this time in

the Pl'esent prob-lem, is not fully closed. lhe \'lave reflection

takes place as shovm in Fiqure 2•1 (b) and thrn the reflected 1·1ave

70

gets superimposed on the compression wave. The valve becomes fully

closed before the reflection wave reaches the valve. In the

present case as shown above the reflected wave travels 154 ft.

1 from the reflection site at the instunt. the solenoid valve is

ful"ly closed. So pipeline len9th 250- 154 =96ft. is subjE'Ctecl

to the scm1c: maxir::um pressure rs i·f tlie valve closed instantv.ncously.

Figure 24(c!) sho:ts the pressure ~l"istory at tine 2L/a, \·Jhen

the head of the l'eflccted 1·1ave re,Khes the solenoid valve. /\t

the solenoid v2lve the expansion wave is reflected as an expansion

v;ave and it travels bad in the pipel·ine. The minirnur:1 pressure

reached is the vapor pressure of the liquic' and pressu·te history

for this case is represented by F·i~ure 24 (c-f). The head of

the 1·1a.vc cJets renee ted fi'Onl the pr·i1:wry reflection site and

travels alor1~1 the pipe. .'At t-ime 4L/a the v,rhole p·ipel ine

reaches the~ sar11e conditions as they \·Jere at time t "' 0. The

cycle is repeated in the same fashion and the pressure drops in

subsequent cycles and reaches the finc:·l ste0dy state: condHions.

i'l

c::

r:c

-·

r I

fU

I!

I >

ti C 1 0 < l

II J: C;

1: ;;: ....

( ij :: -·

t (j) - ro t'

• . J -, {'"; _ J r:1

'-J ~

v .. v ( '-J

f1.l cv ""-, 1·- •r-! - :.~ :

>t y --1

r ;:.-,. • J I "';;:.

<'i ' \.]

rv I -- I I f1.l ,. - V> ( .... J

;~~ t· • ..J f- - : .'·,

"' l i . .l 1';.\ .. _ J

I I '

I t.:...: I { ~

LJ QJ '+- c;. ' "" ~ '/; c . I

.-::z: (._)

C.:. ! !J ..... -c :;;..;.><

Cl LU . _J

~ > -LU -~ ~ . . I •::( 0 ::--> ( . t -'i C )

c:l t ; r- r·· - - - ,- -··

I ,. (.(:

I C.J

\ 1- -(/)

·-~ <

\ \ Cj

1- - I ··-w

\ ~')

0 -<

l i \

r ,

~-> t r •

I :>

tU \ ~.~ .....

A _ J

I N \ (/")

f1.l >I > { .

(/)

i 1. '

\ (·,_, t . ~

ro 1- - {)_

' - -1 _ J \ \' r .... :

:.::: -0 J; n;l 10 " N t-·' t-:,S \ -1-- -- l- _, \ f -· (..) J <!)

LL1 \ ·-- t

_ J

A y \ >~, \ LL

u... i...LJ \ 0.: LtJ \ f-- .,..

:;- ...... > f1.l \ C(; Vl -~

-'

t .-_-,: \ :._..._ l .

' ·o

c·:: rv .0 \ u 0..