pipe flow book donald.pdf

8/22/2019 Pipe Flow Book Donald.pdf

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11EXPANSIONS

Flow through '' sudde n or ab rupl expansion in a pipingsys1cm (see f ig. 11.1) gives rise to an increase in staticp ressure :;11 lbc cxpcase o f : ; ~ drop in kinc1ic energy. A-potential" core rorros in the expanded Initiallythe ac;bieved :t dist:tnce ratios US on lhc orde r of 12-16.

Using a divergent connecting passage.or ton1:1kc the trM .silion from :t p:t.ssagc of .snwllcr crt:m.ssec{ ion to a passage of larger cross section can substau-tia ll)' reduce cxpunsiun losses. The p rimury purpo.-;c ofadiffuser is to CO R\'ert kine ticenergy ornow (o r dynamichc.ad) into Static pressure (o r static head) with minimumloss of to tnl pressure. Much dal:'l, iotim:;a tely rc i:Hcd tothe prestn(e or absence or llow separation . or stall. are

; ~ v a i a b l e in the literature on tbe performance anddes ignofstraight.two-.rcssure. or the loss coefficient. of th ree-dimensional diffusers., with ( ) r wi lho ut appreciableThe in fonMltjon prese tHed in this chapter is basedon im.x)mpressibh: tluw. The informatiun is b a . . ~ c d unsymmetrictl l inlet conditions be1ween the ex tremes oruniform tmd or fully developed turbule-nt Rowac rhc in lcl to chc > = ' ~ m m e reasonablelength of dow nst ream Stl'aight 1>pe. Data on a number,,r inlet anti o utlet Rnw c()ntlition.s nHI \ be fnuntl in theliterature (see Ihe section ..Further R e.ad ing . at the endof this chapte-r) .

A sudden :txh;ymmctric expansion is shown in FigureL .2. 11te energy, momen tum . attd continuity equationsare applied to predict > . ~ s c s through the sudden cxpimsion.Whi le focus is d irected to (l single ci rcul;:.r p;:.ssag,c.th is treatmen t is general and app lies to both single andmulliplc passngc CX I)ans i


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II.& EXPA.SSIOSS

'"--""- ....- ~ ....c-..-=-=- = _ ~ - ~ ~ ~ ~ - - r- , , - - - - - ~ ~ ~ ---o,

H GURE U.l. SuddC"n c x p l l n ~ i u n

f ------ - - ' ~~ ,. ' ~ ..' ,, :'-' ~- - - - - - - - - - - - - - - - - - djt'IGVRE 11.2. Sudden cxpunsion.

In J')l':tclicc, the \'Ciocily discriblHiOil d o m uni fonnor flat 0\CI the (TOSS sec.-tion. fo r the mo1nen1. assumeuniform ~ c k . K ' I I y profile a1 rhe 1nle1 and c:dt of theroncrol \Oiume. The n e ~ balance is &I\ en by:

P, Pz. Vl- +-=-+-+ 11,.p. Zg p. 2gSoiYi ng rothe head loss f l, g.i\cs:

( 11.2)

Ntbselt ji)J>rO\ed expcrimenlally !hat f01 subsoic Rowthe prc. . ~ u r c on the o w n s t r c ~ m fa: of the enlargementis equaJ lo tbe S-talic pressure in I he .s1renm ;ust prior tonpaMiun. Auumin! lhal lhc h ) ' l i r c ~ t ~ o t a t l c pt."Murcs: P 1and p. Me C:\cnly distributed O\'er 1hc unl fell' lhc effect of ' 'a ri mLs ax isymmetric (n r nununifor m) inlc1Yc locily dis lributions on diffu11cr lo-.s (sec"f'unher Jtcn \ling . .sec:lion).

Th e ll(l l'da - (."nmol cqunrion eanno1 be 1li>Piicd wilhaccuracy 10 e o m p e ~ b l e Oow where the Mac.h numberat th e nltl ISgreater than aboul 0.2.1lenedict c1 al. (lJ&he gtnenlized ana.l)1ical solutions fo r inrontpress1ble.

s u ~ n 1 c , and upen.ritcal flow across an ahrupl tn largement. Bened1ct ct 0'11. also presem expe:nmcntal

\ ' ~ r i f i c : a t i O n o ( the,: SOlutiOn. in duding h : - ~ b IR\'OI\'Inghgh bca m110:t.

l 1.2 STRA I(; HT CONICAL OIFfUSE IIA diffu,.cr gr.adu;lly expanding section lh:"ll i : l : - . . ~ n g e to alarxerone I I ( t--hown in F1gurc IJ 3.l'hc pnm:.ry pufJlOSc lk lnb d .t ... c ~ d ) dlr'.d lllb suddca npaa,:xMI ~ ~ ~ 1 f t h8paper. flloiMr. he' d t l um-d lfw: IQ t it tJtrc i l l ,. ' t .wrpd

c s ~ l wilh bquN l i i J W f ~ ._.Poe ia til


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, , a J, -' ,,J' JGU RE 11.3. Straight conknJ dlffutoc:r.

of Ia diffuser IS IO ft'CO\'Cf flu1d ~ l l l l k : pti!'"SllfC '\\rithnuninl:'llloss of total pressure bile redudn' the flowvt"loatyTbe mcreast in the crOSSstl.'1ional area of thc:dlffU5er causes a drop in th e aHrage fto "doci1y. anda J>OI110n of kinetic ene-rgy of the now '" r o n H ~ r t t " dinto the pn lcnti:.l energy of prc!.'surc. An eflicicm diffusc:r i;; one t ha i con,etts the lughcst pito.sihlc petctntngc of ki ncuc energy into p ressure energy within a giventi n1it lt tion on diffuser ength I or divcrgcnee ( 01indu ded)nnglv r;..1'hc pcrfornwncc level of : difruscr intimatelyrel.au:d to 1hc presence Of absem:c of now se para tion(or $l:l ll).ltcgion.s ofst:dlcd ftow in l!lffu.."cr block: thenow. caus(' I0\1. pressure ret."'\'ery. and 111:'1)' result i.o$\C:fC ftC)W Mymmrtry. SC\'CfC' Un.\ICIIdlnCSS. M both .Consequenll).much susdy bas centered o n the preseaceor ~ n t - e of !>tall. ratbCT than diredhon the mportantde"'' " consideration. :u least '" Ppin ' design, ofrnn:.:imum press ure recover)'.The mzin geometric con:-;dcmt iu no: (l f cu ni CJt l diffu:ulence prodm . i : l l l o s , ~ c ~ grcntcr than fur a d t . l c n l tmenl. Where, in the desig n of bydn,ulic passa!!i!S. i l is

e c ~ a r y fur thc"c va lues u( diffuser onglc to beexceeded. a sudden enlargement or seclion will &i"c amOrC' trfic.icnt and 3otUd) transfonnauon or tncrsy thanv.ill:. comcal 1 f r u ~ rThe loss coefficient of ronicaJ diffusers dCJ}ends onmany paramctm hc!iidel) d i v c r ~ n c e ang.lc a :md be taratio /J. H on the bouodary layer thi cknc.;s atthe entrance: the lihllpe of the \elocit)' profile the

n t o c c ~ the degree or now turbulence at the c n t r n n c c ~the Row IC!tirue: t111d the length of .struig.ht duwnst ream pipe.A thicke1boundary Iaye t at the cntm ncc 10 tlt diffu :o;cr tend.!! w 111(1'Cil$C the h>..'>:'i codficicnt. Nununrform\'Ciocily profile nt the cntra.nce. parti4!ularly if It is disIOrttd. c.:an c : m ~ c earlier onset O( fluw Scparahon frumth e all 11nd creatly increase th e loss c()(!'maent. Acon' ex or pointed \tlocity protilt. such as in laminarOow, w1th ml\tmum ' 'ekJcity at the ctntcr and reduced,eiOI.iliC!i nt th e wa lls. agsravates lhe onliet and theextent uf Oow M:flllfBtiOn .Swirl {or tnngenti:.l ro tation or now) IS !IOMCL illlCSpresent ln corlie11 diffusers as a re.s uh of l'lUallnp.m:.chincl')'. or l o s c c o u p d bows or bend!;. Sw irl hn!'l.iulc d fcl.)l on the p o ~ ; s i h l c recover)': the longer length is l'equl rc llat higher di ffuse r flrl!}les. Desisn m tluH mn)'imp rQ\'C pcrform:mcc arc lhc usc (I ( 'IICilpcdand two -stage diffusers. These mu lthlagt dif

f u ~ c n ; :arc trcmed in the next scdiun.A great deal of data on Row in dherging paJSagcshas be.:n am'b..-.cd m the tOO yean.. l u c h of d:11ab:we been on twodimensional Md rec::umguiM i f r u ~ c "and mosl data hn\e ~ n on diffuse-rS whic.h dsch:trgeinto 11 J:u-gc plenum (free diseharg_c). Some d:ua are


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116 EXPANSIONSava ila ble Cor 1:onil'".al diffusets whi(h ac t sole.ly as expansio n.:: be tween constant :m.-n circu h1r p a s s n g ( . ~ T hc:;cdata at e developed here because it ap plies to pipittgsystem di ffuser applications.In Gibson's c b . 1 s s i ~ l (4 ,5] . conica l di ffus t.rs with ups tream an d down$trt.in(a/ 2) ( I 1.7)Equal on l l J was deve lopc.d fot coMergi ng Row. bu t i1sccm 1) to work just llli we ll for dive rging llow. In thisc:nsc.the friction fa cto r }'is the o rdinary friJ'' -0. 170-

3.28(0 .0625-,B')J"-20"}(1-,6' )' + f(I-{J'') ,4(1 Ssin(a / 2)(20's as 60')(0 /1 ~ n K w:clh. frk tiun ~ l l m < at tht: di !fusn ink ill

hQm o.mo to 0.1)2() were then e ~ ~ t i m : d fmnl tlk MQOd)Dia._.u am,


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MULTISTAGE. CON ICAL DIFFUSERS JL9TAULE II . I. t oss Codlld\ 'nt K,.,, ((lr Opdonu:u Sttl )p'->d Dlfluser.i u 1 p a r ~ d 1(1 l. .t)SS Coefticl


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uo EXPANSIONSTABLE U.2. Oin .rgt!lllc:e AJgle (l.,., fur Optinn.un Oitfusc:r-1! C.m1mred 11 O c m ~ ;\ngle ( t fo r t r ~ t i g b t ( (m.iculOirfuSt.-"" ( I ( E1nul Lt.'flglla urul O''L"'"all Arc:a R11fi(l

A reo Hatio If 0.5 1.0 2.0 3.0 4.00.00 180 180 180 IS() 180

II. / l .U 12.4 11.20.05 147.9 120.1 8 1.9 60.111.5 13.4 11.4 II./0 .10 130.4 94.5 56.8 39.6

II .Y 13.5 11.3 11.90.15 115.4 76.7 4J.2 29.511.3 13.7 /1J 11.80.20 102.0 63.J 34.3 23.312.7 J.i.S 12.1 11.50.25 90.0 .:B. I 28.1 Ht9

13.2 1.1.9 11 .0 11.2OJO 79.1 44 .9 2..1.3 15 .1

13.6 14.0 11.6 9 o.J s 69.2 38.1 19.6 13.1J.l,l) /4 .0 11.3 Q,.'l0.40 60 .1 J2.4 16.5 11.1

14.4 1.1.9 11.8 8.70.45 52.J 27.6 14.0 9_)

14.7 13.7 112 8.10.50 45.0 2>.4 I I.$ 7.91.5.(} 13.3 9.5 730.55 38.4 11.0 6.6

15.1 /Z ,7 8.7 6.50.60 .l2 . WAlL l l lf l'US!; I{I t wou ld appear that ! l trumpct-slwpcd pw-.sngc maywell give mi nimum loss for a given length nnd given


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. /... .1 .. "., 1 . ..:.... .,A, ,,/1 ! ' I ~ ~ ........_

IIIGU HE U.S. (..'urved woll diffuicr.

r:1110 of n l ~ r g c m e n (sec Fig. I UH. A d1ffu:Jer in whichthe r e ~ u r e gradient rernaiosoonstarttalons he passagt(dp 'd.r CQn,lant) ma y be the bcJt chOtec.ldel'chik (61 presents an equation for dp/d.f =tfUH fu r th e boundary "all uf a cuf\'cd wnll dtf(u.ser of:1 circular (o r cross section.

y y,~ + ((y, I y, ) ' - I .

PIPE R E O U C E R ; E . X P A . ~ O ~ G Il lTh e t:uru:d \loall difrust.'T appears to be generaU) mo rerffie1cn1 than the: ~ u d d c n expansion and th e !'INUC.hlco nical dirruser. Jt owe\er. except at high atea ratios.it d oe s no t 1ppc11r tn be as effective as thediffuser. Th cl'c OO tlsiderab le uoceru,mty ~ s o dwilh llll' cnlculnted luss coemcients i n Table l l.: t J'"utur ttest s :uul cv:lluntlon" could ch:ngc the rc l!i ult.s. B \'Cilthell. l lc curvod wull diffuser may no t be a vini>ICcho icebcc;n1,t;c 1he hn p ru\emcnt m:y be slight und no t worthlh e ex tr:1 cfrol'l in volvcd in designing and fa bricntng th ec urved wnll,

U.S PIP J; REDUCER: EXPANDt'IGSta ndard bunweld p 1 ~ fiumg.s. ANSI' reducc:rJ'.. arc:used to join pipe sections of diiTcrenl d m m e t e ~ (fig.11.9). l)ptcnlly. th e fi thngs are g e n c r o tuundtd IUthe in te rsection of tho co nica l an d cylind rical !'urfacc..'\,(u 1he e n ~ ~ . : of u t r m re ducers (Stl' Sectio n 10.7).rounding c 1 U i y red uces e ne rgy loss thmugh the lhling.1-1owever. rOlHI(Iing, has liu le e ffe(:t in de


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12:! EXPAN SION STAULE 11.3. m p a . - . t i ~ Effe


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( the htst tetm) in Eq uatio ns I I.S-11.10. as appt'Op riatc.Surfncc fri ction fnctor:> f, :md J! at d 1 11nd assumi nga surface rough ess of 0.00015 f l . we re assign ed usingvon K 2- 1/2 0 .77 22 X 16 0.172 X 1114 0.16 8x6 6 0 .1 5 22 )( 14 0.3.12 x I 0 .49 8x5 0.40 x21 20 0.002 x 3!4 0.67 S x4 0.(.7 2.! X 20 0.0.12112 x2 3.5 0.03 Sx: l-112 0.73 24 x iR 0.151112 )( J. J/1 0.19 O x 7 0 .10 24 X 16 O lO2 112 X 0.38 10 )( 6 0.47 26 x 24 24 0.0 12 1/2 X I 0.6 .1 lOx 5 0.67 2(1 ( ll 0.033 x2 112 3.5 0.05 lOx 4 0.76 26 x10 0.103


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0.35

0,30

0.25

0.20K,

0. 15

0. 10

0.05

0

K,

0

t I-I $'Equation 11 .8Id,it l" )a I - $:::::::> ::(: .f 0.03 .~ . .

! ~ \ I I I I I / ~ / JA- Rtio /32 = 0 -1- . ::(: +r . 0.1 7 v.! ~ '0 t03'' I ' 0.4' . . ..eft . 0 .-I- I I 11- 0.6 'T'!t - 0.7 0.8 T T TI -:0 91=I ' '0 5 10 15 20

Oi, ergenc:::.e Ang le a {degrees)OIAGR1.M .11.5. Loss e o d f \ l : :1 > n i c : l l l f u s e r - a = 2"' h) (j= 0.03).

0 5

0.3 -0.4 .+ ~ s

I+

1-1

0 .6T+ ++ -1-

10 15Di\ergcnce Ang lea (degrees)

tI'T+I

0.8 lo9J.I20

DIAGRAM 11.6. Loss eodfie ieot \)f 1 OOJlitttl dirfu-ser-a = 2 . tlJ 20" ( j = ().04).


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0.35 I 11 ll -U ii- I-;/-I 0.05 7 I1-- . t-l :f+ ~ -;'' J Area Ratio p- - 0 ...L v- - I- - - 0.10.30

0.25

0.20- - I . 0.2 ..\ ' :--.. I .Y r / I'-- ~ : : s c 0.3~ : s : 0.4 :tr . . +

~ " " I I I I I 0.5::P"i' - =I= 0. >I I I 0.7::+- '

0. 15

0. 10

0.05

~ ! . ! 0.90 0 5 10 15 20

Divergence Anglea - degreesOIAGRA!\1 11.7. I..S$ ,)f:. r o n i e : ~ l Jiffu-s:tr--a = 2 h) 20" if= 0.05).

I1.0

-" .. ~ a [1,rc:aRat io = 1 ~ = 0 l"r-c 0.1 [ = 0.05 ---------/; / I r- om.8 :>!" / / = 0.0 1 .........................

, ff / /0}/r / 0.3.6

K, h.-' '/ V ~ u i a o c . e due ltl# / d " ~ surface tiiction// v: -#'/-"' / 0.4/ / , /_.ffl'

.4/ li" ' // 05.. / / ,__...... / - 0.6/.2

/ ' - 0.7, .. . 0.8(I 20 25 30 40 50 60 70 80 90 120 ISO 180

Oivt gence Ang lea (deg.rees)OlA(; RAM ll.S. Loss coeflldentor a n i < : diffuscr-2()'llo 180'.


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U ~ EXPANSIONSREFERENCESI. Nu!iseh. W . The pre!isure in the annulus of pipe-s with nsuddC1 increase in CfOSS scctkln for high \'clocity ai1llow.

F o r . ~ c l n m g tmf ulcwt:nt d!i!S F1uides pa r ldOriftec:s des Vases. M trmoin:: de l'tt(ri! sl!ibh: 0()w in c')ni-


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McDonald. A. T.. R. W. F\)X. M d R. V. V:\ll Dcwocst i1H:. Efrccts.of w i r l i inl et Oow on . : : ~ u r e r ~ C O \ ' C l ) in cl.mkt) di(.fuscr!\.AIIU'rico" /tr)'l il fl lt' o fAt'I'OJWtttics a1ul Astronllluics) Qumal.9( 10) . 1971. 2014-2 018.

Runchul. A. K Mass tr ansfer in\'estigat io n in turbulen t flowdownst ream of a sudde n e-nlargement of a cin:ular pipefo r high Schmid t numbers. lmemar/t'mal Jmmwl of1 /t : l l l 1 ! 1 ( / l d n N ~ Tr(ln4t.,., 14, 1971 . 781- 79 I.Back. L. 11. and E. J. Roschkc . Shea r-la)cr llow 1t"gimcs and

W ~ 1 v o : 3ud 1"!:lUachment l.::ngths d < H \ ' 1 \ S r ~ : n uol' an abrupl circ.uhuchannc l a n s i o n . / o u m u l o fApplily ofMt.'t'lwnical Engimws. IOJ. 1979. 224-229.

Webb. A. I. C., Ih:ad lo:ss l'r a sudden Xpansion. /tuWirl in "sw irl ing sudden expansion now. Joumal of Pluids ngil l&!riiiJ,:. Tn>n.Utc:l;dnJ ) f t l t e All lf!Jicau Sodet,\' elf Met lumicui t ~ g l e t : n 110. 19RS, 155 - 160.

Sticglmcir. M . C. Tropea. N. Weise and WNit sche:. Expcrimcntlkl ilw.::st igmion of th e flow through axis)mmctrici ' n ~ O n $ , Jcumwl tJ fFluids E n g i r e e e r i t ~ g Tran.sutlitJns (Jf

t h f! Ameri

pipe flow book donald.pdf

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