",.:~. SyslI:m C'alc(~I;\ti()n aLld pump or ran selcI:tion 22 4.4. Tolerance on calcubllco values 26 -1-.5. Iil!ilaHed pump capacity 26

5. / Inlernal Flows 21' 5.1. ILltl'uductioll 27 5.2. Straight pipes and passages 27

5.2.1. introduClion 27 5.2.2. I'"low dcvelopment 28 5.2.3. Circul~r cronscelion, 29 5.2.'1. Nuncircular cros~ ~eetions 29 5.2.5. Surfaec roughness 32

5.3. Strai){ht wall diITuscrs 34 5.3.1. Ddinitions S4-5.3.2. flo II' ch~rllctcrislies 5G 5.S.3. SIal! aud outlet tondilionl 37 5.S.4. Inlet conditions 38 5.3.5. Cronseclional shape 39 5.3.G. Melhoth of iml'roving petCormllncc 39 5.3.7. Pre.C'ntation of diffuler pcrformillce 40 5.3.8. 1110w 'tallililY 42

5'+. Turning how 43 5.4.1. Introduction 43 5.4.2. St'ClJlII.h.ry nnw. 43 5.4.3. c.:orY;l.lure effectl on tur1l1.llenee 45 5..1..1. [)istrihotio;\ of louu beLween 11 bend and lu outle~ ip-5.4.5,. ShurL oullel pipes or poassagcs 47 . 5.4.6. Cronstelional shape 49 . 5.4.7. lltnd "c:rfClr'manec ehartl and Reynolds "u.mber eHceti 49

5.5.' Interactions between bends 50 5.G. Cmnbined 111r1linH .~nd dirfusing llow 51 5.7. Combining anc.l dividing now 55

5.7.1. Inlrodoclion 55 5.7.2. Combiulllg'T' junctions 55 5.7.!!. Diviliing 'T' jUnclionii 57 5.7.4. Manifold.!i9

5.S. Abrupt c11angcs in area and direction 61 5.9. Laminar now 137

. ,. .

6. Cavitation 70

lIiii

G.I. Inll'tl

11.1. Introduction 165 11.2. Inlct conditions and Rcynolds numbcrs 165 11.3. Conica! diffuscrs 167

11.3.1. Frcc dischargc - Rcynolds numbcr lo6 and thick inlct boundary laycr (clars 1) 169 11.3.2. Frcc dirchargc - Rcynolds numbcr o f l o 6 and thin inlct boundary laycr (claas 1) 169 11.3.3. Long outlct pipc - Rcynolds numhcr o f l o6 a n d a thick inlct boundary layer (clirs 1) 169 11.3.4. Long outlct pipc - thin inlct boundary laycr (clars 2) 170 11.3.5. Short outlct pipc - O t o 4 diamctcrs (claas 2) 170 11.3.6. Examplc 171

1 1.4. Rectangular diffitscrs 172 11.4.1. Frcc dischargc - Rcynolds numbcr l o6 and thick inlet boundary laycr

(clars 1 for A S >0.5) 173 11.4.2. Outlct passagc 173 11.4.3. Asymmctric diffusion 173

11.5. Annular diffuscrs (class 3) 174 11.6. Curvcd wall diffuscrs (class 2) 174

11.6.1. Diffuscrs witli a frcc dirchargc'175 11.6.2. Diffuscrs with an outlct pipc or parsagc 175

11.7. Croppcd diffusers 176 11.7.1. Examplc 176

11.8. Circular t o rectangular diffusing transitions 177 11.9. Vancd Diffuscrs (class 2) 178

11.9.2. Examplc 180 S1 U@ ,CQS.

*,0n~UL 12. Cornbined Turning and Diffusing Flow 183

12.1. Introcluctioti 183 12.1.1. Rcynolds numbcr (clars 3) 184

12.2. Dcsign rccommcndations for curvcd diffuscrs 185 12.2.1 Baris for dcsigns 185 12.2.2. Minimum Icngths and loss cocfficicnts (clara 2) 185 12.2.3. Diffuscr arrangcmcnts 187

12.3. 9O0 bcnd-spaccr-diffuscr combinations (class 1) 189 12.4. Effcct o i hend radius ratio (class 2) 198 12.5. Eficct o bcnd anglc (class 2) 200 12.6. Effcct o i inlct boundary Iaycr thickncss (class 2) 203 12.7. Effcct of aspcct ratio (class 3) 203 12.8. Diffuscr-spaccr-hcnd combinations (class 2) 208 12.9. Compositc turning diffuscrs (class 2) 210 12.10. Curvcd wall diffuscrs 212 12.1'1. Corrcctions for bciid-difluscr and diffuscr-bend'intcractions 216'

12.11.1. Dcnd-,pacer-dlffurcr lntcraction corrcctlon f a c t o ~ r Cb.d (class 2) 216 12.1 1.2. Diffuscr-spaccr-bcnd intcraction corrcction factors Cd.b (class 2) 218

13. Dividing and Cornbining Flow 220 13.1. lntroduction 220

13.1.1. Rcynolds numbcr (class 3) 221 13.1.2. Cross-scctional rhapc (clars 2) 221

13.2. Sliarp-cdged combining 'T's (class 1) 221 13.2.1. lnlct and outlct condiiions (clasr 2) 228

13.3. Effcct of radii on combining 'T' cocfficients (class 2) 228 13.4. Symmctrical combining junctioris 230 13.5. Sharp-cdgcd dividing "i"s (class 2) 233

13.5.1. lnlct and outlct conditions (clasa 2) 233 13.5.2. Examplc 234

13.6. I.:I'I'cci oI' raclii oii ilividiiip "1" c i icf l ic ie i i~s 237 13.7. l inp r t~vcd pcrlr>riiiaiicc dividiiig "r's (class 2) 237

. . 13.8. Syniinciric:il diuidiiig juiict ions 237 . . , ., , . . . 13.3. 4-\v;iy dividiiig juiiciioii (class 2) 239

13.10. C:Ticct ol crt>ss-scctiotinl shiipe (i:lass 2) 240 ' : 13.1 1. Loss cocrficieiiis for Iiolcs iii pipe wails (class 2) 246

. . . . . . 13.1 1.1. Suriioii flow inio pipc 246 j . . . . 13.1 1.2. Dirchiir~c from a pipr 246

. .

13. I Y . hI;iiiiI'oIyriiiiIi sciils 2 8 5 : . . . : . i . : . . . ; . . . . . 15.8. ' F l o ~ i i i ~ ~ u r c i i i c i i i 285

. . . . . . . . .

15.9. "Fhw ilirougli tube 1)undles 285 . . 1 . . . t . ~. i.

15.10. Adrdiioii 2 8 5 J '

x;; i 1

-

.. ,

. , .., . .... 1 , , . . , . . ,

Symbols, Non -dimensional Ratios and Pressure Definitions

Flow distribution parameters In fliiid mcchanics thc practicc is to reduce al1 gcomctric and llow paramctcrs t o non- dimensional lorm by thc sclcction of appropriatc and gcncrally acccptcd non-dimcnsionalising lactors. I r is oiily wlicn a calculation lor a p:irticiilar flow situation is rcqiiircd that dimensional quantitics are involvcd.

Prcscnt undcrstanding o l intcrnal flows is not sufficient to dcvclop non-dimensional paramctcrs wliicli describe flow distributions iii such a manner tha t thcy can be uscd lor componciit pcrlormancc prcdictions. ln ordcr t o havc somc gcncral and 'cnginccringly' uscful dcscriptions o l flow distril~utiOns thc lollo\ving nrc uscd in tlic tcxt:

Tliin boiindary laycr - A flow which is ncarly Lnc-dimensional. such as thai obscrvcd in the lirst diametcr r ~ l pipc lollowing a smooth contraction.

Tliick Boundary 1;iycr - A non-swirling flow which has a vclocity and turbulcncc distri- b i~ t ion appropriatc to a pipc flow 10diatiietcrs alter a component siich as a contraction, orilicc platc, suridcn cxpansion, scrccn or otlicr ricvicc which dcics no t crcatc strong swirl.

Devclopcd flow . A flow wliich h a trivcrscd 3 0 or morc diamctcrs o l straigli! pipc.

Symbols Thc gcncral practicc is t o define symbols as they occur in cach cliaptcr. 0:afy thc principlc symbols are listcd bclow.

A cross-scctional arca A constant a prcssurc w;ivc vclocity B constant b width or brcadih C corrcction factor C constant Cp prcssiirc rccovcry

c~>cll icicnt D hydraiilic diamctcr

(4 AlJ'J rl pipc diamctcr / I'rictioii coclliciciit g gravity II total hcad II* non-dimcrisional hc:i

Pressure definitions . . . .

. .

Ilydi-aulic cnginccrs oftcn ded witli fluids flowing from one level to another, for eiainplefrom; oiie rescrvoir i o uioilirr. In dcscribing systcin performance i t is appropriate to use differcnces' in lcvcls iir Iiciiils cxpi-csscd iii meires of fluid. Low speed i r aiid gas lows are usually. I I I L . : L S L I ~ C ~ usiiig !iiaiii~iriciers which register pressures e. a displaccment of a fluid column. For ilirsr illid titlier reuoiis it Iias become common, in'certain branches of luid mechanics, t o use ~ I i c reriii toi:il 1ic:icl :ind vclocity Iiead in preferente to total prcssurc w d velocity prcssure. 'Cliis p rx i i ce is followed i r i tlic presciit text as it reduces the monotonous use of pressure terms :uid folli)ws ilie coiivciitioii adopted witli SI units of continuiiig, whcrc appropriate, t o rxpress . prcssurcs i i i lcriiis o " ~ T C S S U I < : hcads". l t sliould always bc remeinbcred that although heads. are iakcii as ilic Iiciglii o l a licluid colunin tlie units of head are those of pressure, N/m2 (Iieight

liquid c~luii i i i (in) X density (kgjin3) X gravity (in/s".): . . . siaiic prcssui-c - thr prcssure acting equaily in ail directions a t a point in a fluid. velocity prcssurc - givrri by 1 / 2 p U , d s o called !he kinetnatic pressure., total prcssiire - tlie sum o the sedtic .and velocity pressures. ioral pi-cssurc loss - tlic differeiice in total prrssures betwccn two points related to a '

commoii daturn. As al1 prcssurc lossrsin ihe text are total prcssure losscs the words t o t d 2nd presk"'re are often dropped.

~ : i ~ i g ~ p r c s u ~ - ilie pressurr above or below local atinoipheric pressure, if the gauge pressure is lcss than-a~mos~her ic prcssure it is aisp cailed tlir vacuum pressure.

:il)solulc pressurc . ilie pressurr above zero. given b y the sum of the locd atmospheric prrssure md tlle gaugc pressure. , .

atmosphcl-ic (baronictric) ~>rcssurc - tlie local pressure measuredwi tha . . . . barometer. siaiidard aiiiiosplicric ; , , . I

. . . . . .

prcssurc - cqiiiil to 101.325 k~ / i i i " vapour ~ i ~ e s s i ~ r e - ihe pressure cxertrd whcii a liquid is in cquilibriuni with its own

vapour. piezi>inciric o r Iiydr.iulic - tlie Iicad &ovc a daturn t o which fluid rises in a tube connccted Iicad to a tapping in a pipe p r passage, or the water level in a reservoir. velocity Iieacl - givcn by U / 2 g !I

' i towl heiid - the sum of the piezometric and velocity hcad. ioial 1ic;id loss - ihe diffrrence in total head betwecn two points related t o a

coninion daturn. As al1 hcad losses in the t rx t are total head j/ losscs the words total w d head are often dropprd. !! lnnnp Iicad - Iicad ycneraird by a yump given b y thc picromrtric head

diffrrencc across tlie pump plus the diffcrence in velocity heads ,

betwecii outlct and inlct. !! ' v i ~ ~ > ~ i u r l i c x l - tlir hcad iri luid exerrcd when a liquid is in equilibrium with its

own vapour.

1. lntroduction

'E: g E...

1.1. TYPES O F FLOW CONSIDERED

Intcrnal flow is conccrned with fluids flowing in pipcs, passagcs, ducts, conduits, ciilvcrts. tuiincls and components such ;L$ bcnds, diffiisers and hcat cxchangcrs. In tlic tcxt, "pipc" is iiscd for flow through circularcross-scctions and "passage" if the cross-scction is non-circular. Conlincd in pipcs m d passagcs most liquids and gasses bchave in a similar manncr so thc gcncral term fluid is appropriate and uscd throiighout thc text.

Exccpt for Chaptcrs G and 7, o n cavitation and transielit ilow, tlie prcscnt work is rcstrictcd to steady flow of a single phasc near Ncwtonian fluid. A Newtbnian fluid is charactcrised b y shc;ir stress bcing proportional t o strain with the constant of proportionality being the absolutc viscosity. Gencrally fully turbulcnt flows arc assumcd except in discussion o n important aspccts o l low Rcynolds numbcr flows. Tiirbulent single phase intcrnal flows are o f coiisidcrablc cnginccring intcrcst but are thc cxception in naturc. For instancc tlic most common forin of pump, thc hcart, opcratcs cyclically in thc laminar to tiirl>ulcnt tr;insition rcgion, pumping a complcx fliiid with living cclls and solids i i i sus\>cnsion, whilst tlic flow of niitricnts ir1 pl;ints occurs wliolly in thc 1;iminar rcgion.

In practicc thc flow of blo

Ilow II:IS ira\~ersctI iliii-ty 01. 1111>re pipc diaiiiricrs. Loss c i i i i s :uirl c;~Icu1aiio11~ o pressurc losses in thc tcxt are for incompressible

Ilow. 'i'liis is i i c i rcs~i-iciion as rcgards liq~iids which havc prcssure wdve velocitics (vclocity oT souiid) o ~ I i c ordrr u l 1000 m/s cornpared t o typical flow velocities of undcr 10 m/s - ecli~ivalciii io hlacli ii~iniber oT 0.0 l . For gasscs the pcrlormancc data can bc uscd directly for Ilo\vs al h.l~cli iiuiiihcr lcss ilim 0.2 . rquivslciit to 70 m/s for air at no rmd temperaiurc and prcssure. I T tlciisicy cliaiigcs arc signiicaiit in loiig systcms, incomprcssiblc cdculations can be inildc by dividiiig up ihc systcin iind using inean densitics and pressures in diffcrent parts of ilic sysiciii. I'riividc