gas dynamic and turbine machine

7/27/2019 Gas Dynamic and Turbine Machine

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Subject : gas Dynamic and Turbine Machine :Weekly Hours : Theoretical:2 UNITS:5 :5:2:

Tutorial : 1 :1Experimental : 1 :1week Contents

1. Principles of thermodynamics .12. Introduction to compressible flow .23. Isentropic flow .34. Choked Isentropic flow .45. Operation of nozzles at variable pressure

ratios .5

6. Normal shock wave .67. Equations of Normal shock wave .78. Oblique shock wave .89. Flow in constant area duct with friction .9

10. Performance of long ducts at variable pressureratios

.1011. Isothermal flow in long ducts .1112. Flow ducts with heating or cooling .1213. = = .1314. = = 14.15. shock wave with change in stagnation

temperature 15.

16. Aerothermodynamics of turbomachinery 16.17. Physical principlea. equation of motion

b. continuity equationc. momentous equation

: . . .17.

18 Turbine momentum notation 1819. Efficiencies 1920 Flow in rotating blades 2021. Axial flow turbine .2122. Velocity triangles 22.23. Impulse turbine 23.24 Reaction turbine 24.25. = = 25.26 Axial flow compressor 25.27 Radial turbine 27.28. Centrifugal compressor 28.


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ChapterOneFundamentalof Ftuid DynamicsIntroduction:Gas dynamics s a branch of fluid mechanicswhich describe he flow ofcompressibleluid.Fluids vhich howappreciableariationn density sa resurts f theflow - such as gases-are called_ ompressiblefluids. The variation n density is duemainly o variationn pressurend emperature.The flow of a compressibrelu.id s govemed y the first rawof thermodynamics,hichrelates o energybalance,and by the second aw of thermodynamics, hich relatesheatinteraction,and neversibirityro entropy.The flow is alsoa-ffected y both kinetic anJdynamiceffects,whicharedescribed y Newton,s awsof motion.An inertial rameofreference hat is, a frame in *'hich NeMon's laws of motionareappricabre-s genera yused. n addition,he low furfils herequirementfconservationimass.These awsare no t dependent n the properties f particularluid, therefore norder o relate he motion o a particurar luid it-is necessaryo usesubsidiaryaws inaddition o theseundamentalrinciples, suchas heequation fstate or perfecigas.p - p R r . . . . . . . ()Ahhoughhemostobvious pplicationf compressibleluid low theoryare nthedesignofhigh speedaircraft, nd his emains n mportantpplicationo thesubject,acknowledgesfcompressibleluid flow theorys requirld n thedisignandoperatioiofmanydevices ommonry ncounteredn engineeringractice. mong hese pplicationare: 1- GasTurbine:he low in thebalding ndnozzles compressible.2- Steamurbine.Here,oo, he low in thenozzles ndbiadesmustbe reated s

compressible.3- Reciprocatingngines,low of gases hrough he valvesand intake andexhaust.4-6-

Naturalgas ransmissionine.Combustion hambersExplosive.I.1 Conserv ation of MassTheprincipleofconservationf mass,when eferredo asystemffixed identity,simplestateshat the mass f thesystems constanl onsiieran arbitrarycontr6ivolumehrough hich luidstreamsig,l.we wish o derivehe ormof the aw ofconservationfmassas t appliedo thiscontrolvolume. owever,n order o applythe aw, we mustbeginwitha system f fixed dentity, ndsowe "nn"a our ryitl"rias he luid whichsomenstant occupieshecontrol olume.Next, we considerwhat happens uring the succeedingime intervaldt. Bydefinition,hecontrol olumeemainsixed n space,ut hesystemmovesn thegeneral irectionfthestream.line.he woposition f rhe ystemre hownn fig.1by dashedines.For conveniencen analyiis,we consideihree egionof spicedeooted t I,II,III in fig. . At timer thesystem ccupiespacesand iI, andatitme/+d/ it occupiespace and 1L hus, incehemasiof rhesystems conserved,ewrite-


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tn t r + nJr t = f r t na , f r u , *a , . . . . . . . . . . . - . . . . . .2where m11means he mass of ttre fluid in space at time t, and so on . A simplerearrangementhengives.

fr / t+dt - frt t fru!, fru,*adtttThe first term representhe time rateof change f masswithin spaceL But as d/ goes ozerospace.Icoincide ith the contrcl olume, ndso n the imit.

f f i t , * a t - f r t t A .dt ' at \" 'c u lwheremn denoted he instantaneous asswithin the controlvolume.The hird rermmay be wrinen.* l * A*" ldt - /2q:I kd r=l.u,r'tt-,-a _ fa.^., ________-Sa t a t d t ) - '

where6m1,*4lepresenthe amountof mass rossinghe elementary urfacedlo4 duringthe time dr. The ratio \mp*4/dt is called he out going flux of masscross he areadAol,Or the mass ate of flow and s denotedor conveniencey dmoulsimilar easoning ields or inlet.m,, , ,.:, = ldn,. _______________6dIandso the conservationaw may now be expresseds!o".t= a^,.-a,..,for detailedcomputationwe note hatajtany nstant^,.=Ja.""=oo,wheredv is an element fcontrol volume, is the ocalmassdensity fthat element ndthe integral s to be takenover the entirecontrolvolume.d n " " 0 r r 0 o .- = , I Pav = | :-4vdt d t 4v r , d lwith thehelpof fig.1 we may expresshemass ateof flow in the orm.

dmo, _Myt-a _ p(dA-)(V"dt) = pyd4",,fur,*u, -------- 10dt dtwhere p is the local instantaneousmassdensity n the neighbourhoodof dA.,r andy, isthe conesponding ocal instantaneous omponent f velocity normal to dAo",.,with theforgoingexpressionquation maynowwriften.Lp'= [cv.a't,.-nv.a*a formwhich s usually alled he equationfcontinuitv.When the flow is steady, he identityof the fluid within the control;volumechangesontinuously,ut the otalmassemainsonstant r mathematical \api6t s zerofor eachelementof controrvorume Thenequation r state hat the incomingandoutgoingmass ateof flow are dentical.

t 1


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jov"ae,,lor/"ae". I 2For one dimensional teadystate flow equationbecome.P,v,A, prvrA, l 3

fi5.(1) Flow through a contol volume(conrinuieequation)

of 9.2 Onedimensionalf ow

.fig..1.flov,hroughcontrol yolumewith obstacle(momentum quation)

12 for rhe inlet and outletcondition

T


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Example:Ten kglsecof air enters tankof r0m3 n volumewhire2 kg/secs dischargerom thetankasshow n fig. Ifthe temperaturefthe air insidehe ank emainsonrtu-nt t300Ko.Find he rateofpressure ise nside he ank.Solution:Applingcontinuityequationn An| :4r= lpv"dA'- lpy-d,r , 8 r

t A " l ' - r 6 . \et* = 287 300**:68880 pysecOI lUExample:2

A tank I ml in volume containsair at an initial pressure f6 atrrr 606.95 kpa) and aninitial temperatureof25"c. Air is dischargedsothermallyFom the ank at the rate of0.lm3ls.Assuming that thedischargedair has he samedensityas batofthe air in the tanlgfind aa expression or the time rate of changeof density of the air in tl'e tank whatwould be tle rate of pressuredrop i_nhe tank a.fter5 seconds?solution:Applingonrinuiryquation 9k, = lpr/"A,. lpV-d-A^r ' o t

dp1 . 0 ; : - 0 . 1 0or

d pa r : - o ' t o

Separating variables and integrating gives:/ " . \p : p , 2 _ 0 . 1 r :[ _ - t _ l e _ o . r ,\ ^ 1 1 /where subscript I refers to initial conditions in the tank pressure change may beexpressed n terms of density cha-nge ccordingto the relation

P : P R T

dp dpA : R r A : x z ( - 0 . l p )

butp= pRT 6p --6oSO -1 = .(1 --1ot dI

so that:


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-n 1 p, y -!)-- o-0 tr" ' " " R T , '- 0 .1p,s -o ' t

Substitutingumericalalues ives:

#: - O.tX 606.95e-o.r(s)-102.3pa/s

1.2-M omen um conse ation th eorem.The undamentalrinciple f dynamicss Newton's aw of motion,andaccordingto this aw the resultantfforceappliedo a particlewhich may be at restor in motion sequal to the rate of chargeof momentumof the particle in the direction of the resultantforce. Newton's secondaw is vectorrelation.Consider he x-direction we write for thesystem.Tr'=!!hY"l -----------t4d t '

Where he eft hand ide epresenthe algebraicumofthe X-force actingon thesystemduring he ime nterval /,and he ighthandside epresenthe imeof change f thetotalmomentumf thesystemee ig.3.d , . , , , , , ( ^v , ) , , .0 ,+ (nv , )n , * ,(mv, ) , , (mv, ) , r ,*\mt/,)=ff i -----15(mv,)' ,. , - (mV,),, --- as dl goes o zero this term represent he time rate ofdlchange f theX-momentum ithin hecontrol olume.= 9f^r,1""otso hat

a .fr, =fit*v,t,.+Jv.a^.,-v,a.,.or

l o

fi, =fa'!t+ !or.v.at",,!nv,v,a,,--------t7


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Example:3Air flowing isentropicarlyn a nozzre trikesa stationaryblade vhen t leaves henozzle shown n fig. Dererminel- The magnitude f the reaction n the x-directionand in the y-direction eededohold heblade n place.2- The magnitude fthe reaction n thex-direction nd n they-direction f thebrademoves owed the nozz.leat g0m/sec.Solut ion:

/ p , \ t r - t | t , / I \ 0 . / r . 4n : n l ; l : 3 0 8 { , r r / - 2 1 4 . 3 KThe eas vlocity at this scction is obtained from the enerEy equalion:

v2 tr2, , ' 1 , , / 2nt -f -l - - n2 + -t*Therefore:

r r 2 t r |1 - = a P \ t t - t t ) - - i .

: IOOO(3O8rroa*$from which V2: 266.46 m/s. The mass rate of flow is:

. i - p t A r r r - ( r r L \ n , n ,/ 1.5 l .or3x lo j \: | _______;\1_;;;_ l(25 x r0-.X60)\ 2 8 . . - - - , l

: 0.258kr,/sApplying the nroalenturn qua tion o the control volume showngiees:

&- i(Vt, - Y2) : O.258(V3cos30 + ,'2): 0.258(266.a6 os30 + 266.46) 128.28Nand

\ - ,i '(Vt, - V7): O.25a(n sin30 - 0)- 0.258(266.46 in 30) - 34.3?N

Y '-\\


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(b) When ti.l1e lade 6oves toward the nozzle, the relative velocity is 26646+3A:296.46 m/s. The mass sttikinB the blade per ullit time now becomes:/ 296.46 \i :0.258 | ... ^. l= 0.281 E/ s

Fromthe velocirydiagram shown:V1,: 256.'14 r.ls ar,d Vr: 148.23m./s

The momeatum equatioo then gives:&: i(Vz. - Vu \ -0281(256 t4 + 266'46) 149'7

andRr : ;(v3r - vzlr: 0.287(148.23 0) : 42 54N

Example:4An airplane s traveling at a constantspeed f200 m/s. Air entershejetengjne'snletatthe rate of40 kg/s while tlle combustionproductsaredischarged t an erit velocityof600 m/s relative o the airplane.The intakearea s0.3 m2and he exit area 6 nP' Theambient ressures 0.? atm,ald the pressure t the exit s 0.72 am,- alculatehenet:hrustdevelopedby the engine.Assumeuniform steadyconditions t he nletand exitplanesand the properties of the productsof combustion o be lh sameas thoseofair.

Solution:consider he et engineas a controlvolumeas n fig. theair entershe enginewith a speedof 200m,/s.assuminghorizontal flight and neglectinghe momentum f thefuel, the net force opposite to thrust is:Appling momentumequation:

Rr ;(v3r - vzlr: 0.287(148.230) 42.54

since the case s steady state thus mean that 6pl&:0 therefore he momentum quationbecomeF: (pzAz ;Vz , - ( pLA . t mVt )

: l ( O . ' 1 2O . ? ) l . o l 3 l d x 0 . 6 + 4 0 x 6 0 0 1 ( 0 + 4 0 x 2 0 o ): 1 7 , 2 1 5 . 6

Vz- 2Oonl s V: - 600 m/s

- r d d v r , , , t ,I ,= L-; ;* JV,dm., , -V,dm,"

A: ' 0' 6 mrP2 - O-72attn


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1.3TheFirst La theEnergy is conveyeda cross the boundaryof controlvolume n he form of heatand

(Er,*a, En,*a,) (Et , + Ent )

work. Consider he flow through he control olumewith of fig., with the systemdefinedas the materialocccuping he control olumeat time . we consideiwhathappensduring the time interval dt. passingthrough he controlsurfaceare astationarystrut and a rotating shaftattachedo a turbo-machine,erhaps compressoror turbine.The energyequationn a simple ormcanbewritten s ollowing.Q _ d E + 5 Wdt dt dtRateof changeof total energyE:dt

dEdt

dtt

B- work Donebv Shearstresses: hisworkmaybeconvenientlyivided nto twocategoriesi) the work doneby the partof tle shaft nsidehesystem n the partoutsidehe systemtowingo the orquen the otating haft esultingrom the shearstresses.ii) the shearwork doneat theboundariesfthe systemn adjacentluidwhich s in motion.Thereforehe ate hangefrvork anbewrittenas oilow.# =r,* +W"0",,!nvd^.,,+[nrd^,,The otal luidenergy ermasslowe s

dE_Et,*a,-Er,.* lrpr__ 116l_d t d t J d t J d t6dE ,aE, | , t ,a= \ a ) -+ Jeamou,Jeamnd E 1 d e p d v s , Ia,=L a * )edm." ' -edm.Rate of work done.Omitting from our considerationcapiliary,magnetic,and electrical force, thework done during the processes s the resultof normal and shearstressesat themovingboundaries fthe system.A- Work Done by Normal Stresses.Taking the normal stressat theboundary f the system s hehydrostatic ressure.the work done by the systemowing to normal orceat an elementof areadAoul spdA""dx, where d: is the componentof distancemovednormal o dAo,,.BuI dAodxis the volume of the masselementdr27,17,.hichvolumemaybe writeenas v6m11,*4,-The total rate of work done by normalstresses uring he processmay now be setdown,with theaidof the foregoing, s

r\t ._ Jpvbm,,,*o,Jpvdmtt,= [rvd*",,- nd,,,

Total fluid energ)': internalenergy+ kineticenergy potentialenergy


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yze = u + - + g zu = h - p v = h - PpSubstitutehese quationsnto he energy quationesults

Q = r r , . r * . * 1 o e d v , " r t 2 t t 2d t . ea r , . . . sh@, ,L d t . ) { h+V+g4am. , , - f h+L+gz1dm, "

I .4Ihe rqe4d aw of Thermodynamics:In afixed-massystemntropyhangeccurJasresult f irreversibrevents r asaresultof intemction it h heenvironmentn which here s heat ransfer.t dQ , as , | ,1,7=t;l- * J'd,,,* 'd,,"4 *, | 4!!+d spvd.t.trfor steady one dimensionlow. rlnrn ( s , s , ) 2 { }for adiabaticlow dO=0 herefores, -s , )0 or ds>0 for sentropiclow ds=0 and low adiabaticrreversiblelowds>0

Formost roblemn gas ynamics,heassumptionf perfect as aw is sufficientlyin accordwith hepropertiesfreal gases s o bea acciptable.we shalr herefore etdownherehe pecialhermodynamicselationshichaiply to perfect as.1- Eauation fstate:pv = - = Kl = -1 - - - - - - - - - - - - - -- o J {Where.Ts heabsoluteemperarureK1, R is rhegas onstant(ykg.mol.Ko),isthe univenal asconsrantnd s equal o g134.3J&g.mol.d" La ful i, tt "


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molecularweight kgkg.mol. For atmosphericair berw,--.eenand i00 km,i\.{:28.966,hereforeheair gasconstants 287.04Jikg.K'\!hen a perfectgasundergoes thermodynamic rocessbetween o equilibriumstate. r ' ' - " - h , = f ' c n . d r, - u t = t r . c v . d l n a - . f , .

,6u du .Ah. dho = i) , = dT anacP=\ *) e= 7 lor perlect asC p _ C u = 4 1 du _ d ( u + v )_ d u - d T r )' d T d T d t d T d TC p - C v= RThe pecif iceat atio s 7= 94 11.r.1or. p=-fR- andCu= R,C v y - l 7 - 1Changesf Entropy Applyinghe specialelation f a perfect as o the generalrelation etween z,y we get. du pdv ^ dT -dvd s = - + - = C v - + R -T T T vand,upon ntegrations, -.9,=cv nA +.Rnh =cvtn(21.1'.,,uT , , , ' T r ' ' v r 'Altematively,emayeliminateither or v from hisexpresshe aidof pv=RT,andsoobtainS, -S, = 6'v1n : + Cph!! = Crln(P' (" ),P t V t A V tS . - S , = C p l n 2 - R l n 4 = C v l n ( t z l r P z ; t ' tt t Pr l r Pt

f - ' ' l | / - t IlLh)=Cp(r. r,)=Cp,l (;) - t |=Cpr,l"]1, - t IL r r ) l h l

The sentropic. ften he sentropic rocesss taken as a model or as a Iimit for realadiabatic rocesses.f entropy s constant t eachstep of the processes,t followsfrom equation ha*Tand v,p andv, andT andp areconnectedwith eachother duringtheprucessesy the ollowing aws: /+ , - t P T ' - )l v = C O n S I . p v = L = C O n S t - - = C O n S t .p pFor sentropiclow processhe enthalpy hanges important. t is calculatedn termsof the nitial emperaturend hepressureatioas ollows:

l 0


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ChapterT*'o WavePropagationn Compressiblelorv?.1Lntroduction:The erm ompressiblelo$, mplies ar iat ionn densityhroughhe ieldof f lou.These.ar iat lonsre, n manycases,he result r incipal l l , f pressurehangesromone point o anolher. he rateof change f densit l ' \ ' i th respecto pressurgS'therefore.n mportanr arametern-theanal; 's is f compressiblelorv'and,asueshal l ee. r is ciosely.connectedi th the velocit l ,ofpropagat ionf snral l ressure

d is tu rbance ..e . i t h t he e loc i t y f sound .2.2 \'ale Propagationn Elastic4edia:ler us er.mlne rr .hat appenshen a sol id elast icobjectsuchas steelbar issub jec i'do a sudden njformdistr ibuledonrpressivetress ppl!ed t oneend. nrhe irsi nsranr f r inte. thin Jal,er e.r t o thepointof appl icat icns compressecj.uhi le he emainderf the bar s unaffected.hiscompressions hen ransmit tedorhenesr aler .andsoon dor in he bar.Thusa disturbancereated t he ef ts ide ser,entual lr .senselt heopposite nd.ThecontpressionaYe nit ia led t he eft sjdeof rhe bar rakes f ini te ime to trarel to rhe r ight side. he rrare reloci ir beinSdepenienrn heelast iciq 'ncl ensi i l ' f themedia'Gas:s and l iquid also are elast icsubstance nd longitudinal\ 'a\e can bepropagatedhroughhesemedian thesa:re ral ' that \ 'aves ropasatedhrough -ol id 'L. , 'u g., becor l ined n a long ubeu i tha piston t he efthand. hepistons girer la sudcienush o t lre ighr. n the irst nstant layer f gaspi lesupnert o thepls1onand s compressed.he remirderof the -sass unaffected he compresslonrarecreared;, t i re isron henmo'es hroughhegasunt i leventual lyl l hegas s able os.ns.rh. *o, , . i r rent f rhepisron. f the mpulse iven o thegas s inf ini tesimall ; 'small . heuare is cal ied sound 'aveand he resultant ompression\ave mo\erhroughhegas tvelocity qualo hevelocity f sound.Let the pressurehange crosshe u avebe dp andJet he correspondingensityand,.rp.rrrur. changebe dp and dT respectively. he gas into $hich the $'ave ispropagareds assrlmedo be at rest.The *'ave *,ill then inducea gas 'e)ocit; dl'.tenirrJ t as t move hrough he gas.The changes cross he .'are are, herefore ssboru in fig.2.2. n order o analyzehe lorv hrough heuave and hus o deternline(a). i is co-nr.eniento usea coordinated)slem hat s anachedo the$ave. .e, smovingNirh he$'aYe.n thiscoordinate) 'stem,he rvave ' i l l ofcourse e at restand hJgas i1l ef fecr i 'e ly, f lorvhrought * l ih the elocit , , .a. head f the 'aveanda .'elocil'. a-til/, behind he rvare ln this coordinate ystem. hen, the changesthroughie \\ar .e re hortn n f ig 2 3.Thepressure.emperaturenddensity hange'ofcourse.ndependentf thecoordinateysteln sed

i---+-' D::t, - a : = l llF-)--lh ',,"xtI'

l l


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Thecont inuit l ' ndmqm-enfumquat ion reappl iedo a control olumeof un;tarea:cross he \,a\:e s indicatedn fig. For steady tate he conrinuityequarion or thecontrol olumes:nt' - Sn (p + dp)(a- dV) ------------------2.1r 'herem is hemasso*' rateperunit areahroughheu,a'e.Since hecase f a 'e11*eak is being onsider.he second rder erm.dpdl l ihatar isesn equat ionan beneg)ectednd hisequat ionhen. ires:

nd p = ' ; 1 ' - " ' - - - - - - - - - - - - - - ' - - - - - - - llQCons:n i : ; . :of ; t ro; , . rentums nert considei-ed.heonJl, or .ce cl i rgorr heco;t t roi'o iumear: ihe pressureorce.The mor)tentur) tqrar ion or sreacivtatebecorne:p . . 1(p+d t1 .1= , , , ' l@-d t , ) - o ] - - - - - - - - - - - - - - - - - - - 2 . j

u hlch eacio:,4dp= n1'r11to, dp = padl ' - - - - - - - - -2.1Subsr i : u tequa i i on.2 n toequar ion.4s i ves :;- G.T=o . o , = . :+ __- . -___ t . 5a p \ d pIn orcer o e'aluare using beaboreequai ion.t is necessaryokno*' rheprocesslhat hesasundergoesn passinghrough he *,a 'e. Because 'erv *eak *are isbeins onsidered.he ernperarurend'elocity chaneeshrough he *,a'e are er 'small nd i :esradient f iemperaturend'e)ocit1,* irhinheu,ave emain mall .Forih is eason.eat ransfernd' iscous ffect or f lorv hroushheu,ave reassunredobenee) ig ib le .ence .s pass ingh rouehhe ra re . hesa i s assumed0underso nisentropic;ocess.he lot ' throughheu'are is. herefore.ssumedo sat isf_\ , :tL = co tis!. -------------- - - --2.6p

puuinehis nro ogar irhlnicorm.anddif ferenr iat ingheequar ion;) n p - ; r l n p = 6 6 p 5 1 .lp do du .D..L = -', --Lp p d p p

not ins hat he lujd s compressiblend s perfect as, herefore=pRf subst i tur ingt h i s r t o e " ' r : ' i n n 7 : n d e n " : r j o n1 . 5 ./t,,o = * =. / ;1R __--__________2.8dp

t ' )


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2.3Pressure ieldCreated y a Moving PointDisrurbance:In order o illustrare le "ifeJt of thevelJcityof rhebodi'relariveo the speed fsoundon the f low R"tc, consio.,

he smallbodl ' , e, essent ial lypointsou.rcefdisturbance,o be movlns ar a uniforrn inervelocity ' hroughhe gasand. let he;p:; ;?"t i l " ; " ; ; " ; ; t be a Although he bodv s essenr ial lvmit ted arecont lnuously, serres r *,are emitred t ime nten'al $i l l beconsider. ince heU"it ' " " - ; ; f "g rhrough tt" gut ' the or i-eirof . these a'es * ' i l l be cont iniJal l- r '.hrnoino \ \ rave enerate dr ir ie 0 t '2t ' anJ t u i l l beconsideredirst ' onsiderhe:#i i ;:;. ;; 'pi. i " i ,r," u"iv is 'ei1 mat)omparedo he peedfsound'.The;;;; ' ;;;;;;" '- irhich exjststinf in' i"nts thenound l superpositionf all hepressure ulses r l l ich tt t ' " p"t iousl l ernjt ted' ig shorrs ereral ressure ulsen a t t e l n f o i d i f f e r e n t r ' a | u e o f r h e s p e e d o | t h e s o u r c e c o m p a r e d r r i t h t ] r e s p e e J o-.oun in rhe luid.

l ' )

Fie(2.3) .Pressureieldproducedy apointsource fdislurbanceor ngatuniformsPeedeftuards'(a ) IncomPressible luid (I//c : 0) 'iui Subsodicmoriot (V c = rr57-ic j Traosooic otion 7/c : l)f;i S;;eff";'i-"";ot]on', iti*ttt'tiog Karman'sthree rulesof supersoniclow(v /c : 2).*- Incompressible lorv: \ ihen hemediurns ncompressib)ef ig2 3a or rrhen hespeed frhe movingpointai ' i ' i lunt" is small omparedr i th hespeed fsound' hepressureulse pread niforrnl i n al ldirect ion'

I J


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*- Subsonic lo*: \ \ ihen hesourcemo'e at subsonicpeeds. ig.2.3b.hepressuredisturbances 'eJrn al l d irecr ion ndaral lpoints n spaie, ut h i pressurearrems no)on er lmmeir ical.*- Supersoniclo iv:Forsupersonicpeed ig.2.3dndicateshal hephenomenareentirel-vifferentrom those t subsonic peed. ll thepressure jsturLancere ncludedin a cone hich has hepoint ource fdisturbance.hecone ' i th in ' i r ich hedislurbancesre onf ineds cal led heMachcone. ig.2.3c hoq,shepressure enem ttheboundar-r 'enr en ubsonicnd upersonic.hat s- or hecase r Lerehesi.eamr eJocir ls deni icalvirh he onjc e)ocity: ere heu,ave ront s a plane.Karman'sRuJesfSupersonic ro* : Fig2.3 i lusrrateshe hree ules fsupersonici iorvproposedr \r2nKarr lan's1- TheRules fForbidden- ignars. heeffecl fpressurehanse roduced y arodr r t rornsara speedasterhan ound annol each ointarreadf rheboir , .2 . -Thezone f Ac t ion nd heZoneo fs i l ences . s la r iona ryo in r ou rcen asupersoi icrream roducesf lect ni 'on J:ointhat ieon or insidehe 4achcone r:endingoNnsireamlor:r hepoir ' r tource.onverselr ' .bepressurenc. lrelocit r ra:rarbit rarv ointofthe slream anbe nf luenced,.r lv 'disrurbun.":cr ine ipoint hat ieson or nside cone rtending psrrearnioni hepoinrconsideiedndhavinehesante errex ngle s he ,Jach one., t - TheRu le fConcen t ra tedc t jon . hep ressu rei s ru rbances la ree lvconcenlratedn theneighbourhoodfrhe \4achone hat or:ns he-oureri ; r i r ofihezone f act ion.

2..1 be lach Numberand the \ lachAngle:I i *as sho*n hat henature fthe f lorvpaltern ependsn rhecomparar i 'enrasnirudesfrheslreanrelocir1, nd hesonic'elocit l , . he at iooft i iese . ,or elocit l ,scel ledhe 4ach .-umber.hus.I.' -2.9aTliesemi-anglef rhe,\4achone s relaredo he r4ach unrber ,r ,

Noie hat hemachansle sExample:

. is : : l d = - - - - - - - 2 - 1 0.\.!imaginarvor subsonc fou,.

An obsenern hegroundinds hatanairplanefying or izontal lytanalt i tudeof 5000m has raveled2 km f rom rheoverhead ositionbefore hesoundof rheairp)anes irstheard.Estimate hespeed r u,hich heairplanes flying.

1 t


16/77

SolutionI t is assumedhat henetdis turbance roCuced y the ai rcraf t s weak, .e. . hat ,as ndicated -v hewording f thequest ion. as icai ly'hat s being nvest igatedshow far rheai icraf t i l l have raveledrom theoverhead osi t ion 'hen he soundwavesemitted by the aircraft are first heard by th e obsen'er. f the discussionoflvlach s'avesgivenabove s considered,t tt'i l l be seen hat, as indicated in Fig.E3.9, he aircraft vill irst be heardby the obsen'erwhen the Mach \r'aveemanai-ins f rom the noseof theai rcraf t eacheshe obsener.

Now, sincc hc lamperulure 'aries hrolgh the a',nospheie,he spccdofsounC arjes as the sou;rd r'alesp!ss lsun through lhe almcsphere hichme!ns that tbe \{ach 'eres irom lhc aitcrafl arc aclua:l! cur\:d. This ef; ' :ct s,ho\1c!ar, nal l and r\ i l l f 'a neglectcd ere, h. sounC Paed l th. avetala tem-p3ratrre bai\rc.n :h c gio,Jid an d lhe sirciafi bcitg used o d:icrr be the \ ' lachNo* as discuss3,jn Erimple i. l, fo r all ituC.s, H. cf f :on 0m (se"- ;91;l . \ e l ) Lo l l 019 Ihe icm le ra iu t c : l ! : a a lm osphcres g i ren b1 I =: ( ( l 6 - l ' 0 / A( r / sL1 : l : . c Fe r : r i : i : u i : c f l j i l : : . t h : t e : rpe : . : t - : : s

: 33 .16 00065x l i l : { = l i l . 9 K . Hen .e , henca r spe3d fsc r r , d s g i r : n b1 :

o= jlnr = \,Tlll$ or " r,-rg= 330.6m;s,_ -From heabore iguret * i i1 beseenhat f o is he r lach ng le ased n lher.:]eanpeeC f sound hen

tano = 50CO/ i200= 0.117But s ince i i ro= l / -V, i t fo i lo *s ha t vns=1 /v$ lz i 5s

HeDce. t fol)o$ s lhat:| c loc i tyof a i rc ra f r l 6 ' l jO 6 . 859 m. 's

Problem: 4 Vl . I Air at a temperaluref 25"C s f louingnith a velocity f 180mis. A project i leis i red nto heairstream vitha velocity f 800m/s n the opposite irect ion othatof theair fow.Calculateheangle hat heMachwaves rom theproject i lemake o the direct ion f mot ion.2.2 An obsen'er l seaevel oes ot hearan aircraft hat s fl; ' ingal an altitudeof?000m until it is a distance f 13km from the observer. stimate he Machnumberat which heaircraft s fying. n arrivingat theanswer, ssumehat theaverageemperaturef theair betrveenea eveland 7000m is - l0'C.

An obsener n theground inds hat an airplanel)ing horizontally t an/ ' ) al t i tude f2500m as raveledkm from heoverheadosit ion efore hesound f theairplanes irstheard. ssuminghat,overall,he aircraft reatesa smalldisrurbance,stimalche speed t u'hich he airplane s fl;ing. Theavcrage ir lempratureetweenhe groundand the altitudeat $hjch theairplanes fyings I0"C.Explainhcassumptionsouhavemade n arrivingat theanswcr.

( l / ' 0 . { ) r -

l 5


17/77

In rheabsent f electromagnet icorce nd t j th r ict ionneg) igib1e.heonly orceact ingon the controlsuriace repressureorce.,Assumehata pressurep-dp/2actson the sidesu face frhecontrololume.dp.p.-1 (p + z)d,4 - (p + dp)(A+ dA)= QA V ( + d t/ - V ,aSimpliffng ields.dp=pl 'dv=0The energl, quation ith noextemal eat ransfer ndno tr ork. for stadyone-d mensionallol ' beconre., r )l (h+;)( nt/d'a)0 ---------------

J_-o r d h + d _ = 0lAn expressionor hesecondas'of hermodl 'nanr ics siven.. Jp clpTds= dh- z :n,.i , 'or seniiopiclou ds=0 .thereforedh = -: -p p

Conrb in inghesequa t ione ob la jn :dt) Ir = - d - - c r d p - p l d l = 0 r rh i c h s t h es a n t e s l t e r t o r t t e t r : u t nq J 3 1 i (t .p J

3.3 Isentropiclou'Tluou-eh\/ar1ngAreaCharurel.Combininghe ont inui i l 'and omenium quat ionor sentropiclou'result n.. . . t dp. d. t fuu -1- pt II p A JButY = at fherefore.or sentrop:clorr 'ep

)^ / lJ I ldp pl ' t ( - "P. - l i t = 0 rnd -i l = -' p2: .1 adp(t-\1')= p'=+ --------------------3.3,1

Equat ion .3demonstrateshe nf luencef \4ach umber n that lorv.ForV< i ,subsonjclou,. he errn /--1./s positive. herefore, n ncreasen area esult n anincreasen pressurend romequat ion.2a decreasen velocity. iker l ise. decreasenarea esultsn decreasenpressurendan ncreasen veiocity. orsupersoniclou' , hererm -1t42n equat ion.3 snegat i le, ndoppositear iat ion ccur. he:-esultl lustratein f ig har,eamif icat ions.ubsoniclo* 'cannot eacceleratedo avelocity 'greaterhant lre e loci ty fsoundn a convergingozzle. his s rue rrespect ivef lhe pressuredifferencemposed n rhe lon, hroughhenozzle. f it js desiredo accelerate strearrfronrnegl igib)eelocit l , toupersonicelocit l ' .A convergenl-diversenthannel rust eusedas shou,n fi e.

3.2

17


18/77

S T J B S O X T CD I F . F t , S E R

F l o ' 6 d c c r . o r . t : - : : - : +*a i v t?. , , .k , r l ' ,>, ) ' j l i " ",44' z1'7 ' .t n- I///{" ,,rlrg}j,. /


19/77

3.5 FlorvperUnit Area.Ne\t \ \e u i l l der ive useful elat ion etueenhetemperature.ressure nd 4achnumber or perlect as.con t inu i t y. remake he o l l o r rnga r ra r r ce rnen ls :

nt ' . . D . . pl ' t / iT, l- - - - . - r j - :.1 RT .,7Rf \ ,t I I \j4Subst i tutequat ion .4 for adiabat iclorvr ; " - " r_ : L= . : : , _ L . \ 1 . : ' t _ , ' . \ l - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ 1 . ?I | R l r I l " ' a ' aTo f inda c.rnr nt ionalormula or hemass los'peruni larean err ls f \1. u ee l i n r i na tein heequa t io r rbo re 1 ' n reansf the sen t rop ican ' r e l : t i ono r su ' os t i t u teeq a ton3 .5 .

D l t ' ! l )r l r D" ' - \ ' "

j\l

florvpe runitarea. tag-nationStaning , irh heequar ionf

---------------3.8r - l, t / - t l . / l \ : r ' - l )' )

i .6 \4axi;r runrFlot ' per Unit Area: To f ind hecondit ion f mar inrunrorr peru r : i t l ea re cou ld i f i e ren t ia tequa t ion1 .Sr th es lec to { and e t h i s e r i r t i rec .ua lo ze lo . l t t h i s ond i i i on .e u ou ld lnd ha t 1 -1 .The re fb reo ind r r . i r , . - \ eneed n l l se t 1= l i n equa t ion.8 . t hus re ind .,' ' ' , - ' ' ' - Er j*r# t= -----------------3.9. , . ) . . , - . ; = r , _ ( - l , . , = -. l . + \ K 7 + ) . , 1 o

Fora givengas. herefore.henraximu;:rlo*,peru;r i t rea ependsnlyon herztto .AT". Fora girenvalue f thestagnat ionressurend tagnat ionemperaturendfor a passagel i th minimum rea. quat ion .9shou'sbatmaximumlo*'uhich canbepasseds reJat ivel,vargeor gases fhigh molecular eisht nd elat iveJl 'srnal lorgasesof lou'molecular'eight.Doubl inghepressureevel oubleshemarinum lou' .\ he reas cub ) ingheabso lu teen lpe ra tu ree re l educehe ra r imumlo$ b1 ,a ou t 9pe r en t . o ra l r r t h ; ' =1 . - lndR=187 kg .Kohe na r imum asslo r r e run i ta rea: :

= 0.0.10-12Thepanicular alue f the emperature.ressurenddensityat ios t hecr i t ical tate( i .eat he r inimu;n rea) re ound y sef i ingM=l in equat ions.4. 3.5. .6, Weu,i l lrei 'ero hecr i t ical ropert iesy superscr iptster isk*) .T' ')- = ( - l l O r a l I = U . d J - )T" " / +1 't1 /L =1 - \ t - t l n r 2 i r=0 i tS j\ . )p . 7 + t

^ l= ( ; ) ' - ' fo ra i r=0.6339y + r


20/77

t e { (3.7 TheareaRat io.Just s \ 'ehave ound r conveniento uork rr i th hedimensionlessrat jo/poelc'i t i scon l ' en ien t to in t r oducead imens ion ]essa rea ra t io ' ob r ' i ous l l , t heapp rop r ta te;; ;" ;""; t .^ is ' . andso \e computero:nequai ion 8 and 9 the ormula'l ' l- "1:-::z - : -I - - , . . - ' r - '- ' 1 7 - r r - - - - - - - - - - - - - - 3 . 1 0r - ' l

graterhanuiritl ' .and oran;' eivenalueV. one or subsonc flo*' and heoiher or

l ' [ "

r . oo.5

. ;*

o.oo5

1 t

5u - ' b : ' t ' . 1 ( l

r )A _ " r , 1 _ _ , l 1 - _ 1 1 1r / l . , , 1A l t l t . 1 : t 1 L / / - |Thearea at ion s a)rralsa)rr vscorrespondu'o alue f -f lorr .

i.8 \\ iorkingCharts ndTablesfor sentroPic Iou'Sincehe ormulashus aeder ired eadoted ious umer i ca la l cu la t i ono f rheo fat ia l-ero na: ral.pracr ical ont utal ionregreat)1'aci l i tated 1 u oikingchart nd

Chart or IsentroPic Io*.Fig. epresentn graphicalorm hevar ious imensionlessat io oe sentropicf lsu, ' l i th ' { as ndependent'ar i lb le 'Sincechangesf f lu id propeniesn isentropictlorv-are roughta bout hrougbchangencross-sect ionaliea. hekeycun'eon hischad srhatof A/A' . Theeffects fchangein area notherpropenies a1'ea.qi lyefoundby racin-s'hecun e of { l '1 ' ,keepi:ein mind hat.1' . o etc. real l constantreferertcealue or a givenproblem' orexaniple.n ncreasen area t subsonicspeed roduces decreasen velocit l ' ' nincreasenp. f , P.

of A'/A' heresupersonc\-

0.5 r.oM

is avai lable.ists hevar iousindependentrgumeni

1 0 0

o. ltol " = f t e , r .

{

l \

l L -o. l,00

\\rorkingTables.Foraccurater extensivealculat ionablesisentropicuncrion or y= .4 s ith \4achnumber s

0.52.0M p/P. TiTo p/po -4"4

0 813 0.95238 0.8893 1.3398 112950. 2i8 0.55s56 0-2300 1.6875 0'2I567

: T\-i" tTj\4 .\ iIfI

\ - -v- -l 1L +ih z \+ Il+ I

-|jti , r\ r 't lI I

+ n

20


21/77

3.9 sentropic lou'in Convergent ozzle:Coniidera fluid storedn a large esen,oirs to be dischargehrougha convergingnozzleo region .hereheback ressurea s control lable1 'means f a valve.Foraconstantesenoirpressure o t is desiredo study ireeffecrs f thevariationsn backpressure n therat of mass lorv hroughhe nozzle.hepressure istribution long hepassage ndon the exit-plane ressure . . These ffectareportraled graphicallf n Fig,anrb. and . respect ivel) .

Vo=P o 'To=

ppo

o ic o n s tconst F t o w Pe E x h o u s t e r

(a)

PEYo

U Pa/Po(c)

[*-R egimeu --J.- negime-' ]l-.-frp"( v ) ( i v ) i ( i i )

( i i )( t )

PB P o(b)

Fig. operation fconverging ozzle tvarious ackpressure'

Ps( v o r i o b l e )

vF

V o l v eR e g i m eIR e g i m eII

i ) ( i L -( i i i ) -

D i s l o n c e A l o n g N o z z e

Regimerfi r:-,,

2 l


22/77

To beginu,ith.supposeharPb/Po=, shou'nascondition i) in fig.. Thepressureis thenconstanthrough he nozzle, nd here s no floiv. f P6 s now reducedo a valueslightly ess hanPoasshorvn y condition ii), there vill be florv*'ith a constantlydecreasingressurehrough henozzle.Becausehe exit flow is subsonic,heexit-pJanpressure e mustbe he same s hebackpressure l, A funher eductionn Pt 1ocondit ion( i i i )cts o ncreasehe low rate nd o changehepressurejstr ibut ionbutrheres noqual i tat ivehangen performance.imilar onsjderat ionpplyunt i lcondit ion(v ) is reach t u,hichpoinrPb/Po qual he criticalpressureatioand hevalueof lvleequal ni ty. Furthereduct jonn PbiPo. a1'o condit ionv).cannot roduceunherchaneen condit ion i ih in henozzle.or hevalueof Pe/Po annot emade ess hanthecrr t jcal ressureat iounlessheres a throal pstreamfthe exit sect ion i t isassumedere hat hestrea:ni1ls hepassage).onsequentJyt condit ion(r ' ) .hepressuredistr ibut ion i thin henozzle.hev alue f PetPo.anddre lo\ \ rateareal l dent ical l ihihecorrespondinguant i l jesor condit ioniv) . \ /hen he lorv eachhecondi i ionhef lou' iscal led o bechocked.To summarizeheproceedir teiscussion.he rvodif ferentl peof f lorr . i l l .cdenoted s egime and egi ire I . Theseeginresraybecompared s ol lor is.

Ree ime Res imeI

c)$'

PblPo > P*iPoPelPo=Pb/Po

t\.{flR*. >:Ai r llows isentropically throughA 6nvergingnozzl-elAt he inlet of the nozzle th epr" . r r r"pr : :ab tpa, the empirature1 is 55-0K, the ve)oci ty Z1 is 200 m'us '' . t d ,h" . ror=-r"c t ionalarea,41s 9.3X l0-3 m2' Considerai r to be an idealgas\! it h y : I .4 and find:(a) The stagnat ion emperelure ndpressure 'iUi fh" sonic veloc i tyand he Mach numbetat the io leL(c) The area,Pressure,emperature,nd e)oci ty at the exi t i f 'L | : I at ex i t '

o ' \ t


29/77

t

J-6(d) Draw graphsof G, M, 4 aad./ vcrsusprcssurc,ndicating the values at theinlet and exit of thc nozzlc.3.15, Superheated stcam expards isentropicallyn a.convcrgcut{iverge_lrt-ozzle froma-n nitial state n which th e pressurcs 2. OMpa and hc superhcdts 37 g K to apre5sure.of680 kPa The rate of flow s 0.5 kgls.(a) Find the velocity ofthe steamand hecrosesectiooa!aJeaofthe nozzlc at thesectionswhere thc pressures re 1.0 Mpa and 1.2 Mpa(b ) Determine the pressure, elocity,andcrosssectional res at the tbjoat.(c) Determine the vclocity ard cross-sectionalrea at discharge.Assume : o.r .Po3.16. A convergentnozzle eceives tcamat a pressuref3.4 Mpa anda temperature f64 0 K with negligible elocity.The nozzle ischargcsntoa chamberar which thepressures mainta ined t 1.36MPa. f the throat rea f thenozzle s2.3 X l0-1m2 and the discharge harnbcra-reas 0.056nt', find(a ) The velocity at the throat.(b ) Thc mass rate of flow.

^',Assumc a : 0.55 and the flo*, is isentrooic-Po3.17. Air flows isenrropically kough rhe convergent-divcrgenrozzle shown in Fig3.24 The inlet pressure s 80 kPa, he inlcr temperature 95 K, and the back

4 . 1 . 0 m i ' "a "

FIGURE J1pressure l.0l 3 kPa. What shouldbe the exjt diameterof t ie nozzle whichcorresponds o the maximum obtainable alue f Mach numberat the exit? WhataJe he mass rate of flo w, the exitMach number, :rd he exit temperature?

3.18. A rocket motor is fitted with a convertenr-divergentozzle having a throatdiameter 2.5 cm, If the chamberprcssures I MPa and hc chamber emperatureis 22OOK, determine:(a) The mass low rate through he nozzle.(b) Th e Mach number at the exit (ps..1 101.3kPa).(c ) The thrust developedat sea evel.Assume that the products of combustion chave ike a perfect gas (7:1.4,.R: 540-J/kg K) and the expansionhrough be nozzle s isentropic.

3. 9. Air is flowing hrougha section f a straigltcodvergentozzle.At tlteentrance othe nozzle section the area is 4 X lO-3 n? , the velocity is 100 m./s, he airpressure s 68 0 kPa, and the air temperatures 365 K. At the exit of the sectionthe area is 2 X l0-3 r#. Assume reversible diabatic low. Calculate th emagnitude and direction of the for ce exened y the fluid upon th e gjven nozzlesection.

i.',rl ''.1,t \

I t


30/77

i l. i "ChapterFourIntroduction:

NormalShockWavesThe shockprocess epresentn abrupt hangen fluidproperties,n which finite bAryvariation n pressureemperaturenddensity ccur vera shockhickness omparableo ' "'the mean rie path of the gasmolecules.r hasbeen stablishedhat supersoniclow ^.adjust o theprissureofa body.bymean fsuchshockwave, 'hereas ubsoniclow can \"''adjust by gradual hangen florvproperries.hockimaylsooccur n the flow through 1^:)nozzleor ductandhavea decisive ffect n heselqw.

How ShockWaveTakePlace:Considera piston in a tube and its givena steadlvelociry to the right ofmagnitude v. A soundwave ravels head fthe pistonhrouShhe medium n the tube.Sufpose he piston s now givena secondncrernentfvelocitydv,casinga secondwaveto .ou. into the compressedasbehindhe irst vave. he ocation f thewaveand hepressure istributionn the ubeaftera time aresho*tt n figure.Eachwave ravelat thevelocityof soundwith respecto thegas ntowhich S moving, ince he secondwave smoving nto a gas hat s alreadymovingo he ighrwithvelociry'v. The second ave ismoving into a compressed as havinga slightlyelevatedemperature,herefore he,..ond *,uu" travelwith a greater bsoluteelocityhan he irstwaveandgradually vertake r . A ser ies f this nducedave after tsover ake ach therwil l produce shock\\aveor a sudden hangen pressurend ther rope( iesa l t T3>72>TI therefore a3> 2>

Fig showsoneand wo , threeand heover akeofthesound avepropagatea headof theniston

$to

"tf' t

/b/?tJ\

)rv

30


31/77

ChapterFourNormalShockWaves

t;a c c \ ! t T3>T2>TI therefore a3>a2>a

tto"o?itlXt:"k processepresentnabrupthangen nl'1.00':f;"*.;rt:;g;:$ilt::variationn pressureemperaturenddensity ccurovera shc*:r"#,'l:L::[$f''3*I'"iffitl$Ult'1"*it'T*"'fi:)::1J:ili..l1T;l"l';'."; ;;,r',. *""*

T"""",ToY?;,Il5"':':'t"uu.nat'.'i*l"'.::1*nh'il#oll,"'1i

;:iili*ugnt,iJJule*i:11:i:::i1""'';:"XTi"'.:"'l::iT!Ifl*f;:TfiiJ;:XT*:?::J'J;;:it'f:iti+i{lr'.ffTlj ;::;:tt U" "r "...". ,,r,l':*:fil";t*.:l,:":n'."^'ffifiqist1$::;*:;*":::*;ruil"f i;;-"=;."::"..1t,i#J;:',.ffi''i:",ff:",T"T#?Ji:'"'fiirlJffi:ii-l';;[i:"1,'liin'ioTi.o rvaveftertsoverake"i,'"ir'"'*ili priduce.shockwaveor a sudden hangen pressurendother roperties'

W

Fig showsoneand w


32/77

t"'t'l:?,#,H:[]::,I.fffi ughastationarvr:i,1t,1,"i,],lJi;,".""rm*ll$**:*:t*:m::r,ii'"Ti+:liiilff*;;1""",.*",i""fmass,omert:?,i:1:TtJ,f"l*i?liiili;;iv,"t,.,ipi'"*aWervill refer o theProPenlesdownstreamY Y"'2 i =p ,V ,A ,=P :V iA lTheshock 'ave hicknesssverysmall herefore':'{''

- r/ - ^ L/ ------------ -- 'c ^ . ^ a . P . t o 1 l \r v r f i ! ' r ! - . - - _ =l-r,^F^+ --"'-'4.7

in the flor'vdirectionare he pressure

4 . 1

Sinceheonl l ' force ct ing n hecontrol olumeforce, onsenation f momentums'P,A, P,-A! m'(V, -V')Combine f equation '1 into heabove quation' vherem' = p'V'A- = P'V'A'P, + P,V,2= P, + PrVrt -------4'3Forperfectgas P= P R'-TP ,+ p ,V ,2= P, (1+7M, ' )P,+ p,V,2= P,( l+l M,' )P,(l + rM,' ) = P"(1+ /v I t' ) ----'--'----4'4The lorv hroughhecontrolvolume s adiabatic nd he energy quation ecome'

' ,r , *l --' ,r, *v i =co+r' For diabaticlow hestagnationemperatureoes it

.hung" ulros theshockwave his mean hat To*=Tovr.(r+ v'=, =T,(t+* M,r' ----------------4'5' ^/.Substitute nergyequation4 5 and momentumequation4' 4 into the continuity

equationA . )

;Jfi;*;a^ io "uia"ntr'uto; ';r*ion+'o s he rivialone'M'=M'This olution;i;i;i;;;;il-c' in p'opttti"' n constantrealowconesoondingo isentroptctowand hat snotof interest"' tn" i*"*t;Uf " tinotrnut shock'-Equation4'6 canbesolve

Oisco.rli-

YRT,

to yieldMrin termof M' '

3 l


33/77

l i ')M " + -Y - ll{.' =-# -'-----------------4'7' z / , r 2 1_ l v t r - Ly - lNorv o find the pressureatio after and before he shock,substituteequation4.7 intoequation .4P 2 / . V - ' 0 -D__LP v + lalio to find the emperatureatioaller andbefore he shock, nemaysubstitutequation4.7 ntoequat ion.5- v _,_Eryi_ :t:y:!v) --------------4,9and f *e substitutequation .7 nto equation 'l we can ind he density nd hevelocity atio.

- . ' " t t 2p, y. t r + L) tut _ -___--____--____-- ,+.10--- : -=:=--- ' -=p, Vy 2+(y-1)M,-The atioofstagnationressures a measure fthe irreversibilityn the shockprocess.tmaybe foundbYobservinghat:P P P Po v - y "

, = p p P1 y . r . a xNow /e isgiven yEq.4.8, nd P"/Prand!/P."anaybe oundromEq'3'5'Using

q'4.7 oriheval-uef ,41, egetafteralgebraicimplification'

T, (y+1)M,'

(y+t)M,' ,Lf-l -------------4.1 I2+(y + l )M, '

To evaluatehe entropychangeacross he shock,we employ he perfectgas ormula'

P f t .| t s l r t .P^ lr+t

s,-s, c, l-nnLsubstituteq.4.8and .9 ntoEq.4'12 hen'%-!#"1h,-#). ^l#51- "

- --.1-f - l | ' t l/^l L4.12


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Impossibility f a Rerefaction hockCarfutstudyof Eq'4 12 ndicatehat or gasesvithl


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Convergent-Divergent ozzle:weretumnowtotheproblemoftheoperatingcharacteristicsofconverging-diverging;;;;;;;;.;rru."'ru,io, discussedreviouslvn chapterwo ' Frg' show he;;;;;,;;il;;rfo.mun.. of converglntdivergentnozzlewith variousback pressureothe supplY ressure.FourdifferentegrmesrepossibleIn regime the flow is entirelysubsonic' ndthepassagebehavel ikeaconvent ionalventuretube.Thef lowrateissensit ivetochangein backpressure.t condrtlon ,which orms hedividing ine between and1l theinr.J NirtU", at he hroatsunity'Asregime 1isentered' normalshockappearsa"",, ,t*", ofthe throat, nd heprocessft ofthe shockcomprises ubsonic;;.d*;,i;;. As thebackpressures owered'heshockmovedorvn he nozzleuntil' at"""ajii"i + it' appearsn theexitplane fthe nozzle'n regime11 'as n regime1' heexit;;;; ;t;r;" F; isvirtuallvdentical ith hebackpressure 6' on theotherhand' heflorvrate n regime 1 sconstanind sunaffectedy the backpressure' hi s is in accord

IPo{ConsL)To(Const.)v"'9 t oErhdJstet

eu/ro(b)

!"/ 'o(c)

-.+**F7 6 5 4 3

\.l n 9 a o

z\!\

gtfgo(d)

with the fact thattfuoughoutegime 1 all streampropertiesat the throat sectionareconstant' :ntire nozzle is supersonic'In regime 11L s for condition5, the flow within the t"na tfr. pr.ti".. in the exitplane s lower-thanhe backpressure'The compressionwhichililffiil;."" ""*fai, he iozzlenvolvebliquehockwavewhichcannotetreated n one-dimensionalrounds' ondition is termed he designconditionP-t t".'#i" ,"i"r'r"p."""r" *niition, sinceheexi'planepressures then denticalwith the;ffi'!..",;;;". i reductionn thebackpressureelow.ihat orrespondingo condition6has no effectwhatsoevern the flow patte-rn.wi-thinhe nozzle ln regime / the;##;; ,h. "*it-pl*" pressureo'thebackp-rbssureccursoutside henozzle n

{a) Curves of pressure versus djstsoce along nozzle axis'iu) Edt-p)ene Pressureverstls bsck preseur'(c) Tbroat pressur versus b&cx pltl3ure'iil frl"t" A,j* pBrsrDet. versus ratio of back prdeuro o supPly

0islonca Along Noz2l8(a)

Po

Loctls of / -\.Stotes Down9tr6omof Normol Shock

34


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the form of obliqueexpansionwaveswhich alsocannotbe studiedby one-dimensionalanalysis.-

'In both regimesII and V the florv pattemwithin the nozzle s independent

fbu"kpr"ssur",an i con"rponds o the flow panem or the design ondition'Adjustmentto hebackpressure remadeoutside he nozzle'. For subsonic low, there are an infinite numberof possiblepressure.istanceau.uar.port t . .up".roni"t"gionoff lorv,holvever, thepressure-distancecuryeisunique'i" p", it differenily,n subsonic lorv the pressureatio doesnot depend olelyon theur"u'rutio;n supersoniclow the pressureatiodoesdepend olelyon thearea atio' . .Only overa narrowrangeof back pressureatio,namely' he range overedb' v,.ni." 1, does he florvratedependon the backpressure' or regime I' iil' M' the loirt"i. if i"a.o."a-nr-of th. back'pressure'ince11= at the hroat'may be computedormchokedlorvequation'Converging-DivergingSupersonicDiffuser'- ; di"ffusers a dlvice that causehe staticpressure f a gas o risewhile thegas sa.".t.ruting.Whendecelerations isentropic,he maximumpressurehatcan beattainedis the iseitropic stagnationpressure.Diffusersare ei!her subsonicor supersonlc:i.p""ai"g "^ it," t,tuJnNumber of the approachingtream. n a subsonic iffuser he.ror.-r".iionul area ncreasesn the directlonof flow, while in a supersonic iffuser hecross ectional rea irstdecreasend hen ncreases'- "-- t sup"rronic iffuser s locatedat the inlet o suchair-breathing ngines s,hesuperson ic tu rbo je tand the ramje t , .Theh ighve loc i t ya i r i sdece le ra tedby thed i f f use rU.?o." t is compiessedn the axlal florv compressor f the turbojetor belore t under-eoescombustionn tire amjet.An ide'al upersoniciffuser onsists f

a convergent-divergent;;;;;*", in which h" flo* is shoik-freeand sentropic. eceleration f the flow toilfi;il,il."i ii ioiio*.d by a further ecelerationo subsonicpeedownstreamf,; ih.";. In real application, owever,startingransients nd off-designnterfere_in.ri"urirni"g thedesir'i flow pattem.Themaximum ressurehatcan beachievedn thediffusers he isentropic tagnation ressure. ny loss n available nergy or stagrationo..rru,"l in thediffuserwiil havea harmfuleffecton the operation f the engine sa;;il. i; ; u sup"ooni"diftuser t wouldbe highlydesirableo provideshock reeisentropiclow.--' --To. anyconfiguration f the converging-divergingiffuser,hereare wo valuesof Machnu*b", in *Ihi.h h" flow is sentropicallyompressed,hiswilt called ubsonica"rif ft4u.f-,umber(Mp*6 )and, upersonicesignMachnumber(M2*o)'The

ollowing."r.i * r showhow the flow is eitablishedrom the starting-upo the design lyingMach umber.--- i_ wt.n the flyingMachNumber s belowMp*6value, thismean hat heacrualthroatarea s grater han the criticalarea, hereforehe flow at the throat ssubsonicnd hi flow is continueo compressedt hedivergent artasshow n

{ ic r,- Wil; the flying Mach numberreach he MDebvalue' his mean hat the actualthroatarea s equal to the critical areaofthe flying Mach number' rhereforeheflow at the throat is sonic M=l and the flow is continue o comPressed t thedivergentpartand he exit Mach numberwill be subsoqic ig'b'

35


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-'-\.-)- When the flying Mach number is grater than Mp*6 vdlue, his meanthat theactual hroatarea s less han the critical area his mean hat he hroatarea s too

small to accommodatehe flow. The pressures iristantaneouslyncreased t thethroatareaandpart of the incoming ld4v s divert or spill over he nlet cowl ofthe diffuser as show in fig.c This mean hat as he flying Machnumber ncreasethe different between he throatarea and the requiredarea ncrease nd hencemass pill over s increase.Wl-renhe flying Mach number s grater han one but is less han he Mo',, , inthis case he throat area s less han the criticalareaor therequired rea oaccommodatehe florv. Thereforehe instantaneouslyressureuilt up at thethroat area.A curvedor normalshock s appearsn the frontof the diffuserinlet.The subsoniclorv downstream fthe shock s partially pilledover hediffuser inlet, reducing he mass low th rough he inlet, hi s rvill lower thecombustion ressure nda loss n thrust.When the flying Mach number s equal o the Mp"uo alue, n this case heexistingof the shock vavewill caused f stagnationressureoss.The criticalareabehind he existingshock s increasednd his mean iat the criticalareaupstream f the shock s equal o the hroatarea ut he area ou"n tream f theshock is still grater than the throat area.Therefore he normalshock is stillexistingand he low spill over s continue sshow n fig. d.To over come the existingshock he enginehave o speed ver the designsupersonicMach numbqruntil the shock ocatedat the diffuser nlet. At thiscase he Mach numberdown streamof the shockwave is equal o the M2-6 sothat the machnumberat the throat is equal o sonic.A little increasen speedwill makethe shockwave to swallowedand standat the divergent art of thediffuserasshow n fig. e.To retum back to the design condition the engine haveto slow down to thedesigr supersonic lying Mach number, n this case he shockwave is drawnback toward the throat and it strengthwill reducegraduallyuntil it vanishedatthe throatwhenrhe flying Machnumber s equal o the Mp* asshow n fig.f

n

o-

n

",.^'l*-l \ ^ r , /_. -f Mdr/t-_;;\. / ^ . ' > ^ n \ ?: n',.- M


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P R O B i E M S4.1. Air with initial stagnationconditions of ?0o kPa and 330 K passes hrough.aconvergent-diverge*""J" ttii" t"" of t fgl"' At *t" exit areaof thenozzlethe,t"gnrion pr"*uie is 550 kPa andthestreamPressures 50O kPa-Tbe nozzle isinsulated and there is no irreversibility excpt fcir the "t:g:".n* of a shocL

(a) -What is the nozzle throat area? - - '. . . ' --l'::--l :'"'(b)

whatis h"M";;;;;' i"i.* a"q4t $e s,|9-c5i::" :6ic) Wfrat is rhenozzleareaat the pointof shoekand at the exrtjia; Wt"t is the streamdensity it the'exit?,,-q.2. I'pealstcqsr: t.+i"o*o " "onuersinqivlJiilgnoSrreltfr-aMacl:ybj'of 0.5O and.local pressure and temPeraturevalues of 28O kPa and 280 K'iespectively. The nottl" rhio"t area is 6'5 X lO-a m2 and the nozzle exit area is26x 10-a m2 .The nozzleexit pressures l?0 kPa'

(a) What ale the values of the Mach number and the stream temperatureat theexit?(b) At whatareadoes he shockoccur?iho* you, methodof solutionon a skeletonlow chart'An air nozzlehasan exit area '6 times he hroatarea- f a normalshockoccurs;;;oi;;" where he area s i' 2 times he throat area' find the pressure'i"*p"Lt,r.", and Mach number at the exit' The stagnation emprature ndpr"rrur" before he shockare 310 K and700kPa'Air nters a suPersonic 627ls with inlet conditions4 : 6'5X l0-4 rrP''ii :'1-.e, pt : is kPa,and Tt : 260K' A normalshockoccurs n t}renozzleresultingn an ncreasen entropyof As : I l3 J/kgK' If theMachnumberat the

.. 4.3,.

.t4.4.exit jl{z : 0.3, find:(a) The area of the normal shock l" 'iti frt" Mach numbersbefore and ifter the shock M*' Mr'(c) The pressure at the exitp2.(d) The mass rate of flow Per unit areaat exr!'(e) Show the process on a schematic{low chart and a FanneRayleigh plot'Assurne isentropic flow except for the normal shock'

4.5, An impact (stagnation) tube in an air stream reads 186 kPa' If the locali"tnp"o,"t" L zbS r'-a the local Mach number is 0'8' determine(a) The local Pressure.(b) The mass ite of flow per unit.area'

4.6. A Pitot tube and a thermocouPlegive the ollowing measuremeotspertainingto airflow in a duct:Po : i80 kPa' P - l5 '1 kPa' Io : 1250 K


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la ^-. .

-\4ovingShockWave:-Previoussectionhavedealtrvith the fixed normal shockwave. However, manyphysical ituation rise n rvhich normal hocksmoving.when an explosive ccurs,'ihock propagateshough the atmosphererom the point of .the explosion. As a bluntUoay .-"nt"is the atmo;phererom space, shock ravelsa short distancea headof thebooy.rvnenavalveina-easl ineissuddenlyclosed;ashockpropagatesbackthroughthegas.To ireat hbse ases,-its necessaryo extendheprocedureslreadydevelop or thef icednormal hock ave.. considera normal hockmovingat constantelocity nto still air as show n fig.Le t Vs= absolute hock elocityandV5 velocity f the gases ehind he wave'bothvelocities remeasuredYith especto a fixedobserver' or a fixed observer' he lorv sn o t S t e a d y , s i n c e c o nd i t i o n a t a p o i n t a re d e p e n d e n t o n w he t h e r o r n o t t he s h o c k h a s

passed ver ha tPoint.Norv considerhesame hysical ituation ith an observermovingat the shock-r.vaveelocity.a situation,or nstint,with heobserversittingon the shockwave".Theshock s now fixedrvith especto theobsen'er sshown n fig But this the same asealreadycovered n the normalshocksection. elation avebeen derived and resulttabulaied fo r the fixed normalshock.To apply hese esult to the moving shock'consideration ust be given o the effectof observerelocityon staticand stagnationDroDerties.v; V ' = Q

v; v, v , = V ) - v ' , v , - v ; - v :

{s} t lovirg,*are

Since static pioperties are independent ftransformation f the coordinateystem asno effectproperties n the otherhanddepend n the observerattecteaUy the choice of the coordinate ystem'Tablecoordinateystem nd a moving oordinateystem'

{b} Statiorrary rvare

the observer velocitY, theon staticprope(ies. Stagnationvelocity and consequentlYre4.1 showproperties n a fixed

37


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TABII {.rS/6ic prdperli$i

o - - p ' - -

t " = i

.;rij*.*

]fach autuM.slvi t1- v"u,- ; - -1T/'r Vi . Y,r{',:; =--:-

Sragn4li{a /ro?ei;.ti/ r - r , \r o , = r , \ r * 2 l r )/ v - t - \ro . , - :r \ r * ;u i )/ _ _ r \ r 4 r - I '""-- o" t +J-i ul I* . - \ : /| - - l \ Y " t l - r )n o , - o " l t + - ; : u ] lt t

, , - f , - l rMt." ^ =---.*.t ,,ri t', - vtM i ' ; - "

7"8,'r'

pb|= P;Pdr Pi

( ' r?* ' )I t - t - \l t ' , M i ' l( , , ' - ' . r ' , \ t " ' - "\ ' z " ' JI t - l - \ Y ' ( r r ,l '+ 2 ,4 t ; ' /

' 1 l

\\hen a normalshoik wave ravels n a closed-end.hegasbetweenhe shockwave andthe closedend remainsat rcst'The gasbehined heshock, owever'movesat a velocityvy' as shoo,n n fig. The incidentshock s reflecledat he closedend ofthe tubeandpiop"gut., backthrough he ncominggas.For anobserver ovingwith thewave thei,.iJ"liy upp"", us shoin in fig. Since he gasvelocit-r'ecres crosshe reflectedwave,the ncide;tshockwave sreflectedt beendof the ube sa shock tave'

lDcjcrFnt

Eeilect.d

{.} Va}ecili6 rehriv! tor firgd cooaciirstetyl8!.rl{bl Velocitils .dsiv!tg r Syrttrr rngyingItith th. rEr

r_* v:

38


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Chapter97-1 T-17ranno r rcw

9. "1 NTRODUCTIONAt thes an of Ch3prerlvementionedha tarea hanges,riction,an dhear ranserar ethe most mponmt actors ffecting hepropeniesn a flow system.Up to thispointwe ha le consideredniy oneof t hese actors,ha tof variationsn area.However,wehavealsodiscussedhevariousmechanismsy which a flow adjustso meet mposedboundary onditions l either low direction r pressure qualization.We no w wishto take a look a! the subjectof friction losses.To study onll the effectsof fricrion, we analyze flow in a constant-area uctwithout heat ransferThis conesponds o manypractical flow situadons hat nvolvereasonablyshon ducs- We consider irst the flow of ar arbitrary fluid and discoverthat ts behaviorollows defioite attemwhich s dependentn whetherhe low s nthe subsonicor supersonicegime.Working equationsare developed or the caseofaperfect gas,and he ntroductionof a reference oint allows a lable to be constructed.As before, he ablepemits rapid solutions o manyproblemsof this type which arecalled Fanno flov.

9.2 OBJECTIVESAfter complering his hapter uccessfully,ou shouldbe able o:

1. List theNsumptionsmade o fte analysis f Fanno fow.2. (Optional) Simplify the geoeralequationsofcontinuity, energy,and momen-tum to obtainbasic elationsvalid for any Ruid n Fanno low.3. Sketcha Fanno ine in the ft-u and the lr-s planes. dentify the sonic rrointand regions f subsonicandsugrersoniclow.4. Describehe variationofstatic and stagnationpressure, latic and stagnationtemperarure,atic density,andvelociryas low progresses long aFanno ine.Do for both subsonicand supersonic low,


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5. (OptionalySlrrtingwith basicprinciptes f continuity' nerg) and :loiren-tuor,deriueexpressionsor propeny atios uc has T: Tr ' p:lpr' an'i soonin rermsof lvlach umber rV l an dspecifi hear atio Z fo r Fanno i"\\ N tha perfectgas.

6. Describe incLude -s diagram)horv he Fanno abLes developedrith theus eof a ' relerenceocatton-1 Dei,ne riction facto , equit'olentdiante e ' absolu!e and re cnite nrir


43/77

Energy\\'estaftwith

h, 1+ y' = 7, ' -' uForadiabaticndno work,hi sbecomes

h ' 1= h , , = h r = c o n j tIf rv eneglecthe potentialerm, hismeansha t

V 7/ 7 . = l t + 2 g c = c o n s tSubstituteor thevelocity rom equetion9.2)an dslror' hat

/l r :, + --:- = constP-zEc

(9.3)

(9 .4 )

(9.5)

Now for any given low, he constant, and G areknown.Thus equation 9.5)esiablishesunique elationshipetweenl an dp. Fisure .1 s a plotof thisequarionin the 4-u planefor variousvaluesof G (but all for thesame r). Each curve s calleda Fanno ine andrcpresents ow at a panicular mass lociqt Note carefully that hisis constantG an dnot constant i. Ductsof various izes ouldpass he samemassflow ratebu! would havedifferent massvelocities-

u= U pfigure 9.1 Fannoinesnh-r'pla-ne.

mass elocity


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Once the f luicl s knorvn' "" t '1:r l :oplot l inesof con\t i lntel l t rop) L)n he /r- t '

c l iacrunt. ypicl l curresof ' ' ' ] ct t l l t t 'nt re 'ht" ' n i is cl :Nhet lines n the l igure' I tis nruch rore nstruct lveo p*if t t t t Fa'rnc'i t tes n the anr i l iar r- - rp lene Suclrar l i tgratn -s howll n Figure ' t i t Ut ' tpoint 'a signihcet l t; rctbeconles uiteclear 'Sincerve have rssttmetl h:rt here ' no heet rlrttster r/'s.' 0)' the orrir' * it)' th"ltentropycan be generatet lt int"t*n ir ret 'ersi t ' i l i t ieslsi) ' Thus t l tc ' l t t t t ' 'un otrh'/)/{).grr.tJt)\t(tr(l ttL ((tsi'"3 :t' i"tt ' t' t t'"tny;r'l \Vh'r? ClrLnotl locate the points

oIl t t ; i i ; . ' ; " t entropy ot eact iFannoine n Figr-rre' l lLet us esrnt ineo"t Et"no' i i 'n l " -"t t t t i t t t t t i t ' F i-str re' i sho$'s givetrFirnnoline togethenvith typicalG;;; ii '*t ' lrr point\ t)n this ine represent-s-:::::^"]:fttlie siltttentuss lo$' ,"t" p"""' ir lrel (tnass eiocit)) ' lnd' lte sirtrte tl-qnlttlonen -thrlpy. Dtte to the rreversibt"

'*tu"ot the rictionalettects' he florvcan onl) Pro-

.*.iio th. right' Thus theFanntr ine : divided nto t\r'oclistinctparLs' n upper andn |:rverbranch. vhicharesepxrated'r 'a imit ingpoir l tof t l lx i t r t t tn l entropy'What cloesntuititln tell u-s bouraiiabatic lorf in a constallt-arelducr'l \Ve nor-n ta l t y fee l t ha t f r i c t i ona le t l ec t s \ \ ' i l l sho l r ' t l pasa l l i n le rn l l sene r r t i o t l o f "heaC 'w i tha correspondirrg ecluctiott' i""iti t' 'f he flLrictTo prstihe snnte lorv rate ($'ithconstantarea)'continuilvt"lo't*t theVelo.'ilyo ncrease'Thislt:::llll1"^tlt'energy uust altse decreasenenthalpv'incehe.stagnationnthalpyemalns on-stant.As canbe seen' Fi;;;;-i' tl"t agrees'ith.fiowalong heuppe'r runchoftheFanno ine. t isatso tearnar n this -aseoth hestatic ndstagnotionressurearedecreasing' : * tr,r,o ointson the lowerBtrt what about low alPng he /t)r|erbrunch'!Mtrlbranch ncl ra*'an 'oo i-' 'nAit"teroPer ovementfonq^t!3-f1n1i":^,Yill; ;;;;;;td," tr.'.-'nt'orpvio he nsitvl."-':l"l':J,[?][ JilT# ffi:i. * !q*,i* (9.2)llFrom he isure'vhatshappentngslagnationressure'lir r l^ i'iit!' L tith rrcrenstl'ecrease'orentalnsonstult'

f lgure 9J Two bmnctL5 of a FrnrKr ine


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Trble 9. 1 -\| lJl] 5i i (' f Fr]lt lo Flort firr Fiqure 9 -3Ptrrlt ' I11 Up1'ur r rnc Ll])$ r B[triehtntl.r pvDcr is iL l\!'1,)cit)P lcr \ur , iJ t l l ! i ( }Plcsslt lc (rlu.t t. t l i nr

Nori :e th l r t rn t . r : r ) rver | l r rch.p lopenies Jo t t : t hr . r -n the ntanner redi t - tedh. \ . i t t i l t ; t i i t ] . h-r : h is lu\ l he a f l t t r i ' reginte| i th "r 'h ichwl r re rr( r l en f lnr i l i l r r 'Beibrc ,,r.errre.ti5.rtehe Iirriitirr.l 'oint that sePifiltei these *'o flo'r'regiittes. et usllote thet theie drri! Co hll\e OIte hing io ct-rtttlttot.Recall the srxgnilhonpresstlreenergveqtlttion.STAGI.IATIONRESSUFE-ENERGYAUATIONccnsider the t$.o selrio.r ocatiotrs n theplil-sicalsYstetn h'trvn n Figrrt . .It_*i 1., ,r. disrence e \'.ee1 hr.se rrcatillniapproa--he.'! ',.1'c re dealing* itir an

ininitesimal c!rr)!ml !olult)e t i th the t lrenlrodi'namLc states dif ierentially separated'assho*,:r n Fi_rLrrerlon. Also sh,:ll 'nate hecorre;po[dil)gstagnalioll littes br thesetn r locirions.\\ie rna-v,rrite he follow.ingpropert) ielation betrvien po'rnts and 2:

Figure In nnite in:rl y :eparited tatic lxte! li lh associate'l( '1-cnatlontntes-

T d s = t l h - v d p (. { l)

T:

Note thareven bouglt he strgnalioo tates o not acttlnllyexist, they rePresentlegitimate hemrody arniu- tates,aDd rhrrsany valid property relation or equationmiy be applied o hesepoints.Thus u'e mry alsoappLy quation A'l) betweenstates, and2, :

IIIIt

,''

T, ds, : dh, - rt,dP, (A.2)


46/77

' l

Ho',r-c'r'r.

i l l ) r lr l s : t l q . * t / , c ,

| \ .1)

(.{ -l )

r - \ . i

r { .6 l

r \ .7

{ . \ .g): I lP,l andstagntrtiort rtssu c.-t E g)'

I h i l \ \ t l t t i l ! \ \ l l l d

f i i ( / - s . d5 ' : d l t ' - t '


47/77

Di f' lerentiating,\' eobtaint ' d l /t l h ' d i t + - = 03'

Fromcont inu i t r e h i id bund hr tp l ' = Q : c 0 n 5 1 x n 1

i , J \ ' + V l t = 0"d Dd t = - y -p

Diiferentiit ins th is. !\ e Qbtain

q hich can be solred lbr

/ q -7 1

/ O \

(9.6)

(9.8

(9 . 0)

( 9 . )

r r Q ? \

Introducequatioo9.8)nto 9.6) nd /tor' har. . v2dpdh = _______: (9.9)8c p

Norv recall the property elation

whichca nbe wnttenasT d s : d h - v d p

pSubstitutingor dh from equation9.9) ields

v1dogcp

dpp

We hasren o point our that this exPressions valid or afr,! luid and belweentwo differenfially sepa.ated ointsanyp[ace along he Fanno ine- Norv let's applyequation 9.11) o two adjacent oinls ha tsunound he imitingpoint of ma.rimumentropy. t this ocation = const; hus s = 0, and 9. I I ) becomes

V2dp--: = dp at rrrutpornt

ot


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(#)",,,,.,,,",",' H),.""' (9 . )v' r = g.t__*

This shouldbe a familiar xpressiondp/dp= ) &l ,] an due.recognize ha t tevetoci^. s soricat rt v lintit ingpoini. i{ upperbranch "" ":",ltJ*" :1.::l i.!-ff i;;-";. t; i,o"i ' u'o"i' ' an d he owerbranch s seen o be th esttpersorttc'-ulll " " begin o seea reasonor rh e ailureof our inruirion o.predicrbeha! oror r he lower branch f rh eFanno tn e Fromou r privousstudies t sho\!s hl lf luid beharior n supersonictor" rr-ir.qu.nrty contrary o ou r e


49/77

t i oAV\ t ) - r : t f F t = ( l : l ' . t - - { l : 1 r )g. - .{ . -s , rL r r ' ha tcquJ t ion L L5 cJn bc r r r t en s

r I P ' \ - - D L v ' -o 3 . c t

/ r r v r \ F ; p : l , ll p + - | . = i : + -\ -!. ,/ ,.1 E. ( 9 . l 7 )

( 9 . 1 5 )

(9. 6)

(e.3)

( 9 . 1 8 )

In this onn theequation s not panicularly seful xcept o bringout onesignilicantfacl. For the steady,one-dirne siondl, constcnt-areaote of an'" uid, lhe valueofp + pV1/g, can ot be constant f fr ictional orcesar e present. his fact will berecalledater n the chapterwhenFanno low is compared ith normalshocks.Before eaving his section n fluids n general,vemight saya fervwordsabout

Fanno lorvat ow ivlach numbers.A glanceat Figure9.3 shows hat he upperbranchis asymrotically pproaching e horizontaline of constantoralenthalpy. hus heextremeef ten dof theFanno inervillbenearlyhorizontal. his ndicatesha tEowatvery ow Nlachnumberswill have lmost onstant elocity- hi schecks urpreviouswork, which indicated hat we could treatgasesas ncornpressibleluids if the vlach--numbers ercverysmall.

9 .4 WORKING EOUATIONSFOR PERFECTGASES\\'e havediscovered he general rend ofproperty variations hat occur n Fanno louboth n the subsonic nd supersoniclow regime.No w we wish to developsomespecific orkingequationsor th ecaseofaperfect as.Recall hat hese re elationsbetweeflpropeaties t arbitral . sectionsof a florv sy5lgrnwrinen in terms of N1achnumbers nd he speci6cheal atio.

Energy\l t startwith the energyequation evelopedn Section -3sincehi s leads mmedi-ately o a lemPeratureatio:

*

But for a prfectgcs,enthalpy s a function of temperature nly. Therefore,l t l = t t 2


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No * fo r a perfect aswith constant pecitic eats'

, - , ( r - r E p l ' \' r - . I I l/Hencerhe energyequalion or-Eaano lo* can be wr !eqas

, v - l . \ I : ' - 1 . , ' \r , ' * ' . ' n ' , ' ) = r : ( r+ / I " r r ' : ) ( e t e )TzT1

l + [ ( r - l ) / 2 ] ' ! 1 , :l + t ( y - l ) / 2 1 : v . (9.20)

ContinuitYFrom Section .3 we have

p V : C = c o n s t (9.2)

pg1 = p1V2 (9 21) : i

If we introduce he perfeclgas equationo[ statep = p R T

th edefinitionof NIachnumberV : irlo

and sonicvelocity for a perfectgas^ _ ,Tn:Frr - \ ' q _

equalion 9.21)canbe solved orpz ut ( Tr\t/t (g.zz)a: M1\T)

Canvouobtain hisexpression?ow ntroduceheEmperaturetio from 9 20)andii'irin i"t. ,n. followlngworking elationor staticPressure:


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p . - l t l l l ' 1 , - r l l l / , r 1 r ' :p 1 - r v z t + [ , . , - L t 2 ] u , ) a a l r

caneas i ly e ob ta r : : : : r . rn t q rLar ion9 .20) , 9 .11) . nd rhehe dens i t ; e la t ionperl'ect as a$':

l : _ Nl tpr lvlzt i 1

: - l r l l . ! / L (9 . 1 1

EntropyCha ge\\? start th anexpressionorentropyi.r-.ji harsvalidbetweenny '\ o poin i:

a r 1 - , : c o r r i - n r n 41 . P l ( 1 . 5 i )Equalion 4. 5) ca nbe used o substituteci c- .and venondimensionalizeheequa-tion o

. t . - J l v T . D 1____: ln __: _ ln j_:R y - l T 1 p lI f we no w utilize he expressionsjusterelopedor rhe emperatureatio 9.20)an dthe pressure atio (9.23), heentropychalr!r bicomes

s r - s r | = y h ( l - l t ' / - l ) / 2 ) M t ' \R y - t \ t - [ r _ t ) / 2 ) . v r ! )

(e.25)

, - M t l 1 - [ , v - l ) / 2 ] , 1 / r :\ | r- "'n \t-rt:lttr,'trlSlolr,that rhisentropy hange enveenro lsints n Fanno lo* canbe writtena:

. ' 2 * s r , M 2 l I + [ \ / - l ) / 2 ] r v l r ? \ ( / + r r i ? ( r - r )R M t \ l + [ , r ' - ] t / 2 1 ' v , : / (9.27)Now recall that in Section4.5 we ntegrdt.dd. stagnation ressure-nergy quationfor adiabaticno-work flow of a perfectsa-i. r th the result

U = ,-u aPrt


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9. 5 REFERENCE TATE I']OFANNO A3LE

an dequalion 9.lLrrbtionl.sr t.y:_!C_ = f \,\t .y) (e.:r l)T - T - t ' r - l ) / l l t 1 r " '

\ \ ,esee hatT iT ' - . i t . i l . y ) and \e caneas i l l cons t ruc t lab leg iv ing a lueso it/ ii u.rru, ,rt t ir : grrticular . EclLt:tt ion9 23 ca nbe reatedn a sirrli lar ashion'In th is asep: =) p ,|1:= ,1 (an) vaLue)t1 t :: ? D' . ! /1 .= : I

an dequation 9.11 ;'e.oolesi , _ t ( 0 + t t l l . \ , , = t , . , 1 . r , ( 9 . 1 2 )f - J t \ l - l \ Y - t \ t l l \ l -

The densitl r l i:o .ln be obtainedas a iunction ol N1ach umber an d )/ f 'omeqJon i9.:ir. Thi: is particularly sefulsince I also represents relocity r3tiowhv?

p \ ' . | ( r - ! ,_ t t , l !_ \ ' = r r , rz ,y t (9 .43 )- : = - = - l -F - r - , n \ t 7 + t ) 1 2 /Apply thisame techriques o equation 9 28) and s'toru hr't

f , . . . | \ / , r r I r : \ r Y - l ' / : t / - l l" , , = t ( t + l { y _ D / t t , v t -\ = 1 ru . y t e .44Ja - v \ ( Y + t t l 2 l\\t nolv perform hesame ype of transformarionon equation(9 40)i that is' let

. r l + r M : + M ( a n y a l u e )'1 1 $ 'Y 'vl, :+ L

with th e following .sutt:i ' r - r ' ) / Y l - l \ ' / l - l Y - l ) ' : l t l r \= | - I ln I ___--=:-:-- |D , \ 2 Y / \ ( r + r ) / r /

- L ( L - r ) - / . - l t n M t ( e . 4 5 )l ' \ M = / z YBu t a glance t hePh\ icaldiagram n Figure9 5 shows ha t (r . - r) wilt alwaysbea negalive uanriq:dlus t rs moreconve;ient o change ll signs n equation 9 45 )and simplify it rc


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ll

252 FANNo Lo$// - _ ,I n - I n ( t ' a ' n , , ) = t n n n s r

and hen ifterentiitring,eobtain(9.3

dr , d ( t+ l { 1 ' , t ) / 21 , v2 )7 - t - t r . / - l t 2 l J l r = " (9 .3 i )wh ich ar r eused osubs t i tu teord l / f in (9 .30) .The continuity elation equation9.2)]pu t in termso[ a perfect csbecomes

4: .onr, {9.1: l)r/ TB) ' osar i thmici f fe ren t ia t iontakehenr ru r l l ogar i thmon,J rhend i f te rera te ) . r l lo l rthat

dp d.\l L dT_ _ _ _+ _ o r g t 5 rp t v t 2 Tl\ t ca n nt{oduce quation9.33) o eliminated7 / f, wirh the resulr ha r

dp _ _d,v _ | d ( t + [ (Y - D/2 ) tv2 ) .q ?6 \p t " I 2 l + l Q - t ) / 2 ) 1 2

whichcanbe used o substituteor dp p in (9.301.N'{akehe ndicated ubstitutionsor dp / p anddT /T in fte momentum qucrion,neglecthepotentialerm,an dsrav thatequarion9.30)ca nbe pu t nto he ollowingform:

.d t d( l + lO - t) /2ltd1) div2 z dM' D, | + l t y - l )12 l , r l : l l J y i I I tI d ( t + l \y - t ) / : l , u : )

\ 2 r ' l y ' t 4 1 | - [ ( y - l ) / 2 ] , r / :The ast e|m anbesimplifiedor ntegrarion,v oringhar

r d(r + t(r - 1) /4M2) (y - t) dtr2, , M 2/ M2 | + l (y - I ) /21M2 2y M2(1 , r) d( t + tu - D/?ltvrz)(9.38)2y r+ l (y - r ) /7 ) tv2

The momentumequationcan now be wriften as


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tl.u l . + l d ( 1 + [ ( ] , - l ) i ' l i . t / r )7y I + [( ] , l i i l l . t l :1 c l i l / + l d M ?- t M . - + t l = (9. l9

(9 . .10 l

Equation 9.i9) is restrictedo st.ad) one-dinlensionaLlowof a perfect-gas' ith noheat o. wark tr nsfsr.const3nt rea. nrlneqii-sibleotential hanges'\' e can nowintegratehis eqtration et\!ee n \\ o points n the l!]!\andoblain

, f r : r r ) y * 1 , -D, ?i ,

L ; l r r - 1 r , l l J / , :1 - - 1 1 7 t r ' : l i v l ' l

| . ., -: - 1 i t l , :I I I r ' , ' ' j- t \ l t r i t : J 27 t v t , '

Note that in performing he inte-sratione hareheld he friction aclo( constaltS o m e c o m m e n t s r v i l l b e m a d eo n t h i s i n a l r t e r s ec r i o n ' l f y o u h a v e fo r g o t r e n t h econcepr f eqLtivalenliameter,vo uml,v !an t o rer ew he as tpan of Seclion3 8andequat ion3 .61) .

9 .5 REFEFENCE STATEAND FANNOTABLEThe equationsdereloped n Section9.4 govide the meansof computing he proper-ties at one location n termsof tlose given t someother ocatio-n' he key to problemsolution is predicting the lvlachnumberaf.lhene* location kough the useof equa-cion(9.40). The solution of this equation or 'he unlnown M: Presents messy ask'u, no "^pii"i, relalion is possible.Thus we tum to a technique imilar to thal usedwith isentropic flow in ChaPter.We introduce anorher' tefetencestate, hich is defined n the samemannerasbefore (i.e., that thermodynamicstatewhich\'rouldexist f the fluid reached Machnumbeiofunity bl a partictiar procesr") "rrhiscasewe magine ha twe continueby Fannoow ii.e.,more uc t s added) nril hevelocity eaches {ach Figure9'5shows a physical system ogelher with its f-s diagram or a subsonicFanno flow'1& 'ekr rowtha t i fwecon t inuea long theFanno l ine( remember tha twea lwaysmoveto therigh0, we will eventually each he imit inspointwhere onicvelocity

exists'The dasied lines show a hypotheticalduct of sufiicient ength o enable he flo!\' totraverse he remaining portion of theuPPer rarchand reach he limit point Thisisthe ' referencepointfor Fannofow.The isentrojic * referencepoints havealsobeen ncludedoII the I-s diagram toemphasize he fact that the Finno * reference s a totatly differentthermodynamicstate.One other fact shouldbe mentioned. fdrere is any entroPydifferencebetweentwo points (such as points 1 and 2), their isentropic' reference onditionsare notthe sameand we havi always taliengreatcare o label hem separately s l' and 2''


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----'----_)- Acturlduit-- 1

IIIIl,,1-'

tl 'll t

Hypotheticaldu.tI( r l

,rl I

Howerer. proceedjng rom either point I or point 2 by Fannofow will ultimatelvread o rhessme race hc n i\Iach is reached. hus we do no,i"".,o,"tt oi-il--2' but merel.v in thecaseof Fanno low. Incidenrally.*t y _. ull ,t rJ" " ..f"."n""pointssho*n on th esame orizontaline in Figure .: : iVou ","y na"o to reviewSection .6.)lt'e. nolv rewriri the working equations n terms of the Fanno flow - rlerencecondition- onsiderirsr

T 2 + T

JFigure 9.5 The' reference or Fanno frow.

T I + [ ( y - 1 ) / 2 ] & ! , 2: = - .t r l + t Q - D / z l V z lL"., p:iA2 b an arbirrarypoint in the flow systemand let its Fanno . condition bepoint l. Then

M2+ M (any value)t u l t + l

(9.20)


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an dequltion 9.20)becomesT ( y + I ) / 2T - l + l l r - l ) / 2 ] / ! 1 1= f (N r , y ) (9.-l l

We se e ha fT lT ' = f( l.y) an d \ 'e ca neasilyconstruct tablegi\ing laiuesQf;i i: r;r;;t i' l fo , o poni"ui", 7. Equation 9'23)ca nbe reatedn a simiLarashionIn thiscase

p1 .-> p &1: + M (any value)P t = + P " i / l + l

an dequation9.?l) becomesL = ' ( ! , - ' , ' , / l ^ , , , , ) '= 7 , , " . r 'p . - , ! / \ l + [ ( y - t \ / 2 ) . \ t : (9.,l])

The density atio ca n be obtainedas a functionof N{achnumberan d y from.qu"tion {9.zil. This is panicularlyusefulsince t also represents velocitl ratio'lVhv? #=+:e+;#Y)"'"'Apply the same echniques o equation(9 28) andslzarv hat

# =* (+t#ffi (Y+r)r2o'-r)= @'v)(9.43)

(9.4.+)

\!'e now performthe same ype of transformadonon equation 9 40); that is' lel.r2 =+ r M2 =+ /v/ (anYvalue)1 1 i , r ' M t + l

with the ollowing esuit:

(9.45)ry:(+),^1::#fifv:)- iG- t ) - ' ; - ' , -M 'But aglance t hephysical iagramn Figure9 5 showsh,atr' - x) will always e^ ""g;,i"" q*"ti,y; thus t is moreconveniento change ll signs n equation945)andsimplify t to


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/ r r ' - . { ) l v . t 1 , 1 l t y t 1 / ) l \ t :D , \ 2 v / " \ t r t r y _ ) t l l , t r :(9 .15

The quan!it) (-! ' - .r) representshe amounr f duct hatwouldhr\e to be addedto cause he flow to reach he Fanno reference onclit ion.t can alternlti lelr bevierved s he ma\imumduct ength hatmay be addedwithoutchan-qinSome it),.rcondition. hus heexpression"f t' - r) ir .",.d 4:\D" D"

and s f sted n rable longwirh heotherFanno low parameters: T.. p, ,p- .V V' , an dp, p," ln the nextsecrionwe shallse eho w this able srearl\.implir iesth esolutionoI Fanno lorvproblcm\.Bu t first, om euords about h: deierminlrionof friction rcroo.Dimensional nalysis f the Ruid lorvproblem hows hat he riction actor anbe expressedsJ : f (Re . /D) (9 .17)

whereRe is theReynoldsnumber,

. R.=# ; (e.{s)

t t \* t f , " ' - t ) = f \ l \ r ' Y )

an dtfD = relativeroughness

Typicat valuesof e, the absolne roughness r averageheightof wall inegularities.ar eshown n Table9.2.The relationshipmong/, Re, ands/D is determined xperimenr3llyndplouedon a chart similar ro Figure9.6, rvhichts calleda fuIood.y iagramIf rhe flow rate s known together vith he ductsize and

Table9.2 AbsoluteRoughnessf Common \IaterialsMaterial (ft)Glass, rass, opper,eadSteel,wroughtronCalvanizedronCasthonRiveted teel

smooth 0.00000.000150.00050.000350.03


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TLlrbulent low regjr. le

Lo gReFigure9.6 ivloodydiagramor riction :rctorn circular ucls'

material, he Reynolds umberan d relative oughnessaneasily ecalculated ndih " uulu" of theiriction factor s raken ro m th ediagram The cune in the aminarflow regioncan be rePresentedY 64, - R e

For noncircularcrosssections he eqlrivalentdiameteras describednSection3 8

can be used.4 Ar

(9.49)

(3.61)This equivalentdiametermay b used n the determinationof reladve oughness ndn"ynoiO, nr.U"r, undhencerhe riction factocHowever'caremustbe aken o work*iii ,t " ot*f averagevelocity in all computations'Experiencehasshown hat the"ra oi un aq"i"^l"nc d'iameterworks quite well in the turbulentzone',ln he aminari"*l"gl* ,f.rft ncept is not sufficientand considerationmustalsobe given o theaspect alio of the duct'In some problems the flow rate is not known and thus a rial-and-errorsolutionr"sults.As long as heductsize s given, heproblem s not oo difficult; nexcellent^ppr*fi",i"*" the friclion factor can be madeby taking the valueconespondingto wfrere tree/D curvebegins o lelel ofl This converges apidly to the inal answer'as most engineeringproblemsare *elL into the turbulentrange'

9.6 APPLICATIONSThe following stepsare recommended o devetopgood problem-solring echnique:


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i. Skercb heph_vsicaliruarionincludinghehyporhericalreferencepoinr),2. Labelsectio swhere onditions rekn.rLr or desired_L L i .La l lg i . cn n fo rmr t ioni rh unr r . .4. Compute heequivaient iameter.elxti!. roughness,ndReynolds umber5. Find he ricrion acror rom rhe ilood., Lesram.6. Determineheunkno$ Mach numbe:.7. Calculateheadditional ropenies ;sired.The procedLrrebovema yhave o bealrercd epending n rvhat yp eof infonna

tion sgiven, ndoccasionaliy,rial-and-enorolu:ions re equired. ou should aveno difficult,vncorporatingheseeatures n!-i hebasic traightforwardolurionha sbeenmastered.n complicatedlo w sy5gs6115hrr nr.olvemore han us t Fanno forr,..a 7-s diagranrs frequent)y elpful n solr nuproblems.For the olloiving xamples e ar edeaiing,, rh hesteady ne-dimensionalfo *of air (y : 1,4). vhich anbeEeated sa pedect as,Assume ha tO = 14 = 0 andnegligible otentiaI hanges. he cross-secrionalre aof theduct remains onsranr.FigureE9. is common o Examples .1 hrough .3 .

FigureE9.Example ,1 GirenM1= l.80,pr=40psia-and.y: 1.20.indp2an dJ Lx / D.Since oth lach numbers re nown,wecan olvi mmediatelyor

p ) p ' . ^ | I \p: = - - pt = (0.80!r ( 93 _ J f.lOr 67.9 "i aCheck igure 9.I rosee har

f 3. r _ JL|M _ !L-_,.D D - - - ' = 0 l J l g - 0 0 1 3 6 : 0 1 0 8

Example .2 Given I: = 0.94, l = 400K. and : : 350K, find Mr andp./ pt.To detemjne onditionstsection in Figure 9.1,wemust stablishh9 atio

I/;\I


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r = \ r : - i + o o ) , 1 . s 1 e 5 ; -t o ; :T ' T : T ' \ 1 5 0 /1 l -| - r'om Fanno able at rV1 0 9-lIL GivenLook up ?'/ f ' = t . i65' i in the Fanno !3ble add d! ' te 'minr thal ' '1 l : 0 ls iI h u i p : _ o :p ' = ( i o7 . r . r r. f ) -o l s rP P ' P t \ - : d u r o /

Notice ha t hese xamples onfirrnpre! ausslalemenlsoncernloglilt lc ressure

.fl""g".. f" subsonic lor" lh e staticp;essure ecreases' hereasn supersoniclowih. ri",i" p..rrur. increasesComprrrehestagnation ressureatioan dsholv ha t hefriction osses ause r:/pr t lo decreasen each as eFo r ExamPle l:

(p 1 l p, 1 = 0. 1 6)

(,1i-

P t l

Example 9- 3 Air f lows n a 6-in -diameter,nsulated' alvanizedron duct-nit ialcondit ions

*" oi : ZO psia, T1 = 70'F and V1 = 406 ft'/sec Afrer 70 ft' deermine the final Machnumber.emprlturean dpressureSince the duct is circulaJ we do noi have to compute an equivalentdiameter' From Table

9.2 the absoluie oughnesse is 0.0O05 Thus the telative roughnesse 0 0005D 0,5

\!'e compure he Reynolds number al section I (Figure E9 l) since this is lhe only location\ahere nformJtion l known

- . / ) O \ l l j rt . = # - - f f i = o r o r r b n v i c/ rr : 3.8 x t0-r ttf-secrtil Air prop'rtiesable)

P t tForExample .2 :

Thuso , V 'D rRF, : :-:---i- =l t r8 ,

Fromthe lvloodydiagram ar Re =determinehat he rictionfactor s /neednfo.mation oMachnumberg

( P ' : l P ' t = 0 6 1 l )

\ , ^ ' . t ' l Jt)l '+ i

(0 . t02) ( {06) (0 .5 )= 1 .69 106(1 .8 l0 -7 ) (32 .2 )1.69x 106an de/D = 00O1,we

tLn /yr ,": 0.0198.To use he Fannoable or equa[ons)'we

A - " ^,< . _ \ "5..4t z ?.eX,'y'

+ ^ ) 1 , " ],-/ "'' 1 =z''4*2 -


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at = Qg, .RTt ) t i . t ( l . .1 ) (12 .2 ) (51 . t )153011r1rt 123 y , , " ., v r r = \ = i ! 1 = n . , ^c r l l l S - " - "

Fion th eFanno ib le at r t l t = 0.15,,rehnd,ra.3= . r . 00 . ' ' := r . r os r = == r , r o ,- 7 u

]1:_l?.j: ""T0j.,,:g theproblems nesrablishin-she lachnumbef t he urje(.iJ rhirsoone hrough hefriction en2th:/ A ,r (0 .0193) (70)T = -- ;- - =tt t '

Lookingat rh ephljicalskerch ir isappaienrsrnceI an dD a. econstaors)ha tf ! 1 _ / 1 i " , , _ r J . cD D ' - - = I l s O l - 1 7 7 1 0 1 0 3

\ ! 'e ente. he Fanno ablewilh thiJ r ict ion ength nd ind rhit

Thus

an g- T z T ' - / | \, . = VT, ' =( r . t r36){ . .Gr / r r ro, or " *

In the example above, he friction factor was assumed onstant. n tact, rh.is s_sumptronwa smadewhenequation 9.39)wa s nregratedo oiojn (S.OO),na*i,nthe introduction of the * referenceinheanno,"ur.., r,i,...","""';i:";i:i;i::1....rff11?"f,;11,;lil:li,"i],liReynolds umbers,which n tum drchangeuite,^prdiy;;;;;;;;r:nil# ffJ'I.ll:,i;:i'li.l::,:"J..*.rl:.3-andomparer wirh hatar henf.t f y, = iie fJ*. l* u, joOZ ,r*. ,tBurdon'tdespair.rom onrinuiryeLow that .;;J,;#;.is aJ*ays:"Tr-"::-11.:h^ theontyvariablen Reynoldsumbeis ,fr" i*rii,y E\rremelyrarge emperature ariationsare equired,!o hange heviscosityofa gas rgnilicantly,and thus variations n the Reynoldi1l::r"T"Thil;;;;;;#"ilil::ffi*,T"ii'"i:T"T",H,T;hHhere he rictionfactor s relativelynseositiveoReynolds umte.. +gleu,.a o-tentialeror is involved n theestimation f theduci ;;G;;r;,;il nasamoreignificant ffecron the riction actor


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Exatuple .1 A converging-diver-qingozlh { i th rn alea atio o[ i 4] connectso an u-l!-long;onstanl-aieaectangular uc t (seeFi,:ui: E9'J) The duct s 8 x 4 io i n cro\s sr'ct irnanihcs a irict ion actorof / = 0 01 \\ 'h:t ir lhe ninimum s(agnatioo fessureeeding henozzle t the lorv s supersonichroughoulie entii i ductan d ! exhauslso l4-7 psia?

- ! r = 8 f t

- -+=* i i j r , " .= aTps iaI L I Ii /--'l---'-.- | .,=oo, It / - "C 6 6 6

Figure 9..14 A t : r - j : l = i . J J + l n ,P 1 :( 0 . 0 :S r r l l l- = u . J o

Tobe supefsonicith ArlA: =: 5.42' lt = 316'pi l po = O'Olg5' l P' = 0 1901' ndfLt^""/ D = 0.5582,

J r - 'n 7 , -1 , . . / ! r = 0 . j j3 t _ 0 .36 0 . l9S lD : --ry- oThusM r = t 6 1 3 a n d { = 0 5 1 + 3

an ot t , t l \o . , t u tp t t p ) P D .= ( t ; { : - - L o . r g o l - l ( t 1 . 7 ) ? 2 S p s i . r, , ' - p , t p , p . p , , - \ 0 . 0 1 3 5' - - \ 0 . 5 t 1 J . /

Any pressurebove 88psiawill mainnir he fo$system sspecifiedutwilh expansionwavesoutside he duct. (Recallan undere\Pn-adedozzle Can you envisionwhat wouldhappenf fie inlet tagnltion ressur.el lbelou 33 sia?Recallheoperationf aoover-erpanded ozzle.)

9.7 COFFTELATION ITHSHOCKSAs youhaveprogressedhroughhischapteroumayhave oticed ome imilaritiesb" t*eenra,rno-f lowandnormalShocks.Lelussummarizesomepeninent in|or .malion,

D


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The pointsus tbei.-rrendafteranormal hock epresenttates it h hesamemassllow pe r unirarea. he ame llue of p + p vrg". il;; , ;; ,*_"n",ion n, t utpy.These acis re he esutr f appl.vin-qir eb"ri" ;"*;;;;;;;"i i"",r,-.".n..,"., *oener-qyoan yarbirren iuid.Thisanal_vsisesultedn equarionsfO.Zj, i.:1, an a g.Sl.Fanno ine represenrjtates ith thesamemass ;!v pe ru;i;;J; anq nesamestagnation nrhalp_v.hi s s conErmed I equations


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Sl:ock )\'

/I

Figure 9.8, Combinrt ionoi Fanno low and.no.mal hock.Example9. 5 A large hambef ontains ir ar a tempefature f 100 K and r p.essure f gba rab s FigureE9.5).Th eair enrers con!erging-diverging ozzle r. irhan area atioof 2.4.A constant,areauc t s attachedo the nozzlean d a normal shockstlnds ai the exi! plane.Receiver rcssures 3 barabs.Assume he entire yJtem o be adiabatic nd neglect r ict ion nIh enozzle.Compute he A.(/D for theoucr.

FigureE9.5


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Fof a shock D occu. aj specilied,he duc! f lo!! rnustbe supersonic. hich mians rhrrthe nozzle s opcrating r i l i rh irdcr ir icu lpoint Th e inlercondi(ionr nd nozzle reu lr i i rf ix condit ions r locrrionJ. \\ t can then ind p. at rhe ip of the Fanno ine_ lhenrh e ir i icpjlp' cn nbe computed nd rhe \'{ach umbe.afrer he shock s found rom thaFanno xbla.This solution robabl l.would ot ha\.e ccured to uj ha dwe no tdia*.n he _s uragram noreco-gnizedha tpoint5 ison thessmeFunno ine asJ.4. and'_For Al/At - 1..1,l1r = I. l andpj,/p,r = 0.063-10.\t proceedmmediarelro conrlure

i : =+ r + 4 - 4 = f : ) , , f _ : , ) ( o l r L r= r?o j op ' p . p : \ p i p ' \ s . / \ u . rb i l /From the F:rnno rbl. \r e nnd ther tl r - 0 619.anLl her from th eshcck 3ble..f1r l . lS9Retuming o the Fannoable,/Lra,!,/D = 0.-1099nC Zr,r*/D = 0.2_tSl. huj

= 0 . 1 0 9 90 l i s t = 0 . 1 ; lD D9.8 FRICTIONCHOKINGIn Chapter rv ediscussedheoperation f nozzleshatwere ed by consranrragna-tion nletcanditionsse eFigures .6 and5.8). Ve ound ha tas he eceivcr ressurewas lowered, he flow through the nozzie increased.When the operating pressLtrercrio eached certain aLue,hesection f minimumarea eveloped vlach umberof unity.The nozzlewa s he nsaid o be choked.Further eductionn the pressureratio did not increasehe flow fate. This wasan exampleof area choking.

t

Tlp rl/)

Figure9.9 Co{erginBnozzle ndconsrlnt-arcauctcombinarion.


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4 = constant

sFigure 9.10 T-s diagram or nozzle-doc! ombinalion

The subsqnic anno low situationsquitesimilar'Figure '9 shows given engthof duct fed by a large tank and convergingnozzle lf the receiverpressures belo$the tank p

gas dynamic and turbine machine

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