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Fluid Mechanics - Course 123COMPRESSIBLE FLOW

Flow o f compress ible f lu ids in a p ~ p e invo lves not on lychange o f p ressu re in the downstream d i rec t ion but a lso achange o f both dens i ty o f the f lu id and th e ve loc i ty of f low.The s i t u a t i on i s fu r the r compl ica ted by th e f ac t t h a t hea tmay be t r ans fe r red through the walls o f the p ipe .I f the pipe i s wel l i nsu la t ed the hea t t r an s fe r may beneg l ig ib l e and the changes t he re fo r e a d ia b at ic (but not , o fcourse , i sen t rop ic ) . In sho r t pipes where no spec i f i c provis ioni s made fo r hea t t r an s fe r th e cond i t ions may approximate toad iaba t ic . On the other hand, fo r f lows a t low ve loc i t i e sin long, uninsulated pipes , an appreciable amount of hea t may

be t r ans fe r red through the p ipe walls and if the t empera tu resi ns ide and outs ide th e pipe a re s imi la r th e flow may beapproximate ly i so thermal . This i s so , fo r example in longcompressed-ai r pipe l ine s and in low-veloci ty f lows gene ra l ly .ADIABATIC FLOW

Gas flow through a pipe o r c on sta nt a re a duct i s consideredsUbjec t to the fo l lowing assumpt ions:1) Pe r f e c t gas (constant spec i f i c heats )2) S te ady , on e- di m en s ion al flow3) Adiabat ic flow (no hea t t r an s fe r through th e w alls)4) F ric tio n fa cto r i s cons tan t over th e leng th o f thecondui t5) Changes in e leva t ion a re i n s i gn i f i c an t in r e l a t i on tof r i c t i ona l e f f e c t s .6) No work i s added to , o r e xtra cted from the flowThe con t ro l l ing equat ions are those o f con t inu i ty , energy,momentum and s t a t e .From 1 s t Law o f thermodynamics, in a system where the massi s cons tan t , the amount o f hea t suppl ied to the system i s equalto th e inc rease in energy o f th e system plus a l l the energywhich l eaves the system as work i s done.Thus ~ Q = ~ E + AW

~ Q = hea t added~ W = work done~ E = change in L, k ine t i c , po ten t i a l and i n t e rna l energies

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I f we cons ider work done by a f lu id ~ W i s not the only workdone. Work i s done in overcoming the p re ss ure f or ce s, t h i s ca l ledI f low-work I

ds i s incrementa l dis tanceI f we consider the f low work in spec i f i c energy terms ~ . e .ene rgy /un i t mass we ge t flow work

= P2A2ds 2 - P,A,ds 1dm dm= P2

a- PIe e

Thus the whole equat ion now becomes:Q = + + gZ2) - + + gZI) + U2 - UI + WIQ i s th e hea t added to the f lu id per un i t massWI represen t s the work done by th e f lu id per un i t massu + P may be wri t t en as spec i f i c en tha lpy Ih leThus h + V2 + gZ = cons tan t along a s t reaml ine if no hea t i s

2"added o r s ub tra cted from th e f lu id and no mechanica l work i s done.I f the f lu id i s a pe r f ec t gas h = CpT.I f t he re i s no change in e l eva t ion then cons tan t ho

where ho i s the t o t a l head o r s ta gn atio n enthalpyQm = eAVThus ho = h + 0.5 Qm 2AFor given values of Qm, A and th e s ta gn at io n en tha lpy ho,curves o f h aga ins t ecould be p lo t ted using th e previous equat ion .

For a per fecL

A more s i gn i f i c an t r e la t ionsh ip ex i s t s between h and spec i f i cen t ropy S. Entropy, l ike h & e i s a func t ion o f s t a t e and so maybe determined from values of h & e.

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Sta r t ing from a spec i f i ed s t a t e , po in t 1 in th e diagram,the curve o f h aga in s t S t r a ce s th e s t a t e s through which thesubstance must pass in an adiabat ic proce s s .

h (1)asymptote J ) :b .o_? _------- ~ ~ < 1Curve. for - - ) M=llarger m/A /'

, ) _ - " ' ~ ~ > l

sThis curve i s ca l l ed a Fanno curve (Gino Fanno I t a l i anEng.) All Panna curves show a maximum value o f S. It maybe shown t h a t the spec i f i c en t ropy i s a maximum when th e Machnumber i s uni ty .Mach No. M = r a t i o - ve loc i ty o f the f l u i d to theve loc i ty o f sound in th e f lu id

= Va (dimensionless)The upper pa r t o f th e curve which approaches th e s ta gn at io n

enthalpy ho corresponds to subsonic flow and th e lower branch tosuperson ic f low.Since , fo r ad iaba t ic c on dit io ns th e en t ropy cannot decrease ,f r i c t i on ac t s to inc rease the Mach no. in subsonic flow and r e -duce the Mach no. in superson ic f low. As f r i c t i on invo lves ac on tin u al i nc re as e in e ntro py , s on ic ve loc i ty can only be reacheda t the ex i t of the p ipe , i f a t a l l .I f son ic ve loc i ty i s to be reached in a pa r t i cu l a r pipethen , fo r given i n l e t cond i t ions and ou t l e t pre ssure , a ce r t a i nl eng th i s necessary . I f the ac tua l l eng th i s l e s s than t h i s" l imi t ing" value , son ic cond it ions are no t reached .I f the len gth o f th e pipe i s increased beyond the l im i t i ngvalue , an i n i t i a l l y subsonic flow wi l l be choked, t h a t i s , thera t e o f flow wi l l be reduced so as again to give sonic cond i t ionso f the ou t l e t . An i n i t i a l l y superson ic flow wi l l a lso be adjus tedto give so nic c on d it io n s a t the ex i t ; a normal shock wi l l form

near the end o f th e pipe and th e resu l t ing subsonic flow wi l lacce l e r a t e to son ic con dit ions a t the ex i t . Fur ther inc rease

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in l ength would cause the shock to move towards the i n l e t o f th epipe and then i n to th e nozzle producing th e s up er so nic flow sot ha t the flow would become en t i r e l y sl lhsonic in the pipe . Thusthe maximum f low ra te o cc urs when M = 1.

The fo l lowing t ab le shows how proper t i e s vary fo r ad iaba t icflow with f r i c t i on .

Proper ty

Mach No.Spec i f i c en tha lpyve loc i t yDensi tyTemperaturePressureRe = v ~

]..I

Stagnat ion temp

M

hv

T

P

T

To

Subsonic FlowIncreasesDecreasesIncreasesDecreasesDecreasesDecreasesIncreases

Constant

Supersonic FlowDecreasesIncreasesDecreasesIncreasesIncreasesIncreasesDecreases

Constant

At high ve l oc i t i e s , the ra t e a t which f r i c t i on d i s s i pa t e smechanical energy i s l a rge and supersonic flow in a pipe i sgene ra l ly be t t e r avoided. I f supersonic flow i s subsequen t lyrequi red the gas may be expanded in a convergent - divergentnozz le , where i sentropic f l ~ ~ occurs .

A l ~ t A.- P. -P, p.

In the convergent sec t ion o f a nozz le , the i n i t i a l flow i ssubsonic and th e ve loc i t y i s acce le ra ted as th e gas approachesthe t h roa t which i s the area o f minimum c ross sec t ion . Themaximum ve loc i ty a t the t h roa t occurs when the re i s sonic ve loc i ty

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and t h i s i s known as the c r i t i c a l cond i t i on . The pres su re r a t ioto achieve t h i s i s ca l l ed the c r i t i c a l pres su re r a t io and i s ther a t i o o f the o ressu re a t the t h roa t in r e l a t i on to th e s taanat ion. - - - - - - - - - - - - - oJPressu re . i . e . the pressure with no f low.yy - 1

(y = r a t io o f spec i f i c hea t s ~ )Cvy fo r a i r = 1.4 and c r i t i c a l pres su re r a t io i s 0 .528 , whereas fo r superheated steam y 1 .3 and c r i t i c a l pressr a t io 0.546.I f the pressure o f a vapour f a l l s below sa tura t ion pressurethe vapour no longer performs a t a pe r f e c t gas . y changes andthe equa t ion fo r Pe may no longer be app l i ed .

PoThus the maximum flow r a te i s a funct ion o f th e c r i t i c a lpressure r a t io and th e t h roa t area . No mat ter how the downstream pres su re i s reduced t h i s flow r a t e may no t be exceeded.The nozzle i s now in the "choked" condi t ion . For the flow toa l t e r in response to a change in downstream pres su re a r a r e -fac t ion wave has to t r ave l upstream to the i n i t i a l cond i t ion .However the ve loc i ty o f the r a r e f ac t i on , i s a t maximum, onlyt h a t o f sonic ve loc i ty and obvious ly if the flow a t the t h roa ti s a lre ad y so nic then there i s no way t h a t the upstream pres su remay be s ens i t i ve to th e downstream p re s su r e .The pressure d i s t r i bu t i on in a c on ve rg en t n oz zl e may beseen in the diagram. At ex te rna l pres su re P a there i s no ~ pand no f low. At pres su re Pb the flow i s subsonic a t the t h roa t .At pressure Pc, when the c r i t i c a l pres su re r a t io i s reached , theflow i s sonic . As the o ut si de p re ss ur e i s fur the r d ecre ase d th eflow through the nozzle remains cons t an t and has an ex i t pressureequa l to Pc. Fur the r expansion t akes place ou t s ide the nozzle topressure Pd. In th is s itu a t io n , when the re i s no f u r t h e r responseby flow r a t e , to the lowering o f ac tua l pressu re , th e nozzle i ssa id to be choked.

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~o ----- +-c=::::::>-~,LlL,~ ,* - - _ . ~

I dob----- - - - - - ! - - - - -x Pressure distributions in a convergingnozzle.

S l a g n a ~presSIJ(e P2

Po

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In the divergent sect ion the pressure downstream from thet h roa t var ies with ou t l e t pre ssure . Consider Po to be f ixedbu t the ou t l e t pressure P2 i s va r i ab l e . Consider P 2 = Po thenthe re would be no flow and pressure through the nozzle wouldbe Po represented by l i ne OB. Reduction in Pz causes areduc t ion in the pre ssure a t the end of the nozzle and as theve loc i ty a t the t h roa t i s nowhere near son ic t h i s reduc t ion i ss ee n ups tr eam from th e t h roa t and a pressure l ine ODE i sobta ined.

Fur ther decrease o f ex te rna l pressure in crea se s the ve loc i tya t the t h roa t o f the nozzle and reduces the t h roa t pressure un t i lPz corresponds to point F. At th i s po in t the throat ve loc i ty i sthen sonic and the pressure d i s t r i bu t ion i s represented OCF.

P2 i s fu r the r reduced the condi t ions in the convergent pa r tof the nozzle remain unchanged and th e upstream pre ssure d i s t r i bu t ion fo llow s the s ing le curve OC. I f the ex te rna l pressureP z exac t ly correspond to po in t F then there i s an ad iaba t iccompression fo l lowing curve CF and the ve loc i ty downstream ofthe t h roa t i s en t i r e l y subsonic . Only t h i s value of th e ex t e rna lp res su re , however, al lows a compression from th e po in t C. Itshould be noted t h a t cont inuous i sen t rop ic expansion a t superson ic ve loc i ty i s only poss ib le accord ing to the curve CG.

pressure r a t i o" o f the nozz le .

In a nozz leexpansion OCG i s designed to produce supersonic flow the smooththe i d ea l and the r a t i o !:o. i s the "designp GIt i s apparent t ha t there ex i s t s a range o f ex i t pre ssure sfo r which i sen t rop ic flow through a convergen t -d ivergen t i s notposs ib le . Experiment shows t h a t under t he se c ond itio n s the flowin the nozzle undergoes, a t some po in t in the diverg ing sec t ion ,an abrupt change from supersonic to subsonic ve loc i ty . Thischange i s accompanied by l a rge and abrup t r i s e s in pre ssure ,dens i ty and t empera ture . The zone in which these changes occuri s so t h in t h a t fo r ca lcu la t ions ou ts ide th e zone it may beconsidered as a flow d i scon t inu i ty . This d iscon t inu i ty i sca l l ed a "normal shock wave" ( i . e . a shock wave perpendicula rto the flow d i r e c t ion ) . The th ick ness o f the wave i s around10- m. In t h i s wave the grad ien t s of ve loc i ty and dens i ty areso s teep t h a t viscous ac t io n, hea t conduct ion and mass d i f fus ion

are a l l apprec iable with th e r e su l t t h a t the flow undergoes ala rge ent ropy inc rease as it passes through the wave.The normal shock wave i s only a spec ia l case o f the broaderdass o f flow d i sc o n ti n u it ie s c a ll ed "obl ique shock waves" t h a tare found in most supersonic f lows, both i n t e rna l ( in duc t s ,

p ipes , j e t engine in takes and compressors) and ex te rna l (overthe surfaces o f wings, e t c . )

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For e x te rn a l p re ss ur es , a t the nozz le , between F & G flowcannot take p lace w ith ou t th e formation of a shock wave andconsequent d i s s ipa t i on o f energy. In such circumstances th enozzle i s sa id to be " ov er exp anding ".P 2 i s reduced s l i gh t ly below F a normal shock wave i sformed downstream of the t h roa t . The pressure fol lows curve

CG only as fa r as 8 , then t he re i s an abrup t r i s e o f pressurethrough the shock (8182) and th en s ub so nic dece le ra t ion of th eflow with r i se o f pre ssu re to PH.As th e ex te rna l pressure i s lowered the shock moves

further from the throa t un t i l when P z = Pk l the po in t 51 hasmoved to G and 8 2 has moved to k. For ex i t press l e ss thanPk , the flow with in the en t i r e divergen t sec t ion o f the nozzlei s supersonic and fol lows l i ne CG.I f P2 l i e s between k and G a compression must occur out s idethe nozzle to r a i s e the pressure from PG to th e ex te rna l p re ssure .This c ompre ss io n in vo lv es o bli qu e shock waves which cannot bedea l t with in 'one dimensional ' flow te rms.When th e ex te rna l pressure i s below PG, the nozzle i s sa idto be under e xp andi ng and the expansion to pressure P2 t akesplace outs ide th e n ozzle ag ain by obl ique expansion.

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ASSIGNMENT

1) Discuss how flow i s a f f e c t e d by p r e s s u r e i n a convergentn o z z l e .2) Discuss how supersonic flow may be o b t a i n e d from aconvergent d i v e r g e n t n o z z l e .3) wny i s it i m p o s s i b l e t o o b t a i n s u p e r s o n i c flow i n aconvergent n o z z l e .4) Why do we need t o use n o z z l e s i n e n g i n e e r i n g .

J . I r w i n - C h i l d s

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