(h+ ) = 0, (3.1) s = s 0, (3.2)

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1 Chpter 3 Nozzle Flow Qusistedy idel gs flow in pipes For the lrge vlues of the Reynolds number typilly found in nozzles, the flow is idel. For stedy opertion with negligible body fores the energy nd momentum equtions n be projeted long given stremline to give s l = 0 nd v h+ ) = 0, 3.) l inditing tht the entropy s nd the sum h+v / remin onstnt, i.e., nd s = s 0, 3.) h+ v = h ) If the nozzle is fed from ontiner with uniform thermodynmi properties s nd h, we my evlute the bove epressions by following the stremline upstrem into the ontiner to yield s 0 = s nd h 0 = h. Sine the nozzles re slender, in tht the rtio of their length L nd hrteristi dimeter D stisfies L/D, the resulting stremlines re ligned, with longitudinl nd trnsverse veloity omponents v T /v L D/L, s follows from the ontinuity eqution. Correspondingly, the hrteristi hnges of pressure ross the nozzle setion δ T p ρ vt re muh smller thn the hrteristi pressure vrition long the nozzle δ L p ρ vl ording to δ Tp/δ L p D/L), so tht one my ssumed tht the pressure remins uniform ross eh pipe ross setion. With s = s everywhere in the pipe nd with p = p), where is the distne mesured long the nozzle, it is ler tht ll other thermodynmi vribles e.g., h = hs,p), T = Ts,p), ρ = ρs,p),... ) re lso only funtion of. For instne, hs,p) = hs,p)) = h). This lst result n be used in 3.3) to demonstrte tht v = v). In view of the bove onsidertions, the solution n be determined from 3.) nd 3.3) together with ṁ = ρ)v)a), 3.4)

2 where ṁ is the onstnt mss flow rte irulting in the pipe. Equtions 3.) nd 3.3) n be written with use mde of the equtions of stte in terms of the different stgntion flow vribles, ording to h 0 h) = T ) ) γ )/γ ) γ 0 T) = 0 p0 ρ0 = = = + γ M), 3.5) ) p) ρ) where h 0 = h, T 0 = T, o =, p 0 = p, nd ρ 0 = ρ n be evluted from the ontiner properties. Similrly, it is onvenient to rewrite 3.4) in term of the lol Mh number M) to give ṁ = ρ 0 0 A)M) + γ Subsoni nd supersoni flow in pipes M) γ ). 3.6) It is of interest to omment on the effet of the Mh number on the hrter of the fluid motion. For liquid flow in pipe of vrying ross setion, our intuition tells us tht s the ross setion dereses the veloity must inrese to ommodte the onstnt volume flu Q = v)a), while the orresponding pressure p) must derese to provide the needed elertion, s follows from Bernoulli s eqution p+ρ o v / =onstnt. Things re not lwys quite so intuitive for gs flow, for whih the resulting behvior depends on whether the flow is subsoni or supersoni. To see how hnges in ross-setion re ffet the elertion, we begin by differentiting 3.4) to give dρ ρd + dv v d = da A d. 3.7) On the other hnd, for perfet gs, the ondition of isentropi flow 3.) gives p ρ γ = p 0, 3.8) ρ γ 0 providing dρ ρd = dp ρd upon differentittion, with = γp/ρ. Sine dp ρd = dh d 3.9) 3.0) s follows from the ondition of isentropi flow T ds = dh dp/ρ = 0 nd with obtined by differentiting 3.3), 3.7) finlly yields M ) v dh d = vdv d, 3.) dv d = da A d. 3.) 6

3 For subsoni flow M < ) long onvergent nozzle da/d < 0) the veloity inreses, s n be seen from 3.), nd the enthlpy, temperture, pressure, density, nd veloity of sound derese, s seen in 3.5) nd 3.). With v inresing nd deresing the ssoited Mh number M = v/ ontinuously inreses s the ross setion dereses. Beuse of the presene of the ftor M ) in 3.) the opposite behvior is found for supersoni flow in onvergent nozzle, tht is, the flow deelertes nd the Mh number dereses, while h, T, p, ρ, nd inrese. It is therefore not possible to elerte subsoni strem beyond soni onditions with use mde of onvergent nozzle. Observtion of 3.) revels tht divergent nozzle da/d > 0) is required to elerte supersoni flow. It lso indites tht, if soni onditions M = re rehed t given setion, t tht lotion the ross-setion re must stisfy da/d = 0. These findings suggest tht supersoni flow n be therefore hieved by utilizing onvergent-divergent nozzle so tht the flow elertes through the onverging streth to reh soni onditions t the throt the setion of minimum re), further elerting downstrem into the supersoni regime in the diverging streth. We shll see tht, depending on the onditions, the flow in onvergent-divergent nozzles my inlude number of ompliting fetures, inluding shok nd epnsion wves, to be treted in lter hpter. Critil mgnitudes For stedy gs flow in pipe 3.5) provides the vrition of the different thermodynmi vribles with M. It n be used to determine, for instne, their ritil vlues h 0 h = T 0 T = 0 ) = p0 p ) γ )/γ = ρ0 ρ ) γ = γ ) found t setion where the flow is soni. At tht ritil setion, 3.6) provides ) γ + ṁ = ṁ = ρ 0 0 A γ ), 3.4) whih n be written in the form A A = [ + γ )] M M γ + γ ) 3.5) thereby relting the vlues of A nd M found t given setion with the re A of the setion where soni onditions would be ttined. Flow in onvergent nozzle We begin by nlyzing the dishrge of gs ontiner with known gs properties p, ρ, T,,...) through onvergent nozzle of minimum ross setion AL) t the outlet = L into surrounding gs tmosphere t mbient pressure. The solution, depending on the thermodynmi properties in the ontiner nd on the mbient pressure, n be determined with use mde of 3.5) nd 3.6) with T 0 = T, o =, p 0 = p, nd ρ 0 = ρ. An outflow 7

4 is estblished whenever < p. The flow is strongly subsoni for smll pressure differenes p p, when the gs dishrges into the outer tmosphere s low-mh-number jet with pressure pl) = t the eit setion = L. This lst ondition enbles the Mh number ML) to be omputed from 3.5) ording to ) γ )/γ = + γ ) [ / ML) ML) = γ ) γ )/γ to be substituted into 3.6) to determine the gs flow rte through the nozzle ] /, 3.6) ṁ = ρ AL) γ ) / γ [ )γ γ ] /. 3.7) As the vlue of p / inreses so does the vlue of ML) given by 3.6) nd lso the vlue of ṁ given by 3.7), with vritions given in Fig. 3. for γ =.4. Using the vlue of ṁ, omputed for given vlue of p / from 3.7), in 3.6) enbles the Mh number distribution to be obtined long the nozzle, wheres 3.5) provides the ompnying distributions of pressure, density, nd temperture. Smple shemti profiles re given in Fig γ =.4 M L) m ρ AL) p / Figure 3.: The vrition of the eit Mh number ML) nd mss flow rte ṁ with p / s obtined from 3.6) nd 3.7) for γ =.4. As disussed bove, the Mh number t the eit setion inreses for inresing vlues of p /. The jet solution with pl) = nd the omputtion proedure desribed bove ontinue to hold s long s remins bove ritil hoking vlue p CH t whih the flow rehes soni onditions t the outlet setion. This hoking vlue is determined from 3.6) with ML) = s p p CH = ) γ + γ/γ ) 3.8) giving p /p CH.893 for γ =.4 p CH 0.53p ). For these hoking onditions the mss flow 8

5 rte rehes the ritil vlue ṁ ) γ + ṁ γ ) = ρ AL), 3.9) given in 3.4). The flow nnot elerte pst soni onditions in the onvergent nozzle by deresing the mbient pressure. Insted, the solution beomes hoked, tht is, for vlues of < p CH the flow is soni t the eit, nd beomes independent of the outer pressure. In prtiulr, the mss flow rte, whih for > p CH inreses for inresing vlues of p /, remins equl to ṁ, independent of, for < p CH. In the hoked flow tht emerges for p / > [γ +)/] γ/γ ), the ondition ML) = therefore reples pl) =. The pressure t the eit setion is pl) = p CH >, so tht the jet flow with prllel stremlines found downstrem from the pipe outlet for > p CH is repled for < p CH by omplited flow pttern inluding epnsion wves tht deflet the stremlines outwrds. ρ L p p /p p/p p /p p /p = 0.53 CH M M= M M Figure 3.: Pressure nd Mh number distributions long nozzle for different vlues of /p. In summry, the dishrge of ontiner of pressure p into the tmosphere, t pressure, through onvergent nozzle of minimum ross setion AL) depends on the vlue of p /, so tht Note tht the derese of the mbient pressure below p CH nnot be felt within the nozzle, beuse the flow t the eit is lredy soni, nd therefore the perturbtions nnot trvel upstrem, so tht for < p CH the flow in the nozzle beomes independent of. 9

6 If ) γ/γ ): pl) = ) [ / ) ] γ )/γ / ML) = γ ) / [ )γ ṁ = ρ AL) γ γ γ L p ρ ] / If > ) γ/γ ): ML) = γ/γ ) pl) = p CH = > ṁ = ṁ = ρ AL) γ ) L p ρ Flow in onvergent-divergent nozzles The flow in onvergent-divergent nozzles is more omplited. The nozzle geometry is defined by the vlues of the eit re A e nd throt re A t the setion of minimum ross-setion re). The nozzle is ssumed to be onneted to reservoir, where the gs pressure is p 0, nd dishrges to the tmosphere, where the pressure is. Different solutions emerge depending on the vlue of /p 0, s indited in Fig Let us begin by onsidering the se where the mbient pressure is only slightly smller thn p 0 se in Fig. 3.3). As result of the pressure differene, there ppers slow isentropi stedy outflow. For the resulting subsoni flow, the pressure dereses s the Mh number inreses in the onvergent prt of the nozzle, nd opposite behviors re observed in the divergent streth. The minimum pressure nd the mimum Mh number re therefore rehed t the throt. The flow dishrges to the mbient s subsoni jet. The ondition tht the pressure t the eit equls the mbient vlue n be used to ompute the Mh t the eit setion from p0 ) γ )/γ = + γ Me, 3.0) 30

7 M < t M < e p 0 b M = t M < e.0 p p 0 b M = t M < e 0.58 d e f g d M = t e M = t M > e M.0 f d f M = t M > e b g M = t M > e Figure 3.3: Flow in onvergent-divergent nozzle. 3

8 whih in turn determines the mss flow rte M e ṁ = ρ 0 0 A e M e + γ ) / p0 = ρ 0 0 A e γ γ ) γ [ p0 )γ γ ] /, 3.) with the stgntion properties for the flow being those found in the reservoir. The Mh number t the throt M t > M e n be obtined from A e M e + γ M e γ ) = At M t + γ Mt γ ) 3.) obtined by equting the vlues of ṁ evluted t the throt ṁ = ρ 0 0 A t M t + γ M t γ ) 3.3) nd t the eit. The vlue of M t ontinues to inrese for deresing vlues of /p 0, rehing soni onditions when = p CH se b in Fig. 3.3). The vlue of p CH /p 0 n be determined s funtion of A t /A e by equting the mss flow rte t the throt, ) γ + γ ) ṁ = ρ 0 0 A t 3.4) evluted from 3.3) with M t =, to tht rossing the eit setion, evluted from 3.), yielding ) A ) t γ + γ ) / [ p0 )γ / p0 γ γ = ]. 3.5) A e γ It n be seen tht, for given vlue of A t /A e, this lst eqution is stisfied by two different vlues of /p 0. One of the two solutions, orresponding to the hoking vlue = p CH, is ssoited with n eit Mh number M e <, to be evluted from 3.0) with = p CH. There eists, however, seond isentropi solution with < p CH in whih the flow dishrges s supersoni jet with p e = = p SJ nd M e >. For instne, for A t /A e = the two solutions of 3.5) nd 3.0) re /p 0 = p CH /p 0 = with M e = 0.9 nd /p 0 = p SJ /p 0 = with M e =.5. The results indite tht the onvergent-divergent nozzle n be hoked more esily thn the onvergent nozzle note tht for A t /A e = vlue /p 0 = is suffiient to hoke the flow, to be ompred with the vlue /p 0 = 0.58 required in onvergent nozzles). The flow remins hoked for < p CH, whih implies tht, regrdless of the vlue of, the flow in the onvergent prt of the nozzle is identil, with the mss flow rte tking lwys the onstnt vlue 3.4). In the isentropi solution dishrging s supersoni jet se f in Fig. 3.3) the flow is therefore subsoni in the onvergent streth, soni t the throt, nd supersoni in the 3

9 divergent streth, leving the nozzle with p e =, so tht neither epnsions nor ompressions re needed to dpt the pressure to the mbient vlue. It is of interest to onsider wht hppens when the mbient pressure is below p SJ nd lso when it tkes intermedite vlues in the rnge p SJ < < p CH. For < p SJ se g in Fig. 3.3), the flow onditions t the eit re etly those found for = p SJ, so tht p e = p SJ > underepnded flow). An epnsion is therefore needed outside the nozzle to derese the pressure to the mbient vlue. The epnsion wve tht ppers is lolly plnr ner the nozzle rim nd therefore orresponds there to Prndtl-Meyer epnsion, whih deflets the strem outwrds s it dereses its pressure to while inresing its Mh number beyond tht found t the eit setion. Similrly, when the mbient pressure is slightly bove p SJ se e in Fig. 3.3), the flow onditions t the eit re lso etly those found for = p SJ. In this se, however, the flow is overepnded, nd shok wve is needed to inrese the pressure to the mbient vlue. When is only slightly bove p SJ, n oblique wek shok is suffiient. As the vlue of further inreses, the needed shok wve beomes stronger, nd the ssoited inidene ngle inreses towrds β = π/. There is ritil vlue of for whih the ompression ours through norml shok wve stnding t the eit setion se d in Fig. 3.3). For even lrger vlues of the shok wve migrtes into the divergent streth of the nozzle, so tht downstrem from the throt there is region of supersoni flow tht ends t the inner shok, whih is followed by region of subsoni flow eventully dishrging to the mbient s jet, s indited in Fig These stedy solutions inluding plnr shok wves within the divergent prt of the nozzle re hrd to observe in relity, beuse they re often ffeted by boundry-lyer seprtion immeditely downstrem from the shok, whih my led to signifint hnges in the downstrem flow struture, inluding t times unstedy flow vritions nd symmetri flow ptterns. 33

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