We do not consider too small nozzles, say with chamber size <10 mm and neck size <1 mm, where the effect of boundary layers become predominant.

Transcription

1 NOZZLES Nozzls... 1 Nozzl flow quaions... 4 Chokd flow... 5 Ara raio... 6 Convrging nozzl... 7 Convrging-divrging nozzl... 9 Disconinuiis in nozzl flow: normal and obliqu shocks, xansion fans. Mach diamonds Arosik nozzl NOZZLES A nozzl (from nos, maning 'small sou') is a ub of varying cross-scional ara (usually axisymmric) aiming a incrasing h sd of an ouflow, and conrolling is dircion and sha. Nozzl flow always gnras forcs associad o h chang in flow momnum, as w can fl by handholding a hos and oning h a. In h simls cas of a rock nozzl, rlaiv moion is crad by jcing mass from a chambr backwards hrough h nozzl, wih h racion forcs acing mainly on h oosi chambr wall, wih a small conribuion from nozzl walls. As imoran as h rollr is o shaf-ngin roulsions, so i is h nozzl o j roulsion, sinc i is in h nozzl ha hrmal nrgy (or any ohr kind of high-rssur nrgy sourc) ransforms ino kinic nrgy of h xhaus, and is associad linar momnum roducing hrus. Th flow in a nozzl is vry raid (and hus adiabaic o a firs aroximaion), and wih vry lil fricional loss (bcaus h flow is narly on-dimnsional, wih a favourabl rssur gradin xc if shock wavs form, and nozzls ar rlaivly shor), so ha h isnroic modl all along h nozzl is good nough for rliminary dsign. Th nozzl is said o bgin whr h chambr diamr bgins o dcras (by h way, w assum h nozzl is axisymmric, i.. wih circular cross-scions, in si ha rcangular cross-scions, said wo-dimnsional nozzls, ar somims usd, aricularly for hir as of dircionabiliy). Th mridian nozzl sha is irrlvan wih h 1D isnroic modl; h flow is only dndn on cross-scion ara raios. Ral nozzl flow dars from idal (isnroic) flow on wo ascs: Non-adiabaic ffcs. Thr is a kind of ha addiion by non-quilibrium radical-scis rcombinaion, and a ha rmoval by cooling h walls o k h srngh of marials in longduraion rocks (.g. oraing mraur of cryognic SR-5 rocks usd in Sac Shul is 350 K, abov sl vaorizaion mraur of 3100 K, no jus mling, a 1700 K). Shorduraion rocks (.g. solid rocks) ar no acivly coold bu rly on ablaion; howvr, h nozzl-hroa diamr canno l widn oo much, and rinforcd marials (.g. carbon, silica) ar usd in h hroa rgion. Nozzls 1

2 Thr is viscous dissiaion wihin h boundary layr, and rosion of h walls, wha can b criical if h rosion widns h hroa cross-scion, graly rducing xi-ara raio and consqunly hrus. Axial xi sd is lowr han calculad wih h on-dimnsional xi sd, whn radial ouflow is accound for. W do no considr oo small nozzls, say wih chambr siz <10 mm and nck siz <1 mm, whr h ffc of boundary layrs bcom rdominan. Rsricing h analysis o isnroic flows, h minimum s of inu aramrs o dfin h roulsiv roris of a nozzl (h hrus is h mass-flow-ra ims h xi sd, F = mv ) ar: Nozzl siz, givn by h xi ara, A; h acual ara law, rovidd h nry ara is larg nough ha h nry sd can b nglcd, only modifis h flow insid h nozzl, bu no h xi condiions. Ty of gas, dfind wih wo indndn roris for a rfc-gas modl, ha w ak as h hrmal caaciy raio c/cv, and h gas consan, R Ru/M, and wih Ru=8.314 J/(mol K) and M bing h molar mass, which w avoid using, o rsrv h symbol M for h Mach numbr. If c is givn insad of, hn w comu i from c/cv=c/(c R), having usd Mayr's rlaion, c cv=r. Chambr (or nry) condiions: c and Tc (a rlaivly larg chambr cross-scion, and ngligibl sd, is assumd a h nozzl nry: Ac, Mc 0). Insad of subscri 'c' for chambr condiions, w will us '' for oal valus bcaus h nrgy consrvaion imlis ha oal mraur is invarian along h nozzl flow, and h non-dissiaiv assumion imlis ha oal rssur is also invarian, i.. T=Tc and =c. Discharg condiions: 0, i.. h nvironmnal rssur (or back rssur), is h only variabl of imoranc (bcaus rssur wavs roaga a h local sd of sound and quickly nd o forc mchanical quilibrium, whras h nvironmnal mraur T0 roagas by much slowr ha-ransfr hysical mchanisms). Do no confus discharg rssur, 0, wih xi rssur,, xlaind blow. Th objciv is o find h flow condiions a h xi [,T,v] for a givn s of h abov aramrs, [A,,R, c,tc,,0], so ha: m va va F mv M v = ρ =, =, = (1) RT RT If h nozzl flow is subsonic, hn h xi rssur coincids wih h discharg rssur, =0, a h sady sa (if a an iniial sa hy wr no qual, h im i would ak o qualis is of h ordr of h nozzl lngh dividd by h sound sd), and h ohr variabls would b obaind from h isnroic rlaions, i..: Nozzls

3 1 T v 1 = 0, = = 1+ = 1+ T ct M () Convrging nozzls ar usd o acclra h fluid in subsonic gas srams (and in liquid js), sinc a low sds dnsiy do no vary oo much, and m = ρva = cons can b aroximad by va=cons. Liquid js and low sd gas flows can b sudid wih classical Brnoulli quaion (unil caviaion ffcs aar in liquid flows), bu high-sd gas dynamics is dominad by comrssibiliy ffcs in h liquid. By h way, w do no considrd hr mulihas flow in nozzls. Bu whn h flow is sursonic a som sag (vn jus a h xi), 0, and a mor daild analysis is rquird. Bfor dvloing i, l summaris h rsuls. A convrging nozzl can only bcom sursonic a h xi sag; h sd incrass monoonically along h nozzl. If a convrging nozzl is fd from a consan rssur consan mraur chambr, h flow ra grows as h discharg rssur is bing rducd, unil h flow bcoms sonic (chokd) and h flow ra no longr changs wih furhr dcrasing in discharg-rssur (a s of xansion wavs adjus h xi rssur o his lowr discharg rssur). Exc for old-im urbojs and miliary fighr aircraf, all commrcial j ngins (afr Concord was rird) us convrging nozzls discharging a subsonic sd (boh, h ho cor sram and h coldr fan sram). A convrging-divrging nozzl ('condi' nozzl, or CD-nozzl), is h only on o g sursonic flows wih M>1 (whn chockd). I was dvlod by Swdish invnor Gusaf d Laval in 1888 for us on a sam urbin. Sursonic flow in CD-nozzls rsns a rich bhaviour, wih shock wavs and xansion wavs usually aking lac insid and/or ousid. Svral nozzl gomris hav bn usd in roulsion sysms: 1. Th classical quasi-on-dimnsional Laval nozzl, which has a slndr gomry, wih a raidly convrging shor nranc, a roundd hroa, and a long conical xhaus of som 15º half-con angl (h loss of hrus du o j divrgnc is abou 1.7%). Rarly usd in modrn rocks.. Bll-sha nozzls (or arabolic nozzls), which ar as fficincy as h simls conical nozzl, bu shorr and lighr, hough mor xnsiv o manufacur. Thy ar h rsn sandard in rocks;.g. h Shul main ngin (SME) nozzls yild 99% of h idal nozzl hrus (and h rmaindr is bcaus of wall fricion, no bcaus of wall sha ffc). 3. Annular and linar nozzls, dsignd o comnsa ambin rssur variaion, lik h Arosik nozzl. Thy ar undr dvlomn. W rsn blow h 1D modl of gas flow in nozzls. For mor ralisic dsign, byond his siml modl, a D (or axisymmric) analysis by h mhod of characrisics and boundary layr ffcs should follow, o b comld wih a full 3D nozzl-flow analysis by CFD. Nozzls 3

4 NOZZLE FLOW EQUATIONS L us considr h sady isnroic 1D gas dynamics in a CD-nozzl, wih h rfc gas modl (i.. V=mRT and, aking T=0 K as nrgy rfrnc, h=ct). Consrvaion of mass, momnum, and nrgy, in rms of h Mach numbr, M vc(whr c = RT sands for h sound sd), bcom: ( ) d d d d m = ρva = cons= M RT A T + M + A = 0 (3) RT T M A v h = h+ = cons dv d dh= Tds+ vd dv ρvdv= d + = 0 + dh Ts d = 0 ds= 0 (4) ρ v 1 dt 1 h = h + = = c T + M RT + M + ( ) M M = T cons 1 1 d 0 (5) whr logarihmic diffrniaion has bn rformd. Noic ha, wih his modl, h isnroic condiion can rlac h momnum quaion, so ha diffrniaion of h isnroic rlaions for a rfc gas T/ ( 1)/ =cons, yilds: dt T 1d = (6) Th nrgy balanc ( h=q+w) imlis h consrvaion of oal nhaly and oal mraur (h=ct), and h non-fricion assumion imlis h consrvaion of oal rssur (), wih h rlaions bwn oal and saic valus givn by: 1 v 1 = 1+ = 1+ M = T T ct (7) Noic ha, wih h rfc gas modl, rmains consan hroughou h xansion rocss. Howvr, whn h ngin flow is comosd of ho combusion roducs, ral gas ffcs bcom imoran, and as h gas xands, shifs as a rsul of changs in mraur and in chmical comosiion. Maximum hrus is obaind if h gas comosiion is in chmical quilibrium hroughou h nir nozzl xansion rocss. Choosing h cross-scion ara of h duc, A, as indndn variabl, h variaion of h ohr variabls can b xlicily found from (3)-(6) o b: dt da = M T A (8) d da 1 M = M A (9) dm 1 d 1 A M 1 M = + M A (10) 1 M dv da = v A (11) ( 1 M ) ( 1) ( ) ( ) ( ) Nozzls 4

5 Equaions (8)-(11) show ha: In convrging scions (da<0): o Whn h flow is subsonic (M<1 (1 M )>0): sd incrass (dv>0), Mach-numbr incrass (dm>0), bu rssur and mraur dcras. o Whn h flow is sursonic (M>1 (1 M )<0): sd dcrass (dv<0), Mach-numbr dcrass (dm<0), bu rssur and mraur incras. In divrging scions (da>0): o Whn h flow is subsonic (M<1 (1 M )>0): sd dcrass (dv<0), Mach-numbr dcrass (dm<0), bu rssur and mraur incras. o Whn h flow is sursonic (M>1 (1 M )<0): sd incrass (dv>0), Mach-numbr incrass (dm>0), bu rssur and mraur dcras. Chokd flow Choking is a comrssibl flow ffc ha obsrucs h flow, sing a limi o fluid vlociy bcaus h flow bcoms sursonic and rurbaions canno mov usram; in gas flow, choking aks lac whn a subsonic flow rachs M=1, whras in liquid flow, choking aks lac whn an almos incomrssibl flow rachs h vaour rssur (of h main liquid or of a solu), and bubbls aar, wih h flow suddnly juming o M>1. Going on wih gas flow and laving liquid flow asid, w may noic ha M=1 can only occur in a nozzl nck, ihr in a smooh hroa whr da=0, or in a singular hroa wih disconinuous ara slo (a kink in nozzl rofil, or h nd of a nozzl). Naming wih a '*' variabls h sag whr M=1 (i.. h sonic scion, which may b a ral hroa wihin h nozzl or a som xraolad imaginary hroa downsram of a subsonic nozzl), and ingraing from A o A *, quaions (8)-(10) bcom: T T 1 1+ M = = + 1 T + 1 * 1 T 1 * = T + M T 1 = = * 1 M T * + = 1 T * A A + 1 M = 1 1+ M ( ) (1) (13) (14) whr h xrssions for oal mraur T and oal rssur has bn subsiud o show ha mraur and rssur a h hroa (also known as criical valus), ar jus a funcion of, sinc, for isnroic flows, oal condiions do no chang along h sram. Nozzls 5

6 Alhough h quaions abov aly o all 1D isnroic rfc-gas flows, quaions (1)-(14) mak us of condiions a M=1 (ral or virual), and i is worh analysing scial cass in aricular dail. Ara raio Nozzl ara raio ε (or nozzl xansion raio) is dfind as nozzl xi ara dividd by hroa ara, ε A/A *, in convrging-divrging nozzls, or dividd by nry ara in convrging nozzls. Noic ha ε so dfind is ε>1, bu somims h invrs is also namd 'ara raio' (his conracion ara raio is boundd bwn 0 and 1); howvr, alhough no confusion is ossibl whn quoing a valu (if i is >1 rfrs o A/A *, and if i is <1 rfrs o A * /A), on mus b xlici whn saying 'incrasing ara raio' (w k o ε A/A * >1). To s h ffc of ara raio on Mach numbr, (14) is lod in Fig. 1 for idal monoaomic (=5/3), diaomic (=7/5=1.40), and low-gamma gass as hos of ho rock xhaus (=1.0); gass lik CO and HO hav inrmdia valus (=1.3). Noic ha, o g h sam high Mach numbr,.g. M=3, h ara raio ndd is A * /A=0.33 for =1.67 and A * /A=0.15 for =1.0, i.. mor han doubl xi ara for h sam hroa ara (ha is why sursonic wind unnls ofn us a monoaomic working gas. Fig. 1. Raio A * /A (i.. hroa ara dividd by local ara) vs. Mach numbr M, for =1.0 (big), =1.40 (grn), and =1.67] (rd). Exrcis 1. A sam flow of 0.1 kg/s xands isnroically in a nozzl from a chambr a 300 kpa and 300 ºC o an ousid amoshr a 100 kpa. Find: a) If h nozzl is convrging or convrging-divrging, and h xi Mach numbr. b) Exi ara and minimum ara. Sol.: a) For sam, =1.30 (.g. from c=050 J/(kg K), c/cv=c/(c R)=050/(050-46)=1.9). For isnroic nozzls, =cons and T=consan. Choking mus occur if /<(/(+1) /( 1) =0.55 and in our cas is /=100/300=0.33; hnc, i is a CD-nozzl. Exi mraur is T=T(/) ( 1)/ =445 K (17 ºC). Solving in (13) on finds * =164 kpa and M=1.39. b) Exi ara can b found from (3) sinc, T and M ar known, obaining A=.87 cm. Throa ara can b obaind from (14) wih A=.87 cm and M=1.39, wih a rsul A * =.57 cm. Exi sd is v = M RT =717 m/s. Exrcis. A flow of 100 kg/s of xhaus gass xands in a nozzl wih 0.95 isnroic fficincy from a low-sd nranc a 300 kpa and 400 ºC o an ousid amoshr a 100 kpa. Assuming as avragd valus c=1100 J/(kg K) and =1.35, find: Nozzls 6

7 a) Exi ara, minimum ara, and xhaus Mach numbr and sd, assuming isnroic flow. b) Corrcions du o h sad fficincy. Sol.: a) Choking mus occur if /<(/(+1) /( 1) =0.54 and in our cas is /=100/300=0.33; hnc, i is a CD-nozzl. For isnroic flow, xi mraur is T=T(/) ( 1)/ =506 K, and xi sd ( ) v = c T T =606 m/s, wih Mach numbr M = v RT =1.37. Exi ara can b found from (3) sinc, T and M ar known, obaining A=0.38 m. Throa ara can b obaind from (14) wih A=0.38 m and M=1.37, wih a rsul A * =0.16 m. b) From h dfiniion of isnroic nozzl fficincy, η (h h)/(h hs)= v v, wih h isnroic valus found abov, Ts=506 K and vs=606 m/s, w g Ts=515 K and vs=590 m/s, wih Mach numbr M = v RT =1.33. Exi ara can b found from (3) sinc, T and M ar known, obaining A=0.49 m. Throa ara should b assumd o coincid wih h isnroic valu, A * =0.16 m, sinc viscous dissiaion in h convrging scion would b a small fracion of h oal dissiaion. Convrging nozzl In a convrging nozzl, cross-scion ara smoohly dcrass from a largr valu (usually assumd a lnum chambr wih M 0, c=) o a smallr valu (xi scion A, wih M and ). Th mass flow ra in rms of saic or oal condiions a any sag, wih h isnroic rlaions (7), is: s ( 1) m va MA 1 M MA = ρ = = + (15) R T R T wih m =cons, T=cons, =cons. Whavr h ara law, h flow acclras o a maximum sd a h xi. Two cass may aar: Subsonic xi (M<1). Sonic xi (M=1). For subsonic xi, xi rssur quals ambin rssur (=0), and xi condiions ar: ( 1) = 0, T = T, M 1, = m = 1+ M MA (16) 1 R T valid only if M 1. Th limi M=1 (choking condiions) will b rachd whn: ( 1) ( 1) T =, =, M = 1, m = A = A (17) + 1 T + 1 R T cc whr cc = RT is h sound sd a chambr condiions. In conclusion, if, for givn nry condiions ( and T), ambin rssur is bing lowrd from h no-flow condiion, 0=, firs a subsonic flow dvlos, unil 0= * =(/(+1)) /( 1),.g. 0/=0.53 for =1.4, whr mass flow ra is a a maximum, Nozzls 7

8 and a furhr dcras of ambin rssur has no ffc in nozzl flow (no rssur-informaion can go usram); a fan of obliqu sursonic xansion wavs aars jus a h xi, o accommoda xi rssur (fixd a (17) valu) o ambin rssur 0<, wih a bulging of h xhaus j. Bu nozzl flow bcoms chokd if 0/< * /=(/(+1)) /( 1),.g. if 0/<0.53 for =1.4. This is h yical cas of a high-rssur chambr discharging hrough hol, sinc, unlss h hol is a wlldsignd convrging-divrging nozzl, h flow will sara a h maximum consricion (h hroa), and will bhav as a convrging nozzl. If h fding chambr is a a sady sa (i.. T=cons, =cons), hn h chokd flow is invarian, and h mass-flow-ra a consan, (17), for givn xi ara A, no mar how much h discharg rssur is lowrd. Bu, if h fding chambr is unsady,.g. drssurising bcaus of h scaing mass, hn, vn if h nozzl rmains chokd, h mass flow ra, givn by (17), dcrass wih im, wih h following invarian: m T = cons (18) i.. m changs wih changing nry condiions. Th wo xrm cass of discharg from a gas ank ar: isohrmal (T=cons, so ha m c ), and adiabaic (isnroic, if inrnal dissiaion is ngligibl, i.. T=cons ( 1)/ ( 1) ( ), so ha m c,.g. for air wih =1.4, m c ). Noic ha a gas ank dischargs mor slowly if hrmally isolad han if k isohrmal (in h lar, ha addiion nds o incras rssur and hl h jcion). Exrcis 3. Considr a habiabl saccraf modul of 50 m 3 wih air a 100 kpa and 300 K, and find h im i would ak o drssuriz hrough a 1 cm hol. Sol.: Th air insid scas hrough an ara A=10-4 m (w disrgard h local dails of h hol, which may yically hav a discharg cofficin cd=0.9 (rducing h ffciv ara accordingly). Th discharg is chokd all im sinc xrnal rssur is 0, and hus M=1. W furhr assum ha h small sca ara maks h rocss slow nough o b aroximad as isohrmal. A firs simaion is ha m =ρva ρccca= =0.03 kg/s, so ha h iniial 50 kg of air would sca in abou 50/0.03=1700 s. Mor rcisly, h mass flow ra is givn by (17), ( ) c m = f A c, wih f()=0.805 for air (=1.4), cc = RT =347 m/s, and rssur is roorional o mass, =mrt/v. In rms of rssur, h mass balanc dm/d= m yilds: ( ) d ( ) ( ) V d f A f RTA f RTA = m = = d = 1 x RT d cc cv c cv c i.. rssur dcrass xonnially o 0 wih an iniial ra of 40 Pa/s, i.. wih a characrisic im ccv/(f()rta=(v/(cca))f()/=500 s; his is h im i aks for /1=1/=0.37; consciousnss is los in 3 o 6 minus a his rssur lvl, whr oxygn arial rssur is 7.8 kpa). Nozzls 8

9 Fig. E3. Drssurizaion of a habiabl saccraf modul. If h drssurizaion wr adiabaic, rssur loss would slow down slighly du o h larg mraur dro (.g., afr 000 s, insad of 45 kpa givn by h rvious modl, rssur would b 47 kpa, bu mraur would had fall o 41 K. Convrging-divrging nozzl A convrging-divrging nozzl ('condi' nozzl, or CD-nozzl) mus hav a smooh ara law, wih a smooh hroa, da/dx=0, for h flow o rmain aachd o h walls. Th flow sars from rs and acclras subsonically o a maximum sd a h hroa, whr i may arriv a M<1 or a M=1, as for convrging nozzls. Again, for h nry condiions w us 'c' (for chambr) or '' (for oal), w us '' for h xi condiions, and '*' for h hroa condiions whn i is chokd (M * =1). If h flow is subsonic a h hroa, i is subsonic all along h nozzl, and xi rssur naurally adas o nvironmnal rssur 0 bcaus rssur-wavs ravl usram fasr (a h sd of sound) han h flow (subsonic), so ha /0=1. Bu now h minimum xi rssur for subsonic flow is no longr =(/(+1)) /( 1) (/0=0.53 for =1.4), sinc h choking dos no ak lac a h xi bu a h hroa, i.. i is h hroa condiion ha rmains valid, * =(/(+1)) /( 1),.g. * /0=0.53 for =1.4; now h limi for subsonic flow is,min,sub> * bcaus of h rssur rcovry in h divrging ar. Howvr, if h flow is isnroic all along h nozzl, b i fully subsonic or sursonic from h hroa, h isnroic quaions aly: 1 1 T 1 1 M, M 1 = = + = T 1 A 1 m = ρ va = M 1+ M RT (19) Bu if h flow gs sonic a h hroa, svral downsram condiions may aar. Th conrol aramr is discharg rssur, 0. L considr a fix-gomry CD-nozzl, discharging a givn gas from a rsrvoir wih consan condiions (,T). Whn lowring h nvironmnal rssur, 0, from h no flow condiions, 0=, w may hav h following flow rgims (a lo of rssur variaion along h nozzl is skchd in Fig. ): Subsonic hroa, imlying subsonic flow all along o h xi (voluion a in Fig ). Sonic hroa (no furhr incras in mass-flow-ra whavr low h discharg rssur l b). o Fully subsonic flow xc a h hroa (voluion b). Nozzls 9

10 o Flow bcoms sursonic afr h hroa, bu, bfor xi, a normal shockwav causs a suddn ransiion o subsonic flow (voluion c). I may han ha h flow dachs from h wall (s h corrsonding skch). o Flow bcoms sursonic afr h hroa, wih h normal shockwav jus a h xi scion (voluion d). o Flow bcoms sursonic afr h hroa, and rmains sursonic unil d xi, bu hr, hr cass may b disinguishd: Obliqu shock-wavs aar a h xi, o comrss h xhaus o h highr back rssur (voluion ). Th ys of flow wih shock-wavs (c, d and in Fig. ) ar namd 'ovr-xandd' bcaus h sursonic flow in h divrging ar of h nozzl has lowrd rssur so much ha a rcomrssion is rquird o mach h discharg rssur. Tha is h normal siuaion for a nozzl working a low aliuds (assuming i is adad a highr aliuds); i also occurs a shor ims afr igniion, whn chambr rssur is no high nough. Adad nozzl, whr xi rssur quals discharg rssur (voluion f). Noic ha, as xi rssur only dnds on chambr condiions for a chokd nozzl, a fix-gomry nozzl can only work adad a a crain aliud (such ha 0(z)=). Exansion wavs aar a h xi, o xand h xhaus o h lowr back rssur (voluion ); his is h normal siuaion for nozzls working undr vacuum. This y of flow is namd 'undr-xandd' bcaus xi rssur is no low-nough, and addiional xansion aks lac afr xhaus. Fig.. Nozzl flows for consan nry condiions (,T), as a funcion of discharg rssur 0. As 0 is bing dcrasd, h flow sars bing subsonic (a) all along h nozzl lngh (x), hn i bcoms chokd (and h flow no longr changs in h convrging ar). Bu h flow in h divrging ar may b subsonic (b), or a ransiion from sursonic o subsonic occur wihin (c, from b o d), or h sursonic flow a h xi b followd by comrssion wavs (), b adad (f), or b followd by xansion wavs (g); h lar cas is said undr-xandd, and is yical undr vacuum. Nozzls 10

11 In h convrging-divrging nozzl usd in sursonic aircraf, boh h hroa ara and h xi ara should b oimisd for maximum hrus as a funcion of aliud and fligh sd, bu in racic hr is a singl mchanical adjusmn, using als o achiv a variabl ara nozzl. In rocks, fix-ara nozzls is h rul. Whn h flow is isnroic all along h nozzl, i.. for valus of 0/ from d o g in Fig., h xi Mach numbr M is givn by: A A ( ) 1 * = M 1 1+ M (0) Mind ha, solving (0) for M yilds wo soluions, M,sub<1 and M,su>1, corrsonding o h isnroic xi rssur and mraur airs: 1 T 1 = = 1+ M,sub T,sub,sub (1) 1 T 1 = = 1+ M,su T,su,su () Th sursonic mass-flow-ra and xi sd in h isnroic discharg hrough a nozzl ar: A A m = ρ v A = RT A = = (3) v + 1 ( ) * * * * 1 * * * * * * RT * RT RT RT RT 1 = 1 = (4) 1+ M Noic ha, alhough i is ofn said ha m is consan in a chokd nozzl (criical flow-ra), wha is man is ha h mass-flow-ra dos no dnd on back rssur (rovidd h flow bcoms sursonic), bu m is almos roorional o chambr rssur (and dnds on mraur and gas roris oo; hough hy ar almos invariabl during normal oraion of rocks; s Chambr rssur quaion, blow). Somims, a characrisic sd vc is dfind as chambr rssur () ims hroa ara (A * ) dividd by mass flow ra ( m ), i.. by * vc A m, a modifid sound sd indndn of h xi ara, as can b dducd by subsiuion from (3); whn using such a characrisic sd, a non-dimnsional hrus cofficin is dfind by cf F/(A * )=v/vc+(a/a * )( 0)/, such ha for an adad nozzl i is cf=v/vc. Nozzls 11

12 Noic also ha h maximum xi sd corrsonds o an infini xansion whr all h hrmal nrgy gos o kinic nrgy, is c(t T)=cT=v /;.g. for air a 88 K xanding isnroically o vacuum, v = ct = =760 m/s. Som yical valus of h xhaus gas vlociy for rock ngins burning various rollans ar: For solid rollans and for liquid monorollans v=000 o 3000 m/s;.g. hydrazin caalyic dcomosiion a 7 MPa gnraing gass wih avrag M=0.0 kg/mol and =1. a T=500 K, would yild a maximum of v = ct = =300 m/s undr vacuum. For liquid birollans v=3000 o 4500 m/s;.g. for a cryognic H/O rock gnraing war vaour (and som dissocias) a 0 MPa and 350 K (wih xcss of H), h maximum xi sd is v = ct = =400 m/s. An xaml of CD-nozzl flow comud wih h abov quaions is shown in Fig. 3. Fig. 3. A comud xaml of chokd flow of air (=1.4) in h CD-nozzl gomry shown on o (nozzl hroa a x=1; in scald arbirary dimnsions, ara raio A/A * =.9). Two isnroic soluions xis: on oally subsonic (xc a h hroa, which is sonic), and anohr ha bcoms sursonic afr h hroa. Th los ar: Mach numbr, local rssur rlaiv o chambr valu, local mraur rlaiv o chambr valu, and local sd rlaiv o is hroa valu (sd of sound a hroa condiions). Mor daild nozzl flow simulaions can b found asid. Nozzls 1

13 Looking a h smooh rounding of h nozzl nck (Fig. 3a), i sms amazing h kink in all flow variabls corrsonding o h full subsonic soluion (h xlanaion rsids in h singulariy ha (8)- (11) hav for M=1). Anohr asonishing rsul is h vry raid rssur changs a h nck: in h lngh from x=0.86 o x=1.18 in Fig. 3a, whr h nozzl radius varis from r=0.19 o r=0.19 hrough r=0.18 a h hroa (a 10 % in ara chang from nck o nds), rssur (scald wih h consan oal rssur) varis from /=0.73 a x=0.86 o /=0.3 a x=1.18 if h flow bcoms sursonic (or rcds o h sam subsonic valu, /=0.73 a x=1.18 if i rmains subsonic), assing by * /=0.53 a x=1; i.. bwn h wo scions wih radius 5 % largr han h minimum, rssur dcrass a 56% (in sursonic flow; or i rcovrs, afr dcrasing a 7%). I is also imrssiv how soon h choking occurs whn backrssur is bing dcrasd: a /=1 hr is no flow, and a /=0.97 h nozzl is alrady chokd, wih a fix mass flow ra whavr h valu of /<0.97 (if nry condiions ar mainaind). Bu, if h flow is chokd, how i adas o h changing xi rssur? Through disconinuiis in h flow, braking h isnroic condiion. Exrcis 4. A liquid birollan rock o b usd for h scond sag of a sac launchr, works a 7 MPa and 3300 K in h combusion chambr, gnraing gass wih a man molar mass M=0.00 kg/mol, and =1.30, having a nozzl xi diamr of 1.5 m and nozzl xansion raio of 50. Find: a) Th xi Mach numbr, assuming fully sursonic flow. b) Discharg rssur condiion for adad nozzl. c) Prollans flow ra. d) Exi sd and hrus a 0 km aliud. Sol.: a) Givn h xansion raio, ε A/A *, quaion (0) yilds h xi Mach numbr: A A ( 1) * 1/50 1 ε = M = M M ε 1+ M ( M ) = = 0.0 = = 5.1 b) Th condiion is 0=,su, whr h isnroic xi rssur is obaind by ():,su 1 = 7 MPa = 1.30 M = = 1+ M,su = 7.3 kpa i.. h nozzl would b adad if h nvironmn is a 0=7.3 kpa (z=18. km aliud in ISA modl); if 0<7.3 kpa h flow is undr-xandd (fully sursonic and wih addiional xansion wavs a h xi); howvr, if 0>7.3 kpa, h flow is ovr-xandd (and rcomrssd by shock wavs, ihr a h xi sag, or wihin h nozzl). c) Th mass flow ra of rollans quals h mass flow ra of xhaus gass, which is obaind by (3) ihr a hroa condiions or a h xi sag (i dos no dnd on 0, rovidd h nozzl is chokd): = 1.30 = 7 MPa A = π1.5 4 R= T = 3300 K M = 5.1 A 1 m = ρva = M 1+ M m = 141 kg/s RT Nozzls 13

14 d) W hav sn in b) ha h flow is fully sursonic for 0<7.3 kpa (z>18. km aliud in ISA modl), so ha h xi vlociy is h sam as for an adad nozzl and can b obaind from (4): v = R= T = 3300 K M = 5.1 RT RT 1 = 1 = 1 v 3070 m/s = 1+ M wih =7.3 kpa, Anohr way could hav bn hrough m= ρ va = va ( RT) A=π 1.5 /4=1.77 m, R=Ru/M=8.3/0.00=415 J/(kg K), and T=T/(1+( 1)M /)= 3300/( )=677 K. Th hrus F is: ( ) F = mv + 0 A = ( ) 1.77 = 433 kn +31 kn = 464 kn i.. F=464 kn (433 kn from j momnum, and 31 kn from xcss rssur rlaiv o h nvironmn). Disconinuiis in nozzl flow: normal and obliqu shocks, xansion fans. Mach diamonds Considr h isnroic rssur voluion along h nozzl in Fig.,. Far downsram of h xi sag, h xhaus j (say, mor han a coul of xi diamrs), h j, laving asid a ossibl adjusmn clos o h xi, mus b snsibly qual h xi rssur W hav sn ha, for givn nry condiions, svral cass of nozzl flow aar as a funcion of h imosd discharg rssur 0, o which h xhaus j mus ada, sinc a fr j canno wihsand ransvrsal rssur gradins. This rssur-adaaion may b hrough small (linar) or srong (nonlinar) rssur wavs. By comaring rlaiv xi rssur /, wih rlaiv back rssur 0/, h ossibl flow configuraions ar: If 1>/>,sub/, hn h maching =0 is by acousic wavs ravlling usram from h xi o h nranc, and h flow is subsonic all along h nozzl lngh; for h xaml lod in Fig. 3, 1>/>0.97. If /=,sub/, h flow is chokd bu subsonic xc a h nck (his is h limi of h cas abov). If,sub/>/>,su/, hn h maching =0 is by acousic wavs ravlling usram from h xi o a scion bwn xi and hroa, whr a srong shock aks lac; h locaion of his normal shockwav dvlos som disanc downsram (h furhr down h lowr h backrssur), wih subsonic flow byond, maching h ambin backrssur. For h xaml lod in Fig. 3, his rang is 0.97>/>0.05. This nozzl-flow configuraion is known as ovrxandd (s Fig. ). o Th shockwav is insid h nozzl if,sub/>/>,s/, whr,s is h xi rssur wih xi shock wav; for h xaml lod in Fig. 3, 0.97>/>0.39. Th valu of,s is found by using h normal-shockwav rssur-jum quaion wih,su and M,su as usram condiions. Th suddn comrssion a h normal shock, and h subsqun Nozzls 14

15 unfavourabl rssur-gradin, maks his subsonic flow o dach from h wall, wha is ofn rmd 'grossly ovr-xandd flow', wha yilds oor nozzl rformancs (h yical bhaviour on ground of nozzls dsignd for high-aliud and vacuum oraion). o For,s/>/>,su/, a comlx srucur of obliqu and normal shock wavs dvlo a h nozzl xi, wih succssiv comrssions and xansions rflcing a h fr-j boundary, gradually qualizing h rssur diffrnc bwn h xhaus and h amoshr, wih h aaranc of 'shock diamond' srucurs (Fig. 4, blow). For h xaml lod in Fig. 3, his rang is 0.39>/>0.05. If /=,su/, h sursonic flow arrivs a h xi wih rcisly h ambin-rssur valu, and hr ar no disconinuiis (ohr han h mixing layr around h cylindrical xhaus j (wll, slighly conical, according o h nozzl-slo a h xi). If,su/>/>0, h flow is isnroic all along h nozzl lngh and sursonic from nck o xi scion (as h cas bfor), bu h xi rssur bing highr han h nvironmn (>0), a fan of Prandl-Mayr xansion wavs ss a h xi, and h flow configuraion is calld undrxandd (Fig. g). A good xaml of h occurrnc of h lar condiions was rsn in h Sac Shul Main Engin, which lavs h ad in an ovr-xandd sa (s h rracing xhaus j undr ss, in Fig. 4), bcoms adad (fully xandd) a high aliud, and hn undr xandd as h Shul aroachs h vacuum of sac. Fig. 4. Shock diamonds (blu cons) in Shul's main ngin nozzls during STS-10 launch (NASA), and during ss. Normal shock wihin h nozzl A normal shock gnras nroy and hus lowrs oal rssur (whil graly incrasing saic rssur). Mass flow-ra consrvaion rlas boh valus: oal rssur bfor,, and afr h shock, (xi oal rssur). Alying (15) o hroa and xi condiions (a ach sid of h shock): Nozzls 15

16 1 1 * 1 1 m = 1+ A= 1+ M MA R T R T (5) now w g, insad of (14): * A A + 1 ( ) =, wih = 0 1+ M M 1 1+ M 1 (6) whr w hav o solv for M for givn nry and xi rssur (, 0), and hroa and xi ara (A *,A). Onc h oal rssur loss comud, h acual Mach numbr jus bfor h shock wav, Ms, is found from normal-shock rlaions: M s = M s Ms (7) Solving for his shock-nry Mach numbr, Ms, allows h saial locaion of h fron wihin h nozzl in rms of aras, from (0). In aricular, h shock wav is locad rcisly a h xi scion whn h jum in Mach numbr across yilds a downsram rssur,,s (iminging rssur is,su), qual o h nvironmnal rssur, 0, i.. whn: M + 1 ( 1) * M 1 M, wih M 1 M1 A = + = A whr M1 and M ar h Mach numbrs ahad and bhind h normal shock wav (M1=Ms). Th scific nroy gnraion in h normal chock is sgn=s s1=cln(t/t1) Rln(/1)= Rln(/). (8) Exrcis 5. For air xanding hrough a nozzl of ara raio Axi/Ahroa=4, find: a) Mach numbrs bfor and afr a shock wav, whn h shock gs sabilizd insid a a scion wih As/Ahroa=. b) Mach numbr a h xi scion, and raio of oal rssurs, in h cas abov. c) Discharg rssur o hav h normal shock a h xi. Sol.: a) Th Mach numbr jus ahad of h shock is h sursonic soluion in (0) wih As/A * =, i.. Mahad=.3. Th Mach numbr jus bhind h shock is found from h normal-shock quaions (8), i.. Mbhind=0.55. b) Th xi Mach numbr is h subsonic soluion corrsonding o Mbhind=0.55 and ara raio Axi/As=; i.. M=0.4. Th raio of oal rssurs is obaind from (6), /=0.63. c) To hav h normal shock a h xi, M-=M1=.9 from h sursonic soluion in (0) wih As/A * =4, and finally M=M+=0.48 and 0/=0.30 from (8). Nozzls 16

17 Obliqu shocks a h nozzl xi Shock diamonds (or Mach diamonds) ar arns of sanding wavs ha aars in h sursonic xhaus lum of arosac roulsion sysm (urboj wih os-combusor, solid- or liquid-ful rock, ramj, or scramj), whn orad in an amoshr wih 0> (i.. whn h flow is ovr-xandd in a CD-nozzl. Whn h xhaus flow gs across a normal shock (in rd in Fig. 5a) h abru comrssion causs a suddn mraur incras, wih radicals roducing non-quilibrium chmiluminscnc, which in h cas of LH/LOX rocks is comosd of a wak mission-coninuum in h blu and ulraviol rgions of h scrum, and a 0 ims srongr narrow-band mission a 310 nm, du o xciaion of OH and H radicals and hir rcombinaion o HO. Bsids, his suddn haing may caus h igniion of any rsidual ful rsn in h xhaus, making h Mach disc and rail o glow and bcom visibl lik in Fig. 4. Bhind h normal shock, h rssur is grar han ha of h ambin amoshr, so ha h j xands, rying o qualiz wih h xrnal air (h xanding wavs rflc off h fr j boundary and owards h cnrlin), wha may rquir svral xansions and comrssions. A similar rocss occurs in an undr-xandd flow xiing from a nozzl a high aliud or undr vacuum (Fig. 5b). Th squnc of comrssion and xansion is idnical o ha abov-dscribd for an ovr-xandd nozzl, xc ha i bgins wih h craion of an xansion fan rahr han obliqu shock wavs. This bhaviour causs h flow o billow ouward iniially rahr han sindl inward. a) b) Fig. 5. a) Shock diamonds in an ovr-xandd flow; h firs Mach disc (in rd) is sarad from h xi by xd = ( 3) 0. b) Wav srucurs ha cra shock diamonds in an undrxandd flow (Arosacwb). Nozzls 17

18 Arosik nozzl This is a novl dsign for a fix-gomry nozzl o g adad a all aliuds. Insad of an ouward flow in a bll-sha wall boundary, in h arosik nozzl an annular flow issus radially inward along a dcrasing-diamr innr wall (h sik), wihou xrnal wall (afr a cowl li), s Fig. 6. Th our ambin rssur rgulas h our lum boundary so ha whn <0 (ovr-xansion a low aliuds) h xrnal rssur squzs and maks h lum hinnr, furhr acclraing h xhaus insad of daching i from h walls. Sinc ambin rssur conrols h nozzl xansion, h flow ara a h nd of h arosik changs wih aliud, as if i was a variabl-ara nozzl, and hus, a vry high ara raio nozzl, which rovids high vacuum rformanc, can also b fficinly orad a sa lvl. Th lngh of an idal sik is abou 150 % of a 15 conical nozzl, bu rformancs rduc vry lil if h sik lngh is runcad o h 0 % rang, wih h formaion of a rcirculaing bubbl which, if fd wih a scondary j a h bas, longas h bubbl, forming an arodynamic conour ha rsmbls h runcad orion of h sik (his arodynamic-sha is h rason for h "arosik" rm). Fig. 6. Diffrn flow configuraions in arosik nozzls, and comarison wih bll nozzls. Back o Proulsion Nozzls 18