, 0005 (06) DOI: 0.05/ mecconf/06700005 ICMI 06 Mul-lod Opml Desgn of Burner-nner-lner Under Performnce Index Consrn by Second-Order Polynoml ylor Seres Mehod U Goqo, Wong Chun Nm, Zheng Mn nd ng Kongzheng Lnzhou Unversy of echnology, School of Mechncl nd Elecroncl Engneerng, Lnzhou, Gnsu 70050, Chn Absrc. Usng mxmum expnson pressure of n-decne, he eroengne burner-nner-lner combuson pressure lod s compued. Aerodynmc lods re obned from nernl gs pressure lod nd gs momenum. Mul-lod second-order ylor seres equons re esblshed usng mul-vrn polynomls nd her sensves. Opml desgns re crred ou usng vrous performnce ndex consrns. When 0.5 o 0.8 recfcons of dfferen desgn vrns re mplemened, hey converge under 50 d-norm dfference ro. Inroducon Accordng o he urbne combuson echnque [], mnennce coss on urbne nd combuson chmber ccoun for 60% of he whole rcrf mnennce. herefore, hgh performnce on he opml desgn, operon relbly nd srucurl sfey re demnded on modern eroengne herml componens. As burner nner lner (BIL) s he mor componen n combuson chmber, becomes one of he mos sgnfcn componens of he eroengne. BIL s mellc hn-wll cylnder h conrols he combuson, mxng nd coolng processes. I gudes he combuson chmber cylnder nd roors from herml combuson producs []. rom he mnennce survey [], BIL ccouns for he 6% of he combuson chmber fuls. Combuson chmber cylnder s 5% nd fuel nozzle s %. Snce performnce ndces re deermned by BIL mnly, becomes he reserch focus on cnnulr combuson chmber desgn. In he operon process of BIL, complced pressure lods occurs whch cn be clssfed s follows: ) Due o gs combuson, expnson pressure s exered on s cylnder; ) Pressure lod s genered by he pressure dfferen beween he nner nd ouer cylnder surfces. Mos rdonl opmzon mehods wor on unque lod only one dscplnry nlyss s nvolved. o mprove he performnce of BIL under hese lods n dfferen dscplnry, mul-lod opmzon echnque s mplemened n hs desgn. Mul-lod opmzon echnque llows he mulple responses obecve funcons o be opmzed smulneously under he coupled desgn vrns. hs mehod cers for he complced nercons mong dfferen dscplnry, nd he requremens of vrous performnce ndces. Inequly consrns re mposed by hese ndces. Expnson pressure lod creed n fuel combuson he chnges n recn, produc nd composon mole frcon of n-decne nd von fuel premxed combuson flme re bsclly conssen s ndced by Zeng[]. Alhough he von fuel conns complced ngredens, n-decne cn be used for numercl smulon s lernve of von fuel. Dels of hs proposed recon mechnsm cn descrbe deled dynmc chrcerscs of n-decne premxed combuson. Becuse percenges of C nd H whn hexne hydrogen clsses re he sme, one cn nfer s complee combuson produc proporons re he sme. As resul, n-decne, propne nd cyclohexne possess sme mxmum expnson pressure of 0.86MP. gure. Seconl dmensons n ech rnge. rom hs nown combuson pressure, BIL nsde pressure dsrbuon s obned usng nverse-squre lw P P ( r r) r s n g.. In rnge, 0 0 0 r0r 0.6x0.05 y for 0 x 0.055. r r 0.0700 y for In rnge b, 0 0.055 x 0.095. In rnge c, r r ( y 0.095n( 6)) ( x 0.095) 0.0700 0.095n( 6) 0 he Auhors, publshed by EDP Scences. hs s n open ccess rcle dsrbued under he erms of he Creve Commons Arbuon Lcense.0 (hp://crevecommons.org/lcenses/by/.0/).
, 0005 (06) DOI: 0.05/ mecconf/06700005 ICMI 06 0.095 x 0.5. Usng he dsrbued pressure for on ech rnge compued bove, s hrus lod s compued by equvlen nodl lods ( sn( ) 0 sn( ) 0 sn( ) 0 sn( ) 0 sn( ) 0 ) 8 9 6.980 dn (),, re he xl ngles of rng,,8 respecvely. n s he pressure lod on rng n. he equvlen nodl lod of rng n. 0 n s Aerodynmc lod genered by gs pressure nd gs momenum Accordng o he erodynmc equon, when he gs psses he desgn model (g. ), ol hrus s he gs momenum dfference beween oule nd nle. Se he gs pressure lod nsde s P. ng he hrus drecon s posve, erodynmc force on BIL s PAP AP. Gs momenum s GV ( V ) 0 n ou wh G 50.7 g / s ou. hus we hve he blnce equon PG( V V ) AP 0 AP V n ou n n ou 50 m/ s, V 600 m/ s ou n wh. Usng he con nd recon rule, hrus lod due o erodynmc force s P. Subsung ll compued prs, one hs G( V V ) P ( A ( S S sn )) P A.7907 0 dn () n ou n 0 ou s he gs hole re, Sn s surfce re of rng n. herefore he ol hrus lod s composed of hese mul-lod s.89 0 dn () gure. Desgn model of BIL. Esblshmen of mul-lod opmzon prncple In hs mehod, second-order ylor Seres expnson equon [5] s ulzed o rele he chnges of he mullod responses wh he recfcons of desgn vrns n he desgn model. By nverse compuon, hese recfcons n he desgn vrns re esmed from he BGS lgorhm[6]. he opmzon process s repeed unl specfc ermnon crer re reched. Le m,,, be he -h upded desgn vrn vecor, denoes he fuel nozzle e ngle, gs flow re, fuel combuson pressure nd compressor,,, oule pressure. Menwhle d d d dm s he cul desgn vrn vecor. he seleced ses of n hrus response vlues nd n S hrus response vecors S correspondng o he upded nd cul srucures re denoed by Y A nd Y D respecvely,,, ns Y A, n, S,, S,, S n ns Y D d, d,, d, S d, Sd,, Sd, nd,. In he hrus lod sysem equon, re hrus force due o he fuel combuson, flow re nd r pressure. S re dsrbued sress due o expnson pressure nd dsrbued r pressure on he cylnder surfce. In generl, one cn use second-order ylor seres expnson[7] o obn he chnge of he -h hrus response vecor due o chnge n desgn vrn (,,..., m): m m m S S S ds! () () S s he resdul error vecor n s expnson. or he p-h order expnson of he -h hrus response vlue, one ges m m m d!, (5) n whch s he resdul error vlue n he expnson. ermnon creron s esblshed o conrol he ccurcy of erve process. or he opml desgn of BIL, he obecve funcon s me when he sum of hrus response resdul errors drop o he globl mnmum. As resul, he d-norm d dfference ro s chosen s he
, 0005 (06) DOI: 0.05/ mecconf/06700005 ICMI 06 ermnon ndcor,.e. d d d d 5 0 0 Bsed on he m ordered ses, one cn nerpole m-vre hus response vecor polynoml funcon s n ns s s d d Sd S, s. (6) d K K m m m S S L L Lm m m L s he Lgrnge fcor funcon of he h desgn vrn he h nerpoled sffness vlue, gven by L (7) (,,, K ) (8) Menwhle, s hus response vlue polynoml funcon s K K m m m (9) L L L m m m rs-order dervve erms cn be obned by drec dervve on Eqs. (7), (9) wh respec o. or mulple vrns recfcon, he frs-order hus S response vecor nd vlue dervves [7] cn be expressed s L K K m m S m L Lm m m (0) K K L L L m m L K K. On he oher hnd, he second-order dervve consss of wo erms nmely he repeed dfferenl nd he unrepeed dfferenl. or he unrepeed erm, s m m m m () gven by he drec frs-order dervve of Eqs. 7,9 ccordng o nvolved. Specl cre s gven o he repeed erms drec second-order dervve wh respec o re encounered. or mulple vrns, he second-order hus response vecor nd vlue dervves [7] re expressd s K K L L m m m m m S m K K m m S m L L m m m S L L L K K L L m m m m m m K K L m m L m L m m m L L () ()
, 0005 (06) DOI: 0.05/ mecconf/06700005 ICMI 06 L K K K m m m m m Subsung Eqs. (0), () no Eq. (), second-order ylor seres expnson of hus response vecor s furnshed. Moreover, second-order hus response vlue equon of Eq. (5) s compleed by Eqs. (), (). 5 Normlzon of ylor seres equon usng ccurcy number Here we crry ou he normlzon of he opmzon equons. or he response vecors, we reduce he d precson s smll s possble, whle no ffecng s vron chrcerscs. We mulply he equons n ech dscplne by ccurcy numbers, so h hey re normlzed o he sme precson level. In hs cse, he compuon problems such s he sngulry, ccurcy nd convergency cn be voded. or exmple, ccurces of he dsrbued sress nd pressure re 0 9 6 nd 0 respecvely, hen he normlzon ccurcy numbers 9 6 re [0,0 ]. Under hs remen, fser nd more ccure convergences re ned. When wo desgn vrns, gs flow re nd compressor oule pressure re recfed n 0.5 level, her opmzon nd convergence processes re recorded s follows. rom g., he mnmzon of d s fser up o esmon 0. I becomes slower up o esmon 80, nd sble ferwrds. ermnon creron s me esmon 9 he drops o.8 0. uel nozzle e ngle (rd) 0.5606 0.560 0.5598 0.559 0.5590 gure. Esmon pern of under 0.5 recfcon d 5 6 7 8 9 Esmon Number 6 Opml desgn of BIL under performnce ndex consrn In hs opmzon pproch, nequly consrns re mposed usng vrous performnce ndces. Specfc hrus s consrned n he rnge 60 G80dNs g consumpon ; specfc fuel 0.8 C 600 f S.0 g ( hdn) f 0.08 ; hrus wegh ro s.5 w= Wg.0 nd compressor oule pressure ro s.5 P= P P.5 P s he mospherc pressure. m 5.00 d-norm dfference ro..75.50.5 0.00 gure. Convergence of nd. r ou m 5 6 7 8 9 Esmon Number s d under 0.5 recfcon of Usng dfferen combnons of one o four, vrous 0.5 o 0.8 recfcon cses re esblshed. rom g., rpd dusmen of occurs up o esmon 0. hen becomes sble ferwrds. As s recfcon s relvely smll, wve-le pern hvng pe nd vlley s observed. Gs flow re (g/s) 80 70 60 50 0 5 6 7 8 9 Esmon Number gure 5. Esmon pern of under 0.5 recfcon. Esmon process of s llusred n g.5. Smlr o he pern of d, ncreses rpdly up o esmon 0. Aferwrds, chnges grdully unl ermnon creron s me esmon 9 s percenge error s consrned by he performnce ndces.%. rom he esmon pern of, s rpd dusmen occurs up o esmon 0. hen becomes sble ferwrds. No he sme s, s recfcon cycle s shorer ledng o less sgnfcn pe nd vlley feures. Cler cu pern s observed s n g. 6. As s nomnl vlue s relvely lrge mong he oher vrns, s prory s hgh n he pern. hus s opmzon s
, 0005 (06) DOI: 0.05/ mecconf/06700005 ICMI 06 rpd nd ccure. Usng 0.5 recfcon for sme desgn vrns, 9 esmons re needed. hus, he convergence re s slghly decresed for smller recfcon level. Compressor oule pressure (P) gure 6. Esmon pern of under 0.5 recfcon. When hree desgn vrns,, (m=) re recfed n 0.8 level, her opmzon nd convergence processes re recorded s follows. d-norm dfference ro. gure 7. Convergence of of,,. rom g. 7, d under 0.8 recfcon d s rmped up o he pe 5.5 esmon. hen drops rpdly o.6 nd mnmzes grdully ferwrds. ermnon creron s me esmon 78 drops o.8 0. uel nozzle e ngle (rd).60e+05.0e+05 8.00E+0.00E+0 0.00E+00 60.00 5.00 0.00 5.00 0.00.0 0.9 0.8 0.7 0.6 0.5 5 6 7 8 9 Esmon Number 5 6 7 Esmon Number d 5 6 7 8 Esmon Number gure 8. Esmon pern of under 0.8 recfcon. Esmon process of s llusred n g. 8. Inlly, rmps up rpdly o 7.9rd esmon. Aferwrds, ncreses grdully o he +0.8 recfcon level wh.7 0 % devon. Esmon process of s llusred n g. 9. I ncreses from esmon grdully ner +0.8 recfcon level of 89.g/s s consrned by he performnce ndces.08%. As recfcon s zero, wve-le pern hvng lrge pe nd vlley of 6 mpludes up o 7.50 P s observed n he nl sge. hen drops grdully o he nl level wh 6.9 0 % devon. Gs flow re (g/s) gure 9. Esmon pern of under 0.8 recfcon. Compressor oule pressure (P) 95 85 75 65 55 5.5E+05.E+05 9.0E+0 6.0E+0.0E+0 gure 0. Esmon pern of under 0.8 recfcon. 5 rom g. 0 drops rpdly from.60 P o mnmum level of.6 0. I remns sonry unl esmon 7, nd ncreses grdully o he -0.8 recfcon level wh. 0 % devon. Usng sme desgn vrns wh 0.5 recfcon, 87 esmons re requred. herefore, he convergence re s lower for lrger recfcon level. When m=, 97 esmons re needed n boh 0.5 nd 0.8 recfcon cses. or m=, here s 69 esmons n 0.5 recfcon cse. In he 0.8 recfcon, esmons re requred. When recfcon level s lrger, convergence re s decresed. Menwhle, for he ncrese n m, he convergence re s ncresed. or he llusred cses, m=,, her convergence res follow hese rends. Summry 5 6 7 8 Esmon Number 0.0E+00 5 6 7 8 Esmon Number BIL combuson pressure lod s clculed o be 6.98 0 dn usng 0.86MP n-decne expnson pressure. Aerodynmc lod s obned s.7907 0 dn. Usng developed equons, 0.5 o 0.8 recfcons of one o four desgn vrns consrned by vrous performnce ndces converge 50 d under creron. In generl, lrger recfcon level leds o slower convergence. hs rend s more sgnfcn for lrger m. When m ncreses, s convergence becomes fser. 5
, 0005 (06) DOI: 0.05/ mecconf/06700005 ICMI 06 Acnowledgemens hs reserch s suppored by he Nonl Nurl Scence oundon of Chn under grn numbers 5090508 nd 57555. References. X. Hou, H. J,. Lu, e l., Hgh performnce von gs urbne combuson echnque. Peng: Mlrry Defence Indusrl Press, (00), pp. 5-.. Y. Wng, Aeroengne Prncple. Peng: Behng Unversy Press, (009), pp. 6-.. S. Peng, Aeroengne Combuson Chmber Srucure. Peng: Mlrry Defence Indusrl Press, (978), pp. -7.. W. Zeng, J. Lu, X. Chen, M. Je, Deled recon nec modelng of n-decne premxed combuson, Journl of Aerospce Power, 6(0):58-66. (0) 5. C.N. Wong, A.A. Brhors, Polynoml nerpoled ylor seres mehod for prmeer denfcon of nonlner dynmc sysem, ASME Journl of Compuonl nd Nonlner Dynmcs, ():8-56 (006). 6. C.N. Wong, J. Xong, H. Hung,. Hu, Dmge Deecon of Spce russ usng Second Order Polynoml Mehod wh BGS us-newon Opmzon, Proceedngs of he ASME 00 IDEC/CIE, DEC00-809, (00), pp.75-76. 7. C.N. Wong, J. Xong, H. Hung, Y.J. Zho, A polynoml lgorhm for model updng of engneerng russ, Mechncs Bsed Desgn of Srucures nd Mchnes, 8:-, (00). 6