Optimum Fan Pressure Ratio for Bypass Engines with Separate or Mixed Exhaust Streams

Transcription

1 JOURNAL OF PROPULSION AND POWER Vol. 17, No. 5, September October 2001 Optimum Fn Pressure Rtio for Bypss Engines with Seprte or Mixed Exhust Strems Abhijit Guh University of Bristol, Bristol, Englnd BS8 1TR, United Kingdom The optimum fn pressure rtio is determined both numericlly nd nlyticlly for seprte-strem s well s mixed-strem bypss engines. The results re pplicble to civil nd militry engines. The optimum fn pressure rtio is shown to be predominntly function of the speci c thrust nd wek function of the bypss rtio. Two simple, explicit equtions, one for ech type of engine, hve been derived tht specify the optimum fn pressure rtio. The predictions compre very well ginst numericl optimiztion performed by specilist computer pckge employing itertive nd dvnced serch techniques nd rel gs properties. The nlyticl reltions ccelerte the optimiztion process nd offer physicl insight. It hs been shown tht the optimum fn pressure rtio chieves the conditionv jc /V jh = KE in seprte-strem engine nd the conditionv 163 /V 63 ¼ KE in mixedstrem engine. The condition V 163 /V 63 ¼ KE pplies even under situtions when signi cnt deprtures from the normlly ssumed condition p 016 /p 06 ¼ 1 cn occur. Nomenclture B = bypss rtio c p = speci c het of ir t constnt pressure c pg = speci c het of combustion products t constnt pressure E k = kinetic energy dded by the core engine F N = net thrust OF N = speci c thrust M = Mch number of the ircrft Pm c = mss ow rte through the bypss duct (cold ow) Pm f = mss ow rte of fuel Pm h = mss ow rte through the core engine (hot ow) p = sttic pressure p 0 = totl pressure R = speci c gs constnt of ir T = sttic temperture T = temperture of mbient ir T 0 = totl temperture V = velocity V = forwrd speed of the ircrft V j = jet speed in mixed strem engine V j c = fully expnded jet speed of the cold bypss strem V j h = fully expnded jet speed of the hot core strem V m = men jet speed [given by Eq. (8)] = isentropic index of ir g = isentropic index of combustion products f = isentropic ef ciency of the fn or low pressure compressor KE = ef ciency of energy trnsfer between the core nd bypss ow LPT = isentropic ef ciency of the low pressure turbine NB = isentropic ef ciency of the bypss nozzle Subscript op = optimum vlue Introduction IN this pper simple, explicit reltions hve been derived from fundmentl principles for determining the optimum fn (com- Received 30 My 2000; revision received 20 Februry 2001; ccepted for publiction 29 My Copyright c 2001 by Abhijit Guh. Published by the Americn Institute of Aeronutics nd Astronutics, Inc., with permission. Lecturer, Aerospce Engineering Deprtment, University Wlk; A.Guh@bristol.c.uk pressor) pressurertio in bypss engines. Seprte nlysishs been presented for the mixed-strem nd seprte-strem engines becuse the ow physics is different in the two cses. In both cses, the present theory gives results in good greement with numericl optimiztion clcultions. Determintion of the optimum fn pressure rtio (FPR op ) is importntbecuse, givenll other vriblessuch s the overll pressure rtio (OPR), bypss rtio B, nd turbine entry temperture T 04 the optimum vlue of FPR simultneously ensures mximum speci c thrust nd minimum speci c fuel consumption. Thus, the FPR op chieves the two, usully contrdictory, gols of jet engine design. For civil engines, prticulrlyfor medium to long rnge ircrfts, the FPR op cn be clculted under cruise conditions. For militry engines, optimiztionmy be mission speci c. However, Millhouse et l. 1 hve shown tht, lthough the most fuel-ef cient vlues of such prmeters s bypss rtio nd OPR depend on the integrted effects of the vrious ight ltitudes, Mch numbers, nd thrust required throughout the mission, it is possible to locte heuristiclly the most ef cient FPR to complement other engine prmeters independent of the mission. Thus, the reltions derived in this pper cn be used in the design of both civil nd militry engines. The design wisdom is tht only single-stge fn is used for civil ircrft engine with lrge bypss rtios, nd so the mximum FPR is bout 1.8. Engines with low bypss rtios, for exmple, militry engines, normlly use low-pressure (LP) compressor hving 3 5 stges. In this cse, the mximum possible FPR cn be much higher. The mixing of the two strems in mixed-strem turbofn engine offers performnce gin (lower speci c fuel consumption nd higher speci c thrust), even though mixing incurs loss in totl pressure becuse the enthlpiesof the two strems re redistributed. Moreover, only one rehet system nd only one vrible nozzle re needed. A long ncelle cn contin the fn/compressor noise more effectively.for the sme speci c thrust, the mixed-strem engine hs lower optimum FPR thn seprte-strem engine; thus, the LP turbine my hve one fewer stge if both engines hve the sme bypss rtio. The outer cold strem lso protects the jet pipe fromseveretemperturesresultingfrom fterburning.low-speci c- thrust, high-bypss-rtioengines, on the other hnd, do not employ fterburning,nd mixing of the two stremswould necessitte long ncelle incresing weight, cost, ncelle drg, nd interference effects. These engines, therefore, use seprte-stremcon gurtions. When common nozzle is, however, used for high-bypss-rtioengines,the smll xilextentof the mixingzone is usullyinsuf cient to chieve complete mixing of the two strems. Clcultion nd nlysis of the performnce of turbofn engines re discussed in detils in severl excellent texts. 2 4 In comprison,

2 1118 GUHA the issue of optimized design hs received less ttention, nd nlyticl solutions did not exist. The rel store house of expertise nd knowledge obviously exists with the eroengine mnufcturers who hve ctively pursued the optimiztion of gs turbine performnce for the pst 60 yers. Ruf es 5 shows tht there hs been 50% improvement in the speci c fuel consumption (SFC) of the bre engine, s the engines evolved from the turbojet of 1958 to the high-bypsstrent enginesof moderntime. Improvementsin propulsive ef ciency,componentef ciencynd cycle ef ciencycontribute pproximtely one-third ech to this gin in performnce. The publictions mde by experts from industry (e.g., Jckson, 6 Bennett, 7 Wilde, 8 nd Birch 9 ) contin welth of experience, informtion, nd clcultion results. However, the detils of clcultion method re not known to the reders. The industry would typiclly hve its own, sophistictedcomputer pckges to which the public does not hve ccess. Moreover, the publishedresults often would involve n dopted fmily of engine designs. It is then not lwys obvious to the generl reder how to generlize the results or how to proceed on clen-sheetnlysis. In this pper nlyticlreltionshve been derivedfor clculting FPR op. The nlyticl results not only ccelerte the optimiztion process but lso offer vluble physicl insight. It is shown tht the FPR op is predominntly function of the speci c thrust nd only wekly depends on the bypss rtio. The FPR op monotoniclly increses with incresing speci c thrust t two differentlevels corresponding to the seprte-strem nd mixed-strem turbofn engines. The theory hs been veri ed by compring its predictions with clcultionsperformed by the computer progrm GsTurb, 10 which is generl purpose commercil pckge for the clcultion of design nd off-design performnce of vrious types of gs turbine engines, turbojet, turboshft, seprte- or mixed-strem turbofn, nd geredturbofn,ll with single-or twin-spoolcon gurtions.it uses rel gs propertieswith dissocitionnd hs severl fetures including optimiztion, itertion, trnsient, nd Monte Crlo simultion. Ech numericl point subsequently illustrted hs been clculted by GsTurb by lengthy optimiztion process tht uses dvnced serch techniques to nd the optimum FPR tht minimizes the SFC while, t ech tril, the primry vribles re iterted to give the prescribed speci c thrust. This numericl optimiztion is used not only to verify the theoreticl prediction, but lso to discover wht exctly hppens to the mny vribles when the FPR op is chieved. Bsed on the principles discussed in the present pper, new methodologyfor the thermodynmicoptimiztionof bypssengines (turbofn or dvncedpropulsors) hs been developed by Guh 11 in which the optimum combintion of ll vribles is determined concurrently. The process strts with estblishing n optimum speci c thrust for the engine bsed on n economic nlysis (instlltion constrints, noise regultions,etc., lso need to be considered). The tsk of the optimiztion process is then to nd the combintion of optimum vlues of OPR, B, FPR, nd T 04 concurrently tht minimizes SFC t the xed speci c thrust.this procedureis quite different from the usul prmetric studies of engine performnce, 2;12 in which single prmeteris vried ech time, while keeping ll other prmeters xed, therefore t their nonoptimum vlues. Moreover the usul single-vribleprmetric studies my involve lrge excursion in the vlue of spe cic thrust uncceptble for speci c mission. Guh 11 hs lso discussed t length the determintion of optiml jet velocity, with the derivtion of new nlyticl expression tht performs well ginst numericl optimiztion results. Following industril prctice, the SFC nd the speci c thrust, re expressed here respectively in pounds mss per hour per pounds force nd pounds force per pounds mss per second where 1 lbf/lbm/s D 9:81 m/s nd 1 lbm/hr/lbf D 28: kg/s/n. Seprte-Strem Bypss Engines The schemtic rrngement nd ow structure in seprtestrem bypss engine is shown in Fig. 1. The description of n equivlent turbojet engine is lso shown in Fig. 1 s reference point in the nlysis. The core gs genertor produces, for pr- Turbojet Seprte-strem bypss engine Fig. 1 Schemtic description of seprte-strem turbofn engine nd n equivlent turbojet engine used for nlysis; energy is the sme t point in both engines. ticulr vlue of fuel consumption rte, jet with speed V j;ref in the bsence of ny bypss ow. The kinetic energy dded by the gs genertor is then, neglecting the fuel mss ow rte which is usully very smll proportion of the ir mss ow rte in jet engine, equl to E k D 1 Pm 2 h V 2 j;ref (1) In the bypss engine, the whole of this kinetic energy is, of course, not present in the core jet. The LP turbine extrcts portion of this energy nd turns the fn, which, in turn, dds kinetic energy to the bypss ow. The ef ciency KE of this energy trnsfer between the core nd bypss ow depends on the component ef ciencies of the LP turbine, the fn, nd the bypss nozzle nd lso on the ow lossesin the ductsndmechniclef ciencyof the shfts.assuming the mechnicl ef ciency is close to 1, KE ¼ LPT f NB (2) At the current level of technologies, 4;9 LPT ¼ 0:9, f ¼ 0:9, nd NB ¼ 1. Eqution (2), therefore, suggests tht KE ¼ 0:8. In bypss engine, the kinetic energy given by Eq. (1) is relted to the sum of the kinetic energies of the two strems by E k D 1 Pm 2 h V 2 j;ref D 1 Pm 2 h C 1 2 Pm h.b= KE / j c V 2 j h In Eq. (3) we hve expressedthe mss ow rte of the bypssstrem, Pm c, in terms of Pm h nd B. Similrly the net thrust F N produced by the bypss engine is the sum of the thrusts of the two strems nd is given by (3) F N D Pm h.v j h V / C B Pm h.v jc V / (4) The pressure thrust hs been tken into ccount becuse V jh nd V j c re the fully expnded jet velocities of the two strems. The objectiveof the optimiztionprocess is to determinethe rtio V jc =V j h, which would mximize the net thrust F N while keeping the fuel consumption xed. The condition for mximum net thrust is given F jh D 0 (5) At constnt therml ef ciency, xed fuel consumption mens E k in Eq. (3) is constnt. k jh D 0 (6)

3 GUHA 1119 For constnt bypss rtio B, Eqs. (3 6) cn be combined to give.v j c =V j h / op D KE (7) While evlutingthe prtilderivtivesin Eqs. (5) nd (6), B nd Pm h re ssumed constnt.togetherthey specify constnt inlet ir ow. Becuse the speci c thrustis the rtio of net thrustnd mss ow rte of ir t the inlet, Eq. (5) is lso the condition for mximum speci c thrust. SFC is the rtio of mss ow rte of fuel nd net thrust. Eqution (6) implies xed mss ow rte of fuel; Eq. (5) is, thus, lso the condition for minimum SFC. Becuse Eq. (7) gives both mximum speci c thrust nd minimum SFC, this condition would be used in the nlyticl formultion of the optimiztion process tht follows. A designer cn stisfy Eq. (7) in rel engine by choosing the FPR op, s will be shown. The men jet speed V m cn be expressed, using Eq. (7), s Fig. 3 Comprison of present theory with numericl optimiztion results for FPR op in seprte-strem bypss engines. (Altitude = 11 km, M = 0:82, isentropic ef ciency of compressors nd turbines = 0:9). V m D [1=.1 C B/].B C 1= KE /V jc (8) nd it follows tht OF N D F N =. Pm h C Pm c / D V m V (9) Suppose the rise in totl temperture in the bypss ow for hypotheticl, isentropic compression by the fn, for the sme pressure rtio tht exists cross rel fn, is 1T 0;isen. If the fn nd the bypss nozzle were isentropic, then the sttic temperture of the fully expnded jet would be equl to the mbient temperture. Appliction of the stedy ow energy eqution then gives 1T 0;isen D.1=2c p / jc (10) Equtions (8 10) cn be combined with the isentropic pressure temperture reltion. One then obtins, fter some lgebric mnipultion, the nl expression for the FPR op :. 1/.FPR/ op D 1 C 2 C. 1/M 2 ".1 C B/ 2 OF p N C M.B C 1= KE / 2 RT ¼ 2 M 2 # (11) All quntities in Eq. (11) re nondimensionl numbers. Figure 2 shows grphiclly the prediction of Eq. (11). One interesting feture of Eq. (11) is tht if KE D 1, tht is, in the cse of idel energy trnsfer from the core strem to the bypss strem, the bypss rtio B drops out of the eqution. FPR op would then hve depended only on OF N. At ny rte, Eq. (11) predicts tht, for xed OF N, the dependenceof FPR op on B is wek, especilly t highervluesof B (B > 4). Eqution(11) lso producesthe expected limiting result: lim B! 1 OF N! 0.FPR/ op D 1 For chosen speci c thrust OF N nd bypss rtio B, Eq. (11) immeditely speci es the FPR op. The llowble rnge of FPR is rther restricted. The design wisdom is tht only single-stge fn is used for civil ircrft with lrge-bypss rtios, the mximum Fig. 2 Vrition of FPR op with speci c thrust in seprte-strem bypss engines: prediction of Eq. (11); M = 0:82, T = 216:65 K, KE = 0:81, nd = 1:4. Fig. 4 Dt for vrious current civil turbofn engines under cruise conditions (typiclly 35,000 ft, M = ). FPR is, therefore, restricted to 1.8. A limit of minimum FPR my lso rise due to fn instbilities t prtlod, off-design conditions. According to Rüd nd Lichtfuss, 13 below n FPR of 1.4, the engine will require vrible geometry either vi vrible pitch fn or vrible re bypss nozzle to provide erodynmic fn stbility under prtlod conditions. Therefore, knowing FPR op t n erly stge of the design is n dvntge. If its vlue does not lie within the desired limit, the choice of OF N nd B cn be ltered (of course, nother option would be not to dhere strictly to the optimum vlue of FPR). Eqution (11) hs been tested ginst optimiztion performed by the computer progrm GsTurb. 10 Figure 3 shows the comprison. Ech numericl point hs been clculted by GsTurb by lengthy optimiztion process tht uses dvnced serch techniques to nd the optimum FPR tht minimizes the SFC while, t ech tril, the primry vribles re iterted to give the prescribed speci c thrust. Figure 3 shows tht Eq. (11) performs well ginst sophisticted numericl optimiztion nd cn, therefore, be used for the design of rel engines. To pprecite the current design stndrd, Fig. 4 plots the SFC nd speci c thrust of some current civil turbofn engines of vrious mnufcturers. 14 Speci c thrust dt in Fig. 4 re pproximte becuse lthough net thrust is known t cruise, the mss ow rte is estimted from given vlues t se level sttic conditions by dynmic scling, tht is, ssuming the sme nondimensionloperting point. The speci c thrust dt indictesover which rnge ny theory needs to be pplicble in order to be relevnt for current nd future designs. Figure 4 shows tht for most existing engines, the cruise speci c thrust lies in the bnd lbf/lbm/s. Over the pst 40 yers of civil engine design, the speci c thrust hs reduced signi cntly (producingpprecibleimprovementin propulsiveef ciency) while the bypss rtio hs incresed from 1 2 in the 1960s to 7 9 in the 1990s. 9 The forecst, 9;13 is tht the design driver for future engines would be towrd even lower speci c thrust. A surge in fuel price nd/or the introduction of more stringent noise regultion my necessitte such designs. Future gered turbofn or dvnced ducted propulsors would reduce the speci c thrust signi cntly. Figure 3 thus shows tht Eq. (11) is pplicble to both current nd future bypss engines. Mixed-Strem Turbofn Engines Before developing the theory for FPR op for mixed-strem turbofn engines, the computer pckge GsTurb ws used to determine

4 1120 GUHA the sme numericlly. Figure 5 shows the results of this numericl optimiztion. Clcultions re shown t two bypss rtios. Mixedstrem engines used for militry purposes employ much higher speci c thrust thn tht used for seprte-strem civil pplictions. These engines usully hve smller bypss rtio. Therefore, mximum tolerble turbine entry tempertures cn produce lrge speci c thrust. Of course, such high vlues of speci c thrust would men low propulsive ef ciency giving higher fuel consumption. The reduction in engine size nd weight is, however, more crucil for such pplictions. Figure 6 shows the stndrd sttion numbering for mixed- ow turbofn engine. In the following nlysis these numbers re used s subscripts to denote ow vribles t vrious loctions. The following representtive vlues give resonble pproximtion to the propertiesof ir nd combustionproducts: R D 287 J/kg/K, D 1:4, c p D R=. 1/, g D 1:33, c pg D g R=. g 1/. A simple, nlyticl procedurefor clcultingthe propertiesof combustion products s function of temperture, fuel ir rtio, nd fuel composition is given by Guh. 15 The nlysis cn be formulted more esily if the optimum FPR ndthe speci c thrustre expressedin terms of the primry vribles OPR, B, nd T 04. An energy blnce gives (with T 05 D T 06 nd T 013 D T 016 ).c pg =c p /.T 04 T 06 / D.T 03 T 02 / C B.T 016 T 02 / (12) where the vrious totl tempertures cn be clculted from T 02 =T D 1 C 0:5. 1/M 2 (13) T 03 T 02 D.1= c/t 02 OPR 1 (14) T 016 T 02 D.1= f /T 02 FPR 1 (15) T 04 T 06 D t T 04 1.OPR/FPR/.1 g /= g (16) where c is the isentropic ef ciency of the compression between points 2 nd 3 nd t is the isentropic ef ciency of the expnsion between points 4 nd 6. While writing Eq. (16) it is ssumed tht p 016 =p 06 D 1 when optimum fn pressure rtio is chieved. Strictly speking, this condition is not lwys fully stis ed. However, the vlue of the rtio p 016 =p 06 rises under off-designconditions:if the vlue of this rtio used t the design point is greter thn one, serious mixing losses my result t off-designopertions.therefore, it is prudent to choose the vlue of p 016 =p 06 between 0.95 nd 1 t the design point even if it is slightly suboptimlt tht prticulropertingpoint. Numericlclcultions of Millhouse et l. 1 show tht if the condition p 016 =p 06 ¼ 1 is used t the design point, the computed FPR then gives the optimum vlue independent of speci c mission of the ircrft. Equtions (12 16) cn be solved to determine the optimum FPR. This will, however, require n itertive solution. For smll-bypss rtios, s is usul for mixed-strem militry engines, the second term in the right-hnd side of Eq. (12) is smll compred to the rst term. The index for FPR in Eq. (15) cn then be pproximted by substituting g for. With this pproximtion,the equtionscn be combined to give n explicit reltion for the FPR op : µ FPR. g 1/= g BT02 op C.c pg=c p / t T 04 c pg D f OPR. g 1/= g t T 04 c p 1 T 02 OPR c 1 C BT 02 f (17) If V j is the jet speed nd V is the ircrft speed, the speci c thrust is clculted from where OF N D V j V (18) V j D p 2c pg.t 064 T j / (19) T 064 D T 06 C BT C B (20) T j D T 064.FPR op f M /.1 g/= g (21) f M D [1 C 0:5. 1/M 2 ] =. 1/ (22) Eqution (20) gives the totl temperture of the mixed strem nd Eq. (21) gives the sttic temperture of the jet. While writing the precedingequtions, the losses in the fn, compressorsnd turbines hve been ccounted for, but other losses, for exmple, tht in the vrious ducts, hve been neglected. Figure 7 shows the comprison of the present theory nd the numericl optimiztion results using GsTurb. Ech numericl point hs been clculted by GsTurb to nd the optimum FPR tht minimizes the SFC, while, t ech tril, the primry vriblesre iterted to give the prescribedspeci c thrust. The nozzle re ws lso optimized t ech point so tht it produced fully expndedjet. The convergenceof the numericloptimiztionmy be slow nd sometimes, dependingon the strtingpoint, the numericlserch techniquemy not nd the optimum or my nd locl rther thn the globl optimum. The nlyticlequtions, on the other hnd, redily provide the nswer. Figure 7 shows tht the present theory gives ccurte results: The itertive nlyticl solution [Eqs. (12 16)] grees lmost Fig. 5 FPR op for mixedstrem turbofn engines determined by numericl optimiztion (GsTurb); for ll clcultions, OPR = 17:5, M = 0:82, isentropic component ef ciencies = 0:9 nd mix = 1. Fig. 7 Comprison of the present theory with numericl optimiztion results for FPR op in mixed-strem turbofn engines; for ll clcultions, OPR = 17:5, B = 0:5, M = 0:82, isentropic component ef ciencies = 0:9 nd mix = 1. Fig. 6 Stndrd nomenclture for vrious loctions in mixed-strem turbofn engine (only hlf of the engine is shown).

5 GUHA 1121 Tble 1 Dependence of FPR op on OPR for mixed-strem engines Tble 2 Comprison of theory nd numericl results for optimum V jc /V jh OF N, lbf/lbm/s OPR T 04, K B op FPR op identiclly with GsTurb optimiztion; the explicit eqution (17) lso gives quite cceptble nswer until the speci c thrust becomes very high. The present theory cn be extended to include the effects of incompletemixing by introducing prmeter mix. It is then ssumed tht three seprte strems hving totl tempertures T 016, T 06, nd T 064 expnd through the sme pressure rtio. The totl thrust is the sum of the thrust produced by these individul strems: p.1 /= V j16 D 2c p.t 016 T j16 /; T j16 D T 016.FPR op f M / p.1 g /= g V j6 D 2c pg.t 06 T j6 /; T j 6 D T 06.FPR op f M / OF N D mix V j C.1 mix /.V j6 C BV j16 /=.1 C B/ V (23) Seprte Versus Mixed Strem Engines Figure 5 cn be compred with Figs. 2 nd 3. Three observtions cn be mde. 1) The FPR op for mixed-stremengines rises monotonicllywith the speci c thrust, s it does for seprte strem engines. 2) The FPR op is predominntly function of the speci c thrust nd only wekly depends on the bypss rtio. The dependence on bypss rtio is weker in mixed-stremenginesthn tht in seprtestrems engines. 3) At prticulr vlue of speci c thrust, the FPR op for mixedstrem engines is lower thn tht in the seprte-strems engines. In seprte-strem engine, ltering the vlues of OPR does not lter the vlue of FPR op, so long s the speci c thrust nd bypss rtio re kept xed. This is predicted by Eq. (11) nd borne out by numericl optimiztion. In mixed-strem engine, even t xed speci c thrust, vrious vlues of OPR nd M chnge the vlue of FPR op. Either the nlyticl reltions (12 16) or the numericl optimiztions show this. Tble 1 shows the results of optimiztion performed by GsTurb (both B nd FPR re optimized t xed speci c thrust t two levels of OPR, minimising SFC). Effects of FPR op on the Conditions of Other Vribles The behviorof other ow vribles re now exmined when the FPR op is chieved. Seprte discussion is provided for the mixedstrem nd seprte-stremengines becuse differentphysicl principles re involved. Seprte-Strem Bypss Engines The conditions to which the FPR op correspond hve been determined nlyticlly in this cse. Eqution (7) shows tht there is prticulr rtio of the fully expnded jet velocities of the cold nd hot strem [.V j c =V j h / op D KE ] tht simultneously chieves minimum SFC nd mximum speci c thrust. The designer cn mke the two jet velocities tke this vlue by choosing the FPR op. The computtionl progrm GsTurb ws used to nd the rtio V j c =V j h tht occurs in n engine whose FPR hs been numericlly optimized for minimizing SFC. Tble 2 shows these vlues (for OPR D 30, T 04 D 1200 K, LPT D 0:9, f D 0:9, nd NB D 1), ginst the prediction of Eq. (7). Numericl optimiztions were performed t severl other OPR nd T 04 : V j c =V j h ly between0.77nd 0.82 (for the ssumedcomponent ef ciencies). The numericl clcultionsshow tht the derived nlyticl reltion [Eq. (7)] is pproximtely vlid. Mixed-Strem Bypss Engines The rtio of the totl pressure of the two strems t the beginning of the mixer zone, p 016 =p 06, is n importnt vrible nd the.v j c =V jh / op Bypss Numericl optimiztion.v jc =V j h / op D KE rtio with GsTurb [ KE ¼ LPT f NB ] Tble 3 Vlidity of Eq. (24) for vrious component ef ciencies f LPT. p 016 =p 06 / op.v 163 =V 63 / op KE D f LPT Clculted by numericl optimiztion with GsTurb. Fig. 8 Effects of design mixer Mch number on the two chrcteristic rtios when FPR op is chieved; for ll clcultions, M = 0:82, B = 0:5, ÃF N = 60 lb/lb/s, nd component isentropic ef ciencies = 0:9. FPR controls this. It is generlly recognized, 1;10 tht when FPR op is chieved, the rtio p 016 =p 06 is close to one. The physicl reson is tht, when the totl pressures of the two strems re nerly equl, the loss (entropy genertion) due to mixing would be smll. The computer progrm GsTurb ws used to determine the vlue of FPR (by rndom dptive serch) tht minimizes the SFC t severl speci c thrust levels tht were xed by iterting OPR nd T 04. It ws found tht optimum p 016 =p 06 did not remin xed but vried. Optimum p 016 =p 06 incresed with incresing speci c thrust nd bypss rtios. The optimum vlue of p 016 =p 06 ws found to vry prticulrly strongly with design mixer Mch number M 64, s shown in Fig. 8. However, the present study discovered new reltion between the velocities of the two strems t the mixer inlet under optimum conditions:.v 163 =V 63 / op ¼ KE (24) where KE D f LPT. At the FPR op, the reltion V 163 =V 63 ¼ KE remined nerly vlid t ll vlues of speci c thrust, bypss rtio, nd design mixer Mch number. The similrity of this condition with Eq. (7) derivedfor seprte-strembypssenginesis noticeble.the vrition in the velocity rtio is shown in Fig. 8 (with KE D 0:81). To ssess the ccurcy of Eq. (24), lrge number of numericl experiments were conducted with GsTurb in which the isentropic ef ciencies of vrious components were chnged systemticlly. It ws found tht indeed the ef ciencies of the high-pressure (HP) compressor, HPC, nd the HP turbine, HPT, did not in uence the vlue of.v 163 =V 63 / op, s Eq. (24) suggests. Eqution (24) ws lso stis ed when vrious vlues of the LP compressor nd LP turbine ef ciencies were tested. A few of these numericl optimiztion results re given in Tble 3 to demonstrte this point. (For ll clcultions shown in Tble 3, OF N D 60 1b/1b/s, B D 1, OPR D 17:5, M 64 D 0:3, M D 0:82, nd mix D 1. Similr results re obtined t other vlues of these vribles.) Conclusions The optimum fn (compressor) pressure rtio is determined both numericlly nd nlyticlly for seprte-strem s well s mixedstrem bypss engines. The FPR op is shown to be predominntly function of the speci c thrust nd wek function of the bypss

6 1122 GUHA rtio.for mixed-stremengines,the dependenceof FPR op on bypss rtio is very wek, nd FPR op lso depends on OPR. At the sme speci c thrustndbypssrtio,the FPR op for mixed-stremengine is lower thn tht of seprte-stremengine. Two simple, explicitequtionshvebeenderivedfrom fundmentl principles.eqution(11) gives the FPR op for seprte-stremengines,wheres Eq. (17) [or the equtionset (12 16)] givesthe FPR op for mixed-strem engines. The ccurcy of the nlyticl formuls hs been estblished through extensive veri ction by numericl optimiztion results of the commercil computer pckge GsTurb (Figs. 3 nd 7). The nlyticl results ccelerte the optimiztion process nd offer physicl insight. It hs been shown tht the FPR op chieves the condition V j c =V j h D KE in seprte-strem engine nd the condition V 163 =V 63 ¼ KE in mixed-strem engine. The condition V 163 =V 63 ¼ KE pplies even under situtions when signi - cnt deprtures from the normlly ssumed condition p 016 =p 06 ¼ 1 occur. Acknowledgment The uthor is grteful to the referees for their helpful comments. References 1 Millhouse,P. T., Krmer, S. C., King, P. I., nd Mykytk,E. F., Identifying Optiml Fn Compressor Pressure Rtios for themixed-strem Turbofn Engine, Journl of Propulsion nd Power, Vol. 16, No. 1, 2000, pp Hill, P. G., nd Peterson, C. R., Mechnics nd Thermodynmics of Propulsion, 2nd ed., Addison Wesley Longmn, Reding, MA, 1992, Chp Cohen, H., Roger, G. F. C., nd Srvnmuttoo, H. I. H., Gs Turbine Theory, 4th ed., Longmn, Hrlow, Englnd, U.K., 1996, Chp Cumpsty, N. A., Jet Propulsion, Cmbridge Univ. Press, Cmbridge, Englnd, U.K., Ruf es, P. C., The Future of Aircrft Propulsion, Journl of Mechnicl Engineering Science, Vol. 214, No. C1, 2000, pp Jckson, A. J. B., Some Future Trends in Aero Engine Design for Subsonic Trnsport, Journl of Engineering for Power, Vol. 98, April 1976, pp Bennett, H. W., Aero Engine Development for the Future, Proceedings of the Inst. of Mechnicl Engineers, Series A, Vol. 197, July 1983, pp Wilde, G. L., Future Lrge Civil Turbofns nd Powerplnts, Aeronuticl Journl, Vol. 82, July 1978, pp Birch, N. T., 2020Vision: the Prospects for Lrge Civil Aircrft Propulsion, Aeronuticl Journl, Vol. 104, No. 1038, 2000, pp Kurzke, J., Mnul for GsTurb 7.0 for Windows, MTU, München, Germny, Guh, A., Optimiztion of Aero Gs Turbine Engines, Aeronuticl Journl, Vol. 105, No. 1049, 2001, pp Mttingly, J. D., Elements of Gs Turbine Propulsion, McGrw Hill, New York, 1996, pp Rüd, K., nd Lichtfuss, H. J., Trends in Aero-Engines Development, Aspects of Engine Airfrme Integrtion for Trnsport Aircrft, Proceedings of DLR Workshop, DLR-Mitteilung, Brunschweig, Germny, Mrch Rolls Royce Aero Dt, TS 1491, Issue 9, Derby, Englnd, U.K., Guh, A., An Ef cient Generic Method for Clculting the Properties of Combustion Products, Journl of Power nd Energy, Vol. 215, No. A3, 2001, pp