JOURNAL OF PROPULSION AND POWER Vol.17,No.5, September –October2001

Optimum FanPressure Ratio for Bypass Engines with Separate or Mixed Exhaust Streams

AbhijitGuha ¤ Universityof Bristol,Bristol, England BS8 1TR, United Kingdom

The optimumfan pressure ratiois determinedboth numerically and analytically for separate-stream as well as mixed-streambypass engines. The results are applicableto civil and military engines. The optimumfan pressure ratiois shownto be predominantlya functionof the speciŽc thrustand a weakfunction of the bypassratio. Two simple,explicit equations, one for each type of engine, have been derivedthat specify the optimumfan pressure ratio.The predictionscompare very well againstnumerical optimization performed by a specialistcomputer packageemploying iterative and advanced search techniquesand real gasproperties. The analyticalrelations accelerate the optimizationprocess andoffer physicalinsight. It hasbeen shownthat the optimumfan pressure ratioachieves the condition V /V = ´ ina separate-streamengine and the condition V /V ´ in a mixed- jc jh KE 163 63 ¼ KE streamengine. The condition V163/V63 ´KE applieseven under situations when signiŽ cant departures fromthe normallyassumed condition p /p ¼ 1 can occur. 016 06 ¼

Nomenclature pressor) pressureratio in bypassengines. Separate analysis has been B =bypassratio presentedfor the mixed-stream and separate-stream engines be- c p =speciŽc heatof air at constant pressure causethe  owphysics is different in thetwo cases. In bothcases, c pg =speciŽc heatof combustion products at thepresent theory gives results in good agreement with numerical constantpressure optimizationcalculations. ( ) Ek =kineticenergy added by the core engine Determinationof theoptimum fan pressure ratio FPRop is im- FN = net thrust portantbecause, given all other variables such as theoverallpressure F ratio (OPR),bypassratio B,andturbine entry temperature T04 the ON =speciŽc thrust M =Machnumber of the aircraft optimumvalue of FPR simultaneouslyensures maximum speciŽ c m =mass owrate through the bypass duct (cold  ow) thrustand minimum speciŽ c fuelconsumption. Thus, the FPR op P c m f =mass owrate of fuel achievesthe two, usually contradictory, goals of jetengine design. mP =mass owrate through the core engine (hot  ow) Forcivil engines, particularly for medium to long range aircrafts, P h p =staticpressure the FPRop canbe calculated under cruise conditions. For military p0 =totalpressure engines,optimization may be mission speciŽ c. However,Millhouse R =speciŽc gasconstant of air et al.1 haveshown that, although the most fuel-efŽ cient values of T =statictemperature suchparameters as bypass ratio and OPR dependon theintegrated Ta =temperatureof ambientair effectsof thevarious  ightaltitudes, Mach numbers, and thrust re- T0 =totaltemperature quiredthroughout the mission, it is possible to locateheuristically V = velocity themost efŽ cient FPR tocomplementother engine parameters in- Va =forwardspeed of theaircraft dependentof themission. Thus, the relations derived in thispaper canbe usedin the design of bothcivil and military engines. V j =jetspeed in a mixedstream engine V j c =fullyexpanded jet speed of the cold bypass stream Thedesign wisdom is that only a single-stagefan is usedfor a civilaircraft engine with large bypass ratios, and so themaximum V j h =fullyexpanded jet speed of the hot core stream Vm =meanjet speed [given by Eq. (8)] FPR isabout 1.8. Engines with low bypass ratios, for example, mil- ° =isentropicindex of air itaryengines, normally use a low-pressure (LP) compressorhaving – °g =isentropicindex of combustionproducts 3 5stages.In this case, the maximum possible FPR canbe much higher.The mixing of the two streams in a mixed-streamturbofan ´ f =isentropicefŽ ciency of thefan or low pressurecompressor engineoffers performance gain (lowerspeciŽ c fuelconsumption andhigher speciŽ c thrust ),eventhough mixing incurs a lossin total ´KE =efŽciency of energy transfer between the core andbypass  ow pressurebecause the enthalpies of the two streams are redistributed. Moreover,only one reheat system and only one variable nozzle ´LPT =isentropicefŽ ciency of thelow pressure turbine ´NB =isentropicefŽ ciency of thebypass nozzle areneeded. A longnacelle can contain the fan/ compressornoise moreeffectively. For the same speciŽ c thrust,the mixed-stream en- Subscript ginehas lower optimum FPR thana separate-streamengine; thus, theLP turbinemay have one fewer stage if bothengines have the op= optimumvalue samebypass ratio. The outer cold stream also protects the jet pipe Introduction fromsevere temperatures resulting from afterburning. Low-speciŽ c- thrust,high-bypass-ratio engines, on the other hand, do notemploy Nthispaper simple, explicit relations have been derived from afterburning,and mixingof thetwo streams would necessitate a long I fundamentalprinciples for determining the optimum fan (com- nacelleincreasing weight, cost, nacelle drag, and interference ef- fects.These engines, therefore, use separate-stream conŽ gurations. Received 30May 2000; revision received 20February 2001; accepted Whena commonnozzle is, however, used for high-bypass-ratio en- forpublication 29 May 2001. Copyright c 2001by Abhijit Guha. Pub- lishedby theAmerican Instituteof Aeronautics ° and Astronautics, Inc., with gines,the small axial extent of themixing zone is usually insufŽ cient permission. toachievecomplete mixing of thetwostreams. ¤ Lecturer,Aerospace EngineeringDepartment, University Walk; Calculationand analysis of theperformance of turbofanengines [email protected]. arediscussed in detailsin several excellent texts. 2 4 Incomparison, 1117 1118 GUHA theissue of optimized design has received less attention, and an- alyticalsolutions did not exist. The real store house of expertise andknowledgeobviously exists with the aeroengine manufacturers whohave actively pursued the optimization of gasturbine perfor- mancefor the past 60 years. Ruf es 5 showsthat there has been a 50%improvement in the speciŽ c fuelconsumption (SFC) of the bareengine, as theengines evolved from the turbojet of 1958 to the Turbojet high-bypassTrent engines of moderntime. Improvements in propul- siveefŽ ciency, component efŽ ciency and cycle efŽ ciency contribute approximatelyone-third each to thisgain in performance. The pub- licationsmade by experts from industry (e.g.,Jackson, 6 Bennett,7 Wilde,8 and Birch9 ) containa wealthof experience, information, andcalculation results. However, the details of calculation method arenot known to the readers. The industry would typically have its own,sophisticated computer packages to whichthe public does not haveaccess. Moreover, the published results often would involve an adoptedfamily of enginedesigns. It is then not always obvious to thegeneral reader how to generalize the results or howto proceed onaclean-sheetanalysis. Separate-streambypass engine Inthis paper analytical relations have been derived for calculating FPR .Theanalytical results not only accelerate the optimization Fig.1 Schematicdescription of a separate-streamturbofan engine op andan equivalent turbojet engine used for analysis; energy is the same processbut also offer valuable physical insight. It is shown that at point inboth engines. the FPRop ispredominantly a functionof the speciŽ c thrustand ­­ onlyweakly depends on the bypass ratio. The FPR op monotoni- callyincreases with increasing speciŽ c thrustat two different levels ticularvalue of fuel consumption rate, a jetwith speed V j;ref in the correspondingto the separate-stream and mixed-stream turbofan absenceof anybypass  ow.Thekinetic energy added by the gas engines. generatoris then,neglecting the fuel mass  owrate which is usu- Thetheory has been veriŽ ed bycomparingits predictions with allya verysmall proportion of theair mass  owrate in a jetengine, calculationsperformed by thecomputer program GasTurb, 10 which equal to isageneralpurpose commercial package for the calculation of de- E 1 m V 2 V 2 (1) signand off-design performance of varioustypes of gas turbine en- k D 2 P h j;ref a gines,turbojet, turboshaft, separate- or mixed-stream turbofan, and Inthebypass engine, the whole ¡of thiskinetic ¢ energy is, of course, gearedturbofan, all with single- or twin-spoolconŽ gurations. It uses notpresentin the core jet. The LP turbineextracts a portionof this realgas properties with dissociation and has several features includ- energyand turns the fan, which, in turn,adds kinetic energy to the ingoptimization, iteration, transient, and Monte Carlo simulation. bypass ow.TheefŽ ciency ´KE ofthisenergy transfer between the Eachnumerical point subsequently illustrated has been calculated coreand bypass  owdepends on the component efŽ ciencies of byGasTurbby alengthyoptimization process that uses advanced theLP turbine,the fan, and the bypass nozzle and also on the ow searchtechniques to Ž ndtheoptimum FPR thatminimizes the SFC lossesin theducts and mechanicalefŽ ciency of theshafts. Assuming while,at each trial, the primary variables are iterated to give the themechanical efŽ ciency is close to 1, prescribedspeciŽ c thrust.This numerical optimization is used not onlyto verify the theoretical prediction, but also to discover what ´ ´ ´ ´ (2) KE ¼ LPT f NB exactlyhappens to the many variables when the FPR op isachieved. 4;9 Basedon the principles discussed in the present paper, a new At thecurrent level of technologies, ´LPT 0:9, ´ f 0:9, and ¼ ¼ methodologyfor the thermodynamic optimization of bypassengines ´NB 1. Equation (2),therefore,suggests that ´KE 0:8. ¼ ¼ (turbofanor advanced propulsors ) hasbeen developed by Guha 11 in Ina bypassengine, the kinetic energy given by Eq. (1) is related whichthe optimum combination of all variables is determinedcon- tothesum of thekinetic energies of thetwo streams by currently.The process starts with establishing an optimumspeciŽ c E 1 m V 2 V 2 1 m V 2 V 2 thrustfor the engine based on an economic analysis (installation k D 2 P h j;ref a D 2 P h j h a constraints,noise regulations, etc., also need to beconsidered ). The 1 m¡.B=´ / V¢2 V 2 ¡ ¢ (3) taskof theoptimization process is then to Ž ndthecombination of C 2 P h KE j c a optimumvalues of OPR, B, FPR, and T04 concurrentlythat mini- In Eq. (3) wehave expressed the mass  owrate of thebypass stream, mizesSFC attheŽ xedspeciŽ c thrust.This procedure is quite differ- ¡ ¢ mc , in terms of m h and B.Similarlythe net thrust FN produced by 2;12 entfromthe usual parametric studies of engineperformance, in theP bypass engine P is thesum of thethrusts of thetwo streams and whicha singleparameter is variedeach time, while keeping all other is given by parametersŽ xed,therefore at their nonoptimum values. Moreover theusual single-variable parametric studies may involve a largeex- FN mh .Vj h Va / Bmh .V jc Va / (4) cursionin the value of speŽ cic thrust unacceptable for a speciŽc D P C P mission.Guha 11 hasalso discussed at lengththe determination of Thepressure thrust has beentaken into account because V jh and Vj c optimaljet velocity, with the derivation of anewanalytical expres- arethe fully expanded jet velocities of the two streams. sionthat performs well against numerical optimization results. Theobjective of theoptimization process is to determine the ratio Followingindustrial practice, the SFC andthe speciŽ c thrust, V jc =V j h,whichwould maximize the net thrust FN whilekeeping areexpressed here respectively in pounds mass per hour per thefuel consumption Ž xed.The condition for maximum net thrust poundsforce and pounds force per pounds mass per sec- is given by ondwhere 1 lbf/lbm/s 9:81m/ sand1 lbm/hr/lbf 28:316 6 D D £ @ FN 10 kg/s/N. 0 (5) @V jh D Separate-StreamBypass Engines At constantthermal efŽ ciency, a Žxedfuel consumption means Ek Theschematic arrangement and  owstructure in a separate- in Eq. (3) isconstant. Therefore, streambypass engine is shown in Fig. 1. The description of an @.E / equivalentturbojet engine is also shown in Fig. 1 asa reference k 0 (6) pointin the analysis. The core gas generator produces, for a par- @V jh D GUHA 1119

Fora constantbypass ratio B, Eqs. (3–6) canbe combinedto give

.V =V / ´ (7) j c j h op D KE

Whileevaluating the partial derivatives in Eqs. (5) and (6), B and m h areassumed constant. T ogetherthey specify a constantinlet air ow P . Becausethe speciŽ c thrustis the ratio of net thrustand mass  owrate ofairat theinlet, Eq. (5) isalsothe condition for maximum speciŽ c thrust.SFC isthe ratio of mass  owrate of fuel and net thrust. Equation (6) impliesa Žxedmass  owrate of fuel;Eq. (5) is, thus, alsothe condition for minimum SFC. BecauseEq. (7) gives both maximumspeciŽ c thrustand minimum SFC, thiscondition would beused in the analytical formulation of the optimization process Fig.3 Comparisonof present theorywith numerical optimization re- thatfollows. sultsfor FPR op inseparate-stream bypass engines. (Altitude= 11km, Adesignercan satisfy Eq. (7) inarealengine by choosingthe M = 0:82,isentropicefŽ ciency ofcompressors andturbines = 0 :9). FPRop,aswill be shown. The mean jet speed Vm canbe expressed, using Eq. (7), as

V [1=.1 B/].B 1=´ /V (8) m D C C KE jc andit follows that Fig.4 Datafor various cur- rent civilturbofan engines un- F F m m V V (9) der cruise conditions (typically ON N =. h c / m a D P C P D 35,000 ft, M = 0.8–0.85). Supposethe rise in total temperature in the bypass  owfor a hy- pothetical,isentropic compression by the fan, for the same pressure ratiothat exists across a realfan, is 1T0;isen .Ifthefan and thebypass nozzlewere isentropic, then the static temperature of the fully ex- pandedjet would be equalto the ambient temperature. Application FPR is,therefore, restricted to 1.8. A limitof minimum FPR may ofthesteady  owenergy equation then gives alsoarise due to fan instabilities at partload,off-design conditions. Accordingto R ud¨ andLichtfuss, 13 belowan FPR of1.4, the engine 2 2 1T0;isen .1=2c p/ V V (10) willrequire variable geometry either via a variablepitch fan or a D jc a variablearea bypass nozzle to provide aerodynamic fan stability Equations (8–10) canbe combined ¡with the isentropic ¢ pressure – underpartload conditions. Therefore, knowing FPR op at an early temperaturerelation. One then obtains, after some algebraic manip- stageof thedesign is an advantage.If itsvalue does not lie within ulation,the Ž nalexpression for the FPR op: F B ( thedesired limit, the choiceof ON and canbe altered of course, anotheroption would be notto adhere strictly to the optimum value .° 1/=° .° 1/ .FPR/op 1 ) D C 2 .° 1/M 2 of FPR . C Equation (11) hasbeentested against optimization performed by 10 2 thecomputer program GasTurb. Figure3 showsthe comparison. .1 B/2 F C ON M M 2 (11) Eachnumerical point has beencalculated by GasTurb by alengthy 2 £ ".B 1=´KE/ p° RTa C # optimizationprocess that uses advanced search techniques to Ž nd C » ¼ theoptimum FPR thatminimizes the SFC while,at eachtrial, the Allquantities in Eq. (11) arenondimensional numbers. Figure 2 primaryvariables are iterated to give the prescribed speciŽ c thrust. showsgraphically the prediction of Eq. (11). Figure3 showsthat Eq. (11) performswell against sophisticated Oneinteresting feature of Eq. (11) is that if ´KE 1,thatis, in numericaloptimization and can, therefore, be usedfor thedesign of D thecase of idealenergy transfer from the core stream to the bypass realengines. stream,the bypass ratio B dropsout of the equation. FPR op would Toappreciatethe current design standard, Fig. 4 plotsthe SFC thenhave depended only on F .At anyrate, Eq. (11) predictsthat, ON andspeciŽ c thrustof some current civil turbofan engines of vari- F B 14 for a Ž xed ON ,thedependence of FPR op on isweak, especially at ousmanufacturers. SpeciŽc thrustdata in Fig.4 areapproximate highervalues of B (B > 4). Equation(11)alsoproduces the expected becausealthough net thrust is known at cruise,the mass  owrate limitingresult: isestimatedfrom given values at sea level static conditions by dy- namicscaling, that is, assuming the same nondimensional operating lim .FPR/op 1 B D point.The speciŽ c thrustdata indicates over which range any theory FN! 10 O ! needsto beapplicable in order to be relevantfor currentand future designs. F B ( ) Fora chosenspeciŽ c thrust ON andbypass ratio , Eq. 11 Figure4 showsthat for most existing engines, the cruise spe- immediatelyspeciŽ es the FPR op.Theallowable range of FPR is ciŽc thrustlies in the band 15 –20lbf/ lbm/s.Overthe past 40 years ratherrestricted. The design wisdom is thatonly a single-stagefan ofcivilengine design, the speciŽ c thrusthas reduced signiŽ cantly isused for a civilaircraft with large-bypass ratios, the maximum (producingappreciable improvement in propulsiveefŽ ciency )while thebypass ratio has increased from 1 –2inthe1960s to 7 –9 in the 1990s.9 Theforecast, 9;13 isthat the design driver for future engines wouldbe toward even lower speciŽ c thrust.A surgein fuel price and/orthe introduction of more stringent noise regulation may ne- Fig.2 Variationof FPR op cessitatesuch designs. Future geared turbofan or advancedducted withspeciŽ c thrustin propulsorswould reduce the speciŽ c thrustsigniŽ cantly. Figure 3 separate-streambypass en- thusshows that Eq. (11) isapplicable to both current and future gines:prediction of Eq. (11); bypassengines. M = 0:82, Ta = 216:65 K, ´ = 0:81, and ° = 1:4. KE Mixed-StreamT urbofanEngines

Beforedeveloping the theory for FPR op formixed-stream turbo- fanengines, the computer package GasTurb was used to determine 1120 GUHA thesame numerically. Figure 5 showsthe results of this numerical atthedesign point, the computed FPR thengives the optimum value optimization.Calculations are shown at twobypass ratios. Mixed- independentof aspeciŽc missionof the aircraft. streamengines used for military purposes employ much higher Equations (12–16) canbe solvedto determinethe optimum FPR. speciŽc thrustthan that used for separate-stream civil applications. Thiswill, however, require an iterativesolution. For small-bypass Theseengines usually have a smallerbypass ratio. Therefore, max- ratios,as is usual for mixed-stream military engines, the second imumtolerable turbine entry temperatures can produce a largespe- termin the right-hand side of Eq. (12) issmallcompared to theŽ rst ciŽc thrust.Of course, such high values of speciŽ c thrustwould term.The index for FPR inEq. (15) canthen be approximatedby meanlow propulsive efŽ ciency giving higher fuel consumption. substituting °g for ° .Withthis approximation, the equations can be Thereduction in engine size and weight is, however, more crucial combinedto give an explicitrelation for theFPR op: forsuch applications. Figure6 showsthe standard station numbering for a mixed-ow .°g 1/=°g BT02 .cpg =c p/´t T04 c pg FPRop ´t T04 .°g 1/=°g turbofanengine. In the following analysis these numbers are used ´ f C OPR D cp assubscriptsto denote  owvariables at variouslocations. The fol- µ ¶ ³ ´ lowingrepresentative values give reasonable approximation to the 1 .° 1/=° BT02 propertiesof airand combustion products: R 287 J/kg/K, ° 1:4, T02 OPR 1 (17) ´c C ´ f c ° R=.° 1/, ° 1:33, c ° R=.° D 1/.Asimple,analyt- D p D g D pg D g g £ ¤ icalprocedure for calculating the properties of combustionproducts If V j isthe jet speed and Va isthe aircraft speed, the speciŽ c thrust asafunctionof temperature,fuel –airratio, and fuel composition is iscalculatedfrom 15 givenby Guha. F V V (18) Theanalysis can be formulated more easily if the optimum FPR ON D j a andthespeciŽ c thrustare expressed in termsof theprimary variables where OPR, B, and T04. Anenergy balance gives (with T T and T T ) V j 2cpg.T064 Tj / (19) 05 D 06 013 D 016 D p T06 BT016 .cpg =c p/.T04 T06/ .T03 T02/ B.T016 T02/ (12) T C (20) D C 064 D 1 B wherethe various total temperatures can be calculated from C .1 °g /=°g T j T064.FPRop f M / (21) 2 D T02=Ta 1 0:5.° 1/M (13) 2 ° =.° 1/ D C f M [1 0:5.° 1/M ] (22) .° 1/=° D C T03 T02 .1=´c /T02 OPR 1 (14) D Equation (20) givesthe total temperature of themixed stream and Eq. (21) givesthe static temperature of thejet. While writing the T T .1=´ /T £ FPR.° 1/=° 1¤ (15) 016 02 D f 02 precedingequations, the losses in the fan, compressors and turbines havebeen accounted for, but other losses, for example, that in the T T ´ T 1 .OPR/FPR£ /.1 °g /=°¤g (16) 04 06 D t 04 variousducts, have been neglected. Figure7 showsthe comparison of thepresent theory and the nu- where isthe isentropic efŽ ciency of the compression between ´c £ ¤ mericaloptimization results using GasTurb. Each numerical point points2 and3 and istheisentropic efŽ ciency of theexpansion ´t hasbeen calculated by GasTurb to Ž ndthe optimum FPR thatmini- betweenpoints 4 and6. mizesthe SFC, while,at eachtrial, the primary variables are iterated Whilewriting Eq. (16) itis assumed that p p 1 when opti- 016= 06 togivethe prescribed speciŽ c thrust.The nozzle area was also opti- mumfan pressure ratio is achieved. Strictly speaking, Dthis condition mizedat each point so thatit produceda fullyexpanded jet. The con- isnotalways fully satisŽ ed. However, the value of theratio p p 016= 06 vergenceof thenumerical optimization may be slowand sometimes, risesunder off-design conditions: If the value of thisratio used at the dependingon thestarting point, the numerical search technique may designpoint is greater than one, serious mixing losses may result at notŽ ndtheoptimum or may Ž ndalocalrather than the global op- off-designoperations. Therefore, it isprudentto choose the value of timum.The analytical equations, on theother hand, readily provide p p between0.95 and 1 atthedesign point even if itisslightly 016= 06 theanswer. Figure 7 showsthat the present theory gives accurate re- suboptimalat thatparticular operating point. Numerical calculations sults:The iterative analytical solution [Eqs. (12–16)]agreesalmost ofMillhouseet al. 1 showthat if the condition p =p 1 is used 016 06 ¼

Fig.7 Comparisonof the Fig. 5 FPRop for mixed- present theorywith numer- streamturbofan engines de- icaloptimization results for terminedby numerical op- FPRop inmixed-streamtur- timization (GasTurb); for bofanengines; for all calcu- allcalculations, OPR =17 :5, lations,OPR =17 :5, B = 0:5, M = 0:82,isentropic compo- M = 0:82,isentropic compo- nentefŽ ciencies =0 :9 and nentefŽ ciencies =0 :9 and ´mix = 1. ´mix = 1.

Fig.6 Standardnomenclature for various locations in amixed-streamturbofan engine (onlyhalf of the engineis shown ). GUHA 1121

Table1 Dependence ofFPR op on OPR for Table2 Comparisonof theoryand numerical results mixed-streamengines for optimum Vjc/Vjh

F ,lbf/lbm/sOPR T , K B FPR .V =V / ON 04 op op j c jh op BypassNumerical optimization .Vjc =Vj h /op ´KE 60 17.51454 0.822 4.288 D ratio withGasTurb [´KE ´LPT ´ f ´NB] 60 301548 1 4.896 ¼ 1 0.808 0.81 3 0.791 0.81 identicallywith GasTurb optimization; the explicit equation (17) 6 0.794 0.81 alsogives quite acceptable answer until the speciŽc thrustbecomes very high. Thepresent theory can be extended to include the effects of in- Table3 Validityof Eq. (24) forvarious component efŽ ciencies completemixing by introducing a parameter ´mix.Itis thenassumed a a ´ f ´LPT . p016=p06/op .V163=V63/op ´KE ´ f ´LPT thatthree separate streams having total temperatures T016, T06, and D T064 expandthrough the same pressure ratio. The total thrust is the 0.8 0.8 1.016 0.658 0.640 sumof thethrust produced by theseindividual streams: 0.8 0.9 1.031 0.742 0.720 0.9 0.9 1.044 0.818 0.810 V 2c .T T /; T T .FPR f /.1 ° /=° 0.95 0.95 1.061 0.908 0.903 j16 D p 016 j16 j16 D 016 op M aCalculatedby numerical optimizationwith GasTurb. V p2c .T T /; T T .FPR f /.1 °g /=°g j6 D pg 06 j6 j 6 D 06 op M F p´ V .1 ´ /.V BV /=.1 B/ V (23) ON D mix j C mix j6 C j16 C a Fig.8 Effects ofdesign mixer Machnumber on the twochar- SeparateV ersusMixed Stream Engines acteristic ratioswhen FPR op Figure5 canbe comparedwith Figs. 2 and3.Three observations is achieved;for all calcula- à can be made. tions, M = 0:82, B = 0:5, FN = 60 lb/lb/s,and component isentropic 1) The FPRop formixed-stream engines rises monotonically with thespeciŽ c thrust,as it does for separate stream engines. efŽciencies =0 :9. 2) The FPRop ispredominantlya functionof the speciŽ c thrust andonly weakly depends on thebypass ratio. The dependence on bypassratio is weaker in mixed-streamengines than that in separate- 1;10 streamsengines. FPR controlsthis. It isgenerallyrecognized, thatwhen FPR op is achieved,the ratio p =p isclose to one. The physical reason is 3) At aparticularvalue of speciŽ c thrust,the FPR op for mixed- 016 06 streamengines is lower than that in theseparate-streams engines. that,when the total pressures of thetwo streams are nearly equal, Ina separate-streamengine, altering the values of OPR doesnot the loss (entropygeneration ) dueto mixing would be small. Thecomputer program GasTurb was used to determine the value alterthe value of FPR op,solongas thespeciŽ c thrustand bypass ratioare kept Ž xed.This is predicted by Eq. (11) andborne out of FPR (byrandom adaptive search ) thatminimizes the SFC at bynumerical optimization. In a mixed-streamengine, even at a severalspeciŽ c thrustlevels that were Ž xedby iteratingOPR and ŽxedspeciŽ c thrust,various values of OPR and M changethe value T04.Itwas found that optimum p016=p06 didnot remain Ž xedbut varied.Optimum p016=p06 increasedwith increasing speciŽ c thrust of FPRop.Eitherthe analytical relations (12–16) orthe numerical optimizationsshow this. T able1 showsthe results of optimization andbypass ratios. The optimum value of p016=p06 was found to performedby GasTurb (both B andFPR areoptimized at Ž xed varyparticularly strongly with design mixer Mach number M64, speciŽc thrustat twolevels of OPR, minimisingSFC ). asshown in Fig. 8. However,the present study discovered a new relationbetween the velocitiesof thetwo streams at themixer inlet

Effects ofFPR op onthe Conditionsof Other Variables underoptimum conditions: Thebehavior of other owvariables are now examined when the .V163=V63/op ´KE (24) FPRop isachieved. Separate discussion is provided for the mixed- ¼ streamand separate-stream engines because different physical prin- where ´KE ´ f ´LPT. At the FPRop,therelation V163=V63 ´KE re- ciplesare involved. mainednearly D valid at allvalues of speciŽ c thrust,bypass ¼ratio, and designmixer Mach number. The similarity of this condition with Separate-StreamBypass Engines Eq. (7)derivedfor separate-stream bypass engines is noticeable. The Theconditions to whichthe FPR op correspondhave been deter- variationin thevelocity ratio is shown in Fig. 8 (with ´KE 0:81). minedanalytically in thiscase. Equation (7) showsthat there is a Toassessthe accuracy of Eq. (24),alargenumber of numerical D particularratio of the fully expanded jet velocities of thecold and experimentswere conducted with GasTurb in whichthe isentropic hot stream [.V j c=V j h /op ´KE]thatsimultaneously achieves mini- efŽciencies of various components were changed systematically. It D mumSFC andmaximum speciŽ c thrust.The designer can make the wasfound that indeed the efŽ ciencies of the high-pressure (HP) twojet velocities take this value by choosingthe FPR op. compressor, ´HPC,andthe HP turbine, ´HPT,didnot in uence the Thecomputational program GasTurb was used to Ž ndthe ra- value of .V163=V63/op, as Eq. (24) suggests.Equation (24) was also tio V j c=V j h thatoccurs in an engine whose FPR hasbeen numer- satisŽed when various values of the LP compressorand LP tur- icallyoptimized for minimizing SFC. Table2 showsthese values bineefŽ ciencies were tested. A fewof thesenumerical optimization (for OPR 30, T04 1200 K, ´LPT 0:9, ´ f 0:9, and ´NB 1), resultsare given in T able3 todemonstrate this point. (For all cal- againstthe Dprediction D of Eq. (7). D D D F B culationsshown in T able3, ON 60 1b/1b/s, 1, OPR 17:5, Numericaloptimizations were performed at several other OPR M 0:3, M 0:82, and ´ D1.Similarresults D are obtained D at 64 D D mix D and T04: V j c=V j h laybetween 0.77 and 0.82 (forthe assumed compo- othervalues of thesevariables. ) nentefŽ ciencies ).Thenumerical calculations show that the derived analyticalrelation [Eq. (7)]isapproximatelyvalid. Conclusions Theoptimum fan (compressor) pressureratio is determined both Mixed-StreamBypass Engines numericallyand analytically for separate-stream as well as mixed- Theratio of the total pressure of the two streams at thebegin- streambypass engines. The FPR op isshown to be predominantly ningof themixer zone, p016=p06,isanimportantvariable and the afunctionof thespeciŽ c thrustand a weakfunction of thebypass 1122 GUHA

ratio.For mixed-stream engines, the dependence of FPR op on bypass ofPropulsion ,2nded., Addison Wesley Longman,Reading, MA, 1992, ratiois veryweak, and FPR op alsodepends on OPR. At thesame Chap. 5. 3Cohen,H., Roger, G. F.C., andSaravanamuttoo, H. I.H., Gas Turbine speciŽc thrustand bypassratio, the FPR op fora mixed-streamengine Theory,4thed., Longman, Harlow ,England,U.K., 1996, Chap. 3. islowerthan that of aseparate-streamengine. 4 Twosimple, explicit equations have been derived from fundamen- Cumpsty,N. A., Jet Propulsion ,CambridgeUniv. Press, Cambridge, ( ) England,U.K., 1997. talprinciples. Equation 11 givesthe FPR op forseparate-stream en- 5Rufes, P.C.,“TheFuture of Aircraft Propulsion,” Journalof Mechan- – gines,whereas Eq. (17)[orthe equation set (12 16)]givesthe FPR op icalEngineering Science ,Vol.214, No. C1, 2000, pp. 289 –305. formixed-stream engines. The accuracy of theanalytical formulas 6Jackson,A. J.B.,“ SomeFuture Trends in Aero EngineDesign for Sub- hasbeen established through extensive veriŽ cation by numerical sonicTransport,” Journalof Engineering for Power ,Vol.98, April 1976, optimizationresults of thecommercial computer package GasTurb pp. 281–289. (Figs. 3 and 7).Theanalytical results accelerate the optimization 7Bennett,H. W.,“ Aero EngineDevelopment for the Future,” Proceedings processand offer physical insight. oftheInst. of Mechanical Engineers, Series A ,Vol.197, July 1983, pp. 149 – 157. Ithas been shown that the FPR op achievesthe condition 8 V =V ´ ina separate-streamengine and the condition Wilde,G. L.,“ FutureLarge CivilTurbofans and Powerplants,” Aero- j c j h KE – V V D nauticalJournal ,Vol.82, July 1978, pp. 281 299. 163= 63 ´KE ina mixed-streamengine. The condi- 9Birch,N. T.,“2020Vision: the Prospects for Large CivilAircraft Propul- tion V =¼V ´ applieseven under situations when signiŽ - 163 63 ¼ KE sion,” AeronauticalJournal ,Vol.104, No. 1038, 2000, pp. 347 –352. cantdepartures from the normally assumed condition p016=p06 1 10 ¨ ¼ Kurzke,J., “Manualfor GasTurb 7.0 for Windows,” MTU, M unchen, occur. Germany,1996. 11Guha,A., “ Optimizationof Aero Gas TurbineEngines, ” Aeronautical Journal,Vol.105, No. 1049, 2001, pp. 345 –358. Acknowledgment 12Mattingly,J. D., Elements ofGasTurbine Propulsion , McGraw–Hill, Theauthor is gratefulto the referees for their helpful comments. New York,1996, pp. 240 –460. 13Rud,¨ K., andLichtfuss, H. J.,“ Trendsin Aero-EnginesDevelopment,” Aspects ofEngineAirframe Integration for Transport Aircraft , Proceedings References ofDLRW orkshop ,DLR-Mitteilung,Braunschweig, Germany, March 1996. 1Millhouse,P .T.,Kramer, S.C.,King,P .I.,and Mykytka, E. F.,“ Identify- 14RollsRoyce AeroData ,TS1491,Issue 9,Derby,England, U.K., 1991. ingOptimal Fan Compressor Pressure Ratiosfor the Mixed-StreamTurbofan 15Guha,A., “ AnEfŽ cient Generic Methodfor Calculating the Properties Engine,” Journalof Propulsionand Power ,Vol.16, No. 1, 2000,pp. 79 –86. ofCombustionProducts,” Journalof Power and Energy ,Vol.215, No. A3, 2Hill,P .G.,and Peterson, C. R., Mechanicsand Thermodynamics 2001,pp. 375 –387.