PERFORMANCE CHARACTERIZATION OF TURBOSHAFT ENGINES: WORK POTENTIAL AND SECOND-LAW ANALYSIS

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1 PERFORMANCE CHARACTERIZATION OF TURBOSHAFT ENGINES: WORK POTENTIAL AND SECOND-LAW ANALYSIS Christophr D. Wilson and David W. Riggins Dpartmnt of Mchanical and Arospac Enginring and Enginring Mchanics Univrsity of Missouri Rolla, Rolla, Missouri Bryc Roth and Robrt McDonald School of Arospac Enginring Gorgia Institut of Tchnology, Atlanta, Gorgia ABSTRACT This papr dvlops and dscribs work potntial analysis mthods applicabl to turboshaft ngin flow-filds. Ths mthods ar basd on th scond-law of thrmodynamics and nabl a unifid, comprhnsiv assssmnt of prformanc at th part, componnt, and ngin lvls. Th focus hrin is on using gas spcific powr as a work potntial figur of mrit in analyzing turboshaft ngins. This is shown to b a usful tool for assssing local prformanc potntial in a gas turbin flow-fild. Th fundamntal rlationships btwn hat, work, and irrvrsibility in turboshaft ngins ar dvlopd and th rlationship of flow irrvrsibility to ngin prformanc losss is discussd. Ths thortical idas ar thn formulatd in a mthod that nabls charactrization of prformanc losss in trms of ngin spatial location and loss mchanism. This mthod is thn dmonstratd at th ngin systm lvl via cycl analysis, at th componnt lvl via quasi -D analysis, and at th part lvl via a stator row multi-dimnsional CFD simulation analyzd in trms of irrvrsibl ntropy production. ρ T P R W,w Q,q ghp S,s U M H,h C P γ A c δw δq τ w η J W comp W shaft S irr NOMENCLATURE fluid dnsity fluid tmpratur fluid prssur gas constant work intraction, flow-spcific work hat intraction gas (spcific) powr ntropy, flow-spcific ntropy vlocity Mach nthalpy, flow-spcific nthalpy spcific hat ratio of spcific hats cross sctional ara circumfrnc diffrntial work intraction diffrntial hat intraction wall shar strss scond law ffctivnss (work trm) η for comprssors, /η for turbins comprssor-turbin shaft work pr unit mass powr turbin shaft work pr unit mass hat intraction pr unit mass ntropy gnration pr unit mass Prsntd at th Amrican Hlicoptr Socity 58 th Annual Forum, Montral, Canada, Jun -3,. Copyright by th Amrican Hlicoptr Socity Intrnational. All rights rsrvd. subscripts: i ngin inlt station ngin xit station 4 maximum tmpratur (combustor xit) t total condition INTRODUCTION Th us of work potntial mthods in th dsign and optimization of arospac gas turbin ngins has rcntly rcivd incrasd attntion in th arospac propulsion community [,, 3, 4]. This is bcaus full undrstanding and optimization gas turbin ngin prformanc inhrntly rquirs charactrization and quantification scond-law ffcts on ngin prformanc and oprability [5]. Furthrmor, in ordr to assss ngin prformanc and losss in a truly maningful way, on must valuat and masur th actual prformanc losss rlativ to th prformanc of th bst possibl (idal) ngin of th sam typ [6]. Hnc, th prformanc and prformanc losss of a turboshaft ngin ar most usfully assssd against th idal turboshaft ngin prformanc (in trms of th fundamntal critria of dlivrd shaft powr). Similarly, th prformanc of a turbojt should b assssd against th idal turbojt ngin (in trms of thrust powr producd). Th objctiv of this papr is to show that scond law analysis (or work potntial analysis), whn applid within ths critical constraints, is a consistnt and highly usful assssmnt tool for turboshaft ngin analysis. Th concpt of work potntial has application at all lvls of propulsion systm analysis, ranging from th

2 simplistic flow station rprsntations usd for cycl analysis to highly dtaild flowfild simulations common in componnt dtail dsign. This is illustratd in Figur which shows th various lvls of analysis dtail commonly usd in propulsion systm dsign. In this cas, th comprssor is usd as an xampl to illustrat how an actual systm is typically modld. Th first lvl idalizs th comprssor as two discrt flow stations whrin th avrag proprtis at th comprssor ntranc and xit ar of intrst whil th comprssor itslf is modld as a black box. Work potntial mthods can b usd in conjunction with cycl analysis to stimat total loss insid th comprssor and compar this to losss in othr componnts such that th prformanc of th whol systm can b optimizd. Th scond lvl is a quasi--d flow analysis across th comprssor. In this analysis, th flow is analyzd in trms of its avrag proprtis at ach axial cross-sction. Work potntial analysis at this lvl can yild additional information as to th diffrntial losss occurring at ach stag in th comprssor and is usful for balancing loads across stags to minimiz total componnt loss, tc. Th third and most dtaild lvl is a full simulation of th thrmodynamic procsss undrgon by ach fluid lmnt passing through th comprssor. Th fluid lmnts starting at station a ach hav a uniqu thrmodynamic stat and collctivly form a cloud of points whos cntroid is quivalnt to th avrag station proprtis. Each fluid lmnt undrgos a uniqu thrmodynamic procss, som passing nar th mid-span of th comprssor blads and xprincing littl loss, whil othrs pass nar th blad tip or ar ntraind in th boundary layr, lading to significant loss. Work potntial mthods applid to CFD solutions hlp idntify and liminat thos fluid lmnts that contribut th most to loss. On can thn backtrack to spcific flow intractions insid th comprssor that ld to th loss. Idalizd as Flow Stations Idalizd as Quasi -D Procsss Simulatd in Dtail (CFD) Comprssor Inlt Stn. a Stn. a Stn. a Comprssor Discharg Stn. b Stn. b Stn. b Fig. : Application of work potntial mthods at various lvls of analysis fidlity. This papr dscribs and dmonstrats th application of work potntial mthods at all thr lvls of analysis dtail. Th first part dfins and discusss th thortical undrpinnings of thrmodynamic work potntial (with mphasis on gas spcific powr, or simply gas horspowr) from th standpoint of turboshaft ngin cycl analysis. This is shown to nabl rapid and comprhnsiv nginring assssmnt of losss in all ngin componnts by providing a common currncy to masur loss that applis to any componnt in a turboshaft ngin [7]. Th scond part of this papr focuss on dvlopmnt and application of mthods xplicitly rlating ngin shaft powr and shaft powr losss to flow irrvrsibility. This quasi -D analysis nabls th idntification and quantification of losss in trms of ngin spatial location as wll as in trms of spcific loss mchanism. It is mor dtaild and comprhnsiv than th cycl analysis tchniqu, as it dirctly accounts for downstram intractions and thir ffcts on ngin prformanc within a givn ngin flowfild. Finally, a scond law analysis of a rprsntativ thr-dimnsional stator flow fild is discussd. THERMODYNAMIC WORK POTENTIAL DEFINED Th concpt of thrmodynamic work potntial is quit old but has takn a surprisingly long tim to pntrat into mainstram thrmodynamic knowldg. It is rootd in th scond law of thrmodynamics and th concpt of ntropy. Various work potntial figurs of mrit (FoMs) hav bn dvlopd ovr th yars, th most gnral and bst-known of ths bing xrgy [8, 9, ]. Howvr, if an aircraft ngin is optimizd to produc maximum xrgy output for a givn ful input, th rsult is usually a sub-optimal ngin dsign, as notd by Riggins [7]. Consquntly, xrgy analysis is somwhat sotric if on is a priori constraind to us a particular cycl, notably th Brayton cycl. A mor usful work potntial FoM for analysis of turboshaft ngins is gas horspowr. Gas horspowr (GHP) is dfind as th shaft work that would b obtaind in xpanding a gas from a prscribd tmpratur and prssur to a rfrnc (usually ambint) prssur in an imaginary powr turbin. It is oftn usd to masur thortical powr output of gas gnrators. Exprssd mathmatically: ghp h( Ti, Pi ) h( P = Prf, s = si ) () whr subscript i dnots th thrmodynamic stat of th gas at th initial condition. Gas horspowr is actually a spcial cas of xrgy in which only prssur quilibrium with th nvironmnt is nforcd []. Thrfor, GHP is a thrmodynamic proprty (just lik tmpratur, prssur, nthalpy, tc.) dscribing th maximum work that could b xtractd from a substanc in taking it from a prscribd initial stat at constant vlocity into prssur quilibrium with th nvironmnt. Th GHP loss insid any arbitrary systm can b calculatd by summing th GHP fluxs into and out of th

3 systm. Th diffrnc btwn th fluxs in and out is qual to th sum of th powr output and th GHP loss rat: g hp & in ghp & out = w& out + gh& p. (3) loss Not that whn hat is addd in th combustor, a componnt of this hat nrgy bcoms availabl as gas work potntial. In this cas, Eq. 3 must ncssarily includ anothr trm. This concpt provids a consistnt and comprhnsiv framwork by which to masur th prformanc of turboshaft ngin componnts and systms. TURBOSHAFT CYCLE ANALYSIS INCORPORATING WORK POTENTIAL METHODS Th first application of work potntial analysis mthods will considr a typical turboshaft cycl analysis for an ngin of roughly, shp output and having low and high-prssur comprssors coupld through a singl shaft to a two-stag gas gnrator turbin followd by a two-stag powr turbin. A simplifid mthod for calculating GHP is prsntd and thn usd to calculat GHP at vry ngin flow station. This information is thn usd to calculat GHP loss in ach componnt. Mthod Gas horspowr analysis of turboshaft ngins is a rlativly simpl xrcis in rpatd application of Eqs. and 3 to all stations and componnts in a turboshaft ngin. Th first stp in th procss is to apply Eq. to calculat GHP at vry flow station in th ngin. Sinc gas horspowr is a thrmodynamic proprty mad up of othr proprtis, on nd only hav a comprhnsiv tabulation of th proprtis for ful-air mixturs at various tmpraturs and prssurs. Any numbr of thrmodynamic proprtis softwar packags can b usd to do this. Th most convnint prsntation of th data for us in ngin analysis is to plot contours of mass-spcific gas horspowr as a function of tmpratur, prssur, and ful-air ratio. Fig. and Fig. 3 ar two such contour plots showing gas horspowr pr pound-mass of fluid, xprssd in horspowr pr lbm/s of fluid flow. Fig. shows GHP for Tmpratur (R) Prssur (atm) Fig. : Contours of mass-spcific gas horspowr (HP/pps) for air pur air ovr a rang of tmpratur and prssur, whil Fig. 3 shows GHP for stoichiomtric mixturs of ful and air. Thus, if tmpratur, prssur, flow rat, and ful-air ratio ar known at vry flow station (as from cycl analysis), thn GHP can also b calculatd at vry flow station by simply intrpolating from ths contour plots. A much mor comprhnsiv st of figurs and tabls suitabl for ngin analysis is availabl in Rf. []. Th scond stp is to calculat th total loss in ach componnt. Onc GHP at vry flow station is known, it is fairly simpl to do this using Eq.. Not that this rquirs dtaild knowldg of th cycl modl, cooling flow circuits, powr tak-off, and so on. Th following discussion will illustrat this ida in dtail. Typical Rsults As an illustrativ xampl of how ths idas can b applid to practical analysis of a turboshaft ngin, considr th xampl mntiond in th prvious sction. Th actual ngin is typically idalizd as a collction of flow stations, componnts, and flow circuits, as shown in Figur 4. This cycl modl can b usd to calculat tmpratur, prssur, ful-air ratio, and flow rat at vry station in th ngin, as wll as total ful consumption, shaft powr output, tc. This cycl analysis modl also contains various loss mchanisms, as shown in Figur 4. Ths typically ar du to componnt losss (modld as fficincis ), prssur drops, lakag, cooling losss, and mchanical friction. Th flow station information producd by th cycl modl can b usd in conjunction with th mthod dscribd in th prvious sction to dvlop a comprhnsiv pictur of all losss in th ngin at any flight condition and powr stting. To illustrat, considr th prformanc of a typical turboshaft ngin oprating at sa-lvl static (SLS) conditions on a standard (58 o F) day at full throttl. Th data from th cycl dck can b usd to dirctly calculat th GHP at ach station in th ngin and thrby th GHP loss insid ach componnt. Considr th low-prssur comprssor as an xampl. Th inlt is station and th discharg is station 5. Th flow rat at station is Tmpratur (R) Prssur (atm) Fig. 3: Contours of mass-spcific gas horspowr (HP/pps) for stoichiomtric mixturs of Jt-A and air

4 Primary Flowpath Shaft Cooling Flow Circuit Flow Station Flow Rsistanc Particl Sparator Ovrboard Flow 3.3 GHP 3 Amb,447 HP 5,333 HP HP Shaft Combustor Output Prss. Loss Windag 3 Ful Addition ,89 HP Shaft Powr 5 Rar Fram Prss. Loss 7 9 Comprssor/Turbin Inlt/Nozzl GHP Inlt Prss. Loss - GHP,56 GHP LP Comprssor Loss,47 GHP HP Comp. Loss Lakag 4,833 GHP 377 GHP HPT Non-Ch. Cooling HPT Loss HPT Chargabl Cooling 64 GHP Rar Fram Cooling/Cavity Purg 5, GHP,37 GHP,383 GHP PT Loss GHP 4 GHP 3 GHP GHP Nozzl Loss Fig. 4: Cycl modl schmatic with annotation of major loss mchanisms and stimats on GHP at ach flow station for SLS full powr oprating conditions. lbm/s at 4.3 psia and 59 o R. Fig. shows that ths conditions corrspond to a flow-spcific GHP at station of.7 HP/pps, yilding a total GHP at station of HP. Th flow is dischargd to station 5 at 84.8 psia and 937 o R. Th flow-spcific GHP is found from Fig. as 5 HP/pps. At a flow rat of.5 pps, this yilds a GHP of,56 HP. Th shaft powr into th low-prssur comprssor is calculatd by th cycl dck to b,447 HP. Thrfor, th low prssur comprssor GHP loss is th diffrnc btwn GHP flux in and out of th comprssor:,447 HP + (- HP),56 HP = 8 HP loss. GHP loss in othr componnts can b calculatd in lik mannr, shown in Tabl. Ths rsults illustrat in an intuitiv way th rlativ magnituds of componnt losss. Th largst GHP losss clarly occur in th turbomachinry, though rar fram prssur losss ar high, also. Th thortical GHP addition in th combustor is roughly 3,36 HP, with th nt avrag ngin shaft powr dlivrd bing, HP. Th diffrnc is dstroyd through various loss mchanisms in th ngin. Finally, not that if this procdur wr usd to calculat componnt loss at vry oprating condition, th rsult would b a comprhnsiv dscription of ngin prformanc known as a loss dck [3]. Tabl : GHP losss in a typical turboshaft ngin at SLS standard day conditions. Loss Mchanism Loss (HP) % Total Inlt Prssur Loss 4.3 Particl Sparator Ovrbrd Flow LP Comprssor Loss HP Comprssor Loss 5.3 Combustor Prssur Loss 4.3 HPT Loss LPT Loss HPT Chargabl Cooling 8.6 Rar Fram Prssur Loss 5.3 Nozzl Loss 3 3. Lakag Shaft Windag 9.9 Total Loss,6 Nt Shaft Powr Output, GHP Addition in Combustor +3,36 TURBOSHAFT ENGINE QUASI -D ANALYSIS This sction focuss on th dvlopmnt and application of mthodology and tchniqus for xplicitly rlating incrmntal ngin shaft powr and shaft powr losss to flow irrvrsibility. This mthodology nabls th idntification and quantification of losss in trms of ngin spatial location as wll as in trms of spcific loss mchanism. Th following discussion first dvlops th rational and mthodology rlating flow irrvrsibilitis to losss in shaft powr. This mthod is thn dmonstratd for a simplifid turboshaft flow-fild by utilizing quasi -D flow analysis coupld with scond law concpts. This analysis is basd on application of th stady-flow quasi-on-dimnsional flow quations. Ths quations allow flux-consistnt flow simulations from which ntropy distributions and audits can b asily xtractd for us in th analysis. Th quasi -D comprssibl flow quations in diffrntial form ar: dρ du da + + = (4) ρ u A dp τ c( dx) + udu = w + J w P ρa δ (5) C p dt + udu = δ w + δq (6) dp dρ dt = + (7) P ρ T whr for th comprssor: J = η, and for th turbin: J = /η. Hr δq and δw dnot th diffrntial hat and shaft work pr mass addd to a diffrntial lmnt across dx. Not that th shaft work trm in th momntum quation and th associatd scond-law ffctivnss, η, rprsnt a body forc modl of work intraction [4]. Th dpartur of η from unity thrfor rprsnts irrvrsibilitis associatd with th transfr of nrgy to th flow from an xtrnal shaft. Ths irrvrsibilitis could b proprly subdividd into frictional and thrmal losss if nough information is providd (i.. from multi-dimnsional CFD). Howvr from th standpoint of th quasi -D analysis prsntd in this part of th study, ths irrvrsibilitis ar considrd (and book-kpt) in a sparat lumpd catgory associatd with all work intraction losss.

5 Lost Shaft Work and Irrvrsibilitis In ordr to clarify th rlationship btwn shaft work losss and flow irrvrsibilitis, first considr an axial flow comprssor stag in a turboshaft ngin. A stramtub procssd by that stag has total prssur incrasd by th introduction of swirl kintic nrgy associatd with th angular motion of th rotor blads. From th standpoint of th axial bulk flow through th stag, a portion of this kintic nrgy is ralizd as prssur work within th ovrall stag, hnc raising total prssur. This is also manifstd in th momntum quation (Eq. 5) by an incras in stram thrust (dfind in [5] as ρu A+PA) through th stag. In an idal comprssor, th fluid would xprinc th maximum possibl incras in stram thrust (or total prssur) for a givn amount of xtrnal work intraction to th flow. In a non-idal comprssor (on in which irrvrsibilitis occur) with th sam amount of dlivrd shaft work across th boundary, th fluid would xprinc an incras in stramthrust (or total prssur) du to work intraction lss than that xprincd by th idal comprssor. This dcras in nt stram thrust (or nrgy rcivd by th flow as prssur work) is du to flow irrvrsibilitis that convrt nrgy initially dlivrd from th shaft as work to nrgy rcivd intrnally in th fluid as hat. * This mans that irrvrsibilitis occurring in th ngin ar intimatly rlatd to work potntial and invitably act to dcras th potntial shaft work output availabl from th powr turbin. On can stablish a dirct linkag btwn flow irrvrsibilitis and lost shaft work by xamining th ntropy incras du to irrvrsibilitis in a givn ngin. If th ntropy incras is compard to th diffrnc btwn dlivrd powr turbin shaft work and idal shaft work for a givn ngin (prsuming th sam xtrnal nrgy intractions in th gas turbin cor and with th sam ngin gomtry), on will find that thy ar proportional. Thus, if quasi -D analysis is usd to stimat ntropy chang in ach diffrntial lmnt, this can b dirctly convrtd into a loss in work potntial. This nabls on to book-kp all losss in a turboshaft ngin both in trms of loss mchanism (mass, momntum, or nrgy transfr, nonquilibrium kintics, shocks, tc.) and in trms of ngin location. As an xampl, considr a givn turboshaft ngin flowfild with fixd xit ara and givn xit static prssur. Prsum that a dtaild diffrntial dscription of ntropy incrass associatd with flow irrvrsibilitis is known throughout this flow-fild via quasi -D analysis. Th scond-law analysis bgins with th flow-fild for th idal ngin (flow-path with no irrvrsibilitis). All xtrnal shaft work intractions btwn comprssor and HP turbin as wll as burnr hat intraction ar qual to thos that occur in th actual ngin. Engin xit ara and xit static prssur (P ) ar also th sam btwn idal and actual ngins. This lattr rquirmnt, whn coupld with rvrsibl flow procsss, rsults in additional shaft work obtaind from th powr turbin for th idal ngin as discussd abov. To assss th loss distribution in th ngin, bgin by starting at th ngin xit plan of th actual ngin. Lt us rmov th irrvrsiblity associatd with th first upstram diffrntial stp of th actual ngin flow-fild. Th rsulting powr output from th (now slightly mor idal) ngin can b r-calculatd and compard to th actual ngin. Th diffrnc rprsnts th loss shaft work du to th diffrntial irrvrsibility occurring in th flow ovr th first upstram diffrntial stp. This procss is rpatd succssivly, rmoving irrvrsibilitis progrssivly from th back to th front of th ngin until th idal ngin flow-fild is obtaind (i.. all irrvrsibilitis associatd with th actual ngin flow-fild hav bn consistntly dltd from th actual ngin flow-fild). As th progrssion is mad from th actual ngin to th idal ngin, a summation can b mad of th lost shaft work du to all losss in ach componnt. In addition, a summation is mad of th lost shaft work du to ach sparat loss mchanism. This mthodology can b illustratd using a T-S diagram (Fig. 5) on which both idal and actual ngin conditions ar shown schmatically, including th locus of points dscribing th intrmdiat ngin flow-fild conditions. Not that this tchniqu allows th complt loss analysis of th ngin both in trms of loss mchanism and ngin location sinc th diffrntial ntropy incras for ach mchanism at ach stp is also known. This procss logically dfins a consistnt intgration path in which a family of succssivly mor rvrsibl T r S rvrsibl (idal) ngin actual ngin diffrntial lost shaft work rvrsibl xpansion(s) to back prssur lin - downstram losss rmovd and bookkpt constant prssur lin consistnt intgration path for analyzing losss du to irrvrsibilitis btwn actual and idal turboshaft ngin * Not that scond law considrations also dictat that nrgy addd in th burnr as hat cannot b fully xtractd as shaft work by th powr turbin. This is bcaus th combustion procss is typically highly irrvrsibl, with a consqunt larg loss in work potntial. Th mor irrvrsibl this hat addition procss is, th lss work can ultimatly b producd in th turbin. Fig. 5 T-S diagram showing actual and idal ngin cycls with intrmdiat loss cycls.

6 turboshaft ngin flow-filds is dvlopd, all intrmdiat btwn idal and actual flow-filds. Not that th dirction of th loss-buildup procss is critical (i.. th squntial dltion of irrvrsibilitis from downstram to upstram) and is mandatd hr by th assumption of strong coupling of downstram irrvrsibilitis with upstram irrvrsibilitis. Du to this strong coupling, thr is only on thrmodynamically consistnt intgration path for moving from th actual ngin to th idal ngin and consistntly assigning lost shaft work incrmnts to irrvrsibilitis. Howvr, not that although this mthod provids a snapshot of th lost shaft powr incrmnts associatd with irrvrsibilitis for a givn ngin flow-fild, th ntropy-shaft work rlationship is in fact non-linar and is strongly coupld with any changs in actual fluid dynamics. In othr words, rmoving an irrvrsibility from th middl of th actual ngin (say through a dsign chang) will impact th balanc of shaft work losss associatd with othr irrvrsibilitis occurring lswhr in th ngin. In this sns, th targt for optimization (for givn gas turbin xtrnal intractions) should b maximizing ovrall powr turbin shaft work or quivalntly minimizing ovrall ntropy production throughout th ngin. Analytical Rlationships Btwn Shaft Work and Irrvrsibility in a Turboshaft Engin This sction summarizs th analytical rlationship btwn shaft work dlivrd by th powr turbin and irrvrsibility. Rfrnc [6] givs dtails of th quation dvlopmnt. Figur 6 shows a schmatic of th turboshaft ngin and hat and work nomnclatur usd in this sction. It can b shown from th scond law of thrmodynamics and th govrning quations of fluid dynamics that th total prssur incras from stat i to is dirctly rlatd to th diffrntial work intractions occurring btwn stations i and and th ovrall ntropy incras du solly to irrvrsibility: P P t ti δ w S irr i RTt R =. (8) Hr δw is th diffrntial xtrnal-to-flowpath work intraction pr mass (dfind positiv to th fluid) and S irr is th total irrvrsibility occurring btwn i and. Utilizing this xprssion and applying consrvation of mass and nrgy throughout th ngin, th following i I actual ngin (all losss) C W comp S irr B W shaft (actual) PT Fig. 6 Turboshaft ngin schmatic and nrgy intractions. T quation is obtaind which dirctly rlats shaft work obtaind in th powr turbin to irrvrsibilitis occurring in th ngin flowpath: γ + M Wshaft P A M (9) = + CT CT γ Pi Ai M i M + i whr: γ P γ Wcomp ( ) + CT + () p M P ti i CpT ti P i A i = γ M i S P irr A γ W R comp + CT is th ovrall hat pr unit mass addd in th burnr, and W comp is th shaft work intraction occurring btwn comprssor and HP turbin in th gas turbin cor. P and A ar th xit prssur and ara of th ngin; similarly P i and A i ar th inlt prssur and ara of th ngin. It is assumd for simplicity that thr ar no mchanical losss btwn comprssor and HP turbin. Also thrmodynamic proprtis ar assumd to b constant and th mass of ful is assumd ngligibl rlativ to th mass of air. Ths rstrictions can b rlaxd without modifying th basic loss analysis tchniqu prsntd hr. Not th functional dpndnc of powr turbin shaft powr implicit in ths rlationships: Wshaft P W comp Sirr A () = f, γ,,,, Mi, CT Pi CT CT R A i Ths xprssions rlat th total irrvrsibl ntropy incras in th ngin flow to th shaft work producd by th powr turbin. Th valu of actual shaft work obtaind for a givn irrvrsibility from inlt to xit whn compard to th idal shaft work (also obtaind utilizing this quation with S irr = ) yilds th lost shaft work du to th total irrvrsibility in th ngin. Equation 9 can also b usd to asily facilitat th prviously dscribd loss buildup mthodology by applying it for ach diffrntial irrvrsibl ntropy incras in th ngin in a mannr consistnt with th prvious discussion, and using th intgration path shown in Fig. 5. A simplifid form of th lost shaft work quation can b drivd by assuming that inlt and xit prssurs ar qual and that Mach numbrs ar small: Wcomp ( + CT + ) W p shaft Q ti CT. () = + γ CT CT Sirr γ W R comp + CT Not also th simplifid functional dpndnc for th powr turbin shaft work: W shaft Wcomp S irr = f γ,,,. (3) CT CT CT R Th fundamntal rlationships btwn hat, work, and

7 . irrvrsibility in turboshaft ngins ar analyzd by stting prssur ratios and ara ratios constant in Eqs. 9 and and varying th rmaining paramtrs. As a rsult, two sts of contours ar producd. Th first st of contours (Fig. 7) dpicts constant shaft work curvs for diffrnt combinations of comprssor/hp turbin work intraction and burnr hat addition. Hr th irrvrsibl ntropy incras through th ngin is fixd as indicatd. Includd on this contour plot ar lins that dnot combinations of comprssor/hp turbin work intraction and burnr hat addition that lad to a constant valu of T t(max) (max tmpratur) at th nd of th burnr. Th valus for th irrvrsibl ntropy incras, ara ratio, and inlt Mach numbr ar shown on th figur. By diffrntiating th xprssion rlating maximum tmpratur to work and hat intractions, an quation is obtaind which xprsss th maximum possibl shaft work output as a uniqu combination of comprssor/hp turbin work intraction and burnr hat intraction: C T W + C ptti = Sirr R Comp γ γ W C T Comp. (4) This quation is basd on th following xprssion for th maximum tmpratur in th ngin: Tt (max) Wcomp = + T CT +. (5) C T ti Essntially, this dscribs th rlationship btwn cycl prssur ratio and combustor quivalnc ratio (for a givn maximum allowabl turbin inlt tmpratur) ncssary to achiv th maximum shaft powr output. Figur 8 shows th curv formd by th collction of points gnratd by this xprssion. Th scond st of contours that can b obtaind from Eq. 9 also dpicts lins of constant powr turbin shaft work (s Fig. 9). In this cas, howvr, th ngin maximum tmpratur at th nd of th burnr (T t4 ) was hld constant such that hat addition in th burnr is dtrmind by th comprssor work intraction. Th comprssor/hp turbin work intraction and total irrvrsibility incras through th ngin ar thn varid. Irrvrsibility Dscription for Quasi-On-Dimnsional Modl of Turboshaft Engin In ordr to post-procss an ngin flow-fild and apply th tchniqu dscribd in prvious sctions for analyzing shaft work losss, it is ncssary to dvlop xprssions for th various irrvrsibl ntropy incrass occurring in th flow-fild. Th total diffrntial ntropy incras can b dividd into two contributions ntropy incrass du to rvrsibl hat addition and ntropy incrass du to irrvrsibilitis. Not that for any positiv hat intraction to th flow, no mattr how irrvrsibl, thr is a minimum ntropy incras associatd with that hat addition had it occurrd rvrsibly. Th irrvrsibl ntropy associatd with hat addition is that incrmnt of ntropy gnratd in addition to this rvrsibl ntropy incras. Irrvrsibl ntropy incrass can b dscribd as rsulting from spcific loss mchanisms in th flow. Th most gnral dscription of ths loss mchanisms involvs losss du to mass diffusion (chmical spcis gradints), momntum diffusion (friction and vlocity gradints), thrmal diffusion (hat transfr across tmpratur gradints), and non-quilibrium chmical kintics. Ths mchanisms ar most proprly viwd as nvloping shocks, although dpnding on numrical rsolution issus and modling issus, th Eulr ntropy jump implicit in th govrning quations of fluid dynamics may b tratd as a sparat loss mchanism /C p T ti W Comp /C p T ti S Irr /R=.64 A /A i =.36 M i =. W Shaft /C p T ti 4 3 S Irr /R = T T4 /T ti Fig. 7 Burnr hat addition and comprssor work addition shaft work contours ( W shaft /C p T ti ). Fig. 8 Maximum shaft work vrsus burnr xit total tmpratur ratio for a constant irrvrsibility.

8 - - - S Irr /R W Comp /C p T ti T tmax /T ti = 4.66 A /A i =.36 M i =. Fig. 9 Irrvrsibl ntropy incras and comprssor work addition shaft work contours. Th following rlationships provid a summary of th thr loss mchanisms (irrvrsibilitis) considrd in this papr: hat transfr, circumfrntial friction, and a lumpd loss inclusiv of th xtrnal shaft work intraction with th flow. Not, as prviously discussd in othr sctions, this last loss mchanism is actually composd of friction and hat transfr losss (as wll as possibl shock losss) which must b providd to th quasi-on-dimnsional basd analysis usd hr by highr ordr simulation mthods such as multi-dimnsional computational fluid dynamics. A prviw of th ntropy gnration du to spcific mchanism in an xampl thr-dimnsional stator flow-fild will b providd latr in this papr. Such information could b rapidly foldd into th quasi-on-dimnsional analysis. Hat Transfr ntropy is givn by: δq δq Sirr HT = (6) T Tt Hr T t is th local total tmpratur of th flow and T is th local static tmpratur of th flow. Th hat transfr δq is simply spcifid in th prsnt modl, hnc allowing a Rayligh hat addition procss to modl ful-air hat rlas. Circumfrntial Wall Friction ntropy is givn by: τ wc( dx) Sirr = (7) Fr ρat and shaft work intraction irrvrsibility is givn by: δw S = J (8) irr w ( ) T Cod Dscription A simpl quasi -D cod assuming prfct gas and constant-spcific hats was dvlopd for this study in ordr to simulat th flow-fild through a gnric turboshaft ngin. This cod is basd on th diffrntial quasi--d quations dscribd in Eqs. 4-7 and yilds a complt axial diffrntial dscription of th flow-fild through th ntir ngin rathr than a componnt-only (station-wis) dscription of th flow as providd by cycl analysis. Th diffrntial dscription provids a logical bridg btwn th analysis/mthodology dscribd in this papr and multidimnsional simulations (which can provid vry dtaild dscriptions of losss). Aftr th flow-fild is calculatd using this cod, all irrvrsibilitis and associatd lost shaft work incrmnts ar calculatd and book-kpt in a postprocssing routin basd dirctly on th rlationships and tchniqus dscribd in arlir sctions. Not that th purpos of this diffrntial analysis (DA) cod is intndd solly as proof-of-concpt and hnc is applid to a rlativly simpl ngin flow-fild without provision for dtaild ngin modls such as cooling flow circuits or variabl spcific hats. Howvr, th mthodology dvlopd in this work is compltly gnric and can b applid to any typ of flow-fild. Cod Rsults Tabl provids a dscription of th major faturs of th spcific ngin gomtry and flow conditions modld in ordr to dmonstrat th tchniqu dscribd in th prvious sctions. This rprsnts a gnric ngin flowpath and is not dirctly comparabl to th rsults obtaind arlir using th gas horspowr mthod, which was applid to a mor complx ngin cycl modl with variabl spcific hats, cooling flows, tc. Not also that linar distributions of flow-path cross-sctional aras wr assumd btwn givn axial stations. Also, linar work and hat intractions with axial distanc wr assumd within th rlvant componnts. As discussd abov, th DA cod dtrmins th complt diffrntial dscription of th flow-fild proprtis throughout th ngin. Howvr, th paramtr distribution of intrst for loss analysis is irrvrsibl ntropy incras through th ngin, shown in Fig.. Th distribution is sgmntd by componnt with lins drawn at th inlt/xit of ach componnt. This shows, as xpctd, that ntropy incrass du to irrvrsibl hat addition occur in th burnr. Likwis, irrvrsibilitis du to work addition/xtraction ar only prsnt in th comprssor, gas turbin, and powr turbin. Friction is prsnt in all componnts. Not that th contribution du to th lumpd irrvrsibilitis associatd with work addition/xtraction is significantly gratr than axial friction or hat transfr and thrfor rprsnts th gratst opportunity for improvmnts in prformanc. Th bst way to rsolv this lumpd loss into its constitunt componnts is through dtaild loss analysis using CFD simulations. Ths rsults ar summarizd in Tabl 3. This clarly shows th componnts and spcific loss mchanisms that contribut th most to th total lost shaft work. In this cas, it is again apparnt that th infficincis associatd with th work addition procss in th turbins and comprssors dominat othr losss. Figur prsnts this data in th form of a pi graph showing th various fractions of shaft work lost du to various irrvrsibilitis.

9 Tabl Engin paramtrs. Comprssor prssur ratio = 7.7 Comprssor polytropic fficincy =.9 HP turbin polytropic fficincy =.9 Powr turbin polytropic fficincy =.89 Inlt Mach numbr =. Inlt static tmpratur = 7 o C Inlt static prssur = atm Exit static prssur = atm Maximum total tmpratur = 7 o C Cross-sctional aras Inlt fac =.56 ft Burnr ntranc =.9 ft HP turbin ntranc =.9 ft Powr turbin ntranc =.7 ft Engin xit =.76 ft Work addition (comprssor) = 97 hp Hat addition (burnr) = 85 BTU/lbm Shaft powr dlivrd by powr turbin = hp Total irrvrsibl ntropy incras/r =.64 Idal gas proprtis (constant through ngin) γ =.4 (air proprtis usd throughout) Tabl 3 Componnt/Mchanism Lost Work (hp). Mchanism Work Componnt Total Friction intraction Hat Total Inlt Comprssor Burnr.5.6 HP Turbin 9 8 Powr Turbin S Irr /R Inlt Comprssor Burnr Gas Powr Turbin Turbin Total Friction Work Hat LOSS ANALYSIS APPLIED TO MULTI- DIMENSIONAL CFD-GENERATED FLOW-FIELDS Th gratst potntial bnfit of applying th mthodology dscribd in this papr is to us it in th contxt of multi-dimnsional CFD whr spatially dtaild and comprhnsiv losss ar calculatd with high fidlity. Furthrmor, ths loss distributions can b appropriatly incorporatd into th mthodology/analysis dscribd in this papr. Spcifically, information rlvant to th third loss mchanism (th lumpd work intraction irrvrsibility as tratd in th arlir quasi-on-dimnsional analysis xampl) can b compltly dtrmind. A prliminary xamination of a rprsntativ thrdimnsional CFD gnratd flow-fild in a gas turbin stator row has bn conductd as part of this invstigation. Th flowfild solution usd hr is basd on NASA s Stator Row 37 tst cas [7]. Irrvrsibl ntropy distributions du to friction and hat transfr hav bn xtractd from th flowfild variabls. Th 3-D domain analyzd hr xtnds from stator blad lading dg to trailing dg, from hub to outr casing, and from blad suction surfac to adjacnt blad prssur surfac. Th grid in th stator rgion was 49(axial) by 5(transvrs) by 4(radial). Not that this ffort involvd post-procssing this particular rgion of a largr CFD-gnratd flow-fild. It is analyzd in this study in ordr to dmonstrat rprsntativ audits of irrvrsibl ntropy gnration in a sampl ngin flow-path. This analysis, in fact, follows arlir work dscribd in Rfrncs [3] and [7] in which similar tchniqus wr applid to multi-dimnsional simulations of high-spd (ram/scramjt) flow-filds. Figur shows prssur contours at a slic (constant radius from hub) of th flow-fild across th stator row. Figur 3 provids th rlativ contributions of friction irrvrsibility and hat transfr irrvrsibility to th ovrall irrvrsibl ntropy gnration along th axis of th stator row. Ths irrvrsibilitis ar inclusiv of all friction and hat transfr btwn all adjacnt stramtubs in this multidimnsional simulation. As can b sn, friction dominats hat transfr and is rsponsibl for 8% of all losss through th stator row. Inlt Comprssor Burnr HP Turbin Powr Turbin x/l Fig. Mchanism and componnt irrvrsibl ntropy incrass. Fig. Prcnt of lost shaft work in powr turbin (sortd by componnt).

10 Prssur contours for stator 37 - CFD simulation: constant radius slic Irrvrsibility distribution for stator 37 CFD simulation stator tangntial coordinat (5% radius) Lvl Prssur (atm) scald ntropy gain du to irrvrsibilitis ovrall ntropy gain du to irrvrsibilitis friction hat transfr axial distanc Fig. Prssur contours from stator row 37 (constant radius viw). APPLICATION TO VEHICLE SYSTEMS Though it is not th subjct of this papr, th concpts discussd hrin hav much broadr application than just ngin analysis. Th principls discussd ar compltly gnral and ar thrfor applicabl to analysis of th ntir vhicl itslf [8]. If losss in othr parts of th vhicl ar known, thy can b addd to th data in Tabl 3 to obtain a comprhnsiv pictur of th losss throughout all vhicl systms. For instanc, th ngin losss could b stackd up against rotor drivtrain losss, main rotor parasit powr, inducd powr, airfram drag powr rquird, tail rotor powr rquird, tc. Furthrmor, th total losss in all vhicl systms ar proportional to th total ful consumption of th vhicl [9], maning that thrmodynamic losss can b dirctly rlatd to vhicl mass proprtis. Figur suggsts that th vast majority of ful consumption du to losss in th ngin can b tracd to th powr turbin. Th loss data givn in Fig. can b viwd at th vhicl lvl as bing proportional to th ngin ful consumption incrmnts rquird in ordr to ovrcom th various irrvrsibilitis within th ngin flowpath. Extnsion of work potntial (scond law) mthods byond ngin-only assssmnt would concptually produc a truly optimizd vhicl for a givn mission. For such a vhicl, all vhicl systms and sub-systms ar optimizd with rspct to th fundamntal mission critria subjct only to th physical limits of th vhicl itslf. Finally, appropriat us of scond law mthods may assist ngin/vhicl dsignrs in making rvolutionary advancs in tchnology, allowing thm to think outsid of th box, to brak away from currnt dsign paradigms. Th application of work potntial mthods to turboshaft ngins and hlicoptr airframs could ultimatly nabl: ) th dvlopmnt of rotorcraft vhicls with xtndd rang x(mtrs)alongstatorrow Fig. 3 Entropy distribution du to irrvrsibilitis along stator row 37. and nduranc, ) th dsign of smallr, lightr, and mor powrful powrplants, 3) lowr initial rsarch, dvlopmnt, and production costs, and 4) lowr lif-cycl costs for rotorcraft vhicls. It can provid dirction in planning and quantify ncssary rsourc allocations rgarding dsign changs that may b mandatd by tchnology dvlopmnt. It also nabls th full utilization of numrical simulations in trms of diagnostics and flowfild valuation [7, 4] for both xtrnal and intrnal flows. SUMMARY Th us of scond law mthods in th dsign and valuation of turboshaft ngins can produc significant bnfits in trms of corrctly valuating dsign faturs and flow losss, gnrating dtaild loss audits in trms of flow loss mchanisms and ngin location, and ultimatly rlating ful usag to irrvrsibilitis. This papr dscribd th basis and us of th concpt of work potntial in th form of gas spcific horspowr for assssing th local stat of th flow in th contxt of shaft work potntial. Such a paramtr can b configurd in nginring charts that provid dsignrs/analysts with rlvant information concrning prformanc gains and losss. This information is highly usful bcaus it is a figur of mrit basd on a singl univrsal currncy that is valid across all componnts in th ngin. In addition, mthodology is dvlopd which dirctly rlats lost shaft work to flow irrvrsibilitis for assssing actual ngin flow-filds. This provids a mans for quantifying prformanc losss in a givn ngin. Lastly, irrvrsibl ntropy production from a multidimnsional CFD simulation of flow through a stator row is analyzd in ordr to dtrmin th rlativ contributions of intrnal friction and hat transfr to th ovrall loss. Vry littl scond law information is currntly utilizd from largscal CFD simulations and rlatd post-procssing fforts.

11 Howvr, it can b argud that scond-law information mbddd in xisting and futur high-fidlity simulations is actually th most valuabl information availabl from such simulations and may on day provid th critical tool ndd to nabl continud improvmnts in turbomachin prformanc. ACKNOWLEDGEMENTS This work was fundd by a grant from th AMRDEC (U.S. Army) Rdston Arsnal (Grant numbr NAG3-68 and NAG3-586). Th authors spcifically thank Mr. Jami Kimbl, Mr. Gorg Bobula, and Mr. Doug Thurman for thir support, ncouragmnt, ovrsight, and participation in all phass of this work. Thanks ar also du to Dr. Mik Hathaway (US Army Rsarch Lab) for contributing th CFD simulation of th stator row usd in this work. Univrsity of Missouri-Rolla. [7] Thurman, D., prs. communications rgarding Stator Row 37. [8] Roth, B.A., Mavris, D.N., "A Gnralizd Modl for Vhicl Thrmodynamic Loss Managmnt and Tchnology Concpt Evaluation," SAE [9] Roth, B.A., Mavris, D.N., "Tchnology Evaluation via Loss Managmnt Modls Formulatd in Trms of Vhicl Wight," Prsntd at th 59th Intrnational Confrnc of th Socity of Allid Wight Enginrs, St. Louis, Jun,. SAWE papr numbr 3. REFERENCES [] Roth, B., A Work Potntial Prspctiv of Engin Componnt Prformanc, AIAA Papr -33, July. [] Paulus, D.M., Gaggioli, R., Dunbar, W., Entropy Production as a Prdictiv Prformanc Masur for Turbomachinry, J. of Engr. For Gas Turbins and Powr, Vol. 3, Jan, p7. [3] Gttingr, C. and Riggins, D., Thr-Dimnsional High-Spd Combustor Prformanc Analysis, Procdings, 997 JANNAF CS/PSHS/APS Joint Mting, Octobr 997. [4] Bjan, A., A Rol for Exrgy Analysis and Optimization in Aircraft Systm Dsign, Procdings of th ASME Advancd Enrgy Systms Division 999, S. Acvs, d., ASME, NY, 999. [5] Roth, B. and Mavris, D., A Work Availability Prspctiv of Turbofan Engin Prformanc, AIAA -39, Jan.. [6] Riggins, D.W., Evaluation of Prformanc Loss Mthods for High-Spd Engins and Engin Componnts, Journal of Propulsion and Powr, Vol. 3, No., pp , 997. [7] Riggins, D.W., Thrust Losss in Hyprsonic Engins, Part : Applications, AIAA Journal of Propulsion and Powr, Vol. 3, No., pp , 997. [8] Ahrn, J.E., Th Exrgy Mthod of Enrgy Systms Analysis, Wily, NY, 98. [9] Bjan, A. Advancd Enginring Thrmodynamics, Scond Edition, Wily & Sons, NY, 997. [] Moran, M.J., Availability Analysis, A Guid to Efficint Enrgy Us, ASME Prss, NY, 989 [] Roth, B.A. and Mavris, D.N., A Comparison of Thrmodynamic Loss Modls Suitabl for Gas Turbin Propulsion, Journal of Propulsion and Powr, Vol. 7, No., pp , Mar-April. [] Roth, B. and McDonald, R.A., Working Charts for Estimation of Thrmodynamic Work Potntial in Equilibrium Mixturs of Jt- A and Air, Final Rport, NASA Grant NAG [3] Roth, B., "A Mthod for Comprhnsiv Evaluation of Propulsion Systm Thrmodynamic Prformanc and Loss," AIAA-33. [4] Scott, T. and Riggins, D.W., Work Intraction in Quasi-On- Dimnsional Flows, AIAA Journal of Propulsion and Powr, Vol. 6, No. 6, pp.53-59,. [5] Hisr, W.H., Pratt, D.T., Hyprsonic Airbrathing Propulsion, AIAA Prss, Washington, 994. [6] Wilson, C., Scond-Law Charactrization of Turboshaft Engin Prformanc, MS Thsis in Prparation, August,