THE PRINCIPLE OF HARMONIC COMPLEMENTARITY IN EVALUATION OF A SPECIFIC THRUST JET ENGINE

Transcription

1 U.P.B. S. Bull., Srs D, Vol. 76, Iss., 04 ISSN THE PRINCIPLE OF HARMONIC COMPLEMENTARITY IN EVALUATION OF A SPECIFIC THRUST JET ENGINE Vrgl STANCIU, Crstna PAVEL Th fundamntal da of ths papr s to alulat th spf thrust for of a propulson systm (thrust) [], lass of ar-jts ngn, wth applaton to turbojt smpl flow, basd on th dfnton and us of harmon omplmntary pattrns btwn gas dynams funtons of mpuls and thrust. Kywords: thrust for, omplmntarty, harmony, turbo ngn. Introduton In th followng shall b onsdrd a shmat dagram [] of a gnralzd nozzl, as n Fg. H H V M p T 0 0. S M a T 0(). M M g 0() Fg. Shmat dagram of a gnralzd nozzl whr th haratrst paramtrs natur suh as - mass flow, M ; - p, T, total prssur and tmpratur; - gomtr ara, S, dfn th two stats of th flud, orrspondng to th - ntry to th nozzl, ndx ; - xhaust from th nozzl, ndx. Of ours, th for dvlopd by th nozzl, forward F, onssts of - thrust lmnts, T j ; - propulson lmnts, P k ; F S p T f f V f PhD Prof. Eng., Faulty of Arospa Engnrng, Unvrsty POLITEHNICA of Buharst, Romana, -mal: vvrglstanu@yahoo.om PhD studnt, Eng., Faulty of Arospa Engnrng, Unvrsty POLITEHNICA of Buharst, Romana, -mal: nnapavl@gmal.om

2 4 Vrgl Stanu, Crstna Pavl - ompound lmnts, thrust-propulson C l. As a rsult, th summd for boms [3] n m p F = T + P + C () j k l j= k= l= In prnpl, foundatons of th physal and analytal holst modl (global) valuaton of for prformd by a gnralzd nozzl, F, whh an b rprsntd by any of th omponnts of jt ngn, nlt dv, omprssors (ntrfugal and arodynam), ombuston hambr, turbn, xhaust systm of th turbn and jt nozzl gas, all passd through a workng flud.. Mathmatal bass of a holst modl In gnral [4], th amount of for dvlopd by th nozzl, F, an b xprssd by th formula [5] F = F F, () F symbolzs th urrnt loal for, whr th flud jt mts gas dynams opposton, lk xtrnal atmosphr prssur p H. By dfntons, th urrnt loal for an b wrttn as F = Fv ph S, (3) whr F v rprsnts th loal for of th urrnt n vauum, n th absn of atmosphr. It s known that th loal for of th urrnt n vauum, s th sum of two omponnts, stat, p S and dynam, M V, that s F = M V + p S, (4) v whr V and p ar th absolut loal spd, rsptvly, loal stat prssur of th flud. It s known th xprsson of th loal for of th urrnt n vauum, basd on gas dynams funton of mpuls, z ( λ ), Fv = f M T z λ, (5) th onstant funton f, s k + f = R, (6) k whr k s adabat xponnt of flud voluton, R s gas onstant, and λ s Chaplygn numbr, ountrpart of Mah numbr, rlatv to flow rtal ondtons, mnmum. ( )

3 Th prnpl of harmon omplmntarty n valuaton of a spf thrust jt ngn 5 It taks nto aount that gas dynams funton of mpuls s xprssd [3] by z ( λ) = λ+ (7) λ and an b plottd basd on λ, as n Fg.. Fg. Gas dynams funton of mpuls Gvn th mportant rol playd by flud mass flow n ahvng thrust for, w us th known loal rlatonshp p M = m S q( λ ) T, (8) whr - m s mass flow onstant, m R = k k+ k + k - q ( λ ) th gas dynams funton [5] of th mass flow, k+ k q ( λ) = λ λ. (9) Th graph, basd on λ, th gas dynams funton of th flow has th aspt from Fg. 3. ; k

4 6 Vrgl Stanu, Crstna Pavl Fg. 3 Gas dynams funton of th flow Applyng quaton (5), n th fundamntal stons of th gnralzd nozzl, - and -, and th ondtons gvn by th rlaton (3), thn th urrnt loal fors boms prmary forms [6]. F = f M T z( λ ) ph S (0) and F = f M T z( λ ) ph S. () So, basd on rlaton (), th dvlopd for boms f M T z ( λ ) S F = f M p H S. () f M T z( λ ) S To smplfy wrtng, w dfn th nput offnt X, X X =, X whr X s an arbtrary quantty. As suh, t s statd furthr M - M =, offnt of mass ontrbuton; M T - T =, offnt of tmpratur ontrbuton; T p - p =, offnt of mhanal (or prssur) ontrbuton; p S - S =, offnt of gomtr ontrbuton; S

5 Th prnpl of harmon omplmntarty n valuaton of a spf thrust jt ngn 7 and - f f =, offnt of onstant mpuls; f =, offnt of onstant mass. m - m m Wth ths notatons, rlaton (), an b wrttn as z ( λ ) F = f M f M T ph S( S ). (3) z ( λ ) Rgardng flud mass flow, t an b playd n two man stons by p M = m S ( ) q λ T p M S q. = m T ( λ ) Undr ths ondtons, th offnt of mass nput has th form p q ( ) M λ = m S. (4) T q ( λ ) Assumng that th szs ar known and gvn paramtrs - mass, - gomtral, - thrmal, - mhans, - knmats, n nput ston, thn th problm of dtrmnng th for dvlopd by gnralzd nozzl rturns to th quston of lmnat knmats paramtrs, mor xatly, th offnts of spd n output ston, λ. Obvously, from (3) F + phs( S ) + M f z( λ) = z( λ) (5) f M T and, from (4), M T q( λ) = q( λ). (6) m p S It s, thrfor, nssary to sttl th ondtons whh rprsnt rlatons btwn th two gas dynams funtons to xprss th da of harmon flow out of th gnralzd nozzl ston. Of ours, a smpl, but vry omplatd, s to xprss from (5), th,, λ = f F M T, S... ( )

6 8 Vrgl Stanu, Crstna Pavl and, from (0), (,,,... ) λ = f M T p S thn, by qualzaton, w wll obtan,, F = f M T p, S.... ( ) Th orgnalty of ths papr s to fnd smpl ways of lmnaton, takng z λ and as a bass, th harmony that unts th two gas dynam funtons, ( ) q ( λ ), undr th prnpl of omplmntarty, xprssd by dffrnt laws. 3. Th laws of harmony basd on th prnpl of omplmntarty Mathmatal physs study [7] rvals that thr ar whol aras suh as - ontnuum mhans; - ltromagnt fld thory; - hat transfr, whh an b tratd, wth suss, by th thory of omplx varabl funtons n th omplx plan. In th as of rotatonal movmnts rot V = 0, and f th vloty of th flud, V, s drvd from a potntal, ϕ, V = grad ϕ. Basd on th ontnuty quaton, th potntal vloty s a harmon funton that satsfs Laplas quaton ϕ ϕ Δ ϕ = + = 0. (7) x y Sn th rotor s zro, th vortx has no omponnts, as suh, and urrnt funton s harmon, ψ namly ψ ψ Δ ψ = + = 0. (8) x y On th othr hand, btwn funtonsϕ and ψ ar Cauhys monogn ondtons. Thrfor, th funtons ϕ and ψ, assoatd harmons [8], an b th ral and magnary, of a omplx varabl analyt funtons f z = ϕ x, y + ψ x, y (9) th varabl z s ( ) ( ) ( ) z = x+ y.

7 Th prnpl of harmon omplmntarty n valuaton of a spf thrust jt ngn 9 Transformng Cartsan oordnats to polar oordnats, ρ, θ, thn th two funtons ϕ and ψ an b xprssd n trms of ρ and θ, th varabl z s z = ρ ( osθ + snθ) = ρ θ. Also, n ths oordnat thngs rman unhangd rgardng harmon funtons ϕ ( ρθ, ) and ψ ( ρθ, ). As, any rotatonal movmnt wll b rprsntd by an analytal funton of z, ths s avalabl v vrsa, any analyt funton of omplx varabl z rprsnt an rotatonal movmnt. So, always, an analyt funton f ( z) wll rprsnt omplx potntal of a wll dfnd movmnt. Man funtons ar 3 f ( z) = z; f ( z) = ; 3 f ( z) = z ; 4 f ( z) = z ; z z 5 f ( z) = z+ ; 6 f ( z) = ln z; 7 f ( z) =. z Of ths, th most ntrstng, wth a partular physal sgnfan, s th law 3, f ( z) = z, ommon n fratal gomtry (of natur) [8]. Consdrng that z = x+ y, thn namly f ( z) = ( x+ y), ( ) ( ) f z = x y + xy. (0) Thrfor, ϕ ( x, y) = x y, () and ψ ( x, y) = xy. () Obvously, th abov funtons, ϕ and ψ, shall b harmon baus vrfy th Lapla quaton Δ ϕ = 0 and Δ ψ = 0. In vw of th harmon haratr, onjugat, of th funtons ϕ and ψ, aspts of gas dynam funtons z ( λ ) and q( λ ) as wll as th rlatonshp btwn thm, t an b onludd that th man harmon laws [9] ar: I. x + y = t, lnar law, l l ;

8 30 Vrgl Stanu, Crstna Pavl II. y = t x+ x, parabol law, l p ; III. x y = t, hyprbol law [0], l h. Plottng ths laws mags obtand ar thos from Fg. 4. Fg. 4 Harmon law Ths s why, furthr, t wll dsuss [], to mak a omparson, ah as. 4.. Lnar law Basd on th known [], from varatons of th gas dynam funtons, t s aptabl, for a rang of varaton of th offnt λ 0, 4 λ, that z( λ) + q( λ). (3) Takng nto aount th rlatons (5) and (6), th form of th rsultng thrust for F I s n whh M T ( ) ( ) FI M α M T β S p S γ δ = + + +, (4)

9 Th prnpl of harmon omplmntarty n valuaton of a spf thrust jt ngn 3 f f α = T z ( λ ) f f q ( λ ) β = T m z( λ) p γ = T δ = α+ β H m p q ( λ ). (5) Obvously, baus th spf thrust for, FSP s F FSP =, (6) M thn th nozzl gnralzd spf thrust for, n as I, boms M T F α SP I ( M T ) β = + + ( S ) p S γ + δ. (7) 4.. Parabol law In ths as, rplang th funtons, n harmon parabol law z( λ) = q( λ) + + 3, (8) q( λ ) avalabl n subson rgm, for 0,05 λ. Th valus of onstants, =,, 3 dpndng on th natur of th workng flud. So, - for ar = 0, 5; = 0, 79 ; 3 = 0,005 ; - for burnd gass = 0, 35 ; = 0, 797 ; 3 = 0, 03. Spf for, dvlopd n as II, has th form M T F ( ) ( ) ( SP II = α M T + β γ + S + δ p S ), (9) p S whr

10 3 Vrgl Stanu, Crstna Pavl as h = f and a = m Hyprbol law α = h T 3 β = h T q( λ) p γ = T δ = h T q( λ ) H a p q ( λ ), (30) Ths as s th most ntrstng and, also, th asst, as t prtans to hyprbol law [], z( λ) q( λ), (3) vald for a varaton of th spd offnt n th output 0, λ. Takng nto aount th xprssons (5) and (8), thn, by rmovng th flow rat s obtand Fv v p S, (3) whr v = m f. Undr ths ondtons, th urrnt for, n ralty, boms F v p S ph S. (33) W apply ths xprsson, n th fundamntal stons of th gnralzd nozzl, and obtand, hghlghtng th paramtrs th nput ston, F III M T p H FIII = v ( v p S ) ( S ). (34) m q( λ ) p Spf for, n as III, s wrttn as FSP = ε 3 ( ) 3( ) III v v p S + γ S, (35) T ε3 = v q λ γ m ( ) T H 3 = m q( λ ) p p

11 Th prnpl of harmon omplmntarty n valuaton of a spf thrust jt ngn 33 whr thr whr v v = f ; v =. m Obvously, th valu of λ an b obtand thr from th ondton q z ( λ ) q( λ ) ( λ ) p, M T = p p S = q( λ ). m v, 5. Numral rsults and omparsons btwn modls It s ntrstng, furthr, to mak a omparson btwn th rsults obtand by applyng, n ths thr ass, th modls prsntd, for th sam jt ngn. Th omparson taks nto aount assssmnts of spf thrust fors, F SP, =,,3, undr th sam ondtons assumd (sz, paramtrs, offnts) to ntr nto th systm. It nots that FSP = FSP I ; FSP = FSP II and FSP3 = FSP III, whr spf rlatonshps ar appld to th ntr propulson systm, onsdrd ntgral gnralzd nozzl. All alulatons ar prformd for - stat flyng ondton, H = 0, V = 0 ; - ratd opratng ondtons. Thus, s allowd th followng nput nto th ngn: 5 - p = po =, N/m ; - T = T = 88 K ; o and knmats ondtons q ( λ ) = 0,8and ( ),34 z λ =. Charatrsts of th workng flud ar - xponnts adabat volutons, through systm, ar, k =, 4 ;

12 34 Vrgl Stanu, Crstna Pavl gas, k =, 33 ; - spf flud onstants ar, R = 87 J/kg K ; gas, R = 87,6 J/kg K. Calulatng onstants w hav th followng valus - m = α a = 0, 0404 ; - f = h h = 3,37 ; - v = v =,68 ; - k = k = 4, 753. Allowd, furthr, th followng nput fators, vald for th whol systm, holst: - m ; - f ; - v =, and, rsptvly, - M =, 0 ; - T = 3, ; - p,56, - S =,. Prod to th alulaton of spf thrust fors for ah modl, holst, harmon and omplmntary. 5.. Lnar modl Ar alulatd - α = 888,9 ; - β = 373, 4 ; - γ = 58,38 and - δ = 54, 4. Substtutng n rlatonshp (7), spf thrust for, wll gt FSP 859 m/s. 5.. Parabol modl Coffnts ar - α =,66 ; - β = 9,564 ; - γ = 55,06 and - δ = 50. Substtutng, spf thrust for, from (9) boms FSP 945, 75 m/s. Somtms you an nglt th frst omponnt for, baus α s muh smallr than th othr offnts, th rror that s mad s undr %.

13 Th prnpl of harmon omplmntarty n valuaton of a spf thrust jt ngn Hyprbol modl In ths as, - γ 3 = 43,6 ε 3 = 43,66. Thrfor spf thrust for from rlatonshp (35) s FSP3 80 m/s. 6. Conlusons Basd on th proposd modls and smulatons prformd on ths thr ass, for th thr laws, harmon omplmntary, lnar law, parabol law and hyprbol law, som ntrstng onlusons an b drawn. Among thm, w an rtan followng: - Rgardlss of th modl s notd that, th most mportant ways of ahvmnt of thrust fors (propulson) nluds th us of mass nozzl, whn M vars asndng; thrmal nozzl, whn T nrasng varabl; mhanal nozzl, whn p nrasng varabl; gomtral nozzl, f ston hangs, smpl onvrgnt vrsons, or Laval, onvrgnt-dvrgnt; - Makng a for rqurs th xstn, at th ntran to th nozzl, a flud that has a puls; - Coffnt of thrmal, T, and mhanal ontrbuton, p, ar, usually, orrlatd, as s th as n th systm turbo omprssor; - All modls ar holst, allowng valuaton of th ovrall prforman of th ngn, takng nto aount, th rlatons btwn ngn omponnts; - Aftr th apparan of th thr harmon laws, losst to ralty, whh lads to rasonabl rsults wth th on xstng n ltratur, s hyprbol law. In rlaton wth ths absn, othr laws rrors ar lnar law, 4,9% ; parabol law, 6,7% ; - Th bst harmonzaton law s th hyprbol law, provn by smplty of spf for xprsson rsults, whh onfrmd, on agan, that th tst of truth s smplty; - For hyprbol law obsrv that F = f p, S, SP ( )

14 36 Vrgl Stanu, Crstna Pavl fundamntal prforman of a nozzl or, furthrmor, of a ngn, dpnds only by substantal varaton of total flud prssur; workng hannl ston (xpanson). R E F E R E N C E S [] V. Stanu, E. Rotaru, and A. Bogo, Thory and dsgn of propulson systms, Publshr BREN, Buharst, 00 [] V. Stanu, Modllng thrust propulson systms, Publshr URANIA, Buharst, 00 [3] V. Stanu, R. Sălanu, and B. Pantlmon, Modrn systms to nras thrust and onomy of th turbojt ngns, Publshr Unvrstata Polthna, Buharst, 993 [4] V. Stanu and E. Rotaru, Convntonal propulson systms, Publshr BREN, Buharst, 00 [5] V. Pmsnr, Ar jt ngns, vol. I, Publshr Ddat and Pdagog, Buharst, 984 [6] V. Stanu, Propulson Phlosophy, Publshr PRINTECH, Buharst, 0 [7] V. Stanu and C. Pavl, Orthoturbojt ngn, a hallng of atual ngns, 5 th Europan Confrn for Aronauts and Spa Sns (EUCASS), Munhn, 03 [8] N. Lsmor-Gordon and R. Edny, Introdung Fratal Gomtry, Totm Books, 000 [9] V. Stanu, Rlgon Phlosophy Sn, Publshr PRINTECH, Buharst, 03 [0] T. Toro, Modrn physs and phlosophy, Publshr Edtura Fala, Buharst, 973 [] V. Stanu and C. Pavl, About th n-yologal ntrprtaton of a omplx plan, ICEGD- Intrnatonal Confrn On Engnrng Graphs And Dsgn, Tmoara 03