Rotordynamic coe cients for labyrinth gas seals: single control volume model

Transcription

1 Rotordynamc coe cents for labyrnth gas seals: sngle control volume model R. MALVANO C.N.R. Centro Stud Dnamca Flud F. VATTA and A. VIGLIANI Poltecnco d Torno Dpartmento d Meccanca Corso Duca degl Abruzz, 4 09 Torno Italy Abstract. A numercal code has been developed to determne the rotordynamc coe cents of straght-through labyrnth gas seals n the case of a sngle volume model. Some consderatons are drawn concernng the poston of the sonc secton when the flow s choked. Furthermore the nfluence of knematc vscosty varyng wth pressure has been consdered. The results are compared wth the expermental and theoretcal ones avalable n the lterature. The agreement between results s farly good. Keywords: trbology; labyrnth seals; rotordynamc coe cents c 03 Kluwer Academc Publshers. Prnted n the Netherlands. seals.tex; 4/04/03; 5:7; p.

2 Nomenclature A cavty transverse area N t number of seal strps a, b radal seal dsplacement components p pressure due to ellptcal whrl p r reservor pressure a r,a s dmensonless length p s sump pressure B cavty heght R gas constant C drect dampng coe cent R s seal radus c cross coupled dampng coe cent S r wet permeter on rotor D hydraulc dameter S s wet permeter on stator H radal clearance T absolute temperature th secton ; =0= nlet; t tme = N t = outlet U tangental velocty K drect st ness coe cent U 0/(!R s) nlet velocty rato k cross coupled st ness coe cent cavty geometrc coe cent L cavty length specfc heats rato L T labyrnth seal total length # angular dsplacement M Mach number exponent n the knematc ṁ leakage mass flow rate vscosty-pressure relatonshp per unt crcumference µ flow coe cent m r,n r coe cents for Blasus µ g knetc energy carryover coe cent relaton (rotor) knematc vscosty m s,n s coe cents for Blasus flud densty relaton (stator) r, s shear stress on rotor, stator N c number of labyrnth cavtes! rotor angular speed. Introducton Self excted vbratons represent a serous problem n turbomachnery: they arse n the form of shaft sub synchronous precesson,.e. at angular speed lower than that of the shaft tself. The forces that determne such moton have d erent orgns; one of these s due to the labyrnth seals that are commonly used to reduce flud leakage from hgh pressure regons to low pressure ones. Bascally ther workng prncple les n acceleratng and successvely deceleratng a compressble flow, leadng to dsspaton of knetc energy due to vscous frcton. It s then evdent the mportance of studyng the dynamc characterstcs of labyrnth seals: as a matter of fact they can e ectvely nfluence the motor performances. Labyrnth seals can possess d erent geometry; n the present paper, we wll consder the smplest geometres, known as straght-through seals.tex; 4/04/03; 5:7; p.

3 labyrnth seal. In ths case, the geometrcal parameters of the cavty (A, B, H and L ) are constant along the seal. In ths feld the lterature s vast. It s worth notng that Black [] was the frst to descrbe an annular seal behavor through dynamc coe cents, n analogy wth the studes on lubrcated bearngs. The analyss to determne such coe cents s based on the followng models: sngle control volume multple control volume. The am of the present work s to develop a numercal code to determne st ness and dampng coe cents for a labyrnth seal when adoptng a sngle control volume model. Moreover the authors wll prove that, n the case of sonc flow, the sonc condton can be placed only n correspondence of the outlet secton. Furthermore the work focuses on the nfluence of the hypothess of knematc vscosty varyng wth pressure. To the best of the author s knowledge, these fndngs do not appear to have been reported prevously n lterature. Fnally, n order to test the relablty of the code, the authors compare ther results wth the expermental work of Benckert and Wachter [] n the case of non rotatng shaft and wth numercal results obtaned by Scharrer [3]. 3. Mathematcal model In the followng secton we summarze results already present n the lterature, relatve to the case of a sngle control volume [4]. Wth reference to Fg., showng a generc cavty of a labyrnth seal, the contnuty equaton for the control volume under analyss s ( U A whle the momentum equaton s: R s +ṁ + ṁ = (U where U A R s +U ṁ + U ṁ = A +( r a r s a s ) L () a r = S r L and a s = S s L. seals.tex; 4/04/03; 5:7; p.3

4 4 Fgure. Cavty geometry Shear stresses are descrbed by Blasus model, vald for turbulent flows n smooth ppes. It holds: s = U n s apple U D m s sgn (U ) (3) r = (!R s U ) n r apple!rs U D m r sgn (!R s U ) (4) where D = (B + H )L /(B + H + L ) For a subsonc flow, the flow rate ṁ, accordng to Neumann s emprcal formula [5], s gven by: s p p ṁ = µ g µ H (5) RT where µ s the flow coe cent, whleµ g s the knetc energy carryover coe cent, gven by the followng emprcal expressons: µ = r N t µ g = ( )N t + where = where = p p + 6, 6 H L seals.tex; 4/04/03; 5:7; p.4

5 When the flow s choked (.e. the outlet secton s sonc, as proved n Secton 3), the flow rate, accordng to Flegner s [6], becomes: 5 ṁ Nt =0, 5µ g p Nc H p RT (6) The followng hypotheses hold: the flow thermodynamc evoluton follows the deal gas law temperature s constant along the seal the axal component of velocty can be neglected when computng shear stresses. It s now possble to determne the pressure and the speed n each cavty by substtutng expressons (3 6) n eqs. () and (). The soluton of the of the problem can be obtaned by means of perturbaton technque; let p = p,0 + p 0 H = H,0 + H 0 = H 0 + H 0 U = U,0 + U 0 A = A,0 + L H 0 = A 0 + LH 0 where subscrpt 0 represents statc components (zeroth order), that are constant nsde each cavty but vary from cavty to cavty, whle superscrpt 0 stands for varable components (frst order), that are due to a perturbaton (dsplacement) of the rotor from ts centered poston. The frst order terms are functons both of angular dsplacement and of tme, and are responsble for the forces exerted on the rotor. Wth respect to zeroth order, t holds: ṁ = ṁ + = ṁ 0 (7) ṁ 0 (U,0 U,0 ) L a r r,0 a s s,0 = 0 (8) Gven nlet and outlet pressures, by means of eq. (7) t s possble to compute the pressure dstrbuton along the seal and the flow rate ṁ 0 ; ths result s obtaned adoptng an teratve technque and verfyng whether the flow s choked or not n correspondence of the outlet secton. Fnally, the tangental velocty dstrbuton can be determned through eq. (8). When consderng the frst order terms, we have: seals.tex; 4/04/03; 5:7; p.5

6 G 0 R + G U,0 R + G p,0 R + G 3 p 0 + >< +G 4 p 0 + G 5 p X U,0 R s >: +X 4 p 0 + X 5 H 0 + A,0 R + X U 0 ṁ 0 U 0 + X 3 p 0 + where coe cents G, G, G 3, G 4, G 5, and X, X, X 3, X 4, X 5 are functons of zeroth order varables. Ther expressons are gven n Appendx A. Assumng the shaft perturbaton to be an ellptcal orbt of sem axes a and b, tyelds: (9) H 0 = a [cos (#!t) + cos (# +!t)] b [cos (#!t) cos (# +!t)] (0) Consequently the pressure and velocty fluctuatons are assumed to be n the format: 8 p 0 =p + c cos (# +!t)+p + s sn (# +!t)+p c cos (#!t)+ >< + p s sn (#!t) >: U 0 =U + c cos (# +!t)+u + s sn (# +!t)+u c cos (#!t)+ + U s sn (#!t) () Substtutng eqs. (0) and () nto (9) and groupng lke terms of snes and cosnes elmnates the tme and angular dsplacement dependency; t yelds eght ndependent lnear algebrac equatons per cavty, n the followng unknowns: p + c The resultng system for cavty besdes the above unknowns, depends on the same unknowns expressed as functons of cavty + and ; hence t s a block-trdagonal system that can be easly solved. p + s p c p s U + c U + s U c U s. seals.tex; 4/04/03; 5:7; p.6

7 Beng the system lnear, the unknowns represented by the ampltudes of expressons () are determned by supermposton of the known terms; n partcular, for pressures t yelds: p + s p + c = af + s a = af + c a + bf + s b p s = afa s + bf s b + bf + c b p c = afa c + bf c b where subscrpt a refers to the pressure component obtaned when lettng the condtons a = and b = 0 n eq. (0), whle subscrpt b refers to the pressure component obtaned when lettng the condtons a = 0 and b = n eq. (0). By means of the technque proposed by Scharrer, the st ness and dampng coe cents can be computed as follows: 7 K = R s XN c = F + c a + F c a L k = R s XN c = F + s b F s b L C = R s! XN c = F + s a F s a L c = R s! XN c = F + c b + F c b L 3. Sonc condtons The flud moton nsde the seal n axal drecton can be assumed to behave as the nearly undmensonal flow of a compressble flud n a ppe of constant secton A F. The term nearly undmensonal we refer to the condton of physcal and knematc quanttes varyng only axally, beng constant across the secton. Wth reference to the nternal flow theory [7], t s possble to assume the Mach number relatve varaton due to three terms; t yelds: dm M = M M M M + M daf + M A F M + M df + dt M 0 T 0 () where F s a functon takng nto account the frcton and drag e ects and T 0 s the total temperature. Through the above equaton t s seals.tex; 4/04/03; 5:7; p.7

8 8 possble to analyze the flow n the seal and to prove that, f there s a sonc secton, t can only be the outlet secton of the seal tself. In the case under examnaton, da F s null, as well as dt 0 s null under the hypothess of adabatc flow. Consequently t holds: dm = M + 4 M M df (3) The numerator of eq. (3) s always postve along the axal drecton; hence the sgn of dm depends on the value of M wth respect to one. If we suppose that M< at the nlet secton, eq. (3) gves a postve value for dm, hence M grows along axal drecton. Ths growth can not lead to M = n a secton nsde the seal, because two d erent condtons could take place mmedately downstream ths hypothetcal secton, but ether of them s contradcted by eq. (3): M can not grow (.e. dm > 0), for, from eq. (3), dm s negatve f M>; M can not decrease (.e. dm < 0), for, from eq. (3), dm s postve f M<. Hence,dependng on pressures upstream and downstream the labyrnth seal, only the followng two condtons can take place: the flow s subsonc along the whole seal; the flow becomes sonc at the outlet secton of the seal and then, externally to the seal tself, t suts to downstream pressure through a shock wave; t s the case of chocked flow. These results hold for adabatc flows, whle the analytcal models derved by Iwatsubo, Chlds and Scharrer consder an sothermc flow, so that the leakage flow rate can be expresses as n eq. (5). Hence t s nterestng to nvestgate the consequences determned by the two d erent approaches. Total temperature T 0 depends on statc temperature T accordng to the followng expresson: T 0 = T + M Hence, eq. (), n the sothermc case, becomes: seals.tex; 4/04/03; 5:7; p.8

9 9 M = M 4 + M df + + M + M + After a few smple passages, t yelds: (4) T M T 0 dm dm apple ( ) M M = M + 4 The roots of the left hand sde of eq. (5) are: so eq. (5) can be wrtten as: M = M = M df (5) dm = M + 4 M df M M + (6) It can be mmedately seen that the lmt value that the Mach number can reach at the outlet secton of the seal d ers from unty, and s gven by: M = Therefore, n the sothermc case, the Mach number grows along the seal, however wthout reachng the sonc condton: hence the flow s always subsonc. 4. Influence of pressure on knematc vscosty It s well known that knematc vscosty s strongly nfluenced by pressure when pressure tself undergoes sgnfcant varatons. Hgh pressure ratos (of order 0 or even 30) between nlet and outlet may take place n turbomachnery. In these condtons the assumpton that knematc vscosty s ndependent of pressure does not hold. Consequently, by means of a perturbatve technque, from eq. (3) t holds: seals.tex; 4/04/03; 5:7; p.9

10 s s = s,0 U 0 + H 0 (7) 0 In the most general case, the relatonshp between vscosty and pressure s / 0 =(p/p 0 ), havng supposed constant temperature along the seal. Last term of the rght hand sde of eq. (7) becomes: s 0 s p 0 = ( + m s 0 0 RT = +m s s,0 p 0 p,0 Let s where be s = s,0 + O s () apple +ms O s () = s,0 U 0 + m sd U,0 (B + H) H0 + +m s p,0 p 0 Lkewse, from eq. (4) we have: r = r,0 + O r () where: O r () = r,0 apple +m r U 0 + m rd!r s U,0 (B + H) H0 + +m r p,0 p 0 Hence the e ect of vscosty varyng wth pressure a ects only term X 3 (see Appendx A). When dependence of knematc vscosty on pressure may be neglected (.e. = 0), the above equatons concde wth expressons derved by Chlds [4]. 5. Numercal results and conclusons A computatonal code was developed to determne the dynamc coe - cents of labyrnth seals. We consdered a sngle control volume model and tested the code comparng our results, dentfed by the acronym MVV, wth those obtaned by other researchers. All the seal under examnaton are teeth on stator (T.O.S.) seals. Frstly we compare our results wth those publshed by Benckert e Wachter []: they appear to be n good agreement, as t can be seen seals.tex; 4/04/03; 5:7; p.0

11 Fgure. Comparson of results k? = H k R sl N c (p r p, s) E? =.5 0U 0 p r p s n Fg.. Ths fgure shows also the results publshed by Chlds and Scharrer [4] and by Wllams and Flack [8]. Fgures 3 6 show the comparson between our results and those from Scharrer [3] for all the dynamc coe cents. The e ects of varable knematc vscosty are depcted n fgures 7 0, where we plot the dynamc coe cents versus! both n the case of = const and of = f(p). These plots show that drect st ness and cross coupled dampng coe cents are not a ected by vscosty varatons; moreover, for low pressure ratos, changes of vscosty have neglgble nfluence also on cross coupled st ness and drect dampng coe cents. Wth growng pressure rato, knematc vscosty undergoes relevant varatons and hence t nfluences sgnfcantly the dynamc coe cents wth growng shaft velocty!. Fnally, t s worth notng that the coe cents n and m used to compute the shear stresses hold for smooth ppes; as a matter of fact, real surfaces are rough, hence these coe cents should be sutably adjusted. We verfed that ther numercal values play a fundamental role n evaluatng the dynamc coe cents: n fact, even a slght change causes strong varatons of dynamc coe cent. seals.tex; 4/04/03; 5:7; p.

12 References. Black H. F., Jenssen D. N., Dynamc Hybrd Bearng Characterstcs of Annular Controlled Leakage Seals, Proc. Inst. Mech. Eng, 84, pp.9 00,970.. Benckert H., Wachter J., Flow Induced Sprng Coe cents of Labyrnth Seals for Applcatons n Rotor Dynamcs, NACACP33,pp.89, Scharrer J. K., AComparsonofExpermentalandTheoretcalResultsforRotordynamc Coe cents for Labyrnth Gas Seals, TRC Report No. SEAL--85, Texas A&M Unversty, Chlds D. W., Scharrer J. K., An Iwatsubo based Soluton for Labyrnth Seals: Comparson to Expermental Results, J. Eng. Gas Turbnes and Power, 08, pp.35 33, Neumann K., Zur Frage der Verwendung von Durchblckdchtungen m Dampfturbnenbau, Maschnentechnk,3 (4), John E. A. J., Gas Dynamcs, Wley, Shapro A. H., The Dynamcs and Thermodynamcs of Compressble Flud Flow, vol., Wley, p.30, Wllams B. P., Flack, R. D., Calculaton of Rotor Dynamc Coe cents for Labyrnth Seals, Proc. 7th Congress ISROMAC, Vol.A, Honolulu, pp , 998. seals.tex; 4/04/03; 5:7; p.

13 3 Appendx A Defnton of the frst order contnuty and momentum equaton coe - cents: G = A,0 RT G = p,0 L RT G 3 = ṁ 0 ( G 4 = ṁ 0 " G 5 = ṁ 0 " X = p,0 A,0 RT p,0 p,0 p +,0 + µ +, ,0 p,0 p,0 p,0 p +,0 p,0 p +,0 p,0 p,0 p +,0 p +,0 + µ,0 5 4,0 + µ,0 5 4,0 p + µ +, ,0 p + 0 ) p,0 p,0 p p p +,0 0 # p p + p p 0 # 0 + X = ṁ 0 + +m s U,0 L a s s,0 + +m r!r s U,0 L a r r,0 X 3 = ṁ 0 (U,0 U,0 ) " p,0 p,0 p,0 + µ,0 5 4,0 p p,0 p 0 # + + +m s p,0 L a s s,0 +m r p,0 X 4 = ṁ 0 (U,0 U,0 ) " p,0 p,0 p,0 L a r r,0 + µ,0 5 4,0 p p,0 p 0 # X 5 = ṁ0 H 0 (U,0 U,0 )+ m s D (B + H) L a s s,0 m r D (B + H) L a r r,0 seals.tex; 4/04/03; 5:7; p.3

14 4 Fgure 3. Drect st ness coe cent versus nlet tangental velocty Fgure 4. Cross coupled st ness coe cent versus nlet tangental velocty seals.tex; 4/04/03; 5:7; p.4

15 5 Fgure 5. Drect dampng coe cent versus nlet tangental velocty Fgure 6. Cross-coupled dampng coe cent versus nlet tangental velocty seals.tex; 4/04/03; 5:7; p.5

16 6 Fgure 7. Drect st ness coe cent versus rotor angular speed Fgure 8. Cross-coupled st ness coe cent versus rotor angular speed seals.tex; 4/04/03; 5:7; p.6

17 7 Fgure 9. Drect dampng coe cent versus rotor angular speed Fgure 0. Cross-coupled dampng coe cent versus rotor angular speed seals.tex; 4/04/03; 5:7; p.7