A Novel FE Bulk-Flow Model Improving Predictions of Force Coefficients in Off-Centered Grooved Oil Seals

Transcription

1 Txas A&M Univrsity Mchanical Enginring Dpartmnt Turbomachinry laboratory A Novl FE Bulk-Flow Modl Improving Prdictions of Forc Cofficints in Off-Cntrd Groovd Oil Sals Rsarch Progrss Rport to th Turbomachinry Rsarch Consortium TRC- SEAL-1-8 Adolfo Dlgado Rsarch Assistant Luis San Andrés Mast-Childs Profssor Principal Invstigator Jun 28 TEES projct 32513/1519T7

2 Excutiv Summary TRC-Sal-1-8 A Novl FE Bulk-Flow Modl Improving Prdictions of Forc Cofficints in Off-Cntrd Groovd Oil Sals Oil sals in cntrifugal comprssors rduc lakag of th procss gas into th support barings and ambint. Undr crtain oprating conditions of spd and prssur, oil sals lock, bcoming a sourc of hydrodynamic instability du to xcssivly larg cross coupld stiffnss cofficints. It is a common practic to machin circumfrntial groovs, braking th sal land, to isolat shar flow inducd film prssurs in contiguous lands, and hnc rducing th sal cross coupld stiffnsss. Exhaustiv oil sal tsting prformd by Childs and studnts shows that an innr land groov, shallow or dp, dos not actually rduc th cross-stiffnsss as much as convntional modls prdict. In addition, th tstd groovd oil sals show ovrly larg addd mass cofficints; whil prdictiv modls, basd on classical lubrication thory, miss ntirly th fluid inrtia ffct. A 27 TRC rport introducs a fluid inrtia, bulk-flow modl that proprly accounts for th flow and momnt transport at th groov-land intrfacs. Th novl modl prdicts with xactnss th forc cofficints of multipl-groov laminar flow oil sals at thir cntrd condition. Th analysis rlis on an ffctiv groov dpth, diffrnt from th physical groov dpth, which dlimits th uppr boundary for th flow inducd by dynamic (fluid squzing) motions in th groovd rgion of a sal. Th currnt rport xtnds th bulk-flow analysis to prdict th lakag, raction forcs and dynamic forc cofficints of groovd oil sals oprating at an off-cntrd (ccntric) lockd condition. Prdictions of rotordynamic forc cofficints ar compard to publishd tst cofficints for a smooth land sal and a sal with a singl groov sal with a dpth of 15 tims th land claranc (c=85.9 μm). Th tst data rprsnts opration at 1 krpm and 7 bar oil fd prssur, and four journal ccntricity ratios (/c=,.3,.5,.7). Th nhancd modl prdicts accuratly th smooth and groovd sals lakag, raction forc, and rotordynamic forc cofficints for th lowst ccntricitis (/c=,.3). Th modl yilds modrat to good corrlation with th tst forc cofficints for /c=.5. For th largst journal ccntricity, /c=.7, significant discrpancis btwn th prdictions and xprimntal rsults aris. Th tst data TRC- SEAL-1-8 2

3 tchnical rport dos not offr dtails on oprating conditions lading to larg powr dissipation that may hav affctd th lubricant proprtis and sal claranc. Th modl compltd is a significant improvmnt that prdicts accuratly th rotordynamic forc cofficints of lockd multipl-groovd oil sals. Prior to th currnt rsults, th cross-coupld stiffnss cofficints of groovd oil sals wr largly undr prdictd, and th dirct addd mass cofficints ignord or largly undr prdictd (up 1 tims). Th computational tool intgrats a XLTRC 2 typ GUI to th FEM solution of th flow quations. Not: Th P.I. ditd this rport fiv tims, English and tchnical contnt, prior to its rlas to TRC mmbrs. TRC- SEAL-1-8 3

4 Tabl of Contnts Excutiv Summary... 2 List of Tabls... 5 List of Figurs... 5 Nomnclatur... 7 I Introduction... 8 II Litratur Rviw...9 Analytical work... 9 Exprimntal work... 1 III Analysis Bulk flow formulation Finit lmnt solution Boundary Conditions Zroth Ordr Prssur Fild First Ordr Prssur Filds Effctiv groov dpth... 2 Excl program intrfac IV Modl Prdictions and Validation to Tst Data V Conclusions and Rcommndations VI Rfrncs TRC- SEAL-1-8 4

5 List of Tabls Tabl 1 Oil sal configuration, oprating conditions fluid proprtis and numbr of lmnts for FE msh List of Figurs Figur 1 Oil sal ring assmbly, Rf. [2]... 8 Figur 2 Schmatic viw of groovd annular cavity dividd into flow rgions Figur 3 Viw of rotating and whirling journal and coordinat systm for bulk-flow analysis Figur 4 Coordinat systm and sampl msh for oil sal FEM computational cod Figur 5 FEM msh dpicting nods of intrst for implmntation of boundary conditions Figur 6 a) Schmatic viw of stramlins in axially symmtric groovd annular cavity (ΔP= P s -P d ). b) CFD simulation of prssur drivn stramlins across a 1c and 15c circumfrntial mid-land groov in an oil sal tstd in Rf. [7]. (c= 86 mm, ω=1 RPM, D= 117 mm) Figur 7 Graphical usr intrfac for XLFEGLOSal cod (SI units) Figur 8 Configuration of paralll oil sals tstd in Rf. [6] Figur 9 Masurd journal cntrlin locus for smooth and groovd sal (c g =15c). (7 bar, 1 rpm) [7] Figur 1 Sal raction forcs. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov (c η = 7c) Figur 11 Dirct stiffnss cofficint (K ii ) vrsus ccntricity. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov (c η = 7c) Figur 12 Cross-coupld stiffnss cofficints (K ij ) vrsus ccntricity. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov (c η = 7c) Figur 13 Cross-coupld stiffnss cofficints (K xy ) vrsus shaft spd at two journal ccntricitis (,.3). Exprimnts for smooth sal and sal with innr land groov TRC- SEAL-1-8 5

6 (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov (c η = 7c) Figur 14 Dirct damping cofficints (C ii ) vrsus ccntricity. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov (c η = 7c) Figur 15 Cross-coupld damping cofficints (C ij ) vrsus ccntricity. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov (c η = 7c)... 3 Figur 16 Addd Mass cofficint (M xx, M yy ) vrsus ccntricity. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov using (c η = 7c) Figur 17 Sal lakag vrsus ccntricity. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov using (c η = 7c) TRC- SEAL-1-8 6

7 Nomnclatur C ij c c η d η Dirct damping cofficints [N.s/m] i,j=x,y Claranc [m] Charactristic claranc [m] Charactristic groov dpth [m] Journal ccntricity [m] h Film thicknss [m] L Axial lngth [m] K ij Dirct stiffnss cofficints [N/m] i,j=x,y M ij Addd mass cofficint [Kg] i,j=x,y m Mass flow rats [kg/s] x, z N Numbr of flow rgions in flow domain Nm Numbr of lmnts (FEM msh) n p Numbr of Nods pr lmnt (FEM msh) P Prssur [Pa] P X,P Y First-ordr prssur filds [Pa] i Imaginary numbr ( 1 ) R Journal radius [m] R A V A cρ/μ. Axial flow Rynolds numbr R * ρωc 2 η /12μ. Modifid squz film Rynolds numbr s Zroth ordr prssur axial gradint [Pa/m] t Tim [s] V x, V z Bulk flow vlocitis [m/s] X,Y,Z Inrtial coordinat systm [m] x,z Circumfrntial and axial coordinats [m] Δ Displacmnt amplitud [m] ε Eccntricity ratio (/c) μ Absolut viscosity [Pa.s] ν Kinmatic viscosity [m 2 /s] Ω Rotor rotational spd [rad/s] ω Rotor whirling frquncy [rad/s] ρ Oil dnsity [kg/m 3 ] θ Angular coordinat [dg] ψ Intrpolation functions (FEM) Subscripts Zroth ordr solution xp. Drivd from xprimnts g groov N Last annular cavity sction modl Drivd from prdictions -th annular cavity sction σ X,Y coordinats TRC- SEAL-1-8 7

8 I Introduction Oil sals ar usd in cntrifugal comprssors to rduc lakag of th procss gas into th support oil lubricatd barings as wll as into ambint [1]. An oil sal, shown in Fig. 1, compriss of a floating ring and lastic support that, undr crtain opration conditions, may lock up and act as a hydrodynamic journal baring [3]. Ths sals ar known as potntial sourcs of instability du to th gnration of larg cross-coupld stiffnsss [1, 4]. A common practic to minimiz th dstabilizing ffct of oil sals is to machin circumfrntial groovs to isolat and divid th sal land into shortr lngth lands, thus rducing th hydrodynamic fluid film forcs. To dat, thr ar major discrpancis btwn prdictd and xprimntal forc cofficints obtaind for groovd sals. Exprimntal rsults dtaild in Rfs. [1, 6, 7] show that incorporating circumfrntial groovs do rduc cross-coupld forc cofficints but to a lssr xtnt than prdictions othrwis indicat. Furthrmor, xprimntal addd mass cofficints ar found to b vry larg and not modld in common prdictiv tools. Figur 1 Oil sal ring assmbly, Rf. [2] A prior TRC rport [8] introducs a novl analysis to obtain th forc cofficints of cntrd groovd oil sals. Th bulk-flow formulation dtrmins sal forc cofficints from an amplitud prturbation analysis about a cntrd position. Extnding th original bulk flow analysis, this rport dtails th implmntation of a finit lmnt mthod to obtain th oil sal forc cofficint for journal (rotor) off-cntrd conditions. TRC- SEAL-1-8 8

9 Th following rviw prsnts th analytical and xprimntal work on laminar-flow groovd oil sals. II Litratur Rviw Th archival litratur for laminar flow oil groovd sals is rathr scant [1,9]. Most of th work has bn conductd to analyz turbulnt flow annular prssur sals typical in high prssur comprssors and pumps [9]. This litratur rviw includs th currnt availabl computational tools and xprimntal rsults valid for laminar-flow groovd oil sals, i.. axial flow Rynolds numbr (R A =V a *c/ν) 2, with V a as th axial man flow vlocity, c as th film claranc, and ν is th kinmatic viscosity. Analytical work Smanat and San Andrés [3] prsnt an isoviscous bulk flow analysis to prdict forc cofficints of a groovd oil sal oprating undr ithr laminar or turbulnt flow rgims and not accounting for fluid inrtia trms. Th analysis includs surfac roughnss, fluid inrtia and viscous ffcts at th sal inlt plan, and variabl claranc (i.. taprd sal). Th bulk flow quations ar solvd for small amplitud motions using a prturbation analysis and an itrativ finit diffrnc schm. Th forc cofficints ar prsntd in trms of th ring ccntricity ratio for thr cass: singl land, two lands and thr lands. Th numrical rsults indicat that both th cross-coupld stiffnsss and dirct damping cofficints ar substantially rducd. Ths findings imply that th groovs ffctivly sparat th sal lands, thus rducing th sal forc cofficints. Howvr, th whirl frquncy ratio (WFR), a stability indicator, rmains rlativly constant, and thus thr is not significant improvmnt of th sal rotordynamic stability charactristics. Rgarding th ntranc ffcts du to viscous and inrtia ffcts, th prdictions indicat that such ffcts ar of importanc only for high prssur applications. Bahti and Kirk [4] analyz th dynamic forcd rspons of groovd oil sals including thrmal ffcts but still nglcting fluid inrtia. Th coupld prssur and tmpratur transport quations, dvlopd in trms of a prturbation analysis for small amplitud journal motions, ar modld with a finit lmnt schm. Th study includs arc and squar groov gomtris locatd at th sal mid-land lngth. Th numrical TRC- SEAL-1-8 9

10 rsults show a rduction of 4% in th dirct damping and cross-coupld stiffnss cofficints of an oil sal whn including a squar groov with a groov dpth to claranc ratio (d g /c) qual to 6. On th othr hand, whn using a dpr groov (d g /c = 15) th forc cofficints ar rducd by a factor of 4 by adding a singl groov, and by 1 whn adding two groovs. Thus, for th lattr cas, th rsults indicat that an innr groov ffctivly isolats th prssur distribution of contiguous film lands. San Andrés and Dlgado [8] prsnt a bulk-flow fluid inrtia analysis to prdict th forc cofficints of groovd oils sals and squz film damprs oprating at thir cntrd position. A prturbation analysis yilds zroth and first ordr flow quations at ach individual flow rgion (land and groovs) of constant claranc ( c ). Th authors prsnt an analytical solution for circular cntrd journal orbits of small amplitud. Th modl rlis on dfining an ffctiv groov dpth that rprsnts th actual uppr boundary of th dynamic (squz) flow. A paramtric study using multipl ffctiv groov dpths (d η ) show that thr is an xcllnt corrlation btwn th prdictd and xprimntal forc cofficints of a groovd oil sal [6] for a narrow rang of ffctiv groov dpths. Spcifically, for a short lngth and shallow groov at th mid-land of th sal, prdictions of addd mass, cross-coupld stiffnss, and damping cofficints corrlat bst with xprimntal data [6] whn using a fraction (5% or lss) of th actual groov dpth. Most importantly, th prdictions dmonstrat that th innr groov in th oil sal dos not isolat th adjacnt film lands. Exprimntal work Childs t al. [1] idntify xprimntally th rotordynamic forc cofficints and masur th lakag of smooth land and groovd oil sals oprating undr laminar flow conditions. Th authors aim to quantify th influnc of innr-land groovs on th rotordynamic cofficints of oil sals and to valuat th accuracy of xisting prdictiv modls. Rf. [1] includs a dtaild dscription of smooth and groovd oil sals and thir oprating faturs, and a comprhnsiv litratur rviw of prior work conductd on oil sals oprating in th laminar flow rgim. Particularly, th authors indicat that, prior to thir publication, th only publishd xprimntal work on laminar-flow oil sals was conductd by Kanko [1] on smooth land sals. In Rf. [1], static and dynamic forc cofficints ar idntifid for a smooth sal, a 1-groov sal, and a 3-groov sal (with d g /c = 6). Th xprimntal forc cofficints for th smooth sal corrlat wll with TRC- SEAL-1-8 1

11 prdictions from an annular sal modl givn in Rf. [11], xcpt for th addd mass cofficint that th analysis undrstimats by a factor of about 1. Th authors not that th larg volum of th oil supply cntral groov may xplain th larg discrpancy. Howvr, prssur masurmnts at both th fd groov and xit cavity show no dynamic prssur oscillations. Th xprimntal forc cofficints for th groovd oil sals ar largly undrstimatd by th modl of Smanat and San Andrés [3]. Th tst rsults suggst that, contrary to th accptd assumptions, innr-land groovs ar not dp nough to isolat th hydrodynamic prssurs from contiguous sal film lands. Childs t al. [6] prsnt xprimntal rsults vidncing th ffct of groov dpth on th dynamic forc rspons and lakag of a tst oil sal. Rotordynamic forc cofficints ar idntifid for four sals, including a smooth land sal and thr sals with a singl groov at th middl of th land. Th thr groovd sals prsnt diffrnt groov dpth to claranc ratios; d g /c = 5, 1, and 15. Th idntifid sal forc cofficints and lakag ar prsntd as a function of th sal journal ccntricity for thr spds and thr oil supply prssurs. Th tst rsults indicat that th sal forc cofficints dcras as th groov dpth incrass, xcpt for th addd mass cofficints. Howvr, prdictions basd on Rf. [3] largly undrstimat th groovd sal oil cross-coupld stiffnsss and dirct damping cofficints vn for th tst sal with th dpst groov. Thus, th xprimnts rval that, vn for a groov dpth to claranc ratio as larg as 15, a groov dos not compltly isolat th hydrodynamic prssurs of th two adjacnt sal lands (i.. for th cross-coupld stiffnss cofficints: K xy (2 lands) ¼ K xy (1 land), and th dirct damping cofficints: C xx (2 lands) ¼ C xx (1 land) as prdictions rportd in [3]. In addition, th xprimntal rsults also show rlativly larg addd mass cofficints that incras as th groov dpth incrass. Howvr, a prdiction of addd mass cofficint, basd on a classical lubrication modl [12], yilds only 2.8 kg (i.. 1 tims smallr than th xprimntal valu). On th othr hand, th prdictd addd mass cofficints obtaind with th novl modl in Rf.[8] show a rmarkabl corrlation with th tst data. III Analysis Th litratur rviw rvals th nd for an improvd modl that proprly prdicts th forc cofficints of a (laminar flow) groovd ring sal for ccntric journal opration. TRC- SEAL

12 This dficincy is addrssd by updating th bulk flow modl advancd in Rf. [8] for off-cntrd journal opration using a finit lmnt analysis. Th drivation of th bulk flow quation for th oil sal follows. Bulk flow formulation Th following bulk flow modl is an xtnsion of th original analysis in Rf. [8] to dtrmin th fluid forcs dvlopd in multi-groov annular sal cavitis. Th modl sts a rotating journal whirling with small amplitud motions about an off-cntrd (ccntric) position. As in Rf. [8], th modl applis to both smooth land and groovd oil sals oprating in th laminar flow rgim. Spcifically, th formulation is dtaild for th cas of annular cavitis with axially symmtric groov configurations, including a cntral fding groov, as shown in Fig. 2. This gomtry is slctd to allow dirct comparisons with th xprimntal data in Rf. [7]. Th multipl groov sal is dividd into sparat flow rgions of uniform claranc. In cas of a groov, th dpth is an ffctiv groov dpth (d η ), which diffrs from th actual physical groov dpth. This concpt is xplaind latr whil dscribing th boundary conditions for th FE modl solution. Th following drivation applis to ach individual flow rgion with constant claranc, and with a local coordinat systm whos origin is at th ntranc of th corrsponding groovd or ungroovd flow rgion. Figur 3 dpicts th journal and th coordinat systm usd in th analysis for small journal motions about an off-cntrd position. Oil supply =1,2...Ν Ν 1 2 I II 3 III 4 IV n N n+1 z 1 z II z III z IV z N Axial flow Figur 2 Schmatic viw of groovd annular cavity dividd into flow rgions TRC- SEAL

13 x= θr Δ= journal displacmnt ω= whirl frquncy Ω= rotational spd R h xo yo Δ ω Ω Y journal X Figur 3 Viw of rotating and whirling journal and coordinat systm for bulk-flow analysis Within ach individual flow rgion th mass flow rats in th circumfrntial (x) and axial (z) dirctions ar: m x = ρ h Vx ; m z = ρ hvz = I, II,... N (1) whr h is th film thicknss, ( V, V x z ) ar bulk-flow vlocitis in ach flow rgion, and ρ is th lubricant dnsity. Th bulk-flow continuity and momnt transport quations without fluid advction trms ar [13]: ( m x ) + ( m z ) + ( ρ h ) = (2) x z t ( m x ) P μ ΩR h = kx Vx + x h 2 t (3) ( m z ) P V z h = kz μ + z h t ; = I, II,... N (4) TRC- SEAL

14 with k x =k z =12 for laminar flow. Abov, μ is th lubricant viscosity and P is th prssur at ach flow rgion. Equations (3) and (4) ar rwrittn as: m x ( m x ) ρ ρ ρ = + k μ x k μ t 3 2 h P h hωr x x ( m z ) 3 2 ρh P ρh m z = ; = I, II,... N k μ z k μ t z z 2 ; (5) Diffrntiating with rspct to x, and z with rspct to z, adding both mx m quations, and disrgarding scond ordr trms yilds a Rynolds-lik quation for th film prssur of an incomprssibl fluid [11] P P x x z z t x t = I, II,... N h + h = 12 μ h + 6μRΩ h + ρh h 2 ( ) ( ) ( ) ( ) (6) Th journal dscribs motions of small amplitud (Δ X, Δ Y )<< c η and frquncy ω about an static position ( Xo, Yo ). Th film thicknss (h ) quals [13] { } h = h + Δ cos ( θ ) + Δ sin ( θ ) = h + Δ h ; i = 1; σ = X, Y iωt iωt X Y σ σ with h c X cos θ Y sin θ c h = η + ( ) + ( ) = η + σ σ; hx cos ( θ ), hy sin ( θ ) and c ( c + d ) η η = I, II,... N (7) = = (8) = is th ffctiv claranc ovr th groovd flow rgions. Not that d η is th ffctiv groov dpth, and c η = c in an ungroovd (sal land) flow rgion. Th prssur is xprssd as a suprposition of a zroth ordr fild (P o ) and first ordr (dynamic) filds ( P, P ) X Y iωt P = P + Δ σ P σ ; = I, II,... N (9) Substitution of Eqs. (7) and (9), into Eq. (6) givs th zro th ordr quations for th quilibrium prssur TRC- SEAL

15 P P h h R h x x z z x ( ) = 6μ Ω = I, II,... N (1) and th first ordr quations for journal dynamic displacmnts along th X and Y dirctions P P dh { * } x x z z dx P 2 P 2-3h h 3 h h ; σ σ x x z z σ = X, Y; = I, II,... N 3 σ 3 σ σ h + h = 12iμω 1 + ir hσ + 6μΩR (11) 2 ρω cη whr R* = is a local squz film Rynolds numbr. Onc Eq. (1) and Eq. 12 μ (11) ar solvd, th fluid film raction forcs at a static journal position ( Xo, Yo ) ar N L σ σ θ ; σ= XY, ; = 1 θ F h P R d dz = (12) Th sal forc cofficints ar also obtaind by intgrating th dynamic prssur filds ovr th flow domain [13], K N L 2 σβ - + h P R d dz, XY, hx h ω M σβ Cσβ = σ β σ β= Y = 1 θ ( ) ( ) i ω θ ; ; =cos θ, =sin θ = I, II,... N (13) Finit lmnt solution Th procdur implmntd for solving th Rynolds-lik Eq. (11) with th finit lmnt mthod (FEM) is similar to that in Rfs. [13,14]. Th implmntation of th currnt dvlopmnt into a FORTRAN cod follows from modifying an xisting cod that prdicts forc cofficint for smooth sals without including inrtia ffcts [3]. Without loss of gnrality, th solution is prsntd for a symmtric oil sal with an inlt groov and a singl mid-land groov similar to th sal tstd in Rf. [7]. A similar solution can b applid to multi-groov sals. TRC- SEAL

16 Following th discrtization of th domain into lmnts ( Ω ), as shown in Fig. 4, th static and dynamic prssur filds ar rprsntd as th linar combination of nodal valus P within ach lmnt as i n p n p P = Ψi P, P = i Ψi Pσ ; i σ = X, Y i= 1 i= 1 σ (11) whr ψ ar bilinar intrpolation functions. Th quation to b solvd for ach lmnt follows from th rprsntation of th fluid diffrntial quations (1, 11) in its variational or wak form [13] using th intrpolation functions as wight functions. ` Outlt plan h Ω Γ P=P xit Groovs Inlt Outlt x=θr z z θ Inlt plan 2π P=P supply Figur 4 Coordinat systm and sampl msh for oil sal FEM computational cod Th wak variational form of Eq. (1) for th zroth ordr prssur yilds n p k ij P q i f = + i (12) j=1 with j TRC- SEAL

17 k 3 h Ψ Ψ i j Ψ Ψ i j ij = + dx dz 12μ x x z z (13) Ω Ω R Ψi f = h dx dz (14) x i 2 Ω q i = Ψiqn dγ ; with Γ q n 3 h P h ΩR = + nx (15) 12μ n 2 Similarly for th first ordr prturbd prssur filds, P X and P Y, th st of quations for th nodal prssurs in a finit lmnt ar np np kp ij σ = f ; j σ S i σ P q ij j σi σ= X, Y j= 1 j= 1 (16) whr ΩR Ψ ρω f h h dx dz i h dx dz (17) 2 i 2 σ = i σ + Ψi ω σ i 2 x 12μ Ψ Ω Ω S 2 3h Ψ Ψ i j Ψ Ψ i j σ = + h dx dz ij σ 12μ x x z z (18) Ω q ; = Ψ q dγ σ i i n σ Γ 3 2 h P 3h P R σ Ω qn = h h n σ σ + σ 12μ n 12μ n 2 x (19) with n as th normal vctor to th boundary Γ of an lmnt. Not that Eq. (17) includs th tmporal fluid inrtia trm (scond trm in first intgral). Th intgrals in Eq. (13) through Eq. (19) ar valuatd numrically ovr a mastr lmnt ( ˆΩ ) with normalizd coordinats (isoparamtric lmnt). Rddy and Gartling [15] dtail th coordinat transformation and numrical intgration procdur using Gauss-Lgndr quadratur formulas. Equations (12) and (16), for ach lmnt of th flow domain, ar assmbld to form a linar systm of quations rprsntd as TRC- SEAL

18 [ k] { P } = { Qγ} + { Fγ} Global Global Global Global ; γ : [ ] { γ} = { γ} + { γ} + [ ] { } k P Q F S P ; γ : X, Y Global Global Global Global Global Global (2) whr [ ] Nm [ ] { } Nm, { },{ } Nm k = k Qγ = qγ Fγ = { fγ}, γ:,x,y. (21) Global Global Global = 1 = 1 = 1 Th rsulting global fluidity matrix [ k ] Global is symmtric and can b asily dcomposd into its uppr and lowr triangular form, i.. [ k ] [ ] [ ] [ ] [ ] T G L G U G = L G L G = (22) Thus, onc th L, L T and F matrics ar obtaind, and stating th prssur at th inlt and xit plans of th flow domain, a procss of back- and forward-substitutions rndrs th discrt zroth ordr prssur fild { P } and th zroth ordr prssur fild, th first ordr prssur fild,{ G. Using th rsults from th fluidity matrix P γ } G, ar obtaind. Th slction of th appropriat boundary conditions, basd on th physics of th problm, is ssntial for obtaining th forc cofficints in th groovd sal gomtry. Boundary Conditions Figur 5 shows th msh for ½ a flow domain on an oil sal with an inlt (supply) groov and an innr groov, and th nods of intrst to nforc boundary conditions. Both th zroth and first ordr filds ar priodic in th circumfrntial dirction, P ( θ, z) = P ( θ + 2 π, z) ; γ=, X, Y (23) γ γ Th fluid prssur must b gratr than th lubricant cavitation prssur (P cav ). In th currnt analysis, th Gumbl condition of oil cavitation is nforcd for th zroth and first ordr prssur filds. Not that Eqs. (15) and (19) automatically satisfy th flow continuity at th boundary btwn a smooth land and groov, for xampl. Hnc, no spcial considrations in rgard to flow matching ar rquird. Othr boundary conditions ar: TRC- SEAL

19 Zroth Ordr Prssur Fild A) Constant static prssur at inlt plan (z= ). This condition is st to th lowr nods of lmnts along th inlt plan. P = P (24) z= L supply B) Constant static prssur at xit plan (z= L). This condition is st to th uppr nods of lmnts along th xit plan. P = P (25) z= xit A Outlt plan P=P xit Groov z B x Inlt plan P γ (θ,z)=p γ (θ+2π,z) γ =,x,y Groov P=P supply Figur 5 FEM msh dpicting nods of intrst for implmntation of boundary conditions First Ordr Prssur Filds A) Null dynamic prssur at xit plan (z= L). Whil th static prssur is constant, it is assumd that thr is not gnration of dynamic prssur at th xit (discharg) plan. This condition is st to th uppr nods of lmnts along th xit plan. P γ z= L = (26) B) At th inlt plan (fd or supply cntral groov) th axial flow inducd by th dynamic (fluid squzing) motions is st to zro du to axial symmtry. q = (27) z z= TRC- SEAL

20 This boundary condition implis that th prturbd axial flow dos not cross th middl plan and that thr is a non-zro dynamic prssur fild at this plan. This is a Numann typ B.C. implmntd by trating th nods at inlt plan as intrnal nods. Effctiv groov dpth As advancd in Rf. [8], th laminar flow pattrn at th groov is charactrizd by a rcirculation rgion and a thru flow rgion. Ths rgions ar dividd by a stram lin that is considrd to act as a physical boundary. Figur 5 shows a rprsntation of th stramlins pattrn for a prssur drivn flow through a (symmtric) annular cavity with a supply groov and two mid-land groovs. Th figur also dpicts a clos-up of CFD simulations of th prssur drivn flow at th mid-land groov for two groov dpths (1c and 15c). In this configuration th flow pattrn at th supply and mid-land groovs is charactrizd by two rgions, a rcirculation rgion and a thru flow rgion. Furthrmor, th dividing stramlins for th 1c and 15c groov dpths prsnt a similar pntration dpth. In th proposd analysis, th stramlins dividing th two flow rgions ar assumd to act as physical boundaris dlimiting th domain for th flow inducd du to dynamic (fluid squzing) journal motions. Thus, th fluid film claranc at th groov is rprsntd in trms of an ffctiv claranc c η = (d η + c), with d η as an ffctiv groov dpth and c as th claranc of th ungroovd portion or land. TRC- SEAL-1-8 2

21 a) mid-land groov oil supply, P s >P a fd plnum groov P d P d < P s P s - P d > P d :discharg prssur y z b) 1c 15c 2 mm Figur 6 a) Schmatic viw of stramlins in axially symmtric groovd annular cavity (ΔP= P s -P d ). b) CFD simulation of prssur drivn stramlins across a 1c and 15c circumfrntial mid-land groov in an oil sal tstd in Rf. [7]. (c= 86 mm, ω=1 RPM, D= 117 mm) Excl program intrfac Th MS Excl usr intrfac is namd XLFEGLOSal (Excl Finit lmnt groovd laminar oil sal), as dpictd in Fig. 7. Th usr inputs includ: -Fluid proprtis: Dnsity and viscosity -Oprating conditions: Inlt and outlt prssurs, static journal ccntricity. -Gomtry: Rotor diamtr, claranc, groov dpth, numbr of groovs, inlt and outlt land lngth, intr-groov lngth, groov lngth. Th mid-land groov dpth is st to th actual physical valu for groov dpths lss than (6c), considring that th ffctiv groov dpth and actual groov dpth ar rlativly similar. For dpr short mid-land groovs, as thos found in oil sals, th prssur drivn flow stramlins rmain TRC- SEAL

22 rlativly constant rgardlss of th actual dpth, as shown in Fig. 6. For this cas, a constant ffctiv groov dpth of d η =6c is usd. Th cod outputs dirct and cross-coupld forc cofficints, sal ractions forcs and lakag as a function of shaft spd journal ccntricity. Th program can handl multipl groov configurations. Figur 7 Graphical usr intrfac for XLFEGLOSal cod (SI units) IV Modl Prdictions and Validation to Tst Data This sction prsnts comparisons of xprimntal and prdictd damping, stiffnss and mass cofficints for th oil ring sal tstd by Graviss [7]. Figur 8 dpicts th actual dimnsions of th tst sal, and Tabl 1 lists th sal physical dimnsions, fluid proprtis, oprating conditions, and numbr of lmnts of th FE msh. Figur 9 shows th actual journal locus obtaind from four masurd journal ccntricitis. Considring that th xtrnal load is along th Y dirction, th closnss of th journal cntr to th Y axis (i.. small journal attitud angl) clarly indicats th prsnc of oil cavitation, in particular for th largst ccntricity ratios. As in th tsts, th analysis rports rsults for half of th axially symmtric groovd sal configuration. Publishd tst data and prdictions follow for a smooth land oil sal TRC- SEAL

23 and for th sam sal land with an innr groov of 15c dpth. Following th rsults of th paramtric study in Rf. [8], th ffctiv claranc for th inlt oil supply (cntral) groov is st to 1c. This ffctiv claranc prsnts th bst corrlation with all th forc cofficints obtaind for th smooth sal (i.. no mid-land groov). For th oil sal with a mid-land groov, th inlt oil supply groov ffctiv groov dpth rmains constant and th innr land groov ffctiv claranc is st to 7c. Oil supply 13 mm 4 mm mm 11.4 mm 11.4 mm Mid-land Sal lngth Discharg plnum 136*c 76*c 76*c Journal c (-15)*c Buffr sal Figur 8 Configuration of paralll oil sals tstd in Rf. [6] Tabl 1 Oil sal configuration, oprating conditions fluid proprtis and numbr of lmnts for FE msh Dimnsions Journal Diamtr 117 mm Sal lngth mm Claranc 85.9 μm Inlt groov lngth 17 mm Inlt groov dpth 136c Innr land groov lngth 2 mm Innr land groov dpth -15c Paramtrs Shaft spd 4-1 rpm Oil dnsity 85 kg/m 3 Static ccntricity -.7 Oil viscosity (smooth sal).16 Pa.s (54 C) Oil viscosity (groovd sal).19 Pa.s (49 C) Supply prssur 7 bar FEM msh (lmnts) Circumfrnc 6 Sal land 12 Inlt groov 12 Innr land groov 6 TRC- SEAL

24 .25 Load εy smooth sal Groovd sal -1. ε x Figur 9 Masurd journal cntrlin locus for smooth and groovd sal (c g =15c). (7 bar, 1 rpm) [7] Figur 1 shows th sal raction forcs vrsus th static journal ccntricity. Prdictions and xprimntal rsults prsnt good corrlation for journal ccntricitis up to /c =.5 for th groovd and smooth sals. For th largst journal ccntricity (/c=.7), prdictions ar within 2 % of th xprimntal rsults for th smooth sal. On th othr hand, th raction forc of th groovd sal is undrprdictd by a factor of 2 for th largst journal ccntricity. For th largst journal ccntricitis th oil tmpratur is xpctd to significantly incras du to th small film thicknss (i.. larg shar forcs and powr loss). Thus, th actual sal claranc and oil proprtis for th largst ccntricity may diffr significantly from th nominal valus and hav a larg uncrtainty. Howvr, Rf. [7] dos not dtail information on th xit tmpratur or masurmnts of hot clarancs (immdiatly aftr tsting). Prsntly, prdictions ar compard with tst data for th low to mid-rang journal ccntricitis (i.. ε=,.3,.5) only. TRC- SEAL

25 Forc [N] Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts Static journal ccntricity ratio (/c) Figur 1 Sal raction forcs. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov (c η = 7c) Figurs 11 and 12 dpict th dirct and crossd-coupld stiffnss cofficints (K ij ; i,j=x,y) vrsus th oprating journal ccntricity, rspctivly. Th prdictions corrlat wll with th tst data for th lowr journal ccntricity ratios (ε=,.3). For th 5 % ccntricity ratio thr ar discrpancis. Th diffrncs can b attributd in part to th lack of knowldg in actual claranc and oil xit tmpratur, not rportd in Rf. [7]. TRC- SEAL

26 Stiffnss [MN/m] K xx Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts Static journal ccntricity ratio (/c) 25 Stiffnss [MN/m] Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts K yy Static journal ccntricity ratio (/c) Figur 11 Dirct stiffnss cofficint (K ii ) vrsus ccntricity. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov (c η = 7c) TRC- SEAL

27 15 Stiffnss [MN/m] K xy Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts Static journal ccntricity ratio (/c) 1 Stiffnss [MN/m] K yx Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts Static journal ccntricity ratio (/c) Figur 12 Cross-coupld stiffnss cofficints (K ij ) vrsus ccntricity. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov (c η = 7c) Figur 13 show th cross-coupld stiffnss cofficints vrsus rotor spd for two journal ccntricitis (/c=,.3). Th prdictions ar in good corrlation with th xprimntal rsults. In particular, th modl adquatly prdicts th rduction of th cross-coupld cofficints aftr adding th innr groov to th smooth land sal. TRC- SEAL

28 6 Stiffnss [MN/m] Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts K xy 1 ε o = Rotor spd (RPM) Stiffnss [MN/m] ε o =.3 Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts K xy Rotor spd (RPM) Figur 13 Cross-coupld stiffnss cofficints (K xy ) vrsus shaft spd at two journal ccntricitis (,.3). Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov (c η = 7c) Figurs 14 and 15 prsnt th dirct and cross-coupld damping cofficints (C ij ; i,j=x,y) vrsus static journal ccntricity ratio, rspctivly. Th dirct damping cofficints (C xx, C yy ) show xcllnt corrlation for th all th ccntricity ratios, xcpt for th C xx cofficint of th smooth sal that is 2% undrprdictd for /c=.5. Th cross-coupld cofficints ar much smallr than th dirct damping cofficints and prsnt modrat to good corrlation for th diffrnt tst journal ccntricitis. TRC- SEAL

29 Damping [kn.s/m] C xx Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts Static journal ccntricity ratio (/c) Damping [kn.s/m] C yy Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts Static journal ccntricity ratio (/c) Figur 14 Dirct damping cofficints (C ii ) vrsus ccntricity. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov (c η = 7c) TRC- SEAL

30 Damping [kn.s/m] C xy Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts Static journal ccntricity ratio (/c) Damping [kn.s/m] C yx Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts Static journal ccntricity ratio (/c) Figur 15 Cross-coupld damping cofficints (C ij ) vrsus ccntricity. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov (c η = 7c) Figur 16 dpicts th dirct addd mass cofficints vrsus th static journal ccntricity ratio. Prdictd and xprimntal cross-coupld addd mass cofficints (M xy, M yx ) ar narly null and not prsntd. Th dirct addd mass cofficints (M xx, M yy ) prsnt good corrlation with th xprimntal data. In particular, th analysis prdicts a largr addd mass cofficint for th groovd oil sal as th xprimnts also rval. Not TRC- SEAL-1-8 3

31 that th prdictd addd mass cofficint is narly constant for all th tst journal ccntricitis. 4 Addd Mass [kg] M xx Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts Classical thory [12] Static journal ccntricity ratio (/c) Addd Mass [kg] M yy Classical thory [12] Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts Static journal ccntricity ratio (/c) Figur 16 Addd Mass cofficint (M xx, M yy ) vrsus ccntricity. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov using (c η = 7c) Figur 17 dpicts th sal lakag vrsus static journal ccntricity for opration at 1, rpm and 7 bar fd prssur. Thr is good corrlation btwn xprimnts and prdictions with a variation of lss than ~15 % for both sals. Th good corrlation indicats that th slctd boundary conditions rprsnt wll th physical boundaris of TRC- SEAL

32 th through flow. Not that th xprimnts and prdictions show that th smooth sal laks mor than th groovd sal bcaus its ffctiv viscosity (land claranc) is slightly lowr (largr) du to largr powr losss inducing a tmpratur ris. Lakag [kg/s] Smooth sal- Prdictions Smooth sal- Exprimnts Groovd sal- Prdictions Groovd sal- Exprimnts Static journal ccntricity ratio (/c) Figur 17 Sal lakag vrsus ccntricity. Exprimnts for smooth sal and sal with innr land groov (c g = 15c), 1 rpm, 7 bar [7]. Prdictions for smooth sal and sal with innr land groov using (c η = 7c) V Conclusions and Rcommndations This rport prsnts a bulk-flow formulation to obtain fluid film forcs dvlopd in groovd oil sals and dtails th implmntation of a finit lmnt mthod to obtain th forc cofficint for journal off-cntrd opration. Th currnt analysis xtnds an original bulk-flow modl [8] dvlopd for small amplitud journal motions about a cntrd position. Th prsnt analysis also prdicts addd mass cofficints, largly ignord in prvious analyss of laminar-flow oil sals. Th forc cofficints, lakag and raction forcs of a smooth and groovd oil sal ar prdictd and compard to xprimntal rsults rportd in Rf. [7]. Th tst groovd oil sal includs a rctangular cntral groov locatd at th sal mid-land plan with a TRC- SEAL

33 dpth of 15 tims th sal claranc (c=85.9 μm). Th prdictd paramtrs ar compard to xprimntal rsults for four journal ccntricitis (/c=,.3,.5,.7) at 1, rpm and with a 7 bar oil fd prssur. Prdictd and xprimntal forc cofficints prsnt good corrlation for th dirct forc cofficints for th lowr journal ccntricitis (/c=,.3) and modrat to good corrlation for /c=.5. Th cross-coupld stiffnss cofficints ar also accuratly prdictd for th lowr journal ccntricitis. In particular, th currnt modl accuratly prdicts th rduction of th dirct stiffnss, dirct damping, and cross-coupld stiffnss cofficints whn adding a circumfrntial groov to th sal land. Th addd mass cofficints for both sals ar also prdictd accuratly (within 2 %). Furthrmor, th analysis and xprimntal rsults indicat that a groovd sal shows largr dirct addd mass cofficint than a smooth sal. For journal ccntricity ratios (ε) up to 7% thr ar discrpancis btwn th xprimntal rsults and (currnt) prdictions for som of th forc cofficints. Ths discrpancis ar attributd to (unknown) changs in sal claranc and oil viscosity inducd by thrmal ffcts whn oprating at larg ccntricitis. Nvrthlss, th tst data rportd in Rf. [7] dos not offr nough dtails on oprating conditions and th variation of th lubricant proprtis and sal claranc. Thrfor, th prdictions ar compard with xprimntal rsults only for th low to mid-rang ccntricitis (i.. ε=,.3,.5). Th currnt analysis rprsnts a significant improvmnt ovr th currnt prdictiv tools availabl to analyz groovd oil sals. Mor importantly, th improvd prdictions of th groovd oil sal forc cofficints can lad to a mor accurat stimation of critical spds and stability thrsholds in cntrifugal comprssors. TRC- SEAL

34 VI Rfrncs [1] Childs, D. W. Rodriguz, L. E., Cullotta, V., Al-Ghasm, A., and Graviss, M., 26, Rotordynamic-Cofficints and Static (Equilibrium Loci and Lakag) Charactristics for Short, Laminar-Flow Annular Sals, J. Tribol., 128(2), pp [2] Kirk, R., 1986, Oil Sal Dynamic Considrations for Analysis of Cntrifugal Comprssors, Proc. 15 th Turbomachinry Symposium, Houston, TX, pp [3] Smanat, J., and San Andrés, L., 1993, Analysis of Multi-Land High Prssur Oil Sals, STLE Tribol. Trans., 36(4), pp [4] Bahti, S., and Kirk, R., 1995, Finit Elmnt Thrmo-Hydrodynamic Solution of Floating Ring Sals for High Prssur Comprssors Using th Finit-Elmnt Mthod, STLE Tribol. Trans., 38, pp [5] Allair, P. E., and Kocur, J. A. Jr., 1985, Oil Sal Effcts and Subsynchronous Vibrations in High-Spd Comprssors, NASA Conf. Publ., pp [6] Childs, D. W., Graviss, M., and Rodriguz, L. E., 27, Th Influnc of Groov Siz on th Static and Rotordynamic Charactristics of Short, Laminar-Flow Annular Sals, ASME J. Tribol, 129(2), [7] Graviss, M., 25, Th Influnc of a Cntral Groov on Static and Dynamic Charactristics of an Annular Liquid Sal with Laminar Flow, M.S. Thsis, Txas A&M Univ., Collg Station, TX. [8] San Andrés, L., and Dlgado, A., 27, Paramtr Idntification of an End Sald SFD Part II: Improvd Prdictions of Addd Mass Cofficints for Groovd SFDs and Oil Sals, TRC rport, TRC-SFD-2-6, Jun. [9] Childs, D., 1993, Turbomachinry Rotordynamics, John Wily & Sons, Inc., NY, Chap. 3. [1] Kanko, S., Hori, Y., and Tanaka, M., 1984, Static and Dynamic Charactristics of Annular Plain Sals, Proc. Third IMchE Int. Conf. Vibrations in Rotating Machinry, York, England, Sp., pp [11] Zirklback, N., and San Andrés, L., 1996, Bulk-Flow Modl for th Transition to Turbulnc Rgim in Annular Sals, STLE Tribol. Trans., 39(4), pp [12] Rinhardt, F., and Lund, J. W., 1975, Th Influnc of Fluid Inrtia on th Dynamic Proprtis of Journal Barings, ASME J. Lubr. Tchnol., 97(1), pp [13] San Andrés, L., Modrn Hydrodynamic Lubrication Thory, Tribology Group, Txas A&M Univrsity, [accssd April 28] [14] San Andrés, L., 27, Modrn Lubrication Nots # 7-Analysis of Finit Lngth Barings: Raction Forcs and Dynamic Forc Cofficints Including Tmporal Fluid Inrtia Effcts, Intrnal Communication, Tribology Group [15] Rddy J. N., Gartling, D. K., 21, Th Finit Elmnt Mthod in Hat transfr and Fluid Dynamics, CRC Prss, Florida, Chap. 2. TRC- SEAL