Fluid-induced rotordynamic forces produced by the uid in an annular seal or in the leakage

Transcription

1 Flid Flow Eqations For Rotordynamic Flows In Seals And Leakage Paths C.E. Brennen R.V.Uy California Institte of Technology, Pasadena, California 95 ABSTRACT Flid-indced rotordynamic forces prodced by the id in an annlar seal or in the leakage passage srronding the shrod of a pmp or trbine, are known to contribte sbstantially to the potential excitation forces acting on the rotor. In this paper we explore some of the important featres of the eqations governing blk-ow models of these ows. This in trn sggests methods which might be sed to solve these blk-ow eqations in circmstances where the linearized soltions (sch as those of Childs 987, 989) will no longer be accrate. An example of a nmerical soltion is then presented. Keywords: Rotordynamic forces, trblent annlar seals, blk ow model.. BACKGROUND Over the last few years a sbstantial body of experimental data has been gathered on idindced rotordynamic forces (Brennen 994) generated in narrow, id-lled annli sch as occr in trblent annlar seals (for example, Childs and Dressman 98, Nordmann and Massmann 984) or in the leakage ows srronding the shroded impellers of pmps or trbines (for example, Ginzbrg et al. 994). For example in the context of shroded pmp impellers, the eects of eccentricity, clearance, Reynolds nmber, leakage path geometry, inlet swirl and seals (both at the discharge and the sction) have been examined (Ginzbrg et al. 99, 993, 994, Sivo et al. 995, Uy et al. 997). Moreover, methods of changing these rotordynamic characteristics, for example by the installation of anti-swirl vanes, have also been explored (Sivo et al. 995). This now sbstantial body of data involves many dierent geometric and id ow parameters and it is not always easy for the potential ser to nd his or her particlar choice of parameters within the database. To solve this problem and to allow for greater nderstanding of the nderlying id mechanics it is clearly valable to view this array of data in the context of an accrate analytical model and, if necessary, to tne the frictional and other parameters in the model to provide a reliable tool for the designer. The problem with this strategy is that the available analytical models have not yet shown themselves capable of accrate and reliable predictions. Perhaps the most promising approach has been the blk ow model developed by Childs (987, 989) and sbseqently sed by others (for example, by Ginzbrg 99). This model appears to give reasonable reslts in some cases and nreasonable, even bizarre reslts, in others. Nevertheless, it represents a coherent and rational starting point from which to begin. Before describing some of the inherent problems with this model we smmarize it briey.. BULKFLOW MODELS OF ROTORDYNAMIC FLOWS As presented by Childs (987, 989), the blk ow model depends on the fndamental assmption that the nsteady, three-dimensional, trblent ow in the annls can be accrately approximated in the following ways:

2 . The dimensions of the ow can be redced from three to two (a meridional coordinate and a circmferential coordinate) by assming that the velocity proles within the narrow passage are all self-similar so that the eqations of ow can be averaged over the gap withot nde error. This approximation is, of corse, commonly employed in the Reynolds lbrication eqations bt also has its limitations, especially when frictional stresses are evalated based on the gap-averaged velocities. For example, if the ow separates within the passage so that the velocity close to one wall is in a dierent direction than the gap-averaged velocity, then sing the gap-averaged velocity to evalate the shear stress on that wall will lead to stresses and forces which are qite incorrect. In many, ni-directional lbrication problems this is not a serios concern. Bt in nsteady, mlti-directional ows (sch as can occr in a pmp leakage ow) this cold represent a serios limitation. To examine the potential for sch errors, Sivo et al. (994) sed a laser doppler velocimeter (LDV) to measre the velocity prole of a leakage ow experimentally. Flow reversal close to the rotor shrod was, in fact, observed and was in agreement with three-dimensional comptations performed by Baskharone and Hensel (993). The recirclation occrred at dierent locations in the leakage path for dierent conditions, and seemed to diminish at higher whirl ratios. In some cases, the recirclation regions were observed arond the entire impeller. Sch ow reversals cold lead to a serios error in the blk ow approach.. The Reynolds nmbers of most of these ows are very high so that the ow is trblent. This means that the blk ow model reqires expressions which relate the trblent shear stresses to the gap- averaged velocities. In the crrent form of the blk ow model, the shear stresses on the rotor and the stator are calclated sing friction coecients based on the work of Yamada (96). These are dened by: m h = n () where is the gap-averaged velocity and the m and n are denoted by ms and ns for the stator and mr and nr for the rotor. These expressions are taken from the work of Hirs (973) who does, however, recommend that the coecients m and n be \tted to individal experiments." The frictional coecients are dependent on six physical parameters, inclding the crvatre of the srface, inertial eects, and roghness. Ths, the coecients may not flly accont for the crvatre of the rotor in a particlar annls geometry or the inertial eects de to the crved path of the blk ow. Nor will the roghness parameters be easy to gage. Given the ease with which the frictional factors can be altered in the comptational model, it is reasonable to consider tting them so as to match the experimental data base. Of corse, the se of the above expressions for the trblent shear stresses is sbject to an even more general criticism. They are correlations for steady trblent ows based, primarily, on experimental observations of steady ows. In contrast, the rotordynamic ows of concern here are fndamentally nsteady. The problem is that we know very little abot trblent ows which are nsteady in the sense that the ow is being externally excited. Clearly, at present, this isse can only be resolved by carefl comparison of the experimental and model reslts. 3. Childs treats the rotordynamic ow as a linear pertrbation on the mean ow in the annls. While this may be an accrate assmption for very small eccentricities, there is crrently no way to know at what eccentricity this linearization begins to lose accracy. 4. Thogh the basic eqations appear to be accrate, there is mch more dobt abot the bondary conditions employed by Childs at the inlet to and discharge from the annls. For example, Childs deploys a constant pressre condition as well as a niform swirl velocity condition at inlet. Perhaps the rst shold be a constant total head condition? Moreover, it is assmed that the hydralic loss throgh the orices at inlet and/or discharge are related only to the meridional velocity (or ow rate) and are independent of the swirl velocity. This may not be accrate. So, while the blk ow model proposed by Childs model was a major step forward, there remain many qestions which reqire resoltion before a reliable predictive tool is available.

3 Figre : Sketch of id lled annls between a rotor and a stator. 3. THE BULK FLOW MODEL EQUATIONS Black and his co-workers (Black 969, Black and Jensen 97) were the rst to attempt to identify and model the rotordynamics of trblent annlar seals. Blk ow models (similar to those of Reynolds lbrication eqations) were sed. Several deciencies in this early work cased Childs (983a, 983b) to pblish a revised version of the blk ow model for trblent annlar seals (see also Childs and Dressman 98, 985, Childs and Kim 985, Childs and Scharrer 986) and, later, to extend this model (Childs 987, 989) to examine the rotordynamic characteristics of discharge-tosction leakage ows arond shroded centrifgal pmp impellers. A general geometry is sketched in gre, and is described by coordinates of the meridian of the gap as given by Z(s) and R(s), <s<s, where the coordinate, s, is measred along that meridian. The clearance is denoted by H(s; ;t) where the mean, non-whirling clearance is given by H(s). The eqations governing the blk ow are averaged over the gap. This leads to a continity eqation of the form H t + s (H s)+ R (H )+ H s R = () R s where s and are gap-averaged velocities in the s and directions. The meridional and circmferential momentm eqations are R p s = Ss H + Rs H R p = S H + R H + + t R R s + s t + R s + s s s + s s + s R These are the eqations sed by Childs (987, 989). Note that they inclde not only the viscos terms commonly inclded in Reynolds lbrication eqations (see for example Pinks and Sternlicht 96) bt also the inertial terms (see Fritz 97) which are necessary for the evalation of the rotordynamic coecients. To determine the trblent shear stresses, Childs employed the approach sed by Hirs (973). The trblent shear stresses, Ss and S, applied to the stator by the id in the s and directions are given by: Ss s = S = A S s s + m R s (3) (4) S + (H=) ms (5)

4 and the stresses, Rs and R, applied to the rotor by the id in the same directions: Rs s = ( R R) = A R s s +( R) m R + (H=) mr (6) where the constants A S, A R, m S and m R are chosen to t the available data on trblent shear stresses. Childs (983a) ses typical vales of these constants from simple pipe ow correlations: A S = A R =:664 ; m S = m R = :5 (7) Childs then proceeds to linearize the eqations by dividing the clearance, pressre, and velocities into mean components (sbscript ) that wold pertain in the absence of whirl, and small, linear pertrbations (sbscript ) de to an eccentricity,, rotating at the whirl freqency,!. He develops dierential eqations for the coecients which are fnctions of r only. The present paper will also focs on steady whirl with a constant eccentricity,, rotating at the whirl freqency,!, which is sperimposed on the shaft rotation whose speed is denoted by the radian freqency,. Conseqently, the id ow in a frame of reference rotating at! is steady and it is clearly appropriate to rewrite the eqations and to solve them in this rotating frame. Dening, therefore, a new anglar variable,, and a new anglar velocity,, in this rotating frame sch that it follows that the continity eqation can be written as =!t ; =!R (8) fh g + s frh sg = (9) and this is most easily satised by dening a stream fnction, (s; ) sch that s = RH ; = H s It follows that the total volme ow rate, Q, at any meridional location, s, is given by () Q = (s; ) (s; ) () and this provides a periodic bondary condition on in the direction. In the rotating frame of reference, the eqations of motion are seflly written sing the total pressre, P, instead of the pressre, p, where P = p + ( s + R! ) () The total pressre, P, is sometimes called the rotalpy. The eqations of motion, eqations (3) and (4), then become P = H s (g S + g R ) (3) s P = H s ( +!R)(g S + g R )+Rg R (4) R In these eqations the fnctions, g S and g R, are the shear stress terms for the stator and rotor respectively and are given by g S = A S H H ms s +( +!R) m S+ (5) g R = A R H (6) H mr s +( +!R R) m R+

5 The important qantity,,given by = H Rs (R +!R )+ R (R s) (7) plays a crcial role both in nderstanding the id mechanics of these ows and in the soltion methodology. It is the eective vorticity. A fndamental property of can be discerned by eliminating P from eqations (3) and (4) to obtain the following convection eqation for : s s + R = R( +!R)(g S + g R ) R g R RH s f s(g S + g R )g (8) This demonstrates that, in the absence of viscos eects (g S = g R = ), the vorticity isinvariant along any streamline. Conversely, the shear stresses are alone responsible for any change in along a streamline. If is a coordinate measred along a streamline, then eqation (8) clearly implies that = RH( s + ) R( +!R)(g S + g R ) R g R s f s(g S + g R )g (9) Frthermore, when written in this way, the governing eqations clearly indicate a physically reasonable approach to their nmerical soltion by iterative means. 4. BOUNDARY CONDITIONS AND NUMERICAL METHODS It follows from the above that one method for the nmerical soltion of the eqations for a rotordynamic ow wold be to proceed as follows: () First, for given or gessed vales of the vorticity, (s; ), the Poisson-like eqation (7), rewritten as R!R + = RH () s H s R H mst be solved to obtain the stream fnction, (s; ). From this soltion new vales for (s; ), s (s; ) and (s; ) can then be calclated. Appropriate bondary conditions on for se in the soltion of eqation are: (i) Along s =,we specify an inlet swirl velocity, (;), which, in order to satisfy conservation of anglar momentm, shold normally be pt eqal to the swirl velocity in the reservoir pstream of the inlet. (ii) An appropriate bondary condition at discharge, s = S, wold be that the pressre in the ow exiting the annls shold be niform for all or p s=s = () For later discssion, we observe that the bondary condition = () s=s provides a convenient rst approximation to the condition. (iii) The periodic conditions on bondaries at = and =sch that (s; ) (s; ) = Q (3)

6 Figre : Schematic of the relation between the rotordynamic forces, F n and F t, the rotor center and the circlar whirl orbit. () Second, given the new vales of (s; ), s (s; ) and (s; ), we can proceed to integrate along streamlines to nd new vales for (s; ) sing eqation (9). This reqires evalation of the shear stress fnctions, g R and g S and vales of at the points where the streamlines enter the comptational domain. Clearly this becomes more complicated when there is reverse ow either at inlet or at discharge. Here, we restrict ot attention to the simpler circmstances in which there is no ow reversal and all the streamlines begin at the inlet. Then, assming that the viscos stresses pstream of the inlet are negligible and that the inlet ow is circmferentially niform, eqation (4) reqires that at the inlet bondary, s =, mst take a niform vale given by the denition (7). These two steps are then repeated to convergence. 5. HEAD, PRESSURE AND ROTORDYNAMIC FORCES Having obtained convergence, the total pressre, P, the pressre, p, and the rotordynamic forces can be calclated as follows. The total pressre is most readily obtained by anintegration along the streamline similar to that for the vorticity,. From eqations (3) and (4) it follows that P = ( s + ) R g R s + ( +!R) (g R + g S ) (4) which demonstrates that the total pressre (or energy in the ow) is constant along a streamline in the absence of viscos eects. Since the viscos terms are sally small in these calclations it is sensible to integrate eqation (4) along a streamline in parallel with the integration and so obtain the total pressre everywhere. If one chooses to neglect entrance losses between the pstream reservoir at the inlet plane (s = ), and if the reservoir is circmferentially niform, then this integration begins with the niform vale of P (;) eqal to the total pressre in the reservoir, P res, and this can conveniently be chosen to be zero withot loss of generality. On the other hand if entrance losses are to be inclded then P (;) can be set to a vale smaller than P res byan amont eqal to the entrance loss at that particlar position. Other complications which cold be incorporated inclde a non-niform pstream reservoir (sch as the volte of a pmp operating o-design) which wold imply a circmferentially varying P (;). Having obtained the pressre (and the viscos shear stresses), it only remains to integrate these to obtain the normal and tangential forces acting on the rotor. With the sign convention as dened

7 7 5 6 c 5 M, c 4 3 M c C, k 5 C 5 k K Figre 3: Comparison of the rotordynamic coecients from the experiments of Nordmann and Massmann (984) on a plain, ntapered, smooth seal with varios calclated reslts as follows. Dashed line: Childs' (983a) analytical blk ow model. Dotted line: reslts from nmerical integration of Childs' (983b, 987, 989) pertrbation method for blk ows. Solid line: present blk ow calclation. in gre, it follows that: F n =Z F t = S Z S ( ( dr ds dr ds ) ) Z Z (p cos + R sin ) Rd ds (5) (p sin R cos ) Rd ds (6) In the reslts qoted in this paper the contribtions from the R parts of these integrals are very small and can often be neglected. The rotordynamic coecients are then obtained by tting qadratics to the fnctions, F n (!=) and F t (!=); the forces and coecients are non-dimensionalized as described by Brennen (994). 6. RESULTS For prposes of illstration, we choose to examine the plain, ntapered, smooth axial seal tested by Nordmann and Massmann (984). This has an aspect ratio, S=R = :67, a clearance, H=R = :67 and was tested at a Reynolds nmber, Re = Q=R = 565 (where Q is the volme

8 ow rate throgh the seal). In Figre 3, the non-dimensional rotordynamic coecients from the experiments of Nordmann and Massmann (see Brennen 994) are plotted against a ow coecient, = Q=HR. These are compared with three sets of calclated coecients. The dashed lines are from the analytical expressions obtained by Childs' (983a) as approximate \short seal" soltions to the blk ow. Later Childs (983b) pblished a more accrate \nite length" seal soltion which involved the nmerical integration of more accrate pertrbation eqations. The dotted lines represent the reslts of a similar pertrbation analysis applied to the Nordmann and Massmann seal. The dierences between the two Childs' methods are sbstantial bt similar to the dierences in the examples presented by Childs (983b). They are presmably cased by dierent treatments of the circmferential velocity pertrbations. Similar dierences were noted by Ginzbrg (99), particlarly in the direct added mass, M. Note that all three sets of calclated reslts se an inlet swirl velocity eqal to a half of the rotor tip speed. The solid lines represent reslts sing the present nmerical method. Apart from the direct stiness, K, they are similar to Childs' (983b) more accrate analysis. For reasons which are presently nclear or calclations yield a K which is closer to the short seal vale and to the experimental data. The discrepancies between the more accrate analyses and the experimental data are distrbing and reqire frther stdy. 7. CONCLUSIONS This paper has explored some of the basic characteristics of the blkow model eqations for the trblent ow in a id-lled annls generated by both rotational and whirling motions. The analysis nveils the denition of the appropriate vorticity for these ows and develops evoltionary eqations both for the vorticity and for the total pressre, withot resorting to linearization. Among other featres demonstrated by these eqations is the fact that the changes in vorticity and total pressre along a streamline are entirely de to the shear stresses imposed on the ow. This eqation strctre natrally sggests a way in which nmerical soltions to these eqations might be soght, by iterating between a Poisson-like eqation for the streamfnction sing a preliminary vorticity distribtion and an integration along the streamlines to revise that distribtion. Several sample calclations are sed to illstrate this techniqe. ACKNOWLEDGEMENTS The athors wish to express their gratitde for the spport given to this research by Rocketdyne Division, Boeing North American. REFERENCES. Baskharone, E., and Hensel, S. (993) Flow Field in the Secondary, Seal-Containing Passages of Centrifgal Pmps. ASME J. Flids Eng., 5, 7{79.. Black, H.F. (969). Eects of Hydralic Forces in Annlar Pressre Seals on the Vibrations of Centrifgal Pmp Rotors. J. Mech. Eng. Sci.,, No., 6{3. 3. Black, H.F. and Jensen, D.N. (97). Dynamic Hybrid Properties of Annlar Pressre Seals. Proc. J. Mech. Eng., 84, 9{. 4. Brennen, C.E. (994). Hydrodynamics of Pmps. Concepts ETI and Oxford University Press. 5. Childs, D.W. (983a). Dynamic Analysis of Trblent Annlar Seals Based on Hirs' Lbrication Eqation. ASME J. Lbr. Tech., 5, 49{ Childs, D.W. (983b). Finite Length Soltions for Rotordynamic Coecients of Trblent Annlar Seals. ASME J. Lbr. Tech., 5, 437{ Childs, D.W. (987). Flid Strctre Interaction Forces at Pmp-Impeller-Shrod Srfaces for Rotordynamic Calclations. ASME Symp. on Rotating Machinery Dynamics,, 58{593.

9 8. Childs, D.W. (989). Flid Strctre Interaction Forces at Pmp-Impeller-Shrod Srfaces for Rotordynamic Calclations. ASME J. Vibration, Acostics, Stress and Reliability in Design,, 6{5. 9. Childs, D.W. and Dressman, J.B. (98). Testing of Trblent Seals for Rotordynamic Coecients. Proc. Workshop on Rotordynamic Instability Problems in High-Performance Trbomachinery, NASA Conf. Pbl. 5, 57{7.. Childs, D.W. and Dressman, J.B. (985). Convergent-Tapered Annlar Seals: Analysis and Testing for Rotordynamic Coecients. ASME J. Tribology, 7, 37{37.. Childs, D.W. and Kim, C.-H. (985). Analysis and Testing for Rotordynamic Coecients of Trblent Annlar Seals with Dierent Directionally Homogeneos Srface Roghness Treatment for Rotor and Stator Elements. ASME J. Tribology, 7, 96{36.. Childs, D.W. and Scharrer, J.K. (986). Experimental Rotordynamic Coecient Reslts for Teeth-on-Rotor and Teeth-on-Stator Labyrinth Gas Seals. Proc. Adv. Earth-to-Orbit Proplsion Tech. Conf., NASA Conf. Pbl. 436, 37{ Dietzen, F.J. and Nordmann, R. (987). Calclating Rotordynamic Coecients of Seals by Finite-Dierence Techniqes. ASME J. Tribology, 9, Fritz, R.J. (97). The Aects of an Annlar Flid on the Vibrations of a Long Rotor. Part I Theory and Part II Test. ASME J. Basic Eng., 9, 93{ Ginzbrg, A. (99) Rotordynamic Forces Generated by Discharge-to-Sction Leakage Flows in Centrifgal Pmps. Ph.D. Thesis, Calif. Inst. of Tech. 6. Ginzbrg, A., Brennen, C.E., Acosta, A.J., and Caghey, T.K. (993) The Eect of Inlet Swirl on the Rotordynamic Shrod Forces in a Centrifgal Pmp. ASME J. Eng. for Gas Trbines and Power, 5, 87{ Ginzbrg, A., Brennen, C.E., Acosta, A.J., and Caghey, T.K. (994) Experimental Reslts for the Rotordynamic Characteristics of Leakage Flows in Centrifgal Pmps. ASME J. Flids Eng., 6, {5. 8. Hirs, G.G. (973). A Blk-Flow Theory for Trblence in Lbricant Films. ASME J. of Lbr. Tech., April 973, 37{ Nordmann, R. and Massmann, H. (984). Identication of Dynamic Coecients of Annlar Trblent Seals. Proc. Workshop on Rotordynamic Instability Problems in High Performance Trbomachinery, NASA Conf. Pbl. 338, 95{3.. Pinks, O. and Sternlicht, B. (96). Theory of Hydrodynamic Lbrication. McGraw-Hill, New York.. Sivo, J., Acosta, A.J., Brennen, C.E., Caghey, T.K., Fergson, T., and Lee, G. (994) Laser Velocimeter Measrements in the Leakage Annls of a Whirling Centrifgal Pmp. ASME Laser Anemometry-994, Advances and Applications FED-9, {7.. Sivo, J., Acosta, A.J., Brennen, C.E., and Caghey, T.K. (995) The Inence of Swirl Brakes on the Rotordynamic Forces Generated by Discharge-to-Sction Leakage Flows in Centrifgal Pmps. ASME J. Flids Eng., 7, 4{8. 3. Uy, R.V., Bircmshaw, B.L. and Brennen, C.E. (997). A Parametric Evalation of the Eect of Inlet Swirl on the Rotordynamic Forces Generated by Discharge-to-Sction Leakage Flows in Shroded Centrifgal Pmps. Proceedings of the 997 ASME Flids Engineering Division Smmer Meeting. 4. Yamada, Y. (96). Resistance of Flow Throgh an Annls with an Inner Rotating Cylinder. Bll. JSME, 5, No. 8, 3{3. NOMENCLATURE A r ;A s Empirical constants for rotor and stator respectively C Direct damping coecient, normalized by R L" c Cross-copled damping coecient, normalized by R L" F n Force normal to whirl orbit, normalized by R L"

10 F t Force tangent to whirl orbit, normalized by R L" H Clearance between impeller shrod and hosing K Direct stiness coecient, normalized by R L" k Cross-copled stiness coecient, normalized by R L" L Axial length of the impeller M Direct added mass coecient, normalized by R L" mr;ms Empirical exponent for rotor and stator respectively P Total pressre p Static presse Q Volmetric leakage ow rate R Radis of rotor s Mean leakage inlet path velocity of id Mean leakage inlet swirl velocity of id Eective vorticity dened by Eq. 7 Flid viscosity " Eccentricity of whirl orbit Flid density Leakage ow coecient, s =R Stream fnction, dened by Eq.! Whirl radian freqency Main shaft radian freqency Wall shear stress