Modal Response of Hydraulic Turbine Runners

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1 1 (9) Modal Repone o Hydraulic Turbine Runner Q.W. LIANG Center o Indutrial Diagnotic and Fluid Dynamic (CDIF), Technical Univerity o Catalonia, Barcelona, Spain, quanwei@m.upc.edu C. G. RODRIGUEZ Center o Indutrial Diagnotic and Fluid Dynamic (CDIF), Technical Univerity o Catalonia, Barcelona, Spain, critian.rodriguez@m.upc.edu E. EGUSQUIZA Center o Indutrial Diagnotic and Fluid Dynamic (CDIF), Technical Univerity o Catalonia, Barcelona, Spain, eguquiza@m.upc.edu X. ESCALER Center o Indutrial Diagnotic and Fluid Dynamic (CDIF), Technical Univerity o Catalonia, Barcelona, Spain, ecaler@m.upc.edu F. AVELLAN Laboratory or Hydraulic Machine (LMH), Swi Federal Intitute o Technology, Lauanne, Switzerland, rancoi.avellan@epl.ch Key word: Dynamic Repone, Hydraulic Turbine, Runner. Abtract The mechanical deign o hydraulic turbine i conditioned by the dynamic repone o the runner that i uually etimated by a computational model. Neverthele, the runner ha complex boundary condition that are diicult to include in the computational model. One o thee boundary condition i the water in which the runner i ubmerged. The eect o the added ma and damping o water can modiy coniderably the natural requencie o the runner. In order to analyze thi eect on a Franci turbine runner, an experimental and a numerical invetigation in a reduced cale model wa carried out. The experimental invetigation wa baed on modal analyi. Several impact tet with the runner in air and in water were done. The repone wa meaured with accelerometer located in dierent poition o the runner. Special attention wa taken to determine the mot uitable poition o meaurement and impact. From the modal analyi, the natural requencie, damping ratio, and mode hape were determined. The imulation o the ame runner wa alo carried out uing a FEM method. Firt, ome tet including a enitivity analyi were done to check the accuracy o the numerical reult. Second, the runner wa imulated and the requencie and mode hape were calculated both in air and in water like in the experiment. The imulation wa compared with the experimental reult to determine it accuracy epecially regarding the added ma eect. Similar mode hape and requency reduction ratio were obtained o the imulation gave rather good reult. In the paper, the requencie, damping and mode hape obtained in air and in water both rom experiment and imulation are indicated. The ame mode hape obtained in air were

2 2 (9) obtained in water but with lower natural requencie and higher damping ratio. The dierence in the natural requencie i hown to be dependent baically on the added ma eect o the water and not on it added damping. Thi dierence alo depend on the geometry o the mode preenting dierent value or dierent mode hape. Uing nondimenional value, the reduction in the natural requencie can be extrapolated to other Franci runner preenting imilar geometrical characteritic. Introduction The power concentration in hydraulic turbine i contantly being increaed when deigning new machine or upgrading exiting one. A a conequence the hydraulic excitation orce on the turbine runner increae due to higher head and luid velocitie. Moreover, the operation range i widened to atiy the demand o larger regulation capacity. The operation at o-deign condition alo reult in larger orce that combined with a reduced thickne/weight ratio in runner can provoke even higher vibration level. Thi dynamic excitation i the reponible o atigue damage. A complete analyi o the modal behavior o the runner i neceary to prevent thi type o damage. Since the runner are working ubmerged in water, the eect due to the preence o thi heavy luid mut be taken into account. Thereore, thi analyi ha to conider the tudy o the excitation orce and o the runner repone conidering the water eect. Particularly, the current paper i concentrated only on the repone o a hydraulic runner. Although experimental and numerical modal analyi have been extenively ued to tudy the dynamic repone o tructure, mot o them have been limited to imple geometrie (Re 1, 2, 3, 4, 5). Furthermore, ew work have conidered the eect o urrounding luid. In the cae o hydraulic turbine runner, it i very diicult to tet the real operation condition rom the experimental point o view. And rom the numerical point o view, the main challenge are due to the complexity o the geometry, epecially conidering the meh compatibility between the luid and tructure on the interace. In act, the current reearch on turbine runner ubmerged in water ha been limited to ome impliied theoretical model only validated againt ew experimental reult (Re 6, 7, 8, 9). In ummary, the reult o the modal repone o a Franci turbine runner both in air and in water (Re 10) are dicued in detail in thi paper. The natural requencie, mode hape and requency reduction ratio due to the water added ma eect obtained with an experimental modal analyi and with a numerical imulation with inite element method are compared howing a very good accuracy. Theoretical background In theoretical conideration, in order to take into account the added ma eect due to the urrounding luid the ytem ha to be treated a a luid-tructure interaction problem. In that cae, the tructural dynamic equation i coupled with the equation o the luid, which can be decribed a the dicretized equation (1) a preented in (Re 11):

3 3 (9) where ii i M 0 C 0 u u K K u F M M + + = ii 0 C i 0 K p 0 p p M = tructure ma matrix; C = tructure damping matrix; (1) K = tructure tine matrix; F = applied load vector; u = nodal diplacement; M = luid equivalent ma matrix; C = luid equivalent damping matrix; K = luid equivalent tine matrix; F = luid load produced by tructure diplacement at the interace; p = preure on the node; M = ρ R, deined a equivalent coupling ma matrix; K = R, deined a equivalent coupling tine matrix. Thi equation conider the ollowing aumption (Re 12): The luid i lightly compreible (denity change due to preure variation); The luid i non-vicou (no vicou diipation); The low i irrotational; There i no mean luid low; Change o mean denity and preure in dierent area o the luid domain remain mall. Thereore, or a problem involving luid-tructure interaction the luid element generate all the matrice with ubcript in addition to the coupling matrice ρ R and R. The matrice with ubcript are generated by the compatible tructural element ued in the model. Experimental etup and numerical model Experimental etup The teted model runner wa contructed ollowing the IEC (International Electrotechnical Commiion) Standard or international tet acceptance. It i a replica at a reduced cale o 1:10 o a Franci turbine runner with a peciic peed o The model ha 17 blade and a diameter o 409 mm. It hape i hown in Figure 1. The material ued i a bronze alloy with denity o 8300 kg/m 3 and Young Modulu o 110 Gpa. To get the runner ytem parameter a a ree body with no contrain, the runner wa upended with a lexible tring. For the tet in water, the runner wa ubmerged in water contained in a quared tank. A erie o impact tet were applied on 16 poition in the band and on 6 poition in each blade, giving a total o 118 impact poition. Six impact were done on each poition. The detailed reult obtained with thee tet have been publihed by Rodriguez et al. (Re 13). Numerical model Baed on the cyclic ymmetrical characteritic o the tructure, one ector including one blade

4 4 (9) and covering an angle o 360/17 degree wa ued to do the imulation. Then the reult were expanded to the whole runner (Re 14, 15). The meh, a hown in Figure 1, wa built up by dicretizing the CAD model uing hexahedral element, which provided more accuracy. Figure 1 Dicretized inite element meh with hexahedral element A enitivity analyi wa carried out to determine the inluence o the meh denity. Five mehe with dierent element denitie were imulated, each one including approximately a double number o element than the previou one. In Figure 2, the natural requency value are plotted or all o them. It can be een that the reult converge when the element denity i increaed. Thereore, the meh with 6133 element per ector, previouly hown in Figure 1, wa elected or the imulation. Figure 2 Reult o the meh enitivity analyi To do the imulation in water, the model o the runner wa urrounded by a cylindrical luid domain conidering the cyclic ymmetric characteritic o the runner. The boundary condition were careully treated according to the experimental tet. In Figure 3, the drat o the model and FEM meh are hown.

5 5 (9) Figure 3 Fluid domain (blue) around the runner (pink) Reult The modal parameter o the runner, both in air and ubmerged in water, have been well determined with the experiment and the imulation. Thee reult have been claiied baed on the mode hape becaue they have been ound to be imilar between the tructure in air and in water. Due to the cyclic ymmetric characteritic o Franci turbine runner, the vibration mode can be claiied according to the number o nodal diameter (ND) appearing on the band part. Deined by the condition ND=0, the mode are inglet and natural requencie are ditinct. The mode with ND 0 are doublet; they have a pair o mode hape with the ame natural requency (Re 16). Since in our cae it ha been oberved that the crown generally ha mall diplacement compared to the band and the blade, the mode hape have been decribed employing the number o nodal diameter (ND) on the band a the criterion. A hown in Figure 5 and Figure 6, the 0ND mode correpond to a torion mode without band deormation and the 1ND mode correpond to a lexion mode with the band winging like a pendulum. For the mode with 2 or more ND, the band behave bending with deormation in the radial direction. The natural requencie ound or the ame mode hape are lited in Table 1 and plotted in Figure 4. Table 1 Natural requencie o the runner in air and in water(hz) 0ND 1ND (Torion) (Flexion) 2ND 3ND 4ND 5ND Exp. Air Exp. Water Sim. Air Sim. Water Note: 1. Exp.=experiment, Sim.=imulation; 2. For all doublet mode, average value o 2 requencie are lited in table.

6 6 (9) Figure 4 Natural requencie in air and in water (T: torion, F: lexion) The damping value obtained with the experiment or the runner in air and ubmerged in water are lited in Table 2. Table 2 Damping obtained by experiment 0ND (T) 1ND (F) 2ND 3ND 4ND 5ND In air In water The mode hape o the band or the runner in air are hown in Figure 5 and Figure 6 rom a bottom view. It mut be noted that the ame mode have been ound with the experiment and the imulation. Moreover, the mode hape obtained in water are jut avoided to be repeated ince they are imilar to thoe in air. Figure 5 Mode hape obtained rom the experiment in air (ordered by number o ND rom let to right)

7 7 (9) Figure 6 Mode hape obtained by imulation in air (ordered by number o ND rom let to right) In order to check the level o accuracy o the imulation compared to the experiment, the dierence in natural requency value have been calculated with equation (2). They are lited in Table 3. where (%) ( ) ( im ) % = U U / U 100. exp. exp. i the dierence in percent, and U im. and U exp. are the natural requency value obtained by imulation and by experiment repectively. Table 3 Deviation o imulation reult compared with experiment one (%) 0ND (T) 1ND (F) 2ND 3ND 4ND 5ND Natural requency (in air) Natural requency (in water) It can be clearly noticed that the maximum dierence i o about ±3.6%. Moreover the mode hape, both in air and in water, alo repreent good conitency between imulation and experiment. Thereore, the imulation ha been ound to be accurate enough to calculate the modal characteritic o uch a turbine runner tructure and, in turn, to correctly determine the added ma eect. Dicuion Added ma eect Comparing the reult in air and in water, a igniicant decreae o the natural requencie can be oberved. Thi reduction i due to the eect o the added ma induced by the urrounding water. The added ma eect can be quantiied by calculating the requency reduction ratio δ o each mode hape, deined with equation (3): δ = ( )/ (3) a w a (2) where a and w are the natural requencie in air and in water repectively. In Table 4, the requency reduction ratio obtained are lited. Table 4 Frequency reduction ratio o the runner mode o vibration 0ND (T) 1ND (F) 2ND 3ND 4ND 5ND Sim Exp

8 8 (9) It can be clearly noticed that the natural requencie are coniderably reduced by the preence o luid. The requency reduction ratio how a igniicant variation rom 0.10 to 0.39 depending on the correponding mode hape. In order to extrapolate the reult about added ma eect, a non-dimenional added ma actor or each mode can be deined with equation (4): (Re 10) 2 ai, ρ γ i = 1 (4) wi, ρ where γ i i the non-dimenional added ma actor, which remain contant or a peciied mode (i), claiied by the eature o the mode-hape (ND, etc.). The non-dimenional added ma depend only on the geometrical characteritic o the tructure, which i determined by the deign o the runner. In other word, the relationhip between natural requencie in air and in water, obtained by the method employed in thi invetigation, can alo be valid or any other Franci turbine runner with geometrical imilarity. Thi can be practically ued to etimate the natural requencie o other geometrically imilar turbine runner contructed with dierent material and dimenion. Damping The damping ound ha low value in all the mode. Thi i the typical behavior or a tructure vibrating with mall amplitude and high requencie in tagnant luid in abence o wave radiation. It can be een that the damping are increaed by the preence o water, varying depending on the mode hape, but they are not high enough to aect igniicantly the value o the natural requencie. Concluion The modal characteritic o a reduced cale model o a Franci turbine, in air and in water, have been invetigated experimentally and numerically. Comparion o reult indicate that the imulation reult how good agreement with the experimental one. Meanwhile the mode hape in water are imilar to thoe in air, the preence o water reduce coniderably the natural requencie while the damping are increaed. The added ma eect o the urrounding till water ha been quantiied by calculating the requency reduction ratio, which varie in a range o 0.10~0.39 depending on the geometry o the mode hape. A non-dimenional added ma actor ha been derived which can be ued to extrapolate the natural requencie o geometrically imilar runner o dierent material. Thereore, the methodology ued in thi invetigation ha been proved to be valid o that it can be extended to tudy the dynamic behavior o any other hydraulic turbine runner. Acknowledgement Thi reearch ha been carried out in the HYDRODYNA project. The author would like to acknowledge the LMH-EPFL and Voith Siemen Hydro Power Generation or their collaboration.

9 9 (9) Reerence Re 1 Lindholm, U.S., et al. Elatic vibration characteritic o cantilever plate in water. Journal o Ship Reearch 1965;9(1): Re 2 Gladwell, G.M.L. and Maon, V. Variational inite element calculation o the acoutic repone o a rectangular panel. Journal o Sound and Vibration 1971;14(1): Re 3 Rao, P.S., Sinha, G., and Mukhopadhyay, M. Vibration o ubmerged tiened plate by the inite element method. International Shipbuilding Progre 1993;40(423): Re 4 Liang, C.C., et al. The ree vibration analyi o ubmerged cantilever plate. Ocean Engineering 2001;28(9): Re 5 Ergin, A. and Ugurlu, B. Linear vibration analyi o cantilever plate partially ubmerged in luid. Journal o Fluid and Structure 2003;17(7): Re 6 Duba, M. and Schuch, M. Static and dynamic calculation o a ranci turbine runner with ome remark on accuracy. Computer and Structure 1987;27(5): Re 7 Du, J.B., He, S.J., and Wang, X.C. Dynamic analyi o hydraulic turbine runner and balde ytem (ii) - analyi o example. Journal o Tinghua Univerity (Sci&Tech) 1998;38(8): Re 8 He, S.J., Du, J.B., and Wang, X.C. Dynamic analyi o hydraulic turbine runner and balde ytem (i) - mechanic model and ormulation. Journal o Tinghua Univerity (Sci&Tech) 1998;38(8): Re 9 Xiao, R.F., et al. Study on dynamic analyi o the ranci turbine runner. Journal o Large Electric Machine and Hydraulic Turbine 2001;7(41-43). Re 10 Q. W. Liang, et al. Numerical imulation o luid added ma eect on a ranci turbine runner. Computer & Fluid Accepted. Re 11 Woyjak, D.B. Acoutic and luid tructure interaction, a reviion 5.0 tutorial. Houton: Swanon Analyi Sytem, Inc; Re 12 Kinler, L.E., et al. Fundamental o acoutic. New York: John Wiley and Son; Re 13 Rodriguez, C.G., et al. Experimental invetigation o added ma eect on a ranci turbine runner in till water. Journal o Fluid and Structure 2006;22(5): Re 14 Thoma, D.L. Dynamic o rotationally periodic tructure. International Journal or Numerical Method in Engineering 1979;14(1): Re 15 Zienkiewicz, O.C. and Scott, F.C. On the principle o repeatability and it application in analyi o turbine and pump impeller. International Journal or Numerical Method in Engineering 1972;4(3): Re 16 Kim, M., Moon, J., and Wickert, J.A. Spatial modulation o repeated vibration mode in rotationally periodic tructure. Journal o Vibration and Acoutic, Tranaction o the ASME 2000;122(1):62-68.

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