Method for Hydrodynamic Coupling of Concentric Cylindrical Shells and Beams

Transcription

1 ethod for Hydrodymc Couplg of Cocetrc Cyldrcl Shells d Bems tthew D. Syder Westghouse Electrc Compy, LLC Abstrct Dymc lyses of pressurzed wter rector vessels, core support structures, d terls requre ccurte represetto of dymc couplg due to flud ul betwee cocetrc cyldrcl shells. I prctce, the structure s frequetly modeled usg bem elemets wth Frtz s hydrodymc mss mtrx s s mplemeted usg the FLUID38 elemet. I dvced rector desgs, t s desrble to hve comprehesve lyss model tht cludes shell modes whle remg suffcetly smple for log oler trset dymc lyses. The m of ths pper s to descrbe strghtforwrd method tht c be used to represet hydrodymc mss effects models of cocetrc cyldrcl vessels costructed of shell elemets. I the method, shell dsplcemet d hydrodymc ertl forces re expressed s Fourer seres. The Fourer seres re etered to ASYS usg ler costrt equtos tht relte dsplcemets of ech shell ode to dsplcemets t supplemetry sets of Fourer odes. Hydrodymc mss s the etered usg STIF7 elemets coectg Fourer odes ssocted wth correspodg odes o er d outer shells. The method llows the Frtz bem mode mss to be etered seprtely from mss mtrces represettve of ech shell mode. The method lso pples whe ether the er or outer shell or both my be represeted s bem. Exmples of the method re preseted wth dscusso of techques for evlutg mss mtrces for shell modes bsed o theory d expermetl dt. Applcto of the method s demostrted for rregulr meshes d vrble-thckess or folded-plte structures of er cyldrcl form. Exmples lso demostrte tht the method s ehced by ts comptblty wth usg superelemets (ATRIX50) to represet more complex shell structures. Itroducto The structurl system of Pressurzed Wter Rector (PWR) cludes flud-coupled cocetrc cyldrcl vessels ( See Fgure ) whose fuctos re cotg the rector core, drectg flow, d provdg support for the rector core d terls. For mproved relblty d mtblty, the ext geerto of PWR s hs corported some sgfct desg chges these structures (Refereces d ). Of prtculr mportce re modfctos of bolts, welds, te rods, hold-dow sprgs, d other mes used to jo or support the terl structures d to prevet excessve reltve movemet respose to dymc forces.

2 Fgure. Fetures of the AP000 Structurl Desg I desg of erler geertos of PWR s, dymc lyss of lterl respose hs bee frequetly performed usg lumped mss bem models wth mss mtrx to represet flud couplg. For more ccurte evluto of the ewer desgs d order to evlute d fully pply scle model test dt from other plts, structurl model tht cluded both bem d shell mode respose ws eeded. The purpose of ths pper s to descrbe oe reltvely smple wy to exted models to clude shell mode resposes wth flud-structure couplg resobly szed model. I ths pper, the terms bem d shell mode refer to vbrto modes of the cyldrcl shells. Whe vewed from the ed, the vbrto modes of the cyldrcl shell my cosst of y umber of crcumferetl wves. The umber of crcumferetl wves s deoted by. Sce t represets clsscl bem bedg deformto wthout dstorto of the cross secto of the cylder, the mode = s clled the bem mode. odes for re clled the shell modes d the mode = 0 s clled the brethg mode.

3 Descrpto of Structurl Problem Alyses descrbed ths pper re for the Westghouse AP000 "Advced Pssve" rector plt desg (See Fgure ). The AP000 rector vessel terls clude my fetures preset other Westghouse desgs (Referece ). However, most otble of dffereces from prevous desgs s the replcemet of bffle-former structure wth core shroud. Both structures drectly surroud rector core; however bffle-former structure s bolted to the core brrel wheres core shroud s welded structure tht s bolted to the lower core plte. The core shroud, core brrel, rector vessel d ulr spces of wter betwee them re represettve of the types of flud coupled cocetrc cyldrcl vessels tht re the subject of ths pper. For dymc lyses of erler rector desgs, 3-D bem models hve bee cosdered cceptble methodology to clculte overll structurl respose of the mjor compoets such s rector vessel, core brrel, d core support pltes. The motos of these supportg structures re the used more detled lyss of terls structures tht clude the rry of fuel ssembles tht forms the rector core. For y desg, there must be some cocer bout the ccurcy of usg bem models to represet cyldrcl shells whose legth to dmeter rto s much smller th those for whch Tmosheko bem theory s geerlly cosdered pplcble. For shells s short s the core shroud, ths s of prtculr cocer. Expermetl mesuremet of structures smlr to the core shroud show tht sher deformto leds to sgfct dstorto of the cross-secto eve the fudmetl bem bedg mode. To some extet ths could be compested for by the use of pproprte sher fctors; however lttle work hs bee doe to refe sher fctors from usul bem vlues. Wth ths md, t ws decded tht the core brrel d core shroud would be modeled of the AP000 plt s usg combto of shell d sold elemets. To represet the hydrodymc couplg the shell elemet models severl optos were cosdered: The wter could be modeled usg elemets such s FLUID30 or FLUID80. odels hve bee costructed usg these elemets d hve prove useful for the egevlue problem. Ths c be prtculrly useful evlutg cses where the flud ul re o crculr, however these models hve bee foud to be mprctcl for most log trset dymc lyses. To vod the cost of flud elemet modelg, models were tlly costructed wth rdl mss mtrces dstrbuted betwee ech er d ts outer shell. Bsed o the Frtz (Referece 3) mss, the mtrces were ssged vlues so tht the bem mode frequecy s reproduced reltvely well. Whe represeted ths wy, the shell mode frequeces were ucceptbly uderestmted. Ths should hve bee expected s wll be ppret from dscusso of the bem mode hydrodymc mss tht follows. I the method whch s the focus of ths pper, flud dsplcemets d ertl forces were represeted usg Fourer seres. Ths permts mss mtrces to be ssged depedetly for the bem d shell modes. The degrees of freedom used to represet the flud moto re reduced to smll set of Fourer mpltudes. The dvtge of ths pproch s eve greter whe these degrees of freedom re used wth superelemet represettos. ethodology for Added ss of Bem d Shell odes Added ss Couplg for the Bem ode For reltve lterl moto of two log cocetrc cylders, the hydrodymc mss s the Frtz (Referece 3) mss mtrx whch s the bss of the ASYS FLUID38 elemet. Ths mss mtrx s bsed o the ssumpto tht the cross sectos of the cylders do ot deform s they move lterlly,.e. dsplcemet the bem mode, =. The Frtz mss mtrx, s well s the mss mtrces gve lter for shell modes ssume tht flud motos re smll, tht the flud s compressble, d tht flud veloctes re much less th speed of soud.

4 The equtos for the Frtz mss re repeted here order to estblsh the otto for use wth shell modes >. h ( + h) [ ] = () ( + h) + + h For two cocetrc cylders: = π Lρ s the mss of flud dsplced by the er cylder. = πblρ s the mss of flud tht could fll the outer cylder bsece of the er cylder b + h = b = rdus of er cylder b = rdus of outer cylder ρ = flud desty L = xl legth of cylders. Ths my be wrtte s: s the hydrodymc mss ssocted wth reltve ccelertos of the cylders. πlρ ( + b ) b πlρ ( + r ) = = [ ] () b b b( + b) r b( + r ) where r =/b. I most stutos of terest the flud ulus s th reltve to the rdus of the cylders. For th cyldrcl ulus ( >> c=b-) c π ρ h L πlρ b [ ] = b (4) c b (3) All of the bove equtos pply to cses where xl flow duced by reltve lterl moto of the cylders s sgfct ether becuse the flow s blocked t the eds or becuse the cylders re log eough to gore ed effects. For commo cses such s the dowcomer ulus (See Fgure -- Fetures of the AP000 Structurl Desg), xl flow s blocked t oe ed ope t other. For these stutos t s commo to employ the followg pproxmto tht pples oly to the th ulr cse.

5 Fgure. Shell Elemet odel of Core Shroud Cosderg fte legth th cyldrcl ulus ope t oe ed d mposg reltve lterl ccelerto whle permttg oly xl flow results dded xl hydrodymc mss: 3 h xl = πρl (5) 3 c Ths xl mss effectvely cts seres wth the crcumferetl mss gve prevously so tht the two my be put together to gve the hydrodymc mss for the fte legth cse: 3 h xl h L h = πρ fte h + xl h c + 3 L The fl bem mss mtrx the hs the form: h ( + ) fte h fte [ ] = (7) fte ( + h ) fte + + h fte Equto (7) represets the totl hydrodymc mss for the bem mode. It s dstrbuted to odes over the heght of the model usg ether STIF7 or FLUID38 elemets. (6)

6 Added ss Couplg for Shell odes The flud dsplcemets d resultt forces from pressure ctg o the structures wll be defed by Fourer seres expso. Ths defes set of geerlzed forces, F d F,d geerlzed dsplcemets, A d A A&& () t U&& t A t A t 0 r( θ, ) = + && ( )cos( θ) + && ( )s( θ) (8) = = F () t F t F t F t 0 r( θ, ) = + ( )cos( θ) + ( )s( θ) (9) = = ext, the shell mode mss mtrces, [ ] c be defed s reltg the geerlzed force d ccelertos: F A&& = [ ] (0) F A b && b F A&& = [ ] () F A b && b where subscrpts d b re used for the er d outer shells. The flud mss mtrces re defed usg solutos of the equtos for compressble flow whereby the coustc pressure dstrbuto wth the flud ulus c be obted usg seprto of vrbles. The complete soluto, gve Referece 5, expresses the rdl pressure dstrbuto usg Bessel fuctos, the crcumferetl dstrbuto s the se d cose terms, d the xl dstrbuto form Fourer seres. The full soluto s somewht complex but c be smplfed by otg tht for the rge of structurl frequeces of terest, the xl legth of the cylders s smll compred to the wvelegth of coustc wves d the soluto from Referece 5 c be wrtte s: + + πlρ ( + b ) b [ ] = + + b ( ) b b ( + b ) πlρ ( + r ) br = () ( r ) br b ( + r ) where r = / b It s emphszed tht use of ths equto mples the sme ssumptos s used for the Frtz mss of FLUID 38: The motos re smll wth respect to ulus wdth. The flud s compressble (flud veloctes re less th 0% of speed of soud the flud.) The xl legth of the chel s smll reltve to wvelegth of propgtg dsturbces. To compre shell mode d bem mode msses t s helpful to expd the terms equto s follows:

7 + r log(/ r) r = + r r + r + O ( log ( ) log ( ) log ( ) [ ] (3) d r 7 3 r log(/ r) = ( log ( r) + log ( r) log ( r) + O[ ] (4) The, for /3 x log (r) << : + r r log(/ r) (5) Smlrly for equto 4: r r log(/ r) (6) These pproxmtos re suffcetly ccurte for the cses of geerl terest, e.g. for r 0.8 the errors the pproxmtos re less th {%, 7%, 5%, 6%} for = {,,3,4}. Therefore resoble pproxmto for th flud ul s: πlρ b [ ] == (7) log( b/ ) b b Ths gves the very smple result tht the shell mode mss mtrces s defed bove re equl to / tmes the bem mode mss mtrx. The / result pples oly to the crcumferetl prt of the mss. For fte legth cylder, dded mss for flow the xl drecto wll be proporto to the crcumferetl hlf wvelegth, λ θ = π /. Therefore equto 5 c be geerlzed for to be: h xl 3 = πρl 3 c (8) The xl mss vres proporto to / versus the / behvor of the crcumferetl mss. Therefore, sce the xl mss cts seres wth the crcumferetl mss, the djustmet for fte legth hs much less sgfcce for hgher modes; evertheless the xl legth correcto c stll be pproxmted s descrbed dervg equto 6.

8 Approch to Represetg Shell ss trx ASYS The crcumferetl vrto of rdl dsplcemets t y elevto of the cyldrcl shell c be expressed s Fourer seres gve equto 8. Smlr expressos c be wrtte for tgetl d vertcl dsplcemets but they wll ot be eeded sce hydrodymc forces re oly rdl. The =0 (or A o ) term s the brethg mode; = terms re the bem mode terms d terms re shell modes. The cose terms re symmetrc bout the θ =0 ple d the se terms re tsymmetrc. The brethg modes for these shells hve very hgh frequecy so tht ths term s ot cluded models dscussed lter; however, t s cluded here for completeess. kg use of orthogolty, the Fourer coeffcets re: π A = Ur( θ)cos( θ) d θ for 0 (9) π 0 π A = Ur( θ)s( θ) d θ for (0) π 0 If the shell s represeted by eqully spced ( θ=π/) odes usg shell elemets roud the crcumferece, the prevous tegrls my be pproxmted by the followg sums: where U r s the rdl dsplcemet. A = U cos( θ ) for 0 () r = A = U s( θ ) for () r = Equtos d c be etered to ASYS s costrt equtos. Ths s doe by cretg ew odes whch wll be referred to s "Fourer" odes d settg the dsplcemet of these Fourer odes to the bove reltos,.e. U = U cos( θ ) for 0 (3) x r = U = U s( θ ) for (4) y r = For ueqully spced odes, θ=(θ + - θ - )/, d equtos d become: θ+ θ A = U cos( θ ) for 0 (5) r π = θ+ θ r θ π = A = U s( ) for (6) Equtos 3 d 4 re modfed smlrly. These re the equtos used for the o-crculr d rregulr mesh of the core shroud. Typclly, the =0 mode s ot eeded d o more th the bem d frst 3 shell modes re sgfct. Therefore, 4 odes (t ech shell elevto) re creted ech wth degrees of freedom (U x d U y ). ote tht the degrees of freedom t the ew odes re the Crtes x d y drectos. The z drecto s ssumed to be the xs of the cylders.

9 Equtos 3 d 4 re form tht ASYS defes s ler costrt equtos. If these costrts re etered to ASYS, the dsplcemets clculted t the "Fourer" odes wll be equl to the coeffcets of the Fourer expso of rdl dsplcemets. For the cse of =, these coeffcets re equl to the bem mode dsplcemets the Crtes x d y drectos. Fourer odes re creted o both the er shell d outer shell. The hydrodymc mss s etered s STIF7 mss mtrces coectg the Fourer odes ( ote: The cse tht ether the er or the outer cylder s modeled s bem s dscussed below.) F x ( ) + h U && x b = b F ( h) x h U&& x F y ( ) + h U && y b = b F ( h) y h U&& y (7) (8) where d b correspod to Fourer odes of the er d outer shell respectvely. ote tht the mss mtrces wll be dfferet for ech mode, but t s ssumed tht they c ll be wrtte ths form. The mss mtrx for = wll be tke s the usul Frtz mss. As dscussed prevously tl estmte mss for c be obted by multplyg the msses for = by /,.e. use /4, /9, d /6 for modes, 3, d 4. The rector vessel s modeled s bem whch s equvlet to ssumg tht ts shell modes re "rgd". Therefore the mss mtrces couple the rector vessel d core brrel for = oly. For = the mss mtrx s plced betwee the Fourer ode of the core brrel d the ode of the bem elemet. For >, the mss mtrx s betwee the Fourer ode of the core brrel d fcttous fxed (grouded) ode. Added ss for ocrculr Flud Aul The problem of dded mss for o-crculr ul hs bee studed expermetlly Referece 4. Addtol ureported studes hve bee performed usg models tht mke used of 3-D flud elemets. evertheless ths rems re whch further reserch s eeded. I the preset study, the bem mode mss for o-crculr ul, such s tht betwee the core shroud d core brrel, ws determed by usg equvlet crculr rdus. The equvlet crculr rdus ws tke to be tht whch gves the sme flud cross sectol re s the ctul geometry. Expermetl mesuremets hve show ths to be resoble for the bem mode. For hgher modes the / djustmet to the bem mode mss ws used. Although for these structures ths produced resoble greemet wth vlble dt d models usg 3-dmesol flud elemets, ths should ot be used s geerl rule for ocrculr ul wthout further study or expermetl dt. Procedure I summry the methodology s ppled s the followg steps: ) Crete "Fourer" odes t ech elevto of ech shell. If the frst m shell modes re cosdered, there wll be m seprte odes per ech shell per ech elevto t whch the hydrodymc mss s to be dstrbuted to m Fourer degrees of freedom. ote tht odl coordtes o the shells re rotted to the cyldrcl coordte system whle Fourer odes rem Crtes odl coordte system. b) Eter costrt equtos to defe the Fourer ode dsplcemets. For ech Fourer ode there wll be "x" d "y" drecto costrt. Terms of the costrt equto re esly etered mcro wth do-loops over elevto, drecto d shell mode.

10 c) Eter the x hydrodymc msses betwee djcet ( elevto d drecto) Fourer odes. It s recommeded tht: ) Frst ru the model for -r frequeces before ddg the Fourer odes d costrts. Add the Fourer odes d costrt equtos but set mss mtrces to smll vlues. Verfy tht the -r modes d frequeces re ot ffected by the costrts d Fourer odes. ) Add the hydrodymc msses for the bem mode d verfy tht the bem mode s correct d tht -r shell modes re ot ffected. 3) Add msses for shell modes to complete problem. 4) Estmte the hydrodymc mss usg the / d / rules dscussed bove. These vlues should be treted s tl estmtes wth refemet bsed o expermetl dt. Alyss A full dscusso of the AP000 structurl model d results s beyod the lmts of ths pper; however bref descrpto of the model wll expl the beefts of usg the methods dscussed bove to obt ecoomcl model. The full model of the AP000 system cluded over 5000 elemets wth over ctve degrees of freedom (DOF). The mjorty of the elemets d DOF were ssocted wth the core shroud whch hd over 000 shell elemets d DOF (See Fgure Shell Elemet odel of Core Shroud). The core shroud ws represeted s superelemet (ATRIX50) wth 40 mster DOF. The mster DOF cluded the Fourer odes used for represetg the hydrodymc mss for the core shroud to core brrel ulus for the bem d frst 3 shell modes. Usg the superelemet, the umber of ctve DOF the lyss ru ws reduced from pproxmtely 85,000 to 9,000. I geerl the lyss models cluded smll umber (< 00) of oler gp, cotct d sldg elemets t supports of the vessels. Therefore most lyses of the respose of the AP000 structures due to turbulet forces were performed usg o-ler trset dymc lyss wth ppled pseudordom force-tme hstores. Followg the tme-hstory lyss, the super elemet soluto ws expded t selected tmes. Flly, submodels were lyzed to clculte pek stresses loclzed regos of the core shroud d core brrel. Cocluso The lyses performed for the AP000 plt demostrted tht the method of cludg hydrodymc mss for shell modes wll reduce models to prctcl szes eeded for o-ler trset dymc lyss. Hd the sme models bee prepred usg sold flud elemets, they would hve bee order of mgtude more costly to ru f eve possble. The method reported here hs sgfct dvtges ts comptblty wth usg superelemets to represet portos of the structure; wheres models wth flud elemets typclly hve couplg tht mkes dvso to superelemets much more dffcult. For ths d smlr dymc lyses of flud-structure tercto problems, (Referece 6) ASYS costrt equtos hve bee demostrted to be useful mes to mmze the umber of elemet tht would be eeded to represet couplg of compressble fluds. The exteso of models to clude shell modes s essetl more ccurte ssessmet of flow duced vbrto (FIV). These ssessmets re ecessry tht reltve movemets of structures tht surroud the rector core could led to core dmge or terferece wth cotrol rod movemet. Flow-duced vbrtos my lso be cuse of some cses of loose prts d dcto of other problems the system. Flly, lthough FIV s ot geerlly the cotrollg lodg for stress levels these structures, the ssessmet of flow duced vbrto s requred evlutg orml opertg stress d ftgue dmge. The tl pplcto of the methodology descrbed ths pper hs bee for ssessmet of structurl vbrtos resultg from lterl forces due to turbulet flow; however, the models hve lso bee used

11 lyss of structurl respose due to vertcl forces. The methods my lso be used future lyses for other dymc lodgs cludg pump-duced coustc wves, sesmc, d Loss-of-Coolt-Accdet (LOCA) lyses. Refereces ). D. Syder, et.l., AP000 Rector Iterls Flow-Iduced Vbrto Progrm, 003 Itertol Cogress o Advces ucler Power Plts (ICAPP 03), y 4-7, 003, Córdob, SPAI. ) Yosho yke d Hrosh uk, For the Log Term Stble Supply of Electrc Eergy, tsubsh Hevy Idustres, Ltd. Techcl Revew, Vol. 40, o., Februry ) R. J. Frtz, The Effect of Lquds o the Dymc oto of Immersed Solds, Jourl of Egeerg for Idustry, Trsctos of the ASE, Februry 97. 4) S. kur, et.l., Bsc Test of Added ss d Added Dmpg of Rgd Body Eclosed coter rrow Pssge, Proceedgs of ICOE 8, 8th Itertol Coferece o ucler Egeerg, Bltmore, D., Aprl -6,000. 5). K.Au-Yg, Dymcs of Coupled Flud-Shells, Jourl of Vbrto, Acoustcs, Stress, d Relblty Desg,, Trsctos of the ASE, Vol. 08, July ) R.E. Schwr, D.R. Forsyth,. D. Syder, et.l., "The Bffle-Brrel-Boltg Alyss Progrm: Evoluto & Techcl Accomplshmets", th Itertol Coferece o ucler Egeerg, ICOE-9, Prs, Aprl, 00.