Application of spectral tuning on the dynamic model of the reactor VVER 1000 support cylinder

Transcription

1 Appled and Computatonal Mechancs 1 (007) Applcaton of spectral tunng on the dynamc model of the reactor VVER 1000 support cylnder A. Musl a, *, Z. Hlaáč a a Faculty of Appled Scences, UWB n Plsen, Unerztní, Plzeň, Czech Republc Receed 10 September 007; receed n resed form 1 September 007 Abstract The paper deals wth the optmzaton of parameters of the dynamc model of the reactor VVER 1000 support cylnder. Wthn the model of the whole reactor, support cylnder appears to be a sgnfcant subsystem for ts modal propertes hang domnant nfluence on the behaour of the reactor as a whole. Relate sensttes of egenfrequences to a change of the dscrete parameters of the model were determned. Obtaned alues were appled n the followng spectral tunng process of the (selected) dscrete parameters. Snce the past calculatons hae shown that spectral tunng by the changes of mass parameters s not effecte, the presented paper demonstrates what results are acheed when the set of the tunng parameters s extended by the geometrc parameters. Tunng tself s then formulated as an optmzaton problem wth nequaltes. 007 Unersty of West Bohema. All rghts resered. Keywords: reactor VVER 1000, support cylnder, spectral tunng 1. Introducton The tunng of the dynamc model was based on the alues of egenfrequences that were computed on the FEM model, and could also be dentfed on the dynamc model. The computaton model, that has been deeloped n Cosmos/M programmng code, contans support cylnder wth core barrel housng. The egenfrequences obtaned by modal analyss of ths model represent the obecte alues for the egenfrequences of the tuned dynamc model of the support cylnder. The tunng of the dscrete parameters of the dynamc model of the support cylnder body s the frst step of dentfcaton of the parameters of the complete VVER 1000 reactor model. It was necessary to use dentcal boundary condtons for both models so that modal propertes of the support cylnder would not be nfluenced by any other factor. In case of ths model, t means that all stffness parameters, ncludng the gland flange stffness, the bearng support stffness and the stffness of the torodal ppes, were not consdered. The only stffness element whch was consdered was the contact stffness of the support cylnder wth the flange of the pressure essel. As the past calculatons [3] showed, spectral tunng carred out only by changes of mass parameters has not appeared to be effecte. Therefore t was necessary to add some other parameters for a new tunng process. It was found reasonable that geometrc parameters (thckness of the support cylnder wall and ts dameter) could be the sutable ones to extend the set of tunng parameters. The fact that the total weght of the new dynamc model should be approxmately the same as the weght of the FEM model s also taken nto account. Ths requrement s represented by two nequalty constrants. Further, the changes of all dscrete pa- * Correspondng author. Tel.: , e-mal: amusl@kme.zcu.cz. 193

2 A. Musl et. A. al Musl / Appled et al. / and Appled Computatonal and Computatonal Mechancs Mechancs 1 (007) rameters shouldn t be radcal. Ths s proded by mposng approprate lower and upper bound for each tunng parameter. After the tunng of the dscrete parameters of the support cylnder the dynamc model wll be tested by modal analyss. Prodng the desred results are acheed, the stffness parameters mentoned aboe could be ncluded. At the meantme, the core contanng 163 fuel cassettes wll be mplemented, and next tunng processes wll then follow.. Spectral tunng The am of tunng of a mathematcal model of a mechancal system s to change the alues of parameters (mass, stffness, geometrc) to new alues that prode acheng requred characterstc alues. In case of spectral tunng they are represented by egenfrequences. Mechancal system s assumed to be undamped. Mass, stffness, or geometrc parameters, that wll be changed to tune the mathematcal model, consttute a ector p = [p ] where = 1,, s of tunng parameters. Characterstc alues, whch are selected egenfrequences n ths case, and whch are mposed some requrements, consttute a ector l = [l ] where = 1,, k. Ths ector s called a ector of tunng. Selected tuned quanttes depend (by means of coeffcent matrces M and K of the mathematcal model) on the selected tunng parameters. Thus, we may put l = l(p) s k where l(p) s a transformaton from a space R nto a space R. Hence, the tunng problem can be formulated as a problem of fndng a ector p * whch satsfes l ( p ) = l, (1) where l * s a ector of requred alues of egenfrequences. Dmenson of ths ector s k. Let s assume that the transformaton l(p) s dfferentable n the neghbourhood of the startng pont p 0. Let s defne a Jacob matrx of the transformaton l(p) as ( ) L p ( p ) T grad l 1 0 l =... = ( p0 ), p T grad l k ( ) p0 = 1,..., k; = 1,..., s. The Jacob matrx L s called a matrx of senstty. Generally, t s a rectangular matrx of dmenson (k,s) where ts element n the -th row and n the -th column determnes the rate of a change of a quantty l wth respect to a unt change of a quantty p. It expresses the senstty of a change of the -th tuned quantty (egenfrequency Ω ) to a change of the -th tunng parameter. Thus, we deal wth absolute senstty hang ts dmenson correspondng to the dmenson of the model parameters. Howeer, more useful nformaton s proded when relate senstty s used. The adantage of relate senstty s that t s related to unt frequency and unt parameter p, therefore t enables the comparson of partcular sensttes. For the relate senstty l of the changes of the -th egenfrequency to a change of the -th parameter we may wrte l Ω = p p Ω. () (3) 194

3 A. Musl et. A. al Musl / Appled et al. / and Appled Computatonal and Computatonal Mechancs Mechancs 1 (007) Tunng as an optmzaton problem l = k l = The (selected) tuned parameters form a ector of tunng [ ] 1 s p =, the selected tunng parameters form a ector = [ ] 1 p, and the requrements consttute the ector l * = l *. Accordng to [1], the tunng problem can be formulated by the k-crteron functon expressed = 1 as ψ ( p) ( p) k l = g 1, * = 1 l (4) where g s a weght of the -th parameter. Each tunng parameter p must le n a permssble close range. Values of p (lower bound of the -th tunng parameter), and p (upper bound d of the same parameter), are chosen approprate whle the startng pont must le n ths regon. Incdence wth a permssble parameter range s determned by s nequaltes d h p p p, = 1,..., s. (5) From the defnton (4), t s apparent that ψ ( p) 0. See that f all requrements are met, then the alue of the obecte functon s zero. If k-multplcty of the functon (4) s not sutable for k > 1, then t s possble to choose a preferable (most mportant) 0 -th requrement of whch fulflment s to be optmzed, and the remanng requrements are ncluded n the nequalty lmtatons. Then the obecte functon of the optmzaton problem s gen by ψ ( p) l = ( p) 0 1. * l 0 In addton to the lmtaton (5), the permssble regon wll also be specfed by the lmtatons h ( ), d whle the bounds l d and l h are sutably chosen. l l p l = 1,..., k;, (7) 0 h k (6) 4. Dynamc model of the reactor VVER 1000 support cylnder At the Department of Mechancs of the Unersty of West Bohema n Plsen, the mathematcal model of the reactor VVER 1000 nstalled n the Temeln NPP has been deeloped. The eoluton of the methodology whch has been appled n the modellng of the bratons of the reactor or the reactor components s documented n [4]. Wthn ths model, the reactor s composed of eght subsystems whch are: pressure essel, support cylnder and core wth fuel cassettes, gude tube block, upper block support system, control rod dres housngs, block of electromagnets, and dre assembles. Each subsystem s modelled n a sutably chosen confguratonal space of one of other subsystems, and couplng stffnesses are dscretzed by translatonal and rotatonal stffness elements. Support cylnder (SC) s modelled n a confguratonal space of pressure essel (PV). Its model conssts of two stff bodes (SC1 and SC3) whch are connected by one-dmensonal beam-type contnuum (SC). The body SC1 s dened relate transerse dsplacements at the locaton of ts hangng on the pressure essel flange, and relate rotatons about the re- 195

4 A. Musl et. A. al Musl / Appled et al. / and Appled Computatonal and Computatonal Mechancs Mechancs 1 (007) actor axs relate to pressure essel. SC1 can both moe longtudnally (ertcally), and rotate by bendng moton generalzed coordnates y 1 x 1 z 1. SC s dded nto two elements by a node at ts centre. Plane secton passng through the node s enabled general spatal moton - generalzed coordnates x, y, z x y z. SC3 can also moe by general spatal moton - generalzed coordnates x 3, y 3, z 3 x 3 y 3 z 3. The model of the support cylnder has 15 degrees of freedom. Vector of generalzed coordnates of the support cylnder then has a followng form: q = y, x, y, z, x, y, z. (8) SC 1 x1 z1 x y z x3 y3 z3 Fg. 1. Scheme of the dynamc model of the support cylnder. leel A emplacement of the pressure essel n the reactor concrete pt leel B hangng of the support cylnder on the pressure essel flange.1. Geometrc parameters of the model AB = 4.7 m dstance between A and B, a SC(1) = 4.0 m dstance between the upper node of the tube SC and the horzontal plane passng through A, a SC(3) =.56 m dstance between the lower node of the tube SC and the horzontal plane passng through A, 196

5 A. Musl et. A. al Musl / Appled et al. / and Appled Computatonal and Computatonal Mechancs Mechancs 1 (007) r SC = 1.8 m dstance between the centre of the contact surface of the hangng of the support cylnder on the pressure essel flange and the reactor axs, r tp = 1.8 m dstance between the contact cure of the torodal ppes wth the support cylnder and the reactor axs... Mass parameters of the model m 1 = kg weght of SC1, I 1 = kgm moment of nerta of SC1 about the transersal axs passng through the centre of graty, m 3 = kg weght of SC3, I 3 = kgm moment of nerta of SC3 about the transersal axs passng through the centre of graty, I o3 = kgm moment of nerta of SC3 about the ertcal axs of the reactor..3. Stffness parameters of the model The stffness parameters whch are the axal gland flange stffness k gf, the axal and the torsonal bearng support stffness k xbs and k xxbs, the stffness of the torodal ppes durng the ertcal moton, durng the bendng moton respectely, k ytp, k xxtp respectely, were not consdered so that boundary condtons can be dentcal wth those that were consdered n the FEM model. The alue of the contact stffness, the bendng stffness respectely, of the hangng of the support cylnder on the pressure essel flange k ysc, k xxsc respectely, was taken suffcently hgh to represent support. 5. Senstty of egenfrequences to a change of dscrete parameters The behaour of the conserate mathematcal model of the support cylnder s determned by the matrx equaton of moton Mq + Kq = 0. (9) Concrete form of the symmetrc, postely defnte mass matrx M and symmetrc matrx K s dered from the knetc and deformaton energy of the system after the substtuton nto the Lagrange equatons of the nd order Senstty of egenfrequences to a change of mass parameters Accordng to [4], for the relate senstty of the -th egenfrequency Ω to a change of the -th mass parameter (whch has no nfluence on the stffness matrx K), we can wrte l m =, (10) m where s an egenector related to an egenfrequency Ω, and whch s normalzed by matrx M, so t satsfes the equaton T M =1. (11) 197

6 A. Musl et. A. al Musl / Appled et al. / and Appled Computatonal and Computatonal Mechancs Mechancs 1 (007) M The quadratc forms contanng the matrx hae domnant role n the expresson (10). m Wthn the mathematcal model, such matrx can be expressed ery easly. For the dscrete parameters of the support cylnder, we obtan 3 = 1 + b1 ( + 3 ), (1) m 1 = + 3, (13) I 1 = b3 ( + 3 ) b3 ( ), m (14) = ( ), (15) I 3 = 14. (16) I 5.. Senstty of egenfrequences to a change of geometrc parameters 03 Generally, the relate senstty of the -th egenfrequency Ω to a change of the -th parameter s determned by the followng expresson: l 1 p T K = p. (17) Ω p p Snce the nestgated geometrc parameters nfluence only the segments of the matrces M and K whch are related to those two elements modellng the mddle part of the support cylnder, one may wrte X = p x T e T e, (18) p T e e= 1 where X = M, K, and the correspondng x e = m e, k e are mass and stffness matrces of the e-th shaft element of the annular cross-secton wth 1 degrees of freedom. T e s a matrx of transformaton between the global and the local coordnates of the e-th shaft element. The detal structure of the matrx T e s descrbed n [4]. 6. FEM model of the reactor VVER 1000 support cylnder In fg., there s a computaton model of the support cylnder whch was deeloped n Cosmos/M programmng code as a thn-shell tank, and to whch the fuel cassettes wll be mplemented n the future. The support cylnder body conssts of 14 segments of arous dameters and thcknesses. The nlets for the coolng loop ppelnes that are located n two dfferent zones of the support cylnder were not modelled. Instead of that, the thcknesses n the support cylnder segments whch contan these nlets were reduced n the rate of area of the wall wth the nlets to the area of the full wall. Usng ths smplfcaton that was already appled n the calculaton [], the bendng stffness should not change dramatcally. 198

7 A. Musl et. A. al Musl / Appled et al. / and Appled Computatonal and Computatonal Mechancs Mechancs 1 (007) Core barrel, of whch total heght s 4.07 m, conssts of two horzontal slabs and sx rngs. The shape of ther outer shell enables to nsert 163 fuel cassettes of regular hexagonal outer cross-secton wth the dameter of nscrbed crcle equal to 0.36 m. Fe upper rngs are m hgh whle the bottom rng s m hgh. These sx rngs are bolted together by sx hollow screws (outer dameter 0.15 m, nner dameter 0.1 m) wth the thread n the slab. Twele bolts by whch the bottom rng s anchored to the lower slab were not ncluded n the model. The thckness of the upper and lower slab of the core barrel s 0.1 m. The model also contans 163 ppes connectng the lower slab of the core barrel wth the bottom of the support cylnder. The support cylnder s nserted n the pressure essel n the ertcal axal drecton (y axs), and hung on the flange. At the locaton of the contact whch s between the frst and the second segment of support cylnder wall, the correspondng crcumferental nodes were mposed the zero dsplacement for all axal drectons. The reactor VVER 1000 support cylnder s made of the steel materal 08X18H10T. In accordance wth the past calculatons, the aerage operatonal temperature of the support cylnder s 305 o C. The correspondng alue of Young s modulus of elastcty E = Posson s rato and materal densty do not practcally depend on the temperature. The materal propertes for modal analyss are summarzed n tab. 1. Densty Young s modulus Posson s ρ [kg/m 3 ] E [Pa] rato υ Tab. 1. Materal propertes. Fg.. FEM model of the support cylnder. 7. The tunng process The alues of the frst eght egenfrequences computed on both models are lsted n tab.. Those egenfrequences that were found the most sutable for the tunng process are 199

8 A. Musl et. A. al Musl / Appled et al. / and Appled Computatonal and Computatonal Mechancs Mechancs 1 (007) bold marked (the frequences of the FEM model defne the obecte functons). They represent the frst bendng, torsonal, and longtudnal mode. As was already mentoned, the spectral tunng carred out by changes of mass parameters has not proded satsfyng results. Therefore n the next step, the geometrc parameters were also ncluded n the set of tunng parameters. The relate sensttes of the domnant frequences to a change of the mass parameters of the reactor VVER 1000 support cylnder are lsted n tab. 3, and to the change of the geometrc parameters (together wth the sensttes to the materal constants) are lsted n tab. 4. DYNAMIC MODEL FEM MODEL Order of Egenfrequency Order of Egenfrequency Mode shape Mode shape frequency f [Hz] frequency f [Hz] bendng bendng 5.84 bendng bendng torsonal torsonal longtudnal longtudnal bendng core barrel bendng bendng torsonal bendng longtudnal shell Tab.. Egenfrequences of the reactor VVER 1000 support cylnder. Order of frequency m 1 I 1 m 3 I 3 I o Tab. 3. Relate senstty to a change of the mass parameters of the reactor VVER 1000 support cylnder. Order of frequency r h ρ E Tab. 4. Relate senstty to a change of the geometrc parameters and materal constants of the reactor VVER 1000 support cylnder. Accordng to the alues of the relate sensttes, the selected tunng parameters were: weght of SC3 m 3, moment of nerta of SC3 about the ertcal axs of the reactor I o3, and newly nner radus of the SC secton of the support cylnder r, and thckness of the SC wall. Densty of the support cylnder materal ρ, and Young s modulus E reman unchanged. 00

9 A. Musl et. A. al Musl / Appled et al. / and Appled Computatonal and Computatonal Mechancs Mechancs 1 (007) Whle performng the tunng process, the alue of the total weght of the support cylnder should be respected. Snce the total weght found on the FEM model s tones, the followng two nequalty constrants hae been mposed: 135t m1 + m + m3 145t. (19) What remans to be done s to determne approprate alues of the lower and upper bounds of the tunng parameters. The computatons hae shown that the most conenent results can be obtaned when the bounds of the tunng parameters are as follows: (see tab. 5) Parameter Orgnal alue Lower bound Upper bound m 3 [kg] I o3 [kgm ] r [m] h [m] Tab. 5. Lower and upper bounds of the tunng parameters The results of the tunng of the dynamc model of the support cylnder shown n tab. 6 where s the comparson of the egenfrequences of the model wth the egenfreqeunces of the orgnal model and the FEM model (obecte alues). The comparson of the new alues of the tunng parameters wth the orgnal alues s shown n tab 7. Order of frequency Mode shape Frequency f [Hz] Orgnal dynamc model New dynamc model 1 bendng bendng torsonal longtudnal FEM model Tab. 6. Effect of the spectral tunng on the egenfrequences of the reactor VVER 1000 support cylnder. Parameter Orgnal alue New alue m 3 [kg] I o3 [kgm ] r [m] h [m] Tab. 7. The changes of the tunng parameters. Fnally, let us check the total weght of the new model: m SC = m + m + m = = (0) 1 3 t 01

10 8. Concluson A. Musl et. A. al Musl / Appled et al. / and Appled Computatonal and Computatonal Mechancs Mechancs 1 (007) Ths work s an ntroducton to the research of whch am s to mplement the model of the support cylnder wth core and fuel cassettes to the dynamc model of the whole reactor. Includng of the geometrc parameters to the tunng process brought a sgnfcant mproement when the desrable modal propertes were practcally acheed. On the other hand, the acheng of ths result was possble only by dramatc changes of the mass parameters of the dynamc model of the support cylnder. Therefore, applcaton of ths method n ths case stll remans questonable. Another approach, and more sutable, s to execute the condensaton of the FEM model and to mplement the condensed model nto the dynamc model of the reactor. Ths wll be the task for the next phase of the research. Acknowledgement The work has been supported by the grant proect MSM of the Mnstry of Educaton, Youth and Sports of the Czech Republc. References [1] Z. Hlaáč, Dynamc synthess and optmzaton, textbook, Unersty of West Bohema, Plzeň, [] P. Marko, J. Maer, Creatng of the computatonal models and the calculaton of the frequency-modal characterstcs of the man components of the reactor VVER 1000, research document Škoda ZES, Plzeň, [3] A. Musl, Z. Hlaáč, Identfcaton of parameters of the dynamc model of the reactor VVER 1000 support cylnder by spectral tunng, Proceedngs of the Internatonal Engneerng Mechancs Conference, Sratka, Czech Republc, 007. [4] V. Zeman, Z. Hlaáč, Modellng of braton of the reactor VVER 1000 usng a decomposton method, Proceedngs of the Internatonal Engneerng Mechancs Conference, Sratka, Czech Republc,