In the near-past we introduced a simplified linearized model for the fluid-coupled vibratory responses of nuclear fuel racks, neglecting three-dimensional flow effects and assuming small gaps between the fuel assemblies. In this paper, using the same basic approach, we generalize the above-mentioned model to account for nonlinear squeeze-film and dissipative flow effects. The proposed methodology can be automatically implemented on a symbolic computer environment. Numerical simulations highlight the significance of nonlinear flow effects at high vibration amplitudes, yielding more realistic predictions

Antunes, J.

Moreira, Miguel

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Flow Induced Vibration, de Langre & Axisa ed. Ecole Polytechnique, Paris, 6-9th July 2004A NONLINEARMODEL FOR FLUID-COUPLED VIBRATIONSOF SPENT NUCLEAR RACKSMiguel MoreiraMathematical Department, College of Technology, Setúbal Polytechnic , PortugalJosé AntunesApplied Dynamics Laboratory, ITN, Lisbon, PortugalABSTRACTIn the near-past we introduced a simplified linearizedmodel for the fluid-coupled vibratory responses of nu-clear fuel racks, neglecting three-dimensional flow ef-fects and assuming small gaps between the fuel as-semblies. In this paper, using the same basic ap-proach, we generalize the above-mentioned model toaccount for nonlinear squeeze-film and dissipativeflow effects. The proposed methodology can be au-tomatically implemented on a symbolic computer en-vironment. Numerical simulations highlight the sig-nificance of nonlinear flow effects at high vibrationamplitudes, yielding more realistic predictions.1. INTRODUCTIONSpent fuel storage racks are welded honeycomb stain-less steel structures (generally described as blockswith rectangular walls) with large surfaces (the areaof a rack wall can be of the order of 10 m2). Theyare used to accommodate spent nuclear fuel assem-blies. The spent fuel racks are placed on a fuel poolfreely on the floor and are separated by small watergaps (sometimes down to the order of 10 mm). Thefuel pool is filled with water several meters above thetop of the racks. Fluid effects induce strong couplingbetween immersed nuclear fuel racks, when they aresubjected to earthquake excitations [see for instanceBroc et al (2000); Stabel and Ren (2001); Zhao etal (1996); Hinderks et al (2001)]. Therefore, dur-ing a seismic event, spent fuel storage racks maybend, slide, twist and uplift. Note that, racks are de-signed with short aspect ratios so that they would nottilt over. Undoubted understanding the complex dy-namic behaviour of immersed spent fuel assembliesstorage racks under earthquake is of prime impor-tance for the safety of nuclear plant facilities.In the near-past [Moreira and Antunes (2002)] weintroduced a simplified linearized model for the fluid-coupled vibratory responses of nuclear fuel racksbased on the following main simplifying assump-tions: (i) Three-dimensional flow effects were ne-glected; (ii) Gaps between the fuel assemblies and be-tween these and the container were small when com-pared with the longitudinal length-scales; (iii) Dy-namical fluid effects were linearized. From theseassumptions, we postulated a simplified flow insidethe channels, such that the gap-averaged velocity andpressure fields were described in terms of a singlespace- coordinate, for each fluid channel. Time-domain simulations of the system responses to seis-mic excitations were also produced and, despite thesimplifications introduced, the model yielded qual-itatively similar predictions when compared withother recently published work. However, nonlinearsqueeze-film and dissipative effects connected withvery large amplitude responses cannot be properlymodeled unless assumption (iii) is relaxed. Such isthe aim of the present paper. Here, using the samebasic approach, we generalize the above-mentionedmodel to account for nonlinear flow effects. Al-though algebraically involved, the proposed method-ology can be automatically implemented on a sym-bolic computer environment, leading to a system ofDAE’s which is then solved through adequate time-step integration solver.2. MODEL FORMULATION2.1. Fluid FormulationConsider a pool withM ×N nuclear spent fuel racksarranged in M lines and N columns, which will bedescribed using matrix notation.The dimensions along the principal directions ofeach rack cross-section are LX and LY . The X- andY -direction channels (between each pair of racks orbetween a wall and a rack) are denoted asHXj , 1 ≤ i ≤M + 1, (1)HYj , 1 ≤ j ≤ N + 1. (2)In Figure 1 one can see the main geometrical para-meters, for a quite general system configuration.ij 1+ij11 ++ jiji 1+HYj 1+rackXLYLijXijY0Y0XHXi 1+Figure 1: Main geometrical parameters.The absolute positions of each rack(X˜ij (t) , Y˜ij (t))can be defined asX˜ij (t) = X0ij +Xij (t) , (3)Y˜ij (t) = Y0ij + Yij (t) , (4)where(X0ij, Y0ij)are the coordinates of the geomet-ric centers with respect to the pool container, thatis, (Xij (t) , Yij (t)) are the local coordinates of eachrack. So, the actual X-direction hXand Y -directiongaps hY can be defined ashXij = HXi + Yi−1j − Yij,{1 ≤ i ≤M + 11 ≤ j ≤ N(5)andhYij = HYj +Xij −Xij−1,{1 ≤ i ≤M1 ≤ j ≤ N + 1(6)whereY0j = YM+1j = Xi0 = XiN+1 = 0by definition. Note also thath˙Xij = Y˙i−1j − Y˙ij,{1 ≤ i ≤M + 11 ≤ j ≤ N, (7)h˙Yij = X˙ij − X˙ij−1,{1 ≤ i ≤M1 ≤ j ≤ N + 1(8)andh¨Xij = Y¨i−1j − Y¨ij,{1 ≤ i ≤M + 11 ≤ j ≤ N, (9)h¨Yij = X¨ij − X¨ij−1,{1 ≤ i ≤M1 ≤ j ≤ N + 1.(10)Following Antunes and Piteau (2000), a sim-plified flow model inside the X-direction and Y -direction channels can be developed from the above-mentionned assumptions. With this approach thegap-averaged velocity and the pressure fields are de-scribed in terms of a single space coordinate and thecontinuity and momentum equations, in each chan-nel, can be formulated as∂h∂t+∂∂z(uh) = 0 (11)andρ[∂∂t(uh) + h∂∂z(u2)]+ ρu |u| f + h∂p∂z= 0(12)where u = u (z, t) , h = h (t), p = p (z, t) and zstand respectively for the fluid velocity, channel gap,pressure and single spatial coordinate along the chan-nel length. The parameter f represents an appropriateskin-friction coefficient accounting for the flow/wallsdistributed stresses. Typically f changes with theflow Reynolds number as f = aReb for adequate pa-rameters a and b [see, for instance Hirs (1973)].Integration of the continuity equation (11), leadsto the general velocity field for the X-direction andY -direction channels asuXij (x, t) = −h˙XijhXijx+ CXij , (13)with 1 ≤ i ≤M + 1, 1 ≤ j ≤ N , anduYij (y, t) = −h˙YijhYijy + CYij , (14)with 1 ≤ i ≤M and 1 ≤ j ≤ N +1, where CXij andCYij are functions of time stemming from the integra-tion.The following forms of equation (12) can be de-duced for the X-direction (with 1 ≤ i ≤ M + 1and 1 ≤ j ≤ N ) and Y -direction channels (with1 ≤ i ≤M and 1 ≤ j ≤ N + 1)∂pXij∂x= −ρ1hXij∂∂t(uXijhXij)+2uXij∂∂x(uXij)+ 1hXijuXij∣∣∣uXij ∣∣∣ f, (15)∂pYij∂y= −ρ1hYij∂∂t(uYijhYij)+2uYij∂∂y(uYij)+ 1hYijuYij∣∣∣uYij∣∣∣ f, (16)where pXij and pYij are the pressures alongX-directionand Y -direction channels.Integration of equations (15)-(16) yields for 1 ≤i ≤M + 1 and 1 ≤ j ≤ NpXij (x, t) =ρ2(h¨XijhXij− 2(h˙XijhXij)2)x2+ρ(h˙XijhXijCXij − C˙Xij)x+ρ3∣∣∣∣∣h˙XijhXijx−CXij∣∣∣∣∣3h˙Xijf−ρ3|CXij |3h˙Xijf + pXij (0, t),(17)and for 1 ≤ i ≤M and 1 ≤ j ≤ N + 1pYij (y, t) =ρ2(h¨YijhYij− 2(h˙YijhYij)2)y2+ρ(h˙YijhYijCYij − C˙Yij)y+ρ3∣∣∣∣∣h˙YijhYijx−CYij∣∣∣∣∣3h˙Yijf−ρ3|CYij (t)|3h˙Yijf + pYij (0, t).(18)The X- and Y - direction fluid forces acting (perunit length) on each rack can be found as follows:FXij (t) =∫ LY /2−LY /2(pYij (y, t)− pYij+1 (y, t))dy,F Yij (t) =∫ LX/2−LX/2(pXi+1j (x, t)− pXij (x, t))dx,for 1 ≤ i ≤M and 1 ≤ j ≤ N , that is,FXij (t) =ρ24(h¨YijhYij− 2(h˙YijhYij)2)L3Y− ρ24(h¨Yij+1hYij+1− 2(h˙Yij+1hYij+1)2)L3Y+ ρ12hYij(h˙Yij)2TYi,j (t) f− ρ12hYij+1(h˙Yij+1)2TYi,j+1 (t) f+ρ3(|CYij+1|3h˙Yij+1−|CYij |3h˙Yij)fLY+(pYij (0, t)− pYij+1 (0, t))LY,(19)with T Yi,j (t) =∣∣∣∣ h˙YijhYij LY2 − CYij∣∣∣∣3(h˙YijhYijLY2 −CYij)+∣∣∣∣ h˙YijhYij LY2 + CYij∣∣∣∣3(h˙YijhYijLY2 + CYij)andFYij (t) =ρ24(h¨Xi+1jhXi+1j− 2(h˙Xi+1jhXi+1j)2)L3X− ρ24(h¨XijhXij− 2(h˙XijhXij)2)L3X+ ρ12hXi+1j(h˙Xi+1j)2TXi+1,j (t) f− ρ12hXij(h˙Xij)2TXi,j (t) f+ρ3(|CXij |3h˙Xij−|CXi+1j|3h˙Xi+1j)fLX+(pXi+1j (0, t)− pXij (0, t))LX,(20)with T Xi,j (t) =∣∣∣∣ h˙XijhXij LX2 − CXij∣∣∣∣3(h˙XijhXijLX2 − CXij)+∣∣∣∣ h˙XijhXij LX2 +CXij∣∣∣∣3(h˙XijhXijLX2 + CXij) .Note that in equations (17)-(20)X¨ij = X˙ij = 0, if i = 0 or i = M + 1 andY¨ij = Y˙ij = 0, if j = 0 or j = N + 1,as (Xij (t) , Yij (t)) are the relative coordinates ofeach rack.2.2. Formulation of the coupled systemAssuming that the racks are linear systems with struc-tural massMs, damping Cs and stiffnessKs, all theseparameters being per unit length, one can deduce thefollowing fluid-structure model:MsX¨ij +CsX˙ij +KsXij = FXij + FXij,aut,(21)MsY¨ij + CsY˙ij +KsYij = FYij + FYij,aut,(22)for 1 ≤ i ≤ M and 1 ≤ j ≤ N where FXij , FYij ,FXij,aut and FYij,aut represent, respectively, the above-deduced fluid forces and the external autonomousforces per unit length. Here, the structural parame-ters have been assumed identical for both directions.However, dealing with asymmetrical systems bringno further difficulties whatsoever.2.3. Formulation of the complete systemNote that the 2 ×M × N equations (21)-(22) gen-erated by this approach are not sufficient to find allthe corresponding unknowns which are summarizedin Table 1.However, between rack or rack/wall positions ij,ij + 1, i + 1j and i + 1j + 1, one can establishthe additional equations we need, namely, (M + 1)×(N + 1)− 1 linearly independent equations of com-patibility of flow (mass conservation for all nodes butone), 4×M×N−(M − 1)×(N − 1) linearly inde-pendent equations of compatibility of pressure (in allcorners of each rack except (M − 1)× (N − 1) cor-ners) and finally one last equation setting a referencefor the pressure.Unknowns NumberXij (t) MNYij (t) MNCXij (t) (M + 1)NCYij (t) M (N + 1)pXij (0, t) (M + 1)NpYij (t) M (N + 1)Total 6MN + 2 (M +N)Table 1: Total number of unknowns.To complete the model formulation, a number ofadditional algebraic equations must be stated in orderto enforce the flow continuity and pressure compati-bility between flow-cells. Here are the compatibilityequations of the flow (or continuity in each corner)h˙Xi+1j+1LX + 2CXi+1j+1hXi+1j+1+h˙Yij+1LY + 2CYij+1hYij+1+h˙Xi+1jLX − 2CXi+1jhXi+1j+h˙Yi+1j+1LY − 2CYi+1j+1hYi+1j+1= 0, (23)where 0 ≤ i ≤ M and 0 ≤ j ≤ N . Note thatwe must disregard one of these equations in order toobtain a set of (M + 1) × (N + 1) − 1 linearly in-dependent equations. In fact, the flow in a corner iscompletely determined by the flow in the remainingcorners.The equations of compatibility of pressure estab-lish the following 4×M ×N − (M − 1)× (N − 1)relationspXij (−LX/2, t) = pYij (−LY /2, t) , (24)pXij (LX/2, t) = pYij+1 (−LY /2, t) , (25)pXi+1j (−LX/2, t) = pYij (LY /2, t) , (26)where 1 ≤ i ≤M , 1 ≤ j ≤ N andpXi+1j (LX/2, t) = pYi+1j+1 (−LY /2, t) , (27)where{i = M1 ≤ j ≤ Nand{1 ≤ i ≤M − 1j = N.Finallyi=M+1,j=N∑i=1,j=1pXij (0, t) +i=M,j=N+1∑i=1,j=1pYij (0, t) = 0,(28)sets a reference for the pressure.Observe that at the pool walls one has in equations(23)-(27)X¨ij = X˙ij = 0, if i = 0 or i = M + 1 andY¨ij = Y˙ij = 0, if j = 0 or j = N + 1,whenever In Table 2 we summarize the above-mentioned equations defining our linearized modelfor the fluid-coupled vibratory responses of the sys-tem.Equation Number of Equations(21) MN(22) MN(23) (M + 1) (N + 1)− 1(24)-(27) 4MN − (M − 1) (N − 1)(28) 1Total 6MN + 2 (M +N)Table 2: Equations of the nonlinear model for thefluid-coupled vibratory response of the system.All these equations represent a set of differential-algebraic equations (DAE’s). That is, among thoseequations, some of them are pure algebraic constrainsbetween unknowns. Clearly, this is the case of equa-tion (28). Note that this class of equations arise natu-rally in many applications but present numerical andanalytical difficulties which do not occur with sys-tems of ordinary differential equations Brenan et al(1996). In our case the DAE’s developed can be clas-sified as a nonlinear differential-algebraic system ofequationsNote that these equations can be written and estab-lished for generic systems of M × N racks entirelyon a symbolic computer environment as it was donehere for the ilustrative computations.3. NUMERICAL SIMULATIONSDefine the set of differential-algebraic equations cor-responding to our model asF (v˙,v,t) = 0 (29)where v is the vector of unknowns. The simplestfirst order backward difference formula is the implicitEuler methodF(vn+1 − vntn+1 − tn,vn+1,tn+1)= 0 (30)in which equation (29) is approximated by finite dif-ferences [Brenan et al (1996)]. In the present workwe used a fourth and fifth-order generalization of (30)coded in MATLAB [Roberts (1998)].In order to validate our nonlinear model, sev-eral numerical simulations were performed over fuelpools with one centered single rack and fuel poolswith six racks regularly stored in 2 lines and 3columns. The present numerical simulations agreedwith our previous work, for low vibration amplitudes(with respect to the average gaps), when the lineariza-tion assumptions were fulfilled [Moreira and Antunes(2002)]. However when nonlinear flow effects aresignificant, our new nonlinear model yields muchmore realistic predictions. Consider two fuel pools(Pool A and Pool B), using dissimilar fluid gaps,each one with six racks regularly stored in 2 linesand 3 columns. Their response to the same seismicexcitation is presented next. The seismic excitationused was the east-west component of the Loma Pri-eta earthquake in October, 1989, applied along theX-direction (also an axis of symmetry of the storagepools with the racks).Pool A Pool BLX (m) 2 2LY (m) 2 2HXi 0.2 0.03HYj 0.2 0.03Table 3: Main geometrical parameters for the numer-ical simulations.All numerical simulations were performed with themain geometrical, physical and structural modal pa-rameters presented in Tables 3 and 4. The time-step used t = 0.005, was one order of magnitudesmaller than 1/(2fmax), with fmax ≈ 20 Hz (maxi-mum frequency of interest assuming a dominant fre-quency in the seismic spectrum below 10 Hz). In or-der to provide an appropriate spectral resolution thesenumerical simulations were performed over an inter-val of 35 s.4. RESULTS AND DISCUSSIONIn Figures 2 and 3 we display the spectral responsein both directions of the rack in position (1, 1)when all the fuel storage pools were submitted tothe Loma Prieta earthquake applied along the X-direction. Note that in Pool A the inter-rack gaps andStructural mass,Ms (kg) 32000Structural damping, Cs (N s/m) 8000Structural stiffness,Ks (N/m) 5× 106Modal frequency in air, fs (Hz) 2Reduced damping in air, ζ 0.01Rack density, (ρs/ρwater) 8fricton factor, f 0.01Table 4: Physical and structural modal parametersfor the numerical simulations.wall/rack gaps are 0.2 m, while in Pool B are only0.03 m, so that nonlinear squeeze-film and dissipa-tive effects are enhanced.In spite of the excitation being the same on eachrack and only along the direction X responses dis-played in Figures 2 and 3 exhibit motions perpen-dicular to this direction. This is accounted for thepresence of a strong fluid-structure interaction, whichcoupleds all racks and the two motions directions.In Figure 2 one can observe the spectra responseconcentrated around 1.5 Hz. Note that the in air, themodal frequency of the structure was 2 Hz. This dif-ference is obviously due to the fluid added mass ef-fect.0 0.5 1 1.5 2 2.5 30123456789x 10-3 Rack (1,1)|FFT(X)|0 0.5 1 1.5 2 2.5 300.511.522.533.544.55x 10-3|FFT(Y)|Frequency [Hz]Figure 2: Pool A. Spectral response of rack (1, 1) tothe seismic excitation applied along directionX: (a)responses along directionX and (b) responses alongdirection Y .In Figure 3 the spectral response energy of rack(1, 1), in both directions, is now diluted over a largerfrequency interval (between 0.4Hz and 1.4Hz). Thisenergy spread is related to the nonlinear flow forces,as a result of the smaller inter-rack and rack/wallgaps, which enhanced the squeeze-film and dissipa-tive effects. Additionally one observe also that theincreased fluid added mass lowered even further theresponse frequencies of our system.0 0.5 1 1.5 2 2.5 300.511.522.533.544.55x 10-3 Rack (1,1)|FFT(X)|0 0.5 1 1.5 2 2.5 300.20.40.60.811.21.4x 10-3|FFT(Y)|Frequency [Hz]Figure 3: Pool B. Spectral response of rack (1, 1) tothe seismic excitation applied along directionX: (a)responses along directionX and (b) responses alongdirection Y .5. CONCLUSIONSIn this paper, we propose a generalization of ourprevious linearized approach [Moreira and Antunes(2002)], introducing a nonlinear model for the fluid-coupled vibratory responses of nuclear fuel racks, ne-glecting three-dimensional flow effects and assum-ing small gaps between the fuel assemblies. Theproposed methodology can be automatically imple-mented on a symbolic computer environment. Nu-merical simulations highlighted the significance ofnonlinear flow effects at high vibration amplitudes (orsmall inter-rack and wall/rack gaps). Under such con-ditions, the proposed approach yields more realisticpredictions than our previous work.6. REFERENCESAntunes, J. and Piteau, P., 2001, A Nonlinear Modelfor Squeeze-film Dynamics Under Axial Flow, Pro-ceedings of the ASME Pressure Vessel and PipingConference, Atlanta, USA, July 2001, 420-2: 53-62.Blevins, R., 1984, Fluid Dynamics, Van NostrandReinhold Company, New York.Brenan, K. E., Campbell, S. L. and Pezold, L. R.,1996, Numerical Solution of Initial-Value Problemsin Differential-Algebraic Equations, SIAM.Broc, D., Queval J and Chaudat, T, 2000, Fluid Struc-ture Interaction For Nuclear Spent Fuel racks, Pro-ceedings of PVP , Emerging Technologies in Fluids,Structures and Fluid/Structure Interactions. PressureVessel and Piping Conference, Seattle, USA, July2000, PVP 414-2: 171-177.Hinderks, M., Ungoreit, H. and Kremer, G., 2001,Improved method to demonstrate the structural in-tegrity of high density fuel storage racks. NuclearEngineering and Design 206: 177-184.Hindmarsh, A. C. and Petzold, L. R., 1988, Numeri-cal Methods for Solving Ordinary Differential Equa-tions and Differential/Algebraic Equations, EnergyTech. Rev., September, 23-36, 1988.Hirs, G., 1973, A Bulk Flow Theory for Turbulencein Lubrificant Films, ASME Journal of LubrificationTechnology, 95: 137-146.Moreira, M. and Antunes, J., 2002, A simplifiedLinearized Model for the fluid-coupled vibrations ofspent nuclear fuel racks. Journal of Fluids and Struc-tures 16(7): 971-987.Roberts, A. J., 1998, Differential - Algebraic equa-tions solver DAE, http://www.mathworks.com /sup-port /ftp /diffeqv5.shtml.Stabel, J. and Ren, M., 2001, Fluid-structure-interaction for the analysis of the dynamics of fuelstorage racks in the case of seismic loads. NuclearEngineering and Design 206: 167:176.Zhao, Y., Wilson, P. R. and Stevenson, 1996, Nonlin-ear 3-D dynamic time history in the reracking modifi-cations for a nuclear power plant. Nuclear Engineer-ing and Design 165: 199-211.

A nonlinear model for the fluid-coupled vibrations of  spent nuclear racks

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A nonlinear model for the fluid-coupled vibrations of spent nuclear racks

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