Seismic and stress qualification of LMFR fuel rod and simple method for determination of LBE added mass effect

Seismic and stress qualification of LMFR fuel rod and simple method for determination of LBE added mass effect 1 2 first-author https://orcid.org/0000-0003-0841-5641 1 corresp jianwei.chen@fds.org.cn jianwei.chen@fds.org.cn jianwei.chen@fds.org.cn https://orcid.org/0000-0003-0242-2445 1 2 1 1 1 Key Laboratory of Neutronics and Radiation Safety, Institute of Nuclear Energy Safety Technology, Chinese Academy of Sciences, Hefei 230031, China Key Laboratory of Neutronics and Radiation Safety, Institute of Nuclear Energy Safety Technology, Chinese Academy of Sciences, Hefei 230031, China University of Science and Technology of China, Hefei 230027, China University of Science and Technology of China, Hefei 230027, China In this study, two different designs of liquid metal fast reactor (LMFR) fuel rods wire-wrapped and non-wire-wrapped (bare) are compared with respect to different parameters as a means of considering the optimum fuel design. Nuclear seismic rules require that systems and components that are important to safety must be capable of bearing earthquake effects, and that their integrity and functionality should be guaranteed. Mode shapes, natural frequencies, stresses on cladding, and seismic aspects are considered for comparison using ANSYS. Modal analysis is compared in a vacuum and in lead-bismuth eutectic (LBE) using potential flow theory by considering the added mass effect. A simple and accurate approach is suggested for the determination of the LBE added mass effect and is verified by a manually calculated added mass, which further proved the usefulness of potential flow theory for the accurate estimation of the added mass effect. The verification of the hydrodynamic function (τ) over the entire frequency range further validated the finite element method (FEM) modal analysis results. Stresses obtained for fuel rods against different loading combinations revealed that they were within the allowable limits with maximum stress ratios of 0.25 (bare) and 0.74 (wire-wrapped). In order to verify the structural integrity of cladding tubes, stresses along the cladding length were determined during different transients and were also calculated manually for static pressure. The manual calculations could be roughly compared with the ANSYS results and the two showed a close agreement. Contact analysis methodology was selected and the most appropriate analysis options were suggested for establishing contact between the wire and cladding for the wire-wrapped design grid-independence analysis, which proved the accuracy of the results, confirmed the selection of the appropriate procedure, and validated the use of the ANSYS mechanical APDL code for LMFR fuel rod analysis. The results provided detailed insight for the structural design of LMFR fuel rods by considering different structural configurations (i.e., bare and wire-wrapped) in the seismic loading; this not only provides a FEM procedure for LMFR fuel with complex configuration, but also guides the reference design of LMFR fuel rods. LMFR Fuel rod Added mass Seismic analysis Contact analysis 1. Introduction 1 The Chinese Academy of Sciences (CAS) has launched an engineering program for nuclear transmutation by developing an accelerator driven subcritical (ADS) system. The design proposed by the Institute of Nuclear Energy Safety Technology (INEST), was selected as the reference reactor [ 1 1 ][ 2 2 ][ 3 3 ][ 4 4 ][ 5 5 ] . Initially, the design of a 10 MW th lead-bismuth cooled research reactor was under consideration [ 6 6 ][ 7 7 ][ 8 8 ][ 9 9 ][ 10 10 ] . The ADS system is an advanced stage nuclear energy system for the transmutation of long-lived radioactive waste and fission fuel breeding [ 8 8 ][ 9 9 ][ 10 10 ][ 11 11 ][ 12 12 ][ 13 13 ][ 14 14 ] . Nuclear fuel is the most critical and important component that bears safety Class-III, ASME Class-I, and seismic Class-I code components. To accomplish the requirements laid down in the codes, seismic qualification of the fuel rod is mandatory with respect to operational basis earthquakes (OBEs) and safe shutdown earthquakes (SSEs). Researchers have studied the potential failures of fuel rod cladding and have shown that the mechanical failure of fuel rods encompasses cladding collapse, fatigue fracture, and rupture, and that these excessive stresses on cladding are usually caused by different operational transients during the lifetime of a plant [ 15 15 ] . The authors proposed the RODSIS rod design model, whereby the fuel rods were fastened using a stiff middle-grid and softer end-grids that were made of HT-9 due to high coolant density. Finite element analysis (FEA) was conducted using ANSYS for thermal and mechanical analyses of the middle grid only, and validated that for a peak power of the fuel rod, adequate margins exist against static stresses, fuel melting, clad oxidation, and fuel cladding chemical interaction (FCCI) without taking into consideration the seismic aspect. Another study highlighted the mechanical response, deformation, and bowing of wire-wrapped fuel rods and their effect on the wrapper by considering their neutronic and thermal environments for liquid metal fast breeder reactor (LMFBR) fuel assembly [ 16 16 ][ 17 17 ][ 18 18 ] . A sub-channel deformation analysis code (SHADOW) for wire-wrapped fuel assemblies was developed and applied to study the deformation due to thermal bowing of 169 fuel pins of a prototype fuel assembly of an LMFBR. This code was considered to be an effective tool for thermal and structural analysis of fuel assembly, but it lacks the capability to accommodate dynamic effects in the event of an earthquake event. The flow path of the coolant between the rods within the fuel assembly is maintained either by tightly wrapped thin wires around the rods or by grid spacers. The Fermi reactor in the U.S. and the demonstration and prototype plants in Germany and the U.K. used grid spacers, whereas a wire-wrapped design has been adopted for the fuel assembly design by other plants [ 19 19 ][ 20 20 ] , which focused on the computational fluid dynamics (CFD) analysis of a 19-pin, wire-wrapped, LBE cooled fuel assembly of a GEN-IV research reactor. A CFD and sensitivity analysis model was developed to describe and validate the experimental test section [ 20 20 ] . This code was validated by the experimental data of a similar geometry cooled by sodium by considering only the heat transfer effects. A steel wire of a usually circular cross-section is welded to the extreme ends of cladding and wrapped around the cladding with a quantified axial pitch. The formerly used grid-spacer design usually consists of a steel web-like structure containing rods, which is attached to the fuel assembly duct wall at designated axial levels depending upon the length of the assembly. Wire-wrapped designs are widely preferred for their several benefits, including that fabrication is easy and less expensive when compared to the grid-spacer design that was usually employed in the previous generation of reactors [ 20 20 ] . Due to the use of wire-wrapped technology, contact with neighboring cladding occurs at six axial locations for every pitch of wire, which helps to minimize cladding mechanical vibrations. The grid design requires several grids to provide a similar structural stability, thus resulting in an unnecessary pressure drop. On the other hand, the added advantage of the wire-wrapped design is that it enables a better heat mixing of the coolant due to an increase in the local turbulence of the coolant. Generally speaking, mostly of the studies that have been conducted so far in the research and development of LMFRs relate to heat transfer and fluid dynamics. To some extent, literature exists regarding the justification of the seismic design of reactor components as a whole (fuel assembly or core), but is limited to individual fuel rods or a comparison between different designs of fuel rods. The present study highlights the comparison of two different designs of fuel rods that are under consideration for the CLEAR-I (China Lead-Based Research Reactor) and presents their qualification in detail by considering different loads (e.g., dead weight (DW)), design and operating pressures (DP, OP), earthquakes (as response spectrum), different plant conditions, and combinations of the aforesaid loads according to code requirements using ANSYS mechanical APDL code (ANSYS parametric design language). The present work deals with the seismic and stress analyses of the Class-I component that should be confirmed "by analysis" instead of "by rules" [ 21 21 ][ 22 22 ] . 2. Structural Design and Modeling 1 The fuel rod comprised a 15-15Ti stainless-steel cladding holding UO 2 pellets [ 23 23 ] enclosed by upper and lower end caps, along with reflectors, gas chambers, and a ballast for compensating the buoyancy caused by the high density of the LBE. Based on the CLEAR-I design, each fuel assembly consisted of 61 fuel rods [ 24 24 ][ 25 25 ] . The two designs under consideration differed in geometry in terms of one having a tightly wrapped wire around cladding and being welded at extreme ends of the cladding. The remaining features of the two designs are completely identical. Figure 1 Figure 1 shows the schematic of two different fuel rod designs for the CLEAR-I. The fuel rods in most LMFR’s assemblies are separated by wire that is usually of a very small diameter, which is wrapped helically around the fuel-rod cladding (wire spacers) along its axis. These wire-wrapped fuel rods were initially designed to maintain a constant gap among the fuel rods, but it was later found that they also provide additional turbulence and cause rotation of the liquid metal coolant within the assembly as it flows through the core [ 26 26 ] . In consideration of the service conditions and design load cases, the preliminary structural design parameters and material properties of fuel rods are listed in Tables 1 Tables 1 and 2 2 . 1 1 Component 1 1 Wire-wrapped 1 1 Bare 1 1 Total length (mm) 1 1 1675 1 1 1675 1 1 Cladding length (mm) 1 1 1630 1 1 1630 1 1 Cladding outside diameter (mm) 1 1 12 1 1 12 1 1 Cladding thickness (mm) 1 1 0.4 1 1 0.4 1 1 Total length of wire (mm) 1 1 1843.62 1 1 - 1 1 Diameter of wire (mm) 1 1 1.6 1 1 - 1 1 Number of loops 1 1 3 1 1 - 1 1 Component 1 1 Material 1 1 S u (MPa) 1 1 S y (MPa) 1 1 S m (MPa) 1 1 E (MPa) 1 1 ρ (Kg/m 3 ) 1 1 υ (-) 1 1 Fuel rod 1 1 316L 1 1 360 1 1 149 1 1 120 1 1 1.65e5 1 1 7778 1 1 0.3 1 1 Wire 1 1 316L 1 1 360 1 1 149 1 1 120 1 1 1.65e5 1 1 7778 1 1 0.3 During assembling, the pellets are piled within the cladding to the required height. A compressive spring is then installed on the top of the fuel column, and the upper and lower end plugs are inserted and welded onto the ends of the clad tube. In order to minimize clad stresses and clad flattening due to the coolant operating pressure, the fuel rod is internally pressurized with helium during the plug welding process. An initial rod pressure is selected to slow down the pellet-clad mechanical interaction and to avoid the probability of rod flattening. Modeling of the wire around cladding requires special care and attention because the wire should remain in contact with the cladding along the whole length of the cladding. Wire surrounds the cladding in complete full 03 loops or turns. By choosing a correct angle and performing orientation calculations for modeling, the ANSYS model is able to attain the geometry that is closest to the actual geometry. The cladding was divided into equal numbers after exact calculations of wire turns and radius were performed. A helical path was made around the rod centerline according to the number of cladding divisions. A circular cross-section equal to the wire diameter was created for the wire, which was away from and exactly equal to the radius of wire, on the bottom horizontal plane. This was then simplified radially to reduce the cell count followed by sweeping the cross-section along the helical path. The purpose of the geometric estimation in this method is to keep the wire perfectly horizontal instead of normal to the helical path in order to resemble the real geometry because of the selection of the sweep path and the orientation of the plane on which the sweep sketch was made. By using accurate wire dimensions, this problem is hardly evident: the angle between the wire normal plane and the horizontal plane is smaller, so the variation in the sweep is considerably small and can therefore be ignored [ 27 27 ] . ANSYS 3-D models of both designs and wire are shown in Fig. 2 Fig. 2 . 3. Methodology of Analysis and Assumptions 1 DP (0.9 and 3.75 MPa) and OP (0.6 and 2.5 MPa) were applied on the outside and inside areas of the cladding. Only plant operating conditions that involve seismic loadings were considered. Thus, the service levels with a loading combination other than seismic loads were overlooked, as listed in Table 3 Table 3 . Moreover, in the absence of nozzles, service levels B and C are the same. 1 1 Service level 1 1 Operating condition 1 1 Load combination 1 1 Level 0 1 1 Design 1 1 DP + DW 1 1 Level A 1 1 Normal 1 1 OP + DW 1 1 Level B 1 1 Upset 1 1 OP + DW + OBE 1 1 Level D 1 1 Faulted 1 1 OP + DW + SSE 3.1 Floor Response Spectrum 2 The U.S. NRC RG 1.60 spectra was used as the site response spectrum. The design response spectra for SSEs was characterized by RG 1.60 spectra, the horizontal component of which was scaled to a maximum ground acceleration of 0.3 g and the vertical component was scaled to a maximum ground acceleration of 0.217 g, which are considered to be fit for a typical site in China for an LMFR. The response spectra for horizontal and vertical components of OBEs were obtained by dividing the corresponding values of the SSE spectra by 2. Figure 3 Figure 3 shows the acceleration versus frequency curves for OBEs and SSEs. The spectrum amplification factors for SSEs and OBEs are all in accord with RG 1.60. Since the fuel rods first-order natural frequencies ( Table 4 Table 4 ) are 8.07 Hz (bare) and 7.91 Hz (wire-wrapped), they are less than the zero-period acceleration of 33 Hz; hence, the seismic calculation of the equivalent static method could not be applied. The FEM based software ANSYS was used in this analysis. The methodology is presented in a flow chart in Fig. 4 Fig. 4 . 1 1 Model type 1 1 Element type 1 1 Total elements 1 1 Total nodes 1 1 Bare 1 1 Solid95 1 1 658,560 1 1 1320976 1 3 Wire-wrapped 1 1 Solid95 1 3 1178846 1 3 2282429 1 1 Targe170 1 1 Contac174 Following assumptions were made before undergoing any analysis: The fluid (LBE) was viscous, incompressible, stationary and Newtonian gravity effects were neglected. The rod bundle did not undergo any deformation, and the pitch between the rods remained constant in the fuel assembly. The effective density method was used for the analysis, which included the density of all integral components of the fuel rod to form a cumulative mass for the fuel rod. In a stationary fluid, added stiffness is typically quite small compared to the structural stiffness, and can therefore be ignored. The model was then investigated by static (pressure and weight) and dynamic (OBE and SSE) analyses, whereby loads were applied and results were compared with the allowable limits defined by code. 3.2 Mesh Detail and Mesh independency study 2 After modeling in 3-D with the necessary details, the models were discretized with a higher-order 20-node 3-D solid element (SOLID95) to tolerate irregular configurations of geometry without loss of certainty. SOLID95 is compatible with displacement shapes and is appropriate for curved borders [ 28 28 ] . The type of elements used and the total number of elements and nodes used in this analysis are listed in Table 4 Table 4 , and the 3-D meshed view of the fuel rod and the wire are shown in Fig. 5 Fig. 5 . The authenticity of the FEM results is based upon the convergence of solution, i.e., the results should become independent to the mesh or grid size. The grid independency procedure provides the basis for selecting the optimum mesh-size for better and accurate results with minimal processing time. The meshed fuel rods (bare and wire-wrapped) were subjected to a calculated stress with three different meshed states: (i) A coarse mesh of (0.262075, bare) and (1.923422, wire-wrapped) million total nodes; (ii) A medium mesh of (1.320976, bare) and (2.282429, wire-wrapped) million total nodes; (iii) A fine mesh of (1.51195, bare) and (3.311429, wire-wrapped) million total nodes. The analysis results revealed that the solution became independent to the mesh size for both models when the total number of nodes were (1.320976, bare) and (2.282429, wire-wrapped) million, as can be seen in Fig. 6 Fig. 6 , which plots the average stress intensity ( S INT ) against the number of nodes. 4. Contact Analysis Strategy for Wire-wrapped Design 1 ANSYS FEA has a huge library of elements. These elements range from those that are simple, complex, sophisticated, and of general purpose. Selection of the best choice of contact element and solution options can greatly affect the results and system performance. By considering the geometry of the cladding and by keeping in view the layout of the wire-wrapped around the cladding, surface-to-surface contact elements TARGE170 and CONTA174 are best suited to this analysis because they overcome most of the restrictions or limitations of other elements. This is due to the fact that 3-D systems with rigid or flexible target faces can be treated, sliding is permitted, higher-order solid (e.g., Solid95) element faces are compatible, and stiffness can be assigned automatically for a closed gap. The software assigns default values of parameters for contact elements but still provides the user with a selection of different options: stiffness of a closed gap (FKN), formulation methods (penalty method or penalty plus LaGrange multipliers), allowable penetration tolerance (FTOLN), pinball region size (PINB), friction/sliding behavior, and initial closure (ICONT). For this analysis the default values assigned by the software were used. 4.1 ASYMMETRIC Pattern of CONTACT Elements 2 Asymmetric contact pattern defines the selection of target and contact surfaces and assigns all the contact elements on a contact surface and the target elements on a target surface. This is also called 'one pass contact’ and is the most efficient way of modeling surface-to-surface contact. ANSYS provides the following guidelines for defining the contact and target surfaces in a model for TARGE170/CONTA174 elements: If convex and flat or concave surfaces come into contact, the concave/flat surface would be the target surface. If fine and coarse mesh surfaces come into contact, the coarse mesh would be the target surface. If stiffer and softer surfaces come into contact, the stiffer surface would be the target surface. If surfaces with higher and lower order elements come into contact, the lower order element surface would be the target surface. The general layout for surface-to-surface contact is shown in Fig. 7 Fig. 7 . Keeping in view the aforementioned guidelines, the wire-wrapped around the cladding was treated as the target surface because the solid wire of 1.6 mm diameter was much stiffer and the cladding was treated as the contact surface in this analysis. The cladding surface was meshed with CONTA174 and the wrapped wire surface was meshed with TARGE170 elements. While using the TARGE170/CONTA174 element pair meshing, the following elements from the KEYOPT and SOLUTION settings were considered after a detailed review of available literature and the ANSYS guidelines to expedite the solution: KEYOPT,3,2,1, KEYOPT,3,6,0, KEYOPT,3,7,0, KEYOPT,3,8,1, KEYOPT,3,9,1, KEYOPT,3,11,0, KEYOPT,n,12,5; SOLCON, OFF, OFF, NSUBST, 1, 1, 1, NEQIT, 1, LNSRCH, OFF, NLGEOM, OFF. "Always bonded" contact was found to be a very useful feature. The use of contact elements is quite ideal when compared to the constraint equations to connect different meshes because the contact elements function is appropriate for large deformation problems, whereas the constraint equations are limited to small deformations. 5. Results and Discussion 1 Since fuel pellets are not a solitary integral piece, they do not contribute during the bending moment. The fuel column was therefore divided into small pieces (pellets), which did not defy the bending of the fuel rod [ 26 26 ][ 27 27 ] ; hence, it can be fairly assumed that the pellets contributed only to the mass of the fuel rod and not to the stiffness, and that the mass per unit length collectively included the fuel and cladding [ 28 28 ] . The lower end cap was constrained in all degree of freedom (DOF), whereas the upper-end cap was constrained in horizontal directions, thereby allowing room for thermal expansion of the fuel rod in the vertical direction. 5.1 Modal Analysis in A Vacuum and in LBE 2 Modal analysis of the two designs was compared in a vacuum and in LBE by means of potential flow theory using the added mass effect (i.e., the mass of LBE displaced by the fuel rod volume). The added mass effect in the system due to LBE (higher density in comparison to the vacuum), was inversely proportional to the natural frequencies. The results obtained from the modal analysis showed very little difference in natural frequencies and mode shapes of the two designs. The wire-wrapped design showed a small lag in both mode shapes and natural frequencies in comparison to the bare design because of an increase in mass, as exhibited in Fig. 8 Fig. 8 . Referring to recent numerical studies, the hydrodynamic function (τ) for a circular cylinder and a thin rectangular beam are approximately identical. Furthermore, it has been reported that the hydrodynamic functions for a rectangular beam and a circular cylinder possess the same asymptotic forms, and the difference in the results never exceeds 15% over the entire frequency range [ 29 29 ] . <math display="block" id="M1"><mrow><msub><mtext>ω</mtext><mrow><mtext>lbe</mtext></mrow></msub><msub><mrow/><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msub><mo>=</mo><msub><mtext>ω</mtext><mrow><msub><mrow><mtext>vac</mtext></mrow><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msub></mrow></msub><mo> </mo><msup><mrow><mrow><mo>[</mo><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mi>π</mi><mi>l</mi><msub><mi>ρ</mi><mrow><mtext>lbe</mtext></mrow></msub></mrow><mrow><mn>4</mn><mi>t</mi><msub><mi>ρ</mi><mrow><mtext>rod</mtext></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow><msub><mi>τ</mi><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msub></mrow><mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math> where, <math id="M2"><mrow><msub><mtext>ω</mtext><mrow><mtext>lbe</mtext></mrow></msub></mrow></math> and <math id="M3"><mrow><msub><mtext>ω</mtext><mrow><mtext>vac</mtext></mrow></msub></mrow></math> are frequencies in LBE and the vacuum, <math id="M4"><mrow><msub><mi>ρ</mi><mrow><mtext>lbe</mtext></mrow></msub></mrow></math> and <math id="M5"><mrow><msub><mi>ρ</mi><mrow><mtext>rod</mtext></mrow></msub></mrow></math> are densities of LBE and the rod, and <math id="M6"><mi>l</mi></math> and <math id="M7"><mi>t</mi></math> are the length and thickness of the fuel rod. Substituting values gives a constant hydrodynamic function (τ) over the entire frequency range, which further validates the FEM modal analysis results. The vacuum and LBE frequency comparison results for the verification of the hydrodynamic function are listed in Table 5 Table 5 . 1 1 Medium 1 1 Design 2 1 Mode 1 1 1 st 1 1 2 nd 1 1 3 rd 1 1 4 th 1 1 5 th 1 1 6 th 1 1 7 th 1 1 8 th 1 1 9 th 1 1 10 th 1 2 Vacuum 1 1 Bare 2 1 Frequency (Hz) 1 1 8.07 1 1 8.27 1 1 22.2 1 1 22.76 1 1 43.44 1 1 44.53 1 1 71.63 1 1 73.44 1 1 106.73 1 1 109.43 1 1 Wire-wrapped 2 1 Frequency (Hz) 1 1 7.91 1 1 8.35 1 1 21.53 1 1 22.09 1 1 42.01 1 1 42.93 1 1 69.45 1 1 71.03 1 1 103.99 1 1 106.57 1 2 LBE 1 1 Bare 2 1 Frequency (Hz) 1 1 7.57 1 1 7.76 1 1 20.84 1 1 21.36 1 1 40.77 1 1 41.79 1 1 67.23 1 1 68.93 1 1 100.18 1 1 102.71 1 1 Wire-wrapped 2 1 Frequency (Hz) 1 1 7.38 1 1 7.78 1 1 20.07 1 1 20.6 1 1 39.17 1 1 40.02 1 1 64.75 1 1 66.23 1 1 96.95 1 1 99.36 3 1 Hydrodynamic function (τ) 1 1 Bare 1 1 9e-4 1 1 9e-4 1 1 9e-4 1 1 9e-4 1 1 9e-4 1 1 9e-4 1 1 9e-4 1 1 9e-4 1 1 9e-4 1 1 9e-4 1 1 3 1 Hydrodynamic function (τ) 1 1 Wire-wrapped 1 1 1e-3 1 1 1e-3 1 1 1e-3 1 1 1e-3 1 1 1e-3 1 1 1e-3 1 1 1e-3 1 1 1e-3 1 1 1e-3 1 1 1e-3 1 1 5.2 Verification of LBE Added Mass Effect for Natural Frequencies 2 The calculation of the added mass effect generally involves varying engineering judgments regarding the considerations of geometry, adjacent members, and certain irregularities amongst others. These factors vary significantly from one situation to another and in some cases a preliminary analysis must be performed. The potential flow theory accurately provides the added mass effect values. The effect of added mass for single isolated members has been thoroughly investigated both analytically and experimentally. Theoretically, the potential flow theory has been quite successful for determining the added mass effect. The natural frequency of a component or structure submerged in, or in-contact with, a fluid decreases considerably in comparison to that in a vacuum. This phenomenon is termed the fluid-structure interaction (FSI). A lot of work has been undertaken to find the approximate solutions for determining the added mass effect in order to approximately calculate the change in the natural frequency of a solid body vibrating in a liquid [ 30 30 ] . When a cylinder vibrates in a liquid it induces an acceleration in the liquid, which in turn produces a force on the cylinder. This extra force can be fairly assumed as the mass of fluid displaced, which can be approximated as being equal to the volume of the cylinder using potential flow theory. In order to determine the natural frequencies of a cylinder vibrating in a liquid, the solution of stiffness and mass matrices have to be obtained using high level computer resources [ 31 31 ] . On the other hand, for a cylinder of uniform thickness, the reverse calculations (presented later in this section) would be time-saving and accurate, and could be verified manually by existing research. To support the previous statements, the volume of the cylinder was calculated manually and then verified through software computation; the mass of the fluid was calculated later by assuming an equal amount of fluid volume displaced by the cylinder. Furthermore, the natural frequencies of the cylinder were then calculated in a vacuum and in fluid. Subsequently, the derivation presented below was used to verify the added mass over the entire calculated frequency range. The uniform added mass can therefore obtained to show the validity of this reverse method for any order of natural frequencies. The dynamic equation of motion can be written as follows: <math display="block" id="M8"><mrow><mi>m</mi><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>ω</mi></mrow><mrow><mtext>d</mtext><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mi>k</mi><mi>ω</mi><mo>=</mo><mn>0</mn><mi>，</mi></mrow></math> where m and k are the mass and stiffness of cylinder, respectively. Natural frequency in a vacuum, ω vac , is given as: <math display="block" id="M9"><mrow><msub><mtext>ω</mtext><mrow><msub><mrow><mtext>vac</mtext></mrow><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msub></mrow></msub><mo>=</mo><mn>2</mn><mi>π</mi><msub><mi>f</mi><mrow><mtext>vac</mtext></mrow></msub><mo>=</mo><msqrt><mrow><mfrac><mi>k</mi><mi>m</mi></mfrac></mrow></msqrt><mi>．</mi></mrow></math> When the body is immersed in a fluid, the corresponding equation of motion due to the added mass effect of the liquid can be written as follows: <math display="block" id="M10"><mrow><mrow><mo>(</mo><mrow><mi>m</mi><mo>+</mo><msub><mi>m</mi><mrow><mtext>add</mtext></mrow></msub></mrow><mo>)</mo></mrow><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>ω</mi></mrow><mrow><mtext>d</mtext><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mi>k</mi><mi>ω</mi><mo>=</mo><mn>0</mn><mi>，</mi></mrow></math> where m add is the added mass of the fluid. Furthermore, the natural frequency of the structure vibrating in a fluid can be expressed as: <math display="block" id="M11"><mrow><msub><mtext>ω</mtext><mrow><mtext>lbe</mtext></mrow></msub><mo>=</mo><mn>2</mn><mi>π</mi><msub><mi>f</mi><mrow><mtext>lbe</mtext></mrow></msub><mo>=</mo><msqrt><mrow><mfrac><mi>k</mi><mrow><mi>m</mi><mo>+</mo><msub><mi>m</mi><mrow><mtext>add</mtext></mrow></msub></mrow></mfrac></mrow></msqrt><mi>．</mi></mrow></math> Comparing Eq. (3) and Eq. (5), we get the natural frequency as: <math display="block" id="M12"><mrow><mfrac><mrow><msub><mtext>ω</mtext><mrow><mtext>lbe</mtext></mrow></msub></mrow><mrow><msub><mi>ω</mi><mrow><mtext>vac</mtext></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>f</mi><mrow><mtext>lbe</mtext></mrow></msub></mrow><mrow><msub><mi>f</mi><mrow><mtext>vac</mtext></mrow></msub></mrow></mfrac><mo>=</mo><msqrt><mrow><mfrac><mi>m</mi><mrow><mi>m</mi><mo>+</mo><msub><mi>m</mi><mrow><mtext>add</mtext></mrow></msub></mrow></mfrac></mrow></msqrt><mi>，</mi></mrow></math> <math display="block" id="M13"><mrow><mfrac><mrow><msub><mi>f</mi><mrow><mtext>lbe</mtext></mrow></msub></mrow><mrow><msub><mi>f</mi><mrow><mtext>vac</mtext></mrow></msub></mrow></mfrac><mo>=</mo><msqrt><mrow><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mo stretchy="false">(</mo><msub><mi>m</mi><mrow><mtext>add</mtext></mrow></msub><mo>/</mo><mi>m</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></msqrt><mi>．</mi></mrow></math> Since the term m add / m in Eq. (7) is greater than zero, the non-dimensional frequency, ω lbe / ω vac , is always less than unity. This indicates how the liquid reduces the natural frequency. It is evident that the dynamic behavior is dependent upon the added mass for a single rigid body in an ideal fluid. The manually calculated volume and volume obtained from FEM were approximately identical; the mass of the rod was calculated to be about ~ 2054.4 g (bare) and 2083.03 g (wire-wrapped), and the calculated added mass ( m cal ) of the LBE was approximated to be ~277.1 g (bare) and 314.44 g (wire-wrapped). The frequencies were then calculated by using this added mass effect for an LBE environment. Equation (7) can be re-written in the following form for the calculation of the added mass effect: <math display="block" id="M14"><mrow><msub><mi>m</mi><mrow><mtext>add</mtext></mrow></msub><mo>=</mo><mi>m</mi><mrow><mo>[</mo><mrow><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mrow><mi>f</mi><msub><mi>n</mi><mrow><mtext>vac</mtext></mrow></msub></mrow><mrow><mi>f</mi><msub><mi>n</mi><mrow><mtext>lbe</mtext></mrow></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup><mo>−</mo><mn>1</mn></mrow><mo>]</mo></mrow><mi>，</mi></mrow></math> where, <math id="M15"><mrow><mi>f</mi><msub><mi>n</mi><mrow><mtext>vac</mtext></mrow></msub></mrow></math> and <math id="M16"><mrow><mi>f</mi><msub><mi>n</mi><mrow><mtext>lbe</mtext></mrow></msub></mrow></math> are the n th order natural frequencies of the rod in a vacuum and in LBE, respectively. By substituting frequencies values from Table 5 Table 5 , a constant and uniform added mass of LBE was found over the entire frequency range, as presented in Table 6 Table 6 . Hence, for simple geometries, the added mass effect for single isolated members can be easily predicted with great accuracy. 1 1 Design 1 1 Mode 1 1 1 st 1 1 2 nd 1 1 3 rd 1 1 4 th 1 1 5 th 1 1 6 th 1 1 7 th 1 1 8 th 1 1 9 th 1 1 10 th 1 3 Bare 1 1 m add 1 1 280.35 1 1 278.911 1 1 276.886 1 1 278.129 1 1 277.894 1 1 278.229 1 1 277.708 1 1 277.628 1 1 277.425 1 1 277.621 1 1 m cal 1 1 277.104 1 1 277.104 1 1 277.104 1 1 277.104 1 1 277.104 1 1 277.104 1 1 277.104 1 1 277.104 1 1 277.104 1 1 277.104 1 1 % Diff 1 1 1.17 1 1 0.65 1 1 0.07 1 1 0.36 1 1 0.28 1 1 0.4 1 1 0.21 1 1 0.18 1 1 0.11 1 1 0.18 1 3 Wire-wrapped 1 1 m add 1 1 309.932 1 1 316.407 1 1 314.085 1 1 312.23 1 1 313.009 1 1 313.943 1 1 313.377 1 1 312.876 1 1 313.501 1 1 313.277 1 1 m cal 1 1 314.442 1 1 314.44 1 1 314.44 1 1 314.44 1 1 314.44 1 1 314.44 1 1 314.44 1 1 314.44 1 1 314.44 1 1 314.44 1 1 % Diff 1 1 1.43 1 1 0.62 1 1 0.11 1 1 0.7 1 1 0.45 1 1 0.15 1 1 0.33 1 1 0.49 1 1 0.29 1 1 0.37 5.3 Stress Distribution and Evaluation of Results 2 The maximum stressed nodes, as sorted for ease of understanding the results, provide a comparison for the two different designs of LMFR fuel rods for different plant conditions. The stress distribution or stress contours has been compared along the most stressed nodes of the fuel rods at different loading conditions. As shown in Fig. 9 Fig. 9 , the most stressed nodes for the bare design are at the joint of the cladding with the end caps or cladding plugs, whereas they are at the extreme welding ends for the wire-wrapped design due to the discontinuity in geometry (i.e., the connection of the wire with the cladding). The contours were elaborated in terms of stress intensity (i.e., twice the maximum shear stress), which was defined as the difference between the algebraically largest and smallest principal stresses at a given position. To evaluate the results, the calculated stresses were compared with the allowable stress limits defined by the code, which showed that they were within the allowable limits. Stress classification was performed to identify the "primary" (P) and "secondary" (Q) stresses. Primary stresses relate to equilibrium equations, while secondary stresses are linked to compatibility equations. In general, these stresses come from mechanical and thermal loadings, respectively. The integrity of the cladding is of the main concern of this study because it is the thinnest and longest section and bears all the mechanical loads. In the FEM, when continuum elements are used, the total stress distribution is obtained. Therefore, to calculate the membrane ( P m ) and bending stresses ( P b ), the stress distribution should be linearized across the thickness [ 26 26 ] . To check the stress limits and gain a better understanding of the behavior of the cladding for the two different designs, paths were defined along the full length of the cladding. The selection of the same path distance was considered for both models in order to get a true picture of the results. The stresses were then linearized and compared with the allowable limits[ 32 32 ]. For ease in understanding and for simplification, only four paths (the lowest and uppermost, and middle two paths of cladding) were selected and the ratio was also determined between calculated and allowable stress, for P m + P b whichever is maximum (inside, center and outside of the cladding), listed in Table 7 Table 7 . 1 1 Service level/ Allowable stress 1 1 Distance (mm) 1 1 4 1 Bare 4 1 Wire-wrapped 1 1 1 1 1 1 Path 3 1 Calculated stress (MPa) 1 1 Ratio 3 1 Calculated stress (MPa) 1 1 Ratio 1 1 1 1 1 1 3 1 P m + P b 1 1 3 1 P m + P b 1 1 1 1 1 1 1 1 1 1 Inside 1 1 Center 1 1 Outside 1 1 1 1 Inside 1 1 Center 1 1 Outside 1 1 1 4 Design (1.5 S m = 180) 1 1 0 1 1 1 1 1 1.39 1 1 1.33 1 1 1.27 1 1 0.008 1 1 79.16 1 1 106.4 1 1 133.8 1 1 0.74 1 1 510 1 1 7 1 1 38.8 1 1 41.45 1 1 44.3 1 1 0.25 1 1 39.94 1 1 41.36 1 1 44.19 1 1 0.24 1 1 1105 1 1 14 1 1 38.69 1 1 41.44 1 1 44.29 1 1 0.25 1 1 38.88 1 1 41.36 1 1 43.98 1 1 0.24 1 1 1630 1 1 20 1 1 22.76 1 1 15.55 1 1 10.51 1 1 0.13 1 1 13.1 1 1 8.4 1 1 7.49 1 1 0.07 1 4 Normal (3 S m = 360) 1 1 0 1 1 1 1 1 0.81 1 1 0.77 1 1 0.74 1 1 0.002 1 1 69.02 1 1 97.14 1 1 125.5 1 1 0.35 1 1 510 1 1 7 1 1 26.07 1 1 27.63 1 1 29.54 1 1 0.08 1 1 27.39 1 1 28.29 1 1 29.46 1 1 0.08 1 1 1105 1 1 14 1 1 25.82 1 1 27.63 1 1 29.53 1 1 0.08 1 1 25.94 1 1 27.57 1 1 29.33 1 1 0.08 1 1 1630 1 1 20 1 1 15.21 1 1 10.41 1 1 7.01 1 1 0.04 1 1 8.42 1 1 5.61 1 1 4.67 1 1 0.02 1 4 Upset (3 S m = 360) 1 1 0 1 1 1 1 1 2.8 1 1 2.69 1 1 2.59 1 1 0.008 1 1 104.5 1 1 103.9 1 1 115.2 1 1 0.32 1 1 510 1 1 7 1 1 25.77 1 1 27.63 1 1 29.53 1 1 0.08 1 1 25.69 1 1 27.57 1 1 29.46 1 1 0.08 1 1 1105 1 1 14 1 1 25.74 1 1 27.63 1 1 29.52 1 1 0.08 1 1 25.85 1 1 27.57 1 1 29.32 1 1 0.08 1 1 1630 1 1 20 1 1 9.04 1 1 4.83 1 1 7.22 1 1 0.02 1 1 17.81 1 1 23.56 1 1 30.57 1 1 0.08 1 4 Faulted (3.6 S m = 432) 1 1 0 1 1 1 1 1 1.96 1 1 1.95 1 1 1.93 1 1 0.004 1 1 71.34 1 1 101.5 1 1 132.1 1 1 0.3 1 1 510 1 1 7 1 1 26.08 1 1 27.63 1 1 29.54 1 1 0.07 1 1 27.03 1 1 27.93 1 1 29.46 1 1 0.06 1 1 1105 1 1 14 1 1 25.82 1 1 27.63 1 1 29.53 1 1 0.07 1 1 25.91 1 1 27.57 1 1 29.32 1 1 0.06 1 1 1630 1 1 20 1 1 15.22 1 1 10.44 1 1 7.02 1 1 0.03 1 1 13.23 1 1 11.34 1 1 10.47 1 1 0.03 A graphical comparison was prepared to provide a better understanding of the two designs for different plant conditions ( Fig. 10 Fig. 10 ). Twenty paths were selected on the full length of the cladding by bearing in mind the same path distance for the two models in order to make a clear and true comparison between the two designs. It is interesting to note that the behavior of the two models for different plant conditions were opposite to each other at the beginning of the cladding length (0 mm distance) because of the discontinuity in geometry due to the attached wire. In structural discontinuity, there is an obvious increase in stress that can be observed due to the compatibility between linking parts. For the remaining length of the cladding, both models behaved in almost the same way for different loading combinations. The stresses remained uniform throughout the cladding length except for a small difference between the two models at the top section of the cladding ( Fig. 10 Fig. 10 ). 6. Analytical Solution for Static Pressure 1 The prime concern in relation to the fuel rod is cladding. For the static analytical solution, the cladding can be treated as a thin cylinder since it satisfies the following laws of thin and thick cylinders: od > 10t or id / t > 20, where od is outside diameter, t is thickness, and id is the inside diameter of the cladding. When r o = 6 mm, t = 0.4 mm, and id = 11.6 mm, the criteria that the cladding could be treated as a thin cylinder are satisfied. The method for determining the hoop or tangential stresses ( δ t ), which are considered to be uniform throughout the wall thickness, and the radial stresses ( δ r ), which are insignificant in comparison to the hoop stress, at any thickness of a cylinder against the applied pressure was presented by the French electrician Gabriel Lame in (1833). This applies to a cylinder with a given inside radius ( r i ) and outside radius ( r o ) subjected to a uniformly distributed internal ( P i ) and external pressure ( P o ) [ 33 33 ]. When considering a thin shell of the radius ( r ) for the thickness (d r ), the tangential stress in this shell is δ t , the radial stress on the inner surface is δ r , and that on the outer surface is δ r + d δ r , where d δ r is the increment in δ r due to the variation of pressure across the cylinder wall. The radial stresses are assumed to be tensile, so a negative result for δ r will denote compression. Hence, for equilibrium; the vertical summation of forces must be zero: <math display="block" id="M17"><mrow><mrow><mo>(</mo><mrow><msub><mi>δ</mi><mtext>r</mtext></msub><mo> </mo><mo>+</mo><mo> </mo><mtext>d</mtext><msub><mi>δ</mi><mtext>r</mtext></msub></mrow><mo>)</mo></mrow><mo> </mo><mo>×</mo><mo> </mo><mn>2</mn><mrow><mo>(</mo><mrow><mi>r</mi><mo> </mo><mo>+</mo><mo> </mo><mtext>d</mtext><mi>r</mi></mrow><mo>)</mo></mrow><mo>−</mo><mo> </mo><msub><mi>δ</mi><mtext>r</mtext></msub><mrow><mo>(</mo><mrow><mn>2</mn><mi>r</mi></mrow><mo>)</mo></mrow><mo>−</mo><mo> </mo><mn>2</mn><msub><mi>δ</mi><mtext>t</mtext></msub><mtext>d</mtext><mi>r</mi><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></mrow></math> The final equation obtained from solving Eq. (9) gives the following general expression for δ t and δ r at any point: <math display="block" id="M18"><mrow><msub><mtext>δ</mtext><mtext>t</mtext></msub><mo>=</mo><mfrac><mrow><msub><mi>r</mi><mtext>i</mtext></msub><msup><mrow/><mn>2</mn></msup><msub><mi>P</mi><mtext>i</mtext></msub><mo>−</mo><mo> </mo><msub><mi>r</mi><mtext>o</mtext></msub><msup><mrow/><mn>2</mn></msup><msub><mi>P</mi><mtext>o</mtext></msub></mrow><mrow><msub><mi>r</mi><mtext>o</mtext></msub><msup><mrow/><mn>2</mn></msup><mo>−</mo><msub><mi>r</mi><mtext>i</mtext></msub><msup><mrow/><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mi>r</mi><mtext>i</mtext></msub><msup><mrow/><mn>2</mn></msup><msub><mi>r</mi><mtext>o</mtext></msub><msup><mrow/><mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>P</mi><mtext>i</mtext></msub><mo>−</mo><mo> </mo><msub><mi>P</mi><mtext>o</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>r</mi><mtext>o</mtext></msub><msup><mrow/><mn>2</mn></msup><mo>−</mo><msub><mi>r</mi><mtext>i</mtext></msub><msup><mrow/><mn>2</mn></msup></mrow><mo>)</mo></mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac><mo>,</mo></mrow></math> <math display="block" id="M19"><mrow><msub><mi>δ</mi><mtext>r</mtext></msub><mo>=</mo><mfrac><mrow><msub><mi>r</mi><mtext>i</mtext></msub><msup><mrow/><mn>2</mn></msup><msub><mi>P</mi><mtext>i</mtext></msub><mo>−</mo><mo> </mo><msub><mi>r</mi><mtext>o</mtext></msub><msup><mrow/><mn>2</mn></msup><msub><mi>P</mi><mtext>o</mtext></msub></mrow><mrow><msub><mi>r</mi><mtext>o</mtext></msub><msup><mrow/><mn>2</mn></msup><mo>−</mo><msub><mi>r</mi><mtext>i</mtext></msub><msup><mrow/><mn>2</mn></msup></mrow></mfrac><mo>−</mo><mfrac><mrow><msub><mi>r</mi><mtext>i</mtext></msub><msup><mrow/><mn>2</mn></msup><msub><mi>r</mi><mtext>o</mtext></msub><msup><mrow/><mn>2</mn></msup><mo stretchy="false">(</mo><msub><mi>P</mi><mtext>i</mtext></msub><mo>−</mo><mo> </mo><msub><mi>P</mi><mtext>o</mtext></msub><mo stretchy="false">)</mo></mrow><mrow><mrow><mo>(</mo><mrow><msub><mi>r</mi><mtext>o</mtext></msub><msup><mrow/><mn>2</mn></msup><mo>−</mo><msub><mi>r</mi><mtext>i</mtext></msub><msup><mrow/><mn>2</mn></msup></mrow><mo>)</mo></mrow><msup><mi>r</mi><mn>2</mn></msup></mrow></mfrac><mo>.</mo></mrow></math> Substituting values for all variables and different values for ' r ’ against the design and operating pressures in Eq. (10), we obtain the following results in comparison to the ANSYS calculations ( Table 8 Table 8 ). The results were roughly compared with the ANSYS exact calculations and the two showed a close agreement. 1 2 Designs 1 2 Radius (mm) 3 1 Design condition 3 1 Normal operation 1 1 ANSYS (MPa) 1 1 Manual (MPa) 1 1 % Diff. 1 1 ANSYS (MPa) 1 1 Manual (MPa) 1 1 % Diff. 1 3 Bare 1 1 5.6 (I) 1 1 38.8 1 1 40.47 1 1 4.12 1 1 26.07 1 1 26.98 1 1 3.37 1 1 5.8 (C) 1 1 41.45 1 1 38.97 1 1 5.98 1 1 27.63 1 1 25.98 1 1 5.97 1 1 6 (O) 1 1 44.3 1 1 37.62 1 1 15.07 1 1 29.54 1 1 25.08 1 1 15.09 1 3 Wire-wrapped 1 1 5.6 (I) 1 1 39.94 1 1 40.47 1 1 1.3 1 1 27.39 1 1 26.98 1 1 1.49 1 1 5.8 (C) 1 1 41.36 1 1 38.97 1 1 5.77 1 1 28.29 1 1 25.98 1 1 8.16 1 1 6 (O) 1 1 44.19 1 1 37.62 1 1 14.86 1 1 29.46 1 1 25.08 1 1 14.86 It should be mentioned that for thin-walled cylinders, radial stresses are usually neglected because they are negligibly small. Due to internal pressure, the deformation of the cladding takes place, which results in stresses in the cladding wall. Each wall element is subjected to circular and axial expansion and radial compression that is governed by Eqs. (12) and (13). The inside wall deformation at r = r i is given as: <math display="block" id="M20"><mrow><mi>Δ</mi><msub><mi>r</mi><mtext>i</mtext></msub><mo>=</mo><mfrac><mrow><msub><mi>r</mi><mtext>i</mtext></msub></mrow><mi>E</mi></mfrac><mrow><mo>[</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>P</mi><mtext>i</mtext></msub><mo>−</mo><msub><mi>P</mi><mtext>o</mtext></msub></mrow><mo>)</mo></mrow><mfrac><mrow><msub><mi>r</mi><mtext>o</mtext></msub><msup><mrow/><mn>2</mn></msup><mo>+</mo><msub><mi>r</mi><mtext>i</mtext></msub><msup><mrow/><mn>2</mn></msup></mrow><mrow><msub><mi>r</mi><mtext>o</mtext></msub><msup><mrow/><mn>2</mn></msup><mo>−</mo><msub><mi>r</mi><mtext>i</mtext></msub><msup><mrow/><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mi>ν</mi></mrow><mo>]</mo></mrow><mo>.</mo></mrow></math> The outside wall deformation at r = r o is expressed as: <math display="block" id="M21"><mrow><mtext>Δ</mtext><msub><mi>r</mi><mtext>o</mtext></msub><mo>=</mo><mfrac><mrow><msub><mi>r</mi><mtext>o</mtext></msub></mrow><mi>E</mi></mfrac><mrow><mo>[</mo><mrow><mrow><mo>(</mo><mrow><msub><mi>P</mi><mtext>i</mtext></msub><mo>−</mo><msub><mi>P</mi><mtext>o</mtext></msub></mrow><mo>)</mo></mrow><mfrac><mrow><mn>2</mn><msub><mi>r</mi><mtext>i</mtext></msub><msup><mrow/><mn>2</mn></msup></mrow><mrow><msub><mi>r</mi><mtext>o</mtext></msub><msup><mrow/><mn>2</mn></msup><mo>−</mo><msub><mi>r</mi><mtext>i</mtext></msub><msup><mrow/><mn>2</mn></msup></mrow></mfrac></mrow><mo>]</mo></mrow></mrow></math> By substituting values into Eqs. (12) and (13) we obtain an estimated inside wall deformation <math id="M22"><mrow><mi>Δ</mi><msub><mi>r</mi><mi>i</mi></msub><mo>=</mo><mn>0.00157</mn><mo> </mo><mtext>mm</mtext></mrow></math> and an outside wall deformation <math id="M23"><mrow><mo> </mo><mtext>Δ</mtext><msub><mi>r</mi><mtext>o</mtext></msub><mo>=</mo><mo> </mo><mn>0.00135</mn><mo> </mo><mtext>mm</mtext></mrow></math> . 7. Conclusion 1 A comparison of two different designs of lead-based reactor fuel rods was established for different plant conditions. ANSYS contact analysis methodology was developed for the wire-wrapped design and most appropriate KEYOPT and SOLUTION settings were suggested. A simple yet accurate method for the determination and verification of the added mass effect was presented and verified using a manually calculation to show the usefulness of the potential flow theory for simple geometries. Modal frequencies were compared in a vacuum and in LBE, and a constant hydrodynamic function (τ) validated the FEM results. The use of wrapped-wire provided thermal mixing of the coolant due to increased local turbulence in the coolant flow, although this resulted in high-stress ratios on the extreme welded locations of the cladding. In comparison to the bare design, the wrapped-wire design yielded better stress results but with some compromise in other aspects. Wire-wrapped designs are widely preferred due to several benefits including the fact that fabrication is easy and less expensive when compared to the grid-spacer design that was usually employed in the previous generation of reactors. Secondly, due to the use of wire-wrapped technology, the contact with neighboring cladding occurred at six axial locations for every pitch of wire, which helped to minimize cladding mechanical vibrations and reactivity oscillations. Furthermore, the results were verified numerically and provided a close approximation to the ANSYS results. A mesh dependency test was also performed to determine the accuracy of the meshed model and to further validate the use of the APDL code for LMFR fuel rods analysis. The stresses remained uniform throughout the cladding length with a small difference at the top section of the cladding. The maximum stress ratios of 0.25 (bare) and 0.74 (wire-wrapped) that were computed on the cladding of the fuel rods in terms of the design condition were within the allowable limits defined by code. Hence, it is concluded that both fuel rod designs are adequate to bear the anticipated loads and will perform functions in all plant conditions. These results provide detailed insight for the design of LMFR fuel rods in consideration of different technical aspects and provide guidelines for further analysis. References Y. Wu , Design and R&D progress of China lead-based reactor for ADS research facility . J. Eng . 2 : 124 - 131 ( 2016 ). https://doi.org/10.1016/J.ENG.2016.01.023 https://doi.org/10.1016/J.ENG.2016.01.023 Design and R&D progress of China lead-based reactor for ADS research facility H. Shi , H. Ouyang , S. Wang et al ., RF tuning and beam commissioning of CW RFQ for China-ADS Injector-I . Nucl. Sci. Tech . 29 : 142 ( 2018 ). https://doi.org/10.1007/s41365-018-0478-x https://doi.org/10.1007/s41365-018-0478-x RF tuning and beam commissioning of CW RFQ for China-ADS Injector-I Z. L. Zhao , Y. W. Yang , H. Y. Meng et al ., Preparation and verification of mixed high-energy neutron cross-section library for ADS . Nucl. Sci. Tech . 29 : 140 ( 2018 ). https://doi.org/10.1007/s41365-018-0487-9 https://doi.org/10.1007/s41365-018-0487-9 Preparation and verification of mixed high-energy neutron cross-section library for ADS Y. Wu , Y. Bai , Y. Song et al ., Development strategy and conceptual design of China lead based research reactor . Ann. Nucl. Energ . 87 : 511 - 516 ( 2016 ). https://doi.org/10.1016/J.ENG.2016.01.023 https://doi.org/10.1016/J.ENG.2016.01.023 Development strategy and conceptual design of China lead based research reactor Y. Wu , J.P. Qian , J. N. Yu , The fusion-driven hybrid system and its material selection . J. Nucl. Mat . 3 : 1629 - 1636 ( 2002 ). https://doi.org/10.1016/S0022-3115(02)01272-2 https://doi.org/10.1016/S0022-3115(02)01272-2 The fusion-driven hybrid system and its material selection Y. Wu , Y. Bai , Y. Song et al ., Conceptual design of China lead based research reactor CLEAR-1 . Nucl. Sci. Eng . 34 : 201 - 208 ( 2014 ). https://doi.org/10.3969/j.issn.0258-0918.2014.02.009 https://doi.org/10.3969/j.issn.0258-0918.2014.02.009 Conceptual design of China lead based research reactor CLEAR-1 Y. Wu , M. H. Wang , Q. Huang et al ., Development status and prospects of lead-based reactors . Nucl. Sci. Eng . 35 : 213 - 221 ( 2015 ). https://doi.org/CNKI:SUN:HKXY.0.2015-02-004 https://doi.org/CNKI:SUN:HKXY.0.2015-02-004 Development status and prospects of lead-based reactors M. Wang , C. Lian , Y. Li et al ., Preliminary conceptual design of a lead-bismuth cooled small reactor (CLEAR-SR) . Int. J. Hydrogen Energ . 40 ( 44 ): 15132 - 15136 ( 2015 ). https://doi.org/10.1016/j.ijhydene.2015.03.097 https://doi.org/10.1016/j.ijhydene.2015.03.097 Preliminary conceptual design of a lead-bismuth cooled small reactor (CLEAR-SR) M. Zeng , Y. Song , X. Ding , Kinematics analysis and verification of in-vessel fuel handling system for China lead based research reactor . Ann. Nucl. Energ . 85 : 301 - 305 ( 2015 ). https://doi.org/10.1016/j.anucene.2015.05.031 https://doi.org/10.1016/j.anucene.2015.05.031 Kinematics analysis and verification of in-vessel fuel handling system for China lead based research reactor Q. Han , Q. Wu , J. Chen et al ., Analysis of fixation method of fuel assembly for lead alloy cooled reactor . Nucl. Sci. Tech . 26 : 050601 ( 2015 ). https://doi.org/10.13538/j.1001-8042/nst.26.050601 https://doi.org/10.13538/j.1001-8042/nst.26.050601 Analysis of fixation method of fuel assembly for lead alloy cooled reactor Y. Wu , FDS Team , CLEAR-S: An integrated non-nuclear test facility for China lead based research reactor , Int. J. Energ. Res . 40 ( 14 ): 1951 - 1956 ( 2016 ). https://doi.org/10.1002/er.3569 https://doi.org/10.1002/er.3569 CLEAR-S: An integrated non-nuclear test facility for China lead based research reactor Y. Wu , S. Zheng , X. Zhu et al ., Conceptual design of the fusion driven subcritical system FDS-I . Fus. Eng. Des . 81 : 1305 - 1311 ( 2006 ). https://doi.org/10.1016/j.fusengdes.2005.10.015 https://doi.org/10.1016/j.fusengdes.2005.10.015 Conceptual design of the fusion driven subcritical system FDS-I M. Barzegari , M. Aghaie , A. Zolfaghari , Assessment of fuel-rod meltdown in a severe accident at Bushehr nuclear power plant (BNPP) . Nucl. Sci. Tech . 30 : 55 ( 2019 ). https://doi.org/10.1007/s41365-019-0589-z https://doi.org/10.1007/s41365-019-0589-z Assessment of fuel-rod meltdown in a severe accident at Bushehr nuclear power plant (BNPP) K. L. Shi , S. Z. Li , X. L. Zhang et al ., Partial flow blockage analysis of the hottest fuel assembly in SNCLFR-100 reactor core . Nucl. Sci. Tech . 29 : 16 ( 2018 ). https://doi.org/10.1007/s41365-017-0351-3 https://doi.org/10.1007/s41365-017-0351-3 Partial flow blockage analysis of the hottest fuel assembly in SNCLFR-100 reactor core W. C. Nam , H. W. Lee , S. Hwang , Fuel design study and optimization for PEACER development . Nucl. Eng. Des . 237 : 316 - 324 ( 2007 ). https://doi.org/10.1016/j.nucengdes.2006.04.032 https://doi.org/10.1016/j.nucengdes.2006.04.032 Fuel design study and optimization for PEACER development A. E. Aboanber , A. A. Nahla , A. M. Edress , Developed mathematical technique for fractional stochastic point kinetics model in nuclear reactor dynamics . Nucl. Sci. Tech . 29 : 132 ( 2018 ). https://doi.org/10.1007/s41365-018-0467-0 https://doi.org/10.1007/s41365-018-0467-0 Developed mathematical technique for fractional stochastic point kinetics model in nuclear reactor dynamics Z. L. Zhao , Y. W. Yang , H. Y. Meng et al ., Preparation and verification of mixed high-energy neutron cross-section library for ADS . Nucl. Sci. Tech . 29 : 140 ( 2018 ). https://doi.org/10.1007/s41365-018-0487-9 https://doi.org/10.1007/s41365-018-0487-9 Preparation and verification of mixed high-energy neutron cross-section library for ADS K. MiKi , Analytical method of fuel pin deformations in LMFBR assembly . J. Nucl. Sci. Technol . 14 ( 11 ): 791 - 804 ( 1977 ). https://doi.org/10.1080/18811248.1977.9730840 https://doi.org/10.1080/18811248.1977.9730840 Analytical method of fuel pin deformations in LMFBR assembly X. Yang , Y. T. Gao , Y. Zhong et al ., Stress analysis of the TMSR graphite component under irradiation conditions . Nucl. Sci. Tech . 29 : 173 ( 2018 ). https://doi.org/10.1007/s41365-018-0516-8 https://doi.org/10.1007/s41365-018-0516-8 Stress analysis of the TMSR graphite component under irradiation conditions R. Marinari , D. Piazza , CFD Pre-test analysis of the rod bundle experiment in the HLM facility NACIE-UP . ENEA, ADPFISS-LP 2 - 110 ( 2015 ). doi: nureth-16/data/papers/13290.pdf http://doi.org/nureth-16/data/papers/13290.pdf CFD Pre-test analysis of the rod bundle experiment in the HLM facility NACIE-UP H. S. Kang , K. N. Song , H. K. Kim et al ., Modal analysis of a PWR fuel rod supported by the newly designed spacer grid . Transaction, SMiRT - 16 ( 2001 ). doi: 1840.20/30513/C1425.pdf http://doi.org/1840.20/30513/C1425.pdf Modal analysis of a PWR fuel rod supported by the newly designed spacer grid R. L. Frano , A. Sanfiorenzo , G. Forasassi et al ., Structural dynamic evaluation of the effects caused by the core compaction . CERSE-UNIPI RL , 1 ( 12 ): 16154 ( 2014 ). Structural dynamic evaluation of the effects caused by the core compaction G. Grasso , C. Petrovich , D. Mattiol et al ., The core design of ALFRED, a demonstrator for the European lead cooled reactors . Nucl. Eng. Des . 278 : 287 - 301 ( 2014 ). https://doi.org/10.1016/j.nucengdes.2014.07.032 https://doi.org/10.1016/j.nucengdes.2014.07.032 The core design of ALFRED, a demonstrator for the European lead cooled reactors Y. Wu , FDS Team , Conceptual design of China fusion power plant FDS-II . Fus. Eng. Des . 83 : 1683 - 1689 ( 2008 ). https://doi.org/10.1016/j.fusengdes.2008.06.048 https://doi.org/10.1016/j.fusengdes.2008.06.048 Conceptual design of China fusion power plant FDS-II Y. Wu , Z. Chen , L. Hu et al ., Identification of safety gaps for fusion demonstration reactors, Nat . Energ . 1 ( 12 ): 16154 ( 2016 ). https://doi.org/10.1038/nenergy.2016.154 https://doi.org/10.1038/nenergy.2016.154 Identification of safety gaps for fusion demonstration reactors, Nat A. Shams , F. Roelofs , E. Baglietto et al ., High-fidelity numerical simulations of an infinite wire wrapped fuel assembly . NURETH - 16 ( 2015 ). https://doi.org/10.1016/j.nucengdes.2018.06.012 https://doi.org/10.1016/j.nucengdes.2018.06.012 High-fidelity numerical simulations of an infinite wire wrapped fuel assembly E. Sosnovsky , E. Baglietto , S. Keijers et al ., CFD simulations to determine the effects of deformations on liquid metal cooled wire wrapped fuel assemblies . NURETH - 16 : 2747 - 2761 ( 2015 ). http://glc.ans.org/nureth-16/data/papers/13595.pdf http://glc.ans.org/nureth-16/data/papers/13595.pdf CFD simulations to determine the effects of deformations on liquid metal cooled wire wrapped fuel assemblies S. Hussain , M. Rafique , A. Ahmad et al ., CNPP fuel rod vibration analysis using finite element method . Tech. J. Uni. Eng. Tech. Vibration analysis issue : 23 - 33 ( 2012 ). CNPP fuel rod vibration analysis using finite element method E. O. Tuck , Calculation of unsteady flows due to small motion of cylinders in a viscous fluid . J. Eng. Math . 3 ( 1 ): 29 - 44 ( 1969 ). https://doi.org/10.1007/BF01540828 https://doi.org/10.1007/BF01540828 Calculation of unsteady flows due to small motion of cylinders in a viscous fluid K. H. Jeong , M. J. Jhung , Added mass estimation of square sections coupled with a liquid using finite element method . Nucl. Eng. Tech . 49 : 234 - 244 ( 2017 ). https://doi.org/10.1016/j.net.2016.07.010 https://doi.org/10.1016/j.net.2016.07.010 Added mass estimation of square sections coupled with a liquid using finite element method P. Ray , S. Han , A simple and accurate added mass model for hydrodynamic fluid structure interaction analysis . J. Frank. Inst . 6 : 929 - 945 ( 1996 ). https://doi.org/10.1016/0016-0032(96)00043-9 https://doi.org/10.1016/0016-0032(96)00043-9 A simple and accurate added mass model for hydrodynamic fluid structure interaction analysis H. Y. Lee , S. K. Son , M. G. Won et al ., Risks of non-conservative design according to ASME B31.1 for high-temperature piping subjected to long-term operation in the creep range . Nucl. Sci. Tech . 30 : 77 ( 2019 ). https://doi.org/10.1007/s41365-019-0604-4 https://doi.org/10.1007/s41365-019-0604-4 Risks of non-conservative design according to ASME B31.1 for high-temperature piping subjected to long-term operation in the creep range X. Y. Wang , S. F. Zhu , X. Wang et al ., Stress and thickness calculation of a bolted flat cover with double metal sealing rings . Nucl. Sci. Tech . 29 : 120 ( 2018 ). https://doi.org/10.1007/s41365-018-0457-2 https://doi.org/10.1007/s41365-018-0457-2 Stress and thickness calculation of a bolted flat cover with double metal sealing rings The authors are grateful to other members of the FDS Team for their great help in this research. 10.1007/s41365-019-0721-0 This work was supported by the National Key R&D Program of China (No. 2018YFB1900601) and National Natural Science Foundation of China (No. 11772086). 2019-05-30 2019-11-19 2019-11-21 2019-12-21 2020-01-01 2021-04-10 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2020 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-019-0721-0 http://dx.doi.org/10.1007/s41365-019-0721-0 . 2020 Nuclear Science and Techniques 1 31