Analysis of the axial fitting clearance between the fuel rod and the end seat in CIADS

Analysis of the axial fitting clearance between the fuel rod and the end seat in CIADS 1 first-author 1 corresp Corresponding author. E-mail address: yangyongwei@impcas.ac.cn Corresponding author. E-mail address: yangyongwei@impcas.ac.cn yangyongwei@impcas.ac.cn 1 1 1 1 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China The fuel assembly is key structure in China Initiative Accelerator Driven System (CIADS), and the axial fitting clearance (AFC) for the fuel assembly design is an essential subject of study. In this paper, different methods are used to calculate critical stress in cylindrical shells. Because the thermal expansion of fuel assembly outer tube is larger than that of the cladding of fuel rod, enough space should be reserved between the upper end plug and upper seat slot. The collapse critical compressive stress of the cladding is obtained numerically through ANSYS simulation calculation. The AFC range between the fuel rod cladding and the end seat due to the displacement of thermal expansion is given by the theoretical formulas and ANSYS buckling analysis. These provide a reference for the AFC design of the reactor fuel assembly. Thermal expansion Critical stress AFC calculation ANSYS simulation 1 Introduction 1 Cylindrical shell, a common mechanical structure, has been wide used in aviation, aerospace, nuclear power and other industries. In a nuclear reactor, fuel cladding and outer sleeve can be simplified as a cylindrical shell model, and axial fitting clearance (AFC) design of the fuel cladding and outer sleeve is important in China Initiative Accelerator Driven System (CIADS) [ 1 1 ] . Stability of cylindrical shells is also a hot spot of study [ 2 2 - 4 4 2-4 ] . Chen [ 5 5 ] and Zhao [ 6 6 ] did numerical buckling analysis of cylindrical shell under uniform axial compression in their thesis. Imani et al . [ 7 7 ] studied pellet cladding interaction (PCI). As the fast reactor fuel rods are designed with high precision requirements for manufacturing and assembling [ 8 8 ] , the thermal expansion of the fuel cladding and the end seat should be considered, and the structural thermal expansion gap is particularly important. Ishiwatari et al . [ 9 9 ] and Williamson et al . [ 10 10 ] studied the fuel behavior in different working conditions. Hofmeister et al . [ 11 11 ] laid a foundation for the fuel assembly design. In order to prevent the transition extrusion and the fuel rod buckling collapse from occurring, a certain compensation gap between the fuel rod end and its seat must be reserved. In this paper, theoretical formulas to solve the cylindrical shell buckling problem are discussed for performing numerical simulations on the fuel cladding-end seat gap, towards engineering applications of the fuel rods in CIADS. 2 Model analysis 1 2.1. Problem description 2 In order to prevent buckling of the pellet cladding due to different thermal expansion of the materials, a certain gap ( x g ) is reserved between a fuel rod and its outer sleeve assembly, as shown in Fig.1 Fig.1 . 2.2. Theoretical calculation of the critical stress 2 For cylindrical shell, a dimensionless shell length parameter χ = l 2 /( Rt ) is incorporated, where, R is radius of the cylindrical shell middle surface, l is the cylindrical shell length, and t is the thickness. Axial cylindrical shells are classified into three categories, [ 5 5 ] i.e. slender cylindrical shell ( χ ≤3.018), middle length ( χ > 3.018), and short cylindrical shell ( χ ≥8.069 R 2 / t 2 ). When the fuel rod axially fixed, it is subjected to axial force. For the three kinds of cylindrical shells with axial compression model, the critical stress can be calculated as follows: For the slender cylindrical shell, the critical pressure load σ cr can be obtained by the formula of Euler beam, <math display="block" id="M1"><mrow><msub><mi>σ</mi><mrow><mtext>cr</mtext></mrow></msub><mo>=</mo><mtext> </mtext><msup><mi>π</mi><mn>2</mn></msup><mi>E</mi><mi>I</mi><mo>/</mo><mrow><mo>[</mo><mrow><mi>A</mi><msup><mrow><mrow><mo>(</mo><mrow><mi>μ</mi><mi>l</mi></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow><mo>]</mo></mrow><mo>,</mo></mrow></math> where, E is elastic or Young's modulus; I = π( D 4 − d 4 )/64 is the cross sectional moment of inertia, with d and D being the internal and external diameters of cylindrical shells, respectively; A is the cross-sectional area of cylindrical shell; and μ is length factor. The critical load of the cylindrical shell can be calculated by Eq.(2) [ 12 12 ], <math display="block" id="M2"><mrow><msub><mi>σ</mi><mrow><mtext>cr</mtext></mrow></msub><mo>=</mo><mi>E</mi><msup><mrow><mrow><mo>(</mo><mrow><mi>t</mi><mo>/</mo><mi>R</mi></mrow><mo>)</mo></mrow></mrow><mn>3</mn></msup><mo>/</mo><mrow><mo>[</mo><mrow><mn>4</mn><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msup><mi>υ</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mo>.</mo></mrow></math> For middle or short length cylindrical shell, Eq.(3) can be applied based on the Donnell linear theory [ 3 3 ] , <math display="block" id="M3"><mrow><mi>D</mi><mo>'</mo><msup><mtext>d</mtext><mn>4</mn></msup><mi>w</mi><mo>/</mo><mtext>d</mtext><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><msub><mi>σ</mi><mi>x</mi></msub><mi>t</mi><mo>+</mo><msup><mrow><mtext> d</mtext></mrow><mn>2</mn></msup><mi>w</mi><mo>/</mo><mtext>d</mtext><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>E</mi><mi>t</mi><mi>w</mi><mo>/</mo><msup><mi>R</mi><mn>2</mn></msup><mo>=</mo><mtext> </mtext><mn>0</mn><mo>,</mo></mrow></math> where, D '= Et 3 /[12(1− υ 2 )] is bending stiffness of cylindrical shell, with E being the elastic modulus and υ the Poisson' ratio; w is the lateral displacement, x is the axial coordinate, t is thickness of the cylindrical shell, R is the medium radius, and σ x is the axial stress. By solving Eq.(3), the critical stress σ cr is obtained. <math display="block" id="M4"><mrow><msub><mi>σ</mi><mrow><mtext>cr</mtext></mrow></msub><mo>=</mo><mi>E</mi><mrow><mo>(</mo><mrow><mi>t</mi><mo>/</mo><mi>R</mi></mrow><mo>)</mo></mrow><mo>/</mo><msup><mrow><mrow><mo>[</mo><mrow><mn>3</mn><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><msup><mi>υ</mi><mn>2</mn></msup></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup></mrow></math> However, the theoretical and experimental results are of great error [ 3 3 ] . Generally, an empirical formula is used to calculate the critical stress of a cylindrical shell of medium length: <math display="block" id="M5"><mrow><msub><mi>σ</mi><mrow><mtext>cr</mtext></mrow></msub><mo>=</mo><mtext> </mtext><mn>0.25</mn><mi>E</mi><mi>t</mi><mo>/</mo><mi>R</mi><mo>.</mo></mrow></math> 3 Case study 1 A fuel rod referred to the design for China fast reactor [ 13 13 ] in the CIADS is shown schematically in Fig.2 Fig.2 . Its active region consists of fuel pellets stacked like "dish", top and bottom reflection layer, bob-weight, bottom end plug, gas chamber, gas gap, cladding structure. For LMFBR, 316L is usually used as a candidate material for the cladding [ 14 14 ] . According to the ASME code, Section 2, part D [ 15 15 ] , the 316L physical parameters can be obtained in Table 1 Table 1 . 1 1 Temperature (°C) 1 1 250 1 1 275 1 1 300 1 1 325 1 1 350 1 1 375 1 1 400 1 1 425 1 1 Elastic modulus (GPa) 1 1 179 1 1 —— 1 1 176 1 1 —— 1 1 172 1 1 —— 1 1 169 1 1 —— 1 1 Coefficient of thermal expansion   (×10 −6   ) 1 1 17.4 1 1 17.5 1 1 17.7 1 1 17.8 1 1 17.9 1 1 18.0 1 1 18.1 1 1 18.2 1 1 Thermal conductivity (W) 1 1 17.6 1 1 17.9 1 1 18.3 1 1 18.7 1 1 19.0 1 1 19.4 1 1 19.7 1 1 20.1 3.1. Temperature distribution of fuel cladding 2 The following formulas can be derived by the steady state heat transfer equation [ 16 16 ] , <math display="block" id="M6"><mrow><msub><mi>q</mi><mtext>L</mtext></msub><mi>L</mi><mo>=</mo><mi>Q</mi><msub><mi>C</mi><mrow><mtext>LBE</mtext></mrow></msub><mrow><mo>(</mo><mrow><msub><mi>T</mi><mtext>o</mtext></msub><mo>−</mo><msub><mi>T</mi><mtext>i</mtext></msub></mrow><mo>)</mo></mrow><mo>,</mo><mo> </mo><mo> </mo><mi>R</mi><mo>'</mo><mi>Δ</mi><mi>T</mi><mo>/</mo><mi>Φ</mi><mo>=</mo><mrow><mo>[</mo><mrow><mi>l</mi><mi>n</mi><mrow><mo>(</mo><mrow><msub><mi>T</mi><mtext>o</mtext></msub><mo>/</mo><msub><mi>T</mi><mtext>i</mtext></msub></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mo>/</mo><mrow><mo>(</mo><mrow><mn>2</mn><mi>π</mi><msub><mi>λ</mi><mtext>c</mtext></msub><mi>L</mi></mrow><mo>)</mo></mrow><mo>,</mo></mrow></math> where, q L is the fuel rod linear power density (W/mm) calculated by reactor physics analysis; Q = ρAV is mass flow (g/s), with ρ , A and V being fluid density, the single flow channel area and the flow velocity, respectively; C LBE is the specific heat capacity (J·g −1 ·°C) of coolant (lead bismuth eutectic) at constant pressure; λ c is the thermal conductivity (W·mm −1 ·°C −1 ) of 316L; R ' is the thermal resistance (°C/W); Φ is the heat flux (W); T o is the outlet temperature (°C) and T i is the inlet temperature (°C). By Eq.(6), the temperature distribution of coolant can be calculated by: <math display="block" id="M7"><mrow><msub><mi>T</mi><mtext>o</mtext></msub><mo>=</mo><msub><mi>q</mi><mtext>L</mtext></msub><mi>L</mi><mo>/</mo><mrow><mo>(</mo><mrow><mi>Q</mi><msub><mi>C</mi><mtext>p</mtext></msub></mrow><mo>)</mo></mrow><mtext> </mtext><mo>+</mo><msub><mi>T</mi><mrow><mtext>i</mtext><mo>.</mo></mrow></msub></mrow></math> The heat balance equation is written as: <math display="block" id="M8"><mrow><msub><mi>q</mi><mtext>L</mtext></msub><mi>L</mi><mo>=</mo><mi>α</mi><mi>Ω</mi><mo>·</mo><mi>Δ</mi><msub><mi>T</mi><mtext>θ</mtext></msub><mo>.</mo></mrow></math> So, Δ T θ = q L L /( αΩ ), where, Δ T θ is the film temperature pressure of coolant and cladding, α is the convective heat transfer coefficient, and Ω is the heat transfer area (mm 2 ) of the heat exchange. And thus the temperature of outer surface of the shell is obtained by: <math display="block" id="M9"><mrow><msub><mi>T</mi><mrow><mtext>os</mtext></mrow></msub><mo>=</mo><mi>T</mi><mo>+</mo><mi>Δ</mi><msub><mi>T</mi><mtext>θ</mtext></msub><mo>,</mo></mrow></math> where, T is the temperature of coolant [ 17 17 ] . Through the discrete solution of Eq.(9), the temperature distribution of the cladding from Eq.(9) is shown in Fig.3 Fig.3 . By Eq.(6), the temperature difference between inner and outer surface of fuel cladding can be derived as <math display="block" id="M10"><mrow><mi>Δ</mi><msub><mi>T</mi><mtext>c</mtext></msub><mo>=</mo><mi>R</mi><mo>'</mo><mi>Φ</mi><mo>=</mo><mrow><mo>[</mo><mrow><msub><mi>q</mi><mtext>L</mtext></msub><mo>/</mo><mrow><mo>(</mo><mrow><mn>2</mn><mi>π</mi><msub><mi>λ</mi><mtext>c</mtext></msub></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mi>l</mi><mi>n</mi><mrow><mo>(</mo><mrow><msub><mi>R</mi><mtext>o</mtext></msub><mo>/</mo><msub><mi>R</mi><mtext>i</mtext></msub></mrow><mo>)</mo></mrow><mtext> </mtext><mo>.</mo></mrow></math> From Eq.(10), the temperature difference is 2°C to 3°C. Thus, it can be neglected, and the average temperature of inner and outer surface of fuel cladding can be used as the cladding temperature. The temperature function of T = 0.16 x + 296.3 is obtained by fitting the data in Fig.3 Fig.3 . 3.2. Calculation of the gap for axial expansion of cladding 2 Radial deformation of the 0.5-mm thick cladding is negligible, and the deformation of cladding is just axial deformation. If the shell deformation by thermal expansion is more than the reserve space of the upper end, the cladding can produce extrusion stress. The gap calculation is discussed as follows. Thermal expansion coefficients of 316L at different temperatures in Table 1 Table 1 can be fitted by K ( T ) = 4×10 −9 T + 16.5×10 −6 . In this paper, the thermal expansion is obtained from ASME code, section 2 [ 15 15 ] . The calculation length of the cladding and the outer sleeve is L =1675 mm. For Δ x , the thermal expansion can be written as Δ δ = K ( T ) Δ x Δ T , where Δ T = T −25 is the temperature changes by setting reference temperature at 25°C.. For the active region, the total deformation is <math display="block" id="M11"><mrow><msub><mi>δ</mi><mrow><mtext>active</mtext></mrow></msub><mo>=</mo><msub><mo>∫</mo><mn>0</mn></msub><msup><mrow/><mi>L</mi></msup><mrow><mo>(</mo><mrow><mi>K</mi><mo>·</mo><mi>Δ</mi><mi>T</mi></mrow><mo>)</mo></mrow><mtext>d</mtext><mi>x</mi><mo>.</mo></mrow></math> On substituting the parameters into Eq.(11), we have δ active = 4.562 mm. In addition to active region of the fuel rod, the lengths of upper and lower non-active region of the cladding are calculated to be L lower = L upper = 575 mm. In the active region, the cladding temperature change is so small that the expansion can be calculated based on the average temperatures, <math display="block" id="M12"><mtable columnalign="left"><mtr><mtd><mi>Δ</mi><msub><mi>L</mi><mrow><mtext>lower</mtext></mrow></msub><mo>=</mo><msub><mi>K</mi><mrow><mn>300</mn></mrow></msub><msub><mi>L</mi><mrow><mtext>lower</mtext></mrow></msub><mrow><mo>(</mo><mrow><mn>300</mn><mo>−</mo><mn>25</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>2.081</mn><mtext> mm </mtext><mo>,</mo></mtd></mtr><mtr><mtd><mi>Δ</mi><msub><mi>L</mi><mrow><mtext>upper</mtext></mrow></msub><mo>=</mo><msub><mi>K</mi><mrow><mn>421</mn></mrow></msub><msub><mi>L</mi><mrow><mtext>upper</mtext></mrow></msub><mrow><mo>(</mo><mrow><mn>421</mn><mo>−</mo><mn>25</mn></mrow><mo>)</mo></mrow><mo>=</mo><mn>1.982</mn><mtext> mm </mtext><mo>,</mo></mtd></mtr><mtr><mtd><mi>Δ</mi><msub><mi>L</mi><mrow><mtext>total</mtext></mrow></msub><mo>=</mo><msub><mi>δ</mi><mrow><mtext>active</mtext></mrow></msub><mo>+</mo><mi>Δ</mi><msub><mi>L</mi><mrow><mtext>lower</mtext></mrow></msub><mo>+</mo><mi>Δ</mi><msub><mi>L</mi><mrow><mtext>upper</mtext></mrow></msub><mo>=</mo><mn>8.625</mn><mtext> mm</mtext><mo>,</mo></mtd></mtr></mtable></math> where, K 300 and K 421 is the thermal expansion coefficient of the lower and upper parts. By applying a fixed constraint to the lower end of cladding and the temperature function in Fig.3 Fig.3 , the displacement of cladding in the active region can be calculated by the ANSYS code ( Fig.4 Fig.4 ). Comparing simulation results in Fig.5 Fig.5 with the calculation results, we can see that the radial thermal deformation has a greater influence than the axial thermal deformation, and thus <math display="block" id="M13"><mrow><mi>Δ</mi><msub><mi>L</mi><mrow><mtext>total</mtext></mrow></msub><mo>=</mo><mn>6.40</mn><mo>+</mo><mi>Δ</mi><msub><mi>L</mi><mrow><mtext>lower</mtext></mrow></msub><mo>+</mo><mi>Δ</mi><msub><mi>L</mi><mrow><mtext>upper</mtext></mrow></msub><mo>=</mo><mn>10.46</mn><mtext> mm</mtext><mo>.</mo></mrow></math> From thermal hydraulic calculation, the outer sleeve temperature distributes linearly, and temperature changes of the outer sleeve can be obtained by Eq.(13), <math display="block" id="M14"><mrow><mi>T</mi><mo>'</mo><mo>=</mo><mn>0.072</mn><mi>x</mi><mo>+</mo><mn>250</mn><mtext> </mtext><mo>.</mo></mrow></math> Taking Eq.(13) into Eq.(11), the thermal deformation is 5.107 mm. Thermal deformations of the outer sleeve calculated by the ANSYS code are shown in Fig.5 Fig.5 . Through linear buckling analysis using the ANSYS code, critical stresses of the cladding under axial compression are shown in Fig.6 Fig.6 . 3.3. Analysis and discussion of the result 2 Because of the calculation results of ANSYS taking into account the influence of the radial to the axial direction, the total deformation of the cladding is 2.09 mm larger than that of the outer sleeve. Tuckling critical stresses calculated using Eq.(1) and APDL ANSYS code are 9.33 and 1.04 MPa, respectively; while the critical stress of cladding collapse calculated using Eq.(2) is 37.7 MPa The three calculation methods above can obtain critical stress of cladding failure, with the empirical formula of Eq. (2) giving the largest result. In the reactor, the fuel rod load in addition to the axial pressure include the coolant and chamber gas, so the balance of radial load increases the structure stability. Therefore, the gap is calculated using Eq. (2). According to the critical stress, the average strain can be estimated at <math display="block" id="M15"><mrow><mi>ε</mi><mo>=</mo><msub><mi>σ</mi><mrow><mtext>cr</mtext></mrow></msub><mo>/</mo><mi>E</mi><mo>=</mo><mn>1.38</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>4</mn></mrow></msup><mo>.</mo></mrow></math> Then the allowed compress value of the fuel rods is Δ L = εL =0.23 mm. Therefore, to prevent the fuel rod buckling, the gap range is 1.86~2.09 mm. 4 Conclusion 1 In this paper three kinds of critical stress calculation methods about cylindrical shells under different shell length is analyzed and obtained. The temperature distribution of cladding is calculated which is based on nuclear reactor physics and heat transfer calculation for CIADS fuel rod. The temperature difference between inlet and outlet is about 120 °C. Finally, the thermal expansion gap reserved is obtained, which is vital of importance for the design of fuel rod assembly. Besides, the calculation method of thermal expansion gap is studied, the results of three calculation methods are compared and analyzed. Through the analysis it is found that the optimal method to calculating the reserved clearance is the second method in table 2. the gap between the end plug and the end seat is 1.86~2.09 mm. References W.L. Zhan , H.S. Xu , Advanced Fission Energy Program - ADS Transmutation System . Bulletin of Chinese Academy of Sciences . 27 , 375 - 383 ( 2012 ). doi: 10.3969/j.issn.1000-3045.2012.03.017 http://doi.org/10.3969/j.issn.1000-3045.2012.03.017 Advanced Fission Energy Program - ADS Transmutation System M.S. Ismail , J. Purbolaksono , A. Andriyana et al ., The use of initial imperfection approach in design process and buckling failure evaluation of axially compressed composite cylindrical shells . Eng. Fail. 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( High Education Press , Beijing, China , 2007 ), pp. 34 - 53 Heat transfer W.L. Huang , R.Y. Sa , D.N. Zhou , et al ., Experimental study on fragmentation behaviors of molten LBE and water contact interface . Nucl. Sci. Tech . 2015 , 26 : 060601 . doi: 10.13538/j.1001-8042/nst.26.060601 http://doi.org/10.13538/j.1001-8042/nst.26.060601 Experimental study on fragmentation behaviors of molten LBE and water contact interface 10.1007/s41365-016-0143-1 This work was supported by the "Strategic Priority Research Program" of Chinese Academy of Sciences (No.XDA03030102) . 2016-03-10 2016-06-20 2016-06-22 2016-10-17 2016-12-20 2021-04-11 © 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. 2016 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/s41365-016-0143-1 http://dx.doi.org/10.1007/s41365-016-0143-1 . 2016 Nuclear Science and Techniques 06 27