1 Introduction
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]. Stability of cylindrical shells is also a hot spot of study [2-4]. Chen[5] and Zhao[6] did numerical buckling analysis of cylindrical shell under uniform axial compression in their thesis. Imani et al.[7] studied pellet cladding interaction (PCI).
As the fast reactor fuel rods are designed with high precision requirements for manufacturing and assembling [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] and Williamson et al.[10] studied the fuel behavior in different working conditions. Hofmeister et al.[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
2.1. Problem description
In order to prevent buckling of the pellet cladding due to different thermal expansion of the materials, a certain gap (xg) is reserved between a fuel rod and its outer sleeve assembly, as shown in Fig.1.
2.2. Theoretical calculation of the critical stress
For cylindrical shell, a dimensionless shell length parameter χ = l2/(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] i.e. slender cylindrical shell (χ ≤3.018), middle length (χ >3.018), and short cylindrical shell (χ ≥8.069R2/t2). 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 σcr can be obtained by the formula of Euler beam,
where, E is elastic or Young's modulus; I = π(D4−d4)/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],
For middle or short length cylindrical shell, Eq.(3) can be applied based on the Donnell linear theory [3],
where, D'= Et3/[12(1− υ2)] 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 σx is the axial stress.
By solving Eq.(3), the critical stress σcr is obtained.
However, the theoretical and experimental results are of great error [3]. Generally, an empirical formula is used to calculate the critical stress of a cylindrical shell of medium length:
3 Case study
A fuel rod referred to the design for China fast reactor [13] in the CIADS is shown schematically in 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]. According to the ASME code, Section 2, part D[15], the 316L physical parameters can be obtained in Table 1.
| Temperature (°C) | 250 | 275 | 300 | 325 | 350 | 375 | 400 | 425 |
|---|---|---|---|---|---|---|---|---|
| Elastic modulus (GPa) | 179 | —— | 176 | —— | 172 | —— | 169 | —— |
| Coefficient of thermal expansion (×10−6 ) | 17.4 | 17.5 | 17.7 | 17.8 | 17.9 | 18.0 | 18.1 | 18.2 |
| Thermal conductivity (W) | 17.6 | 17.9 | 18.3 | 18.7 | 19.0 | 19.4 | 19.7 | 20.1 |
3.1. Temperature distribution of fuel cladding
The following formulas can be derived by the steady state heat transfer equation [16],
where, qLis 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; CLBE is the specific heat capacity (J·g−1·°C) of coolant (lead bismuth eutectic) at constant pressure; λc is the thermal conductivity (W·mm−1·°C−1) of 316L; R' is the thermal resistance (°C/W); Φ is the heat flux (W); To is the outlet temperature (°C) and Ti is the inlet temperature (°C).
By Eq.(6), the temperature distribution of coolant can be calculated by:
The heat balance equation is written as:
So, ΔTθ= qLL /(αΩ), where, ΔTθ is the film temperature pressure of coolant and cladding, α is the convective heat transfer coefficient, and Ω is the heat transfer area (mm2) of the heat exchange.
And thus the temperature of outer surface of the shell is obtained by:
where, T is the temperature of coolant [17].
Through the discrete solution of Eq.(9), the temperature distribution of the cladding from Eq.(9) is shown in Fig.3.
By Eq.(6), the temperature difference between inner and outer surface of fuel cladding can be derived as
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 = 0.16x + 296.3 is obtained by fitting the data in Fig.3.
3.2. Calculation of the gap for axial expansion of cladding
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 can be fitted by K(T) = 4×10−9T + 16.5×10−6.
In this paper, the thermal expansion is obtained from ASME code, section 2[15]. The calculation length of the cladding and the outer sleeve is L=1675 mm. For Δx, the thermal expansion can be written as Δδ=K(T) ΔxΔT, where ΔT = T−25 is the temperature changes by setting reference temperature at 25°C..
For the active region, the total deformation is
On substituting the parameters into Eq.(11), we have δactive= 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 Llower = Lupper = 575 mm. In the active region, the cladding temperature change is so small that the expansion can be calculated based on the average temperatures,
where, K300 and K421 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, the displacement of cladding in the active region can be calculated by the ANSYS code (Fig.4).
Comparing simulation results in 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
From thermal hydraulic calculation, the outer sleeve temperature distributes linearly, and temperature changes of the outer sleeve can be obtained by Eq.(13),
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. Through linear buckling analysis using the ANSYS code, critical stresses of the cladding under axial compression are shown in Fig.6.
3.3. Analysis and discussion of the result
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
Then the allowed compress value of the fuel rods is ΔL=εL=0.23 mm. Therefore, to prevent the fuel rod buckling, the gap range is 1.86~2.09 mm.
4 Conclusion
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.
Advanced Fission Energy Program - ADS Transmutation System
. Bulletin of Chinese Academy of Sciences. 27, 375-383(2012). doi: 10.3969/j.issn.1000-3045.2012.03.017The use of initial imperfection approach in design process and buckling failure evaluation of axially compressed composite cylindrical shells
. Eng. Fail. Anal. 51, 20-28(2015). doi: 10.1016/j.engfailanal.2015.02.017Experiment and numerical analysis of axial elastic and plastic buckling of welded cylindrical shells, Ph. D. Thesis
,The analysis on elastic and plastic stability of cylindrical shell, Master Thesis
,Numerical buckling analysis and optimization of cylindrical shell under uniform axial compression, Ph. D. Thesis
,Stability design of axially compressed the cylindrical shells, Eng
. Mech. 20, 116-126(2003). doi: 10.3969/j.issn.1000-4750.2003.06.021Numerical study of fuel-clad mechanical interaction during long-term burnup of WWER1000
. Ann. Nucl. Energy. 80, 267-278(2015). doi: 10.1016/j.anucene.2015.01.036European lead fast reactor—ELSY
, Nucl. Eng. Des. 241, 3470-3480(2011). doi: 10.1016/j.nucengdes.2011.03.029Fuel rod behavior under normal operating conditions in Super Fast Reactor with high power density
, Nucl. Eng. Des. 289, 166-174(2015). doi: 10.1016/j.nucengdes.2015.04.037Multidimensional multiphysics simulation of nuclear fuel behavior
, J. Nucl. Mater. 423,149-163(2012). doi: 10.1016/j.jnucmat.2012.01.012Fuel assembly design study for a reactor with supercritical water
, Nucl. Eng. Des. 237, 1513-1521(2007). doi: 10.1016/j.nucengdes.2007.01.008Fuel design for the experimental ADS MYRRHA
,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