The Coupling State Equations and the Symplectic Algorithm for Control Rod Drop and Fluid Flow
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摘要:
针对核反应堆内控制棒下落问题,提出了描述控制棒下落与流体流动的耦合非线性状态方程。该状态方程对于落棒过程内不同的流体状态,具有统一的表达形式,可以很方便地处理不同工况下的落棒问题。为高效分析落棒过程,准确捕捉落棒过程内流动状态的突变,并保证时程积分的数值稳定,提出了一种基于时间步长自适应的保辛算法。数值算例表明,提出的数值模型可以采用较大的时间步长精确计算控制棒在下落过程中的位移、速度、加速度、落棒时间等关键数据,计算结果与商业软件所得结果高度吻合。
Abstract:The nonlinear state equations describing the coupling between control rod drop and fluid flow were proposed to solve the problem of control rod drop in nuclear reactors. The state equations have a uniform format for different fluid states in the process of control rod drop, which can conveniently deal with the problem for different working conditions. To efficiently analyze the falling process, accurately capture the sudden change of flow state and ensure the numerical stability of time integral, an adaptive time step-based symplecticity-preserving algorithm was proposed. Numerical examples show that, the proposed numerical model can accurately calculate the key data such as the displacement, the velocity, the acceleration and the falling time of the control rod in the falling process with a large time step, and the calculated results are in good agreement with those obtained the by the commercial software.
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表 1 小堆的主要输入参数
Table 1. Main input parameters for a small reactor
parameter value unit reactor in-core temperature 312.22 ℃ reactor in-core pressure 15.5 MPa control rod length 2.5756 m control rod diameter 0.009675 m control rod mass 11.407 kg control rod absolute roughness 3.0 × 10−8 m guide tube average diameter 0.01124 m guide tube absolute roughness 4.0 × 10−7 m control rod initial insertion depth 0.2874 m spring preload 1876.4 N spring stiffness 123200 N/m 表 2 本文模型在不同时间步长下与商业软件关于T5和T5 + T6的比较
Table 2. The comparison of T5 and T5 + T6 between the commercial software and the proposed model for different initial time steps
commercial software initial time steps of this paper $\Delta {t_0}/{\rm{s}}$ 0.001 0.005 0.01 0.015 T5/s 2.140 2.140
(0%)2.1506
(0.50%)2.1469
(0.32%)2.1488
(0.41%)(T5 + T6)/s 3.560 3.5575
(0.07%)3.5675
(0.21%)3.5706
(0.30%)3.5559
(0.12%) -
[1] YOON K H, KIM J Y, LEE K H, et al. Control rod drop analysis by finite element method using fluid-structure interaction for a pressurized water reactor power plant[J]. Nuclear Engineering and Design, 2009, 239(10): 1857-1861. doi: 10.1016/j.nucengdes.2009.05.023 [2] 肖聪, 罗英, 杜华, 等. 基于动网格技术的单根控制棒落棒行为仿真分析[J]. 核动力工程, 2017, 38(2): 103-107 doi: 10.13832/j.jnpe.2017.02.0103XIAO Cong, LUO Ying, DU Hua, et al. Simulation and analysis of single control rod dropping behavior based on dynamic grid technique[J]. Nuclear Power Engineering, 2017, 38(2): 103-107.(in Chinese) doi: 10.13832/j.jnpe.2017.02.0103 [3] RAJAN BABU V, THANIGAIYARASU G, CHELLAPANDI P. Mathematical modelling of performance of safety rod and its drive mechanism in sodium cooled fast reactor during scram action[J]. Nuclear Engineering and Design, 2014, 278: 601-617. doi: 10.1016/j.nucengdes.2014.08.015 [4] 刘新, 陈先龙, 张修, 等. 控制棒下落时间计算模型[J]. 核技术, 2014, 37(11): 68-74 doi: 10.11889/j.0253-3219.2014.hjs.37.110604LIU Xin, CHEN Xianlong, ZHANG Xiu, et al. Calculation model of controlling rod drop time[J]. Nuclear Technology, 2014, 37(11): 68-74.(in Chinese) doi: 10.11889/j.0253-3219.2014.hjs.37.110604 [5] 刘言午, 黄炳臣, 冉小兵, 等. 反应堆控制棒落棒时间计算方法分析[J]. 核动力工程, 2014, 35(6): 106-110 doi: 10.13832/j.jnpe.2014.06.0106LIU Yanwu, HUANG Bingchen, RAN Xiaobing, et al. Analysis of calculation method of reactor control rod drop time[J]. Nuclear Power Engineering, 2014, 35(6): 106-110.(in Chinese) doi: 10.13832/j.jnpe.2014.06.0106 [6] 王栋. 算子分裂法及其在解抛物型方程中的应用[D]. 硕士学位论文. 长春: 吉林大学, 2009.WANG Dong. Operator-splitting method and its application for solving parabolic equations[D]. Master Thesis. Changchun: Jilin University, 2009. (in Chinese) [7] 孔珑. 工程流体力学[M]. 北京: 中国电力出版社, 2001.KONG Long. Engineering Fluid Mechanics[M]. Beijing: China Electric Power Press, 2001. (in Chinese) [8] WU F, GAO Q, ZHONG W X. Fast precise integration method for hyperbolic heat conduction problems[J]. Applied Mathematics and Mechanics (English Edition) , 2013, 34(7): 791-800. doi: 10.1007/s10483-013-1707-6 [9] 邢誉峰, 杨蓉. 动力学平衡方程的Euler中点辛差分求解格式[J]. 力学学报, 2007, 39(1): 100-105 doi: 10.3321/j.issn:0459-1879.2007.01.013XING Yufeng, YANG Rong. Application of Euler midpoint symplectic integation method for the solution of dynamic equilibrium equations[J]. Acta mechanica Sinica, 2007, 39(1): 100-105.(in Chinese) doi: 10.3321/j.issn:0459-1879.2007.01.013 [10] 吴锋, 姚征, 孙雁, 等. 位移浅水内孤立波[J]. 计算力学学报, 2016, 36(3): 297-303.WU Feng, YAO Zheng, SUN Yan, et al. Displacement shallow water internal solitary wave[J]. Chinese Journal of Computational Mechanics, 2016, 36(3): 297-303. (in Chinese) [11] 钟万勰. 离散动力学数值积分应该保辛近似[J]. 北京工业大学学报, 2016, 42(12): 12-14.ZHONG Wanxie. Symplectic conservative approximation for discrete dynamics integration[J]. Journal of Beijing University of Technology, 2016, 42(12): 12-14. (in Chinese) [12] 钟万勰. 离散动力学只能说保辛[J]. 应用数学和力学, 2016, 37(8): 775-777ZHONG Wanxie. Only symplectic conservation is characteristic of discrete dynamics[J]. Applied Mathematics and Mechanics, 2016, 37(8): 775-777.(in Chinese) [13] 高强, 彭海军, 吴志刚, 等. 非线性动力学系统最优控制问题的保辛求解方法[J]. 动力学与控制学报, 2010, 8(1): 1-7.GAO Qiang, PENG Haijun, WU Zhigang, et al. Symplectic method for solving optimal control problem of nonlinear dynamical systems[J]. Journal of Dynamics and Control, 2010, 8(1): 1-7. (in Chinese) [14] 王昕炜, 彭海军, 钟万勰. 具有潜伏期时滞的时变SEIR模型的最优疫苗接种策略[J]. 应用数学和力学, 2019, 40(7): 701-712WANG Xinwei, PENG Haijun, ZHONG Wanxie. Optimal vaccination strategies of time-varying SEIR epidemic model with latent delay[J]. Applied Mathematics and Mechanics, 2019, 40(7): 701-712.(in Chinese) [15] 钟万勰, 吴锋, 孙雁, 等. 保辛水波动力学[J]. 应用数学和力学, 2018, 39(8): 855-874ZHONG Wanxie, WU Feng, SUN Yan, et al. Symplectic water wave dynamics[J]. Applied Mathematics and Mechanics, 2018, 39(8): 855-874.(in Chinese) [16] 吴锋, 钟万勰. 浅水问题的约束Hamilton变分原理及祖冲之类保辛算法[J]. 应用数学和力学, 2016, 37(1): 1-13 doi: 10.3879/j.issn.1000-0887.2016.01.001WU Feng, ZHONG Wanxie. The constrained Hamilton variational principle for shallow water problems and the Zu-type symplectic algorithm[J]. Applied Mathematics and Mechanics, 2016, 37(1): 1-13.(in Chinese) doi: 10.3879/j.issn.1000-0887.2016.01.001