NUCLEAR TECHNIQUES, Volume. 47, Issue 10, 100602(2024)
Design method of high-flux lead-bismuth cooled reactor neutron flux maximization based on BP neural network
Figures & Tables(17)
Fig. 1. Cross-sectional views of a multifunctional ultra-high flux lead-bismuth cooled reactor(a) Cross section of core along X-Y axis, (b) Cross section of core along X-Z axis, (c) Cross section of fuel assembly, (d) Cross section of fuel rod
Fig. 3. Learning curve of BP neural network prediction model(a) φmax neural network prediction model, (b) keff neural network prediction model
Fig. 4. Flow chart of sensitivity analysis method for core design parameters based on Sobol index method
Fig. 5. Flowchart of optimization method based on BP neural network dynamic surrogate model
Fig. 6. Operation flow of high-flux lead-bismuth cooled reactor optimization design platform
Fig. 7. Sensitivity index of core design parameters to maximum neutron flux (color online)
Fig. 8. Iterative optimization results of maximum neutron flux density in the core
Table 1. Multi-functional ultra high flux lead-bismuth cooled reactor design parameters
View tableView in ArticleTable 1. Multi-functional ultra high flux lead-bismuth cooled reactor design parameters
设计参数Design parameters 数值Numerical value 热功率Thermal power / MW 150 换料周期Refueling cycle / d 90 冷却剂材料Coolant material 208Pb-Bi 反射层材料Reflecting layer material 208Pb 包壳材料Cladding material T91 燃料棒间隙填充气体Fuel rod gap filler gas He 入口温度Inlet temperature / ℃ 170 出口温度Outlet temperature / ℃ 536.5 燃料装载量/235U装载量Fuel loads/235U loads / kg 779/175.3 堆芯活性区等效直径Equivalent diameter of active zone / cm 58.14 堆芯活性区高度Height of active zone / cm 50 燃料棒内/外直径Fuel rod inner/outer diameter / cm 4/4.6 燃料棒气隙宽度Fuel rod gas gap width / mm 0.1 包壳厚度Cladding thickness / mm 0.2 栅距Grid pitch / mm 5.2 栅径比P/D Grid diameter ratio P/D 1.130 4 反射层轴向/径向厚度Reflective layer axial/radial thickness / cm 80/120
Table 2. Multi-functional ultra-high flux lead-bismuth cooled reactor optimization variable value intervals
View tableView in ArticleTable 2. Multi-functional ultra-high flux lead-bismuth cooled reactor optimization variable value intervals
设计变量Design variables 取值区间Value interval 栅径比Grid diameter ratio [1.00, 1.50] 燃料芯块直径Fuel diameter / cm [0.3, 1.5] 堆芯活性区高度Height of core active area / cm [40, 200] 反射层厚度Reflective layer thickness / cm [ 20 , 220]
Table 3. Physical parameters of lead-bismuth alloy
View tableView in ArticleTable 3. Physical parameters of lead-bismuth alloy
参数Parameters 值/计算式Value/Equation 熔点Melting point / K TM=398.0 沸点Boiling point / K TB=1 927.0 表面张力Surface tension / N‧m-1 σ=(448.5–0.08T)×10-3 密度Density / kg‧m-3 ρ=11 065–1.293T 等压比热Constant pressure specific heat / J‧(kg‧K)-1 Cp=164.8–3.94×10-2T+1.25×10-5T2–4.56×105T2 动力粘度Viscosity of dynamics / Pa‧s μ=4.94×10-4exp(754.1/T) 热导率Heat conduction / W‧(m‧K)-1 λ=3.284+1.617×10-2T–2.305×10-6T2
Table 4. High-flux lead-bismuth cooled reactor training database
View tableView in ArticleTable 4. High-flux lead-bismuth cooled reactor training database
样本
Sample
size
设计变量
Design variable
目标函数响应值
Objective function
response value
约束条件响应值
Constraint response
value
栅径比
Grid pitch ratio
燃料芯块直径
Fuel diameter
/ cm
堆芯活性区高度
Height of core
active zone / cm
反射层厚度
Reflective layer
thickness / cm
φmax / n·cm–2·s–1 keff 1 1.493 776 0.429 575 185.214 6 162.842 2 1.662 1×1015 1.215 37 2 1.315 769 0.677 587 91.307 3 185.845 7 1.548 1×1015 1.327 71 3 1.495 511 0.741 313 164.465 9 152.645 8 7.189 8×1014 1.391 04 4 1.427 046 0.884 549 72.922 3 133.951 0 1.218 5×1015 1.334 17 5 1.420 833 1.470 295 55.927 6 193.398 0 6.153 2×1014 1.398 67 6 1.236 895 1.407 295 68.267 4 60.641 8 5.943 2×1014 1.461 60 7 1.240 646 1.489 613 100.568 1 109.038 4 3.461 9×1014 1.528 40 8 1.227 713 0.751 283 162.399 0 115.514 5 7.351 6×1014 1.437 19 9 1.174 609 1.465 010 187.868 3 143.498 9 1.858 2×1014 1.592 46 10 1.485 631 0.663 466 143.914 9 177.673 0 9.966 5×1014 1.349 12 … … … … … … … 1 600 1.183 904 1.387 367 85.603 8 152.429 0 4.558 4×1014 1.516 41
Table 5. Hyperparameter space setting
View tableView in ArticleTable 5. Hyperparameter space setting
超参数Super-parameter 数值Numerical value 隐藏层层数Number of hidden layers [ 1 ,2 ,3 ,4 ,5 ]学习率Learning rate [1×10–4, 1×10–3, 1×10–2, 1×10–1] 训练批次Batch size [32, 48, 96, 128, 256, 512, 1 024] L2正则化系数L2 regularizer coefficient [1×10–3, 1×10–2, 1×10–1]
Table 6. Neural network model architecture setup
View tableView in ArticleTable 6. Neural network model architecture setup
参数
Parameters
φmax神经网络模型
φmax neural network model
keff神经网络模型
keff neural network model
输入参数
Input parameters
燃料芯块直径、栅距、活性区高度、反射层厚度
Fuel diameter, grid pitch, active zone height, reflective layer thickness
输出参数
Output parameters
堆芯最大快中子通量
Maximum fast neutron flux in the core
有效增殖因数
Effective multiplication factor
学习率Learning rate 0.001 0.01 训练次数Epochs 2 000 2 000 训练批次Batch size 32 512 隐藏层层数Number of hidden layers 1 3 隐藏层神经元个数Number of neurons per hidden layer 100 100/100/100 激活函数Activation function ReLu ReLu 损失函数Loss function Mean_squared_error Mean_squared_error 优化器Optimization algorithm Adam Adam 正则化Regularization L2(0.000 1) L2(0.000 1)
Table 7. Prediction accuracy of neural network models
View tableView in ArticleTable 7. Prediction accuracy of neural network models
神经网络预测模型
Neural network prediction model
MSE / 10–4 R2 / 10–2 φmax神经网络φmax neural network 9.974 4 0.999 1 keff神经网络keff neural network 1.093 9 0.998 5
Table 8. Comparison of results between neural network predicted values and RMC calculated values
View tableView in ArticleTable 8. Comparison of results between neural network predicted values and RMC calculated values
堆芯设计参数
Core design parameters
数值Value 第一组Group 1 第二组Group 2 第三组Group 3 栅径比Grid pitch ratio 1.096 2 1.095 4 1.091 5 燃料芯块直径Fuel diameter / cm 0.391 4 0.405 4 0.405 1 堆芯活性区高度Height of core active zone / cm 44.313 8 40.340 9 40.381 1 反射层厚度Reflective layer thickness / cm 212.711 1 207.437 4 211.449 3 φmax
/1015 n·cm–2·s–1
BP神经网络预测值BP NN predicted value 8.832 0 9.107 0 9.128 8 RMC计算值RMC calculated value 8.827 3 9.105 0 9.127 5 相对误差Relative error / % 0.053 3 0.021 8 0.014 3 keff BP神经网络预测值BP NN predicted value 1.008 2 1.003 0 1.003 8 RMC计算值RMC calculated value 1.008 7 1.002 8 1.004 0 相对误差Relative error / % 0.047 0 0.022 5 0.019 4
Table 9. Optimal program design parameters
View tableView in ArticleTable 9. Optimal program design parameters
堆芯方案及参数Core program and parameters 初始方案Initial program 优化方案Optimization solutions 最大中子通量密度Maximum neutron flux density / n·cm–2·s–1 7.979 8×1015 9.209 8×1015 栅径比Grid pitch ratio 1.108 7 1.090 6 燃料芯块直径Fuel diameter / cm 0.4 0.402 7 堆芯活性区高度Height of core active zone / cm 50 40.477 7 反射层厚度Reflective layer thickness / cm 80 214.182 0 初始keff Initial keff 1.005 9 1.001 5 换料周期Refuel cycle / d ＞90 ＞90 包壳最高温度Maximum cladding temperature / °C 533.63 525.43 芯块最大温度Maximum fuel pellet temperature / °C 992.67 969.10
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Tong WANG, Zijing LIU, Pengcheng ZHAO, Yingjie XIAO. Design method of high-flux lead-bismuth cooled reactor neutron flux maximization based on BP neural network[J]. NUCLEAR TECHNIQUES, 2024, 47(10): 100602
Paper Information
Category: NUCLEAR ENERGY SCIENCE AND ENGINEERING
Received: Jan. 31, 2024
Accepted: --
Published Online: Dec. 13, 2024
The Author Email: LIU Zijing (LIUZijing)
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