Author Affiliations
1College of Nuclear Science and Technology, University of South China, Hengyang 421001, China2Hunan Digital Reactor Engineering and Technology Research Center, University of South China, Hengyang 421001, Chinashow less
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. 2. Data distribution plot of 1 600 design variables
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
| 设计参数Design parameters | 数值Numerical value |
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| 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 |
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Table 1. Multi-functional ultra high flux lead-bismuth cooled reactor design parameters
| 设计变量Design variables | 取值区间Value interval |
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| 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] |
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Table 2. Multi-functional ultra-high flux lead-bismuth cooled reactor optimization variable value intervals
| 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 |
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Table 3. Physical parameters of lead-bismuth alloy
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 |
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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 |
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Table 4. High-flux lead-bismuth cooled reactor training database
| 超参数Super-parameter | 数值Numerical value |
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| 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] |
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Table 5. Hyperparameter space setting
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 |
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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) |
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Table 6. Neural network model architecture setup
神经网络预测模型 Neural network prediction model | MSE / 10–4 | R2 / 10–2 |
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| 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 |
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Table 7. Prediction accuracy of neural network models
堆芯设计参数 Core design parameters | 数值Value |
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| 第一组Group 1 | 第二组Group 2 | 第三组Group 3 |
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| 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 |
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Table 8. Comparison of results between neural network predicted values and RMC calculated values
| 堆芯方案及参数Core program and parameters | 初始方案Initial program | 优化方案Optimization solutions |
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| 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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Table 9. Optimal program design parameters
