Abstract：[Objective] In high-temperature gas-cooled reactors (HTGR), experimental results show that thermal radiation plays a significant role in the heat transfer process and it is highly related to the inherent safety of the nuclear reactor. The HTGR core is a dense pebble bed of single-sized fuel spheres. It is a fundamental and very challenging task to discuss the particle-scale radiative heat transfer in a pebble bed. In this study, using a machine learning approach, an artificial intelligent regression model is developed and trained on a large dataset to predict the obstructed view factor between fuel spheres in a nuclear pebble bed.[Methods] Comparing the numerical and experimental results, the radiative transfer equation (RTE) method significantly underestimates the radiative heat flux in dense pebble beds. The Lagrangian discrete element method (DEM) is often applied in pebble flow simulations. Thus, particle-particle interactions in the DEM framework are used in this present study. The view factor in the base model is calculated using an explicit analytical expression with the elliptic integral function. The model reasonably describes the effect of the distance between spheres and the average obstruction contribution of the surroundings. The obstruction function is proposed by fitting the numerical results. The analytical base model is used to efficiently obtain the dominant parts of the view factor. Furthermore, for the machine learning model, a large dataset of the view factor is established by the particle-scale DEM packing of the HTR-PM, an HTGR demonstration plant, and the thermal ray tracing method. Following preprocessing, the dataset contains a total of 16.6 million records under various conditions in a pebble bed. The gradient boosting decision tree (GBDT) model is used to learn the rules for view factor regression. The model input is the Cartesian coordinates of the sphere and its surrounding ones. The model output is the difference between the ground truth of the view factor and the analytical base model's prediction. 80% of the dataset is used for training, and 20% is left for validation. The mean square error (MSE) and coefficient of determination are selected to evaluate the machine learning model. The GBDT model was trained using the open-source software LightGBM, and the hyperparameter tuning was performed in the FLAML platform to find the best model parameters.[Results] The MSE of the trained machine learning model decreases gradually as the model complexity increases. The analytical base model provides a generally satisfactory forecast of the view factor in the pebble bed, and the gradient boosting decision tree model trained by big data greatly improved the prediction accuracy. With the base model, the coefficients of determination of the trained machine learning model are greater than 0.999.[Conclusions] This study presents an efficient artificial intelligent model for obstructed view factor prediction for heat transfer research, parameter optimization, and thermal hydraulic analysis of the nuclear pebble bed. The trained machine learning model can also be used in effective thermal conductivity analysis, and it is feasible to be coupled with the CFD-DEM simulations of conduction and heat convection in large-scale nuclear pebble beds.
吴浩, 牛风雷. 高温球床辐射传热中的机器学习模型[J]. 清华大学学报（自然科学版）, 2023, 63(8): 1213-1218.
WU Hao, NIU Fenglei. Machine learning model of radiation heat transfer in the high-temperature nuclear pebble bed. Journal of Tsinghua University(Science and Technology), 2023, 63(8): 1213-1218.
[1] 张作义, 吴宗鑫, 王大中, 等. 我国高温气冷堆发展战略研究[J]. 中国工程科学, 2019, 21(1):12-19. ANG Z Y, WU Z X, WANG D Z, et al. Development strategy of high temperature gas cooled reactor in China[J]. Strategic Study of CAE, 2019, 21(1):12-19. (in Chinese) [2] 姜胜耀, 桂南, 杨星团, 等. 球床式高温气冷堆球流及球床辐射的理论与实验研究进展[J]. 原子能科学技术, 2019, 53(10):1918-1929. JIANG S Y, GUI N, YANG X T, et al. Theoretical and experimental research progress on pebble flow and effective thermal conductivity in pebble-type HTGR[J]. Atomic Energy Science and Technology, 2019, 53(10):1918-1929. (in Chinese) [3] PENG Z B, DOROODCHI E, MOGHTADERI B. Heat transfer modelling in discrete element method (DEM)-based simulations of thermal processes:Theory and model development[J]. Progress in Energy and Combustion Science, 2020, 79:100847. [4] ROUSSEAU P G, DU TOIT C G, VAN ANTWERPEN W, et al. Separate effects tests to determine the effective thermal conductivity in the PBMR HTTU test facility[J]. Nuclear Engineering and Design, 2014, 271:444-458. [5] WU Y Y, REN C, YANG X T, et al. Repeatable experimental measurements of effective thermal diffusivity and conductivity of pebble bed under vacuum and helium conditions[J]. International Journal of Heat and Mass Transfer, 2019, 141:204-216. [6] 任成, 杨星团, 李聪新, 等. 高温堆球床等效导热系数测量实验加热系统设计及可行性验证[J]. 清华大学学报(自然科学版), 2014, 54(8):1068-1072. REN C, YANG X T, LI C X, et al. Heating system design and validation for the pebble bed effective thermal conductivity experiment in a high temperature gas-cooled reactor[J]. Journal of Tsinghua University (Science and Technology), 2014, 54(8):1068-1072. (in Chinese) [7] GUSAROV A V. Statistical approach to radiative transfer in the heterogeneous media of thin-wall morphology-II:Applications[J]. Journal of Heat Transfer, 2019, 141(1):012701. [8] GUI N, HUANG X L, YANG X T, et al. HTR-PM-based 3D pebble flow simulation on the effects of base angle, recirculation mode and coefficient of friction[J]. Annals of Nuclear Energy, 2020, 143:107442. [9] MEHRABIAN R, SHIEHNEJADHESAR A, SCHARLER R, et al. Multi-physics modelling of packed bed biomass combustion[J]. Fuel, 2014, 122:164-178. [10] ZHOU Z Y, YU A B, ZULLI P. Particle scale study of heat transfer in packed and bubbling fluidized beds[J]. AIChE Journal, 2009, 55(4):868-884. [11] JOHNSON E F, TARI I, BAKER D. Radiative heat transfer in the discrete element method using distance based approximations[J]. Powder Technology, 2021, 380:164-182. [12] KE G L, MENG Q, FINLEY T, et al. LightGBM:A highly efficient gradient boosting decision tree[C]//Proceedings of the 31st International Conference on Neural Information Processing Systems. Long Beach, USA:ACM, 2017:3149-3157. [13] HUGHES M T, KINI G, GARIMELLA S. Status, challenges, and potential for machine learning in understanding and applying heat transfer phenomena[J]. Journal of Heat Transfer, 2021, 143(12):120802. [14] GAO Z Y, SHI L. Thermal hydraulic calculation of the HTR-10 for the initial and equilibrium core[J]. Nuclear Engineering and Design, 2002, 218(1-3):51-64. [15] WANG C, WU Q, WEIMER M, et al. FLAML:A fast and lightweight AutoML library[C]//Proceedings of 4th Machine Learning and Systems Conference. San Jose, USA, 2021:434-447. [16] ZHANG Z Y, DONG Y J, LI F, et al. The Shandong Shidao Bay 200 MWe high-temperature gas-cooled reactor pebble-bed module (HTR-PM) demonstration power plant:An engineering and technological innovation[J]. Engineering, 2016, 2(1):112-118. [17] HOWELL J R, DAUN K J. The past and future of the Monte Carlo method in thermal radiation transfer[J]. Journal of Heat Transfer, 2021, 143(10):100801.
2023, Vol. 63 
