Prediction and Analysis of Heat Transfer Characteristics of Supercritical Fluids Based on Interpretable Machine Learning
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摘要: 超临界流体在拟临界温度附近发生剧烈的物性变化，传热特性难以准确预测。本研究采用可解释机器学习的研究方法预测并分析超临界流体传热特性。使用粒子群优化算法（PSO）搜索反向传播神经网络（BPNN）模型最优超参数，建立了超临界流体传热预测模型，并与传统经验关联式进行了精度比较。使用SHAP可解释算法对BPNN模型进行了全局和局部解释，根据不同工况下特征重要度的变化发现超临界相关机理现象。结果显示，所建立的神经网络模型在测试集上的平均绝对百分比误差（MAPE）为1.4%，决定系数R2为0.9992，与经验关联式相比，该模型具有更高的预测精度；对于垂直向上流动，浮升力效应在传热恶化工况中明显具有更高的特征重要度，是传热恶化行为发生的主要因素。因此，本研究建立的基于可解释机器学习的研究方法对进一步研究超临界流体传热特性具有一定的参考意义。Abstract: The physical properties of supercritical fluids change drastically near the pseudo-critical temperature, making it challenging to accurately predict heat transfer characteristics. In this study, the method of interpretable machine learning was used to predict and analyze the heat transfer characteristics of supercritical fluids. The particle swarm optimization algorithm (PSO) was used to search for the optimal hyperparameters of the back propagation neural network (BPNN) model, the supercritical fluid heat transfer prediction model was established, and its accuracy was compared with the traditional empirical correlation. The global and local interpretation of the BPNN model was carried out by using the SHAP interpretable algorithm, and the supercritical correlation phenomenon and mechanism were found according to the change of feature importance under different conditions. The results show that the MAPE of the established neural network model on the test set is 1.4%, and the coefficient of determination R2 is 0.9992, which has higher prediction accuracy compared with the empirical correlation formula. For vertical upward flow, buoyancy effect obviously has higher feature importance in heat transfer deterioration condition, which is the main factor of heat transfer deterioration behavior. Therefore, the research method based on interpretable machine learning established in this study has certain reference significance for further study of the heat transfer characteristics of supercritical fluids.
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图 2 Spearman相关系数绝对值矩阵图
Figure 2. Absolute Value Matrix of Spearman Correlation Coefficient
图 3 各特征与Nu的相关系数绝对值
Figure 3. Absolute Value of the Correlation Coefficient between Each Feature and Nu
图 5 BPNN模型建立流程图
train—训练集子集；validation—验证集；f(x)—适应度函数；Fold—划分的每一折数据；Model—五折交叉验证中每次训练出的模型
Figure 5. Establishment Process of BPNN Model
图 8 经验关联式与BPNN模型在不同工况上预测结果
Figure 8. Empirical Correlations and BPNN Model Prediction Results under Different Conditions
图 11 工况1下Tw与Nu分布以及特征重要度变化
Figure 11. Tw and Nu Distribution and Feature Importance Change under Condition 1
图 12 工况2下Tw与Nu分布以及特征重要度变化
Figure 12. Tw and Nu Distribution and Feature Importance Change under Condition 2
表 1 特征选择前的无量纲数
Table 1. Dimensionless Number before Feature Selection
序号 无量纲数 序号 无量纲数 1 $R{e_{\text{b}}}$ 12 $\dfrac{{{c_{p,{\text{b}}}}}}{{{c_{p,{\text{w}}}}}}$ 2 $P{r_{\text{b}}}$ 13 $\dfrac{{{{\bar c}_p}}}{{{c_{p,{\text{b}}}}}}$ 3 $R{e_{\text{w}}}$ 14 $\dfrac{{{{\bar c}_p}}}{{{c_{p,{\text{w}}}}}}$ 4 $R{e_{\text{f}}}$ 15 $\dfrac{{{\rho _{\text{b}}}}}{{{\rho _{{\text{pc}}}}}}$ 5 $P{r_{\text{w}}}$ 16 $\dfrac{{{k_{\text{b}}}}}{{{k_{{\text{pc}}}}}}$ 6 ${\overline {Pr} _{\text{b}}}$ 17 $\dfrac{{{c_{p,{\text{b}}}}}}{{{c_{p,{\text{pc}}}}}}$ 7 ${\overline {Pr} _{\text{w}}}$ 18 $ Gr_{\text{b}}=\dfrac{\left(\rho_{\text{b}}-\rho_{\text{w}}\right)gd^3}{\rho_{\text{b}}\mathit{\upsilon}_{\text{b}}^2} $ 8 $\dfrac{{{\mu _{\text{b}}}}}{{{\mu _{\text{w}}}}}$ 19 $B{u_1} = \dfrac{{G{r_{\text{b}}}}}{{Re_{\text{b}}^2}}$ 9 $\dfrac{{{k_{\text{b}}}}}{{{k_{\text{w}}}}}$ 20 $B{u_2} = \dfrac{{G{r_{\text{b}}}}}{{Re_{\text{b}}^{2.7}}}$ 10 $\dfrac{{{\rho _{\text{b}}}}}{{{\rho _{\text{w}}}}}$ 21 $B{u_3} = \dfrac{{G{r_{\text{b}}}}}{{Re_{\text{b}}^{2.625}Pr_{\text{w}}^{{\text{0}}{\text{.4}}}}}{\left( {\dfrac{{{\rho _{\text{b}}}}}{{{\rho _{\text{w}}}}}} \right)^{0.5}}\left( {\dfrac{{{\mu _{\text{w}}}}}{{{\mu _{\text{b}}}}}} \right)$ 11 $\dfrac{{{\nu _{\text{b}}}}}{{{\nu _{\text{w}}}}}$ 
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表 2 超参数搜索空间及搜索结果
Table 2. Hyper Parameters Search Space and Results
超参数名称 搜索范围 搜索结果 隐藏层数 1~4 2 隐藏层节点数 1~300 119、196 初始学习率 0.00009~0.009 3.0521×10−3 批数量 2~128 115 迭代次数 500~3000 2352
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表 3 5%和10%预测误差对应数据占比
Table 3. Proportion of Data Corresponding to the 5% and 10% Prediction Errors
数据集 ${\eta _{5\text{%} }}$ ${\eta _{10\text{%} }}$ 训练集 98.74% 99.76% 测试集 95.31% 98.12% ${\eta _{5\text{%} }}$—预测误差小于5%的数据比例；${\eta _{10\text{%} }}$—预测误差小于5%的数据比例
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表 4 数据集误差
Table 4. Dataset Error
数据集 MAPE/% RMSE R2 训练集 0.8 7.0 0.99991 测试集 1.4 18.8 0.99921
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表 6 不同模型的对比结果
Table 6. Comparison of Different Models
模型 训练时间/s MAPE/% RMSE R2 SVR 13.46 3.9 50.9 0.994 RFR 24.63 3.5 82.3 0.984 BPNN 129.01 1.4 18.8 0.999
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