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The prediction of heat transfer coefficients or wall temperatures of heat exchanger tubes is an important research topic in supercritical heat transfer, which is extremely significant for the application of supercritical fluids in industrial production and the design of the entire thermal system. At present, the empirical correlation method is the most widely adopted prediction method, but its predicted heat transfer coefficient still has significant difference from the actual data near the pseudo-critical temperature. Therefore, some scholars proposed using artificial neural networks to predict the heat transfer performance of supercritical fluids in tubes. On the basis of previous researches, this work further explores the effectiveness of artificial neural network in predicting supercritical heat transfer, focusing on the influence of input parameters on neural network prediction results and the influence of genetic algorithm optimization on the prediction results. In this research, a neural network prediction model for supercritical R1234ze(E) cooled in horizontal straight tubes is established and compared with the modified D-B heat transfer correlation. The result shows that the input parameter has great influence on the prediction accuracy of BPNN, and not all BPNN input parameter combinations can bring better prediction results than heat transfer correlation. The combination of $ {{Re} _{\text{b}}} $ ,$ {Pr _{\text{b}}} $ ,$ {{{\rho _{\text{b}}}} \mathord{\left/ {\vphantom {{{\rho _{\text{b}}}} {{\rho _{\text{w}}}}}} \right. } {{\rho _{\text{w}}}}} $ ,$ {{{{\overline C }_{\text{p}}}} \mathord{\left/ {\vphantom {{{{\overline C }_{\text{p}}}} {{C_{{\text{pw}}}}}}} \right. } {{C_{{\text{pw}}}}}} $ ,$ {{{\lambda _{\text{b}}}} \mathord{\left/ {\vphantom {{{\lambda _{\text{b}}}} {{\lambda _{\text{w}}}}}} \right. } {{\lambda _{\text{w}}}}} $ ,$ {{{\mu _{\text{b}}}} \mathord{\left/ {\vphantom {{{\mu _{\text{b}}}} {{\mu _{\text{w}}}}}} \right. } {{\mu _{\text{w}}}}} $ features the best prediction performance. The AAD and Errormax of the prediction result for the trial set are only 2.02% and 9.34%, which are far lower than the prediction deviation of the heat transfer correlation, and the predictions of the trend of h in the high temperature region, the maximum value of h and the position of the peak value of h are more precise than heat transfer correlation. Moreover, this research compares GA-BP model with BP model under two different fitness value calculation methods to reveal the effectiveness of GA-BP in enhancing the prediction accuracy of supercritical heat transfer, concluding that when the same dataset is adopted for network training and fitness value calculation, over-fitting will occur and the GA-BP cannot further improve the prediction accuracy; when different datasets are used to train the network and calculate the fitness value, the generalization ability of the network will be strengthened, and the root mean square deviation and the maximum deviation of the prediction result can be further reduced.This work will provide a practical tool for predicting the cooling convection heat transfer of supercritical R1234ze(E) in horizontal tubes, laying the foundation for its application in trans-critical heat pump systems, and providing inspiration for potential research directions of ANN in supercritical heat transfer prediction. 
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Keywords:
- R1234ze(E) /
- supercritical heat transfer /
- heat transfer prediction /
- neural network
[1] Dittus F W, Boelter L M K 1930 Univ. California Publicat. Eng. 2 443
[2] Gnielinski V 1976 Int. J. Chem. Eng. 16 359
[3] Bae Y Y, Kim H Y 2009 Exp. Therm. Fluid Sci. 33 329
Google Scholar
[4] Saltanov E 2015 Ph. D. Dissertation (University of Ontario Institute of Technology
[5] Kim J K, Jeon H K, Lee J S 2007 Nucl. Eng. Des. 237 1795
Google Scholar
[6] Kuang G, Ohadi M, Dessiatoun S 2008 HVACR Res. 14 861
Google Scholar
[7] Zhang S J, Xu X X, Liu C, Liu X X, Ru Z P, Dang C B 2020 Int. J. Heat Mass Transfer 149 119074
Google Scholar
[8] Wang L, Pan Y C, Lee J D, Wang Y, Fu B R, Pan C 2020 Int. J. Heat Mass Transfer 159 120136
Google Scholar
[9] Liu S H, Huang Y P, Liu G X, Wang J F, Leung L K H 2017 Int. J. Heat Mass Transfer 106 1144
Google Scholar
[10] Kim D E, Kim M H 2010 Nucl. Eng. Des. 240 3336
Google Scholar
[11] Kumar R, Nikam K, Jilte R 2020 Applied Computer Vision and Image Processing Advances in Intelligent Systems and Computing (Singapore: Springer
[12] Ahmadi M H, Ghazvini M, Maddah H, Kahani M, Pourfarhang S, Pourfarhang A, Heris S Z 2020 Physica A 546 124008
Google Scholar
[13] Aghayari R, Maddah H, Ahmadi M H, Yan W M, Ghasemi N 2018 Energies 11 1190
Google Scholar
[14] Mensah R A, Jiang L, Asante-Okyere S, Xu Q, Jin C 2019 J. Therm. Anal. Calorim. 138 3055
Google Scholar
[15] Ye K, Zhang Y L, Yang L L, Zhao Y R, Li N, Xi C K 2019 Appl. Therm. Eng. 150 686
Google Scholar
[16] Zhu B, Zhu X, Xie J, Xu J L, Liu H 2021 J. Therm. Sci. 30 1751
Google Scholar
[17] Prasad K S R, Krishna V, Bharadwaj M S, Ponangi B R 2022 J. Heat Transfer 144 011802
Google Scholar
[18] Ma D, Zhou T, Chen J, Qi S, Shahzad M A, Xiao Z J 2017 Nucl. Eng. Des. 320 400
Google Scholar
[19] Chang W, Chu X, Fareed A F B S, Pandey S, Luo J Y, Weigand B, Laurien E 2018 Appl. Therm. Eng. 131 815
Google Scholar
[20] Sun F, Xie G, Li S 2021 Appl. Soft Comput. 102 107110
Google Scholar
[21] Sun F, Xie G, Song J, Li S L, Markides C N 2021 Appl. Therm. Eng. 194 117067
Google Scholar
[22] Jiang Y R, Hu P, Ibrahim A 2020 Int. J. Heat Fluid Flow 85 108650
Google Scholar
[23] Dang C B, Hihara E 2012 Int. J. Refrigeration 35 1130
Google Scholar
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图 1 传热数据集的全部数据点分布 (a)不同管径下的传热系数; (b) 不同质量通量下的传热系数; (c) 不同热流密度下的传热系数; (d) 不同压力下的传热系数
Figure 1. Distributions of all data points for the heat transfer dataset: (a) Heat transfer coefficients under different tube diameters; (b) heat transfer coefficients under different mass fluxes; (c) heat transfer coefficients under different heat fluxes; (d) heat transfer coefficients under different pressures.
图 2 具有一层隐藏层的BPNN结构
Figure 2. BPNN structure with one hidden layer.
图 3 BPNN训练流程
Figure 3. BPNN training process.
图 4 关联式和神经网络对试验集预测结果的对比 (a)工况1; (b)工况2; (c) 工况3; (d)工况4
Figure 4. Comparison of prediction results of correlation and neural network on the trial set: (a) Case 1; (b) Case 2; (c) Case 3; (d) Case 4.
图 5 关联式和神经网络对试验集努塞尔数预测结果的对比
Figure 5. Comparison of Nusselt prediction results of correlation and neural network on the trial set.
图 6 GA-BP神经网络流程图
Figure 6. GA-BP neural network flowchart.
图 7 (a) BP网络与(b) GA-BP网络对试验集预测结果的对比
Figure 7. Comparison of prediction results of (a) BP network and (b) GA-BP network on the trial set.
图 8 (a) BP网络与(b) GA-BP-I网络对试验集预测结果的对比
Figure 8. Comparison of prediction results of (a) BP network and (b) GA-BP-I network on the trial set.
图 9 BP网络与GA-BP-I网络对试验集177个数据点的预测偏差分布图
Figure 9. Distributions of prediction bias of BP network and GA-BP-I network for 177 data points of the trial set.
表 1 CFD模拟数据集全部工况
Table 1. CFD simulation dataset for all operating conditions.
Case d/mm G/(kg·m–2·s–1) q /(kW·m–2) P/MPa Tb /K 1—12 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14320 –40 4.0 370—420 13—24 6 160, 200, 240, 280, 320, 360,
400, 440, 480, 520, 560, 600–40 4.0 370—420 25—34 6 320 –10, –20, –30, –40, –50, –60,
–70, –80, –90, –1004.0 370—420 35—44 6 320 –40 3.8, 4.0, 4.2, 4.4,
4.6, 4.8, 5.0, 5.2,370—420 
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表 2 BPNN模型参数的设置
Table 2. Parameters of the BPNN model.
序号 项目 值或选择 1 隐藏层神经元数目 由测试集的预测结果确定 2 隐藏层传递函数 tansig 3 输出层传递函数 purelin 4 训练函数类型 由测试集的预测结果确定 5 学习函数类型 learngdm 6 最大迭代次数 20000 7 训练目标误差 10–10 8 网络学习速率 0.1 9 验证集最大确认失败数 6
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表 3 4种工况参数设置
Table 3. Parameters of four cases.
d/mm G/(kg·m–2·s–1) q/(kW·m–2) P/MPa Case 1 8 250 –75 3.9 Case 2 5 460 –55 3.9 Case 3 8 220 –65 4.5 Case 4 7 300 –25 4.5
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表 4 输入参数对试验集预测结果的影响
Table 4. Impacts of input parameters on the prediction results of the trial set
输入参数 试验集预测结果 AAD/% RMSE/% Errormax/% 传热关联式 — 5.58 7.72 29.40 BP神经网络 1) $ {{Re} _{\text{b}}} $, $ {Pr _{\text{b}}} $, $ {{{\rho _{\text{b}}}} \mathord{\left/ {\vphantom {{{\rho _{\text{b}}}} {{\rho _{\text{w}}}}}} \right. } {{\rho _{\text{w}}}}} $, $ {{{{\overline C }_{\text{p}}}} \mathord{\left/ {\vphantom {{{{\overline C }_{\text{p}}}} {{C_{{\text{pw}}}}}}} \right. } {{C_{{\text{pw}}}}}} $, $ {{Gr} \mathord{\left/ {\vphantom {{Gr} {{Re} _{\text{b}}^{2}}}} \right. } {{Re} _{\text{b}}^{2}}} $ 8.24 12.40 43.98 2) $ {{Re} _{\text{b}}} $, $ {Pr _{\text{b}}} $, $ {{{\rho _{\text{b}}}} \mathord{\left/ {\vphantom {{{\rho _{\text{b}}}} {{\rho _{\text{w}}}}}} \right. } {{\rho _{\text{w}}}}} $, $ {{{{\overline C }_{\text{p}}}} \mathord{\left/ {\vphantom {{{{\overline C }_{\text{p}}}} {{C_{{\text{pw}}}}}}} \right. } {{C_{{\text{pw}}}}}} $ 3.51 5.07 15.19 3) $ {{Re} _{\text{b}}} $, $ {Pr _{\text{b}}} $, $ {{{\rho _{\text{b}}}} \mathord{\left/ {\vphantom {{{\rho _{\text{b}}}} {{\rho _{\text{w}}}}}} \right. } {{\rho _{\text{w}}}}} $, $ {{{{\overline C }_{\text{p}}}} \mathord{\left/ {\vphantom {{{{\overline C }_{\text{p}}}} {{C_{{\text{pw}}}}}}} \right. } {{C_{{\text{pw}}}}}} $, $ {{{\lambda _{\text{b}}}} \mathord{\left/ {\vphantom {{{\lambda _{\text{b}}}} {{\lambda _{\text{w}}}}}} \right. } {{\lambda _{\text{w}}}}} $, $ {{{\mu _{\text{b}}}} \mathord{\left/ {\vphantom {{{\mu _{\text{b}}}} {{\mu _{\text{w}}}}}} \right. } {{\mu _{\text{w}}}}} $ 2.02 3.01 9.34 4) G, q, d, Tb, $ {\rho _{\text{b}}} $ 9.89 15.52 73.24 5) G, q, d, Tb, $ {\rho _{\text{b}}} $, $ {C_{{\text{pb}}}} $, $ {\lambda _{\text{b}}} $, $ {\mu _{\text{b}}} $, 4.58 5.35 11.35 6) G, q, d, Tb, $ {\rho _{\text{b}}} $, $ {{{\rho _{\text{b}}}} \mathord{\left/ {\vphantom {{{\rho _{\text{b}}}} {{\rho _{\text{w}}}}}} \right. } {{\rho _{\text{w}}}}} $, $ {{{{\overline C }_{\text{p}}}} \mathord{\left/ {\vphantom {{{{\overline C }_{\text{p}}}} {{C_{{\text{pw}}}}}}} \right. } {{C_{{\text{pw}}}}}} $, $ {{{\lambda _{\text{b}}}} \mathord{\left/ {\vphantom {{{\lambda _{\text{b}}}} {{\lambda _{\text{w}}}}}} \right. } {{\lambda _{\text{w}}}}} $, $ {{{\mu _{\text{b}}}} \mathord{\left/ {\vphantom {{{\mu _{\text{b}}}} {{\mu _{\text{w}}}}}} \right. } {{\mu _{\text{w}}}}} $ 47.24 63.17 224.89 7) G, q, d, Tb, $ {\rho _{\text{b}}} $, $ {C_{{\text{pb}}}} $, $ {\lambda _{\text{b}}} $, $ {\mu _{\text{b}}} $, $ {{{\rho _{\text{b}}}} \mathord{\left/ {\vphantom {{{\rho _{\text{b}}}} {{\rho _{\text{w}}}}}} \right. } {{\rho _{\text{w}}}}} $, $ {{{{\overline C }_{\text{p}}}} \mathord{\left/ {\vphantom {{{{\overline C }_{\text{p}}}} {{C_{{\text{pw}}}}}}} \right. } {{C_{{\text{pw}}}}}} $, $ {{{\lambda _{\text{b}}}} \mathord{\left/ {\vphantom {{{\lambda _{\text{b}}}} {{\lambda _{\text{w}}}}}} \right. } {{\lambda _{\text{w}}}}} $, $ {{{\mu _{\text{b}}}} \mathord{\left/ {\vphantom {{{\mu _{\text{b}}}} {{\mu _{\text{w}}}}}} \right. } {{\mu _{\text{w}}}}} $ 52.52 63.15 151.79
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[1] Dittus F W, Boelter L M K 1930 Univ. California Publicat. Eng. 2 443
[2] Gnielinski V 1976 Int. J. Chem. Eng. 16 359
[3] Bae Y Y, Kim H Y 2009 Exp. Therm. Fluid Sci. 33 329
Google Scholar
[4] Saltanov E 2015 Ph. D. Dissertation (University of Ontario Institute of Technology
[5] Kim J K, Jeon H K, Lee J S 2007 Nucl. Eng. Des. 237 1795
Google Scholar
[6] Kuang G, Ohadi M, Dessiatoun S 2008 HVACR Res. 14 861
Google Scholar
[7] Zhang S J, Xu X X, Liu C, Liu X X, Ru Z P, Dang C B 2020 Int. J. Heat Mass Transfer 149 119074
Google Scholar
[8] Wang L, Pan Y C, Lee J D, Wang Y, Fu B R, Pan C 2020 Int. J. Heat Mass Transfer 159 120136
Google Scholar
[9] Liu S H, Huang Y P, Liu G X, Wang J F, Leung L K H 2017 Int. J. Heat Mass Transfer 106 1144
Google Scholar
[10] Kim D E, Kim M H 2010 Nucl. Eng. Des. 240 3336
Google Scholar
[11] Kumar R, Nikam K, Jilte R 2020 Applied Computer Vision and Image Processing Advances in Intelligent Systems and Computing (Singapore: Springer
[12] Ahmadi M H, Ghazvini M, Maddah H, Kahani M, Pourfarhang S, Pourfarhang A, Heris S Z 2020 Physica A 546 124008
Google Scholar
[13] Aghayari R, Maddah H, Ahmadi M H, Yan W M, Ghasemi N 2018 Energies 11 1190
Google Scholar
[14] Mensah R A, Jiang L, Asante-Okyere S, Xu Q, Jin C 2019 J. Therm. Anal. Calorim. 138 3055
Google Scholar
[15] Ye K, Zhang Y L, Yang L L, Zhao Y R, Li N, Xi C K 2019 Appl. Therm. Eng. 150 686
Google Scholar
[16] Zhu B, Zhu X, Xie J, Xu J L, Liu H 2021 J. Therm. Sci. 30 1751
Google Scholar
[17] Prasad K S R, Krishna V, Bharadwaj M S, Ponangi B R 2022 J. Heat Transfer 144 011802
Google Scholar
[18] Ma D, Zhou T, Chen J, Qi S, Shahzad M A, Xiao Z J 2017 Nucl. Eng. Des. 320 400
Google Scholar
[19] Chang W, Chu X, Fareed A F B S, Pandey S, Luo J Y, Weigand B, Laurien E 2018 Appl. Therm. Eng. 131 815
Google Scholar
[20] Sun F, Xie G, Li S 2021 Appl. Soft Comput. 102 107110
Google Scholar
[21] Sun F, Xie G, Song J, Li S L, Markides C N 2021 Appl. Therm. Eng. 194 117067
Google Scholar
[22] Jiang Y R, Hu P, Ibrahim A 2020 Int. J. Heat Fluid Flow 85 108650
Google Scholar
[23] Dang C B, Hihara E 2012 Int. J. Refrigeration 35 1130
Google Scholar
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