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Abstract
The forms and constant terms of heat transfer correlations can be determined simultaneously with symbolic regression with prescribed precision. In this paper, a genetic programming (GP) technique developed by Silva is adopted as the tool of symbolic regression. However, the test results indicate that when the undetermined functions contain constant terms, symbolic regression results based on the code of Silva usually have increasing sizes. It is difficult to employ them for practical applications when the correlations are not concisely enough. In order to solve this problem, some new function modules are inserted into the GP of Silva, including (a) a function structure simplification module, (b) a constants optimization module, (c) an expansion rate reduction module with “self-swap” genetic operator, and (d) a small term search intensity enhancement module with “intro-new” genetic operator. The statistical performance demonstrates that all of these four new modules are beneficial to improve performance of symbolic regression, especially when the constants optimization module is added. Furthermore, when different modules are added simultaneously, the improvements are more remarkable. Finally, applications of deriving heat transfer correlations for a shell-and-tube heat exchanger with continuous helical baffles and a single-row heat exchanger with helically finned tubes are performed. The results indicate that heat transfer correlations obtained in this paper are proven to be better than the power-law-based correlations.
NOMENCLATURE
| ADR | = | average deviation rate: |
| ANOVA | = | analysis of variance |
| C | = | constant in EquationEq. (1) |
| C | = | set of constants |
| d | = | depth of a node or a parse tree |
| da | = | tube diameter in air side, m |
| e | = | experimental value |
| F | = | fitness function |
| F | = | set of operators |
| G | = | number of current generation |
| Gr | = | number of generations in the run |
| GP | = | genetic programming |
| GPLAB | = | genetic programming toolbox for MATLAB |
| i | = | individual |
| j | = | jth input value |
| m | = | constant in EquationEq. (1) |
| MDR | = | maximum deviation rate: |
| n | = | constant in EquationEq. (2) |
| N | = | number of experimental sets |
| Noi | = | node number of optimal individuals |
| Nu | = | Nusselt number |
| p | = | predicted value |
| P | = | probability |
| Pr | = | Prandtl number |
| Re | = | Reynolds number |
| Rec | = | Reynolds number based on the velocity in the minimum cross section |
| st | = | size of parse tree |
| STHE | = | shell-and-tube heat exchanger |
| T | = | dimensionless oil temperature |
| T’ | = | oil temperature, °C |
| T | = | set of terminals |
| vE | = | dimensionless Engler viscosity |
| vE′ | = | Engler viscosity, °E |
| xj | = | variable in EquationEq. (4) |
| Xt | = | transversal tube pitch, m |
Subscripts
| 1, 2, 3 | = | variable labels in Figure 1 and Figure 3 |
| c | = | crossover |
| i | = | “intro-new” genetic operator |
| m | = | mutation |
| max | = | maximum |
| n | = | node |
| oi | = | optimal individual |
| r | = | reproduction |
| s | = | “self-swap” genetic operator |
| t | = | parse tree |
Additional information
Notes on contributors
Yan Liu
Yan Liu is a Ph.D. student in the School of Energy and Power Engineering, Xi’an Jiaotong University, China, since 2010. He received his bachelor's degree in the School of Science from Xi’an Jiaotong University, China, in 2010. His main research interests are computational fluid dynamics, waste heat recovery, and application of computational intelligence in thermal engineering.
Zhi-Long Cheng
Zhi-long Cheng is a Ph.D. student in the School of Energy and Power Engineering, Xi’an Jiaotong University, China, since 2012. He received his bachelor's degree from Northwestern Polytechnical University, China, in 2012. His main research interests are combustion in porous media and biomass fuel.
Jing Xu
Jing Xu is an engineer in Suzhou Nuclear Power Research Institute (SNPI), China. She received her master's degree in engineering thermophysics from Xi’an Jiaotong University, China, in 2011. Her main research interest is application of computational intelligence in thermal engineering.
Jian Yang
Jian Yang is an associate professor in the School of Energy and Power Engineering, Xi’an Jiaotong University. He received his Ph.D. of power engineering and engineering thermophysics from Xi’an Jiaotong University, China, in 2010. His main research interests are heat and fluid flow in porous media, and heat transfer enhancement.
Qiu-Wang Wang
Qiu-wang Wang is a professor at the School of Energy and Power Engineering, Xi’an Jiaotong University, China. He received his Ph.D. in engineering thermophysics from Xi’an Jiaotong University, China, in 1996. He then joined the faculty of the university and took the professor post in 2001. His main research interests include computational fluid dynamics and numerical heat transfer, heat transfer enhancement, transport phenomena in porous media, compact heat exchangers, building energy savings, and indoor air quality. He has also been author or co-author of 4 books and more than 100 journal papers, and his H-index is 21. He has obtained 16 China Invent Patents and two U.S. patents.