Hybrid Models for Solving the Colebrook–White Equation Using Artificial Neural Networks
This study proposes hybrid models to solve the Colebrook–White equation by combining explicit equations available in the literature to solve the Colebrook–White equation with an error function. The hybrid model is in the form of <inline-formula><math xmlns="http://www.w3.org/1998/Math/...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-06-01
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| Series: | Fluids |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2311-5521/7/7/211 |
| _version_ | 1797406467790209024 |
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| author | Muhammad Cahyono |
| author_facet | Muhammad Cahyono |
| author_sort | Muhammad Cahyono |
| collection | DOAJ |
| description | This study proposes hybrid models to solve the Colebrook–White equation by combining explicit equations available in the literature to solve the Colebrook–White equation with an error function. The hybrid model is in the form of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mi>H</mi></msub><mo>=</mo><msub><mi>f</mi><mi>o</mi></msub><mo>−</mo><msub><mi>e</mi><mi>A</mi></msub><mo>.</mo><mo> </mo><msub><mi>f</mi><mi>H</mi></msub><mo> </mo></mrow></semantics></math></inline-formula> is the friction factor value <i>f</i> predicted by the hybrid model, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mi>o</mi></msub></mrow></semantics></math></inline-formula> is the value of <i>f</i> calculated using several explicit formulas for the Colebrook–White equation, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>e</mi><mi>A</mi></msub></mrow></semantics></math></inline-formula> is the error function determined using the neural network procedures. The hybrid equation consists of a series of hyperbolic tangent functions whose number corresponds to the number of neurons in the hidden layer. The simulation results showed that the hybrid models using five hyperbolic tangent functions could produce reasonable predictions of friction factors, with the maximum absolute relative error (MAXRE) around one tenth, or ten times lower than that produced by the corresponding existing formula. The simplified hybrid models are also given using four and three tangent hyperbolic functions. These simplified models still provide accurate results with <i>MAXRE</i> of less than 0.1%. |
| first_indexed | 2024-03-09T03:26:51Z |
| format | Article |
| id | doaj.art-e327991b31ee4b5b8268285e1593305c |
| institution | Directory Open Access Journal |
| issn | 2311-5521 |
| language | English |
| last_indexed | 2024-03-09T03:26:51Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fluids |
| spelling | doaj.art-e327991b31ee4b5b8268285e1593305c2023-12-03T15:02:01ZengMDPI AGFluids2311-55212022-06-017721110.3390/fluids7070211Hybrid Models for Solving the Colebrook–White Equation Using Artificial Neural NetworksMuhammad Cahyono0Faculty of Civil and Environmental Engineering, Bandung Institute of Technology, Jalan Ganesa No. 10, Bandung 40132, IndonesiaThis study proposes hybrid models to solve the Colebrook–White equation by combining explicit equations available in the literature to solve the Colebrook–White equation with an error function. The hybrid model is in the form of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mi>H</mi></msub><mo>=</mo><msub><mi>f</mi><mi>o</mi></msub><mo>−</mo><msub><mi>e</mi><mi>A</mi></msub><mo>.</mo><mo> </mo><msub><mi>f</mi><mi>H</mi></msub><mo> </mo></mrow></semantics></math></inline-formula> is the friction factor value <i>f</i> predicted by the hybrid model, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>f</mi><mi>o</mi></msub></mrow></semantics></math></inline-formula> is the value of <i>f</i> calculated using several explicit formulas for the Colebrook–White equation, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>e</mi><mi>A</mi></msub></mrow></semantics></math></inline-formula> is the error function determined using the neural network procedures. The hybrid equation consists of a series of hyperbolic tangent functions whose number corresponds to the number of neurons in the hidden layer. The simulation results showed that the hybrid models using five hyperbolic tangent functions could produce reasonable predictions of friction factors, with the maximum absolute relative error (MAXRE) around one tenth, or ten times lower than that produced by the corresponding existing formula. The simplified hybrid models are also given using four and three tangent hyperbolic functions. These simplified models still provide accurate results with <i>MAXRE</i> of less than 0.1%.https://www.mdpi.com/2311-5521/7/7/211Colebrook–White equationfriction factorexplicit approximationartificial neural networkshybrid model |
| spellingShingle | Muhammad Cahyono Hybrid Models for Solving the Colebrook–White Equation Using Artificial Neural Networks Fluids Colebrook–White equation friction factor explicit approximation artificial neural networks hybrid model |
| title | Hybrid Models for Solving the Colebrook–White Equation Using Artificial Neural Networks |
| title_full | Hybrid Models for Solving the Colebrook–White Equation Using Artificial Neural Networks |
| title_fullStr | Hybrid Models for Solving the Colebrook–White Equation Using Artificial Neural Networks |
| title_full_unstemmed | Hybrid Models for Solving the Colebrook–White Equation Using Artificial Neural Networks |
| title_short | Hybrid Models for Solving the Colebrook–White Equation Using Artificial Neural Networks |
| title_sort | hybrid models for solving the colebrook white equation using artificial neural networks |
| topic | Colebrook–White equation friction factor explicit approximation artificial neural networks hybrid model |
| url | https://www.mdpi.com/2311-5521/7/7/211 |
| work_keys_str_mv | AT muhammadcahyono hybridmodelsforsolvingthecolebrookwhiteequationusingartificialneuralnetworks |