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Applying different types of artificial neural network for modeling thermal conductivity of nanofluids containing silica particles

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Abstract

Nanofluids are widely applicable in thermal devices with porous structures. Silica nanoparticles have been dispersed in different heat transfer fluids in order to increase their thermal conductivity and heat transfer capability. In this study, group method of data handling (GMDH) and multilayer perceptron artificial neural networks are applied for determining thermal conductivity of nanofluids with silica particles and different base fluids such as ethylene glycol, glycerol, water and ethylene glycol–water mixture. For cases with multilayer perceptron models, trained by applying scaled conjugate gradient (SCG) and Levenberg–Marquardt (LM) have been tested as two different training algorithms. The outputs of the applied models have good agreement with the values obtained in experimental studies. The values of \({R}^{2}\) in the optimum conditions of using GMDH, LM and SCG are 0.9997, 0.9991 and 0.9998, respectively. In addition, the MSE values of the mentioned methods are approximately 0.000010, 0.000032 and 0.0000078, respectively.

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Acknowledgement

The authors would like to appreciate M. Afshrzadeh for her support and help.

Author information

Authors and Affiliations

  1. Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran

    Akbar Maleki

  2. Department of Mechanical Engineering, University of California, Riverside, USA

    Arman Haghighi

  3. School of Mechanical Engineering, University of Tehran, Tehran, Iran

    Misagh Irandoost Shahrestani

  4. Institute of Research and Development, Duy Tan University, Da Nang, 550000, Vietnam

    Zahra Abdelmalek

  5. Faculty of Medicine, Duy Tan University, Da Nang, 550000, Vietnam

    Zahra Abdelmalek

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  1. Akbar Maleki
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  2. Arman Haghighi
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  3. Misagh Irandoost Shahrestani
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Correspondence to Akbar Maleki or Zahra Abdelmalek.

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Appendix

Appendix

$$Thermal\;conductivity = 0.00126014 - N46*N2*97.3843 + N46^2*48.6909 + N2*0.993462 + N2^2*48.699$$

where

$$N2 = 9.37646 * {10}^{-5} - N363 * 0.0335857 + N3 * 1.03339$$
$$N3 = -0.000874636 + N221 * 0.122798 + N221 * N4 * 11.3205 - N221^2 * 5.74673 + N4 * 0.882377 - N4^2 * 5.57863$$
$$N4 = -0.00215872 - N23 * 1.62829 - N23 * N7 * 2.48184 + N23^2 * 2.47217 + N7 * 2.63815$$
$$N7 = 0.000763705 + N230 * 0.304117 + N230 * N13 * 6.8786 - N230^2 * 3.84272 + N13 * 0.693187 - N13^2 * 3.03338$$
$$N13 = -2.22059 * {10}^{-5} + N19 * 0.521767 + N23 * 0.478278$$
$$N19 = 0.00745006 - N406 * 0.325665 - N406 * N44 * 2.88803 + N406^2 * 1.75851 + N44 * 1.29031 + N44^2 * 1.16405$$
$$N44 = -0.00373522 + N90 * 2.62726 + N90 * N111 * 81.2924 - N90^2 * 42.2947 - N111 * 1.61017 - N111^2 * 39.0116$$
$$N90 = 0.00439835 + N140 * N152 * 52.0459 - N140^2 * 25.4458 + N152 * 0.985753 - N152^2 * 26.586$$
$$N152 = 0.00164091 + N260 * 4.11528 + N260 * N267 * 135.676 - N260^2 * 69.9834 - N267 * 3.11957 - N267^2 * 65.6878$$
$$N267 = -0.00647904 + N358 * 1.04195 - N358 * N568 * 6.13814 + N358^2 * 3.00401 + N568^2 * 3.0572$$
$$N568 = -3.84502 * {10}^{-11} + N576$$
$$N140 = 0.0808505 - {x}_{2} *0.00549205 - {x}_{2} * N301 * 0.0550408 +{x}_{2}^2 * 0.000522728 + N301 * 0.968932 + N301^2 * 1.74424$$
$$N301 = -0.0144666 - N389 * 0.282082 + N389 * N504 * 1.90272 + N504 * 1.35027 - N504^2 * 1.97036$$
$$N389 = 0.14449 - {x}_{4} * 0.00484756 + {x}_{4} * N505 * 0.0108732 + {x}_{4}^2 * 1.97213 * {10}^{-5}+ N505 * 0.654134 - N505^2 * 0.101332$$
$$N505 = -0.0488979 +\sqrt[3]{{x}_{1}} * \sqrt[3]{{x}_{3}} * 0.599567 + \sqrt[3]{{x}_{1}}^2 * 0.963904 - \sqrt[3]{{x}_{3}} * 0.580766 +\sqrt[3]{{x}_{3}}^2 * 0.121747$$
$$N406 = -0.0120488 - N483 * 56.7433 - N483 * N485 * 37520.8 + N483^2 * 18822.1 + N485 * 57.7807 + N485^2 * 18698.6$$
$$N485 = 0.0771466 + N510 * 0.61926 + N510 * N522 * 6.46372 - N510^2 * 3.19209 - N522^2 * 2.84671$$
$$N522 = -0.493982 - {x}_{4} * 0.0135231 +{x}_{4} * \sqrt[3]{{x}_{1}} * 0.0212663 +\sqrt[3]{{x}_{1}} * 1.19902$$
$$N510 = -11.2593 + \sqrt[3]{{x}_{2}} * 7.12025 +\sqrt[3]{{x}_{2}} * \sqrt[3]{{x}_{3}} * 0.111236 - \sqrt[3]{{x}_{2}}^2 * 1.06188 -\sqrt[3]{{x}_{3}} * 0.44152 +\sqrt[3]{{x}_{3}}^2 * 0.106266$$
$$N483 = 0.07833 + N512 * 0.612841 + N512 * N520 * 6.5401 - N512^2 * 3.22536 - N520^2 * 2.88274$$
$$N230 = -0.0006912 + N271 * 0.322697 + N271 * N340 * 0.705158 + N340 * 0.67916 - N340^2 * 0.705715$$
$$N340 = -0.110704 + N456 * 1.54893 - N456 * N571 * 5.64152 + N456^2 * 2.71745 + N571^2 * 2.33681$$
$$N456 = 0.0665178 + N508 * 0.546738 + N508 * N516 * 6.96973 - N508^2 * 3.38871 + N516 * 0.116385 - N516^2 * 3.19363$$
$$N271 = -0.00622633 + N353 * 1.04014 - N353 * N569 * 6.01031 + N353^2 * 2.93825 + N569^2 * 2.99795$$
$$N569 = -2.67753 * {10}^{-11} + N575$$
$$N353 = 0.192405 -\sqrt[3]{{x}_{3}} * 0.302692 +\sqrt[3]{{x}_{3}} * N516 * 0.317988 + \sqrt[3]{{x}_{3}}^2 * 0.123219 + N516 * 0.5975 + N516^2 * 0.0640845$$
$$N516 = -1.69737 + {x}_{2} * 0.126947 + {x}_{2} * {x}_{4} * 0.000155269 - {x}_{2}^2 * 0.00175107 -{x}_{4}^2 * 2.86687 * {10}^{-5}$$
$$N23 = 0.00447349 - N153 * N50 * 78.5724 + N153^2 * 38.8351 + N50 * 0.977106 + N50^2 * 39.7578$$
$$N50 = -0.00391508 + N84 * 1.99388 + N84 * N111 * 50.3034 - N84^2 * 26.4171 - N111 * 0.97457 - N111^2 * 23.9045$$
$$N111 = 0.00345229 + N384 * 0.521188 + N384 * N147 * 29.5477 - N384^2 * 15.4699 + N147 * 0.481536 - N147^2 * 14.0842$$
$$N384 = 0.20894 - \sqrt[3]{{x}_{3}} * 0.323349 + \sqrt[3]{{x}_{3}} * N520 * 0.31765 +\sqrt[3]{{x}_{3}}^2 * 0.13443 + N520 * 0.562513 + N520^2 * 0.0980371$$
$$N84 = 0.000661737 + N143 * N153 * 69.385 - N143^2 * 34.1426 + N153 * 1.00566 - N153^2 * 35.2489$$
$$N153 = 0.0016409 + N260 * 4.11528 + N260 * N266 * 135.675 - N260^2 * 69.9831 - N266 * 3.11957 - N266^2 * 65.6875$$
$$N266 = -0.00647904 - N576 * N358 * 6.13814 + N576^2 * 3.0572 + N358 * 1.04195 + N358^2 * 3.00401$$
$$N576 = 6.28532 + {x}_{1} * 9.35319 + x1 * \sqrt[3]{{x}_{1}} * 21.8155 - {x}_{1}^2 * 18.2776 -\sqrt[3]{{x}_{1}} * 11.9925 - \sqrt[3]{{x}_{1}}^2 * 7.78641$$
$$N221 = 3.45931 * {10}^{-5} + N315 * 1.98289 + N315 * N363 * 9.78517 - N315^2 * 5.79152 - N363 * 0.98315 - N363^2 * 3.98824$$
$$N315 = -0.0353646 - N417 * 0.312089 - N417 * N504 * 6.73671 + N417^2 * 4.34277 + N504 * 1.47506 + N504^2 * 2.21611$$
$$N417 = 0.046616 - {x}_{3} * 0.0227056 + {x}_{3} * N523 * 0.0936959 + {x}_{3}^2 * 0.00319128 + N523 * 0.773515 + N523^2 * 0.0944211$$
$$N523 = 0.554784 -\sqrt[3]{{x}_{1}} * 0.446589 + \sqrt[3]{{x}_{1}} * \sqrt[3]{{x}_{4}} * 0.735148 - {\text{``}}x4, cubert{\text{''}} * 0.468161$$
$$N363 = -0.113544 + N480 * 1.56256 - N480 * N571 * 4.71348 + N480^2 * 2.28464 + N571^2 * 1.82954$$
$$N571 = -1.05339 * {10}^{-12}+ N573$$
$$N573 = -0.619777 +{x}_{2} * 0.0558848 + {x}_{2} * {\text{``}}x1, cubert{\text{''}} * 0.0469653 -{x}_{2}^2 * 0.00128743 -\sqrt[3]{{x}_{1}} * 0.799765 + \sqrt[3]{{x}_{1}}^2 * 0.423977$$
$$N480 = 0.0793283 + N507 * 0.606649 + N507 * N520 * 6.5024 - N507^2 * 3.19284 - N520^2 * 2.87022$$
$$N520 = -0.0731976 + {x}_{1} * 1.57015 + {x}_{1} * {x}_{4} * 0.01282 - {x}_{1}^2 * 0.98239 - {x}_{4} * 0.00321404$$
$$N507 = -1.62014 +{x}_{2} * 0.123157 +{x}_{2} * {x}_{3} * 0.00118382 -{x}_{2}^2 * 0.00162357$$
$$N46 = -0.00384249 + N87 * 2.23248 + N87 * N112 * 15.7771 - N87^2 * 9.40143 - N112 * 1.21432 - N112^2 * 6.39333$$
$$N112 = 0.00122818 + N402 * 0.497913 + N402 * N147 * 23.2124 - N402^2 * 12.2215 + N147 * 0.510862 - N147^2 * 11.0014$$
$$N147 = 0.0151096 - N517 * 0.733569 + N517 * N248 * 1.09936 + N248 * 1.66087 - N248^2 * 1.01956$$
$$N248 = -0.0312296 + {x}_{1} * 0.215028 - {x}_{1} * N358 * 7.49913 +{x}_{1}^2 * 4.13888 + N358 * 0.946609 + N358^2 * 3.23154$$
$$N402 = 0.0376697 -{x}_{3} * 0.0223321 +{x}_{3} * N521 * 0.0940964 + {x}_{3}^2 * 0.00297722 + N521 * 0.814425 + N521^2 * 0.053195$$
$$N521 = 0.180556 + {x}_{1} * 0.565333 + {x}_{1} * \sqrt[3]{{x}_{4}} * 0.443784 - {x}_{1}^2 * 0.970109 - \sqrt[3]{{x}_{4}} * 0.111981$$
$$N87 = 0.000661738 + N143 * N151 * 69.385 - N143^2 * 34.1426 + N151 * 1.00566 - N151^2 * 35.2488$$
$$N151 = 0.00164091 + N260 * 4.11528 + N260 * N269 * 135.676 - N260^2 * 69.9834 - N269 * 3.11957 - N269^2 * 65.6878$$
$$N269 = -0.00647904 - N575 * N358 * 6.13814 + N575^2 * 3.0572 + N358 * 1.04195 + N358^2 * 3.00401$$
$$N358 = 0.188766 - \sqrt[3]{{x}_{3}} * 0.299711 +\sqrt[3]{{x}_{3}} * N519 * 0.317394 +\sqrt[3]{{x}_{3}}^2 * 0.121656 + N519 * 0.608858 + N519^2 * 0.0534157$$
$$N519 = -11.1969 +\sqrt[3]{{x}_{2}} * 7.37547 + \sqrt[3]{{x}_{2}} * \sqrt[3]{{x}_{4}} * 0.172616 -\sqrt[3]{{x}_{2}}^2 * 1.17692 - \sqrt[3]{{x}_{4}} * 0.451075$$
$$N575 = -4.05032 -\sqrt[3]{{x}_{1}} * 3.8942 - \sqrt[3]{{x}_{1}} * \sqrt[3]{{x}_{2}} * 2.24393 + \sqrt[3]{{x}_{1}}^2 * 8.41247 + \sqrt[3]{{x}_{2}} * 3.32065 - \sqrt[3]{{x}_{2}}^2 * 0.239694$$
$$N260 = -0.0159074 - {x}_{1} * N341 * 7.38292 + {x}_{1}^2 * 4.28392 + N341 * 1.07114 + N341^2 * 3.08084$$
$$N341 = 0.191389 - \sqrt[3]{{x}_{3}} * 0.304664 + \sqrt[3]{{x}_{3}} * N517 * 0.320767 + \sqrt[3]{{x}_{3}}^2 * 0.123665 + N517 * 0.606339 + N517^2 * 0.0525463$$
$$N517 = -12.2238 -{x}_{4} * 0.0130871 + {x}_{4} * \sqrt[3]{{x}_{2}} * 0.00500479 +\sqrt[3]{{x}_{2}} * 7.77345 - \sqrt[3]{{x}_{2}}^2 * 1.17885$$
$$N143 = 2.65296 -\sqrt[3]{{x}_{2}} * 2.27093 -\sqrt[3]{{x}_{2}} * N285 * 1.69357 + \sqrt[3]{{x}_{2}}^2 * 0.497524 + N285 * 4.57061 + N285^2 * 1.71161$$
$$N285 = -0.0129764 - N383 * 0.270567 + N383 * N504 * 1.89115 + N504 * 1.33173 - N504^2 * 1.95175$$
$$N504 = -0.438416 + {x}_{2} * 0.0407094 - {x}_{2} * N512 * 0.0702555 + N512 * 0.204885 + N512^2 * 2.79313$$
$$N512 = -1.50849 + {x}_{2} * 0.119589 + {x}_{2} * \sqrt[3]{{x}_{3}} * 0.0034988 -{x}_{2}^2 * 0.00160038 -\sqrt[3]{{x}_{3}} * 0.201899 +\sqrt[3]{{x}_{3}}^2 * 0.105077$$
$$N383 = 1.00566 - \sqrt[3]{{x}_{4}} * 0.494159 +\sqrt[3]{{x}_{4}} * N508 * 0.391485 + \sqrt[3]{{x}_{4}}^2 * 0.0549278 - N508 * 0.153024 - N508^2 * 0.189057$$
$$N508 = -0.240191 - {x}_{3} * 0.10489 + {x}_{3} * \sqrt[3]{{x}_{1}} * 0.187864 +\sqrt[3]{{x}_{1}}^2 * 1.2194.$$

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Maleki, A., Haghighi, A., Irandoost Shahrestani, M. et al. Applying different types of artificial neural network for modeling thermal conductivity of nanofluids containing silica particles. J Therm Anal Calorim 144, 1613–1622 (2021). https://doi.org/10.1007/s10973-020-09541-x

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  • DOI: https://doi.org/10.1007/s10973-020-09541-x

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