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Thermal conductivity modeling of nanofluids with ZnO particles by using approaches based on artificial neural network and MARS

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Abstract

Nanofluids are attractive alternatives for the current heat transfer fluids due to their remarkably higher thermal conductivity which leads to the improved thermal performance. Nanofluids are applicable in porous media for improving their heat transfer. Proposing accurate models for forecasting this feature of nanofluids can facilitate and accelerate the design and modeling of nanofluids’ thermal mediums with porous structure. In the present study, three methods including MARS, artificial neural network (ANN) with Levenberg–Marquardt for training and GMDH are employed for thermal conductivity of the nanofluids containing ZnO particles. The confidence of the models is compared according to various criteria. It is observed that the most accurate model is obtained by using ANN with Levenberg–Marquardt followed by GMDH and MARS. R2 of the mentioned models are 0.9987, 0.9980 and 0.9879, respectively. Finally, sensitivity analysis is performed to find the importance of the input variables and it is concluded that the thermal conductivity of the base fluids has the highest importance followed by volume fraction of solid phase, size of particles and temperature.

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Authors and Affiliations

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

    Akbar Maleki

  2. Department of Mechanical Engineering, University of Bonab, Bonab, Iran

    Milad Elahi

  3. Department of Sustainable and Renewable Energy Engineering, University of Sharjah, Sharjah, United Arab Emirates

    Mamdouh El Haj Assad

  4. Department of Renewable Energies, Faculty of New Science and Technologies, University of Tehran, Tehran, Iran

    Mohammad Alhuyi Nazari

  5. CORIA Lab./CNRS UMR-6614, University and INSA of Rouen, Normandie University, 76000, Rouen, France

    Mostafa Safdari Shadloo

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

    Narjes Nabipour

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  1. Akbar Maleki
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  2. Milad Elahi
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  4. Mohammad Alhuyi Nazari
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  5. Mostafa Safdari Shadloo
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Appendix 1

Appendix 1

$$\begin{aligned} & TC = - 0.0176657 + N319*0.219462 + N319*N2*0.307774 - N319^{2} \\ & \quad *0.427988 + N2*0.873413 \\ \end{aligned}$$
$$N2 = 7.69955*10^{ - 5} - N70*0.162689 + N3*1.16249$$
$$N3 = - 4.95526*10^{ - 5} + N15*0.261551 + N4*0.738579$$
$$N4 = 0.000136022 - N99*0.298267 + N6*1.29791$$
$$\begin{aligned} & N6 = - 0.00482405 - N11*1.19431 - N11*N16*4.63991 + N11^{2} *4.61091 \\ & \quad + N16*2.21842 \\ \end{aligned}$$
$$\begin{aligned} & N16 = 0.00260313 + N50*0.983077 - N50*N18*825.582 + N50^{2} *410.751 \\ & \quad + N18^{2} *414.853 \\ \end{aligned}$$
$$N18 = - 0.000153167 + N34*0.494953 + N47*0.505449$$
$$N47 = - 0.0223358 + N145*1.11586 + N145*N121*109.727 - N145^{2} *55.9648 - N121^{2} *53.8994$$
$$\begin{aligned} & N121 = - 0.0514928 + N184*3.59611 + N184*N148*11.7972 - N184^{2} *10.9249 \\ & \quad - N148*2.3335 - N148^{2} *1.19407 \\ \end{aligned}$$
$$\begin{aligned} & N34 = 0.0162676 - N322*0.307286 - N322*N101*0.331167 \\ & \quad + N322^{2} *0.494972 + N101*1.24354 - N101^{2} *0.106404 \\ \end{aligned}$$
$$N50 = - 0.0021794 + N78*1.01062 + N78*N101*1.19449 - N78^{2} *1.20683$$
$$N101 = 0.0142793 + N143*2.3709 + N143*N176*4.17943 - N143^{2} *4.09344 - N176*1.44218$$
$$N176 = - 0.0592844 + N204*1.30927 - N204*N221*351.323 + N204^{2} *175.057 + N221^{2} *175.866$$
$$N204 = - 0.225366 - N238*N268*4.47015 + N238^{2} *3.10376 + N268*2.12858$$
$$\begin{aligned} & N143 = - 0.106234 + N185*3.4398 + N185*N169*172.435 - N185^{2} *91.4176 \\ & \quad - N169*1.87794 - N169^{2} *81.7228 \\ \end{aligned}$$
$$\begin{aligned} & N78 = - 0.0562197 + N178*3.50652 - N178*N148*9.15795 \\ & \quad - N148*2.22181 + N148^{2} *8.81021 \\ \end{aligned}$$
$$N148 = - 0.0729269 - N175*N262*2.91955 + N175^{2} *2.45894 + N262*1.37206$$
$$N175 = 0.0769239 + N185*0.789345 - N238*0.1795 + N238^{2} *0.476052$$
$$N238 = - 0.366614 + N245*1.96107 - N245*N328*3.85139 + N245^{2} *0.723865 + N328*1.19108$$
$$N11 = 0.000236778 + N23*0.998401 - N23*N30*167.397 + N23^{2} *82.9438 + N30^{2} *84.4522$$
$$N30 = - 0.0163455 + N145*1.07923 + N145*N63*169.572 - N145^{2} *86.0041 - N63^{2} *83.6506$$
$$N63 = - 0.0735931 + N314*0.377191 - N314^{2} *0.462206 + N69*0.994821$$
$$N314 = 1.17334 - N324*5.16795 + N324*N326*15.9779 - N326^{2} *7.89645$$
$$N326 = 3.46561 + x_{3} *0.0400498 - x_{3}^{2} *0.00573401 - N328*18.3688 + N328^{2} *26.5217$$
$$N324 = - 0.103354 + x_{3} *0.171453 - x_{3} *N325*0.396155 - x_{3}^{2} *0.00265483 + N325^{2} *3.06177$$
$$N23 = - 0.0145464 - N132*N60*356.781 + N132^{2} *178.502 + N60*1.06685 + N60^{2} *178.197$$
$$N60 = - 0.0683454 + N262*0.720575 - N262*N70*1.71076 + N70*0.628952 + N70^{2} *1.27853$$
$$N99 = - 0.00524059 + N182*2.42504 - N182*N138*5.7139 - N138*1.39644 + N138^{2} *5.67566$$
$$N182 = 0.377424 - x_{4} *0.00213238 + x_{4} *N284*0.00947956 - x_{4}^{2} *5.43079*10^{ - 6} - N284*0.90302 + N284^{2} *1.97389$$
$$N284 = - 0.165314 + x_{3} *0.0116425 - x_{3}^{2} *0.000667677 + N300*1.66512 - N300^{2} *0.723299$$
$$N15 = - 0.00295291 + N24*1.01447 - N24*N27*57.5092 + N24^{2} *28.0308 + N27^{2} *29.4603$$
$$N27 = - 0.0157849 + N145*0.79061 + N145*N55*181.895 - N145^{2} *91.8971 + N55*0.286356 - N55^{2} *90.0791$$
$$N55 = 0.0238995 + N67*5.95806 + N67*N69*123.969 - N67^{2} *67.2278 - N69*5.08027 - N69^{2} *56.5903$$
$$N69 = 0.0158676 - N138*3.28471 - N138*N141*9.629 + N138^{2} *9.72383 + N141*4.20539$$
$$N141 = - 0.0330155 - N169*0.671093 + N169*N178*3.57479 + N178*1.83897 - N178^{2} *3.77977$$
$$N169 = - 0.037664 - N185*N298*2.63904 + N185^{2} *2.41529 + N298*1.18464$$
$$N67 = 0.000840653 + N142*2.17137 + N142*N177*3.57604 - N142^{2} *3.56919 - N177*1.17608$$
$$N177 = 0.0643097 + N209*0.687415 - N209*N217*308.398 + N209^{2} *154.537 + N217^{2} *154.208$$
$$N217 = - 0.680632 + N237*1.96497 - N237*N317*5.03311 + N237^{2} *1.49805 + N317*2.81549 - N317^{2} *1.64736$$
$$N317 = 0.808557 - x_{3} *N322*0.046462 + x_{3}^{2} *0.00358317 - N322*3.5375 + N322^{2} *6.33062$$
$$N209 = - 0.571488 + N237*2.86135 - N237*N318*4.21518 + N318*1.27117$$
$$N318 = 1.13386 - N322*2.67351 + N322*N325*8.96291 - N325*2.75097$$
$$N237 = 0.0506727 - N245*0.416666 + N245*N283*2.37448 + N283*1.14746 - N283^{2} *2.02719$$
$$N145 = - 0.0267507 + x_{1} *0.668661 - x_{1} *N178*7.8786 + x_{1}^{2} *3.50955 + N178*0.531387 + N178^{2} *4.07859$$
$$N24 = - 0.0243146 + N132*1.1126 - N132*N48*319.664 + N132^{2} *158.385 + N48^{2} *161.149$$
$$N48 = - 0.0130361 - N80*N106*120.224 + N80^{2} *60.6373 + N106*1.06104 + N106^{2} *59.5158$$
$$N106 = 0.0188887 - N132*3.49404 - N132*N142*9.51607 + N132^{2} *9.62785 + N142*4.40009$$
$$N142 = - 0.0292551 - N170*0.648922 + N170*N178*3.49316 + N178*1.79744 - N178^{2} *3.67412$$
$$N170 = 0.00652534 - N185*N297*2.09961 + N185^{2} *2.14504 + N297*0.962813$$
$$N297 = 0.096511 + x_{1} *1.08962 - N322*0.241671$$
$$N80 = 0.0548859 - N138*3.94016 - N138*N147*220.435 + N138^{2} *116.202 + N147*4.66 + N147^{2} *104.569$$
$$N147 = 0.00347901 - N181*0.296463 - N181*N262*7.55137 + N181^{2} *5.18356 + N262*1.28156 + N262^{2} *2.3753$$
$$N262 = 0.332581 + N286*1.90067 - N286*N321*4.90236 + N286^{2} *1.27135 - N321*2.77042 + N321^{2} *6.17904$$
$$N321 = 1.06442 - x_{2} *0.0335236 + x_{2} *N325*0.102003 - N325*2.37618$$
$$N286 = 0.299854 - x_{1} *N328*1.47893 + x_{1}^{2} *2.22396$$
$$N181 = 0.194506 - x_{4} *0.00180935 + x_{4} *N268*0.00883743 - x_{4}^{2} *6.60581*10^{ - 6} + N268^{2} *0.897453$$
$$N132 = 0.00961835 + N161*0.958064 - N161*N223*36.7956 + N161^{2} *17.9033 + N223^{2} *18.926$$
$$N223 = - 0.0936659 + N239^{2} *0.807559 + N268*1.45895 - N268^{2} *1.34759$$
$$N268 = - 0.00769765 + N283*1.08486 - N283*N325*0.167757$$
$$N161 = - 0.168237 + N300*1.03256 - N300*N184*7.53604 + N300^{2} *2.59234 + N184*0.81681 + N184^{2} *3.89938$$
$$N70 = 0.0251658 - N138*4.27282 - N138*N140*166.714 + N138^{2} *89.6171 + N140*5.14603 + N140^{2} *77.2453$$
$$N140 = - 0.00535533 - N172*0.674549 + N172*N178*3.52513 + N178*1.70163 - N178^{2} *3.55777$$
$$N178 = - 0.0151785 + x_{3} *N242*0.0210041 + N242*0.969038 + N242^{2} *0.0678718$$
$$N242 = 0.417166 - x_{4} *0.00285149 + x_{4} *N300*0.0106948 - x_{4}^{2} *2.44755*10^{ - 6} - N300*1.04339 + N300^{2} *2.09903$$
$$N172 = 0.0575751 - N185*N277*7.60877 + N185^{2} *4.8825 + N277*0.706673 + N277^{2} *3.07754$$
$$N277 = - 0.0401214 + N300*1.62702 - N300*N328*1.35934$$
$$N185 = 0.195967 - x_{4} *0.00186632 + x_{4} *N283*0.00896443 - x_{4}^{2} *6.63628*10^{ - 6} + N283^{2} *0.891171$$
$$N138 = - 0.00840279 + N158*1.04734 - N158*N221*43.0623 + N158^{2} *20.9581 + N221^{2} *22.0316$$
$$N221 = 0.0432807 - N239*0.596357 + N239^{2} *1.46592 + N266*1.36626 - N266^{2} *1.17206$$
$$N266 = 0.298198 + N283*1.02025 - N328*1.45799 + N328^{2} *1.71157$$
$$N283 = 0.139504 + x_{1} *0.238172 + x_{1}^{2} *1.10886 + x_{3} *0.0107559 - x_{3}^{2} *0.000384995$$
$$N239 = - 0.616025 + N245*3.03362 - N245*N325*4.85465 + N325*1.45416$$
$$N158 = - 0.197164 + N184*0.771435 - N184*N298*6.94272 + N184^{2} *3.65626 + N298*1.22415 + N298^{2} *2.06504$$
$$N298 = - 0.0185617 + N300*1.16495 - N300*N322*0.300698$$
$$N322 = 0.0127235 + x_{2} *0.0251002 + x_{2} *x_{4} *9.80065*10^{ - 5} - x_{2}^{2} *0.000390176 - x_{4} *0.00208712 - x_{4}^{2} *2.61385*10^{ - 5}$$
$$N300 = 0.185992 + x_{1} *0.367112 - x_{1} *x_{2} *0.00486533 + x_{1}^{2} *1.14827 - x_{2} *0.00247947 + x_{2}^{2} *5.45705*10^{ - 5}$$
$$N184 = - 0.0282892 + x_{3} *N245*0.0221675 + N245*1.0279$$
$$N245 = 0.541455 - x_{1} *1.79012 + x_{1} *x_{4} *0.0104767 + x_{1}^{2} *3.3013 - x_{4} *0.00235751 - x_{4}^{2} *3.03253*10^{ - 6}$$
$$N319 = 3.1631 - N325*15.7802 + N325*N328*35.1175 + N325^{2} *5.1452 - N328^{2} *18.0806$$
$$N328 = 0.41085 - x_{3} *0.0343551 + x_{3}^{2} *0.00504291 + x_{4} *0.00115375 - x_{4}^{2} *1.70035*10^{ - 5}$$
$$N325 = - 0.133685 + x_{2} *0.0283308 - x_{2} *x_{3} *0.000703991 - x_{2}^{2} *0.000356603 + x_{3}^{2} *0.00566806$$

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Maleki, A., Elahi, M., Assad, M.E.H. et al. Thermal conductivity modeling of nanofluids with ZnO particles by using approaches based on artificial neural network and MARS. J Therm Anal Calorim 143, 4261–4272 (2021). https://doi.org/10.1007/s10973-020-09373-9

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