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

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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Correspondence to Akbar Maleki or Narjes Nabipour.

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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 (2020). https://doi.org/10.1007/s10973-020-09373-9

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Keywords

  • ZnO particles
  • Nanofluid
  • Artificial neural network
  • Thermal conductivity