# Prediction method of thermal conductivity of nanofluids based on radial basis function

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## Abstract

Accurately predicting the thermal conductivity of nanofluids under various thermodynamic conditions is of great importance to promote the industrial application of nanofluids. Unfortunately, the accuracy and applicability of the current theoretical or empirical models cannot meet the demand, due to the inherent complex of nanofluids. In this study, an intelligent model, named radial basis function artificial neural network (RBF–ANNs), is developed to predict the thermal conductivity of nanofluids under various conditions. Five parameters including nanoparticle volume concentration, temperature, nanoparticle diameter, thermal conductivity of nanoparticle and thermal conductivity of base fluid are selected as the input variables. A total of 1444 experimental data samples are collected to optimize the structure of model. The RBF model is compared with six theoretical models and three intelligent models through statistical and graphical analyses. Also, trend analysis and sensitivity analysis are conducted to evaluate the influencing mechanism of nanoparticle concentration, temperature, nanoparticle size, thermal conductivities of base fluids and nanoparticle on the thermal conductivity of nanofluids. Meanwhile, the quality of the experimental data is evaluated by means of leverage algorithm. Results indicate the superiority of the RBF, especially when the data size is large. The overall correlation coefficient (*R*^{2}), average absolute relative deviation (AARD%) and root-mean-squared error of the developed model are 0.9931, 2.715 and 0.0316, respectively. Among the five input parameters, the volume fraction of nanoparticles has the greatest impact on the thermal conductivity of nanofluid. The results of outlier detection demonstrate that the proposed RBF model and data samples are statistically valid.

## Keywords

Effective thermal conductivity Nanofluid Radial basis function Neural network Sensitivity analysis Outlier detection## Notes

### Acknowledgements

This work was supported by Yunnan Provincial Department of Education Science Research Fund Project (Grant No. 2018JS551, 2019J0025) and Scientific Research Foundation of Kunming Metallurgy College (Grant No. Xxrcxm201802).

## References

- 1.S.U.S. Choi and J.A. Eastman. Enhancing thermal conductivity of fluids with nanoparticles, presented at international mechanical engineering congress and exhibition, November 1995, San Francisco, CA.Google Scholar
- 2.Yang L, Xu JY, Du K, Zhang XS. Recent developments on viscosity and thermal conductivity of nanofluids. Powder Technol. 2017;317:348–69.CrossRefGoogle Scholar
- 3.Arthur O, Karim MA. An investigation into the thermophysical and rheological properties of nanofluids for solar thermal applications. Renew Sustain Energy Rev. 2016;55(4):739–55.CrossRefGoogle Scholar
- 4.Bahiraei M, Hangi M, Saeedan M. A novel application for energy efficiency improvement using nanofluid in shell and tube heat exchanger equipped with helical baffles. Energy. 2015;93:2229–40.CrossRefGoogle Scholar
- 5.Rostami Z, Rahimi M, Azimi N. Using high-frequency ultrasound waves and nanofluid for increasing the efficiency and cooling performance of a PV module. Energy Convers Manag. 2018;160:141–9.CrossRefGoogle Scholar
- 6.Kleinstreuer C, Feng Y. Experimental and theoretical studies of nanofluid thermal conductivity enhancement: a review. Nanoscale Res Lett. 2011;6:229.PubMedPubMedCentralCrossRefGoogle Scholar
- 7.Zhang HY, Wang SX, Lin YX. Stability, thermal conductivity, and rheological properties of controlled reduced graphene oxide dispersed nanofluids. Appl Therm Eng. 2017;119:132–9.CrossRefGoogle Scholar
- 8.Kumar N, Sonawane SS, Sonawane SH. Experimental study of thermal conductivity, heat transfer and friction factor of Al
_{2}O_{3}based nanofluid. Int Commun Heat Mass Transfer. 2018;80:1–10.CrossRefGoogle Scholar - 9.Alade O, Oyehan TA, Popoola IK, Olatunji SO, Bagudu A. Modeling thermal conductivity enhancement of metal and metallic oxide nanofluids using support vector regression. Adv Powder Technol. 2018;29:157–67.CrossRefGoogle Scholar
- 10.Hemmati-Sarapardeha A, Varamesh A, Huseinc MM, Karan K. On the evaluation of the viscosity of nanofluid systems: modeling and data assessment. Renew Sustain Energy Rev. 2018;81:313–29.CrossRefGoogle Scholar
- 11.Nadooshan A. An experimental correlation approach for predicting thermal conductivity of water-EG based nanofluids of zinc oxide. Physica E. 2017;87:15–9.CrossRefGoogle Scholar
- 12.Keyvani M, Afrand M, Toghraie D, Reiszadeh M. An experimental study on the thermal conductivity of cerium oxide/ethylene glycol nanofluid: developing a new correlation. J Mol Liq. 2018;266:211–7.CrossRefGoogle Scholar
- 13.Shukla KN, Koller TM, Rausch MH, Fröba AP. Effective thermal conductivity of nanofluids: a new model taking into consideration Brownian motion. Int J Heat Mass Transf. 2016;99:532–40.CrossRefGoogle Scholar
- 14.Chon CH, Kihm KD, Lee SP, Choi SUS. Empirical correlation finding the role of temperature and particle size for nanofluid (Al
_{2}O_{3}) thermal conductivity enhancement. Appl Phys Lett. 2005;87(153107):1–3.Google Scholar - 15.Godson L, Raja B, Lal DM, Wongwises S. Experimental investigation on the thermal conductivity and viscosity of silver-deionized water nanofluid. Exp Heat Transf. 2010;23:317–32.CrossRefGoogle Scholar
- 16.Hamilton RL, Crosser OK. Thermal conductivity of heterogeneous two-component systems. Ind Eng Chem Fundam. 1962;1:187–91.CrossRefGoogle Scholar
- 17.Timofeeva EV, Gavrilov AN, McCloskey JM, Tolmachev YV. Thermal conductivity and particle agglomeration in alumina nanofluids: experiment and theory. Phys Rev Lett. 2007;76:061203–16.Google Scholar
- 18.Mintsa HA, Roy G, Nguyen CT, Doucet D. New temperature dependent thermal conductivity data for water-based nanofluids. Int J Therm Sci. 2009;48:363–71.CrossRefGoogle Scholar
- 19.Prasher RS, Bhattacharya P, Phelan PE. Thermal conductivity of nanoscale colloidal solutions (nanofluids). Phys Rev Lett. 2005;94:025901.PubMedCrossRefPubMedCentralGoogle Scholar
- 20.Aminian A. Predicting the effective thermal conductivity of nanofluids for intensification of heat transfer using artificial neural network. Powder Technol. 2016;301:288–309.CrossRefGoogle Scholar
- 21.Ahmadloo E, Azizi S. Prediction of thermal conductivity of various nanofluids using artificial neural network. Int Commun Heat Mass Transfer. 2016;74:69–75.CrossRefGoogle Scholar
- 22.Bahiraei M, Heshmatian S, Moayedi H. Artificial intelligence in the field of nanofluids: a review on applications and potential future directions. Powder Technol. 2019;353:276–301.CrossRefGoogle Scholar
- 23.Heshmatian S, Bahiraei M. Numerical investigation of entropy generation to predict irreversibilities in nanofluid flow within a microchannel: effects of Brownian diffusion, shear rate and viscosity gradient. Chem Eng Sci. 2017;172:52–65.CrossRefGoogle Scholar
- 24.Bahiraei M, Heshmatian S. Optimizing energy efficiency of a specific liquid block operated with nanofluids for utilization in electronics cooling: a decision-making based approach. Energy Convers Manag. 2017;154:180–90.CrossRefGoogle Scholar
- 25.Bahiraei M, Heshmatian S, Keshavarzi M. Multi-criterion optimization of thermohydraulic performance of a mini pin fin heat sink operated with ecofriendly graphene nanoplatelets nanofluid considering geometrical characteristics. J Mol Liq. 2019;276:653–66.CrossRefGoogle Scholar
- 26.Longo GA, Zilio C, Ceseracciu E, Reggiani M. Application of Artificial Neural Network (ANN) for the prediction of thermal conductivity of oxide-water nanofluids. Nano Energy. 2012;1:290–6.CrossRefGoogle Scholar
- 27.Vafaei M, et al. Evaluation of thermal conductivity of MgO-MWCNTs/EG hybrid nanofluids based on experimental data by selecting optimal artificial neural networks. Physica E. 2017;85:90–6.CrossRefGoogle Scholar
- 28.Esfea MH, Bahiraei M, Hajmohammad MH, Afrand M. Rheological characteristics of MgO/oil nanolubricants: experimental study and neural network modeling. Int Commun Heat Mass Transfer. 2017;86:245–52.CrossRefGoogle Scholar
- 29.Papari MM, Yousefi F, Moghadasi J, Karimi H, Campo A. Modeling thermal conductivity augmentation of nanofluids using diffusion neural networks. Int J Therm Sci. 2011;50:44–52.CrossRefGoogle Scholar
- 30.Hojjat M, Etemad SGh, Bagheri R, Thibault J. Thermal conductivity of non-Newtonian nanofluids: experimental data and modeling using neural network. Int J Heat Mass Transf. 2011;54:1017–23.CrossRefGoogle Scholar
- 31.Zhao NB, Wen XY, Yang JL, Li SY, Wang ZT. Modeling and prediction of viscosity of water-based nanofluids by radial basis function neural networks. Powder Technol. 2015;281:173–83.CrossRefGoogle Scholar
- 32.Zhao NB, Li ZM. Viscosity prediction of different ethylene glycol/water based nanofluids using a RBF neural network. Appl Sci. 2017;7:409–25.CrossRefGoogle Scholar
- 33.Sayahi T, Tatar A, Bahrami M. A RBF model for predicting the pool boiling behavior of nanofluids over a horizontal rod heater. Int J Therm Sci. 2016;99:180–94.CrossRefGoogle Scholar
- 34.Zendehboudi A, Tatar A. Utilization of the RBF network to model the nucleate pool boiling heat transfer properties of refrigerant-oil mixtures with nanoparticles. J Mol Liq. 2017;247:304–12.CrossRefGoogle Scholar
- 35.Barati-Harooni A, Najafi-Marghmaleki A. An accurate RBF-NN model for estimation of viscosity of nanofluids. J Mol Liq. 2016;224:580–8.CrossRefGoogle Scholar
- 36.Murshed SMS, Leong KC, Yang C. Investigations of thermal conductivity and viscosity of nanofluids. Int J Therm Sci. 2008;47:560–8.CrossRefGoogle Scholar
- 37.Patel HE, Sundararajan T, Das SK. An experimental investigation into the thermal conductivity enhancement in oxide and metallic nanofluids. J Nanopart Res. 2010;12:1015–31.CrossRefGoogle Scholar
- 38.Żyła G, Fal J. Experimental studies on viscosity, thermal and electrical conductivity of aluminum nitride–ethylene glycol (AlN–EG) nanofluids. Thermochim Acta. 2016;637:11–6.CrossRefGoogle Scholar
- 39.Yiamsawasd T, Dalkilic AS, Wongwises S. Measurement of the thermal conductivity of titania and alumina nanofluids. Thermochim Acta. 2012;545:48–56.CrossRefGoogle Scholar
- 40.Esfe MH, et al. Experimental study on thermal conductivity of ethylene glycol based nanofluids containing Al
_{2}O_{3}nanoparticles. Int J Heat Mass Transf. 2015;88:728–34.CrossRefGoogle Scholar - 41.Mostafizur RM, Bhuiyan MHU, Saidur R, Abdul-Aziz AR. Thermal conductivity variation for methanol based nanofluids. Int J Heat Mass Transf. 2014;76:350–6.CrossRefGoogle Scholar
- 42.Mostafizur RM, Saidur R, Abdul-Aziz AR, Bhuiyan MHU. Thermophysical properties of methanol based Al
_{2}O_{3}nanofluids. Int J Heat Mass Transf. 2015;85:414–9.CrossRefGoogle Scholar - 43.Mahbubul IM, Saidur R, Amalina MA. Influence of particle concentration and temperature on thermal conductivity and viscosity of Al
_{2}O_{3}/R141b nanorefrigerant. Int Commun Heat Mass Transfer. 2013;43:100–4.CrossRefGoogle Scholar - 44.Pryazhnikov MI, Minakov AV, Rudyak V Ya, Guzei DV. Thermal conductivity measurements of nanofluids. Int J Heat Mass Transf. 2017;104:1275–82.CrossRefGoogle Scholar
- 45.Esfe MH, Saedodin S, Mahian O, Wongwises S. Thermal conductivity of Al
_{2}O_{3}/water nanofluids. J Therm Anal Calorim. 2014;117:675–81.CrossRefGoogle Scholar - 46.Garg J, et al. Enhanced thermal conductivity and viscosity of copper nanoparticles in ethylene glycol nanofluid. J Appl Phys. 2008;103:074301.CrossRefGoogle Scholar
- 47.Jiang HF, et al. Temperature dependence of the stability and thermal conductivity of an oil–based nanofluid. Thermochim Acta. 2014;579:27–30.CrossRefGoogle Scholar
- 48.Aberoumand S, Jafarimoghaddam A. Experimental study on synthesis, stability, thermal conductivity and viscosity of Cu–engine oil nanofluid. J Taiwan Inst Chem Eng. 2017;71:315–22.CrossRefGoogle Scholar
- 49.Jiang WT, Ding GL, Peng H. Measurement and model on thermal conductivities of carbon nanotube nanorefrigerants. Int J Therm Sci. 2009;48:1108–15.CrossRefGoogle Scholar
- 50.Esfe MH, et al. Experimental investigation and development of new correlations for thermal conductivity of CuO/EG–water nanofluid. Int Commun Heat Mass Transfer. 2015;65:47–51.CrossRefGoogle Scholar
- 51.Ebrahimi S, Saghravani SF. Experimental study of the thermal conductivity features of the water based Fe
_{3}O_{4}/CuO nanofluid. Heat Mass Transf. 2017;54:999–1008.CrossRefGoogle Scholar - 52.Esfe MH, Saedodin S, Mahian O, Wongwises S. Heat transfer characteristics and pressure drop of COOH − functionalized DWCNTs/water nanofluid in turbulent flow at low concentrations. Int J Heat Mass Transf. 2014;73:186–94.CrossRefGoogle Scholar
- 53.Afrand M, Toghraie D, Sina N. Experimental study on thermal conductivity of water − based Fe
_{3}O_{4}nanofluid: development of a new correlation and modeled by artificial neural network. Int Commun Heat Mass Transfer. 2016;75:262–9.CrossRefGoogle Scholar - 54.Amiri A, Sadri R, Shanbedi M, et al. Performance dependence of thermosyphon on the functionalization approaches: an experimental study on thermophysical properties of graphene nanoplatelet − based water nanofluids. Energy Convers Manag. 2015;92:322–30.CrossRefGoogle Scholar
- 55.Esfe MH, Saedodin S, Mahian O, Wongwises S. Thermophysical properties, heat transfer and pressure drop of COOH − functionalized multi walled carbon nanotubes/water nanofluids. Int Commun Heat Mass Transfer. 2014;58:176–83.CrossRefGoogle Scholar
- 56.Duangthongsuk W, Wongwises S. Measurement of temperature—dependent thermal conductivity and viscosity of TiO2—water nanofluids. Exp Thermal Fluid Sci. 2009;33:706–14.CrossRefGoogle Scholar
- 57.Said Z, et al. Performance enhancement of a flat plate solar collector using titanium dioxide nanofluid and polyethylene glycol dispersant. J Clean Prod. 2015;92:343–53.CrossRefGoogle Scholar
- 58.Suganthi KS, Vinodhan VL, Rajan KS. Heat transfer performance and transport properties of ZnO–ethylene glycol and ZnO–ethylene glycol–water nanofluid coolants. Appl Energy. 2014;135:548–59.CrossRefGoogle Scholar
- 59.Pastoriza-Gallego MJ, Lugo L, Cabaleiro D, Legido JL, Piñeiro MM. Thermophysical profile of ethylene glycol–based ZnO nanofluids. J Chem Thermodyn. 2014;73:23–30.CrossRefGoogle Scholar
- 60.Lee GJ, et al. Thermal conductivity enhancement of ZnO nanofluid using a one − step physical method. Thermochim Acta. 2012;542:24–7.CrossRefGoogle Scholar
- 61.Jeong J, et al. Particle shape effect on the viscosity and thermal conductivity of ZnO nanofluids. Int J Refrig. 2013;36:2233–41.CrossRefGoogle Scholar
- 62.Broomhead DS, Lowe D. Radial basis functions, multi-variable functional interpolation and adaptive networks. Compl Syst. 1988;2:321–55.Google Scholar
- 63.Yu L, Lai KK, Wang S. Multistage RBF neural network ensemble learning for exchange rates forecasting. Neurocomputing. 2008;71:3295–302.CrossRefGoogle Scholar
- 64.Zendehboudi A, Saidur R. A reliable model to estimate the effective thermal conductivity of nanofluids. Heat Mass Transf. 2018;55(2):397–411.CrossRefGoogle Scholar
- 65.Ayatollahi S, Hemmati-Sarapardeh A, Roham M, Hajirezaie S. A rigorous approach for determining interfacial tension and minimum miscibility pressure in paraffin-CO
_{2}systems: application to gas injection processes. J Taiwan Inst Chem Eng 2016: https://doi.org/10.1016/j.jtice.2016.02.013.CrossRefGoogle Scholar - 66.Jung S, Kwon S. Weighted error functions in artificial neural networks for improved wind energy potential estimation. Appl Energy. 2013;111:778–90.CrossRefGoogle Scholar
- 67.Mohammadi H, Eslamimanesh A, Gharagheizi F, Richon D. A novel method for evaluation of asphaltene precipitation titration data. Chem Eng Sci. 2012;78:181–5.CrossRefGoogle Scholar
- 68.Mohagheghian E, Zafarian-Rigaki H, Ghahfarrokhi YM, Hemmati-Sarapardeh A. Using an artificial neural network to predict carbon dioxide compressibility factor at high pressure and temperature. Korean J Chem Eng. 2015;32(10):2087–96.CrossRefGoogle Scholar
- 69.Bruggeman DAG. Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen. Ann Phys. 1935;416:636–64.CrossRefGoogle Scholar
- 70.Wasp FJ. Solid–liquid slurry pipeline transportation. Berlin: Trans. Tech; 1977.Google Scholar
- 71.Vatani A, Woodfield PL, Dao DV. A survey of practical equations for prediction of effective thermal conductivity of spherical–particle nanofluids. J Mol Liq. 2015;211:712–33.CrossRefGoogle Scholar
- 72.Baghban A, Habibzadeh S, Ashtiani F. Toward a modeling study of thermal conductivity of nanofluids using LSSVM strategy. J Therm Anal Calorim. 2018. https://doi.org/10.1007/s10973-018-7074-5.CrossRefGoogle Scholar
- 73.Meybodi M, Naseri S, Shokrollahi A, Daryasafar A. Prediction of viscosity of water-based Al
_{2}O_{3}, TiO_{2}, SiO_{2}, and CuO nanofluids using a reliable approach. Chemometr Intell Lab Syst. 2015;149:60–9.CrossRefGoogle Scholar - 74.Mehrabi M, Pesteei S, Pashaee T. Modeling of heat transfer and fluid flow characteristics of helicoidal double-pipe heat exchangers using Adaptive Neuro-Fuzzy Inference System (ANFIS). Int Commun Heat Mass Transfer. 2011;38:525–32.CrossRefGoogle Scholar
- 75.Selimefendigil F, Öztop H. Magnetic field effects on the forced convection of CuO-water nanofluid flow in a channel with circular cylinders and thermal predictions using ANFIS. Int J Mech Sci. 2018;146–147:9–24.CrossRefGoogle Scholar
- 76.Aminossadati S, Kargar A, Ghasemi B. Adaptive network-based fuzzy inference system analysis of mixed convection in a two-sided lid-driven cavity filled with a nanofluid. Int J Therm Sci. 2012;52:102–11.CrossRefGoogle Scholar
- 77.Tatar A, Barati-Harooni A, Najafi-Marghmaleki A, et al. Predictive model based on ANFIS for estimation of thermal conductivity of carbon dioxide. J Mol Liq. 2016;224:1266–74.CrossRefGoogle Scholar