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Thermal conductivity prediction of nanofluids containing CuO nanoparticles by using correlation and artificial neural network

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Abstract

Nanofluids are employed in different thermal devices due to their enhanced thermophysical features which lead to noticeable heat transfer augmentation. One of the major reasons of the heat transfer improvement by using the nanofluids is their increased thermal conductivity. Several methods have been applied to estimate this property of nanofluids such as correlations and artificial neural networks (ANNs). In the present paper, group method of data handling (GMDH) and a mathematical correlation are proposed for forecasting the thermal conductivity of nanofluids containing CuO nanoparticles. The inputs of the both models are the base fluids’ thermal conductivities, concentration, temperature and nanoparticle dimension. Comparison of the forecasted data by these two approaches revealed more favorable performance of GMDH. The values of R-squared in the cases where polynomial and ANN were utilized were 0.9862 and 0.9996, respectively. Moreover, the average absolute relative deviation values were 5.25% and 0.881% for the indicated methods, respectively. According to these statistical values, it is concluded that employing the ANN-based regression leads to more confident model for forecasting the TC of the nanofluids containing CuO nanoparticles.

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Abbreviations

\(a_{\text{i}}\) :

\(i\)th coefficient

\(x\) :

Input vector

\(x_{1}\) :

Temperature of nanofluid (°C)

\(x_{2}\) :

Volume fraction of solid phase (%)

\(x_{3}\) :

Size of particles (nm)

\(x_{4}\) :

Thermal conductivity of the base fluid (W m−1 K−1)

\(y\) :

Output vector

\(y_{\text{i}}^{\text{experimental}}\) :

Measured data in experiment

\(y_{\text{i}}^{\text{predicted}}\) :

Predicted value by the model

AARD:

Average absolute relative deviation

ANN:

Artificial neural network

GMDH:

Group method of data handling

MSE:

Mean square error

RD:

Relative deviation

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Authors and Affiliations

  1. Department of Civil, Geo and Environmental Engineering, Technical University of Munich, Munich, Germany

    Ali Komeilibirjandi

  2. School of Mechanical Engineering, Purdue University, West Lafayette, IN, USA

    Amir Hossein Raffiee

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

    Akbar Maleki

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

    Mohammad Alhuyi Nazari

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

    Mostafa Safdari Shadloo

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  1. Ali Komeilibirjandi
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  3. Akbar Maleki
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  4. Mohammad Alhuyi Nazari
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  5. Mostafa Safdari Shadloo
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Appendix

Appendix

$$\begin{aligned} N156 & = 0.0638187 + N250*2.30293 \\ & \quad + N250 * N264*23.5634 - \left( {N250} \right)^{2} *13.7886 \\ & \quad - N264*1.68318 - \left( {N264} \right)^{2} *9.33539 \\ N264 & = 0.0432462 + N377*0.0323191 \\ & \quad + N377*N315*0.403573 + N315*0.482274 \\ & \quad + \left( {N315} \right)^{2} *0.363301 \\ N315 & = - 0.806049 + N345*0.801919 \\ & \quad + N345*N380*0.592981 - \left( {N345} \right)^{2} *0.0493165 \\ & \quad + N380*3.74782 - \left( {N380} \right)^{2} *4.28298 \\ N250 & = - 0.00466728 - N335*0.446895 \\ & \quad - N335*N357*27.6001 + \left( {N335} \right)^{2} *15.9285 \\ & \quad + N357*1.43754 + (N357)^{2} *11.6728 \\ N323 & = 0.00163454 - N350*5.08638 \\ & \quad - N350*N353*32.6077 + \left( {N350} \right)^{2} *32.5894 \\ & \quad + N353*6.08958 \\ N350 & = 0.293485 - \sqrt[3]{{x_{4} }} *1.64301 + \left( {\sqrt[3]{{x_{4} }}} \right)^{2} *2.37782 \\ & \quad + N377*0.182398 \\ N60 & = 0.0214562 - N197*N77*4.09698 \\ & \quad + \left( {N197} \right)^{2} *2.28313 + N77*0.854643 \\ & \quad + \left( {N77} \right)^{2} *1.96742 \\ N77 & = - 0.0235764 - N332*0.243521 - N332*N185*54.0984 \\ & \quad + (N332)^{2} *26.1746 + N185*1.40135 \\ & \quad + \left( {N185} \right)^{2} *27.7079 \\ N185 & = 0.000925156 + N266*0.717883 + N266*N284*9.61885 \\ & \quad - \left( {N266} \right)^{2} *5.14743 + N284*0.272103 - \left( {N284} \right)^{2} *4.43229 \\ N284 & = 0.0131173 + \sqrt[3]{{x_{2} }}*0.0105856 + \sqrt[3]{{x_{2} }}*N320*0.0680691 \\ & \quad + N320*0.782995 + \left( {N320} \right)^{2} *0.140752 \\ N320 & = - 1.18285 + \sqrt[3]{{x_{3} }}*0.809801 + \sqrt[3]{{x_{3} }}*N334*0.122827 \\ & \quad - \left( {\sqrt[3]{{x_{3} }}} \right)^{2} *0.13057 + N334*0.458137 + \left( {N334} \right)^{2} *0.0931282 \\ N266 & = 0.740175 - \sqrt[3]{{x_{4} }}*4.94909 - \sqrt[3]{{x_{4} }}*N335*14.4033 \\ & \quad + \left( {\sqrt[3]{{x_{4} }}} \right)^{2} *7.61967 + N335*6.04715 + \left( {N335} \right)^{2} *6.09104 \\ N335 & = - 0.878954 + N356*1.16337 - N356*N380*0.405506 \\ & \quad + \left( {N356} \right)^{2} *0.0176024 + N380*3.83443 - \left( {N380} \right)^{2} *4.01012 \\ N197 & = - 0.947113 + N380*4.34746 + N380*N302*0.391864 \\ & \quad - \left( {N380} \right)^{2} *4.90199 + N302*0.896456 - \left( {N302} \right)^{2} *0.0559059 \\ N302 & = 0.00439602 - N339*1.49386 - N339*N357*67.8386 \\ & \quad + \left( {N339} \right)^{2} *37.3589 + N357*2.47456 + \left( {N357} \right)^{2} *30.4492 \\ N339 & = 0.00924584 + N345*0.716211 + N345*N375*0.256732 \\ & \quad + \left( {N345} \right)^{2} *0.179385 + N375*0.0486508 \\ N129 & = - 0.0617473 - N344*N217*27.9366 + \left( {N344} \right)^{2} *13.1372 \\ & \quad + N217*1.40366 + \left( {N217} \right)^{2} *14.3235 \\ N217 & = 0.0372683 - N298*2.21564 - N298*N299*3.82076 \\ & \quad + \left( {N298} \right)^{2} *3.83466 + N299*2.97188 + \left( {N299} \right)^{2} *0.242383 \\ N298 & = - 0.00461328 - N338*0.605155 - N338*N357*37.0172 \\ & \quad + \left( {N338} \right)^{2} *20.7828 + N357*1.65003 + \left( {N357} \right)^{2} *16.1369 \\ N357 & = 0.224904 - \sqrt[3]{{x_{4} }}*1.55475 - \sqrt[3]{{x_{4} }}*N375*0.51028 \\ & \quad + \left( {\sqrt[3]{{x_{4} }}} \right)^{2} *2.47322 + N375*0.443408 \\ N375 & = - 1.3152 + \sqrt[3]{{x_{2} }}*\sqrt[3]{{x_{3} }} *0.0459091 + \sqrt[3]{{x_{3} }}*1.11126 - \left( {\sqrt[3]{{x_{3} }}} \right)^{2} *0.186381 \\ N338 & = 0.0129124 + N345*N356*43.7902 - \left( {N345} \right)^{2} *21.8442 \\ & \quad + N356*0.911248 - \left( {N356} \right)^{2} *21.7936 \\ N344 & = - 0.477016 + N353*1.12265 - N353*N379*0.265719 \\ & \quad + N379*2.10893 - \left( {N379} \right)^{2} *2.25038 \\ N353 & = - 0.0220018 + N356*0.994495 + N377*0.0584774 \\ N48 & = - 0.000454786 - N69*N71*0.870543 \\ & \quad + \left( {N69} \right)^{2} *0.868246 + N71*1.00227 \\ N71 & = 0.0109597 - N139*N214*50.1732 + \left( {N139} \right)^{2} *25.6696 \\ & \quad + N214*0.921561 + \left( {N214} \right)^{2} *24.5636 \\ N214 & = - 0.0144738 + N331*1.08299 - N331*N269*19.656 \\ & \quad + \left( {N331} \right)^{2} *7.87523 + \left( {N269} \right)^{2} *11.6828 \\ N269 & = - 0.285722 + N380*1.126 - N380*N303*0.586986 \\ & \quad - \left( {N380} \right)^{2} *1.0272 + N303*1.25418 \\ N303 & = 0.00379619 + \sqrt[3]{{x_{2} }}*0.0252342 + \sqrt[3]{{x_{2} }}*N334*0.0472854 \\ & \quad + N334*0.771192 + \left( {N334} \right)^{2} *0.17568 \\ N69 & = - 0.0119576 + N273*1.07403 - N273*N139*29.6595 \\ & \quad + \left( {N273} \right)^{2} *13.7071 + \left( {N139} \right)^{2} *15.8416 \\ N139 & = 1.31772 - \sqrt[3]{{x_{4} }}*7.24065 - \sqrt[3]{{x_{4} }}*N268*15.7399 \\ & \quad + \left( {\sqrt[3]{{x_{4} }}} \right)^{2} *9.82624 + N268*6.84983 + \left( {N268} \right)^{2} *6.10165 \\ N268 & = - 0.9162 + N380*4.16699 + N380*N336*0.265379 \\ & \quad - \left( {N380} \right)^{2} *4.69022 + N336*0.98972 - \left( {N336} \right)^{2} *0.0976737 \\ N336 & = 0.0171419 + \sqrt[3]{{x_{2} }}*0.0106666 + \sqrt[3]{{x_{2} }}*N345*0.0751379 \\ & \quad + N345*0.748366 + \left( {N345} \right)^{2} *0.170354 \\ N273 & = - 0.0680299 + N380*0.191049 - N380*N304*0.643235 \\ & \quad + N304*1.20139 + \left( {N304} \right)^{2} *0.0835203 \\ N304 & = - 0.0193152 + N334*0.777182 + N334*N377*0.252188 \\ & \quad + \left( {N334} \right)^{2} *0.109905 + N377*0.0983817 \\ N118 & = 0.0155467 - N136*0.337627 - N136*N223*22.2581 \\ & \quad + \left( {N136} \right)^{2} *12.3103 + N223*1.23063 + \left( {N223} \right)^{2} *10.046 \\ N223 & = - 0.0208803 - N286*2.33259 + N286*N299*41.8394 \\ & \quad - \left( {N286} \right)^{2} *15.7271 + N299*3.48829 - \left( {N299} \right)^{2} *26.2913 \\ N299 & = - 0.0212474 + N331*0.802107 + N331*N376*0.275152 \\ & \quad + \left( {N331} \right)^{2} *0.0724277 + N376*0.0897678 \\ N331 & = - 0.676945 + \sqrt[3]{{x_{4} }}*2.13489 - \left( {\sqrt[3]{{x_{4} }}} \right)^{2} *0.948832 \\ & \quad - N334*0.303469 + \left( {N334} \right)^{2} *0.92214 \\ N286 & = 0.0171763 + N332*0.675503 + N332*N376*0.481898 \\ & \quad + \left( {N332} \right)^{2} *0.116863 + N376*0.0098905 \\ N376 & = 3.96757*10^{ - 14} + N377*1 \\ N332 & = - 0.254285 + N334*1.1415 - N334*N380*0.346046 \\ & \quad + N380*0.997576 - \left( {N380} \right)^{2} *0.884078 \\ N334 & = 0.665183 - \sqrt[3]{{x_{1} }}*0.151091 + \sqrt[3]{{x_{1} }}*\sqrt[3]{{x_{4} }}*0.212101 \\ & \quad + \left( {\sqrt[3]{{x_{1} }}} \right)^{2} *0.00618198 - \sqrt[3]{{x_{4} }}*1.99001 + \left( {\sqrt[3]{{x_{4} }}} \right)^{2} *2.16703 \\ N136 & = 0.0109802 - N247*0.896005 - N247*N346*19.3321 \\ & \quad + \left( {N247} \right)^{2} *11.8366 + N346*1.81279 + \left( {N346} \right)^{2} *7.56095 \\ N346 & = - 0.483901 + N354*1.11989 - N354*N379*0.267761 \\ & \quad + N379*2.11768 - \left( {N379} \right)^{2} *2.224 \\ N379 & = - 8.47397 + \sqrt[3]{{x_{3} }}*3.10228 - \sqrt[3]{{x_{3} }}*N380*2.08197 \\ & \quad - \left( {\sqrt[3]{{x_{3} }}} \right)^{2} *0.336827 + N380*16.7518 - \left( {N380} \right)^{2} *9.85072 \\ N354 & = - 1.37254*10^{ - 13} + N356*1 \\ N356 & = 0.325471 + \sqrt[3]{{x_{2} }}*0.0275276 - \sqrt[3]{{x_{4} }}*1.60857 + \left( {\sqrt[3]{{x_{4} }}} \right)^{2} *2.35855 \\ N247 & = - 0.818308 + N380*3.6748 - \left( {N380} \right)^{2} *4.05446 \\ & \quad + N329*1.06662 - (N329)^{2} *0.060733 \\ N329 & = 0.0216857 + N345*0.60242 + N345*N377*0.531008 \\ & \quad + \left( {N345} \right)^{2} *0.173633 + N377*0.0189504 \\ N377 & = 0.0244881 + \sqrt[3]{{x_{1} }}*0.0503244 - \sqrt[3]{{x_{1} }}*\sqrt[3]{{x_{2} }}*0.0191276 + \sqrt[3]{{x_{2} }}*0.274573 \\ N345 & = 1.09654 + \sqrt[3]{{x_{3} }}*\sqrt[3]{{x_{4} }}*0.427947 - \left( {\sqrt[3]{{x_{3} }}} \right)^{2} *0.0443035 \\ & \quad - \sqrt[3]{{x_{4} }}*3.60702 + \left( {\sqrt[3]{{x_{4} }}} \right)^{2} *2.68179 \\ N380 & = 0.967148 - \sqrt[3]{{x_{1} }}*1.74116 + \sqrt[3]{{x_{1} }}*\sqrt[3]{{x_{3} }}*0.435543 + \\ & \quad \left( {\sqrt[3]{{x_{1} }}} \right)^{2} *0.0204996 + \sqrt[3]{{x_{3} }}*1.57949 - \left( {\sqrt[3]{{x_{3} }}} \right)^{2} *0.452209. \\ \end{aligned}$$

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Komeilibirjandi, A., Raffiee, A.H., Maleki, A. et al. Thermal conductivity prediction of nanofluids containing CuO nanoparticles by using correlation and artificial neural network. J Therm Anal Calorim 139, 2679–2689 (2020). https://doi.org/10.1007/s10973-019-08838-w

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