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A reliable model to estimate the effective thermal conductivity of nanofluids

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Abstract

The thermal conductivity is a key parameter to study the applicability of nanofluids for heat transfer enhancement of flowing liquids. This paper is an effort on implementing various methods to model the effective thermal conductivity of 26 nanofluids under different situations and evaluate the authenticity of the reported experimental data in the open literature. The most influential physical properties of nanofluids, such as the nanoparticle volume fraction, nanoparticle diameter, thermal conductivity of base fluid, temperature, and thermal conductivity of solid particle are considered as the input variables. With the purpose of introducing a comprehensive and pragmatic model with desired accuracy, a Multilayer Perceptron-Artificial Neural Network (MLP-ANN) approach is constructed and tested using data generated from 993 experiments. To appraise the creditability of the MLP-ANN model, a comparison with other 10 alternative techniques is carried out. The predictions made by the MLP-ANN yield excellent match with the experimentally generated samples against those of the other approaches. The coefficient of determination and relative root mean squared error are found to be 0.994 and 1.534%, respectively. Likewise, the results of the data analysis and the outlier detection method have proved that some of the data samples are significantly inconsistent with the remainder of the data set.

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Authors and Affiliations

  1. Department of Building Science, School of Architecture, Tsinghua University, Bei**g, 100084, China

    Alireza Zendehboudi

  2. Research Centre for Nano-Materials and Energy Technology (RCNMET), School of Science and Technology, Sunway University, No. 5, Jalan Universiti, Bandar Sunway, 47500, Petaling Jaya, Selangor Darul Ehsan, Malaysia

    R. Saidur

  3. American University of Ras Al Khaimah, Ras Al Khaimah, United Arab Emirates

    R. Saidur

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  1. Alireza Zendehboudi
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  2. R. Saidur
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Correspondence to Alireza Zendehboudi.

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Appendix 1

Appendix 1

Figure 9 shows the architecture of the MLP-ANN developed in this study, including the input layer, hidden layers, and output layer. To make the outcomes of the model reproducible, the matrices of the weights and biases of all layers of the optimized trained model are presented.

Fig. 9
figure 9

The optimized MLP-ANN architecture

Full size image
$$ {\displaystyle \begin{array}{l}{iw}_{\left\{1,1\right\}}=\left(\begin{array}{lllll}0.39082& -0.014084& -10.6324& 0.26459& 0.84956\\ {}-3.0686& 1.7539& -0.3762& 4.4476& -0.40305\\ {}-6.2119& -0.59865& 8.0815& 3.1959& 2.8045\\ {}4.604& 0.032355& -0.41081& 4.0842& -4.3598\\ {}-1.8269& -0.21067& 14.0708& -5.1867& -5.2764\\ {}3.9682& 0.028767& -2.3525& 0.49441& 3.9595\\ {}-14.7481& 0.022126& -0.30675& 17.16& 1.8299\\ {}4.1637& 0.13297& 2.2652& 1.5591& 6.3607\\ {}6.5973& 0.17642& -7.7412& 0.87903& -2.8476\\ {}-5.5843& 0.038979& -4.2519& 4.1328& 1.7722\\ {}16.4519& 0.13228& 0.21144& 8.4349& -3.0043\\ {}-1.9754& 0.19147& -4.5379& -15.0046& 2.8383\end{array}\right)\\ {}{iw}_{\left\{2,1\right\}}=\left(\begin{array}{llllllllllll}9.3909& 5.2654& -5.0754& 2.0546& 4.9026& 0.10895& 5.8691& -2.3513& 2.7339& 1.873& -4.6554& -1.9204\\ {}-2.3822& 3.1975& -2.894& -2.1243& -6.7652& 1.6242& 3.8816& 0.41683& -2.9673& 2.6467& 4.6923& -3.4879\\ {}0.31545& 1.1402& 0.21657& 1.2344& 4.6941& -0.43117& 1.9072& 0.25659& 1.9737& -5.0242& -6.0346& 1.0776\\ {}2.6388& 6.633& 0.57383& -0.40435& 4.3536& 2.5118& 9.6553& -1.3428& 2.8852& -8.4108& -2.802& -3.8271\\ {}7.7157& 6.3849& -4.1712& 1.3367& 0.13122& 2.649& -2.7644& -3.5906& 3.5364& 6.7589& -2.332& -1.7197\\ {}-12.1882& 0.66339& 6.865& -0.065198& -6.2056& 1.1504& -7.9341& -1.0004& 1.2303& 0.51055& 5.7287& 4.174\\ {}13.211& 1.4139& -2.1939& 1.2398& 2.3873& -2.4558& 7.2297& 2.8206& 7.6622& -6.4518& -8.8091& -0.35605\\ {}4.0242& -3.912& 0.17356& 1.8337& 5.7274& 9.1018& 6.9179& -5.3676& 1.3547& 1.2528& -1.7182& 11.8254\\ {}8.6342& -1.9964& -8.5127& -2.57& -7.4611& 1.3305& 2.968& -1.098& -2.9124& 3.2901& -2.7158& 5.1096\\ {}1.8004& -5.711& 1.3356& -0.64724& -7.3865& -2.7065& -0.049451& 2.2235& 0.28367& -5.7362& -1.3295& -2.2794\\ {}1.3055& -8.0506& -1.4746& -4.0037& 2.207& -7.8095& -2.418& 5.267& -2.6655& -1.4461& 3.6883& -8.5931\\ {}-0.76732& 3.3217& -1.6655& -3.2034& 3.8284& -0.80539& -3.6202& 0.11302& 0.34143& 2.269& 2.6533& -4.565\end{array}\right)\\ {}{iw}_{\left\{3,2\right\}}=\left(-7.2558\kern0.5em 8.2687\kern0.5em 9.1859\kern0.5em 4.6823\kern0.5em 7.1299\kern0.5em 11.3354\kern0.5em 4.6121\kern0.5em -16.6478\kern0.5em 8.8137\kern0.5em -10.5927\kern0.5em 5.4952\kern0.5em -10.5162\right)\\ {}{b}_{\left\{1\right\}}=\left(\begin{array}{l}-9.5135\\ {}-6.1122\\ {}1.8874\\ {}2.3953\\ {}4.2362\\ {}-0.21758\\ {}-5.9346\\ {}4.5738\\ {}-5.7288\\ {}-3.0885\\ {}10.7221\\ {}-8.9511\end{array}\right)\\ {}{b}_{\left\{2\right\}}=\left(\begin{array}{l}-10.2081\\ {}0.086536\\ {}4.5056\\ {}-0.51204\\ {}-8.6412\\ {}1.7373\\ {}0.11254\\ {}-5.564\\ {}3.3595\\ {}7.221\\ {}7.3636\\ {}1.8576\end{array}\right)\\ {}{b}_{\left\{3\right\}}=\left(-3.4709\right)\end{array}} $$

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Zendehboudi, A., Saidur, R. A reliable model to estimate the effective thermal conductivity of nanofluids. Heat Mass Transfer 55, 397–411 (2019). https://doi.org/10.1007/s00231-018-2420-5

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