_{2}/Water Nanofluids in the Framework of a Machine Learning Approach

_{2}/Water Nanofluids in the Framework of a Machine Learning Approach

_{2}/Water Nanofluids in the Framework of a Machine Learning Approach

In this study, comparing multiple models of machine learning, a multiple linear regression (MLP), multilayer feed-forward artificial neural network (BP) model, and a radial-basis feed-forward artificial neural network (RBF-BP) model are selected for the optimization of the thermal properties of TiO_{2}/water nanofluids. In particular, the least squares support vector machine (LS-SVM) method and radial basis support vector machine (RB-SVM) method are implemented. First, curve fitting is performed by means of multiple linear regression in order to obtain bivariate correlation functions for thermal conductivity and viscosity of the nanofluid. Then the aforementioned models are used for a predictive analysis of the dependence of its thermal conductivity and viscosity on temperature and volume fraction. The results show that the least squares support vector machine (LS-SVM) has a prediction accuracy higher than the other models. The model predicts the thermal conductivity of TiO_{2}/water MSE = 1.0853 × 10^{−6}, R^{2 }= 0.99864, MAE = 0.00092, RMSE = 0.00104, and the viscosity of TiO_{2}/water MSE = 8.1397 × 10^{−6}, R^{2 }= 0.99995, MAE = 0.00074, RMSE = 0.0009.

Scholars and researchers have proposed several methods to improve the heat transfer of fluids [

Forecasting is a method of predicting the future based on existing information. In recent decades, artificial intelligence has advanced with the times due to the rapid development of computers. Machine learning has been widely used in engineering research, especially in predicting systems with nonlinear behavior. Several artificial intelligence-based model prediction methods exist, including artificial neural networks (ANN) [

Moreover, the large amount of data generated by experimental studies are also challenging to model with conventional techniques. In a generalized scheme of model prediction, the relationship between control factors and response variables is established. Traditional analytical methods have poor predictive power and poor coupling ability. Artificial neural networks to predict the behavior of nanofluids predicting thermal conductivity and viscosity have been the subject of several studies [

Ahammed et al. [_{2} nanofluid on thermal properties. As the doping rates and concentrations of different materials were investigated in the temperature range of

In some studies, researchers have used artificial neural network methods and described the details of the algorithms. Sharma et al. [_{3}O_{4}-MWCMT mixed nanofluids and established a model to predict the mixed nanofluids. Models using gene expression programming and adaptive neural fuzzy influence systems (ANFIS) were used to predict Fe_{3}O_{4}-coated mixed nanofluids, and GEP and ANFIS predicted thermal properties well. Ariana et al. [_{2}O_{3} water-based nanofluids. They predicted thermal conductivity data for 285 nanofluids at 0.0013–0.1 vol%, particle size 8–283 nm, and _{2}O_{3} nanoparticles to study the effects of temperature and concentration on their stability, viscosity, and thermal conductivity. The experimental results show that when _{2}O_{3}/water, SiO_{2}/water ethylene glycol (50:50), ZnO-Ag/water, and Al_{2}O_{3}/ethylene glycol nanofluids [

According to the above literature review, the majority of studies at present have high costs and little experimental data. Secondly, all machine learning methods are used to study the single thermal properties of a particular nanofluid. This method cannot compare the advantages of different machine learning methods in predicting the thermal properties of nanofluid, and it may have a different excellent effect in predicting other thermal properties. With this in mind, this paper investigates the accuracy of different machine-learning models in predicting nanofluids’ thermal conductivity and viscosity based on a small amount of data. The highlight of this study lies in designing various optimized machine learning methods to predict the two thermal properties and compare their effects. Secondly, the neural network selected in many studies can only be applied to predicting large data volumes. This study uses a small data volume model to predict nanofluids. This paper uses several machine learning models to predict the thermal conductivity and viscosity of TiO_{2}/water nanofluids with 0.25–2 vol%. In this study, the grid search algorithm and cross-validation are applied to the machine learning model selected in this paper. The model is optimized to ensure its high accuracy and wide application.

The experimental preparation of TiO_{2}/water nanofluids in the laboratory was chosen as the material for study in the paper [_{2}) has excellent optical and electronic properties, low cost, high photocatalytic activity, chemically stable lines, non-toxicity, antibacterial properties, UV protection, and environmental cleanliness. Moreover, the thermal conductivity of TiO_{2}/water nanofluid was increased by 22% compared to other essential fluids [_{2}/water nanofluid was prepared by selecting spherical TiO_{2} with a diameter of 20 nm and using deionized water as the base fluid. Choose spherical TiO_{2} particles with a diameter of 20 nm, use deionized water as the base solution, and add TiO_{2} in different proportions. Then put the liquid into the ultrasonic cell and vibrate to ensure that the nanoparticles are fully dispersed in the base liquid to prepare TiO_{2}/water nanofluids. Forty sets of TiO_{2}/water nanofluid thermal conductivity and viscosity data are shown in

In this section, we briefly introduce the selected machine learning models and the model accuracy evaluation criteria and highlight their advantages and disadvantages from a practical point of view. In this paper, we program the models using python and choose the most adaptable version3.6. The sklearn library is called to write five machine-learning models and evaluation metrics, and the matplotlib library is used to plot the images.

Artificial intelligence-based multiple linear regression and artificial neural networks are models for testing a single nanofluid’s thermal conductivity and viscosity. Therefore, in this paper, the most widely used multiple linear regression (MLR) model, the multilayer feedforward artificial neural network (BP) model, and the radial basis feedforward artificial neural network (RBF-BP) model of artificial neural networks are selected. Subsequently, two models, the Radial Basis Support Vector Machine (RB-SVM) and Least Squares Support Vector Machine (LS-SVM), which are more popular and have high accuracy at this stage, are developed.

In real-world problems, changes in the dependent variable are often influenced by several important factors when it is necessary to use two or more influencing factors as independent variables to explain the changes in the dependent variable. If this relationship is linear, the linear multiple regression model can be used to describe it. The mathematical model of linear regression is

BP model is a kind of artificial neural network multilayer perceptron, which was proposed by a scientific group headed by Rumelhart and Hinton in 1986 [

The BP neural network is improved by combining the BP neural network, which can better predict the unknown samples, and the RBF neural network, which can nonlinearly approximate any data set. RBF-BP composite neural network algorithm is a two-layer implicit layer neural network system. The RBF neural network is the first-level hidden layer, and the BP neural network is the second-level hidden layer.

Support vector machine is a supervised learning binary classification model that maps the feature vectors of the training set to some points in space, which the neural network classifies in an optimal line. Vapnik and Chervonenkis first proposed SVM in 1963, and the current version was modified by Hearst et al. [

The least squares support vector machine is a refinement and modification of the support vector machine that simplifies the solution process by solving a linear system of equations instead of the quadratic optimization problem in the SVM. The LS-SVM model consists of a regularization parameter (c) and a kernel parameter (σ^2). The kernel function defines the magnitude of the impact of a single training sample, with smaller values having a more significant impact and larger values having a minor impact.

This study uses a total of four evaluation metrics, including mean square error (MSE), root means square error (RMSE), mean absolute error (MAE), and regression coefficient (R^{2}), to evaluate the models [^{2} are mathematically by

where

It is worth mentioning that when the MSE, RMSE, and MAE values converge to 0, and the model with R^{2} is close to 1, it is considered the most accurate model. The RMSE is mainly for outliers with large deviations, while the MAE is for all individual differences for the mean. In addition to these metrics, it is sometimes necessary to consider the model’s size when searching for the best machine-learning model.

In machine learning models, hyperparameters are the parameters that must perform well. These include the number of neurons per layer and the number of hidden layers in the artificial neural network. If the hyperparameters are not selected correctly, the models will not perform well. Therefore, there are two ways to select hyperparameters: one is to fine-tune them empirically, and the other is to select different size parameters to bring into the model and pick the best ones. However, the above method requires manual debugging, which wastes much time and leads to failure to find the optimal hyperparameters. So cross-validation using grid search is the best method. The grid search is a parameter search. It is to adjust the parameters sequentially according to the set steps within the specified parameter range, train the model with the adjusted parameters, and compare the accuracy to find the best parameters.

Normalization refers to a linear variation of the initial data set, which results in a result mapped between 0 and 1. Data normalization speeds up gradient descent to find the optimal solution and improves accuracy using

Data normalization where

This section summarizes and compares the machine learning models for the selected area. It compares the accuracy of various models with different parameters, selects the most accurate model from them, and analyzes the results.

The evaluation metrics are filtered and validated by training and test set data using grid search CV to select the optimal number of neurons and parameters. All models are trained with 20 iterations and report only the best-selected model result, thus eliminating the effect of randomness on the performance of the developed models.

_{2}/water, according to the MLR sub-correlation. Hence, the design of a one-time multiple linear regression model was more suitable for predicting the thermal conductivity model.

A curve-fitting model function _{2}/water, a function of temperature and particle ratio, was fitted using a linear regression method. _{2}/water. T is the nanofluid temperature in °C and

TiO_{2} prediction data |
Database | Sensitivity accuracy analysis | |||
---|---|---|---|---|---|

MSE | R^{2} |
MAE | RMSE | ||

Thermal conductivity | Train | 1.7910 * e^{−5} |
0.98611 | 0.00332 | 0.00423 |

Test | 2.6231 * e^{−5} |
0.98611 | 0.00424 | 0.00512 | |

Viscosity | Train | 0.00089 | 0.95029 | 0.02536 | 0.02990 |

Test | 0.00098 | 0.94265 | 0.02639 | 0.03137 |

^{2}, MAE, RMSE) in _{2}/water thermal conductivity model is the most suitable model when the number of neurons in the hidden layer is up to 66. The TiO_{2}/water viscosity model is the best model when the number of neurons in the hidden layer is up to 45.

TiO_{2} prediction data |
Optimal number of neurons | Database | Sensitivity accuracy analysis | |||
---|---|---|---|---|---|---|

MSE | R^{2} |
MAE | RMSE | |||

Thermal conductivity | 66 | Train | 0.00029 | 0.78030 | 0.01350 | 0.01707 |

Test | 0.00019 | 0.75829 | 0.01223 | 0.01387 | ||

Viscosity | 45 | Train | 0.00202 | 0.88871 | 0.03186 | 0.04496 |

Test | 0.00130 | 0.92447 | 0.02721 | 0.03600 |

Based on the grid search CV, _{2}/water thermal conductivity and viscosity.

TiO_{2} prediction data |
Optimal number of neurons | Database | Sensitivity accuracy analysis | |||
---|---|---|---|---|---|---|

MSE | R^{2} |
MAE | RMSE | |||

Thermal conductivity | 10 | Train | 1.5144 * e^{−6} |
0.99886 | 0.00102 | 0.00123 |

Test | 1.7951 * e^{−6} |
0.9977 | 0.00099 | 0.00134 | ||

Viscosity | 7 | Train | 6.8713 * e^{−5} |
0.99622 | 0.00691 | 0.00829 |

Test | 2.9337 * e^{−5} |
0.99829 | 0.00415 | 0.00542 |

The RB-SVM model selects the radial basis as the kernel function and the regularization parameter (c) according to the kernel function.

TiO_{2} prediction data |
Regularization parameter | Database | Sensitivity accuracy analysis | |||
---|---|---|---|---|---|---|

MSE | R^{2} |
MAE | RMSE | |||

Thermal conductivity | 0.00138 | Train | 1.8737 * e^{−5} |
0.98587 | 0.00305 | 0.00433 |

Test | 2.3435 * e^{−5} |
0.97054 | 0.00388 | 0.00484 | ||

Viscosity | 0.03126 | Train | 0.00088 | 0.95153 | 0.02544 | 0.02967 |

Test | 0.00103 | 0.94011 | 0.02627 | 0.03206 |

According to the model parameters ^{2} = 3.24 and use grid search CV to select the most appropriate regularization parameter kcp.

TiO_{2} prediction data |
Optimal nuclear parameter kcp | Optimal nuclear parameter | Database | Sensitivity accuracy analysis | |||
---|---|---|---|---|---|---|---|

MSE | R^{2} |
MAE | RMSE | ||||

Thermal conductivity | 0.67615 | 94.7322 | Train | 2.0793 * e^{−6} |
0.99843 | 0.00108 | 0.00144 |

Test | 1.0853 * e^{−6} |
0.99864 | 0.00092 | 0.00104 | |||

Viscosity | 0.86569 | 4155.4553 | Train | 5.1781 * e^{−6} |
0.99972 | 0.00182 | 0.00228 |

Test | 8.1397 * e^{−6} |
0.99952 | 0.00230 | 0.00285 |

This study aims to determine the most accurate model to predict the thermal conductivity of TiO_{2} nanofluid using small data volumes and therefore compares the prediction accuracy of multiple models. The optimal parameter results for BP, RBF-BP, RB-SVM, and LS-SVM applied in this paper are reported in ^{2}, MAE, and RMSE.

TiO_{2} prediction data |
Machine learning | Database | Sensitivity accuracy analysis | |||
---|---|---|---|---|---|---|

MSE | R^{2} |
MAE | RMSE | |||

Thermal conductivity | BP | Train | 0.00029 | 0.78030 | 0.01350 | 0.01707 |

Test | 0.00019 | 0.75829 | 0.01223 | 0.01387 | ||

RBF-BP | Train | 1.5144 * e^{−6} |
0.99886 | 0.00102 | 0.00123 | |

Test | 1.7951 * e^{−6} |
0.9977 | 0.00099 | 0.00134 | ||

RB-SVM | Train | 1.8737 * e^{−5} |
0.98587 | 0.00305 | 0.00433 | |

Test | 2.3435 * e^{−5} |
0.97054 | 0.00388 | 0.00484 | ||

LS-SVM | Train | 2.0793 * e^{−6} |
0.99843 | 0.00108 | 0.00144 | |

Test | 1.0853 * e^{−6} |
0.99864 | 0.00092 | 0.00104 | ||

Viscosity | BP | Train | 0.00202 | 0.88871 | 0.03186 | 0.04496 |

Test | 0.00130 | 0.92447 | 0.02721 | 0.03600 | ||

RBF-BP | Train | 6.8713 * e^{−5} |
0.99622 | 0.00691 | 0.00829 | |

Test | 2.9337 * e^{−5} |
0.99829 | 0.00415 | 0.00542 | ||

RB-SVM | Train | 0.00088 | 0.95153 | 0.02544 | 0.02967 | |

Test | 0.00103 | 0.94011 | 0.02627 | 0.03206 | ||

LS-SVM | Train | 5.1781 * e^{−6} |
0.99972 | 0.00182 | 0.00228 | |

Test | 8.1397 * e^{−6} |
0.99952 | 0.00230 | 0.00285 |

It can be concluded from this that the BP artificial neural network is unsuitable for small data volume TiO_{2}/water nanofluid compared to the support vector machine. When predicting two sets of data with the RB-SVM model, the parameters fluctuate wildly, which indicates that its predictive classification of two sets of data will be biased due to fuzzy classifications of some data, resulting in poor accuracy as a result. The LS-SVM model shows better prediction accuracy when predicting both data sets, but the LS-SVM model has higher accuracy in comparison. Therefore, LS-SVM is the most accurate neural network model for predicting the thermal conductivity of TiO_{2} nanofluid with a small amount of data. In contrast, the other models are less accurate in comparison.

Nanofluids are famous heat and mass transfer materials in various fields at this stage. Thermal conductivity and viscosity are the most important thermophysical properties, and nanofluids operating temperature, volume fraction, particle morphology, and particle size directly affect their thermal conductivity and viscosity. In this study, focusing on experimental data, the effects of TiO_{2} concentration and temperature on nanofluids’ thermal conductivity and viscosity were investigated by curve fitting, artificial neural network, and support vector machine methods. We propose a simple bivariate correlation using curve fitting to show the relationship between the parameters. Then, four machine learning models are selected to predict thermal conductivity and viscosity, with temperature and concentration as input variables. Based on the MSE = 1.82 * e^{−6} and MSE = 0.4942 [

The results show that a better and more accurate model can better predict the model. For future research, it is necessary to examine the universal application of the model. This includes the influence of different input conditions on the model as well as the possibility that the model can still be applied after replacing the nanofluid. It is also necessary to develop a database of the model with high accuracy and strong applicability. As a result, this area requires further research.

Artificial neural network

Multiple linear regression

Multilayer feedforward artificial neural network

Radial basis feedforward artificial neural network

Support vector machines

Radial basis support vector machine

Least squares support vector machine

Mean square error

Root mean square error

Mean absolute error

^{2}

Regression coefficient

Temperature (

Volume fraction (vol%)

_{i}

Related variables

^{T}

Variable coefficient

Estimate

^{s}

Actual value

Regularization parameter

^{2}

Nuclear parameter

Experimental data

Forecast data

Dynamic viscosity

Thermal conductivity

This research is financially supported by the

The authors declare that they have no conflicts of interest to report regarding the present study.

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