Development of a neural architecture to predict the thermal conductivity of nanofluids

The present study proposes a comprehensive and accurate artificial neural network (ANN) model for correctly estimating the thermal conductivity (K) of an extensive range of nanofluids. The ANN model was designed using the Levenberg–Marquardt (L–M) algorithm based on 800 experimental data containing spherical nanoparticles of TiO2, ZnO, CuO, Al2O3, ZrO2, Fe2O3, Fe3O4, SiO2, CeO2, MgO, Fe, Al, Cu, Ag, SiC and diamond in various fluids of oil, ethylene glycol, water, and radiator cooling. The nanoparticle and base fluid thermal conductivity, volume fraction (0.4–0.4%), and particle diameter (4–150 nm) of the nanoparticles, and temperature (10−80 °C) were considered as effective input variables, while the thermal conductivity of nanofluid was defined as the target variable. According to the results, R and MSE using 5–13–1 topology for all data, and training sub-set were founded to be about 0.9975 and 0.000238, and 0.9976 and 0.000229, respectively, indicating the proper ability of the designed ANN model. In addition, the developed model showed an excellent ability for predicting the thermal conductivity for oil and radiator cooling-based nanofluids with MSE of 0.000037 and 0.000042, respectively. The validation of the ANN model was successfully confirmed by achieving a low error between experimental and predicted data. These findings prove the comprehensive and accurate function of the developed ANN model. An artificial neural network model using 5–13-1 topology for predicting thermal conductivity of nanofluids


Introduction
Benefiting from a high-efficiency of heat transfer medium is one of the basic needs of today's industries [1].Heat transfer media is usually composed of fluids such as water, ethylene glycol, or oil.As a result, it is important to use more efficient heat transfer fluids with higher thermal conductivity to provide proper heat delivery in industries [2].Several methods can be used to promote the heat delivery efficiencies, such as the use of microchannels and microtubes, creating more turbulence and reducing the thermal resistance of the circulating fluid [3].In this regard, the use of efficient nanofluids, as a new generation of fluids with proper potential in industrial applications, has been recommended to improve the heat exchange process [4,5].Nanofluids are colloidal suspensions consisting of nanometer-scale solid particles suspended in a base fluid [6].Many researchers have also found that dispersing nanoparticles into a base fluid can improve thermal conductivity and enhance the heat delivery rate [7].These particles are conventionally carbon nanotubes, metal particles, metal oxides, and non-metallic solid particles, whereas water, engine oils, and ethylene glycol are used as common main fluids [8][9][10].
Thermal conductivity is considered the most important parameter to investigate the potential of nanofluid to increase heat transfer so that even low concentrations of nanoparticles are quite noticeable in heat transfer performance.Therefore, knowing the thermal conductivity coefficient of nanofluid is of immense importance to present a proper heat transfer model for a nanofluid-based thermal medium.However, the experimental calculation of this thermophysical property in different operating conditions may be faced many limitations [11].Hence, presenting the proper models to predict and determine the thermal conductivity coefficient seems to be necessary [12].In this regard, the theoretical, analytical, and experimental models for the prediction of the thermal conductivity of nanofluids were widely developed in recent years [13,14].Maxwell initially presented a model based on the classical theories of two-phase mixtures which took the size, shape, and distribution of particles into consideration [15].This model suffered from a high error for non-spherical particles [16].Considering the empirical shape factor and sphericity of nanoparticles, another experimental-based model was then proposed by Hamilton and Crosser [15].The modified Maxwell model was then suggested which takes into consideration the Brownian motion and particle-fluid system thermal diffusivity [17].However, the theoretically developed models were not able to estimate and predict this thermophysical property of nanofluids properly.
Thereafter, analytical models were developed to estimate the nanofluid's thermal conductivity, which was based on the effects of some parameters such as the type, size, and shape of nanoparticles [18,19].Some of these proposed models have been derived based on macro models and did not able to show good agreement with laboratory data [20].
To overcome these problems, theories, and mechanisms were proposed that showed more suitable adaptation to laboratory results.These mechanisms often took into account the effect of Brownian motion.Recently, developing new methods such as ANN models, which are based on using artificial intelligence algorithms, has attracted the attention of researchers in this field [21].Using neural networks provides a better understanding of the thermophysical properties of nanofluids and can lead to the prediction of these properties with higher accuracy than other methods [22].An acceptable ANN model to estimate the thermal conductivity of nanofluids containing carbon nanotubes with an average absolute relative deviation of 3.26 was proposed by Papari et al. [23].Also, an accurate ANN model was designed by Ariana et al. [24] for predicting the thermal conductivity of alumina-water-based nanofluids.Ahmadlou and Azizi [25] presented a semi-experimental ANN model to predict the thermal conductivity of nanofluids using 800 different experimental data related to different fluids of water, ethylene glycol, and oil in the range of 10-71 °C, volume fractions of 0.1-1%, and nanoparticles diameter of 5-150 nm.According to their results, the R and MSE were found to be 0.993 and 0.000527.In addition, Hemmati-Sarapardeh et al. [26] introduced an accurate and intelligent model to predict the nanofluid's thermal conductivity using an extensive data set.However, it seems necessary to develop a comprehensive and accurate model to estimate the thermal conductivity of nanofluids that can cover a wide range of nanoparticles, base fluids, and operating conditions.This work intended to present a comprehensive and accurate ANN model to estimate the thermal conductivity coefficient of a wide range of nanofluids.This model was developed using the Marquardt (L-M) backpropagation algorithm based on 800 experimental data and considering effective parameters of thermal conductivity of the main fluid and nanoparticles, the volume fraction of the nanoparticles, temperature, and particle size.

Description of artificial neural network
Artificial neural networks as a part of deep learning are a set of connected units called nodes that are modeled based on the human brain.These computational systems are modern systems that can easily mimic the complex relationships between input and output and utilize this computational ability for parallel data processing.Based on this computational ability in handling complex nonlinear relations, these networks have been developed for applications such as properties prediction, simulating dynamical systems, image processing, personality recognition, language processing, stock market prediction, and weather forecasting [27].
One of the most basic neural models is the multi-layer perceptron (MLP) model, which simulates the transfer function of the human brain to solve the regression problem [28].The basic topology of ANNs has been constructed from three sequent connected layers, namely, input, hidden, and output, as well as many intralayer processing elements called neurons.A random number generator is responsible for assigning the weight value in a typical neural network [29].Each neural in the MLP network is determined through a nonlinear activation function, as indicated in Eq. (1).Except for the input neuron, each neuron receives an input vector (P i ) from each node in the previous layer, multiplies it by the corresponding weight between the neurons j and node i (W ij ) in the previous layer, and then passes the sum multiplies (P i W ij ) plus the bias of the neurons j (b j ) through the activation function to the other neurons [30].
While neurons in the hidden layer will convert into target value in the output layer through a linear activation function.The common transfer functions for the hidden layers in the MLP network are sigmoid logarithmic and hyperbolic sigmoid tangent [31].ANN can analyze and detect the accurate relationship among variables and target values in complex systems through a proper algorithm using a training process, validation, and test of data.Training process in the neural network is performed with the aim of finding the best values of weight and bias for learning relationship between inputs to outputs, according to optimizing the network performance [25].The Levenberg-Marquardt (L-M) backpropagation algorithm is conventionally used as a training function due to its suitable ability to quickly converge to an optimal solution and minimize a function [30,32].

Data collection
To present a suitable semi-experimental model for accurately estimating the nanofluid's thermal conductivity, we need a high number of experimental data, with different nanoparticles and base fluids in various conditions such as temperature ranges and volume fractions.Table 1 shows the database of used nanofluids and their experimental conditions.In this research, 800 experimental points and corresponding variables have been extracted from reported experimental results (1)

ANN model architecture
A 5-input ANN model was designed using 800 experimental data points for accurately estimating the thermal conductivity of nanofluids.The neural network toolbox of MATLAB R2019 was used to architecture this ANN model.The Levenberg-Marquardt (L-M) algorithm as a fast backpropagation tool was used for the training process and to converge the network output and the target points [31].Here, variables of K bf , K p , φ, T, and d np were presented as the first input to the fifth input, respectively.The thermal conductivity of nanofluid was also introduced as the target variable.As schematically exhibited in Fig. 1, an input layer, a hidden layer, and an output layer formed the main construction of the developed ANN model, so that the tangent sigmoid activation function and linear activation function were utilized to determine the output of the network in the hidden and output layers.Network training and validation were performed using 70% (560 experimental points) and 15% (120 experimental points) of the entire data points, and the remaining 120 unused data (15% of the entire data points) were also chosen for network testing.It should be mentioned that the data points in the sub-sets of training, validation, and testing were randomly selected and divided by the network itself.Due to the notable effect of the number of neurons in the hidden The same database was used in our previous study [33]  layer on the performance and accuracy of created network, the training process was performed using various neurons from 1 to 17.The optimal number of neurons is conventionally obtained using the trial-and-error technique.
Here, two other learning algorithms including Scaled Conjugate Gradient (SCG), and Bayesian Regularization (BR) with same architecture were used to compare and validate the developed 5-input ANN model.
Here, the statistical indexes of correlation coefficient (R) and mean square error (MSE) between estimated values and related experimental values were applied to address the accuracy and performance of the designed ANN model [30], as follows: Here, K exp,m and K pre,m demonstrate the m-th element in the laboratory values and the corresponding m-th element in predicted values by the ANN model, respectively.N represents the number of entire data.Also, K exp and K pre are the average of the laboratory data, and the mean of the values obtained from the ANN model, respectively. (2) 3 Results and discussion

ANN model using variables of K bf , K p , φ, T, and d np
To address the accuracy and performance of the presented ANN model, the results of experimental data were compared with the results of the predicted thermal conductivity using the ANN model.The variables of K p , K bf , φ, T, and d np were, respectively, defined as input variables, and thermal conductivity obtained from laboratory data was applied as the objective function in this modeling.As reported in Table 2, the number of neurons gradually increased from 1 to 17 in each step to obtain a proper and acceptable result.
Here, increasing the number of neurons up to 13 led to a higher correlation coefficient and lower mean squared error for all data.
The progress details of 5-input model including iteration, gradient, number of hidden layers and node in hidden layer are reported in Table 3.
Also, the results of the presented network using 13 neurons for all data and training, validation, and test sub-sets are given in Table 4.As is clear, the best correlation coefficient and the lowest mean square error were related to the network training sub-set, about 0.9976 and 0.000229, respectively.
Figure 2 shows the comparison between the estimated nanofluid's thermal conductivity using the presented ANN model (output) and actual data (target), for all data and subsets.Accordingly, a good fifing was observed for all data and each sub-sets, indicating proper accuracy and performance of the presented ANN model.
Also, the results of the ANN model include MSE and ± maximum of the error (max error) from laboratory data (K exp,m − K ANN,m ) for water, oil, ethylene glycol, and RC base nanofluids are reported in Table 5.As is clear, the ANN has a more appropriate ability for predicting the thermal conductivity of nanofluids based on radiator cooling, oil, and ethylene glycol.However, the most positive and negative errors from laboratory data and the highest MSE are + 0.077, − 0.063, and 0.000359, respectively, which are related to water-based nanofluids.
The error histogram graph was used to further evaluate the amount of error between values predicted by the ANN model and experimental data.The results in Fig. 3, illustrated an approximately narrow distribution of error so that most of the error between values predicted by 5-the parameter ANN model and used experimental data are less than ± 0.02, confirming the good performance and proper accuracy of the designed 5-input ANN model.The maximum and minimum errors are approximately + 0.077 and − 0.063 and include a limited amount of data.
The details of the developed 5-13-1 topology in the current ANN model including generated weights and biases in hidden and output layers are given in Table 6.These reported details are needed to predict the thermal conductivity of the nanofluid in similar conditions and systems.Here, parameters of i, j, and z represent the number of input variables equal to 5, the number of optimal neurons equal to 13, and the number of elements of the output layer equals 1.

Comparison and validation of presented ANN model
The results of 5-input ANN model were compared using two other learning algorithms of the Scaled Conjugate Gradient (SCG) and Bayesian Regularization (BR) algorithms.All 800 data point were also used in these models.In addition, these neural network models were performed using a same architecture (5-13-1 topology).Also, 70, 15, and 15% of entire data points were chosen to develop neural network model using these algorithms.Figure 4 and Table 7 represent comparison of 5-input ANN model with results of SCG and BR algorithms.As is clear, the developed ANN model indicates an almost better MSE compared to model based on SCG algorithm for both training sub-set and all data.Furthermore, the Bayesian Regularization algorithm showed a lower ability to predict the thermal conductivity coefficient of nanofluids, so that the R and MSE using BR algorithm were obtained about 0.9933, and 0.000659 for all data, and 0.9930, and 0.000664 for training, respectively.These comparison confirmed the proper performance of 5-ANN input model which is based on Levenberg-Marquardt learning algorithm.
Figure 5 indicate the comparison between 5-input ANN model (using L-M algorithm), and obtained results from SCG, and BR algorithms for all data to estimate nanofluid's thermal conductivity as predicted data versus experimental data.Accordingly, the developed 5-input ANN model shows a better fifing for all data, especially for ethylene glycol-based and oil-based nanofluids, compared to network performed by BR algorithm.These results shows the proper performance of Levenberg-Marquardt algorithm.
Also, the results of the semi-experimental model, presented by Ahmadlou et al. were compared with the proposed 5-input ANN model [25].They developed an ANN model to predict the thermal conductivity of nanofluids using 800 different experimental data points related to different nanoparticles and main fluids of water, ethylene glycol, and oil in the range of 10-71 °C, volume fractions of 0.1-1%, and nanoparticles diameter of 5-150 nm.According to their results, the R and MSE were found to be 0.993 and 0.000527.However, our developed model showed an appropriate performance to predict the thermal conductivity coefficient in nanofluids, which can cover a wide range of nanofluids with different conditions.Ali Aminian [28] also investigated and predicted the thermal conductivity of nanofluids using an ANN model over various nanofluids.They defined the parameters of T, ϕ, d np , K bf, and K p as input variables of ANN.The presented model by them was performed in the range of 0-200 °C, a nanoparticle diameter of 9-150 nm, and a nanoparticle volume fraction of 0.005-14%.The calculated correlation coefficient was about 0.9309, and the mean squared error was 3.06% for all data.
In addition, Hemmat Esfe et al. [72] proposed an ANN model to predict the thermal conductivity of nanofluids using input data of temperature, the diameter of particles, and solid volume fraction, and reported a maximum error of 2% between experimental and predicted data.While the 5-input ANN model presented in the current study, showed a better mean squared error of 0.000238 and a further correlation coefficient of 0.9975, confirming the accuracy and appropriate performance of the proposed 5-input ANN model in predicting the thermal conductivity of nanofluids.Jamei et al. [73] used genetic programming (GP) to predict the thermal conductivity of ethylene glycol-based nanofluids.The performance criteria of the GP model exhibited a correlation coefficient of about 0.950 and a root mean square error of 0.0225, while the 5-input ANN model in the current study indicated a more appropriate performance to predict the thermal conductivity of ethylene glycol-based nanofluids with correlation coefficient of about 0.9973.To evaluate the accuracy of the ANN model to estimate the thermal conductivity of nanofluids, the cross validation of the presented model was done with the test of 104 other experimental data points, which were collected from some previous research.The details of the used database for validation of the 5-input ANN model are reported in Table 8.
The results of the comparison between predicted values of thermal conductivity (output) versus corresponding actual laboratory data (target) are represented in Fig. 5.As is clear, the results of the developed ANN model indicated a proper agreement with the experimental data, confirming the proper ability of the presented model.The maximum deviation between values predicted by the ANN model and corresponding experimental data for nanofluids based on oils and ethylene glycol is less than + 1.2% and − 0.5%, respectively.However, the maximum error for nanofluids based on water is about + 3.7%, which shows an acceptable and satisfactory amount.The outcomes proved the accuracy and proper strength of the developed 5-input ANN model to predict the thermal conductivity of nanofluids that are in the limitation of reported operating conditions in the current study.These results confirms the accurate and excellent performance of novel 5-input ANN model to estimate the thermal conductivity of nanofluids.

Conclusion
A comprehensive and accurate artificial neural network model using 800 laboratory data was presented to estimate the thermal conductivity of nanofluids in this study.Here,

Fig. 1
Fig. 1 Schematic of a neural network with an input layer consisting of variables K bf , K p , φ, T, and d np

Fig. 2 5 Fig. 3
Fig. 2 Comparison between the thermal conductivity predicted by the ANN model and experimental data for training, validation, testing, and entire data

Fig. 4 7 646
Fig. 4 Thermal conductivity of predicted data versus experimental data using a 5-input ANN model (Levenberg-Marquardt algorithm), b CSG algorithm, and c BR algorithm

Fig. 5
Fig. 5 Comparison between the K nf obtained from the ANN model and experimental data for a all validation data, b oil-based nanofluids, and c EG-based nanofluids

Table 1
in previous studies.The thermal conductivity of the base fluids and nanofluids was collected and reported at different temperatures due to changes in the thermal conductivity with temperature.Several nanofluids including different spherical nanoparticles of Al 2 O 3 , SiO 2 , ZnO, TiO 2 , ZrO 2 , CuO, CeO 2 , MgO, Fe 2 O 3 , Fe 3 O 4 Cu, Al, Ag, Sic, Fe, and diamond in various base fluids of ethylene glycol, water, and radiator cooling (RC, 50% water and 50% ethylene glycol), engine oil, kerosene, transmission oil, gear oil, and TH-66 were used to design the presented ANN model.The thermal conductivity of the used nanoparticles (K p ), the thermal conductivity of the main fluid (K bf ), temperature (T), the volumetric concentration of used nanoparticles (φ), and nanoparticle diameter (d np ) were also considered as effective parameters on enhancing the thermal conductivity property in the nanofluids.The temperature range, nanoparticle volume fractions, and nanoparticles diameter in the nanofluids used to architect the current ANN model are 10-80 °C, 0.4-0.4%, and 4.150 nm, respectively.

Table 2
The results of neural networks with different numbers of neurons for total data

Table 4
The results of the developed ANN model using 13 neurons