Comprehensive study and scientific process to increase the accuracy in estimating the thermal conductivity of nanofluids containing SWCNTs and CuO nanoparticles using an artificial neural network

This investigation aimed to evaluate the thermal conductivity ratio (TCR) of SWCNT-CuO/Water nanofluid (NF) using experimental data in the T range of 28–50 ℃ and solid volume fraction range of SVF = 0.03 to 1.15% by an artificial neural network (ANN). MLP network with Lundberg-Marquardt algorithm (LMA) was utilized to predict data (TCR) by ANN. In the best case, from the set of various structures of ANN for this nanofluid, the optimal structure was chosen, which consists of 2 hidden layers, the first layer with the optimal structure consisting of 5 neurons and the second layer containing 7 neurons. Eventually, for the optimal structure, the R 2 coefficient and MSE are 0.9999029 and 6.33377E-06, respectively. Based on all ANN information, MOD is in a limited area of − 3% < MOD < + 3%. Comparison of test, correlation yield, and ANN yield display that ANN evaluates laboratory information more exactly.


Introduction
Nowadays, polymer nanofibers and metal oxides are prepared with the help of electrospinning (E-spin) techniques.Nanoparticles (NPs) produced by the E-spin technique are used in fields such as sensor development, decontamination, energy storage, biomedicine, nanofluids and catalysts, etc. [1][2][3].The suspension liquid, in which the constituent solid particle is less than 100 nm is nanofluid (NF) [4].Due to the improved properties of the dispersion of nanoparticles in liquids, the published papers in the past two decades in the field of NFs are increasing sharply.The thermal conductivity (TC) that is acquired using adding nanoparticles to the base fluid [5][6][7][8][9][10], the basis for many innovations is in heat transfer intensification [11][12][13].This is used in various parts of the industry such as cooling and heating, electricity generation, and transportation [14][15][16][17][18][19].The impact of NF on heat inverters can be reducing the size of the heating system and reducing the amount of fluid in circulation [20].The thermal attributes and efficiency of NFs HT with different combinations were studied by many scientists.Table 1 lists some of the research on NF heat transfer.Esfe et al. [12] researched properties of TC, dynamic viscosity and Nusselt number of MgO/water NF.TC is increased up to 23%.
To model attributes like viscosity and TC of NFs, mathematical correlations can be presented based on the obtained experimental data.But recently, software and coding techniques such as ANNs were used for this purpose [27][28][29][30][31][32][33][34].ANN has different applications in different industries and different scientific disciplines.Some applications of ANNs are shown in Fig. 1.
Today, researchers are looking for a way to achieve the best results in the shortest time and at the lowest cost.Many studies in various scientific and engineering fields were carried out by ANN [35][36][37].Malika et al. [38] investigated the efficiency of a sonophotocatalytic reactor and the removal of toxic particles from industrial wastewater with Al(OH) 3 -MWCNT HNF suspension with Ti + 4 coating using RSM and ANN methods.The ANN with multi-layer perceptron method and R 2 value = 0.999 confirmed the success of the experimental findings.Due to the significant differences related to the properties of different nanofluids and time-consuming experimental experiments, the need for methods such as ANN became more apparent.Table 2 shows some modeling of ANNs for TC of nanofluids.
Today, the need to use the ANN to estimate and forecast the relatively complex attributes of nanofluids is not hidden from anyone.Efforts to optimize the ANN or other post-processing methods could be the basis for the next generation of research by researchers to address the widespread use of nanofluids.Different types of ANNs have targeted and imitated only a part of the learning and adaptation capabilities of the human brain, including multilayer perceptron model (MLP), radial neural network (RBF), support vector machines (SVM) and Hopfield ANN.However, one of the most basic models used for data estimation is the multi-layer perceptron network, which simulates the transfer function of the human brain.In this study, experimental data were extracted from reference [44].Therefore, Fig. 1 Some applications of ANNs in the current investigations, the focus was on increasing the accuracy and optimization of the ANN structure in estimating the TC of NFs.For this purpose, ANN training was performed using experimental data [45] on TC of SWCNT (25)-CuO (75%)/Water NF based on changes in SVF and temperature, and then ANN output with data.
Experiments were compared and its deviation was carefully evaluated.To increase the accuracy, about 400 species of different ANN structures have been studied and based on the results, the best structure was selected and its output was based on the analysis of the current investigation (See Fig. 2).The experimental data set includes 40 data, but to predict the viscosity of the nanofluid in the ANN with two hidden layers and 10 neurons in each layer, a set of 400 neural network structures is designed.The results of this research will be used to develop the available data for the exploitation of hybrid nanofluids in industrial equipment.The results show the importance of ANN optimization in estimating properties and its much higher accuracy than the classical model.

About ANN
ANNs are mathematical tools that can model complex functions by mimicking the behavior of biological nervous networks.ANNs are utilized in various scientific courts because of their high ability to reproduce and model nonlinear processes.One of the applications of ANNs is the ability to model multivariate problems to solve complex problems [46,47].ANNs are one of the best nonlinear methods, due to the high accuracy method.ANNs are utilized to solve challenges of intricate modeling like estimating and template diagnosis.
With advances in software engineering, scientists use artificial intelligence software like ANNs to model viscosity and TC [48,49].Past research review displays that providing an experimental relationship and designing data using ANN is the appropriate method.These methods can replace laboratory methods.Therefore, in this study, for the first time, the TCR of the SWCNT-CuO (25:75)/Water nanofluid versus temperature and SVF [45] was designed by ANNs.In this method, ANNs with lots of neurons and various transfer functions have been evaluated and optimal ANN is chosen.The most common type of used ANN to solve the regression problem is the MLP [50,51].Utilized ANN in current investigation is an MLP and the utilized algorithm to train this network is LMA.Functions of sigmoid transfer were considered on neurons in the latent layers.Figure 3 displays best structure from 400 structures investigated to forecast TCR.The ANN topology was proposed in Fig. 4 to design and TCR of nanofluid.Generally, 40 experimental information was utilized for ANN training.The MLP algorithm was also used for ANN and the TCR was selected as the ANN output.
Different ANN structures were analyzed to choose the best ANN structure.Various structure's accuracy of ANNs has been presented in Table 3.The best value  of R 2 is 0.999903, which is related to the fourteenth structure in Table 3.

Results and discussion
Figure 5 displays the trend of MSE changes related to TCR in the selected structure from among the 400 ANN structures for different stages.In Fig. 5, the MSE has the lowest MSE in the test phase compared to other phases, which is 6.33377E-06 according to Eq. 1, As mentioned, a value of R 2 close to 1 demonstrates a close communication between TCR laboratory information and predicted information from ANN.According to correlation curves for different stages in 4 separate sections in Fig. 6, it can be seen that the amount of R 2 for all information in the current ANN is higher than 0.9.Regression coefficient results in Fig. 6a are more (1) Fig. 3 The best ANN Fig. 4 Flowchart to achieve the optimal ANN important for all data, which belongs to the last ( 14) proposed topology and is equal to 0.9999029.
In Fig. 7, the experimental data set and TCR data according to different temperatures and SVFs are used to model the ANN.Examination of forecasted information of ANN with outcomes of tests in 4 antiseptic parts shows that based on this figure, there is good agreement between ANN outputs and TCR tests and modeled data accurately predicts tests.
Figure 8 displays ANN adaptation outcomes obtained from training, test stages, and validation with laboratory tantamount information set in SVF = 0.03% to 1.15% through the ANN procedure.As displayed in Fig. 8, all points forecasted using ANN match experimental points that indicate high precision and suitable performance of ANNs in forecasting laboratory information.According to Fig. 8, there is a close correlation between train stages and all data, which shows predicted TCR data accuracy.
The comparison among the experimental information for TCR and the cases forecasted using ANN, simultaneously due to the changes of temperature and SVF, is better shown in Fig. 9.As you can see in Fig. 9, all forecasted points using ANN correlate well with a laboratory data point, but based on the contour of the visible changes in

Fig. 6 Regression coefficient in latent layer neurons terms
Figure 10 displays the error values in predicting TCR amounts of the hybrid nanofluid at different Ts.According to Fig. 10, the maximum error is between 0.033 and − 0.045.This displays high precision of TCR of the information predicted from the ANN model.
Figure 11 shows the histogram of TCR data errors for the three steps in the ANN.In Fig. 11, most of the data errors are near the zero line and in the ± 0.02 range.The least error with the highest frequency belongs to the training stage is − 0.001422943.In Eq. 2, the formula for computing MOD of forecasted ANN information and laboratory information is provided.These data MOD at various SVFs are displayed in Fig. 12. MOD doesn't exceed 3.1 to − 3.8.The validation stage has the lowest MOD.

Experimental Data Neural Network Output
(2) To evaluate the TCR of tNF, an experimental relation based on measured parameters for the TCR is presented [52]: In Fig. 13, two methods of data forecasting, suggested new relationships, and ANN design are compared according to forecasting laboratory information.This comparison was performed at T = 50-33 ℃ and various SVFs.As displayed, the ANN procedure has a greater ability to forecast information.With increasing SVF and T, the TCR of computational data was more distant from experimental information, but ANN information is consistent with laboratory data ANN simulation is more precise than computational data.

Conclusion
In this study, the use of an ANN to evaluate the TCR of SWCNT (25)-CuO (75)/Water hybrid NF is presented.Use of ANN was done at SVF = 0.03% and T = 28-50 ℃ to 1.15%.SVF and T were selected as input variables of ANN, while TCR values were selected as output.To design the ANN, the MLP model with LMA training with 2 transfer functions has been used.To obtain the best topology, many analyses of different structures of MLP were performed.It was found that the developed ANN provides a precise forecast of TCE amounts with R 2 and (3)

Fig. 2
Fig. 2 Choosing the best structure for ANNs

9 Fig. 7
Fig. 7 Examination of forecasted information of ANN with outcomes of tests in four antiseptic parts

Fig. 9 Fig. 10
Fig. 9 Investigation of simultaneous T and SVF effect on TCR

Table 1
Summary of research that investigated the NFs heat transfer properties

Table 2
Some modeling of ANNs for TC of nanofluids

Table 3
Characteristics of top 14 ANN structures MSE in terms of latent layer neuronsthis figure, the black square area marked on the figure has the highest correlation between the data, which is related to different temperatures and SVF, and more precisely the SVFs between 0.23% -0.95% and temperatures between 28 and 38 C make up this area.
MSE of 0.9999029 and 6.33377E-06, respectively.The highest frequency of MOD values was in the range of less than − 3% < MOD < + 3%.A comparison of the TCR test with ANN data and mathematical calculation data output shows the high capability and precision of ANN in modeling TCR information.From the comparison and analysis and examination of proposed model data compared to experimental data, it can be concluded that there is a good match between data and to save money and time, ANN can determine TCR of NFs with high accuracy.