Predicting the accuracy of nanofluid heat transfer coefficient's computational fluid dynamics simulations using neural networks

This research presents a neural network algorithm to identify the best modeling and simulation methods and assumptions for the most widespread nanofluid combinations. The neural network algorithm is trained using data from earlier nanofluid experiments. A multilayer perceptron with one hidden layer was employed in the investigation. The neural network algorithm and data set were created using the Python Keras module to forecast the average percentage error in the heat transfer coefficient of nanofluid models. Integer encoding was used to encode category variables. A total of 200 trials of different neural networks were taken into consideration. The worst‐case error bound for the chosen architecture was then calculated after 100 runs. Among the eight models examined were the single‐phase, discrete‐phase, Eulerian, mixture, the mixed model of discrete and mixture phases, fluid volume, dispersion, and Buongiorno's model. We discover that a broad range of nanofluid configurations is accurately covered by the dispersion, Buongiorno, and discrete‐phase models. They were accurate for particle sizes (10–100 nm), Reynolds numbers (100–15,000), and volume fractions (2%–3.5%). The accuracy of the algorithm was evaluated using the root mean square error (RMSE), mean absolute error (MAE), and R2 performance metrics. The algorithm's R2 value was 0.80, the MAE was 0.77, and the RMSE was 2.6.


| INTRODUCTION
Nanofluids were coined by Choi at Argonne in 1995. 1 They are colloids of solids in liquids where the size of solids ranges between 5 and 200 nm. They are called nanoparticles because of their size and particle form. Examples of nanoparticles include Al 2 O 3 , CuO, Cu, SiO 2 , TiO 2 , ZnO, and carbon nanotubes. These small particles are suspended in fluid, and the fluid is commonly used in industries to transfer heat. Examples of these fluids include water, oils, and ethylene glycol. These fluids, where the nanoparticles are submerged, are called base fluids. These colloids are preferentially suspended stably and uniformly for greater potency of the nanofluids. [2][3][4][5][6][7][8] A nanofluid model is a representation of the behavior of nanofluids. This can be in terms of their flow and heat transfer properties. One of the benefits of developing models is a reduction in the design and operational costs of industries and research groups. Besides, they lead to improvements in carrying out studies and the design of new technology or products. For example, Toyota uses models to generate car designs. The accuracy of this process can lead to good cars, or otherwise, it can be very disastrous. 9 The process of modeling involves analytical methods, 10 experimental methods, and/or statistical methods. 8,[11][12][13] At the heart of nanofluid modeling is the computational fluid dynamics (CFD) approach, which serves to approximate nanofluid behavior to a certain accuracy. This accuracy depends on the human factor, the level of discretization of the flow domain, and the underlining numerical scheme. In terms of heat transfer, the best model is one that accurately represents the thermal properties of nanofluid. This is so since we favor accurate models over inaccurate ones. There are different configurations of nanofluid as there are different needs and uses in terms of its heat transfer characteristics. The nanofluids could vary according to the size of the nanoparticles, the volume fraction/concentration, the base fluid type, the nanoparticle types, heat, and the flow conditions of their application. Many researchers have modeled nanofluids using various modeling strategies based on different nanofluid physics. Other researchers just modeled nanofluids based on trial and error. For example, some researchers have modeled nanofluid flow, which has been understood as a particulate flow, using the volume of a fluid model that is based on the mixture of two immiscible fluids. 14 Yet, they reported high accuracy in predicting the volume of a fluid model in terms of predicting the heat transfer coefficient of nanofluid. 14 Furthermore, due to the numerous simulation studies that have been conducted for nanofluid flows in different conduits, researchers can now easily identify the best strategy for simulating a nanofluid configuration of interest Artificial intelligence (AI), which has been recently tied to machine learning (ML), has been applied to solve various engineering problems like self-driving cars, facial recognition, security, and trading in the financial markets. AI has shown extreme accuracy in understanding patterns that humans may or may not notice. Hence, AI is applied to assist humans in solving problems, whether they are everyday problems or specific engineering problems. In the next decade, AI has been said to be in everything and cut through the fiber of human existence, and help in predicting the future. 15,16 Examples of algorithms that are used in AI include neural networks, support vector machines, the k-nearest neighbors, the k-means, decision trees, and ensemble methods. One method in AI is the artificial neural network (ANN). In this paper, this ANN method is applied.

| MODELS FOR SIMULATION OF NANOFLUIDS
When modeling nanofluids, it is critical to first decide whether the thermophysical properties should be considered constant (CONST) properties that do not change with temperature or variable (VAR) properties that do change with temperature. The following step is to consider the models to be used. Both steps are referred to as the selection of modeling assumptions. The various models for simulating nanofluids are presented in this section.
First, the single-phase model (SPM) is discussed thus: The basic assumption in the conventional SPM is that the nanofluid is taken as a homogeneous fluid flow with enhanced transport properties. It also assumes that the liquid and particle phases move together with the same flow velocity and are in thermal equilibrium. Bianco et al. 17 studied the developing laminar flow of nanofluid under forced convection numerically. They applied the common SPM in their study. The nanofluid they studied was the Al 2 O 3 -water nanofluid in a cylindrical pipe with constant (CONST) wall heat flux boundary conditions. The results they obtained were then compared with a two-phase (discrete-phase) model. They reported in their paper that there was a maximum deviation of 11% from each other and that the heat transfer coefficient was higher for the cases where the transport properties were assumed to vary with the fluids' temperature. Besides, they stated that the single phase with constant transport property assumption deviated from the experimental data by 17% of the maximum value. Although the experimental data was carried out with constant wall temperature settings, they gave a correction of a 20% increase in the Nusselt number concerning the constant wall heat flux boundary condition. 18 Rostamani et al. 19 were interested in the turbulent flow characteristics of nanofluids. They studied different nanoparticles such as alumina, cupric oxide, and titania with various nanoparticle concentrations. The cylindrical pipe configuration was used in their geometric setup. The control volume approach was applied to solve the governing equations with varying transport properties, and constant wall heat flux was set up as the wall thermal boundary condition. They discovered that the resulting Nusselt number from their studies agreed with the results of the correlations presented by Pak and Cho 20 and Maïga et al. 21 The limitations of the common SPM are the reliance on selecting the correct thermophysical properties' correlations, which makes this model dependent. Hence, using the common SPM ONYIRIUKA | 3391 comes with the need to have accurate correlations to represent the nanofluids' transport properties. 18,22 Second, in the dispersion model, it is assumed that sedimentation and dispersion exist together in a nanofluid flow. This is coupled with the Brownian force, the friction force, and the gravity force on the base fluid and nanoparticles. These bring about a difference in velocities between the nanoparticle and the host fluid, which is assumed to be a nonnegligible quantity. Again, the nanoparticle random motion enhances the thermal dispersion in nanofluids. This leads to higher heat flow. Mojarrad et al. 23 used the control volume approach to study the heat transfer of a nanofluid made from α-alumina nanoparticles with water as the host fluid. Their interest was in the rounded pipe's entrance region. They compared the results they got with experimental data in the open literature. They concluded that the dispersion model does well to predict the heat transfer of nanofluid despite its simplicity. Other researchers 24,25 also agree with these findings. 18 Next two-phase flows are discussed, several factors are involved in this, including forces due to Brownian motion, friction, thermophoresis (a force due to temperature gradient), and gravity, which lead some researchers to classify and treat nanofluids as two-phase flows. 26,27 This has led to analytical and empirical equations for representing mixtures of solids and liquids. Furthermore, governing equations have been developed to accommodate this kind of model: the most common two-phase models used in nanofluid studies are the Eulerian, Eulerian-Lagrangian, the volume of a fluid, a combined model of the mixture, and Eulerian-Lagrangian, and the simplified Eulerian model (known as the mixture model). 26,27 All these two-phase models have the inherent drawback of high computational cost. Hence, as we are treating the flow with two different phases, we have at least one extra equation to solve for the second phase/particulate phase, which then updates the continuous phase. It is worth noting that all the two-phase assumptions technically do not require the effective property models of nanofluids to be known. 18 From the two-phase flows considered in this study the Eulerian-Lagrangian is presented first; in this model, the dispersed phase or particulate phase is tracked in the Lagrangian frame, and the fluid phase is updated in terms of interaction with the particles represented as a source term in both momentum and energy equations. This model has a limit of a 10% concentration of nanoparticles since any value above that threshold will lead to particle-particle interactions and hence void the model's makeup. But nanofluids never really get up to that volume fraction in practice due to the diminishing effect it has on the fluid's potential as the volume fraction increases above a 5%-6% concentration. It was also noted by Xu et al. 28 and Safaei et al. 29 that the model takes a large amount of time to solve and is best at a volume fraction of less than 1%. Behroyan et al. 30 were interested in the turbulent flow regime (Reynolds' number of 10,000-25,000) and a 0%-2% concentration of copper nanoparticles. They found that the discrete-phase model (DPM)-also known as the Lagrangian-Eulerian model-was more accurate than the other two models studied, namely, the Eulerian and mixture models. The Eulerian model was found to give incorrect results, excluding volume fraction (ϕ) = 0.5%. A maximum error of 15% was observed for the mixture model. The Newtonian SPM and the DPM were the suggested models for future investigations. 18 Second, the Eulerian-Eulerian model is considered. In this model, the governing equations are solved for each phase while pressure is assumed to be equal for all the phases. Chen et al. 31 were interested in forced convection in the laminar and turbulent flow regimes. They studied the flow using both the mixture and Eulerian models. Their finding was that the mixture model led to a better agreement with the experimental results. 18 Third, the mixture model is presented. This model assumes that all phases share a single pressure, that the dispersed phase interactions are negligible, and that the phase slip is used to solve the dispersed phase equation. This model is popular for its simplicity and low cost of computation. Naphon and Nakharintr 32 studied nanofluid flow in a three-dimensional minichannel heat sink under laminar convection heat transfer. Coupled with this, they also carried out experiments to validate their model. They observed that the mixture model was in better agreement with experimental outcomes than the SPM. 18 Fourth, the volume of fluid (VOF) is discussed here. In this model, the continuity equation for the second phase is solved to get the volume fraction of all phases for the complete flow domain. To obtain the components of velocity, one momentum equation set is solved for all the phases. Average weighting is used to calculate the physical properties of the different phases in line with their volume fraction in each control volume. A study was carried out by Naphon and Nakharintr 32 where they employed the SPM, mixture model, and VOF models along with the k-ε turbulence model. They also carried out experiments to complement the study. It was discovered that the two-phase models agreed with the experimental studies, but the SPM could not predict the Nusselt number as well as the others. And they attributed this to the Brownian motion and disordered distribution of the nanoparticles in their host fluid not captured by the SPM. They carried out a grid independence search and had the highest percentage deviation of 1.02% from other grids. This means their numerical setup was not a function of the grid chosen. 18 Lastly, the combined model of the discrete phase and mixture phase is presented. This model involves the combination of the DPM and mixture models where the flow and energy equations are solved for the host fluid, and then the Brownian force and particle heat transfer are implemented, and the discrete phase is solved for only one iteration after which the particle concentration is stored, the temperature gradient is calculated, fluid properties are evaluated, and the energy equation source terms are implemented. The mixture model step is then completed. 33 Mahdavi et al. 33 had a 10% error in pressure drop by using this model. They also reported good agreement with heat transfer data from experiments.
There are a lot of other models applied to multiphase studies. Some of the important ones as applied to nanofluids are presented here.
First, the lattice-Boltzmann method was used to study nanofluids, replacing microscopic and macroscopic views with molecular dynamics. The assumption in this model is that nanoparticles are microscopically located at the lattice site and are treated according to Boltzmann's method. The advantages of the model are first the use of uniform algorithms to solve multiphase flows and second, the ability to deal with complex boundaries. This model is applied to free, mixed, and forced convection of nanofluid. Karimipour et al. 34 used the double population thermal lattice-Boltzmann model (LBM) method for a 100 nm diameter and a 2%-4% concentration of Cu-water nanofluid flow in a microchannel with heat flux boundary condition. Their results showed the applicability of the LBM in the microflow of nanofluid. The Nusselt number increases as the slip coefficient increases and the solid volume fraction decreases and this increase is found to be more significant as the Reynolds number increases. With published literature, they had an error of 0.2% and 1.9%. 34 They also carried out grid independence, which means their results did not vary with the grid size chosen and gave stable results. 18 Second, the nonhomogeneous two-component model (Buongiorno et al.'s transport model) is presented here. In Buongiorno et al., 35 seven slip mechanisms, including gravity, Brownian, thermophoresis, Magnus effect, inertia, diffusiophoresis, and fluid drainage, are studied. He discovered by dimensionless analysis that the most important ones in terms of flow and heat transfer were the Brownian and thermophoresis mechanisms. The governing equations were based on the following assumptions: negligible external forces, incompressible flow, no chemical reactions, negligible viscous dissipation, a dilute mixture (a volume fraction far less than 1%), neglecting radiative heat transfer, a local thermal equilibrium of the nanoparticles, and the base fluid. Sheikhzadeh et al. 36 studied the natural convection of an Al 2 O 3 -water nanofluid in a square cavity and made comparisons between predictions of this transport model and the homogeneous model. This comparison revealed that the transport model was more in agreement with the experimental results in contrast to the homogeneous model. 18 Third, the optimal homotopy analysis method (OHAM) is presented. This method involves the conversion of the nonlinear partial differential equations into nonlinear ordinary differential equations and is solved analytically using OHAM. In some cases, 37 in just one iteration, OHAM gives the exact solutions but depends upon selecting a forcing function, whether in full or in part. Furthermore, numerical solutions are returned in good congruency with the exact solutions. Besides, small perturbations, discretization, and linearization are not needed for OHM. Hence, the computations required are greatly reduced, as detailed in Safaei et al. 18 and Mufti et al. 37 As a way of summary, the work of Hanafizadeh et al. 38 is cited. In their study, they compared the single-phase and two-phase models using the Fe 3 O 4 -water nanofluid flowing in a circular constant wall heat flux pipe. They studied both the developed and developing regions for 0.5%-2% volume concentration in the laminar flow regime. They observed that for an increase in the Reynolds number and volume concentration of the fluid, the heat transfer coefficient averaged over the length of the geometry was enhanced. Further, they observed that increasing the number of dispersed nanoparticles in the host fluid in the developed region reduces the error of the applied numerical schemes. For low Reynolds numbers in the developing region, the increase in volume concentration of the nanofluid led to decreased accuracy of the applied numerical schemes. The reverse was the case for moderate and high Reynolds numbers. Further, the mixture model was found to have the least deviation from experimental studies within the studied volume fraction, and they suggested that the mixture model can estimate the average heat transfer coefficient in all Reynolds numbers, ranging from 300 to 1200. The paper, however, did not provide any grid independence studies. 18 Vaferi et al. 39 presented the prediction of the heat transfer coefficient using ANNs. They considered a circular tube from experimental data with different wall conditions under different flow regimes. They compared the performance of the proposed approach with reliable correlations given in different studies. They stated that their model was better in performance than the other published works. [40][41][42] Their study was focused on predicting the heat transfer coefficient of nanofluids flowing through circular conduits. This study is focused on predicting the best model to apply to accurately simulate nanofluid flow in various flow geometries, including circular tubes. Furthermore, this study shifts the focus from predicting the thermal behavior of nanofluids to focusing on deciding the best assumptions to make about the physics of the flow and heat characteristics of nanofluids of different types. The resulting algorithm can therefore be used to select the best model for simulating nanofluids' flow. By this, the accuracy of nanofluids simulation is increased. The ANN is a tool that has been shown by researchers to be highly effective in making predictions.
This study aims to come up with an algorithm that determines the best model for nanofluid representation under the most common setups and applicative uses. To the best of our knowledge, no researcher has carried out this study. 39,43,44 There exist disagreements among various researchers in the literature regarding which model is best for nanofluids. Hence, this paper also aims to harmonize the results of researchers regarding the accuracy of nanofluid models for different cases. It also seeks to aid in the design process of nanofluids by making an exhaustive search of optimal nanofluid parameters and models possible. By coming up with a neural network algorithm created from data gotten from researchers' reports on eight different models, the most suitable model for the most configurations of nanofluid, conventional setups, and applicative uses is determined. It is worthwhile to state that the paper is focused on the error in predicting the heat transfer coefficient by the different models. A recommendation of the models for different case setups to be used and their corresponding accuracy will also be available from this work.

| Method
The algorithm that determines the suitable modeling assumption, which produces the least prediction error, for carrying out CFD simulations of nanofluids for different nanofluid configurations and flow geometry has been developed. The most common conduits in the literature will be used as case studies along with the most common models for CFD simulations of multiphase flows being considered. Data on models, nanofluid configurations, and percentage errors in predicting heat transfer coefficients will be collected. The input to the algorithm will be the model types, nanofluid configurations, and the geometry of flow. The output of the algorithm is the percentage error in predicting the heat transfer coefficient for each model (the percentage error is the difference between the simulation heat transfer coefficient and its corresponding experimental heat transfer coefficient gotten from the literature that used any of the models covered in this study). Specifically, an ANN is designed to predict the model that will have the least error for a particular nanofluid simulation. The objectives of this chapter are as follows: First, collect relevant data from the literature; second, preprocess the data into a form useful for neural network development; and third, train, evaluate, and verify the neural network algorithm prediction using an unseen test data set. Finally, we present the results for commonly encountered scenarios in nanofluid simulations.

| Data collection
The data set (102 rows and 10 columns) is summarized in Table 1 and was collected from the literature. 17,19,21,26,30,31,33,41,[44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61][62][63] These pieces of literature were chosen because they used any of the various models that were of interest in this study. This study was interested in commonly used models. Each row in the data set represents a given record in the data set. And it reflects the nanofluid configuration, flow regime, experimental setup, and nanofluid models along with their corresponding percentage error between the model prediction and the experimental results. From Figure 1A The predictors were normalized to 0 and 1. This was done to control the gradients in the neural network computations so that we could obtain an optimal solution. 64 Table 1 shows the correlation plot of each variable. We can observe that the highest correlation with the percentage deviation was a positive correlation of 0.5, obtained for the property assumption variable. The others were the particle size and Reynolds number, which had a negative correlation of 0.4 each. This also implies that these variables have the most impact on the accuracy of different models in modeling nanofluid flows. The neural network was chosen due to its ability to handle these kinds of problems where the correlations of variables are low. 64 The collected data had the following features as shown in Table 2a. Furthermore, the complexity of the problem we have set out to solve requires that we consider the computational schemes used in obtaining the solutions for the different models. Hence, we present Table 2b.  Table 2b shows the different computational schemes used for each model. From this table, we can observe the model with a different computational scheme was the VOF model, which has a Pressure-Implicit with Splitting of Operators. We can assume that, in terms of the computational schemes used, the models have been properly adjusted such that the numerical error is not a dominant part of the model's prediction. We also point out that we collected simulation results from authors that used grid independence methods to show their results were not dependent on the mesh.

| Method-artificial neural network
ANN has been shown to handle difficult problems, like, handwriting recognition, facial recognition, currency trading, and self-driving cars. Although it is a very simple concept, it is based on the way the human brain functions. The basic neural network is made of neurons. For example, in the multilayer perceptron (MLP) architecture, layers of neurons are typical, with input and output layers where information flows in one direction T A B L E 1 Correlation plot of the data set collected from the literature.
(feedforward). That is, one node receives data from other nodes in the layer under it and sends data to nodes in the layers above. A detailed description of the MLP is discussed in the paper by Tangri et al. 65

| The neural network learning process
Vafaei et al. 66 studied the steam distillation process using neural networks. They used ANN and an adaptive neurofuzzy interference system (ANFIS) to investigate the distillate recoveries using a collection of data. The models were able to estimate with a minimum error in the yield of the distillates. While Salehi et al. 67  algorithm for nanofluid in a closed thermosyphon, in their study, they also made an MLP network (a backpropagation network). As stated in their paper, the MLP is termed a Universal Approximation because, with its simple structure, it can map any nonlinear input/ output interaction. It is made up of an input layer, a hidden layer, and an output layer. They found the MLP to correctly predict the experimental data, likewise the genetic algorithm neural network. Bahiraei 44 reviewed the algorithms of AI applied to study nanofluids and the potential of nanofluids with the challenges ahead in the field of nanofluids and their applications. However, they did not mention the use of AI-or ML-related tools to determine the best model for nanofluid simulation, which is lacking in the literature. They probably did not mention it because there was no paper on the subject. It can, therefore, be concluded that this application is new and novel. Also, they mentioned the most commonly used algorithms for nanofluids studies, listed as MLP, radial basis function (RBF), FUZZY LOGIC, and optimization methods, like, genetic algorithm, particle swarm optimization, artificial bee algorithm, and ANFIS. They also mentioned the various activation functions for AI in nanofluids studies, listed as sigmoid, hyperbolic tangent, inverse tangent, threshold, Gaussian radial basis, and linear functions. In their paper, they pointed out that for thermal conductivity and viscosity predictions, the Levenberg-Marquardt approach and the Bayesian-based regularization were the best. The MLP networks are made up of three different layers: the input, hidden, and output layers. Information from the predictor variables is fed into the input layer according to a mathematical procedure. They are then sent to the hidden layer where they are processed and sent to the output layer. The output neuron gives the value of the target variable. The output value of each neuron can be calculated by Equation (1): Equation (1) shows that the b j (which represents the biases) is summed with the results from the product of the information (x r ) and their corresponding weight coefficient (w jr ). The transfer function f is the function that receives the output of each neuron. They include various types of transfer functions, including the hyperbolic tangent sigmoid, RBFs, rectified linear units (ReLUs), logarithmic sigmoids, and linear. But in this study, the ReLU has been used for the input, hidden layer. The output layer is left without activation with a single unit, and it is a linear layer since this is a regression problem, and we are trying to predict a single continuous value. Its formulation is given in Equation (2):

| Description of the ANN model
The ANN model is developed from the following inputs: These inputs represent the main determinants in the modeling of nanofluids. The inputs are particle size, particle volume fraction, wall thermal condition, thermophysical property assumption, model, particle type, host fluid, and geometry. The output variable is the average percentage error of the heat transfer coefficient (h) predicted by the model used for simulation. The lack of literature data that considers nanoparticle shape to be an important factor in nanofluid simulation prevents it from being considered.
The ANN model consists of nine input variables and one target variable. It consists of one hidden layer, as Hornik et al. 68 have shown that the MLPs with only one hidden layer are universal approximators and that they do well even for small data, meaning they can accurately estimate any multivariate function if their hidden units have a nonlinear transfer function. This single hidden layer consists of 170 neurons. This number was selected based on the lowest root mean square error (RMSE) and R 2 value after 200 runs of different networks with different numbers of neurons, as shown in Figure 2A,B. However, other researchers use Equation (3). Between the size of the input layer and the size of the output layer, there should be an appropriate number of hidden neurons. Heaton 69 stated that 2/3 the size of the input layer plus the size of the output layer should be the number of hidden neurons or less than twice as many hidden neurons as the input layer should be present.
where N is the number of and α a random scaling factor, usually between 2 and 10.
The RMSprop algorithm 70,71 is used as the optimizer in this ANN model. These settings were implemented in Python using the Keras module. The schematic of a simple ANN model and the Keras neural network algorithm created is shown in Figure 3 for clarity. The neural network in Keras shown in Figure 1B was run for 1000 epochs with a learning rate reduced to a plateau and with batch normalization. The loss function was the mean square error (MSE), while the error metric was the mean absolute error (MAE). A loss function is a mathematical formula that converts an event or the values of several variables to a real number that, inferentially, represents some "cost" related to the occurrence. When MSE is employed as a loss function, this has the effect of "punishing" models more for higher errors. Similarly, when the RMSE is used, it has the same effect since they both have the square of the error. The MAE has a neutral effect when used as a loss function. An error score is a discrepancy between a person's measured or scored results and their anticipated results. It can be represented by MSE, RMSE, MAE, R 2 , mean absolute deviation (MAD), mean absolute percentage error (MAPE), and  standard error (SE). The MAE measures the average error between the predicted error and the true error in heat transfer coefficients of nanofluids. The RMSE measures the square root of the average value of the square of the error between the true error and the predicted error in the heat transfer coefficient of nanofluids. While the R 2 quantified the degree of fit between the predicted error and the true error of the heat transfer coefficient of nanofluids, The MAD represents the average of the collected data's absolute departures from its mean. An MAPE is a statistic that measures how well a modeling approach predicts the future. While for error standard deviation, the SE is a statistical model that assesses how accurately a sample distribution represents a population. In this study, the following error score criteria were applied: MAE, RMSE, and R 2 . This mix of performance criteria accesses different characteristics of the performance of the model in this study, as highlighted above.

| Performance of the ANN model
The data set was split into training, validation, and testing sets by a 70%:15%:15% split, that is, 71 of them were used for training, 15 for validation, and 16 for testing. The subset for training was used for determining the unknown weights and biases and finding the best network. The ability of the ANN model to predict the heat transfer coefficient is examined and validated by the testing data set, and this can be evaluated statistically using the MSE and the R 2 values. Equations (4)-(9) 39 depict the statistical definition of these quantities.
where n is the number of samples, h i is the observed heat transfer coefficient in the data, and h i pred is the predicted heat transfer coefficient by the ANN model. The MAE and RMSE of this model were found to be 2.6 and 0.77, while the R 2 was 0.80. Because the RMSE gives the root of the squared error, a value of 2.6 is acceptable. The MAE also gives the mean of the difference between real and predicted values, so this is also an acceptable value for the range of error seen throughout this study. An R 2 of 0.80 indicates that the true error and predicted error are well fit.

| Results and discussion of the resulting model
For the 16 test data which serve as the verification set to verify the neural network, the true percentage error of the models in predicting the heat transfer coefficient is the red solid line in Figure 4A. The green dashed line is the prediction of the neural network for each model. A good correlation can be observed, and this verifies the R 2 value of 0.80. This value implies that the algorithm can explain the variance in the data set. However, the areas of mismatch in Figure 4A indicate the algorithm prediction error. The lowest error encountered in the test data was 0.4%, which was for the combined model of discrete and mixture phases of nanofluid with a volume fraction of 1% and a 41 nm particle size in the laminar flow regime, while the largest error was 5.9% for the mixture model in the laminar flow regime for a nanofluid of 0.2% volume fraction and 20 nm particle size.
To validate the performance of the prediction with the data, the same training procedure is performed with a different division of the data into the training, the validating, and the testing, and the overall error for the trained neural network is calculated with the variance. This is repeated 81 times. The prediction performance is shown in Figure 4B. Figure 4B shows the plot of MAE for each test data set over the model trained with different randomly selected training data. It illustrates that for a σ 1 , the maximum error in the worst case was 6%. This also shows that the modeling technique is valid and suitable for the problem to which it has been applied. From the plot, the errors ranged from high values to lower values.
The model prediction percentage error is examined concerning different Reynolds Numbers. Sixteen different models, whose specific definitions can be found in the abbreviation section, are considered in Figure 5ia-d,iia-d. models, there is a significant difference between the results obtained under the constant properties assumption and the variable or temperature-dependent properties assumption. The constant properties assumption is best for all the models concerning the simulation of nanofluids. This observation is because the temperature is always changing in the variable properties case and hence the values of the related variables change during the generation of the solution for the different field variables. These changes may not be properly handled by the numerical solver, resulting in a loss of solution accuracy. There are some highly accurate models; they are dispersion, Buongiorno, and DPM. They all have an accuracy within 5%, with the dispersion model being the best for both constant and temperature-dependent property assumptions. The DPM model seems to do better in turbulent flow than in laminar flow, while the Buongiorno models do better under laminar flow conditions. This is because the DPM model caters to turbulent eddies while Buongiorno's model makes assumptions and does not agree that all turbulent features are important for nanofluid modeling.
Similarly, Figure 5iia-d shows no significant difference from Figure 5ia-d. This is because the wall boundary conditions have no real difference under fully developed flows. That is, regardless of the wall boundary condition, constant heat flux, or constant wall temperature, the results of heat transfer are the same for fully developed flow. So, henceforth, only constant wall heat flux calculation is considered. Figure 6A-D shows the plot of percentage error with particle size for all models under a range of particle sizes. As the particle size increases, all the models give acceptable accuracies. This agrees with the fact that the bigger the size of the particles, the easier it is for the models to capture the underlining physics. As the sizes grow bigger, they enter the micrometer range that has been comfortably modeled. The dispersion, Buongiorno, and DPM models do well for all particle sizes considered. They also maintain high accuracy for constant property assumptions, as shown in Figure 5ia-d,iia-d. Figure 7A-D shows the plot of percentage error with volume fractions for all models under a range of volume fractions. It can be observed that there exists an optimal volume fraction that gives the best results for all models. This range can be observed to be the 2%-3.5% volume fraction. The too-low volume fraction is as difficult to model as the too-high volume fraction. This is because the physics of the particle interactions is better captured by these models at this range than at other ranges for nanofluid modeling. The dispersion, Buongiorno, and DPM F I G U R E 6 (A-D) A plot of percentage error versus particle size for all three-dimensional pipe flow models with constant heat temperature. BUO, Buongiorno; CDM, combined model of discrete phase and mixture phase; CONST, constant thermophysical properties assumption; DIS, dispersion model; DPM, discrete-phase model; EUL, Eulerian-Eulerian model; MIX, mixture model; SPM, single-phase model; VAR, variable thermophysical properties assumption; VOF, volume of fluid. ONYIRIUKA | 3405 models do well for all volume fractions considered. They also maintain high accuracy for constant property assumptions, as shown in Figure 5ia-d,iia-d.

| CONCLUSION
The objective of this study was to find the best model to simulate nanofluids under different conditions and give a summary of their strengths and weaknesses regarding nanofluid representation. And, to come up with an algorithm to quickly test out the best models and assumptions out of the plethora of models that should be applied for a particular nanofluid simulation setup. The neural network algorithm showed a good fit. It was observed from the study that the algorithm had an 80% fit between true error and predicted error.
From this study, the major findings are that the dispersion model, Buongiorno model, and DPM can simulate a very wide coverage of nanofluid configurations with high accuracy. They were found to be accurate for all Reynolds numbers (60-59,300) considered in this study; all particle sizes (20-150) and volume fractions ranged from 2% to 3.5%. The SPM is not a good model for nanofluids and should only be used where there is no other option. It is the suggestion from this work that the mixture model and the SPM should be improved by incorporating accurate nanofluid properties into the model.

CONFLICT OF INTEREST STATEMENT
The author declares no conflict of interest.

DATA AVAILABILITY STATEMENT
All data used were sourced from the open literature, and there are references in this material anywhere they appear.