Prediction of Nanofluid Characteristics and Flow Pattern on Artificial Differential Evolution Learning Nodes and Fuzzy Framework.

A combination of a fuzzy inference system (FIS) and a differential evolution (DE) algorithm, known as the differential evolution-based fuzzy inference system (DEFIS), is developed for the prediction of natural heat transfer in Cu-water nanofluid within a cavity. In the development of the hybrid model, the DE algorithm is used for the training process of FIS. For this purpose, first, the case study is simulated using the computational fluid dynamic (CFD) method. The CFD outputs, including velocity in the y-direction, the temperature of the nanofluid, and the nanoparticle content (Ø), are employed for the learning process of the DEFIS model. By choosing the optimum number of inputs and the number of population, the underlying DEFIS variable parameters are studied. After reaching the high value of DEFIS intelligence, in the learning step, a variety of Ø values (e.g., 0.5, 1, and 2) are reviewed. For the full intelligence of DEFIS, the velocity of the nanofluid is predicted in further nodes of the cavity domain. Finally, the velocity of the nanofluid is predicted by using the data at Ø = 0.15, which are absent in the DEFIS process.

Introduction

Open cavities have different uses, including solar thermal receivers and solar collectors with insulated strips, and are used for heat convection from prolonged layers in heat exchangers.[1−3] Basak and Chamkha[4] investigated natural convection in nanofluids confined in a cavity with different boundary conditions. Some important related works of nanofluid flow, as well as heat transfer in cavities are investigated in ref (5−7).

Numerical investigations on open cavities were commonly carried out by many researchers. Scientists assessed gaping isothermal square cavities using an aspect ratio of 1.[8,9] They performed a numerical investigation on square geometry with adiabatic bottom and up walls and the isothermally heated side.[3,10] Scientists evaluated normal laminar convection in inclined open superficial openings within the inclination between 0 and 45° with an interval of 15°.[11−13] It was reported that the angle of the hot plate could change the heat transfer, and also volumetric flow in the domain.[14] In an experimental analysis, an inclined open square enclosure is assessed, and the same results are found with preceding studies.[15] The cooling is one of the most dominant usages of the open enclosure. The fluids like water and ethylene glycol are conventionally utilized for heat transfer uses having a relatively low thermal conductivity. They are not able to act as effective heat transfer agents.[16,17] Using nanoparticles is identified as an effective way to improve the main fluids’ thermal conductivity. Fluids possessing suspended nanoparticles are titled nanofluids. Hence, nanofluids at a very low nanoparticle concentration involve an anomalous high thermal conductivity.[18,19] Numerous experimental, theoretical, and numerical studies were carried out on nanofluids’ natural convection flow in various forms. The buoyancy-motivated heat transfer improvement in a two dimensional (2D) geometry was numerically assessed by applying nanofluids.[20] Putra et al.[21] evaluated the nanofluids’ heat transfer features in natural convection within a horizontal cylinder cooled and heated from both sides. It is reported that for forced convection, or dissimilar conduction, a certain and systematic decline in natural convection occurred in the existence of nanoparticles, and this decline relies on the concentration, density of particles, and the cylinder’s aspect ratio. Kim et al.[22] systematically studied the convective instability compelled by nanofluids’ heat transfer and buoyancy features using theoretical simulations. The SiO2/water nanofluid’s impacts are assessed on the pure fluid. The nano-fluid’s thermal conductivity was analyzed based on the theoretical formulations and the experimental findings on natural convection for the laminar regime within a square geometry.[23] They numerically simulated natural convective heat transfer in an open area possessing two thin vertical heat sources exposed to a nanofluid.[24]

Currently, powerful numerical techniques for computational fluid dynamics (CFD) are utilized for simulating the 2D and 3D frost layer’s growth, providing detailed data on frost features.[25−27] Although an appropriate CFD arrangement can precisely predict the growth of the frost layer, the procedure is challenging and time-consuming. Furthermore, the CFD requires computer resources for performing these complex multiphase modeling. Thus, while the innovative experimental methods and CFD instrument are very dominant for exactly calculating the frost layer thickness, considering the problems of these approaches, like being costly, complicated, challenging, and time-consuming, employing the intelligence process can be an appropriate alternative for solving the multiple problems. In the literature, a small number of intelligent techniques were used to approximate the thickness of the frost layer.[28] The support vector machine (SVM) method was offered by Cao et al.[29] to model the thickness of the frost layer over the cold flat plates. Scientists proposed different intelligent methods, such as multiple linear regression, artificial neural network (ANN),[28] adaptive neuro-fuzzy inference system (ANFIS),[30,31] and least squares SVM to predict the hydrodynamic parameters of the CFD simulation cases.[32−34] They stated that the best results were obtained using the ANFIS method. In another investigation, a multilayer perceptron-ANN was used, to approximate the thickness of the layer and frost density on parallel and horizontal surfaces, and the findings were compared to those of the recognized models.[35] Here, the input factors included air velocity and temperature, wall temperature, time, and relative humidity.

The evolution of soft computing methods for different multiphase flow problems and various physics has shown that machine learning can stand beside complicated numerical techniques to explore optimum values in physics. In this case, machine learning with the results of computational nodes can simulate physics and, in short, computational time can find optimum values in the domain. In previous works, the ANFIS method was extensively used to mimic the physics and flow characteristics, but still, there is a lack of information about different learning methods besides the fuzzy structure mechanism. For example, a differential evolution (DE) learning framework can be an excellent candidate to examine the learning stage in machine learning methods instead of other conventional learning algorithms, such as the neural network. This method has been investigated in previous work, and the potential part of this method in hybridization and accuracy was shown, and it can be defined as a powerful mathematical toolbox and adaptable evolutionary optimization method for the continuous parameter spaces.[36−38]

To the best of our knowledge, differential, the evolution learning method, and fuzzy system have not thoroughly been examined to simulate flow characteristics and thermal distribution in the domain. This study can be a case study for simulation of the thermal dataset with the machine learning method, and it can be extended for complicated thermal problems.

In this research, first, a nanofluid cavity is simulated using a CFD method. The nanofluid velocity and temperature in different nanoparticle volume fractions are obtained using the CFD simulations. For training, 60% of the data are used for the learning algorithm, and after the training method, all the data sets (i.e., 100%) are compared with the CFD data. All data have been normalized as the input and output matrix, and they are used during either the training or testing process. These prediction criteria enable us to evaluate the ability of the method during the prediction time, and also the testing period. After the evaluation of the model in the testing period, enough understanding of the model process can be obtained. Hence, the method is able to predict other conditions where there are no data set. The important note is that the prediction of the process is valid when the thermal and flow regime is unique. By changing the flow regime in the domain, the new training process should be considered. The combination of machine learning and the CFD can basically define a new way to optimize the process, which represents an optimization of the local value for each computational node.

Results and Discussion

Natural convection of the copper–water nanofluid is simulated using the CFD method. The results of this simulation, such as the nanofluid velocity in the x and y directions, the nanofluid temperature for different nanoparticle volume fractions, are extracted. The CFD process needs a long computational time and also complex equations are need to be solved. To avoid the limitation of numerical methods, artificial intelligence (AI) methods are employed to solve the physical process.

This method is called a differential evolution-based fuzzy inference system (DEFIS). DEFIS is used for the global optimizing method over continuous domains. This method is used for the prediction or the optimization process. Learning data set through DEFIS can improve the prediction process in a low computational time. The CFD is implemented in the MATLAB code using the single processing code. Figure shows the CFD code and thermal/flow analysis of the domain. The CFD outputs are considered as inputs and outputs of the DEFIS method. In this way, x, y (i.e., the locations of the nodes in the nanofluid domain), Ø (e.g., 0.5, 1, and 2), and the nanofluid temperature are selected as the inputs, while the nanofluid velocity in the y-direction is selected as the DEFIS output. FIS and DE have some parameters that need to be set before starting the DEFIS process.

Figure 1

CFD code and thermal/flow analysis of the domain.

Table shows subtractive clustering and DE algorithm parameters including the no. of data, the no. of inputs, the maximum epoch, the percentage of data used in the training process (P), the number of population (DE algorithm), the type of subtractive clustering, the cluster influence range (CIR) as a parameter in subtractive clustering. Determining DEFIS’s variable parameters and considering x and y as inputs and the nanofluid velocity in the y-direction as the output, the DEFIS training process is started. For increasing the accuracy of the DEFIS prediction, the number of population is changed to different values of 5, 15, 25, and 35. The DEFIS training process is repeated for these values. Figure shows that the training and testing root mean square error (RMSE) values are 0.0120 and 0.0119, respectively, when the no. of population is equal to 5. Increasing the no. of population to 35, the RMSE values for both training and testing are decreased. No increase is seen in DEFIS intelligence. Changing the number of inputs from 2 to 3 and adding Ø as the third input, the learning process that included training and testing steps is repeated for a variety of no. of population.

Table 1

Subtractive Clustering and DE Algorithm Parameters

subtractive clustering parameters[39]	CIR	0.2, 0.3, 0.4, 0.5
	data scale	“auto”
	squash factor	1.25
	accept ratio	0.5
	reject ratio	1.5
DE algorithm parameters	number of population	5, 15, 25, 35
	cross over	0.5
	lower bound of scaling factor	0.2
	upper bound of scaling factor	0.8

Figure 2

(a) DEFIS training processes RMSE error with changes in the number of population and number of inputs. The detailed information about evaluation criteria, and the comparison of the CFD findings and the AI results can be found in the Supporting Information. (b) DEFIS testing processes RMSE error with changes in the number of population and the number of inputs. The detailed information about evaluation criteria, and the comparison of the CFD findings and the AI results can be found in the Supporting Information.

According to Figure , the RMSE value for the training and testing process decreases, when there are three inputs in the DEFIS learning processes. The training and testing RSME values are 0.0110 and 0.0113, respectively. This means that there is 9 and 5% reduction in the RMSE values of the training and testing, respectively. A high level of intelligence is achieved by increasing the number of population to 15, 25, and 35. The results revealed that the increase in the number of population makes a little growth in DEFIS intelligence. At this level, the nanofluid temperature is added as the 4th input, and the training/testing process for a different number of population is carried out. The RMSE value decreases significantly for both testing and training steps. For a population of 5, the training and testing RMSE values are 0.002892 and 0.002894, respectively. Changing the number of the population from 5 to 15, 25, and finally 35, the DEFIS intelligence is positively affected.

In Figure , correlation determination (R2) for the training step is 0.99774, and R2 for the testing is equal to 0.99729. Therefore, DEFIS intelligence is more than 99%. By using this capability of AI, the nanofluid velocity in the y-direction is predicted, as indicated in Figure . The prediction surfaces in Figure let us predict more nodes in the first domain. In other words, surfaces are predicted by using 7500 nodes data in the learning process. These surfaces included more than 100,000 node data; also, there is a capability to increase the number of node data to more than 100,000 node data.

Figure 3

(a) DEFIS training data. The number of inputs is four, the number of population = 35. (b) DEFIS testing data. The number of inputs is four, CIR = 0.2, and the number of population = 35.

Figure 4

DEFIS prediction, variety number of input, CIR = 0.2, number of population = 35, output1 is the nanofluid velocity in the y-direction, input 1 is the x-direction, input 2 is the y-direction, input 3 is the percentage of nanoparticles in the nanofluid, and input 4 is the nanofluid temperature.

In learning processes, Ø= (0.5, 1, and 2) is used as the input. In this study, the nanofluid velocity in the y-direction when Ø = 1.5 is predicted, and these data are absent in learning processes. Figure shows a comparison between the CFD output (absent in learning processes) and the DEFIS prediction of the nanofluid velocity in the y-direction. The 2D and 3D plots in Figure show that there is a perfect correlation between the CFD output and DEFIS prediction. Figure shows the quiver and contour plots of nanofluid velocity in the y-direction with DEFIS prediction in Ø = 1.5, which is also a lucrative feature of the DEFIS method to predict data in nodes without using CFD outputs.

Figure 5

2D and 3D comparison of velocity in the y-direction between the DEFIS prediction and the CFD output, Ø = 0.15.

Figure 6

Comparison of the velocity direction between the DEFIS prediction and the CFD output, the x-axis is the x-direction, and the y-axis is the y-direction.

The initial population values are based on the sensitivity study and the appropriate prediction values. In machine learning methods, there are several tuning parameters such as the population values, which change the prediction behavior. This prediction behavior should be tuned based on the exact physical behavior. Hence, all initial values can be changed when the flow or thermal regime is changed. Indeed, the constant population values for each physical regime can be observed, and it is found that the population equal to 35 is the best.

The prediction of the process is valid when the thermal and flow regime is unique. By changing the flow regime from laminar to turbulence, a new training process should be considered with new information. For example, more complex turbulence behavior such as the eddy length scale or turbulence spectrum requires the finer training method. The combination of machine learning and the CFD can basically define a new way to optimize the data set, which shows the optimization of the local value at each computing mesh. The method of learning can only learn the data set and mathematics behind that data. This method itself cannot feel physics, and we can consider it as a “non-sense” training method and prediction framework. However, this nonsense learning section can easily avoid numerical instability and convergence issues. Therefore, moving from different flow regimes that represent new physics can be a stopping point for the prediction process.

During the training method, we consider several volume fractions of nanoparticles that represent the influence of nanomaterials on thermal behavior. Adding more data set in the training system can only increase the training time and not accuracy. The machine learning method can specifically visualize different fractions of nanomaterials based on their own understanding, which is achieved through the learning framework. This method can represent the local value flow or thermal characteristics after the training section.

Conclusions

In this study, natural convection of the copper–water nanofluid inside a cavity is simulated using the CFD method. A kind of AI called the DE algorithm is used in the training step of the fuzzy inference system (FIS) method and created a DEFIS method. DEFIS intelligence is evaluated using the data extracted from the CFD method. After achieving a full DEFIS intelligence, the DEFIS method predicted the velocity of the nanofluid in the y-direction in more than 100,000 nodes. This study’s achievement is the velocity prediction of the nanofluid for Ø equal to 0.15. There was no information about the cavity when Ø = 0.15. As a result, the DEFIS method predicted the velocity of the nanofluid in the y-direction in a situation that the CFD outputs are not considered as the DEFIS input parameters. For future studies, carrying out the investigations changing some parameters such as the percentage of data in the training process (P), the CIR, and the maximum epoch and influence of these parameters in the DEFIS intelligence is recommended.

The prediction of flow in the domain by machine learning can be connected with physics and the existing data set. Changing the regime of flow in the machine learning model without having numerical or experimental data set in the learning process cannot be a good representative of the prediction tools for that regime. However, this type of modeling is always an assisting tool to probe into the particular process and optimize that process. Machine learning methods can also find the meaningful correlation between many input and output parameters, which is very difficult in conventional statistical models. When they contain a high rate of R2 and they mimic the fluid flow pattern, including velocity and thermal distributions, they can accurately predict the process within the domain of the training data set, and the mathematical correlation of AI can be trustable for future studies.

Computational Methods

CFD Method

A square cavity is used in a simulation in which the cavity’s right and left vertical walls are subjected to constant temperatures, and their various nondimension values are 1. There is a lower temperature on the left wall compared to the right wall, while there is no heat transfer in the top and bottom walls (see Figure ). The solid walls are set to constant circumstances, and the bottom and topsides are maintained in adiabatic circumstances. The nanofluid includes Cu–water. CIP is used initially in this work to minimize the numerical diffusion for a high-order Navier–Stokes equation in 2D problems.

Figure 7

Boundary conditions of the cavity. No heat transfer in the top and bottom walls, left boundary is cold wall and the right wall is hot.

Velocity and energy equations were defined in terms of dimensionless analysis where the thermal diffusivity is given as follows[40]

A fluid’s operative density with suspended particles at a given temperature can be expressed as[39]where ρf, φ, and ρs represent the pure fluid density, suspended particles’ volume fraction, and particle density, respectively. Brinkman offers the operative viscosity of a nanofluid consisting of water possessing the viscosity and a dilute suspension of solid, small, and spherical particles[41]

Wasp initially proposed the operative stagnant thermal conductivity for the solid–liquid mixture[42] as follows

The CIP method is utilized for obtaining solution of the advection term to solve the velocity, and further details on the numerical scheme are reported in ref (40).

Fuzzy Inference System

FIS is a common calculating scheme in terms of the conceptions of fuzzy reasoning, fuzzy set theory, and fuzzy If-Then rules.[43,44] Three distinct sorts of fuzzy reasoning exist for which Takagi and Sugeno established If-Then rules in the FIS architecture.[45] In this work, x direction, y direction, percentage of nanoparticles in the nanofluid, and nanofluid temperature are used to achieve the velocity of the nanofluid in the y-direction as the output. In this approach, the function of the ith rule iswhere w represents an outcome signal of the node of the 2nd layer, and μA, μB, μC, and μD show incoming signals from the applied membership functions on inputs, x coordination (x), y coordination (y), percentage of particles in the nanofluid (Ø), and nanofluid temperature (t) to the node of the second layer.

In the third layer, the comparative value of the firing strength of each rule is estimated (see Figure ). This corresponds to each layer’s weight over the overall quantity of firing strengths of all rules[39]where shows the normalized firing strengths.[45]

Figure 8

Schematics of the FIS layers with two inputs.[46]

Therefore, the node function can be explained as[31,32,47,48]where p, q, r, s, and l show the parameters of If-Then rules and are known as the resulting elements.[46]

The RMSE equation and the coefficient of determination (R2) can be defined as[46]

In the above equations, N is the number of test data.

DE Algorithm

DE was initially recommended by Price and Storn,[49] and includes three control search factors. There are different methods, such as binominal and exponential crossover. In this work, binominal crossover is used for the whole study. The DE algorithm was used in the FIS training process to achieve a high level of FIS intelligence. Detailed information on the DE algorithm can be found in numerous articles,[50−54] where suggested value ranges are provided for CR, the crossover control parameter, F, the mutation control parameter, NP, and the population size.[55]

Through the introduction of two novel parameters (tolerances) and considering the differences in the population, the crossover control parameter and the value of the mutation parameter may be altered for improving the algorithm’s efficiency and the solution quality.[56]

A self-adaptive method for the mutation increase, DE factors, and the crossover factor is provided in ref (50). The effect of three control factors on the action of DE is confirmed separately by performing tests on functions of the test in ref (52), revealing that the techniques of enhancing the efficiency and strength of DE can be obtained by discovering a more efficient method for adjusting the values of control search parameter of DE.