Introduction

Recently, an attempt has been made to improve the efficiency of heat transfer fluids by adding inert solid particles (e.g. alumina) to the fluid for heat transfer applications. If the added particles are at nano size, the fluid can be recognized as nanofluid. These nanofluids are novel heat transfer fluids that are made by dispersion of particles in nanometer sizes in a base working fluid such as refrigerants, oil, water, etc. Utilizing the metallic nanoparticle suspension (such as copper, silver, silicon, and aluminum) has shown more augmentation of the nanofluid conductivity than their conventional base fluids1,2,3,4. So, the nanofluids have made a promising area for increasing the heat transfer efficiency in many kinds of applications. Comparing micro- and milli- sized particles, the suspension of the nanoparticles could minimize the drawback of clogging5.

The mentioned advantages of the nanofluids as working fluids took the attention of researchers and as a result, a progressive trend has been made in heat transfer investigations for nanofluids both experimentally and computationally as well. The numerical study conducted in6 deals with assessing the thermal and hydrodynamic behaviors of completely established turbulent flow of alumina + water nanofluids. Behroyan et al.7,8 analyzed the importance of the single and two-phase modeling on the numerical investigations of the nanofluid convective flows. They reported that the single-phase model could be considered for modeling the nanofluid flow if the thermos-physical properties of the nanofluid are adopted precisely. Bahmani et al. 9 have modeled the heat transfer of the turbulent flow of a nanofluid in a heat exchanger. The increase of the outlet and the wall temperatures by the nanoparticle volume fraction was reported in their study. Bianco et al.10 numerically simulated and reported the laminar convection of the alumina water based nanofluid in a channel with a rectangular cross-section. It was reported that using the nanofluid, the wall of the channel is cooler than that of using the base fluid. However, the nanofluids impose more pumping power as a result of the more pressure drop than water. Benkhedda et al.11 reported the influence of solid particles shape on heat transfer efficiency as well as pressure loss. According to the results, the most rate of heat transfer was related to the nanoparticle in the blade shape, while the maximum pressure drop is for the shape of the platelet. Sharma et al.12 highlighted the importance of the nanoparticle material and the base fluid properties on heat transfer efficiency. Zainon and Azmi13 reported the heat transfer improvement for the suspension of the hybrid nanoparticles in the mixture of water and Bio-Glycol.

Computational fluid dynamics (CFD) is known as a precise and reputable approach in engineering design and troubleshooting purposes. The predicted thermal and hydrodynamic parameters of fluid flows could help engineers and researchers in the trial and error expenses of the experiments. Although the CFD could predict all variables of the fluid flow, the connections between such variables with each other could not be found by this approach. There are some fluid flow variables related to each other implicitly and the mathematical equations or functions for connecting them cannot be achieved easily. In this case, the machine learning technique of the artificial intelligence (AI) algorithm is a way for finding the data patterns and their changes. The AI algorithms cannot reach the maximum intelligence level (the best intelligence) unless the connections of all learned data are found from each other, and the optimum structure needs to be specified. Although the AI algorithms have been used for data analysis for years in different concepts14,15,16,17,18,19,20,21, a few studies have recently shown the application of the artificial intelligence algorithms in simplification of the CFD modeling22,23,24,25,26,27,28,29. The critical literature review on numerical simulating nanofluid flows shows a deep gap in the applications of artificial intelligence in combination of CFD modeling to establish hybrid simulation methodology. The most studies are simple presentations of the AI algorithm of the adaptive network-based fuzzy interface system (ANFIS) for CFD data capture dealing with fluid flow and transport phenomena30.

For addressing such a research gap, this study reports a novle methodology on the CFD simulating of nanofluid flowing in a heated pipe at turbulent regime. The genetic algorithm based fuzzy interface system, known as GAFIS, is used to learn the CFD results. This investigation tries to develop a correlation relating the temperature distribution of the nanofluid flow to fluid flow nodes position, the inlet temperature, and the turbulent kinetic energy using the GAFIS artificial intelligence. This approach would be a sample idea for finding the implicit function between the fluid flow variables.

Methodology

CFD approach

In the current work, a cylindrical tube is taken into account in a horizontal position (L = 1 m and DI = 0.01 m). Constant heat flux (85 kW/m2) is used for the tube wall. The inlet velocity of the nanofluid is 0.91 m/s, while the inlet temperature changes by different values of 300, 305, 310, 315, and 320 K. The main equations used here are7,31,32:

Continuity equation:

$$\nabla .\left({\rho }_{eff}\overrightarrow{U}\right)=0.$$
(1)

Momentum equation:

$$\nabla .\left({\rho }_{eff}\overrightarrow{U}\overrightarrow{U}\right)=-\nabla P+\nabla .({\mu }_{eff}\nabla \overrightarrow{U}-{\rho }_{eff}\overline{UU)}.$$
(2)

Energy equation:

$$\nabla .\left({\rho }_{eff}{C}_{p.eff}\overrightarrow{U}T\right)=\nabla .\left(({k}_{eff}+{k}_{turb})\nabla T\right).$$
(3)

The relevant equations for the \(k-\varepsilon\) turbulence model is written as 7,31,33,34,30:

$$\nabla .(\rho_{eff} KU) = \nabla .[(\frac{{\mu_{t} }}{{\sigma_{k} }})\nabla (K)] + G_{k} - \rho_{eff} \varepsilon ,$$
(4)
$$\nabla .(\rho_{eff} \varepsilon U) = \nabla .[\frac{{\mu_{turb} }}{{\sigma_{\varepsilon } }}\nabla \varepsilon ] + \frac{\varepsilon }{k}(C_{1\varepsilon } G_{k} - C_{2\varepsilon } \rho_{eff} \varepsilon ),$$
(5)
$$G_{k} = \mu_{turb} (\nabla U + (\nabla U)^{turb} ), \, \mu_{turb} = \rho_{eff} C_{\mu } \frac{{K^{2} }}{\varepsilon } ,$$
(6)
$$C_{\mu } = 0.09,\sigma_{k} = 1.00,\sigma_{\varepsilon } = 1.30,C_{1\varepsilon } = 1.44,C_{2\varepsilon } = 1.92$$

The temperature-dependent correlations for water properties are given as follows35,29:

Density 36:

$$\rho_{water} = 2446 - 20.674T + 0.11576T^{2} - 3.12895 \times 10^{ - 4} T^{3} + 4.0505 \times 10^{ - 7} T^{4} - 2.0546 \times 10^{ - 10} T^{5},$$
(7)

Viscosity 37:

$$\mu_{water} = \alpha 10^{{\left( {\frac{\beta }{T - \delta }} \right)}},$$
(8)

where, α = 2.414 × 10–5, β = 247.8, and δ = 140.

Specific heat 36:

$$\left( {C_{p} } \right)_{water} = \exp \left( {\frac{8.29041 - 0.012557T}{{1 - \left( {1.52373 \times 10^{ - 3} } \right)T}}} \right) .$$
(9)

The properties of the nanofluid are estimated using 35,36,37:

$${\rho }_{eff}=\left(1-\varnothing \right){\rho }_{water}+\varnothing {\rho }_{s},$$
(10)
$${c}_{p,eff}=\frac{(1-\varnothing )(\rho {c}_{p}{)}_{water}+\varnothing (\rho {c}_{p}{)}_{s}}{(1-\varnothing ){\rho }_{water}+\varnothing {\rho }_{s}},$$
(11)

For the nanofluid viscosity, the correlation suggested by Maiga et al.6 is employed.

$$\mu_{eff} = (1 + 7.3\phi + 123\phi^{2} )\mu_{water},$$
(12)

The thermal conductivity could be also obtained as Chon et al. 37 recommended:

$$k_{eff} /k_{water} = 1 + 64.7\left( \phi \right)^{0.7460} \left( {\frac{{d_{water} }}{{d_{s} }}} \right)^{0.3690} \left( {\frac{{k_{water} }}{{k_{s} }}} \right)^{0.7476} Pr^{0.9955} Re_{np}^{1.2321},$$
(13)
$$\left\{\begin{array}{c}294<T<344\\ 11nm<ds<150nm,\end{array}\right.,$$
$$Re_{np} = \frac{{\rho_{water} B T}}{{3\pi \mu_{water}^{2} \gamma }},$$
(14)

where γ = 0.17 nm is defined as the mean free path between water molecules and B is the constant of Boltzmann. This correlation is recommended for Al2O3/water nanofluid and the temperature of 294 K to 344 K.

Grid dependency test

The meshing process was carried out using the design modeller tool existing in ANSYS. Two different mesh densities were selected for testing the independence of the models from the mesh size (i.e. 1,074,537 nodes and 6,751,125 nodes). The temperatures obtaining from both mesh sizes were the same. So, all CFD simulations have been done based on the first mesh density.

CFD validation

For verification of the CFD results, the average Nusselt number of the dilute Al2O3/water nanofluid (i.e. 0.08 volume fraction) of this simulation is compared with the measured results taken from the literature. Table 1 illustrates the sample results of both studies with an acceptable agreement.

Table 1 CFD verification test.
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GAS (genetic algorithms)

Essentially, genetic algorithms are for searching in terms of natural genetics and nature’s mechanics (such as natural selection, the fittest survival)39,40. Indeed, GAS analyzes the encodings of the parameters rather than the real ones. These factor sets are decreased to some signs from an arbitrary, yet operative alphabet and the solutions are assessed in terms of the given symbol structures. A population of solution structures is maintained by GAS through the procedure; hence, they are not restricted by selecting the primary single point solution guesses. Only the encoding techniques and objective function should be defined within the programmer in the case of GAS39,40,41.

Fuzzy inference system (FIS)

FIS is based on training process, and is capable to utilize other algorithms as a trainer for example using ant colont optimization algorithm called ACOFIS and using artificial neural network as a trainer called ANFIS30. There are three types of fuzzy sets proposed and implemneted by Takagi and Sugeno 42 where the rth rule function is stated as 42,43:

$${w}_{r}={\mu }_{Ar}\left({\theta}\right) {\mu }_{Br}\left(r\right){\mu }_{Cr}(z) {\mu }_{Dr}(Tin){\mu }_{Er}(TKE),$$
(15)

where \({w}_{r}\) represents the weight of each rule in the fuzzy structure, and \(\mu\) indicates the membership functions (MF) incoming signals based on each inputs, e.g. angle (θ), radius (r), z-direction (z), inlet temperature (Tin), and turbulence kinetic energy (TKE) in this work.

The amount of firing strength is determined for each rule42,44:

$$\stackrel{-}{{w}_{r}}=\frac{{w}_{r}}{\sum \left({w}_{r}\right)},$$
(16)

where \(\stackrel{-}{{w}_{r}}\) is known as normalized firing strengths. The function of a consequence if–then rule presented by Takagi and Sugeno is applied 42.

Consequently, the function of the nodes is 42,45,29:

$$\stackrel{-}{{w}_{r}}{f}_{r}=\stackrel{-}{{w}_{r}}({o}_{r}{\theta}+{p}_{r}r+{q}_{r}z+{r}_{r}Tin+{s}_{r}TKE+{u}_{r}),$$
(17)

where \({{o}_{r} , {p}_{r} , q}_{r}\), \({r}_{r}\), \({s}_{r}\), and \({u}_{r}\) show the parameters of the if–then rules and are termed consequent parameters. We have developed and implemented this approach in our previous works for simulation of physical systems21,24,45,46,47,48,49,29.

Results and discussion

After finding the intelligence, the implicit function between the flow variables could be developed. This study is intended to show how the genetic algorithm with a fuzzy interface system (GAFIS) can be used for this purpose. The turbulent nanofluid flow of Al2O3/water in a circular tube is the case for CFD modeling. The single-phase theory (i.e. homogeneous distribution of nanoparticles in the based fluid) is considered for the nanofluid modeling. Figure 1 illustrates all steps for setup and prediction of the target variables of the GAFIS. The GAFIS trains the inlet temperature, the cylindrical coordinates (i.e. r, ϴ, and z) of the nodes in the pipe domain and their corresponding turbulent kinetic energy (TKE), as inputs, and the temperature of the nodes, as the output. The number of inputs increases step by step until the best intelligence condition is met32.

Figure 1
figure 1

Flowchart of GAFIS method.

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The fuzzy C-means clustering (FCM) is selected for generating inertia FIS in this simulation case study29. For setting FIS parameters, the number of data, the number of iterations, and the percentage of data in the training are determined equal to 24,164, 150, and 75% respectively. As the FCM parameter, 10 clusters are considered for each input. Different numbers of exponents (i.e. 2, 3, 4, and 5), as the FCM parameter, are checked during the modeling so that the best intelligence is derived. The population size as one of the genetic algorithm parameters are also checked for the values of 5, 10, 20, and 30 until the best intelligence is found. After the definition of all parameters, a genetic algorithm based on FIS begins the training of the data. The temperature of the nanofluid predicted by the GAFIS is compared by that predicted by the CFD. The more regression number, the best intelligence is achieved.

Figure 2 shows the domain of the nodes learned by the GAFIS. According to Fig. 2, the r is between 0 to 5(mm), the ϴ is between ± 2 (rad), and the z is between 0.1 to 0.9 (m). The cross-section view of the nodes is also shown in Fig. 3.

Figure 2
figure 2

Learning data which is CFD output.

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Figure 3
figure 3

The cross-section of learning nodes.

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Figure 4 shows the regression number changes by adding the inputs. The highest regression number (i.e. 0.97) is obtained by considering all 5 inputs (i.e. r, θ, z, inlet temperature, and TKE). It should be noted that by adding the turbulent kinetic energy as the fifth input the regression value increases from 0.95 to 0.98. This means that considering the turbulent kinetic energy the GAFIS reaches a higher level of intelligence by distinguishing the more difference between the learned data.

Figure 4
figure 4

Learning process with changes in number of inputs from one to five.

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According to Figs. 5 and 6, the highest regression number (i.e. 0.979) is related to the exponent of 3 and the population size of 30. So, the best intelligence can be found for 5 inputs, the exponent of 3, and the population size of 30.

Figure 5
figure 5

Learning process with changes in exponent as FCM clustering parameter (2,3,4,5).

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Figure 6
figure 6

Learning process with changes in population size as GA parameter (5,10,20,30).

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Schematic diagram of the GAFIS structure is shown in Fig. 7. The number cluster for each input, the number of rules, and the number of membership functions of the output all are equal to 10. The Gaussian function is considered for membership function in this case. The shape of the Gaussian function is schematically shown on the left side of Fig. 7. The temperatures of the CFD prediction are compared with those of the GAFIS. A high compatibility exists between both predictions as depicted in Fig. 8. Figure 9 illustrates the temperature distribution of the nodes for different inlet temperatures versus the turbulent kinetic energy and the dimensionless positions of the nodes. As expected, the TKE is zero on the wall. This is because there is not any slip velocity on the wall. The TKE is maximum in the vicinity of the wall and within the hydrodynamic boundary layer. As getting closer to the pipe centerline and far from the wall, the TKE decreases.

Figure 7
figure 7

GAFIS structure in the best result of learning.

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Figure 8
figure 8

(a) Prediction of temperature and its validation with CFD results using inputs 1 and 3. (b) Prediction of temperature and its validation with CFD results using inputs 1 and 4. (c) Prediction of temperature and its validation with CFD results using inputs 2 and 4.

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Figure 9
figure 9

The temperature distribution for different inlet temperatures versus the turbulent kinetic energy and the dimensionless positions of the nodes.

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Finally, a general formula is developed to determine the nanofluid temperature in domain depending on the inputs. According to Eq. (18), once the optimum intelligence of the model is established, the consequent factors and the parameters of the Gaussian function can be found32. Table 2 shows the general equation of the Gaussian function and its parameters (i.e. C, and σ). Table 3 indicates the Gaussian function parameters for each cluster in each input. So, there are 10 sets of data, based on 10 clusters, for each of 5 inputs. Table 4 also shows 10 sets of the consequent parameters (i.e. om , pm, qm, rm, sm, and um) for all 10 clusters. In this way, the temperature of the nanofluid can be found relating to any values of the inlet temperature and anywhere in the domain without CFD modeling.

Table 2 Gaussian membership function equation in GAFIS learning process.
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Table 3 Inputs membership functions parameters in GAFIS intelligence learning process.
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Table 4 GAFIS method consequent parameters for predicting temperature.
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$$temperature=\frac{\sum_{i=1}^{10}\sum_{j=1}^{10}\sum_{k=1}^{10}\sum_{l=1}^{10}\sum_{n=1}^{10}\left({\mu }_{1i}\times {\mu }_{2j}\times {\mu }_{3k}\times {\mu }_{4l}\times {\mu }_{5n}\right)\times \left({o}_{m}\theta \times {p}_{m}r\times {q}_{m}z\times {r}_{m}{T}_{inlet}\times {s}_{m}TKE\times {u}_{m}\right)}{\sum_{i=1}^{10}\sum_{j=1}^{10}\sum_{k=1}^{10}\sum_{l=1}^{10}\sum_{n=1}^{10}\left({\mu }_{1i}\times {\mu }_{2j}\times {\mu }_{3k}\times {\mu }_{4l}\times {\mu }_{5n}\right)},$$
(18)

In which:

$$\mu_{1i} = e\frac{{ - (x - c_{j} )^{2} }}{{(2\sigma^{2} )}},\mu_{2j} = e\frac{{ - (x - c_{j} )^{2} }}{{(2\sigma^{2} )}},\mu_{3k} = e\frac{{ - (x - c_{j} )^{2} }}{{(2\sigma^{2} )}},\mu_{4l} = e\frac{{ - (x - c_{j} )^{2} }}{{(2\sigma^{2} )}}\quad {\text{and}}\;\mu_{5n} = e\frac{{ - (x - c_{j} )^{2} }}{{(2\sigma^{2} )}},$$
(19)

Conclusion

This work was aimed to provide a facile approach to connect the fluid flow characteristics resulted from the computational fluid dynamic (CFD) predictions. The CFD is able to predict all hydro-thermal parameters of fluid flows. But there is no way to find the connections such parameters with each other using the CFD. Machine learning by the artificial intelligence (AI) algorithm could intelligently record the data patterns. However, such a pattern record does not exist for the CFD data. For this purpose, the simulation of convection of Al2O3/water nanofluid in a tube was simulated at the turbulence flow regime. The artificial intelligence of the genetic algorithm with the fuzzy interface system was used for integration with the CFD. The cylindrical coordinates (i.e. r, ϴ, and z) of the CFD nodes, the inlet temperature, and the turbulent kinetic energy (TKE) were learned as inputs to predict the temperature of the nodes in the system of interest. Different input numbers, population sizes, and exponents were used in order to get the best intelligence of GAFIS. The results discovered that the best intelligence of GAFIS was obtained for 5 inputs, the population size of 30, and the exponent of 3. Adding the turbulent kinetic energy as the fifth input the regression value increases from 0.95 to 0.98. This means that considering the turbulent kinetic energy the GAFIS reaches a higher level of intelligence by distinguishing the more difference between the learned data. At the best intelligence, the predicted temperatures by the GAFIS were the same as those predicted by the CFD. The regression number for this condition was around 0.98. Then, using the GAFIS, a correlation was developed to relate the temperature of the node to the inputs (i.e. cylindrical coordinates, inlet temperature, and TKE).