1 Introduction

In recent decades, the role of nanotechnology in the development of engineering sciences has been significant and abundant, and there have always been many and interesting researches on various branches of nanotechnology (Zhang et al., 2015; Zhang et al., 2015; Gao et al., 2019; Dwijendra et al., 2022; Kazemi and Shiri, 2022; Gupta, 2022; Rakshe et al., 2021; Azin and Pourghobadi, 2021; Asif, 2018; Athab et al., 2015). Also, fluids, as a main component in various branches of engineering sciences, perform many applications in the field of lubrication, heat transfer, fluid transfer, etc., and research on various fluid parameters has always been one of the most important engineering challenges (Wang et al., 2016; Zhang et al., 2022; Zhang et al., 2022; Li and Zhao, 2022). In recent decades, nanoparticles play important role in all branches of sciences (Yang et al., 2017; Desai et al., 2021; Mohammed et al., n.d.). Nanofluids are prepared by dispersing the particles (oxide ceramics, polymers, glass-ceramics, nitrides metals non-metals, etc.) in nano sizes and diameters<100 nm in the base fluids including water, ethylene glycol, and oil (Hemmat Esfe et al., 2017; Alirezaie et al., 2018; Hemmat Esfe et al., 2018; Ghorabaee et al., 2019; Hemmat Esfe et al., 2017; Esfe et al., 2019; Doust et al., 2015; Esfe and Motallebi, 2019; Esfe et al., 2019; Abbasian Arani et al., 2016; Hemmat Esfe et al., 2018; Ahmadi et al., 2019; Mahian et al., 2019; Mahian et al., 2019; Hemmat Esfe et al., 2017; Rostamian et al., 2017; Alirezaie et al., 2017; Hemmat Esfe et al., 2018; Kherbeet et al., 2015). Some of the main applications of nanofluids in the industry are heat exchangers, engine coolants, electronic devices, nuclear plants, and oil recovery due to the amazing properties of these kinds of fluids (Esfe et al., 2018; Hemmat Esfe and Abbasian Arani, 2018; Esfe et al., 2019; Dastmalchi et al., 2015; Esfe et al., 2018; Abbasian Arani et al., 2012; Hemmat and Davoodi, 2019; Abbasian Arani et al., 2017; Gharibshahi et al., 2018; Abbasian Arani et al., 2014; Hemmat Esfe et al., 2017; Hemmat Esfe et al., 2017; Hemmat Esfe et al., 2017; Hemmat Esfe et al., 2017; Mansoury et al., 2019). So, it is important to identify their characteristics. One of these characteristics is ${\mathrm{\mu }}_{\mathit{nf}}$ that plays an essential role in its application in chemical and oil engineering. Viscosity is an effective parameter in fluid lubrication, pumping, heat transfer and enhanced oil recovery (Esfe et al., 2019; Ghasemi and Karimipourm, 2018; Hemmat Esfe, 2018; Hemmat Esfe et al., 2017; Hemmat Esfe and Esfandeh, 2018; Hemmati-Sarapardeh et al., 2018; Lebon and Hatim, 2018; Ruhani et al., 2019a,b; Esfe, 2015). Lots of studies were carried out on the effect of $\phi$ , temperature, nanoparticle size, nanoparticles types and base oils on ${\mathrm{\mu }}_{\mathit{nf}}$ (Aghaei et al., 2018; Hemmat Esfe et al., 2015; Hemmat Esfe et al., 2016; Hemmat Esfe et al., 2017; Hemmat Esfe et al., 2017; Abdul Hamid et al., 2018; Gholami et al., 2018). The best way of investigating ${\mathrm{\mu }}_{\mathit{nf}}$ is conducting different experiments and measurements with special apparatus, but this method is expensive and time-consuming. Since the equipment is not calibrated and is worn out, there might be some errors. For this reason, scientists are interested in predicting ${\mathrm{\mu }}_{\mathit{nf}}$ using theoretical and empirical correlations. All the proposed models have their limitations. Some of them are only used for spherical nanoparticles, and the application of most of them is limited by the temperature range (Esfe, ,2017; Izadi et al., 2018; Hemmat Esfe et al., 2018; Asadi and Asadi, 2016; Esfe, ,2017; Asadi et al., 2016). Another way for saving time and reducing the cost of experiments is using prediction tools like ANN for predicting the thermophysical properties of thermal systems working with nanofluids. ANN method was designed in the 1980 s, inspired by the human brain, and nowadays is applied in modeling and simulation of processes (Huai et al., 2021; Tian, 2021). Neurons are an important part of an ANN. Lots of researchers have worked on modeling properties of nanofluids (Zaferani et al., 2019; Hemmat Esfe et al., 2017; Esfe et al., 2016; Hemmat Esfe et al., 2017; Hemmat Esfe et al., 2018; Esfe et al., 2019; Hemmat Esfe et al., 2018; Esfe et al., 2018). Maharabi et al. (Mehrabi et al., 2013) presented a model for the prediction of ${\mathrm{\mu }}_{\mathit{nf}}$ employing an adaptive neuro-fuzzy inference system (FCM-ANFIS) based on FCM and a group of experimental data. They have chosen effective ${\mathrm{\mu }}_{\mathit{nf}}$ as the goal parameter size of nanoparticle and temperature and $\phi$ as input variables. They compared predicted viscosities with experimental data for four different nanofluids of CuO, Al₂O₃, SiO₂, and TiO₂ and in which water was used as the base fluid. Results obtained from the proposed FCM-ANFIS model fit the measured data very well. Hemmat Esfe Research Group is one of the leading groups in nanofluid studies (Esfe et al., 2015; Esfe et al., 2015; Esfe et al., 2015). They have done many studies on various nanoparticle compounds, especially in the field of hybrid nanofluids. They opened new perspectives in the optimization of nanofluids attributes and also were able to control the unwanted increase in ${\mathrm{\mu }}_{\mathit{nf}}$ after expanding NPs in BF (Esfe et al., 2018). The rheological behavior of MWCNT-MgO in oil base nano-lubricants was investigated by Alirezaei et al. (Alirezaie et al., 2016) at various $\dot{\mathrm{\gamma }}$ ; $\phi$ and temperatures. They carried out their experiments at $\phi$  = 0.0625–1 %, T = 25–50 ˚C and $\dot{\mathrm{\gamma }}=$ 670–8700 s⁻¹, and found out that ${\mathrm{\mu }}_{\mathit{nf}}$ falling with raising the temperature. In addition, the nanofluid shows slightly non-Newtonian behavior, but at higher temperatures, its behavior was Newtonian. So, ${\mathrm{\mu }}_{\mathit{nf}}$ increases with increasing $\phi$ . Changing $\dot{\mathrm{\gamma }}$ between 666.5 and 8664.5 s⁻¹ caused an increase (grow to 10 %) in ${\mathrm{\mu }}_{\mathit{nf}}$ . But increasing the temperature from T = 25 to 50 ˚C decreases the ${\mathrm{\mu }}_{\mathit{nf}}$ about 75 %. An equation with three variables was presented to evaluate the experimental data; experimental results were also modeled with ANN. Toghraie et al. (Toghraie et al., 2019) predicted the dynamic viscosity of Ag/Ethylene glycol nanofluid in different temperatures (25–50 °C) with v $\phi$  = 0.2 %–2% by ANN. They modeled an ANN by using 6 to 31 neurons, and simulated 30 times for each neuron. The model they presented has the accuracy parameters of MSE and SSE with values of 6.9456E −5 and 0.0030, respectively. In another study, Maddah et al. (Maddah et al., 2018) explored and anticipated the relative viscosity of MWCNT/carbon (60/40)/SAE 10 W40/SAE 85 W90 (50/50) at various temperatures (25–50 °C) and $\phi$  = 0–1 % by applying ANNs based on experimental data. The Levenberg–Marquardt (LM) training algorithm was applied for updating ANN weights. The mean square error, correlation coefficient and standard deviation were equal to 2.0193E-008, 1 and 0.00021082, respectively. Also, Ahmadi et al. (Ahmadi et al., 2019) proposed three algorithms including multivariable polynomial regression (MPR), ANN–multilayer perceptron (ANN-MLP) and multivariate adaptive regression splines (MARS) are applied to model the dynamic viscosity of silver (Ag)/water nanofluid. The R² values for the concentrated models are 0.9998, 0.9997 and 0.9996 for the ANN-MLP, MARS and MPR algorithms, respectively. Also, the prediction of some properties of nanofluids using different algorithms by other researchers is presented in Table 1.

Based on the above literature survey, adequate tools for predicting the thermophysical properties are not available due to presented existing limitations. In addition, obvious and sufficient limitations for a correlation do not exist. For having a correlation that has an obvious condition with any limitation, a correlation based on the ANN method using RSM is proposed. Used data in referred correlation were measured by the present investigator, and for this reason, their limitations are considered in the proposed correlation. In this research, ${\mathbf{\mu }}_{\mathit{n}\mathit{f}}$ of MWCNT (30 %)-TiO₂ (70 %)/SAE50 hybrid nano-lubricant was first measured experimentally. Then by using RSM and ANN methods, the parameters affecting the ${\mathbf{\mu }}_{\mathit{n}\mathit{f}}$ were investigated and the ${\mathbf{\mu }}_{\mathit{n}\mathit{f}}$ was modeled. Obtained results from the proposed correlation of ANN were evaluated. A summary of the investigation on μ_nf that was used from the ANN method is presented in Table 2.

2 Experimentation

MWCNT-TiO₂ / SAE50 hybrid nano-lubricant samples were prepared with SVFs of (30:70) in SAE50 base oil with $\phi$  = 0.0625 %, 0.125 %, 0.25 %, 0.5 %, 0.75 % and 1 %. For the preparation of nano-lubricant samples, nanoparticles were purchased from Kimia Nano Danesh Company in Iran (Specifications of nanoparticles are presented in Table 3).

Then, Eq. (1) was used for the calculation of $\phi$ . Nanoparticles were then dispersed in SAE50 base oil using a two-step method,

X-ray diffraction patterns of carbon nanotubes and TiO₂ nanoparticles are shown in Fig. 1. This spectrometer shows the range of particle sizes. Morphological properties of nanoparticles were studied using X-ray diffraction and are shown in Fig. 1. The average size of TiO₂ is calculated using XRD images and the Debye-Scherrer equation (Holzwarth and Gibson, 2011);

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Fig. 1

X-ray diffraction pattern images of nanoparticles.

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3 Results and discussion

3.1

3.1 Effect of parameters on ${\mathrm{\mu }}_{\mathit{nf}}$

If the shear stress varies nonlinear changes with the $\dot{\mathrm{\gamma }}$ , fluid has non-Newtonian behavior. One of the branches of non-Newtonian behavior is Pseudo-plastic behavior. Curve fitting of the experimental data has resulted in a model for nano lubricants behavior. So, curve fitting of the shear stress versus the $\dot{\mathrm{\gamma }}$ results in the power-law Ostwald de Waele (Andersson et al., 1992) that can predict the nano-lubricant behavior (Eq. (3)).

(3)

$${\mathrm{\tau }}_{\mathrm{y}\mathrm{x}}=m{\left({\dot{\gamma }}_{\mathit{yx}}\right)}^n$$

${\mathrm{\mu }}_{\mathit{nf}}$ is in the form of Eq. (4) based on power-law (or Ostwald de Waele):

(4)

$${\mathrm{\mu }}_{\mathit{nf}}=\frac{\tau }{\dot{\gamma }}=m{\left(\dot{\gamma }\right)}^{n-1}$$

In Eq.4, m and n are determined experimentally by curve fitting of the shear stress versus the $\dot{\mathrm{\gamma }}$ and are consistency index and power-law index, respectively. In this equation, when n < 1, fluid has pseudo-plastic characteristics (shear-thinning). For n = 1, fluid is Newtonian behavior and about n > 1, fluid's behavior is dilatant (shear-thickening) and as n value is lower, the order of pseudo-plasticity is higher. The most time-independent behavior in non-Newtonian fluids belongs to the pseudo-plastic (shear-thinning) behavior, which is identified by decreasing ${\mathrm{\mu }}_{\mathit{nf}}$ with the $\dot{\mathrm{\gamma }}$ . Fig. 2 shows a power-law index of nano-lubricants against the temperature at $\phi$ . Considering this figure, since the power-law index is less than unity in all temperatures and $\phi$ , nano-lubricants are non-Newtonian and of pseudo-plastic type. Also, with temperature enhancement of nano lubricant, the non-Newtonian behavior will continue. Fig. 3a shows the ${\mathrm{\mu }}_{\mathit{nf}}$ versus the temperature. Temperature increase causes a drop-in resistance of nano-lubricant versus the exerted forces and the ${\mathrm{\mu }}_{\mathit{nf}}$ decline of nano-lubricant. Results reveal that the ${\mathrm{\mu }}_{\mathit{nf}}$ growth by 76 % with a shoot in the temperature at $\dot{\mathrm{\gamma }}=$ 3999 s⁻¹ and $\phi$  = 1 %. Increment of nanoparticles to the base fluid causes the formation and growth of nanoclusters, these nanoclusters prohibit the sliding of fluid layers on each other and increase the ${\mathrm{\mu }}_{\mathit{nf}}$ . Fig. 3b shows ${\mathrm{\mu }}_{\mathit{nf}}$ versus $\phi$ . It is apparent that the effect of nanoparticle addition on the enhancement of the ${\mathrm{\mu }}_{\mathit{nf}}$ is lower at higher temperatures. It is necessary to mention that, because of the economic aspects like the cost of MWCNT and TiO₂, and the evaluation of $\phi$ at low values (up to 1 %), $\phi$ is used in low values. Since most of the researchers investigated the thermophysical properties of nanofluids at $\phi$  = 1 to 5 %, evaluating the ${\mathrm{\mu }}_{\mathit{nf}}$ and the behavior of nanofluids in the presence of low amounts of nanoparticles can be expressed as another novelty of the present study.

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Fig. 2

Power-law index in terms of $\phi$ and temperatures.

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Fig. 3

a) ${\mathrm{\mu }}_{\mathit{nf}}$ versus temperature. b) ${\mathrm{\mu }}_{\mathit{nf}}$ versus at different $\phi$ .

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Fig. 4 shows the viscosity enhancement percent of MWCNT-TiO₂ (30:70)/SAE50 hybrid nano-lubricant versus $\phi$ at various temperatures. In low $\phi$ , a decrease in the ${\mathrm{\mu }}_{\mathit{nf}}$ and in $\phi$  > 0.5 %, has created an increase in the ${\mathrm{\mu }}_{\mathit{nf}}$ . Fig. 4 shows the maximum ${\mathrm{\mu }}_{\mathit{nf}}$ reduction (-18 %) that was observed at T = 40 ˚C and $\phi$  = 0.0625 %. By increasing the number of nanoparticles in the base fluid, the number of collisions between particles and base fluid molecules noticeably goes up and this avoids sliding of fluid layers on each other which results in shoot up the ${\mathrm{\mu }}_{\mathit{nf}}$ . Maximum ${\mathrm{\mu }}_{\mathit{nf}}$ increase (+11 %) occurs at T = 25 ˚C and $\phi$  = 1 %. So, if we need a fluid with a lower ${\mathrm{\mu }}_{\mathit{nf}}$ , it is better to use nano-lubricant with low $\phi$ .

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Fig. 4

Viscosity Enhancement versus $\phi$ at different temperatures.

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3.2

3.2 Simulation of ${\mathrm{\mu }}_{\mathit{nf}}$ with RSM

Choosing a model for the prediction of nano-lubricant properties can reduce the cost of repeating experiments. RSM evaluates suitable operational conditions using experimental data. RSM is a method for optimization of processes by mathematical and statistical correlations and can be used when different inputs affect the measurement and evaluation of a process (Bezerra et al., 2008). Different methods are used to evaluate the accuracy of a linear regression model (Wang et al., 2021), like least-squares estimation. In this investigation, Design of experiment software was used to model the viscosity of nanofluid. In this model, if n > k is assumed, it can be seen that variable answers will be available. It can be said ( $y_1,y_2,\cdots ,y_n$ ). Along with each observed response $y_i$ , we will have an observation on each regressor variable, let $x_{\mathit{ij}}$ denote the observation or level of variable $x_j$ . The matrix notation of the observations may be written as $y=X\beta +\epsilon$ , where $X$ is the levels of the independent variables, $\beta$ , is the regression coefficients, and $\epsilon$ is random errors. An unbiased estimator of parameters in multiple linear regression models is generated using the least-squares method. The summation of squares of residuals is a vital parameter and is calculated from Eq.5,

(5)

$$S{S}_E=\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2=\sum _{i=1}^n{e}_i^2=e^{T}e$$

There are n × 1 vector of residuals, obtained from the difference of considerations ( $y_i$ ) and fitted values ( ${\widehat{y}}_i$ ) that is denoted by:

(6)

$$e_i=y_i-{\widehat{y}}_i$$

Since $X^{T}Xb=X^{T}y$ the formula for computing SS_E may be shown as:

(7)

$$S{S}_E=y^{T}y-b^T{X}^{T}y$$

Eq.7 is known as error or residual sum of squares.

(8)

$$S_T=y^{T}y-\frac{{\left({\sum }_{i=1}^n{y}_i\right)}^2}{n}=\sum _{i=1}^n{y}_i^2-\frac{{\left({\sum }_{i=1}^n{y}_i\right)}^2}{n}$$

Eq.8 is presented for the calculation of the total sum of squares. The determination coefficient (R²) is calculated by Eq.9 (Palta et al., 1982);

(9)

$$R^2=1-\frac{S{S}_E}{S{S}_T}$$

Evaluation of Eq. (9) shows that R² values are in the range of 0 to 1; if its value is close to 1, the accuracy of the model is higher. In some studies, $R_{\mathit{adj}}^2$ is also investigated and is calculated from Eq.10 (Palta et al., 1982);

(10)

$$R_{\mathit{adj}}^2=1-\frac{S{S}_E/n-p}{S{S}_T/n-1}=1-\frac{n-1}{n-p}(1-R^2)$$ where p is the regression coefficient and n is the number of measurements. $R_{\mathit{adj}}^2$ will be decreased by adding too much of the parameter. As the parameters are added, the accuracy of the correlation will be decreased. Some of these parameters are not important, and $R_{\mathit{adj}}^2$ can recognize and remove them. This is the best way of adding unnecessary variables to the model without changing R² (Simpson et al., 1998). ANOVA method was employed to analyze the used model (Table 4) and prove its validity.

Table 4 ANOVA for.. ${\mu }_{\mathit{nf}}$ .

Source	Sum of Squares	df	Mean Square	F Value	p-value Prob > F
Model	2,757,867	19	145150.9	9888.233	< 0.0001
A- $\phi$	7269.028	1	7269.028	495.1939	< 0.0001
B-T	74072.14	1	74072.14	5046.076	< 0.0001
C- $\dot{\gamma }$	322.2865	1	322.2865	21.95538	< 0.0001
AB	3110.403	1	3110.403	211.8925	< 0.0001
AC	14.11681	1	14.11681	0.961691	0.3283
BC	199.1713	1	199.1713	13.56831	0.0003
A2	4474.296	1	4474.296	304.8061	< 0.0001
B2	9574.162	1	9574.162	652.2284	< 0.0001
C2	10.49573	1	10.49573	0.715009	0.3991
ABC	31.60646	1	31.60646	2.153153	0.1443
A2B	1584.072	1	1584.072	107.913	< 0.0001
A2C	1.27553	1	1.27553	0.086894	0.7686
AB2	1135.7	1	1135.7	77.36822	< 0.0001
AC2	7.81384	1	7.81384	0.532309	0.4667
B2C	185.7014	1	185.7014	12.65068	0.0005
BC2	0.072301	1	0.072301	0.004925	0.9441
A3	22.43328	1	22.43328	1.528241	0.2183
B3	2794.823	1	2794.823	190.394	< 0.0001
C3	81.60985	1	81.60985	5.559574	0.0196
Residual	2260.59	154	14.67916
Cor Total	2,760,128	173
Standard deviation = 3.8313
R2 (Adequate) = 0.9991, R2 (Predicted) = 0.9989, R2 (Adjusted) = 0.999

RSM was used to estimate the ${\mathrm{\mu }}_{\mathit{nf}}$ in various temperatures, $\phi$ and $\dot{\gamma }$ . Modeling with RSM results in Eq.11 for ${\mathrm{\mu }}_{\mathit{nf}}$ based on $\phi$ , temperature, and $\dot{\gamma }$ . Based on the p-value, it was decided which terms are included in the category of important sentences and which terms are excluded in the correlation equation. The criterion (critical) value for p-value is 0.05 and the terms of the equation that have a p-value>0.05 are of little importance and can be removed.

(11)

$${\mathrm{\mu }}_{\mathit{nf}}=1901.53+731.13\phi -97.42\mathrm{T}-0.042\dot{\mathrm{\gamma }}-20.28\phi \mathrm{T}+0.0013\mathrm{T}\dot{\mathrm{\gamma }}-302.39{\phi }^2+1.83{\mathrm{T}}^2+4.79{\phi }^2\mathrm{T}+0.14\phi {\mathrm{T}}^2-1.51\times {10}^{-5}{\mathrm{T}}^2-0.012$$

The proposed model can predict the ${\mathrm{\mu }}_{\mathit{nf}}$ of MWCNT (30 %)-TiO₂ (70 %)/SAE50 hybrid nano-lubricant at $\phi$  = 0.0625 to 1 %, T = 25 to 50 ˚C and $\dot{\mathrm{\gamma }}=$ 666.5 and 7998 s⁻¹. The importance of the regression model can be figured out from the high F-value of 9888.233. P-values can determine statistical aspects of the model, when the p-value is<0.05, that parameter is significant but parameters with p-values higher than 0.05 are less significant. So, non-significant parts of the model will be omitted. R² adj. of used model for the ${\mathrm{\mu }}_{\mathit{nf}}$ is 0.999. It should be noted that R² adj. shows the quality of the output. As its value be closer to unity, the proposed model has higher accuracy. After correlating ${\mathrm{\mu }}_{\mathit{nf}}$ with RSM, the conformism of predicted data with experimental data and residual values were checked to evaluate the quality of the model, and the results are presented in Fig. 5. Good conformity and low values of residuals (-12, +9) imply the high accuracy of the proposed model.

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Fig. 5

a) Conformity of proposed model with experimental data b) Residuals versus predicted results.

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Three-dimensional ${\mathrm{\mu }}_{\mathit{nf}}$ diagram against the $\dot{\mathrm{\gamma }}$ and temperature in $\dot{\mathrm{\gamma }}=$ 6710.03 s⁻¹ and versus temperature and $\dot{\mathrm{\gamma }}$ at $\phi$  = 0.51 % are depicted in Fig. 6. Fig. 6 shows that in low temperatures and $\dot{\gamma }$ and high $\phi$ , the ${\mathrm{\mu }}_{\mathit{nf}}$ has a higher value.

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Fig. 6

3-Dimensional curves of a) ${\mu }_{\mathit{nf}}$ versus temperature and $\dot{\mathit{\gamma }}$ b) ${\mu }_{\mathit{nf}}$ versus temperature and φ.

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Fig. 7 presents the effects of each parameter on the ${\mathrm{\mu }}_{\mathit{nf}}$ . It shows that the temperature has the maximum effect and $\dot{\gamma }$ quality of the output. As its value be closerhas minimum effect on the ${\mathrm{\mu }}_{\mathit{nf}}$ . Also, the temperature and $\dot{\gamma }$ have opposite effects but $\phi$ has a direct effect on the ${\mathrm{\mu }}_{\mathit{nf}}$ .

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Fig. 7

Effect of a) solid volume fraction b) shear rate c) temperature parameter on.. ${\mu }_{\mathit{nf}}$

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A computer with RAM 4.00 GB & CPU core i5 was used. The CPU time to reach the results at different ${\mathrm{\mu }}_{\mathit{nf}}$ ( $\dot{\mathrm{\gamma }}$  = 5332 s⁻¹, T = 35 °C) was reported in Table 5. The information in Table 5 is based on the information extracted from the design of experiment software and refers to the CPU time required to study the input data of the software based on the historical data method.

Table 5 CPU Time for different.. ${\mu }_{\mathit{nf}}$ .

${\mathbf{\mu }}_{\mathit{n}\mathit{f}}$	260.22	275.36	286.44	290.55
CPU time (second)	4.10	5.25	6.11	7.08

4 Simulation of ${\mathit{\mu }}_{\mathit{nf}}$ with ANN

In a complex information structure, in which the forecast utilizing ordinary techniques is lumbering and testing, the ANN gives a superior choice. The ANN can gain proficiency with very perplexing non-linear designed information. ANN is a computational model which learns like a human mind through epitomes. There are two main features that engineers have to face within natural processes. In the first step, they depend on many variables, in the second step, very complicated relations exist between the parts, so their analysis is so hard. This problem always affects the modeling accuracy. In this research, ANN was chosen between different modeling methods. Its main idea was adopted from the ANN of the human brain. The clearest likeness between a neural network and the brain is the presence of neurons as the most essential unit of the nervous system. However, how neurons take input in the two cases is unique. The creation of new structures for processing systems is the key parameter of this idea. To clarify process parameters, many data exist in this system that is called neurons that connect continuously. Synapse (electrochemical communication) is used for data transfer to coordinate them with each other. These ANNs can learn, and learning is a comparative concept for them. In other words, the weight of the synapse will be changed by using these samples and the system will give correct results in several ways including increasing hidden layers, changing activation function, changing activation function in the output layer, increasing the number of neurons, weight initialization, more data, and normalizing/scaling data (Behrang et al., 2010; Pedro and Maas, 2005; Yilmaz and Kaynar, 2011). Dropout is a regularization method that avoids overfitting the neural network. During training, it randomly releases neurons from the neural network, which is equivalent to training different neural networks. Different networks will excel differently, so the net effect of the regularization method will be to reduce redundant connections so that our model will be good for predictive analytics (Santos and Papa, 2022). The conventional method of solving Multilayer Perceptron (MLP) is the ANN. Two transfer functions that are used in the hidden layer and output layer are tangent sigmoid and logistic sigmoid. The levenberg-Marquardt algorithm was used for the determination of a suitable ANN structure. This algorithm changes the weight of neurons between real and target output, based on error. Several hidden layers are chosen depending on the complication level of the problem. First, the number of hidden layers is determined, and then in the next step, the number of neurons in hidden layers in the trial and error method is different to get the target output of the functions. This trend continues to find the number of neurons that result in a minimum error; this value will be reported as the optimum number of neurons in the hidden layer. There are several methods for determining the correct number of neurons to use in hidden layers:

• Between the size of the input layer and the size of the output layer.

• Two-thirds of the size of the input layer plus the size of the output layer.

• Less than twice the size of the input layer (Yu and Yan, 2019).

The output of multilayer perceptron ANN is mentioned below:

100 structures of ANN with two hidden layers, each one containing 1 to 5 neurons and tangent sigmoid transfer function and logistic sigmoid transfer function were designed for each layer in this modeling. To determine the optimum structure for ANN, the fourth item in Table 6 which has two hidden layers with five and two neurons and logistic sigmoid transfer function for the first layer and tangent sigmoid transfer function for the second layer were chosen to consider calculated error values. Since the values of errors for this item were acceptable (MSE = 0.7663 and MAE = 1.3367) and R was close to unity (0.999957), some neurons were suitable too. The structure of the designed ANN is presented in Fig. 8.

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Fig. 8

Topology of a typical ANN structure.

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The accuracy of the proposed structure was calculated from the below equations:

In the above equations, T_ij is the experimental value and P_ij is the estimated value from the model. $\overline{T}$ and $\overline{p}$ are the mean experimental values and values estimated by the model for n data.

As Fig. 9 shows, all data are spread around the oblique line. Minimum and maximum differences between values of experimental data and predicted data by the ANN method were ± 4 %. Analysis shows that the proposed ANN model is a valid and exact model for the prediction of ${\mathrm{\mu }}_{\mathit{nf}}$ .

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Fig. 9

a) Curve of compliance between ANN data with experimental data b) Residuals ANN results.

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Fig. 10 shows an analogy among the empirical data and predicted results by training, test, validation, and all data of ANN. According to this figure, predicted results by ANN could properly cover the experimental data. This shows that ANN was trained properly which resulted in an acceptable accuracy of the test, validation, and all data.

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Fig. 10

ANN Prediction performance on a) training b) test c) validation d) all data.

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5 Comparison of ANN model with RSM

In this part, a comparison is done between the presented model by RSM (Eq. (11)) and the ANN method to reach the best model. As it is evident in Fig. 11, both models could predict empirical data with good approximation in different $\phi$ and temperatures.

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Fig. 11

Comparison of ANN prediction with the RSM result using experimental data for a) φ = 1 % b) φ = 0.75 % c) φ = 0.5 % d) φ = 0.25 % e) φ = 0.125 % f) φ = 0.0625 %.

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Fig. 12 shows the margin of deviation for the proposed model by RSM (Eq. (11)) and ANN method at different $\phi$ . The margin of deviation is calculated from Eq. (21),

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Fig. 12

Comparison of error for ANN and RSM model in a) φ = 1 % b) φ = 0.75 % c) φ = 0.5 % d) φ = 0.25 % e) φ = 0.125 % f) φ = 0.0625 %.

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As it is obvious, ANN has the least value of MOD (%) in all $\phi$ . Based on the results obtained from graphs 5 and 9 mentioned above. So, it results that the ANN model can estimate the experimental data more accurately with less error.

6 Conclusion

In this paper, experimental data obtained for ${\mathrm{\mu }}_{\mathit{nf}}$ of MWCNT (30 %)-TiO₂ (70 %)/SAE50 hybrid nano-lubricant were modeled using RSM and ANN methods. By comparing the accuracy parameters for ANN and RSM models, it was concluded that the proposed ANN model predicts the empirical data more accurately. Statistical regression analysis of training and test data (R² = 0.9999), Mean Square Error (MSE) = 1.3367 and Mean Absolute Error (MAE = 0.7663) have shown the ability of improved ANN in the prediction of ${\mathrm{\mu }}_{\mathit{nf}}$ . The following results were also obtained:

• Since the power-law index is less than one at all temperatures and $\phi$ , nano-lubricants are non-Newtonian and pseudo-plastic (shear-thinning).

• R² in the proposed model by RSM method for prediction of ${\mathrm{\mu }}_{\mathit{nf}}$ was 0.9991 which shows the high accuracy of the correlation.

• Analysis reveals that temperature and $\dot{\mathrm{\gamma }}$ have the maximum and minimum effects on ${\mathrm{\mu }}_{\mathit{nf}}$ , respectively.

• Maximum and minimum difference between experimental and predicted data by ANN was ± 4 % which shows the acceptable correctness of the model.

• Comparison of MOD of two presented models revealed that MOD of ANN has a lower value at all $\phi$ . This implies that ANN is more accurate in the prediction of ${\mathrm{\mu }}_{\mathit{nf}}$ .