Introduction

Engine oil is responsible for lubricating the moving parts of the engine, minimizing the friction and wear, reducing the heat, and absorbing the dirt and sludge deposits from combustion. Since engine oil must perform several complex tasks simultaneously, its thermo-physical properties are of particular importance. Therefore, researchers have tried to improve its thermal properties through the addition of nanomaterials. In general, researchers have reported that the addition of nanomaterials, such as metal oxides and carbon nanomaterials, to conventional fluids, where a nanofluid is produced, can increase thermal conductivity and thereby improve heat transfer [1,2,3,4,5,6,7,8,9,10]. Therefore, it is expected that thermo-physical properties are also affected with the addition of such substances to the fluid [11,12,13,14,15,17]. In the meantime, viscosity of the nanofluid is a key parameter that is widely examined [18,19,20,23].

Oils are produced with different viscosities for different weather conditions. The use of high-viscosity oil in winter delays lubrication until the engine is warmed up, and the oil will not reach all parts of the engine before that. The use of low-viscosity oil in the summer causes wear in the engine. Therefore, when nanomaterials are added to the oil, knowing their rheological behavior is necessary for better performance of the engine. As such, in recent years researchers have paid attention to the study of viscosity of oils containing nanomaterials, which are called nanolubricant [24,25,26,27,28,29,30,31,32]. For example, Afrand et al. [29] measured the viscosity of SiO2-MWCNTs/SAE40 hybrid nanolubricant at different temperatures, shear rates and volume fractions. Their experiments indicated that the nanolubricant had Newtonian behavior. Experimental results also showed that the maximum increase in viscosity of the nanolubricant was 37.4%. They also compared experimental and theoretical viscosities and found that existing theoretical models are powerless to estimate the viscosity of the nanolubricant. Sensitivity analysis and presentation of the correlation were also done by them. Dardan et al. [30] examined the rheological behavior of MWCNTs-Al2O3/SAE40 hybrid nanolubricant. They measured the viscosity of all nanolubricant samples at different temperatures and shear rates. They found that all samples had Newtonian behavior. The results from the sensitivity analysis indicated that the viscosity sensitivity to temperature changes is slight. They also observed an increase of 46% in viscosity. Sepyani et al. [31] dispersed ZnO nanoparticles in SAE50 oil with the volume fractions of 0.125–1.5% at the first. Then, they measured the viscosity of samples at various shear rates under different temperatures. They observed a Newtonian behavior for all nanolubricant samples. Their results revealed that the viscosity marginally augmented with increasing quantity of ZnO nanoparticles (about 12%). Finally, they proposed an accurate correlation for predicting the viscosity of ZnO/SAE50 nanolubricant using experimental data. Hemmat Esfe et al. [32] experimentally investigated the effect of hybrid nanoadditives composed of ZnO and MWCNTs on the rheological behavior of SAE40 oil. They performed viscosity measurements at temperatures ranging from 25 to 60 °C at different shear rates. Their results showed that all nanolubricant had Newtonian behavior. Calculation of relative viscosity specified that the maximum increase in viscosity was 33.3%. They also proposed an experimental correlation for estimating viscosity of the nanolubricant.

Despite all the laboratory research in the field of nanolubricants, it should be noted that several tests are needed to determine their thermo-physical properties in different conditions, which will be time-consuming and expensive. Thus, to eschew the cost of testing, computational techniques are used to estimate the thermo-physical properties of nanofluids. Across these techniques, curve fitting, fuzzy logic, artificial neural network (ANN) and genetic algorithms are the most extensively used techniques. In this context, a number of works on thermal conductivity and viscosity of nanofluids have been performed using computational techniques. For example, Esfe et al. [33,34,35,36,37] designed a neural network that could predict thermal conductivity of nanofluid by using experimental data to train the artificial neural network and could propose acceptable correlations between inputs and outputs. Afrand et al. [38] studied the thermal conductivity of a magnetic nanofluid dispersing into the water with different solid volume fractions with different temperature ranges and predicted the experimental results by using an artificial neural network.

In the other work, Afrand et al. [39] also studied the relative viscosity of MWCNTs-SiO2/SAE40 nanolubricant by using experimental data and designed an optimal artificial neural network that could predict data with minimum error. Vafaei et al. [40] studied the thermal conductivity ratio of MgO-MWCNTs/EG hybrid nanofluids and by changing different neuron numbers in the hidden layer of the ANN, generated an optimal neural network including 12 neurons in the hidden layer. Hemmat Esfe et al. [41] studied the effects of temperature and solid volume fraction on thermal conductivity of CNTs-Al2O3/water nanofluids with various solid volume fractions and various fluid temperatures and discovered that the thermal conductivity of nanofluid depends on the solid volume fraction and finally proposed correlations for different temperatures for the experimental data that could predict thermal conductivity of nanofluid. Mehrabi et al. [42] employed a FCM-ANFIS based on experimental data to predict the effective viscosity of water-based nanofluids containing Al2O3, CuO, TiO2 and SiO2 nanoparticles. They used temperature, size and volume fraction of nanoparticles as the design parameters. Their evaluations showed that the predicted values agreed with the experimental outcomes. Karimi et al. [43] predicted the viscosity of nanofluids by using a neural network based on genetic algorithm. In fact, they used the genetic algorithm for optimizing the neural network parameters. Input parameters included temperature and concentration of nanoparticles. Their findings revealed that the suggested model was in compliance with experiments.

As mentioned above, much research has been done on the properties of nanofluids. However, no study reported on estimates the viscosity of SiO2-MWCNTs (90:10%)/10W40 nanolubricant. Hence, we tried to predict the rheological behavior of SiO2-MWCNTs (90:10%)/10W40 hybrid nanolubricant using artificial neural network (ANN). In this work, temperature, shear rate and solid volume fraction are as input variables and viscosity of the nanolubricant is output parameter.

Experimental

One hundred and fifty-one package experimental data including temperature, volume fraction, shear rate and viscosity of SiO2-MWCNTs (90:10%)/10W40 hybrid nanolubricant have been used for developing ANN procedure [44]. The hybrid nanolubricant consisted of a combination of MWCNTs and SiO2 dispersed in SAE10W40 engine oil. The specifications of the nanoadditives and engine oil used for nanolubricant data are presented in Tables 1–3.

Table 1 Physicochemical property of SiO2 nanoparticle [44]
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Table 2 Physicochemical characteristics of MWCNTs [44]
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Table 3 Engine oil (10W40) characteristics [44]
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ANN modeling

In order to predict viscosity data of SiO2-MWCNTs (90%:10%)/10W40, Einstein [45] and Wang et al. [46] theoretical equations were utilized, viscosity data at 5 °C are depicted in Fig. 1. It is clearly observable that none of the equations was able to forecast the viscosity data (Table 2).

Fig. 1
figure 1

Comparison between experimental data and theoretical models

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The main objective of the study of this section was to obtain the pattern of nanofluid dynamic viscosity changes based on variables of temperature, volume fraction and shear rate by modeling the empirical data. By using the available empirical data of different volume fractions of nanoparticles, various temperature points and different shear rates, the most appropriate technique for modeling of the empirical data, which has an excellent precision, is using neural networks. Many researches have been done to find optimal topology design of artificial neural networks [47–52] in order to use neural networks, the input data must be inserted first; the networks divide the data into three sets of training data, validation data, and test data after the data are preprocessed. In order to put data into the first-layer neurons, a mass and a bias value are determined. After passing the data through each layer’s neurons, the extracted values are compared with the empirical values of viscosity. If the error value is not within the acceptable range, this process is iterated with different values of mass. Otherwise, training stops and makes the network structure fixed. Figure 2 shows a summary of this process.

Fig. 2
figure 2

ANN modeling flowchart

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Figure 3 illustrates the accuracy value of the predicted data for all variables along with the empirical data. According to the figure, fewer errors are observed at low magnitudes of apparent viscosity, but in the whole it can be obtained that almost all of data are laid on the bisector line or in its proximity as mean squared error (MSE) value is 0.000194.

Fig. 3
figure 3

Predicted data with ANN based on empirical data for all variables

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Error values are depicted in Fig. 4 in terms of number of data. According to this figure the highest error is 0.08 and it occurred near 1.5. Most of the data have errors within the range of 0–0.03.

Fig. 4
figure 4

Calculated error values

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The specifications of the neural network which was used in the present paper are shown in Table 4. Regarding this table, the neural networks exhibited an excellent accuracy as R2 value was 99.56%.

Table 4 Modeling specifications conducted by the neural network
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Errors of test, train and validation processes are illustrated in Fig. 5. 70% of the data were used to train, and the remaining 30% were used in test and validation processes. Regarding this figure each one of the processes has an appropriate accuracy and most of the modeling results were in accordance with empirical data. As it can be seen in the figure, the highest deviations in train and test curves occurred in low dynamic viscosity values (nearly at 1.15–1.20).

Fig. 5
figure 5

Predicted data with ANN based on empirical data for training, test and validation processes

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In Fig. 6 values of output data and viscosity of SiO2-MWCNTs (90:10%)/10W40 nanolubricant are compared for a constant shear rate. In the diagram of relative viscosity–temperature, the accuracy of data prediction was very high and the errors are negligible. Also in relative viscosity–concentration diagram, the data are in accordance with each other. For both of the diagrams, neural network errors of data prediction occurred at low temperature (5 °C) and in viscosity ranges of 1.15–1.25. This is clearly observable in train and test curves shown in Fig. 6.

Fig. 6
figure 6

Comparison of relative viscosity data with neural network modeling

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Conclusions

In this work, the viscosity of SiO2-MWCNTs (90:10%)/10W40 hybrid nanolubricant has been evaluated using artificial neural network (ANN). For this purpose, one hundred and fifty-one package experimental data including temperature, volume fraction, shear rate and viscosity of the hybrid nanolubricant have been used for developing ANN procedure. Solid volume fraction, temperatures and shear rate were considered as input variables for ANN, and relative viscosity was output parameter. Experimental data were compared with data obtained from previous classical models. The result showed that none of the previous classical models were able to predict the viscosity data. Hence, neural networks were utilized for predicting viscosity data. A neural network was designed so that its outputs were in highest achievable accordance with empirical data, as R-squared and mean squared error values were 0.9948 and 0.000194, respectively. The extension of this paper and our previous studies results afford engineers a good option for nanofluid in applications like electronics, automotive and heat transfer improvement goals [53–70].