Introduction

Nanofluids have superior thermal conducting properties due to the presence of nanoparticles (size < 100 nm) in a base fluid. Ethylene glycol (EG), water, engine oil, propylene glycol (PEG), refrigerants etc., have been used as the base fluid to prepare the nanofluid [1,2,3,4]. Due to the small size of nanoparticles, the homogeneous distribution of nanoparticles in the base fluid can be obtained. The significance of nanofluids utilization in heat transfer has attracted many researchers. With the present advances in nanotechnology, the preparation of nanoparticles can be accomplished easily by various methods [5]. As a result, the possibility of adding nanoparticles in a commonly used base fluid for improving thermophysical properties has been experimentally exploited by various researchers [6,7,8,9,10]. Thermal conductivity of the fluid is the most important property for heat transfer studies. It is also known that the larger surface area of the particles results in enhanced thermal conductivity in the base fluid. Nanoparticles of metals, metal oxides, non-metals and non-metal oxides have been used to study the thermal properties and applications in heat exchangers [11,12,13]. Various nanofluids have been used to carry out the studies on heat exchangers such as double pipe heat exchanger [14], shell and tube heat exchanger [15,16,17], helical coil heat exchanger [18, 19], plate heat exchanger [20, 21] etc.

Eastman et al. [22] used water and HE 200 oil as a base fluid to disperse the Al2O3, CuO and Cu nanoparticles and found thermal conductivity enhancement of 60% for the volume concentration of 5%. Further studies were carried out using EG as base fluid using Cu particles (with less than 10 nm of size) and reported 40% of enhancement for 0.3% of volume fraction [23]. Yu et al. [24] investigated the thermal conductivity of ZnO nanoparticles dispersed in EG. Results showed that thermal conductivity was strongly dependent on particle concentration and temperature.

Ethylene glycol and water (EG:W) mixture gained a lot of attention due to its low freezing point and high boiling temperature compared to water. EG:W-based nanofluids have attracted studies for applications such as car radiator, heat exchangers, chillers etc. [25]. Vajja and Das [26] reported a significant improvement in thermal conductivity for EG:W-based Al2O3 and CuO nanofluids. Their studies reported 69% of enhancement for 6% volume concentration of CuO nanoparticles and 10% of volume concentration for Al2O3 nanoparticles in EG:W base fluid. Kole and Dey [27] have carried out the thermal conductivity measurement on graphene oxide nanoparticles dispersed in a mixture of EG and DI water (70:30 v/v). Thermal conductivity was measured with varying the graphene oxide concentration (0.041–0.395 vol%) and temperature (10–70 °C). An enhancement of 15% for adding only 0.395 vol% of graphene oxide was observed at room temperature. Reddy et al. [28] performed thermal conductivity study of TiO2 nanoparticles dispersed in EG:W (40:60) and reported an enhancement of 5% at 70 °C for 1% of volume concentration. Sundar et al. [29] prepared EG:W (50:50)-based Al2O3 and CuO nanofluid and examined thermal conductivity at different temperatures. Both the nanofluids were found to have enhanced thermal conductivity with respect to concentration and temperature compared with the base fluid of EG:W. However, enhancement of EG:W–CuO nanofluid was higher compared to EG:W–Al2O3 nanofluids. They extended their studies further for different ratios of EG:W (20:80%, 40:60% and 60:40% in mass) to study the effect of EG:W ratio, concentration of Al2O3 nanoparticles and temperature [30]. Maximum enhancement of 32.26% was obtained for a 20:80 ratio of EG:W at 1.5% concentration. Serebryakov et al. [31] used 90% of EG and 10% of water with Al2O3 nanoparticles to measure the thermal conductivity. Thermal conductivity of SiO2 nanoparticles in EG:W (at different volume ratios) was investigated for a mass concentration of 0.3% nanoparticles with a temperature range of (25–45  °C) [32]. Thermal conductivity of EG:W–SiO2 nanofluids decreased with the increase in EG content. Researchers have performed experimental investigations using oxides of metals such as alumina (Al2O3), copper oxide (CuO) and titanium oxide (TiO2) as they are cheaper compared to the corresponding metal nanopowder. All the literature reports have shown enhancement in the thermal conductivity by the addition of nanoparticles. Table 1 summarizes the literature work reported on maximum enhancement in thermal conductivity.

Table 1 Thermal conductivity enhancement reported in the literature for different ratios of EG and water mixture
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Table 2 presents mathematical models for thermal conductivity for various nanofluids. The developed models such as Maxwell and Hamilton–Crosser consider basic mixture of solid particles in liquid. Some of the correlations developed in the literature are based on their experimental results as a function of temperature and concentration. However, the development of empirical model using regression method for prediction with a multiple input variables is a difficult task due to large deviation in prediction. To overcome this disadvantage, artificial neural network (ANN) modelling can be employed for accurate prediction with several input parameters. Many researchers have performed the prediction of effective thermal conductivity (knf/kbf) using ANN modelling considering the factors such as the concentration of nanofluid, temperature, particle size, etc. Hojjat et al. [41] performed ANN modelling to predict thermal conductivity for 0.5 mass% CMC-based Al2O3, CuO and TiO2 nanofluids using feedforward artificial neural network (FF-ANN) considering concentration, temperature and thermal conductivity of nanoparticles as input variables. The prediction has shown good agreement with experimental results. Ariana et al. [42] carried out ANN modelling to predict the thermal conductivity of water-based Al2O3 nanofluids. Thermal conductivity data for various particle sizes of Al2O3 were collected at different concentrations and temperatures. They used FF-ANN model of two hidden layers having 14 neurons to predict the data and results showed satisfactory prediction with a regression coefficient (R2) of 0.971, absolute average relative deviation (AARD %) of 1.27%, and mean square error (MSE) of 4.73 × 10−4. Esfe et al. [43] developed an empirical correlation and carried out FF-ANN modelling to predict the thermal conductivity of water-based Al2O3 nanofluid. The correlation and ANN modelling performed well in predicting the thermal conductivity of nanofluid. One more study was carried out by them, in which ANN modelling proved better than correlation for water/EG mixture (60:40)-based MgO nanofluids [44].

Table 2 Correlations proposed for the effective thermal conductivity of nanofluids from the literature
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Tahani et al. [45] performed ANN modelling for thermal conductivity of deionized water-based graphite oxide nanoplatelets considering mass concentration and temperature as input variables. In their study, two hidden layers with eight neurons were used to predict the thermal conductivity. From the predicted data, root mean square error (RMSE), mean absolute percentage error and R2 were determined to evaluate the performance of ANN. Results showed the accurate prediction of the thermal conductivity with experimental data. Ahmadloo and Azizi [46] conducted ANN modelling using several inputs to predict the effective thermal conductivity of nanofluids. Similar work considering EG-based metals and metal oxides nanofluids was reported by Wang et al. [47]. Their results have shown that ANN modelling tool can be used to predict the effective thermal conductivity with minimum deviation. Esfe et al. [48] developed ANN modelling to predict the effective thermal conductivity for EG-based MgO nanofluids using experimental data and results obtained by modelling are in good agreement with the measured data. Zhao et al. [49] adopted radial basis function (RBF)-ANN approach to predict the thermal conductivity of water-based Al2O3 nanofluids considering concentration and temperature as an input variables. Their results concluded that ANN modelling can be used as an effective method to predict the thermophysical properties of nanofluids with an error of ± 2%. Esfe [50] developed RBF-ANN modelling to predict the thermal conductivity EG-based TiO2 nanofluids. The results showed better than the correlation approach.

From the literature, it can be observed that there are studies reported on thermal conductivity of EG:water-based nanofluids, considering nanoparticles of CuO, Al2O3 and TiO2 at lower concentration (< 0.5% volume concentration). Table 1 gives an account of studies. Lower concentration of nanoparticles can provide a limited enhancement in thermal conductivity. On the other hand, due to high concentration higher viscosity of nanofluids was reported. However, the effective use of nanofluids in the applications such as heat exchangers can deliver significant improvement in heat transfer. As observed from the literature review, there have been a number of experimental and modelling studies on EG:W mixture using various nanopowders. However, it can be seen that the modelling work pertained only to the respective experimental studies. No generalized models have been reported. In this work, a generalized correlation and ANN model have been developed to predict the effective thermal conductivity of nanofluids using the experimental results (35:65 (V/V)) from the present study and literature data for EG:W-based nanofluids.

Materials and methods

Preparation of nanofluid

The mixture of ethylene glycol and water 35:65 (volume/volume) was used as base fluid in the present study. Table 3 shows the thermophysical properties of EG:W from ASHRAE [51]. Nanoparticles of CuO, Al2O3 and TiO2 were used to prepare the EG:W-based nanofluids separately. Table 4 shows the details of the nanopowders.

Table 3 ASHRAE data for 35:65 EG:water mixture at different temperatures [51]
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Table 4 Details of nanoparticles used in the present study
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The required amount of nanopowder for preparing nanofluids of mass concentration of 0.2%, 0.5%, 1%, 1.5% and 2% is calculated using Eq. (1).

$$100\, \emptyset_{\text{m}} = \frac{{m_{\text{g}} }}{{m_{\text{g}} + m_{\text{EG/w}} }}$$
(1)

where \(m_{\text{g}}\) \(m_{\text{EG/w}}\)  = mass of graphite nanoparticles and base fluid, respectively. \(\emptyset_{\text{m}}\)  = Mass fraction of nanopowder.

Before adding the nanoparticles to 40 mL of the base fluid, 0.2% mass concentration of sodium dodecyl benzene sulphonate (SDBS) surfactant was added to base fluid and stirred to maintain a stable dispersion and to provide long-term stabilization of nanofluid. The required amount of nanoparticles for each concentration was added and stirred for 2 h in magnetic stirrer at 750–800 rpm. Later, the stirred sample was ultrasonicated for 2 h using Hielscher UP200H (200 W), at a frequency of 24 kHz to breakdown of agglomeration and to ensure the uniform and stable mixture of required concentration. As the nanofluid was found to be stable (observed visually and by thermal conductivity measurement), this method was followed for the preparation of nanofluids required for the study. Thereafter, nanofluids of different concentrations of CuO, Al2O3 and TiO2 in EG:W mixture were prepared separately.

Measurement of thermal conductivity of nanofluid

Thermal conductivity of EG:W–CuO, Al2O3 and TiO2 nanofluids was determined using KD2 Pro thermal properties analyser (Decagon Devices, Inc., USA). The instrument meets the requirements of ASTM D5334 and IEEE 442-1981 standards and has been used by various researchers [1,2,3,4, 29, 35, 36, 52, 53]. This instrument consists of a microcontroller and a KS-1 sensor needle with a size of 1.3 mm diameter and 6 cm long, capable of measuring the thermal conductivity in the range of 0.02–2.00 W m−1 K−1. Prepared samples of specific nanofluid (volume = 30 mL) were taken in a glass tube of 30 mm diameter, which was equipped with a small opening (slightly larger than the sensor) through which the sensor needle was placed in it. The sensor was inserted into the fluid, oriented centrally and vertically inside the container without touching the side walls of the container as displayed in Fig. 1. Each nanofluid sample of CuO, Al2O3 and TiO2 of different concentrations (0.2–2%) was taken into 30-mL glass bottle after preparation. Thermal conductivity at 30, 35, 40, 45, 50, 55 and 60 °C was measured in a constant temperature bath (TEMPO SM-1014). For each sample, five readings were taken by allowing 15 min for each reading for the temperature to equilibrate. The average of these readings was used for reporting the thermal conductivity of nanofluids. Figure 2 shows the comparison of thermal conductivity measurement with the ASHRAE data. The measured values show a maximum deviation of ± 2.5%.

Fig. 1
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Experimental arrangement for thermal conductivity measurement of nanofluids

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Fig. 2
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Comparison of experimental results for k with ASHRAE data for EG:W mixture

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Correlation for effective thermal conductivity

Considering experimental data from the present study

Thermal conductivity of nanofluids is dependent on the material of nanoparticle, base fluid, nanoparticle concentration, temperature and particle size. In the present study, the following correlation is proposed for predicting the effective thermal conductivity (ratio of knf/kbf) of nanofluids (three nanopowders separately; diameter for each material is constant) as a function of nanofluid concentration and temperature based on the experimental data.

$$k_{\text{eff}} = \frac{{k_{\text{nf}} }}{{k_{\text{bf}} }} = A\left( {\emptyset_{\text{m}} } \right)^{\text{B }} \left( {\frac{{T_{\text{nf}} }}{{T_{\text{r}} }}} \right)^{\text{C}}$$
(2)

where knf and kbf represent thermal conductivity of nanofluids and EG:water, respectively. Tnf and Tr represent the temperature of nanofluids and reference temperature (273.15 K). ϕm is mass fraction of nanofluids. A, B and C are the constants to be obtained after regression using experimental data.

Considering literature data

To develop more generalized correlations considering the experimental results from the present study and the results from literature reports, the following correlations (Eqs. (3) and (4)) are proposed. The effective thermal conductivity (keff) model of EG:W-based CuO, Al2O3 and TiO2 proposed in Eq. (2) considers only temperature and mass concentration. The other parameters, viz., the volume ratio of EG:W mixture, temperature and size of nanoparticles which affect the thermal conductivity were not considered as they are kept constant. Two correlations, one material specific [Eq. (3)] and the other one [Eq. (4) ] more general, considering the three materials, all the effecting variables and literature data, are proposed for keff.

$$k_{\text{eff}} = a \left( {\emptyset_{\text{m}} } \right)^{\text{b}} \left( {V_{\text{r}} } \right)^{\text{c}} \left( {\frac{{T_{\text{nf}} }}{{T_{\text{ref}} }}} \right)^{\text{d}} \left( {\frac{{d_{\text{np}} }}{{d_{\text{w}} }}} \right)^{\text{e}}$$
(3)
$$k_{\text{eff}} = a_{1} \left( {\emptyset_{\text{m}} } \right)^{{b_{1} }} \left( {V_{\text{r}} } \right)^{{{\rm c}_{1} }} \left( {\frac{{T_{\text{nf}} }}{{T_{\text{ref}} }}} \right)^{{{\rm d}_{1} }} \left( {\frac{{{\rm d}_{np} }}{{{\rm d}_{\text{w}} }}} \right)^{{{\rm e}_{1} }} \left( {\frac{{k_{\text{np}} }}{{k_{\text{bf}} }}} \right)^{{{\rm f}_{1} }}$$
(4)

where keff is effective thermal conductivity, \(\emptyset_{\text{m}}\)  = mass fraction of EG:W-based nanofluids, Vr = EG/water ratio (volume/volume), Tnf = the temperature of nanofluid in K, Tref = reference temperature in K (273 K), \(d_{\text{np}}\) = nanoparticle diameter (nm), \(d_{\text{w}}\) = water molecule size in nm, \(k_{\text{np}}\) = thermal conductivity of nanoparticles (W m−1 K−1), \(k_{\text{bf}}\) = thermal conductivity of the base fluid (W m−1 K−1).

Artificial neural network (ANN) modelling

ANN modelling is a computational model based on the structure and functions of biological neural networks. Due to the variation of data in a nonlinear pattern, achieving the output data prediction by the conventional method is quite difficult. ANN can provide better output by learning the nonlinear pattern of data through a modelled network. ANN modelling consists of three different layers with a number of neurons in each layer. Input neurons are the first layer (input layer), these neurons send data to the second layer (hidden layer), then output neuron to the third layer (output layer). The neurons in all layers are interconnected with each other with a mass coefficient as shown in Fig. 3a, b. Similar topology was used for the multiple variable input as used in Eqs. (3) and (4). Each neuron multiples these mass coefficients with a received input and adds up to get the output using a transfer function as shown in Fig. 3b. The processing of data takes place until the difference between the successive outputs data reaches a minimum. This can be represented in terms of ANN characterization parameters like masses (m), biases (b) and a function (f). The processing of output followed by the equation,

$$Y_{\text{i}} = f\left( {\mathop \sum \limits_{i = 1}^{n} w_{\text{ij}} x_{\text{i}} + b_{\text{i}} } \right)$$
(5)

where \(Y_{\text{i}}\), \(x_{\text{i}}\) and n are the output, input and number of neurons that connect to the ith neuron, respectively. \(w_{\text{ij}}\) and \(b_{\text{i}}\) are the mass coefficients and bias, respectively. In this study, MLP-FF backpropagation ANN neural network containing three-layer perceptron was used. In order to optimize the results, hidden layers and neurons were varied to achieve a better prediction. This modelling process contains three objectives. (1) To compare the experimental data with the proposed Eq. (2) and ANN modelling (2) to optimize the ANN model to predict the effective thermal conductivity (material specific) considering the literature data and present experimental data, and comparing with regression Eq. (3) and (3) finally, to check the prediction of effective thermal conductivity considering present data and literature data of all materials (Table 1) together with Eq. (4) and ANN modelling.

Fig. 3
figure 3

ANN topology for the modelling of effective thermal conductivity

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Modelling was performed using NNTOOL in MATLAB software. The formulation of ANN modelling follows three steps, training, testing and validation. Training is the process of predicting appropriate masses and biases in order to recognize the specific relationship between the target and input functions. Levenberg–Marquardt learning algorithm was selected for learning process. Neural transfer functions used for hidden layer and output layer are tan-sigmoid and pure line, respectively. In this study, 75% of data (input) were randomly taken for training and the remaining 25% of data used for testing. The performance of model and regression equation prediction was evaluated using mean absolute deviation (MAD), mean square error (MSE) and mean absolute percentage error (MAPE) as follows.

$${\text{MAD}} = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left| {Y_{\text{e}} - Y_{\text{p}} } \right|$$
(6)
$${\text{MSE}} = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left( {Y_{\text{e}} - Y_{\text{p}} } \right)^{2}$$
(7)
$${\text{MAPE}} = \frac{1}{N}\mathop \sum \limits_{i = 1}^{N} \left| {\frac{{Y_{\text{e}} - Y_{\text{p}} }}{{Y_{\text{e}} }} \times 100} \right|$$
(8)

where \(Y_{\text{e}}\), \(Y_{\text{p}}\) and N are the experimental value, predicted value from the ANN and number of data points.

Results and discussion

Effective thermal conductivity of nanofluids

In this study, thermal conductivity measurements at various temperatures (30, 35, 40, 45, 50, 55 and 60 °C) were carried out for nanoparticles of different materials (CuO, Al2O3 and TiO2) with five different mass concentrations from 0 to 2%. Each measurement was repeated three times and the average value of the measurements was considered. The enhancement of thermal conductivity is presented in the form of effective thermal conductivity (knf/kbf).

Figure 4a, b shows the effect of temperature and concentration on the effective thermal conductivity of EG:W–CuO nanofluids at different temperatures and mass concentrations. It can be observed from Fig. 4a that thermal conductivity increases to a maximum of 2.3% for 0.2% of concentration at 30 °C, and at 60 °C showed 6.16% enhancement compared to the base fluid. On the other hand, 2% mass concentration at 30 °C showed 8.53% of enhancement, and at 60 °C, it resulted in 14.53% enhancement.

Fig. 4
figure 4

a Effect of temperature on effective thermal conductivity for EG:W–CuO nanofluid, b effective thermal conductivity of EG:W–CuO nanofluids at a different mass concentrations

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Figure 5a, b shows the effective thermal conductivity increase as a function of temperature for EG:W–Al2O3 nanofluids. From these figure, it can be observed that thermal conductivity enhancements were 1.84% and 4.40% for 0.2% concentration at 30 °C, and at 60 °C, respectively. On the other hand, nanofluid at 2% mass concentration showed an enhancement of 7.14% and 9.69% at 30 °C and 60 °C, respectively.

Fig. 5
figure 5

a Effect of temperature on effective thermal conductivity for EG:W–Al2O3 nanofluid, b effective thermal conductivity of EG:W–Al2O3 nanofluid at a different mass concentrations

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Figure 6a, b shows the effect of temperature and concentration on effective thermal conductivity with a temperature range of 30–60 °C for EG:W–TiO2 nanofluids. The maximum thermal conductivity enhancement is 3.07% for 0.2 mass% mass concentration at 60 °C compared with base fluid (EG:W). Nanofluid of 2% mass concentration showed an enhancement of 3.22% and 6.35% at 30 °C and 60 °C, respectively.

Fig. 6
figure 6

a Effect of temperature on effective thermal conductivity for EG:W–TiO2 nanofluid, b effective thermal conductivity of EG:W–TiO2 nanofluids at a different mass concentrations

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The results showed a significant improvement in effective thermal conductivity due to the addition of nanoparticles. Thermal conductivity of EG:W-based nanofluid is higher than that of the base fluid at all concentrations. Thermal conductivity of nanofluids increases with temperature as in the case of the base fluid. Temperature and concentration of all nanofluids (CuO, Al2O3 and TiO2) showed a significant effect on effective conductivity. Nevertheless, maximum enhancement was found at a higher concentration due to the higher thermal conductivity of solid particles. It can be observed from Figs. 4–6 that effective thermal conductivity increases with the temperature and concentration for all the nanofluids. A similar trend was also well reported in the literature for the CuO, Al2O3 and TiO2 nanofluids [28, 29, 34, 35, 54].

Studies in literature indicate that aggregation of nanoparticles in suspensions influences thermal conductivity. To minimize the formation of agglomeration, sodium dodecyl benzene sulphonate (SDBS) surfactant was added in the present study. The addition of a surfactant is intended to help particles dispersed in base fluids. As a result of the addition of a surfactant, the particle agglomeration is reduced by the formation of a nanoparticle chain in the base fluid [55]. The enhancement of thermal conductivity with increased nanoparticle loading was due to the collisions between the particles. Brownian motion of nanoparticles increases at high fluid temperatures and the viscosity of nanofluid also decreases. With an increased Brownian motion, the role of micro-convection in heat transport increases which results in enhancement of thermal conductivity of nanofluids [29, 30, 35]. Another possible reason could be the nanolayer formation between the solid–liquid interfaces. The liquid molecules are known to form layered structures very near to the particle surface and behave more like solid. The formation of these layer structures further creates a thermal bridge between the solid particles and liquid [36].

The effective thermal conductivity was maximum for the CuO nanofluids compared to Al2O3 and TiO2 nanofluids as can be seen in Table 5 and Fig. 7. Al2O3 nanofluid showed better enhancement than the TiO2 nanofluid. This can be attributed to the solid particles of the respective materials possessing higher thermal conductivity. However, it can also be observed that EG:W–Al2O3 nanofluid gives an enhancement of less than 4% as compared to the EG:W CuO nanofluids at 2 mass% of concentration. Due to its lower density, EG:W–Al2O3 can provide better stability compared to EG:W–CuO nanofluids. The selection between these two materials has to be done considering all the above factors.

Table 5 Thermal conductivity enhancement (percentage) of EG:W-nanofluids for a temperature range of 30–60 °C
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Fig. 7
figure 7

Effect of three oxide nanopowders on effective thermal conductivity with respect to mass concentration at 30 °C

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Development of correlation

Considering present experimental data

Regression was performed using experimental data to determine the correlation constants of A, B and C in Eq. (2). Correlation constants and R2 values are given in Table 6. Performance of the model [Eq. (2)] was evaluated using the experimental data from the present study. The maximum and minimum deviations for the CuO, Al2O3 and TiO2 nanofluids are reported in Table 7.

Table 6 A, B, C and R2 values for Eq. (2)
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Table 7 Performance of Eq. (2) in terms of deviation from experimental results
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Validation of developed correlation

For predicting the results of the present study, the performance of the proposed correlation [Eq. (2)] was compared with the literature correlations for CuO, Al2O3 and TiO2 nanoparticles in EG:W base fluid and with few classical mixture models. The comparison is shown in Figs. 8–10. It can be observed from Fig. 8 that at lower concentration Sundar et al. [29] correlation could not predict the effective thermal conductivity of EG:W–CuO which is used in the present study. This is due to the fact that the correlation developed by them was for higher concentration and 50:50 EG:W mixture. Figure 9 shows the comparison of keff, for Al2O3 nanoparticles. It can be seen that the model of Chiam et al. [35] performs better than the other two models. This may be attributed to the closer ratios of EG:W mixture (35:65 in the present study and 40:60 in Chiam et al.). Figure 10 shows the effective thermal conductivity comparison for EG:W–TiO2 nanofluids. The studies by Hamid et al. [34] and Reddy et al. [28] involved different particle size, base fluid ratio and concentration. The experimental results from the present study showed higher thermal conductivity than the models proposed by Maxwell for all mass concentrations (Figs. 8–10). It has been observed that thermal conductivity increases with an increase in mass concentration in both experimental data and developed models. However, the calculated thermal conductivity for all nanofluids by the Maxwell model is lower than the current experimental data. This is due to the fact that the classical model does not consider other factors affecting thermal conductivity, such as the interaction between the particle and the liquid, size of the particle, the Brownian motion of the particles and the aggregation of the particles.

Fig. 8
figure 8

Comparison of effective thermal conductivity of CuO nanofluid from the present study and literature models at 30 °C

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Fig. 9
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Comparison of effective thermal conductivity of Al2O3 nanofluid from the present study and literature models at 30 °C

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Fig. 10
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Comparison of effective thermal conductivity of TiO2 nanofluid from the present study and literature models at 30 °C

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Considering literature data

The correlation proposed previously [Eq. (2)] considers temperature and concentration only. However, thermal conductivity also depends on the factors such as volume ratio of EG:W, nanoparticle size and thermal conductivity of nanoparticles. Therefore, considering all the factors regression was carried out for Eq. (3) using the experimental results from the present study and literature, for the three materials separately. Regression constants and correlation coefficients are given in Table 8. Subsequently, a generalized equation was developed for all the materials (Eq. (4); Table 9). It can be seen from Tables 8 and 9 that, the fit is better for individual materials than the combined correlation (as indicated by R2 values).

Table 8 Regression constants and R2 value for Eq. 3
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Table 9 Regression constants and R2 value for Eq. 4
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ANN modelling

Considering present experimental data

The regression for Eq. (2) gave a regression coefficient of about 0.95. For a better prediction, MLP-ANN modelling was carried out to predict the effective thermal conductivity. Experimental results were used as a target for training and validating with the input data. Neural networks with a single hidden layer with 6, 8 and 10 neurons were tried for modelling the experimental results. Table 10 shows the ANN modelling results for the aforementioned neural network. The modelling coefficient (R) between the yield model and the training set, validation set, testing set and finally the entire experimental data are shown in Table 10 for evaluation criteria. It can be observed from the tables that, ANN modelling with less number of neurons in a single hidden layer predicts effective thermal conductivity for all materials with limited data. However, eight neurons in a single hidden layer structure were considered to present the ANN modelling result due to the optimum value of the MSE and modelling coefficient as shown in Table 10.

Table 10 ANN modelling results for EG:W-based nanofluids considering present experimental results
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The prediction of effective thermal conductivity of CuO, Al2O3 and TiO2 nanofluids from ANN modelling is shown in Figs. 11–13, respectively, for eight hidden neurons in a single hidden layer of neural network. The predicted data of effective thermal conductivity obtained from ANN are compared with the experimental data and Eq. (2) as shown in Figs. 14–16. From these figures, it can be seen that prediction of the results by ANN is significantly better. ANN modelling and Eq. (2) are further compared in terms of different parameters of error such as MAD, MSE and MAPE in Table 11. It can be seen from Table 11 that ANN modelling provides a minimum MAD, MSE and MAPE compared to Eq. (2) for all the nanofluid.

Fig. 11
figure 11

Optimized ANN modelling regression for EG:W–CuO nanofluid

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Fig. 12
figure 12

Optimized ANN modelling regression for EG:W–Al2O3 nanofluid

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Fig. 13
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Optimized ANN modelling regression for EG:W–TiO2 nanofluid

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Fig. 14
figure 14

Effective thermal conductivity of ANN and proposed correlation comparison for EG:W–CuO nanofluid

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Fig. 15
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Effective thermal conductivity of ANN and proposed correlation comparison for EG:W–Al2O3 nanofluid

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Fig. 16
figure 16

Effective thermal conductivity of ANN and proposed correlation comparison for EG:W–TiO2 nanofluid

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Table 11 Statistics for ANN model and Eq. (2) considering present experimental results
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Considering present experimental data and literature data

Modelling was carried out previously using experimental results from the present study only, considering non-dimensional temperature and concentration as input variables. Equation (3) was proposed for performing the regression considering the literature data with each material separately. The values of the regression coefficient for Eq. (3) obtained for all three materials are significantly less than 1 (shown in Table 8) indicating scope for a better model. Artificial neural network (ANN) modelling was carried out considering the results from the present study and literature data.

Table 12 shows the results for different configurations used to determine the optimum neural network for predicting the thermal conductivity of EG:W nanofluids for each material separately. Table 12 shows the results for different hidden layer neurons for EG:W nanofluids of three materials. The single hidden layer with neurons 10, 12 and 6 (CuO, Al2O3 and TiO2) gave optimized results with minimum MSE. The corresponding optimized ANN results are shown in Figs. 17–19. These results were compared with the experimental, literature results and Eq. (3) as shown in Figs. 20–22. It can be observed from the figure that ANN modelling performs better prediction than Eq. (3) for EG:W nanofluids with large amount of data.

Table 12 ANN modelling results for EG:W-based nanofluids considering literature data
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Fig. 17
figure 17

Optimized ANN modelling of EG:W–CuO nanofluids considering literature data

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Fig. 18
figure 18

Optimized ANN modelling of EG:W–Al2O3 nanofluids considering literature data

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Fig. 19
figure 19

Optimized ANN modelling of EG:W–TiO2 nanofluids considering literature data

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Fig. 20
figure 20

Comparison of ANN model and Eq. (3) with present data and literature data of EG:W–CuO nanofluid

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Fig. 21
figure 21

Comparison of ANN model and Eq. (3) with present data and literature data of EG:W–Al2O3 nanofluid

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Fig. 22
figure 22

Comparison of ANN model and Eq. (3) with present data and literature data of EG:W–TiO2 nanofluid

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Considering literature data and present data of all materials together

The generalized Eq. (4) developed previously from the regression is not satisfactory since the R2 value is 0.693 (Table 9). Therefore, ANN modelling was carried out considering the same input parameters in a non-dimensional form as used in Eq. (4) and effective thermal conductivity as output. In this case, it is important to optimize the hidden layer neurons as the input variables are more compared to previous case. Hence, the optimization of neural network structure was carried out using both single and two hidden layers with varying the neurons. Table 13 gives the details of the hidden layer with different neurons. The neural network with two hidden layers containing twelve neurons was found to have the least MSE and therefore the most favourable structure for modelling the effective thermal conductivity. Figure 23 shows the performance of optimized number of neurons from ANN modelling. ANN results so obtained are compared with Eq. (4) in Fig. 24. ANN modelling shows better prediction than Eq. (4). The effect of five parameters on effective thermal conductivity for a wide range of data (literature and current data) can be obtained from the optimized neural network. Thus, the neural network structure of the MLP offers an efficient way to predict the properties of nanofluids over a wide range of conditions.

Table 13 ANN modelling results for EG:W-based nanofluids considering literature data and present study for the three nanofluids
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Fig. 23
figure 23

Optimized ANN modelling of EG:W nanofluids considering literature data of three materials

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Fig. 24
figure 24

Comparison of ANN and correlation developed considering literature data and present study

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Conclusions

In this study, the effective thermal conductivity of CuO, Al2O3 and TiO2 nanoparticles dispersed in EG:W mixture of 35/65 volume by volume was investigated. Results showed that thermal conductivity of EG:W nanofluids is higher compared to the base fluid. Thermal conductivity of the nanofluid follows the same trend as that of the EG:W mixture and increases with temperature. The maximum enhancement of thermal conductivity for CuO, Al2O3 and TiO2 nanoparticles in EG:W mixture was 8.53%, 7.14% and 3.22% at 30 °C and 14.54%, 9.69% and 6.35% at 60 °C, respectively, for 2% of mass concentration. Empirical correlations for CuO, Al2O3 and TiO2 in 35:65 ratio of EG:W mixture were developed. The proposed correlation predicted the experimental results with a deviation of ± 1.16%. However, ANN modelling improved the prediction compared to the correlation. Subsequently, literature data for EG:W mixture-based CuO, Al2O3 and TiO2 nanofluids were considered to develop the correlation separately for the three materials. Further, a generalized correlation (for three materials) was developed considering literature data which gave R2 values of 0.846, 0.831 and 0.663 for CuO, Al2O3 and TiO2 nanofluids, respectively, and generalized correlation for all materials R2 of 0.692. Prediction was significantly improved for all the above cases by ANN modelling. Hence, the effect of several influencing factors on effective thermal conductivity can be modelled using optimized neural network for a wide range of data. Therefore, the neural network structure of the MLP can be considered as a reliable tool for modelling experimental data of nanofluids.