Using Neural Network for Determination of Viscosity in Water-TiO 2 Nanofluid

Abstract

Using nanofluids is a novel solution to enhance heat transfer. This study tries to extract the model of viscosity changes in water-TiO₂ nanofluid through examining the effect of temperature and volume fraction on the viscosity. Results were recorded and analyzed within temperature range of 25 to 70°C with increments of five for 0.1, 0.4, 0.7, and 1% volume fractions. The obtained results demonstrated that the viscosity of this nanofluid decreases by increasing the temperature and increases by raising the volume fraction. The results show that conventional correlations are unable to properly predict nanofluid viscosity especially at high volume fractions. A model was developed by the data obtained from experiments to estimate viscosity of water-TiO₂ nanofluid based on two variables of temperature and volume fraction using neural network. The proposed model was qualified as highly competent for determination of nanofluid viscosity.

1. Introduction

Various techniques are applied to enhance heat transfer. Increasing the surface area of heat transfer is a common method which is used for example by fins. Vibration of heated surfaces, injection, or suction of fluid and creation of magnetic field are other ways of enhancing heat transfer. It goes without saying that implementation of these techniques in general applications is both difficult and expensive. Thus, researchers are looking for new alternative methods to enhance heat transfer of fluids.

Fluids like water, ethylene glycol, engine oil, and so forth which have industrial applications show rather poor thermal properties in comparison with metals and metal oxides. Therefore, these are expected to show better transport properties including thermal conductivity by suspension of solid particles in liquid phase. Numerous studies have emphasized on thermal conductivity of nanofluids. However, very few studies have been performed on the viscosity of nanofluids whereas viscosity of nanofluids is as significant as their thermal conductivity.

Viscosity is described as internal resistance of the fluid to movement and it is an important property in heat transfer for all applications of fluids. Pumping power and Convection Heat Transfer Coefficient are dependent on fluid viscosity, while pressure loss is directly related to viscosity in a laminar flow. Viscosity is as important as thermal conductivity in engineering systems including fluid flow [1]. Numerous studies have been recently done on nanofluids but most of them have taken into account their heat transfer characteristics. In this regard, heat transfer enhancement [2–4] which was along with previous work of the authors of this paper [5], measurement of thermal conductivity [6, 7], effective thermal conductivity [8–13], and estimation of thermal conductivity [14] can be mentioned. However, some review papers have just emphasized on thermal conductivity [15, 16]. Although viscosity is an important property for nanofluid, few studies have been implemented in this context so far. Das et al. have concentrated on the significance of studying viscosity of nanofluids [17], while Murshed et al. have demonstrated the importance of volumetric concentration effect on viscosity [18]. Ramesh and Prabhu [19] have summarized several earlier studies on viscosity of nanofluids in their review paper. Another review paper by Mahbubul et al. [20] has tried to assess the previous works done on viscosity of nanofluids more completely.

Most of the existing studies have investigated the effect of concentration on viscosity of nanofluids and just a few of them have launched to evaluate the effect of temperature. Prasher et al. [21] demonstrated that viscosity is increased by raising the concentration of particles. They reported a 10% increase in the viscosity of nanofluids. Chevalier et al. [22] measured the viscosity of SiO₂-ethanol nanofluid at room temperature for three different particle sizes and in volume fractions of 1.4 to 7%. The viscosity was considerably increased with raising the concentration of nanofluid in their work. Pak and Cho [23] used a Brookfield viscometer to measure the viscosity of water-alumina and water-TiO₂ nanofluids at room temperature. The obtained results showed that the viscosity tends to rise by increasing the volume concentration. They also found that the relative viscosity of water-TiO₂ and water-alumina nanofluids were 3 and 200 in 10 vol.%, respectively, with the obtained viscosity being much greater than the value estimated by Batchelor's equation [24]. He et al. [25] studied heat transfer and flow characteristics of water-TiO₂ nanofluid at 22°C using Bohlin CVO rheometer (Malvern Instrument). The nanoparticles employed were TiO₂ of 95 nm size. Their results showed that the viscosity is increased by raising the volume fraction, similar to the trend observed for increased size of nanoparticles. Murshed et al. [26] studied viscosity and thermal conductivity of nanofluids from suspended TiO₂ and alumina in deionized water and ethylene glycol as well as aluminum nanoparticles suspended in ethylene glycol and engine oil both experimentally and theoretically. For viscosity measurements, a rheometer was used at temperatures of 20 to 60°C. Their results demonstrated that both thermal conductivity and viscosity were higher than those of the base fluid, while they rise by increase in the volume fraction.

A few studies have taken into account the effect of temperature on the viscosity of nanofluids. Yang et al. [27] measured the viscosity of a nanofluid comprised of graphite nanoparticles at four different temperatures (35, 43, 50, and 70°C). They showed that kinematic viscosity is decreased with raising the temperature. Nguyen et al. evaluated changes in the viscosity of Al₂O₃-water and CuO-water nanofluids versus temperature for 21–75°C. They reported that the viscosity of nanofluids is reduced by increasing the temperature [28, 29]. Chen et al. [30] examined the effect of temperature on the viscosity in nanofluid of water-carbon nanotubes experimentally. Temperature was varied in the range of 5 to 65°C and they observed that the relative viscosity is noticeably increased over 55°C. Kole and Dey [1] studied the viscosity of engine oil-alumina nanofluid in temperatures of 10 to 50°C for 0.1 to 1.5% volume fractions. They finally concluded that the viscosity of nanofluid is decreased by increasing the temperature. Duangthongsuk and Wongwises [31] examined the viscosity of water-TiO₂ nanofluid in various temperatures and volume fractions. The size of nanoparticles was selected 21 nm in their research, while 0.2, 0.6, 1.0, 1.5, and 2.0 volume fractions were used in pH values of 7.5, 7.1, 7, 6.8, and 6.5 at temperatures between 15 to 35°C. An ultralow viscosity rotary rheometer (Gemini 200 HR Nano) was utilized to test the viscosity. The results were indicative of smaller and greater viscosity values with increasing temperature and volume concentration of particles, respectively.

However, various studied are rather controversial. For example, some studies report a Newtonian behavior while some others indicate a non-Newtonian one. The work conducted by Prasher et al. demonstrated that the viscosity changes linearly versus volume fraction [21], while Nguyen et al. proposed an exponential correlation between viscosity of nanofluid and volume fraction of particles [28]. Furthermore, some authors have claimed that existing correlations can predict the viscosity of nanofluid, while some others have declared that the existing correlations are unable to predict the viscosity of nanofluid and they need to develop new correlations for the nanofluids. It seems that more experimental works are required to identify the effect of temperature and volume fraction on viscosity of nanofluid.

Modeling amazing characteristics of brain in an artificial system has always been challenging. Artificial neural networks which are inspired from structure of men's brain are recently being developed and advanced in terms of quality, quantity, and capability. Number of various techniques for neural calculations is growing in recent years, while its scientific and practical activities have been developed in engineering and technical issues including control systems, signal processing, identifying and making models. Analytical computer programs need huge calculations to generate accurate answers which are time-consuming and entail utilization of powerful computers. When the number of parameters under study is increased, time and volume of calculations will grow exponentially. Thus, neural networks can be used with the capability of learning different models instead of complicated analytical correlations. Time and volume of calculations can be significantly reduced by application of this tool in CFD. Artificial neural networks are used in various fields of engineering. This tool has been previously utilized in modeling and prediction of Stainless Steel Clad Bead [32], heat exchangers for cooling applications [33], and analysis on performance of heat pumps [34]. Neural network was also used in one earlier work by the authors of this article to predict the quality of bread in cooking process [35]. However, just a few studies have recently adopted this method in their estimations for nanofluids. Papari et al. benefited from neural network to predict thermal conductivity of several nanofluids which contain single walled and multiple walled carbon nanotubes [36]. Hojjat et al. used it to model thermal conductivity for the non-Newtonian nanofluid of Carboxymethyl Cellulose (CMC) aqueous solution containing nanoparticles of aluminum oxide, copper oxide, and TiO₂ [37]. Santra et al. predicted convection heat transfer of water-copper nanofluid in laminar flow by neural network. They assumed the nanofluid as non-Newtonian and used resilient-propagation (RPROP) algorithm to train the neural network. They extracted data required for training neural network from the results of numerical simulation by finite volume method and SIMPLER algorithm. They finally demonstrated that neural network predicts heat transfer of the nanofluids properly [38].

This study was implemented to experimentally investigate the viscosity of water-TiO₂ nanofluid in addition to find a model to determine the viscosity of nanofluid. The viscosity was measured in terms of temperature and volume fraction in a laboratory setup and then applied in the neural network. It will be shown that trained neural network is able to successfully correlate the viscosity in terms of temperature and volume fraction.

2. Description of Setup

Viscosity measurement setup is formed of three main units, namely, viscometer, constant temperature bath, and temperature controlling unit (Figure 1).

A glass viscometer made by Witeg (Germany) was considered for this purpose. The selected viscometer was a Canon-Fenske reverse flow type working at 0.4 to 2 cP.

Constant temperature bath unit was comprised of a glass container which keeps the viscometer, in addition to a pump and a container in which heaters are located. The constant temperature bath was considered to study the effect of temperature on viscosity of the nanofluid. Two electric heaters (100 W, 12 v; 1500 W, 220 v) were utilized. The preheating heater automatically turns off after reaching the desired temperature and the other heater performs controlling task. Temperature of the water inside bath is measured in certain time steps and the heater is automatically adjusted by the controlling system in order to keep the temperature inside the bath constant. Since the temperature of water pumped from the tank might be reduced after passing through the pipes. Thus, a $p t - 100$ temperature sensor was devised inside the bath in order to control the temperature based on the temperature inside the bath (and not the tank).

Temperature controlling unit includes a temperature sensor and a heater controller, which is connected to the data acquisition system and finally controlled by computer.

Figure 1:

Schematic view of viscosity measurement setup.

3. Calculation of Viscosity from Experiments

Dynamic and kinematic viscosities are given by

\begin{matrix} μ = c ρ t, \\ ν = c t, \end{matrix}

(1)

where μ and ν are dynamic and kinematic viscosities. c is calibration coefficient obtained from calibrating the setup with a fluid of given viscosity content. The time measured for passing of the fluid from specific part of viscometer in each test is recorded and thereby, the viscosity is calculated.

The time can be recorded twice during each run in this viscometer. Each of the bubbles has its own specific constant. The constants of this viscometer for first and second bubbles were found as 0.0025443 and 0.0018690, respectively.

4. Nanofluid Preparation

The water-TiO₂ nanofluid for this experiment was prepared by two-step method in volume fractions of 0.1, 0.4, 0.7, and 1%. For this purpose, the required amount of nanoparticles was calculated first and added to water then. Afterwards, the nanofluid was poured into a magnetic stirrer (Heidolph, Germany) to make a homogenous mixture. After mixing, the nanofluid was put in an ultrasonic device (up 200 s ultrasonic homogenizer) to disintegrate agglomerates of particles and also to improve stability of the nanofluid using ultrasonic waves. At the end, stable nanofluids were produced after mixing and application of ultrasonic waves, without any sign of agglomeration or precipitation of particles even several months after preparing the nanofluid. The viscosity of these stable nanofluids was independent of time for prolonged periods.

5. Viscosity Measurement of Nanofluid

As mentioned previously, viscosity is an important property of a nanofluid. The number of published papers about nanofluid viscosity was noticeably smaller than that of thermal conductivity. The measurements were performed at volume fractions of 0.1, 0.4, 0.7, and 1% and temperatures of 25, 30, 35, 40, 45, 50, 55, 60, 65, and 70°C. Experiments were replicated three times and the average of these three runs was taken due to the very small difference between tests.

In this experiment, uncertainty of the experimental data is due to parameters such as time, temperature, and mass of nanoparticles (since the volume fraction depends on the mass). The uncertainty was calculated according to the method that has been used by Xuan and Li [39] in their experiments. The error of the temperature reading from calibrated pt-100 sensor was less than 0.1%. The measurement error of time and mass was less than 1% and 2%, respectively. Hence, the uncertainty of the measured viscosity was estimated less than 3%.

6. Artificial Neural Network

Multilayer perceptron neural network was used to model the viscosity of nanofluid. This kind of neural network is formed of several layers, with some neurons in each layer. Each of the neurons in a layer is connected to neurons of the next layer via coefficients called weight coefficients. The neural network determines the correlation between input and output variables by updating weights and biases. Values of these weight coefficients change during training the network. Moreover, an activation function is defined on the neurons of each layer which are applied on the sum of the weighted inputs and the bias of each neuron to generate the neuron output. Making an artificial neural network for modeling includes the three following steps: producing required data for training the network, evaluating different structures of the neural network to choose optimal structure, and eventually, testing the neural network using the data not used previously for training the network. Although the neural network might fit the training data properly, it would be exposed to great error when network validity is assessed by new data. This problem is called overfitting. It is possible to prevent this problem and to improve network generalization by modifying the performance function.

Temperature and volume fraction were taken as input, while viscosity of nanofluid was considered as output of the network (Figure 2).

Figure 2:

Neural network structure.

Backpropagation algorithm, Quasi-Newtonian training, and regularization technique were adopted in this modeling. The neural network learns with the backpropagation algorithm via updating the weight coefficients and biases. This study benefits from Mean Relative Error (MRE) and coefficient of determination ( $R^{2}$ ) to study performance of the neural network. Moreover, tan-sigmoid activation function was used in the hidden layers while linear activation function was employed in the output layer. Quasi-Newtonian method is based on Newtonian method but there is no need to calculate second derivatives. An approximate Hessian matrix is updated in each iterating of the algorithm. The update is calculated as a function of gradient.

Performance function is modified in regularization technique in order to improve generalization of the network. Performance function is used to calculate error during training, the most common type of which is applied for feedforward neural networks and is called mean sum of squares of the network errors given below:

\begin{matrix} mse = \frac{1}{n} \sum_{i = 1}^{n} {(e_{i})}^{2}, \end{matrix}

(2)

where e_i is the difference between desired value and value obtained by the model for each pattern.

A term including mean of the sum of squares of the network weights is added to the performance function to make the new function as follows:

\begin{matrix} mse new = γ mse + (1 - γ) msw, \end{matrix}

(3)

where

\begin{matrix} msw = \frac{1}{n} \sum_{j = 1}^{n} w_{j}^{2} \end{matrix}

(4)

and $w_{j} s$ are the weights of network and γ is called performance ratio.

Using this performance function causes the neural network to have smaller weights and bias, such that the network will generate smoother answers and thus network generalization will be improved.

7. Results and Discussion

7.1. Results Analysis

Prior to measurement tests on the viscosity of nanofluid, viscosity of distilled water was measured and compared with the reference data presented in the standard textbook [40], in order to ensure validity of the results obtained from this setup. The comparison is illustrated in Figure 3 for 40, 50, 60, and 70°C temperatures. As can be observed in Figure 3 and Table 1, the results show a good consistency between measured values and reference values. Therefore, it can be claimed that the setup has been successful to measure the viscosity with small error.

Table 1:

Comparison between measured and reference values of viscosity.

Temperature (°C)	Reference viscosity (m²/s ×10⁶)	Measured viscosity (m²/s ×10⁶)	Error (%)

40	0.658	0.662	0.61
50	0.548	0.557	1.64
60	0.475	0.481	1.26
70	0.413	0.427	3.39

Figure 3:

Comparison between measured and reference values of viscosity for distilled water.

Having ensured valid performance of the setup, now it is time to measure the viscosity of water-TiO₂ nanofluid in the mentioned volume fractions and temperatures. Considering Figure 4, it can be observed that the viscosity of nanofluid is significantly dependent on temperature, such that the viscosity is reduced by raising the temperature. Variations in the viscosity of nanofluid versus temperature represent a descending trend almost similar to that of water.

Figure 4:

Viscosity of nanofluid versus temperature for various volume fractions.

Values of average and maximum increase in the viscosity of nanofluid have been summarized in Table 2 as compared with water for various volume fractions and at 10 different temperatures. It can be seen that the greatest increase was 26.1% which has occurred in the volume fraction of 1%.

Table 2:

Average and maximum values of increase in the viscosity of nanofluid as compared with water.

φ (%)	Ave. (%)	Max. (%)

0.1	3.4	5.3
0.4	8.0	11.2
0.7	13.8	18.0
1.0	21.0	26.1

The viscosity of nanofluid approximately experiences 50% decrease by increasing the temperature from 25 to 70°C for the volume fractions under study. For instance, it is decreased from 0.000989 Kg/ms to 0.000500 Kg/ms in volume fraction of 1%.

Figure 5 depicts variations of the viscosity for water-TiO₂ nanofluid versus volume fraction at various temperatures. The viscosity of nanofluid averagely rises 21% by increasing the volume fraction from 0.1 to 1.0 vol.%.

Figure 5:

Viscosity of nanofluid versus volume fraction for various temperatures.

Correlations of Einstein [41], Brinkman [42], and Batchelor [24] are the most important equations used to predict the viscosity of suspensions. Einstein's equation is one of the first correlations developed in this field, and other equations coming next were derived from it. Einstein's equation is applicable for low volume fractions (less than 2%):

\begin{matrix} μ_{eff} = μ_{bf} (1 + 2.5 ϕ) . \end{matrix}

(5)

Here, μ_eff is the viscosity of suspension; μ_bf is the viscosity of base fluid and φ is the volume fraction of particles in the base fluid.

This is a linear correlation for viscosity versus volume fraction. Numerous studies have been done so far to modify this equation. For example, Brinkman modified Einstein's equation for being used in moderate volume fractions [42]. This correlation is more acceptable than Einstein's equation among relevant researchers. Brinkman's equation which is applicable for volume fractions less than 4% is given below:

\begin{matrix} μ_{eff} = \frac{μ_{bf}}{{(1 - ϕ)}^{2.5}} . \end{matrix}

(6)

Lundgren suggested the following equation in the form of Taylor series [43]:

\begin{matrix} μ_{eff} = μ_{bf} (1 + 2.5 ϕ + \frac{25}{4} ϕ^{2} + f (ϕ^{3})) . \end{matrix}

(7)

Taking into account the effect of Brownian motion of particles on bulk stress of an almost isotropic suspension of rigid and spherical particles, Batchelor proposed the following formula [24]:

\begin{matrix} μ_{eff} = μ_{bf} (1 + 2.5 ϕ + 6.5 ϕ^{2}) . \end{matrix}

(8)

It is evident from the two abovementioned equations that they will be transformed into Einstein's equation if second and higher orders are neglected.

The results of current experiments have been depicted in Figure 6 for 25 and 70°C temperatures as compared with those from equations of Einstein, Batchelor, and Brinkman. As can be seen, the results from these three formulae almost overlap, while being rather different from experimental results obtained here since they are unable to predict the viscosity of nanofluid especially for high volume fractions. Furthermore, these equations showed a small dependency of the viscosity to volume fraction, whereas the experimental results of current work for nanofluid revealed a much greater dependency on it.

Figure 6:

Comparison of experimental results with those from conventional correlations for: (a) 25°C; (b) 70°C.

The average difference between Batchelor's equation and experimental results of current research has been listed in Table 3 for various volume fractions in percents. It is evident that this difference is intensified by increasing the volume fraction of nanoparticles. However, Batchelor's equation can be used with a small error for this nanofluid at lower volume fractions. Because, as can be seen at 0.1% volume fraction the difference is 3.1% which is relatively small but this value is greater at higher volume fractions such that neither the equation of Batchelor nor the equations of Einstein and Brinkman could be applied to predict the viscosity of nanofluid. Therefore, it seems necessary to develop new correlations for nanofluids. Due to the very close results of the three abovementioned equations, Batchelor's equation has been represented for the purpose of comparison.

Table 3:

Average difference between Batchelor's equation and results of current study.

φ (%)	0.1	0.4	0.7	1
Difference (%)	3.1	6.2	10.3	14.6

Comparison between results of this study and those provided by Duangthongsuk and Wongwises [31] on the viscosity of water-TiO₂ nanofluid has been illustrated in Figure 7 for 25 and 35°C. The results are close to each other and the small difference between these two studies can be attributed to their different size of particles. They conducted their tests for volume fractions of 0.2, 0.6, 1.5, and 2 vol.%. The differences between their results and ours were 1.2 and 1.3% at 25 and 35°C, respectively. Their results were also indicative that viscosity increases by increasing the volume fraction and decreases by raising the temperature.

Figure 7:

Comparison of experimental results with those referenced in [31].

7.2. Modeling

7.2.1. Producing the Required Data

This is one of the most important parts of modeling using neural network. As mentioned earlier, the tests were conducted at 10 temperatures and 5 volume fractions (0, 0.1, 0.4, 0.7, and 1 vol.%). Taking into consideration the viscosity recorded in any condition, 50 data series are produced totally. Data were divided into training and test parts, the former were used for training the neural network while the later were applied for validation of the network once the training was over. 45 data series out of these 50 data series were employed for training while the remaining 5 data series were applied to test the network. All data were preprocessed and scaled within range of [−1 1] in order to increase the accuracy.

7.2.2. Model of Viscosity

Since the number of layers and neurons are of significant importance in extraction of the optimal structure of the neural network, various structures were evaluated to obtain a neural network model with the ability to accurately determine the viscosity of nanofluid. The neurons within each layer varied from 2 to 14 with increments of two. Thus, 14 networks were totally assessed. Having assessed the networks, the one having 1 hidden layer and 12 neurons in this layer was chosen as the best network, which was selected based on the minimum difference between outputs of the network and results of experiments (Table 4). This model offers the viscosity with $R^{2}$ and MRE values of 0.9988 and 1.06, respectively for the testing data set. Maximum relative errors in this model were reported as 3.7% and 1.7% for test data and training data, respectively. Table 4 summarizes values of $R^{2}$ and MRE for different structures.

Table 4:

Performance of different structures of neural network.

Number of hidden layers	Number of neurons in each layer	MRE (%)		$R^{2}$
Test data	Training data	Test data	Training data

1	2	2.58	2.04	0.978	0.985
1	4	1.79	1.65	0.989	0.996
1	6	2.48	2.04	0.979	0.986
1	8	2.33	1.80	0.989	0.994
1	10	2.17	1.73	0.982	0.990
1	12	1.06	1.01	0.998	0.999
1	14	1.28	1.24	0.996	0.999
2	2	2.50	2.24	0.980	0.989
2	4	2.41	2.14	0.984	0.990
2	6	2.54	2.45	0.981	0.989
2	8	2.09	1.89	0.990	0.995
2	10	2.28	1.97	0.988	0.991
2	12	2.34	1.96	0.992	0.994
2	14	2.13	1.91	0.992	0.995

Experimental data and results obtained by the neural network have been compared in Figures 8 and 9 for the test data. It can be seen in Figure 8 that the experimental data and results of the proposed model are properly consistent. Meanwhile, it is evident from Figure 9 that the model with $R^{2}$ value close to unit is able to estimate the viscosity of nanofluid for the testing data set.

Figure 8:

Comparison of experimental results with results obtained from the proposed model for the testing data set.

Figure 9:

Correlation between experimental value and model output for the testing data set.

Furthermore, the comparison between experimental results and those obtained from neural network for the training data set is depicted in Figures 10 and 11.

Figure 10:

Comparison of experimental results with results obtained from the proposed model for the training data set.

Figure 11:

Correlation between experimental value and model output for the training data set.

It can be observed that the proposed model was able to determine the viscosity of nanofluid properly and accurately. Meanwhile, errors from training and test data sets were very close which is indicative of both appropriate division of the data into two test and training parts, and proper generalization of the network. This behavior can be attributed to modification implemented on the performance function. Values of MRE for training and test data were reported as 1.01 and 1.06%, while $R^{2}$ values for these two data sets were found 0.9986 and 0.9988, respectively, which are specifically close to each other. In spite of the relatively accurate model developed for the viscosity of this nanofluid, further studies are still needed to reach comprehensive models which are applicable for different nanofluids.

8. Conclusion

The viscosity of water-TiO₂ nanofluid was assessed in this study versus volume fraction of nanoparticles and temperature. Results were recorded and analyzed within temperature range of 25 to 70°C with increments of five for 0.1, 0.4, 0.7, and 1 vol.% concentrations. The obtained results demonstrated that the viscosity of nanofluid decreases by raising the temperature and increases by raising the volume fraction, such that the viscosity of nanofluid is averagely increased 21% with raising the concentration of nanofluid from 0.1 to 1 vol.%. It was also shown that the conventional correlations are unable to properly predict the viscosity of nanofluid, especially at high volume fractions. Therefore, it seems necessary to develop new correlations for nanofluids. A model was proposed using the data obtained from experiments to correlate the viscosity of water-TiO₂ nanofluid in terms of temperature and volume fraction based on neural network. The suggested model was shown successful to determine the viscosity of nanofluid with a great accuracy and MRE value of 1.06% for the testing data set.

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