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Predicting the effective viscosity of nanofluids for the augmentation of heat transfer in the process industries
Predicting the effective viscosity of nanofluids for the augmentation of heat transfer in the process industries Ali Aminian PII: DOI: Reference:
S0167-7322(16)32762-3 doi:10.1016/j.molliq.2016.12.071 MOLLIQ 6754
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
15 September 2016 26 November 2016 19 December 2016
Please cite this article as: Ali Aminian, Predicting the effective viscosity of nanofluids for the augmentation of heat transfer in the process industries, Journal of Molecular Liquids (2016), doi:10.1016/j.molliq.2016.12.071
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ACCEPTED MANUSCRIPT Predicting the effective viscosity of nanofluids for the augmentation of heat transfer in the process industries
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Ali Aminian*
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Faculty of Chemical, Petroleum and Gas Engineering, Semnan University, PO Box 35195-63, Semnan, Iran *
Email: [email protected]; Tel: 98 912 431 9323
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Abstract
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Effective viscosity is helpful for the thermal design tasks in process industries, especially those related to pumping devices. Therefore, rheological behavior of
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nanofluids is important from power consumption point of view. In this study, an
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artificial neural network (ANN) model is developed to predict the effective viscosity
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of nanofluids based on a high number of experimental data available in the literatures. The effects of temperature, nanoparticles volume fraction, and the size of nanoparticles on the dynamic viscosity of nanofluids determined over wide ranges of
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operating conditions. The results indicate that the presented model can accurately predict the dynamic viscosity of nanofluids compared to the most important models for the dynamic viscosity of nanofluids. Eight different types of nanofluids, namely, Al2O3-water, CuO-water, TiO2-water, SiC-water, MWCNT-water, Fe3O4-water, Niwater, and Ag-water are used to evaluate the accuracy of the proposed ANN model. The predicted effective dynamic viscosities of the nanofluids are in excellent agreement with experimental data with the AAD of 6.66% and R2-value of 0.9842. Keywords: ANN; Dynamic viscosity; Nanofluids; Nanoparticles; Prediction
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ACCEPTED MANUSCRIPT 1. Introduction Dynamic viscosity is a key feature of nanofluids, which can be used to assess the
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efficiency of a nanofluid for heat transfer applications. Traditional heat transfer fluids
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such as water, oil and ethylene glycol are suffering from poor thermal conductivity. In
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recent years, researchers are trying to intensify the heat transfer capability of traditional fluids in passive and/or active modes. Nanofluids are finely dispersed
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metal or metal-oxide nanoparticles in a base fluid in which the thermal properties of the resulting nanofluid is significantly higher than that of base fluid. However, there
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are substantial research areas requiring the knowledge about the various aspects of these outstanding heat transfer fluids, including thermal and fluid science of
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nanofluids [1, 2].
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It is well known that by adding solid particles into a base fluid the dynamic viscosity will increase. On the other hand, it is important to assure that nanofluids will exhibit
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Newtonian behavior over the entire range of nanoparticle volume fractions without the need for extra pumping power [3]. Several researchers recognized that high
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thermal conductivity of nanofluids can be attained even when the concentration of suspended nanoparticles is lower than 5 Vol.% [4-7]. Therefore, this work aims to assess the effects of particle concentration and temperature on the viscosity of studied nanofluids. Some nanofluids such as TiO2-water exhibit non-Newtonian behavior at higher loadings of about 10 Vol.%, while Al2O3-water nanofluids show at 3 Vol.% [8]. Anoop et al. [5] measured viscosity of three types of nanofluids including aluminawater, alumina-ethylene glycol, and copper oxide-ethylene glycol. They concluded that Newtonian behavior can be expected for both water-based and ethylene glycolbased nanofluids at particle concentrations ranging from 0.5 to 6 Vol.%. The 2
ACCEPTED MANUSCRIPT influence of temperature and particle size on the dynamic viscosities of Al2O3-water and CuO-water nanofluids was investigated experimentally [6]. The most influencing
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parameters affecting the viscosity of the nanofluids were temperature and particle
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volume fraction for temperature ranging from 22 to 75 C and particle volume
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fractions varied from 1% to 9.4%. The results showed that the viscosities of the alumina-water nanofluids were approximately equal for 47 and 36 nm particles at volume fractions of less than 4%. Also, CuO-water nanofluid was shown to exhibit
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the highest viscosity values among the other alternatives. Finally, it was found that the
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Einstein's formula [9], the Batchelor [10] formula, and the Brinkman [11] formula did not suitable for the nanofluids, especially those containing intermediate to high
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particle concentrations.
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Although there are several works related to the modeling of dynamic viscosity of single or hybrid nanofluids, however, there is a need for a more general model
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covering wide ranges of operating conditions for various types of nanofluids. Literature survey reveals that there is not any reported work about modeling of
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effective dynamic viscosity of different nanofluids by using an ANN model based on a high number of experimental data. For example, Afrand et al. [12] developed an ANN model for predicting the dynamic viscosity of MWCNTs-SiO2/AE40 nano lubricant comprising 48 data points. Dalkilic et al. [13] used experimental data on graphite nanofluids to predict the dynamic viscosity based on 26 experimental data points. Also, 182 experimental data applied to an ANN model to predict the viscosity of CuO/propylene glycol+water, CuO/ethylene glycol+water, SiO2/water, SiO2/ ethanol, Al2O3/water, and TiO2/ water [14]. Moreover, 30 experimental data were used to develop an ANN model for the dynamic viscosity of MWCNT/water nanofluids with 24 and 6 data points used for training and testing ANN model,
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ACCEPTED MANUSCRIPT respectively [15]. However, sufficient amount of data for validation and generalizeability purposes are needed to make every model for the effective dynamic viscosity
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of nanofluids being reliable.
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Since theoretical models underestimate the effective viscosity of nanofluids and the
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existing semi-empirical models used for computing the nanofluids effective viscosity may have slight errors in the process design of heat transfer equipments, this work aims to predict the effective viscosity of nanofluids, including Al2O3-water, CuO-
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water, TiO2-water, SiC-water, MWCNT-water, Fe3O4-water, Ni-water, and Ag-water
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at various particle volume concentrations, particle diameters, and temperatures. Moreover, the predicted data of the effective viscosity of nanofluids compared to the
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corresponding values of the existing theoretical and empirical models. The predicted
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process equipments.
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effective viscosity of nanofluids can be utilized for proper design of heat transfer in
2. Artificial neural network modeling
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2.1. ANN background
An ANN consists of an input layer, hidden layer(s) and an output layer, while the activation function of the hidden neurons can be selected as tan-hyperbolic, Log-sig, or radial basis function. In fact, an ANN model is a static or dynamic mapping between its input and output [16]. During the learning phase, each input data multiplied by a weighting factor and the results added by an activation threshold to produce an input signal into a transfer function. For a MLP network, a typical output of each node in the hidden layer is computed from the following equation: n y j f w ji xi b j j 1
(1)
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ACCEPTED MANUSCRIPT where yj is the output of the jth neuron, f denotes transfer function, bj is the jth neuron bias, wji represents the weight factor from the ith neuron to the jth neuron, xi is the ith
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input vector to the jth neuron, and n defines the number of neurons in the hidden
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layer. At first, by initializing the parameters of the network and moving toward the
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last layer the output vector of the network is computed, where the corresponding error vector between estimated and target ones can be used for optimizing the values of the network parameters (by steepest gradient, for example). The error function can be
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Oi N
i 1
ti
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E
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expressed as:
(2)
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where N is the number of elements in the output vector, Oi is the i-th element of the
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overall network output and ti represents the target value. The optimization of the
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parameters can be accomplished according to a backpropagation algorithm in which a differential optimization method is utilized for updating the new value of the network parameters.
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The activation function and the number of hidden layer(s)/neurons should be selected carefully due to the problem of overfitting. In the case of an overfitted network, the training dataset error is driven to a very low value, but when new data is subject to the network the error is high [17]. Therefore, experimental data should be divided into three sets: the training, the validation and the testing sets. 2.2. Training result In order to predict the effective viscosity of studied nanofluids, the required data points are taken from a wide variety of references [3, 5-8, 18-25, 28-30], which are used for training, validation and testing the ANN model. The nanoparticles volume fraction, temperature, nanoparticles density and diameter have been selected as input 5
ACCEPTED MANUSCRIPT variables for the effective viscosity. Experimental datasets chosen with volume fraction range from 0.01 to 11.22 Vol.%, temperature vary from 10 to 90 C and the
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nanoparticles size ranging from 7 to 100 nm. The total number of data points is 622
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for the effective viscosity, where 70% of datasets used to train the ANN model in
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order to capture the nonlinear relationships between the most influencing parameters and the target one. The model performance is also tested with 15% of data points known as validation datasets. The choice among the most commonly used transfer
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functions is a crucial task. The overall error between the network output and target
the minimum validation error.
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ones should be monitored for a set of activation functions by selecting the one with
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Simultaneously, the number of hidden layers and hidden neurons can be optimized for
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each transfer function, once the minimum error for the validating datasets reached. Furthermore, from generalize-ability point of view, unseen data are implemented to
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test the accuracy of the model for the remaining 15% of experimental data known as testing datasets, which are completely kept unseen to the developed model.
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The learning phase is carried out by using the Levenberg-Marquardt backpropagation algorithm, while the tan-sigmoid activation function is found to be the proper activation function. It is found that one hidden layer comprising nine neurons gives the minimum error between the predicted and target ones by using a trial-and-error strategy. The optimal network architecture for predicting the dynamic viscosity of the selected nanofluids as functions of state variables, namely, the nanoparticles concentration, temperature, the nanoparticles density and size is shown in Fig. 1. The ranges and numbers of the input variables for the effective viscosity of nanofluids are presented in Table 1.
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ACCEPTED MANUSCRIPT The training result of the proposed ANN model is displayed in Fig. 2. As shown, there is a good agreement between the experimental data and the trained ones with the
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correlation coefficient (R2-value) of 0.9805. The parity charts of the trained network
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for validation and testing data can be seen in Figs. 3 and 4. The correlation
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coefficients for validation and testing calculations are 0.9898 and 0.9868, suggesting that the proposed network is well trained and validated. The optimum calculated
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values of network parameters to be used in simulations are given in Table 2. 3. Results and discussions
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In order to compare the results of the presented model with other alternative models, different theoretical and empirical well-known viscosity models employed in this
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section and shown in Table 3. Different types of nanofluids are currently recognized
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by their particle density; therefore the particle density has been added as an input
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variable. In this study, different types of viscosity models for nanofluids are employed to test the accuracy of the presented ANN model, as shown in Table 3. The proposed neural network model is compared with Duangthongsuk and
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Wongwises [7, 23], Wang et al. [34], Nguyen et al. [6], Batchelor [10], Esfe et al. [25], Godson et al. [29, 30], Thomas and Muthukumar [35], Chandreasekar et al. [36], Tseng and Chen [37], Syam Sundar et al. [38], Vand [39], Khanafer and Vafa [40] models regarding the prediction of dynamic viscosity of Ni-water nanofluid at nanoparticle concentration of 0.6 Vol.% over the temperature range of 20-60 C in Fig. 5. As shown in this figure, the theoretical and empirical models have dynamic viscosity prediction errors. As a result, the underpredictions and overpredictions may cause the problems of heat transfer performance and fluidity of the system. In contrary, using the ANN model leads to a much more accurate prediction of the dynamic viscosity of Ni-water nanofluid. 7
ACCEPTED MANUSCRIPT The predicted and measured dynamic viscosities of magnetite (Fe3O4)-water nanofluid [19] are compared with those of the other models in Fig. 6. As shown, the comparison
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performs concerning the experimental data from [19] with nanoparticle concentrations
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at 1 and 2 Vol.%. It can be inferred from Fig. 6 that the experimental viscosity of
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ferrofluid is much higher than that predicted values by theoretical and empirical models, which is due to the strong interactions between nanoparticles in the base fluid medium. The semi-empirical correlations, such as Goharkhah and Ashjaee [19], are
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limited to the type of the nanofluid system. A few viscosity models considered the
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effect of temperature, while they applied to a specific nanofluid system. Hence, the need for a more general model is essential, which is taking into account the effects of
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different influencing parameters. Therefore, as shown, the proposed neural network
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model can be utilized as a tool for prediction purposes with high accuracy compared to the existing correlations. The percentage average absolute deviation (AAD%) of
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the ANN model regarding dynamic viscosity of Fe3O4-water nanofluid over the concentration range of 1-2 Vol.% is about 5%, showing the reasonable accuracy of the
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developed model.
The predicted dynamic viscosities by using the presented model are compared with experimental data for Al2O3-water nanofluid of 30 nm particles [6] at different volume concentrations and temperatures, as shown in Fig. 7. The predicted dynamic viscosities of Al2O3-water nanofluid system are in good agreement with the experimental data as shown in Fig. 7. Also, Fig. 8 shows the variation in Al2O3-water nanofluid viscosities of 50 nm particles [5] with temperature and volume concentration. It may be seen that the viscosity of the nanofluid decreases with an increase in temperature. Therefore, operating at certain temperature and volume concentration can reduce the side effects
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ACCEPTED MANUSCRIPT of flow behavior of nanofluids and power consumption. Since estimating the viscosity of nanofluids requires the proper knowledge about the size of aggregates and the
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viscous forces present in the solution, the presented ANN model can successfully
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capture these nonlinear relationships among different parameters.
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Furthermore, Fig. 9 reveals the dynamic viscosity of Al2O3-water nanofluid at different temperature and nanoparticle concentrations [6]. It can be concluded that at higher concentrations, higher temperature is needed to reduce the viscosity of the
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solution to the viscosity of water at 20 C. Moreover, the ANN model outperforms the
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correlations concerning the prediction of Al2O3-water nanofluid system with an average diameter size of 36 and 47 nm.
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Fig. 10 shows the variation of viscosity against temperature at different particle
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volume concentrations for CuO-water nanofluid. The figure shows that the Esfe et al.'s equation [25] and Nguyen et al.'s model [6] underestimate the dynamic viscosity,
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whereas the proposed ANN model accurately predicts the dynamic viscosities than those from other alternative formulas. As can be seen from this figure, at higher
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nanoparticle concentrations, the deviations between the experimental data and the empirical correlations become more obvious. The AAD of about 5% shows the accuracy of the proposed ANN model for predicting the dynamic viscosity of CuOwater nanofluid over the temperature and concentration ranges of 20-50 C and 1-9 Vol.%, respectively. Fig. 11 shows the experimental viscosity values from CuO-water nanofluid system plotted against volume fraction. From Fig. 11, the aggregation model developed by Pastoriza-Gallego et al. [21] exhibits deviation from experimental data at higher temperatures and volume concentrations. Thus, both the aggregation phenomena from the electric double layer repulsion along with the particle size distribution function of
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ACCEPTED MANUSCRIPT each class size in nanofluid systems should be considered for better understanding the rheological behavior of the nanofluids. As shown, it can be seen that the presented
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ANN model can be used successfully over the temperature range of 20-55 C and
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concentration between 0.15% and 1.7% with AAD of about 3.15%.
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Fig. 12 shows the comparison between measured dynamic viscosity of TiO2-water nanofluid [23, 24] and the ANN model result with the predicted values from wellknown correlations. The results show that there are a relatively good agreement
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between the experimental data and the predicted values by using the ANN model. The
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AAD of about 6.5% is achieved for TiO2-water nanofluid in the temperature range of 10-70 C, concentration ranging from 0.24 to 11.22%, and nanoparticle size of 21 and
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76 nm. On the other hand, the Krieger-Dougherty theory [32], which considered the
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aggregates formation underestimates the dynamic viscosity of TiO2-water nanofluid. From Fig. 12, experimental dynamic viscosity of TiO2-water nanofluid is much higher
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than that predicted by the classical dynamic viscosity models, proofing the strong interactions among the nanoparticles.
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Also, Fig. 13 demonstrates the dynamic viscosity of TiO2-water nanofluid [3, 24] as a function of particle concentration. The mean diameter of TiO2 nanoparticles are 21 and 76 nm [3, 24]. From this figure, the Batchelor's model [10], the Krieger and Dougherty model [32], and the Nielsen's model [33] are in disagreement with the experimental values for the higher nanoparticles volume fractions of ϕ>1%, while the aforementioned models underestimate the dynamic viscosity data for ϕ>1%. However, the trend observed in the proposed ANN model show a good agreement with the experimental results. The variation of dynamic viscosity of the MWCNT-water nanofluid [25] versus nanoparticle volume fraction at 25 and 55 C is depicted in Fig. 14. It is evident from
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ACCEPTED MANUSCRIPT Fig.14 that the dynamic viscosity of MWCNT-water nanofluid decreases as temperature increases. In addition, the relative viscosity increases with the increase in
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the nanoparticles volume fraction. Comparisons made between the experimental data
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for MWCNT-water nanofluid and the predictions by the ANN model with the well-
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known models for the viscosity of nanofluids. As can be seen from Fig.14, the large deviations between experimental data and the results of the theoretical and empirical models indicate the incapability of the existing models for the dynamic viscosity of
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MWCNT-water nanofluid. The AAD of about 2.72% proofs the accuracy of the
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presented ANN model for predicting the dynamic viscosity of MWCNT-water nanofluid in the temperature range of 25-55 C and volume fractions up to 1%.
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The viscosity of SiC-water nanofluid [28] is plotted in Fig. 15. Various models have
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been used to examine the accuracy of the ANN model regarding the prediction of the dynamic viscosity of SiC-water nanofluid at different temperatures and particle
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volume fractions. As shown in Fig. 15, the models of Batchelor, Nguyen et al., Wang et al., Duangthongsuk and Wongwises and so on underpredict significantly the
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dynamic viscosity of the nanofluid, even at very low volume fractions and higher temperatures. On the other hand, the AAD of 7.9% represents a reasonable agreement between the results of the ANN model and the experimental data over the temperature range of 30-70 C and the nanoparticle concentration ranging from 1 to 3 Vol.%. Figs. 16 & 17 reveal the comparisons between the experimental dynamic viscosity of Ag-water nanofluids and prediction results by the ANN model and the other existing theoretical and empirical models, such as the model proposed by Godson et al. [29]. As a whole, the AAD of about 3.92% suggesting the accuracy of the ANN model for predicting the dynamic viscosity of Ag-water nanofluids over a wide range of operating conditions. As suggested by several researchers, the viscosity depends on
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ACCEPTED MANUSCRIPT various parameters from the material properties to random micro-convective motions as well as the stability of the prepared nanofluid. Therefore, the need for a unified
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model based on the accurate experimental data can be useful as a foundation for
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principal design of heat recovery/removal systems.
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As reported by several researchers, the results indicated that there are some differences among the measured dynamic viscosity data by researchers. It may be contributed to the various influencing factors such as particle size, particle preparation
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(solution chemistry), the measurement technique or even different particle sources [7].
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As a result, the uncertainty related to the measurement techniques and nanofluid preparation methods are responsible for the discrepancies between the ANN models
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and real data. On the other hand, number of experimental data is a key point affecting
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the ANN models. However, the results contend that the presented ANN model is markedly superior to the existing correlations for dynamic viscosity of the studied
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nanofluids. Fig. 18 demonstrates the parity plot for the whole dataset used for developing the artificial neural network model. The R2-value of 0.9842 and AAD of
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6.66% suggest the accuracy and generalizeability of the proposed model for accurate predicting the effective viscosity of the studied nanofluids. 4. Conclusions
A feedforward neural network model has been developed for accurate predicting the effective viscosity of nanofluids over wide ranges of temperature and nanoparticles concentration. The effective viscosities of different nanofluids, namely, Al2O3-water, CuO-water, TiO2-water, SiC-water, MWCNT-water, Fe3O4-water, Ni-water, and Agwater with nanoparticles diameter in the range of 7 nm to 100 nm are predicted by using an artificial neural network model. The effects of different influencing factors such as nanoparticles concentration, size and type of the particles, and temperature on 12
ACCEPTED MANUSCRIPT the viscosities are investigated and compared to the other models. As nanoparticle concentration increases, the error between the predictions of the most important
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existing models and the real data rise. The underestimation problems dealing with the
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aforementioned theoretical and semi-empirical models make them unreliable for
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prediction purposes. By using the proposed model and proper determining the effective viscosity of nanofluids as functions of particle volume fraction, particle size and temperature, the pressure drop problems associated with nanofluid systems can be
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mitigated.
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ACCEPTED MANUSCRIPT References [1] S.S. Mallick, A. Mishra, L. Kundan, An investigation into modelling thermal
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conductivity for alumina-water nanofluids, Powder Technol. 233 (2013) 234-244.
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[2] L. Chen, W. Yu, H. Xie, Enhanced thermal conductivity of nanofluids containing
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Ag/MWNT composites, Powder Technol. 231 (2012) 18-20.
[3] L. Fedele, L. Colla, S. Bobbo, Viscosity and thermal conductivity measurements
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of water-based nanofluids containing titanium oxide nanoparticles, Int. J. Refrig. 35 (2012) 1359-1366.
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[4] P.K. Namburu, D.P. Kulkarni, O. Misra, D.K. Das, Viscosity of copper oxide nanoparticles dispersed in ethylene glycol and water mixture, Exp. Thermal Fluid Sci.
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32 (2007) 397- 402.
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[5] K. B. Anoop, S. Kabelac, T. Sundararajan, Sarit K. Das, Rheological and flow characteristics of nanofluids: Influence of electroviscous effects and particle
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agglomeration, J. Appl. Phys. 106 (2009) 034909-7. [6] C.T. Nguyen, F. Desgranges, G. Roy, N. Galanis, T. Mare´, S. Boucher, H. Angue
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Mintsa, Temperature and particle-size dependent viscosity data for water-based nanofluids-Hysteresis phenomenon, Int. J. Heat Fluid Flow 28 (2007) 1492-1506. [7] W. Duangthongsuk, S. Wongwises, Measurement of temperature-dependent thermal conductivity and viscosity of TiO2-water nanofluids, Exp. Therm. Fluid Sci. 33 (2009) 706-714. [8] B.C. Pak, Y. I. Cho, Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles, Exp. Heat Transfer 11 (1998) 151-170. [9] A. Einstein, Eine neue Bestimmung der Moleküldimensionen. Annalen der Physik 19 (1906) 289-306.
14
ACCEPTED MANUSCRIPT [10] G.K. Batchelor, The effect of Brownian motion on the bulk stress in a suspension of spherical particles, J. Fluid Mech. 83 (1977) 97-117.
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[11] H.C. Brinkman, The viscosity of concentrated suspensions and solution, J. Chem.
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Phys. 20 (1952) 571-581.
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[12] M. Afrand, K. Nazari Najafabadi, N. Sina, M.R. Safaei, A.Sh. Kherbeet, S. Wongwises, M. Dahari, Prediction of dynamic viscosity of a hybrid nano-lubricant by an optimal artificial neural network, Int. Commun. Heat Mass Transfer 76 (2016)
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209-214.
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[13] A.S. Dalkilic, A. Çebi, A. Celen, O. Yıldız, O. Acikgoz, C. Jumpholkul, M. Bayrak, K. Surana, S. Wongwises, Prediction of graphite nanofluids' dynamic
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viscosity by means of artificial neural networks, Int. Commun. Heat Mass Transfer 73
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(2016) 33-42.
[14] F. Yousefi, H. Karimi, M.M. Papari, Modeling viscosity of nanofl uids using
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diffusional neural networks, J. Molecular Liquids 175 (2012) 85-90. [15] M. Afrand, A. Ahmadi Nadooshan, M. Hassani, H. Yarmand, M. Dahari,
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Predicting the viscosity of multi-walled carbon nanotubes/water nanofluid by developing an optimal artificial neural network based on experimental data, Int. Commun. Heat Mass Transfer 77 (2016) 49-53. [16] D. Graupe, Principles of Artificial Neural Networks, Second ed., WSPC, USA, 2007. [17] S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice-Hall, Upper Saddle River, NJ, 1999. [18] L.S. Sundar, M.K. Singh, I. Bidkin, A.C.M. Sousa, Experimental investigations in heat transfer and friction factor of magnetic Ni nanofluid flowing in a tube, Int. J. Heat Mass Transfer 70 (2014) 224-234.
15
ACCEPTED MANUSCRIPT [19] M. Goharkhah, M. Ashjaee, Experimental investigation on heat transfer and hydrodynamic behavior of magnetite nanofluid flow in a channel with recognition of
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the best models for transport properties, Exp. Thermal Fluid Sci. 68 (2015) 582-592.
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[20] M.H. Kayhani, M. Nazari, H. Soltanzadeh, M.M. Heyhat, F. Kowsary,
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Experimental analysis of turbulent convective heat transfer and pressure drop of Al2O3 /water nanofluid in horizontal tube, Micro Nano Letters 7 (2012) 223-227. [21] M.J. Pastoriza-Gallego, C. Casanova, J.L. Legido, M.M. Pineiro, CuO in water
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nanofluid: Influence of particle size and polydispersity on volumetric behaviour and
MA
viscosity, Fluid Phase Equilibria 300 (2011) 188-196. [22] D. Kwek, A. Crivoi, F. Duan, Effects of temperature and particle size on the
D
thermal property measurements of Al2O3-water nanofluids, J. Chem. Eng. Data 55
TE
(2010) 5690-5695.
[23] W. Duangthongsuk, S. Wongwises, An experimental study on the heat transfer
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performance and pressure drop of TiO2-water nanofluids flowing under a turbulent flow regime, Int. J. Heat Mass Transfer 53 (2010) 334-344.
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[24] A. Turgut, I. Tavman, M. Chirtoc, H. P. Schuchmann, C. Sauter, S. Tavman, Thermal conductivity and viscosity measurements of water-based TiO2 nanofluids, Int. J. Thermophys. 30 (2009) 1213-1226. [25] M. H. Esfe, S. Saedodin, O. Mahian, S. Wongwises, Thermophysical properties, heat transfer and pressure drop of COOH-functionalized multi walled carbon nanotubes/water nanofluids, Int. Commun. Heat Mass Transfer 58 (2014) 176-183. [26] H. Xie, W. Yu, W. Chen, MgO nanofluids: higher thermal conductivity and lower viscosity among ethylene glycol-based nanofluids containing oxide nanoparticles, J. Exp. Nanosci. 5 (2010) 463-472.
16
ACCEPTED MANUSCRIPT [27] M.J. Pastoriza-Gallego, L. Lugo, D. Cabaleiro, J.L. Legido, M.M. Piñeiro, Thermophysical profile of ethylene glycol-based ZnO nanofluids, J. Chem.
T
Thermodynamics 73 (2014) 23-30.
IP
[28] S.W. Lee, S.D. Park, S. Kang, I.C. Bang, J.H. Kim, Investigation of viscosity and
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thermal conductivity of SiC nanofluids for heat transfer applications, Int. J. Heat Mass Transfer 54 (2011) 433-438.
[29] L. Godson, D. Mohan Lal, S. Wongwises, Experimental investigation on the
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thermal conductivity and viscosity of silver-deionized water nanofluid, Exp. Heat
MA
Transfer, 23 (2010) 317-332.
[30] L. Godson, D. Mohan Lal, S. Wongwises, Measurement of thermo physical
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properties of metallic nanofluids for high temperature applications, Nanosci.
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Microsci. Therm. Eng. 14 (2010) 152-173. [31] M. Goharkhah, A. Salarian, M. Ashjaee, M. Shahabadi, Convective heat transfer
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characteristics of magnetite nanofluid under the influence of constant and alternating magnetic field, Powder Technol. 274 (2015) 258-267.
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[32] I.M. Krieger, T.J. Dougherty, A mechanism for non-Newtonian flow in suspensions of rigid spheres, Trans. Soc. Rheol. 3 (1959) 137-152. [33] L.E. Nielsen, Generalized equation for the elastic moduli of composite materials, J. Appl. Phys. 41 (1970) 4626-4627. [34] X. Wang, X. Xu, S.U.S. Choi, Thermal conductivity of nanoparticle- fluid mixture, J. Thermophys. Heat Transf. 13 (1999) 474-480. [35] C.U. Thomas, M. Muthukumar, Three-body hydrodynamic effects on viscosity of suspensions of spheres, J. Chem. Phys. 94 (1991) 5180.
17
ACCEPTED MANUSCRIPT [36] M. Chandrasekar, S. Suresh, A. Chandra Bose, Experimental investigations and theoretical determination of thermal conductivity and viscosity of Al2O3/water nano
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fluid, Exp. Thermal Fluid Sci. 34 (2010) 210-216.
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[37] W.J. Tseng, C.N. Chen, Effect of polymeric dispersant on rheological behavior
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of nickel-terpineol suspensions, Mater. Sci. Eng. A Struct. Mater. Prop. Microstruct. Process. 347 (2003) 145-153.
[38] L. Syam Sundar, M.K. Singh, A.C.M. Sousa, Investigation of thermal
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conductivity and viscosity of Fe3O4 nanofluid for heat transfer applications, Int.
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Commun. Heat Mass Transfer, 44 (2013) 7-14.
[39] V. Vand, Viscosity of solutions and suspensions, I Theory, J. Phys. Colloid
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Chem. 52 (1948) 277-299.
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[40] K. Khanafer, K.A. Vafai, A critical synthesis of thermophysical characteristics of
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nano fluids, Int. J. Heat Mass Transfer 54 (2011) 4410-4428.
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average radius of the single particles
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average radius of the aggregates
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error function
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transfer function
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bias value of the j-th neuron
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hidden nodes number
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number of elements in the input-output vectors
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overall network output for the i-th element
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correlation coefficient
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temperature (C)
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target values of the i-th element
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net output from j-th neuron
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nanoparticle volume fraction dynamic viscosity; learning rate
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density
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effective volume fraction of solid spheres
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particle
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Fe3O4-Water
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TiO2- Water CuO- Water
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systems
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Table 1. The ranges of experimental conditions for different nanofluid systems studied in this work. Ref.
[18]
[19]
0.25-9.4
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[5], [6], [8], [20], [22]
13, 21, 25, 76
0.211.22
114
[3], [7], [23], [24]
11, 29, 30
0.15-9
130
[6], [21]
7
0.01-1
20
[25]
100
0.1-3
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[28]
60, 63
0.3-1.2
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[29], [30]
MWCNT-Water
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Ag-Water
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SiC-Water
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-1.0377
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0.8458
1.941 -4.586 1.1836 -1.0514 -0.0966 -2.1029 -1.9222 0.4339 -2.2837
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-3.2755 1.5984 -0.0057 1.1138 4.0947 0.4089 0.8949 -1.2789 -4.484
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, j=1…,9; i=1…,4
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Chandreasekar et al. 2.8 1 5200 bf 1
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[37]
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[39]
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S0167-7322(16)32762-3 doi:10.1016/j.molliq.2016.12.071 MOLLIQ 6754
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
15 September 2016 26 November 2016 19 December 2016
Please cite this article as: Ali Aminian, Predicting the effective viscosity of nanofluids for the augmentation of heat transfer in the process industries, Journal of Molecular Liquids (2016), doi:10.1016/j.molliq.2016.12.071
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ACCEPTED MANUSCRIPT Predicting the effective viscosity of nanofluids for the augmentation of heat transfer in the process industries
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Ali Aminian*
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Faculty of Chemical, Petroleum and Gas Engineering, Semnan University, PO Box 35195-63, Semnan, Iran *
Email: [email protected]; Tel: 98 912 431 9323
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Abstract
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Effective viscosity is helpful for the thermal design tasks in process industries, especially those related to pumping devices. Therefore, rheological behavior of
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nanofluids is important from power consumption point of view. In this study, an
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artificial neural network (ANN) model is developed to predict the effective viscosity
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of nanofluids based on a high number of experimental data available in the literatures. The effects of temperature, nanoparticles volume fraction, and the size of nanoparticles on the dynamic viscosity of nanofluids determined over wide ranges of
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operating conditions. The results indicate that the presented model can accurately predict the dynamic viscosity of nanofluids compared to the most important models for the dynamic viscosity of nanofluids. Eight different types of nanofluids, namely, Al2O3-water, CuO-water, TiO2-water, SiC-water, MWCNT-water, Fe3O4-water, Niwater, and Ag-water are used to evaluate the accuracy of the proposed ANN model. The predicted effective dynamic viscosities of the nanofluids are in excellent agreement with experimental data with the AAD of 6.66% and R2-value of 0.9842. Keywords: ANN; Dynamic viscosity; Nanofluids; Nanoparticles; Prediction
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ACCEPTED MANUSCRIPT 1. Introduction Dynamic viscosity is a key feature of nanofluids, which can be used to assess the
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efficiency of a nanofluid for heat transfer applications. Traditional heat transfer fluids
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such as water, oil and ethylene glycol are suffering from poor thermal conductivity. In
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recent years, researchers are trying to intensify the heat transfer capability of traditional fluids in passive and/or active modes. Nanofluids are finely dispersed
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metal or metal-oxide nanoparticles in a base fluid in which the thermal properties of the resulting nanofluid is significantly higher than that of base fluid. However, there
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are substantial research areas requiring the knowledge about the various aspects of these outstanding heat transfer fluids, including thermal and fluid science of
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nanofluids [1, 2].
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It is well known that by adding solid particles into a base fluid the dynamic viscosity will increase. On the other hand, it is important to assure that nanofluids will exhibit
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Newtonian behavior over the entire range of nanoparticle volume fractions without the need for extra pumping power [3]. Several researchers recognized that high
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thermal conductivity of nanofluids can be attained even when the concentration of suspended nanoparticles is lower than 5 Vol.% [4-7]. Therefore, this work aims to assess the effects of particle concentration and temperature on the viscosity of studied nanofluids. Some nanofluids such as TiO2-water exhibit non-Newtonian behavior at higher loadings of about 10 Vol.%, while Al2O3-water nanofluids show at 3 Vol.% [8]. Anoop et al. [5] measured viscosity of three types of nanofluids including aluminawater, alumina-ethylene glycol, and copper oxide-ethylene glycol. They concluded that Newtonian behavior can be expected for both water-based and ethylene glycolbased nanofluids at particle concentrations ranging from 0.5 to 6 Vol.%. The 2
ACCEPTED MANUSCRIPT influence of temperature and particle size on the dynamic viscosities of Al2O3-water and CuO-water nanofluids was investigated experimentally [6]. The most influencing
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parameters affecting the viscosity of the nanofluids were temperature and particle
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volume fraction for temperature ranging from 22 to 75 C and particle volume
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fractions varied from 1% to 9.4%. The results showed that the viscosities of the alumina-water nanofluids were approximately equal for 47 and 36 nm particles at volume fractions of less than 4%. Also, CuO-water nanofluid was shown to exhibit
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the highest viscosity values among the other alternatives. Finally, it was found that the
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Einstein's formula [9], the Batchelor [10] formula, and the Brinkman [11] formula did not suitable for the nanofluids, especially those containing intermediate to high
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particle concentrations.
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Although there are several works related to the modeling of dynamic viscosity of single or hybrid nanofluids, however, there is a need for a more general model
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covering wide ranges of operating conditions for various types of nanofluids. Literature survey reveals that there is not any reported work about modeling of
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effective dynamic viscosity of different nanofluids by using an ANN model based on a high number of experimental data. For example, Afrand et al. [12] developed an ANN model for predicting the dynamic viscosity of MWCNTs-SiO2/AE40 nano lubricant comprising 48 data points. Dalkilic et al. [13] used experimental data on graphite nanofluids to predict the dynamic viscosity based on 26 experimental data points. Also, 182 experimental data applied to an ANN model to predict the viscosity of CuO/propylene glycol+water, CuO/ethylene glycol+water, SiO2/water, SiO2/ ethanol, Al2O3/water, and TiO2/ water [14]. Moreover, 30 experimental data were used to develop an ANN model for the dynamic viscosity of MWCNT/water nanofluids with 24 and 6 data points used for training and testing ANN model,
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of nanofluids being reliable.
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Since theoretical models underestimate the effective viscosity of nanofluids and the
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existing semi-empirical models used for computing the nanofluids effective viscosity may have slight errors in the process design of heat transfer equipments, this work aims to predict the effective viscosity of nanofluids, including Al2O3-water, CuO-
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water, TiO2-water, SiC-water, MWCNT-water, Fe3O4-water, Ni-water, and Ag-water
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at various particle volume concentrations, particle diameters, and temperatures. Moreover, the predicted data of the effective viscosity of nanofluids compared to the
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corresponding values of the existing theoretical and empirical models. The predicted
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process equipments.
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effective viscosity of nanofluids can be utilized for proper design of heat transfer in
2. Artificial neural network modeling
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2.1. ANN background
An ANN consists of an input layer, hidden layer(s) and an output layer, while the activation function of the hidden neurons can be selected as tan-hyperbolic, Log-sig, or radial basis function. In fact, an ANN model is a static or dynamic mapping between its input and output [16]. During the learning phase, each input data multiplied by a weighting factor and the results added by an activation threshold to produce an input signal into a transfer function. For a MLP network, a typical output of each node in the hidden layer is computed from the following equation: n y j f w ji xi b j j 1
(1)
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ACCEPTED MANUSCRIPT where yj is the output of the jth neuron, f denotes transfer function, bj is the jth neuron bias, wji represents the weight factor from the ith neuron to the jth neuron, xi is the ith
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input vector to the jth neuron, and n defines the number of neurons in the hidden
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layer. At first, by initializing the parameters of the network and moving toward the
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last layer the output vector of the network is computed, where the corresponding error vector between estimated and target ones can be used for optimizing the values of the network parameters (by steepest gradient, for example). The error function can be
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parameters can be accomplished according to a backpropagation algorithm in which a differential optimization method is utilized for updating the new value of the network parameters.
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The activation function and the number of hidden layer(s)/neurons should be selected carefully due to the problem of overfitting. In the case of an overfitted network, the training dataset error is driven to a very low value, but when new data is subject to the network the error is high [17]. Therefore, experimental data should be divided into three sets: the training, the validation and the testing sets. 2.2. Training result In order to predict the effective viscosity of studied nanofluids, the required data points are taken from a wide variety of references [3, 5-8, 18-25, 28-30], which are used for training, validation and testing the ANN model. The nanoparticles volume fraction, temperature, nanoparticles density and diameter have been selected as input 5
ACCEPTED MANUSCRIPT variables for the effective viscosity. Experimental datasets chosen with volume fraction range from 0.01 to 11.22 Vol.%, temperature vary from 10 to 90 C and the
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nanoparticles size ranging from 7 to 100 nm. The total number of data points is 622
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for the effective viscosity, where 70% of datasets used to train the ANN model in
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order to capture the nonlinear relationships between the most influencing parameters and the target one. The model performance is also tested with 15% of data points known as validation datasets. The choice among the most commonly used transfer
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functions is a crucial task. The overall error between the network output and target
the minimum validation error.
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ones should be monitored for a set of activation functions by selecting the one with
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Simultaneously, the number of hidden layers and hidden neurons can be optimized for
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each transfer function, once the minimum error for the validating datasets reached. Furthermore, from generalize-ability point of view, unseen data are implemented to
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test the accuracy of the model for the remaining 15% of experimental data known as testing datasets, which are completely kept unseen to the developed model.
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The learning phase is carried out by using the Levenberg-Marquardt backpropagation algorithm, while the tan-sigmoid activation function is found to be the proper activation function. It is found that one hidden layer comprising nine neurons gives the minimum error between the predicted and target ones by using a trial-and-error strategy. The optimal network architecture for predicting the dynamic viscosity of the selected nanofluids as functions of state variables, namely, the nanoparticles concentration, temperature, the nanoparticles density and size is shown in Fig. 1. The ranges and numbers of the input variables for the effective viscosity of nanofluids are presented in Table 1.
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ACCEPTED MANUSCRIPT The training result of the proposed ANN model is displayed in Fig. 2. As shown, there is a good agreement between the experimental data and the trained ones with the
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values of network parameters to be used in simulations are given in Table 2. 3. Results and discussions
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In order to compare the results of the presented model with other alternative models, different theoretical and empirical well-known viscosity models employed in this
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section and shown in Table 3. Different types of nanofluids are currently recognized
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by their particle density; therefore the particle density has been added as an input
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variable. In this study, different types of viscosity models for nanofluids are employed to test the accuracy of the presented ANN model, as shown in Table 3. The proposed neural network model is compared with Duangthongsuk and
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Wongwises [7, 23], Wang et al. [34], Nguyen et al. [6], Batchelor [10], Esfe et al. [25], Godson et al. [29, 30], Thomas and Muthukumar [35], Chandreasekar et al. [36], Tseng and Chen [37], Syam Sundar et al. [38], Vand [39], Khanafer and Vafa [40] models regarding the prediction of dynamic viscosity of Ni-water nanofluid at nanoparticle concentration of 0.6 Vol.% over the temperature range of 20-60 C in Fig. 5. As shown in this figure, the theoretical and empirical models have dynamic viscosity prediction errors. As a result, the underpredictions and overpredictions may cause the problems of heat transfer performance and fluidity of the system. In contrary, using the ANN model leads to a much more accurate prediction of the dynamic viscosity of Ni-water nanofluid. 7
ACCEPTED MANUSCRIPT The predicted and measured dynamic viscosities of magnetite (Fe3O4)-water nanofluid [19] are compared with those of the other models in Fig. 6. As shown, the comparison
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performs concerning the experimental data from [19] with nanoparticle concentrations
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at 1 and 2 Vol.%. It can be inferred from Fig. 6 that the experimental viscosity of
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ferrofluid is much higher than that predicted values by theoretical and empirical models, which is due to the strong interactions between nanoparticles in the base fluid medium. The semi-empirical correlations, such as Goharkhah and Ashjaee [19], are
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limited to the type of the nanofluid system. A few viscosity models considered the
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effect of temperature, while they applied to a specific nanofluid system. Hence, the need for a more general model is essential, which is taking into account the effects of
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different influencing parameters. Therefore, as shown, the proposed neural network
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model can be utilized as a tool for prediction purposes with high accuracy compared to the existing correlations. The percentage average absolute deviation (AAD%) of
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the ANN model regarding dynamic viscosity of Fe3O4-water nanofluid over the concentration range of 1-2 Vol.% is about 5%, showing the reasonable accuracy of the
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developed model.
The predicted dynamic viscosities by using the presented model are compared with experimental data for Al2O3-water nanofluid of 30 nm particles [6] at different volume concentrations and temperatures, as shown in Fig. 7. The predicted dynamic viscosities of Al2O3-water nanofluid system are in good agreement with the experimental data as shown in Fig. 7. Also, Fig. 8 shows the variation in Al2O3-water nanofluid viscosities of 50 nm particles [5] with temperature and volume concentration. It may be seen that the viscosity of the nanofluid decreases with an increase in temperature. Therefore, operating at certain temperature and volume concentration can reduce the side effects
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ACCEPTED MANUSCRIPT of flow behavior of nanofluids and power consumption. Since estimating the viscosity of nanofluids requires the proper knowledge about the size of aggregates and the
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viscous forces present in the solution, the presented ANN model can successfully
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capture these nonlinear relationships among different parameters.
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Furthermore, Fig. 9 reveals the dynamic viscosity of Al2O3-water nanofluid at different temperature and nanoparticle concentrations [6]. It can be concluded that at higher concentrations, higher temperature is needed to reduce the viscosity of the
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solution to the viscosity of water at 20 C. Moreover, the ANN model outperforms the
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correlations concerning the prediction of Al2O3-water nanofluid system with an average diameter size of 36 and 47 nm.
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Fig. 10 shows the variation of viscosity against temperature at different particle
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volume concentrations for CuO-water nanofluid. The figure shows that the Esfe et al.'s equation [25] and Nguyen et al.'s model [6] underestimate the dynamic viscosity,
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whereas the proposed ANN model accurately predicts the dynamic viscosities than those from other alternative formulas. As can be seen from this figure, at higher
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nanoparticle concentrations, the deviations between the experimental data and the empirical correlations become more obvious. The AAD of about 5% shows the accuracy of the proposed ANN model for predicting the dynamic viscosity of CuOwater nanofluid over the temperature and concentration ranges of 20-50 C and 1-9 Vol.%, respectively. Fig. 11 shows the experimental viscosity values from CuO-water nanofluid system plotted against volume fraction. From Fig. 11, the aggregation model developed by Pastoriza-Gallego et al. [21] exhibits deviation from experimental data at higher temperatures and volume concentrations. Thus, both the aggregation phenomena from the electric double layer repulsion along with the particle size distribution function of
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ACCEPTED MANUSCRIPT each class size in nanofluid systems should be considered for better understanding the rheological behavior of the nanofluids. As shown, it can be seen that the presented
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ANN model can be used successfully over the temperature range of 20-55 C and
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concentration between 0.15% and 1.7% with AAD of about 3.15%.
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Fig. 12 shows the comparison between measured dynamic viscosity of TiO2-water nanofluid [23, 24] and the ANN model result with the predicted values from wellknown correlations. The results show that there are a relatively good agreement
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between the experimental data and the predicted values by using the ANN model. The
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AAD of about 6.5% is achieved for TiO2-water nanofluid in the temperature range of 10-70 C, concentration ranging from 0.24 to 11.22%, and nanoparticle size of 21 and
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76 nm. On the other hand, the Krieger-Dougherty theory [32], which considered the
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aggregates formation underestimates the dynamic viscosity of TiO2-water nanofluid. From Fig. 12, experimental dynamic viscosity of TiO2-water nanofluid is much higher
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than that predicted by the classical dynamic viscosity models, proofing the strong interactions among the nanoparticles.
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Also, Fig. 13 demonstrates the dynamic viscosity of TiO2-water nanofluid [3, 24] as a function of particle concentration. The mean diameter of TiO2 nanoparticles are 21 and 76 nm [3, 24]. From this figure, the Batchelor's model [10], the Krieger and Dougherty model [32], and the Nielsen's model [33] are in disagreement with the experimental values for the higher nanoparticles volume fractions of ϕ>1%, while the aforementioned models underestimate the dynamic viscosity data for ϕ>1%. However, the trend observed in the proposed ANN model show a good agreement with the experimental results. The variation of dynamic viscosity of the MWCNT-water nanofluid [25] versus nanoparticle volume fraction at 25 and 55 C is depicted in Fig. 14. It is evident from
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ACCEPTED MANUSCRIPT Fig.14 that the dynamic viscosity of MWCNT-water nanofluid decreases as temperature increases. In addition, the relative viscosity increases with the increase in
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the nanoparticles volume fraction. Comparisons made between the experimental data
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for MWCNT-water nanofluid and the predictions by the ANN model with the well-
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known models for the viscosity of nanofluids. As can be seen from Fig.14, the large deviations between experimental data and the results of the theoretical and empirical models indicate the incapability of the existing models for the dynamic viscosity of
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MWCNT-water nanofluid. The AAD of about 2.72% proofs the accuracy of the
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presented ANN model for predicting the dynamic viscosity of MWCNT-water nanofluid in the temperature range of 25-55 C and volume fractions up to 1%.
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The viscosity of SiC-water nanofluid [28] is plotted in Fig. 15. Various models have
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been used to examine the accuracy of the ANN model regarding the prediction of the dynamic viscosity of SiC-water nanofluid at different temperatures and particle
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volume fractions. As shown in Fig. 15, the models of Batchelor, Nguyen et al., Wang et al., Duangthongsuk and Wongwises and so on underpredict significantly the
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dynamic viscosity of the nanofluid, even at very low volume fractions and higher temperatures. On the other hand, the AAD of 7.9% represents a reasonable agreement between the results of the ANN model and the experimental data over the temperature range of 30-70 C and the nanoparticle concentration ranging from 1 to 3 Vol.%. Figs. 16 & 17 reveal the comparisons between the experimental dynamic viscosity of Ag-water nanofluids and prediction results by the ANN model and the other existing theoretical and empirical models, such as the model proposed by Godson et al. [29]. As a whole, the AAD of about 3.92% suggesting the accuracy of the ANN model for predicting the dynamic viscosity of Ag-water nanofluids over a wide range of operating conditions. As suggested by several researchers, the viscosity depends on
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ACCEPTED MANUSCRIPT various parameters from the material properties to random micro-convective motions as well as the stability of the prepared nanofluid. Therefore, the need for a unified
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model based on the accurate experimental data can be useful as a foundation for
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principal design of heat recovery/removal systems.
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As reported by several researchers, the results indicated that there are some differences among the measured dynamic viscosity data by researchers. It may be contributed to the various influencing factors such as particle size, particle preparation
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(solution chemistry), the measurement technique or even different particle sources [7].
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As a result, the uncertainty related to the measurement techniques and nanofluid preparation methods are responsible for the discrepancies between the ANN models
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and real data. On the other hand, number of experimental data is a key point affecting
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the ANN models. However, the results contend that the presented ANN model is markedly superior to the existing correlations for dynamic viscosity of the studied
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nanofluids. Fig. 18 demonstrates the parity plot for the whole dataset used for developing the artificial neural network model. The R2-value of 0.9842 and AAD of
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6.66% suggest the accuracy and generalizeability of the proposed model for accurate predicting the effective viscosity of the studied nanofluids. 4. Conclusions
A feedforward neural network model has been developed for accurate predicting the effective viscosity of nanofluids over wide ranges of temperature and nanoparticles concentration. The effective viscosities of different nanofluids, namely, Al2O3-water, CuO-water, TiO2-water, SiC-water, MWCNT-water, Fe3O4-water, Ni-water, and Agwater with nanoparticles diameter in the range of 7 nm to 100 nm are predicted by using an artificial neural network model. The effects of different influencing factors such as nanoparticles concentration, size and type of the particles, and temperature on 12
ACCEPTED MANUSCRIPT the viscosities are investigated and compared to the other models. As nanoparticle concentration increases, the error between the predictions of the most important
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existing models and the real data rise. The underestimation problems dealing with the
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aforementioned theoretical and semi-empirical models make them unreliable for
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prediction purposes. By using the proposed model and proper determining the effective viscosity of nanofluids as functions of particle volume fraction, particle size and temperature, the pressure drop problems associated with nanofluid systems can be
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mitigated.
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ACCEPTED MANUSCRIPT References [1] S.S. Mallick, A. Mishra, L. Kundan, An investigation into modelling thermal
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conductivity for alumina-water nanofluids, Powder Technol. 233 (2013) 234-244.
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[2] L. Chen, W. Yu, H. Xie, Enhanced thermal conductivity of nanofluids containing
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Ag/MWNT composites, Powder Technol. 231 (2012) 18-20.
[3] L. Fedele, L. Colla, S. Bobbo, Viscosity and thermal conductivity measurements
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of water-based nanofluids containing titanium oxide nanoparticles, Int. J. Refrig. 35 (2012) 1359-1366.
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[4] P.K. Namburu, D.P. Kulkarni, O. Misra, D.K. Das, Viscosity of copper oxide nanoparticles dispersed in ethylene glycol and water mixture, Exp. Thermal Fluid Sci.
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32 (2007) 397- 402.
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[5] K. B. Anoop, S. Kabelac, T. Sundararajan, Sarit K. Das, Rheological and flow characteristics of nanofluids: Influence of electroviscous effects and particle
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agglomeration, J. Appl. Phys. 106 (2009) 034909-7. [6] C.T. Nguyen, F. Desgranges, G. Roy, N. Galanis, T. Mare´, S. Boucher, H. Angue
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Mintsa, Temperature and particle-size dependent viscosity data for water-based nanofluids-Hysteresis phenomenon, Int. J. Heat Fluid Flow 28 (2007) 1492-1506. [7] W. Duangthongsuk, S. Wongwises, Measurement of temperature-dependent thermal conductivity and viscosity of TiO2-water nanofluids, Exp. Therm. Fluid Sci. 33 (2009) 706-714. [8] B.C. Pak, Y. I. Cho, Hydrodynamic and heat transfer study of dispersed fluids with submicron metallic oxide particles, Exp. Heat Transfer 11 (1998) 151-170. [9] A. Einstein, Eine neue Bestimmung der Moleküldimensionen. Annalen der Physik 19 (1906) 289-306.
14
ACCEPTED MANUSCRIPT [10] G.K. Batchelor, The effect of Brownian motion on the bulk stress in a suspension of spherical particles, J. Fluid Mech. 83 (1977) 97-117.
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[11] H.C. Brinkman, The viscosity of concentrated suspensions and solution, J. Chem.
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Phys. 20 (1952) 571-581.
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[12] M. Afrand, K. Nazari Najafabadi, N. Sina, M.R. Safaei, A.Sh. Kherbeet, S. Wongwises, M. Dahari, Prediction of dynamic viscosity of a hybrid nano-lubricant by an optimal artificial neural network, Int. Commun. Heat Mass Transfer 76 (2016)
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209-214.
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[13] A.S. Dalkilic, A. Çebi, A. Celen, O. Yıldız, O. Acikgoz, C. Jumpholkul, M. Bayrak, K. Surana, S. Wongwises, Prediction of graphite nanofluids' dynamic
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viscosity by means of artificial neural networks, Int. Commun. Heat Mass Transfer 73
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(2016) 33-42.
[14] F. Yousefi, H. Karimi, M.M. Papari, Modeling viscosity of nanofl uids using
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diffusional neural networks, J. Molecular Liquids 175 (2012) 85-90. [15] M. Afrand, A. Ahmadi Nadooshan, M. Hassani, H. Yarmand, M. Dahari,
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Predicting the viscosity of multi-walled carbon nanotubes/water nanofluid by developing an optimal artificial neural network based on experimental data, Int. Commun. Heat Mass Transfer 77 (2016) 49-53. [16] D. Graupe, Principles of Artificial Neural Networks, Second ed., WSPC, USA, 2007. [17] S. Haykin, Neural Networks: A Comprehensive Foundation, Prentice-Hall, Upper Saddle River, NJ, 1999. [18] L.S. Sundar, M.K. Singh, I. Bidkin, A.C.M. Sousa, Experimental investigations in heat transfer and friction factor of magnetic Ni nanofluid flowing in a tube, Int. J. Heat Mass Transfer 70 (2014) 224-234.
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ACCEPTED MANUSCRIPT [19] M. Goharkhah, M. Ashjaee, Experimental investigation on heat transfer and hydrodynamic behavior of magnetite nanofluid flow in a channel with recognition of
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the best models for transport properties, Exp. Thermal Fluid Sci. 68 (2015) 582-592.
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[20] M.H. Kayhani, M. Nazari, H. Soltanzadeh, M.M. Heyhat, F. Kowsary,
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Experimental analysis of turbulent convective heat transfer and pressure drop of Al2O3 /water nanofluid in horizontal tube, Micro Nano Letters 7 (2012) 223-227. [21] M.J. Pastoriza-Gallego, C. Casanova, J.L. Legido, M.M. Pineiro, CuO in water
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nanofluid: Influence of particle size and polydispersity on volumetric behaviour and
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viscosity, Fluid Phase Equilibria 300 (2011) 188-196. [22] D. Kwek, A. Crivoi, F. Duan, Effects of temperature and particle size on the
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thermal property measurements of Al2O3-water nanofluids, J. Chem. Eng. Data 55
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(2010) 5690-5695.
[23] W. Duangthongsuk, S. Wongwises, An experimental study on the heat transfer
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performance and pressure drop of TiO2-water nanofluids flowing under a turbulent flow regime, Int. J. Heat Mass Transfer 53 (2010) 334-344.
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[24] A. Turgut, I. Tavman, M. Chirtoc, H. P. Schuchmann, C. Sauter, S. Tavman, Thermal conductivity and viscosity measurements of water-based TiO2 nanofluids, Int. J. Thermophys. 30 (2009) 1213-1226. [25] M. H. Esfe, S. Saedodin, O. Mahian, S. Wongwises, Thermophysical properties, heat transfer and pressure drop of COOH-functionalized multi walled carbon nanotubes/water nanofluids, Int. Commun. Heat Mass Transfer 58 (2014) 176-183. [26] H. Xie, W. Yu, W. Chen, MgO nanofluids: higher thermal conductivity and lower viscosity among ethylene glycol-based nanofluids containing oxide nanoparticles, J. Exp. Nanosci. 5 (2010) 463-472.
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ACCEPTED MANUSCRIPT [27] M.J. Pastoriza-Gallego, L. Lugo, D. Cabaleiro, J.L. Legido, M.M. Piñeiro, Thermophysical profile of ethylene glycol-based ZnO nanofluids, J. Chem.
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Thermodynamics 73 (2014) 23-30.
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[28] S.W. Lee, S.D. Park, S. Kang, I.C. Bang, J.H. Kim, Investigation of viscosity and
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thermal conductivity of SiC nanofluids for heat transfer applications, Int. J. Heat Mass Transfer 54 (2011) 433-438.
[29] L. Godson, D. Mohan Lal, S. Wongwises, Experimental investigation on the
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thermal conductivity and viscosity of silver-deionized water nanofluid, Exp. Heat
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Transfer, 23 (2010) 317-332.
[30] L. Godson, D. Mohan Lal, S. Wongwises, Measurement of thermo physical
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properties of metallic nanofluids for high temperature applications, Nanosci.
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Microsci. Therm. Eng. 14 (2010) 152-173. [31] M. Goharkhah, A. Salarian, M. Ashjaee, M. Shahabadi, Convective heat transfer
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characteristics of magnetite nanofluid under the influence of constant and alternating magnetic field, Powder Technol. 274 (2015) 258-267.
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[32] I.M. Krieger, T.J. Dougherty, A mechanism for non-Newtonian flow in suspensions of rigid spheres, Trans. Soc. Rheol. 3 (1959) 137-152. [33] L.E. Nielsen, Generalized equation for the elastic moduli of composite materials, J. Appl. Phys. 41 (1970) 4626-4627. [34] X. Wang, X. Xu, S.U.S. Choi, Thermal conductivity of nanoparticle- fluid mixture, J. Thermophys. Heat Transf. 13 (1999) 474-480. [35] C.U. Thomas, M. Muthukumar, Three-body hydrodynamic effects on viscosity of suspensions of spheres, J. Chem. Phys. 94 (1991) 5180.
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ACCEPTED MANUSCRIPT [36] M. Chandrasekar, S. Suresh, A. Chandra Bose, Experimental investigations and theoretical determination of thermal conductivity and viscosity of Al2O3/water nano
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fluid, Exp. Thermal Fluid Sci. 34 (2010) 210-216.
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[37] W.J. Tseng, C.N. Chen, Effect of polymeric dispersant on rheological behavior
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of nickel-terpineol suspensions, Mater. Sci. Eng. A Struct. Mater. Prop. Microstruct. Process. 347 (2003) 145-153.
[38] L. Syam Sundar, M.K. Singh, A.C.M. Sousa, Investigation of thermal
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conductivity and viscosity of Fe3O4 nanofluid for heat transfer applications, Int.
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Commun. Heat Mass Transfer, 44 (2013) 7-14.
[39] V. Vand, Viscosity of solutions and suspensions, I Theory, J. Phys. Colloid
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Chem. 52 (1948) 277-299.
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[40] K. Khanafer, K.A. Vafai, A critical synthesis of thermophysical characteristics of
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nano fluids, Int. J. Heat Mass Transfer 54 (2011) 4410-4428.
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ACCEPTED MANUSCRIPT Nomenclature
average radius of the single particles
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a
average radius of the aggregates
dp
average particle diameter
E
error function
f
transfer function
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bias value of the j-th neuron
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bj
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AAD% percentage average absolute deviation defined as
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aa
hidden nodes number
N
number of elements in the input-output vectors
Oi
overall network output for the i-th element
R2
correlation coefficient
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temperature (C)
ti
target values of the i-th element
wji
weight value between the i-th neuron and the j-th neuron
xi
i-th input to the network
yj
net output from j-th neuron
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n
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ACCEPTED MANUSCRIPT Greeks ϕ
nanoparticle volume fraction dynamic viscosity; learning rate
ρ
density
ϕa
effective volume fraction of aggregates
ϕm
effective volume fraction of solid spheres
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Subscript base fluid
exp
experimental value nanofluid
p
particle
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nf
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bf
predicted value
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pred
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η
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dp (nm)
ϕ(%)
No. of data points
Magnetic NiWater
75
0.2-0.6
52
Fe3O4-Water
30
1-2
Al2O3-Water
13, 25, 30, 47, 50
TiO2- Water CuO- Water
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systems
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Table 1. The ranges of experimental conditions for different nanofluid systems studied in this work. Ref.
[18]
[19]
0.25-9.4
152
[5], [6], [8], [20], [22]
13, 21, 25, 76
0.211.22
114
[3], [7], [23], [24]
11, 29, 30
0.15-9
130
[6], [21]
7
0.01-1
20
[25]
100
0.1-3
37
[28]
60, 63
0.3-1.2
90
[29], [30]
MWCNT-Water
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Ag-Water
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SiC-Water
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27
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Table 2 . The optimum calculated values of the neural network model parameters
0.389 -1.4256 -1.8088 -1.3278 -1.7873 -1.022 -2.5784 1.1698 1.6924
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0.3205
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-1.0377
-0.3644 5.3024 -0.6042 -0.5722 1.6869 1.6242 -1.0589 0.8556 -2.9068 , j=1; i=1…,9 -0.0916 2.5411 -3.2348 , j=1 0.3895
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0.8458
1.941 -4.586 1.1836 -1.0514 -0.0966 -2.1029 -1.9222 0.4339 -2.2837
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-3.2755 1.5984 -0.0057 1.1138 4.0947 0.4089 0.8949 -1.2789 -4.484
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, j=1…,9; i=1…,4
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-0.5788
, j=1…,9 -0.8478 0.0154 1.21 0.0989 1.8152 0.2263 0.1065 -1.0605 0.3509
-1.0495
0.3751
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2.5m
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1 a bf m
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nf
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Table 3. Theoretical and empirical dynamic viscosity correlations of nanofluids. Effective dynamic viscosity model Reference Krieger and Dougherty
Duangthongsuk and Wongwises nf 1.013 0.092 0.015 2 bf
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Wang et al.
[32]
[7] [34]
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Nguyen et al.
Batchelor
bf
[33]
1.1296 38.15 0.0017357T
nf
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1 1.5 e / 1m
Esfe et al.
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nf
Nielsen
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bf
[10]
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nf
[6]
bf
Godson et al.
1.005 0.497 0.1149 2
Thomas and Muthukumar
[25]
[30] [35]
Chandreasekar et al. 2.8 1 5200 bf 1
nf
Tseng and Chen
[36]
[37]
Syam Sundar et al. [38] Vand
[39]
Khanafer and Vafa
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[40]
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Pastoriza-Gallego et al. 1.2
1.5125
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aa 1 bf 0.605 a
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[19]
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Goharkhah and Ashjaee
[27]
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Fig. 1. Proposed neural network model architecture
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Viscosity
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R2=0.9805
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Predicted dynamic viscosity (mPa.s)
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Fig. 2. Neural network training result for the dynamic viscosity of nanofluids. The
this phase.
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nonlinear relationships between dependent and independent variables are extracted in
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R2=0.9898
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Experimental dynamic viscosity (mPa.s) Fig. 3. Neural network validation result for the dynamic viscosity of nanofluids
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employing %15 of experimental data during the model development.
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Fig. 4. Neural network testing result for the dynamic viscosity of nanofluids when
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presenting the completely unseen datasets to the model.
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ANN model Wang et al. Batchelor Godson et al. Syam Sundar et al. Khanafer and Vafa Tseng and Chen
Fig. 5. Comparison among the proposed neural network model and experimental data
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Fig. 7. Comparison between the predicted dynamic viscosity and experimental data
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Fig. 8. Comparison between the predicted dynamic viscosity and experimental data
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for Al2O3-water nanofluid [5] with the prediction results of different models.
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4 3
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Viscosity, mPa.s
5
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T, C
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65
Exp., Vol.%=9.1, 36 nm Exp., Vol.%=4, 47 nm Nguyen et al. Godson et al. Batchelor Vand Chandreasekar et al.
TE
D
MA
Exp., Vol.%=9.4, 47 nm Exp., Vol.%=7, 36 nm ANN model Wang et al. Duangthongsuk and Wongwises Thomas and Muthukumar Khanafer and Vafa Esfe et al.
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Fig. 9. Comparison between the predicted dynamic viscosity and experimental data
AC
for Al2O3-water nanofluid [6] with the prediction results of different models.
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7
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6 5 4
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3 2 1 15
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T, C
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Exp., Vol.%= 7 Exp., Vol.%=1 Esfe et al. Godson et al. Batchelor Vand Chandreasekar et al.
TE
D
MA
Exp., Vol.%=9 Exp., Vol.%=4.5 ANN model Wang et al. Duangthongsuk and Wongwises Thomas and Muthukumar Khanafer and Vafa Nguyen et al.
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CE P
Fig. 10. Comparison between experimental and predicted values of different models regarding viscosity of CuO-water nanofluid with 29 nm particles [6].
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T IP
1.5
SC R
Viscosity, mPa.s
2
1
0 0.002
0.004
0.006
0.008 0.01 0.012 0.014 0.016 0.018 Volume fraction Exp., T=10 C Exp., T=25 C Exp.,T=35 C Exp., T=50 C ANN model Esfe et al. Nguyen et al. Wang et al. Godson et al. Duangthongsuk and Wongwises Thomas and Muthukumar Vand Khanafer and Vafa Chandreasekar et al. Pastoriza-Gallego et al. Batchelor
TE
D
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Fig. 11. Dynamic viscosity variation with volume fraction at different temperatures
AC
for CuO-water nanofluid [21] as well as prediction results of different models.
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Viscosity, mPa.s
2.5
1
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0.5 0 0
10
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T, C
TE
D
Exp., Vol.%=0.24, 76 nm Exp., Vol.%=3, 21 nm Nguyen et al. Godson et al. Batchelor Vand Chandreasekar et al. Nielsen
Exp., Vol.%=2.54, 76 nm ANN model Wang et al. Duangthongsuk and Wongwises Thomas and Muthukumar Khanafer and Vafa Krieger and Dougherty Esfe et al.
CE P
Fig.12. Comparison of the TiO2-water nanofluid dynamic viscosity between measured
AC
data [23, 24] and calculated values from other models.
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3 2.5
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Viscosity, mPa.s
3.5
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1 0.5
8
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Exp., Vol.%=11.22, 76 nm Exp., Vol.%=2, 76 nm Nguyen et al. Godson et al. Batchelor Vand Chandreasekar et al. Nielsen
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58
68
Exp., Vol.%=5.54, 76 nm ANN model Wang et al. Duangthongsuk and Wongwises Thomas and Muthukumar Khanafer and Vafa Krieger and Dougherty Esfe et al.
Fig. 13. Experimental viscosity of TiO2-water nanofluids as a function of temperature
AC
at different volume fraction and nanoparticle size [3, 24].
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T IP
1.4 1.2
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Viscosity, mPa.s
1.6
1 0.8
0.4 0
0.002
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0.006
0.008
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0.012
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Volume fraction Exp.,T= 55 C Esfe et al. Wang et al. Duangthongsuk and Wongwises Thomas and Muthukumar Khanafer and Vafa
Fig. 14.
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TE
D
Exp., T=25 C ANN model Nguyen et al. Godson et al. Batchelor Vand Chandreasekar et al.
Dynamic viscosity variation versus solid volume fraction at various
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temperatures for MWCNT-water nanofluid [25].
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1.2 1 0.8
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Viscosity, mPa.s
1.4
0.6 0.4
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0.2
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65
75
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T, C
CE P
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Exp., Vol.%=3 Exp., Vol.%=1 Wang et al. Duangthongsuk and Wongwises Thomas and Muthukumar Khanafer and Vafa Esfe et al.
Exp., Vol.%=2 ANN model Godson et al. Batchelor Vand Chandreasekar et al. Nguyen et al.
Fig.15. Dynamic viscosity-Temperature curves of SiC-water nanofluid [28].
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Comparison between the ANN and different models.
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Exp., Vol.%=0.9, 60 nm Exp., Vol.%=0.3, 60 nm Esfe et al. Wang et al. Duangthongsuk and Wongwises Thomas and Muthukumar Khanafer and Vafa
Exp., Vol.%=0.6, 60 nm ANN model Nguyen et al. Godson et al. Batchelor Vand Chandreasekar et al.
Fig. 16. Comparison of the measured dynamic viscosity of Ag-water nanofluid [29]
AC
with those obtained from various models.
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T IP
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SC R
Viscosity, mPa.s
0.8
0.2 55
65
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45
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0.4
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TE
D
Exp., Vol.%=1.2, 63 nm Exp., Vol.%=0.4, 63 nm Esfe et al. Wang et al. Duangthongsuk and Wongwises Thomas and Muthukumar Khanafer and Vafa
75
85
95
Exp., Vol.%=0.8, 63 nm ANN model Nguyen et al. Godson et al. Batchelor Vand Chandreasekar et al.
Fig. 17. Comparison of the measured dynamic viscosity of Ag-water nanofluid [30]
AC
with those obtained from various models.
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R2=0.9842 10
IP
T
8
SC R
6 4
2 0 2
4
6
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0
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Predicted dynamic viscosity (mPa.s)
12
8
10
12
Experimental dynamic viscosity (mPa.s)
D
Fig. 18. Comparison of the proposed neural network model predictions and the whole
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CE P
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experimental dynamic viscosity of the nanofluids.
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ACCEPTED MANUSCRIPT Highlights:
T
An artificial neural network has been presented for the effective viscosity of
IP
nanofluids.
SC R
The ANN model is superior than the existing models over wide ranges of operating conditions.
The predictions have been performed based on the most influencing parameters.
AC
CE P
TE
D
MA
NU
The proposed model can be utilized as a useful correlation for design tasks.
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