international communications in heat and mass...

International Communications in Heat and Mass Transfer 68 (2015) 50–57

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International Communications in Heat and Mass Transfer

j ourna l homepage: www.e lsev ie r .com/ locate / ichmt

Designing an artificial neural network to predict thermal conductivityand dynamic viscosity of ferromagnetic nanofluid☆

Mohammad Hemmat Esfe a,⁎, Seyfolah Saedodin b, Nima Sina a, Masoud Afrand a,⁎, Sara Rostami aa Department of Mechanical Engineering, Faculty of Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Isfahan, Iranb Faculty of Mechanical Engineering, Semnan University, Semnan, Iran

☆ Communicated by W.J. Minkowycz⁎ Corresponding authors.

E-mail addresses: [email protected] (M. [email protected] (M. Afrand).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.06.010735-1933/© 2015 Elsevier Ltd. All rights reserved.

a b s t r a c t
a r t i c l e i n f o
Available online 26 June 2015

Keywords:Artificial neural networkNanofluidsDynamic viscosityThermal conductivity

This paper focuses on designing an artificial neural networkwhich can predict thermal conductivity and dynamicviscosity of ferromagnetic nanofluids from input experimental data including temperature, diameter of particles,and solid volume fraction. The experimental data were extracted and they were used as learning dataset to trainthe neural network. Tofind a proper architecture for network, an iterationmethodwas used. Based on the results,therewasnoover-fitting indesigned neural network and the neural networkwas able to track the data. ANNout-puts showed that themaximum errors in predicting thermal conductivity and dynamic viscosity are 2% and 2.5%,respectively. Based on the ANN outputs, two sets of correlations for estimating the thermal conductivity and dy-namic viscosity were presented. The comparisons between experimental data and the proposed correlationsshowed that the presented correlations were in an excellent agreement with experimental data.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

For many years, common fluids such as water, ethylene glycol, or oilhave been used as the operating fluid in industry and engineering de-signs. The main problem is that the thermal conductivity of these fluidsis small and thus the heat transfer rate is low. This is one of the limita-tions of this type of fluid that has led engineering design to have certainlimitations. Recently, in order to increase the thermal conductivity ofthe fluid, nano-sized particles will be added to the fluid. In this way,the heat transfer of nanofluids within the enclosure and the influenceof thermo-physical characteristics of nanofluids (such as thermalconductivity, thermal expansion coefficient, and dynamic viscosity)has been much studied [1–5].

In the last decade, several studies have been conducted on the ther-mal conductivity nanofluids. Wook Oh et al. [6] presented experimentaldata on the thermal conductivity enhancement in Al2O3 nanoparticle dis-persed in DI water and EG as base fluids using the modified 3-omega(3ω) method. They showed that for DI water-based nanofluids, thermalconductivity incremental data agreed well with those of Wang et al. [7],which show higher increment compared to the results of Lee et al. [8]and Das et al. [9]. Also, they showed that EG-based nanofluids had rela-tively low thermal conductivity values comparedwith those of Lee et al.[8] and Wang et al. [7]. Experimental data on enhancement of thermalconductivity of ethylene glycol based silver nanofluids are reported

mat Esfe),

3

by Sharma et al. [10]. They illustrated thermal conductivity of silvernanofluids enhanced to 10%, 16%, and 18% as the amounts of silverparticles in nanofluid were 1000, 5000, and 10,000 ppm, respectively.The effect of solid volume fraction of ethylene glycol based copperoxide fluids is investigated by Lee et al. [11]. Their experimental resultsdemonstrated that these nanofluids, containing a small amount ofnanoparticles, have significantly higher thermal conductivities thanthe same liquids without nanoparticles. In another study focusing onthe solid concentration and shape of the nanoparticles, Xie et al. [12]studied silicon carbide nanoparticles into two spherical and cylindricalforms added to water and ethylene glycol. They observed that thethermal conductivity is further increased when nanoparticles are cylin-drical. Althoughmany researchers have not considered the effect of tem-perature on thermal conductivity, recent studies show that nanofluidstemperature effect on the properties of nanofluids is very important.Das et al. [13] studied the behavior of CuO–water and Al2O3–waternanofluidswith temperature. They concluded thatwith increasing tem-perature, the thermal conductivity of nanofluids increases. In anotherstudy, Karthik et al. [14] investigated the thermal conductivity ofCuO–DI water nanofluids experimentally. Their study also showedthat temperature has a significant influence on the thermal conductivityof nanofluids.

To access the thermophysical characteristics of nanofluids in differ-ent concentrations, particle diameters, and temperatures, we need todo several experiments which are time consuming and expensive. Inrecent decade, in order to avoid such costs, there has been an interestin using soft computing methods to predict the behavior of nanofluidswhich are known as neural networks, fuzzy logic, and genetic algorithms.Among these, artificial neural networks are good tools to solve complex

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Fig. 1. Structure of the neural network.

51M. Hemmat Esfe et al. / International Communications in Heat and Mass Transfer 68 (2015) 50–57

problems in different application with a considerable reduction in timeand cost. However, there is a little reportedwork aboutmodeling of ther-mal conductivity and viscosity of nanofluids using artificial neural net-work. In this regard, Mehrabi et al. [15] developed a model to predictthe effective viscosity of nanofluids using an FCM-based adaptiveneuro-fuzzy inference system (FCM-ANFIS) and a set of experimentaldata. They selected as the design parameters the size of the nanoparticles,volume concentration, and temperature to predict the effective viscosityof nanofluids. To model the viscosity, experimental data from literaturewere divided into two sets: a train and a test data set. The predictedviscosities were compared with experimental data for four nanofluids,which were Al2O3, CuO, TiO2, and SiO2, and with water as base fluid.The predicted results agreed with the experimental measurement.

An accurate and efficient artificial neural network based on geneticalgorithm (GA) for predicting of nanofluids viscosity was developedby Karimi et al. [16]. They used genetic algorithm (GA) for optimizingthe neural network parameters. Temperature and nanoparticle volumefraction were used as input data. The obtained results demonstratedthat the GA-NN model was in good agreement with the experimentaldata. Papari et al. [17] used the neural network method to predict thethermal conductivity of different nanofluids, including single-walledcarbon nanotubes in epoxy and polymethylmethacrylate and alsomulti-walled carbon nanotubes suspended in oil, decene, water, ethyl-ene glycol. The comparisons between predicted and experimental datashowed good agreement. Thermal conductivity modeling of γ-Al2O3,TiO2, and CuO nanoparticles in a 0.5 wt% of carboxymethyl cellulose(CMC) aqueous solution using a three-layer feed-forward neural net-work is exhibited by Hojjat et al. [18]. They proposed neural networkmodels to report the thermal conductivity as a function of the volumefraction of nanoparticles, temperature, and the thermal conductivity ofthe nanoparticles. Proposed models were in good agreement with theexperimental values. Longo et al. [19] presented two neural networkmodels for predicting the thermal conductivity of Al2O3–water andTiO2–water nanofluids by considering the temperature, volume frac-tion, nanoparticle diameter, and particle thermal conductivity as theinput variables. Both models revealed that reasonable predicted dataare in good agreement with the experimental data; but the 4-inputmodel showed better performance. Recently, Hemmat Esfe et al. [20]modeled the thermal conductivity of MgO/EG nanofluids using experi-mental data and artificial neural network. Feedforward multilayerperceptron neural network was used to predict the thermal conductiv-ity of the MgO/EG nanofluid. They considered the volume fraction, par-ticle size, and temperature as the input data. The predicted results are ingood agreement with the measured data.

Literature survey reveals that there is not any reported work aboutmodeling of thermal conductivity and dynamic viscosity of Fe/EGnanofluids using artificial neural network (ANN). Therefore, in thiswork, ANN is designed using obtained experimental thermal conductiv-ity and dynamic viscosity values of Fe/EG nanofluids depending on thedifferent temperatures, diameters of particles, and volume fractions.

2. Architecture of artificial neural network

In complicated systemswith several effective input parameters, arti-ficial neural network (ANN) can predict output data. ANN is inspiredfrom the human brain in order to process data and information. It in-cludes integrated process units called neurons that can process inputdata. The multi-layer perceptron neural network includes input layer,hidden layer, and output layer. In this article, theANN is used formodel-ing the behavior of Fe-EG nanofluid. The thermal conductivity andviscosity of Fe-EG nanofluid is modeled by ANN. The number of inputparameters is three, which includes temperature (T), diameter ofparticles (dp), and volume fraction (ϕ). Output values are the thermalconductivity (k) and dynamic viscosity (μ).

The basic unit of neural network is the neuron. In each neuron, thesum of input values are weighted and added with a parameter called

bias, and the sum is passed through a function which is called transferfunction or activation function (see Fig. 1). The transfer function calcu-lates the output of a neuron from its input. Some ANN includes severallayers. Each layer includes several neurons and performs a simple pro-cess on data. In this ANN, the input dataset is divided into 3 sets ran-domly: train data, validation data, and test data. 70% of data set isregarded as train data. 15% of dataset is regarded as validation, and15%of data is regarded for test data. During training process, the weightand biases of each neuron is generated. Validation data are used duringtraining and it tunes parameters of classifier and defines stoppingcriteria also it prevents the network from over-fitting. Over-fitting oc-curs when the error on the training dataset reaches a very small value,but when new data are imported to the ANN, the new error is notsmall. It means that the ANN has memorized only the training exam-ples, but the network is not able to generalize and predict new inputvalues correctly. The test dataset shows how good the ANN is trained.In this simulation, the ANN includes two hidden layer and an outputlayer.

Feedforward multilayer perceptron neural network is used to pre-dict the thermal conductivity enhancement and dynamic viscosity ofthe Fe/EG nanofluid. The first hidden layer includes seven neuronswith tan-sigmoid transfer function (Eq. (1)). The second layer includes5 neurons with the same previous transfer function.

nj ¼2

1þ e2Z −1 ð1Þ

Where nj is the output of the jth neuron and z is given by

z ¼Xr

i¼1wijpi þ bj ð2Þ

Where wij are the weights of connections from the i-th neurons inthe previous layer to the j-th neurons, pi denotes the output from thei-th neuron, b refers to the bias parameter, and r represents the numberof neurons in the previous layer.

Finally, the output layer should have only two neurons because thenumber of outputs is two. The designed transfer function for outputlayer is linear. The number of input data is 72, which is extracted fromexperimental results. The selected learning algorithm is LevenbergMarquardt. In this network, during learning process, maximumnumberof iteration is 1000, maximum number of fail in validation check is 20,and the performance of network is mean square error (MSE). Thevalue of MSE is calculated from Eq. (3).

MSE ¼Xn

i¼1 ti−aið Þ2

nð3Þ

Fig. 2. Proposed algorithm to find the optimal neural network.

52 M. Hemmat Esfe et al. / International Communications in Heat and Mass Transfer 68 (2015) 50–57

Where n is the number of data, ti denotes the i-th target value, and aiis the predicted value.

In some cases, ANN is not able to predict the output value pre-cisely. In order to select the best architecture of ANN, differentstructures were used to obtain the optimal neural network whichcan predict data correctly. Different structures were made by per-mutation. Two limitations for the number of neurons were selected.The lower band was 10 and the upper band was 59. The proposed

network is selected between 1675 different structures which wereobtained by permutation. This optimum architecture was selectedbased on the smallest difference between predicted values (outputsof ANN) and experimental dataset. In order to decide which archi-tecture is the best one and considering the fact that dividing inputdata into three main sets is random in each run of the program, tojudge about output and performance of the network, it is better toconsider a mean approximation after several iteration. Because the


data were randomly selected, for every structure, several times the net-work was run. In this work, this inner iteration is 20. After performingpermutation, the best network architecture with highest accuracy andwithout over-fitting was selected. Comparing final results indicatedthat an ANN with 7 neurons in the first hidden layer and 5 neurons inthe second hidden layer is the best architecture for predicting the ther-mal conductivity and viscosity of Fe-EG nanofluid by presented specifi-cations. It can be seen that the network outputs have a very good

Fig. 3. Correlation between experimental values and predicted outputs for thermalconductivity. (a) training data set, (b) validation data set, (c) test data set.

agreement with experimental data. Both specification of Fe-EGnanofluid is predicted precisely. The proposed algorithm to find theoptimal neural network is illustrated in Fig. 2.

3. Results and discussions

The artificial neural networkwhich can predict thermal conductivityand dynamic viscosity of Fe/EG nanofluids from input experimental

Fig. 4. Comparison between experimental results and predicted outputs for thermalconductivity. (a) training data set, (b) validation data set, (c) test data set.


data including temperature, diameter of particles, and volume fractionis designed. The experimental data set is divided into 3 main subsetsincluding train, validation, and test. In the present study, the numberof total experimental data were 72 which 50 of them is regarded astraining data, 11 of them is considered as validation data and remaindata referred to test data.

Fig. 3 shows correlation between experimental values and predictedthermal conductivity for training data set, validation data set and test

Fig. 5. Correlation between experimental values and predicted outputs for dynamicviscosity. (a) training data set, (b) validation data set, (c) test data set.

data set. As can be observed, most of the data are on the bisector or inits vicinity which presents a proper correlation between experimentaldata and predicted outputs. This figure indicates the closeness amongthe experimental data and the predicted results using the ANN. InFig. 3, the maximum error (error is difference between experimentaland predicted values) and MSE are 2% and 0.00016, respectively. Inaddition, it is shown in Fig. 4 that there is a good agreement betweenexperimental data and predicted results.

Fig. 6. Comparison between experimental results and predicted outputs for dynamicviscosity. (a) training data set, (b) validation data set, (c) test data set.

Fig. 7. Effect of solid volume fraction and temperature on the variation of (a) thermalconductivity and (b) dynamic viscosity (dp is nanoparticle diameter).


A correlation between experimental data and predicted dynamicviscosity for training data set, validation data set, and test data set isillustrated in Fig. 5. It can be observed that there is an appropriate corre-lation between experimental values and predicted data. The maximumerror and MSE are, respectively, 2.5% and 0.00026 as shown in Fig. 5. Toverify ANNmodel of dynamic viscosity of Fe/EG nanofluid, another dia-gram based on experimental findings predicted, data and data numberis displayed in Fig. 6 which shows that experimental data and predictedoutputs have a good agreement with each other.

Table 1Constant values of proposed correlation to estimate thermal conductivity of Fe/EG nanofluid.

dp(nm) A B C D

40 0.5519 0.0102 −6.1066 × 10−5 6.170 0.5861 0.0094 −6.0211 × 10−5 5.5100 0.8156 0.0035 −1.5484 × 10−5 2.6

Table 2Constant values of proposed correlation to estimate dynamic viscosity of Fe/EG nanofluid.

dp(nm) A B C D

40 −2.4030 0.0709 −0. 4564 × 10−3 5170 −0.9722 0.0313 −0. 1100 × 10−3 35100 −1.9525 0.0585 −0. 3488 × 10−3 46

Fig. 7 illustrates the effect of solid volume fraction and temperatureon the thermal conductivity and dynamic viscosity for different diame-ters of nanoparticles. In this figure, the adaptation of the neural networkmodel on experimental results is clearly shown. As can be seen thefitted curve corresponds well with experimental results. The compari-son between Fig. 7a and b reveal that ANN errors in the estimation ofthe thermal conductivity are lower than the dynamic viscosity whichis confirmed by MSE value in Figs. 3 and 5.

It can be observed that the ANN is able to predict the thermalconductivity and dynamic viscosity accurately. Also, both of errors inpredicting thermal conductivity and dynamic viscosity are small. Thisis due to the use of different number of neurons in different hiddenlayers to achieve an optimal network (see Fig. 2).

Following, due to lack of appropriate correlation to estimate thethermal conductivity and dynamic viscosity of Fe/EG nanofluid, basedon ANN outputs, two sets of correlations with the same mathematicalformula for different particle sizes are separately presented. In thesecorrelations T and φ are temperature (°C) and solid volume fraction(%), respectively.

Eqs. (4) and (5) estimate respectively the thermal conductivity anddynamic viscosity of Fe/EG nanofluids as a function of the temperatureand solid volume fraction. The constant values for different particlesizes are presented in Tables 1 and 2. The proposed correlations canbe used in temperature ranging from 26 °C to 55 °C and volume frac-tions range of 0.125–3.0%.

To evaluate the accuracy of Eqs. (4) and (5), the comparisons amongexperimental data and results obtained by the Eqs. (4) and (5) aredisplayed in Figs. 8 and 9. These figures show that these correlationshave very high accuracy. Also, calculations illustrate the maximum dif-ference between the experimental results and correlations outputs is2%. Consequently, these correlations can predict the thermal conductiv-ity and dynamic viscosity of Fe/EG nanofluids at temperature rangingfrom 26 °C to 55 °C for solid volume fractions range of 0.125% to 3%.

kn fkb f

¼ Aþ BTþ CT2 þ DTþ Eφþ Fφ2 þ G φ

T

� �2þHφ3 ð4Þ

μnfμb f

¼ Aþ BTþ CT2 þ DTþ Eφþ Fφ2 þ G φ

T

� �2þHφ3 ð5Þ

4. Conclusion

The artificial neural network using 72 experimental data to predictthermal conductivity and dynamic viscosity of Fe/EG nanofluids isdesigned. Predicting is based on three input variables including tempera-ture, diameter of particles, and volume fraction. The thermal conductivityand dynamic viscosity are regarded as ANN outputs. The main aim ofANN designing is reaching the smallest error in predicting thermal

E F G H

300 0.1526 −0.0106 −9.5280 0.0017241 0.1258 −0.0077 −8.9830 0.0025285 0.0861 −0.0113 −7.1067 0.0050

E F G H

.4963 0.1885 −0.0514 7.5851 0.0074

.0313 0.1642 −0.0446 7.1096 0.0050

.7555 0.0589 0.0194 5.2864 −0.0057

Fig. 8. Comparison between experimental results and correlation outputs for thermalconductivity for different particle sizes. (a) 40 nm, (b) 70 nm, (c) 100 nm.

Fig. 9. Comparison between experimental results and correlation outputs for dynamicviscosity for different particle sizes. (a) 40 nm, (b) 70 nm, (c) 100 nm.


conductivity and dynamic viscosity of Fe/EG Nanofluids. ANN outputsshowed that maximum error in predicting thermal conductivity and dy-namic viscosity are 2% and 2.5%, respectively. Based on the ANN outputs,two sets of correlations for predicting the thermal conductivity and dy-namic viscosity were proposed. The comparisons have been performedbetween experimental data with the proposed correlations in this study

and itwas found that the proposed correlationswere in awell agreementwith experimental data. The extension of the present work and otherworks [21–24] related to the heat transfer and thermophysical propertiesof nanofluid affords engineers a good option for nanofluid in applicationslike electronics, automotive, and nuclear applications where improvedheat transfer or efficient heat dissipation is required.


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Designing an artificial neural network to predict thermal conductivity and dynamic viscosity of ferromagnetic nanofluid1. Introduction2. Architecture of artificial neural network3. Results and discussions4. ConclusionReferences

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