Experiment and Artificial Neural Network Prediction of Thermal

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May 19, 2017 - Keywords: nanofluids; thermal conductivity; viscosity; artificial neural network; experimental data. 1. ... water, oil, ethylene glycol, refrigerant, etc.

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Experiment and Artificial Neural Network Prediction of Thermal Conductivity and Viscosity for Alumina-Water Nanofluids Ningbo Zhao and Zhiming Li * College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China; [email protected] * Corespondence: [email protected]; Tel.: +86-0451-8251-9647 Academic Editor: Thomas Fiedler Received: 15 April 2017; Accepted: 12 May 2017; Published: 19 May 2017

Abstract: To effectively predict the thermal conductivity and viscosity of alumina (Al2 O3 )-water nanofluids, an artificial neural network (ANN) approach was investigated in the present study. Firstly, using a two-step method, four Al2 O3 -water nanofluids were prepared respectively by dispersing different volume fractions (1.31%, 2.72%, 4.25%, and 5.92%) of nanoparticles with the average diameter of 30 nm. On this basis, the thermal conductivity and viscosity of the above nanofluids were analyzed experimentally under various temperatures ranging from 296 to 313 K. Then a radial basis function (RBF) neural network was constructed to predict the thermal conductivity and viscosity of Al2 O3 -water nanofluids as a function of nanoparticle volume fraction and temperature. The experimental results showed that both nanoparticle volume fraction and temperature could enhance the thermal conductivity of Al2 O3 -water nanofluids. However, the viscosity only depended strongly on Al2 O3 nanoparticle volume fraction and was increased slightly by changing temperature. In addition, the comparative analysis revealed that the RBF neural network had an excellent ability to predict the thermal conductivity and viscosity of Al2 O3 -water nanofluids with the mean absolute percent errors of 0.5177% and 0.5618%, respectively. This demonstrated that the ANN provided an effective way to predict the thermophysical properties of nanofluids with limited experimental data. Keywords: nanofluids; thermal conductivity; viscosity; artificial neural network; experimental data

1. Introduction Considering the higher thermal conductivity of many solid materials, including Cu, CuO, TiO2 , ZnO, Fe3 O4 , MgO, Al2 O3 and graphite, dispersing solid particles in the conventional coolants (such as water, oil, ethylene glycol, refrigerant, etc.) is one of the most efficient ways to enhance the heat transfer process [1]. However, a large number of experimental results indicated that the lower suspension stability of large particles seriously limited the practical application of the traditional liquid-solid mixture. In the 1990s, the idea of nanofluids consisting of nanoparticles and base fluid was firstly introduced by Choi [2]. Due to the potential advantages in flow and heat transfer performance, nanofluids become a focus in the field of thermal science [3]. Thermal conductivity and viscosity are the most important physical parameters and play crucial roles for studying nanofluids. Over the last two decades, various experimental investigations have been published to evaluate the effects of nanoparticles on thermal conductivity and the viscosity characteristics of base fluids. References [4–13] respectively reviewed the experimental and theoretical developments of various nanofluids’ thermophysical parameters. According to their analysis, it could be found that the addition of nanoparticles did enhance the thermal conductivity and viscosity of base fluids in varying degrees. However, it was unfortunate that there were still many differences in the Materials 2017, 10, 552; doi:10.3390/ma10050552

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measurements of thermal conductivity and viscosity due to the effects of nanofluids manufacturing and measuring technologies [14]. In addition, considering the complex mechanisms, including nanoparticle heat transport [15,16], nano-shells at the interface between liquid and particle [17–20], Brownian motion [21,22], and clustering of nanoparticles [23,24], the thermal conductivity and viscosity of nanofluids are very difficult to predict accurately using the traditional model-based approach. For these cases, it is very valuable to further study the experimental characteristics and predictive modeling of nanofluids’ thermal conductivity and viscosity. In recent years, with the development of artificial intelligence technology, various data-driven modeling approaches have been put forward to solve the thermal science problem [25,26]. Inspired by the biological brain, an artificial neural network (ANN) can effectively establish the relationship between the input and output variables without considering the detailed physical process, which attracts increasing attention in terms of predicting the thermophysical properties of nanofluids. Hojjat et al. [27] firstly analyzed the application of a three-layer feed forward neural network on the thermal conductivities prediction of various non-Newtonian nanofluids. They found that the ANN predicted values were in agreement with the experimental data. The average and maximum errors were only 1.6% and 5.8%, respectively. On this basis, Longo et al. [28], Mehrabi et al. [29,30], Ariana et al. [31], Esfe et al. [32–38], and Vakili et al. [39] successively designed different ANN models (such as a feed forward neural network, adaptive neuro-fuzzy inference system, diffusional neural networks, etc.) to further verify the effectiveness of ANN in the modeling and prediction of nanofluid thermal conductivity. All of their results demonstrated that ANN was an effective tool in comparison with the traditional model-based approach for describing the enhancement behavior of nanofluid thermal conductivity. In addition, attracted by the better nonlinear mapping and recognition abilities of ANN, Yousefi et al. [40], Mehrabi et al. [41], Zhao et al. [42,43], and Heidari et al. [44] also extended the ANN based modeling approach to the prediction of nanofluid viscosity. As reported in their analysis, ANN could be used for predicting the viscosity of nanofluids with satisfactory accuracy. Up to now, much valuable literature has demonstrated the effectiveness of different ANN models for modeling and predicting the thermalphysical properties of nanofluids. However, considering the data characteristics of nanofluids’ properties and the modeling process of an ANN, there are still many difficulties or obstacles to be resolved. For example, most of the developed ANN had only one output (thermal conductivity, viscosity, or density) and were usually trained by using a large number of samples, which were obtained from different experiments. Fewer publications discussed the multiple parameter modeling and prediction performance of ANNs, especially with limited experimental data. This may means that the application of ANNs in nanofluids is still in its infancy. Based on the above background, this study presents a further investigation into the prediction of Al2 O3 -water nanofluid thermal conductivity and viscosity by using ANN and the limited experimental data. With the influences of nanoparticle volume fraction (1.31%, 2.72%, 4.25%, and 5.92%) and temperature (from 296 to 313 K), four different Al2 O3 -water nanofluids were prepared and measured. On this basis, a RBF neural network with multiple outputs was constructed and verified through the experimental data. 2. Experimental Methods 2.1. Materials and Method of Preparing Nanofluids In this study, a two-step method is used to manufacture the Al2 O3 -water nanofluids with different nanoparticle volume fraction. The spherical Al2 O3 particles (Xuan Cheng Jing Rui New Material Co., Ltd, Xuancheng, China) with an average diameter of 30 nm, a purity of 99.9%, a density of 3.6 g/cm3 , and the specific surface area of 15 m2 /g are selected. During the manufacturing, the measurement of the masses of the nanoparticles is carried out by using an electronic balance with an accuracy of 1 mg. The volume fractions of the nanoparticles (1.31%, 2.72%, 4.25%, and 5.92%) are calculated according to the following function:

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   ff  pp p  (1)  p   p  ρf φp p  p p (1) f       f p p p ϕp = p (1) ρ p + ρ f φp − ρ pis φpthe weight fraction of the nanoparticle, where  p is the volume fraction of the nanoparticle, where  p is the volume fraction of the nanoparticle,  pp is the weight fraction of the nanoparticle, where is thevolume fraction of the nanoparticle, φp is the weight fraction of the nanoparticle, and and  ϕp p and f are the densities of the nanoparticle and base fluid, respectively. and are the densities of the nanoparticle base fluid, respectively.  pρ and  f densities ρ p and are the of the nanoparticle and baseand fluid, respectively.

To fkeep the stability of the nanofluids, the sodium dodecylbenzene sulfonate (SDBS) from To keep keep the the stability stability of of the the nanofluids, nanofluids, the the sodium sodium dodecylbenzene dodecylbenzene sulfonate (SDBS) from Guangzhou Chemical Reagent Factory (Guangzhou, China) is added as the dispersing agent. The Guangzhou ChemicalReagent ReagentFactory Factory (Guangzhou, China) is added as dispersing the dispersing The Guangzhou Chemical (Guangzhou, China) is added as the agent.agent. The mass mass of the SDBS is quantitatively determined with the electronic balance and equal to that of mass theisSDBS is quantitatively determined with the balance electronic balance to that of of the of SDBS quantitatively determined with the electronic and equal toand thatequal of nanoparticle, nanoparticle, based on the existing experiment investigations [45,46]. Moreover, periodical magnetic nanoparticle, based onexperiment the existinginvestigations experiment investigations [45,46]. Moreover, periodical magnetic based on the existing [45,46]. Moreover, periodical magnetic stirring and stirring and ultrasonic oscillating are applied to prepare the nanofluids, considering that the fact that stirring and ultrasonicare oscillating areprepare appliedthe to nanofluids, prepare the considering nanofluids, considering that ultrasonic oscillating applied to that the fact that the fact the stability process technologies of nanofluids in different studies are not very unified. The stability times of the stability process technologies ofin nanofluids different studies not very unified. The times of process technologies of nanofluids different in studies are not veryare unified. The times of periodical periodical magnetic stirring and ultrasonic oscillating are usually determined according to the actual periodical magnetic stirring and ultrasonic are oscillating are usually determined according toconditions. the actual magnetic stirring ultrasonic usually determined according to the actual conditions. In theand present study,oscillating the corresponding times of the above stability process technologies conditions. In the present study, the corresponding times of the above stability process technologies In the corresponding times of the above stability process technologies are 12 h and arethe 12 present h and 5 study, h, respectively. are h and 5 h, respectively. 5 h, 12 respectively. Figure 1 shows the manufactured Al2O3-water nanofluids after different standing times (0 h, 36 h, Figure 2O33-water nanofluids after standing times (0 h, theseen manufactured -water nanofluids after different different standing h, 36 36 h, h, and 72 h). 1Itshows can be that thereAlis2 no obvious sedimentation for the abovetimes four (0 different and 72 72h). h).It It can be seen that there is no obvious sedimentation for the above fournanofluids, different can be seen that there is no obvious sedimentation for the above four different nanofluids, which means that they were manufactured successfully. In addition, Figure 2 presents nanofluids, which means that they were manufactured successfully. In addition, Figure presents which means that they were manufactured successfully. addition, Figure 2 presents the2Scanning the Scanning Electron Microscope (SEM) image of the AlIn 2O3 nanoparticle with a volume fraction of the Scanning Electron Microscope (SEM) of the Al2O3with nanoparticle with a volume fraction of Electron (SEM) image of the Alimage a volume of and 5.92% in water. 2 O3 nanoparticle 5.92% in Microscope water. It is clearly observed that most of the nanoparticles can befraction spherical have good 5.92% in water. It is clearly observed that most of the nanoparticles can have be spherical and have good It is clearly observed that most of the nanoparticles can be spherical and good dispersion. dispersion. dispersion.

(a) (b) (c) (a) (b) (c) Figure 1. The manufactured Al2O3-water nanofluids with different nanoparticle volume fractions Figure 1. nanofluids with different nanoparticle volume fractions after 2 O2O 3 -water 3-water nanofluids with different nanoparticle volume fractions Figure 1. The Themanufactured manufacturedAlAl after different standing times; (a) 0 h; (b) 36 h; (c) 72 h. different standing times; (a) 0 h; (b) 36 h; (c) 72 h. after different standing times; (a) 0 h; (b) 36 h; (c) 72 h.

Figure 2. Scanning Electron Microscope (SEM) image of the Al2O3 nanoparticle with a volume fraction Figure 2. Scanning Electron Microscope (SEM) image of the Al2O3 nanoparticle with a volume fraction Figure 2. in Scanning of 5.92% water. Electron Microscope (SEM) image of the Al2 O3 nanoparticle with a volume fraction of 5.92% in water. of 5.92% in water.

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2.2. Method Method of of Investigating Investigating the the Thermal Thermal Conductivity Conductivity and and Viscosity Viscosity 2.2. 2.2. Method of Investigating the Thermal Conductivity and Viscosity To effectively -water nanofluids, nanofluids, aa transient transient hot-wire hot-wire To effectively measure measure the the thermal thermal conductivity conductivity of of Al Al22O O33-water To effectively measure the thermal conductivity of Al 2O3-water nanofluids, a transient hot-wire apparatus designed by Xi’an Xiatech Electronic Technology Co., Ltd (Xi’an, China) is applied. apparatus designed by Xi’an Xiatech Electronic Technology Co., Ltd (Xi’an, China) is applied. The apparatus designed by Xi’an Xiatech Electronic Technology Co., Ltd (Xi’an, China) is applied. The The measuring accuracy this apparatus is in ±2%–3% in the thermal conductivity range of measuring accuracy of this of apparatus is ±2%–3% the thermal conductivity range of 0.001–20 W/m measuring accuracy of this apparatus is ±2%–3% in the thermal conductivity range of 0.001–20 W/m 0.001–20 W/m · K and temperature range of 113–423 K. Considering the constant temperature K and temperature range of 113–423 K. Considering the constant temperature requirement, an K and temperature range of 113–423 K. Considering the constant an requirement, an external temperature-controlled bath isin used, as shown in Figure 3. requirement, For the viscosity external temperature-controlled bath is used, as shown Figure 3. Fortemperature the viscosity measurement of external temperature-controlled bath is used, as shown in Figure 3. For the viscosity measurement of measurement of Al2 O3 -water the apparatus (Figure 4) a Kinexus proRotation + Super Al 2O3-water nanofluids, the nanofluids, apparatus (Figure 4) including a including Kinexus pro + Super Al2O3-water nanofluids, the Instruments apparatus (Figure 4)UK) including aa Kinexus pro + Super(Shanghai Rotation Rotation Rheometer (Malvern Ltd, Malvern, UK) Silent AirCompressor Compressor (Shanghai Rheometer (Malvern Instruments Ltd, Malvern, and and a Silent Air Rheometer (Malvern Instruments Ltd, Malvern, UK) and a Silent Air Compressor (Shanghai Dynamic Industry Co., Ltd, Shanghai, China), is applied in the present experiment. Since the viscosities Dynamic Industry Co., Ltd, Shanghai, China), is applied in the present experiment. Since the Dynamic Industry Co., Ltd, Shanghai, China), applied in thethepresent experiment. Since the of Al2 O3 -water usually not high,isnot the Peltierhigh, Cylinder Cartridge is selected. The size viscosities of Alnanofluids 2O3-water are nanofluids arevery usually very Peltier Cylinder Cartridge is viscosities of Al2Oof 3-water nanofluids arediameter usually of not very the Cartridge is of the cup is size C14 (DIN standard). The the bobhigh, is 14of mm. Both themm. cup Both and bob are selected. The the cup is C14 (DIN standard). The diameter thePeltier bob isCylinder 14 the cup selected. The size of the cup is C14 (DIN standard). The diameter of the bob is 14 mm. Both the cup sandblasted to reduce slippage. angular the rheometer is rangedisfrom 10 from nrad/s and bob are sandblasted to reduceThe slippage. Thevelocity angularofvelocity of the rheometer ranged 10 and bob are sandblasted to reduce slippage. The angular velocity of the rheometer is ranged from to 500 rad/s. The temperature resolution of this viscosity measuring equipment is 0.01 K in the nrad/s to 500 rad/s. The temperature resolution of this viscosity measuring equipment is 0.01 K 10 in nrad/s to 500range rad/s. temperature of this information viscosityinformation measuring equipment isprocedure 0.01 K in temperature ofThe 233–473 K. More detailed devices and the experimental the temperature range of 233–473 K.resolution More detailed devices and the experimental the viscosity temperature rangemeasurement of are 233–473 K. detailed devices information and the experimental for referenced in [47–49]. procedure formeasurement viscosity areMore referenced in [47–49]. procedure for viscosity measurement are referenced in [47–49]. Transient hot-wire apparatus Transient hot-wire apparatus

External temperaturecontrolled bath External temperaturecontrolled bath

Figure 3. Measuring equipment of thermal conductivity for Al2O3-water nanofluids. Figure -water nanofluids. nanofluids. Figure 3. 3. Measuring Measuring equipment equipment of of thermal thermal conductivity conductivity for for Al Al22O O33-water Kinexus pro + Super Rotation pro Rheometer Kinexus + Super Rotation Rheometer

Silent Air Compressor Silent Air Compressor

Figure 4. Measuring equipment of viscosity for Al2O3-water nanofluids. Figure 4. Measuring equipment of viscosity for Al2O3-water nanofluids. Figure 4. Measuring equipment of viscosity for Al2 O3 -water nanofluids.

3. Modeling Method Based on ANN 3. Modeling Method Based on ANN 3. Modeling Method Based on ANN As an effective data-driven modeling approach, an ANN is put forward based on the As an effective data-driven modeling approach, anmechanism. ANN is put forward based the inspiration from the human brain’s structure and activity Nowadays, areon many As an effective data-driven modeling approach, an ANN is put forward based onthere the inspiration inspiration from for the various human brain’s structure and activity mechanism. Nowadays, there predictive are many different ANNs applications. the fields of curve-fitting nonlinear from the human brain’s structure and activityInmechanism. Nowadays, thereand are many different ANNs different ANNs for various applications. In the fields of curve-fitting and nonlinear predictive modeling, RBF neural In network exhibits better ability in comparison with others [43]. for variousthe applications. the fields of curve-fitting and nonlinear predictive modeling, the RBF modeling, the RBF neural network exhibits better ability in comparison with others [43]. neural network exhibits better ability in comparison with others [43]. 3.1. RBF Neural Network Theory 3.1. RBF Neural Network Theory In general, an RBF neural network (as shown in Figure 5) is constituted by an input layer, In layer, general, RBF neural network Figure 5) is constituted by an and input layer, hidden andanoutput layer. The input (as andshown outputinlayer correspond to the dendrite synapse hidden layer, and output layer. The input and output layer correspond to the dendrite and synapse

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Theory

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In general, an RBF neural network (as shown in Figure 5) is constituted by an input layer, hidden of biological neurons, respectively. Similarly to the function of cyton, the hidden layer plays a role of layer, and output layer. The input and output layer correspond to the dendrite and synapse of biological intermediation to process the input-output information and deliver it to the output layer. The neurons, respectively. Similarly to the function of cyton, the hidden layer plays a role of intermediation connections between different layers are established through a series of artificial neurons and to process the input-output information and deliver it to the output layer. The connections between weights. Theoretically, the modeling process of an RBF neural network is to solve the mapping from different layers are established through a series of artificial neurons and weights. Theoretically, q q (n, q ≥ of X n modeling , q  1 ). Assuming Xn, to to Y ( nprocess input vector is ofto ansolve RBF the neural network is X theYresponse the of an RBF the neural network mapping from 1). q Assuming theininput vector of an RBF neural network is X, theusing response of kth neuron in the output the output layer ( yk  be obtained the following linear weighting k th neuron Y ) can layer (yk [50]: ∈ Y q ) can be obtained using the following linear weighting function [50]: function m

m

yk ∑ωjkjkRRj (j X ( X),), ((kk =  1, yk = 1,2,2, · ·, ·q ), q)

(2) (2)

j=1j 1

where theconnection connectionweight weightbetween betweenthe thejth jhidden th hidden layer neuron the output  jk isisthe k th output where ω layer neuron and and the kth layer jk and are the numbers ofin neurons in the corresponding layer, respectively. layer neuron neuron and mand and m q are theqnumbers of neurons the corresponding layer, respectively. Input layer

Hidden layer

Output layer

R1 wjk

wij

x1



y1

R2 x2

···



Rm

y2

···

···

R3

···

··· xn



yq

Figure neural network. network. Figure 5. 5. Typical Typical architecture architecture of of an an radial radial basis basis function function (RBF) (RBF) neural

Different from those of many other ANN, the responses of RBF neural networks’ j th hidden Different from those of many other ANN, the responses of RBF neural networks’ jth hidden layer layer neuron are usually determined by the RBF. When it selects the Gauss function, the neuron are usually determined by the RBF. When it selects the Gauss function, the corresponding corresponding R j ( X ) can be defined as: R j ( X ) can be defined as: 2

X−c ck2j kX exp( RjR ( Xj ()X=) exp (− 2σ2j 2 ), ), 2j

j 1, 1,2, 2,  ) ) ( j( = · · ·, m, m

(3) (3)

j

j th center thethe Euclidean distance between input vector X andX the jththe neuron c j center and σj isc jwidth where kk is is Euclidean distance between input vector and neuron and of the jth neuron.  j is width of the j th neuron. Analyzing Equations (2) and (3), it can be easily found that the key to RBF neural network Analyzing Equations (2) and (3), it can be easily found that the key to RBF neural network training is how to determine ω jk , c j , and σj . In the past decades, different unsupervised and supervised training is how to determine  jk , c j , and  j . In the past decades, different unsupervised and algorithms have been developed to solve the above problem [51]. In this study, the network parameters supervised have been least developed solve the above In this study, are updatedalgorithms by using a orthogonal square to (OLS) approach, for problem which the[51]. minimizing functionthe is network parameters are updated by using a orthogonal least square (OLS) approach, for which the shown in Equation (4). More detailed information about OLSs can be found in [52]. minimizing function is shown in Equation (4). More detailed information about OLSs can be found q in [52]. minJ = ∑ (|ynk − ydk |2 ) (4) q

min J   ( ynk  ydk ) (4) k 1 where ynk and ydk are the network output and desired output of the kth output layer node, respectively. where ynk and ydk are the network output and desired output of the k th output layer node, k =1

2

respectively. 3.2. Implementing Procedure In the present investigation, a typical three layer RBF neural network is developed. For the Al2O3-water nanofluids with the determined nanoparticle size, nanoparticle volume fraction and

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3.2. Implementing Procedure In the present investigation, a typical three layer RBF neural network is developed. For the Materialsnanofluids 2017, 10, 552 6 of 17 Al2 O3 -water with the determined nanoparticle size, nanoparticle volume fraction and Materials 2017, 10, 552 6 of 17 temperature are the most important factors for influencing the thermal conductivity and viscosity. temperature are the most important factors for influencing the thermal conductivity and viscosity. Therefore, both input and output layers the RBFneural neural network consist ofneurons, two neurons, as temperature are the important factors forof the thermal conductivity and viscosity. Therefore, the both themost input and output layers ofinfluencing the RBF network consist of two as illustrated in Figure 6. The neurons in the hidden layer and others are determined in the training Therefore, both the input and output layers of the RBF neural network consist of two neurons, as illustrated in Figure 6. The neurons in the hidden layer and others are determined in the training illustrated in7Figure 6. The in the hidden layer and others are in the training process. 7 presents the detailed procedure forfor implementing thedetermined modelling and prediction of process. FigureFigure presents theneurons detailed procedure implementing the modelling and prediction process. Figure 7 presents the detailed procedure for implementing the modelling and prediction of nanofluids based on the RBF neural network. To improve the training accuracy, all the input and of nanofluids based on the RBF neural network. To improve the training accuracy, all the input and nanofluids based on RBF neural network. To improve the training accuracy, all the input and output variables arethe normalized. output variables are normalized. output variables are normalized. xx−  xxminmin x'  x0 = (5) (1) xmax xmaxxmin −xmin x xmin x'  x

(1)

x

max min where xwhere is the xoriginal is the xnormalized value, and xmaxand andxmax xminand are xthe corresponding is the value, originalx 0value, value, are the ' is the normalized min x minimum is themaximum original value, where x ' is ofthex . normalized value, and xmax and xmin are the maximum and of x.and corresponding minimum

corresponding maximum and minimum of x . Hidden layer Hidden layer

Input layer Nanoparticle Input layer volume fraction Nanoparticle volume fraction

Output layer Thermal Output layer conductivity ratio  Thermal conductivity ratio



··· ···

Temperature



Temperature



Viscosity ratio Viscosity ratio

Figure 6. The RBF neural network developed in this study. Figure 6. The RBF neural network developed in this study.

Figure 6. The RBF neural network developed in this study. Samples data Samples data Data preprocessing Data preprocessing Training samples

Testing samples

Training samples

Testing samples

Initial determine hidden layer node Initialnumber determine m=1 hidden layer node number m=1 RBF neural network training RBF neural network training Are errors acceptable? Are errors acceptable? No Adjustment No of hidden layer node Adjustment of number m=m+1 hidden layer node number m=m+1

Yes Yes

Trained RBF neural network Trained RBF neural network Prediction results Prediction results

Figure 7. Implementing process of the RBF neural network for modeling and prediction. Figure 7. Implementing process of the RBF neural network for modeling and prediction.

Figure 7. Implementing of the RBF neural modeling andfour prediction. To effectively evaluate process the predictive accuracy of network the RBF for neural network, important To effectively evaluate the squared predictive accuracy the RBF neural network,error four(MAPE), important parameters, namely root mean error (RMSE),ofmean absolute percentage sum 2),error parameters, namely root mean squared error (RMSE), mean absolute percentage (MAPE), sum of squared error (SSE), and statistical coefficient of multiple determination (R are used. To effectively evaluate the predictive accuracy of the RBF neural network, four important of squared error (SSE), and statistical coefficient of multiple determination (R2), are used.

parameters, namely root mean squared error (RMSE), mean absolute percentage error (MAPE), sum of squared error (SSE), and statistical coefficient of multiple determination (R2 ), are used.

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RMSE = (

1 t | Pl − Ql |2 ) t l∑ =1

1/2

(6)

100% t Pl − Ql MAPE = P t l∑ l =1

(7)

t

SSE =

∑ ( Pl − Ql )2

(8)

l =1

t

2 ∑ ( Pl − Ql )

R2 = 1 −

l =1

(9)

t

2 ∑ ( Pl )

l =1

where P is the desired value, Q is the network output value, and t is the number of samples. 4. Results and Discussion 4.1. Enhancement of Thermal Conductivity To verify the effectiveness of the above thermal conductivity measuring apparatus, water is measured first. Considering the temperature balance of the testing sample and the fluid in bath, the testing temperature can be determined when it remains constant for 20 min. Every experimental data is the average value of five measurements with a frequency interval of 5 min. Table 1 presents the experimental thermal conductivity of water in the temperature range of 288–318 K. According to the comparison, it is concluded that the thermal conductivity apparatus has good precision for the present study. Based on the experimental data, the measurement uncertainty of thermal conductivity is less than 5% for water. Table 1. Thermal conductivity measurement and analysis for water. Temperature (K)

Reference Values (W/(m·K))

Measure Values (W/(m·K))

Deviation (%)

288 293 298 303 308 313 318

0.5916 0.6003 0.6088 0.6173 0.6245 0.6318 0.6379

0.595 0.6019 0.6111 0.6185 0.6252 0.6292 0.6374

0.5747 0.2665 0.3778 0.1944 0.1121 −0.4115 −0.0784

Figure 8 presents the change of the thermal conductivity ratio (k n f /k b f ) between Al2 O3 -water nanofluids and water with different nanoparticle volume fractions at room temperature. From Figure 8, it is found that the thermal conductivity of water can be enhanced obviously with the increase of the Al2 O3 nanoparticle. For example, at a nanoparticle volume fraction of 1.31%, the enhancement of water thermal conductivity is 9.4%. When the volume fraction of the Al2 O3 nanoparticle increases to 5.92%, the k n f /k b f can change to 1.231. In addition, Figure 8 compares the present measurements with many experimental data obtained from the existing publications. The results show that they are in good agreement with both the qualitative and quantitative aspects. This may mean that both the sample preparation and thermal conductivity measurements are successful. In addition, it is also clearly observed from Figure 8 that the enhancement of the Al2 O3 nanoparticle on water thermal conductivity cannot be described accurately by using the well-known Maxwell model and the Yu and Choi model due to the complex influence mechanisms such as the interfacial layer, nanoparticle Brownian motion, and clustering.

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Thermal conductivity ratio, knf /kbf

1.8

Pak and Cho[53], 13nm Wang et al.[54], 28nm Li and Peterson[55], 36nm Das et al.[56], 38.4nm Lee et al.[2], 38.4nm Chandrasekar et al.[57], 43nm Murshed et al.[58], 80nm Present measurements, 30nm Maxwell model[59] Yu & Choi model[60]

1.6

1.4

1.2

8 of 17 1.8 1.0

0

1

2

3

4

5

6

7

Thermal conductivity ratio, knf /kbf

Pak and Cho[53], 13nm volume fraction (%) WangNanoparticle et al.[54], 28nm Li and Peterson[55], 36nm Figure 8. Variations of 1.6 Al2O3-water nanofluids k nf / kbf with nanoparticle volume fraction at room Das et al.[56], 38.4nm Figure 8. Variations of Al2 O3 -water nanofluids k n f /k b f with nanoparticle volume fraction Lee et al.[2], 38.4nm temperature [2,53–60]. temperature [2,53–60]. Chandrasekar et al.[57], 43nm Murshed et al.[58], 80nm 1.4 Maxwell model [59] Present measurements, 30nm Maxwell model[59] k Choi k p model[60]  2kbf  2 p (k p  kbf ) Maxwell model [59] Yunf&  1.2

at room

(10)

k n f kbfk p +k p2k p (k p − k b f )  2bkfbf+  2ϕ p ( k p  kbf ) = kb f k p + 2k b f − ϕ p (k p − k b f )

(10)

where k nf , k p , and k bf are the thermal conductivity of nanofluids, the nanoparticle, and base 1.0 thermal conductivity of nanofluids, the nanoparticle, and base where k n f fluid, , k p ,respectively. and k b f are the 0 1 2 3 4 5 6 7 Yu & Choi model [60] fluid, respectively. Nanoparticle volume fraction (%) Yu & Choi model [60] of Al2O3-water knf nanofluids k pl  2kbf k2(k/plk kbfwith )(1 nanoparticle  )3  p Figure 8. Variations volume fraction at room

kbf

temperature [2,53–60].

kn f

Maxwell model [59]k

=

bf



nf

bf

k pl  2kbf  (k pl  kbf )(1   )3  p

(11)

3

k pl + 2k b f + 2(k pl − k b f )(1 + β) ϕ p 3 [2(1   )  (1   ) (1  2 )]

3  kp kkplpl + 2k k pl− k b f2)( (1bf − ) ((1 )3 (1  )1 + β ) ϕ p

(12)

knf k p  2kbf  2 p (k p  kbf )  (10)of where   kl / k p , kl is the thermalk conductivity of interfacial layer,   h / rp , h is thickness  2k(bf1 + k3p (1kbf+) 2γ pβ() [2(1bf− γk)p + )]γ interfacial layer, and rp is kp k plthe=radius of nanoparticle. thermal where k nf , k p , and k bf are the −( 1 − γconductivity ) + (1 + β)of3 (nanofluids, 1 + 2γ) the nanoparticle, and base 1.40

(11)

(12)

Thermal conductivity ratio, knf /kbf

where γ =fluid, k l /krespectively. of interfacial layer, β = h/r p , h is thickness of interfacial p , k l is the thermal conductivity Volume fraction of 1.31% Yu & Choi model [60]1.35 Volume fraction of 2.72% layer, and r p is the radius of nanoparticle. fraction of 4.25% knf Volume k pl ranging 2kbf  2(k plfrom  k )(1   )3  p Considering the effects of temperature K, Figure 9 presents(11) the variation 1.30 Volume fraction of 5.92% bf 296 to 313  kbf k pl  2kbf  (k pl  kbf )(1   )3  p of k n f /k b f with various volume fractions. It can be found that, for any volume fraction of the Al2 O3 1.25 nanoparticle, the corresponding k n f /k b f can [2(1linearly   )  (1  improve )3 (1  2 )]with the increase of temperature, which k pl  kp (12) 1.20 (1   )  (1   )3 (1  2 ) is usually explained by the enhancement of nanoparticle Brownian motion. In the present study, taking 1.15 the nanofluids Al kO fraction of 2.72% as an example, k n f /k 3 volume wherewith the thermal conductivity of interfacial layer,  the ofb f of 1.283   kan h / rmaximum l is l / kp , 2 p , h is thickness is obtainedinterfacial at the nanoparticle volume fraction of 5.92% and a temperature of 313 K. layer, and rp is1.10 the radius of nanoparticle. 1.05 1.40295

Thermal conductivity ratio, knf /kbf

1.35

Figure 9. Variations of k nf / kbf 1.30

300

305

310

315

Volume fraction of 1.31% (K) Temperature Volume fraction of 2.72% Volume fraction of 4.25% with temperature for different nanoparticle volume fractions of Volume fraction of 5.92%

Al2O3-water nanofluids. 1.25 1.20 1.15 1.10 1.05 295

300

305

310

315

Temperature (K)

Figure 9. Variations of k nf / kbf with temperature for different nanoparticle volume fractions of

Figure 9. Variations of k n f /k b f with temperature for different nanoparticle volume fractions of Al2O3-water nanofluids. Al2 O3 -water nanofluids.

variation of k nf / kbf with various volume fractions. It can be found that, for any volume fraction of the Al2O3 nanoparticle, the corresponding k nf / kbf can linearly improve with the increase of temperature, which is usually explained by the enhancement of nanoparticle Brownian motion. In the present study, taking the nanofluids with an Al2O3 volume fraction of 2.72% as an example, the maximum a k nf 552 / kbf of 1.283 is obtained at the nanoparticle volume fraction of 5.92% and Materials 2017, 10, 9 of 16 temperature of 313 K. 4.2. Viscosity Investigation 4.2. Viscosity Investigation Before experimentally analyzing the viscosity of Al O -water nanofluids, it is also necessary to Before experimentally analyzing the viscosity of Al22O33-water nanofluids, it is also necessary to evaluate the apparatus’s effectiveness by selecting water as a sample. Both the measuring frequency evaluate the apparatus’s effectiveness by selecting water as a sample. Both the measuring frequency and data analysis method are same as those for thermal conductivity. From the contrastive analysis and data analysis method are same as those for thermal conductivity. From the contrastive analysis shown in Table 2, it can be inferred that the measurements of viscosity are effective, with a maximum shown in Table 2, it can be inferred that the measurements of viscosity are effective, with a deviation of 0.988% in the temperature ranges of 288–318 K. In addition, the experimental analysis maximum deviation of 0.988% in the temperature ranges of 288–318 K. In addition, the experimental indicates that the measurement uncertainty of water viscosity is less than 5% using the above mentioned analysis indicates that the measurement uncertainty of water viscosity is less than 5% using the approach when the shear rate changes. above mentioned approach when the shear rate changes. Table 2. Viscosity measurement and analysis for water. Table 2. Viscosity measurement and analysis for water. Temperature (K) (K) Reference Values (mPa ·s) Temperature Reference Values (mPa·s) 288 1.1426 288 1.1426 1.0094 293 293 1.0094 298 298 0.8938 0.8938 303 303 0.8029 0.8029 308 308 0.7226 0.7226 313 313 0.6634 0.6634 318 0.6008 318 0.6008

Measure Values ·s) Values (mPa (mPa·s) 1.1517 1.1517 0.9998 0.8943 0.7958 0.7182 0.7182 0.6693 0.6693 0.6043 0.6043

Deviation Deviation (%)(%) −0.7955 −0.7955 0.9517 0.9517 −0.0658 −0.0658 0.8791 0.8791 0.6106 0.6106 −0.8866 −0.8866 −0.5876 −0.5876

To investigate investigate the the influence influence of of Al Al22O33-water -water nanofluids, nanofluids, Figure Figure 10 presents the relationship between the the shear shearrate rateand andnanofluid nanofluid viscosity temperature of 298 K. The results viscosity at at thethe temperature of 298 K. The results showshow that, that, with − 1 − 1 −1 −1 with the increase the shear to 126.2 , theviscosities viscositiesofofAl Al22O O33-water -water nanofluids the increase of theofshear rate rate fromfrom 6.3266.326 s sto 126.2 s s, the with different Al22O33 volume volume fractions fractions do do not not change change significantly. significantly. This may mean that the viscosities of the Al22O O33-water -waternanofluids nanofluids obtained obtained in in the the present present study study display display Newtonian Newtonian behavior. behavior. 2.2

Volume fraction of 1.31% Volume fraction of 2.72% Volume fraction of 4.25% Volume fraction of 5.92%

Viscosity (mPa.s)

2.0

1.8

1.6

1.4

1.2

1.0

0

20

40

60

80

100

120

140

Shear stress (s-1)

Figure 10. Rheological behaviors of Al2O3-water nanofluids at room temperature. Figure 10. Rheological behaviors of Al2 O3 -water nanofluids at room temperature.

Considering the influence of the Al2O3 volume fraction at the temperature of 298 K, the Considering the influence the Al fraction the temperature 298 K, 2 O3 volume experimental viscosities of the Al2Oof3-water nanofluids are given and at compared with much of published the experimental of the Al2that O3 -water nanofluidsofare and compared with much data in Figure 11. viscosities All the results show the suspension Al2given O3 nanoparticles can increase the published data in Figure 11. All the results show that the suspension of Al2 O3 nanoparticles can increase the viscosity of water, and there is a slight non-linear relationship between the viscosity of nanofluids and nanoparticle volume fraction. Moreover, a careful inspection of Figure 11 reveals that the theoretical viscosities obtained by the classical Brinkman model are significantly lower than the corresponding measurements. Compared to the Brinkman model, the Corcione model can effectively describe the effect of nanoparticle volume fraction on viscosity, but its prediction precision is not very ideal. This is because the viscosity of nanofluids depends strongly on many known and unknown factors.

viscosity of water, and there is a slight non-linear relationship between the viscosity of nanofluids and nanoparticle volume fraction. Moreover, a careful inspection of Figure 11 reveals that the theoretical viscosities obtained by the classical Brinkman model are significantly lower than the corresponding measurements. Compared to the Brinkman model, the Corcione model can effectively describe the effect of nanoparticle volume fraction on viscosity, but its prediction precision is not Materials 2017, 552 This is because the viscosity of nanofluids depends strongly on many known and 10 of 16 very 10, ideal. unknown factors. 8

Pak and Cho[53], 13nm Chen et al.[61], 28nm Chen et al.[61], 36nm Nguyen et al.[62], 36nm Nguyen et al.[62], 47nm Anoop et al.[63], 95nm Present measurements, 30nm Brinkman model[64] Corcione model[65]

Viscosity (mPa.s)

6

4

2

0

0

2

4

6

8

10

12

14

Nanoparticle volume fraction (%)

Figure 11. Variations of Al2O3-water nanofluids’ viscosity with nanoparticle volume fraction at room

Figure 11. Variations of Al2 O3 -water nanofluids’ viscosity with nanoparticle volume fraction at room temperature [53,61–65]. temperature [53,61–65]. The Brinkman model [64] is as follows:

The Brinkman model [64] is as follows: nf µn fbf



1 (1   p ) 2.5

(13)

1

=

(13)

where  nf and  bf are the viscosity ofµnanofluids bf (1 −and ϕ p base )2.5 fluid, respectively. The Corcione model [65] is as follows:

where µn f and µb f are the viscosity of nanofluids and base fluid, respectively. nf 1  The Corcione model [65] is as follows: 0.3 1.03 bf

1  34.87(d p / d f )

p

µn f 1 1/3  6M  =  d µb f   −0.3 ϕ p 1.03 f 1 − 34.87 p /d f ) N(d 

f0

(14)

(15)



1/3 23 1 where M is the molar mass of base fluid molecule, 6M N  6.022 10 mol is avogadro’s number, d f = of 293 K, and d p is the diameter of nanoparticle.  f 0 is the density of base fluids at temperature

"

(14)

#

Nπρ f 0

(15)

Figure 12 presents the variation of the viscosity ratio, nf / bf , between nanofluids and water 1 is avogadro’s number, ρ mol − where M molar mass of base fluid molecule, N = 6.022 × 1023 as is thethe functions of temperature and nanoparticle volume fraction. From Figure 12, it is observed f0 that, forof the manufactured Al2O3-water nanofluids this study, temperature enhanced effect is the density base fluids at temperature of 293 K,inand d p is the diameterhas ofan nanoparticle. on viscosity in the the temperature ranges ofviscosity 296–313 K.ratio, At theµnanoparticle volume fraction of 4.25%, Figure 12 presents variation of the n f /µb f , between nanofluids and water as the nf / bf fractions are respectively 1.605, 1.664, 1.687, and 1.694 when the temperatures are 298 K, the functions of temperature and nanoparticle volume fraction. From Figure 12, it is observed that, 303 K, 308 K, and 313 K. for the manufactured Al2 O3 -water nanofluids in this study, temperature has an enhanced effect on viscosity in the temperature ranges of 296–313 K. At the nanoparticle volume fraction of 4.25%, the µn f /µb f fractions are respectively 1.605, 1.664, 1.687, and 1.694 when the temperatures are 298 K, 303 K, 308 K, and 3132017, K. 10, 552 Materials 11 of 17 2.4

Viscosity ratio, μnf /μbf

2.2

2.0

Volume fraction of 1.31% Volume fraction of 2.72% Volume fraction of 4.25% Volume fraction of 5.92%

1.8

1.6

1.4

1.2 295

300

305

310

315

Temperature (K)

Figure 12. Variations of nf / bf with temperature for different nanoparticle volume fractions of Figure 12. Variations of µn f /µ b f with temperature for different nanoparticle volume fractions of Al2O3-water nanofluids. Al2 O3 -water nanofluids.

4.3. Predictive Analysis of RBF Neural Networks Based on the above experiment, the limited experimental data (40) are used to discuss the modeling and prediction processes of the RBF neural network for the thermal conductivity and viscosity of Al2O3-water nanofluids. Among them, the ratio of training and testing samples is 3:1. For the RBF neural network, Spread is usually a very important factor for influencing the

1.2 295

300

305

310

315

Temperature (K)

Figure 12. Variations of nf / bf with temperature for different nanoparticle volume fractions of Al22017, O3-water nanofluids. Materials 10, 552

11 of 16

4.3. Predictive Analysis of RBF Neural Networks 4.3. Predictive Analysis of RBF Neural Networks Based on the above experiment, the limited experimental data (40) are used to discuss the Basedand on the above experiment, experimental (40)thermal are used to discuss and the modeling prediction processes of the the limited RBF neural network data for the conductivity modeling and prediction processes of the RBF neural network for the thermal conductivity and viscosity of Al2O3-water nanofluids. Among them, the ratio of training and testing samples is 3:1. viscosity Amongisthem, the ratio of training andfactor testing is 3:1.the For of theAlRBF neuralnanofluids. network, Spread usually a very important forsamples influencing 2 O3 -water For the RBF neural is usually a very important influencing training training process. Figurenetwork, 13 showsSpread the relationships of mean squarefactor errorfor (MSE) and thethe number of process. Figure 13 shows the relationships of mean square error (MSE) and the number of hidden hidden layer neurons with different values of Spread. Table 3 lists the effect of Spread on network layer neurons with different values of Spread. Table lists thereported effect of in Spread on13network modeling modeling accuracy. Comprehensively analyzing the3results Figure and Table 3, it is accuracy. Comprehensively analyzing the results reported in Figure 13 and Table 3, it is found that found that both the network structure and modeling performance cannot be changed significantly both the network structure and modeling performance cannot be changed significantly with different with different values of Spread in this study. Therefore, the network structure of 2-8-2 neurons with values of Spread study. Therefore, the network structure of 2-8-2 with the can Spread of 0.1 the Spread of 0.1in is this used. The related weights and biases of a 2-8-2 RBFneurons neural network be found is used. The related weights and biases of a 2-8-2 RBF neural network can be found in Table 4. in Table 4. -1.5

Spread=0.1 Spread=0.3 Spread=0.5 Spread=0.7 Spread=1

-2.0

log10(MSE)

-2.5 -3.0 -3.5 -4.0

Target value -4.5 0

1

2

3

4

5

6

7

8

9

Number of hidden layer neurons

Figure 13. 13. Relationships Relationships of of mean mean square square error error (MSE) (MSE) and and the the number number of of hidden hidden layer layer neurons neurons with with Figure different values of Spread. different values of Spread. Table 3. Performance evaluation of the RBF neural network for the total samples with different values of Spread.

Parameters

Evaluation Criteria

Spread 0.1

0.3

0.5

0.7

1

8.572 × 0.5177 2.939 × 10−3 0.999944

10−3

6.797 × 0.4803 1.848 × 10−3 0.999965

10−3

4.140 × 0.2872 6.857 × 10−4 0.999987

10−3

knf /kbf

RMSE MAPE(%) SSE R2

10−3

4.346 × 0.3197 7.556 × 10−4 0.999986

4.043 × 10−3 0.2866 6.538 × 10−4 0.999988

µnf /µbf

RMSE MAPE(%) SSE R2

1.423 × 10−2 0.5618 8.094 × 10−3 0.999913

2.311 × 10−2 1.3862 2.137 × 10−2 0.999770

1.624 × 10−2 0.8233 1.055 × 10−2 0.999887

1.381 × 10−2 0.7169 7.634 × 10−3 0.999918

1.658 ×10−2 0.8280 1.100 × 10−2 0.999882

Figures 14 and 15 compare the RBF predicted thermophysical properties of Al2 O3 -water nanofluids with the corresponding experimental data. Table 5 lists the predictive evaluation criteria of the RBF neural network for the training and testing samples. As shown in Figure 14 and Table 5, the RBF neural network has a high accuracy for modeling the thermal conductivity and viscosity of Al2 O3 -water nanofluids with limited experimental data. All the prediction errors of thermal conductivity and nearly 92.5% of those of viscosity are within the ±2% error band. It is worth noting that there is a higher accuracy for the testing dataset but not the training ones. This is because the samples of the testing dataset are very few in this study. In addition, the results analysis of Figure 15 reveals

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that the effects of nanoparticle volume fraction and temperature on the above two thermophysical properties can be effectively extracted in the data-driven prediction of the RBF neural network. All of the above investigations demonstrate that a RBF neural network provides a successful alternative to the traditional model-based prediction approach for the thermal conductivity and viscosity of Al2 O3 -water nanofluids. Table 4. Weight and bias coefficients of the developed RBF neural network.

Neuron

Hidden Layer

Output Layer

Weights wij and Biases

Weights wjk and Biases

Nanoparticle Volume Fraction 1 2 3 4 5 6 7 8 Materials 2017, 10, 552

Temperature

Biases

Thermal Conductivity

Viscosity

Biases

1 1 1 1 0.9930 0.9841 1 1

1.1894 1.1894 1.1894 1.1894 1.1894 1.1894 1.1894 1.1894

−61.8011 −4.4132 24.4808 92.2675 −76.4725 36.6988 −61.8011 −4.4132

−143.4012 −18.1112 49.7229 218.6915 −178.5244 86.1722 −143.4012 −18.1112

−4.7283 −6.1187

0.7184 0 1 0.4595 0.2208 0 0.7184 0

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Materials 2017, 10, 552 2.1 2.1

1.4 1.4

Training samples Training samples Testing samples Testing samples

2% 2% 1.8 1.8

Training samples Training samples Testing samples Testing samples

2% 2% -2% -2%

RBFpredicted predictedμ μ/μ /μ bf nf RBF nf bf

-2% -2%

R B F predicted k nf /k bf R B F predicted k nf /k bf

1.3 1.3 1.2 1.2

1.5 1.5

1.1 1.1

1.2 1.2

1.0 1.0 0.9 0.9 0.9 0.9

1.0 1.0

1.1 1.1

1.2 1.2

Experimental knf /kbf Experimental knf /kbf

1.3 1.3

1.4 1.4

0.9 0.9 0.9 0.9

1.2 1.2

(a) (a)

1.5 1.5

Experimental μnf /μbf Experimental μnf /μbf

1.8 1.8

2.1 2.1

(b) (b)

Figure 14. Scatter plots of (a) k nf / k b f and (b)  nf /  bf for the RBF predicted results and and (b) (b) µnnff /µ Figure Scatter plots plots of of (a) (a) kknnff /k andand Figure 14. 14.Scatter the RBF RBF predicted predictedresults results / bbff for the / kbbff and experimental data. experimental data. experimental data. 1.4 1.4

2.4 2.4

Experimental data Experimental data RBF predictive values RBF predictive values 5.92% 5.92% 4.25% 4.25% 2.72% 2.72% 1.31% 1.31%

1.2 1.2 1.1 1.1

2.1 2.1

Viscosity ratio, μ nf /μ bf Viscosity ratio, μ nf /μ bf

Thermal conductivity ratio, k nf /kbf Thermal conductivity ratio, k nf /k bf

1.3 1.3

1.8 1.8

0.9 294 0.9 294

297 297

300 300

303 303

306 306

309 309

Temperature (K) Temperature (K)

(a) (a)

312 312

315 315

4.25% 4.25% 2.72% 2.72%

1.5 1.5

0% 0%

1.0 1.0

Experimental data Experimental data RBF predictive values RBF predictive values 5.92% 5.92%

1.31% 1.31%

1.2 1.2

318 318

0.9 294 0.9 294

0% 0% 297 297

300 300

303 303

306 306

309 309

Temperature (K) Temperature (K)

312 312

315 315

318 318

(b) (b)

Figure 15. Comparisons of the experimental and RBF predicted results for (a) k nf / k b f and (b) and (b) Figure 15. Comparisons of the experimental and RBF predicted results for (a) k / k

Figure of the experimental and RBF predicted results for (a) k n f /k b fnf andb f(b) µn f /µb f . .  nf /15.  Comparisons  nf / bfbf . Table 5. Performance evaluation of the RBF neural network for the training and testing samples. Table 5. Performance evaluation of the RBF neural network for the training and testing samples. Parameters Evaluation Criteria Training Samples Testing Samples Parameters Evaluation Criteria Training Samples Testing Samples 3.974 × 10−3−3 RMSE 4.464 × 10−3−3 3.974 × 10 RMSE 4.464 × 10 MAPE (%) 0.3230 0.3098

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Table 5. Performance evaluation of the RBF neural network for the training and testing samples. Parameters

Evaluation Criteria

Training Samples

Testing Samples

knf /kbf

RMSE MAPE (%) SSE R2

10−3

4.464 × 0.3230 5.977 × 10−4 0.999985

3.974 × 10−3 0.3098 1.579 × 10−4 0.999988

µnf /µbf

RMSE MAPE (%) SSE R2

1.419 × 10−2 0.7472 6.040 × 10−3 0.999913

1.263 × 10−2 0.6261 1.594 × 10−3 0.999932

5. Conclusions In this paper, the experiments on Al2 O3 -water nanofluid preparation and thermophysical properties measurement are performed to obtain the effects of nanoparticle volume fraction and temperature on thermal conductivity and viscosity. All the experimental results showed that both thermal conductivity and viscosity could be enhanced with the increase of the Al2 O3 nanoparticle volume fraction and temperature. On this basis, considering the advantage of a RBF neural network in modeling, a case study was investigated to discuss the application of a RBF neural network on the prediction of nanofluids’ thermal conductivity and viscosity with 40 sets of experimental data. By comparing the RBF predictive values and the experimental data, it was demonstrated that RBF neural network not only exhibited good modeling accuracy (thermal conductivity: RMSE = 8.572 × 10−3 , MAPE = 0.5177%, SSE = 2.939 × 10−3 , R2 = 0.999944; viscosity: RMSE = 1.423 × 10−2 , MAPE = 0.5618%, SSE = 8.094 × 10−3 , R2 = 0.999913), but also could effectively extract the influences of nanoparticle volume fraction and temperature on Al2 O3 -water nanofluids’ thermal conductivity and viscosity. Acknowledgments: The authors acknowledge the financial support from the Fundamental Research Funds for the Central Universities (No. HEUCFJ170304) and are thankful for Fengchen Li and Tongzhou Wei for their experimental support. Author Contributions: Ningbo Zhao prepared the Al2 O3 -water nanofluids, and performed the measurements of thermophysical properties of nanofluids. Zhiming Li provided the program codes of RBF neural network and wrote the basic theory of the RBF neural network. Conflicts of Interest: The authors declare no conflict of interest.

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28. 29.

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