Journal of Thermal Analysis and Calorimetry https://doi.org/10.1007/s10973-017-6886-z

Connectionist intelligent model estimates of convective heat transfer coefficient of nanofluids in circular cross-sectional channels Alireza Baghban1 • Fathollah Pourfayaz2 • Mohammad Hossein Ahmadi3 • Alibakhsh Kasaeian2 Seyed Mohsen Pourkiaei2 • Giulio Lorenzini4

•

Received: 16 July 2017 / Accepted: 3 December 2017  Akade´miai Kiado´, Budapest, Hungary 2017

Abstract Nanofluids are kind of fluids, which have a wide range of applications in different fields such as industry or engineering systems. The present study efforts to find accurate relationships between the convective heat transfer coefficient of the nanofluids containing the silica nanoparticles as a function of Reynolds number, Prandtl number, and mass fraction nanofluid. To that end, a number of seven different models including adaptive neuro-fuzzy inference system (ANFIS), artificial neural network (ANN), support vector machine (SVM), least square support vector machine (LSSVM), genetic programming (GP), principal component analysis (PCA), and committee machine intelligent system (CMIS) have been implemented according to experimental databases designed for measuring the convective heat transfer coefficient of nanofluid in circular cross-sectional channels. Results indicated the satisfactory capability of suggested models, especially CMIS model in order to estimate the convective heat transfer coefficient of nanofluid. The obtained statistical analyses such as the mean square error and R-squared (R2) for the ANFIS, ANN, SVM, LSSVM, PCA, GP, and CMIS were 380.6671 and 0.9946, 215.062 and 0.9969, 335.748 and 0.9951, 298.88 and 0.9959, 1601.336 and 0.977, 1891.861 and 0.973, and 205.366 and 0.9970 correspondingly. We expect that these suggested models can help engineers who deal with heat transfer phenomenon to have great predictive tools for estimating convective heat transfer coefficient of nanofluid. Keywords Convective heat transfer  Reynolds number  Prandtl number  Intelligent models  Optimization

Introduction

& Fathollah Pourfayaz [email protected] & Mohammad Hossein Ahmadi [email protected]; [email protected] 1

Department of Chemical Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran

2

Department of Renewable Energies, Faculty of New Sciences and Technologies, University of Tehran, Tehran, Iran

3

Faculty of Mechanical Engineering, Shahrood University of Technology, Shahroud, Iran

4

Dipartimento di Ingegneria e Architettura, Universita` degli Studi di Parma, Parco Area delle Scienze 181/A, 43124 Parma, Italy

Nanofluids are complex of nanoparticles in a fluid context such as water and oil. This new kind of fluids has a wide range of applications in different fields such as industry or engineering systems [1, 2]. One of the most valuable characteristics of nanofluids is thermal conductivity which makes them useful in cooling and heating systems. Many studies have been performed on this property of nanofluids [3–8]. Nasiri et al. [9] experimentally have investigated the heat transfer of Al2O3/H2O and TiO2/H2O nanofluid through an annular duct for turbulent flow regime. They have reported a significant increase in heat transfer in both nanofluids. Nasrin and Alim [10] have presented a semiempirical correlation for forced convective analysis through a solar collector. They have reported that the heat transfer rate had an increase about 26% for nanofluid. Sahin et al. [11] have experimentally investigated the steady-state turbulent convective heat transfer of Al2O3–

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water nanofluid. They have reported that increasing the concentration of nanoparticles more than 1 vol% increases the thermal conductivity. Saeedinia et al. [12] have experimentally investigated the heat transfer of nanofluid flow and presented two empirical correlations for predicting the experimental data in an error range of (± 0.2). Moghadassi et al. [13] have studied the effect of the thermal conductivity of nanofluids. They have presented a novel model which could predict the effective thermal conductivity of nanofluids. Barbes et al. [14] have studied empirically the thermal properties and specific heat capacity of Al2O3–water and Al2O3–ethylene glycol in various concentrations and temperatures. One of the most interesting phenomena of nanofluids is heat conductivity which is concerned as the most important feature of nanofluids in many industrial fields and engineering applications [15]. Many investigations have been carried out on calculating the convective heat transfer coefficients of nanofluids [15, 16]. Vajjha et al. [17] have studied the forced convective heat transfer of nanofluids in the fully developed turbulent regime and presented a new relation for the convective heat transfer. Moreover, investigations showed that the wetting kinetics and rheological properties greatly affect the convective heat transfer coefficient of the nanofluids [18, 19]. Yang et al. [20] have investigated a theoretical model of thermal conductivity of nanofluids with particles in a cylindrical shape by anisotropy analysis. They have presented a thermal conductivity model by taking into account of the thermal conductivity coefficient transformation of equivalent anisotropic material. Valinataj-Bahnemiri et al. [21] have modeled and optimized the heat transfer of two phases of nanofluid by applying an artificial intelligence technique, bee colony. The results showed that volume fraction has a direct effect on the thermal–hydraulic performance factor. In addition, an optimum rate of Reynolds number was set up to obtain the maximum thermal–hydraulic performance factor. Islam et al. [22] have studied the utilizing of nanofluids in proton exchange membrane fuel cell cooling systems. They have reviewed the convective heat transfer and other thermophysical properties of nanofluids as well. Recently, artificial intelligence has been turned into an attractive topic and the focus of many studies and investigations [23–25]. The parallel answer to a series of input data, no need for memory, no complex computational tool, and simple utilization are some of the basic artificial intelligence systems advantages. These systems mainly subdivide into some well-known categories such as the artificial neural network (ANN), fuzzy logic system, adaptive neuro-fuzzy inference system (ANFIS), support vector machine (SVM), and evolutionary algorithms. Many researchers have been used ANN to model and estimate the

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thermophysical characteristics of nanofluids [26–29]. Bhoopal et al. [30] have studied the applying of ANN in order to forecast the effective thermal conductivity. They have reached reliable and accurate prediction of their network output in comparison with actual data. In this study, a number of seven different models including adaptive neuro-fuzzy inference system (ANFIS), artificial neural network (ANN), support vector machine (SVM), least square support vector machine (LSSVM), genetic programming (GP), principal component analysis (PCA), and committee machine intelligent system (CMIS) are employed to predict the convective heat transfer coefficient of the nanofluids in circular cross-sectional channels.

Theory In the current study, we used the seven different strategies to estimate heat transfer coefficient (H) based on given Reynolds number (Re), Prandtl number (Pr), and mass fraction of nanofluid (M). These strategies are the adaptive neuro-fuzzy inference system (ANFIS), artificial neural network (ANN), support vector machine (SVM), least square support vector machine (LSSVM), genetic programming (GP), principal component analysis (PCA), and committee machine intelligent system (CMIS). Their concepts are presented in the current section.

Theory of ANFIS The fuzzy logic concept was firstly well known by Zade is an intelligent, creative, and impressive method that diminishes the complexity of the issues in order to solve with great precision. In fact, the fuzzy logic is an extension of classical fuzzy regarding conclusions. In classical logic, conclusion part is similar to the binary Boolean system with the value of one or zero, but fuzzy logic can conclude with a degree of value and introduces new concepts such as partial truth [24]. The adaptive neuro-fuzzy inference system (ANFIS) is a combination of the neural network and fuzzy logic system, and it was first introduced by Jang [31, 32]. An ANFIS approach, which is shown in Fig. 1, is usually designed from the five layers. Some nodes exist in each layer, which are specified by their node functions. There are internal connections between each of the two layers. The output of particular preceding layer is connected to the next ahead layer. In more details, consider specific inputs contain two terms with the notification x and y and also fi as an output parameter. ANFIS strategy is composed of fuzzy if–then rules based on Takagi–Sugeno–Kang (TSK) type fuzzy

Connectionist intelligent model estimates of convective heat transfer coefficient… LAYER1

INPUT

LAYER2

LAYER3

X

A1 ∏

x

ω1

N

LAYER5

LAYER4 Y

ϖ1

ϖ 1f1

A2

Σ B1

fout

ϖ 21f2 ∏

y

OUTPUT

N

ω2

ϖ2

B2

X Y

Fig. 1 Construction of typical ANFIS

system [33]. As shown in the following expressions, the definition of applied rules in the first-order fuzzy inference system is given as follows [34]: First rule

The validity of defined statements in previous sections (firing strength) is performed in the second layer. The following formulation indicates this validity: o2;i ¼ xi ¼ lAi ð xÞ  lBi ð yÞ

If x is A1 and y is B1 then z ¼ f 1ðx; yÞ •

If x is A2 and y is B2 then z ¼ f 2ðx; yÞ The inputs characterized by x and y, and A and B refer to the fuzzy sets, and fi (x, y) employs as the outputs of fuzzy inference system. According to Fig. 1, five layers in the ANFIS are defined as follows: First layer

In this layer, x and y are the input nodes, A and B refer to the linguistic parameters, and membership functions are also characterized by l(x) and l(y). O1;i  lAi ð xÞ O1;i  lAi2 ð xÞ

for i ¼ 1; 2 for i ¼ 3; 4

ð1Þ ð2Þ

Owning to the acceptable application of Gaussian membership functions in modeling and solving nonlinear relations between the effective parameters of systems, this study used this function and it is indicated as follows:   ðx  zÞ O1i ¼ exp 0:5 ð3Þ r2 Here, O refers to the output of layer, z stands for the Gaussian center, and the variance is represented by r2 . To obtain a precise and robust estimation, the Gaussian function parameters are optimized by the ANFIS. •

Second layer:

ð5Þ

Here, xi stands for the rules’ firing strength.

Second rule

•

for i ¼ 1; 2

Third layer

In the third layer, the aim is to normalize the firing strengths presented in the previous layer. The following expression shows the normalizing process as: xi xi O3;i ¼ xi ¼ P ¼ for i ¼ 1; 2 ð6Þ xi x1 þ x2

•

Fourth layer

In the fourth layer, the linguistic characterizations of output term are supplied. This layer is well known as resulting layer. Based on the following rules which are present in the total output, this task is performed. O4;i ¼ xi  fi

• •

for i ¼ 1; 2

ð7Þ

First rule: if x is A1 and y is B1 then f1 ¼ p1 x þ q1 y þr1 Second rule: if x is A2 and y is B2 then f2 ¼ p2 x þ q2 y þr2 Here, pi, qi and ri refer to the consequent terms.

•

Fifth layer

This layer uses an averaging method in order to gather all specified rules for individual output and their summation as particular geometric outputs. The above calculation is conducted as follows:

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Stop: Display result

NO

Target value YExp

Comparison:

YES

Error funtion (MSE) between neuron output and target value is much lesser than GOAL

Update weights and bias

bk (bias)

P1

Wk1

Neuron k Output

INPUTS

Activation function P2

Wk2

Pm

Wkm

ϕ

Sk = Σ(PjWkj+bk)

YANN

yk = ϕ (Sk)

Synaptic weights

Fig. 2 Configuration of multilayer perceptron artificial neural network

O5;i ¼ fout ¼

X

xi  fi

ð8Þ

i

The following expression indicates the overall output, which is the linear combination of the consequent terms [35]. x1 x1 cfout ¼ -1 f1 þ -2 f2 ¼ f1 þ x1 þ x2 x1 þ x2 ð9Þ f2 ¼ ð-1 xÞp1 þ ð-1 yÞq1 þ ð-1 Þr1

the next layer. Moreover, weight and bias terms are introduced as two critical parameters in the ANN. Generally, the linear function, sigmoid function, and hyperbolic tangent function are the commonly used transfer functions in the ANN structure, which are formulated as follows. •

Linear function: f ðxÞ ¼ x

•

Sigmoid function: f ðxÞ ¼

•

Hyperbolic tangent function: f ðxÞ ¼

þ ð-2 xÞp2 þ ð-2 yÞq2 þ ð-2 Þr2

Theory of artificial neural network (ANN) ANN is a machine learning approach which behaves like biological nervous system. The multilayer perceptron, selforganizing map, multilayer recurrent, and cellular neural networks are the widely used constructions of ANN [36, 37]. The multilayer perceptron (MLP) ANN is an amazing structure which is broadly applications in modeling. Figure 2 indicates MLP construction of ANN. This structure comprises from three layers, namely the input layer, output layer, and hidden layer. There are some nodes in each layer named neurons that are linked to the following neurons in

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ð10Þ 1 1 þ ex

ð11Þ ex  ex ex þ ex

ð12Þ

Based on above-mentioned assumptions, the output of ANN can be expressed as follows [38]:  n  X 1 Z¼ w3i ð13Þ þ b3 1 þ eðxi wi Þ i¼1 where wi refers to the weight vector values in the hidden layer neurons, n represents the number of hidden layer neurons, wi,3 stands for the weight vector of neurons in the output layer, b3 denotes the bias, and Z indicates the overall output.

Connectionist intelligent model estimates of convective heat transfer coefficient…

Moreover, the typical structure of the ANN is trained usually by the Levenberg–Marquardt algorithm to determine optimal construction. The training procedure is carried out in order to minimize the difference between the actual and estimated values with adjusting the weight and bias values [39, 40]. The following equation indicates the error function as: X XX p E¼ EðxÞ ¼ ðri  Op;l ð14Þ i Þ p

p

i

where p denotes the number of data points at training stage, Op;l i refers to the output for ith neuron of the first layer, and p ri represents the ith actual output of pth data point.

Theory of support vector machine Support vector machines have been applied to the credit rating problem for the past decade. The SVM is sometimes referred to as an optimum margin classifier [37] and widely used for data analyzing and pattern recognizing. Considering the problem of approximating the set of training vectors belonging to two separate classes, {ðx1  y1 Þ    ðxl  yl Þ} , X 9 R, where X denotes the space of the input patterns with a nonlinear function [40]: f ð x Þ ¼ ðw  ; ð x ÞÞ þ b

ð15Þ

where w and b represent the weight vector and bias terms and ;ð xÞ represents the nonlinear function that maps x into n-dimensional feature space and carries out the linear regression. In order to lower the model complication of SVM and also to improve the speed of SVM, the least squares SVM classifiers were proposed by Suykens and Vandewalle to develop a better technique [41]. The standard SVM is solved using quadratic programming methods. However, these methods are often time-consuming and are difficult to implement adaptively [42]. To make the algorithm work for nonlinearly separable data sets as well as be less sensitive to outliers, we reformulate our optimization as follows: N X 1 jjwjjT w þ C ni þ ni 2 i¼1 8 < yi  ðw; ;ðxi ÞÞ  b  e þ ni Subject to: ðw; ;ðiÞÞ þ b  yi  e þ ni : ni ; ni  0

ð16Þ

ð17Þ

where e refers to the sensitivity and ni , ni are slack variables that determine the amount of deviation of the output from the positive and negative classes. In addition, The Euclidean norm which is represented by the ||x||2 is the regularization parameter.

The Lagrangian formulation can be written as: l  l  X  X  1 ni þ ni  gi ni þ gi ni Lsvm ¼ kwk2 þC 2 i¼1 i¼1 l X ai ðe þ ni  yi þ ðw; xi Þ þ bÞ  i¼1



l X

  ai e þ ni þ yi  ðw; xi Þ  b

ð18Þ

i¼1

The Lagrangian multipliers are shown by the gi , gi , ai , characterizations. The disadvantage of SVM is its high computational for the constrained optimization programming [43]. Least square support vector machine (LSSVM) is preferred, especially for a large-scale problem which solves linear equations instead of quadratic programming problem, and resolves the aforementioned disadvantage of the SVM. Minimizing the empirical risk functional in the feature space with a squared loss leads to the following primal optimization problem. ai

N 1 1 X minjðw  eÞwbe ¼ kwk2 þ c e2 2 2 i¼1 i

Subject to

y i ¼ ð w  ;ð x i Þ Þ þ b þ e i

ð19Þ i ¼ 1. . .N

ð20Þ

The relative importance of these terms is determined by the positive real constant called c. the above formulation is related to ridge regression. This problem is easily solved by setting the partial derivatives as: ojðw  eÞwbe ¼ 0 and oðwÞ

ojðw  eÞwbe ¼0 oð e Þ

ð21Þ

To solve the optimization problem in the dual space, the Lagrangian formulation in the LSSVM structure is defined as: N N X 1 1 X Llssvm ¼ kwk2 þ c e2i  ai fðw  ;ðxi ÞÞ þ b þ ei  yi g 2 2 i¼1 i¼1

ð22Þ where ek refers to the regression error at training phases. The solution is given by saddle point of Lagrangian with Lagrange multipliers ai [ R (called support vectors): 8 N X > oLlssvm > > ¼ 0 ! w ¼ ai ;ð x i Þ > > > ow > i¼1 > > > N X > > < oLlssvm ¼ 0 ! ai ¼ 0 ob i¼1 > > oLssvm > > > ¼ 0 ! ai ¼ cei i ¼ 1. . .N > > oei > > > oL > > : ssvm ¼ 0 ! ðw  ;ðxi ÞÞ þ b þ ei  yi ¼ 0 i ¼ 1. . .N oai ð23Þ

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The following expression indicates set of linear equations: 2 32 3 2 3 w 0 I 0 0 zT 6 0 0 0 yT 76 b 7 6 0 7 6 76 7 6 7 4 0 0 cI I 54 e 5 ¼ 4 0 5 ~ a 1 z y I 0 Terms in above matrix can be defined as: h i Z ¼ ;ðx1 ÞT y1 . . .;ðxN ÞT yN Y ¼ ½y1 . . .yN  ~ 1 ¼ ½1; . . .; 1 e ¼ ½e1 . . .eN  a ¼ ½a1 . . .aN 

where X ¼ zzT The following Mercer’s condition is applied as: Xil ¼ yk yl ;ðxl Þ ¼ yk yl K ðxi  xl Þ

ð25Þ

Equations (27, 28) have been identified as the RBF and polynomial kernel functions, respectively. ! ð27Þ

ð28Þ

where r2 is the squared variance of the RBF and d is the polynomial degree. Therefore, in the LSSVM case, every data point is a support vector. This is immediately clear from the condition for optimality. ai ¼ cei ;

123

i ¼ 1. . .n

DetðVÞ ¼

P

ðaj  ai Þ

1  i\j  n

ð29Þ

ð31Þ

The aim in the PCA approach is to determine the coefficients of the polynomial in the Vandermonde matrix [44]. Pn ðxÞ ¼ a0 þ a1 x þ a2 x2 þ    þ an xn

Here, K ðxi  xl Þ is the kernel function. Many kernel functions such as the linear, polynominal, radial basis function (RBF), and sigmoid are mentioned. However, most widely used kernel functions are RBF and polynominal. 9 8 linear Xi  X l > > > > > > d = polynomial   < ðcXi Xl þ CÞ ð26Þ K xi ; xj ¼ 2 exp cjXi  Xl j RBF > > > > > > ; : tanhðcXi  Xl þ C Þ sigmoid

 d xTk xl K ðx i  x l Þ ¼ 1 þ C

The principal component analysis (PCA) is an innovative method in order to find a simple and accurate relationship between involved parameters [44]. Vandermonde matrix is identified as a matrix that uses a geometric improvement in its rows [45]: 3 2 1 x0 x20 x30 . . . xn0 6 1 x1 x2 x3 . . . xn 7 1 1 17 6 6 1 x2 x2 x3 . . . xn 7 2 2 27 6 ð30Þ 6: : : : ... : 7 7 6 5 4: : : : ... : 1 xn x2n x3n . . . xnn The following equation indicates determinant of above square matrix is as [46]:

One solution is yield after elimination of w  e given as follows.      b 0 0 yT ¼ ~ ð24Þ 1 y X þ c1 I a

kx i  x l k2 K ðxi  xl Þ ¼ exp  r2

Principal component analysis (PCA)

ð32Þ

Interpolation of n ? 1 points by this matrix ((x0, y0), (x1, y1), …, (xn, yn)) and a collection of linear equations are represented as follows. Pn ðx0 Þ ¼ y0 ) a0 þ a1 x0 þ a2 x20 þ    þ an xn0 ¼ y0

ð33Þ

Pn ðx1 Þ ¼ y1 ) a0 þ a1 x1 þ a2 x21 þ    þ an xn1 ¼ y1

ð34Þ

: :

)::

Pn ðxn Þ ¼ yn ) a0 þ a1 xn þ a2 x2n þ    þ an xnn ¼ yn

ð35Þ

Above expressions can be written in matrix construction as [45]: 32 3 2 3 2 a0 y0 1 x0 x20 x30 . . . xn0 6 1 x1 x2 x3 . . . xn 76 a1 7 6 y1 7 1 1 1 76 7 7 6 6 6 1 x2 x2 x3 . . . xn 76 a2 7 6 y2 7 2 2 2 76 7¼6 7 6 ð36Þ 6: : 6 7 6 7 : : ... : 7 76 : 7 6 : 7 6 4: : : : . . . : 54 : 5 4 : 5 2 3 an yn 1 xn xn xn . . . xnn The coefficients of Vandermonde matrix which is indicated by a = (a0, a1, a2, …) can be obtained by solving the system as: V a¼y

ð37Þ

Connectionist intelligent model estimates of convective heat transfer coefficient…

Genetic programming (GP) GP is introduced as a symbolic regression which relates the inputs to output through mathematical expressions [47]. In the symbolic regression, the coefficients and functions are computed analytically and it is a distinction of this technique against classical regression approaches such as the ANN, fuzzy, and SVM. These mathematical expressions are selected according to widespread nonstop educating guided search in growing search space. GP is well known as an evolutionary mathematical approach that is effectively employed in numerous fields of engineering applications. In the GP, the probable solutions are symbolizes based on tree illustration, while the common optimization techniques such as the genetic algorithm (GA) give the solutions as numbers [48]. GP has some genes in its structure which contain specific functions and terminals. These functions contain regular mathematics operations such as {?, -, *, /, Sqrt, Exp, Log, Ln, Sin, Cos, …}, logical operations such as {and, or, not, nor, …}, and some arbitrary functions such as {PLn, PSqrt, Pdivide, …}. These arbitrary functions are expressed as follows: PLnðxÞ ¼ Lnðj xjÞ; PLnð0Þ ¼ 0

ð38Þ

PdivideðxÞ ¼ divideðxÞ; Pdivideð1Þ ¼ 0

ð39Þ

PSqrtðxÞ ¼ Sqrtðj xjÞ

ð40Þ

In addition, terminals consider as the inputs of the model that can be indicated by inclusive terms and numerical coefficients. The schematic flow diagram of typical GP algorithm is demonstrated in Fig. 3. In this figure, the procedure of GP begins with a procreation of the first population randomly. To that end, grow, half and full initialization ways are employed. Afterward, based on the proposed fitness functions present in the literature, fitness of each individual is tested and this procedure continues till stopping criteria are met. This criterion finds the best solution along the progress. GP basically works by three vital operators called reproduction, crossover, and mutation in its configuration that are employed on the tree arrangements of the individuals.

Start

Generate initial population

Evalute fitness of each

individual

G=G+1

Terminated criteria satis fied?

End

Apply genetic operators

Reproduction, crossover, motation

New population

Fig. 3 Flow diagram of GP model

dynamic. We use the static construction in the present work, namely the ensemble averaging. Outputs of other predictive approaches are utilized in order to obtain the best solution. Selecting an appropriate type of averaging methods is vitally important in order to combine the outputs of other predictive approaches [51, 52]. A simple averaging is the linear combination outputs obtained from the other techniques [53]. It is worth mentioning that the best predictive method has the most contribution and higher weight. Coefficients of linear equations should be determined by using optimization techniques such as the least square optimizer and genetic algorithm. In this study, we used the least square methods to determine these coefficients. These coefficients are determined through minimization of the MSE.

Committee machine intelligent system (CMIS) Committee machine (CM) was firstly presented by Nilsson [49]. CM is well known as a supervised approach that simplifies tasks in order to solve them rapidly. These tasks are departed into some subtasks, and then, through solving these subtasks and combining of their solutions the final solution is obtained. This technique uses a number of predictive tools in order to solve the tasks. Truly, the CM is formed from the hybrid of intelligent schemes [50]. CM has mainly two constructions such as the static and

Experimental methodology and data preparation Nanofluids with different volume concentrations of SiO2 nanoparticles were prepared by adding specific amounts of the nanoparticles to the distilled water. The silica nanoparticles were provided by Fadak Group (Iran). Figure 4 shows the schematic of the experimental setup used to measure the convective heat transfer coefficients. The setup is consist of circulating pumps (9 2), reservoirs

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Data acquisition

Reservoir

Manometer

Manometer

Thermocouples

T in

Insulating

T1 T2 T3 T4 T5 T6 T7 T8 T9

AC power

T out

Flow meter

Coppertube Heater

Cooling unit Reservoir Pump

Fig. 4 Schematic of the experimental setup used to measure the convective heat transfer coefficients

(9 3), Reynolds valve, heat transfer channel (1 m), fluid flowmeter, heat exchanger, dimer (In order to achieve constant heat flux), sensors (9 11), data logger and pipes at the beginning and end of the channel (for pressure drop measurement). In the first step, the reservoir number one is being filled by the fluid. In the next step, the fluid is sent to the second reservoir by the circulating pump. After filling the second reservoir, the Reynolds valve directs the fluid flows to the horizontal channel. Subsequently, the fluid passes through the circular cross-sectional heat transfer channel with a constant heat flux. The channel wall temperature is measured by nine sensors, while the inlet and outlet temperatures are measured by two other sensors. Finally, in order to keep the inlet temperature constant, the fluid enters the cooling heat exchanger. The flow loop consists of a test section, a fluid reservoir, pump, flow measuring apparatus, calming section, cooling unit, riser section, and thermocouples. The test section consists of a copper tube with inner diameter of 8 mm and thickness of 1 mm and length of 1000 mm. The working fluid is circulated with a flow rate of 5800 L h-1 by a MULTI 5800 SICCE pump. The calming section is employed to exclude the entrance effect and provide fully developed laminar flow. The test section is insulated by glass wool to reduce the heat loss. Two thermocouples are installed for inlet and outlet flow temperatures measurement. Furthermore, nine SMT-160 thermocouples are installed along the test section to measure wall temperatures. The time that the graduated cylinder is filled by the fluid is measured, in order to calculate the flow rate. To measure the pressure drop through the test section, two manometers were utilized. A heat exchanger is placed

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to reduce the fluid temperature and fix the initial temperature [54]. The ranges of Reynolds number, Prandtl number, and mass fraction of nanofluid used in this study for development of our suggested models are summarized in Table 1.

Implementation of models Preprocessing procedure In this study, seven models with different concepts including the artificial neural network, adaptive neurofuzzy inference system, support vector machine, least square support vector machine, genetic programming, principal component analysis, and committee machine intelligent approaches have been implemented in MATLAB 2014 in order to determine heat transfer coefficient as a function of Prandtl number, Reynolds number, and mass fraction of nanofluid. Introducing the output and input variables of models is the next step after data preparation. Heat transfer coefficient (H) is the output variable, and Prandtl number, Reynolds number, and mass fraction of nanofluid are the three input parameters. In addition, overall data points must be basically divided into two categories. The first class, which builds the training set, contains 177 data points (e.g., 75% of the whole data), and the remaining 59 data points build the testing set which is employed to examine the efficiency of the suggested models. In order to homogenize variables, data points are normalized within the ranges of - 1 and ? 1 as expressed in below equation.

Connectionist intelligent model estimates of convective heat transfer coefficient…

Range of Re

Range of H

No. of samples

700–1013

4.5289–6.1528

93.8975–952.6543

54

0.0005

617–990

4.2201–5.4195

79.3459–979.3821

72

0.0007

632–961

4.1937–5.6726

99.6735–1074.8

63

0.002

694–870

3.614–4.8865

137.347–1168.3

45

D  Dmin 1 Dmax  Dmin

ð41Þ

Here DN and D represent the normalized and actual data points, respectively. Dmin and Dmax refer to the minimum and maximum values of data points.

Model developing Model based on the ANN algorithm uses the multilayer perceptron (MLP) in its structure [55, 56]. Moreover, training process was carried out by the Levenberg–Marquardt algorithm. In the present ANN model, log-sigmoid functions have been applied in the hidden layer and linear function was employed in the output layer. In addition, hidden layer contains eight neurons. So, the number of adjusted parameters in its structure is 41. The obtained values of weight and bias are summarized in Table 2. The optimizing procedure is continued till the mean square error is minimized. Figure 5 indicates the performance of the Levenberg–Marquardt algorithm in order to obtain optimal ANN structure. Moreover, dependent membership function parameters in the adaptive neuro-fuzzy inference system have been optimally determined by the PSO algorithm. The Gaussian function has been utilized as the membership function. This function composed of two parameters. A total number of ANFIS parameters are determined based on the number of clusters and variables of model accompanied with the type of membership functions. As formulated in below equation, the total number of parameters is 80. The number of Table 2 Bias and weight values for the MLP-ANN model

Range of Pr

0

Neuron

Mean square error (MSE)

DN ¼ 2

Mass fraction of nanofluid

10 0

10 –2

10 –4 0

400

200

600

0.25

0.2

0.15

0.1

0.05

0 0

200

400

600

800

1000

Fig. 6 Training ANFIS model using PSO at different iterations

Output layer

Pr

M

Bias

Weight

Bias

B1

H

B2

1

- 1.91327

- 1.16286

- 1.16048

0.848921

0.59668

2 3

- 0.30709 - 0.30368

- 1.31005 - 22.569

- 2.29775 44.63736

1.895293 33.50896

- 5.60177 - 2.04258

4

0.210419 - 0.04579

6

- 0.07353

7

0.230242

8

0.029183

1200

Iteration

Weight

5

1000

Fig. 5 ANN performance during different iterations

Hidden layer

Re

800

Epochs

RMSE

Table 1 Ranges of experimental data used in this study

6.232576 - 0.75548 2.051787 16.2814 0.908214

15.86349 1.621466 - 3.625 8.470157 - 4.99851

- 10.7979 3.427626 - 1.22499 - 9.63925 - 6.61316

67.60499

- 0.87737 - 62.7858 - 3.32378 2.55509 - 38.9664

123

A. Baghban et al. Fig. 7 Trained parameters of fuzzy inference system for input variables

1

Degree of membership

Cluster1 0.8

Cluster2 Cluster3 Cluster4

0.6

Cluster5 Cluster6 0.4

Cluster7 Cluster8 Cluster9

0.2

Cluster10 0 –1

– 0.5

0

0.5

1

Normalized Re

Degree of membership

1

Cluster1

0.8

Cluster2 Cluster3 0.6

Cluster4 Cluster5 Cluster6

0.4

Cluster7 Cluster8 0.2

Cluster9 Cluster10

0 –1

– 0.5

0

0.5

1

Normalized Re

Degree of membership

1

Cluster1

0.8

Cluster2 Cluster3

0.6

Cluster4 Cluster5 Cluster6

0.4

Cluster7 Cluster8 Cluster9

0.2

Cluster10 0 –1

– 0.5

0

Normalized Re

123

0.5

1

Connectionist intelligent model estimates of convective heat transfer coefficient…

Data Preparation (model inputs and output; Dividing into training and testing sets)

Input Introduce Training dataset

Introduce Training dataset

Random division of data into training and testing

Selecting kernel function

Training Data

Selecting user defined variables (C, ε and kernel funtion’s parameter) Implement GA and select σ 2,γ

Undergoing the SVM algorithm (model training)

Training Data

Employ feature subset (σ 2,γ )

Model performance assessment using statistical criteria

Construct LSSVM model No

No

Criteria? Evaluate the model by training and testing data

Meet stopping criterion?

Yes Obtain The Optimized Model

Yes Determine Optimum σ 2,γ

Fig. 8 Schematic illustration of proposed SVM model

Return the LSSVM using optimum feature

variables is 4 (i.e., Pr, Re, M, H), and the initial number of clusters was adjusted to 10. NT ¼ Nc NV Nmf

GA-LSSVM

ð42Þ

In above equation, Nc , NV , and Nmf stand for the number of clusters, a number of variables, and a number of membership function parameters, respectively. Based on the PSO algorithm for obtaining ANFIS parameters, the RMSE between the experimental and estimated values was considered as performance criteria. The values of RMSE are shown in Fig. 6 at different iterations. Trained parameters of fuzzy inference system are demonstrated for the input variables in Fig. 7. In this figure, the input values are normalized within the range of - 1 and ? 1. SVM comprises of three parameters such as the penalty factor (C), e value in the e-insensitive loss function, and kernel parameter (K). Specifying these parameters is essential before starting the training process. Kernel function used in the SVM model is based on Gaussian function due to its more applicability. Schematic diagram of proposed SVM model is shown in Fig. 8. These parameters are manually optimized by trial and error procedures. The applied cost function in this strategy is the mean square error (MSE) between the predicted and real data points. Moreover, the LSSVM approach uses two parameters in its structure including the regularization parameters (c) and the kernel parameter (r2). In addition, the radial basis function (RBF) is employed as the kernel function of the LSSVM due to its

model

Fig. 9 Schematic diagram of combination of the LSSVM with GA

wonderful capability. As the number of support vectors increase, the elapsed time for training process increases. To determine LSSVM parameters, this topology uses the genetic algorithm (GA) as a remarkable evolutionary algorithm. Figure 9 illustrates a diagram of proposed LSSVM model. Moreover, details of proposed ANN, ANFIS, SVM, and LSSVM models are presented in Table 3. The model based on principal component analysis (PCA) concepts explains as follows. Firstly, heat transfer coefficient (H) is correlated as a function of the Reynolds number (Re) and Prandtl number (Pr) based on Eq. (43). 2

H ¼ a þ bRe þ c Pr þdðRe  PrÞ þ e Pr

ð43Þ

Then, five coefficients of above equation are correlated as a function of the mass fraction of nanofluid (M) based on the cubic polynomial equations present as follows. a ¼ A1 þ B1 M þ C1 M 2 þ D1 M 3 2

b ¼ A2 þ B2 M þ C2 M þ D2 M

3

ð44Þ ð45Þ

123

A. Baghban et al. Table 3 Details of the trained proposed models for the prediction of convective heat transfer coefficient ANN

ANFIS

No. input neuron layer

3

Type

Value/comment

No. hidden neuron layer

7

Membership function

Gaussian

No. output neuron layer

1

No. of MF parameters

80

Hidden layer activation function

Logsig

No. of clusters

10

Output layer activation function

Purelin

Number of data used for training

176

Optimization method

Levenberg–Marquardt

Number of data used for testing

58

Number of data used for training

176

Population size

50

Number of data used for testing

58

Iteration

1000

Number of max iterations

1000

C1

1

C2

2

SVM

LSSVM

No. of training data

176

Kernel function

RBF

No. of testing data

58

c

32,513.545

Kernel function

Gaussian

r2

0.8102

C

10,000

Number of data used for training

176

e

0.01

Number of data used for testing

58

c

0.6

Population size

70

Table 4 Tuning parameters of PCA model

123

Parameter

Value

A1

8042.938422

Iteration

1500

C1

1

C2

2

c ¼ A3 þ B3 M þ C3 M 2 þ D3 M 3

ð46Þ

d ¼ A4 þ B 4 M þ C 4 M 2 þ D 4 M 3

ð47Þ

2

e ¼ A5 þ B5 M þ C5 M þ D5 M

3

ð48Þ

B1

13,663.4847

C1

9834.401177

D1

12,199.66555

A2

- 0.939976607

B2

- 0.972194662

C2

- 1.112971084

D2

- 1.111274031

A3

- 3203.444703

B3

- 6002.487329

order to determine new coefficients for Eq. (39) to predict heat transfer coefficient. These coefficients are optimally determined by the least square approach. Table 4 summarizes tuning coefficients present in Eqs. (44–48). Briefly, following steps should be repeated in order to obtain the correlation’s coefficients as:

C3

- 4224.521941

•

D3

- 6110.585298

A4

0.236189843

B4

0.278356787

•

C4 D4

0.306292651 0.374513539

•

A5

317.3198098

B5

654.9301928

C5

450.5185234

D5

757.8207723

The above-mentioned cubic polynomials are used in

Correlate the heat transfer coefficient (H) as a function of Reynolds number (Re) and Prandtl number for a given mass fraction of nanofluid (M). Repeat above the stage for other mass fraction of nanofluid (M). Correlate corresponding polynomial coefficients, which are written in the past steps against mass fraction of nanofluid (M), a = f(M), b = f(M), c = f(M), d = f(M) [see Eqs. (44–48)].

Accordingly, Eq. (41) refers to the principal equation in which five coefficients (a, b, c, d, e) are utilized in order to calculate heat transfer coefficient (H) as a function of

Connectionist intelligent model estimates of convective heat transfer coefficient… Fig. 10 Obtained best RMSE by GP algorithm

120

Best root mean squared error

100

80

60

40

20

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 211 221 231 241 251 261 271 281 291

0

Iteration

Fig. 11 Average RMSE obtained by GP algorithm against different generation

300

RMSE RMSE+STD

Root mean squared error

250

RMSE-STD

200

150

100

50

1 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191 201 211 221 231 241 251 261 271 281 291

0

Generation

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gen 3 :  1036 M  Pr ðPr þ 2:573Þ

Reynolds number (Re) and Prandtl number for a given mass fraction of nanofluid (M), where the corresponding coefficients are summarized in Table 4. Another model which is based on genetic programming (GP) uses seven Gens with some function nodes, which are represented by the following equations. We used the functions and mathematics operations such as {?, -, *, /, Sqrt, Exp, Ln, PLog, PSqrt} in these nodes.

ð51Þ pffiffiffiffiffi Gen 4 : 22:55PSqrtðRe  6:908Þ þ 22:55 Pr ðPr þ MÞ

Gen 1 : 4:849e5  31499 expðexpðexpðMÞÞÞ ð49Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gen 2 : 561:4Pr  280:7 PLnðMÞ þ 280:7PLnðPrÞ

ð54Þ

ð52Þ rffiffiffiffiffiffiffiffiffiffi! 1181 Gen 5 :  2178 Pr þ PLnðMÞ þ 477 qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi Gen 6 : 2:008e6 M Pr pffiffiffiffiffi Gen 7 : 16:01PLnðPLnðMÞÞ Pr ðPr þ MÞ

ð53Þ

ð55Þ

ð50Þ

123

A. Baghban et al. Fig. 12 Experimental versus predicted conductive heat transfer coefficient by the ANFIS and ANN models

Train Exp.

Train ANFIS

Test Exp.

Test ANFIS

1400 1200

H value

1000 800 600 400 200 0 0

50

100

150

200

250

Data index

Train Exp.

Train ANN

Test Exp.

Test ANN

1400 1200

H value

1000 800 600 400 200 0 0

50

100

150

200

250

Data index

Simplified overall GP expressions can be written as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi H ¼ 2:008e6 M Pr  1617Pr  280:7 PLnðMÞ  31500ExpðExpðExpðMÞÞÞ þ 22:55PSqrtðRe  6:908Þ pffiffiffiffiffi þ 280:7LnðPrÞ  2178PLnðMÞ þ 22:55 Pr ðPr þ MÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1036 M  Pr ðPr þ 2:573Þ pffiffiffiffiffi þ 16:01PLnðPLnðMÞÞ Pr ðPr þ MÞ þ 4:8833e5

ð56Þ Figure 10 illustrates the best values of mean square error during the period of running the suggested GP model. The best fitness is the fitness of the best individual in a specified

123

population that is found over the course of the run. In addition, the mean fitness that refers to the mean of the fitness values of the whole population is indicated in Fig. 11. The comprehensive model which considers all above models in its structure has been formulated in Eq. (57). As mentioned, the coefficients of CMIS technique were obtained by the least square optimization algorithm. HCMIS ¼ 0:2403  0:0891HANFIS  0:4151HANN þ 0:6142HLSSVM þ 0:1289HSVM

ð57Þ

 0:0595HGP  0:0116HPCA In above equation, HCMIS is the obtained heat transfer coefficient by the CMIS model and HANFIS, HANN,

Connectionist intelligent model estimates of convective heat transfer coefficient… Fig. 13 Experimental versus predicted conductive heat transfer coefficient by the SVM and LSSVM models

Train Exp.

Train LSSVM

Test Exp.

Test LSSVM

1400 1200

H value

1000 800 600 400 200 0 0

50

100

150

200

250

Data index

Train Exp.

Train SVM

Test Exp.

Test SVM

1400 1200

H value

1000 800 600 400 200 0 0

50

100

150

200

250

Data index

HLSSVM, HSVM, HGP, and HPCA refer to obtained heat transfer coefficient by the ANFIS, ANN, LSSVM, SVM, GP, and PCA models, respectively.

The following formulations show the mathematical representations of aforementioned statistical analyses. N 1X ðHexp  Hcal Þ2 N i¼1   N  Hexp  Hcal  100 X ARDð%Þ ¼ N i¼1 Hexp

MSE ¼

Performance evaluation Performance and capability of suggested models can be determined by some well-known statistical approaches including the mean squared errors (MSEs), percentage of average relative deviation (ARD %), standard deviations (STD), root mean square error (RMSE), and the square of correlation coefficient (r2) between the experimental and estimated values of heat transfer coefficient by the models.

STDerror ¼

N 1 X ðerror  errorÞ2 N  1 i¼1

ð58Þ

ð59Þ !0:5 ð60Þ

123

A. Baghban et al. Fig. 14 Experimental versus predicted conductive heat transfer coefficient by the PCA and GP models

Train Exp.

Train GP

Test Exp.

Test GP

1400 1200

H value

1000 800 600 400 200 0 0

50

100

150

200

250

Data index

Train Exp.

Train PCA

Test Exp.

Test PCA

1400 1200

H value

1000 800 600 400 200 0 0

50

100

150

200

250

Data index

N 1X ðHexp  Hcal Þ2 N i¼1

RMSE ¼

N P

r2 ¼ 1 

Results and discussion ð61Þ

ðHexp  Hcal Þ2

i¼1 N P

!0:5

ð62Þ ðHexp  H exp Þ2

i¼1

where N is the total number of data points. In addition, Hexp and Hcal stand for the experimental and estimated values of heat transfer coefficient correspondingly and an average of experimental values of heat transfer coefficient was indicated by H exp .

123

The present study was carried out to determine heat transfer coefficient as a function of Prandtl and Reynolds numbers using different kinds of techniques such as the artificial neural network (ANN), adaptive neuro-fuzzy inference system (ANFIS), support vector machine (SVM), least square support vector machine (LSSVM), principal component analysis (PCA), genetic programming (GP), and committee machine intelligent system (CMIS). Various graphical approaches have been employed in order to evaluate the reliability and success of the suggested models. In order to achieve a superior graphical background about the accuracy of the suggested models, the experimental and estimated heat transfer coefficients

Connectionist intelligent model estimates of convective heat transfer coefficient… Fig. 15 Experimental versus predicted conductive heat transfer coefficient by the CMIS model

Train Exp.

Train CMIS

Test Exp.

Test CMIS

1400 1200

H value

1000 800 600 400 200 0 0

50

100

150

200

250

Data index Train Data

Fig. 16 Regression plot between experimental versus predicted conductive heat transfer coefficients by the ANN and ANFIS models

Test Data

Linear (Train Data)

Linear (Test Data)

1400

Test: y = 0.9777x + 1.526, R² = 0.9914

Experimental H value

1200

Train: y = 1.0005x - 0.2402, R² = 0.9957

1000 800 600 400 200 0 0

200

400

600

800

1000

1200

1400

Predicted H value by the ANFIS

Train Data

Test Data

Linear (Train Data)

Linear (Test Data)

1400

Train: y = 1x - 0.0047, R² = 0.9973

Experimental H value

1200

Test: y = 0.9827x + 4.5754, R² = 0.9959 1000 800 600 400 200 0 0

200

400

600

800

1000

1200

1400

Predicted H value by the ANN

123

A. Baghban et al. Fig. 17 Regression plot between experimental versus predicted conductive heat transfer coefficients by the SVM and LSSVM models

Train Data

Test Data

Linear (Train Data)

Linear (Test Data)

1400

Train: y = 1.0012x - 0.5151, R² = 0.998

1200

Experimental H value

Test: y = 0.9885x + 1.8425, R² = 0.9846 1000 800 600 400 200 0 0

200

400

600

800

1000

1200

1400

Predicted H value by the LSSVM

Train Data

Test Data

Linear (Train Data)

Linear (Test Data)

1400

Train: y = 0.9909x + 1.4478, R² = 0.9977

Experimental H value

1200

Test: y = 0.9715x + 8.2052, R² = 0.9897 1000 800 600 400 200 0 0

200

400

600

800

1000

1200

1400

Predicted H value by the SVM

are demonstrated in Figs. 12–15 simultaneously versus the arrangement of data points for both training and testing phases. It is obvious from above figures that the values of heat transfer coefficients obtained by the aforementioned models properly follow the trend of their corresponding experimental values. However, this accommodation is poor for PCA and GP models as is clear from deviations in the trend of estimated values from their outcomes. Figure 16 indicates the predicted heat transfer coefficients by the ANN and ANFIS models versus corresponding experimental values. This figure shows that most of the data points perch nearby the Y = X line (45 line) as can be obvious from high strangulation of data points around the unit slope line, which proves that the experimental and estimated heat transfer coefficients by these two

123

models are in a respectable agreement. In addition, Fig. 17 shows similar observation for the LSSVM and SVM models to estimate heat transfer coefficients. The SVM and LSSVM models have also acceptable results, and most of their data points perch nearby the Y = X line. As shown in Fig. 18, the regression plots for the PCA and GP models indicate their poorer performance than other models. The regression plot for the CMIS model which was a linear combination of above models is shown in Fig. 19. As expected, the output of CMIS model is in a satisfactory agreement with experimental heat transfer coefficients. The obtained equation of linear regressions based on the mathematical outlook can be written for the testing phase of ANFIS, ANN, SVM, LSSVM, PCA, GP, and CMIS models, respectively, as:

Connectionist intelligent model estimates of convective heat transfer coefficient… Fig. 18 Regression plot between experimental versus predicted conductive heat transfer coefficients by the PCA and GP models

Train Data

Test Data

Linear (Train Data)

Linear (Test Data)

1400

Train: y = 1x + 2E-05, R ² = 0.9791

1200

Experimental H value

Test: y = 0.9503x + 14.984, R ² = 0.9714 1000 800 600 400 200 0 0

200

400

600

800

1000

1200

Predicted H value by the GP

Train Data

Test Data

Linear (Train Data)

Linear (Test Data)

1400

Train: y = 1.0067x - 0.6139, R ² = 0.9728

Experimental H value

1200

Test: y = 0.9757x + 3.2629, R ² = 0.9737

1000 800 600 400 200 0 0

200

400

600

800

1000

1200

Predicted H value by the PCA

y ¼ 0:9777x þ 1:526;

R2 ¼ 0:9914

ð63Þ

y ¼ 0:9827x þ 4:5754;

R2 ¼ 0:9959

ð64Þ

y ¼ 0:9715x þ 8:2052;

R2 ¼ 0:9897

ð65Þ

y ¼ 0:9885x þ 1:8425;

R2 ¼ 0:9846

ð66Þ

y ¼ 0:9757x þ 3:2629;

2

ð67Þ

2

R ¼ 0:9737

y ¼ 0:9503x þ 14:984;

R ¼ 0:971

ð68Þ

y ¼ 0:9901x þ 2:2107;

R2 ¼ 0:9918

ð69Þ

In order to specify the accuracy of the suggested models, Figs. 20–23 are prepared to plot the relative deviations between outputs of models and experimental heat transfer

coefficients. In these figures, relative deviations of the ANN, ANFIS, LSSVM, SVM, and CMIS models are higher concentrated nearby the zero deviation line than GP and PCA models, which represents that those models are in great reliability with corresponding experimental data. The percentage of average relative deviations of ANFIS, ANN, LSSVM, SVM, PCA, GP, and CMIS models are 5.371382, 4.618381, 3.383514, 3.177146, 4.462264, 4.645614, 2.502826, respectively. A huge number of data points can be seen in the zone constrained by relative deviations of ± 10%. Moreover, statistical analyses confirmed this fact that the proposed models have the pleasing capability in order to predict heat transfer coefficients. As previously

123

A. Baghban et al. Fig. 19 Regression plot between experimental versus predicted conductive heat transfer coefficients by the CMIS model

Train Data

Test Data

Linear (Train Data)

Linear (Test Data)

1400

Train: y = x, R ² = 0.9985

Experimental H value

1200

Test: y = 0.9901x + 2.2107, R ² = 0.9918

1000 800 600 400 200 0 0

200

400

600

800

1000

1200

1400

Predicted H value by the CMIS

Fig. 20 Relative deviations of experimental and predicted conductive heat transfer coefficient for: a ANN and b ANFIS

(a)

30

Train Data 20

Test Data

Relative deviation%

10

0 0

200

400

600

800

1000

1200

1400

–10

–20

–30

–40

(b)

H value

25 20

Train Data

15

Test Data

Relative deviation/%

10 5 0

0

200

400

600

–5 –10 –15 –20 –25 –30

123

H value

800

1000

1200

1400

Connectionist intelligent model estimates of convective heat transfer coefficient…

(a)

15

Train Data

Relative deviation/%

10

Test Data

5

0 0

200

400

600

800

1000

1200

1400

–5

–10

–15

–20

H value

(b)

30 25

Train Data

20

Test Data

Relative deviation/%

15 10 5 0 0

200

400

600

800

1000

1200

1400

–5 –10 –15 –20 –25

H value

Fig. 21 Relative deviations of experimental and predicted conductive heat transfer coefficient for: a LSSVM and b SVM

formulated, the employed statistical approaches include the mean square errors (MSEs), mean relative errors (MREs), standard deviations (STD), root mean square errors (RMSEs), and R-squared (R2) that are obtained for all models and presented in Table 5 for testing, training, and total data points. Obtained statistical results are also showed the satisfactory accuracy of suggested models. Another graphical analysis was conducted to discover about behaving of suggested models, and we can decide wisely regarding reasonability and applicability of our models by the histogram of residuals which are the differences between the experimental and estimated outcomes. The histograms of proposed models for testing

stages are illustrated in Figs. 24–26. The shape of histograms is similar to bell shape, and this fact is due to the normal behavior of suggested models.

Outlier detection The experimental measurements, used in order to implement the model, significantly affect its credibility [57]. In this study, we measured heat transfer coefficients of nanofluid at different Reynolds and Prandtl numbers, and then, these measurements have been employed for developing suggested models. The data points which are far away from the entire trend of almost data points are

123

A. Baghban et al. Fig. 22 Relative deviations of experimental and predicted conductive heat transfer coefficient for: a GP and b PCA

(a)

100

Train Data

80

Test Data

Relative deviation/%

60

40

20

0 0

200

400

600

800

1000

1200

1400

–20

–40

–60

H value

(b)

40

Train Data

30

Test Data

20

Relative deviation/%

10 0 0

200

400

600

800

1000

1200

1400

–10 –20 –30 –40 –50 –60

H value

introduced as the outliers. Hence, applying systematic approaches for detecting these outliers within the data points has really high importance to improve the satisfactory design of suggested model [58]. In the present analysis, two approaches have been employed such as the Leverage mathematical technique and standardized residuals to determine outliers. The Leverage technique performs this task by determining a Hat matrix of input data points which is expressed as follows [59].

123

H ¼ XðX T XÞ1 X T

ð70Þ

Here X denotes the matrix with a size m  n in which m and n refer to the total number of measured heat transfer coefficients and number of variables of model, respectively. The main diagonal of Hat matrix gives us the hat values. Hence, based on William’s plot, we can fully determine the outlier data points. This figure shows the variation in standardized residuals versus corresponding hat values. Since the CMIS is a comprehensive model and considers all outcomes of suggested models in its structure by linear combination of them, we illustrated William’s

Connectionist intelligent model estimates of convective heat transfer coefficient… Fig. 23 Relative deviations of experimental and predicted conductive heat transfer coefficient for CMIS model

20

Train Data

15

Test Data

Relative deviation/%

10

5

0 0

200

400

600

800

1000

1200

1400

–5

–10

–15

–20

H value

Table 5 Performance evaluations of proposed models by the statistical analyses Model

R2

MSE

MRE/%

RMSE

STD

ANFIS Train

0.991442

593.2841

5.901591

23.1954

24.35742

Test

0.995701

310.6002

5.196654

17.6238

17.62385

Total

0.99456

380.6671

5.371382

19.42032

19.51069

Train

0.995931

264.9499

3.718892

16.11402

16.27728

Test

0.99725

198.6222

4.468111

14.09334

14.09334

Total

0.996906

215.0624

4.618381

14.65403

14.66501

Train

0.984643

925.9274

5.404771

30.30769

30.42905

Test

0.998046

141.2574

3.000867

11.88517

11.88517

Total

0.995149

335.7482

3.383514

18.31107

18.32343

SVM Train

0.989692

671.6658

5.127711

25.72605

25.91652

Test

0.997723

176.0257

2.720269

13.06028

13.26747

Total

0.995948

298.8767

3.177146

17.10133

17.28805

ANN

LSSVM

PCA Train

0.971423

1875.858

43.03701

43.31118

Test

0.979085

1510.868

11.44128 4.712989

38.86988

38.86988

Total

0.976972

1601.336

4.462264

39.99851

40.01669

Train

0.973672

1639.356

39.96465

40.48896

Test

0.972766

1975.073

4.525457

44.39022

44.44179

Total

0.972621

1891.861

4.645614

43.49553

43.49553

GP 10.82088

CMIS Train

0.991803

494.8805

4.289618

22.1804

22.24591

Test

0.998478

109.9581

1.63785

10.48609

10.48609

Total

0.997035

205.3662

2.502826

14.32436

14.3306

123

A. Baghban et al.

(a) 8

8

7

7

6

6

Frequency

Frequency

(a) 9

5 4 3

5 4 3 2

2

1

0

0 –110 –100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90 100 110

–110 –100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90 100 110

1

Residual

Residual

6

12

5

Frequency

(b) 7

14

Frequency

(b)16

10 8 6 4

4 3 2 1

0

0 –110 –100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90 100 110

–110 –100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90 100 110

2

Residual

Residual

(c) 9

(c) 7

8

6 5

6

Frequency

Frequency

7 5 4 3

4 3 2

2 1

1 –110 –100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90 100 110

–110 –100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90 100 110

0

0

Residual

Residual

Fig. 24 Histogram of residual data points for the a ANFIS, b ANN, and c LSSVM models

plot for this model in Fig. 27. The following formulation is applied to specify the value of critical leverage value (H  ) as: H  ¼ 3ðn þ 1Þ=m

ð71Þ

In William’s plot, the green line points to the leverage limit, and accordingly, those data points with higher hat values than critical hat value are detected as outliers. Furthermore, red lines indicate satisfactory boundary for the standard deviations.

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Fig. 25 Histogram of residual data points for the a SVM, b GP, and c PCA models

Sensitivity analysis Sensitivity analysis is a great approach to determine effective input variables over our target variable. Relevancy factor (r) is the term which can perform this task. Its value ranges between - 1 and ? 1, and higher absolute value of r for each input emphasizes on its higher effect on the outcome of the model. In addition, positive and negative r values indicate that as the input increases, the outcome value increases and decreases correspondingly. The relevancy factor can be expressed as follows [58]:

Connectionist intelligent model estimates of convective heat transfer coefficient… 12 0.7

0.638955008

10 0.6

Relevency factor

Frequency

8 6 4

0.5 0.4 0.3 0.2

0.136731865

0.124156034

0.1

2

0 –110 –100 –90 –80 –70 –60 –50 –40 –30 –20 –10 0 10 20 30 40 50 60 70 80 90 100 110

0 Re

Residual

Mass

Fig. 28 Sensitivity analysis of CMIS model

Fig. 26 Histogram of residual data points for the CMIS model

Fig. 27 William’s plot for CMIS model

Pr

Valid Data Leverage limit Suspected limit Suspected limit Suspected Data

5 4

Standardized residual

3 2 1 0 –1 –2 –3 –4 –5 0

0.01

0.02

0.03

0.04

0.05

0.06

Hat value

Pn i¼1 ðXk;i  Xk ÞðYi  YÞ ffi r ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 Pn 2 ðX  X Þ ðY  YÞ k;i k i¼1 i¼1 i

ð72Þ

In above equation, Xk , Yi and Y refer to the ith input value, the average value of the kth input, the ith output value, and the average value of output correspondingly and n is the total number of data points. As shown in Fig. 28, there is a proportional relationship between the heat transfer coefficient and other input variables such as the Reynolds and Prandtl numbers and mass fraction of nanofluid. Prandtl number has the main effect on the heat transfer coefficient by 0.64 relevancy factor, and the mass fraction of nanofluid has minimum effect on the heat transfer coefficient with 0.12 relevancy factor.

Conclusions The present research was aimed to establish a relationship between heat transfer coefficient and Reynolds number, Prandtl number, and mass fraction of nanofluid. To that end, a number of seven different models including adaptive neuro-fuzzy inference system (ANFIS), artificial neural network (ANN), support vector machine (SVM), least square support vector machine (LSSVM), genetic programming (GP), principal component analysis (PCA), and committee machine intelligent system (CMIS) have been implemented according to experimental databases designed for measuring the convective heat transfer coefficient of nanofluid. Moreover, some popular statistical analyses including the mean square errors (MSEs), mean relative errors

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(MREs), standard deviations (STD), root mean square errors (RMSEs), and R-squared (R2) received from the outcome of models were determined. Above-mentioned analyses indicated that all the models have a great capability of predictions for determining heat transfer coefficients with low deviation from the corresponding experimental samples. Among the proposed models, we used the CMIS model due to consideration of all outcomes of models in its structure for analysis of outlier detection and sensitivity analysis of variables. Hence, William’s plot of the results produced by the CMIS model was illustrated. We believe these models are valuable for other conditions and/or other kinds of nanofluid; accordingly, the input indispensable data points must be identified to these suggested models in order to obtain expectable outputs. As a final point, evaluations the performance of other optimization approaches can be our future purposes in order to create better structures of intelligent techniques. Acknowledgements The authors would like to thank the reviewers for their valuable comments, which have been utilized in improving the quality of the paper.

Compliance with ethical standards Conflict of interest The authors declare that they have no conflict of interests.

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