International Communications in Heat and Mass Transfer 39 (2012) 1647–1653

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International Communications in Heat and Mass Transfer

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Prediction of thermal and fluid flow characteristics in helically coiled tubes usingANFIS and GA based correlations☆

Reza Beigzadeh, Masoud Rahimi ⁎CFD Research Center, Chemical Engineering Department, Razi University, Kermanshah, Iran

☆ Communicated by W.J. Minkowycz.⁎ Corresponding author at: Chemical Engineering Depa

Bostan, Kermanshah, Iran.E-mail address: masoudrahimi@yahoo.com (M. Rahim

0735-1933/$ – see front matter © 2012 Elsevier Ltd. Allhttp://dx.doi.org/10.1016/j.icheatmasstransfer.2012.10.0

a b s t r a c t

a r t i c l e i n f o

Available online 19 October 2012

Keywords:Helically coiled tubeAdaptive Neuro-Fuzzy Inference SystemGenetic AlgorithmHeat transferFriction factor

This study introduces the ability of Adaptive Neuro-Fuzzy Inference System (ANFIS) and genetic algorithm(GA) based correlations for estimating the hydrodynamics and heat transfer characteristics in coiled tubes.The experimental data related to the heat transfer and pressure drop in helically coiled tubes with deferentgeometrical parameters (coil diameter and pitch) were used. In the experiments, hot water was passed inthe coiled tubes, which were placed in a cold bath. Two ANFIS models were developed for predicting theNusselt number (Nu) and friction factor (f) in the coiled tubes and the geometric parameters were employedas input data. Moreover, empirical correlations for estimating the Nu and f were developed by a phenomeno-logical argument in the form of classical power–law correlations and their constants were found using the GAtechnique. The mean relative errors (MRE) of the developed ANFIS models for estimation of Nu and f are6.24% and 3.54%, respectively. On the other hand, for empirical correlations, a MRE of 8.06% was found forprediction Nu while MRE of 5.03% was obtained for f. The results show that the ANFIS models can predictNu and f with the higher accuracy than the developed correlations.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The use of coiled pipes is of considerable engineering interest and issignificant in practical applications, such as heat exchangers, steamgen-erators, chemical reactors, membrane separations, and piping systems[1–3]. Using of helically coiled tubes for heat transfer augmentation isa passive method. In a coiled tube, the centrifugal force generated bythe curvature of the tubes lead to a secondary flow which cause higherheat transfer rate compared with a straight tube. Moreover, employingthe coiled tubes lead to the size of the heat exchanger to be considerablydecreased. Vashisth et al. [3] reviewed the potential industrial applica-tions of curved tubes for single-phase and two-phase flow. Moreover,they presented a collection of the available correlations for estimationof friction factor and heat andmass transfer coefficients in curved tubes.

Due to the wide researches on heat transfer and flow characteristicsin helically coiled tubes and also many applications in industrial pro-cesses, providing accurate numericalmethods appears to be very useful.In some studies, numerical modeling was done for this purpose [4–6]and the effects of dimensionless geometrical parameters of the helicallycoiled tubes such as curvature ratio and coil pitch on the hydrodynam-ics and heat transfer characteristics were investigated. Artificial Neural

rtment, Razi University, Taghe

i).

rights reserved.11

Network (ANN), Adaptive Neuro-Fuzzy Inference System (ANFIS), andGenetic Algorithm (GA) are capable artificial intelligence methodsfor modeling, predicting and optimizing the performance of thermalsystems. Belanger and Gosselin [7] used ANNs in the context of mate-rials selection for thermofluid design. Mehrabi et al. [8] employedANFIS to predict the heat transfer rate and pressure drop in helically-coiled double-pipe heat exchangers. They showed ANFIS network is asuitable method for modeling more complicated systems in whichthere is no clear relationship between variables. Mehrabi and Pesteei[9] used ANFIS network for modeling the convection heat transfer ofturbulent supercritical carbon dioxide flow in a vertical circular tube.Many investigators used the adaptive neuro-fuzzy network as a propertechnique for modeling of the complex systems [10–12].

Genetic algorithm is a stochastic numerical search technique whichtaking inspiration from evolutionary processes. This method can beused in energy and cost optimization [13] or developing the correlationsby optimizing the correlation constants [14,15]. Artificial neural net-workswere developed tomodelingheat transfer andflow characteristicin coiled tubes in previous work [16].

This study aims to show the ability of ANFIS and GA in predictingNusselt number (Nu) and friction factor (f) in the coiled tubes. Theheat transfer rate and pressure drop were measured in three coppercoiled tubes with different coil diameter and also various pitch foreach tubes (total of nine cases). Two ANFIS models were developedfor predicting Nusselt number and friction factor in the coiled tubes.Dimensionless geometrical parameters (curvature ratios and coil

Nomenclature

A Heat transfer area (m2)CP Specific heat capacity (kJ/kg K)Ci Constant/Gaussian membership function centerD Tube diameter (m)D Coil diameter (m)H Heat transfer coefficient (W/m2 K)F Friction factorH Pitch (m)K Thermal conductivity (W/m K)L Tube length (m)M Mass flow rate (kg/s)N Number of data pointsNu Nusselt numberP Pressure, N/m2

Pr Prandtl numberQ Heat transfer rate (W)T Target dataRe Reynolds numberT Temperature (K)V Velocity (m s−1)Y Predicted value

Greek symbolsΔ Curvature ratio, di/DΓ Coil pitch, H/πDΝ Kinematic viscosity (m2/s)Ρ Density (kg m−3)Σ Gaussian membership function width

SubscriptsB MeanI Inlet/inner/input layerM Number of input variablesO Outlet/outerW Local wall

1648 R. Beigzadeh, M. Rahimi / International Communications in Heat and Mass Transfer 39 (2012) 1647–1653

pitches) of the coiled tubes were employed as input data. Moreover,the GA estimator used in search for global optimal estimation of thenew power–law correlation coefficients. Finally, characteristics ofthe methods were analyzed.

2. Experimental setup and procedure

Experiments were performed in a loop rig shown in Fig. 1. A sche-matic viewof a helical coil with itsmain geometrical parameters is illus-trated. As shown in this figure, D is the coil diameter (measuredbetween the centers of the pipes), and pitch H, is the distance betweentwo adjacent turns. Moreover, di and do are the inner and outer tubediameter, respectively. Three copper coiled tubes with different coildiameter and pitch was used in the experiments. The dimensions ofthe employed coiled tubes are presented in Table 1. In order to charac-terize the tubes dimensions, two dimensionless parameters of thehelically coiled tube, the curvature ratio (δ=di/D) and the coil pitch(γ=H/πD) were employed.

Cold water was passed upon a coiled tube with a mass flow rate of0.26 kg/s and temperatures between of 15.2 and 16.5 °C. The hotwater was supplied in a tank by an electric heater and diverted insidethe coiled tubes. A pump circulated the hot fluid with temperaturesbetween of 59.6 and 62.2 °C from the tank toward the test section.All the requisite parameters such as the inlet and outlet temperatures

of the fluids, pressure drop and flow rate of the fluids were measuredby means of appropriate instruments. Two pressure transducers mea-sured the pressure drops across the coiled tube. The inlet and outlettemperatures of the hot and cold fluid were measured using K-typethermocouples. For evaluating the average Nusselt number, tempera-tures at 10 different positions on the external surface of the coiledtube were measured. All ten temperature-sensing probes wereconnected to a data logger set made by Letron, BTM-4208SD.

3. Data reduction

The Nusselt number and friction factor were determined frommeasured temperatures and pressure drop across the helically coiledtubes. The Reynolds number based on the inner diameter of the coiledtubes is obtained by:

Re ¼ Vdiν

ð1Þ

The average heat transfer coefficients are evaluated from the mea-sured temperatures. The thermal resistance of the tube wall was as-sumed to be negligible. Through heat transferred to the cold fluid (Q)and the temperature difference between of wall and fluid (Tw−Tb),average heat transfer coefficient obtained from the experimental dataas follows:

Q ¼ mCp Ti−Toð Þ ð2Þ

h ¼ Q

A Tb−~TW

� � ð3Þ

Where the subscripts ‘i’ and ‘o’ refer to the tube inlet and outlet,respectively. h is the convection heat transfer rate coefficient, A isthe internal surface of the coiled tubes, Tb is the bulk temperatureand ~TW is the arithmetic mean temperature of the 10 measuringpoints:

Tb ¼ Ti þ To

2and ~TW ¼ ∑TW

10ð4Þ

The average Nusselt number is written as:

Nu ¼ hdi

kfð5Þ

The friction factor is evaluated as follows:

f ¼ 2ΔPdiρV2L

ð6Þ

Where V is the velocity, ν is the kinematic viscosity, ΔP is the pres-sure drop, ρ is the density, and L is the coiled tube length. All the fluidproperties were determined at mean bulk temperature.

4. Modeling

The main goal of the present work is to investigate the ability ofadaptive neuro-fuzzymodeling andGA based correlations for predictingthe heat transfer and friction characteristics in helically coiled tubes. Themodeling methods used in this study are explained in the subsequentsections.

4.1. Adaptive Neuro-Fuzzy Inference System

The artificial neural networks do not given any explicit knowledge orcausal relationships for the system and this is themajor drawback of the

Fig. 1. The experimental setup.

1649R. Beigzadeh, M. Rahimi / International Communications in Heat and Mass Transfer 39 (2012) 1647–1653

ANNs [17]. Jang [18] developed a fuzzy systembyneural networkswhichovercome the shortcomings of ANNs and fuzzy systems. This techniquewas called adaptive Neuro-Fuzzy Inference System (ANFIS). ANFIS em-ploys two fuzzy logic and ANN approaches. This combination may leadto results that will comprise the both abilities.

The ANFIS structure consists of two parts, introductory and conclud-ing, which are linked together by a set of fuzzy rules. Fig. 2 illustrates atwo input Sugeno type fuzzy system which consisting of two inputs i.e.x and y and an output or f which is related with the fuzzy rules:

Rule 1 : If x is A1ð Þ and y is B1ð Þ then f1 ¼ p1xþ q1yþ r1 ð7Þ

Rule 2 : If x is A2ð Þ and y is B2ð Þ then f2 ¼ p2xþ q2yþ r2 ð8Þ

Where pi, qi and ri are adjustable parameters and determined duringthe training process. Ai, Bi and fi are fuzzy sets and system's output, re-spectively. As shown in Fig. 2, ANFIS structure consists of five distinctlayers, a fuzzy layer, a product layer, a normalized layer, a defuzzylayer, and a total output layer. The subsequent sections explained thelayers and relationship between the input and output of the layers inthe ANFIS architecture.

Table 1Dimensions of the employed helically coiled tubes.

Coilnumber

Casenumber

di,mm

do,mm

D,mm

H,mm

L,cm

δ γ

Coil 1 1 7.5 10 60 20 140 0.1250 0.10612 7.5 10 60 45 140 0.1250 0.2387

Coil 2 3 5 8 58 15 185 0.0862 0.08234 5 8 58 25 185 0.0862 0.13725 5 8 58 60 185 0.0862 0.3293

Coil 3 6 5.5 8 110 15 180 0.0500 0.04347 5.5 8 110 25 180 0.0500 0.07238 5.5 8 110 52 180 0.0500 0.15059 5.5 8 110 75 180 0.0500 0.2170

As shown in Fig. 2, layer 1 named fuzzy layer. In this layer, each nodei is an adaptive node with specific membership function form in thefuzzy set. The output of each node is equal to the input variable mem-bership grade of a fuzzy set. In this study, Gaussian curve membershipfunctions were used which in this function maximum is 1 and mini-mum is 0, as follows [18]:

μAixð Þ ¼ exp −1

2x−ciσ i

� �2� �ð9Þ

Where x is the input value of the node, and Ci and σi are the mem-bership function parameter set which explain Gaussian membershipfunction center and Gaussianmembership functionwidth, respectively.

The second layer is the product layer that input signal values prod-uct the firing strength of a rule. The output of layer 2, Q2,i, is calculatedas follows:

Q2;i ¼ wi ¼ μAi xð ÞμBi yð Þ; fori ¼ 1;2 ð10Þ

μAi is the membership grade of x in Ai fuzzy set and μBi is the member-ship of y in fuzzy set of Bi.

Layer 3 is the normalized layer in which the ith node evaluates theratio of the ith firing strength of rules to the sum of all rule's firingstrengths. The output is the normalized firing strength of ith rule.

Q3;i ¼ wi ¼wi

w1 þw2; for i ¼ 1;2 ð11Þ

The fourth layer is the defuzzy layer, in which the output, Q4,i,achieved from multiplication of the normalized firing strength byfirst order of Sugeno fuzzy rule as follows:

Q4;i ¼ �wifi ¼ �wi pixþ qiyþ ri Þ; for i ¼ 1;2ð ð12Þ

Layer 1 (Fuzzy layer)

Layer 2 (Product layer)

Layer 3 (Normalized layer)

Layer 4 (Defuzzy layer)

M

M

x

y

A1

A2

B1

B2

N

N

x y

x y

f1

f2

Layer 5 (Total output layer)

f

Forward pass

Backward pass

Fig. 2. The ANFIS architecture.

1650 R. Beigzadeh, M. Rahimi / International Communications in Heat and Mass Transfer 39 (2012) 1647–1653

The fifth layer is the total output layer and the output of this layeris the overall output:

Q5;i ¼ ∑iwif i ¼

∑iwif i∑iwi

ð13Þ

Hence, ANFIS has two types of parameters, premise and consequentparameters, which were adjusted during the learning process. The aimof the modeling with ANFIS is determining the optimum structure ofANFIS and also evaluating the best parameters (premise and conse-quent) in order to achieve the accurate prediction. ANFIS uses twolearning algorithms, such as backpropagation and hybrid method,which seeks to minimize error function [19]. The parameters are evalu-ated in two passes. Consequent parameters are calculated forward andpremise parameters are obtained backward.

Due to the flexibility of the ANFIS structure and parameters, themodel appears to be capable in prediction of complex relationships.In the present work two ANFIS model were developed for predictingNu and f in coiled tubes. The first ANFIS had four input parameters in-cludes Reynolds number, Prandtl number, curvature ratio, and coilpitch (Nu=f {Re, Pr, δ, γ}). In the ANFIS for modeling f, three inputparameters were employed, Reynolds number, curvature ratio, andcoil pitch (f=f {Re, Pr, δ, γ}). The structure of the ANFIS networkswere considered by trial-and-error method and the parameters ofthe models were obtained using hybrid-learning algorithm.

4.2. Genetic algorithm

Genetic algorithm (GA) is the other field of artificial intelligence andis a stochastic numerical search technique. GA is motivated by Darwin'sevolution theory and inspired by evolutionary processes. A comprehen-sive discussion about GA was presented by Goldberg [20]. GA usuallyis a suitable choice for estimation based on regression [14,20]. Thismethod can achieved to the optimum answer (minimum ormaximum)rapidly, after searching a small part of the search space. In this study, GAwas applied for solving problems on which traditional techniques notsucceed to return the global minimum/maximum and in particularwhen the function characteristics are not known.

Based on the experimental results, a correlation between theNusseltnumber and Reynolds number (Re), Prandtl number (Pr), and dimen-sionless curvature ratio (δ) and coil pitch (γ) is obtained by means ofGA optimization. For this purpose, the following functional relationshipis assumed:

Nu ¼ C1ReC2PrC3δC4γC5 ð14Þ

The fitness function from experimental and predicted values canbe defined as:

E C1;C2;C3;C4;C5ð Þ ¼ 100N

XNi ¼1

NuExpi �NuPred

i

��� ���NuExp

i

0@

1A ð15Þ

Where N is the number of data, GA can be used for minimizing theabove error function.

A similar procedure was considered for friction factor as follows:

f ¼ C1ReC2δC3γC4 ð16Þ

EðC1;C2;C3;C4Þ ¼100N

XNi¼1

fExpi −fPredi

��� ���fExpi

0@

1A ð17Þ

By applying searchingmethod of GA, the optimum constant values(Ci) of assumed power–law correlations can be obtained.

5. Results and discussion

The heat transfer rate and pressure drop in helically coiled tubeswere experimentally measured and the effects of the coiled tube geo-metrical parameters on them were investigated. The Nu and f werecalculated from measured temperatures, pressures and flow rates.Fig. 3 shows the variations of Nu and f for the all examined coiledtubes. As was expected, different geometrical parameters of coiledtubes lead to different values of heat transfer rate and friction factor.As can be found in this figure, higher Nu and f values were obtained inhigher curvature ratio and the lower coil pitch.

Table 3Parameters of Gaussian membership functions for predicting f.

Membershipfunction

Input 1 Re Input 2 δ Input 3 γ

MF1 n 18,050 0.02895 0.1215m 4810 0.0543 0.04356

MF2 n 18,050 0.01236 0.1205m 47,320 0.1377 0.3294

n indicates Gaussian MFs width and m represents Gaussian MFs center.

Table 4Fuzzy rule base of the optimum ANFIS structure for predicting Nu.

Rulenumber

Rule description

1 If (Re is Re MF1) and (δ is δMF1) and (γ is γMF1) and (Pr is Pr MF1) then(Nu=0.0076×Re+349.6×δ−33.8×γ+167.1×Pr−582.4)

2 If (Re is Re MF1) and (δ is δMF2) and (γ is γMF2) and (Pr is Pr MF1) then(Nu=0.0066×Re+1504×δ−69.92×γ−87.26×Pr+144.1)

3 If (Re is Re MF2) and (δ is δMF1) and (γ is γMF1) and (Pr is Pr MF2) then(Nu=0.0110×Re+2137×δ−411×γ+368.4×Pr−1586)

4 If (Re is Re MF2) and (δ is δMF2) and (γ is γMF2) and (Pr is Pr MF2) then(Nu=0.0054×Re+483.3×δ−379.1×γ+261.1×Pr−761.3)

Table 5Fuzzy rule base of the optimum ANFIS structure for predicting f.

Rule number Rule description

1 If (Re is Re MF1) and (δ is δ MF1) and (γ is γ MF1) then(Nu=−1.871×10−6×Re+0.1861×δ+0.028×γ+0.04684)

2 If (Re is Re MF1) and (δ is δ MF1) and (γ is γ MF2) then(Nu=−1.72×10−6×Re−0.1045×δ+0.0743×γ+0.04109)

3 If (Re is Re MF1) and (δ is δ MF2) and (γ is γ MF1) then

Fig. 3. Variation of Nusselt number and friction factor with Reynolds number for all theemployed coiled tubes.

1651R. Beigzadeh, M. Rahimi / International Communications in Heat and Mass Transfer 39 (2012) 1647–1653

As far as the effects of the geometric parameters on heat transferand pressure drop in the coiled tubes are quite complicated, twoANFIS models were developed to estimate Nu and f in them. Ninevarious coiled tubes were employed in the experiments. A total of71 experimental data of Nu and 84 experimental data points for fwithin a Reynolds number ranging from 4000 to 48,000 were used.

These data are divided randomly into two sections, the first datagroup (two-third of all of the data set of Nu and f) was considered fortraining the networks and the second data group (remaining data)was used for validation of the ANFIS networks. Optimum structures ofthe ANFIS networks were obtained by trial-and-error method. Hybridmethod was used to achieve the best parameters of the networks thatlead to minimum deviations between the predicted values and targetdata. Input variables were fuzzified with different membership func-tions, which were named MF1 and MF2 with Gaussian membership

Table 2Parameters of Gaussian membership functions for predicting Nu.

Membershipfunction

Input 1 Re Input 2 δ Input 3 γ Input 4 Pr

MF1 n 16180 0.0155 0.1312 0.3408m 4283 0.04758 0.04673 3.001

MF2 n 16180 0.01084 0.1269 0.3407m 42380 0.1345 0.3261 3.8

n indicates Gaussian MFs width and m represents Gaussian MFs center.

function. The parameters of these membership functions are listed inTables 2, 3 for Nu and f, respectively. The rule based on first-orderSugeno inference system reflecting the physical property of the devel-oped models along with membership functions is presented inTables 4, 5. Moreover, the optimal consequent parameters acquiredafter the ANFIS training are given in the tables.

Most important applied deviations to evaluate modeling methodsare mean relative errors (MRE), the mean square errors (MSE), andabsolute fraction of variance (R2), which are presented as follows:

MRE %Þ ¼100N

XNi ¼1

ti�yij jti

� � ð18Þ

MSE ¼ 1N

XNi ¼1

ti�yið Þ2 ð19Þ

R2 ¼ 1�

XNi ¼1

ti�yið Þ2

XNi ¼1

tið Þ2ð20Þ

(Nu=−5.299×10−6×Re+0.0137×δ+0.0128×γ+0.1104)4 If (Re is Re MF1) and (δ is δ MF2) and (γ is γ MF2) then

(Nu=7.57×10−8×Re+0.00658×δ+0.0101×γ+0.05237)5 If (Re is Re MF2) and (δ is δ MF1) and (γ is γ MF1) then

(Nu=−1.09×10−6×Re+0.08116×δ+0.013×γ+0.07527)6 If (Re is Re MF2) and (δ is δ MF1) and (γ is γ MF2) then

(Nu=−8.1×10−7×Re+0.1415×δ+0.01996×γ+0.05013)7 If (Re is Re MF2) and (δ is δ MF2) and (γ is γ MF1) then

(Nu=−3.1×10−6×Re+0.02317×δ+0.01849×γ+0.1855)8 If (Re is Re MF2) and (δ is δ MF2) and (γ is γ MF2) then

(Nu=9.47×10−8×Re+0.00322×δ+0.00153×γ+0.026)

Fig. 4. Predicted versus experimental Nusselt number values.Fig. 5. Predicted versus experimental friction factor values.

1652 R. Beigzadeh, M. Rahimi / International Communications in Heat and Mass Transfer 39 (2012) 1647–1653

Where N is the number of data points, t is the target (experimen-tal) data, and y is the predicted value. The mean relative errorsEq. (18) of the developed ANFIS models for estimation of Nu and fare 6.24% and 3.54%, respectively.

Using GA searching method new empirical correlations for esti-mating the Nu and f were developed by a phenomenological argu-ment in the form of classical power–law correlations Eqs. (14), (16).After employing the experimental data for estimating Nu and f, the

Table 6MRE, MSE, and R2 values for prediction of Nu and f.

Method Nu f

MRE (%) MSE R2 MRE (%) MSE R2

GA based correlations 8.06 135 0.9923 5.03 1.19×10−5 0.9926ANFIS 6.24 37 0.9986 3.54 4.27×10−6 0.9979ANN [16] 2.46 12 0.9993 1.26 4.36×10−7 0.9997

1653R. Beigzadeh, M. Rahimi / International Communications in Heat and Mass Transfer 39 (2012) 1647–1653

correlation constants (Ci) were obtained and the following relationswere found with MRE of 8.06% and 5.03% for Nu and f, respectively:

Nu ¼ 0:359Re0:781Pr0:016δ0:933γ−0:172 ð21Þ

f ¼ 2:32Re−0:311δ0:467γ−0:074 ð22Þ

In Fig. 4 it is tried to show the Nusselt number prediction validity.Turning to Fig. 4a, it reveals a comparison between the measured andANFIS-predicted values of Nu for the training and testing data set. Thefigure indicates that the ANFIS predicted values for all data points areclose to the experimental values. Furthermore, acceptable differencein error values between the train and test data set proves the validityof the model. Fig. 4b illustrates a comparison between the experimen-tal and predicted using Eq. (21) Nusselt numbers. Moreover, in Fig. 4cthe results of ANN modeling reported in previous work [16] is shownin order to show that the both present models are able to predict theNusselt number with almost at a same order of precision predicted byANN. However, in these newmodels the number of training data is 71while 228 data points used for Nu prediction in the ANN model [16].

With a similar approach, the models validation for friction factor isillustrated in Fig. 5. Fig. 5a shows a plot of the experimental data andthe predicted f using ANFIS. The high accuracy of friction factor pre-diction for testing data indicates the verification of the method. More-over, the accuracy of the developed correlation for friction factorEq. (22) is illustrated in Fig. 5b. Moreover, similar to above discussionfor Nusselt number, the results from previous work [16] shown inFig. 5c reveals that the new developed models can predict the frictionfactor values with a same precision as ANN. However, a lower numberof training data, 84 points, was used in the present work in compari-son with 198 data points used in the ANN model [16].

In order to compare the predicted results from the developedmodels, the amount of MRE, MSE, and R2 for Nu and f predictionusing ANFIS models and GA based correlations are given in Table 6.The results show that the developed ANFIS models are more accuratethan the empirical correlations for Nu and f prediction. Moreover, thistable gives the accuracy of the ANN models developed in previouswork [16] with more data points. The ANN model leads to the lowererror in prediction of Nu and f compared with ANFIS models and GAbased correlations. However, although the ANN appears more accuratebut theANFISmodel used lower number of data values.Moreover, thereis no ambiguity in ANFIS, unlike the ANNs [21,22]. Furthermore, theANFIS learning duration is very short in comparison with ANNs [21].On the other hand, the developed correlations using GA can be used topredict the Nu and f in a more efficient way compared with ANFIS andANN.

6. Conclusions

This investigation attempts to applied ANFIS and GA techniques tomodel and predict the thermal and fluid flow characteristics in helicallycoiled tubes. The experimental data were measured and were employedas input data for themodels. Nusselt numberwas determined as functionof Re, Pr, δ,γ, and also friction factorwas considered as functionof Re, δ,γ.Two ANFIS models were proposed for estimating of Nu and f, and two

power–law correlations were developed for predicting them using GAmethod. The results show that both two techniques are accurate and re-liablemethods for this purpose. The results reveal that the accuracy of thepredicted values fromANFISmodels aremore than the GA based correla-tions. However, it is obvious that applying the correlations is easier,which can be treated as an advantage for GA based correlations. The pre-cision of the methods was compared with ANN's results in the previousinvestigation [16]. From this study, it can be concluded that all threemethods are appropriate and capable techniques in modeling of thermaland fluid flow characteristics in helically coiled tubes.

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