International Communications in Heat and Mass Transfer 39 (2012) 1279–1285

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International Communications in Heat and Mass Transfer

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Prediction of heat transfer and flow characteristics in helically coiled tubes usingartificial neural networks☆

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

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

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

0735-1933/$ – see front matter © 2012 Elsevier Ltd. Alldoi:10.1016/j.icheatmasstransfer.2012.06.008

a b s t r a c t

a r t i c l e i n f o

Available online 19 June 2012

Keywords:Helically coiled tubeArtificial neural networkHeat transferFriction factor

In this study, Artificial Neural Network (ANN) models were developed to predict the heat transfer and frictionfactor in helically coiled tubes. The experiments were carried out with hot fluid in coiled tubes which placedin a cold bath. Coiled tubes with various curvature ratios and coil pitches (nine Layouts) were used. Theoutput data of the ANNs were Nusselt number and friction factor. The validity of the method was evaluatedthrough a test data set, which were not employed in the training stage of the network. Moreover, the perfor-mance of the ANN model for estimating the Nusselt number and friction factor in the coiled tubes wascompared with the existing empirical correlations. The results of this comparison show that the ANN modelshave a superior performance in predicting Nusselt number and friction factor in the coiled tubes.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Heat transfer processes are widely used in a variety of industrialapplications and heat transfer enhancement (HTE)methods are subjectof many recent investigations [1–3]. Hence, reliable predictionmethodsare absolutely essential for design appropriate industrial equipmentssuch as heat exchangers.

Heat transfer enhancement techniques can be classified into threemain groups: active, passive and a combination of active and passivemethods. Helically-coiled tube is treated as one of the passive methodsin heat transfer enhancement. Due to high heat transfer and lowvolumeoccupation, helical coil heat exchangers are widely used inmany indus-trial applications [4]. The centrifugal force generated by the curvatureof the pipe, leads to a secondary flow and this causes an enhancementin heat transfer rate compared with a straight tube. However, morepressure drops are disadvantage of this enhancement technique [5–7].Ito [8] proposed the most well‐known relationship for the frictionfactor of the curved channel. The heat transfer coefficients under theboundary conditions of constantwall temperaturewere experimentallyinvestigated in some other research works [9–12].

Modeling of the hydrodynamics and heat transfer characteristics ofthe coiled tubes was considered in different literature. Moreover, thenumerical modeling was performed in various literatures [13–16]and effects of dimensionless geometrical parameters of the helically

Department, Razi University,

mi).

rights reserved.

coiled tubes such as curvature ratio and coil pitch on the heat transferand flow characteristics were investigated.

Artificial neural networks by using a large number of parameters(weights and biases) are able to estimate target data of thermalsystems in engineering applications with a high accuracy. Mehrabiet al. [17] used Adaptive neuro-fuzzy inference system (ANFIS) tomodel and predict the heat transfer and pressure drop in Helically-coiled double-pipe heat exchangers. They employed the geometricalparameters as input data of the ANFIS model. Bélanger and Gosselin[18] studied application of neural networks to evaluate the designspace of the heat transfer problems with a number of choices ofmaterials. Jafari Nasr et al. [19] employed ANNs to predict Nusseltnumber and friction factor inside tube with wire coil inserts. They com-pared their results with corresponding power–law regressions and asuperior performance of the ANN compared with those methods wasfound. Islamoglu and Kurt [20] used ANNs for heat transfer analysisof air flowing in corrugated channels. In another study, Deng et al.[21] used the ANN method to estimate the heat flux transmitted to aworkpiece from a flame gun during thermal spray preheating process.

This study introduces a feed forward ANN model trained by theLevenberg–Marquardt algorithm [22–24] to predict Nusselt numberand friction factor in helically coiled tubes. A data set was obtainedexperimentally and prepared for processing with the use of ANNs.Coiled tubes with different geometrical parameters (nine Layouts)like curvature ratio and coil pitch were used and these dimensionlessgeometrical parameters were considered as input data of the ANNs. Inaddition, the empirical correlations developed by Ito [8], and Xin andEbadian [25] were used for estimating the friction factor and Nusseltnumber in the coiled tubes. The predicted results from these correla-tions were compared with those obtained from ANN models.

Nomenclature

A heat transfer area (m2)bj biasCp specific heat capacity (kJ/kg K)d tube diameter (m)D coil diameter (m)h heat transfer coefficient (W/m2 K)f friction factorF transfer functionH pitch (m)k thermal conductivity (W/m K)L tube length (m)m mass flow rate (kg/s)N number of turns/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)wij weighty predicted value

Greek symbolsδ curvature ratio, di/Dγ coil pitch, H/πDν kinematic viscosity (m2/s)ρ density (kg m−3)

Subscriptsb meani inlet/inner/input layerj hidden layerk output layerm number of input variablesn number of neuronsN non-coiled tubeo outlet/outerw local wall

1280 R. Beigzadeh, M. Rahimi / International Communications in Heat and Mass Transfer 39 (2012) 1279–1285

2. Experimental work

Fig. 1 illustrates a schematic diagram of the experimental setup. Theexperimental rig includes a coiled tube which was placed into a bathand measuring temperature and pressure instruments. The cold waterwas passed upon the coiled tube with a mass flow rate of 0.26 kg/sand temperatures between of 15.2 and 16.5 °C. The piping connectionsin the test section were designed so that the parts can be replacedconveniently. A schematic view of a helical coil and its geometricalparameters are also shown in Fig. 1. As illustrated in the figure, d isthe diameter of the tube and ‘i’ and ‘o’ refer to the inner and outer,respectively. Pitch, H, is the distance between two adjacent turns andcoil diameter D is the distance between the centers of the pipe in oneloop. Two dimensionless parameters of the coiled tube, coil pitch (γ)and curvature ratio (δ), are commonly used and expressed as follows:

γ ¼ HπD

ð1Þ

δ ¼ diD

ð2Þ

The experiments were performed using three copper coiled tubeswith curvature ratios of 0.125, 0.0862, and 0.05. In addition, variouspitches for each coiled tube (two pitches for coil 1, three pitches forcoil 2 and four pitches for coil 3) were employed. Table 1 gives thedetails of the coiled tubes that used in the experiments.

The hot water was supplied in a tank by using an electric heater.The hot fluid with a temperature between 59.6 and 62.2 °C wascirculated from the tank through the test section using a pump. Inorder to control the flow rate of the hot and cold water, flow meters,temperature recorder, and controller apparatuses were used. Fur-thermore, the pressure drops across the coiled tubes for differenthot water flow rates were measured using two pressure transducers.Temperatures at the inlet and outlet of the cold and hot fluid weremeasured using K‐type thermocouples. In order to calculate theaverage Nusselt number, the temperatures at 10 different positionson the external surface of the coiled tube were measured. The 10temperature-sensing probes were connected to a data logger set forthis purpose.

3. Data acquisition and analysis

The Nusselt number (Nu) and friction factor (f) were evaluatedfrom measured temperatures and pressure drop across the tube. Forthis purpose, in the first step the heat transfer rate of the hot fluidin the coiled tubes was obtained from following equation:

Q ¼ mCp T i−Toð Þ ð3Þ

where Q is the heat transfer rate, m is the mass flow rate, Cp is thespecific heat capacity, T is the temperature, and the subscripts ‘i’ and‘o’ refer to the tube inlet and outlet, respectively.

The heat transfer rate to the cold fluid around the tube was calcu-lated as follows:

Q ¼ hA Tb−~TW

� �ð4Þ

where h is the convection heat transfer rate coefficient, A is the inter-nal surface of the coiled tubes; Tb is the bulk temperature and ~TW isthe arithmetic mean temperature of the 10 measuring points:

Tb ¼ T i þ Tb

2and T̃W ¼ ∑TW

10ð5Þ

where Tw is the local wall temperature and is measured at the exter-nal wall surface of the coiled tubes.

The average heat transfer rate coefficient, Nusselt number, Reynoldsnumber, and friction factor were calculated from following equations:

h ¼ mCp T i−Toð ÞA Tb−~TW

� � ð6Þ

Nu ¼ hdikf

ð7Þ

Re ¼ Vdiν

ð8Þ

f ¼ 2ΔPdiρV2L

ð9Þ

where V is the velocity, ν is the kinematic viscosity, ΔP is the pressuredrop, ρ is the density, and L is the coiled tube length. All the fluidproperties were determined at mean bulk temperature.

Fig. 1. The experimental setup.

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4. Artificial neural network modeling

The ANNs by using a large number of parameters (weights andbiases) can learn and recognize complicated non-linear functions, sowidely used in a variety of the fields of chemical engineering [26]. TheANNs are including of an input and an output layer and a number ofhidden layers that each comprises neurons. In theory, an ANN with onehidden layer with the appropriate number of neurons is able to estimateapproximately any type of non-linear functions [27]. However, some-times using more hidden layers leads to superior performance. The neu-rons in the each layer are linked to the adjacent layers and the strengthof this interconnected configuration is determined by the weights. Theinterconnected structure is conducive to a logical relationship betweeninput and output parameters. The number of neurons for the input andoutput layers is equal to the input and output variables, respectively.Nevertheless, in the hidden layer different numbers of neurons can beemployed and it is significant for optimization of the network.

In the network, information is transferred passing through theconnections of the neurons in the adjacent layers. The final outputof the ANN obtained from the following relation:

Y ¼ FpXnj¼1

Wkj FtXmi¼1

WjiXi þ bj

!" #þ bk

8<:

9=; ð10Þ

Table 1Details of the helically coiled tubes.

Coilnumber

Layoutnumber

di,mm

do,mm

D,mm

H,mm

L,cm

δ γ Na

Coil 1 1 7.5 10 60 20 140 0.1250 0.1061 6.52 7.5 10 60 45 140 0.1250 0.2387 6.5

Coil 2 3 5 8 58 15 185 0.0862 0.0823 9.54 5 8 58 25 185 0.0862 0.1372 9.55 5 8 58 60 185 0.0862 0.3293 9.5

Coil 3 6 5.5 8 110 15 180 0.0500 0.0434 4.57 5.5 8 110 25 180 0.0500 0.0723 4.58 5.5 8 110 52 180 0.0500 0.1505 4.59 5.5 8 110 75 180 0.0500 0.2170 4.5

a Number of turns.

where Y is the final answer of the network, X is input value of the net-work, W is the weight, b is the bias, n is the number of neurons, m isthe number of input variables, ‘i’, ‘j’ and ‘k’ refer to the input, hidden,and output layer, respectively. F is the transfer function that is used toget the normalized output values from the neurons. In this study, hy-perbolic tangent sigmoid transfer function was considered for hiddenlayers and purelin transfer function was used for the output layer.These functions can be defined as follows:

Ft xð Þ ¼ ex−e−x

ex þ e−x ð11Þ

Fp xð Þ ¼ x ð12Þ

In this study, two ANNs were developed to estimate the Nu and fin the helically coiled tubes. The ANN for estimating Nusselt numberhad four input parameters includes Reynolds number, Prandtl num-ber, curvature ratio, and coil pitch:

Nu ¼ f Re; Pr; δ;γð Þ ð13Þ

In the ANN for predicting friction factor, three input parameterswere considered which were Reynolds number, curvature ratio, andcoil pitch:

f ¼ f Re; δ;γð Þ ð14Þ

As far as in training procedure of the neural networks, the inputand output data had various physical units and range sizes, all datawere normalized in the 0–1 range to avoid any computational diffi-culty using following relation:

Normalized data ¼ data value−minimum valuemaximum value−minimum value

ð15Þ

The Levenberg–Marquardt back propagation algorithm was ap-plied for training of the ANNs. In this method, weights and biasesiteratively adjust to reduce deviation of the predicted values ofthe network from the desired values according to the Levenberg–Marquardt [22–24] optimization procedure.

1282 R. Beigzadeh, M. Rahimi / International Communications in Heat and Mass Transfer 39 (2012) 1279–1285

In order to evaluate the validity of the model, all input data pointswere divided into two sections: train and test data set. Two-thirdsof the data points were selected for training to develop the neuralnetwork and the remaining data were considered as the test dataset. Moreover, trial-and-error method was used to determined appro-priate number of neurons in the hidden layer.

Fig. 2. Nusselt number versus Reynolds number for all the coiled tubes (solid lines:trend line). (a) Coil 1. (b) Coil 2. (c) Coil 3.

Fig. 3. Friction factor versus Reynolds number for all the coiled tubes (solid lines: trendline). (a) Coil 1. (b) Coil 2. (c) Coil 3.

5. Results and discussion

In this study, two ANNs were developed to estimate the Nusseltnumber and friction factor in the helically coiled tubes. For thispurpose, a large number of experiments were carried out using coiledtubes with different geometrical parameters. Obviously, changing the

Table 2Range of Re and Pr, and number of data points for each coiled tubes.

Layoutnumber

Nu f

Range Number ofdata points

Range Number ofdata points

Re Pr Re

1 4270–40986 3.02 -3.60 22 4270 -44207 242 3360–39555 2.97–3.61 21 5722–46920 273 4851–38127 2.99–3.63 32 5839–45876 154 4873–37631 3.17–3.90 28 6508–46352 195 4215–37077 3.00–3.66 28 5473–47279 236 5957–41236 3.09–3.58 27 5425–48652 187 4974–41999 3.13–3.94 24 4487–48662 278 5230–41968 3.02–3.62 26 5275–46449 249 4859–42256 3.02–3.68 20 5306–44792 21

Table 4Parameters (weight and bias) of the ANN for prediction of Nusselt number.

Neuron Wji bj bk=0.926

δ γ Pr Re Wkj

1 3.1374 −0.8046 −0.5463 −6.5230 −2.1420 −0.18012 −0.4586 −4.3414 −7.5719 −3.9718 8.8503 0.07323 −2.8473 2.1886 1.4239 −0.2967 1.3803 −0.17164 4.9383 1.6013 5.0009 2.2095 −8.1626 0.11385 0.1037 0.3153 0.4899 3.1770 −1.0868 0.16146 −9.8239 22.966 −5.1752 −15.685 3.1476 −0.01567 0.3657 1.1905 −0.4851 1.8275 −2.2182 1.27048 0.1749 1.3566 −0.9760 1.7168 −2.0516 −1.08269 2.7657 2.1488 10.739 2.3809 −2.2625 −0.6105

Table 5Parameters (weight and bias) of the ANN for prediction of friction factor.

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geometric parameters in the coiled tubes leads to different valuesof heat transfer rate and pressure drop. The variation of Nusseltnumber and Friction factor at various Reynolds numbers for all thehelically coiled tubes are shown in Figs. 2 and 3. Range of Re and Prvalues, and number of data points for each coiled tubes are listed inTable 2.

Two developed ANNs, which employed to estimate Nu and f,had one hidden layer. In general, there is no theoretical techniqueto evaluate suitable number of neurons in the hidden layer beforetraining. On the other hand, a large number of neurons in the hiddenlayer can cause “overfitting”. Overfitting typically happens whenthe model is complicated (i.e. increase the number of weights andbiases). In this work, various numbers of neurons were used forhidden layer and the optimal number of neurons was evaluated. Thedeviations, which were used for selecting the best ANN architecture,are the mean square errors (MSE), mean relative errors (MRE) andabsolute fraction of variance (R2) defined as follows:

MSE ¼ 1N

XNi¼1

ti−yið Þ2 ð16Þ

MRE %ð Þ ¼ 100N

XNi¼1

ti−yij jti

� �ð17Þ

R2 ¼ 1−

PNi¼1

ti−yið Þ2

PNi¼1

tið Þ2ð18Þ

where N is the number of data points, t is the target (experimental)data, and y is the predicted value. The deviation values of differentANN configurations for prediction of Nusselt number are presentedin Table 3. According to this table the best ANN configuration hadone hidden layer consist of nine neurons. TheMRE,MSE, and R2 values

Table 3MRE, MSE and R2 values of different ANN configurations for Nu and f prediction.

Nu f

No. ofneurons

MRE MSE R2 No. ofneurons

MRE MSE R2

4 3.24 16.38 0.999 3 2.07 1.24×10−6 0.99927 2.97 14.54 0.9992 6 1.50 6.71×10−7 0.99969 2.46 12.95 0.9993 8 1.40 6.98×10−7 0.999511 2.51 12.87 0.9993 10 1.37 7.59×10−7 0.999514 2.53 11.58 0.9994 12 1.26 4.36×10−7 0.999718 2.66 13.75 0.9991 14 1.33 5.54×10−7 0.999620 2.57 13.28 0.9993 20 1.34 5.94×10−7 0.9996

of the ANN for predicting Nusselt number were 2.46, 12.95, and 0.9993,respectively.

In addition, comparison of deviations by different ANN configura-tions for friction factor prediction is presented in Table 3. According tothis table, ANN configuration with one hidden layer consist of twelveneurons had the best performance. The MRE, MSE, and R2 valuesof the ANN for predicting friction factor were 1.26, 4.36×10−7, and0.9993, respectively. The parameters (weight and biases) of the bestselected networks have been given in Tables 4 and 5.

The performance of the developed ANNs was assessed throughusing a test data set consisting of one-thirds of the data points, whichnot previously considered for training of the networks. Training andtest data set for predictingNusselt numberwere 152 and 76 data points,respectively. These values for predicting friction factor were 132 and66 data samples, respectively.

The results show that testing deviation values for predictingNuwereMRE=3.13%, MSE=18.58, and for predicting f were MRE=1.58%,MSE=7.08×10−7. The comparison of Nu and f predicted values resultsof the developed neural network and the experimental data points fortraining and testing data sets are shown in Fig. 4. The solid line indicatesthe perfect fit (predicted equal target data). The figure illustrates a goodagreement between the network predicted results and target values.Moreover, low MRE and MSE values of the test data set, and also anacceptable difference between deviation values for train and test dataset indicate the verification of the ANN models.

In order to illustrate the ability of the developed ANN models, thepredicted results from this study for Nusselt number were comparedwith those predicted from Xin and Ebadian [25] correlation. In thatwork, the authors performed an experimental investigation on heattransfer in helically coiled tubes and proposed the following correlation:

Nu ¼ 0:00619Re0:92Pr0:4 1þ 3:455δð Þ ð19Þ

Fig. 5a illustrates the Nusselt number versus Reynolds number forcoiled tubes with lowest coil pitch and the prediction results of the

Neuron Wji bj bk=5.502

δ γ Re Wkj

1 2.8683 0.3644 −7.6228 −2.6784 4.38502 −2.5981 0.0777 −12.582 −1.5847 4.33973 −1.5537 −7.1539 1.3669 3.7914 0.71474 1.8951 7.7897 −1.3126 −4.2368 0.69835 2.0637 12.223 −3.3056 0.9496 −0.01116 −20.249 2.0457 12.949 8.0409 −0.05007 3.2435 0.4471 −8.3945 −2.8684 −3.38568 6.1441 8.1724 4.2216 −5.5644 0.00819 −1.8918 −0.1085 9.0935 −2.3413 −0.044310 5.8123 0.1714 −5.5396 3.0029 0.033411 −0.1833 5.9676 7.2182 −7.0787 −0.009112 11.312 3.0688 −13.755 2.1184 0.0174

Fig. 4. The ANNs model validation. (a) Nusselt number. (b) Friction factor.Fig. 5. An evaluation of prediction methods for Layouts 3 and 6.

1284 R. Beigzadeh, M. Rahimi / International Communications in Heat and Mass Transfer 39 (2012) 1279–1285

Eq. (19) and the ANNmodel. As can be seen in the figure, the ANN hassuperior performance for predicting the Nusselt number.

On the other hand, the friction factor measurements in this studywere compared with the empirical correlations of Ito [8] for the coiledtubes in turbulent flow regime. His correlation was as follows:

f ¼ 0:304 Re−0:25 þ 0:029ffiffiffiδ

pð20Þ

This comparison is illustrated in Fig. 5b, which shows the frictionfactor versus Reynolds number for coiled tubes with lowest coilpitch. The results indicate that the developed ANN is able to predictfriction factor with a high accuracy.

6. Conclusions

This work reports the advantages of using the ANN modelingtechnique for predicting heat transfer rate and flow characteristic inhelically coiled tubes. The experimental data were collected usingnine coiled tubes with various dimensionless geometrical parametersincluding; curvature ratio and coil pitch. Three-layer feed forwardANNs were developed to predict Nusselt number and friction factor inthe studied coiled tubes. Optimal neural network configurations were

evaluated by trial-and-error method. An ANN with 9 hidden neuronswas selected for predicting of Nusselt number and a network with 12hidden neurons was employed to predict friction factor in the tubes.Moreover, the predicted Nusselt number from ANNs was comparedwith experimental results as well as those obtained from correlationdeveloped by Xin and Ebadian [25]. In addition, the ANN predictedfriction factor was compared with measured values and those obtainedfrom empirical correlation proposed by Ito [8]. For both cases, thesuperior performance of developed neural network was proved.

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