Heat Transfer Coefficient and Friction Factor

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Heat Transfer Coefficient and Friction Factor Prediction of Corrugated Tubes Combined With Twisted Tape Inserts Using Artificial Neural Network Mohammad Reza Jafari Nasr a; Ali Habibi Khalaj a a Petrochemical Research & Technology Company (NPC-RT), Affiliated of National Petrochemical Company, Tehran, Iran Online Publication Date: 01 January 2010

To cite this Article Nasr, Mohammad Reza Jafari and Khalaj, Ali Habibi(2010)'Heat Transfer Coefficient and Friction Factor Prediction

of Corrugated Tubes Combined With Twisted Tape Inserts Using Artificial Neural Network',Heat Transfer Engineering,31:1,59 — 69 To link to this Article: DOI: 10.1080/01457630903263440 URL: http://dx.doi.org/10.1080/01457630903263440

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Heat Transfer Coefficient and Friction Factor Prediction of Corrugated Tubes Combined With Twisted Tape Inserts Using Artificial Neural Network MOHAMMAD REZA JAFARI NASR and ALI HABIBI KHALAJ Petrochemical Research & Technology Company (NPC-RT), Affiliated of National Petrochemical Company, Tehran, Iran

In the research described here, artificial neural network (ANN) approach has been utilized to characterize the thermohydraulic behavior of corrugated tubes combined with twisted tape inserts in a turbulent flow regime. The experimental data sets were extracted from 57 tubes, 9 and 3 spirally corrugated tubes with varying geometries combined with 5 and 4 twisted tapes with different pitches. The tests were carried out with Reynolds numbers ranging from 3000 to 60,000. The experimental data sets have been utilized in training and validation of the ANN in order to predict the heat transfer coefficients and friction factors inside the corrugated tubes combined with twisted tape inserts, and the results were compared to the experimental data. The mean relative errors between the predicted results and experimental data were less than 2.9% for the heat transfer coefficients and less than 0.36% for the friction factor. The performance of the neural networks was found to be superior in comparison with the models correlated in the form of mathematical functions with their own assumptions. The results of this study suggested that ANN can be considered as a powerful tool and can be easily utilized to predict the performance of thermal systems in engineering applications.

INTRODUCTION

[1]. An extensive literature survey of research on all types of HTE techniques is given in Webb [2] and Bergles [3]. Generally, HTE can be classified in three broad categories:

Heat transfer enhancement (HTE) is typically considered as a means to improve and to intensify the thermal performance of heat transfer systems, such as heat exchangers, evaporators, thermal power plants, air-conditioning equipment, and refrigerators, and lately they have been applied widely in industrial applications. Recently, large numbers of attempts have been made to develop HTE techniques to reduce the size and costs of heat exchangers in order to improve their overall performance

(a) Active method: Active augmentation involves some external power input to bring about the desired flow modification for HTE and has not shown much potential owing to complexity in design. Furthermore, external power is not easy to provide in several applications. (b) Passive method: This method does not need any external power input, and the additional power needed to enhance the heat transfer is taken from the available power in the system. Tube insert devices including twisted tape inserts, wire coil inserts, extended surface inserts, and wire mesh inserts are considered the most important techniques of this group. (c) Compound method: A compound method is a hybrid method in which both active and passive methods are used in combination. The compound method involves complex designs and hence it has limited applications.

The authors would like to express their gratitude toward National Petrochemical Company-Research and Technology (NPC-RT) and Iranian Fuel Conservation Company (IFCO) without whose help and support this paper wouldn’t have been completed successfully. A simple word of thanks goes to Mr. Saeed Mousavi who kindly read and refined the final draft of this paper from linguistic point of view. Address correspondence to Mr. Ali Habibi Khalaj, National Petrochemical Company, Petrochemical Research & Technology Company, PO Box 14358, Tehran, Iran. E-mail: [email protected]

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M. R. JAFARI NASR AND A. H. KHALAJ

Surface roughness has been utilized to reduce the thickness of the boundary layer close to the surface and to introduce better fluid mixing. In contrast to small roughness elements and ribs, the twisted tape inserts cause alteration in the entire flow pattern, creating rotating and/or secondary flows [4]. In general, HTE is attributed to several mechanisms: increased flow path length, increased flow velocity/reduced hydraulic diameter, tape-generated swirl motion, and tape fin effects [5, 6]. It is well known that two or more of the existing techniques can be utilized simultaneously to produce an enhancement larger than that produced by one technique alone. The combination of different techniques acting simultaneously is known as compound enhancement. Interactions between different enhancement techniques contribute to greater values of heat transfer coefficients compared to the sum of the corresponding values for the individual techniques utilized alone [7, 8]. Preliminary studies on compound passive enhancement technique were very encouraging; some examples include rough tube with a twisted tape [9] and grooved rough tube with a twisted tape [10]. Recent articles [11, 12] reported on an experimental investigation to see whether or not heat transfer can be enhanced by the multiplicative effect of a corrugated tube combined with a twisted tape. A simple mathematical model to predict the friction factor and heat transfer coefficients for the case of a fully developed single-phase turbulent flow in a corrugated tube combined with a twisted tape insert has been developed by Zimparov [11, 12]. Despite the fact that comprehensive studies were conducted on a variety of corrugated tubes combined with twisted tapes, lack of sufficient knowledge about the flow mechanism does not permit us to predict the friction factors and heat transfer coefficients either by analytical methods or even by mathematical models requiring mathematical functional from assumptions, which limit their accuracy. To address these limitations, techniques that can effectively overcome the complexity of the problem without ad hoc assumptions are needed. One of these techniques is the artificial neural network (ANN), inspired by the biological network of neurons in the brain.

APPLICATION OF ANNS IN THERMAL SYSTEMS Although ANNs have flourished recently, their applications in the thermal science literature are limited. In this section, several articles are described in order to exemplify how ANNs can be implemented successfully in the heat transfer research area. The fundamentals of ANNs have been clearly explored by Zurada [13], Haykin [14], and Gupta et al. [15]. Thibault and Grandjean [16] were among the first authors to show the application of ANNs in heat transfer data analysis. They applied three-layer, feed-forward neural networks to solve these three different heat transfer problems: a thermocouple lookup table, a series of correlations between Nusselt and Rayleigh numbers for the free convection around horizonheat transfer engineering

tal smooth cylinders, and the problem of natural convection along slender vertical cylinders with variable surface heat flux. The authors concluded that neural networks can be utilized efficiently to model and correlate heat transfer data without the burden of finding appropriate model structures to fit experimental data. Sen and Yang [17] described the scope of ANNs and genetic algorithm techniques in thermal science applications including an exhaustive bibliography. The authors presented two interesting examples that used ANNs to predict the performance of compact heat exchangers. The first heat exchanger was a singlerow, fin-tube, cross-flow air-to-water type, and the second one was similar but with more tube rows, air-side condensation, and with variable fin spacing. The main purpose of that study was to compare the mathematical models of heat transfer with the ANNs model. The study also demonstrated a successful application of ANNs for transient analysis of first set of the two heat exchangers. Sen and Yang [17] proved that in both cases the ANN approach yields more accurate results. Sablani [18] developed an ANN model as a non-iterative procedure to calculate the Biot number/heat transfer coefficient in fluid–particle systems with time-dependent boundary conditions using measured fluid and particle temperatures. The model was able to predict Biot number satisfactorily, with a mean relative error (MRE) of less than 2%. Liu et al. [19] developed a multilayer feed-forward neural network (MLFNN) with back-propagation (BP) learning algorithm to evaluate and predict boiling HTE using additives. The effects of 30 additives tested by the authors and other researchers in the field of boiling heat transfer augmentation were analyzed with the ANN model. The results showed that the evaluation of all 30 additives was consistent with the experimental data. Training accuracy of their model was 100% and its prediction accuracy was over 90%. Therefore, they concluded that their model is reliable in order to predict the boiling HTE using additives. Islamoglu [20] utilized an MLFNN with BP learning algorithm to predict heat transfer rates of a wire-on-tube type heat exchanger that is widely utilized in small refrigeration systems. Islamoglu proposed a network with 12 input nodes, describing heat exchanger geometry and fluid flow rates, one output node corresponding to the heat flux, and one hidden layer with five nodes. Mean absolute relative error between experimental and ANN model was less than 3%. The results pointed out that the ANN approach could be applied in order to successfully predict the heat transfer rate. Ghajar et al. [21] utilized ANNs to significantly improve heat transfer correlations in the transition region for a circular tube with three different inlet configurations. The utilized network had five input nodes, one hidden layer with 11 nodes, and one output node. A separate training process was used for each tube inlet configuration. Zdaniuk et al. [22] correlated experimentally determined Colburn j -factors and Fanning friction factors for water flowing in straight tubes with internal helical fins using an ANN vol. 31 no. 1 2010



approach. The authors tested various network architectures and suggested that the feed-forward network with log-sigmoid node functions in the first layer and a linear node function in the output layer is the most advantageous architecture in order to predict helically finned tube performance. Sozbir and Ekmekci [23] utilized ANNs for the prediction of unsteady heat transfer in a rectangular duct in transient heat transfer. The authors employed an MLFNN with BP learning algorithm to predict the temperature distribution in a duct with a periodically varying inlet temperature in hydrodynamically developed and thermally developing unsteady laminar convection. The mean absolute relative error between experimental and ANN’s results was less than 39%. Xie et al. [24] applied ANN for heat transfer analysis of shelland-tube heat exchangers with segmental baffles or continuous helical baffles. In their experimental study three heat exchangers were used and some experimental data were obtained for training and testing the neural network configurations using BP algorithm. In their study, different network configurations were also used to aid the search for a relatively better network for prediction, and the maximum deviation between the predicted results and experimental data was less than 2%. Ermis et al. [25] applied an MLFNN with BP learning algorithm for heat transfer analysis of a phase-change process in a latent heat thermal energy storage system with finned tube. The authors verified the ANN model by comparing the numerical results and experimental data. They found that the ANN model can provide better agreement with the experimental data for both laminar and turbulent flows in heat storage systems. What is worthwhile is that their network had a mean absolute relative error of 5.58%, while the mean absolute relative error of numerical model was 14.99%. The authors finally concluded that the results obtained through ANN are more accurate than for the numerical model.

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processing elements (or nodes) that produce a dynamic response to external input or stimuli. ANNs were originally developed as approximations of the capabilities exhibited by biological neural systems, and they are based on a connectionist structure and mathematical functions that imitate the architecture and functions of the human brain [13–15]. The MLFNN with BP learning algorithm has already been applied to directly correlate the experimental data of heat transfer coefficients as a function of the working conditions among the different ANN architectures [16–25], and it seems to be very effective as a universal approximation of continuous functions in a compact domain. In the MLFNN architecture, as shown in Figure 1a, the input information is fed forward recursively to the higher hidden layers, and finally to the output layer, which is why the networks have also been called “propagation networks.” Since the input–output relationship of a MLFNN is described by static algebraic manipulations, the neural outputs are computed in a straightforward manner [15]. The BP training algorithm is an iterative gradient algorithm, designed to minimize the mean square error between the predicted output and the desired output. The BP algorithm of training a neuron is shown graphically in Figure 1b. The summarized BP algorithm [13–15] for training the ANN is depicted in the flow chart as shown in Figure 2. The BP algorithm, presented in Figure 2, has been summarized for the network with three layers of neurons: an input layer, a hidden layer, and an output layer. MATLAB software is utilized in order to implement the MLFNN with BP learning algorithm.

ARTIFICIAL NEURAL NETWORK DEVELOPMENT Algorithms for analytic computer codes in engineering systems are commonly complicated, because of involving the solution of complex differential equations. These programs usually require a large computer power and need a considerable amount of time in order to achieve accurate predictions. Despite having complex rules and mathematical routines, ANNs are able to learn the key information patterns within multidimensional information [26]. ANN is widely accepted as a technology offering an alternative way to tackle complex problems in actual situations. The advantage of ANN in comparison with other models is its ability to model a multivariable problem given by the complex relationships between the variables and its ability to extract the implicit nonlinear relationships among these variables by means of “learning” with training data [15]. An ANN is a computational structure, consisting of a number of highly interconnected heat transfer engineering

Figure 1 (a) The layout of the MLFNN architecture. (b) BP algorithm for the adaptation of the weights of the neuron (i, j ).

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Figure 3 Geometrical characteristics of a corrugated tube and twisted tape insert [7, 8].

the network, and weights were adjusted to minimize the error between the network output and the actual value.


EXPERIMENTAL DATA In this study a novel model of heat transfer coefficients and friction factor inside the corrugated tubes combined with twisted tape inserts has been developed by applying an ANN method in the extracted experimental data from the references [7, 8, 11, 12]. The geometrical characteristics of the corrugated tube and twisted tape inserts are shown in Figure 3. The characteristic parameters of the corrugated tubes are defined as pitch of corrugation p, height of corrugation e, spiral angle β, and dimensionless parameters e/Di , p/e, and βa . The twisted tape is comprised of a heat-treated brass tape with a thickness of 0.8 mm and a width of 12.5 mm. As shown in Table 1, in the experimental program, 9 (tubes 1–9) and 3 (tubes 10–12) spirally corrugated tubes with varying geometries combined with 5 and 4 twisted tapes with different pitches were used, respectively. The tests were carried out with Reynolds numbers ranging from 3000 to 60,000.

Figure 2 Flow chart of the BP learning algorithm.

In this study, two MLFNNs with the BP learning algorithm were utilized to predict the heat transfer coefficients and friction factors inside corrugated tubes combined with twisted tape inserts, and the results were compared to the experimental data. Input–output pairs, from experimental data, were presented to

RESULTS AND DISCUSSION The MLFNN has become the most popular among the various types of neural network for various applications, and the

Table 1 Characteristic dimension of the corrugated tubes and twisted tapes [7, 8, 11, 12] Tube number 1 2 3 4 5 6 7 8 9 10 11 12

Dimension of corrugated tubes Di [mm]

e [mm]

p [mm]

β [◦ ]

e/Di [—]

p/e [—]

βa [—]

Dimension of different twisted tapes: H /Di [—]

13.9 12.44 13.39 13.15 13.66 13.53 13.73 13.68 13.38 13.68 13.65 13.59

0.312 0.515 0.497 0.593 0.622 0.507 0.781 0.557 0.581 0.315 0.44 0.464

5.76 4.48 5.77 5.06 8.12 4.55 5.82 5.97 5.08 6.67 6.01 8.55

82.4 83.4 82.2 83 79.3 72.2 68 67.4 70.1 90 90 90

0.0224 0.0414 0.0371 0.0451 0.0456 0.0375 0.0569 0.0407 0.0434 0.023 0.0322 0.0341

18.46 8.7 11.61 8.53 13.05 8.97 7.45 10.73 8.74 21.17 13.66 18.43

0.916 0.927 0.913 0.922 0.881 0.802 0.755 0.749 0.779 1.000 1.000 1.000

15.11 - 12.09 - 7.63 - 5.76 - 4.75 16.88 - 13.50 - 8.52 - 6.43 - 5.23 15.68 - 12.54 - 7.91 - 5.97 - 4.85 15.97 - 12.77 - 8.06 - 6.08 - 4.94 15.38 - 12.31 - 7.77 - 5.86 - 4.76 15.52 - 12.42 - 7.83 - 5.91 - 4.80 15.30 - 12.24 - 7.72 - 5.83 - 4.73 15.35 - 12.28 - 7.75 - 5.85 - 4.75 15.69 - 12.55 - 7.92 - 5.98 - 4.86 15.35 - 12.28 - 7.75 - 5.85 15.38 - 12.30 - 7.76 - 5.86 15.45 - 12.36- 7.80 - 5. 89

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BP network is the most common technique for training the MLFNN. This is due to the fact that there is a mathematically strict learning scheme to train the network and guarantee the mapping between inputs and outputs [17]. In this study, an ANN modeling for prediction of heat transfer coefficients and friction factors inside corrugated tubes combined with twisted tape inserts has been utilized. Therefore, an MLFNN with BP is utilized for the training and validation processes. Since in training of the ANNs the variables of input–output data have different physical units and range sizes, it is advantageous to normalize all the input–output data by considering the largest and smallest values of each data set. Therefore, to avoid any computational difficulty, all of the input–output pairs were normalized between 0.1 and 0.9 to the restriction of sigmoid function. All experimental data can be normalized in order to use the values in this study through Actual value − Smallest value Largest value − Smallest value × (High value − Low value) + Low value

(1)

The smallest and largest values refer to the minimum and maximum data value, respectively. High and low values that are commonly utilized are 0.9 and 0.1, respectively [16–25]. The ANN utilized to predict the heat transfer coefficients contains five input parameters in the input layer, Re, e/Di , p/e, βa , and H /Di , while the output term in the output layer is considered as Nu/Pr0.4 , where Nu/Pr0.4 = Nu(Re, e/Di , p/e, βa , H /Di ). Similarly, the structure of ANN to predict the friction factor consists of five input parameters in the input layer, Re, e/Di , p/e, βa , and H /Di , while the output term in the output layer is considered as f , where f is a function of Re, e/Di , p/e, βa , and H /Di . In spite of the fact that corresponding networks contain the same input structure with a different output, the experiments were carried out with different Reynolds numbers for each model. Thus, although ANN models have the same input structure with the same dimensionless parameters, utilization of the different values and numbers of Reynolds number (544 and 570 different experimental data for heat transfer coefficients and friction factors modeling) varies the input of corresponding networks. Therefore, two different ANN models are utilized to characterize thermo-hydraulic behavior of corrugated tubes combined with twisted tape inserts. The weights, biases, and hidden node numbers are altered to minimize the error between the output values and the current data. In order to obtain the least error convergence, the configurations of the ANN are set by selecting the number of hidden layers and nodes. Model validation is the process by which the input vectors from input–output data sets on which the ANN was not trained are presented to the trained model, to see how well the trained model predicts output values of the corresponding data set. heat transfer engineering

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A sufficient number of data points should be used to guarantee good ANN training. There is still no formula to estimate the number of data points required to train a neural network. Depending on the complexity of the problem and the quality of the data, the number can vary greatly, but many neural networks have been trained successfully with a smaller number of data points. An optimization of the number of data sets that is really needed is still a challenge in the field of neural networks [15]. Although there is no formula to estimate the number of data points required to train a neural network, previous works [16–25] utilized different numbers of data sets for training the ANN (50–70% of the data set). Therefore, in this work 60% of all of the data set is utilized for training the networks. In total, 544 and 570 experimental samples with a Reynolds number ranging from 3000 to 60,000 were formed out of 57 tubes for heat transfer coefficients and friction factors modeling, respectively. These samples are divided into two data set groups; the first data group (60% of all of the data set for both heat transfer coefficients and friction factors) is utilized for training the networks and the second data group (remaining data) is utilized for verification of the ANN models. The performance of various ANN configurations was compared using MRE and standard deviations in the relative errors (STD). Moreover, the coefficient of determination, R 2 , of the linear regression line between the predicted values from the neural network model and the desired output was (absolute fraction of variance) utilized as a measure of performance. The three error-measuring parameters used to compare the performance of the various ANN configurations are: 1 ABS(Er) n i=1 n

MRE =

STD =

n (Er − Er)2 i=1 n−1 n

R = 1− 2

i=1 (yi n

− ai )2

(2)

(3)

(4)

(ai )2

i=1

where Er = (yi − ai )/ai

(5)

The parameter yi represents the predicted output from the neural network model for a given input, while ai is the desired output (i.e., exact data) from the same input that was produced by setup. vol. 31 no. 1 2010

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Table 2 Comparison of errors by different ANN configurations for prediction of heat transfer coefficients ANN configuration 5 5 5 5 5 5 5 5 5 5

31 71 11 1 13 1 14 1 13 1 13 3 13 5 13 6 13 5

1 1 1 1 21

Training error

Validation error

MRE (%)

STD (%)

MRE (%)

STD (%)

R2

8.7394 6.2603 5.0421 3.7707 4.1171 3.0183 2.5270 2.1788 1.8560 2.8436

11.5159 9.1197 7.4842 5.1449 6.7824 3.9991 3.7289 3.0097 2.6692 3.6791

8.5716 5.2924 4.3849 3.4343 3.7394 3.5129 3.3789 2.8013 3.4305 3.5425

11.0337 6.9116 5.5504 4.2890 4.6857 4.4335 4.4802 3.7483 4.3140 4.5124

0.9912 0.9917 0.9937 0.9942 0.9961 0.9947 0.9942 0.9966 0.9940 0.9941


Boldface type indicates chosen network for prediction of heat transfer coefficient.

Optimal ANN Configuration for Prediction of Heat Transfer Coefficients In this section, the selection method of an optimal ANN configuration for prediction of heat transfer coefficients of corrugated tubes combined with twisted tapes is described. Given the fact that the performance of an ANN depends on its architecture, in this study, more than 40 different ANN configurations have been studied for prediction of the heat transfer coefficients in order to find a relatively good configuration. The associated MRE, STD, and R 2 with ANN configurations in the training and validation processes are listed in Table 2. In order to illustrate the rate of change of the evaluation functions, and also to show the selecting process of the optimal network based on minimum value of corresponding statistical functions, some of the important networks are elaborated in Table 2. The 331 experimental points obtained from 57 tubes have been utilized in the training process, and the remaining 213 experimental points have been utilized in the validation process for generalization of the ANN model. In order to evaluate the optimal ANN configuration, prediction results of Table 2 are given for the training and validation data sets. Since the performance of the ANN versus validation data set is important for generalization, the optimal ANN configuration has been selected according to the validation errors. In the training and validation processes of the ANN, when the number of hidden nodes increased to 13, MRE for the validation data set became smaller than other configurations for the network with three-layer configuration. This indicates that adding more hidden nodes may not improve the predicted results. Thus, the ANN with configuration 5 13 1 (a network with 5 input neurons, 13 neurons in a single hidden layer, and 1 output neuron) is selected for adding another hidden layer. For four layers, when the number of hidden nodes in the second hidden layer is increased to five, MRE becomes smaller than heat transfer engineering

Figure 4 Comparison of predicted values using ANN and the experimental values for validation data set.

for other configurations. Thus, the ANN with configuration 5 13 5 1 is selected for adding another hidden layer. By doing so, MRE increases; therefore, the configuration with four layers has a higher accuracy of prediction than those with five layers. It should also be noted that adding more hidden layers may not improve the prediction. Therefore, in this case, the ANN with configuration 5 13 5 1 is selected for prediction, with the smallest MRE = 2.8013% and STD = 3.7483% and R 2 = .9966. Figure 4 shows the comparison of predicted values of ANN model and experimental values of system for validation data set. Figure 5 shows the error of each pattern in validation process. The performance of the ANN configuration for prediction of the heat transfer coefficients is depicted in Figure 6. As shown in Figure 6, nonlinear degree of the model increases when Nu/Pr0.4 becomes large, which increases the complexity of the model and consequently leads to an increase in the deviation. Figures 4, 5, and 6 imply that the heat transfer coefficients within a wide flow range (3000 ≤ Re ≤ 60,000) have been

Figure 5 Error percent of each sample in validation process.

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W4,3 = [ 1.72 ⎡


Figure 6 Scatter diagram showing the performance of ANN with configuration 5 13 5 1.

predicted with an MRE of less than 2.9% and an STD of less than 3.8% by means of the ANN method. They are highly comparable with the corresponding values given by the mathematical models proposed by Zimparov [8], which leads to a relative difference of less than ±10% for the 268 points, ±10–15% for 170 points, and ±15–20% for the remaining 106 points. The weight and threshold values (biases) of the ANN with configuration 5 13 5 1 for prediction of the heat transfer coefficients are given here: ⎡

W2,1

−0.93 ⎢ ⎢−0.50 ⎢ ⎢ 0.68 ⎢ ⎢ ⎢−0.16 ⎢ ⎢−1.26 ⎢ ⎢ ⎢ 0.24 ⎢ =⎢ ⎢ 0.71 ⎢ ⎢ 1.01 ⎢ ⎢ 0.19 ⎢ ⎢ ⎢ 0.87 ⎢ ⎢−0.48 ⎢ ⎢ ⎣−0.75 1.32 ⎡

W3,2

6.75 0.67

54.05 3.85

−14.95 22.94

−1.03 0.10

−3.28 0.14

−6.04 −4.96

2.58 2.23

7.35 8.97

20.64 8.61

−0.42

−6.83

−5.47

−3.55 −3.07

−0.48 1.37

−4.72 21.20

12.61 −0.86

14.74 6.69

−22.27 31.50

2.60 6.54

−0.92 2.20

3.77 16.91

−0.38 2.05

3.09 −2.21

3.72 9.51 9.10

6.44 0.11 10.81 −4.60 4.58 12.65

⎢ ⎢ −6.69 ⎢ =⎢ ⎢ 4.98 ⎢ ⎣ −1.47 −13.51

4.88 −2.29

⎤ −12.76 ⎥ −0.81⎥ ⎥ −0.18⎥ ⎥ ⎥ 0.25⎥ ⎥ 2.50⎥ ⎥ ⎥ 0.29⎥ ⎥ −1.68⎥ ⎥ ⎥ 3.71⎥ ⎥ −0.91⎥ ⎥ ⎥ −8.72⎥ ⎥ −8.82⎥ ⎥ ⎥ −3.30⎦ 0.90 0.12

65

−14.37

−1.53

14.81

−0.68

3.05]

⎤

⎥ ⎢ ⎢−22.72⎥ ⎥ ⎢ ⎢ 7.58⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 5.25⎥ ⎥ ⎢ ⎢−22.35⎥ ⎥ ⎢ ⎥ ⎢ ⎢−15.19⎥ ⎥ ⎢ ⎥ θ2 = ⎢ ⎢ 9.82⎥ ⎥ ⎢ ⎢ 3.33⎥ ⎥ ⎢ ⎢−17.97⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 8.31⎥ ⎥ ⎢ ⎢−25.97⎥ ⎥ ⎢ ⎥ ⎢ ⎣ −1.11⎦ −18.04

⎤ 2.89 ⎥ ⎢ ⎢−1.53⎥ ⎥ ⎢ ⎥ θ2 = ⎢ ⎢ 3.432⎥ ⎥ ⎢ ⎣ 3.14⎦ −9.57 ⎡

θ2 = [3.31]

Optimal ANN Configuration for Prediction of Friction Factor In this section, an optimal ANN configuration for prediction of the friction factor of corrugated tubes combined with twisted tapes is presented. As demonstrated in the previous section, the performance of an ANN depends on its architecture. Thus, more than 50 different ANN configurations have been studied for prediction of the friction factor. In addition, performances of the utilized networks have been evaluated with three statistical functions (MRE, STD, R 2 ). In order to show the rate of change of the evaluation functions, and also to show the selecting process of the optimal network, some of the important networks are shown in Table 3. Training and validation processes include 342 and 228 experimental data samples, respectively. By following the same procedure as described in the previous section, the network with 5 9 8 1 configuration with the smallest MRE = 0.3582% and STD = 0.6961% and R 2 = 1.0000 is the optimal ANN configuration for prediction of the friction factor inside corrugated tubes with twisted tape inserts. A comparison of predicted values using an ANN model and the experimental values of system for validation data set is depicted in Figure 7. The error of each sample in the validation process is shown in Figure 8. The performance of the ANN configuration to predict the friction factor is depicted in Figure 9.

0.81 −1.34 −0.45

−1.31 −0.06

−0.12 −0.42 −0.67

⎤

⎥ 14.45 −1.13 −1.86 13.78 −2.94 0.16 −12.10 20.61 1.06⎥ ⎥ 0.50 −3.80 −5.62 15.63 −13.02 0.59 −15.91 24.84 8.87⎥ ⎥ ⎥ 1.52 4.16 0.51 −0.98 −2.30 −0.12 −1.11 −3.00 −9.15⎦ 8.21 −5.63 5.07 0.39 2.27 0.14 0.63 4.75 1.18 5.87


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Table 3 Comparison of errors by different ANN configurations for prediction of friction factor Training error

Validation error

ANN configuration

MRE (%)

STD (%)

MRE (%)

STD (%)

R2

5 5 5 5 5 5 5 5 5 5

12.3920 5.2493 2.7329 1.3806 1.5822 1.1857 0.5244 0.3208 0.4295 0.5290

18.2001 6.7063 3.7507 1.9205 2.0350 1.5961 0.7594 0.5369 0.5986 0.7698

12.0692 5.0277 2.6968 1.3366 1.4941 1.1103 0.5416 0.3582 0.5409 0.5421

16.4467 6.2805 3.5960 1.7692 1.9416 1.4199 0.8422 0.6961 0.8701 0.8564

0.9121 0.9989 0.9996 0.9998 0.9994 0.9982 1.0000 1.0000 1.0000 1.0000

11 41 71 91 10 1 921 951 981 991 9821

Figure 7 Comparison of predicted values using ANN and the experimental values for validation data set.


Boldface type indicates chosen network for prediction of friction factor.

The friction factor within a wide flow range has been pre- W4,3 dicted with an MRE of less than 0.36% and an STD of less than 0.7% by means of ANN method, as shown graphically in = [−2.96 5.55 −10.63 −10.28 −9.36 4.28 −7.26 11.26] Figures 7, 8, and 9. They are comparable with the correspond⎤ ⎡ ing values that were predicted by mathematical model proposed −5.86 ⎤ ⎡ −1.35 by Zimparov [7]. Zimparov’s model led to a relative difference ⎥ ⎢ ⎢ 12.42⎥ ⎢−4.68⎥ of less than ±10% for 395 points, ±10–15% for 105 points, ⎥ ⎢ ⎥ ⎢ ⎢ 7.46⎥ ⎥ ⎢ and ±15–20% for the remaining 51 points. The weight and ⎥ ⎢ ⎢ 17.73⎥ ⎥ ⎢ ⎥ ⎢ threshold values (biases) of ANN with configuration 5 9 8 1 ⎢ 12.03⎥ ⎢ 0.34⎥ ⎥ ⎢ ⎥ ⎢ for prediction of friction factor are ⎥ θ2 = ⎢ ⎥ θ2 = [26.77] ⎢ −6.47⎥ θ2 = ⎢ ⎥ ⎢ −8.94 ⎥ ⎢ ⎥ ⎢ ⎢ 13.69⎥ ⎥ ⎢ ⎥ ⎢ ⎢ 1.66⎥ ⎢−14.26⎥ ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ 0.31 3.43 2.13 1.43 1.61 ⎣ 5.20⎦ ⎥ ⎢ ⎥ ⎢ ⎣ −2.36⎦ 1.13 −10.3 0.73⎥ ⎢−0.07 −6.86 −4.15 ⎥ ⎢ −5.72 ⎥ ⎢−0.01 9.60 −12.2 −6.66 −9.16⎥ ⎢ ⎥ ⎢ 2.45 −14.75 0.19⎥ ⎢−0.03 −0.15 Although maximum deviations between the ANN results and ⎥ ⎢ ⎥ 0.24 3.63 2.44 1.86 1.65 W2,1 = ⎢ experimental data for prediction of the heat transfer coefficients ⎥ ⎢ ⎥ ⎢ and friction factors are approximately ±10% and ±5%, respec⎢ 0.007 −4.99 −2.34 −6.97 −9.57⎥ ⎥ ⎢ tively, performances of the ANN models are evaluated based on ⎢ 0.03 −12.13 −10.38 30.02 3.24⎥ ⎥ ⎢ the statistical evaluation functions presented above (correlations ⎥ ⎢ 1.81 0.92 −1.47 0.008⎦ ⎣ 2.52 2–4), as commonly utilized [16–24]. 0.04

14.53

3.053 −2.803 −0.213

⎡

W3,2

⎤ 1.33 0.22 2.19 1.08 −1.50 −2.33 −1.23 −0.02 −0.27 ⎢ 7.51 −11.26 1.55 1.45 −10.22 −4.05 8.13 −1.74 −17.30⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1.25 −0.08 −2.64 1.46 −1.72 2.90 1.67 0.08 −16.84⎥ ⎢ ⎥ ⎢ 0.92 −0.70 1.41 3.01 −1.25 1.60 0.68 0.91 5.08⎥ ⎢ ⎥ =⎢ ⎥ ⎢−0.32 −0.30 0.06 1.23 0.72 −0.13 0.33 −10.56 0.69⎥ ⎢ ⎥ ⎢−1.80 −0.89 −1.27 0.64 2.32 −1.13 −1.20 0.11 −3.02⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 0.47 −0.18 −0.43 0.61 −0.53 −0.44 0.34 −0.10 −5.53⎦ 1.60

−0.12 −1.75 2.10

−2.13

1.81

1.70


0.11

4.44

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learning algorithm were utilized to train and generalize the networks to predict the thermal and hydraulic performances. The performances of the ANNs were evaluated with the experimental data, and the results pointed out that the ANNs approach is able to learn the training data set in order to predict the output of unseen validation data set accurately. Also, performances of the ANNs proved their superiority over the mathematical models formed with assumptions. Since ANNs are able to learn the key information patterns within a multidimensional information domain, it is recommended to utilize the ANNs to simulate the behavior of the thermal systems. NOMENCLATURE


Figure 8 Error percent of each sample in validation process.

ai ANN BP Di e f fh ( ) fk ( )

Figure 9 Scatter diagram showing the performance of ANN with configuration 5 9 8 1.

The errors associated with the models presented by Zimparov [7, 8] for prediction of friction factors and heat transfer coefficients are higher than the errors associated with the ANN approach to correlate the database. Figures 4–9 clearly show that the corresponding networks work very well in order to predict the heat transfer coefficients and friction factor inside corrugated tubes combined with twisted tape inserts. Based on this visual inspection and the low error values shown in Tables 2 and 3, it can be concluded that given the experimental uncertainty, the prediction error associated with these ANNs can be safely ignored. The current results show that ANNs perform extremely well, and they are able to learn the training data set and accurately predict the output of an unseen validation data set. Thus, using the ANN to predict the performance of thermal systems in engineering applications is recommended.

h hi H HTE k MLFNN MRE n Nu p Pr R2 Re S STD T t U Wj i Wkj X Y yi

CONCLUSIONS In this article, the ANNs have been applied to predict friction factors and heat transfer coefficients inside corrugated tubes combined with twisted tape inserts. Two MLFNNs with BP heat transfer engineering

experimental value artificial neural network back propagation tube diameter (m) height of corrugation (m), or error in Figure 1 Fanning friction factor (2τW /(ρU2m )) (dimensionless) logistic sigmoid activation function from input layer to hidden layer logistic sigmoid activation function from hidden layer to output layer vector of hidden-layer neurons heat transfer coefficient (W/m2 K) pitch of the twisted tape (m) heat transfer enhancement thermal conductivity (W/m K) multilayer feed-forward neural network mean relative error number of the data points Nusselt number (hi Di /k) (dimensionless) Pitch of corrugation (m) Prandtl number Absolute fraction of variance Reynolds number (Um Di /v) (dimensionless) output of linear combiner in each neuron standard deviations in the relative errors target activation of output layer time period axial velocity component (m/s) weights connecting the ith input node to the j th hidden layer node weights connecting the j th hidden layer node to the kth output layer input value of ANN output value of ANN predicted output of ANN for a given input

Greek Symbols α

learning rate vol. 31 no. 1 2010

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β βa δj δk η θj θk ν ρ τw


spiral angle (◦ ) β/90 vector of errors for each hidden layer neuron vector of errors for each output neuron momentum factor threshold between the input and hidden layers threshold connecting the hidden and output layers kinematic viscosity (m2 /s) fluid density (kg m−3 ) shear stress (Pa)

[11]

[12]

[13] [14]

Subscripts


[15]

h i j k m n

hidden layer input node hidden layer node output layer node number of layers, mean value in correlation of Fanning friction factor number of the data

[16]

[17]

[18]

REFERENCES [1] Jafari Nasr, M. R., and Polley, G. T., Should You Use Enhanced Tubes? Chemical Engineering Progress, April, pp. 44–50, 2002. [2] Webb, R. L., Principles of Enhancement Heat Transfer, John Wiley & Sons, New York, 1994. [3] Bergles, A. E., Techniques to Augment Heat Transfer, In Handbook of Heat Transfer Application, ed. W. M. Rosenhow, McGraw-Hill, New York, pp. 3–80, 1985. [4] Smithberg, E., and Landis, F., Friction and Forced Convection Heat Transfer Characteristics in Tubes With Twisted Tape Swirl Generators, Transactions of the ASME, Journal of Heat Transfer, vol. 86, pp. 39–49, 1964. [5] Manglik, R. M., and Bergles, A. E., Heat Transfer Enhancement and Pressure Drop in Viscous Liquid Flows in Isothermal Tubes With Twisted-Tape Inserts, Warme. Stoffubertrag, vol. 27, pp. 249–257, 1992. [6] Manglik, R. M., and Bergles, A. E., Heat Transfer and Pressure Drop Correlations for Twisted-Tape Inserts in Isothermal Tubes, Part II, Transition and Turbulent Flows, Transactions of the ASME, Journal of Heat Transfer, vol. 115, pp. 890–896, 1993. [7] Zimparov, V. D., Prediction of Friction Factors and Heat Transfer Coefficients for Turbulent Flow in Corrugated Tubes Combined With Twisted Tape Inserts. Part 1: Friction Factors, International Journal of Heat and Mass Transfer, vol. 47, pp. 589–599, 2004. [8] Zimparov, V. D., Prediction of Friction Factors and Heat Transfer Coefficients for Turbulent Flow in Corrugated Tubes Combined With Twisted Tape Inserts. Part 2: Heat Transfer Coefficients, International Journal of Heat and Mass Transfer, vol. 47, pp. 385–393, 2004. [9] Bergles, A. E., Lee, R. A., and Mikic, B. B., Heat Transfer in Rough Tubes With Tape-Generated Swirl Flow, Transactions of the ASME, Journal of Heat Transfer, vol. 91, pp. 443–445, 1969. [10] Usui, H., Sano, Y., Iwashita, K., and Isozaki, A., Enhancement of Heat Transfer by a Combination of an Internally Grooved Rough


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Ali Habibi Khalaj is working as an energy committee director of Petrochemical Research and Technology Company with the aim of identifying and supervising the core business projects in order to conserve energy in the petrochemical complexes. He received his master’s degree from K. N. Toosi University of Technology, Tehran, Iran, in the Energy System Engineering Division in 2008. His main research interests include energy conservation in oil refineries and petrochemical complexes, optimization of thermal systems, and applications of artificial intelligence in oil, gas and petrochemical industries.


Mohammad Reza Jafari Nasr is working as an R&T director of Petrochemical Research and Technology Company as an affiliated company of National Iranian Petrochemical Company and also as a lecturer for university students and the plant engineers at oil refineries and petrochemical complexes. He has more than 20 years of experience in research and development of pilot plant design particularly for gas and petrochemical processing. In the area of research his major interests are design and optimization of thermal equipment, energy integration of oil, gas and petrochemical industries, and also gas conversion and gas processing. He published more than 80 papers in national and international journals and presented many oral lectures for the related conferences.


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