A. rahman, zhang 2018 prediction of oscillatory heat transfer coefficient for a thermoacoustic heat exchanger through artificial neural

1. Prediction of oscillatory heat transfer coefficient for a thermoacoustic heat exchanger through artificial neural network technique Anas A. Rahman, Xiaoqing Zhang ⇑ Department of Refrigeration & Cryogenics, School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China a r t i c l e i n f o Article history: Received 18 January 2018 Received in revised form 14 March 2018 Accepted 8 April 2018 Available online 24 April 2018 Keywords: Thermoacoustic heat exchanger Oscillatory heat transfer Artificial neural network Standing wave thermoacoustic refrigerator Oscillating frequency Mean pressure a b s t r a c t Heat exchangers under oscillatory flow condition in thermoacoustic devices are quite different with the traditional ones in heat transfer and flow behavior of thermo-viscous fluid. As a result, one cannot directly apply the heat transfer correlations for the steady flow to design thermoacoustic heat exchangers, otherwise, significant deviation will arise. However, some correlations of heat transfer for the oscillatory flow have not been well established yet. This study involves the application of artificial neural network (ANN) as a new approach to predict oscillatory heat transfer coefficient of one thermoacoustic heat exchanger under some operating conditions. One ANN model for the oscillatory heat exchanger used in one standing wave thermoacoustic refrigerator has been developed based on the published experimental data. This proposed ANN model has three layers with the configuration of 2-10-1, namely one input layer with two neurons representing two operating parameters, oscillating frequency and mean pressure, one hidden layer with optimal ten hidden neurons and one output layer with one neuron representing the oscillatory heat transfer coefficient as response. Moreover, a statistical analysis has been provided for studying the influence strength of these two input parameters on the oscillatory heat transfer coefficient. This ANN model had been proven to be desirable in accuracy for predicting oscillatory heat transfer coefficient by comparing ANN model results with both experimental results and calculated results by several other correlations from the published literature at the same operating conditions. This research work provides a new and accurate modeling approach based on ANN technique for the research of thermoacoustic heat exchangers and solving heat transfer problems related with oscillatory flow condition. Ó 2018 Elsevier Ltd. All rights reserved. 1. Introduction Thermoacoustic technology deals with the conversion between thermal energy and acoustic energy (a kind of mechanical work) utilizing thermoacoustic effect, underlying which thermoacoustic devices can be developed to generate electricity or pump heat [1,2], with special advantages including using environmentally- friendly working substance such as air or inert gases, fewer or no moving parts and low maintenance, as well as being capable of using low grade thermal energy which will provide new opportu- nities for energy conservation [3]. Oscillating flow is a typical operating condition and physical feature encountered in thermacoustic devices where the gas parcels or liquids follow a periodic oscillatory motion, and a heat exchanger under oscillatory flow has been considered as one of the crucial components affecting the whole performance of thermoacoustic devices besides the thermoacoustic stack or regenerator. With the advancement of research and development of thermoacoustic technology, the heat transfer in oscillatory heat exchangers at two ends of stack/regenerator has been found to become a bottle neck to transfer the thermoacoustic heat flow in thermoacoustic devices. However, the fundamental theory and some practical correlations for the oscillatory heat transfer have not been well established yet. Therefore, the modeling, theoretical and experimental research concerning oscillatory flow and heat transfer is a necessary and urgent task for some applications under oscillatory flow environment, such as thermoacoustic engines or refrigerators, as well as the heat transfer enhancements using oscillating flow. Typical design procedures for traditional compact heat exchangers generally refer to the heat transfer theory for a steady and unidirectional flow of fluid while thermoacoustic heat exchangers need to be designed based on oscillatory flow condition usually with zero mean velocity. The heat transfer characteristics under an oscillatory flow would be significantly different from those for unidirectional steady flow, due to the following facts that oscillating flow has two thermal entrance regions which make it enhance heat transfer [4]; Another hypothesis for the oscillatory https://doi.org/10.1016/j.ijheatmasstransfer.2018.04.035 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: zhangxq@mail.hust.edu.cn (X. Zhang). International Journal of Heat and Mass Transfer 124 (2018) 1088–1096 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

2. flow is that the entrance length in a laminar reciprocating flow would be shorter than that in unidirectional steady flow; Addition- ally, the velocity profiles in an oscillatory flow, which nearly do not change in the entrance region, tend to be flatter than those of a steady flow [5]. These phenomena have not been fully understood yet due to the scarcity of experimental data and some complexities in mathematical modeling [4]. Because of these differences above, conventional heat transfer correlations for the steady flow cannot be directly applied to estimate the heat transfer coefficient between gas and solid walls for thermoacoustic heat exchangers, otherwise significant deviation in accuracy will arise [6]. In the present study, one artificial neural network (ANN) as a new approach was applied to predict oscillatory heat transfer coefficient for one thermoacoustic heat exchanger. Oscillating frequency and mean pressure are considered as two important factors influencing oscillatory heat transfer which will be investigated in this paper. One ANN model is considered a purely data driven model, and it learns from examples where inputs and outputs are introduced to the network sequentially and repeatedly. The computation in ANN model is distributed over simple several units called neurons which are interconnected and operated in parallel. Hence, it is a simple, rapid and accurate model. In recent years, ANN technique as efficient problem-solving paradigms has been applied to many science and engineering fields, because of its significant advantages in some aspects, such as recognizing and learning the underlying relations between inputs and outputs with no need for any explicit physical relation, regardless of the problem dimensionality and system nonlinearity; and the high tolerance to data containing noise and measurement errors due to the distributed processing within the network [7,8]. On the other hand, neural networks, in particular the multi-layer feed forward neural networks, have become an effective alternative to more traditional statistical techniques for function approximation and data fitting purposes. More- over, unlike other statistical techniques, the multi-layer networks make no prior assumptions concerning the data distribution as they can highly model non-linear functions through accurate input-output data mapping and hence can be generalized for new unseen data. These advantageous features make them attrac- tive alternatives to statistical approaches and also to numerical models. Actually, in the last two decades, it was found from the review of literature that artificial neural networks had been widely applied to solve many complex thermal science problems [9], such as heat transfer enhancement, multi-phase flow and phase change problems in the evaporators and condensers for traditional refrigerators and air conditioning systems, as well as compact heat exchangers used in many industrial applications. Recently, the research of oscillatory heat exchangers has attracted more attention where many studies on oscillatory flows were focused on the understanding of fluid flow and heat transfer mechanisms [1,10–13]. More recently, researchers had proposed different approaches to overcome the limitations of steady flow and characterize the heat transfer of an oscillatory heat exchanger. Particularly, some experimental work [14–17], empirical correlations [6,11,14], analytical models [1,18] and numerical studies [19–21] were carried out for hydrodynamic analyses and thermal calculations for studying the thermal behavior and measuring the performance of the heat exchanger based on oscillatory flow condition instead of using steady flow approaches in the design of thermoacoustic devices [4]. It is noteworthy that the oscillatory heat transfer is a complex process in thermoacoustic devices, such as, a standing-wave thermoacoustic refrigerator, where the heat transfer occurs between gas particles with high-amplitude acoustic standing waves and solid materials within heat exchangers located at each end of the stack, and dynamic pressure, velocity and temperature changes in the acoustic standing wave would cause a net thermoacoustic heat flow up a mean temperature gradient along the direction of sound wave propagation in the thermoacoustic core. Accordingly, the performance of heat exchangers at the two ends of stack/regenerator is considered a significant limiting factor for thermoacoustic heat flow and thus the performances of thermoacoustic systems. Hence, some researchers focused on the oscillatory heat transfer of a finned heat exchanger as an ambient heat exchanger either for thermoacoustic refrigerators [14] or pulse tube refrigerators [6] in order to estimate the Nusselt number (Nu) or oscillatory heat transfer coefficient (h) as a measure for the performance of ambient heat exchanger, subjected to different operating conditions (parameters) such as oscillating frequency and mean pressure. The experimental work conducted in [6,14] was correlated in terms of Nusselt number, Reynolds number and either Prandtl number or Valensi number in order to obtain some correlations for the heat transfer coefficient of the ambient heat exchanger. All these researches above undoubtedly provide some useful correlations as well as theoretical fundamentals of the oscillatory heat transfer for the development of thermoacoustic devices. At the same time, the results from these existing correlations with some uncertainties and lack of generalization stimulate our research interests to explore a new approach to investigate the complex oscillatory heat transfer for thermoacoustic applications. One related work [22] was to predict the Nusselt number through ANN for one oscillating annular flow with reference to four parameters, namely, kinetic Reynolds number, dimensionless amplitude, filling heights, and heat flux. The network structure in [22] was optimized to 4-5-1 configuration, and the network predictions of Nusselt number by ANN were closer to the experimental values with a deviation of about 5%. After the successful application mentioned above to estimate the cycle-averaged Nusselt number in an annular channel subjected to oscillating flow, ANN is confirmed as a promising approach for engineer’s preliminary estimation of complex heat transfer problems in the present study. Hence, one artificial neural network model will be introduced to map the relationship between the operating parameters and performance of thermoacoustic heat exchanger, and thus to provide a more accurate and practical approach to predict the response of oscillatory heat transfer coefficient to the oscillating frequency and mean pressure for one typical standing wave thermoacoustic refrigerator conducted in [14]. 2. Artificial neural network model for predicting oscillatory heat transfer coefficient of one thermoacoustic heat exchanger 2.1. Physical model For comparison, the experimental setup system with the ambient heat exchanger of the finned-tube structure in [14], was used as the physical model in the present research. It is a standing wave thermoacoustic refrigerator with a k/4 resonator tube, driven by an acoustic driver at one end, and closed rigidly with a cone-shaped Acoustic driver Stack Ambient heat exchanger Cold heat exchanger Cone-shaped buffer volume Resonator Fig. 1. Schematic diagram of a standing wave thermoacoustic refrigerator with a k/ 4 resonator tube. A. A. Rahman, X. Zhang / International Journal of Heat and Mass Transfer 124 (2018) 1088–1096 1089

3. buffer volume at the other end as shown in Fig. 1. Helium gas was used as the working substance. The major components are stack, heat exchangers, resonator containing the working gas and acoustic driver. The resonator was constructed from aluminum tube but with plastic tube at the inner diameter portion to reduce heat loss by conduction. The stack was fabricated from a thermoplastic material to reduce thermal conduction along the stack plates [14]. The function of heat exchangers in the thermoacoustic system is to transfer time-averaged heat to/from the oscillating gas from/ to an external heat source or sink. The finned-tube heat exchanger as a thermal sink presented in [14] had been used as the physical model in the present study (called as ambient heat exchanger here) as shown in Fig. 2. In this heat exchanger, water flows in the tube, which forms the circumference and the working gas oscillates across the fins outside the tube. This heat exchanger was located at the ambient temperature side of the stack whilst another electri- cal resistance heater arrangement was located at the cold side of the stack to supply the variable load for the refrigerating system. 2.2. Correlations for oscillatory heat transfer coefficient of the ambient heat exchanger Nsofor et al. [14] conducted one experimental work on the heat transfer for such ambient heat exchanger of the thermoacoustic refrigerator mentioned above. Based on 24 experimental measure- ments, they obtained one correlation in terms of Nusselt number Nu, Prandtl number Pr and Reynolds number Re under different operating conditions of frequency and mean pressure as given by [14] Nu ¼ hdh k ¼ 0:61Re0:31 rms Pr0:11 ð1Þ where dh is the hydraulic diameter representing the flow gap formed by the fin spacing in the heat exchanger while k is the thermal conductivity of the working fluid. The root mean square Rey- nolds number Rerms is defined as, Rerms ¼ Remax= ffiffiffi 2 p , where Remax is the maximum Reynolds number. Hence, Eq. (1) can be expressed as Nu ¼ hdh k ¼ 0:548Re0:31 maxPr0:11 : ð2Þ Various approaches such as Eq. (2) were obtained from many literature research [1,6,11]. Swift [1] suggested an expression for Nusselt number based on a boundary layer conduction heat transfer model and can be expressed as Nu ¼ hdh k ¼ 1 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Va Á Pr p ð3Þ where h % 1ffiffi 2 p k dk and dk is the thermal penetration depth of the working fluid. The correlation of Swift shows the effect of Valensi number (Va) including frequency on oscillatory heat transfer where Va ¼ xd2 h m and both x and m are the angular frequency and kinematic viscosity of the working fluid, respectively. In Swift’s correlation, the effect of maximum Reynolds number (Remax) was ignored compared with Nsofor’s correlation mentioned previously. Zhao and Cheng [11] correlated the Nusselt number to both the kinetic Reynolds number (i.e., Valensi number, Va) and the dimensionless oscillation amplitude of fluid corresponding to Remax. Their correlation was developed for a pipe subjected to a laminar reciprocating flow and was given by Nu ¼ hdh k ¼ 0:036Re0:85 maxVaÀ0:27 : ð4Þ Tang et al. [6] developed a new correlation based on the work of Zhao and Cheng [11] mentioned previously. For Tang’s correlation, a heat exchanger similar to that used by Nsofor et al. [14] was tested where Nusselt number was a function of both Remax (corresponding to the velocity amplitude of the gas in fin spacing) and the Va (corresponding to the oscillating frequency). Remax and Va were varied in the range of 200 < Remax < 1200 and 150 < Va < 350, and the correlation obtained was expressed as Nu ¼ hdh k ¼ 0:43Re0:0876 max Va0:405 : ð5Þ In the present study, the oscillatory heat transfer coefficient from Nsofor’s correlation in Eq. (2) will be compared with the results from ANN model and other correlations mentioned above in Eqs. (3)–(5). 2.3. ANN model 2.3.1. Data preparation In our research work, the experimental values for the two operating parameters (frequency and mean pressure) and the response representing the oscillatory heat transfer coefficient, were extracted from the experimental work conducted in [14]. It is noticed that 24 data samples were used in our present research for studying the effect of frequency and mean pressure on the oscillatory heat transfer coefficient. The data sample values of frequency range from 300 Hz to 450 Hz while the values of mean pressure range from 3.03 bar to 8.1 bar. It can be noticed from related work [23] that there is an optimal frequency to achieve the maximal oscillating heat transfer coefficient for a given mean pressure. At the same time, the mean pressure has also some influence on the oscillatory heat transfer coefficient. Therefore, both frequency and mean pressure are considered two of the most important parameters influencing the oscillatory heat transfer coefficient at the ambient heat exchanger of thermoacoustic refrigerator, and thus were determined as the input data in our ANN model. 2.3.2. Determination of samples, training algorithm and structure for ANN model Sufficient training sample size and representation of the training sample for the environment of interest are the first and most important step in designing of ANN model. In our ANN model, the frequency and mean pressure were considered as the input parameters as studied in [14], whilst the output parameter was the oscillatory convective heat transfer coefficient h as one measure of ambient heat exchanger performance which needs to be predicted by ANN model.Fig. 2. Finned-tube ambient heat exchanger. 1090 A. A. Rahman, X. Zhang / International Journal of Heat and Mass Transfer 124 (2018) 1088–1096

4. For the sake of convenience of comparison, the 24 experiments from [14] as shown in Table 1 would be used as the data samples for building the ANN model where only oscillating frequency and mean pressure were mainly taken into account as the input parameters for a determined structure of heat exchanger in the present study. The geometrical parameters of heat exchanger are also important parameters influencing the oscillatory heat transfer, but were not considered in the present study due to unavailable experimental data for comparison. Of course, ANN model including the geometrical parameters will be investigated in the analyses and optimization design of oscillatory heat exchanger in our future research. For our ANN model, the 24 data samples were divided into two groups. The first group includes 20 data samples for the purpose of building and learning the ANN model to predict the oscillatory heat transfer coefficient responding to the operating parameters. The second group including the remaining 4 data samples, were used for the purpose of verifying the prediction ability of our ANN model to recognize the heat transfer coefficient with new operating parameters (not used previously in building the model), as well as comparing these predicted results by ANN model with corresponding results from other correlations given by Eqs. (2)–(5). Optimization of the neural network aims at minimization of the objective function (i.e., error function) during the training process through tuning the values of weights and biases of the network. In our case the mean square error (MSE) is commonly used as the objective function for feed-forward neural networks which gives the average squared error between the network outputs (o) and the target (actual) outputs (t) as given: MSE ¼ 1 N XN i¼1 ðeiÞ2 ¼ 1 N XN i¼1 ðti À oiÞ2 ð6Þ The optimization of the network performance is often subjected to many constraints such as the structure of neural network, the amount of both training and testing data sets, learning parameters including both of learning rate and momentum coefficient, as well as the type of activation functions. Moreover, the appropriate training algorithm needs to be decided for the optimization process in order to accurately obtain the optimum weights and biases in the direction of the global minimum of the error function. Hence, these constraints mentioned above should be determined carefully for a successful neural network. In the present study, some of these constraints such as learning parameters, were taken as the default values in the Neural Network Toolbox while the others were varied or decided based on suitability for the application such as the training algorithm, and activation functions for hidden and output neurons. For convenience, the training and testing data sets were taken with the 85%:15% ratio noting that the training data sets include the validation data sets. Specifically, the 20 data samples in the first group used for learning, were divided into three data sets: 70% for training, 15% for validation and the remaining 15% for testing. The training data set is used only for learning (i.e., to fit the weights of the network) whilst the validation data set is used to minimize over-fitting problem that may occur in the training process and the test data set is used only to assess the generalization performance of the trained neural network. These three data sets were randomized and then introduced sequentially to our model after the determination of suitable training algorithm and network structure. A typical feed-forward neural network structure with back- propagation (BP) algorithm as shown in Fig. 3 was adopted in the present study. It is most commonly used in many researches [7,8], and includes multiple layers, namely input, hidden and output layers with specific number of neurons in each layer, initial weights and biases as well as neuron functions. The neuron in each layer performs two functions, summation and activation functions respectively to sum the weighted inputs then squash this summation to produce the output. The information is fed and flows between the neurons in the forward direction where each layer receives signals from one neuron and passes its output to another subsequent layer. In BP algorithm, the cumulative network error between its final output and the actual one is back-propagated to adjust the weights values in the whole network mechanism as shown in Fig. 3. Structurally, one hidden layer is commonly adopted in majority of applications [24,25] but the optimum number of neurons required in the hidden layer is a problem dependent. This number Table 1 Experimental and correlated oscillatory heat transfer coefficient versus frequency (f) and mean pressure (pm) (Data based on Ref. [14]). No. Frequency [Hz] Mean pressure [Â 105 Pa] For Nsofor’s correlation hexp [W/m2 K] Remax Pr Nu hcorr [W/m2 K] 1 300 3.03 32.09727 0.663 1.535155 83.01207 90 2 350 3.03 35.24971 0.663 1.580394 85.45832 79 3 400 3.03 37.77411 0.663 1.614646 87.31047 93 4 450 3.03 40.07154 0.663 1.644471 88.92325 101.5 5 300 4.053 37.56731 0.663 1.6119 87.16202 74 6 350 4.053 41.72641 0.663 1.665231 90.04582 71.5 7 400 4.053 44.33077 0.663 1.696781 91.75184 81.5 8 450 4.053 45.56055 0.663 1.711235 92.53344 106.67 9 300 5.066 42.93195 0.663 1.679999 90.8444 102.5 10 350 5.066 46.41834 0.663 1.721158 93.07005 72 11 400 5.066 49.45259 0.663 1.755277 94.91497 88.5 12 450 5.066 52.59304 0.663 1.789101 96.74397 91 13 300 6.079 45.6137 0.663 1.711853 92.56689 80 14 350 6.079 50.06758 0.663 1.762015 95.27933 83 15 400 6.079 53.62152 0.663 1.799874 97.32654 77 16 450 6.079 56.27605 0.663 1.827037 98.79534 114 17 300 7.093 50.09152 0.663 1.762276 95.29345 89 18 350 7.093 54.76785 0.663 1.811716 97.96684 88 19 400 7.093 58.39474 0.663 1.848089 99.93371 89.5 20 450 7.093 60.97289 0.663 1.873007 101.2811 84 21 300 8.1 53.62787 0.663 1.79994 97.33011 104 22 350 8.1 58.37374 0.663 1.847883 99.92256 92 23 400 8.1 61.92188 0.663 1.881996 101.7672 95 24 450 8.1 66.41561 0.663 1.923317 104.0016 107.5 A. A. Rahman, X. Zhang / International Journal of Heat and Mass Transfer 124 (2018) 1088–1096 1091

5. depends on the complexity of input and output mapping, the amount of noise in the data and the amount of training data available. If the number of neurons in the hidden layer is too small, the back-propagation algorithm will fail to converge to a minimum error during training. Conversely, too many neurons will result in network over-fitting training data which will lead to poor generalization performance [26]. Hence, the number of neurons in the hidden layer needs to be accurately optimized. In order to find the optimum number of neurons in the hidden layer, sensitivity analysis of the addressed model versus number of hidden neurons should be demonstrated. For the present study, the number of hidden neurons was changed iteratively from 2 to 20 with a gradual step of 2. Then, during training the network, the mean square error (MSE) of the network was checked. This mean square error obtained from Eq. (6) evaluates the network performance through showing how close or far the network outputs are with respect to the actual ones. Fig. 4 illustrates that the minimum mean square error was achieved at 10 hidden neurons. On the other hand, two indicators were used to confirm the optimum number of hidden neurons including (i) the correlation coefficient (R2 ) between targets (actual outputs) and predicted outputs from the network as shown in Eq. (7), and (ii) the average prediction error when introducing new inputs to the trained network as illustrated in Eqs. (8) and (9). R2 ¼ 1 À Pn i ðoi À tiÞ2 Pn i ðtiÞ2 ! ð7Þ 0 2 4 6 8 10 12 14 16 18 20 22 0.84 0.86 0.88 0.90 0.92 0.94 0.96 Correlationcoefficient(R 2 ) Number of hidden neurons Fig. 5. Variation of correlation coefficient between network and actual outputs versus number of hidden neurons. Fig. 6. Average prediction error of oscillatory heat transfer coefficient [%] versus number of hidden neurons. 0 2 4 6 8 10 12 14 16 18 20 22 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Meansquareerror(MSE) Number of hidden neurons Fig. 4. Sensitivity of the mean square error (MSE) for the network model versus number of hidden neurons. ∙∙ ∙ Error back propagation and weight updating mechanism(BPL) Input layer ∙ Hidden layer Output layer Cummulative network error Set of target output Fig. 3. A multi-layer feed-forward ANN with a back-propagation algorithm. 1092 A. A. Rahman, X. Zhang / International Journal of Heat and Mass Transfer 124 (2018) 1088–1096

6. Prediction error % ¼ Predicted result À Experimental result Experimental result

14. ð8Þ Average prediction error % ¼ Pn i¼1 Prediction error % n ð9Þ In general, the R2 values varies between 0 and +1, where the values of R2 close to +1 indicate a robust positive linear correlation between the network outputs and targets (actual outputs), while the values near to 0 indicate a very weak correlation [27,28]. Fig. 5 illustrates that the maximum R2 was obtained at 10 hidden neurons as the optimum value. Finally, new unseen data of frequency and mean pressure were introduced to these different network structures for verifying their prediction ability. As shown in Fig. 6, the average prediction error of oscillatory heat transfer coefficient was minimized in the network structure with 10 hidden neurons. Hence, the structure of our network used in the present study is 2-10-1 as shown in Fig. 7, namely 2 neurons representing oscillating frequency and mean pressure in the input layer, optimum 10 hidden neurons in the single hidden layer and one neuron representing the oscillatory heat transfer coefficient in the output layer. 3. Results and discussions In order to validate the performance of established network, regression plots for the network outputs were presented in Fig. 8 with respect to the targets (actual outputs) in the phases of training, validation and testing. These plots indicated a very good fit between the network results and actual outputs due to higher values of regression (R). For further validation on the network performance, an error histogram plot illustrated in Fig. 9 was introduced to show how the error sizes are reasonably distributed. Typically, when most errors are near zero, this indicates a better trained model. This histogram presents an indication of outliers i.e., the data points where the fit is significantly worse than the majority of data. In the present case as shown in Fig. 9, the majority of errors in training, validation and Fig. 8. Regression plots for training, testing and validation phases. Input layer Output layerOne hidden layer with optimal 10 neurons Mean pressure Frequency Oscillatory heat transfer coefficient Fig. 7. A three-layer feed-forward ANN model for oscillatory heat exchanger. A. A. Rahman, X. Zhang / International Journal of Heat and Mass Transfer 124 (2018) 1088–1096 1093

15. testing phases are near the zero error line which confirms the well- trained network and its higher performance for data prediction. For verifying the prediction ability of the proposed ANN model using the 4 data samples (experimental data) of the second group mentioned previously, which were not still used in the model, the predicted results from ANN model were compared with both experimental results and other correlations given by Eqs. (2)–(5) under the same operating conditions. The predicted results for the oscillatory heat transfer coefficient were shown in Table 2. It was observed that the predicted results from ANN model were in a very good agreement with the experimental results and more accurate than the predicted results from the other correlations. Compared to the experimental results, the average prediction error percentage of the ANN model was found to be 3.2% which implies the ability of ANN model to predict the oscillatory heat transfer coefficient values for any input values inside the covered range with a good degree of accuracy. On contrary, the relevant correlations predictions had revealed significant deviation from the experimental values as shown in Fig. 10 at the same conditions of Remax. Moreover, among the four correlations mentioned previously, the correlation obtained by Nsofor et al. [14] as shown in Eq. (2) had shown a better agreement to the experimental results than Eq. (3)–(5). Nevertheless, further experiments should be considered in order to verify the applicability of these proposed correlations. It can be noticed from Fig. 10 that Zhao and Cheng’s correlation [11] was likely to fail in matching the experimental results in the present study with under-predicted values compared with the experimental ones where the negative exponent of the Va leads to decreasing in Nu and hence decreasing the oscillatory heat transfer coefficient (h). It can be also noticed from Fig. 10 that both Swift’s [1] and Tang’s [6] correlations achieved over-predicted results compared with the experimental ones and the results from these two correlations are near from each other as stated in [6]. The over-predicted results obtained from Swift’s correlation [1] indicate the strong effect of Va corresponding to the oscillating frequency, on the enhancement of heat transfer due to the reduction in the boundary layer thickness which increases the oscillatory heat transfer coefficient (h). On the other hand, the over- predicted results obtained from Tang’s correlation [6] indicate the high influence of both Remax and Va on enhancing the heat transfer for the oscillatory heat exchanger. In general, the previous comparisons imply that the above-mentioned correlations have different limitations when evaluating the performance of oscillatory heat exchangers. In addition to that, other criteria besides Rey- nolds number, Prandtl number and Valensi number for the oscillatory flow may affect the oscillatory heat transfer to some extent in a limited range. Generally, in the research of correlations, different criteria num- bers referring to different parameters would be considered having significant influence on oscillatory heat transfer coefficient, for example, Valensi number Va is related to the oscillating frequency, and Remax is related to the maximum acoustic velocity, and in turn to the maximum acoustic pressure which is a mean pressure dependent at a given driving ratio in a thermoacoustic refrigerator. Hence, operating parameters mentioned above have different influences on the oscillatory heat transfer coefficient. Accordingly, in order to select appropriate parameters with good perspective for building either the ANN model or the correlation in the future research, it is necessary to roughly analyze the degree of influence of oscillating frequency and mean pressure on the oscillatory heat transfer coefficient in a statistical form in the present study. Statistically, the experimental data considered in the present study approximately follow a normal distribution as shown in Fig. 11 and the normal plot of residuals which confirms the error -15 -10 -5 0 5 10 0 2 4 6 8 10 12 14 16 Instances Errors = Targets - Outputs Training Validation Test Zero Error Fig. 9. Error histogram of ANN model. Table 2 Comparison of predicted results by ANN model with both the experimental and predicted results by the correlations for heat transfer coefficient (Note: experimental data were extracted from Ref. [14]). Input parameters Responses of (h) [W/m2 K] Frequency [Hz] Mean pressure [Â105 Pa] Experimental work ANN model Nsofor’s correlation Swift’s correlation Zhao & Cheng’s correlation Tang’s correlation 350 4.053 71.5 70.8 90 446.5 9.1 369.1 350 5.066 72 70.5 93.1 446.5 10 372.5 400 4.053 81.5 76 91.8 477.3 9.3 391.7 400 6.079 77 79.3 97.3 477.3 10.9 398.3 40 42 44 46 48 50 52 54 0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 hexp hANN hNsofor hSwift hZhao & Cheng hTang Oscillatoryheattransfercoefficient(h)[W/m 2 K] Maximum Reynolds number (Remax ) Fig. 10. Oscillatory heat transfer coefficient (h) versus Remax based on experimental work, ANN model and other correlations. 1094 A. A. Rahman, X. Zhang / International Journal of Heat and Mass Transfer 124 (2018) 1088–1096

16. normality, is illustrated in Fig. 12 based on the following equation [29]: pk ¼ ½k À ð1=2ÞŠ N k ¼ 1; 2; . . . ; N ð10Þ Moreover, the plot for residuals versus fitted values is shown in Fig. 13, from which error independency and variance constancy can be noticed. Hence, from ANOVA table [29,30] as illustrated in Table 3 with confidence level of 95% to analyze the influence degree of the parameters in consideration in the present study, it can be noticed that the oscillating frequency has a significant effect on the oscillatory heat transfer coefficient with the contribution (influence) percentage of 39%. On contrary, the mean pressure has an insignificant effect with the contribution percentage of 18.6%, lower than that of the oscillating frequency. From these results mentioned above, one can see the degrees to which these two operating parameters influence the oscillatory heat transfer coefficient, are reasonable and consistent with theoretical analyses and practical considerations. 4. Conclusions A novel modeling approach for oscillatory thermoacoustic heat exchanger based on an artificial neural network technique, had been presented in this paper. A three-layer feed-forward network with a back-propagation algorithm and a configuration of 2-10-1 was adopted for this work. In addition, from statistical point of view, the strength of influence of the operating parameters on oscillatory heat transfer coefficient had been analyzed. The average error percentage for predicted oscillatory convective heat transfer coefficient compared to experimental values was 3.2% which indicates the high accuracy of our model better than the results predicted from other correlations in the literature, and proves that well-trained neural network models can provide fast, accurate and consistent results. The comparison between the correlated results and experimental ones has shown some uncertainties from over-prediction or under-prediction. This comparison also implies that there are still some work to do for improving the generalization of studied correlations, even they would be possibly good enough for their specific research conditions. Moreover, in order to extend the knowledge of oscillatory heat transfer under oscillatory flow condition, further experimental work is required for the investigation into the effects of heat exchanger geometry, different operating pressure ratios, and different working fluids as well as the effects of additional components such as stack, on the heat transfer performance. ANN is found to be a new flexible modeling tool with a high degree of accuracy for linear or nonlinear heat transfer problems related with thermoacoustic heat exchangers operating under oscillatory flow condition. The present work is our first attempt in the application of artificial neural network to model one thermoacoustic heat exchanger. The present work gives authors significant confidence in ANN ability for data prediction. Further applied research will be carried out in modeling complex thermoacoustic problems in the near future, including mapping complex 12011010090807060 99 95 90 80 70 60 50 40 30 20 10 5 1 Oscillatory heat transfer coefficient [W/m^2 K] Percent Mean 89.76 StDev 11.72 N 24 Probability Plot of Oscillatory Heat Transfer Coefficient Normal Fig. 11. Probability plot of oscillatory heat transfer coefficient. Table 3 ANOVA for oscillatory heat transfer coefficient. Source DF SS MS F p Contributiona [%] Oscillating frequency 3 1230.74 410.248 4.58 0.018 39 Mean pressure 5 586.09 117.218 1.31 0.312 18.6 Error 15 1342.39 89.493 42.4 Total 23 3159.22 100 DF, degrees of freedom; SS, sum of squares; MS, mean squares; F, F-value; p, p-value. a Contribution ½%Š ¼ ðSSfactor=SStotalÞ Ã 100: 20100-10-20 99 95 90 80 70 60 50 40 30 20 10 5 1 Residual (1-Pk)*100[%] Normal Probability Plot of the Residuals (response is oscillatory heat transfer coefficient) Fig. 12. Normal plot of residuals. 110100908070 15 10 5 0 -5 -10 -15 Fitted Value Residual Residuals versus the Fitted Values (response is oscillatory heat transfer coefficient) Fig. 13. Residuals versus fitted values. A. A. Rahman, X. Zhang / International Journal of Heat and Mass Transfer 124 (2018) 1088–1096 1095

17. relationship between the performance and significant influencing factors, identifying parameters, as well as the optimization for nonlinear thermoacoustic systems. Conflict of interest The authors declare that there is no conflict of interest. Acknowledgement This work was supported by the National Natural Science Foun- dation of China (NSFC) (Grant No. 51476062). References [1] G.W. Swift, Analysis and performance of a large thermoacoustic engine, J. Acoust. Soc. Am. 92 (3) (1992) 1551–1563. [2] T. Jin, J. Huang, Y. Feng, R. Yang, K. Tang, R. Radebaugh, Thermoacoustic prime movers and refrigerators: thermally powered engines without moving components, Energy 93 (2015) 828–853. [3] X. Zhang, J. Chang, S. Cai, J. Hu, A multi-stage travelling wave thermoacoustic engine driven refrigerator and operation features for utilizing low grade energy, Energy Convers. Manage. 114 (2016) 224–233. [4] M.T. Pamuk, A new heat transfer correlation for oscillating fluid flow, Therm. Sci. 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Zhang / International Journal of Heat and Mass Transfer 124 (2018) 1088–1096

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