Factor Analysis of Convective Heat Transfer for a Horizontal Tube in the Turbulent Flow Region Using Artificial Neural Network

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1 COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE EPMESC X, Aug. -3, 6, Sanya, Hainan,China 6 Tsinghua University ess & Sringer-Verlag Factor Analysis of Convective Heat Transfer for a Horizontal Tube in the Turbulent Flo gion Using Artificial Neural Netork H. K. Tam *, S. C. Tam, A. J. Ghaar 3, L. M. Tam. Deartment of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, China. Deartment of Mathematics, Faculty of Science and Technology, University of Macau, China 3. School of Mechanical and Aerosace Engineering, Oklahoma State University, Stillater, Oklahoma, USA hktam@umac.mo Abstract Artificial neural netork (ANN) has shon its suerior redictive oer comared to the conventional aroaches in many studies. Hoever, it is alays treated as a black box because it rovides little exlanation to the relative influence of the indeendent variables in the rediction rocess. Ghaar et al. [] used the ANN method to develo an emirical correlation for the heat transfer data in a horizontal tube ith a reentrant inlet under uniform all heat flux boundary condition in the transition region. In their ork, the least and the most imortant variables ere examined using the coefficient matrices based on a single training. Hoever, the method as only alied to one set of exerimental data. The alicability of their method to other data sets is not knon. In this study, the method roosed in the revious study is modified and a ne set of exerimental data for different inlet configurations (square-edged and bell-mouth) from the ork of Ghaar and Tam [] in the turbulent region are used to further verify this method. An index of contribution is defined in this study. Furthermore, the gradient method used and the number of neurons and iterations for each training are carefully examined. Using the revised method and the index of contribution defined in this study, an ANN correlation is established and the ynolds number () and the andtl number () are observed as the most imortant arameters. The length-to-diameter ratio () and the viscosity ratio (µ b /µ ).4 are found to be the least imortant arameters. Key ords: convective heat transfer, artificial neural netork, turbulent flo, index of contribution *Student aer cometition INTRODUCTION Heat transfer inside horizontal tubes in the transitional and turbulent flo regimes have been studied exerimentally by various researchers in the ast. Usually, the research results are resented in the form of heat transfer correlations. The form of the correlations is based either on different theoretical models or they are comletely emirical. The coefficients of the correlations are usually determined by the conventional least squares method. Kakac et al. [3] documented some of the most ell acceted correlations in the transition and turbulent flo regions. cently, Ghaar et al. [] roosed a ne correlation in the transition region based on the method of artificial neural netork (ANN) ith excellent accuracy. In their aer, it as mentioned that ANN can also be used in the determination of the most and least imortant variables using the coefficient matrices obtained from the eight and bias matrices of the ANN correlation. Hoever, there are some unansered questions regarding this technique, such as () alicability of this technique to other data sets and () besides the most imortant variables, i.e., the normalized ynolds and Grashof numbers, and least imortant variables, i.e., the normalized Sieder and Tate factor (µ b /µ ).4, the imortance of the normalized andtl number can not be seen. Therefore, this method is modified and then verified by using a different set of exerimental data, the turbulent heat transfer data for uniformly heated horizontal tube fitted ith different inlet configurations of Ghaar and Tam []. Furthermore, the gradient method selected, the number of neurons and iterations used for training ill be analyzed and defined systematically in this study. Based on the defined netork arameters, the index of contribution for each indeendent variable ill be found from the ANN correlation by the revised method.

2 EXPERIMENTAL DATASET The heat transfer exerimental data used in this study, along ith a detailed descrition of the exerimental aaratus and rocedures used, ere reorted by Ghaar and Tam []. A schematic of the overall exerimental setu used for heat transfer measurements is shon in Figure. In this aer, only a brief descrition of the exerimental setu and rocedures ill be rovided. The local forced and mixed convective measurements ere made in a horizontal, electrically heated, stainless steel circular straight tube ith reentrant, square-edged, and bell-mouth inlets under a uniform all heat flux condition. The ie had an inside diameter of.58 cm and an outside diameter of.9 cm. The total length of the test section as 6. m, roviding a maximum length-inside diameter ratio of 385. A uniform all heat flux boundary condition as maintained by a dc arc elder. Thermocoules (T-tye) ere laced on the outer surface of the tube all at close intervals near the entrance and at greater intervals further donstream. Tenty-six axial locations ere designated, ith four thermocoules laced at each location. The thermocoules ere laced 9 degrees aart around the erihery. From the local eriheral all temerature measurements at each axial location, the inside all temeratures and the local heat transfer coefficients ere calculated. In these calculations, the axial conduction as assumed negligible ( > 4, in all cases), but eriheral and radial conduction of heat in the tube all ere included. In addition, the bulk fluid temerature as assumed to increase linearly from the inlet to the outlet. As reorted by Ghaar and Tam [], the uncertainty analyses of the overall exerimental rocedures shoed that there is a maximum of 9% uncertainty for the heat transfer coefficient calculations. Moreover, the heat balance error for each exerimental run indicates that in general, the heat balance error is less than 5%. For ynolds numbers loer than here the flo is strongly influenced by secondary flo, the heat balance error is slightly higher (5 8%) for that articular ynolds number range. To ensure a uniform velocity distribution in the test fluid before it entered the test section, the flo assed through calming and inlet sections. The calming section had a total length of 6.6 cm and consisted of a 7.8 cm diameter acrylic cylinder ith three erforated acrylic lates, folloed by tightly acked soda stras sandiched beteen galvanized steel mesh screens. Before entering the inlet section, the test fluid assed through a fine mesh screen and floed undisturbed through 3.5 cm of a 6.5 cm-diameter acrylic tube before it entered the test section. The inlet section had the versatility of being modified to incororate a reentrant or bell mouth inlet (see Figure ). The reentrant inlet as simulated by sliding.93 cm of the tube entrance length into the inlet section, hich as otherise the square-edged (sudden contraction) inlet. For the bell-mouth inlet, a fiberglass nozzle ith a contraction ratio of.7 and a total length of 3.5 cm as used in lace of the inlet section. In the exeriments, distilled ater and mixtures of distilled ater and ethylene glycol ere used. The exeriments covered the local bulk ynolds number range of 8 to 49, the local bulk andtl number range of 4 to 58, the local bulk Grashof number range of to.5 5, and the local bulk Nusselt number range of 3 to 58. The all heat flux for the exeriments ranged from 4 to 67 kw/m. As mentioned in the revious section, only the turbulent heat transfer data is considered in this study. Because of that, Grashof number makes no contribution to Nusselt number. Therefore, the correlation only consists of five variables, hich are Nu,,,, and µ b /µ. and the range of the variables is summarized as follos: 57.4 Nu 35.3, , , ,. µ b /µ.54 A total of 48 data oints is used. Out of them, 73 data oints are for the reentrant inlet, 43 data oints are for the square-edged inlet, and data oints are for the bell-mouth inlet. Ghaar and Tam [] correlated their data in the turbulent flo region in the folloing form: Nu = µ () t.3 () ( µ b/ ) The accuracy of the correlation is described in Table. Figure 3 shos that the maority of the turbulent data ere redicted by this correlation ithin ±%. The next sections ill be devoted to ho the variables in Equation () contribute to the Nusselt number.

3 Table : ediction sults for Equation () Inlet Configurations Number of Data ithin ±% Number of Data ithin ±5% Abs. Mean Dev. (%) Range of Dev. Abs.(%) Total data oints (48 ts.) % to.5% entrant (73 ts.) % to.5% Square-edged (43 ts.) % to 6.37% Bell-mouth ( ts.) % to.8% Figure : Schematic diagram of exerimental setu Figure : Schematic of the three different inlet configurations

4 +% -% Nu cal entrant Square-edged Bell-mouth Nu ex Figure 3: Comarison beteen exerimental Nusselt numbers for three different inlet configurations and those redicted by the turbulent region heat transfer correlation, Equation () ANN CORRELATION AND THE INDEX OF CONTRIBUTION ANALYSIS To correlate the turbulent heat transfer data, the ANN ith single hidden layer is emloyed. Figure 4 shos a tyical examle of ANN model of this kind. It has been shon that any continuous correlation can be modeled by the netork [4]. The eight and the bias of the otimal ANN model are usually determined by the back roagation algorithms [5]. In order to determine the contribution of each indeendent variable to the correlation, the matrix form of the otimal ANN model has to be examined. R f ( + b ) = R f ( + b ) ) [,, ] R = L = S + N(, L, b () M R f ( b ) S + S = here (, L, R ) are the R inuts, S is the number of hidden neurons, f(t) = e t is the transfer function and s and b s are the eights and biases of the ANN resectively. The contribution of the indeendent variables to the outut of the k th R neuron in the hidden layer f( = + is simly k bk ) k and the contribution of the k th neuron outut to the ANN model is Ratio _Wk (3) k = S k = k Therefore, the contribution of the indeendent variables to the ANN model is oduct_w S = ratio_w (4) k = k k

5 Finally, in order to comare ith the other indeendent variables, the index of contribution of is defined to be index( oduct _W ) = % R oduct _W = (5) Hence, the most significant indeendent variable ould have the largest index of contribution. On the other hand, the variable ith small index aears to be less imortant., 3 b,, a b b R S,R,S Inut Layer b S Hidden Layer Outut Layer Figure 4: An ANN ith S neurons in its hidden layer RESULTS AND DISCUSSION Before the usage of the index of contribution defined in Equation (5), it is orthhile to evaluate the traditional least squares Equation (). According to the form of Equation (), it is obvious that ynolds and andtl numbers are both imortant. Hoever, it is not ossible to tell hich one is more imortant than the other by simly udging the exonents of them. It is also obvious that, and µ b /µ are the least imortant variables. According to the range of the data, the value of the terms () -.54 and (µ b /µ ).4 are forced to a value very close to one, hence they made less contribution to the Nusselt number. Theoretically, for turbulent flo, the thermal entry length is usually very short and the entrance effect is insignificant. Moreover, from Sieder and Tate [6], the data in the turbulent region sho little variation beteen µ b and µ, firstly, because fluids hich give turbulent flo seldom have a large temerature coefficient of viscosity and secondly, because the heat transfer rates are high, reventing large temerature differences. Therefore, both the terms () -.54 and (µ b /µ ).4 act as correction factors to make the correlation more accurate. Coming back to the ANN analysis, using the suervised three-layer feedforard neural netork ith fully eighted connection and the algorithm described in the revious section to determine the index of contribution for each indeendent variable, e are able to decide (a) the gradient method to be used, and (b) the reasonable number of iterations and neurons used for each netork training. For the gradient method to be used, according to Ghaar et al. [,7], the Levenberg-Marquart algorithm (LM) as adoted as the gradient method in back roagation. According to Hagan and Menha [8], this method can seed u the netork training. Hoever, it is hard to tell hether this method orks ell ith the index of contribution analysis. Therefore, in this study, the classical gradient method, hich is the sloer stee descent algorithm (SDA) is considered. The turbulent heat transfer data is correlated using the LM and SDA methods. The number of neurons is varied from 5 to 8 and the iteration for each training is fixed at,. The index of contribution based on the algorithm described in the revious section using different number of neurons is calculated and shon in Figure 5. Each data oint shon in the figure is an average of trainings. From Figure 5, it is obvious that the SDA method gave more consistent information regarding the contribution from each indeendent variable irregardless of ho many neurons ere used. Moreover, the findings based on SDA agree ith the findings according to the traditional least squares equation, here and contribute the most and () -.54 and (µ b /µ ).4 contribute less. When the LM method is considered (see Figure 5b), the influence of (µ b /µ ).4 is large and the same observations can not be seen. Since (µ b /µ ).4 is roved to be a correction factor in the turbulent region, the LM method cannot clearly identify the contribution of each variable. Therefore, the SDA method should be selected as the gradient method in this study.

6 (a) µ b /µ (b) µ b /µ Neurons Neurons Figure 5: A comarison of the ercent contribution by using to different gradient descent algorithms. (a) Sloer algorithm SDA method; (b) Faster algorithm LM method. Each data oint shon is the average value from trainings. For the reasonable number of iterations and neurons to be used for each netork training, the folloing stes ere taken. The ANN correlation and the index of contribution calculation is comuted based on 5, 6, 7 and 8 neurons. The number of iterations for each training ranged from, to, times. The increment of the iterations is set as, hen the number of iterations is from, to,. The increment of the iterations is set as, hen the number of iterations is from, to,. The index of contribution for different number of neurons and iterations are shon in Figure 6. Again, each data oint shon in these figures are the average of trainings. According to these figures, it is imortant to find out that the contribution from each variable is not ell established hen the number of iterations is less than,. When the number of iterations is more than,, the trend for the contribution from each variable is highly consistent for different number of neurons. The contributions from and are alays close to % and the other variables contribute less than %. Therefore, the number of iterations based on Figure 6 is selected as,. Since the trend for the contribution from each variable is relatively insensitive to the number of neurons used, the number of neurons is arbitrarily chosen as 6.

7 (a) (b) µ b /µ µ b /µ Iterations Iterations 45 (c) (d) 35 5 µ b /µ µ b /µ Iterations Iterations Figure 6: The ercent contribution from ANN training for different dimensionless numbers to Nusselt number by adusting the iterations and the number of neurons, (a) 5 neurons, (b) 6 neurons, (c) 7 neurons, (d) 8 neurons. Each data oint is the average value from trainings. In summary, based on the observations made above, the gradient method is selected as the SDA, the number of iterations used is selected as, and the number of neurons used is selected as 6. Moreover, the form of the ANN correlation is given by Equation (). With the establishment of the ANN correlation, the index of contribution according to the above mentioned criteria can no be comuted. For reliability uroses, ninety ercent of the total of 48 data oints ere used for training and the remaining data is for verification. The initial value of the free arameters (eights and biases) is randomly selected ithin ±. For satisfying the log-sigmoid transfer function, the normalized inut variables, ynolds number, andtl number, length-to-diameter ratio, and viscosity ratio are arranged into the inut vector, : = normal normal min x = x x D normal D D µ min b µ µ µ b b normal µ µ min [ min ] / [ max min ] [ ] / [ ] / / max min x x D max D min µ µ b b µ max µ min

8 In Equation (), the deendent outut is the Nusselt number for the turbulent heat transfer data. The,, b, b terms used in Equation () are constant matrices or scalars. Their numerical values are shon in the folloing matrices: = , = [ ], b =, b = As shon in Table, all the exerimental data are redicted ithin -% to.38%. The absolute deviation of all the redictions is 3.%. About 98% of the data (4 data oints) are redicted ithin ±% deviation. About 77% of all the data (33 data oints) are redicted ithin ±5% deviation. The rediction of bell-mouth data oints is most accurate, i.e. all the data oints are redicted ithin ±% deviation. As comared to the revious correlation, significant imrovement is observed. Table : ediction sults for the Imroved Correlation by Using ANN Inlet Configurations Number of Data ithin ±% Number of Data ithin ±5% Abs. Mean Dev. (%) Range of Dev. Abs.(%) Total data oints (48 ts.) % to.38% entrant (73 ts.) % to.38% Square-edged (43 ts.) % to.% Bell-mouth ( ts.) % to 7.67% For the calculation of the index of contribution for each variable, Equations (3) to (5) are emloyed according to the eight matrices, and shon above. The comutation rocess is as follos: i. For each hidden neuron k, the absolute value of the hidden-outut layer connection eight is divided by the summation of the hidden-outut layer connection eight. Hidden Hidden Hidden Hidden Hidden Hidden Neuron Neuron Neuron 3 Neuron 4 Neuron 5 Neuron 6 Ratio _W k ii. For each hidden neuron k, multily the Ratio _W k by the absolute value of the hidden-inut layer connection eight. Then, sum u the values to obtain the oduct_w for each inut variable. µ b /µ oduct_w iii. Comute the index of contribution in ercentage by dividing oduct_w by the sum of the oduct_w corresonding to each inut variable. Finally, the index of contribution for each variable is established as: µ b /µ index( ) 4.4 % 38.8 % 3.6 % 6. % Through the comutation, the contribution of each variable can be obviously seen. The result demonstrates that the andtl number and ynolds number are the most imortant variables since they contribute more to the Nusselt number. On the other hand, the and viscosity ratio are the least imortant variables since they contribute less to the Nusselt number. That is absolutely identical to the observations made from the conventional heat transfer correlations.

9 CONCLUSIONS AND RECOMMENDATIONS Develoment of correlations using ANN is considered as a black box aroach. Hoever, in this study, besides the establishment of the ANN heat transfer correlation, the contribution from each variable is evaluated by using the index of contribution defined in this study. Moreover, the gradient method used, the number of neurons emloyed and the number of iterations adoted in this study ere carefully examined. Based on the numerical ork done in this study, it can be concluded that not only the ANN method can be used in accurately correlating the data, but also can be used to determine the contribution from each variable. Unlike the analysis for the most and least imortant variables based on the traditional least squares correlation, hich can only determine hat is imortant, the roosed method in this study can clearly quantify the contribution for different variables. Therefore, this method not only can evaluate the contributions qualitatively, but also quantitatively. Hoever, it is still recommended that more exerimental data sets should be examined to verify this interesting method. REFERENCES. Ghaar, A. J., Tam, L. M., Tam, S. C., A Ne Heat Transfer Correlation in the Transition gion for a Horizontal Pie ith a entrant Inlet Using Artificial Neural Netork, oceeding of the th International Heat Transfer Conference, vol., 89-94, Grenoble, France, August 8-3,.. Ghaar, A. J. and Tam, L. M., Heat Transfer Measurements and Correlations in the Transition gion for a Circular Tube ith Three Different Inlet Configurations, Exerimental Thermal and Fluid Science, Vol. 8, No.,. 79-9, Kakac, S., Shah, R. K., and Aung, W., Handbook of Single-Phase Convective Heat Transfer, Wiley, Ne York, Hornik K., Aroximation Caabilities of Multilayer Feedforard Netorks, Neural Netorks, Vol. 4, No.,. 5-57, Rumelhart, D. E., Hinton, G. E., and Williams, R. J., Learning Internal resentations by Error oagation, Parallel Distributed ocessing: Exlorations in the Microstructure of Cognition, (Eds: D. E. Rumelhart, and J. L. McClelland) Vol., MIT ess, Cambridge, Mass., , Sieder, E. N., and Tate, G. E., Heat Transfer and essure Dro in Liquids in Tubes, Ind. Eng. Chem., vol. 8, , Ghaar, A. J., Tam, L. M., Tam, S. C., Imroved Heat Transfer Correlation in the Transition gion for a Circular Tube ith Three Inlet Configuration Using Artificial Neural Netork, Heat Transfer Engineering, Vol. 5, No.,. -, Hagan, M. T., and Menha, M. B., Training Feedforard Netorks ith Marquardt Algorithm, IEEE Transactions on Neural Netorks, Vol. 5, No.6, , 994.