On Heat Transfer in case of a Viscous Flow over a Plane Wall with Periodic Suction by Artificial Neural Network U. K. Tripathy #1, S. M.

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1 On Heat Transfer in case of a Viscous Flow over a Plane Wall with Periodic Suction b Artificial Neural Network U. K. Tripath #, S. M. Patel * # td. Professor & Head Dept of Mathematics V.S.S.Universit, Burla-7687, Sambalpur, India. * Lecturer, Department of Mathematics Sundargarh Engineering School, Sundargarh. 7773, India Abstract: This paper shows that the problem of heat transfer in case of a viscous flow over a plane wall with periodic suction has been studied taking into account the effect of viscous dissipative terms. The solution of the equation of heat balance has been obtained approximatel b perturbation method choosing ε, the amplitude of the periodic suction velocit to be perturbation parameter the artificial neural networks method. The effect of the Prtl number, Eckert number nold s number on the correction factor (F) to the rate of heat transfer from the wall has been compared for the both tpe of solutions. The effect of the parameters on F are exactl same as the numerical calculation of which is the correction to the quasi two-dimensional rate of heat transfer (Nusselt number ) from the wall. The results obtain b artificial neural network (ANN) gives a better approximation then that of the numerical technique the most important part of this paper is ANN technique can easil be hle with a large number of data in a short time. Hence the ANN model which provides an exact, quick reliable result than the conventional time consuming numerical method. Ke words: artificial neural network, back-error propagation, correction to the quasi two-dimensional rate of heat transfer.. INTRODUCTION The problem of a non-newtonian fluid passing through a porous plate under the influence of a magnetic field has attached the interest of different research communities because of its various applications. In view of these applications, the problem of flow heat transfer of a viscous incompressible liquid along a plan wall with periodic suction has been solved b []. A similar tpe of problem is solved b [] taking into account the viscous dissipative terms. The problem described here is solved b following [3]. The boundar laer flow of heat mass transfer over a stretching surface under different phsical situations has been done b [4], [5], [6], [7]. Likewise, man others as [8], [9], [], [] have examined the MHD flow of non-newtonian viscoelastic flow, heat mass transfer under different phsical situations. The numerical solution for the MHD flow problems of power law fluids has been widel reported in the literature [], [3], [4]. These tpes of problems are intrinsicall non linear like the most fluid mechanics problems do not have an exact solution. Again related to this article variable viscosit thermal conductivit on stead free convection heat transfer of micro-polar fluid flow over a porous hot vertical plate with constant heat flux b [5] MHD radiation effect on heat transfer in a non-newtonian Maxwell fluid over an unstead stretching sheet with heat source/sink b [6]. The solved paper Estimation of MHD boundar laer slip flow over a permeable stretching clinder in the presence of chemical reaction through numerical ANN modeling b [7]. The solved paper Estimation of the flow heat transfer in MHD flow of a power law fluid over a porous plate ANNs b [8]. As described b [8] we have solved the present problem. This paper has been arranged as follows: Section the heat transfer of a viscous flow over a plane wall with periodic suction are presented. In section 3 formulations of the equations solutions are presented. Section 4 contains the importance the discussion of ANN method the Back Propagation algorithm. Section 5, concluded the achievement of this stud.. THE HEAT TRANSFER OF A VISCOUS FLOW OVER A PLANE WALL WITH PERIODIC SUCTION The asmptotic solution for the flow of an incompressible viscous liquid past an infinite porous flat plate has been discussed b [9]. Here the heat transfer along a plane wall with periodic suction of a viscous flow is discussed whose solution is available in []. The same problem is solving b backpropagation neural networks its result is compared with that of the solution avaible in the above reference. Problems of this tpe are quite important in the theor of laminar flow control (LFC) sstem. Assuming the wall to be the x z plane ISSN: Page 94

2 axis to be normal to it, the suction velocit is assumed to be of the form where v, l is the wave length of the periodic suction velocit distribution is the amplitude of suction velocit variation. In the paper cited above the authors have solved the energ equation neglecting the viscous dissipation. Our aim in this note is to solve the energ equation including the viscous dissipative term compare its results with that of the artificial neural networks solution following [7] [8]. The ANN method has not been used or tested for heat transfer analsis of fluid flow with vicious dissipative term. Therefore this stud primaril focuses on the applicabilit of ANN of the above said heat transfer analsis. In the present stud, the effects of different parameters of the heat flow analsis are compared with the solution of the ANN method. Appropriate ANN is applied to the problem training procedure for the ANN presented here. A back error propagation training algorithms is emploed to train the network find the best weight the performance is examined. 3. FORMULATION OF THE EUQATIONS AND SOLUTIONS Denoting w v, w, to be the components of the fluid velocit an point x,, z, Gersten Gross[] have found out the expressions for these as follows: u v w u, v w are the dimensionless U v vo velocit components, U is the free stream velocit in the x-direction, vl v is the, nold s number, where v is the co-efficient of kinematic viscosit. The equation of heat balance in the dimensionless from including viscous dissipative terms given b where C Pr, E U C ( T T ) Eckert number respectivel. dimensionless quantit, w are the Prtl number v L U T Tw is T T w is a the dimensionless temperature function. The quantities c are the specific heat the thermal conductivit respectivel. The boundar conditions to which equation (3.4) is simplified as : : (3.5) When (the case of strong suction) equation (3.4) is simplified as In the above expression l z z l are the normalized space variables, The solution of this equation when Pr is ISSN: Page 95

3 where, Since, we assume the solution for when Pr, in general as We further assume that First of all we substitute equations (3.), (3.), (3.3) (3.8) into equation (3.4) collect terms of the same order then in the resulting equation for for we substitute the expression as suggested in equation (3.9) obtain When Pr = proceeding in the same manner as above we can find out the solution for as From equations (3.5), (3.8) (3.9) the derived boundar conditions for equation (3.) are : : where (3.) Pr The solution of equation (3.) when under the boundar conditions (3.) is ISSN: Page 96

4 (RMSE) the absolute fraction of variance determined from. are The expressions for the Nusselt number, Nu d d which is at for the case Pr Pr= with the help of equations (3.8), (3.9), (3.) (3.3) are respectivel The function F() is given b where are the desired (target) output output of neural network values respectivel for the ith output neuron M is total number of data sets. The absolute fraction of variance ranges between. The values closer to indicate a ver good fit, while the values closer to indicate a poor fit. In this stud, ANN structure has been designed trained using the MATLAB Neural Network Toolbox. The Back-Error Propagation (BEP) training algorithm has been used in feed-forward with one hidden laer. In the structural network (shown in figure.), the inputs are the output is The function F(), signifies the correction to the quasi two-dimensional rate of heat transfer Nusselt number (Nu) from the wall. 4. ANN METHOD The training of the neural network is accomplished b adjusting the weights is carried out through a large number of training sets training ccles (epochs). The purpose of the learning procedure is to find the optimal set of weights, which is an ideal case would produce the correct output for an relative input. The output of the network is compared with a desired response to determine an error. The performance of the multilaer perceptron (MLP) is measured in terms of the desired signal the criterion for convergence. For training, validation, test all samples, the root mean square error A sigmoid function has been used as the activation function of artificial neurons training has been done using a large number of epochs. The total 3 numerical results were used to train, validated test the artificial neural network (ANN) model. The data set were used for the training set, 5 data set were used for validate rest of the data were used for testing of the model. The performance of of the proposed ANN model is shown in figure. (a), (b), (c) (d) respectivel. The value of are considered as the x-axis that of on - axis. ISSN: Page 97

5 E Pr Input Laer F () Output Unit Y=T 3 Bias unit Hidden Laer ANN Figure. Schematic diagram of a multi laer (a). Training Table. Skin friction data of F() for with error. Sl. No Pr E % Error Y=T 3 (b). Validation Y=T.5.5 ISSN: Page 98

6 (d). All (c). Test Figure.(a),(b),(c),(d) Graphical representation of F() 5. CONCLUSION... Y=T 3 The function, signifies the correction to the quasi two-dimensional rate of heat transfer (Nusselt number ) from the wall. The effect of (the Eckert number), (the Prtl number) (the nolds number) on is discussed. It is seen that when increases for an value of increases in its value. As increases the quasi two-dimensional correction increases. For large nolds number, as is seen from the table., the effect of is appreciabl significant. In the present work the ANN approach is developed successfull to simulate the quasi twodimensional error in a problem of heat transfer along a plane wall with periodic suction. This ANN structure has been trained, validated tested using the MATLAB environment. Table.shows effect of the parameters on F are exactl same as the numerical calculation of which is the correction to the quasi two-dimensional rate of heat transfer (Nusselt number ) from the wall. The bias function, the number of epochs (96) have been used in this ANN training model. The prediction of through ANN model is in good agreement with a numerical data calculated in this chapter. The percentage of error between is varing within 7% barring few data. Hence the ANN model which provides an exact, quick reliable result than the conventional time consuming numerical method. REFERENCES [] K. Gersten J.F. Gros, ZAMP, Vol 5, (974) 399. [] S. Padhi U.K. Tripath, Proc. Seminar on cent advances in applied mathematics its applications, IIT Kharagpur, Ma (978). [3] J. Freeman D.M.Skapura, Neural Networks:Algorithms, application, programming Technique, Addison Wesle, New York, (99). [4] N. Bachok, A. Ishak, I. Pop, Unstead boundar laer flow heat transfer of a nano fluid over a permeable streaching/shrinking, Int. J. Heat Mass Transf. 55()- 9. [5] K. Bhattachra, S. Mulkhopadha, G.C. Laek, I. Pop, Effect of thermal radition on micropolar fluid flow heat transfer a porous shrinking sheet, Int. J. Heat Mass Transf. 55() [6] A.S. Chethan, G.N. Sekhar, P.G. Siddheshwar, Flow heat transfer of an exponential stretching sheet in viscoelastic liquid with Navier slip boundar comdition, J.Appl. Fluid Mech. 8() (5)3-9. [7] R. Cortell, Viscous flow heat transfer over a nonlinearl stretching sheet, Appl. Math. Comput. 84() (7) [8] M.S. Able N. Mahesha, Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivit, non-uniform heat source radiation, Applied Mathematical Modeling, 3()(8) [9] M.S. Able M.M. Neppanavar, Heat transfer in MHD viscoelastic boundar laer flow over a stretching sheet with non-uniform heat source/sink, Communications in Nonlinear Science Numerical Simulation, 4(5) (9)-3. [] M.K. Chowdhur M.N. Islam, MHD free convection flow of visco-elastic fluid past an infinite vertical porous plate, Heat Mass Transfer, 36(5)() [] S.K. Khan, M.S. Able R. Sonth, Visco-elastic MHD flow, heat mass transfer over a porous stretching sheet with dissipation of energ stress work, Heat Mass Transfer, 4(-)(3) [] P.S. Datti, K.V. Prasad, M.S. Able A. Joshi, MHD viscoelastic flow over a non-isothermal stretching sheet, International Journal of Engineering Science, 4(8-9)(4) [3] K.V. Prasad, M.S. Able S.K. Khan, Momentum heat transfer in visco-elastic fluid flow in a porous medium over a non-isothermal stretching sheet, International Journal of Numerical Methods for Heat Fluid Flow, (8)() [4] K.V. Prasad K. Vajravelu, Heat transfer in the MHD flow of a power law fluid over a non-isothermal stretching sheet, International Journal of Heat Mass Transfer, 5(- )(9) [5] G.C. Hazarika, P. Hatimota, Variable viscosit thermal conductivit on stead free convection heat transfer of micro-polar fluid flow over a porous hot vertical plate with constant heat flux, IJMTT, (4) Volume-5 Number- [6] S.Vijaa Lakshmi, T. Amaranatha dd, M. S. dd, MHD radiation effect on heat transfer in a non-newtonian Maxwell fluid over an unstead stretching sheet with heat source/sink, IJMTT, (6) Volume-4 Number-4 [7] P.B.A. dd R. Das, Estimation of MHD boundar laer slip flow over a permeable stretching clinder in the presence of chemical reaction through numerical ANN modeling, Engineering Science Technolog, an International Journal, (6) doi:. [8] M.F. Shahri A.F. Nezhad, Estimation of the flow heat transfer in MHD flow of a power law fluid over a porous plate ANNs, Middle East J. Sci. s. (9)(4)4-49 [9] F.W. Meridith A.A. Griffith, (955) A.R.C. 35 ISSN: Page 99