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  • International Journal of Advanced Research in Engineering RESEARCH IN ENGINEERING INTERNATIONAL JOURNAL OF ADVANCED and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online)TECHNOLOGY (IJARET) pp. 06-15, © IAEME AND Volume 5, Issue 2, February (2014), ISSN 0976 - 6480 (Print) ISSN 0976 - 6499 (Online) Volume 5, Issue 2, February (2014), pp. 06-15 © IAEME: www.iaeme.com/ijaret.asp Journal Impact Factor (2014): 4.1710 (Calculated by GISI) www.jifactor.com IJARET ©IAEME FUZZY FINITE ELEMENT ANALYSIS OF A CONDUCTION HEAT TRANSFER PROBLEM Ajeet Kumar Rai*, Gargi Jaiswal** *Department of Mechanical Engineering, SHIATS-DU Allahabad, India **Department of Applied Mechanics, IIT Delhi, India ABSTRACT In the heat transfer analysis, the material properties are taken as crisp values. Since the input data are imprecise, no matter what techniques are used, the solution will not be reliable. In fuzzy finite element heat transfer analysis, the material properties are considered fuzzy parameters in order to take uncertainty into account. In the present study, a circular rod made up of iron has been considered. The end of the bar is insulated and heat transfer is taking place through the periphery. Here variation in material properties are considered as fuzzy and this requires the consideration of complex interval or fuzzy arithmetic in the analysis and the problem is discretized into finite number of elements and interval / fuzzy arithmetic is applied to solve this problem. The values of temperature at different points (x = 0, x = 1.25, x = 2.5, x = 3.75, x = 5) of a fin is calculated by using different methods. Corresponding results are given and compared with the known results in the special cases. Keywords: Finite Element Method, Triangular Fuzzy Number & Alpha Cut. INTRODUCTION The fuzzy finite element method combines the well-established finite element method with the concept of fuzzy numbers, the latter being a special case of a fuzzy set. The advantage of using fuzzy numbers instead of real numbers lies in the incorporation of uncertainty (on material properties, parameters, geometry, initial conditions, etc.) in the finite element analysis. In the fuzzy finite element method uncertain geometrical, material and loading parameters are treated as fuzzy values. The modeling of uncertain parameters as fuzzy values is necessary when it is not possible to uniquely and reliably specify these parameters either deterministically or 6
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME stochastically. Often in such cases, only a limited number of samples are available or the reproduction conditions for generating sample elements vary. The parameters possess informal or lexical uncertainty, which may be modeled as fuzziness. Physical parameters possessing fuzziness with regard to external loading or material, geometrical and model parameters may occur at all points of a structure. An exhaustive work has been reported by various scholars showing the use of fuzzy logic combining with the results of FEM. Nicolai, De Baerdemaeker [1] has used finite element perturbation method for heat conduction problem with uncertain physical parameters. They found the temperature in heat conduction problem for randomly varying parameters with respect to time. B.M, Nicolai [2] has used a method for the direct computation of mean values and variances of the temperature in conduction-heated objects with random variable thermo physical properties. Method is based on a Taylor expansion of the finite element formulation of the heat conduction equation and offers a powerful alternative to the computationally expensive Monte Carlo method. Both steadystate and transient problems are considered. The simulations indicate that the variability of the thermo physical properties may cause a considerable variability of the temperature within the heated object. Rao S.S, Sawyer J.P. [3] has applied the concepts of fuzzy logic to a static finite element analysis. The basic concepts of the theory of fuzzy logic are described. Later, the static finite element analysis will be extended to dynamic analyses. Naidoo P. [4] has studied the application of intelligent technology on a multi-variable dynamical system, where a fuzzy logic control algorithm was implemented to test the performance in temperature control. Jose et al. [5] has used four different global optimization algorithms for interval finite element analysis of (non)linear heat conduction problems: (i) sequential quadratic programming (SQP), (ii) a scatter search method (SSm), (iii) the vertex algorithm, and (iv) the response surface method (RSM). Their performance was compared based on a thermal sterilization problem and a food freezing problem. The RSM fuzzy finite element method was identified as the fastest algorithm among all the tested methods. It was shown that uncertain parameters may cause large uncertainties in the process variables. Majumadar Sarangam and Chakraverty S. [6] have proposed new methods to handle fuzzy system of linear equations. The material properties are actually uncertain and considered to vary in an interval or as fuzzy. And in that case; complex interval arithmetic or fuzzy arithmetic was considered in the analysis. Identification of the Problem Fig 1. Schematic diagram of a circular rod used as a fin In the present communication calculation of temperature at different points (x = 0 , x = 1.25 , x = 2.5 , x = 3.75 , x = 5) of a fin is done by using analytical method, Finite difference method, and Finite element method. The end of the fin is insulated and heat is convected through periphery. 7
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME Diameter was taken as 2 cm, length as 5 cm, heat transfer coefficient as 5 W/cm2K, and thermal conductivity as 70 W/cm-K. One dimensional steady state condition was assumed to solve the problem. Corresponding results are given and compared with the known result in special cases. MATERIALS AND METHODS 1. Analytical method 2. Finite difference method (FDM) Qleft + Qright + qm .A. x = 0 Where, qm is the energy generation at node m, and m = 1, 2,3……m-1. 3. Finite element method (FEM) Steps: (i). Discretization of the problem Divide in small sub domain/ element. Each and every element has unique number. (ii). Finite element approximation ………………..+ Where T1, T2 = temperature at respective nodes of the element N1, N2 = Shape functions (iii). Elemental equation for heat transfer (iv). Assembly (global equation) = 8
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME (v). Imposition of boundary conditions MODIFIED MATRIX (vi). Nodal solution Solving the matrix obtained by using MATLAB. Here the problem is solved by fem firstly by increasing nodes and secondly by increasing element. Six cases will arise with two different number of nodes (i.e. 2 and 3) and three different number of elements ( i.e. 1, 2 and 3). Each case is solved and comparison of all results is done with the exact solution. 4. Fuzzy finite element method In this method the value of thermal conductivity K is considered as TFN [ 69, 70, 71] and heat transfer coefficient h is considered as TFN[ 4, 5,6 ] . The corresponding values of temperature in this interval are plotted with the help of fuzzy. Triangular Fuzzy Number (TFN) and alpha cut Let X denote a universal set. Then, the membership function by which a fuzzy set A is where [0,1] denotes the interval of real numbers from 0 usually defined as the form , to 1. Such a function is called a membership function and the set defined by it is called a fuzzy set. A fuzzy number is a convex, normalized fuzzy set which is piecewise continuous and has the functional value (x) =1 , where at precisely one element. Different types of fuzzy numbers are there. These are triangular fuzzy number, trapezoidal fuzzy number and Gaussian fuzzy number etc. Here we have discussed the said problem using triangular fuzzy number only. The membership function for triangular fuzzy number is as below. Fig.2 Triangular Fuzzy Number (TFN) [a, b, c] Alpha cut is an important concept of fuzzy set. Given triangular fuzzy number may be written as [a, b, c] may be written as 9 then the alpha cut for above ].
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME RESULTS AND DISCUSSION S.NO. DISCRETE POINTS TEMPERATURE ANALYTICAL FDM FEM 2N1E 2N2E 2N3E 3N1E 3N2E 3N3E 1 At x= 0 T1(given ) 140 140 140 140 140 140 140 140 2 At x= 1.25 T2 104.54 131.8075 119.622 110.89 106.673 106.489 104.555 105.6 3 At x = 2.5 T3 83.75 124.3149 99.245 81.78 85.116 83.6753 83.822 84.6459 4 At x =3.75 T4 72.91 117.4077 78.867 74.59 73.1395 71.55 72.9751 74.54 5 At x = 5.00 T5 69.55 107.787 58.49 67.4 68.53 70.1321 69.6341 70.6384 Temperature in celsius T at x =1.25 150 100 50 T at x =1.25 0 EXACT FDM 2N 1 E 2N 2 E 2N 3 E 3N 1 E 3N 2 E 3N 3 E Different methods Fig.3 Value of temperature at x=1.25, T2 Temperature in Celsius T at x= 2.5 150 100 50 T at x= 2.5 0 exact FDM 2N 1E 2N 2E 2N 3E 3N 1E 3N 2E Different methods Fig.4 Value of temperature at x=2.5, T3 10 3N 3E
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME T at x =3.75 Temperature in celsius 150 100 50 T at x =3.75 0 exact FDM 2N 1E 2N 2E 2N 3E 3N 1E 3N 2E 3N 3E Different methods Fig.5 Value of temperature at x=3.75, T4 Temperature in celsius T at x = 5 150 100 50 T at x = 5 0 exact FDM 2N 1E 2N 2E 2N 3E 3N 1E 3N 2E 3N 3E Different methods Fig.6. Value of temperature at x= 5, T5 Representation of FEM Results by Fuzzy Plot For Temperature at point 2 . T2 Alpha values Fig7. 2 noded 1 element Fig.8. 2 noded 2 element 11
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME Alpha values Fig.9. 2 noded 3 element Fig. 10. 3 noded 1 element Fig.12. 3 noded 3 element Fig.11. 3 noded 2 element Alpha values For temperature at point 3 T3 Alpha values Fig.12. 2 noded 1 element Fig.13. 2 noded 2 element Fig.14. 2 noded 3 element Fig.15. 3 noded 2 element Alpha values 12
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME For temperature at point 4 T4 Alpha values Fig.15. 2 noded 1 element Fig.15. 2 noded 2 element Fig.15. 2 noded 3 element Fig.15. 3 noded 1 element Fig.15. 3 noded 2 element Fig.15. 3 noded 2 element Alpha values Alpha values For temperature at point 5 T5 Alpha values 2 noded 2 element 2 noded 1element 13
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME Alpha values 3 noded 1 element 2 noded 3 element Alpha values 3 noded 2 element 3 noded 3 element Here heat transfer problem is solved by Finite Difference Method and Finite Element Method. FDM shows a large variation in comparison to the exact solution or analytical solution. FEM results are near to the exact solution. In FEM we have taken 6 cases, by increasing the element and by increasing the node. By increasing the node we get better result. But results of increasing the number of elements are very close to the exact solution. So, it is clear that while increasing the node we get approximate solutions to the exact solution. By increasing node, we get less complication to solve the problem. Considering the uncertain parameters as fuzzy we have presented the result of the said problem in fuzzy plot. When the value of the alpha becomes zero the fuzzy results changes to interval form & for the value of alpha as one, the result changes into crisp form. We get a series of narrow & peak distribution of temperatures which reflect better solution for the said problem & are very close to the solution obtained from the traditional finite element method with crisp parameters. CONCLUSION The present work an attempt has been made to find the solution of a one dimensional steady state heat conduction problem by using different techniques with and without consideration of uncertainty in the material properties. Results obtained through different techniques are compared and the effect of varying number of nodes and number of elements on the results obtained by FEM is also studied. Uncertain parameters are considered as fuzzy and results are presented in triangular fuzzy plot. 14
  • International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 – 6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 2, February (2014), pp. 06-15, © IAEME REFERENCES 1) De Baerdemaeker, J., Computation of heat conduction in materials with random variable thermophysical properties. Int. J. Numer. Methods Eng. 36,(1993), 523–536. 2) Bart M. Nicolai, Jose A. Egea, NicoScheerlinck, Julio R. Banga, Ashim K. Datta, (2011) Fuzzy Finite Element Analysis of Heat Conduction Problems with Uncertain Parameters, Journal of Food Engineering 103, 38–46. 3) Rao, S. S., Sawyer, P., (1995) "Fuzzy Finite Element Approach for Analysis of Imprecisely Defined Systems", AIAA Journal, v. 33, n. 12, pp. 2364-2370. 4) Naidoo P. (2003) The Application of Intelligent Technology on a Multi-Variable Dynamical System, Meccanica, 38(6):739-748. 5) Bart M. Nicolai, Jose A. Egea, NicoScheerlinck, Julio R. Banga, Ashim K. Datta, Fuzzy Finite Element Analysis of Heat Conduction Problems with Uncertain Parameters, Journal of Food Engineering 103 (2011) 38–46. 6) Majumadar Sarangam and Chakraverty S. may (2012) fuzzy finite element method for one dimensional steady state heat conduction problem. 7) Ajeet Kumar Rai and Mustafa S Mahdi, “A Practical Approach to Design and Optimization of Single Phase Liquid to Liquid Shell and Tube Heat Exchanger”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 3, Issue 3, 2012, pp. 378 - 386, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. 8) Ajeet Kumar Rai and Ashish Kumar, “A Review on Phase Change Materials & Their Applications”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 3, Issue 2, 2012, pp. 214 - 225, ISSN Print: 0976-6480, ISSN Online: 0976-6499. 9) N.G.Narve and N.K.Sane, “Heat Transfer and Fluid Flow Characteristics of Vertical Symmetrical Triangular Fin Arrays”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 2, 2013, pp. 271 - 281, ISSN Print: 0976-6480, ISSN Online: 0976-6499. 10) Ajeet Kumar Rai, Shahbaz Ahmad and Sarfaraj Ahamad Idrisi, “Design, Fabrication and Heat Transfer Study of Green House Dryer”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 4, 2013, pp. 1 - 7, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. 15