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```					 INTERNATIONAL JOURNAL OF ADVANCED and Technology (IJARET), ISSN 0976 –
International Journal of Advanced Research in Engineering RESEARCH IN ENGINEERING
AND Volume 5, Issue 2, February (2014),
6480(Print), ISSN 0976 – 6499(Online)TECHNOLOGY (IJARET) pp. 06-15, © IAEME

ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
IJARET
Volume 5, Issue 2, February (2014), pp. 06-15
Journal Impact Factor (2014): 4.1710 (Calculated by GISI)
www.jifactor.com

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

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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 steady-
state 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.

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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)\

=

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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
usually defined as the form ,                where [0,1] denotes the interval of real numbers from 0
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             then the alpha cut for above
triangular fuzzy number may be written as [a, b, c] may be written as                            ].

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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.   DISCRET-                             TEMPER-        ANALY-         FDM
E POINTS                              ATURE          TICAL
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

T at x =1.25
150
Temperature in celsius

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

T at x= 2.5
150
Temperature in Celsius

100

50
T at x= 2.5
0
exact       FDM     2N 1E     2N 2E      2N 3E       3N 1E      3N 2E       3N 3E
Different methods

Fig.4 Value of temperature at x=2.5, T3

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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
150
Temperature in celsius
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

T at x = 5
150
Temperature in celsius

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

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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

Alpha
values

Fig.12. 3 noded 3 element                    Fig.11. 3 noded 2 element

For temperature at point 3 T3

Alpha
values

Fig.12. 2 noded 1 element                   Fig.13. 2 noded 2 element

Alpha
values

Fig.14. 2 noded 3 element                   Fig.15. 3 noded 2 element

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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

Alpha
values

Fig.15. 2 noded 3 element                     Fig.15. 3 noded 1 element

Alpha
values

Fig.15. 3 noded 2 element                      Fig.15. 3 noded 2 element

For temperature at point 5 T5

Alpha
values

2 noded 1element                            2 noded 2 element

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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

2 noded 3 element                                3 noded 1 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.

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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.

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