# Comparative Analysis of Power-Law Fin-Type Problems Using Variational Iteration Method  and Finite Element Method

Authors: Mehmet Tarık Atay; Safa Bozkurt Coşkun
Journal: Mathematical Problems in Engineering
 (2008)

---

## Abstract

Variational iteration method is applied to examine the temperature
distribution within a single fin with a one-dimensional steady-state nonlinear heat conduction
equation. Variation of temperature due to different levels of nonlinearities is analyzed. The
results obtained by means of variational iteration method are compared with the results obtained
from finite element method. A fourth iteration variational iteration solution is used in all cases
considered. An error analysis is also conducted to evaluate the performance of proposed solution
technique. The results have shown that variational iteration method is a powerful solution
technique in the analysis of power-law fin-type problems.


---

## Body

## 1. Introduction

Heat
conduction problems for the fins and finned surfaces with a non-uniform heat
transfer coefficient along the fin have been studied extensively
[1]. Heat
transfer coefficient depends on the local temperature difference between the
fin surface and the surrounding medium and can be expressed as a power law.The
temperature distribution along a fin of constant cross-sectional area and the
thermal conductance at the fin base can be modeled as a one-dimensional
steady-state heat conduction equation, which in dimensionless form is given by
[2, 3](1.1)d2θdx2−Nθm=0,0≤x≤1.Subjected to following boundary
conditions:(1.2)θ′(0)=0,θ(1)=1,where the differentiation is with
respect to the dimensionless coordinate x, measured along the fin length from
its adiabatic and θ is the dimensionless temperature, N is the
convective-conductive parameter of the fin and the exponent m depends on the
heat transfer mode. Physical values of the power m of some practical interest
are related to free convection, nucleate boiling and radiation. Some physical
cases are defined in [2] with the geometry of the problem. For example, for the convecting fin problem m=1,N=(2hL2/bk)1/2,θ=(T−Te)/(T−Tb), x=X/L, and for the fin radiating to the zero environment temperature m=4, N=(2ETb3L2/bk)1/2, θ=T/Tb, x=X/L
where E is emissivity, h is heat transfer coefficient, L is fin length, T is temperature, Tb is fin base temperature, Te environment temperature, b is fin thickness, k is thermal conductivity, and X is distance from fin tip. The problem is depicted
in Figure 1.Figure 1
Geometry of
one-dimensional heat conduction problem in a straight fin.The
role of the fin-tip heat transfer has been comprehensively studied in [4] and hence
to avoid additional parameters in the current
problem, only the case of a fin with an insulated tip, see (1.2), is
considered in this paper.The
problem governed by (1.1)-(1.2) possesses an analytical solutions for certain
values of m [3, 5–9]. The analytical solution given in [5] includes a
three-parameter hyper geometric function, and also a Dawson’s integral in [3].Previously,
fin-type problems have been solved by using different methods of
analysis. Yu and Chen [10] used Taylor
transformation method to optimize rectangular fins with variable thermal
parameters. Yu and Chen [11] used differential transformation method for
circular fins with variable thermal parameter. Chiu and Chen [12] used
decomposition method for the analysis of circular fins. Arslantürk [13, 14]
applied Adomian decomposition method successfully for the analysis of
convective fins and space radiators. Lesnic and Heggs [15] also used Adomian
decomposition method for power-law fin-type problems.In recent years a
solution technique called variational iteration method (VIM) [16] has been
given great importance for solving nonlinear differential equations. VIM is a
kind of variational-based analytical technique in efficient solution of
nonlinear differential equations including boundary value and initial value
problems, nonlinear system of differential equations, nonlinear partial
differential equations [17–19].
Coşkun and Atay [20, 21]
used VIM for the analysis of convective straight and radial fins with
temperature-dependent thermal conductivity and examined the effect of
temperature-dependent thermal conductivity considering various cases. These
successful applications of the method to the fin-type problems are the reasons
for choosing VIM in present problem.VIM directly gives the solution of corresponding
equation which is one of its advantages when compared to ADM. The formulation
and solution process of VIM is much easier when compared to decomposition
methods, in this respect; VIM is an easy-to-apply method for the analysis of
nonlinear problems in engineering.In this study,
power-law fin-type problems will be analyzed using VIM. In the analysis, the effects
of convective-conductive parameter of the fin (N) and exponent of the power-law
fin-type (m) which corresponds to heat transfer mode will be studied in detail.
An error analysis will also be conducted by comparing VIM results with the
finite element solution of the same equation.


## 2. VIM Formulation of the Problem

According to VIM, the following
differential equation may be considered:(2.1)Lu+Nu=g(x),where L is a linear operator, N is
a nonlinear operator, and g(x) is an inhomogeneous term.Based on VIM, a correct functional
can be constructed as follows:(2.2)un+1(x)=un(x)+∫0xλ(ξ){Lun(τ)+Nu˜n(τ)−g(τ)}dτ,where λ is a general Lagrangian multiplier, which can
be identified optimally via the variational theory, the subscript n denotes the
nth-order approximation, u˜ is considered as a restricted variation, that
is, δu˜=0.
By using this formulation, a differential equation for obtaining the Lagrangial
multiplier λ(ξ) is(2.3)λ′′(ξ)=0with corresponding conditions,(2.4)λ(ξ)|ξ=x=0,[1−λ′(ξ)]|ξ=x=0.By solving (2.3) with respect to
boundary conditions in (2.4), Lagrangial multiplier λ(ξ) is obtained as follows:(2.5)λ(ξ)=ξ−x.If the above VIM formulation is
applied to  (1.1), following iteration
formula can be obtained accordingly:(2.6)θn+1(x)=θn(x)+∫0xλ(ξ){θn′′(ξ)−Nθ˜nm(ξ)}dξ,where N is previously defined as nonlinear operator in
(2.1). Lagrange multiplier λ is obtained as follows by assuming that  L=d2/dξ2 with the restricted variation δu˜n=0.The iteration
formula given in (2.6) is a simple approximation when compared to ADM formulation
of the same problem [14]. This is due to the fact that new formulation directly
gives the expression instead of summing recursive approximations which is a
tedious work.


## 3. Solutions for Fin Temperature Distribution

As
a starting approximation for VIM solution,θ is assumed as constant, which was assumed as
constant also in ADM solution [14]. For simplicity, first few iterations of VIM
corresponding to N=1 with m=2,3,4 are presented.ForN=1, m=2, first three
iterations are
(3.1a)θ0(x)=B,(3.1b)θ1(x)=B+B2x22,(3.1c)θ2(x)=B+B2x22+B3x412+B4x6120,(3.1d)θ3(x)=B+B2x22+B3x412+B4x672+B5x8560+11B6x1064800+B7x1295040+B8x142620800.ForN=1, m=3, first three
iterations are
(3.2a)θ0(x)=B,(3.2b)θ1(x)=B+B3x22,(3.2c)θ2(x)=B+B3x22+B5x48+B7x640+B9x8448,(3.2d)θ3(x)=B+B3x22+B5x48+3B7x680+23B9x82240+83B11x1033600+101B13x12197120+⋯.ForN=1, m=4, first three
iterations are
(3.3a)θ0(x)=B,(3.3b)θ1(x)=B+B4x22,(3.3c)θ2(x)=B+B4x22+B7x46+B10x620+B13x8112+B16x101440,(3.3d)θ3(x)=B+B4x22+B7x46+13B10x6180+17B13x8560+1789B16x10151200+701B19x12166320+⋯.Analysis of the
problem is conducted forN=0.1,0.5,1,2,5 with m=2,3,4 for each N up
to fourth iteration of θ(θ4(x)). By using these
parameters, the effect of increasing convective-conductive parameter of the fin
(N) and exponent of the power-law fin-type (m) which corresponds to heat
transfer mode is studied.
These parameters cause nonlinearity in (1.1). Hence, with the performed analysis,
the effect of degree of nonlinearity in the solution of this equation is
examined. For this purpose an error criterion is defined and calculated for
each case by using the finite element solution of the equation and VIM solution
with fourth iteration. Error criterion is defined as(3.4)Error=(∑I=1n|θFEM−θVIM|2∑I=1n|θFEM|2)1/2,where θFEM is the finite element
solution, θVIM is the fourth iteration
VIM solution, and n is the number of points where the solution is considered.


## 4. Numerical Results

Proposed solution
technique is applied for different cases explained in previous section. These
different parameters represent the degree of nonlinearity in the corresponding
equation. In this study, finite element solutions of (1.1) are obtained using the
software FlexPDE Version 5. In finite element analysis quadratic basis
functions are used and a modified Newton-Raphson algorithm is employed with a
root mean square error criterion of order 10−7 or less to solve
nonlinear set of resulting equations.First,
the problem is solved for small values of convective-conductive parameterN
with varying m values. To this aim, N is chosen as 0.1 and 0.5.Figure2 shows the variation of θ with
respect to x for N=0.1. From the
figure it seems that VIM solution is in
excellent agreement with FEM solutions. From Figure 8, a very small error of
order 10−8 can be observed.Figure 2
Dimensionless temperature distribution for convective-conductive parameterN=0.1.ForN=0.5, the results produced by using both methods are
shown in Figure 3.
Again, an excellent agreement can be seen from this figure with the same order
of errors as in
previous case.Figure 3
Dimensionless temperature distribution for convective-conductive parameterN=0.5.At
the next stage, the analysis is conducted forN=1 and N=2.
Figures 4 and 5
show the results obtained for these cases. From the graphs the results obtained
from both methods are still in very good agreement, however as shown
in Figures 7  and 8
errors are increased gradually for these cases when compared to first
two cases.Figure 4
Dimensionless temperature distribution for convective-conductive parameterN=1.Figure 5
Dimensionless temperature distribution for convective-conductive parameterN=2.Last
case study is performed forN=5 with the same m values. The results are shown
in Figure 6. From the analysis, it can be observed that VIM and FEM results are
still in good agreement but maximum errors are obtained as it can be seen form
Figures 7 and 8
for this case when
compared to previous cases.Figure 6
Dimensionless temperature distribution for convective-conductive parameterN=5.Figure 7
Variation of errors with respect to
heat transfer mode (m) for different convective-conductive parameters (N).Figure 8
Variation of errors with respect to
convective-conductive parameter (N) for different heat transfer modes (m).Case
studies have shown that increasing nonlinearity due to the increasing
parametersN and m resulted in increasing error values. However, obtained VIM
results are still in good agreement with the FEM results even for the maximum
values of N and m considered.In
order to show the effect of nonlinearity in the solutions obtained, two
different error analyses are
conducted with respect toN and m parameters. In all cases considered, a fourth
iteration solution in VIM analysis is used. From Figure 7, it can be observed
that errors increase with increasing m for each N values. Also, maximum errors
are obtained for maximum N value (N=5).
Figure 8
describes variation of error
with respect to N for different m values. From the graph, it can be observed
that error increases with increasing N even though N is a constant value. Equation
(1.1)  includes a nonlinear term multiplied by this constant N.
Figure 8  shows
that N has a significant effect on the solution of the governing equation given
in (1.1). Hence, with the increasing multiplier for the nonlinear term in (1.1),
more iteration may be required for an accurate analysis. However, fourth
iteration VIM solution used in this study is accurate enough for the analysis
of the problem with the values considered in this study.


## 5. Conclusions

In this paper,
power-law fin-type problems have been considered. The nonlinear fin equation
has been solved by using the VIM which provides an analytical expression
converging to the solution of the governing equation. These results were
compared with the FEM results of the same equation and the effect of
nonlinearity due to the convective-conductive parameter of the fin (N) and
exponent of the power-law fin-type (m) has been analyzed. The results have
shown that VIM can be efficiently used in the analysis of power-law fin-type
problems and another remarkable point is that the analysis with VIM requires not
much effort with respect to formulation of the problem and iteration required.
An additional important point is that the error obtained has been increased
with the parameters affecting the nonlinearity of the governing equation.
However, VIM handles the nonlinearity successfully even with less iteration for
the cases considered. Another advantage of the VIM is that the method gives an
analytical expression as a solution. This is a superiority when compared to
mesh-based numerical methods.



---

*Source: 635231-2008-07-21.xml*