EXACT SOLUTION OF FIN PROBLEM WITH LINEAR TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY
A.H. Abdel Kader, M.S. Abdel Latif, H.M.
Nour
Mathematics and Engineering Physics
Department, Faculty of Engineering
Mansoura University, Egypt
Leaderabass87@gmail.com, m_gazia@mans.edu.eg, hanour@mans.edu.eg
Received: 20
September 2016; accepted: 15 November 2016
Abstract: In
this paper, we obtain the general exact solution of a nonlinear fin equation
which governs heat transfer in a rectangular fin with linear
temperature-dependent thermal conductivity using the partial
Noether method. The
relationship between the fin efficiency and the
thermo-geometric fin parameter is obtained. Additionally, we obtained the relationship among the fin effectiveness,
the thermo-geometric fin parameter and the Biot number.
Keywords: exact solution, fin
equation, fin efficiency, thermal conductivity
1. Introduction
In this paper, we
assume that the rectangular fin subjected to some assumptions such as steady
state heat transfer operation with no heat generation, the fin tip
is insulated, and the heat transfer is one dimensional. Under these
assumptions,
the energy balance equation of rectangular fin is given by [1-11]
![](2016_4/art_06/image001.png) | (1) |
where,
is the fin temperature,
is the axial distance
measured from the fin tip,
is the cross-sectional
area of the fin,
is the fin
perimeter,
is the thermal
conductivity of the fin,
is the heat transfer
coefficient and
is the ambient
temperature.
Here, we take the
heat transfer coefficient
as a
constant and the thermal conductivity
as a linear function
of temperature [1-9]
where
is the thermal
conductivity of the fin at the ambient temperature
,
is
a constant.
Substituting (2) into (1), we obtain
To make Eq. (3) dimensionless; the following
transformations are introduced
[1-14]
where,
is the length of fin,
is the temperature of the heat source where
the fin is attached and the parameter
is called thermo geometric fin parameter.
Using the transformations (4), Eq. (3) becomes
Equation (5) can be rewritten as
The boundary
conditions are:
At the fin tip
, since the fin tip is
insulated, so
Using the transformation (4), Eq. (7) becomes
At the fin base
the fin temperature is
the same temperature as the heat source ![](2016_4/art_06/image016.png)
Using the transformation (4), Eq. (9) becomes
Approximate solutions of Eq. (6) with
boundary conditions (8) and (10) are
investigated using the Parameterized Perturbation
method in [1], by using optimal homotopy asymptotic method in [2], by using the
homotopy analysis method
in [3, 4], by using the Residue minimization technique in [5], by using
the variational iteration method in [6] and by using the decomposition method
in [7, 8].
The homotopy analysis method is widely used in investigating many fin problems
in [12-14]. In this paper, we will obtain the exact solution of Eq. (6) using
the partial Noether method.
The paper will be organized as follows: In
section 2, the exact solution of
Eq. (6) is obtained using the partial Noether method. In section 3, the fin
efficiency will be discussed. In section 4, the fin effectiveness is studied.
In section 5, we will discuss the obtained results in this paper.
2. Partial Noether Method
Definition [15, 16]. A Lie operator
of a form
is called a partial Noether operator
corresponding to a partial Lagrangian
, if there exists
a function
, such that
where
is the total differentiation with respect to
and
is called the Euler-
-Lagrange operator, which are defined as,
Theorem [15, 16]. If the Lie operator (11) is a partial Noether operator corresponding
to a partial Lagrangian
of Eq. (6), then the
first integral
of (6) is
given by
which is satisfied by the conservation law
where,
is a Noether operator
which is defined as:
where,
Consider the partial Lagrangian of Equation (6)
[15, 16]
where,
To obtain the partial Noether operator of Eq.
(6), we will substitute (18) and (19) into the condition (12) to obtain
Substituting (11) into (20), we obtain the
determining equation
Let,
and
, the determining equation (21) becomes
Equating the coefficients of the derivatives of
with zero, we obtain
The solution of system (23)-(26) is given by
Substituting (18) into (16), we obtain
Substituting (27) and (28) into (14), we obtain
Suppose the first integral
, hence, we obtain
where
is a constant. Using
the boundary condition (8), we can determine
the constant
as follows
where,
is the temperature of
fin at the fin tip
Substituting (31) into
(30), we obtain
Let,
Hence, Eq. (32) becomes
Integrating Eq. (33), we obtain
Hence, we obtain the following exact implicit
solution of Eq. (5)
where,
is the
incomplete elliptic integral of the second kind, which is defined as [17]
is the incomplete elliptic integral of the first kind, which is defined as
[17]
and ![](2016_4/art_06/image076.png)
The solution (35) has an unknown parameter
namely
. This parameter can be
easily determined with the help of the boundary condition
as follows:
Equation (36) shows the relation between the
temperature at fin tip
and the
thermo-geometric parameter
and ![](2016_4/art_06/image079.png)
Figure 1 shows the effect of the
thermo-geometric parameter
on the fin tip
temperature
. We find that the fin
tip temperature
decreases with
increasing
Figure 2 shows the distribution of fin temperature
along the fin. We find that
the fin temperature decreases with increasing ![](2016_4/art_06/image089.png)
3. Fin efficiency
The fin efficiency
is the ratio of the
actual heat transfer rate
from the fin to ideal
heat transfer
rate from the fin if
the entire fin were at base temperature
[3-11]
Using Eq. (5), Eq. (37) becomes
Using the boundary conditions (8) and (10), Eq. (38)
becomes
From (32), when
we obtain
Substituting (40) into (39), we obtain
Using the relations (41) and (36), we can plot
the relation between the efficiency
and the thermo-geometric fin parameter
(see Figure 3).
3. Fin effectiveness
Fin effectiveness ϵ is the ratio of
heat transferred from the fin area
to the heat which
would be transferred if entire fin area was at base temperature
[11]
where,
is a parameter
which depends on the fin geometry.
From (37), we find
Substituting (41) into (43), we obtain
The parameter
can be rewritten in the form
where,
is the Biot number.
Substituting (45) into (44), we obtain
Using the relations (45), (46) and (36), we
can plot the relation between the fin effectiveness
and the parameters
and
(Figures 4
and 5).
Figure 3 shows the effect of the
thermo-geometric parameter
on fin efficiency
. We find that the fin efficiency
decreases with increasing
Figure 4 shows
the effect of the parameter
on fin effectiveness
. We find that fin effectiveness
increases with increasing ![](2016_4/art_06/image122.png)
![Kader_Fig-5.jpg](2016_4/art_06/image123.jpg)
Fig. 5. Plot the
relation between the fin effectiveness
and
when
for various values of ![](2016_4/art_06/image117.png)
Figure 5 shows the effect of the Biot number on
fin effectiveness
We find that
fin effectiveness
decreases with increasing
.
5. Discussions and concluding remarks
In this paper, we obtain the general exact
solution (35) of the fin equation (5) which is subjected to the boundary
conditions (8) and (10). The solution is valid for all values of the thermo-geometric
fin parameters
and
We observe in Figure 1 that the fin tip temperature
decreases with an
increase in the thermo-geometric parameter
. Figure 2 shows that
the fin temperature
increases with an
increasing
(in other words, the temperature increases when approaching a heat source). The
relation between the fin efficiency
and the parameters
and
is obtained.
Figure 3 shows that the fin efficiency
decreases with
increasing the thermo-geo-
metric parameter
The relation between the fin effectiveness
and the parameters,
and Biot number
is obtained. Figures 4 and 5 show that the fin effectiveness
increases when increasing
and decreases when increasing the Biot number
.
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