Exact solution of fin problem with linear temperaturedependent thermal conductivity
A.H. Abdel Kader
,M.S. Abdel Latif
,H.M. Nour
Journal of Applied Mathematics and Computational Mechanics 
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EXACT SOLUTION OF FIN PROBLEM WITH LINEAR TEMPERATUREDEPENDENT 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 temperaturedependent thermal conductivity using the partial Noether method. The relationship between the fin efficiency and the thermogeometric fin parameter is obtained. Additionally, we obtained the relationship among the fin effectiveness, the thermogeometric 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 [111]
(1) 
where, is the fin temperature, is the axial distance measured from the fin tip, is the crosssectional 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 [19]


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 [114]


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


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 [1214]. 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
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 thermogeometric parameter and
Fig. 1. Plot of the relation (36) between and for various values of 
Fig. 2. Plot of the relation (35) between and for various values of when 
Figure 1 shows the effect of the thermogeometric 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
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 [311]


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 thermogeometric 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).
Fig. 3. Plot the relation between the efficiency and the thermogeometric parameter for various values of 
Fig. 4. Plot the relation between the effectiveness of fin and when for various values of 
Figure 3 shows the effect of the thermogeometric 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
Fig. 5. Plot the relation between the fin effectiveness and when for various values of
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 thermogeometric fin parameters and We observe in Figure 1 that the fin tip temperature decreases with an increase in the thermogeometric 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 thermogeo 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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