Steady Thermal Analysis of Two-Dimensional Cylindrical Pin Fin with a Nonconstant Base Temperature

Abstract

Steady heat transfer through a pin fin is studied. Thermal conductivity, heat transfer coefficient, and the source or sink term are assumed to be temperature dependent. In the model considered, the source or sink term is given as an arbitrary function. We employ symmetry techniques to determine forms of the source or sink term for which the extra Lie point symmetries are admitted. Method of separation of variables is used to construct exact solutions when the governing equation is linear. Symmetry reductions result in reduced ordinary differential equations when the problem is nonlinear and some invariant solution for the linear case. Furthermore, we analyze the heat flux, fin efficiency, and the entropy generation.

1. Introduction

Fins play an important role in increasing the efficiency of heating systems which is achieved by increased (extended) surface area. In particular, fins are used in power generators, air conditioning, semiconductors, refrigeration, cooling of computer processor, exothermic reactors, and many other devices in which heat is generated and must be transported. Theory, solutions and applications of problems on extended surfaces may be found in texts such as [1].

One-dimensional steady state numerical analysis have been considered, for example, by [2–5], and exact solutions were constructed via symmetry techniques in [6]. The transient one-dimensional fin problem has attracted sizeable interest from the Lie symmetry analysts (see, e.g., [7–11]). Two-dimensional transient analysis have been carried out for fins without heat source or sink in [12]. Solutions for two-dimensional fin models exist for the constant thermal conductivity (see e.g., [13–16]). Recently, heat transfer and entropy generation in two-dimensional orthotropic pin fin has been studied in [17]. In [18], the authors combined the Laplace transformation and the finite difference methods to determine solutions for the two-dimensional pin fins with nonconstant base heat flux. A search for exact and numerical solutions for heat transfer in extended surfaces continues to be of scientific interest. perhaps the interest is instilled by frequent encounters in many engineering applications. To cite a few, some other contributions of heat flow particularly in pin fins may be found, for example, in [19, 20].

In this paper, we consider the steady heat flow through a two-dimensional pin fin with a temperature-dependent internal heat generating or extracting function and thermal conductivity. Furthermore, heat is transferred at the boundary through the temperature-dependent heat transfer coefficient. Symmetry analysis is employed to determine all possible forms of the source or sink term for which the problem is at least reducible to ordinary differential equations. The paper is arranged as follows: in Section 2, we provide mathematical formulation of the problem. Section 3 deals with the symmetry analysis, and exact solutions are constructed in Section 4. The fin efficiency and heat flux are given in Section 5. Entropy analysis is carried out in Section 6. Lastly, we provide the discussions in Section 7 and concluding remarks in Section 8.

2. Mathematical Formulation

We consider a two-dimensional pin fin with length 𝐿 and radius 𝑅. The fin is attached to a base surface of temperature (𝑇_𝑏−𝑇_∞)𝑔(𝑅) and extended into the fluid of temperature 𝑇_∞. The tip of the fin is insulated (i.e., heat transfer at the tip is negligibly small). The fin is measured from the tip to the base. A schematic representation of a pin fin is given in Figure 1. We assume that the heat transfer coefficient along the fin is nonuniform and temperature dependent and that the fin has internal heat source or sink. Furthermore, the temperature-dependent thermal conductivity is assumed to be the same in both radial and axial directions. The base temperature is assumed to be nonconstant (see, e.g., [19, 20]). The energy balance equation is written as1𝑅𝜕𝜕𝑅𝑅𝐾(𝑇)𝜕𝜕𝑅𝑇+𝜕𝜕𝑋𝐾(𝑇)𝜕𝜕𝑋𝑇=𝑆(𝑇).(2.1) The boundary conditions are𝜕𝑇𝜕𝑋=0,𝑋=0,0≤𝑅≤𝑅^𝑎𝑇−𝑇_∞,=𝑇𝜕𝑇_𝑏−𝑇_∞𝑔(𝑅),𝑋=𝐿,0≤𝑅≤𝑅_𝑎,𝐾(𝑇)𝜕𝑇𝜕𝑅=0,𝑅=0,0≤𝑋≤𝐿,𝜕𝑅=−𝐻(𝑇)𝑇−𝑇^∞,𝑅=𝑅_𝑎,0≤𝑋≤𝐿.(2.2) Here, 𝑅_𝑎 is radial distance from the center to the surface of the pin fin.

Figure 1 

Schematic representation of pin fin.

Introducing the dimensionless variables𝜃=𝑇−𝑇^∞𝑇_𝑏−𝑇_∞,𝑥=𝑋𝐿,𝑟=𝑅𝑅_𝑎,𝑘=𝐾𝐾_𝑎,𝐸²=𝐿𝑅_𝑎²𝑠(𝜃)=𝐿₂,𝑆(𝑇)𝐾_𝑎𝑇_𝑏−𝑇_∞,ℎ=𝐻ℎ_𝑏,(2.3) we obtain𝐸²1𝑟𝜕𝜕𝑟𝑟𝑘(𝜃)𝜕𝜕𝜃𝑟+𝜕𝜕𝑥𝑘(𝜃)𝜕𝜕𝑥𝜃=𝑠(𝜃),(2.4) and the boundary conditions are𝜕𝜃𝜃=𝜕𝜕𝑥𝜃𝑔=(𝑟)0,,𝑥𝑥==0,1,𝑘(𝜃)𝜕𝜃𝜕𝑟=0,𝑟=0,𝜕𝑟=−𝐵𝑖ℎ(𝜃)𝜃,𝑟=1,(2.5) where 𝐸 is the fin extension factor, ℎ_𝑏 is the fin base heat transfer coefficient, and 𝐵𝑖 is the Biot number given by 𝐵𝑖=ℎ^𝑏𝑅_𝑎𝐾_𝑎.(2.6)

Three physically realistic functions of thermal conductivity are (i) 𝐾 linearly dependent on temperature, (ii) the power law case, and (iii) the exponential case. We focus on two cases: (i) 𝐾(𝑇) depending linearly on temperature; that is, 𝐾(𝑇)=𝐾_𝑎1+𝛽𝑇−𝑇_∞,(2.7) and (ii) 𝐾(𝑇) is given by the power law (nonlinear) 𝐾(𝑇)=𝐾_𝑎𝑇𝑇_𝑏−−𝑇𝑇^∞_∞^𝑛.(2.8) In dimensionless variables, we have (i)𝑘(𝜃)=1+𝐵𝜃,(ii)𝑘(𝜃)=𝜃^𝑛,(2.9) where 𝐵=𝛽(𝑇_𝑏−𝑇_∞). Here, 𝐾_𝑎 is the thermal conductivity of the fin at ambient temperature, 𝛽 is the thermal conductivity gradient, 𝑛 is an exponent, and 𝐵 is the thermal conductivity parameter. Applying the Kirchoff's transformation (see, e.g., [16]),𝑤(𝑥,𝑟)=^𝜃0𝑘𝜃_∗𝑑𝜃_∗.(2.10) The boundary value problem reduces to𝐸²1𝑟𝜕𝜕𝑟𝑟𝜕𝜕𝑤𝑟+𝜕²𝑤𝜕𝑥²𝜕𝑤=𝑠(𝑤),(2.11)𝑤(1,𝑟)=𝑔(𝑟)+𝐵𝜕𝑥=0,𝑥=0,(2.12)2(𝑔(𝑟))²=𝐹(𝑟),given𝑘=1+𝐵𝜃,(2.13)

or𝑤(1,𝑟)=𝑔(𝑟)^𝑛+1𝑛+1=𝐺(𝑟),given𝑘=𝜃^𝑛𝜕𝑤,𝑛≠−1,(2.14)𝜕𝑤𝜕𝑟=0,𝑟=0,(2.15)𝜕𝑟=−𝐵𝑖𝑤,𝑟=1,(2.16) where in ℎ(𝜃)=1+𝐵ℎ(𝜃)=𝜃_𝑛2𝜃,when𝑘(𝜃)=1+𝐵𝜃,𝑛+1,when𝑘(𝜃)=𝜃^𝑛,𝑛≠−1.(2.17) Note that 𝑤=ln𝜃 when 𝑛=−1 so that (2.4) holds.

3. Classical Lie Point Symmetry Analysis

In brief, a symmetry of a differential equation is an invertible transformation of the dependent and independent variables that leaves the original equation invariant (unchanged). Furthermore, a symmetry of differential equations maps an arbitrary solution to another solution of the same differential equation. Symmetries depend continuously on a parameter and form a group, the one-parameter group of transformations. The classical Lie groups of point invariance transformations act on the system's graph space that is coordinated by the independent and dependent variables. This group may be determined algorithmically (by Lie's classical method), and there are a number of computer algebraic packages developed to construct symmetries (see, e.g., [21–23]), or one may use interactive packages such as REDUCE [24].

Differential equations arising in modelling real-world problems often involve one or more functions depending on either the independent variable or on the dependent variables. It is possible by symmetry techniques to determine the cases which allow the equation in question to admit extra symmetries. The exercise of searching for the forms of arbitrary functions for which extra symmetries are admitted is called group classification. The notion of group classification is pioneered by Lie [25].

The theory and applications of symmetry analysis may be found in excellent text such as those of [26–29]. We adopt the direct methods in [27] (which exclude explicit equivalence transformation analysis) to determine possible forms of the source term for which (2.11) admits extra point symmetries.

In essence, determining classical Lie point symmetries for the governing Equation (2.11) implies seeking transformation of the form𝑥𝑟_∗∗=𝑟+𝜖𝜉¹(𝑟,𝑥,𝑤)+𝑂𝜖²,𝑤_∗=𝑥+𝜖𝜉²(𝑟,𝑥,𝑤)+𝑂𝜖²,=𝑤+𝜖𝜂(𝑟,𝑥,𝑤)+𝑂𝜖²,(3.1) generated by the vector fieldΓ=𝜉¹(𝑟,𝑥,𝑤)𝜕𝜕𝑟+𝜉²(𝑟,𝑥,𝑤)𝜕𝜕𝑥+𝜂(𝑟,𝑥,𝑤)𝜕𝜕𝑤,(3.2) which leaves the governing equation invariant. Note that we seek symmetries that leave the single (2.11) invariant rather than the entire boundary value problem. This is because the number of symmetries admitted by the governing equation and the imposed boundary conditions is less than for those admitted by the single governing equation. One may apply the boundary condition to the obtained invariant solutions.

The action of Γ is extended to all the derivatives appearing in the governing equation through the second prolongationΓ^[2]=Γ+𝜁^𝑥𝜕𝜕𝑤_𝑥+𝜁^𝑟𝜕𝜕𝑤_𝑟+𝜁^𝑥𝑥𝜕𝜕𝑤_𝑥𝑥+𝜁^𝑟𝑟𝜕𝜕𝑤_𝑟𝑟+⋯,(3.3) where𝜁𝜁^𝑥_𝑟=𝐷_𝑥(𝜂)−𝑤_𝑟𝐷_𝑥𝜉¹−𝑤_𝑥𝐷_𝑥𝜉²,𝜁_𝑥𝑥=𝐷_𝑟(𝜂)−𝑤_𝑟𝐷_𝑟𝜉¹−𝑤_𝑥𝐷_𝑟𝜉²,𝜁_𝑟𝑟=𝐷_𝑥(𝜁^𝑥)−𝑤_𝑥𝑟𝐷_𝑥𝜉¹−𝑤_𝑥𝑥𝐷_𝑥𝜉²,=𝐷_𝑟(𝜁^𝑟)−𝑤_𝑟𝑟𝐷_𝑟𝜉¹−𝑤_𝑟𝑥𝐷_𝑟𝜉²,(3.4) with 𝐷_𝑟=𝜕𝜕𝑟+𝑤^𝑟𝜕𝜕𝑤+𝑤^𝑟𝑟𝜕𝜕𝑤_𝑟+𝑤_𝑥𝑟𝜕𝜕𝑤_𝑥𝐷+⋯,_𝑥=𝜕𝜕𝑥+𝑤^𝑥𝜕𝜕𝑤+𝑤^𝑥𝑥𝜕𝜕𝑤_𝑥+𝑤_𝑥𝑟𝜕𝜕𝑤_𝑟+⋯,(3.5) being the operators of total derivatives. The generator Γ is a Lie point symmetry of (2.11), ifΓ^[2](Equation(2.11))∣_{Equation(2.11)}=0.(3.6)

The invariance condition (3.6) yields the determining equations−𝐸²1𝑟²𝑤_𝑟𝜉¹+𝐸²1𝑟𝜁^𝑟+𝐸²𝜁^𝑟𝑟+𝜁^𝑥𝑥=𝜂𝑠^(𝑤),(3.7) on solutions of (2.11). Here prime implies differentiation with respect to 𝑤. Since the coefficient of Γ (the infinitesimals) does not involve the derivatives of the dependent variable, we can separate (3.7) with respect to these derivatives and solve the resulting overdetermined system of linear homogeneous partial differential equations. Further calculations are omitted at this stage since they were facilitated by the freely available interactive computer algebra package REDUCE [24].

In the initial symmetry analysis of (2.11) where 𝑠 is arbitrary, we obtained nothing beyond translation in 𝑥. The cases of the sink or source term for which the principal Lie algebra is extended are listed in Table 1. Wherever they appear in Table 1, 𝑎,𝑏,𝑝, and 𝑞 are arbitrary constants.

Table 1 

Extra point symmetries admitted by (2.11).

4. Exact Solutions

In this section we construct exact solutions, first using method of separation of variables and secondly using symmetry techniques. Note that 𝑠=0 and 𝑠=𝑤 renders (2.11) linear and hence solvable by method of separation of variable. In fact, the solutions have been constructed, for example, in [16] when 𝑠=0 (source or sink term is neglected).

4.1. Exact Solutions by Separation of Variables

We consider (2.11) with a linear source term. Three cases arise for the separation constant, 𝜎. Note that 𝜎=0 leads to trivial solutions.

4.1.1. Case 𝜎=−𝜆², 𝜆>0

Exact solution to (2.11) is given by𝑤(𝑥,𝑟)=^∞_𝑚₌₁𝑑_𝑚cosh𝜆_𝑚𝑥𝐽₀𝛿_𝑚𝑟,(4.1) where eigenvalues 𝜆_𝑚 satisfy𝛿_𝑚𝐽₁𝛿_𝑚=−𝐵𝑖𝐽₀𝛿_𝑚,(4.2) with 𝛿_𝑚=𝜆^2𝑚−1/𝐸,𝜆^2𝑚≠1, for all 𝑚=1,2,3,…, and𝑑_𝑚=∫¹⁰𝑟𝐹(𝑟)𝐽₀𝜆_𝑚𝑟𝑑𝑟cosh𝜆_𝑚∫¹⁰𝑟𝐽²⁰𝜆_𝑚𝑟𝑑𝑟,(4.3) given boundary condition (2.13); otherwise, 𝐹(𝑟) may simply be replaced by 𝐺(𝑟) if condition (2.14) is given. Here, 𝐽₀ and 𝐽₁ are Bessel functions of order 0 and 1, respectively [30]. Solution (4.1) is depicted in Figures 2, 3, 4, 5, 6, 7, 8, and 9. In Figures 2–5, we have plotted solution (4.1) given thermal conductivity which is linearly dependent on temperature, whereas in Figures 6–9, we considered a power law thermal conductivity. Note that for all 𝜆^2𝑚=1, we obtain trivial solutions.

Figure 2 

Temperature distribution using 550 terms of solution (4.1) with 𝐸=2,𝐵𝑖=0.2 and 𝑔(𝑟)=𝑟2. Here, 𝑘 is linear in 𝜃.

Figure 3 

Temperature profile at 𝑟=0 using 550 terms of the series (4.1) with 𝐸=2, 𝐵𝑖=0.2 and 𝑔(𝑟)=𝑟2. Here, 𝑘 is linear in 𝜃.

Figure 4 

Temperature profile at 𝑥=0 using 550 terms of the series (4.1). Here, we used parameters 𝐸=2, 𝐵𝑖=0.2 and 𝑔(𝑟)=𝑟2. Here, 𝑘 is linear in 𝜃.

Figure 5 

Temperature profile at 𝑥=1 using 550 terms of the series (4.1) with 𝐸=2, 𝐵𝑖=0.2 and 𝑔(𝑟)=𝑟2. Here, 𝑘 is linear in 𝜃.

Figure 6 

Temperature distribution using 550 terms of solution (4.1) with 𝐸=2, 𝐵𝑖=0.2 and 𝑔(𝑟)=𝑟2. Here, 𝑘 is given by the power law.

Figure 7 

Temperature profile at 𝑟=0 using 550 terms of the series (4.1) with 𝐸=2, 𝐵𝑖=0.2 and 𝑔(𝑟)=𝑟2. Here, 𝑘 is given by the power law.

Figure 8 

Temperature profile at 𝑥=1 using 550 terms of the series (4.1) with 𝐸=2,𝐵𝑖=0.2 and 𝑔(𝑟)=𝑟2. Here, 𝑘 is given by the power law.

Figure 9 

Temperature profile at 𝑥=0 using 550 terms of the series (4.1). Here, we used parameters 𝐸=2,𝐵𝑖=0.2 and 𝑔(𝑟)=𝑟2. Here, 𝑘 is given by the power law.

4.1.2. Case 𝜎=𝜆², 𝜆>0

In this case, we obtain the exact solution𝑤(𝑥,𝑟)=^∞_𝑚₌₁𝑐_𝑚cos𝜆_𝑚𝑥𝐼₀𝛼_𝑚𝑟,(4.4) where eigenvalues 𝜆_𝑚 satisfy𝛼_𝑚𝐼₁𝛼_𝑚=−𝐵𝑖𝐼₀𝛼_𝑚,(4.5) with 𝛼_𝑚=1+𝜆^2𝑚/𝐸 and𝑐_𝑚=∫¹⁰𝑟𝐹(𝑟)𝐼₀𝜆_𝑚𝑟𝑑𝑟cos𝜆_𝑚∫¹⁰𝑟𝐼²⁰𝜆_𝑚𝑟𝑑𝑟.(4.6) Here, 𝐼₀ and 𝐼₁ are modified Bessel functions of order 0 and 1, respectively [30]. We omit further analysis of solution 25, since similar observations to solution 24 are obtained. Note that in terms of the original variables, we obtain solutions𝜃=−1±√1+2𝐵𝑤𝜃=𝑤𝐵_1/(𝑛+1)given𝑘(𝜃)=1+𝐵,𝜃g=ive𝑒n_𝑤𝑘(𝜃)=𝜃^𝑛,𝑛≠−1,𝑛=−1.(4.7) with 𝑤 given in Sections 4.1.1 and 4.1.2.

4.2. Symmetry Reductions and Invariant Solutions

If a differential equation is invariant under a Lie point symmetry, then one can reduce the order of the ordinary differential equation or the number of variables of the partial differential equation by one. The reduced equation may or may not be solved exactly. The exact (similarity) solutions obtained via symmetries are referred to as invariant solutions. In this section, we consider two cases as illustrative examples.

4.2.1. Linear Sink Term

We consider the linear combination of the symmetry generators Γ₁ and Γ₂; namely,Γ₁±𝛼₁Γ₂=𝜕𝜕𝑥±𝛼¹𝑤𝜕𝜕𝑤,𝛼¹≠0.(4.8)

The basis for the invariants is constructed by the corresponding characteristic equations in Pfaffian form 𝑑𝑤𝑤=±𝛼¹𝑑𝑥1=𝑑0𝑟.(4.9) Thus, we obtain the functional form of the solution 𝑤=𝑒^±𝛼1𝑥𝑈(𝑟),(4.10) where 𝑈(𝑟) satisfy 𝐸²𝑈^+𝐸²1𝑟𝑈^+±𝛼²¹−1𝑈=0,(4.11) where prime indicates the derivative with respect to 𝑟. The general exact solution to (4.11) is given by 𝑈(𝑟)=𝑎₁𝐽₀⎛⎜⎜⎜⎝±𝛼²¹−1𝐸𝑟⎞⎟⎟⎟⎠+𝑎²𝑌₀⎛⎜⎜⎜⎝±𝛼²¹−1𝐸𝑟⎞⎟⎟⎟⎠,(4.12) where 𝑎₁ and 𝑎₂ are arbitrary constants and 𝐽₀ and 𝑌₀ are Bessel 𝐽 and 𝑌 functions of order 0, respectively. In terms of the transformed variable 𝑤, we obtain the general exact solution𝑤=𝑎₁𝑒^±𝛼1𝑥𝐽₀(𝛾𝑟),with𝛾𝐽₁(𝛾)+𝐵𝑖𝐽₀(𝛾)=0,𝛾=±𝛼²¹−1𝐸,(4.13) for (2.11) with 𝑠=𝑤. We observe that this solution does not satisfy the boundary conditions at 𝑥=0 and 𝑥=1. One may consider semifinite fins where the temperature gradient vanishes at large 𝑥 values and with a appropriate choice of 𝐹, 𝑎₁ may be given by an exponential constant. Note that 𝛼₁=1 leads to trivial solutions.

4.2.2. Nonlinear Sink Term

The power law source term 𝑠=𝑤^𝑝, 𝑝≠1 renders (2.11) nonlinear and separation of variables is inapplicable. Following the techniques outlined in Section 4.2.1, we observe that the symmetry generator Γ₂ listed in Table 1 leads to the reduction𝑤=𝑥^2/1−𝑝𝐺(𝛾),where𝛾=𝑥1+𝐸₂𝑟and𝐺satisfytheO.D.E.(4.14)𝛾²𝐺^+𝐸²𝛾+41−𝑝𝛾1𝐺^+21−𝑝1−2𝑝−1𝛾1²𝐺−𝛾1²𝐺^𝑝=0,𝑝≠1.(4.15) We observe that (4.15) is harder to solve exactly. Furthermore, the boundary conditions are not invariant under Γ₂ given a nonlinear source term.

5. Fin Efficiency and Heat Flux

5.1. Heat Flux

The heat transfer from the fin base may be constructed by evaluating heat conduction rate at the base (see, e.g., [31])𝑞_𝑏=2𝜋^𝑅𝑎₀𝐾(𝑇)𝜕𝑇𝜕𝑋|||_𝑋=𝐿𝑅𝑑𝑅=2𝜋𝑅^2𝑎𝑇_𝑏−𝑇_∞𝐾_𝑎𝐿¹⁰𝑘(𝜃)𝜕𝜃𝜕𝑥|||_𝑥=1𝑟𝑑𝑟.(5.1)

The dimensionless heat transfer rate from the base of the fin is defined by [31]𝑄=𝑞^𝑏𝐿2𝜋𝐾_𝑎𝑇_𝑏−𝑇_∞𝑅^2𝑎=¹⁰𝑘(𝜃)𝜕𝜃𝜕𝑥|||_𝑥=1𝑟𝑑𝑟.(5.2)

5.2. Fin Efficiency

Fin efficiency (overall fin performance) is defined as the ratio of the actual heat transferred from the fin surface to the surrounding fluid to the heat which would be transferred if the entire fin area were kept at the base temperature [2, 32]. For the pin fin, analogous to the definition in [33], the local fin efficiency is defined by𝜂=𝑞^𝑏𝑄_𝑖=2𝜋𝑅^2𝑎𝑇_𝑏−𝑇_∞𝐾_𝑎/𝐿∫¹⁰𝑘(𝜃)(𝜕𝜃/𝜕𝑥)∣_𝑥=1𝑟𝑑𝑟2𝜋𝑅_𝑎ℎ_𝑏𝑇_𝑏−𝑇_∞𝐿,(5.3) or simply𝜂=1𝐸²𝐵𝑖¹⁰𝑘(𝜃)𝜕𝜃𝜕𝑥|||_𝑥=1𝑟𝑑𝑟.(5.4)

5.2.1. Flux and Fin Efficiency Given (4.1)

Given the solution (4.1) with linear thermal conductivity 𝑘=1+𝐵𝜃 as an example, we obtain heat flux for 𝑤𝑄=¹⁰𝜕𝑤𝜕𝑥|||_𝑥=1𝑟𝑑𝑟=^∞_𝑚₌₁𝑑_𝑚𝜆_𝑚sinh𝜆_𝑚𝐽₁𝛿_𝑚𝛿_𝑚,(5.5) and fin efficiency𝜂=1𝐸²𝐵𝑖¹⁰𝜕𝑤𝜕𝑥|||_𝑥=1𝑟𝑑𝑟=1𝐸²𝐵𝑖^∞_𝑚=1𝑑_𝑚𝜆_𝑚sinh𝜆_𝑚𝐽₁𝛿_𝑚𝛿_𝑚.(5.6)

6. Entropy Generation Analysis

Entropy generation results from the nonequilibrium conditions arising due to the exchange of energy within the fluid (in case of a flow between two plates) and the solid boundaries [34]. In fact, entropy generation analysis has been limited to vertical cylindrical annulus (or channels) (see, e.g., [35–37]), and studies in pure conduction may be found in the literature such as in [38–40]. The local volumetric rate of entropy generation is given in dimensionless variable (see, e.g., [17])𝑁_𝐿=𝜕𝜃𝜕𝑟²+𝜕𝜃𝜕𝑥².(6.1) The total dimensionless entropy generated in a pin fin is given by [17]𝑁_𝑇=¹⁰𝜕𝜕𝜃𝑟²+𝜕𝜃𝜕𝑥²𝑟𝑑𝑟𝑑𝑥.(6.2) Given 𝑘=1+𝐵𝜃, we have in terms of 𝑤𝑁_𝐿=11+2𝐵𝑤𝜕𝜕𝑤𝑟²+𝜕𝑤𝑁_𝑇𝜕𝑥²,=¹⁰11+2𝐵𝑤𝜕𝜕𝑤𝑟²+𝜕𝑤𝜕𝑥²𝑟𝑑𝑟𝑑𝑥.(6.3)

7. Discussions

We follow the analysis in [17]. The number of eigenvalues required to calculate the temperature distribution, heat flux, and fin efficiency accurately depend on the Biot number 𝐵𝑖. We observe in Table 2 below that Biot number is inversely proportional to the eigenvalues. The expression for the temperature distribution is given explicitly in (4.1) and (4.4). However, in further analysis, we focus on solution (4.4). The temperature distribution depends on a number of variables including 𝐵𝑖, eigenvalues, and the arbitrary function of 𝑟 describing the temperature at the base of the fin. We may choose any function 𝑔 such that 𝑔^(0)=0. In fact, the nonuniform base temperature is modeled by cosine function of 𝑟, namely, 1+𝑝cos𝑟 for some parameter 𝑝 [13], and base temperature may be given in general [41] by the power law 1+𝑝cos^𝜅𝑟, 𝜅 being an exponent (see also Chapter 15 in [1]). In Figure 2, not surprisingly, we observe that the temperature is higher at the center of the pin, that is, at 𝑟=0. Figure 3 depicts the temperature profile along the 𝑟=0, and temperature decreases from the base to the tip of the fin. We observe in Figures 4 and 5 that the temperature is much higher at the center of the pin than at its surface. Similar results are recorded in the literature (see, e.g., [17]). Similar profiles are observed in Figures 6–9, wherein thermal conductivity is assumed to be given by the power law. We note the difference at surface of the fin tip in Figures 4 and 9. In both cases, the temperature drops even to the negative values when 𝑘 is linear in temperature. The drop in the temperature is due to the presence of the sink term.

Table 2 

Eigenvalues 𝜆_𝑛 for different values of the Biot number with 𝐸=2.

The classical Lie point symmetry analysis resulted in a number of admitted symmetries. In general, symmetries yield self-similar or similarity solutions known also as invariant solutions. It is more difficult to construct invariant (similarity) solutions for the boundary value problem defined by characteristic length. However, one may consider the semiinfinite fins (see also [1]), whereby either temperature or temperature gradient vanishes at large spatial variable. In our case, at large 𝑥. Note that construction of invariant solutions for steady nonlinear one-dimensional problems is easier (see, e.g., [6]). On the other hand, symmetry techniques may be use to reduce the boundary value problem in partial differential equation to the boundary value problem in ordinary differential equations, as such the reduced problem may be solve exactly or easily by numerical schemes.

8. Concluding Remarks

We have considered a steady state problem describing heat dissipation in a pin fin. We have successfully applied the Kirchoff transformation to partly linearize the resulting nonlinear diffusion equation. To the best of our knowledge, symmetry methods have not yet been applied to two-dimensional fin problems. Some cases of the source term for which the governing equation admits extra symmetries have been obtained. Unfortunately, admitted symmetries do not leave the entire boundary value problem invariant. Some new solutions are constructed by separation of variables. Heat transfer analysis is carried out following the work in [17]. Further analysis into the influence of a larger number of terms may reveal a more in depth understanding of the underlying dynamics of the system under consideration.

Acknowledgments

R. J. Moitsheki wishes to thank the National Research Foundation of South Africa under Thuthuka program for the continued generous financial support. The authors thank the anonymous referees on pointing out a number of references and for the valuable comments which led to clarifications and improvements to this paper.