Steady Heat Transfer through a Two-Dimensional Rectangular Straight Fin

Abstract

Exact solutions for models describing heat transfer in a two-dimensional rectangular fin are constructed. Thermal conductivity, internal energy generation function, and heat transfer coefficient are assumed to be dependent on temperature. We apply the Kirchoff transformation on the governing equation. Exact solutions satisfying the realistic boundary conditions are constructed for the resulting linear equation. Symmetry analysis is carried out to classify the internal heat generation function, and some reductions are performed. Furthermore, the effects of physical parameters such as extension factor (the purely geometric fin parameter) and Biot number on temperature are analyzed. Heat flux and fin efficiency are studied.

1. Introduction

Fins are extended surfaces used to increase the heat transfer rate between a hot body and its surroundings. There are a variety of uses such as in air conditioning systems, compressors, and cooling of electronic components. The theory on heat transfer in extended surface may be found in texts such as [1, 2]. Few exact solutions exist even for the one dimensional fin problem with constant thermal conductivity and heat transfer coefficient [3]. Series solutions for one-dimensional fin problem with constant heat transfer coefficient and temperature-dependent thermal conductivity are given in [4]. Furthermore, analytical and exact solutions for one dimensional fins models with temperature-dependent thermal conductivity and heat transfer coefficient were obtained for example, in [5–7]. A compendium of heat transfer in all types of one dimensional fins is given in [8]. Exact steady-state solutions exist for two-dimensional models with constant thermal conductivity and heat transfer coefficient, with no internal heat generation [9–15], and with internal heat generation function depending on a spatial variable [16, 17]. Solutions for transient heat transfer in fins are constructed in [18].

In this paper, we apply the Kirchoff transformation to the two-dimensional steady-state heat balance equation in Section 2. The resulting model contains the arbitrary source term. In Section 3, we construct exact solutions which satisfy the physical boundary conditions. Symmetry analysis is carried out for the single governing equation, wherein group classification is performed in Section 4. Heat transfer results including fin efficiency and heat flux are analyzed in Section 5. Lastly, we provide concluding remarks in Section 6.

2. Mathematical Model

We consider a two-dimensional rectangular fin of length 𝐿 as shown in Figure 1. The fin is mounted to a base surface of temperature 𝑇_𝑏𝑔(𝑦) and extended into its surrounding of temperature 𝑇_∞. The heat flow is assumed to be symmetric along the line 𝑦=0. We assume that the heat transfer coefficient along the fin is nonuniform and temperature dependent. Also, the internal heat generation is nonzero and temperature dependent.

Figure 1 

Schematic representation of a rectangular straight fin.

The two-dimensional heat balance equation is given by (see, e.g., [16, 17])𝜕𝜕𝑦₁𝐾(𝑇)𝜕𝜕𝑦𝑇₁+𝜕𝜕𝑥₁𝐾(𝑇)𝜕𝜕𝑥𝑇₁=𝑠(𝑇).(2.1) The imposed boundary conditions are𝐾(𝑇)𝜕𝑇𝜕𝑥₁𝑇𝐿,𝑦=₁−𝐻(𝑇)𝑇0,𝑦₁−𝑇_∞,𝑥₁=0,=𝜕𝑇𝑇_𝑏−𝑇_∞𝑔𝑦₁,𝑥₁=𝐿,𝜕𝑦₁𝐾(𝑇)𝜕𝑇=0,𝑦₁=0,𝜕𝑦₁=−𝐻(𝑇)𝑇𝑥₁,𝑙2−𝑇^∞,𝑦¹=𝑙2.(2.2)

Here, 𝑇 is the dimensionless temperature, 𝑇_𝑏 is the fin base temperature, 𝐻 is the heat transfer coefficient, 𝑥₁ is the longitudinal coordinate, 𝑦₁ is the transverse coordinate, 𝑆 is the internal heat generation function, and 𝐾 is the thermal conductivity. Several authors have considered the two-dimensional problem with 𝑆=0 and thermal conductivity being a constant (see, e.g., [19, 20]) and the case 𝑆=0 with a temperature-dependent thermal conductivity [21].

Introducing the dimensionless variables𝜃=𝑇−𝑇^∞𝑇_𝑏−𝑇_∞,𝑥=𝑥¹𝐿,𝑦=𝑙𝑦/2¹,𝑘(𝜃)=𝐾𝐾(𝑇_𝑎)ℎ(𝜃)=𝐻(𝑇),ℎ_𝑏,𝐸²=𝐿𝑙/2²,𝑆(𝜃)=𝑠(𝑇)(𝑙/2)²𝐾_𝑎𝑇_𝑏−𝑇_∞,(2.3)

we obtain𝐸²𝜕𝜕𝑦𝑘(𝜃)𝜕𝜕𝜃𝑦+𝜕𝜕𝑥𝑘(𝜃)𝜕𝜕𝑥𝜃=𝐸²𝑆(𝜃).(2.4)

The corresponding dimensionless boundary conditions are𝑘(𝜃)𝜕𝜃𝜕𝑥=−Bi^𝜙𝜃(1,𝜕𝑦𝜃)=𝑔(𝑦ℎ)(,𝜃)𝜃𝑥,=𝑥1,=0,𝑘(𝜃)𝜕𝜃𝜕𝑦=0,𝑦=0,𝜕𝑦=−Bi^𝜓ℎ(𝜃)𝜃,𝑦=1,(2.5) where 𝐸 is the fin extension factor (purely geometric parameter), and Bi_𝜙=ℎ_𝑏𝐿/𝐾_𝑎 and Bi_𝜓=ℎ_𝑏(𝑙/2)/𝐾_𝑎 are the Biot numbers. 𝐸 is the reciprocal to aspect ratio (see, e.g., [16]). ℎ_𝑏 and 𝑘_𝑎 are the heat transfer at the base and thermal conductivity of the fin at the ambient temperature, respectively.

3. Exact Solutions

In this section, we construct exact solutions for the boundary value problem (BVP) (2.4)–(2.5). The problem is simplified by the introduction of the Kirchoff transformation. The Kirchoff's transformation (see, e.g., [21])𝜔(𝑥,𝑦)=^𝜃𝜃₀𝑘𝜃^∗𝑑𝜃^∗,(3.1) with 𝜃₀ being an arbitrary constant, reduces the BVP (2.4)–(2.5) to𝐸²𝜕²𝜔𝜕𝑦²+𝜕²𝜔𝜕𝑥²=𝐸²𝑆(𝜔),(3.2) subject to the conditions𝜕𝜔𝜕𝑥+Bi^𝜙𝜕𝜔𝜔=0,𝑥=0,(3.3)𝜕𝑦+Bi^𝜓𝜕𝜔𝜔=0,𝑦=1,(3.4)𝜔𝜕𝑦(1=,𝑦0),=𝐺𝑦=(𝑦0),.((33..56))

We consider two cases of the thermal conductivity. Note that in (3.3) and (3.4), we required the product of the heat transfer coefficient and temperature to match the integral of thermal conductivity. In fact, in a one-dimensional case, ℎ must be the differential consequence of 𝑘 (see, e.g., [7]). 𝐺(𝑦) is quadratic in 𝑔(𝑦) when 𝐾 is linear, and 𝐺 is given by the power law 𝑔(𝑦)^𝑛+1 when 𝐾 is nonlinear.

3.1. Case 1. Linear Thermal Conductivity

Thermal conductivity is assumed to be a linear function of temperature for many engineering applications [3]. We assume thermal conductivity to be linear function of temperature (see also [21, 22]) 𝐾(𝑇)=𝑘_𝑎1+𝛽𝑇−𝑇_∞.(3.7)

𝛽 is the parameter that describes temperature dependency [3, 4]. In dimensionless variables, 𝑘(𝜃)=1+𝐵𝜃 where 𝐵=𝛽(𝑇_𝑏−𝑇_∞). This case of 𝑘 requires ℎ(𝜃)=(1+(𝐵/2)𝜃)−(𝜃₀+(𝐵/2)𝜃²⁰)𝜃⁻¹, so that BVP (3.2)–(3.6) hold. Note that 𝜃=0 renders heat transfer coefficient to be singular. However, one can remove singularity by choosing, without loss of generality, 𝜃₀=0. Assuming the internal heat generation to be linearly dependent on temperature, then the governing equation becomes the modified Helmholtz type equation𝐸²𝜕²𝜔𝜕𝑦²+𝜕²𝜔𝜕𝑥²=𝐸²𝜔.(3.8) We seek exact solutions of (3.8) subject to (3.3)–(3.6). Using method of separation of variables, we obtain the nontrivial exact solutions for two cases of the separation constant 𝜎. Note that 𝜎=0 leads to trivial solutions.

3.1.1. 𝜎<0;𝑠𝑎𝑦𝜎=−𝜆²,𝜆>0

𝜔(𝑥,𝑦)=𝛾cosh(𝜆𝑦)cos𝐸√𝜆²−1𝑥−Bi^𝜙𝐸√𝜆²−1sin𝐸√𝜆²−1𝑥,𝜆²≠1,(3.9) where 𝛾 is an arbitrary constant and 𝜆 satisfies 𝜆tanh(𝜆)=−Bi_𝜓. Also,𝜔(𝑥,𝑦)=𝛾cosh(𝑦)1+Bi_𝜙𝑥,𝜆²=1.(3.10)

3.1.2. 𝜎>0;𝑠𝑎𝑦𝜎=𝜆²

𝜔(𝑥,𝑦)=^∞_𝑛=1𝑘_𝑛cos𝜆_𝑛𝑦⎡⎢⎢⎢⎣cosh𝐸𝜆^2𝑛+1𝑥−Bi^𝜙𝐸𝜆^2𝑛+1sinh𝐸𝜆^2𝑛+1𝑥⎤⎥⎥⎥⎦,(3.11) where 𝜆_𝑛 is satisfied by𝑘_𝑛𝜆_𝑛tan𝜆_𝑛=Bi_𝜓,=2⎡⎢⎢⎢⎣cosh𝐸𝜆^2𝑛+1−Bi^𝜙𝐸𝜆^2𝑛+1sinh𝐸𝜆^2𝑛+1⎤⎥⎥⎥⎦⁻¹¹⁰cos𝜆_𝑛𝑦𝐺(𝑦)𝑑𝑦.(3.12) The temperature profile for solution (3.11) is given in Figures 2(a) and 2(b). In Figures 3(a) and 3(b), we plot the temperature profile at the boundaries 𝑦=1 and 𝑥=0, respectively. For simplicity, we allowed Bi_𝜙 and Bi_𝜓 to be equal. We list the first five eigenvalues of (3.12) for various values of the Biot number in Table 2.

(a) Parameters used are Bi𝜙=Bi𝜓=0.01,0.5,2 from bottom to top sheet, respectively. 𝐸 is fixed at 2

(b) Parameters used are 𝐸=2,5,10 from top to bottom sheet, respectively. Both Bi𝜙 and Bi𝜓 are fixed at 0.2

Figure 2 

Temperature profile for the first twenty terms of the series given in (3.11).

(a) Temperature profile at 𝑦=1. Parameters used are Bi𝜙=Bi𝜓=0.02,𝐸=2

(b) Temperature profile at 𝑥=0. Parameters used are Bi𝜙=Bi𝜓=0.02,𝐸=2

Figure 3 

Graphical representation of temperature distribution at the fin boundaries.

In terms of the original temperature variable, we obtain the exact solution𝜃=−1±√1+2𝐵(𝜔+𝛿)𝐵,(3.13)

where 𝛿=𝜃₀+(𝐵/2)𝜃²⁰.

3.2. Case 2: Nonlinear Thermal Conductivity

Assuming the power law temperature-dependent thermal conductivity𝐾(𝑇)=𝑘_𝑎𝑇𝑇_𝑏−−𝑇𝑇^∞_∞^𝑛,(3.14)

which is given in dimensionless variables as 𝑘(𝜃)=𝜃^𝑛, requires ℎ(𝜃)=𝜃^𝑛𝑛+1−(𝑛𝜃+1^0𝑛+1)𝜃,𝑛≠−1.(3.15)

Again, one may, without loss of generality, assume, 𝜃₀=0. We obtain in terms of original variables the exact solution𝜃=𝜃[(𝑛=+ex1p)𝜔(𝜔]),𝑛^1/(𝑛+1),=𝑛−≠1,−1,(3.16)

where 𝜔 is given in Sections 3.1.1 and 3.1.2.

4. Symmetry Analysis

In this section, we analyze (3.2) with an arbitrary source term. We employ symmetry techniques. A symmetry group of a system of differential equations is a group of transformations mapping any arbitrary solution to another solution of the system. Such groups depend on continuous parameters. Given a continuous one-parameter symmetry group, in most practical cases, one may reduce the number of independent variables by one. The most familiar symmetry is the rotational symmetry that enables one to reduce the variables (𝑥,𝑦) to a single radial variable 𝑟. Recent accounts on this theory may be found in many excellent texts such as those of [23, 24]. We use the methods in [24] (which exclude the use of equivalence transformations) to determine possible cases of 𝑆 for which extra symmetries are obtained.

In the initial symmetry analysis where 𝑆 is arbitrary, we obtain the translations of the spatial variables 𝑥 and 𝑦 and the rotational symmetry. Cases for which extra symmetries are obtained are listed in Table 1. Note that 𝑆=0 reduces (3.2) to the steady two-dimensional thermal diffusion equation (the Laplace equation which has been already analyzed).

Table 1 

Extra point symmetries admitted by (3.2).

Table 2 

The first five eigenvalues of (3.12).

4.1. Symmetry Reductions: Illustrative Example

Given the internal heat generation term as the power law 𝑠(𝑤)=𝜔^𝑚,𝑚≠1 (as listed in Table 1), 𝑋₄ leads to the functional form 𝜔=𝑥^2/(1−𝑚)𝐹(𝜉),𝜉=𝑥𝑦,(4.1)

where 𝐹 satisfies a nonlinear ordinary differential equation (ODE)1−𝐸²𝜉²𝐹^+2𝐸²𝜉+1−4𝑚1𝜉𝐹′+1−2𝑚1−2𝑚−11𝜉𝐹−𝐸𝜉²𝐹^𝑚=0.(4.2) The ODE (4.2) is harder to solve exactly. Furthermore, the boundary conditions are not invariant under 𝑋₄. In fact, the BVP (2.4)–(2.5) is not invariant under all the admitted symmetries listed. One may seek numerical solutions when internal heat generation term is nonlinear. We omit numerical analysis in this paper. However, the obtained exact solutions in Section 3 may be used as benchmarks for the numerical schemes.

5. Heat Transfer Results

The number of eigenvalues required to calculate the temperature distribution, heat flux, and fin efficiency accurately depends on the Biot number Bi_𝜓. We observe in Table 2 below that Biot number is directly proportional to the eigenvalues. Similar results are obtained for heat transfer in orthotropic convective pin fin [14]. The expression for the temperature distribution is given explicitly in (3.9) and (3.11). However, in further analysis, we focus on solution (3.11). The temperature distribution depends on a number of variables including Bi_𝜙, Bi_𝜓, eigenvalues, and the arbitrary function of 𝑦 describing the temperature at the base of the fin. We are free to choose any function 𝐺. Note that temperature distribution is proportional to both Bi_𝜙 and Bi_𝜓.

We observe in Figure 2(a) that the Biot number is directly proportional to the temperature distribution in the fin. Also, in Figure 2(b), we notice that temperature decreases with the increase of the extension factor. Clearly, if the length of the fin is increased, temperature is at its lowest value, or increased width of the fin results in increased temperature distribution. Figures 3(a) and 3(b) depict the temperature distribution at the boundaries of the fin. We note that, in Figure 3(a), there is a significant reduction in temperature along the boundary and toward the tip of the fin (one may recall that we assumed that the fin is measured from the tip to the base). Figure 3(b) shows temperature variation at the tip of the fin.

5.1. Fin Efficiency and Heat Flux

5.1.1. Heat Flux

The heat transfer from the fin base may be constructed by evaluating heat conduction rate at the base (see, e.g., [12])𝑞_𝑏=^0ℓ/2𝐾(𝑇)𝜕𝑇𝜕𝑥₁||||_𝑥1=𝐿𝑑𝑦₁=𝐾^𝑎𝑇_𝑏−𝑇_∞𝑙/2𝐿¹⁰𝑘(𝜃)𝜕𝜃𝜕𝑥|||_𝑥=1𝑑𝑦.(5.1)

The dimensionless heat transfer rate from the base of the fin is defined by [12]𝑄=𝑞^𝑏𝐿𝐾_𝑎𝑇_𝑏−𝑇_∞𝑙/2=¹⁰𝑘(𝜃)𝜕𝜃𝜕𝑥|||_𝑥=1𝑑𝑦.(5.2)

5.1.2. Fin Efficiency

Fin efficiency (overall fin performance) is defined as the ratio of the actual heat transfer from the fin rate of heat that would be ideally transferred if the entire fin was at the temperature of the fin base [25]. The local fin efficiency is defined by [13]𝜂=𝑞^𝑏𝑄_𝑖=𝐾^𝑎𝑇_𝑏−𝑇_∞𝑙/2/𝐿∫¹⁰𝑘(𝜃)(𝜕𝜃/𝜕𝑥)||_𝑥=1𝑑𝑦ℎ_𝑏𝑇_𝑏−𝑇_∞𝑙/2,(5.3) or simply𝜂=1𝐸Bi_𝜓¹⁰𝑘(𝜃)𝜕𝜃𝜕𝑥|||_𝑥=1𝑑𝑦.(5.4)

5.1.3. Flux and Fin Efficiency Given (3.11)

Given the solution (3.11), we obtain heat flux in terms of 𝑤, 𝑄=¹⁰𝜕𝑤𝜕𝑥|||_𝑥=1𝑑𝑦=^∞_𝑛=1𝑘_𝑛sin𝜆_𝑛𝜆_𝑛𝐸𝜆^2𝑛+1sinh𝐸𝜆^2𝑛+1−Bi_𝜙cosh𝐸𝜆^2𝑛+1,(5.5) and fin efficiency𝜂=1𝐸Bi_𝜓¹⁰𝜕𝑤𝜕𝑥|||_𝑥=1=1𝑑𝑦𝐸Bi_{𝜓∞𝑛=1}𝑘_𝑛sin𝜆_𝑛𝜆_𝑛𝐸𝜆^2𝑛+1sinh𝐸𝜆^2𝑛+1−Bi_𝜙cosh𝐸𝜆^2𝑛+1.(5.6)

Fin efficiency (5.6) and heat flux (5.5) are depicted in Figures 4(a) and 4(b), respectively. Wherein both heat flux and fin efficiency are plotted against the extension factor. Here, Bi_𝜓 is fixed at 0.2. In Figure 4(a), we observe that fin performance decreases with increased extension factor. Moreover, the increased Biot number Bi_𝜓 yields decreased fin efficiency. In our entire analysis, we have assumed a nonuniform internal heat generation. Internal heat generation function is assumed to be a uniform in [26]. In Figure 4(b), we observe that Biot number Bi_𝜓 is inversely proportional to the heat flux. However, heat flux is directly proportional to the extension factor (implying that longer fins result in higher heat flux (see also [14])). The observations in Figures 4(a) and 4(b) are consistent with the results in the literature (see, e.g., [26, 27] or Chapter 15 in [1]). In [26], finite elements methods were used to determine, among others, the effects of uniform internal heat generation. Here, internal energy generation is given as linear function of temperature, and base temperature is quadratic in 𝑦. Sikka and Iqbal [27] provided the fin efficiency for two-dimensional pin fin. The observations in Figure 4(b) are consistent with those of [27].

(a) Fin efficiency

(b) Heat flux

Figure 4 

Graphical representation of the heat flux and the fin performance in (5.5) and (5.6), respectively. Here, the first twenty terms of the series in (3.11) are plotted. Bi𝜓 is fixed at 0.2, and Bi𝜙 varies as indicated in the graph.

6. Concluding Remarks

We have successfully applied the Kirchoff’s transformation to reduce the nonlinearity of (2.4) (the source term remained arbitrary). Exact solutions for two-dimensional rectangular fin with temperature-dependent thermal conductivity and heat transfer coefficient, furthermore with internal energy generation function, which depend linearly on temperature are constructed. In the analysis, we allowed the temperature of the fin base to be quadratic in 𝑦. The forms of the internal energy generation term for which extra symmetries are admitted were obtained. Reduction of the single PDE (given nonlinear source term) to the ODE is achieved. As far as we know, symmetry methods have not yet been employed to two-dimensional fin problems. However, the entire BVP is not invariant under the admitted symmetries. Constructed exact solutions have provided insight into the heat transfer processes in a rectangular straight fin and may be used as benchmarks for the numerical schemes, particularly when thermal conductivity, heat transfer coefficient, and internal thermal energy generation function are all temperature dependent.

Nomenclature

𝑇:	Temperature distribution
𝑇𝑏:	Fin base temperature
𝑇∞:	Surrounding temperature
𝜃:	Dimensionless temperature
𝜔:	Transformed temperature variable
𝐾:	Thermal conductivity
𝐾𝑎:	Thermal conductivity of the fin at the ambient temperature
𝑘:	Dimensionless thermal conductivity
𝐻:	Heat transfer coefficient
ℎ𝑏:	Heat transfer coefficient at the fin base
ℎ:	Dimensionless heat transfer coefficient
𝑦1:	Transverse coordinate
𝑥1:	Longitudinal coordinate
𝑥:	Spatial variable
𝑦:	Spatial variable
𝐿:	Length of the fin
𝑙/2:	Half width
𝑠(𝑇):	Internal energy generation function
𝑆(𝜃):	Dimensionless internal energy generation function
Bi𝜓=(ℎ𝑏(𝑙/2))/𝐾𝑎:	Biot number
Bi𝜙=(ℎ𝑏𝐿)/𝐾𝑎:	Biot number
𝐸:	Extension factor (purely geometric parameter)
𝑞𝑏:	Heat flux at the fin base
𝑄:	Heat flux
𝑄𝑖:	Overall heat flux for an ideal fin
𝜂:	Fin efficiency
𝐵,𝛽,𝜃0,𝛿,𝑛,𝑚,𝛼1,𝛼2:	Constants given in the paper.