Thermal Characterizations of Exponential Fin Systems

Abstract

Exponential fins are mathematically analyzed in this paper. Two types are considered: (i) straight exponential fins and (ii) pin exponential fins. The possibility of having increasing or decreasing cross-sectional areas is considered. Different thermal performance indicators are derived. The maximum ratio between the thermal efficiency of the exponential straight fin to that of the rectangular fin is found to be 1.58 at an effective thermal length of 2.0. This ratio is even larger when exponential fins are compared with triangular and parabolic straight fins. Moreover, the maximum ratio between the thermal efficiency of the exponential pin fin to that of the rectangular pin fin is found to be 1.17 at an effective thermal length of 1.5. However, exponential pin fins thermal efficiencies are found to be lower than those of triangular and parabolic pin fins. Moreover, exponential joint-fins may transfer more heat than rectangular joint-fins especially when differences between their senders and receivers portions dimensionless indices are very large. Finally, it is found that increasing the joint-fin exponential index may cause straight exponential joint-fins to transfer more heat than rectangular joint-fins.

1. Introduction

Enhancing heat transfer between solids and the adjoining fluids is one of the most important objectives in thermal engineering. Therefore, many methods were proposed to achieve this goal. Bergles [1, 2] classified these methods to active and passive methods. Active methods are those requiring external power to maintain their enhancement such as well stirring the fluid or vibrating the solid surface [3, 4]. On the other hand, the passive methods do not require external power to maintain the enhancement effect as when fins are utilized. Fins are widely used in industry, especially in heat exchanger and refrigeration industries [5–10]. Moreover, fins are used in cooling of large heat flux electronic devices as well as in cooling of gas turbine blades [11].

According to design aspects, fins can have simple designs such as rectangular, triangular, parabolic, hyperbolic, annular, and pin fins [9]. Complicated designs of fins such as spiral fins have been utilized [12, 13]. In addition, fins can be arranged uniformly on the solid surface [10]. In contrast, they can be arranged on the solid surface in complex networks as can be seen in the works of Alebrahim and Bejan [14], Almogbel and Bejan [15] and Khaled [16]. Moreover, fins can be further classified based on the number of the adjoining fluids interacting with their surfaces. Examples of works including fins surrounded by more than one adjoining fluid can be found in the works of Khaled [17, 18]. In addition, fins are usually attached to solid surfaces [5–13] but they may have roots in the solid walls [19]. To the best knowledge of the author, thermal characterization of exponential fin systems received almost negligible attention in the literature. Perhaps, this is due to the difficulty associated with manufacturing them in the past. However, the recent advancements in manufacturing technologies, which led to accurate micro- and nanosystems fabrications, may increase the opportunities of these passive systems to be implemented in industry.

In this paper, fins with exponentially varying cross-sectional areas are modeled and mathematically analyzed. Two types were considered: (i) exponential straight fins and (ii) exponential pin fins. The appropriate energy equations are solved, and the temperature distributions are found. Accordingly, different thermal performance indicators are calculated. The analysis is expanded to account for exponential joint-fins. Extensive parametric study is performed for the various controlling parameters in order to evaluate these kinds of systems.

2. Problem Formulation

It should be mentioned before starting the analysis that the following assumptions are considered: (i)one-dimensional heat transfer analysis,(ii)conduction and convection heat transfer rates being governed by the Fourier law and the Newtons law of cooling, respectively,(iii)having, for exponential pin fins, (𝑑𝑟/𝑑𝑥)²≪1.0,(iv)uniform heat transfer coefficient between the fin and the fluid stream.

2.1. Straight Fins with Exponentially Varying Widths

Consider a rectangular fin having a uniform thickness t that is much smaller than its width H(x) and length L as shown in Figure 1. The fin width varies along the fin centerline axis (x-axis) according to the following relationship: 𝐻(𝑥)≡𝐻(𝑥/𝐿)𝐻_𝑏=𝑒^{−𝑏𝐿𝑥}, (2.1) where 𝑥=𝑥/𝐿 and b is a real number named as the exponential index. The quantity 𝐻_𝑏 represents the fin half-width at its base (𝑥 = 0). Note that, when 𝑏>0, the analysis corresponds to the right portion of the joint-fin shown in Figure 1 while it corresponds to the left fin portion of the joint-fin when 𝑏<0.

Figure 1 

Schematic diagram for a straight exponential fin and exponential joint-fin and the system coordinates.

The application of the energy equation [20] on a fin differential element results in the following differential equation: 𝑑𝑑𝑥(𝐴_c𝑑𝑇ℎ𝑑𝑥)−𝑘(𝑑𝐴_s𝑑𝑥)(𝑇−𝑇_∞)=0, (2.2) where T, 𝑇_∞, k, and h are the fin temperature, free stream temperature, fin thermal conductivity, and the convection heat transfer coefficient between the fin and the fluid stream, respectively. The quantities𝐴_𝑐 and 𝐴_𝑠are the cross-sectional and the surface areas of the fin, respectively. Equation (2.2) has the following dimensionless form: 𝑑𝑑𝑥(𝐻𝑑𝜃𝑑𝑥)−(𝑚𝐿)²𝐻𝜃=0, (2.3) where 𝑚=2ℎ/(𝑘𝑡) and 𝜃=(𝑇(𝑥)−𝑇_∞)/(𝑇_𝑏−𝑇_∞). The quantity m is called the fin index while 𝑇_𝑏 is the fin temperature at its base. Equation (2.3) prescribes the following general solution: 𝜃(𝑥)=𝐶₁𝑒^𝑠1𝑥+𝐶₂^𝑠2𝑥, (2.4) where 𝑠₁and s₂ are equal to𝑠_1,2=𝑏𝐿2[1∓4𝑋²𝑠+1]if𝑏>0,_1,2=𝑏𝐿2[1±4𝑋²+1]if𝑏<0, (2.5) where 𝑋=𝑚/𝑏. The quantity X is named as the dimensionless exponential fin parameter. It represents the ratio of the fin index, m, to the exponential index, b. When 𝑋≪1.0, cross section gradients near the base are expected to be larger than the nearby temperature gradients. The opposite scenario occurs when 𝑋≫1. The boundary conditions for an adiabatic fin tip are given by𝜃(𝑥=0)=1.0,𝜕𝜃𝜕𝑥|_𝑥=1=0.0. (2.6) As such, the dimensionless temperature distribution has the following form:𝜃(𝑒𝑥)=^𝑠1𝑥−[(𝑠₁𝑒^𝑠1)/(𝑠₂𝑒^𝑠2)]𝑒^𝑠2𝑥1−(𝑠₁𝑒^𝑠1)/(𝑠₂𝑒^𝑠2). (2.7)

The rate of heat transfer through the fin is called the fin heat transfer rate. For this case, it is equal to𝑞_𝑓=−𝑘𝐴_𝑐𝑑𝑇|𝑑𝑥_𝑥=0=𝐻_𝑏𝑡𝑘𝑏{−1+4𝑋²+1}(𝑇_𝑏−𝑇_∞)Φ₁(𝑠₁,𝑠₂);𝑏>0,−𝐻_𝑏𝑡𝑘𝑏{1+4𝑋²+1}(𝑇_𝑏−𝑇_∞)Φ₂(𝑠₁,𝑠₂);𝑏<0, (2.8) where Φ₁and Φ₂factors are smaller than unity. They are equal toΦ₁(𝑠₁,𝑠₂)=1−Exp(−𝑏𝐿4𝑋²+1)1−((1−4𝑋²+1)/(1+4𝑋²+1))Exp(−𝑏𝐿4𝑋²+1), (2.9)Φ₂(𝑠₁,𝑠₂)=1−Exp(𝑏𝐿4𝑋²+1)1−((1+4𝑋²+1)/(1−4𝑋²+1))Exp(𝑏𝐿4𝑋²+1). (2.10)

Utilizing (2.9) and (2.10), the fin lengths 𝐿=(𝐿_∞)₁ and 𝐿=(𝐿_∞)₂ that make Φ₁and Φ₂equal to 0.99, respectively, can be approximated by𝑚(𝐿_∞)₁≅𝑋4𝑋²1+1(5.293−ln[1+4𝑋²+1]),𝑏>0,𝑚(𝐿_∞)₂≅−𝑋4𝑋²1+1(5.293−ln[1−4𝑋²+1]),𝑏<0, (2.11) where quantity mL_∞ is called the effective thermal length. This is because the fin material exists after 𝑥=𝐿_∞ encounters negligible heat transfer rates and should be removed. The fin thermal efficiency 𝜂_𝑓 is defined as the fin heat transfer rate divided by the fin heat transfer rate if the fin temperature is kept at 𝑇_𝑏. For this case, it can have the following forms: 𝜂_𝑓≡0.99(𝑞_𝑓)_max4ℎ(𝑇_𝑏−𝑇_∞)∫^𝐿∞₀=𝐻𝑑𝑥0.99{−1+4𝑋²+1}2𝑋²[1−Exp(−𝑏(𝐿_∞)₁)],𝑏>0,0.99{1+4𝑋²+1}2𝑋²[Exp(−𝑏(𝐿_∞)₂)−1],𝑏<0, (2.12) where (𝑞_𝑓)_max is the fin heat transfer rate when 𝐿≫𝐿_∞. Now, define the fin performance indicator 𝛾 as the ratio of the fin heat transfer rate when 𝐿>𝐿_∞ to the fin heat transfer rate for a rectangular fin having a uniform width of 2𝐻_𝑏, uniform thickness t and an infinite length.  As such, γ is equal to𝛾≡(𝑞_𝑓)_max(𝑞_𝑓)|_{𝐻=𝐻𝑏}=(𝑞_𝑓)_max2𝐻_𝑏√2ℎ𝑘𝑡(𝑇_𝑏−𝑇_∞)=(−1+4𝑋²+1(2𝑋),𝑏>0,1+4𝑋²+12𝑋),𝑏<0. (2.13)

2.2. Pin Fins with Exponentially Varying Radii

Consider a pin fin of radius 𝑟(𝑥), as shown in Figure 2, that varies exponentially along the x-axis according to the following relationship: 𝑟(𝑥)≡𝑟(𝑥)𝑟_𝑏=𝑒^−𝑏𝑥, (2.14) where 𝑏>0. As such, (2.2) changes to: 𝑑𝑑𝑥(𝑒^−2𝑏𝑥𝑑𝜃𝑑𝑥)−𝑚²𝑒^−𝑏𝑥𝜃=0, (2.15) where 𝑚=2ℎ/(𝑘𝑟_𝑏). It can be shown that the general solution of (2.15) is𝜃(𝑥)=𝑒^{𝑚𝑥/𝑋}{𝐶₁𝐼₂(2𝑋𝑒^{𝑚𝑥/(2𝑋)})+𝐶₂𝐾₂(2𝑋𝑒^{𝑚𝑥/(2𝑋)})}, (2.16) where 𝑋=𝑚/𝑏.As such, the fin heat transfer rate is𝑞_f=−𝑘𝜋𝑟²_𝑏(𝑇_𝑏−𝑇_∞)𝑏{𝐶₁[𝐼₂(2𝑋)+0.5𝑋(𝐼₁(2𝑋)+𝐼₃(2𝑋))]+𝐶₂[𝐾₂(2𝑋)−0.5𝑋(𝐾₁(2𝑋)+𝐾₃(2𝑋))]}. (2.17)

Figure 2 

Schematic diagram for an exponential pin fin with $b<0$ and the system coordinate.

For a fin with an infinite length (𝐿→∞), the constants C₁ and C₂ are given by 𝐶₁=0,𝐶₂=1𝐾₂(2𝑋). (2.18) This is because 𝑒^{𝑚𝑥/(2𝑋)} approaches infinity as x approaches infinity; hence 𝐼₂(2𝑋𝑒^{𝑚𝑥/(2𝑋)}) approaches infinity. Thus, C₁ is equal to zero.

For adiabatic fin tips, boundary conditions given by (2.6) should be satisfied. Accordingly, the constants C₁ and C₂ are equal to𝐶₁=(𝐼₂(2𝑋)−𝐾₂(2𝑋)𝐶₃)⁻¹,𝐶₃=𝐼₂(2𝑋𝑒^{𝑚𝐿/(2𝑋)})+0.5𝑋𝑒^{𝑚𝐿/(2𝑋)}{𝐼₁(2𝑋𝑒^{𝑚𝐿/(2𝑋)})+𝐼₃(2𝑋𝑒^{𝑚𝐿/(2𝑋)})}𝐾₂(2𝑋𝑒^{𝑚𝐿/(2𝑋)})−0.5𝑋𝑒^{𝑚𝐿/(2𝑋)}{𝐾₁(2𝑋𝑒^{𝑚𝐿/(2𝑋)})+𝐾₃(2𝑋𝑒^{𝑚𝐿/(2𝑋)},𝐶)}₂=−𝐶₁𝐶₃. (2.19)

The fin efficiency η_f for 𝑏>0 can be found to be equal to𝜂_𝑓≡0.99(𝑞_𝑓)_max2𝜋𝑟_𝑏ℎ(𝑇_𝑏−𝑇_∞)∫^𝐿∞₀=𝑟(𝑥)𝑑𝑥0.99(1−𝑒^{−𝑚𝐿∞/𝑋})(1𝑋²𝑋){2[𝐾₁(2𝑋)𝐾₂+𝐾(2𝑋)³(2𝑋)𝐾₂(2𝑋)]−1},𝑏>0, (2.20) where 𝑚𝐿_∞ is obtained from the solution of the following equation:(𝑞_𝑓)_Ins.Tip(𝑞_𝑓)_𝐿→∞=𝜙(𝑋,𝑚𝐿_∞𝑋)=0.99. (2.21)

The fin performance indicator 𝛾 for this case is defined as the ratio of the fin heat transfer rate when 𝐿≫𝐿_∞ to that of a rectangular pin fin having a uniform radius of r_b and an infinite length. It is equal to the following:𝛾₁≡(𝑞_𝑓)_max(𝑞_𝑓)|_{𝑟=𝑟𝑏}=(𝑞_𝑓)_max𝜋𝑟_𝑏2ℎ𝑘𝑟_𝑏(𝑇_𝑏−𝑇_∞)1=(𝑋𝑋){2[𝐾₁(2𝑋)𝐾₂+𝐾(2𝑋)³(2𝑋)𝐾₂(2𝑋)]−1},𝑏>0. (2.22)

For cases when 𝑏<0; X is replaced with −X, and the constants C₁ and C₂ for a fin with infinite length are replaced by𝐶₁=1𝐼₂(−2𝑋),𝐶₂=0,𝑏<0. (2.23) As such, the fin thermal efficiency 𝜂_𝑓 and the indicator 𝛾 when 𝑏<0 change to𝜂_𝑓=0.99(𝑒^{−𝑚𝐿∞/𝑋}(1−1)𝑋²){−𝑋2[𝐼₁(−2𝑋)𝐼₂+𝐼(−2𝑋)³(−2𝑋)𝐼₂𝛾(−2𝑋)]+1},𝑏<0,₂≡𝑞_𝑓(𝑞_𝑓)|_{𝑟=𝑟𝑏}=(−1𝑋){−𝑋2[𝐼₁(−2𝑋)𝐼₂+𝐼(−2𝑋)³(−2𝑋)𝐼₂(−2𝑋)]+1},𝑏<0. (2.24)

2.3. Pin Fins with Exponentially Decaying Temperature Distribution

Consider a pin fin having a given fin temperature distribution that varies exponentially with x according to the following relationship: 𝜃(𝑥)≡𝑇(𝑥)−𝑇_∞𝑇_𝑏−𝑇_∞=𝑒^{−𝑎𝑥𝑜𝑥}, (2.25) where 𝑥=𝑥/𝑥_𝑜 and a is the exponential index. The dimensionless form of the energy equation has the following form 𝑑𝑑𝑥(𝑟²(𝑥)𝑑𝜃𝑑𝑥)−(𝑚𝑥_𝑜)²𝑟(𝑥)𝜃=0, (2.26) where 𝑚=2ℎ/(𝑘𝑟_𝑏) and 𝑟(𝑥)=𝑟(𝑥)/𝑟_𝑏. By substituting (2.25) in (2.26), a differential equation of first order constructed. It has the form 𝑑𝑟𝑟−𝑋²=(𝑎𝑥₀)2𝑑𝑥, (2.27) where 𝑋=𝑚/𝑎. The solution to (2.27) is given by𝑟(𝑥)=𝑋²+[1−𝑋²]Exp(𝑎𝑥₀2𝑥). (2.28)

For engineering problems, 𝑟(𝑥) cannot be negative and it should intersect with fin centerline at 𝑥=1 when 𝑋>1.0. As such, 𝑥₀ is found to be equal to 𝑚𝑥₀𝑋=(2𝑋)ln[²𝑋²−1],𝑋>1.0. (2.29) In situations when 0<𝑋<1.0, 𝑚𝑥₀ is minimally equal to 4.605𝑋 (𝑥₀= 4.605/a; 𝑋<1.0). Under this constraint, the heat transfer rate at the fin tip (𝑥=𝑥₀) is always 0.01 times the fin heat transfer rate. The rate of heat transfer through the fin base is equal to 𝑞_𝑓=−𝑘_𝑓𝐴_𝑐𝑑𝑇|𝑑𝑥_𝑥=0=𝜋𝑟²_𝑏𝑘𝑎(𝑇_𝑏−𝑇_∞). (2.30)

As such, the fin thermal efficiency and the performance indicator for this case are equal to𝜂_f=𝑞_𝑓2𝜋𝑟_𝑏𝑥₀ℎ(𝑇_𝑏−𝑇_∞)∫¹₀𝑟(𝑥)𝑑𝑥=12𝑋²{𝑋²𝑋ln[²𝑋²−1]−1}⁻¹1,𝑋>1.0,2𝑋²(9.0−6.697𝑋²)𝛾,0≤𝑋≤1.0,₁=𝑞_𝑓𝜋𝑟_𝑏2ℎ𝑘𝑟_𝑏(𝑇_𝑏−𝑇_∞)=1𝑋. (2.31)

2.4. Exponential Joint-Fins

Consider an infinite exponential fin joining two different fluid streams separated by a wall of negligible thickness such as a pipe wall. The convection coefficient between the fin and the fluid stream of the heat source side (side with maximum free stream temperature 𝑇_∞1) is h₁. This coefficient is h₂ for the heat sink side (side with 𝑇_∞2<𝑇_∞1) as illustrated in Figure 1. The joint-fin portion on the source side is named as the “joint-fin receiver portion” while the other portion is named as the “joint-fin sender portion”. The heat transfer rates through a straight exponential joint-fin (𝑞_𝑓)_𝑠, pin exponential joint-fin (𝑞_𝑓)_𝑃, and the pin joint-fin with exponential decaying temperature (𝑞_𝑓)_𝑇 are given by the following equations: (𝑞_𝑓)_𝑠=−𝛾₂2𝐻_𝑏2ℎ₁𝑘𝑡(𝑇_𝑏−𝑇_∞1)=𝛾₁2𝐻_𝑏2ℎ₂𝑘𝑡(𝑇_𝑏−𝑇_∞2),(𝑞_𝑓)_𝑃=−𝛾₂𝜋𝑟_𝑏2ℎ₁𝑘𝑟_𝑏(𝑇_𝑏−𝑇_∞1)=𝛾₁𝜋𝑟_𝑏2ℎ₂𝑘𝑟_𝑏(𝑇_𝑏−𝑇_∞2),(𝑞_𝑓)_𝑇=−𝛾₁𝜋𝑟_𝑏2ℎ₁𝑘𝑟_𝑏(𝑇_𝑏−𝑇_∞1)=𝛾₁𝜋𝑟_𝑏2ℎ₂𝑘𝑟_𝑏(𝑇_𝑏−𝑇_∞2), (2.32)(2.33)(2.34) where the exponential index for the joint-fin receiver portion is considered to be negative, 𝑏<0, while that for the sender portion is positive, 𝑏>0. This is only for cases represented by (2.32) and (2.33).

By solving (2.32)–(2.34), the temperature at the joint-fin base (𝑥=0) can be calculated. They are equal to(𝑇_𝑏)_𝑠=⌊4𝑋²₁+1+1⌋𝑇_∞1+⌊4𝑋²₂+1−1⌋𝑇_∞24𝑋²₁+1+4𝑋²₂+1,(𝑇_𝑏)_𝑃=𝑀×𝑇_∞1+𝑁×𝑇_∞2𝑀+𝑁,(𝑇_𝑏)_𝑇=𝑇_∞1+𝑇_∞22, (2.35)(2.36)(2.37) where M and N are given by𝑋𝑀=[(¹𝑋₂𝐼){¹(2𝑋₁)2𝐼₂(2𝑋₁)+𝐼₃(2𝑋₁)2𝐼₂(2𝑋₁)1}+𝑋₂𝐾],𝑁=[¹(2𝑋₂)2𝐾₂(2𝑋₂)+𝐾₃(2𝑋₂)2𝐾₂(2𝑋₂)−1𝑋₂]. (2.38) By substituting (2.35)–(2.37) in (2.32)–(2.34), the joint-fin heat transfer rates reduce to the following forms:(𝑞_𝑓)_𝑠=2𝐻_𝑏2ℎ₁𝑘𝑡(𝑇_∞1−𝑇_∞2)⌊4𝑋²₁+1+1⌋⌊4𝑋²₂+1−1⌋2𝑋₁{4𝑋²₁+1+4𝑋²₂,+1}(𝑞_𝑓)_𝑠=𝜋𝑟_𝑏2ℎ₁𝑘𝑟_𝑏(𝑇_∞1−𝑇_∞2)𝑁(𝑋𝑀(𝑀+𝑁)𝑋²₁),(𝑞_𝑓)_𝑇=𝜋𝑟_𝑏2ℎ₁𝑘𝑟_𝑏(𝑇_∞1−𝑇_∞2)2𝑋₁. (2.39)

Define the joint-fin performance indicator 𝛾₃ as the ratio of the joint-fin maximum heat transfer rate to maximum heat transfer rate through a rectangular joint-fin with uniform cross-section (𝑏=0). It is mathematically defined as 𝛾₃≡𝑞_𝑓[(𝑞_𝑓)|_{𝐴𝑐=constant}]_Joint-fin. (2.40) The heat transfer rate through the joint fin when 𝑏=0 is obtainable from [17]. It is equal to[(𝑞_𝑓)|_{𝐻=𝐻𝑏}]_Joint-fin=ℎ₁𝑘𝑃𝐴_𝑐1+ℎ₁/ℎ₂(𝑇_∞1−𝑇_∞2) (2.41) As such, 𝛾₃ can be written in the following forms: (𝛾₃)_𝑠=12(1𝑋₁+1𝑋₂)⌊4𝑋²₁+1+1⌋⌊4𝑋²₂+1−1⌋{4𝑋²₁+1+4𝑋²₂,+1}(𝛾₃)_𝑃𝑁={𝑋𝑀(𝑀+𝑁)}(²𝑋₁+1),(𝛾₃)_𝑇=𝑋₂+𝑋₁2𝑋₁𝑋₂. (2.42)

3. Discussion of the Results

Figure 3 illustrates the effects of the fin dimensionless parameter X on the effective thermal length 𝑚𝐿_∞ for a straight exponential fin. When 𝑏>0, 𝑚𝐿_∞ increases as X increases. It also increases as X increases for the other case (𝑏>0) until X reaches almost unity. For both cases, 𝑚𝐿_∞ approaches to an asymptotic value of 2.65 as 𝑋→∞. Similar findings can be noticed for exponential pin fins except that, when 𝑏<0,𝑚𝐿_∞ increases as X increases until X reaches almost the value of 1.7 as shown in Figure 4. On the other hand, 𝑚𝑥₀ decreases as X increases for pin fins with exponential decaying temperature when 𝑋>1.0. For exponential pin fins, the effective thermal lengths 𝑚𝐿_∞ values shown in Figure 4 are correlated to the parameter X by the following correlations: 𝑚𝐿_∞𝑋=0.8233(^0.8804−0.00470.2945𝑋^0.8934+0.2428),𝑏>0,𝑚𝐿_∞𝑋=0.7237(^0.6906+3.3301𝑋^1.4019+0.3939X^0.6902−0.03020.9547𝑋^1.3923+𝑒^{−0.6311X1.9309}−0.7425),𝑏<0. (3.1)(3.2) These correlations were obtained using the least square method by utilizing a specialized iterative statistical software. The maximum percentage error between correlations (3.1), and (3.2) and the results shown in Figure 4 are found to be 7.5% and 13% at 𝑋=0.01 when 𝑏>0, and 𝑏<0, respectively.

Figure 3 

Effect of the fin dimensionless parameter X on the effective thermal length $m{L}_{\infty }$ for straight exponential fins with $b>0$ and $b<0$ ($m=\sqrt{2h/(kt)}$).

Figure 4 

Effect of the fin dimensionless parameter X on the effective thermal length mL∞ for exponential pin fins with $b>0\text{,}$$b<0$ and $L_{\infty }=x_0$ ($m=\sqrt{2h/(k{r}_{\text{b}})}$).

Figure 5 shows the relation between the effective thermal length 𝑚𝐿_∞ on the fin thermal efficiency 𝜂_f for a straight exponential fin. It is seen that 𝜂_𝑓 when 𝑏>0 is greater than the fin thermal efficiency of rectangular, triangular, and parabolic straight fins having the same thermal length. However, the latter thermal efficiencies are greater than the fin thermal efficiency for the straight exponential fin when 𝑏<0. It can be shown using Figure 5 that the maximum ratio between the thermal efficiency of the exponential straight fin to that of the rectangular fin is 1.58 at an effective thermal length of 2.0. For pin exponential fins with 𝑏>0, 𝜂_𝑓 is found to be higher than 𝜂_𝑓 for the rectangular pin fins and lower than those for triangular and parabolic pin fins having the same thermal length as shown in Figure 6. It is recommended to operate pin exponential fins, 𝑏<0, at smaller values of 𝑚𝐿_∞ as their efficiencies increase as 𝑚𝐿_∞ decreases as can be seen from Figure 6. In addition, the maximum ratio between the thermal efficiency of the exponential pin fin to that of the rectangular pin fin is found to be 1.17 at an effective thermal length of 1.5.

Figure 5 

Effect of the fin dimensionless parameter $m{L}_{\infty }$ on the fin efficiency ${\eta }_f$ for straight exponential fins with $b>0$ , or $b<0$ ($m=\sqrt{2h/(kt)}$; other than exponential fin $m{L}_{\infty }$ is replaced with mL).

Figure 6 

Effect of the fin dimensionless parameter $m{L}_{\infty }$ on the fin efficiency ${\eta }_f$ for exponential pin fins with $b>0\text{,}$$b<0$ and $L_{\infty }=x_0$ ($m=\sqrt{2h/(k{r}_b)}$; other than exponential fin $m{L}_{\infty }$ is replaced with mL).

Exponential straight or pin fins having increasing cross-sectional areas (𝑏<0) always exhibit higher fin heat transfer rates relative to rectangular straight or pin fins as can be seen from Figures 7 and 8. However, 𝛾₁ values for those having decreasing cross-sectional areas (𝑏>0) are always smaller than unity as shown in Figures 7 and 8. Exponential joint-fins are found to transfer more heat than rectangular joint-fins fins at smaller values of X₁ and larger values of X₂ as can be seen from Figures 9 and 10. On the other hand, pin joint-fins with exponentially decaying temperatures were found to be preferable over rectangular pin joint-fins at smaller X₁ and X₂ values as shown in Figure 11.

Figure 7 

Effect of the fin dimensionless parameter $m{L}_{\infty }$ on the performance indicator $\gamma$ for straight exponential fins with $b>0$ and $b<0$ ($m=\sqrt{2h/(kt)}$).

Figure 8 

Effect of the fin dimensionless parameter $m{L}_{\infty }$ on the performance indicator $\gamma$ for exponential pin fins with $b>0\text{,}$$b<0$ and $L_{\infty }=x_0$ ($m=\sqrt{2h/(k{r}_b)}$).

Figure 9 

Effect of the parameters $X_1\mathrm{\hspace{0.17em}\hspace{0.17em}}$ and $X_2$ on the performance indicator ${({\gamma }_3)}_s$ for a straight exponential joint-fin with one side having $b<0$ and the other side having $b>0$.

Figure 10 

Effect of the parameters $X_1\mathrm{\hspace{0.17em}\hspace{0.17em}}$ and $X_2$ on the performance indicators ${({\gamma }_3)}_P$ for an exponential pin joint-fin with one side having $b<0$ and the other side having $b>0$.

Figure 11 

Effect of the parameters $X_1\mathrm{\hspace{0.17em}\hspace{0.17em}}$ and $X_2$ on the performance indicators ${({\gamma }_3)}_T$ for an exponential pin joint-fin having an exponential decaying temperature distribution.

The effect of increasing the exponential index b on 𝛾₃ can be illustrated using Figures 9 and 10, for example, increasing b by a factor of 10 while maintaining the other parameters results in reductions in both X₁ and X₂ values by a factor of 0.1, for example, if 𝑋₁=10 and 𝑋₂=10. This produces (𝛾₃)_𝑠=1.52 and (𝛾₃)_𝑃=0.857. Increasing b by factor of 10 changes the joint-fin performance indicators (𝛾₃)_𝑠=0.894 and (𝛾₃)_𝑃=0.167 which are smaller than the initial values. In contrast, initially selecting 𝑋₁=0.1 and 𝑋₂=10 which produce (𝛾₃)_𝑠=9.22 and (𝛾₃)_𝑃=414 results in final 𝑋₁=0.01 and 𝑋₂=1.0 which lead to (𝛾₃)_𝑠=38.6 and (𝛾₃)_𝑃=10.91. As such, we can conclude that only (𝛾₃)_𝑠 may increase as b increases when 𝑋₂−𝑋₁ is relatively large.

4. Conclusions

Exponential fin systems were modeled and mathematically analyzed in this work. The possibility of having decreasing or increasing cross-sectional areas was considered. Rectangular and circular cross-sectional areas are considered. Special thermal performance indicators were derived. The maximum ratio between the thermal efficiency of the exponential straight fin to that of the rectangular fin was found to be 1.58 at an effective thermal length of 2.0. This ratio was found to be larger when the exponential fin was compared with triangular and parabolic fins. Meanwhile, the maximum ratio between the thermal efficiency of the exponential pin fin to that of the rectangular pin fin was found to be 1.17 at an effective thermal length of 1.5. However, exponential pin thermal efficiency was found to be lower than those of triangular and parabolic pin fins. In addition, exponential joint-fins may transfer more heat than rectangular joint-fins especially when differences between their senders and receivers portions dimensionless indices are very large. Finally, the summary of the closed-form solutions and correlations reported in this work as compared to those of rectangular, triangular, and parabolic fin systems are summarized in Table 1.

Table 1 

Efficiencies of exponential fins compared to efficiencies of common fins.

Nomenclature

a, b:	Exponential functions indices
H:	Half-fin width
𝐻𝑏:	Half-fin width at its base
h:	Convection heat transfer coefficient between the fin and the fluid stream
h1:	Convection heat transfer coefficient for the joint-fin source side
h2:	Convection heat transfer coefficient for the joint-fin sink side
In(x):	Modified Bessel functions of the first kind of order n
Kn(x):	Modified Bessel functions of the second kind of order n
k:	Fin thermal conductivity
L:	Fin length
𝐿∞:	Effective fin length
m:	Fin thermal index
𝑞𝑓:	Fin heat transfer rate
r:	Pin fin radius
𝑟𝑏:	Pin fin radius at its base
T:	Fin temperature
𝑇𝑏:	Fin base temperature
𝑇∞:	Free stream temperature of the adjoining fluid
𝑇∞1:	Free stream temperature of the source side adjoining fluid
𝑇∞2:	Free stream temperature of the sink side adjoining fluid
t:	Fin thickness
X:	Dimensionless exponential fin parameter
X1:	Dimensionless exponential parameter of the receiver fin portion
X2:	Dimensionless exponential parameter of the sender fin portion
x:	Coordinate axis along the fin centerline
𝑥0:	Pin fin length for exponential fins with exponentially decaying temperature
𝑥:	Dimensionless x-coordinate.

Greek Symbols

𝜃:	Dimensionless fin temperature
𝛾1,2:	Fin second thermal performance indicators
𝛾3:	Joint-fin thermal performance indicator
𝜂𝑓:	Fin thermal efficiency.

Acknowledgment

The author acknowledges the support of this work by King Abdulaziz City for Science and Technology (KACST).