# Thermal Characterizations of Exponential Fin Systems

Authors: A.-R. A. Khaled
Journal: Mathematical Problems in Engineering
 (2010)

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## 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.


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## Body

## 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 thicknesst 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:
(2.1)H̅(x̅)≡H(x/L)Hb=e-bLx̅,
where x̅=x/L and b is a real number named as the exponential index. The quantity Hb  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:
(2.2)ddx(AcdTdx)-hk(dAsdx)(T-T∞)=0,
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 quantitiesAc and Asare the cross-sectional and the surface areas of the fin, respectively. Equation (2.2) has the following dimensionless form: (2.3)ddx̅(H̅dθdx̅)-(mL)2H̅θ=0,
where m=2h/(kt) and θ=(T(x)-T∞)/(Tb-T∞). The quantity m is called the fin index while Tb is the fin temperature at its base. Equation (2.3) prescribes the following general solution: (2.4)θ(x̅)=C1es1x̅+C2s2x̅,
where s1ands2 are equal to(2.5)s1,2=bL2[1∓4X2+1]ifb>0,s1,2=bL2[1±4X2+1]ifb<0,
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 by(2.6)θ(x̅=0)=1.0,∂θ∂x̅|x̅=1=0.0.
As such, the dimensionless temperature distribution has the following form:(2.7)θ(x̅)=es1x̅-[(s1es1)/(s2es2)]es2x̅1-(s1es1)/(s2es2).The rate of heat transfer through the fin is called the fin heat transfer rate. For this case, it is equal to(2.8)qf=-kAcdTdx|x=0={Hbtkb{-1+4X2+1}(Tb-T∞)Φ1(s1,s2);b>0,-Hbtkb{1+4X2+1}(Tb-T∞)Φ2(s1,s2);b<0,
where Φ1and Φ2factors are smaller than unity. They are equal to(2.9)Φ1(s1,s2)=1-Exp(-bL4X2+1)1-((1-4X2+1)/(1+4X2+1))Exp(-bL4X2+1),(2.10)Φ2(s1,s2)=1-Exp(bL4X2+1)1-((1+4X2+1)/(1-4X2+1))Exp(bL4X2+1).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 by(2.11)m(L∞)1≅X4X2+1(5.293-ln[1+14X2+1]),b>0,m(L∞)2≅-X4X2+1(5.293-ln[1-14X2+1]),b<0,
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  ηf  is defined as the fin heat transfer rate divided by the fin heat transfer rate if the fin temperature is kept at Tb. For this case, it can have the following forms: (2.12)ηf≡0.99(qf)max4h(Tb-T∞)∫0L∞Hdx={0.99{-1+4X2+1}2X2[1-Exp(-b(L∞)1)],b>0,0.99{1+4X2+1}2X2[Exp(-b(L∞)2)-1],b<0,
where (qf)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 2Hb, uniform thickness t and an infinite length.  As such, γ is equal to(2.13)γ≡(qf)max(qf)|H=Hb=(qf)max2Hb2hkt(Tb-T∞)={(-1+4X2+12X),b>0,(1+4X2+12X),b<0.


### 2.2. Pin Fins with Exponentially Varying Radii

Consider a pin fin of radiusr(x), as shown in Figure 2, that varies exponentially along the x-axis according to the following relationship:
(2.14)r̅(x)≡r(x)rb=e-bx,
where b>0. As such, (2.2) changes to:
(2.15)ddx(e-2bxdθdx)-m2e-bxθ=0,
where m=2h/(krb). It can be shown that the general solution of (2.15) is(2.16)θ(x)=emx/X{C1I2(2Xemx/(2X))+C2K2(2Xemx/(2X))},
where X=m/b.As such, the fin heat transfer rate is(2.17)qf=-kπrb2(Tb-T∞)b{C1[I2(2X)+0.5X(I1(2X)+I3(2X))]+C2[K2(2X)-0.5X(K1(2X)+K3(2X))]}.Figure 2
Schematic diagram for an exponential pin fin withb<0 and the system coordinate.For a fin with an infinite length (L→∞), the constants C1 and C2 are given by
(2.18)C1=0,C2=1K2(2X).
This is because emx/(2X) approaches infinity as x approaches infinity; hence I2(2Xemx/(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 to(2.19)C1=(I2(2X)-K2(2X)C3)-1,C3=I2(2XemL/(2X))+0.5XemL/(2X){I1(2XemL/(2X))+I3(2XemL/(2X))}K2(2XemL/(2X))-0.5XemL/(2X){K1(2XemL/(2X))+K3(2XemL/(2X))},C2=-C1C3.The fin efficiencyηf for b>0 can be found to be equal to(2.20)ηf≡0.99(qf)max2πrbh(Tb-T∞)∫0L∞r̅(x)dx=0.99(1-e-mL∞/X)(1X2){X2[K1(2X)K2(2X)+K3(2X)K2(2X)]-1},b>0,
where mL∞ is obtained from the solution of the following equation:(2.21)(qf)Ins.Tip(qf)L→∞=ϕ(X,mL∞X)=0.99.The fin performance indicatorγ  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 rb and an infinite length. It is equal to the following:(2.22)γ1≡(qf)max(qf)|r=rb=(qf)maxπrb2hkrb(Tb-T∞)=(1X){X2[K1(2X)K2(2X)+K3(2X)K2(2X)]-1},b>0.For cases whenb<0; X is replaced with −X, and the constants C1 and C2 for a fin with infinite length are replaced by(2.23)C1=1I2(-2X),C2=0,b<0.
As such, the fin thermal efficiency  ηf  and the indicator γ  when b<0 change to(2.24)ηf=0.99(e-mL∞/X-1)(1X2){-X2[I1(-2X)I2(-2X)+I3(-2X)I2(-2X)]+1},b<0,γ2≡qf(qf)|r=rb=(-1X){-X2[I1(-2X)I2(-2X)+I3(-2X)I2(-2X)]+1},b<0.


### 2.3. Pin Fins with Exponentially Decaying Temperature Distribution

Consider a pin fin having a given fin temperature distribution that varies exponentially withx according to the following relationship:
(2.25)θ(x̅)≡T(x)-T∞Tb-T∞=e-axox̅,
where x̅=x/xo and a is the exponential index. The dimensionless form of the energy equation has the following form
(2.26)ddx̅(r̅2(x)dθdx̅)-(mxo)2r̅(x)θ=0,
where m=2h/(krb) and r̅(x)=r(x)/rb. By substituting  (2.25) in (2.26), a differential equation of first order constructed. It has the form
(2.27)dr̅r̅-X2=(ax0)2dx̅,
where X=m/a. The solution to (2.27) is given by(2.28)r̅(x̅)=X2+[1-X2]Exp(ax02x̅).For engineering problems,r̅(x̅) cannot be negative and it should intersect with fin centerline at x̅=1 when X>1.0. As such, x0 is found to be equal to
(2.29)mx0=(2X)ln[X2X2-1],X>1.0.
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
(2.30)qf=-kfAcdTdx|x=0=πrb2ka(Tb-T∞).As such, the fin thermal efficiency and the performance indicator for this case are equal to(2.31)ηf=qf2πrbx0h(Tb-T∞)∫01r̅(x̅)dx̅={12X2{X2ln[X2X2-1]-1}-1,X>1.0,12X2(9.0-6.697X2),0≤X≤1.0,γ1=qfπrb2hkrb(Tb-T∞)=1X.


### 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 temperatureT∞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 (qf)s, pin exponential joint-fin (qf)P, and the pin joint-fin with exponential decaying temperature (qf)T are given by the following equations: (2.32)(qf)s=-γ22Hb2h1kt(Tb-T∞1)=γ12Hb2h2kt(Tb-T∞2),(2.33)(qf)P=-γ2πrb2h1krb(Tb-T∞1)=γ1πrb2h2krb(Tb-T∞2),(2.34)(qf)T=-γ1πrb2h1krb(Tb-T∞1)=γ1πrb2h2krb(Tb-T∞2),
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(2.35)(Tb)s=⌊4X12+1+1⌋T∞1+⌊4X22+1-1⌋T∞24X12+1+4X22+1,(2.36)(Tb)P=M×T∞1+N×T∞2M+N,(2.37)(Tb)T=T∞1+T∞22,
where M and N are given by(2.38)M=[(X1X2){I1(2X1)2I2(2X1)+I3(2X1)2I2(2X1)}+1X2],N=[K1(2X2)2K2(2X2)+K3(2X2)2K2(2X2)-1X2].
By substituting (2.35)–(2.37) in (2.32)–(2.34), the joint-fin heat transfer rates reduce to the following forms:(2.39)(qf)s=2Hb2h1kt(T∞1-T∞2)⌊4X12+1+1⌋⌊4X22+1-1⌋2X1{4X12+1+4X22+1},(qf)s=πrb2h1krb(T∞1-T∞2)NM(M+N)(X2X1),(qf)T=πrb2h1krb(T∞1-T∞2)2X1.Define the joint-fin performance indicatorγ3  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
(2.40)γ3≡qf[(qf)|Ac=constant]Joint-fin.
The heat transfer rate through the joint fin when b=0 is obtainable from [17]. It is equal to(2.41)[(qf)|H=Hb]Joint-fin=h1kPAc1+h1/h2(T∞1-T∞2)
As such, γ3  can be written in the following forms: (2.42)(γ3)s=12(1X1+1X2)⌊4X12+1+1⌋⌊4X22+1-1⌋{4X12+1+4X22+1},(γ3)P={NM(M+N)}(X2X1+1),(γ3)T=X2+X12X1X2.


## 2.1. Straight Fins with Exponentially Varying Widths

Consider a rectangular fin having a uniform thicknesst 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:
(2.1)H̅(x̅)≡H(x/L)Hb=e-bLx̅,
where x̅=x/L and b is a real number named as the exponential index. The quantity Hb  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:
(2.2)ddx(AcdTdx)-hk(dAsdx)(T-T∞)=0,
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 quantitiesAc and Asare the cross-sectional and the surface areas of the fin, respectively. Equation (2.2) has the following dimensionless form: (2.3)ddx̅(H̅dθdx̅)-(mL)2H̅θ=0,
where m=2h/(kt) and θ=(T(x)-T∞)/(Tb-T∞). The quantity m is called the fin index while Tb is the fin temperature at its base. Equation (2.3) prescribes the following general solution: (2.4)θ(x̅)=C1es1x̅+C2s2x̅,
where s1ands2 are equal to(2.5)s1,2=bL2[1∓4X2+1]ifb>0,s1,2=bL2[1±4X2+1]ifb<0,
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 by(2.6)θ(x̅=0)=1.0,∂θ∂x̅|x̅=1=0.0.
As such, the dimensionless temperature distribution has the following form:(2.7)θ(x̅)=es1x̅-[(s1es1)/(s2es2)]es2x̅1-(s1es1)/(s2es2).The rate of heat transfer through the fin is called the fin heat transfer rate. For this case, it is equal to(2.8)qf=-kAcdTdx|x=0={Hbtkb{-1+4X2+1}(Tb-T∞)Φ1(s1,s2);b>0,-Hbtkb{1+4X2+1}(Tb-T∞)Φ2(s1,s2);b<0,
where Φ1and Φ2factors are smaller than unity. They are equal to(2.9)Φ1(s1,s2)=1-Exp(-bL4X2+1)1-((1-4X2+1)/(1+4X2+1))Exp(-bL4X2+1),(2.10)Φ2(s1,s2)=1-Exp(bL4X2+1)1-((1+4X2+1)/(1-4X2+1))Exp(bL4X2+1).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 by(2.11)m(L∞)1≅X4X2+1(5.293-ln[1+14X2+1]),b>0,m(L∞)2≅-X4X2+1(5.293-ln[1-14X2+1]),b<0,
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  ηf  is defined as the fin heat transfer rate divided by the fin heat transfer rate if the fin temperature is kept at Tb. For this case, it can have the following forms: (2.12)ηf≡0.99(qf)max4h(Tb-T∞)∫0L∞Hdx={0.99{-1+4X2+1}2X2[1-Exp(-b(L∞)1)],b>0,0.99{1+4X2+1}2X2[Exp(-b(L∞)2)-1],b<0,
where (qf)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 2Hb, uniform thickness t and an infinite length.  As such, γ is equal to(2.13)γ≡(qf)max(qf)|H=Hb=(qf)max2Hb2hkt(Tb-T∞)={(-1+4X2+12X),b>0,(1+4X2+12X),b<0.


## 2.2. Pin Fins with Exponentially Varying Radii

Consider a pin fin of radiusr(x), as shown in Figure 2, that varies exponentially along the x-axis according to the following relationship:
(2.14)r̅(x)≡r(x)rb=e-bx,
where b>0. As such, (2.2) changes to:
(2.15)ddx(e-2bxdθdx)-m2e-bxθ=0,
where m=2h/(krb). It can be shown that the general solution of (2.15) is(2.16)θ(x)=emx/X{C1I2(2Xemx/(2X))+C2K2(2Xemx/(2X))},
where X=m/b.As such, the fin heat transfer rate is(2.17)qf=-kπrb2(Tb-T∞)b{C1[I2(2X)+0.5X(I1(2X)+I3(2X))]+C2[K2(2X)-0.5X(K1(2X)+K3(2X))]}.Figure 2
Schematic diagram for an exponential pin fin withb<0 and the system coordinate.For a fin with an infinite length (L→∞), the constants C1 and C2 are given by
(2.18)C1=0,C2=1K2(2X).
This is because emx/(2X) approaches infinity as x approaches infinity; hence I2(2Xemx/(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 to(2.19)C1=(I2(2X)-K2(2X)C3)-1,C3=I2(2XemL/(2X))+0.5XemL/(2X){I1(2XemL/(2X))+I3(2XemL/(2X))}K2(2XemL/(2X))-0.5XemL/(2X){K1(2XemL/(2X))+K3(2XemL/(2X))},C2=-C1C3.The fin efficiencyηf for b>0 can be found to be equal to(2.20)ηf≡0.99(qf)max2πrbh(Tb-T∞)∫0L∞r̅(x)dx=0.99(1-e-mL∞/X)(1X2){X2[K1(2X)K2(2X)+K3(2X)K2(2X)]-1},b>0,
where mL∞ is obtained from the solution of the following equation:(2.21)(qf)Ins.Tip(qf)L→∞=ϕ(X,mL∞X)=0.99.The fin performance indicatorγ  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 rb and an infinite length. It is equal to the following:(2.22)γ1≡(qf)max(qf)|r=rb=(qf)maxπrb2hkrb(Tb-T∞)=(1X){X2[K1(2X)K2(2X)+K3(2X)K2(2X)]-1},b>0.For cases whenb<0; X is replaced with −X, and the constants C1 and C2 for a fin with infinite length are replaced by(2.23)C1=1I2(-2X),C2=0,b<0.
As such, the fin thermal efficiency  ηf  and the indicator γ  when b<0 change to(2.24)ηf=0.99(e-mL∞/X-1)(1X2){-X2[I1(-2X)I2(-2X)+I3(-2X)I2(-2X)]+1},b<0,γ2≡qf(qf)|r=rb=(-1X){-X2[I1(-2X)I2(-2X)+I3(-2X)I2(-2X)]+1},b<0.


## 2.3. Pin Fins with Exponentially Decaying Temperature Distribution

Consider a pin fin having a given fin temperature distribution that varies exponentially withx according to the following relationship:
(2.25)θ(x̅)≡T(x)-T∞Tb-T∞=e-axox̅,
where x̅=x/xo and a is the exponential index. The dimensionless form of the energy equation has the following form
(2.26)ddx̅(r̅2(x)dθdx̅)-(mxo)2r̅(x)θ=0,
where m=2h/(krb) and r̅(x)=r(x)/rb. By substituting  (2.25) in (2.26), a differential equation of first order constructed. It has the form
(2.27)dr̅r̅-X2=(ax0)2dx̅,
where X=m/a. The solution to (2.27) is given by(2.28)r̅(x̅)=X2+[1-X2]Exp(ax02x̅).For engineering problems,r̅(x̅) cannot be negative and it should intersect with fin centerline at x̅=1 when X>1.0. As such, x0 is found to be equal to
(2.29)mx0=(2X)ln[X2X2-1],X>1.0.
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
(2.30)qf=-kfAcdTdx|x=0=πrb2ka(Tb-T∞).As such, the fin thermal efficiency and the performance indicator for this case are equal to(2.31)ηf=qf2πrbx0h(Tb-T∞)∫01r̅(x̅)dx̅={12X2{X2ln[X2X2-1]-1}-1,X>1.0,12X2(9.0-6.697X2),0≤X≤1.0,γ1=qfπrb2hkrb(Tb-T∞)=1X.


## 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 temperatureT∞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 (qf)s, pin exponential joint-fin (qf)P, and the pin joint-fin with exponential decaying temperature (qf)T are given by the following equations: (2.32)(qf)s=-γ22Hb2h1kt(Tb-T∞1)=γ12Hb2h2kt(Tb-T∞2),(2.33)(qf)P=-γ2πrb2h1krb(Tb-T∞1)=γ1πrb2h2krb(Tb-T∞2),(2.34)(qf)T=-γ1πrb2h1krb(Tb-T∞1)=γ1πrb2h2krb(Tb-T∞2),
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(2.35)(Tb)s=⌊4X12+1+1⌋T∞1+⌊4X22+1-1⌋T∞24X12+1+4X22+1,(2.36)(Tb)P=M×T∞1+N×T∞2M+N,(2.37)(Tb)T=T∞1+T∞22,
where M and N are given by(2.38)M=[(X1X2){I1(2X1)2I2(2X1)+I3(2X1)2I2(2X1)}+1X2],N=[K1(2X2)2K2(2X2)+K3(2X2)2K2(2X2)-1X2].
By substituting (2.35)–(2.37) in (2.32)–(2.34), the joint-fin heat transfer rates reduce to the following forms:(2.39)(qf)s=2Hb2h1kt(T∞1-T∞2)⌊4X12+1+1⌋⌊4X22+1-1⌋2X1{4X12+1+4X22+1},(qf)s=πrb2h1krb(T∞1-T∞2)NM(M+N)(X2X1),(qf)T=πrb2h1krb(T∞1-T∞2)2X1.Define the joint-fin performance indicatorγ3  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
(2.40)γ3≡qf[(qf)|Ac=constant]Joint-fin.
The heat transfer rate through the joint fin when b=0 is obtainable from [17]. It is equal to(2.41)[(qf)|H=Hb]Joint-fin=h1kPAc1+h1/h2(T∞1-T∞2)
As such, γ3  can be written in the following forms: (2.42)(γ3)s=12(1X1+1X2)⌊4X12+1+1⌋⌊4X22+1-1⌋{4X12+1+4X22+1},(γ3)P={NM(M+N)}(X2X1+1),(γ3)T=X2+X12X1X2.


## 3. Discussion of the Results

Figure3 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 untilX 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: (3.1)mL∞=0.8233(X0.8804-0.00470.2945X0.8934+0.2428),b>0,(3.2)mL∞=0.7237(X0.6906+3.3301X1.4019+0.3939X0.6902-0.03020.9547X1.3923+e-0.6311X1.9309-0.7425),b<0.
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 parameterX on the effective thermal length mL∞  for straight exponential fins with b>0 and b<0 (m=2h/(kt)).Figure 4
Effect of the fin dimensionless parameterX on the effective thermal length mL∞ for exponential pin fins with b>0, b<0 and L∞=x0 (m=2h/(krb)).Figure5 shows the relation between the effective thermal length mL∞ on the fin thermal efficiency ηf for a straight exponential fin. It is seen that ηf 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, ηf  is found to be higher than ηf 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 parametermL∞ on the fin efficiency ηf for straight exponential fins with b>0, or b<0 (m=2h/(kt); other than exponential fin mL∞ is replaced withmL).Figure 6
Effect of the fin dimensionless parametermL∞ on the fin efficiency ηf for exponential pin fins with b>0,  b<0 and L∞=x0 (m=2h/(krb); other than exponential fin mL∞ is replaced withmL).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 parametermL∞ on the performance indicator γ for straight exponential fins with b>0 and b<0 (m=2h/(kt)).Figure 8
Effect of the fin dimensionless parametermL∞ on the performance indicator γ for exponential pin fins with b>0, b<0 and L∞=x0 (m=2h/(krb)).Figure 9
Effect of the parametersX1 and X2 on the performance indicator (γ3)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 parametersX1 and X2 on the performance indicators (γ3)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 parametersX1 and X2 on the performance indicators (γ3)T for an exponential pin joint-fin having an exponential decaying temperature distribution.The effect of increasing the exponential indexb 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 Table1.Table 1
Efficiencies of exponential fins compared to efficiencies of common fins.
Fin typeCross-sectional areamL∞ηfPerimeterRectangular straight or pin fins with insulatedtips(i) [20]2Ht4H, πrb22πrb2.65tanh(mL)(mL)Triangular straightfin(i) [20]H0.5to{1-x/L}2H—1(mL)I1(2mL)I0(2mL)Triangular pinfin(ii) [20]πrb2{1-(xL)}22πrb{1-(xL)}—2(mL)I2(2mL)I1(2mL)Parabolic straightfin(i) [20]H0.5to{1-(x/L)2}2H—2(4(mL)2+1)0.5+1
fParabolic pinfin(ii) [20]πrb2{1-(xL)2}22πrb{1-(xL)2}—2([4/9](mL)2+1)0.5+1Exponential straightfin(i)(1)2Hbe-bxt4Hbe-bx(X4X2+1)(5.293-ln[1+(14X2+1)]);b>0,(-X4X2+1)(5.293-ln[1-(14X2+1)]);b<00.99{-1+4X2+1}2X2[1-Exp(-b(L∞)1)],b>00.99{1+4X2+1}2X2[Exp(-b(L∞)2)-1],b<0Exponential pinfin(ii)(1)πrb2e-2bx2πrbe-bx0.8233(X0.8804-0.00470.2945X0.8934+0.2428);b>0,0.7237(X0.6906+3.3301X1.4019+0.3939X0.6902-0.03020.9547X1.3923+e-0.6311X1.9309-0.7425);b<00.99(1-e-mL∞/X)×1X2(X2){[K1(2X)K2(2X)+(K3(2X)K2(2X))]}-1,b>00.99(e-mL∞/X-1)×1X2{-X2[I1(-2X)I2(-2X)+I3(-2X)I2(-2X)]+1},b<0Exponential decaying temperature-pinfin(ii)(2)πrb2{X2+[1-X2]×e-0.5axox}22πrb{X2+[1-X2]×e-0.5axox}(2X)ln[X2(X2-1)];X>1.04.605X;X≤1.012X2{X2ln[X2(X2-1)]-1}-1,X>1.012X2(9.0-6.697X2),X≤1.0m(i)=2h/(kt); m(ii)=2h/(krb); X(1)=m/b; X(2)=m/a.



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*Source: 765729-2010-06-07.xml*