Thermal Characterizations of Exponential Fin Systems

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.


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 .

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.

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: 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. The application of the energy equation 20 on a fin differential element results in the following differential equation: 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 s are the cross-sectional and the surface areas of the fin, respectively.Equation 2.2 has the following dimensionless form: 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: where s 1 and s 2 are equal to 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 θ x 0 1.0, ∂θ ∂x x 1 0.0.

2.6
As such, the dimensionless temperature distribution has the following form:

2.7
The rate of heat transfer through the fin is called the fin heat transfer rate.For this case, it is equal to where Φ 1 and Φ 2 factors are smaller than unity.They are equal to .

2.10
Utilizing 2.9 and 2.10 , the fin lengths L L ∞ 1 and L L ∞ 2 that make Φ 1 and Φ 2 equal to 0.99, respectively, can be approximated by 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: 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 to

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:  where m 2h/ kr b .It can be shown that the general solution of 2.15 is where X m/b.As such, the fin heat transfer rate is

2.17
For a fin with an infinite length L → ∞ , the constants C 1 and C 2 are given by This is because e mx/ 2X approaches infinity as x approaches infinity; hence I 2 2Xe mx/ 2X approaches infinity.Thus, C 1 is equal to zero.
For adiabatic fin tips, boundary conditions given by 2.6 should be satisfied.Accordingly, the constants C 1 and C 2 are equal to

2.19
The fin efficiency η f for b > 0 can be found to be equal to where mL ∞ is obtained from the solution of the following equation:

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:

2.22
For cases when b < 0; X is replaced with −X, and the constants C 1 and C 2 for a fin with infinite length are replaced by As such, the fin thermal efficiency η f and the indicator γ when b < 0 change to 2.24

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:

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, x 0 is found to be equal to In situations when 0 < X < 1.0, mx 0 is minimally equal to 4.605X x 0 4.605/a; X < 1.0 .Under this constraint, the heat transfer rate at the fin tip x x 0 is always 0.01 times the fin heat transfer rate.The rate of heat transfer through the fin base is equal to

2.30
As such, the fin thermal efficiency and the performance indicator for this case are equal to 2.31

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 h 1 .This coefficient is h 2 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: 32 33 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 35 36 where M and N are given by

2.38
By substituting 2.35 -2.37 in 2.32 -2.34 , the joint-fin heat transfer rates reduce to the following forms:

2.39
Define the joint-fin performance indicator γ 3 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
The heat transfer rate through the joint fin when b 0 is obtainable from 17 .It is equal to As such, γ 3 can be written in the following forms:

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, mx 0 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 ∞ 0.8233 X 0.8804 − 0.0047 0.2945X 0.8934 0.2428 , b > 0, 3.1 mL ∞ 0.7237 X 0.6906 3.3301X 1.4019 0.3939X 0.6902 − 0.0302 0.9547X 1.3923 e −0.6311X 1.9309 − 0.7425 , b < 0.

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 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.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, γ 1 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 X 1 and larger values of X 2 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 X 1 and X 2 values as shown in Figure 11.X 1 0.01 The effect of increasing the exponential index b on γ 3 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 X 1 and X 2 values by a factor of 0.1, for example, if X 1 10 and X 2 10.This produces γ 3 s 1.52 and γ 3 P 0.857.Increasing b by factor of 10 changes the joint-fin performance indicators γ 3 s 0.894 and γ 3 P 0.167 which are smaller than the initial values.In contrast, initially selecting X 1 0.1 and X 2 10 which produce γ 3 s 9.22 and γ 3 P 414 results in final X 1 0.01 and X 2 1.0 which lead to γ 3 s 38.6 and γ 3 P 10.91.As such, we can conclude that only γ 3 s may increase as b increases when X 2 − X 1 is relatively large.

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 closedform solutions and correlations reported in this work as compared to those of rectangular, triangular, and parabolic fin systems are summarized in Table 1.

Nomenclature a, b: Exponential functions indices H:
Half-fin width H b : Half-fin width at its base h: Convection heat transfer coefficient between the fin and the fluid stream h 1 : Convection heat transfer coefficient for the joint-fin source side h 2 : Convection heat transfer coefficient for the joint-fin sink side I n (x): Modified Bessel functions of the first kind of order n K n (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 q f : Fin heat transfer rate r: Pinfinradius r b : Pin fin radius at its base T: Fin temperature T b : 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 X 1 : Dimensionless exponential parameter of the receiver fin portion X 2 : Dimensionless exponential parameter of the sender fin portion x: Coordinate axis along the fin centerline x 0 : Pin fin length for exponential fins with exponentially decaying temperature x: Dimensionless x-coordinate.

Figure 1 :
Figure 1: Schematic diagram for a straight exponential fin and exponential joint-fin and the system coordinates.

Figure 2 :
Figure 2: Schematic diagram for an exponential pin fin with b < 0 and the system coordinate.

Figure 3 :
Figure 3: Effect of the fin dimensionless parameter X on the effective thermal length mL ∞ for straight exponential fins with b > 0 and b < 0 m 2h/ kt .

Exponential pin L ∞ x 0 ExponentialFigure 4 :Figure 5 :
Figure 4: Effect of the fin dimensionless parameter X on the effective thermal length mL ∞ for exponential pin fins with b > 0, b < 0 and L ∞ x 0 m 2h/ kr b .

Figure 6 :ExponentialFigure 7 :Figure 8 :Figure 9 :Figure 10 :
Figure 6: Effect of the fin dimensionless parameter mL ∞ on the fin efficiency η f for exponential pin fins with b > 0, b < 0 and L ∞ x 0 m 2h/ kr b ; other than exponential fin mL ∞ is replaced with mL .

Figure 11 :
Figure 11: Effect of the parameters X 1 and X 2 on the performance indicators γ 3 T for an exponential pin joint-fin having an exponential decaying temperature distribution.

Table 1 :
Efficiencies of exponential fins compared to efficiencies of common fins.