تجزیه و تحلیل عملکرد و بهینه سازی باله های مخروطی مستقیم با ضریب انتقال حرارت متغیر

In the present paper, the thermal analysis and optimization of straight taper fins has been addressed. With the help of the Frobenius expanding series the temperature profiles of longitudinal fin, spine and annular fin have been determined analytically through a unified approach. Simplifying assumptions like length of arc idealization and insulated fin tip condition have been relaxed and a linear variation of the convective heat transfer coefficient along the fin surface has been taken into account. The thermal performance of all the three types of fin has been studied over a wide range of thermo-geometric parameters. It has been observed that the variable heat transfer coefficient has a strong influence over the fin efficiency. Finally, a generalized methodology has been pointed out for the optimum design of straight taper fins. A graphical representation of optimal fin parameters as a function of heat duty has also been provided.

Fins or extended surfaces are widely used to augment the rate of heat transfer from the primary surface to the ambient medium in a large variety of thermal equipment. An accurate analysis of heat transfer in fins has become crucial with the growing demand of high performance of heat transfer surfaces with progressively smaller weights, volumes, initial and running cost of the system. Over the years different fin shapes have been evolved depending upon the application and the geometry of the primary surface. Kern and Kraus [1] have identified three main fin geometries. These are longitudinal fins, radial or circumferential fins and pin fins or spines. For any of the above geometry, fins with straight profile or constant thickness are a common choice as they can be manufactured easily. The thermal design of a constant thickness fin is also relatively simple. However, in any fin the temperature difference reduces from the fin base to fin tip. Accordingly, a saving of fin material can be obtained by progressively narrowing down the fin section. This has initiated a lot of exercises for the determination of optimum fin shapes so that the fin volume is minimum for a given rate of heat dissipation or the rate of heat dissipation is maximum for a given fin volume. The criteria for optimum fin profile under convective conditions was first proposed by Schmidt [2] based on a physical reasoning. Later on Duffin [3] proved Schmidt criteria using calculus of variation. Both Schmidt [2] and Duffin [3] estimated the fin surface area neglecting the profile curvature. This has formed a major assumption in further exercises of fin optimization and is known as length of arc idealization (LAI) in literature. LAI was used for optimizing fin shapes under convecting, radiating, convective-radiating condition [4], for fins with heat generation [5] and for variable thermal conductivity. Maday [6] in his pioneering analysis proposed the correct formulation for the optimization of longitudinal fin with the elimination of LAI and obtained a profile much different from Duffin [3]. Guceri and Maday [7] further extended this analysis for radial fins. However, fin shapes determined by the above procedure are complex and difficult to manufacture. These fins have structurally weak slender tips, which do not substantially contribute to the overall heat dissipation. This has resulted in a parallel effort to design optimum fins where the fin shape is specified a priori and fin dimensions are determined to give maximum heat dissipation for a given fin volume. Aziz [8] in his review paper thoroughly discussed the state of the art of fin optimization when fin shape is specified. Based on the assumptions proposed by Murray [9] and Gardner [10], he has presented the optimum dimensions of longitudinal fins (straight, triangular and concave parabolic profiles), spines (cylindrical, conical, concave and convex parabolic profiles) and radial fins (straight, trapezoidal and triangular profiles). Finally he discussed the effect of tip heat loss, variable heat transfer coefficient, temperature dependent thermal properties and internal heat generation on the optimal dimensions of a few specific fins. Chung and Kan [11] considered the effect of profile curvature on the optimum dimensions of longitudinal fins of triangular, concave and convex parabolic profile. While they have proposed an analytical solution for triangular fin they had to take the resort of numerical techniques for parabolic fins. Razelos and Satyaprakash [12] presented an analysis for optimum longitudinal fin of trapezoidal section based on an assumption of negligible heat loss from the fin tip and negligible surface curvature effect and finally suggested a correlation for the optimality criteria. Based on a diameter dependent convective heat transfer coefficient, Chung [13] improved the design of optimum cylindrical pin fins originally proposed by Sonn and Bar-Cohen [14]. Chung and Kan [11] determined the optimum dimensions of spines having different profiles (cylindrical, conical, concave and convex parabolic) from a generalized formulation using a numerical procedure. They reported a profound influence of profile curvature on the optimum dimensions of the spine. Razelos [15] analyzed the heat transfer from convective spines of different profiles assuming negligible surface curvature and no tip loss. Using the Lagrangian multiplier technique the author derived the thermo-geometric criteria for the optimum spines. On the other hand Ulman and Kalman [16] solved the conduction equation for radial fins of different profiles (straight, hyperbolic, triangular and parabolic) numerically to find out the rate of heat dissipation. They obtained the optimum fin dimensions for each of the profiles for the maximum value of heat dissipation under a volume constraint. Most of the analytical works on fin design and optimization carried out till date are based on the assumption of constant convective heat transfer coefficient along the fin length. However, existence of a non-uniform heat transfer coefficient has been established theoretically and observed experimentally. Kraus [17] has thoroughly discussed the results of the investigations, which have considered non-uniform heat transfer coefficient along the fin surface. He has concluded that non-uniformities have an impact on the rate of heat dissipation by the fins. By a unique experiment Ghai [18] demonstrated that heat transfer coefficient increases towards the fin tip with a minimal value at the fin base. Gardner [19] showed that the variation of heat transfer coefficient along the fin length could be expressed in the form of an equation using Ghai's experimental results. Kraus et al. [20] discussed the findings of Ghai and Gardner in some details. This has triggered a number of investigations considering linear [21], power law [22] and exponential variation [23] of heat transfer coefficient. However, the actual nature of variation of the heat transfer coefficient can be obtained by a conjugate analysis of conduction in the fin along with the convection in the adjacent fluid. Such studies have been taken up by a number of researchers. Stachiewicz [24] reported a general increase of heat transfer coefficient from fin base to fin tip with a marked dip at about 75% of the fin length. Sparrow and Acharya [25] observed a decrease in the heat transfer coefficient near the fin base and a subsequent increase in the down stream for fins under natural convection for a wide range of conditions. Simultaneous, solution of the problem for convection in the fluid and conduction in the fin has also been tried by Advani and Sukhatme [26] and Garg and Velusamy [27]. In a parallel effort, variation of local heat transfer coefficient as a function of local temperature excess has been considered by Unal [28] and Yeh [29]. Such studies have been taken up for their particular relevance to nucleate boiling. The exact nature of variation of the heat transfer coefficient is not yet established. Nevertheless, efforts have been made for finding out the optimum fin dimensions assuming typical variations of heat transfer coefficients a priori. Razelos and Imre [30] considered the effect of variable heat transfer coefficient and variable thermal conductivity on the optimum dimensions of radial fins with trapezoidal cross section. The authors solved the equation optimality numerically for a length dependent heat transfer coefficient and temperature dependent thermal conductivity. Netrakanti and Huang [31] employed a method of invariant embedding to solve the identical problem. Razelos [15] used the Pontryagin's minimum criteria for finding out minimum mass convective fins. Using variational principle Natarajan and Shenoy [32] determined the optimum profile of conical spines. They considered a power law type dependence of heat transfer coefficient with diameter for different convective conditions. Using hypergeometric function, Yeh [29], [33], [34] and [35] optimized straight fins, spine and fin assemblies for temperature dependent local heat transfer coefficients. In the present paper, a method has been suggested for optimizing longitudinal, radial and pin fin with straight taper (trapezoidal profile) based on a generalized approach. Simplifying assumptions like LAI and insulated fin tip have been eliminated. A variable convective heat transfer coefficient at the fin surface has been considered. Finally, the fin performance has been obtained in an analytical form so that classical techniques can be adopted for optimization.

A methodology for the thermal analysis and optimization of straight taper fins has been discussed in the present paper. The technique adopted for the thermal analysis is a generalized one as it presents a common formulation, set of boundary conditions and method of solution for the three main types of fins, namely longitudinal, spine and annular. The analysis does not make simplifying assumptions like LAI and insulated fin tip. Moreover, it considers a linear variation of heat transfer coefficient along the fin length. Using the Frobenius expanding series, the generalized fin equation has been solved analytically and a single expression has been provided for the temperature distribution in the fins. Based on the thermal analysis the optimum design of the straight taper fins has been obtained from the classical derivative technique. The derived condition of optimality gives an open choice to the designer. For a specified fin volume, optimum fins can be designed to have the maximum rate of heat transfer; alternatively, fins of minimum volume can be designed for a specified heat duty. It has been seen that the ratio of base to tip heat transfer coefficient (ε) has a profound effect on the fin performance. In general, the efficiency of all the three types of fins increases with the increase of ε. However, the rate of increase is higher in the intermediate range of ε. For a given fin volume the rate of heat dissipation increases also from an optimally designed fin for higher values of ε. The merit of the technique lies in the fact that its application is not limited to only linear variation of fin thickness and heat transfer coefficient. The method is applicable for fins with arbitrary variation of fin thickness and heat transfer coefficient along the fin length.