**1. Introduction**

A substantial amount of research endeavors have been carried out to determine the best dimensions of the annular fins so that the rate of heat transfer can be minimized for a given fin volume or the fin volume can be maximized for a specified heat duty. However, the use of fins with optimum profile is restricted due to the associated difficulty of manufacturing.

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A detailed review of literature on optimum design of fins was carried out starting with Gardener [1] where upon using a set of idealizing assumptions, the efficiency of various straight fins and spines have been reported. Duffin [2] gave a method for carrying out the minimum weight design of a fin using a rigorous mathematical method based on Variational calculus and assumed constant thermal conductivity of a fin material and a constant heat transfer coefficient along the fin surface. For purely conductive and convective fins, the criterion for the optimal problem was first proposed by Schmidt [3]. Murray [4] presented equations for the temperature gradient and the effectiveness of annular fins with constant thickness with a symmetrical temperature distribution around the base of the fin. Carrier and Anderson [5] discussed straight fins of constant thickness and annular fins of constant crosssectional area, presenting equations for fin efficiency of each. Duffin and McLain [6] solved the optimization problem of straight-based fins assuming that the minimum weight fin had a linear temperature distribution along its length. Brown [7] reported the optimum dimensions of uniform annular fin by relating fin dimensions to the heat transfer and thermal properties of the fin and heat transfer coefficient between the fin and its surroundings. Smith and Sucec [8] derived analytically the efficiency of circular fins of triangular profile by using Frobenius method.

Maday [9] found the optimum fin thickness variation along the fin. The optimization of fins is generally based on two approaches: one is to minimize the volume or mass for a given amount of heat dissipation and the other is to maximize the heat dissipation for a given volume or mass. Ullmann and Kalman [10] adopted the first way and determined the efficiency and optimum dimensions of annular fins with triangular, exponential, and hyperbolic profiles using numerical techniques. Dhar and Arora [11] described the methods of carrying out the minimum weight design of finned surfaces of specific type by first obtaining the optimum surface profile of a fin required to dissipate a certain amount of heat from the given surface, with no restriction on the fin height and then extended their study for the case when fin height is given. Mikk [12] found the optimum fin thickness variation along the fin. This type of fin shape is complex and involves manufacturing problems. Mikk [13] further worked for convective fins of minimum mass.

In a recent work, Arauzo et al [14] reported a ten-term power series method for predicting the temperature distributions and the heat transfer rates of annular fins of hyperbolic profiles. Assuming fixed fin volume, Arslanturk [15] reported simple correlation equations for opti‐ mum design of annular fins with uniform cross sections to obtain the dimensionless geomet‐ rical parameters of the fin with maximum heat transfer rates. These simple correlation equations can help the thermal design engineers for carrying out the study on optimum design of annular fins of uniform thickness. In their recent work, Kundu and Das [16] reported the performance analysis and optimization of concentric annular fins with a step change in thickness using Lagrange multiplier. Performance of annular fin of rectangular profile having functionally graded materials (FGM) was reported by Aziz and Rahman [17].

In a recent work, Acosta-Iborra and Campo [18] reported that approximate analytic tempera‐ ture profiles and heat transfer rates of good quality are easily obtainable without resorting to the exact analytic temperature distribution and heat transfer rate based on modified Bessel functions. Kang [19] reported the optimum performance and fin length of a rectangular profile annular fin using variations separation method. Theory for FGM for the temperature-depend‐ ent material properties with multiobjective optimization was carried out by Goupee and Vel [20]. In this work, the thermal conductivity varies inversely with the square of the radius. Aziz and Fang [21] presented alternative solutions for different tip conditions of longitudinal fins having rectangular, trapezoidal, and concave exponential profiles and reported relationship between dimensionless heat flux, fin parameter, and dimensionless tip temperature for all the geometries. Aziz and Khani [22] presented an analytical solution for thermal performance of annular fins of rectangular and different convex exponential profiles mounted on a rotating shaft, losing heat by convection to its surroundings. In their work, convection heat transfer coefficient was assumed to be a function of radial coordinate and shaft speed.

In an experimental study, heat transfer rate and efficiency for circular and elliptical annular fins were analyzed for different environmental conditions by Nagarani [23] and high efficiency was reported for elliptical fins as compared to circular ones. In a recent work, Aziz and Fang [24] derived analytical expressions for the temperature distribution, tip heat flow, and Biot number at the tip and reported thermal performance of the annular fin under both cooling and heating conditions.

In the present work, investigation has been reported for variation of thickness and thermal conductivity of material along the radius of the fin keeping the weight of the fin equal to that of a standard rectangular fin. The constraint of constant weight is imposed to compare variation in thickness with rectangular profile as there are several applications where weight is a very crucial parameter to decide the fin selection. Also without imposing the constraint of constant weight, there is always a possibility that selection of fin has been done based on larger surface area having higher weight.

It is a well-known fact that the temperature gradient is higher at the base, hence for the maximum heat transfer higher thermal conductivity should be provided at the base as compared to other part of fin; this can be achieved with functional grading of thermal conductivity of fin material. The original concept of functional grading of material was proposed to take the advantage of excellent thermal performance of ceramics with the toughness of metals. This gave way to the idea of gradient-based varying of microstructure from one material to another material. This transition is usually based on power series. Aerospace industry, chip manufactures, engine and energy component manufacturers are most interested in evolution of new graded materials that can withstand high thermal gradients.
