**2. Mathematical formulation**

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

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

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

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

functionally graded materials (FGM) was reported by Aziz and Rahman [17].

method.

convective fins of minimum mass.

50 Advances in Functionally Graded Materials and Structures

It is assumed that there are no temperature gradients along the thickness of the fins. It is also assumed that the effect of external environment on the surface convection is negligible and hence a constant convective heat transfer coefficient has been adopted for the fin material. The second-order differential equation for the heat transfer through the fins has to be developed new graded materials that can withstand high thermal gradients.

Using above equation the following equation can be arrived at

having higher weight.

to find the temperature profile. This correlation equation (second-order differential equation) has been solved using computational algorithm in MATLAB software and the computed information has been analyzed. For calculating the heat balance, the details for a control volume of length '*dr*' of a fin is shown in Figure 1. Applying the law of conservation of energy or thermal energy balance: hence a constant convective heat transfer coefficient as been adopted for the fin material. The second order differential equation for the heat transfer through the fins has to be developed to find the temperature profile. This correlation equation (second order differential equation) has been solved using computational algorithm in MATLAB software and the computed information has been analyzed. For calculating the heat balance, the details for a control volume of length '*dr'*' of a fin is shown in **Fig. 1**. Applying the law of conservation of energy or thermal energy balance:

It is assumed that there are no temperature gradients along the thickness of the fins. It is also assumed that the effect of external environment on the surface convection is negligible and

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 on the basis of larger surface area

It is 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

$$Q\_r = Q\_{r+dr} + Q\_{annv} \tag{1}$$

**Figure 1. Fig. 1:** Elemental strip in fin of length ' Elemental strip in fin of length '*drdr*' and a comparison of constant weight uniform and exponential profile. ' and a comparison of constant weight uniform and exponential profile Normalization of the problem is carried out using the following equations Using the above equation, the following equation can be arrived at

$$\frac{d}{dr}\left(2\pi.r.k.\delta.\frac{d\theta}{dr}\right)dr = 4\pi.r.h.\theta.dr.\left(1 + \left(d\delta \wedge 2dr\right)^2\right)^{0.5} \tag{2}$$

$$\overline{\delta} = \frac{\delta}{\delta\_0}, \phi = \frac{\theta}{\theta\_0}, \chi = \frac{R - R\_0}{R\_1 - R\_0}, R\_{f'} = \frac{R\_1}{R\_0}$$

and the fin parameter, *mf* =(2*h* / *aδ*0)0.5

upon introducing the normalized variables, the governing equation becomes

$$\frac{d^2\phi}{dx^2} + A\_1 \frac{d\phi}{dx} + A\_2 \phi = 0\tag{3}$$

where

Performance of Functionally Graded Exponential Annular Fins of Constant Weight http://dx.doi.org/10.5772/63100 53

$$A\_1 = \left[\frac{1}{\overline{\mathcal{S}}} . \frac{d\overline{\mathcal{S}}}{d\mathfrak{x}} + \frac{1}{k} . \frac{dk}{d\mathfrak{x}} + \frac{L}{R\_0 + L\mathfrak{x}}\right]$$

Here in*A***1**, the first term represents the variation in thickness and the second term represents the variation of thermal conductivity along the length of the fin. It means for rectangular profile with isotropic materials only the third term of *A*1 exists and that makes equation (3) more generalized for handling complex geometry with FGM.

$$A\_2 = -\frac{2hL^2}{k\delta\_0\overline{\delta}}\sqrt{1 + \frac{\delta\_0^2}{4L^2}(\frac{d\overline{\delta}}{dx})^2}$$

Assuming the geometry variation and thermal conductivity variation of FGM material as follows

$$\delta = \delta\_0 \left( e^{-\mathbf{n} \cdot \mathbf{x}^{-n}} \right) \text{ or } \delta = \left( e^{-\mathbf{n} \cdot \mathbf{x}^{-n}} \right)$$

and *k* =*ar <sup>b</sup>*

to find the temperature profile. This correlation equation (second-order differential equation) has been solved using computational algorithm in MATLAB software and the computed information has been analyzed. For calculating the heat balance, the details for a control volume of length '*dr*' of a fin is shown in Figure 1. Applying the law of conservation of energy

Qr = Qr+dr + Qconv. (1)

**Figure 1. Fig. 1:** Elemental strip in fin of length ' Elemental strip in fin of length '*drdr*' and a comparison of constant weight uniform and exponential profile. ' and a comparison of constant weight uniform and exponential profile

( ( ) )

0 1

0 0 10 0 ,, , *<sup>f</sup>*

f


è ø (2)

 d

*R R <sup>R</sup> x R RR R* 0.5 <sup>2</sup>

(3)

Normalization of the problem is carried out using the following equations

2 . . . . 4 . . . . . 1 /2 *d d r k dr r h dr d dr*

 q

 q

 pq

 æ ö ç ÷ = +

Using the above equation, the following equation can be arrived at

q

d

2

 f d

upon introducing the normalized variables, the governing equation becomes

*dx dx* f

<sup>2</sup> 1 2 <sup>0</sup> *d d A A*

 f+ +=

d

*dr dr*

and the fin parameter, *mf* =(2*h* / *aδ*0)0.5

where

pd

It is assumed that there are no temperature gradients along the thickness of the fins. It is also assumed that the effect of external environment on the surface convection is negligible and hence a constant convective heat transfer coefficient as been adopted for the fin material. The second order differential equation for the heat transfer through the fins has to be developed to find the temperature profile. This correlation equation (second order differential equation) has been solved using computational algorithm in MATLAB software and the computed information has been analyzed. For calculating the heat balance, the details for a control volume of length '*dr'*' of a fin is shown in **Fig. 1**. Applying the law of conservation of energy or thermal energy

new graded materials that can withstand high thermal gradients.

Using above equation the following equation can be arrived at

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 on the basis of larger surface area

It is 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

*QQ Q r r dr conv* <sup>+</sup> = + (1)

or thermal energy balance:

balance:

**Mathematical Formulation:**

52 Advances in Functionally Graded Materials and Structures

having higher weight.

Using the above relation, equation (3) becomes

$$\begin{aligned} \frac{d^2 \phi}{dx^2} + [-m.n.x^{m-1} + \frac{L}{(Lx + R\_0)}(1+b)]\frac{d\phi}{dx} \\ -\frac{L^2 m\_f^2}{\bar{\sigma}\_\* \left(Lx + R\_0\right)^b} \left[\sqrt{\frac{\delta\_0^2}{4L^2} \left(m.n.e^{-\mathbf{x}^m} \cdot \mathbf{x}^{m-1}\right)^2} + 1\right] \phi = 0 \end{aligned} \tag{4}$$

Equation (4) is solved using the following boundary conditions:

\*\*i.\*\* 
$$\phi = 1 \quad \text{at } \mathbf{x} = 0$$
 
$$\begin{array}{ll} \mathbf{ii.} & \frac{d\phi}{d\mathbf{x}} = 0 \text{ } at \text{ } \mathbf{x} = 1 \text{ } i. \text{e.e.} \text{ tip is insulated} \end{array}$$

Similarly, efficiency and effectiveness of the fin is obtained from the general equation as follows:

$$
\eta = \frac{-\left(\dot{\phi}\right)\_{\text{x}=0}}{\frac{L^2 m^2}{R\_0^{b+1}} \frac{1}{\phi} \left[ \left(Lx + R\_0\right) \sqrt{\left(\frac{\delta\_0 mm}{2L}\right)^2 e^{-2\alpha x^m} x^{2m-2}} + 1 \right]} \tag{5}
$$

$$
\varepsilon = \frac{-aR\_0^b \left(\dot{\phi}\right)\_{\text{x}=0}}{hL} \tag{6}
$$

If the temperature distribution or if the first derivative of the temperature (*ϕ*' )*x*=0 at the fin base is known, then equations (5) and (6) enable the fin efficiency and effectiveness to be calculated. The first derivative of temperature at the fin base for different shapes of fin has been calculated using the solution of general second-order differential equation (4).

The effect on fin performance has been carried out for the following cases:

**Case I:** By varying the geometry parameters *n* and *m* and keeping all other parameters constant.

**Case II:** By varying coefficient of thermal conductivity, *a* is observed with geometry parame‐ ters, keeping grading parameter *b* constant.

**Case III:** The geometry is kept constant with variation in grading parameters *a* and *b*.
