**3. Results and discussion**

The dimensionless temperature **ϕ** is a function of normalized variables, *n* and *m* due to chosen fin shape, grading parameter, *a* and *b* for thermal conductivity variation**,** *mf* , *R<sup>f</sup>* and *x* due to the fin geometry. Considering *x* as the only independent variable and keeping the other variables constant, equations (4) and (5) have been solved for the various values of *n* ranging between 0 and 0.9, *m* ranging between 0.5 and 1.5, grading parameter *b* ranging between −2 and 2, coefficient of thermal conductivity *a* ranging between 5 and 25, and for all the cases the values of *R<sup>f</sup>* are kept constant, i.e., 3. The numerical values of system properties are *h* = 25 W/ m<sup>2</sup> , **δ**0 = 0.01 m, and *R*0 = 0.1 m.

The first derivative of temperature at the fin base, i.e.**,ϕ ′***x***=0** helps to calculate the fin efficiency and effectiveness and is calculated by solving second-order differential equation in MATLAB. The values of **ϕ′***x***=0** can be obtained for different values of *Rf* , *n*, *m*, and *b*. Temperature variation along the radius is depicted in Figure 2 for the rectangular annular fin, i.e., *n* = 0 for different values of *mf* at grading parameter *b* = −1, it has been observed that the results obtained from the numerical coding are having good agreement with the Aziz and Rahman's study [18].

Figure 3 clearly shows the variation of efficiency and effectiveness with the parameter *m* for different values of geometry parameter *n* keeping the other variable constant (i.e., *b* = −1 and*mf* = 20). It is evident from Figure 3 that efficiency first decreases and then increases with increase in parameter *m* and a minima has been observed near the value of *m* = 0.6. Effectiveness increases monotonously with the parameter *m* and tends to effectiveness of rectangular fins, i.e., *n* = 0, this is due to the constraint of constant weight. Both efficiency and effectiveness decrease sharply with geometry parameter *n* as shown in Figure 4, and similar trends have also been obtained for different values of geometry parameter *m* as shown in Figure 4.

Variation of efficiency and effectiveness with coefficient of thermal conductivity *a* with different values of *n* keeping other variables constant (i.e., *m* and *b*) is depicted in Figure 5. The efficiency and effectiveness increase sharply with increase in *a*, which is due to the fact that as *a* increases*mf* decreases.

Performance of Functionally Graded Exponential Annular Fins of Constant Weight http://dx.doi.org/10.5772/63100 55

due to the constraint of constant weight. Both efficiency and effectiveness decreases sharply with

geometry parameter *n* as shown in Fig. 4 and similar trends have also been obtained for different **Figure 2.** Excess temperature over radial co-ordinate for rectangular annular fin.

values of geometry parameter *m* as shown in Fig. 4.

If the temperature distribution or if the first derivative of the temperature (*ϕ*'

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

using the solution of general second-order differential equation (4).

ters, keeping grading parameter *b* constant.

54 Advances in Functionally Graded Materials and Structures

**3. Results and discussion**

, **δ**0 = 0.01 m, and *R*0 = 0.1 m.

The values of **ϕ′***x***=0** can be obtained for different values of *Rf*

decreases.

constant.

values of *R<sup>f</sup>*

that as *a* increases*mf*

m<sup>2</sup>

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

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

**Case II:** By varying coefficient of thermal conductivity, *a* is observed with geometry parame‐

The dimensionless temperature **ϕ** is a function of normalized variables, *n* and *m* due to chosen

the fin geometry. Considering *x* as the only independent variable and keeping the other variables constant, equations (4) and (5) have been solved for the various values of *n* ranging between 0 and 0.9, *m* ranging between 0.5 and 1.5, grading parameter *b* ranging between −2 and 2, coefficient of thermal conductivity *a* ranging between 5 and 25, and for all the cases the

The first derivative of temperature at the fin base, i.e.**,ϕ ′***x***=0** helps to calculate the fin efficiency and effectiveness and is calculated by solving second-order differential equation in MATLAB.

along the radius is depicted in Figure 2 for the rectangular annular fin, i.e., *n* = 0 for different values of *mf* at grading parameter *b* = −1, it has been observed that the results obtained from the numerical coding are having good agreement with the Aziz and Rahman's study [18].

Figure 3 clearly shows the variation of efficiency and effectiveness with the parameter *m* for different values of geometry parameter *n* keeping the other variable constant (i.e., *b* = −1 and*mf* = 20). It is evident from Figure 3 that efficiency first decreases and then increases with increase in parameter *m* and a minima has been observed near the value of *m* = 0.6. Effectiveness increases monotonously with the parameter *m* and tends to effectiveness of rectangular fins, i.e., *n* = 0, this is due to the constraint of constant weight. Both efficiency and effectiveness decrease sharply with geometry parameter *n* as shown in Figure 4, and similar trends have also been obtained for different values of geometry parameter *m* as shown in Figure 4.

Variation of efficiency and effectiveness with coefficient of thermal conductivity *a* with different values of *n* keeping other variables constant (i.e., *m* and *b*) is depicted in Figure 5. The efficiency and effectiveness increase sharply with increase in *a*, which is due to the fact

are kept constant, i.e., 3. The numerical values of system properties are *h* = 25 W/

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

fin shape, grading parameter, *a* and *b* for thermal conductivity variation**,** *mf*

)*x*=0 at the fin base

, *R<sup>f</sup>*

, *n*, *m*, and *b*. Temperature variation

and *x* due to

**Fig. 3:** Variation of efficiency and effectiveness with parameter *m* for different values of *n* (case I) **Figure 3.** Variation of efficiency and effectiveness with parameter *m* for different values of *n* (case I).

Variation of efficiency and effectiveness with coefficient of thermal conductivity *a* with different

**Fig.4:** Variation of efficiency and effectiveness with *n* for different values of *m* at *mf*=20 and *b*=-1 (Case I) **Figure 4.** Variation of efficiency and effectiveness with *n* for different values of *m* at *mf* = 20 and *b* = −1 (Case I).

**Fig. 5:** Variation of efficiency and effectiveness with *a* for different values of *n* (Case II)

Similar trends has also been observed for variation of efficiency and effectiveness with coefficient of thermal conductivity *a* with different value of *m* keeping other variable constant (i.e. *n* and *b* ) is depicted in Fig. 6 but the effect of variation in parameter m is not significance.

**Fig. 6:** Variation of efficiency and effectiveness with *a* for different values of *m* (Case II)

**Fig.4:** Variation of efficiency and effectiveness with *n* for different values of *m* at *mf*=20 and *b*=-1 (Case I)

**Fig. 5:** Variation of efficiency and effectiveness with *a* for different values of *n* (Case II) **Figure 5.** Variation of efficiency and effectiveness with *a* for different values of *n* (Case II).

Similar trends has also been observed for variation of efficiency and effectiveness with coefficient of thermal conductivity *a* with different value of *m* keeping other variable constant (i.e. *n* and *b* ) is depicted in Fig. 6 but the effect of variation in parameter m is not significance. Similar trends have also been observed for variation of efficiency and effectiveness with coefficient of thermal conductivity *a* with different values of *m*, keeping other variables constant (i.e., *n* and *b*), is depicted in Figure 6 but the effect of variation in parameter *m* is not significant. **Fig. 5:** Variation of efficiency and effectiveness with *a* for different values of *n* (Case II) Similar trends has also been observed for variation of efficiency and effectiveness with

coefficient of thermal conductivity *a* with different value of *m* keeping other variable constant (i.e. *n* and *b* ) is depicted in Fig. 6 but the effect of variation in parameter m is not significance.

**Figure 6.** Variation of efficiency and effectiveness with **Fig. 6:** Variation of efficiency and effectiveness with *a* for different values of *a* for different values of *m* (Case II). *m* (Case II)

Variation of excess temperature *θ* along the radial co-ordinate has also been analyzed in Figure 7 and Figure 8 with coefficient of thermal conductivity *a* and grading parameter *b* keeping the geometry of annular fin constant (i.e., *n =* 0.5 and *m =* 0.5). The performance of fin decreases sharply as the grading parameter shifted from negative value to positive value and it is highest for when it has been kept −2. This is because of higher thermal conductivity that is available at the base (as the temperature of base is higher) to dissipate more heat. The variation of temperature along the radius with *a* falls exponentially, and performance of fin increases with increase in *a* value for a fixed geometry of the exponential profile fin.

Performance of Functionally Graded Exponential Annular Fins of Constant Weight http://dx.doi.org/10.5772/63100 57

**Figure 7.** Excess temperature over the radial surface for different grading parameter *b* (Case III).

**Figure 8.** Excess temperature over the radial surface for different grading parameters *a* at *n* = 0.5, *m* = 0.5, and *b* = −1 (Case III).

#### **4. Conclusion**

**Fig.4:** Variation of efficiency and effectiveness with *n* for different values of *m* at *mf*=20 and *b*=-1 (Case I)

**Fig.4:** Variation of efficiency and effectiveness with *n* for different values of *m* at *mf*=20 and *b*=-1 (Case I)

**Fig. 5:** Variation of efficiency and effectiveness with *a* for different values of *n* (Case II)

**Figure 5.** Variation of efficiency and effectiveness with *a* for different values of *n* (Case II).

56 Advances in Functionally Graded Materials and Structures

significant.

Similar trends has also been observed for variation of efficiency and effectiveness with coefficient of thermal conductivity *a* with different value of *m* keeping other variable constant (i.e. *n* and *b* ) is depicted in Fig. 6 but the effect of variation in parameter m is not significance.

**Fig. 5:** Variation of efficiency and effectiveness with *a* for different values of *n* (Case II)

Similar trends has also been observed for variation of efficiency and effectiveness with coefficient of thermal conductivity *a* with different value of *m* keeping other variable constant (i.e. *n* and *b* ) is depicted in Fig. 6 but the effect of variation in parameter m is not significance.

Similar trends have also been observed for variation of efficiency and effectiveness with coefficient of thermal conductivity *a* with different values of *m*, keeping other variables constant (i.e., *n* and *b*), is depicted in Figure 6 but the effect of variation in parameter *m* is not

**Fig. 6:** Variation of efficiency and effectiveness with *a* for different values of *m* (Case II)

Variation of excess temperature *θ* along the radial co-ordinate has also been analyzed in Figure 7 and Figure 8 with coefficient of thermal conductivity *a* and grading parameter *b* keeping the geometry of annular fin constant (i.e., *n =* 0.5 and *m =* 0.5). The performance of fin decreases sharply as the grading parameter shifted from negative value to positive value and it is highest for when it has been kept −2. This is because of higher thermal conductivity that is available at the base (as the temperature of base is higher) to dissipate more heat. The variation of temperature along the radius with *a* falls exponentially, and performance of fin increases with

**Figure 6.** Variation of efficiency and effectiveness with **Fig. 6:** Variation of efficiency and effectiveness with *a* for different values of *a* for different values of *m* (Case II). *m* (Case II)

increase in *a* value for a fixed geometry of the exponential profile fin.

The performance of exponential annular fins made of FGM is reported. During the analysis, the weight of the different geometry fin is kept constant to the rectangular fin. The study is carried out for different values of geometry parameters *n* and *m* and grading parameters *a* and *b*. It is observed that both effectiveness and efficiency decrease as*mf* increases for any geometry and grading. Also the variation in efficiency and effectiveness with geometry parameter *m* is almost constant but decreases sharply with *n*, and the effect of variation of *n* on efficiency and effectiveness is direct in nature. The effect of grading parameter *b* on fin performance is also investigated, and it is observed that both efficiency and effectiveness decrease as *b* value shifted from negative to positive and it is highest for *b* = −2. Therefore, it can be concluded that highest thermal conductivity is required at the base of the fin to dissipate more amount of heat (due to the highest temperature difference available). Also grading of thermal conductivity should provide inversely square of the length from the base of the fin to obtain the highest performance of the fin compared to the isotropic material (i.e., *b* = 0). Performance of the fin increases with increase of coefficient of thermal conductivity *a* due to higher thermal conductivity material of the fin.
