Performance & Optimal Design of Slender Fins

1. Performance & Optimal Design of Slender Fins P M V Subbarao Professor Mechanical Engineering Department IIT Delhi Weigh the benefits critically… Before You Invest….

2. Cost – Benefit Analysis of Fins • The benefit of a fin is due to an extra heat transfer. • This benefit is defined as effectiveness of a fin. • An ideal fin will have highest value of effectiveness. • An ideal fin is the one whose temperature is equal to temperature of the surface. • This is possible only if the thermal conductivity of fin material is infinitely high. • The effectiveness of an actual fin material is always lower than an ideal fin. • The relative performance of a given fin is defined as efficiency of a fin. • Provision of fins on a surface requires more material and hence more capital cost. • A judicial decision is necessary to select correct factors of fin design. • Best fin design should have higher benefits with a lower amount of material.

3. Effectiveness (Benefit) of A Fin : Strip Fin • The fin effectiveness,efin , is defined as the ratio of the heat dissipation with fin to the heat dissipation without a fin.

4. Size of Fin Vs Effectiveness : Adiabatic Tip Effectiveness Size of Fin : mb

5. Efficiency of Strip Fin The fin efficiency, h, is defined as the ratio of the actual heat dissipation to the ideal heat dissipation if the entire fin were to operate at the base temperature excess

6. For infinitely long strip fin: For Adiabatic strip fin:

7. Strip Fin: Infinitely Long

8. Strip Fin: Adiabatic tip

9. Longitudinal Fin of Rectangular Profile: Adiabatic tip • Temperature Excess Profile • Heat Dissipated • = Heat Entering Base • Fin Efficiency

10. Size of Fin Vs Fin Efficiency : Adiabatic Tip Fin Efficiency Size of A fin : mb

11. Strip Fin of Least Material • The heat flux is not constant throughout the fin surface area. • It decreases as some function of distance from the fin base. • Two models are possible: • For a constant heat flux, the cross-section of the fin must also decrease as some function of distance from the base. • Schmidt reasoned that the problem reduced to the determination of a fin width function, d(x), that would yield minimum profile area.

12. Longitudinal Fin of Least Material Constant Heat Flux Model Consider With Acsa function of x. Then For a constant heat flux (with k a constant by assumption): and which is a linear temperature excess profile

13. Strip Fin of Least Material : OPTIMUM SHAPES • Least profile area for a given rate of heat transfer can be modified as maximum rate of heat transfer for a given profile area Ap • For a Longitudinal fin of Rectangular Cross Section with L = 1: ( L=1) , let With Hence

14. Optimum Shapes : Strip Fin Find the point where and get Solving iteratively gives bR=1.4192 As the optimum dimensions for a given Ap

15. Performance of Optimum Profiles : Strip Fin (L=1) Heat dissipated Optimum fin width (mb=1.4192)

16. Performance of Optimum Profiles Substitute  in qb And solve for Ap[with optimal value of mb tanh (1.4192) = 0.8894 ]

17. T & h d Tb b Pin Fins : Profile Optimization The objective function is to maximize heat dissipation for a given volume. With So that

18. Optimization of Pin Fin Profile We find the point where: The results is the transcendental equation Where

19. Trial and error method of root finding, gives: Volume of maximum heat dissipating pin fin: Or

21. Comparison of Optimal Fins : Constant C.S. Area For Strip fin: For pin fin: