2. Optimal Fin Shapes & Profiles P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi Geometrical Optimization is the Basic Goal of Optimal Design ….

3. CYLINDRICAL SPINE (Pin Fins) b d T  & h  TbTb Pin fin with adiabatic tip and corrected height:

4. Pin Fins : Profile Optimization Sonn and Bar-Cohen (1981) developed an optimization method based on minimization of the spine volume. The objective function is to maximize heat dissipation for a given volume. With So that b d T  & h  TbTb

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

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

8. For Strip fin: For pin fin:

9. LONGITUDINAL FIN OF TRIANGULAR PROFILE The differential equation for temperature excess : L x=0 b x=b x qbqb

10. The differential equation for temperature excess is a form of Bessel’s equation: The particular solution foris: The fin heat dissipation is: The fin efficiency is:

11. Optimum Shapes : Triangular Fin L=1 With This makes

12. Optimum Shapes Iterative solving yields  T =2.6188 and

13. Comparison of Longitudinal Fins Rectangular Profile: Triangular Profile: For the same material, surrounding conditions and which is basically the user’s design requirement. Triangular profile requires only about 68.8% as much metal as rectangular profile.

14. Capacity Enhancement of Fins In both (stip and triangular) fins, profile area varies as the cube of To double the heat flow, you use two fins or make one fin eight times as large. There is a virtue in using more number of small fins. In pin fin, profile volume varies as To double the heat flow, you use two fins or make one fin 3.17 times as large.

15. Pentium III Pentium II Pentium IV

16. Design and Optimization of Fin Arrays P M V Subbarao Mechanical Engineering Department IIT Delhi Millions of Ants are more Powerful than a Single Cobra……

17. Geometry of Fin Array tftf S b

18. Determination of Heat Transfer Coefficient S

19. Optimum spacing b

20. Optimum Natural Convection Array  From Elenbaas (1942):  For an array of optimally spaced fins:

21. Towards an Optimum Array of Optimum Fins Heat flow from each optimum fin: With the h for Optimum Spacing: With the Interfin Spacing

22. Industrial Practice Define the thermal resistance of the heat sink is given by

23. Selection Curves

24. Forced Convection Heat Sinks  Analytical modeling  Maximization of heat dissipation  Least-material optimization  Design for manufacture

25. Design Calculations for Fin Arrays – Thermal Resistance In order to select the appropriate heat sink, the thermal designer must first determine the maximum allowable heat sink thermal resistance. To do this it is necessary to know the maximum allowable module case temperature, T case, the module power dissipation, P mod, and the thermal resistance at the module-to- heat sink interface, R int. The maximum allowable temperature at the heat sink attachment surface, T base, is given by

26. The maximum allowable heat sink resistance, R max, is then given by The thermal resistance of the heat sink is given by parameters the gap, b, between the fins may be determined from

27. Constant air velocity

28. Constant volumetric flow rate

29. Heat Sink Pressure Drop To determine the air flow rate it is necessary to estimate the heat sink pressure drop as a function of flow rate and match it to a curve of fan pressure drop versus flow rate. A method to do this, using equations presented here. As in the previous article, the heat sink geometry and nomenclature used is that shown Figure 1.

30. Pressure Drop Curves

31. Effect of number of fins and fin height

32. Thermal Resistance

33. Closure a fan with a different fan curve is employed, the flow rates will change and the optimum heat sink design point may change as well. The important point is that to determine how a heat sink will perform in a given application both its heat transfer and pressure drop characteristics must be considered in concert with the pressure-flow characteristics of the fan that will be used. It should also be noted that an underlying assumption is that all the flow delivered by the fan is forced to go through the channels formed between the heat sink fins. Unfortunately this is often not the case and much of the air flow delivered by the fan will take the flow path of least resistance bypassing the heat sink. Under such circumstances the amount of flow bypass must be estimated in order to determine the heat sink performance.