2. One Dimensional Non-Homogeneous Conduction Equation P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi Another simple Mathematical modification….. But finds innumerable number of Applications….

3. Homogeneous ODE How to obtain a non-homogeneous ODE for one dimensional Steady State Heat Conduction problems? Blending of Convection or radiation effects into Conduction model. Generation of Thermal Energy in a solid body. GARDNER-MURRAY Ideas.

4. Nature know More More Mathematics !!!

5. Amalgamation of Conduction and Convection Heat Conduciton in Heat Conduciton out Heat Convection In/out

6. Innovative Fin Designs

7. Single Fins :Shapes Longitudinal Radial Pins

8. profile PROFILE AREA cross-section CROSS-SECTION AREA

9. Anatomy of A LONGITUDINAL FIN thickness x xx

10. Involves conduction through a solid medium as well as convection and/or radiation energy transfer Goal is to enhance heat transfer between a solid and a fluid. Possibilities: increase heat transfer coefficient increase surface temperature decrease fluid temperature increase surface area The most common way to enhance heat transfer is by increasing the surface area for convection via an extension from a solid medium: fins Heat Transfer from Extended Surfaces

11. GARDNER-MURRAY ANALYSIS ASSUMPTIONS  Steady state one dimensional conduction Model.  No Heat sources or sinks within the fin.  Thermal conductivity constant and uniform in all directions.  Heat transfer coefficient constant and uniform over fin faces.  Surrounding temperature constant and uniform.  Base temperature constant and uniform over fin base.  Fin width much smaller than fin height.  No heat flow from fin tip.  No bond resistance between fin base and prime surface.  Heat flow off fin proportional to temperature excess.

12. Slender Fins thickness x xx

13. Steady One-dimensional Conduction through Fins QxQx Q x+dx Q conv Conservation of Energy: Where

14. Substituting and dividing by  x: Taking limit  x tends to zero and using the definition of derivative: Substitute Fourier’s Law of Conduction:

15. For a constant cross section area:

16. Define: At the base of the fin:

17. Tip of A Fin