One Dimensional Non-Homogeneous Conduction Equation

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

2. Further Mathematical Analysis : 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.

3. Continuous Convection or Radiation heat transfer to/from fin surface Conduction heat transfer to /from body. Blending of Convection or Radiation in Conduction Equation Extended surface Body to gain or loose heat Conduction through the fin is strengthened or weakened by continuous convection or radiation from/to fin surface.

4. An optimum body size is essential for the ability to regulate body temperature by blood-borne heat exchange. For animals in air, this optimum size is a little over 5 kg. For animals living in water, the optimum size is much larger, on the order of 100 kg or so. Mathematical Ideas are More Natural This may explain why large reptiles today are largely aquatic and terrestrial reptiles are smaller.

5. Mathematical Ideas are More Natural • Reptiles like high steady body temperatures just as mammals and birds. • They have sophisticated ways to manage flows of heat between their bodies and the environment. • One common way they do this is to use blood flow within the body to facilitate heat uptake and retard heat loss. • Blood flow is not effective as a medium of heat transfer everywhere in the body. • Body shape also enters into the process. • It also helps expalin the odd appendages like crests and sails that decorated extinct reptiles like Stegosaurus or mammal-like reptiles like Dimetrodon. • Theoretical Biologists did Calculations to show these structures could act as very effective heat exchange fins. • These fins are allowing animals with crests to heat their bodies up to high temperatures much faster than animals without them.

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

7. Basic Geometric Features of Fins profile PROFILE AREA cross-section CROSS-SECTION AREA

8. Innovative Fin Designs

9. Single Fins :Shapes Longitudinal or strip Radial Pin Fins

10. Anatomy of A STRIP FIN Dx thickness x Direction of Heat Convection Flow Direction Direction of Heat Conduction

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

12. Dx thickness x Slender Fins

13. Steady One-dimensional Conduction through Fins qconv or qradiation qx qx+dx Conservation of Energy for CV: OR

14. Where OR

15. Substituting qconv or qrad and dividing by Dx: Taking limit Dx tends to zero and using the definition of derivative: Substitute Fourier’s Law of Conduction: