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One Dimensional Non-Homogeneous Conduction Equation

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…. Homogeneous ODE.

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One Dimensional Non-Homogeneous Conduction Equation

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  1. 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….

  2. 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. Nature know More More Mathematics !!!

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

  5. Innovative Fin Designs

  6. Single Fins :Shapes Longitudinal Pins Radial

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

  8. Dx thickness x Anatomy of A LONGITUDINAL FIN

  9. Heat Transfer from Extended Surfaces • 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

  10. 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.

  11. Dx thickness x Slender Fins

  12. Qconv Qx Qx+dx Steady One-dimensional Conduction through Fins Conservation of Energy: Where

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

  14. For a constant cross section area:

  15. Define: At the base of the fin:

  16. Tip of A Fin

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