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Unsteady State Heat Conduction

Unsteady State Heat Conduction. Scope. One dimensional conduction Transient condition (Unsteady state) Temperature as a function of time and position Analysis Lumped Systems Average temperature analysis

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Unsteady State Heat Conduction

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  1. Unsteady State Heat Conduction

  2. Scope • One dimensional conduction • Transient condition (Unsteady state) • Temperature as a function of time and position • Analysis • Lumped Systems • Average temperature analysis • Transient heat conduction in infinite and semi-infinite solids, Convective boundary conditions

  3. One dimensional transient heat conduction • Thermal diffusivity • T(x,t), 2 BC and 1 IC • Analytical method using Method of Separation of Variables • Numerically by Finite difference or Finite element method • By Charts and Graphs • Lumped analysis: T(t)

  4. Lumped System Analysis • Bodies behave like a “lump” whose interior temp. remains uniform at all time during the heat transfer process • Example • A hot copper ball coming from an oven

  5. Validity of Lumped Analysis • Assumed to cases where temperature gradient within the solid is small • Occurs if the thermal resistance within the solid is very small compared to the external thermal resistance • Solids with very high thermal diffusivity • Applicable if Biot Number is less than 0.1

  6. s is the thermal time constant • A product of the resistance to convection heat transfer and lumped thermal capacitance

  7. Heat transferred Derive Q for body which lump system analysis is valid…

  8. Biot number and Characteristic length

  9. Compute the Biot Number

  10. Compute the Biot Number

  11. Derive the following equation: Fourier Number (dimensionless “time”)

  12. Fourier Number (Fo)

  13. Time of death analysis Ans. 12.2 hr What if the BIOT number is greater than 0.1?

  14. Average Temperature Analysis slab Infinite long cylinder Sphere Sphere (McCabe and Smith, 5th ed)

  15. Heat transferred using the average temperature Slab Infinite long cylinder Sphere

  16. Local Temperature Analysis Temperature (position,time)

  17. Geometric and thermal symmetry

  18. Transient Temperature Charts Local temperature is a function of position and time (Heisler and Grober charts). Check for such charts in ChE Handbook/Unit Ops book… Always check the parameters to be used in the chart …

  19. Sample Heisler charts Chart

  20. Analysis for semi-infinite solids • When solids are heated and cooled such that the temperature changes in the solids are found in the region near one surface only • e.g., thick wall, earth surface

  21. Complementary error function

  22. (McCabe and Smith, 5th ed)

  23. Penetration distance • The distance where the temperature change has reached about 1% of initial change in surface temp.

  24. Heat transferred

  25. Minimum burial depth of water pipes to avoid freezing Answer: about 0.80 m Determine the penetration distance for this condition…. (about 4 m?)

  26. Transient heat conduction in multi-dimensional system

  27. Superposition approach (Product solution) The solution for a multi-dimensional geometry is the product of the solutions of the one dimensional geometries whose intersection is the multi-dimensional body.

  28. Total transient heat transfer

  29. Exercises Answer: about 15 minutes

  30. Exercises

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