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

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

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

10. Biot number and Characteristic length

11. Compute the Biot Number

12. Compute the Biot Number

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

14. Fourier Number (Fo)

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

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

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

19. Temperature (position,time)
Local Temperature Analysis
Temperature (position,time)

20. Geometric and thermal symmetry

21. 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 …

22. Sample Heisler charts
Chart

23. 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

24. Complementary error function

27. (McCabe and Smith, 5th ed)

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

29. Heat transferred

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

32. Transient heat conduction in multi-dimensional system

33. 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.

38. Total transient heat transfer

39. Exercises
Answer: about 15 minutes

40. Exercises