1. Transient Heat Conduction

2. Chapter 4
The Temperature is usually changing with time as well as position.
T = T(x,y,z,t) for transient 3-dimensional HT.
T = T(z,y,z) for steady 3-dimensional HT.
In the previous lectures, we discussed the steady state heat transfer.
In this chapter we discuss the heat conduction as a function of time in one dimension.

3. Objectives
We will start with the analysis of lumped systems in which the temperature of a solid varies with time but remains uniform throughout the solid at any time.
Then, we consider the variation of T with time and position for one dimensional heat conduction in walls, cylinders, and spheres.

8. The rate of heat convection between the body and the environment is
The Total Heat Transfer is
The Maximum Heat transfer is

9. Define the Characteristic length
Define Biot Number
Lumped system analysis is applicable if

16. In this topic, we consider the variation of temperature with time and position in one dimension.
Consider a plane wall of thickness 2L, along cylinder of radius ro, and a sphere of radius ro initially at a uniform temperature Ti as shown below.

18. Temperature profiles

19. Solution of the problem
1. Analytical solution
Be careful of L in Biot number

20. One-term approximate solution

21. Coefficients used in the solution

23. Heat transfer

24. 2. Graphical solution
Temperature at the center

25. Temperature at a point other than the center

26. Heat Transfer

27. Conditions of using the one-term and graphical solutions
The body is initially at a uniform temperature.
T and h of the environment are constant and uniform.
No energy generation in the body.

29. Solution

33. Since Bi=1/45.8=0.022 < 0.1, we can use the lumped system analysis: