1. Transient Conduction: Spatial Effects and the Role of Analytical Solutions

2. Solution to the Heat Equation for a Plane Wall with
Symmetrical Convection Conditions
If the lumped capacitance approximation can not be made, consideration must
be given to spatial, as well as temporal, variations in temperature during the
transient process.
For a plane wall with symmetrical convection
conditions and constant properties, the heat
equation and initial/boundary conditions are:
(5.26)
(5.27)
(5.28)
(5.29)
Existence of seven independent variables:
(5.30)
How may the functional dependence be simplified?

3. See Appendix for first four roots (eigenvalues ) of Eq. (5.39c)
Plane Wall (cont.)
Non-dimensionalization of Heat Equation and Initial/Boundary Conditions:
The Biot Number:
Exact Solution:
(5.39a)
(5.39b,c)
See Appendix for first four roots (eigenvalues ) of Eq. (5.39c)

4. Approximation Analytical and graphical solution
Plane Wall (cont.)
Approximation Analytical and graphical solution
The One-Term Approximation :

6. Approximation Analytical and graphical solution
Plane Wall (cont.)
Approximation Analytical and graphical solution
Variation of midplane temperature (X= 0) with time :

7. Graphical Representation of the One-Term Approximation
Heisler Charts
Graphical Representation of the One-Term Approximation
The Heisler Charts,
The Maximum heat that a body can gain (Qmax)

9. Heisler Charts (cont.)

10. Heisler Charts (cont.)

11. Heisler Charts (cont.)

12. Heisler Charts (cont.)

13. Heisler Charts (cont.)

14. Heisler Charts (cont.)

24. 7.1 0C and the temperature at centre found at 10 0C
After 2 hours from starting cooling process the surface temperature recorded
7.1 0C and the temperature at centre found at 10 0C
r =0.04 m
8 tomato are exposed this cooling process