1. Transient Conduction: Spatial Effects and the Role of Analytical Solutions
Chapter 5
Sections 5.4 to 5.8

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

4. The One-Term Approximation :
Plane Wall (cont.)
The One-Term Approximation :
Variation of midplane temperature (x*= 0) with time :
(5.41)
Variation of temperature with location (x*) and time :
(5.40b)
Change in thermal energy storage with time:
(5.43a)
(5.46)
(5.44)
Can the foregoing results be used for a plane wall that is well insulated on one
side and convectively heated or cooled on the other?
Can the foregoing results be used if an isothermal condition is
instantaneously imposed on both surfaces of a plane wall or on one surface of
a wall whose other surface is well insulated?

5. Graphical Representation of the One-Term Approximation
Heisler Charts
Graphical Representation of the One-Term Approximation
The Heisler Charts
Midplane Temperature:

6. Temperature Distribution:
Heisler Charts (cont.)
Temperature Distribution:
Change in Thermal Energy Storage:

7. Radial Systems Long Rods or Spheres Heated or Cooled by Convection.
One-Term Approximations:
Long Rod: Eqs. (5.49) and (5.51)
Sphere: Eqs. (5.50) and (5.52)
Graphical Representations:
Long Rod: Figs. D.4 – D.6
Sphere: Figs. D.7 – D.9

8. The Semi-Infinite Solid
A solid that is initially of uniform temperature Ti and is assumed to extend
to infinity from a surface at which thermal conditions are altered.
Special Cases:
Case 1: Change in Surface Temperature (Ts)
(5.57)
(5.58)

9. Semi-Infinite Solid (cont.)
Case 2: Uniform Heat Flux
(5.59)
Case 3: Convection Heat Transfer
(5.60)

10. Multidimensional Effects
Solutions for multidimensional transient conduction can often be expressed
as a product of related one-dimensional solutions for a plane wall, P(x,t),
an infinite cylinder, C(r,t), and/or a semi-infinite solid, S(x,t). See Equations
(5.64) to (5.66) and Fig
Consider superposition of solutions for two-dimensional conduction in a
short cylinder:

11. Problem: Thermal Energy Storage
Problem 5.66: Charging a thermal energy storage system consisting of
a packed bed of Pyrex spheres.

12. Problem: Thermal Energy Storage

13. Problem: Thermal Energy Storage

14. Problem: Thermal Response Firewall
Problem: 5.82: Use of radiation heat transfer from high intensity lamps
for a prescribed duration (t=30 min) to assess
ability of firewall to meet safety standards corresponding to
maximum allowable temperatures at the heated (front) and
unheated (back) surfaces.

15. Problem: Thermal Response of Firewall

16. Problem: Thermal Response of Firewall