Transient Heat Conduction Through Planer Wall
Kevin R. Guenther
October 7, 1999
Numerical Analysis for Engineering
PROPOSAL
Numerical solution for transient heat conduction through a planer wall with heat
convection applied to both sides of the wall and no internal heat generation.
BACKGROUND
T(x,0)=Ti
T,h
T,h
L
L
X
Figure 1: Planer Wall
Figure 1 represents the system which will be analyzed. It is a uniform, semi-infinite wall
with a total thickness of 2L. The fact that the ratio of the width of the wall to the height
(y/x) is small allows for the assumption that any conduction will take place in only the x
direction. This allows the problem to be analyzed as a one dimensional heat transfer
problem.
The wall is at an initial, uniform temperature of Ti. At t=0+ both faces of the wall are
subjected to a uniform environmental temperature, T, and constant, convective heat
transfer coefficient. Over time, the wall will heat up maintaining a symmetrical
temperature profile about the x axis until the wall reaches a uniform temperature of T.
For this problem (also assuming that the material is uniform, hence constant thermal
conductivity), the heat transfer equation reduces to:
 2 T T

x 2 t
(1)
The derivation of Equation 1 will be shown in the final paper.
Given the initial condition of uniform temperature:
T(x,0) = Ti
(2)
and the following boundary conditions:
T
x
k
T
x
0
(3)
x 0
 hT ( L, t )  T 
(4)
xL
an exact solution has been obtained for Equation 1:

 *   C n exp(  n2 Fo) cos( n x * )
(5)
n 1
The terms in the Equation 1 have been non-dimensionalized. They, and how they were
derived will be discussed in the final paper.
PROJECT
Numerically solve equation 1 using various numerical solution techniques at a specific
time and location. Compare the numerical solution with the closed-form solution to
determine the accuracy of the solvers. Compare the solution time of the solvers. Based
on accuracy and solution speed, determine the most practical solver to be used with a
similar problem.
REFERENCES
Burden, R., and Faires, J.D., Numerical Analysis, Brooks/Cole Publishing Company,
Pacific Grove, CA, 1997
DeWitt, D., and Incropera, F., Fundamentals of Heat and Mass Transfer, John Wiley & Sons, New
York, NY, 1996