Homework #2
Joseph Ortoleva
Feb 3, 2000
Conduction Heat Transfer
#2-8) Given:
x0
y0
z0
T = To
T=0
T=0
T=0
at entire region
At
x=0
At
y=0
At
z=0
when t = 0
when t  0
when t  0
when t  0
The solution for this multidimensional problem can be written down as the product of the
solutions of one-dimensional problems.
T(x,y,z,t) = T1(x,t) T2(y,t) T3(z,t)
The initial condition is expressible as a product:
To = 1  1  To
The solution of each one-dimensional problem was examined I the textbook in Ex 2-3
p57, thus the solutions are:
 x 
T 1( x, t )  erf 

 4 t 
 y 
T 2( y, t )  erf 

 4 t 
 z 
T 3( z , t )  To erf 

 4 t 
are from Text p57 – 59
Therefore, the solution to the 3-D problem becomes:
T(x,y,z,t) = T1(x,t)T2(y,t)T3(z,t)
 x   y   z 
T ( x, y, z, t )  (To)erf 
erf 
erf 

 4 t    t    t 