Jeff Bode
MEAE 6630 Conduction Heat Transfer
HW#4
P4.5
Obtain an expression for the temperature distribution T(r,,t) in a solid sphere of radius r = b that is initially
at temperature F(r,) and for times t > 0 the boundary surface at r = b is kept insulated.
The governing equation is:
 2T 2 T 1  
T  1 T

 2
(1   2 )  

2
r r r  
   t
r
in 0  r  b
T (r ,  )  R(r ) M (  )
Define a new variable V(r,,t) as
V  r1 / 2T
The governing equation becomes:
 2V 1 V 1 V
1  
V  1 V


 2
(1   2 )


2
2
r
r r 4 r
r  
   t
in 0  r < b,  <  < , t > 0
V
 0 at r = b, t > 0
r
V  r1 / 2 F (r ,  ) for t = 0 in the sphere
Using equations 4-11, we know the following:
(t ) : e t
2
R(r ) : J n 1 / 2 (r )
R(r ) : Yn1 / 2 (r )
M (  ) : Pn (  )
M (  ) : Qn (  )
But only the following 3 elementary solutions are possible:
(t ) : e  t
R(r ) : J n 1 / 2 (r )
2
M (  ) : Pn (  )
This leads to the complete solution of V(r,,t) being


V (r ,  , t )   cnpe
n  0 p 1
2npt
J n 1 / 2 (npr ) Pn ( )
These separated leads to:
d 2 R 2 dR n(n  1)


R0
dr 2 r dr
r2
d 
dM 
(1   2 )
 n(n  1) M  0

d 
d 
and their elementary solutions are:
R(r):
rn and r-n-1
M( Pn() and Qn()