Page 471 - Modelling in Transport Phenomena A Conceptual Approach

10.2. ENERGY TRANSPORT 45 1

The initial and the boundary conditions associated with Eq. (10.2-92) are
at t=O T=To (10.2-93)
a-
-0
at r=O -- (10.2-94)
dr
dT
at r=R k-=((h)(T,-T) (10.2-95)
dT
Introduction of the dimensionless quantities
Tw -T
8= (10.2-96)
Tw - To
at
r=- ( 10.2-97)
R2
(10.2-98)

(10.2-99)
reduces Eqs. (10.2-92)-(10.2-95) to

(10.2-100)

at r=O 0=1 (10.2-101)
(10.2-102)

(10.2-103)

10.2.2.1 Solution for 0.1 <BiH < 40

Note that the transformation
U (10.2-104)
O=T
converts the spherical geometry into the rectangular geometry. Substitution of Eq.
(10.2-104) into Eq. (10.2-100) leads to

au - a2u (10.2-105)
_- -
67 at2

which is identical with Eq. (10.2-69). Therefore, the solution is
u = e- "7 [A sin(Xc) + B cos(~t)l ( 10.2- 106)