Page 486 - Modelling in Transport Phenomena A Conceptual Approach

466 CHAPTER io. UNSTEADY mmoscoprc BAL. WITHOUT GEN.

Introduction of the dimensionless quantities

(10.3-43)

z
t=- (10.3-44)
L
(10.3-45)

reduces Eqs. (10.3-39)-(10.3-42) to

(1 0.3-46)

at r=O 6=1 (10.3-47)
ae
at (=O -- (10.3-48)
at -O
at (=1 @=O (10.3-49)

The transformation
U
(10.3-50)
converts the spherical geometry into the rectangular geometry. Substitution of En.
(10.3-50) into E!q. (10.3-46) leads to
azc
8%
-- -- (10.3-51)
ar at2
which is identical with Eq. (10.2-69). Therefore, the solution is

= e- A% [A sin(X) + B cos(A<)] (10.3-52)

or,
e=e-AaT [.- sin( At)
6 +B=l 6 (10.3-53)
The boundary condition defined by Eq. (10.3-48) indicates that B = 0. Application
of Eq. (10.3-49) yields

sinA=O + A,,=nr n=l,2,3, ... (10.3-54)

Therefore, the general solution is

(10.3-55)
n=l h