Page 403 - Modelling in Transport Phenomena A Conceptual Approach

9.4. MASS TRANSFER WITHOUT CONVECTION 383

reduces Eqs. (9.426)-(9.428) to
--(F"-)-A28=0 (9.433)
d8
d
1
E24 4
(9.434)

at (=1 8=1 (9.435)
Problems in spherical coordinates are converted to rectangular coordinates by
the use of the following transformation

(9.436)

From Eq. (9.436), note that

(9.437)

(9.438)

(9.439)

Substitution of Eqs. (9.436) and (9.439) into Eq. (9.433) yields

(9.440)

On the other hand, the boundary conditions, Eqs. (9.434) and (9.435), become

at (=0 u=O (9.441)
at (=1 u=l (9.442)

The solution of Eq. (9.440) is
8 = KI sinh(A() + K2 cosh(A() (9.443)
where Kl and Kz are constants. Application of the boundary conditions, Eqs.
(9.441) and (9.442), gives the solution as
sinh(A5)
21= (9.444)
sinh A
or,
(9.445)