Page 452 - Modelling in Transport Phenomena A Conceptual Approach

432 CHAPTER 10. UNSTEADY MICROSCOPIC BAL. WITHOUT GEN.

at E=O 0=1 (10.1-18)

at <=l 8=0 (10.1-19)
Since the boundary condition at [ = 0 is not homogeneous, the method of sepaxa-
tion of variables cannot be directly applied to obtain the solution. To circumvent
this problem, propose a solution in the form
E)
0(r,E) = QOo(5) - @t(r, (10.1-20)
in which OM(() is the steady-state solution, i.e.,

(10.1-21)

with the following boundary conditions
at t=O Bm=l (10.1-22)
at [=l 0,=0 (10.1-23)

The steady-state solution is
8,=1-E (10.1-24)
which is identical with Eq. (8.1-12). On the other hand, the transient contribution
Bt(r, E) satisfies Eq. (10.1-16), Le.,

(10.1-25)

Erom Eqs. (10.1-20) and (10.1-24), et = 1 - 5 - 8. Therefore, the initial and the
boundary conditions associated with Eq. (10.1-25) become

at T=O Ot=1-< (10.1-26)
at [=O Ot=0 (10.1-27)
at 5=1 0,=0 (10.1-28)
Note that the boundary conditions at = 0 and [ = 1 are now homogeneous
and this parabolic partial differential equation can be solved by the method of
separation of variables as described in Section B.6.1 in Appendix B.
The separation of variables method assumes that the solution can be represented
as a product of two functions of the form

et (7,5) = F(4 G(6) (10.1-29)
Substitution of Eq. (10.1-29) into Eq. (10.1-25) and rearrangement gives

(10.1-30)