Page 502 - Modelling in Transport Phenomena A Conceptual Approach

482 CHAPTER 11. UNSTEADY MICROSCOPIC BUANCES WTH GEN.

Substitution of &. (11.2-1) into &. (11.2-5) gives the governing equation for
temperature as

(11.2-6)

All physical properties are assumed to be independent of temperature in the devel-
opment of Eq. (11.2-6). The initial and boundary conditions associated with Eq.
(11.2-6) are

at t=O T=T, (1 1.2-7)
at z=O T=T1 (11.2-8)
at a=L T=Tl (1 1.2-9)
Introduction of the dimensionless quantities

(11.2-10)

.z
<=- (1 1.2- 11)
L
at
7-z- ( 1 1.2-12)
L2
reduces Eqs. (11.2-6)-(11.2-9) to

(11.2-13)

at T=O 0=1 (1 1.2-14)
at <=O 0=0 (11.2-15)
at <=l O=O (11.2-16)

where
( 1 1.2- 17)
Since Eq. (11.2-13) is not homogeneous, the solution is proposed in the form

e(T,<) = &d<) - et(T,<) (11.218)
in which Boo is the steady-state solution, i.e.,

(11.2-19)

with the following boundary conditions
at <=O 8,=0 (11.2-20)