Page 462 - Modelling in Transport Phenomena A Conceptual Approach

442 CHAPTER 10. UNSTEADY MICROSCOPIC BAL. WITHOUT GEN.

10.2.1.1 Macroscopic equation

Integration of the governing equation, Eq. (10.2-7), over the volume of the system
gives

[:A" Jlo" pep dxdydz = JL J" J" k dxdgdz (10.2-33)
-L 0 0 az2
Evaluation of the integrations yields

(10.2-34)

L
\
T
Rate of accumulation of energy Rate of energy entering
from surfaces at z = f L
Note that Eq. (10.2-34) is the macroscopic energy balance by considering the
rectangular slab as a system. The rate of energy entering into the slab, 0, can be
calculated from Eq. (10.2-34) as
( El,=,>
Q=2WH k-

2 WHk (Ti - To) 89
=- (10.235)
L
Substitution of Eq. (10.2-32) into Eq. (10.2-35) gives

(10.2-36)

The amount of heat transferred can be calculated from
t L2
Q=/ 0 Qdt=--Jd Qdr (10.2-37)

Substitution of Eq. (10.2-36) into Eq. (10.2-37) yields

(10.2-38)
-

where QOO is the amount of heat transferred to the slab when it reaches steady-
state, i.e.,
Q- = 2LWHp ep(T1 -To) (10.2-39)
Mass of the slab