Page 458 - Modelling in Transport Phenomena A Conceptual Approach

438 CHAPTER 10. UNSTEADY MICROSCOPIC BAL. WITHOUT GEN.

Note that z = 0 represents a plane of symmetry across which there is no net flux,
i.e., aT/dz = 0. Therefore, it is also possible to express the initial and boundary
conditions as
at t = 0 T = To for all z
aT
-0
at z=O -- t>O (10.2-9)
dz
at z=L T=T1 t>O
The boundary condition at z = 0 can also be interpreted as an insulated surface. As
a result, Eqs. (10.2-7) and (10.2-9) also represent the following problem statement:
“A slab of thickness L is initially at a uniform temperature of To. One side of the
slab is perfectly insulated while the other surface is kept at a constant temperature
of TI with TI > To for t > 0.”
The physical significance and the order of magnitude of the terms in Eq. (10.2-
7) are given in Table 10.2.

Table 10.2 The physical significance and the order of magnitude of the terms in
Eq. (10.2-7).

Term Physical Significance Order of Magnitude

k Rate of conduction k (TI - To)
dz2 L2
- dT
PCP dt Rate of energy PGQl - To)
accumulation t

Therefore, the ratio of the rate of conduction to the rate of energy accumulation is
given by

Rate of conduction - k (TI - To)/L2 at
-
=-
Rate of energy accumulation pep(Tl - To)/t L2 (10.2-10)
In the literature, the term d/L2 is usually referred to as the Fourier number, Fo .
Introduction of the dimensionless quantities

(10.211)

E=- Z (10.2-12)
L
at
T=- (10.213)
L2
reduces Eqs. (10.2-7) and (10.2-8) to

(10.2-14)