Page 461 - Modelling in Transport Phenomena A Conceptual Approach

10.2. ENERGY TRANSPORT 441

Example 10.1 A copper slab (a = 117x m2/s) of lOcm thick is initially at
a temperature of 20°C. If it is dipped in a boiling water at atmospheric pressure,
estimate the time it takes for the center of the slab to reuch 40°C. Repeat the
problem also for a lOcm thick stainless steel slab (a = 3.91 x m2/s).

Solution
The use of Eq. (10.2-32) gives

TI - Tc
-- O0 (-1)n n2at
Tl -To -2 n=O E (n+a)r
where Tc is the temperature at the center, i.e., the value of T at ( = 0.

Copper:
Substitution of the values into Eq. (1) gives

The value of t can be calculated as 4.5s.
Stainless steel:
Substitution of the values into Eq. (1) gives
1
x
100 - 40 00 ~~(3.91 t
1oo-20=2 n=O

The value of t can be calculated as 136s.
Comment: For an unsteady-state conduction problem, the ratio of the rate of
accumulation to the rate of conduction is equal to the Fourier number. Let sub-
scripts c and s represent wpper and stainless steel, respectively. Then, it is possible
to equate the Fourier numbers, i.e.,
(qC= (4)
(g)
LCh

Since Lch is the same for both cases, Eq. (4) simplifies to

= 4.5 El:) = 135s

which is almost equal to the exact value.