Page 212 - Modelling in Transport Phenomena A Conceptual Approach

192 CHAPTER 7. UNSTEADY-STATE MACROSCOPIC BALANCES

iii) The steady-state period
The concentration in the tank reaches its steady-state value, CA,, as t + 00. In
this case, the exponential term in Eq. (15) vanishes and the result is

Note that Eq. (16) can also be obtained from Eq. (12) by letting dcA/dt = 0. The
time required for the concentration to reach 99% of its steady-state value, t,, is
{
t, = t* + - 100 [1- (.> 1+kr [l - exp(--kr)]]} (17)
In
7-
1+kr
When kr << 1, Le., a slow first-order reaction, Eq. (17) simplifies to
t, - t* = 4.67 (18)

Example 7.4 A sphere of naphthalene, 2cm in diameter, is suspended in air at
90°C. Estimate the time required for the diameter of the sphere to be reduced to
one-half its initial value it
a) The air is stagnant,
b) The air is flowing past the naphthalene sphere with a velocity of 5 m/ s.
Solution

Physical properties

{ psat - 11.7mmHg
MA = 128
For naphthalene (species A) at 90°C (363K) : p2 = 1145 kg/ m3
-
A
Diffusion coeficient of species A in air (species B):

-
DAB)^ = (0.62 x (E3”” = 8.25 x m2/s

For air at 90 “C (363 K) : v = 21.95 x m2/ s
The Schmidt number is
v
sc = -
DAB
21.95 x
-
- = 2.66
8.25 x 10-6
Assumptions
1. Pseudo-steady-state behavior.