Page 248 - Modelling in Transport Phenomena A Conceptual Approach

228 CHAPTER 7. UNSTEADY-STATE MACROSCOPIC BALANCES

where r is the residence time. Using

show that Eq. (6) reduces to
dz
---p.~=k
dt

Note that CA, in Eq. (7) represents the steady-state concentration satisfying the
equation
kc;. -I- 7 = -
CA,
CA,
7
Solve Eq. (8) and obtain
CA = CAS + 1 (11)
[(.I - cA,)-' + (k/p)] exP [p(t -t*>] - (k/p)

where ci and t* represent the concentration and time at the end of the filling
period, respectively.

7.12 For creeping flow, i.e., Re << 1, a relationship between the friction factor
and the Reynolds number is given by Stokes' law, Eq. (4.3-7).
a) Substitute Eq. (4.3-7) into Eq. (7.47) and show that

b) Show that the time required for the sphere to reach 99% of its terminal velocity,
t,, is given by
D'
(pp
t, = - + 0.5~)
3.9 /A
and investigate the cases under which initial acceleration period is negligible.
c) Show that the distance travelled by the particle during unsteady-state fall is
given by

where ut is the terminal velocity of the falling particle and is defined by