Page 642 - Bird R.B. Transport phenomena

622 Chapter 20 Concentration Distributions with More Than One Independent Variable

t == 0 t> 0
z, Semi-infinite medium z ,
in region z > 0. The
mass transfer surface
Sit) changes with time
uniformly. ^
^ ^ / ^

уУ^ У ^ y ^ ^
C A ~ c A0
atz = 0
atz = 0
Fig. 20.1-4. Time-dependent diffusion across a mass transfer interface
Sit) that is changing with time. The liquid B, in the region above the
plane z = 0, has a velocity distribution v x = +\ax, v y = +\ay, and
v 2 = -az, where a = d In S/dt.

Since Eq. 20.1-62 is solved by the method of combination of variables, the same technique can
be tried here. We postulate

(20.1-66)

Substitution of this trial solution into Eq. 20.1-65 gives

(20.1-67)

If we set the expression within the brackets equal to unity, then we accomplish two things: (i)
we obtain an equation for g that has the same form as Eq. 4.1-9, to which the solution is
known; (ii) we get an equation for 8 as a function of t:

(20.1-68)

This equation may be integrated to give
г sum)
d(SSf = № S4t)dt (20.1-69)
Jo AB
The lower limit on the left side is chosen so as to ensure that c A = 0 initially throughout the
fluid. This choice then leads to

2
S(t) =J43) ] [S(t)/S(t)] dt (20.1-70)
AB
and we get finally for the concentration profiles

£*- = !-erf Z - (20.1-71)
C
A0 2
M f [S(t)/SU)] dt
The interfacial mass flux is then obtained by differentiating Eq. 20.1-71 to get
Л-1/2
= c A (20.1-72)