Page 634 - Bird R.B. Transport phenomena

614 Chapter 20 Concentration Distributions with More Than One Independent Variable

SOLUTION For this system the equation of continuity for the mixture, given in Eq. 19.1-12, becomes
dv*
-^ = 0 (20.1-1)

in which v* is the z-component of the molar average velocity. Integration with respect to z gives
v* = i;* (0 (20.1-2)
0
Here and elsewhere in this problem, the subscript "0" indicates a quantity evaluated at z = 0.
According to Eq. (M) of Table 17.8-1, this velocity can be written in terms of the molar fluxes
of A and В as
N B
vZ = (20.1-3)

However, N is zero because of the insolubility of species В in liquid A. Then use of Eq. (D)
Bz0
of Table 17.8-2 gives finally
V AB dx fi
v- = - 1 - x dz (20.1-4)
A0
in which x is the interfacial gas-phase concentration, evaluated here on the assumption of
A0
interfacial equilibrium. For an ideal gas mixture this is just the vapor pressure of pure A di-
vided by the total pressure.
The equation of continuity of Eq. 19.1-17 then becomes

z
d x
dX A (20.1-5)
dt - x A \ dZ
This is to be solved with the initial and boundary conditions:
I.C.: at t = 0, x = 0 (20.1-6)
A
B.C.I: at z = 0, x = x A0 (20.1-7)
A
B.C. 2: at z = x A = 0 (20.1-8)
We can try the same kind of combination of variables used in Example 4.1-1; namely, X =
X X a n
A/ AO ^ 2 = z/\/4£b ABt. However, since Eq. 20.1-5 contains the parameter x A0, we can an-
ticipate that X will depend not only on Z but also parametrically on x A0.
In terms of these dimensionless variables, Eq. 20.1-5 can be written as
2
d X
(20.1-9)
dZ
Here the quantity
dX
(20.1-10)
z=o
is a dimensionless molar average velocity, <p = v*\/t/%b AB, as can be seen by comparing Eqs.
20.1-10 and 20.1-4. The initial and boundary conditions in Eqs. 20.1-6 to 8 now become
B.C. 1: atZ -o, X = 1 (20.1-11)
B.C. 2 and I.C.: atZ = 00 X = 0 (20.1-12)

Equation 20.1-9 can be attacked by first letting dX/dZ = Y. This gives a first-order differential
equation for У that can be solved to obtain

= C x p [ - ( Z - 0 ] - ^ (20.1-13)
2
i e