Page 646 - Bird R.B. Transport phenomena

626 Chapter 20 Concentration Distributions with More Than One Independent Variable

Note that we use different thicknesses, 8 and 8 , for the velocity and concentration boundary
C
layers. In order to relate this problem to that of Example 12.4-1, we introduce the quantity
A = 8 /8, which in this case is a function of x because of the chemical reaction occurring. We
c
restrict the discussion to A ^ 1, for which the concentration boundary layer lies entirely
within the velocity boundary layer. We can also neglect the interfacial velocity v 0 = v l ,
}/ /=0
which is small here because of the small solubility of A. Insertion of these expressions into
Eqs. 20.2-5 and 7 then gives the differential equations
^ f (20.2-10)

for the boundary layer thicknesses 8 and 8 = 6A.
C
Equation 20.2-10 is readily integrated to give
5= / l 2 ^ (20.2-12)

Insertion of this result into Eq. 20.2-11 and multiplication by -8A/vc gives
A0
^- = х^-Д 3 + Д 3 + n\ " Z X 1A 2 (20.2-13)
k C
|
as the differential equation for A. Thus A depends on the Schmidt number, Sc = /х/рЯЬ and
АВ/
on the dimensionless position coordinate shown in the square brackets. The bracketed quan-
6
tity is 1/(M 4- 1) times the first Damkohler number based on the distance x.
When no reaction is occurring, k'" is zero, and Eq. 20.2-13 becomes a linear first-order reac-
3
tion for A . When that equation is integrated, we get
3
Д = ^ + ^ (20.2-14)
in which С is a constant of integration. Because A does not become infinite as x —> 0, we obtain
in the absence of chemical reaction (cf. Eq. 12.4-15):
A = Sc" 1/3 A<1 (20.2-15)
That is, when there is no reaction and Sc > 1, the concentration and velocity boundary layer
thicknesses bear a constant ratio to one another, dependent only on the value of the Schmidt
number.
When a slow reaction occurs (or when x is small), a series solution to Eq. 20.2-13 can be ob-
tained:
1/3
A = Sc" (1 + a£ + a 2£ 2 + • • •) (20.2-16)
in which

Substitution of this expression into Eq. 20.2-13 gives
1/3
2/3
a } = -^Sc , a 2 = +i^Sc , etc. (20.2-18)
Because a } is negative, the concentration boundary layer thickness is diminished by the chem-
ical reaction.

6
G. Damkohler, Zeits.f. Electrochemie, 42, 846-862 (1936); W. E. Stewart, Chem. Eng. Prog. Symp.
Series, #58, 61,16-27 (1965).