Page 616 - Bird R.B. Transport phenomena

596 Chapter 19 Equations of Change for Multicomponent Systems

This is to be solved with the boundary conditions
B.C.1: atz = 0, c = c (19.4-21)
A A0
B.C. 2: at r = R, c A = 0 (19.4-22)
B.C. 3: at r = 0, c A = finite (19.4-23)
For short distances z into the reactor, the concentration c A differs from c A0 only near the wall,
where the velocity profile is practically linear. Hence we can introduce the variable у = R - r,
neglect curvature terms, and replace B.C. 3 by a fictitious boundary condition at у = oo (see
Example 12.2-2 for a detailed discussion of this method of treating the entrance region of the
tube).
The reformulated problem statement is then
д
У дСА С А
2v z ma x 2 (19.4-24)
' R dz ду
with the boundary conditions
B.C. 1: at z = 0, С А0 (19.4-25)
B.C. 2: at у = 0, <м = 0 (19.4-26)
B.C. 3: at у = oo, С А = С А0 (19.4-27)
This problem can be solved by the method of combination of independent variables by seeking
2
1/3
a solution of the form c /c A0 = /(17), where 77 = (y/R)(2z; zmax R /92)^ z) . One thus obtains the
s
A
ordinary differential equation/" + З77/' = 0, which can be integrated to give (see Eq. C.I-9)
2
f — — —
3
Jo exp(-77 exp(-rj )drj (19.4-28)
C A0
Jo
This problem is mathematically analogous to the Graetz problem of Problem 12D.4, в of that
problem being analogous to 1 - (c /c ) here.
A0
A
Experiments of the type described here have proved useful for obtaining mass transfer
data at high Schmidt numbers. 2 A particularly attractive reaction is the reduction of ferri-
cyanide ions on metallic surfaces according to the reaction
Fe(CN) 2 > Fe(CN) " 4 (19.4-29)
6 6
in which ferricyanide and ferrocyanide take the place of Л and В in the above development.
This electrochemical reaction is quite rapid under properly chosen conditions. Furthermore,
since it involves only electron transfer, the physical properties of the solution are almost en-
tirely unaffected. The forced diffusion effects neglected here may be suppressed by the addi-
tion of an indifferent electrolyte in excess. ' 3 4

Figure 19.4-3 shows schematically how oxygen and carbon monoxide combine at a catalytic
EXAMPLE 19.4-3
surface (palladium) to make carbon dioxide, according to the technologically important
Catalytic Oxidation reaction 5
of Carbon Monoxide O, + 2CO -> 2CO, (19.4-30)

2
D. W. Hubbard and E. N. Lightfoot, bid. Eng. Chem. Fundam., 5, 370-379 (1966).
3
J. S. Newman, Electrochemical Systems, 2nd edition, Prentice-Hall, Englewood Cliffs, N.J. (1991), §1.10.
J. R. Selman and C. W. Tobias, Advances in Chemical Engineering, 10, Academic Press, New York,
4
N.Y. (1978), pp. 212-318.
В. С Gates, Catalytic Chemistry, Wiley, New York (1992), pp. 356-362; C. N. Satterfield,
5
Heterogeneous Catalysis in Industrial Practice, McGraw-Hill, New York, 2nd edition (1991), Chapter 8.