C A T A L Y T I C SURFACE R E A C T I O N S Equation 36 is clearly nonlinear and is best solved numerically because the infinite seriesexpansion has, in this case, a finite radius of convergence. Thus, if t
z23/2
it follows from Equation 11 that Equation 36 can be approximated by
Nomenclature d x ) , c n ( x ) , c h ) = dimensionlesssurface
concentrations of reacting specie A,B,C = symbols for reactants D = diffusion coefficient = distance along surface from leading edge
~ ( x ) = local surface shear stress {(x) = defined by Equation 5 = defined by Equation 7 +(z)
Ax,ax = coefficients defined by Equations 12 and 26, respectively, and tabulated in Table I 2 = defined by Equation 22, 28, or =
t
33 z3/2
k ~ , k z = reaction rate constants K = defined by Equation 31 = defined by Equation 27 Q
[i
1
n-1
-
( A t ~ 2 / 8 n l , 3 b ] ~ a l f 1 2 - i (n
literature Cited
- w1+m = l c A ( m ) A h m (37) 9
from which cA(n) can be calculated. Table I V shows a few numerical results, with p = 1, for the quantity ( C J U ) ~ which, in this special case, is proportional to the local rate of the surface reaction. Again a comparison is made with the results of the approximate method. The numerical results of this section illustrate the applicability of the mathematical formulas to some simple problems. One might be tempted to argue, after examining the results of examples 1 and 3, that the approximate method appears to be rather accurate even though it leads to considerable errors in example 2. However, this viewpoint is believed to be incorrect. The approximate method has, strictly speaking, no foundation. It can be justified in each particular case only by comparison with the results of the exact method given here. The fact that the comparison is as favorable in the examples considered is encouraging, but nevertheless coincidental, because counter examples can be constructed, where the approximate method breaks down. Thus for more complicated surface reactions and especially for nonisothermal problems the approximate method can be very unreliable. For example, even in the isothermal first-order consecutive surface reaction
Y u V
a
b
R(c) p,v,p
Sc
= distance from surface = velocity,component in x- direction = velocity component in y- direction = free stream concentration of reactant A = free stream concentration Of reactant B = kinetic law for rate of surface reaction = viscosity, kinematic viscosity, and density of fluid, respectively = Schmidt number. v / D = free stream velocity past a flat plate I
U,
(1) Chambr6, P. L., Appl. Sci.Research, A 6 , 97 (1956). (2) Chambr6, P. L., Acrivos, A., J Appl. Phys. 27, 1322 (1956). (3) Chapman, D. R., Rubesin, M. W., J. Aeronaut. Sci. 16, 547 (1949). (4) Frank-KamenetskiI, D. A., “Diffusion and Heat Exchange in Chemical Kinetics,” Chap. 11, Princeton University Press, Princeton, N. J., 1955. ( 5 ) Lighthill, M. J., Proc. Roy. SOC. A202, 359 (1950). (6) Schlichting, H., “Boundary Layer Theory,” McGraw-Hill, New York, 1955. (7) Tifford, A. N., Second Midwestern Conference Ohio, on Fluid Columbus, 1952. Mechanics, (8) Wagner, C., J . Math. Phys. 32, 289 (1954).
,
RECEIVED for review December 7, 1956 ACCEPTEDMarch 1, 1957
for
Correction I n the article on “Motion of a Sphere and Fluid in a Cylindrical Tube” [John Happel and B. J. Byrne, IND. ENG. CHEM.46, 1181 (1954)] the equations should be corrected as follows : The second term on the right side of Equation 20 should read:
J + zaJo 2(. . . * )
In Equation 36 substitute BaT a4 2
for
4
4
24
Ba’ a(
and substitute In Equation 27 substitute
Ba7a4 1
2
4
for for
Ba7a2 12 Ro4
A+B+C+D
the local rate of production of substance D on the surface of a flat plate, as determined by the approximate method, can be in error by as much as 120%.
I n Equation 32 substitute
In Equation 48 substitute
+zzJw 4 R2 y d a
5 Ba6iX2 12
for
for 10 BasiX2
24
Table IV.
0
Numerical Results for
= 1, 2(At)2/a/3 = 0.29240 (cA/~)’cx~c~
1 5 10 15
0.6244 0.3730 0.2767 0.2285
(CA/~)’*PIXOX.
0.5655 0.3311 0.2446 0.2051
The first term on right of Equation 35 should read
% Error
-
9.4 -11.2 -11.6 -10.2
Equation 49 should read [uz],,.
I n Equation 35 substitute
= 2.105 5 [(Uo - U ) -
Ro
Ba*]
- 3.582
as
[Uo- U ]
+
aRB 0.138 R > [Uo- v] VOL. 49, NO. 6
JUNE 1957
+ . .. 1029
