Page 639 - Bird R.B. Transport phenomena

§20.1 Time Dependent Diffusion 619

The average rate of absorption up to time t is then

(20.1-39)
erfVy/2)^
Hence the average rate up to time t is just twice the instantaneous rate.

EXAMPLE 20.1-3 When species A diffuses in a liquid medium В and reacts with it irreversibly (A + В —> С) ac-
cording to a pseudo-first-order reaction, then the process of diffusion plus reaction is de-
Unsteady Diffusion scribed by
with First-Order
Homogeneous дш А (v • VOJ A ) = - k'"(o A (20.1-40)
Reaction 58

provided that the solution of A is dilute and that not much С is produced. Here k'" is the rate
constant for the homogeneous reaction. Equation 20.1-40 is frequently encountered with the
initial and boundary conditions

I.C. at t = 0: a> A = w Al (x, y, z) (20.1-41)
B.C. at bounding surfaces: o) A = o) A0 (x, y, z) (20.1-42)
and with a velocity profile independent of time. For such problems show that the solution is

o) = g exp(-k"'t) + Г ехр(-*Г0 -^-f(x, у, z, t'W (20.1-43)
Jo at
A
Here / is the solution of Eqs. 20.1-40 to 42 with k'" = 0 and a> = 0, whereas g is the solution
Al
with k"' = 0 and со = 0.
А0
SOLUTION This problem is linear in ш . It may, therefore, be solved by a superposition of two simpler
А
problems:
(20.1-44)
])
with (o described by the equations
A
2
P.D.E.: —f- + (v • V O = ® V a>« ] - k?a> l) l (20.1-45)
dt AB A A
I.C. at t = 0: co A = <o Al {x,y,z) (20.1-46)
])
B.C. at surfaces: «JJ> = 0 (20.1-47)

5 P. V. Danckwerts, Trans. Faraday Soc., 47,1014-1023 (1951). Peter Victor Danckwerts (1916-1984)
was bomb disposal officer for the Port of London during "the Blitz' and was wounded in a mine field in
7
Italy during WWII; while teaching at Imperial College in London and at Cambridge University he
directed research on residence-time distribution, diffusion and chemical reaction, and the role of
diffusion in gas absorption.
6 A. Giuliani and F. P. Foraboschi, Atti. Acad. Sci. Inst. Bologna, 9,1-16 (1962); F. P. Foraboschi, ibid.,
11,1-14 (1964); F. P. Foraboschi, AIChE journal 11, 752-768 (1965).
7 E. N. Lightfoot, AIChE Journal, 10, 278-284 (1964).
8
W. E. Stewart, Chem. Eng. Sci., 23, 483^87 (1968); corrigenda, ibid., 24,1189-1190 (1969). There this
approach was generalized to time-dependent flows with homogeneous and heterogeneous first-order
reactions.