Page 712 - Bird R.B. Transport phenomena

692 Chapter 22 Interphase Transport in Nonisothermal Mixtures

at the interface. Because of the coupling, it is convenient to use the method of Laplace trans-
form. First, however, we restate the problem in dimensionless form, using f = r/R, т =
9) t/R 2 f C = c /c , C, = (mc , + b)/c , and N = k R/m9) . Eqs. 22.3-20 and 24 become
0
5
AB
c
AB
As
0
A
асЛ ac
~ ~s i и I У2 5 I s s = NQ (22.4-25,26)
with C s finite at the sphere center, C s = C, at the sphere surface, and C s = 1 throughout the
sphere initially.
When we take the Laplace transform of this problem, we get
*С
= NQ (22.4-27,28)
with Q finite at the sphere center, and C = Q at the sphere surface. The solution of Eqs. 22.4-
s
27 (which is a nonhomogeneous analog of Eq. C.I-6a) and 28 is
— N sinh \v£ i
s = 7= F 7= 1- + ^ (22.4-29)
p[Vp coshVp - (1 - N) sinhV^] f ^
The Laplace transform of M A, the total amount of A within the sphere at any time t, is

(22.4-30)
3
2
4тгЯ с h s p (Vp cothVp - (1 - N)) p 2 3 P
0
1
Inversion by using the Heaviside partial fractions expansion theorem for repeated roots gives
2
* • = 6 2 B exp(-Al® t/R ) (22.4-31)
AB
n
The constants A,, and B are found to be, for finite k (or N),
c
n
А cot A,, - (1 - N) = 0; B = Щ ^ - ^ — (22.4-32,33)
и n
A?, (A,, -sinA cosA, )
n z
and for infinite k (or N),
c
( 1 V
(22.4-34,35)
Note that we have succeeded in getting the total amount of A transferred across the interface,
M (t), without finding the expression for the concentration profile in the system. This is an
A
advantage in using the Laplace transform.
We may now define two overall mass transfer coefficients: (i) the correct overall coeffi-
cient for this system based on the solid phase
Щ^ _к i dM* .
к = ( 2 2 4 3 6 )
С АЬ 3 M dt
A
where c Ab is the volume-average concentration of A in the solid phase, and (ii) an approximate
overall coefficient, based on the separately calculated behavior of the two phases, calculated
by Eq. 22.4-13,
- ° + ^ (22.4-37)
1 3 M
R dM° /dt К
A
where the superscript 0 indicates "zero external resistance" and k is the liquid-phase transfer
c
coefficient.
1
A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of Integral Transforms, McGraw-
Hill (1954), p. 232, Formula 21.