Page 420 - Modelling in Transport Phenomena A Conceptual Approach

400 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

The boundary conditions associated with J3q. (9.588) are

at z=O CA = CA, (9.589)

at x =O CA = C (9.590)
:

at x=S -- -0 (9.591)
acA
ax
It is assumed that the liquid has a uniform concentration of CA, for z < 0. At
the liquid-gas interface, the value of cTq is determined from the solubility data, i.e.,
Henry’s law. Equation (9.591) indicates that species A cannot diffuse through the
wall.
The problem will be analyzed for two cases, namely, for long and short contact
times.

9.5.2.1 Long contact times

The solution of Eq. (9.588) subject to the boundary conditions given by Eqs.
(9.589)-(9.5-91) is first obtained by Johnstone and Pigford (1942). Their series
solution expresses the bulk concentration of species A at z = L as
‘> -
= 0.7857e-5.’213~+0.1001 e-39*318’1+0.03599 e- lo5aM9+... (9.592)
c> - CA,
where
(9.593)

As an engineer we are interested in expressing the results in the form of a mass
transfer correlation. For this purpose it is first necessary to obtain an expression
for the mass transfer coefficient.
For a rectangular differential volume element of thickness Ax, length Az and
width W, as shown in Figure 9.20, the conservation statement given by Eq. (9.583)
is also expressed as

[QCAbI~+fC(c>-cAb)WAz] -&cAbl,z+Al=O (9.594)
Dividing Eq. (9.594) by Az and taking the limit as At --t 0 gives

(9.595)

or,
dCAb
Q - kc (cT~ - CA~) W (9.596)
=
dz
Equation (9.596) is a separable equation and rearrangement gives
(9.597)