Page 126 - Separation process principles 2

3.4 Molecular Diffusion in Laminar Flow 91

This equation was solved by Johnstone and Pigford [33] For mass transfer, a composition driving force replaces
and later by Olbrich and Wild [34], for the following bound- AT. As discussed later in this chapter, because composition
ary conditions: can be expressed in a number of ways, different mass-
transfer coefficients are defined. If we select AcA as the dri-
CA = CA, at z = 0 for y > 0
ving force for mass transfer, we can write
CA = CA~ at y = 0 for 0 < z < 6
acA/az=O atz=6 forO<y<L n~ = kc A AcA (3-105)

where L = height of the vertical surface. The solution of which defines a mass-transfer coefficient, kc, in
Olbrich and Wild is in the form of an infinite series, giving time-area-driving force, for a concentration driving force.
CA as a function of z and y. However, of more interest is the Unfortunately, no name is in general use for (3-105).
average concentration at y = L, which, by integration, is For the falling laminar film, we take AcA = cA, - ZA,
which varies with vertical location, y, because even though
CA, is independent of y, the average film concentration, CA,
increases with y. To derive an expression for kc, we equate
For the condition y = L, the result is (3-105) to Fick's first law at the gas-liquid interface:

+
.
.
+ 0.036093-'~~.~~~
.
where Although this is the most widely used approach for defin-
ing a mass-transfer coefficient, in this case of a falling film it
fails because (acA/az) at z = 0 is not defined. Therefore, for
this case we use another approach as follows. For an incre-
P
Nsc = Schmidt number = mental height, we can write for film width W,
(3-101)
momentum diffusivity, b/p
- - . ..
mass diffusivity, DAB This defines a local value of kc, which varies with distance y
NPeM = NReNSc = Peclet number for mass transfer because CA varies with y. An average value of kc, over a
46Uy (3- 102) height L, can be defined by separating variables and inte-
-- grating (3-107):
-
DAB
The Schmidt number is analogous to the Prandtl number, S: kc d~ i,6 SCAL [dCA/(cA, - ?A)]
CAo
-
-
used in heat transfer: kcavg = -
L L
CP P (PIP) momentum diffusivity iy8 $4, - CA~ (3- 108)
Npr=-=------ - - -
k (kip Cp) thermal diffusivity
L CA. - ?A,
The Peclet number for mass transfer is analogous to the
In general, the argument of the natural logarithm in
Peclet number for heat transfer:
(3-108) is obtained from the reciprocal of (3-99). For values
of 7 in (3-100) greater than 0.1, only the first term in (3-99)
is significant (error is less than 0.5%). In that case,
Both Peclet numbers are ratios of convective transport to
molecular transport.
The total rate of absorption of A from the gas into the
liquid film for height L and width W is
Since In ex = x,
n~ = iyGW(EAL - cAo) (3-103)
Mass-Transfer Coefficients
In the limit, for large 'q, using (3-100) and (3-102), (3-110)
Mass-transfer problems involving fluids are most often
becomes
solved using mass-transfer coefficients, analogous to heat-
transfer coefficients. For the latter, Newton S law of cooling
defines a heat-transfer coefficient, h:
In a manner suggested by the Nusselt number,
Q=hAAT (3- 104)
NNu = h6lk for heat transfer, where 6 = a characteristic
where
length, we define a Sherwood number for mass transfer,
Q = rate of heat transfer
which for a falling film of characteristic length 6 is
A = area for heat transfer (normal to the direction of
heat transfer)
AT = temperature-driving force for heat transfer