Page 128 - Separation process principles 2

3.4 Molecular Diffusion in Laminar Flow 93

general solution for intermediate values of 7 is not available Thus,
in closed form. Similar limiting solutions for large and small
values of appropriate parameters, usually dimensionless
groups, have been obtained for a large variety of transport
and kinetic phenomena, as discussed by Churchill [35]. Solving for FA,,
Often the two limiting cases can be patched together to pro- - -
FAL = CA, - (CA, - cb) exp ( ;;;)
vide a reasonable estimate of the intermediate solution, if a
single intermediate value is available from experiment or the
general numerical solution. The procedure is discussed by
Churchill and Usagi [361. The general solution of Emmert
and Pigford [37] to the falling, laminar liquid film problem is
included in Figure 3.13.

Thus, the exiting liquid film is saturated with C02, which implies
EXAMPLE 3.13 equilibrium at the gas-liquid interface. From (3-103),
Water (B) at 25OC, in contact with pure C02 (A) at 1 atm, flows as
nA = 0.0486(1.15 X 10-'l(3.4 X = 1.9 x low7 kmous
a film down a vertical wall 1 m wide and 3 m high at a Reynolds
number of 25. Using the following properties, estimate the rate of
adsorption of COz into water in kmol/s: Boundary-Layer Flow on a Flat Plate

Consider the flow of a fluid (B) over a thin, flat plate parallel
with the direction of flow of the fluid upstream of the plate,
Solubility of C02 in water at 1 atm and 25°C = 3.4 x as shown in Figure 3.14. A number of possibilities for mass
mol/cm3. transfer of another species, A, into B exist: (1) The plate
might consist of material A, which is slightly soluble in B.
SOLUTION (2) Component A might be held in the pores of an inert solid
plate, from which it evaporates or dissolves into B. (3) The
From (3-93),
plate might be an inert, dense polymeric membrane, through
NRe~ 25(0.89)(0.001)
r=-- - = 0.00556- kg which species A can pass into fluid B. Let the fluid velocity
4 4 m-s profile upstream of the plate be uniform at a free-system ve-
locity of u,. As the fluid passes over the plate, the velocity ux
From (3-101),
in the direction of flow is reduced to zero at the wall, which
establishes a velocity profile due to drag. At a certain dis-
tance z, normal to and out from the solid surface, the fluid ve-
locity is 99% of u,. This distance, which increases with
From (3-92),
increasing distance x from the leading edge of the plate, is
arbitrarily defined as the velocity boundary-layer thickness,
6. Essentially all flow retardation occurs in the boundary
layer, as first suggested by Prandtl [38]. The buildup of this
From (3-90) and (3-91), iiy = (2/3)uy,,, . Therefore, layer, the velocity profile in the layer, and the drag force can
be determined for laminar flow by solving the equations
- 2 (1.0)(1,000)(9.807)(1.15 x
3 [
Uy = - 2(0.89)(0.001) 1 = 0.0486 mls of continuity and motion (Navier-Stokes equations) for the
x-direction. For a Newtonian fluid of constant density and
viscosity, in the absence of pressure gradients in the x- and
From (3- 1 OO),
-
Free
I
I I I I I stream
I I I __----
I I
Therefore, (3-1 11) applies, giving -u0 I 1 ..- /--- uo
- I I Uo t- ->
I
I
3.41(1.96 x 10-~)(10-') I
kcavg = = 5.81 x lop5 mls
1.15 x lo-'
To determine the rate of absorption, FA, must be determined. From
(3-103) and (3-113), - - - - - - -.
-x
Flat plate
(CAL - CA~
n~ = Uy6W(FAL - c&) = kcWgA Figure 3.14 Laminar boundary-layer development for flow across
ln[(c~, - CA~)/(CA, CAL )I a flat plate.
-