Page 125 - Separation process principles 2

90 Chapter 3 Mass Transfer and Diffusion

3.4 MOLECULAR DIFFUSION Thus, the maximum liquid velocity, which occurs at z = 0, is
IN LAMINAR FLOW
Many mass-transfer operations involve diffusion in fluids in
laminar flow. The fluid may be a film flowing slowly down a
The bulk-average velocity in the liquid film is
vertical or inclined surface, a laminar boundary layer that
forms as the fluid flows slowly past a thin plate, or the fluid
may flow through a small tube or slowly through a large pipe
or duct. Mass transfer may occur between a gas and a liquid Thus, the film thickness for fully developed flow is indepen-
film, between a solid surface and a fluid, or between a fluid
dent of location y and is
and a membrane surface.

Falling Liquid Film
where r = liquid film flow rate per unit width of film, W.
Consider a thin liquid film, of a mixture of volatile A and
For film flow, the Reynolds number, which is the ratio of
nonvolatile B, falling in laminar flow at steady state down
the inertial force to the viscous force, is
one side of a vertical surface and exposed to pure gas, A, on
the other side of the film, as shown in Figure 3.12. The sur-
face is infinitely wide in the x-direction (normal to the page).
In the absence of mass transfer of A into the liquid film, the
where r~ = hydraulic radius = (flow cross section)/(wetted
liquid velocity in the z-direction, u,, is zero. In the absence of
perimeter) = (W6)l W = 6 and, by the equation of continu-
end effects, the equation of motion for the liquid film in fully
ity,r =iyp6.
developed laminar flow in the downward y-direction is
As reported by Grimley [32], for NR, < 8 to 25, depend-
ing on the surface tension and viscosity, the flow in the film
is laminar and the interface between the liquid film and the
gas is flat. The value of 25 is obtained with water. For 8 to
25 < NRe < 1,200, the flow is still laminar, but ripples and
Usually, fully developed flow, where uy is independent of the
waves may appear at the interface unless suppressed by the
distance y, is established quickly. If 6 is the thickness of the
addition of wetting agents to the liquid.
film and the boundary conditions are uy = 0 at z = 6 (no-
For a flat liquid-gas interface and a small rate of mass
slip condition at the solid surface) and duy/dz = 0 at z = 0
transfer of A into the liquid film, (3-88) to (3-93) hold and
(no drag at the liquid-gas interface), (3-88) is readily inte-
the film velocity profile is given by (3-89). Now consider a
grated to give a parabolic velocity profile:
mole balance on A for an incremental volume of liquid film
of constant density, as shown in Figure 3.12. Neglect bulk
flow in the z-direction and axial diffusion in the y-direction.
Then, at steady state, neglecting accumulation or depletion
of A in the incremental volume,

Rearranging and simplifying (3-94),

(3-95)
In the limit, as Az -+ 0 and Ay -+ 0,

Substituting (3-89) into (3-96),

Pigure 3.12 Mass transfer from a gas into a falling, laminar
liquid film.