Page 129 - Separation process principles 2

94 Chapter 3 Mass Transfer and Diffusion

y- (normal to the x-z plane) directions, these equations for If mass transfer begins at the leading edge of the plate and if
the region of the boundary layer are the concentration in the fluid at the solid-fluid interface is
constant, the additional boundary conditions are
CA = CA, at x = 0 for z > 0,
CA = CA~ at z = 0 for x > 0,
and CA = CA, at z = co for x > 0
If the rate of mass transfer is low, the velocity profiles are
The boundary conditions are undisturbed. The solution to the analogous problem in heat
transfer was first obtained by Pohlhausen [42] for Np, > 0.5,
as described in detail by Schlichting [40]. The results for
mass transfer are
The solution of (3-124) and (3-125) in the absence of heat
and mass transfer, subject to these boundary conditions, was
first obtained by Blasius [39] and is described in detail by
where
Schlichting [40]. The result in terms of a local friction factor,
f,, a local shear stress at the wall, 7wr, and a local drag coef-
ficient at the wall, CDI , is
and the driving force for mass transfer is CA, - CA,.
The concentration boundary layer, where essentially all
of the resistance to mass transfer resides, is defined by
where

and the ratio of the concentration boundary-layer thickness,
Thus, the drag is greatest at the leading edge of the plate, a,, to the velocity boundary thickness, 6, is
where the Reynolds number is smallest. Average values of
the drag coefficient are obtained by integrating (3-126) from
x = 0 to L, giving Thus, for a liquid boundary layer, where Ns, > 1, the concen-
tration boundary layer builds up more slowly than the veloc-
ity boundary layer. For a gas boundary layer, where Nsc x 1,
the two boundary layers build up at about the same rate. By
analogy to (3-130), the concentration profile is given by
The thickness of the velocity boundary layer increases with
distance along the plate:

Equation (3-132) gives the local Sherwood number. If
this expression is integrated over the length of the plate, L,
A reasonably accurate expression for the velocity profile
the average Sherwood number is found to be
was obtained by Pohlhausen [411, who assumed the empiri-
cal form u, = Clz + c2z3.
If the boundary conditions,
where

are applied to evaluate C1 and C2, the result is
EXAMPLE 3.14
Air at 100°C, 1 atm, and a free-stream velocity of 5 m/s flows over
a 3-m-long, thin, flat plate of naphthalene, causing it to sublime.
This solution is valid only for a laminar boundary layer,
which by experiment persists to NRe, = 5 x lo5. (a) Determine the length over which a laminar boundary layer
persists.
When mass transfer of A into the boundary layer occurs,
the following species continuity equation applies at constant (b) For that length, determine the rate of mass transfer of naphtha-
lene into air.
diffusivity :
(c) At the point of transition of the boundary layer to turbulent
ilow, determine the thicknesses of the velocity and concentra-
tion boundary layers.