Page 655 - Bird R.B. Transport phenomena

§20.3 Steady-State Boundary Layer Theory for Flow Around Objects 635

This is a linear, first-order equation for 8 , which has to be solved with the boundary
A
condition 8 = 0 at x = 0. Integration of Eq. 20.3-11 gives
A
2
/Э Г h h v x
л в x z 5
8 (x, z) = 2 J J Q ^ (20.3-12)
Ты
A
as the thickness function for the diffusional boundary layer. Since Eq. 20-3-9 and the bound-
ary conditions then contain 77 as the only independent variable, the postulated combination
of variables is valid, and the concentration profiles are given by the solution of Eq. 20.3-9:
fit]) = 1 - - | = f exp (-^ ) d^ = 1 - erfrj (20.3-13)
V
2
Equations 20.3-12 and 13 are the solution to the problem at hand.
Next, we combine this solution with Fick's first law to evaluate the molar flux of
species A at the interface:

y=0

2
= +Э с Л0 4 = (exp (-г, ) ^ 11 = c A0 1=^. — ^ (20.3-14)
лв
J J htf
tfvjx
This result shows the same dependence of the mass flux on the \-power of the diffusivity
that arose in Eq. 18.5-17, for the much simpler gas absorption problem solved there. In
fact, if we set the scale factors h x and h z equal to unity and replace v s by v , we recover
max
Eq. 18.5-17 exactly.

Linear Velocity Profile Near the Mass-Transfer Surface
This velocity function is appropriate for mass transfer at a solid surface (see Example
12.4-3) when the concentration boundary layer is very thin. Here v x depends linearly on
у within the concentration boundary layer, and v y can be obtained from the equation of
continuity. Consequently, when the net mass flux through the interface is small, the ve-
locity components in the concentration boundary layer are
v x = p(x, z)y (20.3-15)

& £ - w I (203 16)
-
in which у depends on x and z. Substituting these expressions into Eq. 20.3-2 gives the
diffusion equation for the liquid phase
2
Pydc A 2 dc A d c A
which is to be solved with the boundary conditions
B.C. 1: at x = 0, с = О (20.3-18)
л
B.C. 2: at у = 0, с = c (20.3-19)
л A0
B.C.3: asy->oo, c A -* 0 (20.3-20)
Once again, we use the method of combination of variables, by setting c /c AQ = /(17),
A
where 17 = y/8 (x f z).
A
When the change of variables is made, the diffusion equation becomes
d2 f , W / 8 Э8 , _
e2 А