Consider a tank of some volatile liquid, *A*, which is
vented to the atmosphere through a chimney that has a stiff breeze
blowing by it. Once we have reached steady state, there will be a
mixture of *A* and air (which we will call *B*) within
the chimney. The *A* is evaporating and flowing (diffusing) up
through the column until it is ultimately blown away by the wind.
The *B*, on the other hand, is completely stagnant (not moving
up the chimney nor down the chimney).

This is clear if you consider that, at steady state with no
accumulation, any *B* that moves down the chimney must exit
the bottom and that any that moves up the chimney must be
*replaced* at the bottom (if either of these is not met,
there *will* be accumulation). Also note that this is
*not* the case for the *A* since there is a steady
stream of *A* being supplied by the liquid in the tank. In
this case there is no net flux of *B*
(*N*_{B} = 0), but there *is* a non-zero
flux of *A* ($N_A \neq 0$).

By definition, *N*_{A} is given by

$\displaystyle{N_A = -c_{total}D\nabla y_A + c_AV}$

where the molar-averaged velocity *V*, we can written for a
binary system as

$\displaystyle{V = \frac{c_Av_A + c_Bv_B}{c_A+c_B} = \frac{N_A +N_B}{c_{total}}}$

In this particular problem (diffusion through a stagnant gas) we
have already noted that *N*_{B} = 0, so

$\displaystyle{V = \frac{N_A + 0}{c_{total}}=\frac{N_A}{c_{total}}}$

We can have a non-zero bulk flow even in a "diffusion" problem
(as long as the total fluxes do not cancel out). We must
*determine* if there is a flow or not from examining the
total mass flux term itself! (This is the most important difference
between heat and mass transfer.)

Plugging in our expression for *V*, we reduces our
expression for *N*_{A} to

$\displaystyle{N_A = -c_{total}D\nabla y_A + y_AN_A}$

we can solve this for *N*_{A} to give

$\displaystyle{N_A = -\frac{c_{total}D\nabla y_A}{1-y_A}}$

which in one dimension becomes

$\displaystyle{N_A = -\frac{c_{total}D\frac{dy_A}{dz}}{1-y_A}}$

We can then solve for the flux by integrating between
*z*=0,*L* and note that the corresponding concentrations
are *y*_{A} =
*y*_{Asat}, 0

$\displaystyle{N_A\int_0^L dz = -c_{total}D\int_{y_{A_{sat}}}^0 \frac{dy_A}{1-y_A}}$

(Here we note that $N_A \neq F(z)$ by analogy to our previous
example as well as heat transfer...if it *did* change with
*z*, we would have accumulation!)

Integrating gives

$N_A L = c_{total}D(\ln{[1-0]} - \ln{[1-y_{A_{sat}}]}) = -c_{total}D\ln{[1-y_{A_{sat}}]}$

so that *N*_{A} is

$\displaystyle{N_A = -\frac{c_{total}D}{L}\ln{[1- y_{A_{sat}}]}}$

which we note is not a function of *z*, as we
stipulated.

Calculate the magnitude of diffusive mass flow/flux through a planar stagnant film