Page 327 - Numerical Methods for Chemical Engineering

316 6 Boundary value problems

6.C.2. Modify the program of problem 6.C.1 to account for external mass transfer resistance,
given the effective binary diffusivities D jf of each species in the external fluid, the Sherwood
number Sh, and the bulk concentrations c jb . Report your results for the parameter values of
problem 6.C.1, replacing the known surface concentrations with those computed by external
mass transfer with

2
D jf = 10 −5 cm /s Sh = 10
(6.248)
c Ab = 1M c Bb = 1M c Cb = c Db = 0
6.C.3. Solve problem 6.C.1. by FEM using the PDE toolkit or FEMLAB TM .

6.C.4. Instead of assuming a given Sherwood number for the external mass transfer coeffi-
cients, use FEMLAB TM to compute the laminar velocity profile around the cylinder to directly
model the effect of forced convection on the external mass transfer rate. Report your results
−4
−2
−1
for Reynolds’ numbers of 10 , 10 , 10 , 1, 10, using a viscosity µ = 10 −3 Pa s and
3
3
density ρ = 10 kg/m .
6.C.5. A common technique to remove a minor component from a gas stream is reactive
absorption (Cussler & Varma, 1997). A gas phase containing a minor component A is passed
through a column where it contacts a liquid phase containing a species B that reacts with
A. The depletion of A through reaction with B allows the removal of more A from the gas
stream than would be possible from considerations of solubility alone.
Let us consider the problem of reactive absorption into a falling film (Figure 6.32) from
a large volume of gas with a constant partial pressure of A, p A = 10 −4 atm. At the inter-
face, the concentration of A is in equilibrium with the gas according to Henry’s law, H A =
3
10 M/atm. We assume that the amount of A entering the film has a negligible effect on the
viscosity µ and density ρ (use the values for water) and that the film travels so slowly that the
flow is laminar. Within the liquid, A reacts with B according to a reversible first-order reac-
3
tion A + B ⇔ AB at the rate r R = k(c A c B − K −1 c AB ), K = 10 and k = 10 −2 L/(mol s).
The feed stream has c B0 = 1M , c A0 = c AB0 = 0, and model B and AB as being nonvolatile
by using no-flux BCs at the liquid–gas interface. For a dilute solution of A and B, we neglect
the heat of absorption and assume a uniform temperature.
Propose a model for this system in the form of a BVP, specifying both the set of PDEs
and the appropriate boundary conditions. Assume a film of thickness b = 1 mm, length
◦
L = 50 cm, inclined at θ = 80 . Use effective binary diffusivities in the film of D j =
2
10 −5 cm /s, j = A, B, AB. Compute the steady-state concentration profile of each species
within the film and the average absorption rate per unit area. Then, decrease the rate constant
to zero to see what the mass transfer rate would be without reaction.