Page 245 - Adsorption Technology & Design, Elsevier (1998)

222 Selected adsorption processes

OL-~z2 -- U-~Z + u~dq" dZ - 0 (7.14)

where e is the bed voidage, c and q are fluid and solid phase concentrations
(each on the basis of moles per unit mass) at a point z in the column, u~ and u
are the solid and interstitial fluid flow rates and DL is the axial dispersion
coefficient.
Assuming the rate of adsorption dq/dt (equivalent to - u~ dq/dz) can be
represented by a linear driving force k (q* -q) and that a linear equilibrium
relationship q* - Kc holds, the defining equation becomes

DL-S-~_ 2 -- u-- -- k(Kc-q) = 0 (7.15)
dz dz
A steady state material balance between a plane z in the column and the inlet
yields

(1 - e) u~ (q - qo) - eu (c - co) (7.16)
Substitution of this mass balance into the differential equation for the
column and application of the usual Dankwerts boundary condition at the
column fluid entrance (z - 0) and zero change of concentration flux at the
column exit (z = L) leads to a formal solution for the concentration ratio.
When plug flow prevails and mass transfer resistance is small, CL/CO is given
by
/
C....~L
= ~1 ~ (1-qo/Kco)ye St(l-r) + yqo/Kco-1 I (7.17)
co ?'- 1
J
where the Stanton number St - kL/u and y - (1 -e) Kudeu. For a four-
column countercurrent adsorptive fractionating system there are four such
equations. In conjunction with four mass balances over each of the four beds
and two additional mass balances at the feed point and over the fluid
recirculating stream, ten equations define the total system enabling ten
unknown fluid and adsorbent concentrations to be found. Given values for
the Peclet and Stanton numbers (Pe and St), the two equilibrium constants
KA and KB, the bed voidage e and the dispersion coefficient DL, any given
adsorptive fractionation of a binary feed may be completely described.
As an alternative to the continuous countercurrent model an equilibrium
stage model of the adsorptive fractionation system exists (Ching et al. 1985).
In effect the concentration ratio CL/Co for each column is represented by the
well-known equation (Kremser 1930, Souders and Brown 1932):
~,n+l
CO- CL -- ~'
= (7.18)
co- cL/K yn§ _ 1