Page 245 - Chiral Separation Techniques

9.3 Modeling of SMB Processes 223

competitive adsorption. A constant selectivity factor model, e.g., multicomponent
Langmuir isotherm, usually is used to describe this behavior. However, for many chi-
ral separations the selectivity factor decreases with the increase of the concentration
of chiral species and a concentration-dependent selectivity factor model is needed,
such as the linear + Langmuir or the bi-Langmuir adsorption isotherm.
The question concerning the degree of equivalence between SMB and TMB
strategies of modeling should be addressed. Three cases are analyzed for the SMB
system: SMB4, constituted by four columns, one in each section; SMB8, with eight
columns, two per section; and SMB12, with three columns per section. The mathe-
matical models developed are based on the following assumptions: axial dispersed
plug flow for the fluid; plug flow for the countercurrent solid flow in the TMB
approach; the adsorbent particles are considered as homogeneous and mass transfer
between fluid and solid is described by the LDF model.

9.3.1 The SMB Model

In the SMB operation, the countercurrent motion of fluid and solid is simulated with
a discrete jump of injection and collection points in the same direction of the fluid
phase. The SMB system is then a set of identical fixed-bed columns, connected in
series. The transient SMB model equations are summarized below, with initial and
boundary conditions, and the necessary mass balances at the nodes between each
column.
Mass balance in a volume element of the bed k:
2
∂c ∂ c ∂c 1 ( − ε)
ik = D ik − ν * ik − kq −( * q ) (1)
∂t L k ∂z 2 k ∂z ε ik ik
where the subscripts i (i = A, B) refers to the species in the mixture, and k is the
column number, c and q are the fluid and average adsorbed phase concentra-
ik ik
tions of species i in column k of the SMB unit, respectively, z is the axial coordi-
*
nate, t is the time variable, ε is the bed porosity, ν is the interstitial fluid veloc-
k
th
ity in the k SMB column, D is the axial dispersion coefficient, and k is the
L k
intraparticle mass transfer coefficient.
Mass balance in the particle:
∂q ik = kq −( * q ) (2)
∂t ik ik
*
where q is the adsorbed phase concentration in equilibrium with c .
ik ik
Initial conditions:
t = 0: c = q = 0 (3)
ik ik