Page 247 - Chiral Separation Techniques

9.3 Modeling of SMB Processes 225

the ratio between solid and fluid volumes, 1− ε (11)
ε

ν * ν *
the ratio between fluid and solid interstitial velocities, γ = k = k (12)
*
k *
u L t
/
s k
ν * L
the Peclet number, Pe = kk (13)
k
D
Lk
the number of mass transfer units, α = kL k = kt * (14)
k
u
s
where u is the interstitial solid velocity in the equivalent TMB model. Due to the switch
s
of inlet and outlet lines, each column plays different functions during a whole cycle,
depending on its location (section). The boundary conditions for each column change
after the end of each switch time interval. This time-dependent boundary conditions
leads to a cyclic steady state, instead of a real steady state as in the TMB model. This
means that, after cyclic steady state is reached, the internal concentration profiles vary
during a given cycle, but they are identical at the same time for two successive cycles.

9.3.2 The TMB Model

In the TMB model, the adsorbent is assumed to move in plug flow in the opposite
direction of the fluid, while the inlet and outlet lines remain fixed. As a consequence,
each column plays the same function, depending on its location. An equivalence
between the TMB and the SMB models can be made by keeping constant the liquid
velocity relative to the solid velocity, i.e., the liquid velocity in the TMB is:
*
ν = ν – u (15)
j j s
where ν is the interstitial liquid velocity in the TMB. Also, the solid interstitial
j
velocity in the TMB model u must be evaluated from the value of the switch time
s
interval t*of the SMB model, as
u = L /t * (16)
s c
where L is the length of one SMB column. The equations for the transient TMB
c
model are:
Mass balance in a volume element of the bed j:
∂c ij = D ∂c ij − ν ∂c ij − 1 ( − ε) kq −( * q ) (17)
∂t L j ∂z 2 j ∂z ε ij ij
Mass balance in the particle:
∂q ∂q (18)
ij = u ij + kq −( * q )
∂t s ∂z ij ij