Page 254 - Chiral Separation Techniques

232 9 Modeling and Simulation in SMB for Chiral Purification

Qc >1 ; Qc >1 and Qc <1 ; (29)
II BII
IBI
II AII
Qq BI Qq AII Qq BII
S
S
S
Qc >1 and Qc <1 ; Qc <1
III BIII
III AIII
IV AIV
Qq AIII Qq BIII Qq AIV
S
S
S
where Q , Q , Q , Q are the volumetric liquid flow rates in the various sections of
I II III IV
the TMB, Q is the solid flow rate, c , c are the concentrations of species A and B
S Aj Bj
in the liquid phase and q , q are the adsorbed concentrations of components A and
Aj Bj
B, in section j. The same constraints can be expressed in terms of fluid and solid
interstitial velocities. Defining the dimensionless parameter:
Γ = ε γ c ij (30)
ij − ε j
1 q
ij
the constraints become Γ >1; Γ >1 and Γ < 1; Γ > 1 and Γ < 1; Γ < 1.
BI AII BII AIII BIII AIV
For the case of a binary system with linear adsorption isotherms, very simple for-
mulas can be derived to evaluate the better TMB flow rates [19, 20]. For the linear
case, the net fluxes constraints are reduced to only four inequalities, which are
assumed to be satisfied by the same margin β (β > 1) and so:
Q I = β ; Q II = β ; Q III = 1 ; Q IV = 1 (31)
QK B QK A QK B β QK A β
S
S
S
S
where K and K are the coefficients of the linear isotherms for the less and more
A B
retained species, respectively. The flow rates for TMB operation are then: Q =
E
2
2
2
(αβ – 1) Q , Q = (α – 1) β Q , Q = (α – β ) Q and Q = (α – 1) Q where
RF X RF F RF R RF
Q , Q , Q , and Q are the eluent, extract, feed, and raffinate volumetric flow rates,
E X F R
respectively. The volumetric flow rate in the section IV is the recycling flow rate,
Q = K Q /β and α = K /K is the selectivity factor of the binary linear system. The
RF A S B A
total inlet or outlet volumetric flow rate is given by Q + Q = Q + Q = (α – 1)
E
F
X
R
2
(1 + β ) Q . The specification of β and the solid flow rate (or, alternatively, one
RF
of the liquid flow rates) defines all the flow rates throughout the TMB system. The
β parameter has a higher limit, since the feed flow rate must be higher than zero,
—
1 < β < √α. The case of β = 1 corresponds to the situation where dilution of species
is minimal, and the extract and raffinate product concentrations approach the feed
concentrations. In fact, for β = 1, we obtain Q = Q = Q = Q = (α – 1) Q = (K
E X F R RF B
– K ) Q .
A S
In the case of complete separation, the concentrations of the component A in the
R
F
raffinate and of the component B in the extract are, respectively, C = C Q /Q =
F
A
A
R
2
F
2
2
F
F
X
C (α – β )/(α – 1) and C = C Q /Q = C (α – β )/(α – 1)β . Following the
A B B F X B
equivalence of internal flow rates, it results that the inlet and outlet flow rates are the
same for the two operating modes, and

1
A
Q * = Q + ε Q =   K A + ε   Q = 1 + ( − ε ) K  ε V c (32)


RF RF ( − ε ) S  β ( − ε )  S  ε β  t *
1
1
where Q* is the recycling flow rate in the SMB operation.
RF