Page 286 - Chiral Separation Techniques

10.3 SMB as a Development Tool 265

It is convenient to define a parameter γ≥ 1 and to replace Equation (11) by:

1
m IV = K ⋅ 1 m III = K ⋅ γ
γ
A
B
(12)
m = K ⋅γ m = K ⋅γ
II
I
B
A
γ can be considered as a safety factor, if it is equal to 1, the system works at its opti-
mum productivity, but it will be very sensitive to any deviation regarding the
flowrates. If it is greater than 1, the system is less efficient in term of specific pro-
ductivity or eluent consumption, but it is less sensitive to possible perturbations.
Taking into account Equation (12) and the definition of the normalized flow rates,
one can derive:
A)
⋅(
Q = Q F Q Ext = Q K − K ⋅γ
B
1
K ⋅ γ − K ⋅γ (13)
B
A
⋅(
⋅
Q Raf = Q K − K A) γ / Q = Q K ⋅γ
B
B
l
As the feed flowrate is known, all the TMB flowrates are calculated from Equation
(13), for a given γ value. From Equation (13), it follows that γ must fall in this range:
1≤ γ < K B (14)
K A
There are no general rules allowing selection of the correct γ value. The correct
selection is a result of technico-economical optimization; however, for a first guess
γ can be set to γ = 1.02.
At this point, all the flowrates are known and only the total number of plates
required has to be estimated. This estimation is determined by numerical simulation.
Experience shows that SMB equivalent to 500 plates solves almost all problems.

Nonlinear case: The calculation of the flowrates is much more complex, and it is
beyond the scope of this chapter to present it in detail. However, as a useful tool, Mor-
bidelli and coworkers [48–50, 63], applied the solutions to the equations of the equi-
librium theory (when all the dispersion phenomena are neglected) to a four-zone TMB.
The solutions are explicit for m and m
I IV
m > ( ) = K B (15)
m
I
I min
A A (
 K + m − 2λ + K C m − m ) 
˜
F
m <( m ) = 1  A III III II 2   + λ (16)
A [

A(
A A (
IV
˜
λ
IV max
˜
F
2  − K + m − 2λ + K C m − m )] − 4 NK m − )  
III

III
II
III