Page 246 - Chiral Separation Techniques

224 9 Modeling and Simulation in SMB for Chiral Purification

Boundary conditions for column k:

D L dc
z = 0 : c − k ik = c (4)
,
ik ν * dz ik 0
k
where c is the inlet concentration of species i in column k.
ik,0
z = L :
k
For a column inside a section and for extract and raffinate nodes
(5a)
c = c
ik ik+10,

For the eluent node c = ν * I (5b)
ik ν * c ik+10 ,
IV
ν * ν
For the feed node c = III c − F c F (5c)
ik ν * II ik+10 , ν * II i

Global balances:
*
Eluent node ν = ν * + ν (6a)
I IV E
*
*
Extract node ν = ν – ν (6b)
II I X
*
Feed node ν * = ν + ν (6c)
III II F
Raffinate node ν * = ν * – ν (6d)
IV III R
Multicomponent adsorption equilibrium isotherm:
q* = f (c , c ) and q* = f (c , c ) (7)
Ak A Ak Bk Bk B Ak Bk
*
*
Introducing the dimensionless variables x = z/L and θ = t/t , where t is the switch
k
time interval, and L is the length of one SMB column, the model equations
k
become:
2
∂c  1 ∂ c ∂c  (1 − ε)
*
ik = γ  ik − ik  − α q −( * q ) (8)
∂θ k  Pe k ∂x 2 ∂x  ε k ik ik
∂q
ik = α q −( * q ) (9)
∂θ k ik ik
The initial and boundary conditions are the same presented before and, for x = 0
(z = 0), Equation (9.4) becomes:

c − 1 dc ik = c (10)
ik ik 0,
Pe dx
k
The model parameters, in addition to the adsorption equilibrium parameters, are: